Nonlinear Continuum Mechanics and Large Inelastic Deformations
SOLID MECHANICS AND ITS APPLICATIONS Volume 174
Serie...

Author:
Yuriy I. Dimitrienko

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Nonlinear Continuum Mechanics and Large Inelastic Deformations

SOLID MECHANICS AND ITS APPLICATIONS Volume 174

Series Editors:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For other titles published in this series, go to www.springer.com/series/6557

Yuriy I. Dimitrienko

Nonlinear Continuum Mechanics and Large Inelastic Deformations

123

Prof. Yuriy I. Dimitrienko Bauman Moscow State Technical University 2nd Baumanskaya St. 5 105005 Moscow Russia [email protected]

ISSN 0925-0042 ISBN 978-94-007-0033-8 e-ISBN 978-94-007-0034-5 DOI 10.1007/978-94-007-0034-5 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010938719 c Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Nonlinear continuum mechanics is the kernel of the general course ‘Continuum Mechanics’, which includes kinematics of continua, balance laws, general nonlinear theory of constitutive equations, relations at singular surfaces. Moreover, in the course of nonlinear continuum mechanics one also considers the theory of solids at finite (arbitrary) deformations. This arbitrariness of deformations makes the equations describing the behavior of continua extremely complex – nonlinear (so that sometimes the term ‘strongly nonlinear’ is used), as the relationships contained in them cannot always be expressed in an explicit analytical way. If we drop the condition of the arbitrariness of continuum deformations and consider only infinitesimal deformations – usually the deformations till 1%, then the situation changes: the equations of continuum mechanics can be linearized. Hence for solving the applied problems one can exploit the wide range of analytical and numerical methods. However, many practical tasks demand the analysis not of the infinitesimal, but just the arbitrary (large) deformations of bodies, for example, such tasks include the rubber structure elements design (shock absorbers, gaskets, tires) for which the ultimate deformations can reach 100% and even higher. The various tasks of metal working under high pressure also belong to that class of problems, where large plastic deformations play a significant role, as well as the dynamical problems of barrier breakdown with a striker (aperture formation in the metal barrier while the breakdown is an example of large plastic deformations). Within this class of problems one can also find many problems of ground and rock mechanics, where there usually appears the need to consider large deformations, and modelling the processes in biological systems such as the functioning of human muscular tissue, and many others. The theory of infinitesimal deformations of solids appeared in the XVII century in the works by Robert Hooke, who formulated one of the main assumptions of the theory: stresses are proportional to strains of bodies. Translating the assertion into mathematical language, this means that relations between stresses and displacements gradients of bodies are linear. Nowadays the theory of infinitesimal deformations is very deeply and thoroughly elaborated. On the different parts of this theory such as elasticity theory, plasticity theory, stability theory and many others there are many monographs and textbooks.

v

vi

Preface

But the well-known Hooke’s law does not hold for finite (or large) deformations: the basic relations between stresses and displacement gradients become ‘strongly non-linear’, and they cannot always be expressed analytically. The basis of finite deformations theory was laid in the XIX century by the eminent scientists A.L. Cauchy, J.L. Lagrange, L. Euler, G. Piola, A.J.C. Saint-Venant, G.R. Kirchhoff, and then developed by A.E.H. Love, G. Jaumann [28], M.A. Biot, F.D. Murnaghan [41] and other researchers. The works by M. Mooney and R.S. Rivlin written in the 1940s of the XX century contributed much to the formation of finite deformations theory as an independent part of continuum mechanics. The fundamental step was made in 1950–1960s of the XX century by the American mechanics school, first of all by B.D. Coleman [9], W. Noll and C. Truesdell [43, 54–56], who considered the nonlinear mechanics from the point of view of the formal mathematics. According to D. Hilbert, they introduced the axiomatics of nonlinear mechanics which structured the system of accumulated knowledge and made it possible to formulate the main directions of the further investigations in this theory. Together with R.S. Rivlin and A.J.M. Spencer [10, 13, 51] they elaborated the special mathematical apparatus for formulation of relationships, generalizing Hooke’s law for finite deformations, namely the theory of nonlinear tensor functions. And also the tensor analysis widely used in continuum mechanics was considerably adapted to the problems of nonlinear mechanics. Equations of continuum mechanics got the invariant (i.e. independent of the choice of a reference system) form. The further development of this direction was made by A.C. Eringen, A.E. Green, W. Zerna, J.E. Adkins and others [1–7, 11, 12, 14–27, 29, 30, 32–35, 38–40, 42, 47–50, 52, 53, 57–60]. The role of Russian mechanics school in the development of contemporary nonlinear continuum mechanics principles is also quite substantial. In 1968 the first edition of the fundamental two-part textbook ‘Continuum Mechanics’ by L.I. Sedov was published, which is still one of the most popular books on continuum mechanics in Russia. Outstanding results in the theory of finite elastic deformations were obtained by A.I. Lurie [36, 37], who wrote the principal monograph on the nonlinear theory of elasticity and systemized in it the problem classes of the theory of finite elastic deformations allowing for analytical solutions. Also the considerable step was made by K.F. Chernykh [8], who developed the theory of finite deformations for anisotropic media and elaborated the methods for solving the problems of nonlinear theory of shells and nonlinear theory of cracks. One can also mention the works by mechanics scientists: B.E. Pobedrya, V.I. Kondaurov, V.G. Karnauhov, A.A. Pozdeev, P.V. Trusov, Yu.I. Nyashin and many others who made considerable contributions to the theory of viscoelastic, elastoplastic and viscoplastic finite deformations. This book is based on the lectures which the author has been giving for many years in Moscow Bauman State Technical University. The book has several fundamental traits: 1. It follows the mathematical style of course exposition, which assumes the usage of axioms, definitions, theorems and proofs. 2. It applies the tensor apparatus, mostly in the indexless form, as the latter combined with the special skills is very convenient in usage, and does not shade

Preface

3.

4.

5.

6. 7.

vii

the physical essence of the laws, and permits proceeding to any appropriate coordinate system. It uses the divergence form of dynamic equations of deformation compatibility, that made it at last possible to write the complete system of balance laws of nonlinear mechanics in a single generalized form. The theory of constitutive equations being the key part of nonlinear mechanics is for the first time exposed with the usage of all energetic couples of tensors, which were established by R. Hill [26] and K.F. Chernykh [8] and ordered by the author [12], and also with quasi-energetic couples of tensors found by the author [12]. To derive constitutive equations of nonlinear continuum mechanics, the author applied the theory of nonlinear tensor functions and tensor operators, elaborated by A.J.M. Spencer, R.S. Rivlin, J.L. Ericksen, V.V. Lokhin, Yu.I. Sirotin, B.E. Pobedrya, the author of this book and others. The bases of theories of large elastic, viscoelastic and plastic deformations are explored from the uniform position. The book uses a ‘reader-friendly‘ style of material exposition, which can be characterized by the presence of quite detailed necessary mathematical calculations and proofs.

The axiomatic approach used in this book differs a bit from the analogous ones suggested by C. Truesdell [56] and other authors. The system of continuum mechanics axioms in the book is composed so as to minimize their total number, and give each axiom a clear physical interpretation. That is why the axioms by C. Truesdell connected with the logic relations between bodies are not included in the general list, the axioms on the bodies’ mass are united into one axiom, the mass conservation law, and analogously the axioms on the existence of forces and inertial reference systems are united into one axiom, the momentum balance law. Though, the last axiom is split into the two parts: first the Sect. 3.2 considers the case of inertial reference systems, and then Sect. 4.10 deals with non-inertial ones. Unlike the axiomatics by C. Truesdell [56], in this book the axiom system includes so called principles of constitutive equations construction which play a fundamental role in the formation of a continuum mechanics equations system. The axiomatic approach to the exploration of continuum mechanics possesses at least one merit – it permits the separation of all the values into two categories: primary and secondary. These are introduced axiomatically and consequently within the continuum mechanics there is no need to substantiate their appearance. The secondary category includes combinations of the first category’s values. The axiomatic approach allows us also to distinguish from continuum mechanics statements between the definitions and corollaries of them (theorems); this is extremely useful for the initial acquaintance with the course. To get acquainted with the specific apparatus of tensor analysis the reader is recommended to use the author’s book ‘Tensor Analysis and Nonlinear Tensor Functions’ [12], which uses the same main notations and definitions. All the references to the tensor analysis formulas in the text are addressed to the latter book.

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Preface

This book covers the fundamental classical parts of nonlinear continuum mechanics: kinematics, balance laws, constitutive equations, relations at singular surfaces, the basics of theories of large elastic deformations, large viscoelastic deformations and large plastic deformations. Because of limits on space, important parts such as the theory of shells at large deformations, and the theory of media with phase transformations were not included in the book. I would like to thank Professor B.E. Pobedrya (Moscow Lomonosov State University), Professor N.N. Smirnov (Moscow Lomonosov State University) and Professor V.S. Zarubin (Bauman Moscow State Technical University) for fruitful discussions and valuable advice on different problems in the book. I am very grateful to Professor G.M.L. Gladwell of the University of Waterloo, Canada, who edited the book and improved the English text. I also thank my wife, Dr. Irina D. Dimitrienko (Bauman Moscow State Technical University), who translated the book into English and prepared the camera-ready typescript. I hope that the book proves to be useful for graduates and post-graduates of mathematical and natural-scientific departments of universities and for investigators, academic scientists and engineers working in solid mechanics, mechanical engineering, applied mathematics and physics. I hope that the book is of interest also for material science specialists developing advanced materials. Russia

Yuriy Dimitrienko

Contents

1

Introduction: Fundamental Axioms of Continuum Mechanics . . . . . . . . . . .

1

2 Kinematics of Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1 Material and Spatial Descriptions of Continuum Motion.. . . . . . . . . . . . . 2.1.1 Lagrangian and Eulerian Coordinates: The Motion Law .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.2 Material and Spatial Descriptions.. . . . . . . . . .. . . . . . . . . . . . . . . . . ı 2.1.3 Local Bases in K and K . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.4 Tensors and Tensor Fields in Continuum Mechanics . . . . . . . ı 2.1.5 Covariant Derivatives in K and K . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.6 The Deformation Gradient . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.7 Curvilinear Spatial Coordinates.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2 Deformation Tensors and Measures . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.1 Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.2 Deformation Measures . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.3 Displacement Vector.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.4 Relations Between Components of Deformation Tensors and Displacement Vector .. . . . . . . . . 2.2.5 Physical Meaning of Components of the Deformation Tensor . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.6 Transformation of an Oriented Surface Element .. . . . . . . . . . . 2.2.7 Representation of the Inverse Metric Matrix in terms of Components of the Deformation Tensor .. . . . . . . 2.3 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3.1 Theorem on Polar Decomposition . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3.2 Eigenvalues and Eigenbases . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3.3 Representation of the Deformation Tensors in Eigenbases . 2.3.4 Geometrical Meaning of Eigenvalues . . . . . .. . . . . . . . . . . . . . . . . 2.3.5 Geometric Picture of Transformation of a Small Neighborhood of a Point of a Continuum . . . . . . . 2.4 Rate Characteristics of Continuum Motion.. . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.2 Total Derivative of a Tensor with Respect to Time . . . . . . . . .

5 5 5 9 9 11 13 14 16 24 24 25 25 26 28 30 33 36 36 40 42 44 45 49 49 50 ix

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Contents

2.4.3 2.4.4 2.4.5

Differential of a Tensor .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Properties of Derivatives with Respect to Time .. . . . . . . . . . . . The Velocity Gradient, the Deformation Rate Tensor and the Vorticity Tensor .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.6 Eigenvalues of the Deformation Rate Tensor . . . . . . . . . . . . . . . 2.4.7 Resolution of the Vorticity Tensor for the Eigenbasis of the Deformation Rate Tensor .. . . . . . . . . . . . . . . . 2.4.8 Geometric Picture of Infinitesimal Transformation of a Small Neighborhood of a Point . . . . . . . 2.4.9 Kinematic Meaning of the Vorticity Vector . . . . . . . . . . . . . . . . . 2.4.10 Tensor of Angular Rate of Rotation (Spin) .. . . . . . . . . . . . . . . . . 2.4.11 Relationships Between Rates of Deformation Tensors and Velocity Gradients . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.12 Trajectory of a Material Point, Streamline and Vortex Line .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.13 Stream Tubes and Vortex Tubes . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Co-rotational Derivatives .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.1 Definition of Co-rotational Derivatives .. . . .. . . . . . . . . . . . . . . . . 2.5.2 The Oldroyd Derivative (hi D ri ) . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.3 The Cotter–Rivlin Derivative (hi D ri ) . . . . .. . . . . . . . . . . . . . . . . 2.5.4 Mixed Co-rotational Derivatives .. . . . . . . . . . .. ................ ı 2.5.5 The Derivative Relative to the Eigenbasis pi of the Right Stretch Tensor .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.6 The Derivative in the Eigenbasis (hi D pi ) of the Left Stretch Tensor . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.7 The Jaumann Derivative (hi D qi ). . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.8 Co-rotational Derivatives in a Moving Orthonormal Basis . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.9 Spin Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.10 Universal Form of the Co-rotational Derivatives.. . . . . . . . . . . 2.5.11 Relations Between Co-rotational Derivatives of Deformation Rate Tensors and Velocity Gradient . . . . . . .

53 54

3 Balance Laws .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1 The Mass Conservation Law .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1.1 Integral and Differential Forms . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1.2 The Continuity Equation in Lagrangian Variables . . . . . . . . . . 3.1.3 Differentiation of Integral over a Moving Volume .. . . . . . . . . 3.1.4 The Continuity Equation in Eulerian Variables . . . . . . . . . . . . . 3.1.5 Determination of the Total Derivatives with respect to Time . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1.6 The Gauss–Ostrogradskii Formulae . . . . . . . .. . . . . . . . . . . . . . . . . 3.2 The Momentum Balance Law and the Stress Tensor .. . . . . . . . . . . . . . . . . 3.2.1 The Momentum Balance Law. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.2.2 External and Internal Forces . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

89 89 89 90 91 92

2.5

56 58 59 60 62 63 65 73 75 77 77 79 80 81 81 82 83 83 84 85 85

93 94 95 95 97

Contents

xi

3.2.3 3.2.4 3.2.5 3.2.6

3.3

3.4

3.5

3.6

Cauchy’s Theorems on Properties of the Stress Vector .. . . . 98 Generalized Cauchy’s Theorem .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .101 The Cauchy and Piola–Kirchhoff Stress Tensors . . . . . . . . . . .102 Physical Meaning of Components of the Cauchy Stress Tensor .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .103 3.2.7 The Momentum Balance Equation in Spatial and Material Descriptions .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .107 The Angular Momentum Balance Law .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .109 3.3.1 The Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .109 3.3.2 Tensor of Moment Stresses . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .110 3.3.3 Differential Form of the Angular Momentum Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .111 3.3.4 Nonpolar and Polar Continua . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .112 3.3.5 The Angular Momentum Balance Equation in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .113 The First Thermodynamic Law . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .114 3.4.1 The Integral Form of the Energy Balance Law.. . . . . . . . . . . . .114 3.4.2 The Heat Flux Vector .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .116 3.4.3 The Energy Balance Equation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .117 3.4.4 Kinetic Energy and Heat Influx Equation . .. . . . . . . . . . . . . . . . .118 3.4.5 The Energy Balance Equation in Lagrangian Description .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .119 3.4.6 The Energy Balance Law for Polar Continua . . . . . . . . . . . . . . .121 The Second Thermodynamic Law . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .124 3.5.1 The Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .124 3.5.2 Differential Form of the Second Thermodynamic Law . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .126 3.5.3 The Second Thermodynamic Law in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .127 3.5.4 Heat Machines and Their Efficiency .. . . . . . .. . . . . . . . . . . . . . . . .128 3.5.5 Adiabatic and Isothermal Processes. The Carnot Cycles . . .132 3.5.6 Truesdell’s Theorem .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .136 Deformation Compatibility Equations .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .141 3.6.1 Compatibility Conditions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .141 3.6.2 Integrability Condition for Differential Form . . . . . . . . . . . . . . .142 3.6.3 The First Form of Deformation Compatibility Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .142 3.6.4 The Second Form of Compatibility Conditions .. . . . . . . . . . . .143 3.6.5 The Third Form of Compatibility Conditions .. . . . . . . . . . . . . .145 3.6.6 Properties of Components of the Riemann–Christoffel Tensor . . . . . . . . .. . . . . . . . . . . . . . . . .146 3.6.7 Interchange of the Second Covariant Derivatives .. . . . . . . . . .148 3.6.8 The Static Compatibility Equation.. . . . . . . . .. . . . . . . . . . . . . . . . .148

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3.7

3.8 3.9

Dynamic Compatibility Equations .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .149 3.7.1 Dynamic Compatibility Equations in Lagrangian Description.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .149 3.7.2 Dynamic Compatibility Equations in Spatial Description ..151 Compatibility Equations for Deformation Rates . . . . . .. . . . . . . . . . . . . . . . .152 The Complete System of Continuum Mechanics Laws .. . . . . . . . . . . . . . .155 3.9.1 The Complete System in Eulerian Description.. . . . . . . . . . . . .155 3.9.2 The Complete System in Lagrangian Description . . . . . . . . . .156 3.9.3 Integral Form of the System of Continuum Mechanics Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .157

4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .161 4.1 Basic Principles for Derivation of Constitutive Equations.. . . . . . . . . . . .161 4.2 Energetic and Quasienergetic Couples of Tensors . . . .. . . . . . . . . . . . . . . . .162 4.2.1 Energetic Couples of Tensors . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .162 I

4.2.2

The First Energetic Couple .T; ƒ/ . . . . . . . . . .. . . . . . . . . . . . . . . . .164

4.2.3

The Fifth Energetic Couple .T; C/ . . . . . . . . .. . . . . . . . . . . . . . . . .165

4.2.4

The Fourth Energetic Couple . T ; .U E// . . . . . . . . . . . . . . . . .166

4.2.5

The Second Energetic Couple .T; .E U1 // . . . . . . . . . . . . .167

4.2.6 4.2.7

The Third Energetic Couple . T ; B/ . . . . . . . .. . . . . . . . . . . . . . . . .167 General Representations for Energetic Tensors of Stresses and Deformations .. . . . .. . . . . . . . . . . . . . . . .168 Energetic Deformation Measures . . . . . . . . . . .. . . . . . . . . . . . . . . . .173 Relationships Between Principal Invariants of Energetic Deformation Measures and Tensors . . . . . . . . . . .175 Quasienergetic Couples of Stress and Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .176

4.2.8 4.2.9 4.2.10

V

IV

II

III

I

4.2.11

The First Quasienergetic Couple .S; A/ . . . .. . . . . . . . . . . . . . . . .177

4.2.12 4.2.13

The Second Quasienergetic Couple .S; .E V1 // . . . . . . . .178 The Third Quasienergetic Couple .Y; TS /.. . . . . . . . . . . . . . . . .178

4.2.14

The Fourth Quasienergetic Couple . S ; .V E// . . . . . . . . . . .179

4.2.15 4.2.16 4.2.17 4.2.18

The Fifth Quasienergetic Couple .S; J/ . . . . .. . . . . . . . . . . . . . . . .180 General Representation of Quasienergetic Tensors . . . . . . . . .180 Quasienergetic Deformation Measures . . . . .. . . . . . . . . . . . . . . . .182 Representation of Rotation Tensor of Stresses in Terms of Quasienergetic Couples of Tensors .. . . . . . . . . . . .183 Relations Between Density and Principal Invariants of Energetic and Quasienergetic Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .185 The Generalized Form of Representation of the Stress Power . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .186

4.2.19

4.2.20

II

IV

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4.2.21

4.3

4.4

4.5

4.6

4.7

4.8

Representation of Stress Power in Terms of Co-rotational Derivatives .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .187 4.2.22 Relations Between Rates of Energetic and Quasienergetic Tensors and Velocity Gradient . . . . . . . . .188 The Principal Thermodynamic Identity . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .196 4.3.1 Different Forms of the Principal Thermodynamic Identity . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .196 4.3.2 The Clausius–Duhem Inequality .. . . . . . . . . . .. . . . . . . . . . . . . . . . .198 4.3.3 The Helmholtz Free Energy .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .198 4.3.4 The Gibbs Free Energy .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .199 4.3.5 Enthalpy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .201 4.3.6 Universal Form of the Principal Thermodynamic Identity . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .202 4.3.7 Representation of the Principal Thermodynamic Identity in Terms of Co-rotational Derivatives .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .203 Principles of Thermodynamically Consistent Determinism, Equipresence and Local Action . . . . . . . .. . . . . . . . . . . . . . . . .205 4.4.1 Active and Reactive Variables . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .205 4.4.2 The Principle of Thermodynamically Consistent Determinism .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .206 4.4.3 The Principle of Equipresence . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .208 4.4.4 The Principle of Local Action . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .208 Definition of Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .209 4.5.1 Classification of Types of Continua . . . . . . . .. . . . . . . . . . . . . . . . .209 4.5.2 General Form of Constitutive Equations for Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .210 The Principle of Material Symmetry . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .213 4.6.1 Different Reference Configurations.. . . . . . . .. . . . . . . . . . . . . . . . .213 4.6.2 H -indifferent and H -invariant Tensors . . . .. . . . . . . . . . . . . . . . .215 4.6.3 Symmetry Groups of Continua . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .219 4.6.4 The Statement of the Principle of Material Symmetry.. . . . .220 Definition of Fluids and Solids. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .221 4.7.1 Fluids and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .221 4.7.2 Isomeric Symmetry Groups .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .222 4.7.3 Definition of Anisotropic Solids . . . . . . . . . . . .. . . . . . . . . . . . . . . . .226 4.7.4 H -indifference and H -invariance of Tensors Describing the Motion of a Solid . . . . . . . . . . .. . . . . . . . . . . . . . . . .228 4.7.5 H -invariance of Rate Characteristics of a Solid . . . . . . . . . . . .231 Corollaries of the Principle of Material Symmetry and Constitutive Equations for Ideal Continua . . . . . . . .. . . . . . . . . . . . . . . . .236 4.8.1 Corollary of the Principle of Material Symmetry for Models An of Ideal (Elastic) Solids . . . . . . . . .236 4.8.2 Scalar Indifferent Functions of Tensor Argument.. . . . . . . . . .237 4.8.3 Producing Tensors of Groups . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .239

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4.8.4 4.8.5

Scalar Invariants of a Second-Order Tensor .. . . . . . . . . . . . . . . .240 Representation of a Scalar Indifferent Function in Terms of Invariants .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .243 4.8.6 Indifferent Tensor Functions of Tensor Argument and Invariant Representation of Constitutive Equations for Elastic Continua . . . . . . . . . . . . .244 4.8.7 Quasilinear and Linear Models An of Elastic Continua . . . .249 4.8.8 Constitutive Equations for Models Bn of Elastic Continua . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .253 4.8.9 Corollaries to the Principle of Material Symmetry for Models Cn and Dn of Elastic Continua . . . . .254 4.8.10 General Representation of Constitutive Equations for All Models of Elastic Continua . . . . . . . . . . . . . .262 4.8.11 Representation of Constitutive Equations of Isotropic Elastic Continua in Eigenbases .. . . . . . . . . . . . . . . .265 4.8.12 Representation of Constitutive Equations of Isotropic Elastic Continua ‘in Rates’ . . . .. . . . . . . . . . . . . . . . .271 4.8.13 Application of the Principle of Material Symmetry to Fluids . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .276 4.8.14 Functional Energetic Couples of Tensors.. .. . . . . . . . . . . . . . . . .283 4.9 Incompressible Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .287 4.9.1 Definition of Incompressible Continua . . . . .. . . . . . . . . . . . . . . . .287 4.9.2 The Principal Thermodynamic Identity for Incompressible Continua . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .288 4.9.3 Constitutive Equations for Ideal Incompressible Continua .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .289 4.9.4 Corollaries to the Principle of Material Symmetry for Incompressible Fluids . . . . . . .. . . . . . . . . . . . . . . . .291 4.9.5 Representation of Constitutive Equations for Incompressible Solids in Tensor Bases .. . . . . . . . . . . . . . . . .292 4.9.6 General Representation of Constitutive Equations for All the Models of Incompressible Ideal Solids . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .295 4.9.7 Linear Models of Ideal Incompressible Elastic Continua.. .296 4.9.8 Representation of Models Bn and Dn of Incompressible Isotropic Elastic Continua in Eigenbasis.. .298 4.10 The Principle of Material Indifference.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .300 4.10.1 Rigid Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .300 4.10.2 R-indifferent and R-invariant Tensors .. . . . .. . . . . . . . . . . . . . . . .301 4.10.3 Density and Deformation Gradient in Rigid Motion.. . . . . . .302 4.10.4 Deformation Tensors in Rigid Motion .. . . . .. . . . . . . . . . . . . . . . .303 4.10.5 Stress Tensors in Rigid Motion . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .304 4.10.6 The Velocity in Rigid Motion .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .306 4.10.7 The Deformation Rate Tensor and the Vorticity Tensor in Rigid Motion.. .. . . . . . . . . . . . . . . . .306

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4.10.8 4.10.9 4.10.10 4.10.11

Co-rotational Derivatives in Rigid Motion .. . . . . . . . . . . . . . . . .307 The Statement of the Principle of Material Indifference.. . .312 Material Indifference of the Continuity Equation .. . . . . . . . . .313 Material Indifference for the Momentum Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .313 4.10.12 Material Indifference of the Thermodynamic Laws . . . . . . . .316 4.10.13 Material Indifference of the Compatibility Equations . . . . . .318 4.10.14 Material Indifference of Models An and Bn of Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .319 4.10.15 Material Indifference for Models Cn and Dn of Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .320 4.10.16 Material Indifference for Incompressible Continua .. . . . . . . .322 4.10.17 Material Indifference for Models of Solids ‘in Rates’ . . . . . .322 4.11 Relationships in a Moving System . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .324 4.11.1 A Moving Reference System . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .324 4.11.2 The Euler Formula .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .326 4.11.3 The Coriolis Formula .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .327 4.11.4 The Nabla-Operator in a Moving System . .. . . . . . . . . . . . . . . . .329 4.11.5 The Velocity Gradient in a Moving System . . . . . . . . . . . . . . . . .330 4.11.6 The Continuity Equation in a Moving System . . . . . . . . . . . . . .330 4.11.7 The Momentum Balance Equation in a Moving System . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .331 4.11.8 The Thermodynamic Laws in a Moving System .. . . . . . . . . . .331 4.11.9 The Equation of Deformation Compatibility in a Moving System . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .332 4.11.10 The Kinematic Equation in a Moving System . . . . . . . . . . . . . .335 4.11.11 The Complete System of Continuum Mechanics Laws in a Moving Coordinate System . . . . . . . . . .335 4.11.12 Constitutive Equations in a Moving System . . . . . . . . . . . . . . . .335 4.11.13 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .338 4.12 The Onsager Principle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .339 4.12.1 The Onsager Principle and the Fourier Law.. . . . . . . . . . . . . . . .339 4.12.2 Corollaries of the Principle of Material Symmetry for the Fourier Law . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .341 4.12.3 Corollary of the Principle of Material Indifference for the Fourier Law .. . . . . . . . . . .. . . . . . . . . . . . . . . . .343 4.12.4 The Fourier Law for Fluids . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .344 4.12.5 The Fourier Law for Solids . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .345 5 Relations at Singular Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .347 5.1 Relations at a Singular Surface in the Material Description .. . . . . . . . . .347 5.1.1 Singular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .347 5.1.2 The First Classification of Singular Surfaces.. . . . . . . . . . . . . . .347 5.1.3 Axiom on the Class of Functions across a Singular Surface.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .350

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5.2

5.3

5.4

5.1.4

The Rule of Differentiation of a Volume Integral in the Presence of a Singular Surface.. . . . . . . . . . . . . .352

5.1.5 5.1.6

Relations at a Coherent Singular Surface in K . . . . . . . . . . . . . .355 Relation Between Velocities of a Singular

ı

ı

Surface in K and K . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .357 Relations at a Singular Surface in the Spatial Description.. . . . . . . . . . . .358 5.2.1 Relations at a Coherent Singular Surface in the Spatial Description . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .358 5.2.2 The Rule of Differentiation of an Integral over a Moving Volume Containing a Singular Surface . . . . .360 Explicit Form of Relations at a Singular Surface . . . . .. . . . . . . . . . . . . . . . .362 5.3.1 Explicit Form of Relations at a Surface of a Strong Discontinuity in a Reference Configuration .. . .362 5.3.2 Explicit Form of Relations at a Surface of a Strong Discontinuity in an Actual Configuration . . . . . .363 5.3.3 Mass Rate of Propagation of a Singular Surface .. . . . . . . . . . .363 5.3.4 Relations at a Singular Surface Without Transition of Material Points . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .365 The Main Types of Singular Surfaces . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .366 5.4.1 Jump of Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .366 5.4.2 Jumps of Radius-Vector and Displacement Vector.. . . . . . . . .367 5.4.3 Semicoherent and Completely Incoherent Singular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .369 5.4.4 Nondissipative and Homothermal Singular Surfaces . . . . . . .369 5.4.5 Surfaces with Ideal Contact . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .370 5.4.6 On Boundary Conditions .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .372 5.4.7 5.4.8

ı

Equation of a Singular Surface in K . . . . . . . .. . . . . . . . . . . . . . . . .372 Equation of a Singular Surface in K . . . . . . . .. . . . . . . . . . . . . . . . .374

6 Elastic Continua at Large Deformations . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .377 6.1 Closed Systems in the Spatial Description . . . . . . . . . . . .. . . . . . . . . . . . . . . . .377 6.1.1 RU VF -system of Thermoelasticity .. . . . . .. . . . . . . . . . . . . . . . .377 6.1.2 RVF -, RU V -, and U V -Systems of Dynamic Equations of Thermoelasticity . . . . . . . . . . . . . . . . .381 6.1.3 T RU VF -system of Dynamic Equations of Thermoelasticity .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .384 6.1.4 Component Form of the Dynamic Equation System of Thermoelasticity in the Spatial Description . . . . .385 6.1.5 The Model of Quasistatic Processes in Elastic Solids at Large Deformations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .388 6.2 Closed Systems in the Material Description.. . . . . . . . . .. . . . . . . . . . . . . . . . .390 6.2.1 U VF -system of Dynamic Equations of Thermoelasticity in the Material Description . . . . . . . . . . . .390

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U V - and U -systems of Thermoelasticity in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .394 6.2.3 T U VF -system of Thermoelasticity in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .395 6.2.4 The Equation System of Thermoelasticity for Quasistatic Processes in the Material Description . . . . . .396 Statements of Problems for Elastic Continua at Large Deformations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .399 6.3.1 Boundary Conditions in the Spatial Description .. . . . . . . . . . .399 6.3.2 Boundary Conditions in the Material Description . . . . . . . . . .402 6.3.3 Statements of Main Problems of Thermoelasticity at Large Deformations in the Spatial Description . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .405 6.3.4 Statements of Thermoelasticity Problems in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .408 6.3.5 Statements of Quasistatic Problems of Elasticity Theory at Large Deformations .. . . . . . . . . . . . . . . .410 6.3.6 Conditions on External Forces in Quasistatic Problems .. . .412 6.3.7 Variational Statement of the Quasistatic Problem in the Spatial Description . . . . . . . . .. . . . . . . . . . . . . . . . .413 6.3.8 Variational Statement of Quasistatic Problem in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .417 6.3.9 Variational Statement for Incompressible Continua in the Material Description .. . . . . .. . . . . . . . . . . . . . . . .417 The Problem on an Elastic Beam in Tension .. . . . . . . . .. . . . . . . . . . . . . . . . .421 6.4.1 Semi-Inverse Method .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .421 6.4.2 Deformation of a Beam in Tension . . . . . . . . .. . . . . . . . . . . . . . . . .421 6.4.3 Stresses in a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .422 6.4.4 The Boundary Conditions .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .424 6.4.5 Resolving Relation 1 k1 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .424 6.4.6 Comparative Analysis of Different Models An . . . . . . . . . . . . .425 Tension of an Incompressible Beam . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .430 6.5.1 Deformation of an Incompressible Elastic Beam . . . . . . . . . . .430 6.5.2 Stresses in an Incompressible Beam for Models Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .431 6.5.3 Resolving Relation 1 .k1 / . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .432 6.5.4 Comparative Analysis of Models Bn . . . . . . .. . . . . . . . . . . . . . . . .432 6.5.5 Stresses in an Incompressible Beam for Models An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .436 Simple Shear .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .438 6.6.1 Deformations in Simple Shear . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .438 6.6.2 Stresses in the Problem on Shear . . . . . . . . . . .. . . . . . . . . . . . . . . . .439 6.6.3 Boundary Conditions in the Problem on Shear . . . . . . . . . . . . .441 6.2.2

6.3

6.4

6.5

6.6

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6.6.4

6.7

6.8

Comparative Analysis of Different Models An for the Problem on Shear . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .442 6.6.5 Shear of an Incompressible Elastic Continuum . . . . . . . . . . . . .443 The Lam´e Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .446 6.7.1 The Motion Law for a Pipe in the Lam´e Problem . . . . . . . . . .446 6.7.2 The Deformation Gradient and Deformation Tensors in the Lam´e Problem . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .448 6.7.3 Stresses in the Lam´e Problem for Models An . . . . . . . . . . . . . . .449 6.7.4 Equation for the Function f . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .450 6.7.5 Boundary Conditions of the Weak Type .. . .. . . . . . . . . . . . . . . . .451 6.7.6 Boundary Conditions of the Rigid Type .. . .. . . . . . . . . . . . . . . . .453 The Lam´e Problem for an Incompressible Continuum . . . . . . . . . . . . . . . .454 6.8.1 Equation for the Function f . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .454 6.8.2 Stresses in the Lam´e Problem for an Incompressible Continuum . . . . . . . . . .. . . . . . . . . . . . . . . . .454 6.8.3 Equation for Hydrostatic Pressure p. . . . . . . .. . . . . . . . . . . . . . . . .456 6.8.4 Analysis of the Problem Solution .. . . . . . . . . .. . . . . . . . . . . . . . . . .456

7 Continua of the Differential Type . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .461 7.1 Models An and Bn of Continua of the Differential Type . . . . . . . . . . . . . .461 7.1.1 Constitutive Equations for Models An of Continua of the Differential Type .. . . . . . .. . . . . . . . . . . . . . . . .461 7.1.2 Corollary of the Onsager Principle for Models An of Continua of the Differential Type .. . .. . . . . . . . . . . . . . . . .463 7.1.3 The Principle of Material Symmetry for Models An of Continua of the Differential Type .. . . . . . .465 7.1.4 Representation of Constitutive Equations for Models An of Solids of the Differential Type in Tensor Bases . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .467 7.1.5 Models Bn of Solids of the Differential Type .. . . . . . . . . . . . . .470 7.1.6 Models Bn of Incompressible Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .471 7.1.7 The Principle of Material Indifference for Models An and Bn of Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .472 7.2 Models An and Bn of Fluids of the Differential Type . . . . . . . . . . . . . . . . .473 7.2.1 Tensor of Equilibrium Stresses for Fluids of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .473 7.2.2 The Tensor of Viscous Stresses in the Model AI of a Fluid of the Differential Type .. . . . . .. . . . . . . . . . . . . . . . .474 7.2.3 Simultaneous Invariants for Fluids of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .476 7.2.4 Tensor of Viscous Stresses in Model AV of Fluids of the Differential Type .. . . . . . . . . .. . . . . . . . . . . . . . . . .478

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xix

Viscous Coefficients in Model AV of a Fluid of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .479 7.2.6 General Representation of Constitutive Equations for Fluids of the Differential Type . . . . . . . . . . . . . . .480 7.2.7 Constitutive Equations for Incompressible Viscous Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .481 7.2.8 The Principle of Material Indifference for Models An and Bn of Fluids of the Differential Type . . . . . . . . . . . . . . .481 Models Cn and Dn of Continua of the Differential Type .. . . . . . . . . . . . .482 7.3.1 Models Cn of Continua of the Differential Type .. . . . . . . . . . .482 7.3.2 Models Cnh of Solids with Co-rotational Derivatives . . . . . . .484 7.3.3 Corollaries of the Principle of Material Symmetry for Models Cnh of Solids . . . . . . . .. . . . . . . . . . . . . . . . .485 7.3.4 Viscosity Tensor in Models Cnh . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .488 7.3.5 Final Representation of Constitutive Equations for Model Cnh of Isotropic Solids . . . . . . . . . . . . . . . .489 7.3.6 Models Dnh of Isotropic Solids . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .491 The Problem on a Beam in Tension.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .492 7.4.1 Rate Characteristics of a Beam . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .492 7.4.2 Stresses in the Beam .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .492 7.4.3 Resolving Relation .k1 ; kP1 / . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .493 7.4.4 Comparative Analysis of Creep Curves for Different Models Bn . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .493 7.4.5 Analysis of Deforming Diagrams for Different Models Bn of Continua of the Differential Type .. . . . . . . . . . .496 7.2.5

7.3

7.4

8 Viscoelastic Continua at Large Deformations . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .497 8.1 Viscoelastic Continua of the Integral Type . . . . . . . . . . . .. . . . . . . . . . . . . . . . .497 8.1.1 Definition of Viscoelastic Continua.. . . . . . . .. . . . . . . . . . . . . . . . .497 8.1.2 Tensor Functional Space . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .498 8.1.3 Continuous and Differentiable Functionals . . . . . . . . . . . . . . . . .499 8.1.4 Axiom of Fading Memory . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .503 8.1.5 Models An of Viscoelastic Continua . . . . . . .. . . . . . . . . . . . . . . . .505 8.1.6 Corollaries of the Principle of Material Symmetry for Models An of Viscoelastic Continua . . . . . . . .506 8.1.7 General Representation of Functional of Free Energy in Models An . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .507 8.1.8 Model An of Stable Viscoelastic Continua .. . . . . . . . . . . . . . . . .510 8.1.9 Model An of a Viscoelastic Continuum with Difference Cores . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .511 8.1.10 Model An of a Thermoviscoelastic Continuum .. . . . . . . . . . . .513 8.1.11 Model An of a Thermorheologically Simple Viscoelastic Continuum . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .514 8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua .. . .516 8.2.1 Principal Models An of Viscoelastic Continua .. . . . . . . . . . . . .516

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Principal Model An of an Isotropic Thermoviscoelastic Continuum .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .518 8.2.3 Principal Model An of a Transversely Isotropic Thermoviscoelastic Continuum . .. . . . . . . . . . . . . . . . .519 8.2.4 Principal Model An of an Orthotropic Thermoviscoelastic Continuum .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .520 8.2.5 Quadratic Models An of Thermoviscoelastic Continua . . . .522 8.2.6 Linear Models An of Viscoelastic Continua . . . . . . . . . . . . . . . .522 8.2.7 Representation of Linear Models An in the Boltzmann Form .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .525 8.2.8 Mechanically Determinate Linear Models An of Viscoelastic Continua . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .528 8.2.9 Linear Models An for Isotropic Viscoelastic Continua.. . . .530 8.2.10 Linear Models An of Transversely Isotropic Viscoelastic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .531 8.2.11 Linear Models An of Orthotropic Viscoelastic Continua .. .532 8.2.12 The Tensor of Relaxation Functions .. . . . . . .. . . . . . . . . . . . . . . . .533 8.2.13 Spectral Representation of Linear Models An of Viscoelastic Continua . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .535 8.2.14 Exponential Relaxation Functions and Differential Form of Constitutive Equations.. . . . . . . . . . .539 8.2.15 Inversion of Constitutive Equations for Linear Models of Viscoelastic Continua . . . . . . . . . . .. . . . . . . . . . . . . . . . .542 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .549 8.3.1 Models An of Incompressible Viscoelastic Continua .. . . . . .549 8.3.2 Principal Models An of Incompressible Isotropic Viscoelastic Continua .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .549 8.3.3 Linear Models An of Incompressible Isotropic Viscoelastic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .551 8.3.4 Models Bn of Viscoelastic Continua . . . . . . .. . . . . . . . . . . . . . . . .552 8.3.5 Models An and Bn of Viscoelastic Fluids . .. . . . . . . . . . . . . . . . .554 8.3.6 The Principle of Material Indifference for Models An and Bn of Viscoelastic Continua .. . . . . . . . . . .556 Statements of Problems in Viscoelasticity Theory at Large Deformations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .558 8.4.1 Statements of Dynamic Problems in the Spatial Description . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .558 8.4.2 Statements of Dynamic Problems in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .562 8.4.3 Statements of Quasistatic Problems of Viscoelasticity Theory in the Spatial Description .. . . . . . .564 8.4.4 Statements of Quasistatic Problems for Models of Viscoelastic Continua in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .566 8.2.2

8.3

8.4

Contents

8.5

8.6

xxi

The Problem on Uniaxial Deforming of a Viscoelastic Beam . . . . . . . . .568 8.5.1 Deformation of a Viscoelastic Beam in Uniaxial Tension .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .568 8.5.2 Viscous Stresses in Uniaxial Tension .. . . . . .. . . . . . . . . . . . . . . . .569 8.5.3 Stresses in a Viscoelastic Beam in Tension . . . . . . . . . . . . . . . . .569 8.5.4 Resolving Relation 1 .k1 / for a Viscoelastic Beam . . . . . . . .570 8.5.5 Method of Calculating the Constants B . / and . / . . . . . . . .571 8.5.6 Method for Evaluating the Constants m, N l1 , l2 and ˇ, m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .574 8.5.7 Computations of Relaxation Curves .. . . . . . .. . . . . . . . . . . . . . . . .575 8.5.8 Cyclic Deforming of a Beam. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .577 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .579 8.6.1 The Problem on Dissipative Heating of a Beam Under Cyclic Deforming .. . . . . . .. . . . . . . . . . . . . . . . .579 8.6.2 Fast and Slow Times in Multicycle Deforming . . . . . . . . . . . . .580 8.6.3 Differentiation and Integration of Quasiperiodic Functions.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .580 8.6.4 Heat Conduction Equation for a Thin Viscoelastic Beam . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .581 8.6.5 Dissipation Function for a Viscoelastic Beam . . . . . . . . . . . . . .582 8.6.6 Asymptotic Expansion in Terms of a Small Parameter . . . . .583 8.6.7 Averaged Heat Conduction Equation .. . . . . .. . . . . . . . . . . . . . . . .584 8.6.8 Temperature of Dissipative Heating in a Symmetric Cycle. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .585 8.6.9 Regimes of Dissipative Heating Without Heat Removal . . .585 8.6.10 Regimes of Dissipative Heating in the Presence of Heat Removal . . . . . . . . . . .. . . . . . . . . . . . . . . . .586 8.6.11 Experimental and Computed Data on Dissipative Heating of Viscoelastic Bodies .. . . . . . . . . . . . .588

9 Plastic Continua at Large Deformations . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .591 9.1 Models An of Plastic Continua at Large Deformations .. . . . . . . . . . . . . . .591 9.1.1 Main Assumptions of the Models. . . . . . . . . . .. . . . . . . . . . . . . . . . .591 9.1.2 General Representation of Constitutive Equations for Models An of Plastic Continua .. . . . . . . . . . . . . .594 9.1.3 Corollary of the Onsager Principle for Models An of Plastic Continua . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .597 9.1.4 Models An of Plastic Yield. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .598 9.1.5 Associated Model of Plasticity An . . . . . . . . . .. . . . . . . . . . . . . . . . .599 9.1.6 Corollary of the Principle of Material Symmetry for the Associated Model An of Plasticity . . . . . .603 9.1.7 Associated Models of Plasticity An for Isotropic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .605

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9.1.8 9.1.9

9.2

9.3

The Huber–Mises Model for Isotropic Plastic Continua . . .607 Associated Models of Plasticity An for Transversely Isotropic Continua . . . . . . . .. . . . . . . . . . . . . . . . .610 9.1.10 Two-Potential Model of Plasticity for a Transversely Isotropic Continuum .. . .. . . . . . . . . . . . . . . . .612 9.1.11 Associated Models of Plasticity An for Orthotropic Continua . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .614 9.1.12 The Orthotropic Unipotential Huber–Mises Model for Plastic Continua.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .616 9.1.13 The Principle of Material Indifference for Models An of Plastic Continua . . . . . . . . .. . . . . . . . . . . . . . . . .618 Models Bn of Plastic Continua . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .623 9.2.1 Representation of Stress Power for Models Bn of Plastic Continua . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .623 9.2.2 General Representation of Constitutive Equations for Models Bn of Plastic Continua .. . . . . . . . . . . . . .627 9.2.3 Corollaries of the Onsager Principle for Models Bn of Plastic Continua . . . . . . . . .. . . . . . . . . . . . . . . . .629 9.2.4 Associated Models Bn of Plastic Continua . . . . . . . . . . . . . . . . .631 9.2.5 Corollary of the Principle of Material Symmetry for Associated Model Bn of Plasticity . . . . . . . . . .632 9.2.6 Associated Models of Plasticity Bn with Proper Strengthening . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .635 9.2.7 Associated Models of Plasticity Bn for Isotropic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .635 9.2.8 Associated Models of Plasticity Bn for Transversely Isotropic Continua . . . . . . . .. . . . . . . . . . . . . . . . .637 9.2.9 Associated Models of Plasticity Bn for Orthotropic Continua . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .638 9.2.10 The Principle of Material Indifference for Models Bn of Plastic Continua . . . . . . . . .. . . . . . . . . . . . . . . . .639 Models Cn and Dn of Plastic Continua . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .640 9.3.1 General Representation of Constitutive Equations for Models Cn of Plastic Continua .. . . . . . . . . . . . . .640 9.3.2 Constitutive Equations for Models Cn of Isotropic Plastic Continua .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .643 9.3.3 General Representation of Constitutive Equations for Models Dn of Plastic Continua . . . . . . . . . . . . . .646 9.3.4 Constitutive Equations for Models Dn of Isotropic Plastic Continua .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .650 9.3.5 The Principles of Material Symmetry and Material Indifference for Models Cn and Dn .. . . . . . . . . . . . . .651

Contents

9.4

9.5

9.6

9.7

9.8

9.9

xxiii

Constitutive Equations of Plasticity Theory ‘in Rates’ . . . . . . . . . . . . . . . .652 9.4.1 Representation of Models An of Plastic Continua ‘in Rates’ . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .652 Statements of Problems in Plasticity Theory .. . . . . . . . .. . . . . . . . . . . . . . . . .655 9.5.1 Statements of Dynamic Problems for Models An of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .655 9.5.2 Statements of Quasistatic Problems for Models An of Plasticity . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .657 The Problem on All-Round Tension–Compression of a Plastic Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .659 9.6.1 Deformation in All-Round Tension–Compression .. . . . . . . . .659 9.6.2 Stresses in All-Round Tension–Compression .. . . . . . . . . . . . . .660 9.6.3 The Case of a Plastically Incompressible Continuum . . . . . .661 9.6.4 The Case of a Plastically Compressible Continuum . . . . . . . .662 9.6.5 Cyclic Loading of a Plastically Compressible Continuum .664 The Problem on Tension of a Plastic Beam . . . . . . . . . . .. . . . . . . . . . . . . . . . .666 9.7.1 Deformation of a Beam in Uniaxial Tension .. . . . . . . . . . . . . . .666 9.7.2 Stresses in a Plastic Beam . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .667 9.7.3 Plastic Deformations of a Beam . . . . . . . . . . . .. . . . . . . . . . . . . . . . .668 9.7.4 Change of the Density . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .670 9.7.5 Resolving Equation for the Problem .. . . . . . .. . . . . . . . . . . . . . . . .671 9.7.6 Numerical Method for the Resolving Equation . . . . . . . . . . . . .672 9.7.7 Method for Determination of Constants H0 , n0 , and s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .675 9.7.8 Comparison with Experimental Data for Alloys . . . . . . . . . . . .676 9.7.9 Comparison with Experimental Data for Grounds .. . . . . . . . .677 Plane Waves in Plastic Continua .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .679 9.8.1 Formulation of the Problem .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .679 9.8.2 The Motion Law and Deformation of a Plate . . . . . . . . . . . . . . .680 9.8.3 Stresses in the Plate. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .681 9.8.4 The System of Dynamic Equations for the Plane Problem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .682 9.8.5 The Statement of Problem on Plane Waves in Plastic Continua.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .684 9.8.6 Solving the Problem by the Characteristic Method . . . . . . . . .685 9.8.7 Comparative Analysis of the Solution for Different Models An . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .689 9.8.8 Plane Waves in Models AIV and AV . . . . . . . .. . . . . . . . . . . . . . . . .691 9.8.9 Shock Waves in Models AI and AII . . . . . . . .. . . . . . . . . . . . . . . . .693 9.8.10 Shock Adiabatic Curves for Models AI and AII . . . . . . . . . . . .695 9.8.11 Shock Adiabatic Curves at a Given Rate of Impact .. . . . . . . .697 Models of Viscoplastic Continua . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .699 9.9.1 The Concept of a Viscoplastic Continuum .. . . . . . . . . . . . . . . . .699

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9.9.2 9.9.3 9.9.4 9.9.5 9.9.6

Model An of Viscoplastic Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .699 Model of Isotropic Viscoplastic Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .701 General Model An of Viscoplastic Continua .. . . . . . . . . . . . . . .702 Model An of Isotropic Viscoplastic Continua .. . . . . . . . . . . . . .703 The Problem on Tension of a Beam of Viscoplastic Continuum of the Differential Type . . . . . . . .703

References .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .707 Basic Notation . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .708 Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .713

Chapter 1

Introduction: Fundamental Axioms of Continuum Mechanics

Continuum mechanics, including nonlinear continuum mechanics, studies the behavior of material bodies or continua. We can mathematically define a body as follows: it is a set B consisting of elements M called material points. The concept of a material point in continuum mechanics is primary, i.e. axiomatic, as is the concept of a geometrical point in elementary geometry. The mathematical description of a body B in continuum mechanics starts from the following definition. Definition 1.1. A material body B, for which there is a one-to-one correspondence e between each material point M 2 B and its image in some metric space X , i.e. W e W B ! W.B/ e W X ; or e a D W.M/;

M 2 B;

a 2 X;

(1.1)

is called a continuum. The set of all continua B is called the universe U. The definition of one-to-one correspondence one can find, for example, in [31]. A metric space X is characterized by the presence of a distance function l.M; N /, with the help of which one can measure the distance between any two points M and N of a body B [31]. The one-to-one correspondence between material points and points of a metric space X allows us to consider not the material body itself but only its image. Below, we draw no distinction between a material point and its image. Definition 1.1 should be complemented with three fundamental axioms. e Axiom 1 (on continuity). The image W.B/ of a body B is a continuous set (a continuum) in space X . The concept of a continuum was considered in [31]. Axiom 1 introduces the main model of continuum mechanics, namely a continuous set, that is an idealization of real bodies consisting of discrete atoms and molecules. Physically, there is a limiting distance lmin such that for l 6 lmin the neighborhood of a material point M 2 B is empty. However, in a continuum (and in its image), due to the properties of a continuous set [31], any infinitesimal "-neighborhood U" .A/ of a point A 2 W.B/ X contains an infinite number Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 1, c Springer Science+Business Media B.V. 2011

1

2

1 Introduction: Fundamental Axioms of Continuum Mechanics

of other points of this medium. In this sense, a continuum is only a model of a real body; and computed results in continuum mechanics referred to real bodies for distances between points l 6 lmin may prove to be incorrect. Nevertheless, Axiom 1 is of great importance: it allows us to apply the methods of tensor analysis in metric spaces. Axiom 2 (on an Euclidean space). As a metric space X , in which continua are considered, we can choose a three-dimensional Euclidean metric space E3a , i.e X D E3a . An Euclidean point space E3a (called also an affine Euclidean space) is a set of points M; N ; : : : ; where there exists a mapping, uniquely assigning each ordered pair of points M; N to an element (vector) y of Euclidean vector space E3 adjoined ! to E3a (it is also denoted by MN D y). In Euclidean space E3 , there are operations of addition of elements (vectors) x C y, multiplication by a real number y, 2 R and scalar product of vectors [31], that can be given by a metric matrix gij relative to some basis ei in E3 : x y D gij x i y j , where x i and y j are coordinates of the vectors with respect to the same basis: x D x i ei , y D y i ei . Moreover, in E3 we can introduce the vector product of vectors a D ai ei and b D b i ei as follows: cDabD

1 p g ijk ai b j ek D p ijk ai bj ek : g

(1.2)

Here ijk and ijk are the Levi–Civita symbols [12] (they are zero, if at least two of the indices i; j; k are coincident; and they are equal to 1, if the indices i; j; k form an even permutation, and are equal to (1) if the indices form an odd permutation), and g D det .gij / is the determinant of the metric matrix. Thus, Axiom 2 allows us to describe materials points of a continuum with the help of instruments of Euclidean point spaces: in the space E3a we can introduce a rectangular Cartesian coordinate system O eN i , being common for all continua and consisting of some point O (the origin of the coordinate system) and an orthonormal basis eN i . In this system O eN i , every material point M is uniquely assigned to ! its radius-vector x D OM. The distance l.M; N / between points M and N is ! measured by the length of vector MN ! ! ! l.M; N / D jMN j D .MN MN /1=2 D jyj D .y y/1=2 : The length l.M; N / specifies a metric in the space E3a ; such a space is called a metric space. In a metric space we define the concept of a domain V , and in an Euclidean point space, the concepts of a plane and a straight line. In addition, in a metric space there are concepts of convergence of a point sequence, continuity and differentiability of functions etc.; Axiom 2 allows us to apply these concepts of mathematical analysis to continua. Due to isomorphism (one-to-one correspondence) of Euclidean point spaces of the same dimension as a space E3a , we can always consider the space of elementary

1 Introduction: Fundamental Axioms of Continuum Mechanics

3

e .B/ in E3a Fig. 1.1 A real body B and its image W geometry E3a , where points are usual geometric points, and vectors are directed straight-line segments in the space. The space E3a allows us to show different objects of continuum mechanics geometrically. Example 1.1. Figure 1.1 shows schematically a real material body B and its image e W.B/ in the space E3a . t u Consider the pair .M; t/, where M 2 B, and t 2 RC0 is some nonnegative real number. This pair is an element of the Cartesian product of the sets B RC0 . Axiom 3 (on the existence of absolute time). For every body B there exists a mapping W W B RC0 ! VN E3a ; in the form of a function A D W.M; t/;

M 2 B;

t 2 RC0 ;

A 2 VN E3a :

(1.3)

The parameter t is called the absolute time (or simply time). Notice that both Axioms 1 and 3 establish relations between points M and A. To avoid the ambiguity one should assume that the following consistency condition is satisfied: the mapping (1.1) coincides with (1.3) at some time t D t1 e W.M/ D W.M; t1 / 8M 2 B:

(1.4)

Axiom 3 allows us to describe the motion of a body B, which is defined as ! changes of the radius-vectors x D OM of material points M of the body in a coordinate system O eN i being common for all M 2 B with time t. For the same material point M at different times t1 and t2 we have, in general, distinct radius-vectors x1 and x2 , respectively, in the system O eN i . Example 1.2. Figure 1.2 shows the motion of a real body B and its images W.B; t1 / and W.B; t2 / in the space E3a at times t1 and t2 , respectively. t u The absolutism of time means that the time t is independent of the radius-vector x of a point M in the coordinate system O eN i , i.e., in physical terms, the time varies

4

1 Introduction: Fundamental Axioms of Continuum Mechanics

Fig. 1.2 The motion of a body B

in the same course for all material points M. In a physical sense, this axiom remains valid only for the motion of bodies with speeds which are considerably smaller than the speed of light; otherwise, relativistic effects become essential, and Axiom 3 ceases to describe actual processes adequately. Relativistic effects are not considered in this work.

Chapter 2

Kinematics of Continua

2.1 Material and Spatial Descriptions of Continuum Motion 2.1.1 Lagrangian and Eulerian Coordinates: The Motion Law Let us consider a continuum B. Due to Axiom 2, at time t D 0 there is a oneto-one correspondence between every material point M 2 B and its radius-vector ! ı x D OM in a Cartesian coordinate system O eN i . Denote Cartesian coordinates of ı ı ı the radius-vector by x i (x D x i eN i ) and introduce curvilinear coordinates X i of the same material point M in the form of some differentiable one-to-one functions ı

ı

x i D x i .X k /: ı

(2.1)

ı

Since x D x i eN i , the relationship (2.1) takes the form ı

ı

x D x.X k /:

(2.2)

Let us fix curvilinear coordinates of the point M, and then material points of the continuum B are considered to be numbered by these coordinates X i . For any motion of the continuum B, coordinates X i of material points are considered to remain unchanged; they are said to be ‘frozen’ into the medium and move together with the continuum. Coordinates X i introduced in this way for a material point M are called Lagrangian (or material). Due to Axiom 3, at every time t there is a one-to-one correspondence between ! every point M 2 B with Lagrangian coordinates X i and its radius-vector x D OM with Cartesian coordinates x i , where x and x i depend on t. This means that there is a connection between Lagrangian X i , and the Cartesian x i coordinates of point M and time, i.e. there exist functions in the form (1.3) x i D x i .X k ; t/

8X k 2 VX :

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 2, c Springer Science+Business Media B.V. 2011

(2.3)

5

6

2 Kinematics of Continua

These functions determine a motion of the material point M in the Cartesian coordinate system O eN i of space E3a . The relationships (2.3) are said to be the law of the motion of the continuum B. Coordinates x i in (2.3) are called Eulerian (or spatial) coordinates of the material point M. Since x D x i eN i and the coordinate system O eN i is the same for all times t, the equivalent form of the motion law follows from (2.2): x D x.X k ; t/:

(2.4)

Since the consistency conditions (1.4) must be satisfied, from (2.2) and (2.4) we get the relationships ı

x.X k ; 0/ D x.X k /;

ı

x i .X k ; 0/ D x i .X k /:

(2.5)

Here the initial time t D 0 is considered as the time t1 in (1.4), because just at time t D 0 we introduced Lagrangian coordinates X i of point M. Unless otherwise stipulated, functions (2.3) are assumed to be regular in the domain VX R3 for all t, thus there exist the inverse functions X k D X k .x i ; t/ 8x i 2 Vx R3 : ı

The closed domain V D W.B; 0/ in a fixed coordinate system O eN i , which is occupied by continuum B at the initial time t D 0, is called the reference configuı

ration K, and the domain V D W.B; t/ occupied by the same continuum B at the time t > 0 is called the actual configuration K. Figure 2.1 shows a geometric picture of the motion of a continuum from the ı

reference configuration K to the actual one K at time t in space E3a .

Fig. 2.1 The motion of a continuum: positions of continuum B and material point M in reference and actual configurations

2.1 Material and Spatial Descriptions of Continuum Motion

7

It should be noticed that if the continuum motion law (2.3) (or (2.4)) is known, then one of the main problems of continuum mechanics (to determine coordinates of all material points of the continuum at any time) will be resolved. However, in actual problems of continuum mechanics this law, as a rule, is unknown and must be found by solving some mathematical problems, whose statements are to be formulated. One of our objectives is to derive these statements. Example 2.1. Let us consider a continuum B, which at time t D 0 in the reference ı

ı

ı

configuration K is a rectangular parallelepiped (a beam) with edge lengths h1 , h2 ı

and h3 , and in an actual configuration K at t > 0 the continuum is also a rectangular parallelepiped but with different edge lengths: h1 , h2 and h3 . We assume that corresponding sides of both the parallelepipeds lie on parallel planes, and for one of the sides, which for example is situated on the plane .x 2 ; x 3 /, points of diagonals’ ı

intersection in K and in K are coincident (Fig. 2.2). Then the motion law (2.3) for this continuum takes the form x ˛ D k˛ .t/ X ˛ ; ı

˛ D 1; 2; 3;

(2.6)

ı

i.e. coordinates x i ; x i D X i of any material point M in K and K are proportional, ı

and k˛ .t/ D h˛ .t/=h˛ is the proportion function. The motion law (2.6) is called the beam extension law. t u ı

Example 2.2. In K, let a continuum B be a rectangular parallelepiped oriented as shown in Fig. 2.3; its motion law (2.3) has the form

Fig. 2.2 Extension of a beam

Fig. 2.3 Simple shear of a beam

8

2 Kinematics of Continua

8 1 1 2 ˆ ˆ <x D X C a.t/X ; x2 D X 2; ˆ ˆ :x 3 D X 3 ;

(2.7)

ı

where x i D X i , a.t/ is a given function. In K this continuum B has become a parallelepiped, all cross-sections of which are planes orthogonal to the Ox 3 axis and are the same parallelograms. This motion law is called simple shear; the tangent of the shear angle ˛ is equal to a. t u ı

Example 2.3. Consider a continuum B, which in K is a rectangular parallelepiped (a ı

beam) shown in Fig. 2.4; under the transformation from K to K this parallelepiped changes its dimensions without a change in its angles (as in Example 2.1) and rotates by an angle '.t/ in the plane Ox 1 x 2 around the point O (Fig. 2.4). The motion law for the continuum is called the rotation of a beam with extension. In this case Eq. (2.1) have the form ı

ı

x i D F0ij x j ;

xj D X j ;

(2.8)

where the matrix F0ij is the product of two matrices, the rotation matrix O0 and the stretch matrix U0 : F0ij D O0i k U0kj ;

U0ij

0 1 k1 0 0 D @ 0 k2 0 A ; 0 0 k3

O0i j

F0ij

0 cos ' sin ' D @ sin ' cos ' 0 0

1 0 0A ; 1

0 1 k1 cos ' k2 sin ' 0 D @ k1 sin ' k2 cos ' 0 A ; 0 0 k3

and k˛ .t/ D h˛ .t/= h0˛ are the proportion functions characterizing the ratio of ı

lengths of the beam edges in K and K (as in Example 2.1).

Fig. 2.4 Rotation of a beam with extension

t u

2.1 Material and Spatial Descriptions of Continuum Motion

9

2.1.2 Material and Spatial Descriptions In continuum mechanics, physical processes occurring in bodies are characterized by a certain set of varying scalar fields D .M; t/, vector fields a D a.M; t/, and tensor fields of the nth order n .M; t/. We will consider tensors and tensor fields in detail in Sect. 2.1.4 (see also [12]). Since in the Cartesian coordinate system O eN i a material point M corresponds to both Lagrangian coordinates X i and Eulerian coordinates x i , varying scalar and vector fields can be written as follows: .X i ; t/ D .X i .x j ; t/; t/ D e .x j ; t/ D e .x; t/; a.X i ; t/ D e a.x j ; t/ D e a.x; t/;

(2.9)

With the help of the motion law (2.3) (or (2.4)), we can pass from functions of Lagrangian coordinates to functions of Eulerian coordinates in formulae (2.9). In continuum mechanics the tilde e is usually omitted (we will do this below). For a fixed time t in (2.9), we obtain stationary scalar and vector fields. If in (2.9) a material point M is fixed, and time t changes within the interval 0 6 t 6 t 0 , then we get an ordinary scalar function D .M; t/ and vector function a D a.M; t/ depending on time. According to relationships (2.9), there are two ways to describe different physical processes in continua. In the material (Lagrangian) description of a continuum, all tensor fields describing physical processes are considered as functions of X i and t. In the spatial (Eulerian) description, all tensor fields describing physical processes are functions of x i and t. Both the descriptions are equivalent. It should be noted that for solids we more often use the material description, where it is convenient to fix coordinates X i of a material point M and to observe its motion at different times t. For gaseous and fluid continua, Eulerian description is more convenient; when an observer fixes a geometric point with coordinates x i and monitor the material points M passing through this point x i at different times t.

ı

2.1.3 Local Bases in K and K Using the motion law (2.4) and relationship (2.1), at every material point M with coordinates X i in the actual and reference configurations we can introduce its local basis vectors: rk D

@x @x i eN i D Qik eN i ; D k @X @X k

ı

rk D

ı

ı

ı @x @x i eN i D Qik eN i ; D k k @X @X

(2.10)

10

2 Kinematics of Continua

Fig. 2.5 Local basis vectors in reference and actual configurations

where ı

ı

Q ik D @x i [email protected] k ;

Qik D @x i [email protected] k ;

P ik D @X i [email protected] k ;

P ik D @X i [email protected] k

ı

ı

(2.11)

are Jacobian matrices and inverse Jacobian matrices. ı Here and below all values referred to the configuration K will be denoted by ı superscript ı . As follows from the definition (2.11), local bases vectors rk and rk are directed tangentially to corresponding coordinate lines X k (Fig. 2.5). ı

ı

In K and K introduce metric matrices gkl , g kl and inverse metric matrices g kl ,

ı kl

g

as follows: j

gkl D rk rl D Qik Q l ıij D ı

ı

ı

ı

g kl D rk rl D

@x i @x j ıij ; @X k @X l

ı

@x i @x j k ıij ; g kl glm D ım ; @X k @X l

ı

ı

k g kl g lm D ım ;

(2.12)

and also vectors of reciprocal local bases ri D g i m rm ;

ı

ı

ı

ri D g i m rm ;

(2.13)

which satisfy the reciprocity relations ri rj D ıi j ;

ı

ı

ri rj D ıi j ;

(2.14a)

q ı ı rn rm D g nmk rk ;

(2.14b)

and also the following relations: rn r m D

p

k

g nmk r ;

ı

ı

2.1 Material and Spatial Descriptions of Continuum Motion

11

With the help of the mixed multiplication, i.e. sequentially applying scalar and vector products to three different local bases vectors, we can determine the volumes ı

jV j and jV j constructed by these vectors: q q ˇ ˇ ı ı ı ˇ ˇ jV j D r1 .r2 r3 / D g D det .g ij / D ˇ@x k [email protected] i ˇ; p jV j D r1 .r2 r3 / D g D [email protected] k [email protected] i j: ı

ı

ı

ı

(2.15)

ı

It should be noted that although local bases ri and ri have been introduced in ı

different configurations K and K, they correspond to the same coordinates X i (if one consider the same point M); therefore each of the bases can be carried as a ı

rigid whole into the same point in K or in K. Due to this, we can resolve any vector ı ı field a.M/ for each of the bases ri , ri , ri and ri : ı ı

ı ı

a D ai ri D ai ri D ai ri D ai ri :

(2.16) ı

If curvilinear coordinates X i are orthogonal, then the vectors ri are orthogonal as ı ı ı ı well: (ri rj D ıij ), and matrices g ij and g ij are diagonal; hence we can introduce q ı ı Lam´e’s coefficients H ˛ D g ˛˛ (˛ D 1; 2; 3) and the physical orthonormal basis ı b ı ı ı ı r˛ D r˛ =H ˛ D r˛ H ˛ :

(2.17)

Components of a vector a with respect to this basis are called physical: ı b ı b a D ai ri :

(2.18) ı

The actual basis ri is in general not orthogonal even if the basis ri is orthogonal; therefore we cannot introduce the corresponding physical basis in K. One can introduce a physical basis in K not with the help of ri , but with the help of another special basis (see Sect. 2.1.7).

2.1.4 Tensors and Tensor Fields in Continuum Mechanics ı

ı

For different local bases ri , ri , ri , ri or eN i at every point M, and with the help of formulae given in the work [12] we can introduce different dyadic (tensor) bases: ı ı ı ı ı ri ˝rj , ri ˝ rj , ri ˝rj , ri ˝ rj , ri ˝rj etc., which are equivalence classes of vector sets consisting of 2 3 D 6 vectors (for example, r1 ˝ rj D Œr1 rj r2 0r3 0, where Œ is the notation of an equivalence class), and ˝ is the sign of tensor product. A field

12

2 Kinematics of Continua

of second-order tensor T.M/ can be represented as a linear combination of dyadic basis elements: ı

ı

ı ı ı ı T D T ij ri ˝ rj D T ij ri ˝ rj D TN ij eN i ˝ eN j D T ij ri ˝ rj :

(2.19)

During the passage from one basis to another, tensor components T ij are transformed by the tensor law: ı

ı

ı

TN ij D P ik P jl T kl D P ik P jl T kl : ı

(2.20)

ı

Metric matrices g i m , gi m , g i m and g i m are components of the unit (metric) tensor E with respect to different bases: ı

ı

ı

ı

ı

ı

E D gi m ri ˝ rm D g i m ri ˝ rm D gi m ri ˝ rm D g i m ri ˝ rm :

(2.21)

For a second-order tensor T, in continuum mechanics one often uses the transpose tensor TT D T ij rj ˝ ri and the inverse tensor T1 , where T1 T D E. The inverse tensor exists only for a nonsingular tensor (when det T ¤ 0). The determinant of a tensor is defined by the determinant of its mixed components matrix: det T D det T ij . Besides second-order tensors, in continuum mechanics one sometimes uses tensors of higher orders [12]. To introduce the tensors, we define polyadic bases by induction: ri1 ˝ : : : ˝ rin ; the bases are equivalence classes of vector sets consisting of n 3 D 3n vectors. A field of nth order tensor n .M/ can be represented by a linear combination of polyadic basis elements: n

ı

ı

ı

D i1 :::in ri1 ˝ : : : ˝ rin D i1 :::in ri1 ˝ : : : ˝ rin ; ı

where i1 :::in and i1 :::in are components of the nth order tensor with respect to the corresponding polyadic basis. For fourth-order tensors, analogs of the tensor E are the first, second and third unit tensors defined as follows: I D ei ˝ ei ˝ ek ˝ ek D E ˝ E; II D ei ˝ ek ˝ ei ˝ ek ; III D ei ˝ ek ˝ ek ˝ ei ;

(2.22)

and also the symmetric fourth-order unit tensor D

1 .II C III /: 2

D ijkl ei ˝ ej ˝ ek ˝ el ;

ijkl D

1 i k jl .ı ı C ı i l ı jk /: 2

(2.22a)

2.1 Material and Spatial Descriptions of Continuum Motion

13

We can transpose fourth-order tensors as follows: 4

.m1 m2 m3 m4 / D i1 i2 i3 i4 rim1 ˝ rim2 ˝ rim3 ˝ rim4 ;

where .m1 m2 m3 m4 / is some substitution, for example, 4 .4321/ D i1 i2 i3 i4 ri4 ˝ ri3 ˝ ri2 ˝ ri1 :

ı

2.1.5 Covariant Derivatives in K and K ı

Introduce the following nabla-operators in configurations K and K, respectively: ı

@ ; @X k

r D rk ˝

ı

r D rk ˝

@ : @X k

(2.23)

Applying the nabla-operators to a vector field, we get the gradients of a vector in ı

K and K: r ˝ a D rk ˝ ı

ı

r ˝ a D rk ˝

ı

@a @X k

D rk ai rk ˝ ri ; ı

ı

ı ı

ı

ı

ı ı

ı ı

ı

D r k ai rk ˝ ri D r k ai rk ˝ ri D r k ai rk ˝ ri ; (2.24)

@a @X k

where we have denoted the following covariant derivatives in different tensor bases ı

in configurations K and K: ı

ı

r k ai D rk ai D

ı

@a i @X k @ai @X k

ı

ı

ı

ı

m a ; ik m

r k ai D

m a ; ik m

rk ai D

ı

@ai @X k @ai @X k

ı

ı

C ikm am C ikm am :

(2.25) ı

ı

Here ijm and m ij are the Christoffel symbols in configurations K and K. For the Christoffel symbols the following relations (see [12]) hold:

@gkj @gki @gij C i j @X @X @X k

m ij

1 D g km 2

ı

0 1 ı ı ı @g ij 1 ı km @ @g kj @g ki A: D g C 2 @X i @X j @X k

m ij

;

(2.26)

ı

Contravariant derivatives in K and K are introduced as follows: ı

ı

ı

ı

ı

r k ai D g km r m ai ;

r k ai D gkm rm ai :

(2.27)

14

2 Kinematics of Continua ı

The covariant derivatives (2.25) are components of the second-order tensors r ˝ a and r ˝ a, therefore during the passage from the local basis ri to another one they are transformed by the tensor law (2.20). ı

ı

The nabla-operators r and r in K and K can be applied to a tensor field n .X i / of nth order: r ˝ n D rk ˝

ı ı @ n ı ı ı D r k i1 :::in rk ˝ ri1 ˝ : : : ˝ rin ; k @X

r ˝ n D rk ˝

@ n D rk i1 :::in rk ˝ ri1 ˝ : : : ˝ rin ; @X k

ı

ı

ı

(2.28) ı

ı

where rk i1 :::in and r k i1 :::in are the covariant derivatives in K and K, respectively: n ı ı @ ı i1 :::in X ı is ı i1 :::is Dm:::in i1 :::in rk D C mk : (2.29) @X k sD1 In the same way we can define operations of scalar product of the nabla-operator in ı

K (the divergence of a tensor): ı

ı

r n D rk

ı ı @ n ı ı D r k ki2 :::in ri2 ˝ : : : ˝ rin ; k @X

(2.30)

ı

and vector product of the nabla-operator in K (the curl of a tensor): ı

ı

r n D rk

@ n 1 ijk ı ı ı ı ı D q r i j i2 :::in rk ˝ ri2 ˝ : : : ˝ rin : ı @X k g

(2.31)

2.1.6 The Deformation Gradient Consider how a local neighborhood of a point M is transformed during the passage ı

ı

from configuration K to K. Take an arbitrary elementary radius-vector d x connectı

ing in K two infinitesimally close points M and M0 (Fig. 2.6). In configuration K, these material points M and M0 are connected by the elementary radius-vector d x. ı The vectors d x and d x can always be resolved for local bases: ı

d x.X k / D

@x @x ı ı dX k D rk dX k ; d x.X k / D dX k D rk dX k : k k @X @X

(2.32)

2.1 Material and Spatial Descriptions of Continuum Motion

15

Fig. 2.6 Transformation of an elementary radius-vector during the passage from the reference configuration to the actual one

ı

On multiplying the first equation by rm and the second – by rm , we get rm d x D rm rk dX k D dX m ;

ı

ı

ı

ı

rm d x D rm rk dX k D dX m :

(2.33) ı

Substitution of dX m (2.33) into the first equation of (2.32) yields d x D rk ˝ rk d x. Changing the order of the tensor and scalar products (that is permissible by the ı tensor analysis rules), we get the relation between d x and d x: ı

ı

d x D F d x:

(2.34)

Here we have denoted the linear transformation tensor ı

F D rk ˝ rk ;

(2.35)

called the deformation gradient. As follows from (2.34), the deformation gradient ı connects elementary radius-vectors d x and d x of the same material point M in ı

configurations K and K. Definition (2.19) allows us to give a geometric representation of the deformation ı gradient: if ri are considered as the left vectors and ri – as the right vectors, then, by formulae of Sect. 2.1.4 (see [12]), the tensor F takes the form ı

ı

ı

ı

F D ri ˝ ri D Œr1 r1 r2 r2 r3 r3 : According to the geometric definition of a tensor (see Sect. 2.1.4), the tensor ı F can be represented as equivalence class of the ordered set of six vectors ri ; ri (Fig. 2.7).

16

2 Kinematics of Continua

Fig. 2.7 Geometric representation of the deformation gradient

Besides F, in continuum mechanics one often uses the transpose tensor FT , the inverse tensor F1 and the inverse to the transpose tensor F1T : ı

ı

FT D rk ˝ rk D rk ˝

@x @X k

ı

ı

D r ˝ x; ı

F1T D .rk ˝ rk /T D rk ˝ rk D rk ˝

ı

F1 D rk ˝ rk ; ı

@x @X k

ı

D r ˝ x:

(2.35a)

It follows from (2.35) that ı

ı

F ri D rk ˝ rk ri D rk ıik D ri :

(2.36)

i.e. the deformation gradient transforms local bases vectors of the same material ı

point M from K to K. Theorem 2.1. The transpose deformation gradient FT connects gradients of an arı

bitrary vector a in K and K: ı

r ˝ a D FT r ˝ a;

ı

r ˝ a D F1T r ˝ a:

(2.37)

H To derive formulae (2.37), we apply the definitions (2.24) and (2.35): r ˝ a D ri ˝

ı @a @a @a ı ı D rj ıji ˝ D rj ˝ rj ri ˝ D F1T r ˝ a: N (2.38) i i i @X @X @X

2.1.7 Curvilinear Spatial Coordinates Notice that the choice of Cartesian basis O eN i as a fixed (immovable) system in the spatial (Eulerian) description of the continuum motion is not a necessary condition. For some problems of continuum mechanics it is convenient to consider a moving ! system O 0 eN 0i with the origin at a moving point O 0 (x0 D OO 0 ) and a moving orthonormal basis eN 0i (Fig. 2.8), which is connected to eN i by the orthogonal tensor Q: eN 0i D Q eN i :

(2.39)

In this case, instead of Cartesian coordinates x of a point M in the basis eN i one consider its Cartesian coordinates e x i in basis eN 0i : i

e x D x x0 D e x i eN 0i :

(2.40)

2.1 Material and Spatial Descriptions of Continuum Motion

17

Fig. 2.8 Moving bases eN0i ande ri and curvilinear spatial coordinates e X i in moving system O 0 eN 0i

Let Q ij be components of the tensor Q with respect to the basis eN i : Q D Qij eN i ˝ eN j ;

(2.41)

then relation (2.39) takes the form eN 0i D Qji eN j ;

(2.42)

and coordinates e x i and x i are connected as follows: j e x i eN 0i D e x i Q i eN j ; x D x x0 D .x i x0i /Nei D e

x j ; @x i [email protected] x j D Qij ; x i x0i D Qij e

@e x j [email protected] i D P ji :

(2.43)

Instead of Cartesian coordinates e x i , we can consider special curvilinear coordinates k 0 e X with the origin at point O : e k /; x i .X e xi D e

(2.44)

which, due to (2.27), are connected to x i by the relations e k / x i .X e k ; t/ or X ej D X e j .x i ; t/: x i D x0i .t/ C Qij .t/e x j .X

(2.45)

The dependence on t in the relations is defined by functions x0i .t/ and Q ij .t/ (i.e. only by the motion of system O 0 eN 0i ), which are assumed to be known in continuum mechanics. e i are no longer Lagrangian (material): at different times they Coordinates X correspond to different material points. However, it is often convenient to choose

18

2 Kinematics of Continua ı

e i coincident with X i in the reference configuration K. In this case we coordinates X have the relations ı e j ; 0/: x i .X i / D x i .X j ; 0/ D x i .X

(2.46)

With the help of transformation (2.45) we can use the spatial description in coore i as well when consider the functions dinates X e i ; t/I a.X a D a.x i ; t/ D e

(2.47)

e i are called curvilinear spatial coordinates. therefore coordinates X Introduce local vectors e ri D

@x @x j D eN : ei ei j @X @X

(2.48)

x D x, and In particular, the basis eN 0i may be fixed (Fig. 2.9), then eN 0i D eN i , e ej D X e j .x i / are independent of t; the basis e curvilinear spatial coordinates X ri is independent of t as well, and from (2.46) and (2.48) it follows that the basis ı coincides with ri : ı @x j @x j ı e Nj D ri D (2.49) e eN j D ri : ei @X i @X ı

When the basis eN 0i is moving, basese ri and ri are no longer coincident. e i and defined The vectors e ri are directed tangentially to the coordinate lines X simultaneously with ri at every point M at any time t > 0. A change of vectorse ri in time is defined only by the motion of basis eN 0i , because from (2.42), (2.43), and (2.48) it follows that j eN 0i D Q i

@X ek ek ek @X @x j @X e e e r r rk ; D D k k @x j @e x i @x j @e xi

(2.50)

e k [email protected] e k @X x i is independent of t according to (2.44). and the matrix P i

Fig. 2.9 Curvilinear spatial coordinates e X i and Lagrangian coordinates X i for the fixed basis 0 eNi D eN i

2.1 Material and Spatial Descriptions of Continuum Motion

19

The bases vectors ri and e ri are connected as follows: ri D

ek ek @x @x @X @X e D D rk : e k @X i @X i @X i @X

(2.51)

Just as in Sect. 2.1.2, we define the metric matrix e g ij and the inverse metric matrix e g ij : e g ij D e ri e rj ; e g ij e g jk D ıki ; (2.52) and the reciprocal basis vectors e g i ke ri D e rk D

ei ei @X @X j N D e eN 0k : @x j @e xk

(2.53)

According to formulae (2.51) and (2.52), we find the relation between matrices gij and e gkl : gij D ri rj D

el el e k @X e k @X @X @X e e r D e g kl : r k l @X i @X j @X i @X j

(2.54)

The inverse matrix g ij is found from (2.54) by the rule of matrix product inversion (see Exercise 2.1.13): gij D

@X i @X j kl e g : e k @X el @X

(2.55)

From (2.51), (2.53) and (2.55) we can find the relation between vectors of reciprocal bases ri and e ri : ri D g ij rj D

e ri :

em @X i @X j kl @X @X i k e e r r : e g D m e k @X el ek @X j @X @X

(2.56)

Let there be a tensor n , then it can be resolved for the basis ri and for the basis n

ei1 :::ine D i1 :::in ri1 ˝ : : : ˝ rin D rin : ri1 ˝ : : : ˝e

(2.57)

On substituting (2.51) into (2.57), we derive transformation formulae for tensor ei : components during the passage from coordinates X i to X e i1 e in e i1 :::in D j1 :::jn @X : : : @X : @X j1 @X jn

(2.58)

e of covariant differentiation in coordinates X ei : Introduce the nabla-operator r @ e De r ri e @X i

(2.59)

20

2 Kinematics of Continua

and contravariant derivatives of components e a i of a vector a D e aie ri in coordii e : nates X @e ai @e ai i eke eke ai D e m e a ; r a D Ce ikme am : (2.60) r m ik ek ek @X @X ei The Christoffel symbols e m g ij by the relaij in coordinates X are connected to e tions which are similar to (2.26). e i and X i (in Theorem 2.2. The results of covariant differentiation in coordinates X the configuration K) are coincident: e ˝ n ; r ˝ n D r

e n ; r n D r

e n : r n D r

(2.61)

H Prove the first formula in (2.61). Due to (2.23), we have r ˝ n D ri ˝

el @ n @X i k @ n @X @ n e ˝ n : (2.62) e D D ıkl rk ˝ Dr r ˝ i i k l e e @X el @X @X @X @X

The remaining two formulae in (2.61) can be derived in the same way (see Exercise 2.1.8). N Going to components of a tensor n with respect to bases ri and e ri , from (2.58) we get the relation between the covariant derivatives: ei e j1 :::jn : ri j1 :::jn D r

(2.63)

e i into X j : Determine the tensor H transforming coordinates X ı e ij e rj D H rj D HN ij eN i ˝ eN j : ri ˝e H D rj ˝e

(2.64)

Then we get the relations (see Exercises 2.1.10 and 2.1.11): ı

ej e ri D H e ri D H i rj ;

ı

e ij /1e ri D H1T e r i D .H rj :

(2.65)

e i are often chosen orthogonal, then the bases e The coordinates X ri and e ri are ij orthogonal as well, and the matrices e g ij and e g are diagonal; and we can introduce the physical (orthonormal) basis: b e ˛; e r˛ D e r˛ = H

(2.66)

p e g ˛˛ are Lam´ qe’s coefficients, which are in general not coincident ı ı e r˛ with the coefficients H ˛ D g ˛˛ . Tensor components with respect to the basis b are called physical: b eijb e e TDT ri ˝ b rj : (2.67) e˛ D where H

Relations between physical and covariant components of a tensor are determined by the known formulae (see [12]).

2.1 Material and Spatial Descriptions of Continuum Motion

21

Exercises for 2.1 2.1.1. With the help of formulae (2.10), (2.12), (2.13), and (2.17) show that if the motion law of a continuum describes extension of a beam (2.6) (see Example 2.1), ı then the local basis vectors ri and the metric matrices have the forms ı

ı

ri D ei ; r˛ D k˛ eN ˛ ;

ri D ei ;

r˛ D .1=k˛ /Ne˛ ;

ı

˛ D 1; 2; 3;

ı

g ij D ıij ; g ij D ı ij ; 1 1 0 2 0 2 k1 0 0 k1 0 0 .gij / D @ 0 k22 0 A ; .g ij / D @ 0 k22 0 A ; 0 0 k32 0 0 k32 i.e.

g˛ˇ D k˛2 ı˛ˇ ; g ˛ˇ D k˛2 ı˛ˇ ; H˛ D

p

g˛˛ D k˛ ; b r˛ D e ˛ :

2.1.2. Show that if the motion law of a continuum describes a simple shear (see Example 2.2), then the local basis vectors and the metric matrices have the forms ı

ri D ei ;

ı

r i D ei ;

ı

ı

g ij D ıij ;

r1 D eN 1 ; r2 D aNe1 C eN 2 ;

gij D ı ij ; r3 D eN 3 ;

r1 D eN 1 aNe2 ; r2 D eN 2 ; r3 D eN 3 ; 0 0 1 1 a 0 1 C a2 a ij 2 gij D @a 1 C a 0A ; g D @ a 1 0 0 0 0 1

1 0 0A : 1

2.1.3. Show that if the motion law describes rotation of a beam with extension (see Example 2.3), then with introducing the rotation tensor O0 and the stretch tensor U0 : 3 X O0 D O0i j eN i ˝ eN j ; U0 D k˛ eN ˛ ˝ eN ˛ ; ˛D1

we can rewrite the beam motion law in the tensor form ı

x D F0 x;

F0 D O0 U0 :

Show that the local basis vectors and metric matrices for this problem have the forms ri D F0ki eN k ;

ı

ri D eN i ;

22

2 Kinematics of Continua

gij D F0ki F0lj ıkl

0 2 1 k1 cos2 ' C k22 sin2 ' .k12 k22 / cos ' sin ' 0 D @ .k12 k22 / cos ' sin ' k12 sin2 ' C k22 cos2 ' 0 A ; 0 0 k32 g D k1 k2 k3 ;

1 k22 sin2 ' C k12 cos2 ' .k12 k22 / cos ' sin ' 0 gij D @ .k12 k22 / cos ' sin ' k22 cos2 ' C k12 sin2 ' 0 A : 0 0 k32 0

2.1.4. Using the property (2.14) of reciprocal basis vectors, show that the following relations hold: @X i @X i k ı ri D ı eN k ; ri D P ik eN k D eN : @x k @x k 2.1.5. Show that F, FT , F1 and F1T in the Cartesian coordinate system take the forms FD

@x m ı

@x i

eN m ˝ eN i ;

FT D

@X k ı

eN i ˝

@x i

@x m @x m i N D e eN ˝ eN m ; m ı @X k @x i

ı

F

1

ı

@x m @X k i @x m N N D ˝ D e e eN m ˝ ei ; m @x i @x i @X k ı

F1T D

ı

@X k i @x m @x m i N N ˝ D e e eN ˝ em : m @x i @x i @X k

2.1.6. Substituting (2.54), (2.52), and (2.55) into (2.12), derive formula (2.55). 2.1.7. Prove that

ı

ri D F1T ri :

2.1.8. Derive the third formula of (2.61). ı

2.1.9. Prove that for any scalar function '.X i / its gradients in K and K are connected by the relationship ı

r ' D F1T r ': 2.1.10. Show that formulae (2.65) follow from (2.64). 2.1.11. Using (2.47), show that in formulae (2.64) the tensor H has the following components with respect to bases eN i and e ri : ı ı k i ek ei @x i @X @x i @X k e i @x k @X i i 1 N e ij /1 D @x @X : N ; . H / D ; H D ; . H H jD k j j e k @xı j e j @xı k @X j @x k @X @x j @X @X

2.1 Material and Spatial Descriptions of Continuum Motion

23

ı

2.1.12. Introducing the notation F ij for components of the deformation gradient F ı

ı

ı

ı

ı

ı

ı

with respect to basis ri : F D F ij ri ˝ rj D F ij ri ˝ rj ; show that formula (2.36) yields ı

ı

rj D F ij ri : 2.1.13. Show that the Levi-Civita symbols are connected by the relations ijk ijk D 6; p

ijk i lm D ıjl ıkm ıkl ıjm ;

p g ijk D .1= g/ mnl gmi gnj glk :

ijk ijl D 2ıil ; ijk T jk D 0

where T jk are components of an arbitrary symmetric tensor: T jk D T kj . 2.1.14. Using relations (2.14a), show that the local bases vectors are connected by the relations q ı ı p ı ı r˛ rˇ D g r ; r˛ rˇ D g r ; ˛ ¤ ˇ ¤ ¤ ˛: 2.1.15. Show that the unit fourth-order tensors I , II and III defined by formulae (2.22) have the following properties: I T D I1 .T/E; I1 .T/ D T E; II T D TT ; III T D T;

TD

1 .T C TT /; 2

and I 4 D E ˝ E 4 ; III 4 D 6 ;

II 4 D .2134/ ;

4 D

1 .2134/ C /; . 2

for arbitrary second-order tensor T and fourth-order tensor 4 . As follows from these formulae, the tensor III is the ‘true’ unit fourth-order tensor. 2.1.16. Show that components of the symmetric unit fourth-order tensor with respect to a tetradic basis have the form D

1 .ei ˝ el ˝ ei ˝ el C ei ˝ el ˝ el ˝ ei / D ijkl ei ˝ ej ˝ ek ˝ el : 2 ijkl D

1 i k jl .ı ı C ı i l ı jk /: 2

2.1.17. Show that for any second-order tensor T and for any vector a the following formula of covariant differentiation hold: r .T a/ D T .r ˝ a/T C a r T:

24

2 Kinematics of Continua

2.2 Deformation Tensors and Measures 2.2.1 Deformation Tensors Besides F, important characteristics of the motion of a continuum are deformation tensors, which are introduced as follows: 1 gij 2 1 AD gij 2 1 ı ij g ƒD 2 1 ı ij JD g 2 CD

ı ı ı ı ı g ij ri ˝ rj D "ij ri ˝ rj ; ı g ij ri ˝ rj D "ij ri ˝ rj ; ı ı ı ı gij ri ˝ rj D "ij ri ˝ rj ; gij ri ˝ rj D "ij ri ˝ rj :

(2.68)

Here C is called the right Cauchy–Green deformation tensor, A – the left Almansi deformation tensor, ƒ – the right Almansi deformation tensor, and J – the left Cauchy–Green tensor. As follows from the definition of the tensors, covariant components of C and A are coincident, but they are defined with respect to different tensor bases. Components "ij are called covariant components of the deformation tensor. Contravariant components of the tensors ƒ and J are also coincident and called contravariant components "ij of the deformation tensor, but they are defined with respect to different tensor bases of the tensors ƒ and J. Notice that the deformation tensor components "ij D

1 ı gij g ij ; 2

"ij D

1 ı ij g gij ; 2

(2.69)

have been defined independently of each other, therefore the formal rearrangement of indices is not permissible for these components, i.e. "Mkl D "ij gi k g jl ¤ "kl ;

"Mkl D "ij gi k gjl ¤ "kl :

(2.70)

Thus, when there is a need to obtain contravariant components from "ij and covariant components from "ij , one should use the notation "M kl and "Mkl . We will also use the notation ı ı ı ı ı ı "kl D "ij g i k g jl ; "kl D "ij g i k g jl : (2.71) Theorem 2.3. The deformation tensors C, A, ƒ and J are connected to the deformation gradient F as follows: C D 12 .FT F E/;

A D 12 .E F1T F1 /;

ƒ D 12 .E F1 F1T /;

J D 12 .F FT E/:

(2.72)

2.2 Deformation Tensors and Measures

25 ı

H Let us derive a relation between C and F. Having used the definitions of gij , g ij and F, we get 1 ıi 1 1 ı ı ı r ˝ ri rj ˝ rj E D .FT F E/: .ri rj /ri ˝ rj E D 2 2 2 (2.73) The remaining relations of (2.72) can be proved in the same way: 1 1 ı ı AD E ri ˝ ri rj ˝ rj D E F1T F1 ; 2 2 1 1 ı ı E ri ˝ ri rj ˝ rj D E F1 F1T ; ƒD 2 2 1 1 ıi ı j ri ˝ r r ˝ rj E D F FT E : N JD (2.73a) 2 2

CD

2.2.2 Deformation Measures Besides the deformation tensors, we define deformation measures: the right Cauchy– Green measure G and the left Almansi measure g: ı

ı

G D gij ri ˝ rj D FT F D E C 2C; ı

g D g ij ri ˝ rj D F1T F1 D E 2A;

(2.74)

and also the left Cauchy–Green measure g1 and the right Almansi measure G1 : ı

g1 D g ij ri ˝ rj D F FT D E C 2J; ı

ı

G1 D gij ri ˝ rj D F1 F1T D E 2ƒ:

(2.75)

2.2.3 Displacement Vector Introduce a displacement vector u of a point M from the reference configuration to the actual one as follows (Fig. 2.10): ı

u D x x:

(2.76)

Theorem 2.4. The deformation tensors and the deformation gradient are connected to the displacement vector u by the relations ı

F D E C .r ˝ u/T ; ı

FT D E C r ˝ u;

F1 D E .r ˝ u/T ; F1T D E r ˝ u;

(2.77)

26

2 Kinematics of Continua

Fig. 2.10 The displacement vector of a point M from the reference configuration to the actual one

and also CD AD ƒD JD

ı ı ı 1 ı T T r ˝uCr ˝u Cr ˝ur ˝u ; 2 1 r ˝ u C r ˝ uT r ˝ u r ˝ uT ; 2 1 r ˝ u C .r ˝ u/T r ˝ uT r ˝ u ; 2 ı ı ı 1 ı r ˝ u C r ˝ uT C r ˝ uT r ˝ u : 2

(2.78)

H The definition (2.76) of the displacement vector and the properties (2.35) of the deformation gradient yield ı

ı

ı

FT D r ˝ x D r ˝ .x C u/ ı

D ri ˝

ı

ı ı ı @x ı ı C r ˝ u D ri ˝ ri C r ˝ u D E C r ˝ u: i @X

Then the tensor C takes the form ı ı 1 T CD .E C r ˝ u/ .E C r ˝ u / E 2 ı ı ı 1 ı T T r ˝uCr ˝u Cr ˝ur ˝u : D 2

(2.79)

(2.80)

In a similar way, we can prove the remaining relations of the theorem. N

2.2.4 Relations Between Components of Deformation Tensors and Displacement Vector ı

The displacement vector u can be resolved for both bases ri and ri : ı ı

u D ui ri D ui ri :

(2.81)

2.2 Deformation Tensors and Measures

27

The derivative with respect to X i can be determined in both the bases as well: ı ı ı @u k k D r i u rk D ri u rk : @X i

(2.82)

Then the displacement vector gradients take the forms ı

ı

r ˝ u D ri ˝

ı ı ı ı ı ı @u ı ı k i i k D r u r ˝ r D r u ri ˝ rk ; i k @X i

@u D ri uk ri ˝ rk D r i uk ri ˝ rk : @X i Substitution of these expressions into (2.77) gives r ˝ u D ri ˝

ı ı ı ı ı ı ı F D ıik C r i uk rk ˝ ri D F ki rk ˝ ri :

(2.83)

(2.84)

(2.85)

Here we have introduced components of the deformation gradient in the reference configuration: ı ı

ı

F ki D ıik C r i uk :

(2.86)

The transpose gradient FT has the components ı

ı

ı

ı

ı

ı

FT D F ki ri ˝ rk D F ik rk ˝ ri ; ı

ı

ı

F ik D ıik C r k ui ; ı

(2.87)

(2.88)

ı

where .F ik /T D F ki . In a similar way, one can find the expression for the inverse gradient k F1 D ıik ri uk rk ˝ ri D F 1 i rk ˝ ri

(2.89)

and for the inverse–transpose gradient F1T D .F 1 /ki ri ˝ rk D .F 1 /ik rk ˝ ri ;

(2.90)

where their components with respect to the actual configuration are expressed as follows: .F 1 /ik D ıik r k ui ; (2.91) .F 1 /ki D ıik ri uk : Thus, we have proved the following theorem.

(2.92)

28

2 Kinematics of Continua

Theorem 2.5. Components of the deformation gradients F, FT , F1 and F1T in ı

local bases of configurations K and K are connected to components of the displacement vector u by relations (2.86), (2.87), (2.91), and (2.92). On substituting formulae (2.83) and (2.84) into (2.78) for C and A and comparing them with (2.68), we get ı ı ı ı ı ı 1 ı ı "ij D r i u j C r j u i C r i uk r j uk ; 2 1 "ij D ri uj C rj ui ri uk rj uk ; 2

(2.93)

– The expressions for covariant components of the deformation tensor in terms of ı

components of the displacement vector with respect to K and K. In a similar way, substituting (2.83) and (2.84) into (2.78) for ƒ and J, we obtain ı ı ı ı ı ı 1 ı i ıj r u C r j u i C r k u i r k uj ; 2 1 r i uj C r j u i r k u i rk uj "ij D 2 "ij D

(2.94)

– The relations between contravariant components of the deformation tensor and ı

components of the displacement vector in K and K. Then with using relations (2.69), (2.93) and (2.94), we can find the connection between the metric matrices and displacement components: ı

ı ı

ı

ı

ı ı

ı

ı ı

ı

ı

ı

ı

ı

ı

ı

gij D g ij C r i uj C r j ui C r i uk rj uk D g ij Cri uj Crj ui ri uk rj uk ; (2.95) ı ı

g ij D g ij C r i uj C r j ui C r k ui r k uj D g ij Cr i uj Cr j ui r k ui rk uj : (2.96) Thus, we have proved the following theorem. Theorem 2.6. Components of the deformation tensor "ij , "ij and metric matrices gij , g ij are connected to components of the displacement vector u by relations (2.93)–(2.96).

2.2.5 Physical Meaning of Components of the Deformation Tensor Let us clarify now a physical meaning of components of the deformation tensor: 1 1 ı ı ı gij g ij D ri rj ri rj : "ij D (2.97) 2 2

2.2 Deformation Tensors and Measures

29

By the definition of the scalar product (see [12]), we have "˛ˇ ı

where ı

˛ˇ

and

1 D jr˛ jjrˇ j cos 2

ı

ı

jr˛ jjrˇ j cos

˛ˇ

ı

;

˛ˇ

(2.98) ı

˛ˇ

ı

are the angles between basis vectors r˛ , rˇ and r˛ , rˇ in K and

K, respectively.

ı

ı

Consider elementary radius-vectors d x and d x in configurations K and K, and ı introduce their lengths ds and d s, respectively: ds 2 D d x d x;

ı

ı

ı

d s 2 D d x d x:

(2.99)

ı

Since d x is arbitrary, we can choose it to be oriented along one of the basis vectors ı r˛ . Then d x will be directed along the corresponding vector r˛ as well, because ı under this transformation r˛ becomes r˛ for the same material point M with Lagrangian coordinates X k . In this case we have ˇ @xı ˇ ı ˇ ˇ jd xj D d s ˛ D ˇ ˛ dX ˛ ˇ D jr˛ j dX ˛ ; @X ˇ @x ˇ ˇ ˇ jd xj D ds˛ D ˇ ˛ dX ˛ ˇ D jr˛ j dX ˛ : @X ı

ı

(2.100)

Hence ı

ı

ds˛ =d s ˛ D jr˛ j=jr˛ j D ı˛ C 1;

(2.101)

where ı˛ is called the relative elongation. Formula (2.101) yields ı

jr˛ j D jr˛ j.1 C ı˛ /:

(2.102)

On substituting this expression into (2.98), we get "˛ˇ D

1 ı ı jr˛ jjrˇ j .1 C ı˛ /.1 C ıˇ / cos 2

Consider the case when ˛ D ˇ, then

˛ˇ

D

ı ˛ˇ

˛ˇ

cos

ı ˛ˇ

:

(2.103)

D 0 and

ı

"˛˛ D

g 1 ı 2 jr˛ j .1 C ı˛ /2 1 D ˛˛ .1 C ı˛ /2 1 : 2 2

(2.104)

30

2 Kinematics of Continua ı

Let coordinates X i be coincident with Cartesian coordinates x i , then g ˛ˇ D ı˛ˇ ; and for infinitesimal values of the relative elongation, when ı˛ 1, we obtain "˛˛ ı˛ ;

(2.105)

i.e. "˛˛ is coincident with the relative elongation. In general, "˛˛ is a nonlinear function of corresponding elongations. Consider ˛ ¤ ˇ and assume that X i D x i , then we get

ı

˛ˇ

D =2; and from (2.103)

1 ı ı jr˛ jjrˇ j.1 C ı˛ /.1 C ıˇ / cos ˛ˇ 2q q 1 ı ı D g ˛˛ g ˇˇ .1 C ı˛ /.1 C ıˇ / sin ˛ˇ 2 1 D .1 C ı˛ /.1 C ıˇ / sin ˛ˇ ; 2

"˛ˇ D

(2.106)

ı

where ˛ˇ D ˛ˇ ˛ˇ D . =2/ ˛ˇ is the change of the angle between basis vectors r˛ and rˇ . For small relative elongations, when ı˛ 1, and small angles

˛ˇ 1, from (2.106) we get "˛ˇ ˛ˇ =2;

(2.107)

i.e. "˛ˇ is a half of the misalignment angle of the basis vectors.

2.2.6 Transformation of an Oriented Surface Element In actual configuration K consider a smooth surface †, which contains two coordinate lines X ˛ and X ˇ . Then we can introduce the normal n to the surface † as follows: 1 n D p r˛ rˇ : e g

(2.108)

Here e g D det .e g ˛ˇ /, and e g ˛ˇ is the two-dimensional matrix of the surface (˛; ˇ D 1; 2): (2.109) e g ˛ˇ D r˛ rˇ (it is not to be confused with the metric matrix gij D ri rj ). In configuration K consider a surface element d† constructed on elementary radius-vectors d x˛ , which are directed along local basis vectors, i.e. d x˛ D r˛ dX ˛ (Fig. 2.11). The value p d† D e g dX ˛ dX ˇ (2.110)

2.2 Deformation Tensors and Measures

31

Fig. 2.11 Introduction of oriented surface element n d†

is called the area of the surface element d† constructed on vectors d x˛ and d xˇ . Then formula (2.108) takes the form n d† D r˛ dX ˛ rˇ dX ˇ D d x˛ d xˇ ;

(2.111)

where n d† is called the oriented surface element. Show that the normal n defined by formula (2.108) is a unit vector. According to the property (2.14b) of the vector product of basis vectors and the results of Exercise 2.1.13, we can rewrite Eq. (2.111) in the form p n d† D r˛ rˇ dX ˛ dX ˇ D g˛ˇ r dX ˛ dX ˇ p D .1= g/ ijk g˛i gˇj rk dX ˛ dX ˇ

(2.112)

(there is no summation over ˛; ˇ). Thus, n d† n d† D

p

1 g ˛ˇ r p ijk g˛i gˇj rk /.dX ˛ dX ˇ /2 g j

D .˛ˇ k ijk g˛i gˇj /.dX ˛ dX ˇ /2 D.ı˛i ıˇ ıˇi ı˛j /g˛i gˇj .dX ˛ dX ˇ /2 2 D .g˛˛ gˇˇ g˛ˇ /.dX ˛ dX ˇ /2 D e g.dX ˛ dX ˇ /2 D d†2 :

(2.113) Hence, n n D 1:

ı

ı

The surface element d† in K corresponds to the surface element d † in K, which ı ı is constructed on elementary radius–vectors d x˛ and d xˇ : ı

ı

ı

ı

ı

ı

n d † D r˛ dX ˛ rˇ dX ˇ D r˛ rˇ dX ˛ dX ˇ : ı

(2.114)

ı

Here n is the unit normal to d †. ı Since r D F1T r , we get p ı n d† D g ˛ˇ F1T r dX ˛ dX ˇ q ı ı ı D g=g F1T r˛ rˇ dX ˛ dX ˇ q ı ı ı D g=g F1T n d †: Thus, we have proved the following theorem.

(2.115)

32

2 Kinematics of Continua ı

ı

ı

Theorem 2.7. The oriented surface elements n d † and n d† in K and K are connected by the relation q n d† D

ı ı

g=g n F

1

q ı ı ı d † D g=g F1T n d †: ı

(2.116)

With the help of the deformation measures we can derive formulae connecting ı the normals n and n to the surface element containing the same material points in K ı

and K. Multiplying Eq. (2.116) by itself and taking the formula n n D 1 into account, we get d†2 D

ı ı g ı ı 1 1T ı 2 1 ı 2 . . n F F n/ d † D n G n/ d † : ı ı g g

g

Thus,

q

ı

d†=d † D

ı

ı

ı

g=g .n G1 n/1=2 :

(2.117)

(2.118)

ı

On the other hand, expressing n from (2.116) and then multiplying the obtained relation by itself, we obtain ı

ı

d †2 D

ı

g g .n F FT n/ d†2 D .n g1 n/ d†2 ; g g

Thus, we find that

q

ı

d †=d† D

ı

g=g .n g1 n/1=2 :

(2.119)

(2.120)

On introducing the notation ı

ı

ı

ı

ı

k D .n G1 n/1=2 D .n F1 F1T n/1=2 ; k D .n g1 n/1=2 D .n F FT n/1=2 ;

(2.121)

relations (2.119) take the form ı

d †=d† D

q

q ı ı g=g k D g=g .1=k/: ı

Thus, we get

(2.122)

ı

k D 1=k:

(2.123)

Substitution of (2.119) and (2.120) into (2.116) gives the desired relations ı

ı

kn D F1T n;

ı

k n D FT n:

(2.124)

2.2 Deformation Tensors and Measures

33

2.2.7 Representation of the Inverse Metric Matrix in terms of Components of the Deformation Tensor Components gij of the metric matrix are connected to components of the deformation tensor "ij by relation (2.69). In continuum mechanics, one often needs to know the expression of the inverse metric matrix gij in terms of "ij (but not in terms of "ij ). To derive this relation, we should use the connection between components of a matrix and its inverse (see [12]): g ij D

1 imn jkl gmk gnl : 2g

(2.125)

ı

For g ij , we have the similar formula ı

g ij D

1 ı

2g

ı

ı

imn jkl g mk g nl :

(2.126)

ı

On substituting the relations (2.69) between gmn , g mn and "mn into (2.126), we get 1 imn jkl ı ı .g mk C 2"mk /.g nl C 2"nl / 2g 1 imn jkl ı ı ı ı .g mk g nl C 2g mk "nl C 2g nl "mk C 4"mk "nl /: D 2g

gij D

(2.127)

Removing the parentheses, modify four summands in (2.127) in the following way. The first summand with taking formula (2.126) into account gives the maı ı trix g ij .g=g/. To transform the second and the third summands, we should use the formulae q q ı ı ı ı ı jkl 1= g D g t sp g jt g ks g lp : (2.128) ı

ı

ı

ı

ı

ı

ı

ı

.1=g/ imn jkl g mk D imn t sp g jt g ks g lp gmk D imn t mp g jt g lp ı

ı

ı

ı

ı

ı

D .ıti ıpn ıpi ıtn /g jt g lp D g ij gnl g j n g i l : ı

ı

ı

ı

ı

ı

.1=g/imn jkl gmk "nl D .gij g nl g j n g i l /"nl :

(2.129)

(2.130)

Formula (2.128) follows from the relation p

p g ijk D .1= g/ mnl gmi gnj glk

(see [12]), and relationship (2.129) has been obtained by using formula (2.128) and the properties of the Levi-Civita symbols (see Exercise 2.1.13).

34

2 Kinematics of Continua

On substituting formula (2.130) into (2.127), we get ı

g g ij D g

! ı ı 2 ı ı g C 2 g ij g nl 2gi l gj n "nl C ı imn jkl "mk "nl : g ı ij

(2.131)

Finally, we should express the determinant g D det .gij / in terms of "ij . To do this, we multiply relation (2.131) by gij and take formula (2.69) into account: ı

ı ij

ı ij ı nl

3g D g g C 2.g g

ı il ıjn

g g /"nl C

2 ı

!

imn jpl

g

ı

"mp "nl .g ij C 2"ij /: (2.132)

Thus, we get ı ı ı ı 3g D g 3 C 4gnl "nl C .2=g/imn jpl g ij "mp "nl

ı ı ı ı ı ı C 2g ij "ij C 4.g ij g nl gi l gj n /"nl "ij C .4=g/ imn jpl "ij "mp "nl : (2.133)

Modifying the third summand on the right-hand side by formula (2.130) and introducing the notation ı

I1" D g nl "nl ; I2" D

1 ı ij ı nl ı i l ı j n ı g g g g "ij "nl ; I3" D det ."ij gi k /; (2.134) 2

from (2.133) we get the desired formula ı

g D g.1 C 2I1" C 4I2" C 8I3" /:

(2.135)

Here we have taken account of formula (2.125) for the matrix determinant and also ı the relation I3" D .1=g/det ."ij /. Thus, we have proved the following theorem. Theorem 2.8. The inverse metric matrix gij is expressed in terms of components ı "ij of the deformation tensor and g ij by formulae (2.131) and (2.135). Formulae (2.131) and (2.135) allow us to find the expression of contravariant components "ij of the deformation tensor in terms of "ij . It follows from (2.69) that ı g gı gı ı ı ı 1 ı ı "ij D.1=2/ g ij g ij Dg ij g ij g nl g i l g j n "nl imn jkl "mk "nl : 2g g g (2.136) Substitution of formulae (2.93) or (2.94) into (2.131) and (2.136) gives the expresı sion for components "ij in terms of components of the displacement vector ui or ui .

2.2 Deformation Tensors and Measures

35

Exercises for 2.2 2.2.1. Using the results of Exercise 2.1.1, show that the deformation gradient F and its inverse F1 for the problem on a beam in tension (see Example 2.1) have the forms F D FT D

3 X

k˛ eN ˛ ˝ eN ˛ ;

F1 D F1T D

˛D1

3 X 1 eN ˛ ˝ eN ˛ : k ˛D1 ˛

For this problem, the deformation tensors are determined by the formulae CDƒD

3 1X 2 .k 1/Ne˛ ˝ eN ˛ ; 2 ˛D1 ˛

ADƒD

3 1X .1 k˛2 /Ne˛ ˝ eN ˛ ; 2 ˛D1

the deformation measures are determined as follows: G D g1 D

3 X

k˛2 eN ˛ ˝ eN ˛ ;

g D G1 D

˛D1

3 X

k˛2 eN ˛ ˝ eN ˛ ;

˛D1

and components of the deformation tensor take the forms "˛ˇ D

1 2 .k 1/ı˛ˇ ; 2 ˛

"˛ˇ D

1 .1 k˛2 /ı˛ˇ : 2

2.2.2. Using the results of Exercise 2.1.2, show that for the problem on a simple shear we have the following formulae for the deformation gradient: ı

F D F ij eN i ˝ eN j D E C aNe1 ˝ eN 2 ; F1 D E aNe1 ˝ eN 2 ; i.e.

FT D E C aNe2 ˝ eN 1 ;

F1T D E aNe2 ˝ eN 1 ;

0

ı

F ij

1 1a0 D @0 1 0A ; 001

det F D 1;

for the deformation tensors: C D .a=2/O3 C .a2 =2/Ne22 ;

A D .a=2/O3 .a2 =2/Ne22 ;

ƒ D .a=2/O3 .a2 =2/Ne21 ;

J D .a=2/O3 C .a2 =2/Ne21 ;

and for components of the deformation tensor: 1 0 a=2 0 ."ij / D @a=2 a2 =2 0A ; 0 0 0 0

1 0 a=2 0 ."ij / D @a=2 a2 =2 0A : 0 0 0 0

36

2 Kinematics of Continua

Here we have introduced the notation O3 eN 1 ˝ eN 2 C eN 2 ˝ eN 1 ;

eN 2˛ eN ˛ ˝ eN ˛ ;

˛ D 1; 2; 3:

2.2.3. Using the formulae from Example 2.3 (see Sect. 2.1.1), show that for the problem on rotation of a beam with extension, the deformation gradient has the form F D F0ij eN i ˝ eN j D cos '

2 X

k˛ eN ˛ ˝ eN ˛ C k3 eN 3 ˝ eN 3 C sin 'k2 .Ne2 ˝ eN 1 eN 1 ˝ eN 2 /:

˛D1

2.2.4. Using formulae (2.36), (2.85)–(2.92) and the results of Exercise 2.1.7, show that local basis vectors are connected to displacements by the relations ı

ı

ı

ri D rk .ıik C r k ui /;

ı

ri D rk .ıik ri uk /:

2.2.5. Using formulae (2.104) and q (2.106), show that the physical components of b ı ı ı the deformation tensor "˛ˇ D "˛ˇ = g ˛˛ g ˇˇ are connected to relative elongations ı˛ and angles ˛ˇ by the relations 1 b ı "˛˛ D ..1 C ı˛ /2 1/; 2

1 b ı "˛ˇ D .1 C ı˛ /.1 C ıˇ / sin ˛ˇ : 2

e i the expression 2.2.6. Show that in the basis e ri of curvilinear coordinate system X 1 of the tensor F in terms of r ˝ u can be rewritten in the form similar to (2.89)– (2.92): u De uke rk ;

e 1 /ki e F1 D .F ri ; rk ˝e

eie e 1 /ki D ıik r .F uk :

2.2.7. Using formula (2.34), show that the following relationships hold: ı

ı

jd xj2 D d x G d x;

ı

jd xj2 D d x g d x:

2.3 Polar Decomposition 2.3.1 Theorem on Polar Decomposition According to (2.36), the tensor F can be considered as a tensor of the linear transı ı formation from the basis ri to the basis ri . Since the vectors ri and ri are linearly independent, the tensor F is nonsingular. Then for this tensor the following theorem is valid.

2.3 Polar Decomposition

37

Theorem 2.9 (on the polar decomposition). Any nonsingular second-order tensor F can be represented as the scalar product of two second-order tensors: FDOU

or F D V O:

(2.137)

Here U and V are the symmetric and positive-definite tensors, O is the orthogonal tensor, and each of the decompositions (2.137) is unique. H Prove the existence of the decomposition (2.137) in the constructive way, i.e. we should construct the tensors U, V and O. To do this, consider the contractions of the tensor F with its transpose: FT F and F FT . Both the tensors are symmetric, because .FT F/T D FT .FT /T D FT F and .F FT /T D .FT /T FT D F FT ; (2.138) and positive-definite: a .FT F/ a D .a FT / .F a/ D .F a/ .F a/ D b b D jbj2 > 0

(2.139)

for any non-zero vector a, where b D F a. Since any symmetric positive-definite tensor has three real positive eigenvalues [12], eigenvalues of tensors FT F and ı

F FT can be denoted as 2˛ and 2˛ . These tensors are diagonal in their eigenbases, i.e. they have the following forms: FT F D

3 X

ı

ı

ı

2˛ p˛ ˝ p˛ ;

F FT D

˛D1

3 X

2˛ p˛ ˝ p˛ :

(2.140)

˛D1

ı

Here p˛ are the eigenvectors of the tensor FT F and p˛ – of the tensor F FT , which are real-valued and orthonormal: ı

ı

p˛ pˇ D ı˛ˇ ;

p˛ pˇ D ı˛ˇ :

(2.141)

The right-hand sides of (2.140) are the squares of certain tensors U and V defined as UD

3 X

ı

ı

ı

ı

˛ p˛ ˝ p˛ ; ˛ > 0I

VD

˛D1

3 X

˛ p˛ ˝ p˛ ; ˛ > 0:

(2.142)

˛D1

Here signs at ˛ are always chosen positive. In this case, the following relations are valid: F T F D U2 ;

F FT D V2 :

(2.143)

38

2 Kinematics of Continua

The constructed tensors V and U are symmetric due to formula (2.142) and positivedefinite, because for any nonzero vector a we have aUaD

3 X

ı

ı

ı

˛ a p˛ ˝ p˛ a D

˛D1

3 X

ı

ı

˛ .a p˛ /2 > 0;

(2.144)

˛D1

ı

as ˛ > 0. In a similar way, we can prove that the tensor V is positive-definite. Both the tensors V and U are nonsingular, because, under the conditions of the theorem, the tensor F is nonsingular. And from (2.143) we get .det U/2 D det U2 D det .FT F/ D .det F/2 ¤ 0:

(2.145)

Then there exist inverse tensors U1 and V1 , with the help of which we can construct two more new tensors ı

O D F U1 ;

O D V1 F;

(2.146)

which are orthogonal. Indeed, ı

ı

OT O D .F U1 /T .F U1/ D U1 FT F U1 D U1 U2 U1 D E: (2.147) ı

According to [12], this means that the tensor O is orthogonal. In a similar way, we can show that the tensor O is orthogonal as well. ı

Thus, we have really constructed the tensors U and O, and also V and O, the product of which, due to (2.146), gives the tensor F: ı

F D O U D V O:

(2.148) ı

Here U and V are symmetric, positive-definite tensors, O and O are orthogonal tensors. Show that each of the decompositions (2.148) is unique. By contradiction, let there be one more resolution, for example ı

But then

ee U: FDO

(2.149)

FT F D e U2 D U 2 ;

(2.150)

hence, e U D U, because the decomposition of the tensor FT F for its eigenbasis is ı

ı

˛ are chosen positive by the condition. The coincidence of unique. Signs at ˛ and e ı

ı

e and O are coincident as well, because U and e U leads to the fact that O ı

ı

e DFe O U1 D F U1 D O:

(2.151)

2.3 Polar Decomposition

39

This has proved uniqueness of the decomposition (2.148). We can verify uniqueness of the decomposition F D V O in a similar way. ı

Finally, we must show that the orthogonal tensors O and O are coincident, i.e. formula (2.137) follows from (2.148). To do this, we construct the tensor ı

ı

ı

F OT D O U OT :

(2.152)

Due to (2.148), this tensor satisfies the following relationship: ı

ı

ı

O U OT D V O OT :

(2.153)

ı

The tensor O OT is orthogonal, because ı

ı

ı

ı

ı

ı

.O OT /T .O OT / D O OT O OT D O OT D E:

(2.154)

Then the relationship (2.153) can be considered as the polar decomposition of the ı

ı

tensor O U OT . This tensor is symmetric, because ı

ı

ı

ı

ı

ı

.O U OT /T D .OT /T .O U/T D O U OT :

(2.155)

Then the formal equality ı

ı

ı

ı

O U OT D O U OT

(2.156)

is one more polar decomposition. However, as was shown above, the polar decomposition is unique; hence, the following relationships must be satisfied: ı

ı

ı

V D O U OT and O OT D E: ı

(2.157) ı

Thus, the orthogonal tensors O and O are coincident: O D O. N The tensors U and V are called the right and left stretch tensors, respectively, and O is the rotation tensor accompanying the deformation. The tensor F has nine independent components, the tensor O – three independent components, and each of the tensors U and V – six independent components. Remark 1. Since the rotation tensor O is unique in the polar decomposition, from formula (2.157) we get that the stretch tensors U and V are connected by means of the tensor O: V D O U OT ; U D OT V O: (2.157a) t u

40

2 Kinematics of Continua

Theorem 2.10. The Cauchy–Green and Almansi deformation tensors can be expressed in terms of the stretch tensors U and V as follows: 1 2 1 .U E/; A D .E V2 /; 2 2 1 1 ƒ D .E U2 /; J D .V2 E/: 2 2 CD

(2.158)

H To see this, let us substitute the polar decomposition (2.137) into (2.72), and then we get the relationships (2.158). N

2.3.2 Eigenvalues and Eigenbases Theorem 2.11. Eigenvalues of the tensors U and V defined by (2.142) are coincident: ı

˛ D ˛ ;

˛ D 1; 2; 3;

(2.159)

ı

and eigenvectors p˛ and p˛ are connected by the rotation tensor O: ı

p˛ D O p˛ :

(2.159a)

H To prove the theorem, we use the definition (2.142) and the first formula of (2.157a): VD

3 X

˛ p˛ ˝ p˛ D O U OT D

˛D1

where

3 X

ı

ı

ı

˛ O p˛ ˝ .O p˛ / D

˛D1

3 X

ı

˛ p0˛ ˝ p0˛ ;

˛D1 ı

p0˛ D O p˛ :

According to the relationship, we have obtained two different eigenbases of the tensor V and two sets of eigenvalues, that is impossible. Therefore, ı

ı

p0˛ D O p˛ D p˛ and ˛ D ˛ ; as was to be proved. N Due to (2.141), both the eigenbases are orthogonal. Therefore, reciprocal vectors ı of the eigenbases do not differ from p˛ and p˛ : p˛ D p˛ ;

ı

ı

p˛ D p˛ :

(2.160)

2.3 Polar Decomposition

41 ı

The important problem for applications is to determine ˛ , p˛ and p˛ by the given deformation gradient F. To solve the problem, one should use the following method: 1. Construct the tensor U2 D FT F (or V2 D F FT ) and find its components in some basis being suitable for a considered problem; for example, in the Cartesian basis eN i : U2 D .UN 2 /i eN i ˝ eN j and V2 D .VN 2 /i eN i ˝ eN j : j

j

2. Find eigenvalues of the matrix .UN / j by solving the characteristic equation 2 i

det .U2 2˛ E/ D 0;

(2.161)

which in the basis eN i takes the form det ..U 2 /ij 2˛ ıji / D 0:

(2.161a)

ı

3. Find eigenvectors p˛ of the tensor U and eigenvectors p˛ of the tensor V from the equations ı

ı

U2 p˛ D 2˛ p˛ ; V2 p˛ D 2˛ p˛ ;

(2.162)

written, for example, in the basis eN i : ı

b j˛ D 0; ..UN 2 /ij 2˛ ıji /Q

b j˛ D 0; ..VN 2 /ij 2˛ ıji /Q

(2.162a)

ı

b j˛ are Jacobian matrices of the eigenvectors: b j˛ and Q where Q ı

ı

b j˛ eN j ; p˛ D Q

b j˛ eN j : p˛ D Q

(2.163)

ı

b j˛ and Q b j˛ , one should consider only independent To determine the matrices Q equations of the system (2.162a) and the normalization conditions (2.141): jp˛ j D 1;

ı

jp˛ j D 1;

(2.164)

which are equivalent to the quadratic equations ı

ı

b i˛ Q b j˛ ıij D 1; Q

b j˛ ıij D 1: b i˛ Q Q

(2.164a)

42

2 Kinematics of Continua

4. Write the dyadic products (2.142) and find resolutions of the tensors U and V for the eigenbases; for example, for the Cartesian basis eN i : UD

3 X

ı

ı

b i˛ Q b j˛ eN i ˝ eN j ; V D ˛ Q

˛D1

3 X

b i˛ Q b j˛ eN i ˝ eN j : ˛ Q

˛D1

Exercises 2.3.2–2.3.4 show examples of determination of the tensors U and V. Remark 2. Notice that a solution of the quadratic equations (2.164a) may be not ı

b i˛ , this ambigub i˛ and Q unique due to the choice of signs of matrix components Q ity is resolved by applying one more additional condition, namely the condition of ı coincidence of the vectors p˛ and p˛ when t ! 0C : ı

t ! 0C ) p˛ .t/ D p˛ .t/;

˛ D 1; 2; 3:

ı

b i˛ , the ambiguity of the sign choice remains. However, if there For the matrix Q ı

is a field of eigenvectors p˛ .x; t/, then this ambiguity may be retained only at one point x0 at one time, for example, t D 0; and for the remaining x and t, a sign at b i˛ is chosen from the continuity condition of the vector field pı ˛ .x; t/ (for continQ ı

uous motions). If the eigenvector field p˛ .x0 ; 0/ contains the vectors eN ˛ , then the ı remaining ambiguity is resolved by the condition p˛ .x0 ; 0/ D eN ˛ : The ambiguity of a solution of the system (2.162a), (2.164a) may also appear, if at some time t1 at a point x the eigenvalues ˛ .t1 / prove to be triple. In this case, ı

b i˛ .t1 / are determined, as a rule, by passage to b i˛ .t1 / and Q values of the matrices Q the limit: b i˛ .t/; b i˛ .t1 / D lim Q Q t !t1

ı

ı

b i˛ .t1 / D lim Q b i˛ .t/; ˛ D 1; 2; 3: Q t !t1

In the case of double eigenvalues ˛ , these formulae are applied only to their correı

b i˛ and Q b i˛ . sponding matrix components Q

t u

2.3.3 Representation of the Deformation Tensors in Eigenbases ı

ı

Theorem 2.12. In the tensor bases p˛ ˝pˇ and p˛ ˝ pˇ , the Cauchy–Green tensors C and J, the Almansi tensors A and ƒ, and the deformation measures G, g1 and G1 , g have the diagonal form: CD

3 X 1 2 ı ı .˛ 1/p˛ ˝ p˛ ; 2 ˛D1

ƒD

3 X 1 ı ı .1 2 ˛ /p˛ ˝ p˛ ; 2 ˛D1

2.3 Polar Decomposition

43

3 X 1 .1 2 ˛ /p˛ ˝ p˛ ; 2 ˛D1

AD

JD

3 X 1 2 . 1/p˛ ˝ p˛ I (2.165a) 2 ˛ ˛D1

and GD

3 X

ı

ı

G1 D

2˛ p˛ ˝ p˛ ;

˛D1

g

1

D

3 X

3 X

ı

ı

2 ˛ p˛ ˝ p˛ ;

˛D1

2˛ p˛

˝ p˛ ;

gD

˛D1

3 X

2 ˛ p˛ ˝ p˛ :

(2.165b)

˛D1

H On substituting formulae (2.142) into (2.158), we get (2.165a). Formulae (2.165b) follow from (2.165a) and (2.74), (2.75). N Similarly to formulae (2.165), we can introduce new deformation tensors by deı ı termining their components with respect to the bases p˛ ˝ pˇ or p˛ ˝pˇ as follows: 3 X

ı

MD

ı

ı

f .˛ /p˛ ˝ pˇ ;

MD

˛D1

3 X

f .˛ /p˛ ˝ pˇ ;

(2.166)

˛D1

where f .˛ / is a function of ˛ . If f .1/ D 0, then we get the deformation tensors; and if f .1/ D 1, then we get the deformation measures. Among the tensors (2.166), the logarithmic deformation tensors and measures ı

HD

P3

ı ˛D1 lg ˛ p˛ ı ı H1 D H

ı

˝ pˇ ;

HD

C E;

P3

˛D1

lg ˛ p˛ ˝ pˇ ;

H1 D H C E;

(2.167)

are the most widely known; they are called the right and left Hencky tensors, and also the right and left Hencky measures, respectively. ı With the help of the eigenvectors p˛ and p˛ we can form the mixed dyads 3 X ˛D1

ı

p˛ ˝ p˛ D

3 X

p˛ ˝ p˛ O D

˛D1

3 X

! p˛ ˝ p

˛

O D E O:

(2.168)

˛D1

Here we have used the properties (2.159a) and (2.160), and the representation of the unit tensor E in an arbitrary mixed dyadic basis. Thus, the rotation tensor O accompanying the deformation can be expressed in the eigenbasis as follows: OD

3 X ˛D1

ı

ı

p˛ ˝ p˛ D pi ˝ pi :

(2.169)

44

2 Kinematics of Continua

On substituting (2.169) and (2.142) into (2.137) and taking (2.141) into account, we get the following expression of the deformation gradient in the tensor eigenbasis: FDOUD

3 X

ı

p˛ ˝ p˛

3 X

ı

ı

ı

ˇ pˇ ˝ pˇ D

ˇ D1

˛D1

3 X

ı

˛ p˛ ˝ p˛ :

(2.170)

˛D1

According to (2.170), the transpose FT and inverse F1 gradients are expressed as follows: F D T

3 X

ı

˛ p˛ ˝ p˛ ;

F

1

D

˛D1

3 X

ı

1 ˛ p˛ ˝ p˛ :

(2.171)

˛D1

2.3.4 Geometrical Meaning of Eigenvalues ı

ı

Vectors of eigenbases p˛ and p˛ are connected by the transformation (2.159a). In K ı ı take elementary radius-vectors d x˛ oriented along the eigenbasis vectors p˛ , then in K they correspond to radius-vectors d x˛ : ı

ı

ı

ı

d x˛ D p˛ jd x˛ j;

d x˛ D F d x˛ :

(2.172)

Substitution of (2.170) into (2.172) yields d x˛ D

3 X

ı

ı

ı

ˇ pˇ ˝ pˇ p˛ jd x˛ j D ˛ jd x˛ jp˛ ;

(2.173)

ˇ D1

i.e. the elementary radius-vectors d x˛ in K will be also oriented along the corresponding eigenbasis vectors p˛ . ı ı Denote lengths of the vectors d x˛ and d x˛ by d s ˛ and ds, respectively, and derive relations between them: ı

ı

ı

ı

ı

ds˛2 D d x˛ d x˛ D d x˛ FT F d x˛ D jd x˛ j2 p˛ FT F p˛ ı ı

ı

ı

D d s 2˛ p˛ G p˛ D d s 2˛ 2˛ :

(2.174)

Here we have used Eqs. (2.165b) and (2.172). Formula (2.174) proves the following theorem. Theorem 2.13. Eigenvalues ˛ (principal stretches) are the elongation ratios for material fibres oriented along the principal (eigen-) directions: ı

˛ D ds˛ =d s ˛ :

(2.175)

2.3 Polar Decomposition

45

2.3.5 Geometric Picture of Transformation of a Small Neighborhood of a Point of a Continuum ı

In K, consider a small neighborhood of the material point M contained in a conı tinuum; then every point M0 , connected to M by the elementary radius-vector d x (Fig. 2.12), will be connected to the same point M by radius-vector d x in K. These radius-vectors are related as follows: ı

d x D F d x:

(2.176) ı

The relation can be considered as the transformation of arbitrary radius-vector d x into d x. Rewrite the relation (2.176) in Cartesian coordinates: ı dx i D FNmi d x m ;

(2.177)

where FNmi are components of the deformation gradient with respect to the Cartesian basis (see Exercise 2.1.5): ı (2.178) FNmi D [email protected] i [email protected] m /; ı

which depend only on coordinates x m of the point M, but they are independent of ı coordinates d x m of its neighboring points M0 . Therefore the transformation (2.177) ı is a linear transformation of coordinates d x m into dx i , i.e. this is an affine transformation. As follows from the general properties of affine transformations, straight lines ı

and planes contained in a small neighborhood in K will be straight lines and planes in actual configuration K. Parallel straight lines and planes are transformed into ı

parallel straight lines and planes. Therefore if a small neighborhood in K is chosen

Fig. 2.12 Transformation of a small neighborhood of the point contained in a continuum

46

2 Kinematics of Continua

to be a parallelogram, then in K the neighborhood will be a parallelogram as well (although angles between its edges, edge lengths and orientation of planes in space may change). ı

Since a second-order surface in K (and, in general, a surface specified by an algebraic expression of arbitrary nth order) is transformed into a surface of the same ı

order in K, a small spherical neighborhood in K is transformed into an ellipsoid in actual configuration K (Fig. 2.12). ı As follows from formula (2.101), the ratio of lengths ds˛ =d s ˛ of an arbitrary ı

vector (or of elementary radius-vector d x in K and K) is independent of the initial ı ı length d s ˛ of the vector (because the relative elongation ı˛ is independent of d s ˛ ). According to the polar decomposition (2.137), the transformation (2.176) from ı

K to K can always be represented as the superposition of two transformations: ı

d x D O d x0 ;

ı

ı

d x0 D U d x;

(2.179)

realized with the help of the stretch tensor U and the rotation tensor O, or d x D V d x0 ;

ı

d x0 D O d x:

(2.180) ı

The stretch tensor U, which has three eigendirections p˛ , transforms a small neighborhood of the point M with compressing or extending the neighborhood ı along these three directions p˛ . The tensor O rotates the neighborhood deformed ı ı along p˛ as a rigid whole until the direction of p˛ becomes the direction of p˛ . If ı

ı

one use the left stretch tensor V, so rotation of axes p˛ in K till their coincidence with p˛ is first realized, and then compression or tension of the neighborhood occurs along the direction p˛ . The result will be the same as for U. ı If a point M˛ is connected to M by radius-vector d x˛ oriented along the ı eigendirection p˛ (which is unknown before deformation), then in K the point M˛ will be connected to M by radius-vector d x˛ oriented along the corresponding eigendirection p˛ . ı

If a small neighborhood of point M is chosen to be a sphere in K (see Fig. 2.12), then in K the sphere becomes an ellipsoid with principal axes oriented along the eigendirections p˛ . Thus, the transformation of a small neighborhood of every point M contained in a continuum under deformation can always be represented as a superposition of tension/compression along eigendirections and rotation of the neighborhood as a rigid whole, and also displacement as a rigid whole.

2.3 Polar Decomposition

47

Exercises for 2.3 2.3.1. Using the formula (2.157a), show that the following relations between V and U hold: Vm D O Um OT ; Um D OT Vm O for all integer m (positive and negative). 2.3.2. Using the results of Exercises 2.1.1 and 2.2.1, show that for the problem on tension of a beam, eigenvalues ˛ are ˛ D k˛ ;

˛ D 1; 2; 3:

The stretch tensors U and V are coincident and have the form UDVD

3 X

k˛ e˛ ˝ e˛ ;

˛D1 ı

and eigenvectors p˛ and p˛ coincide with e˛ : ı

p˛ D p˛ D e˛ ;

˛ D 1; 2; 3:

The rotation tensor O for this problem is the unit one: O D E. 2.3.3. Using the results of Exercises 2.1.2, 2.2.2 and Remark 2, show that for the problem on simple shear (see Example 2.2 from Sect. 2.1.1), the tensors U2 and V2 are expressed as follows: U2 D FT F D E C aO3 C a2 e2i D .UN 2 /ij eN i ˝ eN j ; V2 D E C aO3 C a2 e2i D .VN 2 /ij eN i ˝ eN j ; 0 0 1 1 a 0 1 C a2 a 2 i 2 i .UN / j D @a 1 C a2 0A ; .VN / j D @ a 1 0 0 0 0 1

1 0 0A ; 1

eigenvalues ˛ are 2˛ D 1 C b˛ jaj; ˛ D 1; 2I b1 D

a p C 1 C a2 =4; 2

b2 D

3 D 1;

a p 1 C a2 =4; 2

ı

eigenvectors p˛ and p˛ (a > 0) are 1 ı ı p˛ D p .Ne1 C b˛ eN 2 /; p3 D eN 3 ; 2 1 C b˛

48

2 Kinematics of Continua

1 1 p1 D q .b1 eN 1 C eN 2 /; p2 D q .b2 eN 1 C eN 2 /; p3 D eN 3 ; 2 1 C b1 1 C b22 the stretch tensors U and V are U D UN ij eN i ˝ eN j D U0 eN 21 C U1 O3 C U2 eN 22 C eN 23 ; V D VN ij eN i ˝ eN j D U2 eN 21 C U1 O3 C U0 eN 22 C eN 23 ; 0

UN ij

1 U0 U 1 0 B C D @U1 U2 0A ;

VN ij

0 0 1 b1ˇ

Uˇ D

0 1 U2 U1 0 D @U1 U0 0A ; 0 0 1

p

p 1 C b1 a b2ˇ 1 C b2 a C ; 1 C b12 1 C b22

ˇ D 0; 1; 2;

and the rotation tensor O has the form O D ON ij eN i ˝ eN j D cos '.Ne21 C eN 22 / C sin '.Ne1 ˝ eN 2 eN 2 ˝ eN 1 /; 0

ON ij cos ' D

1 cos ' sin ' 0 D @ sin ' cos ' 0A ; 0 0 1

b2 b1 ; 1 C b12 1 C b22

sin ' D

b12 b22 : 2 1 C b1 1 C b22

Show that functions b1 .a/ and b2 .a/ satisfy the following relationships: b1 C b2 D a;

b1 b2 D 1;

b12 C b22 D 2 C a2 :

Show that at a D 0 for the considered problem the following equations really hold: b1 D 1;

b2 D 1;

1 D 2 D 3 D 1;

1 1 ı ı p1 D p1 D p .Ne1 C eN 2 /; p2 D p2 D p .Ne1 eN 2 /: 2 2 2.3.4. Using the results of Exercise 2.2.3, show that for the problem on rotation of a beam with tension (see Example 2.3 from Sect. 2.1.1), eigenvalues ˛ have the form ˛ D k˛ ;

˛ D 1; 2; 3;

and eigenvectors ı

p˛ D eN ˛ ;

p˛ D O0 eN ˛ ;

˛ D 1; 2; 3:

2.4 Rate Characteristics of Continuum Motion

49

Using formulae from Exercise 2.1.3 and data from Example 2.3, show that tensors U, V, O, and also C, A, ƒ and J have the form U D U0 D

3 X

k˛ eN ˛ ˝ eN ˛ ;

O D O0 D O0i j eN i ˝ eN j ;

˛D1

V D O0 U0 OT0 D V0 eN 21 C V1 O3 C V2 eN 22 C k3 eN 23 D V0 ij eN i ˝ eN j ; 0 1 V0 V 1 0 i V 0 j D @ V1 V 2 0 A ; 0 0 k3 V0 D k1 cos2 ' C k2 sin2 ';

V1 D .k1 k2 / cos ' sin ';

V2 D k1 sin2 ' C k2 cos2 '; 3 X 1 2 1 2 .k 1/Ne˛ ˝ eN ˛ ; C D .U0 E/ D 2 2 ˛ ˛D1

ƒD AD

3 X 1 1 / D .E U2 .1 k˛2 /Ne˛ ˝ eN ˛ ; 0 2 2 ˛D1

1 1 .E V2 / D .ı ij gij /Nei ˝ eN j ; 2 2

JD

1 2 1 .V E/ D .gij ıij /Nei ˝ eN j ; 2 2

where metric matrices gij and g ij are determined by formulae from Exercise 2.1.3. We should take into consideration that the tensors C and ƒ do not feel the beam rotation – they are coincident with the corresponding tensors for the problem on pure tension of the beam. Show that if we change the sequence of transformations (i.e. we first rotate and then extend the beam), then the tensors A and J do not feel the rotation.

2.4 Rate Characteristics of Continuum Motion 2.4.1 Velocity The velocity (vector) of the motion of a material point M with Lagrangian coordinates X i is determined as the partial derivative of the radius-vector x.X i ; t/ with respect to time at fixed values of X i : v.X i ; t/ D

@x i ˇˇ .X ; t/ˇ i : X @t

(2.181)

50

2 Kinematics of Continua

Velocity components vN i with respect to the basis eN i have the form v D vN i eN i D

@x i eN i ; @t

vN i D

@x i j .X ; t/: @t

(2.182)

2.4.2 Total Derivative of a Tensor with Respect to Time Any vector field a.x; t/ (and also scalar or tensor field) varying with time, which describes some physical process in a continuum, can be expressed in both Eulerian and Lagrangian descriptions with the help of the motion law (2.3): a.x; t/ D a.x.X j ; t/; t/:

(2.183)

Determine the derivative of the function with respect to time at fixed X i (i.e. for a fixed point M): @a ˇˇ @a ˇˇ @a @x j ˇˇ (2.184) ˇ i D ˇ iC j ˇ : @t X @t x @x @t X i Definition 2.1. The partial derivative of a varying vector field a (2.183) with respect to time t at fixed coordinates X i is called the total derivative of the function (2.183) with respect to time: da @a ˇˇ aP (2.185) D ˇ : dt @t X i According to formulae (2.182), (2.11), and (2.23), the second summand on the right-hand side of (2.184) can be rewritten as follows: @a @a @x j @a @a D vN i eN i eN j P kj ˝ k D vrk ˝ k D vr ˝a: (2.186) D vN j P kj j k @x @t @X @X @X Then the relationship (2.184) yields da @a D C v r ˝ a; dt @t

(2.187)

where we have introduced the notation for the partial derivative with respect to time which will be widely used below: @a i ˇˇ @a D .x ; t/ˇ i : x @t @t

(2.188)

In formula (2.187) the vector a is considered as a function a.x j ; t/. It is evident that if a is considered as a function of .X j ; t/, then from the definition (2.185) we get ˇ da i @ ˇ .x ; t/ D a.X j ; t/ˇ i : X dt @t

(2.189)

2.4 Rate Characteristics of Continuum Motion

51

The total derivative .d a=dt/ is also called the material (substantial, individual) derivative with respect to time, [email protected][email protected]/ in (2.187) is the partial (local) derivative with respect to time, and v r ˝ a is the convective derivative. The material derivative d a=dt characterizes a change of the vector field a in a fixed material point M, the local derivative determines a change of values of a in time at a fixed point x in space, and from (2.186) we get that the convective derivative characterizes a change of the field due to transfer of the material particle M from a point x to a point x C vdt in space. If we choose the vector v as a, then the relationship between the displacement u and the velocity v vectors has the form vD

dx du @u D D C v r ˝ u: dt dt @t

(2.190)

Similarly to formula (2.185), we can define the total derivative of the nth-order tensor n with respect to time: n

ˇ n P D d n .x i ; t/ D @ .X i ; t/ˇˇ : Xi dt @t

(2.191)

Theorem 2.14. The total derivative (2.191) of a varying tensor field n .x i ; t/ can be written as a sum of local and convective derivatives: d n @ n C v r ˝ n : D dt @t

(2.192)

H Proof of the theorem is similar to the proof of the relationship (2.187). Details are left as Exercise 2.4.6 N Let us consider now the question on components of the total derivative tensor. P are connected with Theorem 2.15. Components of the total derivative tensor n ı the corresponding components of a tensor n with respect to stationary bases ri , eN i and e ri and a moving basis ri as follows: ˇ d ı i1 :::in @ ı ˇ D i1 :::in .X i ; t/ˇ i ; X dt @t

(2.193)

@ N i1 :::in i @ N i1 :::in i d N i1 :::in D .x ; t/ C vN k k .x ; t/; dt @t @x

(2.194)

@ e i1 :::in e i d e i1 :::in ek e i1 :::in .X e i ; t/; D .X ; t/ Ce vk r dt @t

(2.195)

n X @ d i1 :::in D i1 :::in .X i ; t/ C .i1 :::k:::in rk vi˛ /.X i ; t/; dt @t ˛D1

(2.196)

52

2 Kinematics of Continua

where n

ı P D d N i1 :::in .x i ; t/Nei1 ˝ : : : ˝ eN in D d i1 :::in .X i ; t/rıi1 ˝ : : : ˝ rıin dt dt d e i1 :::in e i d i1 :::in i D .X ; t/e ri1 ˝ : : : ˝e rin D .X ; t/ri1 ˝ : : : ˝ rin : dt dt (2.197)

In formula (2.196), the component i1 :::k:::in as the ˛th superscript has index k in place of i˛ . H To prove the theorem, we resolve the tensor n for different bases: n

ı

N i1 :::in .x i ; t/Nei1 ˝ : : : ˝ eN in D i1 :::in .X i ; t/rı i1 ˝ : : : ˝ rıin D e i1 :::in .X e i ; t/e D ri1 ˝ : : : ˝e rin D i1 :::in .X i ; t/ri1 ˝ : : : ˝ rin ;

(2.198)

and choose arguments of components of the tensor n as in formula (2.198). ı Then, substituting the resolution (2.198) for the basis ri into the definition ı (2.191), we get the expression (2.193), because d ri =dt D 0. On substituting the resolution (2.198) for the basis eN i into the relationship (2.192), we obtain n

N i1 :::in @ N i1 :::in m P D @ eN i1 ˝ : : : ˝ eN in C vN k eN k m eN ˝ eN i1 ˝ : : : ˝ eN in : (2.198a) @t @x

It is evident that formula (2.194) follows from (2.198a). In a similar way, substituting the resolution (2.198) for the basis e ri into the relae, and also the tion (2.192) and using the property (2.61) of nabla-operators r and r equation @e ri [email protected] D 0 (for the stationary basise ri see Sect. 2.1.7), we get n

e i1 :::in P D @ em e i1 :::ine e ri1 ˝ : : : ˝e rk r rm ˝e rin Ce vke ri1 ˝ : : : ˝e rin I (2.199) @t

and formula (2.195) follows from (2.199) at once. Finally, substituting the resolution (2.198) for the moving basis ri into the definition of the total derivative (2.192), we obtain n i1 :::in ˇ X @ri ˇˇ ˇ P D @ i1 :::i˛ :::in ri1 ˝ : : : ˛ ˇ i : : : ˝ rin : ˇ i ri1 ˝ : : : ˝ rin C X @t @t X ˛D1 (2.200) Due to the definition (2.10) of local bases vectors and the definition (2.181) of the velocity, we have n

@ri˛ j ˇˇ @2 x ˇˇ @v ˇˇ .X ; t/ˇ j D D ˇ ˇ D ri˛ vk rk : X @t @[email protected] i˛ X j @X i˛ X j

(2.201)

2.4 Rate Characteristics of Continuum Motion

53

On substituting (2.201) into (2.200) and then collecting components at the same elements of the polyadic basis, we derive the formula (2.196). N It should be noted that arguments of the resolutions (2.198) and of the derivatives of tensor components (2.193)–(2.196) have been chosen in the specific way.

2.4.3 Differential of a Tensor Definition 2.2. For a tensor field n .x i ; t/, the following object d n D

d n dt dt

(2.202)

is called the differential of a tensor field (or the differential of a tensor) n .x i ; t/. According to formula (2.192) for the total derivative of a tensor with respect to time, we get that the differential of a tensor can be written in the form d n .x i ; t/ D

@ n C v r ˝ n dt: @t

(2.203)

According to (2.190), the relation (2.203) takes the form d n D

@ n dt C d x r ˝ n @t

(2.204)

When a tensor field is stationary (i.e. @ n [email protected] D 0), the differential of the tensor field has the form b d n D d x r ˝ n : (2.205) For stationary tensor fields b d n D d n , but in general these differentials are not coincident. According to Theorem 2.15, components of the tensor d n with respect to the ı fixed basis ri are written as follows: n

ı

d D d

j1 :::jn ı

ı

rj1 ˝ : : : ˝ rjn ;

ı

d

ı

j1 :::jn

d j1 :::jn dt: D dt

(2.206)

From (2.202) and (2.187) we get the following expression for the differential of a vector: da @a i d a.X ; t/ D dt D C v r ˝ a dt; (2.207) dt @t and from (2.205) we have db a D .v r ˝ a/dt D .r ˝ a/T d x:

(2.208)

54

2 Kinematics of Continua ı

In particular, if a D x, then, by formulae (2.208) and (2.35a), we obtain

or

b ı ı d x D .r ˝ x/T d x D F1 d x;

(2.209)

b ı d x D F d x:

(2.210)

On comparing formulae (2.210) with (2.34), we find that the elementary radiusı vector d x, introduced in Sect. 2.1 and connecting two infinitesimally close material b ı points M and M0 , coincides with the vector d x in the notation (2.205).

2.4.4 Properties of Derivatives with Respect to Time Let us establish now important properties of partial and total derivatives of vector fields with respect to time. Theorem 2.16. The partial derivative of the vector product of basis vectors with respect to time has the form @rˇ @r˛ @ .r˛ rˇ / D rˇ C r˛ : @t @t @t

(2.211)

H Determine the derivative of the vector product of two local basis vectors with respect to time: @ @ @ j j .r˛ rˇ / D .Qi˛ eN i Q ˇ eN j / D .Qi˛ Q ˇ /Nei eN j @t @t @t j @Q ˇ @Qi˛ eN i Qjˇ eN j C Qi˛ eN i eN j : D @t @t With use of relation (2.10) we really get (2.211). N Theorem 2.17. For arbitrary continuously differentiable vector fields a.x; t/ D aN i .x k ; t/Nei and b.x; t/ D bN i .x k ; t/Nei , we have the formulae @a @b @ .a b/ D bCa ; @t @t @t

(2.212)

@ @a @b .a ˝ b/ D ˝bCa˝ ; @t @t @t

(2.213)

@a @b @ .a b/ D bCa : @t @t @t H A proof is similar to the proof of Theorem 2.16. N

(2.214)

2.4 Rate Characteristics of Continuum Motion

55

Theorem 2.18. The total derivatives of the vector and scalar products of two arbitrary vector fields a.x; t/ and b.x; t/ with respect to time have the forms d da db .a b/ D bCa ; dt dt dt

(2.215)

da db d .a b/ D bCa : dt dt dt

(2.216)

H To prove formula (2.215), one should use the property of the total derivative (2.187): @ d .a b/ D .a b/ C v r ˝ .a b/: dt @t Modify the first summand by formula (2.212) and the second summand – by the formula r ˝ .a b/ D .r ˝ a/ b .r ˝ b/ a [12], then we get d @a @b .a b/ D b a C v .r ˝ a/ b v .r ˝ b/ a: dt @t @t Collecting the first summand with the third one and the second summand with the fourth one, and using the property (2.187) of the total derivative of a vector, we obtain d da db da db .a b/ D b aD bCa : dt dt dt dt dt Formula (2.216) can be proved in a similar way. N p Theorem 2.19. The total derivative of g with respect to time is connected to the divergence of the velocity v by d p p p g D g ri vi D g r v: dt

(2.217)

H Let us differentiate the second relation of (2.15) with taking formula (2.211) into account: d d p gD r1 .r2 r3 / dt dt 2 @2 x @2 x @ x D .r2 r3 / C r1 r3 C r1 r2 : @[email protected] 1 @[email protected] 2 @[email protected] 3 (2.218) Since

@v @2 x D D ri v D ri vj rj ; @[email protected] i @X i

56

2 Kinematics of Continua

we get d p p g D r1 v g r1 C r1 .r2 v r3 / C r1 .r2 r3 v/: dt

(2.219)

Here we have used the relations from Exercise 2.1.14. According to the definition of the vector product (1.2), we obtain r1 r2 v r3 D r1

p

g ijk r2 vi ı3j rk D

p

g i 31 r2 vi D

p g r2 v2 :

(2.220)

On substituting (2.220) into (2.219), we really get formula (2.217). N

2.4.5 The Velocity Gradient, the Deformation Rate Tensor and the Vorticity Tensor ı

Consider elementary radius-vectors d x and d x connecting two infinitesimally close ı

points M and M0 in configurations K and K, respectively. Determine the velocity of the point M0 relative to the configuration connected to the point M. To do this, determine the velocity differential b d v: T 2 2 ı @ x @v T ı @ x @ ıi ıi ı ı i b d x D r ˝ v d x: d v D d xD i dX D i ˝ r d xD r ˝ @t @X @t @X @t @X i (2.221) Here we have used the second equation of (2.33), the definition of the gradient (2.24) and formula (2.181). In a similar way, using the first equation of (2.33): dX i D d v: ri d x, we get one more expression for the vector b b d v D .r ˝ v/T d x:

(2.222)

The second-order tensor .r ˝ v/T is called the velocity gradient, which connects the relative velocity b d v of an elementary radius-vector d x to the vector d x itself: b d v D L d x;

L D .r ˝ v/T :

(2.223)

Just as any second-order tensor (see [12]), the tensor L can be represented by a sum of the symmetric tensor D and the skew-symmetric tensor W: L D D C W:

(2.224)

The symmetric deformation rate tensor D is determined as follows: DD

1 .r ˝ v C r ˝ vT /: 2

This tensor has six independent components.

(2.225)

2.4 Rate Characteristics of Continuum Motion

57

The skew-symmetric vorticity tensor W is determined as follows: WD

1 .r ˝ vT r ˝ v/: 2

(2.226)

Since the tensor W is skew-symmetric and has three independent components, we can put the tensor W in correspondence with the vorticity vector ! connected to the tensor (see [12]) as follows: !D

1 W "; 2

W D ! E:

(2.227)

where " is the Levi-Civita tensor, which has the third order (see [12]). This tensor is determined as follows: 1 " D p ijk ri ˝ rj ˝ rk : g

(2.228)

On substituting (2.224)–(2.227) into (2.222), we prove the following theorem. Theorem 2.20 (Cauchy–Helmholtz). The velocity v.M0 / of an arbitrary point M0 in a neighborhood of the material point M consists of the translational motion velocity v.M/ of the point M, the velocity ! d x of rotation as a rigid whole and the deformation rate D d x, i.e.

or

b dv D ! dx C D dx

(2.229)

v.M0 / D v.M/ C ! d x C D d x C o .jd xj/:

(2.229a)

Example 2.4. Determine the tensor L for the problem on tension of a beam (see Example 2.1), substituting (2.182) into (2.223): LT D eN i

3 3 X @ X P ˛ @ i N N ˝ v D e ˝ X D e k kP˛ eN ˛ ˝ eN ˛ D L: ˛ ˛ @X i @X i ˛D1 ˛D1

Since the velocity gradient L in this case proves to be a symmetric tensor, from (2.225) and (2.226) it follows that D D L;

W D 0:

Thus, in this case ! D 0.

t u

Example 2.5. Determine the tensor L for the problem on simple shear (see Example 2.2), substituting formula (2.182) into (2.223): LT D eN i ˝

@v @Nvj i @Nv1 N N e eN 2 ˝ eN 1 D aN D ˝ e D P e2 ˝ eN 1 : j @X i @X i @X 2

58

2 Kinematics of Continua

According to formulae (2.225) and (2.226), we get D D .a=2/.N P e1 ˝ eN 2 C eN 2 ˝ eN 1 /; j

j

i i W D .a=2/.N P e1 ˝ eN 2 eN 2 ˝ eN 1 / D .a=2/.ı P ei ˝ eN j : 1 ı2 ı2 ı1 /N

Using formula (2.227), we determine the vorticity vector !D

aP aP aP 1 W " D .ı1i ı2j ı2i ı1j /j i k eN k D .21k 12k /Nek D eN 3 ; 2 4 4 2 t u

which is orthogonal to the shear plane.

2.4.6 Eigenvalues of the Deformation Rate Tensor Just as any symmetric tensor, the deformation rate tensor D has three orthonormal real-valued eigenvectors and three real positive eigenvalues (see [12]). Denote the eigenvectors by q˛ (these vectors, in general, are not coincident with p˛ ) and the eigenvalues – by D˛ . Then the tensor D can be resolved for its dyadic eigenbasis as follows: 3 X D˛ q˛ ˝ q˛ ; q˛ qˇ D ı˛ˇ : (2.230) DD ˛D1

Take in the actual configuration K an elementary radius-vector d x˛ , connecting points M and M0 , so that the vector is oriented along the eigenvector q˛ of the tensor D; then, similarly to (2.172), we can write d x˛ D q˛ jd x˛ j;

jd x˛ j D .d x˛ d x˛ /1=2 :

(2.231)

Apply the Cauchy–Helmholtz theorem (2.229) to the elementary radius-vector: b d v ˛ D ! d x˛ C D d x˛ :

(2.232)

Multiplying the left and right sides of the equation by d x˛ and taking account of the property of the mixed derivative d x˛ .! d x˛ / D 0, we get d x˛ b d v ˛ D d x ˛ D d x˛ :

(2.233)

Substituting in place of D its expression (2.230) and in place of d x˛ their expressions (2.231), we obtain d x˛ b d v˛ D jd x˛ j2

3 X

Dˇ q˛ qˇ ˝ qˇ q˛ D D˛ jq˛ j2 :

ˇ D1

Here we have used the property (2.230) of orthonormal vectors q˛ .

(2.234)

2.4 Rate Characteristics of Continuum Motion

59

Modify the scalar product on the left-hand side as follows: @ 1 @ @ d x˛ b d v˛ D d x˛ d x ˛ D .d x˛ d x˛ / D jd x˛ j jd x˛ j: @t 2 @t @t

(2.235)

On comparing (2.234) with (2.235), we obtain the following theorem. Theorem 2.21. Eigenvalues D˛ of the deformation rate tensor D are the rates of relative elongations of elementary material fibres oriented along the eigenvectors q˛ : D˛ D

1 @ jd x˛ j: jd x˛ j @t

(2.236)

2.4.7 Resolution of the Vorticity Tensor for the Eigenbasis of the Deformation Rate Tensor Modify the right-hand side of (2.232) as follows: b d v˛ D ! d x˛ C D d x˛ D .! q˛ C D˛ q˛ /jd x˛ j;

(2.237)

and the left-hand side of (2.232) with taking (2.236) into account: @ @ @jd x˛ j @q˛ @q˛ b : q˛ C jd x˛ j Djd x˛ j D˛ q˛ C d v˛ D d x˛ D .jd x˛ jq˛ /D @t @t @t @t @t (2.238) On comparing (2.237) with (2.238), we get the following theorem. Theorem 2.22. The vorticity tensor W (or the vorticity vector !) connects the rate of changing the eigenvectors q˛ to the vectors q˛ themselves: qP ˛ D

@q˛ D ! q˛ D W q˛ : @t

(2.239)

Using formula (2.239), we can resolve the tensor W for the eigenbasis q˛ of the deformation rate tensor as follows: WD

3 X ˛D1

qP ˛ ˝ q˛ D qP i ˝ qi :

(2.240)

60

2 Kinematics of Continua

2.4.8 Geometric Picture of Infinitesimal Transformation of a Small Neighborhood of a Point If in configuration K at time t we consider an elementary radius-vector d x connecting two infinitesimally close material points M and M0 , then for infinitesimal time dt the radius-vector is transformed into radius-vector d x0 in configuration K.t C dt/ (Fig. 2.13): d x D x0 .t/ x.t/;

d x0 D x0 .t C dt/ x.t C dt/;

(2.241)

where x.t/ and x0 .t/ are radius-vectors of the points M and M0 in configuration K.t/, respectively; and x.t C dt/ and x0 .t C dt/ – in configuration K.t C dt/. Displacements of points M and M0 for infinitesimal time are defined by the velocity vectors v.M/ and v.M0 /, respectively: x.t C dt/ x.t/ D v.M/ dt;

x0 .t C dt/ x0 .t/ D v.M0 / dt:

(2.242)

Formulae (2.241), (2.242), and simple geometric relations (see Fig. 2.13) give v.M0 /dt v.M/dt D d x0 d x:

(2.243)

On substituting (2.243) into (2.229a), we obtain the relation between elementary radius-vectors d x0 and d x: d x0 D d x C dt! d x C dtD d x C dt o.jd xj/:

(2.244)

The relation (2.244) can be considered as the transformation of coordinates dx i ! dx 0i in a small neighborhood of the point contained in a continuum. Since dt! and dtD are independent of d x and d x0 , so the transformation is

Fig. 2.13 Infinitesimal transformation of an elementary radius-vector

2.4 Rate Characteristics of Continuum Motion

61

linear, i.e. affine. The relation (2.244) can be represented as a superposition of two transformations up to an accuracy of o.jd xj/: d x00 D AD d x; d x0 D Q! d x00 ;

AD D E C dtD;

(2.245)

Q! D E C dt! E:

(2.246)

The tensor AD is symmetric and has three eigendirections, which are coincident with the eigendirections q˛ of the deformation rate tensor D. So just as the tensor U, the tensor AD transforms a small neighborhood of a point M by extending or compressing the neighborhood along the principal directions q˛ . The material segments jd x00˛ j oriented along the eigendirections q˛ retain their orientation under the transformations (2.245), but their lengths vary as follows: d x˛ D jd x˛ jq˛ ;

d x00˛ D .1 C D˛ dtjd x˛ j/q˛ D .1 C D˛ dt/d x˛ :

The tensor Q! (2.246) is orthogonal up to an accuracy of values .dt/2 , because Q! QT! D .E C dtW/ .E C dtWT / D E .dt/2 W2 :

(2.247)

Here we have taken into account that the vorticity tensor W is skew-symmetric. Thus, the transformation (2.246) determined by the tensor Q! is rotation of the M-point neighborhood as a rigid whole for infinitesimal time dt. The vorticity vector ! forming the tensor Q! can be considered as instantaneous angular rate of rotation of the small neighborhood as a rigid whole, or as instantaneous angular rate of rotation of the eigentrihedron q˛ of the deformation rate tensor relative to the fixed basis eN i . This fact will be considered in detail in Sect. 2.5.7. On uniting the properties of the transformations (2.245) and (2.246), we can make the following conclusion. Theorem 2.23. The infinitesimal transformation of a small neighborhood of the point contained in a continuum is a superposition of tension-compression of the neighborhood along the eigendirections q˛ and rotation of the axes q˛ as a rigid whole about the axis with the direction vector !. ı

Thus, we have the certain analogy between the eigendirections p˛ of the tensor U ı and the directions q˛ of the tensor D: elementary material fibres oriented along p˛ and along q˛ remain mutually orthogonal and undergo only tension-compression. ı

ı

The axes p˛ remain mutually orthogonal under any finite transformations from K to K, but q˛ – only under infinitesimal transformations from K.t/ to K.t C dt/.

62

2 Kinematics of Continua

2.4.9 Kinematic Meaning of the Vorticity Vector Just as any orthogonal tensor, the orthogonal tensor Q! of infinitesimal rotation from K.t/ to K.t C dt/ can be represented in the form (see [12]) Q! D E cos.d'/ C .1 cos.d'//e ˝ e e E sin.d'/;

(2.248)

where d' is the infinitesimal angle of rotation of the trihedron q˛ about the axis with the direction vector e. Since values of d' are infinitesimal, we have Q! D E e Ed':

(2.249)

On comparing (2.249) with (2.246), we get !D

d' e; dt

j!j D

d' ; dt

(2.250)

i.e. the vorticity vector ! is really oriented along the instantaneous rotation axis e, and the length j!j is equal to the instantaneous angular rate of rotation of the trihedron q˛ of the deformation rate tensor. Let us consider now the question: relative to what system the vorticity vector ! defines the rotation rate. To answer the question, we introduce another orthogonal rotation tensor OW D qi ˝ eN i ;

(2.251)

which transforms the Cartesian trihedron eN i as a rigid whole into the orthonormal trihedron qi : qi D OW eN i :

(2.252)

The tensor OW is a function of time t, because qi D qi .t/. According to (2.240) and (2.252), the vorticity tensor W takes the form P W OT : W D qP i ˝ qi D O W

(2.253)

With the help of (2.253) we can represented the orthogonal tensor Q! as follows: P W OT : Q! D E C dtW D E C dt O W

(2.254)

Thus, at each time t two orthogonal tensors OW and Q! connect local neighı

borhoods of a point M in the reference K and actual configurations K.t C dt/. ı

If in K we consider an elementary radius-vector d x00 , then in K we find its

2.4 Rate Characteristics of Continuum Motion

63

ı

corresponding radius-vector d x00 obtained with the help of the rotation tensor OW , and in K.t C dt/ – radius-vector d x0 : ı

d x00 D OW d x00 ;

d x0 D Q! d x00 :

A fixed observer connected to the Cartesian trihedron eN i sees both the transformations: finite rotation for time t, which is described by the tensor OW , and instantaneous rotation of a local neighborhood for time dt, which is described by the infinitesimal rotation tensor Q! Thus, the vorticity vector ! is the vector of instantaneous angular rate of rotation of the trihedron q˛ relative to the trihedron eN i . Comparing (2.246) with (2.254) (or (2.253) with (2.227)), we get P W OT D ! E: O W

(2.255)

2.4.10 Tensor of Angular Rate of Rotation (Spin) P W OT , where OW is the orthogonal In Sect. 2.3 we have introduced the tensor O W P QT can be set up for any orthogonal tensor tensor of rotation. Such tensor D Q Q depending on time t. P QT is skew-symmetric, because The tensor Q

i.e.

P QT C Q Q P T D .Q QT / D .E/ D 0; Q

(2.256)

P QT /T D Q Q P T D Q P QT D : T D .Q

(2.257)

This tensor characterizes the angular rate of rotation of the orthonormal trihedron hi formed with the help of Q: hi D Q eN i ; (2.258) relative to the Cartesian trihedron eN i . P QT we can form the tensor (2.254): Indeed, with the help of the tensor Q P QT ; Q! D E C dt Q

(2.259)

which, according to (2.247), is the orthogonal tensor of infinitesimal rotation; and this tensor can be represented in the form (2.248) or (2.249): Q! D E d'e E;

(2.260)

where d' is the infinitesimal angle of rotation of the trihedron hi about the axis with the direction vector e. Comparing (2.259) with (2.260), we get the expression P QT D d' e E; Q dt

(2.261)

64

2 Kinematics of Continua

P T characterizes the instantaneous which makes clear the sentence that the tensor QQ angular rate d'=dt of rotation of the trihedron hi about the axis e. P QT is called the tensor of angular rate of rotation or the spin. The tensor Q Expressing the tensor Q from (2.258) in terms of the bases hi and eN i (h˛ D h˛ , ˛ D 1; 2; 3, as the vectors are orthonormal): Q D hi ˝ eN i ;

(2.262)

we get another representation of the spin: P QT D hP i ˝ hi : Q

(2.263)

Thus, we have proved the following theorem. Theorem 2.24. The spin connects the rates hP i and the vectors hi defined by formula (2.258) as follows: P QT / hi : hP i D .Q (2.264) T P Q is a skew-symmetric tensor, we can introduce the correSince the spin Q sponding vorticity vector !h : P QT D !h E Q

(2.265)

(from formulae (2.261) and (2.265) it follows that !h D .d'=dt/e); then formula (2.264) takes the form hP i D !h hi : (2.266) j

Resolving the vector !h for the orthonormal basis hi : !h D ! h hj ; we get one more representation of formula (2.264): j j hP i D ! h hj hi D j i k ! h hk :

(2.267)

This formula can also be rewritten in the form hP ˛ D !hˇ h ;

˛ ¤ ˇ ¤ ¤ ˛:

(2.268)

Taking different orthogonal tensors (or orthonormal bases) as Q (or hi ), we obtain different spins: ı

(1) If we choose eigenvectors of the stretch tensor U as hi , i.e. hi D pi , then, according to (2.263), the corresponding spin U takes the form ı

ı

P U OT D pi ˝ pi ; U D O U and formula (2.264) yields

P ı ı pi D U pi :

ı

OU D pi ˝ eN i ;

(2.269)

(2.270)

2.4 Rate Characteristics of Continuum Motion

65

(2) If hi D pi , then the corresponding spin V and the rotation tensor OV have the forms P V OT D pP i ˝ pi ; OV D pi ˝ eN i ; V D O (2.271) V pP i D V pi :

(2.272)

(3) If hi D qi , then the corresponding spin W coincides with the vorticity tensor W (see (2.253)): P W OT D qP i ˝ qi D W; W D O W

(2.273)

qP i D W qi :

(2.274)

(4) If we take the rotation tensor O accompanying deformation as Q, then, as shown ı in (2.159a), the tensor O connects two moving bases pi and pi : ı

pi D O pi :

(2.275)

The tensor O can be expressed in terms of OV and OU as follows: ı

ı

O D pi ˝ pi D pi ˝ eN i eN j ˝ pj D OV OTU :

(2.276)

The corresponding spin has the form P O T D .O P V OT C OV O P T / OU O T DO U U V T T T P P D OV OV C OV OU OU OV D V OV U OTV :

(2.277)

Unlike the cases (1)–(3), the spin tensor characterizes the angular rate of roı tation of the trihedron pi relative to the moving trihedron pi , but not relative to the trihedron eN i being fixed. Therefore, for the cases (1)–(3) the spins characterize the total angular rate, and for the case (4) – the relative rate.

2.4.11 Relationships Between Rates of Deformation Tensors and Velocity Gradients In continuum mechanics, one often needs the relations between rates of the deformation tensors (and also measures) and the velocity gradients L D .r ˝ v/T and ı T ı L D r ˝v : Let us derive these relations.

(2.278)

66

2 Kinematics of Continua

Theorem 2.25. The rates of varying the gradient FP and the inverse gradient .F1 / ı

are connected to L and L by the relations FP D L F;

.F1 / D F1 L;

ı

FP D L; ı

.F1 / D F1 F1T L:

(2.279)

H Differentiating the relationships (2.35a) with respect to time t and taking the definition of the velocity (2.181) into account, we get ı

FP T D r ˝ v D FT r ˝ v;

FP D .r ˝ v/T F:

(2.279a)

ı

According to the definitions of tensors L (2.223) and L (2.278), from (2.279a) we obtain formulae (2.279). Differentiating the identity .F F1 / D EP D 0; we find that FP F1 D F .F1 / I whence we get .F1 / D F1 FP F1 :

(2.280)

On substituting the first two formulae of (2.279) into (2.280), we obtain .F1 / D F1 .r ˝ v/T ;

.F1T / D .r ˝ v/ F1T ;

(2.281)

i.e. the third and the fourth relationships of (2.279) hold as well. N According to formulae (2.225), (2.279a), and (2.281), we find that the rate of the deformation gradient is connected to the deformation rate tensor D by the relations DD

1 P 1 .F F C F1T FP T /; 2

1 D D .F FP 1 C FP 1T FT /: 2

(2.282)

Here and below we will use the notation FP 1 .F1 / . P A, P ƒ, P JP and deformation Theorem 2.26. The rates of the deformation tensors C, ı P gP , .G1 / and .g1 / are connected to the velocity gradients L and L measures G, by the relationships 8 0, It is invariant relative to any transformation of coordinates (2.1) and relative to

any motion (2.3). Due to the third property, the mass in any actual configuration remains constant: M.B; t/ D const:

(3.1)

Remark. The mass conservation law is valid only for a continuum containing the same material points for a considered time interval Œ0; t. If a continuum B loses or acquires material points as time goes on (in this case phase transformations are said to occur in the continuum (see Sect. 5.1)), then the mass conservation law in the form (3.1) is no longer valid. The mass conservation law does not hold as well, if we exclude Axiom 3 and consider the motions with speeds close to the light speed; however, relativistic phenomena are not considered in classical continuum mechanics. t u The law (3.1) can be written in another form dM=dt D 0:

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 3, c Springer Science+Business Media B.V. 2011

(3.2)

89

90

3 Balance Laws

Since the mass is additive, M may be expressed as follows: Z M D d m;

(3.3)

V

where d m is the mass of an elementary volume dV involving a material point M belonging to the considered continuum V . Definition 3.1. The ratio D d m=dV

(3.4)

is called the density of substance at the point M. Since the mass M and volume dV are positive, the mass d m and density are always positive as well: > 0; d m > 0: (3.4a) On substituting (3.3) and (3.4) into (3.2), we can rewrite the mass conservation law (3.2) in the form Z d dV D 0: (3.5) dt V

Having applied this relationship to an elementary volume, we get ı

ı

dV D d V D const;

(3.6) ı

ı

ı

where and are the densities in configurations K and K, respectively, and d V is ı

the elementary volume in the configuration K. The relationship (3.5) is called the mass conservation law in the integral form, and (3.6) – in the differential form.

3.1.2 The Continuity Equation in Lagrangian Variables ı

ı

Consider in K an elementary volume d V constructed on elementary radius-vectors ı ı oriented along the local basis vectors d x˛ D r˛ dX ˛ (see Sect. 2.2.6). This volume corresponds to a volume dV constructed on vectors r˛ dX ˛ in the actual configuraı

tion K. The volumes d V and dV are determined by formulae (2.15): q ˇ @xı k ˇ ı ˇ ˇ 1 2 3 d V D r1 .r2 r3 /dX dX dX D g dX dX dX D ˇ i ˇdX 1 dX 2 dX 3 ; @X ˇ @x k ˇ p ˇ ˇ dV D r1 .r2 r3 /dX 1 dX 2 dX 3 D g dX 1 dX 2 dX 3 D ˇ i ˇdX 1 dX 2 dX 3 : @X (3.7) ı

ı

ı

ı

1

2

3

On substituting (3.7) into (3.6), we get the following theorem.

3.1 The Mass Conservation Law

91 ı

Theorem 3.1. A change of the density in going from K to K can be determined by one of the following equations: ı

D

s

g ı

D

g

ˇ @x k ˇ ˇ ˇ D ˇ ı ˇ D det F: ı j n i [email protected] [email protected] j @x [email protected] k [email protected] i j

(3.8)

Unlike (3.6), these equations are called the continuity equations in Lagrangian variables. ı The ratio of the elementary volumes in configurations K and K, which follows from (3.7), is also widely used: ı

q

dV =d V D

ı

g=g:

(3.9)

3.1.3 Differentiation of Integral over a Moving Volume Consider a vector field a.x i ; t/, which is a continuously differentiable function of x i and t in a domain V .t/, 8t > 0, where the domain V .t/ contains the same material points (such a domain is usually called a moving volume). i Let us integrate the field R a.x ; t/ over the domain V .t/ and determine the derivad tive of the integral: dt V .t / a dV: ı

To do this, we should change variables in the integral: x i ! x i , where x i 2 V , ı

ı

ı

i k i x q 2 V , and, according to (3.8), the Jacobian of the transformation is [email protected] [email protected] j D ı g=g. Then we get Z Z s ı d g d a dV : a dV D (3.10) ı dt dt g V .t /

ı

V

Since this change in variables means the transformation from the configuration K ı

ı

to K, where the domain V is time–independent, the derivative with respect to t may be introduced under the integral sign: d dt

Z

Z a dV D V .t /

ı

V

1 Z p ı d B g C ı 1 d p p da gC g dV : a q @ q aA d V D dt ı ı dt dt g g ı 0

V

(3.10a)

92

3 Balance Laws

On substituting (2.187) and (2.217) into (3.10a), we get d dt

Z q

Z a dV D

ı

V .t /

ı @a g=g ar v C C v r ˝ a dV @t ı

V

Z s

D

g ı

ı

g

@a C r .v ˝ a/ @t

ı

Z

dV D

@a C r .v ˝ a/ @t

dV:

V

V

(3.11) ı

The last equality we have obtained by the reverse substitution x i ! x i . Thus, we have proved following theorem. Theorem 3.2 (The Rule of Differentiation of the Integral over a Moving Volume). For any vector field a.x i ; t/ specified in V .t/ 8t > 0 and being a continuously differentiable function of x i and t, the following relationship holds: d dt

Z

Z a.x i ; t/ dV D

@a Cr v˝a @t

dV:

(3.12)

V

V .t /

Taking in formula (3.12) '.x i ; t/Ne˛ as a vector field a.x i ; t/, where eN ˛ is one of the Cartesian basis vectors, and '.x i ; t/ is a scalar field, and factoring eN ˛ outside the integral sign on the left and right sides, we get the formula for differentiation of the integral of a scalar field: d dt

Z

Z '.x i ; t/ dV D

@' C r .'v/ @t

dV:

(3.13)

V

V .t /

3.1.4 The Continuity Equation in Eulerian Variables Putting ' D in formula (3.13) and using the mass conservation law (3.5), we obtain Z @ C r v dV D 0 (3.14) @t V

– the mass conservation law in Eulerian variables. Unlike the law (3.5) which is formulated for a volume V of a continuum containing at different times t > 0 the same material points, the law (3.14) holds for an arbitrary geometric domain V of space E3a , which at different times t may involve different material points. Since formula (3.14) holds true for an arbitrary domain V E3a , the integrand must vanish. So we have proved the following theorem.

3.1 The Mass Conservation Law

93

Theorem 3.3. If the function , satisfying the mass conservation law (3.5), and the velocity v are continuously differentiable in V .t/ for all considered times t > 0, then at every point x of the domain V E3a of a continuum the following relation holds 8t > 0: @ C r v D 0: (3.15) @t Equation (3.15) is called the continuity equation in Eulerian variables. According to the definition (2.187) of the total derivative with respect to time, Eq. (3.15) can be rewritten in the form @ d C v r C r v D C r v D 0; @t dt

(3.16)

1 d D r v: dt

(3.17)

or

Due to (3.17), the continuity equation takes the form

d dt

1 D r v:

(3.18)

3.1.5 Determination of the Total Derivatives with respect to Time With the help of the continuity equation (3.15) we will prove several relationships which are frequently used in continuum mechanics. Theorem 3.4. If there is a tensor field n .x; t/ of the nth order, which is continuously differentiable in a domain V .t/ 8t > 0, then the following relations hold: @ n d n D C r .v ˝ n /; dt @t @ d .n / D . n / C r .v ˝ n /: dt @t

(3.19) (3.20)

H According to the definition of the total derivative with respect to time (2.192), we have n d n @ n D C v r ˝ dt @t @ n @ n 2 D 2 .v r ˝ n / C v r ˝ n : @t @t (3.21)

94

3 Balance Laws

Collecting the second and the third summands and taking the following consequence of the continuity equation into account: [email protected][email protected]/ C v r D r v, we finally get d n @ n @ n C r .v ˝ n /: (3.22) D C .r v/ n C v r ˝ n D dt @t @t To prove formula (3.20), it is sufficient to make the substitution n ! n in (3.19). N

3.1.6 The Gauss–Ostrogradskii Formulae In continuum mechanics, the Gauss–Ostrogradskii formulae are frequently used for tensor fields k .x/ of arbitrary kth order in reference and actual configurations: Z

Z

ı

ı

ı

n ˝ k d † D ı

†

Z

ı

r ˝ k d V ; ı

Z

V ı

ı

ı

†

Z

n k d † D

ı

Z

ı

r k d V ;

Z

r ˝ k dV: (3.23) V

n k d† D

ı

†

Z n ˝ k d† D

†

V

r k dV: V

According to the second formula of (3.23), Eq. (3.12) takes the form Z Z Z @a d a dV D dV C n v ˝ a d†: dt @t V .t /

(3.24)

V

(3.25)

†

Exercises for 3.1 3.1.1. Show that the continuity equation (3.8) can be written in the differential form: p d d g D p : dt g dt 3.1.2. Using formulae (3.10), (3.19), and (3.20), show that the following integral relations (called the rule of differentiation of the integral of a tensor field over a moving volume) hold: Z Z Z n d n @ d n n dV D dV D C r .v ˝ / dV dt dt @t V .t /

V .t /

Z

D V

@ n dV C @t

Z

V

n .v ˝ n / d†; †

3.2 The Momentum Balance Law and the Stress Tensor

d dt

Z

Z n dV D V .t /

d n dV D dt

D

95

@ n . / C r .v ˝ n / @t

dV

V

V .t /

Z

Z

@ n . / dV C @t

V

Z n .v ˝ n / d†:

†

3.1.3. Using formulae (3.19), show that for an arbitrary vector field a.X k ; t/ the following relation holds:

d dt

d a D a C a.r v/: dt

3.1.4. Similarly to formula (3.10), prove that for a tensor field k .x/ of arbitrary kth order the following formula of change of variables is valid: Z

Z k

k

dV D

ı

ı

d V :

ı

V

V

3.2 The Momentum Balance Law and the Stress Tensor 3.2.1 The Momentum Balance Law Definition 3.2. The vector Z

Z v dm D

ID V

v dV;

(3.26)

V

is called the momentum of a continuum. Remark 1. As in the mass conservation law (3.5), a domain V .t/ contains the same material points for all considered time interval. t u Remark 2. It should be noted that since the velocity v D d x=dt of a material point M, due to (2.181), has been introduced in a certain coordinate system O eN i , where x.M/ is the radius-vector of the point M, the momentum vector I is connected to this coordinate system. If we have chosen another coordinate system RO 0 e0i , then, applying formula (3.26), we obtain another momentum vector I0 D V v0 dV, where v0 D d x0 =dt, and x0 .M/ is the radius-vector of the same material point M t u but in the system O 0 e0i .

96

3 Balance Laws

This remark is of great importance for understanding the following axiom. Axiom 5 (The momentum balance law). For any two continua B and B1 at any time t, there exists a vector function F D F .B; B1 ; t/ (i.e. the transformation F W U U RC0 ! E3a ), which is possible to be zero-valued (i.e. F D 0). This vector function is called the force of interaction of the bodies B and B1 , and has the following properties: It is additive:

F .B 0 C B 00 ; B1 ; t/ D F .B 0 ; B1 ; t/ C F .B 00 ; B1 ; t/; F .B; B10 C B100 ; t/ D F .B; B10 ; t/ C F .B; B100 ; t/; where B D B 0 [ B 00 ; B1 D B10 [ B100, The rate of changing the momentum I of a continuum B at any time t is equal to the vector F .B; t/ D F .B; B e ; t/ which is the summarized vector of external forces, acting onto the body B (where B e D U n B is the outside of the body B): d I=dt D F :

(3.27)

The relationship (3.27) is called the momentum balance law. Remark 3. Since the vector I has been introduced in a certain coordinate system O eN i , the momentum balance law (3.27) is written for this coordinate system O eN i . In another coordinate system O 0 e0i the vector d I0 =dt may be different from d I=dt, and the law (3.27) also changes its form. The momentum balance law for such systems will be given in Sect. 4.10.11. The coordinate systems, in which the momentum balance law has the form (3.27), are called inertial (the system O eN i is also inertial). The existence of inertial coordinate systems is declared by the first Newton law, and Eq. (3.27) expresses the second Newton law. In continuum mechanics both these laws are introduced by Axiom 5 together with Axioms 1–3 on the existence of the special coordinate system O eN i . In more detail, noninertial coordinate systems will be considered in Sect. 4.9. u t As a body B, consider the elementary volume dV of a continuum involving a point M. Then, by Axiom 5, the summarized vector of external forces denoted by d F acts on dV. Here the following two cases are possible: (1) M is an interior point of the domain V . (2) M is on the surface † of the domain V (in this case the intersection of the surface † with closure of the domain d V is denoted by d†). Definition 3.3. The vector fD

dF dF D dm dV

(3.28)

3.2 The Momentum Balance Law and the Stress Tensor

97

is called the specific mass force and the vector s D d F=d†

(3.29)

is called the specific surface force. Here d m is the mass of an elementary volume dV of a continuum. Since the vector F is additive, we have the following relations for the whole continuum occupying the volume V in K: Z Fm D V

F D F m C F †; Z Z f d m D f dV ; F † D s d†: V

(3.30) (3.31)

†

The vector F m is called the summarized vector of external mass forces acting on the considered continuum, and F † is called the summarized vector of external surface forces. Due to (3.30), the momentum balance law (3.27) takes the integral form: Z Z Z d v dV D f dV C s d†: (3.32) dt V

V

†

3.2.2 External and Internal Forces Mass f and surface s forces are external forces for a continuum V , if they are induced by objects not belonging to the considered continuum V (i.e. by external objects). External mass forces may be of the following types: (1) the gravity force f D g† eN , where g† is the acceleration in a free fall at a surface of a planet (for the Earth g† 9:8 kg=m s2 ), and eN is the normal vector to the planet surface; (2) external forces of inertia caused by the motion of a body relative to a moving coordinate system (see Sect. 4.11); (3) electromagnetic forces. External surface forces are forces of interaction of two continua contacting with each other, for example, at impact of two solid bodies. Besides external forces, there are internal forces considered in continuum mechanics. Modify Eq. (3.32). According to the rule (3.12) and the continuity equation (3.15), we have d dt

Z

Z v dV D V

@ v C r .v ˝ v/ @t

dV

V

Z

D V

@ @v v C C .r v/v C v r ˝ v @t @t

Z dV D

dv dV : dt

V

(3.33)

98

3 Balance Laws

Fig. 3.1 Internal surface forces on the surface †0

Then Eq. (3.32) takes the form Z D

Z dv f dV C s d† D 0: dt

V

(3.34)

†

Hence, the acceleration .d v=dt/ is the specific mass force but it is an internal force caused by inertia effects. Therefore, it is also called the internal inertia force. Consider an example of internal surface forces. Take an arbitrary continuum volume V and divide the volume by a surface †0 into two parts V1 and V2 (Fig. 3.1). Let n be a normal vector to †0 at a point M 2 †0 , and the vector be directed outwards from V1 . Then each of volumes V1 and V2 may be considered as a separate continuum undergoing the action of the external forces F 1 and F 2 . If we consider the surface element d† 2 †0 , which contains the point M, then for the domain V1 , according to (3.29), the surface element d† undergoes the action of the surface force d F 1 (or the specific surface force s1 ), and for the domain V2 the same surface element d† undergoes the action of the surface force d F 2 (or the specific surface force s2 ). Denote these specific surface forces as follows: tn D s1 D d F 1 =d†

and tn D s2 D d F 2 =d†:

(3.35)

The vectors tn and tn are called the stress vectors. They are specific internal surface forces relative to the whole volume V of the continuum (because they are defined for interior points M of the continuum).

3.2.3 Cauchy’s Theorems on Properties of the Stress Vector The split of forces into external and internal ones is relative: the same forces may be internal or external with respect to different subdomains of a considered continuum. Let us write the momentum balance equation (3.34) for the whole volume V and for its separate parts V1 and V2 :

3.2 The Momentum Balance Law and the Stress Tensor

V1

Z

99

Z Z dv f1 dV C s1 d† C tn d† D 0; dt

Z

†1

†0

Z Z dv dV C s2 d† C tn d† D 0; f2 dt

V2

†2

(3.36)

(3.37)

†0

where fi and si are the forces acting in the volumes Vi and on their surfaces †i , i.e. f D fi in Vi

and s D si on †i :

Since all the functions si and fi are continuous, so having subtracted Eqs. (3.36) and (3.37) from (3.34), we get Z .tn C tn / d† D 0:

(3.38)

†0

Since the surface †0 is arbitrary, we get tn C tn D 0. Thus, we have proved the following theorem. Theorem 3.5 (The first Cauchy theorem – on continuity of the stress vector). For the same point M, which is an interior point of a volume V , the stress vectors defined relative to the surface elements nd†0 and .nd†0 / differ only in sign: tn D tn ;

(3.39)

i.e. the field tn .x/ is continuous in the volume V . Remark. The result (3.39) is a consequence of the assumption that there are no discontinuities of the functions, which was made at the derivation of Eq. (3.34). When the functions are not continuous (for example, for shock waves in a continuum), Eq. (3.39) does not hold. t u Let us consider one more important property of the stress vector. At any point M we may construct an elementary volume dV in the form of a tetrahedron (Fig. 3.2),

Fig. 3.2 The properties of the internal stresses

100

3 Balance Laws

whose edges are oriented along vectors dx˛ D r˛ dX ˛ . Let d†˛ be areas of three sides lying on the coordinate planes, and d†0 be the area of the inclined side of the tetrahedron. The exterior normal vector to the surface element d†0 is denoted by n. On the surfaces d†˛ , the normal vectors are defined as .r˛ =jr˛ j/, because the reciprocal basis vectors r˛ are orthogonal to the planes d†˛ . Areas of the sides d†0 and d†˛ are connected by the relation n d†0

3 X r˛ d†˛ D 0: jr˛ j ˛D1

(3.40)

This relation follows, for example, from the equation Z

Z n d† D †

Z n E d† D

†

r E dV D 0;

(3.41)

V

applied to the tetrahedron, where † and V are the total area and the total volume of the tetrahedron, respectively. Here we have used the Gauss–Ostrogradskii formula (3.23). The scalar product of (3.40) by r˛ yields jr˛ jn r˛ d†0 D d†˛ :

(3.42)

Having applied Eq. (3.34) to the tetrahedron, we get tn d†0 C

3 X ˛D1

dv dV D 0; t˛ d†˛ C f dt

(3.43)

where t˛ is the stress vector on the surface element d†˛ with the normal .r˛ =jr˛ j/, and t˛ – with the normal .r˛ =jr˛ j/. Taking formulae (3.39) and (3.42) into account, we have d v jdV j tn n r˛ jr jt˛ D f : dt d†0 ˛D1 3 X

˛

(3.44)

Since dV=d†0 is an infinitesimal value, we get the following theorem. Theorem 3.6 (The second Cauchy theorem). The stress vector tn on an arbitrary surface element with a normal n is expressed in terms of stress vectors t˛ on three coordinate areas as follows: tn D

3 X ˛D1

n r˛ jr˛ jt˛ :

(3.45)

3.2 The Momentum Balance Law and the Stress Tensor

101

Theorems 3.5 and 3.6 have been proved by Cauchy in another form. Since p jr˛ j D .r˛ r˛ /1=2 D g ˛˛ ;

(3.46)

Equation (3.45) takes the form tn D n T;

(3.47)

where T is the second-order tensor called the Cauchy stress tensor: TD

3 X

r˛ ˝ t˛ D ri ˝ ti ;

˛D1

t ˛ t˛

p g ˛˛ :

(3.48) (3.49)

Thus, Theorem 3.6 may be enunciated in another way. Theorem 3.6a (The Cauchy theorem). For a continuous in V [ † stress vector field s.x/ D tn .x/ satisfying Eq. (3.32), there always exists a tensor field T.x/ satisfying the relation (3.47) in V [ †.

3.2.4 Generalized Cauchy’s Theorem Theorem 3.6a may be stated for an arbitrary vector or even tensor field ˆ.x; t/ satisfying the equation, which is similar to (3.32). Theorem 3.6b. Let there be a continuously differentiable tensor field m A.x; t/, m > 0, and a continuous tensor field m C.x; t/ in a volume V , and there also be a continuous in V [ † vector field m Bn .x; t/ depending on the choice of normal vector field n.x; t/, which satisfy the equation d dt

Z

Z A dV D e V

Z C dV C

m

m

e V

m

e V; Bn d† 8V

(3.50)

e †

e is the boundary of the domain V e, then the field m Bn .x; t/ may depend where † on the field n.x; t/ only linearly, i.e. there exists a tensor field mC1 B.x; t/ of the .m C 1/th order, such that in V [ † the following relation holds: m

Bn D n

mC1

B:

(3.51)

H A proof of this theorem is similar to the proof of Theorem 3.6, where as the e we should take the tetrahedron, whose edges are oriented along the local domain V basis vectors ri (see Exercise 3.2.3). N Equation (3.50) is called the balance equation.

