Nonlocal Continuum Field Theories

Nonlocal Continuum Field Theories A. Cemal Eringen SPRINGER Nonlocal Continuum Field Theories Springer New York He...

Author: A. Cemal Eringen

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Nonlocal Continuum Field Theories

A. Cemal Eringen

SPRINGER


Springer New York Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo

A. Cemal Eringen


With 72 Figures

A. Cemal Eringen Emeritus Professor, Princeton University 15 Red Tail Drive Littleton, CO 80126-5001 USA

Library of Congress Cataloging-in-Publication Data Eringen, A. Cemal. Nonlocal continuum field theories/A. Cemal. Eringen. p. cm. Includes bibliographical references and index. ISBN 0-387-95275-6 (alk. paper) 1. Field theory (Physics). 2. Continuum mechanics. I. Title. QC173.7.E76 2001 530–dc21

200102044

© 2002 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Timothy Taylor; manufacturing supervised by Erica Bresler. Typeset by by TEXniques, Inc., Cambridge, MA. Printed and bound by Sheridan Books, Inc., Ann Arbor, MI. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 ISBN 0-387-95275-6

SPIN 10833382

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Preface

This book is devoted to the development of the nonlocal continuum field theories of material bodies where nonlocal intermolecular attractions are important. There exists a large class of problems in classical physics and continuum mechanics (classical field theories) that fall outside their domain of applications. Fracture of solids, stress fields at the dislocation core and at the tips of cracks, singularities present at the point of application of concentrated loads (forces, couples, heat, etc.), sharp corners and discontinuities in bodies, the failure in the prediction of short wavelength behavior of elastic waves, and several decades of the viscosity rise in fluids flowing in microscopic channels are but few major anomalies that defy classical treatment. In the field of electromagnetic theory, polaritons, gyrotropic effects, and superconductivity cannot be treated by the classical field theories. For example, several branches of high-frequency waves are not predicted and short wavelength regions deviate grossly from experimental observations. To cast light on the materials behavior in these areas, physicists often seek help from atomic lattice dynamics. The main purpose of this book is to present a unified foundation for the development of the basic field equations of nonlocal continuum field theories. To this end, we have relied on the natural extensions of the two fundamental laws of physics to nonlocality: (i) the energy balance law is postulated to remain in global form; and (ii) a material point of the body is considered to be attracted by all points of the body, at all past times. By means of these two natural generalizations of the corresponding local principles, theories of nonlocal elasticity, fluid dynamics, and electromagnetic field theories are formulated that include nonlocality in both space and time (memory-dependence).

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Preface

Solutions of the field equations are given for many critical problems in these fields, displaying unexpected favorable agreements with the atomic theories and/or the experimental observations. Presently, there exists no published treatise in this field, except some papers and limited reviews that are scattered in the literature. We believe this book offers rich material for research ideas in nonlocal solid and fluid mechanics, electromagnetic theory, and the nonlocal theories of media with microstructure and nonlocal theory of liquid crystals. It provides mathematical methods for the exploration of many failures of classical field theories and for the discoveries of new physical phenomena and/or explanations of the old ones. It can be used for graduate study in the departments of applied physics, applied mathematics, engineering sciences, and for research. We express our thanks to Jennifer Page who typed the nonmathematical parts of the manuscript. My daughter Meva who freed much time for me by undertaking many chores of life and life itself. To her goes my everlasting love. Littleton, Colorado December 2000

A. Cemal Eringen

Introduction

Nonlocal continuum field theories are concerned with the physics of material bodies whose behavior at a material point is influenced by the state of all points of the body. Following the classical notions, material points of a body are considered to be continuous and are assigned some physically independent objects (variables) (e.g., mass, charge, electric field, magnetic field). The state of the body, at a material point, is described by the relations of the response objects that constitute another class (e.g., stress, internal energy, heat) as functions of the independent objects. These relations are called constitutive equations. The nonlocal theory generalizes the classical field theory in two respects: (i) the energy balance law is considered valid globally (for the entire body); and (ii) the state of the body at a material point is described by the response functionals. This means that we need complete knowledge of the independent variables at all points of the body to describe the state of the body at each point. In this book the terminology of nonlocal field theories is to be understood in this sense. In fact, we have also included, in this description, nonlocality in time which is known as memory-dependence. As a subclass, the present definition includes gradient theories which possess limited nonlocality. Here various order gradients of the independent objects, at a material point and their time rates, enter into response functions at the same material points, Eringen [1966]. A third category of nonlocality involves constitutive ersatzs between the spatial statistical moments of dependent and corresponding sets of independent variables. Such a group of theories are known as director theories, micromorphic, microstretch, and micropolar theories. These theories are explored elsewhere, Eringen [1999], [2001].

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Introduction

The present book is devoted to the development of the nonlocal field theories described by the fundamental notions (i) and (ii) above. The question arises, why nonlocality? The domain of the applicability of classical field theories is intimately connected to the length and time scales. If L denotes the external characteristic length (e.g., crack length, wavelength, a length over which applied loads are smooth) and l the internal characteristic length (e.g., granular distance, lattice parameter), then in the region L/ l 1, classical field theories predict sufficiently accurate results. On the other hand, when L/ l ∼ 1, local theories fail and we must resort to either atomic or nonlocal theories that can account for the long-range interatomic attractions. For the dynamical case, there will be a similar scale T /τ where T is the external characteristic time (e.g., the time scale of the applied loads) and τ is the internal characteristic length (e.g., the time scale of transmission of signal from one molecule to the next). Again, classical theories fail when T /τ ∼ 1. Thus we see that the physical phenomenon in space–time requires nonlocality and memory effects scaled by L/ l and T /τ . The concept of nonlocality is inherent in solid state physics where the nonlocal attractions of atoms are prevalent. Here the material is considered to consist of discrete atoms connected by distant forces from other neighboring atoms. Hence, we expect relations between the nonlocal field theories and the lattice dynamics (see Section 6.9). Real materials do not resemble perfect crystal lattices or lattices with few imperfections (e.g., dislocations, foreign atoms, etc.) which enable a physicist to describe the state of the body. Unfortunately, amorphous and engineering materials do not lend themselves to such ideal descriptions, while nonlocal continuum theories make it possible to describe material properties very well. In fact, this description is faithful in microscopic scales all the way to the size of the lattice parameter (see Sections 6.10 to 6.17). Thus, nonlocal continuum theory should be considered a viable discipline in physics and applied science. Prior to the 1960s some elements of continuum theory had entered into the discussion of various physical phenomena that required mathematical descriptions in the forms of the integro-partial differential equations. For example, Chandrasekhar’s [1950] radiative transfer equation is of this type. Other historical references in this category are given in several articles in Continuum Physics, Vol. IV, edited by Eringen [1976]. The formulation of the nonlocal elastic constitutive equations by means of lattice dynamics, was given by Krumhansl [1965], [1968], Kröner and Datta [1966], and Kunin [1966]. The continuum approach to memory-dependent nonlocal elasticity was made by Eringen [1966] and to nonlocal micropolar elasticity by Eringen [1965]. These early theories did not include the aspect (i), namely, the global nature of the balance laws. This aspect along with the nonlocal constitutive equations was formulated by Eringen and Edelen [1972] and Eringen [1972a,b]. A review and development of the theory, and historic reference up to 1976, are contained in Continuum Physics, Vol. IV, edited by Eringen [1976].

Introduction

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The theory of nonlocal elasticity has attracted the attention of many writers, perhaps because of its early success in resolving an old problem in fracture mechanics. It is well known that classical elasticity predicts infinite stress at the tip of a sharp crack. This led researchers to invent several fracture criteria (e.g., Griffith, fracture toughness, energy release rate, J -integral, etc.) independent of the state of stress at the tip of a crack. The nonlocal elasticity solution of Eringen and Kim [1974a,b] and Eringen et al. [1977] showed that the stress at the tip of a crack is finite, it rises to a maximum and then diminishes with the distance from the crack tip (cf. Section 6.14). Similarly, the nonlocal solution of the discreet dislocation problem (Eringen [1977a,b]) leads to vanishing stress at the eye of the dislocation, rising to a maximum in the interior of the body and approaches the classical solution with the distance. These results led to the introduction of a natural fracture criterion, based on the maximum stress hypothesis and the accurate prediction of the cohesive stress (cf. Sections 6.12 to 6.17). Moreover, nonlocal solutions also lead to dispersion relations for harmonic waves that are faithful to the lattice dynamical results throughout the entire Brillouin zone. The nonlocal continuum theory of fluent media, including the microstructural effects, were discussed by Eringen [1972b], [1973b], applications to turbulence by Speziale and Eringen [1981], the diffusion of gases by Demiray and Eringen [1978], and magnetohydrodynamics by Eringen [1986]. McCay and Narasimhan [1981] discussed theory of nonlocal electromagnetic fluids, and Narasimhan and McCay [1981] applied their theory to dispersive waves in dielectric fluids. Eringen [1973c], [1984a], [1984b], [1984c], [1986], [1990], [1991] introduced the theories of nonlocal electromagnetic elastic solids and gave solutions of several problems: point charges, optical waves, polaritons, Eddy currents, gyrotropic media, superconductivity, Alfven waves, etc. A book by Agranovich and Ginsburg [1984] covers many aspects of the nonlocal effects on crystal optics. The nonlocality here appears under the terminology of “Spatial Disperson." In spite of its age, the literature on nonlocal field theories is not extensive. There exist several reviews, conference symposia, and reports. Among them we mention: Continuum Physics, Vol. IV, edited by A.C. Eringen, Academic Press, 1976. This book contains accounts of the polar and nonlocal polar theories of elastic solids. Nonlocal Theories of Material Media, The Polish Academy, 1976. This book contains reviews on nonlocal elasticity. Nonlocal Theory of Material Media, CISM Courses and Lectures, No. 268, edited by D. Rogula, Springer-Verlag, 1982. Here, lattice defects, nonlocal elasticity, and some soluble problems are discussed by several authors. Elastic Media with Microstructure, Vols. I and II, by I.A. Kunin, Springer-Verlag, 1982/1983. In these books the lattice dynamical approach is employed for the discussion of microstructural problems.

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Introduction

Mechanics, Electromagnetic Materials, and Structures, AMD, Vol. 61, MD-Vol. 42, edited by J.S. Lee, G.A. Maugin, and Y. Shindo, American Society of Mechanical Engineers, 1993. This is a collection of papers presented at a scientific meeting. The articles are mostly in classical fields, except one article entitled “Vistas of Nonlocal Electrodynamics" by Eringen. EUROMECH Colloquium, Vol. 378, Mulhouse, France 20–22, April 1998, organized and published by Universite de Mulhouse CNRS-CSI. This report contains several articles on micromechanics, nonlocal elasticity, and the solutions of some problems. Report No. 39 by E.S. Suhubi, ¸ March 1987. This report was used for the lectures by Professor Suhubi ¸ at the University of Calgary Canada. I am indebted to Professor Suhubi ¸ for sending me a copy of this report. The unified formulation of the theories presented in this book is new and has not appeared before in the literature. The solutions to a plethora of problems are based mostly on the author’s, and his collaborator’s, work. References to these and works of other authors appear in the appropriate sections. It is not feasible to make a comprehensive survey of the fields that include all the published work. Needless to say, there are many other published works that may have escaped my attention. For this failure, I express my apologies to those authors whose work I missed to give credit. The main purpose of this book is to present a unified approach to the formulations of the nonlocal field theories for memory-dependent elastic solids, viscous fluids, electromagnetic solids, and fluids that conduct heat. Nonlocal balance laws are obtained by using the axiom of the Galilean invariance of the energy balance law (Chapter 2). Nonlocal constitutive equations are then constructed, based on some postulates and the thermodynamic conditions of admissibility. The resulting equations are nonlinear and include memory and nonlocality. Chapter 3 presents the constitutive equations of nonlinear, nonlocal, memory-dependent thermoelastic solids, and thermoviscous fluids. In Chapter 4, we obtain the nonlocal balance laws of the electromagnetic theory and in Chapter 5 the constitutive equations of nonlinear, nonlocal, memory-dependent electromagnetic thermoelastic solids and thermoviscous fluids. In principle, the basic formulations are now complete. In Chapter 6 on nonlocal elasticity, we begin to develop the linear theories and we present a multitude of applications to the theory of different fields. The solutions of some critical problems show unequivocally the power and potential of the nonlocal theory. For example, the stress field at the core of a discrete dislocation tends to vanish, reaching a maximum outside the core of dislocation. When this maximum is equated to the cohesive stress that holds the atomic bonds together, we obtain the celebrated Griffith criterion. Alternatively, we are able to predict the cohesive stress exactly, by setting the internal characteristic length to the one based on phonon dispersion (cf. Sections 6.10 to 6.14). The solution of the crack problems, rigid stamp on a half-plane leads us to a Griffith-like fracture criterium

Introduction

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with a Griffith constant determined. The crack-tip singularity prevalent in the classical solution is eliminated by the nonlocal solution. The dispersion relations of the plane harmonic waves in nonlocal elasticity are identical to those obtained in lattice dynamics, throughout the entire Brillouin zone. The nonlocal fluid dynamics problems treated in Section 7.3 clearly display the nonlocal effects on the mean viscosity. The nonlocal linear electromagnetic theory, presented in Chapter 8, explains the nature of the dielectric tensor with regard to longitudinal and transverse electromagnetic waves, in perfect accord with lattice dynamics. Optical waves, polaritons, Eddy currents, gyrotorpic media, and Alfven waves and their dispersive characters are explained. Remarkably enough, we are able to construct a general theory of superconductivity that includes nonlocality similar to the one proposed by Pippard [1965] in a completely different way. Memory-dependent nonlocal thermoelastic solids, thermofluids, electromagnetic elastic solids, and electromagnetic thermofluids are the subjects of Chapters 9 to 12. With Chapter 13, we begin to develop the theory of nonlocal micromorphic thermoelastic solids, microstretch thermoelastic solids, and micropolar thermoelastic solids. The uniqueness and reciprocal theorem are given. Variational principles are presented and several problems solved. Here the solution of the plane harmonic waves in nonlocal elastic solids leads to four branches of the dispersion curves with a faithful representation in the entire Brillouin zone to those known atomic lattice dynamics (see Section 13.15). The field equations of memory-dependent nonlocal micropolar electromagnetic elastic solids are obtained in Chapter 14. Finally, in Chapter 15, we present a nonlocal electromagnetic continuum theory of liquid crystals. Nonlocal field theories contain very interesting physics, in fact, all physics, excluding quantum effects and elementary particle physics. This can be extended further to include the nonlocal mixture theory, diffusion, and other allied phenomena (Eringen [1998a]). It could provide bases for rheological, biological, and neural systems. It is my hope that future generations will continue on this path as the destination offers great rewards and is very rich in the world of science.

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Contents

Preface

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Introduction 1 Motion and Deformation 1.0 Scope . . . . . . . . . . . . . 1.1 Motion . . . . . . . . . . . . . 1.2 Strain Measures and Rotations 1.3 Strain Invariants . . . . . . . . 1.4 Time-Rate of Tensors . . . . . 1.5 Objective Tensors . . . . . . . 1.6 Compatibility Conditions . . . Problems . . . . . . . . . . . . . .

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2 Stress 2.0 Scope . . . . . . . . . . . . . . 2.1 Balance of Energy . . . . . . . . 2.2 Second Law of Thermodynamics 2.3 Dissipation Potential . . . . . . Problems . . . . . . . . . . . . . . .

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3 Constitutive Equations 31 3.0 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Constitutive Axioms . . . . . . . . . . . . . . . . . . . . . . . . 31

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3.2 Memory-Dependent Nonlocal Thermoelastic Solids . . . . . . . . 38 3.3 Memory-Dependent Nonlocal Thermofluids . . . . . . . . . . . . 43 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Nonlocal Electromagnetic Theory 4.0 Scope . . . . . . . . . . . . . . . . . . . . 4.1 Electromagnetic Balance Laws . . . . . . . 4.2 Electromagnetic Force, Couple, and Power . 4.3 Electromagnetic Force, Couple, and Power . 4.4 Mechanical Balance Laws . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . .

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49 49 49 53 55 56 59

5 Constitutive Equations of Nonlocal Electromagnetic Media 5.0 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Electromagnetic Thermoelastic Solids . . . . . . . . . . 5.2 Electromagnetic Thermoviscous Fluids . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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61 61 62 66 70

6 Nonlocal Linear Elasticity 6.0 Scope . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Linear Constitutive Equations . . . . . . . . . . . 6.2 Lattice Dynamical Foundations of Linear Elasticity 6.3 Material Stability . . . . . . . . . . . . . . . . . . 6.4 Field Equations of Nonlocal Linear Elasticity . . . 6.5 Uniqueness Theorem . . . . . . . . . . . . . . . . 6.6 Power and Energy . . . . . . . . . . . . . . . . . . 6.7 Reciprocal Theorem . . . . . . . . . . . . . . . . . 6.8 Variational Principles . . . . . . . . . . . . . . . . 6.9 Approximate Models . . . . . . . . . . . . . . . . 6.10 Screw Dislocation . . . . . . . . . . . . . . . . . . 6.11 Edge Dislocation . . . . . . . . . . . . . . . . . . 6.12 Dislocation in Nonlocal Hexagonal Elastic Solids . 6.13 Continuous Distribution of Dislocations . . . . . . 6.14 Nonlocal Stress Field at the Griffith Crack . . . . . 6.15 Line Crack Subject to Shear . . . . . . . . . . . . 6.16 Interaction of a Dislocation with a Crack . . . . . . 6.17 Interaction Between Defects and Dislocation . . . . 6.18 Straight Wedge Disclination . . . . . . . . . . . . 6.19 Somigliana-Type Representation . . . . . . . . . . 6.20 Fundamental Solution . . . . . . . . . . . . . . . . 6.21 Nonlocal Elastic Half-Plane . . . . . . . . . . . . 6.22 Rigid Stamp on a Nonlocal Elastic Half-Space . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . .

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71 71 73 78 82 82 87 91 93 95 98 106 112 116 123 132 138 144 153 161 164 165 166 171 175

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Contents

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7 Nonlocal Fluid Dynamics 7.0 Scope . . . . . . . . . . . . . . . . . . . . . 7.1 Constitutive Equations . . . . . . . . . . . . 7.2 Field Equations of Nonlocal Fluid Dynamics 7.3 Channel Flow . . . . . . . . . . . . . . . . . 7.4 Lubrication in Microscopic Channels . . . . . 7.5 Lubricant Film Flow on a Rotating Disk . . . Problems . . . . . . . . . . . . . . . . . . . . . .

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177 177 178 179 181 183 189 192

8 Nonlocal Linear Electromagnetic Theory 8.0 Scope . . . . . . . . . . . . . . . . . . 8.1 Balance Laws . . . . . . . . . . . . . . 8.2 Nonlocal Electromagnetic Elastic Solids 8.3 Electromagnetic Solid Media . . . . . . 8.4 Models for the Dielectric Tensor . . . . 8.5 Point Charge . . . . . . . . . . . . . . 8.6 Optical Waves . . . . . . . . . . . . . . 8.7 Polaritons . . . . . . . . . . . . . . . . 8.8 Eddy Currents . . . . . . . . . . . . . . 8.9 Gyrotropic Media . . . . . . . . . . . . 8.10 Superconductivity . . . . . . . . . . . . 8.11 Nonlocal Theory . . . . . . . . . . . . 8.12 Alfven Waves . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . .

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193 193 194 195 199 200 204 205 206 209 210 212 214 220 221

9 Memory-Dependent Nonlocal Thermoelastic Solids 9.0 Scope . . . . . . . . . . . . . . . . . . . . . . . 9.1 Linear Constitutive Equations . . . . . . . . . . 9.2 Boundary-Initial Value Problems . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . .

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223 223 223 228 229

10 Memory-Dependent Nonlocal Fluids 10.0 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Nonlocal Thermoviscous Fluids . . . . . . . . . . . . . 10.2 Field Equations of Memory-Dependent Nonlocal Fluids . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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231 231 231 234 235

11 Memory-Dependent Nonlocal E-M Elastic Solids 11.0 Scope . . . . . . . . . . . . . . . . . . . . . 11.1 Linear Constitutive Equations . . . . . . . . 11.2 Field Equations . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . .

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237 237 237 243 245

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12 Memory-Dependent Nonlocal E-M Thermofluids 247 12.0 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 12.1 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . 247 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

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13 Nonlocal Microcontinua 13.0 Scope . . . . . . . . . . . . . . . . . . 13.1 Kinematical Preliminaries . . . . . . . 13.2 Time-Rate of Tensors . . . . . . . . . . 13.3 Mass, Inertia, Kinetic Energy . . . . . . 13.4 Stress, Stress Moments, Energy Balance 13.5 Balance Laws . . . . . . . . . . . . . . 13.6 Second Law of Thermodynamics . . . . 13.7 Micromorphic Elastic Solids . . . . . . 13.8 Microstretch Elastic Solids . . . . . . . 13.9 Micropolar Elastic Solids . . . . . . . . 13.10Nonlocal Micropolar Elasticity . . . . . 13.11Uniqueness Theorem . . . . . . . . . . 13.12Reciprocal Theorem . . . . . . . . . . . 13.13Variational Principles . . . . . . . . . . 13.14Nonlocal Micropolar Moduli . . . . . . 13.15Propagation of Plane Waves . . . . . . 13.16Displacement Potentials . . . . . . . . . 13.17Two Vector Fields . . . . . . . . . . . . 13.18Fundamental Solutions . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . .

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253 253 254 261 264 267 271 276 278 286 293 296 301 306 308 311 316 320 321 323 324

14 Memory-Dependent Nonlocal E-M Elastic Solids 14.0 Scope . . . . . . . . . . . . . . . . . . . . . 14.1 Constitutive Equations of E-M Elastic Solids 14.2 Field Equations . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . .

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325 325 325 335 336

15 Nonlocal Continuum Theory of Liquid Crystals 15.0 Scope . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Description of Liquid Crystals . . . . . . . . . . . 15.2 Balance Laws . . . . . . . . . . . . . . . . . . . . 15.3 Nonlocal E-M Liquid Crystals . . . . . . . . . . . 15.4 Constitutive Equations of Nematic Liquid Crystals 15.5 Field Equations . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . .

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337 337 337 340 340 343 347 349

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

Appendix A

351

References

357

Index

365

Errata

375

1 Motion and Deformation

1.0 Scope This chapter is devoted to discussions of the kinematics of continuous media. In Section 1.1 we present the motion of a material point X in the body as a oneparameter mapping from X to a spatial point x. The inverse motion is the inverse one-to-one mapping from x to X. Deformation gradients are introduced and used to determine area and volume changes with the deformation. In Section 1.2 we introduce material and spatial strain measures and obtain their expressions in terms of the displacement vector. Section 1.3 gives a short account of strain invariants. In Section 1.4 we obtain the time-rate of deformation gradients, deformation tensors, area and volume changes with deformation, which are relevant to the dynamics of continua. Section 1.5 discusses the principle of objectivity, fundamental to the constitutive theory. Compatibility conditions are obtained in Section 1.6. This chapter is fundamental to the mathematical developments presented in later chapters.

1.1 Motion A material body B is a collection of material particles with a volume V and surface S, embedded in Euclidean space E 3 . As a reference frame, we employ Cartesian coordinates XK , K = 1, 2, 3, or P = X, at the reference state, at time t = 0, to locate a particle P in V (Figure 1.1.1). At a later time, t, the body may occupy a new configuration b in space, with P now located at p with position vector p = x or Cartesian coordinates xk . The coordinate system XK and xk may be selected to have different origins and orientations.

2


The motion of the point P ∈ B to p ∈ b, at time t, is described by the mapping xk = xk (X1 , X2 , X3 , t)

or

xk = xk (X, t),

k = 1, 2, 3,

(1.1.1)

and the inverse motion by XK = Xk (x1 , x2 , x3 , t)

or

XK = XK (x, t).

(1.1.2)

For brevity we also express these in the forms x = x(X, t),

X = X(x, t).

(1.1.3)

Thus, according to (1.1.1), the motion takes a material particle P ∈ B at time t = 0, to a spatial position p ∈ b at time t. The inverse motion (1.1.2) states the converse. We assume that the mappings (1.1.3) are single valued and that they possess continuous partial derivatives with respect to their arguments. Moreover, they are a unique inverse of each other. This is guaranteed by ∂xk > 0, (1.1.4) J ≡ det ∂XK where J is the Jacobian determinant. If we determine the spatial position xk , of every point XK ∈ B at time t, we will know the new shape and the position of the body at each time t relative to that at t = 0. Thus, the theory of continuous media deals, in essence, with the determination of (1.1.1) given the initial data at t = 0 and the bounary data. Through the equations of motion (1.1.3) we have dxk = xk,K dXK ,

dXK = XK,k dxk .

(1.1.5a)

Alternatively, dP = ck dxk , ck (x, t) ≡ XK,k IK ,

dp = CK dXK , CK (X, t) ≡ xk,K ik ,

(1.1.5b)

where IK and ik are Cartesian base vectors (Figure 1.1.1). Here, and henceforth, repeated indices denote summation over the range of summed indices and an index following a comma denotes the partial derivative, e.g., xk,K ≡

∂xk , ∂XK

XK,k ≡

∂XK . ∂xk

(1.1.6)

These two sets of quantities, defined by (1.1.6), are called deformation gradients. Sometimes (1.1.6) is also denoted by F = ∇ x,

FkK ≡ xk,K .

(1.1.7)

1.1 Motion

3

Figure 1.1.1. Coordinate systems. In this case (1.1.6)2 is expressed as F−1 . Through the chain rule of differentiation, we have xk,K XK,l = δkl ,

XK,k xk,L = δKL .

(1.1.8)

Each of the nine quantities xk,K or XK,k can be solved from either of these linear equations. Using the Cramer rule of determinants we can determine XK,k in terms of xk,K : 1 1 XK,k = cofactor xk,K = (1.1.9) εKLM klm xl,L xm,M , J 2J where KLM and klm are the permutation symbols and J = det xk,K = 16 KLM klm xk,K xl,L xm,M .

(1.1.10)

By differentiation (1.1.10), we obtain (J XK,k ),K = 0, (J −1 xk,K ),k = 0, ∂J = cofactor xk,K = J XK,k . ∂(xk,K )

(1.1.11) (1.1.12)

From (1.1.5b) it can be seen that C1 dX1 , C2 dX2 , and C3 dX3 represent the edge vectors of a parallelepiped built on the vectors dp and c1 dx1 , c2 dx2 , and c3 dx3 on that built on dP (Figure 1.1.2). Consequently, the area vector for the surface, built on edge vectors C1 dX1 and C2 dX2 , is given by da3 = C1 × C2 dX1 dX2 , which is the deformed image of the area dA3 based on I1 dX1 and I2 dX2 , i.e., dA3 = dX1 dX2 . Thus, da3 = C1 × C2 dX1 dX2 = xk,1 , xl,2 ik × il dX1 dX2 .

(1.1.13)

4


Figure 1.1.2. (a) Deformation of a parallelepiped (Lagrangian representation); (b) A rectangular parallelepiped in the deformed body was a rectilinear one in undeformed body (Eulerian representation).

Similar expressions are obtained for da1 and da2 so that an element of the area vector is given by da = da1 + da2 + da3 = J XK,k dAK ik

(1.1.14)

1.2 Strain Measures and Rotations

5

and (da)2 = J 2 XK,k XL,l dAK dAL .

(1.1.15)

By taking the inner product of da3 with C3 dX3 , we obtain the deformed volume element dv = J dV . (1.1.16)

1.2 Strain Measures and Rotations Elements of arc lengths dS and ds, in reference and spatial configurations B and b, are, respectively, given by dS 2 = dXK dXK ,

ds 2 = dxk dxk .

(1.2.1)

Upon using (1.1.5) in these expressions we obtain dS 2 = ckl dxk dxl ,

ds 2 = CKL dXK dXL ,

(1.2.2)

where ckl (x, t) = δKL XK,k XL,l = XK,k XK,l , CKL (X, t) = δkl xk,K xl,L = xk,K xk,L ,

(1.2.3) (1.2.4)

are, respectively, the Cauchy deformation tensor and the Green deformation tensor. Both of these tensors are symmetric tensors, i.e., CKL = CLK , ckl = clk . Sometimes also used are the inverse of these tensors: −1 c kl −1 C KL

≡ bkl = δKL xk,K xl,L ,

(1.2.5)

≡ BKL = δkl XK,k XL,l ,

(1.2.6)

which are known, respectively, as the Finger and Piola deformation tensors. Note that ckr brl = δkl , CKR BRL = δKL . (1.2.7) Using (1.2.1) and (1.2.2) we have ds 2 − dS 2 = 2EKL (X, t) dXK dXL = 2ekl (x, t) dxk dxl ,

(1.2.8)

where 2EKL = CKL (X, t) − δKL ,

2ekl = δkl − ckl (x, t),

(1.2.9)

are known as the Lagrangian and Eulerian strain tensors, respectively. From (1.2.8) it follows that EKL = ekl xk,K xl,L ,

ekl = EKL XK,k XL,l .

(1.2.10)

6


An alternative way of introducing the deformation tensors is through the Cauchy decomposition theorem which states that a matrix F may be decomposed as products of two matrices, one of which is orthogonal and the other is a symmetric tensor. For the proof, see Eringen [1967, p. 42] or [1980, p. 46]. Thus, F = RU = VR, with U2 = FT F = C,

(1.2.11) −1

V2 = FFT = c = b,

(1.2.12)

where a superscript T denotes transpose. Here R is orthogonal. This is the rotation tensor. U and V are called the right and left stretch tensors. Commensurate with this terminology, C and b are also called the right and left Cauchy–Green deformation tensors. We may express the strain components in terms of the displacement vector u, which extends from P to p (Figure 1.1.1): u = x − X + 0.

(1.2.13)

Here 0 is the vector connecting the origin of the coordinate frames. Substituting (1.2.13) into (1.2.3) and (1.2.4), we obtain 2EKL = CKL − δKL = UK,L + UL,K + δMN UM,K UN,L , 2ekl = δkl − ckl = uk,l + ul,k − δmn um,k un,l ,

(1.2.14) (1.2.15)

where we employed u = UL IL = ul il ,

(1.2.16)

with UL representing the component of u in the reference frame B, and ul representing those in the spatial frame b. KL and KL and The linear strain tensors E ekl , and the linear rotation tensors R rkl , are introduced by KL = 1 (UK,L + UL,K ) ≡ U(K,L) , E ekl = 21 (uk,l + ul,k ) ≡ u(k,l) , 2 KL = 1 (UK,L − UL,K ) ≡ U[K,L] , R rkl = 21 (uk,l − ul,k ) ≡ u[k,l] , 2

(1.2.17)

where subscript parentheses enclosing indices denote the symmetric part of the indices enclosed and brackets denote the antisymmetric part. In three-dimensional space an antisymmetric tensor can be represented by an axial vector so that 2 rk = klm rml ,

ML . K = KLM R 2R

(1.2.18)

K represents the linear rotation vectors. The finite rotation vectors Here rk and R follow from (1.2.11): R = FU−1 = V−1 F. (1.2.19)

1.3 Strain Invariants

1.3

7

Strain Invariants

From (1.2.2) we have 2(n) ≡

ds dS

2 = CKL NK NL ,

NK =

dXK , dS

(1.3.1)

where (n) is called the stretch. But through (1.2.1), NK is subject to δKL NK NL = 1.

(1.3.2)

The stationary value of the stretch is obtained by differentiating 2(n) : ∂ [CKL NK NL − C(δKL NK NL − 1)] = 0, ∂NM

(1.3.3)

where C is a Lagrange multiplier. This gives three linear equations (CKL − CδKL )NL = 0.

(1.3.4)

The solution for N may exist if the characteristic determinant vanishes, i.e., det(CKL − CδKL ) = 0.

(1.3.5)

Expansion of this determinant gives −C 3 + IC C 2 − IIC C + IIIC = 0,

(1.3.6)

where IC = CKK ,

2 − C2 − C2 , IIC = C11 C22 + C22 C33 + C33 C11 − C12 23 31 IIIC = det CKL , (1.3.7) are the invariants of the deformation tensor CKL . The characteristic equation (1.3.6) possesses three roots Cα , α = 1, 2, 3, called the eigenvalues of CKL . It can be shown that Cα are real, and the proper directions N1 and N2 corresponding to any two proper numbers C1 and C2 , are orthogonal. We note that three other sets of invariants related to (1.3.7) exist, e.g.,

I1 = tr C,

I2 = tr C2 ,

I3 = tr C3 ,

(1.3.8)

with relations I1 = IC ,

2 − 2II , I2 = IC C

3 − 3I II + 3III , I3 = IC C C C

IC = I1 ,

IIC = 21 (I12 − I22 ),

IIIC = 16 (2I3 − 3I1 I2 + I13 ).

(1.3.9)

For the proofs of these and other related interesting topics, see Eringen [1967 or 1980, Section 1.10].

8


1.4 Time-Rate of Tensors Definition 1. The material time-rate of a function f (x, t) is defined by ∂f Df ˙ , = f = Dt ∂t X

(1.4.1)

where the subscript X accompanying a bar denotes that X is held constant. If f = f (X, t), then ∂f (X, t) . (1.4.2) f˙ = ∂t If f = f (x, t), then Df ∂f ∂f f˙ = x˙k . (1.4.3) = + Dt ∂t ∂xk For example, the velocity vector v and the acceleration vector a are defined by v=

∂x(X, t) = x˙ , ∂t

a = v˙ =

∂ 2 x(X, t) . ∂t 2

(1.4.4)

If we substitute X = X(x, t) into v, we obtain v(X(x, t), t) ≡ vˆ (x, t).

(1.4.5)

In this case, vˆ represents the velocity of a material point at x. However, the identity of the particle at x is not known. For the acceleration of a material point x we have ∂ vˆ + vˆ ,k vˆk . a = v˙ˆ = ∂t Henceforth, we drop the circumflex (ˆ) for brevity and express this as a = v˙ =

∂v Dv = + v,k vk . Dt ∂t

(1.4.6)

Lemma 1. The material derivative of xk,K is given by D (xk,K ) = vk,l xl,K Dt

or

D (dxk ) = vk,l dxl . Dt

(1.4.7)

The proof of this follows from the fact that the operators D/Dt and ∂/∂XK commute. Corollary. The material derivative XK,k is given by D (XK,k ) = −XK,l vl,k . Dt

(1.4.8)

1.4 Time-Rate of Tensors

9

This result follows from differentiating XK,k xk,L = δK,L and using (1.4.7). Theorem 1. The material time derivative of the deformation tensor CKL is given by C˙ KL = 2dkl xk,K xl,L , (1.4.9) where dkl (called the deformation-rate tensor) is defined by dkl = 21 (vk,l + vl,k ).

(1.4.10)

This is obtained by differentiating (1.2.4) and using (1.4.7). Theorem 2 (Killing). The necessary and sufficient condition for the motion of a body to be locally rigid is dkl = 0. Proof . For rigid motions we must have D (ds 2 ) = 0. Dt Using (1.2.2)2 and (1.4.9), this gives D (ds 2 ) = 2dkl dxk dxl = 0. Dt

(1.4.11)

This must be true for every pair of points in the body, i.e., for arbitrary dx. Hence, we must have dkl = 0. The deformation-rate tensor d is fundamental to the development of the constitutive equations of the fluent media. Lemma 2. The material time derivative of the Jacobian is given by DJ = J vk,k . Dt

(1.4.12)

Proof . We have DJ D ∂J ∂J D(xk,K ) = (det xk,K ) = = vk,l xl,K , Dt Dt ∂(xk,K ) Dt ∂(xk,K ) where we used (1.4.7). But according to (1.1.12): ∂J = J XK,k . ∂(xk,K ) Hence, there follows (1.4.12).

10


Definition 2. Spin tensor wkl is defined by wkl = v[k,l] = 21 (vk,l − vl,k ).

(1.4.13)

From (1.4.10) and (1.4.13), by addition, we obtain vk,l = dkl + wkl .

(1.4.14)

Equivalent to the spin tensor is the vorticity vector wk defined by wk = klm wml = klm vm,l

or

w ≡ curl v.

(1.4.15)

Theorem. The material time derivative of the element of area is given by D (dak ) = vr,r dak − vr,k dar . Dt

(1.4.16)

To prove this we calculate the material derivative of (1.1.14): D DJ D (J XK,k dAK ) = XK,k + J (XK,k ) dAK . Dt Dt Dt

Upon using (1.4.8) and (1.4.12) we obtain (1.4.16). Similarly, from (1.1.16) and (1.4.12), it follows that D (dv) = ∇ · v dv. Dt

(1.4.17)

1.5 Objective Tensors It is intuitively clear that the material properties of a body, measured in a rigidly moving frame of references, must remain unaltered. In the formulation of the response functions, it is desirable to employ quantities that are independent of the motion of the observer. Such quantities are called objectives or material frame-indifferent. Definition 1. Two motions xk (X, t) and xk (X, t) are called objectively equivalent if and only if xk (X, t) = Qkl (t)xl (X, t) + bk (t),

t = t − a,

(1.5.1)

where a is a constant time-shift, b(t) is a time-dependent translation, and {Q(t)} are time-dependent full orthogonal transformations, i.e., Qkl Qml = Qlk Qlm = δkm ,

det Qkl = ±1.

(1.5.2)

Here, Qkl consists of all rigid rotations det Q = +1, and inversions det Q = −1.

1.5 Objective Tensors

11

Two objectively equivalent motions differ only in the relative frame and time. For a fixed frame and time the two motions may be made to coincide by a superposition of an arbitrary rigid motion and a shifting origin of time. Definition 2 (Objectivity). Any tensorial quantity that obeys the tensor transformation law under (1.5.1) is called objective or material frame-indifferent. For example, a vector qk and a tensor tkl are objective since they obey the transformation laws t) = Qkl (t)ql (X, t), qk (X, tkl (X, t) = Qkm (t)Qln (t)tmn (X, t).

(1.5.3)

Vectors and tensors that do not depend on time are objective. For time-dependent vectors and tensors this is not always true. Consider, for example, the velocity vector v = x˙ . From (1.5.1), we have D xk ˙ kl xl + b˙k , = Qkl x˙l + Q Dt or

(1.5.4) ˙ kl xl + b˙k . vk = Qkl vl + Q

This is not in the form (1.5.3)1 . Hence the velocity vector is not objective. The same is also true for the acceleration vector. Theorem. The deformation-rate tensor dkl is objective but the spin tensor wkl is not objective. Proof . From (1.5.4) we have vk,l = Qkm vm,n

∂xn ˙ km ∂xm . +Q ∂ xl ∂ xl

(1.5.5)

xl we solve for xn from (1.5.1), by multiplying it by Qkn and To calculate ∂xn /∂ using (1.5.2): xn = Qkn ( xk − bk ). Thus

∂xn = Qln . ∂ xl

(1.5.6)

Substituting this in (1.5.5) we obtain ˙ km Qlm . vk,l = Qkm Qln vm,n + Q

(1.5.7)

Interchanging the indices k and l and adding, we obtain dkl = Qkm Qln dmn ,

(1.5.8)

12


where we used

˙ km Qlm + Qkm Q ˙ lm = 0, Q

(1.5.9)

which follows from (1.5.2). Equation (1.5.8) shows that dkl is an objective tensor. Upon using (1.5.7), for the spin tensor we obtain w kl = Qkm Qln wmn + kl ,

(1.5.10)

where kl is the relative angular velocity of the two frames, defined by ˙ km Qlm . kl ≡ Q Equation (1.5.10) indicates that the spin tensor is not objective.

(1.5.11)

1.6 Compatibility Conditions According to (1.2.15) the six components of ckl and ekl are expressed in terms of the three components of the displacement vector uk , i.e., 2ekl = δkl − ckl = uk,l + ul,k − δmn um,k un,l .

(1.6.1)

When the displacement vector possesses a continuous partial derivative, by substituting u into this equation we can calculate ekl and ckl . On the other hand, if ekl or ckl is given, is it possible to obtain single-valued continuous displacement fields corresponding to ekl and ckl ? This statement poses a question for the integrability of the six partial differential equations (1.6.1). This is answered by a theorem of Riemann. To prepare for this theorem, we recall expressions (1.2.1) and (1.2.2) of the elements of arc lengths, namely, dS 2 = δKL dXK dXL = ckl dxk dxl , ds 2 = CKL dXK dXL = δkl dxk dxl .

(1.6.2)

Both the deformed and undeformed bodies are embedded in a Euclidean three-dimensional space. Thus, if we consider the motion xk = xk (X1 , X2 , X3 , t), k = 1, 2, 3, as a coordinate transformation from the rectangular coordinates XK to the curvilinear coordinates xk at a fixed time, then ckl plays the role of a metric tensor in the curvilinear coordinates xk , and the same is valid for CKL for the inverse motion. Now in Euclidean space, any six quantities ckl cannot be a metric tensor, unless they satisfy a theorem of Riemann, namely: For a symmetric tensor akl to be a metric tensor for a Euclidean space, it is necessary and sufficient that akl be a nonsingular, positive-definite tensor, and the (a) Riemann–Christoffel tensor Rklmn formed from it vanishes identically.1 1 See Appendix A.

1.6 Compatibility Conditions

13

(a)

Rklmn ≡ 21 (akn,lm + alm,kn − akm,ln − aln,km ) −1

+ a rs ([lm, s][kn, r] − [ln, s][kn, r]),

(1.6.3)

where [kl, m] ≡ 21 (akm,l + alm,k − akl,m ), −1 a ns

−1

asl = ans a sl = δnl .

(1.6.4)

Both CKL and ckl are nonsingular symmetric, positive-definite tensors of the Euclidean three-dimensional space. Hence, we must have2 (C)

RKLMN = 0,

(c)

Rklmn = 0.

(1.6.5)

Of the 81 components of each of these tensors only six are algebraically independent and nonvanishing. By substituting C and c for a in (1.6.5) we obtain the compatibility conditions for CKL and ckl or EKL and ekl . We give below one of these: ekn,lm + elm,kn − ekm,ln − eln,km −1

− c rs [(ekr,n + ern,k − ekn,r )(els,m + ems,l − elm,s ) − (ekr,m + emr,k − ekm,r )(els,n + ens,l − eln,s )] = 0.

(1.6.6)

For the small strains we drop the products of ekl to obtain the compatibility conditions of the linear strain tensor ekn,lm + elm,kn − ekm,ln − eln,km = 0.

(1.6.7)

When these conditions are satisfied, then the single-valued integral of

exists and is given by

ekl = 21 (uk,l + ul,k )

(1.6.8)

kl xl + bk , uk = u0k + R

(1.6.9)

kl is a skew-symmetric tensor independent of where u0k is any solution of (1.6.8), R xk , and bk is a vector independent of xk . Thus, the displacement field is determined uniquely to within an arbitrary time-dependent rigid motion. If the compatibility conditions are satisfied, then there will not be a single-valued solution for the displacement field corresponding to ekl . In this case, the body will contain dislocations, cracks, inclusions, or other discontinuous fields. 2 For a brief account on tensor calculus, see Eringen [1980, Appendix C]. See also Eringen [1971, Section 3.7].

14


Problems 1.1 Prove equations (1.1.11) and (1.1.12). KL follows 1.2 Show that for small deformations the linear rotation tensor R from the finite rotation R given by (1.2.19). 1.3 Show that the proper directions Nα are mutually orthogonal. 1.4 Prove the Cayley–Hamilton theorem for ckl : c3 − Ic c2 + IIc c − IIIc 1 = 0. 1.5 Obtain the material time-rate c˙kl of ckl . Is c˙kl an objective tensor? 1.6 The velocity field of a continuum is described by x˙ = f (y) − yg(r),

y˙ = xg(r),

z˙ = 0,

r = (x 2 + y 2 )1/2 .

(i) Determine the deformation rate tensor and its invariants. (ii) Obtain the expression of the vorticity vector. 1.7 Show that the linear compatibility equations (1.6.7) can also be derived by using the expression of the strain tensor ekl . 1.8 Study the literature and give the proof of the Riemann theorem (1.6.5).

2 Stress

2.0 Scope Chapter 2 is concerned with the fundamental concept of stress. The energy balance law for the entire body is postulated in Section 2.1. The global equation of the conservation of mass is given. By means of the transport and Green–Gauss theorem, the global laws of mass and energy are localized. An important theorem shows that the invariance of the energy balance law, under each Galilean group, leads to a balance law. Localization involves residuals, which are discussed in the remainder of this section. We have shown that the localization residuals can be incorporated into the concepts of the stress and energy leaving the new residuals only in the jump conditions. In this way, new concepts of stress and energy involving long-range interactions can be made part of the classical concepts. In Section 2.2 we present the second law of thermodynamics. The Clausius– Duhem (C–D) inequality, fundamental to the development of the constitutive equations, is introduced. Section 2.3 discusses the concept of dissipation. A theorem shows how one may obtain the solution of the C–D inequality. Onsager reciprocal relations are discussed. The solution of the C–D inequality is extended to the memory functionals relevant to the discussions of the dissipative media.

2.1 Balance of Energy In a material body, two types of loads are considered to act: Extrinsic body loads and surface loads. The first of these act at the mass points of the body. This may be called volume or body loads. Examples of these are the force of gravity, electromagnetic loads, and heat sources.

16

2 Stress

The surface loads arise from the action of one part of the body on another part through the common surface. The surface loads give rise to the concept of the stress tensor, electromagnetic surface loads, and the heat vector. These loads depend on the orientation of the surface on which they act. Denoting the external unit normal of the surface by n, the surface loads per unit surface area may be expressed as a linear function of n, e.g., t(n)k = tlk nl ,

q(n) = qk nk .

(2.1.1)

Of these, the first one is the surface traction per unit area and the second one is the heat load per unit area. Here tlk is the stress tensor and qk is the heat vector that act on a surface. The body loads and surface loads possess energies that are balanced by the internal energy stored in the body, and the kinetic energy. The law of the energy balance is then postulated as d 1 (tkl vl + qk )nk da ρ + 2 v · v dv = dt V −σ ∂ V −σ + ρ(fk vk + h) dv. V −σ

(2.1.2)

The left-hand side represents the time-rate of change of the total internal and kinetic energies. On the right-hand side, the surface integral expresses the energies of the total surface tractions and heat and the volume integral, the energies of the applied body force f, and other loads (e.g., heat input). As usual, ρ is the mass density. Equation (2.1.2) is the global equation of the energy balance law, valid for the entire body having volume V and surface ∂V, excluding the points of a discontinuity surface σ which may be sweeping the body with its own velocity u in the direction of the unit normal of σ · V − σ and ∂V − σ abbreviate V − σ = V − V ∩ σ,

∂V − σ = ∂V − ∂V ∩ σ.

(2.1.3)

We note that an internal energy density is postulated to exist. This is the stored energy in the body. If there are other loads, beside thermomechanical loads, then the energies of these loads are to be added to the right-hand side of (2.1.2). Adjoined to (2.1.2) is the equation of the conservation of mass which reads d ρ dv = 0. dt V −σ

(2.1.4)

For the evaluation of the time-rate of integrals and the conversion of surface integrals, we need some theorems:

2.1 Balance of Energy

Transport Theorems: d ∇ · v) dv + [ϒ(v − u)] · da, ϒ dv = (ϒ˙ + ϒ∇ dt V −σ V −σ σ ∂q d ∇ · q · da + curl(q × v) + v∇ q · da = dt S −γ S −γ ∂t [q × (v − u)] · dx; +

17

(2.1.5a)

(2.1.5b)

γ

Green–Gauss Theorem. A · n da = ∂ V −σ

V −σ

∇ · A dv +

[A] · n da;

Stokes’ Theorem. ∇ × A) · da + [A] · h ds = (∇ S −γ

(2.1.6a)

σ

γ

∂ S −γ

A · dx,

(2.1.6b)

where ϒ is any tensor and q and A are vectors, all being continuously differentiable in the star-shaped domains. The bold brackets [ ] denote the jump of its enclosure across the discontinuity surface σ or the discontinuity line γ · σ (γ ) may be sweeping the body (surface) with its own velocity u (see Figures 2.1.1 and 2.2.2). For proof of these theorems, see Eringen [1967, 1980, Section 2.5]. By means of expressions (2.1.5a) and (2.1.6a) the energy balance laws (2.1.2) may be transformed to

∇ · v) + (tkl,k + ρfl − ρ v˙l )vl −( + 21 v · v)(ρ˙ + ρ∇ V −σ −ρ ˙ + tkl vl,k + qk,k + ρh dv + [ − ρ( + 21 v · v)(vk − uk ) σ

+ tkl vl + qk ]nk da = 0. Similarly, equation (2.1.4) of the conservation of mass becomes ∇ · v) dv + [ρ(vk − uk )]nk da = 0. (ρ˙ + ρ∇ V −σ

(2.1.7)

(2.1.8)

σ

These equations are valid globally, i.e., for the entire body. Equation (2.1.8) may be expressed in local form as ∇ · v = ρˆ ρ˙ + ρ∇ [ρ(vk − uk )]nk = Rˆ

in V − σ,

(2.1.9a)

on σ,

(2.1.9b)

ˆ called the mass density residuals, are subject to where ρˆ and R,

18

2 Stress

d a = n da

σ ∂V – σ V–σ

Figure 2.1.1. Domain of application of equations (2.1.5a) and (2.1.6a). V − σ and ∂V − σ excludes points of σ .

V

ρˆ dv +

V −σ

Rˆ da = 0.

(2.1.10)

Here ρˆ and Rˆ represent, respectively, the body and surface mass creations. For chemically inert bodies they vanish, i.e., ∇ ·v =0 ρ˙ · ρ∇ [ρ(v − u)] · n = 0

in V − σ,

(2.1.11a)

on σ.

(2.1.11b)

Thus, in chemically inert bodies the mass is conserved locally. In order to obtain the local balance laws, we postulate that the energy balance law is invariant under the Galilean group of transformations. Intuitively, this is clear since the balance laws must be valid in a frame of reference that may be undergoing rigid motions. Axiom. The energy balance law is form-invariant under the Galilean group of transformations. We now employ this axiom to obtain the balance laws for thermomechanics.


19

n

γ h S–γ

∂S – γ

Figure 2.1.2. Surface of application of equations (2.1.5b) and (2.1.6b). S − γ and ∂S − γ excludes points of γ . Theorem. For each Galilean group of the energy balance law, there is a balance law of thermomechanics. Proof . For a time-dependent rigid motion of the frame of reference we have xk = Qkl (t)xl + bk (t),

(2.1.12)

where Qkl is an orthogonal tensor, i.e., QQT = QT Q = 1,

det Q = 1.

(2.1.13)

The velocity field in the new fram is given by ˙ kl xl + b˙k . vk = Qkl vl + Q

(2.1.14)

The angular velocity is expressed by (see (1.5.11): ˙ km Qlm . kl = Q

(2.1.15)

Suppose that, at time t, the body is brought back to the original orientation (i.e., Q = 1) having only constant translation and angular velocities (i.e., and b˙ are constants). Then, at time t, vk = vk + kl xl + b˙k ,

˙ kl . kl = Q

(2.1.16)

Under these transformations, the mass density ρ, the internal energy density , the stress tensor tkl , and the heat vector qk do not change, since the motion is rigid. But the body force fk must be accommodated by accelerations, i.e., f − v˙ = f − v˙ .

(2.1.17)

20

2 Stress

The new kinetic energy densities in the x- and x-frames are given by K = 21 v · v,

= 1 ˙ v. ˙ K 2v ·

(2.1.18)

Subtracting the energy balance (2.1.2) in the x-frame from that in the x-frame, we obtain d − K) dv = ρ(K tkl ( vl − vl )nk da dt V −σ ∂ V −σ ρ(fk vk − fk vk ) dv. (2.1.19) + V −σ

Applying the transport and Green–Gauss theorem, this becomes

˙ − K) − K)(ρ˙ + ρ∇ ˙ + tkl,k ( ∇ · v) − ρ(K vl − vl ) −(K V −σ vl − vl ),k + ρ(fk vk − fk vk ) dv + tkl ( − K)(vk − uk ) + tkl ( + [ − ρ(K vl − vl )]nk da = 0. (2.1.20) σ

Using (2.1.16) to (2.1.18), we have − K = vk (km xm + b˙k ) + 1 (kl xl + b˙k )(km xm + b˙k ), K 2 ˙ − K˙ = K v˙ · v − v˙ · v, vk − vk = kl xl + b˙k , v˙ k − v˙k = kl vl , fk vk − fk vk = fk ( vk − vk ) + vk ( v˙ k − v˙k ). (2.1.21) Substituting these into (2.1.20) we obtain

∇ · v) dv − vk (km xm + b˙k ) + 21 (kl xl + b˙k )(km xm + b˙k ) (ρ˙ + ρ∇ V −σ [(tkl,k + ρfl − ρ v˙l )(lr xr + b˙l ) + tkl lk ] dv + V −σ [tkl − ρvl (vk − vk )]nk (b˙l + lr xr ) da + σ − 21 (lr xr + b˙l )(lm xm + b˙l )[ρ(vk − uk )]nk da. σ

For chemically inert bodies, in view of (2.1.11), the first volume integral and the last surface integral vanish. For arbitrary b˙k and lk the second volume integral and the first surface integral may be localized to give tkl,k + ρ(fl − v˙l ) = ρ fˆl [tkl − ρvl (vk − uk )]nk = Fˆl tkl = tlk

in V − σ,

(2.1.22a)

on σ,

(2.1.22b)

in V − σ,

(2.1.23)


where the residuals ρ fˆ and Fˆ are subject to ρ fˆl dv + Fˆl da = 0. V −σ

21

(2.1.24)

σ

Using these results in the energy balance law (2.1.7) we obtain −ρ ˙ + tkl vl,k + qk,k + ρh = −ρ ˆ [ − ρ( +

1 2v

· v)(vk − uk ) + tkl vl + qk ]nk = −Eˆ

in V − σ,

(2.1.25a)

on σ,

(2.1.25b)

where the energy residuals are subject to ρ(fˆl vl − ˆ ) dv − Eˆ da = 0. V −σ

(2.1.26)

σ

If (2.1.24) and (2.1.26) are assumed to be true for any arbitrary surface, we would have ˆ Fˆl da = 0, ρ fl dv = 0, (2.1.27) V −σ σ Kˆ dv = 0, Eˆ da = 0, (2.1.28) V −σ

where

σ

Kˆ = ρ(fˆl vl − ˆ ).

(2.1.29)

However, in general, interactions between the surface and body load dictates that (2.1.24) and (2.1.26) represent the true situation. For chemically inert bodies, equations (2.1.11) and (2.1.22) to (2.1.26) constitute the balance of nonlocal continuum mechanics. They result from the invariance of the global energy balance under translations and the rigid rotations of the frame of reference. Hence the proof of the theorem.

A. Nature of Nonlocal Residuals The presence of nonlocal residuals in the balance laws is due to the fact that only global balance laws are posited to be true. In the special case where these residuals vanish, the balance laws become local. Classically, this is obtained by positing that the global (integral) balance laws are valid for every volume and every surface element of the body. This is known as localization. Accordingly, local balance laws are considered to be valid for each infinitesimal volume element isolated from the body. The fact that classical (local) continuum mechanics has been rather successful in its predictions, except for some singular situations (e.g., concentrated loads, sharp cracks, corners, etc.), owes its success to the extremely small zones of cohesion. The interatomic attractions of a reference atom are known to extend a few atomic distances, after becoming extremely small. With distance, interatomic

22

2 Stress

attractions die out fast. In lattice dynamical calculations, Born–Kármán approximation is often used, which extends only to the nearest neighbors. Seldom more than 10 neighbors are considered, which in crystal lattices corresponds roughly ◦

to 10 lattice parameter’s distance (or approximately 10 A= 10−7 cm for Nacl crystal). Clearly then the cohesive distance is rc ≤ 10−7 cm for Nacl crystals. In continuum mechanics, it is permissible to consider much larger cohesive radii (e.g., granular media). Let us define rc for each material as the characteristic radius at the end of which the cohesive force reduces to 0.1% of its value at the reference point. If a sphere with radius rc is considered to be the arbitrary volume element for the nonlocal body, then the nonlocal residuals can be considered to vanish. Therefore, the effects of the nonlocal residuals would be important only in situations where the cohesive zone is penetrated by the nature of the physical phenomena. This would be the situation, for example, if a concentrated load is applied to a single atom, or a crack tip is only one to two atomic distances sharp. Such mechanical experiments are nearly impossible, even with the 10 atomic distances. In any case, in such rare cases, nonlocal theory will also fail, because of the quantum considerations, and one may have to resort to atomic theories. This discussion does not imply that we are back into local continuum mechanics. Indeed, the nonlocal properties of materials will be brought into play through the constitutive equations. However, the balance laws will remain in their local forms to a very high degree of accuracy. Thus, dropping the residuals we have the local balance laws: Conservation of Mass. ∇ ·v =0 ρ˙ + ρ∇ [ρ(vk − uk )]nk = 0

in V − σ,

(2.1.30a)

on σ.

(2.1.30b)

Balance of Momentum. tkl,k + ρ(fl − v˙l ) = 0 [tkl − ρvl (vk − uk )]nk = 0

in V − σ,

(2.1.31a)

on σ.

(2.1.31b)

Balance of Moment of Momentum. tkl = tlk

in V − σ.

(2.1.32)

Balance of Energy. −ρ ˙ + tkl vl,k + qk,k + ρh = 0 [tkl vl + qk − ρ( +

1 2v

· v)(vk − uk )]nk = 0

in V − σ,

(2.1.33a)

on σ.

(2.1.33b)


23

B. Reduction of Nonlocal Balance Laws By a new interpretation of the stress tensor and the internal energy density, the nonlocal balance laws can be reduced to local form. To this end we define ρ fˆl = −tˆkl,k , tˆkl = tˆlk , −ρ ˆ = ρ ˙ − tˆkl vl,k − ρ h,

Tkl = tkl + tˆkl , l = Fˆl + [tˆkl ]nk , F ρH = ρ(h + h). (2.1.34)

E = + ,

= −Eˆ + [tˆkl vl − ρ H (vk − uk )]nk ,

Using the transport and Green–Gauss theorems, the balance laws (2.1.22) to (2.1.26) may be expressed as Tkl,k + ρ(fl − v˙l ) = 0, l , [Tkl − ρvl (vk − uk )]nk = F Tkl = Tlk , ˙ −ρ E + Tkl vl,k + qk,k + ρH = 0, 1 . [ − ρ(E + 2 v · v)(vk − uk ) + Tkl vl + qk ]nk = H

(2.1.35a) (2.1.35b) (2.1.36) (2.1.37a) (2.1.37b)

l and the surface energy H , these Except for the presence of the surface traction F equations are similar to their local forms (2.1.31) to (2.1.33). The foregoing exercise shows that if we regard the stress and internal energy as nonlocal fields, then the classical balance laws in V − σ are not altered. Only the jump conditions on σ may have additional effects arising from the creation of new surfaces. Consequently, for economy in notations, we may keep the old notations (i.e., we use tkl for the new Tkl , for E, and h for H ). Thus, with these interpretations, the nonlocal balance laws read ∇ ·v =0 ρ˙ + ρ∇ [ρ(vk − uk )]nk = 0 tkl,k + ρ(fl − v˙l ) = 0 l [tkl − ρvl (vk − uk )]nk = F [ − ρ( +

1 2v

in V − σ,

(2.1.38a)

on σ, in V − σ,

(2.1.38b) (2.1.39a)

on σ,

(2.1.39b)

−ρ ˙ + tkl vl,k + qk,k + ρh = 0 in V − σ, · v)(vk − uk ) + tkl vl + qk ]nk = H on σ.

(2.1.40a) (2.1.40b)

With the new definitions (2.1.34) we no longer need the auxiliary conditions (2.1.24) and (2.1.26). However, we reproduce here the relations of these two sets l da = 0, (2.1.41) tˆkl nk da − F ∂ V −σ σ d da = 0. (2.1.42) ˆ ˆ tkl nk da + H ρ dv − ρ h dv − dt V −σ ∂ V −σ ∂ V −σ σ These new definitions of the stress tensor Tkl , energy E, and heat input H by (2.1.34), introduce new concepts. These quantities must satisfy the equations of

24

2 Stress

motion and energy in the local forms (2.1.35a) and (2.1.37a). In effect, these equations are identical to those of the classical (local) theory, except for the appearance l and H in the jump conditions (2.1.39b) and (2.1.40b). Of course, Tkl , E, and of F H include the nonlocal effects in their constitutions. l and H may be explained as follows: The presence of F We may imagine that there exists an ideal continuum of an infinite extent in all directions, with no internal stresses and energy. When there are no applied loads (force, couple, heat), this ideal continuum is at rest in its natural state. We shall call such a body a perfect body or a perfect continuum. For our study, if we isolate a subbody with a surface, by cutting a portion from the perfect body, this subbody will acquire internal stresses and stored energy, because of the creation of new surfaces. Also, some energy is imparted through the cutting process. In order to l and a restore the subbody to the natural state, we need to supply a surface traction F . Generally, these loads are small since the residual intermolecular surface energy H attractions are confined to a very thin boundary layer adjacent to the surface and they die out rapidly within a short distance from the surface. However, it is clear that, in order to return the subbody to the natural state, it is necessary to consider l and the surface energy H . These loads may become important the surface load F in some cases, e.g., phase transitions, and some critical surface phenomena. The new definition of the stress tensor tkl and energy possess nonlocality in their constitutions, i.e., we must express them with nonlocal constitutive equations. Indeed, this is also indicated by noting that, in atomic lattice models, the nonlocal effects are brought in through distant spring forces from all the other atoms to the reference atom, and the equations of motion involve no residuals.

C. A Second Alternative If we insist on keeping the local definition of stress, as in classical mechanics, then the alternative to the characterization of nonlocal residuals is through a constitutive ersatz for the residuals. This process requires constitutive equations for both volume and surface residuals. In this way, surface physics is brought into the domain of continuum physics. Nonlocal residuals may become important in extremely small (atomic scale) bodies and within a few atomic layer near surfaces. In this book we shall develop the first point of view, based on (2.1.38a) to l and H , (2.1.40b). Here too the surface physics is indicated by the residuals F when needed. Based on these observations and the short coherence length of the interatomic forces, we conclude that it is safe to discard nonlocal residuals except, possibly, atomic scale phenomena.

2.2 Second Law of Thermodynamics The global form of the second law of thermodynamics has the classical form 1 ρh d ρη dv − qk nk da − dv ≥ 0, (2.2.1) dt V −σ ∂ V −σ θ V −σ θ

2.2 Second Law of Thermodynamics

25

where η is the entropy density and (θ > 0, inf θ = 0) is the absolute temperature. By means of the transport theorem (2.1.5a) and the Green–Gauss theorem (2.1.6a), equation (2.2.1) is converted to ρ η˙ −

q k

θ

,k

[ρη(vk − uk ) −

−

ρh − γˆ ≥ 0 θ

qk ]ηk − ˆ ≥ 0 θ

in V − σ,

(2.2.2a)

on σ,

(2.2.2b)

where the entropy residuals γˆ and ˆ are subject to γˆ dv + ˆ da = 0. V −σ

(2.2.3)

σ

For an arbitrary and independent choice of σ , (2.2.3) implies that γˆ dv = 0, ˆ da = 0. V −σ

(2.2.4)

σ

We introduce the Helmholtz free energy by ψ = − θη

(2.2.5)

and eliminate from (2.2.40a), to cast the energy equation into the form −ρ(ψ˙ + ηθ ˙ + ηθ˙ ) + tkl vl,k + qk,k + ρh = 0. Substituting ρh from this into (2.2.2a) we obtain the (C–D) inequality 1 1 ˙ + tkl dkl + qk θ,k dv ≥ 0. −ρ(ψ˙ + ηθ) θ V −σ θ

(2.2.6)

(2.2.7)

Since tkl = tlk , we have tkl vl,k = tkl dkl where dkl is the deformation-rate tensor, defined by (1.4.10). We decompose the constitutive-dependent variables η and tkl into static (equilibrium, reversible) and dynamics parts denoted, respectively, by the left subscripts R and D, i.e., η = R η + D η, tkl = R tkl + D tkl . (2.2.8) With the introduction of η in (2.2.5), ψ is considered to be a fully static variable and, of course, the heat q has no static part. Definition (Thermodynamic Equilibrium). The state of body is said to be in thermodynamic equilibrium if, in C–D inequality (2.2.7), the dynamic contributions vanish, hence 1 ˙ + R tkl dkl ] dv = 0. (2.2.9) [−ρ(ψ˙ + R ηθ) V −σ θ

26

2 Stress

Subtracting (2.2.9) from (2.2.7) we obtain 1 1 −ρD ηθ˙ + D tkl dkl + qk θ,k dv ≥ 0. θ V −σ θ

(2.2.10)

This is the C–D inequality not to be violated by the dynamic parts of the constitutivedependent variables D η, D t, and q. Inequality (2.2.10) has the form V −σ

1 J · Y dv ≥ 0, θ

(2.2.11)

where J and Y are the ordered sets defined by J = (−ρD η, D tkl , qk /θ), Y = (θ˙ , dkl , θ,k ).

(2.2.12)

Here, the collection J is called the thermodynamic flux and Y is called the thermodynamic force. Thus, the dissipation of energy represented by (2.2.10) is expressible as the inner product of two vectors in 10-dimensional space.

2.3

Dissipation Potential

In the order listed, we may consider J and Y as vectors in 10-dimensional space, whose scalar product (2.2.11) must be nonnegative. The general solution of an inequality of the form (2.2.11) was given by Edelen [1993]. Theorem (Edelen). Let C 1,0 denote the collection of all scalar-valued functions of the arguments {Y, ω} that are of class C 1 in Y and continuous in ω , and let ˆ Cˆ 1,0 denote all collections {J (Y, ω ), K(Y, ω) | 1 ≤ < M} such that each of 1,0 ˆ ∈ Cˆ 1,0 that satisfy J , K} these M + 1 functions belong to C . All systems {J ˆ Y · J (Y, ω ) ≥ K(Y, ω ),

(2.3.1)

ˆ ω ) is subject to where K(Y,

ˆ K(Y, ω ) dV = 0

(2.3.2)

V

are given by J (Y, ω ) = ∇ ,Y (Y, ω ) + U(Y, ω ), 1 ˆ Y, ω )] dτ + ϕ(ω ω ), [P (τ Y, ω ) + K(τ (Y, ω ) = τ 0 where

ˆ P (0, ω ) + K(0, ω ) = 0,

(2.3.3) (2.3.4)

(2.3.5)

2.3 Dissipation Potential

27

and P (Y, ω ) is any element of C 1,0 such that P (Y, ω ) ≥ 0,

(2.3.6)

and U (Y, ω ) is any vector-valued function such that each component belongs to C 1,0 and Y · U = 0. (2.3.7) We shall call (Y, ω ) the dissipation potential and U (Y, ω ) the constitutive residual. For proof, we refer the reader to Edelen. The concept underlying the above representation can be deduced from ˆ Y · J (Y, ω ) = Y · ∇ ,Y = P + K.

(2.3.8)

A theorem due to Poincaré requires that any closed differential form on a starshaped domain is an exact differential form. More explicitly, if a vector-valued function J is curlless, in the sense that ∂J ∂J , = ∂Y ∂Y then J admits a potential in a star-shaped domain. This potential can be computed by a line integral of J ·dY along a straight line from the origin of the domain to the point Y. Hence is given by (2.3.4). From (2.3.7), it is clear that the vector U(Y, ω ) does not contribute to the dissipation inequality. If J (Y, ω ) is of C 2,0 , then any solution of the inequality (2.3.1) satisfies the symmetry relations ∂ ∂ [J (Y, ω ) − U (Y, ω )] = [J (Y, ω ) − U (Y, ω )]. ∂Y ∂Y

(2.3.9)

These reduce to Onsager reciprocity relations ∂J ∂J = ∂Y ∂Y

(2.3.10)

if and only if U(Y, ω ) = 0. The case Kˆ = 0 is of interest. ˆ ω ) = 0, then the satisfaction of the strong form, Y ·J J (Y, ω ) ≥ Corollary. If K(Y, ω 0 of the fundamental inequality, implies that (Y, ) is given by

1

(Y, ω ) =

P (τ Y, ω ) 0

P (Y, ω) = Y · J (Y, ω)

dτ ω ), + ϕ(ω τ (2.3.11)

28

2 Stress

and Y · U(Y, ω ) = 0, J (Y, ω ) = ∇ ,Y (Y, ω ) + U(Y, ω ), J (0, ω ) = U(0, ω) = 0, P (0, ω ) = 0,

(2.3.12) (2.3.13)

where (0, ω ) is the absolute minimum of (Y, ω ) with respect to Y at each ω , and the symmetry relations (2.3.9) hold if each of the components of J (Y, ω ) belongs to C 2,0 . It is clear that (0, ω ) = min (Y, ω ) Y

(2.3.14)

and that (Y, ω ) is a nondecreasing function on all rays in the vector space Y that emanates from the origin 0. Edelen gives an example for the residual U in terms of Hall current in the constitutive equation for the current J = σ E + µH E × B,

(2.3.15)

which shows that U·E=0 since µH (E × B) · E = 0. This latter relation shows that Y · U = 0, where Y = (E, B) and U = µH E × B. It is well known that, in the atomic scale, with the time reversal, B changes sign, so that in (2.3.15) the nonlinear term in B alone occurs. Thus, all higher-order approximations will involve second and higher degrees in Y. Consequently, U(λY, ω ) = O(λ2 ).

(2.3.16)

J (Y, ω ) + O(λ2 ), J (λY, ω ) = λJ

(2.3.17)

In this case, then and the Onsager relation (2.3.9) is satisfied to within O(λ2 ). In this book, we shall be concerned mostly with constitutive equations that are linear in Y , so that we shall take U = 0 in the sense of (2.3.16). Alternatively, we shall use: Onsager’s Postulate. Onsager reciprocal relations are posited to be valid for constitutive equations that are linear in the thermodynamic force Y, with proper consideration of the time reversal hypothesis. We note that for large thermodynamics forces, the constitutive residuals U can make important contributions, as is exemplified by the Hall current, nonlinear memory-dependent materials, the second sound (observed in He II), etc.

2.3 Dissipation Potential

29

Memory Functionals For memory-dependent nonlocal media, J is a functional of the thermodynamic forces over the space occupied by body V and all past time histories. In this case, the dissipation inequality is of the form ˆ Y(x, t) · J [Y(x , t ), ω ] ≥ K,

x ∈ V,

0 ≤ t < ∞.

(2.3.18)

Here Y(x , t ) is a collection of thermodynamic forces at all points x of the domain occupied by the body and at all present and past times. If z is a member of J at x and t, then we would like to find z as a functional of Y in the form z = F [Y(x , t ); Y(x, t)],

−∞ < t ≤ t,

x ∈ V,

(2.3.19) Y(x , t )

where we omit showing the dependence of z on ω . Now, if we treat as a parameter and consider z as a function of Y(x, t), then the solution (2.3.3) again applies.

Problems 2.1 Prove the validity of equations (2.1.1). 2.2 In some physical problems (e.g., plasticity) the time-rate of the stress tensor t˙kl becomes important. Is t˙kl objective? If not, what other tensor can be used instead of t˙kl ?

in the two rectangular 2.3 Give the relations of the stress tensors tkl and tkl

coordinates xk and xk that are obtained by rigid rotation about an axis not coincident with any coordinates.

2.4 Calculate the surface traction t(n)k on a surface x3 = f (x1 , x2 ) in terms of tkl referred to rectangular coordinates. 2.5 Determine the components of the stress tensor in spherical coordinates in terms of those in rectangular coordinates. 2.6 Carry out, in all detail, the proof of the theorem to obtain all the balance laws of thermomechanics. 2.7 Write expressions for the balance laws in spherical coordinates. 2.8 In a body in thermodynamic equilibrium, show that ψ = (ρ, θ, CKL ),

Rη

=−

∂ , ∂θ

tkl = ρ

∂ xk,K xl,L . ∂CKL

2.9 Navier–Stokes fluids have constitutive equations described by tkl = −pδkl + λvr,r δkl + µ(vk,l + vl,k ),

qk = kθ,k .

What conditions λ, µ, and k must be subjected in order not to violate the Clausius–Duhem inequality?


3 Constitutive Equations

3.0 Scope In Section 3.1, we present eight axioms that are fundamental to the development of the constitutive equations. Since the inception of these axioms in 1966, their universality became clear in many diverse applications in many different fields. Here we postulate and discuss such axioms separately and give mathematical formulations. Later chapters contain many diverse applications to nonlocal memory-dependent elastic solids, fluids, electromagnetism, microcontinuum field theories, etc., as a testimony of their fundamental nature. In Section 3.2, using these axioms, constitutive equations are obtained for memory-dependent nonlocal thermoelastic solids, and in Section 3.3 for memorydependent, nonlocal thermoviscous fluids. Thus far, these developments are exact and applicable to nonlinear media undergoing finite deformations and motions.

3.1 Constitutive Axioms The fundamental laws of nonlocal continuum mechanics consist of five partial differential equations (2.1.38a) to (2.1.40a) and one inequality (2.2.7). Given the external loads f and h there are 14 scalar variables ρ, vk , tkl , θ, and qk . Clearly, the systems of basic equations are highly indeterminate. We need nine more equations to close the system. This is also clear from the fact that the five partial differential equations of the basic laws are valid for all continua irrespective of their material constitutions. They are valid for elastic solids, viscous fluids, oil, blood, plastics, etc.

32


The additional nine equations needed must somehow reflect the material properties of bodies. In continuum mechanics these relations are called constitutive equations. Constitutive equations for elastic bodies are often called stress–strain relations. For viscous fluids they express the stress deformation-rate relations and for heat conditions the Fourier law. Here, we shall present a general setting for the construction of such relations, within the context of the nonlocal heat-conducting nonlinear continua from which various order theories can be extracted. It is possible to construct constitutive equations by examining interatomic attractions and using statistical mechanics. A simple example will be given in Section 6.2. However, this method of approach soon ceases to be tractable when material composition is complicated. For example, amorphous or porous solids, with memory effects and/or undergoing finite deformation or displaying nonlocal effects, do not lend themselves to such approaches. On the other hand, when the continuum approach is used there will be left some unknown material functions or constants (e.g., the Young modulus, viscosity moduli) which must be determined either by experimental means, or by comparing the constitutive equations with those obtained from the atomic approaches (if available). Examples of these are to be found in Sections 6.2, 8.4, and 13.14. Fundamental to the continuum approach to constitutive equations are the following eight basic axioms, which were proposed by Eringen [1966], [1967]: (i) Axiom of Causality. (ii) Axiom of Determinism. (iii) Axiom of Equipresence. (iv) Axiom of Objectivity. (v) Axiom of Material Invariance. (vi) Axiom of Neighborhood. (vii) Axiom of Memory. (viii) Axiom of Admissibility. (i) Axiom of Causality. The motions, temperatures and charges of the material points of a body are the cause of all physical phenomena. The remaining variables (other than those derivable from motion, temperature and charges) that enter the expressions of the Clausius–Duhem (C–D) inequality are the response functions (or constitutive-dependent variables). Thus, the independent mechanical variables are x = x(X , t ),

θ = θ(X , t ) > 0,

X ∈ V,

inf θ = 0, −∞ < t ≤ t.

(3.1.1)

3.1 Constitutive Axioms

33

Electromagnetically active media, the presence and motion of charges, cause the electric field E and magnetization B so that the electromagnetic (E-M) independent variables are joined to (3.1.1): E = E(X , t ),

B(X , t )

X ∈ V

− ∞ < t ≤ t.

(3.1.2)

Thus Y = x(X , t ), θ(X , t), E (X , t ), B(X , t ) , X ∈ V , −∞ < t ≤ t,

(3.1.3)

constitutes the set of independent variables. The constitutive-dependent variables are then, the internal energy density , the entropy η, the stress tensor tkl , and the heat vector qk . For the electromechanically active media to these we adjoin the polarization P, the magnetization M , the current vector J (see Section 5.2), i.e., Z = (, η, tkl , qk , Pk , Mk , Jk ).

(3.1.4)

(ii) Axiom of Determinism. The constitutive-dependent variables at a material point X, at time t, are functionals of the independent variables over the entire material points X of the body, at all past times t up to and including the present time t. Symbolically, this is expressed by Z(X, t) = F[Y(X , t )],

X ∈ V ,

−∞ < t ≤ t.

(3.1.5)

(iii) Axiom of Equipresence. At the outset, all constitutive-dependent variables must be expressed as functionals of the same list of independent constitutive variables until the contrary is deduced. This simply states that each member of Z is a functional of all members of Y. It is an axiom for a precautionary measure not to forget some variable or favor others. (iv) Axiom of Objectivity. Constitutive equations must be form-invariant with respect to rigid motions of the spatial frame of reference. It is evident that the material properties cannot depend on the motion of the observer. This imposes the objectivity (or material frame-indifference) on the constitutive equations (see Section 1.5). (v) Axiom of Material Invariance. Constitutive equations must be form-invariant with respect to the symmetry group of the material points.

34


The crystallographic orientations of the material points in a body give rise to certain symmetries represented by the crystallographic point group in the reference state. Under this group of transformations, of the material frame of reference, restrictions are imposed on the constitutive equations. For the electromechanical media the magnetic group enlarges 32 members of the crystallographic point group to 90 members. The constitutive equations of E-M elastic crystals are discussed in a book by Kiral and Eringen [1990]. (vi) Axiom of Neighborhood. The values of the independent constitutive variables at distant material points X from the reference point X do not appreciably affect the value of the constitutive-dependent variables at X. It is well known that the interatomic attractions on a reference atom from distant atoms are weak. In fact, for most materials, the atomic neighborhood influencing a reference atom is less than 10 atomic distances. This influence region may be called a cohesive zone. Beyond the cohesive zone the interatomic attractions can be neglected. Within the cohesive zone the influence of the neighboring atoms decreases drastically with the distance from the reference atom. In continuum theory this may be expressed in two different ways. (a) Smooth Neighborhood. Suppose that in a region V0 ∈ V , appropriate to each material body, the independent variables admit Taylor series expansions in X − X in V0 for all t ≤ t0 , terminating with gradients of order P , Q, etc.,

x(X , t ) = x(t ) + (XK − XK1 )x,K1 (t ) 1 1

+··· + (X − XK1 ) · · · (XK − XKp )x,K1 ···KP , p P ! K1 and similarly for the other variables θ(X , t ), E (X , t ), and B(X , t ). If the response functionals are sufficiently smooth so that they can be approximated by the functionals in the field of real functions

x(t ), x,K1 (t ), . . . , x,K1 ···KP (t ), θ(t ), θ,K1 (t ), . . . , θ,K1 ···KQ (t ),

(3.1.6)

we say that the material at X for all t ≤ t satisfies a smooth neighborhood hypothesis. Material of this type, for P > 1, Q > 1 are called nonsimple materials of gradient type. (b) Attenuating Neighborhood Hypothesis (Eringen [1966], [1975]). For some materials, smooth neighborhood hypotheses fail to apply. Actually, some of the gradients of the constitutive-dependent variables may not exist. We can still formulate a hypothesis which introduces a bias in favor of the points near the reference point X. To this end, we introduce an influence function Hr (|X − X|) > 0, or order r as Hr (0) = 1,

lim

|X −X|=max

|X − X|r Hr (|X − X|) = 0,

r > 3, (3.1.7)


35

monotonically for large |X − X|. An example of such a function is α(t ) > 0,

H∞ = exp(−αR 2 ),

(3.1.8)

where R is the distance of X from X, in rectangular coordinates, i.e., R = |X − X|.

(3.1.9)

Other possibilities exist. For example, Hr = [1 + β(t )R 1+p ]−1 ,

3 < r < p.

(3.1.10)

A generalized function, with finite support (e.g., a Dirac-delta sequence), may also serve this purpose. The influenced distance between x(X , t ) and x(t ) ≡ x(X, t ) or the H -norm of x(X , t ) − x(t ) may be defined as

1/2 |x(X , t ) − x(X, t )| H (|X − X| dV (X ) .

|x(X , t ) − x(X, t )|H =

2

V

(3.1.11)

An axiom of attenuating neighborhood can now be stated as: Axiom ((vi) b) (Attenuating Neighborhood). The response functionals are continuous in a neighborhood of X in the function space normed according to (3.1.11) for all t ≤ t. Alernatively, for every e there exists an a > 0 such that Z[x(X , t )] − Z[x(t )] < e, (3.1.12) whenever

|Y(X , t ) − Y(t )|H < a,

(3.1.13)

for all times t ≤ t where denotes the magnitudes of each member of Z, and e, a may be a set of positive numbers appropriate to each member of Z and Y, respectively. Accordingly, when the distances of the members of the set Y(X , t ) differ very little from those of Y(t ) at X, then the corresponding members of the response functional set Z are close together. Thus the presence of the influence function H makes the influence of X near X feel much stronger than those from distant points. (vii) Axiom of Memory. The values of the constitutive-independent variables at distants past the present do not appreciably affect the values of the constitutive functionals at the present time. This axiom is the counterpart of the Axiom of Neighborhood in the time domain. Again, two alternative formulations exist.

36


(a) Smooth Memory. Suppose that there exists a time τ0 < t, appropriate to each material such that the independent variable set Y(X , t ) possesses a Taylor series expansion at t = t for all X ∈ V , terminating some numbers p, q . . ., e.g., (3.1.14) Y(X ) ≡ Y(X , t) = x(X , t), x˙ (X , t), . . . , x(p) (X , t),

(q)

˙ , t), . . . , θ (X , t), etc. . . . . θ(X , t), θ(X If the constitutive functionals are sufficiently smooth, so that the dependence on Y(X , t ) can be replaced by (3.1.14), we say that they satisfy the Axiom of Smooth Memory. Formally

)], Z(X , t ) = Z[Y(X

(3.1.15)

where dependence on the present time t is not shown. Such materials are called the rate-type materials. (b) Fading Memory (Coleman and Noll [1961]). The independent variable set Y may not admit a Taylor series expansion about t, and/or the response function set Z may not be smooth enough to be expressible in terms of the rates (3.1.14). In this case, we can formulate a fading-memory hypothesis by introducing influence functions in time such that the set Y near the present time is favored. Let Ip (t −t ) be such an influence function of order p, defined over −∞ < t ≤ t such that Ip (0) = 1, lim (t − t )p Ip (t − t ) = 0, (3.1.16) t→−∞

p > 1 monotonically for large

t − t .

Examples of the influence functions are I∞ (t − t ) = exp[−α(t − t )], α > 0,

p+1 + 1]−1 , r < p. Ir (t − t ) = [k(X )(t − t )

(3.1.17)

The normed distance between x(X , t ) and x(X , t) of the I-norm is then defined by

|x(X , t ) − x(X , t)| =

t

−∞

I (t − t )|x(X , t ) − x(X , t)| dt 2

1/2 (3.1.18)

Axiom ((vii) b) Fading Memory. The response functionals are continuous from below in a neighborhood of t of the independent variable set Y(X , t ), in the function space normed according to (3.1.18) for all X ∈ V .


37

Mathematically, for every e > 0 there exists an a > 0 such that

whenever

Z[x(X , t )] − Z[x(X , t)] < e,

(3.1.19)

|Y[x(X , t ) − Y[x(X, t)]| < a,

(3.1.20)

for all times t ≤ t, X ∈ V · e, and a may be different for each member of Z and Y. Thus, if the histories of Y(X , t) differ very little from those at t ≤ t, then the response functionals are close together. The influence regions may differ for the different members of Z. For example, the E-M influence region may be different than the influence of deformations. (viii) Axiom of Admissibility. All constitutive equations must be consistent with the balance laws and the entropy inequality. This seemingly obvious postulate, which is certainly valid for all scientific disciplines, helps to simplify the constitutive equations and provides thermodynamic admissibility. For example, mass density ρ(X , t ) must obey the law of the conservation of mass, ρ0 /ρ(X , t ) = det xk,K (X , t ). (3.1.21) Whenver xk,K (X , t ) becomes a member of the constitutive-independent variable set Y, we can discard ρ(X , t ) as an independent variable (e.g., elastic solids). More importantly, all members of the constitutive response functional set Z are subject to the second law of thermodynamics which must not be violated. This restriction eliminates some members of Y and/or it places restrictions on them. Restrictions from Rigid Body Motions It is well known that rigid motions do not alter the shape of the body or cause any response in the body, e.g., no stress is created. Consequently, a constitutive equation of the type (3.1.5) is too general to be of any utility. We now proceed to replace x(X , t ), a member of (3.1.3), with a more suitable variable that is not affected by the rigid motion. From (1.2.11) we have F = RU

or

1/2

xk,K = RkL C LK .

(3.1.22)

In rectangular coordinates, by integration, we have xk (X , t ) − xk (X, t) =

X

1/2

RkL (X, t ) C LK (X, t ) dXK .

(3.1.23)

X

This result shows that we may replace the functional dependence x(X , t )−x(X, t) in the set Y by R(X , t ) and C(X , t ). But the dependence of the set Z on R is not permissible on account of the Axiom of Objectivity, unless we employ an objective

38


tensor constructed purely in terms R(X , t ). For nonpolar materials, Axiom (v) eliminates this dependence so that (3.1.3) is replaced by Y(X , t ) = C(X , t ), θ(X , t ), E (X , t ), B(X , t ) . (3.1.24) With the introduction of the Helmholtz free energy ψ by (2.2.5) and and η being functionals of the Y(X , t ), we are free to consider ψ, as a static dependent variable, i.e., ψ = [C(X ), θ(X ), E (X ), B(X )], (3.1.25) where, here and henceforth, the dependence of C, θ, E , and B on t is suppressed. The axiom of thermodynamic admissibility for the mechanical systems requires that ψ, η, tkl , and qk must not violate the C–D inequality (2.2.7), namely, 1 ˙ ˙ −ρ(ψ + ηθ ) + tkl dkl + qk θ,k dv ≥ 0, (3.1.26) θ V since θ > 0, inf θ = 0. For the electromechanical system the C–D inequality is given in Section 4.4. In the following sections, we explore the consequences of these axioms in the development of the constitutive equations.

3.2 Constitutive Equations of Memory-Dependent Nonlocal Thermoelastic Solids1 We introduce the material tensors TKL and QK by TKL =

ρ0 tkl XK,k XL,l , ρ

QK =

ρ0 qk XK,k . ρ

(3.2.1)

Substituting these into (3.1.26), using (1.4.9) and ρ dv = ρ0 dV , inequality (3.1.26) is transformed to 1 ˙ + 1 TKL C˙ KL + QK θ,K dV ≥ 0. −ρ0 (ψ˙ + ηθ) (3.2.2) 2 θ V Nonlocal elastic solids are defined by the free energy of the form (3.1.25). In order ˙ we need to introduce a function space. For simplicity, we confine to calculate ψ, ourselves to the class of linear functionals in the sense of Friedman and Katz [1966]. Let C denote the class of bounded real-valued continuous functions defined in V. Define the norm of a function f (x) ⊂ C by f = max |f (x)|. x∈V

1 Theory was given by Eringen [1974b].

(3.2.3)

3.2 Memory-Dependent Nonlocal Thermoelastic Solids

39

Suppose that the functional F(f ) is continuous and bounded by the norm (3.2.3). This is additive in the sense that for two functions {f1 , f2 } ⊂ C: F(f1 + f2 ) = Fg (f1 ) + Fg (f2 ),

(3.2.4)

where Fg (f ) = F(f + g) − F(g) for {f, g} ⊂ C. Under these conditions, the theorem of Friedman and Katz states that F(f ) = K[f (x ), x ] dv(x ). (3.2.5) V

We identify F with the free energy function and write ρ0 ψ(X, t) = F [C(X ), θ(X ), X ; C(X), θ(X), X] dV (X ).

(3.2.6)

V

The total free energy of the body is given by ρ0 ψ(X, t) dV = F (C , θ , X ; C, θ, X) dV dV , V

V

(3.2.7)

V

where we put C = C(X ), θ = θ(X ), and dV = dV (X ). Since the order of integration does not affect the total free energy, we assume that F is a symmetric function of its arguments, i.e., S

F (C , θ , X ; C, θ, X) = F (C, θ, X; C , θ , X ) ≡ F ,

(3.2.8)

where, and henceforth, a superscript (S) indicates the symmetrization, i.e., the interchange of X and X . We assume that F is continuously differentiable with respect to C , θ , C, and θ. Using (3.2.6) we calculate ∂F ∂F ∂F S ∂F S

˙ ˙ ·C + θ˙ dV + D, (3.2.9) ρ0 ψ = + + ∂C ∂C

∂θ ∂θ

V where

D= V

∂F ˙

·C − ∂C

Clearly,

∂F ∂C

S

˙ + ∂F θ˙ − ·C ∂θ

∂F ∂θ

S θ˙ dV .

(3.2.10)

D dV = 0.

(3.2.11)

V

We decompose the constitutive-independent variables into static (recoverable) and dynamic parts η = R η + D η,

T = R T + D T,

Q = D Q,

(3.2.12)

40


where the left subscripts R and D denote, respectively, the static and dynamic parts. The heat vector Q possesses no static part. With these, the C–D inequality (3.2.2) becomes ∂F S ∂F

˙ + dV θ − ρ0R η + ∂θ ∂θ

V V ∂F ∂F S

1 ˙ + 2 RT − dV · C + ∂C

V ∂C 1 1 ˙ ˙ − ρ0D ηθ + 2 D T · C + Q · ∇ θ dV ≥ 0. (3.2.13) θ ˙ it is clear that we must have Because R η, R T, and F do not depend on θ˙ and C, ∂F 1 ∂F S dV , + Rη = − ρ0 V ∂θ ∂θ

∂F ∂F S dV , + RT = 2 ∂C

V ∂C

(3.2.14)

and −ρ0D ηθ˙ +

1 ˙ 2 DT · C +

1 Q · ∇ θ − Kˆ ≥ 0. θ

(3.2.15)

The last inequality is the dissipation inequality which has the form (2.3.1) with Kˆ satisfying (2.3.2). Hence the solution of (3.2.15) is given by (2.3.3), i.e., ∂ + U η, ∂ θ˙ ∂ 1 + UT , 2 DT = ˙ ∂C ∂ Q = + UQ , ∇ θ) θ ∂(∇

−ρ0D η =

(3.2.16)

where is the dissipation potential and U η , UT , and UQ are the constitutive residuals, subject to ˙ + UQ · ∇ θ = 0. U η θ˙ + UT · C Thus, they do not contribute to the dissipation of energy.

(3.2.17)

3.2 Memory-Dependent Nonlocal Thermoelastic Solids

41

Let the collection of the dependent variables be denoted by J, the independent variables by Y and ω , and the collection of the constitutive residuals by U, i.e.,2 Q 1 , J = −ρ0D η, 2 D T, θ ˙ Y(X, t) = {θ˙ (X, t), C(X, t), ∇ θ(X, t)}, ω (X, t) = {C(X, t), θ(X, t)}, U = {U η , UT , UQ }. (3.2.18) Then the C–D inequality (3.2.15) may be expressed in the form of a scalar product J · Y − Kˆ ≥ 0.

(3.2.19)

The collections Y(X, t) and J(X, t) are, respectively, called the thermodynamic force and the thermodynamic flux. In order to characterize the dissipation potential we introduce the difference histories of Y: Y(t) (X, τ ) = Y(X, t − τ ) − Y(X, t),

0 ≤ τ < ∞.

(3.2.20)

Note that with the change of variable t − τ = s, this becomes Y(t) (X, t − s) = Y(X, s) − Y(X, t),

−∞ < s ≤ t,

(3.2.21)

which justifies the terminology difference history. The dissipation potential will depend on the histories of Y at all points of the body. We express as a functional of Y(t) : ∞ ∞ = dτ dτ G[Y(t) (X , τ ); Y(t) (X, τ )] dV

(3.2.22) 0

0

where G may also depend on Y and ω. The total dissipation in the body is given by ∞ ∞

W = dV = dτ dτ G dV dV . V

0

0

V

(3.2.23)

V

From this it is clear that only the symmetric part of G in (X , τ ) and (X, τ ) contributes to W . Hence, we stipulate that G[Y(t) (X , τ ); Y(t) (X, τ )] = G[Y(t) (X, τ ); Y(t) (X , τ )], lim G = 0. Y(t) →0

(3.2.24)

In addition, we stipulate: 2Although it is more natural to place Q as a member of J and ∇ θ/θ as that of Y(X, t), for memory-dependent materials the present organization (3.2.18) eliminates much complications. Moreover, the constitutive equations obtained for Q/θ upon multiplication by θ obtains the constitutive equations for Q. Thus, both results must be equivalent.

42


Postulate of Dissipation The total dissipation W in the body is nonnegative for all arbitrary and independent variations of the members of Y(t) , throughout the body V , and at all times −∞ < t ≡ t − τ ≤ t. The dynamic part of the constitutive equations (3.2.16) is expressed in a compact form ∂ J= + U. (3.2.25) ∂Y Upon substituting from (2.2.5) into (2.1.40a) of the energy balance, we have ˙ + tkl dkl + qk,k + ρh = 0. −ρ(ψ˙ + ηθ ˙ + ηθ)

(3.2.26)

Using (3.2.1) and (1.4.9) this equation takes its material form −ρ0 (ψ˙ + ηθ ˙ + ηθ˙ ) + 21 TKL C˙ KL + QK,K + ρ0 h = 0.

(3.2.27)

If we now use (3.2.14), this is reduced to −ρ0 θ(R η˙ + D η) ˙ +

1 ˙ 2 D TKL CKL

+ QK,K + ρ0 h = 0.

(3.2.28)

Nonlocal Elastic Solids without Memory3 ˙ and ∇ θ, we obtain the ˙ C, By dropping the dependence on the histories of θ, constitutive equations of nonlocal thermoelastic solids ˙ = G[θ˙ (X , t), ∇ θ(X , t); θ(X, t), ∇ θ(X, t)] dV , V S (3.2.29) dV ≥ 0, G=G. V

The static portion of the constitutive equation remains unchanged. Hence, ∂F ∂G ∂F S

−ρ0 η = dV dV + U η , + +

∂θ ∂θ V V ∂ θ˙ S ∂F ∂F 1 dV , + 2T =

∂C ∂C V 1 ∂G (3.2.30) Q= dV + UQ , ∇θ θ V ∂∇ with

U η θ˙ + UQ · ∇ θ = 0.

(3.2.31)

3 The theory without heat conduction was given by Eringen and Edelen [1972] and Eringen [1972a] and with the heat conduction by Eringen [1974].

3.3 Memory-Dependent Nonlocal Thermofluids

43

The spatial forms of tkl and qk follow from (3.2.1): ρ xk,K xl,L TKL , ρ0 ρ xk,K QK . qk = ρ0

tkl = 2

(3.2.32)

3.3 Constitutive Equations of Memory-Dependent Nonlocal Thermofluids Fluids possess no natural states. Consequently, response functionals cannot depend on the independent variables referred to the natural state. But they must be functionals of a set of independent variables referred to the spatial frame at the present time t. If we let X → x in (1.2.4) we see that CKL → δKL ,

θ,K → θ,k .

Consequently, CKL cannot be an independent variable for ψ, but θ(x, t) and θ,k (x, t) can be. For fluids, then we must try C˙ KL as X → x. In this case, from (1.4.9), we have C˙ KL → dkl δkK δlL . Thus, the deformation-rate tensor dkl , which is shown to be objective, may be considered a proper independent variable for fluids. At rest, fluids possess pressure. But with X → x, the density ρ (at time t), which is related to the rest density ρ0 by ρ0 = det xk,K ρ

(3.3.1)

is eliminated, i.e., ρ0 = ρ. Consequently, we must reinstate the mass density ρ into the list of constitituve independent variables Y = {ρ(x , t ), dkl (x , t ), θ,k (x , t ), θ(x, t)}, x ∈ V, −∞ < t ≤ t.

(3.3.2)

The constitutive dependent variables are Z = {, tkl , qk }.

(3.3.3)

The free energy ψ = − θη can be considered to be independent of the memory, as discussed before. However, , η, tkl , and qk may depend on the list of variables Y. For ψ we propose a functional of the form F [ρ(x ), θ(x ); ρ(x), θ(x), |x − x|] dv , (3.3.4) ρψ = V

44


where the dependence of the constitituve variables on the present time is surpressed, e.g., we wrote ρ(x ) for ρ(x , t). We assume that S

F =F .

(3.3.5)

The stress tensor and entropy are decomposed to static and dynamic parts, as usual, η = R η + D η,

tkl = R tkl + D tkl ,

R tkl

= −πδkl ,

where π is the thermodynamic pressure. For the C–D inequality (2.2.7) we have 1 ˙ ˙ ˙ −ρ(ψ + R ηθ + D ηθ ) + R tkl dkl + D tkl dkl + qk θ,k dv ≥ 0. θ V

(3.3.6)

(3.3.7)

Using (3.3.4) and (2.1.38a), we calculate S S ∂F

∂F ˙ ρ ψ = vk,k −ρ + F + F dv

− ρ

∂ρ ∂ρ V ∂F ∂F ∂F S

dv + + κ˙ dv + D, + θ˙

∂θ V ∂θ V ∂κ where κ = |x − x| and D=

We note that

∂F ∂ρ

S

∂F

v − ∂ρ k,k V S ∂F

˙ − θ + F vk,k − F vk,k dv . ∂θ ρ

vk,k − ρ

∂F ∂θ

S

θ˙

V

D dv = 0.

Substituting ρ ψ˙ into (3.3.7) we arrange this as S S ∂F ∂F + ρ

−π + ρ − F − F dv vk,k dv ∂ρ ∂ρ V V ∂F ∂F S

˙ − ρR η + dv θ dv + ∂θ

V V ∂θ ∂F 1

˙ − κ˙ dv − ρ D ηθ + D tkl dkl + qk θ,k dv ≥ 0. (3.3.8) θ V V ∂x Since F is independent of vk,k , θ˙ , and κ, ˙ for arbitrary and independent variations of these quantities throughout V, the inequality cannot be maintained in one sign.


45

Hence, we have the necessary and sufficient conditions for the second law of thermodynamics not to be violated ∂F dv = 0, (3.3.9) ∂κ V S S ∂F

∂F π = ρ + ρ

− F − F dv , ∂ρ ∂ρ V ∂F S ∂F ρR η = − + dv , ∂θ

V ∂θ and −ρ D ηθ˙ + D tkl dkl +

1 qk θ,k − Kˆ ≥ 0, θ

where constitutive residual Kˆ is subject to Kˆ dv = 0.

(3.3.10) (3.3.11)

(3.3.12)

(3.3.13)

V

Thus, the thermodynamic pressure and the static part of the entropy are determined through the knowledge of the free energy density F . However, equation (3.3.9) places a severe restriction on F , since it must be true for all x. Hence we must have ∂F /∂κ = 0. In this case, the integration of (3.3.4) over V implies that ψ is a function of ρ(x), θ(x) and the surface mass and heat on ∂V. The surface energy arising from the surface tension and surface heat constitute such effects that are already accounted for in the jump conditions (2.1.39b) and (2.1.40b). Hence we take ψ = [ρ(x), θ(x)]. (3.3.14) This then gives, for the thermodynamic pressure and static part of the entropy, the classical (local) thermodynamic pressure and entropy, namely, π = ρ2

∂ , ∂ρ

Rη

=−

∂ . ∂θ

(3.3.15)

Hence, we conclude that: In fluid continuum the pressure and static entropy do not possess nonlocality. The C–D inequality (3.3.12), for the dynamical parts, is cast in the form (2.3.1) with Kˆ satisfying (2.3.2) by introducing vectors in the 10-dimensional space DY

= {θ˙ , dkl , θ,k },

DJ

Then (3.3.12) reads DJ

= {−ρ D η, D tkl , qk /θ}.

· D Y − Kˆ ≥ 0.

(3.3.16)

(3.3.17)

46


Consequently, from (2.3.3) we obtain the dynamic constitutive equations ∂ + U η, ∂ θ˙ ∂ + Uklt , D tkl = ∂dkl 1 ∂ q + Uk , qk = θ ∂θ,k

−ρ D η =

(3.3.18) q

where is the dissipation potential and U η , Uklt , and Uk are the constitutive residuals subject to ˙ η + Ut · d + Uq · ∇ θ = 0. θU

(3.3.19)

This means that the constitutive residuals do not cause dissipation. Parallel to the development in Section 3.2, (equivalent to (3.3.16)) we introduce the notation J = {−ρ D η, D t, q/θ} , Y(x, t) = {θ˙ (x, t), d(x, t), ∇ θ(x, t)}, ω(x, t) = {ρ, θ},

U = {U η , Ut , U q },

(3.3.20)

and Y(t) (x, τ ) = Y(x, t − τ ) − Y(x, t).

(3.3.21)

With these, (3.3.18) and (3.3.19) read ∂ + U, ∂Y

J=

U · Y = 0.

(3.3.22)

The dissipation potential will depend on the history of its independent variables at all points of the body, i.e., it is a space–time functional. We express in the form

∞

=

∞

dτ 0

dτ 0

V

G[Y(t) (x , τ ); Y(t) (x, τ )] dv ,

(3.3.23)

where G may also depend on ω {ρ, θ}. With the use of (3.3.6) and (3.3.15), equation (3.2.26) of the energy balance is reduced to −ρθ(R η˙ + D η) ˙ + D tkl dkl + qk,k + ρh = 0.

(3.3.24)


47

Thermoviscous Fluids without Memory4 For thermoviscous fluids that do not depend on the history of the independent variables we drop the memory dependence. In this case, we have =

V

˙ G(θ˙ , dkl , θ,k ) dv ,

S

G=G.

The dynamic parts of the constitutive equations are then given by ∂G

dv + U η , −ρ D η = V ∂ θ˙ ∂G t = dv + Uklt , D kl ∂d kl V 1 ∂G q dv + Uk , qk = θ V ∂θ,k

(3.3.25)

(3.3.26)

with the constitutive residuals subject to θ˙ U η + d · Ut + Uq · ∇ θ = 0.

(3.3.27)

The static parts (3.3.15) of the constitutive equations remain unchanged. For nonviscous fluids G does not depend on dkl so that D tkl = 0. For isothermal inviscid fluids D t, D η, and q are absent. Such fluids are called ideal fluids. By selecting polynomials for G in terms of its variables, special constitutive equations can be constructed involving various order terms up to some specified degree, e.g., linear, quadratic degrees. We shall be concerned mostly with the linear viscous effects.

Problems 3.1 For isotropic thermoelastic solids without memory determine the final forms of the nonlocal, nonlinear constitutive equations, assuming that the constitutive residuals vanish. 3.2 For nonlocal thermoviscous fluids without memory the dissipation potential depends on the invariants of d and ∇ θ. Determine the final forms of the constitutive equations, assuming that the constitutive residuals vanish. 3.3 For local, linearly memory-dependent fluids, investigate the consequences of the second law of thermodynamics. 4 The theory was given by Eringen [1972b]. In this paper the inclusion of the density difference ρ(x ) − ρ(x) as a constitutive independent variable in the free energy ψ led to surface tension effect which may not be valid for homogeneous fluids except near the surface of fluids.

48


3.4 A gas with constitutive equations for pressure p given by p=

Rθρ − aρ 2 1 − bρ

is called a Van der Waals gas. Here R, b, and a are constants. Show that = cv (θ) dθ − aρ. Express ρ and p as a function of θ only. 3.5 (Short Term Paper.) Write a short account on the entropy concept based on statistical mechanics.

4 Nonlocal Electromagnetic Theory

4.0 Scope The interaction of electromagnetic (E-M) fields with deformable bodies requires new concepts. Here we need to extend the Maxwell equations to nonlocal media, i.e., to write global balance laws involving E-M fields. This is achieved in Section 4.1. The localization involves some E-M residuals. Here too we are able to incorporate the nonlocal residuals into the E-M fields. In this way, there remains one surface residual only, relevant to the jump conditions on the magnetic field. Electromagnetic force, couples, and power are discussed in Section 4.2, where we also give the expressions of the stress tensor, the E-M momentum, and the Poynting vector that enter into the mechanical balance equations. In order to account for jump discontinuities at a moving discontinuity surface σ through the body, we need the jump discontinuities of the E-M force, couples, and power. This is elaborated on in Section 4.3. We are now ready to write the expressions for the mechanical balance laws for electromechanically active media. This is done in Section 4.4. The energy balance equation and the Clausius–Duhem (C–D) inequality are obtained for the development of the constitutive equation’s E-M elastic solids.

4.1 Electromagnetic Balance Laws The E-M balance laws can be obtained by applying the Lorentz invariance group to the energy balance laws, since they are relativistically invariant. Here we are concerned with nonrelativistic approximation. It is simpler to approach the subject

50


Figure 4.1.1. Moving discontinuity surface σ (t). by the statement of the Maxwell equations in global form. To this end, we need to write the balance laws over a volume and on an open surface. Let V(t) be a simply connected open material region of E 3 with a regular boundary ∂V(t). A discontinuity surface σ (t) is sweeping V with its own velocity u, Figure 4.1.1. Then two of the Maxwell equations, in nonrelativistically moving matter in V − σ , may be stated as: Gauss’ Law

∂ V −σ

D · da −

V −σ

qe dv = 0.

(4.1.1)

Conservation of Magnetic Flux ∂ V −σ

B · da = 0,

(4.1.2)

where D, B, and qe are, respectively, the dielectric displacement vector, the magnetic flux vector, and the free charge density in a fixed laboratory frame RG . The second set of the Maxwell equations are concerned with balance laws on an open material surface S(t) with boundary ∂S(t). A discontinuity line γ (t) may be sweeping S. The exterior unit normal of S is denoted by n, Figure 4.1.2. The global statements of the remaining two E-M balance laws are: Faraday’s Law

1 d B · da + E · dx = 0. c dt S −γ ∂ S −γ

Ampère’s Law 1 d 1 D · da + J · da − H · dx = 0, c dt S −γ c S −γ ∂ S −γ

(4.1.3)

(4.1.4)

where c is the speed of light in vacuum. E and H are, respectively, the electric field vector and the magnetic field vector in a co-moving reference frame RC with

4.1 Electromagnetic Balance Laws

51

Figure 4.1.2. Material surface with discontinuity line γ (t).

the body. We use capital letters to denote the E-M fields in RG and script capital letters for the E-M field in RC . The two sets are related to each other by

1 E = E + v × B, c 1 B = B − v × E, c

1 H = H − v × D, c J = J − qe v,

(4.1.5)

when the surface charges and currents exist the balance laws become more involved. For the most general case, we refer the reader to Eringen and Maugin [1989, Section 3.9].

By means of the transport and the Green–Gauss and Stokes’ theorem (2.1.5) and (2.1.6), the balance laws (4.1.1) to (4.1.4) are converted to

∇ · D − qe ) dv + n · [D] da = 0, (∇ V −σ σ ∇ · B dv + n · [B] da = 0, V −σ

σ

52


S −γ

S −γ

1 ∂B ∇ ∇ + v∇ · B + × E · da c ∂t 1 n × [E + u × B] · ds = 0, + c ∂ S −γ 1 ∂D 1 ∇ · D − ∇ × H + (J − qe v) · da + v∇ c ∂t c 1 n × [H − u × D] · ds = 0. + c ∂ S −γ

(4.1.6)

Note that we accept the nonexistence of the magnetic charge and the associated current. Consequently, the localization residuals for the equations (4.1.2) of the magnetic flux and (4.1.3) for the Faraday law are null. With this, the balance laws (4.1.1) to (4.1.4) are localized to ∇ · D − qe = −qˆ

ˆ n · [D] = Q V −σ

on σ,

qˆ dv = 0,

∇ ·B=0 n · [B] = 0 1 ∂B +∇ ×E = 0 c ∂t 1 n × [E + u × B] = 0 c 1 1 ∂D − ∇ × H + (J − Jˆ − qe v) = 0 c c ∂t 1 ˆ n × [H − u × D] = [n × H] c S −γ

in V − σ,

Jˆ · da = 0,

ˆ da = 0, Q

σ

in V − σ, on σ, in V − σ, on σ, in V − σ, on σ,

ˆ · ds = 0. [n × H]

γ

(4.1.7) The arguments advanced in Section 2.1, regarding the balance law residuals, are valid here too. In fact, here the influence zone of the E-M fields in the atomic scale is even smaller. The derivation of the microscopic Maxwell equations from the Lorentz theory of electrons shows that the residuals do not appear in the balance laws (see Eringen and Maugin [1989, p. 38], see also De Groot and Suttorp [1972, Chapter 1]). In the atomic scale we will have to resort to discrete atomic theories. However, in slightly larger scales, the microscopic balance laws are free of the residuals. Hence, for length scales of the orders of 5 to 10 lattice parameters it should be safe to dispense with the balance law residuals.

4.2 Electromagnetic Force, Couple, and Power

53

Alternatively, in a similar fashion to Secton 2.1-B, it is possible to reduce the nonlocal balance laws to local forms, by redefining ˆ qˆ = ∇ · D, ˆ D = D + D,

∨ ˆ − 1 u × D, ˆ H=H c ˆ ∂D Jˆ = − , ∂t

J = J − qe v, ˆ ˆ + n · [D], we = Q

(4.1.8)

and then writing D for D and J for J, the E-M balance laws (4.1.8) become ∇ · D − qe = 0 n · [D] = we ∇ · B = 0, n · [B] = 0 1 ∂B +∇ ×E = 0 c ∂t 1 n × [E + u × B] = 0 c 1 ∂D 1 −∇ ×H + J = 0 c ∂t c ∨ 1 n × [H − u × D] = n × [ H ] c

in V − σ,

(4.1.9a)

on σ, in V − σ,

(4.1.9b) (4.1.10a)

on σ,

(4.1.10b)

in V − σ,

(4.1.11a)

on σ,

(4.1.11b)

on V − σ,

(4.1.12a)

on σ.

(4.1.12b)

∨

Here we represents the surface charge and n × [ H ] the magnetic jump due to the creation of a surface σ . Except for this last term, the Maxwell equations keep their classical local forms. With the new definitions, the dielectric displacement D and the current vector J are nonlocal in character. Hence, we need to write nonlocal constitutive equations for the E-M response functions.

4.2 Electromagnetic Force, Couple, and Power Macroscopic expressions of the E-M force FE , the couple CE , and power W E are obtained by using the theory of electrons of Lorentz. A detailed account of this is given by De Groot and Suttorp [1972] (see also Eringen and Maugin [1989, Section 3.5] and a brief derivation by Eringen [1980, Section 10.6]: 1 ∇ E) · P + (∇ ∇ B) · M FE = qe E + J × B + (∇ c 1 1 ∂ + ∇ · (vP × B) + (P × B), c c ∂t 1 CE = P × E + M × B + v × (P × B), c ∂B ∂P E W = J·E+ ·E−M· + ∇ · [v(P · E)], ∂t ∂t

(4.2.1) (4.2.2) (4.2.3)

54


where qe , P, and M are, respectively, the free charge density, the polarization vector, and the magnetization vector. The latter two are related to D and B by D = E + P,

B = H + M.

(4.2.4)

The E-M force, the couple, and power given above are expressed in terms of the fields in the fixed laboratory frame RG . In a co-moving reference frame RC , then can be expressed by using (4.1.5): ∗ 1 J + P ) × B + (P · ∇ )E E + (∇ ∇ B) · M , FE = qeE + (J c CE = P × E + M × B, E · (P/ρ)• − M · B˙ + J · E W E = FE · v + ρE

= FE · v + w E ,

(4.2.5) (4.2.6) (4.2.7)

where 1 M = M + v × P, c ∗ ˙ ∇ · v). P = P − (P · ∇ )v + P(∇

(4.2.8)

For some purposes, it may prove convenient to express the E-M force, the couple, and power in the form of the mechanical type of balance equations: ∂G = FE , ∂t E = CkE , klm tlm ∂ 1 2 ∇ · [v(E · P) − S] − (E + B 2 ) = W E , ∂t 2 ∇ · tE −

(4.2.9) (4.2.10) (4.2.11)

where tE , G, and S are, respectively, the E-M stress tensor, the E-M momentum, and the Poynting vector. They are defined by tE = P ⊗ E − B ⊗ M + E ⊗ E + B ⊗ B M · B)1, − 21 (E 2 + B 2 − 2M 1 G = E × B, c S = cE × H.

(4.2.12) (4.2.13) (4.2.14)

Expressions (4.2.9) to (4.2.11) are identities. They may be verified by substituting (4.2.12) to (4.2.14) into (4.2.9) to (4.2.11) and using the Maxwell equations. In the co-moving reference frame RC , the E-M momentum, and Poynting vectors are defined as 1 G = E × B, c E × H. S = cE

(4.2.15) (4.2.16)

4.3 Electromagnetic Force, Couple, and Power

55

These are related to G and S. Using (4.2.11), we can express the power W E in the alternative form: d 1 E 2 2 W = −ρ (4.2.17) (E + B ) + ∇ · [(tE + v ⊗ G) · v − S ]. dt 2ρ We emphasize that the pondermotive force, the couple, and power, which are deduced from the Lorentz theory of electrons, are basic notions. The E-M stress tensor, the E-M momentum, and the Poynting vector are notions that are derived from these, through identifies (4.2.9) to (4.2.11). Nevertheless, they are useful expressions in evaluating the E-M force, the couple, and power at a discontinuity surface.

4.3 Electromagnetic Force, Couple, and Power at a Discontinuity Surface The E-M loads (E-M force, couple, and power) possess jump discontinuities across a singular surface σ , which may be sweeping the body at a velocity u in the positive direction of the normal n of σ . In order to obtain their discontinuities, we evaluate the volume integrals of these loads. For example, for the E-M force FE we write, symbolically, FE dv = FE dv + Fˆ E da. (4.3.1) V

V −σ

σ

Employing (4.2.9) and introducing G = ρg, into the first integral on the right-hand side of (4.3.1), we have E ∇ · (tE + ρv ⊗ g) dv. F dv = − ρ g˙ dv + (4.3.2) V −σ

V −σ

V −σ

Using the transport and the Green–Gauss theorems (2.1.5b) and (2.1.6a) this may be transformed to read d FE dv = − G dv + n · (tE + v ⊗ G) da dt V −σ V −σ V −σ E n · [t + u ⊗ G]da. (4.3.3) − σ

From (4.3.3) it is clear that the E-M force acquires a jump discontinuity given by Fˆ = [Fˆ kE ]nk ,

E Fˆ kE ≡ (tkl + uk Gl )il .

(4.3.4)

Similarly, by integrating x × FE + CE and W E and using (4.2.2) and (4.2.3) we obtain the jump discontinuities on the E-M couple and the E-M power: ˆ E = [CE Ê C Wˆ E = [Wˆ kE ]nk , k ]nk = x × [Fk ]nk , E Wˆ kE ≡ (tkl + uk Gl )vl − Sk − 21 (E 2 + B 2 )(vk − uk ).

(4.3.5)

These jump discontinuities must be incorporated into those of the mechanical ones.

56


4.4 Mechanical Balance Laws The mechanical balance laws in V − σ , given by (2.1.38a) to (2.1.40b), remain valid if we consider that the applied loads consist of a sum of mechanical and E-M loads, i.e., ρf + FE , ρh + wE , (4.4.1) where f and h are of purely mechanical origin and FE and wE are of E-M origin. Note that only wE ≡ W E − FE · v enters into the energy equation, since FE · v cancels with the balance of linear momentum. The E-M loads are given by (4.2.1) to (4.2.3) which also include the E-M couple CE . The associated jump conditions to the mechanical balance laws are modified by the addition of the E-M jump conditions obtained in Section 4.3. Consequently, the balance laws for electromechanically active continua take the following forms: Conservation of Mass ∇ ·v =0 ρ˙ + ρ∇ [ρ(v − u)] · n = 0

in V − σ,

(4.4.2a)

on σ.

(4.4.2b)

Balance of Momentum tkl,k + ρ(fl − v˙l ) + FlE = 0 E l [tkl + tkl + uk Gl − ρvl (vk − uk )]nk = F

in V − σ,

(4.4.3a)

on σ.

(4.4.3b)

Balance of Angular Momentum t[kl] = E[k Pl] + B[k Ml] .

(4.4.4)

Balance of Energy E · (P/ρ)• ρ ˙ − tkl vl,k − ∇ · q − ρh − ρE ˙ − J · E = 0 in V − σ, + M·B + uk Gl )vl + qk − Sk on σ. − ρ + 21 v 2 + 21 (E 2 + B 2 ) (vk − uk )]nk = H

(4.4.5a)

E [(tkl + tkl

(4.4.5b)

We note that, in the case of Angular Momentum Balance, the jump condition is satisfied automatically and t[kl] is determined in terms of the E-M fields. The energy balance law (4.4.5a) is obtained by replacing ρh by ρh + w E in (2.1.33a), where wE is given by (4.2.7). In fact, all local balance laws follow from the global (integral) balance laws that include the E-M loads (force, couple, and energy). For this, we refer the reader to Eringen and Maugin [1989, Section 3.10]. The collections of the mechanical balance laws and the Maxwell equations constitute the complete set of balance laws of the electromagnetically active media.

4.4 Mechanical Balance Laws

57

Second Law of Thermodynamics The second law of thermodynamics has the same form as given by (2.2.2), namely, [ρ η˙ − (qk /θ),k − ρh/θ ] dv ≥ 0. (4.4.6) V

The energy balance law (4.4.5a), upon introducing the generalized free energy = − θη − ρ −1E · P,

(4.4.7)

takes the form ˙ + θ˙ η + θ η) ρ( ˙ − tkl vl,k − qk,k − ρh + Pk E˙k + Mk B˙ k − Jk Ek = 0. (4.4.8) Carrying ρh from this into (4.4.6), we obtain a generalized C–D inequality 1 1 ˙ + θ˙ η) + tkl vl,k + qk θ,k − Pk E˙k − Mk B˙ k + Jk Ek dv ≥ 0. −ρ( θ V θ (4.4.9) This inequality is fundamental to the development of the constitutive equations. The inequality appears to dictate that Ek and Bk are the E-M independent variables. However, this is not necessary. Other choices are possible, simply by introducing a Legendre transformation for a new choice of . In this book we select Ek and Bk as our E-M independent variables. From (4.4.4) it is clear that the stress tensor tkl is not a symmetric tensor. We introduce a symmetric tensor by t kl = t(kl) + E(k Pl) + B(k Ml) = t lk .

(4.4.10)

With this it is possible to express the energy balance law in terms of objective quantities. To this end, we replace in (4.4.5a) by = − ρ −1M · B.

(4.4.11)

˙ , we Upon substituting these into (4.4.5a), and using (4.2.8) to replace P˙ and M obtain ∗

∗

˙ + θ˙ η + θ η) J · E = 0, (4.4.12) ρ(ψ ˙ − t kl dkl − ∇ · q − ρh − E · P −B· M −J by where we also introduced the free energy ψ = ψ − θη.

(4.4.13)

Substituting ρh/θ, solved from (4.4.12), into the entropy inequality (4.4.6) we obtain ∗ ∗ 1 qk ˙ ˙ J · E dv ≥ 0. (4.4.14) −ρ(ψ + ηθ ) + t kl dkl + θ,k + E · P +B· M +J θ V θ

58


From this inequality, it is clear that the E-M independent variables are now P and M. We observe that this inequality is constructed with the product terms whose constituents are objective, i.e., (t kl , qk , E , B ), as well as their factors (dkl , θ,k /θ, ∗

∗

P , M being objective vectors and tensors. Expressions (4.4.12) and (4.4.14) make it possible to express the energy and the C–D inequality in the reference state variables, defined by ρ0 ρ0 T KL = t (kl) XK,k XL,l , QK = qk XK,k , ρ ρ ρ0 ρ0 K = Pk XK,k , MK = Mk XK,k , ρ ρ BK = Bk xk,K . (4.4.15) EK = Ek xk,K , We calculate the time-rate of K and MK by using (1.4.8), (2.1.38a), and (4.2.8): ρ ∗ ρ0 ∗ ˙ K = 0 P k XK,k , M˙ K = (4.4.16) Mk XK,k . ρ ρ Using (4.4.15), (4.4.16), and dv = (ρ0 /ρ) dV in (4.4.12) and (4.4.14) we obtain the material form of the energy equation ˙ + ηθ˙ + ηθ ˙K ˙ ) − 21 T KL C˙ KL − QK,K − ρ0 h − EK ρ0 (ψ − BK M˙ K − JK EK = 0,

(4.4.17)

and the material form of the entropy inequality 1 1 ˙ + ηθ˙ ) + 1 T C˙ ˙K −ρ0 (ψ KL KL + QK θ,K + EK 2 θ V θ ˙ + BK MK + JK EK dV ≥ 0.

(4.4.18)

The use of the E-M independent variables EK and BK provide some simplicity in the field equations. Consequently, we introduce − ψ =ψ

1 1 EK K − BK MK , ρ0 ρ0

(4.4.19)

to express (4.4.17) and (4.4.18) in the forms ˙ − 21 T KL C˙ KL − QK,K − ρ0 h ρ0 (ψ˙ + ηθ˙ + ηθ) + K E˙K + MK B˙ K − JK EK = 0, 1 1 ˙ + 1 T KL C˙ KL + QK θ,K −ρ0 (ψ˙ + ηθ) 2 θ V θ − K E˙K − MK B˙ K + JK EK dV ≥ 0.

(4.4.20)

(4.4.21)

In this form, the C–D inequality (4.4.21) simplifies the development of the constitutive equations for solid media.

4.4 Mechanical Balance Laws

59

Problems 4.1 The E-M fields in a fixed laboratory frame and in a moving body are related by (4.1.5). Obtain these relationships from the relativistic invariance of these fields. 4.2 The E-M force, couple, and energy are given by (4.2.1) to (4.2.3). Show that (4.2.5) to (4.2.7) are equivalent to these. 4.3 Derive the identities (4.2.9) to (4.2.11). 4.4 Obtain the expression (4.4.5a) of the energy. 4.5 Obtain the expression (4.4.9) of the entropy inequality. 4.6 For nonlocal elastic solids the energy and entropy, given by (4.4.17) and (4.4.18), are suitable for the development of constitutive equations. Obtain these equations from (4.4.8) and (4.4.9). 4.7 Show that the Maxwell equations can be expressed as ∇ ×E +

1 ∗ P = 0, c

∇ ×H −

4.8 Obtain material forms of the Maxwell equations.

1 ∗ 1 D= J . c c


5 Constitutive Equations of Nonlocal Electromagnetic Media

5.0 Scope The formulation of the constitutive equations of electromagnetic (E-M) elastic solids must employ a special type of Helmholtz free-energy functions and a dissipation functional. To this end, we need to develop constitutive equations in the material frame of reference. Thus, the energy equations and the Clausius–Duhem (C–D) inequality are expressed in a material frame. Employing the additive functional, introduced in Section 3.2, we develop separate constitutive equations for the static and dynamic members of the constitutive-dependent variables. This is done by using the C–D inequality. The dynamic members of the constitutive equations require the introduction of the difference histories of the dependent variables. The resulting constitutive equations are nonlocal both in space and time. Section 5.2 develops the constitutive equations of the memory-dependent nonlocal E-M thermoviscous fluids. Here the development is in the spatial domain. From the constitutive equations obtained in this chapter many special results may be deduced by omitting memory, nonlocality, thermal effects, E-M effects, etc.

62


5.1 Constitutive Equations of Memory-Dependent Nonlocal Electromagnetic Thermoelastic Solids1 The primitive independent variables of E-M thermoelastic solids consist of the array θ, xk,K , θ,k , Ek , Bk (5.1.1) and the dependent variables η, tkl , qk , Pk , Mk .

(5.1.2)

The internal energy will depend only on the independent invariants of the set (5.1.1) which are given by θ, CKL = xk,K xk,L ,

θ,K = θ,k xk,K ,

EK = Ek xk,K ,

(5.1.3)

at all points X of the body at all past times −∞ < t ≤ t. The free energy can only depend on these quantities (excluding θ,K ) at all points X of the body at time t. Thus, we propose ρ0 ψ(X, t) = F [C(X ), θ(X ), EK (X ), BK (X ), X ; V

C(X), θ(X), EK (X), BK (X), X] dV ,

(5.1.4)

where we suppressed the dependence on t, e.g., we wrote C(X ) ≡ C(X , t). The free energy is a symmetric functional, so that S

F =F .

(5.1.5)

Expressions of the energy (4.4.20) and the C–D inequality (4.4.21) may be expressed in a compact form by introducing the following ordered sets: Static Set = −ρ0R η, 21 R T KL , −R K , −R MK , R Y = θ, CKL , EK , BK .

(5.1.6)

= −ρ0D η, 21 D T KL , QK /θ, −D K , −D MK , JK , ˙ KL , θ,K , E˙K , B˙ K , EK . D Y = θ˙ , C

(5.1.7)

RJ

Dynamic Set DJ

The collections D J and D Y are called thermodynamic fluxes and thermodynamic forces, respectively. We shall refer to R J and R Y as thermostatic forces and fluxes. 1 The theory was first given by Eringen [1973a] and further elaborated on in Eringen [1984a,b,c], [1986], [1990], [1991].

5.1 Electromagnetic Thermoelastic Solids

63

With these, (4.4.20) and (4.4.21) are expressed in the form of scalar products, in 35-dimensional Euclidean space ˙ + D J · D Y + θ(QK /θ),K + ρ0 h = 0, (5.1.8) ˙ + D ηθ ˙ ) + RJ · RY −ρ0 (ψ˙ + R ηθ

˙ + D J · D Y dV ≥ 0. −ρ0 ψ˙ + R J · R Y

(5.1.9)

V

Since θ > 0, we also omitted the factor θ −1 from (5.1.9). The static part of the thermostatic fluxes R J can be obtained from the static part of (5.1.9). Using the notation (5.1.6) the free energy ψ, given by (5.1.4), is expressed by

F R Y(X ), X ; R Y(X), X dV . (5.1.10) ρ0 ψ = V

˙ We calculate ψ: ρ0 ψ˙ =

V

where

D≡ V

∂F + ∂RY

∂F ∂ R Y

∂F ˙ − · RY ∂ R Y

S

∂F ∂ R Y

˙ dV + D, · RY

(5.1.11)

˙ · R Y dV ,

(5.1.12)

S

where we used a prime to denote the dependence on X , e.g., R Y = R Y(X ). Also, S

a superscript S denotes symmetrization, i.e., interchange X and X . Since F = F , it follows that D dV = 0.

(5.1.13)

V

˙ we Substituting (5.1.11) into (5.1.9), and noting that F does not depend on R Y, obtain ∂F ∂F S dV . (5.1.14) + RJ = ∂ R Y

V ∂RY With this C–D inequality, (5.1.9) reduces to D J · D Y dV − D ≥ 0.

(5.1.15)

V

This inequality is in the form (2.3.1). From (2.3.3), it follows that DJ

=

∂ + U, ∂DY

(5.1.16)

64


where is the dissipation functional and U is the constitituve residual that does not contribute to the dissipation of energy, i.e., DY

· U = 0.

(5.1.17)

In principle, (5.1.16) determines the dynamical parts of the constitutive equations. However, we need to specify the form of the dissipation potential . For nonlocal memory dependence, it is appropriate to introduce difference histories of the variable set D Y, i.e., D Y(t) (X, τ )

= D Y(X, t − τ ) − D Y(X, t).

(5.1.18)

The dissipation potential can then be expressed as

∞

=

∞

dτ 0

dτ 0

G[D Y(t) (X , τ ); D Y(t) (X, τ )] dV ,

(5.1.19)

V

where G is a symmetric function with respect to D Y(t) (X , τ ) and D Y(t) (X, τ ), i.e., G[D Y(t) (X , τ ); D Y(t) (X, τ )] = G[D Y(t) (X, τ ); D Y(t) (X , τ )].

(5.1.20)

Using (5.1.6) and (5.1.14) we have, for the specific forms of the static constitutive equations,

∂F S −ρ0R η = dV ,

∂θ V S ∂F ∂F 1 dV , +

2 R TKL = ∂CKL V ∂CKL ∂F ∂F S −R PK = dV , +

∂EK V ∂EK ∂F ∂F S −R MK = dV , +

∂BK V ∂BK and the dynamic parts (5.1.16):

∂F + ∂θ

(5.1.21)

5.1 Electromagnetic Thermoelastic Solids

−ρ D η = 1 2 D T KL

=

1 QK = θ −D K = −D MK = JK =

∂ + U η, ∂ θ˙ ∂ T + UKL , ∂ C˙ KL ∂ Q + UK , ∂θ,K ∂ + UKP , ∂ E˙K ∂ + UKM , ∂ B˙ K ∂ + UKJ . ∂EK

65

(5.1.22)

The passage to the spatial forms of the constitutive equations is provided through (4.4.15), namely, t kl = qk = Pk = Mk = Ek =

ρ T KL xk,K xl,L , ρ0 ρ QK xk,K , ρ0 ρ K xk,K , ρ0 ρ MK xk,K , ρ0 EK XK,k , Bk = BK XK,k .

(5.1.23)

Upon using (5.1.21), the energy takes the form −ρ0 (R ηθ ˙ + D ηθ ˙ + D ηθ˙ ) +

1 ˙ 2 D T KL CKL

+ QK,K + ρ0 h − D K E˙K − D MK B˙ K + JK EK = 0.(5.1.24)

Nonlocal Electromagnetic Solids without Memory2 We ignore the effects of the past motions, the thermal gradients, and the E-M fields. This implies that the functional ψ retains its form (5.1.10) but is reduced to

˙ ), θ,K (X ), X ; θ(X), ˙ = G θ(X θ,K (X), X dV , V

dV ≥ 0, V 2 Eringen [1974].

S

G = G,

(5.1.25)

66


and vanishes with θ˙ and θ,K /θ. The constitutive equations now become ∂F ∂F S −ρ0R η = dV , +

∂θ ∂θ V S ∂F ∂F 1 dV , +

2 R T KL = ∂CKL V ∂CKL ∂F ∂F S −R K = dV , +

∂E ∂E K V K ∂F ∂F S −R MK = dV , +

∂BK V ∂BK ∂G ∂G S −ρ D η = dV , +

˙ ˙ ∂ θ ∂ θ V  S  QK ∂G ∂G   dV . + =

θ ∂θ,K ∂θ,K V

(5.1.26)

The spatial forms follow using (5.1.23).

5.2 Constitutive Equations of Nonlocal Memory-Dependent Electromagnetic Thermoviscous Fluids For nonlocal, memory-dependent fluids, to the constitutive-dependent variables (3.3.2), are adjoined the E-M independent variables, so that list is now Y = ρ(x , t ), dkl (x , t ), θ,k (x , t ), θ(x , t ), Ek (x , t ), Bk (x , t ) , x ∈ V, −∞ < t ≤ t. (5.2.1) The constitutive-dependent variables are Z = {, tkl , qk , Pk , Mk , Jk }.

(5.2.2)

With the introduction of the free energy, is replaced by ψ and η, where ψ is not memory-dependent, but η is. As in Section 3.3, we may select a linear functional in the sense of Friedman and Katz (see Section 3.2): ρψ = F [ρ(x ), θ(x ), Ek (x ), Bk (x ); V

ρ(x), θ(x), Ek (x), Bk (x), |x − x|],

(5.2.3)

5.2 Electromagnetic Thermoviscous Fluids

67

where F is a symmetric function, i.e., S

F =F .

(5.2.4)

As usual, the superscript (S) denotes the symmetrization, i.e., the interchange of x and x in the argument functions of F . Again, we introduce the static and dynamic parts of the response functions by the left subscripts R and D: tkl = R tkl + D tkl , Pk = R Pk + D Pk ,

η = R η + D η, Mk = R Mk + D Mk .

(5.2.5)

The static parts of the constitutive equations can be obtained from the C–D inequality. In fact, it is free for our disposal. But we select them so that the dynamics part of the C–D inequality does not contain the static response functions. As in Section 3.3, we arrive at the condition ∂F (5.2.6) dv = 0, ∂κ V where κ = |x − x|. Since (5.2.6) must be true for all x, similar to the arguments leading to (3.3.4), we must have ∂F /∂κ = 0. This then naturally leads to ψ = [ρ(x, t), θ(x, t), Ek (x, t), Bk (x, t)], ∂ π = ρ2 , R tkl = −πδkl , ∂ρ ∂ , Rη = − ∂θ ∂ , R Pk = − ∂Ek ∂ . R Mk = − ∂Bk

(5.2.7)

With this, the C–D inequality (4.4.9) reduces to −ρ D ηθ˙ + D tkl vl,k +

1 qk θ,k − D Pk E˙k − D Mk B˙ k + Jk Ek − Kˆ ≥ 0. θ

(5.2.8)

With CE given by (4.2.6), from (4.4.4) we have tkl = t(kl) + t[kl] , t[kl] = E[k Pl] + B[k Ml] .

(5.2.9)

Substituting this into the C–D inequality (5.2.8), we express it as 1 ˙ −ρ D ηθ + D t kl dkl + qk θ,k − D Pk E k −D Mk B k +Jk Ek dv ≥ 0, θ V (5.2.10)

68


where D t kl Ek

= D t(kl) + E(k Pl) + B(k Ml) = D t lk ,

= E˙k + El vl,k ,

B k = B˙ k + Bl vl,k .

(5.2.11)

The C–D inequality (5.2.10) is expressed in terms of the scalar products of the

objective variables {θ,k , dkl , E k , B k , Ek } and {qk /θ, express it in the compact local form DJ · DY

D t kl ,

−Pk , −Mk , Jk }. We

− Kˆ ≥ 0,

(5.2.12)

where DJ

= {−ρ D η,

DY

= {θ˙ , dkl ,

D t kl , qk /θ, −D Pk , θ,k , E k , B k , Ek }.

−D Mk , Jk }, (5.2.13)

The inequality (5.2.12) agrees with (2.3.1) so that the dynamic constitutive equations follow from (2.3.3), i.e., DJ

=

∂ + U, ∂DY

(5.2.14)

· U = 0.

(5.2.15)

where DY

Here is the dissipation function which vanishes with D Y. For the memory-dependent media, we introduce the difference histories D Y(t) (x, τ )

= D Y(x, t − τ ) − D Y(x, t).

(5.2.16)

We then propose for a linear functional

∞

=

∞

dτ 0

dτ 0

V

G[D Y(t) (x , τ ); D Y(t) (x, τ )] dv ,

(5.2.17)

where G is a symmetric function of D Y(t) (x , τ ) and D Y(t) (x, τ ), as described by (5.1.20): S G = G, dv ≥ 0, lim = 0. (5.2.18) V

G may also depend on {ρ, θ, Ek , Bk }.

D Y→0

5.2 Electromagnetic Thermoviscous Fluids

69

The dynamic parts of the constitutive equations memory-dependent, nonlocal thermofluids are then given by (5.2.14). More explicitly, ∞ ∞ ∂G

−ρ D η = dv + U η , dτ dτ

˙ ∂ θ V 0 ∞ 0 ∞ ∂G

dτ dτ dv + Ut , Dt = V ∂d 0 0 ∞ ∞ ∂G 1 dv + Uq , dτ dτ

q= ∇ θ) θ V ∂(∇ 0 0 ∞ ∞ ∞ ∂G

dτ dτ −D P = dv + U P , 0 0 0 ∂ ∞ E ∞ ∞ ∂G

−D M = dτ dτ

dv + U N , 0 0 0 ∂ ∞ B ∞ ∞ ∂G

dτ dτ

(5.2.19) dv + U J . J = E ∂E 0 0 0 Upon using (5.2.7), the energy equation (4.4.8) becomes ˙ + D tkl dkl + qk,k + ρh ˙ + D ηθ ˙ + D ηθ) −ρ(R ηθ − D Pk E˙k − D Mk B˙ k + Jk Ek = 0.

(5.2.20)

Nonlocal Thermoviscous Fluids without Memory3 In the case of the absence of the memory static parts of the constitutive equations, (5.2.7) remains unchanged. The dynamic part (5.2.19) is modified by dropping the dependence on difference histories. In this case, we have

˙ dkl , θ,k , Ek ) dv , G(θ˙ , dkl , θ,k , Ek ; θ, (5.2.21) = V

and we obtain

−ρ D η = D tkl

V

=

V 1 qk = θ V Jk = V

∂G

dv + U η , ∂ θ˙ ∂G dv + Uklt , ∂dkl ∂G q dv + Uk , ∂θ,k ∂G

dv + UkJ . ∂Ek

(5.2.22)

The constitutive residuals are subject to U η θ˙ + Uklt dkl + Uk θ,k + UkJ Ek = 0. q

3 The theory, without E-M effects, was given by Eringen [1972b].

(5.2.23)

70


Clearly, from the foregoing sets of constitutive equations, one may obtain special and approximate theories. For example: (a) For nonviscous fluids we drop D t and the dependence on d. (b) For the isothermal case we omit q and the dependence on ∇ θ. (c) In the purely E-M case we ignore D η, t, q, and the corresponding independent variables θ˙ , d, and ∇ θ. (d) For the linear constitutive equations, for F and G, we write second-degree polynomials in terms of their independent variables. (e) For local theories (we express F and G in the form F = F0 δ(x )). G = G0 δ(x ), where δ(x ) is a three-dimensional delta function. Clearly many other possibilities exist.

Problems 5.1 Obtain the constitutive equations of isotropic nonlocal E-M elastic solids (without memory), in terms of the invariants of the independent variables. 5.2 Obtain the constitutive equations of nonlocal fluids (without memory) as functionals of the invariants of the independent variables. 5.3 For local linear, memory-dependent fluids, find the restrictions arising from the second law of thermodynamics. 5.4 For isothermal solid bodies (no deformation) express the nonlocal isotropic E-M constitutive equations in terms of the invariants of the independent variables up to and including the second degree. 5.5 For E-M fluids obtain the pressure as functions of the mechanical and E-M variables. 5.6 (Short Term Paper.) Study the literature and give a discussion on the group properties of E-M elastic crystals.

6 Nonlocal Linear Elasticity

6.0 Scope In Chapter 3, we obtained the general constitutive equations of the memorydependent nonlocal elastic solids. With practical applications in mind, we construct here specific constitutive equations for the nonlocal linear elastic solids. To this end, in Section 6.1, we introduce a measure for the order of magnitude evaluation. With a linear strain measure ekl and small temperature changes T from a constant ambient temperature T0 , we write a quadratic free-energy functional in terms of ekl and T , and a dissipation functional in terms of T,k . The constitutive equations for the nonlocal anisotropic elastic solids are then obtained by using the equations given in Chapter 3. Afterward, we pass to the constitutive equations for the isotropic solids. Various different, but equivalent, forms of these equations are presented. In Section 6.2 we show how one may obtain the same constitutive equations through the lattice dynamical approach. This way relations are established between the nonlocal constitutive moduli based on the continuum and atomic lattice dynamics. This then leads us to determine material moduli for crystal classes. For the isotropic case, generalized functions and Dirac-delta sequences offer a rich field for the selection of the material moduli. In fact, it turns out that, we can construct nonlocal elastic moduli that give exactly the same displacement at the atomic sites but not necessarily between atoms. All these material moduli obey the axioms of the attenuating neighborhood. Material stability is dealt with in Section 6.3.

72


In Section 6.4 we give the field equations of nonlocal thermoelasticity, and formulate mixed boundary-initial value problems. The formulation by means of convolution offers many facilities in proving some important theorems. This is formulated in this section. In Section 6.5 we prove the uniqueness theorem for mixed boundary-initial value problems. Theorems on power and energy are the subject of Section 6.6. In Section 6.7 the reciprocal theorem is proved. This theorem is useful in constructing solutions for the boundary-initial value problems from those of the fundamental solutions. Section 6.8 discusses the most general, and a special form, of the variational principles. These are important for constructing the approximate solutions of the boundary-initial value problems. Section 6.9 contains various approximate models for the reduction of the integro– partial differential equations to partial differential equations. By means of some operators, suggested by lattice dynamics, generalized functions (with finite support), Dirac-delta sequences, and other considerations, it has been possible to achieve such reductions. These are all explored in Section 6.9. In Section 6.10 we begin to solve some critical problems to demonstrate the power and potential of the nonlocality. Screw dislocations in full- and half-planes are discussed in Section 6.10. Here, for the first time, we find that at the core of dislocation the stress vanishes and with it the classical singularity. The maximum stress appears slightly apart from the core, thus allowing the establishment of a maximum stress hypothesis as a criterium of fracture. In fact, the cohesive stress calculated agrees with that observed experimentally. All these predictions are supported by the solutions of many other problems treated later in this chapter. These include edge dislocation (Section 6.11), screw dislocation in nonlocal hexagonal solids (Section 6.12), nonlocal stress fields at the Griffith crack (Section 6.14), and line cracks subject to shear (Section 6.15). The continuous distribution of dislocation is dealt with in Section 6.13. A Peach– Koehler-type formula is obtained and applied to several special distributions, e.g., finite line distribution and the uniform distribution of screws along a circle. Again, no stress singularity is predicted. The interaction of a dislocation with a crack is discussed in Section 6.16 and the interaction between defect and dislocation in Section 6.17. Section 6.18 discusses the case of straight wedge disclinations. In Section 6.19, we obtain a Somigliana-type representation of nonlocal elasticity. In the following section, we apply this representation to obtain the solution of the fundamental problems, namely the concentrated load in infinite space. A nonlocal elastic half-plane under a concentrated force is presented in Section 6.21. Chapter 6 ends with the treatment of a rigid stamp on a nonlocal elastic half-plane. From all these solutions one reaches the conclusions: (a) Nonlocal solutions eliminate singularities predicted by the classical (local) theory of elasticity.

6.1 Linear Constitutive Equations

73

(b) Maximum stress can be used to calculate the cohesive stress that holds atomic bonds together. (c) Classical elasticity is obtained when the nonlocal moduli become Diracdelta functions. (d) The cohesive stress calculated agrees very well with the lattice dynamical results and experimental observations. These overwhelming results should convince the fracture mechanics community that there is no need for all those ersatz regarding fracture criteria (e.g., energy, fracture toughness, J -integral, etc.).