102

3 Balance Laws

3.2.5 The Cauchy and Piola–Kirchhoff Stress Tensors The Cauchy stress tensor T defined by formulae (3.47)–(3.49), as a second-order tensor, can be resolved for any tensor basis, for example: T D T ij ri ˝ rj D Tij ri ˝ rj :

(3.52)

The definition (3.48) allows us to give a geometric representation of the Cauchy stress tensor. Indeed, if we take vectors ri as the left vectors and consider ti as the right vectors, then we can represent the tensor T in terms of equivalency classes (see Sect. 2.1.4 and [12]): T D ri ˝ ti D Œr1 t1 r2 t2 r3 t3 : (3.53) According to the geometric definition of a tensor (see [12]), the tensor T can be represented as the ordered set of six vectors ri ; ti with the common origin at a considered point M (Fig. 3.3), where the basis vectors ri are defined. The Cauchy stress tensor T is defined on a deformable surface element d† in the actual configuration. We can determine the stress tensor on the corresponding ı

nondeformed surface element d † in the reference configuration as well. To do this, ı

write out the relation (2.116) which connects oriented surface elements in K and K: q n d† D

ı

ı ı

g=g n F1 d †0 ;

(3.54)

and consider the stress vector tn on the surface element d†: q tn d† D n T d† D

ı ı

ı

ı

ı

ı

ı

g=g n F1 T d † D n P d † D tn d †:

(3.55)

Here we have introduced the tensor q ı P D g=g F1 T;

Fig. 3.3 Geometric representation of the Cauchy stress tensor

(3.56)

3.2 The Momentum Balance Law and the Stress Tensor

103

Fig. 3.4 Geometric representation of the Piola–Kirchhoff stress tensor P

called the first Piola–Kirchhoff stress tensor, which is defined on the nondeformed ı

surface element d †. ı The vector tn is called the Piola–Kirchhoff stress vector; this vector is connected to the tensor P by the Cauchy formula (3.47): ı tn

ı

D n P:

(3.57)

With the help of the relations (2.35) and (3.48), the expression (3.56) can be represented in terms of equivalency classes (see Sect. 2.1.4) ı

ı

ı ı ı ı ı ı

P D ri ˝ ti D Œr1 t1 r2 t2 r3 t3 ;

(3.58)

where the following vectors have been denoted ı ˛

t D t˛

q

ı

g˛˛ g=g:

(3.59)

With the help of this expression we can also give a geometric representation of the ı Piola–Kirchhoff stress tensor in the basis ri of the reference configuration (Fig. 3.4).

3.2.6 Physical Meaning of Components of the Cauchy Stress Tensor Formulae (3.53) and (3.58) give a geometric representation of the tensors T and P. bij of the Cauchy stress tensor Let us explain a physical meaning of components T with respect to an orthonormal basis. Let there be a local basis ri in K, then, using the orthogonalization process, we can always construct an orthogonal basis e ri , and, normalizing the vectors e ri , we obtain the orthonormal basis b ri called the physical basis. In this basis the Cauchy bij , which are coincident with T bij : stress tensor (3.52) has components T bijb TDT ri ˝b rj :

(3.60)

104

3 Balance Laws

Fig. 3.5 Physical meaning of components of the Cauchy stress tensor

Take an arbitrary material point M 2 V and consider an elementary volume dV 2 V involving this point and having the form of a cube whose sides are orthogonal to the vectorsb r˛ (Fig. 3.5). Denote the area of the cube side orthogonal tob r˛ (˛ D 1; 2; 3) by d†˛ . Since we consider the elementary volume dV, the stress tensor T.x/ is the same at every point x 2 dV and x 2 d†˛ (˛ D 1; 2; 3). Each of the sides d†˛ undergoes the action of the surface force d F ˛ connected to the stress vector tn on this side by the relation (3.35): on d†˛ W

dF˛ D tn D n T: d†˛

(3.61)

ri : Resolving the vectors d F ˛ and n for the basisb d F ˛ D d F˛i b ri ;

n Db ni b ri ;

(3.62)

from (3.61) we get on d†˛ W

d F˛i T ji D b Db nj b T ˛i ; d†˛

˛ D 1; 2; 3;

(3.63)

because on d†˛ : b n˛ D 1, b nˇ D b n D 0, ˛ ¤ ˇ ¤ ¤ ˛. bij of Formula (3.63) allows us to explain the physical meaning of components T the Cauchy stress tensor: ˇ

˛ b˛˛ D d F˛ ; T b˛ˇ D d F˛ ; T b˛ D d F˛ ; ˛ ¤ ˇ ¤ ¤ ˛: on d†˛ W T d†˛ d†˛ d†˛ (3.64) ˛˛ b is the ratio of the corresponding normal component d F˛˛ The normal stress T of the surface force d F ˛ , acting on the surface element d†˛ , to the area of this b˛ˇ , b surface element; and the tangential stresses T T ˛ are the ratios of tangential ˇ components d F˛ , d F˛ of the same force d F ˛ , acting on the surface element d†˛ , to the area of the same surface element.

Remark 1. Since the tensor T is the same in the whole cube dV, by Theorem 3.5, the forces d F ˛ on the opposite sides of the cube differ only in sign. But the normals n

3.2 The Momentum Balance Law and the Stress Tensor

105

on these sides differ also only in sign, therefore relations (3.61) on these sides are the same as well as the relations (3.64). t u Remark 2. Due to Remark 1, there are three different relations in formulae (3.64) on each of three sides d†˛ , i.e. we have nine different relations: for three normal b˛˛ and six tangential stresses T b˛ˇ , ˛ ¤ ˇ. The pair tangential stresses stresses T b˛ˇ and T bˇ ˛ , defined on different sides d†˛ and d†ˇ , in general, are not coinciT dent (Fig. 3.6). Below, we will know that for many problems of continuum mechanics the stress tensor proves to be symmetric T D TT (due to additional assumptions); it is the b˛ˇ D T bˇ ˛ . only case when pair tangential stresses coincide: T t u b ij of the Piola– Let us clarify now the physical meaning of components P b ı Kirchhoff stress tensor P with respect to the orthonormal basis ri of the reference configuration: b ı ı b ijb ri ˝ rj : (3.65) PDP ı

ı

Take a material point M in K and consider an elementary volume d V being a cube ı b ı and containing this point. Let sides of the cube d †˛ be orthogonal to the vectors r˛ (Fig. 3.7).

Fig. 3.6 Difference of pair tangential stresses

Fig. 3.7 Physical meaning of components of the Piola–Kirchhoff stress tensor

106

3 Balance Laws ı

In configuration K a distorted volume dV corresponds to the cube d V . According to the geometric representation of transformation of a small neighborhood of a material point (see Sect. 2.3.5), the volume dV is a parallelepiped, in general, with inclined sides d†˛ remaining plane. ı

ı

For elementary volumes d V and dV the stress tensors T.x/ and P.x/ are the same ı

ı

at every point x 2 d V and x 2 d V . The deformed (but plane) side d†˛ undergoes the action of the surface force ı

ı

d F ˛ . Carry the force in the parallel way in K onto the corresponding side d †˛ , ı

then the relations (3.55) on the side d †˛ take the form 0 1 0 1 d F ˛ @ d†˛ A d† ı ˛ D tn @ ı A D n P: ı d†˛ d †˛ d †˛

ı

on d †˛ W

(3.66)

b ı ı Resolving the vectors d F ˛ and n for the basis ri : ı b ı d F ˛ D d F i˛ ri ;

b ı ı ı b n D ni ri ;

from (3.66) we get ı

ı

on d †˛ W

d F i˛ b ı b ij D P b ˛i ; D ni P d†˛

˛ D 1; 2; 3I

(3.67)

ı b b b ı ı ı because on †˛ : n˛ D 1, and nˇ D n D 0. As follows from (3.67), ı

ı

ı

b ˛ D d F ˛ ; ˛ ¤ ˇ ¤ ¤ ˛: on d †˛ W P ı d †˛ (3.68) ˛˛ b Thus, according to (3.68), the normal stress P is the ratio of normal compo˛ b ˛˛ D d F ˛ ; P ı d †˛

ı

ˇ b ˛ˇ D d F ˛ ; P ı d †˛

ı

nent d F ˛˛ of the surface force d F ˛ , acting on deformed surface element d†˛ , to ı

the area of corresponding undistorted surface element d †˛ . The tangential stresses ı ı b ˛ˇ , P b ˛ are the ratios of tangential components d F ˇ˛ , d F ˛ of the same surface P force d F ˛ , acting on the deformed surface element d†˛ , to the area of undistorted ı

surface element d †˛ . Remarks 1 and 2 for the tensor P are also valid with the exception of the fact that the tensor P remains nonsymmetric even if T is symmetric.

3.2 The Momentum Balance Law and the Stress Tensor

107

3.2.7 The Momentum Balance Equation in Spatial and Material Descriptions Substituting the notation (3.35) for internal surface forces and Cauchy’s formula (3.47) into (3.32), we get one more widely used form of the momentum balance law: Z

Z

d dt

v dV D V

Z n T d† C

†

f dV:

(3.69)

V

Having substituted (3.35) and (3.47) into (3.34): Z Z dv f dV D n T d†; dt V

(3.70)

†

and then having transformed the surface integral into the volume one by the Gauss– Ostrogradskii formula (3.24), we obtain Z dv f r T dV D 0: dt

(3.71)

V

Since the volume V is arbitrary, the integrand must always vanish. Thus, we have proved the following theorem. Theorem 3.7. If the functions F, v, T and f, which satisfy the momentum balance law (3.69) and depend on x i ; t, are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V .t/ the following momentum balance equation in Eulerian description (i.e. in K) holds:

dv D r T C f: dt

(3.72)

According to the property (3.20) of the total derivative, we can rewrite the momentum balance equation in the divergence form: @v C r v ˝ v D r T C f: @t

(3.73)

Modify the momentum balance equation (3.70) with taking (3.55) into account: Z ı

V

dv f dt ı

ı

Z

dV

ı

ı

n P d † D 0: ı

†

(3.74)

108

3 Balance Laws

With the help of the Gauss–Ostrogradskii theorem (3.24), we can rewrite this equation in the form Z ı ı ı dv f r P d V D 0: dt

(3.75)

ı

V ı

Since the volume V is arbitrary, we obtain the following theorem. Theorem 3.8. If the conditions of Theorem 3.7 are satisfied, then at every point ı

M 2 V for all considered times t > 0 the following momentum balance equation ı

in Lagrangian description (i.e. in K) holds: ı dv

ı

ı

D f C r P:

dt

(3.76)

Since in Lagrangian description the total derivative with respect to time coincides with the partial one (see (2.189)), the divergence form of the momentum balance ı

equation in K coincides with (3.76): ı @v

@t

ı

ı

D r P C f:

(3.77)

Exercises for 3.2 3.2.1. Using formulae (2.35a) and (3.56), show that the Piola–Kirchhoff tensor P ı ı has the components ..=/T ij / with respect to the mixed dyadic basis ri ˝ rj : ı

ı

P D .=/T ij ri ˝ rj : ı

3.2.2. Using the results of Exercises 3.2.1 and 2.1.12, show that components P ij of the tensor P with respect to the reference configuration basis can be written in the form ı

ı

ı

P D P ij ri ˝ rj ;

ı

ı

ı

F D F jm rj ˝ rm ;

ı

ı

ı

P ij D .=/T i m F jm :

3.2.3. Prove Theorem 3.6b by the same method as we used in the proof of Theorem 3.6.

3.3 The Angular Momentum Balance Law

109

3.3 The Angular Momentum Balance Law 3.3.1 The Integral Form Consider the next fundamental law of continuum mechanics. Definition 3.4. The vectors k0 D

Z V

0m 0†

Z x v dm D

Z D

V

Z

D

x v dV; Z

x f dm D V

x f dV;

(3.78)

V

x tn d†; 0 D 0m C 0† ;

†

are called as follows: k0 is the angular momentum vector of a continuum, 0m is the vector of mass moments of a continuum, 0† is the vector of surface moments of a continuum, 0 is the vector of moments of a continuum. Axiom 6 (The angular momentum balance law). For any continuum B there are two additive vector-functions: k00 .B; t/ called the own angular momentum vector and 00 .B; t/ called the vector of own moments, and also the positive constant s > 0 called the spin constant, such that 8t > 0 the following equation holds: dk D ; dt

(3.79)

where k is the vector of the total angular momentum of the continuum, and is the total moment vector: k D k0 C

1 00 k ; s

D 0 C 00 :

Remark. Axiom 6 postulates the existence of two new vectors: k00 and 00 , similarly to Axiom 5 introducing the force vector F . Physical interpretation of the vectors k00 and 00 is more complicated than for the force F (for example, effects caused by the gravity force or the inertia force are intuitively clear and well known). The appearance of the vectors k00 and 00 can be caused by electromagnetic effects in continua, which have an ordered magnetic structure (for example, ferromagnetic materials). However, the vectors k00 and 00 can appear for some continua, where there are no electromagnetic effects (such continua are often called the Cosserat continua). It should be noted that although the spin constant s in the law (3.79) seems to be unnecessary (in place of .1=s /k00 we could introduce the vector k00 ), its appearance is very essential; this will be explained in Sect. 3.4.6. t u

110

3 Balance Laws

Since the vectors k00 and 00 are additive, so, according to Sect. 3.2.1, we obtain the following classification. Definition 3.5. The specific own angular momentum is the vector km , the specific own mass moment is the vector hm , and the specific own surface moment is the vector h† , which are defined at a point M of a continuum as follows: d k00 ; dm

km D

hm D

d 00 ; dm

h† D

d 00 : d†

(3.80)

Since the vectors k00 and 00 are additive, for the whole considered volume of a continuum we have Z (3.81) k00 D km dV; 00 D 00m C 00† ; V

where the following notation has been introduced: 00m

Z D

hm dV;

00†

V

Z D

h† d†;

(3.81a)

†

On substituting (3.78) and (3.81) into (3.79), the angular momentum balance law takes the integral form d dt

Z Z Z x v C km dV D .x f C hm / dV C .x tn C h† / d†: s

V

V

†

(3.82)

3.3.2 Tensor of Moment Stresses Equation (3.82) has the form of (3.50), if we introduce the following notation: A D x v C km ;

C D x f C hm

and

B D x tn C h† :

Then to Eq. (3.82) we can apply Theorem 3.6b, which asserts the existence of a e t/, defined in V [ †, such that the following relation second-order tensor field M.x; being an analog of (3.47) holds:

With new notation

e h† C x tn D n M:

(3.83)

e C T x; MDM

(3.84)

this result can be written in another form.

3.3 The Angular Momentum Balance Law

111

Theorem 3.9. For vector fields tn and h† being continuous in V [ † and satisfying Eqs. (3.32) and (3.82), there exists a second-order tensor field M.x; t/ such that in V [ † the following relation holds: h† D n M:

(3.85)

The tensor M is called the tensor of moment stresses.

3.3.3 Differential Form of the Angular Momentum Balance Law ı

At first, consider the left-hand side of (3.82) and pass from V to V : d dt

Z ı

ı 1 ı x v C km d V s

V

d D dt Z D V

Z ı

Z ı 1 1 d km ı ı d x v C km d V D dV .x v/ C s dt s dt ı

ı

V

V

Z dv dv 1 d km 1 d km vvCx dV D x dV ; C C dt s dt dt s dt V

(3.86) as v v D 0. Modify the right-hand side of Eq. (3.82) by using the Gauss–Ostrogradskii theorem (3.24) and formula (3.47): Z Z Z Z x tn d† D x .n T/d† D n .T x/ d† D r .T x/ dV : †

†

†

V

(3.87)

The nabla-operator r in (3.87) is determined by the formula @ .T x/ D .r T/ x C ri T ri @X i D x .r T/ C " T:

r .T x/ D ri

(3.88)

Substitution of (3.88) into (3.87) yields Z

Z x tn d† D

†

Z x r T dV

V

" T dV: V

(3.89)

112

3 Balance Laws

Having substituted (3.89) and (3.86) into (3.82), we get Z V

Z dv 1 d km x f r T dV C dV dt s dt V Z Z Z D hm dV C r M dV " T dV: V

V

(3.90)

V

Here we have used the definition of the tensor of moment stresses M (3.84). Due to formula (3.72), Eq. (3.90) takes the final form Z V

d km hm r M C " T dV D 0: s dt

(3.91)

Since the volume V is arbitrary, we get the following theorem. Theorem 3.10. If the functions F, v, T, f, satisfying Eq. (3.69), and also the functions km , hm , M, satisfying Eq. (3.90), are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V .t/ the following angular momentum balance equation holds: d km D hm C r M " T: s dt

(3.92)

Thus, the angular momentum balance equation includes only one classical mechanical characteristic, namely the stress tensor T, the remaining functions km , hm and M, as it was mentioned above, either have a nonmechanical nature or are caused by nonclassical mechanical properties.

3.3.4 Nonpolar and Polar Continua For classical continua, for which there are not any nonmechanical values, we have km D 0;

hm D 0;

M D 0:

(3.93)

Such continua are called nonpolar, and for them it follows from (3.93) and (3.92) that 3 X p "T D g .T ˛ˇ T ˇ ˛ /r D 0; (3.94) ˛D1

˛¤ˇ¤ ¤˛

and hence T D TT ; i.e. the Cauchy stress tensor for nonpolar continua is symmetric.

(3.95)

3.3 The Angular Momentum Balance Law

113

It should be noted that the corresponding Piola–Kirchhoff tensor is not symmetric even for nonpolar continua; that follows from its definition (3.56). Thus, for nonpolar continua the angular momentum balance law reduces to the condition (3.95) that the Cauchy stress tensor T is symmetric. A continuum, for which values of km , hm and M are different from identically zero: km ¤ 0; hm ¤ 0; M ¤ 0; (3.96) is called polar. For polar continua, the Cauchy stress tensor T is not symmetric.

3.3.5 The Angular Momentum Balance Equation in the Material Description The integral form of the angular momentum balance equation (3.82) can be represented in the material description. To do this, we should modify the left-hand side of formula (3.82) according to (3.86): d dt

Z V

Z ı 1 1 d ı x v C km dV D x v C km d V s dt s ı

V

ı dv 1 d km x dV C dt s dt

Z

ı

D ı

(3.97)

V

ı

ı

and introduce Lagrangian tensor of moment stresses M in K similarly to the Piola–Kirchhoff stress tensor: ı

ı

ı

n M d† D n M d †;

(3.98)

ı

ı

and the specific own surface moment h† in K: ı

ı

ı

h† D n M:

(3.99) ı

Then the surface integral in (3.82) can be written in K as follows: Z

Z .x tn C h† / d† D †

Z n .T x C M/ d† D

†

ı

ı

ı

n .P x C M/ d †; ı

†

(3.100)

114

3 Balance Laws

and Eq. (3.82) takes the form Z ı d 1 ı x v C km d V dt s ı

V

Z

Z

ı

ı

.x f C hm / d V C ı

ı

ı

ı

n .P x C M/ d †:

(3.101)

ı

V

†

Exercises for 3.3 3.3.1. Show that from (3.101) we can get the angular momentum balance equation in the material description ı

ı ı d km ı D hm C r M " .F P/: s dt

3.3.2. Show that for nonpolar continua the angular momentum balance equation in the material description (see Exercise 3.3.1) takes the form " .F P/ D 0

F P D PT F T

or

and coincides exactly with (3.96). 3.3.3. Using equation (3.98) and the formula of transformation of an oriented surı

face element, show that the tensors M and M are connected by the relation q ı ı M D g=g F1 M:

3.4 The First Thermodynamic Law 3.4.1 The Integral Form of the Energy Balance Law The mass, momentum and angular momentum balance laws describe the motion of a continuum. If we need thermal effects to be taken into account, then we must use thermodynamic laws. Let us consider nonpolar continua, for which the relations (3.93) are satisfied. Definition 3.6. The scalar function Z Z jvj2 vv KD dm D dV; 2 2 V

V

is called the kinetic energy of a continuum V .

jvj2 D v v

(3.102)

3.4 The First Thermodynamic Law

115

Definition 3.7. The power of external mass forces Wm and the power of external surface forces W† are the following scalar functions: Z

Z

Wm D

f v dm D V

Z f v dV;

W† D

V

tn v d†:

(3.103)

†

Axiom 7 (The first thermodynamic law (or the energy balance law)). For any continuum B there are two scalar additive functions: U.B; t/ called the internal energy of the continuum and Q.B; t/ called the heating rate for the continuum such that 8t > 0 the following equation is satisfied: dE D W C Q; dt

(3.104)

where E is called the total energy of the continuum and consists of U and K: E D U C K;

W D W m C W† :

(3.105)

Remark. There are different statements of the thermodynamic law, and the above is called Truesdell’s statement [54, 55]. The statement is convenient, because its form corresponds to Axioms 4–6. Moreover, this statement (unlike others) is really universal, i.e. it is independent of the type of a continuum. t u Since the functions U and Q are additive, similarly to Sect. 3.2.1 we can introduce the corresponding specific functions. Definition 3.8. The specific internal energy is the function e, the specific heat flux from mass sources is the function qm and the specific heat flux from surface sources is the function q† , which are defined at every point M of a continuum as follows: eD

dU ; dm

qm D

dQ ; dm

q† D

dQ : d†

(3.106)

Since the functions Q and U are additive, for a whole continuum we have Z Qm D V

Q D Qm C Q † ; Z Z qm d m D qm dV; Q† D q† d†; V

Z

Z e dm D

U D V

(3.107)

†

e dV: V

Substitution of Eqs. (3.102), (3.103), (3.105)–(3.107) into (3.104) yields the energy balance law in the integral form

116

3 Balance Laws

d dt

Z

Z Z jvj2 eC dV D .f v C qm / dV C .tn v C q† / d†: (3.108) 2

V

V

†

Using on the left-hand side of (3.108) the differentiation rules for a volume integral (see Exercise 3.1.2): d dt

Z

Z jvj2 de dv eC dV D dV; Cv 2 dt dt

V

(3.109)

V

we obtain the following form of the energy balance law: Z

Z d 1 e C jvj2 C f v C qm dV C .tn v C q† / d† D 0: (3.110) dt 2

V

†

3.4.2 The Heat Flux Vector Equation (3.108) has the form (3.50), if in (3.50) we put A D e C jv2 j=2;

C D f v C qm

and B D tn v C q† :

Then we can apply Theorem 3.6b to this equation; the theorem claims the existence of the vector .q/ such that in V [ † the following relation holds: q† D n q:

(3.111)

The vector q is called the heat flux vector. Prove the analog of Theorem 3.6 allowing us to determine components of the vector q. As was made in Sect. 3.2.3, at a point M we construct an elementary volume dV in the form of a tetrahedron, whose edges are oriented along vectors r˛ dX ˛ . Then for the volume all formulae (3.40)–(3.42) hold. Applying Eq. (3.110) to the tetrahedron, when there is no action of any mass or surface forces on the continuum (i.e. tn D 0, f D 0), we get q† d†0 C

d jvj2 q˛ d†˛ C qm eC dV D 0: dt 2 ˛D1 3 X

(3.112)

Here q˛ is the heat flux from surface sources on the surface element d†˛ with the normal .r˛ =jr˛ j/, and q† – on the surface element d†0 .

3.4 The First Thermodynamic Law

117

Then with taking (3.42) into account, we have d jvj2 dV q† C n r˛ jr jq˛ D qm : eC dt 2 d† 0 ˛D1 3 X

˛

(3.113)

Since the value dV=d†0 is infinitesimal, we get the following theorem. Theorem 3.11. On every surface element with a normal n, the heat flux from surface sources q† is expressed in terms of heat fluxes on three coordinate surface elements: 3 X p n r˛ g ˛˛ q˛ : (3.114) q† D ˛D1

Comparing (3.114) with (3.111), we find components of the heat flux vector: qD

3 X

r˛

p g ˛˛ q˛ D ri q i ;

q ˛ D q˛

p g ˛˛ :

(3.114a)

˛D1

3.4.3 The Energy Balance Equation Let us consider now formula (3.110) and transform the surface integrals by the Gauss–Ostrogradskii formula (3.24) with taking (3.111) into account as follows: Z Z Z q† d† D n q d† D r q dV; Z

†

Z

tn v d† D †

†

V

Z

n .T v/ d† D †

(3.115) r .T v/ dV:

V

Substituting the expression (3.115) into (3.108), we obtain another integral form of the energy balance law: d dt

Z V

Z Z jvj2 eC dV D .f v C qm / dV C n .T v q/ d†: (3.116) 2 V

†

Substitution of formulae (3.115) into (3.110) yields Z d jvj2 =2 de f v qm C r q r .T v/ dV D 0: (3.117) C dt dt V

Since the volume V is arbitrary, we get the following theorem.

118

3 Balance Laws

Theorem 3.12. If the functions F, v, T, f, qm , q and e, satisfying Eq. (3.116), are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V .t/ the energy balance equation has the following differential form: d dt

jvj2 eC D f v C qm r q C r .T v/: 2

(3.118)

According to formula (3.20), we can rewrite the energy balance equation in the divergence form @ C r .v/ D r q C r .T v/ C f v C qm ; @t

DeC

jvj2 ; (3.119) 2

where is the specific total energy of the continuum.

3.4.4 Kinetic Energy and Heat Influx Equation The scalar product of the momentum balance equation (3.72) by v yields

dv v D f v C .r T/ v: dt

(3.120)

Using the property of covariant differentiation (see Exercise 2.1.17), we can reduce this equation to the form 1 d jvj2 D f v C r .T v/ T .r ˝ v/T : 2 dt

(3.121)

Definition 3.9. The scalar function Z W.i / D

T .r ˝ v/T dV

(3.122)

V

is called the power of internal surface forces of a continuum V . Integrating Eq. (3.121) over a volume V and using Definitions 3.6, 3.7, 3.9, and the rule of differentiation of an integral over a moving volume (see Exercise 3.1.2), we get the following theorem. Theorem 3.13 (The kinetic energy balance equation). A change of the kinetic energy of a continuum is equal to the summarized power of external and internal forces acting on the continuum: dK D W C W.i / : dt

(3.123)

3.4 The First Thermodynamic Law

119

Subtracting the kinetic energy equation (3.121) from the energy balance equation (3.118), we obtain the differential heat influx equation

de D T .r ˝ v/T C qm r q: dt

(3.124)

This equation can be rewritten in the divergence form @e C r .ev/ D r q C T .r ˝ v/T C qm : @t

(3.125)

Subtracting Eq. (3.123) from (3.104), we get the heat influx equation in the integral form d (3.126) U C W.i / D Q: dt

3.4.5 The Energy Balance Equation in Lagrangian Description Let us write now the energy balance equation in Lagrangian description. Use the equation in the integral form (3.108) and transform the surface integrals as follows: Z

Z .tn v C q† / d† D

†

Z q D

n .T v q† / d† † ı

ı ı

g=g n .F1 T v F1 q† / d † D

ı

Z

ı ı n P v q d†:

ı

†

†

(3.127) Here we have used the definition (3.56) of the Piola–Kirchhoff tensor and introduced the heat flux vector in the reference configuration: q

ı

qD

ı

g=g F1 q:

(3.128)

ı

Theorem 3.14. Heat flux vectors q and q are connected by the relation ı

ı

ı

ı

ı

q n d † D q n d†:

(3.129)

H Indeed, multiplying (3.128) by n d † and using formula (2.116), we obtain ı

ı

ı

q n d† D

q

ı

ı

ı

g=g q F1T n d † D q n d†: N

120

3 Balance Laws

Then Eq. (3.108) in the reference configuration takes the form Z ı ı ı ı jvj2 ı d ı ı eC f v qm r .P v/ C r q d V D 0: dt 2

(3.130)

ı

V

ı

Since the volume V is arbitrary, the integrand vanishes. Thus, we have proved the following theorem. ı

Theorem 3.15. Under the conditions of Theorem 3.12, at every point M 2 V for all considered times t > 0 we have the energy balance equation in Lagrangian description ı

d dt

ı ı ı jvj2 ı ı eC D f v C qm C r .P v/ r q: 2

(3.131)

Similarly to formula (3.120), we can write the kinetic energy balance equation in the reference configuration ı dv

dt

ı

ı

v D f v C v .r P/;

(3.132)

with the help of which Eq. (3.131) takes the form ı

e

ı ı ı d ı D P .r ˝ v/T C qm r q: dt

(3.133)

This equation is called the heat influx equation in material (Lagrangian) description. Since in Lagrangian description the total derivative coincides with the partial one, the divergence form of the heat influx equation does not differ from (3.133): ı

ı @e

ı

ı

ı

D r q C P .r ˝ v/T C qm :

@t

(3.134) ı

Integrate the heat influx equation (3.133) over the domain V and take into account that the internal energy U and the heating rate Q, according to (3.107), can also be represented in Lagrangian description: Z U D Z Qm D

Z e dV D

V

ı

Q D Q m C Q† ;

ı

Z

qm dV D V

ı

e d V ; V

ı

ı

qm d V ; ı

V

Z Q† D

Z q† d† D

†

ı

ı

q † d †; ı

†

(3.135)

3.4 The First Thermodynamic Law

121

here we have used Eq. (3.129). As a result, from (3.133) we get the heat influx equation in the integral form (3.126) again, where the power of internal surface forces is defined by the following relations in Eulerian and Lagrangian descriptions, respectively: Z W.i / D

Z T .r ˝ v/T dV D

ı

ı

P .r ˝ v/T d V :

(3.136)

ı

V

V

According to relation (3.55), we can represent the power of external forces (3.103) in Lagrangian description as follows: Z Wm D

Z f v dV D

Z

ı

ı

f v d V ; W† D ı

V

Z tn v d† D

ı

v d †: (3.137)

ı

†

V

ı tn

†

3.4.6 The Energy Balance Law for Polar Continua Let us consider now polar continua, for which the relations (3.96) hold, i.e. the values of km , hm and M are different from identically zero. The energy balance law for polar continua differs from the corresponding law (3.104), (3.105) for nonpolar continua. Similarly to (3.102), introduce the kinetic energy of the own motion of a continuum: Z Z jkm j2 Ks D dm D jkm j2 dV; jkm j2 D km km : (3.138) 2s s V

V

Similarly to (3.103), introduce the power of mass moments Wms and the power of surface moments W†s : Z Wms D

hm km dV; V

Z W†s

D

Z

h† km d† D †

(3.139)

n M km d†:

(3.140)

†

The energy balance law for a polar continuum has the universal form (3.104), but the total energy E and the summarized power W of a continuum are defined not by (3.105) but by the formulae E D U C K C Ks ;

W D Wm C Wms C W† C W†s :

(3.141)

122

3 Balance Laws

As a result, we obtain the following form of the energy balance law for a polar continuum: d .U C K C Ks / D Wm C Wms C W† C W†s C Q: dt

(3.142)

From (3.142) with the help of (3.138)–(3.140) we get the integral form of the law: Z Z d jvj2 jkm j2 eC C dV D .f v C hm km C qm / dV dt 2 2s V V Z C .t† v C h† km C q† / d†: (3.143) †

Similarly to Sect. 3.4.2, the heat flux vector q for polar continua is introduced by formulae (3.111) and (3.114). Thus, in Eq. (3.143) we can transform the surface integral to the volume one, then we obtain the following differential form of the energy balance law for a polar continuum (the energy equation):

d dt

jvj2 jkm j2 eC D f v C hm km C qm r q C r .T v C M hm /: C 2 2s (3.144)

Remark. The energy equation (3.143) clarifies the meaning of introduction of the spin constant s in the angular momentum balance law (3.79): if in (3.79) or in (3.92) we eliminated the constant s by introducing the vector e km D km =s instead of km , then the energy equation (3.144) would include the constant s all the same. This constant s plays a role of the energetic equivalent between the kinetic energy of a body and the kinetic energy of the own body motion caused by the own angular momentum for polar continua. t u According to relation (3.20), the energy equation (3.144) takes the divergence form @ " C r .v" C q T v M km / D f v C hm km C qm ; @t jvj2 jkm j2 : "DeC C 2 2s

(3.145)

Let us formulate now an analog of the kinetic energy balance equation (3.123) for the kinetic energy of own motion Ks . Consider the angular momentum balance equation (3.92) and multiply the equation by km : d km km D hm km C km r M km 3 " T: s dt

(3.146)

According to the property of the nabla-operator from Exercise 2.1.17, we can reduce Eq. (3.146) to the form

3.4 The First Thermodynamic Law

123

d jkm j2 D hm km C r .M km / M .r ˝ km /T km 3 " T: (3.147) s dt Integrating this equation over V and using the Gauss–Ostrogradskii formula and the differentiation rule of an integral over a moving volume (3.12), from (3.147) we get dKs D Wms C W†s C W.is / C Ws : dt

(3.148)

Here we have introduced the following notation: Z W.is / D M .r ˝ km /T dV

(3.149)

V

– the power of internal surface moments and Z Ws D

km 3 " T dV

(3.150)

V

– the power of internal surface forces for the own body motion. Relation (3.148) is called the kinetic energy balance equation for the own body motion. Subtracting Eqs. (3.123) and (3.148) from the energy balance law (3.142), we get the integral heat influx equation for polar continua: dU D Q W.i / W.is / Ws : dt

(3.151)

Subtracting Eqs. (3.147) and (3.121) from the energy equation (3.144), we obtain the differential heat influx equation for polar continua:

de D T .r ˝ v/T C M .r ˝ km /T C km 3 " T r q C qm : (3.152) dt

If we assume km D 0, M D 0 and hm D 0, then the heat influx equation in the forms (3.151) and (3.152) coincides with the corresponding equations (3.124) and (3.126) for nonpolar continua. ı

According to the definition (3.97) of Lagrangian tensor of moment stresses M, we can rewrite the heat influx equation (3.152) in the material description ı de

dt

ı

ı

ı

ı

ı

ı

D P .r ˝ v/T C M .r ˝ km /T C km 3 " .F P/ r q C qm : (3.153)

This assertion should be proved as Exercise 3.4.1.

124

3 Balance Laws

Exercises for 3.4 3.4.1. Prove the relation (3.153).

3.5 The Second Thermodynamic Law 3.5.1 The Integral Form To state the second thermodynamic law, we introduce a new local characteristic of a continuum which is called a temperature. Axiom 8 (on the existence of an absolute temperature). For every material point M of any continuum B for all t > 0 there is a scalar positive function .X i ; t/ D .x i ; t/ > 0;

(3.154)

called an absolute temperature. The axiom on the existence of an absolute temperature is sometimes called the zero thermodynamic law. It should be noted that we have introduced a temperature as local but not integral characteristic of a continuum (unlike mass M or angular momentum). Although we R can always introduce the integral temperature ‚, for example, as ‚ D V d m, but this value does not appear in any fundamental laws of continuum mechanics. However, there are integral characteristics containing a temperature, which are of great importance for formulating the second thermodynamic law. Definition 3.10. The entropy production by external mass sources is the scalar function QN m and the entropy production by external surface sources is the scalar function QN † , which are defined for a volume V as follows: QN m D

Z V

qm dm D

Z V

qm dV;

QN † D

Z

q† d† D

†

Z

nq d†: (3.155)

†

Remark. If by analogy with (3.106) we consider the entropy production by external mass sources d QN m , supplied to elementary volume dV by mass d m, then from (3.155) and (3.106) we get d QN m D qm d m= D dQ=;

(3.155a)

i.e. d QN m is the heat influx due to external mass sources divided by temperature of the volume dV heated. Thus, d QN m characterizes the efficiency of heating the volume dV, and d QN † characterizes the efficiency of heating the surface element d†. u t

3.5 The Second Thermodynamic Law

125

Axiom 9 (The second thermodynamic law). For any continuum B there are two scalar additive functions: H.B; t/ called the entropy of the continuum and QN .B; t/ called the entropy production by internal sources, such that for all t > 0 the following equation holds: dH D QN C QN ; (3.156) dt where we have denoted the summarized entropy production by external sources as follows: (3.157) QN D QN m C QN † ; and a value of QN is always nonnegative (the Planck inequality): QN > 0:

(3.158)

Since the functions H and QN are additive, we can introduce the corresponding specific functions at every point M 2 V . Definition 3.11. The specific entropy and the specific internal entropy production q are defined for every point M of a continuum as follows: D

dH ; dm

q D

d QN : dm

(3.159)

As follows from the Planck inequality (3.158), and also the inequalities (3.154) and (3.4a), the function q is always nonnegative: q > 0:

(3.158a)

This inequality is also called the Planck inequality. Since the functions H and QN are additive for a whole continuum, we have QN D

Z

q dm D

V

Z

q dV; H D

V

Z

Z dm D V

dV:

(3.160)

V

Two relations (3.156) and (3.158) are equivalent to the Clausius inequality N dH=dt > Q:

(3.161)

On substituting (3.155) and (3.160) into (3.156), we obtain the integral form of the second thermodynamic law d dt

Z

Z dV D

V

V

.qm C q / dV

Z †

nq d†:

(3.162)

126

3 Balance Laws

3.5.2 Differential Form of the Second Thermodynamic Law Transform the surface integral in (3.162) by the Gauss–Ostrogradskii formula (3.24): Z Z Z q r q 1 nq d† D dV: (3.163) d† D r Cqr †

V

V

Then Eq. (3.162) takes the form Z q d .qm C q / dV D 0: Cr dt

(3.164)

V

Since the volume V is arbitrary, we get the following theorem. Theorem 3.16. If the functions , q, q , qm and , satisfying Eq. (3.162), are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V the second thermodynamic law has the following differential form (the entropy balance equation):

q d qm C q C D r : dt

(3.165)

According to the property (3.20), we can rewrite Eq. (3.165) in the divergence form @ qm C q q D C r v C : (3.165a) @t According to the equation r

q

D

1 1 r q 2 q r ;

formulae (3.165) and (3.165a) can be rewritten in the form

d D r q C qm C w ; dt

(3.166)

or

@ C r v D .1=/r q C .1=/.qm C w /: @t Here we have introduced the function w D q C

q r ;

(3.166a)

(3.167)

called the dissipation function. Besides the Planck inequality (3.158), there is one more important inequality in continuum mechanics.

3.5 The Second Thermodynamic Law

127

Axiom 10. At every point M of a continuum for all t > 0 the Fourier inequality holds: q r 6 0: (3.168) Rewrite (3.167) in the form q D w

q r :

(3.169)

Equation (3.169) means that the specific internal entropy production is caused by two factors: dissipation (i.e. irreversible conversion of part of mechanical energy to heat energy) and nonuniform heating of a continuum (when there are irreversible processes of heat transfer from hotter parts of the continuum to its colder ones). The first cause corresponds to the function w , and the second one corresponds to the function 1 q r wq : (3.170) A mathematical expression for the dissipation function w will be given in Chap. 4; however, it should be noted here that if w is independent of the temperature gradient r (in Chap. 4 we will show that this assumption always holds), then, ı

considering the motion of a continuum from K to K only with uniform temperature field D .t/ (r 0 in V ), from (3.169) and the Planck inequality we get the dissipation inequality w > 0: (3.171) Since the function w is independent of r , the inequality (3.171) holds under nonuniform heating as well.

3.5.3 The Second Thermodynamic Law in the Material Description In the material description, Eq. (3.162) takes the form Z

ı d

ı

dt

ı

Z

dV D ı

V

ı

.qm C q / ı dV

V

Z ı

ı

ı

nq ı d †;

(3.172)

†

then with the help of the usual manipulations we get the second thermodynamic law in the material description ı d

dt

ı

D r

ı

q

!

ı qm

C

C q :

(3.173)

128

3 Balance Laws

This equation can be rewritten in the form ı d

dt

ı

ı

ı

ı

D r q C qm C w ;

where

ı

q ı r > 0

ı

ı

(3.174)

w D q C

(3.175)

ı

is the dissipation function in K. The Clausius inequality (3.161) in the material (Lagrangian) description retains its form, but entropy H and entropy production QN by external sources, defined by (3.160), in Lagrangian description take the forms Z H D

Z dV D

Z V

QN D QN m C QN † ;

ı

V

QN m D

ı

ı

d V ;

qm dV D

Z ı

V

QN D

V ı

qm ı dV ;

QN † D

Z

q† d† D

†

Z

q dV D

V

Z ı

Z ı

ı

q† ı d †; (3.176)

†

ı

q ı dV :

V

3.5.4 Heat Machines and Their Efficiency The present-day heat machines are facilities, which convert heat into mechanical work (for example, internal-combustion engines, jet and turbo-jet engines, gasturbine engines, nuclear engines etc.), or, on the contrary, convert mechanical work into heat (for example, some heating and cooling devices), or use both the conversions in turn (for example, diesel engines). The operation of all the heat machines is based on the thermodynamic laws. Conventionally, the structure of a heat machine consists of a vessel with solid walls, which is filled with a working body of volume V (it is usually gas or fluid). Heat machines can be split into the three main types: (1) Machines, for which the domain V .t/ is varying but material points of a ı

working body are the same, i.e. V D const. (2) Machines, for which V remains unchanged with time but material points of a working body, occurring in V , are different. (3) Machines, for which the domain V is varying and material points occurring in V are different.

3.5 The Second Thermodynamic Law

129

Fig. 3.8 The scheme of three main types of heat machines

These three types of heat machines are shown in Fig. 3.8. The first type involves cooling and heating facilities, and also machines operating on the Carnot cycle (see below). The second type includes air-jet and rocket engines etc. The third type involves gas-turbine, turbo-jet, internal-combustion engines etc. Lagrangian description proves to be more convenient for computations of heat machines of the first type, and Eulerian description – for the ones of the second and third types. Further, we will consider heat machines only of the first type; machines of the second and third types can be considered in a similar way (see Exercise 3.5.2). All heat machines are characterized by their efficiency. To introduce the concepts of heat, mechanical work and the efficiency, we should return to the integral statements (3.126) and (3.161) of the thermodynamic laws. Consider an arbitrary time interval t1 6 t 6 t2 and integrate Eqs. (3.126) and (3.161) with respect to t from t1 to t2 ; as a result, we obtain U C A.i / D C;

(3.177)

H > CN :

(3.178)

Here we have introduced the notation U D U.t2 / U.t1 /; Zt2 A.i / D

Zt2 Z

Zt2

Zt2 Z

ı

ı

ı

V

ı

Zt2 Z

f v d V dt C t1

ı

V

(3.179)

P r ˝ vT d V dt; t1

W .t/ dt D t1

ı

W.i / .t/ dt D t1

A.e/ D

H D H.t2 / H.t1 /I

(3.180) ı

ı

tn v d † dtI

t1

ı

†

130

3 Balance Laws

Zt2 C D

CN D

Zt2 Z Q.t/ dt D

qm d V dt C ı

t1

t1

Zt2

Zt2 Z

N Q.t/ dt D

t1

Zt2 Z

ı

ı

t1

t1

V

ı

qm ı d V dt C

ı

ı

ı

†

Zt2 Z t1

V

ı

q † d † dt; (3.181)

ı

q† ı d † dt;

ı

†

where we used Lagrangian representation of the integral values U , H , W.i / , Q and QN (see (3.135), (3.136), and (3.176)). The value U is called the change of the internal energy of a body B, H – the change of the entropy of a body B, A.i / – the mechanical work done by a body B (or simply the work), C – the heat supplied to a body (or simply the heat), CN – the entropy influx to a body B, A.e/ – the work done on a body B (or the work done by external forces). ı

Let us use Lagrangian description in variables X i ; t, where X i 2 V .t/, t 2 ı

Œt1 ; t2 , and divide the four-dimensional integration domain V .t1 ; t2 / of volume ı

integrals into subdomains V t ˛ such that: (1) in each of these subdomains a sign of the scalar function qm remains unchanged, i.e. either qm > 0, or qm < 0, and (2) ı

ı

ı

[V t ˛ D V .t1 ; t2 /. In a similar way, divide the integration domain † .t1 ; t2 / ˛

ı

ı

of surface integrals into subdomains †t ˛ such that: (1) in each †t ˛ a sign of the ı

ı

ı

ı

function q † remains unchanged, i.e. either q † > 0, or q † < 0, and (2) [†t ˛ D ˛

ı

† .t1 ; t2 /. Then the heat C supplied to the body can be split into two nonnegative parts: C D CC C ; Zt2 Z C˙ D t1 ˙ qm

ı ˙ qm

Zt2 Z

ı

d V dt C

ı

t1

V

D .jqm j ˙ qm /=2 > 0;

ı q˙ †

(3.182) ı

ı

q˙ † d † dt > 0;

(3.183)

ı

† ı

ı

D .jq † j ˙ q † /=2 > 0:

(3.184)

Here CC is the heat absorbed by the body (or supplied to the body), and C is the heat released by the body (or withdrawn from the body). Since the temperature of a body is nonnegative, we can also split the entropy influx CN to the body into two nonnegative parts: CN D CN C CN ; ı ˙ ı ı Rt2 R q Rt2 R qı ˙ † m CN ˙ D d V dt C d † dt > 0: t1

ı

V

t1

ı

†

(3.185) (3.186)

3.5 The Second Thermodynamic Law

131

Substituting (3.182) into (3.177), we can rewrite the heat influx equation in the form C D CC A.i / U:

(3.187)

Assume that CC ¤ 0, then we can introduce the ratio of the mechanical work done by the body B to the heat supplied: kB D A.i / =CC ;

(3.188)

that is called the efficiency of the body B for the time interval Œt1 ; t2 considered. According to the axiom (3.154), in each body B for any time interval Œt1 ; t2 the conditions 0 < .X i ; t/ < C1 are satisfied; then we can introduce minimum and maximum values of the temperature of a body: ı

0 < min 6 .X i ; t/ 6 max < C1; 8X i 2 V ; 8t 2 Œt1 ; t2 :

(3.189)

From relations (3.189) we obtain ı

ı

ı

C C qm = > qm =max ı

ı

ı

C qC † = > q † =max ı

ı

ı

and qm = 6 qm =min ; ı

and q † = 6 q † =min ;

(3.190)

ı

C , qC because qm † , qm , q † , and are nonnegative. Substituting (3.190) into the integrals (3.183), (3.186) and taking into account that the integrands are nonnegative, we find that

CN C > CC =max ;

CN 6 C =min :

(3.191)

From formulae (3.178), (3.182), and (3.191) we get H > CN D CN C CN >

CC C : max min

(3.192)

Substitution of the expression (3.187) into (3.192) yields H > CC

1 max

1 min

C

1 .A.i / C U /: min

(3.193)

Dividing this relation by CC and multiplying by min (both the values are nonnegative), we obtain A.i / U Hmin min C C 1 6 : (3.194) CC CC max CC According to the definition of the efficiency kB (3.188), we get the final Truesdell estimate for the efficiency:

132

3 Balance Laws

kB 6 1

min U min H : max CC

(3.195)

Notice that since a sign of the third summand on the right-hand side of (3.195) is unknown, we cannot claim that in any process for each body kB < 1. This assertion, which is well known from the course of physics, holds for special processes of motion and heating of a body. Consider them in the next paragraph.

3.5.5 Adiabatic and Isothermal Processes. The Carnot Cycles One can say that in a body B there occurs a locally adiabatic process in the subdoı

ı

main V t ˛ , if the heat influx due to mass sources into V t ˛ is zero: ı

qm 0 8.X i ; t/ 2 V t ˛ :

(3.196)

ı

If in some subdomain †t ˛ the heat influx due to surface sources is zero: ı

ı

q † 0 8.X i ; t/ 2 †t ˛ ;

(3.197) ı

then one can say that a locally adiabatic process occurs in the subdomain †t ˛ of the body B. If in a body B within the time interval Œt1 ; t2 the following conditions are satisfied: ı ı ı (3.198) qm 0 8X i 2 V ; q † 0 8X i 2 †; then one can say that there occurs an adiabatic process in the restricted sense in the body B. Finally, if in a whole body B within the time interval Œt1 ; t2 the following conditions are satisfied: ı

qm 0 8X i 2 V ;

ı

ı

q 0 8X i 2 V ;

(3.199)

then one can say that there occurs an adiabatic process in the body B in the broad sense (or simply an adiabatic process). Since (3.196) and (3.197) follow from the condition (3.199) (but not conversely), one can say that if in a body B an adiabatic process in the broad sense occurs, then in this body there occurs also an adiabatic process in the restricted sense (but not conversely). One can say that in a body B there occurs a locally isothermal process in the ı

subdomain V t ˛ , if the temperature in this subdomain remains constant: ı

.X i ; t/ const 8.X i ; t/ 2 V t ˛ :

(3.200)

3.5 The Second Thermodynamic Law

133

If the temperature remains constant within a time interval Œt1 ; t2 in a whole body B: ı

.X i ; t/ const 8X i 2 V ; 8t 2 Œt1 ; t2 ;

(3.201)

then one can say that in the body B there occurs an isothermal process. One can say that in a body B there occurs a thermodynamic cycle within the time interval Œt1 ; t2 , if the following conditions of periodicity are satisfied simultaneously: U D U.t2 / U.t1 / D 0; H D H.t2 / H.t1 / D 0; CC ¤ 0;

(3.202)

ı

.X i ; t2 / .X i ; t1 / 0 8X i 2 V : Notice that not for every continuum we may ensure that the conditions (3.202) be satisfied, and, thus, not for every body a thermodynamic cycle can be realized. However, there are whole classes of bodies, for which the conditions (3.202) are satisfiable. Example 3.1. Consider a continuum B, where the specific internal energy e and the specific entropy uniquely depend only on the density and temperature : e D e.; /;

D .; /:

(3.203)

In Chap. 4 we will show that such a model of a continuum describes an ideal fluid or gas. If relations (3.203) hold, then the conditions (3.202) can easily be satisfied by giving such external heat and mechanical actions that the fields of temperature .X i ; t/ and density .X i ; t/ in the body B be the same at t D t1 and t D t2 : .X i ; t1 / D .X i ; t2 /, .X i ; t1 / D .X i ; t2 /. Indeed, in this case the following conditions are satisfied: e.X i ; t1 / D e..X i ; t1 /; .X i ; t1 // D e..X i ; t2 /; .X i ; t2 // D e.X i ; t2 /; and hence

Z U.t1 / D

ı

ı

Z

e.X ; t1 / d V D i

ı

ı

ı

e.X i ; t2 / d V D U.t2 /; ı

V

V

i.e. U D 0. The condition H D 0 can be satisfied in a similar way. A thermodynamic cycle, occurring in an ideal fluid (gas), at every material point t u X i is assigned to some closed curve on the plane .; / (Fig. 3.9). Example 3.2. Consider a continuum B, where e and are one-valued functions of temperature and the deformation gradient F: e D e.F; /;

D .F; /:

(3.204)

134

3 Balance Laws

Fig. 3.9 A thermodynamic cycle at the material point X i of an ideal fluid

ı

Since density is a function of F ( D =det F due to (3.8)), the relations (3.203) are a particular case of (3.204). If dependences (3.204) cannot be reduced to (3.203), then they correspond to the model of an ideal solid (these models will be considered in detail in Chap. 8). For an ideal solid, one can readily ensure the conditions (3.202) to be satisfied by using relations (3.204). For this, it is sufficient that the following periodicity conditions for the cycle be satisfied: ı

F.X i ; t1 / D F.X i ; t2 /; .X i ; t1 / D .X i ; t2 / 8X i 2 V : A thermodynamic cycle, occurring in an ideal solid, at every material point X i is also assigned to some closed curve in the generalized seven-dimensional space .FN ij ; t/, where FN ij are components of the tensor F with respect to the Cartesian basis eN i . t u One can say that a generalized Carnot cycle occurs in a body B, if the following conditions are satisfied simultaneously: (1) In the body B there occurs a thermodynamic cycle within Œt1 ; t2 . ı

ı

(2) The domains V Œt1 ; t2 and † Œt1 ; t2 of the body B consist only of the subı

ı

domains V t ˛ and †t ˛ , where there occur locally adiabatic or locally isothermal processes. (3) Locally isothermal processes may be only of the two types: either with the minimum temperature min and heat absorbtion: ı

in V t ˛ W ı

on †t ˛ W

.X i ; t/ D min ; qm < 0; (3.205) ı

.X ; t/ D min ; q † < 0; i

or with the maximum temperature max and heat release: ı

in V t ˛ W ı

on †t ˛ W

.X i ; t/ D max ; qm > 0; ı

.X i ; t/ D max ; q † > 0:

(3.206)

3.5 The Second Thermodynamic Law

135

Fig. 3.10 The typical picture of location of subdomains in the generalized Carnot cycle

ı

ı

The subdomains V t ˛ and †t ˛ , where there occur locally isothermal processes, have no common points (and even their closures have no common points too) due ı

to continuous differentiability of the temperature .X i ; t/ within V Œt1 ; t2 . ı

ı

Figure 3.10 shows a typical location of the subdomains V t ˛ and †t ˛ in the generalized Carnot cycle. One can say that in a body B there occurs an uniform thermomechanical process within the time interval Œt1 ; t2 , if fields of its mechanical and thermodynamic values are uniform, i.e. they are independent of coordinates X i but may depend on time t: ı

.X i ; t/ D .t/ 8X i 2 V ; 8t 2 Œt1 ; t2 ; D f ; q; F; T; P; qm ; e; ; ; q g:

(3.207)

(Notice that the velocity v and displacement u vectors of material points in an uniform process may depend on the coordinates X i .) The generalized Carnot cycle, which occurs in a body under the conditions of a uniform thermomechanical process, is called simply the Carnot cycle. The Carnot cycle, according to the definitions given above, consists only of isothermal and adiabatic processes in a body B, which alternate with each other; here within the time subinterval T0 2 Œt1 ; t2 , when the process is adiabatic in the whole body, the rate of heating Q, according to (3.198) and (3.107), is zero; within the time subinterval T 2 Œt1 ; t2 , when the temperature in the whole body reaches its minimum value min , the rate of heating Q is negative; and within the time subinterval TC 2 Œt1 ; t2 , when the temperature in the whole body has its maximum value max , the rate of heating Q is positive:

136

3 Balance Laws

Fig. 3.11 The elementary Carnot cycle

T0 W TC W T W

Q.t/ D 0;

D max ; Q.t/ > 0; D min ; Q.t/ < 0:

(3.208)

Figure 3.11 shows the elementary Carnot cycle. This cycle consists of four processes: two isothermal and two adiabatic ones. Other Carnot cycles for the same body B can differ from the elementary Carnot cycle only by the number of isothermal sections.

3.5.6 Truesdell’s Theorem Theorem 3.16a (of Truesdell). (1) If for the body B there exists a thermodynamic cycle with conditions (3.202) and given values min and max in this cycle, then the efficiency reaches its maximum possible value min kBmax D 1 < 1; (3.209) max only in generalized Carnot cycles. (2) Among all bodies B, for which the thermodynamic cycle (3.202) is possible under the conditions of an uniform thermomechanical process (3.207), the efficiency reaches its maximum value (3.209) for such bodies, in the cycle of which there is no entropy production by internal sources: QN 0

8t 2 Œt1 ; t2 :

(3.210)

H (1) To prove the first assertion of Truesdell’s Theorem we consider an arbitrary thermodynamic cycle; for this cycle, relationships (3.187) and (3.188) yield A.i / D CC C ; and hence kB D 1

C : CC

(3.211)

(3.212)

3.5 The Second Thermodynamic Law

137

As follows from (3.211) and (3.209), in order for the efficiency to reach its maximum value kB D kBmax in the thermodynamic cycle, it is necessary and sufficient that the following condition be satisfied: C =CC D min =max :

(3.213)

Since, according to (3.202), for any thermodynamic cycle H D 0, relation (3.192) with use of (3.213) takes the form CC C D 0I max min

0 > CN C CN > that is possible only if

(3.214)

CN C D CN :

(3.215)

Due to this relation, the system of two inequalities (3.191) becomes C CC CC 6 CN C D CN 6 D : max min max

(3.216)

Since the left-hand and right-hand sides of (3.216) coincide, the inequality signs in this relation should be replaced by the equality sign. Thus, the following two equations must hold: C CC D CN C ; D CN ; (3.217) max min or with use of the notations (3.183) and (3.186) Zt2 Z t1

ı

1 1 max

ı

ı

qm

1 min

1

Zt2 Z

ı

dV dt C t1

V

Zt2 Z t1

ı C qm

ı

ı

1 1 max

ı

d† dt D 0;

†

Zt2 Z

dV dt C t1

V

ı

qC †

ı

q †

1 min

ı

1

(3.218) ı

d† dt D 0:

†

Since all integrands in these equations are non-negative, the sum of integrals is zero only if the integrands vanish; i.e. the following equations must hold: 1 1 D 0; max 1 1 qm D 0; min

C qm

8X i 2 V ;

1 1 D 0; max 1 1 ı q† D 0; min ıC

q†

8t 2 Œt1 ; t2 :

(3.219)

138

3 Balance Laws

These equations are satisfied if at least one of the cofactors vanishes, i.e. if in ı

subdomains V t ˛ the following conditions are satisfied: ı

in V t ˛ W ı

qm > 0; D max ; or qm < 0; D min ; or qm D 0; ı

ı

ı

at †t ˛ W q † > 0; D max ; or q † < 0; D min ; or q † D 0:

(3.220) (3.221)

But, according to (3.205) and (3.206), this means that the thermodynamic cycle considered is a generalized Carnot cycle. Thus, we have shown that the requirement that the efficiency reach its maximum kB D kBmax in a thermodynamic cycle leads to the requirement that this cycle be a Carnot cycle, and vice versa. (2) To prove the second assertion of the theorem we use the fact established above that the efficiency reaches its maximum value at the generalized Carnot cycles; hence, under the conditions of uniform thermomechanical processes it is sufficient to prove only that the relation (3.210) holds for simple Carnot cycles. Take into account that for a thermodynamic cycle H D 0, and represent H as a sum of three terms: Zt2 0 D H D H.t2 / H.t1 / D

HP dt D

Z

HP dt C

T0

t1

Z

HP dt C

TC

Z

HP dt; (3.222)

T

where the subsets T0 , TC and T are defined by the conditions (3.208) in an uniform thermomechanical process. Within TC , according to (3.208), D max and Q > 0, then due to the Clausius inequality (3.161), we have within TC W

HP > QN D max Q > 0:

(3.223)

e C of the set TC , the Assume that within some finite time interval being a subset T strict inequality in (3.223) holds: HP > max Q. Then, integrating (3.223) over the whole TC , we obtain the strict inequality Z

HP dt > max

TC

Z Q dt D max CC :

(3.224)

TC

(The case, when the inequality HP > max Q is satisfied only at an isolated point, is excluded, because we assume that all the functions H.t/, Q.t/ and .t/ are continuous.) Due to the Clausius inequality (3.161) and (3.208), Z T

HP dt >

Z T

QN dt D min

Z

T

Qdt D min C :

(3.225)

3.5 The Second Thermodynamic Law

139

(Here we have usedR the notations (3.183) R t and (3.184), which for uniform processes reduce to C D T Qdt D .1=2/ t12 .jQj Q/dt.) Addition of two inequalities (3.224) and (3.225) yields Z

HP dt C

TC

Z

HP dt > max CC min C D 0;

(3.226)

T

because we consider the Carnot cycle, for which the conditions (3.213) are satisfied. As follows from (3.222) and (3.226), the integral over T0 must be negative: Z

HP dt < 0:

(3.227)

T0

P However, due to the R Clausius inequality (3.161), within T0 the conditions H > P N Q D Q D 0 and T0 H dt > 0 must be satisfied; therefore, the inequality (3.227) e C : HP > max Q is not true, is impossible. Thus, the made assumption that within T and from (3.223) it follows that within the whole TC the following equality must hold: HP D max Q: (3.228) TC W In a similar way, one can prove that within T W

HP D min Q:

(3.229)

Then, integrating (3.228) over TC and (3.229) over T and substituting the results into (3.222), we obtain Z 0 D H D

HP dt C max CC min C D

T0

Z

HP dt:

(3.230)

T0

Here we have taken into account once more that the condition (3.213) is satisfied for Carnot’s cycle. As noted above, due to the Clausius inequality (3.161), within T0 we have HP > 0; then (3.230) yields T0 W HP D 0 D Q: (3.231) Collecting the three relations (3.228), (3.229), (3.231), and taking (3.208) into account, we find that in the Carnot cycle with the maximum efficiency for the whole time interval considered the following relation holds: HP D Q D QN

8t 2 Œt1 ; t2 :

(3.232)

140

3 Balance Laws

This relationship, according to (3.156), means that there is no entropy production QN by internal sources, i.e. the condition (3.210) is really satisfied. N Remark 3. Since the assertion (2) of Truesdell’s theorem holds for uniform thermomechanical processes, for which, by (3.207), the gradient of a temperature field is zero in a whole body B: ı

ı

r 0 8t 2 Œt1 ; t2 ;

8X i 2 V I

(3.233)

so it follows from (3.175), (3.176), and (3.210) that the body B, satisfying the assertion (2), must be nondissipative, i.e. in this body, at least, within the interval Œt1 ; t2 the dissipation function is zero: w 0 8t 2 Œt1 ; t2 ;

ı

8X i 2 V :

(3.234)

The processes, for which in a whole body the conditions (3.233) and (3.234) are satisfied simultaneously, i.e. the specific internal entropy production q is identically zero, are often called reversible; and the processes, in which at least one of these conditions is violated, are called irreversible. t u

Exercises for 3.5 3.5.1. Using formula (3.128) and the result of Exercise 2.1.9, show that the dissipaı tion functions w and w , defined by (3.167) and (3.175), are connected by ı ı

w D .=/w : 3.5.2. For heat machines of the second and third types, by analogy with (3.179)– (3.181) introduce the concepts of H , A.i / , C and CN and prove that Truesdell’s estimate (3.195) is valid. Introduce the concepts of adiabatic and isothermal processes by analogy with the definition from Sect. 3.5.5 and also the concept of a thermodynamic cycle for Œt1 ; t2 , having replaced the last relation in (3.202) by the two ones V .x i ; t1 / D V .x i ; t2 /;

.x i ; t1 / .x i ; t2 / D 0 8x i 2 V;

(which mean the coincidence of domains V occupied by a body at times t1 and t2 and coincidence of temperature fields in the domains). Prove Truesdell’s theorem for thermodynamic cycles occurring in heat machines of the second and third types.

3.6 Deformation Compatibility Equations

141

3.6 Deformation Compatibility Equations 3.6.1 Compatibility Conditions So we have completed the formulation of balance laws of continuum mechanics. However, the conditions of continuity of the motion of bodies, which have been used in Chap. 2, can also be rewritten in the form of some formal ‘conservation law’. This formal law is of great importance for closing the equation system in continuum mechanics; thus, it will be consider in Sects. 3.6–3.8. Formulate the requirement for continuity of a motion as follows: let it be known ı

that in configuration K we can put every material point M of a continuum in one-toı one correspondence with its radius-vector x.X k / in the Cartesian coordinate system O eN i ; then we must define the conditions, under which there exists a one-valued function (the displacement vector u.X k ; t/) connecting positions of the point M in ı

K and K. If in K some discontinuities appear (cracks, pores, etc.) (Fig. 3.12), then the motion is no longer continuous. Definition 3.12. Necessary and sufficient conditions of the existence of a onevalued vector-function u.X k ; t/ for a continuum V are called deformation compatibility conditions (equations) for the continuum V . ı

If there is no one-valued vector-function u.X k ; t/ for all X k 2 V , this means that in K we cannot introduce one-valued radius-vector x.X k ; t/. Thus, the configuration K does not belong to Euclidean space E3a . The converse statement holds as well. So we have proved the following theorem. Theorem 3.17. The deformation compatibility conditions for a continuum V are equivalent to the condition that an actual configuration of the continuum belong to Euclidean space E3a . Remark. At first sight it seems strange that an usual medium with cracks does not belong to an Euclidean space. But this continuum is not quite usual, because it is

Fig. 3.12 The example of violation of the compatibility conditions

142

3 Balance Laws

defined in actual configuration K in such a way that in the reference configuration ı

K this medium is assigned to a continuum already without discontinuities. If we defined a considered continuum with cracks in another way, namely as belonging ı

e then each material point of the continto some new reference configuration K, uum would have its individual radius-vector and such a continuum would belong to Euclidean space E3a . t u

3.6.2 Integrability Condition for Differential Form There are two types of compatibility equations: dynamic and static. In this paragraph we consider static compatibility equations. To derive the equations we should consider some differential form 3 X

A˛ dX ˛ ;

(3.235)

˛D1

where A˛ D A˛ .X i / are smooth functions of variables X j . As known from the course of mathematical analysis, this form gives a total differential dA if and only if the following integrability conditions hold @Aˇ @A˛ D ; ˇ @X ˛ @X

˛; ˇ D 1; 2; 3:

(3.236)

A˛ dX ˛ :

(3.237)

In this case we have the representation dA D

3 X ˛D1

This expression can be considered as an equation in differentials. The equation can be resolved for A if and only if the conditions (3.236) are satisfied.

3.6.3 The First Form of Deformation Compatibility Conditions ı

Let A be radius-vectors x and x of the same material point M. For them the relation (3.237) is written as follows: d x D ri dX i ;

ı

ı

d x D ri dX i :

(3.238)

3.6 Deformation Compatibility Equations

143

These relations can be considered from different points of view. On the one hand, ı

if the motion law is known and in K there is a common Cartesian coordinate system, then the following relations hold ı

ı

x D x.X j /; x D x.X j ; t/;

(3.239)

which are smooth functions of their arguments. Then Eqs. (3.238) are consequences of the relations (3.239). This approach has been used above. On the other hand, if basis vectors ri are known and radius-vector x is unknown ı (by the condition, the radius-vector x is always known, and it is a one-valued function), then, similarly to the expression (3.237), the first equation of (3.238) is an equation in differentials for radius-vector x. A solution of this equation always exists, because the integrability conditions (3.236) are satisfied. Indeed, @rˇ @r˛ @x @x D D D : ˇ ˇ ˛ ˛ ˇ @X ˛ @X @X @X @X @X

(3.240)

Thus, if there exist local basis vectors at every point X i in K, then there also exist radius-vectors of these points in K. If every material point M with coordinates X i in K may be uniquely assigned to ı its radius-vector x, this means that there exist one-valued functions u.X k ; t/ D x x ı

of displacements of the points from K to K. Conversely, if there exist one-valued displacements u.X k ; t/ of material points ı

ı

from K into K, then in K we can always introduce the radius-vector x D x C u and the local basis vectors ri D @[email protected] i . Thus, we have proved the following theorem. Theorem 3.18. The deformation compatibility conditions for a continuum are satisfied if and only if in K there exist local basis vectors ri , i.e. functions having the following properties: they are linearly independent at every point X i ; they are one-valued and smooth 8X i 2 V ; they have a vector potential x: ri D @[email protected] i .

3.6.4 The Second Form of Compatibility Conditions ı

As A, we choose local bases vectors ri and ri . For them, the relation (3.237) becomes ı ı @ri ı j m j m d ri D dX j : dX D .

r / dX ; d r D

r (3.241) i ij m ij m @X j

144

3 Balance Laws

If the motion law is known (the second equation in (3.239)) and the deformation compatibility conditions are satisfied, then relations (3.241) follow from the motion law (3.239). If the Christoffel symbols ijm are known, then (3.241) are equations for functions ri . The integrability conditions (3.236) for these equations have the form (the indices j and n change places) @ @ . m rm / D . m rm /: @X n ij @X j i n

(3.242)

Removing the parentheses, we obtain @ ijm

@ imn k rm C imn mj rk ; @X j k m k m C ij kn i n kj rm D 0:

(3.243)

@ imn m m C ijk kn ikn kj D 0; @X j

(3.244)

k r C ijm mn rk D n m

@X m @ ij

@ imn @X n @X j

Since rm are arbitrary, we have m Rnji

@ ijm @X n

m is some notation for the present. where Rnji Using the condition (3.244), we can formulate the following theorem.

Theorem 3.19. The deformation compatibility conditions are satisfied if and only if in K there exist Christoffel symbols ijm satisfying the integrability conditions (3.244). H Indeed, if there exist ijm satisfying the conditions (3.244), then the integrability conditions (3.242) remain valid. Hence the first form in (3.241) has a differential and there exist local basis vectors ri . Then, according to Theorem 3.18, the deformation compatibility conditions will be satisfied. Conversely, if the compatibility conditions are satisfied, then, according to Theorem 3.18, there exist vector functions ri . Performing all manipulations (3.241)– (3.244), we verify that the conditions (3.244) hold. N ı

ı

ı

Notice that in K there always exist vectors x and ri . Hence we can perform similar manipulations for the second form in (3.241) too; as a result, we obtain the equations ı ı ı ı @ @ m m

ij rm D

i n rm (3.245) @X n @X j and

ı

ı

Rm nji

@ m ij @X n

ı

ı ı ı ı @ m k m in C kij m kn i n kj D 0: j @X

(3.246)

3.6 Deformation Compatibility Equations

145

ı

m Values Rnji and Rm nji are called components of the Riemann–Christoffel fourthı

ı

order tensors 4 R and 4 R in configurations K and K, respectively. We will verify that these are really components of tensors in paragraph 3.6.6.

3.6.5 The Third Form of Compatibility Conditions ı

Introduce the Christoffel symbols of the first kind ijk and ijk , which are connected ı

to ijm and m ij , called the Christoffel symbols of the second kind, by the relations (see [12])

ijk D gkm ijm ;

ı

ı

ı

ijk D g km m ij :

(3.247)

The Christoffel symbols of the first kind and the metric matrix are connected by the formula @gkj 1 @gi k @gij

ijk D C ; (3.248) 2 @X j @X i @X k which follows from (2.26). Interchanging indices i and k in (3.248) and summing the result with ijk , we get

ijk C kj i D

@gi k 1 C @X j 2

@gkj @gkj @gij @gij C i k k @X @X i @X @X

D

@gi k ; @X j

(3.249)

Purely covariant components of the Riemann–Christoffel tensor are defined as follows: @. ijl g ml / @. ikl g ml / m Rnjik D Rnji gmk D gmk C gsl . ijl snk inl sjk /: (3.250) @X n @X j According to the equation gmk

@gml @gmk D gml D gml . mnk C knm /; @X n @X n

(3.251)

we can rewrite Rnjik in the form Rnjik D

@ ijk @ i nk ijl . mnk C knm /g ml @X n @X j C g ml inl . kjm C mjk / C g sl . ijl snk inl sjk /

D

@ ijk @ i nk gml . inl kjm ijl knm /: @X n @X j

(3.252)

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3 Balance Laws

Substituting in place of derivatives of ijk their expressions (3.248) in terms of the metric matrix components, we get Rnjik D 2 @2 gkj @2 gij @2 gi k @2 gkn @2 gi n @ gi k 1 C C 2 @X j @X n @X i @X n @X k @X n @X n @X j @X i @X j @X k @X j C gml . inl kjm ijl knm /; or Rnjik

1 D 2

@2 gkj @2 gij @2 gi n @2 gkn C @X i @X n @X k @X n @X i @X j @X k @X j

C g ml . inl kjm ijl knm / D 0:

(3.253) ı

In the same way we can find the expression for Rnjik in terms of components of ı

the metric matrix g ij : ı

Rnjik

1 0 ı ı ı ı @2 g ij 1 @ @2 g kj @2 g kn @2 g i n A D C 2 @X i @X n @X i @X j @X k @X j @X k @X n ı ı ı ı ı C g ml inl kjm ijl knm D 0:

(3.254)

Theorem 3.19a. The deformation compatibility conditions are satisfied if and only if in K there exists a metric matrix gij satisfying the Eqs. (3.253). H If the matrix gij satisfying Eqs. (3.253) is known, then their equivalent relations (3.244) hold. As shown above, the relations (3.244) define a space to be Euclidean; and, according to Theorem 3.19, this means that the deformation compatibility conditions are satisfied too. Conversely, if the deformation compatibility conditions are satisfied, then relations (3.244) hold and in Euclidean space the coefficients ijm have a potential being gij ; thus, relations (3.248) hold too. Then their equivalent relations (3.253) remain valid as well. N

3.6.6 Properties of Components of the Riemann–Christoffel Tensor We can check that functions Rnjik are symmetric in pairs of indices n; j and i; k: Rnjik D Ri knj ;

(3.255)

3.6 Deformation Compatibility Equations

147

and also skew-symmetric in indices n; j and i; k: Rnjik D Rnjki ;

Rnjik D Rj ni k :

(3.256)

Thus, among 81 components Rnjik there are only six independent ones. They are usually chosen as follows: R1212 ; R2323 ; R3131 ; R1223 ; R1231 ; R2331 ;

(3.257)

the remaining components either are equal to zero or can be expressed in terms of ı

components (3.257). Functions Rnjik have the same properties. ı

With the help of components Rnjik and Rnjik we can set up the following fourthorder tensors: 4

R D Rnjik rn ˝ rj ˝ ri ˝ rk ;

4

ı

ı

ı

ı

ı

ı

R D Rnjik rn ˝ rj ˝ ri ˝ rk ;

(3.258)

ı

which are called the Riemann–Christoffel tensors in K and K. ı

We can verify that Rnjik and Rnjik are components of a tensor, for example, in the following way. ı ı Consider an arbitrary vector a D ak rk D ak rk and determine its covariant derivative @ak C iks as (3.259) ri ak D @X i and the second covariant derivative rj ri ak D

@ iks s @ @2 ak k k m m k .r a / C

r a

r a D C a i i m jm ji @X j @X j @X i @X j ! m @a @ak @as m s k C iks j C jkm C

a C ms as : jmi is i @X @X @X m (3.260)

Interchange indices i and j and write the difference: ! k k @

@

js m is C jkm ims ikm js as D Rj i sk as : rj ri ak ri rj ak D @X j @X i (3.261) It follows from (3.261) that Rjki s are components of a tensor, because as and the left-hand side of (3.261) are tensor components.

148

3 Balance Laws

3.6.7 Interchange of the Second Covariant Derivatives In Euclidean space E3a , the following relations have been established: ı

Rkji s D Rjki s D 0;

(3.262)

(see formulae (3.253) and (3.254)); then from (3.261) it follows that the second covariant derivatives may be interchanged. So we have proved the following theorem. Theorem 3.20. If the deformation compatibility conditions (3.253), and also (3.254) are satisfied, then the second covariant derivatives may be interchanged: ı

ı ı

ı

ı

ı

rj ri ak D ri rj ak and r j r i ak D r i r j ak :

(3.263)

This differentiation rule remains valid for tensors of any order.

3.6.8 The Static Compatibility Equation Just as in Sect. 3.6.5, we assume that in K there is a metric matrix gij satisfying ı

Eq. (3.253), and in K the continuity conditions (i.e. conditions (3.254)) are satisfied. Then in place of gij we can consider components of the deformation tensor: "ij D

1 ı .gij g ij / 2

(3.264)

and express the deformation compatibility conditions in terms of "ij . Subtracting (3.254) from (3.253), we obtain ı

Rnjik Rnjik D

@2 "kj @2 "kn @2 "i n @2 "ij C @X i @X n @X i @X j @X k @X j @X k @X n ı

ı

ı

ı

ı

Cgml . inl kjm ijl knm / g ml . inl kjm ijl knm / D 0; (3.265) where

ı

ijk D ijk C "ijk ;

"ijk D

@"jk @"i k @"ij C : j i @X @X @X k

(3.266)

The inverse metric matrix g ml can be expressed in terms of deformation tensor ı

ı

components according to formulae (2.131). Since all the functions inl and g ml in the reference configuration are assumed to be known, the relations (3.265), with

3.7 Dynamic Compatibility Equations

149

taking (3.266) and (2.131) into account, state the system of six scalar equations (because among components Rnjik there are only six independent ones) for six scalar functions "ij . These equations are called static equations of compatibility (or deformation compatibility equations). Thus, we have proved the following theorem. Theorem 3.21. The deformation compatibility conditions for a continuum are satisfied if and only if in K the deformation tensor components "ij satisfy Eqs. (3.265). Finally, let us formulate one more important theorem. Theorem 3.22. The deformation compatibility equations (3.265) have the solution ı ı ı ı ı ı ı 1 ı ı kl r i uj C r j ui C r i uk r j ul g "ij D : 2

(3.267)

Sometimes this result is said as follows: a solution of Eqs. (3.265) admits a potential, i.e. six functions "ij are expressed in terms of covariant derivatives of three ı

functions ui . H To prove the theorem we can replace the covariant derivatives by the partial ones: ı ı

ı

ı

ı

r i uj D uj= i m ij um ;

(3.268)

and, substituting the expressions (3.268) and (3.267) into Eq. (3.265), verify that this equation will be identically satisfied. However, this method is very clumsy. So let us prove the theorem in another way. Let there be a solution of Eqs. (3.265) (functions "ij ). According to Theorem 3.21, this means that the deformation compatibility equations are satisfied. Then, by ı Definition 3.12, there exists a one-valued vector-function u.X k ; t/ D x x beı ing the displacement vector of material points. By components of this vector ui , according to formulae (2.93), we can always find components e "ij , which coincide with "ij (3.267). Indeed, ife "ij ¤ "ij , then this means that there exist two distinct solutions of the deformation compatibility equations (3.265) and their corresponding, by Definition 3.12, different displacement vectors; that is impossible due to uniqueı

ness of displacements u.X k ; t/ of a continuum under the transformation from K to K. Thus,e "ij D "ij , and formula (3.267) really holds. N

3.7 Dynamic Compatibility Equations 3.7.1 Dynamic Compatibility Equations in Lagrangian Description The deformation compatibility equations can be written in one more equivalent form, namely in terms of the velocity.

150

3 Balance Laws ı

Consider Eq. (2.77) connecting F to r ˝ uT , and differentiate the equation with respect to t, taking the definition (2.181) of the velocity v into account: ı d ı @2 u @v d FT ı ıi D : r ˝ u D ri ˝ r ˝ D r ˝ v D dt @X i @t @X i dt

(3.269)

Then we get the dynamic compatibility equation in Lagrangian description: ı d FT D r ˝ v: dt

(3.270)

Theorem 3.23. The deformation compatibility conditions are satisfied if and only if in K there is a tensor field F.X i ; t/ satisfying the following conditions: det F ¤ 0 at every point X i ; F.X i ; 0/ D E at t D 0; the field F has a vector potential, i.e. there exists a vector field v such that equa-

tion (3.270) holds 8t > 0 and 8X i 2 V . H Let the deformation compatibility conditions be satisfied, then, by Definition 3.12, ı

there is a displacement vector u.X i ; t/ with its gradient r ˝ u. On making the manipulations (3.269), we verify that Eq. (3.270) remains valid. Prove that the converse assertion is valid. Let there R tbe a vector-function v satisfying Eq. (3.270). Consider the function e u.X i ; t/ D 0 v.X i ; / d . This function satisfies the equation ı

ı

Zt

r ˝e uDr˝

Zt v d D

0

Zt r ˝ v d D

0

d FT d D FT E: d

(3.271)

0 ı

Then with the help of this function we can construct the radius-vector e x D x Ce u. As a result, the tensor F takes the form ı

ı

u D ri ˝ FT D E C r ˝ e

ı

@.x C e u/ ı D ri ˝e ri ; i @X

where e ri D @e [email protected] i . But this means that the vector e u is the desired displacement vector u, and F is the deformation gradient, because they satisfy all the kinematic relations: (2.35), (2.77), etc. Thus, the displacement vector really exists u, and hence the deformation compatibility conditions are satisfied. N

3.7 Dynamic Compatibility Equations

151

3.7.2 Dynamic Compatibility Equations in Spatial Description At first, let us prove the following auxiliary statement. Theorem 3.24. If the continuity equation (3.15) holds, then the deformation gradient satisfies the following relationship: r .F/ D 0:

(3.272)

H Represent the deformation gradient in the dyadic basis: F D F ij ri ˝ rj :

(3.273)

Let us write the following formula for the divergence of a tensor (see [12]): ı

@ 1 @ p g F ij rj D p r .F/ D p i g @X g @X i

q

ı ıi

gr

:

(3.274)

q ı ı Here we have used the continuity equation D g=g, and also the evident relations ı

ı

F i k rk D F jk ri rj ˝ rk D ri F D .ri rk / ˝ rk D ri :

(3.275)

Differentiating (3.274) by parts, we get 1 0 q ı q q ıi ı q g @ B ıi ı @r C ı ı s ıi ı ı s ıi r .F/ D p @ r C g g

r g

r D 0: D p A is is g @X i @X i g ı

(3.276) Here we have used the properties of the Christoffel symbols (see [12]). N Modify Eq. (3.270) with use of (2.37): d FT D FT r ˝ v; dt

(3.277)

and use the continuity equation (3.15) multiplied by FT : @ T F C FT r v D 0: @t

(3.278)

Multiply (3.277) by and use the definition (2.187) of the total derivative with respect to time:

d FT @FT D C v r ˝ FT D FT r ˝ v: dt @t

(3.279)

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3 Balance Laws

On summing Eqs. (3.278) and (3.279), we get @FT C r .v ˝ FT / FT r ˝ v D 0: @t

(3.280)

Tensor multiplication of Eq. (3.272) by v (i.e. r .F/ ˝ v D 0) and summation of the obtained expression with (3.280) give the dynamic compatibility equation in the spatial description: @FT C r v ˝ FT F ˝ v D 0: @t

(3.281)

In order for the deformation compatibility conditions be satisfied in K, it is necessary and sufficient that Eq. (3.281), as well as (3.270), hold. Transforming the first and the second summands in (3.281) by formula (3.20), we can rewrite the dynamic compatibility equation in the form

d FT D r .F ˝ v/: dt

(3.282)

Exercises for 3.7 3.7.1. Show that the dynamic compatibility equation (3.270) can be written in the divergence form ı

d FT =dt D r .E ˝ v/: 3.7.2. Using the relation F1T F1 D E, show that the dynamic compatibility equation (3.270) can be rewritten as follows: d F1T =dt D .r ˝ v/ F1T :

3.8 Compatibility Equations for Deformation Rates Equations (3.270) and (3.281) express a condition of the existence of displacements u, when a field of the deformation gradient F is given. The deformation compatibility conditions can also be formulated as conditions for a field of the deformation rate tensor D. ı

Theorem 3.25. If a continuum does not contain discontinuities in K and the symmetric deformation rate tensor D (2.225) is given in K, then the continuum does not contain discontinuities in K if and only if the following compatibility conditions for deformation rates are satisfied:

3.8 Compatibility Equations for Deformation Rates

Ink D D 0:

153

(3.283)

Here the differential operator of incompatibility (see [12]) has been introduced as follows: Ink D D r .r DT / D .1=g/ ijk mnl ri rm Dj n rk ˝ rl ;

(3.284)

or in the component form .Ink D/kl D .1=g/ ijk ˛ˇ l .ri r˛ Djˇ ri rˇ Dj˛ /; ˛ ¤ ˇ ¤ l:

(3.285)

H Show the necessity of (3.283). If a continuum in K does not contain discontinuities, then there exist one-valued functions of coordinates: a radius-vector x, a displacement vector u and a velocity v. According to the definitions of the tensor D (2.225) and of the operator Ink (3.284), we get Ink D D

1 ijk mnl .ri rm rj vn C ri rm rn vj /rk ˝ rl D 0: 2g

(3.286)

Here we have used the property (3.263) of interchange of covariant derivatives ri rj and rm rn , and also the property of contraction of the Levi-Civita symbols with components of an arbitrary symmetric tensor (see Exercise 2.1.13). Prove the sufficiency of the conditions (3.283). Let the symmetric tensor D, satisfying the conditions (3.283), be given. Then we can write the following differential form similar to (3.235): d ! D .r D/ d x D .r D ri / dX i ;

(3.287)

which is a total differential if and only if the following integrability conditions of the type (3.236) are satisfied: @ @ .r D ri / D .r D rj /: j @X @X i

(3.288)

These conditions with use of the definition (2.31) of the tensor curl .r D/ take the form 1 1 rj p psk rp Dsi rk D ri p psk rp Dsj rk : (3.289) g g p Due to Ricci’s theorem (see [12]), the metric matrix gij and g can be placed outside the covariant derivative sign; therefore the relation (3.289) is rewritten as follows: psk .rp rj Dsi rp ri Dsj / D 0: (3.290)

154

3 Balance Laws

On comparing formulae (3.290) with (3.285), we obtain that condition (3.288) is equivalent to the condition r .r D/T D 0 (3.291) (formula (3.283)). Thus, the conditions (3.288) are always satisfied, because the condition (3.291) is assumed to be satisfied and hence the integral form (3.287) is a total differential, and ! is a one-valued function !.x/. Then the total differential can be written in the form d ! D r ˝ !T d x: (3.292) It follows from (3.287) and (3.292) that the curl of the tensor D under the condition (3.291) can always be represented as the transpose gradient of some vector !: r D D r ˝ !T :

(3.293)

With the help of the vector ! by formula (2.47) we can form the second-order tensor p W D ! E D g mi k ! m ri ˝ rk ; (3.294) which is skew-symmetric, because mi k ! m D mki ! m . With the help of D and W we can write one more differential form: d v D .D C W/ d x:

(3.295)

Similarly to the integrability conditions (3.291) for the form (3.287), the integrability conditions for the form (3.295) can be represented as follows: r .D C W/T D 0:

(3.296)

But this condition is always satisfied, because r .D C W/T D r ˝ !T r W D r ˝ !T r .! E/ D r ˝ !T C .r !/E r ˝ !T C ! r ˝ E ! ˝ r E D r ˝ !T r ˝ !T D 0: (3.297) Here we have used formula (3.294) and the property of the operator r .! E/ (see [12]), and also taken into account that r ˝ E D 0, r E D 0 and r ! D 0 due to r m ! m D r ˝ !T E D r D E 1 1 D p ijk ri Djl rk ˝ rl rm ˝ rm D p ijk ri Djk D 0: g g (3.298) Thus, the form (3.295) is a total differential, and there exists a vector v, being a one-valued function of x, hence by (2.222) we can represent its differential in the form d v D .r ˝ v/T d x: (3.299)

3.9 The Complete System of Continuum Mechanics Laws

155

On comparing (3.299) with (3.295), we get that r ˝ vT can be represented as a sum of symmetric D and skew-symmetric W tensors: r ˝ vT D D C W;

(3.300)

but this decomposition is unique, therefore by (2.225) and (2.226) we get D D .1=2/.r ˝ vT C r ˝ v/; W D .1=2/.r ˝ v r ˝ vT /;

(3.301) (3.302)

that proves the existence of the vector v, satisfying Eq. (3.301) and hence being the velocity. N As known from Sect. 2.4.5, the tensor W, defined by (3.294) and (3.302), is called the vorticity tensor, and ! – the vorticity vector.

3.9 The Complete System of Continuum Mechanics Laws 3.9.1 The Complete System in Eulerian Description Consider a nonpolar continuum. The system of continuum mechanics laws in the spatial description (3.18), (3.72), (3.118), (3.165), and (3.282) can be rewritten in the generalized form

d AN˛ D r BN˛ C C˛ ; dt

˛ D 1; : : : ; 6;

(3.303)

where the following generalized vectors are denoted: 1 1= C B v C B C B 2 =2 e C jvj C B AN˛ D B C; C B C B A @ u FT 0

0

1 v B T C B C B C T v q B C BN ˛ D B C; B q= C B C @ A 0 F ˝ v

1 0 C B f C B C B B f v C qm C C˛ D B C: B.qm C q /= C C B A @ v 0 (3.304)

Here index ˛ D 1 corresponds to the continuity equation, ˛ D 2 – to the momentum balance equation, ˛ D 3 – to the energy balance equation, ˛ D 4 – to the entropy balance equation, ˛ D 5 – to the kinematic equation, ˛ D 6 – to the dynamic compatibility equation.

0

156

3 Balance Laws

Equation (3.303) at ˛ D 5 has been obtained from (2.190) with the help of its multiplication by : du D v: (3.305) dt This relation is called the kinematic equation. The system (3.303) is called the universal system of continuum mechanics laws in total differentials. This system can be reduced to the divergence form. To do this, we can collect the corresponding divergence forms of Eqs. (3.15), (3.73), (3.119), (3.165a), and (3.281), or can transform the left-hand side of the system (3.303) with the help of the continuity equation as follows:

@AN˛ C v r ˝ AN˛ C @t

@AN˛ @ C r v AN˛ D C r v ˝ AN˛ ; @t @t ˛ D 2; : : : ; 6: (3.306)

Then together with the continuity equation, the following complete system of continuum mechanics laws in the spatial description in the divergence form holds: @ A˛ C r .v ˝ A˛ B˛ / D C˛ ; @t where

0

1 1 B C v B C B C 2 Be C jvj =2C A˛ D B C; B C B C @ A u T F

˛ D 1; : : : ; 6;

(3.307)

0

1 0 B T C B C B C BT v qC B˛ D B C: B q= C B C @ A 0 F ˝ v

(3.308)

In particular, the kinematic equation (3.305) has the divergence form @u C r v ˝ u D v: @t

(3.309)

Thus, we have proved the following theorem. Theorem 3.26. The complete system of continuum mechanics laws in Eulerian description can be represented in the universal form in total differentials (3.303) and in the equivalent divergence form (3.307).

3.9.2 The Complete System in Lagrangian Description In Lagrangian description, the system of continuum mechanics laws (3.8), (3.76), (3.131), (3.173), (2.190), and (3.270) can also be written in the universal form

3.9 The Complete System of Continuum Mechanics Laws ı ı d A˛

dt

ı

ı

ı

D r B ˛ C C˛ ;

157

˛ D 1; : : : ; 6;

(3.310)

where two more generalized vectors appear: 0

0

1 ı .=/ det F B C v B C B C ı 2 B e C jv j=2 C A˛ D B C; B C B C @ A u T F

1

0 P

B C B C B ıC ı BP v q C C: B˛ D B ı B q= C B C B C 0 @ A ı E ˝ v

(3.311)

Since in a Lagrangian coordinate system the total derivative d=dt coincides with the partial derivative @[email protected], the equation set (3.310) already has the divergence form Remark. It should be noted, that the first equation in the system (3.310) at ˛ D 1 has been obtained by differentiation of the continuity equation (3.8) with respect to Lagrangian variables. In this case we must complement this differential equation with the initial condition det F D 1. Then Eq. (3.310) at ˛ D 1 with this initial conı dition always has a solution, which is the relation (3.8): .=/ det F D 1. Hence, as ı

ı

function A1 in (3.311) we can always use its actual value: A1 D 1. This means that ı

ı

the generalized vectors A˛ in (3.311) and A˛ in (3.308) are coincident: A˛ D A˛ ; ˛ D 1; : : : ; 6: t u

3.9.3 Integral Form of the System of Continuum Mechanics Laws The system of continuum mechanics laws in the integral form (3.5), (3.69), (3.116), and (3.162) can be written in the universal form as well. Theorem 3.27. The system of continuum mechanics laws in the differential form (3.307) in the spatial description is equivalent to the integral form d dt

Z

Z A˛ dV D V

Z n B˛ d† C

†

C˛ dV;

˛ D 1; : : : ; 6:

(3.312)

V

H To prove the theorem for ˛ D 1; : : : ; 4, it is sufficient to substitute the generalized vectors A˛ , B˛ and C˛ into (3.312) and to write Eq. (3.312) for each of ˛ D 1; : : : ; 4. As a result, we obtain the relations (3.5), (3.69), (3.116), and (3.162) proved above. For ˛ D 5; 6, we should integrate Eqs. (3.309) and (3.281) over V and then apply the rule of differentiation of an integral over a moving volume (see Exercise 3.1.2) and the Gauss–Ostrogradskii theorem (3.24). N

158

3 Balance Laws

In the same way we can prove the following theorem (see Exercise 3.9.2). Theorem 3.28. The system of continuum mechanics laws in the differential form in the material description (3.310) is equivalent to the integral form d dt

Z

ıı

ı

Z

A˛ d V D ı

ı

ı

ı

n B˛ d † C ı

V

Z

†

ı

ı

C˛ d V ;

˛ D 1; : : : ; 6:

(3.313)

ı

V

In spite of the above-mentioned equivalence of differential and integral forms of the continuum mechanics laws, there is a considerable difference between them: in the integral form the angular momentum balance equation (3.82) even for nonpolar continua is not identically satisfied, while the corresponding equation in the differential form (see Sect. 3.3), reduced to symmetry of the stress tensor T, has been excluded from the system (3.303). Thus, the integral form (3.312) must be complemented with the law (3.82); in other words, the index ˛ ranges from 1 to 7 for the generalized vectors A˛ , B˛ and C˛ in (3.312): 1 0 0 1 1 1 0 0 C B B T C B C v f C B B B C C Be C jvj2 =2C BT v qC B fvCq C C B B B C m C C B B B C C A˛ D B C ; B˛ D B q= C ; C˛ D B.qm C q /= C : (3.314) C B B B C C C B B B C C u 0 v C B B B C C A @ @ F ˝ v A @ A FT 0 xv T x xf 0

The integral form (3.313) in the material description must consist of seven equaı

ı

tions as well; and the generalized vectors A˛ and B ˛ have the forms 1 ı .=/ det F C B v C B C B 2 j=2 e C jv C B ı C B A˛ D B C; C B C B u C B T A @ F xv 0

Exercises for 3.9 3.9.1. Prove Theorem 3.27. 3.9.2. Prove Theorem 3.28.

0

0 P

1

C B C B B ıC BP v q C ı C B ı C B˛ D B B q= C : C B 0 C B C Bı @ E ˝ v A P x

(3.315)

3.9 The Complete System of Continuum Mechanics Laws

159

3.9.3. Consider an arbitrary varying domain VQ .t/ V , whose boundary points Q Q †.t/ move with velocity c D d x† =dt, x† 2 †.t/ (see Sect. 5.1.6), and show with use of formula (2.12) that in this case the complete system of continuum mechanics laws in the divergence form (3.307) yields one more integral formulation d dt

Z

Z A˛ dV C

VQ

Z n ..v c/ ˝ A˛ B˛ / d† D

Q †

C˛ dV:

(3.316)

VQ

In the particular case when the domain VQ is fixed, the last formula reduces to d dt

Z

Z A˛ dV C

VQ

Z n .v ˝ A˛ B˛ / d† D

Q †

C˛ dV: VQ

(3.317)

Chapter 4

Constitutive Equations

4.1 Basic Principles for Derivation of Constitutive Equations Consider nonpolar continua. The equation system (3.307) consists of 18 scalar equations (each of the vector equations in (3.307) is equivalent to three scalar equations, and each of the tensor ones is equivalent to nine scalar equations), but involves 29 scalar unknowns: ; v; u; T; e; ; ; q; F; q :

(4.1)

The functions qm and f are usually assumed to be known when electromagnetic effects are absent. Thus, the set (3.307) is incomplete. To close the equation system (3.307) we need additional relations. These relations are called constitutive equations, because just they specify the type of a continuum (the universal balance laws do not differ types of continua, they are the same for all bodies). If constitutive equations are given in any way, then a continuum model is said to be specified. Derivation of constitutive equations is based on some additional principles, i.e. on physical assumptions of a general character, which cannot be formulated in the form of partial differential equations. We will accept the following basic principles:

the principle of thermodynamically consistent determinism the principle of local action the principle of equipresence the principle of material indifference the principle of material symmetry the Onsager principle

Besides them, other additional principles may be formulated for special models of a continuum.

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 4, c Springer Science+Business Media B.V. 2011

161

162

4 Constitutive Equations

4.2 Energetic and Quasienergetic Couples of Tensors 4.2.1 Energetic Couples of Tensors Before formulating the above-mentioned principles, let us give definitions of special stress and deformation tensors, which are of great importance for the theory of constitutive equations in continuum mechanics. Consider the power of internal forces W.i / (3.122) and introduce the stress power w.i / and the elementary work d 0 A of stresses: Z w.i / D T .r ˝ v/T ;

W.i / D

V

w.i / dV ;

d 0 A D w.i / dt:

(4.2)

ı

According to relation (2.37) between the gradients r ˝ v and r ˝ v, and also the dynamic compatibility equation (3.270), we can find one more representation for w.i / : ı

w.i / D T r ˝ vT D T .r ˝ vT F1 / D .F1 T/

dF : dt

(4.3)

Here we have used the formula of rearrangement of three tensors in the scalar product (see [12]): A .B C/ D .C A/ B D .A B/ C:

(4.4)

The following properties of contraction of two arbitrary tensors (see [12]) will be frequently used below: A B D AT BT ;

(4.5)

1 .A B C AT BT /; 2

(4.6)

A B D AS BS C AK BK :

(4.7)

ABD

Here AS , BS are symmetric tensors and AK , BK are skew-symmetric tensors: AS D .A C AT /=2;

AK D .A AT /=2:

According to the definition (3.56) of the Piola–Kirchhoff tensor and formula (4.3), we obtain the following expression for w.i / in terms of the tensor P: ı

w.i / D .=/ P

dF : dt

(4.8)

4.2 Energetic and Quasienergetic Couples of Tensors

163

The representation (4.8) of the stress power as the contraction of some stress tensor and the rate of some tensor describing deformation proves to be not unique. There are several different such representations. Since they are of great importance for derivation of constitutive equations in continuum mechanics, let us find the representations. To derive the representations, we use the definition (4.2) and formula (4.7) with substitutions A ! T; B ! L D r ˝ vT ; then we get w.i / D TS D C TK W;

(4.9)

where D is the deformation rate tensor (2.225), W is the vorticity tensor (2.226) introduced by the decomposition (2.224) of the velocity gradient L; TS and TK are symmetric and skew-symmetric parts of the tensor T: TS D

1 .T C TT /; 2

TK D

1 .T TT /: 2

(4.10)

For nonpolar continua, the tensor T is symmetric, and Eq. (4.9) takes the form w.i / D T D:

(4.11)

Theorem 4.1. There exists a denumerable set of couples of symmetric tensors .n/ .n/

. T ; C/, with the help of which the stress power w.i / can be expressed in the form d .n/ C C TK W; dt

.n/

w.i / D T

n 2 Z:

(4.12)

If the tensor T is symmetric, then the expression (4.12) reduces to the form .n/

w.i / D T

d .n/ C; dt

n 2 Z:

(4.13) .n/

.n/

Tensors T are called the energetic tensors of stresses, and C – the energetic tensors of deformations. Here Z is the set of integers, and the index n may be positive or negative integer, or zero. To emphasize the special status of the energetic couples we introduce the following notation: particular values of index n will be denoted by Roman numerals: I, II, III, IV, V etc., or I, II, III etc. The first five couples of energetic tensors I

I

II II

III III

IV IV

V V

are the most important, namely .T; C/, .T; C/, .T; C/, . T; C/ and .T; C/; they will be called principal, because these couples are most often used in solving different problems, and these energetic tensors may be expressed explicitly in terms of the stress and deformation tensors introduced before (see Table 4.1). For the symmetric tensor T, the couples at n D I, III, IV and V were derived by Hill [26], and the couple at n D II was found by K.F. Chernykh [8]. The special

164

4 Constitutive Equations

Table 4.1 Principal energetic couples of tensors Energetic deformation Energetic stress Number n .n/ .n/ of couple tensors T tensors C I FT TS F ƒ D 12 .E U2 / TS O C OT TS F/

II

1 .FT 2

III

OT TS O

IV

1 .F1 TS O C 2 1 S 1T

V

F

.n/

Energetic deformation .n/

measures G 12 G1

E U1

U1

B

G

III

OT TS F1T / U E

T F

CD

1 .U2 2

U E/

1 G 2

.n/

.n/

notation T and C and the method of ordering of these tensors, when each tensor C is expressed in terms of the corresponding power of the right stretch tensor U, were suggested by the author [12]. Prove the theorem for each of the energetic couples separately.

I

4.2.2 The First Energetic Couple .T; ƒ/ Consider the expression TS D and substitute in place of the tensor D its represenP tation (2.282) in terms of the deformation gradient rate F: DD

1 P 1 .F F C F1T FP T /: 2

(4.14)

Then .i /

wS D TS D D

1 S T .FP F1 C F1T FP T /: 2

(4.15)

Differentiation of the identities F F1 D E and FT F1T D E gives FP F1 D F FP 1 ;

F1T FP T D FP 1T FT :

(4.16)

On substituting these expressions into (4.15), we get .i /

wS D

1 S 1 T F FP 1 C TS FP 1T FT : 2 2

(4.17)

Rewrite the tensors FP 1T and FP 1 as the scalar products of themselves by the metric tensor E: FP 1T D E FP 1T D .F F1 / FP 1T;

FP 1 D FP 1 E D FP 1 .F1T FT/: (4.18)

4.2 Energetic and Quasienergetic Couples of Tensors

165

Substitution of these expressions into (4.17) yields / w.i S D

1 S 1 T .F F1 FP 1T / FT C TS .F FP 1 F1T / FT : 2 2

(4.19)

Applying the rule (4.4) of rearrangement of three arbitrary tensors in the scalar product, we obtain 1 T S 1 .F T F/ .F1 FP 1T / C .FT TS F/ .FP 1 F1T / 2 2 1 1 P 1T P 1 1T P D FT TS F CF F F F D .FT TS F/ ƒ: 2 (4.20)

/ w.i S D

Here we have used the expression (2.72) for the right Almansi deformation tensor ƒD

1 1 .E F1 F1T / D .E U2 /; 2 2

and P D ƒ

1 P 1 1T F F C F1 FP 1T : 2

(4.21)

(4.22)

Introducing the new tensor I

T D FT TS F;

(4.23)

from (4.9) and (4.20) we obtain I

P C TK W: w.i / D T ƒ

(4.24)

Thus, the representation (4.12) really exists, and the first energetic couple of tensors I

I

I

is .T; C/ D .T; ƒ/:

V

4.2.3 The Fifth Energetic Couple .T; C/ Introduce the following new tensor: V

T D F1 TS F1T

(4.25)

166

4 Constitutive Equations

and rewrite the expression (4.15) as follows: .i /

wS D D

1 1 S P .F T F C TS F1T FP T / 2 1 1 S 1T FT FP C F1 TS F1T FP T F/ .F T F 2 V

D T

V 1 T P P F F C FP T F D T C: 2

(4.26)

Here we have used the property (4.4) of the scalar product of tensors, have written the tensor TS in the form TS D TS E D TS F1T FT ;

TS D E TS D F F1 TS ;

(4.27)

and have taken account of the expression for the derivative of the right Cauchy– Green deformation tensor C (2.287): P D 1 FT FP C FP T F : C 2

(4.28)

Thus, there is one more energetic couple of tensors: V

V

V

.T; C/ D .T; C/:

IV

4.2.4 The Fourth Energetic Couple .T ; .U E// V

Rewrite the expression (4.26), passing from T to TS by formula (4.25) and from C to U by (2.158). Then we get V

/ P w.i S D TCD

1 1 S 1T P CU P U/: .F T F / .U U 2

(4.29)

Use the rules (4.4) of rearrangement of three tensors in the scalar product: / w.i S D

1 1 S 1T P C 1 U F1 TS F1T U: P UU F T F 2 2

(4.30)

On taking the polar decomposition (2.137) into account: F D O U;

F1 D U1 OT ;

F1T D O U1 ;

(4.31)

4.2 Energetic and Quasienergetic Couples of Tensors

167

we finally get / w.i S D

IV IV 1 1 S P D T C : F T O C OT TS F1T U 2 IV

Here we have introduced the fourth energetic stress tensor T: IV

TD

1 1 S F T O C OT TS F1T ; 2 IV

(4.32) IV

P coupled for which is the tensor C D UE. We have taken into account that C D U, P as E D 0. II

4.2.5 The Second Energetic Couple .T; .E U1 // Rewrite the first energetic couple (4.24), replacing the tensor ƒ by its expression (2.158): ƒ D 12 .E U2 /. Then from (4.24) we get 1 1 1 T S / P CU P 1 U1 U w.i S D F T F U 2 1 T S P 1 1 U1 FT T F U P 1 : D F T F U1 U 2 2 According to the property (4.31) of the polar decomposition, we obtain II

II

/ w.i S D TC :

Here we have introduced the second energetic stress tensor II

TD

1 T S F T O C OT TS F ; 2 II

(4.33)

II

coupled for which is the tensor .E U1 / D C, as C D .U1 / .

III

4.2.6 The Third Energetic Couple . T ; B/ We can derive one more energetic couple from Eq. (4.30), if in place of F1 and F1T substitute their polar decompositions (4.31): .i /

wS D

1 1 P C 1 U U1 OT TS O U1 U: P U OT TS O U1 U U 2 2

168

4 Constitutive Equations

Changing the order in the scalar product, we get .i /

wS D

1 T S P U1 C 1 OT TS O U1 U: P O T OU 2 2

Hence, there exists the third energetic couple III

/ P w.i S D T B; III

where T is the third energetic stress tensor: III

T D OT TS O;

(4.34)

III

and B D C is the the third energetic deformation tensor determined by its derivative and its initial value at t D 0 (see (2.317)).

4.2.7 General Representations for Energetic Tensors of Stresses and Deformations .n/

Theorem 4.2. Each of the energetic deformation tensors C can be expressed in terms of the corresponding power of the right stretch tensor as follows: .n/

CD

1 .UnIII E/; 8n 2 Z; n ¤ III; .n III/

(4.35)

.n/

and the energetic stress tensors T satisfy the equations IV

TD

nIV X .n/ 1 UnkIV T Uk ; n III

if n > IVI

(4.35a)

IIn 1 X nIICk .n/ k U T U ; III n

if n 6 II:

(4.35b)

kD0

II

TD

kD0

Here and below in such expressions at particular values of n we must replace Roman numerals by corresponding Arabic ones and then perform arithmetic operations. For example, the relation (4.35) at n D I has the form I

CD

1 1 .U13 E/ D .E U2 /: 13 2

4.2 Energetic and Quasienergetic Couples of Tensors

169

H (1) A proof of formula (4.35) for n D I, II, IV, and V is evident, if we write out the formula for these values of n and compare the obtained expressions with the .n/

tensors C from Table 4.1. .n/

.n/

(2) In order to show that the tensors C (when n > V) with tensors T also constitute .n/

energetic couples of the form (4.12), we should determine the derivative of C : .n/

C D

1 1 P UnV C : : : P UnIV C U U .U : : : U/ D .U n III „ƒ‚… n III nIII

P U C UnIV U/ P D : : : C UnV U

nIV X 1 P UnIVk : Uk U n III kD0

.n/

So, if there exists energetic stress tensor T , then it satisfies the equation .n/

.n/

T C D

nIV 1 .n/ X k P T U U UnIVk n III kD0

D

nIV X .n/ 1 UnIVk T Uk n III

! P U:

kD0

IV

Since the tensor .U E/ D C is assigned to the fourth energetic stress tensor IV

IV

.n/

T, the expression in parentheses should form the tensor T. Thus, for all T (when n > V) the formula (4.35a) really holds. While n D V, formula (4.35a) takes the form IV

TD

IV

V V 1 .U T C T U/: 2

V

Since the tensors T and T have been introduced before according to Table 4.1, we should verify the last formula. Substitution of expressions (4.32) and (4.25) into this formula yields 1 1 S 1 .F T O C OT TS F1T / D .U F1 TS F1T C F1 TS F1T U/: 2 2 This formula is valid due to the polar decomposition (2.137) as OT D U F1 . Hence formula (4.35a) really holds when n D V.

170

4 Constitutive Equations .n/

(3) To prove formula (4.35b) we determine the derivative of tensors C when n < I: .n/

C D

IIn 1 1 X 1 k .U1 „ƒ‚… : : : U1 / D .U / .U1 / .U1 /IInk : n III n III kD0

IIIn

Then .n/

.n/

T C D

IIn 1 X nIICk .n/ k U T U III n

! .U1 / :

kD0

II

II

Since the tensor .E U1 / D C is assigned to the second energetic stress tensor T, II

the expression in parentheses just forms the tensor T. Hence formula (4.35b) holds for all n < I. When n D I, formula (4.35b) takes the form II

TD

I 1 1 I .U T C T U1 /: 2

We can verify this formula by substituting expressions (4.33) and (4.23) into the last equation. N .n/

A general relation between T and T is more complicated than the relation (4.35) .n/

.n/

between C and U; however, the tensors T and TS are evident to be connected by the linear relation .n/

.n/

T D 4 E 1 TS ;

n 2 Z;

(4.36)

.n/

where 4 E 1 are the inverse tensors of energetic equivalence (fourth-order tensors), whose expressions for the principal energetic couples are given in Table 4.2. These representations should be derived as Exercise 4.2.11. The notation .FT ˝ F/.1432/ for the transpose fourth-order tensor has been introduced in Sect. 2.1.4. .n/

Expressions for the tensors of energetic equivalence 4 E can be found by inverting the relations (4.36): .n/

.n/

TS D 4 E T ; Table 4.2 Inverse tensors of energetic equivalence

n 2 Z:

(4.37)

.n/

E 1 .FT ˝ F/.1432/

Number n I

4

II III IV V

˝ O C OT ˝ F/.1432/ .O ˝ O/.1432/ 1 .F1 ˝ O C OT ˝ F1T /.1432/ 2 .F1 ˝ F1T /.1432/ 1 .FT 2 T

4.2 Energetic and Quasienergetic Couples of Tensors

171

Substituting the relationship (4.35a) into formula (4.37) where n D IV, we obtain .n/

IV

the expression for tensors 4 E (when n > V) in terms of the tensor 4 E: 4

nIV 1 4 IV X nkIV E .U ˝ Uk /.1432/ ; n III

.n/

E D

n > V:

(4.38a)

kD0

.n/

In a similar way, we can find the expression for tensors 4 E when n 6 I: 4

IIn 1 4 II X nIIk E .U ˝ Uk /.1432/ ; III n

.n/

E D

n 6 I:

(4.38b)

kD0

.n/

ı

Components of all the tensors 4 E with respect to the eigenbases pi and pi can be expressed in terms of only eigenvalues ˛ . .n/

Theorem 4.3. The tensors of energetic equivalency 4 E , connecting the tensors TS .n/

and T with the help of the linear relations (4.37), have the form 4

3 X

.n/

E D

.n/

ı

ı

E ˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

n 2 Z;

(4.38)

˛;ˇ D1

where I

E ˛ˇ D

II III IV V 2˛ ˇ 1 2 ; E ˛ˇ D ; E ˛ˇ D 1; E ˛ˇ D ; E ˛ˇ D ˛ ˇ ; ˛ ˇ ˛ C ˇ ˛ C ˇ .n/

E ˛ˇ

nIV X 2˛ ˇ D ˛nkIV kˇ ; .˛ C ˇ /.n III/

n > V;

kD0

.n/

E ˛ˇ

IIn X 2 D nIICk k ˛ ˇ ; ˛ C ˇ

n 6 I:

(4.39)

kD0

.n/

H (a) Consider the case when n D I, and represent the tensors F, O, TS and T in ı the eigenbases pi and pi (see formulae (2.169)–(2.171)): FD

3 X

ı

˛ p˛ ˝ p˛ ; FT D

˛D1

3 X ˛D1

TS D

3 X ˛;ˇ D1

.p/

3 X

ı

˛ p˛ ˝ p˛ ; O D

T˛ˇ p˛ ˝ pˇ ;

ı

p˛ ˝ p˛ ;

˛D1 .n/

T D

3 X .n/ı

T

˛;ˇ D1

.p/ ı ˛ˇ p˛

ı

˝ pˇ ;

(4.40)

172

4 Constitutive Equations .n/ı

.p/

where T˛ˇ are components of the tensor TS with respect to the basis p˛ , and T .n/

.p/ ˛ˇ

ı

are components of the tensor T with respect to the basis p˛ . On substituting these representations into formula (4.36) at n D I, we get I

TD

3 X

Iı

ı

.p/ ı

T ˛ˇ p˛ ˝ pˇ D FT TS F

˛;ˇ D1

D

3 X

ı

ı

.p/ ˛ p˛ ˝ p˛ T! p ˝ p! ˇ pˇ ˝ pˇ D

3 X ˛;ˇ D1

˛;ˇ;;!D1

.p/ ı

ı

˛ ˇ T˛ˇ p˛ ˝ pˇ :

Here we have taken into account that the basis p˛ is orthonormal (see (2.141)). Hence, Iı

.p/ T .p/ D T˛ˇ ˛ ˇ : ˛ˇ

Use once again the representation of the tensor TS in the basis p˛ and take into account that Iı

ı

I

ı

T .p/ D p˛ T pˇ ; ˛ˇ then TS D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

.p/ T˛ˇ p˛ ˝ pˇ D

3 X ˛;ˇ D1

Iı 1 p˛ ˝ T .p/ ˝ pˇ ˛ˇ ˛ ˇ

I ı 1 ı p˛ ˝ p˛ T pˇ ˝ pˇ ˛ ˇ I 1 ı ı .p˛ ˝ p˛ ˝ pˇ ˝ pˇ /.1432/ T ˛ ˇ I 1 ı ı .p˛ ˝ pˇ ˝ pˇ ˝ p˛ / T: ˛ ˇ

I

Thus, the representation (4.37) with the tensor 4 E (4.38), (4.39) really holds at n D I. (b) Consider the case when n D II. According to the representations (4.40), we can rewrite relation (4.36) for n D II as follows:

4.2 Energetic and Quasienergetic Couples of Tensors 3 X

II

TD

IIı

ı

1 T .F T O C OT T F/ 2

ı

T .p/ p ˝ pˇ D ˛ˇ ˛

˛;ˇ D1

D

1 2

3 X

173

ı

ı

.p/ .˛ p˛ ˝ p˛ T! p ˝ p! pˇ ˝ pˇ

˛;ˇ;;!D1 ı

3 1 X ı .p/ ı .˛ C ˇ /T˛ˇ p˛ ˝ pˇ : 2

ı

.p/ C p˛ ˝ p˛ T! p ˝ p! ˇ pˇ ˝ pˇ / D

˛;ˇ D1

Thus, we get IIı

.p/

T ˛ˇ D

1 .p/ .˛ C ˇ /T˛ˇ : 2

The further proof is analogous to the proof for n D I. The theorem for n D III; IV and V can be proved in the same way (see Exercise 4.2.18). (c) To prove the theorem for the cases when n > V and n 6 I, one should use formulae (4.38), (4.39) at n D II and IV, and also the expression (2.142) for the tensor U and substitute them into (4.38a) and (4.38b). .n/ı

In proving the theorem, we have established that components T .n/

ı

.p/ ˛ˇ

of the tensor

.p/

T with respect to the basis p˛ and components T˛ˇ of the tensor T with respect to the basis p˛ are connected by the relations Iı

.p/ T .p/ D ˛ ˇ T˛ˇ ; ˛ˇ IIIı

T

.p/ ˛ˇ

.p/ D T˛ˇ ;

IVı

T

.p/ ˛ˇ

D

IIı

T .p/ D ˛ˇ

1 .p/ ; .˛ C ˇ /T˛ˇ 2

1 1 1 .p/ T ; C 2 ˛ ˇ ˛ˇ

Vı

T .p/ D ˛ˇ

1 T .p/ ; ˛ ˇ ˛ˇ

(4.41)

or .n/

.p/

.n/ı .p/ ˛ˇ ;

T˛ˇ D E ˛ˇ T

8n 2 Z: N

4.2.8 Energetic Deformation Measures .n/

Besides the energetic tensors C, introduce the energetic deformation measures .n/

GD

1 UnIII ; n III

8n 2 Z;

n ¤ III;

which are the corresponding powers of the right stretch tensor U.

(4.42)

174

4 Constitutive Equations .n/

.n/

One can readily establish the relation between G and C : .n/

.n/

CDG

1 E: n III

(4.43) .n/

.n/

Since the derivative of the metric tensor is zero, the derivative tensors G and C are coincident: .n/

.n/

C D G :

(4.44)

Introduce the third energetic deformation measure III

III

G D E C C:

(4.43a)

Due to (2.317), the measure satisfies the equation III

1 dG D dt 2

dU dU ; U1 C U1 dt dt

III

G.0/ D E;

(4.43b)

and also satisfies the general equation (4.44). Due to (4.44), the relation (4.12) for w.i / takes the form .n/

.n/

w.i / D T G C TK W;

8n 2 Z:

(4.45)

If the tensor T is symmetric, then this formula reduces to the form .n/

w.i / D T

d .n/ G; dt

8n 2 Z;

(4.46) .n/ .n/

i.e. besides the couples of the energetic stress and deformation tensors . T ; C/, there .n/ .n/

are couples of the energetic stress tensors and deformation measures . T ; G/ (see Table 4.1). .n/

It should be noted that besides the tensors G we can introduce a set of tensors of .n/ P 0, which also satisfy the form G C N, where N is an arbitrary constant-tensor: N .n/

the relation (4.44). However, only the tensors G, which were already used above, have some physical meaning: I II IV V 1 1 G D G1 ; G D U1 ; G D U; G D GI 2 2

(4.47)

4.2 Energetic and Quasienergetic Couples of Tensors

175

they are the right Cauchy–Green and Almansi deformation measures, and also the .n/ .n/

.n/

right stretch tensor and its inverse. All the tensors T , C and G are symmetric. The principal energetic deformation measures for n D I; : : : ; V are given in Table 4.1. For the first time, formulae (4.35) and (4.42), and also the existence of energetic deformation measures have been derived in the systematized form by the author in [12].

4.2.9 Relationships Between Principal Invariants of Energetic Deformation Measures and Tensors In continuum mechanics, three principal invariants of a second-order tensor (see [12]) are frequently used; they are determined as follows: .n/

.n/

.n/

I1 . C/ D E C; I2 . C / D

.n/ .n/ .n/ 1 2 .n/ .I1 . C / I1 . C 2 //; I3 . C / D det. C /: 2

(4.48)

The concept of an invariant will be considered in detail in Sect. 4.8.4. .n/

Now, let us derive relations between principal invariants I˛ . C / (˛ D 1; 2; 3) of .n/

.n/

the energetic tensors C and principal invariants I˛ . G/ of the energetic deformation measures. .n/

.n/

Theorem 4.4. The principal invariants of the tensors C and G are connected by the relations .n/

.n/

3 ; n III .n/ .n/ .n/ 2 3 ; I2 . C / D I2 . G/ I1 . G/ C n III .n III/2 I1 . C / D I1 . G/

.n/

.n/

.n/

.n/

I2 . G/ 1 I1 . G/ ; I3 . C / D I3 . G/ n III .n III/2 .n III/3 .n/ .n/ 3 ; I1 . G/ D I1 . C/ C n III .n/ .n/ .n/ 2 3 I2 . G/ D I2 . C / C ; I1 . C / C n III .n III/2 .n/

.n/

.n/

.n/

I1 . C / I2 . C / 1 I3 . G/ D I3 . C / C C C ; n III .n III/2 .n III/3 n D I; II; IV; V:

(4.49)

176

4 Constitutive Equations

H To prove the theorem, one should express the first, second and third powers of the .n/

.n/

tensors C in terms of powers of the tensors G: .n/

.n/

CDG

.n/

1 E; n III

.n/

2 .n/ 1 GC E; n III .n III/2 .n/ .n/ .n/ 3 .n/2 3 1 C 3 D G3 G C G E: 2 n III .n III/ .n III/3 C 2 D G2

.n/

.n/

.n/

Write out the following invariants of the tensors: I1 . G/, I1 . G 2 / and I1 . G 3 /, and then, according to the formulae (see [12]) 1 3 I1 .T/ 3I1 .T/I1 .T2 / C 2I1 .T3 / ; 6 1 I1 .T3 / I13 .T/ C 3I1 .T/I2 .T/ ; I3 .T/ D 3

I3 .T/ D det .T/ D

(4.50)

.n/

we get the desired expressions for the first, second and third invariants of C . The derivation should be performed in detail as Exercise 4.2.10. N

4.2.10 Quasienergetic Couples of Stress and Deformation Tensors .n/

With the help of the left stretch tensor V, we can introduce couples of the tensors S .n/

and A as well. However, in this case, the stress power w.i / depends, in addition, on .n/ .n/

the derivative of the rotation tensor OT . Therefore, such couples . S ; A/ are called quasienergetic. .n/

.n/

Definition 4.1. The tensors S and A, defined by Table 4.3, are called the principal quasienergetic stress and deformation tensors, respectively. Theorem 4.5. The stress power w.i / (4.2) can always be expressed in terms of each of the quasienergetic tensor couples: .n/

w.i / D S

d .n/ ı d T A C S O C TK W; 8n 2 Z: dt dt

(4.51)

4.2 Energetic and Quasienergetic Couples of Tensors

177

Table 4.3 Principal quasienergetic couples of tensors Quasienergetic Quasienergetic deformation Number n .n/ .n/ tensors A of couple stress tensors S AD EV

V2 /

I II

V TS V 1 .V TS C TS V/ 2

III IV V

TS Y 1 1 S S 1 .V T C T V / V E 2 J D 12 .V2 E/ V1 TS V1

1 .E 2 1

Quasienergetic deformation .n/

measures g 12 g1 1 V III

g V 1 g 2

If the tensor T is symmetric, then the expression for w.i / takes the form .n/

w.i / D S

d .n/ ı d T A CS O ; dt dt

8n 2 Z:

(4.51a) ı

.n/ .n/

Here S , A are symmetric second-order quasienergetic tensors; and the tensor S is the same for all the couples, and it is called the rotation tensor of stresses: ı

SD

1 .V TS V1 V1 TS V/ O: 2

(4.52)

Let us prove the theorem for each of the principal quasienergetic couples separately.

I

4.2.11 The First Quasienergetic Couple .S; A/ Consider the first energetic couple (4.24) and pass from the tensor U2 to V2 with the help of the relations U2 D OT V2 O (see Exercise 2.3.1): I I 1 T S / S P 2 w.i S D T D D T ƒ D F T F U 2 1 P : P T V2 O C OT V P 2 O C OT V2 O D FT TS F O 2 (4.53)

According to the polar decomposition (2.137) and the rules (4.4) of rearrangement of tensors in the scalar product, we obtain / w.i S D

1 1 S P T C V TS V V P 2 V T VOO 2 P OT : CV TS V1 O

(4.54)

178

4 Constitutive Equations

Taking into account that O OT D E and, hence, P OT D O O P T; O

(4.55)

we obtain the following final expression of the form (4.51): / w.i S

1 2 ı P T DS V CSO : 2 I

I

I

(4.56) I

Here the first quasienergetic tensors S, A are defined as follows: S D V TS V ı

I

and A D A D 12 .E V2 / D 12 .E F1T F1 /, and the tensor S is defined by formula (4.52). Thus, we have proved the existence of the first quasienergetic couple I

I

I

.S; A/ D .S; A/:

II

4.2.12 The Second Quasienergetic Couple .S; .E V1 // P 2 to V P 1 in (4.55), we get Going from the derivative V ı P T 1 V TS V V1 V P 1 C V P 1 V1 w.i / D S O 2 ı 1 T P .V TS C TS V/ V P 1 : D SO 2

(4.57)

Here we have used the rule (4.4) again. As a result, we find the second quasienergetic couple: ı

II

P T; w.i / D S .E V1 / C S O

(4.58)

II

where the second quasienergetic stress tensor S has been introduced as follows: II

SD

1 .V TS C TS V/; 2

(4.59)

II

and the second quasienergetic deformation tensor is A D .E V1 /.

4.2.13 The Third Quasienergetic Couple .Y; TS / P Consider the third energetic couple (see Sect. 4.2.6) and pass from the tensor U P to V: 1 T S / 1 P P w.i C U1 U/ S D O T O .U U 2

4.2 Energetic and Quasienergetic Couples of Tensors

D

179

1 T S P T V O C OT V P OT V1 O P O C OT V O/ O T O .O 2 1 P T V O C OT V P P O C OT V O/: C OT TS O OT V1 O.O 2

Changing the order in the scalar product of tensors in each of the summands, we get .i /

wS D

1 S P T C V1 TS V P OT P C V1 TS V O .T O O 2 P T C TS V1 V P OT /: (4.60) P C TS O CV TS V1 O O

According to the relation (4.55), we see that the first and the last terms are canceled in pair. Thus, Eq. (4.60) yields ı

/ S PT P w.i S DT YCSO :

(4.61)

where we have introduced the new tensor Y similar to the tensor B, which is defined by its derivative (see (2.318)): P V1 C V1 V/; P D 1 .V P Y 2

Y.0/ D 0:

(4.62) III

III

Thus, we have proved the existence of the third quasienergetic couple: .A; S/ D .Y; TS /:

IV

4.2.14 The Fourth Quasienergetic Couple . S ; .V E// Due to (4.62), formula (4.61) takes the form / w.i S D

ı 1 1 S P TI P CSO .V T C TS V1 / V 2

(4.63)

that proves the existence of the fourth quasienergetic couple: IV

IV

ı

/ PT w.i S D S A CSO ;

where the fourth quasienergetic tensors are defined as follows (see Table 4.3): IV

SD

1 1 S .V T C TS V1 /; 2

IV

A D V E:

(4.63a)

180

4 Constitutive Equations V

4.2.15 The Fifth Quasienergetic Couple .S; J/ Rewrite the relation (4.63) as follows: / w.i S D

1 1 S P C TS V1 V P V V1 / .V T V1 V V 2 ı ı P T D V1 TS V1 1 .V V P T: P CV P V/ C S O CS O 2

Hence, there exists the fifth quasienergetic couple: ı

V

/ PT P w.i S DSJCSO ;

where the fifth quasienergetic stress and deformation tensors are defined as follows: V

S D V1 TS V1 ;

V

ADJD

1 2 1 .V E/ D .F FT E/: 2 2

(4.64)

For the first time, the principal quasienergetic couples have been derived by the author in [12].

4.2.16 General Representation of Quasienergetic Tensors .n/

Theorem 4.6. Each of the quasienergetic deformation tensors A can be expressed in terms of the corresponding power of the left stretch tensor V as follows: .n/

AD

1 .VnIII E/; .n III/

8n 2 Z;

n ¤ III;

(4.65)

nIV X .n/ 1 VnkIV S Vk ; n III

if n > IVI

(4.65a)

IIn 1 X nIICk .n/ k V S V ; III n

if n 6 II:

(4.65b)

.n/

and quasienergetic stress tensors S satisfy the equations IV

SD

kD0

II

SD

kD0

H The theorem for n DI, II, IV and V can be proved by immediate comparison of .n/

formula (4.65) with the tensors A from Table 4.3, and for n > V and n < I – by the same method as was used in the proof of Theorem 4.2. N

4.2 Energetic and Quasienergetic Couples of Tensors Table 4.4 Inverse tensors of quasienergetic equivalency

181

Number n of couple I II III IV V

4

.n/ 1

Q .V ˝ V/.1432/ 1 .V ˝ E C E ˝ V/.1432/ 2 1 .V1 ˝ E C E ˝ V1 /.1432/ 2 .V1 ˝ V1 /.1432/

.n/

The quasienergetic stress tensors S can be represented in the general form (see Exercise 4.2.17) .n/

.n/

S D 4 Q 1 TS ;

(4.66)

.n/

where 4 Q 1 are the inverse tensors of quasienergetic equivalency, whose expressions for n D I; : : : ; V are given in Table 4.4. .n/

Expressions for the tensors of quasienergetic equivalency 4 Q can be obtained by inverting the relations (4.66): .n/

.n/

TS D 4 Q S :

(4.67)

To invert them, we use the representations of the tensors in the eigenbasis pi of the left stretch tensor V, as it was made in Sect. 4.2.7. .n/

Theorem 4.7. The tensors of quasienergetic equivalency 4 Q, connecting the tensors .n/

TS and S with the help of the linear relations (4.67), have the form 4

.n/

QD

3 X

.n/

E ˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

n 2 Z;

(4.68)

˛;ˇ D1 .n/

where E ˛ˇ are expressed by formulae (4.38), (4.39). H Consider only the case when n D I; for the remaining cases a proof is analogous and should be performed as Exercise 4.2.19. .n/

Expressing the tensors S in the eigenbasis pi : .n/

S D

3 X ˛;ˇ D1

.n/ .p/ S ˛ˇ p˛

˝ pˇ ;

(4.69)

182

4 Constitutive Equations

and using the representations (4.40) for TS and (2.142) for V, we can rewrite the relation (4.66) for n D I as follows: 3 X

I

SD

I

˛;ˇ D1

S .p/ p ˝ pˇ D V TS V ˛ˇ ˛

3 X

D

3 X

.p/ ˛ p˛ ˝ p˛ T! p ˝ p! ˇ pˇ ˝ pˇ D

˛;ˇ D1

˛;ˇ;;!D1

.p/

˛ ˇ T˛ˇ p˛ ˝ pˇ ;

I

.p/ .p/ i.e. S ˛ˇ D ˛ ˇ T˛ˇ : Then

T D S

3 X

.p/ T˛ˇ p˛

˝ pˇ D

˛;ˇ D1

D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

3 X ˛;ˇ D1

I 1 p˛ ˝ S .p/ ˝ pˇ ˛ˇ ˛ ˇ

I 1 p˛ ˝ p˛ S pˇ ˝ pˇ ˛ ˇ I 1 .p˛ ˝ pˇ ˝ pˇ ˝ p˛ / S: N ˛ ˇ

Remark. It follows from the performed proof and formula (4.41) that components I

.p/

Iı

I

.p/

I

S ˛ˇ and T ˛ˇ of the tensors S and T are coincident. The same result is valid for all .n/

.n/

the tensors S and T (see Exercises 4.2.18 and 4.2.19): .n/ S .p/ ˛ˇ

.n/ı

D T

.p/ : ˛ˇ

(4.70)

4.2.17 Quasienergetic Deformation Measures .n/

Similarly to the measures G, let us introduce quasienergetic deformation .n/

measures g : .n/

g D

1 VnIII ; n III

n 2 Z;

n ¤ III;

(4.71)

being the corresponding powers of the left stretch tensor V. The third quasienergetic III

deformation measure g is introduced as follows: III

III

g D E C A:

(4.71a)

4.2 Energetic and Quasienergetic Couples of Tensors

183

Due to (4.62), the measure satisfies the equation III

dg 1 D dt 2 .n/

dV dV 1 1 ; V CV dt dt

III

g .0/ D E:

(4.71b)

.n/

The tensors g and A are connected by the relation .n/

.n/

AD g

1 E; n III

(4.72)

(when n D III the multiplier (nIII) is replaced by 1), therefore for all the couples .n/

.n/

A D g :

(4.73)

According to this relation, the power (4.51) can be expressed in another form .n/

ı

.n/

P T C TK W; w.i / D S g C S O

n 2 Z:

(4.74)

For the symmetric tensor T, this expression reduces to .n/

.n/

ı

P T; w.i / D S g C S O

n 2 Z:

(4.74a)

.n/

The tensors g at n D I, II, IV, V are the left Cauchy–Green and Almansi deformation measures, and also the left stretch tensor and its inverse: 1 I g D g1 ; 2

II

g D V1 ;

IV

g D V;

V

gD

1 g: 2

(4.75)

All the tensors are symmetric.

4.2.18 Representation of Rotation Tensor of Stresses in Terms of Quasienergetic Couples of Tensors ı

Theorem 4.8. The rotation tensor of stresses S (4.52) can be represented in terms of the quasienergetic couples of the stress and deformation tensors as follows: ı

.n/ .n/

.n/ .n/

S D . A S S A / O;

8n 2 Z;

n ¤ III:

(4.76)

184

4 Constitutive Equations

H We first consider the principal quasienergetic couples when n D I; II; IV; V. ı

The definition of the tensor S (4.52) ı

SD

1 .V TS V1 V1 TS V/ O 2

(4.77)

can be rewritten in the following four equivalent forms: ı

1 ..V TS V/ V2 V2 .V TS V// O; 2 ı 1 S D ..V TS C TS V/ V1 V1 .V TS C TS V// O; 2 ı 1 S D .V .V1 TS C TS V1 / .V1 TS C TS V1 / V/ O; 2 ı 1 2 S D .V .V1 TS V1 / .V1 TS V1 / V2 / O: (4.78) 2 SD

.n/

Comparison of these relations with the definition of the quasienergetic tensors S .n/

and A (see Table 4.3) gives the formulae ı

I I I 1 I .S .E 2A/ .E 2A/ S/ O; 2 ı II II II 1 II S D .S .E A/ .E A/ S/ O; 2 ı IV IV IV 1 IV S D ..A C E/ S S .E C A// O; 2 ı V V V V 1 S D ..E C 2A/ S S .E C 2A// O: 2

SD

(4.79) I

I

Since the scalar product by the unit tensor is always commutative: S E D E S etc.; so from (4.79) we obtain the desired relation (4.76). Let us consider now the case when n > V. Since for n D IV formula (4.76) has already been proved, we can use it and substitute relationship (4.65a) into this formula: ı

IV IV

IV IV

IV

IV

S OT D A S S A D V S S V .n/ .n/ .n/ .n/ 1 V .VnIV S C VnV S VC : : : CV S VnV C S VnIV / D n III .n/ .n/ .n/ .n/ 1 .VnIV S CVnV S VC : : : CV S VnV C S VnIV / V: nIII

4.2 Energetic and Quasienergetic Couples of Tensors

185

If we remove parentheses in this formula, all the terms are canceled there except the first and last ones; therefore, we get ı

S OT D .n/

.n/ .n/ .n/ .n/ .n/ .n/ 1 .VnIII S S VnIII / D A S S A ; n III

.n/

as E S D S E. Thus, formula (4.76) really holds for n > V. The theorem for the case when n < I can be proved in a similar way with the help of formula (4.65b). N From this theorem we can get the following significant corollary, which will be used below. ı

Theorem 4.9. The rotation tensor of stresses S is zero-tensor if and only if the ten.n/

.n/

sors S and A commutate with each other.

4.2.19 Relations Between Density and Principal Invariants of Energetic and Quasienergetic Deformation Tensors Relations between the density and principal invariants of the energetic and quasienergetic deformation tensors and measures will be frequently used below. Let us establish them. ı

Theorem 4.10. For every continuum the ratio of densities = is a one-valued function of the third principal invariant of the energetic and quasienergetic measures .n/

.n/

I3 . G/ and I3 . g /, and also a function of principal invariants of the energetic and .n/

.n/

quasienergetic deformation tensors I˛ . C / and I˛ . A /: .n/

.n/

I3 . G/ D I3 . g / D

1 ı .=/IIIn ; .n III/3

(4.80)

ı

.n/

.n/

.n/

ı

.n/

.n/

.n/

= D .1 C .n III/I1 . C/ C .n III/2 I2 . C / C .n III/3 I3 . C //1=.IIIn/ ; (4.81) = D .1 C .n III/I1 . A / C .n III/2 I2 . A / C .n III/3 I3 . A//1=.IIIn/ : (4.82) To prove formula (4.80), one should use the continuity equation (3.8) and the polar decomposition (2.137): ı

= D det F D det .O U/ D det U D det V:

(4.83)

186

4 Constitutive Equations .n/

Using the definition (4.42) of the measures G and formula (4.83), we verify that .n/

formula (4.80) for I3 . G/ is valid:

.n/

I3 . G/ D det

1 UnIII n III

D

1 1 ı .detU/nIII D .=/IIIn : 3 .n III/ .n III/3 (4.84)

Similarly, according to the definition (4.75) of quasienergetic measures and formula (4.83), we can prove the remaining relations in formula (4.80): .n/

I3 . g / D det

1 VnIII n III

D

1 1 ı .detV/nIII D .=/IIIn : 3 3 .n III/ .n III/ (4.85)

On substituting the sixth formula of (4.50) into (4.80), we obtain the relations .n/

ı

(4.81) between .=/ and I˛ . C /. Since there are connections between the invariants .n/

.n/

I˛ . g / and I˛ . A /, which are similar to (4.49) (see Exercise 4.2.8), so values of ı

.n/

.=/ and I˛ . A/ are also connected by relations (4.82) similar to (4.81). N

4.2.20 The Generalized Form of Representation of the Stress Power Consider only nonpolar continua, when the tensor T is symmetric; then TS D T

and

TK 0;

(4.86)

and, according to formulae (4.13), (4.46), (4.51a), and (4.74a), the stress power w.i / (4.2) can be expressed in the following equivalent forms: w.i / w.i /

.n/

.n/

dC D T ; dt .n/

n 2 Z;

(4.87)

n 2 Z;

(4.88)

.n/

dG ; D T dt .n/

w.i /

ı dA d OT D S CS ; dt dt

w.i /

ı dg d OT D S CS ; dt dt

.n/

.n/

n 2 Z;

(4.89)

n 2 Z:

(4.90)

.n/

The stress power w.i / is said to be written in the form An , Bn , Cn or Dn , if we use expressions (4.87), (4.88), (4.89) or (4.90), respectively.

4.2 Energetic and Quasienergetic Couples of Tensors

187 .n/

Introduce the generalized energetic stress tensors T G , the generalized energetic .n/

.n/

deformation tensors C G , the generalized energetic deformation measures G G and the generalized rotation tensor of stresses SG as follows:

.n/

TG

.n/

GG

8 .n/ ˆ ˆ T; ˆ ˆ ˆ ˆ .n/ ˆ 0 there exists a vector F .i / .B; t/, called the body inertia force, which in the inertial system O eN i for configuration K has the form KW

F .i / D d I=dt;

(4.620a)

where I is the momentum vector of the body (3.26), and the vector F .i / is transformed during the passage to K0 in the R-indifferent way: K0 W

F 0.i / D F .i / Q:

(4.620b)

(2) For any pair of bodies B and B1 8t > 0 there exists a vector of bodies interaction force F .B; B1 ; t/, which may be zero-valued and has the following properties: the vector is additive:

F .B 0 C B 00 ; B1 ; t/ D F .B 0 ; B1 ; t/ C F .B 00 ; B1 ; t/; F .B; B10 C B100 ; t/ D F .B; B10 ; t/ C F .B; B100 ; t/; where B D B 0 [ B 00 , B1 D B10 [ B100 ;

316

4 Constitutive Equations

the vector is R-indifferent during the passage to K0 :

F 0 D F QI the vector is inertial, i.e. the summarized vector of external forces acting on

the body B, F .B; t/ D F .B; B e ; t/ (where B e D U n B is the surroundings of the body B) in K is equal to the body inertia force vector: K W F .i / D F :

(4.621)

A corollary of this axiom is the following theorem. Theorem 4.53a. If Axiom 5a is assumed to be valid, then under the transformation from configuration K to K0 the momentum balance law retains its form: K0 W F 0.i / D F 0 :

(4.621a)

This theorem is an integral analog of Theorem 4.53. Axiom 5a ensures that the conditions involved in statements of Theorems 4.53 and 4.53a be satisfied. Thus, due to the acceptance of Axiom 5a, the main purpose of this section (to ensure that the momentum balance equations satisfy the principle of material indifference) has been achieved. Axiom 5 introduced in Sect. 3.2.1 is entirely contained in the generalized Axiom 5a; therefore, all the conclusions, obtained above according to Axiom 5, remain valid.

4.10.12 Material Indifference of the Thermodynamic Laws The following axiom asserts material indifference of the thermodynamic functions. Axiom 7a (Objectivity of thermodynamic functions). The thermodynamic functions U , Q, H , QN and temperature are R-indifferent scalars: U 0 D U;

Q0 D Q;

H 0 D H;

QN 0 D QN ;

0 D :

(4.622)

It follows from (4.622) that the specific internal energy and specific entropy are R-indifferent scalars: 0 D I (4.622a) e 0 D e; and from (4.622), (3.106), and (3.159) we get that the heat fluxes from mass and surface sources, and also the specific internal entropy production are R-indifferent scalars: 0 qm D qm ;

0 q† D q† ;

q 0 D q ;

(4.622b)

4.10 The Principle of Material Indifference

317

because the mass m is independent of the rigid motion and d† is always defined in the R-indifferent way. The normal n and relations (3.111) are R-indifferent (see Sect. 4.10.5), then we obtain that the heat flux vector q is R-indifferent: q0 D q Q;

(4.623)

because the following relation holds: 0 D n0 q0 D n Q q0 D n q D q† : q†

Theorem 4.54. The principle of material indifference holds for the first thermodynamic law, i.e. from the energy balance equation (3.118), written for K in the form

de dt

C v a r .T v/ C r q qm f v D 0;

(4.624)

we get the following energy balance equation in K0 : 0

de 0 dt

0 0 f0 v0 D 0; C v0 a0 r 0 .T0 v0 / C r 0 q0 0 qm

(4.625)

where a0 is defined by (4.617). H Consider transformations of separate summands on the left-hand side of equation (4.625). Due to Axiom 5a, the momentum balance equation (4.619) holds in K0 ; then, multiplying the left and right sides of this equation by v0 , we get 0 v0 a0 D v0 r 0 T0 C 0 f0 v0 ;

(4.626a)

or 0 v0 a0 r 0 .T0 v0 / C T0 .r 0 ˝ v0 /T 0 f0 v0 D 0:

(4.626b)

Due to Axiom 7a and the results of Exercises 4.10.2 and 4.10.5, we have r 0 q0 D r q;

0

de 0 de D : dt dt

(4.627)

Since the tensor T0 is symmetric (see Exercise 4.10.6), the results of Exercise 4.10.4 and Theorem 4.38 yield T0 .r 0 ˝ v0 /T D T0 D0 D T D D T .r ˝ v/T :

(4.628)

318

4 Constitutive Equations

Modify the left-side expression in (4.625) with taking formulae (4.626b), (4.627) and (4.628) into account: 0

de 0 dt

0 C v0 a0 r 0 .T0 v0 / C r 0 q0 0 f0 v0 0 qm

D

de 0 T0 .r 0 ˝ v0 /T C r 0 q0 qm dt

D

de T .r ˝ v/T C r q qm : dt

(4.629)

Since Eq. (4.624) is satisfied, the heat influx equation (3.124) being its consequence holds as well. On comparing (4.629) with (3.124), we conclude that the expression (4.629) must vanish, and therefore the equation (4.625) really holds. N Theorem 4.55. The entropy balance equation (the second thermodynamic law) satisfies the principle of material indifference, i.e. from (3.303) at ˛ D 4 we get the equation 0 0 q 0 0 0 d 0 .q C q 0 / D 0: (4.630) r dt 0 0 m H A proof of the theorem should be performed as Exercise 4.10.7. N

4.10.13 Material Indifference of the Compatibility Equations Let us verify whether equations (3.303) at ˛ D 5 and 6 always satisfy the principle of material indifference. Theorem 4.56. The kinematic equation (3.303) at ˛ D 5 and the dynamic compatibility equation (3.303) at ˛ D 6 always satisfy the principle of material indifference. ı

H Since u0 D x0 x, we have d u0 =dt D d x0 =dt D v0 ; and hence the kinematic equation is always satisfied in K0 : 0 .d u0 =dt/ D 0 v0 :

(4.631)

Let the compatibility equation (3.303) at ˛ D 6 be satisfied in K:

d FT r .F ˝ v/ D 0: dt

(4.632)

4.10 The Principle of Material Indifference

319

Consider the similar expression in K0 : 0

d FT0 r 0 .0 F0 ˝ v0 / dt d FT P 0 FT0 r 0 ˝ v0 .r 0 0 F0 / ˝ v0 : (4.633) D 0 Q C FT Q dt

Here we have used formula (4.559). With taking this equation and (4.578) into account, the third summand takes the form P 0 FT0 r 0 ˝ v0 D FT Q QT r ˝ v Q C FT Q QT Q T T P D F r ˝ v Q C F Q:

(4.634)

The fourth summand in (4.633) is zero, because r 0 0 F0 D r0i

@0 F0 @F D ri Q Q T D r F D 0: i @X @X i

(4.635)

The last equality in (4.635) is valid due to Theorem 3.24. On substituting (4.634) and (4.635) into (4.633), we finally get d FT T r . F ˝ v / D F r ˝ v .r F/ ˝ v Q dt dt d FT D r .F ˝ v/ Q D 0; (4.636) dt 0 dF

T0

0

0 0

0

because Eq. (4.632) is satisfied by condition. N

4.10.14 Material Indifference of Models An and Bn of Ideal Continua Consider the constitutive equations (4.165) and (4.167) of models An of ideal continua (both solids and fluids) in actual configuration K. Then in configuration K0 these equations, according to the principle of material indifference (4.603)–(4.605), must be written in the form 8 .n/ .n/ .n/ ˆ ˆ ˆ T 0 D [email protected] [email protected] C / F . C 0 ; 0 /; < .n/ (4.637) D . C 0 ; 0 /; ˆ ˆ ˆ : D @ [email protected]; where

.n/

and F . C 0 ; 0 / are the same tensor functions as in (4.165) and (4.167).

320

4 Constitutive Equations

But Eqs. (4.637) really hold, if the constitutive equations (4.165) and (4.167) are .n/

.n/

.n/

.n/ .n/

.n/

satisfied, because all the tensors T and C are R-invariant, i.e. T 0 D T , C 0 D C ; and the temperature 0 , due to (4.622), is also R-invariant: 0 D . For models Bn , the situation is analogous. Thus, we can formulate the following theorem. Theorem 4.57. For models An and Bn of ideal continua (both solids and fluids), the principle of material indifference is always identically satisfied for any potentials .n/

. C ; / and

.n/

. G; /.

4.10.15 Material Indifference for Models Cn and Dn of Ideal Continua The situation is different when the principle of material indifference is applied to models Cn of ideal continua. Let in configuration K the relations (4.171) and (4.174) be satisfied, then in configuration K0 , according to the principle of material indifference (4.603)–(4.605), the following equations must hold: 8.n/ .n/ .n/ ˆ ˆ S 0 D [email protected] [email protected] A 0 / ˆ. A 0 ; O0 ; /; ˆ ˆ ˆ ˆ .n/ < D . A 0 ; O0 ; /; ˆ ˆ D @ [email protected]; ˆ ˆ ˆ ˆı ı .n/ : S0 D @ [email protected]O0 ˆ. A 0 ; O0 ; /;

(4.638)

ı

where ˆ, ˆ and

are the same tensor functions as in (4.171) and (4.174). .n/

.n/

Since the tensors S and A are R-indifferent (see Theorems 4.36 and 4.37) and the tensor O is transformed during the passage from K to K0 by formula (4.568), ı

so the relations (4.638) hold if and only if the functions ˆ, ˆ, and following equations for any orthogonal tensor Q: 8 .n/ .n/ ˆ ˆQT ˆ. A ; O; / Q D ˆ.QT A Q; QT O; /; ˆ ˆ ˆ ˆ ı .n/ ı .n/ ˆ 0:

(4.728)

The value q is nonnegative due to the Planck inequality (3.158a). Axiom 16 (The Onsager Principle). In order for the Planck inequality to hold, the specific internal entropy production q must be representable in the quadratic form q D

X

Qˇ Xˇ D

ˇ D1

X

Lˇ Xˇ X > 0:

(4.729)

ˇ; D1

Here Qˇ D

X

Lˇ X

(4.730)

ˇ D1

are functions called thermodynamic fluxes, and Xˇ are functions called thermodynamic forces. As follows from (4.729), the matrix Lˇ is symmetric: Lˇ D Lˇ :

(4.730a)

Remark. As shown in Sect. 4.4.3, the principles of thermodynamically consistent determinism and equipresence result in the fact that all the active variables ƒ included in constitutive equations (4.158)–(4.161) are independent of the temperature gradient r . Hence, the dissipation function w being among the active variables is independent of r as well. Thus, if we assume r 0 8M 2 V and 8t > 0, then

340

4 Constitutive Equations

the value of w remains unchanged and the summand q r vanishes; and from (4.728) we obtain w > 0:

(4.731)

This inequality is known (see (3.171)) as the dissipation inequality. The reasoning has proved the fact that the dissipation inequality is a consequence of the Planck inequality (4.728) and principles of thermodynamically consistent determinism and equipresence. For ideal continua, w 0, then the Planck inequality (4.728) yields q r 6 0;

(4.732)

this relation is called the Fourier inequality. For nonideal continua, the Fourier inequality (4.732) does not follow from the Planck inequality (4.728) and it should be assumed to be an independent axiom (see Axiom 10 in Sect. 3.5.2). t u The Onsager principle P is applied as follows: expression (4.728) for q is represented as the sum ˇ Qˇ Xˇ ; for example, for ideal continua we choose the temperature gradient X1 D r as a thermodynamic force X1 and the value Q1 D .1=/q as a corresponding thermodynamic flux. Then, according to the Onsager principle, the thermodynamic flux must be a linear function of X1 , i.e.

1 q D Q1 D L11 X1 D L11 r :

(4.733)

The coefficient L11 in this case is a second-order tensor, and the product L11 D is called the heat conductivity tensor; then relation (4.733) takes the form q D r ;

(4.734)

which is called the Fourier law in the spatial description. Due to the Fourier inequality, the heat conductivity tensor is positive-definite: r r > 0:

(4.735)

The Fourier law can be formulated in the material description as well. To do this, ı one should replace the vector q in (4.734) by the vector q according to formula ı

(3.128) and the gradient r by r according to the formula similar to formulae ı

(2.23): r D F1T r . Then we obtain the relation ı

ı

ı

q D r ;

(4.736)

4.12 The Onsager Principle

341

ı

where is the heat conductivity tensor in a reference configuration: q ı D g=g F1 F1T : ı

(4.737)

The Fourier law (4.734) is included in the general system of constitutive equations (4.158), (4.159), (4.160) or (4.161) of a continuum. This law can also be written in the form of operator relation (4.156), where the vector q is an active variable and the vector r is a reactive one. Then, depending on the considered model An , Bn , Cn or Dn , the heat conducı

tivity tensors and should be assumed to be operators of the corresponding set of reactive variables R: .n/

M C ; /; models An W D . .n/

M G; /; models Bn W D . .n/

M A ; O; /; models Cn W D . .n/

M g ; O; /; models Dn W D .

ı ıM .n/ D . C; ı ıM .n/ D . G; M .n/ ı ı D . A ; O; ı ıM .n/ D . g ; O;

/; /; /; /:

(4.737a)

4.12.2 Corollaries of the Principle of Material Symmetry for the Fourier Law ı

Consider the Fourier law (4.736) in a reference configuration K with use of one of the models An ; : : : ; Dn ; for definiteness, we choose the model Bn . Since the Fourier law as well as the constitutive equations (4.158)–(4.161) describes some physical properties of a continuum, so this law must satisfy the principle of material ı

ı

symmetry (4.180)–(4.181) if the vector q is considered as ƒ and the vector r – as ı

R. In other words, if Eq. (4.736) holds in a configuration K, then there exists a symı

ı

metry group G s such that for each H -transformation from this group (H W K ! K) the form of function (4.736) remains unchanged: ı

ı .n/

ı

q D . G; / r :

(4.738a)

and

ı .n/

q D . G ; / r :

(4.738b)

342

4 Constitutive Equations

Theorem 4.63. The principle of material symmetry (4.738a), (4.738b) holds for the ı

Fourier law in the material description if and only if in K there exists a symmetry ı

group G s such that the heat conductivity tensor is transformed as follows: ı .n/

. G; / D

q ı .n/ ı ı g=g H1 . G ; / H1T 8H 2 G s :

(4.739)

H Since the vector q and temperature are defined in K, they are H -invariant. Then ı

ı

q and r under H -transformations change as follows:

q

qD

1

g=g F

r D ri

q q ı 1 q D g=g H F q D g=g H q;

ı @ i ı ı k @ 1T D r ˝ r r D H r : i @X i @X k

(4.740)

Let the principle of material symmetry be valid, then relations (4.738a) and (4.738b) hold. On substituting (4.740) into (4.738b), we obtain

or

q ı ı ı ı g=g H q D H1T r ;

(4.741)

q ı ı ı q D g=g .H1 H1T / r :

(4.742)

ı

Comparing (4.738a) with (4.742), we see that in this case the relation (4.739) ı

must hold for the heat conductivity tensor . Prove the converse assertion. Let the condition (4.739) be satisfied. Substituting the condition into (4.736), we obtain formula (4.742). Performing the transformations in the reverse order, we get formulae (4.738a) and (4.738b). N Theorem 4.63 still holds for the remaining models An , Cn and Dn . For them the relation (4.739) has the forms q ı .n/ ı g=g H . C ; / HT ; q ı .n/ ı .n/ ı . A ; O ; / D g=g H . A ; O; / HT ; q ı .n/ ı .n/ ı . g ; O ; / D g=g H . g ; O; / HT : ı .n/

. C ; / D

(4.743)

Theorem 4.64. The Fourier law (4.734) in the spatial description identically satisfies the principle of material symmetry for each symmetry group Gs . ı

H Indeed, for any H -transformation of a reference configuration K ! K, since the vectors q and r are always H -invariant, relation (4.734) retains its form with the same heat conductivity tensor for each tensor H. N

4.12 The Onsager Principle

343 ı

Certainly, there is no discrepancy between relation (4.739) for the tensor durı

ing the passage from K to K and invariability of the tensor , because formula (4.739) together with (4.737) and formula (4.188) of transformation of the deformation gradient F ensure that the tensor be independent of the choice of reference

configuration K: D

q q q ı ı ı ı g=g F FT D g=g F H HT FT D g=g F FT D ./ :

Here ./ is the tensor defined in configuration K, which, as it was expected,

coincides with the tensor itself (it is not to be confused with , which is the ı

tensor defined in configuration K).

4.12.3 Corollary of the Principle of Material Indifference for the Fourier Law ı

ı

The heat flux vector q and temperature gradient r are defined in a reference conı

figuration K, therefore they remain unchanged at rigid motions from K to K0 (i.e. ı

ı

ı

ı

q0 D q, .r /0 D r ). Hence, the Fourier law (4.736) in the material description also remains unchanged under the transformation from K to K0 : ı

ı

ı

q0 D .r /0

(4.744)

for any orthogonal rotation tensor Q. This means that relation (4.736) of the Fourier law satisfies the principle of material indifference (4.603), (4.605) with the same ı

heat conductivity tensor , i.e.

ı

ı

0 D :

(4.745)

For the Fourier law (4.734) in the spatial description, the situation is different. Theorem 4.65. The Fourier law (4.734) in the spatial description satisfies the principle of material indifference (4.603), (4.605) (i.e. the relation q0 D .r /0

(4.746)

follows from (4.734) for any rigid motion K ! K0 ) if and only if the heat conductivity tensor is both R-invariant and R-indifferent: 0 D ;

0 D QT Q:

(4.747)

344

4 Constitutive Equations

Remind that tensors satisfying the condition (4.747) for any orthogonal tensor Q, according to (4.290), are called indifferent relative to the full orthogonal group. H The heat flux vector q is R-indifferent (just as the tensor T). To prove the fact, one should use relation (3.111) and take into account that the scalar function of heat 0 influx due to surface sources q† is R-invariant: q† D q† , and the normal vector n is R-indifferent, then 0 q† D q0 n0 D q0 QT n D q† D q n:

(4.748)

q0 D QT q:

(4.749)

Thus, we obtain

The gradient of a scalar is also R-indifferent: .r /0 D ri 0

@ @ D Q T ri D QT r : @X i @X i

(4.750)

Multiplying the relation (4.734) by QT , we obtain QT q D QT r D QT Q QT r ;

(4.751)

q0 D 0 r 0 ;

(4.752)

0 D QT Q:

(4.753)

or

As follows from (4.752), the principle of material indifference holds if and only if the tensor is indifferent, i.e. 0 D . N Notice that relations (4.747) are consistent with (4.745): substituting formulae (4.747), (4.559), and (4.560) into (4.737), we obtain q q ı ı 10 ı 0 0 1T0 D g=g F F D g=g F1 Q 0 QT F1T q ı ı D g=g F1 F1T D ; (4.754) that is exactly the relation (4.745).

4.12.4 The Fourier Law for Fluids For fluids, one usually use the Fourier law in the spatial description (4.734). According to Theorem 4.64, the relation (4.734) for any tensor satisfies idenı

tically the principle of material symmetry for each symmetry group G s , including ı

the unimodular group G s D U corresponding to a fluid.

4.12 The Onsager Principle

345

To satisfy the principle of material indifference, the tensor in (4.734) must be indifferent relative to the full orthogonal group Gs D I , i.e. the relations (4.747) must hold. As shown in Sect. 4.8.3 (see (4.293)), Eq. (4.747) for the group I has an unique solution being the spherical tensor : D E;

> 0;

(4.755)

where is the constant called the heat conductivity coefficient. Substitution of (4.755) into (4.734) gives the Fourier law for fluids q D r :

(4.756) ı

With the help of Eq. (4.737) we can write the heat conductivity tensor for fluids as follows: q q ı ı ı D g=g F1 F1T D g=g G1 : (4.757) This tensor is not spherical, but it satisfies Eq. (4.739). The Fourier law (4.736) in the material description for fluids has the form q ı ı ı q D g=g G1 r :

(4.758)

4.12.5 The Fourier Law for Solids For solids, one usually apply the Fourier law in the material description (4.736). ı

As noted in Sect. 4.12.3, for any tensor this relation satisfies identically the principle of material indifference. To satisfy the principle of material symmetry, by ı

ı

Theorem 4.63, the tensor must be indifferent relative to a group G s corresponding to the symmetry group of a considered solid, i.e. the relation (4.739) must hold. According to the results of Sect. 4.8.3 (see formulae (4.293)), we find that: ı

ı

if G s D I (an isotropic solid), then the tensor has the only independent comı

ponent :

ı

ı

D E;

(4.759)

ı

b3 (a transversely isotropic solid with the transverse isotropy vectorb if G s D T c3 ), ı

ı

ı

then the tensor has two independent components – 1 and 2 : ı

ı

ı

c23 C 2 E; D 1b

(4.760)

346

4 Constitutive Equations ı

b (an orthotropic solid with the principal orthotropy basis b c ), then the if G s D O ı

ı

tensor has three independent components – : ı

D

3 X

ı

c2 : b

(4.761)

D1

To find the tensor for solids one should substitute formulae (4.759)–(4.761) into (4.737). For example, for an isotropic continuum the tensor has the form q D

q ı ı ı ı g=g F FT D g=g F FT :

(4.762)

On substituting expressions (4.759)–(4.761) into (4.736), we get the Fourier law for isotropic solids ı

ı

q D E;

(4.763)

the Fourier law for transversely isotropic solids ı

ı

ı

q D .1b c23 C 2 E/;

(4.764)

and the Fourier law for orthotropic solids ı

qD

3 X D1

ı

b c2 :

(4.765)

Chapter 5

Relations at Singular Surfaces

5.1 Relations at a Singular Surface in the Material Description 5.1.1 Singular Surfaces Up to now we considered the case when all functions appearing in the balance laws: , u, v, T, F, f etc. are continuously differentiable functions of coordinates X i (or of x i ) and time t. However, in practice one often meets with problems, where this condition is violated. For example, for the phenomena of impact, explosion, combustion etc., a part of the indicated functions in a domain V considered can suffer a jump discontinuity across some surface S (Fig. 5.1). As a singular surface we can consider an interface S between two contacting media (Fig. 5.2). Besides differential equations in a domain V , mathematical formulation of problems in continuum mechanics should involve conditions on the surface bounding the domain V (boundary conditions). This boundary can also be considered as a singular surface for the functions. The integral balance laws still hold for a volume V containing a singular surface S ; but differential equations of the corresponding laws are not valid across such surface S , because in deriving the equations we have essentially used the condition of continuous differentiability of the functions inside V . Therefore, we need to establish appropriate consequences of the balance laws for the case when there is a singular surface for the functions. This is a purpose of the chapter.

5.1.2 The First Classification of Singular Surfaces Consider a continuum occupying a volume V in K. We assume that there is a surface S dividing the volume V into two subdomains VC and V such that inside VC and V all functions (, v, F etc.) considered are continuously differentiable. However, during the passage from VC to V across S some of the functions can suffer a jump discontinuity. Consider the set Mt of material points belonging to the singular Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 5, c Springer Science+Business Media B.V. 2011

347

348

5 Relations at Singular Surfaces

Fig. 5.1 The singular surface S in domain V

Fig. 5.2 The interface S between two media VC and V

Fig. 5.3 There is no transition of material points across a singular surface ı

surface S.t/ at a time t > 0. This set Mt in a reference configuration K belongs to ı

ı

a surface S.t/, which, in general, depends on t and divides V into two subdomains VC and V . At different times t (t1 6 t 6 t2 ), positions of the singular surface S.t/ in the actual configuration may be distinct, and there is an alternative: Sets Mt , corresponding to surfaces S.t/ 8t 2 Œt1 ; t2 , coincide Not all sets Mt , corresponding to surfaces S.t/ at considered times t (t 2 Œt1 ; t2 ),

are coincident In the first case, one can say that material points do not cross the singular surface S.t/ 8t 2 Œt1 ; t2 ; in the second case, there occurs a transition of material points across the singular surface. If there is no transition of material points across the interface S.t/, 8t 2 Œt1 ; t2 , ı

ı

then the position of the surface S .t/ in K 8t 2 Œt1 ; t2 remains unchanged (Fig. 5.3). While material points cross a singular surface at t 2 Œt1 ; t2 , the location of the ı

surface S.t/ in the reference configuration varies (Fig. 5.4). Let us give now a classification of singular surfaces (Fig. 5.5).

5.1 Relations at a Singular Surface in the Material Description

349

Fig. 5.4 Material points cross a singular surface

Fig. 5.5 Classification of singular surfaces

Definition 5.1. A singular surface S.t/, across which unknown functions , u, v, T, F etc. suffer jump discontinuities, is called a surface of a strong discontinuity. If these functions remain continuous across S.t/ and only their first derivatives with respect to time t and coordinates (r , r ˝u, r ˝v etc.) suffer jump discontinuities, then S.t/ is called a surface of a weak discontinuity. The transition of material points can occur only across surfaces of a strong discontinuity. Definition 5.2. A surface of a strong discontinuity S.t/, across which there is no transition of material points 8t 2 Œt1 ; t2 , is called a surface of a contact discontinuity if constitutive equations in domains VC .t/ and V .t/ are the same at all times t considered; otherwise, it is called a surface of contact. Definition 5.3. A surface of a strong discontinuity S.t/, across which there is a transition of material points at t 2 Œt1 ; t2 , is called a surface of a shock wave if constitutive equations are the same at all times t considered; otherwise, S.t/ is called a surface of phase transformation. Besides the classification introduced above, singular surfaces can be split into coherent and incoherent ones.

350

5 Relations at Singular Surfaces

Definition 5.4. A singular surface S.t/, across which the radius-vectors x of material points M 2 S.t/ suffer jump discontinuities, is called incoherent; otherwise, the surface is called coherent. For an incoherent surface S.t/, a local neighborhood dV of every point M, belonging to the surface S.t/ at some time t, at the time t C t is divided into two half-neighborhoods dVC and dV , moving apart one from another, because if ı

ı

in K the radius-vector x.M/ of the point M is continuous, then the radius-vector x.M; t C t/ of this point is no longer continuous: ı

Œx D 0;

Œx ¤ 0:

(5.1)

For a coherent surface S.t/, local half-neighborhoods dVC and dV of its every point M 2 S.t/ at any time t do not move apart one from another, and the jump of ı the radius-vector x is zero if Œx D 0: ı

Œx D 0;

Œx D 0:

(5.2)

The example of an incoherent surface is a surface of contact (an interface) between an ideal fluid and a solid: an ideal fluid slips along a solid without grip. If at time t we choose two contacting points MC and M of an incoherent surface S.t/, then their radius-vectors xC and x coincide but the radius-vectors ı

ı

ı

xC and x in K may be not coincident, i.e. instead of (5.1) the following relations may hold ı

Œx ¤ 0;

Œx D 0:

It is evident that an incoherent surface is a surface of a strong discontinuity.

5.1.3 Axiom on the Class of Functions across a Singular Surface ı

ı

Let us consider now a surface of a strong discontinuity S .t/ dividing a volume V in ı

ı

ı

K into two subdomains V C and V . ı

ı

Definition 5.5. Let there be a function A.x; t/ defined in a domain V which conı

tains a singular surface S . The value ˇ ˇ ŒA D AC ˇ ı A ˇ ı ; S

S

(5.3)

5.1 Relations at a Singular Surface in the Material Description

where

ˇ A˙ ˇ ı D S

ı

351

ı

A˙ .x; t/;

lim ı

x!x† ı ı ı ı x2V ˙ ; x † 2S ı

is called the jump of the function across the surface S . If a surface S.t/ is coherent, then from (5.2) it follows that the functions ı

x i .X i ; t/ do not suffer jump discontinuities across the surface S .t/: ˇ ˇ x i .X j ˇ ı ; t/ D x i .X j ˇ ı ; t/: SC

S

ı

Then in (5.3) we can always pass from coordinates x i to x i and consider the jump of a function across a corresponding singular surface S.t/ in K: ˇ ˇ ŒA D AC ˇS A ˇS ; where

ˇ A˙ ˇ S D

lim

i x i !x† x i 2V˙ ; x i 2S.t / †

(5.4)

A.x i ; t/:

ı

ı

Axiom 17. For a continuum containing in K a singular surface S.t/, which divides ı

ı

ı

ı

ı

ı

the volume V into two parts V C and V , the functions , A˛ and B ˛ appearing in ı

ı

the balance equation system (3.310) are assumed to be smooth in V C and V : ı

A˛ D

8ı 0; and their solution is written as follows: D .x/;

F D F.x/;

u D u.x/:

(6.39)

Here the right-hand sides of the relations can be found after solving the system (6.35), (6.36).

Exercises for 6.1 6.1.1. Show that for the quasilinear models AI (see Remark 1 or Sect. 4.8.7) the constitutive equations (6.8a) have the explicit analytic form similar to (6.16b): T D F1T .4 M ƒ/ F1 ; ƒD

1 .r ˝ u C r ˝ uT r ˝ uT r ˝ u/; 2

F1 D E r ˝ uT ;

or T D .E r ˝ u/

4

M

1 E r ˝ uT r ˝ u .E r ˝ uT /: 2

6.1.2. Using formula (4.322), show that for isotropic continua the relations (6.9a) can be written as follows: T D F1T . 1

C 2ƒ C D '1 C '2 I1 C '3 I2 ; 3

D '3 ;

/ F1 ; D '2 C I1 '3 ;

3ƒ

1E

2

2

I˛ D I˛ .ƒ/; ˛ D 1; 2; 3:

Using formula (4.337), show that for the linear model AI of isotropic continua this relation takes the form T D J.l1 I1 .ƒ/F1T F1 C 2l2 F1T ƒ F1 /;

390

6 Elastic Continua at Large Deformations

or

1 T T D J l1 r u r ˝ u r ˝ u .E r ˝ u/ .E r ˝ uT / 2 1 C 2l2 .E r ˝ u/ .E r ˝ uT / r ˝ u .E r ˝ u/T ; 2 where l1 and l2 are the constants. 6.1.3. Show that constitutive equations (6.16b) for the model AI in basise ri have the component form eij D T

r X D1

e I .s/ ms

D

ems ; @I.s/ [email protected]ƒ

.s/ e 1 /m .F e 1 /se ' e g i ke g jl .F k l I ms ;

' D [email protected] [email protected].s/ /;

D

e ms /; /; .I.s/ .ƒ

ese eke eke eke eme e 1 /m D ı m r e ms D 1 .r us C r um r um r us /; .F um : ƒ k k 2 Using the result of Exercise 6.1.1, show that for the quasilinear model AI this component representation takes the form e e 1 /p .F e 1 /q f ems ; T ij D e g i ke gjl .F k l M pqms ƒ and for the model AI of isotropic continua – the form e 1 /p .F e 1 /q eij De g i ke g jl .F T k l

g pq C 1e

e ms C g pme g qs ƒ 2e

mu esv e e g e g e g : ƒ ƒ 3 pm qs uv

6.2 Closed Systems in the Material Description 6.2.1 U VF -system of Dynamic Equations of Thermoelasticity in the Material Description For the material description, we should consider the balance law system (3.310) having the explicit form ı

D det F1 ; ı

ı

(6.40a) ı

[email protected][email protected]/ D r P C f; ı

ı

ı

ı

[email protected][email protected]/ D r q C qm ;

(6.40b) (6.40c)

ı

@FT [email protected] D r ˝ v;

(6.40d)

@[email protected] D v:

(6.40e)

6.2 Closed Systems in the Material Description

391

Here we have used the entropy balance equation (3.174) in place of the energy balance equation. The system (6.40) as well as (6.1) becomes closed after complementing it with the constitutive equations consisting of the two equations ı

ı

ı

q D r ;

D @ [email protected];

(6.41)

and the universal constitutive equations (6.3) and (6.4) written for the Piola– Kirchhoff tensor: .n/

P D F ıG .F; /;

.s/

D

.n/

.I G . C G .F//; /

.F; /:

(6.42)

Here we have introduced the tensor function .n/ F ıG .F; /

.n/

ı

.=/F1 F G .F; /:

(6.43)

The system (6.40)–(6.42) contains 16 scalar unknowns (the density is assumed to be expressed in terms of F): ; u; v; F k X i ; t;

(6.44)

which are functions of Lagrangian coordinates X i and time t, and consists of 16 scalar equations (after substitution of (6.41) and (6.42) into (6.40)). This system is called the U VF -system of thermoelasticity in the material description. For solids, this system proves to be more preferable, because a domain of defı

inition of the functions (6.44) is known: this is V Œ0; tmax . The exceptions are ı

problems with phase transformations, where V varies with time and is determined in the process of solving. However, in this case the system (6.40)–(6.42) also proves to be more preferable than the corresponding system (6.1)–(6.6) in the spatial description. .n/

Remark 1. Although the definition (6.43) of the tensor function F ıG includes F1 , this function can be represented as a function of F. To do this, we should use the representation of F1 in the eigenbasis (see (2.171)) and the expression (6.4) of the .n/

function F G ; then we obtain .n/

P D F ıG .F; / D

r X D1

4

.n/

.n/

E ıG D F1 4 E G D

3 X

ı

ı

ı

.n/

' 4 E ıG I.s/ G; ı

ı

E ˛ˇ p˛ ˝ pˇ ˝ .hG pˇ ˝ p˛ C .1 hG /pˇ ˝ p˛ /;

˛;ˇ D1 ı

(6.45)

E ˛ˇ D E˛ˇ =˛ ;

ı

ı

' D .=/' ; G D A; B; C; D:

392

6 Elastic Continua at Large Deformations .n/

.s/

The form of the tensors I G coincides with (6.9c). The tensors 4 E ıG are called Lagrangian tensors of energetic equivalence; they connect the Piola–Kirchhoff .n/

stress tensor to T G :

.n/

.n/

P D 4 E ıG T G :

(6.46) .n/

.n/

For example, choosing the tensor functions T G D F G . C G ; / for the models An .n/

in the form (6.8), we obtain tensor function F ıG in the form .n/

.n/

.n/

.n/

.n/

P D 4 E ı .4 M C C C 2 C 6 L . C ˝ C //; .n/

(6.47)

.n/

where 4 E ı D F1 4 E G . Notice that although the equation system (6.40) contains the Piola–Kirchhoff stress tensor P, the final (after solving the system (6.40) with boundary and initial conditions) analysis of a stress field in a solid usually needs the Cauchy stress tensor T to be known; therefore, besides constitutive equations (6.45), the material description involves the relations (6.4). t u Remark 2. Although the tensor function (6.45) has a rather complex form, which in general cannot be expressed analytically, this form may be simplified for two exceptional models AV and BV , because ı

PD

ı

ı

V 1 V F T D F1 .F T FT / D T FT :

(6.48)

V

On substituting the expression (6.4) for the tensor function T D F A .F; / into (6.48), we obtain the following tensor function having the explicit analytical representation: V

P D F ıA .F; / D

r X

T I.s/ F ;

(6.49)

D1 ı

ı

' D [email protected] [email protected].s/ /;

D

.s/ .I.s/ .C/; /; I.s/ D @I [email protected];

C D .1=2/.FT F E/: V

V

The substitution C ! G in (6.49) yields the tensor function F ıB .F; / for the model BV . For the models AV and BV , there is no need to determine eigenbases and eigenvalues in constructing the functions (6.49). The mentioned advantages discern the models AV and BV among others while using the material description as well as the

6.2 Closed Systems in the Material Description

393

models AI and BI while using the spatial description (see Remark 3 in Sect. 6.1). These models are the most frequently applied to computations of problems of elasticity theory at large deformations. The models AI and BI are of a higher level of complexity in the material description. For the model AI , constitutive equations (6.49) are replaced by the relations I

P D F ıA .F; / D ı

ı

r X

ı

1 ' F1 F1T I.s/ F ;

(6.50)

D1

' D [email protected] [email protected].s/ /;

.s/ D .I.s/ .ƒ/; /; I.s/ D @I [email protected]ƒ; 1 ƒ D .E F1 F1T /: 2

For this model, although there is no need to determine eigenvalues and eigenvectors, however, in comparison with the models AV and BV , one should invert the tensor F1 . Just as in the spatial description, the models AII , BII and AIV , BIV are the most complicated for computations. t u The entropy balance equation in the system (6.40) can be modified just as in Sect. 6.1.1 by using the principal thermodynamic identity in the material description (4.125): ı ıd ı d (6.51) C P r ˝ vT D 0: dt dt Differentiating the identity with respect to and using the second relation of (6.41), we obtain the following expression for the rate of changing the specific entropy in the material description: ı @

@t

2 ı@

D

@ 2

@P ı @ r ˝ vT : @t @

(6.52)

On substituting (6.52) and (6.41), (6.42) into the entropy balance equation of the system (6.40), we get the desired heat conduction equation for an elastic continuum in the material description: ı

c"

ı

.n/ ı ı ı @ .n/ ı D r . r / C .4 E ıG T G / r ˝ vT C qm ; @t

(6.53)

where the heat capacity c" is determined by (6.11a). Here we have taken into account that equations (6.45) give the following expression for the derivative with respect to : ı

ı r X .n/ .n/ @' .s/ @ .n/ı .n/ @P D I G D 4 E ıG T G ; F G .F; / D 4 E ıG @ @ @ D1

(6.53a)

394

6 Elastic Continua at Large Deformations .n/

where the tensor T G has been introduced by formula (4.423): .n/

T G D

r X @' .s/ I : @ G D1

(6.53b)

6.2.2 U V - and U -systems of Thermoelasticity in the Material Description Return to the general system (6.40)–(6.42) and notice that the deformation gradient F can be excluded from the system with the help of Eq. (2.77): ı

F D E C r ˝ uT :

(6.54)

Then we obtain the U V -system of thermoelasticity in the material description: ı @v

ı

c"

@t

ı

ı

.n/

ı

D r F ıG ..E C r ˝ uT /; / C f;

ı ı ı ı ı @ @ .n/ı ı F G .E C r ˝ uT ; / r ˝ vT C qm ; D r . r / C @t @ @[email protected] D v;

(6.55)

consisting of seven equations for seven scalar unknowns: ; u; v k X i ; t:

(6.56)

The density does not appear in this system explicitly, and it can always be evaluated with the help of the continuity equation (6.17): ı

ı

D det .E C r ˝ uT /:

(6.57)

Since in the material description the velocity v is connected to the displacement vector u by an explicit kinematic relation, we can eliminate the velocity v between the unknowns. As a result, we obtain the U -system of thermoelasticity in the material description: 2 ı@ u

@t 2

ı

.n/

ı

ı

D r F ıG ..E C r ˝ uT /; / C f;

ı ı ı ı ı @ @ .n/ı @u ı D r . r / C C qm ; c" F G ..E C r ˝ uT /; / r ˝ @t @ @t ı

(6.58)

6.2 Closed Systems in the Material Description

395

for four scalar unknowns: ; u k X i ; t:

(6.59)

In particular, for the exceptional model AV , the U -system (6.58) with account of (6.49) and (2.78) is written in the form 2 ı@ u

ı

ı

D r P C f; 0 1 r X ı ı ı ı ı @ ı T T A c" ' I.s/ D r . r / C @ C .F r ˝ v / C qm ; @t D1

@t 2

@2

ı

' D PD

r X D1

CD

ı

@@I.s/

.s/

' I C F T ;

D

;

ı

ı

' D

@ @I.s/

(6.60)

;

.s/

.I.s/ .C/; /;

I C D @I.s/ [email protected];

ı ı ı 1 ı .r ˝ u C r ˝ uT C r ˝ u r ˝ uT /; 2

ı

F D E C r ˝ uT :

6.2.3 T U VF -system of Thermoelasticity in the Material Description .n/

Using the relations (6.46) between the tensors P and T G and complementing the system (6.40) with constitutive equations (4.442) ‘in rates’, we obtain the T U VF system of thermoelasticity in the material description: ı @v

ı

c"

@t

ı

.n/

.n/

ı

D r .4 E ıG T G / C f;

.n/ .n/ ı ı ı ı @ ı ı D r . r / C .=/.4 E ıG T G / r ˝ vT C qm ; @t

@[email protected] D v;

ı

(6.61)

@[email protected] D r ˝ vT ;

.n/

.n/ .n/ .n/ .n/ ı @ @TG D 4 P ıGh .F; / r ˝ vT C ZGh T G C T G ZGh C T G ; @t @t

for 22 scalar unknowns: .n/

T G ; ; u; v; F k X i ; t:

(6.62)

396

6 Elastic Continua at Large Deformations

Here we have introduced the notation for the fourth-order tensor 4

.n/ P ıGh

.n/

.4 P .1243/ F1T /.1243/ : Gh

(6.63)

The remaining notations are the same as in Sect. 6.1.3. The heat conduction equation in (6.61) has been written with the help of (6.53). Component forms of equation systems of elasticity theory in the material deı

ı

scription are usually used in the basis ri of a reference configuration K (see Exercise 6.2.2 and 6.2.3).

6.2.4 The Equation System of Thermoelasticity for Quasistatic Processes in the Material Description The model of quasistatic processes introduced in Sect. 6.1.5 can be considered for the material description as well. In this case the equation system (6.40)–(6.42) with account of the assumption (6.34) takes the form ı

ı

r P C f D 0; ı

c"

ı

@ ı D r . r / C qm ; @t .n/

(6.64)

ı

(6.65)

ı

P D F ıG .E C r ˝ u; /:

(6.66)

The system consists of four scalar equations for four scalar unknown functions: ; u k X i ; t;

(6.67)

and it is called the quasistatic equation system of thermoelasticity in the material description. In this case the dynamic compatibility equation and the kinematic equation in system (6.40) are satisfied only approximately, and they are not considered. These equations are satisfied exactly for static processes starting from some time t0 .

Exercises for 6.2 6.2.1. Show that constitutive equations (6.49) for the quasilinear model AV can be written in the form ı

P D .=/.4 M C/ FT ;

T D F .4 M C/ FT ;

6.2 Closed Systems in the Material Description

397

or in terms of the displacement gradient:

ı

PD

4

ı ı 1ı T M E C r ˝ u r ˝ u .E C r ˝ u/: 2

6.2.2. Using the results of Exercise 6.1.1, show that for isotropic media the relations (6.49) have the forms PD.

ı 1E ı

ı

C

2C ı

1

C

ı 3C

2

ı

/ FT ;

T D JF .

ı

D ' 1 C ' 2 I1 C ' 3 I2 ; ı

ı

' ˛ D [email protected] [email protected]˛ /;

ı

ı 1

ı 2C

ı

ı

2

C

D ' 2 C I1 ' 3 ;

I˛ D I˛ .C/;

C

ı

ı 3C

2

/ FT ;

ı

D '3;

3

˛ D 1; 2; 3:

Show that for the linear model AV of isotropic continua these relations take the forms P D l1 I1 .C/FT C 2l2 C FT ;

T D J.l1 I1 .C/F FT C 2l2 F C FT /;

or in terms of the displacement gradient: ı ı ı 1ı T P D l1 r u C r ˝ u r ˝ u .E C r ˝ u/ 2 ı

ı

ı

ı

ı

Cl2 .r ˝ u C r ˝ uT C r ˝ u r ˝ uT / .E C r ˝ u/: 6.2.3. Using the result of Exercise 6.2.1, show that the constitutive equations for ı the quasilinear model AV in basis ri have the component form ı

ı

j

ı

ı

ı

P i D .=/M jkms "ms F lk g li I and for the model AV of isotropic media: ı

P ji D .

ı

ı jk 1g

T ij D

ı

C

ı ij 1g

ı

ı j m ı ks g "ms 2g

C

ı

C

ı i m ı js 2 g g "ms

ı

ı ı ı j m ı ks ı uv g g "mu "sv /F lk g li ; 3g

C

ı

ı i m ı js ı uv 3 g g g "mu "sv : ı

6.2.4. Show that the U VF -system (6.40)–(6.42) and (6.45) in basis ri in the material description has the component form

398

6 Elastic Continua at Large Deformations

8 ı ı ı ı ıı ˆ [email protected] [email protected]/ D r j P j i C f i ; ˆ ˆ ˆ ˆ ı ı ı ı ı ı ı ı 0; f > 0; k > 0:

(6.201)

Since F is diagonal, one can readily find the stretch tensors U and V and the rotation tensor O: U D V D F; O D E: (6.202) With the help of formulae (4.25) and (4.42) we find the energetic deformation ten.n/

.n/

sors C and deformation measures G: .n/

CD

1 ..f 0 /nIII 1/er ˝ er n III

ı C..f =r/nIII 1/e' ˝ e' C .k nIII 1/ez ˝ ez ; (6.203)

.n/

GD

1 0nIII ı f er ˝ er C .f =r/nIII e' ˝ e' C k nIII ez ˝ ez : n III

6.7.3 Stresses in the Lam´e Problem for Models An Assume that constitutive equations of the cylinder correspond to the linear model An (6.135) of an isotropic elastic continuum. Then substituting the expression (6.203) into (6.135), we find the energetic stress tensors .n/

.n/

.n/

.n/

T D T r er ˝ er C T ' e' ˝ e' C T z ez ˝ ez ; ı

.n/

r Tr D .l1 I1 C 2l2 .f 0nIII 1//; .n III/f 0 f k ı

.n/

T' D .n/

r ı .l1 I1 C 2l2 ..f =r/nIII 1//; .n III/f 0 f k ı

TzD

r .l1 I1 C 2l2 .k nIII 1//; .n III/f 0 f k ı

I1 D .f 0 /nIII C .f =r/nIII C k nIII 3:

(6.204)

450

6 Elastic Continua at Large Deformations .n/

Since the tensors F and T are diagonal, from Eqs. (6.137), (6.200), and (6.204) we obtain the expression for the Cauchy stress tensor .n/

T D r er ˝ er C ' e' ˝ e' C z ez ˝ ez D FnIII T ;

(6.205)

ı

r D

' D

r .f 0 /nIII1 .l1 I1 C 2l2 .f 0nIII 1//; .n III/f k

1 ı ı .f =r/nIII1 .l1 I1 C 2l2 ..f =r/nIII 1//; .n III/f 0 k ı

z D

rk nIII1 .l1 I1 C 2l2 .k nIII 1//: .n III/f 0 f

Components of the stress tensor in this problem prove to depend on the coordinate ı r and time t, therefore, the equilibrium equations are not automatically satisfied. In this case, as a rule, it is convenient to use the equilibrium equations (6.100) in the material description. To do this, we should calculate the Piola–Kirchhoff stress tensor by formulae (3.56): PD

.n/ 1 1 1 ı ı ı F T D FnIII1 T D r er ˝ er C ' e' ˝ e' C z ez ˝ ez ; (6.206) J0 J

where ı

r D

.f 0 /nIII1 ı ..l1 C 2l2 /f 0nIII C l1 .f =r/nIII .3l1 C 2l2 / C l1 k nIII /; n III ı

' D

ı

z D

1 ı ı .f =r/nIII1 ..l1 C 2l2 /.f =r/nIII n III Cl1 f 0nIII .3l1 C 2l2 / C l1 k nIII /;

(6.207)

k nIII1 ı ..l1 C 2l2 /k nIII C l1 .f 0nIII C .1=r/nIII / .3l1 C 2l2 //: n III

6.7.4 Equation for the Function f ı

Writing components of the divergence r P with respect to the physical basis er ; e' ; ez (see [12]), we can represent the equilibrium equation projected onto the axis Oer as follows: ı

@ r ı

@r

ı

C

ı

r ' ı

r

D 0:

(6.208)

6.7 The Lam´e Problem

451 ı

ı

ı

ı

ı

Here we have taken into account that r , ' and z are independent of ' and z. The other two projections of the equilibrium equation onto the axes Oe' and Oez are satisfied identically. On substituting the expressions (6.207) into (6.208), we obtain an ordinary difı ı ferential equation of the second order for the function f .r ; t/. The function f .r; t/ is determined up to the two constants of integration C1 .t/ and C2 .t/ being functions only of time; to evaluate them one should use boundary conditions.

6.7.5 Boundary Conditions of the Weak Type Boundary conditions at inner and outer surfaces of the cylinder have the form (6.80) (gas or fluid pressures pe1 and pe2 are given). According to formula (6.206) and ı ı the fact that at the surfaces r D r ˛ (˛ D 1; 2) the normal vectors have the form ı n D er , the boundary condition (6.80) for this problem becomes ı

ı

r D r˛ W

ı

ı

r D pe˛

f .r ˛ ; t/k ı

;

˛ D 1; 2:

(6.209)

r˛ ı

Boundary conditions at the end surfaces z D 0; h3 serve for determining the function k. For example, if at the surface z D 0 the symmetry conditions are given, ı and at z D h03 – the pressure pe3 , then we have ı ˇ uz D .z z/ˇızD0 D 0;

ı

zD0W ı

ı

z D h3 W

z D pe3 :

(6.210) (6.211)

On substituting the expression (6.194) and (6.205) into (6.210) and (6.211), we verify that the first boundary condition is satisfied identically, and the second one is an additional differential equation for the function f and the constant k. The solution found above cannot satisfy this boundary condition completely, therefore we should consider other variants of boundary conditions. One of such variants weakens the boundary condition (6.211) by replacing it with the integral one ı

ı

z D h3 W

Z

r2 r1

1

z rdr D pe3 .r22 r12 /: 2

(6.212)

One can say that such a condition is of the weak type. The method of replacing exact boundary conditions by the integral ones is called the Saint-Venant method.

452

6 Elastic Continua at Large Deformations ı

According to (6.195), we have rdr D f df D ff 0 d r. Then, substituting the expression (6.205) for z into (6.212) and passing to the reference configuration, we rewrite the condition (6.212) as follows: k nIII1 2l1 ı n III r 2 rı 2 2 1

Z

ı

r2

0 nIII

.f /

ı

C

nIII ! ı ı r dr

f ı

r

r1

!

C.l1 C 2l2 /k

nIII

ı

.3l1 C 2l2 / D pe3

ı

f 2 .r 2 / f 2 .r 1 / ı

ı

r 22 r 21

! :

(6.213) Expressing the function k by Eq. (6.213) and substituting the result into (6.207) and then formulae (6.207) into (6.208) and (6.209), we obtain an integro-differential ı equation for determining the function f .r; t/. For the case when n D IV (John’s model AIV ), Eqs. (6.208) and (6.209) have a simple analytical solution: ı

ı

f .r; t/ D C1 .t/r C

C2 .t/ ı

:

(6.214)

r

Indeed, in this case Eqs. (6.207) become ı

r D 2.l1 C l2 /C1 2l2 ı

' D 2.l1 C l2 /C1 C 2l2

C2 ı

r2 C2 ı

r2

C l1 k .3l1 C 2l2 /; (6.215) C l1 k .3l1 C 2l2 /;

and relations (6.205) take the forms ı

r D

r2 ı

l1 .2C1 C k/ C 2l2 C1

r2

C2

.C1 r 2 C C2 /k ı

' D

l1 .2C1 C k/ C 2l2 C1 C

ı

.C1 r 2 C2 /k

ı

r2

C2 ı

r2 !

!

!

.3l1 C 2l2 / ; !

.3l1 C 2l2 / ; (6.216)

ı4

z D

r ı

.C12 r 4 C22 /

.l1 .2C1 C k/ C 2l2 k .3l1 C 2l2 // :

Substituting the expressions (6.215) into (6.208), one can readily verify that the equilibrium equation is identically satisfied.

6.7 The Lam´e Problem

453

Substitution of (6.215) into the boundary conditions (6.209) and (6.211) yields 2.l1 C l2 /C1

2l2

2.l1 C l2 /C1

2l2

ı

r 21 ı

r 22

C2 C l1 k .3l1 C 2l2 / D pe1 C1 C

C2

C2 C l1 k .3l1 C 2l2 / D pe2 C1 C

C2

! ;

ı

r 21 ı

r 22

!

(6.217) :

After substitution of (6.214), the boundary condition (6.213) for the model AIV takes the form ! C22 2 2C1 l1 C .l1 C 2l2 /k .3l1 C 2l2 / D pe3 C1 ı ı : (6.218) r 21 r 22 Solving the system of three algebraic equations (6.217), (6.218), we find C1 , C2 , and k.

6.7.6 Boundary Conditions of the Rigid Type In place of (6.212) we can consider another boundary condition, namely the condition of the rigid type when displacements uez along the axis Oz are given: ı

ı

z D h3 W

ı ˇ uz D .z z/ˇı

ı

zDh3

D uez :

(6.219)

Then we obtain the following simple expression for k: ı

k D 1 C .uez =h3 /;

(6.220)

which is similar to the corresponding expression of (6.142) in the problem on a beam in tension. For the model AIV , this relation takes the place of the condition (6.218) and the system (6.217), (6.218) for C1 , C2 , and k becomes linear; its solution has the form C1 D C2 ;

ı2

C2 D r 1

1 2.k 1/ ; .1 C .1 2/e p1 / .1 2/.1 e p1 /

e p1 e p 2 ˇ02 1 C ˇ02

D ; e p2 e p1

ı

ı

ˇ0 D r 1 = r 2 :

Here e p ˛ D pe˛ =2l2, and the Poisson ratio (6.146) has been introduced.

(6.221)

454

6 Elastic Continua at Large Deformations

Expressions for outer r2 and inner r1 radii of the cylinder in K can be found with the help of formulae (6.194) and (6.214): ı

ı

ı

ı

r˛ =r ˛ D f .r ˛ ; t/=r ˛ D C1 C .C2 =r 2˛ /;

˛ D 1; 2:

(6.222)

For other models An , we can also consider the boundary condition (6.219), which yields the expression (6.220) for k. Then, substituting the expressions (6.207) into (6.208) and (6.209), we obtain one nonlinear differential second-order equation for the function f with two boundary conditions.

6.8 The Lam´e Problem for an Incompressible Continuum 6.8.1 Equation for the Function f Consider the Lam´e problem on a cylindrical pipe under internal and external pressures (see Sect. 6.7), but let the pipe be made of isotropic incompressible material described by linear models Bn (see (4.532) or (6.160)). In this case the motion law for the pipe is also sought by the semi-inverse method in the form (6.194) and (6.195); therefore, all the relations (6.197)–(6.203) hold. From the condition of incompressibility of a material considered det F D 1 and from (6.201) it follows that ı the function f .r; t/ must satisfy the equation ı

f 0 f k D r: ı

(6.223) ı

Rewriting this equation in the form f df D k1 r d r ; one can easily find its solution (choosing a positive root) ı r2 f2 D C C; (6.224) k where C is the constant of integration.