6.1 Linear Constitutive Equations1 In Section 3.2 we obtained the constitutive equations of the nonlocal thermoelastic solids without memory. For the linear theory, constitutive residuals are ignored (Onsager postulate). We ignore the dependence on θ˙ which may be important in rigid conductors. With this, the general constitutive equations (3.2.30) and (3.2.32) take the form 1 ∂F S ∂F η=− + dV , ρ0 V ∂θ ∂θ

S ∂F ρ ∂F tkl = 2 xk,K xl,L dV , +

ρ0 ∂CKL V ∂CKL 1 ∂G ρ xk,K dV . (6.1.1) qk = θ ρ0 ∂θ ,k V Of course D T and D η vanish. When the deformation and thermal gradients are small, we can derive a linear theory from (6.1.1). To this end, we introduce the displacement vector u and temperature change T by u = x − X, T = θ − T0 ,

T0 |T |,

T0 > 0

(6.1.2)

where T0 is a constant ambient temperature of the order of room temperature. To have systematic approximation we let W = (u,k , T,k , T ),

(6.1.3)

= W = (W · W)1/2

(6.1.4)

with norm

1 Eringen [1972a], [1974].

74


and we assume that, in some neighborhood of = 0, there exist positive constant K0 and positive integer n such that O( n ) ≤ K0 n ,

K0 > 0,

n > 0.

(6.1.5)

With this measure, for the first order (linear) approximation, we have xk,K = (δkl + uk,l )δlK + O( 2 ),

vk =

∂uk + O( 2 ), ∂t

XK,k = (δkl − ul,k )δlK + O( 2 ), ρ = 1 − uk,k + O( 2 ), θ,k = T,k + O( 2 ). (6.1.6) ρ0 We introduce the Lagrangian and Eulerian strain tensors EKL and ekl by EKL = 21 (CKL − δKL ) = ekl xk,K xl,L ,

(6.1.7)

which upon substitution from (6.1.6) for xk,K gives EKL = ekl δkK δlL + O( 2 ),

ekl ≡ 21 (uk,l + ul,k ).

For the linear theory, (6.1.1) takes the form ∂F ∂F S 1 dv , + η=− ρ0 V ∂T ∂T

∂F ∂F S tkl = dv , +

∂ekl V ∂ekl 1 ∂G qk = dv . T0 V ∂T,k

(6.1.8)

(6.1.9)

Here F and G are expressed as second-degree polynomials in their variables ρ0 C

T T − 21 βkl (T ekl + T ekl ) + U, T0

T0 G = G0 + 21 kij (T,i T,j + T,i T,j ) + 21 hij k (eij T,k + eij T,k ), (6.1.10) F = F0 −

where U is the strain energy density, defined by

1 ρ0 C T T + λij kl eij ekl . U=2 T0

(6.1.11)

The material moduli F0 , G0 , and ρ0 are constants, and the remaining material functions are subject to the following symmetry regulations, as can be deduced S

S

from (6.1.10) and the fact that F = F and G = G: C(x, x ) = C(x , x), βkl (x, x ) = βkl (x , x) = βlk (x, x ), λij kl (x, x ) = λij kl (x , x) = λj ikl (x, x ) = λij lk (x, x ), kij (x, x ) = kij (x , x) = kj i (x, x ), hij k (x, x ) = hij k (x , x) = hj ik (x, x ).

(6.1.12)


75

The material moduli (6.1.12) are also restricted by the material symmetry conditions and the condition of the material stability. The former is expressed by the symmetry group and the latter by the thermodynamic requirements that both the total strain energy and the total dissipation must be nonmagnetic, i.e., U= U dv dv ≥ 0, (6.1.13) V V = G dv dv ≥ 0. (6.1.14) V

V

and T , it cannot be But, because is a linear functional with respect to eij ,k maintained in one sign unless hij k = 0, and we have:

T0 G = G0 + 21 kij (T,i T,j + T,i T,j ).

(6.1.15)

We now substitute F from (6.1.10) and G from (6.1.15) into (6.1.9) to obtain the linear constitutive equations of the nonlocal, anisotropic, thermoelastic solids2 T (x ) 1 C(x , x) η= + βkl (x , x)ekl (x ) dv , T0 ρ0 V tkl = [−βkl (x , x)T (x ) + λij kl (x , x)eij (x )] dv , V kkl (x , x)T,l (x ) dv . (6.1.16) qk = V

Isotropic Solids For the homogeneous media, F0 , βkl , λij kl , and kij are functions of x − x, e.g., λij kl = λij kl (x − x),

(6.1.17)

and for homogeneous and isotropic solids they are isotropic functions of κ = x −x, i.e., C = C(|x − x|), βkl = βδkl + β1 κk κl , λij kl = λδij δkl + µ(δik δj l + δil δj k ) + λ1 (κi κj δkl + κk κl δij ) + λ2 κi κj κk κl , kkl = kδkl + k1 κk κl ,

(6.1.18)

where the material moduli β, β1 , λ, µ, λ1 , λ2 , k, and k1 are functions of κ = |κκ | = |x − x|, e.g., k1 = k1 (|x − x|). (6.1.19) λ = λ(|x − x|), 2 Eringen [1972a], [1974].

76


The material moduli β1 , λ1 , λ2 and k1 are interatomic contributions to the isotropy. They are absent in classical (local) elasticity. Because of the products of these moduli with κi = xi − xi , we expect that these moduli are generally smaller in magnitude than β, λ, µ, and k. Moreover, lattice dynamical observations have shown that interatomic forces die out with distance, (attenuating neighborhood hypothesis, Section 3.1). Hence, within a first-order approximation we drop these terms T (x ) β(|x − x|)

η= C(|x − x|) + ekk (x ) dv , T0 ρ0 V

tkl = −β(|x − x|)T (x )δkl + λ(|x − x|)err (x )δkl V + 2µ(|x − x|)ekl (x ) dv , qk = k(|x − x|)T,k (x ) dv . (6.1.20) V

We observe that these constitutive equations reduce to those of the classical (local) elasticity, by letting {C(κ), β(κ), λ(κ), µ(κ), k(κ)} → {C0 , β0 , λ0 , µ0 , k0 }δ(κ),

(6.1.21)

where δ(κ) is the Dirac-delta measure. It is well known that interatomic attractions die out with distance (attenuation neighborhood). This implies that the material functions C, β, λ, µ, and k must attenuate rapidly with distance, i.e., lim {C(κ), β(κ), λ(κ), µ(κ), k(κ)} → 0.

κ→∞

(6.1.22)

A great deal of simplification would result if we assume that the degree of attenuation for all material moduli is the same, i.e., C(|x − x|) β(|x − x|) λ(|x − x|) µ(|x − x|) = = = C0 β0 λ0 µ0

k(|x − x|) = α(|x − x|), (6.1.23) = k0 where C0 , β0 , λ0 , µ0 , and k0 are the material constants of the local (classical) theory, i.e., β0 ekk , ρ0 σkl = −β0 T δkl + λ0 err δkl + 2µ0 ekl , C qk = k0 T,k . Cη

= C0 T +

(6.1.24)

Here, C η is the local entropy density, σkl is the Hookean stress tensor, and C qk is the local heat vector. The kernel function α(|x −x|) is normalized over the volume of the body, i.e., V

α(|x |) dv = 1.

(6.1.25)


With these constitutive equations, (6.1.20), are abbreviated to η= α(|x − x|) C η(x ) dv , V tkl = α(|x − x|)σkl (x ) dv , V qk = α(|x − x|) C qk (x ) dv . V

77

(6.1.26)

From (6.1.26) it is clear that the kernel function α(|x − x|) has the dimension length−3 . Consequently, it must depend on an internal characteristic length. This then indicates that, unlike the classical (local) elasticity, in the nonlocal theory of elasticity, predictions are influenced by an internal characteristic length. This will also be indicated through the solution of some problems presented in later sections. In fact, it is possible to fix this characteristic length with respect to the external scales involved in problems. We shall show that by means of the nonlocal theory of elasticity, it is possible to discuss problems all the way to the atomic scales. Remark. Clearly, the constitutive equations (6.1.16) may be expressed in forms that display the local constitutive equations explicitly, i.e.,

T 1 0 1

T (x )

η = C0 + βkl ekl + C (x, x ) + βkl (x, x )ekl (x ) dv , T0 ρ0 T0 ρ0 V 0

tkl = −βkl T + λ0ij kl eij + [−βkl (x, x )T (x ) + λ ij kl (x, x )eij (x )] dv , V 0

T,l + kkl (x, x )T,l (x ) dv . (6.1.27) qk = kkl V

In fact, originally they were derived in this form, even including the initial fields in the natural state, Eringen [1972a]. This form follows by simply expressing C(x, x) = C0 δ(x − x ) + C (x, x ), 0

βkl (x, x ) = βkl δ(x − x ) + βkl (x, x ),

λij kl (x, x ) = λ0ij kl δ(x − x ) + λ ij kl (x, x ), 0

kkl (x, x ) = kkl δ(x − x ) + kkl (x, x ), 0, C0 , βkl

λ0ij kl ,

(6.1.28)

0 kkl

where and are constant moduli of the local theory and δ(x) is the Dirac-delta measure. Clearly this format is also valid for the homogeneous and isotropic case (6.1.20), and similarly obtained T β0 T (x ) β (|x − x |) η = C0 + ekl + C (|x − x |) + ekk (x ) dv , T0 ρ0 T0 ρ0 V tkl = −β0 T + λ0 err δkl + 2µ0 ekl + [−β (|x − x |)T (x )δkl V

+ λ (|x − x |)err (x )δkl + 2µ (|x − x |)ekl (x )] dv , qk = k0 T,k + k (|x − x |)T,k (x ) dv . V

(6.1.29)

78


6.2 Lattice Dynamical Foundations of Linear Elasticity Here, I would like to consider a simple lattice to illustrate how we can obtain the constitutive equations of the nonlocal isothermal elastic solids. According to lattice dynamics, a perfect crystal consists of discrete atomic mass points attached to each other by springs (Figure 6.2.1). Let u(n) denote the displacement vector of an atom located at a discrete point marked by n from the equilibrium position x0 (n). Let ek denote a set of base vectors and ek their reciprocal, so that ek · el = δ kl ,

Figure 6.2.1. Atomic model.

6.2 Lattice Dynamical Foundations of Linear Elasticity

79

where δ kl is the Kronecker delta. We may locate the undeformed position x0 (n) and the deformed position x(n) of an atom by (Figure 6.2.1b): x0 (n) = nk ek ,

x(n) = x0 (n) + u(n).

(6.2.1)

For a net consisting of atoms that are located at the knots (corners) of a parallelepiped with equal edge lengths parallel to ek , nk may be taken as integers. Such a lattice is called a simple lattice or a Bravais lattice. The potential energy (internal energy) of such a lattice is a function of the distance between atoms, i.e., V = U [|x(n) − x(n )|]. (6.2.2) n,n

A mechanical analogy to this is a spring where the restoring force is a function of the change of the length of the spring. In harmonic approximation, it is simple to show that a quadratic approximation to V prevails. Thus, we may express V as V = 21 uk (n)Ukl (n, n )ul (n ), (6.2.3) n,n

where a constant potential energy corresponding to the equilibrium position is discarded and (6.2.4) Ukl (n, n ) = U,kl (|x(n) − x0 (n )|). The Lagrangian of the system is given by ˙ ˙ L = 21 m(n)u(n) · u(n) n

−

1 2

u(n) · U(n, n ) · u(n ) −

n,n

f(n) · u(n),

(6.2.5)

n

where m(n) is the mass of the nth atom and f(n) is the external force applied to the atom. Lagrange’s equation reads: m(u)u¨ + U(n, n ) · u(n ) − f(n) = 0. (6.2.6) n

This is the basic equation, underlying the nonrelativistic lattice dynamics. We would now like to pass to nonlocal representation, sometimes also called quasi-continuum representation. To this end, one may employ the concept of the sampling function introduced by Shannon [1949]. According to this theorem, if the Fourier transform f (k) of a function f (x) vanishes outside a region |k| > k0 , then this function is determined uniquely from its discrete values f (n) = f (x) at x = n. In fact, if f (k) = v0 f (n)ein·k (6.2.7) n

80


is the Fourier transform of f (x), then the inverse transform is given by f (x) = v0 f (n)δB (x − n),

(6.2.8)

n

where δB (x) = (2π)−3

eik·x dk,

(6.2.9)

B

where v0 is the volume of an elementary cell constructed on ek . For a simple lattice with lattice parameter a, |ek | = a, v0 = a 3 , and the domain of k is 0 < |k| < π/a. It is clear that δB (0) = v0−1 and δB (n) = 0 for all other n = 0. It is important to note that f (x), given by (6.2.8), coincides exactly with f (n) at discrete points x = n, however, being completely arbitrary elsewhere. Thus, we expect that the predictions of the nonlocal theory will agree with those of the lattice dynamics at the atomic sites, differing possibly in between. Using Shannon’s theorem, we have the nonlocal representation of (6.2.3) and (6.2.6): V = 21 u(x) · U(x, x ) · u(x ) dv(x ), (6.2.10) V V ρ u¨ + U(x, x ) · u(x ) dv(x ) − ρf(x) = 0, (6.2.11) V

where ρ(x) is the mass density and U kl (x, x ) = U lk (x , x).

(6.2.12)

The invariance of U under rigid translations and rotations shows that the Fourier transform can be expressed as U (k, k ) = λ kl

ikj l

(k , k)ki kj ,

(6.2.13)

where the overbar denotes the Fourier transform. Substituting the Fourier inversion of the operator (6.2.13) into (6.2.10) and (6.2.11) we obtain V = 21 λij kl (x , x)eij (x)ekl (x ) dv dv , (6.2.14) V

V

t kl,k + ρ(f l − u¨ l ) = 0, where t kl =

δV = λij kl (x, x )eij (x ) dv , δekl V eij = 21 (ui,j + uj,i ).

(6.2.15)

(6.2.16) (6.2.17)

Here δV /δekl denotes the functional partial derivative. If the material properties vary slowly over a distance d, then λij kl (x, x ) = λ0 δB (x − x), ij kl

(6.2.18)

6.2 Lattice Dynamical Foundations of Linear Elasticity

81

ij kl

where λ0 are the elastic moduli of the zeroth order with respect to d. In the limit, as the internal characteristic length a approaches zero, δB becomes a Diracdelta measure and (6.2.16) gives the classical Hooke’s law. Comparing (6.2.16) with (6.1.16)2 and setting βkl = 0 (the isothermal case) we see that we have the identical constitutive equation obtained through continuum theory. The method presented here suggests that we can determine the material functions such as λij kl by comparing the nonlocal results of calculations with those of lattice dynamics. In fact, δB (x) has been determined for some crystals. For cubic crystals, δB has the simple form

δB (x − x ) = π

−3

3 i=1

π(xi − xi ) 1 sin ,

xi − xi a

(6.2.19)

where a is the lattice parameter. Another simple case is the so-called Debye continuum, which corresponds to the case of a spherical Brillouin zone, with radius κ = π/a. For this case, we have (cf. Kunin [1983, p. 37]): √ sin(κr) κ δB (x) = (6.2.20) − cos(κr) , r = x · x, 2 2 2π r κr for the three-dimensional case, and δB (x) =

κ J1 (κr) 2πr

(6.2.21)

for the two-dimensional case. There are other cases for the isotropic media. In fact, the nonlocal kernel α(|x −x |) can also be determined as a solution of a differential equation (see Sections 6.9 and 6.13). For engineering materials, it is useful to select other kernel functions to give a better match with the dispersion curves in the neighborhood of k = 0 (long wavelength). We give a summary of the important observations: (a) In nonlocal elasticity, the total internal energy is a functional of the strain. (b) The internal energy density coincides with those of the lattice dynamics at the discrete points occupied by the atoms. (c) The stress field predicted at the atomic sites are identical in both the nonlocal elasticity and the lattice dynamics. (d) Nonlocal elastic moduli can be determined from the interatomic potential. (e) In the limit when the internal characteristic length (e.g., lattice parameter a) goes to zero, nonlocal elasticity reduces to classical (local) elasticity.

82


The development given above is for the harmonic approximation and nondissipative systems. In general, the interatomic potential is a nonlinear function of the atomic spacings. The harmonic approximation is valid only in the neighborhood of the minimum of the interatomic potential. When dislocations, holes, and impurities are present (as in the case of real materials) dissipations play a major role. In such situations, it is very difficult (if not impossible) to construct a nonlocal theory based on lattice dynamics. A continuum approach, on the other hand, is possible, as shown in Section 3.2. Thus, the nonlocal continuum theory makes sense as an independent discipline.

6.3

Material Stability

As the strains and temperature increase (decrease), it is expected that the strain energy U increases (decreases). Also the thermodynamic stability requires that the dissipation of energy must be nonnegative. Hence, we must have ρ0 C

U = 21 T T + λij kl eij ekl dv dv ≥ 0, (6.3.1) T0 V V kij

= 21 T T dv dv ≥ 0. (6.3.2) 2 ,i ,j V V T0 These conditions place restrictions on the material functions C, λij kl , and kij . Since the thermal effects are not coupled with the strains, from (6.3.1) it follows that ρ0 UT = CT T dv dv ≥ 0, (6.3.3) 2T0 V V

UE = 21 λij kl eij ekl dv dv ≥ 0. (6.3.4) V

V

Thus, we must have UT ≥ 0,

UE ≥ 0,

≥ 0,

(6.3.5)

for any arbitrary temperature, strain, and temperature gradient distributions.

6.4 Field Equations of Nonlocal Linear Elasticity For nonlocal linear elastic solids, we have the two surviving balance laws (2.1.39a) and (2.1.40a), namely tkl,k + ρ(fl − u¨ l ) = 0, −ρ ˙ + tkl vl,k + qk,k + ρh = 0,

(6.4.1) (6.4.2)

since the density is considered to be constant to a first degree approximation. Upon introducing the Helmholtz free energy ψ = −θη into (6.4.2), the energy equation becomes −ρ(ψ˙ + ηθ ˙ + ηθ˙ ) + tkl vl,k + qk,k + ρh = 0. (6.4.3)

6.4 Field Equations of Nonlocal Linear Elasticity

Since no viscous dissipation is present, by introducing η = ψ = (ρ, θ, ekl ), the energy equation (6.4.3) becomes

R η, tkl

=

−ρθ η˙ + qk,k + ρh = 0,

83

R tkl ,

and

(6.4.4)

which also follows from (3.2.28) upon using (3.2.32). The constitutive equations for homogeneous elastic solids are given by (6.1.16): T (x ) 1 η= C(x − x) + βkl (x − x)ekl (x ) dv(x ), T0 ρ V

tkl = −βkl (x − x)T (x ) + λij kl (x − x)eij (x ) dv(x ), V qk = kkl (x − x)T,l (x ) dv(x ). (6.4.5) V

We recall that the constitutive moduli C, βkl , λij kl , and kij are symmetric functions of x − x described by C(x − x) = C(x − x ), βkl (x − x) = βkl (x − x ) = βlk (x − x), λij kl (x − x) = λij kl (x − x ) = λj ikl (x − x) = λij lk (x − x), kij (x − x) = kij (x − x ) = kj i (x − x). (6.4.6) The field equations are obtained by substituting (6.4.5) into (6.4.1) and (6.4.4) and using ekl = 21 (uk,l + ul,k ). (6.4.7) This then gives ∂ (−βkl T + λij kl u i,j ) dv + ρ(fl − u¨ l ) = 0, ∂xk V ∂

˙ − (ρC T + T0 βkl u˙ k,l ) dv + kkl T,l dv + ρh = 0. ∂xk V V

(6.4.8)

We transform partial derivative of integrals in these equations by noting the identities of the type ∂ ∂

−βkl (x − x)T (x ) dv(x ) =

[βkl (x − x)]T (x ) dv(x ) ∂xk V ∂x V k ∂ ∂T (x )

= [β (x − x)T (x )] dv(x ) − β (x − x) dv(x ). kl kl

∂xk

V ∂xk V Upon using the Green–Gauss theorem in the first integral we obtain ∂

−βkl (x − x)T (x ) dv(x ) = βkl (x − x)T (x ) dak (x ) ∂xk V ∂V ∂T (x ) − βkl (x − x) dv(x ). (6.4.9) ∂xk

V

84


Using this type of identity for other terms in (6.4.8), we obtain −

(6.4.10)

kkl T,l dak

(kkl T,lk − ρC T˙ − T0 βkl u˙ k,l ) dv + ρh = 0. +

(6.4.11)

V

−

(−βkl T + λij kl u i,j ) dak

+ (−βkl T,k + λij kl u i,j k ) dv + ρ(fl − u¨ l ) = 0,

∂V

∂V

V

These are the field equations of nonlocal, homogeneous thermoelastic solids for the determination of the displacement field uk (x, t) and the temperature field T (x, t). We observe that nonlocality also occurs over the surface of the body ∂V. A mixed boundary-initial value problem requires the solution of these field equations (with fl and h given), under the boundary and initial conditions. Boundary Conditions Let V denote a regular region of the Euclidean space occupied by the body whose boundary is ∂V. The interior of V is denoted by V, and the exterior normal to ∂V by n. Let S1 to S4 denote the subsets of ∂V such that S 1 ∪ S2 = S 3 ∪ S4 = ∂V,

S1 ∩ S2 = S3 ∩ S4 = 0,

(6.4.12)

The boundary conditions of these subsurfaces, at the time interval T + = [0, ∞), may be expressed as uk = uˆ k (x, t) T = Tˆ (x, t)

on S 1 × T + , on S 3 × T + ,

tkl nk = tˆl (x, t)

on S2 × T + ,

qk nk = q(x, ˆ t)

on S4 × T + , (6.4.13)

l and where quantities carrying a caret (ˆ) are prescribed. Here tˆl = tˆkl nˆ k + F . qˆ = qˆk nk + H Initial Conditions The initial conditions usually consisting of Cauchy data, are expressed by uk (x, 0) = u0k (x),

u˙ k (x, 0) = vk0 (x),

T (x, 0) = T 0 (x) in V,

(6.4.14)

wher quantities carrying a superscript (0) are prescribed throughout V. Clearly other possibilities exist.

6.4 Field Equations of Nonlocal Linear Elasticity

85

A. Isotropic Solids For homogeneous and isotropic solids, the constitutive equations (6.4.5) reduce to (6.1.20), namely

T (x ) β(|x − x|)

η= C(|x − x|) + uk,k (x ) dv(x ), T0 ρ0 V tkl = [−β(|x − x|)T (x ) + λ(|x − x|)ui,i (x )]δkl V + µ(|x − x|)[uk,l (x ) + ul,k (x )] dv(x ), k(|x − x|)T,k (x ) dv(x ). qk =

V

(6.4.15)

The field equations (6.4.10) and (6.4.11) then become −

∂ V

[−β(|x − x|)T + λ(|x − x|)u i,i ]δkl + µ(|x − x|)(u k,l + u l,k ) dak

−β(|x − x|)T,l + [λ(|x − x| + µ (|x − x|)]u k,lk +µ(|x − x|)u l,kk dv + ρ(fl − u¨ l ) = 0,

− k(|x − x|)T,k dak + [k(|x − x|)T,kk − ρC(|x − x|)T˙

+

V

∂V

(6.4.16)

V

− T0 β(|x − x|)u˙ k,k ] dv + ρh = 0.

(6.4.17)

If we further call for the attenuating neighborhood hypothesis and employ (6.1.23), the constitutive equations take the form η= tkl = qk =

V

V V

α(|x − x|) C η(x ) dv , α(|x − x|)σkl (x ) dv , α(|x − x|) C qk (x ) dv ,

(6.4.18)

and the field equations (6.4.16) and (6.4.17) reduce to

α(|x − x|)σkl,k (x ) dv + ρ(fl − u¨ l ) = 0, −S q + α(|x − x|)(C qk,k − ρT0 C η) ˙ dv = 0,

−S tl +

V

V

(6.4.19) (6.4.20)

where C η, σkl , and C q are given by (6.1.24). Here S tl is the surface tension and S q is the surface heat arising from elimination of the symmetry of the intermolecular

86


attractions by the creation of the surface ∂V. They are given by α(|x − x|)σkl (x )nk (x ) da , S tl = ∂V α(|x − x|)T,k (x )nk (x ) da . S q = k0 ∂V

(6.4.21) (6.4.22)

We note that the kernel function α in these expressions will not be a function |x − x| but (x , x), unless the intermolecular attractions are taken into account from both inside and outside the body. These effects, however, are confined to a thin boundary layer of few atomic distances in the neighborhood of the surface ∂V.

B. Formulation by Means of Convolution Definition (Convolution). Let φ(x, t) and ψ(x, t) be in V × T + ,

{φ(x, t), ψ(x, t)} ∈ C 0,0

T + = [0, ∞),

(6.4.23)

then the function θ(x, t), defined by

t

θ(x, t) = φ ∗ ψ =

φ(x, t − τ )ψ(x, τ ) dτ,

(6.4.24)

0

is called the convolution of φ and ψ. Some elementary properties of the convolution can be deduced from its definitions: (a) Commutative property: φ ∗ ψ = ψ ∗ φ. (b) Associative property: φ ∗ (ψ ∗ ω) = (φ ∗ ψ) ∗ ω = φ ∗ ψ ∗ ω. (c) Distributive property: φ ∗ (ψ + ω) = φ ∗ ψ + φ ∗ ω. (d) Titchmarch’s theorem: φ ∗ ψ = 0 on V × T + implies that either φ = 0 or ψ = 0 on V × T + . We can show that

t

1∗φ =

φ dt, 0

˙ 0) − φ(x, 0). t ∗ φ¨ = φ − t φ(x,

(6.4.25)

Theorem. Let uk ∈ C 0,2 , tkl ∈ C 1,0 , and η ∈ C 0,1 , then uk , η, tkl , and qk satisfy the equations of motion and the initial conditions if and only if t ∗ tkl,k + Fl = ρul , −ρT0 η + 1 ∗ qk,k + R = 0,

(6.4.26) (6.4.27)

6.5 Uniqueness Theorem

87

where F = t ∗ ρf(x, t) + ρ[tv 0 (x) + u0 (x)], R = 1 ∗ ρh + ρT0 η(x, 0).

(6.4.28)

Proof . Equations (6.4.26) and (6.4.27) follow from (6.4.1) and (6.4.4). Conversely, differentiating twice (6.4.26) and once (6.4.27) with respect to t we obtain (6.4.1) and (6.4.4). The initial conditions on u and η are also included in the formulation. Theorem (Equivalence). The mixed boundary-initial value problems of nonlocal thermoelastic solids are equivalent to the system (6.4.26), (6.4.27), the constitutive equations (6.4.5), and the boundary conditions (6.4.13).

6.5 Uniqueness Theorem3 The uniqueness theorem of the linear theory of nonlocal elasticity, for the static case, was elaborated by Chirita [1976]. The dynamic case was considered by Altan [1984] and Cracium [1996]. The present proof differs from those given by these authors. The basic equations of the linear theory of nonlocal thermoelasticity for homogeneous solids are given by: Equations of Motion tkl,k + ρfl = ρ u¨ l ,

tkl = tlk .

(6.5.1)

The Energy Equation −ρT0 η˙ + qk,k + ρh = 0. Constitutive Equations 1 1

C(x − x)T (x ) + βkl (x − x)ekl (x ) dv , η= ρ V T0

tkl = −βkl (x − x)T (x ) + λij kl (x − x)ui,j (x ) dv(x ), V qk = kkl (x − x)T,l (x ) dv(x ), V

(6.5.2)

(6.5.3)

where C(x −x), βkl (x −x), λij kl (x −x), and kkl (x −x) are symmetric functions of x −x as described by (6.4.6). A mixed boundary-initial value problem is expressed by the boundary conditions (6.4.13) and by the initial conditions (6.4.14), namely: 3 Not published before.

88


Boundary Conditions uk = uˆ k (x, t) T = Tˆ (x, t)

on S 1 × T + , on S 3 × T + ,


on S2 × T + ,

qk nk = q(x, ˆ t)

on S4 × T + . (6.5.4)

Initial Conditions uk (x, 0) = u0k (x),

u˙ k (x, 0) = vk0 (x),

T (x, 0) = T 0 (x) in V,

(6.5.5)

where V is the interior of the body and S1 to S4 are partial surfaces on the surface of the body ∂V: S 1 ∪ S2 = S 3 ∪ S4 = ∂V,

S1 ∩ S2 = S3 ∩ S4 = 0.

(6.5.6)

The following continuity requirements are assumed: uk (x, t) ∈ C 1,2 , {tkl (x, t), qk (x, t)} ∈ C 1,0 , T (x, t) ∈ C 1,0 in V × T + , {uˆ k (x, t), tˆkl (x, t), Tˆ (x, t), q(x, ˆ t)} ∈ C 0,0 {u0k (x), vk0 (x), T 0 (x)} ∈ C 0 ,

(6.5.7)

on ∂V × T + , (6.5.8)

where T + = [0, ∞) and C i,j denotes the continuous ith partial derivatives with respect to x and the continuous j th partial derivative with respect to time t. We now establish some results that are useful. We define 1 K(s, τ ) = 2 ρ u˙ k (s)u˙ k (τ ) dv, V ρC

1 U(s, τ ) = 2 T (s)T (τ ) + λij kl eij (s)ekl (τ ) dv dv, V V T0 ρh(τ ) ˙ )+ P (s, τ ) = ρf(s) · u(τ T (s) dv T0 V 1 ˙ ) + q(τ )T (s) da. + t(s) · u(τ (6.5.9) T0 ∂V We note that K(t, t) ≡ K(t) is the total kinetic energy, U(t, t) ≡ U(t) is the total strain energy, and P (t, t) ≡ P (t) is the power of the applied body and surface loads. Theorem 1. U(t) − K(t) =

1 2

t

[P (t + s, t − s) − P (t − s, t + s] ds

0

+ U(0, 2t) − K(0, 2t).

(6.5.10)


Proof . We introduce the notation E(s, τ ) = [tkl (s)e˙kl (τ ) + ρT (s)η(τ ˙ )] dv. V

89

(6.5.11)

From (6.5.9) and (6.5.3) it can be verified that ˙ E(t, t) ≡ E(t) = U.

(6.5.12)

Using (6.5.11), we can show that E(t − s, t + s) − E(t + s, t − s) = 2

d U(t − s, t + s). ds

(6.5.13)

By means of the equations of motion (6.5.1), (6.5.2), and 2ekl = uk,l + ul,k , we evaluate 1 E(t − s, t + s) = tkl (t − s)u˙ l,k (t + s) + qk,k (t + s)T (t − s) T0 V ρ + h(t + s)T (t − s) dv, (6.5.14) T0 using the identity fg,l = (fg),l − gf,l , and the Green–Gauss theorem (6.5.14) is transformed to 1 E(t − s, t + s) = tk (t − s)u˙ k (t + s) + T (t − s)q(t + s) da T0 ∂V 1 − tkl,k (t − s)u˙ l (t + s) + T,k (t − s)qk (t + s) T 0 V ρ + T (t − s)h(t + s) dv. T0 Upon substituting tkl,k and qk from (6.5.3), this becomes ρ u¨ k (t − s)u˙ k (t + s) dv E(t − s, t + s) = P (t − s, t + s) − V 1 − kkl T,k (t − s)T,l (t + s) dv dv. (6.5.15) T0 V V By replacing s by −s, we obtain the expression of E(t + s, t − s). Substituting these two expressions into (6.5.13) we obtain 2

d [U(t −s, t +s)−K(t −s, t +s)] = P (t −s, t +s)−P (t +s, t −s). (6.5.16) ds

90


Integration of this relation, from 0 to τ , gives 2U(t − τ, t + τ ) − 2K(t − τ, t + τ ) − 2U(t) + 2K(t) τ = [P (t − s, t + s) − P (t + s, t − s)] ds. (6.5.17) 0

Setting t = τ leads to the desired result (6.5.10). Lemma 1. The functionals U(t) and K(t) are given by 2U(t) = U(0) + K(0) + U(0, 2t) − K(0, 2t) t 1 [P (t + s, t − s) − P (t − s, t + s) + 2P (s, s)] ds +2 0 t 1 − ds kkl T,k (s)T,l (s) dv dv, (6.5.18) T0 0 V V 2K(t) = U(0) + K(0) − U(0, 2t) − K(0, 2t) t 1 −2 [P (t + s, t − s) − P (t − s, t + s) − 2P (s, s)] ds 0 t 1 − ds kkl T,k (s)T,l (s) dv dv, (6.5.19) T0 0 V V for all t ∈ T + [0, ∞). Proof . Recalling (6.5.12) and setting s = 0 in (6.5.15) we obtain 1 U˙ + K˙ = P (t) − kkl T,k T,l dv dv. T0 V V Integration gives t P (s, s) ds U + K = U(0) + K(0) + 0 t 1 − ds kkl T,k (s)T,l (s) dv dv. T0 0 V V

(6.5.20)

Using this equation and (6.5.10) we obtain (6.5.17) and (6.5.18), thus completing the proof. Theorem (Uniqueness). If ρ is strictly positive, kkl is positive semidefinite, and C is strictly positive (or negative), then the boundary-initial value problems of nonlocal linear thermoelasticity have at most one solution. Proof . Suppose that the contrary is valid, and two solutions u(α) , T (α) , α = 1, 2, exist. Let T = T (1) − T (2) . u = u(1) − u(2) ,

6.6 Power and Energy

91

Then clearly u and T satisfy (6.5.1) to (6.5.5), with f = h = uˆ = tˆ = Tˆ = qˆ = u0 = v0 = T 0 = 0. This means that we have homogeneous equations and boundary and initial conditions. With these, (6.5.19) reduces to V

ρ u˙ k u˙ k dv +

1 T0

t

ds 0

V

V

kkl T,k T,l dv dv = 0.

(6.5.21)

By the hypothesis of the theorem then

ds 0

u˙ = 0

t

V

V

in V × T + ,

kkl T,k T,l dv dv = 0,

t ∈ T +.

(6.5.22)

Since initially u vanishes we have u = 0.

(6.5.23)

In view of this and (6.5.22)2 , (6.5.18) gives

ρ CT T dv dv = 0. T V V 0

This implies that T = 0 in V × T + . Hence the proof of the theorem.

(6.5.24)

6.6 Power and Energy Definitions. The ordered set of functions S{uk , ekl , tkl , qk , T }, that satisfy the continuity requirements (6.5.7) and (6.5.8), is called an admissible thermoelastic state. S is considered to be a linear function space, i.e., S1 + S2 = {u1 + u2 , e1 + e2 , t1 + t2 , q1 + q2 , T 1 + T 2 }, αS = {αu, αe, αt, αq, αT }.

(6.6.1)

An admissible state may be restricted by imposing additional requirements. For example, if S meets the constitutive equations (6.5.3) and the strain displacement relations ekl = (uk,l + ul,k )/2, we call this kinematically admissible. If kinematically admissible states satisfy equations of motion (6.5.1) and (6.5.2) and the boundary and initial conditions (6.5.4) and (6.5.5) this is called the solution of the mixed problems.

92


Theorem (Power and Energy). If S{uk , ekl , tkl , qk , T } is a solution of the nonlocal thermoelasticity, then4 h 1 tˆl vl0 da + ρ f · u˙ + T dv + qˆ Tˆ da T0 T0 ∂ V V ∂V = ρ u˙ · u¨ dv V CT T 1 + + λij kl u i,j u˙ k,l + kkl T,l T,k dv dv. (6.6.2) T0 T0 V V Proof . We multiply (6.5.1) by u˙ l and integrate over V: tkl,k u˙ l dv + ρfl u˙ l dv = ρ u¨ l u˙ l dv. V

V

V

(6.6.3)

Using the identity tkl,k u˙ l = (tkl u˙ l ),k − tkl u˙ l,k , and the Green–Gauss theorem, we express the first term in (6.6.3): tkl,k u˙ l dv = tkl e˙kl dv, tˆk vk0 da − V

∂V

V

(6.6.4)

where we also used (6.5.4) and (6.5.5). Carrying tkl from the constitutive equations, the second term on the right-hand side becomes tkl e˙kl dv = (βkl T u˙ k,l − λij kl u i,j u˙ k,l ) dv dv. (6.6.5) − V

V

V

But from the energy equations (6.5.2) and (6.5.3)1 , we have 1 1 βkl T u˙ k,l dv dv = − CT T˙ dv dv + qk,k T dv T0 V V T0 V V V 1 + ρhT dv. (6.6.6) T0 V With the use of the Green–Gauss theorem we have qk,k T dv = [(qk T ),k − qk T,k ] dv V V = qˆ Tˆ da − qk T,k dv. ∂V

V

Consequently (6.6.6) becomes 1

βkl T u˙ k,l dv dv = − (CT T˙ + kkl T,k T,l ) dv dv T0 V V V V 1 1 + ρhT dv + qˆ Tˆ da. (6.6.7) T0 V T0 ∂ V 4 For an isotropic body, see Altan [1990].

6.7 Reciprocal Theorem

93

Carrying this into (6.6.5), we have C ˙ 1 − tkl e˙kl dv = − T T + λij kl u i,j u˙ k,l + kkl T,k T,l dv dv T0 V V V T0 1 1 + ρhT dv + qˆ Tˆ da. (6.6.8) T0 V T0 ∂ V When we substitute this into (6.6.4) and then the result to (6.6.3) we obtain (6.6.2) which completes the proof.

6.7 Reciprocal Theorem The reciprocal theorem was considered for the isothermal case by Ie¸san [1977] and for the isotropic case by Altan [1990]. Here we present this theorem in the style of the reciprocal theorem of the local case (of Eringen [1998, Section 5.8]). Consider the two external systems L(α) = {f (α) , h(α) , tˆ(α) , qˆ (α) , u0(α) , v0(α) , η0(α) }, S (α) = {u(α) , T (α) , t(α) , q(α) , η(α) },

(6.7.1)

of which S (α) is the solution of the nonlocal thermoelasticity corresponding to L(α) . Lemma. Let

1 1 ∗ q (α) (τ ) ∗ T (β) (s) da T0 ∂V 1 (β) (β) (α) (α) + Fk (τ ) ∗ uk (s) + t ∗ 1 ∗ qk (τ )T,k (s) T0 V 1 (β) (α) − t ∗ R (α) ∗ T (β) − ρuk (τ ) ∗ uk (s) dv, T0 α, β = 1, 2, τ, s ∈ T + , (6.7.2)

Wαβ (τ, s) = t ∗

(α)

(β)

tl (τ ) ∗ ul (s) −

where Fk and R are given by (6.4.28) and (∗) denotes the convolution product (see Section 6.4). We will prove that Wαβ (τ, s) = Wβα (s, τ ).

(6.7.3)

Proof . By means of the Green–Gauss theorem, we convert surface integrals to volume integrals (β) (α) (α) (α) (α) (β) Wαβ (τ, s) = [(t ∗ tkl,k + Fl − ρul ) ∗ ul ] dv + t ∗ tkl ul,k V V 1 1 (α) (β) (α) (β) dv. (6.7.4) − t ∗ 1 ∗ qk,k ∗ T − t ∗R ∗T T T0

94


The first integral vanishes, on account of the equation of motion (6.4.26). Replacing (α) 1 + qk,k by its expressions (6.4.27) we obtain (β) (α) Wαβ (τ, s) = t ∗ (tkl ∗ ekl − ρη(α) ∗ T (β) ) dv. (6.7.5) V

Substituting for

(α) tkl

and

η(α)

from the constitutive equation (6.4.5) we have (β)

(α) (β) −βkl T (α) ekl + λij kl eij ekl Wαβ (τ, s) = t ∗ V V ρ

(α)

(α) (β) (β) dv. (6.7.6) − βkl ekl ∗ T − CT ∗T T0

Hence the proof of the lemma. Expression (6.7.2) may be decomposed into Wαβ (τ, s) = Rαβ (τ, s) + Sαβ (τ, s), where

(β) (α) tl (τ ) ∗ ul (s) Rαβ (τ, s) = Rβα (s, τ ) = t ∗ ∂V 1 (α) (β) − 1 ∗ q (τ ) ∗ T (s) da T0 1 (β) (α) (α) (β) + Fk (τ ) ∗ uk (s) − t ∗ R (τ ) ∗ T (s) dv, T0 V Sαβ (τ, s) = Sβα (s, τ ) 1 (β) (β) (α) (α) = 1 ∗ t ∗ qk (τ ) ∗ T,k (s) − ρuk (τ ) ∗ uk (s) dv. V T0

(6.7.7)

(6.7.8)

(6.7.9)

The symmetry of Sαβ may be seen by carrying the expression qk from (6.5.3). Theorem (Reciprocal). If the S (α) , α = 1, 2, is a solution of the linear nonlocal thermoelasticity systems, then 1 (1) (2) (1) (2) tl ∗ ul − 1 ∗ q ∗ T da t ∗ T0 ∂V 1 (1) (2) (1) (2) + Fk ∗ uk − t ∗ R ∗ T dv T0 V 1 (2) (1) =t∗ tl ∗ ul − 1 ∗ q (2) ∗ T (1) da T 0 ∂V 1 (2) (1) (2) (1) + Fk ∗ uk − t ∗ R ∗ T dv. (6.7.10) T0 V The proof follows from (6.7.8) by setting τ = s = t.

6.8 Variational Principles

95

6.8 Variational Principles5 A variational principle for the isothermal case was given by Ie¸san [1977], and for the isotropic solids by Altan [1990]. Here we give the most general case of the homogeneous and anisotropic nonlocal thermoelastic case, in a treatment similar to the local case presented in Eringen [1998]. Theorem. Let K be the set of all admissible states and let S ∈ K. Define the functional t (S) on K, for each t ∈ T + , by

1

λklmn t ∗ emn ekl dv dv + 2 ρuk ∗ uk − t ∗ tkl ∗ ekl V V V 1 − (t ∗ tkl,k + Fl ) ∗ ul dv − kkl 1 ∗ t ∗ gl ∗ gk dv dv 2T0 V V ρ

− t ∗ βkl ekl ∗ T dv dv − Ct ∗ T ∗ T dv dv 2T 0 V V V V 1 + t ∗ [1 ∗ qk ∗ gk + (R + 1 ∗ qk,k ) ∗ T ] dv T0 V + t ∗ tk ∗ uˆ k da + t ∗ (tk − tˆk ) ∗ uk da S1 S2 1 1 ˆ − t ∗ 1 ∗ q ∗ T da + t ∗ 1 ∗ (q − q) ˆ ∗ T da, (6.8.1) T0 S3 T0 S4

(t) (S) =

1 2

where S1 ∪ S2 = S3 ∪ S4 = ∂V, S1 ∩ S2 = S3 ∩ S4 = 0, and q = qk nk , tk = tlk nl , R = 1 ∗ ρh + ρT0 η(x, 0).

Fk = t ∗ ρfk + ρ(tvk0 + u0k ), (6.8.2)

Then t (S) is stationary, i.e., for every S + S ∈ K: δt (S) ≡

d = 0 on K, t (S + S) d =0

t ∈ T+

(6.8.3)

if and only if S is a solution of the mixed problem. Proof . Let S{u, T , e, t, q, g} ∈ K be an arbitrary variation in the state S. The variation of (6.8.1) reads 5 Not published before.

96


δt (S) =

V

V

λklmn t ∗ emn ∗ ekl dv dv +

V

[ρuk ∗ uk − t ∗ (tkl ∗ ekl

− t kl ∗ ekl ) − (t ∗ tkl,k + Fl ) ∗ ul − t ∗ t kl,k ∗ ul ] dv 1 − kkl 1 ∗ t ∗ gl ∗ g k dv dv T0 V V

− t ∗ βkl (ekl ∗ T + ekl ∗ T ) dv dv V V ρ 1

− Ct ∗ T ∗ T dv dv + t ∗ [1 ∗ (q k ∗ gk + qk ∗ g k ) T0 V V T0 V + (R + 1 ∗ qk,k ) ∗ T + 1 ∗ q k,k ∗ T ] dv + t ∗ t k ∗ uˆ k da + t ∗ [(tk − tˆk ) ∗ uk + t k ∗ uk ] da S1 S2 1 − t ∗ 1 ∗ q ∗ Tˆ da T0 S3 1 + [t ∗ 1 ∗ [(q − q) ˆ ∗ T + q ∗ T ] da = 0. T0 S4 By means of the Green–Gauss theorem we have t ∗ ul ∗ t kl,k dv = t ∗ [(ul ∗ t kl ),k − ul,k ∗ t kl ] dv V V t ∗ ul ∗ t l da − ul,k ∗ t kl dv. = ∂V

V

(6.8.4)

(6.8.5)

Using these, and collecting the coefficients of u, T , . . . we arrive at

(λklmn emn − βkl T ) dv − tkl ∗ ekl dv δt (S) = t ∗ V V − (t ∗ tkl,k + Fl − ρul ) ∗ ul dv V

−t∗ ekl − 21 (uk,l + ul,k ) ∗ t kl dv V 1 1 + ρCT

∗t ∗ −kkl gl dv + qk ∗ g k dv + t ∗ T0 T0 V V V V 1

+ −T0 βkl ekl dv + (R + 1 ∗ qk,k ∗ T dv + t ∗ 1 ∗ (gk T0 V V − T,k ) ∗ q k dv + t ∗ (uˆ k − uk ) ∗ t k da + t ∗ (tk − tˆk ) ∗ uk da S1 S 2 t t + ∗1∗ (T − Tˆ ) ∗ q da + ∗1∗ (q − q) ˆ ∗ T da. (6.8.6) T0 T 0 S3 S4


97

In view of (6.4.5), (6.4.7), (6.5.4), (6.5.5), (6.4.26) and (6.4.27), it is clear that if S is a solution of the mixed problem, then δt (S) = 0. This proves the sufficiency condition. The proof of necessity requires a subtler approach, namely: we must stipulate that the members of S can be selected completely arbitrarily and independently from one another. Then δt (S) requires that coefficients of the members of S must vanish separately. This completes the proof. This theorem constitutes the most general variational principles for the mixed boundary-initial value problems in that the admissible states meet none of the relations satisfied by the field variables, except some smoothness and symmetry regulations. By restricting the set of admissible states to a smaller set one can construct other variational principles of less complexity. Definition. An admissible state S is called kinematically admissible, if its meets the strain-displacement relations (6.4.7), the constitutive relations (6.4.5), and the boundary conditions (6.4.13) on uk and T . Theorem. Let U be the set of all kinematically admissible fields and let u, T ∈ U and for each t ∈ T + define a functional (U ) by (U ) = 21 {t ∗ [tkl ∗ ekl + ρ(η − η0 ) ∗ T ] + ρuk ∗ uk − 2Fk ∗ uk } dv V 1 1 + t ∗ −ρ(η − η0 ) ∗ T + 1 ∗ ρh ∗ T − 1 ∗ qk ∗ T,k dv T0 2T0 V 1 − t ∗ tˆk ∗ uk da + t ∗ 1 ∗ T ∗ qˆ da, (6.8.7) T0 S4 S2 where η0 = η(x, 0). Then δ(U ) = 0

on U,

t ∈ T +,

(6.8.8)

if and only if U is a field corresponding to the solution of the mixed problem. Proof . Because of the fact that the constitutive equations are satisfied, the variation of gives δ(U ) = − (t ∗ tkl,k + Fl − ρul ) ∗ ul dv V 1 +t ∗ −ρ(η − η0 ) + 1 ∗ (ρh + qk,k ) ∗ T dv T0 V 1 + (tk − tˆk ) ∗ uk da + t ∗ 1 ∗ T ∗ (qˆ − q) da. (6.8.9) T0 S2 S4 From (6.4.26), (6.4.27), and the boundary conditions, it is clear that if a kinematically admissible field is a solution of the mixed problem, then (6.8.8) is fulfilled.

98


Conversely, for arbitrary and independent variations of u and T , the equations of motion and boundary conditions are satisfied leading to (6.8.8). Thus, (6.8.8) constitutes both the necessary and sufficient conditions for the solution of the mixed problem.

6.9 Approximate Models A. Slowly Varying Fields If the fields are varying slowly over a characteristic distance d, we can replace the integral operators with differential operators. This simply means that we can expand the Fourier transforms of the constitutive moduli C, λij kl , βij , and kij into power series of the wave number ki , e.g., √ λij kl (k) = λij klτ1 ···τr (ik1 )(ik2 ) · · · (ikr ), i = −1, (6.9.1) with λij klτ1 ···τr being real constants, and the stress constitutive equations, such as λij kl (x − x)eij (x ) dv(x ), tkl = V

gives a couple stress theory involving r derivatives of eij .

B. Matching a Dispersion Curve with Atomic Models A more useful case involves matching the Fourier transforms of constitutive moduli (such as λij kl ) in the wave number space with the dispersion curves based on the atomic models. For example, according to the Born–Kármán lattice model 2 ωj2 (k)/ω0j = (2κ/πk) sin2 (πk/2κ),

(6.9.2)

where 2 ω0j = k 2 cj2 ,

c12 = (λ0 + 2µ0 )/ρ,

c22 = µ0 /ρ.

(6.9.3)

Here ω01 and ω02 are, respectively, the classical circular frequencies of the irrotational and equivoluminal waves. Here κ is the upper limit of k (the boundary of the Brillouin zone), e.g., for the one-dimensional lattice κ = π/a, where a is the lattice parameter. Thus, we see the introduction of an internal characteristic length a. For an infinite isothermal solid with fl = 0, taking the Fourier transform of (6.4.19) and substituting the Fourier transform of σkl , given by (6.1.24), we obtain ωj2 (k)/cj2 = k 2 α(k),

j = 1, 2,

(6.9.4)

where α is the Fourier transform of α(|x − x|) and c1 and c2 are, respectively, the phase velocities of the irrotational and equivoluminal waves in classical elasticity. A polynomial form is most convenient for the approximation of (6.9.4): 1/α = 1 + βi (k/κ)2 + δj (k/κ)4 .

(6.9.5)

6.9 Approximate Models

99

At k = 0 this gives the classical frequency, and at k = κ it will satisfy the condition dω/dk = 0, with δj = 1. The latter condition implies that, at the end of the Brillouin zone, the phase velocity vanishes (no wave progagation)—a well-known result in lattice dynamics. Hence we have α = [1 + βj (k/κ)2 + (k/κ)4 ]−1 , k ω/ωjd = [1 + βj (k/κ)2 + (k/κ)4 ]−1/2 , κ

(6.9.6) (6.9.7)

where ωjd = κcj is the Debye frequency. The parameter βj is related to the ratio of the boundary frequency ωj (κ) and to the Debye frequency ωjd : ωj (κ)/ωjd = (2 + βj )1/2 .

(6.9.8)

−2 < βj < ∞.

(6.9.9)

From this it follows that In Figure 6.9.1, we display the dispersion curves based on (6.9.7), along with the one based on the Born–Kármán model (see also Kunin [1983, p. 39]). For engineering purposes, perhaps it may be more practical to leave βj and δj free for better curve-fitting at low wave numbers relevant to the macroscopic problems. If we set α = (1 + 2 k 2 + γ 4 k 4 )−1 , (6.9.10) then the nonlocal stress constitutive equation (6.1.26)2 , for the infinite media, gives (1 + 2 k 2 + γ 4 k 4 )t kl = σ kl ,

(6.9.11)

where t kl and σ kl are the Fourier transforms of tkl and σkl . The inverse Fourier transform of (6.9.11) is (1 − 2 ∇ 2 + γ 4 ∇ 4 )tkl = σkl .

(6.9.12) βj = –1

0

ωj

Born-Kármán βj = –1

ωjd

5

k/x

Figure 6.9.1. Dispersion curves.

100


In terms of (6.1.26)2 , this implies that (1 − 2 ∇ 2 + γ 4 ∇ 4 )α(|x − x|) = δ(|x − x|).

(6.9.13)

If we take the divergence of (6.9.12) and use (6.4.1), we obtain σkl,k + (1 − 2 ∇ 2 + γ 4 ∇ 4 )(ρfl − ρ u¨ l ) = 0,

(6.9.14)

or using (6.1.24)2 with β0 = 0 and (6.4.7), we have the field equations (λ + µ)uk,lk + µul,kk + (1 − 2 ∇ 2 + γ 4 ∇ 4 )(ρfl − ρ u¨ l ) = 0.

(6.9.15)

These equations replace the Navier equations of classical elasticity, that are singularly perturbed. We note that, for static problems and vanishing body forces, (6.9.14) reduces to the equations of equilibrium in classical elasticity and (6.9.15) reduces the Navier equations. However, since the real stress is not σkl but tkl , in order to determine the stress fields, we must invert (6.9.11). Alternatively, when the Hookean stress σkl is known, we can use (6.9.12) to determine the stress field. An expression of the Laplacian of the stress tensor tkl , in curvilinear coordinates, is given in the Appendix. Thus, we can integrate (6.9.12), or a simpler equation (1 − 2 ∇ 2 )t = σ (6.9.16) to obtain the stress tensor t in the curvilinear coordinate (cf. Section 6.18).

C. Linear Chains Consider a one-dimensional linear chain of atoms lying along an x-axis. Equally spaced atoms are attached to each other by springs of spring constant K (Figure 6.9.2). The differential equation of motion for the nth atom is M u¨ n = K(un+1 + un−1 − 2un ),

(6.9.17)

where K is related to the potential energy U (R). If U (R0 ) is the potential energy between two atoms at their equilibrium position, then the spring constant K is given by 2 d U K= . dR 2 R=R0 M

Un –1

K

Un

Figure 6.9.2. Linear chain.

Un +1


101

ωa/c1

2

ka

Figure 6.9.3. Dispersion curve in one Brillouin zone. Equation (6.9.17) may be solved by introducing un = u exp(inka − iωt),

(6.9.18)

leading to the dispersion relations 4 ω2 = 2 sin2 2 a c1

ka 2

,

−

π π ≤k≤ , a a

(6.9.19)

√ where c1 = a K/M. Within one Brillouin zone (6.9.19) is displayed in Figure 6.9.3. Accordingly, the nonlocal kernel α is given by (Born–Kármán model): 4 2 ka α(k) = 2 2 sin . (6.9.20) k a 2 The inverse Fourier transform of this gives  |x − x| 1   1− ,  a a

α(|x − x|) =     0,

|x − x| ≤ 1, a |x − x| > 1. a

(6.9.21)

This function is displayed in Figure 6.9.4. Here we found a triangular kernel with a finite support.6 6 See also Eringen [1972a], [1974c].

102

6 Nonlocal Linear Elasticity αa

x/a

Figure 6.9.4. Nonlocal elastic modulus (one-dimensional chain). For more complicated linear chains, (6.6.20) is replaced by7 α(k) =

N n2 sin2 (nka/2) 1 Kn , (λ + 2µ)a (nka/2)2

(6.9.22)

n=1

where N is the number of atoms in the cohesion zone and Kn are the force constants. The inversion of this gives  N   |x − x| n 1  1 − K , |x − x| ≤ na, n 2 a na α(|x − x|) = λ + 2µ n=1    0, |x − x| > na. (6.9.23) For isotropic solids this model may be generalized to the two- and three-dimensional case by 

n  A 1 − |x − x| , n = 2, 3, |x − x| ≤ a, an α(|x − x|) = (6.9.24) 

0 |x − x| > a. cf. Artan [1996]. The constant A is obtained by the normalization of α(|x − x|), i.e., V

α(|x|) dv = 1.

This gives A = 2/πa 2 A = 3/2πa 3 7 Nowinski [1990].

in two-dimensions, in three-dimensions.

(6.9.25)


103

D. Other Kernels Clearly many other kernels can be created by matching lattice dynamical or experimental dispersion curves with the nonlocal results. In fact, continuous kernels that are of a Dirac-delta sequence offer simpler calculations. In the early days, one such kernel, used by Eringen in his calculations, was 2 k (6.9.26) α(|x − x|) = α0 exp − 2 |x − x|2 , where α0 and k are constants, being the constant internal characteristic length. Expression (6.9.26) is normalized by α(|x|) dv = 1, (6.9.27) V

giving

α0 = π −n/2 (k/)n

(6.9.28)

for n dimensions (n = 1, 2, 3). Clearly (6.9.26) is a delta sequence since → 0 gives a Dirac-delta distribution. The constant k can be fixed by a suitable curve fitting with the lattice dynamical results. For example, for k = 1.65, the dispersion curve obtained from (6.9.26) for plane waves in infinite elastic media is indistinguishable from that of the onedimensional Born–Kármán model. As we shall see when we come to the discussion of crack and dislocation problems, the brittle fracture criteria is another way of determining the constant k. This method is valid for static deformations.

E. α(|x − x|) is a Green Function of a Linear Differential Operator By matching the dispersion curves of plane waves with those of lattice dynamics we obtained the differential equation (6.9.13). This equation is in the form Lα(|x − x|) = δ(|x − x|),

(6.9.29)

where L is the differential operator 1 − 2 ∇ 2 + γ 4 ∇ 4 . However, other differential operators may also be admissible. If such an operator can be found representing the state of the body faithfully, then applying L to (6.1.26)2 , we obtain Ltkl = σkl .

(6.9.30)

In particular, if L is a differential operator with constant coefficients, then (Ltkl ),k = Ltkl,k and (6.4.1) gives (Eringen [1983]): σkl,k + L(ρfl − ρ u¨ l ) = 0.

(6.9.31)

104


In this case, we have partial differential equations to solve, instead of integropartial differential equations. If ρf −ρ u¨ = 0, as in the static case, then the classical equation of equilibrium is to be satisfied σkl,k = 0.

(6.9.32)

This then leads to the Navier equations of classical elasticity for the displacement field (λ + µ)uk,lk + µul,kk = 0. (6.9.33) Once the solution for u is obtained then one can determine the stress field by solving the differential equations (6.9.30). As an extension of the differential operator, we consider a parabolic differential operator (diffusion-type), subject to the initial conditions, e.g., ∂tkl ∂ tkl = 0, tkl (x, 0) = σkl (x). (6.9.34) −P ∂τ ∂xr This is a Cauchy problem for the stress tensor with, initially, the stress tensor being equal to the local stress tensor in rectangular coordinates. The nonlocal stress tensor tkl is considered to be a function of x and a parameter τ . This parameter represents the internal characteristic length . Consequently, for the vanishing internal characteristic length, the stress tensor is taken to be the Hookean (classical) stress tensor. The Green function α(x, τ ) for the Cauchy problem (6.9.34) is the solution of ∂α ∂ α = 0, α(x, 0) = δ(x). (6.9.35) −P ∂τ ∂xr The solution of the Cauchy problem (6.9.34) is then given by tkl = α(x, τ ) ∗ σkl (x),

(6.9.36)

where the asterisk (∗) denotes the convolution. Alternatively, applying the Laplace transform to (6.9.35) we have ∂ st kl − P t kl = σkl (x). (6.9.37) ∂xr When the solution of this equation is obtained for tkl under the appropriate boundary conditions, by obtaining the inverse Laplace transform of t kl , we obtain the nonlocal stress tensor in terms of the local stress tensor. Thus, either way, we have a solution for tkl given by the inversion of (6.9.30), i.e., −1

tkl = L σkl .

(6.9.38)

We demonstrate this procedure by a simple example in one dimension 1 α(x, τ ) = √ exp(−x 2 /4τ ) 2 πτ

(6.9.39)


105

is the Green functions of the heat equation, i.e., the solution of the initial value problem ∂α ∂ 2α α(x, 0) = δ(x). (6.9.40) − 2 = 0, ∂t ∂x The solution of the Cauchy problem for tkl is given by (6.9.36), i.e., ∞ 1 (x − ξ )2 α(ξ ) dξ. (6.9.41) exp − tkl = α ∗ σkl = √ 4τ 2 πτ −∞ Similarly, 1 sin(νx) , 0 < ν < ∞, (6.9.42) π x is the delta sequence in one dimension arising from the differential equation α(x, ν) =

∂ ∂α (xα) − ν = 0, α(x, 0) = δ(x). (6.9.43) ∂x ∂ν Here we have a variable coefficient differential operator. The solution for the Green function is (6.9.42). This function is illustrated in Figure 6.9.5. For the two-dimensional case, the Green function of the operator L = 1 − 2 ∇ 2 leads to

α(|x|) = (2π2 )−1 K0 (|x|/),

(6.9.44) 0 < < ∞.

(6.9.45)

where K0 (z) is the Bessel function of imaginary argument. This is also a Dirac-delta sequence. Thus, we have many possibilities. The question arises, what is rational for the selection of the most appropriate kernel? In quantum mechanics this selection is made through the wave function which satisfies the Schrödinger equation. Here our only guidance seems to be with the minimum number of parameters (such as ) to obtain a close fit with the lattice dynamical and/or experimental dispersion curves. As we shall see through various solutions, in fact, even with one parameter matching, excellent agreements are obtained with the predictions of experiments and lattice dynamics.

Figure 6.9.5. α(x, ν) =

1 sin(νx) π x .

106


Figure 6.10.1. Screw dislocation.

6.10 Screw Dislocation8 A screw dislocation is envisioned as a constant discontinuity in the antiplane displacement component, at a point x2 = 0. This is obtained by cutting a solid along the plane x2 = 0, x1 ≥ 0 and introducing a constant displacement discontinuity u3 = b, where b is called the Burger’s vector (Figure 6.10.1). For the case of antiplane strain u1 = u2 = 0 and u3 = w(x1 , x2 ), the equation of equilibrium tkl,k = 0 (6.10.1) is satisfied if we take t13 = φ,2 ,

t23 = −φ,1 .

(6.10.2)

Equations (6.9.30), with the kernel (6.9.44), take the form (1 − 2 ∇ 2 )φ,2 = σ13 ,

(1 − 2 ∇ 2 )φ,1 = −σ23 .

(6.10.3)

σ23 = µu3,2 ,

(6.10.4)

Hooke’s law reads σ13 = µu3,1 ,

which must satisfy the compatibility condition σ13,2 = σ23,1 .

(6.10.5)

(1 − 2 ∇ 2 )∇ 2 φ = 0.

(6.10.6)

Using (6.10.3) this gives An appropriate solution of (6.10.6), which makes the stress vanish as r = (x12 + x22 )1/2 → ∞, is φ = φ0 + C1 ln r + C2 K0 (r/ l), 8 Eringen [1977b], see also [1983a], [1987a], [1987b], [1990a].

(6.10.7)

6.10 Screw Dislocation

107

where K0 (z) is the Bessel function of the imaginary argument and (r, θ) are the plane polar coordinates x1 = r cos θ,

x2 = r sin θ.

(6.10.8)

In these coordinates we have tzr = t31 cos θ + t32 sin θ, tzθ = −t31 sin θ + t32 cos θ,

(6.10.9)

and

1 ∂φ ∂φ , tzθ = − . (6.10.10) r ∂θ ∂r The jump discontinuity u3 (2π) − u3 (0) = −b gives C1 = −b/2π. At the eye of the dislocation r = 0, tzθ vanishes if C2 = −µb/2π. Thus, we have the solution of the problem tzr =

µb φ = C0 − [ln(r/ l) + K0 (r/ l)], 2π r µb tzθ = 1 − K1 (r/ l) , tzr = 0. 2πr l The strain energy per unit length is given by R U µb2 1 t23 (x1 , 0)b dx1 = =2 [ln(R/r0 ) L 4π r0 + K0 (R/ l) − K0 (r0 / l)].

(6.10.11) (6.10.12)

(6.10.13)

From (6.10.12) and (6.10.13) we see that neither the stress nor the strain energy display singularities at the eye of the dislocation r = r0 = 0. As expected, for large R, the strain energy depends on the size of the solid. In Figure 6.10.2, we give a plot of the nondimensional stress T = (2π/b)tzθ /µ = ρ −1 [1 − ρK1 (ρ)],

ρ ≡ r/ l.

(6.10.14)

The maximum shear stress occurs at ρ = 1.1: tzθ max /µ ∼ = 0.3993

b . 2πl

(6.10.15)

By use of this we can determine the cohesive strength tc of a perfect crystal by 0.39a where a is the lattice equating tzθ max to tc . Based on phonon dispersion, √ parameter. For fcc materials and b/a = 1/ 2, (6.10.15) gives tc /µ = 0.12.

(6.10.16)

This result compares well with the value 0.11 for Al(fcc); W , α − Fe(bcc), and 0.12 for Nacl and MgO (cf. Lawn and Wilshaw [1975, p. 160]).

108


Figure 6.10.2. Nondimensional shear stress in screw dislocation. After Eringen [1983].

If the crystal is ductile near the eye of the dislocation at tc > taθ max , new dislocations will be produced. Thus, the region 0 ≤ ρ < ρc is a dislocation free zone, i.e., dislocations will emerge at ρ = ρc and will pile up in a region ρ ≥ ρc . This prediction of the theory is supported by observations made on electron microscopy (cf. Ohr and Chang [1982]). According to the theory, the rupture or dislocation initiation does not begin at the eye (ρ = 0) of the dislocation but at ρ = ρc > 0. This is against our previous understanding of dislocation and fracture mechanism but is supported by experiment.


109

A. Distribution of Dislocations Beside the unimportant constant C0 (which can be dropped) φ is the Green’s function for the differential equation (6.10.6). For an arbitrary distribution of the Burger’s vector b(ξξ ) over an area S the stress field can be calculated by superposition ∂G(|x − ξ |) b(ξξ ) da(ξξ ), t13 = µ ∂x2 S ∂G(|x − ξ |) b(ξ ) da(ξξ ), (6.10.17) t23 = −µ ∂x1 S where G(|x − ξ |) = −

1 [ln(|x − ξ |/ l) + K0 (|x − ξ |/ l)]. 2π

(6.10.18)

For a line distribution the stress field is given by

∂G(|x − ξ |) b(ξξ ) ds(ξξ ), ∂x2 C ∂G(|x − ξ |) = −µ b(ξξ ) ds(ξξ ). ∂x1 C

t13 = µ t23

(6.10.19)

In polar coordinates these read

1 ∂G b(ξξ ) ds(ξξ ), C r ∂θ ∂G = −µ b(ξξ ) ds(ξξ ). C ∂r

tzr = µ tzθ

(6.10.20)

Using these, we can calculate the stress fields when the dislocation density b(ξξ ) is given. As an example, for a uniform distribution of screw dislocations along a line segment, (6.10.19) gives

x+1 T2 ≡ t32 /td = ln x−1

+ K0 (γ |x + 1|) − K0 (γ |x − 1|),

(6.10.21)

where td = µb/2π = µb0 N/2πL,

x = x1 /L,

γ = L/ l.

(6.10.22)

Here b0 is the atomic Burger vector and N is the number of dislocations over an interval of half-length L. The stress field (6.10.21) is continuous everywhere along the x-axis (Figure 6.10.3).

110


Figure 6.10.3. Shear stress distribution for uniformly distributed screws in a line segment. After Eringen [1990a].

B. Screw Dislocation in a Half-Plane9 The stress distribution arising from a screw dislocation in a half-plane x3 = 0, located at x1 = L, x2 = 0, with the constant Burger’s vector b, can be calculated by means of a superposition. We superpose the stress field of a dislocation located at x1 = −L, x2 = 0 with the Burger’s vector −b to that located at x1 = L, x2 = 0 (Figure 6.10.4): µb 1 1 1 1 tzθ = − + K1 (r2 / l) − K1 (r1 / l) , (6.10.23) 2π r1 r2 l l where r1 = (r 2 − 2rL cos θ + L2 )1/2 ,

r2 = (r 2 + 2rL cos θ + L2 )1/2 .

(6.10.24)

We observe that at the boundary x1 = 0, r1 = r2 , t13 vanishes. Hence (6.10.23) is the correct solution of this problem. Suppose we slice the x1 -axis from x1 = 0 to the point of application of the screw x1 = L. A displacement of the upper part, with respect to the lower part, of 9 Eringen [1990a].


111

Figure 6.10.4. Screw dislocation in a half-plane. this cut by an amount of the Burger’s vector requires an energy per unit length L U 2L µb2 2L t23 (x1 , 0)b dx1 = − = 21 ln + K0 − C , (6.10.25) L 4π 0 where C = 0.577215 . . . is the Euler constant. The force required per unit length is given by F 2L ∂ µb2 2L U = =− 1− K1 . (6.10.26) L ∂L L 4πL As → 0, this is identical to the classical result Fc µb2 =− . L 4πL

(6.10.27)

According to the classical solution, the force on the dislocation becomes very large with decreasing L, so that the dislocation will move to the free boundary x1 = 0. However, the nonlocal solution (6.10.26) shows that, as L → 0, the force vanishes, so that while dislocation will tend to move toward the boundary it will never get there. Near the free boundary the friction force will balance the force of dislocation. This implies that between the free surface x1 = 0 and the stopping point of dislocation there will be a dislocation free zone. In fact, this prediction was observed by Ohr et [1982] for perfect crystals.

112


Figure 6.11.1. Edge dislocation.

Screw dislocation in nonlocal hexagonal elastic crystals was treated by Eringen and Balta [1978].

6.11 Edge Dislocation10 A straight-edge dislocation may be envisioned by slicing a circular cylinder along a radial plane (0 < r ≤ R, θ = 0, z) and shifting the lower surface outward in the radial plane with respect to the upper face (Figure 6.11.1). The problem is two dimensional, so that to determine the classical stress fields σij we need to solve the biharmonic equation ∇ 4 ψ = 0 which in polar coordinates reads 2 1 ∂ ∂ 1 ∂2 r + 2 2 ψ = 0, r ∂r ∂r r ∂θ

(6.11.1)

where ψ(r, θ) is the stress function. An appropriate solution, relevant to the edge dislocation, is given by

ψ = A r ln r − 21 βr 3 sin θ. 10 Eringen [1977a].

(6.11.2)

6.11 Edge Dislocation

113

The Hookean stress field is calculated by σrr σθ θ σrθ

1 1 1 − βr sin θ, = 2 ψ,θ θ + ψ,r = A r r r 1 − 3βr sin θ, = ψ,rr = A r 1 ∂ 1 ψ,θ = −A − βr cos θ, =− r ∂r r

where A and β are constants. The nonlocal stress field is given by

(k) (k ) t () = α(|x − x|)σ (l ) (x )δ l l δ kk dv , V

(6.11.3)

(6.11.4)

where δ l l and δ kk are the cosine directors of the cylindrical coordinates at x with respect to those at x: x1 = r cos θ, x1 = r cos θ ,

x2 = r sin θ,

x3 = z,

x2 = r sin θ ,

x3 = z .

(6.11.5)

Hence trr tθ θ

α(|x − x|) 21 (σrr + σθ θ ) ± (σrr − σθ θ ) cos 2(θ − θ) V

∓ σrθ sin 2(θ − θ) dv

= α(|x − x|) 21 (σrr − σθ θ ) sin 2(θ − θ) V

+ σrθ cos 2(θ − θ) dv , (6.11.6)

=

trθ

where |x − x| = [r 2 + r 2 − 2rr cos(θ − θ) + (z − z)2 ]1/2 , dv = r dr dθ dz .

(6.11.7)

For the kernel α(|x − x|) we select α(|x − x|) = π −3/2 (k/ l)3 (6.11.8) 2 2 2

2 · exp{−(k/ l) [r + r − 2rr cos(θ − θ) + (z − z) ]}. This is the same kernel given by (6.9.26), expressed in polar coordinates. Substituting (6.11.3) and (6.11.8) into (6.11.6) we perform some tedious integrations in the order z , θ , and r , using the tables given by Gradshteyn and Ryzhik

114


[1965], to obtain A {1 − βr 2 − (l/kr)2 [1 − exp(−k 2 r 2 / l 2 )]} sin θ, r A = {−1 − 3βr 2 + [2 + (l/kr)2 ][1 − exp(−k 2 r 2 / l 2 )]} sin θ, r A (6.11.9) = − {1 − βr 2 − (l/kr)2 [1 − exp(−k 2 r 2 / l 2 )]} cos θ. r

trr = tθ θ trθ

The displacement fields ur and uθ are identical to the classical forms that may be obtained by integrating the stress–strains relations of two-dimensional elasticity err = = eθ θ = = erθ =

1 ∂ur = [(1 − ν)σrr − νσθ θ ] ∂r 2µ A [1 − 2ν − β(1 − 4ν)r 2 ] sin θ, 2µr ur 1 1 ∂uθ + = [−νσrr + (1 − ν)σθ θ ] r ∂θ r 2µ A [1 − 2ν − β(3 − 4ν)r 2 ] sin θ, 2µr ∂uθ uθ σrθ A 1 1 ∂ur + − = =− (1 − βr 2 ) cos θ, (6.11.10) 2 r ∂θ ∂r r 2µ 2µr

where ν is the Poisson ratio and µ is the shear modulus. Integration (6.11.10) gives ur =

A

(1 − 2ν) ln r sin θ − 2(1 − ν)θ cos θ 2µ

− 21 β(1 − 4ν)r 2 sin θ] + B cos θ + C sin θ, A

uθ = (1 − 2ν) ln r cos θ + 2(1 − ν)θ sin θ + cos θ 2µ + 21 β(5 − 4ν)r 2 cos θ − Dr − B sin θ + C cos θ,

(6.11.11)

where B, C, and D represent the rigid body displacement. We are now ready to determine the constants A and β from the boundary conditions trr = trθ = 0 at r = R, ur (r, 2π) − ur (r, 0) = b.

(6.11.12)

This gives

2 β0 ≡ βl 2 /k 2 = 1 − P −2 (1 − e−P ) P −2 , µb kR kr A=− , P ≡ , ρ≡ . 2π(1 − ν) l l

(6.11.13)

6.11 Edge Dislocation

115

The nonlocal stress field is obtained to be µbk 2 [ρ −1 − β0 ρ − ρ −3 (1 − e−ρ )] sin θ, 2π(1 − ν)l µbk 2 [−ρ −1 − 3β0 ρ + ρ −1 (2 + ρ −2 )(1 − e−ρ )] sin θ, =− 2π(1 − ν)l µbk 2 [ρ −1 − β0 ρ − ρ −3 (1 − e−ρ )] cos θ. = (6.11.14) 2π(1 − ν)l

trr = − tθ θ trθ

The elastic strain energy U is calculated by 1 U = 2 (trr err + tθ θ eθ θ + 2trθ erθ ) dv. V

(6.11.15)

Substituting from (6.11.10) and (6.11.14), and integrating over the regions 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ L, we obtain U =

−2 µb2 L 2 P (1 − e−P ) − 1 − 2(1 − ν)[Ei(−P 2 ) − C − ln P 2 ] 2 16π(1 − ν) −P 2 2 4 + β0 (3 − 4ν)(1 − e − 2P + β0 P ) , (6.11.16)

where C is the Euler constant and Ei(x) is the exponential integral function x t e C = 0.5772 . . . , Ei(x) = dt, x < 0. (6.11.17) −∞ t In classical elasticity the stress field for the edge (Volterra) dislocation of a hollow cylinder with inner radius r0 and outer radius R is given by (cf. Lardner [1974, p. 77]: r02 1 r − 2 − 3 sin θ, trr = A r R r r02 1 3r tθ θ = A − 2 − 3 sin θ, r r R r02 1 r trθ = −A (6.11.18) − 2 − 3 cos θ. r R r For the full cylinder r0 = 0, and along the axis of the cylinder, the stress field becomes infinite as 1/r. Thus, a core region 0 < r ≤ r0 had to be excluded from the calculations. The nonlocal result does not contain this singularity. In fact, we observe that: (i) the stress field is not singular at r = 0 but depends on the internal characteristic length l = 0;

116


(ii) when l → 0 the solution (6.11.14) reverts to the classical solution (6.11.18); (iii) the stored energy U has no singularity for P = 0 unless either the cylinder has infinite radius or l = 0; and (iv) the maximum shear stress occurs at ρ = ρm , the root of exp(ρ 2 ) =

3 + 2ρ 2 . 3 − ρ 2 − β0 ρ 4

(6.11.19)

For a cylinder of infinite radius β0 = 0, and the root of (6.11.19) is given by ρm 1.46595, aside ρ = 0. At this, trθ max is obtained to be trθ max ≡ τc =

µbk µbk ρm 0.2009 . π(1 − ν)l 3 + 2ρm π(1 − ν)l

(6.11.20)

If the nonlocal effects at n atomic distance r = na are assumed to diminish to 1% of its value at r = 0, then k = 2.146/n and we obtain τc = 0.4311

µb . π(1 − ν)an

(6.11.21)

√ For a single atomic dislocation of face centered metals b = a/ 6, and we obtain τc /µ = 0.08/n,

(6.11.22)

where we took ν = 0.3. For n = 2 this gives τc /µ = 0.04 and for n = 3, τc /µ = 0.027. These values are in the accepted range known in solid state physics. For example, for aluminum and cooper (at 20 ◦ C) the tabulated values of τc /µ = 0.039 (Kelly [1966]) for n = 2.05, the present theory gives exactly τc /µ = 0.039.

6.12 Screw Dislocation in Nonlocal Hexagonal Elastic Solids11 Here we present another solution of the screw dislocation in order to demonstrate the situation for an anisotropic solid, namely a nonlocal elastic solid with hexagonal symmetry. For such a solid the nonlocal kernel α(x − x) has a form different than an isotropic solid. A useful one is α(x − x) = π −3/2 k12 k2 l −3 exp[−(k1 / l)2 (xβ − xβ )(xβ − xβ ) − (k2 / l)2 (x2 − x2 )2 ],

β = 1, 3.

(6.12.1)

Note that the degree of attenuation in the x2 -direction is different than in the x1 and x2 -directions. 11 Eringen and Balta [1978]. Edge dislocation in nonlocal hexagonal elastic crystals was discussed by Eringen and Balta [1979a].

6.12 Dislocation in Nonlocal Hexagonal Elastic Solids

117

In hexagonal solids, Hooke’s law is in the form:



    σ11 0 0 c11 c12 c13 0 e11  σ22   c12 c22 c12 0   0 0       e22   σ33   c13 c12 c11 0   0 0   e33     .  σ23  =  0   0 0 c44 0 0       e23   σ31   0   e31  0 0 0 c55 0 0 0 0 0 0 c44 σ12 e12

(6.12.2)

A pure screw dislocation in the z-direction, in the basal plane of a hexagonal crystal, is possible (Figure 6.12.1). Referred to rectangular coordinates, the nonzero components of the displacement and the Hookean stress components are given by

Figure 6.12.1. Screw dislocation in hexagonal crystals.

118


(cf. Hirth and Lothe [1968, p. 426]): √ b tan−1 (x2 /x1 B), 2π Ab x2 =− , 2 2π Bx1 + x22 ABb x1 = , 2π Bx12 + x22

u3 = σ13 σ23

(6.12.3)

where b = (0, 0, b) is the Burger’s vector and A and B are constants related to the elastic moduli cij by A = [c44 (c11 − c13 )/2]1/2 ,

B = 2c44 /(c11 − c13 ).

(6.12.4)

The elastic energy per unit length of a cylinder, with inner and outer radii r0 and R, is given by Uc Ab2 R . (6.12.5) = ln L 4π r0 Hence we notice the stress and energy singularities as r0 → 0. This is a troublesome state, well known in classical elasticity. In cylindrical coordinates the stress field is given by σθ z = −σ13 sin θ + σ23 cos θ =

Ab 1 , 2π r

A(B − 1)b 1 tan θ . (6.12.6) 2π r B + tan2 θ The expression of the nonlocal stress is obtained, similar to (6.11.4), or by coordinate transformation, as trz = α(x − x)[−σθ z (x ) sin(θ − θ) + σrz (x ) cos(θ − θ)] dv(x ), V tθ z = α(x − x)[σθ z (x ) cos(θ − θ) + σrz (x ) sin(θ − θ)] dv(x ), (6.12.7) σrz =

V

with α(x − x) given by α(x − x) = −π −3/2 k12 k2 l −3 exp[−(k1 / l)2 (z − z)2 ] · exp[−(k1 / l)2 r 2 cos2 θ − (k2 / l)2 r 2 sin2 θ] · exp[−(k1 / l)2 (r 2 cos2 θ − 2rr cos θ cos θ ) − (k2 / l)2 (r 2 sin2 θ − 2rr sin θ sin θ )].

(6.12.8)

Since α is an even function of θ − θ, the integrals (6.12.7) take the form A(B − 1)b ∞ ∞ 2π tan θ cos(θ − θ)

trz = dz dr α(x − x) dθ , 2π B + tan2 θ

−∞ 0 0 Ab ∞ ∞ 2π tθ z = α(x − x) cos(θ − θ) dz dr 2π −∞ 0 0 tan θ

+(B − 1) sin(θ − θ) dθ . (6.12.9) B + tan2 θ


119

Integrations over r and z are carried out to obtain Akκb 2π sin θ cos θ

(B − 1) cos(θ − θ)f (ρ, θ , θ) dθ , trz = 4π 3/2 l 0 1 + (B − 1) cos2 θ

Akκb 2π cos(θ − θ) tθ z = 4π 3/2 l 0 sin θ cos θ

+ (B − 1) sin(θ − θ) f (ρ, θ , θ) dθ , (6.12.10) 1 + (B − 1) cos2 θ

where ρ = kr/ l,

k = k1 ,

κ = k2 /k1 ,

f (ρ, θ, θ ) (cos θ cos θ + κ 2 sin θ sin θ )2 2 2 2 = exp −ρ 1 + (κ − 1) sin θ − 1 + (κ 2 − 1) sin2 θ

cos θ cos θ + κ 2 sin θ sin θ

× 1 + erf ρ [1 + (κ 2 − 1) sin2 θ ]1/2 (6.12.11) · [1 + (κ 2 − 1) sin θ ]−1/2 . The displacement field is given by ur = uθ = 0,

uz =

b tan−1 (B −1/2 tan θ), 2π

(6.12.12)

so that the nonzero component of the strain tensor is eθ z =

B 1/2 b [1 + (B − 1) cos2 θ]−1 . 4πr

The total strain energy per unit length is calculated by B 1/2 b 2π R U/L = [1 + (B − 1) cos2 θ]−1 tθ z (r, θ) dr dθ. 4π 0 0

(6.12.13)

(6.12.14)

The nondimensional shear stress τθ z = tθ z /t0 ,

t0 ≡

Akb , 2πl

(6.12.15)

is displayed in Figure 6.12.2, for various hexagonal crystals. κ has been selected to be c22 /c11 , with the consideration that the attenuation in a given direction is probably proportional to the elastic modulus in that direction. The elastic constant used (taken from Hearmon [1969]) are shown in Table 6.12.1. Also listed in this table are κ = c22 /c11 , the maximum value of τθ z and its location ρ = ρm . It is clear that the maximum value of the shear stress and its location is greatly affected by the anisotropy. In particular, for Cd and Zn, the maximum shears are nearly

120


Figure 6.12.2. Nondimensional shear stress τθ z versus ρ. After Eringen and Balta [1978]. Table 6.12.1. Maximum shear stress. c11 Material Zn Mg Cd Apatite Ice

16.5 5.93 11.4 16.7 1.34

Elastic Constants c12 c13 c22 × 1011 dyn/cm2 5.0 3.1 6.2 2.14 2.57 6.15 4.0 3.94 5.08 6.6 1.31 14.0 0.53 0.65 1.45

c44 3.96 1.64 2.0 6.63 0.313

κ= (c22 /c11 ) 0.376 1.037 0.446 0.838 1.082

τθ z = (tθ z /t0 ) 0.3370 0.6555 0.3911 0.5871 0.6824

ρm = (krm /a) 2.65 1.08 2.25 1.29 1.03

one-half of that of the isotropic solids. In Figures 6.12.3 and 6.12.4, the angular dependence of τθ z and τrz are displayed for Mg. Finally, in Figure 6.12.5, the ratio of the shear stress tθ z to that of the isotropic solids is displayed for various materials. These curves may be used to give estimates of other hexagonal crystals with different κ.


121

Table 6.12.2. tθ z /c44 . Material Zn Mg Cd Apatite Ice

1.00 0.0698 0.1056 0.0850 0.1007 0.1140

k 1.25 0.0872 0.1320 0.1062 0.1258 0.1425

1.50 0.1049 0.1584 0.1275 0.1510 0.1710

Figure 6.12.3. Nondimensional shear stress τθ z = tθ z /t0 versus θ for Mg. After Eringen and Balta [1978].

Figure 6.12.4. Nondimensional shear stress τrz = trz /t0 versus θ. After Eringen and Balta [1978]. Unlike the results given in classical theory, the shear stresses do not display any singularity but acquire maxima. Equating the maxima to the cohesive shear stress we obtain the condition that produces a dislocation of the single atomic distance.

122


Figure 6.12.5. Nondimensional shear stress τθ z to shear stress for isotropic bodies. After Eringen and Balta [1978]. Table 6.12.2 lists the ratio tθ z /c44 for various k, b/a = 1. The value 1.50 makes the dispersion curve for shear waves, obtained theoretically by using (6.12.1), nearly coincident with those obtained from experiments by Joynson [1954]. It appears that the theoretical shear stress calculated is about twice the value, based on the lattice dynamics (Kelly [1966, p. 19]) for Zn. Considering the inaccuracies involved in estimating the interatomic force laws, it seems that the present prediction is in the right range. With the smaller adjustments of k and κ, it is possible to lower this prediction. The total energy given by (6.12.14) may be put into the form U = 2Ab2 LU0 ,

(6.12.16)

where U0 depends on κ, B, and P = kR/ l. For various materials U0 is calculated and listed in Table 6.12.3. The case of isotropic materials agrees very well with the result given by Eringen [1977b] and with that given in Section 6.10. Of course, U0 grows with the radius R becoming infinite at R = ∞, as expected.

6.13 Continuous Distribution of Dislocations

123

Table 6.12.3. Coefficient U0 of total energy; U0 = U/2Ab2 L. Material Isotropic, Eringen [1977b] Isotropic (Present: B = κ = 1) Zn Mg Cd Apatite Ice

1 0.01585

2 0.03914

P = kR/a 3 5 0.05520 0.07552

7 0.08891

10 0.10301

0.01586

0.03916

0.05521

0.07551

0.08890

0.10300

0.00610 0.01633 0.00714 0.01366 0.01688

0.01831 0.03987 0.02107 0.03548 0.04069

0.03027 0.05593 0.03412 0.05139 0.05676

0.04895 0.07624 0.05341 0.07167 0.07707

0.06210 0.08962 0.06660 0.08505 0.09045

0.07617 0.10380 0.08069 0.09923 0.10463

6.13 Continuous Distribution of Dislocations12 A small neighborhood n(x) of x, in a distorted body V, may be relaxed to a small neighborhood N(X) of the image point X of x, in an undistorted (or natural) configuration V , by releasing the constraints exerted by the rest of the body. We can express a line element dx at x ∈ n(x) in terms of its image dX ∈ N(X) by dx = A dX,

(6.13.1)

where A(X) is called the elastic distortion. We assume that A(X) is continuously differentiable and possesses a unique inverse −1

dX = A dx.

(6.13.2)

Consider a smooth surface S in V bounded by a closed curve C. The Burger’s vector b of the dislocations piercing through S is defined by * * −1 b= dX = an da, (6.13.3) A dx = C

C

S

where n is the unit normal to S. The positive sense of C is counterclockwise, when sighting along n. Here a is called the true dislocation density −1

a = curl A

or

−1

aj k = kmn A j n,m .

(6.13.4)

−1 A kl

(6.13.5)

For small distortions we can take Akl = δkl + αkl ,

δkl − αkl ,

so that aj k = kmn αj m,n . 12 Eringen [1984].

(6.13.6)

124


From this it follows that aj k,k = 0.

(6.13.7)

For the linear strain tensors and spins wkl we have ekl = 21 (αkl + αlk ),

(6.13.8)

wkl =

(6.13.9)

1 2 (αkl

− αlk ).

The incompatibility of the strain tensor is expressed as ij k lmn ein,j m = ηkl ,

(6.13.10)

where ηkl is called the incompatibility tensor. This is given by ηkl = 21 (kmn anl,m + lmn ank,m ).

(6.13.11)

So far these results are well known in the classical theory of dislocation (cf. Teodosiu [1982]). For nonlocal, homogeneous, and isotropic elastic solids the stress–strain relations are given by (6.1.20)2 . Under mild assumptions (with γ = 0), regarding the kernel α(|x − x|), by (6.9.12) we have (1 − 2 ∇ 2 )tkl = λerr δkl + 2µekl .

(6.13.12)

From this equation we solve for ekl : ekl =

1 ν (1 − 2 ∇ 2 ) tkl − trr δkl , 2µ 1+ν

(6.13.13)

where ν = λ/2(λ + µ) is the Poisson ratio. Substituting (6.13.13) into (6.13.10), we obtain 1 2 2 2 2 (trr,kl − ∇ trr δkl ) = 2µηkl . (6.13.14) (1 − ∇ ) ∇ tkl + 1+ν These equations must be solved to determine the stress field tkl which is subject to the equilibrium condition tkl,k = 0. (6.13.15) Following the classical approach (cf. Kröner [1954]), we take tkl /2µ = ∇ 2 χkl +

1 (χrr,kl − ∇ 2 χrr δkl ), 1−ν

(6.13.16)

where χkl = χlk is a stress function subject to χkl,l = 0.

(6.13.17)

Substituting (6.13.16) into (6.13.14) we obtain (1 − 2 ∇ 2 )∇ 4 χkl = ηkl .

(6.13.18)


125

Thus, if the dislocation density function akl is given through (6.13.11) we can calculate the incompatibility tensor ηkl . The solution of (6.13.18) will then give χkl . The stress field follows from (6.13.16). As l → 0, (6.13.18) reduces to the well-known classical equation of the local theory. When l = 0, (6.13.18) is perturbed singularly. In order to obtain the solution of (6.13.18), we introduce the Green’s tensor Gklmn (x, ξ ) which must satisfy (1 − l 2 ∇ 2 )∇ 4 Gklmn = δ(x − ξ )δkl δmn ,

(6.13.19)

when Gklmn is determined, then the solution of (6.13.18) is given by χkl = Gklmn (x, ξ )ηmn (ξξ ) dv(ξξ ). V

(6.13.20)

We now determine the Green’s function for a solid of infinite extent. For the infinite space we seek the solution of (6.13.19) that depends on |x − ξ | only, i.e., (1 − l 2 ∇ 2 )∇ 4 G = δ(x − ξ ).

(6.13.21)

The operators 1 − l 2 ∇ 2 and ∇ 4 commute, so that we may replace (6.13.21) by (1 − l 2 ∇ 2 G) = H,

∇ 4 H = δ(x − ξ ).

(6.13.22)

For the infinite space, H is given by H = −|x − ξ |/8π.

(6.13.23)

In spherical coordinates using the operator d d ∇ 2 = r −2 r2 , dr dr the solution of (6.13.21) is obtained to be |x − ξ | l2 exp(−|x − ξ |/ l) − , 4π|x − ξ | 8π |x − ξ | , l = 0, G(|x − ξ |) = − 8π G(|x − ξ |) =

l = 0,

where we have determined a constant to render tkk regular at x = ξ . The solution of (6.13.18) for an infinite medium is given by χkl = G(|x − ξ |)ηkl (ξξ ) dξξ , V

(6.13.24)

(6.13.25)

which satisfies (6.13.17) on account of (6.13.7) and (6.13.11). Upon substituting from (6.13.11), (6.13.25) may be expressed as ∂G ∂G χkl (x) = 21 ij k aj l (ξξ ) dv(ξξ ) + 21 ij l aj k (ξξ ) dv(ξξ ), (6.13.26) ∂xi ∂xi V V

126


where we used the Green–Gauss theorem and set a surface term at infinity equal to zero. Equation (6.13.26), in the case of a line distribution of dislocations along a closed curve C, takes the form * * ∂G ∂G χkl = 21 ij k bj dsl + 21 lij bj dsk , (6.13.27) ∂x ∂x i i C C where bj is the Burger’s vector per unit length of C and dsl is the element of arc. The stress fields arising from a line distribution of dislocations are obtained by substituting (6.13.27) into (6.13.16): * 1 tkl /2µ = 2 rij bj ∇ 2 G,i (δrk dsl + δrl dsk ) 2 2 + (G,kli − ∇ G,i δkl ) dsr . (6.13.28) 1−ν This is identical to the Peach–Koehler formula (cf. Hirth and Lothe [1968], Teodosiu [1982]) with the modification that here G is the nonlocal Green’s function (6.13.24) with l = 0. The most interesting feature of (6.13.28) is that it does not exhibit unbounded stress fields and energy at any point.

A. Plane Strain In the case of plane strain, we introduce the Airy’s stress function (x1 , x2 ) by t11 = ,22 ,

t22 = ,11 ,

t12 = −,12 .

(6.13.29)

In this case (6.13.18) is replaced by (1 − l 2 ∇ 2 )∇ 4 = 2µη,

(6.13.30)

η = η33 = a23,1 − a13,2 , a23 = α21,2 − α22,1 , a13 = α11,2 − α12,1 .

(6.13.32)

where η is given by (6.13.31)

and depends on x1 and x2 only. For a two-dimensional infinite plane the Green’s function can be found similarly using the decomposition (6.13.22), with ∇ 2 and H given by 1 d d 1 ∇2 = r , H =− ln |x − ξ |. (6.13.33) r dr dr 2π Hence we obtain (x − ξ ) · (x − ξ ) 1 K0 (|x − ξ |/ l) − ln(|x − ξ |/ l), = 0, 2π 8πl 2 (x − ξ ) · (x − ξ ) ln(|x − ξ |), = 0, (6.13.34) G(|x − ξ |) = − 8π G(|x − ξ |) =


where K0 (z) is the modified Bessel’s function. The Airy’s stress function is then found to be ∂G ∂G a13 (ξξ ) da(ξξ ), a23 (ξξ ) − (x) = 2µ ∂x2 S ∂x1

127

(6.13.35)

where we used the Green–Gauss theorem and disregarded a line integral at infinity. For a line distribution in the (x3 = 0)-plane, (6.13.35) is replaced by ∂G ∂G (x) = −2µ (6.13.36) b2 (ξξ ) dξ1 + b1 (ξξ ) dξ2 . ∂x2 C ∂x1 The stress field is calculated by using (6.13.29): t11 = −2µ (G,122 b2 dξ1 + G,222 b1 dξ2 ), C t22 = −2µ (G,111 b2 dξ1 + G,211 b1 dξ2 ), C t12 = 2µ (G,112 b2 dξ1 + G,212 b1 dξ2 ), C

(6.13.37)

where the indices after the comma denote partial derivatives, with respect to xk , e.g., ∂ 3G G,122 = ,... . ∂x1 ∂x22

B. Antiplane Strain In the case of antiplane strain the equations of equilibrium are satisfied by introducing ∂φ ∂φ t13 = , t23 = − . (6.13.38) ∂x2 ∂x1 In this case (6.13.30) is replaced by (1 − l 2 ∇ 2 )∇ 2 φ = µa33 ,

(6.13.39)

a33 = α31,2 − α32,1 .

(6.13.40)

where In this case the Green’s function is found to be 1 [ln(|x − ξ |/ l) + K0 (|x − ξ |/ l)], 2π 1 G(|x − ξ |) = − ln(|x − ξ |), l = 0. 2π G(|x − ξ |) = −

The stress function φ is given by φ(x) = µ

l = 0, (6.13.41)

S

G(|x − ξ |)a33 (ξξ ) da.

(6.13.42)

128


For the stress tensor we obtain

∂G b(ξ ) dξ1 dξ2 , S ∂x2 ∂G = −µ b(ξ ) dξ1 dξ2 . ∂x 1 S

t13 = µ t23

(6.13.43)

For a line distribution of dislocation on the (x3 = 0)-plane the stress field reads ∂G t13 = µ b(ξξ ) ds, C ∂x2 ∂G t23 = −µ b(ξξ ) ds. (6.13.44) ∂x 1 C In polar coordinates we have

1 ∂G b(ξξ ) ds, C r ∂θ ∂G = −µ b(ξξ ) ds. C ∂r

tzr = µ tzθ

(6.13.45)

With these formulas developed we can calculate the stress field for any type of distributions of dislocations.

C. Stress Fields for Some Special Distributions of Dislocations Here we consider some special distributions of dislocations and give the resulting stress fields. (i) Edge Dislocation Along a Line Segment Along a line segment −L < x1 < L, x2 = 0, the edge dislocation with the Burger’s vector (b1 = const., b2 = 0), the stress field follows from (6.13.37) by noting that G,2 = ∂G/∂x2 = −∂G/∂ξ2 , is t11 = 2µb1 [G,22 (ρ1 ) − G,22 (ρ2 )], t22 = 2µb1 [G,11 (ρ1 ) − G,22 (ρ2 )], t12 = −2µb1 [G,12 (ρ1 ) − G,12 (ρ2 )],

(6.13.46)

where ρ1 = [(x1 − L)2 + x22 ]1/2 / l,

ρ2 = [(x1 + L)2 + x22 ]1/2 / l,

and the Green’s function G is given by (6.13.34).

(6.13.47)


129

(ii) Screw Dislocations Along a Line Segment For the uniformly distributed screw dislocations along a line segment −L < x1 < L, x2 = 0, through (6.13.43) we find that t23 = µb[G(ρ1 ) − G(ρ2 )],

(6.13.48)

where G is given by (6.13.41) and ρ1 and ρ2 by (6.13.47). If there are N dislocations over a distance L, with the atomic Burger’s vector b0 , then for the macroscopic Burger’s vector b we have b = b0 N/2L.

(6.13.49)

The shear stress (6.10.48) may be expressed in a nondimensional form T2 = t23 /td = ln

|x + 1| + K0 (γ |x + 1|) − K0 (γ |x − 1|), |x − 1|

(6.13.50)

where

µb µb0 N x1 L = , x= , γ = . 2π 2πL L This result is identical to (6.10.21) given in Section 6.10. In the case of a single screw, we take td =

b(ξξ ) = b0 δ(ξξ )

(6.13.51)

(6.13.52)

in (6.10.44) and obtain µb0 ∂G =− ∂x2 2π ∂G µb0 = −µb0 = ∂x1 2π

t13 = µb0 t23

x2 r x1 r

r 1 − Kl (r/ l) , l r 1 − Kl (r/ l) , l

(6.13.53)

where r = (x12 + x22 )1/2 . This result is also identical to that given in Section 6.10, as expected. From (6.13.53) we obtain Tθ (ρ) = (2πl/µb0 )tzθ = ρ = r/ l.

1 [1 − ρK1 (ρ)], ρ (6.13.54)

Based on the results and discussions given before, we now state a fracture criterion: Fracture Criterion. Fracture occurs when the maximum shear stress reaches the value of the cohesive stress that holds atomic bonds together. With this criterion we see that the maximum stress hypothesis, practiced for macroscopic fracture, is restored for the atomic phenomena as well. In the case of the line distribution of screws the nondimensional stress T2 , given by (6.13.50), has been displayed previously in Figure 6.10.3. Here we observe that

130


Table 6.13.1. Maximum shear stress and its location for a uniform distribution of screw dislocations along a straight-line segment. γ = x= T2 max

1 1.446 0.7478

1.5 1.197 1.0501

2 1.103 1.3008

3 1.039 1.6851

5 1.000 2.3026

10 1.000 2.9957

the maximum shear stress is located outside the segment, not x1 = ±L, x2 = 0 (see Table 6.13.1). Contrary to the classical elasticity solution, there is no singularity at x1 = ±L, x2 = 0. The behavior of T2 is governed basically by the first term in (6.13.50) except near x = ±1. At x = 1 we have µB µb0 N ln γ = , 2πl γ 2πh B ≡ b0 N, h ≡ lγ / ln γ .

t23 =

(6.13.55)

Equation (6.13.55) may be interpreted in terms of the atomic layers of distance h by one macroscopic dislocation of the Burger’s vector B. The ratio of the cohesive stress tcl , for the line distribution to that of a single screw tc , is given by tcl N ln γ = . tc 0.3993 γ

(6.13.56)

This gives the shear stress reduction, due to the presence of 2N dislocations, uniformily distributed along the line segment of length 2L. Since tcl ≤ tc the maximum number of dislocations is obtained as Nmax = 0.3993

γ . ln γ

(6.13.57)

When the distribution is nonuniform this number is modified. (iii) Uniform Distribution of Screw Dislocations Along a Circle We consider uniformly distributed screw dislocations along a circle of radius R. In polar coordinates, we have x1 = r cos θ, x2 = r sin θ, ξ1 = R cos φ, ξ2 = R sin φ, 2 2 |x − ξ | = [r + R − 2rR cos(φ − θ)]1/2 .

(6.13.58)

The Green’s function is given by G(|x − ξ |) = −

1 [ln(|x − ξ |/ l) + K0 (|x − ξ |/ l)]. 2π

(6.13.59)


131

In order to calculate the stress fields tzr and tzθ we need to evaluate two integrals 2π +θ ln{l −1 [r 2 + R 2 − 2rR cos(φ − θ)]1/2 } dφ, (6.13.60) I1 =

θ 2π +θ

I2 =

K0 {l −1 [r 2 + R 2 − 2rR cos(φ − θ)]1/2 } dφ.

(6.13.61)

θ

To evaluate (6.13.60) we write ln{l −1 [r 2 + R 2 − 2rR cos(φ − θ)]1/2 } = ln[|r + R|/ l] + ln[1 − α0 cos2 (ψ/2)]1/2 , where α0 = 4rR/(r + R)2 ,

φ − θ = ψ.

The second term involving cos2 (ψ/2) can be integrated by writing x = cos(ψ/2) and consulting Gradshteyn and Ryzhik [1965, p. 562]: 2π ln(r/ l), r > R, (6.13.62) I1 = 2π ln(R/ l), r < R. By using the same reference [1965, p. 741], we evaluate I2 : 2πI0 (R/ l)K0 (r/ l), r > R, I2 = 2πI0 (r/ l)K0 (R/ l), r < R.

(6.13.63)

Consequently, I = 0

=−

2π

G(|x − ξ |) ds

ln(r/ l) + I0 (R/ l)K0 (r/ l), ln(R/ l) + I0 (r/ l)K0 (R/ l),

r > R, r < R,

(6.13.64)

where I0 (z) and K0 (z) are the Bessel’s functions. The stress field is given by µbR ∂I = 0, r ∂θ ∂I µb(R/ l)[(l/r) − I0 (R/ l)K1 (r/ l)], = = µbR µb(R/ l)I1 (r/ l)K0 (R/ l), ∂r

tzr = tzθ

In the special case when R → 0 and 2πRb = b0 we obtain r r µb0 1 − K1 , tzθ = 2πr l l

r > R, (6.13.65) r < R.

(6.13.66)

which is identical to our previous result (6.13.54) for a screw dislocation with the Burger’s vector b0 . For the uniform distribution of screws along a circle the stress field given by (6.13.65) may be expressed in the nondimensional form 1 − κI0 (κ)K1 (κρ), ρ > 1, (6.13.67) T = tzθ /µb = ρ κK0 (κ)I1 (κρ), ρ < 1,

132


Figure 6.13.1. Shear stress (circular distribution). After Eringen [1984]. where ρ = r/R and κ = R/ l. This is displayed in Figure 6.13.1. For κ ≥ 50 the maxima of T occurs near ρ = 1, 0.324 ≤ Tmax ≤ 1. Clearly, by means of the nonlocal Peach–Koehler formula (6.13.28), one can evaluate stress fields for the distribution of dislocations on any curve, once the Green’s function G is determined. In fact, Povstenko [1995] gave the solution for the circular dislocation loops. Screw dislocation in a nonlocal bimedium was discussed by Gao [1990]. In all cases, nonphysical singularities do not appear.

6.14 Nonlocal Stress Field at the Griffith Crack An elastic plate with a line crack, located at −L < x < L, y = 0 and subject to a uniform tension tyy = t0 at infinity, is known as the Griffith crack (Figure 6.14.1). The classical elasticity solution of this problem leads to infinite stresses at the tips of the line crack. Consequently, no fracture criterion, based on the maximum stress hypothesis, could be given. In fact, Griffith proposed that fracture will occur when the applied stress reaches a critical value given by t02 L = Kγ ,

(6.14.1)

where K is a material constant and γ is the surface energy. In previous sections we have shown that in nonlocal theory, by equating the maximum stress to the cohesive stress that hold atomic bonds together, it is possible to introduce a natural fracture criterion. Also, we have found that the maximum nonlocal stress predicted by the nonlocal theory occurs slightly away from the crack tip. According to a lattice

6.14 Nonlocal Stress Field at the Griffith Crack

133

Figure 6.14.1. Griffith crack subject to uniform tension. model, Elliott [1947], and with computer simulations, Gohar [1979], this result is also confirmed. The nonlocal elasticity solutions for the Griffith crack was first given by Eringen and Kim [1974] and Eringen et al. [1977]. This problem was also treated in a more general way by Ari and Eringen [1983]. Here we follow the latter works. Equations of equilibrium follow from (6.4.19): L α(|x − x|)σkl,k (x ) dx dy − α(|x − x|)σ2l (x , 0) dx = 0. (6.14.2) −L

R

The region R covers the entire plane z = 0, excluding the crack line. When an incision is made in the undeformed body, the body will in general deform under the attractions of long-range interatomic forces. Moreover, it will become inhomogeneous in the neighborhood of the crack line because of the lack of attractions eliminated from the crack surface. Thus, the kernel α will fail to be a function of |x − x| in a thin boundary layer around the crack. Here we disregard these small changes and consider the body to be in its natural state. The stress field is given by α(|x − x|)σkl (x ) dx dy , (6.14.3) tkl = R

where σkl is the Hookean stress σkl = λerr δkl + 2µekl ,

(6.14.4)

ekl = 21 (uk,l + ul,k ).

(6.14.5)

and ekl is the strain tensor

134


For the nonlocal kernel α we have found several choices in Section 6.9. For a two-dimensional case an appropriate kernel was given by (6.9.45), i.e., α(|x − x|) = (2πl 2 )−1 K0 (|x − x|/ l).

(6.14.6)

According to (6.9.12) this kernel with γ = 0 relates tkl to σkl by (1 − l 2 ∇ 2 )tkl = σkl .

(6.14.7)

The boundary conditions at the crack line x2 = 0 require that tyx = 0, y = 0 ∀ x, tyy (x, 0) = −t0 , y = 0, |x| < L, v = 0, y = 0, |x| > L, {u, v} → 0,

as

(x 2 + y 2 )1/2 → ∞.

(6.14.8)

From (6.14.7) it follows that these conditions are equivalent to y = 0 ∀ x, σyx = 0, σyy (x, 0) = −t0 , y = 0, |x| < L, v = 0, y = 0, |x| > L, {u, v} → 0

as

(x 2 + y 2 )1/2 → ∞.

(6.14.9)

Note that for nonuniform loading, (6.14.8) will lead to more complicated expressions for σkl . The equation’s equilibrium tkl,k = 0 through (6.14.7) gives σkl,k = 0. Hence we have the field equations (λ + µ)(u,xx + v,xy ) + µ∇ 2 u = 0, (λ + µ)(u,xy + v,yy ) + µ∇ 2 v = 0.

(6.14.10)

These are identical to the Navier equations in classical elasticity. Using the Fourier transform ∞ 1/2 f (ξ, y) = (2π) f (x, y) exp(iξ x) dx, (6.14.11) −∞

the general solutions of (6.14.10) is obtained −1/2

∞

−∞ ∞

u = (2π)

v = (2π)−1/2

−∞

λ + 3µ i |ξ |A(ξ ) + (|ξ |y − B(ξ ) exp(−|ξ |y − iξ x) dξ, ξ λ+µ [A(ξ ) + yB(ξ )] exp(−|ξ |y − iξ x) dξ.

(6.14.12)


135

The Hookean stress components are given by ∞ 2λ + 3µ 2µ σxx = √ |ξ |A(ξ ) + |ξ |y − B(ξ ) λ+µ 2π −∞ × exp(−|ξ |y − iξ x) dξ, ∞ −µ 2µ A(ξ )|ξ | + + |ξ |y B(ξ ) σyy = − √ λ+µ 2π −∞ × exp(−|ξ |y − iξ x) dξ, ∞ i|ξ | λ + 2µ 2µ −iξ A(ξ ) + −iξy + · B(ξ ) σyx = √ ξ λ+µ 2π −∞ × exp(−|ξ |y − iξ x) dξ. (6.14.13) The boundary condition σyx (x, 0) = 0 gives B(ξ ) =

ξ2 λ + µ A(ξ ). |ξ | λ + 2µ

(6.14.14)

Since σyy (x, y) = σyy (−x, y) and v(x, 0) = v(−x, 0) we have A(−ξ ) = A(ξ ) and the remaining boundary conditions (6.14.9) reduce to ∞ A(ζ )ζ cos(ζ x) dζ = T0 , x < 1, 0 ∞ A(ζ ) cos(ζ x) dζ = 0, x ≥ 1, (6.14.15) 0

where A(ξ ) = A(ζ ), ξ L = ζ, √ 2π λ + 2µ T0 = t0 L2 . 4µ λ + µ

x = x/L, (6.14.16)

The solutions of the dual integral equations (6.14.15) are well known in fracture mechanics (cf. Sneddon [1951, p. 424]: A(ξ ) = T0 J1 (ζ )/ζ,

B(ξ ) =

λ+µ |ζ |A(ζ ). λ + 2µ

(6.14.17)

Through (6.14.12) and (6.14.13) we can calculate u, v, σxx , σyy , and σxy . The stress tensor tkl is calculated by (6.14.3). Here we are interested in the behavior of the hoop stress along the crack line. After evaluating the integral over x and y in (6.14.3) we obtain ∞ H (ζ )J1 (ζ ) cos(ζ x) dζ, (6.14.18) tyy (x, 0)/t0 = − 0

where H (ζ ) = [(1 + 2 ζ 2 )1/2 + ζ ]−1 + ζ [(1 + 2 ζ 2 )1/2 + ζ ]−2 , ≡ l/L.

(6.14.19)

136


The integral in (6.14.18) is to be evaluated numerically. However, it is more convenient to employ the asymptotic expansion of J1 (ζ ): J1 (ζ ) ∼ ζ −1/2 cos(ζ − 3π/4) + O(ζ −3/2 ),

|ζ | → ∞.

Using this in (6.14.18), to the order ( 0 ), tyy (x, 0) is obtained: √ + χ tyy (x, 0)/t0 = π/2 (I1/4 K3/4 + I3/4 K1/4 ) √ χ + (I1/4 K3/4 − I5/4 K7/4 + I3/4 K1/4 − I7/4 K5/4 ) , 2 χ = x − 1, (6.14.20) where Iν and Kν are modified Bessel’s functions of order ν they are evaluated at χ/2. At the crack tip, x = 1, and we obtain tyy (1, 0)/t0 = 0.57446 −1/2 .

(6.14.21)

However, hoop stress is not maximum at the crack tip. Table 6.14.1 gives the location and value of the hoop stress near the crack tip. From Table 6.14.1 it is clear that tyy max = 1.12tyy (1, 0),

(6.14.22)

x = x/L = 1 + 0.5(l/L).

(6.14.23)

and this maximum occurs at

From (6.14.21) and (6.14.22) we obtain a Griffith-like fracture criterion by equating tyy max to the cohesive tensile stress tc , i.e., t02 L = Kc ,

Table 6.14.1. Hoop stress near the crack tip. χ/2 0.0 0.1 0.2 0.25 0.3 1.0 1.5 2.0

tyy (x, 0)/tyy (1, 0) 1.00 1.09 1.10 1.12 1.10 0.89 0.75 0.66

(6.14.24)


137

Figure 6.14.2. Hoop stress near the crack tip. After Ari and Eringen [1983]. where Kc = 0.531tc2 a.

(6.14.25)

The normal stress tyy along the crack line given by (6.14.20) is plotted in Figure 6.14.2 as a function of x = x/ l for = 0.22a/L, where a = l is taken as the lattice parameter. This graph also gives the results obtained by Elliott [1947] and Eringen et al. [1977]. 2L/a = 1000 and ν = 0.25 used by Elliott is also used here. Quantitative agreement is very good considering the fact that we are dealing with atomic size cracks and using a nonlocal continuum theory.

138


Figure 6.15.1. Line crack under shear.

6.15 Line Crack Subject to Shear13 Here we present the solution of another important crack problem via nonlocal elasticity. Here a line crack, sliced in an isotropic elastic solid along the x-axis at −L < x < L, y = 0 and the crack surface, is subjected to a constant shear stress (Figure 6.15.1). In this case, the boundary conditions can be reduced to σyx = τ0 , σyy = 0, u(x, 0) = 0, {u, v} → 0

y = 0,

|x| < L,

y = 0 ∀ x, y = 0, |x| > L, as

(x 2 + y 2 )1/2 → ∞.

(6.15.1)

The solution of this problem was given by Eringen [1978]. The general solutions of the Navier equations are the same as given by (6.14.12) and (6.14.13), with A(ξ ) and B(ξ ) to be determined by use of the first three boundary conditions. Using σyy (x, 0) = 0, B(ξ ) is determined as

B(ξ ) =

λ+µ |ξ |A(ξ ). µ

(6.15.2)

Noting that A(−ξ ) = −A(ξ ) the displacement and stress fields are reduced to the forms 13 Eringen [1978].

6.15 Line Crack Subject to Shear

u(x, y) v(x, y) σxx (x, y) σyy (x, y) σyx (x, y)

139

1/2 λ + 2µ λ+µ ∞ 2 − ξy e−ξy cos(ξ x) dξ, A1 (ξ ) = λ+µ π µ 0 1/2 λ+µ ∞ µ 2 A1 (ξ ) + ξy e−ξy sin(ξ x) dξ, = π µ λ+µ 0 1/2 ∞ 2 = −2 (λ + µ) A1 (ξ )(2ξ − ξ 2 y)e−ξy sin(ξ x) dξ, π 0 1/2 ∞ 2 = −2 (λ + µ) A1 (ξ )ξ 2 ye−ξy sin(ξ x) dξ, π 0 1/2 ∞ 2 (λ + µ) A1 (ξ )ξ(1 − ξy)e−ξy cos(ξ x) dξ,(6.15.3) = −2 π 0

where we wrote A(ξ ) = iA1 (ξ ). The remaining two boundary conditions lead to , 2 ∞ C(ζ )ζ 1/2 cos(ζ x) dζ = −T0 , π 0 ∞ C(ζ )ζ −1/2 cos(ζ x) dζ = 0,

0 < x < 1, x > 1,

(6.15.4)

0

where ζ −1/2 C(ζ ) = A(ξ ), x = x/L, 2 T0 = τ0 L /2(λ + µ).

ζ = ξ L, (6.15.5)

The solution of this dual integral equation is given in classical fracture mechanics , π C(ζ ) = − (6.15.6) T0 ζ −1/2 J1 (ζ ), 2 where J1 (ζ ) is the Bessel’s function. With σkl given by (6.15.3) the nonlocal stress tensor can be calculated by tkl = α(|x − x|)σkl (x ) da(x ). (6.15.7) R

Eringen [1978] uses a nonlocal kernel of the type α(|x − x|) =

1 (β/a)2 exp[−(β/a)2 (x − x) · (x − x)], π

(6.15.8)

where β is an arbitrary constant length and a is the lattice parameter. With this tyy (x, 0) is calculated through (6.15.7) and σyx is given by (6.15.3): ∞ tyx /τ0 = K(ζ )J1 (ζ ) cos(ζ x) dζ, (6.15.9) 0

140


Figure 6.15.2. The behavior of K(ζ ) versus ζ . After Eringen [1978].

where 2 K(ζ ) = (1 + 2ζ 2 )[1 − (ζ )] − √ ∈ ζ exp(− 2 ζ 2 ), π ≡ a/2βL. (6.15.10) Here (z) is the error function defined by (z) = 2π −1/2

z

exp(−t 2 ) dt.

0

K(ζ ) is displayed in Figure 6.15.2 as a function of ζ . Calculations of the stress field (6.15.9) have been carried out on computers. The results are displayed in Figures 6.15.3 to 6.15.5 for = 1/20, 1/50, 1/100, and 1/200. It can be seen that satisfaction of the boundary condition t21 (x, 0) = τ0 for |x| < L is not very good for large values of . However, for = 1/100 and the lower boundary condition on tyx (x, 0) for |x| < L is satisfied in a strong approximate sense. The relative error is less than 1.5%. Thus, we conclude that the classical solution (6.15.6) for C(ζ ) leads to satisfactory results for 2βL/a ≥ 100. The stress concentration occurs near the crack tip, and is given by tyx (L, 0)/τ0 = c −1/2 ,

(6.15.11)


141

Figure 6.15.3. t21 /τ0 versus x1 /L for = 1/20 and = 1/50. After Eringen [1978]. where c converges to about −0.30, i.e., c −0.30.

(6.15.12)

The following significant observations may be made: (i) The maximum shear stress occurs near the crack tip with a finite value. (ii) The shear stress at the crack tip becomes infinite as the internal characteristic length a → 0. This is the classical elasticity limit of the square root singularity. (iii) When tyx max = tc = (cohesive shear stress), fracture will occur. In the case τ02 L = KG ,

(6.15.13)

where KG is a constant given by KG = (a/2βc2 )tc2 .

(6.15.14)

Expression (6.15.13) is none other than the Griffith criterion for brittle fracture.

142


Figure 6.15.4. t21 /τ0 versus x1 /L for = 1/100. After Eringen [1978]. We have arrived at this criterion via a maximum stress hypothesis unifying the classical macroscopic concept of fracture in the strength of materials. (iv) The cohesive stress may be estimated if one employs the Griffith definition of surface energy in terms of which KG was given originally, namely KG = 4µγ /π(1 − ν). Equating this to (6.15.14) we have tc2 l = Kc γ ,

(6.15.15)

Kc = 8µc2 β/π(1 − ν).

(6.15.16)

where Employing the values of γ and the elastic constants γ = 1975 CGS, ν = 0.291,

µ = 6.92 × 1011 CGS, ◦

l = a = 2.48 A,


143

Figure 6.15.5. t21 /τ0 versus x1 /L for = 1/200. After Eringen [1978]. where a is the atomic lattice parameter, we obtain + tc /µ = 0.19287 β.

(6.15.17)

Suppose that at a radial n atomic distance, r = na, the nonlocal effects attenuate to 1% of its value at r = 0. Using (6.15.8) we find that √ β = 2.146/n, tc /µ = 0.2825/ n. (6.15.18) For n = 4 and 6 this gives tc /µ = 0.141, tc /µ = 0.115,

n = 4, n = 6.

These results are in the right range, well accepted by materials scientists. For example, Kelly [1966] gives tc /µ = 0.11. The solution of the problem “line crack,” subject to antiplane shear in a nonlocal elastic solid, was given by Eringen [1979].

144


Figure 6.16.1. Crack subject to antiplane shear (Mode III).

6.16 Interaction of a Dislocation with a Crack14 A homogeneous, isotropic elastic solid of infinite extent is weakened by a crack located at −L < x1 < L, x2 = 0, −∞ < x3 < ∞, where xk are rectangular coordinates (Figure 6.16.1). A dislocation intersects the plane x3 = 0 at S (x1 = ξ , x2 = η). The solid is subject to a constant antiplane shear at x2 = ±∞. The classical solution of this problem was given by Louat [1965]. Here we obtain the solution of this in a form better for our purposes, eliminating notational problems and taking limits. The problem is two dimensional. The classical stress field is given by ∗L f (t) σ23 − iσ13 = A dt, (6.16.1) z −t −L √ where z = x1 − ix2 , i = −1, and f (t) is the distribution function which is the solution of the equation of equilibrium of the forces acting on the crack surface ∗L f (t) dt = σd (x) + σ0 , A = µλ0 /2π. (6.16.2) A t −x −L Here the integral denotes a Cauchy principal value, µ is the shear modulus, λ0 is the displacement vector of a unit positive dislocation, and σ0 and σd (x) are the stress fields at the crack surface due to the applied load and the dislocation, respectively. The solution of the integral equation (6.16.2) is well known, Tricomi [1951]: ∗L + dt 1 Q L2 − t 2 [σ0 + σd (t)] , f (x) = − +√ √ 2 2 2 2 t −x π A L − x −L L − x2 (6.16.3) 14 Eringen [1983].

6.16 Interaction of a Dislocation with a Crack

145

where Q is a constant to be determined from the condition that

L −L

f (x) dx = n.

(6.16.4)

Here nλ0 is the total dislocation content of the distribution function f (x). The stress field σd (t) is given by σd (t) =

µb(t − ξ ) . 2π[(t − ξ )2 + η2 ]

(6.16.5)

Substituting this into (6.16.3) and integrating we obtain     . +   2   2 ζ −L 1 σ0 x b  ζ 2 − L2  f (x) = √ − 1 + Q , − +   πλ0 2(ζ − x) 2(ζ − x) L2 − x 2   πA  (6.16.6) where ζ = ξ + iη, ζ = ξ − iη. Using (6.16.4) we get Q = n/π and carrying f (x) into (6.16.1), after some tedious integrations, we obtain + 1 z ζ 2 − L2 bA σ23 − iσ13 = σ0 + 1− + −1 − 2λ0 z − ζ z2 − L2 z2 − L2   . 2 ζ − L2  1  b A + +n + . (6.16.7)  + 1 − + λ0 z−ζ z2 − L2 z2 − L2 The forces acting on the dislocations at (ξ, η), due to the crack, are given by F1 = bσ23 ,

F2 = bσ13 ,

(x1 = ξ, x2 = η).

(6.16.8)

If the dislocation is located along the x1 -axis and the crack is free of traction we set η = 0 and add bA σ0 + σd (x1 ) = σ0 + (6.16.9) λ0 (z − ξ ) to the right-hand side of (6.16.7) to obtain the stress field at any point outside of the crack σ23 − iσ13 + σ0 + σd (x1 ) + 1 b bA ξ 2 − L2 σ0 z + A = + +n + ,(6.16.10) λ0 λ0 z − ξ z2 − L2 with the body loaded at x2 = ±∞ with a constant shear σ23 = ±σ0 . If the crack contains no dislocations then n = 0. Two special cases are important:

146


(i) No Crack and σ0 = 0. In this case the Hookean (classical) stress field is given by µb σ = . (6.16.11) 2πz The dislocation is now at the origin of the coordinates. (ii) No Dislocation. In this case A = 0 and we have σ =+

σ0 z z2 − L2

.

(6.16.12)

Both of these results are well known in the literature. The nonlocal stress field is obtained by solving the differential equation (1 − l 2 ∇ 2 t) = σ,

(6.16.13)

where t = t23 − it13 ,

σ = σ23 − iσ13 + σ0 + σd (x1 ).

(6.16.14)

Since ∇ 2 σ = 0, t = σ is a special solution of (6.16.13), the complimentary solution of (6.16.13) is tc = Kν (r/ l)(Aν eiνθ + Bν e−iνθ ),

(6.16.15)

where Aν , Bν , and ν are constants, Kν (p) is a modified Bessel’s function of order ν, and (r, θ) are the polar coordinates. The boundary conditions on the crack surface is t23 = 0. Taking the origin r = 0 of the coordinates at the right-hand crack tip and writing r = r1 , θ = θ1 in (6.16.15), the boundary condition is fulfilled if we set ν = 1/2, since all other solutions lead to singularities at r1 = 0. The classical stress field σ is singular at the screw dislocation x1 = ξ , x2 = 0. The surface traction trz on the edge surface of the dislocation must also vanish, in accordance with the boundary conditions. To fulfill this condition we take ν = 1 and move the origin to the coordinates to x1 = ξ , x2 = 0. This is expressed by writing r = rd , θ = θ3 and rd eiθ3 = r1 eiθ1 − x0 .

(6.16.16)

The general solution of (6.16.13), appropriate to the present problem, is then given by t = (πl/2r1 )1/2 e−r1 / l (C1 eiθ1 /2 + C2 e−iθ1 /2 ) + K1 (rd / l)(C3 eiθ3 + C4 e−iθ3 ) + σ.

(6.16.17)

In order to determine Ci we express the stress field in polar coordinates tθ z − itrz = (t23 − it13 )e−iθ1 .

(6.16.18)


147

We imagine the crack tip as the limit of a small circle with radius r1 = approaching zero. For small we have, approximately, z = L + z1 = −L + z2 = ξ + z3 = L + x0 + z3 ,

z1 = eiθ1 .

(6.16.19)

Using these in (6.16.10) we obtain L 1/2 iθ1 /2 nλ0 µb 2L 1/2 µb σ = σ0 + . 1+ − 1+ e 2r1 2πL b 2πL x0 (6.16.20) Equation (6.16.17) now gives nλ0 µb 2L 1/2 L 1/2 −iθ1 /2 µb tθ z − itrz = σ0 + 1+ − 1+ e 2r1 2πL b 2πL x0 1/2 πl e−r1 / l (C1 e−iθ1 /2 + C2 e−3iθ1 /2 ) + 2r1 r1 r1 + K1 (|x0 |/ l) C3 eiθ1 − C3 + C4 e−iθ1 − C4 e−iθ1 . (6.16.21) x0 x0 The boundary condition on trz requires that lim trz = 0.

r1 →0

This condition is fulfilled approximately (x0 / l 1) if C2 = 0 and 1/2 nλ0 µb 2L 1/2 L µb C1 = − σ0 + . (6.16.22) 1+ − 1+ πl 2πL b 2πL x0 Considering the internal characteristic length l = a = atomic lattice parameter, x0 /a 1 is certainly fulfilled if the dislocation is located at several atomic parameters away from the crack tip. At the core of dislocation, as rd → 0, trz must vanish. This gives C4 = 0 and µb . (6.16.23) 2πL The general solutions satisfying all the boundary conditions, are then given by πl 1/2 −r1 / l e C1 eiθ1 /2 + K1 (rd / l)C3 eiθ3 + σ, (6.16.24) t= 2r1 C3 = −

with C1 and C3 given by (6.16.22) and (6.16.23). In polar coordinates this reads πl 1/2 −r1 / l e C1 e−iθ1 /2 + K1 (rd / l)C3 ei(θ3 −θ1 ) tθ z − itrz = 2r1 µb nλ0 + (r1 r2 )−1/2 ei(θ2 −θ1 )/2 σ0 re−iθ + 1+ 2π b 1/2 µb . (6.16.25) + (r1 e−iθ1 − x0 )−1 [x0 (x0 + 2L) 2π

148


The special cases (i) and (ii) can now be obtained from this result. (i) No Crack tθ z − itrz =

µb 1 [1 − ρK1 (ρ)], 2πL ρ

(6.16.26)

with the origin of coordinates at the dislocation point, so that ρ = rd / l. (ii) No Dislocation, but Crack 2 1/2 2r L 1/2 i(−θ +θ2 /2) −r1 / l e−iθ1 /2 . (6.16.27) e −e tθ z − itrz = σ0 2r1 Lr2 It is clear that tθ z acquires its maximum near the crack tips. tθ z , along the crack line r1 ≥ 0, is given by γρ −1/2 −1/2 −ρ tθ z /σ0 = (2γρ) (1 + γρ) 1 + , (6.16.28) −e 2 where ρ = r1 / l, γ = l/L. tθ z = 0 at the crack tip ρ = 0, and has a maximum at ρ = ρc which is the root of γρ −3/2 . e−ρ (1 + 2ρ) = 1 + 2

(6.16.29)

Since γ 1 (γ ≤ 10−6 ), we see that ρc is independent of L and is given by ρc 1.2565.

(6.16.30)

The maximum stress is given by tθ z max = σ0

−1/2 −1 + l 1 2ρc + √ . L 2ρc

(6.16.31)

From (6.16.28) we also observe that tθ z → σ0 as ρ → √ ∞, as it should be. Following the classical tradition, if we express KIII = πLσ0 , then (6.16.28) may be put into the form √ √ Tθ (ρ) = γ tθ z /σ0 = πltθ z /KIII γρ −1/2 (6.16.32) = (2ρ)−1/2 1 + γρ 1 + − e−ρ . 2 This is displayed in Figure 6.16.2 in the vicinity of the crack tip. The classical result is also shown by a dotted line. From the figure, it is clear that classical shear deviates considerably from the nonlocal one, in the region 0 ≤ ρ < 5. In fact, it becomes infinite at the crack tip.


149

Figure 6.16.2. Nondimensional shear (no dislocation). After Eringen [1979]. A perfect crystal will rupture when tθ z max = ty , where √ ty is the cohesive yield stress. At this value KIII takes its critical value Kc = πlσ0c so that + √ Kc 1 3.9278 l. = (πl)1/2 2ρc + √ (6.16.33) ty 2ρc In (6.16.33) we may set l = e0 a, where a is the lattice parameter, and e0 is determined by matching the plane shear wavelength at the end of the Brillouin zone with that of the lattice dynamics, as discussed in Section 6.9 (e0 = (π 2 − 4)1/2 /2π 0.39). √ Table 6.16.1 gives a comparison of Kc = 4µγs (classical), K (experiments of Ohr and Chang [1982], and Kg (present nonlocal). Classical estimates, based on the surface energy γs , are expected to be inaccurate, considering the fact that even with the best present-day techniques availabe, γs could not be measured to an accuracy better than a factor of 2. On the other hand, the Ohr and Chang experiments required the measurement of the length of the plastic zone among the other constants. This implies the existence of dislocations. Thus, the nonlocal results are expected to be more faithful to the real values.

150


Table 6.16.1. Critical stress intensity factors.

Material Al (fcc) Cu (fcc) Ni (fcc) Fe (bcc)

a (10−8 cm)∗ 4.05 3.61 3.52 2.87

∗ Kittel [1974].

∗∗ Rice and Thomson [1974]. # Kelly et al. [1967]. ## Ohr and Chang [1982].

µ (1011 cgs)∗∗ 2.51 4.05 7.48 6.9

γs (cgs)∗∗ 840 1688 1725 1975

ty (1011 cgs)# 0.262 0.137 0.274 0.71

Classical Kc /ty (10−3 cm1/2 ) 1.11 3.86 2.62 1.04

Experiment## K/ty (10−3 cm1/2 ) 0.31 0.66 0.66 0.23

Present Kg /ty (10−3 cm1/2 ) 0.49 0.47 0.46 0.42


151

Figure 6.16.3. Nondimensional shear stress due to dislocation interaction. After Eringen [1983].

Dislocation and Crack The stress field is given by (6.16.25) along the crack line x1 ≥ L, x2 = 0 gives trz = 0 and with r = 0: tθ z = t c + t dc ,

(6.16.34)

where γρ −1/2 −ρ (1 + γρ) 1 + , −e t γ /σ0 ≡ T1 = (2ρ) 2 1/2 √

−1/2 t dc γ γρ 2 e−ρ − 1− 1+ = T2 = (2ρ)−1/2 1 + σ0 β 2 γ x0 1/2 2ρ − sgn(ρ − x 0 ) K1 (|ρ − x 0 |) γ −1 γρ −1/2 2 1/2 ρ + 1+ , 1+ −1 2 γ x0 x0 c√

−1/2

x 0 = x0 /e0 a,

β=

µb . 2πLσ0

(6.16.35)

152


When the dislocation is absent t dc = 0. Consequently, t dc is the shear stress due to the interaction of the dislocation with the crack. At the crack tip ρ = 0 and we have (6.16.36) tθ z /σ0 β = γ −1 K1 (x 0 ). This shows that for positive dislocation (b > 0) the shear stress is positive and the crack tip will tend to close up for b > 0. For b < 0 the crack will open. However, the stress given by (6.16.36) tends to zero quickly with distance. It becomes large when the dislocation is very close to the crack tip. Figure 6.16.3 displays T2 (ρ) for several values of x 0 = (1.7, 3.1, 4.1, 6.1, and 10.1) keeping γ = 10−8 fixed. The crack tip is not stress free. The maximum of T2 occurs close to dislocation. For example, in the case x 0 = 3.1, the maximum occurs at x = 4.4 and in the case x 0 = 10.1 at x = 11.2. In order to get an idea of the combined effects, in Figure 16.6.4, the ratio tθ z /σ (for the combined stress) is plotted with β = 10−4 . When dislocation is located close to the maximum of T1 , the combined effect is large. For example, while T1 max 0.451 for the crack alone, for the combined effect Tmax = 0.63, and Kg tot /Kg1 0.71.

(6.16.37)

This suggests that a dislocation, located at a distance of two lattice parameters away from the crack tip, reduces the fracture toughness by about 30%.

Figure 6.16.4. Nondimensional shear stress due to dislocation interaction with crack. After Eringen [1983].

6.17 Interaction Between Defects and Dislocation

153

A more realistic picture requires the interaction of a large number of dislocations with the crack (dislocation pile-up near the crack).

6.17 Interaction Between Defects and Dislocation Crystalline solids contain dislocations and point defects whose interactions with solids are responsible for the many properties of imperfect crystals. Materials scientists have studied these interactions via the vehicle of classical (local) continuum mechanics. In fact, these topics occupy several chapters in several books on dislocations (cf. Nabarro [1967], Hirth and Lothe [1968], Teodosiu [1982]). However, most of the previous calculations were carried out within the framework of classical linear elasticity, which led to divergent interaction energies with the decreasing distance between the point defects and dislocation lines. Recently, Vörös and Kovács [1995] have studied some of these interactions by means of the nonlocal elasticity. Here we give an account of the subject. As we know, a dislocation is a discontinuity of the displacement field along a cut surface. The volume defect is characterized by the total volume change of the body caused by a displacement. In classical elasticity, sources (such as displacement fields) are characterized by the centers of force dipoles. These dipoles are considered localized to the place of the defect.

A. Point Defect Suppose that a displacement field in a nonlocal elastic solid is discontinuous along a surface S. We can determine the displacement field elsewhere, by solving the equations of equilibrium λiklm,i (x − x )ul,m (x ) dv + ρfk = 0. (6.17.1) V

The stress field is given by tik =

V

λiklm (x − x )ul,m (x ) dv .

The Green’s function for (6.17.1) must satisfy λiklm,i (x − x )Glj,m (x − x

) dv + δj k δ(x − x

) = 0, V

(6.17.2)

(6.17.3)

where Glj is the Green’s function, δ(x) is the Dirac-delta measure, and the subscript

. Multiplying (6.17.3) by u (x

), and m denotes differentiation with respect to xm k integrating over the volume, we obtain uj (x) = − λiklm,i (x − x )Glj,m (x − x

)uk (x

) dv dv

. (6.17.4) V

V

154


Using the symmetry property of the Green’s function and exchanging x and x

, this equation can be written in the form uj (x) = − λiklm,i (x

− x )Glj,m (x − x)uk (x

) dv dv

. (6.17.5) V

V

This volume integral may be transformed to a surface integral by disclosing from the total volume the surface S, by a closed surface consisting of surfaces S + and S − that are parallel to S. Integrating by parts and using the Green–Gauss theorem, in the limit S + , S − → S, we obtain uj (x) = (6.17.6) λiklm (x

− x )Glj,m (x − x )[uk (x

)] dv dai

, S

where

V

[uk (x

)] =

lim

S + ,S − → S

uk (x

)S + − uk (x

)S −

(6.17.7)

is the jump of the displacement vector along the surface S. For example, if [u(x

)] = b = const., then (6.17.6) gives the displacement field of a dislocation. Suppose that the defect is located inside S at a point P with the position vector x0 (Figure 6.17.1), then x

can be written as

where r varies along S and

x

= x0 + r,

(6.17.8)

|r| |x0 − x |.

(6.17.9)

Substituting (6.17.8) into (6.17.6), to a first-order approximation we have uj (x) = λiklm (x0 − x )Glj,m (x − x ) dv

[uk (x0 + r)] dair V S

1 = 2 λiklm (x0 − x )Glj,m (x − x ) dv {[uk (x0 + r)] dair V

+ [ui (x0 + r)] dakr },

S

where we used the symmetry regulation λiklm = λkilm . The quantity aki (x0 ) = 21 {[uk (x0 + r)] dair + [ui (x0 + r)] dakr } S

(6.17.10)

(6.17.11)

is called the kinematical tensor of the defect. This can be used to characterize the basic properties of the point defects. Substituting (6.17.11) in (6.17.10), we have uj (x, x0 ) = Plm (x , x0 )Glj,m (x − x ) dv , (6.17.12) V

6.17 Interaction Between Defects and Dislocation

155

z

S

r x0 x″

y

x

Figure 6.17.1. Small surface of discontinuity. where

Plm (x , x0 ) = λiklm (x0 − x )aik (x0 )

(6.17.13)

characterizes the sources off elastic displacement due to a given lattice defect. These arise from the force dipoles and they may be named elastic poles. In the special case λiklm (x0 − x ) = λ0iklm δ(x0 − x ), where λ0iklm is constant, (6.17.12) reduces to the classical form obtained by Kröner [1981] and by Gairola [1982]. The strength of a pole is defined by 0 Plm = Plm (x , x0 ) dv = λ0iklm aik (x0 ) = Plm (6.17.14) V

and is the same for local as well as nonlocal theory, since λiklm (x0 − x ) dv = λ0iklm . V

The Fourier transform of (6.17.3) is given by ki km Gij (k)λiklm (k) + δij = 0. In the Fourier space an isotropic quasi-continuum may be defined by 0 λiklm , |k| ≤ k0 , λiklm (k) = 0, |k| > kB ,

(6.17.15)

(6.17.16)

156


where λ0iklm is the constant classical elastic modulus and kB is the wave number at the boundary of the first Brillouin zone. From (6.17.15) and (6.17.16) it follows that in the region |k| ≤ kB the Green’s function G and GQ of the local- and quasi-continuum are the same, i.e., G0ij (k) = Gik (k).

(6.17.17)

The Fourier transform of (6.17.12), upon using (6.17.13) and (6.14.14), gives uj (k) = −ikm λ0iklm Glj (k)aik (x0 ) exp(ik · x0 ).

(6.17.18)

Hence, in the first Brillouin zone |k| ≤ kB , the displacement field uj (k), due to the defect, is the same for both the local- and quasi-continuum, and is determined once the kinematical tensor of the defect is known. The inverse Fourier transform of (6.17.18) determines the displacement field in the physical domain. From these it follows that the strain and stress tensors are given by Q eik (k) = eik (k), when |k| ≤ kB , Q tik (k) = σik (k), Q

eik (k) = 0,

Q

tik (k) = 0

when

|k| > kB ,

(6.17.19)

where eik and σik denote, respectively, the strain and stress tensors arising from the defect in the local continuum. Using (6.17.13) and (6.17.14) we can also determine the displacement field, from (6.17.18), of an elastic pole in the Fourier space uj (k) = −ikm Plm Glj (k),

|k| ≤ kB ,

(6.17.20)

where 0 (x0 ) exp(ik · x0 ), Plm (k) = λ0iklm aik (x0 ) exp(ik · x0 ) = Plm |k| ≤ kB . (6.17.21)

B. Interaction Energy Interaction energy between the two defects A and B are given by 1 ∗A A B B elm tlm dv = tlm (k) e lm (k) d 3 k, UK = 3 (2π) V |k| 0 upon whose boundary y = 0 is pressed a rigid punch with a flat surface, Figure 6.22.1. All z = const.-planes are considered to be equivalent so that the problem is two-dimensional and all quantities are independent z-coordinates. The boundary conditions are tyx = 0, v = v0 , tyy = 0,

y = 0, y = 0, y = 0,

u, v → 0,

−∞ < x < ∞, |x| < L, |x| > L,

(x 2 + y 2 )1/2 → ∞,

(6.22.1)

where v0 = const. These conditions are equivalent to19 σyx = 0, v = v0 , σyy = 0,

y = 0, y = 0, y = 0,

u, v → 0,

−∞ < x < ∞, |x| < L, |x| > L,

(x 2 + y 2 )1/2 → ∞.

(6.22.2)

In Section 6.21, we have shown that the displacement field of a nonlocal elastostatic field is identical to that of the classical elasticity. The general solution of the Navier equation of classical elasticity was given by (6.14.13). Using the first of the boundary conditions σyx (x, 0) = 0 in (6.14.13) we obtain B(ξ ) =

λ + µ ξ2 A(ξ ). λ + 2µ |ξ |

(6.22.3)

Substituting this into the expression v(x, y) given by (6.14.12)2 and into σyy (x, y) given by (6.14.13)2 , we obtain, for the remaining boundary conditions,

∞

0

ζ 2 A1 (ζ ) cos(ζ x) dζ = 0,

∞

ζ A1 (ζ ) cos(ζ x) dζ =

0 18 Eringen and Balta [1979]. 19 Eringen and Balta have used (6.22.1).

+

π/2v0 ,

x > 1, 0 < x < 1,

(6.22.4)

172


y

L

L

x

O

v0

P

Figure 6.22.1. Rigid stamp on nonlocal elastic half-space. where A(ξ ) = A1 (ζ )ζ L,

ξ L = ζ,

x/L = x.

(6.22.5)

The dual integral equations (6.22.4) are identical to those obtained in classical elasticity (cf. Sneddon [1951, p. 434]). The solution of (6.22.4) is then given by A1 (ζ ) = Kζ −2 J0 (ζ ),

(6.22.6)

where J0 (ζ ) is the Bessel’s function of order zero and K is a constant. In fact, substituting A1 (ζ ) into (6.22.4)1 we find that it is satisfied identically. Also, the derivative of (6.22.4)2 with respect to x is null so that it is also satisfied. K is related to v0 . However, we would like to determine K in terms of the total load P (per unit depth) exerted by the punch. The true stress field in nonlocal media is given by α(|x − x |)σkl (x ) dv(x ), (6.22.7) tkl = V

6.22 Rigid Stamp on a Nonlocal Elastic Half-Space

173

where α is a function of |x − x | and x, since near the surface, y = 0, the half-space looses its homogeneity. This is because the intermolecular attractions are missing from the y < 0− side of the surface. An approximate attenuating kernel is given by α(|x − x |, y ) = α0 (y) exp[−(k/a)2 (x − x) · (x − x)], (6.22.8) subject to

V

α(|x − x|, y) dv(x ) = 1,

which gives for α0 : α0 (y) = 2(k/a)3 π −3/2 [1 + erf(ky/a)]−1 .