6.8.2 Stresses in the Lam´e Problem for an Incompressible Continuum On substituting Eq. (6.223) into (6.203), we find 1 0 nIII ı !nIII 1 @ r f GD er ˝ er C ı e' ˝ e' C k nIII ez ˝ ez A : n III fk r

.n/

(6.225)

6.8 The Lam´e Problem for an Incompressible Continuum

455

Having substituted this expression int the constitutive equation (6.160), we obtain .n/

that the energetic stress tensors T have the diagonal form (6.204) in the case of an incompressible continuum too, but their components are different from the ones for compressible materials:

.n/

T r D p

fk ı

T ' D p

C .n III/ 1 C ˇ C .1 ˇ/

r f

T z D pk

nIII

! Ck

nIII

;

r !nIII

C .n III/ 1 C ˇ C .1 ˇ/

.n/

f ı

r ı

.n/

nIII

nIII

C .n III/ 1 C ˇ C .1 ˇ/

ı

r fk

ı

r fk

nIII

nIII

C

! C k nIII f

;

nIII ! ;

ı

r

n D I; II; IV; V: (6.226) For the Cauchy stress tensor, relations (6.205) hold as well, and their components are written as follows:

r ; ' D p C e

' ; z D p C e

z;

r D p C e

(6.227)

where ı

r e

r D .n III/ fk e

' D .n III/

f

!nIII

1 C ˇ C .1 ˇ/

f

nIII

ı

! Ck

nIII

;

r

nIII

1 C ˇ C .1 ˇ/

ı

r

e

z D .n III/k nIII 1 C ˇ C .1 ˇ/

ı

r fk

ı

r fk

nIII

nIII

C

! C k nIII f ı

;

nIII ! :

r

(6.228) With the help of the relation P D F tensor: ı

r D r

fk ı

r

;

1

T, we find components of the Piola–Kirchhoff ı

' D

ı

r

' ; f

ı

z D

z : k

(6.229)

456

6 Elastic Continua at Large Deformations

6.8.3 Equation for Hydrostatic Pressure p Substitution of the Piola–Kirchhoff tensor components (6.229) into the equilibrium equation (6.208) yields an ordinary differential equation for unknown function ı p.r; t/ being the hydrostatic pressure in an incompressible continuum:

pf k

0

ı

C

ı

p.f 2 k r 2 /

D

ı

r 2f

r

ı

e h

@ r e h D ı .f ke

r/ e

': f @r

; ı r

(6.230)

Rearranging the left-hand side of Eq. (6.230) with the help of (6.223) and (6.224), we reduce the equation to the form dp f k ı

dr

ı

r

D

e h ı

:

(6.231)

r

Integration of this equation yields

Z ı h ı 1 r e p D p0 C (6.232) d r: ı k r1 f Here p0 is the constant of integration, which together with C can be found from the boundary conditions (6.209) after substitution of expressions (6.229) and (6.232) into them: 1 p0 C k

Z

ı

r2 ı

r1

e h ı ı ı

r .r 2 /; p0 D pe1 C e

r .r 1 /: d r D pe2 C e f

(6.233)

For the constant C , from (6.233) we obtain the nonlinear algebraic equation: 1 F .C / e

r .r 2 / e

r .r 1 / k ı

ı

Z

ı

r2 ı

r1

ı e h dr p D 0; p D pe1 pe2 ; (6.234) f

into which the expressions (6.224), (6.228), and (6.230) for e

r , f and e h should be substituted.

6.8.4 Analysis of the Problem Solution Figure 6.27 shows the graph of the function F .C /. In the general case, for all n and ˇ the equation F .C / D 0 may have two roots C1 and C2 , among which one should choose the least root C1 , because just the one satisfies the normalization condition F .C1 / D 0 at k D 1 and p D 0 when there is no loading and deformations of the cylinder. For this case

6.8 The Lam´e Problem for an Incompressible Continuum

C D 0;

457 ı

f D r;

' D e

z 0 D .n III/.3 ˇ/ D const; e

r D e p D p0 D const; p0 D pe1 C 0 ; ı

ı

(6.235)

ı

r D ' D z D p0 C 0 D pe1 D const: The stresses r , ' and z in the cylinder are zero when pe1 D pe2 D 0; and they are equal to each other but are nonzero when pe1 D pe2 ¤ 0. When k D 1 and p > 0 (excess internal pressure), the least root C1 of the function F .C / is positive; and when p < 0 (excess external pressure) the root C1 is negative (Fig. 6.27). With growing the value k > 1 (longitudinal extension), the root C1 is displaced into the domain of negative values (there occurs transverse compression of the cylinder); and with decreasing the value k < 1 (longitudinal compression), on the contrary, the root C1 is displaced into the domain of positive values (there occurs lateral dilatation of the cylinder). For the function F .C /, a peculiarity of interest is the existence of the ultimate value p : while p > p , there are no roots of the function F .C / (see Fig. 6.27). This means that at such values of p the nonlinear Lam´e problem has no solution (unlike the Lam´e problem in linear elasticity theory, which has a solution at all values of p). If we consider the process of monotone increasing the pressure difference p from 0 up to p , then the cylinder radius also monotonically grows; and when p > p a solution does not exist. This effect is called the loss of stability of a material in tension. ı ı Figure 6.28 shows the dependence of the dilatation coefficient yR D .r1 r 1 /=r 1 of the cylinder upon the dimensionless pressure difference p= (when p > 0)

Fig. 6.27 The function F .C /

Fig. 6.28 Dependence of the coefficient of relative dilatation of a thin-walled cylinder on internal excess pressure for different models of incompressible materials (k D 1, r2 =r1 D 1.01)

458

6 Elastic Continua at Large Deformations ı ı

ı

ı

ı

for the cylinder with a very thin wall: h=r 1 D 0.01, where h D r 2 r 1 is the initial thickness of the wall. The function p.yR / is nonlinear and exists only if ı h.r/ on p 6 p . A fact of interest is a very weak dependence of the function e the coefficient ˇ when k D 1; due to this, the functions F .C / and p.yR / are also practically independent of the values of ˇ within the interval Œ1; 1 . Moreover, values of the function p.yR / at n D I and V, and also at n D II and IV are practically not distinguishable in pairs. Therefore, when k D 1 there are only two essentially distinct functions p.yR / at n D I and II (Fig. 6.28). Their corresponding ultimate ı ı

values p = are 0.04 and 0.0072 while h=r 1 D 0.01. Ultimate magnitudes of the dilatation coefficient yR are 218 and 41%. If we consider the cylinder with a thicker ı

prove to be smaller. wall h, then the ultimate values yR In the case of compression when p < 0, the root C1 is negative, and from Eq. (6.224) it follows that there exists a limiting value ı

C1 D r 21 =k;

(6.236)

such that the root C1 cannot take on values smaller than C1 . Hence there also exists a limiting negative value of the pressure difference p such that when p < p there is no solution of the Lam´e problem. Graphs of the function p.yR / in compression are shown in Fig. 6.29. Just as in tension, for the case when k D 1, the functions p.yR / at different values of the parameter ˇ are practically coincident, and they are distinct only for the models n D I; V and n D II; IV. ı ı

In the case when h=r 1 D 0.01, we found the limiting value p = D 3.2 for n D I; V, and for n D II; IV: p = D 0.33. It should be noted that for real thin-walled structures at essentially smaller values of pressure in compression such that jpj= jp j=, there occurs a loss of stability of the structure itself; thus, the values p = are usually not realized. Figures 6.30 and 6.31 exhibit distributions of the stresses r = and ' = versus the cylinder thickness at different values of the parameter ˇ and different n when ı ı pe2 D 0. The radial stress r .Nr /, where rN D r=r 1 , depends weakly on ˇ and

Fig. 6.29 Dependence of the coefficient of relative compression of a thin-walled cylinder on external excess pressure for different models of incompressible materials (k D 1, r2 =r1 D 1.01)

6.8 The Lam´e Problem for an Incompressible Continuum

a

459

b

Fig. 6.30 Distribution of the radial stresses versus the thickness of a thick-walled cylinder for the model BI at different values of ˇ (a) and for different models of incompressible material .ˇ D 1/ (b)

Fig. 6.31 Distribution of the tangential stresses versus the thickness of a thick-walled cylinder for different models of incompressible material and different values of the parameter ˇ

n; it monotonically decreases from the value pe1 to 0. The tangential stress ' .r/ depends considerably more on ˇ and n especially for thin-walled cylinders. So when r2 =r1 D 2, for the model BI and ˇ D 1 the stress ' is positive everywhere, and it reaches its minimum value at the interior point r=r1 1.2 of the cylinder. When ˇ 6 0.6, the stress ' on the inner surface of the cylinder becomes negative, and when ˇ D 1, the stress ' is negative in the whole cylinder. For the models BII , BIV and BV , the stress ' is always positive. For the model BII , when ˇ D 1,

460

6 Elastic Continua at Large Deformations

Fig. 6.32 Distributions of radial and tangential stresses versus the thickness of a thin-walled cylinder for different models of incompressible material .ˇ D 1/

the stress ' reaches its maximum on the outer surface of the cylinder; and for the models BIV and BV , at all ˇ the function ' .Nr / is always monotonically decreasing and reaches its maximum on the outer surface. For thin-walled cylinders, the stress ' is positive and practically constant versus the cylinder thickness; it is almost independent of ˇ and n, and its value is close to the value determined by the theory of thin linear-elastic shells at small deformations: ı

ı

' p r 1 =h (Fig. 6.32).

Chapter 7

Continua of the Differential Type

7.1 Models An and Bn of Continua of the Differential Type 7.1.1 Constitutive Equations for Models An of Continua of the Differential Type Let us consider now nonideal continua. Practically all real bodies are nonideal media, and they can be considered as ideal ones in a certain approximation. According to the general theory of constitutive equations stated in Sect. 4.4.2, a continuum is nonideal if its operator constitutive equations (4.156) include the dissipation function w being nonzero. Models of nonideal materials, which are widely used in practice, are models of continua of the differential type. Definition 7.1. A continuum is called a continuum of the differential type, if corresponding operator relations (4.156) are usual functions of active variables R.t/ and P their derivatives R.t/, i.e. P ƒ.t/ D f R.t/; R.t/ :

(7.1)

Functions (7.1) are assumed to be continuously differentiable. One can say that this is the model An of a continuum of the differential type, if some model An of a continuum has been chosen and its corresponding operator constitutive equations (4.156) are simply functions of the arguments indicated above and their rates. In particular, the Helmholtz free energy has the form .t/ D

.n/

.n/

. C .t/; C .t/; .t//:

(7.2)

(In continuum mechanics, the set of arguments of the function (7.2) does not involve the derivative P .)

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 7, c Springer Science+Business Media B.V. 2011

461

462

7 Continua of the Differential Type

The total derivative of

with respect to time has the form .n/

.n/

d2 C @ d d dC @ @ C D C : 2 .n/ .n/ dt dt @ dt dt @C @C .n/

(7.3)

.n/

The partial derivatives @ [email protected] C and @ [email protected] C are symmetric second-order tensors. The remaining constitutive equations (4.158), connecting the active variables to the reactive ones, for the model An of a continuum of the differential type have the forms .n/ .n/ .n/

.n/

T D T . C; C ; /;

.n/ .n/

D . C ; C ; /;

.n/ .n/

w D w . C ; C ; /:

(7.4)

By analogy with the tensor function connecting the energetic tensors of stresses and deformations, introduce the two tensor functions .n/

.n/

.n/ .n/

T e . C ; / D T . C; 0; /;

(7.5)

and .n/

.n/

.n/

T v D T T e;

(7.6)

called the function of equilibrium stresses and the function of viscous stresses, respectively. Substituting the expressions (7.3) and (7.6) into PTI (4.121) and collecting like terms, we get 0

1

.n/ .n/ .n/ .n/ Te C @ @ 1 [email protected] C d C .w T v C / dt D 0: @ .n/ A d C C .n/ d C C @ @C @C (7.7) .n/

.n/

.n/

Since the differentials d C , d C , d and dt are mutually independent, the identity (7.7) is equivalent to the equation system 8 .n/ .n/ .n/ ˆ ˆ ˆ T e D [email protected] [email protected] C/ F . C ; /; ˆ ˆ ˆ ˆ .n/ ˆ < @ [email protected] C D 0; ˆ ˆ ˆ D @ [email protected]; ˆ ˆ ˆ ˆ .n/ ˆ : .n/ w D T v C :

.7:8a/ .7:8b/ .7:8c/ .7:8d/

7.1 Models An and Bn of Continua of the Differential Type

463 .n/

Just the fact that terms within the first parentheses of (7.7) are independent of C .n/

.n/

ensures that the differentials d C and d C be mutually independent. In turn, the .n/

.n/

presence of T e instead of T within the first parentheses of (7.7) and the validity of .n/

Eq. (7.8b) ensure that terms within these parentheses be independent of C . Thus, continua of the differential type have the following properties: (1) They are dissipative (i.e. nonideal), because for them the dissipation function w is not identically zero. .n/

.n/

(2) The tensor function of equilibrium stresses T e (but not T ) is quasipotential (see Sect. 4.5.2). (3) The quasipotential

.n/

depends only on C and : D

.n/

. C ; /;

(7.9)

.n/ .n/

.n/

the remaining functions , T v , w depend on C , C and . According to the property (2), for models of the differential type it is not sufficient to specify only one function (7.9); in addition we need the viscous stresses function (7.6) to be given: .n/ .n/

.n/

T v D F v . C ; C ; /:

(7.10)

According to formulae (7.4)–(7.6), this function depends also on the deformation .n/

tensor rates C . Relationships (7.8)–(7.10) are constitutive equations for models An of continua of the differential type.

7.1.2 Corollary of the Onsager Principle for Models An of Continua of the Differential Type The tensor function of viscous stresses (7.10) is not quite arbitrary: it satisfies the conditions (7.5) and (7.6), i.e. .n/

F v . C ; 0; / D 0:

(7.11)

In addition, the function must satisfy the Onsager principle (see Sect. 4.12.1, Axiom 16). This principle is applied to continua of the differential type as follows.

464

7 Continua of the Differential Type

First, we should form the scalar function (4.728) being the specific internal entropy production: q D w

.n/ .n/ .n/ q q r D F v . C ; C ; / C r > 0:

(7.12)

Here we have used expression (7.8d) for the dissipation function w of continua of the differential type. Then we represent the expression (7.12) in the form (4.729) with thermodynamic forces .n/

X2 D C

X1 D r ;

(7.13)

(where X1 is a vector, and X2 is a second-order tensor) and thermodynamic fluxes .n/ .n/

.n/

1 Q1 D q;

Q2 D T v D F v . C ; C ; /:

(7.14)

According to the Onsager principle, thermodynamic fluxes Qˇ must be linear (tensor-linear) functions of Xˇ in the form (4.730), i.e. 8 ˆ 0; .n/

.n/

.n/

w D F v C D C 4 Lv C > 0;

(7.19) (7.20)

7.1 Models An and Bn of Continua of the Differential Type

465

which are the Fourier inequality and the dissipation inequality for continua of the differential type, respectively. From these inequalities it follows that the tensors and 4 Lv are nonnegative-definite. In addition, from (7.20) it follows that the dissipation function w for continua of the differential type is a quadratic scalar function .n/

of C , and the viscosity tensor 4 Lv has the following symmetry in components: 4

b Lijkl D b Lijlk ;

Lv D b Lijklb cj ˝b ck ˝b cl ; ci ˝b

b Lijkl D b Lj i kl ;

b Lijkl D b Lklij ;

(7.21)

whereb ci is an orthonormal basis. The total number of components of the tensor 4 Lv is 81, and the number of independent components, according to Eqs. (7.21), does not exceed 21 (see [12]). Notice that the viscosity tensor 4 Lv and its components, in general, depend on .n/

.n/

the tensor C and also on C (the Onsager principle (7.18) does not prohibit this dependence, it only requires that F v be a quasilinear tensor function (see Sect. 4.8.7 and [12])). .n/

.n/

If the tensor 4 Lv is independent of C and depends only on C , then the function of viscous stresses F v is pseudopotential, i.e. satisfies the relation .n/

.n/ .n/

T v D F v . C; C ; / D

.n/ 1 [email protected] [email protected] C /: 2

(7.22)

Such function F v is also called potential with respect to the second tensor argument. Equation (7.22) can readily be verified by differentiating the scalar function w .n/

(7.20) with respect to the argument C .

7.1.3 The Principle of Material Symmetry for Models An of Continua of the Differential Type Notice that the principle of material symmetry (Axiom 14), and also Definitions 4.5 and 4.6 of fluids and solids, respectively, have been formulated for arbitrary operator constitutive equations (4.156); i.e. they can be applied not only to ideal materials but also to continua of the differential type and other types of nonideal materials. According to this principle, models An of continua of the differential type with bs constitutive equations (7.8)–(7.10) in some undistorted reference configuration K ı

ı

(as this configuration we will choose K) have a symmetry group G s such that ı

ı

for each transformation tensor H 2 G s (H W K ! K/ the constitutive equations

(7.8)–(7.10) in the reference configuration K have the form

466

7 Continua of the Differential Type

8 .n/ .n/ .n/ ˆ ˆ T D F . C ; / [email protected] [email protected] C /; ˆ e ˆ ˆ ˆ ˆ ˆ .n/ ˆ ˆ ˆ D . C ; /; ˆ ˆ < D @ [email protected]; ˆ ˆ ˆ ˆ .n/ .n/ ˆ ˆ ˆ / D T C ; .w ˆ v ˆ ˆ ˆ ˆ .n/ .n/ ˆ.n/ : T v D F v . C ; C ; /:

.7:23a/ .7:23b/ .7:23c/ .7:23d/ .7:23e/

Let us consider first the models An of solids of the differential type, whose symı

metry group G s is a subgroup of the full orthogonal group I . For these materials, the further construction is the same as in Sect. 4.8 for ideal solids: since the tensors ı

.n/

.n/

.n/

.n/

T and C are H -indifferent relative to the group G s , tensors T e (7.5), T v (7.6) .n/

and C are also H -indifferent, according to Theorem 4.21. Then Eqs. (7.23) are equivalent to the relations

.n/

.n/

QT F . C; / Q D F .QT C Q; /;

.n/

. C ; / D

.n/

.QT C Q; /;

.n/ .n/

(7.24a)

.n/

(7.24b)

.n/

QT F v . C ; C ; / Q D F v .QT C Q; QT C Q; /;

(7.24c)

ı

for any Q 2 G s . Since .n/

.n/

.n/

.n/

.n/

.n/

.w / D T v C D QT T v Q QT C Q D T v C D w ;

(7.25)

Eq. (7.23d) is always satisfied. Theorem 7.1. For models An of solids of the differential type, the principle of material symmetry (7.24) holds if and only if the two conditions (7.24b) and (7.24c) for the function and the viscous stresses function F v are satisfied. H The conditions (7.24b) and (7.24c) are necessary, because if the principle of material symmetry holds, then all conditions (7.24) are satisfied. Prove that the conditions (7.24b) and (7.24c) are sufficient. If the condition (7.24b) is satisfied, then (7.24a) follows from (7.24b); a proof of this fact is similar to the proof of formula (4.280). N

7.1 Models An and Bn of Continua of the Differential Type

467

7.1.4 Representation of Constitutive Equations for Models An of Solids of the Differential Type in Tensor Bases In comparison with ideal continua, for solids of the differential type not only the condition (7.24b), being the condition of indifference of the scalar function ı

.n/

. C ; / relative to some group G s (see Definition 4.9), must be satisfied but also .n/ .n/

the condition (7.24c) for the tensor function F v . C ; C ; / of two tensor arguments. .n/

For the scalar function . C ; /, the same representations as were derived in Sect. 4.8 remain valid; in particular, these are representations (4.304) as functions ı

.n/

of invariants I.s/ . C / relative to group G s : D

.n/

.n/

.I1.s/ . C/; : : : ; Ir.s/ . C /; /;

(7.26) .n/

and also representations (4.311) and (4.324) for the tensor function F . C ; / (7.8a). For example, for an isotropic solid of the differential type, from (4.322) we obtain the expression for equilibrium stresses .n/

Te D

1E

C

.n/

2C

.n/

C

3C

2

;

(7.27)

where are determined by formulae (4.322a). By analogy with Definition 4.13 (see Sect. 4.8.6) for a function of one tensor ar.n/ .n/

gument, the function F v . C ; C ; / of two tensor arguments, which satisfies the ı

condition (7.24c), is called indifferent relative to the group G s (see [12]). The following theorem gives representations of this function in tensor bases. Theorem 7.2. Any tensor function of viscous stresses (7.10), which is ı

indifferent relative to some orthogonal group G s I , i.e. satisfies the condition

(7.24c), .n/

quasilinear in the second argument C , i.e. satisfies the conditions (7.18) and

(7.20), ı

can be represented in the tensor basis of the corresponding group G s : .n/

.n/ .n/

.n/

T v D F v . C ; C ; / D .1 E ˝ E C 22 / C ;

(7.28)

or .n/

.n/

.n/

T v D 1 I1 . C /E C 2 C ;

(7.29)

468

7 Continua of the Differential Type ı

— for an isotropic continuum of the differential type .G s D I /, .n/

T v D .1 E ˝ E C 2b c3 C 3 .E ˝b c23 Cb c23 ˝ E/ c23 ˝b .n/

C 4 .O1 ˝ O1 C O2 ˝ O2 / C 25 / C

(7.30)

ı

— for a transversely isotropic continuum of the differential type .G s D T3 /, .n/

Tv D

3 X

.˛b c˛ C 3C˛ .b c2ˇ ˝b c2 Cb c2 ˝b c2ˇ / c2˛ ˝b

˛D1 .n/

C6C˛ O˛ ˝ O˛ / C ; ˛ ¤ ˇ ¤ ¤ ˛; ˛; ˇ; D 1; 2; 3; (7.31) ı

— for an orthotropic continuum of the differential type .G s D O/. The tensorsb c2˛ and O˛ are determined by formulae (4.292) and (4.315). H Substituting relation (7.8) of quasilinearity of the viscous stresses function F v into (7.24c), we obtain that the following condition must be satisfied:

.n/

.n/

.n/

QT .4 Lv C / Q D 4 Lv .QT C Q/ 8 C :

(7.32)

Hence the viscosity tensor 4 Lv must satisfy the equation (see Exercise 7.1.1) 4

Lv D 4 Lv .Q ˝ Q ˝ Q ˝ Q/.57312468/

ı

8Q 2 G s :

(7.32a)

This condition is an analog of the condition (4.289) for fourth-order tensors. Therefore, a fourth-order tensor 4 Lv satisfying Eq. (7.32a) is called indifferent relative to ı

the group G s . All indifferent fourth-order tensors are formed with the help of the operations of tensor and scalar products by producing tensors of a corresponding group ı

G s (see Sect. 4.8.3) and can be resolved for the tensor basis 4 Os. / by analogy with P the resolution (4.291) of second-order tensors: 4 Lv D k D1 a 4 Os. / . The number k of elements of this basis coincides with the number of independent components of ı

the tensor 4 Lv ; it is equal to 2, 5, and 9 for the groups G s D I; T3 ; O, respectively (see [12]). The elastic moduli tensors 4 M defined by formulae (4.337), (4.341), and ı

(4.344) are also indifferent relative to the groups G s D I; T3 ; O, respectively; and the representations (4.337), (4.341), and (4.344) are the desired resolutions for the tensor bases, being the same for each group [12]. Hence, the viscosity tensor can alı

ways be represented in the similar form for the corresponding group G s D I; T3 ; O too. Thus, the representations (7.28)–(7.31) actually hold. N

7.1 Models An and Bn of Continua of the Differential Type

469

Remark. The coefficients in representations (7.28)–(7.31) are, in general, scalar .n/

.n/

functions of corresponding invariants of the tensors C and C : .n/ .n/

D . C ; C ; /:

(7.33)

But since we have assumed everywhere that remain unchanged under H transformations in a corresponding group, so the coefficients must satisfy the relations

.n/ .n/

.n/

.n/

ı

. C; C ; / D .QT C Q; QT C Q; / 8Q 2 G s :

(7.34)

Such scalar functions are called simultaneous invariants of two tensor arguments ı

relative to a group G s considered. According to the theorem proved in [12], for ı

each group G s there is a functional basis of independent simultaneous invariants .s/

.n/ .n/

J . C; C / ( D 1; : : : ; z), where z D 9 for the full orthogonal group I , z D 11 for the transverse isotropy group T3 , z D 12 for the orthotropy group O. .n/

.n/

As a functional basis of simultaneous invariants J.s/ . C ; C / of two tensors we can choose the following sets [12]: ı

for the isotropy group G s D I : .n/

.n/

.I / J3C˛ D I˛ . C /; ˛ D 1; 2; 3;

J˛.I / D I˛ . C /; .n/

.n/

J7.I / D C C ;

.n/

.n/

J8.I / D C 2 C ;

.n/

.n/

J9.I / D C . C /2 I

(7.35)

ı

for the group G s D T3 : .n/

J˛.3/ D I˛.3/ . C /; ˛ D 1; : : : ; 5; .n/

.n/

.n/

J5Cˇ D Iˇ . C /; ˇ D 1; : : : ; 4; .3/

.3/

.n/

.n/

c23 / C / .b c23 C /; J11 D C C 2J10 J2 J8 I J10 D ..E b (7.36) .3/

.3/

.3/

.3/

.3/

ı

for the group G s D O: .n/

J˛.O/ D I˛.O/ . C /; ˛ D 1; : : : ; 6I .n/

.n/

.O/ D .b c22 C/ .b c23 C /; J10

.n/

.O/ J6Cˇ D Iˇ.O/ . C /; ˇ D 1; 2; 3; 6I .n/

.n/

.O/ J11 D .b c21 C / .b c23 C /:

(7.37)

470

7 Continua of the Differential Type .s/

Invariants I˛ of corresponding groups are determined by formulae (4.295)–(4.297). Then the viscous coefficients can always be represented as functions of simultaneous invariants: .n/ .n/

.n/ .n/

D .J1 . C ; C /; : : : ; Jz.s/ . C ; C /; /: .s/

(7.38)

The dissipation function (7.20) for an isotropic medium of the differential type, according to (7.29), has the form .n/

.n/

w D 1 I12 . C / C 2 I1 . C 2 /:

(7.39)

7.1.5 Models Bn of Solids of the Differential Type For solids of the differential type, models Bn can be obtained formally from corre.n/

.n/

sponding models An by replacing the tensors C with the measures G. In particular, constitutive equations (7.8)–(7.10) become 8 .n/ .n/ .n/ ˆ ˆ T e D F . G; / D [email protected] [email protected] G/; ˆ ˆ ˆ ˆ ˆ .n/ ˆ < D . G; /; .n/ ˆ ˆw D .n/ ˆ T v G ; ˆ ˆ ˆ ˆ .n/ .n/ ˆ :.n/ T v D F v . G; G ; /:

(7.40)

For models Bn of isotropic continua of the differential type, the tensors of equilibrium and viscous stresses have the forms .n/

Te D

.n/

C

.n/

2

(7.41)

T v D 1 I1 . G /E C 22 G ;

(7.42)

.n/

.n/

where the coefficients of invariants:

C

;

1E

2G

3G

.n/

are expressed by formulae (4.322), and are functions .n/ .n/

.n/ .n/

D .J1 . G; G ; /; : : : ; J10 . G; G ; //:

(7.43)

In a similar way, from (7.28) and (7.29) we get equations for models Bn of transversely isotropic and orthotropic materials of the differential type.

7.1 Models An and Bn of Continua of the Differential Type

471

7.1.6 Models Bn of Incompressible Continua of the Differential Type Similarly to models Bn for elastic incompressible continua (see Sect. 4.9), we can introduce models Bn for incompressible continua of the differential type. As shown in Sect. 4.9, for these models the potential in (7.40) depends only on r 1 linear .n/

and quadratic invariants I˛.s/ . G/. In particular, for an isotropic incompressible continuum of the differential type,

.n/

.n/

.n/

depends only on I1 . G/ and I2 . G/, and T e is a

.n/

quasilinear function of G: .n/

T D

D

.n/

p .n/1 G C n III

.n/

.I1 . G/; I2 . G/; /;

1

1E

.n/

C

.n/

.n/

2G

C 1 I1 . G /E C 22 G ;

2

D '2 ; '˛ D [email protected] [email protected]˛ /:

ı

D ' 1 C ' 2 I1 ;

(7.44) For the simplest model Bn of isotropic incompressible continua of the differential type, we assume that the viscous coefficients are connected by the relation 2 1 D 2 : 3

(7.45)

Then the constitutive equation (7.44) takes the form .n/

T D

p .n/1 G C n III

1E

C

.n/

2G

.n/

C 22 dev G ;

(7.46)

where .n/ .n/ 1 .n/ (7.47) dev G D G I1 . G/E 3 is the deviator of the tensor (the more detailed information on deviators can be found in Sect. 8.2.13 and [12]). For the simplest models Bn , the first principal invariants of the stress tensor .n/

.n/

T and the deformation measures G are connected by the elastic relations (see Exercise 7.1.4) .n/

I1 . T / D

.n/ p I1 . G 1 / C 3 n III

being independent of the deformation rates.

1

C

.n/

2 I1 . G/;

(7.48)

472

7 Continua of the Differential Type

When solving problems in practice, one usually applies models Bn of incompressible continua of the differential type with steady creep, where the potential .n/

is independent of the invariants I˛ . G/: .n/

T D

.n/ p .n/1 G C 22 dev G ; n III

D

./;

(7.49)

This model describes the phenomenon of creep being a change in deformation of a body with time at constant stresses (see Sect. 7.4). Remark. The consistency conditions (4.328), which must be satisfied by constitutive equations at a natural unstressed state, for materials of the differential type should be complemented by the requirement that the rates of the deformation ten

sors and measures vanish in K: ı

KW

.n/

.n/

T D S D T D 0; .n/

.n/

A D C D 0;

.n/

.n/

A D C D 0;

.n/

.n/

.n/

g DGD

.n/

g D G D 0; D 0 ;

1 E; n III D

0:

(7.50)

All the constitutive equations (7.29), (7.30), (7.31), (7.41), (7.42), (7.44), (7.46), and (7.49) derived above satisfy these conditions. t u

7.1.7 The Principle of Material Indifference for Models An and Bn of Continua of the Differential Type .n/ .n/

.n/

Since all the energetic tensors T , C and G are R-invariant, the constitutive equations (7.8)–(7.10) and also (7.28)–(7.31) for solids of the differential type are the same in actual configurations K and K0 obtained one from another by a rigid motion. Therefore, the principle of material indifference for models An and Bn of solids of the differential type (as well as for elastic continua) is satisfied identically.

Exercises for 7.1 7.1.1. Show equivalence of relations (7.32) and (7.32a).

7.2 Models An and Bn of Fluids of the Differential Type

473

7.1.2. Show that the component representations of functional bases of simultaneous invariants (7.35)–(7.37) in the basis b c˛ have the forms for the transverse isotropy ı

group G s D T3 : b b .b n/ .b n/ .n/ .b n/ .b n/ .n/ .b n/ .b n/ J˛.3/ D f C 11 C C 22 ; C 33 ; C 213 C C 223 ; C 211 C C 222 C 2 C 212 ; det C; .b n/

.b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ C 11 C C 22 ; C 33 ; . C 13 /2 C . C 23 /2 ; . C 11 /2 C . C 22 /2 C 2. C 12 /2 ;

b b b .n/ .b n/ .n/ .b n/ .b n/ .b n/ .n/ .b n/ .b n/ C 13 C 13 C C 23 C 23 ; C 11 C 11 C C 22 C 22 C 2 C 12 C 12 gI

.b n/

ı

for the orthotropy group G s D O: b .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .n/ .b n/ .b n/ J˛.O/ D f C 11 ; C 22 ; C 33 ; C 223 ; C 213 ; C 12 C 13 C 23 ; C 11 ; C 22 ; C 33 ; b b .b .n/ .b n/ .b n/ .b n/ .n/ n/ C 23 C 23 ; C 13 C 13 ; C 12 C 13 C 23 g:

.b n/

7.1.3. Show that for the simplest model Bn of isotropic incompressible continua of the differential type (7.46), the relation (7.48) between the first principal invariants holds.

7.2 Models An and Bn of Fluids of the Differential Type 7.2.1 Tensor of Equilibrium Stresses for Fluids of the Differential Type Let us use the principle of material symmetry (7.23) to derive constitutive equations for fluids of the differential type. ı

For fluids, the symmetry group G s , relative to which Eqs. (7.23) hold, is the ı

.n/ .n/

.n/

unimodular group G s D U . Tensors T , C and C are no longer H -indifferent V

I

I

relative to this group, except T and the tensor G D C .1=2/E (see Table 4.9); ı

therefore, relations (7.24) for G s D U do not hold and one should use their general form (7.23).

474

7 Continua of the Differential Type .n/

Due to Theorem 4.31, from these relations it follows that the function F . C; / has the form (4.443a) .n/

.n/

T e D F . C; / D

p .n/1 G ; n III

(7.51)

.n/

p D p.I3 . G/; /;

(7.52)

where p is the pressure being a scalar function only of the third invariant of the ten.n/

.n/

sor G. Thus, for fluids, the tensor C appears in models An only in the combination .n/

.n/

1 G D C C nIII E; i.e. in fact, models An and Bn are coincident. For the tensor of viscous stresses, the situation is analogous.

7.2.2 The Tensor of Viscous Stresses in the Model AI of a Fluid of the Differential Type I

I

Let us consider further only the models AI and AV . The tensors T and G are trans

formed during the passage to configuration K according to formulae (4.201) and (4.203a). On substituting these formulae into (7.23e), we obtain that for the group ı

G s D U the following equation must hold: I

I

I

I

H1T F v .G; G ; / H1 D F v .H G HT ; H G HT ; / 8H 2 U: (7.53) I

I

I

Here we have gone from the argument C to G, because just the measure G satisfies

I

the relation (7.53), and the tensor C is transformed during the passage to K in another way, namely according to (4.201a), and its expression involves the additional term: .1=2/.H HT E/. I

I

The tensor function F v .G; G ; / satisfying the condition (7.53) is called AI -unimodular or AI -indifferent relative to the group U . Notice that the group U contains a subgroup being the full orthogonal group I ; therefore, if H D OT and Q 2 I , then formula (7.53) becomes (7.24c). In other words, if a tensor function is AI -unimodular, then the function is isotropic (i.e. indifferent relative to the group I ). Then, since the Onsager principle holds for all I

I

groups and the function F v .G; G ; / is quasilinear: I

I

I

I

Tv D F v .G; G ; / D 4 Lv G ;

(7.54)

7.2 Models An and Bn of Fluids of the Differential Type

475

so for this function one of the representations of an isotropic function in the tensor basis (7.29) with two independent constants must hold. But such a representation may be not AI -unimodular; in particular, the representation (7.29) does not satisfy the condition (7.53). The following isotropic tensor function proves to be appropriate: I

Tv D F v D

I I I I 1 .1 I1 G1 C 22 G1 G G1 /; 4

(7.55)

where I

I

I

I

I1 .G1 G / D G1 G :

(7.56)

We can verify that the function (7.55) is AI -unimodular, i.e. satisfies the Eq. (7.53): I

I

I

I

I

I

I

I

4F.G ; G ; / D 1 .G1 G /G1 C 22 G1 G G1 I

I

I

D 1 .H1T G1 H1 H G HT /H1T G1 H1 I

I

I

C 22 H1T G1 H1 H G HT H1T G1 H1 I

I

I

D H1T .1 J1 G1 C 22 G1 G G1 / H1 I

I

D 4H1T F .G; G ; / H1 :

(7.57)

As follows from (7.57), the function (7.55) satisfies also the condition (7.24c) for Gs D I , i.e. it is isotropic. Thus, the isotropic tensor function (7.55) constructed is AI -unimodular. Thus, we obtain the following theorem. Theorem 7.3. Any quasilinear AI -unimodular tensor function of viscous stresses (7.54) can be represented in the form (7.55) or, with the help of the Cauchy stress tensor, in the form (7.58) Tv D 1 I1 .D/E C 22 D; where D is the tensor of deformation rates (2.225). H The first assertion of the theorem (formula (7.55)) has been proved above. Let us prove that the representations (7.55) and (7.58) are equivalent. I

Go from the energetic tensor of viscous stresses Tv to the tensor of viscous stresses T: I

Tv D F1T Tv F1 :

(7.59)

Substitution of (7.55) into (7.59) yields I I I I 1 1T F .1 G1 J1 C 22 G1 G G1 / F1 4 1 P 1 FT /: D .21 J1 E 42 F G 4

Tv D

(7.60)

476

7 Continua of the Differential Type I

I

Here we have taken into account that G D 12 G1 D 12 F1 F1T and G1 D I

2G D 2FT F: According to formula (2.284), we can express the tensor G in terms of the deformation rate tensor D: I

P 1 D 2F1 D F1T : 2G D G Taking into account that I

I

I

I

P 1 I1 .G1 G / D G1 G D G G D 2FT F F1 D F1T D 2D E D 2I1 .D/; (7.61) from (7.60) we really get formula (7.58). N

7.2.3 Simultaneous Invariants for Fluids of the Differential Type I

I

Notice that the coefficients 1 and 2 in formula (7.55) may depend on G and G : I

I

D .G; G ; /:

(7.62)

However, they cannot vary under unimodular transformations: I

I

I

I

.G; G ; / D .H G HT ; H G HT ; /;

8H 2 U;

(7.63)

i.e. must be scalar AI -unimodular functions of two tensor arguments. Such functions are called simultaneous AI -invariants relative to the group U . Theorem 7.4. A functional basis of independent simultaneous AI -invariants .U /

I

I

I

I

J .G; G / of the tensors G and G relative to the unimodular group U consists of not more than five elements, which can be chosen as follows: I

I

I

I

J.U / D I .G1 G /; D 1; 2; 3I J4.U / D I3 .G/; J5.U / D I3 .G /: (7.64) I

I

.U / H Each simultaneous invariant J .G; G / is an AI -unimodular function of two tensor arguments, i.e. it satisfies Eq. (7.63). Then the invariant is an isotropic tensor function of two arguments if H D QT and Q 2 I U ; i.e. it is a simultaneous invariant relative to the group I .

7.2 Models An and Bn of Fluids of the Differential Type

477

In the group I , a functional basis of simultaneous invariants can be formed by I

I

I

I

contractions of powers of the tensors: G1 G , .G1 /2 G etc. However, among all the contractions, only contractions of the tensor I

I

G D G1 G

(7.65)

are invariants relative to the group U (see Exercise 7.2.1). The number r of independent invariants of this tensor cannot exceed three (invariants of the tensor relative to the group U are its invariants relative to the group I too, and for the group I : r D 3). As these invariants we can choose I .G /, D 1; 2; 3. Moreover, each I

I

of the tensors G and G has one unimodular invariant (see Theorem 4.31), these I

I

are det G and det G , respectively. There are no other independent simultaneous I

I

invariants of the tensors G and G relative to this group. N Remark 1. Notice that in the theorem the number r does not exceed 5; but the invariI

I

ant I3 .G / is not independent: it may be expressed in terms of I3 .G/ and I3 .G /. Let us show this fact. .U / Consider the representation (7.58), then simultaneous AI -invariants J (7.64) can be written as functions of the principal invariants of the tensor D and of the density : I

I

J1.U / .G; G / D I1 .G / D 2I1 .D/; .U /

J2

I

1 2 .I .G / I1 .2G // D 4I2 .D/; 2 1

.U /

P 1 D 8I3 .D/; .G; G / D det G D det G det G 1 ı ı .U / J4.U / D .=/2 ; J5 D .=/2 I3 .D/: 8

I

.G; G / D I2 .G / D J3

.U /

Hence J5

I

I

.U /

is not independent, because it is expressed in terms of J3 .U /

J5

.U /

D J3

.U /

J3

:

(7.66) .U /

and J4

:

(7.67)

Due to Theorem 7.4, the viscous coefficients (7.62) can be represented as functions of the simultaneous invariants (7.64): D .J1.U / ; : : : ; J4.U / ; /;

(7.68)

or in the form D .I .D/; ; /;

D 1; 2; 3:

(7.69)

478

7 Continua of the Differential Type

7.2.4 Tensor of Viscous Stresses in Model AV of Fluids of the Differential Type Let us consider now the model AV .

V

V

The tensors T and G are transformed during the passage to K by formulae (4.200a) and (4.203). On substituting these formulae into (7.23e), we obtain that the function of viscous stresses must satisfy the relation V

V

V

V

H F v .G; G ; / HT D F v .H1T G H1 ; H1T G H1 ; / V

(7.70)

V

8H 2 U . Such a tensor function F v .G; G ; / is called AV -unimodular or AV -indifferent relative to the group U . According to the Onsager principle, this function must be quasilinear: V

V

V

V

Tv D F v .G; G ; / D 4 Lv G :

(7.71)

The condition for AV -unimodularity (7.70) goes into the condition for the isotropy of a tensor function when H D QT and Q 2 I U ; therefore, for the function (7.71) one of the representations of an isotropic function in the tensor basis (7.29) with two independent constants must hold. The following isotropic tensor function is appropriate: V

Tv D F v D V

V V V V 1 .1 I1 G1 C 2 G1 G G1 /; 4

(7.72)

V

where I1 D I1 .G1 G /: We can immediately verify that this function is AV -unimodular, i.e. satisfies the relation (7.70) (Exercise 7.2.3). Thus, the following theorem holds. Theorem 7.5. Any quasilinear AV -unimodular tensor function of viscous stresses (7.71) can be represented in the form (7.72) or, with the help of the Cauchy stress tensor, in the form (7.58). H The first part of the theorem has been proved above, therefore we will show only equivalence of representations (7.72) and (7.58). V

Going from Tv to Tv by the formula V

Tv D F Tv FT ;

(7.73)

7.2 Models An and Bn of Fluids of the Differential Type

479

from (7.72) we obtain V V V V 1 F .1 G1 I1 C 22 G1 G G1 / FT 4 1 P F1 /: D .21 I1 E C 42 F1T G 4

Tv D

V

(7.74)

V

Here we have taken into account that G D 12 G D 12 FT F and G1 D 2G1 D P in terms of D: 2F1 F1T : With the help of formula (2.284) we can express G V

P D 2G D 2FT D F: G

(7.75)

Using the relations V

V

P D 2F1 F1T FT D F D 2I1 .D/; I1 .G1 G / D G1 G

(7.76)

from (7.74) and (7.75) we actually obtain representation (7.58). N

7.2.5 Viscous Coefficients in Model AV of a Fluid of the Differential Type V

V

Coefficients in (7.72) are functions of G and G : V

V

D .G; G ; /

(7.77)

and remain unchanged under unimodular transformations: V

V

V

V

.G; G ; / D .H1T G H1 ; H1T G H1 ; /;

8H 2 U I (7.78)

i.e. they are scalar AV -unimodular functions. In other words, they are simultaneous V

V

AV -invariants of the tensors G and G relative to the group U . For these invariants, Theorem 7.4 and Remark 1 still remain valid. Therefore, (7.77) can be represented as functions D .J1.U / ; : : : ; J4.U / ; /

(7.79)

of simultaneous AV -invariants formed by formulae (7.64): V

V

J.U / D I .G1 G /;

D 1; 2; 3;

V

J4.U / D det G:

(7.80)

480

7 Continua of the Differential Type

These invariants can be expressed in terms of the principal invariants of the tensor D and : V

V

J1.U / .G1 G / D 2I1 .D/; J2.U / D 4I2 .D/; .U /

J3

.U /

D 8I3 .D/; J4

D

1 ı .=/2 ; 8

(7.81)

hence the viscous coefficients (7.79) can be written in the form (7.69) as well.

7.2.6 General Representation of Constitutive Equations for Fluids of the Differential Type Since for both the models AI and AV constitutive equations can be written in the single generalized form (7.58) and (7.68), it is appropriate to give a common name for the models of fluids of the differential type. Definition 7.2. The models AI and AV (and also BI and BV coincident with them) of fluids of the differential type, whose constitutive equations (7.51), (7.55), and (7.72) can be represented in the single form (7.58), (7.68), i.e. 8 ˆ T D Te C Tv ; ˆ ˆ ˆ ˆ 0:

(7.147)

The elongation function ı1 .t/ D k1 .t/ 1 (measured in experiments) for a beam of material of the differential type under such loading is usually called the creep

494

7 Continua of the Differential Type

Fig. 7.1 The stepwise process of loading (a) and the typical curve of creep (b)

a

b

Fig. 7.2 The process of loading–unloading (a) and creep curve in this process (b)

a

b

curve. A typical creep curve for materials, whose properties depend on the deformation rate, is shown in Fig. 7.1. There are three typical sections along this curve: (1) the initial section, (2) the section of steady creep, and (3) the section of unsteady creep. The value of elongation ı1 .0C / at t ! 0C is called the instantly elastic elongation. The creep effect is observed for metals and alloys under high temperatures. If at the time t1 unloading occurs (Fig. 7.2): 11 .t/ D 0 .h.t/ h.t t1 //;

(7.148)

then the elongation ı1 .t/ decreases, and for materials of the differential type the elongation does not return to its initial zero value as t ! 1 (Fig. 7.2), i.e. there appears a residual deformation of creep ı1 .1/. If the process of loading 11 .t/ is given, then Eq. (7.144) is a nonlinear ordinary differential equation for the functions k1 . Solving the equation, we obtain k1 .t/ D H

1

Z

t 0

! 11 . / d ; 2

2 H.k1 / 3

Z

k1 1

.n/ 2

L .k/ d k: k

(7.149)

On substituting (7.146) into (7.149), we find that under a constant load the elongation of the beam grows with time: ı1 .t/ D H 1 . 0 t =2 / 1;

(7.150)

and under unloading the elongation remains constant and does not decrease: ı1 .t/ D ı1 .t1 / at t > t1 .

7.4 The Problem on a Beam in Tension

495

Fig. 7.3 Creep curves for nickel alloy (solid curves are computations, dashed lines are experimental data)

Figure 7.3 shows computed and experimental creep curves ı1 .t/ for nickel alloy at temperature 1100ıC in compression for different values of the stress 0 6 0. Theoretical curves have been computed according to formula (7.149) for different models Bn , and the creep curves ı1p .t/ are the difference in elongation between the experimental values ı1 .t/ and the initial value: ı1p .t/ D ı1 .t/ ı1 .0C /. One of the experimental creep curves ı1p .t/ at the smallest value 0 was used for determining the constant 2 , that was calculated by minimizing the mean-square distance between computed and experimental results at N points being times ti : !1=2 N ı1 .ti / 2 1 X ! min: (7.151) 1

D N ı1p .ti / i D1

Table 7.1 shows values of the constant 2 for nickel alloy, computed by the method mentioned above for different models Bn . The model BII exhibits the best approximation to the experimental data for the case considered (Fig. 7.3). It should be noted that for many metals, values of the high-temperature creep elongation ı1p .t/ at sufficiently great t considerably exceed instantly elastic values ı1 .0C /, therefore the last ones are often neglected in computations of creep problems. The considered models Bn of continua of the differential type with steady creep fall into this class of models.

496

7 Continua of the Differential Type Table 7.1 Values of the constant 2 and the relative error ı of approximation to the creep curves for models Bn of continua of the differential type for nickel alloy at temperature 1100ı C n I II IV V 2 , GPas 30 30 32 35 ı, % 18 17 19 20

Fig. 7.4 Diagrams of deforming for nickel in compression

7.4.5 Analysis of Deforming Diagrams for Different Models Bn of Continua of the Differential Type Consider one more regime of deforming, when the deforming process of a beam is given by the law ( bt 2 =2t1 ; t < t1 ; k1 .t/ D 1 C (7.152) b.t t1 =2/; t > t1 : where b D const is the rate of deforming, and t1 is the beginning of deforming with a constant rate (the initial interval 0 6 t 6 t1 is necessary for the consistency conditions (7.50) to be satisfied). The rate kP1 is determined by the expression kP1 D and

p

.2k1 b=t1 /

kP1 D b

when t < t1

when t > t1 :

Substituting (7.152) into (7.144), we obtain the relation 11 .k1 / called the diagram of deforming for the beam considered. Figure 7.4 exhibits the experimental diagram of deforming for nickel in compression at 1100 ı C and b D 0:00025 s1 and computed diagrams, for which the viscosity coefficient 2 has been evaluated with the help of the creep curves (see Sect. 7.4.4 and Table 7.1). The model BII gives the best approximation to the experimental data.

Chapter 8

Viscoelastic Continua at Large Deformations

8.1 Viscoelastic Continua of the Integral Type 8.1.1 Definition of Viscoelastic Continua Besides models of the differential type considered in Chap. 7, in continuum mechanics there are other types of nonideal media. One widely uses models of viscoelastic materials, which are also called continua of the integral type, or hereditarily elastic continua. Models of viscoelastic continua most adequately describe the mechanical properties of polymer materials, composites based on polymers, different elastomers, rubbers and biomaterials, in particular, human muscular tissues. Further, viscoelastic continua of the differential type will be called continua of the differential type, and viscoelastic continua of the integral type – simply viscoelastic materials. Definition 8.1. A medium is called a viscoelastic continuum of the integral type (or simply a viscoelastic continuum), if any of the models An , Bn , Cn or Dn is assumed for the medium, and corresponding operator constitutive equations (4.156) or (4.158)–(4.161) are functionals of time t: Dt

ƒ.t/ D f .R.t/; Rt .//;

(8.1)

D0

i.e. values of active variables ƒ.t/ depend not only on values of reactive variables R.t/ at the same instant of time but also on their prehistory Rt ./ R.t /, i.e. on their values at all preceding instants of time 0 < 6 t, starting from initial one D 0. Due to such specific dependence, viscoelastic continua are also called continua with memory. For viscoelastic continua: 1. “The present can depend only on the past but not on the future”; therefore, all the functionals (8.1) depend only on their previous history (prehistory) R.t /, 0 < 6 t. Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 8, c Springer Science+Business Media B.V. 2011

497

498

8 Viscoelastic Continua at Large Deformations

2. “The past is not infinite”, i.e. the times > t do not affect the functionals (8.1). This means that R./ 0 at < 0I

Rt ./ D R.t / 0 at t > :

(8.2)

8.1.2 Tensor Functional Space To perform operations with functionals (8.1), we need some additional knowledge of functional analysis. Consider the set of previous histories of k-order tensors k Tt ./ D k T.t / (0 < 6 t), and define for two arbitrary prehistories k Tt1 and k Tt2 their scalar product as follows: Zt .

k

Tt1 ; k Tt2 /t

k

D

Tt1 ./ „ƒ‚… : : : .k Tt2 .//.k:::1/ 2 ./ d :

(8.3)

k

0

The function ./ is called the function of memory. This function is positive, continuous, monotonically decreasing, defined within the interval Œ0; C1/ and squared-integrable, i.e. Z1 2 ./ d D 02 < C1: (8.4) 0

Since the memory function is monotonically decreasing, quantities of the prehistory k Tt ./ D k T.t / at small values of give a greater contribution to the scalar product (8.3) than quantities of k Tt ./ at large values of . In other words, a continuum better remembers events having occurred at times closer to the current instant t than the ones at more remote times. The functionals (8.1) are assumed to have this property too; therefore, viscoelastic continua of the integral type are also called continua with fading memory. Let us consider now the set k Ht of processes of changing the tensor k T./ (0 < 6 t) and assign to each process the pair .k T.t/; k T.// consisting of values of the tensor k T.t/ at time t and its prehistory k T./ D k T.t / (0 < 6 t). Then we can introduce the scalar product of processes k T1 ./ and k T2 ./ included in k Ht :

where

k

k

T1 ; k T2

t

D

k

T.t/; k T.t/ C k Tt1 ; k Tt2 ; t

.k ::: 1/ T1 .t/; k T2 .t/ D k T1 .t/ „ƒ‚… : : : k T2 .t/ k

is the scalar product of k-order tensors.

(8.5)

(8.6)

8.1 Viscoelastic Continua of the Integral Type

499

The set k Ht of all processes of changing the tensor k T./ (0 6 6 t), for which the scalar product (8.5) exists and which at each fixed are elements of the tensor space T3k .E3 / with the operations of addition and multiplication by a number, is called the tensor functional space k Ht . The space k Ht is a Hilbert space, because we can always go into a Cartesian basis, where the components TN i1 :::ik ./ of tensors included in k Ht are squared.m/ integrable functions, i.e. they belong to the function space L2 Œ0; t (where m D 3k ), which is known as a Hilbert space. Due to the property (8.2), the scalar product of processes included in k Ht (8.5) can be written in the form Z1 k

.

Tt1 ; k Tt2 /t

D

k

Tt1 ./ „ƒ‚… : : : .Tt2 .//.k:::1/ 2 ./ d < C1;

(8.7)

k

0

which often proves to be convenient for analysis of the models of a viscoelastic continuum. In the space k Ht , there is a natural norm of the process k T./: k k T kD .k T; k T/1=2 t ;

(8.8)

where . /t is the scalar product (8.5).

8.1.3 Continuous and Differentiable Functionals Using the norm (8.8), we can introduce the concept of a continuous functional in the form (8.1) m

t

S D m F .k T.t/; k Tt .//; D0

(8.9)

which is considered as the mapping of the domain U contained in the space k Ht into the domain V of the space m Ht : m

FW

U k Ht ! V m Ht :

(8.10)

Definition 8.2. The functional (8.10) is called continuous in a domain U k Ht , if for each process k T 2 U the following condition is satisfied: 8" > 0 9ı > 0 such T, where .k T C k e T/ 2 U and that for every process k e T kt < ı; k ke

(8.11)

500

8 Viscoelastic Continua at Large Deformations

values of the operators in the norm (8.8) of the space k Ht are sufficiently close: t

kmF

k

D0

t T.t/; k Tt ./ C k e Tt ./ m F .k T.t/; k Tt .// kt < ": T.t/ C k e D0

(8.12)

The functional (8.10) is called linear if it satisfies the two conditions

t

F

k

D0

T1 .t/ C k T2 .t/; k T1 ./ C k T2 ./ t

D F

D0

t

F

D0

k

t T1 .t/; k T1 ./ C F k T2 .t/; k T2 ./ ; D0

t s k T.t/; s k T./ D s F k T.t/; k T./ ; D0

(8.13)

(8.14)

for all processes k T1 ./ and k T2 ./ included in k Ht and for every real number s. In space k Ht we can use Riesz’s theorem that any linear functional (8.9) can be represented as the scalar product of a fixed element from k Ht and an arbitrary process k T./ from k Ht ; so a scalar linear functional in 2 Ht has the form e.t; t/ TT .t/ C f .T.t/; T .// D

Zt

t

e .t; t / TT .t / 2 ./ d ; (8.15a)

0

e.t; t/ is the instantaneous value at D t of where e .t; t / is the prehistory and e the fixed process .t; / (0 6 6 t) for the given functional f (the appearance of one more argument t for the process e .t; / means that this process can vary with changing the time interval considered). Replacing the variable t D y and introducing the notation e T .t; y/ 2 ./ D T .t; y/, 0 D .t; t/, after the reverse substitution y ! we obtain another representation of the linear scalar functional Zt f D 0 T.t/ C

.t; / T./ d ;

(8.15b)

0

called Volterra’s representation. For viscoelastic continua, it is convenient to use the Dirac ı-function ı.t/ having the following main property: (

Zt B.t; /ı.t0 / d D 0

for any continuous tensor process B.t; /.

B.t; t0 /; t0 2 Œ0; t; 0;

t0 … Œ0; t;

(8.16)

8.1 Viscoelastic Continua of the Integral Type

501

According to (8.16), the linear functional (8.15b) can be written in the form Zt A.t; / T./ d ;

f D

(8.15c)

0

A.t; / D 0 ı.t / C .t; /:

(8.15d)

Definition 8.3. The functional (8.9) is called Fr´echet–differentiable at point m T 2 U of a domain U m Ht , if there exist two functionals @F and ıF having the following properties: they are defined over the Cartesian product of the space Ht

@m F W

k

Ht k Ht ! m Ht I

ımF W

k

Ht k Ht ! m Ht ;

(8.17)

they can be written in the form similar to (8.9) m

t ˇ T.t// P1 D @m F .k T.t/; k Tt ./ˇk e D0

D

@

t

F .k T.t/; k Tt .// : : : k T.k:::1/ .t/;

@k T.t/ D0

m

t ˇ t T .//; P2 D ı m F .k T.t/; k Tt ./ˇk e

(8.18) (8.19)

D0

(the vertical line separates two different arguments of the process), they are linear and continuous in the second argument, they satisfy the condition: 8" > 0 9ı such that for any process .k e T.t/; k e Tt .//,

for which .k T1 .t/; k Tt1 .// U and k k T kt < ı;

(8.20)

k m F kt 6 " k k e T kt ;

(8.21)

the following inequality holds:

where t

t

m F D m F .k T1 .t/; k Tt1 .// m F .k T.t/; k Tt .// D0

D0

t ˇ ˇ t T.t// ı F .k T.t/; k Tt ./ˇk e T .//; @m F .k T.t/; k Tt ./ˇk e t

D0

D0

(8.22) k

. T1 .t/;

k

Tt1 .//

ke

k et

. T.t/ C T.t/; T ./ C T .//: k

k

t

(8.23)

502

8 Viscoelastic Continua at Large Deformations

The operator (8.19) is called the Fr´echet–derivative; and the right-hand side of expression (8.18) is the partial derivative of F (considered as a tensor function of T.t/) with respect to the tensor argument T.t/. If the functional (8.9) is Fr´echet–differentiable, then it is continuous (see Exercise 8.1.2). The set of processes k T.t/ 2 k Ht 0 having the first and second continuous derivaP R and k T.t/, which belong to k Ht 0 , will be denoted tives with respect to time t: k T.t/ by Ut 0 . Theorem 8.1. Let the functional (8.9) be Fr´echet–differentiable in k Ht 0 , then there exists such t (t 2 .0; t 0 /) that for all processes k T./ 2 Ut the process m S.t/ is differentiable with respect to t and the following rule of differentiation of the functional with respect to time holds: d dt

m

S.t/ D

@ @

t

F .k T.t/; k Tt .// : : :

k T.t/ D0

d k .k:::1/ T .t/ dt

t ˇ C ı F .k T.t/; k Tt ./ˇk TP t .//: D0

Here k

TP t D

(8.24)

d d k T.t / D k Tt ./: d.t / dt

H A proof of the theorem can be found in [31]. N Remark. The theorem gives the possibility to calculate the Fr´echet–derivatives of the operators (8.9) by evaluating the ordinary derivative of the functions S.t/ with respect to t according to formula (8.24). t u Example 8.1. Determine the Fr´echet–derivatives of the linear operator (8.15b) for the case when .t; / D .t / and .t; t/ D .0/. According to formula (8.24), we calculate the ordinary derivative with respect to time t by the rule of differentiation of an integral with a varying upper limit: d dT f D 0 .t/ C .0/ T.t/ C dt dt

Zt

0 .t / T./ d ;

(8.25)

0

where 0 .y/ D @.y/[email protected] Comparing (8.25) with (8.24), we find the partial derivative and the Fr´echet–derivative: d @F T.t/; D 0; dt @T Zt ıF D .0/ T.t/ C 0 .t / T./ d : @F D 0

0

(8.26)

8.1 Viscoelastic Continua of the Integral Type

503

8.1.4 Axiom of Fading Memory For viscoelastic continua, in addition we assume the following axiom. Axiom 18 (of fading memory). The functionals (8.1) occurring in constitutive equations of viscoelastic continua are Fr´echet–differentiable and hence satisfy the rule of differentiation with respect to time (8.24): t t ˇ t d @ d P .//: (8.27) f .R.t/; Rt .// R.t/ C ı f .R.t/; Rt ./ˇR ƒ.t/ D dt @R.t/ D0 dt D0

The interconnection between Fr´echet–differentiability and fading memory of functionals is clarified by the theorem on relaxation. Let there be a process R./ (0 6 6 t), which is arbitrary up to some time t0 < t, and when t0 > > t it remains constant: R./ D R.t0 /;

t0 > > t:

(8.28)

Such process R./ is called a process with the constant extension (Fig. 8.1). In addition, consider a static process

R./ D R.t0 / D const;

0 6 6 t:

(8.29)

Then the following theorem can be formulated. Theorem 8.2. Let the functional f (8.1) be Fr´echet–differentiable, then its partial derivative @f and Fr´echet–derivative ıf , and also the function ƒ.t/ for any process R./ with the constant extension at fixed time t0 have the limits as t ! C1, which

Fig. 8.1 For Theorem 8.2

504

8 Viscoelastic Continua at Large Deformations

coincide with values of the derivatives @f , ı f and ƒ , respectively, in correspond

ing static processes R./: lim

t0

t

f .R.t/; Rt .// D ƒ f .R.t0 /; Rt0 .//;

t !C1 D0

D0

t0 ˇ ˇ P D @f @ f .R.t0 /; Rt0 ./ˇ0/; lim @ f .R.t/; Rt ./ˇR.t// t

t !C1 D0

D0

t0 ˇ ˇ lim ı f .R.t/; Rt ./ˇRP t .// D ı f ı f .R.t0 /; Rt0 ./ˇ0/: t

t !C1 D0

(8.30)

D0

In simple words, for a process R./ such that starting from some time t0 the process reaches a constant level, in a certain time interval a viscoelastic continuum forgets the process R./ up to the time t0 , because the continuum response, expressed by the functionals f , @f and ıf , at sufficiently great values of t differs little from its response to a static process. H Consider the process

e R./ D R./ R./;

0 6 6 t:

(8.31)

Since e R./ 0

at t0 > > t;

(8.32)

so we have e kR

k2t

e2

Zt

e2

Zt

R .t / ./ d D

D R .t/ C

2

0

Zt0 D

e 2 ./ 2 .t / d R

0

e 2 ./ 2 .t / d 6 c R

Zt 2 ./ d :

(8.33)

t t0

0

Due to the property (8.4) and monotone decreasing the function ./, we find that e kt ! 0 at t ! C1. kR Since a Fr´echet–differentiable functional f is continuous as well as @f and ıf , the condition (8.12) yields the inequality t

t

e e t .// f .R.t/; Rt .// kt < "I Rt ./ C R k f .R.t/ C R.t/; D0

(8.34)

D0

this means that the first limit of (8.30) exists. In a similar way, we can prove that the second and third limits of (8.30) exist too. N

8.1 Viscoelastic Continua of the Integral Type

505

8.1.5 Models An of Viscoelastic Continua For models An of viscoelastic continua, the free energy is a functional in the form (8.1), and as reactive variables one should choose the set (4.148): D

.n/

t

!

.n/ t

t

C .t/; .t/; C ./; ./ :

(8.35)

D0

According to the rule (8.24) of differentiation of a functional, we obtain the expression for the total derivative of with respect to t: .n/

d dC @ @ d D C Cı : .n/ dt dt @ dt @C

(8.36)

Substituting this expression into PTI (4.121) and collecting like terms, we get 0

.n/

1

.n/ TC [email protected] @ .n/ A d C C @C

@ w C d C Cı dt D 0: @

.n/

(8.37)

.n/

When the prehistories C t , t and the current values C .t/ and .t/ are fixed, the .n/

increments d C , d and dt can vary arbitrarily, therefore the identity (8.37) holds when and only when the coefficients of these increments vanish. As a result, we obtain the equation system 8 .n/ .n/ .n/ t .n/ ˆ ˆ ˆ T D [email protected] [email protected] C / D F . C .t/; .t/; C t ./; t .//; < D0 (8.38) D @ [email protected]; ˆ ˆ ˆ : w D ı ; that together with (8.35) is a system of constitutive equations for models An of viscoelastic continua. Just as for ideal continua, for viscoelastic materials it is sufficient to give only the free energy functional (8.35), then the remaining relations are determined by its differentiation according to formulae (8.38). Notice that although relations (8.38) are formally similar to the corresponding relations (4.168) for models An of ideal continua, they essentially differ by the fact .n/

that in (8.38) there is a functional dependence on C and . Moreover, viscoelastic continua are dissipative: for them the dissipation function w is not identically zero.

506

8 Viscoelastic Continua at Large Deformations

8.1.6 Corollaries of the Principle of Material Symmetry for Models An of Viscoelastic Continua According to the principle of material symmetry (Axiom 14), for each viscoelastic b Just as in continuum there exists an undistorted reference configuration K. ı

Sect. 4.7.4, for simplicity, let the reference configuration K be undistorted. Then ı

ı

for K there is a subgroup G s U of the unimodular group U such that for each ı

ı

transformation tensor H 2 G s (H W K ! K) the constitutive equations (8.38) writı

ten for K are transformed during the passage to K as follows: 8.n/ .n/ .n/ t .n/ ˆ t t ˆ T D F . C ; ; C ; / D [email protected] [email protected] C /; ˆ ˆ D0 ˆ ˆ ˆ .n/ t .n/ < . C ; ; C t ; t / ; D D0 ˆ ˆ ˆ D @ [email protected]; ˆ ˆ ˆ ˆ ı : .w / D .ı / ; 8H 2 G s :

(8.39)

.n/ .n/

For viscoelastic solids, due to H -indifference of all the tensors T , C .t/ and .n/

.n/

C t ./ D C.t / (see Sect. 4.7.4), relations (8.39) take the forms

.n/

t

.n/

t

.n/

.n/

QT F . C ; ; C t ; t / Q D F .QT C Q; ; QT C t Q; t /; D0

t

D0

.n/

D0

t

. C ; / D

.n/

.n/

(8.40a)

ı

.QT C Q; ; QT C t Q; t / 8Q 2 G s ;

(8.40b)

D0

ı

Dı

:

(8.40c)

Theorem 8.3. The principle of material symmetry in the form (8.40) holds for models An of viscoelastic solids if and only if the condition (8.40b) for is satisfied. H It is evident that the condition (8.40b) is necessary. Prove that this condition is sufficient. Let the condition (8.40b) be satisfied. Then, since the functional F is a tensor function being the derivative of with respect to .n/

C .t/, so with the help of the method used in the proof of Theorem 4.23 we can prove that the condition (8.40b) yields (8.40a). To prove Eq. (8.40c) we should use

and rewrite it in K:

formula (8.36) for ı

.n/

ı

d 1 .n/ d C @ d T : D dt dt @ dt

(8.41)

8.1 Viscoelastic Continua of the Integral Type

Since

D

507

ı

and the passage from K to K is independent of t, so d =dt D d .n/

.n/

.n/

=dt.

.n/

Due to H -indifference of the tensors T and C , we have the relation T .d C =dt/D .n/

.n/

T .d C =dt/, and also @ [email protected] D @ [email protected]. Thus, the right-hand side of Eq. (8.41) coincides with ı , and hence .ı / D ı ; i.e. Eq. (8.40c) actually holds. N

8.1.7 General Representation of Functional of Free Energy in Models An Scalar functional

(8.35) satisfying the condition (8.40b) is called functionally ı

indifferent relative to the group G s . Let us find a general representation of such a ı

.n/

functional in terms of invariants of tensor C of the corresponding group G s . In Sect. 8.1.3 we have derived a general representation of a linear scalar functional in the form (8.15d). Similarly to (8.15d), define a quadratic scalar functional as a double integral in the form Zt Zt 2

4

D 0

.n/

.n/

A.t; 1 ; 2 / . C .1 / ˝ C.2 //.4321/ d 1 d 2 ;

(8.42)

0

where 4 A.t; 1 ; 2 / is a fixed fourth-order tensor called the core of the functional. We also define a n-fold scalar functional: Zt m

D

Zt :::

0

e m .t; 1 ; : : : ; m / d 1 : : : d m ;

(8.43)

0

where .n/

.n/

e m .t; 1 ; : : : ; m / D 2n A.t; 1 ; : : : ; m / : : : . C .1 /˝: : :˝ C .m //.m;m1;:::;2;1/ : „ƒ‚… 2m

(8.44)

Here 2n A.t; 1 ; : : : ; m / is the core of this functional (it is a fixed tensor of order .2n/ depending on m C 1 arguments). Theorem 8.4 (Stone–Weierstrass). Any continuous scalar functional (8.35) in space Ht can be uniformly approximated by n-fold scalar functionals (8.43): D

t

.n/

.n/ t

. C .t/; .t/; C ./; .// D

D0

t

1 X

m;

(8.45)

mD1

where the equality means that the partial sum uniformly converges in the norm (8.8).

508

8 Viscoelastic Continua at Large Deformations

H A proof of the theorem for the space Ht can be found in [9]. N Let us consider the integrand n .t; 1 ; : : : ; n / of the n-fold functional (8.43). At any fixed set of values t; 1 ; : : : ; m , this expression is a scalar function (but not .n/

.n/

a functional!) of n tensor arguments C.i / C i (i D 1; : : : ; m): em .t; 1 ; : : : ; m / D em .t; 1 ; : : : ; m ; C1 ; : : : ; Cm /:

(8.46)

Here while values of t; 1 ; : : : ; m vary, the number and the form of tensor arguments of the functions remain unchanged. On substituting the representation (8.45) into the condition (8.40b) of functional indifference of , we find that the functions e m (8.44) at each fixed value t; 1 ; : : : ; m must satisfy the condition em .t; 1 ; : : : ; m ; C1 ; : : : ; Cm /

D em .t; 1 ; : : : ; m ; QT C1 Q; : : : ; QT Cm Q/;

(8.47)

ı

i.e. they must be H -indifferent scalar functions relative to the group G s . Theorem 8.5. Every scalar function (8.46) of m tensor arguments C1 ; : : : ; Cm , ı

which is indifferent relative to a group G s , can be represented as a function of finite number z (z 6 6m) of simultaneous invariants J.s/ D J.s/ .C1 ; : : : ; Cm /;

D 1; : : : ; z;

(8.48)

ı

relative to the group G s in the form em D e m .t; 1 ; : : : ; m ; J.s/ .C1 ; : : : ; Cm //:

(8.49)

H By analogy with simultaneous invariants of two tensors, which have been considered in Sect. 7.1.4, we now introduce simultaneous invariants of m tensors relative ı

to a group G s . The simultaneous invariants are scalar functions J (8.48), which are ı

H -indifferent relative to the group G s ; i.e. they satisfy the relations

J.s/ .C1 ; : : : ; Cm / D J.s/ .QT C1 Q; : : : ; QT Cm Q/ 8Q 2 Gs :

(8.50)

Their functional basis consists of z simultaneous invariants, where z is a finite number, which cannot exceed the total number of components of all the tensors, i.e. z 6 6m. Moreover, since J.s/ form a basis, any other H -indifferent scalar funcı

tion relative to the same group G s can be expressed in this basis. But the function e (8.40) is just such a function due to (8.47), therefore the relation (8.49) actually holds. N

8.1 Viscoelastic Continua of the Integral Type

509

Substitution of the expression (8.49) into (8.43) and then into (8.45) yields the following general representation of the continuous functional (8.35), which is funcı

tionally indifferent relative to a group G s : D

1 Z X mD1 0

Zt

t

:::

.n/

.n/

e m .t; 1 ; : : : ; m ; J . C .1 /; : : : ; C.m /// d 1 : : : d m :

0

(8.51)

In deriving this formula we have used representation (8.15c) of linear functionals with the help of ı-function. Let us perform now the inverse operation: segregate the ı-type constituent from the cores e m , that allows us to separate the instantaneous .n/

value of the tensor C .t/ from its prehistory. Formula (8.15d) can be generalized for functions of m C 1 arguments t; 1 ; : : : ; m as follows: m X

e m .t; 1 ; : : : ; m ; J.s/ / D

ı.t 1 / : : : ı.t k /

kC1 ; : : : ; m ; J.s/ /:

mk .t;

kD0

(8.51a)

We assume that at k D m the argument kC1 D mC1 does not appear among .s/ arguments of the function mk : mm D mk .t; J /. On substituting (8.51a) into (8.51), we get D

1 X m Z X

Zt

t

.n/

::: „ƒ‚…

mD1 kD0 0

mk

mk 0

.n/

.n/

.n/

t; kC1 ; : : : ; m ; J.s/ C.t/; : : : ; C .t/; ƒ‚ … „ k

!!

C.kC1 /; : : : ; C.m /

d kC1 : : : d m :

(8.52)

Rearrange summands in the expression (8.52) and take into account that simulta.n/

neous invariants J.s/ of m tensors, among which there are k tensors C.t/, can always be expressed in terms of simultaneous invariants of .m k C 1/ ten.n/

.n/

.n/

sors C .kC1 /; : : : C.m / and C.t/. Then we finally obtain the general form of the functional (8.35) .n/

D '0 .t; J.s/ . C .t/// C

1 Z X mD1 0

Zt

t

::: „ƒ‚… m

.n/ 'm t; 1 ; : : : ; m ; J.s/ . C .t/;

0

.n/ C.1 /; : : : ; C.m // d 1 : : : d m ;

.n/

(8.53)

510

8 Viscoelastic Continua at Large Deformations ı

which is functionally indifferent relative to the group G s . Here we have introduced the notation: '0 – the instantly elastic part and 'm – cores of the functional: 1 X

'0 D

mm .t;

J.s/ .C.t///;

mD1

'm D

1 X kD0

mCk;k .t;

.n/

.n/

1 ; : : : ; m ; J.s/ . C.t/ : : : C .t/; C.1 /; : : : ; C.m ///: ƒ‚ … „ k

.s/

(8.54)

.n/

.s/

.n/

Simultaneous invariants J . C .t// of one tensor are simply invariants I . C.t// ı

of the tensor relative to the same group G s . The expression (8.53) is the desired general representation of the functional (8.35) for models An .

8.1.8 Model An of Stable Viscoelastic Continua Definition 8.4. One can say that this is the model An of a stable viscoelastic continuum, if the functional of the model is invariant relative to a shift of the pro.n/

cess of deforming and heating in time; i.e. if there are two processes . C ./; .// .n/

C ./; e .//, 0 6 6 t1 , such that they are different from each other only by a and e shift in time: 8 < .n/ .n/ e e . C ./; .// D . C . t0 /; . t0 //; t0 < 6 t1 ; (8.55) :.0; /; 0 6 6 t0 I 0 then the corresponding values of the functionals and e are different only by a shift in time as well (Fig. 8.2): ( .t t0 /; t0 < t 6 t1 ; eD (8.56) .0/; 0 6 t 6 t0 : Remark. Since the functional

for stable continua is invariant relative to a shift .n/

in time, its partial derivatives with respect to C .t/ and .t/ and also the Fr´echet– derivative ı have this property; hence the constitutive equations (8.38) are invariant relative to a shift in time too. Due to the property of invariance, stable continua are also called non-aging, and their constitutive equations do not change with time themselves when there are no deformations and variations of temperature. t u Let us consider two important models of stable continua.

8.1 Viscoelastic Continua of the Integral Type

511

Fig. 8.2 For the definition of a stable viscoelastic continuum

8.1.9 Model An of a Viscoelastic Continuum with Difference Cores One can say that this is the model An of a viscoelastic continuum with difference cores, if in the general representation of the functional (8.53) there is no explicit dependence of cores 'm on the times t and i , but there is a dependence only on their difference t i or on the temperatures .t/ and .i /: .n/

.n/

.n/

'm D 'm .t1 ; : : : ; tm ; .t/; .1 /; : : : ; .m /; J.s/ . C .t/; C.1 /; : : : ; C.m ///; (8.57) and '0 does not depend explicitly on temperature: .n/

'0 D '0 .I.s/ . C .t//; .t//:

(8.58)

Functions 'm are assumed to satisfy the following conditions of normalization and symmetry with respect to any permutations of the first m arguments: 'm .t 1 ; : : : ; t m ; 0 ; : : : ; 0 ; 0; : : : ; 0/ D 0;

(8.59)

'm .y1 ; : : : ; yn ; : : : ; yl ; : : : ; ym ; ; 1 ; : : : ; m ; J.s/ / D 'm .y1 ; : : : ; yl ; : : : ; yn ; : : : ; ym ; ; 1 ; : : : ; m ; J.s/ /; where 0 D .0/, n D .n /, yn D t n , and 1 6 n; l 6 m. Simultaneous invariants J.s/ can always be chosen to satisfy the normalization conditions J.s/ .0; : : : ; 0/ D 0;

D 1; : : : ; z:

(8.60)

512

8 Viscoelastic Continua at Large Deformations

For the model with difference cores, the functional (8.53) has the form .t/ D

.n/ '0 .I.s/ . C .t//;

1 Z X

.t// C

Zt

t

:::

mD1 0

'm d 1 : : : d m ;

(8.61)

0

where 'm are determined by formula (8.57). Theorem 8.6. The model An of a viscoelastic continuum with difference cores is stable. .n/

H Let the first process be of the form C ./, 0 6 6 t, then the corresponding .n/

C ./ when 6 t0 functional .t/ has the form (8.61). Since the second process e is identically zero, so, due to the normalization conditions (8.59) and (8.60), we have 'm 0 when 0 6 i 6 t0 (i D 1; : : : ; m); therefore, the lower limits of the integrals in the expression (8.61) for functional e.t/ can be determined as t0 : .n/ e e.t/ D '0 I.s/ . e C .t//; .t/ C

1 Z X mD1 t

Zt

t

:::

' m t 1 ; : : : ; t m ; e .t/; e .1 /; : : : ; e .m /;

t0

0

.n/ .n/ .n/ J.s/ . e C .t/; e C.1 /; : : : ; e C.m // d 1 : : : d m : (8.62) .n/

.n/

C .i / by e C.i t0 / when i > t0 and then substituting i t0 D e i , we Replacing e obtain (when t > t0 ) .n/

e.t/ D '0 .I.s/ . C .t t0 //; .t t0 // tZt0 1 tZt0 X ::: 'm t t0 e 1 ; : : : ; t t0 e m ; .t t0 /; C mD1 0

0

.n/ .n/ .n/ .e 1 /; : : : ; .e C .t t0 /; C .e m /; J.s/ . e 1 /; : : : ; C.e m // de 1 : : : de m:

(8.63) Comparing (8.61) and (8.63), we verify that e.t/ and the time-shift t ! .t t0 /, because at t D t0 e.t0 / D '0 .0; .0// D i.e. the relation (8.56) holds. N

.0/;

.t/ are connected only by

8.1 Viscoelastic Continua of the Integral Type

513

On substituting the functional (8.61) into (8.38), we get the general form of constitutive equations for stable continua: .n/

T D

z X D1

.s/ @'0 @I

@I.s/ D

w D '10 C

.n/

C

1 Zt X

1 Zt X

@'0 C @ mD1

:::

0

:::

0

Zt

.s/ @'m @J

@J.s/

.n/

! d 1 : : : d m ;

@ C .t/

@'m d 1 : : : d m ; @.t/

0

Zt

Zt

mD1 0

:::

mD1 0

@ C .t/

1 X

Zt

@'m 0 C 'mC1 @t

d 1 : : : d m :

(8.64)

0

Here the cores 'm are functions in the form (8.57), and @'m [email protected] is the partial derivative of the function when its first arguments .t 1 /; : : : ; .m / vary and the arguments J.s/ are fixed (i.e. there is no differentiation with respect to J.s/ ). We have introduced the following notation for a value of the function 'mC1 (8.57) at mC1 D t: 0 D 'mC1 t 1 ; : : : ; t m ; 0; .t/; .1 /; : : : ; .m /; .t/; 'mC1 .n/ .n/ .n/ .n/ J.s/ . C .t/; C.1 /; : : : ; C.m //; C .t/ : In deriving the expression for the dissipation function we have used the conditions (8.59).

8.1.10 Model An of a Thermoviscoelastic Continuum Let us consider the most widely used method to take the dependence of cores 'm (8.57) on temperature into account. For the model An of a thermoviscoelastic continuum with difference cores, temperature appears in simultaneous invariants J.s/ , i.e. .n/

.n/

.n/

'm D 'm t 1 ; : : : ; t m ; J.s/ C .t/; C .1 /; : : : ; C .m / where .n/

.n/

ı

C ./ D C./ "./;

Z./ "./ D ˛.e /d e : ı

0

!! ; (8.65)

(8.66)

514

8 Viscoelastic Continua at Large Deformations ı

The tensor " is called the tensor of heat deformation, and ˛ – the tensor of heat expansion. Both the tensors are symmetric and H -indifferent relative to a considered ı

group G s :

QT ˛ Q D ˛

ı

8Q 2 G s ;

(8.67) ı

.n/

therefore, the functions J.s/ of C are also H -indifferent relative to the group G s . Taking the dependence of constitutive equations upon temperature as the difference between the deformation tensor and the heat deformation tensor (8.66) is called the Duhamel–Neumann model. In a similar way, the Duhamel–Neumann model describes the dependence of the function '0 on temperature: ! .n/

'0 D '0 .I.s/ C .t/ ; .t//:

(8.68)

.n/

Since @ C [email protected] D ˛, the derivatives with respect to .t/ in (8.64) for this model have the form z .s/ X @0 '0 @'0 @'0 @I ˛C D ; @ @I .n/ @.t/ D1 @C

z .s/ X @'m @'m @J ˛..t//; D @.t/ @J .n/ D1 @ C .t/ (8.68a)

where @0 [email protected] means the derivative with respect to the second argument in formula (8.68). .n/

.n/

.n/

.n/

Taking into account that @I.s/ [email protected] C D @I.s/ [email protected] C and @J.s/ [email protected] C D @J.s/ [email protected] C and substituting (8.68a) into (8.64), we obtain the expression for the specific entropy D

.n/ @0 '0 1 C ˛ T: @.t/

(8.69)

8.1.11 Model An of a Thermorheologically Simple Viscoelastic Continuum One can say that this is the model An of a thermorheologically simple viscoelastic continuum, if the cores 'm (8.57) in constitutive equations (8.53) and (8.64) depend on temperature in the functional way through the so-called reduced time: .n/ .n/ 0 ; J.s/ . C .t/; C.1 /; : : : ; 'm D 'm t 0 10 ; : : : ; t 0 m C.m // a ..1 // : : : a ..m //;

.n/

(8.70)

8.1 Viscoelastic Continua of the Integral Type

515

where 0

Zt

t D

Zi

i0

a ..e // de ;

D

0

a ..e // de

(8.71)

0

is the reduced time being a functional of the function a ./ called the function of the temperature-time shift. The functions 'm (8.70) and a satisfy the normalization conditions 0 'm 0; : : : ; 0; J.s/ D 0; 'm t 0 10 ; : : : ; t 0 m ; 0 D 0; a .0 / D 1: (8.72) .n/

If the process C ./ is considered with respect to the reduced time 1 0 Z .n/ e A D C ./; C. 0 / D e C @ a de

.n/

.n/

0

then, since d i0 D a ..i // d i , the functional (8.53) with the core (8.70) can be written with respect to the reduced time 0

.t / D '0

.n/ I.s/ . C .t 0 //;

!

t0

1 Z X

0

.t / C

Zt 0 :::

mD1 0 .n/

.n/

.n/

0 // J.s/ . C .t 0 /; C.10 /; : : : ; C.m

0 'm t 0 10 ; : : : ; t 0 m ;

0

!

0 d 10 : : : d m :

(8.73)

Substituting the functional (8.73) into (8.38) and using the differentiation rule (8.27), we obtain constitutive equations for a thermorheologically simple viscoelastic continuum 1 0 Zt 0 1 Zt .s/ X @I.s/ @'m @J @' 0 0 0 @ T D ::: C d 1 : : : d m A ; .s/ .s/ .n/ .n/ @I @J 0 D1 mD1 0 @C @ C .t / 0 z X

.n/

0

.n/

D [email protected] '0 [email protected]/ C .1=/˛ T ;

w D

a '10

C a

0 1 Zt X

mD1 0

Zt 0 ::: 0

Here we have used that @[email protected] D a [email protected][email protected] 0 /.

@'m 0 0 C 'mC1 d 10 : : : d m : @t 0

(8.74)

516

8 Viscoelastic Continua at Large Deformations

Notice that with the help of (8.38) and (8.74) the dissipation function w can be represented in another equivalent form .n/

w D T

0 .n/ d @ '0 d .n/ d C ˛ T ; C dt dt @ dt

(8.75)

which proves to be useful for cyclic loading. Theorem 8.7. A thermorheologically simple continuum is stable. H The reduced time (8.71) for the shifted process of heating e ./ D . t0 / with use of the normalization condition (8.72) can be represented in the form 0

Zt0

t D

Zt a d C

0

Zt e a ./ d D t0 C a .. t0 // d

t0

t0 tZt0

D t0 C

a ..e // de ; 0

0

t Z 0

D t0 C

a ..e // de ;

(8.76)

0

when t0 < < t. The further proof is the same as the one in Theorem 8.6 (see Exercise 8.1.1). N

Exercises for 8.1 8.1.1. Complete the proof of Theorem 8.7. 8.1.2. Using Definitions 8.2 and 8.3, prove that if a functional is Fr´echet– differentiable, then it is continuous.

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua 8.2.1 Principal Models An of Viscoelastic Continua Constitutive equations containing multiple integrals of the type (8.53), (8.61) or (8.73) are very awkward, and their application in practice is considerably difficult. Therefore, special models of viscoelastic materials, in which one may retain a finite number of integrals, are widely used.

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

517

For the principal model An of a thermoviscoelastic continuum with difference cores, the sum (8.61) contains only one integral .m D 1/, i.e. in this model has the form D

.n/ '0 .I.s/ . C /;

Zt

.n/

.n/

'1 .t ; J.s/ . C .t/; C .// d :

/

(8.77)

0

Here '0 and '1 are functions of the arguments indicated, and the function '1 is chosen with the negative sign that can always be done by simple renaming the functions. Constitutive equations for the continuum considered have the form (8.64), where m should be assumed to be equal to 1: .n/

T D

z X D1

'0 J.s/ C

Zt

'1 J.s/ d : C

(8.78)

0

Here we have denoted the partial derivatives of '0 and '1 .n/

'0 .J˛.s/ . C .t/// D [email protected]'0 [email protected].s/ /; '1 .t ;

.n/ .n/ J˛.s/ . C .t/; C .///

D

D 1; : : : ; r;

[email protected]'1 [email protected].s/ /;

(8.79)

D 1; : : : ; z; .n/

and also the partial derivative tensors of J.s/ with respect to C .t/: .n/

.n/

.s/ .s/ J.s/ C D @J [email protected] C .t/ D @J [email protected] C .t/; .s/

.n/

D 1; : : : ; z:

(8.80)

.n/

The simultaneous invariants J . C .t/; C .// can be chosen so that the first r .n/

ı

ones form a functional basis of invariants I.s/ . C .t// in the same group G s . We will assume below that J.s/ are ordered in such a way; then the following equations hold: .s/

.n/

J C D @I.s/ [email protected] C .t/; D 1; : : : ; rI

'0 0; D r C 1; : : : ; z:

(8.81)

These equations have been used for deriving the relation (8.78). According to (8.64) and (8.69), the dissipation function w and the specific entropy for the principal models An have the forms .n/

.n/

w D '1 .0; J.s/ . C .t/; C .t//C

Zt

.n/ .n/ @ '1 .t ; J.s/ . C .t/; C .// d > 0; @t

0

D

0

.n/ 1 @ '0 C ˛ T: @

(8.82)

518

8 Viscoelastic Continua at Large Deformations

Here @[email protected] is the partial derivative of '1 with respect to the first argument, and @0 [email protected] is the partial derivative of '0 with respect to the second argument. Simultaneous invariants J.s/ of two tensors can be written by analogy with the ones for continua of the differential type (see Sect. 7.1.4).

8.2.2 Principal Model An of an Isotropic Thermoviscoelastic Continuum For the principal model An of an isotropic thermoviscoelastic continuum with difference cores, the functional basis of simultaneous invariants J.I / consists of 9 invariants, which can be chosen as follows (see (7.35)): .n/

.n/

.I / D I˛ . C .//; ˛ D 1; 2; 3; J˛.I / D I˛ . C .t//; J˛C3 .I /

J7

.n/

.n/

D C ./ C .t/;

.I /

.n/

.n/

.I /

J8 D C 2 ./ C .t/; r D 3 and z D 9:

J9

(8.83)

.n/

.n/

D C ./ C 2 .t/;

The derivative tensors of these invariants are calculated by the formulae (see [12]) .I / J1C D E;

.n/

.I /

.n/

J2C D EI1 . C .t// C .t/; .I /

.I /

.n/

.n/

.I /

J C3;C D 0; D 1; 2; 3I

.I /

.n/

.n/

J8C D C 2 ./;

J7C D C ./; .n/

.n/

.n/

.I /

J3C D C 2 .t/ I1 C .t/ C EI2 ;

.n/

J9C D C ./ C .t/ C C .t/ C ./:

(8.84)

Substituting these expressions into (8.77) and collecting terms with the same tensor powers, we obtain constitutive equations for the principal model An of an isotropic thermoviscoelastic continuum: .n/

.n/

.n/

T D 'M 1 E C 'M 2 C C 'M3 C 2 :

(8.85)

Here we have denoted the functionals Zt 'M1 '01 C '02 I1 .t/ C '03 I2 .t/

.'11 C '12 I1 .t/ C '13 I2 .t// d ; 0

.n/

.n/

Zt

.n/

.n/

'M2 C .'02 C '03 I1 .t// C .t/ ..'12 C '13 I1 .t// C .t/ '17 C .// d ; 0

(8.86)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua .n/

519

.n/

'M3 C 2 '03 C 2 .t/ Z t .n/ .n/ .n/ .n/ .n/ .n/ '13 C 2 .t/C'18 C 2 ./C'19 . C .t/ C ./C C ./ C .t// d : 0

Relation (8.85) is formally similar to the corresponding relation (4.322) for an isotropic elastic continuum, but in (8.85) 'M1 , 'M2 and 'M3 are no longer functions .n/

of invariants of the tensor C ; they are functionals in the form (8.86).

8.2.3 Principal Model An of a Transversely Isotropic Thermoviscoelastic Continuum For the principal model An of a transversely isotropic (relative to the group T3 ) thermoviscoelastic continuum with difference cores, the functional basis of simul.3/ taneous invariants J consists of 11 invariants, which can be chosen as follows (see (7.36)): .n/ .n/ .3/ J.3/ D I.3/ C .t/ ; D 1; : : : ; 5I J5C D I.3/ . C .//; D 1; : : : ; 4I .n/ .n/ .3/ D ..E b c23 / C .t// b c23 C ./ ; J10 .n/

.n/

.3/ .3/ J11 D C .t/ C ./ 2J10 J2.3/ J7.3/ ;

r D 5;

z D 11:

(8.87) Here the invariants I.3/ are determined by formulae (4.297). The partial derivatives J.3/ C of these invariants have the forms (see [12]) .n/ 1 .O1 ˝ O1 C O2 ˝ O2 / C .t/; 2 .n/ 1 D 24 O3 C .t/; 4 O3 .O1 ˝ O1 C O2 ˝ O2 / b c23 ˝b c23 ; 2

.3/ .3/ .3/ D E b c23 ; J2C Db c23 ; J3C D J1C .3/ J4C

.3/

.n/

.n/

.3/

.3/

.3/

.3/

J5C D C 2 .t/ I1 C .t/ C EI2 ; J6C D J7C D J8C D J9C D 0; .3/ J10C D

.n/ 1 .O1 ˝ O1 C O2 ˝ O2 / C ./; 4

.n/

.3/ J11C D 4 O3 C ./:

(8.88)

Substituting these expressions into (8.77) and rearranging the summands, we obtain constitutive equations for the principal model An of a transversely isotropic thermoviscoelastic continuum: .n/

.n/

.n/

.n/

c23 C .O1 ˝ O1 C O2 ˝ O2 / 'M3 C C 'M4 C C 'M5 C 2 : (8.89) T D 'M1 E C 'M 2b

520

8 Viscoelastic Continua at Large Deformations

Here we have denoted the functionals Zt 'M 1 '01 C '05 I2

.'11 C '15 I2 .t// d ; 0

'M 2 '02 '01

2'04 I2.3/

Zt

.'12 '11 2'14 I2.3/ .t/ 2'1;11 I2.3/ .// d ;

0 .n/

.n/ 1 1 'M 3 C .'03 2'04 / C 2 2

! .n/ .n/ Zt '1;10 '13 '14 C .t/C '1;11 C ./ d ; 2 2 0

Zt .n/ .n/ .2'14 '15 I1 .t// C .t/ C '1;11 C ./ d ; 'M4 C .2'04 '05 I1 / C .n/

.n/

0 .n/

'M5 C 2

!

Zt

.n/

'15 d C 2 .t/:

'05

(8.90)

0

8.2.4 Principal Model An of an Orthotropic Thermoviscoelastic Continuum For the principal model An of an orthotropic thermoviscoelastic continuum with difference cores, the functional basis of simultaneous invariants consists of 12 invariants, which can be chosen as follows (see (7.37)): .n/

J.O/ D I.O/ . C .t//; D 1; : : : ; 6I

.n/

.O/

J C6 D I.O/ . C .//; D 1; 2; 3; 6I

.n/ .n/ .n/ .n/ .O/ .O/ c22 C .t/ b c23 C ./ ; J11 D b c21 C .t/ b c23 C ./ ; J10 D b r D 6;

z D 12:

(8.91) This set should be complemented by two more invariants (being dependent) in order to obtain relations symmetric with respect to the vectors b c2˛ : .n/

.n/

.O/ J13 D .b c21 C .t// .b c22 C .//; .O/ J14

D

.n/ I7.O/ . C .t//

.n/

(8.92) .n/

D .b c21 C .t// .b c22 C .t//:

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

521

The partial derivative tensors of these invariants have the forms (see [12]) J.O/ c2 ; D 1; 2; 3I C Db .n/

.O/

.n/ 1 .O ˝ O / C .t/; D 1; 2I 2

.O/

J C3;C D .n/

.O/

J6C D 3 6 Om C .t/ ˝ C .t/; .O/ J10C .O/ J13C

.O/

J C6;C D J12C D 0; D 1; 2; 3I

.n/ 1 D .O1 ˝ O1 / C ./; 4 .n/ 1 D .O3 ˝ O3 / C ./; 4

.n/ 1 D .O2 ˝ O2 / C ./; 4 .n/ 1 .O/ J14C D .O3 ˝ O3 / C .t/; 2

(8.93)

.O/ J11C

where the tensor 6 Om is determined by formula (4.316). Substituting these expressions into (8.77) and grouping like terms, we obtain constitutive equations for the principal model An of an orthotropic thermoviscoelastic continuum: .n/

T D

3 X

.n/

.n/

.n/

.'M b c2 C O ˝ O 'M3C C / C 'M 7 6 Om C ˝ C :

(8.94)

D1

Here we have denoted the functionals Zt 'M '0

'1 t ; J˛.O/ d ; D 1; 2; 3;

0

.n/

'M3C C

Zt

.n/ 1 1 '0; 3C C 2 2

.n/ 1 '1; 9C t ; J˛.O/ C ./ 2

0

C'1; 3C t ;

J˛.O/

Zt 'M 7 3'06 3

.n/ C .t/ d ; D 1; 2;

'16 t ; J˛.O/ d ;

0 .n/

.n/ 1 1 'M 6 C '0;14 C 2 2

Zt

.n/ 1 '1;13 t ; J˛.O/ C ./ 2

0

.n/ C'1;14 t ; J˛.O/ C .t/ d :

(8.95)

522

8 Viscoelastic Continua at Large Deformations

8.2.5 Quadratic Models An of Thermoviscoelastic Continua For the quadratic model An of a thermoviscoelastic continuum with difference cores, we retain two integrals in the sum (8.61), i.e. m D 1; 2. A form of constitutive equaı

tions for specific symmetry groups G s becomes considerably more complicated, because there appear double integrals and we need to consider simultaneous invariants J.s/ of three tensors. Therefore, one usually considers the particular case of the quadratic model when m D 1 and 2, but simultaneous invariants J.s/ of only two tensors appear there just as in the principal model: .n/ I.s/ . C .t//;

D '0

!

Zt

'1 .t

; J.s/

.n/

C

.n/

.n/

d

!

'2 .t 1 ; t 2 ; J.s/ C .1 /; C .2 / 0

!

C .t/; C .1 /

0

Zt Zt

.n/

d 1 d 2 :

(8.96)

0

Here '0 , '1 and '2 are functions of the arguments indicated. Since the core '2 in .n/

.n/

this model is independent of C .t/, so @'2 [email protected] C .t/ 0, and constitutive equations prove to be coincident with (8.78); and hence they coincide with (8.85), (8.89) and (8.94) too. The distinction between the principal and quadratic models consists only in the forms of the functional and the dissipation function w . Such a situation is typical for viscoelastic continua when distinct functionals of the free energy correspond to the same relations between the tensors of stresses and deformations. Comparing the principal model (8.77) with the quadratic one (8.96), we can also notice that the principal model has only one core '1 appearing also in relation (8.78), and the quadratic model has two cores '1 and '2 , one of which is not included in relation (8.78) between stresses and deformations. Thus, for the principal model we can restore the functional of the free energy by Eqs. (8.78) up to the entropy term @'0 [email protected] and the constant '0 .0; 0 / D 0 . Models of viscoelastic continua having such a property are called mechanically determinate. The quadratic model is not mechanically determinate: due to the presence of the core '2 we cannot restore the form of by relations (8.78) between stresses and deformations. Nevertheless, this model is also used in practice due to its quadratic structure being typical for thermodynamic potentials.

8.2.6 Linear Models An of Viscoelastic Continua The quadratic model An of a thermoviscoelastic continuum with difference cores (8.96), where the functions '0 , '1 and '2 depend linearly upon the quadratic

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua .s/

523 .s/

invariants J and quadratically upon the linear invariants J , is called linear (cubic invariants do not occur in this model): '0 D 1 X

0C

r1 1 X

lˇ I.s/ .t/Iˇ.s/ .t/ C

ı

2

;ˇ D1

r1

'1 D

ı

;ˇ D1

.s/

qˇ .t /I.s/ .t/Iˇ ./ C

'2 D

r1 1 X ı

2 ;ˇ D1 1 Cı

2

1

r2 X

ı

X

l I.s/ .t/;

Dr1 C1

r2

ı

Dr1 C1

q .t /J.s/ .t; /; (8.97)

pˇ .t 1 ; t 2 /I.s/ .1 /Iˇ.s/ .2 /

r2 X

p .t 1 ; t 2 /J.s/ .1 ; 2 /:

Dr1 C1

Here lˇ , l are constants; qˇ .t /, q .t / are one-instant cores (they are functions of one argument, being symmetric in and ˇ); pˇ .t 1 ; t 2 / and p .t 1 ; t 2 / are two-instant cores (they are functions of two variables, being symmetric in and ˇ, and also in t 1 and t 2 ). We have introduced the notation ! ! .n/

I.s/ ./ D I.s/ C ./ ;

J.s/ .1 ; 2 / D J.s/

.n/

.n/

C .1 /; C .2 / ;

(8.98)

ı

.n/

where r1 is the number of linear invariants I.s/ . C .t// in group G s .r1 6 r/, and .s/ .r2 r1 / is the number of the quadratic simultaneous invariants J .1 ; 2 / in this group, where r2 6 z. Since not all simultaneous invariants contained in the full basis .n/

.n/

J . C .1 /; C .2 // appear in the expression for the functional in linear models, it is convenient to renumber these invariants in comparison with the bases (8.83), .n/

(8.87), and (8.91) by enumerating first the linear invariants I . C .t// and then the .n/

.n/

quadratic simultaneous invariants J . C .1 /; C .2 //. The principal linear model An of viscoelastic continua can be obtained by applying similar relationships to the functions '0 and '1 of the principal model (8.77) when '2 D 0. The functions '0 and '1 (8.79) for the principal linear model have the forms '0 D J

r1 X

lˇ Iˇ.s/ .t/;

ˇ D1

'0 D J l ;

'1 D J

r1 X

qˇ .t /Iˇ.s/ ./; D 1; : : : ; r1 ;

ˇ D1

'1 D J q .t /; D r1 C 1; : : : ; r2 ; ı

J D =:

(8.99)

524

8 Viscoelastic Continua at Large Deformations

As noted above, constitutive equations (8.77) for both the models coincide; and, for the linear models, they have the form r1 X

.n/

T DJ

;ˇ D1

.s/ lMˇ I.s/ Oˇ C J

r2 X Dr1 C1

.s/ lM I C ;

(8.100)

where we have denoted the linear functionals lMˇ I.s/ lˇ I.s/ .t/

Zt qˇ .t /I.s/ ./ d ;

; ˇ D 1; : : : ; r1 ;

0

.s/ .s/ lM I C l I C .t/

Zt

.s/

q .t /I C ./ d ; D r1 C 1; : : : ; r2 :

(8.101)

0

Here we have taken into account that all the linear invariants have the form I.s/ D .n/

.s/

.s/

C O , where O are producing tensors of the group, and for the quadratic invariants, J.s/ C ./ D

@J .t; / .n/

.s/

D

@ C .t/

1 @I .; / 1 ./; D r1 C 1; : : : ; r2 : (8.102) D I.s/ 2 .n/ 2 C @ C ./ .I /

(For isotropic continua, in order to satisfy this condition, as invariants I one should choose the invariants I1 .C / and I1 .C2 /.) Notice that when the cores qˇ .t/ and q .t/ in (8.101) are absent, then these relations exactly coincide with relations (4.333) of linear models An for ideal continua if in the last ones we assume that m N D 0. For principal linear models An of viscoelastic continua, the dissipation function w (8.82) has the form

w D J

r1 X ;ˇ D1

C 2J

0 @qˇ .0/I.s/ .t/I .s/ .t/ ˇ

Zt C

1 @ .s/ qˇ .t /I.s/ .t/Iˇ ./d A @t

0 r2 X

Dr1 C1

0 @q .0/J.s/ .t; t/ C

Zt

1 @ q .t /J.s/ .t; / d A ; @t

0

(8.103a)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

525

and for linear models An of viscoelastic continua, terms with two-instant cores should be added to the expression (8.103a): 0 Zt r1 X @ .s/ .s/ @ w DJ qˇ .t /I.s/ .t/Iˇ.s/ ./ d qˇ .0/I .t/Iˇ .t/ C @t ;ˇ D1 0 1 Z t Zt 1 @ pˇ .t 1 ; t 2 /I.s/ .1 /Iˇ.s/ .2 / d 1 d 2 A 2 @t 0 0 0 Zt r2 X @ .s/ @2q .0/J .t; t/ C 2 CJ q .t /J.s/ .t; / d @t Dr1 C1 0 1 Zt Zt @ p .t 1 ; t 2 /J .1 ; 2 / d 1 d 2 A : (8.103b) @t 0

0

For linear models An , the specific entropy , according to (8.69) and (8.82), has the form .n/ @ 0 1 D (8.104) C ˛ T; @ where 0 ./ is a function only of the temperature. The dissipation function with the help of formula (8.75) can be represented in the form .n/ .n/ d .n/ d d @ 0 w D T C C ˛ T : (8.104a) dt dt @ dt

8.2.7 Representation of Linear Models An in the Boltzmann Form For linear models An of viscoelastic continua, representations of the free energy in the form (8.96), (8.97), the constitutive equations in the form (8.100), (8.101) and the dissipation function w in the form (8.103b) are called the model An in the Volterra form (the name is connected to the fact that integral expressions of the form (8.101) occurring in this representation were considered for the first time by Volterra in 1909). Let us give another equivalent representation for these models. Introduce new two-instant cores N ˇ .y; z/ and N .y; z/ satisfying the differential equations @2 N ˇ .y; z/ @ N ˇ .y; 0/ D pˇ .y; z/; D qˇ .y/; @[email protected] @y @2 N .y; z/ @ N .y; 0/ D p .y; z/; D q .y/; @[email protected] @y y D t 1 ; z D t 2 : Then the following theorem holds.

N ˇ .0; 0/ D lˇ ; N .0; 0/ D l ; (8.105)

526

8 Viscoelastic Continua at Large Deformations

Theorem 8.8. Let the two-instant cores N ˇ .y; z/ and N .y; z/ 1. Be symmetric functions of their arguments: N ˇ .y; z/ D N ˇ .z; y/;

N .y; z/ D N .z; y/;

(8.105a)

2. Be two times continuously differentiable functions of their arguments within the interval .0; t/, 3. Satisfy the conditions (8.105), then we can pass from the representation of linear model An in the Volterra form (Eqs. (8.96), (8.97), (8.100), (8.101), and (8.103b)) to an equivalent representation of the model An in the Boltzmann form r1 Z t Z t 1 X N ˇ .t 1 ; t 2 / dI.s/ .1 / dI .s/ .2 / 0C ı ˇ 2 ;ˇ D1 0 0

D

C

Zt Zt r2 X

1 ı

Dr1 C1 0 r1 X

.n/

T DJ

.s/ Oˇ

;ˇ D1

N .t 1 ; t 2 / dJ.s/ .1 ; 2 /;

(8.106)

0

Zt rˇ .t

1 / dI.s/ ./

Zt r2 X

CJ

.s/

r .t / dI C ./;

Dr1 C1 0

0

(8.107)

r1 Z t Z t @ N 1 X .s/ .s/ w D J ˇ .t 1 ; t 2 / dI .1 / dIˇ .2 / 2 @t

;ˇ D1 0

J

r2 X

Zt

Dr1 C1 0

0

Zt

@ N .s/ .t 1 ; t 2 / dJ .1 ; 2 /: @t

(8.108)

0

Here we have introduced the notation rˇ .y/ D N ˇ .y; 0/; dI.s/ .1 / D IP1.s/ .1 / d 1 ;

r .y/ D N .y; 0/;

dJ.s/ .1 ; 2 / D J.s/

.n/

.n/

C .1 /; C .2 /

(8.109a) ! d 1 d 2 : (8.109b)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

527

H To prove the theorem, it is sufficient to transform the integrals in (8.106)–(8.108) as follows: 1 0 Zt Zt Zt N ˇ .0; t 2 /I.s/ .t/ @ N ˇ .t 1 ; t 2 / dI.s/ .1 /A dI .s/ .2 / D ˇ 0

0

Zt 0

Zt 0

0

@ N ˇ .t 1 ; t 2 /I.s/ .1 / d 1 dIˇ.s/ .2 / D I.s/ .t/ N ˇ .0; 0/Iˇ.s/ .t/ @1 @ N .s/ ˇ .0; t 2 /Iˇ .2 / d 2 @2

Zt Zt C 0

D

0

Zt 0

@ N ˇ .t 1 ; 0/I.s/ .1 /d 1 Iˇ.s/ .t/ @1

@2 N ˇ .t 1 ; t 2 /I.s/ .1 /Iˇ.s/ .2 / d 1 d 2 @1 @2

.s/ lˇ I.s/ .t/Iˇ .t/

Zt

.s/

.s/

qˇ .t /.I.s/ .t/Iˇ ./ C I.s/ ./Iˇ .t// d 0

Zt

Zt

.s/

pˇ .t 1 ; t 2 /I.s/ .1 /Iˇ .2 / d 1 d 2 :

C 0

(8.110)

0

Here we have taken into account that I.s/ .0/ D 0 and changed the variables @ @ N N ˇ .0; t 2 / D @ N ˇ .0; y/ D qˇ .y/: ˇ .0; t 2 / D @2 @.t 2 / @y In a similar way, we can transform the integrals of N with taking into account that .n/

.n/

J.s/ . C .1 /; C .2 // is a linear function with respect to each tensor argument; therefore, the following relations hold: dJ.s/ .1 ; 2 / D

.n/ @ .s/ .n/ J . C .1 /; C .2 // d 1 d 2 @1

D

.n/ @ .s/ .n/ J . C .1 /; C .2 // d 1 d 2 @2

D

.n/ .n/ @2 J.s/ . C .1 /; C .2 // d 1 d 2 ; @1 @2

528

8 Viscoelastic Continua at Large Deformations

hence Zt Zt 0

N .t 1 ; t 2 / dJ.s/ .1 ; 2 /

0

D

.n/ .n/ l J.s/ . C .t/; C .t//

Zt 2

.n/

.n/

q .t /J.s/ . C .t/; C .// d 0

Zt Zt

.n/

.n/

p .t /J.s/ . C .1 /; C .2 // d 1 d 2 :

C 0

(8.111)

0

On substituting (8.110) and (8.111) into (8.106), we actually obtain the expressions (8.96) and (8.97) for . The representations (8.107) and (8.108) can be proved in a similar way (see Exercise 8.2.1). N Remark. If we consider constitutive equations in the Volterra form (8.96), (8.97), and (8.100) and pass to the limit at t ! 0, then all the integral summands containing the cores qˇ and q vanish. As a result, we get instantly elastic relations which exactly coincide with the corresponding equations (4.331), (4.333), and (4.334) of models An of elastic continua: .0/ D .n/

0 C

r1 1 X ı

2 ;ˇ D1

T .0/ D J

X r1

;ˇ D1

lˇ I.s/ .0/Iˇ.s/ .0/ C

lˇ I.s/ .0/O.s/ ˇ

r2 X

1 ı

Dr1 C1

X r2

CJ

l I.s/ .0/;

Dr1 C1

(8.112)

l I.s/ C .0/:

In order to obtain these relations from the Boltzmann form (8.106), (8.107), one .n/

.n/

should represent the deformation tensors in the form C ./ D C .0/h./, where h./ is the Heaviside function. Then we find that I.s/ ./ D I.s/ .0/h./ and J.s/ .1 ; 2 / D I.s/ .0/h.1 /h.2 /. Substituting these expressions into (8.106) and (8.107) and using (8.105), we obtain the desired relations (8.112) as t ! 0C . t u

8.2.8 Mechanically Determinate Linear Models An of Viscoelastic Continua As noted in Sect. 8.2.5, quadratic models An , including the linear models (8.97), are not mechanically determinate due to the presence of the two-instant cores pˇ .t 1 ; t 2 / and p .t 1 ; t 2 /. However, the models may become

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

529

mechanically determinate after introduction of the additional assumption on a form of the two-instant cores; we assume that they depend on the sum of their arguments: N ˇ .y; z/ D N ˇ .y C z/;

N .y; z/ D N .y C z/:

(8.113)

In this case, the cores N ˇ and N become one-instant, and with the help of formula (8.109a) they can be uniquely expressed in terms of the cores rˇ .y/ and r .y/ included in constitutive equations (8.107): N ˇ .y/ D rˇ .y/;

N .y/ D r .y/:

(8.114)

The functional (8.106) of the free energy takes the form D

0

C

C

r1 Z t Z t 1 X ı

2 ;ˇ D1 0 1

Zt Zt r2 X

ı

Dr1 C1 0

rˇ .2t 1 2 / dI.s/ .1 / dIˇ.s/ .2 /

0

r .2t 1 2 / dJ.s/ .1 ; 2 /:

(8.115)

0

The dissipation function w (8.108) in this model is also determined completely by the functional (8.107) of the constitutive equations: r1 Z t Z t @ J X w D rˇ .2t 1 2 / dI.s/ .1 / dIˇ.s/ .2 / 2 @t

;ˇ D1 0

J

0

Zt Zt r2 X @ r .2t 1 2 / dJ.s/ .1 ; 2 /: @t

Dr1 C1 0

(8.116)

0

The cores rˇ .y/ and r .y/, according to (8.114) and (8.105), are connected to the cores qˇ .y/ and q .y/ by the relations @rˇ .y/ D qˇ .y/; @y

@r .y/ D q .y/; rˇ .0/ D lˇ ; r .0/ D l : @y (8.117)

The cores qˇ .y/ and q .y/ are called the relaxation cores, and the cores rˇ .y/ and r .y/ are called the relaxation functions.

530

8 Viscoelastic Continua at Large Deformations

8.2.9 Linear Models An for Isotropic Viscoelastic Continua Let us derive now constitutive equations for linear models An of viscoelastic continua in the Volterra (8.100) and Boltzmann (8.107) forms for different symı

metry groups G s . For linear models An of viscoelastic isotropic continua, the invariants (8.98) and the derivative tensors I.IC/ (8.102) have the forms r1 D 1;

r D 3; .n/

.I /

.I /

r2 D 2; .n/

.I /

.n/

I1 ./ D I1 . C .//; I2 ./ D J2 .1 ; 2 / D C .1 / C .2 /;

(8.118)

.n/

/ .I / .I / O.I 1 D I1C .t/ D E; I2C .t/ D 2 C .t/:

Then constitutive equations (8.100), (8.106), and (8.107) become ı

1 D 0C 2 ı

Zt Zt

.n/

0

0

Zt Zt C

.n/

r1 .2t 1 2 /dI1 . C .1 // dI1 . C .2 // .n/

.n/

r2 .2t 1 2 / d C .1 / d C .2 /; 0

(8.119)

0 .n/

.n/

T D J.lM1 I1 E C 2lM2 C /:

(8.120)

Here we have denoted the linear functionals .n/

lM1 I1 l1 I1 . C .t//

Zt

Zt

.n/

q1 .t /I1 . C .// d D 0

.n/

.n/

lM2 C D l2 C .t/

.n/

r1 .t / dI1 . C .//; 0

Zt

.n/

Zt

q2 .t / C ./ d D 0

.n/

r2 .t / d C ./: 0

(8.121) Thus, for an isotropic continuum, there are two independent constants l1 ; l2 and two cores q .t / connected to the cores r .y/ by the relations (8.117) @r .y/ D q .y/; @y

r .0/ D l ; D 1; 2:

(8.122) ı

Introducing the fourth-order tensor functional similar to the tensor 4 M (4.337) for elastic continua: 4M R D E ˝ ElM1 C 2lM2 ; (8.123)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

531

we can represent constitutive equations (8.120) in the operator form .n/

.n/

M C; T D J 4R

(8.124)

which is analogous to relations (4.338) for semilinear isotropic elastic continua.

8.2.10 Linear Models An of Transversely Isotropic Viscoelastic Continua For linear models An of viscoelastic transversely isotropic continua, from (8.87) and (8.88) we obtain r D 5;

r1 D 2; .n/

.3/

c23 / C ./; I1 ./ D .E b

r2 r1 D 2; .n/

.3/

I2 ./ D b c23 C ./;

.n/

.3/

.n/

c23 / C .1 // .b c23 C .2 //; J3 .1 ; 2 / D ..E b I˛.3/ ./ D J˛.3/ .; /; ˛ D 3; 4; .n/

.n/

J4.3/ .1 ; 2 / D C .1 / C .2 / 2J3.3/ .1 ; 2 / I2.3/ .1 /I2.3/ .2 /; .3/ ./ D I3C

.n/ 1 .O1 ˝ O1 C O2 ˝ O2 / C ./; 2

.n/

.3/ I4C ./ D 2 4 O3 C ./; (8.125)

where 4 O3 is determined by formula (8.88). Relations (8.100) take the form

.n/

T DJ

.3/ .3/ .3/ .3/ c23 / C lM22 2lM44 I2 C lM12 I1 b c23 lM11 I1 C lM12 I2 .E b ! M .n/ l33 M .n/ C.O1 ˝ O1 C O2 ˝ O2 / l44 C C 2lM44 C : (8.126) 2 .n/

Here the linear operators lMˇ Iˇ.3/ and lM C are determined by expressions (8.101), which can be represented in the Boltzmann form (8.107) .3/ lMˇ Iˇ D

Zt rˇ .t / 0

.3/ dIˇ ./;

.3/ lM J C D

Zt

.3/

r .t / dJ C ./: (8.127) 0

For a transversely isotropic continuum, there are five independent constants l11 , l22 , l12 , l33 , l44 and five cores qˇ .t / or rˇ .t /.

532

8 Viscoelastic Continua at Large Deformations

Introducing the tensor functional being analogous to the tensor of elastic moduli (4.341): 4M c23 ˝b c23e c23 Cb c23 ˝ E R D E ˝ ElM11 Cb lM22 C .lM12 lM11 / E ˝b M l33 M C .O1 ˝ O1 C O2 ˝ O2 / l44 C 2lM44 ; 2 e (8.128) lM D lM 2lM 2lM C lM ; 22

22

44

12

11

we can also represent constitutive equations (8.126) in the form (8.124).

8.2.11 Linear Models An of Orthotropic Viscoelastic Continua For linear models An of viscoelastic orthotropic continua, due to (8.91)–(8.93), the invariants (8.98) and the derivative tensors (8.102) take the forms r D 6;

r1 D 3;

r2 D 6;

.n/

c2˛ C ./; ˛ D 1; 2; 3; I˛.O/ ./ D b .n/ .n/ .O/ 2 2 c2 C .1 / b c3 C .2 / ; J4 .1 ; 2 / D b .O/

J5

.n/ .n/ .1 ; 2 / D b c21 C .1 / b c23 C .2 / ;

.n/ .n/ J6.O/ .1 ; 2 / D b c21 C .1 / b c22 C .2 / ; .n/ 1 .O1 ˝ O1 / C ./; 2 .n/ 1 .O/ I6C ./ D .O3 ˝ O3 / C ./; 2 (8.129) .O/ I4C ./ D

I˛.O/ ./ D J˛.O/ .; /; ˛ D 4; 5; 6I .n/

.O/ I5C ./ D 2.O2 ˝ O2 / C ./;

and the relations (8.100) can be written as follows: 3 X

.n/

T DJ

;ˇ D1

c2 C J lMˇ Iˇ.O/b

3 X

.n/

O .O lM3C;3C C /:

(8.130)

D1

Thus, there are nine independent constants l11 , l22 , l33 , l12 , l13 , l23 , l44 , l55 , l66 and nine cores qˇ .t / or rˇ .t /. Introducing the tensor functional being analogous to (4.344): 4

M D R

3 X ;ˇ D1

b c2 ˝b c2ˇ lMˇ C

3 X

O ˝ O lM3C;3C ;

(8.131)

D1

we can represent constitutive equations (8.130) in the operator form (8.124).

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

533

8.2.12 The Tensor of Relaxation Functions According to the operator form (8.124) of constitutive equations for linear models An of viscoelastic continua, we can introduce the fourth-order tensor 4 R.t/, called the tensor of relaxation functions, by the same formulae as for the elastic modı

uli tensor 4 M in linear models An of elastic continua (see Sect. 4.8.7) if in the corresponding formulae the elastic constants l˛ˇ are replaced by the relaxation functions r˛ˇ .t/. For an isotropic continuum, this tensor has the form 4

R.t/ D r1 .t/E ˝ E C 2r2 .t/I

(8.132)

for a transversely isotropic continuum 4

R.t/ D r11 .t/E ˝ E Ce r 22 .t/b c2 ˝b c23 C .r12 .t/ r11 .t//.E ˝b c23 Cb c23 ˝ E/ 3 1 r33 .t/ r44 .t/ .O1 ˝ O1 C O2 ˝ O2 / C 2r44 .t/; C 2 (8.133) e r 22 .t/ D r11 .t/ C r22 .t/ 2r12 .t/ 2r44 .t/;

and for an orthotropic continuum 4

R.t/ D

3 X

r˛ˇ .t/b c2 ˝b c2ˇ C

˛;ˇ D1

3 X

r3C˛;3C˛ .t/O ˝ O :

(8.134)

˛D1

Then the operator relations (8.124) can be represented as follows: .n/

.n/

M C; T D J 4R

where 4

.n/

M C D R

Zt 4

(8.135)

.n/

R.t / d C ./

(8.136)

0

is a tensor linear functional. For instantaneous loading as t ! 0C , these relations coincide with (8.112) and with the corresponding relations (4.330a) of models An for a linear-elastic continuum, because 4

ı

R.0/ D 4 M:

(8.137)

The tensor 4

K.t/ D

d 4R .t/ dt

(8.138)

534

8 Viscoelastic Continua at Large Deformations ı

is called the tensor of relaxation cores. This tensor for different groups G s has the same form as the tensor 4 R.t/ in (8.132)–(8.134) if in these formulae the substitution r˛ˇ .t/ ! q˛ˇ .t/ has been made. According to (8.137) and (8.138), the constitutive equations (8.135) can be written in the Volterra form .n/

4

ı

.n/

Zt

T D J. M C

.n/

K.t / C ./ d /;

(8.139)

0

that is equivalent to the form (8.100). The operator (8.115) of the free energy for the mechanically determinate model An with the help of the tensor of relaxation functions can be represented in the form (see Exercise 8.2.3) ı

1 D 0C 2 ı

Zt Zt

.n/

.n/

d C .1 / 4 R.2t 1 2 / d C .2 /; 0

(8.140)

0

and the dissipation function (8.116) – in the form J w D 2

Zt Zt

.n/

d C .1 / 0

.n/ d4 R.2t 1 2 / d C .2 /: dt

(8.141)

0

Formula (8.141) gives the following theorem. Theorem 8.9. For mechanically determinate linear models An of viscoelastic continua, the tensors of relaxation cores 4 K.t/ are 1. Nonnegative-definite: h 4 K.t/ h > 0;

8h ¤ 0; 8t > 0;

(8.142)

2. Symmetric in the following combinations of indices: 4

K.t/ D 4 K.1243/ .t/ D 4 K.2134/ .t/ D 4 K.3412/ .t/

8t > 0;

(8.143) ı

(i.e. these tensors have the same symmetry as the elastic moduli tensor 4 M for linear models An of elastic continua). H The dissipation function is always nonnegative (w > 0) and vanishes for vis.n/

coelastic continua only if C ./ 0. Then, choosing the process of deforming in the form of a step-function: .n/

C ./ D h h./;

> 0;

(8.144)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

535

where h./ is the Heaviside function, and h is a symmetric non-zero constant tensor, we obtain .n/

d C ./ D h ı./ d :

(8.144a)

Substituting (8.144a) into (8.141) and using the property (8.16) of the ı-function and formula (8.138), we find that J w D 2

Zt Z t h 4 K.2t 1 2 / hı.1 /ı.2 / d 1 d 2 0

0

J h 4 K.2t/ h > 0; 8t > 0; 2

D

(8.145)

i.e. the tensor 4 K.t/ is nonnegative-definite. The existence of the quadratic form (8.145) and the symmetry of the tensor h lead to symmetry of 4 K.t/ in the first–second and third–fourth indices and also in pairs of the indices, i.e. the relations (8.143) actually hold. N As follows from (8.142) and (8.138), the tensor of relaxation functions 4 R.t/ generates the monotonically non-increasing form h

d 4R .t/ h 6 0; dt

8h ¤ 0; 8t > 0:

(8.146)

ı

And if the elastic moduli tensor M D 4 R.0/ has the symmetry (8.143), then from (8.143) it follows that the tensor 4 R.t/ has the same symmetry 8t > 0: 4

R.t/ D 4 R.1243/ .t/ D 4 R.2134/ .t/ D 4 R.3412/ .t/

8t > 0:

(8.147)

8.2.13 Spectral Representation of Linear Models An of Viscoelastic Continua Let us apply now the theory of spectral decompositions of symmetric second-order tensors (see [12]). According to this theory, for any symmetric tensors, in particular .n/

.n/

for C ./ and T ./, we can introduce their spectral decompositions relative to a ı

symmetry group G s chosen: .n/

T D

nN X ˛D1

P.T/ ˛ ;

.n/

C D

nN X ˛D1

P˛.C/ ; 1 < nN 6 6:

(8.148)

536

8 Viscoelastic Continua at Large Deformations .n/

.n/

.C/ Here P.T/ N are the orthoprojectors of the tensors T and C ˛ and P˛ (˛ D 1; : : : ; n) (their number is denoted by n), N which are symmetric second-order tensors having the following properties: a) they are mutually orthogonal, b) they are linear, c) they ı

are indifferent relative to the group G s : .T/ P.T/ ˛ Pˇ D 0; if ˛ ¤ ˇI .n/

.n/

4 P.T/ ˛ D P˛ . T / D ˛ T ; .n/

.n/

(8.149)

˛ D 1; : : : ; nI N

(8.150)

ı

QT P˛ . T / Q D P˛ .QT T Q/; 8Q 2 G s :

(8.151) ı

Here the fourth-order tensors 4 ˛ are indifferent relative to the group G s , inde.n/

pendent of T and formed only by producing tensors of the group (see Sect. 4.8.3). Among the tensors 4 ˛ .˛ D 1; : : : ; n/, N there are m reducible tensors, i.e. obtained with the help of the tensor product of the second-order tensor a˛ being symmetric ı

and indifferent relative to the same group G s : 4

˛ D

1 a˛ ˝ a˛ ; a˛2 D a˛ a˛ ; ˛ D 1; : : : ; m < n: N a˛2

(8.152)

Expressions for 4 ˛ and a˛ have the following forms (see [12]): ı

for the isotropy group G s D I a1 D E;

4

1 .2/ D E ˝ E; m D 1; nN D 2I 3

(8.153)

ı

for the transverse isotropy group G s D T3 a1 D b c23 ; a2 D E b c23 ; m D 2; nN D 4I 1 1 4 3 D E b c23 ˝ E b c23 b c23 ˝ cN 23 .O1 ˝ O1 C O2 ˝ O2 /; 2 2 1 4 4 D .O1 ˝ O1 C O2 ˝ O2 /I (8.154) 2 ı

for the orthotropy group G s D O c2˛ ; ˛ D 1; 2; 3I a˛ D b

4

˛C3 D

1 O˛ ˝ O˛ ; m D 3; nN D 6: 2

(8.155)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

537

.T/

With the help of the orthoprojectors P˛ , introduce spectral invariants of the .n/

.n/

4 tensor T and denote them by Y˛ . T /. For those P.T/ ˛ , for which ˛ is a reducible tensor, the invariant Y˛ is introduced as follows: .n/

Y˛ . T / D

.n/ 1 a.˛/ T ; a˛

˛ D 1; : : : ; m;

(8.156a)

N and called the spectral linear invariant. For the remaining P.T/ ˛ (˛ D m C 1; : : : ; n), these invariants are introduced by the formula 1=2 .n/ .T/ P Y˛ . T / D P.T/ ˛ ˛

(8.156b)

and called the spectral quadratic invariants. From (8.150) and (8.152) it follows that P.T/ ˛ D

1 Y˛ a.˛/ ; ˛ D 1; : : : ; m: a˛

(8.157)

Notice that for the linear invariants (8.156a), formula (8.156b) also holds. Due to (8.157), the spectral decomposition of the symmetric second-order tensor .n/

T (8.148) can be represented in the form .n/

T D

m n X X .n/ a˛ Y˛ . T / C P.T/ ˛ : a ˛ ˛D1 ˛DmC1

(8.158) ı

For any fourth-order tensor indifferent relative to a group G s , including the tensor of relaxation functions 4 R.t/, we can also introduce the spectral representation 4

R.t/ D

m X

R˛ˇ .t/

˛;ˇ D1

nN X a˛ ˝ a ˇ C R˛˛ .t/4 ˛ ; a˛ aˇ ˛DmC1

(8.159)

where R˛ˇ .t/ and R˛˛ .t/ are the spectral relaxation functions expressed uniquely in terms of r˛ˇ .t/ and r˛˛ .t/ (see Exercise 8.2.6). With the help of the spectral decompositions (8.148) and (8.159) the constitutive equations (8.135) can be represented as relations between the spectral linear invariants and the orthoprojectors (see Exercise 8.2.9) .n/

Y˛ . T / D J

m X

.n/

RM ˛ˇ Yˇ . C /; ˛ D 1; : : : ; mI

ˇ D1

P.T/ ˛

D J RM ˛˛ P˛.C/ ; ˛ D m C 1; : : : ; n; N

(8.160)

538

8 Viscoelastic Continua at Large Deformations

where / RM ˛ˇ P.C D ˇ

Zt

/ R˛ˇ .t / d P.C ./: ˇ

(8.161)

0

Relations (8.160) are called the spectral representation for linear models An of viscoelastic continua. If we introduce the spectral decomposition (8.148) also for the tensor h: hD

nN X

P.h/ ˛ ;

˛D1

then the inequality (8.146) with use of (8.159) takes the form m nN X X d d R˛ˇ .t/Y˛ .h/Yˇ .h/ C R˛˛ .t/Y˛2 .h/ 6 0: dt dt ˛DmC1

(8.162)

˛;ˇ D1

Values of the spectral linear invariants Y˛ .h/ can be assumed to be zero; then, since the spectral invariants are independent, from (8.162) we obtain the condition of monotone non-increasing the spectral relaxation functions: dR˛˛ .t/ 6 0; dt

˛ D 1; : : : ; n: N

(8.163)

With the help of the spectral relaxation functions one can formulate special cases of linear models An for viscoelastic continua. So for the simplest linear model An of an isotropic viscoelastic continuum, one of the two spectral relaxation cores is assumed to be constant: 2 R11 .t/ D R11 .0/ D l1 C l2 D const; 3

@R11 D 0: @t

(8.164)

According to the results of Exercise 8.2.6 and formula (8.122), this condition can be rewritten as the relation between the cores q1 .t/ and q2 .t/ 2 q1 .t/ D q2 .t/; 3

@r˛ D q˛ .t/: @t

(8.165)

Constitutive equations (8.160) in this case take the form 8 .n/ .n/ ˆ < I1 . T / D JR11 .0/I1 . C /; .n/ .n/ Rt ˆ : dev T D J R22 .t / dev @ C ./ d : @ 0

(8.166)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

539

Here we have denoted the orthoprojectors of the tensors relative to the full orthogonal group I , called the deviators .n/ .n/ 1 .n/ dev C D C I1 . C /E; 3

.n/ .n/ 1 .n/ dev T D T I1 . T /E: 3

(8.167)

Equations (8.166) and (8.167) can be rewritten in the form .n/ J T D R11 .0/I1 . C /E C J 3

.n/

Zt

.n/

@ C ./ R22 .t /dev d : @

(8.168)

0

8.2.14 Exponential Relaxation Functions and Differential Form of Constitutive Equations To solve problems analytically or numerically it is convenient to have an analytical form of the spectral relaxation functions R˛ˇ .t/. As established in the preceding section, the property of monotone non-increasing the spectral functions R˛˛ .t/ is a consequence of the dissipation inequality w > 0. Functions having such a property can be approximated by the sum of exponents: 1 C R˛ˇ .t/ D R˛ˇ

t . / B˛ˇ exp . / ; ˛ˇ D1 N X

(8.169)

. / . / where B˛ˇ and ˛ˇ are the constants called the spectrum of relaxation values and 1 the spectrum of relaxation times, respectively, and R˛ˇ is the limiting value of the relaxation functions: 1 lim R˛ˇ .t/ D R˛ˇ ;

t !1

(8.170)

1 D 0. which may be zero: R˛ˇ

. / 1 The constants R˛ˇ and B˛ˇ satisfy the normalization condition at t D 0:

1 C R˛ˇ ı

N X D1

. /

ı

B˛ˇ D C ˛ˇ ;

(8.171)

where C ˛ˇ D R˛ˇ .0/ are the spectral (two-index) elastic moduli under instantaneous loading. There are other methods of analytical approximation to the relaxation functions, however exponential functions have certain merits: (1) choosing a sufficiently large number N of exponents in (8.169), we can approximate practically any function R˛ˇ .t/, (2) constitutive equations (8.160) and (8.161) with exponential cores admit

540

8 Viscoelastic Continua at Large Deformations

their inversion (see Sect. 8.2.15), where cores of the inverse functionals prove to be exponential as well, and (3) the cores (8.169) allow us to represent constitutive equations (8.160), (8.161) or (8.135), (8.136) in the differential form. Indeed, performing the subsequent substitutions (8.169) !(8.159)!(8.138), we find the expression for the tensor of relaxation cores: 4

m X

K.t/ D

K˛ˇ .t/

˛;ˇ D1

K˛ˇ .t/ D

nN X a ˛ ˝ aˇ C K˛˛ .t/4 ˛ ; a˛ aˇ ˛DmC1

. / N X B˛ˇ @R˛ˇ .t/ t exp : D . / . / @t D1 ˛ˇ

(8.172)

(8.173)

˛ˇ

Introduce the second-order tensors . / W˛ˇ

Zt D 0

.n/ t C ./d exp . / ; D 1; : : : ; N: . / ˛ˇ ˛ˇ

(8.174)

/ with respect to t and eliminating the integral, we obtain that Differentiating W. ˛ˇ . /

the tensors W˛ˇ satisfy the first-order differential equations / d W. ˛ˇ

dt

C

/ W. ˛ˇ . / ˛ˇ

.n/

D

C .t/ . / ˛ˇ

; D 1; : : : ; N:

(8.175)

Substituting (8.172) into (8.139) and using the expressions (8.173) and (8.174), we obtain the following representation of constitutive equations: 0 ı

.n/

.n/

T D J @4 M C

N X

1 W. / A :

(8.176)

D1

Here the spectrum of viscous stresses is denoted by W. / being second-order tensors of the form W. / D

m X ˛;ˇ D1

. /

. /

B˛ˇ W˛ˇ

nN X a˛ ˝ aˇ . / . / C B˛˛ W˛˛ 4 ˛ : a˛ aˇ

(8.177)

˛DmC1

Thus, with the help of the exponential cores (8.169) the constitutive equations for the mechanically determinate model An of viscoelastic continua (8.139) can be represented in the differential form (8.175)–(8.177). A result of the passage from integral relations to differential ones is the appearance of additional unknowns, / namely the tensors W. ˛ˇ , for which Eqs. (8.175) have been stated.

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

541

In computations the differential form (8.175)–(8.177), as a rule, proves to be more convenient than the integral form (8.139). Substitution of the expressions (8.172) and (8.138) into (8.141) yields m Z t Zt J X .C / w D K˛ˇ .2t 1 2 / d Y˛.C / .1 / d Yˇ .2 / C 2

˛;ˇ D1 0

J C 2

nN X

0

Zt Zt

.n/

.n/

K˛˛ .2t 1 2 / d C .1 / 4 ˛ d C .2 /:

˛DmC1 0

0

(8.178) .n/

Here we have denoted the linear spectral invariants of the tensor C ./ (see (8.156a)): .n/

Y˛.C / ./ D Y˛ . C .// D

.n/ 1 a˛ C ./; a˛

˛ D 1; : : : ; m:

(8.179)

Substituting the exponential cores (8.173) into (8.178) and modifying the double integrals Zt Zt 0

0

2t 1 2 exp d Y˛.C / .1 / d Yˇ.C / .2 / ˛ˇ Zt

D 0

Zt t 1 t 2 .C / exp d Y˛ .1 / exp d Yˇ.C / .2 / ˛ˇ ˛ˇ 0

D Y˛.C / .t/

Yˇ.C / .t/

Zt

1 ˛ˇ

0

1 ˛ˇ

! t 1 .C / d Y˛ .1 / d 1 exp ˛ˇ

Zt 0

! t 2 .C / exp d Yˇ .2 / d 2 ; ˛ˇ

(8.180)

with use of the notation (8.174) we can represent (8.178) in the form w D

N J X 2 D1

. / / / m X d W. d W. B˛ˇ ˛ˇ ˛ˇ aˇ a˛ a˛ aˇ dt dt ˛;ˇ D1 ! nN . / / X d W. ˛˛ . / d W˛˛ 4 : C B˛˛ dt dt ˛DmC1

(8.181)

542

8 Viscoelastic Continua at Large Deformations

8.2.15 Inversion of Constitutive Equations for Linear Models of Viscoelastic Continua In viscoelasticity theory one often uses constitutive equations inverse to (8.135) or (8.139). To derive the equations we should consider relationship (8.139) as a lin.n/

ear integral Volterra’s equation of the second kind relative to the process C ./, 0 6 6 t. The core K.t/ of this equation is assumed to be continuously differentiable and to satisfy the conditions (8.142) and (8.143). As known from the theory of integral equations, Eq. (8.139) with such core always has a solution, and this solution is written in the same form as the initial equation: .n/

.n/

T C D … C J

Zt

4

.n/ 4

N.t /

T ./ d : J

(8.182)

0

Here 4 … is the tensor of elastic pliabilities, which is inverse of the tensor of elastic ı

moduli 4 M: 4

ı

… 4 M D ;

(8.183)

and 4 N.t/ is the tensor of creep cores having the same form as the tensor 4 K.t/ (8.172): m nN X X a˛ ˝ a ˇ 4 N.t/ D N˛ˇ .t/ C N˛˛ 4 ˛ : (8.184) a˛ aˇ ˛DmC1 ˛;ˇ D1

The functions N˛ˇ .t/ and N˛˛ .t/ are called the spectral creep cores. To find a relation between the cores 4 N.t/ and 4 K.t/, we should substitute (8.139) into (8.182); as a result, we obtain the identity .n/

Zt

.n/

C .t/ D C .t/ C … 4

4

.n/

K.t / C ./ d

0

Zt

4

ı

.n/

N.t / 4 M C ./ d

0

Zt

Zy 4

0

N.t y/

4

.n/

K.y / C ./ d dy:

0

Changing the integration order in the double integral: .0 6 6 y/ .0 6 y 6 t/ ! . 6 y 6 t/ .0 6 6 t/

(8.185)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

543

Fig. 8.3 The integration domain in the double integral

(see Fig. 8.3, where the integration domain is a shaded triangle), from (8.185) we obtain Zt 4

ı

… 4 K.t / 4 N.t / 4 M

0

Zt

4

N.t y/ 4 K.y /dy

9 = ;

.n/

C ./d D 0:

(8.186)

.n/

This equation holds for any C ./ if and only if the expression in braces vanishes. The substitution of variables x D t in braces gives 4

4

4

4

Zt

ı

4

… K.x/ D N.x/ M C

.t y/ 4 K.y C x t/ dy;

0 6 x 6 t:

t x

Then the substitution of variables u D y C x t under the integral sign, where .t x 6 y 6 t/ and .0 6 u 6 x/, yields 4

4

4

4

Zx

ı

4

… K.x/ D N.x/ M C

N.x u/ 4 K.u/ d u; 0 6 u 6 x:

0

Reverting to the initial notation of arguments x ! t and u ! , we obtain the integral relation between the tensor of relaxation cores 4 K.t/ and the tensor of creep cores 4 N.t/: 4

4

4

4

ı

Zt 4

… K.t/ D N.t/ M C

N.t / 4 K./ d :

(8.187)

0

If the core 4 K.t/ is known, then the relation (8.187) is a linear integral Volterra’s equation of the second kind for evaluation of the core 4 N.t/, and vice versa. On substituting the spectral decompositions (8.172) and (8.184) of the tensors ı

K.t/, 4 N.t/ and analogous decompositions of the tensors 4 … and 4 M into (8.187), due to mutual orthogonality of the tensors 4 ˛ , a˛ ˝ aˇ (see [12]), we obtain 4

544 m X

8 Viscoelastic Continua at Large Deformations

0 @…˛ˇ Kˇ" .t/ N˛ˇ .t/Cˇ"

ˇ D1

Zt

1 N˛ˇ .t /Kˇ" ./ d A D 0; ˛; " D 1; : : : ; m;

0

(8.188a) Zt …˛˛ K˛˛ .t/ N˛˛ .t/C˛˛

N˛˛ .t /K˛˛ .t/ d D 0; ˛ D m C 1; : : : ; n; N 0

(8.188b) – the system of scalar integral equations for determining the cores N˛ˇ .t/ in terms of the cores K˛ˇ .t/ or vice versa. By analogy with the tensor of relaxation functions 4 R.t/, introduce the tensor of creep functions 4 ….t/ satisfying the equation d4 ….t/ D 4 N.t/; dt

4

….0/ D 4 …:

(8.189)

Then the inverse constitutive equation (8.182) can be written in the Boltzmann form .n/

.n/

M T C D … J

Zt

4

.n/

4

dT ….t / ./: J

(8.190)

0

The tensors of creep cores and functions have the same properties of symmetry (8.143) and (8.147) as the tensors 4 K and 4 R (see Exercise 8.2.11): 4

….t/ D 4 ….1243/ .t/ D 4 ….2134/ .t/ D 4 ….3412/ .t/; 8t > 0:

(8.191)

For the tensor of creep functions 4 ….t/ as well as for 4 R.t/, we can introduce a spectral representation by formula (8.159): 4

….t/ D

m X

…˛ˇ .t/

˛;ˇ D1

nN X a˛ ˝ a ˇ C …˛˛ .t/4 ˛ ; a˛ aˇ

(8.192)

˛DmC1

where …˛˛ .t/ and …˛ˇ .t/ are the spectral creep functions. Theorem 8.10. If the spectral relaxation cores K˛ˇ .t/ are exponential, i.e. have the form (8.173), then the spectral creep cores N˛ˇ .t/ are exponential too: N˛ˇ .t/ D

. / N X A˛ˇ D1

and vice versa.

. / t˛ˇ

1 t exp @ . / A ; t˛ˇ 0

(8.193)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua . /

545

. /

The constants A˛ˇ and t˛ˇ are called the spectra of creep values and creep times, . /

. /

respectively. They, in general, are not coincident with B˛ˇ and ˛ˇ , respectively; however the number N in (8.192) and (8.173) is the same. H Show that if the cores K˛ˇ .t/ have the form (8.173), then the cores (8.193) are a solution of the integral equation (8.188). Substitution of (8.173) and (8.193) into (8.188) yields m X

0 @

N X

0 @…˛ˇ

D1

ˇ D1

. /

Bˇ"

. / ˇ"

0

N X N X

@

D1 0 D1

1

0

11

0

. /

A˛ˇ

t t exp @ . / A Cˇ" . / exp @ . / AA ˇ" t˛ˇ t˛ˇ

/ . 0 / Z t B A. ˛ˇ ˇ" . / . 0 / t˛ˇ ˇ" 0

0

1

1

0

11

t exp @ . / A exp @ . 0 / A d AA D 0: t˛ˇ ˇ" (8.194)

Calculating the integral in (8.194) Zt 0

0 0 1 0 1 11 . / . 0 / t˛ˇ ˇ" t t t @exp @ 0 A exp @ AA exp @ . / . 0 / A d D . / . 0 / . / . / t˛ˇ ˇ" t˛ˇ ˇ" ˇ" t˛ˇ 0

(8.195) and equating coefficients in (8.194) at the same exponents, we get the system 0 1 0 1 . 0 / . 0 / m N m N X X X A˛ˇ B …˛ˇ X C ˇ" ˇ" @ AB . / D 0; @ AA. / D 0 ˇ" ˛ˇ . / . / . 0 / . / . 0 / . / t t t 0 0 D1 ˇ" D1 ˇ" ˇ D1 ˇ D1 ˛ˇ ˇ" ˛ˇ ˛ˇ (8.196) / . / . / . / and t˛ˇ in terms of the constants Bˇ" and ˇ" for determining the constants A. ˛ˇ (the constants …˛ˇ can always be determined in terms of C˛ˇ and are assumed to be known). . / . / At those values of Bˇ" , ˇ" , …˛ˇ , at which there exists a solution of the system (8.196), the exponential representation of the creep cores (8.193) exists too. N / . / and t˛ˇ in Formulae (8.196) give the method of calculation of the constants A. ˛ˇ

. / . / terms of Bˇ" , ˇ" and vice versa. From (8.188b) we can obtain simpler formulae

/ . / for determining the constants A. ˛˛ and t˛˛ (˛ D m C 1; : : : ; n):

…˛˛ . /

˛˛

D

N X 0 D1

0

. / A˛˛ . /

; . 0 /

˛˛ t˛˛

C˛˛ . /

t˛˛

D

0

N X

. / B˛˛

0 D1

˛˛ t˛˛

. 0 /

. /

:

(8.197)

546

8 Viscoelastic Continua at Large Deformations

On substituting (8.184) and (8.193) into (8.189), we find an expression for the spectral creep functions in the case of exponential cores: …˛ˇ .t/ D …˛ˇ C

N X D1

/ A. ˛ˇ

1 exp

where lim …˛ˇ .t/ D …˛ˇ C

t !C1

N X D1

t

!!

. / t˛ˇ

;

/ A. …1 ˛ˇ : ˛ˇ

(8.198)

(8.199)

Exercises for 8.2 8.2.1. Using the rule of differentiation of an integral with a varying upper limit (see formulae (8.15b) and (8.25)) and calculating the derivative of the functional (8.106) with respect to t, show that PTI (4.121) actually yields formulae (8.107) and (8.108) .n/

for T and w . 8.2.2. Using the definition (8.70), show that for linear models An of thermorheologically simple viscoelastic media, relations (8.106)–(8.108) have the forms r1 Z t Z t 1 X N ˛ˇ .t 0 10 ; t 0 20 /dI˛.s/ .1 / dI .s/ .2 / 0 ./ C ı ˇ 2 ˛;ˇ D1 0 0

D

1

Cı

Z t Zt r2 X ˛Dr1 C1 0

r1 X

.n/

T DJ

O.s/ ˇ

0

r˛ˇ .t

10 /

dI˛.s/ .1 /

CJ

Zt

˛;ˇ D1 0 r2 X

Zt

.s/ r˛˛ .t 0 10 / dI˛C .1 /;

@ N .s/ 0 0 0 0 .s/ ˛ˇ .t 1 ; t 2 / dI˛ .1 / dIˇ .2 / @t

0

Z t Zt

˛Dr1 C1 0

Zt r2 X ˛Dr1 C1 0

0

r1 J X 2

J

0

Zt

˛;ˇ D1

w D

N ˛˛ .t 0 10 ; t 0 20 / dJ˛.s/ .1 ; 2 /;

@ N 0 0 0 0 .s/ ˛˛ .t 1 ; t 2 / dJ˛ .1 ; 2 /; @t

0

where t 0 , 10 and 20 are determined by (8.71). Taking dI˛ ./ D

d d I˛ ./ d D I˛ . 0 / d 0 D dI˛ . 0 / dt d0

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

547

into account, show that these relations can be represented as functions of the reduced time; in particular r1 X

.n/

T DJ

˛;ˇ D1

.s/ Oˇ

Zt 0

r˛ˇ .t 0 10 / dI˛.s/ .10 / C J

Zt 0 r2 X ˛Dr1 C1 0

0

r˛˛ .t 0 10 / dI˛C .10 /: .s/

8.2.3. Using representations (8.132)–(8.134) for the tensor of relaxation functions R.t/, show that representations (8.140) and (8.115) for are equivalent.

4

8.2.4. Substituting formulae (8.132)–(8.134) for the tensor of relaxation functions R.t/ into the expression (8.141) for w , show that formulae (8.141) exactly coincide with (8.116). 4

8.2.5. Show that the condition (8.146) causes monotone non-increasing the relaxation functions @ b˛ˇ ˛ˇ .t/ 6 0; 8t > 0; ˛; ˇ D 1; 2; 3; ˛ ¤ ˇ; R @t

@ b˛˛˛˛ .t/ 6 0; R @t

bijkl .t/ are components of the tensor 4 R.t/ with respect to the basis b where R ci . 8.2.6. Using representations (8.132)–(8.134) for the tensor of relaxation functions 4 R.t/ and its spectral representation (8.159), and also formulae (8.152)–(8.155) for the tensors a˛ and 4 ˛ , show that the functions r˛ˇ .t/ and R˛ˇ .t/ are connected by the following relations for isotropic continua R11 .t/ D r1 .t/ C .2=3/r2 .t/;

R22 .t/ D 2r2 .t/;

and for transversely isotropic continua e3333 .t/ D e R11 .t/ D R r 22 .t/;

b1111 .t/ D r11 .t/ C 2r44 .t/: R22 .t/ C R33 .t/ D 2R

8.2.7. Show that for mechanically determinate linear models An of thermorheologically simple continua, the constitutive equations obtained in Exercise 8.2.2 can be written in the forms (8.136), (8.140), and (8.141) ı

1 D 0 ./ C 2 ı

Z t Zt 0

.n/

.n/

d C .1 / 4 R.2t 0 10 20 / d C .2 /;

0

Zt

.n/

4

T DJ

.n/

R.t 0 0 / d C ./;

0

w D

Ja 2

Z t Zt

.n/

d C .1 / 0

0

.n/ @ 4 R.2t 0 10 20 / d C .2 /: 0 @t

548

8 Viscoelastic Continua at Large Deformations

Show that the constitutive equation (8.139) for this model becomes 0 ı

.n/

.n/

T D J @4 M C .t/

1

Zt 4

.n/

K.t 0 0 / C ./a ./d A; 4 K.t 0 / D d 4 R.t/=dt 0 :

0

8.2.8. Show that for the linear models An of thermorheologically simple media with the exponential cores, the constitutive equations from Exercise 8.2.7 can be written in the form (8.175)–(8.177), (8.181) 0 .n/

ı

.n/

T D J @4 M C

N X

1

. /

WA ;

D1

N Ja X w D 2

D1

d W˛ˇ dt

C

a .t/ . / ˛ˇ

. /

.n/

.W˛ˇ .t/ C .t// D 0;

. / .n/ .n/ m X B˛ˇ a˛ aˇ . / . / C W˛ˇ ˝ C W˛ˇ . / a aˇ ˛

˛;ˇ D1

˛ˇ

.n/ ! nN . / .n/ X B˛˛ . / 4 . / C C W˛˛ ˛ C W˛˛ : . / ˛DmC1 ˛˛ 8.2.9. Prove that the spectral representations (8.148) and (8.159) lead to the spectral form (8.150) of constitutive equations for the linear models An (8.135). 8.2.10. Show that in hydrostatic compression when the Cauchy stress tensor and the deformation gradient are spherical: T D pE; F D kE;

.n/

CD

.n/ 1 .k nIII 1/E; dev C D 0; n III

a viscoelastic continuum, according to the simplest model (8.166), behaves as a purely elastic one: .n/

T D

.n/ J R11 .0/I1 . C /E; 3

i.e. there are no viscoelastic deformations in hydrostatic compression for this model. Many solids actually have such properties up to high pressures p; therefore, the simplest model (8.166) is widely used in continuum mechanics. 8.2.11. Show that the tensors of creep cores and creep functions 4 N.t/ and 4 ….t/ have the same properties of symmetry (8.143), (8.147) as the tensors of relaxation cores and relaxation functions 4 K.t/ and 4 R.t/, and vice versa.

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

549

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids 8.3.1 Models An of Incompressible Viscoelastic Continua For incompressible viscoelastic continua, as usual, there is an additional condition of incompressibility, which can be written in one of the forms (4.487)–(4.492), and the principal thermodynamic identity takes the form (4.495) for models An . For models Bn , its form is analogous. Substituting the functional (8.35) into (4.495) and using the rule (8.27), we get constitutive equations for incompressible viscoelastic continua 8 .n/ .n/ ˆ p .n/1 ı ˆ ˆ T D G C [email protected] [email protected] C .t//; ˆ < n III (8.200) D @ [email protected].t/; ˆ ˆ ˆ ˆ :w D ı : .n/

Since for incompressible continua the number r of independent invariants I.s/ . C .t// is smaller by 1 than that for compressible materials, in each of the representations (8.53), (8.61), (8.73), (8.77), and (8.96) of the free energy functional the subscript .n/

of the function '0 .I.s/ . C/; / takes on r 1 values. In the consistent way, the number z of simultaneous invariants J.s/ occurring in arguments of the cores 'm decreases too.

8.3.2 Principal Models An of Incompressible Isotropic Viscoelastic Continua For the principal models An of incompressible isotropic viscoelastic continua, the functional (8.77) depends only on five simultaneous invariants: J˛.I /

.n/ .n/ .n/ .n/ .I / .I / D I˛ C .t/ ; J3C˛ D I˛ C ./ ; ˛ D 1; 2I J7 D C ./ C .t/; (8.201)

and the constitutive equations (8.200) become .n/

T D

.n/ p .n/1 G C 'M1 E C 'M2 C ; n III

(8.202)

550

8 Viscoelastic Continua at Large Deformations

where Zt 'M 1 E '01 C '02 I1 .t/

.'11 C '12 I1 .t// d ; 0

.n/

Zt

.n/

'M2 C '02 C .t/

(8.203) .n/

.n/

.'12 C .t/ C '17 C .// d ; 0

and '0 and '1 are determined by (8.79). For the principal linear models An of incompressible isotropic viscoelastic continua, the functional has the form (compare this with the potential for elastic incompressible materials (4.526)) ı

Zt .n/ .n/ l1 C 2l2 .n/ 0 N D 0C C m N C I1 . C / q1 .t /I1 . C .//d I1 . C / 2 ı

ı

0

.n/ 2 l2 I2 . C / C

Zt

.n/ .n/ q2 .t / C .t/ C ./ d ;

(8.204)

0

N are the constants, and q1 .t / and q2 .t / are the cores. where l1 , l2 and m The corresponding constitutive equations (8.202) have the form (compare with (4.526) for elastic continua) .n/

T D

.n/ .n/ p .n/1 G C .m N C lM1 I1 . C //E C 2lM2 C : n III

(8.205)

Here we have denoted two linear functionals .n/

lM1 I1 D l1 I1 . C /

Zt

Zt

.n/

q1 .t /I1 . C .// d D 0

.n/

.n/

lM2 C D l2 C

.n/

r1 .t / dI1 . C .//; 0

Zt

.n/

Zt

q2 .t / C ./ d D 0

.n/

r2 .t / d C ./:

(8.206)

0

The constants N 0 and p0 D p.0/ are chosen from the conditions (4.326) and (4.327) according to formulae (4.527), just as for elastic materials: N p0 D p e C m;

N 0 D 0;

(8.207)

where p e is the constant appearing in the initial values of the stress tensors in the e

.n/

natural configuration K: T D p e E.

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

551

8.3.3 Linear Models An of Incompressible Isotropic Viscoelastic Continua For linear models An of incompressible isotropic viscoelastic continua, obtained from quadratic mechanically determinate models An (see Sects. 8.2.6–8.2.9), constitutive equations have the form similar to formulae (8.120) for compressible materials, but the functional should involve the summands linear in the invariant I1 : ı

ı

.n/

ı

C N 0 C mI N 1. C / t t Z Z .n/ .n/ 1 r1 .2t 1 2 / dI1 . C .1 // dI1 . C .2 // C 2

D

0

0

0

Zt Zt C

.n/

.n/

r2 .2t 1 2 / d C .1 / d C .2 /; 0

(8.208)

0

where N 0 and m N are the constants, and r1 .y/ and r2 .y/ are the relaxation functions. According to formulae (8.110) and (8.111) and also Theorem 8.8, the functional can be written in the Volterra form ı

ı

D

.n/

ı

0

C N 0 C mI N 1. C / C

Zt

.n/ l1 2 .n/ I1 . C / C l2 I1 . C 2 / 2

.n/

Zt

.n/

0

1 C 2

.n/

0

Z t Zt

.n/

.n/

p1 .2t 1 2 /I1 . C .1 //I1 . C .2 // d 1 d 2 0

1 C 2

.n/

q2 .t/ C .1 / C .t/ d

q1 .t/I1 . C .// d I1 . C .t//2

0

Z t Zt

.n/

.n/

p2 .2t 1 2 / C .1 / C .2 / d 1 d 2 ; 0

(8.209)

0

where @2 r .y/ D p .y/; @y 2

@r .y/ D q .y/; @y

r .0/ D l :

(8.210)

552

8 Viscoelastic Continua at Large Deformations

By analogy with compressible media (see Sect. 8.2.13), we can consider the simplest model An of isotropic incompressible continua, in which the creep functions q1 .y/ and q2 .y/ are connected by the relation (8.165): 1 q1 .y/ D q.y/; 3

q.y/ D 2q2 .y/;

(8.211)

i.e. this model involves only one core q.y/. Integrating Eq. (8.211) with respect to y and taking the initial condition (8.210) into account, we find the connection between r1 .y/ and r2 .y/ 1 2 r1 .y/ D r2 .y/ C l1 C l2 ; 3 3

r.y/ 2r2 .y/:

(8.211a)

Substituting (8.211) into (8.205) and (8.206) and grouping like terms, we obtain the following constitutive equation (compare with (8.168)): .n/ .n/ p .n/1 T D G C .m N C l1 I1 . C //E C 2l2 C n III

.n/

Zt

.n/

q.t / dev C ./ d ; 0

(8.212)

where .n/ .n/ 1 .n/ dev C D C I1 . C /E 3

(8.213)

.n/

is the deviator of the tensor C (see (8.167)). If q.t/ 0, then Eqs. (8.212) coincide with relations (4.526) for isotropic incompressible elastic continua.

8.3.4 Models Bn of Viscoelastic Continua In models Bn of viscoelastic continua, the free energy D

t

.n/

is a functional in the form

.n/

. G.t/; .t/; G t ./; t .//;

(8.214)

D0

and corresponding constitutive equations can be obtained with the help of the rule (8.27) of differentiation of a functional with respect to time; they have the form 8 .n/ .n/ .n/ t .n/ ˆ t t ˆ ˆ < T D [email protected] [email protected] G.t// F . G.t/; .t/; G ./; .//; D0

D @ [email protected]; ˆ ˆ ˆ : w D ı :

(8.215)

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

All further constructions with functionals

t D0

553

t

and F can be performed for models D0

Bn as well. Special models Bn of viscoelastic continua can be obtained immediately from .n/

.n/

models An , in which one should make the substitution C D G .1=.n III//E. In particular, for linear mechanically determinate models Bn of isotropic incom.n/

.n/

pressible continua, according to the relation C D G , (8.205) and (8.208), we obtain the constitutive equations ı

ı

ı

D

0 C

0

.n/

1 CmI1 . G/C 2

Zt Z t 0

Zt Zt

.n/

0 .n/

.n/

r2 .2t 1 2 / d G.1 / d G.2 /;

C 0

.n/

r1 .2t1 2 / dI1 . G.1 // dI1 . G.2 //

(8.216)

0

Zt Zt

w D

.n/

.n/

q1 .2t 1 2 / dI1 . G.1 // dI2 . G.2 // 0

0

Zt Zt

.n/

.n/

q2 .2t 1 2 / d G.1 / d G.2 /;

C2 0

0 .n/ 1

.n/

p T D G n III

0 C @m C

Zt

(8.217)

1 .n/

r1 .t /dI1 . G.//AE C 2

0

Zt

.n/

r2 .t / d G./: 0

(8.218) For the simplest models Bn , the assumption (8.211a) on the functions r .t/ yields .n/ .n/ p .n/1 T D G C .m C l1 I1 . G//E C 2l2 G n III

.n/

Zt

.n/

q.t / dev G./ d ; 0

(8.219) w D

Zt

Zt

.n/

q.2t 1 2 / dev 0

0

.n/

@ G.1 / @ G.2 / dev d 1 d 2 > 0: @1 @2

(8.220)

Since the function w is nonnegative, we find that q.y/ D @r.y/[email protected] > 0; i.e. the relaxation core q.y/ is always nonnegative.

(8.221)

554

8 Viscoelastic Continua at Large Deformations .n/

e

Passing to the limit as t ! 0, in the natural configuration K, where T .0/ D .n/

p e E, G.0/ D E=.n III/ and p D p0 (see Sect. 4.8.6), from (8.219) we obtain the following relations between the constants 0 , m, l1 , l2 and p0 (see (4.529)): p0 D p e C m C

3l1 C 2l2 ; n III

0

D

3.3l1 C 2l2 / ı

2.n III/2

3m ı

.n III/

:

(8.222)

Notice that relations (8.216) and (8.218) are entirely equivalent to Eqs. (8.208), (8.205); and (8.219) – to Eqs. (8.212) (the constants l1 and l2 in these equations are distinct). We can obtain new models of the class Bn by taking additional assumptions on the constants m, l1 and l2 . For example, if we assume just as in the corresponding elastic models Bn (see (4.530)) that l2 D .1 ˇ/.n III/2 ; 2

l1 C 2l2 D 0;

m D .1 C ˇ/.n III/; (8.223)

where and ˇ are two new independent constants, then from (8.219) we obtain the constitutive equations p .n/1 T D G C .n III/2 n III

.n/

Zt

! ! .n/ .n/ 1Cˇ C .1 ˇ/I1 . G/ E .1 ˇ/ G n III

.n/

q.t / dev G./ d ;

(8.224)

0

which are not equivalent to the corresponding relations (8.212) of models An . Due to (8.223), relations (8.222) become p0 D p e C .3 ˇ/.n III/2 ;

0

ı

D 6 =:

(8.222a)

8.3.5 Models An and Bn of Viscoelastic Fluids Equations (8.214) and (8.215) as well as (8.35) and (8.38) hold true for both solid and fluid viscoelastic continua. However, for fluids, according to the principle of material symmetry, relations (8.39) (and the analogous relations for models Bn ) must be satisfied: .n/

.n/

t

.n/

.n/

T D F . G .t/; .t/; G t ./; t .// D [email protected] [email protected] G /; D0

D

t

.n/

.n/

. G .t/; .t/; G t ./; t .//

8H 2 U;

D0

for any H -transformations included in the unimodular group U .

(8.225) (8.226)

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

555

Theorem 8.11. For models An and Bn (n D I; II; IV; V) of viscoelastic fluids, the constitutive equations (8.214), (8.215) and (8.35), (8.38), satisfying the principle of material symmetry and being continuous functionals in space Ht , can be written as follows: .n/

T D

p .n/1 G ; n III

(8.227a)

0

1 Zt 1 Zt X @' @ @'m t 0 D p ../; .// D 2 @ C d 1 : : : d m A ; p D 2 ::: @.t/ D0 @ @.t/ mD1 0

0

(8.227b) t

D

..t/; .t/; t ./; t .// D '0 ..t/; .t//

D0

C

1 Zt X mD1 0

Zt :::

'm .t; 1 ; : : : ; m ; .t/; .1 /; : : : ; .m // d 1 : : : m ; 0

(8.227c) T D pE:

(8.227d)

H Since is assumed to be a continuous functional, we can apply Theorem 8.4 and expand in terms of n-fold scalar functionals: Zt 1 Zt X D : : : e m .t; 1 ; : : : ; m / d 1 : : : d m ; (8.228) mD1 0

0

whose cores are scalar functions of m tensor arguments: .n/

.n/

e m .t; 1 ; : : : ; m / D e m .t; 1 ; : : : ; m ; G.1 /; : : : ; G.m //:

(8.229)

On substituting the representation (8.228) into (8.226), we get that functions e m must satisfy the relation .n/ .n/ .n/ .n/ em .t; 1 ; : : : ; m ; G 1 ; : : : ; G m / D em t; 1 ; : : : ; m ; G 1 ; : : : ; G m 8H 2 U; (8.230) .n/

.n/

where G i G.i /, i.e. they must be H -indifferent relative to the unimodular group U .

556

8 Viscoelastic Continua at Large Deformations

Applying the same reasoning as we used in proving Theorem 4.31, we can show .n/

that functions of the third invariant of the tensors G i (or, that is the same, of values of the density i D .i / at different times i ) are the only functions ensuring that the condition (8.230) is satisfied: .n/

.n/

em D em .t; 1 ; : : : ; m ; I3 . G 1 /; : : : ; I3 . G m //Dem .t; 1 ; : : : ; m ; 1 ; : : : ; m /: Separating ı-type components from the cores in this expression by analogy with (8.51a), from (8.228) we get the representation (8.227c), where the cores 'm are connected to e m by relations (8.54). Substituting the functional (8.227c) into (8.215) and differentiating with re.n/

spect to G.t/, from (4.448)–(4.453) we actually obtain formulae (8.227a) and (8.227b). Finally, using the transformations (4.455) and (4.456), from (8.227a) we obtain that all the relations (8.227a) for models An and Bn are equivalent to the single relation (8.227d). N Notice that although Eq. (8.227d) for the Cauchy stress tensor is formally the same as the one for an ideal fluid, a viscoelastic fluid is not ideal (it is dissipative), because in this case the pressure p is a functional of the density , and the dissipation function w is not zero due to (8.64):

w D ı

D

'10

1 Z X

Zt

t

mD1 0

:::

@'m 0 C 'mC1 @t

d 1 : : : d m > 0:

0

(8.231)

8.3.6 The Principle of Material Indifference for Models An and Bn of Viscoelastic Continua .n/ .n/

.n/

Since all the energetic tensors T , C and G are R-invariant in rigid motions, all the constitutive equations given in this section for solids and fluids, and also for incompressible continua, are the same in actual configurations K and K0 ; therefore, the principle of material indifference for models An and Bn of viscoelastic media is satisfied identically.

Exercises for 8.3 8.3.1. Using the stepwise loading (8.144) and passing to the limit as t ! 0C , show that the instantly elastic relations obtained from (8.224) coincide with the constitutive equations (4.532) of the model Bn of an elastic isotropic incompressible continuum.

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

557

8.3.2. Using the method applied in Sect. 8.2.14, show that for the simplest model An of isotropic incompressible viscoelastic continua with the exponential core N X t B . / exp . / ; q.t/ D . / D1 the constitutive equation (8.212) takes the form X .n/ .n/ p .n/1 G C .m N C l1 I1 . C //E C 2l2 C W. / B . / ; n III D1 N

.n/

T D

.n/ 1 d W. / D . / .dev C W. / /I dt

and for the simplest model Bn of isotropic incompressible viscoelastic continua with the same exponential core, the constitutive equation (8.224) takes the form .n/

T D

p .n/1 G n III

C .n III/2

! N X .n/ .n/ 1Cˇ W. / B . / ; C .1 ˇ/I1 . G/ E .1 ˇ/ G n III D1

.n/ d W. / 1 D . / dev G W. / : dt

8.3.3. Using the results of Exercise 8.2.7 and Eqs. (8.220) and (8.224), show that for the simplest linear models Bn of isotropic incompressible thermorheologically simple media the following constitutive equations hold: p .n/1 T D G C .n III/2 n III

.n/

Zt

! .n/ .n/ 1Cˇ C .1 ˇ/I1 . G/ E .1 ˇ/ G n III

.n/

q.t 0 0 / dev G./a ./ d ;

0

Zt Zt

w D a

.n/

0

q.2t 0

0

10

20 /

.n/

@ G.1 / @ G.2 / dev dev d 1 d 2 ; @1 @2

where t 0 , 10 and 20 are determined by (8.71).

558

8 Viscoelastic Continua at Large Deformations

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations 8.4.1 Statements of Dynamic Problems in the Spatial Description Statements of problems of viscoelasticity at large deformations can be obtained formally from the corresponding statements of problems of elasticity theory at large deformations (see Sect. 6.1), if in the last ones we replace the generalized constitu.n/

.n/

tive equations of elasticity T G D F G . C ; / by relations of viscoelasticity (for a viscoelastic model chosen from the models considered in Sects. 8.1–8.3). Choosing the most general representations (8.38) and (8.215) for models An and Bn of viscoelasticity and using the method mentioned above and the statement of the dynamic RU VF -problem of elasticity theory (see Sects. 6.1.1 and 6.3.3), we obtain a statement of the dynamic RU VF -problem of thermoviscoelasticity in the spatial description. This statement consists of the equation system in the domain V .0; tmax /: @ C r v D 0; @t

(8.232a)

@v C r v ˝ v D r T C f; @t

(8.232b)

1 qm C w @ C r v D r q C : @t

(8.232c)

@FT C r .v ˝ FT F ˝ v/ D 0; @t

(8.232d)

@u C r .v ˝ u/ D v; @t

(8.232e)

the constitutive equations in the domain VN .0; tmax /: q D r ; .n/

(8.233a)

.n/

T D 4 E G T G; .n/

.n/

t

.n/

(8.233b) .n/

T G D F G . C G .t/; .t/; C tG ./; t .// [email protected] [email protected] C G .t//;

(8.233c)

D0

D @ [email protected].t/; D

t

.n/

.n/

w D ı ;

. C G .t/; .t/; C tG ./; t .//; G D A; B;

D0

(8.233d) (8.233e)

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations

559

which should be complemented by expressions (4.389) and (4.391c) for tensors 4

.n/

.n/

E G and C G : .n/

CG

3 X

1 D n III 4

E D

ı

˝ p˛ hN G E ;

˛D1

3 X

.n/

! ı

nIII p˛ ˛

ı

ı

E˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

˛;ˇ D1 ı

˛ ; p˛ ; p˛ k F;

(8.234)

the boundary conditions (6.68)–(6.76), which for the case when there are no phase transformations take the form n T De tne ; n q D e q ne u D ue ; D e @[email protected] D 0;

v n D 0;

n q D 0;

at †1 ; : : : ; †4 ; †7 ; at †5 ; †6 ; n T ˛ D 0

at †8 ;

(8.235)

and also the initial conditions t D0W

ı

D ; v D v0 ; D 0 ; F D E; u D u0 :

(8.236)

On substituting the constitutive equations (8.233) and (8.234) into (8.232), we get the system of 17 scalar equations for 17 unknown scalar functions: ; ; u; v; F k x; t:

(8.237)

Since the domain V .t/ in the spatial description is unknown, the system obtained should be complemented with either relation (6.89) or Eq. (6.85) for the function f .x; t/ specifying a shape of the surface †.t/ bounding the domain V .t/: @f C v r f D 0; @t t D0W f D f 0 .x/:

(8.238)

In the second case the function f .x; t/ appears among the unknowns (8.237). Remark 1. In viscoelasticity theory, in place of the energy balance equation one often uses the entropy balance equation (3.166a) (the system (8.232) has been written in this way), which contains the dissipation function w explicitly. The entropy balance equation can be represented in the nondivergence form (3.166)

d D r q C qm C w : dt

(8.239)

560

8 Viscoelastic Continua at Large Deformations

When the model An of a thermoviscoelastic continuum with difference cores is considered, then is determined by formula (8.69). If in this formula the derivative @0 '[email protected] is assumed to depend only on , then after substitution of (8.69) and (8.233a) into (8.239) we obtain the following equation of heat conduction for a viscoelastic continuum in the spatial description: 0 0 0 1 11 .n/ .n/ TC T CC @ B [email protected] B c" C v r D r .r / @ @˛ A C v r @˛ AACqm Cw : @t @t (8.240) Here c" D [email protected] '0 [email protected] 2 /

(8.241) t u

is the heat capacity at fixed deformations.

Remark 2. If, for example, we consider the statement of the dynamic RU VF problem for mechanically determinate linear models An of thermorheologically simple viscoelastic continua with exponential cores, then the constitutive equations (8.233) has the form derived in Exercise 8.2.8: 0 ı

.n/

.n/

T D J @4 M C

1

N X

W. / A ;

D1

W. / D

m X ˛;ˇ D1

. / . / B˛ˇ W˛ˇ

/ @W. ˛ˇ

@t

w D

N Ja X 2 D1

nN X a.˛/ ˝ a.ˇ / . / . / C B˛˛ W˛˛ 4 ˛ ; a˛ aˇ ˛DmC1 .n/

Cvr ˝

/ W. ˛ˇ

D a

/ C W. ˛ˇ . /

;

˛ˇ

. / m X B˛ˇ a˛ . / ˛;ˇ D1 ˛ˇ

.n/ .n/ / / aˇ . C W. / ˝ C W. ˛ˇ ˛ˇ a˛ aˇ

! nN . / X .n/ B˛˛ .n/ / 4 . / C C W. : ˛˛ ˛ C W˛˛ . / ˛˛ ˛DmC1

(8.242)

In this case the initial conditions (8.236) are complemented by the additional conditions t D0W

. /

W˛ˇ D 0;

(8.243)

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations

561

and the number of unknown functions (8.237) becomes greater due to adding the functions / W. k x; t; ˛; ˇ D 1; : : : ; nI N D 1; : : : ; N I (8.244) ˛ˇ / (here we always have W. 0 when ˛ ¤ ˇ and ˛; ˇ > m). ˛ˇ

t u

By analogy with the dynamic RU V -, RV - and U -problems of thermoelasticity (see Sect. 6.3.3), we can state the dynamic RU V -problem of viscoelasticity, in which the deformation gradient F is eliminated between the unknowns, the U V problem of viscoelasticity, where in addition the density is eliminated, and the dynamic U -problem of viscoelasticity, where only u and are unknown. Remark 3. Just as the statements of thermoelasticity problems in the spatial description (see Sect. 6.3.3), the statements of thermoviscoelasticity problems mentioned above are strongly coupled, because they cannot be split into heat conduction problems and viscoelasticity problems even if we neglect the entropy term of the .n/

connection (i.e. the term .˛=/ T in the equation of heat conduction (8.240)). To the six causes of the connection in thermoviscoelasticity problems, mentioned in Remark 4 of Sect. 6.3.3, we should add one more factor: the presence of the dissipation function w in the entropy balance equation (8.232c) or in the heat conduction equation (8.240), that is a consequence of non-ideality of viscoelastic continua. In many problems, a contribution of the dissipation function w to the heat conduction equation (8.240) proves to be rather considerable and cannot be neglected in non-isothermal processes. The effect of growing the temperature in viscoelastic materials without heat supplied to a body from the outside but only due to internal heat release in deforming (caused by the presence of the dissipation function w ), is called dissipative heating of the body (see Sect. 8.6). t u Let us pay attention to the sixth cause of the connection mentioned in Remark 4 of Sect. 6.3.3: for viscoelastic continua the dependence of the constitutive equations (8.233c) on temperature can be split into the three constituents: ı

(1) Dependence of the heat deformation " (8.66) when the Duhamel–Neumann model is used. (2) Dependence of the elastic properties on the temperature .t/. (3) Dependence of the viscous properties, i.e. the integral part of Eqs. (8.233c), upon the temperature prehistory t ./. As established in experiments, for most viscoelastic continua, the viscous properties more considerably depend on temperature than the elastic ones. Since the dissipation function w depends on just the viscous properties, it also depends explicitly upon the temperature (in the model An with the exponential cores (8.242) this dependence has the form of function a ..t//). The temperature dependence w ./ leads to the intensification of dissipative heating in viscoelastic materials and, under certain conditions, can cause the effect of heat explosion (see Sect. 8.6.9).

562

8 Viscoelastic Continua at Large Deformations

8.4.2 Statements of Dynamic Problems in the Material Description Using the statement of the dynamic U VF -problem of thermoelasticity in the material description (see Sects. 6.2.1 and 6.3.4) and replacing the constitutive equations (6.42) by viscoelasticity relations in the forms (8.38) and (8.215), for models An and Bn we obtain a statement of the dynamic U VF -problem of thermoviscoelasticity in the material description. This statement consists of the equation system ı

D det F1 ; ı

ı

ı

[email protected][email protected]/ D r P C f; ı

ı

ı

ı

ı

ı

[email protected][email protected]/ D r . r / C qm C w ; ı

@FT [email protected] D r ˝ v; @[email protected] D v

(8.245)

ı

defined in the domain V .0; tmax /, and the constitutive equations in the same doı

main V .0; tmax /: .n/

.n/

TG

.n/

P D 4 E 0G T G ; .n/ .n/ .n/ t D [email protected] [email protected] C G .t// F G C G .t/; .t/; C tG ./; t ./ ;

(8.246a) (8.246b)

D0

ı

ı

D @ [email protected]; w D ı ; G D A; B; .n/ .n/ t D C G .t/; .t/; C tG ./; t ./ ;

(8.246c) (8.246d)

D0

.n/

.n/

which are complemented with expressions for the tensors 4 E 0 and C G : 4

.n/

E0 D

3 X

ı

ı

ı

ı

E ˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

˛;ˇ D1 .n/

CG D

3 1 X nIII ı ı .

p˛ ˝ p˛ hN G E/; n III ˛D1 ˛ ı

˛ ; p˛ ; p˛ k F;

(8.247)

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations

563

the boundary conditions (6.77)–(6.84) taking the forms (when there are no phase transformations) ı

ı

ı

ı

ı

n P D tne ; n q D q ne ı

ı

v n D 0;

ı

ı

ı

ı

D e

u D ue ; u n D 0;

ı

ı

at †1 ; : : : ; †4 ; †7 ; ı

at †5 ; †6 ; ı

ı

q n D 0; n P ˛ D 0 at †8 ;

(8.248)

and the initial conditions t D0W

v D v0 ; u D u0 ; F D E; D 0 :

(8.249)

On substituting the constitutive equations (8.246) and (8.247) into (8.245), we obtain the system for 16 unknown scalar functions being components of the following vectors and tensors: ; u; v; F k X i ; t: Due to the continuity equation, the density can be eliminated between the unknown functions. Just as for problems of thermoelasticity in the material description, the problem ı

(8.245)–(8.250) is formulated for a known domain V , that considerably simplifies its solving. For particular models of viscoelastic continua, formulae (8.246b)–(8.246d) are replaced by appropriate relations derived in Sects. 8.1–8.3. When models An of viscoelastic continua with the difference cores and the Duhamel–Neumann model (8.66) are considered, the specific entropy is determined by formula (8.69). Assuming that @0 '0 [email protected] depends only on temperature, we can rewrite the entropy balance equation of system (8.245) in the form of the heat conduction equation for a viscoelastic medium in the material description: 0 1 .n/ ı ı @ TC ı ı ı @ B ı c" D r r @˛ A C qm C w : @t @t ı

(8.250)

Here the second entropy term on the right-hand side of the equation, as a rule, can ı be neglected in comparison with w . Unlike the statement of the thermoelasticity problem in the material description given in Sect. 6.3.4, the thermoviscoelasticity problem (8.245)–(8.249) is strongly coupled even if there are no phase transformations, that is caused by the presı ence of the dissipation function w . As noted in Sect. 8.4.1, in the general case of ı non-isothermal processes a contribution of the function w to the heat conduction equation may be rather essential and cannot be neglected.

564

8 Viscoelastic Continua at Large Deformations

Using the statements of the dynamic U V -, T U VF - and U -problems of thermoelasticity in the material description (see Sect. 6.3.4), we can formulate the corresponding dynamic problems of thermoviscoelasticity. So the statement of the dynamic U -problem of thermoviscoelasticity in the material description consists of the equation system (6.58): ı

ı

ı

ı

ı

[email protected] [email protected] 2 / D r P C f; ı

ı

ı

(8.251)

ı

c" [email protected][email protected]/ D r . r / C qm C w ı

in the domain V .0; tmax /; constitutive equations (8.246); the expressions for .n/

.n/

tensors 4 E 0 and C G (8.247); the kinematic equation ı

F D E C r ˝ uT I

(8.251a)

boundary conditions (8.248) and initial conditions (8.249). The problem is solved for the four scalar functions: components of the displacement vector u and the temperature .

8.4.3 Statements of Quasistatic Problems of Viscoelasticity Theory in the Spatial Description Statements of quasistatic problems of viscoelasticity theory can be obtained formally from the corresponding statements of quasistatic problems of elasticity theory at large deformations (see Sect. 6.3.5) by replacing the constitutive equations of elasticity with appropriate relations of viscoelasticity derived in Sects. 8.1–8.3. So the statement of the coupled quasistatic problem of thermoviscoelasticity in the spatial description for linear mechanically determinate models An of thermorheologically simple continua has the form (

r T C f D 0; c" [email protected][email protected]/ D r . r / C qm C w in V;

8.n/ .n/ Rt ˆ ˆ T D J 4 R.t 0 0 / d C ./ ˆ ˆ ˆ 0 ˆ ˆ < Rt Rt .n/ d C .1 / w D J.a =2/ ˆ 0 0 ˆ ˆ ˆ .n/ .n/ R ˆ ı ı ˆ ˆ /d e ; : C D C "; " D ˛.e 0

(8.252)

in V [ †; .n/

@ 4 R.2t 0 0 0 / d C . /; 2 1 2 @t 0 .t;/ R .t 0 ; 0 / D a ..e // de ; 0

(8.253)

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations

8 ˇ ˆ n Tˇ† ˆ < ˆ ˆ :

8 .n/ .n/ ˆ 4 ˆ T D E T; ˆ ˆ ˆ ˆ .n/ ˆ P ı ı < C D n 1 III 3˛D1 nIII p˛ ˝ p˛ ; ˛ ˆ .n/ P ˆ ı ı ˆ 4 ˆ E D 3˛;ˇ D1 E˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ; ˆ ˆ ˆ ı :

˛ ; p˛ ; p˛ k F; F D E r ˝ uT in V [ †; ˇ ˇ ˇ D tne ; uˇ† D ue ; n qˇ† D qe ; ˇ† D e ; q

u

(8.254)

u n D 0; n T ˛ D 0 at †8 ; t D0W

565

(8.255)

D 0 :

Here (8.252) is the system of the equilibrium and heat conduction equations, (8.253) are the constitutive equations, (8.254) is the set of kinematic and energetic equivalence relations, (8.255) are the boundary and initial conditions, where † D †1 [ †2 [ †3 [ †4 [ †7 ; †u D †5 [ †6 , †q D †1 [ †2 [ †3 [ †4 [ †6 , † D †5 [ †7 . On substituting Eqs. (8.253) and (8.254) into (8.252), we obtain the system of four scalar equations for the four scalar unknowns: components of the displacement vector and the temperature u; k x; t: (8.256) If the model An with exponential cores is considered, then constitutive equations (8.253) should be replaced by relations (8.242). In particular, one can consider isothermal processes when a temperature field in a body V remains unchanged: .x; t/ D 0 D const; then the heat conduction equation can be excluded from the system (8.252). As a result, we obtain the following statement of the quasistatic problem of viscoelasticity in the spatial description for linear models An : r T C f D 0 in V I

.n/

Zt

T DJ

4

.n/

R.t / d C ./;

0

8 .n/ .n/ ˆ ˆ T D 4 E T; ˆ ˆ ˆ ˆ ˆ P ı ı 0, and in Eq. (8.311) there appears a heat source; therefore, @N [email protected] > 0, i.e. a heat-insulated beam will always be heated with time (while 0 D e and ˛N T D 0). This heating is caused only by energy dissipation; therefore, it is called dissipative heating.

8.6 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming

585

8.6.8 Temperature of Dissipative Heating in a Symmetric Cycle The process of deforming is called a symmetric cycle, if the average value of func.n/

.n/

tion f 1 .k1 / over the oscillation cycle is zero: h f 1 .k1 /i D 0.

.n/

As follows from (8.268), for a symmetric cycle the conditions h f ˛ .k1 /i D 0 (˛ D 1; 2; 3) are satisfied simultaneously. And from (8.312) it follows that the dissipation function depends only on the temperature N : N D a ./q.0/ N wN ./ fN2 ;

fN2

3 X

.n/

h f 2˛ .k1 /i:

(8.313)

˛D1

Then the heat conduction problem (8.311) takes the form ı

c"

@N N fN2 ; D ˛N T .N 0 / C a ./q.0/ @t

t D 0 W N D 0 :

(8.314)

Its solution has the form t D H.N /;

N H./

ZN 0

ı c" d e : 2 N N a ./q.0/f ˛N T .N e /

(8.315)

8.6.9 Regimes of Dissipative Heating Without Heat Removal If there is no heat removal from the beam .˛N T D 0/, then, depending on a form of N two basically distinct regimes of dissipative heating are possible. the function a ./, N is such that the integral (1) If the function a ./ N D H./

Z 0

ı

c" d N .a ./q.0/ fN2

(8.316)

becomes infinite at infinity (i.e. H.N / ! C1 as N ! 1), then the temperature of dissipative heating in the beam gradually grows with no limit (Fig. 8.13, the curve 1). For many real viscoelastic materials, as the function a ./ one frequently uses the Williams–Landel–Ferry dependence a ./ D exp

a1 . 0 / ; a2 C 0

a1 ; a2 const;

(8.317)

N ! C1 as N ! C1 is actually satisfied. for which the condition H./

586

8 Viscoelastic Continua at Large Deformations

Fig. 8.13 Different regimes of dissipative heating for a heat-insulated viscoelastic beam: 1 – unbounded growth of the temperature, 2 – heat explosion, 3 – heat pseudoexplosion

N is bounded at infinity (2) If the dependence a ./ is such that the function H./ N (i.e. H.C1/ < C1), then from (8.316) it follows that the temperature .t/ of dissipative heating reaches infinite values in the finite time t D H.C1/; N has a vertical asymptote as t ! t (Fig. 8.13, in other words, the function .t/ the curve 2). The phenomenon of sharp growth of the temperature at a certain time t is called the heat explosion. If, for example, the function a ./ is exponential (that is typical for some elastomers): a D ea1 . 0 / ;

a1 > 0;

(8.318)

then, calculating the integral in (8.316), we obtain the following expression for the temperature of dissipative heating: ! ı N2 a1 t 1 q.0/ f c" N D 0 lg 1 D : ; t .t/ ı a1 q.0/fN2 a1 c"

(8.319)

From this equation it really follows that N ! C1 as t ! t . (3) In practice sometimes there is an intermediate situation when the condition N with time H.N / ! C1 holds as N ! C1, but the growth of temperature .t/ proves to be so sharp that it becomes similar to the effect of heat explosion: the temperature reaches its ultimate value k , at which there occurs a heat destruction of the material, in comparably small time tk (Fig. 8.13, the curve 3). Such regimes are called the heat pseudoexplosion.

8.6.10 Regimes of Dissipative Heating in the Presence of Heat Removal When there is heat removal .˛T > 0/, the character of dissipative heating of a body N and ˛T .N 0 / are non-negative, the dechanges. Since both the functions ˛ ./ nominator of the integrand in (8.315) at a certain finite value N D 1 < C1 can N D t tends to infinity: N ! 1 as t ! C1. vanish; in this case the function H./ Thus, when there is heat removal, the regimes 1, 2, and 3 are impossible, because

8.6 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming

587

Fig. 8.14 Typical regimes of dissipative heating for a viscoelastic beam in oscillations with heat removal

the temperature always remains bounded, in this case there are other four typical regimes 4, 5, 6, and 7 (Fig. 8.14). (4) The regime 4 is realized when the dissipation function wN is independent of temperature .a D 1/. In this case the heat conduction equation (8.314) yields 2 N d 2 =dt 6 0 (Exercise 8.6.1); i.e. the dissipative heating curve is convex upwards. This is the most wide-spread type of a curve of dissipative heating for real practical problems. At this curve there are two typical sections: the section ‘a’, where the rate of PN heating decreases rapidly from its maximum value .0/ to practically zero PN 0, P and the steady section ‘b’, where N 0. N D 1, then the heat conduction problem (8.314) admits a solution in the If a ./ explicit form (Exercise 8.6.1) ı

q.0/fN2 c" N D 0 C ˛N T

1 exp

˛N T t ı

!! :

(8.320)

c"

N is unbounded as N ! C1 (for example, the exponential function (5) If a ./ (8.318)), then the temperature regime 5 is realized when the curve of dissipative heating can be split into the three typical sections: ‘a’ – initial, ‘b’ – steady, where P const, and ‘c’ – unsteady, where the function N .tN/ is convex downwards (Fig. 8.14) and unbounded as t ! C1. N depends on the temperature but is bounded as N ! C1 and has (6) If a ./ a point of inflection at N D e k (as the function (8.317)), then the temperature regime 6 of dissipative heating occurs (Fig. 8.14), when there are four typical sections: initial – ‘a’, steady – ‘b’, unsteady – ‘c’, followed by steady one – ‘d’. The temperature N .t/ remains bounded as t ! C1. (7) The regime 7 is realized under the same conditions as the regime 6, but at N reaches an the unsteady section ‘c’ the temperature of dissipative heating .t/ ultimate value k such that there occurs a heat destruction of the material (by analogy with the regime 3), and the section ‘d’ is not realizable.

588

8 Viscoelastic Continua at Large Deformations

8.6.11 Experimental and Computed Data on Dissipative Heating of Viscoelastic Bodies Figure 8.15 shows computed and experimental curves of dissipative heating for a polyurethane beam in cyclic deforming by the harmonic law (8.290). The change in temperature D 0 was found by solving the problem (8.311), (8.312) with the help of the implicit difference approximation ı

i C1 D

i C wN t=c" ı

1 C ˛N T =c"

;

(8.321)

where i D .ti / is the value of temperature in the i th node at time ti , and t is the step in time. Values of the relaxation core q.2t 0 / and q.0/ appearing in expression (8.312) for the dissipation function were determined by (8.292), and B . / and . / in this formula – by the relaxation curves with the help of the method given in Sect. 8.5.5. Values of the constants B . / and . / for polyurethane are shown in Table 8.1. The function a ./ was approximated by formula (8.317), and values of the constants in this formula were assumed to be a1 D 21, a2 D 208 K. The ı remaining constants in (8.311) take on the following values: D 103 kg=m3 , c" D 0:8 kJ=.kg K/, ˛N T D 10 kWt=.m3 K/. The mean value kN1 and the oscillation amplitude k10 were expressed in terms of the minimum and maximum deformation values in the cycle (ımin and ımax ): 1 1 kN1 D 1 C .ımax C ımin /; kN1 D .ımax ımin /: 2 2

(8.322)

Figure 8.15 exhibits temperatures of dissipative heating, computed by the method mentioned above for different models Bn at ımax D 50% and ımin D 34%. Under the conditions considered, the model BI gives the best approximation to experimental data. The distinction between different models Bn is considerable: the models BIV

Fig. 8.15 Curves of dissipative heating for polyurethane, computed by models Bn , and experimental curve of dissipative heating .ex/

8.6 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming

589

Fig. 8.16 Curves of dissipative heating for polyurethane, computed by the model BI at different amplitudes of oscillations

and BV lead to a stationary regime of dissipative heating according to the type (4), and the models BI and BII forecast the regime (5) with the presence of unsteady section. Figure 8.16 shows curves of dissipative heating for polyurethane, computed by the model BI at different values of ımin (values of this parameter are given in Fig. 8.16 by numbers at curves); the value ımax D 50% has been fixed. With growing the oscillation amplitude (i.e. in this case with decreasing the value ımin ), the intensity of dissipative heating sharply increases. Notice that the phenomenon of dissipative heating of viscoelastic materials can essentially reduce the durability of structures under cyclic deforming.

Exercises for 8.6 N D const, then a solution of the problem (8.314) is a 8.6.1. Show that if a ./ function being convex upwards, i.e. d 2 N =dt 2 6 0. Prove formula (8.320).

Chapter 9

Plastic Continua at Large Deformations

9.1 Models An of Plastic Continua at Large Deformations 9.1.1 Main Assumptions of the Models While models of viscoelastic continua most adequately describe a behavior of ‘soft’ materials (rubbers, polymers, biomaterials), for simulation of mechanical inelastic properties of ‘stiff’ materials (metals and alloys) one widely uses models of plastic media. There are different models of plastic continua, but we consider only the models falling into the class of models of plastic yield that is frequently used in practice. These models are convenient to be considered in the forms An , Bn , Cn and Dn , .n/

where in place of one should use the Gibbs free energy A (4.126), and the principal thermodynamic identity – in the form (4.133).