(6.22.9)

The constant K in (6.22.6) is determined by the condition that the applied load P by the stamp is balanced with tyy (x, 0), i.e., L P =− tyy (x, 0) dx. (6.22.10) −L

Carrying B(ξ ) from (6.22.3) and A(ξ ) from (6.22.5) into σyy , given by (6.14.13)2 , we obtain 4µ(λ + µ) 1 ∞ σyy = − √ A1 (ζ )ζ 2 (1 + ζ y)e−ζ y cos(ζ x) dζ. (6.22.11) 2π (λ + 2µ) L 0 With this and (6.22.6), tyy becomes tyy

4µ(λ + µ) = −√ α0 (y)KL2 2π (λ + 2µ)

∞

dζ 0

∞ −∞

dx

∞

dy

0

∞ −∞

dz

× exp{− −2 [(x − x)2 + (y − y)2 + (z − z)2 ]}J0 (ζ )(1 + ζ y )e−ζ y cos(ζ x).

(6.22.12)

Integrations with respect to z , y , and x can be performed, resulting in K ∞ tyy (x, y, ) = F (y, ζ, ) cos(ζ x) dζ, (6.22.13) L 0 where

√

2µ(λ + µ) F (y, ζ, ) = − Exp[−(y/2)2 − (ζ )2 ] π(λ + 2µ)[1 + Erf(y/)] √ · 4ζ + π(2 + 2yζ − 4 2 ζ 2 ) 2 y y · Exp − + ζ Erfc − + ζ J0 (ζ ), 2 2 a = 2kL

(6.22.14)

174

6 Nonlocal Linear Elasticity (a)

T (x , )

x T (x , )

(b)

x

Figure 6.22.2. (a) Nondimensional normal stress at the surface. After Eringen and Balta [1980]. (b) Nondimensional normal stress at the surface. After Eringen and Balta [1980].

6.22 Rigid Stamp on a Nonlocal Elastic Half-Space

175

K is determined from (6.22.10): K = −P /2Q, ∞ sin(ζ x) dζ. F (0, ζ, ) Q= ζ 0 The nondimensional stress is then given by ∞ F (y, ζ, ) cos(ζ x) dζ. T (x, y, ) = 2Ltyy /P = −Q−1

(6.22.15)

(6.22.16)

0

Eringen and Balta carried out calculations for the surface stress field tyy (x, 0) and the surface displacement v(x, 0) as functions of x for = 1/50, 1/100, 1/300, 1/500, 1/750, 1/1000. In Figure 6.22.2, T (x, ) is plotted against x for various values of . From these figures, it is clear that for values 1/100, the boundary conditions are satisfied in a strong approximate sense. The classical solution (6.22.6) gives excellent results for < 1/100. A ratio = a/2L greater than this, is probably impossible in practice, since the punch width of the order 10−6 cm is of submicroscopic size. Consequently, the solution is extremely accurate for the macroscopic punch. From these calculations we also have √ (6.22.17) Tmax = max(tyy /p0 ) = Cp / , where Cp = 0.25. This result shows that as a → 0, (6.22.16) produces the same square root singularity well known in the classical theory of elasticity. Even more importantly, by setting tmax = cohesive stress that holds the atomic bonds together, we obtain the critical pressure for the beginning of the penetration of the punch into the half-space 2 pcr (6.22.18) l = tc2 a/2kCp2 . From this we deduce that where P /2l = p0 = pcr , the penetration begins. Such results cannot be obtained through classical elasticity since it predicts tmax = ∞.

Problems 6.1 Give constitutive equations of the nonlocal elastic solids for hexagonal crystals. 6.2 A linear chain of two different types of atoms A and B are attached to each other by linear springs in equal distances, in a sequence ABA. . .. Using harmonic approximation determine the dispersion relations. Then obtain δB for a quasi-continuum approximation. 6.3 For nonlocal isotropic elastic solids determine the inequalities that are to be placed on the material moduli, by the nonnegative character of the strain energy.

176


6.4 Theorem 1 expressed by (6.5.10) is important to the uniqueness theorem. Carry out the analysis indicated in all details. 6.5 In an infinite nonlocal elastic continuum, dislocations are distributed uniformly on a spherical surface in the body. Determine the stress field. 6.6 Derive expression (6.14.18) of the hoop stress in the Griffith crack problem and verify (6.14.19) and (6.14.20). 6.7 Obtain the expression of ∇ 2 tkl in spherical coordinates, in terms of the components of tkl referred to spherical coordinates. 6.8 Give a detailed derivation of (6.19.12) and (6.19.13). 6.9 An infinite nonlocal elastic space is subjected to a couple at x = 0. Determine the stress field in the medium. 6.10 Obtain the nonlocal solution of the elastic half-plane loaded by a couple at its surface.

7 Nonlocal Fluid Dynamics1

7.0 Scope The linear constitutive equations of nonlocal thermoviscous fluids are obtained in Section 7.1. The energy equation is given. Field equations are given in Section 7.2. In Section 7.3 we give the solution of the channel flow problem. Here, the velocity profile come out parabolic, the same as in classical fluid dynamics. However, the shear stress is quite different. In fact, depending on the ratio of the internal characteristic length to the channel depth, the shear stress is reduced to nearly 50% in nonlocal fluids. In Section 7.4 we discuss lubrication problems in microscopic channels. Here, viscosity is affected by the microinertia of the fluid. Moreover, the nonlocal surface stress becomes quite effective, so that these two new elements must be taken into account. Once this is done, the experimental high viscosity and drainage of fluid have agreed very well with the calculations based on the nonlocal theory. Lubricant film flow on a rotating disk is discussed in Section 7.5. Again, agreement with experimental results is very good.

1 Eringen [1972b]. For memory-dependent, orientable nonlocal fluid dynamics, see Eringen [1991a].

178

7 Nonlocal Fluid Dynamics

7.1 Constitutive Equations In Section 3.3 we obtained the general constitutive equations of the nonlocal viscous fluids. They are given by (3.3.15) and (3.3.26), namely, tkl = −πδkl + D tkl , ∂G

dv , D tkl = V ∂dkl

∂ ∂ , , π = ρ2 Rη = − ∂ρ ∂θ ∂G 1 dv . qk = θ V ∂θ,k

(7.1.1)

The free energy , and G, are of the form ψ = (ρ, θ), G = G[dkl (x , t), θ,k (x, t); dkl (x, t), θ,k (x, t), |x − x|],

(7.1.2) (7.1.3)

where G is a symmetric function of its variables, i.e., it is not altered when x and x are interchanged in the argument functions of G. We express this as S

G=G.

(7.1.4)

We would like to develop linear constitutive equations for the dynamic response functions D t and q. To this end, we introduce the temperature deviation T from an ambient temperature T0 by θ = T0 + T ,

T0 0,

|T | T0 .

(7.1.5)

The principle of objectivity requires that G must depend on the invariants of d and ∇ θ. Consequently, for a linear theory, the dissipation potential G may be expressed as 1 G = 21 λ tr d tr d + µ tr(dd ) + kT,i T,i , (7.1.6) 2T0 where λ, µ, and k are functions of |x − x| and a prime denotes the dependence on x , e.g., d = d(x , t). From (7.1.1), it follows that

[λ(|x − x|) tr d δkl + 2µ(|x − x|)dkl ] dv , D tkl = V qk = k(|x − x|)T,k dv . (7.1.7) V

For the deformation tensor dkl we have the usual expression dkl = 21 (vk,l + vl,k ).

(7.1.8)

The nonlocal material moduli λ, µ, and k must obey the attenuating neighborhood hypothesis. This can be expressed as a continuity requirement, e.g., lim {λ(κ), µ(κ), k(κ)}κ 1+ = 0,

κ→0

κ = |x − x|,

> 0.

(7.1.9)

7.2 Field Equations of Nonlocal Fluid Dynamics

179

Any Dirac-delta sequence is a possible candidate for the nonlocal material moduli, e.g., (7.1.10) λ(κ)/λ0 = αv exp(−k02 κ 2 / l 2 ), where, in n dimensions, αv is given by αv = π −n/2 (k0 / l)n ,

(7.1.11)

λ0 , µ0 , k0 , and l are constants. Equally, in two-dimensional space, we have λ(κ)/λ0 = (2πl 2 )−1 K0 (κ/ l),

(7.1.12)

where K0 (z) is the Bessel function (cf. Section 6.9). Also, candidates for λ, µ, and k are generalized functions with finite supports. The appropriate values of constants such as λ0 , k0 , and l will have to be determined, for each viscous fluid, by experimental means. The equation of energy balance is given by (3.3.24) with D η = 0: −ρθ R η˙ + D tkl dkl + qk,k + ρh = 0.

(7.1.13)

Substituting for R η from (7.1.1), this becomes ∇ · v + [λ(|x − x|) tr d tr d + 2µ(|x − x|) tr(d d)] dv

−ρcv T˙ − δ1 π∇ V ∂ + k(|x − x|)T,k dv + ρh = 0, (7.1.14) ∂xk V where

∂ 2 θ ∂π . (7.1.15) , δ1 = ∂θ 2 π ∂θ In general cv , δ1 , and all material moduli will also depend on ρ and T . Equation (7.1.14) is the equation of heat conduction. cv = −θ

7.2 Field Equations of Nonlocal Fluid Dynamics The field equations are obtained by substituting the constitutive equations into the balance laws ∇ ·v =0 ρ˙ + ρ∇ [ρ(vk − uk )]nk = 0

in V − σ,

(7.2.1a)

on σ,

(7.2.1b)

∂

[λ(|x − x|)vr,r δkl + µ(|x − x|)(vk,l + vl,k )] dv

−π,l + ∂xk V + ρ(fl − v˙l ) = 0, V − σ, l on σ, [ − πδkl + D tkl − ρvl (vk − uk )]nk = F

(7.2.2a) (7.2.2b)

180


∇ ·v+ − ρcv T˙ − δ1 π∇

[λ(|x − x|) tr d tr d + 2µ(|x − x|)

V ∂ k(|x − x|)T,k dv + ρh = 0, V − σ, (7.2.3a) × + ∂xk V on σ. [ − ρ(ψ + θη + 21 v · v)(vk − uk ) − πvk + D tkl vl + qk ]nk = H (7.2.3b)

dkl dkl ] dv

Slightly different forms of (7.2.2a) and (7.2.3a) are obtained by replacing such terms as f ϕ,k by (7.2.4) f ϕ,k = (f ϕ),k − f,k ϕ, and then using the Green–Gauss theorem −π,l − s tl + [(λ + µ)vk ,k l + µvl ,k k ] dv + ρ(fl − v˙l ) = 0, (7.2.5) V ˙ ∇ · v + [λ(|x − x|) tr d tr d + 2µ(|x − x|) −ρcv T − δ1 π∇ V

× dkl dkl + k(|x − x|)∇ 2 T ] dv − s q + ρh = 0,

(7.2.6)

where s tl and s q are, respectively, the traction and heat on the surface ∂V of the body. They are given by

[λ(|x − x|) tr d δkl + 2µ(|x − x|)dkl ] dak , s tl = ∂V q = k(|x − x|)T,k dak . (7.2.7) s V

The set of equations (7.2.1a), (7.2.2a), and (7.2.3a) constitute five integro-partial differential equations for the five unknown functions ρ(x, t), v(x, t), and T (x, t). These equations are still nonlinear in density, velocity, and temperature, excluding the dynamical part D tkl of the stress tensor and the heat vector q. Note that the pressure π, entropy η, the acceleration v˙ , and the dissipation contribute nonlinear terms to these equations. It is clear that s tl represents a component of the surface tension since, near the surface, intermolecular attractions lose their symmetry. Similarly, the surface heat s q arises from the nonsymmetry of the temperature l is another part of the gradient near the surface. The surface traction residual F surface tractions arising from the creation of the surface of the body separated from the infinite continuum. If we employ the attenuating neighborhood hypothesis and use the same rate of attenuation for λ and µ, i.e., λ/λv = µ/µv = αv (|x − x|), then the equation of motion is expressed as −π,l − s tl + αv (|x − x|) D σk l ,k (x ) dv + ρ(fl − v˙l ) = 0, V

(7.2.8)

(7.2.9)

7.3 Channel Flow

181

where D σkl (x) is the classical Stokesian dynamic stress tensor D σkl (x)

= λv ∇ · vδkl + µv (vk,l + vl,k ),

(7.2.10)

with λv and µv being constant Stokesian viscosities. In the case of incompressible fluids, π is replaced by an unknown pressure p(x, t) and the field equations become ∇ · v = 0, (7.2.11) ∂

−p,l + µ(|x − x|)(vk,l + vl,k ) dv + ρ(fl − v˙l ) = 0, (7.2.12) ∂xk V

−ρCv T˙ + 21 µ(|x − x|)(vk,l + vl,k )(vk,l + vl,k ) dv

V ∂ + k(|x − x|)T,k dv + ρh = 0. (7.2.13) ∂xk V

7.3 Channel Flow The channel flow of a nonlocal viscous fluid is a simple example to demonstrate the physical nature of nonlocality. We take an x-axis along the axis of the channel located at −h < y < h. The flow is induced by a pressure gradient π,x = P (x). The velocity field is given by v1 = v(y),

v2 = v3 = 0,

|y| < h.

(7.3.1)

The deformation tensor dkl is given by dkl = 21 v,2 (δk1 δl2 + δk2 δl1 ).

(7.3.2)

Consequently, the equation of continuity (7.2.11) is satisfied identically and the equations of motion (7.2.12) give ∂ ∂y

h

−h

αv (|y − y|)

dv(y )

P dy = ,

dy µv

(7.3.3)

where P = π,x = p,x . This is integrated once to

h

−h

αv (|y − y|)

dv

P dy = y + C(x). dy

µv

Using (6.9.13), with γ = 0, this equation becomes dv P y. = dy µv

(7.3.4)

182


The solution, satisfying the boundary conditions v(±h) = 0, is the classical Poiseulle flow profile v(y)/v0 = 1 − (y/ h)2 ),

v0 ≡ −P h2 /2µv .

(7.3.5)

The shear stress is given by tyx = P

h −h

αv (|y − y|)y dy .

(7.3.6)

For the influence function αv , we consider a one-dimensional delta sequence αv (y) =

1 sin(νy), πy

(7.3.7)

where ν −1 = l is the internal characteristic length. Substituting this into (7.3.6) and integrating, we obtain tyx /P h = −

y 2 sin λ sin(λy) + [Si λ(1 + y) + Si λ(1 − y)], πλ π

where Si(x) is the sine integral, defined by x sin t Si(x) = dt, t 0

(7.3.8)

(7.3.9)

and λ = νh = h/ l,

y = y/ h.

(7.3.10)

ν −1

As the internal characteristic length → 0, λ → ∞ and (7.3.8) reduces to the shear stress tcl = P y, as obtained in the classical (local) theory. Since |y| < h, we may consider normalizing αv in the interval (−h, h) instead in (−∞, ∞). In this case αv becomes sin(λy) αv = . (7.3.11) 2hy Si(λ) In this case, (7.3.8) is replaced by 1 1 1 F tyx /P h = − sin λ sin λy + y[Si λ(1 + y) + Si λ(1 − y)] . Si(λ) λ 2 (7.3.12) Again, as λ → ∞ we have tcl = P y. In order to see what happens in the microscopic channel, we consider λ = 20. This corresponds to 2h/ l = 40. Calculating the shear stress at the boundary y = h, from (7.3.8) and (7.3.12), we obtain tyx /P h = 0.479,

F tyx /P h = 0.155,

(7.3.13)

7.4 Lubrication in Microscopic Channels

183

tyx

λ

(a)

F tyx

λ

(b)

Figure 7.3.1. Nondimensional shear stress at walls versus λ. F /P h at which shows more than 50% reduction. The shear stresses tyx /P h and tyx the boundary y = h, are displayed, as a function of λ in Figure 7.3.1a and 7.3.1b. l (appearing However, in channels with microscopic depths, the surface traction F in (7.2.2b)) becomes important. The component of this force in the x-direction will have to be added to tyx as calculated above. There have been models suggested for l , based on theVan der Waal attactions and some experimental measurements. One F such suggestion is given by (7.4.20). Moreover, for fluids that possess suspensions or a microstructure (e.g., polymers), viscosities of the fluid depend on the internal characteristic length of the microstructure, thus altering drastically the viscosity near the rigid boundary (see Section 7.4).

7.4 Lubrication in Microscopic Channels Experimental observations have shown that the viscosity of fluids near rigid surfaces altered drastically as compared to bulk viscosity. In thin films, within a channel depth less than 30 nm, the measured viscosity of PS-cyclohexane is found to be several decades higher than the bulk viscosity (Israelashvili [1986a]). Based on observations, it is conjectured that, near the surface, a thin layer (of the order

184


of 5–10 nm) of fluids becomes rigid, reducing the channel depth. In Section 7.3 we have found that, in microscopic channels, the shear stress near the walls of the channel is reduced, indicating the reduction of the bulk viscosity. How can we explain this discrepancy? A hydrodynamic lubrication problem in a thin film of liquids between two surfaces was considered by several authors (Israeleshvili [1986b], Chan and Horn [1985], Montfort and Hadziioannou [1988]). In these works, the classical Reynolds theory of lubrication was used, with the assumption that the channel depth D is reduced for each boundary surface, by an amount approximately equal to the radius of gyration of the fluids adjacent to the surfaces. Eringen and Okada [1995] have shown that this and related drainage phenomena can be fully explained by use of the nonlocal theory. Essential to this is the consideration that, in very thin films, the microstructural effects play dominant roles. This means that the viscosity of the fluid depends on the molecular shapes and orientations. For orientable fluids such as polymers and suspensions, violent orientation and shape changes in the viscosity of rigid boundaries are not disputed. As a first-order approximation, the fluid viscosity µv may be considered to depend on the microgyration radius Rg of the microstructure. This is justified, since the orientational effect of the molecules can be represented by a secondorder tensor jkl (cf. Section 13.5; see also Eringen [1972c], [1991a]). But the axiom of objectivity requires that scalar viscosities such as µv will depend on only the invariants of jkl . The first invariant jkk is proportional to the square of the radius of gyration Rg of the molecules. Consequently, we may write, approximately, µv /µ0 = 1 + γ (Rg /D)2 ,

(7.4.1)

where µ0 is the bulk viscosity, D is the channel depth, and γ is a material constant. Israelashvili gave a viscosity profile for cyclohexane between two mica surfaces with absorbed polystyrene. His experimental result is based on f =

100D . (µv /µ0 )

(7.4.2)

Corresponding to (7.4.2), Eringen and Okada, using (7.4.1), gave a plot of fE =

100D , 1 + γ (Rg /D)2

(7.4.3)

where γ was selected as γ = 19.5 by matching one experimental point at f = 10−3 cm3 /dyn-s and using Rg = 26 nm, quoted by Israelashvili. On Figure 7.4.1, fE and the experimental results are displayed in the range Rg ≤ D ≤ 300 nm. For D > 300 nm, µv = µ0 . For D < RG = 26 nm, the precise measured value of f was not available to compare with the theoretical curve f . However, we see that df/dD = 0 as D → 0. This agrees with the conjecture that a thin layer (of approximately 26 nm thickness) adjacent to each mica surface may be considered to become a part of the surface, thus reducing the depth of the channel D by 52


185

Figure 7.4.1. Inverse viscosity as a function of channel depth for the cyclohexane between two mica surfaces with adsorbed polystyrene. After Eringen and Okada [1995].

nm. Based on these considerations, it appears that the local viscosity of thin films may be characterized successfully by (7.4.1). Experimentally used crossed cylinder configuration is equivalent to a planesphere geometry (Figure 7.4.2), where the distance D R, where R is the radius of the upper surface. Under this approximation, the gap h is given by h=D+

r2 . 2R

(7.4.4)

Figure 7.4.2. Schematic representation of a thin liquid film between two solid surfaces.

186


For incompressible fluids, the continuity equation reduces to ∂vz 1 ∂ (rvr ) + = 0, r ∂r ∂z

(7.4.5)

where vr and vz are, respectively, velocity components in the cylindrical coordinates (r, θ, z) of Figure 7.4.2. In the lubrication theory, the lateral component of the velocity is large, so that the equation of motion (7.2.2a) reduces to ∂p ∂trz = , ∂z ∂r where

(7.4.6)

∂vr (z )

dz . (7.4.7) ∂z

0 For αv we select the one-dimensional triangular kernel (6.9.21), namely, |z − z| 1

αv (|z − z|) = 1− when |z − z| ≤ Rg , Rg Rg trz =

h

µv αv (|z − z|)

when |z − z| > Rg .

=0

Substituting this into (7.4.7), we obtain z z+Rg µv

vr dz − vr dz . trz = 2 Rg z z−Rg

(7.4.8)

(7.4.9)

With this, (7.4.6) becomes dp µv . [vr (z + Rg ) + vr (z − Rg ) − 2vr (z)] = Rg2 dr

(7.4.10)

Noting that p is a function of r, this difference equation may be integrated under the boundary conditions vr (0) = vr (h) = 0, (7.4.11) to give vr (z) =

1 dp 2 (z − hz). 2µv dr

Carrying this into (7.4.5), and integrating, we obtain 1 ∂ dp z3 z2 r − h(r) , vz = − µv r ∂r dr 6 4

(7.4.12)

(7.4.13)

where we used the boundary condition vz (0) = 0. At z = h, vz is equal to the velocity of the moving surface dD/dt: 1 ∂ dp 3 dD (7.4.14) = r h (r) , dt 12µv r ∂r dr


187

dp/dr being finite at r = 0, (7.4.14) integrates to p(r) = p(∞) −

3µv R dD . h2 (r) dt

(7.4.15)

The boundary traction is given by

∞

FH = −

z )2πr dr, (tzz + F

0

z is the nonlocal traction that appears in the boundary condition (7.2.2b) where F and tzz is the normal stress

h

tzz = −p +

2µv αv (|z − z|)

0

∂vz

dz . ∂z

Employing (7.4.13) and (7.4.8), we obtain 2 ∂ dp 1 2 2 4 2 1 tzz = −p − 2 (6z Rg + Rg ) − 2 h(r)zRg . r Rg r ∂r dr 12

(7.4.16)

(7.4.17)

From (7.4.15) it follows that lim r

r→∞

dp = 0. dr

(7.4.18)

With this, FH becomes

∞

FH = 2π 0

∞

pr dr −

z 2πr dr. F

(7.4.19)

0

There have been models about the nature of the surface force involving the Van der Waal and oscillatory types of attractions. One such experimental force model was suggested by Chan and Horn: z = −Rg A + Be−D/ξ1 cos 2πD F , (7.4.20) 6D 2 ξ2 where A is a Hamaker constant. B, ξ1 , and ξ2 were obtained by fitting them to equilibrium force measurements. Such a force becomes important when Rg ∼ D. For D > 20 nm, experiments show that this force is negligible. Considering only the first term in (7.4.20), the equation of motions of (7.4.19) of the upper surface becomes 2 2 Rg Rg A R dD 6πµ0 1 + γ + = 0. (7.4.21) D D dt 6D 2

188


This has the general solution D 2 + 2γ Rg2 ln D +

Rg A t + C1 , 18πµ0 R 2

(7.4.22)

where C1 is an arbitrary constant. This result shows that the two surfaces of the channel can never come into contact. However, it is possible for the two surfaces to have a gap D = 2Rg , at time t = tg . For this, (7.4.22) reads D 2 − 4Rg2 + 2γ Rg2 ln(D/2Rg ) =

Rg A (tg − t). 18πµ0 R 2

(7.4.23)

Eringen and Okada have solved the equations of motion for the upper surface of the narrow channel numerically by the Runge–Kutta–Gill method for the n hexadecane. The experimental force model (7.4.20) was used with ξ1 adjusted from 0.4 nm to 0.348 nm, as adapted by Chan and Horn. The two parameters Rg and γ associated with the nonlocal viscosity moduli were chosen, respectively, as 0.7 nm and 45.0 nm, the latter resulting from matching the drainage curve with one experimental point. The other parameters being exactly the same, the present theory is found to fit very well with experimental data (see Figure 7.4.3). It must be emphasized that the nonlocal lubrication theory is valid for all times, while the “shear-plane model” of Chan and Horn could not allow the surfaces to come closer than 2Ds = 1.4 nm, which is contrary to their experimental result. Chan and Horn measured the effects of the trace amounts of water on the drainage of octamethylcyclotetrasiloxane (OMCT). The nonlocal theory includes the surface tension effect. Calculations based on the nonlocal theory, once again gave excellent agreement with the experimental observations (see Figure 7.4.4).

Figure 7.4.3. The standing start drainage run in n hexadecane. After Eringen and Okada [1995].

7.5 Lubricant Film Flow on a Rotating Disk

189

Figure 7.4.4. The effect of trace amounts of water on the drainage of OMCTs. After Eringen and Okada [1995].

7.5 Lubricant Film Flow on a Rotating Disk The determination of the flow profile on a rotating disk requires the solution of the equations 1 ∂ ∂u (rv) + = 0, r ∂r ∂z ∂trz ∂p = , ∂z ∂r

(7.5.1) (7.5.2)

under the boundary conditions (Figure 7.5.1) u=0 ∂u =0 ∂z

at z = 0, at z = h(r, t).

(7.5.3)

This problem was treated by Emslie et al. [1958] on the basis of classical fluid mechanics. The present solution, based on the nonlocal theory, was given by Eringen and Okada [1995]. Similar to Section 7.4, the solution is given by

−1/2 νˆ (0) 4ω2 h2 (0)t + , νˆ (t) 3ˆν (t) 3/4 ν(t) 1/2 νˆ (0) 4ω2 h2 (0)t + , r(t) = r(0) ν(0) νˆ (t) 3ˆν (t)

h(t) = h(0)

and νˆ (t) ≡

ν(t) + ν0 , 2

(7.5.4) (7.5.5)

(7.5.6)

190


Figure 7.5.1. Viscous flow model of a thin liquid layer on a rotating disk.

where ν(t) = ν0 + γ [Rg / h(t)]2 is the nonlocal viscosity modulus, as discussed in Section 7.4, ν0 is the bulk viscosity, Rg is the characteristic length of the problem, and h(t) is the thickness profile as a function of time. Equations (7.5.4) and (7.5.5) may be combined to obtain the spatial evolution of the surface profile (Figure 7.5.1): r 2 (t)h3 (t) r 2 (0)h3 (0) . = ν(0) ν(t)

(7.5.7)

The bulk viscosity of perfluropolyether, Demnum S20 film, is 39 cSt. Using Rg = 0.6 nm and γ = 19.85, the corresponding viscosity curve is shown in Figure 7.5.2. With this value of γ , the temporal evolution of the thickness profiles for Demnum

Figure 7.5.2. Apparent viscosity of Demnum S20 films in centifugal flow as a function of the average thickness. After Eringen and Okada [1995].

7.5 Lubricant Film Flow on a Rotating Disk

191

Figure 7.5.3. Evolution of the thickness profile for Demnum S20, initially 17 6 nm thick, when spin ω/2π = 60 Hz. The solid lines were obtained based on the nonlocal lubrication theory. After Eringen and Okada [1995]. S20 are shown in Figure 7.5.3. Without resorting to any artificial viscosity, the nonlocal results agree very well with the experimental profile. Based on the results of Sections 7.4 and 7.5, it can be concluded that the nonlocal theory of lubrication is successful in that: (i) It predicts the viscosity change with the channel depth in complete agreement with experiments in all depths from zero to 300 nm. (ii) It provides a logical basis for the surface attraction. (iii) It gives a drainage curve in perfect agreement with experiments. (iv) It gives an evolution of thickness profile for a thin liquid film on a rotating disk in perfect agreement with experiments. (v) Ad hoc assumptions, such as bonded layers near solid boundaries and surface attractions that have no logical basis, are eliminated. (vi) The effect of a trace amount of water on the drainage of OMCTs between hydrophilic mica surfaces is fully explained. The drainage curves obtained are in good agreement with the existing experimental data. It can be concluded that the physics of the dynamic processes in very thin liquid films can be understood at the molecular level by use of the nonlocal lubrication theory.

192


Problems 7.1 Obtain the nonlocal solution of the pipe flow problem. Give the expression of the stress for an influence function αv given by αv = (1/πr) sin(νr). 7.2 A nonlocal viscous fluid is placed between two coaxial cylinders. The outer cylinder is rotating with an angular velocity and the inner cylinder is at rest. Determine the velocity and stress fields. Use any admissible influence function αv (r). 7.3 In the lubrication problem (Section 7.4) employing the influence function αv (z) = (1/πz) sin(νz), obtain the stress component tzz . 7.4 Construct a boundary layer theory for nonlocal fluid mechanics. 7.5 Obtain the nonlocal solution of the laminar flow around a sphere.

8 Nonlocal Linear Electromagnetic Theory1

8.0 Scope The balance laws of the electromagnetic (E-M) theory consist of the union of the Maxwell equations and the mechanical balance laws modified by the E-M effects. These are given in Section 8.1. In Section 8.2, we develop the linear constitutive equations for the anisotropic and isotropic nonlocal media. For nondeformable media these equations are simplified in Section 8.3 where we give a deeper discussion of the dielectric and magnetic properties. The discussion is extended in Section 8.4 to models for the dielectric and conduction tensors. In this way, a partial differential equation is obtained for the dielectric displacement vector that displays both longitudinal and transverse waves. Micromedia with absorption is considered in Subsection 8.4-C, and nonlocal micromedia with absorption in Subsection 8.4-D. With Section 8.5, we begin to explore the solutions of some problems. Debye screening of electrons results from the solution of the nonlocal theory. Section 8.6 is devoted to the discussion of optical waves, which involve both longitudinal and transverse optic waves. Polaritons is the subject of Section 8.7. Section 8.8 presents an account of the eddy currents in nonlocal media where the penetration depth turns out to display a wavelength dependence. 1A nonlocal electromagnetic theory of elastic solids was first given by Eringen [1973a], [1984a],

[1984b]. See also Eringen [1984c] for the memory-dependent nonlocal E-M elastic continua. Other work includes Eringen [1986], [1990], [1991], [1992], and [1993].

194

8 Nonlocal Linear Electromagnetic Theory

Gyrotropic media (Section 8.9) displays more complicated E-M phenomena. For example, for a constant frequency, three different polaritons are predicted by the nonlocal theory. Superconductivity (Section 8.10) is usually considered to fall into the domain of quantum phenomena. Here we give a formulation based on nonlocal theory. The field equations obtained extend those of London and Pippard, to include the elastic and magnetic effects. Section 8.11 offers a discussion of Alfven waves which turns out to be dispersive against a classical background. A variety of problems is treated here, once again displaying clearly the power and potential of the nonlocal continuum theory in encompassing a large number of physical phenomena that falls outside the domain of classical (local) theories.

8.1 Balance Laws The E-M balance laws were given in Section 4.1. These are the Maxwell equations 1 ∂B = 0, c ∂t 1 ∂D 1 ∇ ×H− = J, c ∂t c ∂qe +∇ ·J = 0 ∂t ∇ ×E+

∇ · D = qe , ∇ · B = 0, in V − σ,

(8.1.1)

where the Lorentz–Heaviside units are used, and E = electric field vector, H = magnetic field vector, J = current vector, c = speed of light.

D = dielectric displacement vector, B = magnetic induction vector, qe = electric charge density,

Accompanying the Maxwell equations are the jump conditions at a discontinuity surface σ which may be sweeping the body with its own velocity u: 1 n × [E + u × B] = 0, c ∨ 1 n × [H − u × D] = n × [ H ], c n · [J] = 0, on σ,

n · [D] = we , n · [B] = 0, (8.1.2) ∨

where n is the unit vector on σ , we is the surface charge density, and H is the nonlocal magnetic field on σ . Polarization P and magnetization M are introduced by D = E + P, B = H + M. (8.1.3)

8.2 Nonlocal Electromagnetic Elastic Solids

195

8.2 Linear Constitutive Equations of Nonlocal Electromagnetic Elastic Solids In Section 5.1, we obtained general constitutive equations of nonlocal elastic solids. Here we give the linear theory. Within the same type norm introduced in Section 6.1, the constitutive equations (5.1.21) and (5.1.22) for the linear theory may be expressed as 1 ∂F ∂F S η=− dv , + ρ V ∂T ∂T

∂F ∂F S tkl = dv , +

∂ekl V ∂ekl 1 ∂G qk = dv , T0 ∂T ,k V ∂F ∂F S Pk = − dv , +

∂E ∂E k V k ∂F ∂F S Mk = − dv , + ∂Bk

V ∂Bk S ∂G

Jk = (8.2.1) dv , G = G, V ∂Ek where the free energy F is expressed as a quadratic polynomial F = F ST + F ET + U S + U E . Here

F ST

and

F ET

(8.2.2)

are given by

ρC

T T − 21 βkl (T ekl + T ekl ), T0 = − 21 ρλTk E (T Ek + T Ek ) − 21 ρλTk B (T Bk + T Bk ),

F ST = F0 − F ET

(8.2.3)

and U S and U E are, respectively, strain energy densities for thermoelastic solids and electromagnetic fields. These are given by ρC

T T + 21 λij kl eij ekl , 2T0

= 21 [χklE Ek El + χklB Bk Bl + λEB kl (Ek Bl + Ek Bl )

US = UE

eB

+ λeE klm (ekl Em + ekl Em ) + λklm (ekl Bm + ekl Bm ).

(8.2.4)

For the dissipation function density G we take G = 21 [σklE Ek El + σklEB (Ek Bl + Ek Bl ) + σklET (Ek T,l + Ek T,l ) eE

+ σklm (ekl Em + ekl Em ) + σklBT (Bk T,l + Bk T,l ) eT

+ kkl (T,k T,l + T,k T,l ) + σklm (ekl T,m + ekl T,m ].

(8.2.5)

196


and e e Three other possible terms involving Bk Bl , Bk elm kl mn are left out from the expression of G, since they do not contribute to the constitutive equations of qk and Jk . The material functions F0 , N0 , C, βkl , λij kl , λTk , . . ., are functions of x and x

and they are symmetric with respect to x and x , e.g.,

βkl (x, x ) = βkl (x , x),

eE

λeE klm (x, x ) = λlkm (x, x ),

λklmn (x, x ) = λklmn (x , x) = λlkmn (x, x ) = λklnm (x , x), . . . . (8.2.6) The same type symmetries are valid for other material functions as is apparent from the construction of (8.2.4) and (8.2.5), symmetric compositions of F and . Substituting (8.2.2) to (8.2.5) into (8.2.1) we obtain the linear constitutive equations of the nonlocal E-M, thermoelastic solids 1 1 η= C(x, x )T (x ) + βkl (x, x )ekl (x ) + λTk E (x, x )Ek (x ) T ρ 0 V + λTk B (x, x )Bk (x ) dv ,

tkl = −βkl (x, x )T (x ) + λij kl (x, x )eij (x ) V

+ λeE (x, x )Em (x ) + λeB klm (x, x )Bm (x ) dv , klm

1 kkl (x, x )T,l (x ) + σlkET (x, x )El (x ) qk = T0 V eT + σlkBT (x, x )Bl (x ) + σlmk (x, x )elm (x ) dv ,

TE ρλk (x, x )T (x ) − χklE (x, x )El (x ) Pk = V

eE

− λeB kl (x, x )Bl (x ) − λlmk (x, x )elm (x ) dv ,

TB ρλk (x, x )T (x ) − χklB (x, x )Bl (x ) Mk = V

eB

− λEB lk (x, x )El (x ) − λlmk (x, x )elm (x ) dv ,

E σkl (x, x )El (x ) + σklEB (x, x )Bl (x ) Jk = V eE + σklET (x, x )T,l (x ) + σlmk (x, x )elm (x ) . (8.2.7) The constitutive equations obtained above contain some terms which are not acceptable, unless combined with the material tensors, they give time-symmetric terms. We recall that B, H, and M are not time-symmetric, i.e., upon the reversal of time (e.g., reversing the direction of electronic spin) they reverse their signs. BT EB EB EB Consequently, the material functions λTk B , λEB klm , σlk , χkl , χlk , and σkl must produce time symmetry for η, qk , Pk , and Jk . Constitutive equations having this property have been discussed by Kiral and Eringen [1990] for the magnetic group. A little scrutiny of these equations indicates that:

8.2 Nonlocal Electromagnetic Elastic Solids

197

EB (a) The terms involving λeE klm and λklm represent, respectively, the nonlocal piezoelectric and piezomagnetic effects.

(b) Heat conduction due to an E-field represented by σlkET denotes a nonlocal Peltier effect. The current due to the temperature gradient, represented by the same coefficient, denotes the nonlocal Seebeck effect. (c) The term with coefficient λTk E produces a polarization with the temperature. This is the nonlocal pyroelectricity. Similarly, λTk B is responsible for producing a magnetic field with temperature (the pyromagnetic effect). (d) The term involving λEB kl in the epxression of Pk denotes the nonlocal magnetoelectric effect. For homogeneous media, the material functions C, βkl , λTk , λij kl . . . are functions of x − x, and for homogeneous and isotropic media they are of the forms C = C(|x − x|),

λTk E = λT1 E (|x − x|)κk ,

χklE = χ E (|x − x|)δkl + χ1E (|x − x|)κk κl ,

eE χklm = χ1eE (|x − x|)κm δkl ,

(8.2.8)

where κk = xk − xk . The time symmetry regulations require that we set λTk B = EB = λEB = 0. All other material moduli have the forms σlkBT = λEB kl = σkl kl patterned by (8.2.8). All terms involving κk are negligibly small as compared to those multiplied by δkl . Moreover, all material moduli C, λT1 E , χ E , . . . attenuate rapidly with a few atomic distances away from the reference point, because the interatomic potential decreases rapidly with distance. This is in accordance with the attenuating neighborhood hypothesis. Consequently, for nonlocal isotropic solids, we arrive at the constitutive equations 1 1

η = η0 + C(|x − x|)T (x ) + β(|x − x|)ekk (x ) dv , ρ V T0

tkl = −β(|x − x|)T (x )δkl + λ(|x − x|)err (x )δkl V + 2µ(|x − x|)ekl (x ) dv , qk = [k(|x − x|)T,k (x ) + σ ET (|x − x|)Ek (x )] dv , V Pk = − χ E (|x − x|)Ek (x ) dv , V Mk = − χ B (|x − x|)Bk (x ) dv , V Jk = [σ E (|x − x|)Ek (x ) + T0−1 σ ET (|x − x|)T,k (x )] dv . (8.2.9) V

We have absorbed the inessential constant T0 in the expression qk into the moduli k and σ ET .

198


When the material functions C, β, λ, µ, κ, σ ET , χ E , χ B , and σ E are considered to be Dirac-delta functions, e.g., λ(|x − x|) = λ0 δ(|x − x|),

(8.2.10)

where λ0 is constant, then equations (8.2.9) reduce to the constitutive equations of the local (classical) E-M thermoelastic solids (see Eringen [1980, p. 469]). As discussed before, the material functions (isotropic and anistropic) are subject to the attenuating neighborhood hypotheses, since they decay quickly about a few ◦

atomic distances away (e.g., 20 or 30 A). Consequently, we can select them as a Dirac-delta sequence, e.g., µ(|x − x|) = µv α(|x − x|),

(8.2.11)

where µv is constant. Here, for simplicity, the kernel function α(|x − x|) will be taken as the same for all material functions with different constants appropriate to each material function. While there may be different attenuation distances for different physical phenomena, this merely fixes the internal characteristic lengths for all. Of course, it is routine to select different kernels for different causes when necessary. The material functions are also subject to thermodynamic restrictions. These restrictions are dictated by the positive semidefinite nature of the stain energy and the dissipation functions Ubody = body =

V

V

V

V

(U S + U E ) dv dv ≥ 0,

(8.2.12)

G dv dv ≥ 0.

(8.2.13)

For the anistropic media, U S and U E are given by (8.2.4) and G by (8.2.5). For the isotropic media we have ρ

T T + 21 (λe eii ejj + 2µe eij eij ), 2T0 = 21 χ E E · E + 21 χ B B · B,

US = UE

(8.2.14)

and for the dissipation potential density we have E · ∇ T + E · ∇ T ) + G = 21 σ E E · E + σ ET (E

1 2

k ∇T · ∇T. T0

(8.2.15)

8.3 Electromagnetic Solid Media

199

8.3 Electromagnetic Solid Media2 Ignoring the thermal effects and deformations, the constitutive equations (8.2.7) for solid media reduce to Pk = − χklE (x , x)El (x ) dv , V Mk = − χklB (x , x)Bl (x ) dv , V Jk = σkl (x , x)El (x ) dv , (8.3.1) V

where for σklE we wrote σkl . The material functions χklE , χklB , and σkl are symmetric functions of x and x, e.g., E χklE (x , x) = χlk (x, x ).

The first two of the constitutive equations (8.3.1) may be expressed in the forms Dk = kl (x , x)El (x ) dv , V

Hk = µ−1 (8.3.2) kl (x, x)Bl (x ) dv . V

From (8.1.3) it follows that kl = δkl δ(x − x) − χklE ,

B µ−1 kl = δkl δ(x − x) + χkl .

(8.3.3)

For homogeneous and isotropic media, kl is of the form kl = 0 δkl + 2 κk κl ,

κ = x − x,

(8.3.4)

and 0 and 2 depend on |x − x|. µ−1 kl has a similar construction. 0 and 2 acquire maxima at κ = 0, and decrease rapidly as |κκ | → ∞. It is more practical to express the material functions in terms of the wave vector k, since the approximate theories are based on approximations around |k| = k = 0 (i.e., the long wavelength λ = 2π/k). Using the three-dimensional Fourier transform we express (8.3.2)1 as D k = kl (k)E l (k),

(8.3.5)

where a superposed bar denotes the Fourier transform. For isotropic media, from App. B, Table 3 of Eringen [1980], we can express kl as kk kl kk kl kl (k) = T (k 2 ) δkl − 2 + L (k 2 ) 2 , k 2 ≡ k · k. (8.3.6) k k 2 Eringen [1993].

200


Expressions of the form (8.3.6) are also valid for µ−1 kl and σkl . For memory-dependent material, the constitutive equations for Dk is of the form3 (see Section 11.1): t ∂Ekl (x , t )

dt kl (x , x, t − t ) dv . (8.3.7) Dk (x, t) = ∂t

V −∞ In the four-dimensional Fourier domain this is equivalent to D k (x, t) = kl (k, ω)E l (k, ω),

(8.3.8)

kk kl kk kl kl = T (k 2 , ω) δkl − 2 + L (k 2 , ω) 2 . (8.3.9) k k Here, ω is the circular frequency, T and L are known as the transverse and longitudinal dielectric moduli. Similar expressions are valid for µ−1 kl and σkl . Given a particular kl (k, ω), the real and imaginary parts of kl are connected by Kramer– Kronig relations. This restriction stems from the upper limit of the integration on t being stopped at t.

where

8.4 Models for the Dielectric Tensor4 The dielectric tensor is determined by considering the interaction of the electric field with the deformation of charges attached to the atoms of crystals (cf. Maradudin et al. [1971]). This is a highly complicated model for the determination of the macroscopic dielectric tensor in terms of the atomic parameters of crystalline solids. Our main interest here is the frequency and wave vector dependence of kl (k, ω). We resort to simple macroscopic models.

A. Spring–Dashpot Model The simplest model is a rigid body model for which = 0 = const. The next simplest model is a simple harmonic oscillator consisting of a mass (atom) attached to a linear spring and dashpot. In this case, the equation of motion is e E, (8.4.1) M where x, e, and E are, respectively, the displacement, charge density, and electric field acting on the mass M. Mγ and MωT2 are, respectively, the damping and spring constants. Substituting E = E0 exp(−iωt) into (8.4.1) we obtain x¨ + γ x˙ + ωT2 x =

x=−

3 See Eringen [1990], also Chapter 11. 4 Eringen [1993].

ω2

e/M E. − iγ ω + ωT2

(8.4.2)

8.4 Models for the Dielectric Tensor

201

Figure 8.4.1. Dielectric constant (oscillator model). The polarization is given by P = −χ ∞ E + eNx. Writing D = E + P = E we obtain ωp2 (ω) Ne2 2 = 1 − , ω ≡ , (8.4.3) p ∞ M ∞ ω2 − iγ ω − ωT2 where N is the number density of oscillators. This result shows that the dielectric constant depends on the frequency ω. A sketch of (ω)/ ∞ is shown in Figure 8.4.1 for γ = 0.

B. Elastic–Solid Model The spring–dashpot model does not give the wave vector dependence. Since the polarization causes separation of charges, next we consider a polarizable microelastic model. A macroelement V in a polarizable elastic solid contains a large number of microelements V V (a microcontinuum model of a large number of atoms). We may consider V as an elastic solid of infinite extent by comparison with the size of V . The field equations of a linear polarizable elastic solid are λij kl uk,li − ρ u¨ j = −eEj ,

(8.4.4)

where uj , λij kl , e, and ρ are, respectively, the displacment vector, elastic constants, charge density, and mass density. The Fourier transform (8.4.4) gives λj k (k, ω)uk = eE j ,

(8.4.5)

202


where a superposed bar denotes the Fourier transform, k is the wave vector, ω is the circular frequency, and λj k (k, ω) = λij kl ki kl − ρω2 δj k .

(8.4.6)

Denoting the inverse of the matrix λj k by λ−1 kj , i.e.,

we solve (8.4.5) for uk :

λ−1 ij λj k = δik ,

(8.4.7)

uk = eλ−1 kl (k, ω)E l .

(8.4.8)

The polarization vector is given by P k = −χklE E l + Neuk .

(8.4.9)

Consequently, the dielectric displacement vector D = E + P is given by D k = kl (k, ω)E l ,

(8.4.10)

∞ kl (k, ω) = kl + Ne2 λ−1 kl (k, ω)

(8.4.11)

∞ ekl = δkl − χklE .

(8.4.12)

where

and

This result clearly displays the nonlocal nature of the dielectric tensor. In the physical domain, (8.4.5) is equivalent to

∂ ∂ ∂2 λij kl − ρδj k 2 ∂xi ∂xl ∂t

∞ (Dk − kr Er ) = −Ne2 Ej .

(8.4.13)

This differential equation replaces the nonlocal constitutive equations (8.3.2)1 for the electrostatic case. It is important to note that (8.4.13) requires additional boundary conditions, whereas (8.3.2)1 does not. Using (8.4.8), we calculate the current vector Jk = σkl∞ El + Neu˙ k .

(8.4.14)

Jk = σ kl (k, ω)E l ,

(8.4.15)

Hence, we have

where the conductivity tensor σ kl is given by ∞ σ kl = σkl∞ + iωkl − iω kl (k, ω).

(8.4.16)

8.4 Models for the Dielectric Tensor

203

Isotropic Solids ∞ are given by For isotropic solids, λij kl and kl

λij kl = λδij δkl + µ(δik δj l + δil δj k ), ∞ kl = ∞ δkl ,

(8.4.17)

where λ and µ are Lamé constants. Carrying these into (8.4.11) we obtain kk kl kk kl kl (k, ω) = T (k , ω) δkl − 2 + L (k 2 , ω) 2 , k k 2

(8.4.18)

where the transverse dielectric modulus T and longitudinal dielectric modulus L are defined by T = ∞ +

∞ ωp2 c22 k 2 − ω

, 2

L = ∞ +

∞ ωp2 c12 k 2 − ω2

.

(8.4.19)

Here c1 , c2 , and ωp are, respectively, the phase velocities of irrotational elastic waves, equivoluminal elastic waves, and the so-called plasma frequency. They are defined by c12 =

λ + 2µ , ρ

c22 =

µ , ρ

ωp2 =

Ne2 . ρ ∞

(8.4.20)

We notice that (8.4.19) agrees with (8.4.3) where cα k 2 is considered constant and γ = 0 (no damping). The dependence of T and L on the wave number explains polariton dispersion which will be discussed in Section 8.7. For isotropic solids, (8.4.13) is reduced to c12∇ ∇ · (D − ∞ E) − c22∇ × ∇ × (D − ∞ E) −

∂2 (D − E ∞ ) = − ∞ ωp2 E. ∂t 2 (8.4.21)

C. Media with Absorption For solid media with absorption these results are modified further. In this case, we consider that microelements depend on the memory of past motions. Then the stress constitutive equations are of the form tkl =

t −∞

λ ij kl (t − t )u˙ k,l (t ) dt .

(8.4.22)

This implies that in all the equations above we must replace λij kl by −iωλij kl (ω).

204


D. Nonlocal Solid Media with Absorption In this case, the stress constitutive equations are of the form (see (9.1.15)): tij =

t −∞

dt

λij kl (x − x , t − t ) V

∂uk,l (x , t ) dv(x ). ∂t

(8.4.23)

This implies that the λij kl in all equations are to be replaced by −iωλij kl (k, ω). With this, kl will be functions of the complex variables k and ω. For the development of the constitutive equations nonlocal media with absorption, see Chapter 11.

8.5 Point Charge5 In the case of a point charge e located at x = 0, the two surviving Maxwell equations read ∇ × E = 0,

∇ · D = eδ(x),

(8.5.1)

where δ(x) is the Dirac-delta function. The first of these equations is satisfied by an electrostatic potential φ(x): ∇ φ. E = −∇

(8.5.2)

The Fourier transform of (8.5.1)2 gives −ik · D = e.

(8.5.3)

Substituting D from (8.4.10) with kl (k, 0) given by (8.4.18), we obtain φ=

1 e , ∞ 2 k + rs−2

(8.5.4)

where rs2 = c12 /ωp2 .

(8.5.5)

The inverse transform of (8.5.4) gives φ=

π 1/2 e 1 exp(−r/rs ), 2 ∞ r

(8.5.6)

where r = |x|. Thus, the electric potential dies out with increasing r. This is the well-known phenomenon, Debye screening of an electron. 5 Eringen [1984b], [1993].

8.6 Optical Waves

205

8.6 Optical Waves6 The Fourier transforms of the Maxwell equations (with qe = 0, J = 0, B = H) are ω k · D = 0, k × E + H = 0, c ω k × H − D = 0, k · H = 0. (8.6.1) c The constitutive equations are D k = kl (k, ω)E l .

(8.6.2)

Eliminating H from (8.6.1) and using (8.6.2) we obtain ω2 2 kl km − k δlm + 2 lm (k, ω) E m = 0. c For nonvanshing E, this gives the dispersion relations ω2 2 det kl km − k δlm + 2 lm (k, ω) = 0. c From (8.6.1)2 it follows that

kl kk E l = 0.

(8.6.3)

(8.6.4)

(8.6.5)

For isotropic media, using (8.3.9), (8.6.4), and (8.6.5), we have ω2 kl km ω2 = 0, (8.6.6) det −1 + 2 2 T δlm + 1 + 2 2 (L − T ) c k c k k2 L (k 2 , ω)E · k = 0. (8.6.7) For E · k = 0, from (8.6.7), it follows that L (k 2 , ω) = 0.

(8.6.8)

This is the dispersion relation of the longitudinal waves. On the other hand, if k · E = 0, the waves are transverse. Then from (8.6.6) it follows that the dispersion relations of the transverse waves are given by ω2 T (k 2 , ω) = 1. c2 k 2

(8.6.9)

The classical E-M theory does not predict longitudinal waves, and transverse waves are not dispersive. The frequency-dependent dielectric moduli are observed in both low- and high-frequency regions. 6 Eringen [1984c], [1990], and [1993].

206


If we use the oscillator model (ω), given by (8.4.3) with γ = 0, (8.6.9) becomes ∞ ωp2 c2 k 2 ∞ = + . ω2 ωT2 − ω2

(8.6.10)

This result is identical to the one obtained in the lattice dynamics for polaritons (cf. Maradudin et al. [1971, p. 261]). Here ωT is considered to be a transverse frequency, and the longitudinal frequence ωL is given by ωL2 = ωT2 + ωp2 .

(8.6.11)

In ionic crystals, transverse optical modes are coupled with polarization. The E-M waves so produced are called polaritons.

8.7 Polaritons7 In ionic crystals (e.g., alkali halides like NaCl, KCl), oscillating transverse electric fields will move oppositely charged ions in opposite directions. The dispersive waves so-created are called polaritons. The simplest model for this is a harmonic oscillator. The equations of motion of two oppositely charged atoms are M1 u¨ 1 = 2K(u2 − u1 ) + eE, M2 u¨ 2 = 2K(u1 − u2 ) − eE.

(8.7.1)

Dividing the first of these by M1 , and the second by M2 , and subtracting one from the other we obtain M u¨ = −2Ku + eE, (8.7.2) where

M −1 = M1−1 + M2−1 ,

Substituting

u = u1 − u2 .

(u, E) = (u, E)e−iωt

(8.7.3) (8.7.4)

into (8.7.2), we obtain eE . 2K − Mω2

(8.7.5)

P = Neu − χ ∞ E,

(8.7.6)

u= The polarization is given by

where N is the number density of oscillators and χ ∞ is the rigid body susceptibility. Writing D = E + P = E we obtain (ω) = ∞ +

7 Eringen [1993].

∞ ωp2 ωT2 − ω2

,

(8.7.7)

8.7 Polaritons

207

Figure 8.7.1. Polariton dispersion (spring model). where ∞ = 1 − χ ∞,

ωp2 = Ne2 /M ∞ ,

ωT2 = 2K/M.

(8.7.8)

Thus, we have arrived at (8.4.3) with γ = 0, but this time, clearly, polarization is caused by the motions of oppositely charged particles, and ωT is the transverse frequency. With this interpretation, (8.6.10) and (8.7.7) become identical. Remembering also ωL2 = ωT2 + ωp2 , ω versus k is displayed in Figure 8.7.1. Three regions are distinguished: (i) 0 ≤ ω < ωT . In this region the polarization is largely electromagnetic. For large k it becomes mechanical vibrations. (ii) ωT < ω < ωL stop band. There are no polaritons. (iii) ωL ≤ ω, mechanical vibrations become E-M waves. Experiments indicate that ionic crystals possess an absorption in the far infrared (λ 10−2 cm). In spite of the crude model used leading to (8.7.7), the wavelength is λ = 2πc/ωT 4.6 × 10−3 cm. (c = speed of light in the material), based on (8.7.8) agrees rather well with the experimental value λ = 6.1 × 10−3 cm for the NaCl atom (Ghatak and Kothari [1972, pp. 413, 436]).

208


It is of interst to use the elastic model for (k, ω). Substituting from (8.4.19)1 into (8.6.9), we obtain ∞ ωp2 c2 k 2 ∞ . = + ω2 c22 k 2 − ω2

(8.7.9)

The replacement ωT2 = c22 k 2 in (8.6.10) makes a drastic change. The root ω2 of (8.7.9) as a function of k is obtained to be 22 = 1 + κ 2 ± (α 2 κ 4 + 2κ 2 + 1)1/2 ,

(8.7.10)

where = ω/ωp , α=

κ = (c22 + c2 / ∞ )1/2

( ∞ c22 /c2 ) − 1 ( ∞ c22 /c2 ) + 1

.

k , ωp (8.7.11)

versus κ, for α 2 = 0.6 is displayed in Figure 8.7.2. Here we notice that for > 1, there are two different wavelengths corresponding to one frequency, i.e., there is no “stop band.” In a more general case, a nonlocal elastic solid with absorption, we have the general relation c2 k 2 /ω2 = (k, ω). (8.7.12) Depending on the forms of (k, ω), the dispersion curves can be much different from Figure 8.7.2. Hopfield [1969] discussed the possibility of polariton modes

Figure 8.7.2. Polariton dispersion (elastic model). After Eringen [1993].

8.8 Eddy Currents

209

even in the absence of transverse E-M fields. The reflectivity measurement of Hopfield and Thomas [1961] on CdS, displays a sharp peak in the vicinity of the lowest energy of exciton. The classical theory, based on wave-numer-independent (ω), fails to explain this peak, but nonlocal theory explains this phenomena.

8.8 Eddy Currents Here we consider the E-M field in conductors subject to a variable magnetic field. Polarization being negligible, the Maxwell equations reduce to 1 ∂B = 0, c ∂t 1 ∇ × H − J = 0, c

∇ ×E+

∇ · E = 0, ∇ · B = 0.

(8.8.1)

J k = σkl (k, ω)E l ,

(8.8.2)

For B and J, we have the constitutive equations B k = µkl (k, ω)H l ,

where µkl is of the form (8.3.9) and σkl is given by (8.4.16). Combining the Fourier transforms (8.8.1) and (8.8.2), we obtain

ω klm rsn µ−1 mr kl ks − i 2 σkn E n = 0. c

(8.8.3)

Substituting µ−1 mr and σkn for the isotropic media, (8.8.3) becomes iω −1 2 2 −µT (k , ω) + 2 2 σT (k , ω) E = 0. c k

(8.8.4)

From this there follows:

−1 ω/c2 k 2 = −i σ T (k 2 , ω)µT (k 2 , ω) .

(8.8.5)

Unlike the prediction of the classical theory, the decay time of the fields depends on the wavelength. It is well known that the magnetic field penetrates into conductors, inducing a variable electric field which, in turn, causes currents to appear. These are known as eddy currents. The penetration depth δ is proportional to δ 2 ∼ c2 [ωµT (k 2 , ω)σT (k 2 , ω)]−1 .

(8.8.6)

This displays the wavelength dependence. It is well known that when δ becomes compatible with the electron mean free path, the classical theory fails. However, the nonlocal theory is still valid and it explains the anomalous skin effect (Eringen [1984b]).

210


8.9 Gyrotropic Media For some optically anistropic media, the dielectric tensor contains both even and odd powers of the wave vector k and frequency ω, as distinguished from the isotropic media. In this case, the discussion of crystal optics and E-M theory become highly involved. However, for long wavelengths, |k| is small and we may expand kl (k, ω) into a power series about k = 0, retaining the first few powers of k: kl (k, ω) = kl (ω) + iγklm (ω)km + αklmn (ω)km kn + iγklmnp (ω)km kn kp + · · · .

(8.9.1)

The inverse of this matrix is also useful in analysis −1 −1 kl (k, ω) = kl (ω) + iδklm (ω)km + βklmn (ω)km kn + iδklmnp (ω)km kn kp + · · · .

(8.9.2)

This tensor appears in the inverted constitutive equations −1 E k = kl (k, ω)D l .

(8.9.3)

For some crystals (e.g., C3v , C4v , and C3h , and D3h ) γklm vanishes so that all the rotations and polarizations of plane waves are determined by the fourth term δklmnp . However, for the point group C3 , C4 , C6 , D3 , D4 , D6 , T , and O, γklm (ω) has nonvanishing elements for the waves propagating along the optic axis. Here we consider one of the simplest cases. From the Maxwell equations we have δkl −

kk kl k2

El −

ω2 D l = 0. c2 k 2

(8.9.4)

Substituting D k = kl (k, ω)E l and using (8.9.1) this leads to

kk kl ω2 δkl − 2 − 2 2 [kl (ω) + iγklm (ω)km ] E l = 0. k c k

(8.9.5)

The dispersion relations for the transverse waves (E · k = 0) are given by ω2 det δkl − 2 2 [kl (ω) + iγklm (ω)km ] = 0. c k

(8.9.6)

For a uniaxial crystal, such as quartz, the nonvanishing components of kl and γklm are 11 = 22 , 33 ,

γ123 = −γ231 ,

γ132 = −γ312 = γ321 = −γ231 , (8.9.7)

8.9 Gyrotropic Media

211

so that the phase velocity v = ω/k and the electric field follow from (8.9.6) and (8.9.5) (see also Maradudin et al. [1971, p. 276]): 2 v± = c2 (11 ± γ123 k3 )−1 ,

1 E ± = √ (1, ∓i, 0)E, 2

(8.9.8)

where the plus and minus signs correspond, respectively, to the left and right circularly polarized waves. Thus, a linearly polarized wave, propagating along the optic axis, upon incidence with the crystal, will decompose into the right and left circularly polarized waves. This is known as the optical activity. For uniaxial crystals with 11 = 22 (but γklm is as described by (8.9.7)), it is convenient to substitute (8.9.3) into (8.9.4). For transverse waves this leads to ω2 −1 kl (k, ω) − 2 2 δkl D l = 0. (8.9.9) c k Selecting the x-axis along D 3 , the condition k · D = 0 is satisfied with D 3 = 0. In the (x1 , x2 )-plane, (8.9.9) gives the dispersion relations 2 2 ω ω −1 −1 2 − I (ω) − II (ω) = δ123 k32 , (8.9.10) c2 k 2 c2 k 2 −1 where I−1 and II−1 are the principal values of kl . This equation shows that the dispersion curves possess four branches for ω, as sketched in Figure 8.9.1.

Figure 8.9.1. Polariton dispersion (gyrotropic media). From Agranovich and Ginsburg [1984, p. 144].

212


From this figure, we observe the possibility of three polaritons corresponding to a single frequency, as indicated by a ω = const. line parallel to the k-axis (see also Agranovich and Ginsburg [1984]).

8.10 Superconductivity In 1911, Kamerling Onnes observed that mercury loses its electrical resistance completely below a critical temperature of 4.2◦ K. Since then, many experiments have been conducted with different materials. In 1986, Bednorz and Muller showed that superconductivity can also exist in high temperatures. Following this discovery, feverish activities produced critical temperatures, as high as 145◦ K. Generally, superconductivity is considered to be a quantum phenomenon. However, F. and H. London [1935, see also 1950] gave a theory based on Lorentz equations of motion for electrons in metals. Here, we present this theory, followed by a nonlocal theory. Equations of motions for a perfect conductor (no resistance) are Newton’s equations of motion: m˙v = eE, (8.10.1) where m, v and e are, respectively, the mass, velocity, and charge of an electron. E is the electric field. Introducing the current density J = nev, where n is the electron density, (8.10.1) becomes ne2 ∂J = E. ∂t m

(8.10.2)

This is usually called the first London equation. We take the curl of this equation and use the Maxwell equations to eliminate ∇ × J and ∇ × E. ne2 µ ∂H ∂ µ ∂ 2 H = − ∇ ×∇ ×H + 2 . ∂t c ∂t 2 mc2 ∂t Integrating this equation with respect to time, we obtain ∇ × ∇ × H + λ−2 L H+

µ ∂ 2 H = K(x), c2 ∂t 2

(8.10.3)

where K(x) is a vector of integration and λL is the London depth. λ−2 L =

ne2 µ . mc2

(8.10.4)

Usually, by reference to the Meissner experiments, K(x) is set equal to zero. The Meissner experiments indicated that the magnetic field does not penetrate into the conductor. However, the solution of (8.10.3), (with K = 0 and the term with factor 1/c2 neglected) displays penetration of the magnetic field into a thin layer near the surface. Some authors state that K = 0 requires a new postulate. The

8.10 Superconductivity

213

vanishing of K can be deduced from (8.10.1), by taking the curl of (8.10.1) and then integrating with respect to time: eµ ∇ × v = − H + K(x). m∇ (8.10.5) c If we now use the initial conditions v(x, 0) = H(x, 0) = 0, we get K = 0. With this, using the curl of the Maxwell equation ∇ ×∇ ×H −

∂ ne ∇ × E) = ∇ × v, (∇ c c ∂t

we obtain

µ ∂ 2 H = −λ−2 (8.10.6) L H. c2 ∂t 2 This is the second London equation of superconductivity. This equation has undergone some heuristic modification by addition of vortex terms. A second change was made by Pippard [1953], by introducing nonlocality to the super current. If we substitute v = J/ne, into (8.10.5), we obtain ∇ ×∇ ×H +

ne2 µ H. (8.10.7) mc Using magnetic potential, H = ∇ × A, subject to London gauge, ∇ · A = 0, (8.10.7) may be integrated, under the boundary condition A · n = 0, leading to ∇ ×J=−

ne2 µ A. (8.10.8) mc Guided by the nonlocal electric-current relations introduced by Chambers [1952], Pippard reasoned that (8.10.8) should have a nonlocal form [A(x ) · R]R −R/ξ0

J(x) = −C e dv , (8.10.9) R4 J=−

where R = x − x , R = |x − x |, and C is obtained by integrating J over the volume and making the result coincide with (8.10.8), C = 3ne2 µ/ξ0 mc. Equation (8.10.9) is supplemented with the expression of fourth Maxwell equation ∂E 1 = J. (8.10.10) c ∂t c Equations (8.10.9) and (8.10.10) constitute Pippard’s theory of superconductivity. From the foregoing analysis, it is clear that this formulation is not in the true spirit of the continuum theory. Equations of motions are in the form of Newton’s equations for a particle. Introduction of a vortex structure is a priori. Nonlocality is introduced only for J, but not for other E-M constitutive equations. Moreover, superconducting media generally involve a mixture consisting of three components: a superconducting fluid, a conducting fluid and an elastic solid. Thus, a continuum mixture theory is suggested. A theory of this nature was given by Eringen [1998a]. Eringen’s theory will be extended here to a mixture with three components with nonlocality. ∇ ×∇ ×A−

214


8.11 Nonlocal Theory8 We consider a mixture consisting of three species with charge densities q i , (i = s, c, e), where superscripts s, c, and e denote, respectively, a superelectron charge, a conduction charge and a charge of an elastic solid. Maxwell’s equations read: ∇ · Di 1 ∂Bi ∇ × Ei + c ∂t ∇ · Bi 1 ∂Di ∇ × Hi − c ∂t ∂q i + ∇ · Ji ∂t

= qi , = 0, = 0, 1 = Ji , c = 0,

(i = s, c, e).

(8.11.1)

Equations of motions of species are Cauchy’s equations: ∇ · ti + ρ i (f i − v˙ i ) + 3 pi + Fi = 0,

(i = s, c, e)

(8.11.2)

where Fi is the E-M force, given by (4.2.5) and 3 pi is the diffusion force, subject to

3 pi = 0.

(8.11.3)

i

Equations of motion of angular momentum lead to (4.4.4) which are nonlinear. Consequently, it may be ignored in a linear theory of constitutive equations. We consider that heat conduction is also negligible and species individually are homogeneous and isotropic, although high Tc -superconductors are generally anisotropic. Here, anisotropy and inhomogeneity are still implied in the sense of the total mixture. The entropy inequality for the total mixture is given by V

−ρ(ψ˙ +ηθ˙ )+

˙ i +3 (ti ·di −Pi ·E i −Mi · B pi ·νν i +J i ·E i dv ≥ 0 (8.11.4)

i

where ν i = vi − ve , dneote diffusion velocities. We have selected ve as the reference velocity, so that ν e = 0. From (8.11.4), in the usual way, we obtain 8 New, not published before.

8.11 Nonlocal Theory

215

linear constitutive equations:

[λe (|x − x |)tree (x )1 + 2µe (|x − x |)ee (x )] dv , tα = −π α 1 + [λαβ (|x − x |)trdβ (x )1 + 2µαβ (|x − x |)dβ (x )] dv ; t = e

V

V

π α = ρρ α

∂ψ , ∂ρ α

{τβi (|x − x |)[vβ (x ) − ve (x )] + κji (|x − x |)E j (x )} dv , Pi = − kji (|x − x |)Ej (x ) dv , V i M =− χji (|x − x |)Bj (x ) dv , V Ji = σji (|x − x |)E j (x ) + κji (|x − x |)[vj (x ) − ve (x )] dv , 3 p = i

V

V

(i = s, c, e), (α, β = s, c),

(8.11.5)

where λe , µe λαβ , µαβ , kji , χji , σji , τβi and κji are the material kernel functions. We observe that the diffusion contributes to the conduction. Eringen’s postulate for superconduction is E s = 0.

(8.11.6)

With this, the Joule heat J s · E s drops out from the C-D inequality (8.11.4) and we are free to introduce a constitutive equation, not restricted by the C-D inequality. We propose Li (|x − x |) × Bi (x ) dv

(8.11.7) Js = V

where Li are material kernel vectors that characterize nonlocal material properties in the superconduction state. As a material property, Li may generally depend on the invariants of electro-mechanical fields, given by {I} = {Ej · Ek , Bj · Bk , θ, tree , ρ α }

(8.11.8)

where ρ α may be replaced by π α . We ignore invariants of second and third order of ee . We note that even though E s vanishes, there will be electric fields in the mixture due to ordinary conduction electrons and the conduction in the elastic solid.

216


We proceed to show that (8.11.7) leads to the London-type equation of superconductivity. To this end, we evaluate ∇ × (Li × B i ) dv

∇ ×Js = V ∇ · Li ) − (B i · ∇ )Li ] dv

= [−B i (∇ V ∇ · Li ) + (B i · ∇ )Li ] dv , = [B i (∇ (8.11.9) V

∇ . We shall show that the second integral vanishes, where we substituted ∇ = −∇ ∂Li (B i · ∇ )Li ] dv = Bj i dv

∂xj V V

i i ∇ · B i )Li dv . = (Bj L ),j − (∇ V

∇

V

· B i

= 0, the second integral vanishes. By means of the Green–Gauss With ∇ theorem, the first integral may be converted to a surface integral which also vanishes, on account of the boundary conditions B · n = 0. Thus, (8.11.9) is reduced to s ∇ ×J =− γ i (|x − x |)B i (x ) dv , (8.11.10) V

where ∇ · Li γ i = −∇

(8.11.11)

Equation (8.11.10) may be localized by setting γ i (x) = γ0i δ(x)

(8.11.12)

δ(x) is the Dirac delta measure. The local form then is given by γ0i Bi . ∇ × Js − qs ∇ × v s = −

(8.11.13)

i

For a single substance (i = s), this gives London’s second equation with vorticity. In this case, γ0s is recognized in terms of the London depth as γ0s = c/µs λ2L .

(8.11.14)

Note that the present theory naturally brings a vorticity term to the London equation. If we use the local form of the constitutive equation Hs = Bs /µs ,

Ds = s Es ,

(s not summed)

and substitute from the fourth Maxwell equation into (8.11.13), we obtain γi s ∂ 2 Bs qs 1 s 0 i B ∇ × ∇ × B + = − + ∇ × vs . c µs c2 ∂t 2 c i i

(8.11.15)


217

This equation shows the influence of the magnetic fields Bc and Be from other species. It is also obvious from (8.11.8) that γ0i and the London depth may depend on the temperature, pressure and the invariants of magnetic field, elactric fields, strain and deformation-rate tensors. The dependence of the transition temperature on the magnetic field was discussed by Bentum et al. [1987], and the pressure dependence by Griessen [1987] and Driessen et al. [1987]. The closure of the theory requires that we supplement (8.11.10) with equations of motions and with the remaining Maxwell equations other than the fourth equations for Hs . We substitute the constitutive equations into the equations of motions to obtain: ∇π + ∇ · −∇ α

∇·

(λαβ trd α 1 + 2µαβ d α ) dv + ρ α (f α − v˙ α ) + [τβα (v β − v e ) + κjα E j ] dv + Fα = 0, V

(λ tre 1 + 2µ e ) dv + ρ e (f e − v˙ e ) + [τβe (v β − v e ) + κje E j ] dv + Fe = 0, e

V

V

e

e e

V

∂ρ α + ∇ · (ρ α vα ) = 0, ∂t

ρ i = ρ,

(α, β = s, c).

(8.11.16)

i

For Maxwell’s equations, we have

∇· E − i

V

kji E j

∇ × Ei +

dv

= qi ,

1 ∂Bi = 0, c ∂t ∇ · Bi = 0,

∇ ×J −q ∇ ×v = − s

s

s

∂q s + q s ∇ · vs = 0, ∂t

V

γ i B i dv , (8.11.17)

∂q r + q r ∇ · vr + ∇ · [σjr E j + κβr (v β − v e )] dv = 0, (r = c, e) ∂t V In the spirit of the linear constitutive theory, we replaced E i and Mi by Ei and Mi , respectively. In this equations, dα , ee , and Fi are to be replaced by α α + vj,i ), dijα = 21 (vi,j

e eij = 21 (uei,j + uej,i ),

218


1 i i i j i j e

q v + [σj E + κj (v − v )] dv × Bi F =q E + c V i i j ∇E ) · ∇ Bi ) · − (∇ kj E dv − (∇ χji B j dv

V V 1 1 i j

i . − ∇· v kj E dv × B − kji E j dv × Bi (8.11.18) c c V V i

i

i

The law of conservation of change, (∂q s /∂t) + q s ∇ · vs = 0, is obtained by taking divergence of (8.11.7) and carrying out by-part integration. Using ∇ ·B = 0, and the boundary condition, Bs · ns = 0, we arrive at this equation. Equations (8.11.16) to (8.11.18) constitute a complete set of field equations of superconductivity in mixtures of superelectron fluid, ordinary conduction fluid and an elastic solid. These equations are fully nonlocal. A count of the number of independent equations will show that we have 38 scalar equations to determine 38 unknown functions vi , ρ α , q i , Ei , B α and Ji , given the body force density. The pressures π α is determined by the equations of state, ψ = (ρ α , {I}). π α = ρρ α

∂ψ0 , ∂ρ α

α is not summed,

where {I} denotes the invariants listed in (8.11.8). Thus, the system is closed. Under the appropriate boundary and initial conditions, this set of integro-partial differential equations may be solved to determine the 38 unknown functions.

A. Local Theory Needless to say, the solution of 38 integro-partial differential equations represents a monumental task. To simplify the matter, we localize these equations by replacing the material property kernels by Dirac delta functions, similar to (8.11.12). With these applied to (8.11.16) through (8.11.18), we obtain Motions: ∇ π α + λα ∇ ∇ · vα − µα ∇ × ∇ × vα + ρ α (f α − v˙ α ) −∇ + τβα (vβ − ve ) + κjα Ej + Fα = 0, λe∇ ∇ · ue − µα ∇ × ∇ × ue + ρ e (f e − v˙ e ) + τβe (vβ − ve ) + κje Ej + Fe = 0, ∂ρ α + ∇ · (ρ α vα ) = 0, ∂t

ρ i = ρ.

(8.11.19)


219

Maxwell equations: ji ∇ · Ej = q i , 1 ∂Bi = 0, c ∂t ∇ · Bi = 0, ∇ × Js − q s ∇ × vs = −γ0i Bi , ∂q s + q s ∇ · vs = 0, ∂t ∇ × Ei +

∂q r + q r ∇ · vr + ∇ · (σjr Ej + κβr (vβ − ve ) = 0, ∂t

(r = c, e)(8.11.20)

where we set ji = δji − kji . We simplify the E-M forces Fc , by noting that, excluding the Lorentz force term, all other terms in (8.11.18) are generally negligible. Hence, 1 Fi = q i Ei + [q i vi + σji Ej + κji (vj − ve )] × Bi , c

i is not summed (8.11.21)

and of course Fs = 0.

B. Pippard Theory The nonlocal equation (8.11.10) of superconductivity can be shown to lead to the Pippard equation (8.10.9). In order to show this, we introduce the magnetic vector potential Bi = ∇ × Ai , ∇ · Ai = 0, (8.11.22) into (8.11.7). Jks =

V

∇ × A i )m dv = klm Lil (∇

V

Lij A ij,k − Lij A ik,j ) dv .

Using by-part integration on each term, we get (Li · A i nk − Li · n A ik ) da + Cji k A ij dv

Jks = ∂V

V

(8.11.23)

where we used the Green–Gauss theorem to take two of the volume integrals to the surface. Cji k is given by Cji k (|x − x |) = ∇ · Li δj k + Lij,k .

(8.11.24)

We select ∇ · Li δj k + Lij,k = 4C i (|x − x |)(Rj Rk /R 4 )e−R/ξ0 ,

R < ξ0 ,

(8.11.25)

220


where R = x − x , R = (R · R)1/2 and ξ0 , and C i (|x − x |) is a material kernel. From (8.11.25), it follows that C i −R/ξ0 e . (8.11.26) R2 With (8.11.25), the surface integral for large |x | can be neglected, and we obtain [Ai (x ) · R]R −R/ξ0

C i (R) e dv . (8.11.27) Js = − R4 V γ i = ∇ · Li =

For a single substance and for the constant C i , this is identical to the expression given by Pippard.

8.12 Alfven Waves In magnetohydrodynamics (MHD) and plasma physics, the importance of longrange intermolecular forces are well known. Here we consider propagation of MHD waves.9 Under the basic assumptions of MHD, isothermal, inviscid plasma, surviving the Maxwell equations are 1 ∂B = 0, ∇ · B = 0, c ∂t 1 ∇ × H − J = 0. c The equations of continuity and motion are given by ∇ ×E+

∇ · v = 0, ρ˙ + ρ∇ 1 ∇ p − ρ v˙ + J × B = 0. −∇ c

(8.12.1)

(8.12.2) (8.12.3)

Here, we ignore the E-M pressure. For a perfect conductor, E + 1c v × B = 0. Using this and the perturbations about the bias field B = B0 + b + O( 2 ),

p = p0 + p + O( 2 ),

ρ = ρ0 + ρ1 + O( 2 ),

v = u + O( 2 ),

(8.12.4)

in (8.12.1) and (8.12.3) we obtain ∂b ∇ × (u × B0 ) − = 0, ∂t 1 ∇ ×b+ ∇ χ B × b dv = J, c V ∂ρ1 + ρ0∇ · u = 0, ∂t ∂u ∇ × b) × B0 + (∇ ∇ χ B × b ) × B0 dv − ρ0 − ∇ p + (∇ = 0. ∂t V 9 Eringen [1986], [1993].

(8.12.5) (8.12.6) (8.12.7) (8.12.8)

8.12 Alfven Waves

221

Considering that the waves are directed in the B0 -direction, the transverse components of these equations become uncoupled. Letting the z-direction be in the direction of the H0 components of (8.12.5) to (8.12.6), in the direction perpendicular to z become ∂u1 ∂b⊥ − = 0, ∂z ∂t ∂χ1B

∂u⊥ b dz − ρ0 = 0, ∂z ⊥ ∂t B0

B0

∂b⊥ + B0 ∂t

∞ −∞

(8.12.9)

where all fields are considered to be functions of z and t and χ1 = χ1 (|z − z |). Eliminating u⊥ between these two equations, we have ∞ 2 B ∂ χ1

ρ0 ∂ 2 b⊥ ∂ 2 b⊥

+ b dz − = 0. (8.12.10) ⊥ 2 ∂z2 ∂t 2 B02 −∞ ∂z For plane harmonic waves b = b0 exp[i(kz − ωt)],

(8.12.11)

2 2 0 {−k 2 [1 + χ B 1 (k)] − (ρ0 ω /B0 )}b⊥ = 0,

(8.12.12)

and this gives

where χB 1 (k) =

∞ −∞

χ1B (z ) exp(−ikz ) dz .

(8.12.13)

Consequently, the phase velocity is obtained to be 2 1/2 A = {[1 + χ B . 1 (k)]B0 /ρ0 }

(8.12.14)

This is the Alfven velocity of a transverse disturbance propagating along the magnetic lines of force. It is uncoupled from the longitudinal acoustic waves. Note that unlike the classical result, these waves are dispersive. The dispersion is important, especially for small wavelengths.

Problems 8.1 Obtain the constitutive equations of the hexagonal crystals. 8.2 Express the nonlocal boundary-value problems for an elastic piezoelectric slab subject to harmonic surface exitation. 8.3 Obtain the field equations replacing (8.4.21) for the micromedia with absorption. 8.4 Charge is distributed uniformly on a spherical surface in an infinite nonlocal solid medium. Determine the electric field inside and outside the spherical surface.

222


8.5 Electric wires carrying high currents are known to explode. Can you explain the mechanism of this failure? 8.6 Determine the displacement of an isotropic wire carrying an electric field. 8.7 Verify equation (8.9.10). 8.8 Obtain the magnetic field in a nonlocal superconducting wire.

9 Memory-Dependent Nonlocal Thermoelastic Solids

9.0 Scope Linear constitutive equations of the memory-dependent nonlocal anistropic thermoelastic solids are obtained in Section 9.1. The isotropic solids and Kelvin–Voigttype models are given as special cases. Mixed boundary and initial value problems are formulated in Section 9.2.

9.1 Linear Constitutive Equations1 Constitutive equations of the thermoviscoelastic solids were given in Section 3.2. The linearization process is similar to that followed in Section 6.1. The static parts of the linear constitutive equations (6.1.16) presented in Section 6.1 remain unchanged, i.e., Rη

=

R tkl

=

V V

C(x, x )

T (x ) 1 + βkl (x, x )ekl (x ) dv , T0 ρ

[−βkl (x, x )T (x ) + λij kl (x, x )eij (x )] dv.

(9.1.1)

The material functions C, βkl , and λij kl are subject to the symmetry conditions given by (6.1.12). For homogeneous materials they are functions of x − x and for homogeneous and isotropic materials the static parts of the constitutive equations 1 Eringen [1974b].

224


have the form given by (6.1.20), namely, T (x ) 1

C(|x − x|) + β(|x − x|)ekk (x ) dv , Rη = ρ T0 V

[−β(|x − x|)T (x )δkl + λ(|x − x|)err (x )δkl R tkl = V

+ 2µ(|x − x|)ekl (x )] dv .

(9.1.2)

The dynamic parts of the constitutive equations can be obtained from the general set given by (3.2.16). For simplicity, we ignore the temperature-rate dependence. Also, for the linear theory, the constitutive residuals vanish (Onsager postulate), so that ∂ ∂ 1 , . qk = xk,K D tkl = 2xk,K xl,L θ ∂θ,K ∂ C˙ KL Using (1.4.9) in the first of these, we obtain the spatial forms D tkl

=

∂ . ∂dkl

(9.1.3)

For the linear theory, we take T = θ − T0 , T0 0 so that 1 ∂ qk = . T0 ∂T,k

(9.1.4)

The dissipation potential is expressed as ∞ ∞ = dτ dτ

G dv , 0

0

(9.1.5)

V

where G is a symmetric quadratic function of the difference histories of the deformation-rate tensor and the temperature gradient. This is given by 2G = µ klij (x , τ ; x, τ )[d(t)ij (x , τ ) + d(t)ij (x, τ )]

(x , τ ; x, τ )[T(t),k (x , τ ) × [d(t)kl (x , τ ) + d(t)kl (x, τ )] + kkl + T(t),k (x, τ )][T(t),l (x , τ ) + T(t),l (x, τ )] + h kli (x , τ ; x, τ )[d(t)kl (x , τ )

+ d(t)kl (x, τ )[T(t),i (x , τ ) + T(t),i (x, τ ].

(9.1.6)

The difference histories are defined similar to (3.2.20), i.e., d(t)kl (x, τ ) = dkl (x, t − τ ) − dkl (x, t), T(t),k (x, τ ) = T(t),k (x, t − τ ) − T,k (x, τ ),

0 ≤ τ < ∞.

(9.1.7)

As mentioned in Section 3.2, upon the change of variables t − τ = s, these take the forms d(t)kl (x, t − s) = dkl (x, s) − dkl (x, t), T(t)k (x, t − s) = T,k (x, s) − T,k (x, t),

−∞ ≤ s ≤ t,

(9.1.8)


225

which display clearly the difference in histories and the present-time values. We mentioned that G is a symmetric function. This means that G remains unchanged when (x , τ ) is interchanged with (x, τ ), respectively. For the total dissipation we have W =

V

dv ≥ 0.

(9.1.9)

, and h

From (9.1.9) and (9.1.6) it is clear that the material moduli µ klij , kkl kli are subject to the following symmetry regulations

µ klij (x , τ ; x, τ ) = µ klij (x, τ ; x , τ ) = µ lkij = µ klj i ,

(x, τ ; x , τ ) = klk , (x , τ ; x, τ ) = kkl kkl

hkli (x , τ ; x, τ ) = hkli (x, τ ; x , τ ) = h lki .

(9.1.10)

Substituting (9.1.6) into (9.1.3) and (9.1.4) we obtain D tkl

= µ0klij (x, t)dij (x, t) + h0kli (x, t)T,i (x, t) ∞ [µ1klij (x, τ )dij (x, t − τ ) + h1kli (x, τ )T,i (x, t − τ )] dτ + 0 + [µ2klij (x , t; x)dij (x , t) + h2kli (x , t)T,i (x , t)] dv

V ∞

+ dτ [µ3klij (x , τ ; x)dij (x , t − τ ) 0

V

+ h3kli (x , τ ; x)T,i (x , t − τ )] dv , 1 0 qk = kkl (x, t)T,l (x, t) + h0ij k (x, t)dij (x, t) T0 ∞ 1 [kkl (x, τ )T,l (x, t − τ ) + h1ij k (x, τ )dij (x, t − τ )] dτ + 0 2 + [kkl (x , t; x)T,l (x , t) + h2ij k (x , t)dij (x , t)] dv

V ∞ 3 + dτ [kkl (x , τ ; x)T,l (x , t − τ ) 0

V

+ h3ij k (x , τ ; x)dij (x , t − τ )] dv , where we have introduced new moduli by ∞ ∞ 0 0 0

dτ dτ {µ klij , kkl , h kli } dv , {µklij , kkl , hkli } = V 0 0 ∞ 1

, h1kli } = − dτ {µ klij , kkl , h kli } dv , {µ1klij , kkl ∞0 ∞V 2 2 2

dτ dτ {µ klij , kkl , h kli }, {µklij , kkl , hkli } = 0 0 ∞ 3

, h3kli } = − {µ klij , kkl , h kli } dτ. {µ3klij , kkl 0

(9.1.11)

(9.1.12)

226


Equations (9.1.11) constitute the linear constitutive equations of the dynamic portions of the response functionals D tkl and qk . We notice four groups of terms in each equation: (i) the first group of two terms, with superscript (0) on moduli, represents the local contribution with no memory; (ii) the second group of terms, with superscript (1) on moduli, is the local contribution with memory; (iii) the third group of terms, with superscript (2) on moduli, is the nonlocal contribution with no memory; and (iv) finally, the last group of terms are the contribution from the memory-dependent nonlocal effects. Clearly, the first three parts can be absorbed into the last part by introducing Dirac-delta measures. Thus, as a generalized function, we can express (9.1.11) in the form ∞ dτ [µklij (x , τ ; x)dij (x , t − τ ) D tkl = 0

V

+ hkli (x , τ ; x)T,i (x , t − τ )] dv , ∞ 1

qk = dτ [kkl (x , τ ; x)T,l (x , t − τ ) T0 V 0 + hij k (x , τ ; x)dij (x , t − τ )] dv ,

(9.1.13)

where µklij = µ0klij δ(x − x)δ(τ ) + µ1klij δ(x − x) + µ2klij δ(τ ) + µ3klij , 0 1 2 3 kkl = kkl δ(x − x)δ(τ ) + kkl δ(x − x) + kkl δ(τ ) + kkl ,

hij k = h0ij k δ(x − x)δ(τ ) + h1ij k δ(x − x) + h2ij k δ(τ ) + h3ij k .

(9.1.14)

By a change of variable t − τ = t , equations (9.1.13) are also expressed in more familiar forms t dt [µklij (x , t − t ; x)dij (x , t ) D tkl = −∞

V

+ hkli (x , t − t ; x)T,i (x , t )] dv , t 1

qk = dt [kkl (x , t − t ; x)T,l (x , t ) T0 V −∞ + hij k (x , t − t ; x)dij (x , t )] dv .

(9.1.15)

The full constitutive equations are the sums of the static and dynamic parts tkl = D tkl + D tkl ,

qk ,

η = R η.

(9.1.16)


227

Isotropic Media For homogeneous and isotropic solids, the material moduli µklij , kkl , and hkli depend on x and x , through |x − x|, and they have the special forms (6.1.18), i.e., µklij = λv δij δkl + µv (δik δj l + δil δj k ) + µ1 (κi κj δkl + κk κl δij ) + µ2 κi κj κk κl , kkl = kδkl + k1 κk κl , hij k = h1 κk δij . (9.1.17) The material moduli λv , µv , µ1 , µ2 , k, k1 , and h1 are functions of κ = |x − x| and t − t , e.g., (9.1.18) µv = µv (|x − x|, t − t ). The effective range of the intermolecular forces is known to extend a few atomic distances. Consequently, the attenuating neighborhood and fading memory hypotheses imply that λv , µv , . . . , h1 will diminish quickly with |x − x| and t − t

to null. Thus, generally, the coefficients of κi in (9.1.17) may also be dropped, and λv , µv , and k may be taken as rapidly diminishing in amplitudes. This situation was fully explored in Section 6.1 for the linear, isotropic nonlocal elastic solids without memory. On this basis, we have µklij = λv (|x − x|, t − t )δij δkl + µv (|x − x|, t − t )(δik δj l + δj k δil ), kkj = k(|x − x|, t − t )δj k , hkli = 0. (9.1.19) With this, the constitutive equations, for homogeneous isotropic materials (9.1.15), take the forms t

t = dt [λv (|x − x|, t − t )dii (x , t )δkl D kl −∞

V

+ 2µv (|x − x|, t − t )dkl (x , t )] dv, t 1 qk = dt

k(|x − x|, t − t )T,k (x , t ) dv . T0 V −∞

(9.1.20)

Kelvin–Voigt Model The nonlocal version of the classical Kelvin–Voigt model is obtained by considering instantaneous material moduli, i.e., λv (|x − x|, t − t ) = λK (|x − x|)δ(t − t ), µv (|x − x|, t − t ) = µK (|x − x|)δ(t − t ), k(|x − x|, t − t ) = k K (|x − x|)δ(t − t ).

(9.1.21)

With these, the full constitutive equations may be expressed as ∂ ∂ err (x )δkl + 2 µ + µK ekl (x ) dv , tkl = λ + λK ∂t ∂t V 1 qk = k K (|x − x|)T,k (x ) dv . (9.1.22) T0 V

228


Other more complicated models, involving the higher-order rates of ekl used in classical viscoelasticity (cf. Eringen [1967, Section 9.5]), can also be generalized to nonlocal forms by assuming special types of generalized functions in the time domain. Since these types of models have limited uses and possess some problems in regard to initial conditions, we shall not pursue the discussion of this topic any further.

9.2 Boundary-Initial Value Problems A class of general boundary-initial value problems in memory-dependent, nonlocal thermoelasticity requires the solutions of the following integro-partial differential equations under some boundary and initial conditions. Equations of Motion tkl,k + ρfl = ρ u¨ l ,

tkl = tlk .

(9.2.1)

The Energy Equation − [ρC(x − x)T˙ (x ) + T0 βkl (x − x)u˙ k,l (x )] dv + qk,k + ρh = 0. (9.2.2) V

Constitutive Equations tkl = [−βkl (x − x)T (x ) + λij kl (x − x)ui,j (x )] dv

V t

+ dt µklij (x − x, t − t )u˙ i,j (x , t ) dv , V −∞ t 1

dt kkl (x − x, t − t )T,l (x , t ) dv . qk = T0 V −∞

(9.2.3)

Boundary Conditions uk = uˆ k (x, t) T = Tˆ (x, t)

on S 1 × T + , on S 3 × T + ,


on S2 × T + ,

qk nk = q(x, ˆ t)

on S4 × T + . (9.2.4)

Initial Conditions uk (x, 0) = u0k (x),

u˙ k (x, 0) = v 0 (x),

T (x, 0) = T 0 (x) in V,

(9.2.5)

where V is the interior of the body V and S1 and S4 are partial surfaces covering the surface ∂V of the body: S 1 ∪ S2 = S3 ∪ S4 = ∂V,

S1 ∩ S2 = S3 ∩ S4 = 0.

(9.2.6)

9.2 Boundary-Initial Value Problems

229

The following smoothness requirements are assumed uk (x, t) ∈ C 1,2 ,

{tkl (x, t), qk (x, t)} ∈ C 1,0 , T (x, t) ∈ C 1,0 in V × T + , {uˆ k (x, t), tˆkl (x, t), Tˆ (x, t), q(x, ˆ t)} ∈ C 0,0 on ∂V × T + .

(9.2.7)

Problems 9.1 Obtain the constitutive equations (9.1.11) giving all the steps. 9.2 An infinite model Kelvin–Voigt elastic bar is subject to harmonic longitudinal waves. Determine the dispersion relations. 9.3 Assuming that material moduli (such as λv , µv , etc.) obey the attenuating neighborhood hypothesis in the space–time (x, t), the field equation can be reduced to partial differential equations. Carry out this reduction for a nonlocal kernel α(|x − x|, t − t ). 9.4 In the case of Problem 9.3 a theorem of correspondence can be proven to obtain the solution of the nonlocal viscoelastic problem from those of the nonlocal elastic problem. Show this.


10 Memory-Dependent Nonlocal Fluids

10.0 Scope In Section 10.1 we obtain the constitutive equations of memory-dependent nonlocal thermoviscous fluids. The mixed boundary-intial value problems are formulated in Section 10.2.

10.1 Linear Constitutive Equations of Memory-Dependent, Nonlocal Thermoviscous Fluids1 The static portions of the constitutive equations of the memory-dependent nonlocal fluids are identical to those given in Section 7.1, namely, R tkl Rη

= −πδkl , =−

π = ρ2

∂ , ∂θ

∂ , ∂ρ (10.1.1)

where is the free energy function, given by ψ = (ρ, θ).

(10.1.2)

1 The theory given here appeared as a special case of the memory-dependent orientable micropolar fluids, see Eringen [1991a].

232


The dynamic portions of the constitutive equations are of the general form (3.3.18). In linear theory, following the Onsager postulate, we drop the constitutive residuals U. Also, we assume that the temperature rate effect is not present. Equation (3.3.18) is then reduced to D η = 0 and ∂ , ∂dkl 1 ∂ . qk = θ ∂θ,k

D tkl

=

(10.1.3)

The dissipation potential is expressed by

∞

=

∞

dτ 0

0

dτ

V

G dv ,

(10.1.4)

where G is given as a quadratic polynomial in terms of the invariants of the difference histories of dkl and T,k , i.e., 2G = λ (x , τ ; x, τ )[d(t)ii (x , τ ) + d(t)ii (x, τ )][d(t)jj (x , τ ) + d(t)jj (x, τ )] + 2µ (x , τ ; x, τ )[d(t)ij (x , τ ) + d(t)ij (x, τ )][d(t)ij (x , τ ) + d(t)ij (x, τ )] + k (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )][T(t),k (x , τ ) + T(t),k (x, τ )], 0 ≤ τ, τ < ∞, (10.1.5) where the difference histories of dkl and T,k are defined, as usual, by (9.1.7). Substituting (10.1.5) into (10.1.3) we get D tkl

= λ0 (x, t)drr (x, t)δkl + 2µ0 (x, t)dkl (x, t) ∞ [λ1 (x, τ )drr (x, t − τ )δkl + 2µ1 (x, τ )dkl (x, t − τ )] dτ + 0 + [λ2 (x , t; x)drr (x , t)δkl + 2µ2 (x , t)dkl (x , t)] dv

V∞ + dτ [λ3 (x , τ ; x)drr (x , t − τ )δkl 0

V

+ 2µ3 (x , τ ; x)dkl (x , t − τ )] dv , ∞ 1 qk = k0 (x, t)T,k (x, t) + k1 (x, τ )T,k (x, t − τ ) dτ T0 0 + k2 (x , t; x)T,k (x , t) dv

V ∞ + dτ

k3 (x , τ ; x)T,k (x , t − τ ) dv, 0

V

(10.1.6)

(10.1.7)

10.1 Nonlocal Thermoviscous Fluids

233

where

∞

{λ0 , µ0 , k0 } =

∞

dτ 0

∞

0

dτ

V

{λ , µ , k } dv ,

dτ {λ , µ , k } dv , {λ1 , µ1 , k1 } = − V 0 ∞ ∞ dτ dτ {λ , µ , k }, {λ2 , µ2 , k2 } = 0 ∞ 0 {λ , µ , k } dτ. {λ3 , µ3 , k3 } = −

(10.1.8)

0

As discussed in Section 9.1, by introducing Dirac-delta measures, (10.1.7) may be expressed as generalized functions. To this end, we define λ = λ0 δ(x − x)δ(τ ) + λ1 δ(x − x) + λ2 δ(τ ) + λ3 , µ = µ0 δ(x − x)δ(τ ) + µ1 δ(x − x) + µ2 δ(τ ) + µ3 , k = k0 δ(x − x)δ(τ ) + k1 δ(x − x) + k2 δ(τ ) + k3 .

(10.1.9)

Since fluids are isotropic, in these equations the material moduli λ, µ, and k will depend on x and x only through |x − x|. Hence, (10.1.6) and (10.1.7) can be written in compact forms ∞ dτ [λ(|x − x|)drr (x , t − τ )δkl D tkl = 0

V

+ 2µ(|x − x|, τ )dkl (x , t − τ )] dv , ∞ 1 qk = dτ

k(|x − x|), τ )T,k (x , t − τ ) dv . T0 V 0

(10.1.10)

Through a change of variable, t −τ = t , these equations take more familiar forms D tkl

=

t −∞

dt

V

[λ(|x − x|, t − t )drr (x , t )δkl

+ 2µ(|x − x|, t − t )dkl (x , t )] dv , t 1 qk = dt

k(|x − x|, t − t )T,k (x , t ) dv , T0 V −∞

(10.1.11) −∞ < t ≤ t.

The material moduli λ, µ, and k are subject to an attenuating neighborhood and a fading memory hypothesis. The attenuating neighborhood hypothesis was expressed in Section 3.1 (see also (7.1.9) to (7.1.12)). The fading memory is formulated in a similar fashion by a continuity requirement lim {λ(|κκ |, t − t ), µ(|κκ |, t − t ), k(|κκ |, t − t )}(t − t )1+δ = 0,

t−t →0

δ > 0.

(10.1.12)

234


Any Dirac-delta sequence is a possible candidate for the nonlocal moduli, e.g., k12 κ 2 k2 t λ(|κκ |, t)/λv = αv exp − 2 − (10.1.13) , κ = |x − x|, l τ where k1 and k2 are appropriate nondimensional scalars, l is an internal characteristic length, τ is an internal characteristic time, and λv is a constant viscosity. αv can be selected to normalize (10.1.13) over space–time αv = π −n/2 (k1 / l)n (τ/k2 ),

(10.1.14)

in n dimensions, over t = [0, ∞). Of course, many such delta sequences exist. The best choice of λ(κ, t) requires either an experimental confirmation or lattice dynamical considerations, as in the case of perfect crystals. Based on the discussions given in nonlocal elasticity, it appears that the specific form of λ(κ, t) does not influence the outcome greatly, so long as parameters such as k1 , l, k2 , and τ are selected properly.

10.2 Field Equations of Memory-Dependent Nonlocal Fluids The field equations of the memory-dependent nonlocal fluids consist of the combination of the balance laws and the constitutive equations. Balance Laws ∇ ·v =0 ρ˙ + ρ∇ [ρ(vk − uk )]nk = 0 − π,l + D tkl,k + ρ(fl − v˙l ) = 0 l [ − πδkl + D tkl − ρvl (vk − uk )]nk = F [ − ρ[ψ + θη +

1 2v

in V − σ, (10.2.1a) on σ,

(10.2.2b)

in V − σ, (10.2.2a) on σ,

(10.2.2b)

− ρθ η˙ + D tkl dkl + qk,k + ρh = 0 in V − σ, (10.2.3a) on σ. · v](vk − uk ) − πvk + D tkl vl + qk ]nk = H (10.2.3b)

Constitutive Equations π = ρ2

∂ , ∂ρ

D tkl =

t −∞

dt

η=− V

∂ , ∂θ

ψ = (ρ, θ),

(10.2.4)

[λ(|x − x|, t − t )drr (x , t )δkl

+ 2µ(|x − x|, t − t )dkl (x , t )] dv , t 1

qk = dt k(|x − x|, t − t )T,k (x , t ) dv . T0 V −∞

(10.2.5) (10.2.6)

10.2 Field Equations of Memory-Dependent Nonlocal Fluids

235

A general mixed boundary-initial value problem is formulated similar to (9.2.4) and (9.2.5), namely: Boundary Conditions vk = v(x, ˆ t) on S 1 × T + , T = Tˆ (x, t) on S 3 × T + ,

tkl nk = tˆl on S2 × T + , qk nk = q(x, ˆ t) on S4 × T + . (10.2.7)

Initial Conditions vk (x, 0) = vk0 (x),

T (x, 0) = T 0 (x) in V,

(10.2.8)

where quantities carrying a hat (ˆ) are prescribed on surfaces S1 to S4 and those carrying a (0) in the body. Surfaces (S 1 , S2 ) and (S 3 , S4 ) make up the total surface ∂V of the body and V is the region occupied by the body S 1 ∪ S2 = S 3 ∪ S4 = ∂V,

S 1 ∩ S2 = S 3 ∩ S4 = 0,

(10.2.9)

T + = [0, ∞) denotes the time interval.

Problems 10.1 Give detailed derivations of the constitutive equations (10.1.11). 10.2 Obtain the solution of the pipe flow problem. 10.3 Set up a boundary layer theory for the memory-dependent nonlocal viscous fluids. 10.4 Formulate a lubrication theory.


11 Memory-Dependent Nonlocal Electromagnetic Elastic Solids

11.0 Scope The linear constitutive equations of the memory-dependent electromagnetic (E-M) elastic solids1 are obtained in Section 11.1. Memory effects include the E-M absorptions and viscous dissipation. Nonlocal continua with no memory, local media with memory, local theory, absence of E-M fields, and rigid media are given as special cases of the general constitutive equations. Field equations are collected in Section 11.2, so that the formulation is now ready for the treatment of a variety of boundary-initial value problems in these fields.

11.1 Linear Constitutive Equations The general forms of the constitutive equations of memory-dependent nonlocal E-M elastic solids were developed in Section 5.1. Using the same scheme as in Section 6.1, equations (5.1.21) are simplified to the spatial form

∂F S −ρ R η = dv , ∂T

V ∂F ∂F S dv , + R tkl =

∂e ∂e kl V kl 1 Eringen [1984c], [1991].

∂F + ∂T

238

11 Memory-Dependent Nonlocal E-M Elastic Solids

∂F S dv , −R Pk =

∂E V k ∂F ∂F S −R Mk = dv . + ∂Bk

V ∂Bk ∂F + ∂Ek

(11.1.1)

We draw attention to the fact that t kl , defind by (4.4.10), in the linear theory becomes t(kl) , since E(k Pl) + B(k Ml) constitute nonlinear terms and therefore can be omitted. This then produces a symmetric stress tensor R tkl . This is also valid for the dynamical part of the stress tensor D tkl . Thus, Voigt’s piezoelectricity is saved by this default. The free energy density F is a quadratic function of T , ekl , Ek , and Bk . We express F as the sum of a purely elastic part F M and an interacting E-M elastic part F E : F = F M + F E.

(11.1.2)

F M is identical to (6.1.10)1 , namely F M = F0 −

ρ

CT T − 21 βkl (T ekl + T ekl ) + U, T0

(11.1.3)

here U being the strain energy density U=

1 2

ρ

CT T + λij kl eij ekl . T0

(11.1.4)

The free energy F E , involving E-M elastic interactions, is of the form 2F E = −ρλTk E (T Ek + T Ek ) − ρλTk B (T Bk + T Bk ) + χklE Ek EL + χklB Bk Bl

+ λEB (Ek Bl + Ek Bl ) + λeE klm (ekl Em + ekl Em )

+ λeB klm (ekl Bm + ekl Bm ),

(11.1.5)

where the constitutive moduli λTk E , . . . , λeB klm are the functions x and x . As usual,

a prime on symbols denotes the dependence on x , e.g., Ek = Ek (x , t). Carrying F , given by (11.1.2) to (11.1.5), into (11.1.1) we obtain the static portions of the linear constitutive equations of memory-dependent nonlocal elastic


239

solids η=

1 1 C(x , x)T (x , x) + βkl (x , x)ekl (x ) ρ V T0

+ λTk E (x , x)Ek (x , t) + λTk B (x , x)Bk (x , t) dv ,

−βkl (x , x)T (x , t) + λij kl (x , x)eij (x , t) R tkl = V

eB

+ λeE klm (x , x)Em (x , t) + λklm (x , x)Bm (x , t) dv ,

TE

ρλk (x , x)T (x , t) − χklE (x , x)El (x , t) R Pk = V

eE

− λEB kl (x , x)Bl (x , t) − λlmk (x , x)elm (x , t) dv ,

TB

ρλk (x , x)T (x , t) − χklB (x , x)Bl (x , t) M = R k V

eB

− λEB lk (x , x)El (x , t) + λlmk (x , x)elm (x , t) dv .

(11.1.6)

The static parts of the constitutive equations display nonlocal piezoelectric and piezomagnetic effects denoted, respectively, by the nonlocal material moduli λeE klm EB and λeB klm . Also, we have a magnetoelectric effect denoted by λkl . Since the term containing this coefficient is not time-symmetric and also axial, at first sight one may think that such effects are forbidden. The existence of the nonlocal magnetoelectric effect depends on the time-symmetry combined by with the material group symmetry. For instance, for isotropic materials, the magnetoelectric effect does not exist. For the material moduli of electromagnetic elastic crystals, under the restrictions of a 90 magnetic group, we refer the reader to Kiral and Eringen [1990]. The dynamic parts of the constitutive equations are given by (5.1.22). We ignore ˙ This gives D η = 0. Following the the dependence on the temperature-rate θ. Onsager postulate, for the linear theory, constitutive residuals are also omitted. The linearization leads to the spatial forms D tkl

=

1 qk = T0 D Pk

=

D Mk

=

Jk =

∂ , ∂dkl ∂ , ∂T,k ∂ − , ∂ E˙k ∂ − , ∂ B˙ k ∂ . ∂Ek

(11.1.7)

240


The dissipation functional is expressed as ∞ ∞ = dτ dτ

G dv , 0

(11.1.8)

V

0

where G is a symmetric quadratic function of the difference histories of the independent dissipative constitutive variables 2G = µ klij (x , τ ; x, τ )[d(t)ij (x , τ ) + d(t)ij (x, τ )][d(t)kl (x , τ ) + d(t)kl (x, τ )] + hkli (x , τ ; x, τ )[d(t)kl (x , τ ) + d(t)kl (x, τ )][T(t),i (x , τ ) + T(t),i (x, τ )] ˙ + µdkliE (x , τ ; x, τ )[d(t)kl (x , τ ) + d(t)kl (x, τ )][E˙(t)i (x , τ ) + E˙(t)i (x, τ )] ˙ + µdkliB (x , τ ; x, τ )[d(t)kl (x , τ ) + d(t)kl (x, τ )][B˙ (t)i (x , τ ) + B˙ (t)i (x, τ )]

+ kkl (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )][T(t),l (x , τ ) + T(t),l (x, τ )] ˙

TE

(x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )][E˙(t)l (x , τ ) + E˙(t)l (x, τ )] + kkl T B˙

+ kkl (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )][B˙ (t)l (x , τ ) + B˙ (t)l (x, τ )] + χkl (x , τ ; x, τ )[E˙(t)k (x , τ ) + E˙(t)k (x, τ )][E˙(t)l (x , τ ) + E˙(t)l (x, τ )] ˙ ˙ + χklE B (x , τ ; x, τ )[E˙(t)k (x , τ ) + E˙(t)k (x, τ )][B˙ (t)l (x , τ ) + B˙ (t)l (x, τ )] ˙

+ χ EE (x , τ ; x, τ )[E˙(t)k (x , τ ) + E˙(t)k (x, τ )][E(t)l (x , τ ) + E(t)l (x, τ )] + γkl (x , τ ; x, τ )[B˙ (t)k (x , τ ) + B˙ (t)k (x, τ )][B˙ (t)l (x , τ ) + B˙ (t)l (x, τ )] ˙ + γklBE (x , τ ; x, τ )[B˙ (t)k (x , τ ) + B˙ (t)k (x, τ )][E(t)l (x , τ ) + E(t)l (x, τ )] + σkl (x , τ ; x, τ )[E(t)k (x , τ ) + E(t)k (x, τ )][E(t)l (x , τ ) + E(t)l (x, τ )] dE

(x , τ ; x, τ )[d(t)kl (x , τ ) + d(t)kl (x, τ )][E(t)i (x , τ ) + E(t)i (x, τ )] + σkli

+ σklT E (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )] × [E(t)l (x , τ ) + E(t)l (x, τ )].

(11.1.9)

The difference histories of the independent variables are defined similar to (9.1.7). The material functions are marked by two superscript indices which show d E˙ is the material moduli for the the independent variable that it belongs to, e.g., γkli products dkl and E˙i . These moduli are symmetric with regard to the interchange of (x , τ ) with (x, τ ). Also, they possess other symmetry relations which are easily deduced from the symmetry of the independent variables, e.g., µ klij = µ lkij = µ klj i ,

˙

˙

dE dE γkli = γlki ,

˙ ˙

˙ ˙

γklE E = γlkE E , . . . .

(11.1.10)

Substituting (11.1.9) into (11.1.7) we obtain the dynamic parts of the constitutive equations. These equations are in the form of (9.1.11) containing local terms, nonlocal terms without memory, local terms with memory, and memory-dependent nonlocal terms (integrals over time and V). To save space we omit writing these expressions and pass to the final form involving the generalized functions. The method of approach to the final form is similar to that described in Section 9.1


241

through the introduction of the Dirac-delta measures for the first three parts of these equations. We also change the variables by t − τ = t , as in Section 9.1, to obtain D tkl

=

1 qk = T0

− D Pk =

− D Mk =

Jk =

t

dt

V −∞ d E˙

+ µkli (x , t dE

(x , t + σkli t

dt

[µklij (x , t − t ; x)dij (x , t ) + hkli (x , t − t ; x)T,i (x , t ) ˙ − t ; x)E˙i (x , t ) + µdkliB (x , t − t ; x)B˙ i (x , t )

− t ; x)Ei (x , t )] dv , [kkl (x , t − t ; x)T,l (x , t ) + hij k (x , t − t ; x)dij (x , t )

V −∞ T B˙

T E˙

+ kkl (x , t − t ; x)E˙l (x , t ) + kkl (x , t − t ; x)B˙ l (x , t ) + σklT E (x , t − t ; x)El (x , t )] dv , t ˙ T E˙

dt [µdijEk (x , t − t ; x)dij (x , t ) + klk (x , t − t ; x)T,l (x , t ) V −∞ ˙ ˙ + χkl (x , t − t ; x)E˙l (x , t ) + χklE B (x , t − t ; x)B˙ l (x , t ) ˙ + χklEE (x , t − t ; x)El (x , t )] dv , t ˙

T B˙

dt [µdijBk (x , t − t ; x)dij (x , t ) + klk (x , t − t ; x)T,l (x , t ) V −∞ E˙ B˙

+ χlk (x , t − t ; x)E˙l (x , t ) + γkl (x , t − t ; x)B˙ l (x , t ) ˙ + γklBE (x , t − t ; x)El (x , t )] dv , t

TE

dt [µdE ij k (x , t − t ; x)dij (x , t ) + σlk (x , t − t ; x)T,l (x , t ) V −∞ ˙ E E˙

+ χlk (x , t − t ; x)E˙l (x , t ) + γlkBE (x , t − t; x)B˙ l (x , t ) + σkl (x , t − t ; x)El (x , t )] dv. (11.1.11)

A close scrutiny of these equations shows that memory-dependent nonlocal temperature gradient rates of electric and magnetic fields cause stress in the body. Conversely, deformation-rate tensors, rates of electric and magnetic fields, and electric fields will produce heat. Polarization, magnetization, and current are induced by the deformation-rate tensor, temperature gradient, rates of electric and magnetic fields, and the electric fields. Of course, the presence of these “cross effects" depends on the material symmetry group and the time symmetry. A discussion of the symmetry group in local E-M elasticity is to be found in Kiral and Eringen [1990]. However, the topic that includes nonlocal and/or memory effects appears not to have been studied presently. We consider several special cases.

242


A. Nonlocal Media with No Memory This case is obtained by simply taking the time parts of the material moduli as Dirac-delta measures, e.g., µklij (x , t − t ; x) = µ1klij (x , x)δ(t − t ).

(11.1.12)

With this, the integrals with respect to t in (11.1.11) disappear and the material moduli are now functions of x and x.

B. Local Media with Memory In this case, we introduce the Dirac-delta measure for the space parts of the material moduli, e.g.,

µklij (x , t − t ; x) = µL klij (t − t )δ(x − x).

(11.1.13)

This eliminates the nonlocality (the volume integral) from (11.1.11) and allows the material moduli to depend on time only.

C. Local Media In this case, we take material moduli in the pattern µklij (x , t − t ; x) = µ0klij δ(x − x)δ(t − t ).

(11.1.14)

Of course, all three cases A, B, and C had appeared before (11.1.11), upon substituting (11.1.9) into (11.1.7) as an intermediate cases similar to (9.1.11) and we skipped writing these equations. It is not difficult for the reader to produce these equations from the master equations (11.1.11).

D. Absence of Electromagnetic Fields In this case, we have purely nonlocal memory-dependent elasticity. In order to obtain the constitutive equations all we need is to set the E-M fields equal to zero. With this, (11.1.6) and (11.1.11) revert to equations (9.1.1) and (9.1.15), respectively.

E. Rigid Media To obtain the constitutive equations of the rigid media, we set the strain tensor eij and its rate equal to zero and ignore the elastic stress tensors. With this (11.1.6)

11.2 Field Equations

243

and (11.1.11) become T (x ) C(x , x) + λTk E (x , x)Ek (x , t) η= T 0 V TB

+ λk (x , x)Bk (x , t) dv ,

− R Pk = −ρλTk E (x , x)T (x ) + χklE (x , x)El (x , t) V

+ λEB kl (x , x)Bl (x , t) dv ,

− R Mk = −ρλTk B (x , x)T (x ) + χklB (x , x)Bl (x , t) V

+ λEB lk (x , x)El (x , t) dv , t

T E˙

− D Pk = klk (x , t − t ; x)T,l (x , t) + χkl (x , t − t ; x)E˙l (x , t ) dt

− D Mk =

Jk =

1 qk = T0

V −∞ ˙ E˙ B˙

+ χkl (x , t − t ; x)B˙ l (x , t ) + χklEE (x , t − t ; x)El (x , t ) dv , t

T B˙

E˙ B˙

klk (x , t − t ; x)T,l (x , t ) + χlk dt (x , t − t ; x)E˙l (x , t ) V −∞ ˙ + γkl (x , t − t ; x)B˙ l (x , t ) + γklBE (x , t − t ; x)El (x , t ) dv , t

TE

˙ EE dt

(x , t − t ; x)E˙l (x , t ) σlk (x , t − t ; x)T,l (x , t ) + χlk V −∞ ˙ + γlkBE (x , t − t ; x)B˙ l (x , t ) + σkl (x , t − t ; x)El (x , t ) dv , t

T E˙

kkl (x , t − t ; x)T,l (x , t ) + kkl dt

(x , t − t ; x)E˙l (x , t ) V −∞ T B˙

+ kkl (x , t − t ; x)B˙ l (x , t ) + σklT E (x , t − t ; x)El (x , t ) dv . (11.1.15)

11.2 Field Equations The field equations consist of the union of the balance laws and the constitutive equations. Upon substituting the constitutive equations into the balance laws we will have the field equations that should be satisfied by the displacement field u(x, t), the temperature field T (x, t), and the E-M independent variables. Balance Laws Conservation of Mass ∇ ·v =0 ρ˙ + ρ∇ [ρ(v − u)] · n = 0

in V − σ, on σ.

(11.2.1a) (11.2.1b)

244


In linear theory, ρ = ρ0 = const., so that no need arises for these equations. Balance of Momentum ((4.4.3a)) tkl,k + ρ(fl − v˙l ) + FlE = 0 E l [tkl + tkl + uk Gl − ρvl (vk − uk )]nk = F

in V − σ,

(11.2.2a)

on σ.

(11.2.2b)

Balance of Energy The energy balance law, in the material frame, is given by ˙ which (5.1.24). In the spatial frame it is obtained from (4.4.8) after inserting ψ, eliminates static contributions from R t, R P, and R M ρθ R η˙ − D tkl dkl − qk,k − ρh + D Pk E˙k + D Mk B˙ k − Jk Ek = 0, E [(tkl + tkl

(11.2.3a)

+ uk Gl )vl + qk − Sk

. − ρ + 21 ρv · v + 21 E · E + 21 B · B (vk − uk )]nk = H

(11.2.3b)

In (11.2.3a) we have dropped the third-order terms involving the dissipation due to the antisymmetric part of the stress tensor D t[kl] which is due to the E-M fields (cf. E , the E-M momentum G, and (5.2.9)). The E-M force FE , the E-M stress tensor tkl the Poynting vector S are, respectively, given by (4.2.12), (4.2.13), and (4.2.16). Gauss’ Law ∇ · D = qe n · [D] = we

in V − σ, on σ.

(11.2.4a) (11.2.4b)

Faraday’s Law ∇ ×E+

1 ∂B =0 c ∂t

1 n × [E + u × B] = 0 c

in V − σ,

(11.2.5a)

on σ.

(11.2.5b)

Magnetic Flux ∇ ·B=0 n · [B] = 0

in V − σ,

(11.2.6a)

on σ.

(11.2.6b)

Ampère’s Law ∇ ×H−

1 ∂D 1 = J c ∂t c

∨ 1 n × [H − u × D] = n × [ H ] c

in V − σ,

(11.2.7a)

on σ.

(11.2.7b)


245

Constitutive Equations The full constitutive equations are the sums of the static and dynamic parts tkl = R tkl + D tkl , Pk = R Pk + D Pk , Mk = R Mk + D Mk , qk , Jk .

(11.2.8)

The static parts are given by (11.1.6) and the dynamic parts (11.1.11). In these equations, 1 E = E + v × B, c J = J − qe v,

1 M = M + v × P, c 1 H = H − v × (E + P), c

(11.2.9)

E , M , H , J ) are the E-M fields in a frame of reference where the script vectors (E comoving with the material point, and the regular vectors (E, B, M, P, H, J) are the E-M fields in the fixed laboratory frame of reference. The relations (11.2.9) are nonrelativistic approximations. It is clear that, by substituting the constitutive equations in (11.2.1) to (11.2.7), we obtain a set of 13 integro-partial differential equations in terms of 13 components of the fields uk , T , Ek , Bk , and Jk , given fl , h, and qe . These equations are to be solved under some appropriate boundary conditions (jump conditions) on the boundary ∂V of the body. Jump conditions are given by (11.2.1b) to (11.2.7b). The initial conditions, usually Cauchy data, which specify v, T , and E-M fields throughout V at t = 0.

Problems 11.1 Carry out a detailed derivation of the constitutive equations (11.1.11). 11.2 Neglecting the memory effect, show that the constitutive equations (11.1.11) reduce to those of the nonlocal E-M elastic solids given in Chapter 8. 11.3 For linear nonlocal rigid media, obtain the expression of dissipation. 11.4 Obtain the heat produced in a coil by an electric field.


12 Memory-Dependent Nonlocal Electromagnetic Thermofluids

12.0 Scope Constitutive equations are obtained for the memory-dependent nonlocal electromagnetic (E-M) fluids1 in Section 12.1. The formulation given here is important to the discussions of various physical phenomena connected with magnetohydrodynamics, plasma physics, and atmospheric ionization.

12.1 Constitutive Equations The general theory of the constitutive equations memory-dependent nonlocal E-M fluid was presented in Section 5.2. The static portions of the constitutive equations are given by (5.2.7): ψ = [ρ(x, t), θ(x, t), Ek (x, t), Bk (x, t)], ∂ , π ≡ ρ2 R tkl = −πδkl , ∂ρ ∂ , Rη = − ∂θ ∂ , R Pk = − ∂Ek ∂ . R Mk = − ∂Bk 1 Eringen [1991a].

(12.1.1)

248

12 Memory-Dependent Nonlocal E-M Thermofluids

We wrote R t kl = R tkl , since in linear theory the E-M part of R t kl is nonlinear and it can be ignored (cf. (4.4.10)). The axiom of objectivity requires that ψ depend on E and B, through their invariants. Hence, a second-degree polynomial for ψ may be expressed as

ψ = ψ0 (ρ, θ) + 21 χ E E · E + 21 χ B B · B.

(12.1.2)

With this, (12.1.1) gives

π = ρ2 R Pk

∂0 , ∂ρ

Rη

= −χ E Ek ,

R Mk

=−

∂0 , ∂θ

= −χ B Bk .

(12.1.3)

The general constitutive equations for the dynamical parts are given by (5.2.14). For the linear theory the constitutive residuals vanish (the Onsager postulate). We also disregard the dependence of the temperature rate, so that D η = 0:

∂ , ∂dkl ∂ D Pk = − , ∂ Ek D t kl

qk ∂ = , T0 ∂T,k ∂ D Mk = − , ∂ Bk

=

Jk =

∂ , ∂Ek

(12.1.4)

where

E k = E˙k + El vl,k ,

B k = B˙ k + Bl vl,k .

(12.1.5)

˙ B= B, ˙ D t kl = t(kl) . Consequently, we must express For the linear theory, E = E, the dissipation potential as a symmetric functional in terms of the difference ˙ histories of the invariants of d, ∇ T , E˙ , and B. We express in the form

∞

=

∞

dτ 0

0

dτ

V

G dv ,

(12.1.6)

12.1 Constitutive Equations

249

where G is a quadratic polynomial in terms of the invariants of the difference ˙ histories of d, ∇ T , E˙ , and B: 2G = λ (x , τ ; x, τ )[d(t)ii (x , τ ) + d(t)ii (x, τ )][d(t)jj (x , t) + d(t)jj (x, τ )] + 2µ (x , τ ; x, τ )[d(t)ij (x , τ ) + d(t)ij (x, τ )][d(t)j i (x , τ ) + d(t)j i (x, τ )] 1 + k (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )][T(t),k (x , τ ) + T(t),k (x, τ )] T0 ˙ + k T E (x , τ ; x, τ )[E˙(t)k (x , τ ) + E˙(t)k (x, τ )][T(t),k (x , τ ) + T(t),k (x, τ )] ˙ + χ E (x , τ ; x, τ )[E˙(t)k (x , τ ) + E˙(t)k (x, τ )][E˙(t)k (x , τ ) + E˙(t)k (x, τ )] ˙ + χ B (x , τ ; x, τ )[B˙ (t)k (x , τ ) + B˙ (t)k (x, τ )][B˙ (t)k (x , τ ) + B˙ (t)k (x, τ )] + σ (x , τ ; x, τ )[E(t)k (x , τ ) + E(t)k (x, τ )][E(t)k (x , τ ) + E(t)k (x, τ )] ˙

+ σ E E (x , τ ; x, τ )[E(t)k (x , τ ) + E(t)k (x, τ )][E˙(t)k (x , τ ) + E˙(t)k (x, τ )] + σ ET (x , τ ; x, τ )[E(t)k (x , τ ) + E(t)k (x, τ )] × [T(t),k (x , τ ) + T(t),k (x, τ )], D tkl

∞

= λ0 drr (x, t)δkl + 2µ0 dkl (x, t) +

(12.1.7)

[λ1 (x, τ )drr (x, t − τ )δkl

0

+ 2µ1 (x, τ )dkl (x, t − τ )] dτ + [λ2 (x , x)drr (x , t)δkl + 2µ2 (x , x)dkl (x , t)] dv

V ∞ dτ [λ3 (x , τ ; x)drr (x , t − τ )δkl + V

0

+ 2µ3 (x , τ ; x)dkl (x , t − τ )] dv ,

∞ 1 ET T E˙ ˙ qk = k0 T,k (x, t) + σ0 Ek (x, t) + k0 Ek (x, t) + [k1 (x, τ )T,k (x, t − τ ) T0 0 ˙ + σ1ET (x, τ )Ek (x, t − τ ) + k1T E (x, τ )E˙k (x, t − τ )] dτ ˙ ˙ , t)] dv

+ [k2 (x , x)T,k (x , t) + σ2ET (x , x)Ek (x , t) + k2T E (x , x)E(x V ∞

+ dτ [k3 (x , τ ; x)T,k (x , t − τ ) V

0

˙

+ σ3ET (x , τ ; x)Ek (x , t − τ ) + k3T E (x , τ ; x)E˙k (x , t − τ )] dv , ∞ ˙ E˙ ˙ T E˙ E E˙ [χ1E (x, τ )E˙k (x, t − τ ) −D Pk = χ0 Ek (x, t) + k0 T,k (x, t) + σ0 Ek (x, t) + 0 ˙ T E˙ E E˙ + k1 (x, τ )T,k (x, t − τ ) + σ1 Ek (x, t − τ )] dτ + [χ2E (x , x)E˙k (x , t) ˙

˙

+ k2T E (x , x)T,k (x , t) + σ2E E (x , x)Ek (x , t)] dv

V

250


∞

+ 0

˙

dτ

˙

V

˙

[χ3E (x , τ ; x)E˙k (x , t − τ ) + k3T E (x , τ ; x)T,k (x , t − τ )

+ σ3E E (x , τ ; x)Ek (x , t − τ )] dv , ∞ ˙ B˙ ˙ χ1B (x, τ )B˙ k (x, t − τ ) dτ −D Mk = χ0 Bk (x, t) + 0

B˙

+ χ2 (x , x)B˙ k (x , t) dv

V ∞ ˙ + dτ

χ3B (x , τ ; x)B˙ k (x , t − τ ) dτ, 0

Jk =

V

˙ σ0 Ek (x, t) + σ0E E E˙k (x, t) + σ0ET T,k (x, t) +

∞

[σ1 (x, τ )Ek (x, t − τ )

0

˙ + σ1E E (x, τ )E˙k (x, t − τ ) + σ1ET (x, τ )T,k (x, t − τ )] dτ ˙ + [σ2 (x , x)Ek (x , t) + σ2E E (x , x)E˙k (x , t)

V

+ σ2ET (x , x)T,k (x , t)] dv

∞ ˙

+ dτ [σ3 (x , τ ; x)Ek (x , t − τ ) + σ3E E (x , τ ; x)E˙k (x , t − τ ) 0

V

+ σ3ET (x , τ ; x)T,k (x , t − τ )] dv ,

(12.1.8)

where the material moduli with index (0) (such as λ0 , µ0 , χ0E , etc.), and those with index 1, 2, and 3 are defined similar to (10.1.8). Here again we notice that the dynamic moduli display local and nonlocal cases (without memory) as special cases. For example, when the integrals are neglected we obtain local classical constitutive equations, and when the integrals over V dropped we have the local memory-dependent fluids. When integration over time is ignored we obtain the constitutive equations nonlocal fluids without memory. Similar to Sections 9.1 and 10.1, here too we can express these constitutive equations in terms of material moduli that constitute Dirac-delta sequences. To this end, the new material moduli, λ, µ, k, . . ., are formed similar to (10.1.9). Also note that the fluids considered here are isotropic, so that dependence on x and x can occur only through |x − x|. We also change the variable τ to t − τ = t . With these considerations, then (12.1.8) can be expressed as t

dt [λ(|x − x|, t − t )drr (x , t )δkl D tkl = −∞

V

+ 2µ(|x − x|, t − t )dkl (x , t )] dv , t 1 qk = dt

k(|x − x, t − t )T,k (x , t ) T0 V −∞ + σ ET (|x − x|, t − t )Ek (x , t ) ˙ + k ET (|x − x|, t − t )E˙k (x , t )] dv ,

12.1 Constitutive Equations

−D Pk =

t

dt

251

˙

[χ E (|x − x|, t − t )E˙k (x , t )

V −∞ T E˙

+ k3 (|x − x|, t E E˙

− t )T,k (x , t )

(|x − x, t − t )Ek (x , t )] dv , ˙ −D Mk = dt [χ B (|x − x, t − t )B˙ k (x , t )] dv , V −∞ t dt [σ (|x − x|, t − t )Ek (x , t ) Jk = +σ t

−∞ E E˙

V

(|x − x|, t − t )E˙k (x , t ) + σ ET (|x − x|, t − t )T,k (x , t )] dv . +σ

(12.1.9)

The complete constitutive equations are the sum of the static and dynamic portions (12.1.3) and (12.1.9), i.e., tkl = −πδkl + D tkl , Pk = R Pk + D Pk ,

R η,

qk ,

Mk = R Mk + D Mk ,

Jk .

(12.1.10)

The dynamic parts of the constitutive equations (12.1.9) display some “cross effects.” In the expression of qk , the second term involving σ ET represents a memorydependent nonlocal Peltier effect, which indicates that an electric field will produce heat generation. Conversely, the last term in the expression of J k , with the same coefficient, denotes the memory-dependent, nonlocal Seebeck effect, which indicates that current will flow due to memory-dependent nonlocal thermal gradients. The memory-dependent nonlocal thermal gradients will also give rise to polarization along with the rate of the electric field. The rate of the electric field also causes electric currents. Nonlocal Electromagnetic Fluids without Memory If we drop the integrals with respect to τ in (12.1.8) we obtain the constitutive equations of nonlocal E-M thermofluids. Further, incorporating the local terms into the volume integrals, by multiplying them with Dirac-delta functions δ(|x − x|), in the usual way, we will have D tkl

=

V

λ(|x − x|)drr (x , t)δkl + 2µ(|x − x|)dkl (x , t)] dv ,

1 qk = [k(|x − x|)T,k (x , t) + σ ET (|x − x|)Ek (x , t) T0 V ˙ + k ET (|x − x|)E˙k (x , t)] dv ,

252


−D Pk =

V

˙ ˙ [χ E (|x − x|)E˙k (x , t) + k T E (|x − x|)T,k (x , t) ˙

+ σ E E (|x − x|)Ek (x , t)] dv , ˙ χ B (|x − x|)B˙ k (x , t) dv , −D Mk = V ˙ Jk = [σ (|x − x|)Ek (x , t) + σ E E (|x − x|)E˙k (x , t) V

+ σ ET (|x − x|)T,k (x , t)] dv .

(12.1.11)

An earlier theory of nonlocal E-M fluids without memory was introduced by McCay and Narasimhan [1981]. These authors employed different E-M variables so that the relation of the present theory to theirs could not be established in a direct way. Narasimhan and McCay [1981] also used their theory to study the dispersion of surface waves in nonlocal dielectric fluids. Field Equations The union of balance laws, given by (11.2.1a) to (11.2.7b) supplemented by the constitutive equations (12.1.10), constitutes the field equations of the memorydependent nonlocal E-M thermofluids. Similar to Section 11.2, here again we have a set of 13 integro-partial differential equations to determine 13 functions uk , T , Ek , Bk , and Jk , given fl , h, and qe .

Problems 12.1 Give the detailed derivations of the constitutive equations (12.1.9). 12.2 Obtain the velocity field and the shear stress in a magnetohydrodynamic channel flow of nonlocal E-M fluids without memory. 12.3 Set up a boundary-layer approximation on nonlocal E-M fluids. 12.4 Obtain the velocity profile and the shear stress in a thin layer of nonlocal viscous fluids (without memory) on a rigid plate rotating with constant angular velocity about an axis perpendicular to the plate.

13 Nonlocal Microcontinua

13.0 Scope Chapters 13 to 15 are written to investigate the effect of nonlocality in the case of continua with microstructure. In Section 13.1 we describe the extra degrees of freedom required by the microstructures. Motion, micromotion, strain, and rotation measures are introduced for the micromorphic, microstretch, and micropolar continua. Compatibility equations for deformation tensors are obtained. In Section 13.2 the kinematics of microcontinua is discussed. These include time-rates of various tensors, e.g., deformation tensors. Section 13.3 continues with the development of the concepts of a mass density microinertia, kinetic energy, and momenta. In Section 13.4 we introduce the stress and microstress concepts and develop balance laws, via the invariance of the global energy conservations subjected to Galilean invariance (Noether’s theorem). In this way, all the balance laws are obtained for micromorphic, microstretch, and micropolar continua (3-M continua). These are listed in Section 13.5 together with the jump conditions. Section 13.6 is concerned with the second law of thermodynamics and the Clausius–Duhem (C–D) inequality, fundamental to the development of the constitutive equations. Constitutive equations are obtained in Section 13.7 for the memory-dependent nonlocal micromorphic solids.

254


In Section 13.8 we develop the constitutive equations of memory-dependent microstretch elastic solids and in Section 13.9 those of the memory-dependent nonlocal micropolar solids. Section 13.10 is concerned with the formulation of the boundary-initial value problems of nonlocal micropolar elasticity. The uniqueness theorem is presented in Section 13.11. The reciprocal theorem is given in Section 13.12, and the variational principles in Section 13.13. In Section 13.14, by matching the solution of the nonlocal micropolar field equations with those of the atomic lattice dynamics in the Fourier domain, we determine some of the micropolar moduli. The propagation of plane waves is discussed in Section 13.15. Here we see the emergence of the four dispersion curves of those observed in the atomic lattice dynamics. These are: (i) longitudinal acoustic branch (LA); (ii) the longitudinal optic branch (LO); (iii) the transverse acoustic branch (TA); and (iv) the transverse optic branch (TO). The dispersion curves are faithful in the entire Brillouin zone. Section 13.16 introduces the displacement potential, which will help to simplify the field equations. In Section 13.17 we develop the Somigliana-type resolution for the displacement and microrotation. In this way, the solution can be achieved once two independent vector fields are determined from two vector partial differential equations, one involving body force and the other body couple. This method of solution is used to obtain the fundamental solutions in Section 13.18.

13.1 Kinematical Preliminaries A material body B (in the reference state) is considered to be a collection of a set of material particles {P }. As distinguished from the classical continuum, the material particles are considered finite but small sizes, so that the deformation of the particle can be taken into account. In this visualization a material point C in P is located by two vectors: The centroidal position C of P , spanned by a vector X and the position vector " relative to C, Figure 13.1.1: X = X + " .

(13.1.1)

In order to pass to a continuum, we imagine that the material particle P is of infinitesimal size, and that its deformation is represented by the mappings X and " to the spatial positions x and ξ at time t, i.e., X → x = xˆ (X, t), " → ξ = ξˆ (X, " , t), x = x + ξ .

(13.1.2) (13.1.3) (13.1.4)

The mapping (13.1.2) is called a macromotion or simply motion, and (13.1.3) the micromotion. We employ capital letters with majuscule indices for quantities referred to the reference state, e.g., XK , "K (K = 1, 2, 3), and minuscule letters and indices for

13.1 Kinematical Preliminaries

255

Figure 13.1.1. Deformation of microelement. those referred to spatial configuration, e.g., xk , ξk (k = 1, 2, 3). When a quantity is associated by both states, they will carry both indices. Material particles are considered to be very small (infinitesimally small) as compared to the scales of the body. Consequently, a linear approximation is permissible for the micromotion, replacing (13.1.3) by ξk = χkK (X, t)"K ,

(13.1.5)

where, and henceforth, the summation convention on repeated indices is understood. We note that ξ (X, 0, t) = 0 since X denotes the position vector of the centroid of the particle. Definition (Micromorphic Continuum).1 A material body is called the micromorphic continuum of grade one (or simply the micromorphic continuum) if its motions are described by (13.1.2) and (13.1.5), which possess continuous partial derivatives with respect to X and t, and they are invertible uniquely, i.e., XK = Xˆ K (x, t), "K = XKk (x, t)ξk ,

(13.1.6) K = 1, 2, 3, k = 1, 2, 3.

(13.1.7)

1 The theory was introduced by Eringen and Suhubi [1964] and Eringen [1964]. The present terminology was introduced by Eringen.

256


Figure 13.1.2. Deformable director. The two point tensors χkK and XKk are called microdeformation and inverse microdeformation tensors (or directors), respectively, Figure 13.1.2. The existence of solutions (13.1.6) is guaranteed by the implicit function theorem which requires the continuous partial derivatives of xˆk with respect to XK and the positive Jacobian J ≡ det(∂xk /∂XK ) > 0. (13.1.8) For (13.1.7) we must also have j ≡ det χkK = 1/ det XKk > 0.

(13.1.9)

The deformation gradients xk,K and XK,k satisfy xk,K XK,l = δkl ,

xk,K XL,k = δKL .

(13.1.10)

Similarly, from (13.1.5) and (13.1.7), we get χkK XKl = δkl ,

χkK XLk = δKL ,

(13.1.11)

where, and henceforth, we omit the caret (ˆ) distinguishing the mapping function from its image, and use a comma to denote the partial derivatives, e.g., xk,K ≡

∂xk , ∂XK

XK,l =

∂XK . ∂xl

The linear equations (13.1.10) and (13.1.11) can be solved for XK,k and XKk : 1 KLM klm xl,L xm,M , 2J 1 KLM klm χlL χmM , = 2j

XK,k = XKk

(13.1.12)


257

where klm and KLM are the permutation symbols and J ≡ det xk,K = 16 KLM klm xk,K xl,L xm,M , j ≡ det χkK = 16 KLM klm χkK χlL χmM .

(13.1.13)

By differentiation, we obtain the following useful results ∂J = cofactor xk,K = J XK,k , ∂xk,K (J XK,k ),K = 0,

∂j = j XKk , ∂χkK

−1

( J xk,K ),k = 0.

(13.1.14) (13.1.15)

Definition (Microstretch Continuum).2 A material body is called a microstretch continuum if 1 (13.1.16) XKk = 2 χkK . j Since j = det χkK represents the microvolume change, we see that a microstretch continuum is a constrained micromorphic continuum that undergoes microrotation and microstretch (expansion and contraction). Definition (Micropolar Continuum).3 A micromorphic continuum is called micropolar if its directors are orthonormal, i.e., χkK χlK = δkl .

(13.1.17)

Consequently, in view of (13.1.16): χkK = XKk ,

j = det χkK = 1.

(13.1.18)

For a micropolar continuum, the directors are rigid. We can represent the motion of the directors as a rigid body rotation with respect to an axis. Theorem (Finite Microrotation Tensor). The finite microrotation tensor χkl is characterized by χkl = χkL δLl = cos φδkl − sin φklm nm + (1 − cos φ)nk nl ,

(13.1.19)

φk , (13.1.20) φ where δLl = IL ·il are called shifters. They are cosine directors of the unit Cartesian vector of the spatial frame with respect to those of the material frame. Here, the unit vector n is the axis of rotation and φ is the angle of rotation, Figure 13.1.3. φ = (φk φk )1/2 ,

nk =

For a proof, see Eringen [1998, Section 1.3]. 2 Microstretch and micropolar continuum were introduced by Eringen [1966a], [1969], [1971a]. 3 Eringen and Suhubi [1964] contains a preliminary version of the micropolar elasticity, under the terminology “Linear theory of couple stress.” Cosserat elasticity is related in principle, but not in full content.

258


Figure 13.1.3. Finite microrotation. It is important to note that the finite microrotation tensor is different from the rotation tensor defined in classical elasticity −1/2 LK

RkK = xk,L C

(13.1.21)

(see Eringen [1980, p. 46]), where CLK is the classical deformation tensor defined by CLK ≡ xk,L xk,K . (13.1.22) In the linear theory, (13.1.21) reduces to Rkm − δkm = 21 (uk,m − um,k ),

(13.1.23)

where uk is the displacement vector.

A. Strain Measures of Micromorphic Continua In micromorphic continua, three different strain measures are introduced: CKL ≡ xk,K XLk ,

CKL ≡ χkK χkL = CLK ,

KLM ≡ XKk χkL,M . (13.1.24)

Here CKL is called the deformation tensor, CKL , the microdeformation tensor, and KLM the wryness tensor. Spatial strain measures are defined by ckl ≡ XK,k χkl ,

çkl ≡ XKk XKl ,

γklm ≡ χkK,M XKl XM,m .

(13.1.25)


259

The displacement vector u is introduced by x − X = u,

(13.1.26)

so that all strain measures can be expressed in terms of φ, n, u, and their gradients. Here we give these expressions only for the linear strain measures. These are obtained by dropping all nonlinear terms. We have xk,K = (δkl + uk,l )δlK , χkK = (δkl + φkl )δlK , XKk = (δkl − φlk )δKl .

(13.1.27) (13.1.28)

Substituting these into (13.1.24) we obtain the linear strain measures CKL − δKL = kl δkK δlL , CKL − δKL = 2ekl δkK δlL , KLM = γklm δkK δlL δmM ,

(13.1.29)

where kl = ul,k − φlk ,

2ekl = φkl + φlk ,

γklm = φkl,m .

(13.1.30)

These are the linear strain measures of the micromorphic continua.

B. Strain Measures of Microstretch Continua Using (13.1.16) in (13.1.24) we obtain CKL = j −2 xk,K χkL ,

CKL = j 2 δKL ,

KLM = j −2 χkK χkL,M . (13.1.31)

While these tensors are possible candidates for the strain measures of microstretch media, they are coupled and they do not constitute an independent set. In fact, there appear 37 components. Strain measures of microstretch continua have much fewer components. An independent set is obtained by letting χkK = j χ kK ,

XKk =

1 XKk , j

(13.1.32)

where χ kK and XKk are subject to χ kK XKl = δkl ,

χ kK XLk = δKL .

(13.1.33)

The deformation tensors of the microstretch continuum are then defined by CKL ≡ xk,K χ kL ,

CKL ≡ j 2 δKL ,

KL ≡ 21 KMN χ kM,L χ kN ,

K =

j,K . j

(13.1.34)

260


These are related to the reduced micromorphic deformation tensor (13.1.31) by CKL = j −1 CKL ,

CKL = j 2 δKL ,

KLM = M δKL −KLR RM . (13.1.35)

The spatial deformation tensors of the microstretch media are defined by ckl = XK,k χ lK , γkl = 21 kmn χ mK χ nK,l ,

ckl = δkl , j,m γm = . j

(13.1.36)

The linear strain measures are obtained by substituting xk,K = (δkl + uk,l )δlk , χkK = ϕδkK + χ kK = (δkl + ϕδkl − klm φm )δlK , j = 1 + ϕ,

(13.1.37)

where ϕ denotes the microstretch and φk the microrotation. Substituting these into (13.1.34) we get the linear strain tensors of the microstretch continua CKL − δKL = kl δkK δlL , K = γk δkK ,

KL = γkl δkK δlL , CKL − δKL = 2eδKL ,

(13.1.38)

where kl = ul,k + lkm φm , γk = ϕ,k ,

γkl = φk,l , e = ϕ.

(13.1.39)

C. Micropolar Continua The strain measures of micropolar continua follow from (13.1.39) by setting the microstretch equal to zero CKL − δKL = kl δkK δlL ,

KL = γkl δkK δlL ,

(13.1.40)

where kl and γkl are the spatial measures, defined by kl = ul,k + lkm φm ,

γkl = φk,l .

(13.1.41)

D. Compatibility Conditions Given a set of deformation tensors CKL , CKL , and KLM , we have 42 first-order differential equations (13.1.24) to determine the unknowns xk and χkK . For the existence of single-valued displacement and director fields, corresponding to the deformation tensors, certain relations must be satisfied among the deformation


261

tensors. These are the integrability conditions of the system of differential equations (13.1.24). They are called the compatibility conditions. Theorem (Compatibility Conditions). For a simply connected micromorphic body, the necessary and sufficient conditions, for the integration of the system of partial differential equations (13.1.24), are KP Q (CP L,Q + CP R LRQ ) = 0, KP Q (LMP ,Q + LRQ RMP ) = 0, CKL,M − (P KM CLP + P LM CKP ) = 0.