Definition 9.1. One can say that this is the model An of a plastic continuum, if for the medium 1. The operator constitutive equations (4.156) are functionals with respect to time t P of reactive variables R and their derivatives R: t

P P t .//I ƒ.t/ D f .R.t/; R.t/; Rt ./; R D0

(9.1)

2. The set of reactive variables R in addition includes a symmetric second-order .n/

tensor C p called the plastic deformation tensor, and the set of active variables .n/

ƒ contains a symmetric second-order tensor C e called the elastic deformation tensor: .n/ .n/

ƒ D f; ; C ; C e ; w g; .n/

.n/

.n/

.n/

R D f T =; C p ; g;

(9.2)

.n/

3. The tensors C e and C p are connected to C by the additive relation .n/

.n/

.n/

C D C e C C p:

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 9, c Springer Science+Business Media B.V. 2011

(9.3) 591

592

9 Plastic Continua at Large Deformations .n/

.n/

Unless the tensors C e and C p were involved in the set of variables R and ƒ, the relation (9.1) could be considered as the single model of viscoelastic continua of the .n/

differential and integral types. However, just the appearance of the tensors C e and .n/

C p leads to new effects which are not usual for the continuum types considered. Just as for continua of the differential type, the dependences (9.1) upon R.t/ and P R.t/ are assumed to be differentiable functions, and the functionals of prehistory P t ./ are assumed to be continuous and Fr´echet–differentiable. Rt ./, R .n/

.n/

The tensors C e and C p can be introduced axiomatically; however, to justify physically the introduction of these tensors one usually considers the following model. ı Introduce a reference configuration K, which is assumed to be unstressed (i.e. ı

T.0/ D 0 in V ), and an actual configuration K, where a stress tensor field T.t/, in general, is different from identically zero. In addition, introduce one more conp

figuration K (it may be not physically realizable), to which the continuum would correspond, if at the time tp > t its stress field were zero again: T.tp / D 0. Moreı

p

p

over, we impose on K the requirement that if the transformations K ! K ! K occur without plastic deformations (i.e. according to the model of an ideal continp

ı

p

uum), then configuration K must coincide with K. Such a configuration K is called unloaded. p ı ı Introduce in each of the configurations K, K and K local basis vectors ri , ri and p r i , respectively: p ı p @x @x @x ı ri D ; ri D ; ri D ; (9.4) @X i @X i @X i ı

p

ı

p

where x, x and x are radius-vectors of a material point M in K, K and K, respectively (Fig. 9.1).

Fig. 9.1 A scheme of transformation of configurations

9.1 Models An of Plastic Continua at Large Deformations

593

ı

p

As before, construct the metric matrices in K, K and K: ı

ı

gij D ri rj ;

ı

g ij D ri rj ; ı

p

p

p

g ij D r i r j ;

(9.5a)

p

and their inverse matrices g ij , gij and g ij , with the help of which introduce the vectors of reciprocal bases ı

ri D gij rj ;

ı

ı

ri D g ij rj ;

p

p

p

r i D g ij r j :

(9.5b)

Introduce the tensors of transformations of a local neighborhood of the point M ı

p

ı

p

from K to K, from K to K and from K to K: ı

p

p

Fp D r i ˝ ri ;

Fe D ri ˝ r i ;

ı

F D ri ˝ ri :

(9.6)

The tensors Fp and Fe are called the gradients of plastic and elastic deformations, respectively. These tensors are connected by the relation F D Fe F p :

(9.7)

Introduce also three right Cauchy–Green deformation tensors: ı

ı

C D "ij ri ˝ rj ;

ı

ı

Cp D "pij ri ˝ rj ;

ı

ı

Ce D "eij ri ˝ rj ;

(9.8)

where we have denoted components of the deformation tensor by "ij , of the plastic deformation tensor – by "pij and of the elastic deformation tensor – by "eij : "ij D

1 ı gij g ij ; 2

"pij D

1 p ı g ij g ij ; 2

"eij D

p 1 gij g ij : 2

(9.9)

In a similar way, introduce three right Almansi deformation tensors: ı

ı

ƒ D "ij ri ˝ rj ;

ı

ı

ƒe D "ij e ri ˝ rj ;

ı

ı

ƒp D "ij p ri ˝ rj ;

(9.10)

where we have denoted contravariant components of the deformation tensor by "ij , of the plastic deformation tensor – by "ij p and of the elastic deformation tensor – by "ij e : 1 ı ij 1 ı ij p ij 1 ı ij g gij ; "ij g g ; "ij g g ij : (9.11) "ij D p D e D 2 2 2 From (9.8)–(9.11) it follows that three right Cauchy–Green tensors and three right Almansi tensors are connected by the additive relations C D Ce C Cp ;

ƒ D ƒe C ƒp :

(9.12)

594

9 Plastic Continua at Large Deformations

From these equations it follows that the relation of additivity (9.3) holds for the I

V

energetic deformation tensors C and C if we assume as usual that I

V

Ce D ƒe ;

I

Ce D C;

V

Cp D ƒp ;

Cp D Cp :

(9.13)

In order to justify the additive relation (9.3) for n D II; IV, we introduce the polar decompositions for the deformation gradients (9.6): Fp D Op Up ;

F D O U;

Fe D Oe Ue ;

(9.14)

and represent the symmetric stretch tensors U and Ue in their eigenbases: UD

3 X

ı

ı

˛ p˛ ˝ p˛ ;

Ue D

˛D1

3 X

e

e

e

˛ p˛ ˝ p˛ :

(9.15)

˛D1

e

Here ˛ and ˛ are eigenvalues of the tensors U and Ue . Then we can introduce the tensors IV

CDUE D

IV

Cp D

ı

.˛ 1/ p˛ ˝ p˛ ;

e e IV P3 ı ı ı ı p ˝ p ; C D 1 p˛ ˝ p˛ ; ˛ ˛ e ˛ ˛ ˛ ˛D1 ˛D1 II

II

ı

˛D1

P3

C D E U1 D Cp D

P3

P3

˛D1

ı ı p˛ ˝ p˛ ; 1 1 ˛

e e II P3 ı ı 1 1 ı 1 ı p ˝ p ; C D p˛ ˝ p˛ ; 1 e ˛ ˛ ˛ ˛ ˛ ˛D1 ˛D1

P3

(9.16)

for which the additive relations (9.3) still hold. Thus, we have shown how the tensors of elastic and plastic deformations satisfying the relation (9.3) can be introduced.

9.1.2 General Representation of Constitutive Equations for Models An of Plastic Continua Consider the principal thermodynamic identity in the form An (4.121): d

.n/

.n/

C d T d C C w dt D 0:

(9.17)

9.1 Models An of Plastic Continua at Large Deformations

595

With the help of the additive relation (9.3) this identity can be rewritten as follows: d

.n/

.n/

.n/

.n/

C d T d C e T d C p C w dt D 0:

(9.18)

Introduce the Gibbs free energy just as in (4.126): .n/

D

T .n/ Ce:

(9.19)

Then for we obtain the identity (the principal thermodynamic identity in the form An ): .n/ .n/ .n/ .n/ (9.20) d C d C C e d T = T d C p C w dt D 0: According to the model An , the Gibbs free energy is a functional in the form (9.1):

t

D D0

.n/ .n/ P t ./ ; R D f T =; C p ; g: P R.t/; R.t/; Rt ./; R

(9.21)

Determine the total differential of this functional by using the rule of differentiation of functionals (8.24) with respect to time: 0 d D P dt D

.n/

1

.n/ @ B T C @ d C [email protected] AC d Cp .n/ .n/ @ @ T = @Cp 0 1

@

.n/

.n/ @ @ P BTC d C C d @ A C d C p C ı dt; (9.22) .n/ .n/ @P @. T =/ @ C p

@

where ı is the Fr´echet–derivative. .n/

The elastic deformation tensor C e is a functional of the same type (9.1): .n/ .n/ t P C e D Ce R.t/; R.t/; Rt ./; RP t ./ ; R D f T =; C p ; g:

.n/

(9.23)

D0

Just as for continua of the differential type (see Sect. 7.1.1), we introduce two new tensor functionals: .n/ C 0e

t D Ce R.t/; 0; Rt ./; RP t ./ ;

(9.24)

D0

.n/ C 1e

.n/

.n/

D C e C 0e :

(9.25)

596

9 Plastic Continua at Large Deformations

P are In (9.24) the arguments corresponding to the rates of changing functions R .n/

.n/

assumed to be zero. The tensors C 0e and C 1e are called the equilibrium elastic deformation tensor and nonequilibrium elastic deformation tensor, respectively. Substituting (9.22), (9.24) and (9.25) into (9.20) and grouping like terms, we obtain the identity

@

C

.n/

.n/ C 0e

@ T = C

@

.n/

.n/

@ T C C d C d @

.n/ d C p

.n/

.n/

C w . T Hp/

.n/ C p

.n/

@

d

.n/

T

@. T =/ C

.n/ C 1e

.n/

T

@ C p

C

@ d @

C ı dt D 0: (9.26)

.n/

Here we have introduced the tensors of strengthening H p and the reduced stress .n/

tensors T H :

.n/

.n/

.n/

H p D [email protected][email protected] C p /; .n/

.n/

.n/

T H D T Hp:

(9.27)

.n/

.n/

Since the differentials d. T =/, d, . T =/ , d , C p and dt are independent, the identity (9.26) holds if and only if coefficients of these differentials vanish; i.e. we have the relations 8 .n/ .n/ ˆ ˆ ˆ C 0e D @[email protected] T =/; .9:28a/ ˆ ˆ ˆ ˆ .9:28b/ < D @[email protected]; .n/

.n/

ˆ @[email protected] T =/ D 0; @[email protected]P D 0; @[email protected] C p D 0; ˆ ˆ .n/ ˆ .n/ .n/ .n/ .n/ ˆ ˆ 1 ˆ : w D . T H p / C p C e T = ı;

.9:28c/ .9:28d/

which are constitutive equations for models An of plastic continua. From these relations it follows that: 1. Plastic media are dissipative (for them w ¤ 0). .n/

.n/

2. The equilibrium deformation tensor C 0e (but not C e ) has the potential . .n/

.n/

3. The potential and hence C e , and are independent of the rates . T =/ , P and .n/ C p : t

P t .//; D .R.t/; Rt ./; R D0

.n/

.n/

R D f T =; C p ; gI .n/

P however, the dissipation function w and tensor C 1e depend on R.t/.

(9.29)

9.1 Models An of Plastic Continua at Large Deformations

597

Thus, the model An of plastic continua is specified by three functionals: the Gibbs free energy (9.29), the nonequilibrium elastic deformation tensor .n/ C 1e

t P P t ./ ; Rt ./; R D C1e R.t/; R.t/;

(9.30)

D0

.n/

and the plastic deformation tensor C p .

9.1.3 Corollary of the Onsager Principle for Models An of Plastic Continua .n/

To construct the functional (9.30) and tensor C p we use the Onsager principle (see Sect. 4.12.1, Axiom 16) and form the specific internal entropy production (4.728) with the help of (9.28d): 0 1 .n/

.n/ .n/ .n/ q q BTC q D w r D T H C p C 1e @ A ı r > 0: (9.31)

In order for this function to be non-negative, according to the Onsager principle, it can be represented in the generalized quadratic form. For this, introduce thermodynamic forces Xˇ : X1 D r;

.n/

X2 D T H ;

.n/

X3 D . T =/

(9.32)

and thermodynamic fluxes Q1 D q=;

.n/

Q2 D C p ;

.n/

Q3 D C 1e :

(9.33)

Then, according to the Onsager principle, the thermodynamic fluxes Qˇ must be linear (or tensor-linear) functions of Xˇ : 8 .n/ .n/ ˆ ˆ ˆ < q= D L11 r C L12 T H C L13 . T =/ ; .n/ .n/ .n/ C p D L12 r C L22 T H C L23 . T =/ ; ˆ ˆ ˆ .n/ .n/ : .n/1 C e D L13 r C L23 T H C L33 . T =/ :

(9.34)

Here L11 is a second-order tensor, L12 and L13 are third-order tensors, and L22 , L23 and L33 are fourth-order tensors, which are, according to the principle of

598

9 Plastic Continua at Large Deformations

equipresence, tensor functionals of the same form as the ones appearing in the general constitutive equations (9.1) of the model An : t

P P t .//: Rt ./; R L˛ˇ D L˛ˇ .R.t/; R.t/;

(9.35)

D0

Relations (9.34) are complementary constitutive equations to the system (8.28) for the model An of plastic continua. The first of the relations is the generalized Fourier law, the second one – the law of changing plastic deformations, and the third one – the law of changing nonequilibrium elastic deformations.

9.1.4 Models An of Plastic Yield Many special models of plastic continua follow from the law of changing plastic deformations (9.34) after introduction of some assumptions on a form of the functionals (9.29) and (9.35). In practice one often uses models An of plastic yield, where the functionals (9.29) and (9.35) are only functions of indicated arguments R and RP and of one scalar functional wp : .n/

.n/

D . T =; C p ; ; wp /; .n/ .n/

.n/

(9.36a)

.n/

L22 4 Lp . T ; C p ; ; T ; C p ; wp /;

(9.36b)

L11 D ./; L12 D 0; L13 D 0; L23 D 0; L33 D 0:

(9.36c)

The dependence on P and in these models is neglected; therefore, instead of the .n/

.n/

argument T = the function L22 can always have the argument T (because D .n/

.n/

.n/

. C / and P can always be expressed in terms of T and C p ). These models do not consider the cross–effects in relations (9.34), and the tensor C1e is identically zero: C1e 0:

(9.37)

The scalar functional wp is usually chosen in the form Z wp D

t .n/ 0

.n/

T ./ C p ./ d ;

(9.38)

called the Taylor parameter (the specific work done by plastic deformations), or in the form

9.1 Models An of Plastic Continua at Large Deformations

Z wp D

t 0

.n/ C p ./

.n/ C p ./

599

!1=2 d

(9.39)

called the Odkwist parameter. In this case relations (9.34) and (9.28d) have the following general form: q D r ; .n/ C p

(9.40a)

.n/

D 4 Lp T H :

.n/

w D . T H

(9.40b)

.n/ @ .n/ T / C p : @wp

(9.40c)

From the inequality (9.31) it follows that the heat conductivity tensor is symmetric and positive-definite, and the tensor 4 Lp is symmetric in pairs of indices .1; 2/ $ .3; 4/ (the symmetry in indices 1 $ 2 and 3 $ 4 follows from symmetry of the .n/

.n/

.n/

tensors C p and T ; H p ), i.e. the tensor is symmetric in the form (7.21), has not more than 21 independent components and is also positive-definite.

9.1.5 Associated Model of Plasticity An

In applications one frequently uses the associated model of plasticity, where the law of plastic yield (9.40b) is connected (or associated) with the concept of a yield surface. Consider this model. .n/

.n/

Let in the six-dimensional space of components T ij of the stress tensor T with ı respect to some basis, for example, with respect to the basis ri , there be a surface whose position is given by the scalar equation system fˇ D 0;

ˇ D 1; : : : ; k:

(9.41)

Here fˇ are functions in the form (9.36a) .n/ .n/

fˇ D fˇ . T ; C p ; ; wp /;

(9.42)

.n/

depending parametrically on C p , and wp , they are called the plastic potentials. Denote the partial derivative of functions (9.42) with respect to time by d 0 fˇ @fˇ .n/ T ; D .n/ dt @T

(9.43)

600

9 Plastic Continua at Large Deformations

that coincides with the total derivative fˇ D dfˇ =dt only if fˇ is independent of .n/

C p , and wp : .n/

fˇ D fˇ . T /:

(9.44)

In this case one can say that the model of an ideally plastic continuum is considered. The associated model (9.42) taking such dependence into account is called the model of a strengthening plastic continuum. We assume axiomatically that: 1. Inside a domain bounded by the yield surface (9.41) plastic deformations remain unchanged, i.e. if all fˇ < 0;

.n/

then C p D 0I

(9.45)

if at least for one value of ˇ the condition dfˇ =dt > 0 is satisfied, then one can say that there occurs active loading, and if all d 0 fˇ =dt 6 0, then there occurs passive loading or unloading. 2. On the yield surface when d 0 fˇ =dt D 0, plastic deformations remain also un.n/

changed (such loading is called neutral), and if d 0 fˇ =dt > 0, then C p vary (one can say that there occurs plastic loading), i.e. .n/

if fˇ D 0; d 0 fˇ =dt D 0; then C p D 0; .n/

if fˇ D 0; d 0 fˇ =dt > 0; then C p ¤ 0:

(9.46) (9.47)

.n/

Notice that the state of a continuum when its tensor T is outside of the yield surface is impossible, because we assume axiomatically that the yield surface moves .n/

.n/

with changing the tensor T if C p ¤ 0; i.e. the condition f > 0 cannot be satisfied. .n/

The specific expression for the rate of plastic deformation C p in the case (9.47) of plastic loading is given by the Drucker model (or the gradient law). According to .n/

this model, the tensor C p is chosen to be proportional to the gradient of the yield surface: k X .n/ .n/ C p D ~P ˛ [email protected]˛ [email protected] T /: (9.48) ˛D1

Here ~P ˛ are the ratio coefficients, which can be written in the convenient form of derivatives with respect to time and which are scalar functions in the form .n/ .n/

.n/

~P ˛ D ~P˛ . T ; C p ; C p ; ; wp /;

˛ D 1; : : : ; k:

(9.49)

9.1 Models An of Plastic Continua at Large Deformations

601

If k 6 6, then one can find these functions from the gradient equation with complementing the tensor equation (9.48) by k equations (9.41) of the yield surface. Thus, we have 6 C k scalar equations (9.48) and (9.41) to determine six components .n/

of the tensor C p and k functions ~˛ (˛ D 1; : : : ; k). Notice that the gradient law (9.48) is written only for plastic loading. In order .n/

to derive an expression for C p under arbitrary loading, we should combine the relations (9.45)–(9.47). This can be done with the help of the Heaviside functions hC .x/ and h .x/: hC .x/ D

x > 0; x < 0;

1; 0;

h .x/ D

1; 0;

x > 0; x 6 0;

(9.50)

and their combinations 0 d fˇ : 1 hC .fˇ /h ˇ D1 dt k

hD1 …

(9.51)

k

Here … is the product. One can easily verify that if the conditions (9.45) or (9.46) ˇ D1

are satisfied, then h D 0, and if the condition (9.47) is satisfied, then h D 1. Thus, .n/

for C p under arbitrary loading we obtain .n/

C p D h

k X

.n/

~P ˛ [email protected]˛ [email protected] T /:

(9.52)

˛D1

This relation must satisfy the corollary of the Onsager principle (9.40b) for models of plastic yield; i.e. the following equation must hold h

k X

.n/

.n/

~P˛ [email protected]ˇ [email protected] T / D 4 Lp T H :

(9.53)

˛D1

Here 4 Lp is some indeterminate symmetric fourth-order tensor. Equation (9.53) means that the functions fˇ called plastic potentials must be quasilinear functions .n/

.n/

of T H p . Remark. For the associated model of plasticity, a part of constitutive equations for .n/

C p is given by Eqs. (9.41) being the implicit forms of components of the tensor

.n/

C p . These equations can also be represented by the expression (9.40b) written in the implicit form .n/

ˆˇ C p 'ˇ D 0;

(9.54)

602

9 Plastic Continua at Large Deformations

where ˆˇ are symmetric second-order tensors, and .n/

.n/

'ˇ D ˆˇ 4 Lp . T H p /:

(9.55)

Indeed, differentiating (9.41) with respect to t, we obtain that .n/

ˆˇ D @f [email protected] C p ;

'ˇ D

@fˇ .n/

.n/

T C

@T

@fˇ P : @

(9.56)

Since relations (9.44) are scalar, the corollary (9.54) of the Onsager principle imposes no constraints on the form of the plastic potentials fˇ , and the relation @fˇ .n/

.n/

T C

@T

.n/ @fˇ P @fˇ 4 Lp T p ; D .n/ @ @Cp

(9.57)

being a consequence of (9.55) and (9.56) and an analog of formula (9.53), can alt u ways be satisfied by the proper choice of indeterminate tensor 4 Lp . .n/

For the elastic deformations C e in the associated model, from (9.28a) and (9.37) we get the relation .n/

.n/

.n/

@

C e D K. T =; C p ; ; wp / D

.n/

:

(9.58)

@ T = The model of a plastic continuum, where the potential does not depend explic.n/

itly on the plastic deformation tensor C p , is called the model An of an elastoplastic continuum; for this model, .n/

.n/

D . T =; ; wp /; H p 0;

.n/

.n/

C e D K. T =; ; wp / D

@ .n/

: (9.59)

@ T = This model is widely used in practice. A model with explicit dependence of upon .n/

C p is usually applied when one needs the effect of deformation anisotropy to be taken into account. This effect consists in the change of a symmetry group for a continuum considered in the process of varying plastic deformations. .n/

The model, where the dependence of upon both C p and wp may be neglected, is called the model of an ideally elastoplastic continuum (it is not to be confused with an ideally plastic continuum, where, according to the definitions (9.44), the .n/

plastic potentials fˇ are independent of C p and wp ).

9.1 Models An of Plastic Continua at Large Deformations

603

.n/

.n/

Finally, if is independent of C p and wp , and depends on T = only quadrati cally, then one can say that this is the model An of a plastic continuum with linear elasticity.

9.1.6 Corollary of the Principle of Material Symmetry for the Associated Model An of Plasticity Apply the principle of material symmetry to constitutive equations (9.53) and .n/

.n/

.n/

(9.58). Since all the tensors C e , C p and T are H -indifferent relative to orthogonal H -transformations, from the principle of material symmetry it follows that the elastic potential (9.36a) must be a function of simultaneous invariants J.s/ ı

.n/

.n/

of the tensors T and C p relative to some group G s in an undistorted reference ı

configuration K:

D J.s/ ; ; wpˇ :

Here J.s/

D

.n/ .n/

J.s/

(9.60)

! D 1; : : : ; z:

T; Cp ;

(9.61)

A proof of this assertion is the same as the one for continua of the differential type (see Sect. 7.1.4). ı

For simplicity, a reference configuration K is assumed to be undistorted; and, ı

as before (see Sects. 4.7.3 and 8.1.6), we will consider groups G s only in this configuration. In place of one scalar functional wp (9.38), the set of arguments of the function (9.60) in general may include functionals wpˇ being integrals of such simultaneous .n/

.s/

.n/

invariants J that contain both the tensors T and C p : wpˇ

Z

t

D 0

.n/

.n/

Jˇ.s/ . T ./; C p .// d ;

ˇ D 1; : : : ; z:

(9.62)

Then the constitutive equations (9.58) can be represented in the tensor basis by analogy with ideal continua: .n/

Ce D

z X D1

' J.s/ T:

(9.63)

604

9 Plastic Continua at Large Deformations .s/

Here ' are scalar functions in the form (9.60), and J T are the derivative tensors: p ' D ' J˛.s/ ; ; wˇ D @[email protected].s/ ;

@J.s/

.s/

J T D

.n/

;

D 1; : : : ; z: (9.64)

@ T = As follows from the principle of material symmetry, the plasticity functions fˇ .s/p (9.42) are also functions of simultaneous invariants J˛ : fˇ D fˇ J˛.s/p ; ; wpˇ ;

ˇ D 1; : : : ; k:

(9.65)

However, according to the corollary (9.53) of the Onsager principle, the derivatives .n/

.n/

.s/p

@fˇ [email protected] T must be quasilinear functions of the tensor T p ; therefore, in (9.65) J˛ .n/

.n/

must be simultaneous invariants of T p and C p , being only linear and quadratic functions of these tensors: .n/

.n/

J˛.s/p D J˛.s/ . T H ; C p /;

˛ D 1; : : : ; z1 6 z:

(9.66)

Then constitutive equations (9.52) in the tensor basis take the form .n/ C p

D

z1 X

.s/p ˛ J˛T ;

(9.67)

˛D1

where ˛

Dh

k X

~Pˇ

ˇ D1

@fˇ .s/p @J˛

;

.n/

.s/p J˛T D @J˛.s/p [email protected] T :

(9.68)

Relations (9.60)–(9.68) are called the representation of the associated model of plasticity in the tensor basis. .s/p .s/ and J˛T are connected by the relation The derivative tensors J˛T .s/p J˛T

D

@J˛.s/p .n/

@TH

.n/

.n/

@. T H p / .n/

.s/p

.n/

D J˛TH . 4 H pT /:

(9.69)

@T

With the help of (9.27) and (9.59) we find an expression for the fourth-order tensor 0 1 .n/ z .s/ X .n/ @ B @ @J C @Hp 4 D H pT @ .s/ .n/ A .n/ .n/ @J D1 @T @T @Cp z ' X ˇ .s/ .s/ .s/ .s/ Jˇ.s/ Cı : ˝ J CJ ˝ J ' J D ˇ T C C TC T p p p 2 ;ˇ D1

(9.70)

9.1 Models An of Plastic Continua at Large Deformations

605

Here we have introduced the notation 'ˇ D

@'

@2

D

.s/

@Jˇ

; .s/

.s/

@Jˇ @J

@2 J.s/

J.s/ TCp D

.n/ .n/

:

(9.71)

@ T @Cp

On substituting (9.68) into (9.67), we finally obtain .n/ C p

D

z1 X

.s/p ˛ J˛TH

.n/

. 4 H pT /:

(9.72)

˛D1 .n/

For the associated model of an elastoplastic continuum, is independent of C p , hence .n/

.n/

H p 0;

J.s/p

D

J.s/ ;

J.s/p T

D

J.s/p TH

4 H pT 0; D

J.s/ T;

(9.73)

J.s/ Cp

J.s/ TCp

D 0;

D 0:

Thus, the invariants J.s/p and J.s/ coincide, and constitutive equations (9.60), (9.63), (9.65), and (9.72) take the forms .n/ D I.s/ . T =/; ; wpˇ ; D 1; : : : ; r; .n/

Ce D

r X D1

.n/ C p

D

z1 X

.s/

(9.74a)

.n/

' I T . T =/;

(9.74b)

.n/ .n/ .s/ ˛ J˛T . T ; C p /;

(9.74c)

˛D1 .n/ .n/

fˇ D fˇ .J˛.s/ . T ; C p /; ; wpˇ /; ' D

@

; .s/

@I

.n/

I.s/ T

D

@I.s/ . T =/ .n/

;

˛

Dh

k X ˇ D1

@ T =

(9.74d)

~Pˇ

@fˇ @J˛.s/

:

(9.74e)

9.1.7 Associated Models of Plasticity An for Isotropic Continua Let us write representations (9.74) for the three main symmetry groups Gs : O, T3 and I , with choosing simultaneous invariants J.s/ in the same way as it was done for viscoelastic continua.

606

9 Plastic Continua at Large Deformations

For an isotropic continuum, the functional basis of simultaneous invariants .n/

.n/

J.I / . T =; C p / consists of nine invariants, which can be chosen as follows (see (7.35) and (8.83)): .n/

.n/

.I / J˛C3 D I˛ . C p /;

J˛.I / D I˛ . T =/; J7.I / D

1 .n/ .n/ T Cp;

J8.I / D

˛ D 1; 2; 3;

1 .n/ .n/2 T C p;

J9.I / D

1 .n/2 .n/ T C p ; (9.75) 2

.s/

Then the derivative tensors J T (9.64) have the forms (see (8.84)) .I / D E; J1T

.n/

.n/

.I / J2T D 1 .EI1 . T / T /;

.I / J3T D

.n/ 1 .T2 2

.n/

.I / .I / J˛C3;T D 0; ˛ D 1; 2; 3I J7T D C p ; .n/ .n/

.I /

.n/

.I /

.n/

I1 T C EI2 /; .n/

J8T D C 2p ;

.n/

J9T D 1 . T C p C C p T /:

(9.76)

Substituting these expressions into (9.63) and collecting terms with the same tensor powers, we obtain the following representation of constitutive equations (9.63) in the tensor basis: .n/

Ce D e ' 1E C

.n/ .n/ .n/ .n/ e ' 2 .n/ '3 .n/2 e ' 6 .n/ .n/ T C 2 T Ce '4 C p C e ' 5 C 2 C . T C p C C p T /: (9.77)

Here we have denoted the scalar functions (compare with (4.322a)) e ' 1 D '1 C ' 2 I1 C ' 3 I2 ;

e ' 2 D '2 C '3 I1 ; e ' 3C D '6C ; D 1; 2; 3: (9.78)

' 5 and e ' 6, For the model of an elastoplastic continuum (9.59), the functions e '4, e due to (9.64), are zero; and we obtain the relation .n/

.n/

.n/

Ce D e ' 1 E C .e ' 2 =/ T C .'3 =2 / T 2 ;

(9.79)

which is analogous to the constitutive equation of an ideally elastic isotropic continuum (4.322) but written in the inverse form. Here ' D

@ ; @I

.n/

.n/

.n/

D .I1 . T =/; I2 . T =/; I3 . T =/; ; wp /;

and there is only one Taylor parameter wp .

(9.80)

9.1 Models An of Plastic Continua at Large Deformations

607

For an elastoplastic continuum, according to the Onsager principle, the set of arguments of the plasticity functions fˇ includes only linear and quadratic invari.n/

.n/

ants, i.e. only J.I / . T ; C p /, D 1; 2; 4; 5; 7, (z1 D 5), among which there is .I / only one simultaneous invariant, namely J7 . On substituting the derivative tensors (9.76) into (9.74c), we obtain .n/

C p D e1 E

where

˛

1,

Dh

2,

and

k X

~Pˇ

ˇ D1

7

.n/

2T

C

.n/

7 Cp;

(9.81)

are determined by formulae (9.68):

@fˇ .n/

;

7

Dh

k X

~P ˇ

ˇ D1

@J˛ . T /

@fˇ .I / @J7

; e1 D

1

C

.n/

2 I1 . T /;

(9.82) .n/

.n/

.n/

.n/

.I /

.n/ .n/

fˇ D f .I1 . T /; I2 . T /; I1 . C p /; I2 . C p /; J7 . T ; C p /; ; wp /:

(9.83)

Formulae (9.78)–(9.82) give the general form of constitutive equations of the associated model of an isotropic elastoplastic continuum.

9.1.8 The Huber–Mises Model for Isotropic Plastic Continua For special models of isotropic elastoplastic continua, one usually accepts additional assumptions on a form of the elastic and plastic potentials and fˇ . As shown in experiments, for many elastoplastic continua, their behavior can be described adequately enough by the Huber–Mises model, where there is only one plastic potential f depending explicitly on the only simultaneous invariant YH : f D f .YH ; ; wp /:

(9.84)

Here the invariant YH has been introduced as contraction of the tensor PH being the .n/

.n/

deviator of the tensor T H C e (see the definition of the deviator (8.167)): YH2 D

3 PH PH ; 2

.n/ .n/ .n/ 1 .n/ PH D . T H C p / I1 . T H C p /E; 3

(9.85)

and H is the strengthening parameter being a scalar function in the form H D H0 Yp2n0 ;

(9.86)

608

9 Plastic Continua at Large Deformations

where H0 and n0 are the constants, and Yp is the invariant of the deviator of the .n/

tensor C p determined similarly to (9.85): Yp2 D

.n/ 1 .n/ Pp D C p I1 . C p /E: 3

3 Pp Pp ; 2

(9.87) .n/

.n/

The invariants YH and Yp are called the intensities of the tensors T H C p and .n/

C p , respectively. .n/

The deviator can be constructed for any tensor; for example, for the tensor T : .n/ 1 .n/ PT D T I1 . T /E; 3

YT2 D

3 PT PT ; 2

(9.88)

.n/

where YT is the intensity of the tensor T . Each deviator of a tensor is orthogonal to the unit (metric) tensor (more detailed information on properties of the deviators can be found in [12]): PH E D 0;

Pp E D 0;

PT E D 0:

(9.89)

The invariants YT and Yp can be expressed in terms of the principal invariants of corresponding tensors (see Exercise 9.1.2): .n/

.n/

.n/

YT2 D I12 . T / 3I2 . T /;

.n/

Yp2 D I12 . C p / 3I2 . C p /:

(9.90)

We can immediately verify that the invariant YH2 can be expressed in terms of the .n/

.n/

invariants I˛ . T /, I˛ . C p / and simultaneous invariant J7.I / : .I /

YH2 D YT2 C H 2 Yp2 3HJ7

.n/

.n/

C HI1 . T /I1 . C p /:

(9.91)

.n/

Determining the derivatives @f [email protected]˛ . T / of the function (9.84): @f @I˛ @f @I1

D

@f @YH @Y˛ @I˛

2 @YH @I˛

D fY .n/

.n/

D fY .2I1 . T / C HI1 . C p //;

; fY

@f @I2

@f 1 2YH @YH

D 3fY ;

@f .I / @I7

;

D 3HfY ; (9.91a)

and substituting them into (9.82), we find that .n/

1

.n/

D ~hf P Y .2I1 . T /CHI1 . C p //;

2

D 3~hf P Y;

7

D 3~hf P Y H: (9.91b)

9.1 Models An of Plastic Continua at Large Deformations

609

Then constitutive equations (9.81) take the form .n/ C p

D 3~hf P Y PH :

(9.92)

Due to the property (9.89) of the deviators, from (9.92) it follows that a continuum described by the Huber–Mises model is plastically incompressible, i.e. .n/ C p

ED0

or

.n/

I1 . C p / D 0:

(9.93)

Therefore, with the help of (9.87) Eq. (9.92) can be written as the following quasilinear relation between the deviators: P Y PH : PP p D 3~hf

(9.94)

The scalar product of Eq. (9.94) by itself yields the following expression for ~P under plastic loading: q PP p PP p ~P D ˙ p : (9.95) 6fY YH After substitution of expression (9.95) into (9.94) the number of independent equations in (9.94) reduces to four, and relationships (9.84), (9.93), and (9.94) form the complete system of constitutive equations (it consists of six independent relationships). If the plastic potential is chosen in the Mises form f D

1 .YH = s /2 1; 3

(9.96)

where s D s .; wp / is the given function of and wp called the yield point or the yield strength, then fY D 1=.3 s2 /, and the final relations of plasticity (9.84), (9.92), and (9.93) become .n/

C p D

f D

.n/ .n/ ~h P .P H C /; I . C p / D 0; T p 1 s2

1 .YH = s /2 1 D 0; 3

(9.97)

.n/ 1 .n/ PT D T I1 . T /E; H D H0 Yp2n0 : 3

The Taylor parameter (9.38), due to plastic incompressibility (9.93), can be written in terms of the deviators: Z t PT PP p d : (9.98) wp D 0

610

9 Plastic Continua at Large Deformations

One can say that this is the model of an isotropic plastic continuum with linear strengthening, if the strengthening parameter H (9.86) is a constant, i.e. H D H0 and n0 D 0:

9.1.9 Associated Models of Plasticity An for Transversely Isotropic Continua For a transversely isotropic continuum, the functional basis of simultaneous invari.n/ .n/

ants J.3/ . T ; C p / consists of 11 invariants, which can be chosen as follows (see (7.36) and (8.87)): .n/

.n/

.3/ J.3/ D I.3/ . T /; D 1; : : : ; 5I J5C D I.3/ . C p /; D 1; : : : ; 4I ! .n/ .n/ .n/ .n/ .3/ .3/ .3/ .3/ .3/ 2 2 c3 T b c3 C p ; J11 D T C p 2J10 J2 J8 : J10 D .E b

(9.99) The derivative tensors .3/ J1T D E b c23 ;

.s/ J T

(9.64) in this case have the forms (see (8.88))

.3/ J2T Db c23 ;

.n/ 1 .O1 ˝ O1 C O2 ˝ O2 / T ; 2

.3/ J3T D

.n/

.n/

.n/

.3/ .3/ J4T D 2 4 O3 T ; J5T D T 2 I1 T C EI2 ; .3/

.3/

.3/

.3/

J6T D J7T D J8T D J9T D 0; .3/ J10T D

.n/ 1 .O1 ˝ O1 C O2 ˝ O2 / C p ; 4

(9.100) .n/

.3/

J11T D 4 O3 C p :

Substituting these expressions into (9.63) and collecting terms with the same tensor powers, we obtain the following representation of constitutive equations (9.63) for a transversely isotropic plastic continuum in the tensor basis: .n/

.n/

Ce D e ' 1E C e ' 2b '3 T c23 C .O1 ˝ O1 C O2 ˝ O2 / .e .n/

.n/

.n/

.n/

' 4 T C '5 T 2 C '11 C p : Ce ' 10 C p / C e

(9.101)

Here we have denoted the scalar functions .n/

.n/

.n/

' 2 D '2 '1 2'4 I2.3/ . T / '11 I2.3/ . C p /; e ' 1 D '1 C '5 I2.3/ . T /; e e '3 D

'3 2

'4 ; e ' 10 D

'10 4

'11 2 ;

.n/

e ' 4 D 2'4 '5 I1 . T /:

(9.102)

9.1 Models An of Plastic Continua at Large Deformations

611

For the model of an elastoplastic continuum (9.58), we have '10 D '11 D 0, and Eq. (9.101) takes the form .n/

.n/

.n/

.n/

c23 C e Ce D e ' 1E C e ' 2b ' 3 .O1 ˝ O1 C O2 ˝ O2 / T C e ' 4 T C '5 T 2 : (9.103)

Here .n/

.n/

D .I1.3/ . T /; : : : ; I5.3/ . T /; ; wp1 ; : : : ; wp3 /;

' D [email protected][email protected].3/ /; (9.104)

and wpˇ are the Taylor parameters (9.62) for a transversely isotropic continuum, and the number of the parameters in this case is equal to three: wp1 D

Z Z

p

w3 D wp2 D

t 0

t .n/

t 0

.n/

Z

b t .n/ 0

b .n/ T 33 C 33 d ;

T C p w2 w1 ;

0

Z

.n/

.n/

.b c23 T /.b c23 C p / d D p

p

.n/

.n/

..E b c23 / T / .b c23 C p / d D

Z 0

t .b n/

.b n/ .b n/ .b n/ . T 13 C 13 C T 23 C 23 / d :

(9.105) For an elastoplastic continuum, according to the Onsager principle, only the .3/

cubic invariant J3 ity functions fˇ :

.n/

D I3 . T / does not appear in the set of arguments of the plastic-

.3/ fˇ D fˇ .J1.3/ ; J2.3/ ; J4.3/ ; : : : ; J11 ; ; wp1 ; : : : ; wp3 /:

(9.106)

.3/ .3/ There are two simultaneous invariants in the set (9.99), namely J10 and J11 . Then, substituting the derivative tensors (9.100) into (9.74c), we obtain .n/

C p D

1E

C.

2

C4 O3 .2

c23 1 /b .n/

4T

1 C .O1 ˝ O1 C O2 ˝ O2 / . 2 .n/

C

11 C p /;

˛

Dh

k X ˇ D1

.n/

3T

C

10

2

.n/

C p/

(9.107) ~Pˇ [email protected]ˇ [email protected]˛.3/ /:

(9.108)

612

9 Plastic Continua at Large Deformations

9.1.10 Two-Potential Model of Plasticity for a Transversely Isotropic Continuum For special models of transversely isotropic elastoplastic continua, we accept an additional assumption on a form of the potentials and fˇ . In the two-potential model, we suppose that there are two plastic potentials, one of which, namely f2 , .3/ c contain components with depends only on those invariants J , which in basis b .b n/ .n/ subscript 3, i.e. T ˛3 and C p˛3 , and f1 depends on the remaining invariants: .n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.3/ . T ; C p /; ; wp1 ; wp2 /; f1 D f1 .I1.3/ . T /; I4.3/ . T /; I1.3/ . C p /; I4.3/ . C p /; J10 .3/ f2 D f2 .I2.3/ . T /; I3.3/ . T /; I2.3/ . C p /; I3.3/ . C p /; J11 . T ; C p /; ; wp3 /:

(9.109) In the transversely isotropic two-potential Huber–Mises model, we assume that each of the potentials f1 and f2 is a function of the simultaneous Huber–Mises invariants for a transversely isotropic continuum Y˛.3/ : f2 D f2 .Y2.3/ ; Y3.3/ ; ; wp1 ; wp2 /;

f1 D f1 .Y1.3/ ; Y4.3/ ; ; wp3 /:

(9.110)

Here .n/

.n/

Y˛.3/ D I˛.3/ . T H˛ C p /;

.n/

0

H˛ D H˛0 .I˛.3/ . C p //n˛ ; (9.111)

˛ D 1; : : : ; 4I

are simultaneous invariants which are uniquely expressed in terms of the invariants (9.99) (see Exercise 9.1.5): .n/

.n/

Y˛.3/ D I˛.3/ . T / H˛ I˛.3/ . C p /;

˛ D 1; 2;

.n/

.n/

.3/ Y3.3/ D I3.3/ . T / C H2 I3.3/ . C p / 2H3 J10 ; .3/

Y4

.3/

.n/

.3/

.n/

(9.112)

.3/

D I4 . T / C H2 I4 . C p / 2H4 J11 ;

and H˛0 and n0˛ are the constants. Calculating the derivatives @f [email protected]˛.3/ of the functions (9.110) (only occurring in the expression (9.67)): .3/

@f1 [email protected]

D f11 ;

.3/

@f1 [email protected]

D f14 ;

.3/

@f1 [email protected] D 2f14 H4 ;

(9.113)

.3/ @f2 [email protected]/ D f22 ; @f2 [email protected]/ D f23 ; @f2 [email protected] D 2H3 f23 ; fˇ @fˇ [email protected].3/ ;

9.1 Models An of Plastic Continua at Large Deformations

613

and substituting them into (9.108), we obtain 1

D ~P 1 f11 h; 10

2

D ~P2 f22 h;

D 2~P 2 f23 H3 h;

3

D ~P2 f23 h;

11

4

D ~P1 f14 h;

D 2~P 1 f14 H4 h:

(9.114)

Then the constitutive equations (9.107) for plastic deformations become .n/

.n/

.n/

C p D ~P1 h.f11 .E b c23 / C 2f14 4 O3 . T H3 C p // C ~P2 h.f22b c23 C

.n/ .n/ f23 .O1 ˝ O1 C O2 ˝ O2 / . T H4 C p //: 2 (9.115)

We can immediately verify (see Exercise 9.1.4) that the tensors .n/

.n/

P1H f11 .E b c23 / C 2f14 4 O3 . T H3 C p /; c23 C P1H f22b

f23 .O1 2

.n/

.n/

˝ O1 C O2 ˝ O2 / . T H4 C p /

(9.116)

are mutually orthogonal: P1H P2H D 0:

(9.117)

Therefore, rewriting Eq. (9.115) with use of notation (9.116) in the form .n/

C p D ~P1 hP1H C ~P 2 hP2H

(9.118)

(this is an analog of the relationship (9.94)) and multiplying the result by P1H and P2H , we get v u .n/ u t C p P1H ; ~P 1 D ˙ P1H P1H

v u .n/ u t C p P2H ~P2 D ˙ P2H P2H

(9.119)

– the expressions for ~1 and ~2 , being similar to the expression (9.95). After substitution of (9.119) into (9.118), for determining all components of the plastic deformation tensor Eq. (9.118) should be complemented by two more scalar ones f1 D 0;

f2 D 0;

where fˇ are expressed by formulae (9.110).

614

9 Plastic Continua at Large Deformations

These functions are usually chosen in a quadratic form being similar to the Mises model (9.96): 2f1 D

2f2 D

Y4H 4s

2

C

jY1H j C Y1H C 2 1s

jY2H j C Y2H C 2 2s

!2

C

!2

C

jY1H j Y1H 2 1s

jY2H j Y2H 2 2s

2

C

Y3H 3s

2

2

1;

1: (9.120)

˙ The functions 1s .; wp3 / are called the yield points (or the yield strengths) in longitudinal tension and compression, respectively, and 4s .; wp3 / – the yield point p p ˙ in shear along the plane of transverse isotropy. The functions 2s .; w1 ; w2 / are p called the yield points in transverse tension and compression, and 3s .; w1 ; wp2 / – the yield point in interlayer shear. These functions are usually determined in experiments. For anisotropic continua, the distinction between the yield points in tension C and ˛s may be and compression is rather considerable; therefore, the functions ˛s essentially distinct. Notice that although the functions fˇ depend on the sign of invariants Y1H and Y2H , they are differentiable everywhere, including the case when Y1H D 0 and Y2H D 0, and their derivatives (9.113) take on the values

f˛˛ D

jY˛H j C Y˛H jY˛H j Y˛H C ; ˛ D 1; 2I C 2 ˛s 2 ˛s

f14 D Y4H = 4s ;

f23 D Y3H = 3s :

(9.121)

9.1.11 Associated Models of Plasticity An for Orthotropic Continua For an orthotropic continuum, the functional basis of simultaneous invariants .n/ .n/

J.O/ . T ; C p / consists of twelve invariants, that should be complemented by two more, in general, dependent invariants in order to obtain the set of invariant, being symmetric relative to all the basis vectorsb c˛ (see (7.37) and (8.91), (8.92)): .n/

.n/

.O/ J.O/ D I.O/ . T /; D 1; ; 6I J.O/ C6 D I . C p /; D 1; 2; 3; 6I .O/

.n/

.n/

.n/

.n/

.O/

.n/

.n/

J10 D .b c22 T / .b c23 C p /; J11 D .b c21 T / .b c23 C p /; .n/

.n/

.n/

.O/ .O/ c21 T / .b c22 C p /; J14 D .b c21 T / .b c22 T / D I7.O/ . T /: (9.122) J13 D .b

9.1 Models An of Plastic Continua at Large Deformations

615

The derivative tensors in this case become (see (8.93)) .O/ J.O/ c2 ; D 1; 2; 3I J C3;T D T Db .n/

.n/ 1 .O ˝ O / T ; D 1; 2I 2

.n/

.O/ .O/ J6T D 3 6 Om T ˝ T ; J C6;T D 0; D 1; 2; 3; 6I

(9.123)

.n/ .n/ 1 1 .O/ D .O1 ˝ O1 / C p ; J11T D .O2 ˝ O2 / C p ; 4 4

.O/ J10T

.O/ D J13T

.n/ 1 .O3 ˝ O3 / C p ; 4

.O/ J14T D

.n/ 1 .O3 ˝ O3 / T ; 4

where the tensor 6 Om is determined by (4.316). Substitution of these expressions into (9.63) yields the following representation of constitutive equations (9.63) for an orthotropic plastic continuum in the tensor basis: ! 3 X .n/ .n/ .n/ .n/ .n/ 1 1 2 Ce D ' 3C T C e 'b ' 6C C p / C3'6 6 Om T ˝ T : c C O ˝ O .e 2 2 D1 (9.124) Here we have denoted the scalar functions e ' 4 D '4 ; e ' 5 D '5 ; e ' 6 D '14 ; e ' 7 D '10 ; e ' 8 D '11 ; e ' 9 D '13 : (9.125) '8 D e ' 9 , and For the model of an elastoplastic continuum (9.59): e '7 D e equations (9.124) take the form .n/

Ce D

3 X

.'b c2 C

D1

.n/ .n/ .n/ e ' 3C .O ˝ O / T / C 3'6 6 Om T ˝ T ; 2

(9.126)

where ' D [email protected][email protected].O/ /;

.O/

D .I1

.n/

.O/

. T /; : : : ; I7

.n/

. T /; ; wp1 ; : : : ; wp6 /: (9.127)

The number of the Taylor parameters (9.62) for an orthotropic continuum is equal to six: Z wp

D 0

t

.b c2

.n/

T /.b c2

.n/ C p /

d ;

D 1; 2; 3I

wp3C

Z

t

D 0

.n/

.n/

.b c2˛ T / .b c2ˇ C p / d ;

˛ ¤ ˇ ¤ ¤ ˛:

(9.128)

616

9 Plastic Continua at Large Deformations

For an orthotropic elastoplastic continuum, the plastic potentials fˇ (9.65) depend .n/

on all simultaneous invariants (9.122) except the cubic invariant J6.O/ D I6.O/ . T /: fˇ D fˇ .J.O/ ; ; wp1 ; : : : ; wp6 /;

D 1; : : : ; 14 and ¤ 6:

(9.129)

.O/ .O/ , J12 and There are three simultaneous invariants in the set (9.122), namely J10 .O/ J13 . Then, having substituted the derivative tensors (9.123) into (9.74c), we obtain the constitutive equation .n/

C p

D

3 X

c2 b

D1

! .n/ .n/ 1 1e e C O ˝ O . 3C T C 6C C p / ; 2 2

(9.130)

where e4 D

4;

e5 D

5;

e6 D

14 ;

Dh

e7 D

k X ˇ D1

~Pˇ

10 ;

@fˇ @J.O/

e8 D

11 ;

e9 D

:

13 ;

(9.131)

9.1.12 The Orthotropic Unipotential Huber–Mises Model for Plastic Continua For special models of orthotropic elastoplastic continua one should accept an additional assumption on a form of the potentials fˇ . The adequacy of a model accepted is verified in experiments. In the orthotropic unipotential Huber–Mises model one assumes that there is only one potential f depending on six simultaneous orthotropic Huber–Mises invariants Y˛.O/ (˛ D 1; : : : ; 6): f D f .Y1.O/ ; : : : ; Y6.O/ ; ; wp1 ; : : : ; wp6 /:

(9.132)

Here .n/

.n/

.n/

.n/

Y˛.O/ D I˛.O/ . T H˛ C p /; ˛ D 1; : : : ; 5; Y6.O/ D I7.O/ . T H6 C p /; .n/

0

.n/

0

H˛ D H˛0 .I˛.O/ . C p //n˛ ; H6 D H60 .I7.O/ . C p //n6 ;

(9.133)

9.1 Models An of Plastic Continua at Large Deformations

617

are the simultaneous invariants, which are uniquely expressed in terms of the invariants (9.122) (see Exercise 9.1.6): .n/

.n/

Y˛.O/ D I˛.O/ . T / H˛ I˛.O/ . C p /;

˛ D 1; 2; 3I

.n/

.n/

.O/ .O/ .O/ .O/ 2 D I3C˛ . T / C H3C˛ I3C˛ . C p / 2H3C˛ J9C˛ ; Y3C˛

˛ D 1; 2I

(9.134)

.n/

.n/

.O/ Y6.O/ D I7.O/ . T / C H62 I7.O/ . C p / 2H6 J13 :

Calculating the derivatives @f [email protected]˛.O/ of the function (9.132) @f [email protected]˛.O/ D f˛ ; ˛ D 1; : : : ; 5I .O/ @f [email protected] D 2H5 f5 ;

.O/

.O/

@f [email protected] D f6 ; .O/

@f [email protected] D 2H6 f6 ;

@f [email protected] D 2H4 f4 ; f˛ @f [email protected]˛.O/

(9.135)

and substituting them into (9.131), we find the non-zero functions 11

D ~hf P ;

D 1; : : : ; 5;

D 2~hH P 5 f5 ;

13

10

D 2~hH P 4 f4 ;

D 2~hH P 6 f6 ;

14

D ~hf P 6:

(9.136)

Then the constitutive equation (9.130) takes the form .n/

C p

D ~h P

3 X D1

fb c2

! .n/ .n/ f3C C O ˝ O . T H3C C p / : 2

(9.137)

Multiplying this equation by itself, we obtain v u .n/ u .n/ u Cp Cp ; ~P D t .O/ PH P.O/ H

(9.138)

where .O/ PH

3 X 1

! .n/ .n/ f 3C O ˝ O . T H3C C p / : c2 C fb 2

(9.139)

The complete equation system for plastic deformations consists of relations (9.137), (9.138) and the following scalar equation of the yield surface: f D 0; where f is expressed by formula (9.132).

(9.140)

618

9 Plastic Continua at Large Deformations

In the quadratic model, the potential (9.132) is chosen in the form 3 X

jY.O/ j C Y.O/

D1

C 2 s

2f D

.O/

Y1

.O/

Y2

.O/

jY

C .O/

12s

!2

Y1

.O/

Y3

13s

.O/

j Y 2 s .O/

Y2

!2 C

.O/ Y3C

!2

C 2 3C;s

.O/

Y3

23s

1:

(9.141)

p

˙ The functions s .; w / are called the yield strengths in tension (or in compression) in the direction ; 3C; s .; wp3C / – the yield strengths in shear along the plane .˛; ˇ/; and ˛ˇ;s .; wp1 ; : : : ; wp3 / – the mixed yield strengths. All the functions are determined in experiments. The derivatives of the function (9.141) have the forms

f D

jY.O/ j C Y.O/ C 2 s

C

jY.O/ j Y.O/ .O/ ; f C3 D Y3C = 3C;s ; D 1; 2; 3: 2 s (9.142)

9.1.13 The Principle of Material Indifference for Models An of Plastic Continua .n/ .n/

.n/

.n/

.n/

.n/

All the energetic tensors T , C , and also C e and C p defined by T and C, are R-invariant. Then all the constitutive equations of models An of plastic continua, stated in Sects. 9.1.2–9.1.6, satisfy the principle of material indifference, because they remain unchanged at the change of actual configuration K ! K0 in rigid motion.

Exercises for 9.1 9.1.1. Show that the scalar invariant of the tensor T, being its intensity Y .T/ determined by (9.88), in any basis has the following component representation: Y 2 .T/ D Y 2 .T/D

3 P P; 2

1 P D T I1 .T/E; 3

1 2 2 2 CT23 CT13 / : .T11 T22 /2 C.T22 T33 /2 C.T33 T11 /2 C6.T12 2

9.1.2. Prove that the invariant Y defined in Exercise 9.1.1 can always be expressed in terms of the principal invariants as follows: Y 2 .T/ D I12 .T/ 3I2 .T/:

9.1 Models An of Plastic Continua at Large Deformations

619

9.1.3. Show that for the model of an isotropic elastoplastic continuum without strengthening, when H 0 (i.e. H0 D 0), Eq. (9.97) takes the form 2 PP p D .~h= P s /PT :

This relation is called the Prandtl–Reuss equation. 9.1.4. Show that the relation of orthogonality (9.117) holds for the tensors (9.116). .3/

9.1.5. Prove relations (9.112) between the simultaneous invariants Y˛ (9.111) and .3/ J (9.99). 9.1.6. Prove relations (9.134). 9.1.7. Consider the plastically compressible Huber–Mises model An of an isotropic continuum, where the constitutive equations (9.81)–(9.83) hold and there is only one plastic potential f depending, unlike (9.84), in addition on the first invariant Y1H : f D f .YH ; Y1H ; ; wp /: Here .n/

.n/

2n1 H1 D H10 Y1p ;

Y1H D . T H1 C p / E;

.n/

Y1p D I1 . C p /;

f1Y @f [email protected] ;

and H10 , n1 are the constants. This model describes the plastic properties of porous media, some grounds and also materials sensitive to the type of loading being volumetric tension or compression (see Sect. 9.6). Show that for this model the constitutive equations (9.92) take the form .n/ C p

1 D 3~h P fY PH C f1Y E : 3

Show that these equations written for the deviators and spherical parts of the tensors of plastic deformations and stresses have the forms (compare them with (9.94) and (9.95)) 8 ˆ PP D 3~hf P Y PH ; ˆ ˆ < P P 1Y ; I1 .Cp / D ~hf r ˆ q ˆ .n/ .n/ ˆ :~P D ˙ C C = f 2 Y 2 C 3f 2 ; f D 0: p p Y H 1Y Show that if the plastic potential f has the form 1 f D 3

YH S

2

C

YC2 T02

C

Y2 1; C02

1 1 1 D 2 2; 02 T T 3 S

1 1 1 D 2 2; 02 C C 3 S

620

9 Plastic Continua at Large Deformations

where S , T , and C are the yield strengths in shear, tension and compression, respectively (depend on and wp ), and YC and Y are invariants of constant signs Y˙ D

1 .jY1H j ˙ Y1H / ; 2

then fY D

1 3 S2

and f1Y D

2YC 2Y C 02 ; T02 C

and the constitutive equations take the form 8 .n/ ˆ 2 < P D .~h= P p S /PH ; ˆ : P .n/ 02 02 I1 . C p / D 2~h..Y P C = / C .Y = //: T

C

9.1.8. Consider the associated model of plasticity An (9.74) with quasilinear elasticity, for which the elastic potential (9.74a) depends only on linear and .n/

quadratic invariants I.s/ of the tensor T = (by analogy with the quasilinear models .n/

An of elastic continua (see Sect. 4.8.7)). Introducing for the functions ' .I.s/ . T =/; p ; wˇ / the representation ı

' D

r1 X

ı

0 lˇ I.s/ ; D 1; : : : ; r1 I

' D l0 ;

D r1 C 1; : : : ; r2 ;

ˇ D1 0 and l0 are functions in the form where lˇ .n/

0 0 lˇ D lˇ .I˛.s/ . T =/; ; wp˛ /;

0 0 lˇ D lˇ ;

show that in this case Eq. (9.74b) may be written as follows: .n/

.n/

C e D 4N T ;

ı

J D =:

Here the tensor 4 N has the form 4

ND

1 0 l1 E ˝ E C 2l20 ; J

0 0 0 l10 D l11 C 2l22 ; l22 D 2l20 ;

9.1 Models An of Plastic Continua at Large Deformations

621

– for isotropic media (r1 D 1; r2 D 2); 4

ND

10 0 0 l 022b c23 C .l12 l11 /.E ˝b c23 Cb c23 ˝ E/ l E ˝ E Ce c23 ˝b J 11 0 l11 0 0 l44 .O1 ˝ O1 C O2 ˝ O2 / C 2l44 ; C 2 0 0 0 0 e l 022 D l22 2.l44 C l12 / l11 ;

– for transversely isotropic media (r1 D 2; r2 D 4); 0 4

ND

[email protected] J

3 X

0 b c2 ˝b lˇ c2ˇ C

;ˇ D1

1

2 X

0 A l3C; 3C O ˝ O

D1

– for orthotropic media (r1 D 3; r2 D 6). Show that there exist inverse relations .n/

.n/

T D 4M C e ;

where the tensors 4 M are inverse of 4 N: 4

M 4 N D ;

and have formally the same structure as the tensor 4 N, but coefficients lˇ of the tensor 4 M are functions in another form .n/

lˇ D lˇ .I˛.s/ . C e /; ; wp˛ /:

9.1.9. For the models An of an elastoplastic continuum (9.79) and (9.80) with linear elasticity, whose potential (9.80) has the quadratic form D 0

ı r1 r2 X ı X .s/ 0 lˇ I.s/ Iˇ l I.s/ ; 2 Dr C1 ;ˇ D1

1

.n/

.s/ .s/ 0 where lˇ are the constants and I D I . T =/, show with use of the results of Exercise 9.1.8 that the tensors 4 N are tensor-constants up to the factor J; and the .n/

.n/

constitutive equations (9.74b) between C e and T take the form .n/

.n/

.n/

T D J.l1 I1 . C e /E C 2l2 C e /;

l1 D

l10 0 2l2 .3l10 C

2l20 /

; l2 D

1 ; 4l20

622

9 Plastic Continua at Large Deformations

– for isotropic materials; .3/ .3/ .3/ .3/ T D J .l11 I1 C l12 I2 /.E b c23 / C ..l22 2l44 /I2 C l12 I1 / b c23 ! .n/ .n/ l33 C l44 .O1 ˝ O1 C O2 ˝ O2 / C e C 2l44 C e ; 2

.n/

.n/

– for transversely isotropic materials (here I.3/ D I.3/ . C e /); 0 .n/

T [email protected]

3 X

lˇ I.O/b c2 C

3 X

1 .n/

l3C;3C O .O C e /A

D1

;ˇ D1

.n/

– for orthotropic materials (I.O/ D I.O/ . C e /). Show that for the models An of elastoplastic isotropic continua with linear elasticity, whose potential has the form ı

.n/ .n/ ı D 0 l10 I12 . T =/ l20 I1 . T 2 =2 /; 2

the constitutive equations (9.74b) become .n/

.n/ 1 0 .n/ .l1 I1 . T /E C 2l20 T /; J l0 l 1 ; l1 D 0 0 1 ; l10 D 2l2 .3l1 C 2l2 / 2l2 .3l1 C 2l20 /

Ce D

2l2 D

1 : 2l20

9.1.10. Using formulae (9.19) and (8.28b), show that the specific internal energy e for the models An of plastic continua is expressed by the formula .n/

eDC

T .n/ @ Ce : @

Show that for models An of an isotropic elastoplastic continuum with linear elasticity the results of Exercise 9.1.9 give 0 1 0 1 ! .n/ .n/ ı 2 l2 .n/ 0 2BTC ı 0 BT C l1 2 .n/ e D e0 C l1 I1 @ A C l2 I1 @ 2 A D e0 C ı I1 C e C ı I1 . C 2e /; 2 2 e0 D 0 [email protected]0 [email protected]/:

9.2 Models Bn of Plastic Continua

623

9.1.11. Show that relations (9.52) can be represented in the form independent explicitly of time: X d~˛ @f˛ d .n/ ; Cp D h d~ d~ .n/ ˛D1 @T k

where ~ D ~1 .

9.2 Models Bn of Plastic Continua 9.2.1 Representation of Stress Power for Models Bn of Plastic Continua Let us consider now models Bn for plastic continua. Constructing these models essentially differs from constructing the models An , because the main additive rela.n/

tion (9.3) for the energetic measures G does not hold. The relation (9.3) is replaced by Eq. (9.7), that is axiomatically assumed in models Bn and justified in Sect. 9.1.1. Since the relation (9.7) is the product of the gradients Fp and Fe , models Bn of plastic continua are called multiplicative, unlike models An which in this case are called additive. Consider the principal thermodynamic identity in the form Bn (4.122) d

.n/

.n/

C d T d G C w dt D 0:

(9.143)

Let us show that for models Bn the additive relation for the stress power derived in the following theorem is an analog of the additive relation (9.3). Theorem 9.1. The stress power w.i / (4.2) can always be represented in the additive form .n/

.n/

.n/

.n/

.n/

.n/

w.i / D T G D T e G e C T p D p ; .n/

(9.144)

.n/

where T e and T p are the symmetric tensors of elastic stresses and yield stresses, .n/

.n/

respectively, G e are the symmetric measures of elastic deformations, and D p are the symmetric energetic measures of plastic deformation rates; their expressions for n D I; : : : ; V are given in Table 9.1. .n/

.n/

.n/

Notice that the tensors T e and G e are entirely analogous to the tensors T and .n/

G and differ from them only by replacing F ! Fe and U ! Ue . The tensor Be , as

624

9 Plastic Continua at Large Deformations .n/

.n/

.n/

.n/

Table 9.1 Expressions for T e , T p , G e and D p at n D I; : : : ; V n

.n/

.n/

.n/

.n/

Te

Ge

Tp

Dp

12 U2 e

I

Te

U1 e

Te

I

FTe T Fe

II

1 .FTe 2

III

OTe T Oe

Be

Te

1 .F1 e 2

V

V

IV

Ue

Te

Dp

V

1T F1 e T Fe

1 2 U 2 e

Te

T Oe C Oe T Fe /

1 .Dp 2 I Dp

I III

T Oe C Oe T Fe1T /

2 T U2 e C Ue Dp /

Dp U1 e C T C U1 D e p Ue /

1 .UTe 2

V

1 .U2e 2

Dp C DTp U2e /

well as the tensor B, is determined by its derivative and initial value B0e (when there are no initial plastic deformations B0e D E): 1 P 1 1 P BP e D .U e Ue C Ue U e /; 2

Be .0/ D B0e :

(9.145)

The plastic deformation rate measure Dp is determined by the relation Dp D FP p F1 p

(9.146)

and is a nonsymmetric tensor. It follows from Table 9.1 that I

II

I

Tp D Tp D Te ; I

III

III

Tp D T e ;

II

IV Dp

Dp D Dp ;

IV Tp

V

V

D Tp D Te ;

V

D Dp :

I

I

H 1. Using the definition of the measure G and introducing the measure Ge by the I

1T similar formula Ge D F1 , with the help of the multiplicative decomposition e Fe (9.7) we obtain I I 1 1 1 1T 1T G D F1 F1T D F1 Fp1T D F1 p Fe Fe p Ge Fp ; 2 2

(9.147)

then I

I

I

I

I

I

1T 1T P 1T T G D T .F1 C FP 1 C F1 p Ge Fp p Ge Fp p Ge Fp /:

(9.148)

9.2 Models Bn of Plastic Continua

625

According to the rule of permutations of tensors in the triple scalar product (4.4), we get I

I

I

I

I

I

1T P 1 T G D .Fp1T T F1 T/ .F1 p / Ge C .Fp p Fp Fp Ge / I

I

T P 1T C .Fp1T T/ F1 p Ge Fp / Fp :

(9.149)

I

1 P 1 P Since Ge D .1=2/U2 e , Fp Fp D Fp Fp D Dp and I

1T T Fp1T T F1 FT T F F1 p D Fp p D F e T Fe ;

(9.150)

so the representation (9.144) really holds at n D I: I I I I 1 1 2 T 2 T G D .Fp1T T F1 p / .Ge C Dp Ue C Ue Dp / 2 2 I

I

D .FTe T Fe / .Ge C Dp /:

(9.151)

I

I

I

I

2. Since the tensors Te and Ge differ from tensors T and G only by the substitutions I

I

F ! Fe and U ! Ue , respectively, the first couple .Te ; Ge / gives all the remaining .n/

.n/

I

I

couples T e ; G e in the same way as the energetic couple .T; G/ does (the tensors Fp do not appear in these relations). 3. We should show only that the first couple gives the third and fifth ones: I

I

III

III

V

V

Te Dp D Te Dp D Te Dp

(9.152)

(the second and fourth couples coincide with the first and fifth ones). I

I

Indeed, according to the definitions of Te and Dp , we have I

I

I

Te Dp D .FTe T Fe / Dp 1 2 T D .Ue OTe T Oe Ue / .Dp U2 e C Ue De / 2 D

III III 1 T 1 T Oe T Oe .Ue Dp U1 e C Ue Dp Ue / D Te Dp 2 (9.153)

626

9 Plastic Continua at Large Deformations

and III

III

Te Dp D

D

1 T 2 1 .O T Oe U1 e / Ue Dp Ue 2 e 1 T 2 C .U1 OTe T Oe U1 e / Dp Ue 2 e V V 1 1 .Fe T Fe1T / .U2e Dp C DTp U2e / D Te Dp : N 2 (9.154)

Theorem 9.2. The yield stress power w.p/ can always be represented as a sum of the powers of plastic stretches and plastic rotations: .n/

.n/

P p C To p : w.p/ D T p D p D TU U

(9.155)

Here TU is the symmetric tensor of plastic stretch stresses, To is the skew-symmetric tensor of plastic rotation stresses, which are defined as follows: TU D

1 1 .F Te Op C OTp TTe Fp1T /; 2 p 1 To D .Te TTe /; 2 Te D F1 e T Fe ;

(9.156a) (9.156b) (9.157)

and p is the skew-symmetric spin tensor of plastic rotation: P p OTp : p D O

(9.158)

H Modify the plastic deformation rate measure Dp (9.146) as follows: 1 T P P Dp D FP p F1 p D .Op Up C Op Up / Up Op T P p OT C Op U P p U1 DO p p Op D p C Dv :

(9.159) Here we have denoted the plastic stretch rate measure T 1 P p U1 P Dv D Op U p Op D Op Up Fp :

(9.160)

Then the first couple in (9.152) can be written in the form I

I

w.p/ D Te Dp D

1I 2 T 2 2 T Te .p U2 e C Ue p C Dv Ue C Ue Dv /: (9.161) 2

9.2 Models Bn of Plastic Continua

627

According to the permutation rule for tensors in the triple scalar product, we obtain w.p/ D

I 1 2 I 1 1 2 I P .Ue Te Te U2 e / p C ..Fp Ue T Op / Up 2 2 I

1T P P C.OTp Te U2 e Fp / Up / D To p C TU Up :

(9.162) Here we have taken into account that the spin p is skew-symmetric and that I

I

2 2 T T 2 U2 e Te Te Ue D Ue Fe T Fe Fe T Fe Ue T 1T D F1 D Te TTe D 2T0 ; e T Fe F e T F e I

I

2 T 2 1 F1 p Ue T Op C Op T Ue Fp 2 T T T 2 1 D F1 p Ue Fe T Fe Op C Op Fe T Fe Ue Fp T T 1T D F1 D 2TU ; p Te Op C Op Te Fp T 1 because U2 e Fe D F e .

.n/

.n/

The fact that other couples T p ; D p give the same result (9.155) follows from equivalence of contractions of the couples (9.152). N On substituting the expression (9.155) into (9.144), we get representations for the stress power .n/

.n/

.n/

.n/

P p C T0 p : w.i / D T G D T e G e C TU U

(9.163)

9.2.2 General Representation of Constitutive Equations for Models Bn of Plastic Continua Substitution of this expression into the principal thermodynamic identity (9.143) yields .n/

.n/

P C P T e G e TU Up To p C w D 0:

(9.164)

Introduce the Gibbs free energy D

.n/ 1 .n/ T e Ge ;

(9.165)

628

9 Plastic Continua at Large Deformations

then for we obtain the principal thermodynamic identity in the form Bn : .n/

.n/

.n/

.n/

P C P C G e . T e =/ T v Up T o p C w D 0I

(9.166)

i.e. a change in the free energy is determined only by changing the functions , .n/

T e =, Up and Op . Therefore, in the model Bn of a plastic continuum, the Gibbs free energy is considered to be a functional in the form (similar to (9.21)) t

P P t .//; D .R.t/; R.t/; Rt ./; R D0

.n/

R D . T e =; Up ; OTp ; /:

(9.167)

.n/

According to the principle of equipresence, the measure G e is a functional (only a tensor one) in the same form .n/

t

P G e D Ge .R.t/; R.t/; Rt ./; RP t .//:

(9.168)

D0

By analogy with (9.24) and (9.25), introduce the equilibrium elastic deformation .n/

.n/

measure G 0e and nonequilibrium elastic deformation measure G 1e as follows: .n/

t

P t .//; G 0e D Ge .R.t/; Rt ./; R D0

.n/ G 1e

.n/

.n/

D G e G 0e :

(9.169)

Substituting the functionals (9.167)–(9.169) into the principal thermodynamic identity (9.166) and grouping like terms, we obtain the identity 0

1

@

.n/ G 0e A

@

C

.n/

@. T e =/ C

.n/

d

Te @ C @

0

@

d C

.n/

@. T e =/

.n/

1

B Te C @ A

@ P p C @ d C @ d U dO p PT @ @Up @O p

.n/

.n/

P p C G 1e . T e =/ C ı e C.w e TU U To p / dt D 0; (9.170) where we have introduced the notation e TU D TU Np ; Np D [email protected][email protected] /; e To D To No ; No D Op [email protected][email protected] /:

9.2 Models Bn of Plastic Continua

629

.n/

.n/

Since the differentials d. T e =/, d, d. T e =/ , d , d Up , d Op and dt are independent, the identity (9.170) is equivalent to the following set of relations: .n/

.n/

G 0e D @[email protected] T e =/;

(9.171)

D @[email protected];

(9.172)

.n/

P Tp D 0; P p D 0; @[email protected] @[email protected] T e =/ D 0; @[email protected]P D 0; @[email protected] .n/

.n/

TU C e To p G 1e . T e =/ ı; w D e

(9.173) (9.174)

which are constitutive equations for models Bn of plastic continua. As follows from relations (9.173), the free energy is independent of the rates P of reactive variables R: t

P t .//; D .R.t/; Rt ./; R D0

.n/

R D . T e =; Up ; OTp ; /:

(9.175)

However, this dependence is observed for the dissipation function w and for the .n/

functional G 1e : .n/

.n/

P G e 1 D G e 1 .R.t/; R.t/; Rt ./; RP t .//:

(9.176)

Thus, the model Bn of a plastic continuum is specified by four functionals: the scalar functional (9.175) for , the tensor functional (9.176) and two more tensor functionals for determining the tensors Up and OTp .

9.2.3 Corollaries of the Onsager Principle for Models Bn of Plastic Continua Just as for models An , we use the Onsager principle in order to construct function.n/

als for the tensors G 1e , Up and Op . Form the specific internal entropy production (4.728) with the help of expression (9.174): 0 q D w

.n/

1

.n/ q q B Te C P p Ce To p G 1e @ r D e TU U A ı r > 0

(9.177)

630

9 Plastic Continua at Large Deformations

and introduce thermodynamic forces Xˇ and fluxes Qˇ as follows: .n/

X1 D r ; X2 D e TU ; X3 D e To ; X4 D . T e =/ ; P p ; Q 3 D p ; Q 4 D Q1 D q=; Q2 D U

.n/ G 1e :

(9.178)

Then, according to the Onsager principle, the following tensor-linear relations between Qˇ and Xˇ hold: .n/

TU C L13 e To C L14 . T e =/ ; q= D L11 r C L12 e .n/

P p D L12 r C L22 e U TU C L23 e To C L24 . T e =/ ; .n/

(9.179)

p D L13 r C L23 e TU C L33 e To C L34 . T e =/ ; .n/

.n/

TU C L34 e To C L44 . T e =/ : G e D L14 r C L24 e Here tensors L˛ˇ are functionals with the general form (9.176) of reactive variables R D fTe =; Up ; OTp ; g; their special forms are given by a model of a plastic continuum considered. P and For models Bn of plastic yield, all L˛ˇ and are only functions of R and R p of the Taylor parameters wˇ : .n/

D . T e ; Up ; OTp ; ; wpˇ /; .n/

.n/

.n/

.n/ T e ;

P p ; wp /; P p; O L22 D 4 LU . T e ; Up ; OTp ; ; T e ; U ˇ L33 D 4 Lo . T e ; Up ; OTp ; ; L11 D ./;

P p ; wp /; P p; O U ˇ

(9.180a) (9.180b) (9.180c)

the remaining Lˇ D 0:

In this model, the cross–effects are not considered, and constitutive equations (9.179) take the form q D r ; P TU ; Up D 4 LU e 4 p D Lo e To ; .n/ G 1e

0:

(9.181a) (9.181b) (9.181c) (9.181d)

The tensor 4 LU is a fourth-order symmetric tensor, and the tensor 4 Lo is skewsymmetric in indices 1, 2 and 3, 4, and symmetric in pairs of indices .1; 2/ $ .3; 4/.

9.2 Models Bn of Plastic Continua

631

9.2.4 Associated Models Bn of Plastic Continua

In the associated model of plasticity Bn , the equations of plastic yield (9.181b) and (9.181c) are connected to the yield surface by the gradient law Pp D h U

k X

~Pˇ

ˇ D1

p D h

k X

@fˇ ; @TU

(9.182a)

@fˇ : @To

(9.182b)

~Pˇ

ˇ D1

Here fˇ are plastic potentials assumed to be functions in the form (9.180a), which can be considered as functions of TU , To and Up , OTp : fˇ D fˇ .TU ; To ; Up ; OTp ; ; wp˛ /:

(9.183)

An equation of the yield surface, just as in the model An , is given by the set of scalar equations (9.41) fˇ D 0; ˇ D 1; : : : ; k: (9.184) The function h, taking on a value 0 or 1 and determining a domain of plastic loading, is evaluated by formula (9.51), where as the partial derivative d 0 fˇ =dt we use an analog of formula (9.43): @fˇ @fˇ d 0 fˇ TP U C TP o : D dt @TU @To

(9.185)

The system of .9 C k/ scalar equations (9.182) and (9.183) (Eq. (9.182a) is equivalent to six scalar equations, (9.182b) – to three scalar equations due to skewsymmetry of the tensors p and To ) allows us to find the 9 C k scalar unknowns: six components of the tensor Up , three components of the tensor Op and k scalar functions ~P ˇ (ˇ D 1; : : : k) which are determined by Eqs. (9.182). The parameters ~Pˇ are functions in the form (9.180b) P p ; ; ; wp /: ~Pˇ D ~P ˇ .TU ; To ; Up ; OTp ; U ˇ

(9.186)

In the model Bn of an elastoplastic continuum, the elastic potential does not depend explicitly on plastic deformations; therefore, for this model the constitutive equations (9.180a) and (9.171) become .n/

.n/

.n/

.n/

D . T e ; ; wpˇ /; N1 0; N0 0; G e D K. T e ; ; wpˇ / D [email protected][email protected] T e /: (9.187)

632

9 Plastic Continua at Large Deformations

9.2.5 Corollary of the Principle of Material Symmetry for Associated Model Bn of Plasticity Apply the principle of material symmetry to the constitutive equations (9.182) and (9.184). To do this we should clarify how the introduced tensors with subscripts p and e are changed under an orthogonal transformation of the reference configuration ı

K ! K. Theorem 9.3. The tensors Oe , Ue , and Ve are H -invariant under orthogonal H ı

.n/

.n/

P p , Vp , T e , T p , TU , transformations: K ! K, and all the tensors Fp , Op , Up , U To , and are H -indifferent. p

H The unloaded configuration K introduced in Sect. 9.1.1 is in accord with the refı

erence configuration K. They must coincide (by definition), if loading is not plastic; for the associated model, this means that loading does not occur outside the yield ı

p

surface. Therefore, the configuration K must be transformed in the same way as K p (the local basis vectors r i must be H -indifferent), and the elastic deformation graı

dient Fe (9.6) must be transformed during the passage K ! K in the same way as F (see (4.188)):

Fe D Fe H;

(9.188)

where

p

Fe D r i ˝ r i ;

p

pi

r D H1T r i ;

p ri

p

D H ri :

(9.189)

Thus, the tensors Oe , Ue and Ve are transformed in the same way as O, U and V

(see (4.256), (4.257)) under orthogonal transformations with the tensor H D QT :

Oe D Oe Q;

Ve D Ve ;

Ue D QT Ue Q:

(9.190)

Figure 9.2 shows a scheme of transformations of different configurations at the ı

change of reference configuration K ! K.

Fig. 9.2 H -transformations of reference and unloaded configurations

9.2 Models Bn of Plastic Continua

633

According to (9.189) and (4.194), the plastic deformation gradient Fp (9.6) under ı

the transformation K ! K takes the form

p

ı

p

Fp D r i ˝ ri D H r i ˝ ri H1 D QT Fp QI

(9.191)

i.e. it is H -indifferent under orthogonal transformations. Using the polar decompo

sition for Fp and Fp , we obtain

Fp D Qp Up D QT Op Up Q D .QT Op Q/ .QT Up Q/:

Since the tensor .QT Op Q/ is orthogonal, the tensor .QT Up Q/ is symmetric and the polar decomposition is unique, we get the following transformation formulae for Op and Up :

Qp D QT Op Q;

Up D QT Up Q:

(9.192)

The tensor Vp is transformed as follows:

Vp D Fp OTp D QT Fp Q QT OTp Q D QT Vp Q;

(9.193)

P p and Vp are H -indifferent under orthogonal then the tensors Tp , Op , Up , U transformations. Due to H -invariance of the tensor T, the tensors of elastic and plastic defor.n/

.n/

mations ( T e and T p , respectively) defined by Table 9.1 are H -indifferent under .n/

.n/

orthogonal transformations as well as the tensors T . Similarly, the tensors G e are H -indifferent too. The spin tensor p (9.158) is H -indifferent as well as the tensors TU (9.156) and To (9.157), because

T 1 T Te D F1 e T Fe D Q Fe T Fe Q D Q Te Q;

TU D

(9.194)

1 T 1 T .Q Fp Q Q Te Q QT Op Q 2

C QT Qp Q QT Te Q QT Fp1T Q/ D QT TU Q: N Applying the principle of material symmetry to Eq. (9.182) and taking H P p , p , TU , To , Up and OTp into account, we obtain indifference of the tensors U that the equations of plastic yield (9.182) must be H -indifferent tensor functions

634

9 Plastic Continua at Large Deformations ı

relative to one or another orthogonal subgroup G s , and the plastic potentials fˇ ı

(9.183) must be scalar H -indifferent functions relative to G s and hence depend on ı

simultaneous invariants J.s/p in this group G s : fˇ D fˇ .J.s/p ; ; wp˛ /;

J.s/p D J.s/ .TU ; To ; Up ; OTp /; D 1; : : : ; z: (9.195)

The scalar functionals wp˛ are integrals of the quadratic simultaneous invariants of P p and To , p : the tensors TU , U Z wp˛

0

Z wp˛

t

D t

D 0

P p .// d ; ˛ D 1; : : : ; r1 ; J˛.s/ .TU ./; U (9.196)

J˛.s/ .To ./;

.// d ; ˛ D r1 C 1; : : : ; r:

Similarly, the elastic potential (9.184a) is a H -indifferent scalar function too; therefore, it can be written in the form .n/

D .J.s/ . T e ; Up ; OTp /; ; wp˛ /:

(9.197)

Then constitutive equations (9.171) and (9.182) can be represented in the tensor basis: .n/

Ge D

z X D1

' D [email protected][email protected].s/ /; Pp D U

z1 X

.s/

' J T ; .s/

(9.198a) .n/

J T D @J.s/ [email protected] T e ;

(9.198b)

.s/ ˛ J˛TU ;

(9.198c)

.s/ ˛ J˛To ;

(9.198d)

˛D1

p D

z1 X ˛D1

˛

Dh

k X ˇ D1

.s/ .s/ ~P ˇ [email protected]ˇ [email protected]˛.s/ /; J˛T D @J˛.s/ [email protected] ; J˛T D @J˛.s/ [email protected] : o U

(9.198e)

9.2 Models Bn of Plastic Continua

635

9.2.6 Associated Models of Plasticity Bn with Proper Strengthening We will consider below only the case of associated models Bn of plasticity with proper strengthening, where the simultaneous invariants J.s/p (9.195) depend only on the tensors TU and To and do not depend explicitly on Up and OTp : J.s/p D J.s/ .TU ; To /:

(9.199)

Since the tensors TU and To defined by formulae (9.156) and (9.158) are combi.n/

nations of the tensors T , Ue , Oe , Up and Op , the model (9.199) allows us to take account of plastic strengthening of a continuum (an increase of the yield strength after the appearance of plastic deformations) but in the special way, namely in terms of the tensors TU and To . Notice that since the tensor To is skew-symmetric, it has only three independent components, and this tensor can be connected uniquely with the vorticity vector (see (2.227)) 1 !0 D " To : (9.200) 2 Then the simultaneous invariants (9.199) are scalar H -indifferent functions of the symmetric tensor TU and the vector !0 : J.s/p D J.s/ .TU ; !0 /:

(9.201)

9.2.7 Associated Models of Plasticity Bn for Isotropic Continua Let us write the constitutive equations (9.198) for the three main symmetry groups ı

G s : O, T3 and I while considering only the case of elastoplastic continua, i.e. while the simultaneous invariants J.s/ in (9.197) coincide with the invariants .n/

I.s/ . T e /. For an isotropic continuum, the functional basis of invariants (9.201) consists of six elements, which can be chosen as follows: J.I / D J .TU /; D 1; 2; 3; J5.I /

D !0 TU !0 ;

J6.I /

J4.I / D j!0 j2 ; D !0

T2U

!0 :

(9.202)

636

9 Plastic Continua at Large Deformations .s/

.s/

The derivative tensors J TU and J To in this case become .I / .I / .I / .I / D E; J2T D EI1 .TU / TU ; J3T D T2U I1 TU C I2 E; J4T D 0; J1T U U U U .I / .I / J5T D !0 ˝ !0 ; J6T D !0 ˝ TU !0 C !0 TU ˝ !0 ; U U .I / .I / J4T D !0 "; J5To D .!0 TU C TU !0 / "; o .I /

.I /

J6To D .!0 T2U C T2U !0 / "; J To D 0; D 1; 2; 3: Here we have used that

1 ": 2

d !0 =d To D

(9.203) (9.204)

The tensors J.IT/o are seen to be skew-symmetric. According to the Onsager principle, Eqs. (9.198c) and (9.198d) must be quasilinear with respect to TU and To , i.e. they must have the forms (9.181b) and (9.181c); therefore, invariants J3.I / , J5.I / and J6.I / must be eliminated between ar.I / guments of the potential fˇ (9.195). Thus, fˇ depends only on three invariants J , D 1; 2; 4. On substituting the expressions (9.202) and (9.203) into (9.198), we obtain the following constitutive equations of the associated model Bn of an isotropic elastoplastic continuum with proper strengthening: 8.n/ .n/ .n/ ˆ ˆ '1E C e ' 2 T e C '3 T 2e ; 1 correspond to all-round compression (pe > 0), and J < 1 – to all-round tension .pe < 0/. When J > 1, for all n the function pe .J / is monotonically increasing; for n D I and II it is convex downwards, but for n D IV and V it is convex upwards. When 0 < J < 1, the function pe .J / has an extremum for n D I; II and IV and pe .0/ D 0; and for n D V: pe ! 1 as J ! 0. Thus, different models An give qualitatively different diagrams pe .J /.

9.6.4 The Case of a Plastically Compressible Continuum

Let us consider now the Huber–Mises model An for a plastically compressible medium (see Exercise 9.1.7). Since for this problem PT D 0 and PH D Pp , the results of Exercise 9.1.7 give P Y Pp ; PP p D 3~hf .n/

IP1 . C p / D ~hf P 1Y ;

(9.313)

f D 0:

(9.314)

The first relation has the zero solution: Pp D 0, and the second one admits a non.n/

trivial solution; hence the tensor C p in this case is spherical: .n/

Cp D

1 Y1p E; 3

.n/

Y1p I1 . C p /:

(9.315)

9.6 The Problem on All-Round Tension–Compression of a Plastic Continuum

663

The simultaneous invariant Y1H in this case, according to (9.305) and (9.315), becomes .n/

.n/

Y1H D . T H1 C p / E D .3e p e C H1 Y1p /; YH 0; 1 2n1 H1 D H10 Y1p ; Y˙ D .jY1H j ˙ Y1H /: 2

(9.316) (9.317)

Equation (9.314) serves for determination of ~, P and the first invariant of the plastic deformation tensor Y1p can be found from the plasticity condition f D 0. If f is assumed to have a quadratic form (see Exercise 9.1.7), then f D

YC2 Y2 C 1 D 0: 02 T C02

(9.318)

On substituting the expressions (9.316) and (9.317) into (9.318), we find ( H1 Y1p D

pe ; C0 3e

. T0

C 3e p e /;

if Y1H < 0;

(9.319)

if Y1H > 0:

Here the yield strengths C0 and T0 , by definition, are assumed to be positive: C0 > 0, T0 > 0, that leads to the proper choice of signs in (9.318) for C0 and T0 : if p e D C0 =3 > 0, and in tension e p e D T0 =3 < 0. Y1p D 0, then in compression e Remind that the equation f D 0 holds if and only if the change in plastic deformation is different from zero. If f < 0 (in the considered case this occurs when .n/

.n/

.n/

Y1H > C0 or Y1H 6 T0 ), then C p D 0, or that is the same: C p .t/ D C p D .n/

.n/

const, where C p D C p .t / is the value reached in the preceding cycle of plastic loading up to time t of the beginning of unloading. Before initial loading, at t D 0 one usually assumes that there are no plastic .n/

deformations, therefore C p .0/ D 0 (Fig. 9.6). Hence, relations (9.319) hold true only under plastic loading. When f < 0, i.e. under unloading or under loading in an elastic domain these relations should be replaced by Y1p .t/ D Y1p . Here Y1p is the value reached in the preceding cycle of plastic loading.

Fig. 9.6 The cycle of plastic loading and subsequent unloading

664

9 Plastic Continua at Large Deformations

On substituting the expression (9.316) for H1 into (9.319), we find that

Y1p

8 0 1 ˆ ˆ p e j 2n1 C1 j C 3e 0 ˆ ˆ sign . C 3e pe / ; if Y1H 6 C0 ; ˆ ˆ H10 < 0 1 D j T C 3pe j 2n1 C1 0 ˆ pe / ; if Y1H > T0 ; sign . T C 3e ˆ ˆ H10 ˆ ˆ ˆ :Y ; if C0 < Y1H < T0 ; 1p (9.320)

where

Y1H D 3e pe C H10 jY1p j2n1 C1 sign Y1p :

Since relations (9.306) and (9.307) for elastic deformations still hold for plastically compressible continuum, so, combining them with (9.315) and (9.320), we obtain .n/

.n/

.n/

C D Ce C Cp D

1 3

pe k IIInC3 Y1p E: K

(9.321)

Then, according to formula (9.299), we get 3 pe IIInC3 Y1p ; .1 k nIII / D k n III K

or pe D K

3.1 k nIII / C Y1p k nIII3 : n III

(9.322)

Combining (9.322) with expression (9.320), we find a nonlinear relation between pe and k. These relations depend on the path of loading of a plastic material. The following example illustrates this fact.

9.6.5 Cyclic Loading of a Plastically Compressible Continuum Let the function e p e pe k IIIn be given in the form of nonmonotone dependence upon t (Fig. 9.7), where e p e .0/ D 0 and Y1p D 0, and H1 > 0. Then from the expression (9.320) for different sections of the function e p e .t/ we get OA W 0 < e p e 6 C0 =3; Y1H D 3e pe > C0 ; Y1p D 0 – elastic compression, p e CY1p H1 / D C0 ; Y1p D . C0 3e p e /=H1 < 0 AB W e p e > C0 =3; Y1H D .3e – plastic compression, BC W T0 =3 6 e p e 6 C0 =3; Y1H D .3e p e C Y1p H1 / > C0 ; Y1p D Y1p 0 DE W e p e 6 T0 =3; Y1H D T0 ; Y1p D . T0 C 3e – plastic tension, EF W e p e > T0 =3; Y1H D .3e p e C Y1p H1 / < T0 ; Y1p D Y1p >0

– unloading. As follows from these relations, after the appearance of plastic deformations at points, where values of e pe are the same, for example, at A and A0 , values of Y1p and k are distinct. In other words, in the plasticity domain the diagrams pe .k/ under loading and unloading are not coincident. Remark. If after preceding plastic compression (sections AB and BC 0 ) (Fig. 9.7) there occurs a tension (the section C 0 C ), then the yield strength in tension changes up to the value T0 =3 in comparison with the value of T0 =3 when there is no preceding plastic compression. This effect really occurs in many plastic materials and is called the Bauschinger effect. t u

666

9 Plastic Continua at Large Deformations

In the model considered, T0 can be found from relation (9.323) at the point C : ˇ Y1H D 3e p e ˇC C Y1p H1 D T0 :

(9.324)

Since Y1p H1 can be evaluated from the conditions (9.323) at the point B:

ˇ Y1p H1 D C0 3e p e ˇB ;

(9.325)

according to (9.324) and (9.325), we obtain ˇ ˇ p e ˇC D T0 C0 C 3e p e ˇB : T0 3e

(9.326)

Thus, the yield strengths in tension T0 and T0 differ from one another by the value . C 3e p e / of excess of the load 3e p e above the yield strength in compression. This result is a typical feature of the Huber–Mises model considered.

9.7 The Problem on Tension of a Plastic Beam 9.7.1 Deformation of a Beam in Uniaxial Tension Consider the above-mentioned classical problem on tension of a beam (see Example 2.1 and Exercises 2.1.1, 2.3.2, and 4.2.13), but for the case when the motion of a beam is described by the model An (the Huber–Mises model) of isotropic elastoplastic continuum, i.e. by Eqs. (9.301) and (9.92). The motion law for the problem is independent of type of a continuum and it is sought in the form (6.131): x ˛ D k˛ .t/ X ˛ ;

˛ D 1; 2; 3:

(9.327) .n/

The deformation gradient F and the energetic deformation tensors C in the problem on tension of a beam (see Exercises 2.2.1 and 4.2.13) have the diagonal forms FD

3 X

k˛ eN ˛ ˝ eN ˛ ;

(9.328)

˛D1 .n/

CD

3 .n/ X CN ˛˛ eN ˛ ˝ eN ˛ ;

(9.329)

˛D1 .n/

CN ˛˛ D

1 .k nIII 1/: n III ˛

(9.330)

9.7 The Problem on Tension of a Plastic Beam

667

We can find a change in the density for this problem as follows: ı

J D = D det F1 D

1 : k1 k2 k3

.n/

(9.331)

.n/

Tensors of elastic and plastic deformations C e and C p are also sought in the diagonal form: .n/

Ce D

3 .n/ X C e˛˛ eN ˛ ˝ eN ˛ ;

(9.332)

˛D1 .n/

Cp D

3 .n/ X C p˛˛ eN ˛ ˝ eN ˛ :

(9.333)

˛D1

Due to the additivity relation (9.3), we get .n/

.n/

.n/

C ˛˛ D C e˛˛ C C p˛˛ ; ˛ D 1; 2; 3:

(9.334)

9.7.2 Stresses in a Plastic Beam Equations (9.301) and (9.302) for the problem also hold true. .n/

.n/

On passing to Cartesian components T ˛ˇ of the tensor T , from (9.301), (9.302), and (9.332) we get that only diagonal components of this tensor are nonzero: .n/

.n/

.n/

T ˛˛ D J.l1 I1 . C e / C 2l2 C e˛˛ /; .n/

T D

˛ D 1; 2; 3:

3 .n/ X T ˛˛ eN ˛ ˝ eN ˛ :

(9.335) (9.336)

˛D1 .n/

A relation between T and T has the form (6.138) .n/

˛˛ D k˛nIII T ˛˛ ; ˛ D 1; 2; 3I

TD

3 X

˛˛ eN ˛ ˝ eN ˛ :

(9.337)

˛D1

The equilibrium equations (9.292), when f D 0, are satisfied identically. Since the lateral surface X ˛ D ˙h0˛ =2 (˛ D 2; 3) of the beam is assumed to be free of loads, from the boundary conditions at this surface (n T D 0), just as for an ideally elastic body, we obtain ˛˛ D 0;

.n/

T ˛˛ D 0; ˛ D 2; 3:

(9.338)

668

9 Plastic Continua at Large Deformations

Substituting the values (9.338) into (9.335) and summing these three relations (9.335), we find that .n/

.n/

I1 . C e / D

T 11 : J.3l1 C 2l2 /

(9.339)

Substitution of the expression (9.339) for the first invariant into (9.335) at ˛ D 1 .n/

.n/

yields the relation between T 11 and C e11 ; and from formula (9.335) at ˛ D 2; 3 we .n/

.n/

.n/

find the relation between C e22 , C e33 , and C e11 : .n/

.n/

T 11 D JE C e11 ;

.n/ C e22

.n/

D C e33 D

(9.340)

.n/ C e11 :

(9.341)

As before, here we have denoted the elastic modulus and Poisson’s ratio at large deformations: ED

.3l1 C 2l2 /l2 ; l1 C l 2

D

l1 : 2.l1 C l2 /

(9.342)

9.7.3 Plastic Deformations of a Beam Let us consider now constitutive equations (9.92) and (9.93) for plastic deformations when an isotropic medium is assumed to be plastically incompressible. Due to the .n/

assumption (9.333) that the tensor C p is diagonal, from (9.92) we get ~h P d .n/p C ˛˛ D dt 3 s2

.n/ .n/ 1 .n/ 1 T ˛˛ T 11 H. C p˛˛ Y1p / ; ˛ D 1; 2; 3: (9.343) 3 3

Here the strengthening parameter H and the first invariant Y1p have the forms H D H0 Yp2n0 ; Yp2 D

.n/

Y1p D I1 . C p /;

(9.344)

.n/ .n/ .n/ .n/ .n/ 1 .n/p .. C 11 C p22 /2 C . C p22 C p33 /2 C . C p11 C p33 /2 /: 2

Notice that Eqs. (9.343) and (9.344) are symmetric in indices ˛ D 2; 3; therefore, .n/

p C 22

.n/

p

D C 33 , and from the condition of plastic incompressibility (9.93) (Y1p D 0) we obtain .n/ .n/ 1 .n/ C p33 D C p22 D C p11 : (9.345) 2

9.7 The Problem on Tension of a Plastic Beam

669

Then the system (9.343) takes the form

d .n/p C dt ˛˛

.n/ .n/ 2 d .n/p ~h P p C 11 D T H C 11 11 ; dt 3 s2 3 .n/ ~h P 1 .n/ p D T 11 H C ˛˛ ; ˛ D 2; 3: 3 s2 3

(9.346)

(9.347)

Equations (9.347) follow from (9.346) and (9.345), therefore, the system (9.346), (9.347) contains only one independent equation, namely (9.346). This equation .n/

.n/ p

allows us to evaluate the parameter of loading ~P if values of T 11 and C 11 are known. .n/ p

To evaluate C 11 we use the yield surface equation (9.96), which for this problem becomes (here we have taken the expression for YH from Exercise 9.1.1 into account): .n/ .n/ .n/ .n/ .n/ 1 .n/ .. T 11 H. C p11 C p22 //2 C . T 11 H. C p11 C p33 //2 2 6 s .n/

.n/

CH 2 . C p22 C p33 /2 / D 1:

(9.348)

Due to (9.345), this equation is simplified: .n/ p 3 .n/ j T 11 H C p11 j D 3 s : 2

(9.349)

Since .n/

.n/

H D H0 Yp2n0 D H0 . C p11 C p22 /2n0 D H0

3 .n/p C 2 11

2n0 ;

(9.350)

Eq. (9.349) takes the final form ˇ.n/ ˇ T 11 H0 .n/

3 .n/p C 2 11

2n0 C1

ˇ p ˇ D 3 s :

(9.351)

To find an expression for C p11 from (9.351), by analogy with formulae (9.318)– (9.320) we should consider separately the cases of plastic tension, plastic compression and leaving the yield surface. Then from (9.351) we get

670

9 Plastic Continua at Large Deformations

.n/

p

C 11

where

8 p .n/ ˆ ˆ .n/ p j T 11 3 s j 1=.2n0 C1/ 2 ˆ ˆ ˆ T 3 / ; sign . 11 s ˆ ˆ3 H0 ˆ ˆ p ˆ ˆ ˆ if Y1H > 3 s ; ˆ < p .n/ D 2 .n/ p j T 11 C 3 s j 1=.2n0 C1/ ˆ ˆ T C 3 / ; sign . 11 s ˆ ˆ3 H0 ˆ ˆ p ˆ ˆ ˆ if Y1H 6 3 s ; ˆ ˆ ˆ p p ˆ :.n/p C 11 ; if 3 s < Y1H < 3 s ;

(9.352)

.n/ .n/ ˇ 3 .n/ ˇ2n C1 Y1H D T 11 H0 ˇ C p11 ˇ 0 sign . C p11 /: 2 .n/

.n/

Here we have taken into account that s > 0, and C 11 p is the value of C p11 reached .n/ p

at the time t of leaving the yield surface (at t D 0: C 11 D 0).

9.7.4 Change of the Density .n/

.n/

According to Eqs. (9.334), (9.341) and (9.345), we can express C 22 in terms of C 11 .n/ p

and C 11 :

.n/

.n/

.n/

p

.n/

C 22 D C e22 C C 22 D C e11

i.e. .n/

p C 22

1 .n/p C ; 2 11

.n/ .n/ 1 p D C 11 C C 11 : 2

(9.353) .n/

.n/

With the help of relations (9.330) we can represent k2 in terms of C 22 and C 11 in terms of k1 : .n/ .n/ 1=.nIII/ 1 D 1 .n III/ C 11 C C p11 k2 D .1 C .n III/ C 22 / 2 .n/ 1=.nIII/ 1 nIII D 1 .k1 1/ .n III/ : (9.354) C p11 2 .n/

1=.nIII/

.n/ p

Thus, we have obtained the expression of k2 in terms of k1 and C 11 . Taking the equality k2 D k3 into account and substituting (9.354) into (9.331), we get .n/ 2=.nIII/ 1 1 nIII J D ı D 1 .k1 C p11 1/ .n III/ : (9.355) k 2 1

9.7 The Problem on Tension of a Plastic Beam

671

Notice that although the continuum considered is plastically incompressible, however, unlike the problem on all-round compression, in this case the density depends on plastic deformations.

9.7.5 Resolving Equation for the Problem .n/

.n/

.n/

Equations (9.340) and (9.334) yield the relation between T 11 , C 11 , and C p11 : .n/

C 11 D

.n/ C p11

.n/

C

T 11 : EJ

(9.356) .n/

1. Consider the case of initial plastic tension, when the conditions C p 11 D 0 and .n/ .n/ p p T 11 3 s are satisfied. Then for the plastic deformation C 11 , from the first line of formula (9.352) we find the expression .n/

.n/

p C 11

D

T

11

p 3 s 1=.2n0 C1/ ; e0 H

e 0 D H0 .3=2/2n0C1 : (9.357) where H .n/

Substituting (9.357) into (9.356) and replacing C 11 by k1 according to formula .n/

(9.330), and T 11 by 11 according to (9.337), we obtain k IIIn 11 2 k1IIIn 11 k1nIII 1 D 1 C e0 n III EJ 3 H

p 3 s 1=.2n0 C1/

:

(9.358)

Combining this equation with expression (9.355) for J , into which formula (9.357) has been substituted: k IIIn 1 1 11 J D 1 .k1nIII 1/ .n III/. / 1 e0 k1 2 H

p

2 3 s 2n01C1 nIII

;

(9.358a)

we find the main resolving equation for this problem in initial plastic tension. This equation has a form of the implicit relation ˆ. 11 ; k1 / D 0 between 11 and k1 . 2. Under initial loading in the elastic domain there is no plastic deformation .n/

( C p11 D 0), and formulae (9.358) and (9.358a) coincide with the resolving equation for models An of elastic continua (6.148): 11 D E

k1nIII1 nIII .k 1/.1 .k1nIII 1//2=.IIIn/ : n III 1

(9.359)

672

9 Plastic Continua at Large Deformations

3. If there is initial plastic loading in the compression domain when the conditions .n/ .n/ p C p 11 D 0 and T 11 3 s are satisfied, then in formula (9.352) we choose the second condition. As a result, for plastic deformation we have .n/

C 11

p

jk1IIIn 11 C D e0 H

!1=.2n0 C1/ p 3 s j

;

(9.360)

and instead of (9.358) and (9.358a), we get the resolving relation k IIIn 11 k1nIII 1 D 1 n III EJ

jk1IIIn 11 C e0 H

p

3 s j

!1=.2n0 C1/ ;

jk IIIn C 1 1 11 J D 1.k1nIII 1/.nIII/. / 1 e0 k1 2 H

(9.361)

p 2 3 s j 2n01C1 IIIn

:

(9.361a)

4. At last, if there occurs unloading after plastic loading in the domain of tension or compression, then from Eqs. (9.355) and (9.356) we obtain the relation between 11 and k1 k IIIn 11 .n/p k1nIII 1 (9.362) D 1 C C 11 ; n III EJ 1 J D k1

1=.nIII/ .n/ 1 p nIII 1/ .n III/. / C 11 : 1 .k1 2

(9.362a)

.n/

Here C p 11 is the maximum value of plastic deformations reached at the time t of the beginning of unloading; this value is calculated by formula (9.357) for preceding plastic tension and by (9.360) – for compression.

9.7.6 Numerical Method for the Resolving Equation Transpose the first summand from the left-hand side of Eq. (9.358) onto the righthand side, and then raise the obtained expression to power 1 C 2n0 . As a result, we find the equation

k nIII 1 k1IIIn 11 A 1 n III EJ

k IIIn 11 D 1 e0 H

p 3 s

;

(9.363)

:

(9.364)

where we have introduced the notation A.k1 ; 11 /

k IIIn k1nIII 1 11 n III J.k; 11 /

2n0

9.7 The Problem on Tension of a Plastic Beam

673

To solve numerically Eq. (9.363) we can apply the method of step-by-step approxifm1g mations, where by given values of k1 and values of 11 at the .m 1/th iteration fmg at the mth iteration: we find a value of 11 e fmg D

1 e 0 Afm1g =J fm1g 1CH

! e 0 Afm1g k nIII p H k nIII ; 3 s C n III

(9.365)

fm1g fm1g where J fm1g D J.k; 11 / and Afm1g D A.k1 ; 11 / are values of the functions at preceding .m 1/th iteration (m D 1; 2; : : :). As an initial value of f0g fmg we can take the value of 11 computed at the preceding iteration cycle for the 11 preceding value of k1 . The method is convergent for values of n0 within the interval 0:5 < n0 < 1. We can solve Eq. (9.361) in the compression domain by a similar way. In the domains of elastic loading and unloading the stress 11 can be determined in the explicit way from Eqs. (9.359) and (9.362). Figure 9.8 shows the functions 11 .k1 / obtained by the above-mentioned numerical method for different models An and at different values of the parameter

a

b

c

d

Fig. 9.8 Diagrams 11 .k1 / for models An of elastoplastic continuum in uniaxial tension (e – the e 0 =E, n0 D 0:1): elastic continuum model, numbers at curves are different values of parameters H (a) – model AI , (b) – model AII , (c) – model AIV , and (d) – model AV

674

9 Plastic Continua at Large Deformations

a

b

c

d

Fig. 9.9 Diagrams of deforming 11 .k1 / for models An of an elastoplastic continuum in uniaxial tension followed by unloading (e – the model of an elastic continuum, D 0:3): (a) – model e 0 =E D 0:2, n0 D 0:1, s =E D 0:001), (b) – model AV (H e 0 =E D 0:05, n0 D 0:1, AI (H e 0 =E D 0:15, n0 D 0:1, s =E D 0:002), and (d) – model AIV s =E D 0:002), (c) – model AII (H e 0 =E D 0:05, n0 D 0:1, s =E D 0:002) (H

e 0 . For the models AI and AII , the functions 11 .k1 / are convex upwards for a H purely elastic continuum and for elastoplastic models; but for the models AIV and AV in the plastic domain (at k > ks , where ks is the elongation at the yield point: .n/ p T 11 .ks / D 3 s ) the functions 11 .k1 / are convex downwards. For all the models An , the appearance of plastic deformations can cause considerable decreasing values of stresses 11 in comparison with an elastic continuum. Figure 9.9 shows diagrams of deforming 11 .k1 / for different models An of an elastoplastic continuum under plastic loading up to some limiting values k and subsequent unloading, which have been computed by Eqs. (9.358) and (9.359). Let us note some important effects caused by large values of plastic deformations: 1. When s =E 6 0:01, diagrams of deforming in the elastic domain at k < ks and under unloading for values k close to 1 (k . 1:1) are practically linear and

9.7 The Problem on Tension of a Plastic Beam

675

have the same slope; however, at higher values of k (k & 1:2) these diagrams are considerably distinct: diagrams of unloading become essentially nonlinear for all models An ; 2. For the models AI and AII , the slope of a tangent to the function 11 .k1 / under unloading decreases with growing k , and for the models AIV and AV it, on the contrary, increases.

9.7.7 Method for Determination of Constants H0 , n0 , and s

The model An of an isotropic plastically-incompressible continuum with the Mises potential (9.301), (9.97) contains five constants: E, , H0 , n0 and s . If the exper.ex/ .ı1 / (diagram of deforming) under active loading imental dependence 11 D 11 and corresponding curves of unloading arep known, then the constant s can be determined by the formula s D k1IIIn 11s = 3. Here 11s is the value of stress 11 at the diagram of deforming, after unloading from which the residual elongation ı1p at 11 D 0 takes on a value given a priori. For metals and alloys, and also for grounds and rocks, one usually assumes that ı1p D 0:2 %, and the corresponding value 11s is denoted by 0;2 (Fig. 9.10). This value is the yield strength. Since p ı1 D 0:002 1, the domain p of elasticity for such materials corresponds to small deformations and s D 11s = 3; the elastic modulus E can be determined as a .ex/ slope of the tangent to the initial section of the experimental diagram 11 .ı1 /. Just as for elastic media, Poisson’s ratio is usually calculated by formula (6.147) as the ratio of transverse elongation of a beam to its longitudinal elongation in the domain of small deformations. e 0 and n0 can be calculated by approximating the experimental The constants H .ex/ .ı1 / under active plastic loading with the help of the diagram of deforming 11 relation 11 .ı1 / (9.358). To do this, one should minimize the functional of meansquare distance between the experimental and theoretical curves at N points: !1=2 N 11 .ı1.i / / ˇˇ2 1 X ˇˇ D ! min: (9.366) ˇ1 .ex/ ˇ N 11 .ı1.i / / i D1

Fig. 9.10 Diagram of deforming for aluminium alloy and method for determination of the yield strength 0;2 : 1 – active loading, 2 – unloading

676

9 Plastic Continua at Large Deformations

9.7.8 Comparison with Experimental Data for Alloys Figure 9.11a shows experimental diagrams of deforming for steel alloy under temperatures 20ı C and 800ı C, and also their approximations by the method, mentioned in Sect. 9.7.6, with the help of different models An according to formula (9.358). e 0 , n0 and s , calculated when D 0:35. Table 9.3 gives values of the constants E, H An accuracy of approximation proves to be high for all the models An ; the model AI exhibits the least value of the deviation . Notice that the value of coefficient n0 is negative, therefore the diagram of deforming is convex upwards in the vicinity of the yield point s . Figures 9.11b and 9.12a show experimental diagrams of deforming for aluminium alloys D16 and AK4 in tension and their approximations by the models An with the help of formula (9.358); and Table 9.3 gives values of the e 0 , n0 , and s . An accuracy of approximation is also sufficiently high constants E, H for all the models An .

a

b

Fig. 9.11 Diagrams of deforming: (a) – for steel alloy under temperatures 20ı C and 800ı C and (b) – for aluminium alloy D16 in tension

e 0 , n0 , s , and in different models An for steel and aluminium alloys Table 9.3 Values of E, H Steel alloy at 20ı C and 800ı C Aluminium alloys D16 and AK4 n I II IV V n I II IV V

E, GPa

e 0 , GPa H n0 s , MPa , %

200 60 4:8 0:72 0:02 0:26 145 46:3 6:4 3:7

200 60 3 0:54 0:08 0:26 145 46:3 2:5 5:4

200 60 1:2 0:18 0:18 0:38 145 46:3 5:4 14

200 60 0:6 0:18 0:3 0:36 145 46:3 5:3 15:2

E, GPa

e 0 , GPa H n0 s , MPa , %

70 67:7 0:63 1:42 0:3 0:08 130 197 2:4 1:7

70 67:7 0:63 0:81 0:28 0:16 130 197 3 1:3

70 67:7 0:42 0:4 0:32 0:24 130 197 4 1:9

70 67:7 0:42 0:61 0:3 0:08 130 197 6:5 7:6

9.7 The Problem on Tension of a Plastic Beam

a

677

b

Fig. 9.12 Diagrams of deforming for aluminium alloy AK4: in tension (a) and in compression (b)

a

b

Fig. 9.13 Diagrams of deforming in compression for sand grounds: wet sand (a) and dry sand (b)

Figure 9.12b shows experimental and computed diagrams of deforming for aluminium alloy AK4 in compression. Computations were performed by formula e 0 , n0 , and s had been evaluated previously by the (9.360), and the constants E, H curve of deforming in tension. In this case the model AI exhibits the best accuracy; the remaining models lead to a considerable error of approximation.

9.7.9 Comparison with Experimental Data for Grounds Figures 9.13 and 9.14 show experimental diagrams of deforming in compression for sand grounds (for dry and wet sands), and also their approximations with the help of relationship (9.361). These diagrams differ from the corresponding compression

678

a

9 Plastic Continua at Large Deformations

b

Fig. 9.14 Approximation of diagrams of deforming for sand grounds in compression by the models with linear strengthening: wet sand (a) and dry sand (b)

a

b

Fig. 9.15 Diagrams of deforming under loading and subsequent unloading (experimental and computed by the models AI and AIV for sand grounds: wet sand (a) and dry sand (b)

diagrams for alloys (see Fig. 9.12b) by the presence of an intense convexity upwards (in absolute values of coordinates); and under unloading the deformation curve goes sharply downwards (Fig. 9.15). For damp sands the accuracy of approximation with the help of all models An is sufficiently high, for dry sands the accuracy is lower. Models AII and AIV show the best approximation results. Unlike metallic alloys, for grounds the constant n0 has positive values, and the ratio of the yield strength to the maximum magnitude of Cauchy stresses ˛s D s = max is considerably smaller than for steels: ˛s 0:001 and ˛s 0:1, respectively. A magnitude of s for grounds is usually such small that the initial elastic stage of deforming is not visible on the diagram of deforming (Figs. 9.13 and 9.15), although values of elastic modulus E under loading and unloading in experiments prove to be close (in the considered models An they are coincident). Table 9.4 gives values e of the constants E, H 0 , n0 and s for the considered types of grounds.

9.8 Plane Waves in Plastic Continua Table 9.4 grounds Dry sand n E, GPa e 0 , GPa H n0 s , MPa , %

679

e 0 , n0 , s , and in models An for different Values of the constants E, H

I 10 0.324 0.09 0.23 21.4

II 10 0.56 0.15 0.23 17.2

IV 10 1.6 0.27 0.23 12.2

V 10 3.1 0.34 0.23 14

Wet sand n E, GPa e 0 , GPa H n0 s , MPa , %

I 10 42 0.6 0.29 55

II 10 45 0.6 0.29 56

IV 10 260 0.8 0.29 55

V 10 600 0.9 0.29 56

Fig. 9.16 Propagation of a plane wave in the plate

Figure 9.14 shows the results of approximation of experimental diagrams of deforming for sand grounds with the help of the models of plastic continua with linear strengthening (see Sect. 9.1.8) when we assume a priori that n0 D 0. The constant H0 for these models can be determined by minimizing the mean-square deviation (9.405). By this method, we have get the following values: for wet sands, H0 D 2 GPa for the models AI and AII ; and for dry sands, H0 D 0:24 GPa for the model AI and H0 D 0:3 GPa for the model AII . The quality of approximation to the experimental diagrams of deforming by the models with linear strengthening is worse than that obtained by the models with power strengthening (9.86) especially for wet sands. However, in some cases this model proves to be more convenient for solving special problems (see Sect. 9.8).

9.8 Plane Waves in Plastic Continua 9.8.1 Formulation of the Problem Let us investigate now the dynamical problem of plasticity theory An (9.288)– (9.290) in the material description and consider the problem on a plane wave in a plate, which is caused by high-speed (not quasistatic) loading on one of its surfaces X 1 D 0 (Fig. 9.16). The end surface X 1 D h01 of the plate is assumed to be free of loads; and on the lateral surfaces X ˛ D ˙h0˛ =2, ˛ D 2; 3, the condition of free slip (the symmetry condition) is given. Thus, boundary conditions for this problem have the form

680

9 Plastic Continua at Large Deformations

X1 D 0 W

X 1 D h01 W ˛

X D

pe .t/ 1 .F /11 ; P12 D P13 D 0; J P˛1 D 0; ˛ D 1; 2; 3I

P11 D

˙h0˛ =2

W

˛

(9.367)

˛

x D X ; P˛1 D 0; ˛ D 2; 3:

Here Pij are Cartesian components of the Piola–Kirchhoff tensor, and .F 1 /11 – of the inverse deformation gradient F1 D .F 1 /ij eN i ˝ eN j . Such boundary conditions approximately simulate the process of impact of a massive rigid slab on the investigated plate of plastic material slipping along a rectangular rigid pipe.

9.8.2 The Motion Law and Deformation of a Plate A law of the motion of the plate is sought in the form 8 1 1 1 ˆ ˆ <x D x .X ; t/; x2 D X 2; ˆ ˆ :x 3 D X 3 ;

(9.368)

where x 1 .X 1 ; 0/ D X 1 . The velocity v has only one non-zero component: vD

@x i eN i D v1 eN 1 ; @t

v1 D @x 1 [email protected]:

The deformation gradient F has the form FD

3 X @x i j N N e ˝ e D k˛ eN 2˛ ; i @X j ˛D1

k1 D k1 .X 1 ; t/ D @x 1 [email protected] 1 ;

(9.369)

k ˛ D @x ˛ [email protected] ˛ D 1; ˛ D 2; 3;

and, as distinct from quasistatic tension of a beam (see (9.328)), here the elongation ratio k1 .X 1 ; t/ depends on X 1 linearly. .n/

The energetic deformation tensors C also contain the only non-zero component: .n/

.n/

C D C 11 eN 21 ;

.n/

C 11 D

1 .k nIII 1/; n III 1

.n/

.n/

C 22 D C 33 D 0;

(9.370)

and changing density is determined by function k1 : ı

J D = D det F1 D 1=k1 :

(9.371)

9.8 Plane Waves in Plastic Continua .n/

681

.n/

Tensors C e and C p are sought in the diagonal form .n/

Ce D

3 .n/ X C e˛˛ eN 2˛ ;

.n/

Cp D

˛D1 .n/

.n/

3 .n/ X C p˛˛ eN 2˛ ;

(9.372)

˛D1

.n/

.n/

C e11 C C p11 D C 11 ;

.n/

.n/ C e33

C e22 C C p22 D 0;

.n/

.n/

C C p33 D 0;

.n/

where all the components C e˛˛ and C p˛˛ depend only on X 1 and t. From the condition of plastic incompressibility we get .n/ C p22

.n/ 1 .n/ D C p33 D C p11 ; 2 .n/ .n/ .n/ 1 p C e22 D C e33 D C 11 ; 2

.n/

(9.373) (9.374)

.n/

i.e. transverse deformations C e22 , C e33 are different from zero only in a plastic domain.

9.8.3 Stresses in the Plate .n/

According to (9.335), the stress tensor T in this problem has the diagonal form .n/

T D

3 .n/ X T ˛˛ eN 2˛ ;

(9.375)

˛D1

where .n/

.n/

.n/

T 11 D J.l1 C 2l2 / C 11 2J l2 C p11 ; .n/

.n/

.n/

(9.376)

.n/

T 22 D T 33 D J.l1 C 11 C l2 C p11 /I

(9.377)

all the values depend only on X 1 and t. According to (9.375) and (9.369), the tensors T and P also have a diagonal form TD

3 X

˛˛ eN 2˛ ;

˛D1

PD

3 X

P˛˛ eN 2˛ D

˛D1 .n/

11 D k1nIII T 11 ; P11 D 11 ;

.n/

1 1 F T; J .n/

22 D 33 D T 22 D T 33 ; 1 P22 D P33 D 22 : k1

(9.378)

682

9 Plastic Continua at Large Deformations

9.8.4 The System of Dynamic Equations for the Plane Problem Since ı

r PD ı

r ˝ vT D 4

3 X @P˛˛ @P11 eN ˛ D eN 1 ; ˛ @X @X 1 ˛D1

@v1 2 eN ; @X 1 1

ı

F1T r ˝ v D ı

.n/

X F1T r ˝ v D k1nIII1

1 @v1 2 eN ; k1 @X 1 1

(9.379)

@v1 2 eN @X 1 1 .n/

(here we have used the result of Exercise 4.2.16 for tensors X), so the first, third, fourth and fifth equations of the system (9.288) in the considered problem take the forms 1 @x 1 @k1 @ 11 @v1 ı @v 1 ; ; ; D D v D @t @X 1 @t @t @X 1 0 1 .n/ .n/ (9.380) @ B T 11 C 2l2 @ C p11 l1 C 2l2 nIII1 @v1 ı : k1 @ AD ı @t @X 1 @t Notice that the last equations of the system (9.380) can be obtained by immediate differentiation of relation (9.376). It should also be noted that the fifth equation of the system (9.288) gives two .n/

.n/

more scalar expressions for @ T 22 [email protected] and @ T 33 [email protected]; however, these are just expressions but not equations, and therefore they do not appear in the general system (9.380). Taking the expression (9.371) for and expression (9.378) for 11 into account, we get 0

.n/

1

k1IIInC1 @ 11 @ B T 11 C III n C 1 @k1 C : 11 k1IIIn @ AD ı ı @t @t @t

(9.381)

The last equation of the system (9.380) after substitution of the expression (9.381) takes the form .n/ p @ 11 .III n C 1/ 11 @v1 2.nIII1/ nIII1 @ C 11 D .l1 C 2l2 /k1 : 2l k 2 1 @t k1 @X 1 @t (9.382)

9.8 Plane Waves in Plastic Continua

683 .n/

p

Just as in other one-dimensional problems considered above, to evaluate @ C 11 [email protected] we use the equation of a yield surface f D 0 (see (9.96)), which, according to the results of Exercise 9.1.1, in this problem has the form .n/ .n/ .n/ 1 .n/ p p . T 11 T 22 H. C 11 C 22 //2 D 1: 2 3 s .n/

(9.383)

.n/

.n/

Substituting expressions (9.377) for T 22 , (9.373) for C p22 , (9.370) for C 11 and (9.350) for H into (9.383), we obtain .n/

.n/

.n/

e 22 H e C 11 j D j T 11 T p

p

3 s ;

(9.384)

where e D 3 H C l2 ; H 2 k1

.n/

nIII .n/ 1/ e 22 D l1 C 11 D l1 .k1 T : k1 .n III/k1

(9.385)

For the model of plasticity with linear strengthening when n0 D 0 and H D H0 , e D 3 H0 C l2 , and Eq. (9.384) has the following analytical solution for we have H 2 k1 .n/

p

C 11 :

.n/ p e 22 .n/ T 3 s e T 11 p e e hC h C C p C 11 D 11 .1 hC /: e e H H Here hC and h are the Heaviside functions: .n/

.n/

(9.386)

e h D hC h ; hC hC C h ; e ( p p 1; Y1H > 3 s ; 1; Y1H < 3 s ; hC D h D p p 0; Y1H < 3 s ; 0; Y1H > 3 s ; (

.n/

(9.387)

.n/

e22 H eCp ; Y1H D T 11 T 11 .n/

and C p 11 is the value of plastic deformation reached at the time t of leaving the yield surface. .n/

.n/

e 22 into (9.386) Substituting the expressions (9.378) and (9.385) for T 11 and T and then differentiating (9.386) with respect to t within differentiability sections, we obtain .n/

p @k1 @ 11 @ C p11 D b0e C ..b1 11 b2 /e : hC h / hC 3 s b3e @t @t @t

(9.388)

684

9 Plastic Continua at Large Deformations

Here we have introduced the following functions of k1 : 2k1IIInC1 2k1IIIn 2l2 ; b1 D III n C ; (9.389) 3H0 k1 C 2l2 3H0 k1 C 2l2 3H0 k1 C 2l2 3H0 k1 .k1nIII 1/ 4l2 2l1 nIII : k ; b D b3 D 2 .3H0 k1 C 2l2 /2 .3H0 k1 C 2l2 /k1 1 .3H0 k1 C 2l2 /.n III/ b0 D

Substituting the expression (9.388) into (9.382) and grouping like terms, we find the equation @ 11 @v1 D .c1 11 c2 / 1 : @t @x

(9.390)

Here we have introduced the following notation for functions of k1 : c0 D 1 C c1 D c2 D

1 c0 k1

4l2e hC ; 3H0 k1 C 2l2

p l1 C 2l2 2.nIII1/ 2l2 nIII1 e h /; k1 C k1 .b2 hC C 3 s b3e c0 c0 ! hC 2l2 4l2e : (9.391) III n C 1 C III n C 3H0 k1 C 2l2 3H0 k1 C 2l2

Notice that for most plastic materials occurring in practice the following condition is satisfied: 11 c2 =c1 1:

(9.392)

Therefore, the term 11 c2 in Eq. (9.390) can be neglected; and we may consider the simpler equation @v1 @ 11 D c12 1 : @t @X

(9.393)

9.8.5 The Statement of Problem on Plane Waves in Plastic Continua Substituting the relation (9.393) in place of the last equation in the set (9.380) and excluding the second equation from (9.380) (because coefficients in the equation set do not depend explicitly on X 1 ), we finally obtain the system 8 ı ˆ ˆ 0. According to the characteristic method, consider differentials of the unknown functions dv D

@v @v dt C dX; @t @X

dk D

@k @k dt C dX; @t @X

dT D

@v @v dt C dX: @t @X (9.397)

The system (9.394) and (9.397) consists of six equations being linear with respect to six unknown functions: vt D @[email protected], vX D @[email protected] , Tt D @T [email protected], TX D @T [email protected] , kt D @[email protected] and kX D @[email protected] . The system can be rewritten in the matrix form 0ı 10 1 0 1 0 vt 0 0 1 0 0 B CB C B C B 0 1 0 0 1 0 C B vX C B 0 C B CB C B C B 0 c1 1 0 0 0 C B Tt C B 0 C (9.398) B CB C D B C: Bdt dX 0 0 0 0 C BTX C B dv C B CB C B C @0 0 0 0 dt dX A @ kt A @ dk A kX dT 0 0 dt dX 0 0 There exists an unique solution of this system if and only if its determinant is different from zero. However, on the plane .X; t/ there are curves (called characteristics) where the solution uniqueness is violated. This case is realized when the determinant of system (9.398) vanishes: ı

det . / D dX 3 C c1 dt 2 dX D 0:

(9.399)

686

9 Plastic Continua at Large Deformations

From (9.399) we find equations of two families of characteristics: dX D ˙a dt; where

(9.400) ı

a2 .k/ c1 .k/=

(9.401)

is the speed of sound in a plastic material considered. Substitution of (9.400) into the first and second equations of system (9.394) yields conditions over the characteristics ı

dv D dT or

dt dT D˙ ; dX a

ı

a d v d T D 0;

dk D dv

dt dv D˙ ; dX a

a d k d v D 0:

(9.402)

Introducing the functions Z

.k/ D 1

k

c1 .k 0 / d k 0 ;

Z

k

'.k/ D

a.k 0 / d k 0 ;

(9.403)

1

for which d D c1 d k, d' D a d k, we can integrate Eq. (9.402):

.k/ T D const;

'.k/ v D const:

(9.404)

Thus, there are two families of characteristics and two conditions over each of the characteristics: 8 ˆ ˆ 3 s : if T 11 T

c0 D 1;

690

9 Plastic Continua at Large Deformations .n/

e 22 and in place Substituting into formula (9.384) the expression (9.385) for T .n/

of T 11 its expressions in terms of k according to (9.378), (9.404), and (9.415): .n/

T 11 D k IIIn T D k IIIn .k/, we find the limiting value k D ks < 1, from which plasticity starts in compression: p 3 s .ksnIII 1/ ksIIIn 2.nIII/1 1/ D : .k 2.n III/ 1 s .1 /.n III/ks l1 C 2l2

(9.418)

Here, as usual, D l1 =.2.l1 C l2 // is the Poisson ratio. In the domain of plastic deformations (in compression) when k 6 ks < 1, relations (9.386), (9.387), and (9.391) become hC D 0; .n/ C p11

D

h D 1;

e hC D 1;

e h D 1;

.n/ p 1 .n/ e 22 C 3 s / . T 11 T e H

l1 C 2l2 D e H c0 D 1 C

k

IIIn

4l2 ; 3H0 k C 2l2

e c 1 .k/ D k 2.nIII1/ C

.k/ D .l1 C 2l2 /

1

k

e c 1 .k/ ; c0 .k/ ! p 3 s b3 b2 ; l1 C 2l2

c1 D .l1 C 2l2 /

1 2 1

p a D a0 e c 1 .k/=c0 .k/; Z

! p .k nIII 1/ 3 s

.k/ ; C .1 /.n III/k l1 C 2l2

k nIII1 q

a0 D

e c 1 .k 0 / d k 0 ; c0 .k 0 /

(9.419)

ı

.l1 C 2l2 /=; Z '.k/ D a0

1

k

s e c 1 .k 0 / d k0 : c0 .k 0 /

Typical dependences of the dimensionless speed of sound a=a0 upon k, determined by functions (9.419), for different models An are shown in Fig. 9.20 (here D 0:3, s =m D 1:5 103 , H=m D 102 when m D l1 C 2l2 ). As follows from these graphs, there is a distinction in kind between the models AI , AII and AIV , AV : for the models AI and AII the speed of sound increases with decreasing k (in compression when k < 1), and for the models AIV and AV it does not grows (for AIV it remains constant, and for AV – decreases). For all models An , the function a.k/ in the case considered has a discontinuity at k D ks due to the assumption on linear strengthening of a continuum (see formula (9.386)).

9.8 Plane Waves in Plastic Continua

691

Fig. 9.20 The speed of sound (dimensionless a=a0 ) versus the ratio k in compression for different models An of elastic materials (e) and plastic materials (p)

9.8.8 Plane Waves in Models AIV and AV Since the function .k/ is negative when k < 1 and monotonically diminishes within the interval of values of k from 1 to 0 for all models An , the function k.t/ D

1 .pe .t// is monotonically decreasing (k.0/ D 1 .pe .0// D 1) (here we have taken into account that p0 .t/ > 0 by the assumption made). Thus, at the front surface X D 0 of the plate the function k.t/ monotonically decreases, and the function a.k/ grows for the models AI , AII and diminishes for the models AIV , AV . Then on the plane .t; X / ‘C’-characteristics in the disturbance domain (as shown above, they are straight lines) have higher values of a slope from the axis OX with growing t for models AIV , AV and smaller values – for models AI , AII (Fig. 9.21). The decrease of a slope of characteristics t D X=a.k/ from the axis OX means that characteristics with different values of k may intersect at the front wave (Fig. 9.21). As a result, the solution becomes ambiguous, that is inadmissible under the assumptions made above on the absence of jumps of the functions k, T and v themselves (only jumps of their first derivatives are admissible). Thus, for the models AI and AII , the solution obtained is inapplicable, and it will be constructed in another way (see Sect. 9.8.9.). For the models AIV and AV , characteristics do not intersect, and the solution (9.415) obtained actually holds. Figure 9.21 shows the graphic method of construction of the solution (9.415) with the help of given values of the function pe .t/.

692

9 Plastic Continua at Large Deformations

Fig. 9.21 Graphic method of construction of the solution (9.415)

a

b

Fig. 9.22 The Riemann waves for the models AIV and AV of plastic continua: p 0 < ps (a) and p 0 > ps (b)

Consider the special case of loading when the load pe has the jump-type form pe D pe0 h.t/;

(9.420)

where h.t/ is the Heaviside function. Then T and k at the face surface of the plate range from 0 and 1 to final values T 0 and k 0 < 1, respectively, and then remain constant for all t > 0. Thus, on the plane .t; X / there appears an angle bounded by the characteristics X D a0 t and X D a.k 0 /t and filled with characteristics X D a.k/t while k 0 < k < 1 (the fan of characteristics). Waves corresponding to these characteristics are called the Riemann waves by analogy with gas dynamics. These waves are characterized by the fact that they propagate without changes in amplitudes of T and v (Fig. 9.22).

9.8 Plane Waves in Plastic Continua

693

If p 0 < ps , where ps is the pressure of the beginning of plasticity in compression (ps D .ks /, here ks is determined by (9.418)), then the wave retains its shape (being a step) (Fig. 9.22). If p 0 > ps , then the shape of the wave spreads: values of k vary from k 0 up to ks , but the maximum value ks remains constant.

9.8.9 Shock Waves in Models AI and AII

Let us return now to the models AI and AII and consider only the case of stepwise loading of the plate (9.420). In this case the system (9.394) and (9.396) admits the trivial solution T D p D const;

k D const;

v D const:

(9.421a)

However, the boundary condition (9.395b) may be satisfied only if we assume that there is a jump discontinuity for the functions, followed by the trivial solution T D 0;

k D 1;

v D 0:

(9.421b)

In other words, for the models AI and AII there is a solution in the form of a shock wave. The function X D XD .t/ separating on the plane .t; X / two solutions (9.421) is an equation of the front of the shock wave. To find this function and also to determine values of k and v (relationships over characteristics do not hold there), we should use relations (5.70) at a surface of a strong discontinuity in the material description. For the considered problem, these relations reduce to the following ones: 8ı ˆ ˆ ˆM v p D 0; < ı

.9:422a/

ı

M .k 1/ C v D 0; ˆ ˆ ı ˆ 2 :M . v C Œe / pv D 0:

.9:422b/ .9:422c/

2

Here we have taken into account that Œv D v, ŒP D P11 D 11 , ŒF D F11 1 D k1, because on one side of the singular surface, according to (9.421b), the medium ı

is quiescent. The mass rate M is determined by (5.57) and (5.12): ı

ı ı

M D D ;

ı

ı

ı

D D c D @x † [email protected] D dXD =dt:

(9.423)

Assume that the temperature jump is zero across the shock wave: Œ D 0, then the result of Exercise 9.1.10 for the internal energy jump Œe gives the expression Œe D e e0 D

.n/ I 2. C e / ı 1

l1

2

C

l2 ı

.n/

I1 . C 2e /:

(9.424)

694

9 Plastic Continua at Large Deformations

Here we have taken into account that, according to (9.421b), e D e0 in a quiescent domain. .n/

From (9.372)–(9.374) we obtain the following expressions for invariants I1 . C e / .n/

and I1 . C 2e /: .n/

.n/

.n/

.n/

.n/

.n/

I1 . C e / D C e11 C 2 C e22 D C e11 C C p11 D C 11 ; .n/ .n/ .n/ .n/ .n/ 1 .n/p p I1 . C 2e / D . C e11 /2 C 2. C e22 /2 D . C 11 C 11 /2 C . C 11 /2 : 2

(9.425)

On substituting (9.425) into (9.424), we find the expression for the internal energy .n/

.n/

jump in terms of deformations C 11 and C p11 : Œe D D

.n/ .n/ .n/ 1 .n/ .l1 C 211 C 2l2 .. C 11 C p11 /2 C . C p11 /2 // 2 2

1

ı

.n/

1 ı

2

.n/

.n/

p

.n/ p

..l1 C 2l2 / C 211 4l2 C 11 C 11 C 3l2 . C 11 /2 /:

(9.426)

.n/

For the case of initial plastic loading of compression when C p 11 D 0, substitution .n/

of formula (9.376) for T 11 into (9.386) yields .n/

C p11 D

1 ek H

.n/ .n/ .n/ p .l1 C 2l2 / C 11 2l2 C p11 l1 C 11 C 3 s k :

(9.427)

.n/

From this equation we can easily express the plastic deformation C p11 as follows: .n/

p

C p11

.n/

1 2 3 s k C b C 11 D ; bD : 3.b C H0 k=2/ 2.1 / .n/

(9.428) .n/

On substituting formula (9.428) for C p11 and formula (9.370) for C 11 into (9.426), we get a jump of internal energy Œe as a function of k: Œe D Œe .k/:

(9.429) .n/

If loading occurs only in the elastic domain, where C p D 0, then Œe .k/ D

l1 C 2l2 ı

2.n III/2

.k nIII 1/2 :

(9.430)

9.8 Plane Waves in Plastic Continua

695

With taking account of expression (9.429), three relationships (9.422) allow us to ı

determine three unknown functions: k, v and M in terms of p. For this purpose, we express v from Eq. (9.422b): ı

ı

v D M .1 k/=;

(9.431) ı

substitute (9.431) into (9.422a) and obtain the expression for M : ı

M D

q ı p=.1 k/:

(9.432)

We have chosen the positive sign of the square root according to the physical meaning of the solution: a shock wave must propagate in the positive direction of the axis OX . Substituting of (9.432) into (9.431) yields q vD

ı

p.1 k/=:

(9.433)

On substituting (9.429), (9.432), and (9.433) into (9.422c), we get the equation for k: p Œe .k/ D ı .1 k/: (9.434) 2 The derived relationship (9.434) with the expression (9.426) for a jump of internal energy allows us to find k as a function of p: k D k.p/. Since p is known from the problem condition, we can determine k from the equaı

tion (9.434) and then find M and v with the help of formulae (9.431) and (9.432). As a result, from the first formula of (9.423) we obtain ı

DD

s

ı

M ı

D

p ı

.1 k/

:

(9.435)

Thus, we have found a complete solution of the problem.

9.8.10 Shock Adiabatic Curves for Models AI and AII The function k D k.p/ (or p D p.k/) expressed by formula (9.434) is called a shock adiabatic curve for an elastoplastic continuum. ı Since k D = is the ratio of densities, we can introduce the specific volı ume V D 1= D k= which is a function of p, i.e. we have a function p D p.V /

696

9 Plastic Continua at Large Deformations

Fig. 9.23 The shock adiabatic curve

(or V D V .p/). This function is well-known in gas dynamics as a shock adiabatic curve for similar problems. If loading occurs only in an elastic domain, then, substituting (9.430) into (9.434), we derive the following equation for the shock adiabatic curve p D p.k/: p 1 D l1 C 2l2 1k

k nIII 1 n III

2 :

(9.436)

Figure 9.23 shows a graph of this function. On substituting (9.426), (9.430), and (9.370) into (9.434), we obtain the equation of the shock adiabatic curve with taking account of plastic deformations: .n/ .n/ .n/ .n/ 1 p D . C 211 b C p11 .4 C 11 C p11 //; l1 C 2l2 1 k1

(9.437)

where

.n/

C 11

p

.n/

3 s k C b C 11 D ; 3.b C H0 k=2/ 1 D .k nIII 1/: n III 1

.n/ C p11

(9.438)

Figure 9.23 shows the shock adiabatic curve p.k/. Figure 9.24 exhibits shock adiabatic curves computed by formula (9.437) for dry and wet sand grounds. Constants E, , s , and H for grounds have been taken from the experimental data for specimens of a beam form in uniaxial high-speed compression. The method of evaluation and values of these constants are given in Sect. 9.7.9; for approximation we have used the plasticity model with linear strengthening. ı

Let us ask the question: what is the speed D of propagation of a shock wave in comparison with the sound speed a0 in a quiescent continuum. To answer the question one should find a value of k when the pressure values are small: p=.l1 C 2l2 / 1. Linearizing the left-hand side of Eq. (9.436) in a neighborhood of the value k D 1, we obtain (9.439) p D .l1 C 2l2 /ı;

9.8 Plane Waves in Plastic Continua

697

Fig. 9.24 Shock adiabatic curves for sand grounds: solid lines correspond to wet sand, dashed lines – to dry sand

Fig. 9.25 Dependence of the speed of a shock wave and the speed of sound upon the compression coefficient k1 in a plastic continuum

where k D 1 C ı, jıj 1, ı 6 0 for both the models AI and AII . Then from (9.435) we find that s s s ı p ı.l1 C 2l2 / l1 C 2l2 D D ı D (9.440) D D a0 ; ı ı ı ı i.e. for the models AI and AII of elastoplastic continua, small disturbances propagate with the speed of sound. ı

As follows from Eq. (9.435), at finite values of p=.l1 C 2l2 / the speed D of a shock wave proves to be supersonic (Fig. 9.25).

9.8.11 Shock Adiabatic Curves at a Given Rate of Impact The solution obtained in Sect. 9.8.10 remains valid for another case of boundary conditions: when on the surface X 1 D 0 of a plate one gives a constant rate v of impact instead of the stress component P11 : X1 D 0 W

v D v0 D const;

P12 D P13 D 0I

(9.441)

the remaining conditions in (9.367) hold without changes. In this case, it is convenient to represent the shock adiabatic curve as a function p D p.v/. For this, we

698

9 Plastic Continua at Large Deformations

Fig. 9.26 Computed and experimental shock adiabatic curves in coordinates .p; v/ for sand grounds: solid curves – wet sand, dashed lines – dry sand

should substitute the function p.k/ expressed by formulae (9.437) and (9.438) into Eq. (9.433). As a result, we get the equation p.k/ D

v2 ı

.1 k/

;

(9.442)

which at a given value of v can be resolved for k. Evaluating a root of the equation (when 0 < k 6 1, this root is unique), for example, by the method of bisecting an interval, we find the function k D k.v/. On substituting this function into Eq. (9.437), we obtain the shock adiabatic curve p D p.k/ D p.k.v// D p.v/ in coordinates .p; v/ (the pressure versus the speed). Figure 9.26 shows graphs of the shock adiabatic curve p D p.v/ for sand grounds, they were computed for the models AI and AII of plastic continua with linear strengthening. The constants E, , s and H0 in the computations were taken also from the results of experiments in uniaxial tension (see Sect. 9.7.9). This figure also exhibits experimental shock adiabatic curves p.ex/ .k/ for dry and wet sand grounds. The computed curves determined by both the models AI and AII satisfactorily approximate the experimental data. The shock adiabatic curve can be represented in another form, namely as a deı

ı

pendence of the speed D on the rate of impact v. To obtain such a function D.v/, we should use formulae (9.422a) and (9.423) and substitute into them the dependence p D p.k/ D p.k.v// D p.v/ derived above; as a result, we obtain the desired form of a shock adiabatic curve: ı p.v/ (9.443) DD ı : v ı

Figure 9.27 shows the shock adiabatic curves D.v/ computed for sand grounds by ı

formula (9.443) and also experimental shock adiabatic curves D .ex/ .v/. The computed and experimental results are sufficiently close.

9.9 Models of Viscoplastic Continua

699

Fig. 9.27 Shock adiabatic curves in coordinates .D o ; v/ for sand grounds: solid curves – wet sand, dashed lines – dry sand

9.9 Models of Viscoplastic Continua 9.9.1 The Concept of a Viscoplastic Continuum In Chap. 7 we considered models of continua of the differential type, and in Sect. 7.4.4 it was shown that these models can be applied to describe the creep effect in some solids. The limitation of models of the differential type consists in the fact that they do not involve elastic deformations; therefore, with their help one cannot determine instantly elastic deformations under loading and subsequent unloading (see Figs. 7.2 and 7.3). In the cases when instantly elastic deformations cannot be neglected in comparison with creep deformations, one should use more complicated models, for example, models of viscoplastic continua, that will be considered in this section. As a rule, a continuum is called viscoplastic if its plastic properties (the appearance of residual deformations) depend on time. The models of associated plastic continua (considered in Sects. 9.1.5–9.1.13) describe pure plastic properties, which are independent of time. Indeed, although the constitutive equations (9.52) of associated models formally involve time t, we can easily exclude the time if we will differentiate not with respect to time but with respect to the loading parameter (see Exercise 9.1.11). From the experimental point of view, this fact means that if we perform experiments in uniaxial tension (see Sect. 9.7) with a constant rate of lengthening: k1 .t/ D bt, then diagrams of deforming in elastic and plastic domains are independent of the lengthening rate b. For viscoplastic continua, this dependence takes place.

9.9.2 Model An of Viscoplastic Continua of the Differential Type Let us consider the simplest viscoplastic model, namely the model An of a viscoplastic continuum of the differential type, which can be formally obtained from

700

9 Plastic Continua at Large Deformations

Eqs. (9.52) if we assume that the expression for plastic deformation holds for all the times at both loading and unloading. In this case, the parameter h should be assumed to be equal to 1: .n/

C v D

k X

.n/

~˛ [email protected]˛ [email protected] T /:

(9.444)

˛D1 .n/

.n/

In this formula we have changed the notation of plastic deformation C p by C v , which is called the viscous deformation, and the functions ~P ˛ are replaced by ~˛ . We can always make this substitution because functions ~P ˛ in (9.48) have been introduced as ratio coefficients of plastic deformations to the yield surface gradient. For the associated models of plastic deformations, the functions ~P ˛ can be found from Eqs. (9.52) with complementing by Eqs. (9.41) of the yield surface, but for a viscoplastic continuum of the differential type Eqs. (9.41) are absent and expressions for ~˛ and fv˛ are given with the help of the additional relations .n/ .n/

.n/ .n/

~˛ D ~˛ . T ; C p ; /; fv˛ D fv˛ . T ; C p ; /;

˛ D 1; : : : ; k:

(9.445)

The functions fv˛ will be called the viscous potentials. .n/

Remark 1. For viscoelastic continua of the differential type the stress tensor T is a sum of equilibrium and viscous stresses (see (7.6)), but for viscoplastic materials .n/

of the differential type the deformation tensor C consists of elastic and viscous deformations, similarly to (9.3): .n/

.n/

.n/

C D C e C C v:

(9.446)

For the unipotential model An of a viscoplastic continuum of the differential type, we assume that k D 1; ~1 D 1, and fv1 D fv ; therefore, .n/

.n/

C v D @fv [email protected] T ;

.n/ .n/

fv D fv . T ; C v ; /:

(9.447)

These relations should be complemented with equations for the elastic deformation .n/

tensor C e , for example, (9.59). The equation system can be rewritten in terms of the invariants as follows: .n/ C v

D

z1 X ˛D1

.s/ ˛ J˛T ;

.n/

Ce D

z X

' I.s/ T;

(9.448)

D1

where ˛ and ' are scalar functions being the derivatives of the plastic and elastic potentials fv and with respect to the invariants:

9.9 Models of Viscoplastic Continua

701 .n/ .n/

.n/

.s/ ˛ D @fv [email protected]˛.s/v ; fv D fv .J˛.s/ ; /; J.s/ D J.s/ . T ; C v /; J˛T D @J˛.s/ [email protected] T ; (9.449) .n/

.n/

.s/ ' D @[email protected].s/ ; D .I.s/ ; /; I.s/ D I.s/ . T =/; I.s/ T D @I [email protected] T =/: (9.450)

Due to the Onsager principle, the plastic potential fv is a quadratic function of the .n/ .n/

linear invariants J.s/ . T ; C v / and a linear function of the quadratic invariants J.s/ .

9.9.3 Model of Isotropic Viscoplastic Continua of the Differential Type For isotropic viscoplastic continua of the differential type, the simultaneous invari.n/ .n/

ants J.s/ . T ; C v / can be chosen in the form (9.75), then relations (9.448) take the forms .n/

.n/

.n/

C v D e 1 E 2 T C 7 C v ;

.n/

.n/

(9.451) .n/

Ce D e ' 1 E C .e ' 2 =/ T C .'3 =2 / T 2 :

Here e 1 D 1 C

.n/

2 I1 . T /, .n/

(9.452)

and the potentials depend on the following invariants: .n/

.n/

.n/

.I /

.n/ .n/

fv D fv .I1 . T /; I2 . T /; I1 . C p /; I2 . C v /; J7 . T ; C v /; /; .n/

.n/

.n/

D .I1 . T =/; I2 . T =/; I3 . T =/; /;

(9.453) (9.454)

If the plastic potential has been chosen in the Huber–Mises form (9.96) fv D

YH2 3 ; YH2 D PH PH ; 3 2

.n/ .n/ .n/ 1 .n/ PH D . T Hv C v / I1 . T Hv C v /E; Hv D Hv0 Yv2nv0 ; 3

Yv2 D

3 Pv Pv ; 2

.n/ 1 .n/ Pv D C v I1 . C v /E; 3

(9.455)

702

9 Plastic Continua at Large Deformations

where Hv0 , nv0 and are the constants, then, performing the manipulations of Sect. 9.1.8, we can rewrite relations (9.450) as follows: .n/ C v

D

.n/ .n/ 1 .PT Hv C v /; I1 . C v / D 0:

(9.456)

For isotropic linear-elastic continua, Eqs. (9.452) with use of the additive relationship (9.3) take the form (see Exercise 9.1.9): .n/

.n/

.n/

.n/

T D J l1 I1 . C /E C 2J l2 . C C v /:

(9.457)

The equation system (9.456), (9.457) is the model An of an isotropic viscoplastic continuum of the differential type with the Huber–Mises potential.

9.9.4 General Model An of Viscoplastic Continua One can say that this is the general model An of viscoplastic continua, if the additive relation (9.3) is replaced by .n/

.n/

.n/

.n/

C D C e C C p C C v;

(9.458)

.n/

i.e. the deformation tensor C in this model is a sum of the three terms: elastic .n/

.n/

.n/

deformation C e , plastic deformation C p and viscous deformation C v . For elastic and plastic deformations we assume the same relations as the ones for pure plastic continua (see Sect. 9.1). For example, for the associated model An of viscoplastic continua, the relations (9.74) hold: .n/

Ce D

.n/ C p

r X D1 z1

X

D

.n/

' I.s/ T . T =/;

(9.459a)

.n/ .n/ .s/ ˛ J˛T . T ; C p /;

(9.459b)

˛D1 .n/

D .I.s/ . T =/; ; wpˇ /; D 1; : : : ; r; .n/ .n/

(9.459c)

p

fˇ D fˇ .J˛.s/ . T ; C p /; ; wˇ /; ' D

@ @I.s/

.n/

;

I.s/ T

D

@I.s/ . T =/ .n/

@ T =

;

˛

Dh

k X ˇ D1

(9.459d) ~Pˇ

@fˇ @J˛.s/

:

(9.459e)

9.9 Models of Viscoplastic Continua

703

And we assume that, just as for viscoplastic continua of the differential type, for viscous deformation Eqs. (9.448) and (9.449) still hold: .n/

C v D

z1 X

.s/

˛ J˛T ;

(9.460)

˛D1 .n/ .n/

.n/

.s/ D @J˛.s/ [email protected] T : ˛ D @fv [email protected]˛.s/ ; fv D fv .J˛.s/ ; /; J.s/ D J.s/ . T ; C v /; J˛T

(9.461)

9.9.5 Model An of Isotropic Viscoplastic Continua Using the results of Sects. 9.1.7 and 9.9.4, from (9.460) and (9.461) we get that the equation system (9.81)–(9.83), (9.451)–(9.454) holds for isotropic viscoelastic continua. Choosing the Huber–Mises model (9.96) and (9.455) for plastic and viscous potentials and applying the linear elasticity model for elastic deformation, from (9.460) and (9.461) we get the equation system .n/

.n/

.n/

.n/

.n/

T D J l1 I1 . C /E C 2J l2 . C C p C v /;

.n/

C p D

.n/ .n/ ~h P .n/ 1 .n/ . T I1 . T /E H C p /; I1 . C p / D 0; 2 s 3

f D .n/

C v D

1 .YH = s /2 1 D 0; 3

(9.462) (9.463)

H D H0 Yp2n0 ;

.n/ .n/ 1 .n/ 1 .n/ . T I1 . T / Hv C v /; I1 . C v / D 0; Hv D Hv0 Yv2nv0 ; 3

(9.464)

where the invariants YH , Yp and Yv are expressed by formulae (9.85), (9.87), and (9.455).

9.9.6 The Problem on Tension of a Beam of Viscoplastic Continuum of the Differential Type As an example, consider the problem on uniaxial tension of a beam of viscoplastic material of the differential type, which is described by the motion law (9.327). The .n/

.n/

deformation gradient F and the deformation tensors C and C e are determined by

704

9 Plastic Continua at Large Deformations .n/

formulae (9.328)–(9.332), (9.341). For viscous deformations C v we have formulae which are similar to Eqs. (9.334), and for plastic deformations – formulae (9.345): .n/

Cv D

3 .n/ X C v˛˛ eN ˛ ˝ eN ˛ :

(9.465)

˛D1 .n/

.n/

.n/

C ˛˛ D C e˛˛ C C v˛˛ ; ˛ D 1; 2; 3: .n/ C v33

(9.466)

.n/ 1 .n/ D C v22 D C v11 : 2

(9.467)

.n/

The stress tensors T are diagonal, for them relations (9.335)–(9.341) hold; in particular, the component 11 is expressed as follows: .n/

.n/

11 D k1nIII JE. C 11 C v11 /:

(9.468)

On substituting (9.465) into (9.456), we get the following equation for viscous de.n/

formation C v11 : 1 d .n/v C D dt 11

.n/ 2 IIIn k1 11 H C v11 ; 3

(9.469)

3 .n/ Hv D Hv0 Yp2nv0 ; Yp2 D . C v11 /2 : 2

(9.470)

Changing density is determined by the same relationship (9.355)

1 J D ı D k 1

2=.nIII/ .n/ 1 nIII v 1/ .n III/. / C 11 : 1 .k1 2

(9.471) .n/

On substituting (9.468) into (9.469) and taking the expression (9.330) for C 11 into .n/

account, we derive the following final equation for viscous deformation C v11 :

.n/ .n/ 3 .n/ 2 1 JE. .k1nIII 1/ C v11 / Hv0 . C v11 /2nv0 C v11 : 3 n III 2 (9.472) If the elongation function k1 .t/ of the beam is given, then, on solving the equation, the stress 11 can be found from Eq. (9.468). If, just as in the creep problem (see Sect. 7.4.4), the stress 11 .t/ is given, then 1 d .n/v C D dt 11

.n/

from Eq. (9.468) we can compute the dependence k1 D k1 . 11 ; C v11 /; and with the

9.9 Models of Viscoplastic Continua

705 .n/

help of Eq. (9.469) we determine the viscous deformation C v11 . A final expression .n/

for k1 .t/ is found by using the dependence k1 . 11 ; C v11 / once again. The considered model of an isotropic viscoplastic continuum of the differential type contains the five material constants: E, , , Hv0 , and nv0 . The elastic modulus is determined by the initial section of the diagram 11 .k1 / at given lengthening k1 .t/, when the influence of viscous deformations can be neglected. Just as for elastic continua, the Poisson ratio is determined by relation (9.341) also for small times, when viscous deformations are small. The remaining three constants: , Hv0 and nv0 can be found with the help of experimental creep curves at the given stress 11 .t/ varying as a step-function (7.146). Figure 9.28 shows the experimental creep curve jı1.ex/ .t/j for Ni-alloy at temperature 1100ıC and its approximation by Eqs. (9.468) and (9.469) for different models An . Constants Hv0 and nv0 in the computations have been chosen to be zero, and the viscous coefficient has been determined by minimization of the mean-square

a

b

c

d

Fig. 9.28 Creep curves for Ni-alloy at temperature 1;100ı C and different values of compressing stress o : dashed curves are experimental data, solid curves are computations by different models An of viscoplastic continua of the differential type: (a) – model AI , (b) – model AII , (c) – model AIV , and (d) – model AV

706

9 Plastic Continua at Large Deformations .ex/

distance between the computational jı1 .t/j D jk1 .t/1j and experimental jı1 .t/j creep curves at o D 20 MPa for several times (see Sect. 7.4.3). We have obtained the following values of the constants: E D 2GPa for all models An ; D 40 GPa s for n D I; D 42 GPa s D 55 GPa s

for n D II; for n D IV;

D 60 GPa s

for n D V:

Figure 9.28 also exhibits computational and experimental creep curves at different values of stress o . The model AII gives the best approximation quality (Fig. 9.28b).

Exercises for 9.9 9.9.1. Show that the model of isotropic viscoplastic continuum (9.455), (9.456), when H0 D 0, can be written as the differential equation .n/ 2l2 .n/ 2J l2 JP I1 . T //E C 2J l2 C T D J.l1 I1 . C / C 3 J

.n/

.n/

!

.n/

T;

JP D r v: J

References

1. Batra, R.C.: Elements of Continuum Mechanics. AIAA Education Series, Reston (2005) 2. Basar, Y., Weichert, D.: Nonlinear Continuum Mechanics of Solids. Springer, Berlin (2000) 3. Bonet, J., Wood, R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge (1997) 4. Borg, S.F.: Matrix Tensor Methods in Continuum Mechanics. World Scientific, London (1990) 5. Bowen, R.M.: Introduction to Continuum Mechanics for Engineers. Plenum Press, New York (1989) 6. Calcote, L.R.: Introduction to Continuum Mechanics. D. Van Nostrand, Princeton (1968) 7. Chadwick, P.: Continuum Mechanics: Concise Theory and Problems. Dover Publications (1999) 8. Chernykh, K.F.: An Introduction to Modern Anisotropic Elasticity, Moscow, Nauka, 1988 (in Russian). Begell Publishing House, New York (1998) 9. Coleman, B.D.: Thermodynamics of materials with memory. Arch. Ration. Mech. Anal. 17(1), 1–46 (1964) 10. Cotter, B.A., Rivlin, R.S.: Tensors associated with time-dependent stress. Quart. Appl. Math. 13(2), 177–182 (1955) 11. Dimitrienko, Yu.I.: Novel viscoelastic models for elastomers under finite strains. Eur. J. Mech. Solid. 21(2), 133–150 (2002) 12. Dimitrienko, Yu.I.: Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers, Dordrecht-Boston-London (2002) 13. Ericksen, J.L., Rivlin, R.S.: Large elastic deformations of homogeneous anisotropic elastic materials. J. Ration. Mech. Anal. 3(3), 281-301 (1954) 14. Eringen, A.C.: Nonlinear Theory of Continuous Media, McGraw–Hill, New York (1962) 15. Eringen, A.C.: Mechanics of Continuum. Wiley, New York (1967) 16. Ferrarese, G., Bini, D.: Introduction to Relativistic Continuum Mechanics (Lecture Notes in Physics). Springer, Berlin (2007) 17. Fung, Y.C.: A First Course in Continuum Mechanics. Prentice Hall, New Jersey (1977) 18. Gladwell, G.M.L.: Contact Problems in the Classical Theory of Elasticity, Sijthoff and Noordhoff, 1980. Chinese Edition published (1991) 19. Green, A.E., Zerna, W.: Theoretical Elasticity. Oxford University Press (1954) 20. Godunov, S.K., Romenskii, E.I.: Elements of Continuum Mechanics and Conservation Laws. Springer, Berlin (2003) 21. Goldstein, R.V., Entov, V.M.: Qualitative Methods in Continuum Mechanics. Longman, London (1994) 22. Gonzalez, O., Stuart, A.M.: A First Course in Continuum Mechanics (Cambridge Texts in Applied Mathematics). Cambridge University Press (2007) 23. Gurtin, M.E.: Introduction to Continuum Mechanics (Mathematics in Science and Engineering). Academic Press, New York (1981) 24. Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin (1999)

707

708

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Basic Notation

A

Left Almansi deformation tensor;

.n/

A aOl , aJ , aCR , aD , ad , aV , aU , and aS B C

.n/

C

Energetic deformation tensors;

.n/

CG

.n/

Quasienergetic deformation tensors; Co-rotational derivatives of Oldroyd, Jaumann, Cotter–Rivlin, left and right mixed, left and right in eigenbasis, and spin; The third energetic deformation tensor; Right Cauchy–Green deformation tensor;

Generalized energetic deformation tensors; .n/

C e and C p b ci

D

Tensors of elastic and plastic deformations; Principal basis of anisotropy (orthonormal) of a solid b in undistorted configuration K; The velocity of a singular surface in a reference configuration; Deformation rates tensor;

D and D

The normal speed of propagation of a singular surface

E E

in configurations K and K; Unit (metric) tensor; Total energy of a body;

ı

c ı

4

.n/

E e eN i F Fe and Fp f G

.n/

G

.n/

GG

ı

Tensors of energetic equivalence; Specific internal energy of a body; Basis of Cartesian coordinate system; Deformation gradient; Gradients of elastic and plastic deformations; Specific mass force vector; Right Cauchy–Green deformation measure; Energetic deformation measures; Generalized energetic deformation measures; 709

710

Basic Notation

g

Left Almansi deformation measure; ı

ı

Metric matrices in configurations K and K; Entropy of a body;

g ij and gij H ı Q and H H

Left and right Hencky logarithmic deformation tensors; Tensor of H -transformation from one reference ı configuration K to another reference configuration K; Momentum vector of a body; Principal invariants of second-order tensor C; Invariants of second-order tensor relative to an

H I I1 .C/, I2 .C/ and I3 .C/ .s/ I1 ./ .n/

.n/

.n/

i A, i B , i J ı J D = K 4 K.t/

C,

.n/

i

D,

ı

K and K L 4 M ı

n and n O P P˛.C/ , ˛ D 1; : : : ; n, N ı p and p p Q QN m and QN † QN Qˇ 4

.n/

Q q qm and q† q 4 R.t/ ı

.n/

i

G,

ı

orthogonal group G s ; i

Different forms of enthalpy; Left Cauchy–Green deformation tensor; Ratio of densities; Kinetic energy of a body; Tensor of relaxation cores; Actual and reference configurations; Velocity gradient; Quasilinear tensor of elasticity;

ı

Normal vectors in configurations K and K; Rotation tensor accompanying the deformation; Piola–Kirchhoff stress tensor; Orthoprojectors of symmetric tensor C; Eigenvectors of stretch tensors V and U; Pressure; Rate of heating; Entropy production by external mass and surface sources; Entropy production by internal sources; Thermodynamic fluxes; Tensors of quasienergetic equivalence; Heat flux vector; Heat influxes due to mass and surface sources; Specific internal entropy production; Tensor of relaxation functions; ı

ri and ri

Vectors of local bases in configurations K and K;

S

Quasienergetic stress tensors;

S SG T

Rotation tensor of stresses; Generalized rotation tensor of stresses; Cauchy stress tensor;

.n/ ı

Basic Notation

711

Th and TH

General notation for co-rotational derivatives of a tensor in covariant and contravariant moving bases hi and hi ;

.n/

T

Energetic stress tensors;

.n/

TG tn U U u V v W Wm and W† W.i / w ı x and x

Generalized energetic stress tensors; Stress vector; Right stretch tensor; Internal energy of a body; Displacement vector; Left stretch tensor; Velocity; Vorticity tensor; Powers of external mass and surface forces; Power of internal surface forces; dissipation function; Radius-vectors of a material point in configurations

X i and x i Xˇ Y˛ .T/

K and K; Lagrangian and Eulerian coordinates; Thermodynamic forces; Spectral invariants of a symmetric second-order tensor;

ı

ı

ı

ijm and m ij ı˛ "ij .n/

.n/

.n/

Christoffel symbols in configurations K and K; Relative elongation; Covariant components of the deformation tensor; .n/

.n/

A, B , C , D , G , ƒ ˛

Different forms of the Gibbs free energy; Specific entropy; Temperature; Right Almansi deformation tensor; Eigenvalues of stretch tensors U and V;

and

Heat conductivity tensors in configurations K and K; Thermodynamic potential;

ı

ı

and ˛ U and V ! ı

r and r H and N

ı

ı

Density in configurations K and K; Unit tangent vectors to a surface S ; Helmholtz free energy; Spin of rotation accompanying the deformation; Spins of the right and left stretch tensors; Vorticity vector; ı

Nabla-operators in configurations K and K; The beginning and the end of a proof of Theorems, respectively; The end of Examples and Remarks.

Index

A Acceleration, 98 Coriolis’s, 327, 331, 338 total, 314, 327 translational, 327, 331, 338 Area of a surface element, 31, 104, 106

B Basis, 2 dyadic, 11, 12, 43, 58, 78, 81, 83, 108, 151, 224, 333 functional, 241, 243, 258, 260, 261, 293, 402, 469, 473, 476, 477, 508, 517–519, 606, 610, 614, 635, 637, 638 local, 9 reciprocal, 10, 79, 80, 100, 303, 448 physical, 103 orthonormal, 11, 20 polyadic, 12, 53 principal (of anisotropy), 226, 228, 238, 239, 336 Body, 1–4, 96, 97, 122, 123, 128, 130–136, 140, 222, 226, 315, 316, 336, 405, 406, 408, 409, 414, 421, 438, 439, 445, 447, 472, 561, 565, 586, 667 Boundary conditions, 347, 372, 381, 399–409, 411, 413, 416, 424, 430–432, 441–442, 444, 451–454, 456, 493, 559, 563, 564, 581, 656–658, 660, 667, 679, 685, 688, 693

C Coeffcient heat conductivity, 345 heat transfer, 582

Lam´e’s, 11, 20 surface tension, 370 viscous, 470, 471, 477, 479–480, 488, 490, 491, 705 Components contravariant, 20, 24, 28, 34, 593 covariant, 20, 24, 28, 145 physical, 11, 20, 28–30, 36, 103–106, 450 Condition boundary, 347, 372, 381, 399–409, 411, 413, 416, 424, 430–432, 441–442, 444, 451–454, 456, 493, 559, 563, 564, 581, 656–658, 660, 667, 679, 685, 688, 693 consistency, 3, 6, 472, 496 deformation compatibility, 141–144, 146, 148–150, 152 of symmetry, 402, 405, 424, 447, 451 Configuration actual, 6, 7, 9, 10, 15, 26, 27, 30, 45, 46, 58, 62, 78, 89, 90, 94, 102, 141, 142, 190, 214, 218, 219, 231, 287, 300, 319, 324, 331, 348, 358, 363, 367, 374, 401, 402, 404, 412, 472, 481, 490, 556, 592, 618, 639, 640, 658 reference, 6, 7, 9, 10, 15, 18, 25–27, 94, 102, 103, 105, 108, 119, 120, 142, 148, 213–215, 219–224, 226, 228, 229, 236, 249, 254, 277, 287, 292, 300, 302, 336, 338, 341–343, 348, 358, 362, 367, 368, 396, 402, 407, 408, 413, 421, 452, 465, 485, 491, 506, 592, 603, 632, 639, 651 undistorted, 222, 224, 226, 228, 229, 234, 236, 238, 254, 292, 465, 506, 603 unloaded, 592, 632

713

714 Constitutive equations, 161–346, 349, 377–380, 382–384, 386, 388–393, 395–398, 402, 405–409, 411, 418, 420, 421, 427, 431, 449, 455, 461–463, 465, 467–473, 480, 481, 483, 489–492, 498, 505, 506, 510, 513–519, 522, 524, 525, 528–534, 538–550, 552–565, 567–569, 581, 591, 594–598, 601, 603–607, 609, 610, 613, 615–622, 627–630, 632, 634–638, 640–656, 658, 668, 699 Continuum, 1 of the differential type, 209, 461–497, 518, 592, 595, 603, 641, 643, 699–706 elastic, 236, 244–276, 287, 292–294, 296–299, 377–460, 472, 519, 528, 530, 531, 533, 534, 550, 552, 568, 620, 671, 673, 674, 702, 705 fluid, 9, 133, 221, 291, 319–321, 554 homogeneous, 209 ideal, 209–213, 236–287, 289, 290, 319–322, 340, 414, 464, 467, 505, 524, 592, 603 incompressible, 287–300, 322, 411, 417–421, 431, 437, 443–446, 454–460, 471–472, 549, 552, 556, 675 inhomogeneous, 209 of the integral type, 209, 497–516 with memory, 497 fading, 498 nondissipative, 211, 212 nonpolar, 112–114, 121, 123, 155, 158, 161, 163, 186 plastic, 591 ideally, 600, 602 strengthening, 600, 610, 635, 650, 679, 683, 696, 698 polar, 112–113, 121–123 simple, 516, 547, 564, 566, 581 solid, 222 anisotropic, 226, 236, 237, 614 isotropic, 237, 261, 271, 272, 346 viscoelastic, 497 with difference cores, 511–513, 519, 522, 560 stable, 510–513 thermorheologically simple, 514–516, 566 viscoplastic, 699 Coordinates Eulerian, 6–9, 16, 74 Lagrangian, 5–9, 17, 18, 49, 74, 79, 80, 157, 301, 328, 369, 391, 447, 449

Index material, 3, 5–7, 17, 29, 49, 73, 74, 135, 143, 209, 227, 327, 447 spatial, 6–9, 16, 74 curvilinear, 16–23, 329 of a vector, 20, 325 Coordinate system Cartesian rectangular, 2 inertial, 96 moving, 16, 17, 79, 81, 97, 329, 332, 334, 335, 338 Core of creep, 542–544, 548 of a functional, 507, 510, 515, 540 of relaxation, 529, 534, 538, 540, 543, 544, 548, 553, 580, 588 Couples of tensors energetic, 163 functional, 283–287 principal, 164, 305 quasienergetic, 176 functional, 285 principal, 177, 180, 184, 305 Curl of a tensor, 14, 154 Cycle Carnot’s, 129, 132–136, 138, 139 elementary, 136 generalized, 134–136, 138 symmetric, 585 thermodynamic, 133, 134, 136–138, 140

D Decomposition polar, 36–49, 71, 166–169, 177, 185, 225, 230, 288, 304, 633, 640, 641 spectral, 535, 537, 538, 543 Deformation measure Almansi left and right, 24, 25, 42, 87, 175, 183, 188, 218, 230, 303, 593 Cauchy-Green left and right, 24, 25, 42, 87, 175, 183, 188, 230, 303, 593 energetic, 173 generalized, 187, 262, 264, 296, 384, 414 Hencky left and right, 43, 76, 268, 307, 308 logarithmic, 43, 267, 269, 270 quasienergetic, 182–183, 185, 189, 192, 196, 230, 262, 280, 288, 304 Density, 90, 91, 133, 134, 185–186, 193, 197, 198, 206, 207, 211, 219, 245, 277, 281, 287, 292, 293, 302–303, 313, 366–367, 383, 389, 391, 394, 477, 480, 556, 561, 563, 660, 667, 670–671, 680, 704

Index Derivative contravariant, 13, 20 convective, 51 co-rotational mixed left and right, 81, 86, 309 Cotter-Rivlin, 80, 81, 87, 88, 232, 308, 309 covariant, 13–14, 20, 147–149, 153 in eigenbasis left, 82–83, 88, 232, 310, 646 right, 81–82, 86, 233, 310, 322, 490 Fr´echet’s, 502, 503, 510, 595 Jaumann’s, 83, 86, 88, 232, 311 material, 51 Oldroyd’s, 79–81, 87, 88, 308 partial with respect to time (local), 50, 51, 54, 74, 77, 355, 599, 642, 651 spin, 84–85, 232, 233, 311 total with respect to time, 50 of a n-th order tensor, 51 Description of a continuum Eulerian, 5–8, 155–156 Lagrangian, 5–8, 50, 156–157 material, 5–23 spatial, 5–23, 155–157, 381, 399–402, 560 Deviator, 471, 492, 539, 552, 607–609, 619, 644, 661 Diagram of deforming 427 in simple shear, 442 Differential of a tensor, 53–54 total, 153, 154, 156, 289, 373, 375, 595 of a vector, 53 Dissipative heating, 561, 579–589 Divergence of a tensor, 14, 151, 324, 334, 416

E Effect Bauschinger’s, 665 Poisson’s, 425, 431 Poynting’s, 442 Efficiency, 124, 128–132, 136–139 Energy free Gibbs’s, 199–200, 204, 286, 591, 595, 597, 627, 628, 641, 648 Helmholtz’s, 198–199, 461, 486 internal, 115, 120, 130, 133, 316, 378, 379, 622, 693, 694 kinetic, 115, 118–123, 331, 332 potential, 414, 417 total, 115, 118, 121, 378 Enthalpy, 201–202, 286

715 Entropy, 125, 126, 128, 130, 133, 155, 316, 318, 377, 380, 381, 391, 393, 403, 409, 410, 514, 522, 525, 559, 561, 563, 656, 657 Entropy production, 124, 125, 127, 128, 136, 140, 316, 339, 352, 369 Equation balance angular momentum, 112–114, 158 energy, 117–123, 155, 317, 332, 377, 380, 388, 403, 407, 410, 559, 656, 657 entropy, 126, 155, 318, 377, 380, 391, 393, 403, 410, 559, 561, 563, 656, 657 momentum, 98, 107–108, 118, 155, 158 compatibility, 141 dynamic, 149–152, 155, 162, 318, 332, 377, 381, 386, 396 static, 142, 148–149 continuity in Eulerian variables, 92–93, 330, 377 in Lagrangian variables, 90–91, 157, 288, 384 heat conduction, 381, 388, 393, 396, 407, 560, 561, 563, 565, 567, 579, 581–582, 584–585, 587 heat influx, 118–121, 123, 131, 318, 332, 380 kinematic, 73, 155, 156, 318, 335, 377, 381, 396, 564 of a singular surface, 372–375 variational, 416, 417, 419

F Field of possible pressures, 417, 419 scalar, 9, 50, 78, 92, 417 tensor, 9 stationary, 53 vector, 9 kinematically admissible, 413 real, 413 Fluid, 221 compressible, 288, 322 linear-viscous, 480, 481 Newtonian, 480 viscous, 480, 481 Force of body inertia, 316 external, 96–98, 115, 118, 124, 130, 316, 412–414, 424, 428

716 of interaction of bodies, 96 internal, 97 inertia, 98 mass, 97, 98, 115, 331, 338, 402, 414, 423, 493, 660 surface, 97, 98, 104, 106, 107, 115, 116, 118, 123, 370, 414 thermodynamic, 340, 464, 630 Formula Coriolis’s, 327–329, 331, 338 Euler’s, 326–327, 332 Gauss-Ostrogradskii, 94–95, 100, 117, 123, 126, 416, 581 Function Dirac’s, 500 dissipation, 126–128, 140, 198, 211, 339, 461, 463, 464, 470, 480, 482, 485–487, 491, 505, 513, 516, 517, 522, 524, 525, 529, 534, 539, 556, 559, 561, 563, 579, 582–584, 587, 588, 596, 629, 643 of equilibrium stresses, 462, 463 Heaviside’s, 493, 528, 535, 576, 601, 683, 692 indifferent relative to a symmetry group, 237–240, 243, 244, 256, 257, 467, 507–510, 514 of memory, 498, 503 pseudopotential, 465 quasiperiodic, 580, 581, 583 quasipotential, 211, 244, 247, 463 relaxation, 529, 533–535, 537–541, 544, 547, 548, 551, 571 rotary-indifferent, 256, 257 scalar isotropic, 243 orthotropic, 243 transversely isotropic, 243 of temperature-time shift, 515 tensor, AI -unimodular, 474–476, 482 of viscous stresses, 462–468, 475, 478, 488 Functional, 497 continuous, 209, 499–503, 509, 555, 592 Fr´echet-differentiable, 501–504, 516, 592 linear, 500, 501, 507, 509, 524, 530, 533, 550 scalar n-fold, 507 quadratic, 507

Index G Gas, 128, 129, 133, 288, 368, 369, 451, 692, 696 Gradient deformation, 15 elastic, 382, 593, 632 plastic, 593, 633, 639 surface, 373, 374, 700 of a vector, 13, 27 velocity, 56–58, 65–73, 81, 85–88, 163, 188–196, 330, 492 H Heat absorbed by a body, 130 explosion, 561, 586 pseudoexplosion, 586 released by a body, 130 Heating dissipative, 561, 579–589 rate, 115, 120 Heat machine, 128–132, 140 I Indeterminate Lagrange multiplier, 417 Inequality Clausius-Duhem, 198 Clausius’s, 125, 128, 138, 139 dissipation, 127, 340, 465, 539 Fourier’s, 127, 340, 465 Planck’s, 125–127, 198, 339, 340, 464 Influx entropy, 130 heat, 118–121, 123, 124, 131, 132, 318, 332, 380 Intensity of a tensor, 608 Invariant, 240 cubic, 241, 249, 250, 523, 611, 616 linear, 241, 250–252, 523, 524, 537, 538, 701 principal, 175–176, 185–186, 192, 193, 242, 243, 245, 260, 261, 266, 273, 274, 277, 280, 283–286, 293, 324, 327, 471, 473, 477, 480, 608, 618 quadratic, 241, 249–251, 471, 607, 620, 701 simultaneous, 469, 470, 476–478, 490, 491, 508–511, 513, 517, 518, 520, 522, 523, 549, 603–608, 611, 612, 614, 616, 617, 619, 634, 635, 637, 638, 650, 651, 656, 663, 701 spectral, 537 linear, 537, 541 quadratic, 537

Index J Jump of a function across a singular surface, 351, 356, 358, 360, 362

L Lagrangian, 5–9, 17, 18, 49, 50, 74, 79, 80, 90–91, 108, 113, 119–121, 123, 128–130, 149–150, 156–157, 288, 301, 328, 369, 384, 391, 392, 414, 415, 419, 447, 449 Law angular momentum balance, 109–114, 122, 412 of changing plastic deformations, 598 conservation of mass, 89–95 Fourier’s, 339–346, 378, 464, 480, 598 gradient, 600, 601, 631, 650 momentum balance, 95–108, 114, 316, 412 motion of a continuum, 5, 6, 24, 74, 114, 121, 127, 213, 300, 301, 324, 367, 410, 414 Stokes’s, 464 thermodynamic first, 114–124, 317 second, 124–140, 318, 332 Length of a vector, 2, 46 Loading active, 600, 675, 685, 689 fixed, 401 neutral, 600 passive, 600 plastic, 600, 601, 609, 631, 663, 672, 674, 675, 694 tracking, 401

M Mass, 24–95, 97, 98, 109, 110, 114–116, 121, 124, 132, 316, 317, 331, 338, 363–366, 399, 402, 403, 414, 423, 493, 660, 693 Material body, 1, 3 point, 1–7, 9, 14–17, 29, 32, 45, 49, 51, 54, 57, 60, 73–75, 89–92, 95, 104–106, 124, 128, 133–135, 141–143, 149, 205, 208, 209, 227, 313, 327, 347–350, 353, 357, 358, 364–368, 370, 375, 400, 401, 406, 411, 447, 592

717 Matrix Jacobian, 10 inverse, 10 metric, 2, 10, 12, 19, 21, 28, 30, 33–36, 49, 145, 146, 148, 153, 191, 192, 214, 448, 593 Method Saint-Venant’s, 451 semi-inverse, 421, 446, 454 Model An , 206, 208–212, 219, 236–238, 243–245, 248–252, 254, 256, 262, 263, 265–267, 270–272, 274–277, 279, 282, 285–287, 289–292, 295–297, 299, 319–320, 322, 336, 339, 341, 342, 378–380, 385, 392, 412, 414, 422, 423, 425–430, 436–439, 442–443, 449–450, 454, 461–484, 497, 505–540, 546–558, 560–571, 573–578, 591–623, 652–659, 662, 671, 673–675, 689–691, 699–703, 705, 706 Bartenev-Hazanovich, 298, 434 Bn , 207, 208, 211, 219, 253–254, 262, 263, 265, 266, 270–272, 274, 275, 279, 282, 285–287, 290, 294, 295, 297–299, 319–320, 322, 337, 339, 341, 378, 379, 385, 414, 420, 431–436, 438, 443, 445, 454, 461–482, 492–497, 549, 552–558, 562, 566, 568–571, 573–577, 581, 588, 589, 623–640 Chernykh’s, 298 Cn , 207, 208, 212, 219, 254–263, 265–267, 270–272, 274, 275, 280–282, 285, 287, 290, 292, 294, 295, 320–323, 337, 339, 341, 342, 378, 379, 385, 414, 482–491, 497, 640–652 Dn , 207, 208, 212, 213, 219, 254–263, 265, 266, 270–272, 274, 275, 282, 285, 291, 295, 298–299, 320–323, 339, 341, 342, 378, 379, 385, 414, 420, 482–491, 497, 640–652 Drucker’s, 600 Duhamel-Neumann, 514, 561, 563 Huber-Mises, 607–610, 612, 616–619, 644, 660, 662, 666, 701, 703 John’s, 251, 452 linear, 250 isotropic, 251, 296, 297 mechanically determinate, 534, 540, 551, 553, 564, 566 Mooney’s, 298 Murnaghan’s, 251

718 of plasticity associated, 599, 606, 610 two-potential, 612 quasilinear, 249–252, 286, 380, 389, 390, 396, 397, 620 semilinear, 251 simplest, 471, 473, 538, 548, 553, 557, 568, 581, 699 Treloar’s, 298, 433, 434 of a viscous fluid, 480, 481 Modulus of volumetric compression, 662 Motion rigid, 300–313. 317, 322, 324, 326, 343, 472, 481, 490, 556, 618, 639, 640, 652 stationary (steady), 74 Moving basis, 17, 18, 51, 52, 65, 78, 80–84, 325, 326, 328, 329, 335 coordinate system, 81, 329, 332, 334, 335, 338 volume, 91–92, 94, 118, 123, 157, 360–362, 414

N Nabla-operator, 13, 14, 19, 52, 111, 122, 303, 329–331 Natural state, 248 unstressed, 249, 472 Neighborhood of a point, 1, 14, 45–49, 60–62 of a singular surface, 355

O Operator, 153, 154, 206–210, 214, 312, 341, 461, 465, 497, 500, 502, 531–534, 591 Oriented surface element, 30–32, 102, 359 Orthoprojector, 536, 537, 539

P Parameter Odkwist’s, 599 of quadratic elasticity, 247 strengthening, 607, 610, 668 Taylor’s, 598, 606, 609, 611, 615, 630, 636, 638, 639 Poisson ratio, 425, 428, 429, 442, 443, 453, 690, 705

Index Potential plastic, 599, 601, 602, 607, 609, 612, 616, 619, 631, 634, 636, 637, 650, 701 viscous, 700 Power of forces, 115, 118, 121, 123, 162 of stresses, 162, 163, 176, 186–188, 197, 204, 268, 271, 298, 380, 415, 487, 490, 623–627, 646–648 Prehistory, 497, 498, 500, 509, 561, 592 Pressure, 276 hydrostatic, 291, 294, 456 Principal axis of anisotropy, 226 of transverse isotropy, 228 Principal thermodynamic identity (PTI), 196–208, 210–213, 288–290, 298, 339, 377, 393, 462, 483, 484, 505, 546, 549, 591, 594, 595, 623, 627, 628, 641–643, 648, 649 Principle of equipresence, 161, 208, 464, 485, 487, 628 of local action, 161, 208–209 of material indifference, 161, 248, 280, 300–324, 343–345, 472, 481, 483, 556, 618, 639–641, 652 of material symmetry, 161, 213–221, 226, 236–287, 291–292, 300, 341–345, 465–466, 473, 485–487, 491, 506–507, 603–605, 632–634, 651–652 of objectivity, 315 Onsager’s, 161, 339–346, 463–465, 474, 478, 484, 488, 491, 597–598, 601, 602, 604, 607, 611, 629–630, 636–638, 644, 649, 701 of thermodynamically consistent determinism, 161, 205–209, 339, 340 variational Lagrange’s, 415, 419 Problem coupled strongly, 407, 410, 561, 563 weakly, 407, 409 Lam´e, 446–460 Process adiabatic, 132, 370 in the broad sense, 132 locally, 132, 134 in the restricted sense, 132 with a constant extension, 503 irreversible, 127, 140

Index isothermal, 133, 410 locally, 132, 134, 135 quasistatic, 388–390, 396–401, 403, 404, 411, 412, 657 reversible, 140 static, 388, 389, 396, 503, 504 uniform thermomechanical, 135, 136, 138 Pseudoinvariant, 256, 258, 259, 261 R Radius-vector, 2, 3, 5, 14, 15, 29, 30, 44–46, 49, 56, 58, 60, 62, 63, 90, 95, 141–143, 150, 153, 214, 215, 300, 301, 315, 325, 326, 350, 353, 357, 367–369, 372, 374, 401, 447, 592 Rate of heating, 115, 120, 135 Relative elongation, 29, 30, 36, 46, 59, 578 Representation Boltzmann’s, 525–528, 531 spectral, 535–539, 544, 547, 548 Volterra’s, 500, 525, 526 Residual deformation of creep, 494 S Shock adiabatic curve, 695–699 Simple shear, 7, 8, 21, 47, 57, 194, 438–446 Space of elementary geometry, 1 Euclidean, 2 point, 2 metric, 1, 2 tensor functional, 498–499 Specific entropy, 125, 133, 316, 381, 393, 514, 525, 563 entropy production, 125, 127, 140, 316, 339, 597, 629, 644 internal energy, 115, 133, 316, 378, 622 mass force, 97, 98, 331 surface force, 97, 98 total energy, 118, 378 Spectrum of relaxation times, 539 of relaxation values, 539 of viscous stresses, 540 Spin, 63–65, 71, 72, 82, 84–86, 109, 122, 232–234, 309–311, 327, 626, 627, 633, 646 Streamline, 73–75 Stress normal, 104, 106, 442, 445 relaxation, 572, 573 tangential, 104–106, 424, 431, 441, 459, 460

719 Surface singular, 347 coherent, 350, 355–361 completely incoherent, 369 of contact, 349 ideal, 370–371 homothermal, 369–370, 399, 402, 404 incoherent, 350 nondissipative, 369–370 of phase transformation, 349, 369, 370 semicoherent, 369 of a shock wave, 369 of a strong discontinuity, 349, 350, 362 of a weak discontinuity, 349 stream, 75 vortex, 75 without singular displacements, 367 Symbols Christoffel of the first kind, 145 of the second kind, 145 Levi-Civita, 2 Symmetry group, 219 continuous, 226, 509 isomeric, 222–225 point, 226

T Temperature absolute, 124 Tensor of angular rate of rotation, 61–65 coaxial, 187, 188, 196, 203, 204, 267, 269, 271, 273, 643, 645, 650 of creep cores, 542, 544, 548 deformation Almansi left and right, 24, 25, 40, 42, 87, 165, 175, 183, 188, 230, 303, 593 Cauchy-Green left and right, 24, 25, 40, 42, 87, 165, 175, 183, 188, 230, 303, 593 generalized energetic, 187, 414 Hencky left and right, 43, 76, 268, 307, 308 logarithmic, 43, 267, 269, 270 principal energetic, 175 quasienergetic, 178, 180, 185–186, 189, 192, 196, 219, 230, 233, 262, 304

720 deformation rate, 56–59, 61, 62, 66, 69, 83, 85–88, 152–155, 163, 306, 475, 476, 653 elasticity quadratic, 247 quasilinear, 247, 379, 620 of energetic equivalence, 170, 263, 264, 392, 444, 656 heat conductivity, 340–343, 345, 407, 599 heat deformation, 514 heat expansion, 514 H -indifferent, 215 absolutely, 229, 231–233, 277, 279 relative to a group, 232, 233, 237, 239 H -invariant, 215–218, 220, 229–235, 239, 255, 262, 274, 280, 282, 336, 338, 486, 491, 632, 651, 652 H -pseudoindifferent, 218 indifferent relative to a symmetry group, 238–240, 243, 256, 345, 467, 468, 474, 478, 507–510, 536, 537 isomeric, 224 Levi-Civita, 2 of moment stresses, 110–113, 123 producing of a group, 239–241, 246, 286, 486, 524, 536 of quasienergetic equivalence, 181 of relaxation cores, 534, 540, 543, 548 of relaxation functions, 533–535, 537, 544, 547, 548 Riemann-Christoffel, 145–147 R-indifferent, 301–312, 314, 320, 321, 323, 324, 343, 481, 483, 490, 491, 639–641, 652 R-invariant, 301–310, 312, 320, 322, 324, 343, 472, 481, 556, 618, 639, 640, 652 rotation accompanying deformation, 39, 43, 65, 191, 225, 230, 304, 379 second-order, 12, 14, 23, 37, 56, 78, 81, 88, 101, 102, 110, 111, 154, 175, 177, 188, 210, 216, 238–243, 273, 275, 283, 302, 324, 340, 462, 464, 468, 535–537, 540, 591, 597, 602 spherical, 70, 234, 248, 276, 282, 345, 400, 619, 659–662 of strengthening, 596 stress Cauchy, 101–104, 112, 113, 190, 218, 248, 263, 281, 291, 304, 392, 400, 406, 423, 436, 439–441, 444, 446, 450, 455, 475, 478, 493, 548, 556 generalized energetic, 262, 264, 296, 384

Index Piola-Kirchhoff, 102–103, 105, 108, 113, 119, 162, 197, 305, 392, 404, 450, 680 principal energetic, 164, 170, 175, 305 quasienergetic, 176, 178, 180, 181, 191, 195, 219, 231, 248, 262, 281, 305, 423, 482, 483, 642 reduced, 596 rotation, 231, 261, 626, 646 stretch left and right, 39, 46, 81–83, 164, 168, 173, 175, 176, 180–183, 230, 232–234, 304, 309, 310, 322, 645, 646 unit (metric), 12, 43, 77, 86, 87, 174, 184, 217, 229, 240, 247, 248, 276, 303, 423, 608 viscosity, 464, 465, 468, 488–489 vorticity, 56–59, 61, 62, 65, 155, 163, 190, 306 Tensor basis, 11, 13, 24, 42, 102, 190–192, 245, 261, 292–295, 467–470, 478, 488, 603, 604, 606 Tensor field, 9 stationary, 53 varying, 51 Tensor law of transformation, 12, 14 Theorem Cauchy-Helmholtz, 57, 58 Cauchy’s, 98–101 Noll’s, 222, 224 on the polar decomposition, 36–40 Stone-Weierstrass, 507–508 Truesdell’s, 136–140 Thermodynamic cycle, 133, 134, 136–138, 140 flux, 339, 340, 464, 597 potential, 202, 203, 213, 522 Time, 3 absolute, 3 fast, 580, 584 reduced, 514–516, 547, 583 slow, 580, 581, 583, 584 Trajectory of a point, 73 Truesdell’s estimate for the efficiency, 131 Tube stream, 75–77 vortex, 75–77 Types of continua, 161, 209–210, 655

U Universe, 1 Unloading, 494, 579, 600, 663, 665, 672–675, 678, 699, 700

Index V Variables active, 205, 206, 208, 210, 212, 214, 220, 312, 339, 341, 461, 462, 484, 497, 591, 648 reactive, 205, 206, 208, 212, 214, 220, 312, 341, 462, 483, 497, 505, 591, 629, 630, 644, 648 Variation of a functional, 414 of a vector field, 413 Vector angular momentum, 109, 110 displacement, 25–28, 34, 36, 135, 141, 149, 150, 153, 367–369, 381, 394, 401, 407, 564, 565, 567 external forces, 96, 316, 426, 428 force, 96, 109, 304, 316 heat flux, 116–117, 119, 122, 317, 331, 339, 343, 344 local, 9–11, 16, 18, 21, 23, 30, 36, 52, 54, 79, 90, 101, 143, 144, 216, 301–303, 329, 448, 592, 632 of mass moments, 109, 110 momentum, 95, 315 stress, 98 Piola-Kirchhoff, 103, 404 of surface moments, 109

721 velocity, 49 of a singular surface, 353, 357 vorticity, 57–59, 61–64, 75, 84, 155, 326, 635, 638 Velocity, 49 of a singular surface, 353 normal, 354 relative, 56, 326, 328, 335 total, 327 translational, 57, 327 Vortex line, 73–75

W Wave plane, 679–699 Riemann, 692 shock, 99, 349, 369, 370, 693–697 Work, 130 done by external forces, 130, 414 elementary of stresses, 162

Y Yield point, 609, 614, 674, 676 strength, 609, 614, 618, 620, 635, 663, 665, 666, 675, 678

SOLID MECHANICS AND ITS APPLICATIONS Volume 174

Series Editors:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For other titles published in this series, go to www.springer.com/series/6557

Yuriy I. Dimitrienko

Nonlinear Continuum Mechanics and Large Inelastic Deformations

123

Prof. Yuriy I. Dimitrienko Bauman Moscow State Technical University 2nd Baumanskaya St. 5 105005 Moscow Russia [email protected]

ISSN 0925-0042 ISBN 978-94-007-0033-8 e-ISBN 978-94-007-0034-5 DOI 10.1007/978-94-007-0034-5 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010938719 c Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Nonlinear continuum mechanics is the kernel of the general course ‘Continuum Mechanics’, which includes kinematics of continua, balance laws, general nonlinear theory of constitutive equations, relations at singular surfaces. Moreover, in the course of nonlinear continuum mechanics one also considers the theory of solids at finite (arbitrary) deformations. This arbitrariness of deformations makes the equations describing the behavior of continua extremely complex – nonlinear (so that sometimes the term ‘strongly nonlinear’ is used), as the relationships contained in them cannot always be expressed in an explicit analytical way. If we drop the condition of the arbitrariness of continuum deformations and consider only infinitesimal deformations – usually the deformations till 1%, then the situation changes: the equations of continuum mechanics can be linearized. Hence for solving the applied problems one can exploit the wide range of analytical and numerical methods. However, many practical tasks demand the analysis not of the infinitesimal, but just the arbitrary (large) deformations of bodies, for example, such tasks include the rubber structure elements design (shock absorbers, gaskets, tires) for which the ultimate deformations can reach 100% and even higher. The various tasks of metal working under high pressure also belong to that class of problems, where large plastic deformations play a significant role, as well as the dynamical problems of barrier breakdown with a striker (aperture formation in the metal barrier while the breakdown is an example of large plastic deformations). Within this class of problems one can also find many problems of ground and rock mechanics, where there usually appears the need to consider large deformations, and modelling the processes in biological systems such as the functioning of human muscular tissue, and many others. The theory of infinitesimal deformations of solids appeared in the XVII century in the works by Robert Hooke, who formulated one of the main assumptions of the theory: stresses are proportional to strains of bodies. Translating the assertion into mathematical language, this means that relations between stresses and displacements gradients of bodies are linear. Nowadays the theory of infinitesimal deformations is very deeply and thoroughly elaborated. On the different parts of this theory such as elasticity theory, plasticity theory, stability theory and many others there are many monographs and textbooks.

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But the well-known Hooke’s law does not hold for finite (or large) deformations: the basic relations between stresses and displacement gradients become ‘strongly non-linear’, and they cannot always be expressed analytically. The basis of finite deformations theory was laid in the XIX century by the eminent scientists A.L. Cauchy, J.L. Lagrange, L. Euler, G. Piola, A.J.C. Saint-Venant, G.R. Kirchhoff, and then developed by A.E.H. Love, G. Jaumann [28], M.A. Biot, F.D. Murnaghan [41] and other researchers. The works by M. Mooney and R.S. Rivlin written in the 1940s of the XX century contributed much to the formation of finite deformations theory as an independent part of continuum mechanics. The fundamental step was made in 1950–1960s of the XX century by the American mechanics school, first of all by B.D. Coleman [9], W. Noll and C. Truesdell [43, 54–56], who considered the nonlinear mechanics from the point of view of the formal mathematics. According to D. Hilbert, they introduced the axiomatics of nonlinear mechanics which structured the system of accumulated knowledge and made it possible to formulate the main directions of the further investigations in this theory. Together with R.S. Rivlin and A.J.M. Spencer [10, 13, 51] they elaborated the special mathematical apparatus for formulation of relationships, generalizing Hooke’s law for finite deformations, namely the theory of nonlinear tensor functions. And also the tensor analysis widely used in continuum mechanics was considerably adapted to the problems of nonlinear mechanics. Equations of continuum mechanics got the invariant (i.e. independent of the choice of a reference system) form. The further development of this direction was made by A.C. Eringen, A.E. Green, W. Zerna, J.E. Adkins and others [1–7, 11, 12, 14–27, 29, 30, 32–35, 38–40, 42, 47–50, 52, 53, 57–60]. The role of Russian mechanics school in the development of contemporary nonlinear continuum mechanics principles is also quite substantial. In 1968 the first edition of the fundamental two-part textbook ‘Continuum Mechanics’ by L.I. Sedov was published, which is still one of the most popular books on continuum mechanics in Russia. Outstanding results in the theory of finite elastic deformations were obtained by A.I. Lurie [36, 37], who wrote the principal monograph on the nonlinear theory of elasticity and systemized in it the problem classes of the theory of finite elastic deformations allowing for analytical solutions. Also the considerable step was made by K.F. Chernykh [8], who developed the theory of finite deformations for anisotropic media and elaborated the methods for solving the problems of nonlinear theory of shells and nonlinear theory of cracks. One can also mention the works by mechanics scientists: B.E. Pobedrya, V.I. Kondaurov, V.G. Karnauhov, A.A. Pozdeev, P.V. Trusov, Yu.I. Nyashin and many others who made considerable contributions to the theory of viscoelastic, elastoplastic and viscoplastic finite deformations. This book is based on the lectures which the author has been giving for many years in Moscow Bauman State Technical University. The book has several fundamental traits: 1. It follows the mathematical style of course exposition, which assumes the usage of axioms, definitions, theorems and proofs. 2. It applies the tensor apparatus, mostly in the indexless form, as the latter combined with the special skills is very convenient in usage, and does not shade

Preface

3.

4.

5.

6. 7.

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the physical essence of the laws, and permits proceeding to any appropriate coordinate system. It uses the divergence form of dynamic equations of deformation compatibility, that made it at last possible to write the complete system of balance laws of nonlinear mechanics in a single generalized form. The theory of constitutive equations being the key part of nonlinear mechanics is for the first time exposed with the usage of all energetic couples of tensors, which were established by R. Hill [26] and K.F. Chernykh [8] and ordered by the author [12], and also with quasi-energetic couples of tensors found by the author [12]. To derive constitutive equations of nonlinear continuum mechanics, the author applied the theory of nonlinear tensor functions and tensor operators, elaborated by A.J.M. Spencer, R.S. Rivlin, J.L. Ericksen, V.V. Lokhin, Yu.I. Sirotin, B.E. Pobedrya, the author of this book and others. The bases of theories of large elastic, viscoelastic and plastic deformations are explored from the uniform position. The book uses a ‘reader-friendly‘ style of material exposition, which can be characterized by the presence of quite detailed necessary mathematical calculations and proofs.

The axiomatic approach used in this book differs a bit from the analogous ones suggested by C. Truesdell [56] and other authors. The system of continuum mechanics axioms in the book is composed so as to minimize their total number, and give each axiom a clear physical interpretation. That is why the axioms by C. Truesdell connected with the logic relations between bodies are not included in the general list, the axioms on the bodies’ mass are united into one axiom, the mass conservation law, and analogously the axioms on the existence of forces and inertial reference systems are united into one axiom, the momentum balance law. Though, the last axiom is split into the two parts: first the Sect. 3.2 considers the case of inertial reference systems, and then Sect. 4.10 deals with non-inertial ones. Unlike the axiomatics by C. Truesdell [56], in this book the axiom system includes so called principles of constitutive equations construction which play a fundamental role in the formation of a continuum mechanics equations system. The axiomatic approach to the exploration of continuum mechanics possesses at least one merit – it permits the separation of all the values into two categories: primary and secondary. These are introduced axiomatically and consequently within the continuum mechanics there is no need to substantiate their appearance. The secondary category includes combinations of the first category’s values. The axiomatic approach allows us also to distinguish from continuum mechanics statements between the definitions and corollaries of them (theorems); this is extremely useful for the initial acquaintance with the course. To get acquainted with the specific apparatus of tensor analysis the reader is recommended to use the author’s book ‘Tensor Analysis and Nonlinear Tensor Functions’ [12], which uses the same main notations and definitions. All the references to the tensor analysis formulas in the text are addressed to the latter book.

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Preface

This book covers the fundamental classical parts of nonlinear continuum mechanics: kinematics, balance laws, constitutive equations, relations at singular surfaces, the basics of theories of large elastic deformations, large viscoelastic deformations and large plastic deformations. Because of limits on space, important parts such as the theory of shells at large deformations, and the theory of media with phase transformations were not included in the book. I would like to thank Professor B.E. Pobedrya (Moscow Lomonosov State University), Professor N.N. Smirnov (Moscow Lomonosov State University) and Professor V.S. Zarubin (Bauman Moscow State Technical University) for fruitful discussions and valuable advice on different problems in the book. I am very grateful to Professor G.M.L. Gladwell of the University of Waterloo, Canada, who edited the book and improved the English text. I also thank my wife, Dr. Irina D. Dimitrienko (Bauman Moscow State Technical University), who translated the book into English and prepared the camera-ready typescript. I hope that the book proves to be useful for graduates and post-graduates of mathematical and natural-scientific departments of universities and for investigators, academic scientists and engineers working in solid mechanics, mechanical engineering, applied mathematics and physics. I hope that the book is of interest also for material science specialists developing advanced materials. Russia

Yuriy Dimitrienko

Contents

1

Introduction: Fundamental Axioms of Continuum Mechanics . . . . . . . . . . .

1

2 Kinematics of Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1 Material and Spatial Descriptions of Continuum Motion.. . . . . . . . . . . . . 2.1.1 Lagrangian and Eulerian Coordinates: The Motion Law .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.2 Material and Spatial Descriptions.. . . . . . . . . .. . . . . . . . . . . . . . . . . ı 2.1.3 Local Bases in K and K . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.4 Tensors and Tensor Fields in Continuum Mechanics . . . . . . . ı 2.1.5 Covariant Derivatives in K and K . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.6 The Deformation Gradient . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.1.7 Curvilinear Spatial Coordinates.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2 Deformation Tensors and Measures . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.1 Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.2 Deformation Measures . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.3 Displacement Vector.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.4 Relations Between Components of Deformation Tensors and Displacement Vector .. . . . . . . . . 2.2.5 Physical Meaning of Components of the Deformation Tensor . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.2.6 Transformation of an Oriented Surface Element .. . . . . . . . . . . 2.2.7 Representation of the Inverse Metric Matrix in terms of Components of the Deformation Tensor .. . . . . . . 2.3 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3.1 Theorem on Polar Decomposition . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3.2 Eigenvalues and Eigenbases . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.3.3 Representation of the Deformation Tensors in Eigenbases . 2.3.4 Geometrical Meaning of Eigenvalues . . . . . .. . . . . . . . . . . . . . . . . 2.3.5 Geometric Picture of Transformation of a Small Neighborhood of a Point of a Continuum . . . . . . . 2.4 Rate Characteristics of Continuum Motion.. . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.2 Total Derivative of a Tensor with Respect to Time . . . . . . . . .

5 5 5 9 9 11 13 14 16 24 24 25 25 26 28 30 33 36 36 40 42 44 45 49 49 50 ix

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2.4.3 2.4.4 2.4.5

Differential of a Tensor .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Properties of Derivatives with Respect to Time .. . . . . . . . . . . . The Velocity Gradient, the Deformation Rate Tensor and the Vorticity Tensor .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.6 Eigenvalues of the Deformation Rate Tensor . . . . . . . . . . . . . . . 2.4.7 Resolution of the Vorticity Tensor for the Eigenbasis of the Deformation Rate Tensor .. . . . . . . . . . . . . . . . 2.4.8 Geometric Picture of Infinitesimal Transformation of a Small Neighborhood of a Point . . . . . . . 2.4.9 Kinematic Meaning of the Vorticity Vector . . . . . . . . . . . . . . . . . 2.4.10 Tensor of Angular Rate of Rotation (Spin) .. . . . . . . . . . . . . . . . . 2.4.11 Relationships Between Rates of Deformation Tensors and Velocity Gradients . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.12 Trajectory of a Material Point, Streamline and Vortex Line .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.4.13 Stream Tubes and Vortex Tubes . . . . . . . . . . . .. . . . . . . . . . . . . . . . . Co-rotational Derivatives .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.1 Definition of Co-rotational Derivatives .. . . .. . . . . . . . . . . . . . . . . 2.5.2 The Oldroyd Derivative (hi D ri ) . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.3 The Cotter–Rivlin Derivative (hi D ri ) . . . . .. . . . . . . . . . . . . . . . . 2.5.4 Mixed Co-rotational Derivatives .. . . . . . . . . . .. ................ ı 2.5.5 The Derivative Relative to the Eigenbasis pi of the Right Stretch Tensor .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.6 The Derivative in the Eigenbasis (hi D pi ) of the Left Stretch Tensor . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.7 The Jaumann Derivative (hi D qi ). . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.8 Co-rotational Derivatives in a Moving Orthonormal Basis . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.9 Spin Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2.5.10 Universal Form of the Co-rotational Derivatives.. . . . . . . . . . . 2.5.11 Relations Between Co-rotational Derivatives of Deformation Rate Tensors and Velocity Gradient . . . . . . .

53 54

3 Balance Laws .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1 The Mass Conservation Law .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1.1 Integral and Differential Forms . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1.2 The Continuity Equation in Lagrangian Variables . . . . . . . . . . 3.1.3 Differentiation of Integral over a Moving Volume .. . . . . . . . . 3.1.4 The Continuity Equation in Eulerian Variables . . . . . . . . . . . . . 3.1.5 Determination of the Total Derivatives with respect to Time . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.1.6 The Gauss–Ostrogradskii Formulae . . . . . . . .. . . . . . . . . . . . . . . . . 3.2 The Momentum Balance Law and the Stress Tensor .. . . . . . . . . . . . . . . . . 3.2.1 The Momentum Balance Law. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 3.2.2 External and Internal Forces . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

89 89 89 90 91 92

2.5

56 58 59 60 62 63 65 73 75 77 77 79 80 81 81 82 83 83 84 85 85

93 94 95 95 97

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xi

3.2.3 3.2.4 3.2.5 3.2.6

3.3

3.4

3.5

3.6

Cauchy’s Theorems on Properties of the Stress Vector .. . . . 98 Generalized Cauchy’s Theorem .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .101 The Cauchy and Piola–Kirchhoff Stress Tensors . . . . . . . . . . .102 Physical Meaning of Components of the Cauchy Stress Tensor .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .103 3.2.7 The Momentum Balance Equation in Spatial and Material Descriptions .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .107 The Angular Momentum Balance Law .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .109 3.3.1 The Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .109 3.3.2 Tensor of Moment Stresses . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .110 3.3.3 Differential Form of the Angular Momentum Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .111 3.3.4 Nonpolar and Polar Continua . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .112 3.3.5 The Angular Momentum Balance Equation in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .113 The First Thermodynamic Law . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .114 3.4.1 The Integral Form of the Energy Balance Law.. . . . . . . . . . . . .114 3.4.2 The Heat Flux Vector .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .116 3.4.3 The Energy Balance Equation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .117 3.4.4 Kinetic Energy and Heat Influx Equation . .. . . . . . . . . . . . . . . . .118 3.4.5 The Energy Balance Equation in Lagrangian Description .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .119 3.4.6 The Energy Balance Law for Polar Continua . . . . . . . . . . . . . . .121 The Second Thermodynamic Law . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .124 3.5.1 The Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .124 3.5.2 Differential Form of the Second Thermodynamic Law . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .126 3.5.3 The Second Thermodynamic Law in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .127 3.5.4 Heat Machines and Their Efficiency .. . . . . . .. . . . . . . . . . . . . . . . .128 3.5.5 Adiabatic and Isothermal Processes. The Carnot Cycles . . .132 3.5.6 Truesdell’s Theorem .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .136 Deformation Compatibility Equations .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .141 3.6.1 Compatibility Conditions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .141 3.6.2 Integrability Condition for Differential Form . . . . . . . . . . . . . . .142 3.6.3 The First Form of Deformation Compatibility Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .142 3.6.4 The Second Form of Compatibility Conditions .. . . . . . . . . . . .143 3.6.5 The Third Form of Compatibility Conditions .. . . . . . . . . . . . . .145 3.6.6 Properties of Components of the Riemann–Christoffel Tensor . . . . . . . . .. . . . . . . . . . . . . . . . .146 3.6.7 Interchange of the Second Covariant Derivatives .. . . . . . . . . .148 3.6.8 The Static Compatibility Equation.. . . . . . . . .. . . . . . . . . . . . . . . . .148

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3.7

3.8 3.9

Dynamic Compatibility Equations .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .149 3.7.1 Dynamic Compatibility Equations in Lagrangian Description.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .149 3.7.2 Dynamic Compatibility Equations in Spatial Description ..151 Compatibility Equations for Deformation Rates . . . . . .. . . . . . . . . . . . . . . . .152 The Complete System of Continuum Mechanics Laws .. . . . . . . . . . . . . . .155 3.9.1 The Complete System in Eulerian Description.. . . . . . . . . . . . .155 3.9.2 The Complete System in Lagrangian Description . . . . . . . . . .156 3.9.3 Integral Form of the System of Continuum Mechanics Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .157

4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .161 4.1 Basic Principles for Derivation of Constitutive Equations.. . . . . . . . . . . .161 4.2 Energetic and Quasienergetic Couples of Tensors . . . .. . . . . . . . . . . . . . . . .162 4.2.1 Energetic Couples of Tensors . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .162 I

4.2.2

The First Energetic Couple .T; ƒ/ . . . . . . . . . .. . . . . . . . . . . . . . . . .164

4.2.3

The Fifth Energetic Couple .T; C/ . . . . . . . . .. . . . . . . . . . . . . . . . .165

4.2.4

The Fourth Energetic Couple . T ; .U E// . . . . . . . . . . . . . . . . .166

4.2.5

The Second Energetic Couple .T; .E U1 // . . . . . . . . . . . . .167

4.2.6 4.2.7

The Third Energetic Couple . T ; B/ . . . . . . . .. . . . . . . . . . . . . . . . .167 General Representations for Energetic Tensors of Stresses and Deformations .. . . . .. . . . . . . . . . . . . . . . .168 Energetic Deformation Measures . . . . . . . . . . .. . . . . . . . . . . . . . . . .173 Relationships Between Principal Invariants of Energetic Deformation Measures and Tensors . . . . . . . . . . .175 Quasienergetic Couples of Stress and Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .176

4.2.8 4.2.9 4.2.10

V

IV

II

III

I

4.2.11

The First Quasienergetic Couple .S; A/ . . . .. . . . . . . . . . . . . . . . .177

4.2.12 4.2.13

The Second Quasienergetic Couple .S; .E V1 // . . . . . . . .178 The Third Quasienergetic Couple .Y; TS /.. . . . . . . . . . . . . . . . .178

4.2.14

The Fourth Quasienergetic Couple . S ; .V E// . . . . . . . . . . .179

4.2.15 4.2.16 4.2.17 4.2.18

The Fifth Quasienergetic Couple .S; J/ . . . . .. . . . . . . . . . . . . . . . .180 General Representation of Quasienergetic Tensors . . . . . . . . .180 Quasienergetic Deformation Measures . . . . .. . . . . . . . . . . . . . . . .182 Representation of Rotation Tensor of Stresses in Terms of Quasienergetic Couples of Tensors .. . . . . . . . . . . .183 Relations Between Density and Principal Invariants of Energetic and Quasienergetic Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .185 The Generalized Form of Representation of the Stress Power . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .186

4.2.19

4.2.20

II

IV

V

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4.2.21

4.3

4.4

4.5

4.6

4.7

4.8

Representation of Stress Power in Terms of Co-rotational Derivatives .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .187 4.2.22 Relations Between Rates of Energetic and Quasienergetic Tensors and Velocity Gradient . . . . . . . . .188 The Principal Thermodynamic Identity . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .196 4.3.1 Different Forms of the Principal Thermodynamic Identity . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .196 4.3.2 The Clausius–Duhem Inequality .. . . . . . . . . . .. . . . . . . . . . . . . . . . .198 4.3.3 The Helmholtz Free Energy .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .198 4.3.4 The Gibbs Free Energy .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .199 4.3.5 Enthalpy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .201 4.3.6 Universal Form of the Principal Thermodynamic Identity . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .202 4.3.7 Representation of the Principal Thermodynamic Identity in Terms of Co-rotational Derivatives .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .203 Principles of Thermodynamically Consistent Determinism, Equipresence and Local Action . . . . . . . .. . . . . . . . . . . . . . . . .205 4.4.1 Active and Reactive Variables . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .205 4.4.2 The Principle of Thermodynamically Consistent Determinism .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .206 4.4.3 The Principle of Equipresence . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .208 4.4.4 The Principle of Local Action . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .208 Definition of Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .209 4.5.1 Classification of Types of Continua . . . . . . . .. . . . . . . . . . . . . . . . .209 4.5.2 General Form of Constitutive Equations for Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .210 The Principle of Material Symmetry . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .213 4.6.1 Different Reference Configurations.. . . . . . . .. . . . . . . . . . . . . . . . .213 4.6.2 H -indifferent and H -invariant Tensors . . . .. . . . . . . . . . . . . . . . .215 4.6.3 Symmetry Groups of Continua . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .219 4.6.4 The Statement of the Principle of Material Symmetry.. . . . .220 Definition of Fluids and Solids. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .221 4.7.1 Fluids and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .221 4.7.2 Isomeric Symmetry Groups .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .222 4.7.3 Definition of Anisotropic Solids . . . . . . . . . . . .. . . . . . . . . . . . . . . . .226 4.7.4 H -indifference and H -invariance of Tensors Describing the Motion of a Solid . . . . . . . . . . .. . . . . . . . . . . . . . . . .228 4.7.5 H -invariance of Rate Characteristics of a Solid . . . . . . . . . . . .231 Corollaries of the Principle of Material Symmetry and Constitutive Equations for Ideal Continua . . . . . . . .. . . . . . . . . . . . . . . . .236 4.8.1 Corollary of the Principle of Material Symmetry for Models An of Ideal (Elastic) Solids . . . . . . . . .236 4.8.2 Scalar Indifferent Functions of Tensor Argument.. . . . . . . . . .237 4.8.3 Producing Tensors of Groups . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .239

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4.8.4 4.8.5

Scalar Invariants of a Second-Order Tensor .. . . . . . . . . . . . . . . .240 Representation of a Scalar Indifferent Function in Terms of Invariants .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .243 4.8.6 Indifferent Tensor Functions of Tensor Argument and Invariant Representation of Constitutive Equations for Elastic Continua . . . . . . . . . . . . .244 4.8.7 Quasilinear and Linear Models An of Elastic Continua . . . .249 4.8.8 Constitutive Equations for Models Bn of Elastic Continua . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .253 4.8.9 Corollaries to the Principle of Material Symmetry for Models Cn and Dn of Elastic Continua . . . . .254 4.8.10 General Representation of Constitutive Equations for All Models of Elastic Continua . . . . . . . . . . . . . .262 4.8.11 Representation of Constitutive Equations of Isotropic Elastic Continua in Eigenbases .. . . . . . . . . . . . . . . .265 4.8.12 Representation of Constitutive Equations of Isotropic Elastic Continua ‘in Rates’ . . . .. . . . . . . . . . . . . . . . .271 4.8.13 Application of the Principle of Material Symmetry to Fluids . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .276 4.8.14 Functional Energetic Couples of Tensors.. .. . . . . . . . . . . . . . . . .283 4.9 Incompressible Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .287 4.9.1 Definition of Incompressible Continua . . . . .. . . . . . . . . . . . . . . . .287 4.9.2 The Principal Thermodynamic Identity for Incompressible Continua . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .288 4.9.3 Constitutive Equations for Ideal Incompressible Continua .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .289 4.9.4 Corollaries to the Principle of Material Symmetry for Incompressible Fluids . . . . . . .. . . . . . . . . . . . . . . . .291 4.9.5 Representation of Constitutive Equations for Incompressible Solids in Tensor Bases .. . . . . . . . . . . . . . . . .292 4.9.6 General Representation of Constitutive Equations for All the Models of Incompressible Ideal Solids . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .295 4.9.7 Linear Models of Ideal Incompressible Elastic Continua.. .296 4.9.8 Representation of Models Bn and Dn of Incompressible Isotropic Elastic Continua in Eigenbasis.. .298 4.10 The Principle of Material Indifference.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .300 4.10.1 Rigid Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .300 4.10.2 R-indifferent and R-invariant Tensors .. . . . .. . . . . . . . . . . . . . . . .301 4.10.3 Density and Deformation Gradient in Rigid Motion.. . . . . . .302 4.10.4 Deformation Tensors in Rigid Motion .. . . . .. . . . . . . . . . . . . . . . .303 4.10.5 Stress Tensors in Rigid Motion . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .304 4.10.6 The Velocity in Rigid Motion .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .306 4.10.7 The Deformation Rate Tensor and the Vorticity Tensor in Rigid Motion.. .. . . . . . . . . . . . . . . . .306

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4.10.8 4.10.9 4.10.10 4.10.11

Co-rotational Derivatives in Rigid Motion .. . . . . . . . . . . . . . . . .307 The Statement of the Principle of Material Indifference.. . .312 Material Indifference of the Continuity Equation .. . . . . . . . . .313 Material Indifference for the Momentum Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .313 4.10.12 Material Indifference of the Thermodynamic Laws . . . . . . . .316 4.10.13 Material Indifference of the Compatibility Equations . . . . . .318 4.10.14 Material Indifference of Models An and Bn of Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .319 4.10.15 Material Indifference for Models Cn and Dn of Ideal Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .320 4.10.16 Material Indifference for Incompressible Continua .. . . . . . . .322 4.10.17 Material Indifference for Models of Solids ‘in Rates’ . . . . . .322 4.11 Relationships in a Moving System . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .324 4.11.1 A Moving Reference System . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .324 4.11.2 The Euler Formula .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .326 4.11.3 The Coriolis Formula .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .327 4.11.4 The Nabla-Operator in a Moving System . .. . . . . . . . . . . . . . . . .329 4.11.5 The Velocity Gradient in a Moving System . . . . . . . . . . . . . . . . .330 4.11.6 The Continuity Equation in a Moving System . . . . . . . . . . . . . .330 4.11.7 The Momentum Balance Equation in a Moving System . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .331 4.11.8 The Thermodynamic Laws in a Moving System .. . . . . . . . . . .331 4.11.9 The Equation of Deformation Compatibility in a Moving System . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .332 4.11.10 The Kinematic Equation in a Moving System . . . . . . . . . . . . . .335 4.11.11 The Complete System of Continuum Mechanics Laws in a Moving Coordinate System . . . . . . . . . .335 4.11.12 Constitutive Equations in a Moving System . . . . . . . . . . . . . . . .335 4.11.13 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .338 4.12 The Onsager Principle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .339 4.12.1 The Onsager Principle and the Fourier Law.. . . . . . . . . . . . . . . .339 4.12.2 Corollaries of the Principle of Material Symmetry for the Fourier Law . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .341 4.12.3 Corollary of the Principle of Material Indifference for the Fourier Law .. . . . . . . . . . .. . . . . . . . . . . . . . . . .343 4.12.4 The Fourier Law for Fluids . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .344 4.12.5 The Fourier Law for Solids . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .345 5 Relations at Singular Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .347 5.1 Relations at a Singular Surface in the Material Description .. . . . . . . . . .347 5.1.1 Singular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .347 5.1.2 The First Classification of Singular Surfaces.. . . . . . . . . . . . . . .347 5.1.3 Axiom on the Class of Functions across a Singular Surface.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .350

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5.2

5.3

5.4

5.1.4

The Rule of Differentiation of a Volume Integral in the Presence of a Singular Surface.. . . . . . . . . . . . . .352

5.1.5 5.1.6

Relations at a Coherent Singular Surface in K . . . . . . . . . . . . . .355 Relation Between Velocities of a Singular

ı

ı

Surface in K and K . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .357 Relations at a Singular Surface in the Spatial Description.. . . . . . . . . . . .358 5.2.1 Relations at a Coherent Singular Surface in the Spatial Description . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .358 5.2.2 The Rule of Differentiation of an Integral over a Moving Volume Containing a Singular Surface . . . . .360 Explicit Form of Relations at a Singular Surface . . . . .. . . . . . . . . . . . . . . . .362 5.3.1 Explicit Form of Relations at a Surface of a Strong Discontinuity in a Reference Configuration .. . .362 5.3.2 Explicit Form of Relations at a Surface of a Strong Discontinuity in an Actual Configuration . . . . . .363 5.3.3 Mass Rate of Propagation of a Singular Surface .. . . . . . . . . . .363 5.3.4 Relations at a Singular Surface Without Transition of Material Points . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .365 The Main Types of Singular Surfaces . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .366 5.4.1 Jump of Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .366 5.4.2 Jumps of Radius-Vector and Displacement Vector.. . . . . . . . .367 5.4.3 Semicoherent and Completely Incoherent Singular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .369 5.4.4 Nondissipative and Homothermal Singular Surfaces . . . . . . .369 5.4.5 Surfaces with Ideal Contact . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .370 5.4.6 On Boundary Conditions .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .372 5.4.7 5.4.8

ı

Equation of a Singular Surface in K . . . . . . . .. . . . . . . . . . . . . . . . .372 Equation of a Singular Surface in K . . . . . . . .. . . . . . . . . . . . . . . . .374

6 Elastic Continua at Large Deformations . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .377 6.1 Closed Systems in the Spatial Description . . . . . . . . . . . .. . . . . . . . . . . . . . . . .377 6.1.1 RU VF -system of Thermoelasticity .. . . . . .. . . . . . . . . . . . . . . . .377 6.1.2 RVF -, RU V -, and U V -Systems of Dynamic Equations of Thermoelasticity . . . . . . . . . . . . . . . . .381 6.1.3 T RU VF -system of Dynamic Equations of Thermoelasticity .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .384 6.1.4 Component Form of the Dynamic Equation System of Thermoelasticity in the Spatial Description . . . . .385 6.1.5 The Model of Quasistatic Processes in Elastic Solids at Large Deformations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .388 6.2 Closed Systems in the Material Description.. . . . . . . . . .. . . . . . . . . . . . . . . . .390 6.2.1 U VF -system of Dynamic Equations of Thermoelasticity in the Material Description . . . . . . . . . . . .390

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U V - and U -systems of Thermoelasticity in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .394 6.2.3 T U VF -system of Thermoelasticity in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .395 6.2.4 The Equation System of Thermoelasticity for Quasistatic Processes in the Material Description . . . . . .396 Statements of Problems for Elastic Continua at Large Deformations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .399 6.3.1 Boundary Conditions in the Spatial Description .. . . . . . . . . . .399 6.3.2 Boundary Conditions in the Material Description . . . . . . . . . .402 6.3.3 Statements of Main Problems of Thermoelasticity at Large Deformations in the Spatial Description . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .405 6.3.4 Statements of Thermoelasticity Problems in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .408 6.3.5 Statements of Quasistatic Problems of Elasticity Theory at Large Deformations .. . . . . . . . . . . . . . . .410 6.3.6 Conditions on External Forces in Quasistatic Problems .. . .412 6.3.7 Variational Statement of the Quasistatic Problem in the Spatial Description . . . . . . . . .. . . . . . . . . . . . . . . . .413 6.3.8 Variational Statement of Quasistatic Problem in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .417 6.3.9 Variational Statement for Incompressible Continua in the Material Description .. . . . . .. . . . . . . . . . . . . . . . .417 The Problem on an Elastic Beam in Tension .. . . . . . . . .. . . . . . . . . . . . . . . . .421 6.4.1 Semi-Inverse Method .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .421 6.4.2 Deformation of a Beam in Tension . . . . . . . . .. . . . . . . . . . . . . . . . .421 6.4.3 Stresses in a Beam . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .422 6.4.4 The Boundary Conditions .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .424 6.4.5 Resolving Relation 1 k1 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .424 6.4.6 Comparative Analysis of Different Models An . . . . . . . . . . . . .425 Tension of an Incompressible Beam . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .430 6.5.1 Deformation of an Incompressible Elastic Beam . . . . . . . . . . .430 6.5.2 Stresses in an Incompressible Beam for Models Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .431 6.5.3 Resolving Relation 1 .k1 / . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .432 6.5.4 Comparative Analysis of Models Bn . . . . . . .. . . . . . . . . . . . . . . . .432 6.5.5 Stresses in an Incompressible Beam for Models An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .436 Simple Shear .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .438 6.6.1 Deformations in Simple Shear . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .438 6.6.2 Stresses in the Problem on Shear . . . . . . . . . . .. . . . . . . . . . . . . . . . .439 6.6.3 Boundary Conditions in the Problem on Shear . . . . . . . . . . . . .441 6.2.2

6.3

6.4

6.5

6.6

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6.6.4

6.7

6.8

Comparative Analysis of Different Models An for the Problem on Shear . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .442 6.6.5 Shear of an Incompressible Elastic Continuum . . . . . . . . . . . . .443 The Lam´e Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .446 6.7.1 The Motion Law for a Pipe in the Lam´e Problem . . . . . . . . . .446 6.7.2 The Deformation Gradient and Deformation Tensors in the Lam´e Problem . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .448 6.7.3 Stresses in the Lam´e Problem for Models An . . . . . . . . . . . . . . .449 6.7.4 Equation for the Function f . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .450 6.7.5 Boundary Conditions of the Weak Type .. . .. . . . . . . . . . . . . . . . .451 6.7.6 Boundary Conditions of the Rigid Type .. . .. . . . . . . . . . . . . . . . .453 The Lam´e Problem for an Incompressible Continuum . . . . . . . . . . . . . . . .454 6.8.1 Equation for the Function f . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .454 6.8.2 Stresses in the Lam´e Problem for an Incompressible Continuum . . . . . . . . . .. . . . . . . . . . . . . . . . .454 6.8.3 Equation for Hydrostatic Pressure p. . . . . . . .. . . . . . . . . . . . . . . . .456 6.8.4 Analysis of the Problem Solution .. . . . . . . . . .. . . . . . . . . . . . . . . . .456

7 Continua of the Differential Type . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .461 7.1 Models An and Bn of Continua of the Differential Type . . . . . . . . . . . . . .461 7.1.1 Constitutive Equations for Models An of Continua of the Differential Type .. . . . . . .. . . . . . . . . . . . . . . . .461 7.1.2 Corollary of the Onsager Principle for Models An of Continua of the Differential Type .. . .. . . . . . . . . . . . . . . . .463 7.1.3 The Principle of Material Symmetry for Models An of Continua of the Differential Type .. . . . . . .465 7.1.4 Representation of Constitutive Equations for Models An of Solids of the Differential Type in Tensor Bases . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .467 7.1.5 Models Bn of Solids of the Differential Type .. . . . . . . . . . . . . .470 7.1.6 Models Bn of Incompressible Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .471 7.1.7 The Principle of Material Indifference for Models An and Bn of Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .472 7.2 Models An and Bn of Fluids of the Differential Type . . . . . . . . . . . . . . . . .473 7.2.1 Tensor of Equilibrium Stresses for Fluids of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .473 7.2.2 The Tensor of Viscous Stresses in the Model AI of a Fluid of the Differential Type .. . . . . .. . . . . . . . . . . . . . . . .474 7.2.3 Simultaneous Invariants for Fluids of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .476 7.2.4 Tensor of Viscous Stresses in Model AV of Fluids of the Differential Type .. . . . . . . . . .. . . . . . . . . . . . . . . . .478

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Viscous Coefficients in Model AV of a Fluid of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .479 7.2.6 General Representation of Constitutive Equations for Fluids of the Differential Type . . . . . . . . . . . . . . .480 7.2.7 Constitutive Equations for Incompressible Viscous Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .481 7.2.8 The Principle of Material Indifference for Models An and Bn of Fluids of the Differential Type . . . . . . . . . . . . . . .481 Models Cn and Dn of Continua of the Differential Type .. . . . . . . . . . . . .482 7.3.1 Models Cn of Continua of the Differential Type .. . . . . . . . . . .482 7.3.2 Models Cnh of Solids with Co-rotational Derivatives . . . . . . .484 7.3.3 Corollaries of the Principle of Material Symmetry for Models Cnh of Solids . . . . . . . .. . . . . . . . . . . . . . . . .485 7.3.4 Viscosity Tensor in Models Cnh . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .488 7.3.5 Final Representation of Constitutive Equations for Model Cnh of Isotropic Solids . . . . . . . . . . . . . . . .489 7.3.6 Models Dnh of Isotropic Solids . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .491 The Problem on a Beam in Tension.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .492 7.4.1 Rate Characteristics of a Beam . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .492 7.4.2 Stresses in the Beam .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .492 7.4.3 Resolving Relation .k1 ; kP1 / . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .493 7.4.4 Comparative Analysis of Creep Curves for Different Models Bn . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .493 7.4.5 Analysis of Deforming Diagrams for Different Models Bn of Continua of the Differential Type .. . . . . . . . . . .496 7.2.5

7.3

7.4

8 Viscoelastic Continua at Large Deformations . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .497 8.1 Viscoelastic Continua of the Integral Type . . . . . . . . . . . .. . . . . . . . . . . . . . . . .497 8.1.1 Definition of Viscoelastic Continua.. . . . . . . .. . . . . . . . . . . . . . . . .497 8.1.2 Tensor Functional Space . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .498 8.1.3 Continuous and Differentiable Functionals . . . . . . . . . . . . . . . . .499 8.1.4 Axiom of Fading Memory . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .503 8.1.5 Models An of Viscoelastic Continua . . . . . . .. . . . . . . . . . . . . . . . .505 8.1.6 Corollaries of the Principle of Material Symmetry for Models An of Viscoelastic Continua . . . . . . . .506 8.1.7 General Representation of Functional of Free Energy in Models An . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .507 8.1.8 Model An of Stable Viscoelastic Continua .. . . . . . . . . . . . . . . . .510 8.1.9 Model An of a Viscoelastic Continuum with Difference Cores . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .511 8.1.10 Model An of a Thermoviscoelastic Continuum .. . . . . . . . . . . .513 8.1.11 Model An of a Thermorheologically Simple Viscoelastic Continuum . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .514 8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua .. . .516 8.2.1 Principal Models An of Viscoelastic Continua .. . . . . . . . . . . . .516

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Principal Model An of an Isotropic Thermoviscoelastic Continuum .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .518 8.2.3 Principal Model An of a Transversely Isotropic Thermoviscoelastic Continuum . .. . . . . . . . . . . . . . . . .519 8.2.4 Principal Model An of an Orthotropic Thermoviscoelastic Continuum .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .520 8.2.5 Quadratic Models An of Thermoviscoelastic Continua . . . .522 8.2.6 Linear Models An of Viscoelastic Continua . . . . . . . . . . . . . . . .522 8.2.7 Representation of Linear Models An in the Boltzmann Form .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .525 8.2.8 Mechanically Determinate Linear Models An of Viscoelastic Continua . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .528 8.2.9 Linear Models An for Isotropic Viscoelastic Continua.. . . .530 8.2.10 Linear Models An of Transversely Isotropic Viscoelastic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .531 8.2.11 Linear Models An of Orthotropic Viscoelastic Continua .. .532 8.2.12 The Tensor of Relaxation Functions .. . . . . . .. . . . . . . . . . . . . . . . .533 8.2.13 Spectral Representation of Linear Models An of Viscoelastic Continua . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .535 8.2.14 Exponential Relaxation Functions and Differential Form of Constitutive Equations.. . . . . . . . . . .539 8.2.15 Inversion of Constitutive Equations for Linear Models of Viscoelastic Continua . . . . . . . . . . .. . . . . . . . . . . . . . . . .542 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .549 8.3.1 Models An of Incompressible Viscoelastic Continua .. . . . . .549 8.3.2 Principal Models An of Incompressible Isotropic Viscoelastic Continua .. . . . . . . . . . . .. . . . . . . . . . . . . . . . .549 8.3.3 Linear Models An of Incompressible Isotropic Viscoelastic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .551 8.3.4 Models Bn of Viscoelastic Continua . . . . . . .. . . . . . . . . . . . . . . . .552 8.3.5 Models An and Bn of Viscoelastic Fluids . .. . . . . . . . . . . . . . . . .554 8.3.6 The Principle of Material Indifference for Models An and Bn of Viscoelastic Continua .. . . . . . . . . . .556 Statements of Problems in Viscoelasticity Theory at Large Deformations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .558 8.4.1 Statements of Dynamic Problems in the Spatial Description . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .558 8.4.2 Statements of Dynamic Problems in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .562 8.4.3 Statements of Quasistatic Problems of Viscoelasticity Theory in the Spatial Description .. . . . . . .564 8.4.4 Statements of Quasistatic Problems for Models of Viscoelastic Continua in the Material Description .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .566 8.2.2

8.3

8.4

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8.5

8.6

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The Problem on Uniaxial Deforming of a Viscoelastic Beam . . . . . . . . .568 8.5.1 Deformation of a Viscoelastic Beam in Uniaxial Tension .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .568 8.5.2 Viscous Stresses in Uniaxial Tension .. . . . . .. . . . . . . . . . . . . . . . .569 8.5.3 Stresses in a Viscoelastic Beam in Tension . . . . . . . . . . . . . . . . .569 8.5.4 Resolving Relation 1 .k1 / for a Viscoelastic Beam . . . . . . . .570 8.5.5 Method of Calculating the Constants B . / and . / . . . . . . . .571 8.5.6 Method for Evaluating the Constants m, N l1 , l2 and ˇ, m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .574 8.5.7 Computations of Relaxation Curves .. . . . . . .. . . . . . . . . . . . . . . . .575 8.5.8 Cyclic Deforming of a Beam. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .577 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .579 8.6.1 The Problem on Dissipative Heating of a Beam Under Cyclic Deforming .. . . . . . .. . . . . . . . . . . . . . . . .579 8.6.2 Fast and Slow Times in Multicycle Deforming . . . . . . . . . . . . .580 8.6.3 Differentiation and Integration of Quasiperiodic Functions.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .580 8.6.4 Heat Conduction Equation for a Thin Viscoelastic Beam . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .581 8.6.5 Dissipation Function for a Viscoelastic Beam . . . . . . . . . . . . . .582 8.6.6 Asymptotic Expansion in Terms of a Small Parameter . . . . .583 8.6.7 Averaged Heat Conduction Equation .. . . . . .. . . . . . . . . . . . . . . . .584 8.6.8 Temperature of Dissipative Heating in a Symmetric Cycle. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .585 8.6.9 Regimes of Dissipative Heating Without Heat Removal . . .585 8.6.10 Regimes of Dissipative Heating in the Presence of Heat Removal . . . . . . . . . . .. . . . . . . . . . . . . . . . .586 8.6.11 Experimental and Computed Data on Dissipative Heating of Viscoelastic Bodies .. . . . . . . . . . . . .588

9 Plastic Continua at Large Deformations . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .591 9.1 Models An of Plastic Continua at Large Deformations .. . . . . . . . . . . . . . .591 9.1.1 Main Assumptions of the Models. . . . . . . . . . .. . . . . . . . . . . . . . . . .591 9.1.2 General Representation of Constitutive Equations for Models An of Plastic Continua .. . . . . . . . . . . . . .594 9.1.3 Corollary of the Onsager Principle for Models An of Plastic Continua . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .597 9.1.4 Models An of Plastic Yield. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .598 9.1.5 Associated Model of Plasticity An . . . . . . . . . .. . . . . . . . . . . . . . . . .599 9.1.6 Corollary of the Principle of Material Symmetry for the Associated Model An of Plasticity . . . . . .603 9.1.7 Associated Models of Plasticity An for Isotropic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .605

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9.1.8 9.1.9

9.2

9.3

The Huber–Mises Model for Isotropic Plastic Continua . . .607 Associated Models of Plasticity An for Transversely Isotropic Continua . . . . . . . .. . . . . . . . . . . . . . . . .610 9.1.10 Two-Potential Model of Plasticity for a Transversely Isotropic Continuum .. . .. . . . . . . . . . . . . . . . .612 9.1.11 Associated Models of Plasticity An for Orthotropic Continua . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .614 9.1.12 The Orthotropic Unipotential Huber–Mises Model for Plastic Continua.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .616 9.1.13 The Principle of Material Indifference for Models An of Plastic Continua . . . . . . . . .. . . . . . . . . . . . . . . . .618 Models Bn of Plastic Continua . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .623 9.2.1 Representation of Stress Power for Models Bn of Plastic Continua . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .623 9.2.2 General Representation of Constitutive Equations for Models Bn of Plastic Continua .. . . . . . . . . . . . . .627 9.2.3 Corollaries of the Onsager Principle for Models Bn of Plastic Continua . . . . . . . . .. . . . . . . . . . . . . . . . .629 9.2.4 Associated Models Bn of Plastic Continua . . . . . . . . . . . . . . . . .631 9.2.5 Corollary of the Principle of Material Symmetry for Associated Model Bn of Plasticity . . . . . . . . . .632 9.2.6 Associated Models of Plasticity Bn with Proper Strengthening . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .635 9.2.7 Associated Models of Plasticity Bn for Isotropic Continua . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .635 9.2.8 Associated Models of Plasticity Bn for Transversely Isotropic Continua . . . . . . . .. . . . . . . . . . . . . . . . .637 9.2.9 Associated Models of Plasticity Bn for Orthotropic Continua . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .638 9.2.10 The Principle of Material Indifference for Models Bn of Plastic Continua . . . . . . . . .. . . . . . . . . . . . . . . . .639 Models Cn and Dn of Plastic Continua . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .640 9.3.1 General Representation of Constitutive Equations for Models Cn of Plastic Continua .. . . . . . . . . . . . . .640 9.3.2 Constitutive Equations for Models Cn of Isotropic Plastic Continua .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .643 9.3.3 General Representation of Constitutive Equations for Models Dn of Plastic Continua . . . . . . . . . . . . . .646 9.3.4 Constitutive Equations for Models Dn of Isotropic Plastic Continua .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .650 9.3.5 The Principles of Material Symmetry and Material Indifference for Models Cn and Dn .. . . . . . . . . . . . . .651

Contents

9.4

9.5

9.6

9.7

9.8

9.9

xxiii

Constitutive Equations of Plasticity Theory ‘in Rates’ . . . . . . . . . . . . . . . .652 9.4.1 Representation of Models An of Plastic Continua ‘in Rates’ . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .652 Statements of Problems in Plasticity Theory .. . . . . . . . .. . . . . . . . . . . . . . . . .655 9.5.1 Statements of Dynamic Problems for Models An of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .655 9.5.2 Statements of Quasistatic Problems for Models An of Plasticity . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .657 The Problem on All-Round Tension–Compression of a Plastic Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .659 9.6.1 Deformation in All-Round Tension–Compression .. . . . . . . . .659 9.6.2 Stresses in All-Round Tension–Compression .. . . . . . . . . . . . . .660 9.6.3 The Case of a Plastically Incompressible Continuum . . . . . .661 9.6.4 The Case of a Plastically Compressible Continuum . . . . . . . .662 9.6.5 Cyclic Loading of a Plastically Compressible Continuum .664 The Problem on Tension of a Plastic Beam . . . . . . . . . . .. . . . . . . . . . . . . . . . .666 9.7.1 Deformation of a Beam in Uniaxial Tension .. . . . . . . . . . . . . . .666 9.7.2 Stresses in a Plastic Beam . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .667 9.7.3 Plastic Deformations of a Beam . . . . . . . . . . . .. . . . . . . . . . . . . . . . .668 9.7.4 Change of the Density . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .670 9.7.5 Resolving Equation for the Problem .. . . . . . .. . . . . . . . . . . . . . . . .671 9.7.6 Numerical Method for the Resolving Equation . . . . . . . . . . . . .672 9.7.7 Method for Determination of Constants H0 , n0 , and s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .675 9.7.8 Comparison with Experimental Data for Alloys . . . . . . . . . . . .676 9.7.9 Comparison with Experimental Data for Grounds .. . . . . . . . .677 Plane Waves in Plastic Continua .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .679 9.8.1 Formulation of the Problem .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .679 9.8.2 The Motion Law and Deformation of a Plate . . . . . . . . . . . . . . .680 9.8.3 Stresses in the Plate. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .681 9.8.4 The System of Dynamic Equations for the Plane Problem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .682 9.8.5 The Statement of Problem on Plane Waves in Plastic Continua.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .684 9.8.6 Solving the Problem by the Characteristic Method . . . . . . . . .685 9.8.7 Comparative Analysis of the Solution for Different Models An . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .689 9.8.8 Plane Waves in Models AIV and AV . . . . . . . .. . . . . . . . . . . . . . . . .691 9.8.9 Shock Waves in Models AI and AII . . . . . . . .. . . . . . . . . . . . . . . . .693 9.8.10 Shock Adiabatic Curves for Models AI and AII . . . . . . . . . . . .695 9.8.11 Shock Adiabatic Curves at a Given Rate of Impact .. . . . . . . .697 Models of Viscoplastic Continua . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .699 9.9.1 The Concept of a Viscoplastic Continuum .. . . . . . . . . . . . . . . . .699

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9.9.2 9.9.3 9.9.4 9.9.5 9.9.6

Model An of Viscoplastic Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .699 Model of Isotropic Viscoplastic Continua of the Differential Type.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .701 General Model An of Viscoplastic Continua .. . . . . . . . . . . . . . .702 Model An of Isotropic Viscoplastic Continua .. . . . . . . . . . . . . .703 The Problem on Tension of a Beam of Viscoplastic Continuum of the Differential Type . . . . . . . .703

References .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .707 Basic Notation . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .708 Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .713

Chapter 1

Introduction: Fundamental Axioms of Continuum Mechanics

Continuum mechanics, including nonlinear continuum mechanics, studies the behavior of material bodies or continua. We can mathematically define a body as follows: it is a set B consisting of elements M called material points. The concept of a material point in continuum mechanics is primary, i.e. axiomatic, as is the concept of a geometrical point in elementary geometry. The mathematical description of a body B in continuum mechanics starts from the following definition. Definition 1.1. A material body B, for which there is a one-to-one correspondence e between each material point M 2 B and its image in some metric space X , i.e. W e W B ! W.B/ e W X ; or e a D W.M/;

M 2 B;

a 2 X;

(1.1)

is called a continuum. The set of all continua B is called the universe U. The definition of one-to-one correspondence one can find, for example, in [31]. A metric space X is characterized by the presence of a distance function l.M; N /, with the help of which one can measure the distance between any two points M and N of a body B [31]. The one-to-one correspondence between material points and points of a metric space X allows us to consider not the material body itself but only its image. Below, we draw no distinction between a material point and its image. Definition 1.1 should be complemented with three fundamental axioms. e Axiom 1 (on continuity). The image W.B/ of a body B is a continuous set (a continuum) in space X . The concept of a continuum was considered in [31]. Axiom 1 introduces the main model of continuum mechanics, namely a continuous set, that is an idealization of real bodies consisting of discrete atoms and molecules. Physically, there is a limiting distance lmin such that for l 6 lmin the neighborhood of a material point M 2 B is empty. However, in a continuum (and in its image), due to the properties of a continuous set [31], any infinitesimal "-neighborhood U" .A/ of a point A 2 W.B/ X contains an infinite number Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 1, c Springer Science+Business Media B.V. 2011

1

2

1 Introduction: Fundamental Axioms of Continuum Mechanics

of other points of this medium. In this sense, a continuum is only a model of a real body; and computed results in continuum mechanics referred to real bodies for distances between points l 6 lmin may prove to be incorrect. Nevertheless, Axiom 1 is of great importance: it allows us to apply the methods of tensor analysis in metric spaces. Axiom 2 (on an Euclidean space). As a metric space X , in which continua are considered, we can choose a three-dimensional Euclidean metric space E3a , i.e X D E3a . An Euclidean point space E3a (called also an affine Euclidean space) is a set of points M; N ; : : : ; where there exists a mapping, uniquely assigning each ordered pair of points M; N to an element (vector) y of Euclidean vector space E3 adjoined ! to E3a (it is also denoted by MN D y). In Euclidean space E3 , there are operations of addition of elements (vectors) x C y, multiplication by a real number y, 2 R and scalar product of vectors [31], that can be given by a metric matrix gij relative to some basis ei in E3 : x y D gij x i y j , where x i and y j are coordinates of the vectors with respect to the same basis: x D x i ei , y D y i ei . Moreover, in E3 we can introduce the vector product of vectors a D ai ei and b D b i ei as follows: cDabD

1 p g ijk ai b j ek D p ijk ai bj ek : g

(1.2)

Here ijk and ijk are the Levi–Civita symbols [12] (they are zero, if at least two of the indices i; j; k are coincident; and they are equal to 1, if the indices i; j; k form an even permutation, and are equal to (1) if the indices form an odd permutation), and g D det .gij / is the determinant of the metric matrix. Thus, Axiom 2 allows us to describe materials points of a continuum with the help of instruments of Euclidean point spaces: in the space E3a we can introduce a rectangular Cartesian coordinate system O eN i , being common for all continua and consisting of some point O (the origin of the coordinate system) and an orthonormal basis eN i . In this system O eN i , every material point M is uniquely assigned to ! its radius-vector x D OM. The distance l.M; N / between points M and N is ! measured by the length of vector MN ! ! ! l.M; N / D jMN j D .MN MN /1=2 D jyj D .y y/1=2 : The length l.M; N / specifies a metric in the space E3a ; such a space is called a metric space. In a metric space we define the concept of a domain V , and in an Euclidean point space, the concepts of a plane and a straight line. In addition, in a metric space there are concepts of convergence of a point sequence, continuity and differentiability of functions etc.; Axiom 2 allows us to apply these concepts of mathematical analysis to continua. Due to isomorphism (one-to-one correspondence) of Euclidean point spaces of the same dimension as a space E3a , we can always consider the space of elementary

1 Introduction: Fundamental Axioms of Continuum Mechanics

3

e .B/ in E3a Fig. 1.1 A real body B and its image W geometry E3a , where points are usual geometric points, and vectors are directed straight-line segments in the space. The space E3a allows us to show different objects of continuum mechanics geometrically. Example 1.1. Figure 1.1 shows schematically a real material body B and its image e W.B/ in the space E3a . t u Consider the pair .M; t/, where M 2 B, and t 2 RC0 is some nonnegative real number. This pair is an element of the Cartesian product of the sets B RC0 . Axiom 3 (on the existence of absolute time). For every body B there exists a mapping W W B RC0 ! VN E3a ; in the form of a function A D W.M; t/;

M 2 B;

t 2 RC0 ;

A 2 VN E3a :

(1.3)

The parameter t is called the absolute time (or simply time). Notice that both Axioms 1 and 3 establish relations between points M and A. To avoid the ambiguity one should assume that the following consistency condition is satisfied: the mapping (1.1) coincides with (1.3) at some time t D t1 e W.M/ D W.M; t1 / 8M 2 B:

(1.4)

Axiom 3 allows us to describe the motion of a body B, which is defined as ! changes of the radius-vectors x D OM of material points M of the body in a coordinate system O eN i being common for all M 2 B with time t. For the same material point M at different times t1 and t2 we have, in general, distinct radius-vectors x1 and x2 , respectively, in the system O eN i . Example 1.2. Figure 1.2 shows the motion of a real body B and its images W.B; t1 / and W.B; t2 / in the space E3a at times t1 and t2 , respectively. t u The absolutism of time means that the time t is independent of the radius-vector x of a point M in the coordinate system O eN i , i.e., in physical terms, the time varies

4

1 Introduction: Fundamental Axioms of Continuum Mechanics

Fig. 1.2 The motion of a body B

in the same course for all material points M. In a physical sense, this axiom remains valid only for the motion of bodies with speeds which are considerably smaller than the speed of light; otherwise, relativistic effects become essential, and Axiom 3 ceases to describe actual processes adequately. Relativistic effects are not considered in this work.

Chapter 2

Kinematics of Continua

2.1 Material and Spatial Descriptions of Continuum Motion 2.1.1 Lagrangian and Eulerian Coordinates: The Motion Law Let us consider a continuum B. Due to Axiom 2, at time t D 0 there is a oneto-one correspondence between every material point M 2 B and its radius-vector ! ı x D OM in a Cartesian coordinate system O eN i . Denote Cartesian coordinates of ı ı ı the radius-vector by x i (x D x i eN i ) and introduce curvilinear coordinates X i of the same material point M in the form of some differentiable one-to-one functions ı

ı

x i D x i .X k /: ı

(2.1)

ı

Since x D x i eN i , the relationship (2.1) takes the form ı

ı

x D x.X k /:

(2.2)

Let us fix curvilinear coordinates of the point M, and then material points of the continuum B are considered to be numbered by these coordinates X i . For any motion of the continuum B, coordinates X i of material points are considered to remain unchanged; they are said to be ‘frozen’ into the medium and move together with the continuum. Coordinates X i introduced in this way for a material point M are called Lagrangian (or material). Due to Axiom 3, at every time t there is a one-to-one correspondence between ! every point M 2 B with Lagrangian coordinates X i and its radius-vector x D OM with Cartesian coordinates x i , where x and x i depend on t. This means that there is a connection between Lagrangian X i , and the Cartesian x i coordinates of point M and time, i.e. there exist functions in the form (1.3) x i D x i .X k ; t/

8X k 2 VX :

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 2, c Springer Science+Business Media B.V. 2011

(2.3)

5

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2 Kinematics of Continua

These functions determine a motion of the material point M in the Cartesian coordinate system O eN i of space E3a . The relationships (2.3) are said to be the law of the motion of the continuum B. Coordinates x i in (2.3) are called Eulerian (or spatial) coordinates of the material point M. Since x D x i eN i and the coordinate system O eN i is the same for all times t, the equivalent form of the motion law follows from (2.2): x D x.X k ; t/:

(2.4)

Since the consistency conditions (1.4) must be satisfied, from (2.2) and (2.4) we get the relationships ı

x.X k ; 0/ D x.X k /;

ı

x i .X k ; 0/ D x i .X k /:

(2.5)

Here the initial time t D 0 is considered as the time t1 in (1.4), because just at time t D 0 we introduced Lagrangian coordinates X i of point M. Unless otherwise stipulated, functions (2.3) are assumed to be regular in the domain VX R3 for all t, thus there exist the inverse functions X k D X k .x i ; t/ 8x i 2 Vx R3 : ı

The closed domain V D W.B; 0/ in a fixed coordinate system O eN i , which is occupied by continuum B at the initial time t D 0, is called the reference configuı

ration K, and the domain V D W.B; t/ occupied by the same continuum B at the time t > 0 is called the actual configuration K. Figure 2.1 shows a geometric picture of the motion of a continuum from the ı

reference configuration K to the actual one K at time t in space E3a .

Fig. 2.1 The motion of a continuum: positions of continuum B and material point M in reference and actual configurations

2.1 Material and Spatial Descriptions of Continuum Motion

7

It should be noticed that if the continuum motion law (2.3) (or (2.4)) is known, then one of the main problems of continuum mechanics (to determine coordinates of all material points of the continuum at any time) will be resolved. However, in actual problems of continuum mechanics this law, as a rule, is unknown and must be found by solving some mathematical problems, whose statements are to be formulated. One of our objectives is to derive these statements. Example 2.1. Let us consider a continuum B, which at time t D 0 in the reference ı

ı

ı

configuration K is a rectangular parallelepiped (a beam) with edge lengths h1 , h2 ı

and h3 , and in an actual configuration K at t > 0 the continuum is also a rectangular parallelepiped but with different edge lengths: h1 , h2 and h3 . We assume that corresponding sides of both the parallelepipeds lie on parallel planes, and for one of the sides, which for example is situated on the plane .x 2 ; x 3 /, points of diagonals’ ı

intersection in K and in K are coincident (Fig. 2.2). Then the motion law (2.3) for this continuum takes the form x ˛ D k˛ .t/ X ˛ ; ı

˛ D 1; 2; 3;

(2.6)

ı

i.e. coordinates x i ; x i D X i of any material point M in K and K are proportional, ı

and k˛ .t/ D h˛ .t/=h˛ is the proportion function. The motion law (2.6) is called the beam extension law. t u ı

Example 2.2. In K, let a continuum B be a rectangular parallelepiped oriented as shown in Fig. 2.3; its motion law (2.3) has the form

Fig. 2.2 Extension of a beam

Fig. 2.3 Simple shear of a beam

8

2 Kinematics of Continua

8 1 1 2 ˆ ˆ <x D X C a.t/X ; x2 D X 2; ˆ ˆ :x 3 D X 3 ;

(2.7)

ı

where x i D X i , a.t/ is a given function. In K this continuum B has become a parallelepiped, all cross-sections of which are planes orthogonal to the Ox 3 axis and are the same parallelograms. This motion law is called simple shear; the tangent of the shear angle ˛ is equal to a. t u ı

Example 2.3. Consider a continuum B, which in K is a rectangular parallelepiped (a ı

beam) shown in Fig. 2.4; under the transformation from K to K this parallelepiped changes its dimensions without a change in its angles (as in Example 2.1) and rotates by an angle '.t/ in the plane Ox 1 x 2 around the point O (Fig. 2.4). The motion law for the continuum is called the rotation of a beam with extension. In this case Eq. (2.1) have the form ı

ı

x i D F0ij x j ;

xj D X j ;

(2.8)

where the matrix F0ij is the product of two matrices, the rotation matrix O0 and the stretch matrix U0 : F0ij D O0i k U0kj ;

U0ij

0 1 k1 0 0 D @ 0 k2 0 A ; 0 0 k3

O0i j

F0ij

0 cos ' sin ' D @ sin ' cos ' 0 0

1 0 0A ; 1

0 1 k1 cos ' k2 sin ' 0 D @ k1 sin ' k2 cos ' 0 A ; 0 0 k3

and k˛ .t/ D h˛ .t/= h0˛ are the proportion functions characterizing the ratio of ı

lengths of the beam edges in K and K (as in Example 2.1).

Fig. 2.4 Rotation of a beam with extension

t u

2.1 Material and Spatial Descriptions of Continuum Motion

9

2.1.2 Material and Spatial Descriptions In continuum mechanics, physical processes occurring in bodies are characterized by a certain set of varying scalar fields D .M; t/, vector fields a D a.M; t/, and tensor fields of the nth order n .M; t/. We will consider tensors and tensor fields in detail in Sect. 2.1.4 (see also [12]). Since in the Cartesian coordinate system O eN i a material point M corresponds to both Lagrangian coordinates X i and Eulerian coordinates x i , varying scalar and vector fields can be written as follows: .X i ; t/ D .X i .x j ; t/; t/ D e .x j ; t/ D e .x; t/; a.X i ; t/ D e a.x j ; t/ D e a.x; t/;

(2.9)

With the help of the motion law (2.3) (or (2.4)), we can pass from functions of Lagrangian coordinates to functions of Eulerian coordinates in formulae (2.9). In continuum mechanics the tilde e is usually omitted (we will do this below). For a fixed time t in (2.9), we obtain stationary scalar and vector fields. If in (2.9) a material point M is fixed, and time t changes within the interval 0 6 t 6 t 0 , then we get an ordinary scalar function D .M; t/ and vector function a D a.M; t/ depending on time. According to relationships (2.9), there are two ways to describe different physical processes in continua. In the material (Lagrangian) description of a continuum, all tensor fields describing physical processes are considered as functions of X i and t. In the spatial (Eulerian) description, all tensor fields describing physical processes are functions of x i and t. Both the descriptions are equivalent. It should be noted that for solids we more often use the material description, where it is convenient to fix coordinates X i of a material point M and to observe its motion at different times t. For gaseous and fluid continua, Eulerian description is more convenient; when an observer fixes a geometric point with coordinates x i and monitor the material points M passing through this point x i at different times t.

ı

2.1.3 Local Bases in K and K Using the motion law (2.4) and relationship (2.1), at every material point M with coordinates X i in the actual and reference configurations we can introduce its local basis vectors: rk D

@x @x i eN i D Qik eN i ; D k @X @X k

ı

rk D

ı

ı

ı @x @x i eN i D Qik eN i ; D k k @X @X

(2.10)

10

2 Kinematics of Continua

Fig. 2.5 Local basis vectors in reference and actual configurations

where ı

ı

Q ik D @x i [email protected] k ;

Qik D @x i [email protected] k ;

P ik D @X i [email protected] k ;

P ik D @X i [email protected] k

ı

ı

(2.11)

are Jacobian matrices and inverse Jacobian matrices. ı Here and below all values referred to the configuration K will be denoted by ı superscript ı . As follows from the definition (2.11), local bases vectors rk and rk are directed tangentially to corresponding coordinate lines X k (Fig. 2.5). ı

ı

In K and K introduce metric matrices gkl , g kl and inverse metric matrices g kl ,

ı kl

g

as follows: j

gkl D rk rl D Qik Q l ıij D ı

ı

ı

ı

g kl D rk rl D

@x i @x j ıij ; @X k @X l

ı

@x i @x j k ıij ; g kl glm D ım ; @X k @X l

ı

ı

k g kl g lm D ım ;

(2.12)

and also vectors of reciprocal local bases ri D g i m rm ;

ı

ı

ı

ri D g i m rm ;

(2.13)

which satisfy the reciprocity relations ri rj D ıi j ;

ı

ı

ri rj D ıi j ;

(2.14a)

q ı ı rn rm D g nmk rk ;

(2.14b)

and also the following relations: rn r m D

p

k

g nmk r ;

ı

ı

2.1 Material and Spatial Descriptions of Continuum Motion

11

With the help of the mixed multiplication, i.e. sequentially applying scalar and vector products to three different local bases vectors, we can determine the volumes ı

jV j and jV j constructed by these vectors: q q ˇ ˇ ı ı ı ˇ ˇ jV j D r1 .r2 r3 / D g D det .g ij / D ˇ@x k [email protected] i ˇ; p jV j D r1 .r2 r3 / D g D [email protected] k [email protected] i j: ı

ı

ı

ı

(2.15)

ı

It should be noted that although local bases ri and ri have been introduced in ı

different configurations K and K, they correspond to the same coordinates X i (if one consider the same point M); therefore each of the bases can be carried as a ı

rigid whole into the same point in K or in K. Due to this, we can resolve any vector ı ı field a.M/ for each of the bases ri , ri , ri and ri : ı ı

ı ı

a D ai ri D ai ri D ai ri D ai ri :

(2.16) ı

If curvilinear coordinates X i are orthogonal, then the vectors ri are orthogonal as ı ı ı ı well: (ri rj D ıij ), and matrices g ij and g ij are diagonal; hence we can introduce q ı ı Lam´e’s coefficients H ˛ D g ˛˛ (˛ D 1; 2; 3) and the physical orthonormal basis ı b ı ı ı ı r˛ D r˛ =H ˛ D r˛ H ˛ :

(2.17)

Components of a vector a with respect to this basis are called physical: ı b ı b a D ai ri :

(2.18) ı

The actual basis ri is in general not orthogonal even if the basis ri is orthogonal; therefore we cannot introduce the corresponding physical basis in K. One can introduce a physical basis in K not with the help of ri , but with the help of another special basis (see Sect. 2.1.7).

2.1.4 Tensors and Tensor Fields in Continuum Mechanics ı

ı

For different local bases ri , ri , ri , ri or eN i at every point M, and with the help of formulae given in the work [12] we can introduce different dyadic (tensor) bases: ı ı ı ı ı ri ˝rj , ri ˝ rj , ri ˝rj , ri ˝ rj , ri ˝rj etc., which are equivalence classes of vector sets consisting of 2 3 D 6 vectors (for example, r1 ˝ rj D Œr1 rj r2 0r3 0, where Œ is the notation of an equivalence class), and ˝ is the sign of tensor product. A field

12

2 Kinematics of Continua

of second-order tensor T.M/ can be represented as a linear combination of dyadic basis elements: ı

ı

ı ı ı ı T D T ij ri ˝ rj D T ij ri ˝ rj D TN ij eN i ˝ eN j D T ij ri ˝ rj :

(2.19)

During the passage from one basis to another, tensor components T ij are transformed by the tensor law: ı

ı

ı

TN ij D P ik P jl T kl D P ik P jl T kl : ı

(2.20)

ı

Metric matrices g i m , gi m , g i m and g i m are components of the unit (metric) tensor E with respect to different bases: ı

ı

ı

ı

ı

ı

E D gi m ri ˝ rm D g i m ri ˝ rm D gi m ri ˝ rm D g i m ri ˝ rm :

(2.21)

For a second-order tensor T, in continuum mechanics one often uses the transpose tensor TT D T ij rj ˝ ri and the inverse tensor T1 , where T1 T D E. The inverse tensor exists only for a nonsingular tensor (when det T ¤ 0). The determinant of a tensor is defined by the determinant of its mixed components matrix: det T D det T ij . Besides second-order tensors, in continuum mechanics one sometimes uses tensors of higher orders [12]. To introduce the tensors, we define polyadic bases by induction: ri1 ˝ : : : ˝ rin ; the bases are equivalence classes of vector sets consisting of n 3 D 3n vectors. A field of nth order tensor n .M/ can be represented by a linear combination of polyadic basis elements: n

ı

ı

ı

D i1 :::in ri1 ˝ : : : ˝ rin D i1 :::in ri1 ˝ : : : ˝ rin ; ı

where i1 :::in and i1 :::in are components of the nth order tensor with respect to the corresponding polyadic basis. For fourth-order tensors, analogs of the tensor E are the first, second and third unit tensors defined as follows: I D ei ˝ ei ˝ ek ˝ ek D E ˝ E; II D ei ˝ ek ˝ ei ˝ ek ; III D ei ˝ ek ˝ ek ˝ ei ;

(2.22)

and also the symmetric fourth-order unit tensor D

1 .II C III /: 2

D ijkl ei ˝ ej ˝ ek ˝ el ;

ijkl D

1 i k jl .ı ı C ı i l ı jk /: 2

(2.22a)

2.1 Material and Spatial Descriptions of Continuum Motion

13

We can transpose fourth-order tensors as follows: 4

.m1 m2 m3 m4 / D i1 i2 i3 i4 rim1 ˝ rim2 ˝ rim3 ˝ rim4 ;

where .m1 m2 m3 m4 / is some substitution, for example, 4 .4321/ D i1 i2 i3 i4 ri4 ˝ ri3 ˝ ri2 ˝ ri1 :

ı

2.1.5 Covariant Derivatives in K and K ı

Introduce the following nabla-operators in configurations K and K, respectively: ı

@ ; @X k

r D rk ˝

ı

r D rk ˝

@ : @X k

(2.23)

Applying the nabla-operators to a vector field, we get the gradients of a vector in ı

K and K: r ˝ a D rk ˝ ı

ı

r ˝ a D rk ˝

ı

@a @X k

D rk ai rk ˝ ri ; ı

ı

ı ı

ı

ı

ı ı

ı ı

ı

D r k ai rk ˝ ri D r k ai rk ˝ ri D r k ai rk ˝ ri ; (2.24)

@a @X k

where we have denoted the following covariant derivatives in different tensor bases ı

in configurations K and K: ı

ı

r k ai D rk ai D

ı

@a i @X k @ai @X k

ı

ı

ı

ı

m a ; ik m

r k ai D

m a ; ik m

rk ai D

ı

@ai @X k @ai @X k

ı

ı

C ikm am C ikm am :

(2.25) ı

ı

Here ijm and m ij are the Christoffel symbols in configurations K and K. For the Christoffel symbols the following relations (see [12]) hold:

@gkj @gki @gij C i j @X @X @X k

m ij

1 D g km 2

ı

0 1 ı ı ı @g ij 1 ı km @ @g kj @g ki A: D g C 2 @X i @X j @X k

m ij

;

(2.26)

ı

Contravariant derivatives in K and K are introduced as follows: ı

ı

ı

ı

ı

r k ai D g km r m ai ;

r k ai D gkm rm ai :

(2.27)

14

2 Kinematics of Continua ı

The covariant derivatives (2.25) are components of the second-order tensors r ˝ a and r ˝ a, therefore during the passage from the local basis ri to another one they are transformed by the tensor law (2.20). ı

ı

The nabla-operators r and r in K and K can be applied to a tensor field n .X i / of nth order: r ˝ n D rk ˝

ı ı @ n ı ı ı D r k i1 :::in rk ˝ ri1 ˝ : : : ˝ rin ; k @X

r ˝ n D rk ˝

@ n D rk i1 :::in rk ˝ ri1 ˝ : : : ˝ rin ; @X k

ı

ı

ı

(2.28) ı

ı

where rk i1 :::in and r k i1 :::in are the covariant derivatives in K and K, respectively: n ı ı @ ı i1 :::in X ı is ı i1 :::is Dm:::in i1 :::in rk D C mk : (2.29) @X k sD1 In the same way we can define operations of scalar product of the nabla-operator in ı

K (the divergence of a tensor): ı

ı

r n D rk

ı ı @ n ı ı D r k ki2 :::in ri2 ˝ : : : ˝ rin ; k @X

(2.30)

ı

and vector product of the nabla-operator in K (the curl of a tensor): ı

ı

r n D rk

@ n 1 ijk ı ı ı ı ı D q r i j i2 :::in rk ˝ ri2 ˝ : : : ˝ rin : ı @X k g

(2.31)

2.1.6 The Deformation Gradient Consider how a local neighborhood of a point M is transformed during the passage ı

ı

from configuration K to K. Take an arbitrary elementary radius-vector d x connectı

ing in K two infinitesimally close points M and M0 (Fig. 2.6). In configuration K, these material points M and M0 are connected by the elementary radius-vector d x. ı The vectors d x and d x can always be resolved for local bases: ı

d x.X k / D

@x @x ı ı dX k D rk dX k ; d x.X k / D dX k D rk dX k : k k @X @X

(2.32)

2.1 Material and Spatial Descriptions of Continuum Motion

15

Fig. 2.6 Transformation of an elementary radius-vector during the passage from the reference configuration to the actual one

ı

On multiplying the first equation by rm and the second – by rm , we get rm d x D rm rk dX k D dX m ;

ı

ı

ı

ı

rm d x D rm rk dX k D dX m :

(2.33) ı

Substitution of dX m (2.33) into the first equation of (2.32) yields d x D rk ˝ rk d x. Changing the order of the tensor and scalar products (that is permissible by the ı tensor analysis rules), we get the relation between d x and d x: ı

ı

d x D F d x:

(2.34)

Here we have denoted the linear transformation tensor ı

F D rk ˝ rk ;

(2.35)

called the deformation gradient. As follows from (2.34), the deformation gradient ı connects elementary radius-vectors d x and d x of the same material point M in ı

configurations K and K. Definition (2.19) allows us to give a geometric representation of the deformation ı gradient: if ri are considered as the left vectors and ri – as the right vectors, then, by formulae of Sect. 2.1.4 (see [12]), the tensor F takes the form ı

ı

ı

ı

F D ri ˝ ri D Œr1 r1 r2 r2 r3 r3 : According to the geometric definition of a tensor (see Sect. 2.1.4), the tensor ı F can be represented as equivalence class of the ordered set of six vectors ri ; ri (Fig. 2.7).

16

2 Kinematics of Continua

Fig. 2.7 Geometric representation of the deformation gradient

Besides F, in continuum mechanics one often uses the transpose tensor FT , the inverse tensor F1 and the inverse to the transpose tensor F1T : ı

ı

FT D rk ˝ rk D rk ˝

@x @X k

ı

ı

D r ˝ x; ı

F1T D .rk ˝ rk /T D rk ˝ rk D rk ˝

ı

F1 D rk ˝ rk ; ı

@x @X k

ı

D r ˝ x:

(2.35a)

It follows from (2.35) that ı

ı

F ri D rk ˝ rk ri D rk ıik D ri :

(2.36)

i.e. the deformation gradient transforms local bases vectors of the same material ı

point M from K to K. Theorem 2.1. The transpose deformation gradient FT connects gradients of an arı

bitrary vector a in K and K: ı

r ˝ a D FT r ˝ a;

ı

r ˝ a D F1T r ˝ a:

(2.37)

H To derive formulae (2.37), we apply the definitions (2.24) and (2.35): r ˝ a D ri ˝

ı @a @a @a ı ı D rj ıji ˝ D rj ˝ rj ri ˝ D F1T r ˝ a: N (2.38) i i i @X @X @X

2.1.7 Curvilinear Spatial Coordinates Notice that the choice of Cartesian basis O eN i as a fixed (immovable) system in the spatial (Eulerian) description of the continuum motion is not a necessary condition. For some problems of continuum mechanics it is convenient to consider a moving ! system O 0 eN 0i with the origin at a moving point O 0 (x0 D OO 0 ) and a moving orthonormal basis eN 0i (Fig. 2.8), which is connected to eN i by the orthogonal tensor Q: eN 0i D Q eN i :

(2.39)

In this case, instead of Cartesian coordinates x of a point M in the basis eN i one consider its Cartesian coordinates e x i in basis eN 0i : i

e x D x x0 D e x i eN 0i :

(2.40)

2.1 Material and Spatial Descriptions of Continuum Motion

17

Fig. 2.8 Moving bases eN0i ande ri and curvilinear spatial coordinates e X i in moving system O 0 eN 0i

Let Q ij be components of the tensor Q with respect to the basis eN i : Q D Qij eN i ˝ eN j ;

(2.41)

then relation (2.39) takes the form eN 0i D Qji eN j ;

(2.42)

and coordinates e x i and x i are connected as follows: j e x i eN 0i D e x i Q i eN j ; x D x x0 D .x i x0i /Nei D e

x j ; @x i [email protected] x j D Qij ; x i x0i D Qij e

@e x j [email protected] i D P ji :

(2.43)

Instead of Cartesian coordinates e x i , we can consider special curvilinear coordinates k 0 e X with the origin at point O : e k /; x i .X e xi D e

(2.44)

which, due to (2.27), are connected to x i by the relations e k / x i .X e k ; t/ or X ej D X e j .x i ; t/: x i D x0i .t/ C Qij .t/e x j .X

(2.45)

The dependence on t in the relations is defined by functions x0i .t/ and Q ij .t/ (i.e. only by the motion of system O 0 eN 0i ), which are assumed to be known in continuum mechanics. e i are no longer Lagrangian (material): at different times they Coordinates X correspond to different material points. However, it is often convenient to choose

18

2 Kinematics of Continua ı

e i coincident with X i in the reference configuration K. In this case we coordinates X have the relations ı e j ; 0/: x i .X i / D x i .X j ; 0/ D x i .X

(2.46)

With the help of transformation (2.45) we can use the spatial description in coore i as well when consider the functions dinates X e i ; t/I a.X a D a.x i ; t/ D e

(2.47)

e i are called curvilinear spatial coordinates. therefore coordinates X Introduce local vectors e ri D

@x @x j D eN : ei ei j @X @X

(2.48)

x D x, and In particular, the basis eN 0i may be fixed (Fig. 2.9), then eN 0i D eN i , e ej D X e j .x i / are independent of t; the basis e curvilinear spatial coordinates X ri is independent of t as well, and from (2.46) and (2.48) it follows that the basis ı coincides with ri : ı @x j @x j ı e Nj D ri D (2.49) e eN j D ri : ei @X i @X ı

When the basis eN 0i is moving, basese ri and ri are no longer coincident. e i and defined The vectors e ri are directed tangentially to the coordinate lines X simultaneously with ri at every point M at any time t > 0. A change of vectorse ri in time is defined only by the motion of basis eN 0i , because from (2.42), (2.43), and (2.48) it follows that j eN 0i D Q i

@X ek ek ek @X @x j @X e e e r r rk ; D D k k @x j @e x i @x j @e xi

(2.50)

e k [email protected] e k @X x i is independent of t according to (2.44). and the matrix P i

Fig. 2.9 Curvilinear spatial coordinates e X i and Lagrangian coordinates X i for the fixed basis 0 eNi D eN i

2.1 Material and Spatial Descriptions of Continuum Motion

19

The bases vectors ri and e ri are connected as follows: ri D

ek ek @x @x @X @X e D D rk : e k @X i @X i @X i @X

(2.51)

Just as in Sect. 2.1.2, we define the metric matrix e g ij and the inverse metric matrix e g ij : e g ij D e ri e rj ; e g ij e g jk D ıki ; (2.52) and the reciprocal basis vectors e g i ke ri D e rk D

ei ei @X @X j N D e eN 0k : @x j @e xk

(2.53)

According to formulae (2.51) and (2.52), we find the relation between matrices gij and e gkl : gij D ri rj D

el el e k @X e k @X @X @X e e r D e g kl : r k l @X i @X j @X i @X j

(2.54)

The inverse matrix g ij is found from (2.54) by the rule of matrix product inversion (see Exercise 2.1.13): gij D

@X i @X j kl e g : e k @X el @X

(2.55)

From (2.51), (2.53) and (2.55) we can find the relation between vectors of reciprocal bases ri and e ri : ri D g ij rj D

e ri :

em @X i @X j kl @X @X i k e e r r : e g D m e k @X el ek @X j @X @X

(2.56)

Let there be a tensor n , then it can be resolved for the basis ri and for the basis n

ei1 :::ine D i1 :::in ri1 ˝ : : : ˝ rin D rin : ri1 ˝ : : : ˝e

(2.57)

On substituting (2.51) into (2.57), we derive transformation formulae for tensor ei : components during the passage from coordinates X i to X e i1 e in e i1 :::in D j1 :::jn @X : : : @X : @X j1 @X jn

(2.58)

e of covariant differentiation in coordinates X ei : Introduce the nabla-operator r @ e De r ri e @X i

(2.59)

20

2 Kinematics of Continua

and contravariant derivatives of components e a i of a vector a D e aie ri in coordii e : nates X @e ai @e ai i eke eke ai D e m e a ; r a D Ce ikme am : (2.60) r m ik ek ek @X @X ei The Christoffel symbols e m g ij by the relaij in coordinates X are connected to e tions which are similar to (2.26). e i and X i (in Theorem 2.2. The results of covariant differentiation in coordinates X the configuration K) are coincident: e ˝ n ; r ˝ n D r

e n ; r n D r

e n : r n D r

(2.61)

H Prove the first formula in (2.61). Due to (2.23), we have r ˝ n D ri ˝

el @ n @X i k @ n @X @ n e ˝ n : (2.62) e D D ıkl rk ˝ Dr r ˝ i i k l e e @X el @X @X @X @X

The remaining two formulae in (2.61) can be derived in the same way (see Exercise 2.1.8). N Going to components of a tensor n with respect to bases ri and e ri , from (2.58) we get the relation between the covariant derivatives: ei e j1 :::jn : ri j1 :::jn D r

(2.63)

e i into X j : Determine the tensor H transforming coordinates X ı e ij e rj D H rj D HN ij eN i ˝ eN j : ri ˝e H D rj ˝e

(2.64)

Then we get the relations (see Exercises 2.1.10 and 2.1.11): ı

ej e ri D H e ri D H i rj ;

ı

e ij /1e ri D H1T e r i D .H rj :

(2.65)

e i are often chosen orthogonal, then the bases e The coordinates X ri and e ri are ij orthogonal as well, and the matrices e g ij and e g are diagonal; and we can introduce the physical (orthonormal) basis: b e ˛; e r˛ D e r˛ = H

(2.66)

p e g ˛˛ are Lam´ qe’s coefficients, which are in general not coincident ı ı e r˛ with the coefficients H ˛ D g ˛˛ . Tensor components with respect to the basis b are called physical: b eijb e e TDT ri ˝ b rj : (2.67) e˛ D where H

Relations between physical and covariant components of a tensor are determined by the known formulae (see [12]).

2.1 Material and Spatial Descriptions of Continuum Motion

21

Exercises for 2.1 2.1.1. With the help of formulae (2.10), (2.12), (2.13), and (2.17) show that if the motion law of a continuum describes extension of a beam (2.6) (see Example 2.1), ı then the local basis vectors ri and the metric matrices have the forms ı

ı

ri D ei ; r˛ D k˛ eN ˛ ;

ri D ei ;

r˛ D .1=k˛ /Ne˛ ;

ı

˛ D 1; 2; 3;

ı

g ij D ıij ; g ij D ı ij ; 1 1 0 2 0 2 k1 0 0 k1 0 0 .gij / D @ 0 k22 0 A ; .g ij / D @ 0 k22 0 A ; 0 0 k32 0 0 k32 i.e.

g˛ˇ D k˛2 ı˛ˇ ; g ˛ˇ D k˛2 ı˛ˇ ; H˛ D

p

g˛˛ D k˛ ; b r˛ D e ˛ :

2.1.2. Show that if the motion law of a continuum describes a simple shear (see Example 2.2), then the local basis vectors and the metric matrices have the forms ı

ri D ei ;

ı

r i D ei ;

ı

ı

g ij D ıij ;

r1 D eN 1 ; r2 D aNe1 C eN 2 ;

gij D ı ij ; r3 D eN 3 ;

r1 D eN 1 aNe2 ; r2 D eN 2 ; r3 D eN 3 ; 0 0 1 1 a 0 1 C a2 a ij 2 gij D @a 1 C a 0A ; g D @ a 1 0 0 0 0 1

1 0 0A : 1

2.1.3. Show that if the motion law describes rotation of a beam with extension (see Example 2.3), then with introducing the rotation tensor O0 and the stretch tensor U0 : 3 X O0 D O0i j eN i ˝ eN j ; U0 D k˛ eN ˛ ˝ eN ˛ ; ˛D1

we can rewrite the beam motion law in the tensor form ı

x D F0 x;

F0 D O0 U0 :

Show that the local basis vectors and metric matrices for this problem have the forms ri D F0ki eN k ;

ı

ri D eN i ;

22

2 Kinematics of Continua

gij D F0ki F0lj ıkl

0 2 1 k1 cos2 ' C k22 sin2 ' .k12 k22 / cos ' sin ' 0 D @ .k12 k22 / cos ' sin ' k12 sin2 ' C k22 cos2 ' 0 A ; 0 0 k32 g D k1 k2 k3 ;

1 k22 sin2 ' C k12 cos2 ' .k12 k22 / cos ' sin ' 0 gij D @ .k12 k22 / cos ' sin ' k22 cos2 ' C k12 sin2 ' 0 A : 0 0 k32 0

2.1.4. Using the property (2.14) of reciprocal basis vectors, show that the following relations hold: @X i @X i k ı ri D ı eN k ; ri D P ik eN k D eN : @x k @x k 2.1.5. Show that F, FT , F1 and F1T in the Cartesian coordinate system take the forms FD

@x m ı

@x i

eN m ˝ eN i ;

FT D

@X k ı

eN i ˝

@x i

@x m @x m i N D e eN ˝ eN m ; m ı @X k @x i

ı

F

1

ı

@x m @X k i @x m N N D ˝ D e e eN m ˝ ei ; m @x i @x i @X k ı

F1T D

ı

@X k i @x m @x m i N N ˝ D e e eN ˝ em : m @x i @x i @X k

2.1.6. Substituting (2.54), (2.52), and (2.55) into (2.12), derive formula (2.55). 2.1.7. Prove that

ı

ri D F1T ri :

2.1.8. Derive the third formula of (2.61). ı

2.1.9. Prove that for any scalar function '.X i / its gradients in K and K are connected by the relationship ı

r ' D F1T r ': 2.1.10. Show that formulae (2.65) follow from (2.64). 2.1.11. Using (2.47), show that in formulae (2.64) the tensor H has the following components with respect to bases eN i and e ri : ı ı k i ek ei @x i @X @x i @X k e i @x k @X i i 1 N e ij /1 D @x @X : N ; . H / D ; H D ; . H H jD k j j e k @xı j e j @xı k @X j @x k @X @x j @X @X

2.1 Material and Spatial Descriptions of Continuum Motion

23

ı

2.1.12. Introducing the notation F ij for components of the deformation gradient F ı

ı

ı

ı

ı

ı

ı

with respect to basis ri : F D F ij ri ˝ rj D F ij ri ˝ rj ; show that formula (2.36) yields ı

ı

rj D F ij ri : 2.1.13. Show that the Levi-Civita symbols are connected by the relations ijk ijk D 6; p

ijk i lm D ıjl ıkm ıkl ıjm ;

p g ijk D .1= g/ mnl gmi gnj glk :

ijk ijl D 2ıil ; ijk T jk D 0

where T jk are components of an arbitrary symmetric tensor: T jk D T kj . 2.1.14. Using relations (2.14a), show that the local bases vectors are connected by the relations q ı ı p ı ı r˛ rˇ D g r ; r˛ rˇ D g r ; ˛ ¤ ˇ ¤ ¤ ˛: 2.1.15. Show that the unit fourth-order tensors I , II and III defined by formulae (2.22) have the following properties: I T D I1 .T/E; I1 .T/ D T E; II T D TT ; III T D T;

TD

1 .T C TT /; 2

and I 4 D E ˝ E 4 ; III 4 D 6 ;

II 4 D .2134/ ;

4 D

1 .2134/ C /; . 2

for arbitrary second-order tensor T and fourth-order tensor 4 . As follows from these formulae, the tensor III is the ‘true’ unit fourth-order tensor. 2.1.16. Show that components of the symmetric unit fourth-order tensor with respect to a tetradic basis have the form D

1 .ei ˝ el ˝ ei ˝ el C ei ˝ el ˝ el ˝ ei / D ijkl ei ˝ ej ˝ ek ˝ el : 2 ijkl D

1 i k jl .ı ı C ı i l ı jk /: 2

2.1.17. Show that for any second-order tensor T and for any vector a the following formula of covariant differentiation hold: r .T a/ D T .r ˝ a/T C a r T:

24

2 Kinematics of Continua

2.2 Deformation Tensors and Measures 2.2.1 Deformation Tensors Besides F, important characteristics of the motion of a continuum are deformation tensors, which are introduced as follows: 1 gij 2 1 AD gij 2 1 ı ij g ƒD 2 1 ı ij JD g 2 CD

ı ı ı ı ı g ij ri ˝ rj D "ij ri ˝ rj ; ı g ij ri ˝ rj D "ij ri ˝ rj ; ı ı ı ı gij ri ˝ rj D "ij ri ˝ rj ; gij ri ˝ rj D "ij ri ˝ rj :

(2.68)

Here C is called the right Cauchy–Green deformation tensor, A – the left Almansi deformation tensor, ƒ – the right Almansi deformation tensor, and J – the left Cauchy–Green tensor. As follows from the definition of the tensors, covariant components of C and A are coincident, but they are defined with respect to different tensor bases. Components "ij are called covariant components of the deformation tensor. Contravariant components of the tensors ƒ and J are also coincident and called contravariant components "ij of the deformation tensor, but they are defined with respect to different tensor bases of the tensors ƒ and J. Notice that the deformation tensor components "ij D

1 ı gij g ij ; 2

"ij D

1 ı ij g gij ; 2

(2.69)

have been defined independently of each other, therefore the formal rearrangement of indices is not permissible for these components, i.e. "Mkl D "ij gi k g jl ¤ "kl ;

"Mkl D "ij gi k gjl ¤ "kl :

(2.70)

Thus, when there is a need to obtain contravariant components from "ij and covariant components from "ij , one should use the notation "M kl and "Mkl . We will also use the notation ı ı ı ı ı ı "kl D "ij g i k g jl ; "kl D "ij g i k g jl : (2.71) Theorem 2.3. The deformation tensors C, A, ƒ and J are connected to the deformation gradient F as follows: C D 12 .FT F E/;

A D 12 .E F1T F1 /;

ƒ D 12 .E F1 F1T /;

J D 12 .F FT E/:

(2.72)

2.2 Deformation Tensors and Measures

25 ı

H Let us derive a relation between C and F. Having used the definitions of gij , g ij and F, we get 1 ıi 1 1 ı ı ı r ˝ ri rj ˝ rj E D .FT F E/: .ri rj /ri ˝ rj E D 2 2 2 (2.73) The remaining relations of (2.72) can be proved in the same way: 1 1 ı ı AD E ri ˝ ri rj ˝ rj D E F1T F1 ; 2 2 1 1 ı ı E ri ˝ ri rj ˝ rj D E F1 F1T ; ƒD 2 2 1 1 ıi ı j ri ˝ r r ˝ rj E D F FT E : N JD (2.73a) 2 2

CD

2.2.2 Deformation Measures Besides the deformation tensors, we define deformation measures: the right Cauchy– Green measure G and the left Almansi measure g: ı

ı

G D gij ri ˝ rj D FT F D E C 2C; ı

g D g ij ri ˝ rj D F1T F1 D E 2A;

(2.74)

and also the left Cauchy–Green measure g1 and the right Almansi measure G1 : ı

g1 D g ij ri ˝ rj D F FT D E C 2J; ı

ı

G1 D gij ri ˝ rj D F1 F1T D E 2ƒ:

(2.75)

2.2.3 Displacement Vector Introduce a displacement vector u of a point M from the reference configuration to the actual one as follows (Fig. 2.10): ı

u D x x:

(2.76)

Theorem 2.4. The deformation tensors and the deformation gradient are connected to the displacement vector u by the relations ı

F D E C .r ˝ u/T ; ı

FT D E C r ˝ u;

F1 D E .r ˝ u/T ; F1T D E r ˝ u;

(2.77)

26

2 Kinematics of Continua

Fig. 2.10 The displacement vector of a point M from the reference configuration to the actual one

and also CD AD ƒD JD

ı ı ı 1 ı T T r ˝uCr ˝u Cr ˝ur ˝u ; 2 1 r ˝ u C r ˝ uT r ˝ u r ˝ uT ; 2 1 r ˝ u C .r ˝ u/T r ˝ uT r ˝ u ; 2 ı ı ı 1 ı r ˝ u C r ˝ uT C r ˝ uT r ˝ u : 2

(2.78)

H The definition (2.76) of the displacement vector and the properties (2.35) of the deformation gradient yield ı

ı

ı

FT D r ˝ x D r ˝ .x C u/ ı

D ri ˝

ı

ı ı ı @x ı ı C r ˝ u D ri ˝ ri C r ˝ u D E C r ˝ u: i @X

Then the tensor C takes the form ı ı 1 T CD .E C r ˝ u/ .E C r ˝ u / E 2 ı ı ı 1 ı T T r ˝uCr ˝u Cr ˝ur ˝u : D 2

(2.79)

(2.80)

In a similar way, we can prove the remaining relations of the theorem. N

2.2.4 Relations Between Components of Deformation Tensors and Displacement Vector ı

The displacement vector u can be resolved for both bases ri and ri : ı ı

u D ui ri D ui ri :

(2.81)

2.2 Deformation Tensors and Measures

27

The derivative with respect to X i can be determined in both the bases as well: ı ı ı @u k k D r i u rk D ri u rk : @X i

(2.82)

Then the displacement vector gradients take the forms ı

ı

r ˝ u D ri ˝

ı ı ı ı ı ı @u ı ı k i i k D r u r ˝ r D r u ri ˝ rk ; i k @X i

@u D ri uk ri ˝ rk D r i uk ri ˝ rk : @X i Substitution of these expressions into (2.77) gives r ˝ u D ri ˝

ı ı ı ı ı ı ı F D ıik C r i uk rk ˝ ri D F ki rk ˝ ri :

(2.83)

(2.84)

(2.85)

Here we have introduced components of the deformation gradient in the reference configuration: ı ı

ı

F ki D ıik C r i uk :

(2.86)

The transpose gradient FT has the components ı

ı

ı

ı

ı

ı

FT D F ki ri ˝ rk D F ik rk ˝ ri ; ı

ı

ı

F ik D ıik C r k ui ; ı

(2.87)

(2.88)

ı

where .F ik /T D F ki . In a similar way, one can find the expression for the inverse gradient k F1 D ıik ri uk rk ˝ ri D F 1 i rk ˝ ri

(2.89)

and for the inverse–transpose gradient F1T D .F 1 /ki ri ˝ rk D .F 1 /ik rk ˝ ri ;

(2.90)

where their components with respect to the actual configuration are expressed as follows: .F 1 /ik D ıik r k ui ; (2.91) .F 1 /ki D ıik ri uk : Thus, we have proved the following theorem.

(2.92)

28

2 Kinematics of Continua

Theorem 2.5. Components of the deformation gradients F, FT , F1 and F1T in ı

local bases of configurations K and K are connected to components of the displacement vector u by relations (2.86), (2.87), (2.91), and (2.92). On substituting formulae (2.83) and (2.84) into (2.78) for C and A and comparing them with (2.68), we get ı ı ı ı ı ı 1 ı ı "ij D r i u j C r j u i C r i uk r j uk ; 2 1 "ij D ri uj C rj ui ri uk rj uk ; 2

(2.93)

– The expressions for covariant components of the deformation tensor in terms of ı

components of the displacement vector with respect to K and K. In a similar way, substituting (2.83) and (2.84) into (2.78) for ƒ and J, we obtain ı ı ı ı ı ı 1 ı i ıj r u C r j u i C r k u i r k uj ; 2 1 r i uj C r j u i r k u i rk uj "ij D 2 "ij D

(2.94)

– The relations between contravariant components of the deformation tensor and ı

components of the displacement vector in K and K. Then with using relations (2.69), (2.93) and (2.94), we can find the connection between the metric matrices and displacement components: ı

ı ı

ı

ı

ı ı

ı

ı ı

ı

ı

ı

ı

ı

ı

ı

gij D g ij C r i uj C r j ui C r i uk rj uk D g ij Cri uj Crj ui ri uk rj uk ; (2.95) ı ı

g ij D g ij C r i uj C r j ui C r k ui r k uj D g ij Cr i uj Cr j ui r k ui rk uj : (2.96) Thus, we have proved the following theorem. Theorem 2.6. Components of the deformation tensor "ij , "ij and metric matrices gij , g ij are connected to components of the displacement vector u by relations (2.93)–(2.96).

2.2.5 Physical Meaning of Components of the Deformation Tensor Let us clarify now a physical meaning of components of the deformation tensor: 1 1 ı ı ı gij g ij D ri rj ri rj : "ij D (2.97) 2 2

2.2 Deformation Tensors and Measures

29

By the definition of the scalar product (see [12]), we have "˛ˇ ı

where ı

˛ˇ

and

1 D jr˛ jjrˇ j cos 2

ı

ı

jr˛ jjrˇ j cos

˛ˇ

ı

;

˛ˇ

(2.98) ı

˛ˇ

ı

are the angles between basis vectors r˛ , rˇ and r˛ , rˇ in K and

K, respectively.

ı

ı

Consider elementary radius-vectors d x and d x in configurations K and K, and ı introduce their lengths ds and d s, respectively: ds 2 D d x d x;

ı

ı

ı

d s 2 D d x d x:

(2.99)

ı

Since d x is arbitrary, we can choose it to be oriented along one of the basis vectors ı r˛ . Then d x will be directed along the corresponding vector r˛ as well, because ı under this transformation r˛ becomes r˛ for the same material point M with Lagrangian coordinates X k . In this case we have ˇ @xı ˇ ı ˇ ˇ jd xj D d s ˛ D ˇ ˛ dX ˛ ˇ D jr˛ j dX ˛ ; @X ˇ @x ˇ ˇ ˇ jd xj D ds˛ D ˇ ˛ dX ˛ ˇ D jr˛ j dX ˛ : @X ı

ı

(2.100)

Hence ı

ı

ds˛ =d s ˛ D jr˛ j=jr˛ j D ı˛ C 1;

(2.101)

where ı˛ is called the relative elongation. Formula (2.101) yields ı

jr˛ j D jr˛ j.1 C ı˛ /:

(2.102)

On substituting this expression into (2.98), we get "˛ˇ D

1 ı ı jr˛ jjrˇ j .1 C ı˛ /.1 C ıˇ / cos 2

Consider the case when ˛ D ˇ, then

˛ˇ

D

ı ˛ˇ

˛ˇ

cos

ı ˛ˇ

:

(2.103)

D 0 and

ı

"˛˛ D

g 1 ı 2 jr˛ j .1 C ı˛ /2 1 D ˛˛ .1 C ı˛ /2 1 : 2 2

(2.104)

30

2 Kinematics of Continua ı

Let coordinates X i be coincident with Cartesian coordinates x i , then g ˛ˇ D ı˛ˇ ; and for infinitesimal values of the relative elongation, when ı˛ 1, we obtain "˛˛ ı˛ ;

(2.105)

i.e. "˛˛ is coincident with the relative elongation. In general, "˛˛ is a nonlinear function of corresponding elongations. Consider ˛ ¤ ˇ and assume that X i D x i , then we get

ı

˛ˇ

D =2; and from (2.103)

1 ı ı jr˛ jjrˇ j.1 C ı˛ /.1 C ıˇ / cos ˛ˇ 2q q 1 ı ı D g ˛˛ g ˇˇ .1 C ı˛ /.1 C ıˇ / sin ˛ˇ 2 1 D .1 C ı˛ /.1 C ıˇ / sin ˛ˇ ; 2

"˛ˇ D

(2.106)

ı

where ˛ˇ D ˛ˇ ˛ˇ D . =2/ ˛ˇ is the change of the angle between basis vectors r˛ and rˇ . For small relative elongations, when ı˛ 1, and small angles

˛ˇ 1, from (2.106) we get "˛ˇ ˛ˇ =2;

(2.107)

i.e. "˛ˇ is a half of the misalignment angle of the basis vectors.

2.2.6 Transformation of an Oriented Surface Element In actual configuration K consider a smooth surface †, which contains two coordinate lines X ˛ and X ˇ . Then we can introduce the normal n to the surface † as follows: 1 n D p r˛ rˇ : e g

(2.108)

Here e g D det .e g ˛ˇ /, and e g ˛ˇ is the two-dimensional matrix of the surface (˛; ˇ D 1; 2): (2.109) e g ˛ˇ D r˛ rˇ (it is not to be confused with the metric matrix gij D ri rj ). In configuration K consider a surface element d† constructed on elementary radius-vectors d x˛ , which are directed along local basis vectors, i.e. d x˛ D r˛ dX ˛ (Fig. 2.11). The value p d† D e g dX ˛ dX ˇ (2.110)

2.2 Deformation Tensors and Measures

31

Fig. 2.11 Introduction of oriented surface element n d†

is called the area of the surface element d† constructed on vectors d x˛ and d xˇ . Then formula (2.108) takes the form n d† D r˛ dX ˛ rˇ dX ˇ D d x˛ d xˇ ;

(2.111)

where n d† is called the oriented surface element. Show that the normal n defined by formula (2.108) is a unit vector. According to the property (2.14b) of the vector product of basis vectors and the results of Exercise 2.1.13, we can rewrite Eq. (2.111) in the form p n d† D r˛ rˇ dX ˛ dX ˇ D g˛ˇ r dX ˛ dX ˇ p D .1= g/ ijk g˛i gˇj rk dX ˛ dX ˇ

(2.112)

(there is no summation over ˛; ˇ). Thus, n d† n d† D

p

1 g ˛ˇ r p ijk g˛i gˇj rk /.dX ˛ dX ˇ /2 g j

D .˛ˇ k ijk g˛i gˇj /.dX ˛ dX ˇ /2 D.ı˛i ıˇ ıˇi ı˛j /g˛i gˇj .dX ˛ dX ˇ /2 2 D .g˛˛ gˇˇ g˛ˇ /.dX ˛ dX ˇ /2 D e g.dX ˛ dX ˇ /2 D d†2 :

(2.113) Hence, n n D 1:

ı

ı

The surface element d† in K corresponds to the surface element d † in K, which ı ı is constructed on elementary radius–vectors d x˛ and d xˇ : ı

ı

ı

ı

ı

ı

n d † D r˛ dX ˛ rˇ dX ˇ D r˛ rˇ dX ˛ dX ˇ : ı

(2.114)

ı

Here n is the unit normal to d †. ı Since r D F1T r , we get p ı n d† D g ˛ˇ F1T r dX ˛ dX ˇ q ı ı ı D g=g F1T r˛ rˇ dX ˛ dX ˇ q ı ı ı D g=g F1T n d †: Thus, we have proved the following theorem.

(2.115)

32

2 Kinematics of Continua ı

ı

ı

Theorem 2.7. The oriented surface elements n d † and n d† in K and K are connected by the relation q n d† D

ı ı

g=g n F

1

q ı ı ı d † D g=g F1T n d †: ı

(2.116)

With the help of the deformation measures we can derive formulae connecting ı the normals n and n to the surface element containing the same material points in K ı

and K. Multiplying Eq. (2.116) by itself and taking the formula n n D 1 into account, we get d†2 D

ı ı g ı ı 1 1T ı 2 1 ı 2 . . n F F n/ d † D n G n/ d † : ı ı g g

g

Thus,

q

ı

d†=d † D

ı

ı

ı

g=g .n G1 n/1=2 :

(2.117)

(2.118)

ı

On the other hand, expressing n from (2.116) and then multiplying the obtained relation by itself, we obtain ı

ı

d †2 D

ı

g g .n F FT n/ d†2 D .n g1 n/ d†2 ; g g

Thus, we find that

q

ı

d †=d† D

ı

g=g .n g1 n/1=2 :

(2.119)

(2.120)

On introducing the notation ı

ı

ı

ı

ı

k D .n G1 n/1=2 D .n F1 F1T n/1=2 ; k D .n g1 n/1=2 D .n F FT n/1=2 ;

(2.121)

relations (2.119) take the form ı

d †=d† D

q

q ı ı g=g k D g=g .1=k/: ı

Thus, we get

(2.122)

ı

k D 1=k:

(2.123)

Substitution of (2.119) and (2.120) into (2.116) gives the desired relations ı

ı

kn D F1T n;

ı

k n D FT n:

(2.124)

2.2 Deformation Tensors and Measures

33

2.2.7 Representation of the Inverse Metric Matrix in terms of Components of the Deformation Tensor Components gij of the metric matrix are connected to components of the deformation tensor "ij by relation (2.69). In continuum mechanics, one often needs to know the expression of the inverse metric matrix gij in terms of "ij (but not in terms of "ij ). To derive this relation, we should use the connection between components of a matrix and its inverse (see [12]): g ij D

1 imn jkl gmk gnl : 2g

(2.125)

ı

For g ij , we have the similar formula ı

g ij D

1 ı

2g

ı

ı

imn jkl g mk g nl :

(2.126)

ı

On substituting the relations (2.69) between gmn , g mn and "mn into (2.126), we get 1 imn jkl ı ı .g mk C 2"mk /.g nl C 2"nl / 2g 1 imn jkl ı ı ı ı .g mk g nl C 2g mk "nl C 2g nl "mk C 4"mk "nl /: D 2g

gij D

(2.127)

Removing the parentheses, modify four summands in (2.127) in the following way. The first summand with taking formula (2.126) into account gives the maı ı trix g ij .g=g/. To transform the second and the third summands, we should use the formulae q q ı ı ı ı ı jkl 1= g D g t sp g jt g ks g lp : (2.128) ı

ı

ı

ı

ı

ı

ı

ı

.1=g/ imn jkl g mk D imn t sp g jt g ks g lp gmk D imn t mp g jt g lp ı

ı

ı

ı

ı

ı

D .ıti ıpn ıpi ıtn /g jt g lp D g ij gnl g j n g i l : ı

ı

ı

ı

ı

ı

.1=g/imn jkl gmk "nl D .gij g nl g j n g i l /"nl :

(2.129)

(2.130)

Formula (2.128) follows from the relation p

p g ijk D .1= g/ mnl gmi gnj glk

(see [12]), and relationship (2.129) has been obtained by using formula (2.128) and the properties of the Levi-Civita symbols (see Exercise 2.1.13).

34

2 Kinematics of Continua

On substituting formula (2.130) into (2.127), we get ı

g g ij D g

! ı ı 2 ı ı g C 2 g ij g nl 2gi l gj n "nl C ı imn jkl "mk "nl : g ı ij

(2.131)

Finally, we should express the determinant g D det .gij / in terms of "ij . To do this, we multiply relation (2.131) by gij and take formula (2.69) into account: ı

ı ij

ı ij ı nl

3g D g g C 2.g g

ı il ıjn

g g /"nl C

2 ı

!

imn jpl

g

ı

"mp "nl .g ij C 2"ij /: (2.132)

Thus, we get ı ı ı ı 3g D g 3 C 4gnl "nl C .2=g/imn jpl g ij "mp "nl

ı ı ı ı ı ı C 2g ij "ij C 4.g ij g nl gi l gj n /"nl "ij C .4=g/ imn jpl "ij "mp "nl : (2.133)

Modifying the third summand on the right-hand side by formula (2.130) and introducing the notation ı

I1" D g nl "nl ; I2" D

1 ı ij ı nl ı i l ı j n ı g g g g "ij "nl ; I3" D det ."ij gi k /; (2.134) 2

from (2.133) we get the desired formula ı

g D g.1 C 2I1" C 4I2" C 8I3" /:

(2.135)

Here we have taken account of formula (2.125) for the matrix determinant and also ı the relation I3" D .1=g/det ."ij /. Thus, we have proved the following theorem. Theorem 2.8. The inverse metric matrix gij is expressed in terms of components ı "ij of the deformation tensor and g ij by formulae (2.131) and (2.135). Formulae (2.131) and (2.135) allow us to find the expression of contravariant components "ij of the deformation tensor in terms of "ij . It follows from (2.69) that ı g gı gı ı ı ı 1 ı ı "ij D.1=2/ g ij g ij Dg ij g ij g nl g i l g j n "nl imn jkl "mk "nl : 2g g g (2.136) Substitution of formulae (2.93) or (2.94) into (2.131) and (2.136) gives the expresı sion for components "ij in terms of components of the displacement vector ui or ui .

2.2 Deformation Tensors and Measures

35

Exercises for 2.2 2.2.1. Using the results of Exercise 2.1.1, show that the deformation gradient F and its inverse F1 for the problem on a beam in tension (see Example 2.1) have the forms F D FT D

3 X

k˛ eN ˛ ˝ eN ˛ ;

F1 D F1T D

˛D1

3 X 1 eN ˛ ˝ eN ˛ : k ˛D1 ˛

For this problem, the deformation tensors are determined by the formulae CDƒD

3 1X 2 .k 1/Ne˛ ˝ eN ˛ ; 2 ˛D1 ˛

ADƒD

3 1X .1 k˛2 /Ne˛ ˝ eN ˛ ; 2 ˛D1

the deformation measures are determined as follows: G D g1 D

3 X

k˛2 eN ˛ ˝ eN ˛ ;

g D G1 D

˛D1

3 X

k˛2 eN ˛ ˝ eN ˛ ;

˛D1

and components of the deformation tensor take the forms "˛ˇ D

1 2 .k 1/ı˛ˇ ; 2 ˛

"˛ˇ D

1 .1 k˛2 /ı˛ˇ : 2

2.2.2. Using the results of Exercise 2.1.2, show that for the problem on a simple shear we have the following formulae for the deformation gradient: ı

F D F ij eN i ˝ eN j D E C aNe1 ˝ eN 2 ; F1 D E aNe1 ˝ eN 2 ; i.e.

FT D E C aNe2 ˝ eN 1 ;

F1T D E aNe2 ˝ eN 1 ;

0

ı

F ij

1 1a0 D @0 1 0A ; 001

det F D 1;

for the deformation tensors: C D .a=2/O3 C .a2 =2/Ne22 ;

A D .a=2/O3 .a2 =2/Ne22 ;

ƒ D .a=2/O3 .a2 =2/Ne21 ;

J D .a=2/O3 C .a2 =2/Ne21 ;

and for components of the deformation tensor: 1 0 a=2 0 ."ij / D @a=2 a2 =2 0A ; 0 0 0 0

1 0 a=2 0 ."ij / D @a=2 a2 =2 0A : 0 0 0 0

36

2 Kinematics of Continua

Here we have introduced the notation O3 eN 1 ˝ eN 2 C eN 2 ˝ eN 1 ;

eN 2˛ eN ˛ ˝ eN ˛ ;

˛ D 1; 2; 3:

2.2.3. Using the formulae from Example 2.3 (see Sect. 2.1.1), show that for the problem on rotation of a beam with extension, the deformation gradient has the form F D F0ij eN i ˝ eN j D cos '

2 X

k˛ eN ˛ ˝ eN ˛ C k3 eN 3 ˝ eN 3 C sin 'k2 .Ne2 ˝ eN 1 eN 1 ˝ eN 2 /:

˛D1

2.2.4. Using formulae (2.36), (2.85)–(2.92) and the results of Exercise 2.1.7, show that local basis vectors are connected to displacements by the relations ı

ı

ı

ri D rk .ıik C r k ui /;

ı

ri D rk .ıik ri uk /:

2.2.5. Using formulae (2.104) and q (2.106), show that the physical components of b ı ı ı the deformation tensor "˛ˇ D "˛ˇ = g ˛˛ g ˇˇ are connected to relative elongations ı˛ and angles ˛ˇ by the relations 1 b ı "˛˛ D ..1 C ı˛ /2 1/; 2

1 b ı "˛ˇ D .1 C ı˛ /.1 C ıˇ / sin ˛ˇ : 2

e i the expression 2.2.6. Show that in the basis e ri of curvilinear coordinate system X 1 of the tensor F in terms of r ˝ u can be rewritten in the form similar to (2.89)– (2.92): u De uke rk ;

e 1 /ki e F1 D .F ri ; rk ˝e

eie e 1 /ki D ıik r .F uk :

2.2.7. Using formula (2.34), show that the following relationships hold: ı

ı

jd xj2 D d x G d x;

ı

jd xj2 D d x g d x:

2.3 Polar Decomposition 2.3.1 Theorem on Polar Decomposition According to (2.36), the tensor F can be considered as a tensor of the linear transı ı formation from the basis ri to the basis ri . Since the vectors ri and ri are linearly independent, the tensor F is nonsingular. Then for this tensor the following theorem is valid.

2.3 Polar Decomposition

37

Theorem 2.9 (on the polar decomposition). Any nonsingular second-order tensor F can be represented as the scalar product of two second-order tensors: FDOU

or F D V O:

(2.137)

Here U and V are the symmetric and positive-definite tensors, O is the orthogonal tensor, and each of the decompositions (2.137) is unique. H Prove the existence of the decomposition (2.137) in the constructive way, i.e. we should construct the tensors U, V and O. To do this, consider the contractions of the tensor F with its transpose: FT F and F FT . Both the tensors are symmetric, because .FT F/T D FT .FT /T D FT F and .F FT /T D .FT /T FT D F FT ; (2.138) and positive-definite: a .FT F/ a D .a FT / .F a/ D .F a/ .F a/ D b b D jbj2 > 0

(2.139)

for any non-zero vector a, where b D F a. Since any symmetric positive-definite tensor has three real positive eigenvalues [12], eigenvalues of tensors FT F and ı

F FT can be denoted as 2˛ and 2˛ . These tensors are diagonal in their eigenbases, i.e. they have the following forms: FT F D

3 X

ı

ı

ı

2˛ p˛ ˝ p˛ ;

F FT D

˛D1

3 X

2˛ p˛ ˝ p˛ :

(2.140)

˛D1

ı

Here p˛ are the eigenvectors of the tensor FT F and p˛ – of the tensor F FT , which are real-valued and orthonormal: ı

ı

p˛ pˇ D ı˛ˇ ;

p˛ pˇ D ı˛ˇ :

(2.141)

The right-hand sides of (2.140) are the squares of certain tensors U and V defined as UD

3 X

ı

ı

ı

ı

˛ p˛ ˝ p˛ ; ˛ > 0I

VD

˛D1

3 X

˛ p˛ ˝ p˛ ; ˛ > 0:

(2.142)

˛D1

Here signs at ˛ are always chosen positive. In this case, the following relations are valid: F T F D U2 ;

F FT D V2 :

(2.143)

38

2 Kinematics of Continua

The constructed tensors V and U are symmetric due to formula (2.142) and positivedefinite, because for any nonzero vector a we have aUaD

3 X

ı

ı

ı

˛ a p˛ ˝ p˛ a D

˛D1

3 X

ı

ı

˛ .a p˛ /2 > 0;

(2.144)

˛D1

ı

as ˛ > 0. In a similar way, we can prove that the tensor V is positive-definite. Both the tensors V and U are nonsingular, because, under the conditions of the theorem, the tensor F is nonsingular. And from (2.143) we get .det U/2 D det U2 D det .FT F/ D .det F/2 ¤ 0:

(2.145)

Then there exist inverse tensors U1 and V1 , with the help of which we can construct two more new tensors ı

O D F U1 ;

O D V1 F;

(2.146)

which are orthogonal. Indeed, ı

ı

OT O D .F U1 /T .F U1/ D U1 FT F U1 D U1 U2 U1 D E: (2.147) ı

According to [12], this means that the tensor O is orthogonal. In a similar way, we can show that the tensor O is orthogonal as well. ı

Thus, we have really constructed the tensors U and O, and also V and O, the product of which, due to (2.146), gives the tensor F: ı

F D O U D V O:

(2.148) ı

Here U and V are symmetric, positive-definite tensors, O and O are orthogonal tensors. Show that each of the decompositions (2.148) is unique. By contradiction, let there be one more resolution, for example ı

But then

ee U: FDO

(2.149)

FT F D e U2 D U 2 ;

(2.150)

hence, e U D U, because the decomposition of the tensor FT F for its eigenbasis is ı

ı

˛ are chosen positive by the condition. The coincidence of unique. Signs at ˛ and e ı

ı

e and O are coincident as well, because U and e U leads to the fact that O ı

ı

e DFe O U1 D F U1 D O:

(2.151)

2.3 Polar Decomposition

39

This has proved uniqueness of the decomposition (2.148). We can verify uniqueness of the decomposition F D V O in a similar way. ı

Finally, we must show that the orthogonal tensors O and O are coincident, i.e. formula (2.137) follows from (2.148). To do this, we construct the tensor ı

ı

ı

F OT D O U OT :

(2.152)

Due to (2.148), this tensor satisfies the following relationship: ı

ı

ı

O U OT D V O OT :

(2.153)

ı

The tensor O OT is orthogonal, because ı

ı

ı

ı

ı

ı

.O OT /T .O OT / D O OT O OT D O OT D E:

(2.154)

Then the relationship (2.153) can be considered as the polar decomposition of the ı

ı

tensor O U OT . This tensor is symmetric, because ı

ı

ı

ı

ı

ı

.O U OT /T D .OT /T .O U/T D O U OT :

(2.155)

Then the formal equality ı

ı

ı

ı

O U OT D O U OT

(2.156)

is one more polar decomposition. However, as was shown above, the polar decomposition is unique; hence, the following relationships must be satisfied: ı

ı

ı

V D O U OT and O OT D E: ı

(2.157) ı

Thus, the orthogonal tensors O and O are coincident: O D O. N The tensors U and V are called the right and left stretch tensors, respectively, and O is the rotation tensor accompanying the deformation. The tensor F has nine independent components, the tensor O – three independent components, and each of the tensors U and V – six independent components. Remark 1. Since the rotation tensor O is unique in the polar decomposition, from formula (2.157) we get that the stretch tensors U and V are connected by means of the tensor O: V D O U OT ; U D OT V O: (2.157a) t u

40

2 Kinematics of Continua

Theorem 2.10. The Cauchy–Green and Almansi deformation tensors can be expressed in terms of the stretch tensors U and V as follows: 1 2 1 .U E/; A D .E V2 /; 2 2 1 1 ƒ D .E U2 /; J D .V2 E/: 2 2 CD

(2.158)

H To see this, let us substitute the polar decomposition (2.137) into (2.72), and then we get the relationships (2.158). N

2.3.2 Eigenvalues and Eigenbases Theorem 2.11. Eigenvalues of the tensors U and V defined by (2.142) are coincident: ı

˛ D ˛ ;

˛ D 1; 2; 3;

(2.159)

ı

and eigenvectors p˛ and p˛ are connected by the rotation tensor O: ı

p˛ D O p˛ :

(2.159a)

H To prove the theorem, we use the definition (2.142) and the first formula of (2.157a): VD

3 X

˛ p˛ ˝ p˛ D O U OT D

˛D1

where

3 X

ı

ı

ı

˛ O p˛ ˝ .O p˛ / D

˛D1

3 X

ı

˛ p0˛ ˝ p0˛ ;

˛D1 ı

p0˛ D O p˛ :

According to the relationship, we have obtained two different eigenbases of the tensor V and two sets of eigenvalues, that is impossible. Therefore, ı

ı

p0˛ D O p˛ D p˛ and ˛ D ˛ ; as was to be proved. N Due to (2.141), both the eigenbases are orthogonal. Therefore, reciprocal vectors ı of the eigenbases do not differ from p˛ and p˛ : p˛ D p˛ ;

ı

ı

p˛ D p˛ :

(2.160)

2.3 Polar Decomposition

41 ı

The important problem for applications is to determine ˛ , p˛ and p˛ by the given deformation gradient F. To solve the problem, one should use the following method: 1. Construct the tensor U2 D FT F (or V2 D F FT ) and find its components in some basis being suitable for a considered problem; for example, in the Cartesian basis eN i : U2 D .UN 2 /i eN i ˝ eN j and V2 D .VN 2 /i eN i ˝ eN j : j

j

2. Find eigenvalues of the matrix .UN / j by solving the characteristic equation 2 i

det .U2 2˛ E/ D 0;

(2.161)

which in the basis eN i takes the form det ..U 2 /ij 2˛ ıji / D 0:

(2.161a)

ı

3. Find eigenvectors p˛ of the tensor U and eigenvectors p˛ of the tensor V from the equations ı

ı

U2 p˛ D 2˛ p˛ ; V2 p˛ D 2˛ p˛ ;

(2.162)

written, for example, in the basis eN i : ı

b j˛ D 0; ..UN 2 /ij 2˛ ıji /Q

b j˛ D 0; ..VN 2 /ij 2˛ ıji /Q

(2.162a)

ı

b j˛ are Jacobian matrices of the eigenvectors: b j˛ and Q where Q ı

ı

b j˛ eN j ; p˛ D Q

b j˛ eN j : p˛ D Q

(2.163)

ı

b j˛ and Q b j˛ , one should consider only independent To determine the matrices Q equations of the system (2.162a) and the normalization conditions (2.141): jp˛ j D 1;

ı

jp˛ j D 1;

(2.164)

which are equivalent to the quadratic equations ı

ı

b i˛ Q b j˛ ıij D 1; Q

b j˛ ıij D 1: b i˛ Q Q

(2.164a)

42

2 Kinematics of Continua

4. Write the dyadic products (2.142) and find resolutions of the tensors U and V for the eigenbases; for example, for the Cartesian basis eN i : UD

3 X

ı

ı

b i˛ Q b j˛ eN i ˝ eN j ; V D ˛ Q

˛D1

3 X

b i˛ Q b j˛ eN i ˝ eN j : ˛ Q

˛D1

Exercises 2.3.2–2.3.4 show examples of determination of the tensors U and V. Remark 2. Notice that a solution of the quadratic equations (2.164a) may be not ı

b i˛ , this ambigub i˛ and Q unique due to the choice of signs of matrix components Q ity is resolved by applying one more additional condition, namely the condition of ı coincidence of the vectors p˛ and p˛ when t ! 0C : ı

t ! 0C ) p˛ .t/ D p˛ .t/;

˛ D 1; 2; 3:

ı

b i˛ , the ambiguity of the sign choice remains. However, if there For the matrix Q ı

is a field of eigenvectors p˛ .x; t/, then this ambiguity may be retained only at one point x0 at one time, for example, t D 0; and for the remaining x and t, a sign at b i˛ is chosen from the continuity condition of the vector field pı ˛ .x; t/ (for continQ ı

uous motions). If the eigenvector field p˛ .x0 ; 0/ contains the vectors eN ˛ , then the ı remaining ambiguity is resolved by the condition p˛ .x0 ; 0/ D eN ˛ : The ambiguity of a solution of the system (2.162a), (2.164a) may also appear, if at some time t1 at a point x the eigenvalues ˛ .t1 / prove to be triple. In this case, ı

b i˛ .t1 / are determined, as a rule, by passage to b i˛ .t1 / and Q values of the matrices Q the limit: b i˛ .t/; b i˛ .t1 / D lim Q Q t !t1

ı

ı

b i˛ .t1 / D lim Q b i˛ .t/; ˛ D 1; 2; 3: Q t !t1

In the case of double eigenvalues ˛ , these formulae are applied only to their correı

b i˛ and Q b i˛ . sponding matrix components Q

t u

2.3.3 Representation of the Deformation Tensors in Eigenbases ı

ı

Theorem 2.12. In the tensor bases p˛ ˝pˇ and p˛ ˝ pˇ , the Cauchy–Green tensors C and J, the Almansi tensors A and ƒ, and the deformation measures G, g1 and G1 , g have the diagonal form: CD

3 X 1 2 ı ı .˛ 1/p˛ ˝ p˛ ; 2 ˛D1

ƒD

3 X 1 ı ı .1 2 ˛ /p˛ ˝ p˛ ; 2 ˛D1

2.3 Polar Decomposition

43

3 X 1 .1 2 ˛ /p˛ ˝ p˛ ; 2 ˛D1

AD

JD

3 X 1 2 . 1/p˛ ˝ p˛ I (2.165a) 2 ˛ ˛D1

and GD

3 X

ı

ı

G1 D

2˛ p˛ ˝ p˛ ;

˛D1

g

1

D

3 X

3 X

ı

ı

2 ˛ p˛ ˝ p˛ ;

˛D1

2˛ p˛

˝ p˛ ;

gD

˛D1

3 X

2 ˛ p˛ ˝ p˛ :

(2.165b)

˛D1

H On substituting formulae (2.142) into (2.158), we get (2.165a). Formulae (2.165b) follow from (2.165a) and (2.74), (2.75). N Similarly to formulae (2.165), we can introduce new deformation tensors by deı ı termining their components with respect to the bases p˛ ˝ pˇ or p˛ ˝pˇ as follows: 3 X

ı

MD

ı

ı

f .˛ /p˛ ˝ pˇ ;

MD

˛D1

3 X

f .˛ /p˛ ˝ pˇ ;

(2.166)

˛D1

where f .˛ / is a function of ˛ . If f .1/ D 0, then we get the deformation tensors; and if f .1/ D 1, then we get the deformation measures. Among the tensors (2.166), the logarithmic deformation tensors and measures ı

HD

P3

ı ˛D1 lg ˛ p˛ ı ı H1 D H

ı

˝ pˇ ;

HD

C E;

P3

˛D1

lg ˛ p˛ ˝ pˇ ;

H1 D H C E;

(2.167)

are the most widely known; they are called the right and left Hencky tensors, and also the right and left Hencky measures, respectively. ı With the help of the eigenvectors p˛ and p˛ we can form the mixed dyads 3 X ˛D1

ı

p˛ ˝ p˛ D

3 X

p˛ ˝ p˛ O D

˛D1

3 X

! p˛ ˝ p

˛

O D E O:

(2.168)

˛D1

Here we have used the properties (2.159a) and (2.160), and the representation of the unit tensor E in an arbitrary mixed dyadic basis. Thus, the rotation tensor O accompanying the deformation can be expressed in the eigenbasis as follows: OD

3 X ˛D1

ı

ı

p˛ ˝ p˛ D pi ˝ pi :

(2.169)

44

2 Kinematics of Continua

On substituting (2.169) and (2.142) into (2.137) and taking (2.141) into account, we get the following expression of the deformation gradient in the tensor eigenbasis: FDOUD

3 X

ı

p˛ ˝ p˛

3 X

ı

ı

ı

ˇ pˇ ˝ pˇ D

ˇ D1

˛D1

3 X

ı

˛ p˛ ˝ p˛ :

(2.170)

˛D1

According to (2.170), the transpose FT and inverse F1 gradients are expressed as follows: F D T

3 X

ı

˛ p˛ ˝ p˛ ;

F

1

D

˛D1

3 X

ı

1 ˛ p˛ ˝ p˛ :

(2.171)

˛D1

2.3.4 Geometrical Meaning of Eigenvalues ı

ı

Vectors of eigenbases p˛ and p˛ are connected by the transformation (2.159a). In K ı ı take elementary radius-vectors d x˛ oriented along the eigenbasis vectors p˛ , then in K they correspond to radius-vectors d x˛ : ı

ı

ı

ı

d x˛ D p˛ jd x˛ j;

d x˛ D F d x˛ :

(2.172)

Substitution of (2.170) into (2.172) yields d x˛ D

3 X

ı

ı

ı

ˇ pˇ ˝ pˇ p˛ jd x˛ j D ˛ jd x˛ jp˛ ;

(2.173)

ˇ D1

i.e. the elementary radius-vectors d x˛ in K will be also oriented along the corresponding eigenbasis vectors p˛ . ı ı Denote lengths of the vectors d x˛ and d x˛ by d s ˛ and ds, respectively, and derive relations between them: ı

ı

ı

ı

ı

ds˛2 D d x˛ d x˛ D d x˛ FT F d x˛ D jd x˛ j2 p˛ FT F p˛ ı ı

ı

ı

D d s 2˛ p˛ G p˛ D d s 2˛ 2˛ :

(2.174)

Here we have used Eqs. (2.165b) and (2.172). Formula (2.174) proves the following theorem. Theorem 2.13. Eigenvalues ˛ (principal stretches) are the elongation ratios for material fibres oriented along the principal (eigen-) directions: ı

˛ D ds˛ =d s ˛ :

(2.175)

2.3 Polar Decomposition

45

2.3.5 Geometric Picture of Transformation of a Small Neighborhood of a Point of a Continuum ı

In K, consider a small neighborhood of the material point M contained in a conı tinuum; then every point M0 , connected to M by the elementary radius-vector d x (Fig. 2.12), will be connected to the same point M by radius-vector d x in K. These radius-vectors are related as follows: ı

d x D F d x:

(2.176) ı

The relation can be considered as the transformation of arbitrary radius-vector d x into d x. Rewrite the relation (2.176) in Cartesian coordinates: ı dx i D FNmi d x m ;

(2.177)

where FNmi are components of the deformation gradient with respect to the Cartesian basis (see Exercise 2.1.5): ı (2.178) FNmi D [email protected] i [email protected] m /; ı

which depend only on coordinates x m of the point M, but they are independent of ı coordinates d x m of its neighboring points M0 . Therefore the transformation (2.177) ı is a linear transformation of coordinates d x m into dx i , i.e. this is an affine transformation. As follows from the general properties of affine transformations, straight lines ı

and planes contained in a small neighborhood in K will be straight lines and planes in actual configuration K. Parallel straight lines and planes are transformed into ı

parallel straight lines and planes. Therefore if a small neighborhood in K is chosen

Fig. 2.12 Transformation of a small neighborhood of the point contained in a continuum

46

2 Kinematics of Continua

to be a parallelogram, then in K the neighborhood will be a parallelogram as well (although angles between its edges, edge lengths and orientation of planes in space may change). ı

Since a second-order surface in K (and, in general, a surface specified by an algebraic expression of arbitrary nth order) is transformed into a surface of the same ı

order in K, a small spherical neighborhood in K is transformed into an ellipsoid in actual configuration K (Fig. 2.12). ı As follows from formula (2.101), the ratio of lengths ds˛ =d s ˛ of an arbitrary ı

vector (or of elementary radius-vector d x in K and K) is independent of the initial ı ı length d s ˛ of the vector (because the relative elongation ı˛ is independent of d s ˛ ). According to the polar decomposition (2.137), the transformation (2.176) from ı

K to K can always be represented as the superposition of two transformations: ı

d x D O d x0 ;

ı

ı

d x0 D U d x;

(2.179)

realized with the help of the stretch tensor U and the rotation tensor O, or d x D V d x0 ;

ı

d x0 D O d x:

(2.180) ı

The stretch tensor U, which has three eigendirections p˛ , transforms a small neighborhood of the point M with compressing or extending the neighborhood ı along these three directions p˛ . The tensor O rotates the neighborhood deformed ı ı along p˛ as a rigid whole until the direction of p˛ becomes the direction of p˛ . If ı

ı

one use the left stretch tensor V, so rotation of axes p˛ in K till their coincidence with p˛ is first realized, and then compression or tension of the neighborhood occurs along the direction p˛ . The result will be the same as for U. ı If a point M˛ is connected to M by radius-vector d x˛ oriented along the ı eigendirection p˛ (which is unknown before deformation), then in K the point M˛ will be connected to M by radius-vector d x˛ oriented along the corresponding eigendirection p˛ . ı

If a small neighborhood of point M is chosen to be a sphere in K (see Fig. 2.12), then in K the sphere becomes an ellipsoid with principal axes oriented along the eigendirections p˛ . Thus, the transformation of a small neighborhood of every point M contained in a continuum under deformation can always be represented as a superposition of tension/compression along eigendirections and rotation of the neighborhood as a rigid whole, and also displacement as a rigid whole.

2.3 Polar Decomposition

47

Exercises for 2.3 2.3.1. Using the formula (2.157a), show that the following relations between V and U hold: Vm D O Um OT ; Um D OT Vm O for all integer m (positive and negative). 2.3.2. Using the results of Exercises 2.1.1 and 2.2.1, show that for the problem on tension of a beam, eigenvalues ˛ are ˛ D k˛ ;

˛ D 1; 2; 3:

The stretch tensors U and V are coincident and have the form UDVD

3 X

k˛ e˛ ˝ e˛ ;

˛D1 ı

and eigenvectors p˛ and p˛ coincide with e˛ : ı

p˛ D p˛ D e˛ ;

˛ D 1; 2; 3:

The rotation tensor O for this problem is the unit one: O D E. 2.3.3. Using the results of Exercises 2.1.2, 2.2.2 and Remark 2, show that for the problem on simple shear (see Example 2.2 from Sect. 2.1.1), the tensors U2 and V2 are expressed as follows: U2 D FT F D E C aO3 C a2 e2i D .UN 2 /ij eN i ˝ eN j ; V2 D E C aO3 C a2 e2i D .VN 2 /ij eN i ˝ eN j ; 0 0 1 1 a 0 1 C a2 a 2 i 2 i .UN / j D @a 1 C a2 0A ; .VN / j D @ a 1 0 0 0 0 1

1 0 0A ; 1

eigenvalues ˛ are 2˛ D 1 C b˛ jaj; ˛ D 1; 2I b1 D

a p C 1 C a2 =4; 2

b2 D

3 D 1;

a p 1 C a2 =4; 2

ı

eigenvectors p˛ and p˛ (a > 0) are 1 ı ı p˛ D p .Ne1 C b˛ eN 2 /; p3 D eN 3 ; 2 1 C b˛

48

2 Kinematics of Continua

1 1 p1 D q .b1 eN 1 C eN 2 /; p2 D q .b2 eN 1 C eN 2 /; p3 D eN 3 ; 2 1 C b1 1 C b22 the stretch tensors U and V are U D UN ij eN i ˝ eN j D U0 eN 21 C U1 O3 C U2 eN 22 C eN 23 ; V D VN ij eN i ˝ eN j D U2 eN 21 C U1 O3 C U0 eN 22 C eN 23 ; 0

UN ij

1 U0 U 1 0 B C D @U1 U2 0A ;

VN ij

0 0 1 b1ˇ

Uˇ D

0 1 U2 U1 0 D @U1 U0 0A ; 0 0 1

p

p 1 C b1 a b2ˇ 1 C b2 a C ; 1 C b12 1 C b22

ˇ D 0; 1; 2;

and the rotation tensor O has the form O D ON ij eN i ˝ eN j D cos '.Ne21 C eN 22 / C sin '.Ne1 ˝ eN 2 eN 2 ˝ eN 1 /; 0

ON ij cos ' D

1 cos ' sin ' 0 D @ sin ' cos ' 0A ; 0 0 1

b2 b1 ; 1 C b12 1 C b22

sin ' D

b12 b22 : 2 1 C b1 1 C b22

Show that functions b1 .a/ and b2 .a/ satisfy the following relationships: b1 C b2 D a;

b1 b2 D 1;

b12 C b22 D 2 C a2 :

Show that at a D 0 for the considered problem the following equations really hold: b1 D 1;

b2 D 1;

1 D 2 D 3 D 1;

1 1 ı ı p1 D p1 D p .Ne1 C eN 2 /; p2 D p2 D p .Ne1 eN 2 /: 2 2 2.3.4. Using the results of Exercise 2.2.3, show that for the problem on rotation of a beam with tension (see Example 2.3 from Sect. 2.1.1), eigenvalues ˛ have the form ˛ D k˛ ;

˛ D 1; 2; 3;

and eigenvectors ı

p˛ D eN ˛ ;

p˛ D O0 eN ˛ ;

˛ D 1; 2; 3:

2.4 Rate Characteristics of Continuum Motion

49

Using formulae from Exercise 2.1.3 and data from Example 2.3, show that tensors U, V, O, and also C, A, ƒ and J have the form U D U0 D

3 X

k˛ eN ˛ ˝ eN ˛ ;

O D O0 D O0i j eN i ˝ eN j ;

˛D1

V D O0 U0 OT0 D V0 eN 21 C V1 O3 C V2 eN 22 C k3 eN 23 D V0 ij eN i ˝ eN j ; 0 1 V0 V 1 0 i V 0 j D @ V1 V 2 0 A ; 0 0 k3 V0 D k1 cos2 ' C k2 sin2 ';

V1 D .k1 k2 / cos ' sin ';

V2 D k1 sin2 ' C k2 cos2 '; 3 X 1 2 1 2 .k 1/Ne˛ ˝ eN ˛ ; C D .U0 E/ D 2 2 ˛ ˛D1

ƒD AD

3 X 1 1 / D .E U2 .1 k˛2 /Ne˛ ˝ eN ˛ ; 0 2 2 ˛D1

1 1 .E V2 / D .ı ij gij /Nei ˝ eN j ; 2 2

JD

1 2 1 .V E/ D .gij ıij /Nei ˝ eN j ; 2 2

where metric matrices gij and g ij are determined by formulae from Exercise 2.1.3. We should take into consideration that the tensors C and ƒ do not feel the beam rotation – they are coincident with the corresponding tensors for the problem on pure tension of the beam. Show that if we change the sequence of transformations (i.e. we first rotate and then extend the beam), then the tensors A and J do not feel the rotation.

2.4 Rate Characteristics of Continuum Motion 2.4.1 Velocity The velocity (vector) of the motion of a material point M with Lagrangian coordinates X i is determined as the partial derivative of the radius-vector x.X i ; t/ with respect to time at fixed values of X i : v.X i ; t/ D

@x i ˇˇ .X ; t/ˇ i : X @t

(2.181)

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2 Kinematics of Continua

Velocity components vN i with respect to the basis eN i have the form v D vN i eN i D

@x i eN i ; @t

vN i D

@x i j .X ; t/: @t

(2.182)

2.4.2 Total Derivative of a Tensor with Respect to Time Any vector field a.x; t/ (and also scalar or tensor field) varying with time, which describes some physical process in a continuum, can be expressed in both Eulerian and Lagrangian descriptions with the help of the motion law (2.3): a.x; t/ D a.x.X j ; t/; t/:

(2.183)

Determine the derivative of the function with respect to time at fixed X i (i.e. for a fixed point M): @a ˇˇ @a ˇˇ @a @x j ˇˇ (2.184) ˇ i D ˇ iC j ˇ : @t X @t x @x @t X i Definition 2.1. The partial derivative of a varying vector field a (2.183) with respect to time t at fixed coordinates X i is called the total derivative of the function (2.183) with respect to time: da @a ˇˇ aP (2.185) D ˇ : dt @t X i According to formulae (2.182), (2.11), and (2.23), the second summand on the right-hand side of (2.184) can be rewritten as follows: @a @a @x j @a @a D vN i eN i eN j P kj ˝ k D vrk ˝ k D vr ˝a: (2.186) D vN j P kj j k @x @t @X @X @X Then the relationship (2.184) yields da @a D C v r ˝ a; dt @t

(2.187)

where we have introduced the notation for the partial derivative with respect to time which will be widely used below: @a i ˇˇ @a D .x ; t/ˇ i : x @t @t

(2.188)

In formula (2.187) the vector a is considered as a function a.x j ; t/. It is evident that if a is considered as a function of .X j ; t/, then from the definition (2.185) we get ˇ da i @ ˇ .x ; t/ D a.X j ; t/ˇ i : X dt @t

(2.189)

2.4 Rate Characteristics of Continuum Motion

51

The total derivative .d a=dt/ is also called the material (substantial, individual) derivative with respect to time, [email protected][email protected]/ in (2.187) is the partial (local) derivative with respect to time, and v r ˝ a is the convective derivative. The material derivative d a=dt characterizes a change of the vector field a in a fixed material point M, the local derivative determines a change of values of a in time at a fixed point x in space, and from (2.186) we get that the convective derivative characterizes a change of the field due to transfer of the material particle M from a point x to a point x C vdt in space. If we choose the vector v as a, then the relationship between the displacement u and the velocity v vectors has the form vD

dx du @u D D C v r ˝ u: dt dt @t

(2.190)

Similarly to formula (2.185), we can define the total derivative of the nth-order tensor n with respect to time: n

ˇ n P D d n .x i ; t/ D @ .X i ; t/ˇˇ : Xi dt @t

(2.191)

Theorem 2.14. The total derivative (2.191) of a varying tensor field n .x i ; t/ can be written as a sum of local and convective derivatives: d n @ n C v r ˝ n : D dt @t

(2.192)

H Proof of the theorem is similar to the proof of the relationship (2.187). Details are left as Exercise 2.4.6 N Let us consider now the question on components of the total derivative tensor. P are connected with Theorem 2.15. Components of the total derivative tensor n ı the corresponding components of a tensor n with respect to stationary bases ri , eN i and e ri and a moving basis ri as follows: ˇ d ı i1 :::in @ ı ˇ D i1 :::in .X i ; t/ˇ i ; X dt @t

(2.193)

@ N i1 :::in i @ N i1 :::in i d N i1 :::in D .x ; t/ C vN k k .x ; t/; dt @t @x

(2.194)

@ e i1 :::in e i d e i1 :::in ek e i1 :::in .X e i ; t/; D .X ; t/ Ce vk r dt @t

(2.195)

n X @ d i1 :::in D i1 :::in .X i ; t/ C .i1 :::k:::in rk vi˛ /.X i ; t/; dt @t ˛D1

(2.196)

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2 Kinematics of Continua

where n

ı P D d N i1 :::in .x i ; t/Nei1 ˝ : : : ˝ eN in D d i1 :::in .X i ; t/rıi1 ˝ : : : ˝ rıin dt dt d e i1 :::in e i d i1 :::in i D .X ; t/e ri1 ˝ : : : ˝e rin D .X ; t/ri1 ˝ : : : ˝ rin : dt dt (2.197)

In formula (2.196), the component i1 :::k:::in as the ˛th superscript has index k in place of i˛ . H To prove the theorem, we resolve the tensor n for different bases: n

ı

N i1 :::in .x i ; t/Nei1 ˝ : : : ˝ eN in D i1 :::in .X i ; t/rı i1 ˝ : : : ˝ rıin D e i1 :::in .X e i ; t/e D ri1 ˝ : : : ˝e rin D i1 :::in .X i ; t/ri1 ˝ : : : ˝ rin ;

(2.198)

and choose arguments of components of the tensor n as in formula (2.198). ı Then, substituting the resolution (2.198) for the basis ri into the definition ı (2.191), we get the expression (2.193), because d ri =dt D 0. On substituting the resolution (2.198) for the basis eN i into the relationship (2.192), we obtain n

N i1 :::in @ N i1 :::in m P D @ eN i1 ˝ : : : ˝ eN in C vN k eN k m eN ˝ eN i1 ˝ : : : ˝ eN in : (2.198a) @t @x

It is evident that formula (2.194) follows from (2.198a). In a similar way, substituting the resolution (2.198) for the basis e ri into the relae, and also the tion (2.192) and using the property (2.61) of nabla-operators r and r equation @e ri [email protected] D 0 (for the stationary basise ri see Sect. 2.1.7), we get n

e i1 :::in P D @ em e i1 :::ine e ri1 ˝ : : : ˝e rk r rm ˝e rin Ce vke ri1 ˝ : : : ˝e rin I (2.199) @t

and formula (2.195) follows from (2.199) at once. Finally, substituting the resolution (2.198) for the moving basis ri into the definition of the total derivative (2.192), we obtain n i1 :::in ˇ X @ri ˇˇ ˇ P D @ i1 :::i˛ :::in ri1 ˝ : : : ˛ ˇ i : : : ˝ rin : ˇ i ri1 ˝ : : : ˝ rin C X @t @t X ˛D1 (2.200) Due to the definition (2.10) of local bases vectors and the definition (2.181) of the velocity, we have n

@ri˛ j ˇˇ @2 x ˇˇ @v ˇˇ .X ; t/ˇ j D D ˇ ˇ D ri˛ vk rk : X @t @[email protected] i˛ X j @X i˛ X j

(2.201)

2.4 Rate Characteristics of Continuum Motion

53

On substituting (2.201) into (2.200) and then collecting components at the same elements of the polyadic basis, we derive the formula (2.196). N It should be noted that arguments of the resolutions (2.198) and of the derivatives of tensor components (2.193)–(2.196) have been chosen in the specific way.

2.4.3 Differential of a Tensor Definition 2.2. For a tensor field n .x i ; t/, the following object d n D

d n dt dt

(2.202)

is called the differential of a tensor field (or the differential of a tensor) n .x i ; t/. According to formula (2.192) for the total derivative of a tensor with respect to time, we get that the differential of a tensor can be written in the form d n .x i ; t/ D

@ n C v r ˝ n dt: @t

(2.203)

According to (2.190), the relation (2.203) takes the form d n D

@ n dt C d x r ˝ n @t

(2.204)

When a tensor field is stationary (i.e. @ n [email protected] D 0), the differential of the tensor field has the form b d n D d x r ˝ n : (2.205) For stationary tensor fields b d n D d n , but in general these differentials are not coincident. According to Theorem 2.15, components of the tensor d n with respect to the ı fixed basis ri are written as follows: n

ı

d D d

j1 :::jn ı

ı

rj1 ˝ : : : ˝ rjn ;

ı

d

ı

j1 :::jn

d j1 :::jn dt: D dt

(2.206)

From (2.202) and (2.187) we get the following expression for the differential of a vector: da @a i d a.X ; t/ D dt D C v r ˝ a dt; (2.207) dt @t and from (2.205) we have db a D .v r ˝ a/dt D .r ˝ a/T d x:

(2.208)

54

2 Kinematics of Continua ı

In particular, if a D x, then, by formulae (2.208) and (2.35a), we obtain

or

b ı ı d x D .r ˝ x/T d x D F1 d x;

(2.209)

b ı d x D F d x:

(2.210)

On comparing formulae (2.210) with (2.34), we find that the elementary radiusı vector d x, introduced in Sect. 2.1 and connecting two infinitesimally close material b ı points M and M0 , coincides with the vector d x in the notation (2.205).

2.4.4 Properties of Derivatives with Respect to Time Let us establish now important properties of partial and total derivatives of vector fields with respect to time. Theorem 2.16. The partial derivative of the vector product of basis vectors with respect to time has the form @rˇ @r˛ @ .r˛ rˇ / D rˇ C r˛ : @t @t @t

(2.211)

H Determine the derivative of the vector product of two local basis vectors with respect to time: @ @ @ j j .r˛ rˇ / D .Qi˛ eN i Q ˇ eN j / D .Qi˛ Q ˇ /Nei eN j @t @t @t j @Q ˇ @Qi˛ eN i Qjˇ eN j C Qi˛ eN i eN j : D @t @t With use of relation (2.10) we really get (2.211). N Theorem 2.17. For arbitrary continuously differentiable vector fields a.x; t/ D aN i .x k ; t/Nei and b.x; t/ D bN i .x k ; t/Nei , we have the formulae @a @b @ .a b/ D bCa ; @t @t @t

(2.212)

@ @a @b .a ˝ b/ D ˝bCa˝ ; @t @t @t

(2.213)

@a @b @ .a b/ D bCa : @t @t @t H A proof is similar to the proof of Theorem 2.16. N

(2.214)

2.4 Rate Characteristics of Continuum Motion

55

Theorem 2.18. The total derivatives of the vector and scalar products of two arbitrary vector fields a.x; t/ and b.x; t/ with respect to time have the forms d da db .a b/ D bCa ; dt dt dt

(2.215)

da db d .a b/ D bCa : dt dt dt

(2.216)

H To prove formula (2.215), one should use the property of the total derivative (2.187): @ d .a b/ D .a b/ C v r ˝ .a b/: dt @t Modify the first summand by formula (2.212) and the second summand – by the formula r ˝ .a b/ D .r ˝ a/ b .r ˝ b/ a [12], then we get d @a @b .a b/ D b a C v .r ˝ a/ b v .r ˝ b/ a: dt @t @t Collecting the first summand with the third one and the second summand with the fourth one, and using the property (2.187) of the total derivative of a vector, we obtain d da db da db .a b/ D b aD bCa : dt dt dt dt dt Formula (2.216) can be proved in a similar way. N p Theorem 2.19. The total derivative of g with respect to time is connected to the divergence of the velocity v by d p p p g D g ri vi D g r v: dt

(2.217)

H Let us differentiate the second relation of (2.15) with taking formula (2.211) into account: d d p gD r1 .r2 r3 / dt dt 2 @2 x @2 x @ x D .r2 r3 / C r1 r3 C r1 r2 : @[email protected] 1 @[email protected] 2 @[email protected] 3 (2.218) Since

@v @2 x D D ri v D ri vj rj ; @[email protected] i @X i

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2 Kinematics of Continua

we get d p p g D r1 v g r1 C r1 .r2 v r3 / C r1 .r2 r3 v/: dt

(2.219)

Here we have used the relations from Exercise 2.1.14. According to the definition of the vector product (1.2), we obtain r1 r2 v r3 D r1

p

g ijk r2 vi ı3j rk D

p

g i 31 r2 vi D

p g r2 v2 :

(2.220)

On substituting (2.220) into (2.219), we really get formula (2.217). N

2.4.5 The Velocity Gradient, the Deformation Rate Tensor and the Vorticity Tensor ı

Consider elementary radius-vectors d x and d x connecting two infinitesimally close ı

points M and M0 in configurations K and K, respectively. Determine the velocity of the point M0 relative to the configuration connected to the point M. To do this, determine the velocity differential b d v: T 2 2 ı @ x @v T ı @ x @ ıi ıi ı ı i b d x D r ˝ v d x: d v D d xD i dX D i ˝ r d xD r ˝ @t @X @t @X @t @X i (2.221) Here we have used the second equation of (2.33), the definition of the gradient (2.24) and formula (2.181). In a similar way, using the first equation of (2.33): dX i D d v: ri d x, we get one more expression for the vector b b d v D .r ˝ v/T d x:

(2.222)

The second-order tensor .r ˝ v/T is called the velocity gradient, which connects the relative velocity b d v of an elementary radius-vector d x to the vector d x itself: b d v D L d x;

L D .r ˝ v/T :

(2.223)

Just as any second-order tensor (see [12]), the tensor L can be represented by a sum of the symmetric tensor D and the skew-symmetric tensor W: L D D C W:

(2.224)

The symmetric deformation rate tensor D is determined as follows: DD

1 .r ˝ v C r ˝ vT /: 2

This tensor has six independent components.

(2.225)

2.4 Rate Characteristics of Continuum Motion

57

The skew-symmetric vorticity tensor W is determined as follows: WD

1 .r ˝ vT r ˝ v/: 2

(2.226)

Since the tensor W is skew-symmetric and has three independent components, we can put the tensor W in correspondence with the vorticity vector ! connected to the tensor (see [12]) as follows: !D

1 W "; 2

W D ! E:

(2.227)

where " is the Levi-Civita tensor, which has the third order (see [12]). This tensor is determined as follows: 1 " D p ijk ri ˝ rj ˝ rk : g

(2.228)

On substituting (2.224)–(2.227) into (2.222), we prove the following theorem. Theorem 2.20 (Cauchy–Helmholtz). The velocity v.M0 / of an arbitrary point M0 in a neighborhood of the material point M consists of the translational motion velocity v.M/ of the point M, the velocity ! d x of rotation as a rigid whole and the deformation rate D d x, i.e.

or

b dv D ! dx C D dx

(2.229)

v.M0 / D v.M/ C ! d x C D d x C o .jd xj/:

(2.229a)

Example 2.4. Determine the tensor L for the problem on tension of a beam (see Example 2.1), substituting (2.182) into (2.223): LT D eN i

3 3 X @ X P ˛ @ i N N ˝ v D e ˝ X D e k kP˛ eN ˛ ˝ eN ˛ D L: ˛ ˛ @X i @X i ˛D1 ˛D1

Since the velocity gradient L in this case proves to be a symmetric tensor, from (2.225) and (2.226) it follows that D D L;

W D 0:

Thus, in this case ! D 0.

t u

Example 2.5. Determine the tensor L for the problem on simple shear (see Example 2.2), substituting formula (2.182) into (2.223): LT D eN i ˝

@v @Nvj i @Nv1 N N e eN 2 ˝ eN 1 D aN D ˝ e D P e2 ˝ eN 1 : j @X i @X i @X 2

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According to formulae (2.225) and (2.226), we get D D .a=2/.N P e1 ˝ eN 2 C eN 2 ˝ eN 1 /; j

j

i i W D .a=2/.N P e1 ˝ eN 2 eN 2 ˝ eN 1 / D .a=2/.ı P ei ˝ eN j : 1 ı2 ı2 ı1 /N

Using formula (2.227), we determine the vorticity vector !D

aP aP aP 1 W " D .ı1i ı2j ı2i ı1j /j i k eN k D .21k 12k /Nek D eN 3 ; 2 4 4 2 t u

which is orthogonal to the shear plane.

2.4.6 Eigenvalues of the Deformation Rate Tensor Just as any symmetric tensor, the deformation rate tensor D has three orthonormal real-valued eigenvectors and three real positive eigenvalues (see [12]). Denote the eigenvectors by q˛ (these vectors, in general, are not coincident with p˛ ) and the eigenvalues – by D˛ . Then the tensor D can be resolved for its dyadic eigenbasis as follows: 3 X D˛ q˛ ˝ q˛ ; q˛ qˇ D ı˛ˇ : (2.230) DD ˛D1

Take in the actual configuration K an elementary radius-vector d x˛ , connecting points M and M0 , so that the vector is oriented along the eigenvector q˛ of the tensor D; then, similarly to (2.172), we can write d x˛ D q˛ jd x˛ j;

jd x˛ j D .d x˛ d x˛ /1=2 :

(2.231)

Apply the Cauchy–Helmholtz theorem (2.229) to the elementary radius-vector: b d v ˛ D ! d x˛ C D d x˛ :

(2.232)

Multiplying the left and right sides of the equation by d x˛ and taking account of the property of the mixed derivative d x˛ .! d x˛ / D 0, we get d x˛ b d v ˛ D d x ˛ D d x˛ :

(2.233)

Substituting in place of D its expression (2.230) and in place of d x˛ their expressions (2.231), we obtain d x˛ b d v˛ D jd x˛ j2

3 X

Dˇ q˛ qˇ ˝ qˇ q˛ D D˛ jq˛ j2 :

ˇ D1

Here we have used the property (2.230) of orthonormal vectors q˛ .

(2.234)

2.4 Rate Characteristics of Continuum Motion

59

Modify the scalar product on the left-hand side as follows: @ 1 @ @ d x˛ b d v˛ D d x˛ d x ˛ D .d x˛ d x˛ / D jd x˛ j jd x˛ j: @t 2 @t @t

(2.235)

On comparing (2.234) with (2.235), we obtain the following theorem. Theorem 2.21. Eigenvalues D˛ of the deformation rate tensor D are the rates of relative elongations of elementary material fibres oriented along the eigenvectors q˛ : D˛ D

1 @ jd x˛ j: jd x˛ j @t

(2.236)

2.4.7 Resolution of the Vorticity Tensor for the Eigenbasis of the Deformation Rate Tensor Modify the right-hand side of (2.232) as follows: b d v˛ D ! d x˛ C D d x˛ D .! q˛ C D˛ q˛ /jd x˛ j;

(2.237)

and the left-hand side of (2.232) with taking (2.236) into account: @ @ @jd x˛ j @q˛ @q˛ b : q˛ C jd x˛ j Djd x˛ j D˛ q˛ C d v˛ D d x˛ D .jd x˛ jq˛ /D @t @t @t @t @t (2.238) On comparing (2.237) with (2.238), we get the following theorem. Theorem 2.22. The vorticity tensor W (or the vorticity vector !) connects the rate of changing the eigenvectors q˛ to the vectors q˛ themselves: qP ˛ D

@q˛ D ! q˛ D W q˛ : @t

(2.239)

Using formula (2.239), we can resolve the tensor W for the eigenbasis q˛ of the deformation rate tensor as follows: WD

3 X ˛D1

qP ˛ ˝ q˛ D qP i ˝ qi :

(2.240)

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2 Kinematics of Continua

2.4.8 Geometric Picture of Infinitesimal Transformation of a Small Neighborhood of a Point If in configuration K at time t we consider an elementary radius-vector d x connecting two infinitesimally close material points M and M0 , then for infinitesimal time dt the radius-vector is transformed into radius-vector d x0 in configuration K.t C dt/ (Fig. 2.13): d x D x0 .t/ x.t/;

d x0 D x0 .t C dt/ x.t C dt/;

(2.241)

where x.t/ and x0 .t/ are radius-vectors of the points M and M0 in configuration K.t/, respectively; and x.t C dt/ and x0 .t C dt/ – in configuration K.t C dt/. Displacements of points M and M0 for infinitesimal time are defined by the velocity vectors v.M/ and v.M0 /, respectively: x.t C dt/ x.t/ D v.M/ dt;

x0 .t C dt/ x0 .t/ D v.M0 / dt:

(2.242)

Formulae (2.241), (2.242), and simple geometric relations (see Fig. 2.13) give v.M0 /dt v.M/dt D d x0 d x:

(2.243)

On substituting (2.243) into (2.229a), we obtain the relation between elementary radius-vectors d x0 and d x: d x0 D d x C dt! d x C dtD d x C dt o.jd xj/:

(2.244)

The relation (2.244) can be considered as the transformation of coordinates dx i ! dx 0i in a small neighborhood of the point contained in a continuum. Since dt! and dtD are independent of d x and d x0 , so the transformation is

Fig. 2.13 Infinitesimal transformation of an elementary radius-vector

2.4 Rate Characteristics of Continuum Motion

61

linear, i.e. affine. The relation (2.244) can be represented as a superposition of two transformations up to an accuracy of o.jd xj/: d x00 D AD d x; d x0 D Q! d x00 ;

AD D E C dtD;

(2.245)

Q! D E C dt! E:

(2.246)

The tensor AD is symmetric and has three eigendirections, which are coincident with the eigendirections q˛ of the deformation rate tensor D. So just as the tensor U, the tensor AD transforms a small neighborhood of a point M by extending or compressing the neighborhood along the principal directions q˛ . The material segments jd x00˛ j oriented along the eigendirections q˛ retain their orientation under the transformations (2.245), but their lengths vary as follows: d x˛ D jd x˛ jq˛ ;

d x00˛ D .1 C D˛ dtjd x˛ j/q˛ D .1 C D˛ dt/d x˛ :

The tensor Q! (2.246) is orthogonal up to an accuracy of values .dt/2 , because Q! QT! D .E C dtW/ .E C dtWT / D E .dt/2 W2 :

(2.247)

Here we have taken into account that the vorticity tensor W is skew-symmetric. Thus, the transformation (2.246) determined by the tensor Q! is rotation of the M-point neighborhood as a rigid whole for infinitesimal time dt. The vorticity vector ! forming the tensor Q! can be considered as instantaneous angular rate of rotation of the small neighborhood as a rigid whole, or as instantaneous angular rate of rotation of the eigentrihedron q˛ of the deformation rate tensor relative to the fixed basis eN i . This fact will be considered in detail in Sect. 2.5.7. On uniting the properties of the transformations (2.245) and (2.246), we can make the following conclusion. Theorem 2.23. The infinitesimal transformation of a small neighborhood of the point contained in a continuum is a superposition of tension-compression of the neighborhood along the eigendirections q˛ and rotation of the axes q˛ as a rigid whole about the axis with the direction vector !. ı

Thus, we have the certain analogy between the eigendirections p˛ of the tensor U ı and the directions q˛ of the tensor D: elementary material fibres oriented along p˛ and along q˛ remain mutually orthogonal and undergo only tension-compression. ı

ı

The axes p˛ remain mutually orthogonal under any finite transformations from K to K, but q˛ – only under infinitesimal transformations from K.t/ to K.t C dt/.

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2 Kinematics of Continua

2.4.9 Kinematic Meaning of the Vorticity Vector Just as any orthogonal tensor, the orthogonal tensor Q! of infinitesimal rotation from K.t/ to K.t C dt/ can be represented in the form (see [12]) Q! D E cos.d'/ C .1 cos.d'//e ˝ e e E sin.d'/;

(2.248)

where d' is the infinitesimal angle of rotation of the trihedron q˛ about the axis with the direction vector e. Since values of d' are infinitesimal, we have Q! D E e Ed':

(2.249)

On comparing (2.249) with (2.246), we get !D

d' e; dt

j!j D

d' ; dt

(2.250)

i.e. the vorticity vector ! is really oriented along the instantaneous rotation axis e, and the length j!j is equal to the instantaneous angular rate of rotation of the trihedron q˛ of the deformation rate tensor. Let us consider now the question: relative to what system the vorticity vector ! defines the rotation rate. To answer the question, we introduce another orthogonal rotation tensor OW D qi ˝ eN i ;

(2.251)

which transforms the Cartesian trihedron eN i as a rigid whole into the orthonormal trihedron qi : qi D OW eN i :

(2.252)

The tensor OW is a function of time t, because qi D qi .t/. According to (2.240) and (2.252), the vorticity tensor W takes the form P W OT : W D qP i ˝ qi D O W

(2.253)

With the help of (2.253) we can represented the orthogonal tensor Q! as follows: P W OT : Q! D E C dtW D E C dt O W

(2.254)

Thus, at each time t two orthogonal tensors OW and Q! connect local neighı

borhoods of a point M in the reference K and actual configurations K.t C dt/. ı

If in K we consider an elementary radius-vector d x00 , then in K we find its

2.4 Rate Characteristics of Continuum Motion

63

ı

corresponding radius-vector d x00 obtained with the help of the rotation tensor OW , and in K.t C dt/ – radius-vector d x0 : ı

d x00 D OW d x00 ;

d x0 D Q! d x00 :

A fixed observer connected to the Cartesian trihedron eN i sees both the transformations: finite rotation for time t, which is described by the tensor OW , and instantaneous rotation of a local neighborhood for time dt, which is described by the infinitesimal rotation tensor Q! Thus, the vorticity vector ! is the vector of instantaneous angular rate of rotation of the trihedron q˛ relative to the trihedron eN i . Comparing (2.246) with (2.254) (or (2.253) with (2.227)), we get P W OT D ! E: O W

(2.255)

2.4.10 Tensor of Angular Rate of Rotation (Spin) P W OT , where OW is the orthogonal In Sect. 2.3 we have introduced the tensor O W P QT can be set up for any orthogonal tensor tensor of rotation. Such tensor D Q Q depending on time t. P QT is skew-symmetric, because The tensor Q

i.e.

P QT C Q Q P T D .Q QT / D .E/ D 0; Q

(2.256)

P QT /T D Q Q P T D Q P QT D : T D .Q

(2.257)

This tensor characterizes the angular rate of rotation of the orthonormal trihedron hi formed with the help of Q: hi D Q eN i ; (2.258) relative to the Cartesian trihedron eN i . P QT we can form the tensor (2.254): Indeed, with the help of the tensor Q P QT ; Q! D E C dt Q

(2.259)

which, according to (2.247), is the orthogonal tensor of infinitesimal rotation; and this tensor can be represented in the form (2.248) or (2.249): Q! D E d'e E;

(2.260)

where d' is the infinitesimal angle of rotation of the trihedron hi about the axis with the direction vector e. Comparing (2.259) with (2.260), we get the expression P QT D d' e E; Q dt

(2.261)

64

2 Kinematics of Continua

P T characterizes the instantaneous which makes clear the sentence that the tensor QQ angular rate d'=dt of rotation of the trihedron hi about the axis e. P QT is called the tensor of angular rate of rotation or the spin. The tensor Q Expressing the tensor Q from (2.258) in terms of the bases hi and eN i (h˛ D h˛ , ˛ D 1; 2; 3, as the vectors are orthonormal): Q D hi ˝ eN i ;

(2.262)

we get another representation of the spin: P QT D hP i ˝ hi : Q

(2.263)

Thus, we have proved the following theorem. Theorem 2.24. The spin connects the rates hP i and the vectors hi defined by formula (2.258) as follows: P QT / hi : hP i D .Q (2.264) T P Q is a skew-symmetric tensor, we can introduce the correSince the spin Q sponding vorticity vector !h : P QT D !h E Q

(2.265)

(from formulae (2.261) and (2.265) it follows that !h D .d'=dt/e); then formula (2.264) takes the form hP i D !h hi : (2.266) j

Resolving the vector !h for the orthonormal basis hi : !h D ! h hj ; we get one more representation of formula (2.264): j j hP i D ! h hj hi D j i k ! h hk :

(2.267)

This formula can also be rewritten in the form hP ˛ D !hˇ h ;

˛ ¤ ˇ ¤ ¤ ˛:

(2.268)

Taking different orthogonal tensors (or orthonormal bases) as Q (or hi ), we obtain different spins: ı

(1) If we choose eigenvectors of the stretch tensor U as hi , i.e. hi D pi , then, according to (2.263), the corresponding spin U takes the form ı

ı

P U OT D pi ˝ pi ; U D O U and formula (2.264) yields

P ı ı pi D U pi :

ı

OU D pi ˝ eN i ;

(2.269)

(2.270)

2.4 Rate Characteristics of Continuum Motion

65

(2) If hi D pi , then the corresponding spin V and the rotation tensor OV have the forms P V OT D pP i ˝ pi ; OV D pi ˝ eN i ; V D O (2.271) V pP i D V pi :

(2.272)

(3) If hi D qi , then the corresponding spin W coincides with the vorticity tensor W (see (2.253)): P W OT D qP i ˝ qi D W; W D O W

(2.273)

qP i D W qi :

(2.274)

(4) If we take the rotation tensor O accompanying deformation as Q, then, as shown ı in (2.159a), the tensor O connects two moving bases pi and pi : ı

pi D O pi :

(2.275)

The tensor O can be expressed in terms of OV and OU as follows: ı

ı

O D pi ˝ pi D pi ˝ eN i eN j ˝ pj D OV OTU :

(2.276)

The corresponding spin has the form P O T D .O P V OT C OV O P T / OU O T DO U U V T T T P P D OV OV C OV OU OU OV D V OV U OTV :

(2.277)

Unlike the cases (1)–(3), the spin tensor characterizes the angular rate of roı tation of the trihedron pi relative to the moving trihedron pi , but not relative to the trihedron eN i being fixed. Therefore, for the cases (1)–(3) the spins characterize the total angular rate, and for the case (4) – the relative rate.

2.4.11 Relationships Between Rates of Deformation Tensors and Velocity Gradients In continuum mechanics, one often needs the relations between rates of the deformation tensors (and also measures) and the velocity gradients L D .r ˝ v/T and ı T ı L D r ˝v : Let us derive these relations.

(2.278)

66

2 Kinematics of Continua

Theorem 2.25. The rates of varying the gradient FP and the inverse gradient .F1 / ı

are connected to L and L by the relations FP D L F;

.F1 / D F1 L;

ı

FP D L; ı

.F1 / D F1 F1T L:

(2.279)

H Differentiating the relationships (2.35a) with respect to time t and taking the definition of the velocity (2.181) into account, we get ı

FP T D r ˝ v D FT r ˝ v;

FP D .r ˝ v/T F:

(2.279a)

ı

According to the definitions of tensors L (2.223) and L (2.278), from (2.279a) we obtain formulae (2.279). Differentiating the identity .F F1 / D EP D 0; we find that FP F1 D F .F1 / I whence we get .F1 / D F1 FP F1 :

(2.280)

On substituting the first two formulae of (2.279) into (2.280), we obtain .F1 / D F1 .r ˝ v/T ;

.F1T / D .r ˝ v/ F1T ;

(2.281)

i.e. the third and the fourth relationships of (2.279) hold as well. N According to formulae (2.225), (2.279a), and (2.281), we find that the rate of the deformation gradient is connected to the deformation rate tensor D by the relations DD

1 P 1 .F F C F1T FP T /; 2

1 D D .F FP 1 C FP 1T FT /: 2

(2.282)

Here and below we will use the notation FP 1 .F1 / . P A, P ƒ, P JP and deformation Theorem 2.26. The rates of the deformation tensors C, ı P gP , .G1 / and .g1 / are connected to the velocity gradients L and L measures G, by the relationships 8 0, It is invariant relative to any transformation of coordinates (2.1) and relative to

any motion (2.3). Due to the third property, the mass in any actual configuration remains constant: M.B; t/ D const:

(3.1)

Remark. The mass conservation law is valid only for a continuum containing the same material points for a considered time interval Œ0; t. If a continuum B loses or acquires material points as time goes on (in this case phase transformations are said to occur in the continuum (see Sect. 5.1)), then the mass conservation law in the form (3.1) is no longer valid. The mass conservation law does not hold as well, if we exclude Axiom 3 and consider the motions with speeds close to the light speed; however, relativistic phenomena are not considered in classical continuum mechanics. t u The law (3.1) can be written in another form dM=dt D 0:

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 3, c Springer Science+Business Media B.V. 2011

(3.2)

89

90

3 Balance Laws

Since the mass is additive, M may be expressed as follows: Z M D d m;

(3.3)

V

where d m is the mass of an elementary volume dV involving a material point M belonging to the considered continuum V . Definition 3.1. The ratio D d m=dV

(3.4)

is called the density of substance at the point M. Since the mass M and volume dV are positive, the mass d m and density are always positive as well: > 0; d m > 0: (3.4a) On substituting (3.3) and (3.4) into (3.2), we can rewrite the mass conservation law (3.2) in the form Z d dV D 0: (3.5) dt V

Having applied this relationship to an elementary volume, we get ı

ı

dV D d V D const;

(3.6) ı

ı

ı

where and are the densities in configurations K and K, respectively, and d V is ı

the elementary volume in the configuration K. The relationship (3.5) is called the mass conservation law in the integral form, and (3.6) – in the differential form.

3.1.2 The Continuity Equation in Lagrangian Variables ı

ı

Consider in K an elementary volume d V constructed on elementary radius-vectors ı ı oriented along the local basis vectors d x˛ D r˛ dX ˛ (see Sect. 2.2.6). This volume corresponds to a volume dV constructed on vectors r˛ dX ˛ in the actual configuraı

tion K. The volumes d V and dV are determined by formulae (2.15): q ˇ @xı k ˇ ı ˇ ˇ 1 2 3 d V D r1 .r2 r3 /dX dX dX D g dX dX dX D ˇ i ˇdX 1 dX 2 dX 3 ; @X ˇ @x k ˇ p ˇ ˇ dV D r1 .r2 r3 /dX 1 dX 2 dX 3 D g dX 1 dX 2 dX 3 D ˇ i ˇdX 1 dX 2 dX 3 : @X (3.7) ı

ı

ı

ı

1

2

3

On substituting (3.7) into (3.6), we get the following theorem.

3.1 The Mass Conservation Law

91 ı

Theorem 3.1. A change of the density in going from K to K can be determined by one of the following equations: ı

D

s

g ı

D

g

ˇ @x k ˇ ˇ ˇ D ˇ ı ˇ D det F: ı j n i [email protected] [email protected] j @x [email protected] k [email protected] i j

(3.8)

Unlike (3.6), these equations are called the continuity equations in Lagrangian variables. ı The ratio of the elementary volumes in configurations K and K, which follows from (3.7), is also widely used: ı

q

dV =d V D

ı

g=g:

(3.9)

3.1.3 Differentiation of Integral over a Moving Volume Consider a vector field a.x i ; t/, which is a continuously differentiable function of x i and t in a domain V .t/, 8t > 0, where the domain V .t/ contains the same material points (such a domain is usually called a moving volume). i Let us integrate the field R a.x ; t/ over the domain V .t/ and determine the derivad tive of the integral: dt V .t / a dV: ı

To do this, we should change variables in the integral: x i ! x i , where x i 2 V , ı

ı

ı

i k i x q 2 V , and, according to (3.8), the Jacobian of the transformation is [email protected] [email protected] j D ı g=g. Then we get Z Z s ı d g d a dV : a dV D (3.10) ı dt dt g V .t /

ı

V

Since this change in variables means the transformation from the configuration K ı

ı

to K, where the domain V is time–independent, the derivative with respect to t may be introduced under the integral sign: d dt

Z

Z a dV D V .t /

ı

V

1 Z p ı d B g C ı 1 d p p da gC g dV : a q @ q aA d V D dt ı ı dt dt g g ı 0

V

(3.10a)

92

3 Balance Laws

On substituting (2.187) and (2.217) into (3.10a), we get d dt

Z q

Z a dV D

ı

V .t /

ı @a g=g ar v C C v r ˝ a dV @t ı

V

Z s

D

g ı

ı

g

@a C r .v ˝ a/ @t

ı

Z

dV D

@a C r .v ˝ a/ @t

dV:

V

V

(3.11) ı

The last equality we have obtained by the reverse substitution x i ! x i . Thus, we have proved following theorem. Theorem 3.2 (The Rule of Differentiation of the Integral over a Moving Volume). For any vector field a.x i ; t/ specified in V .t/ 8t > 0 and being a continuously differentiable function of x i and t, the following relationship holds: d dt

Z

Z a.x i ; t/ dV D

@a Cr v˝a @t

dV:

(3.12)

V

V .t /

Taking in formula (3.12) '.x i ; t/Ne˛ as a vector field a.x i ; t/, where eN ˛ is one of the Cartesian basis vectors, and '.x i ; t/ is a scalar field, and factoring eN ˛ outside the integral sign on the left and right sides, we get the formula for differentiation of the integral of a scalar field: d dt

Z

Z '.x i ; t/ dV D

@' C r .'v/ @t

dV:

(3.13)

V

V .t /

3.1.4 The Continuity Equation in Eulerian Variables Putting ' D in formula (3.13) and using the mass conservation law (3.5), we obtain Z @ C r v dV D 0 (3.14) @t V

– the mass conservation law in Eulerian variables. Unlike the law (3.5) which is formulated for a volume V of a continuum containing at different times t > 0 the same material points, the law (3.14) holds for an arbitrary geometric domain V of space E3a , which at different times t may involve different material points. Since formula (3.14) holds true for an arbitrary domain V E3a , the integrand must vanish. So we have proved the following theorem.

3.1 The Mass Conservation Law

93

Theorem 3.3. If the function , satisfying the mass conservation law (3.5), and the velocity v are continuously differentiable in V .t/ for all considered times t > 0, then at every point x of the domain V E3a of a continuum the following relation holds 8t > 0: @ C r v D 0: (3.15) @t Equation (3.15) is called the continuity equation in Eulerian variables. According to the definition (2.187) of the total derivative with respect to time, Eq. (3.15) can be rewritten in the form @ d C v r C r v D C r v D 0; @t dt

(3.16)

1 d D r v: dt

(3.17)

or

Due to (3.17), the continuity equation takes the form

d dt

1 D r v:

(3.18)

3.1.5 Determination of the Total Derivatives with respect to Time With the help of the continuity equation (3.15) we will prove several relationships which are frequently used in continuum mechanics. Theorem 3.4. If there is a tensor field n .x; t/ of the nth order, which is continuously differentiable in a domain V .t/ 8t > 0, then the following relations hold: @ n d n D C r .v ˝ n /; dt @t @ d .n / D . n / C r .v ˝ n /: dt @t

(3.19) (3.20)

H According to the definition of the total derivative with respect to time (2.192), we have n d n @ n D C v r ˝ dt @t @ n @ n 2 D 2 .v r ˝ n / C v r ˝ n : @t @t (3.21)

94

3 Balance Laws

Collecting the second and the third summands and taking the following consequence of the continuity equation into account: [email protected][email protected]/ C v r D r v, we finally get d n @ n @ n C r .v ˝ n /: (3.22) D C .r v/ n C v r ˝ n D dt @t @t To prove formula (3.20), it is sufficient to make the substitution n ! n in (3.19). N

3.1.6 The Gauss–Ostrogradskii Formulae In continuum mechanics, the Gauss–Ostrogradskii formulae are frequently used for tensor fields k .x/ of arbitrary kth order in reference and actual configurations: Z

Z

ı

ı

ı

n ˝ k d † D ı

†

Z

ı

r ˝ k d V ; ı

Z

V ı

ı

ı

†

Z

n k d † D

ı

Z

ı

r k d V ;

Z

r ˝ k dV: (3.23) V

n k d† D

ı

†

Z n ˝ k d† D

†

V

r k dV: V

According to the second formula of (3.23), Eq. (3.12) takes the form Z Z Z @a d a dV D dV C n v ˝ a d†: dt @t V .t /

(3.24)

V

(3.25)

†

Exercises for 3.1 3.1.1. Show that the continuity equation (3.8) can be written in the differential form: p d d g D p : dt g dt 3.1.2. Using formulae (3.10), (3.19), and (3.20), show that the following integral relations (called the rule of differentiation of the integral of a tensor field over a moving volume) hold: Z Z Z n d n @ d n n dV D dV D C r .v ˝ / dV dt dt @t V .t /

V .t /

Z

D V

@ n dV C @t

Z

V

n .v ˝ n / d†; †

3.2 The Momentum Balance Law and the Stress Tensor

d dt

Z

Z n dV D V .t /

d n dV D dt

D

95

@ n . / C r .v ˝ n / @t

dV

V

V .t /

Z

Z

@ n . / dV C @t

V

Z n .v ˝ n / d†:

†

3.1.3. Using formulae (3.19), show that for an arbitrary vector field a.X k ; t/ the following relation holds:

d dt

d a D a C a.r v/: dt

3.1.4. Similarly to formula (3.10), prove that for a tensor field k .x/ of arbitrary kth order the following formula of change of variables is valid: Z

Z k

k

dV D

ı

ı

d V :

ı

V

V

3.2 The Momentum Balance Law and the Stress Tensor 3.2.1 The Momentum Balance Law Definition 3.2. The vector Z

Z v dm D

ID V

v dV;

(3.26)

V

is called the momentum of a continuum. Remark 1. As in the mass conservation law (3.5), a domain V .t/ contains the same material points for all considered time interval. t u Remark 2. It should be noted that since the velocity v D d x=dt of a material point M, due to (2.181), has been introduced in a certain coordinate system O eN i , where x.M/ is the radius-vector of the point M, the momentum vector I is connected to this coordinate system. If we have chosen another coordinate system RO 0 e0i , then, applying formula (3.26), we obtain another momentum vector I0 D V v0 dV, where v0 D d x0 =dt, and x0 .M/ is the radius-vector of the same material point M t u but in the system O 0 e0i .

96

3 Balance Laws

This remark is of great importance for understanding the following axiom. Axiom 5 (The momentum balance law). For any two continua B and B1 at any time t, there exists a vector function F D F .B; B1 ; t/ (i.e. the transformation F W U U RC0 ! E3a ), which is possible to be zero-valued (i.e. F D 0). This vector function is called the force of interaction of the bodies B and B1 , and has the following properties: It is additive:

F .B 0 C B 00 ; B1 ; t/ D F .B 0 ; B1 ; t/ C F .B 00 ; B1 ; t/; F .B; B10 C B100 ; t/ D F .B; B10 ; t/ C F .B; B100 ; t/; where B D B 0 [ B 00 ; B1 D B10 [ B100, The rate of changing the momentum I of a continuum B at any time t is equal to the vector F .B; t/ D F .B; B e ; t/ which is the summarized vector of external forces, acting onto the body B (where B e D U n B is the outside of the body B): d I=dt D F :

(3.27)

The relationship (3.27) is called the momentum balance law. Remark 3. Since the vector I has been introduced in a certain coordinate system O eN i , the momentum balance law (3.27) is written for this coordinate system O eN i . In another coordinate system O 0 e0i the vector d I0 =dt may be different from d I=dt, and the law (3.27) also changes its form. The momentum balance law for such systems will be given in Sect. 4.10.11. The coordinate systems, in which the momentum balance law has the form (3.27), are called inertial (the system O eN i is also inertial). The existence of inertial coordinate systems is declared by the first Newton law, and Eq. (3.27) expresses the second Newton law. In continuum mechanics both these laws are introduced by Axiom 5 together with Axioms 1–3 on the existence of the special coordinate system O eN i . In more detail, noninertial coordinate systems will be considered in Sect. 4.9. u t As a body B, consider the elementary volume dV of a continuum involving a point M. Then, by Axiom 5, the summarized vector of external forces denoted by d F acts on dV. Here the following two cases are possible: (1) M is an interior point of the domain V . (2) M is on the surface † of the domain V (in this case the intersection of the surface † with closure of the domain d V is denoted by d†). Definition 3.3. The vector fD

dF dF D dm dV

(3.28)

3.2 The Momentum Balance Law and the Stress Tensor

97

is called the specific mass force and the vector s D d F=d†

(3.29)

is called the specific surface force. Here d m is the mass of an elementary volume dV of a continuum. Since the vector F is additive, we have the following relations for the whole continuum occupying the volume V in K: Z Fm D V

F D F m C F †; Z Z f d m D f dV ; F † D s d†: V

(3.30) (3.31)

†

The vector F m is called the summarized vector of external mass forces acting on the considered continuum, and F † is called the summarized vector of external surface forces. Due to (3.30), the momentum balance law (3.27) takes the integral form: Z Z Z d v dV D f dV C s d†: (3.32) dt V

V

†

3.2.2 External and Internal Forces Mass f and surface s forces are external forces for a continuum V , if they are induced by objects not belonging to the considered continuum V (i.e. by external objects). External mass forces may be of the following types: (1) the gravity force f D g† eN , where g† is the acceleration in a free fall at a surface of a planet (for the Earth g† 9:8 kg=m s2 ), and eN is the normal vector to the planet surface; (2) external forces of inertia caused by the motion of a body relative to a moving coordinate system (see Sect. 4.11); (3) electromagnetic forces. External surface forces are forces of interaction of two continua contacting with each other, for example, at impact of two solid bodies. Besides external forces, there are internal forces considered in continuum mechanics. Modify Eq. (3.32). According to the rule (3.12) and the continuity equation (3.15), we have d dt

Z

Z v dV D V

@ v C r .v ˝ v/ @t

dV

V

Z

D V

@ @v v C C .r v/v C v r ˝ v @t @t

Z dV D

dv dV : dt

V

(3.33)

98

3 Balance Laws

Fig. 3.1 Internal surface forces on the surface †0

Then Eq. (3.32) takes the form Z D

Z dv f dV C s d† D 0: dt

V

(3.34)

†

Hence, the acceleration .d v=dt/ is the specific mass force but it is an internal force caused by inertia effects. Therefore, it is also called the internal inertia force. Consider an example of internal surface forces. Take an arbitrary continuum volume V and divide the volume by a surface †0 into two parts V1 and V2 (Fig. 3.1). Let n be a normal vector to †0 at a point M 2 †0 , and the vector be directed outwards from V1 . Then each of volumes V1 and V2 may be considered as a separate continuum undergoing the action of the external forces F 1 and F 2 . If we consider the surface element d† 2 †0 , which contains the point M, then for the domain V1 , according to (3.29), the surface element d† undergoes the action of the surface force d F 1 (or the specific surface force s1 ), and for the domain V2 the same surface element d† undergoes the action of the surface force d F 2 (or the specific surface force s2 ). Denote these specific surface forces as follows: tn D s1 D d F 1 =d†

and tn D s2 D d F 2 =d†:

(3.35)

The vectors tn and tn are called the stress vectors. They are specific internal surface forces relative to the whole volume V of the continuum (because they are defined for interior points M of the continuum).

3.2.3 Cauchy’s Theorems on Properties of the Stress Vector The split of forces into external and internal ones is relative: the same forces may be internal or external with respect to different subdomains of a considered continuum. Let us write the momentum balance equation (3.34) for the whole volume V and for its separate parts V1 and V2 :

3.2 The Momentum Balance Law and the Stress Tensor

V1

Z

99

Z Z dv f1 dV C s1 d† C tn d† D 0; dt

Z

†1

†0

Z Z dv dV C s2 d† C tn d† D 0; f2 dt

V2

†2

(3.36)

(3.37)

†0

where fi and si are the forces acting in the volumes Vi and on their surfaces †i , i.e. f D fi in Vi

and s D si on †i :

Since all the functions si and fi are continuous, so having subtracted Eqs. (3.36) and (3.37) from (3.34), we get Z .tn C tn / d† D 0:

(3.38)

†0

Since the surface †0 is arbitrary, we get tn C tn D 0. Thus, we have proved the following theorem. Theorem 3.5 (The first Cauchy theorem – on continuity of the stress vector). For the same point M, which is an interior point of a volume V , the stress vectors defined relative to the surface elements nd†0 and .nd†0 / differ only in sign: tn D tn ;

(3.39)

i.e. the field tn .x/ is continuous in the volume V . Remark. The result (3.39) is a consequence of the assumption that there are no discontinuities of the functions, which was made at the derivation of Eq. (3.34). When the functions are not continuous (for example, for shock waves in a continuum), Eq. (3.39) does not hold. t u Let us consider one more important property of the stress vector. At any point M we may construct an elementary volume dV in the form of a tetrahedron (Fig. 3.2),

Fig. 3.2 The properties of the internal stresses

100

3 Balance Laws

whose edges are oriented along vectors dx˛ D r˛ dX ˛ . Let d†˛ be areas of three sides lying on the coordinate planes, and d†0 be the area of the inclined side of the tetrahedron. The exterior normal vector to the surface element d†0 is denoted by n. On the surfaces d†˛ , the normal vectors are defined as .r˛ =jr˛ j/, because the reciprocal basis vectors r˛ are orthogonal to the planes d†˛ . Areas of the sides d†0 and d†˛ are connected by the relation n d†0

3 X r˛ d†˛ D 0: jr˛ j ˛D1

(3.40)

This relation follows, for example, from the equation Z

Z n d† D †

Z n E d† D

†

r E dV D 0;

(3.41)

V

applied to the tetrahedron, where † and V are the total area and the total volume of the tetrahedron, respectively. Here we have used the Gauss–Ostrogradskii formula (3.23). The scalar product of (3.40) by r˛ yields jr˛ jn r˛ d†0 D d†˛ :

(3.42)

Having applied Eq. (3.34) to the tetrahedron, we get tn d†0 C

3 X ˛D1

dv dV D 0; t˛ d†˛ C f dt

(3.43)

where t˛ is the stress vector on the surface element d†˛ with the normal .r˛ =jr˛ j/, and t˛ – with the normal .r˛ =jr˛ j/. Taking formulae (3.39) and (3.42) into account, we have d v jdV j tn n r˛ jr jt˛ D f : dt d†0 ˛D1 3 X

˛

(3.44)

Since dV=d†0 is an infinitesimal value, we get the following theorem. Theorem 3.6 (The second Cauchy theorem). The stress vector tn on an arbitrary surface element with a normal n is expressed in terms of stress vectors t˛ on three coordinate areas as follows: tn D

3 X ˛D1

n r˛ jr˛ jt˛ :

(3.45)

3.2 The Momentum Balance Law and the Stress Tensor

101

Theorems 3.5 and 3.6 have been proved by Cauchy in another form. Since p jr˛ j D .r˛ r˛ /1=2 D g ˛˛ ;

(3.46)

Equation (3.45) takes the form tn D n T;

(3.47)

where T is the second-order tensor called the Cauchy stress tensor: TD

3 X

r˛ ˝ t˛ D ri ˝ ti ;

˛D1

t ˛ t˛

p g ˛˛ :

(3.48) (3.49)

Thus, Theorem 3.6 may be enunciated in another way. Theorem 3.6a (The Cauchy theorem). For a continuous in V [ † stress vector field s.x/ D tn .x/ satisfying Eq. (3.32), there always exists a tensor field T.x/ satisfying the relation (3.47) in V [ †.

3.2.4 Generalized Cauchy’s Theorem Theorem 3.6a may be stated for an arbitrary vector or even tensor field ˆ.x; t/ satisfying the equation, which is similar to (3.32). Theorem 3.6b. Let there be a continuously differentiable tensor field m A.x; t/, m > 0, and a continuous tensor field m C.x; t/ in a volume V , and there also be a continuous in V [ † vector field m Bn .x; t/ depending on the choice of normal vector field n.x; t/, which satisfy the equation d dt

Z

Z A dV D e V

Z C dV C

m

m

e V

m

e V; Bn d† 8V

(3.50)

e †

e is the boundary of the domain V e, then the field m Bn .x; t/ may depend where † on the field n.x; t/ only linearly, i.e. there exists a tensor field mC1 B.x; t/ of the .m C 1/th order, such that in V [ † the following relation holds: m

Bn D n

mC1

B:

(3.51)

H A proof of this theorem is similar to the proof of Theorem 3.6, where as the e we should take the tetrahedron, whose edges are oriented along the local domain V basis vectors ri (see Exercise 3.2.3). N Equation (3.50) is called the balance equation.

102

3 Balance Laws

3.2.5 The Cauchy and Piola–Kirchhoff Stress Tensors The Cauchy stress tensor T defined by formulae (3.47)–(3.49), as a second-order tensor, can be resolved for any tensor basis, for example: T D T ij ri ˝ rj D Tij ri ˝ rj :

(3.52)

The definition (3.48) allows us to give a geometric representation of the Cauchy stress tensor. Indeed, if we take vectors ri as the left vectors and consider ti as the right vectors, then we can represent the tensor T in terms of equivalency classes (see Sect. 2.1.4 and [12]): T D ri ˝ ti D Œr1 t1 r2 t2 r3 t3 : (3.53) According to the geometric definition of a tensor (see [12]), the tensor T can be represented as the ordered set of six vectors ri ; ti with the common origin at a considered point M (Fig. 3.3), where the basis vectors ri are defined. The Cauchy stress tensor T is defined on a deformable surface element d† in the actual configuration. We can determine the stress tensor on the corresponding ı

nondeformed surface element d † in the reference configuration as well. To do this, ı

write out the relation (2.116) which connects oriented surface elements in K and K: q n d† D

ı

ı ı

g=g n F1 d †0 ;

(3.54)

and consider the stress vector tn on the surface element d†: q tn d† D n T d† D

ı ı

ı

ı

ı

ı

ı

g=g n F1 T d † D n P d † D tn d †:

(3.55)

Here we have introduced the tensor q ı P D g=g F1 T;

Fig. 3.3 Geometric representation of the Cauchy stress tensor

(3.56)

3.2 The Momentum Balance Law and the Stress Tensor

103

Fig. 3.4 Geometric representation of the Piola–Kirchhoff stress tensor P

called the first Piola–Kirchhoff stress tensor, which is defined on the nondeformed ı

surface element d †. ı The vector tn is called the Piola–Kirchhoff stress vector; this vector is connected to the tensor P by the Cauchy formula (3.47): ı tn

ı

D n P:

(3.57)

With the help of the relations (2.35) and (3.48), the expression (3.56) can be represented in terms of equivalency classes (see Sect. 2.1.4) ı

ı

ı ı ı ı ı ı

P D ri ˝ ti D Œr1 t1 r2 t2 r3 t3 ;

(3.58)

where the following vectors have been denoted ı ˛

t D t˛

q

ı

g˛˛ g=g:

(3.59)

With the help of this expression we can also give a geometric representation of the ı Piola–Kirchhoff stress tensor in the basis ri of the reference configuration (Fig. 3.4).