(13.1.42)

For microstretch continuum, these equations are replaced by KP Q (CP L,Q + MLN N Q CP M ) = 0, KP Q LQ,P + 21 LMN MP N Q = 0.

(13.1.43)

For micropolar continuum, the same equations (13.1.43) are valid. The linear compatibility equations are: (a) Micromorphic kpq (pl,q + γlpq ) = 0, kpq γlmp,q = 0, −2ekl,m + γklm + γlkm = 0.

(13.1.44)

(b) Microstretch and Micropolar −kpq pl,q + γpp δkl − γkl = 0, kpq γlq,p = 0.

(13.1.45)

For the proof of these theorems, see Eringen [1999, Section 1.7]. It must be remarked that if the body contains a defect the compatibility conditions are violated. If the body contains dislocations and/or disclinations these equations will not be satisfied. The dislocations are due to the discontinuities in the displacement fields while disclinations result from the misfits of twists. These are the causes of plastic deformations in the body.

13.2 Time-Rate of Tensors The time-rate of a function f (X, " , t) is defined as df ∂f f˙ = , = dt ∂t X," "

(13.2.1)

262


where the subscripts X and " , accompanying the vertical bar, denote that X and " are fixed. If f = f (X, t), then

If f = f (x, t), then

∂f (X, t) f˙ = . ∂t

(13.2.2)

∂f ∂f f˙ = x˙k . + ∂t ∂xk

(13.2.3)

The velocity v and acceleration a are defined by v≡

∂x(X, t) = x˙ , ∂t

a = v˙ =

∂ 2 x(X, t) . ∂t 2

(13.2.4)

If we substitute X = X(x, t) into v(X, t) we obtain v(X(x, t), t) = vˆ (x, t).

(13.2.5)

The acceleration field a(x, t) at a spatial point is obtained by ∂ vˆ a = v˙ˆ = + vˆ ,k vˆk . ∂t Henceforth we drop the hat (ˆ) for brevity and express a as a=

∂v + v,k vk ∂t

ak =

or

∂vk Dvk + vk,l vl ≡ . ∂t Dt

(13.2.6)

The symbol D/Dt is used universally and is called the material derivative. It is simple to show that D (xk,K ) = vk,l xl,K Dt

or

D (dxk ) = vk,l dxl . Dt

(13.2.7)

The proof follows from the fact that D/Dt and ∂/∂XK commute. Similarly, DXK,k = −XK,l vl,k . Dt

(13.2.8)

This follows by differentiating XK,k xk,L = δKL and using (13.2.7). Definition (Microgryration tensor). νkl is defined by νkl = χ˙ kK XKL .

(13.2.9)

From this it follows that χ˙ kK = νkl χlK ,

˙ Kk = −XKl νlk . X

(13.2.10)


263

Theorem. The material derivatives of strain measures of the micromorphic continuum are given by C˙KL = 2ckl χkK χlL , C˙ KL = akl xk,K XLl , ˙ KLM = bklm XKk χlL xm,M ,

(13.2.11)

where akl , bklm , and ckl are called the deformation-rate tensors. These are defined by akl ≡ vl,k − νlk ,

2ckl ≡ νkl + νlk ,

bklm ≡ νkl,m .

(13.2.12)

Expressions (13.12.11) follow from calculating the material time-rate of (13.1.24) and using (13.2.10). Theorem. The material time-rates of the deformation tensors of the microstretch continuum are given by C˙ KL = (vl,k + lkm νm )xk,K χ lL , ˙ KL = j −2 νk,l xl,L χ kK , ˙ k = ν,k , 2 ˙ CKL = 2j νδKL .

(13.2.13)

To show this, we first calculate ∂(det χkK ) Dj = χ˙ kK = j XKk νkl χlK = j νkk = j ν, Dt ∂χkK

(13.2.14)

since νkk = 0, we call it ν. Next we calculate D (j χ kK ) = j νχ kK + j χ˙ kK = j (νχ kK + ν kl χ lK ) Dt = (νδkl + ν kl )χkK , but, χ˙ kK = νkl χkK . Consequently, we have νkl = νδkl + ν kl .

(13.2.15)

Calculating the time-rates of (13.1.34), and using (13.2.7), (13.2.14), and (13.2.15), we obtain (13.2.13). In the case of the micropolar continuum j = 1, ν = 0, and (13.2.13) gives C˙ KL = (vl,k + lkm νm )xk,K χ lL ,

˙ KL = νk,l xl,L χ kK .

(13.2.16)

In summary, the deformation-rate tensors of all three continua are given by: Micromorphic akl = vl,k − νlk ,

bklm = νkl,m ,

ckl = ν(kl) = 21 (νkl + νlk ). (13.2.17)

264


Microstretch akl = vl,k + lkm νm ,

bkl = νk,l ,

ckl = νδkl .

(13.2.18)

Micropolar bkl = νk,l .

akl = vl,k + lkm νm ,

(13.2.19)

Deformation-rate tensors are fundamental to the development of the constitutive equations of fluent media. For some purposes the time-rate of γkl becomes necessary. This is obtained by differentiating (13.1.36) and using (13.2.10)1 , i.e., D 1 (χj K,L XL,l ) . γ˙kl = 2 kij νir χrK χj K,l + χiK Dt But we have χ˙ j K,L = νj,r χrK,L + νj r,p χrK xp,L , kpq γkl = χpK χqK,l .

(13.2.20)

Using these and (13.2.19), we obtain γ˙kl = bkl + νkr γrl + νlr γkr − γkr alr .

(13.2.21)

13.3 Mass, Inertia, Kinetic Energy A. Micromorphic Continua We consider a material particle P with volume element V in the reference frame, and its image p with volume element v in the spatial frame (Figure 13.3.1). The total masses of these particles are the sum of the masses of their microelements, i.e., ρ0 V = V

ρ0 dV ,

ρ dv ,

ρv =

(13.3.1)

v

where the primed quantities refer to the microelements of P and p. Since " and ξ are the relative position vectors of P and p , from the centers of gravities of C and c of the particles, for the first moments we have ρ0 " dV = 0, ρ ξ dv = 0. (13.3.2) V

v

However, the second moments of ρ0 dV and ρ dv do not vanish and they are given by ρ0 "K "L dV , ρikl v = ρ ξk ξl dv . (13.3.3) ρIKL V = V

v

13.3 Mass, Inertia, Kinetic Energy

265

Figure 13.3.1. Microvolume elements. We assume that the mass of the microelements is conserved during the motion ρ0 dV = ρ dv .

(13.3.4)

This implies that, in the limit V → 0 and v → 0, we have ρ0 dV = ρ dv.

(13.3.5)

This is the law of the conservation of mass. In the limit, (13.3.3) defines the microinertia tensors

ρIKL dV = ρ0 "K "L dV , ρikl = ρ ξk ξl dv . dV

(13.3.6)

dv

Upon using (13.1.5) and (13.1.7), we obtain the law of the conservation of microinertia ikl = IKL χkK χlL , IKL = ikl XKk XLl . (13.3.7) These results are also obtained by defining the probability densities P (X, " ) ≡ ρ0 (X, " )/ ρ0 (X, " ) dV ("),

p (x, ξ ) ≡ ρ (x, ξ )/ ρ (x, ξ ) dv (ξ ), (13.3.8) subject to the conservation law P dV = p dv . Then

(13.3.9)

P "K "L dV = "K "L ,

IKL = V ikl = v

p ξk ξl dv = ξk ξl .

(13.3.10)

266


Definition 1. The kinetic energy density per unit mass is defined by K = 21 (˙x + ξ˙ ) · (˙x + ξ˙ ).

(13.3.11)

Employing ξ˙k = νkl ξl , this gives K = 21 v · v + 21 ikl νmk νml .

(13.3.12)

Definition 2. The spin-inertia per unit mass is defined by σkl ≡ ξ¨k ξl .

(13.3.13)

But, we have ξ˙k = νkl ξl ,

ξk = χkL "L ,

ξ¨k = (˙νkm + νkl νlm )ξm .

(13.3.14)

Using these in (13.3.12) we obtain σkl = iml (˙νkm + νki νim ).

(13.3.15)

B. Microstretch Continua From (13.2.15) we have νkl = νδkl − klm νm .

(13.3.16)

We decompose σkl for the microstretch continua as σkl =

σ δkl − 21 klm σm , 3

(13.3.17)

which introduces the microstretch scaler inertia σ and the microstretch rotatory inertia σm . It is also useful to decompose ikl and IKL as ikl = 21 j0 δkl − jkl , IKL =

1 2 J0 δKL

− JKL ,

jkl = i0 δkl − ikl ,

j0 ≡ jkk ,

JKL = I0 δKL − IKL .

(13.3.18)

Substituting these into (13.3.12) and (13.3.15), we obtain K = 21 v · v + 41 j0 ν 2 + 21 jkl νk νl ,

(13.3.19)

σk = jkl ν˙ l + 2νjkl νl + kmn jmn νl νn , σ = 21 j0 (˙ν + ν 2 ) − jkl νk νl .

(13.3.20)

C. Micropolar Continua In the case of micropolar continua ν = 0, σ = 0, and we have K = 21 v · v + 21 jkl νk νl ,

(13.3.21)

D σk = jkl ν˙ l + klm jmn νl νn = (jkl νl ). Dt

(13.3.22)

13.4 Stress, Stress Moments, Energy Balance

267

13.4 Stress, Stress Moments, Energy Balance A. Micromorphic Continua The concept of stress is well known from classical continuum theory. By isolating one part of the body from other parts, we can replace the effect of the rest of the body on the isolated part as surface forces and couples. For micromorphic bodies, we imagine that the surface tractions arise by taking the mean of the surface tractions acting on the microsurface elements contained in a surface element. Similarly, the surface mean of the microstress moments gives rise to a third-order stress tensor mklm :

ξm 2 . (13.4.1) mklm = tkl Here 2 indicates the surface mean (Figure 13.4.1). Similarly, the volume mean taken over a volume element gives rise to a body force density fk and a body moment density lkl : lkl = fk ξl . (13.4.2) In both (13.4.1) and (13.4.2) the primed quantities are referred to microelements contained in a particle. Thus, micromorphic continua give rise to stress and body moments, in addition to the stress tensor and the body force density. Principle of Energy Balance. The time-rate of the sum of the kinetic and internal energies is equal to the work done by all loads acting on the body, per unit time. Mathematically, this is expressed as d ρ( + K) dv = (tkl vl + mklm νlm + qk ) dak dt V −σ ∂ V −σ + ρ(fk vk + lkl νkl + h) dv. V −σ

(13.4.3)

Here, the left-hand side expresses the time-rate of the total internal and kinetic energies. On the right-hand side, under the surface integral, the three terms denote, respectively, the energies of the surface tractions, surface stress moments, and heat. On the right-hand side, under the volume integral, the three terms represent,

Figure 13.4.1. Traction at microsurface element a .

268


respectively, the energies of the body force, the body moments, and heat input per unit volume. The volume and surface integral excludes the line and surface intersections of a discontinuity surface σ which may be sweeping the body with its own velocity u. This is denoted by V − σ ≡ V − V ∩ σ,

∂V − σ ≡ ∂V − ∂V ∩ σ.

(13.4.4)

The energy law (13.4.3) differs from the classical counterpart in the terms involving mklm and lkl . The origin of these terms may be clarified by a physical picture. Consider a macrosurface element a on the surface of the body with an exterior normal n (Figure 13.4.1). The work of the stress vector tk , acting at a microsurface element a , with unit normal n , upon integration over a, gives the energy

i arising from the tractions on a. For the stress vector we have, classically, tk = tkl l

where il is the Cartesian unit vector and tkl is the microstress tensor. The energy of tk is then given by

tkl (vl + ξ˙l ) dak = (tkl vl + mklm νlm )ak , (13.4.5) a

where we used ξ˙l = νlm ξm and defined the stress moment tensor mklm by (13.4.1), or as a → 0:

mklm ak = tklm ξm dak . (13.4.6) a

Similarly, the work, per unit time, of the body force fk , acting in the microvolume element v contained in v, is given by ρ fk (vk + ξ˙k ) dv = (ρfk vk + ρlkl νkl )v, (13.4.7) v

where lkl is defined in the limit as v → 0: ρlkl v = ρ fk ξl dv .

(13.4.8)

v

B. Microstretch Continua For microstretch continua, we decompose mklm and lkl as mklm = 13 mk δlm − 21 lmr mkr , lkl = 13 lδkl − 21 lkr lr .

(13.4.9)

With these, (13.4.3) gives the energy balance law for the microstretch continua d ρ( + K) dv = (tkl vl + mkl νl + mk ν + qk ) dak dt V −σ ∂ V −σ + ρ(fk vk + lk νk + lν + h) dv, (13.4.10) V −σ

13.4 Stress, Stress Moments, Energy Balance

269

where the kinetic energy K is now given by (13.3.19). Here mkl is the couple stress tensor, mk is a microstretch surface force density, l is an applied scalar microstretch tension, and lk is an applied body couple.

C. Micropolar Continua The energy balance law of micropolar continua follows from (13.4.10) by setting ν = 0, mk = 0, and l = 0: d ρ( + K) dv = (tkl vl + mkl νl + qk ) dak dt V −σ ∂ V −σ + ρ(fk vk + lk νk + h) dv. (13.4.11) V −σ

The kinetic energy K is now given by (13.3.21). Positive directions of stress and couple stress components are shown in Figures 13.4.2, 13.4.3, and 13.4.4.

Figure 13.4.2. Stress tensor.

270


Figure 13.4.3. Couple stress tensor.

Figure 13.4.4. Directions of couple stress.

13.5 Balance Laws

271

13.5 Balance Laws4 A. Balance Laws of Micromorphic Continua We already know two local balance laws, the conservation of mass, and microinertia from Section 13.3: ρ dv = ρ0 dV , (13.5.1) ikl = IKL χkK χlL

IKL = ikl XKk XLl .

or

For the entire body, the global laws are given by d d ρ dv = 0, ρikl XKk XLl dv = 0. dt V −σ dt V −σ By means of the transport theorem (2.1.5a), these are converted to ρˆ dv + Rˆ da = 0, V −σ σ ˆ kl XKk XLl da = 0, (ρi ˆ kl + ρ iˆkl )XKk XLl dv + G V −σ

(13.5.2)

(13.5.3)

(13.5.4) (13.5.5)

σ

where ∇ · v, ρˆ = ρ˙ + ρ∇ Dikl iˆkl = − ikr νlr − ilr νkr , Dt

Rˆ ≡ [ρ(vk − uk )]nk , ˆ kl = [ρikl (vm − um )]nm , G

(13.5.6)

are called nonlocal residuals. In local continuum mechanics, it is postulated that the nonlocal residuals vanish and we obtain the law of the conservation of mass: ∇ ·v =0 ρ˙ + ρ∇ [ρ(v − u)] · n = 0

in V − σ,

(13.5.7a)

on σ,

(13.5.7b)

and the law of the conservation of microinertia: Dikl − ikr νlr − ilr νkr = 0 Dt [ρikl (vm − um )]nm = 0

in V − σ,

(13.5.8a)

on σ.

(13.5.8b)

Since in inert bodies neither mass nor microinertia is created or destroyed we shall assume that (13.5.7) and (13.5.8) remain valid for nonlocal continua. 4 Theories of nonlocal polar elastic continua were introduced by Eringen [1972c]. For the present approach, see Eringen [1992a], [1998]. For a statistical approach to balance laws, see Oevel and Sehröter [1981].

272


The remaining balance laws can be obtained from the energy balance law, by subjecting it to the Galilean invariance. Intuitively, the energy balance equation (13.4.3) should hold in any reference frame undergoing rigid body motion. We postulate this as: Axiom. The energy balance law is form-invariant under the Galilean group of transformations. Theorem. For each Galilean group of transformations of the energy balance law, there is a balance law of microcontinuum mechanics. Consider a time-dependent rigid motion of the frame of reference xk = Qkl (t)xl + bk (t),

(13.5.9)

where Qkl is an orthogonal tensor, i.e., QQT = QT Q = 1,

det Q = 1.

(13.5.10)

The velocity field, in the new frame, is given by ˙ kl xl + b˙k . vk = Qkl vl + Q

(13.5.11)

The angular velocity is calculated by ˙ km Qlm . kl = Q

(13.5.12)

The microinertia tensor, in the new frame, is given by ikl = Qkm imn Qln .

(13.5.13)

The material-time rate of this is Dimn D ikl ˙ km imn Qln . = Qkm Qln + 2Q Dt Dt

(13.5.14)

Suppose that at time t, the body is brought back to its original orientation (i.e., Q = 1, having only constant translational and angular velocities, i.e., and b˙ are constants). Consequently, at time t, we have vk = vk + kl xl + b˙k , νkl = kl + νkl , D ikl Dikl = + 2km iml . Dt Dt

˙ kl , kl = Q

ikl = ikl ,

(13.5.15)

Under these transformations, ρ, tkl , mklm , and qk do not change, since the motion is rigid. But the body force fk and body moments lkl must be accommodated by the corresponding accelerations, i.e., f − v˙ = f − v˙ ,

σkl = lkl − σkl . lkl −

(13.5.16)

13.5 Balance Laws

273

According to (13.3.12) the kinetic energy is given by = 1 K ikl νmk νml . v + 21 2v ·

(13.5.17)

Subtracting the energy balance (13.4.3) in the x-frame from that in the x-frame, we obtain d − K) dv = ρ(K [tkl ( vl − vl ) + mklm ( νlm − νlm )]nk da dt V ∂ V −σ ρ(fk vk − fk vk + lkl νkl − lkl νkl ) dv. (13.5.18) + V −σ

We evaluate D (ikl νmk νml ) = 2σkl νkl , Dt − K = vk (km xm + b˙k ) + 1 (kl xl + b˙k )(km xm + b˙k ) K 2 + ikl νmk ml + 21 ikl mk ml ,

˙ − K˙ = K v˙ k · vk − v˙k vk + ( σkl − σkl ) νkl − σkl kl , vk − vk = kl xl + b˙k , v˙ k − v˙k = kl vl , νkl − νkl = kl , fk vk − fk vk = fk ( vk − vk ) + vk ( v˙ k − v˙k ), lkl νkl − lkl νkl = ( σkl − σkl ) νkl + lkl ( νkl − νkl ). (13.5.19) We apply the transport theorem (2.1.5a) to evaluate the material time-rate and the Green–Gauss theorem to convert the surface integrals to volume integrals in (13.5.18). Upon using (13.5.19) and the conservations of mass and inertia in (13.5.7) and (13.5.8), we obtain ρ[(lm xm + b˙l )fˆl + lm lˆlm ] dv + [(lm xm + b˙l )Fˆl V −σ

σ

+ lm Lˆ lm ] da = 0,

(13.5.20)

in V − σ,

(13.5.21a)

on σ,

(13.5.21b)

ρ lˆlm = mkl,k + tml − sml + ρ(llm − σlm )

in V − σ,

(13.5.22a)

Lˆ lm = [mklm − ρirm νlr (vk − uk )]nk

on σ.

(13.5.22b)

where ρ fˆl = tkl,k + ρ(fl − v˙l ) Fˆl = [tkl − ρvl (vk − uk )]nk

Here, sml is an arbitrary symmetric tensor sml = slm .

(13.5.23)

274


This is introduced, since lm slm = 0 does not contribute to (13.5.20). However, it allows all components of lm to be arbitrary, irrespective of the antisymmetry of lm . For arbitrary and independent variations of b˙l and lm , the coefficients of these quantities in (13.5.20) must vanish. Hence we have (13.5.24) ρ fˆl dv + Fˆl da = 0, σ V −σ (13.5.25) ρ lˆlm dv + Lˆ lm da = 0. V −σ

σ

Equations (13.5.21a) is the balance of momentum and (13.5.22a) is the balance of momentum moments. The accompanying jump conditions are given by, respectively, (13.5.21b) and (13.5.22b). The energy balance laws (13.4.3) may be recast into local form by means of the transport and Green–Gauss theorems −ρ ˆ = −ρ ˙ + tkl (vl,k − νlk ) + skl νlk + mlkm νlm,k + qk,k + ρh in V − σ, −Eˆ = [tkl vl + mklm νlm + qk − (ρ + + Subject to

V −σ

1 2 ρirl νmr νml )(vk

ρ(−ˆ + fˆl vl + lˆlm νlm ) dv −

1 3v

(13.5.26a)

·v

− uk )]nk

on σ.

Eˆ da = 0.

(13.5.26b)

(13.5.27)

σ

This completes the proof of the theorem.

B. Balance Laws of Microstretch Continua A similar analysis, based on (13.4.10), leads to the balance of microstretch continua. Alternatively, by replacing mklm , lkl by (13.4.9) and νkl and σkl by (13.3.16) and (13.3.17), we obtain mathematical expressions for these laws: Conservation of Mass ∇ ·v =0 ρ˙ + ρ∇ [ρ(v − u)] · n = 0

in V − σ, on σ.

Conservation of Microstretch Inertia Djkl − 2νjkl + (kpr jlp + lpr jkp )νr = 0 Dt [ρjkl (v − u)] · n = 0 Dj0 − 2j0 ν = 0 Dt [ρj0 (v − u)] · n = 0

(13.5.28a) (13.5.28b)

in V − σ,

(13.5.29a)

on σ,

(13.5.29b)

in V − σ,

(13.5.29aa)

on σ.

(13.5.29bb)

13.5 Balance Laws

275

Balance of Momentum tkl,k + ρ(fl − v˙l ) = ρ fˆl

[tkl − ρvl (vk − uk )]nk = Fˆl ρ fˆl dv + Fˆl da = 0.

V −σ

in V − σ,

(13.5.30a)

on σ,

(13.5.30b) (13.5.30c)

σ

Balance of Momentum Moments mkl,k + lmn tmn + ρ(ll − σl ) = ρ lˆl [mkl − ρjpl νp (vk − uk )]nk = Lˆ l mk,k

ˆ + t − s + ρ(l − σ ) = ρ l,

in V − σ,

(13.5.31a)

on σ,

(13.5.31b)

in V − σ,

(13.5.31aa)

[mk − − uk )]nk = Lˆ on σ, ρ lˆl dv + Lˆ l da = 0, ρ lˆ dv + Lˆ da = 0 1 2 ρj0 ν(vk

V −σ

σ

V −σ

(13.5.31bb) on σ, (13.5.31c)

σ

where we wrote tkk = t and skk = s. Balance of Energy −ρ ˙ + tkl (vl,k + lkr νr ) + mkl νl,k + mk ν,k + (s − t)ν + qk,k + ρh = −ρ Eˆ

in V − σ, (13.5.32a)

[tkl vl + mkl νl + mk ν + qk − ρ( + 21 v · v V −σ

+ 21 jij νi νj + 41 j0 ν 2 )(vk − uk )]nk = −Eˆ ˆ ˆ ˆ ρ(−ˆ + fl vl + lν + lr νr ) dv − Eˆ da = 0.

on σ,

(13.5.32b) (13.5.32c)

σ

In these equations, constitutive volumes and surface residuals appear. These are denoted by a superposed carat (ˆ) on letters. Minuscule letters (like fˆl , lˆl ) are used for volume residuals and majuscule letters (Fˆl , Lˆ l , etc.) are used for the surface residuals.

C. Balance Laws of Micropolar Continua The balance laws of the micropolar continua are obtained from those of the microstretch continua by discarding equations (13.5.31aa) and (13.5.31bb). Conservation of Mass ∇ · v = 0, ρ˙ + ρ∇ [ρ(v − u)] · n = 0,

in V − σ, on σ.

(13.5.33a) (13.5.33b)

276


Conservation of Microinertia Djkl + (kpr jlp + lpr jkp )νr = 0 Dt [ρjkl (v − u)] · n = 0

in V − σ,

(13.5.34a)

on σ.

(13.5.34b)

in V − σ,

(13.5.35a)

on σ,

(13.5.35b)

Balance of Momentum tkl,k + ρ(fl − v˙l ) = ρ fˆl [tkl − ρvl (vk − uk )]nk = Fˆl ˆ ρ fl dv + Fˆl da = 0. V

(13.5.35c)

σ

Balance of Moment of Momentum mkl,k + lmn tmn + ρ(ll − σl ) = ρ lˆl [mkl − ρjpl νp (vk − uk )]nk = Lˆ ˆ ρ ll dv + Lˆ l da = 0. V −σ

in V − σ,

(13.5.36a)

on σ,

(13.5.36b) (13.5.36c)

σ

Balance of Energy −ρ ˙ + tkl (vl,k + lkr νr ) + mkl νl,k + qk,k + ρh = −ρ ˆ [tkl vl + mkl νl + qk − ρ( + V −σ

1 2v

(13.5.37a)

·v

− uk )]nk = −Eˆ ˆ ˆ ρ(−ˆ + fr vr + lr νr ) dv − Eˆ da = 0. +

in V − σ,

1 2 jij νi νj )(vk

on σ,

(13.5.37b) (13.5.37c)

σ

The balance laws of all three continua are valid for all types of substances (solids, fluids, blood polymers, etc.), irrespective of material constitution. These equations will be closed by the constitutive equations to reflect the material constitution of each class of media. As discussed in Section 2.1, in all balance laws (micromorphic, microstretch, ˆ ρ I, ˆ and ρ ˆ can be absorbed into the and micropolar) the nonlocal residuals ρ f, that stress, stress moments, and energy, leaving only the surface loads F, L, and E ˆ ˆ ˆ replace F, L, and E, in the jump conditions.

13.6 Second Law of Thermodynamics The global form of the second law, given by (2.2.1), is valid here. The local forms are the same as (2.2.2) and (2.2.3) with the introduction of the Helmholtz free energy ψ by ψ = − θη, (13.6.1)

13.6 Second Law of Thermodynamics

277

and an argument similar to the one advanced in Section 2.1 for ˆ , the energy equation (13.5.26a) becomes ˙ + ηθ −ρ( ˙ + ηθ˙ ) + tkl akl + mklm blmk + skl ckl + qk,k + ρh = 0.

(13.6.2)

Substituting ρh from this into (2.2.1), we obtain the C–D inequality 1 1 ˙ − ρηθ˙ ) + tkl akl + skl ckl + mklm blmk + qk θ,k dv ≥ 0. (13.6.3) (−ρ θ V θ

In this, and in (13.6.2), the deformation-rate tensors akl , ckl , and bklm are defined by (13.2.17). We decompose η, tkl , skl , and mklm into static parts (denoted by the left subscript R) and dynamic parts (denotes by a left subscript D): η = R η + D η, skl = R skl + D skl ,

tkl = R tkl + D tkl , mklm = R mklm + D mklm .

(13.6.4)

With the introduction of η by (13.6.1), ψ is considered to be a static variable. In the case of thermostatic equilibrium all the dynamic parts vanish and the C–D inequality reduces to equality (see Definition in Section 2.2), i.e.,

1

˙ + R ηθ˙ ) + R tkl akl + R skl ckl + R mklm blmk dv = 0. −ρ( V θ

(13.6.5)

Substracting this from (13.6.3) we have, for the dynamics part, 1 1 ˙ −ρ D ηθ + D tkl akl + D skl ckl + D mklm blmk + qk θ,k dv ≥ 0. (13.6.6) θ V θ

This is the C–D inequality that is posited not to be violated by the dynamic set of variables D η, D t, D s, D m, and q. The inequality (13.6.6) may be expressed in the form of an inner product 1 J · Y dv ≥ 0, (13.6.7) θ V where J and Y are the ordered sets J = (−ρ D η, D tkl , D skl , D mklm , qk /θ), Y = (θ˙ , akl , ckl , blmk , θ,k ).

(13.6.8)

The collection J is called the thermodynamics flux and Y is called the thermodynamic force. Thus, the dissipation inequality (13.6.6) is expressed as an inner product of two vectors in 43-dimensional space.

278


By substituting (13.3.16) for νkl in (13.6.5) and (13.6.6), we obtain the corresponding expressions for the microstretch continua 1

˙ + R tkl akl + R mkl blk ˙ + R ηθ) −ρ( V θ + R mk ν,k + (R s − R t)ν dv = 0, (13.6.9) 1 −ρ D ηθ˙ + D tkl akl + D mkl blk + D mk ν,k V θ 1 (13.6.10) + (D s − D t)ν + qk θ,k dv ≥ 0. θ From these, by setting νk = ν = 0, we obtain the corresponding expressions for the micropolar continua 1

˙ + R tkl akl + R mkl blk dv = 0, (13.6.11) ˙ + R ηθ) −ρ( θ V 1 1 ˙ −ρ D ηθ + D tkl akl + D mkl blk + qk θ,k dv ≥ 0. (13.6.12) θ V θ For microstretch and micropolar continua, deformation-rate tensors are defined by (13.2.18) and (13.2.19).

13.7 Constitutive Equations of Memory-Dependent Nonlocal Micromorphic Elastic Solids5 For the micromorphic thermoelastic solids, the constitutive independent variables Y and the dependent variables Z are ˙ θ}, Y = {CKL , KLM , CKL , θ,K , θ, Z = {ψ, η, tkl , mklm , skl , qk }.

(13.7.1)

It is convenient to introduce the material forms TKL , MKLM , SKL , and QK by ρ ρ tkl = TKL xk,K XLl , skl = SKL χkK χlL , ρ0 ρ0 ρ ρ mklm = MKLM xk,K XLl χmM , qk = QK xk,K . (13.7.2) ρ0 ρ0 The equations of thermodynamic equilibrium (13.6.5) and the C–D inequality (13.6.6) are put into the forms ρ ˙ + R J · R Y˙ ) dV = 0, (−ρ0 (13.7.3) ρ V 0θ ρ (13.7.4) D J · D Y dV ≥ 0, V ρ0 θ 5 Not published before.

13.7 Micromorphic Elastic Solids

where we introduced the ordered static sets R J = −ρ0 R η, R TKL , R MKLM , RY

1 2 R SKL

279

,

= {θ, CKL , LMK , CKL },

(13.7.5)

and the ordered dynamic sets 1 D J = −ρ0 D η, D TKL , D MKLM , 2 D SKL , QK /θ , ˙ ˙ LMK , C˙KL , θ,K }. D Y = {θ˙ , CKL ,

(13.7.6)

Since ψ is a thermostatic variable, it can depend on only R Y nonlocally. Thus, we propose a constitutive ersatz of the form ρ0 ψ(X, t) = F [R Y(X ), X ; R Y(X), X] dV (X ), (13.7.7) V

where F is a symmetric function of its variables, i.e., S

F = F = F [R Y(X), X; R Y(X ), X ]. The symmetry is with respect to the interchange of X and X. We have ∂F ∂F S ˙ dV + D, ˙ = ρ0 · RY + ∂ R Y

V ∂ RY

where D=

V

∂F ˙ − · RY ∂ R Y

∂F ∂ RY

S

(13.7.8)

(13.7.9)

˙ dV . · RY

Substituting into (13.7.3) we obtain ∂F ∂F S dV . + RJ =

∂ Y ∂ Y R R V

(13.7.10)

(13.7.11)

Explicitly

∂F S −ρ0 R η = dV , ∂θ

V ∂F ∂F S dV , + R TKL = ∂C KL V ∂CKL S ∂F ∂F dV , + R MKLM =

∂LMK V ∂LMK ∂F ∂F S 1 dV . +

2 R SKL = ∂CKL V ∂CKL ∂F + ∂θ

(13.7.12)

280


The dynamical portions of the constitutive equations follow from (13.7.4), upon using (2.3.3): ∂ (13.7.13) = U, DJ = D J · U = 0, ∂ DY where U is the collection of residuals which does not contribute to the dissipation. For the linear theory, U = 0. The dissipation potential is a function of the nonlocal histories of the variable set D Y. We take in the form ∞ ∞

dτ dτ G[D Y(t) (X , τ ), D Y(t) (X, τ )] dV , (13.7.14) = 0

0

V

where D Y(t) is the difference histories of the set D Y, defined as D Y(t) (X, τ )

= D Y(X, t − τ ) − D Y(X, t).

The total dissipation is given by

(13.7.15)

W =

(13.7.16)

dV . V

Consequently, the antisymmetric part of G in its variables do not contribute to the total dissipation. Hence, we stipulate that G is a symmetric function of the form G[D Y(t) (X , τ );

D Y(X, τ )]

lim

D Y(t) →0

= G[D Y(t) (X, τ );

D Y(t) (X

G = 0.

, τ )], (13.7.17)

The volume integral of the energy equation (13.6.2) is given by ˙ + ηθ [−ρ( ˙ + ηθ˙ ) + tkl akl + mklm blmk + skl ckl + qk,k + ρh] dv = 0. V

If we subtract the equilibrium part, we obtain ˙ + D tkl akl + D mklm blmk + D skl ckl [−ρ(R ηθ ˙ + D ηθ ˙ + D ηθ) V

+ qk,k + ρh] dv = 0. (13.7.18)

Upon substituting from (13.7.2) this equation becomes ρ ˙ + D TKL C˙ KL + D MKLM ˙ LMK + 1 D SKL C˙KL [−ρ0 (ηθ ˙ + D ηθ) 2 ρ V 0 + QK,K + ρ0 h] dV = 0. (13.7.19) An alternative compact form of (13.7.19) is ρ [−ρ0 ηθ ˙ + D J · D Y + θ(QK /θ),K + ρ0 h] dV = 0. V ρ0

(13.7.20)

Equations (13.7.18), (13.7.19), and (13.7.20) are three different (but equivalent) forms of the global energy balance law. In the localized forms they serve as the equations of heat conduction.


281

Linear Constitutive Equations For the linear theory, F will be a quadratic symmetric function of the independent variable. Also, linearization, in the scheme of Section 6.1, will show that R J and R Y can be taken in spatial frames, i.e.,

RJ

= {−ρ0 R η,

R tkl , R mklm , R skl },

R Y = {T , kl , γlmk , ekl }.

(13.7.21)

The free energy density F is a quadratic symmetric polynomial in R Y and R Y . We express F as

ρ0

CT T − 21 Akl (T kl + T kl ) − 21 Bkl (T ekl + T ekl ) T0

− 21 Cklm (T γklm + T γklm ) + U, (13.7.22)

F = F0 −

where U is the strain energy density, defined by

2U =

ρ0

CT T + Aklmn kl mn + Bklmn ekl emn + Cklmnpq γklm γnpq T0

+ Eklmn (kl emn + kl emn ) + Fklmnp (kl γmnp + kl γmnp )

γmnp + ekl γmnp ). + Gklmnp (ekl

(13.7.23)

The material moduli (functions) are subject to the symmetry regulations

C(x, x ) = C(x , x), Akl (x, x ) = Akl (x , x), Bkl (x, x ) = Bkl (x , x), Cklm (x, x ) = Cklm (x , x), Aklmn (x, x ) = Amnlk (x , x), Bklmn (x, x ) = Bmnkl (x , x) = Blkmn = Bklnm ,

(13.7.24)

Cklmnpq (x, x ) = Cnpqklm (x , x), Eklmn (x, x ) = Eklmn (x , x) = Eklnm , Fklmnp (x, x ) = Fklmnp (x , x), Gklmnp (x, x ) = Gklmnp (x , x) = Glkmnp .

Substituting (13.7.22) into (13.7.12), and noting through (13.7.2) that in the linear theory TKL → tkl , ρ0 = ρ, SKL → skl , MKLM → mklm , and Qk → qk , we

282


obtain the static portions of the constitutive equations

1 1 1 C(x, x )T (x ) + Akl (x, x )kl (x ) + Bkl (x, x )ekl (x ) T ρ ρ 0 0 0 V 1

+ Cklm (x, x )γklm (x ) dv(x ), ρ 0 [−Akl (x, x )T (x ) + Aklmn (x, x )mn (x ) + Eklmn (x, x )emn (x ) R tkl = Rη =

V

R mklm

+ Fklmnp (x, x )γmnp (x ) dv(x ),

(x ) = [−Clmk (x, x )T (x ) + Clmknpq (x, x )γnpq V

+ Fnplmk (x, x )np (x ) + Gnplmk (x, x )enp (x )] dv(x ), skl = [−Bkl (x, x )T (x ) + Bmnkl (x, x )emn (x ) V

+ Emnkl (x, x )mn (x ) + Gklmnp (x, x )γmnp (x )] dv(x ).

(13.7.25)

For the dynamic parts of the constitutive equations, we express the dissipation potential density G as a quadratic polynomial in terms of its variables. In the linear approximation, D J and D Y are approximated in terms of the spatial variables. We ignore temperature-rate dependence, then D η = 0. Hence,

1 qk , D J = D tkl , D mklm , D skl , T0 D Y = {akl , blmk , ckl , T,k }.

(13.7.26)

Since the natural state is at constant temperature T0 , and free of stress and heat, the dissipation density G cannot have linear terms involving D Y. In order to account for the memory-dependence, we introduce difference histories, e.g.,

a(t)kl (x, τ ) = akl (x, t − τ ) − akl (x, t).

(13.7.27)


283

The difference histories of blmk , ckl , and T,k are similarly constructed

(x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )][a(t)mn (x , τ ) + a(t)mn (x, τ )] G = aklmn

+ bklmn (x , τ ; x, τ )[c(t)kl (x , τ ) + c(t)kl (x, τ )][c(t)mn (x , τ ) + c(t)mn (x, τ )]

+ cklmnpq (x , τ ; x, τ )[b(t)klm (x , τ ) + b(t)klm (x, τ )]

× [b(t)npq (x , τ ) + b(t)npq (x, τ )]

+ eklmn (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )][c(t)mn (x , τ ) + c(t)mn (x, τ )]

+ fklmnp (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )] × [b(t)mnp (x , τ ) + b(t)mnp (x, τ )]

+ gklmnp (x , τ ; x, τ )[c(t)kl (x , τ ) + c(t)kl (x, τ )] × [b(t)mnp (x , τ ) + b(t)mnp (x, τ )] + kkl (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )][T(t),l (x , τ ) + T(t),l (x, τ )] a + kklm (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )][a(t)lm (x , τ ) + a(t)lm (x, τ )] b + kklmn (x , τ ; x, τ )][T(t),k (x , τ ) + T(t),k (x, τ )] × [b(t)lmn (x , τ ) + b(t)lmn (x, τ )] c + kklm (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )]

× [c(t)lm (x , τ ) + c(t)lm (x, τ )],

(13.7.28)

, bklmn , . . . , gklmnp and the heat conduction modwhere the viscosity moduli aklmn c uli kkl to kklm are subject to the symmetry regulations

(x , τ ; x, τ ) = amnkl (x, τ ; x , τ ), aklmn

bklmn (x , τ ; x, τ ) = bmnkl (x, τ ; x , τ ) = blkmn = bklnm ,

cklmnpq (x , τ ; x, τ ) = cnpqklm (x, τ ; x, τ ),

(x , τ ; x, τ ) = eklmn (x, τ ; x , τ ) = eklnm , eklmn

fklmnp (x , τ ; x, τ ) = fklmnp (x, τ ; x, τ ),

(x , τ ; x, τ ) = gklmnp (x, τ ; x , τ ) = glkmnp , gklmnp

kkl (x , τ ; x, τ ) a kklm (x , τ ; x, τ ) b (x , τ ; x, τ ) kklmn c kklm (x , τ ; x, τ )

= kkl (x, τ ; x , τ ) = klk , a = kklm (x, τ ; x , τ ), b = kklmn (x, τ ; x , τ ), c = kklm (x, τ ; x , τ ).

(13.7.29)

The dissipation potential is given by (13.7.14), namely,

∞

=

∞

dτ 0

0

dτ

V

G dv .

(13.7.30)

284


We now substitute this into the constitutive equations (13.7.13) more explicitly ∂ , ∂akl ∂ , D mklm = ∂blmk ∂ , D skl = ∂ckl ∂ qk /T0 = . ∂T,k D tkl

=

(13.7.31)

A little scrutiny of (13.7.28) shows that G has a similar structure fo (11.1.9). Consequently, a similar analysis leads to the dynamic constitutive equation t

t = dt [aklmn (x , t − t ; x)amn (x , t ) D kl −∞

V

+ eklmn (x , t − t ; x)cmn (x , t )

+ fklmnp (x , t − t ; x)bmnp (x , t )

D mklm

a (x , t − t ; x)T,m (x , t )] dv(x ), + kmkl t = dt [clmknpq (x , t − t ; x)bnpq (x , t ) −∞

V

+ fnplmk (x , t − t , x)anp (x , t ) + gnplmk (x , t − t ; x)cnp (x , t ) b + knlmk (x , t − t ; x)T,n (x , t )] dv(x ), t

dt [bklmn (x , t − t ; x)cmn (x , t ) D skl = −∞

V

+ emnkl (x , t − t ; x)amn (x , t ) + gklmnp (x , t − t ; x)bmnp (x , t ) c + kmkl (x , t − t ; x)T,m (x , t )] dv(x ), t 1 qk = dt [kkl (x , t − t ; x)T,l (x , t ) T0 V −∞ a + kklm (x , t − t ; x)alm (x , t ) b + kklmn (x , t − t ; x)blmn (x , t ) c + kklm (x , t − t ; x)clm (x , t ) dv(x ).

(13.7.32)

The complete constitutive equations are given by η = R η,

tkl = R tkl + D tkl , skl = R skl + D skl ,

mklm = R mklm + D mklm , qk .

(13.7.33)

By expressing the static and dynamic material moduli in terms of Dirac-delta measures (as discussed in Section 11.1), it is straightforward to obtain constitutive


285

equations for: (a) nonlocal media with no memory; (b) local media with memory; and (c) local media. As we have seen in Section 9.1, these special equations appear in the equations that follow from (13.7.31) before some Dirac-delta functions are introduced to these terms to obtain the final equations (13.7.32) (cf. (9.1.1)1 ). Isotropic Solids For homogeneous elastic solids, the material moduli will depend on x and x through x − x, e.g., (13.7.34) aklmn = aklmn (x − x, t − t ), and for homogeneous and isotropic solids, they are isotropic functions of κ = x − x, e.g., aklmn = a0 δkl δmn + a1 δkm δln + a2 δkn δlm + a3 κk κl δmn + a4 κm κk δln + a4 κk κn δlm + a6 κl κm δkn + a1 κl κn δkm + a8 κm κn δkl + a9 κk κl κm κn .

(13.7.35)

Similarly, other moduli are constructed. However, since the terms containing κ are negligible, as discussed before (see Section 6.1), we shall not give these expressions; rather we shall produce expressions of the material moduli involving the lowest-order terms only. In this case, all the odd-order tensor moduli vanish and the even-order moduli are given by Bkl = β1 δkl , Akl = β0 δkl , Aklmn = λδkl δmn + (µ + κ)δkm δln + µδkn δlm , Bklmn = (λ + 2ν + τ )δkl δmn + (µ + 2σ + η)(δkm δln + δkn δlm ), Eklmn = (λ + ν)δkl δmn + (µ + σ )(δkm δln + δkn δlm ), Cklmnpq = τ1 (δkl δmn δpq + δkq δlm δnp ) + τ2 (δkl δmp δnq + δkm δlq δnp ) + τ3 δkl δmq δnp + τ4 δkn δlm δpq + τ5 (δkm δln δpq + δkp δlm δnq ) + τ6 δkm δlp δnq + τ7 δkn δlp δmq + τ8 (δkp δlq δmn + δkq δln δmp ) + τ9 δkn δlq δmp + τ10 δkp δln δmq + τ11 δkq δlp δmn ,

(13.7.36)

where β0 , β1 , λ, µ, κ, ν, σ , and τi are functions of |x − x|, e.g., β0 = β0 (|x − x|),

λ = λ(|x − x|),

τi = τi (|x − x|), . . . . (13.7.37)

The dynamic moduli are similar in composition to (13.7.36) and we mark them by a subscript v, indicating that they are viscosity moduli, e.g., λv , µv , etc. Of course, the dynamic moduli also depends on x − x and t − t , e.g., λv = λv (|x − x|), t − t ), . . . .

(13.7.38)

286


13.8 Constitutive Equations of Memory-Dependent Nonlocal Microstretch Elastic Solids6 For the case of microstretch elastic solids, the independent variables are ˙ θ}. Y = {CKL , KL , K , j, θ,K , θ,

(13.8.1)

The response functions (dependent variables) are Z = {, η, tkl , mkl , mk , s − t, qk }.

(13.8.2)

The material forms TKL , MKL , MK , S − T , and QK are introduced by ρ0 tkl XK,k χ lL , ρ ρ0 MK = mk XK,k , ρ

TKL =

ρ0 j 2 mkl XK,k χ lL , ρ ρ0 ρ0 (s − t), QK = qk XK,k . S−T = ρj ρ MKL =

(13.8.3)

Equations of thermodynamic equilibrium (13.6.9) and C–D inequality (13.6.10) are expressed as ρ ρ ˙ dV = 0, ˙ + R J · R Y) (−ρ0 D J · D Y dV ≥ 0, (13.8.4) V ρ0 θ V ρ0 θ where we introduced the ordered static sets (R J, R Y) and the dynamic sets (D J, RJ

= {−ρ0 R η, R TKL , R MKL , R MK , S − T },

RY

= {θ, CKL , LK , K , j },

D Y):

= {−ρ0 D η, D TKL , D MKL , D MK , D S − D T , QK /θ}, ˙ ˙ LK , ˙ K , Dj/Dt, θ,K }. (13.8.5) D Y = {θ˙ , CKL , DJ

Using a constitutive ersatz for ψ in the form (13.7.7), subject to the symmetry expressed by (13.7.8), from (13.8.4) it follows that ∂F ∂F S dV . (13.8.6) + RJ = ∂ R Y

V ∂ RY Similarly, selecting in the form (13.7.14), we obtain ∞ ∞ ∂G ∂

J = dτ dτ = dV , D ∂ DY ∂ Y D V 0 0

(13.8.7)

where G is of the form (13.7.17). In (13.8.7) we dropped the constitutive residual U that does not contribute to the total dissipation. 6 Not published before.

13.8 Microstretch Elastic Solids

287

In a similar fashion to (13.7.18), we obtain the reduced form of the global energy balance law ˙ + D tkl akl + D mkl blk + D mk ν,k [−ρ(R ηθ ˙ + D ηθ ˙ + D ηθ) V

+ (D s − D t)ν + qk,k + ρh] dv = 0.

(13.8.8)

In its localized form, this equation serves as the equation of heat conduction. Linear Constitutive Equations For the linear theory, F is expressed as a quadratic symmetric function of the independent variables at x and x . In linearization j = 1 + ϕ, Dj/Dt = ϕ, ˙ R J, Y, and ρ = ρ can be taken in the spatial frame, x, i.e., R 0 RJ

= {−ρ0 R η, R tkl , R mkl , R mk , R s − R t},

RY

= {T , kl , γlk , γk , ϕ},

(13.8.9)

where kl , γkl , and γk are given by (13.1.39): kl = ul,k + klm φm ,

γkl = φk,l ,

γk = ϕ,k .

(13.8.10)

We express the free energy density F as a symmetric quadratic polynomial in R Y and R Y : 2ρ0

CT T − C1 (T ϕ + T ϕ ) − Dk (T ϕ,k + T ϕ,k ) T0

− Akl (T kl + T kl ) − Bkl (T γkl + T γkl ) + U, (13.8.11)

2F = 2F0 −

where U is the strain energy density, given by 2U =

ρ0

CT T + C s ϕϕ + Cks (ϕ ϕ,k + ϕϕ,k ) T0

s + Askl (ϕ kl + ϕkl ) + Bkl (ϕ γkl + ϕγkl ) s

s

+ Ckl ϕ,k ϕ,l + Aklm (ϕ,k lm + ϕ,k lm ) s

(ϕ,k γlm + ϕ,k γlm ) + Aklmn kl mn + Bklm

+ Bklmn γkl γmn + Cklmn (kl γmn + kl γmn ),

(13.8.12)

where all the material moduli C, C1 , Dk , . . . , Cklmn are symmetric functions of x and x , e.g., Dk (x, x ) = Dk (x , x), (13.8.13) in addition, we note the extra symmetries of s s Ckl = Clk ,

Aklmn = Amnkl ,

Bklmn = Bmnkl .

(13.8.14)

288


Substituting F into (13.8.6) we obtain the static portions of the linear constitutive equations

ρ0 C(x, x )T (x ) + C1 (x, x )ϕ(x ) + Dk (x, x )ϕ,k (x ) V T0

+ Akl (x, x )kl (x ) + Bkl (x, x )γkl (x ) dv(x ), [−Akl (x, x )T (x ) + Askl (x, x )ϕ(x ) + Asmkl (x, x )ϕ,m (x ) R tkl =

ρ0 R η =

V

+A (x, x )mn (x ) + Cklmn (x, x )γmn (x )] dv(x ), klmn s s [−Blk (x, x )T (x ) + Blk (x, x )ϕ(x ) + Bmlk (x, x )ϕ,m (x ) R mkl = V

+B (x, x )γmn (x ) + Cmnlk (x, x )mn (x )] dv(x ), lkmn s [−Dk (x, x )T (x ) + Cks (x, x )ϕ(x ) + Ckl (x, x )ϕ,l (x ) R mk = V

s + As (x, x )lm (x ) + Bklm (x, x )γlm (x )] dv(x ), klm [−C1 (x, x )T (x ) + C s (x, x )ϕ(x ) + Cks (x, x )ϕ,k (x ) RS − RT =

V

s + Askl (x, x )kl (x ) + Bkl (x, x )γkl (x )] dv(x ).

(13.8.15)

The dynamical parts of the constitutive equations require that we express the dissipation density G, in (13.8.7), as a second-degree polynomial in terms of the difference histories D Y(t) , given by (13.7.15), subject to (13.7.16) and (13.7.17). For the linear theory from (13.8.7) we have

∞

DJ =

∞

dτ 0

0

dτ

V

∂G dv . ∂ DY

(13.8.16)

We assume that G does not depend on θ˙ , hence D η = 0. Consequently, D J and D Y, for the linear theory are given by

DJ

= {D tkl ,

DY

= {akl , blk , ν,k , ν, T,k }.

D mkl , D mk , D s

− D t, qk /T0 }, (13.8.17)

Since the natural state is undisturbed, G cannot have any linear terms in D Y. Thus, we write a symmetric second-degree polynomial in terms of the difference


289

histories D Y(t) given by (13.7.15):

G = aklmn (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )][a(t)mn (x , τ ) + a(t)mn (x, τ )]

+ cklmn (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ ][b(t)mn (x , τ ) + b(t)mn (x, τ )]

s + aklm (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )][ν(t),m (x , τ ) + ν(t),m (x, τ )]

s

(x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )][ν(t) (x , τ ) + ν(t) (x, τ )] +akl

a + kklm (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )][T(t),m (x , τ ) + T(t),m (x, τ )]

+ bklmn (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )]

× [b(t)mn (x , τ ) + B(t)mn (x, τ )]

s + bklm (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )][ν(t),m (x , τ ) + ν(t),m (x, τ )]

s

+ bkl (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )][ν(t) (x , τ ) + ν(t) (x, τ )]

b + kklm (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )][T(t),m (x , τ ) + T(t),m (x, τ )]

s

+ ckl (x , τ ; x, τ )[ν(t),k (x , τ ) + ν(t),k (x, τ )][ν(t),l (x , τ ) + ν(t),l (x, τ )] + ck s (x , τ ; x, τ )[ν(t),k (x , τ ) + ν(t),k (x, τ )][ν(t) (x , τ ) + ν(t) (x, τ )]

γ

+ kkl (x , τ ; x, τ )[ν(t),k (x , τ ) + ν(t),k (x, τ )][T(t),l (x , τ ) + T(t),l (x, τ )] + c s (x , τ ; x, τ )[ν(t) (x , τ ) + ν(t) (x, τ )][ν(t) (x , τ ) + ν(t) (x, τ )] + kk ν (x , τ ; x, τ )[ν(t) (x , τ ) + ν(t) (x, τ )][T(t),k (x , τ ) + T(t),k (x, τ )]

+ kkl (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )] × [T(t),l (x , τ ) + T(t),l (x, τ )].

(13.8.18)

All the material moduli are symmetric functions of (x , τ ) and (x, τ ), e.g.,

aklmn (x , τ ; x, τ ) = aklmn (x, τ ; x , τ ).

(13.8.19)

In addition, the following moduli are symmetric in their indices, as indicated,

aklmn = amnkl ,

s

s ckl = clk ,

bklmn = bmnkl ,

kkl = klk .

(13.8.20)

The dynamic parts of the constitutive equations follow by substituting G into (13.8.7), more explicitly, ∞ ∞ ∂G

dτ dτ dv , D tkl = ∂a kl V 0 ∞ 0 ∞ ∂G

dτ dτ dv , D mkl = ∂b lk V 0 ∞ 0 ∞ ∂G

dτ dτ dv , D mk = V ∂ν,k 0 0 ∞ ∞ ∂G

dτ dτ

dv , Ds − Dt = ∂ν V 0 ∞ 0 ∞ ∂G dτ dτ

dv . (13.8.21) qk /T0 = V ∂T,k 0 0

290


The procedure here is similar to Sections 9.1 and 13.7. As in Section 9.1 we first obtain the constitutive equations that are the sum of four types terms: the local terms (without integrals), nonlocal terms involving the volume integrals, the memory-dependent local terms involving integrals with respect to time, and memory-dependent nonlocal terms involving volume and time integrals. By introducing the Dirac-delta measures, as in (9.1.14), we then combine all three types of terms. Finally, a change of variable t − τ = t leads to the constitutive equations for the dynamical parts t

[aklmn (x , t − t ; x)amn (x , t ) t = dt D kl −∞

V

−∞

V

+ cklmn (x , t − t ; x)bmn (x , t ) a s + kklm (x , t − t ; x)T,m (x , t ) + aklm (x , t − t ; x)ν,m (x , t ) s

+ akl (x , t − t ; x)ν(x , t )] dv(x ), t

dt [cmnlk (x , t − t ; x)amn (x , t ) D mkl = + blkmn (x , t − t ; x)bmn (x , t ) s s + blkm (x , t − t ; x)ν,m (x , t ) + blk (x , t − t ; x)ν(x , t ) b + klkm (x , t − t ; x)T,m (x , t )] dv(x ), t

s m = dt [almk (x , t − t ; x)alm (x , t ) D k −∞

V

−∞

V

s + blmk (x , t − t ; x)blm (x , t ) s + ckl (x , t − t ; x)ν,l (x , t ) + cks (x , t − t ; x)ν(x , t ) γ + kkl (x , t − t ; x)T,l (x , t )] dv(x ), t

s s dt [akl (x , t − t ; x)akl (x , t ) + bkl (x , t − t ; x)bkl (x , t ) Ds − Dt =

+ cks (x , t − t ; x)ν,k (x , t ) + cs (x , t − t ; x)ν(x , t ) + kkν (x , t − t ; x)T,k (x , t )] dv(x ), t qk

a = dt [klmk (x , t − t ; x)alm (x , t ) T0 V −∞

b (x , t − t ; x)blm (x , t ) + klmk γ + klk (x , t − t ; x)ν,l (x , t ) + kkν (x , t − t ; x)ν(x , t )

+ kkl (x , t − t ; x)T,l (x , t )] dv(x ).

(13.8.22)

The complete constitutive equations are the sum of the static and dynamic parts tkl = R tkl + D tkl , mkl = R mkl + D mkl , η = R η, mk = R mk + D mk , s − t = R s − R t + D s − D t, qk . (13.8.23) From (13.8.5) and (13.8.8) we can obtain back the constitutive equations of the local and nonlocal elastic solids with and without memory, by taking the constitutive


291

moduli proportional to the Dirac-delta measure. For example, if we take aklmn = a klmn (x , x)δ(t − t ), and similarly other moduli, we will have the nonlocal constitutive equations without memory. Similarly, if we take 0 δ(x − x)δ(t − t ), aklmn = aklmn

and similarly other moduli, we obtain the local (classical) constitutive equations. The constitutive equations for the local memory-dependent media are obtained by setting aklm = aklmn δ(x − x). Isotropic Microstretch Media For homogeneous media, material moduli depend on x and x through x − x , e.g., Aklmn = Aklmn (x − x ), aklmn = aklmn (x − x , t − t ). For homogeneous and isotropic media they will be polynomials in terms of κ = x − x, with coefficients that depend on |κκ | (cf. (13.7.35)). However, as discussed before (cf. Section 6.1), the terms involving powers of κ are negligible as compared to those without κ . It is also important to distinguish polar and axial tensors and even and odd tensors. (tkl , mk , η, s − t, kl , akl , ϕ, ν) are polar tensors and (mkl , γkl , bkl ) are axial tensors. The isotropic constitutive equations for the polar tensors will depend on polar tensorial terms and the axial tensors on axial terms. The existence of the material moduli, obtained above for the anistropic media, depends on the material symmetry group. For anisotropic materials the symmetry group determines the number of independent components of the material tensors. Isotropic materials are the simplest ones for which the group symmetry consists of the full orthogonal group of transformations. Below we give the material moduli for the isotropic media, s Dk = Cks = 0, Akl = β0 δkl , Bkl = Bkl = 0, s s s s Akl = λ0 δkl , Ckl = a0 δkl , Aklm = 0, Bklm = b0 klm ,

Aklmn = λδkl δmn + (µ + κ)δkm δln + µδkn δlm , Bklmn = αδkl δmn + βδkn δlm + γ δkm δln , Cklmn = 0,

(13.8.24)

and for the dynamic moduli aklmn = λv δkl δmn + (µv + κv )δkm δln + µv δkn δlm , a s s kklm = 0, akl = λs δkl , cklmn = 0, bklm = bv klm , s bklmn = αv δkl δmn + βv δkn δlm + γv δkm δln , bkl = 0, b = k b klm , kklm γ kkl = k γ δkl ,

s aklm = 0, a kklm = 0,

s ckl = cv δkl ,

kkl = kδkl ,

cks = 0, kkν = 0.

(13.8.25)

292


The static moduli are functions of |x − x| and the dynamic moduli are functions of |x − x| and t − t , e.g., β0 = β0 (|x − x|),

λv = λv (|x − x|, t − t ).

(13.8.26)

Consequently, for the isotropic media, the static parts of the constitutive equations read

ρ0 C(|x − x |)T (x ) + C1 (|x − x |)ϕ(x ) V T0 + β0 (|x − x|)kk (x ) dv(x ), {[−β0 (|x − x |)T (x ) + λ0 (|x − x |)ϕ(x ) + λ(|x − x |)rr (x )]δkl R tkl =

ρ0 R η =

V

+ [µ(|x − x |) + κ(|x − x |)]kl (x ) + µ(|x − x |)lk (x )} dv(x ), [b0 (|x − x |)mlk ϕ,m (x ) + α(|x − x |)γrr (x )δkl R mkl = V

+ β(|x − x |)γkl (x ) + γ (|x − x |)γlk (x )] dv(x), [a0 (|x − x |)ϕ,k (x ) + b0 (|x − x |)klm γlm (x )] dv(x ), R mk = V [−C1 (|x − x |)T (x ) + C s (|x − x |)ϕ(x ) Rs − Rt = V

+ λ0 (|x − x |)kk (x )] dv(x ).

(13.8.27)

The dynamic parts of the constitutive equations for the isotropic media become D tkl =

t −∞ s

dt

V

{[λv (|x − x |, t − t )arr (x , t )

+ λ (|x − x |, t − t )ν(x , t )]δkl + [µv (|x − x , t − t ) + κv (|x − x , t − t )]akl (x , t ) + µv (|x − x |, t − t )alk (x , t )} dv(x ), t

m = dt {[αv (|x − x |, t − t )brr (x , t )δkl D kl −∞

V

+ βv (|x − x |, t − t )bkl (x , t ) + γv (|x − x |, t − t )blk (x , t ) + lkm [bv (|x − x , t − t )ν,m (x , t ) + k b (|x − x |, t − t )T,m (x , t )]} dv(x ),

13.9 Micropolar Elastic Solids

D mk

=

t −∞

dt

V

293

[bv (|x − x |, t − t )lmk blm (x , t )

+ cv (|x − x |, t − t )ν,k (x , t ) + k γ (|x − x |, t − t )T,k (x , t )] dv(x ), t

s − t = dt [λs (|x − x |, t − t )akk (x , t ) D D −∞ s

V

−∞

V

+ c (|x − x |, t − t )ν(x , t )] dv(x ), t

qk /T0 = dt [k(|x − x |, t − t )T,k (x , t ) + k b (|x − x |, t − t )lmk blm (x , t ) + k γ (|x − x |, t − t )ν,k (x , t )] dv(x ).

(13.8.28)

The field equations are obtained by substituting the strain measures and the deformation-rate tensors into the constitutive equations, and afterward the constitutive equations into the balance laws.

13.9 Constitutive Equations of Memory-Dependent Nonlocal Micropolar Elastic Solids7 The constitutive equations of micropolar elastic solids can be obtained directly from those of microstretch solids by dropping mk , s − t, and setting γk = ν,k = ϕ = ν = 0. Thus, for the linear theory we get F = F0 −

ρ0

CT T − 21 Akl (T kl + T kl ) − 21 Bkl (T γkl + T γkl ) + U, (13.9.1) T0

where U is the strain energy given by 2U =

ρ0

CT T + Aklmn kl mn + Bklmn γkl γmn T0

+ Cklmn (kl γmn + kl γmn ),

to obtain the static parts of the constitutive equations ρ0 R η =

7 Not published before.

ρ0 C(x, x )T (x ) + Akl (x, x )kl (x ) V T0

+ Bkl (x, x )γkl (x ) dv(x ),

(13.9.2)

294


R tkl =

V

[−Akl (x, x )T (x ) + Aklmn (x, x )mn (x )

+C (x, x )γmn (x )] dv(x ), klmn [−Blk (x, x )T (x ) + Blkmn (x, x )γmn (x ) R mkl = V

+ Cmnlk (x, x )mn (x )] dv(x ).

(13.9.3)

For the dynamic parts, the dissipation potential density is given by

(x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )] G = aklmn × [a(t)mn (x , τ ) + a(t)mn (x, τ )]

+ cklmn (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )] × [b(t)mn (x , τ ) + b(t)mn (x, τ )]

a (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )] + kklm × [T(t),m (x , τ ) + T(t),m (x, τ )]

+ bklmn (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )] × [b(t)mn (x , τ ) + b(t)mn (x, τ )]

b (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )] + kklm × [T(t),m (x , τ ) + T(t),m (x, τ )]

+ kkl (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )]

× [T(t),l (x , τ ) + T(t),l (x, τ )].

(13.9.4)

Consequently, the dynamical parts of the constitutive equations are obtained to be t dt [aklmn (x , t − t ; x)amn (x , t ) D tkl = −∞

V

+ cklmn (x , t − t ; x)bmn (x , t ) a + kklm (x , t − t ; x)T,m (x , t )] dv(x ), t

dt [cmnlk (x , t − t ; x)amn (x , t ) D mkl = −∞

V

+ blkmn (x , t − t ; x)bmn (x , t ) b + klkm (x , t − t ; x)T,m (x , t )] dv(x ), t qk a b = dt [klmk (x , t − t ; x)alm (x , t ) + klmk (x , t − t ; x)blm (x , t ) T0 V −∞ + kkl (x , t − t ; x)T,l (x , t )] dv(x ). (13.9.5)

The symmetry regulations of the material are the same as those of the microstretch material moduli. The full constitutive equations are the sum of the corresponding parts of the static and dynamic moduli, i.e., η = R η,

tkl = R tkl + D tkl ,

mkl = R mkl + D mkl ,

qk .

(13.9.6)

13.9 Micropolar Elastic Solids

295

The reduced form of the energy equation follows from that of the microstretch media (13.8.8), by setting ν = 0: ˙ + D tkl akl + D mkl blk + qk,k + ρh] dv = 0. (13.9.7) [−ρ(R ηθ ˙ + D ηθ ˙ + D ηθ) V

Isotropic Micropolar Media Material moduli for anisotropic solids depend on the material symmetry group. The symmetry group of various crystal classes determines the number of independent components of each material tensor. In addition, the polar axial nature of the dependent and independent variables must be taken into account. Here we shall give the independent components of the material moduli only for the isotropic solids. For homogeneous solids, the material functions depend on x and x through κ ≡ x −x . For the homogeneous and isotropic solids, the isotropy group is the full group of the orthogonal transformations of the frame of reference. The independent members of the material moduli can be read from (13.8.24) and (13.8.25): Akl = β0 δkl ,

Bkl = 0,

Aklmn = λδkl δmn + (µ + κ)δkm δln + µδkn δlm ,

Bklmn = αδkl δmn + βδkn δlm + γ δkm δlm , cklmn = 0, aklmn = λv δkl δmn + (µv + κv )δkm δln + µv δkn δlm , a b kklm = 0, kklm = k b klm , kkl = kδkl .

(13.9.8)

The constitutive equations of the isotropic solids are then given by: Static Parts ρ0 C(|x − x |)T (x ) + β0 (|x − x |)rr (x ) dv(x ), ρ0 R η = T V 0 {[−β0 (|x − x |)T (x ) + λ(|x − x |)rr (x )]δkl R tkl = V

+ [µ(|x − x |) + κ(|x − x |)]kl (x ) + µ(|x − x |)lk (x )} dv(x ), m = [α(|x − x |)γrr (x )δkl + β(|x − x |)γkl R kl V

+ γ (|x − x |)γlk (x )] dv(x ).

(13.9.9)

Dynamic Parts D tkl

=

t −∞

dt

V

{λv (|x − x |, t − t )arr (x , t )δkl + [µv (|x − x |, t − t )

+ κv (|x − x |, t − t )]akl (x , t ) + µv (|x − x |, t − t )alk (x , t )} dv(x ),

296


D mkl

=

t −∞

dt

V

[αv (|x − x , t − t )brr (x , t )δkl

+ βv (|x − x |, t − t )bkl (x , t ) + γv (|x − x |, t − t )blk (x , t ) + k b (|x − x |, t − t )lkm T,m (x , t )] dv(x ), t qk = dt [k b (|x − x |, t − t )lmk blm (x , t ) T0 V −∞ + k(|x − x , t − t )T,k (x , t )] dv(x ).

(13.9.10)

The field equations are obtained by substituting the strain and deformation-rate tensors into constitutive equations, and the resulting constitutive equations to the balance laws.

13.10 Nonlocal Micropolar Elasticity8 A. Constitutive Equations When the memory effects are not present, we obtain the nonlocal micropolar constitutive equations by setting aklmn = bklmn = cklmn = 0, ∗

T0 kkl = kkl (x , x)δ(t − t ).

a b kklm = kklm = 0,

(13.10.1)

The constitutive equations (13.9.3) and (13.9.5) reduce to ρ0 C(x, x )T (x ) + Akl (x, x )kl (x ) ρ0 η = V T0 + Bkl (x, x )γkl (x ) dv(x ), tkl = [−Akl (x, x )T (x ) + Aklmn (x, x )mn (x ) V

+C (x, x )γmn (x )] dv(x), klmn mkl = [−Blk (x, x )T (x ) + Blkmn (x, x )γmn (x ) V

+C (x, x )mn (x )] dv(x ), mnlk qk = kkl (x, x )T,l (x ) dv(x ), V

(13.10.2)

∗ , absorbing the inessential constraints T where we dropped the asterisk (∗) on kkl 0 into kkl . 8 Eringen [1965], [1972c], [1973].

13.10 Nonlocal Micropolar Elasticity

297

The strain measures are given by kl = ul,k + lkm φm ,

γkl = φk,l .

(13.10.3)

For homogeneous media, the material moduli depend on x and x through x − x , and for the isotropic media, the constitutive equations reduce to

ρ0 C(|x − x |)T (x ) + β0 (|x − x |)rr (x )] dv(x ), T V 0 tkl = {[−β0 (|x − x |)T (x ) + λ(|x − x |)rr (x )] δkl

ρ0 η =

V

+ [µ(|x − x |) + κ(|x − x |)kl (x ) + µ(|x − x |)lk (x )]} dv(x ), α(|x − x |)γrr (x )δkl + β(|x − x |)γkl (x ) mkl = V

+ γ (|x − x |)γlk (x )] dv(x ), qk = k(|x − x |)T,k (x ) dv(x ).

(13.10.4)

V

B. Field Equations For the linear theory, the density ρ = ρ0 is considered constant. In the case of isotropic media, jkl = j0 δkl so that equation (13.5.34a), of the conservation of microinertia, is satisfied. Therefore, we need to substitute (13.10.4) into equation (13.5.35a) of the balance of momentum, (13.5.36a) of the moment of momentum, and (13.5.37a) of energy. Balance of Momentum ∂ 2 ul tkl,k + ρ fl − 2 = 0 ∂t Balance of Moment of Momentum ∂ 2φ mkl,k + lmn tmn + ρ ll − j0 2 = 0 ∂t

in V − σ.

in V − σ.