3.2.6 Physical Meaning of Components of the Cauchy Stress Tensor Formulae (3.53) and (3.58) give a geometric representation of the tensors T and P. bij of the Cauchy stress tensor Let us explain a physical meaning of components T with respect to an orthonormal basis. Let there be a local basis ri in K, then, using the orthogonalization process, we can always construct an orthogonal basis e ri , and, normalizing the vectors e ri , we obtain the orthonormal basis b ri called the physical basis. In this basis the Cauchy bij , which are coincident with T bij : stress tensor (3.52) has components T bijb TDT ri ˝b rj :

(3.60)

104

3 Balance Laws

Fig. 3.5 Physical meaning of components of the Cauchy stress tensor

Take an arbitrary material point M 2 V and consider an elementary volume dV 2 V involving this point and having the form of a cube whose sides are orthogonal to the vectorsb r˛ (Fig. 3.5). Denote the area of the cube side orthogonal tob r˛ (˛ D 1; 2; 3) by d†˛ . Since we consider the elementary volume dV, the stress tensor T.x/ is the same at every point x 2 dV and x 2 d†˛ (˛ D 1; 2; 3). Each of the sides d†˛ undergoes the action of the surface force d F ˛ connected to the stress vector tn on this side by the relation (3.35): on d†˛ W

dF˛ D tn D n T: d†˛

(3.61)

ri : Resolving the vectors d F ˛ and n for the basisb d F ˛ D d F˛i b ri ;

n Db ni b ri ;

(3.62)

from (3.61) we get on d†˛ W

d F˛i T ji D b Db nj b T ˛i ; d†˛

˛ D 1; 2; 3;

(3.63)

because on d†˛ : b n˛ D 1, b nˇ D b n D 0, ˛ ¤ ˇ ¤ ¤ ˛. bij of Formula (3.63) allows us to explain the physical meaning of components T the Cauchy stress tensor: ˇ

˛ b˛˛ D d F˛ ; T b˛ˇ D d F˛ ; T b˛ D d F˛ ; ˛ ¤ ˇ ¤ ¤ ˛: on d†˛ W T d†˛ d†˛ d†˛ (3.64) ˛˛ b is the ratio of the corresponding normal component d F˛˛ The normal stress T of the surface force d F ˛ , acting on the surface element d†˛ , to the area of this b˛ˇ , b surface element; and the tangential stresses T T ˛ are the ratios of tangential ˇ components d F˛ , d F˛ of the same force d F ˛ , acting on the surface element d†˛ , to the area of the same surface element.

Remark 1. Since the tensor T is the same in the whole cube dV, by Theorem 3.5, the forces d F ˛ on the opposite sides of the cube differ only in sign. But the normals n

3.2 The Momentum Balance Law and the Stress Tensor

105

on these sides differ also only in sign, therefore relations (3.61) on these sides are the same as well as the relations (3.64). t u Remark 2. Due to Remark 1, there are three different relations in formulae (3.64) on each of three sides d†˛ , i.e. we have nine different relations: for three normal b˛˛ and six tangential stresses T b˛ˇ , ˛ ¤ ˇ. The pair tangential stresses stresses T b˛ˇ and T bˇ ˛ , defined on different sides d†˛ and d†ˇ , in general, are not coinciT dent (Fig. 3.6). Below, we will know that for many problems of continuum mechanics the stress tensor proves to be symmetric T D TT (due to additional assumptions); it is the b˛ˇ D T bˇ ˛ . only case when pair tangential stresses coincide: T t u b ij of the Piola– Let us clarify now the physical meaning of components P b ı Kirchhoff stress tensor P with respect to the orthonormal basis ri of the reference configuration: b ı ı b ijb ri ˝ rj : (3.65) PDP ı

ı

Take a material point M in K and consider an elementary volume d V being a cube ı b ı and containing this point. Let sides of the cube d †˛ be orthogonal to the vectors r˛ (Fig. 3.7).

Fig. 3.6 Difference of pair tangential stresses

Fig. 3.7 Physical meaning of components of the Piola–Kirchhoff stress tensor

106

3 Balance Laws ı

In configuration K a distorted volume dV corresponds to the cube d V . According to the geometric representation of transformation of a small neighborhood of a material point (see Sect. 2.3.5), the volume dV is a parallelepiped, in general, with inclined sides d†˛ remaining plane. ı

ı

For elementary volumes d V and dV the stress tensors T.x/ and P.x/ are the same ı

ı

at every point x 2 d V and x 2 d V . The deformed (but plane) side d†˛ undergoes the action of the surface force ı

ı

d F ˛ . Carry the force in the parallel way in K onto the corresponding side d †˛ , ı

then the relations (3.55) on the side d †˛ take the form 0 1 0 1 d F ˛ @ d†˛ A d† ı ˛ D tn @ ı A D n P: ı d†˛ d †˛ d †˛

ı

on d †˛ W

(3.66)

b ı ı Resolving the vectors d F ˛ and n for the basis ri : ı b ı d F ˛ D d F i˛ ri ;

b ı ı ı b n D ni ri ;

from (3.66) we get ı

ı

on d †˛ W

d F i˛ b ı b ij D P b ˛i ; D ni P d†˛

˛ D 1; 2; 3I

(3.67)

ı b b b ı ı ı because on †˛ : n˛ D 1, and nˇ D n D 0. As follows from (3.67), ı

ı

ı

b ˛ D d F ˛ ; ˛ ¤ ˇ ¤ ¤ ˛: on d †˛ W P ı d †˛ (3.68) ˛˛ b Thus, according to (3.68), the normal stress P is the ratio of normal compo˛ b ˛˛ D d F ˛ ; P ı d †˛

ı

ˇ b ˛ˇ D d F ˛ ; P ı d †˛

ı

nent d F ˛˛ of the surface force d F ˛ , acting on deformed surface element d†˛ , to ı

the area of corresponding undistorted surface element d †˛ . The tangential stresses ı ı b ˛ˇ , P b ˛ are the ratios of tangential components d F ˇ˛ , d F ˛ of the same surface P force d F ˛ , acting on the deformed surface element d†˛ , to the area of undistorted ı

surface element d †˛ . Remarks 1 and 2 for the tensor P are also valid with the exception of the fact that the tensor P remains nonsymmetric even if T is symmetric.

3.2 The Momentum Balance Law and the Stress Tensor

107

3.2.7 The Momentum Balance Equation in Spatial and Material Descriptions Substituting the notation (3.35) for internal surface forces and Cauchy’s formula (3.47) into (3.32), we get one more widely used form of the momentum balance law: Z

Z

d dt

v dV D V

Z n T d† C

†

f dV:

(3.69)

V

Having substituted (3.35) and (3.47) into (3.34): Z Z dv f dV D n T d†; dt V

(3.70)

†

and then having transformed the surface integral into the volume one by the Gauss– Ostrogradskii formula (3.24), we obtain Z dv f r T dV D 0: dt

(3.71)

V

Since the volume V is arbitrary, the integrand must always vanish. Thus, we have proved the following theorem. Theorem 3.7. If the functions F, v, T and f, which satisfy the momentum balance law (3.69) and depend on x i ; t, are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V .t/ the following momentum balance equation in Eulerian description (i.e. in K) holds:

dv D r T C f: dt

(3.72)

According to the property (3.20) of the total derivative, we can rewrite the momentum balance equation in the divergence form: @v C r v ˝ v D r T C f: @t

(3.73)

Modify the momentum balance equation (3.70) with taking (3.55) into account: Z ı

V

dv f dt ı

ı

Z

dV

ı

ı

n P d † D 0: ı

†

(3.74)

108

3 Balance Laws

With the help of the Gauss–Ostrogradskii theorem (3.24), we can rewrite this equation in the form Z ı ı ı dv f r P d V D 0: dt

(3.75)

ı

V ı

Since the volume V is arbitrary, we obtain the following theorem. Theorem 3.8. If the conditions of Theorem 3.7 are satisfied, then at every point ı

M 2 V for all considered times t > 0 the following momentum balance equation ı

in Lagrangian description (i.e. in K) holds: ı dv

ı

ı

D f C r P:

dt

(3.76)

Since in Lagrangian description the total derivative with respect to time coincides with the partial one (see (2.189)), the divergence form of the momentum balance ı

equation in K coincides with (3.76): ı @v

@t

ı

ı

D r P C f:

(3.77)

Exercises for 3.2 3.2.1. Using formulae (2.35a) and (3.56), show that the Piola–Kirchhoff tensor P ı ı has the components ..=/T ij / with respect to the mixed dyadic basis ri ˝ rj : ı

ı

P D .=/T ij ri ˝ rj : ı

3.2.2. Using the results of Exercises 3.2.1 and 2.1.12, show that components P ij of the tensor P with respect to the reference configuration basis can be written in the form ı

ı

ı

P D P ij ri ˝ rj ;

ı

ı

ı

F D F jm rj ˝ rm ;

ı

ı

ı

P ij D .=/T i m F jm :

3.2.3. Prove Theorem 3.6b by the same method as we used in the proof of Theorem 3.6.

3.3 The Angular Momentum Balance Law

109

3.3 The Angular Momentum Balance Law 3.3.1 The Integral Form Consider the next fundamental law of continuum mechanics. Definition 3.4. The vectors k0 D

Z V

0m 0†

Z x v dm D

Z D

V

Z

D

x v dV; Z

x f dm D V

x f dV;

(3.78)

V

x tn d†; 0 D 0m C 0† ;

†

are called as follows: k0 is the angular momentum vector of a continuum, 0m is the vector of mass moments of a continuum, 0† is the vector of surface moments of a continuum, 0 is the vector of moments of a continuum. Axiom 6 (The angular momentum balance law). For any continuum B there are two additive vector-functions: k00 .B; t/ called the own angular momentum vector and 00 .B; t/ called the vector of own moments, and also the positive constant s > 0 called the spin constant, such that 8t > 0 the following equation holds: dk D ; dt

(3.79)

where k is the vector of the total angular momentum of the continuum, and is the total moment vector: k D k0 C

1 00 k ; s

D 0 C 00 :

Remark. Axiom 6 postulates the existence of two new vectors: k00 and 00 , similarly to Axiom 5 introducing the force vector F . Physical interpretation of the vectors k00 and 00 is more complicated than for the force F (for example, effects caused by the gravity force or the inertia force are intuitively clear and well known). The appearance of the vectors k00 and 00 can be caused by electromagnetic effects in continua, which have an ordered magnetic structure (for example, ferromagnetic materials). However, the vectors k00 and 00 can appear for some continua, where there are no electromagnetic effects (such continua are often called the Cosserat continua). It should be noted that although the spin constant s in the law (3.79) seems to be unnecessary (in place of .1=s /k00 we could introduce the vector k00 ), its appearance is very essential; this will be explained in Sect. 3.4.6. t u

110

3 Balance Laws

Since the vectors k00 and 00 are additive, so, according to Sect. 3.2.1, we obtain the following classification. Definition 3.5. The specific own angular momentum is the vector km , the specific own mass moment is the vector hm , and the specific own surface moment is the vector h† , which are defined at a point M of a continuum as follows: d k00 ; dm

km D

hm D

d 00 ; dm

h† D

d 00 : d†

(3.80)

Since the vectors k00 and 00 are additive, for the whole considered volume of a continuum we have Z (3.81) k00 D km dV; 00 D 00m C 00† ; V

where the following notation has been introduced: 00m

Z D

hm dV;

00†

V

Z D

h† d†;

(3.81a)

†

On substituting (3.78) and (3.81) into (3.79), the angular momentum balance law takes the integral form d dt

Z Z Z x v C km dV D .x f C hm / dV C .x tn C h† / d†: s

V

V

†

(3.82)

3.3.2 Tensor of Moment Stresses Equation (3.82) has the form of (3.50), if we introduce the following notation: A D x v C km ;

C D x f C hm

and

B D x tn C h† :

Then to Eq. (3.82) we can apply Theorem 3.6b, which asserts the existence of a e t/, defined in V [ †, such that the following relation second-order tensor field M.x; being an analog of (3.47) holds:

With new notation

e h† C x tn D n M:

(3.83)

e C T x; MDM

(3.84)

this result can be written in another form.

3.3 The Angular Momentum Balance Law

111

Theorem 3.9. For vector fields tn and h† being continuous in V [ † and satisfying Eqs. (3.32) and (3.82), there exists a second-order tensor field M.x; t/ such that in V [ † the following relation holds: h† D n M:

(3.85)

The tensor M is called the tensor of moment stresses.

3.3.3 Differential Form of the Angular Momentum Balance Law ı

At first, consider the left-hand side of (3.82) and pass from V to V : d dt

Z ı

ı 1 ı x v C km d V s

V

d D dt Z D V

Z ı

Z ı 1 1 d km ı ı d x v C km d V D dV .x v/ C s dt s dt ı

ı

V

V

Z dv dv 1 d km 1 d km vvCx dV D x dV ; C C dt s dt dt s dt V

(3.86) as v v D 0. Modify the right-hand side of Eq. (3.82) by using the Gauss–Ostrogradskii theorem (3.24) and formula (3.47): Z Z Z Z x tn d† D x .n T/d† D n .T x/ d† D r .T x/ dV : †

†

†

V

(3.87)

The nabla-operator r in (3.87) is determined by the formula @ .T x/ D .r T/ x C ri T ri @X i D x .r T/ C " T:

r .T x/ D ri

(3.88)

Substitution of (3.88) into (3.87) yields Z

Z x tn d† D

†

Z x r T dV

V

" T dV: V

(3.89)

112

3 Balance Laws

Having substituted (3.89) and (3.86) into (3.82), we get Z V

Z dv 1 d km x f r T dV C dV dt s dt V Z Z Z D hm dV C r M dV " T dV: V

V

(3.90)

V

Here we have used the definition of the tensor of moment stresses M (3.84). Due to formula (3.72), Eq. (3.90) takes the final form Z V

d km hm r M C " T dV D 0: s dt

(3.91)

Since the volume V is arbitrary, we get the following theorem. Theorem 3.10. If the functions F, v, T, f, satisfying Eq. (3.69), and also the functions km , hm , M, satisfying Eq. (3.90), are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V .t/ the following angular momentum balance equation holds: d km D hm C r M " T: s dt

(3.92)

Thus, the angular momentum balance equation includes only one classical mechanical characteristic, namely the stress tensor T, the remaining functions km , hm and M, as it was mentioned above, either have a nonmechanical nature or are caused by nonclassical mechanical properties.

3.3.4 Nonpolar and Polar Continua For classical continua, for which there are not any nonmechanical values, we have km D 0;

hm D 0;

M D 0:

(3.93)

Such continua are called nonpolar, and for them it follows from (3.93) and (3.92) that 3 X p "T D g .T ˛ˇ T ˇ ˛ /r D 0; (3.94) ˛D1

˛¤ˇ¤ ¤˛

and hence T D TT ; i.e. the Cauchy stress tensor for nonpolar continua is symmetric.

(3.95)

3.3 The Angular Momentum Balance Law

113

It should be noted that the corresponding Piola–Kirchhoff tensor is not symmetric even for nonpolar continua; that follows from its definition (3.56). Thus, for nonpolar continua the angular momentum balance law reduces to the condition (3.95) that the Cauchy stress tensor T is symmetric. A continuum, for which values of km , hm and M are different from identically zero: km ¤ 0; hm ¤ 0; M ¤ 0; (3.96) is called polar. For polar continua, the Cauchy stress tensor T is not symmetric.

3.3.5 The Angular Momentum Balance Equation in the Material Description The integral form of the angular momentum balance equation (3.82) can be represented in the material description. To do this, we should modify the left-hand side of formula (3.82) according to (3.86): d dt

Z V

Z ı 1 1 d ı x v C km dV D x v C km d V s dt s ı

V

ı dv 1 d km x dV C dt s dt

Z

ı

D ı

(3.97)

V

ı

ı

and introduce Lagrangian tensor of moment stresses M in K similarly to the Piola–Kirchhoff stress tensor: ı

ı

ı

n M d† D n M d †;

(3.98)

ı

ı

and the specific own surface moment h† in K: ı

ı

ı

h† D n M:

(3.99) ı

Then the surface integral in (3.82) can be written in K as follows: Z

Z .x tn C h† / d† D †

Z n .T x C M/ d† D

†

ı

ı

ı

n .P x C M/ d †; ı

†

(3.100)

114

3 Balance Laws

and Eq. (3.82) takes the form Z ı d 1 ı x v C km d V dt s ı

V

Z

Z

ı

ı

.x f C hm / d V C ı

ı

ı

ı

n .P x C M/ d †:

(3.101)

ı

V

†

Exercises for 3.3 3.3.1. Show that from (3.101) we can get the angular momentum balance equation in the material description ı

ı ı d km ı D hm C r M " .F P/: s dt

3.3.2. Show that for nonpolar continua the angular momentum balance equation in the material description (see Exercise 3.3.1) takes the form " .F P/ D 0

F P D PT F T

or

and coincides exactly with (3.96). 3.3.3. Using equation (3.98) and the formula of transformation of an oriented surı

face element, show that the tensors M and M are connected by the relation q ı ı M D g=g F1 M:

3.4 The First Thermodynamic Law 3.4.1 The Integral Form of the Energy Balance Law The mass, momentum and angular momentum balance laws describe the motion of a continuum. If we need thermal effects to be taken into account, then we must use thermodynamic laws. Let us consider nonpolar continua, for which the relations (3.93) are satisfied. Definition 3.6. The scalar function Z Z jvj2 vv KD dm D dV; 2 2 V

V

is called the kinetic energy of a continuum V .

jvj2 D v v

(3.102)

3.4 The First Thermodynamic Law

115

Definition 3.7. The power of external mass forces Wm and the power of external surface forces W† are the following scalar functions: Z

Z

Wm D

f v dm D V

Z f v dV;

W† D

V

tn v d†:

(3.103)

†

Axiom 7 (The first thermodynamic law (or the energy balance law)). For any continuum B there are two scalar additive functions: U.B; t/ called the internal energy of the continuum and Q.B; t/ called the heating rate for the continuum such that 8t > 0 the following equation is satisfied: dE D W C Q; dt

(3.104)

where E is called the total energy of the continuum and consists of U and K: E D U C K;

W D W m C W† :

(3.105)

Remark. There are different statements of the thermodynamic law, and the above is called Truesdell’s statement [54, 55]. The statement is convenient, because its form corresponds to Axioms 4–6. Moreover, this statement (unlike others) is really universal, i.e. it is independent of the type of a continuum. t u Since the functions U and Q are additive, similarly to Sect. 3.2.1 we can introduce the corresponding specific functions. Definition 3.8. The specific internal energy is the function e, the specific heat flux from mass sources is the function qm and the specific heat flux from surface sources is the function q† , which are defined at every point M of a continuum as follows: eD

dU ; dm

qm D

dQ ; dm

q† D

dQ : d†

(3.106)

Since the functions Q and U are additive, for a whole continuum we have Z Qm D V

Q D Qm C Q † ; Z Z qm d m D qm dV; Q† D q† d†; V

Z

Z e dm D

U D V

(3.107)

†

e dV: V

Substitution of Eqs. (3.102), (3.103), (3.105)–(3.107) into (3.104) yields the energy balance law in the integral form

116

3 Balance Laws

d dt

Z

Z Z jvj2 eC dV D .f v C qm / dV C .tn v C q† / d†: (3.108) 2

V

V

†

Using on the left-hand side of (3.108) the differentiation rules for a volume integral (see Exercise 3.1.2): d dt

Z

Z jvj2 de dv eC dV D dV; Cv 2 dt dt

V

(3.109)

V

we obtain the following form of the energy balance law: Z

Z d 1 e C jvj2 C f v C qm dV C .tn v C q† / d† D 0: (3.110) dt 2

V

†

3.4.2 The Heat Flux Vector Equation (3.108) has the form (3.50), if in (3.50) we put A D e C jv2 j=2;

C D f v C qm

and B D tn v C q† :

Then we can apply Theorem 3.6b to this equation; the theorem claims the existence of the vector .q/ such that in V [ † the following relation holds: q† D n q:

(3.111)

The vector q is called the heat flux vector. Prove the analog of Theorem 3.6 allowing us to determine components of the vector q. As was made in Sect. 3.2.3, at a point M we construct an elementary volume dV in the form of a tetrahedron, whose edges are oriented along vectors r˛ dX ˛ . Then for the volume all formulae (3.40)–(3.42) hold. Applying Eq. (3.110) to the tetrahedron, when there is no action of any mass or surface forces on the continuum (i.e. tn D 0, f D 0), we get q† d†0 C

d jvj2 q˛ d†˛ C qm eC dV D 0: dt 2 ˛D1 3 X

(3.112)

Here q˛ is the heat flux from surface sources on the surface element d†˛ with the normal .r˛ =jr˛ j/, and q† – on the surface element d†0 .

3.4 The First Thermodynamic Law

117

Then with taking (3.42) into account, we have d jvj2 dV q† C n r˛ jr jq˛ D qm : eC dt 2 d† 0 ˛D1 3 X

˛

(3.113)

Since the value dV=d†0 is infinitesimal, we get the following theorem. Theorem 3.11. On every surface element with a normal n, the heat flux from surface sources q† is expressed in terms of heat fluxes on three coordinate surface elements: 3 X p n r˛ g ˛˛ q˛ : (3.114) q† D ˛D1

Comparing (3.114) with (3.111), we find components of the heat flux vector: qD

3 X

r˛

p g ˛˛ q˛ D ri q i ;

q ˛ D q˛

p g ˛˛ :

(3.114a)

˛D1

3.4.3 The Energy Balance Equation Let us consider now formula (3.110) and transform the surface integrals by the Gauss–Ostrogradskii formula (3.24) with taking (3.111) into account as follows: Z Z Z q† d† D n q d† D r q dV; Z

†

Z

tn v d† D †

†

V

Z

n .T v/ d† D †

(3.115) r .T v/ dV:

V

Substituting the expression (3.115) into (3.108), we obtain another integral form of the energy balance law: d dt

Z V

Z Z jvj2 eC dV D .f v C qm / dV C n .T v q/ d†: (3.116) 2 V

†

Substitution of formulae (3.115) into (3.110) yields Z d jvj2 =2 de f v qm C r q r .T v/ dV D 0: (3.117) C dt dt V

Since the volume V is arbitrary, we get the following theorem.

118

3 Balance Laws

Theorem 3.12. If the functions F, v, T, f, qm , q and e, satisfying Eq. (3.116), are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V .t/ the energy balance equation has the following differential form: d dt

jvj2 eC D f v C qm r q C r .T v/: 2

(3.118)

According to formula (3.20), we can rewrite the energy balance equation in the divergence form @ C r .v/ D r q C r .T v/ C f v C qm ; @t

DeC

jvj2 ; (3.119) 2

where is the specific total energy of the continuum.

3.4.4 Kinetic Energy and Heat Influx Equation The scalar product of the momentum balance equation (3.72) by v yields

dv v D f v C .r T/ v: dt

(3.120)

Using the property of covariant differentiation (see Exercise 2.1.17), we can reduce this equation to the form 1 d jvj2 D f v C r .T v/ T .r ˝ v/T : 2 dt

(3.121)

Definition 3.9. The scalar function Z W.i / D

T .r ˝ v/T dV

(3.122)

V

is called the power of internal surface forces of a continuum V . Integrating Eq. (3.121) over a volume V and using Definitions 3.6, 3.7, 3.9, and the rule of differentiation of an integral over a moving volume (see Exercise 3.1.2), we get the following theorem. Theorem 3.13 (The kinetic energy balance equation). A change of the kinetic energy of a continuum is equal to the summarized power of external and internal forces acting on the continuum: dK D W C W.i / : dt

(3.123)

3.4 The First Thermodynamic Law

119

Subtracting the kinetic energy equation (3.121) from the energy balance equation (3.118), we obtain the differential heat influx equation

de D T .r ˝ v/T C qm r q: dt

(3.124)

This equation can be rewritten in the divergence form @e C r .ev/ D r q C T .r ˝ v/T C qm : @t

(3.125)

Subtracting Eq. (3.123) from (3.104), we get the heat influx equation in the integral form d (3.126) U C W.i / D Q: dt

3.4.5 The Energy Balance Equation in Lagrangian Description Let us write now the energy balance equation in Lagrangian description. Use the equation in the integral form (3.108) and transform the surface integrals as follows: Z

Z .tn v C q† / d† D

†

Z q D

n .T v q† / d† † ı

ı ı

g=g n .F1 T v F1 q† / d † D

ı

Z

ı ı n P v q d†:

ı

†

†

(3.127) Here we have used the definition (3.56) of the Piola–Kirchhoff tensor and introduced the heat flux vector in the reference configuration: q

ı

qD

ı

g=g F1 q:

(3.128)

ı

Theorem 3.14. Heat flux vectors q and q are connected by the relation ı

ı

ı

ı

ı

q n d † D q n d†:

(3.129)

H Indeed, multiplying (3.128) by n d † and using formula (2.116), we obtain ı

ı

ı

q n d† D

q

ı

ı

ı

g=g q F1T n d † D q n d†: N

120

3 Balance Laws

Then Eq. (3.108) in the reference configuration takes the form Z ı ı ı ı jvj2 ı d ı ı eC f v qm r .P v/ C r q d V D 0: dt 2

(3.130)

ı

V

ı

Since the volume V is arbitrary, the integrand vanishes. Thus, we have proved the following theorem. ı

Theorem 3.15. Under the conditions of Theorem 3.12, at every point M 2 V for all considered times t > 0 we have the energy balance equation in Lagrangian description ı

d dt

ı ı ı jvj2 ı ı eC D f v C qm C r .P v/ r q: 2

(3.131)

Similarly to formula (3.120), we can write the kinetic energy balance equation in the reference configuration ı dv

dt

ı

ı

v D f v C v .r P/;

(3.132)

with the help of which Eq. (3.131) takes the form ı

e

ı ı ı d ı D P .r ˝ v/T C qm r q: dt

(3.133)

This equation is called the heat influx equation in material (Lagrangian) description. Since in Lagrangian description the total derivative coincides with the partial one, the divergence form of the heat influx equation does not differ from (3.133): ı

ı @e

ı

ı

ı

D r q C P .r ˝ v/T C qm :

@t

(3.134) ı

Integrate the heat influx equation (3.133) over the domain V and take into account that the internal energy U and the heating rate Q, according to (3.107), can also be represented in Lagrangian description: Z U D Z Qm D

Z e dV D

V

ı

Q D Q m C Q† ;

ı

Z

qm dV D V

ı

e d V ; V

ı

ı

qm d V ; ı

V

Z Q† D

Z q† d† D

†

ı

ı

q † d †; ı

†

(3.135)

3.4 The First Thermodynamic Law

121

here we have used Eq. (3.129). As a result, from (3.133) we get the heat influx equation in the integral form (3.126) again, where the power of internal surface forces is defined by the following relations in Eulerian and Lagrangian descriptions, respectively: Z W.i / D

Z T .r ˝ v/T dV D

ı

ı

P .r ˝ v/T d V :

(3.136)

ı

V

V

According to relation (3.55), we can represent the power of external forces (3.103) in Lagrangian description as follows: Z Wm D

Z f v dV D

Z

ı

ı

f v d V ; W† D ı

V

Z tn v d† D

ı

v d †: (3.137)

ı

†

V

ı tn

†

3.4.6 The Energy Balance Law for Polar Continua Let us consider now polar continua, for which the relations (3.96) hold, i.e. the values of km , hm and M are different from identically zero. The energy balance law for polar continua differs from the corresponding law (3.104), (3.105) for nonpolar continua. Similarly to (3.102), introduce the kinetic energy of the own motion of a continuum: Z Z jkm j2 Ks D dm D jkm j2 dV; jkm j2 D km km : (3.138) 2s s V

V

Similarly to (3.103), introduce the power of mass moments Wms and the power of surface moments W†s : Z Wms D

hm km dV; V

Z W†s

D

Z

h† km d† D †

(3.139)

n M km d†:

(3.140)

†

The energy balance law for a polar continuum has the universal form (3.104), but the total energy E and the summarized power W of a continuum are defined not by (3.105) but by the formulae E D U C K C Ks ;

W D Wm C Wms C W† C W†s :

(3.141)

122

3 Balance Laws

As a result, we obtain the following form of the energy balance law for a polar continuum: d .U C K C Ks / D Wm C Wms C W† C W†s C Q: dt

(3.142)

From (3.142) with the help of (3.138)–(3.140) we get the integral form of the law: Z Z d jvj2 jkm j2 eC C dV D .f v C hm km C qm / dV dt 2 2s V V Z C .t† v C h† km C q† / d†: (3.143) †

Similarly to Sect. 3.4.2, the heat flux vector q for polar continua is introduced by formulae (3.111) and (3.114). Thus, in Eq. (3.143) we can transform the surface integral to the volume one, then we obtain the following differential form of the energy balance law for a polar continuum (the energy equation):

d dt

jvj2 jkm j2 eC D f v C hm km C qm r q C r .T v C M hm /: C 2 2s (3.144)

Remark. The energy equation (3.143) clarifies the meaning of introduction of the spin constant s in the angular momentum balance law (3.79): if in (3.79) or in (3.92) we eliminated the constant s by introducing the vector e km D km =s instead of km , then the energy equation (3.144) would include the constant s all the same. This constant s plays a role of the energetic equivalent between the kinetic energy of a body and the kinetic energy of the own body motion caused by the own angular momentum for polar continua. t u According to relation (3.20), the energy equation (3.144) takes the divergence form @ " C r .v" C q T v M km / D f v C hm km C qm ; @t jvj2 jkm j2 : "DeC C 2 2s

(3.145)

Let us formulate now an analog of the kinetic energy balance equation (3.123) for the kinetic energy of own motion Ks . Consider the angular momentum balance equation (3.92) and multiply the equation by km : d km km D hm km C km r M km 3 " T: s dt

(3.146)

According to the property of the nabla-operator from Exercise 2.1.17, we can reduce Eq. (3.146) to the form

3.4 The First Thermodynamic Law

123

d jkm j2 D hm km C r .M km / M .r ˝ km /T km 3 " T: (3.147) s dt Integrating this equation over V and using the Gauss–Ostrogradskii formula and the differentiation rule of an integral over a moving volume (3.12), from (3.147) we get dKs D Wms C W†s C W.is / C Ws : dt

(3.148)

Here we have introduced the following notation: Z W.is / D M .r ˝ km /T dV

(3.149)

V

– the power of internal surface moments and Z Ws D

km 3 " T dV

(3.150)

V

– the power of internal surface forces for the own body motion. Relation (3.148) is called the kinetic energy balance equation for the own body motion. Subtracting Eqs. (3.123) and (3.148) from the energy balance law (3.142), we get the integral heat influx equation for polar continua: dU D Q W.i / W.is / Ws : dt

(3.151)

Subtracting Eqs. (3.147) and (3.121) from the energy equation (3.144), we obtain the differential heat influx equation for polar continua:

de D T .r ˝ v/T C M .r ˝ km /T C km 3 " T r q C qm : (3.152) dt

If we assume km D 0, M D 0 and hm D 0, then the heat influx equation in the forms (3.151) and (3.152) coincides with the corresponding equations (3.124) and (3.126) for nonpolar continua. ı

According to the definition (3.97) of Lagrangian tensor of moment stresses M, we can rewrite the heat influx equation (3.152) in the material description ı de

dt

ı

ı

ı

ı

ı

ı

D P .r ˝ v/T C M .r ˝ km /T C km 3 " .F P/ r q C qm : (3.153)

This assertion should be proved as Exercise 3.4.1.

124

3 Balance Laws

Exercises for 3.4 3.4.1. Prove the relation (3.153).

3.5 The Second Thermodynamic Law 3.5.1 The Integral Form To state the second thermodynamic law, we introduce a new local characteristic of a continuum which is called a temperature. Axiom 8 (on the existence of an absolute temperature). For every material point M of any continuum B for all t > 0 there is a scalar positive function .X i ; t/ D .x i ; t/ > 0;

(3.154)

called an absolute temperature. The axiom on the existence of an absolute temperature is sometimes called the zero thermodynamic law. It should be noted that we have introduced a temperature as local but not integral characteristic of a continuum (unlike mass M or angular momentum). Although we R can always introduce the integral temperature ‚, for example, as ‚ D V d m, but this value does not appear in any fundamental laws of continuum mechanics. However, there are integral characteristics containing a temperature, which are of great importance for formulating the second thermodynamic law. Definition 3.10. The entropy production by external mass sources is the scalar function QN m and the entropy production by external surface sources is the scalar function QN † , which are defined for a volume V as follows: QN m D

Z V

qm dm D

Z V

qm dV;

QN † D

Z

q† d† D

†

Z

nq d†: (3.155)

†

Remark. If by analogy with (3.106) we consider the entropy production by external mass sources d QN m , supplied to elementary volume dV by mass d m, then from (3.155) and (3.106) we get d QN m D qm d m= D dQ=;

(3.155a)

i.e. d QN m is the heat influx due to external mass sources divided by temperature of the volume dV heated. Thus, d QN m characterizes the efficiency of heating the volume dV, and d QN † characterizes the efficiency of heating the surface element d†. u t

3.5 The Second Thermodynamic Law

125

Axiom 9 (The second thermodynamic law). For any continuum B there are two scalar additive functions: H.B; t/ called the entropy of the continuum and QN .B; t/ called the entropy production by internal sources, such that for all t > 0 the following equation holds: dH D QN C QN ; (3.156) dt where we have denoted the summarized entropy production by external sources as follows: (3.157) QN D QN m C QN † ; and a value of QN is always nonnegative (the Planck inequality): QN > 0:

(3.158)

Since the functions H and QN are additive, we can introduce the corresponding specific functions at every point M 2 V . Definition 3.11. The specific entropy and the specific internal entropy production q are defined for every point M of a continuum as follows: D

dH ; dm

q D

d QN : dm

(3.159)

As follows from the Planck inequality (3.158), and also the inequalities (3.154) and (3.4a), the function q is always nonnegative: q > 0:

(3.158a)

This inequality is also called the Planck inequality. Since the functions H and QN are additive for a whole continuum, we have QN D

Z

q dm D

V

Z

q dV; H D

V

Z

Z dm D V

dV:

(3.160)

V

Two relations (3.156) and (3.158) are equivalent to the Clausius inequality N dH=dt > Q:

(3.161)

On substituting (3.155) and (3.160) into (3.156), we obtain the integral form of the second thermodynamic law d dt

Z

Z dV D

V

V

.qm C q / dV

Z †

nq d†:

(3.162)

126

3 Balance Laws

3.5.2 Differential Form of the Second Thermodynamic Law Transform the surface integral in (3.162) by the Gauss–Ostrogradskii formula (3.24): Z Z Z q r q 1 nq d† D dV: (3.163) d† D r Cqr †

V

V

Then Eq. (3.162) takes the form Z q d .qm C q / dV D 0: Cr dt

(3.164)

V

Since the volume V is arbitrary, we get the following theorem. Theorem 3.16. If the functions , q, q , qm and , satisfying Eq. (3.162), are continuously differentiable in V .t/ for all considered times t > 0, then at every point M 2 V the second thermodynamic law has the following differential form (the entropy balance equation):

q d qm C q C D r : dt

(3.165)

According to the property (3.20), we can rewrite Eq. (3.165) in the divergence form @ qm C q q D C r v C : (3.165a) @t According to the equation r

q

D

1 1 r q 2 q r ;

formulae (3.165) and (3.165a) can be rewritten in the form

d D r q C qm C w ; dt

(3.166)

or

@ C r v D .1=/r q C .1=/.qm C w /: @t Here we have introduced the function w D q C

q r ;

(3.166a)

(3.167)

called the dissipation function. Besides the Planck inequality (3.158), there is one more important inequality in continuum mechanics.

3.5 The Second Thermodynamic Law

127

Axiom 10. At every point M of a continuum for all t > 0 the Fourier inequality holds: q r 6 0: (3.168) Rewrite (3.167) in the form q D w

q r :

(3.169)

Equation (3.169) means that the specific internal entropy production is caused by two factors: dissipation (i.e. irreversible conversion of part of mechanical energy to heat energy) and nonuniform heating of a continuum (when there are irreversible processes of heat transfer from hotter parts of the continuum to its colder ones). The first cause corresponds to the function w , and the second one corresponds to the function 1 q r wq : (3.170) A mathematical expression for the dissipation function w will be given in Chap. 4; however, it should be noted here that if w is independent of the temperature gradient r (in Chap. 4 we will show that this assumption always holds), then, ı

considering the motion of a continuum from K to K only with uniform temperature field D .t/ (r 0 in V ), from (3.169) and the Planck inequality we get the dissipation inequality w > 0: (3.171) Since the function w is independent of r , the inequality (3.171) holds under nonuniform heating as well.

3.5.3 The Second Thermodynamic Law in the Material Description In the material description, Eq. (3.162) takes the form Z

ı d

ı

dt

ı

Z

dV D ı

V

ı

.qm C q / ı dV

V

Z ı

ı

ı

nq ı d †;

(3.172)

†

then with the help of the usual manipulations we get the second thermodynamic law in the material description ı d

dt

ı

D r

ı

q

!

ı qm

C

C q :

(3.173)

128

3 Balance Laws

This equation can be rewritten in the form ı d

dt

ı

ı

ı

ı

D r q C qm C w ;

where

ı

q ı r > 0

ı

ı

(3.174)

w D q C

(3.175)

ı

is the dissipation function in K. The Clausius inequality (3.161) in the material (Lagrangian) description retains its form, but entropy H and entropy production QN by external sources, defined by (3.160), in Lagrangian description take the forms Z H D

Z dV D

Z V

QN D QN m C QN † ;

ı

V

QN m D

ı

ı

d V ;

qm dV D

Z ı

V

QN D

V ı

qm ı dV ;

QN † D

Z

q† d† D

†

Z

q dV D

V

Z ı

Z ı

ı

q† ı d †; (3.176)

†

ı

q ı dV :

V

3.5.4 Heat Machines and Their Efficiency The present-day heat machines are facilities, which convert heat into mechanical work (for example, internal-combustion engines, jet and turbo-jet engines, gasturbine engines, nuclear engines etc.), or, on the contrary, convert mechanical work into heat (for example, some heating and cooling devices), or use both the conversions in turn (for example, diesel engines). The operation of all the heat machines is based on the thermodynamic laws. Conventionally, the structure of a heat machine consists of a vessel with solid walls, which is filled with a working body of volume V (it is usually gas or fluid). Heat machines can be split into the three main types: (1) Machines, for which the domain V .t/ is varying but material points of a ı

working body are the same, i.e. V D const. (2) Machines, for which V remains unchanged with time but material points of a working body, occurring in V , are different. (3) Machines, for which the domain V is varying and material points occurring in V are different.

3.5 The Second Thermodynamic Law

129

Fig. 3.8 The scheme of three main types of heat machines

These three types of heat machines are shown in Fig. 3.8. The first type involves cooling and heating facilities, and also machines operating on the Carnot cycle (see below). The second type includes air-jet and rocket engines etc. The third type involves gas-turbine, turbo-jet, internal-combustion engines etc. Lagrangian description proves to be more convenient for computations of heat machines of the first type, and Eulerian description – for the ones of the second and third types. Further, we will consider heat machines only of the first type; machines of the second and third types can be considered in a similar way (see Exercise 3.5.2). All heat machines are characterized by their efficiency. To introduce the concepts of heat, mechanical work and the efficiency, we should return to the integral statements (3.126) and (3.161) of the thermodynamic laws. Consider an arbitrary time interval t1 6 t 6 t2 and integrate Eqs. (3.126) and (3.161) with respect to t from t1 to t2 ; as a result, we obtain U C A.i / D C;

(3.177)

H > CN :

(3.178)

Here we have introduced the notation U D U.t2 / U.t1 /; Zt2 A.i / D

Zt2 Z

Zt2

Zt2 Z

ı

ı

ı

V

ı

Zt2 Z

f v d V dt C t1

ı

V

(3.179)

P r ˝ vT d V dt; t1

W .t/ dt D t1

ı

W.i / .t/ dt D t1

A.e/ D

H D H.t2 / H.t1 /I

(3.180) ı

ı

tn v d † dtI

t1

ı

†

130

3 Balance Laws

Zt2 C D

CN D

Zt2 Z Q.t/ dt D

qm d V dt C ı

t1

t1

Zt2

Zt2 Z

N Q.t/ dt D

t1

Zt2 Z

ı

ı

t1

t1

V

ı

qm ı d V dt C

ı

ı

ı

†

Zt2 Z t1

V

ı

q † d † dt; (3.181)

ı

q† ı d † dt;

ı

†

where we used Lagrangian representation of the integral values U , H , W.i / , Q and QN (see (3.135), (3.136), and (3.176)). The value U is called the change of the internal energy of a body B, H – the change of the entropy of a body B, A.i / – the mechanical work done by a body B (or simply the work), C – the heat supplied to a body (or simply the heat), CN – the entropy influx to a body B, A.e/ – the work done on a body B (or the work done by external forces). ı

Let us use Lagrangian description in variables X i ; t, where X i 2 V .t/, t 2 ı

Œt1 ; t2 , and divide the four-dimensional integration domain V .t1 ; t2 / of volume ı

integrals into subdomains V t ˛ such that: (1) in each of these subdomains a sign of the scalar function qm remains unchanged, i.e. either qm > 0, or qm < 0, and (2) ı

ı

ı

[V t ˛ D V .t1 ; t2 /. In a similar way, divide the integration domain † .t1 ; t2 / ˛

ı

ı

of surface integrals into subdomains †t ˛ such that: (1) in each †t ˛ a sign of the ı

ı

ı

ı

function q † remains unchanged, i.e. either q † > 0, or q † < 0, and (2) [†t ˛ D ˛

ı

† .t1 ; t2 /. Then the heat C supplied to the body can be split into two nonnegative parts: C D CC C ; Zt2 Z C˙ D t1 ˙ qm

ı ˙ qm

Zt2 Z

ı

d V dt C

ı

t1

V

D .jqm j ˙ qm /=2 > 0;

ı q˙ †

(3.182) ı

ı

q˙ † d † dt > 0;

(3.183)

ı

† ı

ı

D .jq † j ˙ q † /=2 > 0:

(3.184)

Here CC is the heat absorbed by the body (or supplied to the body), and C is the heat released by the body (or withdrawn from the body). Since the temperature of a body is nonnegative, we can also split the entropy influx CN to the body into two nonnegative parts: CN D CN C CN ; ı ˙ ı ı Rt2 R q Rt2 R qı ˙ † m CN ˙ D d V dt C d † dt > 0: t1

ı

V

t1

ı

†

(3.185) (3.186)

3.5 The Second Thermodynamic Law

131

Substituting (3.182) into (3.177), we can rewrite the heat influx equation in the form C D CC A.i / U:

(3.187)

Assume that CC ¤ 0, then we can introduce the ratio of the mechanical work done by the body B to the heat supplied: kB D A.i / =CC ;

(3.188)

that is called the efficiency of the body B for the time interval Œt1 ; t2 considered. According to the axiom (3.154), in each body B for any time interval Œt1 ; t2 the conditions 0 < .X i ; t/ < C1 are satisfied; then we can introduce minimum and maximum values of the temperature of a body: ı

0 < min 6 .X i ; t/ 6 max < C1; 8X i 2 V ; 8t 2 Œt1 ; t2 :

(3.189)

From relations (3.189) we obtain ı

ı

ı

C C qm = > qm =max ı

ı

ı

C qC † = > q † =max ı

ı

ı

and qm = 6 qm =min ; ı

and q † = 6 q † =min ;

(3.190)

ı

C , qC because qm † , qm , q † , and are nonnegative. Substituting (3.190) into the integrals (3.183), (3.186) and taking into account that the integrands are nonnegative, we find that

CN C > CC =max ;

CN 6 C =min :

(3.191)

From formulae (3.178), (3.182), and (3.191) we get H > CN D CN C CN >

CC C : max min

(3.192)

Substitution of the expression (3.187) into (3.192) yields H > CC

1 max

1 min

C

1 .A.i / C U /: min

(3.193)

Dividing this relation by CC and multiplying by min (both the values are nonnegative), we obtain A.i / U Hmin min C C 1 6 : (3.194) CC CC max CC According to the definition of the efficiency kB (3.188), we get the final Truesdell estimate for the efficiency:

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3 Balance Laws

kB 6 1

min U min H : max CC

(3.195)

Notice that since a sign of the third summand on the right-hand side of (3.195) is unknown, we cannot claim that in any process for each body kB < 1. This assertion, which is well known from the course of physics, holds for special processes of motion and heating of a body. Consider them in the next paragraph.

3.5.5 Adiabatic and Isothermal Processes. The Carnot Cycles One can say that in a body B there occurs a locally adiabatic process in the subdoı

ı

main V t ˛ , if the heat influx due to mass sources into V t ˛ is zero: ı

qm 0 8.X i ; t/ 2 V t ˛ :

(3.196)

ı

If in some subdomain †t ˛ the heat influx due to surface sources is zero: ı

ı

q † 0 8.X i ; t/ 2 †t ˛ ;

(3.197) ı

then one can say that a locally adiabatic process occurs in the subdomain †t ˛ of the body B. If in a body B within the time interval Œt1 ; t2 the following conditions are satisfied: ı ı ı (3.198) qm 0 8X i 2 V ; q † 0 8X i 2 †; then one can say that there occurs an adiabatic process in the restricted sense in the body B. Finally, if in a whole body B within the time interval Œt1 ; t2 the following conditions are satisfied: ı

qm 0 8X i 2 V ;

ı

ı

q 0 8X i 2 V ;

(3.199)

then one can say that there occurs an adiabatic process in the body B in the broad sense (or simply an adiabatic process). Since (3.196) and (3.197) follow from the condition (3.199) (but not conversely), one can say that if in a body B an adiabatic process in the broad sense occurs, then in this body there occurs also an adiabatic process in the restricted sense (but not conversely). One can say that in a body B there occurs a locally isothermal process in the ı

subdomain V t ˛ , if the temperature in this subdomain remains constant: ı

.X i ; t/ const 8.X i ; t/ 2 V t ˛ :

(3.200)

3.5 The Second Thermodynamic Law

133

If the temperature remains constant within a time interval Œt1 ; t2 in a whole body B: ı

.X i ; t/ const 8X i 2 V ; 8t 2 Œt1 ; t2 ;

(3.201)

then one can say that in the body B there occurs an isothermal process. One can say that in a body B there occurs a thermodynamic cycle within the time interval Œt1 ; t2 , if the following conditions of periodicity are satisfied simultaneously: U D U.t2 / U.t1 / D 0; H D H.t2 / H.t1 / D 0; CC ¤ 0;

(3.202)

ı

.X i ; t2 / .X i ; t1 / 0 8X i 2 V : Notice that not for every continuum we may ensure that the conditions (3.202) be satisfied, and, thus, not for every body a thermodynamic cycle can be realized. However, there are whole classes of bodies, for which the conditions (3.202) are satisfiable. Example 3.1. Consider a continuum B, where the specific internal energy e and the specific entropy uniquely depend only on the density and temperature : e D e.; /;

D .; /:

(3.203)

In Chap. 4 we will show that such a model of a continuum describes an ideal fluid or gas. If relations (3.203) hold, then the conditions (3.202) can easily be satisfied by giving such external heat and mechanical actions that the fields of temperature .X i ; t/ and density .X i ; t/ in the body B be the same at t D t1 and t D t2 : .X i ; t1 / D .X i ; t2 /, .X i ; t1 / D .X i ; t2 /. Indeed, in this case the following conditions are satisfied: e.X i ; t1 / D e..X i ; t1 /; .X i ; t1 // D e..X i ; t2 /; .X i ; t2 // D e.X i ; t2 /; and hence

Z U.t1 / D

ı

ı

Z

e.X ; t1 / d V D i

ı

ı

ı

e.X i ; t2 / d V D U.t2 /; ı

V

V

i.e. U D 0. The condition H D 0 can be satisfied in a similar way. A thermodynamic cycle, occurring in an ideal fluid (gas), at every material point t u X i is assigned to some closed curve on the plane .; / (Fig. 3.9). Example 3.2. Consider a continuum B, where e and are one-valued functions of temperature and the deformation gradient F: e D e.F; /;

D .F; /:

(3.204)

134

3 Balance Laws

Fig. 3.9 A thermodynamic cycle at the material point X i of an ideal fluid

ı

Since density is a function of F ( D =det F due to (3.8)), the relations (3.203) are a particular case of (3.204). If dependences (3.204) cannot be reduced to (3.203), then they correspond to the model of an ideal solid (these models will be considered in detail in Chap. 8). For an ideal solid, one can readily ensure the conditions (3.202) to be satisfied by using relations (3.204). For this, it is sufficient that the following periodicity conditions for the cycle be satisfied: ı

F.X i ; t1 / D F.X i ; t2 /; .X i ; t1 / D .X i ; t2 / 8X i 2 V : A thermodynamic cycle, occurring in an ideal solid, at every material point X i is also assigned to some closed curve in the generalized seven-dimensional space .FN ij ; t/, where FN ij are components of the tensor F with respect to the Cartesian basis eN i . t u One can say that a generalized Carnot cycle occurs in a body B, if the following conditions are satisfied simultaneously: (1) In the body B there occurs a thermodynamic cycle within Œt1 ; t2 . ı

ı

(2) The domains V Œt1 ; t2 and † Œt1 ; t2 of the body B consist only of the subı

ı

domains V t ˛ and †t ˛ , where there occur locally adiabatic or locally isothermal processes. (3) Locally isothermal processes may be only of the two types: either with the minimum temperature min and heat absorbtion: ı

in V t ˛ W ı

on †t ˛ W

.X i ; t/ D min ; qm < 0; (3.205) ı

.X ; t/ D min ; q † < 0; i

or with the maximum temperature max and heat release: ı

in V t ˛ W ı

on †t ˛ W

.X i ; t/ D max ; qm > 0; ı

.X i ; t/ D max ; q † > 0:

(3.206)

3.5 The Second Thermodynamic Law

135

Fig. 3.10 The typical picture of location of subdomains in the generalized Carnot cycle

ı

ı

The subdomains V t ˛ and †t ˛ , where there occur locally isothermal processes, have no common points (and even their closures have no common points too) due ı

to continuous differentiability of the temperature .X i ; t/ within V Œt1 ; t2 . ı

ı

Figure 3.10 shows a typical location of the subdomains V t ˛ and †t ˛ in the generalized Carnot cycle. One can say that in a body B there occurs an uniform thermomechanical process within the time interval Œt1 ; t2 , if fields of its mechanical and thermodynamic values are uniform, i.e. they are independent of coordinates X i but may depend on time t: ı

.X i ; t/ D .t/ 8X i 2 V ; 8t 2 Œt1 ; t2 ; D f ; q; F; T; P; qm ; e; ; ; q g:

(3.207)

(Notice that the velocity v and displacement u vectors of material points in an uniform process may depend on the coordinates X i .) The generalized Carnot cycle, which occurs in a body under the conditions of a uniform thermomechanical process, is called simply the Carnot cycle. The Carnot cycle, according to the definitions given above, consists only of isothermal and adiabatic processes in a body B, which alternate with each other; here within the time subinterval T0 2 Œt1 ; t2 , when the process is adiabatic in the whole body, the rate of heating Q, according to (3.198) and (3.107), is zero; within the time subinterval T 2 Œt1 ; t2 , when the temperature in the whole body reaches its minimum value min , the rate of heating Q is negative; and within the time subinterval TC 2 Œt1 ; t2 , when the temperature in the whole body has its maximum value max , the rate of heating Q is positive:

136

3 Balance Laws

Fig. 3.11 The elementary Carnot cycle

T0 W TC W T W

Q.t/ D 0;

D max ; Q.t/ > 0; D min ; Q.t/ < 0:

(3.208)

Figure 3.11 shows the elementary Carnot cycle. This cycle consists of four processes: two isothermal and two adiabatic ones. Other Carnot cycles for the same body B can differ from the elementary Carnot cycle only by the number of isothermal sections.

3.5.6 Truesdell’s Theorem Theorem 3.16a (of Truesdell). (1) If for the body B there exists a thermodynamic cycle with conditions (3.202) and given values min and max in this cycle, then the efficiency reaches its maximum possible value min kBmax D 1 < 1; (3.209) max only in generalized Carnot cycles. (2) Among all bodies B, for which the thermodynamic cycle (3.202) is possible under the conditions of an uniform thermomechanical process (3.207), the efficiency reaches its maximum value (3.209) for such bodies, in the cycle of which there is no entropy production by internal sources: QN 0

8t 2 Œt1 ; t2 :

(3.210)

H (1) To prove the first assertion of Truesdell’s Theorem we consider an arbitrary thermodynamic cycle; for this cycle, relationships (3.187) and (3.188) yield A.i / D CC C ; and hence kB D 1

C : CC

(3.211)

(3.212)

3.5 The Second Thermodynamic Law

137

As follows from (3.211) and (3.209), in order for the efficiency to reach its maximum value kB D kBmax in the thermodynamic cycle, it is necessary and sufficient that the following condition be satisfied: C =CC D min =max :

(3.213)

Since, according to (3.202), for any thermodynamic cycle H D 0, relation (3.192) with use of (3.213) takes the form CC C D 0I max min

0 > CN C CN > that is possible only if

(3.214)

CN C D CN :

(3.215)

Due to this relation, the system of two inequalities (3.191) becomes C CC CC 6 CN C D CN 6 D : max min max

(3.216)

Since the left-hand and right-hand sides of (3.216) coincide, the inequality signs in this relation should be replaced by the equality sign. Thus, the following two equations must hold: C CC D CN C ; D CN ; (3.217) max min or with use of the notations (3.183) and (3.186) Zt2 Z t1

ı

1 1 max

ı

ı

qm

1 min

1

Zt2 Z

ı

dV dt C t1

V

Zt2 Z t1

ı C qm

ı

ı

1 1 max

ı

d† dt D 0;

†

Zt2 Z

dV dt C t1

V

ı

qC †

ı

q †

1 min

ı

1

(3.218) ı

d† dt D 0:

†

Since all integrands in these equations are non-negative, the sum of integrals is zero only if the integrands vanish; i.e. the following equations must hold: 1 1 D 0; max 1 1 qm D 0; min

C qm

8X i 2 V ;

1 1 D 0; max 1 1 ı q† D 0; min ıC

q†

8t 2 Œt1 ; t2 :

(3.219)

138

3 Balance Laws

These equations are satisfied if at least one of the cofactors vanishes, i.e. if in ı

subdomains V t ˛ the following conditions are satisfied: ı

in V t ˛ W ı

qm > 0; D max ; or qm < 0; D min ; or qm D 0; ı

ı

ı

at †t ˛ W q † > 0; D max ; or q † < 0; D min ; or q † D 0:

(3.220) (3.221)

But, according to (3.205) and (3.206), this means that the thermodynamic cycle considered is a generalized Carnot cycle. Thus, we have shown that the requirement that the efficiency reach its maximum kB D kBmax in a thermodynamic cycle leads to the requirement that this cycle be a Carnot cycle, and vice versa. (2) To prove the second assertion of the theorem we use the fact established above that the efficiency reaches its maximum value at the generalized Carnot cycles; hence, under the conditions of uniform thermomechanical processes it is sufficient to prove only that the relation (3.210) holds for simple Carnot cycles. Take into account that for a thermodynamic cycle H D 0, and represent H as a sum of three terms: Zt2 0 D H D H.t2 / H.t1 / D

HP dt D

Z

HP dt C

T0

t1

Z

HP dt C

TC

Z

HP dt; (3.222)

T

where the subsets T0 , TC and T are defined by the conditions (3.208) in an uniform thermomechanical process. Within TC , according to (3.208), D max and Q > 0, then due to the Clausius inequality (3.161), we have within TC W

HP > QN D max Q > 0:

(3.223)

e C of the set TC , the Assume that within some finite time interval being a subset T strict inequality in (3.223) holds: HP > max Q. Then, integrating (3.223) over the whole TC , we obtain the strict inequality Z

HP dt > max

TC

Z Q dt D max CC :

(3.224)

TC

(The case, when the inequality HP > max Q is satisfied only at an isolated point, is excluded, because we assume that all the functions H.t/, Q.t/ and .t/ are continuous.) Due to the Clausius inequality (3.161) and (3.208), Z T

HP dt >

Z T

QN dt D min

Z

T

Qdt D min C :

(3.225)

3.5 The Second Thermodynamic Law

139

(Here we have usedR the notations (3.183) R t and (3.184), which for uniform processes reduce to C D T Qdt D .1=2/ t12 .jQj Q/dt.) Addition of two inequalities (3.224) and (3.225) yields Z

HP dt C

TC

Z

HP dt > max CC min C D 0;

(3.226)

T

because we consider the Carnot cycle, for which the conditions (3.213) are satisfied. As follows from (3.222) and (3.226), the integral over T0 must be negative: Z

HP dt < 0:

(3.227)

T0

P However, due to the R Clausius inequality (3.161), within T0 the conditions H > P N Q D Q D 0 and T0 H dt > 0 must be satisfied; therefore, the inequality (3.227) e C : HP > max Q is not true, is impossible. Thus, the made assumption that within T and from (3.223) it follows that within the whole TC the following equality must hold: HP D max Q: (3.228) TC W In a similar way, one can prove that within T W

HP D min Q:

(3.229)

Then, integrating (3.228) over TC and (3.229) over T and substituting the results into (3.222), we obtain Z 0 D H D

HP dt C max CC min C D

T0

Z

HP dt:

(3.230)

T0

Here we have taken into account once more that the condition (3.213) is satisfied for Carnot’s cycle. As noted above, due to the Clausius inequality (3.161), within T0 we have HP > 0; then (3.230) yields T0 W HP D 0 D Q: (3.231) Collecting the three relations (3.228), (3.229), (3.231), and taking (3.208) into account, we find that in the Carnot cycle with the maximum efficiency for the whole time interval considered the following relation holds: HP D Q D QN

8t 2 Œt1 ; t2 :

(3.232)

140

3 Balance Laws

This relationship, according to (3.156), means that there is no entropy production QN by internal sources, i.e. the condition (3.210) is really satisfied. N Remark 3. Since the assertion (2) of Truesdell’s theorem holds for uniform thermomechanical processes, for which, by (3.207), the gradient of a temperature field is zero in a whole body B: ı

ı

r 0 8t 2 Œt1 ; t2 ;

8X i 2 V I

(3.233)

so it follows from (3.175), (3.176), and (3.210) that the body B, satisfying the assertion (2), must be nondissipative, i.e. in this body, at least, within the interval Œt1 ; t2 the dissipation function is zero: w 0 8t 2 Œt1 ; t2 ;

ı

8X i 2 V :

(3.234)

The processes, for which in a whole body the conditions (3.233) and (3.234) are satisfied simultaneously, i.e. the specific internal entropy production q is identically zero, are often called reversible; and the processes, in which at least one of these conditions is violated, are called irreversible. t u

Exercises for 3.5 3.5.1. Using formula (3.128) and the result of Exercise 2.1.9, show that the dissipaı tion functions w and w , defined by (3.167) and (3.175), are connected by ı ı

w D .=/w : 3.5.2. For heat machines of the second and third types, by analogy with (3.179)– (3.181) introduce the concepts of H , A.i / , C and CN and prove that Truesdell’s estimate (3.195) is valid. Introduce the concepts of adiabatic and isothermal processes by analogy with the definition from Sect. 3.5.5 and also the concept of a thermodynamic cycle for Œt1 ; t2 , having replaced the last relation in (3.202) by the two ones V .x i ; t1 / D V .x i ; t2 /;

.x i ; t1 / .x i ; t2 / D 0 8x i 2 V;

(which mean the coincidence of domains V occupied by a body at times t1 and t2 and coincidence of temperature fields in the domains). Prove Truesdell’s theorem for thermodynamic cycles occurring in heat machines of the second and third types.

3.6 Deformation Compatibility Equations

141

3.6 Deformation Compatibility Equations 3.6.1 Compatibility Conditions So we have completed the formulation of balance laws of continuum mechanics. However, the conditions of continuity of the motion of bodies, which have been used in Chap. 2, can also be rewritten in the form of some formal ‘conservation law’. This formal law is of great importance for closing the equation system in continuum mechanics; thus, it will be consider in Sects. 3.6–3.8. Formulate the requirement for continuity of a motion as follows: let it be known ı

that in configuration K we can put every material point M of a continuum in one-toı one correspondence with its radius-vector x.X k / in the Cartesian coordinate system O eN i ; then we must define the conditions, under which there exists a one-valued function (the displacement vector u.X k ; t/) connecting positions of the point M in ı

K and K. If in K some discontinuities appear (cracks, pores, etc.) (Fig. 3.12), then the motion is no longer continuous. Definition 3.12. Necessary and sufficient conditions of the existence of a onevalued vector-function u.X k ; t/ for a continuum V are called deformation compatibility conditions (equations) for the continuum V . ı

If there is no one-valued vector-function u.X k ; t/ for all X k 2 V , this means that in K we cannot introduce one-valued radius-vector x.X k ; t/. Thus, the configuration K does not belong to Euclidean space E3a . The converse statement holds as well. So we have proved the following theorem. Theorem 3.17. The deformation compatibility conditions for a continuum V are equivalent to the condition that an actual configuration of the continuum belong to Euclidean space E3a . Remark. At first sight it seems strange that an usual medium with cracks does not belong to an Euclidean space. But this continuum is not quite usual, because it is

Fig. 3.12 The example of violation of the compatibility conditions

142

3 Balance Laws

defined in actual configuration K in such a way that in the reference configuration ı

K this medium is assigned to a continuum already without discontinuities. If we defined a considered continuum with cracks in another way, namely as belonging ı

e then each material point of the continto some new reference configuration K, uum would have its individual radius-vector and such a continuum would belong to Euclidean space E3a . t u

3.6.2 Integrability Condition for Differential Form There are two types of compatibility equations: dynamic and static. In this paragraph we consider static compatibility equations. To derive the equations we should consider some differential form 3 X

A˛ dX ˛ ;

(3.235)

˛D1

where A˛ D A˛ .X i / are smooth functions of variables X j . As known from the course of mathematical analysis, this form gives a total differential dA if and only if the following integrability conditions hold @Aˇ @A˛ D ; ˇ @X ˛ @X

˛; ˇ D 1; 2; 3:

(3.236)

A˛ dX ˛ :

(3.237)

In this case we have the representation dA D

3 X ˛D1

This expression can be considered as an equation in differentials. The equation can be resolved for A if and only if the conditions (3.236) are satisfied.

3.6.3 The First Form of Deformation Compatibility Conditions ı

Let A be radius-vectors x and x of the same material point M. For them the relation (3.237) is written as follows: d x D ri dX i ;

ı

ı

d x D ri dX i :

(3.238)

3.6 Deformation Compatibility Equations

143

These relations can be considered from different points of view. On the one hand, ı

if the motion law is known and in K there is a common Cartesian coordinate system, then the following relations hold ı

ı

x D x.X j /; x D x.X j ; t/;

(3.239)

which are smooth functions of their arguments. Then Eqs. (3.238) are consequences of the relations (3.239). This approach has been used above. On the other hand, if basis vectors ri are known and radius-vector x is unknown ı (by the condition, the radius-vector x is always known, and it is a one-valued function), then, similarly to the expression (3.237), the first equation of (3.238) is an equation in differentials for radius-vector x. A solution of this equation always exists, because the integrability conditions (3.236) are satisfied. Indeed, @rˇ @r˛ @x @x D D D : ˇ ˇ ˛ ˛ ˇ @X ˛ @X @X @X @X @X

(3.240)

Thus, if there exist local basis vectors at every point X i in K, then there also exist radius-vectors of these points in K. If every material point M with coordinates X i in K may be uniquely assigned to ı its radius-vector x, this means that there exist one-valued functions u.X k ; t/ D x x ı

of displacements of the points from K to K. Conversely, if there exist one-valued displacements u.X k ; t/ of material points ı

ı

from K into K, then in K we can always introduce the radius-vector x D x C u and the local basis vectors ri D @[email protected] i . Thus, we have proved the following theorem. Theorem 3.18. The deformation compatibility conditions for a continuum are satisfied if and only if in K there exist local basis vectors ri , i.e. functions having the following properties: they are linearly independent at every point X i ; they are one-valued and smooth 8X i 2 V ; they have a vector potential x: ri D @[email protected] i .

3.6.4 The Second Form of Compatibility Conditions ı

As A, we choose local bases vectors ri and ri . For them, the relation (3.237) becomes ı ı @ri ı j m j m d ri D dX j : dX D .

r / dX ; d r D

r (3.241) i ij m ij m @X j

144

3 Balance Laws

If the motion law is known (the second equation in (3.239)) and the deformation compatibility conditions are satisfied, then relations (3.241) follow from the motion law (3.239). If the Christoffel symbols ijm are known, then (3.241) are equations for functions ri . The integrability conditions (3.236) for these equations have the form (the indices j and n change places) @ @ . m rm / D . m rm /: @X n ij @X j i n

(3.242)

Removing the parentheses, we obtain @ ijm

@ imn k rm C imn mj rk ; @X j k m k m C ij kn i n kj rm D 0:

(3.243)

@ imn m m C ijk kn ikn kj D 0; @X j

(3.244)

k r C ijm mn rk D n m

@X m @ ij

@ imn @X n @X j

Since rm are arbitrary, we have m Rnji

@ ijm @X n

m is some notation for the present. where Rnji Using the condition (3.244), we can formulate the following theorem.

Theorem 3.19. The deformation compatibility conditions are satisfied if and only if in K there exist Christoffel symbols ijm satisfying the integrability conditions (3.244). H Indeed, if there exist ijm satisfying the conditions (3.244), then the integrability conditions (3.242) remain valid. Hence the first form in (3.241) has a differential and there exist local basis vectors ri . Then, according to Theorem 3.18, the deformation compatibility conditions will be satisfied. Conversely, if the compatibility conditions are satisfied, then, according to Theorem 3.18, there exist vector functions ri . Performing all manipulations (3.241)– (3.244), we verify that the conditions (3.244) hold. N ı

ı

ı

Notice that in K there always exist vectors x and ri . Hence we can perform similar manipulations for the second form in (3.241) too; as a result, we obtain the equations ı ı ı ı @ @ m m

ij rm D

i n rm (3.245) @X n @X j and

ı

ı

Rm nji

@ m ij @X n

ı

ı ı ı ı @ m k m in C kij m kn i n kj D 0: j @X

(3.246)

3.6 Deformation Compatibility Equations

145

ı

m Values Rnji and Rm nji are called components of the Riemann–Christoffel fourthı

ı

order tensors 4 R and 4 R in configurations K and K, respectively. We will verify that these are really components of tensors in paragraph 3.6.6.

3.6.5 The Third Form of Compatibility Conditions ı

Introduce the Christoffel symbols of the first kind ijk and ijk , which are connected ı

to ijm and m ij , called the Christoffel symbols of the second kind, by the relations (see [12])

ijk D gkm ijm ;

ı

ı

ı

ijk D g km m ij :

(3.247)

The Christoffel symbols of the first kind and the metric matrix are connected by the formula @gkj 1 @gi k @gij

ijk D C ; (3.248) 2 @X j @X i @X k which follows from (2.26). Interchanging indices i and k in (3.248) and summing the result with ijk , we get

ijk C kj i D

@gi k 1 C @X j 2

@gkj @gkj @gij @gij C i k k @X @X i @X @X

D

@gi k ; @X j

(3.249)

Purely covariant components of the Riemann–Christoffel tensor are defined as follows: @. ijl g ml / @. ikl g ml / m Rnjik D Rnji gmk D gmk C gsl . ijl snk inl sjk /: (3.250) @X n @X j According to the equation gmk

@gml @gmk D gml D gml . mnk C knm /; @X n @X n

(3.251)

we can rewrite Rnjik in the form Rnjik D

@ ijk @ i nk ijl . mnk C knm /g ml @X n @X j C g ml inl . kjm C mjk / C g sl . ijl snk inl sjk /

D

@ ijk @ i nk gml . inl kjm ijl knm /: @X n @X j

(3.252)

146

3 Balance Laws

Substituting in place of derivatives of ijk their expressions (3.248) in terms of the metric matrix components, we get Rnjik D 2 @2 gkj @2 gij @2 gi k @2 gkn @2 gi n @ gi k 1 C C 2 @X j @X n @X i @X n @X k @X n @X n @X j @X i @X j @X k @X j C gml . inl kjm ijl knm /; or Rnjik

1 D 2

@2 gkj @2 gij @2 gi n @2 gkn C @X i @X n @X k @X n @X i @X j @X k @X j

C g ml . inl kjm ijl knm / D 0:

(3.253) ı

In the same way we can find the expression for Rnjik in terms of components of ı

the metric matrix g ij : ı

Rnjik

1 0 ı ı ı ı @2 g ij 1 @ @2 g kj @2 g kn @2 g i n A D C 2 @X i @X n @X i @X j @X k @X j @X k @X n ı ı ı ı ı C g ml inl kjm ijl knm D 0:

(3.254)

Theorem 3.19a. The deformation compatibility conditions are satisfied if and only if in K there exists a metric matrix gij satisfying the Eqs. (3.253). H If the matrix gij satisfying Eqs. (3.253) is known, then their equivalent relations (3.244) hold. As shown above, the relations (3.244) define a space to be Euclidean; and, according to Theorem 3.19, this means that the deformation compatibility conditions are satisfied too. Conversely, if the deformation compatibility conditions are satisfied, then relations (3.244) hold and in Euclidean space the coefficients ijm have a potential being gij ; thus, relations (3.248) hold too. Then their equivalent relations (3.253) remain valid as well. N

3.6.6 Properties of Components of the Riemann–Christoffel Tensor We can check that functions Rnjik are symmetric in pairs of indices n; j and i; k: Rnjik D Ri knj ;

(3.255)

3.6 Deformation Compatibility Equations

147

and also skew-symmetric in indices n; j and i; k: Rnjik D Rnjki ;

Rnjik D Rj ni k :

(3.256)

Thus, among 81 components Rnjik there are only six independent ones. They are usually chosen as follows: R1212 ; R2323 ; R3131 ; R1223 ; R1231 ; R2331 ;

(3.257)

the remaining components either are equal to zero or can be expressed in terms of ı

components (3.257). Functions Rnjik have the same properties. ı

With the help of components Rnjik and Rnjik we can set up the following fourthorder tensors: 4

R D Rnjik rn ˝ rj ˝ ri ˝ rk ;

4

ı

ı

ı

ı

ı

ı

R D Rnjik rn ˝ rj ˝ ri ˝ rk ;

(3.258)

ı

which are called the Riemann–Christoffel tensors in K and K. ı

We can verify that Rnjik and Rnjik are components of a tensor, for example, in the following way. ı ı Consider an arbitrary vector a D ak rk D ak rk and determine its covariant derivative @ak C iks as (3.259) ri ak D @X i and the second covariant derivative rj ri ak D

@ iks s @ @2 ak k k m m k .r a / C

r a

r a D C a i i m jm ji @X j @X j @X i @X j ! m @a @ak @as m s k C iks j C jkm C

a C ms as : jmi is i @X @X @X m (3.260)

Interchange indices i and j and write the difference: ! k k @

@

js m is C jkm ims ikm js as D Rj i sk as : rj ri ak ri rj ak D @X j @X i (3.261) It follows from (3.261) that Rjki s are components of a tensor, because as and the left-hand side of (3.261) are tensor components.

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3 Balance Laws

3.6.7 Interchange of the Second Covariant Derivatives In Euclidean space E3a , the following relations have been established: ı

Rkji s D Rjki s D 0;

(3.262)

(see formulae (3.253) and (3.254)); then from (3.261) it follows that the second covariant derivatives may be interchanged. So we have proved the following theorem. Theorem 3.20. If the deformation compatibility conditions (3.253), and also (3.254) are satisfied, then the second covariant derivatives may be interchanged: ı

ı ı

ı

ı

ı

rj ri ak D ri rj ak and r j r i ak D r i r j ak :

(3.263)

This differentiation rule remains valid for tensors of any order.

3.6.8 The Static Compatibility Equation Just as in Sect. 3.6.5, we assume that in K there is a metric matrix gij satisfying ı

Eq. (3.253), and in K the continuity conditions (i.e. conditions (3.254)) are satisfied. Then in place of gij we can consider components of the deformation tensor: "ij D

1 ı .gij g ij / 2

(3.264)

and express the deformation compatibility conditions in terms of "ij . Subtracting (3.254) from (3.253), we obtain ı

Rnjik Rnjik D

@2 "kj @2 "kn @2 "i n @2 "ij C @X i @X n @X i @X j @X k @X j @X k @X n ı

ı

ı

ı

ı

Cgml . inl kjm ijl knm / g ml . inl kjm ijl knm / D 0; (3.265) where

ı

ijk D ijk C "ijk ;

"ijk D

@"jk @"i k @"ij C : j i @X @X @X k

(3.266)

The inverse metric matrix g ml can be expressed in terms of deformation tensor ı

ı

components according to formulae (2.131). Since all the functions inl and g ml in the reference configuration are assumed to be known, the relations (3.265), with

3.7 Dynamic Compatibility Equations

149

taking (3.266) and (2.131) into account, state the system of six scalar equations (because among components Rnjik there are only six independent ones) for six scalar functions "ij . These equations are called static equations of compatibility (or deformation compatibility equations). Thus, we have proved the following theorem. Theorem 3.21. The deformation compatibility conditions for a continuum are satisfied if and only if in K the deformation tensor components "ij satisfy Eqs. (3.265). Finally, let us formulate one more important theorem. Theorem 3.22. The deformation compatibility equations (3.265) have the solution ı ı ı ı ı ı ı 1 ı ı kl r i uj C r j ui C r i uk r j ul g "ij D : 2

(3.267)

Sometimes this result is said as follows: a solution of Eqs. (3.265) admits a potential, i.e. six functions "ij are expressed in terms of covariant derivatives of three ı

functions ui . H To prove the theorem we can replace the covariant derivatives by the partial ones: ı ı

ı

ı

ı

r i uj D uj= i m ij um ;

(3.268)

and, substituting the expressions (3.268) and (3.267) into Eq. (3.265), verify that this equation will be identically satisfied. However, this method is very clumsy. So let us prove the theorem in another way. Let there be a solution of Eqs. (3.265) (functions "ij ). According to Theorem 3.21, this means that the deformation compatibility equations are satisfied. Then, by ı Definition 3.12, there exists a one-valued vector-function u.X k ; t/ D x x beı ing the displacement vector of material points. By components of this vector ui , according to formulae (2.93), we can always find components e "ij , which coincide with "ij (3.267). Indeed, ife "ij ¤ "ij , then this means that there exist two distinct solutions of the deformation compatibility equations (3.265) and their corresponding, by Definition 3.12, different displacement vectors; that is impossible due to uniqueı

ness of displacements u.X k ; t/ of a continuum under the transformation from K to K. Thus,e "ij D "ij , and formula (3.267) really holds. N

3.7 Dynamic Compatibility Equations 3.7.1 Dynamic Compatibility Equations in Lagrangian Description The deformation compatibility equations can be written in one more equivalent form, namely in terms of the velocity.

150

3 Balance Laws ı

Consider Eq. (2.77) connecting F to r ˝ uT , and differentiate the equation with respect to t, taking the definition (2.181) of the velocity v into account: ı d ı @2 u @v d FT ı ıi D : r ˝ u D ri ˝ r ˝ D r ˝ v D dt @X i @t @X i dt

(3.269)

Then we get the dynamic compatibility equation in Lagrangian description: ı d FT D r ˝ v: dt

(3.270)

Theorem 3.23. The deformation compatibility conditions are satisfied if and only if in K there is a tensor field F.X i ; t/ satisfying the following conditions: det F ¤ 0 at every point X i ; F.X i ; 0/ D E at t D 0; the field F has a vector potential, i.e. there exists a vector field v such that equa-

tion (3.270) holds 8t > 0 and 8X i 2 V . H Let the deformation compatibility conditions be satisfied, then, by Definition 3.12, ı

there is a displacement vector u.X i ; t/ with its gradient r ˝ u. On making the manipulations (3.269), we verify that Eq. (3.270) remains valid. Prove that the converse assertion is valid. Let there R tbe a vector-function v satisfying Eq. (3.270). Consider the function e u.X i ; t/ D 0 v.X i ; / d . This function satisfies the equation ı

ı

Zt

r ˝e uDr˝

Zt v d D

0

Zt r ˝ v d D

0

d FT d D FT E: d

(3.271)

0 ı

Then with the help of this function we can construct the radius-vector e x D x Ce u. As a result, the tensor F takes the form ı

ı

u D ri ˝ FT D E C r ˝ e

ı

@.x C e u/ ı D ri ˝e ri ; i @X

where e ri D @e [email protected] i . But this means that the vector e u is the desired displacement vector u, and F is the deformation gradient, because they satisfy all the kinematic relations: (2.35), (2.77), etc. Thus, the displacement vector really exists u, and hence the deformation compatibility conditions are satisfied. N

3.7 Dynamic Compatibility Equations

151

3.7.2 Dynamic Compatibility Equations in Spatial Description At first, let us prove the following auxiliary statement. Theorem 3.24. If the continuity equation (3.15) holds, then the deformation gradient satisfies the following relationship: r .F/ D 0:

(3.272)

H Represent the deformation gradient in the dyadic basis: F D F ij ri ˝ rj :

(3.273)

Let us write the following formula for the divergence of a tensor (see [12]): ı

@ 1 @ p g F ij rj D p r .F/ D p i g @X g @X i

q

ı ıi

gr

:

(3.274)

q ı ı Here we have used the continuity equation D g=g, and also the evident relations ı

ı

F i k rk D F jk ri rj ˝ rk D ri F D .ri rk / ˝ rk D ri :

(3.275)

Differentiating (3.274) by parts, we get 1 0 q ı q q ıi ı q g @ B ıi ı @r C ı ı s ıi ı ı s ıi r .F/ D p @ r C g g

r g

r D 0: D p A is is g @X i @X i g ı

(3.276) Here we have used the properties of the Christoffel symbols (see [12]). N Modify Eq. (3.270) with use of (2.37): d FT D FT r ˝ v; dt

(3.277)

and use the continuity equation (3.15) multiplied by FT : @ T F C FT r v D 0: @t

(3.278)

Multiply (3.277) by and use the definition (2.187) of the total derivative with respect to time:

d FT @FT D C v r ˝ FT D FT r ˝ v: dt @t

(3.279)

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3 Balance Laws

On summing Eqs. (3.278) and (3.279), we get @FT C r .v ˝ FT / FT r ˝ v D 0: @t

(3.280)

Tensor multiplication of Eq. (3.272) by v (i.e. r .F/ ˝ v D 0) and summation of the obtained expression with (3.280) give the dynamic compatibility equation in the spatial description: @FT C r v ˝ FT F ˝ v D 0: @t

(3.281)

In order for the deformation compatibility conditions be satisfied in K, it is necessary and sufficient that Eq. (3.281), as well as (3.270), hold. Transforming the first and the second summands in (3.281) by formula (3.20), we can rewrite the dynamic compatibility equation in the form

d FT D r .F ˝ v/: dt

(3.282)

Exercises for 3.7 3.7.1. Show that the dynamic compatibility equation (3.270) can be written in the divergence form ı

d FT =dt D r .E ˝ v/: 3.7.2. Using the relation F1T F1 D E, show that the dynamic compatibility equation (3.270) can be rewritten as follows: d F1T =dt D .r ˝ v/ F1T :

3.8 Compatibility Equations for Deformation Rates Equations (3.270) and (3.281) express a condition of the existence of displacements u, when a field of the deformation gradient F is given. The deformation compatibility conditions can also be formulated as conditions for a field of the deformation rate tensor D. ı

Theorem 3.25. If a continuum does not contain discontinuities in K and the symmetric deformation rate tensor D (2.225) is given in K, then the continuum does not contain discontinuities in K if and only if the following compatibility conditions for deformation rates are satisfied:

3.8 Compatibility Equations for Deformation Rates

Ink D D 0:

153

(3.283)

Here the differential operator of incompatibility (see [12]) has been introduced as follows: Ink D D r .r DT / D .1=g/ ijk mnl ri rm Dj n rk ˝ rl ;

(3.284)

or in the component form .Ink D/kl D .1=g/ ijk ˛ˇ l .ri r˛ Djˇ ri rˇ Dj˛ /; ˛ ¤ ˇ ¤ l:

(3.285)

H Show the necessity of (3.283). If a continuum in K does not contain discontinuities, then there exist one-valued functions of coordinates: a radius-vector x, a displacement vector u and a velocity v. According to the definitions of the tensor D (2.225) and of the operator Ink (3.284), we get Ink D D

1 ijk mnl .ri rm rj vn C ri rm rn vj /rk ˝ rl D 0: 2g

(3.286)

Here we have used the property (3.263) of interchange of covariant derivatives ri rj and rm rn , and also the property of contraction of the Levi-Civita symbols with components of an arbitrary symmetric tensor (see Exercise 2.1.13). Prove the sufficiency of the conditions (3.283). Let the symmetric tensor D, satisfying the conditions (3.283), be given. Then we can write the following differential form similar to (3.235): d ! D .r D/ d x D .r D ri / dX i ;

(3.287)

which is a total differential if and only if the following integrability conditions of the type (3.236) are satisfied: @ @ .r D ri / D .r D rj /: j @X @X i

(3.288)

These conditions with use of the definition (2.31) of the tensor curl .r D/ take the form 1 1 rj p psk rp Dsi rk D ri p psk rp Dsj rk : (3.289) g g p Due to Ricci’s theorem (see [12]), the metric matrix gij and g can be placed outside the covariant derivative sign; therefore the relation (3.289) is rewritten as follows: psk .rp rj Dsi rp ri Dsj / D 0: (3.290)

154

3 Balance Laws

On comparing formulae (3.290) with (3.285), we obtain that condition (3.288) is equivalent to the condition r .r D/T D 0 (3.291) (formula (3.283)). Thus, the conditions (3.288) are always satisfied, because the condition (3.291) is assumed to be satisfied and hence the integral form (3.287) is a total differential, and ! is a one-valued function !.x/. Then the total differential can be written in the form d ! D r ˝ !T d x: (3.292) It follows from (3.287) and (3.292) that the curl of the tensor D under the condition (3.291) can always be represented as the transpose gradient of some vector !: r D D r ˝ !T :

(3.293)

With the help of the vector ! by formula (2.47) we can form the second-order tensor p W D ! E D g mi k ! m ri ˝ rk ; (3.294) which is skew-symmetric, because mi k ! m D mki ! m . With the help of D and W we can write one more differential form: d v D .D C W/ d x:

(3.295)

Similarly to the integrability conditions (3.291) for the form (3.287), the integrability conditions for the form (3.295) can be represented as follows: r .D C W/T D 0:

(3.296)

But this condition is always satisfied, because r .D C W/T D r ˝ !T r W D r ˝ !T r .! E/ D r ˝ !T C .r !/E r ˝ !T C ! r ˝ E ! ˝ r E D r ˝ !T r ˝ !T D 0: (3.297) Here we have used formula (3.294) and the property of the operator r .! E/ (see [12]), and also taken into account that r ˝ E D 0, r E D 0 and r ! D 0 due to r m ! m D r ˝ !T E D r D E 1 1 D p ijk ri Djl rk ˝ rl rm ˝ rm D p ijk ri Djk D 0: g g (3.298) Thus, the form (3.295) is a total differential, and there exists a vector v, being a one-valued function of x, hence by (2.222) we can represent its differential in the form d v D .r ˝ v/T d x: (3.299)

3.9 The Complete System of Continuum Mechanics Laws

155

On comparing (3.299) with (3.295), we get that r ˝ vT can be represented as a sum of symmetric D and skew-symmetric W tensors: r ˝ vT D D C W;

(3.300)

but this decomposition is unique, therefore by (2.225) and (2.226) we get D D .1=2/.r ˝ vT C r ˝ v/; W D .1=2/.r ˝ v r ˝ vT /;

(3.301) (3.302)

that proves the existence of the vector v, satisfying Eq. (3.301) and hence being the velocity. N As known from Sect. 2.4.5, the tensor W, defined by (3.294) and (3.302), is called the vorticity tensor, and ! – the vorticity vector.

3.9 The Complete System of Continuum Mechanics Laws 3.9.1 The Complete System in Eulerian Description Consider a nonpolar continuum. The system of continuum mechanics laws in the spatial description (3.18), (3.72), (3.118), (3.165), and (3.282) can be rewritten in the generalized form

d AN˛ D r BN˛ C C˛ ; dt

˛ D 1; : : : ; 6;

(3.303)

where the following generalized vectors are denoted: 1 1= C B v C B C B 2 =2 e C jvj C B AN˛ D B C; C B C B A @ u FT 0

0

1 v B T C B C B C T v q B C BN ˛ D B C; B q= C B C @ A 0 F ˝ v

1 0 C B f C B C B B f v C qm C C˛ D B C: B.qm C q /= C C B A @ v 0 (3.304)

Here index ˛ D 1 corresponds to the continuity equation, ˛ D 2 – to the momentum balance equation, ˛ D 3 – to the energy balance equation, ˛ D 4 – to the entropy balance equation, ˛ D 5 – to the kinematic equation, ˛ D 6 – to the dynamic compatibility equation.

0

156

3 Balance Laws

Equation (3.303) at ˛ D 5 has been obtained from (2.190) with the help of its multiplication by : du D v: (3.305) dt This relation is called the kinematic equation. The system (3.303) is called the universal system of continuum mechanics laws in total differentials. This system can be reduced to the divergence form. To do this, we can collect the corresponding divergence forms of Eqs. (3.15), (3.73), (3.119), (3.165a), and (3.281), or can transform the left-hand side of the system (3.303) with the help of the continuity equation as follows:

@AN˛ C v r ˝ AN˛ C @t

@AN˛ @ C r v AN˛ D C r v ˝ AN˛ ; @t @t ˛ D 2; : : : ; 6: (3.306)

Then together with the continuity equation, the following complete system of continuum mechanics laws in the spatial description in the divergence form holds: @ A˛ C r .v ˝ A˛ B˛ / D C˛ ; @t where

0

1 1 B C v B C B C 2 Be C jvj =2C A˛ D B C; B C B C @ A u T F

˛ D 1; : : : ; 6;

(3.307)

0

1 0 B T C B C B C BT v qC B˛ D B C: B q= C B C @ A 0 F ˝ v

(3.308)

In particular, the kinematic equation (3.305) has the divergence form @u C r v ˝ u D v: @t

(3.309)

Thus, we have proved the following theorem. Theorem 3.26. The complete system of continuum mechanics laws in Eulerian description can be represented in the universal form in total differentials (3.303) and in the equivalent divergence form (3.307).

3.9.2 The Complete System in Lagrangian Description In Lagrangian description, the system of continuum mechanics laws (3.8), (3.76), (3.131), (3.173), (2.190), and (3.270) can also be written in the universal form

3.9 The Complete System of Continuum Mechanics Laws ı ı d A˛

dt

ı

ı

ı

D r B ˛ C C˛ ;

157

˛ D 1; : : : ; 6;

(3.310)

where two more generalized vectors appear: 0

0

1 ı .=/ det F B C v B C B C ı 2 B e C jv j=2 C A˛ D B C; B C B C @ A u T F

1

0 P

B C B C B ıC ı BP v q C C: B˛ D B ı B q= C B C B C 0 @ A ı E ˝ v

(3.311)

Since in a Lagrangian coordinate system the total derivative d=dt coincides with the partial derivative @[email protected], the equation set (3.310) already has the divergence form Remark. It should be noted, that the first equation in the system (3.310) at ˛ D 1 has been obtained by differentiation of the continuity equation (3.8) with respect to Lagrangian variables. In this case we must complement this differential equation with the initial condition det F D 1. Then Eq. (3.310) at ˛ D 1 with this initial conı dition always has a solution, which is the relation (3.8): .=/ det F D 1. Hence, as ı

ı

function A1 in (3.311) we can always use its actual value: A1 D 1. This means that ı

ı

the generalized vectors A˛ in (3.311) and A˛ in (3.308) are coincident: A˛ D A˛ ; ˛ D 1; : : : ; 6: t u

3.9.3 Integral Form of the System of Continuum Mechanics Laws The system of continuum mechanics laws in the integral form (3.5), (3.69), (3.116), and (3.162) can be written in the universal form as well. Theorem 3.27. The system of continuum mechanics laws in the differential form (3.307) in the spatial description is equivalent to the integral form d dt

Z

Z A˛ dV D V

Z n B˛ d† C

†

C˛ dV;

˛ D 1; : : : ; 6:

(3.312)

V

H To prove the theorem for ˛ D 1; : : : ; 4, it is sufficient to substitute the generalized vectors A˛ , B˛ and C˛ into (3.312) and to write Eq. (3.312) for each of ˛ D 1; : : : ; 4. As a result, we obtain the relations (3.5), (3.69), (3.116), and (3.162) proved above. For ˛ D 5; 6, we should integrate Eqs. (3.309) and (3.281) over V and then apply the rule of differentiation of an integral over a moving volume (see Exercise 3.1.2) and the Gauss–Ostrogradskii theorem (3.24). N

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3 Balance Laws

In the same way we can prove the following theorem (see Exercise 3.9.2). Theorem 3.28. The system of continuum mechanics laws in the differential form in the material description (3.310) is equivalent to the integral form d dt

Z

ıı

ı

Z

A˛ d V D ı

ı

ı

ı

n B˛ d † C ı

V

Z

†

ı

ı

C˛ d V ;

˛ D 1; : : : ; 6:

(3.313)

ı

V

In spite of the above-mentioned equivalence of differential and integral forms of the continuum mechanics laws, there is a considerable difference between them: in the integral form the angular momentum balance equation (3.82) even for nonpolar continua is not identically satisfied, while the corresponding equation in the differential form (see Sect. 3.3), reduced to symmetry of the stress tensor T, has been excluded from the system (3.303). Thus, the integral form (3.312) must be complemented with the law (3.82); in other words, the index ˛ ranges from 1 to 7 for the generalized vectors A˛ , B˛ and C˛ in (3.312): 1 0 0 1 1 1 0 0 C B B T C B C v f C B B B C C Be C jvj2 =2C BT v qC B fvCq C C B B B C m C C B B B C C A˛ D B C ; B˛ D B q= C ; C˛ D B.qm C q /= C : (3.314) C B B B C C C B B B C C u 0 v C B B B C C A @ @ F ˝ v A @ A FT 0 xv T x xf 0

The integral form (3.313) in the material description must consist of seven equaı

ı

tions as well; and the generalized vectors A˛ and B ˛ have the forms 1 ı .=/ det F C B v C B C B 2 j=2 e C jv C B ı C B A˛ D B C; C B C B u C B T A @ F xv 0

Exercises for 3.9 3.9.1. Prove Theorem 3.27. 3.9.2. Prove Theorem 3.28.

0

0 P

1

C B C B B ıC BP v q C ı C B ı C B˛ D B B q= C : C B 0 C B C Bı @ E ˝ v A P x

(3.315)

3.9 The Complete System of Continuum Mechanics Laws

159

3.9.3. Consider an arbitrary varying domain VQ .t/ V , whose boundary points Q Q †.t/ move with velocity c D d x† =dt, x† 2 †.t/ (see Sect. 5.1.6), and show with use of formula (2.12) that in this case the complete system of continuum mechanics laws in the divergence form (3.307) yields one more integral formulation d dt

Z

Z A˛ dV C

VQ

Z n ..v c/ ˝ A˛ B˛ / d† D

Q †

C˛ dV:

(3.316)

VQ

In the particular case when the domain VQ is fixed, the last formula reduces to d dt

Z

Z A˛ dV C

VQ

Z n .v ˝ A˛ B˛ / d† D

Q †

C˛ dV: VQ

(3.317)

Chapter 4

Constitutive Equations

4.1 Basic Principles for Derivation of Constitutive Equations Consider nonpolar continua. The equation system (3.307) consists of 18 scalar equations (each of the vector equations in (3.307) is equivalent to three scalar equations, and each of the tensor ones is equivalent to nine scalar equations), but involves 29 scalar unknowns: ; v; u; T; e; ; ; q; F; q :

(4.1)

The functions qm and f are usually assumed to be known when electromagnetic effects are absent. Thus, the set (3.307) is incomplete. To close the equation system (3.307) we need additional relations. These relations are called constitutive equations, because just they specify the type of a continuum (the universal balance laws do not differ types of continua, they are the same for all bodies). If constitutive equations are given in any way, then a continuum model is said to be specified. Derivation of constitutive equations is based on some additional principles, i.e. on physical assumptions of a general character, which cannot be formulated in the form of partial differential equations. We will accept the following basic principles:

the principle of thermodynamically consistent determinism the principle of local action the principle of equipresence the principle of material indifference the principle of material symmetry the Onsager principle

Besides them, other additional principles may be formulated for special models of a continuum.

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 4, c Springer Science+Business Media B.V. 2011

161

162

4 Constitutive Equations

4.2 Energetic and Quasienergetic Couples of Tensors 4.2.1 Energetic Couples of Tensors Before formulating the above-mentioned principles, let us give definitions of special stress and deformation tensors, which are of great importance for the theory of constitutive equations in continuum mechanics. Consider the power of internal forces W.i / (3.122) and introduce the stress power w.i / and the elementary work d 0 A of stresses: Z w.i / D T .r ˝ v/T ;

W.i / D

V

w.i / dV ;

d 0 A D w.i / dt:

(4.2)

ı

According to relation (2.37) between the gradients r ˝ v and r ˝ v, and also the dynamic compatibility equation (3.270), we can find one more representation for w.i / : ı

w.i / D T r ˝ vT D T .r ˝ vT F1 / D .F1 T/

dF : dt

(4.3)

Here we have used the formula of rearrangement of three tensors in the scalar product (see [12]): A .B C/ D .C A/ B D .A B/ C:

(4.4)

The following properties of contraction of two arbitrary tensors (see [12]) will be frequently used below: A B D AT BT ;

(4.5)

1 .A B C AT BT /; 2

(4.6)

A B D AS BS C AK BK :

(4.7)

ABD

Here AS , BS are symmetric tensors and AK , BK are skew-symmetric tensors: AS D .A C AT /=2;

AK D .A AT /=2:

According to the definition (3.56) of the Piola–Kirchhoff tensor and formula (4.3), we obtain the following expression for w.i / in terms of the tensor P: ı

w.i / D .=/ P

dF : dt

(4.8)

4.2 Energetic and Quasienergetic Couples of Tensors

163

The representation (4.8) of the stress power as the contraction of some stress tensor and the rate of some tensor describing deformation proves to be not unique. There are several different such representations. Since they are of great importance for derivation of constitutive equations in continuum mechanics, let us find the representations. To derive the representations, we use the definition (4.2) and formula (4.7) with substitutions A ! T; B ! L D r ˝ vT ; then we get w.i / D TS D C TK W;

(4.9)

where D is the deformation rate tensor (2.225), W is the vorticity tensor (2.226) introduced by the decomposition (2.224) of the velocity gradient L; TS and TK are symmetric and skew-symmetric parts of the tensor T: TS D

1 .T C TT /; 2

TK D

1 .T TT /: 2

(4.10)

For nonpolar continua, the tensor T is symmetric, and Eq. (4.9) takes the form w.i / D T D:

(4.11)

Theorem 4.1. There exists a denumerable set of couples of symmetric tensors .n/ .n/

. T ; C/, with the help of which the stress power w.i / can be expressed in the form d .n/ C C TK W; dt

.n/

w.i / D T

n 2 Z:

(4.12)

If the tensor T is symmetric, then the expression (4.12) reduces to the form .n/

w.i / D T

d .n/ C; dt

n 2 Z:

(4.13) .n/

.n/

Tensors T are called the energetic tensors of stresses, and C – the energetic tensors of deformations. Here Z is the set of integers, and the index n may be positive or negative integer, or zero. To emphasize the special status of the energetic couples we introduce the following notation: particular values of index n will be denoted by Roman numerals: I, II, III, IV, V etc., or I, II, III etc. The first five couples of energetic tensors I

I

II II

III III

IV IV

V V

are the most important, namely .T; C/, .T; C/, .T; C/, . T; C/ and .T; C/; they will be called principal, because these couples are most often used in solving different problems, and these energetic tensors may be expressed explicitly in terms of the stress and deformation tensors introduced before (see Table 4.1). For the symmetric tensor T, the couples at n D I, III, IV and V were derived by Hill [26], and the couple at n D II was found by K.F. Chernykh [8]. The special

164

4 Constitutive Equations

Table 4.1 Principal energetic couples of tensors Energetic deformation Energetic stress Number n .n/ .n/ of couple tensors T tensors C I FT TS F ƒ D 12 .E U2 / TS O C OT TS F/

II

1 .FT 2

III

OT TS O

IV

1 .F1 TS O C 2 1 S 1T

V

F

.n/

Energetic deformation .n/

measures G 12 G1

E U1

U1

B

G

III

OT TS F1T / U E

T F

CD

1 .U2 2

U E/

1 G 2

.n/

.n/

notation T and C and the method of ordering of these tensors, when each tensor C is expressed in terms of the corresponding power of the right stretch tensor U, were suggested by the author [12]. Prove the theorem for each of the energetic couples separately.

I

4.2.2 The First Energetic Couple .T; ƒ/ Consider the expression TS D and substitute in place of the tensor D its represenP tation (2.282) in terms of the deformation gradient rate F: DD

1 P 1 .F F C F1T FP T /: 2

(4.14)

Then .i /

wS D TS D D

1 S T .FP F1 C F1T FP T /: 2

(4.15)

Differentiation of the identities F F1 D E and FT F1T D E gives FP F1 D F FP 1 ;

F1T FP T D FP 1T FT :

(4.16)

On substituting these expressions into (4.15), we get .i /

wS D

1 S 1 T F FP 1 C TS FP 1T FT : 2 2

(4.17)

Rewrite the tensors FP 1T and FP 1 as the scalar products of themselves by the metric tensor E: FP 1T D E FP 1T D .F F1 / FP 1T;

FP 1 D FP 1 E D FP 1 .F1T FT/: (4.18)

4.2 Energetic and Quasienergetic Couples of Tensors

165

Substitution of these expressions into (4.17) yields / w.i S D

1 S 1 T .F F1 FP 1T / FT C TS .F FP 1 F1T / FT : 2 2

(4.19)

Applying the rule (4.4) of rearrangement of three arbitrary tensors in the scalar product, we obtain 1 T S 1 .F T F/ .F1 FP 1T / C .FT TS F/ .FP 1 F1T / 2 2 1 1 P 1T P 1 1T P D FT TS F CF F F F D .FT TS F/ ƒ: 2 (4.20)

/ w.i S D

Here we have used the expression (2.72) for the right Almansi deformation tensor ƒD

1 1 .E F1 F1T / D .E U2 /; 2 2

and P D ƒ

1 P 1 1T F F C F1 FP 1T : 2

(4.21)

(4.22)

Introducing the new tensor I

T D FT TS F;

(4.23)

from (4.9) and (4.20) we obtain I

P C TK W: w.i / D T ƒ

(4.24)

Thus, the representation (4.12) really exists, and the first energetic couple of tensors I

I

I

is .T; C/ D .T; ƒ/:

V

4.2.3 The Fifth Energetic Couple .T; C/ Introduce the following new tensor: V

T D F1 TS F1T

(4.25)

166

4 Constitutive Equations

and rewrite the expression (4.15) as follows: .i /

wS D D

1 1 S P .F T F C TS F1T FP T / 2 1 1 S 1T FT FP C F1 TS F1T FP T F/ .F T F 2 V

D T

V 1 T P P F F C FP T F D T C: 2

(4.26)

Here we have used the property (4.4) of the scalar product of tensors, have written the tensor TS in the form TS D TS E D TS F1T FT ;

TS D E TS D F F1 TS ;

(4.27)

and have taken account of the expression for the derivative of the right Cauchy– Green deformation tensor C (2.287): P D 1 FT FP C FP T F : C 2

(4.28)

Thus, there is one more energetic couple of tensors: V

V

V

.T; C/ D .T; C/:

IV

4.2.4 The Fourth Energetic Couple .T ; .U E// V

Rewrite the expression (4.26), passing from T to TS by formula (4.25) and from C to U by (2.158). Then we get V

/ P w.i S D TCD

1 1 S 1T P CU P U/: .F T F / .U U 2

(4.29)

Use the rules (4.4) of rearrangement of three tensors in the scalar product: / w.i S D

1 1 S 1T P C 1 U F1 TS F1T U: P UU F T F 2 2

(4.30)

On taking the polar decomposition (2.137) into account: F D O U;

F1 D U1 OT ;

F1T D O U1 ;

(4.31)

4.2 Energetic and Quasienergetic Couples of Tensors

167

we finally get / w.i S D

IV IV 1 1 S P D T C : F T O C OT TS F1T U 2 IV

Here we have introduced the fourth energetic stress tensor T: IV

TD

1 1 S F T O C OT TS F1T ; 2 IV

(4.32) IV

P coupled for which is the tensor C D UE. We have taken into account that C D U, P as E D 0. II

4.2.5 The Second Energetic Couple .T; .E U1 // Rewrite the first energetic couple (4.24), replacing the tensor ƒ by its expression (2.158): ƒ D 12 .E U2 /. Then from (4.24) we get 1 1 1 T S / P CU P 1 U1 U w.i S D F T F U 2 1 T S P 1 1 U1 FT T F U P 1 : D F T F U1 U 2 2 According to the property (4.31) of the polar decomposition, we obtain II

II

/ w.i S D TC :

Here we have introduced the second energetic stress tensor II

TD

1 T S F T O C OT TS F ; 2 II

(4.33)

II

coupled for which is the tensor .E U1 / D C, as C D .U1 / .

III

4.2.6 The Third Energetic Couple . T ; B/ We can derive one more energetic couple from Eq. (4.30), if in place of F1 and F1T substitute their polar decompositions (4.31): .i /

wS D

1 1 P C 1 U U1 OT TS O U1 U: P U OT TS O U1 U U 2 2

168

4 Constitutive Equations

Changing the order in the scalar product, we get .i /

wS D

1 T S P U1 C 1 OT TS O U1 U: P O T OU 2 2

Hence, there exists the third energetic couple III

/ P w.i S D T B; III

where T is the third energetic stress tensor: III

T D OT TS O;

(4.34)

III

and B D C is the the third energetic deformation tensor determined by its derivative and its initial value at t D 0 (see (2.317)).

4.2.7 General Representations for Energetic Tensors of Stresses and Deformations .n/

Theorem 4.2. Each of the energetic deformation tensors C can be expressed in terms of the corresponding power of the right stretch tensor as follows: .n/

CD

1 .UnIII E/; 8n 2 Z; n ¤ III; .n III/

(4.35)

.n/

and the energetic stress tensors T satisfy the equations IV

TD

nIV X .n/ 1 UnkIV T Uk ; n III

if n > IVI

(4.35a)

IIn 1 X nIICk .n/ k U T U ; III n

if n 6 II:

(4.35b)

kD0

II

TD

kD0

Here and below in such expressions at particular values of n we must replace Roman numerals by corresponding Arabic ones and then perform arithmetic operations. For example, the relation (4.35) at n D I has the form I

CD

1 1 .U13 E/ D .E U2 /: 13 2

4.2 Energetic and Quasienergetic Couples of Tensors

169

H (1) A proof of formula (4.35) for n D I, II, IV, and V is evident, if we write out the formula for these values of n and compare the obtained expressions with the .n/

tensors C from Table 4.1. .n/

.n/

(2) In order to show that the tensors C (when n > V) with tensors T also constitute .n/

energetic couples of the form (4.12), we should determine the derivative of C : .n/

C D

1 1 P UnV C : : : P UnIV C U U .U : : : U/ D .U n III „ƒ‚… n III nIII

P U C UnIV U/ P D : : : C UnV U

nIV X 1 P UnIVk : Uk U n III kD0

.n/

So, if there exists energetic stress tensor T , then it satisfies the equation .n/

.n/

T C D

nIV 1 .n/ X k P T U U UnIVk n III kD0

D

nIV X .n/ 1 UnIVk T Uk n III

! P U:

kD0

IV

Since the tensor .U E/ D C is assigned to the fourth energetic stress tensor IV

IV

.n/

T, the expression in parentheses should form the tensor T. Thus, for all T (when n > V) the formula (4.35a) really holds. While n D V, formula (4.35a) takes the form IV

TD

IV

V V 1 .U T C T U/: 2

V

Since the tensors T and T have been introduced before according to Table 4.1, we should verify the last formula. Substitution of expressions (4.32) and (4.25) into this formula yields 1 1 S 1 .F T O C OT TS F1T / D .U F1 TS F1T C F1 TS F1T U/: 2 2 This formula is valid due to the polar decomposition (2.137) as OT D U F1 . Hence formula (4.35a) really holds when n D V.

170

4 Constitutive Equations .n/

(3) To prove formula (4.35b) we determine the derivative of tensors C when n < I: .n/

C D

IIn 1 1 X 1 k .U1 „ƒ‚… : : : U1 / D .U / .U1 / .U1 /IInk : n III n III kD0

IIIn

Then .n/

.n/

T C D

IIn 1 X nIICk .n/ k U T U III n

! .U1 / :

kD0

II

II

Since the tensor .E U1 / D C is assigned to the second energetic stress tensor T, II

the expression in parentheses just forms the tensor T. Hence formula (4.35b) holds for all n < I. When n D I, formula (4.35b) takes the form II

TD

I 1 1 I .U T C T U1 /: 2

We can verify this formula by substituting expressions (4.33) and (4.23) into the last equation. N .n/

A general relation between T and T is more complicated than the relation (4.35) .n/

.n/

between C and U; however, the tensors T and TS are evident to be connected by the linear relation .n/

.n/

T D 4 E 1 TS ;

n 2 Z;

(4.36)

.n/

where 4 E 1 are the inverse tensors of energetic equivalence (fourth-order tensors), whose expressions for the principal energetic couples are given in Table 4.2. These representations should be derived as Exercise 4.2.11. The notation .FT ˝ F/.1432/ for the transpose fourth-order tensor has been introduced in Sect. 2.1.4. .n/

Expressions for the tensors of energetic equivalence 4 E can be found by inverting the relations (4.36): .n/

.n/

TS D 4 E T ; Table 4.2 Inverse tensors of energetic equivalence

n 2 Z:

(4.37)

.n/

E 1 .FT ˝ F/.1432/

Number n I

4

II III IV V

˝ O C OT ˝ F/.1432/ .O ˝ O/.1432/ 1 .F1 ˝ O C OT ˝ F1T /.1432/ 2 .F1 ˝ F1T /.1432/ 1 .FT 2 T

4.2 Energetic and Quasienergetic Couples of Tensors

171

Substituting the relationship (4.35a) into formula (4.37) where n D IV, we obtain .n/

IV

the expression for tensors 4 E (when n > V) in terms of the tensor 4 E: 4

nIV 1 4 IV X nkIV E .U ˝ Uk /.1432/ ; n III

.n/

E D

n > V:

(4.38a)

kD0

.n/

In a similar way, we can find the expression for tensors 4 E when n 6 I: 4

IIn 1 4 II X nIIk E .U ˝ Uk /.1432/ ; III n

.n/

E D

n 6 I:

(4.38b)

kD0

.n/

ı

Components of all the tensors 4 E with respect to the eigenbases pi and pi can be expressed in terms of only eigenvalues ˛ . .n/

Theorem 4.3. The tensors of energetic equivalency 4 E , connecting the tensors TS .n/

and T with the help of the linear relations (4.37), have the form 4

3 X

.n/

E D

.n/

ı

ı

E ˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

n 2 Z;

(4.38)

˛;ˇ D1

where I

E ˛ˇ D

II III IV V 2˛ ˇ 1 2 ; E ˛ˇ D ; E ˛ˇ D 1; E ˛ˇ D ; E ˛ˇ D ˛ ˇ ; ˛ ˇ ˛ C ˇ ˛ C ˇ .n/

E ˛ˇ

nIV X 2˛ ˇ D ˛nkIV kˇ ; .˛ C ˇ /.n III/

n > V;

kD0

.n/

E ˛ˇ

IIn X 2 D nIICk k ˛ ˇ ; ˛ C ˇ

n 6 I:

(4.39)

kD0

.n/

H (a) Consider the case when n D I, and represent the tensors F, O, TS and T in ı the eigenbases pi and pi (see formulae (2.169)–(2.171)): FD

3 X

ı

˛ p˛ ˝ p˛ ; FT D

˛D1

3 X ˛D1

TS D

3 X ˛;ˇ D1

.p/

3 X

ı

˛ p˛ ˝ p˛ ; O D

T˛ˇ p˛ ˝ pˇ ;

ı

p˛ ˝ p˛ ;

˛D1 .n/

T D

3 X .n/ı

T

˛;ˇ D1

.p/ ı ˛ˇ p˛

ı

˝ pˇ ;

(4.40)

172

4 Constitutive Equations .n/ı

.p/

where T˛ˇ are components of the tensor TS with respect to the basis p˛ , and T .n/

.p/ ˛ˇ

ı

are components of the tensor T with respect to the basis p˛ . On substituting these representations into formula (4.36) at n D I, we get I

TD

3 X

Iı

ı

.p/ ı

T ˛ˇ p˛ ˝ pˇ D FT TS F

˛;ˇ D1

D

3 X

ı

ı

.p/ ˛ p˛ ˝ p˛ T! p ˝ p! ˇ pˇ ˝ pˇ D

3 X ˛;ˇ D1

˛;ˇ;;!D1

.p/ ı

ı

˛ ˇ T˛ˇ p˛ ˝ pˇ :

Here we have taken into account that the basis p˛ is orthonormal (see (2.141)). Hence, Iı

.p/ T .p/ D T˛ˇ ˛ ˇ : ˛ˇ

Use once again the representation of the tensor TS in the basis p˛ and take into account that Iı

ı

I

ı

T .p/ D p˛ T pˇ ; ˛ˇ then TS D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

.p/ T˛ˇ p˛ ˝ pˇ D

3 X ˛;ˇ D1

Iı 1 p˛ ˝ T .p/ ˝ pˇ ˛ˇ ˛ ˇ

I ı 1 ı p˛ ˝ p˛ T pˇ ˝ pˇ ˛ ˇ I 1 ı ı .p˛ ˝ p˛ ˝ pˇ ˝ pˇ /.1432/ T ˛ ˇ I 1 ı ı .p˛ ˝ pˇ ˝ pˇ ˝ p˛ / T: ˛ ˇ

I

Thus, the representation (4.37) with the tensor 4 E (4.38), (4.39) really holds at n D I. (b) Consider the case when n D II. According to the representations (4.40), we can rewrite relation (4.36) for n D II as follows:

4.2 Energetic and Quasienergetic Couples of Tensors 3 X

II

TD

IIı

ı

1 T .F T O C OT T F/ 2

ı

T .p/ p ˝ pˇ D ˛ˇ ˛

˛;ˇ D1

D

1 2

3 X

173

ı

ı

.p/ .˛ p˛ ˝ p˛ T! p ˝ p! pˇ ˝ pˇ

˛;ˇ;;!D1 ı

3 1 X ı .p/ ı .˛ C ˇ /T˛ˇ p˛ ˝ pˇ : 2

ı

.p/ C p˛ ˝ p˛ T! p ˝ p! ˇ pˇ ˝ pˇ / D

˛;ˇ D1

Thus, we get IIı

.p/

T ˛ˇ D

1 .p/ .˛ C ˇ /T˛ˇ : 2

The further proof is analogous to the proof for n D I. The theorem for n D III; IV and V can be proved in the same way (see Exercise 4.2.18). (c) To prove the theorem for the cases when n > V and n 6 I, one should use formulae (4.38), (4.39) at n D II and IV, and also the expression (2.142) for the tensor U and substitute them into (4.38a) and (4.38b). .n/ı

In proving the theorem, we have established that components T .n/

ı

.p/ ˛ˇ

of the tensor

.p/

T with respect to the basis p˛ and components T˛ˇ of the tensor T with respect to the basis p˛ are connected by the relations Iı

.p/ T .p/ D ˛ ˇ T˛ˇ ; ˛ˇ IIIı

T

.p/ ˛ˇ

.p/ D T˛ˇ ;

IVı

T

.p/ ˛ˇ

D

IIı

T .p/ D ˛ˇ

1 .p/ ; .˛ C ˇ /T˛ˇ 2

1 1 1 .p/ T ; C 2 ˛ ˇ ˛ˇ

Vı

T .p/ D ˛ˇ

1 T .p/ ; ˛ ˇ ˛ˇ

(4.41)

or .n/

.p/

.n/ı .p/ ˛ˇ ;

T˛ˇ D E ˛ˇ T

8n 2 Z: N

4.2.8 Energetic Deformation Measures .n/

Besides the energetic tensors C, introduce the energetic deformation measures .n/

GD

1 UnIII ; n III

8n 2 Z;

n ¤ III;

which are the corresponding powers of the right stretch tensor U.

(4.42)

174

4 Constitutive Equations .n/

.n/

One can readily establish the relation between G and C : .n/

.n/

CDG

1 E: n III

(4.43) .n/

.n/

Since the derivative of the metric tensor is zero, the derivative tensors G and C are coincident: .n/

.n/

C D G :

(4.44)

Introduce the third energetic deformation measure III

III

G D E C C:

(4.43a)

Due to (2.317), the measure satisfies the equation III

1 dG D dt 2

dU dU ; U1 C U1 dt dt

III

G.0/ D E;

(4.43b)

and also satisfies the general equation (4.44). Due to (4.44), the relation (4.12) for w.i / takes the form .n/

.n/

w.i / D T G C TK W;

8n 2 Z:

(4.45)

If the tensor T is symmetric, then this formula reduces to the form .n/

w.i / D T

d .n/ G; dt

8n 2 Z;

(4.46) .n/ .n/

i.e. besides the couples of the energetic stress and deformation tensors . T ; C/, there .n/ .n/

are couples of the energetic stress tensors and deformation measures . T ; G/ (see Table 4.1). .n/

It should be noted that besides the tensors G we can introduce a set of tensors of .n/ P 0, which also satisfy the form G C N, where N is an arbitrary constant-tensor: N .n/

the relation (4.44). However, only the tensors G, which were already used above, have some physical meaning: I II IV V 1 1 G D G1 ; G D U1 ; G D U; G D GI 2 2

(4.47)

4.2 Energetic and Quasienergetic Couples of Tensors

175

they are the right Cauchy–Green and Almansi deformation measures, and also the .n/ .n/

.n/

right stretch tensor and its inverse. All the tensors T , C and G are symmetric. The principal energetic deformation measures for n D I; : : : ; V are given in Table 4.1. For the first time, formulae (4.35) and (4.42), and also the existence of energetic deformation measures have been derived in the systematized form by the author in [12].

4.2.9 Relationships Between Principal Invariants of Energetic Deformation Measures and Tensors In continuum mechanics, three principal invariants of a second-order tensor (see [12]) are frequently used; they are determined as follows: .n/

.n/

.n/

I1 . C/ D E C; I2 . C / D

.n/ .n/ .n/ 1 2 .n/ .I1 . C / I1 . C 2 //; I3 . C / D det. C /: 2

(4.48)

The concept of an invariant will be considered in detail in Sect. 4.8.4. .n/

Now, let us derive relations between principal invariants I˛ . C / (˛ D 1; 2; 3) of .n/

.n/

the energetic tensors C and principal invariants I˛ . G/ of the energetic deformation measures. .n/

.n/

Theorem 4.4. The principal invariants of the tensors C and G are connected by the relations .n/

.n/

3 ; n III .n/ .n/ .n/ 2 3 ; I2 . C / D I2 . G/ I1 . G/ C n III .n III/2 I1 . C / D I1 . G/

.n/

.n/

.n/

.n/

I2 . G/ 1 I1 . G/ ; I3 . C / D I3 . G/ n III .n III/2 .n III/3 .n/ .n/ 3 ; I1 . G/ D I1 . C/ C n III .n/ .n/ .n/ 2 3 I2 . G/ D I2 . C / C ; I1 . C / C n III .n III/2 .n/

.n/

.n/

.n/

I1 . C / I2 . C / 1 I3 . G/ D I3 . C / C C C ; n III .n III/2 .n III/3 n D I; II; IV; V:

(4.49)

176

4 Constitutive Equations

H To prove the theorem, one should express the first, second and third powers of the .n/

.n/

tensors C in terms of powers of the tensors G: .n/

.n/

CDG

.n/

1 E; n III

.n/

2 .n/ 1 GC E; n III .n III/2 .n/ .n/ .n/ 3 .n/2 3 1 C 3 D G3 G C G E: 2 n III .n III/ .n III/3 C 2 D G2

.n/

.n/

.n/

Write out the following invariants of the tensors: I1 . G/, I1 . G 2 / and I1 . G 3 /, and then, according to the formulae (see [12]) 1 3 I1 .T/ 3I1 .T/I1 .T2 / C 2I1 .T3 / ; 6 1 I1 .T3 / I13 .T/ C 3I1 .T/I2 .T/ ; I3 .T/ D 3

I3 .T/ D det .T/ D

(4.50)

.n/

we get the desired expressions for the first, second and third invariants of C . The derivation should be performed in detail as Exercise 4.2.10. N

4.2.10 Quasienergetic Couples of Stress and Deformation Tensors .n/

With the help of the left stretch tensor V, we can introduce couples of the tensors S .n/

and A as well. However, in this case, the stress power w.i / depends, in addition, on .n/ .n/

the derivative of the rotation tensor OT . Therefore, such couples . S ; A/ are called quasienergetic. .n/

.n/

Definition 4.1. The tensors S and A, defined by Table 4.3, are called the principal quasienergetic stress and deformation tensors, respectively. Theorem 4.5. The stress power w.i / (4.2) can always be expressed in terms of each of the quasienergetic tensor couples: .n/

w.i / D S

d .n/ ı d T A C S O C TK W; 8n 2 Z: dt dt

(4.51)

4.2 Energetic and Quasienergetic Couples of Tensors

177

Table 4.3 Principal quasienergetic couples of tensors Quasienergetic Quasienergetic deformation Number n .n/ .n/ tensors A of couple stress tensors S AD EV

V2 /

I II

V TS V 1 .V TS C TS V/ 2

III IV V

TS Y 1 1 S S 1 .V T C T V / V E 2 J D 12 .V2 E/ V1 TS V1

1 .E 2 1

Quasienergetic deformation .n/

measures g 12 g1 1 V III

g V 1 g 2

If the tensor T is symmetric, then the expression for w.i / takes the form .n/

w.i / D S

d .n/ ı d T A CS O ; dt dt

8n 2 Z:

(4.51a) ı

.n/ .n/

Here S , A are symmetric second-order quasienergetic tensors; and the tensor S is the same for all the couples, and it is called the rotation tensor of stresses: ı

SD

1 .V TS V1 V1 TS V/ O: 2

(4.52)

Let us prove the theorem for each of the principal quasienergetic couples separately.

I

4.2.11 The First Quasienergetic Couple .S; A/ Consider the first energetic couple (4.24) and pass from the tensor U2 to V2 with the help of the relations U2 D OT V2 O (see Exercise 2.3.1): I I 1 T S / S P 2 w.i S D T D D T ƒ D F T F U 2 1 P : P T V2 O C OT V P 2 O C OT V2 O D FT TS F O 2 (4.53)

According to the polar decomposition (2.137) and the rules (4.4) of rearrangement of tensors in the scalar product, we obtain / w.i S D

1 1 S P T C V TS V V P 2 V T VOO 2 P OT : CV TS V1 O

(4.54)

178

4 Constitutive Equations

Taking into account that O OT D E and, hence, P OT D O O P T; O

(4.55)

we obtain the following final expression of the form (4.51): / w.i S

1 2 ı P T DS V CSO : 2 I

I

I

(4.56) I

Here the first quasienergetic tensors S, A are defined as follows: S D V TS V ı

I

and A D A D 12 .E V2 / D 12 .E F1T F1 /, and the tensor S is defined by formula (4.52). Thus, we have proved the existence of the first quasienergetic couple I

I

I

.S; A/ D .S; A/:

II

4.2.12 The Second Quasienergetic Couple .S; .E V1 // P 2 to V P 1 in (4.55), we get Going from the derivative V ı P T 1 V TS V V1 V P 1 C V P 1 V1 w.i / D S O 2 ı 1 T P .V TS C TS V/ V P 1 : D SO 2

(4.57)

Here we have used the rule (4.4) again. As a result, we find the second quasienergetic couple: ı

II

P T; w.i / D S .E V1 / C S O

(4.58)

II

where the second quasienergetic stress tensor S has been introduced as follows: II

SD

1 .V TS C TS V/; 2

(4.59)

II

and the second quasienergetic deformation tensor is A D .E V1 /.

4.2.13 The Third Quasienergetic Couple .Y; TS / P Consider the third energetic couple (see Sect. 4.2.6) and pass from the tensor U P to V: 1 T S / 1 P P w.i C U1 U/ S D O T O .U U 2

4.2 Energetic and Quasienergetic Couples of Tensors

D

179

1 T S P T V O C OT V P OT V1 O P O C OT V O/ O T O .O 2 1 P T V O C OT V P P O C OT V O/: C OT TS O OT V1 O.O 2

Changing the order in the scalar product of tensors in each of the summands, we get .i /

wS D

1 S P T C V1 TS V P OT P C V1 TS V O .T O O 2 P T C TS V1 V P OT /: (4.60) P C TS O CV TS V1 O O

According to the relation (4.55), we see that the first and the last terms are canceled in pair. Thus, Eq. (4.60) yields ı

/ S PT P w.i S DT YCSO :

(4.61)

where we have introduced the new tensor Y similar to the tensor B, which is defined by its derivative (see (2.318)): P V1 C V1 V/; P D 1 .V P Y 2

Y.0/ D 0:

(4.62) III

III

Thus, we have proved the existence of the third quasienergetic couple: .A; S/ D .Y; TS /:

IV

4.2.14 The Fourth Quasienergetic Couple . S ; .V E// Due to (4.62), formula (4.61) takes the form / w.i S D

ı 1 1 S P TI P CSO .V T C TS V1 / V 2

(4.63)

that proves the existence of the fourth quasienergetic couple: IV

IV

ı

/ PT w.i S D S A CSO ;

where the fourth quasienergetic tensors are defined as follows (see Table 4.3): IV

SD

1 1 S .V T C TS V1 /; 2

IV

A D V E:

(4.63a)

180

4 Constitutive Equations V

4.2.15 The Fifth Quasienergetic Couple .S; J/ Rewrite the relation (4.63) as follows: / w.i S D

1 1 S P C TS V1 V P V V1 / .V T V1 V V 2 ı ı P T D V1 TS V1 1 .V V P T: P CV P V/ C S O CS O 2

Hence, there exists the fifth quasienergetic couple: ı

V

/ PT P w.i S DSJCSO ;

where the fifth quasienergetic stress and deformation tensors are defined as follows: V

S D V1 TS V1 ;

V

ADJD

1 2 1 .V E/ D .F FT E/: 2 2

(4.64)

For the first time, the principal quasienergetic couples have been derived by the author in [12].

4.2.16 General Representation of Quasienergetic Tensors .n/

Theorem 4.6. Each of the quasienergetic deformation tensors A can be expressed in terms of the corresponding power of the left stretch tensor V as follows: .n/

AD

1 .VnIII E/; .n III/

8n 2 Z;

n ¤ III;

(4.65)

nIV X .n/ 1 VnkIV S Vk ; n III

if n > IVI

(4.65a)

IIn 1 X nIICk .n/ k V S V ; III n

if n 6 II:

(4.65b)

.n/

and quasienergetic stress tensors S satisfy the equations IV

SD

kD0

II

SD

kD0

H The theorem for n DI, II, IV and V can be proved by immediate comparison of .n/

formula (4.65) with the tensors A from Table 4.3, and for n > V and n < I – by the same method as was used in the proof of Theorem 4.2. N

4.2 Energetic and Quasienergetic Couples of Tensors Table 4.4 Inverse tensors of quasienergetic equivalency

181

Number n of couple I II III IV V

4

.n/ 1

Q .V ˝ V/.1432/ 1 .V ˝ E C E ˝ V/.1432/ 2 1 .V1 ˝ E C E ˝ V1 /.1432/ 2 .V1 ˝ V1 /.1432/

.n/

The quasienergetic stress tensors S can be represented in the general form (see Exercise 4.2.17) .n/

.n/

S D 4 Q 1 TS ;

(4.66)

.n/

where 4 Q 1 are the inverse tensors of quasienergetic equivalency, whose expressions for n D I; : : : ; V are given in Table 4.4. .n/

Expressions for the tensors of quasienergetic equivalency 4 Q can be obtained by inverting the relations (4.66): .n/

.n/

TS D 4 Q S :

(4.67)

To invert them, we use the representations of the tensors in the eigenbasis pi of the left stretch tensor V, as it was made in Sect. 4.2.7. .n/

Theorem 4.7. The tensors of quasienergetic equivalency 4 Q, connecting the tensors .n/

TS and S with the help of the linear relations (4.67), have the form 4

.n/

QD

3 X

.n/

E ˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

n 2 Z;

(4.68)

˛;ˇ D1 .n/

where E ˛ˇ are expressed by formulae (4.38), (4.39). H Consider only the case when n D I; for the remaining cases a proof is analogous and should be performed as Exercise 4.2.19. .n/

Expressing the tensors S in the eigenbasis pi : .n/

S D

3 X ˛;ˇ D1

.n/ .p/ S ˛ˇ p˛

˝ pˇ ;

(4.69)

182

4 Constitutive Equations

and using the representations (4.40) for TS and (2.142) for V, we can rewrite the relation (4.66) for n D I as follows: 3 X

I

SD

I

˛;ˇ D1

S .p/ p ˝ pˇ D V TS V ˛ˇ ˛

3 X

D

3 X

.p/ ˛ p˛ ˝ p˛ T! p ˝ p! ˇ pˇ ˝ pˇ D

˛;ˇ D1

˛;ˇ;;!D1

.p/

˛ ˇ T˛ˇ p˛ ˝ pˇ ;

I

.p/ .p/ i.e. S ˛ˇ D ˛ ˇ T˛ˇ : Then

T D S

3 X

.p/ T˛ˇ p˛

˝ pˇ D

˛;ˇ D1

D

3 X ˛;ˇ D1

D

3 X ˛;ˇ D1

3 X ˛;ˇ D1

I 1 p˛ ˝ S .p/ ˝ pˇ ˛ˇ ˛ ˇ

I 1 p˛ ˝ p˛ S pˇ ˝ pˇ ˛ ˇ I 1 .p˛ ˝ pˇ ˝ pˇ ˝ p˛ / S: N ˛ ˇ

Remark. It follows from the performed proof and formula (4.41) that components I

.p/

Iı

I

.p/

I

S ˛ˇ and T ˛ˇ of the tensors S and T are coincident. The same result is valid for all .n/

.n/

the tensors S and T (see Exercises 4.2.18 and 4.2.19): .n/ S .p/ ˛ˇ

.n/ı

D T

.p/ : ˛ˇ

(4.70)

4.2.17 Quasienergetic Deformation Measures .n/

Similarly to the measures G, let us introduce quasienergetic deformation .n/

measures g : .n/

g D

1 VnIII ; n III

n 2 Z;

n ¤ III;

(4.71)

being the corresponding powers of the left stretch tensor V. The third quasienergetic III

deformation measure g is introduced as follows: III

III

g D E C A:

(4.71a)

4.2 Energetic and Quasienergetic Couples of Tensors

183

Due to (4.62), the measure satisfies the equation III

dg 1 D dt 2 .n/

dV dV 1 1 ; V CV dt dt

III

g .0/ D E:

(4.71b)

.n/

The tensors g and A are connected by the relation .n/

.n/

AD g

1 E; n III

(4.72)

(when n D III the multiplier (nIII) is replaced by 1), therefore for all the couples .n/

.n/

A D g :

(4.73)

According to this relation, the power (4.51) can be expressed in another form .n/

ı

.n/

P T C TK W; w.i / D S g C S O

n 2 Z:

(4.74)

For the symmetric tensor T, this expression reduces to .n/

.n/

ı

P T; w.i / D S g C S O

n 2 Z:

(4.74a)

.n/

The tensors g at n D I, II, IV, V are the left Cauchy–Green and Almansi deformation measures, and also the left stretch tensor and its inverse: 1 I g D g1 ; 2

II

g D V1 ;

IV

g D V;

V

gD

1 g: 2

(4.75)

All the tensors are symmetric.

4.2.18 Representation of Rotation Tensor of Stresses in Terms of Quasienergetic Couples of Tensors ı

Theorem 4.8. The rotation tensor of stresses S (4.52) can be represented in terms of the quasienergetic couples of the stress and deformation tensors as follows: ı

.n/ .n/

.n/ .n/

S D . A S S A / O;

8n 2 Z;

n ¤ III:

(4.76)

184

4 Constitutive Equations

H We first consider the principal quasienergetic couples when n D I; II; IV; V. ı

The definition of the tensor S (4.52) ı

SD

1 .V TS V1 V1 TS V/ O 2

(4.77)

can be rewritten in the following four equivalent forms: ı

1 ..V TS V/ V2 V2 .V TS V// O; 2 ı 1 S D ..V TS C TS V/ V1 V1 .V TS C TS V// O; 2 ı 1 S D .V .V1 TS C TS V1 / .V1 TS C TS V1 / V/ O; 2 ı 1 2 S D .V .V1 TS V1 / .V1 TS V1 / V2 / O: (4.78) 2 SD

.n/

Comparison of these relations with the definition of the quasienergetic tensors S .n/

and A (see Table 4.3) gives the formulae ı

I I I 1 I .S .E 2A/ .E 2A/ S/ O; 2 ı II II II 1 II S D .S .E A/ .E A/ S/ O; 2 ı IV IV IV 1 IV S D ..A C E/ S S .E C A// O; 2 ı V V V V 1 S D ..E C 2A/ S S .E C 2A// O: 2

SD

(4.79) I

I

Since the scalar product by the unit tensor is always commutative: S E D E S etc.; so from (4.79) we obtain the desired relation (4.76). Let us consider now the case when n > V. Since for n D IV formula (4.76) has already been proved, we can use it and substitute relationship (4.65a) into this formula: ı

IV IV

IV IV

IV

IV

S OT D A S S A D V S S V .n/ .n/ .n/ .n/ 1 V .VnIV S C VnV S VC : : : CV S VnV C S VnIV / D n III .n/ .n/ .n/ .n/ 1 .VnIV S CVnV S VC : : : CV S VnV C S VnIV / V: nIII

4.2 Energetic and Quasienergetic Couples of Tensors

185

If we remove parentheses in this formula, all the terms are canceled there except the first and last ones; therefore, we get ı

S OT D .n/

.n/ .n/ .n/ .n/ .n/ .n/ 1 .VnIII S S VnIII / D A S S A ; n III

.n/

as E S D S E. Thus, formula (4.76) really holds for n > V. The theorem for the case when n < I can be proved in a similar way with the help of formula (4.65b). N From this theorem we can get the following significant corollary, which will be used below. ı

Theorem 4.9. The rotation tensor of stresses S is zero-tensor if and only if the ten.n/

.n/

sors S and A commutate with each other.

4.2.19 Relations Between Density and Principal Invariants of Energetic and Quasienergetic Deformation Tensors Relations between the density and principal invariants of the energetic and quasienergetic deformation tensors and measures will be frequently used below. Let us establish them. ı

Theorem 4.10. For every continuum the ratio of densities = is a one-valued function of the third principal invariant of the energetic and quasienergetic measures .n/

.n/

I3 . G/ and I3 . g /, and also a function of principal invariants of the energetic and .n/

.n/

quasienergetic deformation tensors I˛ . C / and I˛ . A /: .n/

.n/

I3 . G/ D I3 . g / D

1 ı .=/IIIn ; .n III/3

(4.80)

ı

.n/

.n/

.n/

ı

.n/

.n/

.n/

= D .1 C .n III/I1 . C/ C .n III/2 I2 . C / C .n III/3 I3 . C //1=.IIIn/ ; (4.81) = D .1 C .n III/I1 . A / C .n III/2 I2 . A / C .n III/3 I3 . A//1=.IIIn/ : (4.82) To prove formula (4.80), one should use the continuity equation (3.8) and the polar decomposition (2.137): ı

= D det F D det .O U/ D det U D det V:

(4.83)

186

4 Constitutive Equations .n/

Using the definition (4.42) of the measures G and formula (4.83), we verify that .n/

formula (4.80) for I3 . G/ is valid:

.n/

I3 . G/ D det

1 UnIII n III

D

1 1 ı .detU/nIII D .=/IIIn : 3 .n III/ .n III/3 (4.84)

Similarly, according to the definition (4.75) of quasienergetic measures and formula (4.83), we can prove the remaining relations in formula (4.80): .n/

I3 . g / D det

1 VnIII n III

D

1 1 ı .detV/nIII D .=/IIIn : 3 3 .n III/ .n III/ (4.85)

On substituting the sixth formula of (4.50) into (4.80), we obtain the relations .n/

ı

(4.81) between .=/ and I˛ . C /. Since there are connections between the invariants .n/

.n/

I˛ . g / and I˛ . A /, which are similar to (4.49) (see Exercise 4.2.8), so values of ı

.n/

.=/ and I˛ . A/ are also connected by relations (4.82) similar to (4.81). N

4.2.20 The Generalized Form of Representation of the Stress Power Consider only nonpolar continua, when the tensor T is symmetric; then TS D T

and

TK 0;

(4.86)

and, according to formulae (4.13), (4.46), (4.51a), and (4.74a), the stress power w.i / (4.2) can be expressed in the following equivalent forms: w.i / w.i /

.n/

.n/

dC D T ; dt .n/

n 2 Z;

(4.87)

n 2 Z;

(4.88)

.n/

dG ; D T dt .n/

w.i /

ı dA d OT D S CS ; dt dt

w.i /

ı dg d OT D S CS ; dt dt

.n/

.n/

n 2 Z;

(4.89)

n 2 Z:

(4.90)

.n/

The stress power w.i / is said to be written in the form An , Bn , Cn or Dn , if we use expressions (4.87), (4.88), (4.89) or (4.90), respectively.

4.2 Energetic and Quasienergetic Couples of Tensors

187 .n/

Introduce the generalized energetic stress tensors T G , the generalized energetic .n/

.n/

deformation tensors C G , the generalized energetic deformation measures G G and the generalized rotation tensor of stresses SG as follows:

.n/

TG

.n/

GG

8 .n/ ˆ ˆ T; ˆ ˆ ˆ ˆ .n/ ˆ 0 there exists a vector F .i / .B; t/, called the body inertia force, which in the inertial system O eN i for configuration K has the form KW

F .i / D d I=dt;

(4.620a)

where I is the momentum vector of the body (3.26), and the vector F .i / is transformed during the passage to K0 in the R-indifferent way: K0 W

F 0.i / D F .i / Q:

(4.620b)

(2) For any pair of bodies B and B1 8t > 0 there exists a vector of bodies interaction force F .B; B1 ; t/, which may be zero-valued and has the following properties: the vector is additive:

F .B 0 C B 00 ; B1 ; t/ D F .B 0 ; B1 ; t/ C F .B 00 ; B1 ; t/; F .B; B10 C B100 ; t/ D F .B; B10 ; t/ C F .B; B100 ; t/; where B D B 0 [ B 00 , B1 D B10 [ B100 ;

316

4 Constitutive Equations

the vector is R-indifferent during the passage to K0 :

F 0 D F QI the vector is inertial, i.e. the summarized vector of external forces acting on

the body B, F .B; t/ D F .B; B e ; t/ (where B e D U n B is the surroundings of the body B) in K is equal to the body inertia force vector: K W F .i / D F :

(4.621)

A corollary of this axiom is the following theorem. Theorem 4.53a. If Axiom 5a is assumed to be valid, then under the transformation from configuration K to K0 the momentum balance law retains its form: K0 W F 0.i / D F 0 :

(4.621a)

This theorem is an integral analog of Theorem 4.53. Axiom 5a ensures that the conditions involved in statements of Theorems 4.53 and 4.53a be satisfied. Thus, due to the acceptance of Axiom 5a, the main purpose of this section (to ensure that the momentum balance equations satisfy the principle of material indifference) has been achieved. Axiom 5 introduced in Sect. 3.2.1 is entirely contained in the generalized Axiom 5a; therefore, all the conclusions, obtained above according to Axiom 5, remain valid.

4.10.12 Material Indifference of the Thermodynamic Laws The following axiom asserts material indifference of the thermodynamic functions. Axiom 7a (Objectivity of thermodynamic functions). The thermodynamic functions U , Q, H , QN and temperature are R-indifferent scalars: U 0 D U;

Q0 D Q;

H 0 D H;

QN 0 D QN ;

0 D :

(4.622)

It follows from (4.622) that the specific internal energy and specific entropy are R-indifferent scalars: 0 D I (4.622a) e 0 D e; and from (4.622), (3.106), and (3.159) we get that the heat fluxes from mass and surface sources, and also the specific internal entropy production are R-indifferent scalars: 0 qm D qm ;

0 q† D q† ;

q 0 D q ;

(4.622b)

4.10 The Principle of Material Indifference

317

because the mass m is independent of the rigid motion and d† is always defined in the R-indifferent way. The normal n and relations (3.111) are R-indifferent (see Sect. 4.10.5), then we obtain that the heat flux vector q is R-indifferent: q0 D q Q;

(4.623)

because the following relation holds: 0 D n0 q0 D n Q q0 D n q D q† : q†

Theorem 4.54. The principle of material indifference holds for the first thermodynamic law, i.e. from the energy balance equation (3.118), written for K in the form

de dt

C v a r .T v/ C r q qm f v D 0;

(4.624)

we get the following energy balance equation in K0 : 0

de 0 dt

0 0 f0 v0 D 0; C v0 a0 r 0 .T0 v0 / C r 0 q0 0 qm

(4.625)

where a0 is defined by (4.617). H Consider transformations of separate summands on the left-hand side of equation (4.625). Due to Axiom 5a, the momentum balance equation (4.619) holds in K0 ; then, multiplying the left and right sides of this equation by v0 , we get 0 v0 a0 D v0 r 0 T0 C 0 f0 v0 ;

(4.626a)

or 0 v0 a0 r 0 .T0 v0 / C T0 .r 0 ˝ v0 /T 0 f0 v0 D 0:

(4.626b)

Due to Axiom 7a and the results of Exercises 4.10.2 and 4.10.5, we have r 0 q0 D r q;

0

de 0 de D : dt dt

(4.627)

Since the tensor T0 is symmetric (see Exercise 4.10.6), the results of Exercise 4.10.4 and Theorem 4.38 yield T0 .r 0 ˝ v0 /T D T0 D0 D T D D T .r ˝ v/T :

(4.628)

318

4 Constitutive Equations

Modify the left-side expression in (4.625) with taking formulae (4.626b), (4.627) and (4.628) into account: 0

de 0 dt

0 C v0 a0 r 0 .T0 v0 / C r 0 q0 0 f0 v0 0 qm

D

de 0 T0 .r 0 ˝ v0 /T C r 0 q0 qm dt

D

de T .r ˝ v/T C r q qm : dt

(4.629)

Since Eq. (4.624) is satisfied, the heat influx equation (3.124) being its consequence holds as well. On comparing (4.629) with (3.124), we conclude that the expression (4.629) must vanish, and therefore the equation (4.625) really holds. N Theorem 4.55. The entropy balance equation (the second thermodynamic law) satisfies the principle of material indifference, i.e. from (3.303) at ˛ D 4 we get the equation 0 0 q 0 0 0 d 0 .q C q 0 / D 0: (4.630) r dt 0 0 m H A proof of the theorem should be performed as Exercise 4.10.7. N

4.10.13 Material Indifference of the Compatibility Equations Let us verify whether equations (3.303) at ˛ D 5 and 6 always satisfy the principle of material indifference. Theorem 4.56. The kinematic equation (3.303) at ˛ D 5 and the dynamic compatibility equation (3.303) at ˛ D 6 always satisfy the principle of material indifference. ı

H Since u0 D x0 x, we have d u0 =dt D d x0 =dt D v0 ; and hence the kinematic equation is always satisfied in K0 : 0 .d u0 =dt/ D 0 v0 :

(4.631)

Let the compatibility equation (3.303) at ˛ D 6 be satisfied in K:

d FT r .F ˝ v/ D 0: dt

(4.632)

4.10 The Principle of Material Indifference

319

Consider the similar expression in K0 : 0

d FT0 r 0 .0 F0 ˝ v0 / dt d FT P 0 FT0 r 0 ˝ v0 .r 0 0 F0 / ˝ v0 : (4.633) D 0 Q C FT Q dt

Here we have used formula (4.559). With taking this equation and (4.578) into account, the third summand takes the form P 0 FT0 r 0 ˝ v0 D FT Q QT r ˝ v Q C FT Q QT Q T T P D F r ˝ v Q C F Q:

(4.634)

The fourth summand in (4.633) is zero, because r 0 0 F0 D r0i

@0 F0 @F D ri Q Q T D r F D 0: i @X @X i

(4.635)

The last equality in (4.635) is valid due to Theorem 3.24. On substituting (4.634) and (4.635) into (4.633), we finally get d FT T r . F ˝ v / D F r ˝ v .r F/ ˝ v Q dt dt d FT D r .F ˝ v/ Q D 0; (4.636) dt 0 dF

T0

0

0 0

0

because Eq. (4.632) is satisfied by condition. N

4.10.14 Material Indifference of Models An and Bn of Ideal Continua Consider the constitutive equations (4.165) and (4.167) of models An of ideal continua (both solids and fluids) in actual configuration K. Then in configuration K0 these equations, according to the principle of material indifference (4.603)–(4.605), must be written in the form 8 .n/ .n/ .n/ ˆ ˆ ˆ T 0 D [email protected] [email protected] C / F . C 0 ; 0 /; < .n/ (4.637) D . C 0 ; 0 /; ˆ ˆ ˆ : D @ [email protected]; where

.n/

and F . C 0 ; 0 / are the same tensor functions as in (4.165) and (4.167).

320

4 Constitutive Equations

But Eqs. (4.637) really hold, if the constitutive equations (4.165) and (4.167) are .n/

.n/

.n/

.n/ .n/

.n/

satisfied, because all the tensors T and C are R-invariant, i.e. T 0 D T , C 0 D C ; and the temperature 0 , due to (4.622), is also R-invariant: 0 D . For models Bn , the situation is analogous. Thus, we can formulate the following theorem. Theorem 4.57. For models An and Bn of ideal continua (both solids and fluids), the principle of material indifference is always identically satisfied for any potentials .n/

. C ; / and

.n/

. G; /.

4.10.15 Material Indifference for Models Cn and Dn of Ideal Continua The situation is different when the principle of material indifference is applied to models Cn of ideal continua. Let in configuration K the relations (4.171) and (4.174) be satisfied, then in configuration K0 , according to the principle of material indifference (4.603)–(4.605), the following equations must hold: 8.n/ .n/ .n/ ˆ ˆ S 0 D [email protected] [email protected] A 0 / ˆ. A 0 ; O0 ; /; ˆ ˆ ˆ ˆ .n/ < D . A 0 ; O0 ; /; ˆ ˆ D @ [email protected]; ˆ ˆ ˆ ˆı ı .n/ : S0 D @ [email protected]O0 ˆ. A 0 ; O0 ; /;

(4.638)

ı

where ˆ, ˆ and

are the same tensor functions as in (4.171) and (4.174). .n/

.n/

Since the tensors S and A are R-indifferent (see Theorems 4.36 and 4.37) and the tensor O is transformed during the passage from K to K0 by formula (4.568), ı

so the relations (4.638) hold if and only if the functions ˆ, ˆ, and following equations for any orthogonal tensor Q: 8 .n/ .n/ ˆ ˆQT ˆ. A ; O; / Q D ˆ.QT A Q; QT O; /; ˆ ˆ ˆ ˆ ı .n/ ı .n/ ˆ 0:

(4.728)

The value q is nonnegative due to the Planck inequality (3.158a). Axiom 16 (The Onsager Principle). In order for the Planck inequality to hold, the specific internal entropy production q must be representable in the quadratic form q D

X

Qˇ Xˇ D

ˇ D1

X

Lˇ Xˇ X > 0:

(4.729)

ˇ; D1

Here Qˇ D

X

Lˇ X

(4.730)

ˇ D1

are functions called thermodynamic fluxes, and Xˇ are functions called thermodynamic forces. As follows from (4.729), the matrix Lˇ is symmetric: Lˇ D Lˇ :

(4.730a)

Remark. As shown in Sect. 4.4.3, the principles of thermodynamically consistent determinism and equipresence result in the fact that all the active variables ƒ included in constitutive equations (4.158)–(4.161) are independent of the temperature gradient r . Hence, the dissipation function w being among the active variables is independent of r as well. Thus, if we assume r 0 8M 2 V and 8t > 0, then

340

4 Constitutive Equations

the value of w remains unchanged and the summand q r vanishes; and from (4.728) we obtain w > 0:

(4.731)

This inequality is known (see (3.171)) as the dissipation inequality. The reasoning has proved the fact that the dissipation inequality is a consequence of the Planck inequality (4.728) and principles of thermodynamically consistent determinism and equipresence. For ideal continua, w 0, then the Planck inequality (4.728) yields q r 6 0;

(4.732)

this relation is called the Fourier inequality. For nonideal continua, the Fourier inequality (4.732) does not follow from the Planck inequality (4.728) and it should be assumed to be an independent axiom (see Axiom 10 in Sect. 3.5.2). t u The Onsager principle P is applied as follows: expression (4.728) for q is represented as the sum ˇ Qˇ Xˇ ; for example, for ideal continua we choose the temperature gradient X1 D r as a thermodynamic force X1 and the value Q1 D .1=/q as a corresponding thermodynamic flux. Then, according to the Onsager principle, the thermodynamic flux must be a linear function of X1 , i.e.

1 q D Q1 D L11 X1 D L11 r :

(4.733)

The coefficient L11 in this case is a second-order tensor, and the product L11 D is called the heat conductivity tensor; then relation (4.733) takes the form q D r ;

(4.734)

which is called the Fourier law in the spatial description. Due to the Fourier inequality, the heat conductivity tensor is positive-definite: r r > 0:

(4.735)

The Fourier law can be formulated in the material description as well. To do this, ı one should replace the vector q in (4.734) by the vector q according to formula ı

(3.128) and the gradient r by r according to the formula similar to formulae ı

(2.23): r D F1T r . Then we obtain the relation ı

ı

ı

q D r ;

(4.736)

4.12 The Onsager Principle

341

ı

where is the heat conductivity tensor in a reference configuration: q ı D g=g F1 F1T : ı

(4.737)

The Fourier law (4.734) is included in the general system of constitutive equations (4.158), (4.159), (4.160) or (4.161) of a continuum. This law can also be written in the form of operator relation (4.156), where the vector q is an active variable and the vector r is a reactive one. Then, depending on the considered model An , Bn , Cn or Dn , the heat conducı

tivity tensors and should be assumed to be operators of the corresponding set of reactive variables R: .n/

M C ; /; models An W D . .n/

M G; /; models Bn W D . .n/

M A ; O; /; models Cn W D . .n/

M g ; O; /; models Dn W D .

ı ıM .n/ D . C; ı ıM .n/ D . G; M .n/ ı ı D . A ; O; ı ıM .n/ D . g ; O;

/; /; /; /:

(4.737a)

4.12.2 Corollaries of the Principle of Material Symmetry for the Fourier Law ı

Consider the Fourier law (4.736) in a reference configuration K with use of one of the models An ; : : : ; Dn ; for definiteness, we choose the model Bn . Since the Fourier law as well as the constitutive equations (4.158)–(4.161) describes some physical properties of a continuum, so this law must satisfy the principle of material ı

ı

symmetry (4.180)–(4.181) if the vector q is considered as ƒ and the vector r – as ı

R. In other words, if Eq. (4.736) holds in a configuration K, then there exists a symı

ı

metry group G s such that for each H -transformation from this group (H W K ! K) the form of function (4.736) remains unchanged: ı

ı .n/

ı

q D . G; / r :

(4.738a)

and

ı .n/

q D . G ; / r :

(4.738b)

342

4 Constitutive Equations

Theorem 4.63. The principle of material symmetry (4.738a), (4.738b) holds for the ı

Fourier law in the material description if and only if in K there exists a symmetry ı

group G s such that the heat conductivity tensor is transformed as follows: ı .n/

. G; / D

q ı .n/ ı ı g=g H1 . G ; / H1T 8H 2 G s :

(4.739)

H Since the vector q and temperature are defined in K, they are H -invariant. Then ı

ı

q and r under H -transformations change as follows:

q

qD

1

g=g F

r D ri

q q ı 1 q D g=g H F q D g=g H q;

ı @ i ı ı k @ 1T D r ˝ r r D H r : i @X i @X k

(4.740)

Let the principle of material symmetry be valid, then relations (4.738a) and (4.738b) hold. On substituting (4.740) into (4.738b), we obtain

or

q ı ı ı ı g=g H q D H1T r ;

(4.741)

q ı ı ı q D g=g .H1 H1T / r :

(4.742)

ı

Comparing (4.738a) with (4.742), we see that in this case the relation (4.739) ı

must hold for the heat conductivity tensor . Prove the converse assertion. Let the condition (4.739) be satisfied. Substituting the condition into (4.736), we obtain formula (4.742). Performing the transformations in the reverse order, we get formulae (4.738a) and (4.738b). N Theorem 4.63 still holds for the remaining models An , Cn and Dn . For them the relation (4.739) has the forms q ı .n/ ı g=g H . C ; / HT ; q ı .n/ ı .n/ ı . A ; O ; / D g=g H . A ; O; / HT ; q ı .n/ ı .n/ ı . g ; O ; / D g=g H . g ; O; / HT : ı .n/

. C ; / D

(4.743)

Theorem 4.64. The Fourier law (4.734) in the spatial description identically satisfies the principle of material symmetry for each symmetry group Gs . ı

H Indeed, for any H -transformation of a reference configuration K ! K, since the vectors q and r are always H -invariant, relation (4.734) retains its form with the same heat conductivity tensor for each tensor H. N

4.12 The Onsager Principle

343 ı

Certainly, there is no discrepancy between relation (4.739) for the tensor durı

ing the passage from K to K and invariability of the tensor , because formula (4.739) together with (4.737) and formula (4.188) of transformation of the deformation gradient F ensure that the tensor be independent of the choice of reference

configuration K: D

q q q ı ı ı ı g=g F FT D g=g F H HT FT D g=g F FT D ./ :

Here ./ is the tensor defined in configuration K, which, as it was expected,

coincides with the tensor itself (it is not to be confused with , which is the ı

tensor defined in configuration K).

4.12.3 Corollary of the Principle of Material Indifference for the Fourier Law ı

ı

The heat flux vector q and temperature gradient r are defined in a reference conı

figuration K, therefore they remain unchanged at rigid motions from K to K0 (i.e. ı

ı

ı

ı

q0 D q, .r /0 D r ). Hence, the Fourier law (4.736) in the material description also remains unchanged under the transformation from K to K0 : ı

ı

ı

q0 D .r /0

(4.744)

for any orthogonal rotation tensor Q. This means that relation (4.736) of the Fourier law satisfies the principle of material indifference (4.603), (4.605) with the same ı

heat conductivity tensor , i.e.

ı

ı

0 D :

(4.745)

For the Fourier law (4.734) in the spatial description, the situation is different. Theorem 4.65. The Fourier law (4.734) in the spatial description satisfies the principle of material indifference (4.603), (4.605) (i.e. the relation q0 D .r /0

(4.746)

follows from (4.734) for any rigid motion K ! K0 ) if and only if the heat conductivity tensor is both R-invariant and R-indifferent: 0 D ;

0 D QT Q:

(4.747)

344

4 Constitutive Equations

Remind that tensors satisfying the condition (4.747) for any orthogonal tensor Q, according to (4.290), are called indifferent relative to the full orthogonal group. H The heat flux vector q is R-indifferent (just as the tensor T). To prove the fact, one should use relation (3.111) and take into account that the scalar function of heat 0 influx due to surface sources q† is R-invariant: q† D q† , and the normal vector n is R-indifferent, then 0 q† D q0 n0 D q0 QT n D q† D q n:

(4.748)

q0 D QT q:

(4.749)

Thus, we obtain

The gradient of a scalar is also R-indifferent: .r /0 D ri 0

@ @ D Q T ri D QT r : @X i @X i

(4.750)

Multiplying the relation (4.734) by QT , we obtain QT q D QT r D QT Q QT r ;

(4.751)

q0 D 0 r 0 ;

(4.752)

0 D QT Q:

(4.753)

or

As follows from (4.752), the principle of material indifference holds if and only if the tensor is indifferent, i.e. 0 D . N Notice that relations (4.747) are consistent with (4.745): substituting formulae (4.747), (4.559), and (4.560) into (4.737), we obtain q q ı ı 10 ı 0 0 1T0 D g=g F F D g=g F1 Q 0 QT F1T q ı ı D g=g F1 F1T D ; (4.754) that is exactly the relation (4.745).

4.12.4 The Fourier Law for Fluids For fluids, one usually use the Fourier law in the spatial description (4.734). According to Theorem 4.64, the relation (4.734) for any tensor satisfies idenı

tically the principle of material symmetry for each symmetry group G s , including ı

the unimodular group G s D U corresponding to a fluid.

4.12 The Onsager Principle

345

To satisfy the principle of material indifference, the tensor in (4.734) must be indifferent relative to the full orthogonal group Gs D I , i.e. the relations (4.747) must hold. As shown in Sect. 4.8.3 (see (4.293)), Eq. (4.747) for the group I has an unique solution being the spherical tensor : D E;

> 0;

(4.755)

where is the constant called the heat conductivity coefficient. Substitution of (4.755) into (4.734) gives the Fourier law for fluids q D r :

(4.756) ı

With the help of Eq. (4.737) we can write the heat conductivity tensor for fluids as follows: q q ı ı ı D g=g F1 F1T D g=g G1 : (4.757) This tensor is not spherical, but it satisfies Eq. (4.739). The Fourier law (4.736) in the material description for fluids has the form q ı ı ı q D g=g G1 r :

(4.758)

4.12.5 The Fourier Law for Solids For solids, one usually apply the Fourier law in the material description (4.736). ı

As noted in Sect. 4.12.3, for any tensor this relation satisfies identically the principle of material indifference. To satisfy the principle of material symmetry, by ı

ı

Theorem 4.63, the tensor must be indifferent relative to a group G s corresponding to the symmetry group of a considered solid, i.e. the relation (4.739) must hold. According to the results of Sect. 4.8.3 (see formulae (4.293)), we find that: ı

ı

if G s D I (an isotropic solid), then the tensor has the only independent comı

ponent :

ı

ı

D E;

(4.759)

ı

b3 (a transversely isotropic solid with the transverse isotropy vectorb if G s D T c3 ), ı

ı

ı

then the tensor has two independent components – 1 and 2 : ı

ı

ı

c23 C 2 E; D 1b

(4.760)

346

4 Constitutive Equations ı

b (an orthotropic solid with the principal orthotropy basis b c ), then the if G s D O ı

ı

tensor has three independent components – : ı

D

3 X

ı

c2 : b

(4.761)

D1

To find the tensor for solids one should substitute formulae (4.759)–(4.761) into (4.737). For example, for an isotropic continuum the tensor has the form q D

q ı ı ı ı g=g F FT D g=g F FT :

(4.762)

On substituting expressions (4.759)–(4.761) into (4.736), we get the Fourier law for isotropic solids ı

ı

q D E;

(4.763)

the Fourier law for transversely isotropic solids ı

ı

ı

q D .1b c23 C 2 E/;

(4.764)

and the Fourier law for orthotropic solids ı

qD

3 X D1

ı

b c2 :

(4.765)

Chapter 5

Relations at Singular Surfaces

5.1 Relations at a Singular Surface in the Material Description 5.1.1 Singular Surfaces Up to now we considered the case when all functions appearing in the balance laws: , u, v, T, F, f etc. are continuously differentiable functions of coordinates X i (or of x i ) and time t. However, in practice one often meets with problems, where this condition is violated. For example, for the phenomena of impact, explosion, combustion etc., a part of the indicated functions in a domain V considered can suffer a jump discontinuity across some surface S (Fig. 5.1). As a singular surface we can consider an interface S between two contacting media (Fig. 5.2). Besides differential equations in a domain V , mathematical formulation of problems in continuum mechanics should involve conditions on the surface bounding the domain V (boundary conditions). This boundary can also be considered as a singular surface for the functions. The integral balance laws still hold for a volume V containing a singular surface S ; but differential equations of the corresponding laws are not valid across such surface S , because in deriving the equations we have essentially used the condition of continuous differentiability of the functions inside V . Therefore, we need to establish appropriate consequences of the balance laws for the case when there is a singular surface for the functions. This is a purpose of the chapter.

5.1.2 The First Classification of Singular Surfaces Consider a continuum occupying a volume V in K. We assume that there is a surface S dividing the volume V into two subdomains VC and V such that inside VC and V all functions (, v, F etc.) considered are continuously differentiable. However, during the passage from VC to V across S some of the functions can suffer a jump discontinuity. Consider the set Mt of material points belonging to the singular Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 5, c Springer Science+Business Media B.V. 2011

347

348

5 Relations at Singular Surfaces

Fig. 5.1 The singular surface S in domain V

Fig. 5.2 The interface S between two media VC and V

Fig. 5.3 There is no transition of material points across a singular surface ı

surface S.t/ at a time t > 0. This set Mt in a reference configuration K belongs to ı

ı

a surface S.t/, which, in general, depends on t and divides V into two subdomains VC and V . At different times t (t1 6 t 6 t2 ), positions of the singular surface S.t/ in the actual configuration may be distinct, and there is an alternative: Sets Mt , corresponding to surfaces S.t/ 8t 2 Œt1 ; t2 , coincide Not all sets Mt , corresponding to surfaces S.t/ at considered times t (t 2 Œt1 ; t2 ),

are coincident In the first case, one can say that material points do not cross the singular surface S.t/ 8t 2 Œt1 ; t2 ; in the second case, there occurs a transition of material points across the singular surface. If there is no transition of material points across the interface S.t/, 8t 2 Œt1 ; t2 , ı

ı

then the position of the surface S .t/ in K 8t 2 Œt1 ; t2 remains unchanged (Fig. 5.3). While material points cross a singular surface at t 2 Œt1 ; t2 , the location of the ı

surface S.t/ in the reference configuration varies (Fig. 5.4). Let us give now a classification of singular surfaces (Fig. 5.5).

5.1 Relations at a Singular Surface in the Material Description

349

Fig. 5.4 Material points cross a singular surface

Fig. 5.5 Classification of singular surfaces

Definition 5.1. A singular surface S.t/, across which unknown functions , u, v, T, F etc. suffer jump discontinuities, is called a surface of a strong discontinuity. If these functions remain continuous across S.t/ and only their first derivatives with respect to time t and coordinates (r , r ˝u, r ˝v etc.) suffer jump discontinuities, then S.t/ is called a surface of a weak discontinuity. The transition of material points can occur only across surfaces of a strong discontinuity. Definition 5.2. A surface of a strong discontinuity S.t/, across which there is no transition of material points 8t 2 Œt1 ; t2 , is called a surface of a contact discontinuity if constitutive equations in domains VC .t/ and V .t/ are the same at all times t considered; otherwise, it is called a surface of contact. Definition 5.3. A surface of a strong discontinuity S.t/, across which there is a transition of material points at t 2 Œt1 ; t2 , is called a surface of a shock wave if constitutive equations are the same at all times t considered; otherwise, S.t/ is called a surface of phase transformation. Besides the classification introduced above, singular surfaces can be split into coherent and incoherent ones.

350

5 Relations at Singular Surfaces

Definition 5.4. A singular surface S.t/, across which the radius-vectors x of material points M 2 S.t/ suffer jump discontinuities, is called incoherent; otherwise, the surface is called coherent. For an incoherent surface S.t/, a local neighborhood dV of every point M, belonging to the surface S.t/ at some time t, at the time t C t is divided into two half-neighborhoods dVC and dV , moving apart one from another, because if ı

ı

in K the radius-vector x.M/ of the point M is continuous, then the radius-vector x.M; t C t/ of this point is no longer continuous: ı

Œx D 0;

Œx ¤ 0:

(5.1)

For a coherent surface S.t/, local half-neighborhoods dVC and dV of its every point M 2 S.t/ at any time t do not move apart one from another, and the jump of ı the radius-vector x is zero if Œx D 0: ı

Œx D 0;

Œx D 0:

(5.2)

The example of an incoherent surface is a surface of contact (an interface) between an ideal fluid and a solid: an ideal fluid slips along a solid without grip. If at time t we choose two contacting points MC and M of an incoherent surface S.t/, then their radius-vectors xC and x coincide but the radius-vectors ı

ı

ı

xC and x in K may be not coincident, i.e. instead of (5.1) the following relations may hold ı

Œx ¤ 0;

Œx D 0:

It is evident that an incoherent surface is a surface of a strong discontinuity.

5.1.3 Axiom on the Class of Functions across a Singular Surface ı

ı

Let us consider now a surface of a strong discontinuity S .t/ dividing a volume V in ı

ı

ı

K into two subdomains V C and V . ı

ı

Definition 5.5. Let there be a function A.x; t/ defined in a domain V which conı

tains a singular surface S . The value ˇ ˇ ŒA D AC ˇ ı A ˇ ı ; S

S

(5.3)

5.1 Relations at a Singular Surface in the Material Description

where

ˇ A˙ ˇ ı D S

ı

351

ı

A˙ .x; t/;

lim ı

x!x† ı ı ı ı x2V ˙ ; x † 2S ı

is called the jump of the function across the surface S . If a surface S.t/ is coherent, then from (5.2) it follows that the functions ı

x i .X i ; t/ do not suffer jump discontinuities across the surface S .t/: ˇ ˇ x i .X j ˇ ı ; t/ D x i .X j ˇ ı ; t/: SC

S

ı

Then in (5.3) we can always pass from coordinates x i to x i and consider the jump of a function across a corresponding singular surface S.t/ in K: ˇ ˇ ŒA D AC ˇS A ˇS ; where

ˇ A˙ ˇ S D

lim

i x i !x† x i 2V˙ ; x i 2S.t / †

(5.4)

A.x i ; t/:

ı

ı

Axiom 17. For a continuum containing in K a singular surface S.t/, which divides ı

ı

ı

ı

ı

ı

the volume V into two parts V C and V , the functions , A˛ and B ˛ appearing in ı

ı

the balance equation system (3.310) are assumed to be smooth in V C and V : ı

A˛ D

8ı 0; and their solution is written as follows: D .x/;

F D F.x/;

u D u.x/:

(6.39)

Here the right-hand sides of the relations can be found after solving the system (6.35), (6.36).

Exercises for 6.1 6.1.1. Show that for the quasilinear models AI (see Remark 1 or Sect. 4.8.7) the constitutive equations (6.8a) have the explicit analytic form similar to (6.16b): T D F1T .4 M ƒ/ F1 ; ƒD

1 .r ˝ u C r ˝ uT r ˝ uT r ˝ u/; 2

F1 D E r ˝ uT ;

or T D .E r ˝ u/

4

M

1 E r ˝ uT r ˝ u .E r ˝ uT /: 2

6.1.2. Using formula (4.322), show that for isotropic continua the relations (6.9a) can be written as follows: T D F1T . 1

C 2ƒ C D '1 C '2 I1 C '3 I2 ; 3

D '3 ;

/ F1 ; D '2 C I1 '3 ;

3ƒ

1E

2

2

I˛ D I˛ .ƒ/; ˛ D 1; 2; 3:

Using formula (4.337), show that for the linear model AI of isotropic continua this relation takes the form T D J.l1 I1 .ƒ/F1T F1 C 2l2 F1T ƒ F1 /;

390

6 Elastic Continua at Large Deformations

or

1 T T D J l1 r u r ˝ u r ˝ u .E r ˝ u/ .E r ˝ uT / 2 1 C 2l2 .E r ˝ u/ .E r ˝ uT / r ˝ u .E r ˝ u/T ; 2 where l1 and l2 are the constants. 6.1.3. Show that constitutive equations (6.16b) for the model AI in basise ri have the component form eij D T

r X D1

e I .s/ ms

D

ems ; @I.s/ [email protected]ƒ

.s/ e 1 /m .F e 1 /se ' e g i ke g jl .F k l I ms ;

' D [email protected] [email protected].s/ /;

D

e ms /; /; .I.s/ .ƒ

ese eke eke eke eme e 1 /m D ı m r e ms D 1 .r us C r um r um r us /; .F um : ƒ k k 2 Using the result of Exercise 6.1.1, show that for the quasilinear model AI this component representation takes the form e e 1 /p .F e 1 /q f ems ; T ij D e g i ke gjl .F k l M pqms ƒ and for the model AI of isotropic continua – the form e 1 /p .F e 1 /q eij De g i ke g jl .F T k l

g pq C 1e

e ms C g pme g qs ƒ 2e

mu esv e e g e g e g : ƒ ƒ 3 pm qs uv

6.2 Closed Systems in the Material Description 6.2.1 U VF -system of Dynamic Equations of Thermoelasticity in the Material Description For the material description, we should consider the balance law system (3.310) having the explicit form ı

D det F1 ; ı

ı

(6.40a) ı

[email protected][email protected]/ D r P C f; ı

ı

ı

ı

[email protected][email protected]/ D r q C qm ;

(6.40b) (6.40c)

ı

@FT [email protected] D r ˝ v;

(6.40d)

@[email protected] D v:

(6.40e)

6.2 Closed Systems in the Material Description

391

Here we have used the entropy balance equation (3.174) in place of the energy balance equation. The system (6.40) as well as (6.1) becomes closed after complementing it with the constitutive equations consisting of the two equations ı

ı

ı

q D r ;

D @ [email protected];

(6.41)

and the universal constitutive equations (6.3) and (6.4) written for the Piola– Kirchhoff tensor: .n/

P D F ıG .F; /;

.s/

D

.n/

.I G . C G .F//; /

.F; /:

(6.42)

Here we have introduced the tensor function .n/ F ıG .F; /

.n/

ı

.=/F1 F G .F; /:

(6.43)

The system (6.40)–(6.42) contains 16 scalar unknowns (the density is assumed to be expressed in terms of F): ; u; v; F k X i ; t;

(6.44)

which are functions of Lagrangian coordinates X i and time t, and consists of 16 scalar equations (after substitution of (6.41) and (6.42) into (6.40)). This system is called the U VF -system of thermoelasticity in the material description. For solids, this system proves to be more preferable, because a domain of defı

inition of the functions (6.44) is known: this is V Œ0; tmax . The exceptions are ı

problems with phase transformations, where V varies with time and is determined in the process of solving. However, in this case the system (6.40)–(6.42) also proves to be more preferable than the corresponding system (6.1)–(6.6) in the spatial description. .n/

Remark 1. Although the definition (6.43) of the tensor function F ıG includes F1 , this function can be represented as a function of F. To do this, we should use the representation of F1 in the eigenbasis (see (2.171)) and the expression (6.4) of the .n/

function F G ; then we obtain .n/

P D F ıG .F; / D

r X D1

4

.n/

.n/

E ıG D F1 4 E G D

3 X

ı

ı

ı

.n/

' 4 E ıG I.s/ G; ı

ı

E ˛ˇ p˛ ˝ pˇ ˝ .hG pˇ ˝ p˛ C .1 hG /pˇ ˝ p˛ /;

˛;ˇ D1 ı

(6.45)

E ˛ˇ D E˛ˇ =˛ ;

ı

ı

' D .=/' ; G D A; B; C; D:

392

6 Elastic Continua at Large Deformations .n/

.s/

The form of the tensors I G coincides with (6.9c). The tensors 4 E ıG are called Lagrangian tensors of energetic equivalence; they connect the Piola–Kirchhoff .n/

stress tensor to T G :

.n/

.n/

P D 4 E ıG T G :

(6.46) .n/

.n/

For example, choosing the tensor functions T G D F G . C G ; / for the models An .n/

in the form (6.8), we obtain tensor function F ıG in the form .n/

.n/

.n/

.n/

.n/

P D 4 E ı .4 M C C C 2 C 6 L . C ˝ C //; .n/

(6.47)

.n/

where 4 E ı D F1 4 E G . Notice that although the equation system (6.40) contains the Piola–Kirchhoff stress tensor P, the final (after solving the system (6.40) with boundary and initial conditions) analysis of a stress field in a solid usually needs the Cauchy stress tensor T to be known; therefore, besides constitutive equations (6.45), the material description involves the relations (6.4). t u Remark 2. Although the tensor function (6.45) has a rather complex form, which in general cannot be expressed analytically, this form may be simplified for two exceptional models AV and BV , because ı

PD

ı

ı

V 1 V F T D F1 .F T FT / D T FT :

(6.48)

V

On substituting the expression (6.4) for the tensor function T D F A .F; / into (6.48), we obtain the following tensor function having the explicit analytical representation: V

P D F ıA .F; / D

r X

T I.s/ F ;

(6.49)

D1 ı

ı

' D [email protected] [email protected].s/ /;

D

.s/ .I.s/ .C/; /; I.s/ D @I [email protected];

C D .1=2/.FT F E/: V

V

The substitution C ! G in (6.49) yields the tensor function F ıB .F; / for the model BV . For the models AV and BV , there is no need to determine eigenbases and eigenvalues in constructing the functions (6.49). The mentioned advantages discern the models AV and BV among others while using the material description as well as the

6.2 Closed Systems in the Material Description

393

models AI and BI while using the spatial description (see Remark 3 in Sect. 6.1). These models are the most frequently applied to computations of problems of elasticity theory at large deformations. The models AI and BI are of a higher level of complexity in the material description. For the model AI , constitutive equations (6.49) are replaced by the relations I

P D F ıA .F; / D ı

ı

r X

ı

1 ' F1 F1T I.s/ F ;

(6.50)

D1

' D [email protected] [email protected].s/ /;

.s/ D .I.s/ .ƒ/; /; I.s/ D @I [email protected]ƒ; 1 ƒ D .E F1 F1T /: 2

For this model, although there is no need to determine eigenvalues and eigenvectors, however, in comparison with the models AV and BV , one should invert the tensor F1 . Just as in the spatial description, the models AII , BII and AIV , BIV are the most complicated for computations. t u The entropy balance equation in the system (6.40) can be modified just as in Sect. 6.1.1 by using the principal thermodynamic identity in the material description (4.125): ı ıd ı d (6.51) C P r ˝ vT D 0: dt dt Differentiating the identity with respect to and using the second relation of (6.41), we obtain the following expression for the rate of changing the specific entropy in the material description: ı @

@t

2 ı@

D

@ 2

@P ı @ r ˝ vT : @t @

(6.52)

On substituting (6.52) and (6.41), (6.42) into the entropy balance equation of the system (6.40), we get the desired heat conduction equation for an elastic continuum in the material description: ı

c"

ı

.n/ ı ı ı @ .n/ ı D r . r / C .4 E ıG T G / r ˝ vT C qm ; @t

(6.53)

where the heat capacity c" is determined by (6.11a). Here we have taken into account that equations (6.45) give the following expression for the derivative with respect to : ı

ı r X .n/ .n/ @' .s/ @ .n/ı .n/ @P D I G D 4 E ıG T G ; F G .F; / D 4 E ıG @ @ @ D1

(6.53a)

394

6 Elastic Continua at Large Deformations .n/

where the tensor T G has been introduced by formula (4.423): .n/

T G D

r X @' .s/ I : @ G D1

(6.53b)

6.2.2 U V - and U -systems of Thermoelasticity in the Material Description Return to the general system (6.40)–(6.42) and notice that the deformation gradient F can be excluded from the system with the help of Eq. (2.77): ı

F D E C r ˝ uT :

(6.54)

Then we obtain the U V -system of thermoelasticity in the material description: ı @v

ı

c"

@t

ı

ı

.n/

ı

D r F ıG ..E C r ˝ uT /; / C f;

ı ı ı ı ı @ @ .n/ı ı F G .E C r ˝ uT ; / r ˝ vT C qm ; D r . r / C @t @ @[email protected] D v;

(6.55)

consisting of seven equations for seven scalar unknowns: ; u; v k X i ; t:

(6.56)

The density does not appear in this system explicitly, and it can always be evaluated with the help of the continuity equation (6.17): ı

ı

D det .E C r ˝ uT /:

(6.57)

Since in the material description the velocity v is connected to the displacement vector u by an explicit kinematic relation, we can eliminate the velocity v between the unknowns. As a result, we obtain the U -system of thermoelasticity in the material description: 2 ı@ u

@t 2

ı

.n/

ı

ı

D r F ıG ..E C r ˝ uT /; / C f;

ı ı ı ı ı @ @ .n/ı @u ı D r . r / C C qm ; c" F G ..E C r ˝ uT /; / r ˝ @t @ @t ı

(6.58)

6.2 Closed Systems in the Material Description

395

for four scalar unknowns: ; u k X i ; t:

(6.59)

In particular, for the exceptional model AV , the U -system (6.58) with account of (6.49) and (2.78) is written in the form 2 ı@ u

ı

ı

D r P C f; 0 1 r X ı ı ı ı ı @ ı T T A c" ' I.s/ D r . r / C @ C .F r ˝ v / C qm ; @t D1

@t 2

@2

ı

' D PD

r X D1

CD

ı

@@I.s/

.s/

' I C F T ;

D

;

ı

ı

' D

@ @I.s/

(6.60)

;

.s/

.I.s/ .C/; /;

I C D @I.s/ [email protected];

ı ı ı 1 ı .r ˝ u C r ˝ uT C r ˝ u r ˝ uT /; 2

ı

F D E C r ˝ uT :

6.2.3 T U VF -system of Thermoelasticity in the Material Description .n/

Using the relations (6.46) between the tensors P and T G and complementing the system (6.40) with constitutive equations (4.442) ‘in rates’, we obtain the T U VF system of thermoelasticity in the material description: ı @v

ı

c"

@t

ı

.n/

.n/

ı

D r .4 E ıG T G / C f;

.n/ .n/ ı ı ı ı @ ı ı D r . r / C .=/.4 E ıG T G / r ˝ vT C qm ; @t

@[email protected] D v;

ı

(6.61)

@[email protected] D r ˝ vT ;

.n/

.n/ .n/ .n/ .n/ ı @ @TG D 4 P ıGh .F; / r ˝ vT C ZGh T G C T G ZGh C T G ; @t @t

for 22 scalar unknowns: .n/

T G ; ; u; v; F k X i ; t:

(6.62)

396

6 Elastic Continua at Large Deformations

Here we have introduced the notation for the fourth-order tensor 4

.n/ P ıGh

.n/

.4 P .1243/ F1T /.1243/ : Gh

(6.63)

The remaining notations are the same as in Sect. 6.1.3. The heat conduction equation in (6.61) has been written with the help of (6.53). Component forms of equation systems of elasticity theory in the material deı

ı

scription are usually used in the basis ri of a reference configuration K (see Exercise 6.2.2 and 6.2.3).

6.2.4 The Equation System of Thermoelasticity for Quasistatic Processes in the Material Description The model of quasistatic processes introduced in Sect. 6.1.5 can be considered for the material description as well. In this case the equation system (6.40)–(6.42) with account of the assumption (6.34) takes the form ı

ı

r P C f D 0; ı

c"

ı

@ ı D r . r / C qm ; @t .n/

(6.64)

ı

(6.65)

ı

P D F ıG .E C r ˝ u; /:

(6.66)

The system consists of four scalar equations for four scalar unknown functions: ; u k X i ; t;

(6.67)

and it is called the quasistatic equation system of thermoelasticity in the material description. In this case the dynamic compatibility equation and the kinematic equation in system (6.40) are satisfied only approximately, and they are not considered. These equations are satisfied exactly for static processes starting from some time t0 .

Exercises for 6.2 6.2.1. Show that constitutive equations (6.49) for the quasilinear model AV can be written in the form ı

P D .=/.4 M C/ FT ;

T D F .4 M C/ FT ;

6.2 Closed Systems in the Material Description

397

or in terms of the displacement gradient:

ı

PD

4

ı ı 1ı T M E C r ˝ u r ˝ u .E C r ˝ u/: 2

6.2.2. Using the results of Exercise 6.1.1, show that for isotropic media the relations (6.49) have the forms PD.

ı 1E ı

ı

C

2C ı

1

C

ı 3C

2

ı

/ FT ;

T D JF .

ı

D ' 1 C ' 2 I1 C ' 3 I2 ; ı

ı

' ˛ D [email protected] [email protected]˛ /;

ı

ı 1

ı 2C

ı

ı

2

C

D ' 2 C I1 ' 3 ;

I˛ D I˛ .C/;

C

ı

ı 3C

2

/ FT ;

ı

D '3;

3

˛ D 1; 2; 3:

Show that for the linear model AV of isotropic continua these relations take the forms P D l1 I1 .C/FT C 2l2 C FT ;

T D J.l1 I1 .C/F FT C 2l2 F C FT /;

or in terms of the displacement gradient: ı ı ı 1ı T P D l1 r u C r ˝ u r ˝ u .E C r ˝ u/ 2 ı

ı

ı

ı

ı

Cl2 .r ˝ u C r ˝ uT C r ˝ u r ˝ uT / .E C r ˝ u/: 6.2.3. Using the result of Exercise 6.2.1, show that the constitutive equations for ı the quasilinear model AV in basis ri have the component form ı

ı

j

ı

ı

ı

P i D .=/M jkms "ms F lk g li I and for the model AV of isotropic media: ı

P ji D .

ı

ı jk 1g

T ij D

ı

C

ı ij 1g

ı

ı j m ı ks g "ms 2g

C

ı

C

ı i m ı js 2 g g "ms

ı

ı ı ı j m ı ks ı uv g g "mu "sv /F lk g li ; 3g

C

ı

ı i m ı js ı uv 3 g g g "mu "sv : ı

6.2.4. Show that the U VF -system (6.40)–(6.42) and (6.45) in basis ri in the material description has the component form

398

6 Elastic Continua at Large Deformations

8 ı ı ı ı ıı ˆ [email protected] [email protected]/ D r j P j i C f i ; ˆ ˆ ˆ ˆ ı ı ı ı ı ı ı ı 0; f > 0; k > 0:

(6.201)

Since F is diagonal, one can readily find the stretch tensors U and V and the rotation tensor O: U D V D F; O D E: (6.202) With the help of formulae (4.25) and (4.42) we find the energetic deformation ten.n/

.n/

sors C and deformation measures G: .n/

CD

1 ..f 0 /nIII 1/er ˝ er n III

ı C..f =r/nIII 1/e' ˝ e' C .k nIII 1/ez ˝ ez ; (6.203)

.n/

GD

1 0nIII ı f er ˝ er C .f =r/nIII e' ˝ e' C k nIII ez ˝ ez : n III

6.7.3 Stresses in the Lam´e Problem for Models An Assume that constitutive equations of the cylinder correspond to the linear model An (6.135) of an isotropic elastic continuum. Then substituting the expression (6.203) into (6.135), we find the energetic stress tensors .n/

.n/

.n/

.n/

T D T r er ˝ er C T ' e' ˝ e' C T z ez ˝ ez ; ı

.n/

r Tr D .l1 I1 C 2l2 .f 0nIII 1//; .n III/f 0 f k ı

.n/

T' D .n/

r ı .l1 I1 C 2l2 ..f =r/nIII 1//; .n III/f 0 f k ı

TzD

r .l1 I1 C 2l2 .k nIII 1//; .n III/f 0 f k ı

I1 D .f 0 /nIII C .f =r/nIII C k nIII 3:

(6.204)

450

6 Elastic Continua at Large Deformations .n/

Since the tensors F and T are diagonal, from Eqs. (6.137), (6.200), and (6.204) we obtain the expression for the Cauchy stress tensor .n/

T D r er ˝ er C ' e' ˝ e' C z ez ˝ ez D FnIII T ;

(6.205)

ı

r D

' D

r .f 0 /nIII1 .l1 I1 C 2l2 .f 0nIII 1//; .n III/f k

1 ı ı .f =r/nIII1 .l1 I1 C 2l2 ..f =r/nIII 1//; .n III/f 0 k ı

z D

rk nIII1 .l1 I1 C 2l2 .k nIII 1//: .n III/f 0 f

Components of the stress tensor in this problem prove to depend on the coordinate ı r and time t, therefore, the equilibrium equations are not automatically satisfied. In this case, as a rule, it is convenient to use the equilibrium equations (6.100) in the material description. To do this, we should calculate the Piola–Kirchhoff stress tensor by formulae (3.56): PD

.n/ 1 1 1 ı ı ı F T D FnIII1 T D r er ˝ er C ' e' ˝ e' C z ez ˝ ez ; (6.206) J0 J

where ı

r D

.f 0 /nIII1 ı ..l1 C 2l2 /f 0nIII C l1 .f =r/nIII .3l1 C 2l2 / C l1 k nIII /; n III ı

' D

ı

z D

1 ı ı .f =r/nIII1 ..l1 C 2l2 /.f =r/nIII n III Cl1 f 0nIII .3l1 C 2l2 / C l1 k nIII /;

(6.207)

k nIII1 ı ..l1 C 2l2 /k nIII C l1 .f 0nIII C .1=r/nIII / .3l1 C 2l2 //: n III

6.7.4 Equation for the Function f ı

Writing components of the divergence r P with respect to the physical basis er ; e' ; ez (see [12]), we can represent the equilibrium equation projected onto the axis Oer as follows: ı

@ r ı

@r

ı

C

ı

r ' ı

r

D 0:

(6.208)

6.7 The Lam´e Problem

451 ı

ı

ı

ı

ı

Here we have taken into account that r , ' and z are independent of ' and z. The other two projections of the equilibrium equation onto the axes Oe' and Oez are satisfied identically. On substituting the expressions (6.207) into (6.208), we obtain an ordinary difı ı ferential equation of the second order for the function f .r ; t/. The function f .r; t/ is determined up to the two constants of integration C1 .t/ and C2 .t/ being functions only of time; to evaluate them one should use boundary conditions.

6.7.5 Boundary Conditions of the Weak Type Boundary conditions at inner and outer surfaces of the cylinder have the form (6.80) (gas or fluid pressures pe1 and pe2 are given). According to formula (6.206) and ı ı the fact that at the surfaces r D r ˛ (˛ D 1; 2) the normal vectors have the form ı n D er , the boundary condition (6.80) for this problem becomes ı

ı

r D r˛ W

ı

ı

r D pe˛

f .r ˛ ; t/k ı

;

˛ D 1; 2:

(6.209)

r˛ ı

Boundary conditions at the end surfaces z D 0; h3 serve for determining the function k. For example, if at the surface z D 0 the symmetry conditions are given, ı and at z D h03 – the pressure pe3 , then we have ı ˇ uz D .z z/ˇızD0 D 0;

ı

zD0W ı

ı

z D h3 W

z D pe3 :

(6.210) (6.211)

On substituting the expression (6.194) and (6.205) into (6.210) and (6.211), we verify that the first boundary condition is satisfied identically, and the second one is an additional differential equation for the function f and the constant k. The solution found above cannot satisfy this boundary condition completely, therefore we should consider other variants of boundary conditions. One of such variants weakens the boundary condition (6.211) by replacing it with the integral one ı

ı

z D h3 W

Z

r2 r1

1

z rdr D pe3 .r22 r12 /: 2

(6.212)

One can say that such a condition is of the weak type. The method of replacing exact boundary conditions by the integral ones is called the Saint-Venant method.

452

6 Elastic Continua at Large Deformations ı

According to (6.195), we have rdr D f df D ff 0 d r. Then, substituting the expression (6.205) for z into (6.212) and passing to the reference configuration, we rewrite the condition (6.212) as follows: k nIII1 2l1 ı n III r 2 rı 2 2 1

Z

ı

r2

0 nIII

.f /

ı

C

nIII ! ı ı r dr

f ı

r

r1

!

C.l1 C 2l2 /k

nIII

ı

.3l1 C 2l2 / D pe3

ı

f 2 .r 2 / f 2 .r 1 / ı

ı

r 22 r 21

! :

(6.213) Expressing the function k by Eq. (6.213) and substituting the result into (6.207) and then formulae (6.207) into (6.208) and (6.209), we obtain an integro-differential ı equation for determining the function f .r; t/. For the case when n D IV (John’s model AIV ), Eqs. (6.208) and (6.209) have a simple analytical solution: ı

ı

f .r; t/ D C1 .t/r C

C2 .t/ ı

:

(6.214)

r

Indeed, in this case Eqs. (6.207) become ı

r D 2.l1 C l2 /C1 2l2 ı

' D 2.l1 C l2 /C1 C 2l2

C2 ı

r2 C2 ı

r2

C l1 k .3l1 C 2l2 /; (6.215) C l1 k .3l1 C 2l2 /;

and relations (6.205) take the forms ı

r D

r2 ı

l1 .2C1 C k/ C 2l2 C1

r2

C2

.C1 r 2 C C2 /k ı

' D

l1 .2C1 C k/ C 2l2 C1 C

ı

.C1 r 2 C2 /k

ı

r2

C2 ı

r2 !

!

!

.3l1 C 2l2 / ; !

.3l1 C 2l2 / ; (6.216)

ı4

z D

r ı

.C12 r 4 C22 /

.l1 .2C1 C k/ C 2l2 k .3l1 C 2l2 // :

Substituting the expressions (6.215) into (6.208), one can readily verify that the equilibrium equation is identically satisfied.

6.7 The Lam´e Problem

453

Substitution of (6.215) into the boundary conditions (6.209) and (6.211) yields 2.l1 C l2 /C1

2l2

2.l1 C l2 /C1

2l2

ı

r 21 ı

r 22

C2 C l1 k .3l1 C 2l2 / D pe1 C1 C

C2

C2 C l1 k .3l1 C 2l2 / D pe2 C1 C

C2

! ;

ı

r 21 ı

r 22

!

(6.217) :

After substitution of (6.214), the boundary condition (6.213) for the model AIV takes the form ! C22 2 2C1 l1 C .l1 C 2l2 /k .3l1 C 2l2 / D pe3 C1 ı ı : (6.218) r 21 r 22 Solving the system of three algebraic equations (6.217), (6.218), we find C1 , C2 , and k.

6.7.6 Boundary Conditions of the Rigid Type In place of (6.212) we can consider another boundary condition, namely the condition of the rigid type when displacements uez along the axis Oz are given: ı

ı

z D h3 W

ı ˇ uz D .z z/ˇı

ı

zDh3

D uez :

(6.219)

Then we obtain the following simple expression for k: ı

k D 1 C .uez =h3 /;

(6.220)

which is similar to the corresponding expression of (6.142) in the problem on a beam in tension. For the model AIV , this relation takes the place of the condition (6.218) and the system (6.217), (6.218) for C1 , C2 , and k becomes linear; its solution has the form C1 D C2 ;

ı2

C2 D r 1

1 2.k 1/ ; .1 C .1 2/e p1 / .1 2/.1 e p1 /

e p1 e p 2 ˇ02 1 C ˇ02

D ; e p2 e p1

ı

ı

ˇ0 D r 1 = r 2 :

Here e p ˛ D pe˛ =2l2, and the Poisson ratio (6.146) has been introduced.

(6.221)

454

6 Elastic Continua at Large Deformations

Expressions for outer r2 and inner r1 radii of the cylinder in K can be found with the help of formulae (6.194) and (6.214): ı

ı

ı

ı

r˛ =r ˛ D f .r ˛ ; t/=r ˛ D C1 C .C2 =r 2˛ /;

˛ D 1; 2:

(6.222)

For other models An , we can also consider the boundary condition (6.219), which yields the expression (6.220) for k. Then, substituting the expressions (6.207) into (6.208) and (6.209), we obtain one nonlinear differential second-order equation for the function f with two boundary conditions.

6.8 The Lam´e Problem for an Incompressible Continuum 6.8.1 Equation for the Function f Consider the Lam´e problem on a cylindrical pipe under internal and external pressures (see Sect. 6.7), but let the pipe be made of isotropic incompressible material described by linear models Bn (see (4.532) or (6.160)). In this case the motion law for the pipe is also sought by the semi-inverse method in the form (6.194) and (6.195); therefore, all the relations (6.197)–(6.203) hold. From the condition of incompressibility of a material considered det F D 1 and from (6.201) it follows that ı the function f .r; t/ must satisfy the equation ı

f 0 f k D r: ı

(6.223) ı

Rewriting this equation in the form f df D k1 r d r ; one can easily find its solution (choosing a positive root) ı r2 f2 D C C; (6.224) k where C is the constant of integration.

6.8.2 Stresses in the Lam´e Problem for an Incompressible Continuum On substituting Eq. (6.223) into (6.203), we find 1 0 nIII ı !nIII 1 @ r f GD er ˝ er C ı e' ˝ e' C k nIII ez ˝ ez A : n III fk r

.n/

(6.225)

6.8 The Lam´e Problem for an Incompressible Continuum

455

Having substituted this expression int the constitutive equation (6.160), we obtain .n/

that the energetic stress tensors T have the diagonal form (6.204) in the case of an incompressible continuum too, but their components are different from the ones for compressible materials:

.n/

T r D p

fk ı

T ' D p

C .n III/ 1 C ˇ C .1 ˇ/

r f

T z D pk

nIII

! Ck

nIII

;

r !nIII

C .n III/ 1 C ˇ C .1 ˇ/

.n/

f ı

r ı

.n/

nIII

nIII

C .n III/ 1 C ˇ C .1 ˇ/

ı

r fk

ı

r fk

nIII

nIII

C

! C k nIII f

;

nIII ! ;

ı

r

n D I; II; IV; V: (6.226) For the Cauchy stress tensor, relations (6.205) hold as well, and their components are written as follows:

r ; ' D p C e

' ; z D p C e

z;

r D p C e

(6.227)

where ı

r e

r D .n III/ fk e

' D .n III/

f

!nIII

1 C ˇ C .1 ˇ/

f

nIII

ı

! Ck

nIII

;

r

nIII

1 C ˇ C .1 ˇ/

ı

r

e

z D .n III/k nIII 1 C ˇ C .1 ˇ/

ı

r fk

ı

r fk

nIII

nIII

C

! C k nIII f ı

;

nIII ! :

r

(6.228) With the help of the relation P D F tensor: ı

r D r

fk ı

r

;

1

T, we find components of the Piola–Kirchhoff ı

' D

ı

r

' ; f

ı

z D

z : k

(6.229)

456

6 Elastic Continua at Large Deformations

6.8.3 Equation for Hydrostatic Pressure p Substitution of the Piola–Kirchhoff tensor components (6.229) into the equilibrium equation (6.208) yields an ordinary differential equation for unknown function ı p.r; t/ being the hydrostatic pressure in an incompressible continuum:

pf k

0

ı

C

ı

p.f 2 k r 2 /

D

ı

r 2f

r

ı

e h

@ r e h D ı .f ke

r/ e

': f @r

; ı r

(6.230)

Rearranging the left-hand side of Eq. (6.230) with the help of (6.223) and (6.224), we reduce the equation to the form dp f k ı

dr

ı

r

D

e h ı

:

(6.231)

r

Integration of this equation yields

Z ı h ı 1 r e p D p0 C (6.232) d r: ı k r1 f Here p0 is the constant of integration, which together with C can be found from the boundary conditions (6.209) after substitution of expressions (6.229) and (6.232) into them: 1 p0 C k

Z

ı

r2 ı

r1

e h ı ı ı

r .r 2 /; p0 D pe1 C e

r .r 1 /: d r D pe2 C e f

(6.233)

For the constant C , from (6.233) we obtain the nonlinear algebraic equation: 1 F .C / e

r .r 2 / e

r .r 1 / k ı

ı

Z

ı

r2 ı

r1

ı e h dr p D 0; p D pe1 pe2 ; (6.234) f

into which the expressions (6.224), (6.228), and (6.230) for e

r , f and e h should be substituted.

6.8.4 Analysis of the Problem Solution Figure 6.27 shows the graph of the function F .C /. In the general case, for all n and ˇ the equation F .C / D 0 may have two roots C1 and C2 , among which one should choose the least root C1 , because just the one satisfies the normalization condition F .C1 / D 0 at k D 1 and p D 0 when there is no loading and deformations of the cylinder. For this case

6.8 The Lam´e Problem for an Incompressible Continuum

C D 0;

457 ı

f D r;

' D e

z 0 D .n III/.3 ˇ/ D const; e

r D e p D p0 D const; p0 D pe1 C 0 ; ı

ı

(6.235)

ı

r D ' D z D p0 C 0 D pe1 D const: The stresses r , ' and z in the cylinder are zero when pe1 D pe2 D 0; and they are equal to each other but are nonzero when pe1 D pe2 ¤ 0. When k D 1 and p > 0 (excess internal pressure), the least root C1 of the function F .C / is positive; and when p < 0 (excess external pressure) the root C1 is negative (Fig. 6.27). With growing the value k > 1 (longitudinal extension), the root C1 is displaced into the domain of negative values (there occurs transverse compression of the cylinder); and with decreasing the value k < 1 (longitudinal compression), on the contrary, the root C1 is displaced into the domain of positive values (there occurs lateral dilatation of the cylinder). For the function F .C /, a peculiarity of interest is the existence of the ultimate value p : while p > p , there are no roots of the function F .C / (see Fig. 6.27). This means that at such values of p the nonlinear Lam´e problem has no solution (unlike the Lam´e problem in linear elasticity theory, which has a solution at all values of p). If we consider the process of monotone increasing the pressure difference p from 0 up to p , then the cylinder radius also monotonically grows; and when p > p a solution does not exist. This effect is called the loss of stability of a material in tension. ı ı Figure 6.28 shows the dependence of the dilatation coefficient yR D .r1 r 1 /=r 1 of the cylinder upon the dimensionless pressure difference p= (when p > 0)

Fig. 6.27 The function F .C /

Fig. 6.28 Dependence of the coefficient of relative dilatation of a thin-walled cylinder on internal excess pressure for different models of incompressible materials (k D 1, r2 =r1 D 1.01)

458

6 Elastic Continua at Large Deformations ı ı

ı

ı

ı

for the cylinder with a very thin wall: h=r 1 D 0.01, where h D r 2 r 1 is the initial thickness of the wall. The function p.yR / is nonlinear and exists only if ı h.r/ on p 6 p . A fact of interest is a very weak dependence of the function e the coefficient ˇ when k D 1; due to this, the functions F .C / and p.yR / are also practically independent of the values of ˇ within the interval Œ1; 1 . Moreover, values of the function p.yR / at n D I and V, and also at n D II and IV are practically not distinguishable in pairs. Therefore, when k D 1 there are only two essentially distinct functions p.yR / at n D I and II (Fig. 6.28). Their corresponding ultimate ı ı

values p = are 0.04 and 0.0072 while h=r 1 D 0.01. Ultimate magnitudes of the dilatation coefficient yR are 218 and 41%. If we consider the cylinder with a thicker ı

prove to be smaller. wall h, then the ultimate values yR In the case of compression when p < 0, the root C1 is negative, and from Eq. (6.224) it follows that there exists a limiting value ı

C1 D r 21 =k;

(6.236)

such that the root C1 cannot take on values smaller than C1 . Hence there also exists a limiting negative value of the pressure difference p such that when p < p there is no solution of the Lam´e problem. Graphs of the function p.yR / in compression are shown in Fig. 6.29. Just as in tension, for the case when k D 1, the functions p.yR / at different values of the parameter ˇ are practically coincident, and they are distinct only for the models n D I; V and n D II; IV. ı ı

In the case when h=r 1 D 0.01, we found the limiting value p = D 3.2 for n D I; V, and for n D II; IV: p = D 0.33. It should be noted that for real thin-walled structures at essentially smaller values of pressure in compression such that jpj= jp j=, there occurs a loss of stability of the structure itself; thus, the values p = are usually not realized. Figures 6.30 and 6.31 exhibit distributions of the stresses r = and ' = versus the cylinder thickness at different values of the parameter ˇ and different n when ı ı pe2 D 0. The radial stress r .Nr /, where rN D r=r 1 , depends weakly on ˇ and

Fig. 6.29 Dependence of the coefficient of relative compression of a thin-walled cylinder on external excess pressure for different models of incompressible materials (k D 1, r2 =r1 D 1.01)

6.8 The Lam´e Problem for an Incompressible Continuum

a

459

b

Fig. 6.30 Distribution of the radial stresses versus the thickness of a thick-walled cylinder for the model BI at different values of ˇ (a) and for different models of incompressible material .ˇ D 1/ (b)

Fig. 6.31 Distribution of the tangential stresses versus the thickness of a thick-walled cylinder for different models of incompressible material and different values of the parameter ˇ

n; it monotonically decreases from the value pe1 to 0. The tangential stress ' .r/ depends considerably more on ˇ and n especially for thin-walled cylinders. So when r2 =r1 D 2, for the model BI and ˇ D 1 the stress ' is positive everywhere, and it reaches its minimum value at the interior point r=r1 1.2 of the cylinder. When ˇ 6 0.6, the stress ' on the inner surface of the cylinder becomes negative, and when ˇ D 1, the stress ' is negative in the whole cylinder. For the models BII , BIV and BV , the stress ' is always positive. For the model BII , when ˇ D 1,

460

6 Elastic Continua at Large Deformations

Fig. 6.32 Distributions of radial and tangential stresses versus the thickness of a thin-walled cylinder for different models of incompressible material .ˇ D 1/

the stress ' reaches its maximum on the outer surface of the cylinder; and for the models BIV and BV , at all ˇ the function ' .Nr / is always monotonically decreasing and reaches its maximum on the outer surface. For thin-walled cylinders, the stress ' is positive and practically constant versus the cylinder thickness; it is almost independent of ˇ and n, and its value is close to the value determined by the theory of thin linear-elastic shells at small deformations: ı

ı

' p r 1 =h (Fig. 6.32).

Chapter 7

Continua of the Differential Type

7.1 Models An and Bn of Continua of the Differential Type 7.1.1 Constitutive Equations for Models An of Continua of the Differential Type Let us consider now nonideal continua. Practically all real bodies are nonideal media, and they can be considered as ideal ones in a certain approximation. According to the general theory of constitutive equations stated in Sect. 4.4.2, a continuum is nonideal if its operator constitutive equations (4.156) include the dissipation function w being nonzero. Models of nonideal materials, which are widely used in practice, are models of continua of the differential type. Definition 7.1. A continuum is called a continuum of the differential type, if corresponding operator relations (4.156) are usual functions of active variables R.t/ and P their derivatives R.t/, i.e. P ƒ.t/ D f R.t/; R.t/ :

(7.1)

Functions (7.1) are assumed to be continuously differentiable. One can say that this is the model An of a continuum of the differential type, if some model An of a continuum has been chosen and its corresponding operator constitutive equations (4.156) are simply functions of the arguments indicated above and their rates. In particular, the Helmholtz free energy has the form .t/ D

.n/

.n/

. C .t/; C .t/; .t//:

(7.2)

(In continuum mechanics, the set of arguments of the function (7.2) does not involve the derivative P .)

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 7, c Springer Science+Business Media B.V. 2011

461

462

7 Continua of the Differential Type

The total derivative of

with respect to time has the form .n/

.n/

d2 C @ d d dC @ @ C D C : 2 .n/ .n/ dt dt @ dt dt @C @C .n/

(7.3)

.n/

The partial derivatives @ [email protected] C and @ [email protected] C are symmetric second-order tensors. The remaining constitutive equations (4.158), connecting the active variables to the reactive ones, for the model An of a continuum of the differential type have the forms .n/ .n/ .n/

.n/

T D T . C; C ; /;

.n/ .n/

D . C ; C ; /;

.n/ .n/

w D w . C ; C ; /:

(7.4)

By analogy with the tensor function connecting the energetic tensors of stresses and deformations, introduce the two tensor functions .n/

.n/

.n/ .n/

T e . C ; / D T . C; 0; /;

(7.5)

and .n/

.n/

.n/

T v D T T e;

(7.6)

called the function of equilibrium stresses and the function of viscous stresses, respectively. Substituting the expressions (7.3) and (7.6) into PTI (4.121) and collecting like terms, we get 0

1

.n/ .n/ .n/ .n/ Te C @ @ 1 [email protected] C d C .w T v C / dt D 0: @ .n/ A d C C .n/ d C C @ @C @C (7.7) .n/

.n/

.n/

Since the differentials d C , d C , d and dt are mutually independent, the identity (7.7) is equivalent to the equation system 8 .n/ .n/ .n/ ˆ ˆ ˆ T e D [email protected] [email protected] C/ F . C ; /; ˆ ˆ ˆ ˆ .n/ ˆ < @ [email protected] C D 0; ˆ ˆ ˆ D @ [email protected]; ˆ ˆ ˆ ˆ .n/ ˆ : .n/ w D T v C :

.7:8a/ .7:8b/ .7:8c/ .7:8d/

7.1 Models An and Bn of Continua of the Differential Type

463 .n/

Just the fact that terms within the first parentheses of (7.7) are independent of C .n/

.n/

ensures that the differentials d C and d C be mutually independent. In turn, the .n/

.n/

presence of T e instead of T within the first parentheses of (7.7) and the validity of .n/

Eq. (7.8b) ensure that terms within these parentheses be independent of C . Thus, continua of the differential type have the following properties: (1) They are dissipative (i.e. nonideal), because for them the dissipation function w is not identically zero. .n/

.n/

(2) The tensor function of equilibrium stresses T e (but not T ) is quasipotential (see Sect. 4.5.2). (3) The quasipotential

.n/

depends only on C and : D

.n/

. C ; /;

(7.9)

.n/ .n/

.n/

the remaining functions , T v , w depend on C , C and . According to the property (2), for models of the differential type it is not sufficient to specify only one function (7.9); in addition we need the viscous stresses function (7.6) to be given: .n/ .n/

.n/

T v D F v . C ; C ; /:

(7.10)

According to formulae (7.4)–(7.6), this function depends also on the deformation .n/

tensor rates C . Relationships (7.8)–(7.10) are constitutive equations for models An of continua of the differential type.

7.1.2 Corollary of the Onsager Principle for Models An of Continua of the Differential Type The tensor function of viscous stresses (7.10) is not quite arbitrary: it satisfies the conditions (7.5) and (7.6), i.e. .n/

F v . C ; 0; / D 0:

(7.11)

In addition, the function must satisfy the Onsager principle (see Sect. 4.12.1, Axiom 16). This principle is applied to continua of the differential type as follows.

464

7 Continua of the Differential Type

First, we should form the scalar function (4.728) being the specific internal entropy production: q D w

.n/ .n/ .n/ q q r D F v . C ; C ; / C r > 0:

(7.12)

Here we have used expression (7.8d) for the dissipation function w of continua of the differential type. Then we represent the expression (7.12) in the form (4.729) with thermodynamic forces .n/

X2 D C

X1 D r ;

(7.13)

(where X1 is a vector, and X2 is a second-order tensor) and thermodynamic fluxes .n/ .n/

.n/

1 Q1 D q;

Q2 D T v D F v . C ; C ; /:

(7.14)

According to the Onsager principle, thermodynamic fluxes Qˇ must be linear (tensor-linear) functions of Xˇ in the form (4.730), i.e. 8 ˆ 0; .n/

.n/

.n/

w D F v C D C 4 Lv C > 0;

(7.19) (7.20)

7.1 Models An and Bn of Continua of the Differential Type

465

which are the Fourier inequality and the dissipation inequality for continua of the differential type, respectively. From these inequalities it follows that the tensors and 4 Lv are nonnegative-definite. In addition, from (7.20) it follows that the dissipation function w for continua of the differential type is a quadratic scalar function .n/

of C , and the viscosity tensor 4 Lv has the following symmetry in components: 4

b Lijkl D b Lijlk ;

Lv D b Lijklb cj ˝b ck ˝b cl ; ci ˝b

b Lijkl D b Lj i kl ;

b Lijkl D b Lklij ;

(7.21)

whereb ci is an orthonormal basis. The total number of components of the tensor 4 Lv is 81, and the number of independent components, according to Eqs. (7.21), does not exceed 21 (see [12]). Notice that the viscosity tensor 4 Lv and its components, in general, depend on .n/

.n/

the tensor C and also on C (the Onsager principle (7.18) does not prohibit this dependence, it only requires that F v be a quasilinear tensor function (see Sect. 4.8.7 and [12])). .n/

.n/

If the tensor 4 Lv is independent of C and depends only on C , then the function of viscous stresses F v is pseudopotential, i.e. satisfies the relation .n/

.n/ .n/

T v D F v . C; C ; / D

.n/ 1 [email protected] [email protected] C /: 2

(7.22)

Such function F v is also called potential with respect to the second tensor argument. Equation (7.22) can readily be verified by differentiating the scalar function w .n/

(7.20) with respect to the argument C .

7.1.3 The Principle of Material Symmetry for Models An of Continua of the Differential Type Notice that the principle of material symmetry (Axiom 14), and also Definitions 4.5 and 4.6 of fluids and solids, respectively, have been formulated for arbitrary operator constitutive equations (4.156); i.e. they can be applied not only to ideal materials but also to continua of the differential type and other types of nonideal materials. According to this principle, models An of continua of the differential type with bs constitutive equations (7.8)–(7.10) in some undistorted reference configuration K ı

ı

(as this configuration we will choose K) have a symmetry group G s such that ı

ı

for each transformation tensor H 2 G s (H W K ! K/ the constitutive equations

(7.8)–(7.10) in the reference configuration K have the form

466

7 Continua of the Differential Type

8 .n/ .n/ .n/ ˆ ˆ T D F . C ; / [email protected] [email protected] C /; ˆ e ˆ ˆ ˆ ˆ ˆ .n/ ˆ ˆ ˆ D . C ; /; ˆ ˆ < D @ [email protected]; ˆ ˆ ˆ ˆ .n/ .n/ ˆ ˆ ˆ / D T C ; .w ˆ v ˆ ˆ ˆ ˆ .n/ .n/ ˆ.n/ : T v D F v . C ; C ; /:

.7:23a/ .7:23b/ .7:23c/ .7:23d/ .7:23e/

Let us consider first the models An of solids of the differential type, whose symı

metry group G s is a subgroup of the full orthogonal group I . For these materials, the further construction is the same as in Sect. 4.8 for ideal solids: since the tensors ı

.n/

.n/

.n/

.n/

T and C are H -indifferent relative to the group G s , tensors T e (7.5), T v (7.6) .n/

and C are also H -indifferent, according to Theorem 4.21. Then Eqs. (7.23) are equivalent to the relations

.n/

.n/

QT F . C; / Q D F .QT C Q; /;

.n/

. C ; / D

.n/

.QT C Q; /;

.n/ .n/

(7.24a)

.n/

(7.24b)

.n/

QT F v . C ; C ; / Q D F v .QT C Q; QT C Q; /;

(7.24c)

ı

for any Q 2 G s . Since .n/

.n/

.n/

.n/

.n/

.n/

.w / D T v C D QT T v Q QT C Q D T v C D w ;

(7.25)

Eq. (7.23d) is always satisfied. Theorem 7.1. For models An of solids of the differential type, the principle of material symmetry (7.24) holds if and only if the two conditions (7.24b) and (7.24c) for the function and the viscous stresses function F v are satisfied. H The conditions (7.24b) and (7.24c) are necessary, because if the principle of material symmetry holds, then all conditions (7.24) are satisfied. Prove that the conditions (7.24b) and (7.24c) are sufficient. If the condition (7.24b) is satisfied, then (7.24a) follows from (7.24b); a proof of this fact is similar to the proof of formula (4.280). N

7.1 Models An and Bn of Continua of the Differential Type

467

7.1.4 Representation of Constitutive Equations for Models An of Solids of the Differential Type in Tensor Bases In comparison with ideal continua, for solids of the differential type not only the condition (7.24b), being the condition of indifference of the scalar function ı

.n/

. C ; / relative to some group G s (see Definition 4.9), must be satisfied but also .n/ .n/

the condition (7.24c) for the tensor function F v . C ; C ; / of two tensor arguments. .n/

For the scalar function . C ; /, the same representations as were derived in Sect. 4.8 remain valid; in particular, these are representations (4.304) as functions ı

.n/

of invariants I.s/ . C / relative to group G s : D

.n/

.n/

.I1.s/ . C/; : : : ; Ir.s/ . C /; /;

(7.26) .n/

and also representations (4.311) and (4.324) for the tensor function F . C ; / (7.8a). For example, for an isotropic solid of the differential type, from (4.322) we obtain the expression for equilibrium stresses .n/

Te D

1E

C

.n/

2C

.n/

C

3C

2

;

(7.27)

where are determined by formulae (4.322a). By analogy with Definition 4.13 (see Sect. 4.8.6) for a function of one tensor ar.n/ .n/

gument, the function F v . C ; C ; / of two tensor arguments, which satisfies the ı

condition (7.24c), is called indifferent relative to the group G s (see [12]). The following theorem gives representations of this function in tensor bases. Theorem 7.2. Any tensor function of viscous stresses (7.10), which is ı

indifferent relative to some orthogonal group G s I , i.e. satisfies the condition

(7.24c), .n/

quasilinear in the second argument C , i.e. satisfies the conditions (7.18) and

(7.20), ı

can be represented in the tensor basis of the corresponding group G s : .n/

.n/ .n/

.n/

T v D F v . C ; C ; / D .1 E ˝ E C 22 / C ;

(7.28)

or .n/

.n/

.n/

T v D 1 I1 . C /E C 2 C ;

(7.29)

468

7 Continua of the Differential Type ı

— for an isotropic continuum of the differential type .G s D I /, .n/

T v D .1 E ˝ E C 2b c3 C 3 .E ˝b c23 Cb c23 ˝ E/ c23 ˝b .n/

C 4 .O1 ˝ O1 C O2 ˝ O2 / C 25 / C

(7.30)

ı

— for a transversely isotropic continuum of the differential type .G s D T3 /, .n/

Tv D

3 X

.˛b c˛ C 3C˛ .b c2ˇ ˝b c2 Cb c2 ˝b c2ˇ / c2˛ ˝b

˛D1 .n/

C6C˛ O˛ ˝ O˛ / C ; ˛ ¤ ˇ ¤ ¤ ˛; ˛; ˇ; D 1; 2; 3; (7.31) ı

— for an orthotropic continuum of the differential type .G s D O/. The tensorsb c2˛ and O˛ are determined by formulae (4.292) and (4.315). H Substituting relation (7.8) of quasilinearity of the viscous stresses function F v into (7.24c), we obtain that the following condition must be satisfied:

.n/

.n/

.n/

QT .4 Lv C / Q D 4 Lv .QT C Q/ 8 C :

(7.32)

Hence the viscosity tensor 4 Lv must satisfy the equation (see Exercise 7.1.1) 4

Lv D 4 Lv .Q ˝ Q ˝ Q ˝ Q/.57312468/

ı

8Q 2 G s :

(7.32a)

This condition is an analog of the condition (4.289) for fourth-order tensors. Therefore, a fourth-order tensor 4 Lv satisfying Eq. (7.32a) is called indifferent relative to ı

the group G s . All indifferent fourth-order tensors are formed with the help of the operations of tensor and scalar products by producing tensors of a corresponding group ı

G s (see Sect. 4.8.3) and can be resolved for the tensor basis 4 Os. / by analogy with P the resolution (4.291) of second-order tensors: 4 Lv D k D1 a 4 Os. / . The number k of elements of this basis coincides with the number of independent components of ı

the tensor 4 Lv ; it is equal to 2, 5, and 9 for the groups G s D I; T3 ; O, respectively (see [12]). The elastic moduli tensors 4 M defined by formulae (4.337), (4.341), and ı

(4.344) are also indifferent relative to the groups G s D I; T3 ; O, respectively; and the representations (4.337), (4.341), and (4.344) are the desired resolutions for the tensor bases, being the same for each group [12]. Hence, the viscosity tensor can alı

ways be represented in the similar form for the corresponding group G s D I; T3 ; O too. Thus, the representations (7.28)–(7.31) actually hold. N

7.1 Models An and Bn of Continua of the Differential Type

469

Remark. The coefficients in representations (7.28)–(7.31) are, in general, scalar .n/

.n/

functions of corresponding invariants of the tensors C and C : .n/ .n/

D . C ; C ; /:

(7.33)

But since we have assumed everywhere that remain unchanged under H transformations in a corresponding group, so the coefficients must satisfy the relations

.n/ .n/

.n/

.n/

ı

. C; C ; / D .QT C Q; QT C Q; / 8Q 2 G s :

(7.34)

Such scalar functions are called simultaneous invariants of two tensor arguments ı

relative to a group G s considered. According to the theorem proved in [12], for ı

each group G s there is a functional basis of independent simultaneous invariants .s/

.n/ .n/

J . C; C / ( D 1; : : : ; z), where z D 9 for the full orthogonal group I , z D 11 for the transverse isotropy group T3 , z D 12 for the orthotropy group O. .n/

.n/

As a functional basis of simultaneous invariants J.s/ . C ; C / of two tensors we can choose the following sets [12]: ı

for the isotropy group G s D I : .n/

.n/

.I / J3C˛ D I˛ . C /; ˛ D 1; 2; 3;

J˛.I / D I˛ . C /; .n/

.n/

J7.I / D C C ;

.n/

.n/

J8.I / D C 2 C ;

.n/

.n/

J9.I / D C . C /2 I

(7.35)

ı

for the group G s D T3 : .n/

J˛.3/ D I˛.3/ . C /; ˛ D 1; : : : ; 5; .n/

.n/

.n/

J5Cˇ D Iˇ . C /; ˇ D 1; : : : ; 4; .3/

.3/

.n/

.n/

c23 / C / .b c23 C /; J11 D C C 2J10 J2 J8 I J10 D ..E b (7.36) .3/

.3/

.3/

.3/

.3/

ı

for the group G s D O: .n/

J˛.O/ D I˛.O/ . C /; ˛ D 1; : : : ; 6I .n/

.n/

.O/ D .b c22 C/ .b c23 C /; J10

.n/

.O/ J6Cˇ D Iˇ.O/ . C /; ˇ D 1; 2; 3; 6I .n/

.n/

.O/ J11 D .b c21 C / .b c23 C /:

(7.37)

470

7 Continua of the Differential Type .s/

Invariants I˛ of corresponding groups are determined by formulae (4.295)–(4.297). Then the viscous coefficients can always be represented as functions of simultaneous invariants: .n/ .n/

.n/ .n/

D .J1 . C ; C /; : : : ; Jz.s/ . C ; C /; /: .s/

(7.38)

The dissipation function (7.20) for an isotropic medium of the differential type, according to (7.29), has the form .n/

.n/

w D 1 I12 . C / C 2 I1 . C 2 /:

(7.39)

7.1.5 Models Bn of Solids of the Differential Type For solids of the differential type, models Bn can be obtained formally from corre.n/

.n/

sponding models An by replacing the tensors C with the measures G. In particular, constitutive equations (7.8)–(7.10) become 8 .n/ .n/ .n/ ˆ ˆ T e D F . G; / D [email protected] [email protected] G/; ˆ ˆ ˆ ˆ ˆ .n/ ˆ < D . G; /; .n/ ˆ ˆw D .n/ ˆ T v G ; ˆ ˆ ˆ ˆ .n/ .n/ ˆ :.n/ T v D F v . G; G ; /:

(7.40)

For models Bn of isotropic continua of the differential type, the tensors of equilibrium and viscous stresses have the forms .n/

Te D

.n/

C

.n/

2

(7.41)

T v D 1 I1 . G /E C 22 G ;

(7.42)

.n/

.n/

where the coefficients of invariants:

C

;

1E

2G

3G

.n/

are expressed by formulae (4.322), and are functions .n/ .n/

.n/ .n/

D .J1 . G; G ; /; : : : ; J10 . G; G ; //:

(7.43)

In a similar way, from (7.28) and (7.29) we get equations for models Bn of transversely isotropic and orthotropic materials of the differential type.

7.1 Models An and Bn of Continua of the Differential Type

471

7.1.6 Models Bn of Incompressible Continua of the Differential Type Similarly to models Bn for elastic incompressible continua (see Sect. 4.9), we can introduce models Bn for incompressible continua of the differential type. As shown in Sect. 4.9, for these models the potential in (7.40) depends only on r 1 linear .n/

and quadratic invariants I˛.s/ . G/. In particular, for an isotropic incompressible continuum of the differential type,

.n/

.n/

.n/

depends only on I1 . G/ and I2 . G/, and T e is a

.n/

quasilinear function of G: .n/

T D

D

.n/

p .n/1 G C n III

.n/

.I1 . G/; I2 . G/; /;

1

1E

.n/

C

.n/

.n/

2G

C 1 I1 . G /E C 22 G ;

2

D '2 ; '˛ D [email protected] [email protected]˛ /:

ı

D ' 1 C ' 2 I1 ;

(7.44) For the simplest model Bn of isotropic incompressible continua of the differential type, we assume that the viscous coefficients are connected by the relation 2 1 D 2 : 3

(7.45)

Then the constitutive equation (7.44) takes the form .n/

T D

p .n/1 G C n III

1E

C

.n/

2G

.n/

C 22 dev G ;

(7.46)

where .n/ .n/ 1 .n/ (7.47) dev G D G I1 . G/E 3 is the deviator of the tensor (the more detailed information on deviators can be found in Sect. 8.2.13 and [12]). For the simplest models Bn , the first principal invariants of the stress tensor .n/

.n/

T and the deformation measures G are connected by the elastic relations (see Exercise 7.1.4) .n/

I1 . T / D

.n/ p I1 . G 1 / C 3 n III

being independent of the deformation rates.

1

C

.n/

2 I1 . G/;

(7.48)

472

7 Continua of the Differential Type

When solving problems in practice, one usually applies models Bn of incompressible continua of the differential type with steady creep, where the potential .n/

is independent of the invariants I˛ . G/: .n/

T D

.n/ p .n/1 G C 22 dev G ; n III

D

./;

(7.49)

This model describes the phenomenon of creep being a change in deformation of a body with time at constant stresses (see Sect. 7.4). Remark. The consistency conditions (4.328), which must be satisfied by constitutive equations at a natural unstressed state, for materials of the differential type should be complemented by the requirement that the rates of the deformation ten

sors and measures vanish in K: ı

KW

.n/

.n/

T D S D T D 0; .n/

.n/

A D C D 0;

.n/

.n/

A D C D 0;

.n/

.n/

.n/

g DGD

.n/

g D G D 0; D 0 ;

1 E; n III D

0:

(7.50)

All the constitutive equations (7.29), (7.30), (7.31), (7.41), (7.42), (7.44), (7.46), and (7.49) derived above satisfy these conditions. t u

7.1.7 The Principle of Material Indifference for Models An and Bn of Continua of the Differential Type .n/ .n/

.n/

Since all the energetic tensors T , C and G are R-invariant, the constitutive equations (7.8)–(7.10) and also (7.28)–(7.31) for solids of the differential type are the same in actual configurations K and K0 obtained one from another by a rigid motion. Therefore, the principle of material indifference for models An and Bn of solids of the differential type (as well as for elastic continua) is satisfied identically.

Exercises for 7.1 7.1.1. Show equivalence of relations (7.32) and (7.32a).

7.2 Models An and Bn of Fluids of the Differential Type

473

7.1.2. Show that the component representations of functional bases of simultaneous invariants (7.35)–(7.37) in the basis b c˛ have the forms for the transverse isotropy ı

group G s D T3 : b b .b n/ .b n/ .n/ .b n/ .b n/ .n/ .b n/ .b n/ J˛.3/ D f C 11 C C 22 ; C 33 ; C 213 C C 223 ; C 211 C C 222 C 2 C 212 ; det C; .b n/

.b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ C 11 C C 22 ; C 33 ; . C 13 /2 C . C 23 /2 ; . C 11 /2 C . C 22 /2 C 2. C 12 /2 ;

b b b .n/ .b n/ .n/ .b n/ .b n/ .b n/ .n/ .b n/ .b n/ C 13 C 13 C C 23 C 23 ; C 11 C 11 C C 22 C 22 C 2 C 12 C 12 gI

.b n/

ı

for the orthotropy group G s D O: b .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .b n/ .n/ .b n/ .b n/ J˛.O/ D f C 11 ; C 22 ; C 33 ; C 223 ; C 213 ; C 12 C 13 C 23 ; C 11 ; C 22 ; C 33 ; b b .b .n/ .b n/ .b n/ .b n/ .n/ n/ C 23 C 23 ; C 13 C 13 ; C 12 C 13 C 23 g:

.b n/

7.1.3. Show that for the simplest model Bn of isotropic incompressible continua of the differential type (7.46), the relation (7.48) between the first principal invariants holds.

7.2 Models An and Bn of Fluids of the Differential Type 7.2.1 Tensor of Equilibrium Stresses for Fluids of the Differential Type Let us use the principle of material symmetry (7.23) to derive constitutive equations for fluids of the differential type. ı

For fluids, the symmetry group G s , relative to which Eqs. (7.23) hold, is the ı

.n/ .n/

.n/

unimodular group G s D U . Tensors T , C and C are no longer H -indifferent V

I

I

relative to this group, except T and the tensor G D C .1=2/E (see Table 4.9); ı

therefore, relations (7.24) for G s D U do not hold and one should use their general form (7.23).

474

7 Continua of the Differential Type .n/

Due to Theorem 4.31, from these relations it follows that the function F . C; / has the form (4.443a) .n/

.n/

T e D F . C; / D

p .n/1 G ; n III

(7.51)

.n/

p D p.I3 . G/; /;

(7.52)

where p is the pressure being a scalar function only of the third invariant of the ten.n/

.n/

sor G. Thus, for fluids, the tensor C appears in models An only in the combination .n/

.n/

1 G D C C nIII E; i.e. in fact, models An and Bn are coincident. For the tensor of viscous stresses, the situation is analogous.

7.2.2 The Tensor of Viscous Stresses in the Model AI of a Fluid of the Differential Type I

I

Let us consider further only the models AI and AV . The tensors T and G are trans

formed during the passage to configuration K according to formulae (4.201) and (4.203a). On substituting these formulae into (7.23e), we obtain that for the group ı

G s D U the following equation must hold: I

I

I

I

H1T F v .G; G ; / H1 D F v .H G HT ; H G HT ; / 8H 2 U: (7.53) I

I

I

Here we have gone from the argument C to G, because just the measure G satisfies

I

the relation (7.53), and the tensor C is transformed during the passage to K in another way, namely according to (4.201a), and its expression involves the additional term: .1=2/.H HT E/. I

I

The tensor function F v .G; G ; / satisfying the condition (7.53) is called AI -unimodular or AI -indifferent relative to the group U . Notice that the group U contains a subgroup being the full orthogonal group I ; therefore, if H D OT and Q 2 I , then formula (7.53) becomes (7.24c). In other words, if a tensor function is AI -unimodular, then the function is isotropic (i.e. indifferent relative to the group I ). Then, since the Onsager principle holds for all I

I

groups and the function F v .G; G ; / is quasilinear: I

I

I

I

Tv D F v .G; G ; / D 4 Lv G ;

(7.54)

7.2 Models An and Bn of Fluids of the Differential Type

475

so for this function one of the representations of an isotropic function in the tensor basis (7.29) with two independent constants must hold. But such a representation may be not AI -unimodular; in particular, the representation (7.29) does not satisfy the condition (7.53). The following isotropic tensor function proves to be appropriate: I

Tv D F v D

I I I I 1 .1 I1 G1 C 22 G1 G G1 /; 4

(7.55)

where I

I

I

I

I1 .G1 G / D G1 G :

(7.56)

We can verify that the function (7.55) is AI -unimodular, i.e. satisfies the Eq. (7.53): I

I

I

I

I

I

I

I

4F.G ; G ; / D 1 .G1 G /G1 C 22 G1 G G1 I

I

I

D 1 .H1T G1 H1 H G HT /H1T G1 H1 I

I

I

C 22 H1T G1 H1 H G HT H1T G1 H1 I

I

I

D H1T .1 J1 G1 C 22 G1 G G1 / H1 I

I

D 4H1T F .G; G ; / H1 :

(7.57)

As follows from (7.57), the function (7.55) satisfies also the condition (7.24c) for Gs D I , i.e. it is isotropic. Thus, the isotropic tensor function (7.55) constructed is AI -unimodular. Thus, we obtain the following theorem. Theorem 7.3. Any quasilinear AI -unimodular tensor function of viscous stresses (7.54) can be represented in the form (7.55) or, with the help of the Cauchy stress tensor, in the form (7.58) Tv D 1 I1 .D/E C 22 D; where D is the tensor of deformation rates (2.225). H The first assertion of the theorem (formula (7.55)) has been proved above. Let us prove that the representations (7.55) and (7.58) are equivalent. I

Go from the energetic tensor of viscous stresses Tv to the tensor of viscous stresses T: I

Tv D F1T Tv F1 :

(7.59)

Substitution of (7.55) into (7.59) yields I I I I 1 1T F .1 G1 J1 C 22 G1 G G1 / F1 4 1 P 1 FT /: D .21 J1 E 42 F G 4

Tv D

(7.60)

476

7 Continua of the Differential Type I

I

Here we have taken into account that G D 12 G1 D 12 F1 F1T and G1 D I

2G D 2FT F: According to formula (2.284), we can express the tensor G in terms of the deformation rate tensor D: I

P 1 D 2F1 D F1T : 2G D G Taking into account that I

I

I

I

P 1 I1 .G1 G / D G1 G D G G D 2FT F F1 D F1T D 2D E D 2I1 .D/; (7.61) from (7.60) we really get formula (7.58). N

7.2.3 Simultaneous Invariants for Fluids of the Differential Type I

I

Notice that the coefficients 1 and 2 in formula (7.55) may depend on G and G : I

I

D .G; G ; /:

(7.62)

However, they cannot vary under unimodular transformations: I

I

I

I

.G; G ; / D .H G HT ; H G HT ; /;

8H 2 U;

(7.63)

i.e. must be scalar AI -unimodular functions of two tensor arguments. Such functions are called simultaneous AI -invariants relative to the group U . Theorem 7.4. A functional basis of independent simultaneous AI -invariants .U /

I

I

I

I

J .G; G / of the tensors G and G relative to the unimodular group U consists of not more than five elements, which can be chosen as follows: I

I

I

I

J.U / D I .G1 G /; D 1; 2; 3I J4.U / D I3 .G/; J5.U / D I3 .G /: (7.64) I

I

.U / H Each simultaneous invariant J .G; G / is an AI -unimodular function of two tensor arguments, i.e. it satisfies Eq. (7.63). Then the invariant is an isotropic tensor function of two arguments if H D QT and Q 2 I U ; i.e. it is a simultaneous invariant relative to the group I .

7.2 Models An and Bn of Fluids of the Differential Type

477

In the group I , a functional basis of simultaneous invariants can be formed by I

I

I

I

contractions of powers of the tensors: G1 G , .G1 /2 G etc. However, among all the contractions, only contractions of the tensor I

I

G D G1 G

(7.65)

are invariants relative to the group U (see Exercise 7.2.1). The number r of independent invariants of this tensor cannot exceed three (invariants of the tensor relative to the group U are its invariants relative to the group I too, and for the group I : r D 3). As these invariants we can choose I .G /, D 1; 2; 3. Moreover, each I

I

of the tensors G and G has one unimodular invariant (see Theorem 4.31), these I

I

are det G and det G , respectively. There are no other independent simultaneous I

I

invariants of the tensors G and G relative to this group. N Remark 1. Notice that in the theorem the number r does not exceed 5; but the invariI

I

ant I3 .G / is not independent: it may be expressed in terms of I3 .G/ and I3 .G /. Let us show this fact. .U / Consider the representation (7.58), then simultaneous AI -invariants J (7.64) can be written as functions of the principal invariants of the tensor D and of the density : I

I

J1.U / .G; G / D I1 .G / D 2I1 .D/; .U /

J2

I

1 2 .I .G / I1 .2G // D 4I2 .D/; 2 1

.U /

P 1 D 8I3 .D/; .G; G / D det G D det G det G 1 ı ı .U / J4.U / D .=/2 ; J5 D .=/2 I3 .D/: 8

I

.G; G / D I2 .G / D J3

.U /

Hence J5

I

I

.U /

is not independent, because it is expressed in terms of J3 .U /

J5

.U /

D J3

.U /

J3

:

(7.66) .U /

and J4

:

(7.67)

Due to Theorem 7.4, the viscous coefficients (7.62) can be represented as functions of the simultaneous invariants (7.64): D .J1.U / ; : : : ; J4.U / ; /;

(7.68)

or in the form D .I .D/; ; /;

D 1; 2; 3:

(7.69)

478

7 Continua of the Differential Type

7.2.4 Tensor of Viscous Stresses in Model AV of Fluids of the Differential Type Let us consider now the model AV .

V

V

The tensors T and G are transformed during the passage to K by formulae (4.200a) and (4.203). On substituting these formulae into (7.23e), we obtain that the function of viscous stresses must satisfy the relation V

V

V

V

H F v .G; G ; / HT D F v .H1T G H1 ; H1T G H1 ; / V

(7.70)

V

8H 2 U . Such a tensor function F v .G; G ; / is called AV -unimodular or AV -indifferent relative to the group U . According to the Onsager principle, this function must be quasilinear: V

V

V

V

Tv D F v .G; G ; / D 4 Lv G :

(7.71)

The condition for AV -unimodularity (7.70) goes into the condition for the isotropy of a tensor function when H D QT and Q 2 I U ; therefore, for the function (7.71) one of the representations of an isotropic function in the tensor basis (7.29) with two independent constants must hold. The following isotropic tensor function is appropriate: V

Tv D F v D V

V V V V 1 .1 I1 G1 C 2 G1 G G1 /; 4

(7.72)

V

where I1 D I1 .G1 G /: We can immediately verify that this function is AV -unimodular, i.e. satisfies the relation (7.70) (Exercise 7.2.3). Thus, the following theorem holds. Theorem 7.5. Any quasilinear AV -unimodular tensor function of viscous stresses (7.71) can be represented in the form (7.72) or, with the help of the Cauchy stress tensor, in the form (7.58). H The first part of the theorem has been proved above, therefore we will show only equivalence of representations (7.72) and (7.58). V

Going from Tv to Tv by the formula V

Tv D F Tv FT ;

(7.73)

7.2 Models An and Bn of Fluids of the Differential Type

479

from (7.72) we obtain V V V V 1 F .1 G1 I1 C 22 G1 G G1 / FT 4 1 P F1 /: D .21 I1 E C 42 F1T G 4

Tv D

V

(7.74)

V

Here we have taken into account that G D 12 G D 12 FT F and G1 D 2G1 D P in terms of D: 2F1 F1T : With the help of formula (2.284) we can express G V

P D 2G D 2FT D F: G

(7.75)

Using the relations V

V

P D 2F1 F1T FT D F D 2I1 .D/; I1 .G1 G / D G1 G

(7.76)

from (7.74) and (7.75) we actually obtain representation (7.58). N

7.2.5 Viscous Coefficients in Model AV of a Fluid of the Differential Type V

V

Coefficients in (7.72) are functions of G and G : V

V

D .G; G ; /

(7.77)

and remain unchanged under unimodular transformations: V

V

V

V

.G; G ; / D .H1T G H1 ; H1T G H1 ; /;

8H 2 U I (7.78)

i.e. they are scalar AV -unimodular functions. In other words, they are simultaneous V

V

AV -invariants of the tensors G and G relative to the group U . For these invariants, Theorem 7.4 and Remark 1 still remain valid. Therefore, (7.77) can be represented as functions D .J1.U / ; : : : ; J4.U / ; /

(7.79)

of simultaneous AV -invariants formed by formulae (7.64): V

V

J.U / D I .G1 G /;

D 1; 2; 3;

V

J4.U / D det G:

(7.80)

480

7 Continua of the Differential Type

These invariants can be expressed in terms of the principal invariants of the tensor D and : V

V

J1.U / .G1 G / D 2I1 .D/; J2.U / D 4I2 .D/; .U /

J3

.U /

D 8I3 .D/; J4

D

1 ı .=/2 ; 8

(7.81)

hence the viscous coefficients (7.79) can be written in the form (7.69) as well.

7.2.6 General Representation of Constitutive Equations for Fluids of the Differential Type Since for both the models AI and AV constitutive equations can be written in the single generalized form (7.58) and (7.68), it is appropriate to give a common name for the models of fluids of the differential type. Definition 7.2. The models AI and AV (and also BI and BV coincident with them) of fluids of the differential type, whose constitutive equations (7.51), (7.55), and (7.72) can be represented in the single form (7.58), (7.68), i.e. 8 ˆ T D Te C Tv ; ˆ ˆ ˆ ˆ 0:

(7.147)

The elongation function ı1 .t/ D k1 .t/ 1 (measured in experiments) for a beam of material of the differential type under such loading is usually called the creep

494

7 Continua of the Differential Type

Fig. 7.1 The stepwise process of loading (a) and the typical curve of creep (b)

a

b

Fig. 7.2 The process of loading–unloading (a) and creep curve in this process (b)

a

b

curve. A typical creep curve for materials, whose properties depend on the deformation rate, is shown in Fig. 7.1. There are three typical sections along this curve: (1) the initial section, (2) the section of steady creep, and (3) the section of unsteady creep. The value of elongation ı1 .0C / at t ! 0C is called the instantly elastic elongation. The creep effect is observed for metals and alloys under high temperatures. If at the time t1 unloading occurs (Fig. 7.2): 11 .t/ D 0 .h.t/ h.t t1 //;

(7.148)

then the elongation ı1 .t/ decreases, and for materials of the differential type the elongation does not return to its initial zero value as t ! 1 (Fig. 7.2), i.e. there appears a residual deformation of creep ı1 .1/. If the process of loading 11 .t/ is given, then Eq. (7.144) is a nonlinear ordinary differential equation for the functions k1 . Solving the equation, we obtain k1 .t/ D H

1

Z

t 0

! 11 . / d ; 2

2 H.k1 / 3

Z

k1 1

.n/ 2

L .k/ d k: k

(7.149)

On substituting (7.146) into (7.149), we find that under a constant load the elongation of the beam grows with time: ı1 .t/ D H 1 . 0 t =2 / 1;

(7.150)

and under unloading the elongation remains constant and does not decrease: ı1 .t/ D ı1 .t1 / at t > t1 .

7.4 The Problem on a Beam in Tension

495

Fig. 7.3 Creep curves for nickel alloy (solid curves are computations, dashed lines are experimental data)

Figure 7.3 shows computed and experimental creep curves ı1 .t/ for nickel alloy at temperature 1100ıC in compression for different values of the stress 0 6 0. Theoretical curves have been computed according to formula (7.149) for different models Bn , and the creep curves ı1p .t/ are the difference in elongation between the experimental values ı1 .t/ and the initial value: ı1p .t/ D ı1 .t/ ı1 .0C /. One of the experimental creep curves ı1p .t/ at the smallest value 0 was used for determining the constant 2 , that was calculated by minimizing the mean-square distance between computed and experimental results at N points being times ti : !1=2 N ı1 .ti / 2 1 X ! min: (7.151) 1

D N ı1p .ti / i D1

Table 7.1 shows values of the constant 2 for nickel alloy, computed by the method mentioned above for different models Bn . The model BII exhibits the best approximation to the experimental data for the case considered (Fig. 7.3). It should be noted that for many metals, values of the high-temperature creep elongation ı1p .t/ at sufficiently great t considerably exceed instantly elastic values ı1 .0C /, therefore the last ones are often neglected in computations of creep problems. The considered models Bn of continua of the differential type with steady creep fall into this class of models.

496

7 Continua of the Differential Type Table 7.1 Values of the constant 2 and the relative error ı of approximation to the creep curves for models Bn of continua of the differential type for nickel alloy at temperature 1100ı C n I II IV V 2 , GPas 30 30 32 35 ı, % 18 17 19 20

Fig. 7.4 Diagrams of deforming for nickel in compression

7.4.5 Analysis of Deforming Diagrams for Different Models Bn of Continua of the Differential Type Consider one more regime of deforming, when the deforming process of a beam is given by the law ( bt 2 =2t1 ; t < t1 ; k1 .t/ D 1 C (7.152) b.t t1 =2/; t > t1 : where b D const is the rate of deforming, and t1 is the beginning of deforming with a constant rate (the initial interval 0 6 t 6 t1 is necessary for the consistency conditions (7.50) to be satisfied). The rate kP1 is determined by the expression kP1 D and

p

.2k1 b=t1 /

kP1 D b

when t < t1

when t > t1 :

Substituting (7.152) into (7.144), we obtain the relation 11 .k1 / called the diagram of deforming for the beam considered. Figure 7.4 exhibits the experimental diagram of deforming for nickel in compression at 1100 ı C and b D 0:00025 s1 and computed diagrams, for which the viscosity coefficient 2 has been evaluated with the help of the creep curves (see Sect. 7.4.4 and Table 7.1). The model BII gives the best approximation to the experimental data.

Chapter 8

Viscoelastic Continua at Large Deformations

8.1 Viscoelastic Continua of the Integral Type 8.1.1 Definition of Viscoelastic Continua Besides models of the differential type considered in Chap. 7, in continuum mechanics there are other types of nonideal media. One widely uses models of viscoelastic materials, which are also called continua of the integral type, or hereditarily elastic continua. Models of viscoelastic continua most adequately describe the mechanical properties of polymer materials, composites based on polymers, different elastomers, rubbers and biomaterials, in particular, human muscular tissues. Further, viscoelastic continua of the differential type will be called continua of the differential type, and viscoelastic continua of the integral type – simply viscoelastic materials. Definition 8.1. A medium is called a viscoelastic continuum of the integral type (or simply a viscoelastic continuum), if any of the models An , Bn , Cn or Dn is assumed for the medium, and corresponding operator constitutive equations (4.156) or (4.158)–(4.161) are functionals of time t: Dt

ƒ.t/ D f .R.t/; Rt .//;

(8.1)

D0

i.e. values of active variables ƒ.t/ depend not only on values of reactive variables R.t/ at the same instant of time but also on their prehistory Rt ./ R.t /, i.e. on their values at all preceding instants of time 0 < 6 t, starting from initial one D 0. Due to such specific dependence, viscoelastic continua are also called continua with memory. For viscoelastic continua: 1. “The present can depend only on the past but not on the future”; therefore, all the functionals (8.1) depend only on their previous history (prehistory) R.t /, 0 < 6 t. Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 8, c Springer Science+Business Media B.V. 2011

497

498

8 Viscoelastic Continua at Large Deformations

2. “The past is not infinite”, i.e. the times > t do not affect the functionals (8.1). This means that R./ 0 at < 0I

Rt ./ D R.t / 0 at t > :

(8.2)

8.1.2 Tensor Functional Space To perform operations with functionals (8.1), we need some additional knowledge of functional analysis. Consider the set of previous histories of k-order tensors k Tt ./ D k T.t / (0 < 6 t), and define for two arbitrary prehistories k Tt1 and k Tt2 their scalar product as follows: Zt .

k

Tt1 ; k Tt2 /t

k

D

Tt1 ./ „ƒ‚… : : : .k Tt2 .//.k:::1/ 2 ./ d :

(8.3)

k

0

The function ./ is called the function of memory. This function is positive, continuous, monotonically decreasing, defined within the interval Œ0; C1/ and squared-integrable, i.e. Z1 2 ./ d D 02 < C1: (8.4) 0

Since the memory function is monotonically decreasing, quantities of the prehistory k Tt ./ D k T.t / at small values of give a greater contribution to the scalar product (8.3) than quantities of k Tt ./ at large values of . In other words, a continuum better remembers events having occurred at times closer to the current instant t than the ones at more remote times. The functionals (8.1) are assumed to have this property too; therefore, viscoelastic continua of the integral type are also called continua with fading memory. Let us consider now the set k Ht of processes of changing the tensor k T./ (0 < 6 t) and assign to each process the pair .k T.t/; k T.// consisting of values of the tensor k T.t/ at time t and its prehistory k T./ D k T.t / (0 < 6 t). Then we can introduce the scalar product of processes k T1 ./ and k T2 ./ included in k Ht :

where

k

k

T1 ; k T2

t

D

k

T.t/; k T.t/ C k Tt1 ; k Tt2 ; t

.k ::: 1/ T1 .t/; k T2 .t/ D k T1 .t/ „ƒ‚… : : : k T2 .t/ k

is the scalar product of k-order tensors.

(8.5)

(8.6)

8.1 Viscoelastic Continua of the Integral Type

499

The set k Ht of all processes of changing the tensor k T./ (0 6 6 t), for which the scalar product (8.5) exists and which at each fixed are elements of the tensor space T3k .E3 / with the operations of addition and multiplication by a number, is called the tensor functional space k Ht . The space k Ht is a Hilbert space, because we can always go into a Cartesian basis, where the components TN i1 :::ik ./ of tensors included in k Ht are squared.m/ integrable functions, i.e. they belong to the function space L2 Œ0; t (where m D 3k ), which is known as a Hilbert space. Due to the property (8.2), the scalar product of processes included in k Ht (8.5) can be written in the form Z1 k

.

Tt1 ; k Tt2 /t

D

k

Tt1 ./ „ƒ‚… : : : .Tt2 .//.k:::1/ 2 ./ d < C1;

(8.7)

k

0

which often proves to be convenient for analysis of the models of a viscoelastic continuum. In the space k Ht , there is a natural norm of the process k T./: k k T kD .k T; k T/1=2 t ;

(8.8)

where . /t is the scalar product (8.5).

8.1.3 Continuous and Differentiable Functionals Using the norm (8.8), we can introduce the concept of a continuous functional in the form (8.1) m

t

S D m F .k T.t/; k Tt .//; D0

(8.9)

which is considered as the mapping of the domain U contained in the space k Ht into the domain V of the space m Ht : m

FW

U k Ht ! V m Ht :

(8.10)

Definition 8.2. The functional (8.10) is called continuous in a domain U k Ht , if for each process k T 2 U the following condition is satisfied: 8" > 0 9ı > 0 such T, where .k T C k e T/ 2 U and that for every process k e T kt < ı; k ke

(8.11)

500

8 Viscoelastic Continua at Large Deformations

values of the operators in the norm (8.8) of the space k Ht are sufficiently close: t

kmF

k

D0

t T.t/; k Tt ./ C k e Tt ./ m F .k T.t/; k Tt .// kt < ": T.t/ C k e D0

(8.12)

The functional (8.10) is called linear if it satisfies the two conditions

t

F

k

D0

T1 .t/ C k T2 .t/; k T1 ./ C k T2 ./ t

D F

D0

t

F

D0

k

t T1 .t/; k T1 ./ C F k T2 .t/; k T2 ./ ; D0

t s k T.t/; s k T./ D s F k T.t/; k T./ ; D0

(8.13)

(8.14)

for all processes k T1 ./ and k T2 ./ included in k Ht and for every real number s. In space k Ht we can use Riesz’s theorem that any linear functional (8.9) can be represented as the scalar product of a fixed element from k Ht and an arbitrary process k T./ from k Ht ; so a scalar linear functional in 2 Ht has the form e.t; t/ TT .t/ C f .T.t/; T .// D

Zt

t

e .t; t / TT .t / 2 ./ d ; (8.15a)

0

e.t; t/ is the instantaneous value at D t of where e .t; t / is the prehistory and e the fixed process .t; / (0 6 6 t) for the given functional f (the appearance of one more argument t for the process e .t; / means that this process can vary with changing the time interval considered). Replacing the variable t D y and introducing the notation e T .t; y/ 2 ./ D T .t; y/, 0 D .t; t/, after the reverse substitution y ! we obtain another representation of the linear scalar functional Zt f D 0 T.t/ C

.t; / T./ d ;

(8.15b)

0

called Volterra’s representation. For viscoelastic continua, it is convenient to use the Dirac ı-function ı.t/ having the following main property: (

Zt B.t; /ı.t0 / d D 0

for any continuous tensor process B.t; /.

B.t; t0 /; t0 2 Œ0; t; 0;

t0 … Œ0; t;

(8.16)

8.1 Viscoelastic Continua of the Integral Type

501

According to (8.16), the linear functional (8.15b) can be written in the form Zt A.t; / T./ d ;

f D

(8.15c)

0

A.t; / D 0 ı.t / C .t; /:

(8.15d)

Definition 8.3. The functional (8.9) is called Fr´echet–differentiable at point m T 2 U of a domain U m Ht , if there exist two functionals @F and ıF having the following properties: they are defined over the Cartesian product of the space Ht

@m F W

k

Ht k Ht ! m Ht I

ımF W

k

Ht k Ht ! m Ht ;

(8.17)

they can be written in the form similar to (8.9) m

t ˇ T.t// P1 D @m F .k T.t/; k Tt ./ˇk e D0

D

@

t

F .k T.t/; k Tt .// : : : k T.k:::1/ .t/;

@k T.t/ D0

m

t ˇ t T .//; P2 D ı m F .k T.t/; k Tt ./ˇk e

(8.18) (8.19)

D0

(the vertical line separates two different arguments of the process), they are linear and continuous in the second argument, they satisfy the condition: 8" > 0 9ı such that for any process .k e T.t/; k e Tt .//,

for which .k T1 .t/; k Tt1 .// U and k k T kt < ı;

(8.20)

k m F kt 6 " k k e T kt ;

(8.21)

the following inequality holds:

where t

t

m F D m F .k T1 .t/; k Tt1 .// m F .k T.t/; k Tt .// D0

D0

t ˇ ˇ t T.t// ı F .k T.t/; k Tt ./ˇk e T .//; @m F .k T.t/; k Tt ./ˇk e t

D0

D0

(8.22) k

. T1 .t/;

k

Tt1 .//

ke

k et

. T.t/ C T.t/; T ./ C T .//: k

k

t

(8.23)

502

8 Viscoelastic Continua at Large Deformations

The operator (8.19) is called the Fr´echet–derivative; and the right-hand side of expression (8.18) is the partial derivative of F (considered as a tensor function of T.t/) with respect to the tensor argument T.t/. If the functional (8.9) is Fr´echet–differentiable, then it is continuous (see Exercise 8.1.2). The set of processes k T.t/ 2 k Ht 0 having the first and second continuous derivaP R and k T.t/, which belong to k Ht 0 , will be denoted tives with respect to time t: k T.t/ by Ut 0 . Theorem 8.1. Let the functional (8.9) be Fr´echet–differentiable in k Ht 0 , then there exists such t (t 2 .0; t 0 /) that for all processes k T./ 2 Ut the process m S.t/ is differentiable with respect to t and the following rule of differentiation of the functional with respect to time holds: d dt

m

S.t/ D

@ @

t

F .k T.t/; k Tt .// : : :

k T.t/ D0

d k .k:::1/ T .t/ dt

t ˇ C ı F .k T.t/; k Tt ./ˇk TP t .//: D0

Here k

TP t D

(8.24)

d d k T.t / D k Tt ./: d.t / dt

H A proof of the theorem can be found in [31]. N Remark. The theorem gives the possibility to calculate the Fr´echet–derivatives of the operators (8.9) by evaluating the ordinary derivative of the functions S.t/ with respect to t according to formula (8.24). t u Example 8.1. Determine the Fr´echet–derivatives of the linear operator (8.15b) for the case when .t; / D .t / and .t; t/ D .0/. According to formula (8.24), we calculate the ordinary derivative with respect to time t by the rule of differentiation of an integral with a varying upper limit: d dT f D 0 .t/ C .0/ T.t/ C dt dt

Zt

0 .t / T./ d ;

(8.25)

0

where 0 .y/ D @.y/[email protected] Comparing (8.25) with (8.24), we find the partial derivative and the Fr´echet–derivative: d @F T.t/; D 0; dt @T Zt ıF D .0/ T.t/ C 0 .t / T./ d : @F D 0

0

(8.26)

8.1 Viscoelastic Continua of the Integral Type

503

8.1.4 Axiom of Fading Memory For viscoelastic continua, in addition we assume the following axiom. Axiom 18 (of fading memory). The functionals (8.1) occurring in constitutive equations of viscoelastic continua are Fr´echet–differentiable and hence satisfy the rule of differentiation with respect to time (8.24): t t ˇ t d @ d P .//: (8.27) f .R.t/; Rt .// R.t/ C ı f .R.t/; Rt ./ˇR ƒ.t/ D dt @R.t/ D0 dt D0

The interconnection between Fr´echet–differentiability and fading memory of functionals is clarified by the theorem on relaxation. Let there be a process R./ (0 6 6 t), which is arbitrary up to some time t0 < t, and when t0 > > t it remains constant: R./ D R.t0 /;

t0 > > t:

(8.28)

Such process R./ is called a process with the constant extension (Fig. 8.1). In addition, consider a static process

R./ D R.t0 / D const;

0 6 6 t:

(8.29)

Then the following theorem can be formulated. Theorem 8.2. Let the functional f (8.1) be Fr´echet–differentiable, then its partial derivative @f and Fr´echet–derivative ıf , and also the function ƒ.t/ for any process R./ with the constant extension at fixed time t0 have the limits as t ! C1, which

Fig. 8.1 For Theorem 8.2

504

8 Viscoelastic Continua at Large Deformations

coincide with values of the derivatives @f , ı f and ƒ , respectively, in correspond

ing static processes R./: lim

t0

t

f .R.t/; Rt .// D ƒ f .R.t0 /; Rt0 .//;

t !C1 D0

D0

t0 ˇ ˇ P D @f @ f .R.t0 /; Rt0 ./ˇ0/; lim @ f .R.t/; Rt ./ˇR.t// t

t !C1 D0

D0

t0 ˇ ˇ lim ı f .R.t/; Rt ./ˇRP t .// D ı f ı f .R.t0 /; Rt0 ./ˇ0/: t

t !C1 D0

(8.30)

D0

In simple words, for a process R./ such that starting from some time t0 the process reaches a constant level, in a certain time interval a viscoelastic continuum forgets the process R./ up to the time t0 , because the continuum response, expressed by the functionals f , @f and ıf , at sufficiently great values of t differs little from its response to a static process. H Consider the process

e R./ D R./ R./;

0 6 6 t:

(8.31)

Since e R./ 0

at t0 > > t;

(8.32)

so we have e kR

k2t

e2

Zt

e2

Zt

R .t / ./ d D

D R .t/ C

2

0

Zt0 D

e 2 ./ 2 .t / d R

0

e 2 ./ 2 .t / d 6 c R

Zt 2 ./ d :

(8.33)

t t0

0

Due to the property (8.4) and monotone decreasing the function ./, we find that e kt ! 0 at t ! C1. kR Since a Fr´echet–differentiable functional f is continuous as well as @f and ıf , the condition (8.12) yields the inequality t

t

e e t .// f .R.t/; Rt .// kt < "I Rt ./ C R k f .R.t/ C R.t/; D0

(8.34)

D0

this means that the first limit of (8.30) exists. In a similar way, we can prove that the second and third limits of (8.30) exist too. N

8.1 Viscoelastic Continua of the Integral Type

505

8.1.5 Models An of Viscoelastic Continua For models An of viscoelastic continua, the free energy is a functional in the form (8.1), and as reactive variables one should choose the set (4.148): D

.n/

t

!

.n/ t

t

C .t/; .t/; C ./; ./ :

(8.35)

D0

According to the rule (8.24) of differentiation of a functional, we obtain the expression for the total derivative of with respect to t: .n/

d dC @ @ d D C Cı : .n/ dt dt @ dt @C

(8.36)

Substituting this expression into PTI (4.121) and collecting like terms, we get 0

.n/

1

.n/ TC [email protected] @ .n/ A d C C @C

@ w C d C Cı dt D 0: @

.n/

(8.37)

.n/

When the prehistories C t , t and the current values C .t/ and .t/ are fixed, the .n/

increments d C , d and dt can vary arbitrarily, therefore the identity (8.37) holds when and only when the coefficients of these increments vanish. As a result, we obtain the equation system 8 .n/ .n/ .n/ t .n/ ˆ ˆ ˆ T D [email protected] [email protected] C / D F . C .t/; .t/; C t ./; t .//; < D0 (8.38) D @ [email protected]; ˆ ˆ ˆ : w D ı ; that together with (8.35) is a system of constitutive equations for models An of viscoelastic continua. Just as for ideal continua, for viscoelastic materials it is sufficient to give only the free energy functional (8.35), then the remaining relations are determined by its differentiation according to formulae (8.38). Notice that although relations (8.38) are formally similar to the corresponding relations (4.168) for models An of ideal continua, they essentially differ by the fact .n/

that in (8.38) there is a functional dependence on C and . Moreover, viscoelastic continua are dissipative: for them the dissipation function w is not identically zero.

506

8 Viscoelastic Continua at Large Deformations

8.1.6 Corollaries of the Principle of Material Symmetry for Models An of Viscoelastic Continua According to the principle of material symmetry (Axiom 14), for each viscoelastic b Just as in continuum there exists an undistorted reference configuration K. ı

Sect. 4.7.4, for simplicity, let the reference configuration K be undistorted. Then ı

ı

for K there is a subgroup G s U of the unimodular group U such that for each ı

ı

transformation tensor H 2 G s (H W K ! K) the constitutive equations (8.38) writı

ten for K are transformed during the passage to K as follows: 8.n/ .n/ .n/ t .n/ ˆ t t ˆ T D F . C ; ; C ; / D [email protected] [email protected] C /; ˆ ˆ D0 ˆ ˆ ˆ .n/ t .n/ < . C ; ; C t ; t / ; D D0 ˆ ˆ ˆ D @ [email protected]; ˆ ˆ ˆ ˆ ı : .w / D .ı / ; 8H 2 G s :

(8.39)

.n/ .n/

For viscoelastic solids, due to H -indifference of all the tensors T , C .t/ and .n/

.n/

C t ./ D C.t / (see Sect. 4.7.4), relations (8.39) take the forms

.n/

t

.n/

t

.n/

.n/

QT F . C ; ; C t ; t / Q D F .QT C Q; ; QT C t Q; t /; D0

t

D0

.n/

D0

t

. C ; / D

.n/

.n/

(8.40a)

ı

.QT C Q; ; QT C t Q; t / 8Q 2 G s ;

(8.40b)

D0

ı

Dı

:

(8.40c)

Theorem 8.3. The principle of material symmetry in the form (8.40) holds for models An of viscoelastic solids if and only if the condition (8.40b) for is satisfied. H It is evident that the condition (8.40b) is necessary. Prove that this condition is sufficient. Let the condition (8.40b) be satisfied. Then, since the functional F is a tensor function being the derivative of with respect to .n/

C .t/, so with the help of the method used in the proof of Theorem 4.23 we can prove that the condition (8.40b) yields (8.40a). To prove Eq. (8.40c) we should use

and rewrite it in K:

formula (8.36) for ı

.n/

ı

d 1 .n/ d C @ d T : D dt dt @ dt

(8.41)

8.1 Viscoelastic Continua of the Integral Type

Since

D

507

ı

and the passage from K to K is independent of t, so d =dt D d .n/

.n/

.n/

=dt.

.n/

Due to H -indifference of the tensors T and C , we have the relation T .d C =dt/D .n/

.n/

T .d C =dt/, and also @ [email protected] D @ [email protected]. Thus, the right-hand side of Eq. (8.41) coincides with ı , and hence .ı / D ı ; i.e. Eq. (8.40c) actually holds. N

8.1.7 General Representation of Functional of Free Energy in Models An Scalar functional

(8.35) satisfying the condition (8.40b) is called functionally ı

indifferent relative to the group G s . Let us find a general representation of such a ı

.n/

functional in terms of invariants of tensor C of the corresponding group G s . In Sect. 8.1.3 we have derived a general representation of a linear scalar functional in the form (8.15d). Similarly to (8.15d), define a quadratic scalar functional as a double integral in the form Zt Zt 2

4

D 0

.n/

.n/

A.t; 1 ; 2 / . C .1 / ˝ C.2 //.4321/ d 1 d 2 ;

(8.42)

0

where 4 A.t; 1 ; 2 / is a fixed fourth-order tensor called the core of the functional. We also define a n-fold scalar functional: Zt m

D

Zt :::

0

e m .t; 1 ; : : : ; m / d 1 : : : d m ;

(8.43)

0

where .n/

.n/

e m .t; 1 ; : : : ; m / D 2n A.t; 1 ; : : : ; m / : : : . C .1 /˝: : :˝ C .m //.m;m1;:::;2;1/ : „ƒ‚… 2m

(8.44)

Here 2n A.t; 1 ; : : : ; m / is the core of this functional (it is a fixed tensor of order .2n/ depending on m C 1 arguments). Theorem 8.4 (Stone–Weierstrass). Any continuous scalar functional (8.35) in space Ht can be uniformly approximated by n-fold scalar functionals (8.43): D

t

.n/

.n/ t

. C .t/; .t/; C ./; .// D

D0

t

1 X

m;

(8.45)

mD1

where the equality means that the partial sum uniformly converges in the norm (8.8).

508

8 Viscoelastic Continua at Large Deformations

H A proof of the theorem for the space Ht can be found in [9]. N Let us consider the integrand n .t; 1 ; : : : ; n / of the n-fold functional (8.43). At any fixed set of values t; 1 ; : : : ; m , this expression is a scalar function (but not .n/

.n/

a functional!) of n tensor arguments C.i / C i (i D 1; : : : ; m): em .t; 1 ; : : : ; m / D em .t; 1 ; : : : ; m ; C1 ; : : : ; Cm /:

(8.46)

Here while values of t; 1 ; : : : ; m vary, the number and the form of tensor arguments of the functions remain unchanged. On substituting the representation (8.45) into the condition (8.40b) of functional indifference of , we find that the functions e m (8.44) at each fixed value t; 1 ; : : : ; m must satisfy the condition em .t; 1 ; : : : ; m ; C1 ; : : : ; Cm /

D em .t; 1 ; : : : ; m ; QT C1 Q; : : : ; QT Cm Q/;

(8.47)

ı

i.e. they must be H -indifferent scalar functions relative to the group G s . Theorem 8.5. Every scalar function (8.46) of m tensor arguments C1 ; : : : ; Cm , ı

which is indifferent relative to a group G s , can be represented as a function of finite number z (z 6 6m) of simultaneous invariants J.s/ D J.s/ .C1 ; : : : ; Cm /;

D 1; : : : ; z;

(8.48)

ı

relative to the group G s in the form em D e m .t; 1 ; : : : ; m ; J.s/ .C1 ; : : : ; Cm //:

(8.49)

H By analogy with simultaneous invariants of two tensors, which have been considered in Sect. 7.1.4, we now introduce simultaneous invariants of m tensors relative ı

to a group G s . The simultaneous invariants are scalar functions J (8.48), which are ı

H -indifferent relative to the group G s ; i.e. they satisfy the relations

J.s/ .C1 ; : : : ; Cm / D J.s/ .QT C1 Q; : : : ; QT Cm Q/ 8Q 2 Gs :

(8.50)

Their functional basis consists of z simultaneous invariants, where z is a finite number, which cannot exceed the total number of components of all the tensors, i.e. z 6 6m. Moreover, since J.s/ form a basis, any other H -indifferent scalar funcı

tion relative to the same group G s can be expressed in this basis. But the function e (8.40) is just such a function due to (8.47), therefore the relation (8.49) actually holds. N

8.1 Viscoelastic Continua of the Integral Type

509

Substitution of the expression (8.49) into (8.43) and then into (8.45) yields the following general representation of the continuous functional (8.35), which is funcı

tionally indifferent relative to a group G s : D

1 Z X mD1 0

Zt

t

:::

.n/

.n/

e m .t; 1 ; : : : ; m ; J . C .1 /; : : : ; C.m /// d 1 : : : d m :

0

(8.51)

In deriving this formula we have used representation (8.15c) of linear functionals with the help of ı-function. Let us perform now the inverse operation: segregate the ı-type constituent from the cores e m , that allows us to separate the instantaneous .n/

value of the tensor C .t/ from its prehistory. Formula (8.15d) can be generalized for functions of m C 1 arguments t; 1 ; : : : ; m as follows: m X

e m .t; 1 ; : : : ; m ; J.s/ / D

ı.t 1 / : : : ı.t k /

kC1 ; : : : ; m ; J.s/ /:

mk .t;

kD0

(8.51a)

We assume that at k D m the argument kC1 D mC1 does not appear among .s/ arguments of the function mk : mm D mk .t; J /. On substituting (8.51a) into (8.51), we get D

1 X m Z X

Zt

t

.n/

::: „ƒ‚…

mD1 kD0 0

mk

mk 0

.n/

.n/

.n/

t; kC1 ; : : : ; m ; J.s/ C.t/; : : : ; C .t/; ƒ‚ … „ k

!!

C.kC1 /; : : : ; C.m /

d kC1 : : : d m :

(8.52)

Rearrange summands in the expression (8.52) and take into account that simulta.n/

neous invariants J.s/ of m tensors, among which there are k tensors C.t/, can always be expressed in terms of simultaneous invariants of .m k C 1/ ten.n/

.n/

.n/

sors C .kC1 /; : : : C.m / and C.t/. Then we finally obtain the general form of the functional (8.35) .n/

D '0 .t; J.s/ . C .t/// C

1 Z X mD1 0

Zt

t

::: „ƒ‚… m

.n/ 'm t; 1 ; : : : ; m ; J.s/ . C .t/;

0

.n/ C.1 /; : : : ; C.m // d 1 : : : d m ;

.n/

(8.53)

510

8 Viscoelastic Continua at Large Deformations ı

which is functionally indifferent relative to the group G s . Here we have introduced the notation: '0 – the instantly elastic part and 'm – cores of the functional: 1 X

'0 D

mm .t;

J.s/ .C.t///;

mD1

'm D

1 X kD0

mCk;k .t;

.n/

.n/

1 ; : : : ; m ; J.s/ . C.t/ : : : C .t/; C.1 /; : : : ; C.m ///: ƒ‚ … „ k

.s/

(8.54)

.n/

.s/

.n/

Simultaneous invariants J . C .t// of one tensor are simply invariants I . C.t// ı

of the tensor relative to the same group G s . The expression (8.53) is the desired general representation of the functional (8.35) for models An .

8.1.8 Model An of Stable Viscoelastic Continua Definition 8.4. One can say that this is the model An of a stable viscoelastic continuum, if the functional of the model is invariant relative to a shift of the pro.n/

cess of deforming and heating in time; i.e. if there are two processes . C ./; .// .n/

C ./; e .//, 0 6 6 t1 , such that they are different from each other only by a and e shift in time: 8 < .n/ .n/ e e . C ./; .// D . C . t0 /; . t0 //; t0 < 6 t1 ; (8.55) :.0; /; 0 6 6 t0 I 0 then the corresponding values of the functionals and e are different only by a shift in time as well (Fig. 8.2): ( .t t0 /; t0 < t 6 t1 ; eD (8.56) .0/; 0 6 t 6 t0 : Remark. Since the functional

for stable continua is invariant relative to a shift .n/

in time, its partial derivatives with respect to C .t/ and .t/ and also the Fr´echet– derivative ı have this property; hence the constitutive equations (8.38) are invariant relative to a shift in time too. Due to the property of invariance, stable continua are also called non-aging, and their constitutive equations do not change with time themselves when there are no deformations and variations of temperature. t u Let us consider two important models of stable continua.

8.1 Viscoelastic Continua of the Integral Type

511

Fig. 8.2 For the definition of a stable viscoelastic continuum

8.1.9 Model An of a Viscoelastic Continuum with Difference Cores One can say that this is the model An of a viscoelastic continuum with difference cores, if in the general representation of the functional (8.53) there is no explicit dependence of cores 'm on the times t and i , but there is a dependence only on their difference t i or on the temperatures .t/ and .i /: .n/

.n/

.n/

'm D 'm .t1 ; : : : ; tm ; .t/; .1 /; : : : ; .m /; J.s/ . C .t/; C.1 /; : : : ; C.m ///; (8.57) and '0 does not depend explicitly on temperature: .n/

'0 D '0 .I.s/ . C .t//; .t//:

(8.58)

Functions 'm are assumed to satisfy the following conditions of normalization and symmetry with respect to any permutations of the first m arguments: 'm .t 1 ; : : : ; t m ; 0 ; : : : ; 0 ; 0; : : : ; 0/ D 0;

(8.59)

'm .y1 ; : : : ; yn ; : : : ; yl ; : : : ; ym ; ; 1 ; : : : ; m ; J.s/ / D 'm .y1 ; : : : ; yl ; : : : ; yn ; : : : ; ym ; ; 1 ; : : : ; m ; J.s/ /; where 0 D .0/, n D .n /, yn D t n , and 1 6 n; l 6 m. Simultaneous invariants J.s/ can always be chosen to satisfy the normalization conditions J.s/ .0; : : : ; 0/ D 0;

D 1; : : : ; z:

(8.60)

512

8 Viscoelastic Continua at Large Deformations

For the model with difference cores, the functional (8.53) has the form .t/ D

.n/ '0 .I.s/ . C .t//;

1 Z X

.t// C

Zt

t

:::

mD1 0

'm d 1 : : : d m ;

(8.61)

0

where 'm are determined by formula (8.57). Theorem 8.6. The model An of a viscoelastic continuum with difference cores is stable. .n/

H Let the first process be of the form C ./, 0 6 6 t, then the corresponding .n/

C ./ when 6 t0 functional .t/ has the form (8.61). Since the second process e is identically zero, so, due to the normalization conditions (8.59) and (8.60), we have 'm 0 when 0 6 i 6 t0 (i D 1; : : : ; m); therefore, the lower limits of the integrals in the expression (8.61) for functional e.t/ can be determined as t0 : .n/ e e.t/ D '0 I.s/ . e C .t//; .t/ C

1 Z X mD1 t

Zt

t

:::

' m t 1 ; : : : ; t m ; e .t/; e .1 /; : : : ; e .m /;

t0

0

.n/ .n/ .n/ J.s/ . e C .t/; e C.1 /; : : : ; e C.m // d 1 : : : d m : (8.62) .n/

.n/

C .i / by e C.i t0 / when i > t0 and then substituting i t0 D e i , we Replacing e obtain (when t > t0 ) .n/

e.t/ D '0 .I.s/ . C .t t0 //; .t t0 // tZt0 1 tZt0 X ::: 'm t t0 e 1 ; : : : ; t t0 e m ; .t t0 /; C mD1 0

0

.n/ .n/ .n/ .e 1 /; : : : ; .e C .t t0 /; C .e m /; J.s/ . e 1 /; : : : ; C.e m // de 1 : : : de m:

(8.63) Comparing (8.61) and (8.63), we verify that e.t/ and the time-shift t ! .t t0 /, because at t D t0 e.t0 / D '0 .0; .0// D i.e. the relation (8.56) holds. N

.0/;

.t/ are connected only by

8.1 Viscoelastic Continua of the Integral Type

513

On substituting the functional (8.61) into (8.38), we get the general form of constitutive equations for stable continua: .n/

T D

z X D1

.s/ @'0 @I

@I.s/ D

w D '10 C

.n/

C

1 Zt X

1 Zt X

@'0 C @ mD1

:::

0

:::

0

Zt

.s/ @'m @J

@J.s/

.n/

! d 1 : : : d m ;

@ C .t/

@'m d 1 : : : d m ; @.t/

0

Zt

Zt

mD1 0

:::

mD1 0

@ C .t/

1 X

Zt

@'m 0 C 'mC1 @t

d 1 : : : d m :

(8.64)

0

Here the cores 'm are functions in the form (8.57), and @'m [email protected] is the partial derivative of the function when its first arguments .t 1 /; : : : ; .m / vary and the arguments J.s/ are fixed (i.e. there is no differentiation with respect to J.s/ ). We have introduced the following notation for a value of the function 'mC1 (8.57) at mC1 D t: 0 D 'mC1 t 1 ; : : : ; t m ; 0; .t/; .1 /; : : : ; .m /; .t/; 'mC1 .n/ .n/ .n/ .n/ J.s/ . C .t/; C.1 /; : : : ; C.m //; C .t/ : In deriving the expression for the dissipation function we have used the conditions (8.59).

8.1.10 Model An of a Thermoviscoelastic Continuum Let us consider the most widely used method to take the dependence of cores 'm (8.57) on temperature into account. For the model An of a thermoviscoelastic continuum with difference cores, temperature appears in simultaneous invariants J.s/ , i.e. .n/

.n/

.n/

'm D 'm t 1 ; : : : ; t m ; J.s/ C .t/; C .1 /; : : : ; C .m / where .n/

.n/

ı

C ./ D C./ "./;

Z./ "./ D ˛.e /d e : ı

0

!! ; (8.65)

(8.66)

514

8 Viscoelastic Continua at Large Deformations ı

The tensor " is called the tensor of heat deformation, and ˛ – the tensor of heat expansion. Both the tensors are symmetric and H -indifferent relative to a considered ı

group G s :

QT ˛ Q D ˛

ı

8Q 2 G s ;

(8.67) ı

.n/

therefore, the functions J.s/ of C are also H -indifferent relative to the group G s . Taking the dependence of constitutive equations upon temperature as the difference between the deformation tensor and the heat deformation tensor (8.66) is called the Duhamel–Neumann model. In a similar way, the Duhamel–Neumann model describes the dependence of the function '0 on temperature: ! .n/

'0 D '0 .I.s/ C .t/ ; .t//:

(8.68)

.n/

Since @ C [email protected] D ˛, the derivatives with respect to .t/ in (8.64) for this model have the form z .s/ X @0 '0 @'0 @'0 @I ˛C D ; @ @I .n/ @.t/ D1 @C

z .s/ X @'m @'m @J ˛..t//; D @.t/ @J .n/ D1 @ C .t/ (8.68a)

where @0 [email protected] means the derivative with respect to the second argument in formula (8.68). .n/

.n/

.n/

.n/

Taking into account that @I.s/ [email protected] C D @I.s/ [email protected] C and @J.s/ [email protected] C D @J.s/ [email protected] C and substituting (8.68a) into (8.64), we obtain the expression for the specific entropy D

.n/ @0 '0 1 C ˛ T: @.t/

(8.69)

8.1.11 Model An of a Thermorheologically Simple Viscoelastic Continuum One can say that this is the model An of a thermorheologically simple viscoelastic continuum, if the cores 'm (8.57) in constitutive equations (8.53) and (8.64) depend on temperature in the functional way through the so-called reduced time: .n/ .n/ 0 ; J.s/ . C .t/; C.1 /; : : : ; 'm D 'm t 0 10 ; : : : ; t 0 m C.m // a ..1 // : : : a ..m //;

.n/

(8.70)

8.1 Viscoelastic Continua of the Integral Type

515

where 0

Zt

t D

Zi

i0

a ..e // de ;

D

0

a ..e // de

(8.71)

0

is the reduced time being a functional of the function a ./ called the function of the temperature-time shift. The functions 'm (8.70) and a satisfy the normalization conditions 0 'm 0; : : : ; 0; J.s/ D 0; 'm t 0 10 ; : : : ; t 0 m ; 0 D 0; a .0 / D 1: (8.72) .n/

If the process C ./ is considered with respect to the reduced time 1 0 Z .n/ e A D C ./; C. 0 / D e C @ a de

.n/

.n/

0

then, since d i0 D a ..i // d i , the functional (8.53) with the core (8.70) can be written with respect to the reduced time 0

.t / D '0

.n/ I.s/ . C .t 0 //;

!

t0

1 Z X

0

.t / C

Zt 0 :::

mD1 0 .n/

.n/

.n/

0 // J.s/ . C .t 0 /; C.10 /; : : : ; C.m

0 'm t 0 10 ; : : : ; t 0 m ;

0

!

0 d 10 : : : d m :

(8.73)

Substituting the functional (8.73) into (8.38) and using the differentiation rule (8.27), we obtain constitutive equations for a thermorheologically simple viscoelastic continuum 1 0 Zt 0 1 Zt .s/ X @I.s/ @'m @J @' 0 0 0 @ T D ::: C d 1 : : : d m A ; .s/ .s/ .n/ .n/ @I @J 0 D1 mD1 0 @C @ C .t / 0 z X

.n/

0

.n/

D [email protected] '0 [email protected]/ C .1=/˛ T ;

w D

a '10

C a

0 1 Zt X

mD1 0

Zt 0 ::: 0

Here we have used that @[email protected] D a [email protected][email protected] 0 /.

@'m 0 0 C 'mC1 d 10 : : : d m : @t 0

(8.74)

516

8 Viscoelastic Continua at Large Deformations

Notice that with the help of (8.38) and (8.74) the dissipation function w can be represented in another equivalent form .n/

w D T

0 .n/ d @ '0 d .n/ d C ˛ T ; C dt dt @ dt

(8.75)

which proves to be useful for cyclic loading. Theorem 8.7. A thermorheologically simple continuum is stable. H The reduced time (8.71) for the shifted process of heating e ./ D . t0 / with use of the normalization condition (8.72) can be represented in the form 0

Zt0

t D

Zt a d C

0

Zt e a ./ d D t0 C a .. t0 // d

t0

t0 tZt0

D t0 C

a ..e // de ; 0

0

t Z 0

D t0 C

a ..e // de ;

(8.76)

0

when t0 < < t. The further proof is the same as the one in Theorem 8.6 (see Exercise 8.1.1). N

Exercises for 8.1 8.1.1. Complete the proof of Theorem 8.7. 8.1.2. Using Definitions 8.2 and 8.3, prove that if a functional is Fr´echet– differentiable, then it is continuous.

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua 8.2.1 Principal Models An of Viscoelastic Continua Constitutive equations containing multiple integrals of the type (8.53), (8.61) or (8.73) are very awkward, and their application in practice is considerably difficult. Therefore, special models of viscoelastic materials, in which one may retain a finite number of integrals, are widely used.

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

517

For the principal model An of a thermoviscoelastic continuum with difference cores, the sum (8.61) contains only one integral .m D 1/, i.e. in this model has the form D

.n/ '0 .I.s/ . C /;

Zt

.n/

.n/

'1 .t ; J.s/ . C .t/; C .// d :

/

(8.77)

0

Here '0 and '1 are functions of the arguments indicated, and the function '1 is chosen with the negative sign that can always be done by simple renaming the functions. Constitutive equations for the continuum considered have the form (8.64), where m should be assumed to be equal to 1: .n/

T D

z X D1

'0 J.s/ C

Zt

'1 J.s/ d : C

(8.78)

0

Here we have denoted the partial derivatives of '0 and '1 .n/

'0 .J˛.s/ . C .t/// D [email protected]'0 [email protected].s/ /; '1 .t ;

.n/ .n/ J˛.s/ . C .t/; C .///

D

D 1; : : : ; r;

[email protected]'1 [email protected].s/ /;

(8.79)

D 1; : : : ; z; .n/

and also the partial derivative tensors of J.s/ with respect to C .t/: .n/

.n/

.s/ .s/ J.s/ C D @J [email protected] C .t/ D @J [email protected] C .t/; .s/

.n/

D 1; : : : ; z:

(8.80)

.n/

The simultaneous invariants J . C .t/; C .// can be chosen so that the first r .n/

ı

ones form a functional basis of invariants I.s/ . C .t// in the same group G s . We will assume below that J.s/ are ordered in such a way; then the following equations hold: .s/

.n/

J C D @I.s/ [email protected] C .t/; D 1; : : : ; rI

'0 0; D r C 1; : : : ; z:

(8.81)

These equations have been used for deriving the relation (8.78). According to (8.64) and (8.69), the dissipation function w and the specific entropy for the principal models An have the forms .n/

.n/

w D '1 .0; J.s/ . C .t/; C .t//C

Zt

.n/ .n/ @ '1 .t ; J.s/ . C .t/; C .// d > 0; @t

0

D

0

.n/ 1 @ '0 C ˛ T: @

(8.82)

518

8 Viscoelastic Continua at Large Deformations

Here @[email protected] is the partial derivative of '1 with respect to the first argument, and @0 [email protected] is the partial derivative of '0 with respect to the second argument. Simultaneous invariants J.s/ of two tensors can be written by analogy with the ones for continua of the differential type (see Sect. 7.1.4).

8.2.2 Principal Model An of an Isotropic Thermoviscoelastic Continuum For the principal model An of an isotropic thermoviscoelastic continuum with difference cores, the functional basis of simultaneous invariants J.I / consists of 9 invariants, which can be chosen as follows (see (7.35)): .n/

.n/

.I / D I˛ . C .//; ˛ D 1; 2; 3; J˛.I / D I˛ . C .t//; J˛C3 .I /

J7

.n/

.n/

D C ./ C .t/;

.I /

.n/

.n/

.I /

J8 D C 2 ./ C .t/; r D 3 and z D 9:

J9

(8.83)

.n/

.n/

D C ./ C 2 .t/;

The derivative tensors of these invariants are calculated by the formulae (see [12]) .I / J1C D E;

.n/

.I /

.n/

J2C D EI1 . C .t// C .t/; .I /

.I /

.n/

.n/

.I /

J C3;C D 0; D 1; 2; 3I

.I /

.n/

.n/

J8C D C 2 ./;

J7C D C ./; .n/

.n/

.n/

.I /

J3C D C 2 .t/ I1 C .t/ C EI2 ;

.n/

J9C D C ./ C .t/ C C .t/ C ./:

(8.84)

Substituting these expressions into (8.77) and collecting terms with the same tensor powers, we obtain constitutive equations for the principal model An of an isotropic thermoviscoelastic continuum: .n/

.n/

.n/

T D 'M 1 E C 'M 2 C C 'M3 C 2 :

(8.85)

Here we have denoted the functionals Zt 'M1 '01 C '02 I1 .t/ C '03 I2 .t/

.'11 C '12 I1 .t/ C '13 I2 .t// d ; 0

.n/

.n/

Zt

.n/

.n/

'M2 C .'02 C '03 I1 .t// C .t/ ..'12 C '13 I1 .t// C .t/ '17 C .// d ; 0

(8.86)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua .n/

519

.n/

'M3 C 2 '03 C 2 .t/ Z t .n/ .n/ .n/ .n/ .n/ .n/ '13 C 2 .t/C'18 C 2 ./C'19 . C .t/ C ./C C ./ C .t// d : 0

Relation (8.85) is formally similar to the corresponding relation (4.322) for an isotropic elastic continuum, but in (8.85) 'M1 , 'M2 and 'M3 are no longer functions .n/

of invariants of the tensor C ; they are functionals in the form (8.86).

8.2.3 Principal Model An of a Transversely Isotropic Thermoviscoelastic Continuum For the principal model An of a transversely isotropic (relative to the group T3 ) thermoviscoelastic continuum with difference cores, the functional basis of simul.3/ taneous invariants J consists of 11 invariants, which can be chosen as follows (see (7.36)): .n/ .n/ .3/ J.3/ D I.3/ C .t/ ; D 1; : : : ; 5I J5C D I.3/ . C .//; D 1; : : : ; 4I .n/ .n/ .3/ D ..E b c23 / C .t// b c23 C ./ ; J10 .n/

.n/

.3/ .3/ J11 D C .t/ C ./ 2J10 J2.3/ J7.3/ ;

r D 5;

z D 11:

(8.87) Here the invariants I.3/ are determined by formulae (4.297). The partial derivatives J.3/ C of these invariants have the forms (see [12]) .n/ 1 .O1 ˝ O1 C O2 ˝ O2 / C .t/; 2 .n/ 1 D 24 O3 C .t/; 4 O3 .O1 ˝ O1 C O2 ˝ O2 / b c23 ˝b c23 ; 2

.3/ .3/ .3/ D E b c23 ; J2C Db c23 ; J3C D J1C .3/ J4C

.3/

.n/

.n/

.3/

.3/

.3/

.3/

J5C D C 2 .t/ I1 C .t/ C EI2 ; J6C D J7C D J8C D J9C D 0; .3/ J10C D

.n/ 1 .O1 ˝ O1 C O2 ˝ O2 / C ./; 4

.n/

.3/ J11C D 4 O3 C ./:

(8.88)

Substituting these expressions into (8.77) and rearranging the summands, we obtain constitutive equations for the principal model An of a transversely isotropic thermoviscoelastic continuum: .n/

.n/

.n/

.n/

c23 C .O1 ˝ O1 C O2 ˝ O2 / 'M3 C C 'M4 C C 'M5 C 2 : (8.89) T D 'M1 E C 'M 2b

520

8 Viscoelastic Continua at Large Deformations

Here we have denoted the functionals Zt 'M 1 '01 C '05 I2

.'11 C '15 I2 .t// d ; 0

'M 2 '02 '01

2'04 I2.3/

Zt

.'12 '11 2'14 I2.3/ .t/ 2'1;11 I2.3/ .// d ;

0 .n/

.n/ 1 1 'M 3 C .'03 2'04 / C 2 2

! .n/ .n/ Zt '1;10 '13 '14 C .t/C '1;11 C ./ d ; 2 2 0

Zt .n/ .n/ .2'14 '15 I1 .t// C .t/ C '1;11 C ./ d ; 'M4 C .2'04 '05 I1 / C .n/

.n/

0 .n/

'M5 C 2

!

Zt

.n/

'15 d C 2 .t/:

'05

(8.90)

0

8.2.4 Principal Model An of an Orthotropic Thermoviscoelastic Continuum For the principal model An of an orthotropic thermoviscoelastic continuum with difference cores, the functional basis of simultaneous invariants consists of 12 invariants, which can be chosen as follows (see (7.37)): .n/

J.O/ D I.O/ . C .t//; D 1; : : : ; 6I

.n/

.O/

J C6 D I.O/ . C .//; D 1; 2; 3; 6I

.n/ .n/ .n/ .n/ .O/ .O/ c22 C .t/ b c23 C ./ ; J11 D b c21 C .t/ b c23 C ./ ; J10 D b r D 6;

z D 12:

(8.91) This set should be complemented by two more invariants (being dependent) in order to obtain relations symmetric with respect to the vectors b c2˛ : .n/

.n/

.O/ J13 D .b c21 C .t// .b c22 C .//; .O/ J14

D

.n/ I7.O/ . C .t//

.n/

(8.92) .n/

D .b c21 C .t// .b c22 C .t//:

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

521

The partial derivative tensors of these invariants have the forms (see [12]) J.O/ c2 ; D 1; 2; 3I C Db .n/

.O/

.n/ 1 .O ˝ O / C .t/; D 1; 2I 2

.O/

J C3;C D .n/

.O/

J6C D 3 6 Om C .t/ ˝ C .t/; .O/ J10C .O/ J13C

.O/

J C6;C D J12C D 0; D 1; 2; 3I

.n/ 1 D .O1 ˝ O1 / C ./; 4 .n/ 1 D .O3 ˝ O3 / C ./; 4

.n/ 1 D .O2 ˝ O2 / C ./; 4 .n/ 1 .O/ J14C D .O3 ˝ O3 / C .t/; 2

(8.93)

.O/ J11C

where the tensor 6 Om is determined by formula (4.316). Substituting these expressions into (8.77) and grouping like terms, we obtain constitutive equations for the principal model An of an orthotropic thermoviscoelastic continuum: .n/

T D

3 X

.n/

.n/

.n/

.'M b c2 C O ˝ O 'M3C C / C 'M 7 6 Om C ˝ C :

(8.94)

D1

Here we have denoted the functionals Zt 'M '0

'1 t ; J˛.O/ d ; D 1; 2; 3;

0

.n/

'M3C C

Zt

.n/ 1 1 '0; 3C C 2 2

.n/ 1 '1; 9C t ; J˛.O/ C ./ 2

0

C'1; 3C t ;

J˛.O/

Zt 'M 7 3'06 3

.n/ C .t/ d ; D 1; 2;

'16 t ; J˛.O/ d ;

0 .n/

.n/ 1 1 'M 6 C '0;14 C 2 2

Zt

.n/ 1 '1;13 t ; J˛.O/ C ./ 2

0

.n/ C'1;14 t ; J˛.O/ C .t/ d :

(8.95)

522

8 Viscoelastic Continua at Large Deformations

8.2.5 Quadratic Models An of Thermoviscoelastic Continua For the quadratic model An of a thermoviscoelastic continuum with difference cores, we retain two integrals in the sum (8.61), i.e. m D 1; 2. A form of constitutive equaı

tions for specific symmetry groups G s becomes considerably more complicated, because there appear double integrals and we need to consider simultaneous invariants J.s/ of three tensors. Therefore, one usually considers the particular case of the quadratic model when m D 1 and 2, but simultaneous invariants J.s/ of only two tensors appear there just as in the principal model: .n/ I.s/ . C .t//;

D '0

!

Zt

'1 .t

; J.s/

.n/

C

.n/

.n/

d

!

'2 .t 1 ; t 2 ; J.s/ C .1 /; C .2 / 0

!

C .t/; C .1 /

0

Zt Zt

.n/

d 1 d 2 :

(8.96)

0

Here '0 , '1 and '2 are functions of the arguments indicated. Since the core '2 in .n/

.n/

this model is independent of C .t/, so @'2 [email protected] C .t/ 0, and constitutive equations prove to be coincident with (8.78); and hence they coincide with (8.85), (8.89) and (8.94) too. The distinction between the principal and quadratic models consists only in the forms of the functional and the dissipation function w . Such a situation is typical for viscoelastic continua when distinct functionals of the free energy correspond to the same relations between the tensors of stresses and deformations. Comparing the principal model (8.77) with the quadratic one (8.96), we can also notice that the principal model has only one core '1 appearing also in relation (8.78), and the quadratic model has two cores '1 and '2 , one of which is not included in relation (8.78) between stresses and deformations. Thus, for the principal model we can restore the functional of the free energy by Eqs. (8.78) up to the entropy term @'0 [email protected] and the constant '0 .0; 0 / D 0 . Models of viscoelastic continua having such a property are called mechanically determinate. The quadratic model is not mechanically determinate: due to the presence of the core '2 we cannot restore the form of by relations (8.78) between stresses and deformations. Nevertheless, this model is also used in practice due to its quadratic structure being typical for thermodynamic potentials.

8.2.6 Linear Models An of Viscoelastic Continua The quadratic model An of a thermoviscoelastic continuum with difference cores (8.96), where the functions '0 , '1 and '2 depend linearly upon the quadratic

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua .s/

523 .s/

invariants J and quadratically upon the linear invariants J , is called linear (cubic invariants do not occur in this model): '0 D 1 X

0C

r1 1 X

lˇ I.s/ .t/Iˇ.s/ .t/ C

ı

2

;ˇ D1

r1

'1 D

ı

;ˇ D1

.s/

qˇ .t /I.s/ .t/Iˇ ./ C

'2 D

r1 1 X ı

2 ;ˇ D1 1 Cı

2

1

r2 X

ı

X

l I.s/ .t/;

Dr1 C1

r2

ı

Dr1 C1

q .t /J.s/ .t; /; (8.97)

pˇ .t 1 ; t 2 /I.s/ .1 /Iˇ.s/ .2 /

r2 X

p .t 1 ; t 2 /J.s/ .1 ; 2 /:

Dr1 C1

Here lˇ , l are constants; qˇ .t /, q .t / are one-instant cores (they are functions of one argument, being symmetric in and ˇ); pˇ .t 1 ; t 2 / and p .t 1 ; t 2 / are two-instant cores (they are functions of two variables, being symmetric in and ˇ, and also in t 1 and t 2 ). We have introduced the notation ! ! .n/

I.s/ ./ D I.s/ C ./ ;

J.s/ .1 ; 2 / D J.s/

.n/

.n/

C .1 /; C .2 / ;

(8.98)

ı

.n/

where r1 is the number of linear invariants I.s/ . C .t// in group G s .r1 6 r/, and .s/ .r2 r1 / is the number of the quadratic simultaneous invariants J .1 ; 2 / in this group, where r2 6 z. Since not all simultaneous invariants contained in the full basis .n/

.n/

J . C .1 /; C .2 // appear in the expression for the functional in linear models, it is convenient to renumber these invariants in comparison with the bases (8.83), .n/

(8.87), and (8.91) by enumerating first the linear invariants I . C .t// and then the .n/

.n/

quadratic simultaneous invariants J . C .1 /; C .2 //. The principal linear model An of viscoelastic continua can be obtained by applying similar relationships to the functions '0 and '1 of the principal model (8.77) when '2 D 0. The functions '0 and '1 (8.79) for the principal linear model have the forms '0 D J

r1 X

lˇ Iˇ.s/ .t/;

ˇ D1

'0 D J l ;

'1 D J

r1 X

qˇ .t /Iˇ.s/ ./; D 1; : : : ; r1 ;

ˇ D1

'1 D J q .t /; D r1 C 1; : : : ; r2 ; ı

J D =:

(8.99)

524

8 Viscoelastic Continua at Large Deformations

As noted above, constitutive equations (8.77) for both the models coincide; and, for the linear models, they have the form r1 X

.n/

T DJ

;ˇ D1

.s/ lMˇ I.s/ Oˇ C J

r2 X Dr1 C1

.s/ lM I C ;

(8.100)

where we have denoted the linear functionals lMˇ I.s/ lˇ I.s/ .t/

Zt qˇ .t /I.s/ ./ d ;

; ˇ D 1; : : : ; r1 ;

0

.s/ .s/ lM I C l I C .t/

Zt

.s/

q .t /I C ./ d ; D r1 C 1; : : : ; r2 :

(8.101)

0

Here we have taken into account that all the linear invariants have the form I.s/ D .n/

.s/

.s/

C O , where O are producing tensors of the group, and for the quadratic invariants, J.s/ C ./ D

@J .t; / .n/

.s/

D

@ C .t/

1 @I .; / 1 ./; D r1 C 1; : : : ; r2 : (8.102) D I.s/ 2 .n/ 2 C @ C ./ .I /

(For isotropic continua, in order to satisfy this condition, as invariants I one should choose the invariants I1 .C / and I1 .C2 /.) Notice that when the cores qˇ .t/ and q .t/ in (8.101) are absent, then these relations exactly coincide with relations (4.333) of linear models An for ideal continua if in the last ones we assume that m N D 0. For principal linear models An of viscoelastic continua, the dissipation function w (8.82) has the form

w D J

r1 X ;ˇ D1

C 2J

0 @qˇ .0/I.s/ .t/I .s/ .t/ ˇ

Zt C

1 @ .s/ qˇ .t /I.s/ .t/Iˇ ./d A @t

0 r2 X

Dr1 C1

0 @q .0/J.s/ .t; t/ C

Zt

1 @ q .t /J.s/ .t; / d A ; @t

0

(8.103a)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

525

and for linear models An of viscoelastic continua, terms with two-instant cores should be added to the expression (8.103a): 0 Zt r1 X @ .s/ .s/ @ w DJ qˇ .t /I.s/ .t/Iˇ.s/ ./ d qˇ .0/I .t/Iˇ .t/ C @t ;ˇ D1 0 1 Z t Zt 1 @ pˇ .t 1 ; t 2 /I.s/ .1 /Iˇ.s/ .2 / d 1 d 2 A 2 @t 0 0 0 Zt r2 X @ .s/ @2q .0/J .t; t/ C 2 CJ q .t /J.s/ .t; / d @t Dr1 C1 0 1 Zt Zt @ p .t 1 ; t 2 /J .1 ; 2 / d 1 d 2 A : (8.103b) @t 0

0

For linear models An , the specific entropy , according to (8.69) and (8.82), has the form .n/ @ 0 1 D (8.104) C ˛ T; @ where 0 ./ is a function only of the temperature. The dissipation function with the help of formula (8.75) can be represented in the form .n/ .n/ d .n/ d d @ 0 w D T C C ˛ T : (8.104a) dt dt @ dt

8.2.7 Representation of Linear Models An in the Boltzmann Form For linear models An of viscoelastic continua, representations of the free energy in the form (8.96), (8.97), the constitutive equations in the form (8.100), (8.101) and the dissipation function w in the form (8.103b) are called the model An in the Volterra form (the name is connected to the fact that integral expressions of the form (8.101) occurring in this representation were considered for the first time by Volterra in 1909). Let us give another equivalent representation for these models. Introduce new two-instant cores N ˇ .y; z/ and N .y; z/ satisfying the differential equations @2 N ˇ .y; z/ @ N ˇ .y; 0/ D pˇ .y; z/; D qˇ .y/; @[email protected] @y @2 N .y; z/ @ N .y; 0/ D p .y; z/; D q .y/; @[email protected] @y y D t 1 ; z D t 2 : Then the following theorem holds.

N ˇ .0; 0/ D lˇ ; N .0; 0/ D l ; (8.105)

526

8 Viscoelastic Continua at Large Deformations

Theorem 8.8. Let the two-instant cores N ˇ .y; z/ and N .y; z/ 1. Be symmetric functions of their arguments: N ˇ .y; z/ D N ˇ .z; y/;

N .y; z/ D N .z; y/;

(8.105a)

2. Be two times continuously differentiable functions of their arguments within the interval .0; t/, 3. Satisfy the conditions (8.105), then we can pass from the representation of linear model An in the Volterra form (Eqs. (8.96), (8.97), (8.100), (8.101), and (8.103b)) to an equivalent representation of the model An in the Boltzmann form r1 Z t Z t 1 X N ˇ .t 1 ; t 2 / dI.s/ .1 / dI .s/ .2 / 0C ı ˇ 2 ;ˇ D1 0 0

D

C

Zt Zt r2 X

1 ı

Dr1 C1 0 r1 X

.n/

T DJ

.s/ Oˇ

;ˇ D1

N .t 1 ; t 2 / dJ.s/ .1 ; 2 /;

(8.106)

0

Zt rˇ .t

1 / dI.s/ ./

Zt r2 X

CJ

.s/

r .t / dI C ./;

Dr1 C1 0

0

(8.107)

r1 Z t Z t @ N 1 X .s/ .s/ w D J ˇ .t 1 ; t 2 / dI .1 / dIˇ .2 / 2 @t

;ˇ D1 0

J

r2 X

Zt

Dr1 C1 0

0

Zt

@ N .s/ .t 1 ; t 2 / dJ .1 ; 2 /: @t

(8.108)

0

Here we have introduced the notation rˇ .y/ D N ˇ .y; 0/; dI.s/ .1 / D IP1.s/ .1 / d 1 ;

r .y/ D N .y; 0/;

dJ.s/ .1 ; 2 / D J.s/

.n/

.n/

C .1 /; C .2 /

(8.109a) ! d 1 d 2 : (8.109b)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

527

H To prove the theorem, it is sufficient to transform the integrals in (8.106)–(8.108) as follows: 1 0 Zt Zt Zt N ˇ .0; t 2 /I.s/ .t/ @ N ˇ .t 1 ; t 2 / dI.s/ .1 /A dI .s/ .2 / D ˇ 0

0

Zt 0

Zt 0

0

@ N ˇ .t 1 ; t 2 /I.s/ .1 / d 1 dIˇ.s/ .2 / D I.s/ .t/ N ˇ .0; 0/Iˇ.s/ .t/ @1 @ N .s/ ˇ .0; t 2 /Iˇ .2 / d 2 @2

Zt Zt C 0

D

0

Zt 0

@ N ˇ .t 1 ; 0/I.s/ .1 /d 1 Iˇ.s/ .t/ @1

@2 N ˇ .t 1 ; t 2 /I.s/ .1 /Iˇ.s/ .2 / d 1 d 2 @1 @2

.s/ lˇ I.s/ .t/Iˇ .t/

Zt

.s/

.s/

qˇ .t /.I.s/ .t/Iˇ ./ C I.s/ ./Iˇ .t// d 0

Zt

Zt

.s/

pˇ .t 1 ; t 2 /I.s/ .1 /Iˇ .2 / d 1 d 2 :

C 0

(8.110)

0

Here we have taken into account that I.s/ .0/ D 0 and changed the variables @ @ N N ˇ .0; t 2 / D @ N ˇ .0; y/ D qˇ .y/: ˇ .0; t 2 / D @2 @.t 2 / @y In a similar way, we can transform the integrals of N with taking into account that .n/

.n/

J.s/ . C .1 /; C .2 // is a linear function with respect to each tensor argument; therefore, the following relations hold: dJ.s/ .1 ; 2 / D

.n/ @ .s/ .n/ J . C .1 /; C .2 // d 1 d 2 @1

D

.n/ @ .s/ .n/ J . C .1 /; C .2 // d 1 d 2 @2

D

.n/ .n/ @2 J.s/ . C .1 /; C .2 // d 1 d 2 ; @1 @2

528

8 Viscoelastic Continua at Large Deformations

hence Zt Zt 0

N .t 1 ; t 2 / dJ.s/ .1 ; 2 /

0

D

.n/ .n/ l J.s/ . C .t/; C .t//

Zt 2

.n/

.n/

q .t /J.s/ . C .t/; C .// d 0

Zt Zt

.n/

.n/

p .t /J.s/ . C .1 /; C .2 // d 1 d 2 :

C 0

(8.111)

0

On substituting (8.110) and (8.111) into (8.106), we actually obtain the expressions (8.96) and (8.97) for . The representations (8.107) and (8.108) can be proved in a similar way (see Exercise 8.2.1). N Remark. If we consider constitutive equations in the Volterra form (8.96), (8.97), and (8.100) and pass to the limit at t ! 0, then all the integral summands containing the cores qˇ and q vanish. As a result, we get instantly elastic relations which exactly coincide with the corresponding equations (4.331), (4.333), and (4.334) of models An of elastic continua: .0/ D .n/

0 C

r1 1 X ı

2 ;ˇ D1

T .0/ D J

X r1

;ˇ D1

lˇ I.s/ .0/Iˇ.s/ .0/ C

lˇ I.s/ .0/O.s/ ˇ

r2 X

1 ı

Dr1 C1

X r2

CJ

l I.s/ .0/;

Dr1 C1

(8.112)

l I.s/ C .0/:

In order to obtain these relations from the Boltzmann form (8.106), (8.107), one .n/

.n/

should represent the deformation tensors in the form C ./ D C .0/h./, where h./ is the Heaviside function. Then we find that I.s/ ./ D I.s/ .0/h./ and J.s/ .1 ; 2 / D I.s/ .0/h.1 /h.2 /. Substituting these expressions into (8.106) and (8.107) and using (8.105), we obtain the desired relations (8.112) as t ! 0C . t u

8.2.8 Mechanically Determinate Linear Models An of Viscoelastic Continua As noted in Sect. 8.2.5, quadratic models An , including the linear models (8.97), are not mechanically determinate due to the presence of the two-instant cores pˇ .t 1 ; t 2 / and p .t 1 ; t 2 /. However, the models may become

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

529

mechanically determinate after introduction of the additional assumption on a form of the two-instant cores; we assume that they depend on the sum of their arguments: N ˇ .y; z/ D N ˇ .y C z/;

N .y; z/ D N .y C z/:

(8.113)

In this case, the cores N ˇ and N become one-instant, and with the help of formula (8.109a) they can be uniquely expressed in terms of the cores rˇ .y/ and r .y/ included in constitutive equations (8.107): N ˇ .y/ D rˇ .y/;

N .y/ D r .y/:

(8.114)

The functional (8.106) of the free energy takes the form D

0

C

C

r1 Z t Z t 1 X ı

2 ;ˇ D1 0 1

Zt Zt r2 X

ı

Dr1 C1 0

rˇ .2t 1 2 / dI.s/ .1 / dIˇ.s/ .2 /

0

r .2t 1 2 / dJ.s/ .1 ; 2 /:

(8.115)

0

The dissipation function w (8.108) in this model is also determined completely by the functional (8.107) of the constitutive equations: r1 Z t Z t @ J X w D rˇ .2t 1 2 / dI.s/ .1 / dIˇ.s/ .2 / 2 @t

;ˇ D1 0

J

0

Zt Zt r2 X @ r .2t 1 2 / dJ.s/ .1 ; 2 /: @t

Dr1 C1 0

(8.116)

0

The cores rˇ .y/ and r .y/, according to (8.114) and (8.105), are connected to the cores qˇ .y/ and q .y/ by the relations @rˇ .y/ D qˇ .y/; @y

@r .y/ D q .y/; rˇ .0/ D lˇ ; r .0/ D l : @y (8.117)

The cores qˇ .y/ and q .y/ are called the relaxation cores, and the cores rˇ .y/ and r .y/ are called the relaxation functions.

530

8 Viscoelastic Continua at Large Deformations

8.2.9 Linear Models An for Isotropic Viscoelastic Continua Let us derive now constitutive equations for linear models An of viscoelastic continua in the Volterra (8.100) and Boltzmann (8.107) forms for different symı

metry groups G s . For linear models An of viscoelastic isotropic continua, the invariants (8.98) and the derivative tensors I.IC/ (8.102) have the forms r1 D 1;

r D 3; .n/

.I /

.I /

r2 D 2; .n/

.I /

.n/

I1 ./ D I1 . C .//; I2 ./ D J2 .1 ; 2 / D C .1 / C .2 /;

(8.118)

.n/

/ .I / .I / O.I 1 D I1C .t/ D E; I2C .t/ D 2 C .t/:

Then constitutive equations (8.100), (8.106), and (8.107) become ı

1 D 0C 2 ı

Zt Zt

.n/

0

0

Zt Zt C

.n/

r1 .2t 1 2 /dI1 . C .1 // dI1 . C .2 // .n/

.n/

r2 .2t 1 2 / d C .1 / d C .2 /; 0

(8.119)

0 .n/

.n/

T D J.lM1 I1 E C 2lM2 C /:

(8.120)

Here we have denoted the linear functionals .n/

lM1 I1 l1 I1 . C .t//

Zt

Zt

.n/

q1 .t /I1 . C .// d D 0

.n/

.n/

lM2 C D l2 C .t/

.n/

r1 .t / dI1 . C .//; 0

Zt

.n/

Zt

q2 .t / C ./ d D 0

.n/

r2 .t / d C ./: 0

(8.121) Thus, for an isotropic continuum, there are two independent constants l1 ; l2 and two cores q .t / connected to the cores r .y/ by the relations (8.117) @r .y/ D q .y/; @y

r .0/ D l ; D 1; 2:

(8.122) ı

Introducing the fourth-order tensor functional similar to the tensor 4 M (4.337) for elastic continua: 4M R D E ˝ ElM1 C 2lM2 ; (8.123)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

531

we can represent constitutive equations (8.120) in the operator form .n/

.n/

M C; T D J 4R

(8.124)

which is analogous to relations (4.338) for semilinear isotropic elastic continua.

8.2.10 Linear Models An of Transversely Isotropic Viscoelastic Continua For linear models An of viscoelastic transversely isotropic continua, from (8.87) and (8.88) we obtain r D 5;

r1 D 2; .n/

.3/

c23 / C ./; I1 ./ D .E b

r2 r1 D 2; .n/

.3/

I2 ./ D b c23 C ./;

.n/

.3/

.n/

c23 / C .1 // .b c23 C .2 //; J3 .1 ; 2 / D ..E b I˛.3/ ./ D J˛.3/ .; /; ˛ D 3; 4; .n/

.n/

J4.3/ .1 ; 2 / D C .1 / C .2 / 2J3.3/ .1 ; 2 / I2.3/ .1 /I2.3/ .2 /; .3/ ./ D I3C

.n/ 1 .O1 ˝ O1 C O2 ˝ O2 / C ./; 2

.n/

.3/ I4C ./ D 2 4 O3 C ./; (8.125)

where 4 O3 is determined by formula (8.88). Relations (8.100) take the form

.n/

T DJ

.3/ .3/ .3/ .3/ c23 / C lM22 2lM44 I2 C lM12 I1 b c23 lM11 I1 C lM12 I2 .E b ! M .n/ l33 M .n/ C.O1 ˝ O1 C O2 ˝ O2 / l44 C C 2lM44 C : (8.126) 2 .n/

Here the linear operators lMˇ Iˇ.3/ and lM C are determined by expressions (8.101), which can be represented in the Boltzmann form (8.107) .3/ lMˇ Iˇ D

Zt rˇ .t / 0

.3/ dIˇ ./;

.3/ lM J C D

Zt

.3/

r .t / dJ C ./: (8.127) 0

For a transversely isotropic continuum, there are five independent constants l11 , l22 , l12 , l33 , l44 and five cores qˇ .t / or rˇ .t /.

532

8 Viscoelastic Continua at Large Deformations

Introducing the tensor functional being analogous to the tensor of elastic moduli (4.341): 4M c23 ˝b c23e c23 Cb c23 ˝ E R D E ˝ ElM11 Cb lM22 C .lM12 lM11 / E ˝b M l33 M C .O1 ˝ O1 C O2 ˝ O2 / l44 C 2lM44 ; 2 e (8.128) lM D lM 2lM 2lM C lM ; 22

22

44

12

11

we can also represent constitutive equations (8.126) in the form (8.124).

8.2.11 Linear Models An of Orthotropic Viscoelastic Continua For linear models An of viscoelastic orthotropic continua, due to (8.91)–(8.93), the invariants (8.98) and the derivative tensors (8.102) take the forms r D 6;

r1 D 3;

r2 D 6;

.n/

c2˛ C ./; ˛ D 1; 2; 3; I˛.O/ ./ D b .n/ .n/ .O/ 2 2 c2 C .1 / b c3 C .2 / ; J4 .1 ; 2 / D b .O/

J5

.n/ .n/ .1 ; 2 / D b c21 C .1 / b c23 C .2 / ;

.n/ .n/ J6.O/ .1 ; 2 / D b c21 C .1 / b c22 C .2 / ; .n/ 1 .O1 ˝ O1 / C ./; 2 .n/ 1 .O/ I6C ./ D .O3 ˝ O3 / C ./; 2 (8.129) .O/ I4C ./ D

I˛.O/ ./ D J˛.O/ .; /; ˛ D 4; 5; 6I .n/

.O/ I5C ./ D 2.O2 ˝ O2 / C ./;

and the relations (8.100) can be written as follows: 3 X

.n/

T DJ

;ˇ D1

c2 C J lMˇ Iˇ.O/b

3 X

.n/

O .O lM3C;3C C /:

(8.130)

D1

Thus, there are nine independent constants l11 , l22 , l33 , l12 , l13 , l23 , l44 , l55 , l66 and nine cores qˇ .t / or rˇ .t /. Introducing the tensor functional being analogous to (4.344): 4

M D R

3 X ;ˇ D1

b c2 ˝b c2ˇ lMˇ C

3 X

O ˝ O lM3C;3C ;

(8.131)

D1

we can represent constitutive equations (8.130) in the operator form (8.124).

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

533

8.2.12 The Tensor of Relaxation Functions According to the operator form (8.124) of constitutive equations for linear models An of viscoelastic continua, we can introduce the fourth-order tensor 4 R.t/, called the tensor of relaxation functions, by the same formulae as for the elastic modı

uli tensor 4 M in linear models An of elastic continua (see Sect. 4.8.7) if in the corresponding formulae the elastic constants l˛ˇ are replaced by the relaxation functions r˛ˇ .t/. For an isotropic continuum, this tensor has the form 4

R.t/ D r1 .t/E ˝ E C 2r2 .t/I

(8.132)

for a transversely isotropic continuum 4

R.t/ D r11 .t/E ˝ E Ce r 22 .t/b c2 ˝b c23 C .r12 .t/ r11 .t//.E ˝b c23 Cb c23 ˝ E/ 3 1 r33 .t/ r44 .t/ .O1 ˝ O1 C O2 ˝ O2 / C 2r44 .t/; C 2 (8.133) e r 22 .t/ D r11 .t/ C r22 .t/ 2r12 .t/ 2r44 .t/;

and for an orthotropic continuum 4

R.t/ D

3 X

r˛ˇ .t/b c2 ˝b c2ˇ C

˛;ˇ D1

3 X

r3C˛;3C˛ .t/O ˝ O :

(8.134)

˛D1

Then the operator relations (8.124) can be represented as follows: .n/

.n/

M C; T D J 4R

where 4

.n/

M C D R

Zt 4

(8.135)

.n/

R.t / d C ./

(8.136)

0

is a tensor linear functional. For instantaneous loading as t ! 0C , these relations coincide with (8.112) and with the corresponding relations (4.330a) of models An for a linear-elastic continuum, because 4

ı

R.0/ D 4 M:

(8.137)

The tensor 4

K.t/ D

d 4R .t/ dt

(8.138)

534

8 Viscoelastic Continua at Large Deformations ı

is called the tensor of relaxation cores. This tensor for different groups G s has the same form as the tensor 4 R.t/ in (8.132)–(8.134) if in these formulae the substitution r˛ˇ .t/ ! q˛ˇ .t/ has been made. According to (8.137) and (8.138), the constitutive equations (8.135) can be written in the Volterra form .n/

4

ı

.n/

Zt

T D J. M C

.n/

K.t / C ./ d /;

(8.139)

0

that is equivalent to the form (8.100). The operator (8.115) of the free energy for the mechanically determinate model An with the help of the tensor of relaxation functions can be represented in the form (see Exercise 8.2.3) ı

1 D 0C 2 ı

Zt Zt

.n/

.n/

d C .1 / 4 R.2t 1 2 / d C .2 /; 0

(8.140)

0

and the dissipation function (8.116) – in the form J w D 2

Zt Zt

.n/

d C .1 / 0

.n/ d4 R.2t 1 2 / d C .2 /: dt

(8.141)

0

Formula (8.141) gives the following theorem. Theorem 8.9. For mechanically determinate linear models An of viscoelastic continua, the tensors of relaxation cores 4 K.t/ are 1. Nonnegative-definite: h 4 K.t/ h > 0;

8h ¤ 0; 8t > 0;

(8.142)

2. Symmetric in the following combinations of indices: 4

K.t/ D 4 K.1243/ .t/ D 4 K.2134/ .t/ D 4 K.3412/ .t/

8t > 0;

(8.143) ı

(i.e. these tensors have the same symmetry as the elastic moduli tensor 4 M for linear models An of elastic continua). H The dissipation function is always nonnegative (w > 0) and vanishes for vis.n/

coelastic continua only if C ./ 0. Then, choosing the process of deforming in the form of a step-function: .n/

C ./ D h h./;

> 0;

(8.144)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

535

where h./ is the Heaviside function, and h is a symmetric non-zero constant tensor, we obtain .n/

d C ./ D h ı./ d :

(8.144a)

Substituting (8.144a) into (8.141) and using the property (8.16) of the ı-function and formula (8.138), we find that J w D 2

Zt Z t h 4 K.2t 1 2 / hı.1 /ı.2 / d 1 d 2 0

0

J h 4 K.2t/ h > 0; 8t > 0; 2

D

(8.145)

i.e. the tensor 4 K.t/ is nonnegative-definite. The existence of the quadratic form (8.145) and the symmetry of the tensor h lead to symmetry of 4 K.t/ in the first–second and third–fourth indices and also in pairs of the indices, i.e. the relations (8.143) actually hold. N As follows from (8.142) and (8.138), the tensor of relaxation functions 4 R.t/ generates the monotonically non-increasing form h

d 4R .t/ h 6 0; dt

8h ¤ 0; 8t > 0:

(8.146)

ı

And if the elastic moduli tensor M D 4 R.0/ has the symmetry (8.143), then from (8.143) it follows that the tensor 4 R.t/ has the same symmetry 8t > 0: 4

R.t/ D 4 R.1243/ .t/ D 4 R.2134/ .t/ D 4 R.3412/ .t/

8t > 0:

(8.147)

8.2.13 Spectral Representation of Linear Models An of Viscoelastic Continua Let us apply now the theory of spectral decompositions of symmetric second-order tensors (see [12]). According to this theory, for any symmetric tensors, in particular .n/

.n/

for C ./ and T ./, we can introduce their spectral decompositions relative to a ı

symmetry group G s chosen: .n/

T D

nN X ˛D1

P.T/ ˛ ;

.n/

C D

nN X ˛D1

P˛.C/ ; 1 < nN 6 6:

(8.148)

536

8 Viscoelastic Continua at Large Deformations .n/

.n/

.C/ Here P.T/ N are the orthoprojectors of the tensors T and C ˛ and P˛ (˛ D 1; : : : ; n) (their number is denoted by n), N which are symmetric second-order tensors having the following properties: a) they are mutually orthogonal, b) they are linear, c) they ı

are indifferent relative to the group G s : .T/ P.T/ ˛ Pˇ D 0; if ˛ ¤ ˇI .n/

.n/

4 P.T/ ˛ D P˛ . T / D ˛ T ; .n/

.n/

(8.149)

˛ D 1; : : : ; nI N

(8.150)

ı

QT P˛ . T / Q D P˛ .QT T Q/; 8Q 2 G s :

(8.151) ı

Here the fourth-order tensors 4 ˛ are indifferent relative to the group G s , inde.n/

pendent of T and formed only by producing tensors of the group (see Sect. 4.8.3). Among the tensors 4 ˛ .˛ D 1; : : : ; n/, N there are m reducible tensors, i.e. obtained with the help of the tensor product of the second-order tensor a˛ being symmetric ı

and indifferent relative to the same group G s : 4

˛ D

1 a˛ ˝ a˛ ; a˛2 D a˛ a˛ ; ˛ D 1; : : : ; m < n: N a˛2

(8.152)

Expressions for 4 ˛ and a˛ have the following forms (see [12]): ı

for the isotropy group G s D I a1 D E;

4

1 .2/ D E ˝ E; m D 1; nN D 2I 3

(8.153)

ı

for the transverse isotropy group G s D T3 a1 D b c23 ; a2 D E b c23 ; m D 2; nN D 4I 1 1 4 3 D E b c23 ˝ E b c23 b c23 ˝ cN 23 .O1 ˝ O1 C O2 ˝ O2 /; 2 2 1 4 4 D .O1 ˝ O1 C O2 ˝ O2 /I (8.154) 2 ı

for the orthotropy group G s D O c2˛ ; ˛ D 1; 2; 3I a˛ D b

4

˛C3 D

1 O˛ ˝ O˛ ; m D 3; nN D 6: 2

(8.155)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

537

.T/

With the help of the orthoprojectors P˛ , introduce spectral invariants of the .n/

.n/

4 tensor T and denote them by Y˛ . T /. For those P.T/ ˛ , for which ˛ is a reducible tensor, the invariant Y˛ is introduced as follows: .n/

Y˛ . T / D

.n/ 1 a.˛/ T ; a˛

˛ D 1; : : : ; m;

(8.156a)

N and called the spectral linear invariant. For the remaining P.T/ ˛ (˛ D m C 1; : : : ; n), these invariants are introduced by the formula 1=2 .n/ .T/ P Y˛ . T / D P.T/ ˛ ˛

(8.156b)

and called the spectral quadratic invariants. From (8.150) and (8.152) it follows that P.T/ ˛ D

1 Y˛ a.˛/ ; ˛ D 1; : : : ; m: a˛

(8.157)

Notice that for the linear invariants (8.156a), formula (8.156b) also holds. Due to (8.157), the spectral decomposition of the symmetric second-order tensor .n/

T (8.148) can be represented in the form .n/

T D

m n X X .n/ a˛ Y˛ . T / C P.T/ ˛ : a ˛ ˛D1 ˛DmC1

(8.158) ı

For any fourth-order tensor indifferent relative to a group G s , including the tensor of relaxation functions 4 R.t/, we can also introduce the spectral representation 4

R.t/ D

m X

R˛ˇ .t/

˛;ˇ D1

nN X a˛ ˝ a ˇ C R˛˛ .t/4 ˛ ; a˛ aˇ ˛DmC1

(8.159)

where R˛ˇ .t/ and R˛˛ .t/ are the spectral relaxation functions expressed uniquely in terms of r˛ˇ .t/ and r˛˛ .t/ (see Exercise 8.2.6). With the help of the spectral decompositions (8.148) and (8.159) the constitutive equations (8.135) can be represented as relations between the spectral linear invariants and the orthoprojectors (see Exercise 8.2.9) .n/

Y˛ . T / D J

m X

.n/

RM ˛ˇ Yˇ . C /; ˛ D 1; : : : ; mI

ˇ D1

P.T/ ˛

D J RM ˛˛ P˛.C/ ; ˛ D m C 1; : : : ; n; N

(8.160)

538

8 Viscoelastic Continua at Large Deformations

where / RM ˛ˇ P.C D ˇ

Zt

/ R˛ˇ .t / d P.C ./: ˇ

(8.161)

0

Relations (8.160) are called the spectral representation for linear models An of viscoelastic continua. If we introduce the spectral decomposition (8.148) also for the tensor h: hD

nN X

P.h/ ˛ ;

˛D1

then the inequality (8.146) with use of (8.159) takes the form m nN X X d d R˛ˇ .t/Y˛ .h/Yˇ .h/ C R˛˛ .t/Y˛2 .h/ 6 0: dt dt ˛DmC1

(8.162)

˛;ˇ D1

Values of the spectral linear invariants Y˛ .h/ can be assumed to be zero; then, since the spectral invariants are independent, from (8.162) we obtain the condition of monotone non-increasing the spectral relaxation functions: dR˛˛ .t/ 6 0; dt

˛ D 1; : : : ; n: N

(8.163)

With the help of the spectral relaxation functions one can formulate special cases of linear models An for viscoelastic continua. So for the simplest linear model An of an isotropic viscoelastic continuum, one of the two spectral relaxation cores is assumed to be constant: 2 R11 .t/ D R11 .0/ D l1 C l2 D const; 3

@R11 D 0: @t

(8.164)

According to the results of Exercise 8.2.6 and formula (8.122), this condition can be rewritten as the relation between the cores q1 .t/ and q2 .t/ 2 q1 .t/ D q2 .t/; 3

@r˛ D q˛ .t/: @t

(8.165)

Constitutive equations (8.160) in this case take the form 8 .n/ .n/ ˆ < I1 . T / D JR11 .0/I1 . C /; .n/ .n/ Rt ˆ : dev T D J R22 .t / dev @ C ./ d : @ 0

(8.166)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

539

Here we have denoted the orthoprojectors of the tensors relative to the full orthogonal group I , called the deviators .n/ .n/ 1 .n/ dev C D C I1 . C /E; 3

.n/ .n/ 1 .n/ dev T D T I1 . T /E: 3

(8.167)

Equations (8.166) and (8.167) can be rewritten in the form .n/ J T D R11 .0/I1 . C /E C J 3

.n/

Zt

.n/

@ C ./ R22 .t /dev d : @

(8.168)

0

8.2.14 Exponential Relaxation Functions and Differential Form of Constitutive Equations To solve problems analytically or numerically it is convenient to have an analytical form of the spectral relaxation functions R˛ˇ .t/. As established in the preceding section, the property of monotone non-increasing the spectral functions R˛˛ .t/ is a consequence of the dissipation inequality w > 0. Functions having such a property can be approximated by the sum of exponents: 1 C R˛ˇ .t/ D R˛ˇ

t . / B˛ˇ exp . / ; ˛ˇ D1 N X

(8.169)

. / . / where B˛ˇ and ˛ˇ are the constants called the spectrum of relaxation values and 1 the spectrum of relaxation times, respectively, and R˛ˇ is the limiting value of the relaxation functions: 1 lim R˛ˇ .t/ D R˛ˇ ;

t !1

(8.170)

1 D 0. which may be zero: R˛ˇ

. / 1 The constants R˛ˇ and B˛ˇ satisfy the normalization condition at t D 0:

1 C R˛ˇ ı

N X D1

. /

ı

B˛ˇ D C ˛ˇ ;

(8.171)

where C ˛ˇ D R˛ˇ .0/ are the spectral (two-index) elastic moduli under instantaneous loading. There are other methods of analytical approximation to the relaxation functions, however exponential functions have certain merits: (1) choosing a sufficiently large number N of exponents in (8.169), we can approximate practically any function R˛ˇ .t/, (2) constitutive equations (8.160) and (8.161) with exponential cores admit

540

8 Viscoelastic Continua at Large Deformations

their inversion (see Sect. 8.2.15), where cores of the inverse functionals prove to be exponential as well, and (3) the cores (8.169) allow us to represent constitutive equations (8.160), (8.161) or (8.135), (8.136) in the differential form. Indeed, performing the subsequent substitutions (8.169) !(8.159)!(8.138), we find the expression for the tensor of relaxation cores: 4

m X

K.t/ D

K˛ˇ .t/

˛;ˇ D1

K˛ˇ .t/ D

nN X a ˛ ˝ aˇ C K˛˛ .t/4 ˛ ; a˛ aˇ ˛DmC1

. / N X B˛ˇ @R˛ˇ .t/ t exp : D . / . / @t D1 ˛ˇ

(8.172)

(8.173)

˛ˇ

Introduce the second-order tensors . / W˛ˇ

Zt D 0

.n/ t C ./d exp . / ; D 1; : : : ; N: . / ˛ˇ ˛ˇ

(8.174)

/ with respect to t and eliminating the integral, we obtain that Differentiating W. ˛ˇ . /

the tensors W˛ˇ satisfy the first-order differential equations / d W. ˛ˇ

dt

C

/ W. ˛ˇ . / ˛ˇ

.n/

D

C .t/ . / ˛ˇ

; D 1; : : : ; N:

(8.175)

Substituting (8.172) into (8.139) and using the expressions (8.173) and (8.174), we obtain the following representation of constitutive equations: 0 ı

.n/

.n/

T D J @4 M C

N X

1 W. / A :

(8.176)

D1

Here the spectrum of viscous stresses is denoted by W. / being second-order tensors of the form W. / D

m X ˛;ˇ D1

. /

. /

B˛ˇ W˛ˇ

nN X a˛ ˝ aˇ . / . / C B˛˛ W˛˛ 4 ˛ : a˛ aˇ

(8.177)

˛DmC1

Thus, with the help of the exponential cores (8.169) the constitutive equations for the mechanically determinate model An of viscoelastic continua (8.139) can be represented in the differential form (8.175)–(8.177). A result of the passage from integral relations to differential ones is the appearance of additional unknowns, / namely the tensors W. ˛ˇ , for which Eqs. (8.175) have been stated.

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

541

In computations the differential form (8.175)–(8.177), as a rule, proves to be more convenient than the integral form (8.139). Substitution of the expressions (8.172) and (8.138) into (8.141) yields m Z t Zt J X .C / w D K˛ˇ .2t 1 2 / d Y˛.C / .1 / d Yˇ .2 / C 2

˛;ˇ D1 0

J C 2

nN X

0

Zt Zt

.n/

.n/

K˛˛ .2t 1 2 / d C .1 / 4 ˛ d C .2 /:

˛DmC1 0

0

(8.178) .n/

Here we have denoted the linear spectral invariants of the tensor C ./ (see (8.156a)): .n/

Y˛.C / ./ D Y˛ . C .// D

.n/ 1 a˛ C ./; a˛

˛ D 1; : : : ; m:

(8.179)

Substituting the exponential cores (8.173) into (8.178) and modifying the double integrals Zt Zt 0

0

2t 1 2 exp d Y˛.C / .1 / d Yˇ.C / .2 / ˛ˇ Zt

D 0

Zt t 1 t 2 .C / exp d Y˛ .1 / exp d Yˇ.C / .2 / ˛ˇ ˛ˇ 0

D Y˛.C / .t/

Yˇ.C / .t/

Zt

1 ˛ˇ

0

1 ˛ˇ

! t 1 .C / d Y˛ .1 / d 1 exp ˛ˇ

Zt 0

! t 2 .C / exp d Yˇ .2 / d 2 ; ˛ˇ

(8.180)

with use of the notation (8.174) we can represent (8.178) in the form w D

N J X 2 D1

. / / / m X d W. d W. B˛ˇ ˛ˇ ˛ˇ aˇ a˛ a˛ aˇ dt dt ˛;ˇ D1 ! nN . / / X d W. ˛˛ . / d W˛˛ 4 : C B˛˛ dt dt ˛DmC1

(8.181)

542

8 Viscoelastic Continua at Large Deformations

8.2.15 Inversion of Constitutive Equations for Linear Models of Viscoelastic Continua In viscoelasticity theory one often uses constitutive equations inverse to (8.135) or (8.139). To derive the equations we should consider relationship (8.139) as a lin.n/

ear integral Volterra’s equation of the second kind relative to the process C ./, 0 6 6 t. The core K.t/ of this equation is assumed to be continuously differentiable and to satisfy the conditions (8.142) and (8.143). As known from the theory of integral equations, Eq. (8.139) with such core always has a solution, and this solution is written in the same form as the initial equation: .n/

.n/

T C D … C J

Zt

4

.n/ 4

N.t /

T ./ d : J

(8.182)

0

Here 4 … is the tensor of elastic pliabilities, which is inverse of the tensor of elastic ı

moduli 4 M: 4

ı

… 4 M D ;

(8.183)

and 4 N.t/ is the tensor of creep cores having the same form as the tensor 4 K.t/ (8.172): m nN X X a˛ ˝ a ˇ 4 N.t/ D N˛ˇ .t/ C N˛˛ 4 ˛ : (8.184) a˛ aˇ ˛DmC1 ˛;ˇ D1

The functions N˛ˇ .t/ and N˛˛ .t/ are called the spectral creep cores. To find a relation between the cores 4 N.t/ and 4 K.t/, we should substitute (8.139) into (8.182); as a result, we obtain the identity .n/

Zt

.n/

C .t/ D C .t/ C … 4

4

.n/

K.t / C ./ d

0

Zt

4

ı

.n/

N.t / 4 M C ./ d

0

Zt

Zy 4

0

N.t y/

4

.n/

K.y / C ./ d dy:

0

Changing the integration order in the double integral: .0 6 6 y/ .0 6 y 6 t/ ! . 6 y 6 t/ .0 6 6 t/

(8.185)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

543

Fig. 8.3 The integration domain in the double integral

(see Fig. 8.3, where the integration domain is a shaded triangle), from (8.185) we obtain Zt 4

ı

… 4 K.t / 4 N.t / 4 M

0

Zt

4

N.t y/ 4 K.y /dy

9 = ;

.n/

C ./d D 0:

(8.186)

.n/

This equation holds for any C ./ if and only if the expression in braces vanishes. The substitution of variables x D t in braces gives 4

4

4

4

Zt

ı

4

… K.x/ D N.x/ M C

.t y/ 4 K.y C x t/ dy;

0 6 x 6 t:

t x

Then the substitution of variables u D y C x t under the integral sign, where .t x 6 y 6 t/ and .0 6 u 6 x/, yields 4

4

4

4

Zx

ı

4

… K.x/ D N.x/ M C

N.x u/ 4 K.u/ d u; 0 6 u 6 x:

0

Reverting to the initial notation of arguments x ! t and u ! , we obtain the integral relation between the tensor of relaxation cores 4 K.t/ and the tensor of creep cores 4 N.t/: 4

4

4

4

ı

Zt 4

… K.t/ D N.t/ M C

N.t / 4 K./ d :

(8.187)

0

If the core 4 K.t/ is known, then the relation (8.187) is a linear integral Volterra’s equation of the second kind for evaluation of the core 4 N.t/, and vice versa. On substituting the spectral decompositions (8.172) and (8.184) of the tensors ı

K.t/, 4 N.t/ and analogous decompositions of the tensors 4 … and 4 M into (8.187), due to mutual orthogonality of the tensors 4 ˛ , a˛ ˝ aˇ (see [12]), we obtain 4

544 m X

8 Viscoelastic Continua at Large Deformations

0 @…˛ˇ Kˇ" .t/ N˛ˇ .t/Cˇ"

ˇ D1

Zt

1 N˛ˇ .t /Kˇ" ./ d A D 0; ˛; " D 1; : : : ; m;

0

(8.188a) Zt …˛˛ K˛˛ .t/ N˛˛ .t/C˛˛

N˛˛ .t /K˛˛ .t/ d D 0; ˛ D m C 1; : : : ; n; N 0

(8.188b) – the system of scalar integral equations for determining the cores N˛ˇ .t/ in terms of the cores K˛ˇ .t/ or vice versa. By analogy with the tensor of relaxation functions 4 R.t/, introduce the tensor of creep functions 4 ….t/ satisfying the equation d4 ….t/ D 4 N.t/; dt

4

….0/ D 4 …:

(8.189)

Then the inverse constitutive equation (8.182) can be written in the Boltzmann form .n/

.n/

M T C D … J

Zt

4

.n/

4

dT ….t / ./: J

(8.190)

0

The tensors of creep cores and functions have the same properties of symmetry (8.143) and (8.147) as the tensors 4 K and 4 R (see Exercise 8.2.11): 4

….t/ D 4 ….1243/ .t/ D 4 ….2134/ .t/ D 4 ….3412/ .t/; 8t > 0:

(8.191)

For the tensor of creep functions 4 ….t/ as well as for 4 R.t/, we can introduce a spectral representation by formula (8.159): 4

….t/ D

m X

…˛ˇ .t/

˛;ˇ D1

nN X a˛ ˝ a ˇ C …˛˛ .t/4 ˛ ; a˛ aˇ

(8.192)

˛DmC1

where …˛˛ .t/ and …˛ˇ .t/ are the spectral creep functions. Theorem 8.10. If the spectral relaxation cores K˛ˇ .t/ are exponential, i.e. have the form (8.173), then the spectral creep cores N˛ˇ .t/ are exponential too: N˛ˇ .t/ D

. / N X A˛ˇ D1

and vice versa.

. / t˛ˇ

1 t exp @ . / A ; t˛ˇ 0

(8.193)

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua . /

545

. /

The constants A˛ˇ and t˛ˇ are called the spectra of creep values and creep times, . /

. /

respectively. They, in general, are not coincident with B˛ˇ and ˛ˇ , respectively; however the number N in (8.192) and (8.173) is the same. H Show that if the cores K˛ˇ .t/ have the form (8.173), then the cores (8.193) are a solution of the integral equation (8.188). Substitution of (8.173) and (8.193) into (8.188) yields m X

0 @

N X

0 @…˛ˇ

D1

ˇ D1

. /

Bˇ"

. / ˇ"

0

N X N X

@

D1 0 D1

1

0

11

0

. /

A˛ˇ

t t exp @ . / A Cˇ" . / exp @ . / AA ˇ" t˛ˇ t˛ˇ

/ . 0 / Z t B A. ˛ˇ ˇ" . / . 0 / t˛ˇ ˇ" 0

0

1

1

0

11

t exp @ . / A exp @ . 0 / A d AA D 0: t˛ˇ ˇ" (8.194)

Calculating the integral in (8.194) Zt 0

0 0 1 0 1 11 . / . 0 / t˛ˇ ˇ" t t t @exp @ 0 A exp @ AA exp @ . / . 0 / A d D . / . 0 / . / . / t˛ˇ ˇ" t˛ˇ ˇ" ˇ" t˛ˇ 0

(8.195) and equating coefficients in (8.194) at the same exponents, we get the system 0 1 0 1 . 0 / . 0 / m N m N X X X A˛ˇ B …˛ˇ X C ˇ" ˇ" @ AB . / D 0; @ AA. / D 0 ˇ" ˛ˇ . / . / . 0 / . / . 0 / . / t t t 0 0 D1 ˇ" D1 ˇ" ˇ D1 ˇ D1 ˛ˇ ˇ" ˛ˇ ˛ˇ (8.196) / . / . / . / and t˛ˇ in terms of the constants Bˇ" and ˇ" for determining the constants A. ˛ˇ (the constants …˛ˇ can always be determined in terms of C˛ˇ and are assumed to be known). . / . / At those values of Bˇ" , ˇ" , …˛ˇ , at which there exists a solution of the system (8.196), the exponential representation of the creep cores (8.193) exists too. N / . / and t˛ˇ in Formulae (8.196) give the method of calculation of the constants A. ˛ˇ

. / . / terms of Bˇ" , ˇ" and vice versa. From (8.188b) we can obtain simpler formulae

/ . / for determining the constants A. ˛˛ and t˛˛ (˛ D m C 1; : : : ; n):

…˛˛ . /

˛˛

D

N X 0 D1

0

. / A˛˛ . /

; . 0 /

˛˛ t˛˛

C˛˛ . /

t˛˛

D

0

N X

. / B˛˛

0 D1

˛˛ t˛˛

. 0 /

. /

:

(8.197)

546

8 Viscoelastic Continua at Large Deformations

On substituting (8.184) and (8.193) into (8.189), we find an expression for the spectral creep functions in the case of exponential cores: …˛ˇ .t/ D …˛ˇ C

N X D1

/ A. ˛ˇ

1 exp

where lim …˛ˇ .t/ D …˛ˇ C

t !C1

N X D1

t

!!

. / t˛ˇ

;

/ A. …1 ˛ˇ : ˛ˇ

(8.198)

(8.199)

Exercises for 8.2 8.2.1. Using the rule of differentiation of an integral with a varying upper limit (see formulae (8.15b) and (8.25)) and calculating the derivative of the functional (8.106) with respect to t, show that PTI (4.121) actually yields formulae (8.107) and (8.108) .n/

for T and w . 8.2.2. Using the definition (8.70), show that for linear models An of thermorheologically simple viscoelastic media, relations (8.106)–(8.108) have the forms r1 Z t Z t 1 X N ˛ˇ .t 0 10 ; t 0 20 /dI˛.s/ .1 / dI .s/ .2 / 0 ./ C ı ˇ 2 ˛;ˇ D1 0 0

D

1

Cı

Z t Zt r2 X ˛Dr1 C1 0

r1 X

.n/

T DJ

O.s/ ˇ

0

r˛ˇ .t

10 /

dI˛.s/ .1 /

CJ

Zt

˛;ˇ D1 0 r2 X

Zt

.s/ r˛˛ .t 0 10 / dI˛C .1 /;

@ N .s/ 0 0 0 0 .s/ ˛ˇ .t 1 ; t 2 / dI˛ .1 / dIˇ .2 / @t

0

Z t Zt

˛Dr1 C1 0

Zt r2 X ˛Dr1 C1 0

0

r1 J X 2

J

0

Zt

˛;ˇ D1

w D

N ˛˛ .t 0 10 ; t 0 20 / dJ˛.s/ .1 ; 2 /;

@ N 0 0 0 0 .s/ ˛˛ .t 1 ; t 2 / dJ˛ .1 ; 2 /; @t

0

where t 0 , 10 and 20 are determined by (8.71). Taking dI˛ ./ D

d d I˛ ./ d D I˛ . 0 / d 0 D dI˛ . 0 / dt d0

8.2 Principal, Quadratic and Linear Models of Viscoelastic Continua

547

into account, show that these relations can be represented as functions of the reduced time; in particular r1 X

.n/

T DJ

˛;ˇ D1

.s/ Oˇ

Zt 0

r˛ˇ .t 0 10 / dI˛.s/ .10 / C J

Zt 0 r2 X ˛Dr1 C1 0

0

r˛˛ .t 0 10 / dI˛C .10 /: .s/

8.2.3. Using representations (8.132)–(8.134) for the tensor of relaxation functions R.t/, show that representations (8.140) and (8.115) for are equivalent.

4

8.2.4. Substituting formulae (8.132)–(8.134) for the tensor of relaxation functions R.t/ into the expression (8.141) for w , show that formulae (8.141) exactly coincide with (8.116). 4

8.2.5. Show that the condition (8.146) causes monotone non-increasing the relaxation functions @ b˛ˇ ˛ˇ .t/ 6 0; 8t > 0; ˛; ˇ D 1; 2; 3; ˛ ¤ ˇ; R @t

@ b˛˛˛˛ .t/ 6 0; R @t

bijkl .t/ are components of the tensor 4 R.t/ with respect to the basis b where R ci . 8.2.6. Using representations (8.132)–(8.134) for the tensor of relaxation functions 4 R.t/ and its spectral representation (8.159), and also formulae (8.152)–(8.155) for the tensors a˛ and 4 ˛ , show that the functions r˛ˇ .t/ and R˛ˇ .t/ are connected by the following relations for isotropic continua R11 .t/ D r1 .t/ C .2=3/r2 .t/;

R22 .t/ D 2r2 .t/;

and for transversely isotropic continua e3333 .t/ D e R11 .t/ D R r 22 .t/;

b1111 .t/ D r11 .t/ C 2r44 .t/: R22 .t/ C R33 .t/ D 2R

8.2.7. Show that for mechanically determinate linear models An of thermorheologically simple continua, the constitutive equations obtained in Exercise 8.2.2 can be written in the forms (8.136), (8.140), and (8.141) ı

1 D 0 ./ C 2 ı

Z t Zt 0

.n/

.n/

d C .1 / 4 R.2t 0 10 20 / d C .2 /;

0

Zt

.n/

4

T DJ

.n/

R.t 0 0 / d C ./;

0

w D

Ja 2

Z t Zt

.n/

d C .1 / 0

0

.n/ @ 4 R.2t 0 10 20 / d C .2 /: 0 @t

548

8 Viscoelastic Continua at Large Deformations

Show that the constitutive equation (8.139) for this model becomes 0 ı

.n/

.n/

T D J @4 M C .t/

1

Zt 4

.n/

K.t 0 0 / C ./a ./d A; 4 K.t 0 / D d 4 R.t/=dt 0 :

0

8.2.8. Show that for the linear models An of thermorheologically simple media with the exponential cores, the constitutive equations from Exercise 8.2.7 can be written in the form (8.175)–(8.177), (8.181) 0 .n/

ı

.n/

T D J @4 M C

N X

1

. /

WA ;

D1

N Ja X w D 2

D1

d W˛ˇ dt

C

a .t/ . / ˛ˇ

. /

.n/

.W˛ˇ .t/ C .t// D 0;

. / .n/ .n/ m X B˛ˇ a˛ aˇ . / . / C W˛ˇ ˝ C W˛ˇ . / a aˇ ˛

˛;ˇ D1

˛ˇ

.n/ ! nN . / .n/ X B˛˛ . / 4 . / C C W˛˛ ˛ C W˛˛ : . / ˛DmC1 ˛˛ 8.2.9. Prove that the spectral representations (8.148) and (8.159) lead to the spectral form (8.150) of constitutive equations for the linear models An (8.135). 8.2.10. Show that in hydrostatic compression when the Cauchy stress tensor and the deformation gradient are spherical: T D pE; F D kE;

.n/

CD

.n/ 1 .k nIII 1/E; dev C D 0; n III

a viscoelastic continuum, according to the simplest model (8.166), behaves as a purely elastic one: .n/

T D

.n/ J R11 .0/I1 . C /E; 3

i.e. there are no viscoelastic deformations in hydrostatic compression for this model. Many solids actually have such properties up to high pressures p; therefore, the simplest model (8.166) is widely used in continuum mechanics. 8.2.11. Show that the tensors of creep cores and creep functions 4 N.t/ and 4 ….t/ have the same properties of symmetry (8.143), (8.147) as the tensors of relaxation cores and relaxation functions 4 K.t/ and 4 R.t/, and vice versa.

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

549

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids 8.3.1 Models An of Incompressible Viscoelastic Continua For incompressible viscoelastic continua, as usual, there is an additional condition of incompressibility, which can be written in one of the forms (4.487)–(4.492), and the principal thermodynamic identity takes the form (4.495) for models An . For models Bn , its form is analogous. Substituting the functional (8.35) into (4.495) and using the rule (8.27), we get constitutive equations for incompressible viscoelastic continua 8 .n/ .n/ ˆ p .n/1 ı ˆ ˆ T D G C [email protected] [email protected] C .t//; ˆ < n III (8.200) D @ [email protected].t/; ˆ ˆ ˆ ˆ :w D ı : .n/

Since for incompressible continua the number r of independent invariants I.s/ . C .t// is smaller by 1 than that for compressible materials, in each of the representations (8.53), (8.61), (8.73), (8.77), and (8.96) of the free energy functional the subscript .n/

of the function '0 .I.s/ . C/; / takes on r 1 values. In the consistent way, the number z of simultaneous invariants J.s/ occurring in arguments of the cores 'm decreases too.

8.3.2 Principal Models An of Incompressible Isotropic Viscoelastic Continua For the principal models An of incompressible isotropic viscoelastic continua, the functional (8.77) depends only on five simultaneous invariants: J˛.I /

.n/ .n/ .n/ .n/ .I / .I / D I˛ C .t/ ; J3C˛ D I˛ C ./ ; ˛ D 1; 2I J7 D C ./ C .t/; (8.201)

and the constitutive equations (8.200) become .n/

T D

.n/ p .n/1 G C 'M1 E C 'M2 C ; n III

(8.202)

550

8 Viscoelastic Continua at Large Deformations

where Zt 'M 1 E '01 C '02 I1 .t/

.'11 C '12 I1 .t// d ; 0

.n/

Zt

.n/

'M2 C '02 C .t/

(8.203) .n/

.n/

.'12 C .t/ C '17 C .// d ; 0

and '0 and '1 are determined by (8.79). For the principal linear models An of incompressible isotropic viscoelastic continua, the functional has the form (compare this with the potential for elastic incompressible materials (4.526)) ı

Zt .n/ .n/ l1 C 2l2 .n/ 0 N D 0C C m N C I1 . C / q1 .t /I1 . C .//d I1 . C / 2 ı

ı

0

.n/ 2 l2 I2 . C / C

Zt

.n/ .n/ q2 .t / C .t/ C ./ d ;

(8.204)

0

N are the constants, and q1 .t / and q2 .t / are the cores. where l1 , l2 and m The corresponding constitutive equations (8.202) have the form (compare with (4.526) for elastic continua) .n/

T D

.n/ .n/ p .n/1 G C .m N C lM1 I1 . C //E C 2lM2 C : n III

(8.205)

Here we have denoted two linear functionals .n/

lM1 I1 D l1 I1 . C /

Zt

Zt

.n/

q1 .t /I1 . C .// d D 0

.n/

.n/

lM2 C D l2 C

.n/

r1 .t / dI1 . C .//; 0

Zt

.n/

Zt

q2 .t / C ./ d D 0

.n/

r2 .t / d C ./:

(8.206)

0

The constants N 0 and p0 D p.0/ are chosen from the conditions (4.326) and (4.327) according to formulae (4.527), just as for elastic materials: N p0 D p e C m;

N 0 D 0;

(8.207)

where p e is the constant appearing in the initial values of the stress tensors in the e

.n/

natural configuration K: T D p e E.

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

551

8.3.3 Linear Models An of Incompressible Isotropic Viscoelastic Continua For linear models An of incompressible isotropic viscoelastic continua, obtained from quadratic mechanically determinate models An (see Sects. 8.2.6–8.2.9), constitutive equations have the form similar to formulae (8.120) for compressible materials, but the functional should involve the summands linear in the invariant I1 : ı

ı

.n/

ı

C N 0 C mI N 1. C / t t Z Z .n/ .n/ 1 r1 .2t 1 2 / dI1 . C .1 // dI1 . C .2 // C 2

D

0

0

0

Zt Zt C

.n/

.n/

r2 .2t 1 2 / d C .1 / d C .2 /; 0

(8.208)

0

where N 0 and m N are the constants, and r1 .y/ and r2 .y/ are the relaxation functions. According to formulae (8.110) and (8.111) and also Theorem 8.8, the functional can be written in the Volterra form ı

ı

D

.n/

ı

0

C N 0 C mI N 1. C / C

Zt

.n/ l1 2 .n/ I1 . C / C l2 I1 . C 2 / 2

.n/

Zt

.n/

0

1 C 2

.n/

0

Z t Zt

.n/

.n/

p1 .2t 1 2 /I1 . C .1 //I1 . C .2 // d 1 d 2 0

1 C 2

.n/

q2 .t/ C .1 / C .t/ d

q1 .t/I1 . C .// d I1 . C .t//2

0

Z t Zt

.n/

.n/

p2 .2t 1 2 / C .1 / C .2 / d 1 d 2 ; 0

(8.209)

0

where @2 r .y/ D p .y/; @y 2

@r .y/ D q .y/; @y

r .0/ D l :

(8.210)

552

8 Viscoelastic Continua at Large Deformations

By analogy with compressible media (see Sect. 8.2.13), we can consider the simplest model An of isotropic incompressible continua, in which the creep functions q1 .y/ and q2 .y/ are connected by the relation (8.165): 1 q1 .y/ D q.y/; 3

q.y/ D 2q2 .y/;

(8.211)

i.e. this model involves only one core q.y/. Integrating Eq. (8.211) with respect to y and taking the initial condition (8.210) into account, we find the connection between r1 .y/ and r2 .y/ 1 2 r1 .y/ D r2 .y/ C l1 C l2 ; 3 3

r.y/ 2r2 .y/:

(8.211a)

Substituting (8.211) into (8.205) and (8.206) and grouping like terms, we obtain the following constitutive equation (compare with (8.168)): .n/ .n/ p .n/1 T D G C .m N C l1 I1 . C //E C 2l2 C n III

.n/

Zt

.n/

q.t / dev C ./ d ; 0

(8.212)

where .n/ .n/ 1 .n/ dev C D C I1 . C /E 3

(8.213)

.n/

is the deviator of the tensor C (see (8.167)). If q.t/ 0, then Eqs. (8.212) coincide with relations (4.526) for isotropic incompressible elastic continua.

8.3.4 Models Bn of Viscoelastic Continua In models Bn of viscoelastic continua, the free energy D

t

.n/

is a functional in the form

.n/

. G.t/; .t/; G t ./; t .//;

(8.214)

D0

and corresponding constitutive equations can be obtained with the help of the rule (8.27) of differentiation of a functional with respect to time; they have the form 8 .n/ .n/ .n/ t .n/ ˆ t t ˆ ˆ < T D [email protected] [email protected] G.t// F . G.t/; .t/; G ./; .//; D0

D @ [email protected]; ˆ ˆ ˆ : w D ı :

(8.215)

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

All further constructions with functionals

t D0

553

t

and F can be performed for models D0

Bn as well. Special models Bn of viscoelastic continua can be obtained immediately from .n/

.n/

models An , in which one should make the substitution C D G .1=.n III//E. In particular, for linear mechanically determinate models Bn of isotropic incom.n/

.n/

pressible continua, according to the relation C D G , (8.205) and (8.208), we obtain the constitutive equations ı

ı

ı

D

0 C

0

.n/

1 CmI1 . G/C 2

Zt Z t 0

Zt Zt

.n/

0 .n/

.n/

r2 .2t 1 2 / d G.1 / d G.2 /;

C 0

.n/

r1 .2t1 2 / dI1 . G.1 // dI1 . G.2 //

(8.216)

0

Zt Zt

w D

.n/

.n/

q1 .2t 1 2 / dI1 . G.1 // dI2 . G.2 // 0

0

Zt Zt

.n/

.n/

q2 .2t 1 2 / d G.1 / d G.2 /;

C2 0

0 .n/ 1

.n/

p T D G n III

0 C @m C

Zt

(8.217)

1 .n/

r1 .t /dI1 . G.//AE C 2

0

Zt

.n/

r2 .t / d G./: 0

(8.218) For the simplest models Bn , the assumption (8.211a) on the functions r .t/ yields .n/ .n/ p .n/1 T D G C .m C l1 I1 . G//E C 2l2 G n III

.n/

Zt

.n/

q.t / dev G./ d ; 0

(8.219) w D

Zt

Zt

.n/

q.2t 1 2 / dev 0

0

.n/

@ G.1 / @ G.2 / dev d 1 d 2 > 0: @1 @2

(8.220)

Since the function w is nonnegative, we find that q.y/ D @r.y/[email protected] > 0; i.e. the relaxation core q.y/ is always nonnegative.

(8.221)

554

8 Viscoelastic Continua at Large Deformations .n/

e

Passing to the limit as t ! 0, in the natural configuration K, where T .0/ D .n/

p e E, G.0/ D E=.n III/ and p D p0 (see Sect. 4.8.6), from (8.219) we obtain the following relations between the constants 0 , m, l1 , l2 and p0 (see (4.529)): p0 D p e C m C

3l1 C 2l2 ; n III

0

D

3.3l1 C 2l2 / ı

2.n III/2

3m ı

.n III/

:

(8.222)

Notice that relations (8.216) and (8.218) are entirely equivalent to Eqs. (8.208), (8.205); and (8.219) – to Eqs. (8.212) (the constants l1 and l2 in these equations are distinct). We can obtain new models of the class Bn by taking additional assumptions on the constants m, l1 and l2 . For example, if we assume just as in the corresponding elastic models Bn (see (4.530)) that l2 D .1 ˇ/.n III/2 ; 2

l1 C 2l2 D 0;

m D .1 C ˇ/.n III/; (8.223)

where and ˇ are two new independent constants, then from (8.219) we obtain the constitutive equations p .n/1 T D G C .n III/2 n III

.n/

Zt

! ! .n/ .n/ 1Cˇ C .1 ˇ/I1 . G/ E .1 ˇ/ G n III

.n/

q.t / dev G./ d ;

(8.224)

0

which are not equivalent to the corresponding relations (8.212) of models An . Due to (8.223), relations (8.222) become p0 D p e C .3 ˇ/.n III/2 ;

0

ı

D 6 =:

(8.222a)

8.3.5 Models An and Bn of Viscoelastic Fluids Equations (8.214) and (8.215) as well as (8.35) and (8.38) hold true for both solid and fluid viscoelastic continua. However, for fluids, according to the principle of material symmetry, relations (8.39) (and the analogous relations for models Bn ) must be satisfied: .n/

.n/

t

.n/

.n/

T D F . G .t/; .t/; G t ./; t .// D [email protected] [email protected] G /; D0

D

t

.n/

.n/

. G .t/; .t/; G t ./; t .//

8H 2 U;

D0

for any H -transformations included in the unimodular group U .

(8.225) (8.226)

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

555

Theorem 8.11. For models An and Bn (n D I; II; IV; V) of viscoelastic fluids, the constitutive equations (8.214), (8.215) and (8.35), (8.38), satisfying the principle of material symmetry and being continuous functionals in space Ht , can be written as follows: .n/

T D

p .n/1 G ; n III

(8.227a)

0

1 Zt 1 Zt X @' @ @'m t 0 D p ../; .// D 2 @ C d 1 : : : d m A ; p D 2 ::: @.t/ D0 @ @.t/ mD1 0

0

(8.227b) t

D

..t/; .t/; t ./; t .// D '0 ..t/; .t//

D0

C

1 Zt X mD1 0

Zt :::

'm .t; 1 ; : : : ; m ; .t/; .1 /; : : : ; .m // d 1 : : : m ; 0

(8.227c) T D pE:

(8.227d)

H Since is assumed to be a continuous functional, we can apply Theorem 8.4 and expand in terms of n-fold scalar functionals: Zt 1 Zt X D : : : e m .t; 1 ; : : : ; m / d 1 : : : d m ; (8.228) mD1 0

0

whose cores are scalar functions of m tensor arguments: .n/

.n/

e m .t; 1 ; : : : ; m / D e m .t; 1 ; : : : ; m ; G.1 /; : : : ; G.m //:

(8.229)

On substituting the representation (8.228) into (8.226), we get that functions e m must satisfy the relation .n/ .n/ .n/ .n/ em .t; 1 ; : : : ; m ; G 1 ; : : : ; G m / D em t; 1 ; : : : ; m ; G 1 ; : : : ; G m 8H 2 U; (8.230) .n/

.n/

where G i G.i /, i.e. they must be H -indifferent relative to the unimodular group U .

556

8 Viscoelastic Continua at Large Deformations

Applying the same reasoning as we used in proving Theorem 4.31, we can show .n/

that functions of the third invariant of the tensors G i (or, that is the same, of values of the density i D .i / at different times i ) are the only functions ensuring that the condition (8.230) is satisfied: .n/

.n/

em D em .t; 1 ; : : : ; m ; I3 . G 1 /; : : : ; I3 . G m //Dem .t; 1 ; : : : ; m ; 1 ; : : : ; m /: Separating ı-type components from the cores in this expression by analogy with (8.51a), from (8.228) we get the representation (8.227c), where the cores 'm are connected to e m by relations (8.54). Substituting the functional (8.227c) into (8.215) and differentiating with re.n/

spect to G.t/, from (4.448)–(4.453) we actually obtain formulae (8.227a) and (8.227b). Finally, using the transformations (4.455) and (4.456), from (8.227a) we obtain that all the relations (8.227a) for models An and Bn are equivalent to the single relation (8.227d). N Notice that although Eq. (8.227d) for the Cauchy stress tensor is formally the same as the one for an ideal fluid, a viscoelastic fluid is not ideal (it is dissipative), because in this case the pressure p is a functional of the density , and the dissipation function w is not zero due to (8.64):

w D ı

D

'10

1 Z X

Zt

t

mD1 0

:::

@'m 0 C 'mC1 @t

d 1 : : : d m > 0:

0

(8.231)

8.3.6 The Principle of Material Indifference for Models An and Bn of Viscoelastic Continua .n/ .n/

.n/

Since all the energetic tensors T , C and G are R-invariant in rigid motions, all the constitutive equations given in this section for solids and fluids, and also for incompressible continua, are the same in actual configurations K and K0 ; therefore, the principle of material indifference for models An and Bn of viscoelastic media is satisfied identically.

Exercises for 8.3 8.3.1. Using the stepwise loading (8.144) and passing to the limit as t ! 0C , show that the instantly elastic relations obtained from (8.224) coincide with the constitutive equations (4.532) of the model Bn of an elastic isotropic incompressible continuum.

8.3 Models of Incompressible Viscoelastic Solids and Viscoelastic Fluids

557

8.3.2. Using the method applied in Sect. 8.2.14, show that for the simplest model An of isotropic incompressible viscoelastic continua with the exponential core N X t B . / exp . / ; q.t/ D . / D1 the constitutive equation (8.212) takes the form X .n/ .n/ p .n/1 G C .m N C l1 I1 . C //E C 2l2 C W. / B . / ; n III D1 N

.n/

T D

.n/ 1 d W. / D . / .dev C W. / /I dt

and for the simplest model Bn of isotropic incompressible viscoelastic continua with the same exponential core, the constitutive equation (8.224) takes the form .n/

T D

p .n/1 G n III

C .n III/2

! N X .n/ .n/ 1Cˇ W. / B . / ; C .1 ˇ/I1 . G/ E .1 ˇ/ G n III D1

.n/ d W. / 1 D . / dev G W. / : dt

8.3.3. Using the results of Exercise 8.2.7 and Eqs. (8.220) and (8.224), show that for the simplest linear models Bn of isotropic incompressible thermorheologically simple media the following constitutive equations hold: p .n/1 T D G C .n III/2 n III

.n/

Zt

! .n/ .n/ 1Cˇ C .1 ˇ/I1 . G/ E .1 ˇ/ G n III

.n/

q.t 0 0 / dev G./a ./ d ;

0

Zt Zt

w D a

.n/

0

q.2t 0

0

10

20 /

.n/

@ G.1 / @ G.2 / dev dev d 1 d 2 ; @1 @2

where t 0 , 10 and 20 are determined by (8.71).

558

8 Viscoelastic Continua at Large Deformations

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations 8.4.1 Statements of Dynamic Problems in the Spatial Description Statements of problems of viscoelasticity at large deformations can be obtained formally from the corresponding statements of problems of elasticity theory at large deformations (see Sect. 6.1), if in the last ones we replace the generalized constitu.n/

.n/

tive equations of elasticity T G D F G . C ; / by relations of viscoelasticity (for a viscoelastic model chosen from the models considered in Sects. 8.1–8.3). Choosing the most general representations (8.38) and (8.215) for models An and Bn of viscoelasticity and using the method mentioned above and the statement of the dynamic RU VF -problem of elasticity theory (see Sects. 6.1.1 and 6.3.3), we obtain a statement of the dynamic RU VF -problem of thermoviscoelasticity in the spatial description. This statement consists of the equation system in the domain V .0; tmax /: @ C r v D 0; @t

(8.232a)

@v C r v ˝ v D r T C f; @t

(8.232b)

1 qm C w @ C r v D r q C : @t

(8.232c)

@FT C r .v ˝ FT F ˝ v/ D 0; @t

(8.232d)

@u C r .v ˝ u/ D v; @t

(8.232e)

the constitutive equations in the domain VN .0; tmax /: q D r ; .n/

(8.233a)

.n/

T D 4 E G T G; .n/

.n/

t

.n/

(8.233b) .n/

T G D F G . C G .t/; .t/; C tG ./; t .// [email protected] [email protected] C G .t//;

(8.233c)

D0

D @ [email protected].t/; D

t

.n/

.n/

w D ı ;

. C G .t/; .t/; C tG ./; t .//; G D A; B;

D0

(8.233d) (8.233e)

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations

559

which should be complemented by expressions (4.389) and (4.391c) for tensors 4

.n/

.n/

E G and C G : .n/

CG

3 X

1 D n III 4

E D

ı

˝ p˛ hN G E ;

˛D1

3 X

.n/

! ı

nIII p˛ ˛

ı

ı

E˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

˛;ˇ D1 ı

˛ ; p˛ ; p˛ k F;

(8.234)

the boundary conditions (6.68)–(6.76), which for the case when there are no phase transformations take the form n T De tne ; n q D e q ne u D ue ; D e @[email protected] D 0;

v n D 0;

n q D 0;

at †1 ; : : : ; †4 ; †7 ; at †5 ; †6 ; n T ˛ D 0

at †8 ;

(8.235)

and also the initial conditions t D0W

ı

D ; v D v0 ; D 0 ; F D E; u D u0 :

(8.236)

On substituting the constitutive equations (8.233) and (8.234) into (8.232), we get the system of 17 scalar equations for 17 unknown scalar functions: ; ; u; v; F k x; t:

(8.237)

Since the domain V .t/ in the spatial description is unknown, the system obtained should be complemented with either relation (6.89) or Eq. (6.85) for the function f .x; t/ specifying a shape of the surface †.t/ bounding the domain V .t/: @f C v r f D 0; @t t D0W f D f 0 .x/:

(8.238)

In the second case the function f .x; t/ appears among the unknowns (8.237). Remark 1. In viscoelasticity theory, in place of the energy balance equation one often uses the entropy balance equation (3.166a) (the system (8.232) has been written in this way), which contains the dissipation function w explicitly. The entropy balance equation can be represented in the nondivergence form (3.166)

d D r q C qm C w : dt

(8.239)

560

8 Viscoelastic Continua at Large Deformations

When the model An of a thermoviscoelastic continuum with difference cores is considered, then is determined by formula (8.69). If in this formula the derivative @0 '[email protected] is assumed to depend only on , then after substitution of (8.69) and (8.233a) into (8.239) we obtain the following equation of heat conduction for a viscoelastic continuum in the spatial description: 0 0 0 1 11 .n/ .n/ TC T CC @ B [email protected] B c" C v r D r .r / @ @˛ A C v r @˛ AACqm Cw : @t @t (8.240) Here c" D [email protected] '0 [email protected] 2 /

(8.241) t u

is the heat capacity at fixed deformations.

Remark 2. If, for example, we consider the statement of the dynamic RU VF problem for mechanically determinate linear models An of thermorheologically simple viscoelastic continua with exponential cores, then the constitutive equations (8.233) has the form derived in Exercise 8.2.8: 0 ı

.n/

.n/

T D J @4 M C

1

N X

W. / A ;

D1

W. / D

m X ˛;ˇ D1

. / . / B˛ˇ W˛ˇ

/ @W. ˛ˇ

@t

w D

N Ja X 2 D1

nN X a.˛/ ˝ a.ˇ / . / . / C B˛˛ W˛˛ 4 ˛ ; a˛ aˇ ˛DmC1 .n/

Cvr ˝

/ W. ˛ˇ

D a

/ C W. ˛ˇ . /

;

˛ˇ

. / m X B˛ˇ a˛ . / ˛;ˇ D1 ˛ˇ

.n/ .n/ / / aˇ . C W. / ˝ C W. ˛ˇ ˛ˇ a˛ aˇ

! nN . / X .n/ B˛˛ .n/ / 4 . / C C W. : ˛˛ ˛ C W˛˛ . / ˛˛ ˛DmC1

(8.242)

In this case the initial conditions (8.236) are complemented by the additional conditions t D0W

. /

W˛ˇ D 0;

(8.243)

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations

561

and the number of unknown functions (8.237) becomes greater due to adding the functions / W. k x; t; ˛; ˇ D 1; : : : ; nI N D 1; : : : ; N I (8.244) ˛ˇ / (here we always have W. 0 when ˛ ¤ ˇ and ˛; ˇ > m). ˛ˇ

t u

By analogy with the dynamic RU V -, RV - and U -problems of thermoelasticity (see Sect. 6.3.3), we can state the dynamic RU V -problem of viscoelasticity, in which the deformation gradient F is eliminated between the unknowns, the U V problem of viscoelasticity, where in addition the density is eliminated, and the dynamic U -problem of viscoelasticity, where only u and are unknown. Remark 3. Just as the statements of thermoelasticity problems in the spatial description (see Sect. 6.3.3), the statements of thermoviscoelasticity problems mentioned above are strongly coupled, because they cannot be split into heat conduction problems and viscoelasticity problems even if we neglect the entropy term of the .n/

connection (i.e. the term .˛=/ T in the equation of heat conduction (8.240)). To the six causes of the connection in thermoviscoelasticity problems, mentioned in Remark 4 of Sect. 6.3.3, we should add one more factor: the presence of the dissipation function w in the entropy balance equation (8.232c) or in the heat conduction equation (8.240), that is a consequence of non-ideality of viscoelastic continua. In many problems, a contribution of the dissipation function w to the heat conduction equation (8.240) proves to be rather considerable and cannot be neglected in non-isothermal processes. The effect of growing the temperature in viscoelastic materials without heat supplied to a body from the outside but only due to internal heat release in deforming (caused by the presence of the dissipation function w ), is called dissipative heating of the body (see Sect. 8.6). t u Let us pay attention to the sixth cause of the connection mentioned in Remark 4 of Sect. 6.3.3: for viscoelastic continua the dependence of the constitutive equations (8.233c) on temperature can be split into the three constituents: ı

(1) Dependence of the heat deformation " (8.66) when the Duhamel–Neumann model is used. (2) Dependence of the elastic properties on the temperature .t/. (3) Dependence of the viscous properties, i.e. the integral part of Eqs. (8.233c), upon the temperature prehistory t ./. As established in experiments, for most viscoelastic continua, the viscous properties more considerably depend on temperature than the elastic ones. Since the dissipation function w depends on just the viscous properties, it also depends explicitly upon the temperature (in the model An with the exponential cores (8.242) this dependence has the form of function a ..t//). The temperature dependence w ./ leads to the intensification of dissipative heating in viscoelastic materials and, under certain conditions, can cause the effect of heat explosion (see Sect. 8.6.9).

562

8 Viscoelastic Continua at Large Deformations

8.4.2 Statements of Dynamic Problems in the Material Description Using the statement of the dynamic U VF -problem of thermoelasticity in the material description (see Sects. 6.2.1 and 6.3.4) and replacing the constitutive equations (6.42) by viscoelasticity relations in the forms (8.38) and (8.215), for models An and Bn we obtain a statement of the dynamic U VF -problem of thermoviscoelasticity in the material description. This statement consists of the equation system ı

D det F1 ; ı

ı

ı

[email protected][email protected]/ D r P C f; ı

ı

ı

ı

ı

ı

[email protected][email protected]/ D r . r / C qm C w ; ı

@FT [email protected] D r ˝ v; @[email protected] D v

(8.245)

ı

defined in the domain V .0; tmax /, and the constitutive equations in the same doı

main V .0; tmax /: .n/

.n/

TG

.n/

P D 4 E 0G T G ; .n/ .n/ .n/ t D [email protected] [email protected] C G .t// F G C G .t/; .t/; C tG ./; t ./ ;

(8.246a) (8.246b)

D0

ı

ı

D @ [email protected]; w D ı ; G D A; B; .n/ .n/ t D C G .t/; .t/; C tG ./; t ./ ;

(8.246c) (8.246d)

D0

.n/

.n/

which are complemented with expressions for the tensors 4 E 0 and C G : 4

.n/

E0 D

3 X

ı

ı

ı

ı

E ˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ;

˛;ˇ D1 .n/

CG D

3 1 X nIII ı ı .

p˛ ˝ p˛ hN G E/; n III ˛D1 ˛ ı

˛ ; p˛ ; p˛ k F;

(8.247)

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations

563

the boundary conditions (6.77)–(6.84) taking the forms (when there are no phase transformations) ı

ı

ı

ı

ı

n P D tne ; n q D q ne ı

ı

v n D 0;

ı

ı

ı

ı

D e

u D ue ; u n D 0;

ı

ı

at †1 ; : : : ; †4 ; †7 ; ı

at †5 ; †6 ; ı

ı

q n D 0; n P ˛ D 0 at †8 ;

(8.248)

and the initial conditions t D0W

v D v0 ; u D u0 ; F D E; D 0 :

(8.249)

On substituting the constitutive equations (8.246) and (8.247) into (8.245), we obtain the system for 16 unknown scalar functions being components of the following vectors and tensors: ; u; v; F k X i ; t: Due to the continuity equation, the density can be eliminated between the unknown functions. Just as for problems of thermoelasticity in the material description, the problem ı

(8.245)–(8.250) is formulated for a known domain V , that considerably simplifies its solving. For particular models of viscoelastic continua, formulae (8.246b)–(8.246d) are replaced by appropriate relations derived in Sects. 8.1–8.3. When models An of viscoelastic continua with the difference cores and the Duhamel–Neumann model (8.66) are considered, the specific entropy is determined by formula (8.69). Assuming that @0 '0 [email protected] depends only on temperature, we can rewrite the entropy balance equation of system (8.245) in the form of the heat conduction equation for a viscoelastic medium in the material description: 0 1 .n/ ı ı @ TC ı ı ı @ B ı c" D r r @˛ A C qm C w : @t @t ı

(8.250)

Here the second entropy term on the right-hand side of the equation, as a rule, can ı be neglected in comparison with w . Unlike the statement of the thermoelasticity problem in the material description given in Sect. 6.3.4, the thermoviscoelasticity problem (8.245)–(8.249) is strongly coupled even if there are no phase transformations, that is caused by the presı ence of the dissipation function w . As noted in Sect. 8.4.1, in the general case of ı non-isothermal processes a contribution of the function w to the heat conduction equation may be rather essential and cannot be neglected.

564

8 Viscoelastic Continua at Large Deformations

Using the statements of the dynamic U V -, T U VF - and U -problems of thermoelasticity in the material description (see Sect. 6.3.4), we can formulate the corresponding dynamic problems of thermoviscoelasticity. So the statement of the dynamic U -problem of thermoviscoelasticity in the material description consists of the equation system (6.58): ı

ı

ı

ı

ı

[email protected] [email protected] 2 / D r P C f; ı

ı

ı

(8.251)

ı

c" [email protected][email protected]/ D r . r / C qm C w ı

in the domain V .0; tmax /; constitutive equations (8.246); the expressions for .n/

.n/

tensors 4 E 0 and C G (8.247); the kinematic equation ı

F D E C r ˝ uT I

(8.251a)

boundary conditions (8.248) and initial conditions (8.249). The problem is solved for the four scalar functions: components of the displacement vector u and the temperature .

8.4.3 Statements of Quasistatic Problems of Viscoelasticity Theory in the Spatial Description Statements of quasistatic problems of viscoelasticity theory can be obtained formally from the corresponding statements of quasistatic problems of elasticity theory at large deformations (see Sect. 6.3.5) by replacing the constitutive equations of elasticity with appropriate relations of viscoelasticity derived in Sects. 8.1–8.3. So the statement of the coupled quasistatic problem of thermoviscoelasticity in the spatial description for linear mechanically determinate models An of thermorheologically simple continua has the form (

r T C f D 0; c" [email protected][email protected]/ D r . r / C qm C w in V;

8.n/ .n/ Rt ˆ ˆ T D J 4 R.t 0 0 / d C ./ ˆ ˆ ˆ 0 ˆ ˆ < Rt Rt .n/ d C .1 / w D J.a =2/ ˆ 0 0 ˆ ˆ ˆ .n/ .n/ R ˆ ı ı ˆ ˆ /d e ; : C D C "; " D ˛.e 0

(8.252)

in V [ †; .n/

@ 4 R.2t 0 0 0 / d C . /; 2 1 2 @t 0 .t;/ R .t 0 ; 0 / D a ..e // de ; 0

(8.253)

8.4 Statements of Problems in Viscoelasticity Theory at Large Deformations

8 ˇ ˆ n Tˇ† ˆ < ˆ ˆ :

8 .n/ .n/ ˆ 4 ˆ T D E T; ˆ ˆ ˆ ˆ .n/ ˆ P ı ı < C D n 1 III 3˛D1 nIII p˛ ˝ p˛ ; ˛ ˆ .n/ P ˆ ı ı ˆ 4 ˆ E D 3˛;ˇ D1 E˛ˇ p˛ ˝ pˇ ˝ pˇ ˝ p˛ ; ˆ ˆ ˆ ı :

˛ ; p˛ ; p˛ k F; F D E r ˝ uT in V [ †; ˇ ˇ ˇ D tne ; uˇ† D ue ; n qˇ† D qe ; ˇ† D e ; q

u

(8.254)

u n D 0; n T ˛ D 0 at †8 ; t D0W

565

(8.255)

D 0 :

Here (8.252) is the system of the equilibrium and heat conduction equations, (8.253) are the constitutive equations, (8.254) is the set of kinematic and energetic equivalence relations, (8.255) are the boundary and initial conditions, where † D †1 [ †2 [ †3 [ †4 [ †7 ; †u D †5 [ †6 , †q D †1 [ †2 [ †3 [ †4 [ †6 , † D †5 [ †7 . On substituting Eqs. (8.253) and (8.254) into (8.252), we obtain the system of four scalar equations for the four scalar unknowns: components of the displacement vector and the temperature u; k x; t: (8.256) If the model An with exponential cores is considered, then constitutive equations (8.253) should be replaced by relations (8.242). In particular, one can consider isothermal processes when a temperature field in a body V remains unchanged: .x; t/ D 0 D const; then the heat conduction equation can be excluded from the system (8.252). As a result, we obtain the following statement of the quasistatic problem of viscoelasticity in the spatial description for linear models An : r T C f D 0 in V I

.n/

Zt

T DJ

4

.n/

R.t / d C ./;

0

8 .n/ .n/ ˆ ˆ T D 4 E T; ˆ ˆ ˆ ˆ ˆ P ı ı 0, and in Eq. (8.311) there appears a heat source; therefore, @N [email protected] > 0, i.e. a heat-insulated beam will always be heated with time (while 0 D e and ˛N T D 0). This heating is caused only by energy dissipation; therefore, it is called dissipative heating.

8.6 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming

585

8.6.8 Temperature of Dissipative Heating in a Symmetric Cycle The process of deforming is called a symmetric cycle, if the average value of func.n/

.n/

tion f 1 .k1 / over the oscillation cycle is zero: h f 1 .k1 /i D 0.

.n/

As follows from (8.268), for a symmetric cycle the conditions h f ˛ .k1 /i D 0 (˛ D 1; 2; 3) are satisfied simultaneously. And from (8.312) it follows that the dissipation function depends only on the temperature N : N D a ./q.0/ N wN ./ fN2 ;

fN2

3 X

.n/

h f 2˛ .k1 /i:

(8.313)

˛D1

Then the heat conduction problem (8.311) takes the form ı

c"

@N N fN2 ; D ˛N T .N 0 / C a ./q.0/ @t

t D 0 W N D 0 :

(8.314)

Its solution has the form t D H.N /;

N H./

ZN 0

ı c" d e : 2 N N a ./q.0/f ˛N T .N e /

(8.315)

8.6.9 Regimes of Dissipative Heating Without Heat Removal If there is no heat removal from the beam .˛N T D 0/, then, depending on a form of N two basically distinct regimes of dissipative heating are possible. the function a ./, N is such that the integral (1) If the function a ./ N D H./

Z 0

ı

c" d N .a ./q.0/ fN2

(8.316)

becomes infinite at infinity (i.e. H.N / ! C1 as N ! 1), then the temperature of dissipative heating in the beam gradually grows with no limit (Fig. 8.13, the curve 1). For many real viscoelastic materials, as the function a ./ one frequently uses the Williams–Landel–Ferry dependence a ./ D exp

a1 . 0 / ; a2 C 0

a1 ; a2 const;

(8.317)

N ! C1 as N ! C1 is actually satisfied. for which the condition H./

586

8 Viscoelastic Continua at Large Deformations

Fig. 8.13 Different regimes of dissipative heating for a heat-insulated viscoelastic beam: 1 – unbounded growth of the temperature, 2 – heat explosion, 3 – heat pseudoexplosion

N is bounded at infinity (2) If the dependence a ./ is such that the function H./ N (i.e. H.C1/ < C1), then from (8.316) it follows that the temperature .t/ of dissipative heating reaches infinite values in the finite time t D H.C1/; N has a vertical asymptote as t ! t (Fig. 8.13, in other words, the function .t/ the curve 2). The phenomenon of sharp growth of the temperature at a certain time t is called the heat explosion. If, for example, the function a ./ is exponential (that is typical for some elastomers): a D ea1 . 0 / ;

a1 > 0;

(8.318)

then, calculating the integral in (8.316), we obtain the following expression for the temperature of dissipative heating: ! ı N2 a1 t 1 q.0/ f c" N D 0 lg 1 D : ; t .t/ ı a1 q.0/fN2 a1 c"

(8.319)

From this equation it really follows that N ! C1 as t ! t . (3) In practice sometimes there is an intermediate situation when the condition N with time H.N / ! C1 holds as N ! C1, but the growth of temperature .t/ proves to be so sharp that it becomes similar to the effect of heat explosion: the temperature reaches its ultimate value k , at which there occurs a heat destruction of the material, in comparably small time tk (Fig. 8.13, the curve 3). Such regimes are called the heat pseudoexplosion.

8.6.10 Regimes of Dissipative Heating in the Presence of Heat Removal When there is heat removal .˛T > 0/, the character of dissipative heating of a body N and ˛T .N 0 / are non-negative, the dechanges. Since both the functions ˛ ./ nominator of the integrand in (8.315) at a certain finite value N D 1 < C1 can N D t tends to infinity: N ! 1 as t ! C1. vanish; in this case the function H./ Thus, when there is heat removal, the regimes 1, 2, and 3 are impossible, because

8.6 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming

587

Fig. 8.14 Typical regimes of dissipative heating for a viscoelastic beam in oscillations with heat removal

the temperature always remains bounded, in this case there are other four typical regimes 4, 5, 6, and 7 (Fig. 8.14). (4) The regime 4 is realized when the dissipation function wN is independent of temperature .a D 1/. In this case the heat conduction equation (8.314) yields 2 N d 2 =dt 6 0 (Exercise 8.6.1); i.e. the dissipative heating curve is convex upwards. This is the most wide-spread type of a curve of dissipative heating for real practical problems. At this curve there are two typical sections: the section ‘a’, where the rate of PN heating decreases rapidly from its maximum value .0/ to practically zero PN 0, P and the steady section ‘b’, where N 0. N D 1, then the heat conduction problem (8.314) admits a solution in the If a ./ explicit form (Exercise 8.6.1) ı

q.0/fN2 c" N D 0 C ˛N T

1 exp

˛N T t ı

!! :

(8.320)

c"

N is unbounded as N ! C1 (for example, the exponential function (5) If a ./ (8.318)), then the temperature regime 5 is realized when the curve of dissipative heating can be split into the three typical sections: ‘a’ – initial, ‘b’ – steady, where P const, and ‘c’ – unsteady, where the function N .tN/ is convex downwards (Fig. 8.14) and unbounded as t ! C1. N depends on the temperature but is bounded as N ! C1 and has (6) If a ./ a point of inflection at N D e k (as the function (8.317)), then the temperature regime 6 of dissipative heating occurs (Fig. 8.14), when there are four typical sections: initial – ‘a’, steady – ‘b’, unsteady – ‘c’, followed by steady one – ‘d’. The temperature N .t/ remains bounded as t ! C1. (7) The regime 7 is realized under the same conditions as the regime 6, but at N reaches an the unsteady section ‘c’ the temperature of dissipative heating .t/ ultimate value k such that there occurs a heat destruction of the material (by analogy with the regime 3), and the section ‘d’ is not realizable.

588

8 Viscoelastic Continua at Large Deformations

8.6.11 Experimental and Computed Data on Dissipative Heating of Viscoelastic Bodies Figure 8.15 shows computed and experimental curves of dissipative heating for a polyurethane beam in cyclic deforming by the harmonic law (8.290). The change in temperature D 0 was found by solving the problem (8.311), (8.312) with the help of the implicit difference approximation ı

i C1 D

i C wN t=c" ı

1 C ˛N T =c"

;

(8.321)

where i D .ti / is the value of temperature in the i th node at time ti , and t is the step in time. Values of the relaxation core q.2t 0 / and q.0/ appearing in expression (8.312) for the dissipation function were determined by (8.292), and B . / and . / in this formula – by the relaxation curves with the help of the method given in Sect. 8.5.5. Values of the constants B . / and . / for polyurethane are shown in Table 8.1. The function a ./ was approximated by formula (8.317), and values of the constants in this formula were assumed to be a1 D 21, a2 D 208 K. The ı remaining constants in (8.311) take on the following values: D 103 kg=m3 , c" D 0:8 kJ=.kg K/, ˛N T D 10 kWt=.m3 K/. The mean value kN1 and the oscillation amplitude k10 were expressed in terms of the minimum and maximum deformation values in the cycle (ımin and ımax ): 1 1 kN1 D 1 C .ımax C ımin /; kN1 D .ımax ımin /: 2 2

(8.322)

Figure 8.15 exhibits temperatures of dissipative heating, computed by the method mentioned above for different models Bn at ımax D 50% and ımin D 34%. Under the conditions considered, the model BI gives the best approximation to experimental data. The distinction between different models Bn is considerable: the models BIV

Fig. 8.15 Curves of dissipative heating for polyurethane, computed by models Bn , and experimental curve of dissipative heating .ex/

8.6 Dissipative Heating of a Viscoelastic Continuum Under Cyclic Deforming

589

Fig. 8.16 Curves of dissipative heating for polyurethane, computed by the model BI at different amplitudes of oscillations

and BV lead to a stationary regime of dissipative heating according to the type (4), and the models BI and BII forecast the regime (5) with the presence of unsteady section. Figure 8.16 shows curves of dissipative heating for polyurethane, computed by the model BI at different values of ımin (values of this parameter are given in Fig. 8.16 by numbers at curves); the value ımax D 50% has been fixed. With growing the oscillation amplitude (i.e. in this case with decreasing the value ımin ), the intensity of dissipative heating sharply increases. Notice that the phenomenon of dissipative heating of viscoelastic materials can essentially reduce the durability of structures under cyclic deforming.

Exercises for 8.6 N D const, then a solution of the problem (8.314) is a 8.6.1. Show that if a ./ function being convex upwards, i.e. d 2 N =dt 2 6 0. Prove formula (8.320).

Chapter 9

Plastic Continua at Large Deformations

9.1 Models An of Plastic Continua at Large Deformations 9.1.1 Main Assumptions of the Models While models of viscoelastic continua most adequately describe a behavior of ‘soft’ materials (rubbers, polymers, biomaterials), for simulation of mechanical inelastic properties of ‘stiff’ materials (metals and alloys) one widely uses models of plastic media. There are different models of plastic continua, but we consider only the models falling into the class of models of plastic yield that is frequently used in practice. These models are convenient to be considered in the forms An , Bn , Cn and Dn , .n/

where in place of one should use the Gibbs free energy A (4.126), and the principal thermodynamic identity – in the form (4.133).

Definition 9.1. One can say that this is the model An of a plastic continuum, if for the medium 1. The operator constitutive equations (4.156) are functionals with respect to time t P of reactive variables R and their derivatives R: t

P P t .//I ƒ.t/ D f .R.t/; R.t/; Rt ./; R D0

(9.1)

2. The set of reactive variables R in addition includes a symmetric second-order .n/

tensor C p called the plastic deformation tensor, and the set of active variables .n/

ƒ contains a symmetric second-order tensor C e called the elastic deformation tensor: .n/ .n/

ƒ D f; ; C ; C e ; w g; .n/

.n/

.n/

.n/

R D f T =; C p ; g;

(9.2)

.n/

3. The tensors C e and C p are connected to C by the additive relation .n/

.n/

.n/

C D C e C C p:

Y.I. Dimitrienko, Nonlinear Continuum Mechanics and Large Inelastic Deformations, Solid Mechanics and Its Applications 174, DOI 10.1007/978-94-007-0034-5 9, c Springer Science+Business Media B.V. 2011

(9.3) 591

592

9 Plastic Continua at Large Deformations .n/

.n/

Unless the tensors C e and C p were involved in the set of variables R and ƒ, the relation (9.1) could be considered as the single model of viscoelastic continua of the .n/

differential and integral types. However, just the appearance of the tensors C e and .n/

C p leads to new effects which are not usual for the continuum types considered. Just as for continua of the differential type, the dependences (9.1) upon R.t/ and P R.t/ are assumed to be differentiable functions, and the functionals of prehistory P t ./ are assumed to be continuous and Fr´echet–differentiable. Rt ./, R .n/

.n/

The tensors C e and C p can be introduced axiomatically; however, to justify physically the introduction of these tensors one usually considers the following model. ı Introduce a reference configuration K, which is assumed to be unstressed (i.e. ı

T.0/ D 0 in V ), and an actual configuration K, where a stress tensor field T.t/, in general, is different from identically zero. In addition, introduce one more conp

figuration K (it may be not physically realizable), to which the continuum would correspond, if at the time tp > t its stress field were zero again: T.tp / D 0. Moreı

p

p

over, we impose on K the requirement that if the transformations K ! K ! K occur without plastic deformations (i.e. according to the model of an ideal continp

ı

p

uum), then configuration K must coincide with K. Such a configuration K is called unloaded. p ı ı Introduce in each of the configurations K, K and K local basis vectors ri , ri and p r i , respectively: p ı p @x @x @x ı ri D ; ri D ; ri D ; (9.4) @X i @X i @X i ı

p

ı

p

where x, x and x are radius-vectors of a material point M in K, K and K, respectively (Fig. 9.1).

Fig. 9.1 A scheme of transformation of configurations

9.1 Models An of Plastic Continua at Large Deformations

593

ı

p

As before, construct the metric matrices in K, K and K: ı

ı

gij D ri rj ;

ı

g ij D ri rj ; ı

p

p

p

g ij D r i r j ;

(9.5a)

p

and their inverse matrices g ij , gij and g ij , with the help of which introduce the vectors of reciprocal bases ı

ri D gij rj ;

ı

ı

ri D g ij rj ;

p

p

p

r i D g ij r j :

(9.5b)

Introduce the tensors of transformations of a local neighborhood of the point M ı

p

ı

p

from K to K, from K to K and from K to K: ı

p

p

Fp D r i ˝ ri ;

Fe D ri ˝ r i ;

ı

F D ri ˝ ri :

(9.6)

The tensors Fp and Fe are called the gradients of plastic and elastic deformations, respectively. These tensors are connected by the relation F D Fe F p :

(9.7)

Introduce also three right Cauchy–Green deformation tensors: ı

ı

C D "ij ri ˝ rj ;

ı

ı

Cp D "pij ri ˝ rj ;

ı

ı

Ce D "eij ri ˝ rj ;

(9.8)

where we have denoted components of the deformation tensor by "ij , of the plastic deformation tensor – by "pij and of the elastic deformation tensor – by "eij : "ij D

1 ı gij g ij ; 2

"pij D

1 p ı g ij g ij ; 2

"eij D

p 1 gij g ij : 2

(9.9)

In a similar way, introduce three right Almansi deformation tensors: ı

ı

ƒ D "ij ri ˝ rj ;

ı

ı

ƒe D "ij e ri ˝ rj ;

ı

ı

ƒp D "ij p ri ˝ rj ;

(9.10)

where we have denoted contravariant components of the deformation tensor by "ij , of the plastic deformation tensor – by "ij p and of the elastic deformation tensor – by "ij e : 1 ı ij 1 ı ij p ij 1 ı ij g gij ; "ij g g ; "ij g g ij : (9.11) "ij D p D e D 2 2 2 From (9.8)–(9.11) it follows that three right Cauchy–Green tensors and three right Almansi tensors are connected by the additive relations C D Ce C Cp ;

ƒ D ƒe C ƒp :

(9.12)

594

9 Plastic Continua at Large Deformations

From these equations it follows that the relation of additivity (9.3) holds for the I

V

energetic deformation tensors C and C if we assume as usual that I

V

Ce D ƒe ;

I

Ce D C;

V

Cp D ƒp ;

Cp D Cp :

(9.13)

In order to justify the additive relation (9.3) for n D II; IV, we introduce the polar decompositions for the deformation gradients (9.6): Fp D Op Up ;

F D O U;

Fe D Oe Ue ;

(9.14)

and represent the symmetric stretch tensors U and Ue in their eigenbases: UD

3 X

ı

ı

˛ p˛ ˝ p˛ ;

Ue D

˛D1

3 X

e

e

e

˛ p˛ ˝ p˛ :

(9.15)

˛D1

e

Here ˛ and ˛ are eigenvalues of the tensors U and Ue . Then we can introduce the tensors IV

CDUE D

IV

Cp D

ı

.˛ 1/ p˛ ˝ p˛ ;

e e IV P3 ı ı ı ı p ˝ p ; C D 1 p˛ ˝ p˛ ; ˛ ˛ e ˛ ˛ ˛ ˛D1 ˛D1 II

II

ı

˛D1

P3

C D E U1 D Cp D

P3

P3

˛D1

ı ı p˛ ˝ p˛ ; 1 1 ˛

e e II P3 ı ı 1 1 ı 1 ı p ˝ p ; C D p˛ ˝ p˛ ; 1 e ˛ ˛ ˛ ˛ ˛ ˛D1 ˛D1

P3

(9.16)

for which the additive relations (9.3) still hold. Thus, we have shown how the tensors of elastic and plastic deformations satisfying the relation (9.3) can be introduced.

9.1.2 General Representation of Constitutive Equations for Models An of Plastic Continua Consider the principal thermodynamic identity in the form An (4.121): d

.n/

.n/

C d T d C C w dt D 0:

(9.17)

9.1 Models An of Plastic Continua at Large Deformations

595

With the help of the additive relation (9.3) this identity can be rewritten as follows: d

.n/

.n/

.n/

.n/

C d T d C e T d C p C w dt D 0:

(9.18)

Introduce the Gibbs free energy just as in (4.126): .n/

D

T .n/ Ce:

(9.19)

Then for we obtain the identity (the principal thermodynamic identity in the form An ): .n/ .n/ .n/ .n/ (9.20) d C d C C e d T = T d C p C w dt D 0: According to the model An , the Gibbs free energy is a functional in the form (9.1):

t

D D0

.n/ .n/ P t ./ ; R D f T =; C p ; g: P R.t/; R.t/; Rt ./; R

(9.21)

Determine the total differential of this functional by using the rule of differentiation of functionals (8.24) with respect to time: 0 d D P dt D

.n/

1

.n/ @ B T C @ d C [email protected] AC d Cp .n/ .n/ @ @ T = @Cp 0 1

@

.n/

.n/ @ @ P BTC d C C d @ A C d C p C ı dt; (9.22) .n/ .n/ @P @. T =/ @ C p

@

where ı is the Fr´echet–derivative. .n/

The elastic deformation tensor C e is a functional of the same type (9.1): .n/ .n/ t P C e D Ce R.t/; R.t/; Rt ./; RP t ./ ; R D f T =; C p ; g:

.n/

(9.23)

D0

Just as for continua of the differential type (see Sect. 7.1.1), we introduce two new tensor functionals: .n/ C 0e

t D Ce R.t/; 0; Rt ./; RP t ./ ;

(9.24)

D0

.n/ C 1e

.n/

.n/

D C e C 0e :

(9.25)

596

9 Plastic Continua at Large Deformations

P are In (9.24) the arguments corresponding to the rates of changing functions R .n/

.n/

assumed to be zero. The tensors C 0e and C 1e are called the equilibrium elastic deformation tensor and nonequilibrium elastic deformation tensor, respectively. Substituting (9.22), (9.24) and (9.25) into (9.20) and grouping like terms, we obtain the identity

@

C

.n/

.n/ C 0e

@ T = C

@

.n/

.n/

@ T C C d C d @

.n/ d C p

.n/

.n/

C w . T Hp/

.n/ C p

.n/

@

d

.n/

T

@. T =/ C

.n/ C 1e

.n/

T

@ C p

C

@ d @

C ı dt D 0: (9.26)

.n/

Here we have introduced the tensors of strengthening H p and the reduced stress .n/

tensors T H :

.n/

.n/

.n/

H p D [email protected][email protected] C p /; .n/

.n/

.n/

T H D T Hp:

(9.27)

.n/

.n/

Since the differentials d. T =/, d, . T =/ , d , C p and dt are independent, the identity (9.26) holds if and only if coefficients of these differentials vanish; i.e. we have the relations 8 .n/ .n/ ˆ ˆ ˆ C 0e D @[email protected] T =/; .9:28a/ ˆ ˆ ˆ ˆ .9:28b/ < D @[email protected]; .n/

.n/

ˆ @[email protected] T =/ D 0; @[email protected]P D 0; @[email protected] C p D 0; ˆ ˆ .n/ ˆ .n/ .n/ .n/ .n/ ˆ ˆ 1 ˆ : w D . T H p / C p C e T = ı;

.9:28c/ .9:28d/

which are constitutive equations for models An of plastic continua. From these relations it follows that: 1. Plastic media are dissipative (for them w ¤ 0). .n/

.n/

2. The equilibrium deformation tensor C 0e (but not C e ) has the potential . .n/

.n/

3. The potential and hence C e , and are independent of the rates . T =/ , P and .n/ C p : t

P t .//; D .R.t/; Rt ./; R D0

.n/

.n/

R D f T =; C p ; gI .n/

P however, the dissipation function w and tensor C 1e depend on R.t/.

(9.29)

9.1 Models An of Plastic Continua at Large Deformations

597

Thus, the model An of plastic continua is specified by three functionals: the Gibbs free energy (9.29), the nonequilibrium elastic deformation tensor .n/ C 1e

t P P t ./ ; Rt ./; R D C1e R.t/; R.t/;

(9.30)

D0

.n/

and the plastic deformation tensor C p .

9.1.3 Corollary of the Onsager Principle for Models An of Plastic Continua .n/

To construct the functional (9.30) and tensor C p we use the Onsager principle (see Sect. 4.12.1, Axiom 16) and form the specific internal entropy production (4.728) with the help of (9.28d): 0 1 .n/

.n/ .n/ .n/ q q BTC q D w r D T H C p C 1e @ A ı r > 0: (9.31)

In order for this function to be non-negative, according to the Onsager principle, it can be represented in the generalized quadratic form. For this, introduce thermodynamic forces Xˇ : X1 D r;

.n/

X2 D T H ;

.n/

X3 D . T =/

(9.32)

and thermodynamic fluxes Q1 D q=;

.n/

Q2 D C p ;

.n/

Q3 D C 1e :

(9.33)

Then, according to the Onsager principle, the thermodynamic fluxes Qˇ must be linear (or tensor-linear) functions of Xˇ : 8 .n/ .n/ ˆ ˆ ˆ < q= D L11 r C L12 T H C L13 . T =/ ; .n/ .n/ .n/ C p D L12 r C L22 T H C L23 . T =/ ; ˆ ˆ ˆ .n/ .n/ : .n/1 C e D L13 r C L23 T H C L33 . T =/ :

(9.34)

Here L11 is a second-order tensor, L12 and L13 are third-order tensors, and L22 , L23 and L33 are fourth-order tensors, which are, according to the principle of

598

9 Plastic Continua at Large Deformations

equipresence, tensor functionals of the same form as the ones appearing in the general constitutive equations (9.1) of the model An : t

P P t .//: Rt ./; R L˛ˇ D L˛ˇ .R.t/; R.t/;

(9.35)

D0

Relations (9.34) are complementary constitutive equations to the system (8.28) for the model An of plastic continua. The first of the relations is the generalized Fourier law, the second one – the law of changing plastic deformations, and the third one – the law of changing nonequilibrium elastic deformations.

9.1.4 Models An of Plastic Yield Many special models of plastic continua follow from the law of changing plastic deformations (9.34) after introduction of some assumptions on a form of the functionals (9.29) and (9.35). In practice one often uses models An of plastic yield, where the functionals (9.29) and (9.35) are only functions of indicated arguments R and RP and of one scalar functional wp : .n/

.n/

D . T =; C p ; ; wp /; .n/ .n/

.n/

(9.36a)

.n/

L22 4 Lp . T ; C p ; ; T ; C p ; wp /;

(9.36b)

L11 D ./; L12 D 0; L13 D 0; L23 D 0; L33 D 0:

(9.36c)

The dependence on P and in these models is neglected; therefore, instead of the .n/

.n/

argument T = the function L22 can always have the argument T (because D .n/

.n/

.n/

. C / and P can always be expressed in terms of T and C p ). These models do not consider the cross–effects in relations (9.34), and the tensor C1e is identically zero: C1e 0:

(9.37)

The scalar functional wp is usually chosen in the form Z wp D

t .n/ 0

.n/

T ./ C p ./ d ;

(9.38)

called the Taylor parameter (the specific work done by plastic deformations), or in the form

9.1 Models An of Plastic Continua at Large Deformations

Z wp D

t 0

.n/ C p ./

.n/ C p ./

599

!1=2 d

(9.39)

called the Odkwist parameter. In this case relations (9.34) and (9.28d) have the following general form: q D r ; .n/ C p

(9.40a)

.n/

D 4 Lp T H :

.n/

w D . T H

(9.40b)

.n/ @ .n/ T / C p : @wp

(9.40c)

From the inequality (9.31) it follows that the heat conductivity tensor is symmetric and positive-definite, and the tensor 4 Lp is symmetric in pairs of indices .1; 2/ $ .3; 4/ (the symmetry in indices 1 $ 2 and 3 $ 4 follows from symmetry of the .n/

.n/

.n/

tensors C p and T ; H p ), i.e. the tensor is symmetric in the form (7.21), has not more than 21 independent components and is also positive-definite.

9.1.5 Associated Model of Plasticity An

In applications one frequently uses the associated model of plasticity, where the law of plastic yield (9.40b) is connected (or associated) with the concept of a yield surface. Consider this model. .n/

.n/

Let in the six-dimensional space of components T ij of the stress tensor T with ı respect to some basis, for example, with respect to the basis ri , there be a surface whose position is given by the scalar equation system fˇ D 0;

ˇ D 1; : : : ; k:

(9.41)

Here fˇ are functions in the form (9.36a) .n/ .n/

fˇ D fˇ . T ; C p ; ; wp /;

(9.42)

.n/

depending parametrically on C p , and wp , they are called the plastic potentials. Denote the partial derivative of functions (9.42) with respect to time by d 0 fˇ @fˇ .n/ T ; D .n/ dt @T

(9.43)

600

9 Plastic Continua at Large Deformations

that coincides with the total derivative fˇ D dfˇ =dt only if fˇ is independent of .n/

C p , and wp : .n/

fˇ D fˇ . T /:

(9.44)

In this case one can say that the model of an ideally plastic continuum is considered. The associated model (9.42) taking such dependence into account is called the model of a strengthening plastic continuum. We assume axiomatically that: 1. Inside a domain bounded by the yield surface (9.41) plastic deformations remain unchanged, i.e. if all fˇ < 0;

.n/

then C p D 0I

(9.45)

if at least for one value of ˇ the condition dfˇ =dt > 0 is satisfied, then one can say that there occurs active loading, and if all d 0 fˇ =dt 6 0, then there occurs passive loading or unloading. 2. On the yield surface when d 0 fˇ =dt D 0, plastic deformations remain also un.n/

changed (such loading is called neutral), and if d 0 fˇ =dt > 0, then C p vary (one can say that there occurs plastic loading), i.e. .n/

if fˇ D 0; d 0 fˇ =dt D 0; then C p D 0; .n/

if fˇ D 0; d 0 fˇ =dt > 0; then C p ¤ 0:

(9.46) (9.47)

.n/

Notice that the state of a continuum when its tensor T is outside of the yield surface is impossible, because we assume axiomatically that the yield surface moves .n/

.n/

with changing the tensor T if C p ¤ 0; i.e. the condition f > 0 cannot be satisfied. .n/

The specific expression for the rate of plastic deformation C p in the case (9.47) of plastic loading is given by the Drucker model (or the gradient law). According to .n/

this model, the tensor C p is chosen to be proportional to the gradient of the yield surface: k X .n/ .n/ C p D ~P ˛ [email protected]˛ [email protected] T /: (9.48) ˛D1

Here ~P ˛ are the ratio coefficients, which can be written in the convenient form of derivatives with respect to time and which are scalar functions in the form .n/ .n/

.n/

~P ˛ D ~P˛ . T ; C p ; C p ; ; wp /;

˛ D 1; : : : ; k:

(9.49)

9.1 Models An of Plastic Continua at Large Deformations

601

If k 6 6, then one can find these functions from the gradient equation with complementing the tensor equation (9.48) by k equations (9.41) of the yield surface. Thus, we have 6 C k scalar equations (9.48) and (9.41) to determine six components .n/

of the tensor C p and k functions ~˛ (˛ D 1; : : : ; k). Notice that the gradient law (9.48) is written only for plastic loading. In order .n/

to derive an expression for C p under arbitrary loading, we should combine the relations (9.45)–(9.47). This can be done with the help of the Heaviside functions hC .x/ and h .x/: hC .x/ D

x > 0; x < 0;

1; 0;

h .x/ D

1; 0;

x > 0; x 6 0;

(9.50)

and their combinations 0 d fˇ : 1 hC .fˇ /h ˇ D1 dt k

hD1 …

(9.51)

k

Here … is the product. One can easily verify that if the conditions (9.45) or (9.46) ˇ D1

are satisfied, then h D 0, and if the condition (9.47) is satisfied, then h D 1. Thus, .n/

for C p under arbitrary loading we obtain .n/

C p D h

k X

.n/

~P ˛ [email protected]˛ [email protected] T /:

(9.52)

˛D1

This relation must satisfy the corollary of the Onsager principle (9.40b) for models of plastic yield; i.e. the following equation must hold h

k X

.n/

.n/

~P˛ [email protected]ˇ [email protected] T / D 4 Lp T H :

(9.53)

˛D1

Here 4 Lp is some indeterminate symmetric fourth-order tensor. Equation (9.53) means that the functions fˇ called plastic potentials must be quasilinear functions .n/

.n/

of T H p . Remark. For the associated model of plasticity, a part of constitutive equations for .n/

C p is given by Eqs. (9.41) being the implicit forms of components of the tensor

.n/

C p . These equations can also be represented by the expression (9.40b) written in the implicit form .n/

ˆˇ C p 'ˇ D 0;

(9.54)

602

9 Plastic Continua at Large Deformations

where ˆˇ are symmetric second-order tensors, and .n/

.n/

'ˇ D ˆˇ 4 Lp . T H p /:

(9.55)

Indeed, differentiating (9.41) with respect to t, we obtain that .n/

ˆˇ D @f [email protected] C p ;

'ˇ D

@fˇ .n/

.n/

T C

@T

@fˇ P : @

(9.56)

Since relations (9.44) are scalar, the corollary (9.54) of the Onsager principle imposes no constraints on the form of the plastic potentials fˇ , and the relation @fˇ .n/

.n/

T C

@T

.n/ @fˇ P @fˇ 4 Lp T p ; D .n/ @ @Cp

(9.57)

being a consequence of (9.55) and (9.56) and an analog of formula (9.53), can alt u ways be satisfied by the proper choice of indeterminate tensor 4 Lp . .n/

For the elastic deformations C e in the associated model, from (9.28a) and (9.37) we get the relation .n/

.n/

.n/

@

C e D K. T =; C p ; ; wp / D

.n/

:

(9.58)

@ T = The model of a plastic continuum, where the potential does not depend explic.n/

itly on the plastic deformation tensor C p , is called the model An of an elastoplastic continuum; for this model, .n/

.n/

D . T =; ; wp /; H p 0;

.n/

.n/

C e D K. T =; ; wp / D

@ .n/

: (9.59)

@ T = This model is widely used in practice. A model with explicit dependence of upon .n/

C p is usually applied when one needs the effect of deformation anisotropy to be taken into account. This effect consists in the change of a symmetry group for a continuum considered in the process of varying plastic deformations. .n/

The model, where the dependence of upon both C p and wp may be neglected, is called the model of an ideally elastoplastic continuum (it is not to be confused with an ideally plastic continuum, where, according to the definitions (9.44), the .n/

plastic potentials fˇ are independent of C p and wp ).

9.1 Models An of Plastic Continua at Large Deformations

603

.n/

.n/

Finally, if is independent of C p and wp , and depends on T = only quadrati cally, then one can say that this is the model An of a plastic continuum with linear elasticity.

9.1.6 Corollary of the Principle of Material Symmetry for the Associated Model An of Plasticity Apply the principle of material symmetry to constitutive equations (9.53) and .n/

.n/

.n/

(9.58). Since all the tensors C e , C p and T are H -indifferent relative to orthogonal H -transformations, from the principle of material symmetry it follows that the elastic potential (9.36a) must be a function of simultaneous invariants J.s/ ı

.n/

.n/

of the tensors T and C p relative to some group G s in an undistorted reference ı

configuration K:

D J.s/ ; ; wpˇ :

Here J.s/

D

.n/ .n/

J.s/

(9.60)

! D 1; : : : ; z:

T; Cp ;

(9.61)

A proof of this assertion is the same as the one for continua of the differential type (see Sect. 7.1.4). ı

For simplicity, a reference configuration K is assumed to be undistorted; and, ı

as before (see Sects. 4.7.3 and 8.1.6), we will consider groups G s only in this configuration. In place of one scalar functional wp (9.38), the set of arguments of the function (9.60) in general may include functionals wpˇ being integrals of such simultaneous .n/

.s/

.n/

invariants J that contain both the tensors T and C p : wpˇ

Z

t

D 0

.n/

.n/

Jˇ.s/ . T ./; C p .// d ;

ˇ D 1; : : : ; z:

(9.62)

Then the constitutive equations (9.58) can be represented in the tensor basis by analogy with ideal continua: .n/

Ce D

z X D1

' J.s/ T:

(9.63)

604

9 Plastic Continua at Large Deformations .s/

Here ' are scalar functions in the form (9.60), and J T are the derivative tensors: p ' D ' J˛.s/ ; ; wˇ D @[email protected].s/ ;

@J.s/

.s/

J T D

.n/

;

D 1; : : : ; z: (9.64)

@ T = As follows from the principle of material symmetry, the plasticity functions fˇ .s/p (9.42) are also functions of simultaneous invariants J˛ : fˇ D fˇ J˛.s/p ; ; wpˇ ;

ˇ D 1; : : : ; k:

(9.65)

However, according to the corollary (9.53) of the Onsager principle, the derivatives .n/

.n/

.s/p

@fˇ [email protected] T must be quasilinear functions of the tensor T p ; therefore, in (9.65) J˛ .n/

.n/

must be simultaneous invariants of T p and C p , being only linear and quadratic functions of these tensors: .n/

.n/

J˛.s/p D J˛.s/ . T H ; C p /;

˛ D 1; : : : ; z1 6 z:

(9.66)

Then constitutive equations (9.52) in the tensor basis take the form .n/ C p

D

z1 X

.s/p ˛ J˛T ;

(9.67)

˛D1

where ˛

Dh

k X

~Pˇ

ˇ D1

@fˇ .s/p @J˛

;

.n/

.s/p J˛T D @J˛.s/p [email protected] T :

(9.68)

Relations (9.60)–(9.68) are called the representation of the associated model of plasticity in the tensor basis. .s/p .s/ and J˛T are connected by the relation The derivative tensors J˛T .s/p J˛T

D

@J˛.s/p .n/

@TH

.n/

.n/

@. T H p / .n/

.s/p

.n/

D J˛TH . 4 H pT /:

(9.69)

@T

With the help of (9.27) and (9.59) we find an expression for the fourth-order tensor 0 1 .n/ z .s/ X .n/ @ B @ @J C @Hp 4 D H pT @ .s/ .n/ A .n/ .n/ @J D1 @T @T @Cp z ' X ˇ .s/ .s/ .s/ .s/ Jˇ.s/ Cı : ˝ J CJ ˝ J ' J D ˇ T C C TC T p p p 2 ;ˇ D1

(9.70)

9.1 Models An of Plastic Continua at Large Deformations

605

Here we have introduced the notation 'ˇ D

@'

@2

D

.s/

@Jˇ

; .s/

.s/

@Jˇ @J

@2 J.s/

J.s/ TCp D

.n/ .n/

:

(9.71)

@ T @Cp

On substituting (9.68) into (9.67), we finally obtain .n/ C p

D

z1 X

.s/p ˛ J˛TH

.n/

. 4 H pT /:

(9.72)

˛D1 .n/

For the associated model of an elastoplastic continuum, is independent of C p , hence .n/

.n/

H p 0;

J.s/p

D

J.s/ ;

J.s/p T

D

J.s/p TH

4 H pT 0; D

J.s/ T;

(9.73)

J.s/ Cp

J.s/ TCp

D 0;

D 0:

Thus, the invariants J.s/p and J.s/ coincide, and constitutive equations (9.60), (9.63), (9.65), and (9.72) take the forms .n/ D I.s/ . T =/; ; wpˇ ; D 1; : : : ; r; .n/

Ce D

r X D1

.n/ C p

D

z1 X

.s/

(9.74a)

.n/

' I T . T =/;

(9.74b)

.n/ .n/ .s/ ˛ J˛T . T ; C p /;

(9.74c)

˛D1 .n/ .n/

fˇ D fˇ .J˛.s/ . T ; C p /; ; wpˇ /; ' D

@

; .s/

@I

.n/

I.s/ T

D

@I.s/ . T =/ .n/

;

˛

Dh

k X ˇ D1

@ T =

(9.74d)

~Pˇ

@fˇ @J˛.s/

:

(9.74e)

9.1.7 Associated Models of Plasticity An for Isotropic Continua Let us write representations (9.74) for the three main symmetry groups Gs : O, T3 and I , with choosing simultaneous invariants J.s/ in the same way as it was done for viscoelastic continua.

606

9 Plastic Continua at Large Deformations

For an isotropic continuum, the functional basis of simultaneous invariants .n/

.n/

J.I / . T =; C p / consists of nine invariants, which can be chosen as follows (see (7.35) and (8.83)): .n/

.n/

.I / J˛C3 D I˛ . C p /;

J˛.I / D I˛ . T =/; J7.I / D

1 .n/ .n/ T Cp;

J8.I / D

˛ D 1; 2; 3;

1 .n/ .n/2 T C p;

J9.I / D

1 .n/2 .n/ T C p ; (9.75) 2

.s/

Then the derivative tensors J T (9.64) have the forms (see (8.84)) .I / D E; J1T

.n/

.n/

.I / J2T D 1 .EI1 . T / T /;

.I / J3T D

.n/ 1 .T2 2

.n/

.I / .I / J˛C3;T D 0; ˛ D 1; 2; 3I J7T D C p ; .n/ .n/

.I /

.n/

.I /

.n/

I1 T C EI2 /; .n/

J8T D C 2p ;

.n/

J9T D 1 . T C p C C p T /:

(9.76)

Substituting these expressions into (9.63) and collecting terms with the same tensor powers, we obtain the following representation of constitutive equations (9.63) in the tensor basis: .n/

Ce D e ' 1E C

.n/ .n/ .n/ .n/ e ' 2 .n/ '3 .n/2 e ' 6 .n/ .n/ T C 2 T Ce '4 C p C e ' 5 C 2 C . T C p C C p T /: (9.77)

Here we have denoted the scalar functions (compare with (4.322a)) e ' 1 D '1 C ' 2 I1 C ' 3 I2 ;

e ' 2 D '2 C '3 I1 ; e ' 3C D '6C ; D 1; 2; 3: (9.78)

' 5 and e ' 6, For the model of an elastoplastic continuum (9.59), the functions e '4, e due to (9.64), are zero; and we obtain the relation .n/

.n/

.n/

Ce D e ' 1 E C .e ' 2 =/ T C .'3 =2 / T 2 ;

(9.79)

which is analogous to the constitutive equation of an ideally elastic isotropic continuum (4.322) but written in the inverse form. Here ' D

@ ; @I

.n/

.n/

.n/

D .I1 . T =/; I2 . T =/; I3 . T =/; ; wp /;

and there is only one Taylor parameter wp .

(9.80)

9.1 Models An of Plastic Continua at Large Deformations

607

For an elastoplastic continuum, according to the Onsager principle, the set of arguments of the plasticity functions fˇ includes only linear and quadratic invari.n/

.n/

ants, i.e. only J.I / . T ; C p /, D 1; 2; 4; 5; 7, (z1 D 5), among which there is .I / only one simultaneous invariant, namely J7 . On substituting the derivative tensors (9.76) into (9.74c), we obtain .n/

C p D e1 E

where

˛

1,

Dh

2,

and

k X

~Pˇ

ˇ D1

7

.n/

2T

C

.n/

7 Cp;

(9.81)

are determined by formulae (9.68):

@fˇ .n/

;

7

Dh

k X

~P ˇ

ˇ D1

@J˛ . T /

@fˇ .I / @J7

; e1 D

1

C

.n/

2 I1 . T /;

(9.82) .n/

.n/

.n/

.n/

.I /

.n/ .n/

fˇ D f .I1 . T /; I2 . T /; I1 . C p /; I2 . C p /; J7 . T ; C p /; ; wp /:

(9.83)

Formulae (9.78)–(9.82) give the general form of constitutive equations of the associated model of an isotropic elastoplastic continuum.

9.1.8 The Huber–Mises Model for Isotropic Plastic Continua For special models of isotropic elastoplastic continua, one usually accepts additional assumptions on a form of the elastic and plastic potentials and fˇ . As shown in experiments, for many elastoplastic continua, their behavior can be described adequately enough by the Huber–Mises model, where there is only one plastic potential f depending explicitly on the only simultaneous invariant YH : f D f .YH ; ; wp /:

(9.84)

Here the invariant YH has been introduced as contraction of the tensor PH being the .n/

.n/

deviator of the tensor T H C e (see the definition of the deviator (8.167)): YH2 D

3 PH PH ; 2

.n/ .n/ .n/ 1 .n/ PH D . T H C p / I1 . T H C p /E; 3

(9.85)

and H is the strengthening parameter being a scalar function in the form H D H0 Yp2n0 ;

(9.86)

608

9 Plastic Continua at Large Deformations

where H0 and n0 are the constants, and Yp is the invariant of the deviator of the .n/

tensor C p determined similarly to (9.85): Yp2 D

.n/ 1 .n/ Pp D C p I1 . C p /E: 3

3 Pp Pp ; 2

(9.87) .n/

.n/

The invariants YH and Yp are called the intensities of the tensors T H C p and .n/

C p , respectively. .n/

The deviator can be constructed for any tensor; for example, for the tensor T : .n/ 1 .n/ PT D T I1 . T /E; 3

YT2 D

3 PT PT ; 2

(9.88)

.n/

where YT is the intensity of the tensor T . Each deviator of a tensor is orthogonal to the unit (metric) tensor (more detailed information on properties of the deviators can be found in [12]): PH E D 0;

Pp E D 0;

PT E D 0:

(9.89)

The invariants YT and Yp can be expressed in terms of the principal invariants of corresponding tensors (see Exercise 9.1.2): .n/

.n/

.n/

YT2 D I12 . T / 3I2 . T /;

.n/

Yp2 D I12 . C p / 3I2 . C p /:

(9.90)

We can immediately verify that the invariant YH2 can be expressed in terms of the .n/

.n/

invariants I˛ . T /, I˛ . C p / and simultaneous invariant J7.I / : .I /

YH2 D YT2 C H 2 Yp2 3HJ7

.n/

.n/

C HI1 . T /I1 . C p /:

(9.91)

.n/

Determining the derivatives @f [email protected]˛ . T / of the function (9.84): @f @I˛ @f @I1

D

@f @YH @Y˛ @I˛

2 @YH @I˛

D fY .n/

.n/

D fY .2I1 . T / C HI1 . C p //;

; fY

@f @I2

@f 1 2YH @YH

D 3fY ;

@f .I / @I7

;

D 3HfY ; (9.91a)

and substituting them into (9.82), we find that .n/

1

.n/

D ~hf P Y .2I1 . T /CHI1 . C p //;

2

D 3~hf P Y;

7

D 3~hf P Y H: (9.91b)

9.1 Models An of Plastic Continua at Large Deformations

609

Then constitutive equations (9.81) take the form .n/ C p

D 3~hf P Y PH :

(9.92)

Due to the property (9.89) of the deviators, from (9.92) it follows that a continuum described by the Huber–Mises model is plastically incompressible, i.e. .n/ C p

ED0

or

.n/

I1 . C p / D 0:

(9.93)

Therefore, with the help of (9.87) Eq. (9.92) can be written as the following quasilinear relation between the deviators: P Y PH : PP p D 3~hf

(9.94)

The scalar product of Eq. (9.94) by itself yields the following expression for ~P under plastic loading: q PP p PP p ~P D ˙ p : (9.95) 6fY YH After substitution of expression (9.95) into (9.94) the number of independent equations in (9.94) reduces to four, and relationships (9.84), (9.93), and (9.94) form the complete system of constitutive equations (it consists of six independent relationships). If the plastic potential is chosen in the Mises form f D

1 .YH = s /2 1; 3

(9.96)

where s D s .; wp / is the given function of and wp called the yield point or the yield strength, then fY D 1=.3 s2 /, and the final relations of plasticity (9.84), (9.92), and (9.93) become .n/

C p D

f D

.n/ .n/ ~h P .P H C /; I . C p / D 0; T p 1 s2

1 .YH = s /2 1 D 0; 3

(9.97)

.n/ 1 .n/ PT D T I1 . T /E; H D H0 Yp2n0 : 3

The Taylor parameter (9.38), due to plastic incompressibility (9.93), can be written in terms of the deviators: Z t PT PP p d : (9.98) wp D 0

610

9 Plastic Continua at Large Deformations

One can say that this is the model of an isotropic plastic continuum with linear strengthening, if the strengthening parameter H (9.86) is a constant, i.e. H D H0 and n0 D 0:

9.1.9 Associated Models of Plasticity An for Transversely Isotropic Continua For a transversely isotropic continuum, the functional basis of simultaneous invari.n/ .n/

ants J.3/ . T ; C p / consists of 11 invariants, which can be chosen as follows (see (7.36) and (8.87)): .n/

.n/

.3/ J.3/ D I.3/ . T /; D 1; : : : ; 5I J5C D I.3/ . C p /; D 1; : : : ; 4I ! .n/ .n/ .n/ .n/ .3/ .3/ .3/ .3/ .3/ 2 2 c3 T b c3 C p ; J11 D T C p 2J10 J2 J8 : J10 D .E b

(9.99) The derivative tensors .3/ J1T D E b c23 ;

.s/ J T

(9.64) in this case have the forms (see (8.88))

.3/ J2T Db c23 ;

.n/ 1 .O1 ˝ O1 C O2 ˝ O2 / T ; 2

.3/ J3T D

.n/

.n/

.n/

.3/ .3/ J4T D 2 4 O3 T ; J5T D T 2 I1 T C EI2 ; .3/

.3/

.3/

.3/

J6T D J7T D J8T D J9T D 0; .3/ J10T D

.n/ 1 .O1 ˝ O1 C O2 ˝ O2 / C p ; 4

(9.100) .n/

.3/

J11T D 4 O3 C p :

Substituting these expressions into (9.63) and collecting terms with the same tensor powers, we obtain the following representation of constitutive equations (9.63) for a transversely isotropic plastic continuum in the tensor basis: .n/

.n/

Ce D e ' 1E C e ' 2b '3 T c23 C .O1 ˝ O1 C O2 ˝ O2 / .e .n/

.n/

.n/

.n/

' 4 T C '5 T 2 C '11 C p : Ce ' 10 C p / C e

(9.101)

Here we have denoted the scalar functions .n/

.n/

.n/

' 2 D '2 '1 2'4 I2.3/ . T / '11 I2.3/ . C p /; e ' 1 D '1 C '5 I2.3/ . T /; e e '3 D

'3 2

'4 ; e ' 10 D

'10 4

'11 2 ;

.n/

e ' 4 D 2'4 '5 I1 . T /:

(9.102)

9.1 Models An of Plastic Continua at Large Deformations

611

For the model of an elastoplastic continuum (9.58), we have '10 D '11 D 0, and Eq. (9.101) takes the form .n/

.n/

.n/

.n/

c23 C e Ce D e ' 1E C e ' 2b ' 3 .O1 ˝ O1 C O2 ˝ O2 / T C e ' 4 T C '5 T 2 : (9.103)

Here .n/

.n/

D .I1.3/ . T /; : : : ; I5.3/ . T /; ; wp1 ; : : : ; wp3 /;

' D [email protected][email protected].3/ /; (9.104)

and wpˇ are the Taylor parameters (9.62) for a transversely isotropic continuum, and the number of the parameters in this case is equal to three: wp1 D

Z Z

p

w3 D wp2 D

t 0

t .n/

t 0

.n/

Z

b t .n/ 0

b .n/ T 33 C 33 d ;

T C p w2 w1 ;

0

Z

.n/

.n/

.b c23 T /.b c23 C p / d D p

p

.n/

.n/

..E b c23 / T / .b c23 C p / d D

Z 0

t .b n/

.b n/ .b n/ .b n/ . T 13 C 13 C T 23 C 23 / d :

(9.105) For an elastoplastic continuum, according to the Onsager principle, only the .3/

cubic invariant J3 ity functions fˇ :

.n/

D I3 . T / does not appear in the set of arguments of the plastic-

.3/ fˇ D fˇ .J1.3/ ; J2.3/ ; J4.3/ ; : : : ; J11 ; ; wp1 ; : : : ; wp3 /:

(9.106)

.3/ .3/ There are two simultaneous invariants in the set (9.99), namely J10 and J11 . Then, substituting the derivative tensors (9.100) into (9.74c), we obtain .n/

C p D

1E

C.

2

C4 O3 .2

c23 1 /b .n/

4T

1 C .O1 ˝ O1 C O2 ˝ O2 / . 2 .n/

C

11 C p /;

˛

Dh

k X ˇ D1

.n/

3T

C

10

2

.n/

C p/

(9.107) ~Pˇ [email protected]ˇ [email protected]˛.3/ /:

(9.108)

612

9 Plastic Continua at Large Deformations

9.1.10 Two-Potential Model of Plasticity for a Transversely Isotropic Continuum For special models of transversely isotropic elastoplastic continua, we accept an additional assumption on a form of the potentials and fˇ . In the two-potential model, we suppose that there are two plastic potentials, one of which, namely f2 , .3/ c contain components with depends only on those invariants J , which in basis b .b n/ .n/ subscript 3, i.e. T ˛3 and C p˛3 , and f1 depends on the remaining invariants: .n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.n/

.3/ . T ; C p /; ; wp1 ; wp2 /; f1 D f1 .I1.3/ . T /; I4.3/ . T /; I1.3/ . C p /; I4.3/ . C p /; J10 .3/ f2 D f2 .I2.3/ . T /; I3.3/ . T /; I2.3/ . C p /; I3.3/ . C p /; J11 . T ; C p /; ; wp3 /:

(9.109) In the transversely isotropic two-potential Huber–Mises model, we assume that each of the potentials f1 and f2 is a function of the simultaneous Huber–Mises invariants for a transversely isotropic continuum Y˛.3/ : f2 D f2 .Y2.3/ ; Y3.3/ ; ; wp1 ; wp2 /;

f1 D f1 .Y1.3/ ; Y4.3/ ; ; wp3 /:

(9.110)

Here .n/

.n/

Y˛.3/ D I˛.3/ . T H˛ C p /;

.n/

0

H˛ D H˛0 .I˛.3/ . C p //n˛ ; (9.111)

˛ D 1; : : : ; 4I

are simultaneous invariants which are uniquely expressed in terms of the invariants (9.99) (see Exercise 9.1.5): .n/

.n/

Y˛.3/ D I˛.3/ . T / H˛ I˛.3/ . C p /;

˛ D 1; 2;

.n/

.n/

.3/ Y3.3/ D I3.3/ . T / C H2 I3.3/ . C p / 2H3 J10 ; .3/

Y4

.3/

.n/

.3/

.n/

(9.112)

.3/

D I4 . T / C H2 I4 . C p / 2H4 J11 ;

and H˛0 and n0˛ are the constants. Calculating the derivatives @f [email protected]˛.3/ of the functions (9.110) (only occurring in the expression (9.67)): .3/

@f1 [email protected]

D f11 ;

.3/

@f1 [email protected]

D f14 ;

.3/

@f1 [email protected] D 2f14 H4 ;

(9.113)

.3/ @f2 [email protected]/ D f22 ; @f2 [email protected]/ D f23 ; @f2 [email protected] D 2H3 f23 ; fˇ @fˇ [email protected].3/ ;

9.1 Models An of Plastic Continua at Large Deformations

613

and substituting them into (9.108), we obtain 1

D ~P 1 f11 h; 10

2

D ~P2 f22 h;

D 2~P 2 f23 H3 h;

3

D ~P2 f23 h;

11

4

D ~P1 f14 h;

D 2~P 1 f14 H4 h:

(9.114)

Then the constitutive equations (9.107) for plastic deformations become .n/

.n/

.n/

C p D ~P1 h.f11 .E b c23 / C 2f14 4 O3 . T H3 C p // C ~P2 h.f22b c23 C

.n/ .n/ f23 .O1 ˝ O1 C O2 ˝ O2 / . T H4 C p //: 2 (9.115)

We can immediately verify (see Exercise 9.1.4) that the tensors .n/

.n/

P1H f11 .E b c23 / C 2f14 4 O3 . T H3 C p /; c23 C P1H f22b

f23 .O1 2

.n/

.n/

˝ O1 C O2 ˝ O2 / . T H4 C p /

(9.116)

are mutually orthogonal: P1H P2H D 0:

(9.117)

Therefore, rewriting Eq. (9.115) with use of notation (9.116) in the form .n/

C p D ~P1 hP1H C ~P 2 hP2H

(9.118)

(this is an analog of the relationship (9.94)) and multiplying the result by P1H and P2H , we get v u .n/ u t C p P1H ; ~P 1 D ˙ P1H P1H

v u .n/ u t C p P2H ~P2 D ˙ P2H P2H

(9.119)

– the expressions for ~1 and ~2 , being similar to the expression (9.95). After substitution of (9.119) into (9.118), for determining all components of the plastic deformation tensor Eq. (9.118) should be complemented by two more scalar ones f1 D 0;

f2 D 0;

where fˇ are expressed by formulae (9.110).

614

9 Plastic Continua at Large Deformations

These functions are usually chosen in a quadratic form being similar to the Mises model (9.96): 2f1 D

2f2 D

Y4H 4s

2

C

jY1H j C Y1H C 2 1s

jY2H j C Y2H C 2 2s

!2

C

!2

C

jY1H j Y1H 2 1s

jY2H j Y2H 2 2s

2

C

Y3H 3s

2

2

1;

1: (9.120)

˙ The functions 1s .; wp3 / are called the yield points (or the yield strengths) in longitudinal tension and compression, respectively, and 4s .; wp3 / – the yield point p p ˙ in shear along the plane of transverse isotropy. The functions 2s .; w1 ; w2 / are p called the yield points in transverse tension and compression, and 3s .; w1 ; wp2 / – the yield point in interlayer shear. These functions are usually determined in experiments. For anisotropic continua, the distinction between the yield points in tension C and ˛s may be and compression is rather considerable; therefore, the functions ˛s essentially distinct. Notice that although the functions fˇ depend on the sign of invariants Y1H and Y2H , they are differentiable everywhere, including the case when Y1H D 0 and Y2H D 0, and their derivatives (9.113) take on the values

f˛˛ D

jY˛H j C Y˛H jY˛H j Y˛H C ; ˛ D 1; 2I C 2 ˛s 2 ˛s

f14 D Y4H = 4s ;

f23 D Y3H = 3s :

(9.121)

9.1.11 Associated Models of Plasticity An for Orthotropic Continua For an orthotropic continuum, the functional basis of simultaneous invariants .n/ .n/

J.O/ . T ; C p / consists of twelve invariants, that should be complemented by two more, in general, dependent invariants in order to obtain the set of invariant, being symmetric relative to all the basis vectorsb c˛ (see (7.37) and (8.91), (8.92)): .n/

.n/

.O/ J.O/ D I.O/ . T /; D 1; ; 6I J.O/ C6 D I . C p /; D 1; 2; 3; 6I .O/

.n/

.n/

.n/

.n/

.O/

.n/

.n/

J10 D .b c22 T / .b c23 C p /; J11 D .b c21 T / .b c23 C p /; .n/

.n/

.n/

.O/ .O/ c21 T / .b c22 C p /; J14 D .b c21 T / .b c22 T / D I7.O/ . T /: (9.122) J13 D .b

9.1 Models An of Plastic Continua at Large Deformations

615

The derivative tensors in this case become (see (8.93)) .O/ J.O/ c2 ; D 1; 2; 3I J C3;T D T Db .n/

.n/ 1 .O ˝ O / T ; D 1; 2I 2

.n/

.O/ .O/ J6T D 3 6 Om T ˝ T ; J C6;T D 0; D 1; 2; 3; 6I

(9.123)

.n/ .n/ 1 1 .O/ D .O1 ˝ O1 / C p ; J11T D .O2 ˝ O2 / C p ; 4 4

.O/ J10T

.O/ D J13T

.n/ 1 .O3 ˝ O3 / C p ; 4

.O/ J14T D

.n/ 1 .O3 ˝ O3 / T ; 4

where the tensor 6 Om is determined by (4.316). Substitution of these expressions into (9.63) yields the following representation of constitutive equations (9.63) for an orthotropic plastic continuum in the tensor basis: ! 3 X .n/ .n/ .n/ .n/ .n/ 1 1 2 Ce D ' 3C T C e 'b ' 6C C p / C3'6 6 Om T ˝ T : c C O ˝ O .e 2 2 D1 (9.124) Here we have denoted the scalar functions e ' 4 D '4 ; e ' 5 D '5 ; e ' 6 D '14 ; e ' 7 D '10 ; e ' 8 D '11 ; e ' 9 D '13 : (9.125) '8 D e ' 9 , and For the model of an elastoplastic continuum (9.59): e '7 D e equations (9.124) take the form .n/

Ce D

3 X

.'b c2 C

D1

.n/ .n/ .n/ e ' 3C .O ˝ O / T / C 3'6 6 Om T ˝ T ; 2

(9.126)

where ' D [email protected][email protected].O/ /;

.O/

D .I1

.n/

.O/

. T /; : : : ; I7

.n/

. T /; ; wp1 ; : : : ; wp6 /: (9.127)

The number of the Taylor parameters (9.62) for an orthotropic continuum is equal to six: Z wp

D 0

t

.b c2

.n/

T /.b c2

.n/ C p /

d ;

D 1; 2; 3I

wp3C

Z

t

D 0

.n/

.n/

.b c2˛ T / .b c2ˇ C p / d ;

˛ ¤ ˇ ¤ ¤ ˛:

(9.128)

616

9 Plastic Continua at Large Deformations

For an orthotropic elastoplastic continuum, the plastic potentials fˇ (9.65) depend .n/

on all simultaneous invariants (9.122) except the cubic invariant J6.O/ D I6.O/ . T /: fˇ D fˇ .J.O/ ; ; wp1 ; : : : ; wp6 /;

D 1; : : : ; 14 and ¤ 6:

(9.129)

.O/ .O/ , J12 and There are three simultaneous invariants in the set (9.122), namely J10 .O/ J13 . Then, having substituted the derivative tensors (9.123) into (9.74c), we obtain the constitutive equation .n/

C p

D

3 X

c2 b

D1

! .n/ .n/ 1 1e e C O ˝ O . 3C T C 6C C p / ; 2 2

(9.130)

where e4 D

4;

e5 D

5;

e6 D

14 ;

Dh

e7 D

k X ˇ D1

~Pˇ

10 ;

@fˇ @J.O/

e8 D

11 ;

e9 D

:

13 ;

(9.131)

9.1.12 The Orthotropic Unipotential Huber–Mises Model for Plastic Continua For special models of orthotropic elastoplastic continua one should accept an additional assumption on a form of the potentials fˇ . The adequacy of a model accepted is verified in experiments. In the orthotropic unipotential Huber–Mises model one assumes that there is only one potential f depending on six simultaneous orthotropic Huber–Mises invariants Y˛.O/ (˛ D 1; : : : ; 6): f D f .Y1.O/ ; : : : ; Y6.O/ ; ; wp1 ; : : : ; wp6 /:

(9.132)

Here .n/

.n/

.n/

.n/

Y˛.O/ D I˛.O/ . T H˛ C p /; ˛ D 1; : : : ; 5; Y6.O/ D I7.O/ . T H6 C p /; .n/

0

.n/

0

H˛ D H˛0 .I˛.O/ . C p //n˛ ; H6 D H60 .I7.O/ . C p //n6 ;

(9.133)

9.1 Models An of Plastic Continua at Large Deformations

617

are the simultaneous invariants, which are uniquely expressed in terms of the invariants (9.122) (see Exercise 9.1.6): .n/

.n/

Y˛.O/ D I˛.O/ . T / H˛ I˛.O/ . C p /;

˛ D 1; 2; 3I

.n/

.n/

.O/ .O/ .O/ .O/ 2 D I3C˛ . T / C H3C˛ I3C˛ . C p / 2H3C˛ J9C˛ ; Y3C˛

˛ D 1; 2I

(9.134)

.n/

.n/

.O/ Y6.O/ D I7.O/ . T / C H62 I7.O/ . C p / 2H6 J13 :

Calculating the derivatives @f [email protected]˛.O/ of the function (9.132) @f [email protected]˛.O/ D f˛ ; ˛ D 1; : : : ; 5I .O/ @f [email protected] D 2H5 f5 ;

.O/

.O/

@f [email protected] D f6 ; .O/

@f [email protected] D 2H6 f6 ;

@f [email protected] D 2H4 f4 ; f˛ @f [email protected]˛.O/

(9.135)

and substituting them into (9.131), we find the non-zero functions 11

D ~hf P ;

D 1; : : : ; 5;

D 2~hH P 5 f5 ;

13

10

D 2~hH P 4 f4 ;

D 2~hH P 6 f6 ;

14

D ~hf P 6:

(9.136)

Then the constitutive equation (9.130) takes the form .n/

C p

D ~h P

3 X D1

fb c2

! .n/ .n/ f3C C O ˝ O . T H3C C p / : 2

(9.137)

Multiplying this equation by itself, we obtain v u .n/ u .n/ u Cp Cp ; ~P D t .O/ PH P.O/ H

(9.138)

where .O/ PH

3 X 1

! .n/ .n/ f 3C O ˝ O . T H3C C p / : c2 C fb 2

(9.139)

The complete equation system for plastic deformations consists of relations (9.137), (9.138) and the following scalar equation of the yield surface: f D 0; where f is expressed by formula (9.132).

(9.140)

618

9 Plastic Continua at Large Deformations

In the quadratic model, the potential (9.132) is chosen in the form 3 X

jY.O/ j C Y.O/

D1

C 2 s

2f D

.O/

Y1

.O/

Y2

.O/

jY

C .O/

12s

!2

Y1

.O/

Y3

13s

.O/

j Y 2 s .O/

Y2

!2 C

.O/ Y3C

!2

C 2 3C;s

.O/

Y3

23s

1:

(9.141)

p

˙ The functions s .; w / are called the yield strengths in tension (or in compression) in the direction ; 3C; s .; wp3C / – the yield strengths in shear along the plane .˛; ˇ/; and ˛ˇ;s .; wp1 ; : : : ; wp3 / – the mixed yield strengths. All the functions are determined in experiments. The derivatives of the function (9.141) have the forms

f D

jY.O/ j C Y.O/ C 2 s

C

jY.O/ j Y.O/ .O/ ; f C3 D Y3C = 3C;s ; D 1; 2; 3: 2 s (9.142)

9.1.13 The Principle of Material Indifference for Models An of Plastic Continua .n/ .n/

.n/

.n/

.n/

.n/

All the energetic tensors T , C , and also C e and C p defined by T and C, are R-invariant. Then all the constitutive equations of models An of plastic continua, stated in Sects. 9.1.2–9.1.6, satisfy the principle of material indifference, because they remain unchanged at the change of actual configuration K ! K0 in rigid motion.

Exercises for 9.1 9.1.1. Show that the scalar invariant of the tensor T, being its intensity Y .T/ determined by (9.88), in any basis has the following component representation: Y 2 .T/ D Y 2 .T/D

3 P P; 2

1 P D T I1 .T/E; 3

1 2 2 2 CT23 CT13 / : .T11 T22 /2 C.T22 T33 /2 C.T33 T11 /2 C6.T12 2

9.1.2. Prove that the invariant Y defined in Exercise 9.1.1 can always be expressed in terms of the principal invariants as follows: Y 2 .T/ D I12 .T/ 3I2 .T/:

9.1 Models An of Plastic Continua at Large Deformations

619

9.1.3. Show that for the model of an isotropic elastoplastic continuum without strengthening, when H 0 (i.e. H0 D 0), Eq. (9.97) takes the form 2 PP p D .~h= P s /PT :

This relation is called the Prandtl–Reuss equation. 9.1.4. Show that the relation of orthogonality (9.117) holds for the tensors (9.116). .3/

9.1.5. Prove relations (9.112) between the simultaneous invariants Y˛ (9.111) and .3/ J (9.99). 9.1.6. Prove relations (9.134). 9.1.7. Consider the plastically compressible Huber–Mises model An of an isotropic continuum, where the constitutive equations (9.81)–(9.83) hold and there is only one plastic potential f depending, unlike (9.84), in addition on the first invariant Y1H : f D f .YH ; Y1H ; ; wp /: Here .n/

.n/

2n1 H1 D H10 Y1p ;

Y1H D . T H1 C p / E;

.n/

Y1p D I1 . C p /;

f1Y @f [email protected] ;

and H10 , n1 are the constants. This model describes the plastic properties of porous media, some grounds and also materials sensitive to the type of loading being volumetric tension or compression (see Sect. 9.6). Show that for this model the constitutive equations (9.92) take the form .n/ C p

1 D 3~h P fY PH C f1Y E : 3

Show that these equations written for the deviators and spherical parts of the tensors of plastic deformations and stresses have the forms (compare them with (9.94) and (9.95)) 8 ˆ PP D 3~hf P Y PH ; ˆ ˆ < P P 1Y ; I1 .Cp / D ~hf r ˆ q ˆ .n/ .n/ ˆ :~P D ˙ C C = f 2 Y 2 C 3f 2 ; f D 0: p p Y H 1Y Show that if the plastic potential f has the form 1 f D 3

YH S

2

C

YC2 T02

C

Y2 1; C02

1 1 1 D 2 2; 02 T T 3 S

1 1 1 D 2 2; 02 C C 3 S

620

9 Plastic Continua at Large Deformations

where S , T , and C are the yield strengths in shear, tension and compression, respectively (depend on and wp ), and YC and Y are invariants of constant signs Y˙ D

1 .jY1H j ˙ Y1H / ; 2

then fY D

1 3 S2

and f1Y D

2YC 2Y C 02 ; T02 C

and the constitutive equations take the form 8 .n/ ˆ 2 < P D .~h= P p S /PH ; ˆ : P .n/ 02 02 I1 . C p / D 2~h..Y P C = / C .Y = //: T

C

9.1.8. Consider the associated model of plasticity An (9.74) with quasilinear elasticity, for which the elastic potential (9.74a) depends only on linear and .n/

quadratic invariants I.s/ of the tensor T = (by analogy with the quasilinear models .n/

An of elastic continua (see Sect. 4.8.7)). Introducing for the functions ' .I.s/ . T =/; p ; wˇ / the representation ı

' D

r1 X

ı

0 lˇ I.s/ ; D 1; : : : ; r1 I

' D l0 ;

D r1 C 1; : : : ; r2 ;

ˇ D1 0 and l0 are functions in the form where lˇ .n/

0 0 lˇ D lˇ .I˛.s/ . T =/; ; wp˛ /;

0 0 lˇ D lˇ ;

show that in this case Eq. (9.74b) may be written as follows: .n/

.n/

C e D 4N T ;

ı

J D =:

Here the tensor 4 N has the form 4

ND

1 0 l1 E ˝ E C 2l20 ; J

0 0 0 l10 D l11 C 2l22 ; l22 D 2l20 ;

9.1 Models An of Plastic Continua at Large Deformations

621

– for isotropic media (r1 D 1; r2 D 2); 4

ND

10 0 0 l 022b c23 C .l12 l11 /.E ˝b c23 Cb c23 ˝ E/ l E ˝ E Ce c23 ˝b J 11 0 l11 0 0 l44 .O1 ˝ O1 C O2 ˝ O2 / C 2l44 ; C 2 0 0 0 0 e l 022 D l22 2.l44 C l12 / l11 ;

– for transversely isotropic media (r1 D 2; r2 D 4); 0 4

ND

[email protected] J

3 X

0 b c2 ˝b lˇ c2ˇ C

;ˇ D1

1

2 X

0 A l3C; 3C O ˝ O

D1

– for orthotropic media (r1 D 3; r2 D 6). Show that there exist inverse relations .n/

.n/

T D 4M C e ;

where the tensors 4 M are inverse of 4 N: 4

M 4 N D ;

and have formally the same structure as the tensor 4 N, but coefficients lˇ of the tensor 4 M are functions in another form .n/

lˇ D lˇ .I˛.s/ . C e /; ; wp˛ /:

9.1.9. For the models An of an elastoplastic continuum (9.79) and (9.80) with linear elasticity, whose potential (9.80) has the quadratic form D 0

ı r1 r2 X ı X .s/ 0 lˇ I.s/ Iˇ l I.s/ ; 2 Dr C1 ;ˇ D1

1

.n/

.s/ .s/ 0 where lˇ are the constants and I D I . T =/, show with use of the results of Exercise 9.1.8 that the tensors 4 N are tensor-constants up to the factor J; and the .n/

.n/

constitutive equations (9.74b) between C e and T take the form .n/

.n/

.n/

T D J.l1 I1 . C e /E C 2l2 C e /;

l1 D

l10 0 2l2 .3l10 C

2l20 /

; l2 D

1 ; 4l20

622

9 Plastic Continua at Large Deformations

– for isotropic materials; .3/ .3/ .3/ .3/ T D J .l11 I1 C l12 I2 /.E b c23 / C ..l22 2l44 /I2 C l12 I1 / b c23 ! .n/ .n/ l33 C l44 .O1 ˝ O1 C O2 ˝ O2 / C e C 2l44 C e ; 2

.n/

.n/

– for transversely isotropic materials (here I.3/ D I.3/ . C e /); 0 .n/

T [email protected]

3 X

lˇ I.O/b c2 C

3 X

1 .n/

l3C;3C O .O C e /A

D1

;ˇ D1

.n/

– for orthotropic materials (I.O/ D I.O/ . C e /). Show that for the models An of elastoplastic isotropic continua with linear elasticity, whose potential has the form ı

.n/ .n/ ı D 0 l10 I12 . T =/ l20 I1 . T 2 =2 /; 2

the constitutive equations (9.74b) become .n/

.n/ 1 0 .n/ .l1 I1 . T /E C 2l20 T /; J l0 l 1 ; l1 D 0 0 1 ; l10 D 2l2 .3l1 C 2l2 / 2l2 .3l1 C 2l20 /

Ce D

2l2 D

1 : 2l20

9.1.10. Using formulae (9.19) and (8.28b), show that the specific internal energy e for the models An of plastic continua is expressed by the formula .n/

eDC

T .n/ @ Ce : @

Show that for models An of an isotropic elastoplastic continuum with linear elasticity the results of Exercise 9.1.9 give 0 1 0 1 ! .n/ .n/ ı 2 l2 .n/ 0 2BTC ı 0 BT C l1 2 .n/ e D e0 C l1 I1 @ A C l2 I1 @ 2 A D e0 C ı I1 C e C ı I1 . C 2e /; 2 2 e0 D 0 [email protected]0 [email protected]/:

9.2 Models Bn of Plastic Continua

623

9.1.11. Show that relations (9.52) can be represented in the form independent explicitly of time: X d~˛ @f˛ d .n/ ; Cp D h d~ d~ .n/ ˛D1 @T k

where ~ D ~1 .

9.2 Models Bn of Plastic Continua 9.2.1 Representation of Stress Power for Models Bn of Plastic Continua Let us consider now models Bn for plastic continua. Constructing these models essentially differs from constructing the models An , because the main additive rela.n/

tion (9.3) for the energetic measures G does not hold. The relation (9.3) is replaced by Eq. (9.7), that is axiomatically assumed in models Bn and justified in Sect. 9.1.1. Since the relation (9.7) is the product of the gradients Fp and Fe , models Bn of plastic continua are called multiplicative, unlike models An which in this case are called additive. Consider the principal thermodynamic identity in the form Bn (4.122) d

.n/

.n/

C d T d G C w dt D 0:

(9.143)

Let us show that for models Bn the additive relation for the stress power derived in the following theorem is an analog of the additive relation (9.3). Theorem 9.1. The stress power w.i / (4.2) can always be represented in the additive form .n/

.n/

.n/

.n/

.n/

.n/

w.i / D T G D T e G e C T p D p ; .n/

(9.144)

.n/

where T e and T p are the symmetric tensors of elastic stresses and yield stresses, .n/

.n/

respectively, G e are the symmetric measures of elastic deformations, and D p are the symmetric energetic measures of plastic deformation rates; their expressions for n D I; : : : ; V are given in Table 9.1. .n/

.n/

.n/

Notice that the tensors T e and G e are entirely analogous to the tensors T and .n/

G and differ from them only by replacing F ! Fe and U ! Ue . The tensor Be , as

624

9 Plastic Continua at Large Deformations .n/

.n/

.n/

.n/

Table 9.1 Expressions for T e , T p , G e and D p at n D I; : : : ; V n

.n/

.n/

.n/

.n/

Te

Ge

Tp

Dp

12 U2 e

I

Te

U1 e

Te

I

FTe T Fe

II

1 .FTe 2

III

OTe T Oe

Be

Te

1 .F1 e 2

V

V

IV

Ue

Te

Dp

V

1T F1 e T Fe

1 2 U 2 e

Te

T Oe C Oe T Fe /

1 .Dp 2 I Dp

I III

T Oe C Oe T Fe1T /

2 T U2 e C Ue Dp /

Dp U1 e C T C U1 D e p Ue /

1 .UTe 2

V

1 .U2e 2

Dp C DTp U2e /

well as the tensor B, is determined by its derivative and initial value B0e (when there are no initial plastic deformations B0e D E): 1 P 1 1 P BP e D .U e Ue C Ue U e /; 2

Be .0/ D B0e :

(9.145)

The plastic deformation rate measure Dp is determined by the relation Dp D FP p F1 p

(9.146)

and is a nonsymmetric tensor. It follows from Table 9.1 that I

II

I

Tp D Tp D Te ; I

III

III

Tp D T e ;

II

IV Dp

Dp D Dp ;

IV Tp

V

V

D Tp D Te ;

V

D Dp :

I

I

H 1. Using the definition of the measure G and introducing the measure Ge by the I

1T similar formula Ge D F1 , with the help of the multiplicative decomposition e Fe (9.7) we obtain I I 1 1 1 1T 1T G D F1 F1T D F1 Fp1T D F1 p Fe Fe p Ge Fp ; 2 2

(9.147)

then I

I

I

I

I

I

1T 1T P 1T T G D T .F1 C FP 1 C F1 p Ge Fp p Ge Fp p Ge Fp /:

(9.148)

9.2 Models Bn of Plastic Continua

625

According to the rule of permutations of tensors in the triple scalar product (4.4), we get I

I

I

I

I

I

1T P 1 T G D .Fp1T T F1 T/ .F1 p / Ge C .Fp p Fp Fp Ge / I

I

T P 1T C .Fp1T T/ F1 p Ge Fp / Fp :

(9.149)

I

1 P 1 P Since Ge D .1=2/U2 e , Fp Fp D Fp Fp D Dp and I

1T T Fp1T T F1 FT T F F1 p D Fp p D F e T Fe ;

(9.150)

so the representation (9.144) really holds at n D I: I I I I 1 1 2 T 2 T G D .Fp1T T F1 p / .Ge C Dp Ue C Ue Dp / 2 2 I

I

D .FTe T Fe / .Ge C Dp /:

(9.151)

I

I

I

I

2. Since the tensors Te and Ge differ from tensors T and G only by the substitutions I

I

F ! Fe and U ! Ue , respectively, the first couple .Te ; Ge / gives all the remaining .n/

.n/

I

I

couples T e ; G e in the same way as the energetic couple .T; G/ does (the tensors Fp do not appear in these relations). 3. We should show only that the first couple gives the third and fifth ones: I

I

III

III

V

V

Te Dp D Te Dp D Te Dp

(9.152)

(the second and fourth couples coincide with the first and fifth ones). I

I

Indeed, according to the definitions of Te and Dp , we have I

I

I

Te Dp D .FTe T Fe / Dp 1 2 T D .Ue OTe T Oe Ue / .Dp U2 e C Ue De / 2 D

III III 1 T 1 T Oe T Oe .Ue Dp U1 e C Ue Dp Ue / D Te Dp 2 (9.153)

626

9 Plastic Continua at Large Deformations

and III

III

Te Dp D

D

1 T 2 1 .O T Oe U1 e / Ue Dp Ue 2 e 1 T 2 C .U1 OTe T Oe U1 e / Dp Ue 2 e V V 1 1 .Fe T Fe1T / .U2e Dp C DTp U2e / D Te Dp : N 2 (9.154)

Theorem 9.2. The yield stress power w.p/ can always be represented as a sum of the powers of plastic stretches and plastic rotations: .n/

.n/

P p C To p : w.p/ D T p D p D TU U

(9.155)

Here TU is the symmetric tensor of plastic stretch stresses, To is the skew-symmetric tensor of plastic rotation stresses, which are defined as follows: TU D

1 1 .F Te Op C OTp TTe Fp1T /; 2 p 1 To D .Te TTe /; 2 Te D F1 e T Fe ;

(9.156a) (9.156b) (9.157)

and p is the skew-symmetric spin tensor of plastic rotation: P p OTp : p D O

(9.158)

H Modify the plastic deformation rate measure Dp (9.146) as follows: 1 T P P Dp D FP p F1 p D .Op Up C Op Up / Up Op T P p OT C Op U P p U1 DO p p Op D p C Dv :

(9.159) Here we have denoted the plastic stretch rate measure T 1 P p U1 P Dv D Op U p Op D Op Up Fp :

(9.160)

Then the first couple in (9.152) can be written in the form I

I

w.p/ D Te Dp D

1I 2 T 2 2 T Te .p U2 e C Ue p C Dv Ue C Ue Dv /: (9.161) 2

9.2 Models Bn of Plastic Continua

627

According to the permutation rule for tensors in the triple scalar product, we obtain w.p/ D

I 1 2 I 1 1 2 I P .Ue Te Te U2 e / p C ..Fp Ue T Op / Up 2 2 I

1T P P C.OTp Te U2 e Fp / Up / D To p C TU Up :

(9.162) Here we have taken into account that the spin p is skew-symmetric and that I

I

2 2 T T 2 U2 e Te Te Ue D Ue Fe T Fe Fe T Fe Ue T 1T D F1 D Te TTe D 2T0 ; e T Fe F e T F e I

I

2 T 2 1 F1 p Ue T Op C Op T Ue Fp 2 T T T 2 1 D F1 p Ue Fe T Fe Op C Op Fe T Fe Ue Fp T T 1T D F1 D 2TU ; p Te Op C Op Te Fp T 1 because U2 e Fe D F e .

.n/

.n/

The fact that other couples T p ; D p give the same result (9.155) follows from equivalence of contractions of the couples (9.152). N On substituting the expression (9.155) into (9.144), we get representations for the stress power .n/

.n/

.n/

.n/

P p C T0 p : w.i / D T G D T e G e C TU U

(9.163)

9.2.2 General Representation of Constitutive Equations for Models Bn of Plastic Continua Substitution of this expression into the principal thermodynamic identity (9.143) yields .n/

.n/

P C P T e G e TU Up To p C w D 0:

(9.164)

Introduce the Gibbs free energy D

.n/ 1 .n/ T e Ge ;

(9.165)

628

9 Plastic Continua at Large Deformations

then for we obtain the principal thermodynamic identity in the form Bn : .n/

.n/

.n/

.n/

P C P C G e . T e =/ T v Up T o p C w D 0I

(9.166)

i.e. a change in the free energy is determined only by changing the functions , .n/

T e =, Up and Op . Therefore, in the model Bn of a plastic continuum, the Gibbs free energy is considered to be a functional in the form (similar to (9.21)) t

P P t .//; D .R.t/; R.t/; Rt ./; R D0

.n/

R D . T e =; Up ; OTp ; /:

(9.167)

.n/

According to the principle of equipresence, the measure G e is a functional (only a tensor one) in the same form .n/

t

P G e D Ge .R.t/; R.t/; Rt ./; RP t .//:

(9.168)

D0

By analogy with (9.24) and (9.25), introduce the equilibrium elastic deformation .n/

.n/

measure G 0e and nonequilibrium elastic deformation measure G 1e as follows: .n/

t

P t .//; G 0e D Ge .R.t/; Rt ./; R D0

.n/ G 1e

.n/

.n/

D G e G 0e :

(9.169)

Substituting the functionals (9.167)–(9.169) into the principal thermodynamic identity (9.166) and grouping like terms, we obtain the identity 0

1

@

.n/ G 0e A

@

C

.n/

@. T e =/ C

.n/

d

Te @ C @

0

@

d C

.n/

@. T e =/

.n/

1

B Te C @ A

@ P p C @ d C @ d U dO p PT @ @Up @O p

.n/

.n/

P p C G 1e . T e =/ C ı e C.w e TU U To p / dt D 0; (9.170) where we have introduced the notation e TU D TU Np ; Np D [email protected][email protected] /; e To D To No ; No D Op [email protected][email protected] /:

9.2 Models Bn of Plastic Continua

629

.n/

.n/

Since the differentials d. T e =/, d, d. T e =/ , d , d Up , d Op and dt are independent, the identity (9.170) is equivalent to the following set of relations: .n/

.n/

G 0e D @[email protected] T e =/;

(9.171)

D @[email protected];

(9.172)

.n/

P Tp D 0; P p D 0; @[email protected] @[email protected] T e =/ D 0; @[email protected]P D 0; @[email protected] .n/

.n/

TU C e To p G 1e . T e =/ ı; w D e

(9.173) (9.174)

which are constitutive equations for models Bn of plastic continua. As follows from relations (9.173), the free energy is independent of the rates P of reactive variables R: t

P t .//; D .R.t/; Rt ./; R D0

.n/

R D . T e =; Up ; OTp ; /:

(9.175)

However, this dependence is observed for the dissipation function w and for the .n/

functional G 1e : .n/

.n/

P G e 1 D G e 1 .R.t/; R.t/; Rt ./; RP t .//:

(9.176)

Thus, the model Bn of a plastic continuum is specified by four functionals: the scalar functional (9.175) for , the tensor functional (9.176) and two more tensor functionals for determining the tensors Up and OTp .

9.2.3 Corollaries of the Onsager Principle for Models Bn of Plastic Continua Just as for models An , we use the Onsager principle in order to construct function.n/

als for the tensors G 1e , Up and Op . Form the specific internal entropy production (4.728) with the help of expression (9.174): 0 q D w

.n/

1

.n/ q q B Te C P p Ce To p G 1e @ r D e TU U A ı r > 0

(9.177)

630

9 Plastic Continua at Large Deformations

and introduce thermodynamic forces Xˇ and fluxes Qˇ as follows: .n/

X1 D r ; X2 D e TU ; X3 D e To ; X4 D . T e =/ ; P p ; Q 3 D p ; Q 4 D Q1 D q=; Q2 D U

.n/ G 1e :

(9.178)

Then, according to the Onsager principle, the following tensor-linear relations between Qˇ and Xˇ hold: .n/

TU C L13 e To C L14 . T e =/ ; q= D L11 r C L12 e .n/

P p D L12 r C L22 e U TU C L23 e To C L24 . T e =/ ; .n/

(9.179)

p D L13 r C L23 e TU C L33 e To C L34 . T e =/ ; .n/

.n/

TU C L34 e To C L44 . T e =/ : G e D L14 r C L24 e Here tensors L˛ˇ are functionals with the general form (9.176) of reactive variables R D fTe =; Up ; OTp ; g; their special forms are given by a model of a plastic continuum considered. P and For models Bn of plastic yield, all L˛ˇ and are only functions of R and R p of the Taylor parameters wˇ : .n/

D . T e ; Up ; OTp ; ; wpˇ /; .n/

.n/

.n/

.n/ T e ;

P p ; wp /; P p; O L22 D 4 LU . T e ; Up ; OTp ; ; T e ; U ˇ L33 D 4 Lo . T e ; Up ; OTp ; ; L11 D ./;

P p ; wp /; P p; O U ˇ

(9.180a) (9.180b) (9.180c)

the remaining Lˇ D 0:

In this model, the cross–effects are not considered, and constitutive equations (9.179) take the form q D r ; P TU ; Up D 4 LU e 4 p D Lo e To ; .n/ G 1e

0:

(9.181a) (9.181b) (9.181c) (9.181d)

The tensor 4 LU is a fourth-order symmetric tensor, and the tensor 4 Lo is skewsymmetric in indices 1, 2 and 3, 4, and symmetric in pairs of indices .1; 2/ $ .3; 4/.

9.2 Models Bn of Plastic Continua

631

9.2.4 Associated Models Bn of Plastic Continua

In the associated model of plasticity Bn , the equations of plastic yield (9.181b) and (9.181c) are connected to the yield surface by the gradient law Pp D h U

k X

~Pˇ

ˇ D1

p D h

k X

@fˇ ; @TU

(9.182a)

@fˇ : @To

(9.182b)

~Pˇ

ˇ D1

Here fˇ are plastic potentials assumed to be functions in the form (9.180a), which can be considered as functions of TU , To and Up , OTp : fˇ D fˇ .TU ; To ; Up ; OTp ; ; wp˛ /:

(9.183)

An equation of the yield surface, just as in the model An , is given by the set of scalar equations (9.41) fˇ D 0; ˇ D 1; : : : ; k: (9.184) The function h, taking on a value 0 or 1 and determining a domain of plastic loading, is evaluated by formula (9.51), where as the partial derivative d 0 fˇ =dt we use an analog of formula (9.43): @fˇ @fˇ d 0 fˇ TP U C TP o : D dt @TU @To

(9.185)

The system of .9 C k/ scalar equations (9.182) and (9.183) (Eq. (9.182a) is equivalent to six scalar equations, (9.182b) – to three scalar equations due to skewsymmetry of the tensors p and To ) allows us to find the 9 C k scalar unknowns: six components of the tensor Up , three components of the tensor Op and k scalar functions ~P ˇ (ˇ D 1; : : : k) which are determined by Eqs. (9.182). The parameters ~Pˇ are functions in the form (9.180b) P p ; ; ; wp /: ~Pˇ D ~P ˇ .TU ; To ; Up ; OTp ; U ˇ

(9.186)

In the model Bn of an elastoplastic continuum, the elastic potential does not depend explicitly on plastic deformations; therefore, for this model the constitutive equations (9.180a) and (9.171) become .n/

.n/

.n/

.n/

D . T e ; ; wpˇ /; N1 0; N0 0; G e D K. T e ; ; wpˇ / D [email protected][email protected] T e /: (9.187)

632

9 Plastic Continua at Large Deformations

9.2.5 Corollary of the Principle of Material Symmetry for Associated Model Bn of Plasticity Apply the principle of material symmetry to the constitutive equations (9.182) and (9.184). To do this we should clarify how the introduced tensors with subscripts p and e are changed under an orthogonal transformation of the reference configuration ı

K ! K. Theorem 9.3. The tensors Oe , Ue , and Ve are H -invariant under orthogonal H ı

.n/

.n/

P p , Vp , T e , T p , TU , transformations: K ! K, and all the tensors Fp , Op , Up , U To , and are H -indifferent. p

H The unloaded configuration K introduced in Sect. 9.1.1 is in accord with the refı

erence configuration K. They must coincide (by definition), if loading is not plastic; for the associated model, this means that loading does not occur outside the yield ı

p

surface. Therefore, the configuration K must be transformed in the same way as K p (the local basis vectors r i must be H -indifferent), and the elastic deformation graı

dient Fe (9.6) must be transformed during the passage K ! K in the same way as F (see (4.188)):

Fe D Fe H;

(9.188)

where

p

Fe D r i ˝ r i ;

p

pi

r D H1T r i ;

p ri

p

D H ri :

(9.189)

Thus, the tensors Oe , Ue and Ve are transformed in the same way as O, U and V

(see (4.256), (4.257)) under orthogonal transformations with the tensor H D QT :

Oe D Oe Q;

Ve D Ve ;

Ue D QT Ue Q:

(9.190)

Figure 9.2 shows a scheme of transformations of different configurations at the ı

change of reference configuration K ! K.

Fig. 9.2 H -transformations of reference and unloaded configurations

9.2 Models Bn of Plastic Continua

633

According to (9.189) and (4.194), the plastic deformation gradient Fp (9.6) under ı

the transformation K ! K takes the form

p

ı

p

Fp D r i ˝ ri D H r i ˝ ri H1 D QT Fp QI

(9.191)

i.e. it is H -indifferent under orthogonal transformations. Using the polar decompo

sition for Fp and Fp , we obtain

Fp D Qp Up D QT Op Up Q D .QT Op Q/ .QT Up Q/:

Since the tensor .QT Op Q/ is orthogonal, the tensor .QT Up Q/ is symmetric and the polar decomposition is unique, we get the following transformation formulae for Op and Up :

Qp D QT Op Q;

Up D QT Up Q:

(9.192)

The tensor Vp is transformed as follows:

Vp D Fp OTp D QT Fp Q QT OTp Q D QT Vp Q;

(9.193)

P p and Vp are H -indifferent under orthogonal then the tensors Tp , Op , Up , U transformations. Due to H -invariance of the tensor T, the tensors of elastic and plastic defor.n/

.n/

mations ( T e and T p , respectively) defined by Table 9.1 are H -indifferent under .n/

.n/

orthogonal transformations as well as the tensors T . Similarly, the tensors G e are H -indifferent too. The spin tensor p (9.158) is H -indifferent as well as the tensors TU (9.156) and To (9.157), because

T 1 T Te D F1 e T Fe D Q Fe T Fe Q D Q Te Q;

TU D

(9.194)

1 T 1 T .Q Fp Q Q Te Q QT Op Q 2

C QT Qp Q QT Te Q QT Fp1T Q/ D QT TU Q: N Applying the principle of material symmetry to Eq. (9.182) and taking H P p , p , TU , To , Up and OTp into account, we obtain indifference of the tensors U that the equations of plastic yield (9.182) must be H -indifferent tensor functions

634

9 Plastic Continua at Large Deformations ı

relative to one or another orthogonal subgroup G s , and the plastic potentials fˇ ı

(9.183) must be scalar H -indifferent functions relative to G s and hence depend on ı

simultaneous invariants J.s/p in this group G s : fˇ D fˇ .J.s/p ; ; wp˛ /;

J.s/p D J.s/ .TU ; To ; Up ; OTp /; D 1; : : : ; z: (9.195)

The scalar functionals wp˛ are integrals of the quadratic simultaneous invariants of P p and To , p : the tensors TU , U Z wp˛

0

Z wp˛

t

D t

D 0

P p .// d ; ˛ D 1; : : : ; r1 ; J˛.s/ .TU ./; U (9.196)

J˛.s/ .To ./;

.// d ; ˛ D r1 C 1; : : : ; r:

Similarly, the elastic potential (9.184a) is a H -indifferent scalar function too; therefore, it can be written in the form .n/

D .J.s/ . T e ; Up ; OTp /; ; wp˛ /:

(9.197)

Then constitutive equations (9.171) and (9.182) can be represented in the tensor basis: .n/

Ge D

z X D1

' D [email protected][email protected].s/ /; Pp D U

z1 X

.s/

' J T ; .s/

(9.198a) .n/

J T D @J.s/ [email protected] T e ;

(9.198b)

.s/ ˛ J˛TU ;

(9.198c)

.s/ ˛ J˛To ;

(9.198d)

˛D1

p D

z1 X ˛D1

˛

Dh

k X ˇ D1

.s/ .s/ ~P ˇ [email protected]ˇ [email protected]˛.s/ /; J˛T D @J˛.s/ [email protected] ; J˛T D @J˛.s/ [email protected] : o U

(9.198e)

9.2 Models Bn of Plastic Continua

635

9.2.6 Associated Models of Plasticity Bn with Proper Strengthening We will consider below only the case of associated models Bn of plasticity with proper strengthening, where the simultaneous invariants J.s/p (9.195) depend only on the tensors TU and To and do not depend explicitly on Up and OTp : J.s/p D J.s/ .TU ; To /:

(9.199)

Since the tensors TU and To defined by formulae (9.156) and (9.158) are combi.n/

nations of the tensors T , Ue , Oe , Up and Op , the model (9.199) allows us to take account of plastic strengthening of a continuum (an increase of the yield strength after the appearance of plastic deformations) but in the special way, namely in terms of the tensors TU and To . Notice that since the tensor To is skew-symmetric, it has only three independent components, and this tensor can be connected uniquely with the vorticity vector (see (2.227)) 1 !0 D " To : (9.200) 2 Then the simultaneous invariants (9.199) are scalar H -indifferent functions of the symmetric tensor TU and the vector !0 : J.s/p D J.s/ .TU ; !0 /:

(9.201)

9.2.7 Associated Models of Plasticity Bn for Isotropic Continua Let us write the constitutive equations (9.198) for the three main symmetry groups ı

G s : O, T3 and I while considering only the case of elastoplastic continua, i.e. while the simultaneous invariants J.s/ in (9.197) coincide with the invariants .n/

I.s/ . T e /. For an isotropic continuum, the functional basis of invariants (9.201) consists of six elements, which can be chosen as follows: J.I / D J .TU /; D 1; 2; 3; J5.I /

D !0 TU !0 ;

J6.I /

J4.I / D j!0 j2 ; D !0

T2U

!0 :

(9.202)

636

9 Plastic Continua at Large Deformations .s/

.s/

The derivative tensors J TU and J To in this case become .I / .I / .I / .I / D E; J2T D EI1 .TU / TU ; J3T D T2U I1 TU C I2 E; J4T D 0; J1T U U U U .I / .I / J5T D !0 ˝ !0 ; J6T D !0 ˝ TU !0 C !0 TU ˝ !0 ; U U .I / .I / J4T D !0 "; J5To D .!0 TU C TU !0 / "; o .I /

.I /

J6To D .!0 T2U C T2U !0 / "; J To D 0; D 1; 2; 3: Here we have used that

1 ": 2

d !0 =d To D

(9.203) (9.204)

The tensors J.IT/o are seen to be skew-symmetric. According to the Onsager principle, Eqs. (9.198c) and (9.198d) must be quasilinear with respect to TU and To , i.e. they must have the forms (9.181b) and (9.181c); therefore, invariants J3.I / , J5.I / and J6.I / must be eliminated between ar.I / guments of the potential fˇ (9.195). Thus, fˇ depends only on three invariants J , D 1; 2; 4. On substituting the expressions (9.202) and (9.203) into (9.198), we obtain the following constitutive equations of the associated model Bn of an isotropic elastoplastic continuum with proper strengthening: 8.n/ .n/ .n/ ˆ ˆ '1E C e ' 2 T e C '3 T 2e ; 1 correspond to all-round compression (pe > 0), and J < 1 – to all-round tension .pe < 0/. When J > 1, for all n the function pe .J / is monotonically increasing; for n D I and II it is convex downwards, but for n D IV and V it is convex upwards. When 0 < J < 1, the function pe .J / has an extremum for n D I; II and IV and pe .0/ D 0; and for n D V: pe ! 1 as J ! 0. Thus, different models An give qualitatively different diagrams pe .J /.

9.6.4 The Case of a Plastically Compressible Continuum

Let us consider now the Huber–Mises model An for a plastically compressible medium (see Exercise 9.1.7). Since for this problem PT D 0 and PH D Pp , the results of Exercise 9.1.7 give P Y Pp ; PP p D 3~hf .n/

IP1 . C p / D ~hf P 1Y ;

(9.313)

f D 0:

(9.314)

The first relation has the zero solution: Pp D 0, and the second one admits a non.n/

trivial solution; hence the tensor C p in this case is spherical: .n/

Cp D

1 Y1p E; 3

.n/

Y1p I1 . C p /:

(9.315)

9.6 The Problem on All-Round Tension–Compression of a Plastic Continuum

663

The simultaneous invariant Y1H in this case, according to (9.305) and (9.315), becomes .n/

.n/

Y1H D . T H1 C p / E D .3e p e C H1 Y1p /; YH 0; 1 2n1 H1 D H10 Y1p ; Y˙ D .jY1H j ˙ Y1H /: 2

(9.316) (9.317)

Equation (9.314) serves for determination of ~, P and the first invariant of the plastic deformation tensor Y1p can be found from the plasticity condition f D 0. If f is assumed to have a quadratic form (see Exercise 9.1.7), then f D

YC2 Y2 C 1 D 0: 02 T C02

(9.318)

On substituting the expressions (9.316) and (9.317) into (9.318), we find ( H1 Y1p D

pe ; C0 3e

. T0

C 3e p e /;

if Y1H < 0;

(9.319)

if Y1H > 0:

Here the yield strengths C0 and T0 , by definition, are assumed to be positive: C0 > 0, T0 > 0, that leads to the proper choice of signs in (9.318) for C0 and T0 : if p e D C0 =3 > 0, and in tension e p e D T0 =3 < 0. Y1p D 0, then in compression e Remind that the equation f D 0 holds if and only if the change in plastic deformation is different from zero. If f < 0 (in the considered case this occurs when .n/

.n/

.n/

Y1H > C0 or Y1H 6 T0 ), then C p D 0, or that is the same: C p .t/ D C p D .n/

.n/

const, where C p D C p .t / is the value reached in the preceding cycle of plastic loading up to time t of the beginning of unloading. Before initial loading, at t D 0 one usually assumes that there are no plastic .n/

deformations, therefore C p .0/ D 0 (Fig. 9.6). Hence, relations (9.319) hold true only under plastic loading. When f < 0, i.e. under unloading or under loading in an elastic domain these relations should be replaced by Y1p .t/ D Y1p . Here Y1p is the value reached in the preceding cycle of plastic loading.

Fig. 9.6 The cycle of plastic loading and subsequent unloading

664

9 Plastic Continua at Large Deformations

On substituting the expression (9.316) for H1 into (9.319), we find that

Y1p

8 0 1 ˆ ˆ p e j 2n1 C1 j C 3e 0 ˆ ˆ sign . C 3e pe / ; if Y1H 6 C0 ; ˆ ˆ H10 < 0 1 D j T C 3pe j 2n1 C1 0 ˆ pe / ; if Y1H > T0 ; sign . T C 3e ˆ ˆ H10 ˆ ˆ ˆ :Y ; if C0 < Y1H < T0 ; 1p (9.320)

where

Y1H D 3e pe C H10 jY1p j2n1 C1 sign Y1p :

Since relations (9.306) and (9.307) for elastic deformations still hold for plastically compressible continuum, so, combining them with (9.315) and (9.320), we obtain .n/

.n/

.n/

C D Ce C Cp D

1 3

pe k IIInC3 Y1p E: K

(9.321)

Then, according to formula (9.299), we get 3 pe IIInC3 Y1p ; .1 k nIII / D k n III K

or pe D K

3.1 k nIII / C Y1p k nIII3 : n III

(9.322)

Combining (9.322) with expression (9.320), we find a nonlinear relation between pe and k. These relations depend on the path of loading of a plastic material. The following example illustrates this fact.

9.6.5 Cyclic Loading of a Plastically Compressible Continuum Let the function e p e pe k IIIn be given in the form of nonmonotone dependence upon t (Fig. 9.7), where e p e .0/ D 0 and Y1p D 0, and H1 > 0. Then from the expression (9.320) for different sections of the function e p e .t/ we get OA W 0 < e p e 6 C0 =3; Y1H D 3e pe > C0 ; Y1p D 0 – elastic compression, p e CY1p H1 / D C0 ; Y1p D . C0 3e p e /=H1 < 0 AB W e p e > C0 =3; Y1H D .3e – plastic compression, BC W T0 =3 6 e p e 6 C0 =3; Y1H D .3e p e C Y1p H1 / > C0 ; Y1p D Y1p 0 DE W e p e 6 T0 =3; Y1H D T0 ; Y1p D . T0 C 3e – plastic tension, EF W e p e > T0 =3; Y1H D .3e p e C Y1p H1 / < T0 ; Y1p D Y1p >0

– unloading. As follows from these relations, after the appearance of plastic deformations at points, where values of e pe are the same, for example, at A and A0 , values of Y1p and k are distinct. In other words, in the plasticity domain the diagrams pe .k/ under loading and unloading are not coincident. Remark. If after preceding plastic compression (sections AB and BC 0 ) (Fig. 9.7) there occurs a tension (the section C 0 C ), then the yield strength in tension changes up to the value T0 =3 in comparison with the value of T0 =3 when there is no preceding plastic compression. This effect really occurs in many plastic materials and is called the Bauschinger effect. t u

666

9 Plastic Continua at Large Deformations

In the model considered, T0 can be found from relation (9.323) at the point C : ˇ Y1H D 3e p e ˇC C Y1p H1 D T0 :

(9.324)

Since Y1p H1 can be evaluated from the conditions (9.323) at the point B:

ˇ Y1p H1 D C0 3e p e ˇB ;

(9.325)

according to (9.324) and (9.325), we obtain ˇ ˇ p e ˇC D T0 C0 C 3e p e ˇB : T0 3e

(9.326)

Thus, the yield strengths in tension T0 and T0 differ from one another by the value . C 3e p e / of excess of the load 3e p e above the yield strength in compression. This result is a typical feature of the Huber–Mises model considered.

9.7 The Problem on Tension of a Plastic Beam 9.7.1 Deformation of a Beam in Uniaxial Tension Consider the above-mentioned classical problem on tension of a beam (see Example 2.1 and Exercises 2.1.1, 2.3.2, and 4.2.13), but for the case when the motion of a beam is described by the model An (the Huber–Mises model) of isotropic elastoplastic continuum, i.e. by Eqs. (9.301) and (9.92). The motion law for the problem is independent of type of a continuum and it is sought in the form (6.131): x ˛ D k˛ .t/ X ˛ ;

˛ D 1; 2; 3:

(9.327) .n/

The deformation gradient F and the energetic deformation tensors C in the problem on tension of a beam (see Exercises 2.2.1 and 4.2.13) have the diagonal forms FD

3 X

k˛ eN ˛ ˝ eN ˛ ;

(9.328)

˛D1 .n/

CD

3 .n/ X CN ˛˛ eN ˛ ˝ eN ˛ ;

(9.329)

˛D1 .n/

CN ˛˛ D

1 .k nIII 1/: n III ˛

(9.330)

9.7 The Problem on Tension of a Plastic Beam

667

We can find a change in the density for this problem as follows: ı

J D = D det F1 D

1 : k1 k2 k3

.n/

(9.331)

.n/

Tensors of elastic and plastic deformations C e and C p are also sought in the diagonal form: .n/

Ce D

3 .n/ X C e˛˛ eN ˛ ˝ eN ˛ ;

(9.332)

˛D1 .n/

Cp D

3 .n/ X C p˛˛ eN ˛ ˝ eN ˛ :

(9.333)

˛D1

Due to the additivity relation (9.3), we get .n/

.n/

.n/

C ˛˛ D C e˛˛ C C p˛˛ ; ˛ D 1; 2; 3:

(9.334)

9.7.2 Stresses in a Plastic Beam Equations (9.301) and (9.302) for the problem also hold true. .n/

.n/

On passing to Cartesian components T ˛ˇ of the tensor T , from (9.301), (9.302), and (9.332) we get that only diagonal components of this tensor are nonzero: .n/

.n/

.n/

T ˛˛ D J.l1 I1 . C e / C 2l2 C e˛˛ /; .n/

T D

˛ D 1; 2; 3:

3 .n/ X T ˛˛ eN ˛ ˝ eN ˛ :

(9.335) (9.336)

˛D1 .n/

A relation between T and T has the form (6.138) .n/

˛˛ D k˛nIII T ˛˛ ; ˛ D 1; 2; 3I

TD

3 X

˛˛ eN ˛ ˝ eN ˛ :

(9.337)

˛D1

The equilibrium equations (9.292), when f D 0, are satisfied identically. Since the lateral surface X ˛ D ˙h0˛ =2 (˛ D 2; 3) of the beam is assumed to be free of loads, from the boundary conditions at this surface (n T D 0), just as for an ideally elastic body, we obtain ˛˛ D 0;

.n/

T ˛˛ D 0; ˛ D 2; 3:

(9.338)

668

9 Plastic Continua at Large Deformations

Substituting the values (9.338) into (9.335) and summing these three relations (9.335), we find that .n/

.n/

I1 . C e / D

T 11 : J.3l1 C 2l2 /

(9.339)

Substitution of the expression (9.339) for the first invariant into (9.335) at ˛ D 1 .n/

.n/

yields the relation between T 11 and C e11 ; and from formula (9.335) at ˛ D 2; 3 we .n/

.n/

.n/

find the relation between C e22 , C e33 , and C e11 : .n/

.n/

T 11 D JE C e11 ;

.n/ C e22

.n/

D C e33 D

(9.340)

.n/ C e11 :

(9.341)

As before, here we have denoted the elastic modulus and Poisson’s ratio at large deformations: ED

.3l1 C 2l2 /l2 ; l1 C l 2

D

l1 : 2.l1 C l2 /

(9.342)

9.7.3 Plastic Deformations of a Beam Let us consider now constitutive equations (9.92) and (9.93) for plastic deformations when an isotropic medium is assumed to be plastically incompressible. Due to the .n/

assumption (9.333) that the tensor C p is diagonal, from (9.92) we get ~h P d .n/p C ˛˛ D dt 3 s2

.n/ .n/ 1 .n/ 1 T ˛˛ T 11 H. C p˛˛ Y1p / ; ˛ D 1; 2; 3: (9.343) 3 3

Here the strengthening parameter H and the first invariant Y1p have the forms H D H0 Yp2n0 ; Yp2 D

.n/

Y1p D I1 . C p /;

(9.344)

.n/ .n/ .n/ .n/ .n/ 1 .n/p .. C 11 C p22 /2 C . C p22 C p33 /2 C . C p11 C p33 /2 /: 2

Notice that Eqs. (9.343) and (9.344) are symmetric in indices ˛ D 2; 3; therefore, .n/

p C 22

.n/

p

D C 33 , and from the condition of plastic incompressibility (9.93) (Y1p D 0) we obtain .n/ .n/ 1 .n/ C p33 D C p22 D C p11 : (9.345) 2

9.7 The Problem on Tension of a Plastic Beam

669

Then the system (9.343) takes the form

d .n/p C dt ˛˛

.n/ .n/ 2 d .n/p ~h P p C 11 D T H C 11 11 ; dt 3 s2 3 .n/ ~h P 1 .n/ p D T 11 H C ˛˛ ; ˛ D 2; 3: 3 s2 3

(9.346)

(9.347)

Equations (9.347) follow from (9.346) and (9.345), therefore, the system (9.346), (9.347) contains only one independent equation, namely (9.346). This equation .n/

.n/ p

allows us to evaluate the parameter of loading ~P if values of T 11 and C 11 are known. .n/ p

To evaluate C 11 we use the yield surface equation (9.96), which for this problem becomes (here we have taken the expression for YH from Exercise 9.1.1 into account): .n/ .n/ .n/ .n/ .n/ 1 .n/ .. T 11 H. C p11 C p22 //2 C . T 11 H. C p11 C p33 //2 2 6 s .n/

.n/

CH 2 . C p22 C p33 /2 / D 1:

(9.348)

Due to (9.345), this equation is simplified: .n/ p 3 .n/ j T 11 H C p11 j D 3 s : 2

(9.349)

Since .n/

.n/

H D H0 Yp2n0 D H0 . C p11 C p22 /2n0 D H0

3 .n/p C 2 11

2n0 ;

(9.350)

Eq. (9.349) takes the final form ˇ.n/ ˇ T 11 H0 .n/

3 .n/p C 2 11

2n0 C1

ˇ p ˇ D 3 s :

(9.351)

To find an expression for C p11 from (9.351), by analogy with formulae (9.318)– (9.320) we should consider separately the cases of plastic tension, plastic compression and leaving the yield surface. Then from (9.351) we get

670

9 Plastic Continua at Large Deformations

.n/

p

C 11

where

8 p .n/ ˆ ˆ .n/ p j T 11 3 s j 1=.2n0 C1/ 2 ˆ ˆ ˆ T 3 / ; sign . 11 s ˆ ˆ3 H0 ˆ ˆ p ˆ ˆ ˆ if Y1H > 3 s ; ˆ < p .n/ D 2 .n/ p j T 11 C 3 s j 1=.2n0 C1/ ˆ ˆ T C 3 / ; sign . 11 s ˆ ˆ3 H0 ˆ ˆ p ˆ ˆ ˆ if Y1H 6 3 s ; ˆ ˆ ˆ p p ˆ :.n/p C 11 ; if 3 s < Y1H < 3 s ;

(9.352)

.n/ .n/ ˇ 3 .n/ ˇ2n C1 Y1H D T 11 H0 ˇ C p11 ˇ 0 sign . C p11 /: 2 .n/

.n/

Here we have taken into account that s > 0, and C 11 p is the value of C p11 reached .n/ p

at the time t of leaving the yield surface (at t D 0: C 11 D 0).

9.7.4 Change of the Density .n/

.n/

According to Eqs. (9.334), (9.341) and (9.345), we can express C 22 in terms of C 11 .n/ p

and C 11 :

.n/

.n/

.n/

p

.n/

C 22 D C e22 C C 22 D C e11

i.e. .n/

p C 22

1 .n/p C ; 2 11

.n/ .n/ 1 p D C 11 C C 11 : 2

(9.353) .n/

.n/

With the help of relations (9.330) we can represent k2 in terms of C 22 and C 11 in terms of k1 : .n/ .n/ 1=.nIII/ 1 D 1 .n III/ C 11 C C p11 k2 D .1 C .n III/ C 22 / 2 .n/ 1=.nIII/ 1 nIII D 1 .k1 1/ .n III/ : (9.354) C p11 2 .n/

1=.nIII/

.n/ p

Thus, we have obtained the expression of k2 in terms of k1 and C 11 . Taking the equality k2 D k3 into account and substituting (9.354) into (9.331), we get .n/ 2=.nIII/ 1 1 nIII J D ı D 1 .k1 C p11 1/ .n III/ : (9.355) k 2 1

9.7 The Problem on Tension of a Plastic Beam

671

Notice that although the continuum considered is plastically incompressible, however, unlike the problem on all-round compression, in this case the density depends on plastic deformations.

9.7.5 Resolving Equation for the Problem .n/

.n/

.n/

Equations (9.340) and (9.334) yield the relation between T 11 , C 11 , and C p11 : .n/

C 11 D

.n/ C p11

.n/

C

T 11 : EJ

(9.356) .n/

1. Consider the case of initial plastic tension, when the conditions C p 11 D 0 and .n/ .n/ p p T 11 3 s are satisfied. Then for the plastic deformation C 11 , from the first line of formula (9.352) we find the expression .n/

.n/

p C 11

D

T

11

p 3 s 1=.2n0 C1/ ; e0 H

e 0 D H0 .3=2/2n0C1 : (9.357) where H .n/

Substituting (9.357) into (9.356) and replacing C 11 by k1 according to formula .n/

(9.330), and T 11 by 11 according to (9.337), we obtain k IIIn 11 2 k1IIIn 11 k1nIII 1 D 1 C e0 n III EJ 3 H

p 3 s 1=.2n0 C1/

:

(9.358)

Combining this equation with expression (9.355) for J , into which formula (9.357) has been substituted: k IIIn 1 1 11 J D 1 .k1nIII 1/ .n III/. / 1 e0 k1 2 H

p

2 3 s 2n01C1 nIII

;

(9.358a)

we find the main resolving equation for this problem in initial plastic tension. This equation has a form of the implicit relation ˆ. 11 ; k1 / D 0 between 11 and k1 . 2. Under initial loading in the elastic domain there is no plastic deformation .n/

( C p11 D 0), and formulae (9.358) and (9.358a) coincide with the resolving equation for models An of elastic continua (6.148): 11 D E

k1nIII1 nIII .k 1/.1 .k1nIII 1//2=.IIIn/ : n III 1

(9.359)

672

9 Plastic Continua at Large Deformations

3. If there is initial plastic loading in the compression domain when the conditions .n/ .n/ p C p 11 D 0 and T 11 3 s are satisfied, then in formula (9.352) we choose the second condition. As a result, for plastic deformation we have .n/

C 11

p

jk1IIIn 11 C D e0 H

!1=.2n0 C1/ p 3 s j

;

(9.360)

and instead of (9.358) and (9.358a), we get the resolving relation k IIIn 11 k1nIII 1 D 1 n III EJ

jk1IIIn 11 C e0 H

p

3 s j

!1=.2n0 C1/ ;

jk IIIn C 1 1 11 J D 1.k1nIII 1/.nIII/. / 1 e0 k1 2 H

(9.361)

p 2 3 s j 2n01C1 IIIn

:

(9.361a)

4. At last, if there occurs unloading after plastic loading in the domain of tension or compression, then from Eqs. (9.355) and (9.356) we obtain the relation between 11 and k1 k IIIn 11 .n/p k1nIII 1 (9.362) D 1 C C 11 ; n III EJ 1 J D k1

1=.nIII/ .n/ 1 p nIII 1/ .n III/. / C 11 : 1 .k1 2

(9.362a)

.n/

Here C p 11 is the maximum value of plastic deformations reached at the time t of the beginning of unloading; this value is calculated by formula (9.357) for preceding plastic tension and by (9.360) – for compression.

9.7.6 Numerical Method for the Resolving Equation Transpose the first summand from the left-hand side of Eq. (9.358) onto the righthand side, and then raise the obtained expression to power 1 C 2n0 . As a result, we find the equation

k nIII 1 k1IIIn 11 A 1 n III EJ

k IIIn 11 D 1 e0 H

p 3 s

;

(9.363)

:

(9.364)

where we have introduced the notation A.k1 ; 11 /

k IIIn k1nIII 1 11 n III J.k; 11 /

2n0

9.7 The Problem on Tension of a Plastic Beam

673

To solve numerically Eq. (9.363) we can apply the method of step-by-step approxifm1g mations, where by given values of k1 and values of 11 at the .m 1/th iteration fmg at the mth iteration: we find a value of 11 e fmg D

1 e 0 Afm1g =J fm1g 1CH

! e 0 Afm1g k nIII p H k nIII ; 3 s C n III

(9.365)

fm1g fm1g where J fm1g D J.k; 11 / and Afm1g D A.k1 ; 11 / are values of the functions at preceding .m 1/th iteration (m D 1; 2; : : :). As an initial value of f0g fmg we can take the value of 11 computed at the preceding iteration cycle for the 11 preceding value of k1 . The method is convergent for values of n0 within the interval 0:5 < n0 < 1. We can solve Eq. (9.361) in the compression domain by a similar way. In the domains of elastic loading and unloading the stress 11 can be determined in the explicit way from Eqs. (9.359) and (9.362). Figure 9.8 shows the functions 11 .k1 / obtained by the above-mentioned numerical method for different models An and at different values of the parameter

a

b

c

d

Fig. 9.8 Diagrams 11 .k1 / for models An of elastoplastic continuum in uniaxial tension (e – the e 0 =E, n0 D 0:1): elastic continuum model, numbers at curves are different values of parameters H (a) – model AI , (b) – model AII , (c) – model AIV , and (d) – model AV

674

9 Plastic Continua at Large Deformations

a

b

c

d

Fig. 9.9 Diagrams of deforming 11 .k1 / for models An of an elastoplastic continuum in uniaxial tension followed by unloading (e – the model of an elastic continuum, D 0:3): (a) – model e 0 =E D 0:2, n0 D 0:1, s =E D 0:001), (b) – model AV (H e 0 =E D 0:05, n0 D 0:1, AI (H e 0 =E D 0:15, n0 D 0:1, s =E D 0:002), and (d) – model AIV s =E D 0:002), (c) – model AII (H e 0 =E D 0:05, n0 D 0:1, s =E D 0:002) (H

e 0 . For the models AI and AII , the functions 11 .k1 / are convex upwards for a H purely elastic continuum and for elastoplastic models; but for the models AIV and AV in the plastic domain (at k > ks , where ks is the elongation at the yield point: .n/ p T 11 .ks / D 3 s ) the functions 11 .k1 / are convex downwards. For all the models An , the appearance of plastic deformations can cause considerable decreasing values of stresses 11 in comparison with an elastic continuum. Figure 9.9 shows diagrams of deforming 11 .k1 / for different models An of an elastoplastic continuum under plastic loading up to some limiting values k and subsequent unloading, which have been computed by Eqs. (9.358) and (9.359). Let us note some important effects caused by large values of plastic deformations: 1. When s =E 6 0:01, diagrams of deforming in the elastic domain at k < ks and under unloading for values k close to 1 (k . 1:1) are practically linear and

9.7 The Problem on Tension of a Plastic Beam

675

have the same slope; however, at higher values of k (k & 1:2) these diagrams are considerably distinct: diagrams of unloading become essentially nonlinear for all models An ; 2. For the models AI and AII , the slope of a tangent to the function 11 .k1 / under unloading decreases with growing k , and for the models AIV and AV it, on the contrary, increases.

9.7.7 Method for Determination of Constants H0 , n0 , and s

The model An of an isotropic plastically-incompressible continuum with the Mises potential (9.301), (9.97) contains five constants: E, , H0 , n0 and s . If the exper.ex/ .ı1 / (diagram of deforming) under active loading imental dependence 11 D 11 and corresponding curves of unloading arep known, then the constant s can be determined by the formula s D k1IIIn 11s = 3. Here 11s is the value of stress 11 at the diagram of deforming, after unloading from which the residual elongation ı1p at 11 D 0 takes on a value given a priori. For metals and alloys, and also for grounds and rocks, one usually assumes that ı1p D 0:2 %, and the corresponding value 11s is denoted by 0;2 (Fig. 9.10). This value is the yield strength. Since p ı1 D 0:002 1, the domain p of elasticity for such materials corresponds to small deformations and s D 11s = 3; the elastic modulus E can be determined as a .ex/ slope of the tangent to the initial section of the experimental diagram 11 .ı1 /. Just as for elastic media, Poisson’s ratio is usually calculated by formula (6.147) as the ratio of transverse elongation of a beam to its longitudinal elongation in the domain of small deformations. e 0 and n0 can be calculated by approximating the experimental The constants H .ex/ .ı1 / under active plastic loading with the help of the diagram of deforming 11 relation 11 .ı1 / (9.358). To do this, one should minimize the functional of meansquare distance between the experimental and theoretical curves at N points: !1=2 N 11 .ı1.i / / ˇˇ2 1 X ˇˇ D ! min: (9.366) ˇ1 .ex/ ˇ N 11 .ı1.i / / i D1

Fig. 9.10 Diagram of deforming for aluminium alloy and method for determination of the yield strength 0;2 : 1 – active loading, 2 – unloading

676

9 Plastic Continua at Large Deformations

9.7.8 Comparison with Experimental Data for Alloys Figure 9.11a shows experimental diagrams of deforming for steel alloy under temperatures 20ı C and 800ı C, and also their approximations by the method, mentioned in Sect. 9.7.6, with the help of different models An according to formula (9.358). e 0 , n0 and s , calculated when D 0:35. Table 9.3 gives values of the constants E, H An accuracy of approximation proves to be high for all the models An ; the model AI exhibits the least value of the deviation . Notice that the value of coefficient n0 is negative, therefore the diagram of deforming is convex upwards in the vicinity of the yield point s . Figures 9.11b and 9.12a show experimental diagrams of deforming for aluminium alloys D16 and AK4 in tension and their approximations by the models An with the help of formula (9.358); and Table 9.3 gives values of the e 0 , n0 , and s . An accuracy of approximation is also sufficiently high constants E, H for all the models An .

a

b

Fig. 9.11 Diagrams of deforming: (a) – for steel alloy under temperatures 20ı C and 800ı C and (b) – for aluminium alloy D16 in tension

e 0 , n0 , s , and in different models An for steel and aluminium alloys Table 9.3 Values of E, H Steel alloy at 20ı C and 800ı C Aluminium alloys D16 and AK4 n I II IV V n I II IV V

E, GPa

e 0 , GPa H n0 s , MPa , %

200 60 4:8 0:72 0:02 0:26 145 46:3 6:4 3:7

200 60 3 0:54 0:08 0:26 145 46:3 2:5 5:4

200 60 1:2 0:18 0:18 0:38 145 46:3 5:4 14

200 60 0:6 0:18 0:3 0:36 145 46:3 5:3 15:2

E, GPa

e 0 , GPa H n0 s , MPa , %

70 67:7 0:63 1:42 0:3 0:08 130 197 2:4 1:7

70 67:7 0:63 0:81 0:28 0:16 130 197 3 1:3

70 67:7 0:42 0:4 0:32 0:24 130 197 4 1:9

70 67:7 0:42 0:61 0:3 0:08 130 197 6:5 7:6

9.7 The Problem on Tension of a Plastic Beam

a

677

b

Fig. 9.12 Diagrams of deforming for aluminium alloy AK4: in tension (a) and in compression (b)

a

b

Fig. 9.13 Diagrams of deforming in compression for sand grounds: wet sand (a) and dry sand (b)

Figure 9.12b shows experimental and computed diagrams of deforming for aluminium alloy AK4 in compression. Computations were performed by formula e 0 , n0 , and s had been evaluated previously by the (9.360), and the constants E, H curve of deforming in tension. In this case the model AI exhibits the best accuracy; the remaining models lead to a considerable error of approximation.

9.7.9 Comparison with Experimental Data for Grounds Figures 9.13 and 9.14 show experimental diagrams of deforming in compression for sand grounds (for dry and wet sands), and also their approximations with the help of relationship (9.361). These diagrams differ from the corresponding compression

678

a

9 Plastic Continua at Large Deformations

b

Fig. 9.14 Approximation of diagrams of deforming for sand grounds in compression by the models with linear strengthening: wet sand (a) and dry sand (b)

a

b

Fig. 9.15 Diagrams of deforming under loading and subsequent unloading (experimental and computed by the models AI and AIV for sand grounds: wet sand (a) and dry sand (b)

diagrams for alloys (see Fig. 9.12b) by the presence of an intense convexity upwards (in absolute values of coordinates); and under unloading the deformation curve goes sharply downwards (Fig. 9.15). For damp sands the accuracy of approximation with the help of all models An is sufficiently high, for dry sands the accuracy is lower. Models AII and AIV show the best approximation results. Unlike metallic alloys, for grounds the constant n0 has positive values, and the ratio of the yield strength to the maximum magnitude of Cauchy stresses ˛s D s = max is considerably smaller than for steels: ˛s 0:001 and ˛s 0:1, respectively. A magnitude of s for grounds is usually such small that the initial elastic stage of deforming is not visible on the diagram of deforming (Figs. 9.13 and 9.15), although values of elastic modulus E under loading and unloading in experiments prove to be close (in the considered models An they are coincident). Table 9.4 gives values e of the constants E, H 0 , n0 and s for the considered types of grounds.

9.8 Plane Waves in Plastic Continua Table 9.4 grounds Dry sand n E, GPa e 0 , GPa H n0 s , MPa , %

679

e 0 , n0 , s , and in models An for different Values of the constants E, H

I 10 0.324 0.09 0.23 21.4

II 10 0.56 0.15 0.23 17.2

IV 10 1.6 0.27 0.23 12.2

V 10 3.1 0.34 0.23 14

Wet sand n E, GPa e 0 , GPa H n0 s , MPa , %

I 10 42 0.6 0.29 55

II 10 45 0.6 0.29 56

IV 10 260 0.8 0.29 55

V 10 600 0.9 0.29 56

Fig. 9.16 Propagation of a plane wave in the plate

Figure 9.14 shows the results of approximation of experimental diagrams of deforming for sand grounds with the help of the models of plastic continua with linear strengthening (see Sect. 9.1.8) when we assume a priori that n0 D 0. The constant H0 for these models can be determined by minimizing the mean-square deviation (9.405). By this method, we have get the following values: for wet sands, H0 D 2 GPa for the models AI and AII ; and for dry sands, H0 D 0:24 GPa for the model AI and H0 D 0:3 GPa for the model AII . The quality of approximation to the experimental diagrams of deforming by the models with linear strengthening is worse than that obtained by the models with power strengthening (9.86) especially for wet sands. However, in some cases this model proves to be more convenient for solving special problems (see Sect. 9.8).

9.8 Plane Waves in Plastic Continua 9.8.1 Formulation of the Problem Let us investigate now the dynamical problem of plasticity theory An (9.288)– (9.290) in the material description and consider the problem on a plane wave in a plate, which is caused by high-speed (not quasistatic) loading on one of its surfaces X 1 D 0 (Fig. 9.16). The end surface X 1 D h01 of the plate is assumed to be free of loads; and on the lateral surfaces X ˛ D ˙h0˛ =2, ˛ D 2; 3, the condition of free slip (the symmetry condition) is given. Thus, boundary conditions for this problem have the form

680

9 Plastic Continua at Large Deformations

X1 D 0 W

X 1 D h01 W ˛

X D

pe .t/ 1 .F /11 ; P12 D P13 D 0; J P˛1 D 0; ˛ D 1; 2; 3I

P11 D

˙h0˛ =2

W

˛

(9.367)

˛

x D X ; P˛1 D 0; ˛ D 2; 3:

Here Pij are Cartesian components of the Piola–Kirchhoff tensor, and .F 1 /11 – of the inverse deformation gradient F1 D .F 1 /ij eN i ˝ eN j . Such boundary conditions approximately simulate the process of impact of a massive rigid slab on the investigated plate of plastic material slipping along a rectangular rigid pipe.

9.8.2 The Motion Law and Deformation of a Plate A law of the motion of the plate is sought in the form 8 1 1 1 ˆ ˆ <x D x .X ; t/; x2 D X 2; ˆ ˆ :x 3 D X 3 ;

(9.368)

where x 1 .X 1 ; 0/ D X 1 . The velocity v has only one non-zero component: vD

@x i eN i D v1 eN 1 ; @t

v1 D @x 1 [email protected]:

The deformation gradient F has the form FD

3 X @x i j N N e ˝ e D k˛ eN 2˛ ; i @X j ˛D1

k1 D k1 .X 1 ; t/ D @x 1 [email protected] 1 ;

(9.369)

k ˛ D @x ˛ [email protected] ˛ D 1; ˛ D 2; 3;

and, as distinct from quasistatic tension of a beam (see (9.328)), here the elongation ratio k1 .X 1 ; t/ depends on X 1 linearly. .n/

The energetic deformation tensors C also contain the only non-zero component: .n/

.n/

C D C 11 eN 21 ;

.n/

C 11 D

1 .k nIII 1/; n III 1

.n/

.n/

C 22 D C 33 D 0;

(9.370)

and changing density is determined by function k1 : ı

J D = D det F1 D 1=k1 :

(9.371)

9.8 Plane Waves in Plastic Continua .n/

681

.n/

Tensors C e and C p are sought in the diagonal form .n/

Ce D

3 .n/ X C e˛˛ eN 2˛ ;

.n/

Cp D

˛D1 .n/

.n/

3 .n/ X C p˛˛ eN 2˛ ;

(9.372)

˛D1

.n/

.n/

C e11 C C p11 D C 11 ;

.n/

.n/ C e33

C e22 C C p22 D 0;

.n/

.n/

C C p33 D 0;

.n/

where all the components C e˛˛ and C p˛˛ depend only on X 1 and t. From the condition of plastic incompressibility we get .n/ C p22

.n/ 1 .n/ D C p33 D C p11 ; 2 .n/ .n/ .n/ 1 p C e22 D C e33 D C 11 ; 2

.n/

(9.373) (9.374)

.n/

i.e. transverse deformations C e22 , C e33 are different from zero only in a plastic domain.

9.8.3 Stresses in the Plate .n/

According to (9.335), the stress tensor T in this problem has the diagonal form .n/

T D

3 .n/ X T ˛˛ eN 2˛ ;

(9.375)

˛D1

where .n/

.n/

.n/

T 11 D J.l1 C 2l2 / C 11 2J l2 C p11 ; .n/

.n/

.n/

(9.376)

.n/

T 22 D T 33 D J.l1 C 11 C l2 C p11 /I

(9.377)

all the values depend only on X 1 and t. According to (9.375) and (9.369), the tensors T and P also have a diagonal form TD

3 X

˛˛ eN 2˛ ;

˛D1

PD

3 X

P˛˛ eN 2˛ D

˛D1 .n/

11 D k1nIII T 11 ; P11 D 11 ;

.n/

1 1 F T; J .n/

22 D 33 D T 22 D T 33 ; 1 P22 D P33 D 22 : k1

(9.378)

682

9 Plastic Continua at Large Deformations

9.8.4 The System of Dynamic Equations for the Plane Problem Since ı

r PD ı

r ˝ vT D 4

3 X @P˛˛ @P11 eN ˛ D eN 1 ; ˛ @X @X 1 ˛D1

@v1 2 eN ; @X 1 1

ı

F1T r ˝ v D ı

.n/

X F1T r ˝ v D k1nIII1

1 @v1 2 eN ; k1 @X 1 1

(9.379)

@v1 2 eN @X 1 1 .n/

(here we have used the result of Exercise 4.2.16 for tensors X), so the first, third, fourth and fifth equations of the system (9.288) in the considered problem take the forms 1 @x 1 @k1 @ 11 @v1 ı @v 1 ; ; ; D D v D @t @X 1 @t @t @X 1 0 1 .n/ .n/ (9.380) @ B T 11 C 2l2 @ C p11 l1 C 2l2 nIII1 @v1 ı : k1 @ AD ı @t @X 1 @t Notice that the last equations of the system (9.380) can be obtained by immediate differentiation of relation (9.376). It should also be noted that the fifth equation of the system (9.288) gives two .n/

.n/

more scalar expressions for @ T 22 [email protected] and @ T 33 [email protected]; however, these are just expressions but not equations, and therefore they do not appear in the general system (9.380). Taking the expression (9.371) for and expression (9.378) for 11 into account, we get 0

.n/

1

k1IIInC1 @ 11 @ B T 11 C III n C 1 @k1 C : 11 k1IIIn @ AD ı ı @t @t @t

(9.381)

The last equation of the system (9.380) after substitution of the expression (9.381) takes the form .n/ p @ 11 .III n C 1/ 11 @v1 2.nIII1/ nIII1 @ C 11 D .l1 C 2l2 /k1 : 2l k 2 1 @t k1 @X 1 @t (9.382)

9.8 Plane Waves in Plastic Continua

683 .n/

p

Just as in other one-dimensional problems considered above, to evaluate @ C 11 [email protected] we use the equation of a yield surface f D 0 (see (9.96)), which, according to the results of Exercise 9.1.1, in this problem has the form .n/ .n/ .n/ 1 .n/ p p . T 11 T 22 H. C 11 C 22 //2 D 1: 2 3 s .n/

(9.383)

.n/

.n/

Substituting expressions (9.377) for T 22 , (9.373) for C p22 , (9.370) for C 11 and (9.350) for H into (9.383), we obtain .n/

.n/

.n/

e 22 H e C 11 j D j T 11 T p

p

3 s ;

(9.384)

where e D 3 H C l2 ; H 2 k1

.n/

nIII .n/ 1/ e 22 D l1 C 11 D l1 .k1 T : k1 .n III/k1

(9.385)

For the model of plasticity with linear strengthening when n0 D 0 and H D H0 , e D 3 H0 C l2 , and Eq. (9.384) has the following analytical solution for we have H 2 k1 .n/

p

C 11 :

.n/ p e 22 .n/ T 3 s e T 11 p e e hC h C C p C 11 D 11 .1 hC /: e e H H Here hC and h are the Heaviside functions: .n/

.n/

(9.386)

e h D hC h ; hC hC C h ; e ( p p 1; Y1H > 3 s ; 1; Y1H < 3 s ; hC D h D p p 0; Y1H < 3 s ; 0; Y1H > 3 s ; (

.n/

(9.387)

.n/

e22 H eCp ; Y1H D T 11 T 11 .n/

and C p 11 is the value of plastic deformation reached at the time t of leaving the yield surface. .n/

.n/

e 22 into (9.386) Substituting the expressions (9.378) and (9.385) for T 11 and T and then differentiating (9.386) with respect to t within differentiability sections, we obtain .n/

p @k1 @ 11 @ C p11 D b0e C ..b1 11 b2 /e : hC h / hC 3 s b3e @t @t @t

(9.388)

684

9 Plastic Continua at Large Deformations

Here we have introduced the following functions of k1 : 2k1IIInC1 2k1IIIn 2l2 ; b1 D III n C ; (9.389) 3H0 k1 C 2l2 3H0 k1 C 2l2 3H0 k1 C 2l2 3H0 k1 .k1nIII 1/ 4l2 2l1 nIII : k ; b D b3 D 2 .3H0 k1 C 2l2 /2 .3H0 k1 C 2l2 /k1 1 .3H0 k1 C 2l2 /.n III/ b0 D

Substituting the expression (9.388) into (9.382) and grouping like terms, we find the equation @ 11 @v1 D .c1 11 c2 / 1 : @t @x

(9.390)

Here we have introduced the following notation for functions of k1 : c0 D 1 C c1 D c2 D

1 c0 k1

4l2e hC ; 3H0 k1 C 2l2

p l1 C 2l2 2.nIII1/ 2l2 nIII1 e h /; k1 C k1 .b2 hC C 3 s b3e c0 c0 ! hC 2l2 4l2e : (9.391) III n C 1 C III n C 3H0 k1 C 2l2 3H0 k1 C 2l2

Notice that for most plastic materials occurring in practice the following condition is satisfied: 11 c2 =c1 1:

(9.392)

Therefore, the term 11 c2 in Eq. (9.390) can be neglected; and we may consider the simpler equation @v1 @ 11 D c12 1 : @t @X

(9.393)

9.8.5 The Statement of Problem on Plane Waves in Plastic Continua Substituting the relation (9.393) in place of the last equation in the set (9.380) and excluding the second equation from (9.380) (because coefficients in the equation set do not depend explicitly on X 1 ), we finally obtain the system 8 ı ˆ ˆ 0. According to the characteristic method, consider differentials of the unknown functions dv D

@v @v dt C dX; @t @X

dk D

@k @k dt C dX; @t @X

dT D

@v @v dt C dX: @t @X (9.397)

The system (9.394) and (9.397) consists of six equations being linear with respect to six unknown functions: vt D @[email protected], vX D @[email protected] , Tt D @T [email protected], TX D @T [email protected] , kt D @[email protected] and kX D @[email protected] . The system can be rewritten in the matrix form 0ı 10 1 0 1 0 vt 0 0 1 0 0 B CB C B C B 0 1 0 0 1 0 C B vX C B 0 C B CB C B C B 0 c1 1 0 0 0 C B Tt C B 0 C (9.398) B CB C D B C: Bdt dX 0 0 0 0 C BTX C B dv C B CB C B C @0 0 0 0 dt dX A @ kt A @ dk A kX dT 0 0 dt dX 0 0 There exists an unique solution of this system if and only if its determinant is different from zero. However, on the plane .X; t/ there are curves (called characteristics) where the solution uniqueness is violated. This case is realized when the determinant of system (9.398) vanishes: ı

det . / D dX 3 C c1 dt 2 dX D 0:

(9.399)

686

9 Plastic Continua at Large Deformations

From (9.399) we find equations of two families of characteristics: dX D ˙a dt; where

(9.400) ı

a2 .k/ c1 .k/=

(9.401)

is the speed of sound in a plastic material considered. Substitution of (9.400) into the first and second equations of system (9.394) yields conditions over the characteristics ı

dv D dT or

dt dT D˙ ; dX a

ı

a d v d T D 0;

dk D dv

dt dv D˙ ; dX a

a d k d v D 0:

(9.402)

Introducing the functions Z

.k/ D 1

k

c1 .k 0 / d k 0 ;

Z

k

'.k/ D

a.k 0 / d k 0 ;

(9.403)

1

for which d D c1 d k, d' D a d k, we can integrate Eq. (9.402):

.k/ T D const;

'.k/ v D const:

(9.404)

Thus, there are two families of characteristics and two conditions over each of the characteristics: 8 ˆ ˆ 3 s : if T 11 T

c0 D 1;

690

9 Plastic Continua at Large Deformations .n/

e 22 and in place Substituting into formula (9.384) the expression (9.385) for T .n/

of T 11 its expressions in terms of k according to (9.378), (9.404), and (9.415): .n/

T 11 D k IIIn T D k IIIn .k/, we find the limiting value k D ks < 1, from which plasticity starts in compression: p 3 s .ksnIII 1/ ksIIIn 2.nIII/1 1/ D : .k 2.n III/ 1 s .1 /.n III/ks l1 C 2l2

(9.418)

Here, as usual, D l1 =.2.l1 C l2 // is the Poisson ratio. In the domain of plastic deformations (in compression) when k 6 ks < 1, relations (9.386), (9.387), and (9.391) become hC D 0; .n/ C p11

D

h D 1;

e hC D 1;

e h D 1;

.n/ p 1 .n/ e 22 C 3 s / . T 11 T e H

l1 C 2l2 D e H c0 D 1 C

k

IIIn

4l2 ; 3H0 k C 2l2

e c 1 .k/ D k 2.nIII1/ C

.k/ D .l1 C 2l2 /

1

k

e c 1 .k/ ; c0 .k/ ! p 3 s b3 b2 ; l1 C 2l2

c1 D .l1 C 2l2 /

1 2 1

p a D a0 e c 1 .k/=c0 .k/; Z

! p .k nIII 1/ 3 s

.k/ ; C .1 /.n III/k l1 C 2l2

k nIII1 q

a0 D

e c 1 .k 0 / d k 0 ; c0 .k 0 /

(9.419)

ı

.l1 C 2l2 /=; Z '.k/ D a0

1

k

s e c 1 .k 0 / d k0 : c0 .k 0 /

Typical dependences of the dimensionless speed of sound a=a0 upon k, determined by functions (9.419), for different models An are shown in Fig. 9.20 (here D 0:3, s =m D 1:5 103 , H=m D 102 when m D l1 C 2l2 ). As follows from these graphs, there is a distinction in kind between the models AI , AII and AIV , AV : for the models AI and AII the speed of sound increases with decreasing k (in compression when k < 1), and for the models AIV and AV it does not grows (for AIV it remains constant, and for AV – decreases). For all models An , the function a.k/ in the case considered has a discontinuity at k D ks due to the assumption on linear strengthening of a continuum (see formula (9.386)).

9.8 Plane Waves in Plastic Continua

691

Fig. 9.20 The speed of sound (dimensionless a=a0 ) versus the ratio k in compression for different models An of elastic materials (e) and plastic materials (p)

9.8.8 Plane Waves in Models AIV and AV Since the function .k/ is negative when k < 1 and monotonically diminishes within the interval of values of k from 1 to 0 for all models An , the function k.t/ D

1 .pe .t// is monotonically decreasing (k.0/ D 1 .pe .0// D 1) (here we have taken into account that p0 .t/ > 0 by the assumption made). Thus, at the front surface X D 0 of the plate the function k.t/ monotonically decreases, and the function a.k/ grows for the models AI , AII and diminishes for the models AIV , AV . Then on the plane .t; X / ‘C’-characteristics in the disturbance domain (as shown above, they are straight lines) have higher values of a slope from the axis OX with growing t for models AIV , AV and smaller values – for models AI , AII (Fig. 9.21). The decrease of a slope of characteristics t D X=a.k/ from the axis OX means that characteristics with different values of k may intersect at the front wave (Fig. 9.21). As a result, the solution becomes ambiguous, that is inadmissible under the assumptions made above on the absence of jumps of the functions k, T and v themselves (only jumps of their first derivatives are admissible). Thus, for the models AI and AII , the solution obtained is inapplicable, and it will be constructed in another way (see Sect. 9.8.9.). For the models AIV and AV , characteristics do not intersect, and the solution (9.415) obtained actually holds. Figure 9.21 shows the graphic method of construction of the solution (9.415) with the help of given values of the function pe .t/.

692

9 Plastic Continua at Large Deformations

Fig. 9.21 Graphic method of construction of the solution (9.415)

a

b

Fig. 9.22 The Riemann waves for the models AIV and AV of plastic continua: p 0 < ps (a) and p 0 > ps (b)

Consider the special case of loading when the load pe has the jump-type form pe D pe0 h.t/;

(9.420)

where h.t/ is the Heaviside function. Then T and k at the face surface of the plate range from 0 and 1 to final values T 0 and k 0 < 1, respectively, and then remain constant for all t > 0. Thus, on the plane .t; X / there appears an angle bounded by the characteristics X D a0 t and X D a.k 0 /t and filled with characteristics X D a.k/t while k 0 < k < 1 (the fan of characteristics). Waves corresponding to these characteristics are called the Riemann waves by analogy with gas dynamics. These waves are characterized by the fact that they propagate without changes in amplitudes of T and v (Fig. 9.22).

9.8 Plane Waves in Plastic Continua

693

If p 0 < ps , where ps is the pressure of the beginning of plasticity in compression (ps D .ks /, here ks is determined by (9.418)), then the wave retains its shape (being a step) (Fig. 9.22). If p 0 > ps , then the shape of the wave spreads: values of k vary from k 0 up to ks , but the maximum value ks remains constant.

9.8.9 Shock Waves in Models AI and AII

Let us return now to the models AI and AII and consider only the case of stepwise loading of the plate (9.420). In this case the system (9.394) and (9.396) admits the trivial solution T D p D const;

k D const;

v D const:

(9.421a)

However, the boundary condition (9.395b) may be satisfied only if we assume that there is a jump discontinuity for the functions, followed by the trivial solution T D 0;

k D 1;

v D 0:

(9.421b)

In other words, for the models AI and AII there is a solution in the form of a shock wave. The function X D XD .t/ separating on the plane .t; X / two solutions (9.421) is an equation of the front of the shock wave. To find this function and also to determine values of k and v (relationships over characteristics do not hold there), we should use relations (5.70) at a surface of a strong discontinuity in the material description. For the considered problem, these relations reduce to the following ones: 8ı ˆ ˆ ˆM v p D 0; < ı

.9:422a/

ı

M .k 1/ C v D 0; ˆ ˆ ı ˆ 2 :M . v C Œe / pv D 0:

.9:422b/ .9:422c/

2

Here we have taken into account that Œv D v, ŒP D P11 D 11 , ŒF D F11 1 D k1, because on one side of the singular surface, according to (9.421b), the medium ı

is quiescent. The mass rate M is determined by (5.57) and (5.12): ı

ı ı

M D D ;

ı

ı

ı

D D c D @x † [email protected] D dXD =dt:

(9.423)

Assume that the temperature jump is zero across the shock wave: Œ D 0, then the result of Exercise 9.1.10 for the internal energy jump Œe gives the expression Œe D e e0 D

.n/ I 2. C e / ı 1

l1

2

C

l2 ı

.n/

I1 . C 2e /:

(9.424)

694

9 Plastic Continua at Large Deformations

Here we have taken into account that, according to (9.421b), e D e0 in a quiescent domain. .n/

From (9.372)–(9.374) we obtain the following expressions for invariants I1 . C e / .n/

and I1 . C 2e /: .n/

.n/

.n/

.n/

.n/

.n/

I1 . C e / D C e11 C 2 C e22 D C e11 C C p11 D C 11 ; .n/ .n/ .n/ .n/ .n/ 1 .n/p p I1 . C 2e / D . C e11 /2 C 2. C e22 /2 D . C 11 C 11 /2 C . C 11 /2 : 2

(9.425)

On substituting (9.425) into (9.424), we find the expression for the internal energy .n/

.n/

jump in terms of deformations C 11 and C p11 : Œe D D

.n/ .n/ .n/ 1 .n/ .l1 C 211 C 2l2 .. C 11 C p11 /2 C . C p11 /2 // 2 2

1

ı

.n/

1 ı

2

.n/

.n/

p

.n/ p

..l1 C 2l2 / C 211 4l2 C 11 C 11 C 3l2 . C 11 /2 /:

(9.426)

.n/

For the case of initial plastic loading of compression when C p 11 D 0, substitution .n/

of formula (9.376) for T 11 into (9.386) yields .n/

C p11 D

1 ek H

.n/ .n/ .n/ p .l1 C 2l2 / C 11 2l2 C p11 l1 C 11 C 3 s k :

(9.427)

.n/

From this equation we can easily express the plastic deformation C p11 as follows: .n/

p

C p11

.n/

1 2 3 s k C b C 11 D ; bD : 3.b C H0 k=2/ 2.1 / .n/

(9.428) .n/

On substituting formula (9.428) for C p11 and formula (9.370) for C 11 into (9.426), we get a jump of internal energy Œe as a function of k: Œe D Œe .k/:

(9.429) .n/

If loading occurs only in the elastic domain, where C p D 0, then Œe .k/ D

l1 C 2l2 ı

2.n III/2

.k nIII 1/2 :

(9.430)

9.8 Plane Waves in Plastic Continua

695

With taking account of expression (9.429), three relationships (9.422) allow us to ı

determine three unknown functions: k, v and M in terms of p. For this purpose, we express v from Eq. (9.422b): ı

ı

v D M .1 k/=;

(9.431) ı

substitute (9.431) into (9.422a) and obtain the expression for M : ı

M D

q ı p=.1 k/:

(9.432)

We have chosen the positive sign of the square root according to the physical meaning of the solution: a shock wave must propagate in the positive direction of the axis OX . Substituting of (9.432) into (9.431) yields q vD

ı

p.1 k/=:

(9.433)

On substituting (9.429), (9.432), and (9.433) into (9.422c), we get the equation for k: p Œe .k/ D ı .1 k/: (9.434) 2 The derived relationship (9.434) with the expression (9.426) for a jump of internal energy allows us to find k as a function of p: k D k.p/. Since p is known from the problem condition, we can determine k from the equaı

tion (9.434) and then find M and v with the help of formulae (9.431) and (9.432). As a result, from the first formula of (9.423) we obtain ı

DD

s

ı

M ı

D

p ı

.1 k/

:

(9.435)

Thus, we have found a complete solution of the problem.

9.8.10 Shock Adiabatic Curves for Models AI and AII The function k D k.p/ (or p D p.k/) expressed by formula (9.434) is called a shock adiabatic curve for an elastoplastic continuum. ı Since k D = is the ratio of densities, we can introduce the specific volı ume V D 1= D k= which is a function of p, i.e. we have a function p D p.V /

696

9 Plastic Continua at Large Deformations

Fig. 9.23 The shock adiabatic curve

(or V D V .p/). This function is well-known in gas dynamics as a shock adiabatic curve for similar problems. If loading occurs only in an elastic domain, then, substituting (9.430) into (9.434), we derive the following equation for the shock adiabatic curve p D p.k/: p 1 D l1 C 2l2 1k

k nIII 1 n III

2 :

(9.436)

Figure 9.23 shows a graph of this function. On substituting (9.426), (9.430), and (9.370) into (9.434), we obtain the equation of the shock adiabatic curve with taking account of plastic deformations: .n/ .n/ .n/ .n/ 1 p D . C 211 b C p11 .4 C 11 C p11 //; l1 C 2l2 1 k1

(9.437)

where

.n/

C 11

p

.n/

3 s k C b C 11 D ; 3.b C H0 k=2/ 1 D .k nIII 1/: n III 1

.n/ C p11

(9.438)

Figure 9.23 shows the shock adiabatic curve p.k/. Figure 9.24 exhibits shock adiabatic curves computed by formula (9.437) for dry and wet sand grounds. Constants E, , s , and H for grounds have been taken from the experimental data for specimens of a beam form in uniaxial high-speed compression. The method of evaluation and values of these constants are given in Sect. 9.7.9; for approximation we have used the plasticity model with linear strengthening. ı

Let us ask the question: what is the speed D of propagation of a shock wave in comparison with the sound speed a0 in a quiescent continuum. To answer the question one should find a value of k when the pressure values are small: p=.l1 C 2l2 / 1. Linearizing the left-hand side of Eq. (9.436) in a neighborhood of the value k D 1, we obtain (9.439) p D .l1 C 2l2 /ı;

9.8 Plane Waves in Plastic Continua

697

Fig. 9.24 Shock adiabatic curves for sand grounds: solid lines correspond to wet sand, dashed lines – to dry sand

Fig. 9.25 Dependence of the speed of a shock wave and the speed of sound upon the compression coefficient k1 in a plastic continuum

where k D 1 C ı, jıj 1, ı 6 0 for both the models AI and AII . Then from (9.435) we find that s s s ı p ı.l1 C 2l2 / l1 C 2l2 D D ı D (9.440) D D a0 ; ı ı ı ı i.e. for the models AI and AII of elastoplastic continua, small disturbances propagate with the speed of sound. ı

As follows from Eq. (9.435), at finite values of p=.l1 C 2l2 / the speed D of a shock wave proves to be supersonic (Fig. 9.25).

9.8.11 Shock Adiabatic Curves at a Given Rate of Impact The solution obtained in Sect. 9.8.10 remains valid for another case of boundary conditions: when on the surface X 1 D 0 of a plate one gives a constant rate v of impact instead of the stress component P11 : X1 D 0 W

v D v0 D const;

P12 D P13 D 0I

(9.441)

the remaining conditions in (9.367) hold without changes. In this case, it is convenient to represent the shock adiabatic curve as a function p D p.v/. For this, we

698

9 Plastic Continua at Large Deformations

Fig. 9.26 Computed and experimental shock adiabatic curves in coordinates .p; v/ for sand grounds: solid curves – wet sand, dashed lines – dry sand

should substitute the function p.k/ expressed by formulae (9.437) and (9.438) into Eq. (9.433). As a result, we get the equation p.k/ D

v2 ı

.1 k/

;

(9.442)

which at a given value of v can be resolved for k. Evaluating a root of the equation (when 0 < k 6 1, this root is unique), for example, by the method of bisecting an interval, we find the function k D k.v/. On substituting this function into Eq. (9.437), we obtain the shock adiabatic curve p D p.k/ D p.k.v// D p.v/ in coordinates .p; v/ (the pressure versus the speed). Figure 9.26 shows graphs of the shock adiabatic curve p D p.v/ for sand grounds, they were computed for the models AI and AII of plastic continua with linear strengthening. The constants E, , s and H0 in the computations were taken also from the results of experiments in uniaxial tension (see Sect. 9.7.9). This figure also exhibits experimental shock adiabatic curves p.ex/ .k/ for dry and wet sand grounds. The computed curves determined by both the models AI and AII satisfactorily approximate the experimental data. The shock adiabatic curve can be represented in another form, namely as a deı

ı

pendence of the speed D on the rate of impact v. To obtain such a function D.v/, we should use formulae (9.422a) and (9.423) and substitute into them the dependence p D p.k/ D p.k.v// D p.v/ derived above; as a result, we obtain the desired form of a shock adiabatic curve: ı p.v/ (9.443) DD ı : v ı

Figure 9.27 shows the shock adiabatic curves D.v/ computed for sand grounds by ı

formula (9.443) and also experimental shock adiabatic curves D .ex/ .v/. The computed and experimental results are sufficiently close.

9.9 Models of Viscoplastic Continua

699

Fig. 9.27 Shock adiabatic curves in coordinates .D o ; v/ for sand grounds: solid curves – wet sand, dashed lines – dry sand

9.9 Models of Viscoplastic Continua 9.9.1 The Concept of a Viscoplastic Continuum In Chap. 7 we considered models of continua of the differential type, and in Sect. 7.4.4 it was shown that these models can be applied to describe the creep effect in some solids. The limitation of models of the differential type consists in the fact that they do not involve elastic deformations; therefore, with their help one cannot determine instantly elastic deformations under loading and subsequent unloading (see Figs. 7.2 and 7.3). In the cases when instantly elastic deformations cannot be neglected in comparison with creep deformations, one should use more complicated models, for example, models of viscoplastic continua, that will be considered in this section. As a rule, a continuum is called viscoplastic if its plastic properties (the appearance of residual deformations) depend on time. The models of associated plastic continua (considered in Sects. 9.1.5–9.1.13) describe pure plastic properties, which are independent of time. Indeed, although the constitutive equations (9.52) of associated models formally involve time t, we can easily exclude the time if we will differentiate not with respect to time but with respect to the loading parameter (see Exercise 9.1.11). From the experimental point of view, this fact means that if we perform experiments in uniaxial tension (see Sect. 9.7) with a constant rate of lengthening: k1 .t/ D bt, then diagrams of deforming in elastic and plastic domains are independent of the lengthening rate b. For viscoplastic continua, this dependence takes place.

9.9.2 Model An of Viscoplastic Continua of the Differential Type Let us consider the simplest viscoplastic model, namely the model An of a viscoplastic continuum of the differential type, which can be formally obtained from

700

9 Plastic Continua at Large Deformations

Eqs. (9.52) if we assume that the expression for plastic deformation holds for all the times at both loading and unloading. In this case, the parameter h should be assumed to be equal to 1: .n/

C v D

k X

.n/

~˛ [email protected]˛ [email protected] T /:

(9.444)

˛D1 .n/

.n/

In this formula we have changed the notation of plastic deformation C p by C v , which is called the viscous deformation, and the functions ~P ˛ are replaced by ~˛ . We can always make this substitution because functions ~P ˛ in (9.48) have been introduced as ratio coefficients of plastic deformations to the yield surface gradient. For the associated models of plastic deformations, the functions ~P ˛ can be found from Eqs. (9.52) with complementing by Eqs. (9.41) of the yield surface, but for a viscoplastic continuum of the differential type Eqs. (9.41) are absent and expressions for ~˛ and fv˛ are given with the help of the additional relations .n/ .n/

.n/ .n/

~˛ D ~˛ . T ; C p ; /; fv˛ D fv˛ . T ; C p ; /;

˛ D 1; : : : ; k:

(9.445)

The functions fv˛ will be called the viscous potentials. .n/

Remark 1. For viscoelastic continua of the differential type the stress tensor T is a sum of equilibrium and viscous stresses (see (7.6)), but for viscoplastic materials .n/

of the differential type the deformation tensor C consists of elastic and viscous deformations, similarly to (9.3): .n/

.n/

.n/

C D C e C C v:

(9.446)

For the unipotential model An of a viscoplastic continuum of the differential type, we assume that k D 1; ~1 D 1, and fv1 D fv ; therefore, .n/

.n/

C v D @fv [email protected] T ;

.n/ .n/

fv D fv . T ; C v ; /:

(9.447)

These relations should be complemented with equations for the elastic deformation .n/

tensor C e , for example, (9.59). The equation system can be rewritten in terms of the invariants as follows: .n/ C v

D

z1 X ˛D1

.s/ ˛ J˛T ;

.n/

Ce D

z X

' I.s/ T;

(9.448)

D1

where ˛ and ' are scalar functions being the derivatives of the plastic and elastic potentials fv and with respect to the invariants:

9.9 Models of Viscoplastic Continua

701 .n/ .n/

.n/

.s/ ˛ D @fv [email protected]˛.s/v ; fv D fv .J˛.s/ ; /; J.s/ D J.s/ . T ; C v /; J˛T D @J˛.s/ [email protected] T ; (9.449) .n/

.n/

.s/ ' D @[email protected].s/ ; D .I.s/ ; /; I.s/ D I.s/ . T =/; I.s/ T D @I [email protected] T =/: (9.450)

Due to the Onsager principle, the plastic potential fv is a quadratic function of the .n/ .n/

linear invariants J.s/ . T ; C v / and a linear function of the quadratic invariants J.s/ .

9.9.3 Model of Isotropic Viscoplastic Continua of the Differential Type For isotropic viscoplastic continua of the differential type, the simultaneous invari.n/ .n/

ants J.s/ . T ; C v / can be chosen in the form (9.75), then relations (9.448) take the forms .n/

.n/

.n/

C v D e 1 E 2 T C 7 C v ;

.n/

.n/

(9.451) .n/

Ce D e ' 1 E C .e ' 2 =/ T C .'3 =2 / T 2 :

Here e 1 D 1 C

.n/

2 I1 . T /, .n/

(9.452)

and the potentials depend on the following invariants: .n/

.n/

.n/

.I /

.n/ .n/

fv D fv .I1 . T /; I2 . T /; I1 . C p /; I2 . C v /; J7 . T ; C v /; /; .n/

.n/

.n/

D .I1 . T =/; I2 . T =/; I3 . T =/; /;

(9.453) (9.454)

If the plastic potential has been chosen in the Huber–Mises form (9.96) fv D

YH2 3 ; YH2 D PH PH ; 3 2

.n/ .n/ .n/ 1 .n/ PH D . T Hv C v / I1 . T Hv C v /E; Hv D Hv0 Yv2nv0 ; 3

Yv2 D

3 Pv Pv ; 2

.n/ 1 .n/ Pv D C v I1 . C v /E; 3

(9.455)

702

9 Plastic Continua at Large Deformations

where Hv0 , nv0 and are the constants, then, performing the manipulations of Sect. 9.1.8, we can rewrite relations (9.450) as follows: .n/ C v

D

.n/ .n/ 1 .PT Hv C v /; I1 . C v / D 0:

(9.456)

For isotropic linear-elastic continua, Eqs. (9.452) with use of the additive relationship (9.3) take the form (see Exercise 9.1.9): .n/

.n/

.n/

.n/

T D J l1 I1 . C /E C 2J l2 . C C v /:

(9.457)

The equation system (9.456), (9.457) is the model An of an isotropic viscoplastic continuum of the differential type with the Huber–Mises potential.

9.9.4 General Model An of Viscoplastic Continua One can say that this is the general model An of viscoplastic continua, if the additive relation (9.3) is replaced by .n/

.n/

.n/

.n/

C D C e C C p C C v;

(9.458)

.n/

i.e. the deformation tensor C in this model is a sum of the three terms: elastic .n/

.n/

.n/

deformation C e , plastic deformation C p and viscous deformation C v . For elastic and plastic deformations we assume the same relations as the ones for pure plastic continua (see Sect. 9.1). For example, for the associated model An of viscoplastic continua, the relations (9.74) hold: .n/

Ce D

.n/ C p

r X D1 z1

X

D

.n/

' I.s/ T . T =/;

(9.459a)

.n/ .n/ .s/ ˛ J˛T . T ; C p /;

(9.459b)

˛D1 .n/

D .I.s/ . T =/; ; wpˇ /; D 1; : : : ; r; .n/ .n/

(9.459c)

p

fˇ D fˇ .J˛.s/ . T ; C p /; ; wˇ /; ' D

@ @I.s/

.n/

;

I.s/ T

D

@I.s/ . T =/ .n/

@ T =

;

˛

Dh

k X ˇ D1

(9.459d) ~Pˇ

@fˇ @J˛.s/

:

(9.459e)

9.9 Models of Viscoplastic Continua

703

And we assume that, just as for viscoplastic continua of the differential type, for viscous deformation Eqs. (9.448) and (9.449) still hold: .n/

C v D

z1 X

.s/

˛ J˛T ;

(9.460)

˛D1 .n/ .n/

.n/

.s/ D @J˛.s/ [email protected] T : ˛ D @fv [email protected]˛.s/ ; fv D fv .J˛.s/ ; /; J.s/ D J.s/ . T ; C v /; J˛T

(9.461)

9.9.5 Model An of Isotropic Viscoplastic Continua Using the results of Sects. 9.1.7 and 9.9.4, from (9.460) and (9.461) we get that the equation system (9.81)–(9.83), (9.451)–(9.454) holds for isotropic viscoelastic continua. Choosing the Huber–Mises model (9.96) and (9.455) for plastic and viscous potentials and applying the linear elasticity model for elastic deformation, from (9.460) and (9.461) we get the equation system .n/

.n/

.n/

.n/

.n/

T D J l1 I1 . C /E C 2J l2 . C C p C v /;

.n/

C p D

.n/ .n/ ~h P .n/ 1 .n/ . T I1 . T /E H C p /; I1 . C p / D 0; 2 s 3

f D .n/

C v D

1 .YH = s /2 1 D 0; 3

(9.462) (9.463)

H D H0 Yp2n0 ;

.n/ .n/ 1 .n/ 1 .n/ . T I1 . T / Hv C v /; I1 . C v / D 0; Hv D Hv0 Yv2nv0 ; 3

(9.464)

where the invariants YH , Yp and Yv are expressed by formulae (9.85), (9.87), and (9.455).

9.9.6 The Problem on Tension of a Beam of Viscoplastic Continuum of the Differential Type As an example, consider the problem on uniaxial tension of a beam of viscoplastic material of the differential type, which is described by the motion law (9.327). The .n/

.n/

deformation gradient F and the deformation tensors C and C e are determined by

704

9 Plastic Continua at Large Deformations .n/

formulae (9.328)–(9.332), (9.341). For viscous deformations C v we have formulae which are similar to Eqs. (9.334), and for plastic deformations – formulae (9.345): .n/

Cv D

3 .n/ X C v˛˛ eN ˛ ˝ eN ˛ :

(9.465)

˛D1 .n/

.n/

.n/

C ˛˛ D C e˛˛ C C v˛˛ ; ˛ D 1; 2; 3: .n/ C v33

(9.466)

.n/ 1 .n/ D C v22 D C v11 : 2

(9.467)

.n/

The stress tensors T are diagonal, for them relations (9.335)–(9.341) hold; in particular, the component 11 is expressed as follows: .n/

.n/

11 D k1nIII JE. C 11 C v11 /:

(9.468)

On substituting (9.465) into (9.456), we get the following equation for viscous de.n/

formation C v11 : 1 d .n/v C D dt 11

.n/ 2 IIIn k1 11 H C v11 ; 3

(9.469)

3 .n/ Hv D Hv0 Yp2nv0 ; Yp2 D . C v11 /2 : 2

(9.470)

Changing density is determined by the same relationship (9.355)

1 J D ı D k 1

2=.nIII/ .n/ 1 nIII v 1/ .n III/. / C 11 : 1 .k1 2

(9.471) .n/

On substituting (9.468) into (9.469) and taking the expression (9.330) for C 11 into .n/

account, we derive the following final equation for viscous deformation C v11 :

.n/ .n/ 3 .n/ 2 1 JE. .k1nIII 1/ C v11 / Hv0 . C v11 /2nv0 C v11 : 3 n III 2 (9.472) If the elongation function k1 .t/ of the beam is given, then, on solving the equation, the stress 11 can be found from Eq. (9.468). If, just as in the creep problem (see Sect. 7.4.4), the stress 11 .t/ is given, then 1 d .n/v C D dt 11

.n/

from Eq. (9.468) we can compute the dependence k1 D k1 . 11 ; C v11 /; and with the

9.9 Models of Viscoplastic Continua

705 .n/

help of Eq. (9.469) we determine the viscous deformation C v11 . A final expression .n/

for k1 .t/ is found by using the dependence k1 . 11 ; C v11 / once again. The considered model of an isotropic viscoplastic continuum of the differential type contains the five material constants: E, , , Hv0 , and nv0 . The elastic modulus is determined by the initial section of the diagram 11 .k1 / at given lengthening k1 .t/, when the influence of viscous deformations can be neglected. Just as for elastic continua, the Poisson ratio is determined by relation (9.341) also for small times, when viscous deformations are small. The remaining three constants: , Hv0 and nv0 can be found with the help of experimental creep curves at the given stress 11 .t/ varying as a step-function (7.146). Figure 9.28 shows the experimental creep curve jı1.ex/ .t/j for Ni-alloy at temperature 1100ıC and its approximation by Eqs. (9.468) and (9.469) for different models An . Constants Hv0 and nv0 in the computations have been chosen to be zero, and the viscous coefficient has been determined by minimization of the mean-square

a

b

c

d

Fig. 9.28 Creep curves for Ni-alloy at temperature 1;100ı C and different values of compressing stress o : dashed curves are experimental data, solid curves are computations by different models An of viscoplastic continua of the differential type: (a) – model AI , (b) – model AII , (c) – model AIV , and (d) – model AV

706

9 Plastic Continua at Large Deformations .ex/

distance between the computational jı1 .t/j D jk1 .t/1j and experimental jı1 .t/j creep curves at o D 20 MPa for several times (see Sect. 7.4.3). We have obtained the following values of the constants: E D 2GPa for all models An ; D 40 GPa s for n D I; D 42 GPa s D 55 GPa s

for n D II; for n D IV;

D 60 GPa s

for n D V:

Figure 9.28 also exhibits computational and experimental creep curves at different values of stress o . The model AII gives the best approximation quality (Fig. 9.28b).

Exercises for 9.9 9.9.1. Show that the model of isotropic viscoplastic continuum (9.455), (9.456), when H0 D 0, can be written as the differential equation .n/ 2l2 .n/ 2J l2 JP I1 . T //E C 2J l2 C T D J.l1 I1 . C / C 3 J

.n/

.n/

!

.n/

T;

JP D r v: J

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708

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25. Heinbockel, J.H.: Introduction to Tensor Calculus and Continuum Mechanics. Trafford Publishing, Canada (2001) 26. Hill, R.: Constitutive inequalities for isotropic elastic solids under finite strain. Proc. Roy. Soc. Lond. A 314, 1519 (1970) 27. Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, New York (2000) 28. Jaumann, G.: Grundlagen der Bewegungslehre. Springer, Leipzig (1905) 29. Jaunzemis, W.: Continuum Mechanics. The Macmillan Company, New York (1967) 30. Jog, C.S.: Continuum Mechanics. Alpha Science (2007) 31. Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis, Moscow, Nauka, 1976 (in Russian). Dover Publications, New York (1999) 32. Lai, W.M., Rubin, D., Krempl, E.: Introduction to Continuum Mechanics. Butterworth– Heinemann, Pergamon, New York (1993) 33. Lebedev, L.P., Vorovich, I.I., Gladwell, G.M.L.: Functional Analysis: Applications in Mechanics and Inverse Problems. Kluwer Academic Publishers, Dordrecht-Boston-London (1996) 34. Leigh, D.C.: Nonlinear Continuum Mechanics. McGrawHill, New York (1968) 35. Liu, I-Shih: Continuum Mechanics. Springer, Berlin (2002) 36. Lurie, A.I.: Nonlinear Theory of Elasticity. Amsterdam, North–Holland (1990) 37. Lurie, A.I.: Theory of Elasticity. Springer, Berlin (2005) 38. Malvern, L.E.: An Introduction to the Mechanics of a Continuous Media. Prentice–Hall, Englewood Cliffs (1970) 39. Mase, G.E.: Continuum Mechanics for Engineers. CRC Press, Boca Raton (1999) 40. McDonald, P.H.: Continuum Mechanics (PWS Series in Engineering). PWS Publishing, Boston (1995) 41. Murnaghan, F.D.: Finite Deformation of an Elastic Solid. Wiley, New York (1951) 42. Nemat–Nasser, S.: On nonequilibrium thermodynamics of continua. Mechanics Today 2, 94–158 (1975) 43. Noll, W.: The Foundations of Mechanics and Thermodynamics, (Selected papers). Springer, Berlin (1974) 44. Oldroyd, J.G.: On the formulation of rheological equations of State. Proc. Roy. Soc. 200(1063), 523–541 (1950) 45. Onsager, Lars.: Reciprocal relations in irreversible processes. Phys. Rev. 37(2), 405–426 (1931) 46. Onsager, Lars.: Reciprocal relations in irreversible processes. Phys. Rev. Second series 38(12), 2265–2279 (1931) 47. Reddy, J.N.: An Introduction to Continuum Mechanics. Cambridge University Press (2007) 48. Roberts, A.J.: A One-Dimensional Introduction to Continuum Mechanics. World Scientific, London (1994) 49. Roy, M.: Mecanique des millieux continus et deformables. Gauthier–Villars (1950) 50. Smith, D.R.: An Introduction to Continuum Mechanics (Solid Mechanics and Its Applications). Springer, Berlin (1999) 51. Spencer, A.J.M.: Continuum Mechanics. Dover Publications, New York (2004) 52. Talpaert, Y.R.: Tensor Analysis and Continuum Mechanics. Springer, Berlin (2003) 53. Temam, R., Miranville, A.: Mathematical Modeling in Continuum Mechanics. Cambridge University Press (2005) 54. Truesdell, C.A.: The Elements of Continuum Mechanics. Springer–Verlag, New York (1965) 55. Truesdell, C.A.: Rational Thermodynamics. McGrawHill, New York (1969) 56. Truesdell, C.A., Noll, W.: The Nonlinear–Field Theories of Mechanics. Springer, Berlin (2003) 57. Valanis, K.S.: Irreversible Thermodynamics of Continuous Media. Springer, New York (1971) 58. Vardoulakis, I., Eftaxiopoulos, D.: Engineering Continuum Mechanics: With Applications from Fluid Mechanics, Solid Mechanics, and Traffic Flow. Springer, Berlin (2007) 59. Wu Han-Chin: Continuum Mechanics and Plasticity (CRC Series–Modern Mechanics and Mathematics). Chapman & Hall/CRC, Boca Raton (2005) 60. Ziegler, H.: An Introduction to Thermodynamics. Amsterdam, North–Holland (1977)

Basic Notation

A

Left Almansi deformation tensor;

.n/

A aOl , aJ , aCR , aD , ad , aV , aU , and aS B C

.n/

C

Energetic deformation tensors;

.n/

CG

.n/

Quasienergetic deformation tensors; Co-rotational derivatives of Oldroyd, Jaumann, Cotter–Rivlin, left and right mixed, left and right in eigenbasis, and spin; The third energetic deformation tensor; Right Cauchy–Green deformation tensor;

Generalized energetic deformation tensors; .n/

C e and C p b ci

D

Tensors of elastic and plastic deformations; Principal basis of anisotropy (orthonormal) of a solid b in undistorted configuration K; The velocity of a singular surface in a reference configuration; Deformation rates tensor;

D and D

The normal speed of propagation of a singular surface

E E

in configurations K and K; Unit (metric) tensor; Total energy of a body;

ı

c ı

4

.n/

E e eN i F Fe and Fp f G

.n/

G

.n/

GG

ı

Tensors of energetic equivalence; Specific internal energy of a body; Basis of Cartesian coordinate system; Deformation gradient; Gradients of elastic and plastic deformations; Specific mass force vector; Right Cauchy–Green deformation measure; Energetic deformation measures; Generalized energetic deformation measures; 709

710

Basic Notation

g

Left Almansi deformation measure; ı

ı

Metric matrices in configurations K and K; Entropy of a body;

g ij and gij H ı Q and H H

Left and right Hencky logarithmic deformation tensors; Tensor of H -transformation from one reference ı configuration K to another reference configuration K; Momentum vector of a body; Principal invariants of second-order tensor C; Invariants of second-order tensor relative to an

H I I1 .C/, I2 .C/ and I3 .C/ .s/ I1 ./ .n/

.n/

.n/

i A, i B , i J ı J D = K 4 K.t/

C,

.n/

i

D,

ı

K and K L 4 M ı

n and n O P P˛.C/ , ˛ D 1; : : : ; n, N ı p and p p Q QN m and QN † QN Qˇ 4

.n/

Q q qm and q† q 4 R.t/ ı

.n/

i

G,

ı

orthogonal group G s ; i

Different forms of enthalpy; Left Cauchy–Green deformation tensor; Ratio of densities; Kinetic energy of a body; Tensor of relaxation cores; Actual and reference configurations; Velocity gradient; Quasilinear tensor of elasticity;

ı

Normal vectors in configurations K and K; Rotation tensor accompanying the deformation; Piola–Kirchhoff stress tensor; Orthoprojectors of symmetric tensor C; Eigenvectors of stretch tensors V and U; Pressure; Rate of heating; Entropy production by external mass and surface sources; Entropy production by internal sources; Thermodynamic fluxes; Tensors of quasienergetic equivalence; Heat flux vector; Heat influxes due to mass and surface sources; Specific internal entropy production; Tensor of relaxation functions; ı

ri and ri

Vectors of local bases in configurations K and K;

S

Quasienergetic stress tensors;

S SG T

Rotation tensor of stresses; Generalized rotation tensor of stresses; Cauchy stress tensor;

.n/ ı

Basic Notation

711

Th and TH

General notation for co-rotational derivatives of a tensor in covariant and contravariant moving bases hi and hi ;

.n/

T

Energetic stress tensors;

.n/

TG tn U U u V v W Wm and W† W.i / w ı x and x

Generalized energetic stress tensors; Stress vector; Right stretch tensor; Internal energy of a body; Displacement vector; Left stretch tensor; Velocity; Vorticity tensor; Powers of external mass and surface forces; Power of internal surface forces; dissipation function; Radius-vectors of a material point in configurations

X i and x i Xˇ Y˛ .T/

K and K; Lagrangian and Eulerian coordinates; Thermodynamic forces; Spectral invariants of a symmetric second-order tensor;

ı

ı

ı

ijm and m ij ı˛ "ij .n/

.n/

.n/

Christoffel symbols in configurations K and K; Relative elongation; Covariant components of the deformation tensor; .n/

.n/

A, B , C , D , G , ƒ ˛

Different forms of the Gibbs free energy; Specific entropy; Temperature; Right Almansi deformation tensor; Eigenvalues of stretch tensors U and V;

and

Heat conductivity tensors in configurations K and K; Thermodynamic potential;

ı

ı

and ˛ U and V ! ı

r and r H and N

ı

ı

Density in configurations K and K; Unit tangent vectors to a surface S ; Helmholtz free energy; Spin of rotation accompanying the deformation; Spins of the right and left stretch tensors; Vorticity vector; ı

Nabla-operators in configurations K and K; The beginning and the end of a proof of Theorems, respectively; The end of Examples and Remarks.

Index

A Acceleration, 98 Coriolis’s, 327, 331, 338 total, 314, 327 translational, 327, 331, 338 Area of a surface element, 31, 104, 106

B Basis, 2 dyadic, 11, 12, 43, 58, 78, 81, 83, 108, 151, 224, 333 functional, 241, 243, 258, 260, 261, 293, 402, 469, 473, 476, 477, 508, 517–519, 606, 610, 614, 635, 637, 638 local, 9 reciprocal, 10, 79, 80, 100, 303, 448 physical, 103 orthonormal, 11, 20 polyadic, 12, 53 principal (of anisotropy), 226, 228, 238, 239, 336 Body, 1–4, 96, 97, 122, 123, 128, 130–136, 140, 222, 226, 315, 316, 336, 405, 406, 408, 409, 414, 421, 438, 439, 445, 447, 472, 561, 565, 586, 667 Boundary conditions, 347, 372, 381, 399–409, 411, 413, 416, 424, 430–432, 441–442, 444, 451–454, 456, 493, 559, 563, 564, 581, 656–658, 660, 667, 679, 685, 688, 693

C Coeffcient heat conductivity, 345 heat transfer, 582

Lam´e’s, 11, 20 surface tension, 370 viscous, 470, 471, 477, 479–480, 488, 490, 491, 705 Components contravariant, 20, 24, 28, 34, 593 covariant, 20, 24, 28, 145 physical, 11, 20, 28–30, 36, 103–106, 450 Condition boundary, 347, 372, 381, 399–409, 411, 413, 416, 424, 430–432, 441–442, 444, 451–454, 456, 493, 559, 563, 564, 581, 656–658, 660, 667, 679, 685, 688, 693 consistency, 3, 6, 472, 496 deformation compatibility, 141–144, 146, 148–150, 152 of symmetry, 402, 405, 424, 447, 451 Configuration actual, 6, 7, 9, 10, 15, 26, 27, 30, 45, 46, 58, 62, 78, 89, 90, 94, 102, 141, 142, 190, 214, 218, 219, 231, 287, 300, 319, 324, 331, 348, 358, 363, 367, 374, 401, 402, 404, 412, 472, 481, 490, 556, 592, 618, 639, 640, 658 reference, 6, 7, 9, 10, 15, 18, 25–27, 94, 102, 103, 105, 108, 119, 120, 142, 148, 213–215, 219–224, 226, 228, 229, 236, 249, 254, 277, 287, 292, 300, 302, 336, 338, 341–343, 348, 358, 362, 367, 368, 396, 402, 407, 408, 413, 421, 452, 465, 485, 491, 506, 592, 603, 632, 639, 651 undistorted, 222, 224, 226, 228, 229, 234, 236, 238, 254, 292, 465, 506, 603 unloaded, 592, 632

713

714 Constitutive equations, 161–346, 349, 377–380, 382–384, 386, 388–393, 395–398, 402, 405–409, 411, 418, 420, 421, 427, 431, 449, 455, 461–463, 465, 467–473, 480, 481, 483, 489–492, 498, 505, 506, 510, 513–519, 522, 524, 525, 528–534, 538–550, 552–565, 567–569, 581, 591, 594–598, 601, 603–607, 609, 610, 613, 615–622, 627–630, 632, 634–638, 640–656, 658, 668, 699 Continuum, 1 of the differential type, 209, 461–497, 518, 592, 595, 603, 641, 643, 699–706 elastic, 236, 244–276, 287, 292–294, 296–299, 377–460, 472, 519, 528, 530, 531, 533, 534, 550, 552, 568, 620, 671, 673, 674, 702, 705 fluid, 9, 133, 221, 291, 319–321, 554 homogeneous, 209 ideal, 209–213, 236–287, 289, 290, 319–322, 340, 414, 464, 467, 505, 524, 592, 603 incompressible, 287–300, 322, 411, 417–421, 431, 437, 443–446, 454–460, 471–472, 549, 552, 556, 675 inhomogeneous, 209 of the integral type, 209, 497–516 with memory, 497 fading, 498 nondissipative, 211, 212 nonpolar, 112–114, 121, 123, 155, 158, 161, 163, 186 plastic, 591 ideally, 600, 602 strengthening, 600, 610, 635, 650, 679, 683, 696, 698 polar, 112–113, 121–123 simple, 516, 547, 564, 566, 581 solid, 222 anisotropic, 226, 236, 237, 614 isotropic, 237, 261, 271, 272, 346 viscoelastic, 497 with difference cores, 511–513, 519, 522, 560 stable, 510–513 thermorheologically simple, 514–516, 566 viscoplastic, 699 Coordinates Eulerian, 6–9, 16, 74 Lagrangian, 5–9, 17, 18, 49, 74, 79, 80, 157, 301, 328, 369, 391, 447, 449

Index material, 3, 5–7, 17, 29, 49, 73, 74, 135, 143, 209, 227, 327, 447 spatial, 6–9, 16, 74 curvilinear, 16–23, 329 of a vector, 20, 325 Coordinate system Cartesian rectangular, 2 inertial, 96 moving, 16, 17, 79, 81, 97, 329, 332, 334, 335, 338 Core of creep, 542–544, 548 of a functional, 507, 510, 515, 540 of relaxation, 529, 534, 538, 540, 543, 544, 548, 553, 580, 588 Couples of tensors energetic, 163 functional, 283–287 principal, 164, 305 quasienergetic, 176 functional, 285 principal, 177, 180, 184, 305 Curl of a tensor, 14, 154 Cycle Carnot’s, 129, 132–136, 138, 139 elementary, 136 generalized, 134–136, 138 symmetric, 585 thermodynamic, 133, 134, 136–138, 140

D Decomposition polar, 36–49, 71, 166–169, 177, 185, 225, 230, 288, 304, 633, 640, 641 spectral, 535, 537, 538, 543 Deformation measure Almansi left and right, 24, 25, 42, 87, 175, 183, 188, 218, 230, 303, 593 Cauchy-Green left and right, 24, 25, 42, 87, 175, 183, 188, 230, 303, 593 energetic, 173 generalized, 187, 262, 264, 296, 384, 414 Hencky left and right, 43, 76, 268, 307, 308 logarithmic, 43, 267, 269, 270 quasienergetic, 182–183, 185, 189, 192, 196, 230, 262, 280, 288, 304 Density, 90, 91, 133, 134, 185–186, 193, 197, 198, 206, 207, 211, 219, 245, 277, 281, 287, 292, 293, 302–303, 313, 366–367, 383, 389, 391, 394, 477, 480, 556, 561, 563, 660, 667, 670–671, 680, 704

Index Derivative contravariant, 13, 20 convective, 51 co-rotational mixed left and right, 81, 86, 309 Cotter-Rivlin, 80, 81, 87, 88, 232, 308, 309 covariant, 13–14, 20, 147–149, 153 in eigenbasis left, 82–83, 88, 232, 310, 646 right, 81–82, 86, 233, 310, 322, 490 Fr´echet’s, 502, 503, 510, 595 Jaumann’s, 83, 86, 88, 232, 311 material, 51 Oldroyd’s, 79–81, 87, 88, 308 partial with respect to time (local), 50, 51, 54, 74, 77, 355, 599, 642, 651 spin, 84–85, 232, 233, 311 total with respect to time, 50 of a n-th order tensor, 51 Description of a continuum Eulerian, 5–8, 155–156 Lagrangian, 5–8, 50, 156–157 material, 5–23 spatial, 5–23, 155–157, 381, 399–402, 560 Deviator, 471, 492, 539, 552, 607–609, 619, 644, 661 Diagram of deforming 427 in simple shear, 442 Differential of a tensor, 53–54 total, 153, 154, 156, 289, 373, 375, 595 of a vector, 53 Dissipative heating, 561, 579–589 Divergence of a tensor, 14, 151, 324, 334, 416

E Effect Bauschinger’s, 665 Poisson’s, 425, 431 Poynting’s, 442 Efficiency, 124, 128–132, 136–139 Energy free Gibbs’s, 199–200, 204, 286, 591, 595, 597, 627, 628, 641, 648 Helmholtz’s, 198–199, 461, 486 internal, 115, 120, 130, 133, 316, 378, 379, 622, 693, 694 kinetic, 115, 118–123, 331, 332 potential, 414, 417 total, 115, 118, 121, 378 Enthalpy, 201–202, 286

715 Entropy, 125, 126, 128, 130, 133, 155, 316, 318, 377, 380, 381, 391, 393, 403, 409, 410, 514, 522, 525, 559, 561, 563, 656, 657 Entropy production, 124, 125, 127, 128, 136, 140, 316, 339, 352, 369 Equation balance angular momentum, 112–114, 158 energy, 117–123, 155, 317, 332, 377, 380, 388, 403, 407, 410, 559, 656, 657 entropy, 126, 155, 318, 377, 380, 391, 393, 403, 410, 559, 561, 563, 656, 657 momentum, 98, 107–108, 118, 155, 158 compatibility, 141 dynamic, 149–152, 155, 162, 318, 332, 377, 381, 386, 396 static, 142, 148–149 continuity in Eulerian variables, 92–93, 330, 377 in Lagrangian variables, 90–91, 157, 288, 384 heat conduction, 381, 388, 393, 396, 407, 560, 561, 563, 565, 567, 579, 581–582, 584–585, 587 heat influx, 118–121, 123, 131, 318, 332, 380 kinematic, 73, 155, 156, 318, 335, 377, 381, 396, 564 of a singular surface, 372–375 variational, 416, 417, 419

F Field of possible pressures, 417, 419 scalar, 9, 50, 78, 92, 417 tensor, 9 stationary, 53 vector, 9 kinematically admissible, 413 real, 413 Fluid, 221 compressible, 288, 322 linear-viscous, 480, 481 Newtonian, 480 viscous, 480, 481 Force of body inertia, 316 external, 96–98, 115, 118, 124, 130, 316, 412–414, 424, 428

716 of interaction of bodies, 96 internal, 97 inertia, 98 mass, 97, 98, 115, 331, 338, 402, 414, 423, 493, 660 surface, 97, 98, 104, 106, 107, 115, 116, 118, 123, 370, 414 thermodynamic, 340, 464, 630 Formula Coriolis’s, 327–329, 331, 338 Euler’s, 326–327, 332 Gauss-Ostrogradskii, 94–95, 100, 117, 123, 126, 416, 581 Function Dirac’s, 500 dissipation, 126–128, 140, 198, 211, 339, 461, 463, 464, 470, 480, 482, 485–487, 491, 505, 513, 516, 517, 522, 524, 525, 529, 534, 539, 556, 559, 561, 563, 579, 582–584, 587, 588, 596, 629, 643 of equilibrium stresses, 462, 463 Heaviside’s, 493, 528, 535, 576, 601, 683, 692 indifferent relative to a symmetry group, 237–240, 243, 244, 256, 257, 467, 507–510, 514 of memory, 498, 503 pseudopotential, 465 quasiperiodic, 580, 581, 583 quasipotential, 211, 244, 247, 463 relaxation, 529, 533–535, 537–541, 544, 547, 548, 551, 571 rotary-indifferent, 256, 257 scalar isotropic, 243 orthotropic, 243 transversely isotropic, 243 of temperature-time shift, 515 tensor, AI -unimodular, 474–476, 482 of viscous stresses, 462–468, 475, 478, 488 Functional, 497 continuous, 209, 499–503, 509, 555, 592 Fr´echet-differentiable, 501–504, 516, 592 linear, 500, 501, 507, 509, 524, 530, 533, 550 scalar n-fold, 507 quadratic, 507

Index G Gas, 128, 129, 133, 288, 368, 369, 451, 692, 696 Gradient deformation, 15 elastic, 382, 593, 632 plastic, 593, 633, 639 surface, 373, 374, 700 of a vector, 13, 27 velocity, 56–58, 65–73, 81, 85–88, 163, 188–196, 330, 492 H Heat absorbed by a body, 130 explosion, 561, 586 pseudoexplosion, 586 released by a body, 130 Heating dissipative, 561, 579–589 rate, 115, 120 Heat machine, 128–132, 140 I Indeterminate Lagrange multiplier, 417 Inequality Clausius-Duhem, 198 Clausius’s, 125, 128, 138, 139 dissipation, 127, 340, 465, 539 Fourier’s, 127, 340, 465 Planck’s, 125–127, 198, 339, 340, 464 Influx entropy, 130 heat, 118–121, 123, 124, 131, 132, 318, 332, 380 Intensity of a tensor, 608 Invariant, 240 cubic, 241, 249, 250, 523, 611, 616 linear, 241, 250–252, 523, 524, 537, 538, 701 principal, 175–176, 185–186, 192, 193, 242, 243, 245, 260, 261, 266, 273, 274, 277, 280, 283–286, 293, 324, 327, 471, 473, 477, 480, 608, 618 quadratic, 241, 249–251, 471, 607, 620, 701 simultaneous, 469, 470, 476–478, 490, 491, 508–511, 513, 517, 518, 520, 522, 523, 549, 603–608, 611, 612, 614, 616, 617, 619, 634, 635, 637, 638, 650, 651, 656, 663, 701 spectral, 537 linear, 537, 541 quadratic, 537

Index J Jump of a function across a singular surface, 351, 356, 358, 360, 362

L Lagrangian, 5–9, 17, 18, 49, 50, 74, 79, 80, 90–91, 108, 113, 119–121, 123, 128–130, 149–150, 156–157, 288, 301, 328, 369, 384, 391, 392, 414, 415, 419, 447, 449 Law angular momentum balance, 109–114, 122, 412 of changing plastic deformations, 598 conservation of mass, 89–95 Fourier’s, 339–346, 378, 464, 480, 598 gradient, 600, 601, 631, 650 momentum balance, 95–108, 114, 316, 412 motion of a continuum, 5, 6, 24, 74, 114, 121, 127, 213, 300, 301, 324, 367, 410, 414 Stokes’s, 464 thermodynamic first, 114–124, 317 second, 124–140, 318, 332 Length of a vector, 2, 46 Loading active, 600, 675, 685, 689 fixed, 401 neutral, 600 passive, 600 plastic, 600, 601, 609, 631, 663, 672, 674, 675, 694 tracking, 401

M Mass, 24–95, 97, 98, 109, 110, 114–116, 121, 124, 132, 316, 317, 331, 338, 363–366, 399, 402, 403, 414, 423, 493, 660, 693 Material body, 1, 3 point, 1–7, 9, 14–17, 29, 32, 45, 49, 51, 54, 57, 60, 73–75, 89–92, 95, 104–106, 124, 128, 133–135, 141–143, 149, 205, 208, 209, 227, 313, 327, 347–350, 353, 357, 358, 364–368, 370, 375, 400, 401, 406, 411, 447, 592

717 Matrix Jacobian, 10 inverse, 10 metric, 2, 10, 12, 19, 21, 28, 30, 33–36, 49, 145, 146, 148, 153, 191, 192, 214, 448, 593 Method Saint-Venant’s, 451 semi-inverse, 421, 446, 454 Model An , 206, 208–212, 219, 236–238, 243–245, 248–252, 254, 256, 262, 263, 265–267, 270–272, 274–277, 279, 282, 285–287, 289–292, 295–297, 299, 319–320, 322, 336, 339, 341, 342, 378–380, 385, 392, 412, 414, 422, 423, 425–430, 436–439, 442–443, 449–450, 454, 461–484, 497, 505–540, 546–558, 560–571, 573–578, 591–623, 652–659, 662, 671, 673–675, 689–691, 699–703, 705, 706 Bartenev-Hazanovich, 298, 434 Bn , 207, 208, 211, 219, 253–254, 262, 263, 265, 266, 270–272, 274, 275, 279, 282, 285–287, 290, 294, 295, 297–299, 319–320, 322, 337, 339, 341, 378, 379, 385, 414, 420, 431–436, 438, 443, 445, 454, 461–482, 492–497, 549, 552–558, 562, 566, 568–571, 573–577, 581, 588, 589, 623–640 Chernykh’s, 298 Cn , 207, 208, 212, 219, 254–263, 265–267, 270–272, 274, 275, 280–282, 285, 287, 290, 292, 294, 295, 320–323, 337, 339, 341, 342, 378, 379, 385, 414, 482–491, 497, 640–652 Dn , 207, 208, 212, 213, 219, 254–263, 265, 266, 270–272, 274, 275, 282, 285, 291, 295, 298–299, 320–323, 339, 341, 342, 378, 379, 385, 414, 420, 482–491, 497, 640–652 Drucker’s, 600 Duhamel-Neumann, 514, 561, 563 Huber-Mises, 607–610, 612, 616–619, 644, 660, 662, 666, 701, 703 John’s, 251, 452 linear, 250 isotropic, 251, 296, 297 mechanically determinate, 534, 540, 551, 553, 564, 566 Mooney’s, 298 Murnaghan’s, 251

718 of plasticity associated, 599, 606, 610 two-potential, 612 quasilinear, 249–252, 286, 380, 389, 390, 396, 397, 620 semilinear, 251 simplest, 471, 473, 538, 548, 553, 557, 568, 581, 699 Treloar’s, 298, 433, 434 of a viscous fluid, 480, 481 Modulus of volumetric compression, 662 Motion rigid, 300–313. 317, 322, 324, 326, 343, 472, 481, 490, 556, 618, 639, 640, 652 stationary (steady), 74 Moving basis, 17, 18, 51, 52, 65, 78, 80–84, 325, 326, 328, 329, 335 coordinate system, 81, 329, 332, 334, 335, 338 volume, 91–92, 94, 118, 123, 157, 360–362, 414

N Nabla-operator, 13, 14, 19, 52, 111, 122, 303, 329–331 Natural state, 248 unstressed, 249, 472 Neighborhood of a point, 1, 14, 45–49, 60–62 of a singular surface, 355

O Operator, 153, 154, 206–210, 214, 312, 341, 461, 465, 497, 500, 502, 531–534, 591 Oriented surface element, 30–32, 102, 359 Orthoprojector, 536, 537, 539

P Parameter Odkwist’s, 599 of quadratic elasticity, 247 strengthening, 607, 610, 668 Taylor’s, 598, 606, 609, 611, 615, 630, 636, 638, 639 Poisson ratio, 425, 428, 429, 442, 443, 453, 690, 705

Index Potential plastic, 599, 601, 602, 607, 609, 612, 616, 619, 631, 634, 636, 637, 650, 701 viscous, 700 Power of forces, 115, 118, 121, 123, 162 of stresses, 162, 163, 176, 186–188, 197, 204, 268, 271, 298, 380, 415, 487, 490, 623–627, 646–648 Prehistory, 497, 498, 500, 509, 561, 592 Pressure, 276 hydrostatic, 291, 294, 456 Principal axis of anisotropy, 226 of transverse isotropy, 228 Principal thermodynamic identity (PTI), 196–208, 210–213, 288–290, 298, 339, 377, 393, 462, 483, 484, 505, 546, 549, 591, 594, 595, 623, 627, 628, 641–643, 648, 649 Principle of equipresence, 161, 208, 464, 485, 487, 628 of local action, 161, 208–209 of material indifference, 161, 248, 280, 300–324, 343–345, 472, 481, 483, 556, 618, 639–641, 652 of material symmetry, 161, 213–221, 226, 236–287, 291–292, 300, 341–345, 465–466, 473, 485–487, 491, 506–507, 603–605, 632–634, 651–652 of objectivity, 315 Onsager’s, 161, 339–346, 463–465, 474, 478, 484, 488, 491, 597–598, 601, 602, 604, 607, 611, 629–630, 636–638, 644, 649, 701 of thermodynamically consistent determinism, 161, 205–209, 339, 340 variational Lagrange’s, 415, 419 Problem coupled strongly, 407, 410, 561, 563 weakly, 407, 409 Lam´e, 446–460 Process adiabatic, 132, 370 in the broad sense, 132 locally, 132, 134 in the restricted sense, 132 with a constant extension, 503 irreversible, 127, 140

Index isothermal, 133, 410 locally, 132, 134, 135 quasistatic, 388–390, 396–401, 403, 404, 411, 412, 657 reversible, 140 static, 388, 389, 396, 503, 504 uniform thermomechanical, 135, 136, 138 Pseudoinvariant, 256, 258, 259, 261 R Radius-vector, 2, 3, 5, 14, 15, 29, 30, 44–46, 49, 56, 58, 60, 62, 63, 90, 95, 141–143, 150, 153, 214, 215, 300, 301, 315, 325, 326, 350, 353, 357, 367–369, 372, 374, 401, 447, 592 Rate of heating, 115, 120, 135 Relative elongation, 29, 30, 36, 46, 59, 578 Representation Boltzmann’s, 525–528, 531 spectral, 535–539, 544, 547, 548 Volterra’s, 500, 525, 526 Residual deformation of creep, 494 S Shock adiabatic curve, 695–699 Simple shear, 7, 8, 21, 47, 57, 194, 438–446 Space of elementary geometry, 1 Euclidean, 2 point, 2 metric, 1, 2 tensor functional, 498–499 Specific entropy, 125, 133, 316, 381, 393, 514, 525, 563 entropy production, 125, 127, 140, 316, 339, 597, 629, 644 internal energy, 115, 133, 316, 378, 622 mass force, 97, 98, 331 surface force, 97, 98 total energy, 118, 378 Spectrum of relaxation times, 539 of relaxation values, 539 of viscous stresses, 540 Spin, 63–65, 71, 72, 82, 84–86, 109, 122, 232–234, 309–311, 327, 626, 627, 633, 646 Streamline, 73–75 Stress normal, 104, 106, 442, 445 relaxation, 572, 573 tangential, 104–106, 424, 431, 441, 459, 460

719 Surface singular, 347 coherent, 350, 355–361 completely incoherent, 369 of contact, 349 ideal, 370–371 homothermal, 369–370, 399, 402, 404 incoherent, 350 nondissipative, 369–370 of phase transformation, 349, 369, 370 semicoherent, 369 of a shock wave, 369 of a strong discontinuity, 349, 350, 362 of a weak discontinuity, 349 stream, 75 vortex, 75 without singular displacements, 367 Symbols Christoffel of the first kind, 145 of the second kind, 145 Levi-Civita, 2 Symmetry group, 219 continuous, 226, 509 isomeric, 222–225 point, 226

T Temperature absolute, 124 Tensor of angular rate of rotation, 61–65 coaxial, 187, 188, 196, 203, 204, 267, 269, 271, 273, 643, 645, 650 of creep cores, 542, 544, 548 deformation Almansi left and right, 24, 25, 40, 42, 87, 165, 175, 183, 188, 230, 303, 593 Cauchy-Green left and right, 24, 25, 40, 42, 87, 165, 175, 183, 188, 230, 303, 593 generalized energetic, 187, 414 Hencky left and right, 43, 76, 268, 307, 308 logarithmic, 43, 267, 269, 270 principal energetic, 175 quasienergetic, 178, 180, 185–186, 189, 192, 196, 219, 230, 233, 262, 304

720 deformation rate, 56–59, 61, 62, 66, 69, 83, 85–88, 152–155, 163, 306, 475, 476, 653 elasticity quadratic, 247 quasilinear, 247, 379, 620 of energetic equivalence, 170, 263, 264, 392, 444, 656 heat conductivity, 340–343, 345, 407, 599 heat deformation, 514 heat expansion, 514 H -indifferent, 215 absolutely, 229, 231–233, 277, 279 relative to a group, 232, 233, 237, 239 H -invariant, 215–218, 220, 229–235, 239, 255, 262, 274, 280, 282, 336, 338, 486, 491, 632, 651, 652 H -pseudoindifferent, 218 indifferent relative to a symmetry group, 238–240, 243, 256, 345, 467, 468, 474, 478, 507–510, 536, 537 isomeric, 224 Levi-Civita, 2 of moment stresses, 110–113, 123 producing of a group, 239–241, 246, 286, 486, 524, 536 of quasienergetic equivalence, 181 of relaxation cores, 534, 540, 543, 548 of relaxation functions, 533–535, 537, 544, 547, 548 Riemann-Christoffel, 145–147 R-indifferent, 301–312, 314, 320, 321, 323, 324, 343, 481, 483, 490, 491, 639–641, 652 R-invariant, 301–310, 312, 320, 322, 324, 343, 472, 481, 556, 618, 639, 640, 652 rotation accompanying deformation, 39, 43, 65, 191, 225, 230, 304, 379 second-order, 12, 14, 23, 37, 56, 78, 81, 88, 101, 102, 110, 111, 154, 175, 177, 188, 210, 216, 238–243, 273, 275, 283, 302, 324, 340, 462, 464, 468, 535–537, 540, 591, 597, 602 spherical, 70, 234, 248, 276, 282, 345, 400, 619, 659–662 of strengthening, 596 stress Cauchy, 101–104, 112, 113, 190, 218, 248, 263, 281, 291, 304, 392, 400, 406, 423, 436, 439–441, 444, 446, 450, 455, 475, 478, 493, 548, 556 generalized energetic, 262, 264, 296, 384

Index Piola-Kirchhoff, 102–103, 105, 108, 113, 119, 162, 197, 305, 392, 404, 450, 680 principal energetic, 164, 170, 175, 305 quasienergetic, 176, 178, 180, 181, 191, 195, 219, 231, 248, 262, 281, 305, 423, 482, 483, 642 reduced, 596 rotation, 231, 261, 626, 646 stretch left and right, 39, 46, 81–83, 164, 168, 173, 175, 176, 180–183, 230, 232–234, 304, 309, 310, 322, 645, 646 unit (metric), 12, 43, 77, 86, 87, 174, 184, 217, 229, 240, 247, 248, 276, 303, 423, 608 viscosity, 464, 465, 468, 488–489 vorticity, 56–59, 61, 62, 65, 155, 163, 190, 306 Tensor basis, 11, 13, 24, 42, 102, 190–192, 245, 261, 292–295, 467–470, 478, 488, 603, 604, 606 Tensor field, 9 stationary, 53 varying, 51 Tensor law of transformation, 12, 14 Theorem Cauchy-Helmholtz, 57, 58 Cauchy’s, 98–101 Noll’s, 222, 224 on the polar decomposition, 36–40 Stone-Weierstrass, 507–508 Truesdell’s, 136–140 Thermodynamic cycle, 133, 134, 136–138, 140 flux, 339, 340, 464, 597 potential, 202, 203, 213, 522 Time, 3 absolute, 3 fast, 580, 584 reduced, 514–516, 547, 583 slow, 580, 581, 583, 584 Trajectory of a point, 73 Truesdell’s estimate for the efficiency, 131 Tube stream, 75–77 vortex, 75–77 Types of continua, 161, 209–210, 655

U Universe, 1 Unloading, 494, 579, 600, 663, 665, 672–675, 678, 699, 700

Index V Variables active, 205, 206, 208, 210, 212, 214, 220, 312, 339, 341, 461, 462, 484, 497, 591, 648 reactive, 205, 206, 208, 212, 214, 220, 312, 341, 462, 483, 497, 505, 591, 629, 630, 644, 648 Variation of a functional, 414 of a vector field, 413 Vector angular momentum, 109, 110 displacement, 25–28, 34, 36, 135, 141, 149, 150, 153, 367–369, 381, 394, 401, 407, 564, 565, 567 external forces, 96, 316, 426, 428 force, 96, 109, 304, 316 heat flux, 116–117, 119, 122, 317, 331, 339, 343, 344 local, 9–11, 16, 18, 21, 23, 30, 36, 52, 54, 79, 90, 101, 143, 144, 216, 301–303, 329, 448, 592, 632 of mass moments, 109, 110 momentum, 95, 315 stress, 98 Piola-Kirchhoff, 103, 404 of surface moments, 109

721 velocity, 49 of a singular surface, 353, 357 vorticity, 57–59, 61–64, 75, 84, 155, 326, 635, 638 Velocity, 49 of a singular surface, 353 normal, 354 relative, 56, 326, 328, 335 total, 327 translational, 57, 327 Vortex line, 73–75

W Wave plane, 679–699 Riemann, 692 shock, 99, 349, 369, 370, 693–697 Work, 130 done by external forces, 130, 414 elementary of stresses, 162

Y Yield point, 609, 614, 674, 676 strength, 609, 614, 618, 620, 635, 663, 665, 666, 675, 678

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