(13.10.5)

(13.10.6)

Energy Balance The localized form of the energy balance law follows from (13.9.7): ∂η −ρT0 (13.10.7) + qk,k + ρh = 0, ∂t where, in the spirit of the linear theory, the nonlinear terms are also ignored. The field equations are obtained by carrying (13.10.3) into (13.10.4), and the resulting

298


equations into the balance laws. Here we give the explicit forms for the isotropic solids ∂ {[−β0 (|x − x |)T + λ(|x − x |)u r,r ]δkl + [µ(|x − x |) + κ(|x − x |)] ∂xk V ∂ 2 ul

× [u l,k + lkm φm ] + µ(|x − x |)(u k,l + klm φm )} dv + ρ fl − 2 = 0, ∂t ∂

[α(|x − x |)φr,r δkl + β(|x − x |)φk,l + γ (|x − x |)φl,k ] dv

∂xk V ∂ 2 φl + κ(|x − x |)(lmn u n,m − 2φ,l ) dv + ρ ll − j0 2 = 0, ∂t V

∂u ∂T r,r + β0 T0 dv

ρC(|x − x |) − ∂t ∂t V ∂ + k(|x − x |)T,k dv + ρh = 0. (13.10.8) ∂xk V Using the identity ∂f (|x − x |) ∂ ∂g g(x ) = − (fg) + f , ∂x k ∂xk ∂xk and the Green–Gauss theorem to convert volume integrals to surface integrals, we can express (13.10.8) in the forms

−s tl + [−β0 T,k + (λ + µ)u k,kl + (µ + κ)u l,kk + κlkm φm,k ] dv

V ∂ 2 ul +ρ fl − 2 = 0 in V − σ, ∂t

−s ml + [(α + β)φk,kl + γ φl,kk + κlkm u m,k − 2κφl ] dv

V ∂ 2 φl +ρ ll − j0 2 = 0 in V − σ, ∂t

∂u r,r ∂T

− β0 T0 −ρC dv

+ kT,kk −s q + ∂t ∂t V + ρh = 0 in V − σ, (13.10.9) where s tl , s ml , and s q denote the surface loads defined by

[−β0 T + λrr δkl + (µ + κ)kl + µlk ]n k da , s tl ≡ ∂V m ≡ (αγrr δkl + βγkl + γ γlk )n k da , s l ∂ V kT,k da . (13.10.10) sq = ∂V

13.10 Nonlocal Micropolar Elasticity

299

Boundary Conditions Let V denote a regular region of the Euclidean space occupied by a micropolar body whose boundary is ∂V. The interior of V is denoted by V and the exterior normal to ∂V is n. Let Si (i = 1, 2, . . . , 6) denote the subsets of ∂V such that S 1 ∪ S2 = S 3 ∪ S4 = S 5 ∪ S6 = ∂V, S1 ∩ S2 = S3 ∩ S4 = S3 ∩ S6 = 0. The boundary conditions on these surfaces, at the time interval T + = [0, ∞), may be expressed as uk = uˆ k

on S 1 × T + ,

tkl nk = tˆl

on S2 × T + ,

φ = φˆ k

on S 3 × T + ,

mkl nk = m ˆl

on S4 × T + ,

T = Tˆ

on S 5 × T + ,

qk nk = qˆ

on S6 × T + ,

(13.10.11)

where quantities carrying a caret (ˆ) are prescribed l , tˆl = tl + F

l , l = ml + L m

. qˆ = q + H

(13.10.12)

may be ignored. F, L, and H However, for cohesion of the order r ≥ 10−7 cm, Initial Conditions The initial conditions usually consist of the Cauchy data u(x, 0) = u0 (x), φ (x, 0) = φ 0 (x), T (x, 0) = T 0 (x) in V,

˙ u(x, 0) = v0 (x), φ˙ (x, 0) = ν 0 (x), (13.10.13)

where quantities carrying a superscript (0) are prescribed throughout V. A large class of mixed boundary value problems requires solving the field equations (13.10.8) or (13.10.9) under the foregoing boundary and initial conditions. As discussed before, the cohesive regions of intermolecular forces are short, i.e., the intermolecular forces attenuate very rapidly with distance (cf. Section 6.1). We called this the attenuating neighborhood hypothesis. Accepting this also for the micropolar bodies, we can take β0 (|x − x |) β0

λ(|x − x |) µ(|x − x |) κ(|x − x |) = = λ0 µ0 κ0 β(|x − x |) γ (|x − x |) α(|x − x |) = = = α0 β0 γ0 k(|x − x |) C(|x − x |) = = a(|x − x |), (13.10.14) = C0 k0 =

where β 0 , λ0 , . . . , k are constants and a(|x − x |) is a Dirac-delta sequence. Such an identity is probably valid with separate Dirac-delta sequences for (tkl , mkl ) and

300


(qk , η), since the cohesive zones and the attenuations are different for each. Under (13.10.14) the constitutive equations take simpler forms η= tkl = mkl =

V

V

qk =

V V

a(|x − x |) c η(x ) dv , a(|x − x |) c tkl (x ) dv , a(|x − x |) c mkl (x ) dv , a(|x − x |) c qk (x ) dv .

(13.10.15)

The field equations then become ∂ 2 uk = 0, a(|x − x |) c tkl,k (x ) dv + ρ fl − ∂t 2 V −s ml + a(|x − x |)[c mkl,k (x ) + lmn c tmn ] dv

V ∂ 2φ + ρ ll − j0 2 = 0, ∂t ∂ η c −s q + a(|x − x |) −ρT0 + c qk,k dv + ρh = 0, (13.10.16) ∂t V

−s tl +

where c tkl , c mkl , ηc , and c qk denote the classical (local) constitutive equations c tkl

= −β 0 T δkl + λ0 rr δkl + (µ0 + κ0 )kl + µ0 lk ,

c mkl = α0 γrr δkl + β0 γkl + γ0 γlk , c qk = k0 T,k , cη

=

β 1 C0 T + 0 rr , T0 ρ

(13.10.17)

and the surface loads (13.10.10) read s tl = s mk

∂ V

=

sq

=

∂V ∂V

a(|x − x |) c tkl (x ) dak , a(|x − x|) c mkl (x ) dak , a(|x − x |) c q(x ) dak .

(13.10.18)


301

C. Formulation by Means of Convolution Using convolution, as outlined in Section 6.4, we can replace the balance laws with new ones which include the initial conditions t ∗ tkl,k + Fl = ρul , t ∗ (mkl,k + lmn tmn ) + Ll = ρjlk φk , −ρT0 η + 1 ∗ qk,k + R = 0,

(13.10.19) (13.10.20) (13.10.21)

where ρ = ρ0 = const. and F = t ∗ ρf(x, t) + tv0 (x) + u0 (x), Ll = t ∗ ρll (x, t) + tjlk νk0 (x) + jlk φk0 (x), R = 1 ∗ ρh + ρT0 η(x, 0).

(13.10.22)

The proof of these results is straightforward, as in Section 6.4. We remind the reader that

t

1∗φ =

φ(τ ) dτ,

t

t ∗φ =

0

(t − τ )φ(τ ) dτ,

0

˙ 0) − φ(x, 0). t ∗ φ¨ = φ − t φ(x,

(13.10.23)

We state this Theorem. Let (uk , φk ) ∈ C 0,2 , (tkl , mkl , qk ) ∈ C 0,1 , and η ∈ C 0,1 , then uk , φk , tkl , mkl , qk satisfy the equations of motion and the initial conditions if and only if (13.10.19) to (13.10.21) are satisfied.

13.11 Uniqueness Theorem9 The basic equations of the nonlocal linear thermo-micropolar elasticity are: Equations of Motion tkl,k + ρfl = ρ u¨ l mkl,k + lmn tmn + ρll = ρjkl φ¨k −ρT0 η˙ + qk,k + ρh = 0 9 Not published before.

in V,

(13.11.1)

in V − σ, in V,

(13.11.2) (13.11.3)

302


Constitutive Equations tkl = {−Akl (x, x )T (x ) + Aklmn (x, x )[un,m (x ) V

+ nmr φr (x )] + Cklmn (x, x ), φm,n (x )} dv , mkl = {−Blk (x, x )T (x ) + Blkmn (x, x )φm,n (x )

(13.11.4)

V

+ Cmnlk (x, x )[un,m (x ) + nmr φr (x )]} dv , (13.11.5) qk = kkl (x, x )T,l (x ) dv , (13.11.6) V 1 1 η= C(x, x )T (x ) + Akl (x, x )[ul,k (x ) + lkr φr (x )] ρ V T0 1 + Bkl (x, x )φk,l (x )} dv . (13.11.7) ρ a = k b = 0 in the expresThe nonnegative strain energy requires that we set kklm klm sion (13.9.5) of qk . All the material moduli in these equations are symmetric functions of x and x , e.g., Akl (x, x ) = Alk (x , x). A mixed boundary-initial value problem is expressed by the boundary conditions (13.10.11) and the initial conditions (13.10.13), namely,

Boundary Conditions uk = uˆ k

on S 1 × T + ,

tkl nk = tˆl

on S2 × T + ,

φ = φˆ k

on S 3 × T + ,

mkl nk = m ˆl

on S4 × T + ,

T = Tˆ

on S 5 × T + ,

qk nk = qˆ

on S6 × T + .

(13.11.8)

Initial Conditions uk (x, 0) = u0 (x),

u˙ k (x, 0) = vk0 (x), φ˙ k (x, 0) = νk0 (x),

φk (x, 0) = φk0 (x), T (x, 0) = T 0 (x),

in V.

(13.11.9)

We assume the following continuity requirements {uk (x, t), φk (x, t)} ∈ C 1,2 , {tkl (x, t), mkl (x, t), qk (x, t)} ∈ C 1,0 , T (x, t) ∈ C 1,0 , {uˆ k , φˆ k , Tˆ , tˆl , m ˆ l , q} ˆ ∈ C 0,0 in V × T + ,

(13.11.10)

where T + = [0, ∞), and C i,j denotes the continuous ith partial derivative with respect to xk , and the continuous j th partial derivative with respect to time.


303

First we establish some useful results. Let K(s, τ ) = 21 [ρ u˙ k (s)u˙ k (τ ) + ρjkl φ˙ k (s)φ˙ l (τ )] dv, ρ

CT (s)T (τ ) + Aklmn kl (s)mn (τ ) U(s, τ ) = 21 V V T0

+ Bklmn γkl (s)γmn (τ ) + Cklmn [kl (s)γmn (τ )

+ kl (τ )γmn (s) dv dv, ρ ˙ ˙ ) + ρl(s) · φ (τ ) − h(τ )T (s) dv ρf(s) · u(τ P (s, τ ) = T0 V 1 ˙ ˙ ) + m(s) · φ (τ ) − q(τ )T (s) da. (13.11.11) + t(s) · u(τ T0 ∂V We observe that K(t, t) ≡ K(t) is the total kinetic energy of the nonlocal micropolar body, U(t, t) = U(t) is the total strain energy, and P (t, t) ≡ P (t) is the total power of the applied body and surface loads. Theorem 1.

U(t) − K(t) =

1 2

t

[P (t + s, t − s) − P (t − s, t + s)] ds

0

+ U(0, 2t) − K(0, 2t). Proof . We introduce the notation E(s, τ ) = [tkl (s)˙kl (τ ) + mkl (s)γ˙lk (τ ) + ρT (s)η(τ ˙ )] dv. V

(13.11.12)

(13.11.13)

From (13.11.11) and (13.11.4) to (13.11.7), it can be verified that ˙ E(t, t) ≡ E(t) = U(t).

(13.11.14)

Using (13.11.13), we can show that E(t − s, t + s) − E(t + s, t − s) = 2

d U(t − s, t + s). ds

(13.11.15)

By means of the equations of motion (13.11.1) to (13.11.3) and the constitutive equations (13.11.4) to (13.11.7), we evaluate E(t − s, t + s) = [tkl (t − s)˙kl (t + s) + mkl (t − s)γ˙lk (t + s) V

+ ρT (t − s)η(t ˙ + s)] dv. Using the identity fg,k = (fg),k − gf,k

(13.11.16)

304


and the Green–Gauss theorem, (13.11.16) is transformed to E(t − s, t + s) = tk (t − s)u˙ k (t + s) + mk (t − s)φ˙ k (t + s) ∂V 1 + q(t + s)T (t − s) da − [tkl,k (t − s)u˙ l (t + s) T0 V ˙ + mkl,k (t − s)φl (t + s) + klm tkl (t − s)φ˙ m (t + s) 1 ρ − qk (t + s)T,k (t − s) + h(t + s)T (t − s) dv. T0 T0 Upon substituting tkl,k and mkl,k from (13.11.1), and (13.11.2) and qk from (13.11.6), this equation becomes E(t − s, t + s) = P (t − s, t + s) − [ρ u¨ k (t − s)u˙ k (t + s) V

+ ρjkl φ¨k (t − s)φ˙ l (t + s)] dv 1 kkl T,l (t + s)T,k (t − s) dv dv.(13.11.17) − T0 V V By replacing s by −s we obtain the expression of E(t + s, t − s). Substituting these two expressions into (13.11.15) we obtain 2

d [U(t −s, t +s)−K(t −s, t +s)] = P (t −s, t +s)−P (t +s, t −s). (13.11.18) ds

Integrating this relation with respect to s from 0 to τ , we have 2U(t − τ, t + τ ) − 2K(t − τ, t + τ ) − 2U(t) + 2K(t) τ = [P (t − s, t + s) − P (t + s, t − s)] ds. (13.11.19) 0

Setting τ = t leads to the desired resut (13.11.12).

Lemma. The functionals U(t) and K(t) are given by 2U(t) = U(0) + K(0) + U(0, 2t) − K(0, 2t) t + 21 [P (t + s, t − s) − P (t − s, t + s) + 2P (s, s)] ds 0 t 1 − ds kkl T,k T,l dv dv, (13.11.20) T0 0 V V 2K(t) = U(0) + K(0) − U(0, 2t) + K(0, 2t) t − 21 [P (t + s, t − s) − P (t − s, t + s) − 2P (s, s)] ds 0 t 1 − ds kkl T,k T,l dv dv, (13.11.21) T0 0 V V


305

for all t ∈ T + [0, ∞). Proof . Recalling (13.11.14), (13.11.17) with s = 0, can be expressed as 1 kkl T,k T,l dv dv. (13.11.22) U˙ + K˙ = P (t) − T0 V V Integration gives

t

U + K = U(0) + K(0) + 0

1 P (s, s) ds − T0

t

ds 0

V

V

kkl T,k T,l , dv dv.

(13.11.23) Using this equation and (13.11.12), we obtain (13.11.20) and (13.11.21), and complete the proof. Theorem 2 (Uniqueness). If ρ is strictly positive, jkl is positive definite, kkl is positive semidefinite, and C is strictly positive (or negative), then the boundaryinitial value problem of linear nonlocal micropolar thermoelasticity has at most one solution. Proof . Suppose that the contrary is valid, and that the two solutions u(α) , φ (α) , T (α) , α = 1, 2, exist. Let u = u(1) − u(2) ,

φ = φ (1) − φ (2) ,

T = T (1) − T (2) .

(13.11.24)

Then, clearly u, φ , and T satisfy (13.11.1) to (13.11.9), with f = l = 0, h = 0, ˆ = 0, Tˆ = qˆ = 0, u0 = v0 = ν 0 = 0, T 0 = 0. This means that uˆ = tˆ = φˆ = m we have homogeneous equations and boundary and initial conditions. With these, (13.11.21) reduces to t 1 ρ(u˙ k u˙ k + jkl φ˙ k φ˙ l ) dv + ds kkl T,k T,l dv dv = 0. (13.11.25) T 0 0 V V V From the hypothesis of the theorem, it follows that u˙ = 0, φ˙ = 0 in V × T + , t ds kkl T,k T,l dv dv = 0. 0

V

V

(13.11.26)

But u and φ vanish initially, so that we have u = 0,

φ = 0.

In view of this and (13.11.26)3 , (13.11.20) reduces to ρ CT (s)T (τ ) dv dv = 0. V V T0 This implies that T = 0 on V × T + . Hence, the proof of the theorem.

(13.11.27)

306


13.12 Reciprocal Theorem10 We consider two systems (α)

ˆ (α) , q (α) , u0(α) , φ 0(α) , ν 0(α) , η0(α) }, L(α) = {f (α) , l(α) , h(α) , tˆ(α) , φˆ , m S (α) = {u(α) , φ (α) , T (α) , t(α) , m(α) , η(α) , (α) , γ (α) , q(α) }, (13.12.1) of which S (α) is the solution of the nonlocal micropolar equations corresponding to L(α) . Lemma. Let

1 (β) (β) (α) (α) tk (s)uk (τ ) + mk (s)φk (τ ) − q (α) (s)T (β) (τ ) da T0 ∂V 1 (α) (β) (β) (β) (α) (α) + ρfk (s)uk (τ ) + ρlk (s)φk (τ ) + q h (s)T,k (τ ) T 0 V ρ (α) (β) (α) (β) − h (s)T (τ ) dv − [ρ u¨ k (s)uk (τ ) T0 V (β) (α) (s, τ ) ∈ T + , α = 1, 2,(13.12.2) + ρjkl φ¨ (s)φ (τ )] dv,

W αβ (s, τ ) =

l

k

where {q, h, q k } =

t

{q(x, s), h(x, s), qk (x, s)} ds,

(13.12.3)

0

then W αβ (s, τ ) = W βα (τ, s).

(13.12.4)

The quantity W αβ (s, τ ), defined by (13.12.2), may be called the virtual work of the surface, body, thermal, and inertial loads in system L(α) , during the displacements, rotations and temperature changes of S (α) . Proof . Let U

αβ

(s, τ ) =

(α)

(β)

(α)

(β)

tkl (s)kl (τ ) + mkl (s)γlk (τ ) V 1 (α) (α) − [ρh (s) + q k,k (s)]T (β) (τ ) dv. T0

(13.12.5)

By mere substitution of (13.11.4) to (13.11.7) and (13.11.3) we find that U αβ (s, τ ) = U βα (τ, s).

(13.12.6)

Next we substitute the strain measures kl = ul,k + lkr φr , 10 Not published before.

γkl = φk,l ,

(13.12.7)

13.12 Reciprocal Theorem

307

and use the equations of motion (13.11.1) and (13.11.2) in (13.12.5) to obtain 1 (α) (β) (β) (α) (α) (β) αβ U (s, τ ) = tkl (s)ul (τ ) + mkl (s)φl (τ ) − qk (s)T (τ ) T0 V ,k (β) (α) (α) (α) − [ρ u¨ k (s) − ρfk (s)]uk (τ ) − [ρjkl φ¨k (s) 1 (α) (β) (β) (α) − ρll (s)]φl (τ ) + q k (s)T,k (τ ) T0 ρ (α) − h (s)T (β) (τ ) dv. T0

(13.12.8)

Upon using the Green–Gauss theorem we find that U αβ (s, τ ) = W αβ (s, τ ).

(13.12.9)

In view of (13.12.5) this gives (13.12.2), completing the proof.

Theorem (Reciprocal). If S (α) is a solution of the linear, nonlocal thermo-micropolar elastic system, L(α) , α = 1, 2, then 1 (1) (2) (1) (2) (1) (2) da t ∗ tk ∗ uk + mk ∗ φk − 1 ∗ q ∗ T T0 ∂V ρ (1) (2) (1) (2) (1) (2) + ρ fk ∗ uk + ρ lk ∗ φk − 1 ∗ h ∗ T dv T0 V 1 (2) (1) (2) (1) = t ∗ tk ∗ uk + mk ∗ φk − 1 ∗ q (2) ∗ T (1) da T 0 ∂V ρ (2) (1) (2) (1) (2) (1) + ρ fk ∗ uk + ρ lk ∗ φk − t ∗ 1 ∗ h ∗ T dv, (13.12.10) T0 V where ρ fk

(α)

(α) ρ lk

(α)

= t ∗ ρfk =t

(α) ∗ ρlk

0(α)

+ ρ(tvk

0(α)

+ uk

0(α) + ρjkl (tνl

+φ

),

0(α)

).

(13.12.11)

Proof . In (13.12.2) we replace s by τ and τ by t − τ and integrate it from 0 to t. This gives t W 12 (τ, t − τ ) dτ 0 1 (1) (2) (1) (2) = tk ∗ uk + mk ∗ φk − 1 ∗ q (1) ∗ T (2) da T0 ∂V 1 ρ (1) (2) (1) (2) (1) (2) + ρfk ∗ uk + ρlk ∗ φk + 1 ∗ qk ∗ Tk − 1 ∗ h(1) ∗ T (2) dv T0 T0 V (1) (2) (1) (2) − [ρ u¨ k ∗ uk + ρjkl φ¨l ∗ φk ] dv. (13.12.12) V

308


But, in view of (13.12.4), we also have

t

t

W (τ, t − τ ) = 12

0

W 21 (t − τ, τ ) dτ.

(13.12.13)

0

Taking the convolution of (13.12.12) with t and using (α)

t ∗ u¨ k

(α)

0(α)

= uk − tvk

0(α)

− uk

,

(α) 0(α) 0(α) (α) t ∗ φ¨k = φk − tνk − φk ,

we arrive at (13.12.10), and the proof of the theorem.

(13.12.14)

13.13 Variational Principles11 Theorem 1. Let K be a set of all admissible states and let S ∈ K. Define a functional λt (S) ∈ K, for each t ∈ T + , by

[Aklmn kl ∗ mn + Bklmn γkl ∗ γmn + Cklmn (kl ∗ γmn

+ kl ∗ γmn )] dv dv + 21 (ρuk ∗ uk + ρjkl φk ∗ φl ) dv V − t ∗ (tkl ∗ kl + mkl ∗ γlk ) dv − (t ∗ tkl,k + Fl ) ∗ ul dv V V − [t ∗ (mkl,k + lmn tmn ) + Ll ] ∗ φl dv V 1 − t∗ kkl gk ∗ gl dv dv 2T0 V V

− 21 t ∗ [Akl (kl ∗ T + kl ∗ T ) + Bkl (γkl ∗ T + γkl ∗ T )] dv dv V V 1 − t∗ CT ∗ T dv dv 2T0 V V 1 + t ∗ [1 ∗ qk ∗ gk + (R + 1 ∗ qk,k ) ∗ T ] dv T0 V +t ∗ tk ∗ uˆ k da + t ∗ (tk − tˆk ) ∗ uk da S1 S2 +t ∗ mk ∗ φˆ k da + t ∗ (mk − m ˆ k ) ∗ φk da S3 S4 1 1 − t∗ 1 ∗ q ∗ Tˆ da − t ∗ 1 ∗ (q − q) ˆ ∗ T da, (13.13.1) T0 T0 S5 S6

t (S) = 21 t ∗

V

V

11 Not published before.


309

where mk = mlk nl , tk = tlk nl , Fk = t ∗ ρfk + ρ(tvk0 + u0k ),

q = qk nk ,

gk = T,k ,

Lk = t ∗ ρlk + ρjlk (tνl0 + φl0 ), (13.13.2) 1 1 1 R = 1 ∗ ρh + CT (x , 0) + Akl kl (x , 0) + γkl (x , 0)] dv , ρ ρ V T0 then t (S) is stationary, i.e., for every S + S ∈ K: δt (S) =

d t (S + S) = 0 d

on K,

t ∈ T +,

(13.13.3)

if and only if S is a solution of the mixed boundary-initial value problem. Proof . Let S{u, φ , T , , γ , t, m, q, g} ∈ K an arbitrary variation in the state S. The variation of (13.13.1) reads: δt (S) = t ∗

V

V

[Aklmn mn ∗ kl + Bklmn γmn ∗ γ kl

+ Cmnkl mn ∗ γ kl + Cklmn γmn ∗ kl ] dv

+ (ρuk ∗ uk + ρjkl φk ∗ φ l ) dv − t ∗ (t kl ∗ kl + tkl ∗ kl V V + mkl ∗ γlk + mkl ∗ γ lk ) dv − [t ∗ (t kl,k ∗ ul + tkl,k ∗ ul ) + Fl ∗ ul ] dv V − {[t ∗ (mkl,k + lmn tmn ) + Ll ] ∗ φ l + t ∗ (mkl,k + lmn t mn ) ∗ φl } dv V 1

− t∗ kkl gl ∗ g k dv dv − t ∗ [Akl ( kl ∗ T + kl ∗T) T0 V V V V 1 + Bkl (γ kl ∗ T + γkl ∗ T )] dv dv − t ∗ CT ∗ T dv dv T0 V V 1 + t ∗ [1 ∗ (q k ∗ gk + qk ∗ g k ) + R ∗ T + 1 ∗ q k,k ∗ T T0 V + 1 ∗ qk,k ∗ T ] dv + t ∗ t k ∗ uˆ k da S1 +t ∗ (t k ∗ uk + tk ∗ uk − tˆk ∗ uk ) da S 2 ˆ +t ∗ mk ∗ φk da + t ∗ [(mk − m ˆ k ) ∗ φ k + mk ∗ φk ] da S3 S4 1 1 − t∗ 1 ∗ q ∗ Tˆ da − t ∗ [1 ∗ (q ∗ T + q ∗ T − qˆ ∗ T ] da = 0. T0 T 0 S5 S6

310


We evaluate t kl ∗ (ul,k + lkm φm ) dv −t ∗ t kl ∗ kl dv = −t ∗ V V = −t ∗ [(t kl ∗ ul ),k − t kl,k ∗ ul + lkm t kl ∗ φm ] dv V = −t ∗ (t l ∗ ul da + t ∗ (t kl,k ∗ ul + klm t kl ∗ φm ) dv, ∂V

V

where we used the Green–Gauss theorem. Similarly, we obtain −t ∗ mkl ∗ γlk dv = −t ∗ ml ∗ φl dv + t ∗ mkl,k ∗ φl dv, V

∂V

V

and

1 1 t ∗ (R ∗ T + 1 ∗ q k,k ∗ T + 1 ∗ qk,k ∗ T ) dv = t ∗ 1 ∗ q ∗ T da T0 T0 V ∂V 1 + t ∗ [(R + 1 ∗ qk,k ) ∗ T − 1 ∗ q k ∗ T,k ] dv. T0 V

With these, δt (S) is expressed as

δt (S) = t∗ (Aklmn mn + Cklmn γmn − Akl T ) dv − tkl ∗ kl dv V V

+ t∗ (Bklmn γmn + Cmnkl mn − Bkl T ) dv − mlk ∗ γ kl dv V V − (t ∗ tkl,k + Fl − ρul ) ∗ ul dv V − [t ∗ (mkl,k + lmn tmn ) + Ll − ρjkl φk ] ∗ φ l dv V 1 − t ∗1∗ qk − kkl gl dv ∗ g k dv T0 V V CT

−t ∗ ρ + Akl kl + Bkl γkl dv − ρη ∗ T dv T0 V V +t ∗ t k ∗ (uˆ k − uk ) da + t ∗ (tk − tˆk ) ∗ uk da S1 S2 +t ∗ mk ∗ (φˆ k − φk ) da + t ∗ (mk − m ˆ k ) ∗ φ k da S4 S3 1 1 + t ∗1∗ q ∗ (T − Tˆ ) da + t ∗ 1 ∗ (qˆ − q) ∗ T da = 0. T0 T0 S5 S6 In view of (13.10.2), and (13.10.19) to (13.10.22), it is clear that if S is a solution of the mixed problem, then δt (S) = 0. This proves the sufficiency condition.

13.14 Nonlocal Micropolar Moduli

311

The proof of necessity requires that members of S must be selected completely arbitrarily and independently from one another. Then δt (S) = 0 requires that the coefficients of each member of S must vanish separately. This completes the proof. Definition. An admissible state S is called kinematically admissible, if it meets the strain-displacement relations (13.12.7), the constitutive equations (13.11.4) to (13.11.7), and the boundary conditions on u, φ , and T , namely, u = uˆ

on S 1 × T + ,

φ = φˆ

on S 3 × T ,

T = Tˆ

on S5 × T + . (13.13.4)

Theorem 2. Let U be a set of all kinematically admissible fields. Let {u, φ , T } ∈ U and for each t ∈ T + define the functional (U ) by (U ) =

1 2

V

{t ∗ [tkl ∗ kl + mkl ∗ γlk + ρη ∗ T ]

+ ρuk ∗ uk + ρjkl φl ∗ φk − 2Fk ∗ uk − 2Lk ∗ φk } dv 1 1 t ∗ −ρη ∗ T + ∗ ρh ∗ T − 1 ∗ qk ∗ T,k dv + T 2T0 V 0 − t ∗ tˆk ∗ uk da − t ∗m ˆ k ∗ φk da S2 S4 1 + t ∗ 1 ∗ T ∗ qˆ da, (13.13.5) T0 S6 then δ(U ) = 0

over U,

t ∈ T +,

(13.13.6)

if and only if U is a field corresponding to the solution of the mixed problem. The proof is the same as in Eringen [1998, Section 5.9] or Section 6.8.

13.14 Nonlocal Micropolar Moduli12 By comparing the dispersion relations of the planes waves, in a linear nonlocal micropolar elastic solid, with those of one-dimensional lattice dynamics, we can determine some of the micropolar moduli. 12 Eringen [1975a].

312


The field equations of the linear nonlocal isotropic micropolar elastic solids are

[(λ + µ)u k,kl + (µ + κ)u l,kk + κlmk φk,m ] dv

V ∂ 2 ul = 0, + ρ fl = ∂t 2

[(α + β)φk,kl + γ φl,kk + κlkm u m,k − 2κφl

V ∂ 2 φl (13.14.1) −ρ ll − j 2 = 0. ∂t Here we are dealing with the isothermal case (no heat conduction) and for a media extending to infinity in all directions, so that the surface terms s tl = s ml = 0. In fact, we would like to obtain the dispersion relations of one-dimensional shearmicrorotation waves. Thus, let uk = u(x2 , t)δ1k ,

φk = φ(x2 , t)δ3k ,

(13.14.2)

so that the only nonvanishing components of u and φ are u1 = u and φ3 = φ, and they depend on x2 and t only. Substituting (13.14.2) into (13.14.1), we have ∞

[(µ + κ)u ,22 + κφ,2 ] dx2 − ρ u¨ = 0, −∞ ∞

(γ φ,22 − 2κφ − κu ,2 ) dx2 − ρj φ¨ = 0. (13.14.3) −∞

Introducing the Fourier transform ∞ ∞ 1 f (ζ, ω) = f (x2 , t) exp[i(ζ x2 + ωt)] dx2 dt, 2π −∞ −∞ whose inverse is given by ∞ ∞ 1 f (x2 , t) = f (ζ, ω) exp[−i(ζ x2 + ωt)] dζ dω, 2π −∞ −∞

(13.14.4)

(13.14.5)

to (13.14.3), we obtain [ρω2 − (µ + κ)ζ 2 ]u − iκζ φ = 0, iκζ u + (ρj ω2 − γ ζ 2 − 2κ)φ = 0.

(13.14.6)

The determinant of the coefficients of u and φ gives the dispersion relation ω4 − 2αω2 + β 2 = 0,

(13.14.7)


313

where 1 {[γ + (µ + κ)j ]ζ 2 + 2κ}, 2ρj ζ2 β 2 = 2 [(µ + κ)γ ζ 2 + κ(2µ + κ)]. ρ j The roots of (13.14.7) are α=

(13.14.8)

ω2 = α ± (α 2 − β 2 )1/2 ,

(13.14.9)

which gives the two branches of the dispersion curve. From the one-dimensional lattice dynamics, for a linear chain of two different atoms with masses M1 and M2 , alternatively arranged at equidistance a/2 (Figure 13.14.1), we have the same dispersion relations (13.14.7) with 1 1 α = α0 = U1

, β 2 = β02 sin2 (ζ a/2), + M1 M2 β02 ≡ 4U1

2 /M1 M2 ,

−π/a ≤ ζ ≤ π/a,

(13.14.10)

U

where is the interatomic force constant (cf. Brillouin [1946]). If the continuum theory is to give identical dispersion curves to those of the lattice dynamics, we must select α and β 2 , as given by (13.14.10). But, (13.14.8) contains three distinct material moduli µ, κ, and γ . Thus, the system is not determinate. However, we recall that µ+κ/2 corresponds to the shear modulus µ of the nonlocal elasticity, which is already determined before κ κ0 µ + = µ0 + (13.14.11) (ζ a/2)−2 sin2 (ζ a/2). 2 2 Using this, and (13.14.10) and (13.14.8), we solve for κ and γ : sin2 ξ 2 1 − 4m0 sin2 ξ 1 2κ 1 − β1 − 8m0 = ± ρj α0 1 + j0 ξ 2 1 + j0 ξ 2 j0 ξ 2 2 1/2 2 2 2 sin ξ + ξ β1 j0 + 16m0 2 , ξ κ sin2 ξ 1 j0 2γ 1 − − 2m = − κ, (13.14.12) 0 2 2 2 ρj0 α0 a ξ ρj α0 ξ ρj α0 where j0 ≡ j/a 2 ,

β1 ≡ β0 /α0 , ξ ≡ ζ a/2,

−π/2 ≤ ξ ≤ π/2.

M2

′ Vn–2

m0 ≡ (µ0 + κ0 /2)/ρα0 a 2 ,

Un–1

Vn–1

(13.14.13)

M1

Un

Figure 13.14.1. Diatomic crystal.

Vn

Vn+1

314


The inverse transforms of (13.14.12) give κ and γ . We now give an estimate of the values j0 , β1 , and m0 . For β12 from (13.14.10) we have β12 = 4r(1 + r 2 )−2 , r ≡ M1 /M2 . (13.14.14) For a linear chain of equidistant atoms M1 and M2 at intervals a/2, the center of mass x and the moment of inertia j , of the pair about x, are given by 2x/a = (1 + r)−1 ,

j0 = j/a = 41 r(1 + r)−2 .

(13.14.15)

To estimate m0 , we need the value of U1

. From the lattice dynamics of the diatomic chains, in the long wavelength limit, we have (cs /a)2 = U1

/2(M1 + M2 ), where cs is the speed of the shear waves for which we have cs2 = (µ0 + κ0 /2)/ρ. This leads to r m0 = (1 + r)−2 . (13.14.16) 2 Returning to the inversions of (13.14.12) we note that γ in (13.14.12)2 has singularity in the first term. Using the expression of κ, we find that this singularity is eliminated if β12 = 8m0 . (13.14.17) Miraculously this also turns out to be the estimated values of β12 and m0 as given by (13.14.14) and (13.14.16). With these expressions, κ and γ , given by (13.14.12), are reduced to 8κ 1 − 8j0 sin2 ξ 1 sin2 ξ 1/2 2 1 − 16j0 sin ξ 1 + k= = ± , 2µ0 + κ0 1 + j0 ξ 2 1 + j0 ξ 2 ξ2 8γ k sin2 ξ 1 1 − = − 21 j0 k, |ξ | ≤ π/2, (13.14.18) − 4j 0 (2µ0 + κ0 )a 2 ξ2 2 ξ2 where we also substituted ρj α0 = (2µ0 + κ0 )/4, obtained from (13.14.10) and the expression of U1

given just before (13.14.16). The inverse transforms of κ and γ are now given by √ π/2 2 2π a K(y) = κ(x2 ) = k(ξ ) cos(ξy) dζ, 2µ0 + κ0 0 √ π/2 2 2π 1 k(ξ ) (y) = 1− γ (x2 ) = cos(ξy) dζ (2µ0 + κ0 )a ξ2 2 0 √ − 2π j0 1 − 21 |y| [H (y + 2) − H (y − 2)] − 21 j0 K(y),(13.14.19) where k(ξ ) is given by (13.14.18)1 and H (z) is the Heaviside unit function.


315

Figure 13.14.2. Nonlocal micropolar moduli (y) and M(y). The micropolar modulus λ was already given in Section 6.9. Since λ + 2µ + κ in micropolar theory corresponds to λ + 2µ in nonlocal elasticity, we have λ + 2µ + κ 2µ + κ {(y), M(y)} = a ,a λ0 + 2µ0 + κ0 2µ0 + κ0 × 1 − 21 |y| [H (y + 2) − H (y − 2)]. (13.14.20) The determination of the remaining two moduli α and β requires additional considerations. We conjecture that they will in functional forms be similar to γ (x2 ).

Figure 13.14.3. Nonlocal micropolar modulus K(y).

316


Figure 13.14.4. Nonlocal micropolar modulus (y). After Eringen [1975a]. The nondimensional nonlocal micropolar moduli , M, K, and are depicted as functions of y in Figures 13.14.2, 13.14.3, and 13.14.4. From these, it is clear that the maxima of all moduli occur at the origin (infinite wavelength) decaying fast over two atomic distances (period of lattice). For continuum calculations, it should be permissible to employ the straightline approximations over the interval −2 ≤ y ≤ 2 (two atomic distances on both sides of an atom) as in Figure 13.14.2 and take the vanishing K and outside this interval. Alternatively, Figures 13.14.3 and 13.14.4 also suggest the use of Dirac-delta sequences of all moduli, as discussed in Section 6.9.

13.15 Propagation of Plane Waves13 We consider a nonlocal, isothermal, micropolar elastic solid of infinite extent with no applied loads. The equations of motion are mkl,k

tkl,k − ρ u¨ l = 0, + lkj tkj − ρj φ¨l = 0.

(13.15.1)

The constitutive equations are tkl = mkl =

∞

−∞ ∞ −∞

[λu r,r δkl + (µ + κ)(u l,k + lkj φj ) + µ(u k,l + klj φj )] d 3 x

(αφr,r δkl + βφk,l + γ φl,k ) d 3 x,

13 Not published previously.

(13.15.2)

13.15 Propagation of Plane Waves

317

where λ, µ, κ, α, β, and γ are functions of |x − x|, and the quantities with a prime ( ) are functions of x . The three-dimensional and one-dimensional Fourier transforms are defined by ∞ −3/2 f (ξξ , t) = (2π) f (x, t) exp(iξξ · x) d 3 x, −∞ ∞ −1/2 f (x, ω) = (2π) f (x, t) exp(iωt) dt. (13.15.3) −∞

The inverse transforms are given by ∞ f (x, t) = (2π)−3/2 f (ξξ , t) exp(−iξξ · x) d 3ξ , −∞ ∞ f(x, ω) exp(−iωt) dω. f (x, t) = (2π)−1/2

(13.15.4)

−∞

We apply the three-dimensional Fourier transform to (13.15.1) and (13.15.2) and substitute the transform of (13.15.2) into those of (13.15.1) to obtain (λ + µ)(ξξ · u)ξξ + (µ + κ)ξ 2 u + iκξξ × φ + ρ u¨ = 0, φ + ρj φ¨ = 0. (α + β)(ξξ · φ )ξξ + γ ξ 2φ + iκξξ × u + 2κφ

(13.15.5)

We take the scalar and cross products of each equation by ξ and introduce the scalar protentials (σ, τ ) and the vector potentials (U, ), defined by u = ∇ σ + ∇ × U, φ = ∇ τ + ∇ × ,

∇ · U = 0, ∇ · = 0.

(13.15.6)

c32 ξ 2 τ + ω02 τ + τ¨ = 0,

(13.15.7)

The resulting equations are c12 ξ 2 σ + σ¨ = 0, and ¨ = 0, c22 ξ 2 U + 21 ij ω02ξ × + U ¨ = 0, c42 ξ 2 + ω02 + 21 iω02ξ × U +

(13.15.8)

where λ + 2µ + κ µ+κ , c22 = , ρ ρ γ 2κ c42 = , ω02 = . ρj ρj c12 =

c32 =

α+β +γ , ρj (13.15.9)

Further, if we take the Fourier transform with respect to time, (13.15.7) gives dispersion relations for the scalar waves ω12 = c12 ξ 2 ,

ω32 = c32 ξ 2 + ω02 ,

(13.15.10)

318


and (13.15.8) gives 2

jω = 0, U − i 0ξ × (ω2 − c22 ξ 2 ) 2 i 2 = 0. U − (ω2 − ω02 − c42 ξ 2 ) ω ξ × 2 0

(13.15.11)

From (13.15.6) we have = 0. ξ ·

ξ · U = 0,

(13.15.12)

from (13.15.11) we obtain the dispersion relations for the Eliminating U and vector waves ω4 − 2(p 0 + p1 ξ 2 )ω2 + q 0 ξ 2 + q 1 ξ 4 = 0,

(13.15.13)

where 2p0 = ω02 , q 0 = ω02 c22 − j

ω02 4

2p = c22 + c42 , 1 q 1 = c22 c42 .

,

(13.15.14)

The roots of (13.15.13) gives the frequencies of the vector waves ω22 = p0 + p 1 ξ 2 − [(p 0 + p1 ξ 2 )2 − q 0 ξ 2 − q 1 ξ 4 ]1/2 , ω42 = p0 + p 1 ξ 2 + [(p 0 + p1 ξ 2 )2 − q 0 ξ 2 − q 1 ξ 4 ]1/2 .

(13.15.15)

We recall that c1 to c4 are functions of ξ . Consequently, both the scalar and vector waves are dispersive. Adopting the attenuating neighborhood hypothesis, we write 2 a(ξ ), ci2 = ci0

i = 1, 2, 3, 4,

2 ω02 = ω00 a(ξ )

(13.15.16)

2 have the same form as (13.15.9) with λ, µ, κ, α, β, and γ replaced, where ci0 respectively, by λ0 , µ0 , κ0 , α0 , β0 , and γ0 . With this the scalar wave dispersion relations become 2 = a(ξξ )ξ 2 , ω12 /c10

2 ω32 /c30 = a(ξ )[ξ 2 + (ω00 /c30 )2 ]

and the vector wave dispersion relation reads ω42 = {p0 + p1 ξ 2 ± [(p0 + p1 ξ 2 )2 − (q0 ξ 2 + q1 ξ 4 )]1/2 }a(ξξ ). ω22 where 2p00 =

2 ω00 ,

p10 =

2 c20

2 + c40 ,

q00 =

2 ω00

2 c20

(13.15.17)

(13.15.18)

j 2 2 2 + ω00 , q10 = c20 c40 . 4 (13.15.19)

13.15 Propagation of Plane Waves

319

For the discriminant of the vector waves, we have D = (p 0 + p1 ξ 2 )2 − (q 0 ξ 2 + q 1 ξ 4 ). Upon substituting from (13.15.14), we find that 2 2 2 4 2 2 D = 41 {[ω00 + (c40 − c20 )ξ 2 ]2 + j ω00 ξ }a (ξξ ) > 0.

(13.15.20)

Consequently, ω42 and ω22 are real. Thus, we have four distinct waves with frequencies ω1 to ω4 : (i) Clearly ω1 /π is the frequency of a longitudinal displacement wave which is the longitudinal acoustic branch (LA). (ii) ω3 /π is the frequency of a microrotation wave. This is also known as the longitudinal optic branch (LO). (iii) ω2 /π is the frequency of a vector wave. This propagates in a plane normal to ξ . It is called the transverse acoustic branch (TA). (iv) Finally, ω4 /π is the frequency of a vector wave coupled with the TA. This also propagates in a plane normal to ξ and U. It corresponds to a transverse optic branch (TO). In compliance with the lattice dynamics, we select π π 4 − ≤ξ ≤ , sin2 (ξ a/2), ξ 2a2 a a and sketch the dispersion relations. From (13.15.17), (13.15.18), and (13.15.21) we find that a(ξξ ) =

ω1 (0) = 0,

ω1 (π/a) = 2c10 /a, 1/2 ω00 a 2 2 ω3 (π/a) = c30 1 + , a πc30

ω3 (0) = ω00 , ω2 (0) = 0, ω4 (π/a) ω2 (π/a)

(13.15.21)

ω4 (0) = ω00 ,

2 = π ±

1 2 2 ω00

2 2 + 21 (c20 + c40 )

2 2 2 ω00 + (c40 − c20 )

π2 a2 2 2

π a2

4 + j ω00

π2 a2

1/2 1/2 (13.15.22) .

The circular frequencies ωi of the nonlocal micropolar waves versus ξ are sketched in Figure 13.15.1. Here we notice that the nonlocal theory gives a more faithful description of the dispersion relations, as compared to the local miropolar theory (see Eringen [1998, Figure 5.11.2]). For ξ = 0, the LO and TO branches have a common frequency ω00 . This is because we have chosen the kernel function a(ξ ) to be the same for all waves. It is likely that the attentuation of (λ, µ) are different than (κ, α, β, γ ), since the latter moduli arise from a nonlocal microstructure while the former from a macrostructure.

320


ωi

ω00 LO ω3(π/a)

TO

ω4(π/a) ω1(π/a)

2c10 /a LA

ω2(π/a) TA

0

π/a

ξ

Figure 13.15.1. Circular frequency versus wave number of two scalar waves (LA, LO) and two coupled vector waves (TA, TO), within Brilouin zone 0 ≤ ξ ≤ π/a.

13.16 Displacement Potentials In Section 13.15 we saw that displacement potentials in the infinite Fourier space simplify the field equations. The same simplicity is also obtained when the applied loads are present, by introducing potentials for the applied loads, so that we have u = ∇ σ + ∇ × U, φ = ∇ τ + ∇ × ,

∇ · U = 0, ∇ · = 0,

(13.16.1)

f = ∇ g + ∇ × F, l/j = ∇ h + ∇ × L,

∇ · F = 0, ∇ · L = 0.

(13.16.2)

and

When we take the inner and cross products of the Fourier transforms of the field equations (13.15.5) with ξ we obtain (13.15.7) and (13.15.8), modified by the applied load potentials, i.e., c12 ξ 2 σ + σ¨ − g = 0,

c32 ξ 2 τ + ω02 τ + τ¨ − h = 0,

(13.16.3)

13.17 Two Vector Fields

321

and ¨ − F = 0, c22 ξ 2 U + 21 ij ω02ξ × + U ¨ − L = 0, c42 ξ 2 + ω02 + 21 iω02ξ × U +

(13.16.4)

where c1 to c4 and ω0 are given by (13.15.9). From (13.16.2) we have for the Fourier transforms of the applied loads f = −iξξ g − iξξ × F,

l/j = −iξξ h − iξξ × L.

(13.16.5)

The inner and cross products of these, by iξξ , give g = iξξ · f/ξ 2 ,

h = iξξ · l/j ξ 2 ,

F = −iξξ × f/ξ 2 ,

L = −iξξ × l/j ξ 2 .

(13.16.6)

Consequently, given f and l, g, h, F, and L are fully determined. Thus, we have that the solution of the system (13.16.5) is obtained by the inverse Fourier transforms. We determine σ , τ , U, and by, (13.16.3) and (13.16.4). The displacement fields u and φ are then given by (13.16.1). Remark. It is important to note that this is possible only for the infinite domains. However, according to the attenuating neighborhood hypothesis, the cohesive regions of material moduli are a few atomic distances beyond which they vanish. Thus, if we leave a very thin shell of a few atomic distances outside the domain of the body, then the integrals over V can be considered to extend to infinity in all diretions. This then allows us again to use the Fourier transforms. When boundary loads exist, this method will not be valid.

13.17 Resolution of u and φ in Terms of Two Vector Fields14 We consider a nonlocal isotropic elastic solid in an isothermal state, occupying the entire Euclidean space E 3 , and we introduce the operators ∞ {, M, K, A, B, }ϕ = {λ, µ, κ, α, β, γ }ϕ(x , t) d 3 x , (13.17.1) −∞

∂ Xi = , ∂xi

∂ T = , ∂t

X2 = Xi Xi ,

T2 =

∂2 . ∂t 2

(13.17.2)

Then the field equations (13.14.1) may be expressed as ( + M)X(X · u) + (M + K)X2 u + KX × φ + ρf − ρT 2 u = 0, φ + ρj l − ρj T 2φ = 0.(13.17.3) (A + B)X(X · φ ) + X2φ + KX × u − 2Kφ 14 Not published before.

322


By taking the inner and cross products of each of these equations by X we obtain ρX · f ρj X · l , X·φ = − , 3 1 1 X×u = [−ρ3 (X × f) + Kρj X(X · l) + K3 (X2φ )], 2 3 1 X×φ = [KρX(X · f) − ρj 1 (X × l) + K1 (X2 u)], (13.17.4) 1 4 X·u = −

where we put 1 ≡ ( + 2M + K)X2 − ρT 2 ,

2 ≡ (M + K)X2 − ρT 2 ,

3 ≡ (A + B + )X2 − ρj T 2 − 2K,

4 ≡ X2 − ρj T 2 − 2K. (13.17.5)

Substituting (13.17.4) into (13.17.3), and setting 1 , −ρf = 1 (2 4 + K 2 ∇ 2 ) 2 2 2 , −ρj l = 3 (2 4 + K ∇ )

(13.17.6)

we obtain ∇ ∇ · 1 − K3 (∇ ∇ × 2 ), u(x, t) = 1 4 1 − [( + M)4 − K 2 ]∇ 2 ∇∇ · 2 φ (x, t) = 2 3 2 − [(A + B)2 − K ]∇ ∇ × 1 ). − K1 (∇ (13.17.7) Note that 1 to 4 are the wave operators, i.e., 1 ≡ ( + 2M + K)∇ 2 − ρ∂ 2 /∂t 2 ,

2 ≡ (M + K)∇ 2 − ρ∂ 2 /∂t 2 ,

3 ≡ (A + B + )∇ 2 − 2K − ρj ∂ 2 /∂t 2 ,

4 ≡ ∇ 2 − 2K − ρj ∂ 2 /∂t 2 , (13.17.8)

and, according to (13.17.1), the operator products and powers, such as K 2 ϕ, are meaningful, as described by ∞ ∞ κ(|y − x|) d 3 y κ(|x − y|)ϕ(x ) d 3 x . K 2 ϕ(x, t) = K(Kϕ) = −∞

−∞

The solution of (13.17.6) and (13.17.7) determines u(x, t) and φ (x, t). The representations (13.17.7) generalizes those of the micropolar elasticity (cf. Eringen [1998, p. 173]) to the nonlocal micropolar elasticity. In the static case, we set T = 0, and (13.17.6) and (13.17.7) reduce to the generalization of the Galerkin representation to nonlocal elasticity 1 −ρf = ( + 2M + K)∇ 4 [(M + K)∇ 2 − K(2M + K)] −ρj l = [(A + B + )∇ 2 − 2K][(M + K)∇ 4 2 , − K(2M + K)∇ 2 ]

(13.17.9)

13.18 Fundamental Solutions

323

and 1 − [( + M)∇ 2 u(x) = ( + 2M + K)∇ 2 (∇ 2 − 2K) ∇ ∇ · 1 − K[(A + B + )∇ 2 − 2K]∇ ∇ × 2, − K(2 + 2M + K)]∇ 2 2 − [(M + K)(A + B)∇ 2 φ (x) = (M + K)∇ [(A + B + )∇ 2 − 2K] ∇ ∇ · 2 − K( + 2M + K)∇ 2∇ × 1 . − K 2 ]∇

(13.17.10)

13.18 Fundamental Solutions The displacement and rotations of a nonlocal micropolar solid, due to a concentrated force and couple, are called the fundamental solutions. Thus, we have two such solutions, one for a concentrated force alone, and one for a concentrated couple alone. Here we discuss these problems for the static case. While the dynamics case can be treated formally by means of the Fourier transform technique, the inversions of the transforms are highly involved and cannot be carried out on a closed form solution. The three-dimensional Fourier transform is defined by ∞ ϕ(ξξ ) = (2π)−3/2 ϕ(x)e−iξξ ·x d 3 x. (13.18.1) −∞

Applying this to the Galerkin representations (13.17.9) and (13.17.10), we obtain 1 =

ρf , g 1 (ξξ )a2 (ξξ )

2 =

ρj l , g 2 (ξξ )a2 (ξξ )

(13.18.2)

where g 1 (ξξ ) = (λ0 + 2µ0 + κ0 )ξ 4 [(µ0 + κ0 )γ0 ξ 2 + κ0 (2µ0 + κ0 )], g 2 (ξξ ) = [(α0 + β0 + γ0 )ξ 2 + 2κ0 ][(µ0 + κ0 )γ0 ξ 2 + κ0 (2µ0 + κ0 )ξ 2 ],

(13.18.3)

and 1 u(ξξ ) = {(λ0 + 2µ0 + κ0 )ξ 2 (γ0 ξ 2 + 2κ0 ) 2 − [γ0 (λ0 + µ0 )ξ + κ0 (2λ0 + 2µ0 + κ0 )]ξξ (ξξ · 1 ) − κ0 [(α0 + β0 + γ0 )ξ 2 + 2κ 0 ](iξξ × 2 )}a2 (ξξ ), 2 − [(µ0 + κ0 )(α0 + β0 )ξ 2 φ (ξξ ) = {(µ0 + κ0 )ξ 2 [(α0 + β0 + γ0 )ξ 2 + 2κ0 ] + κ02 ]ξξ (ξξ · 2 ) − κ0 (λ0 + 2µ0 + κ0 )ξ 2 (iξξ × 1 )a2 (ξ ).

(13.18.4)

Substituting from (13.18.2) into (13.18.4) we see that u(ξξ ) and φ (ξξ ) do not depend on a(ξξ ). This proves that the fundamental solutions u(x) and φ (x) of the nonlocal elastostatics are identical to those of the local micropolar elastostatics. Hence, we do not need to struggle with the inversions of (13.18.4), and we just borrow

324


the fundamental solutions of the local micropolar elastostatics. These are given elsewhere (cf. Eringen [1998, Section 5.25]). The stress and couple stress tensors are then obtained through the constitutive equations (13.10.4), namely, ∞ a(|x − x |)[λ0 u r,r δkl + (µ0 + κ0 )(u l,k + lkr φr ) tkl = −∞

+ µ0 (u k,l + klr φr )] d 3 x , ∞

δkl + β0 φk,l + γ0 φl,k ) d 3 x . a(|x − x |)(α0 φr,r mkl = −∞

(13.18.5)

Here a(|x − x |) may be chosen from the atomic crystal models.

Problems 13.1 By calculating the square of the arclength (ds 2 ) in the deformed body, justify the strain measures (13.1.24) of the micromorphic continuum. 13.2 Prove the compatibility conditions (13.1.42). 13.3 Obtain expressions (13.2.11) for the time-rates of the deformation tensors. 13.4 Give the derivation of (13.2.21). 13.5 Give the derivations of (13.3.19) to (13.3.22). 13.6 Prove the theorem that states that: The invariance of the energy under each member of the Galilean group of transformations gives a balance law. 13.7 Obtain the constitutive equations of the memory-dependent nonlocal micromorphic elastic solid. 13.8 Obtain the linear constitutive equations of the nonlocal micropolar elastic solids. 13.9 Give a detailed proof of the variational principle for kinematically admissible fields. 13.10 Study the reflections of plane harmonic waves from the surface of a halfspace of the nonlocal micropolar elastic solid for: (a) the incident longitudinal wave; (b) the incident transverse optic wave. 13.11 Obtain the stress field in an infinite nonlocal micropolar elastic solid subject to a concentrated couple at the origin of coordinates (Fundamental Solution II).

14 Memory-Dependent Nonlocal Micropolar Electromagnetic Elastic Solids

14.0 Scope This chapter is concerned with the development of the constitutive equations of memory-dependent nonlocal micropolar electromagnetic (E-M) elastic solids. This is done in Section 14.1. The linear theory of the constitutive equations is obtained for anisotropic and isotropic solids. Section 14.2 displays the field equations along with the jump conditions.

14.1 Constitutive Equations of Memory-Dependent Nonlocal, Micropolar Electromagnetic Elastic Solids1 For the development of the constitutive equations, we need the expressions of energy and entropy. The energy balance law for electrically active micropolar media is obtained by adding the E-M energy wE to the energy equation (13.5.37a) with −ρ ˆ is absorbed into −ρ ˙ , i.e., −ρ ˙ + tkl akl + mkl blk + qk,k + ρh + wE = 0,

(14.1.1)

where wE is given by (4.2.7). Upon substituting wE and introducing the generalized free energy ψ by ψ = − θη + ρ −1 Ek Pk , (14.1.2) 1 This section is new, not published previously.

326


the energy equation is transformed to ˙ + θ˙ η + θ η) −ρ( ˙ + tkl akl + mkl blk + qk,k + ρh − Pk E˙k − Mk B˙ k + Jk Ek = 0. (14.1.3) Eliminating ρh, between (14.1.3) and the entropy inequality (2.2.2), and since θ > 0, inf θ = 0, we have V

1 qk θ,k θ − Pk E˙k − Mk B˙ k + Jk Ek dv ≥ 0.

˙ + ηθ˙ ) + tkl akl + mkl blk + −ρ(

(14.1.4)

For solid media, material forms of these expressions are needed. To this end, we substitute dv = (ρ0 /ρ) dV , ρ0 TKL = tkl XK,k χ lL , ρ ρ0 K = Pk XK,k , ρ EK = Ek xk,K ,

akl = C˙ KL XK,k χ lL , ρ0 MKL = mkl XK,k χ lL , ρ ρ0 MK = Mk XK,k , ρ BK = Bk xk,K ,

bkl = ˙ KL XL,l χ kK , ρ0 QK = qk XK,k , ρ ρ0 JK = Jk XK,k , ρ (14.1.5)

into (14.1.4), to obtain V

ρ ˙ + R ηθ) ˙ + R TKL C˙ KL + R MKL ˙ LK − R K E˙K − R MK B˙ K [−ρ0 ( ρ0 +R K EL xk,K XL,l vl,k + R MK BL xk,K XL,l vl,k ] dV + −ρ D ηθ˙ + D tkl akl + D mkl blk V 1 ˙ ˙ (14.1.6) + qk θ,k − D Pk Ek − D Mk Bk + Jk Ek dv ≥ 0, θ

where we put η = R η + D η, Pk = R Pk + D Pk ,

tkl = R tkl + D tkl , Mk = Mk + D Mk .

mkl = R mkl + D mkl , (14.1.7)

The inequality (14.1.6) is expressed as the sum of two integrals; one over the material volume V which represents the equilibrium part, and the other over the spatial volume V that represents the nonequilibrium part. In fact, the equilibrium part can be set equal to zero. Moreover, in a linear theory, the last two terms in the

14.1 Constitutive Equations of E-M Elastic Solids

327

first integral can be neglected so that ρ ˙ + R ηθ) ˙ + R TKL C˙ KL + R MKL ˙ LK [−ρ0 ( ρ V 0 − R K E˙K − R MK B˙ K ] dV = 0, (14.1.8) 1 −ρ D ηθ˙ + D tkl akl + D mkl blk + qk θ,k θ V ˙ ˙ − D Pk Ek − D Mk Bk + Jk Ek dv ≥ 0. (14.1.9) Both expressions may be written in compact forms, by introducing the thermostatic and thermodynamic fluxes (R J, D J) and forces (R Y, D Y): RJ

= {−ρ0 R η,

RY

= {θ, CKL , LK , EK , BK },

R TKL , R MKL ,

−R K , −R MK }, (14.1.10)

and = {−ρ D η, D tkl , D mkl , qk /θ, −D Pk , −D Mk , Jk }, ˙ k , Ek }. (14.1.11) D Y = {θ˙ , akl , blk , θ,k , E˙k , B DJ

With these, we express (14.1.8) and (14.1.9) in the forms ρ ˙ dV = 0, ˙ + R J · R Y) (−ρ0 V ρ0 and

(14.1.12)

V

D J · D Y dv

≥ 0.

(14.1.13)

Using a constitutive ersatz in the form (13.7.7), subject to the symmetry regulations (13.7.8), from (14.1.12), we derive the static parts of the constitutive equations, in the usual way, ∂F ∂F S dV , (14.1.14) + RJ = ∂ RY

V ∂ RY where F is in the form (13.7.8), i.e., F = F [R Y(X ), X ; R Y(X), X] = F [R Y(X), X; R Y(X ), X ]. From (14.1.13), we obtain the dynamic constitutive equations ∞ ∞ ∂G ∂ dτ dτ

= dv , DJ = ∂ DY V ∂ DY 0 0

(14.1.15)

(14.1.16)

where G is expressed in terms of the difference histories of D Y as a symmetric function G = G[D Y(t) (X , τ ); D Y(t) (X, τ )] = G[D Y(t) (X, τ ); D Y(t) (X , τ )]. (14.1.17)

328


From (14.1.14), we have, explicitly, ∂F ∂F S −ρ0 R η = dV , + ∂θ ∂θ

V  S  ∂F ∂F   dV , + R TKL =

∂CKL V ∂CKL ∂F ∂F S dV , + R MKL =

∂ ∂ LK V LK S ∂F ∂F dV , + −R PK = ∂Ek

V ∂EK ∂F ∂F S −R MK = dV . + ∂Bk

V ∂BK

(14.1.18)

The spatial forms are obtained from (14.1.5) by inversion, e.g., R tkl

=

ρ R TKL xk,K χ lL , ρ0

Pk =

ρ K xk,K . ρ0

In the linearization, the contributions by various quantities to the linear constitutive equations are ρ/ρ0 ∼ 1,

xk,K (δkl + uk,l )δlK , χ kK (δkl − klm φm )δlK ,

∼ {ρ R η, R tkl , R mkl , R Pk , R Mk } ≡ r J, R Y ∼ {θ, kl , γlk , Ek , Bk } ≡ r Y, θ = T0 + T , T0 = const. 0, |T | < T0 . RJ

Hence, for the linear theory, we have ∂F ∂F S −ρ R η = dv , + ∂T

V ∂T ∂F ∂F S dv , + R tkl =

∂kl V ∂kl ∂F ∂F S dv , + R mkl = ∂γlk

V ∂γlk ∂F ∂F S −Pk = dv , + ∂Ek

V ∂Ek ∂F ∂F S −R Mk = dv , + ∂Bk

V ∂Bk

(14.1.19)

(14.1.20)


329

where F is a second-degree polynomial in R Y, i.e., F = F M + F E, ρ

2F M = − CT T − Akl (T kl + T kl ) − Bkl (T γkl + T γkl ) T0

+ Aklmn kl mn + Bklmn γkl γmn + Cklmn (kl γmn + kl γmn ), 2F E = −ρλTk E (T Ek + T Ek ) − ρλTk B (T Bk + T Bk )

+ χklE Ek El + χklB Bk Bl + λEB kl (Ek Bl + Ek Bl )

B

+ λE klm (kl Em + kl Em ) + λklm (kl Bm + kl Bm )

+ λklm (γlk Em + γlk Bm ). + γlk Em ) + λklm (γlk Bm γB

γE

(14.1.21)

All constitutive moduli are symmetric functions of x and x. In addition, Aklmn , Bklmn , χklE , and χklB have the symmetries indicated, e.g., Aklmn (x , x) = Amnkl (x , x), Akl (x , x) = Akl (x, x ), . . . .

E χklE (x , x) = χlk (x, x )

(14.1.22)

Substituting F into (14.1.20), we obtain the static portion of the constitutive equations 1 1 1 η = C(x , x)T (x ) + Akl (x , x)kl (x ) + Bkl (x , x)γkl (x ) R T ρ ρ 0 0 0 V + λTk E (x , x)Ek (x ) + λTk B (x , x)Bk (x ) dv , [−Akl (x , x)T (x ) + Amnkl (x , x)mn (x ) + Cklmn (x , x)γmn (x ) R tkl = V

+ λE (x , x)Em (x ) + λB klm (x , x)Bm (x )] dv , klm [−Blk (x , x)T (x ) + Bmnlk (x , x)γmn (x ) R mkl =

V

+ Cmnlk (x , x)mn (x ) + λklm (x , x)Em (x ) + λklm (x , x)Bm (x )] dv , E

Pk = − [−ρλTk E (x , x)T (x ) + χlk (x , x)El (x ) + λEB kl (x , x)Bl (x ) γE

γB

V γE

+ λE lmk (x , x)lm (x ) + λmlk (x , x)γlm (x )] dv ,

R Mk = −

V

B

[−ρλTk B (x , x)T (x ) + χlk (x , x)Bl (x ) + λEB lk (x , x)El (x )

+ λB lmk (x , x)lm (x ) + λlmk (x , x)γml (x )] dv . γB

(14.1.23)

The dynamic parts of the constitutive equations, upon linearization, read explicitly ∞ ∞ ˙ D tkl , D mkl , qk /T0 , −D Pk , −D Mk , Jk } = dτ dτ

{−ρ D η, 0 0 ∂G ∂G ∂G ∂G ∂G ∂G ∂G dv . , × , , , , , (14.1.24) ˙ k ∂Ek V ∂ θ˙ ∂akl ∂blk ∂T,k ∂ E˙k ∂ B

330


We epxress G as a second-degree symmetric polynomial in the difference histories D Y(t) , e.g., a(t)kl (x, τ ) = akl (x, t − τ ) − akl (x, t), E(t)k (x, τ ) = Ek (x, t − τ ) − Ek (x, t). ˙ we write Excluding the dependence on θ, 2G = µklmn (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )] × [a(t)mn (x , τ ) + a(t)mn (x, τ )]

+ µab klmn (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )] × [b(t)mn (x , τ ) + b(t)mn (x, τ )]

+ µaT klm (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )]

× [T(t),m (x , τ ) + T(t),m (x, τ )] ˙

E (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )] + µaklm × [E˙(t),m (x , τ ) + E˙(t),m (x, τ )] ˙

B (x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )] + µaklm × [B˙ (t)m (x , τ ) + B˙ (t)m (x, τ )] aE

(x , τ ; x, τ )[a(t)kl (x , τ ) + a(t)kl (x, τ )] + σklm × [E(t)m (x , τ ) + E(t)m (x, τ )]

+ αklmn (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )] × [b(t)mn (x , τ ) + b(t)mn (x, τ )] bT (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )] + αklm × [T(t),m (x , τ ) + T(t),m (x, τ )] ˙

bE (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )] + αklm × [E˙(t)m (x , τ ) + E˙(t)m (x, τ )] ˙

bB (x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )] + αklm × [B˙ (t)m (x , τ ) + B˙ (t)m (x, τ )] bE

(x , τ ; x, τ )[b(t)kl (x , τ ) + b(t)kl (x, τ )] + σklm × [E(t)m (x , τ ) + E(t)m (x, τ )] + kkl (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )]

× [T(t),l (x , τ ) + T(t),l (x, τ )] ˙

TE

(x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )] + kkl × [E˙(t)l (x , τ ) + E˙(t)l (x, τ )]

(14.1.25)


331

˙

TB

+ kkl (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )] × [B˙ (t)l (x , τ ) + B˙ (t)l (x, τ )]

+ σklT E (x , τ ; x, τ )[T(t),k (x , τ ) + T(t),k (x, τ )] × [E(t)l (x , τ ) + E(t)l (x, τ )] + χkl (x , τ ; x, τ )[E˙(t)k (x , τ ) + E˙(t)k (x, τ )] × [E˙(t)l (x , τ ) + E˙(t)l (x, τ )] ˙ ˙ + χklB E (x , τ ; x, τ )[B˙ (t)k (x , τ ) + B˙ (t)k (x, τ )] × [E˙(t)l (x , τ ) + E˙(t)l (x, τ )] ˙ + χklEE (x , τ ; x, τ )[E˙(t)k (x , τ ) + E˙(t)k (x, τ )] × [E(t)l (x , τ ) + E(t)l (x, τ )] + γkl (x , τ ; x, τ )[B˙ (t)k (x , τ ) + B˙ (t)k (x, τ )] × [B˙ (t)l (x , τ ) + B˙ (t)l (x, τ )] ˙ + γklBE (x , τ ; x, τ )[B˙ (t)k (x , τ ) + B˙ (t)k (x, τ )] × [E(t)l (x , τ ) + E(t)l (x, τ )] + σkl (x , τ ; x, τ )[E(t)k (x , τ ) + E(t)k (x, τ )]

× [E(t)l (x , τ ) + E(t)l (x, τ )].

(14.1.26)

˙

T E , etc., are symmetric with regard to the The constitutive moduli µklmn , αklmn , kkl

interchange of (x , τ ) with (x, τ ). In addition, we have the usual symmetries

µklmn = µmnkl , χkl = χlk ,

αklmn = αmnkl ,

kkl = klk ,

γkl = γlk ,

σkl = σlk .

(14.1.27)

Dynamic constitutive equations follow from (14.1.24) in the same way as in the case of (11.1.11): t

t = dt [µklmn (x , t − t ; x)amn (x , t ) D kl

D mkl

=

V −∞ ab + µklmn (x , t − t ; x)bmn (x , t )

d E˙

˙

+ µaT klm (x , t − t ; x)T,m (x , t ) + µklm (x , t − t ; x)Em (x , t ) B˙ aE

+ µaklm (x , t − t ; x)B˙ m (x , t )] + σklm (x , t − t ; x)Em (x , t )] dv(x ), t

dt [µab mnlk (x , t − t ; x)amn (x , t ) V −∞ + αlkmn (x , t − t ; x)bmn (x , t ) bT bE˙ +αlkm (x , t − t ; x)T,m (x , t ) + αlkm (x , t − t ; x)E˙m (x , t ) bB˙ bE

+ αlkm (x , t − t ; x)B˙ m (x , t ) + σlkm (x , t − t ; x)Em (x , t )] dv(x ),

332


1 qk = T0

− D Pk =

− D Mk =

Jk =

t

dt

[µaT lmk (x , t − t ; x)alm (x , t ) V −∞ bT (x , t − t ; x)blm (x , t ) + αlmk T E˙

+ kkl (x , t − t ; x)T,l (x , t ) + kkl (x , t − t ; x)E˙l (x , t ) T B˙

+ kkl (x , t − t ; x)B˙ l (x , t ) + σklT E (x , t − t ; x)El (x , t )] dv(x ), t E˙ dt [µalmk (x , t − t ; x)alm (x , t ) V −∞ bE˙ + αlmk (x , t − t ; x)blm (x , t ) T E˙

+ klk (x , t − t ; x)T,l (x , t ) + χkl (x , t − t ; x)E˙l (x , t ) ˙ B˙ E˙

+ χlk (x , t − t ; x)B˙ l (x , t ) + χklEE (x , t − t ; x)El (x , t )] dv(x ), t B˙ dt [µalmk (x , t − t ; x)alm (x , t ) V −∞ bB˙ + αlmk (x , t − t ; x)blm (x , t ) ˙ ˙ T B˙

+ klk (x , t − t ; x)T,l (x , t ) + χklB E (x , t − t ; x)E˙l (x , t ) ˙ + γkl (x , t − t ; x)B˙ l (x , t ) + γklBE (x , t − t ; x)El (x , t )] dv(x ), t aE

dt [σlmk (x , t − t ; x)alm (x , t ) V −∞ bE

(x , t − t ; x)blm (x , t ) + σlmk ˙ EE + σlkT E (x , t − t ; x)T,l (x , t ) + χlk (x , t − t ; x)E˙l (x , t ) ˙ + γlkBE (x , t − t ; x)B˙ l (x , t ) +σkl (x , t − t ; x)El (x , t )] dv(x ). (14.1.28)

The complete constitutive equations consist of the sum of the static and dynamic parts tkl = R tkl + D tkl , mkl = R mkl + D mkl , η = R η, Pk = R Pk + D Pk , Mk = R Mk + D Mk , Jk .

qk , (14.1.29)

Several special cases can be obtained by dropping various elements. To do this, one uses the Dirac-delta measure: (i) Nonlocal Micropolar E-M Elastic Solids. In this case for the material moduli, we set, e.g., µklmn (x , t − t ; x) = µklmn (x , x)δ(t − t ),

(14.1.30)

(ii) Local E-M Micropolar Elastic Solids with Memory (E-M Micropolar Viscoelasticity) µklmn δ(x − x), . . . . µklmn (x , t − t ; x) =

(14.1.31)


333

(iii) Local E-M Micropolar Elastic Solids with No Memory µklmn (x , t − t ; x) = µ0klmn δ(x − x)δ(t − t), . . . .

(14.1.32)

The material moduli of the anisotropic solids are restricted by the symmetry group. The number of independent components are determined by the symmetry group. For various crystal classes these are tabulated. In the case of the E-M elastic solids, 90 classes are distinguished (cf. Kiral and Eringen [1990]). Here we discuss only isotropic solids. In this case, the group symmetry is the full group of orthogonal transformations. The polar response functions are expressed in terms of polar independent variables and axial ones by axial tensors. In addition, the E-M effects depends on the time symmetry. For example, P, E , and J are timesymmetric, while M and B are not. For isotropic materials this eliminates the dependence of P on B and of M on E . In anisotropic material, the combined polarity and time symmetry may produce the dependence of P on B, depending on the group symmetry. For homogeneous materials, material moduli depend on x and x through κ =

x − x, and for homogeneous and isotropic materials on |x − x|. The mechanical parts of the moduli, for the static constitutive equations, are given by (13.9.8). For the E-M parts we have γE

γB

B EB λTk E = λTk B = λE klm = λklm = λklm = λklm = λkl = 0,

χklE = χ E (|x − x|)δkl ,

χklB = χ B (|x − x|)δkl .

(14.1.33)

The static parts of the constitutive equations are reduced to

ρ C(|x − x|)T (x ) + β0 (|x − x|)rr (x ) dv , T V 0 {−β0 (|x − x|)T (x ) + λ(|x − x|)rr (x )δkl R tkl =

ρ Rη =

V

+ [µ(|x − x|) + κ(|x − x|)]kl (x ) + µ(|x − x|)lk (x )} dv , m = [α(|x − x|)γmm (x )δkl + β(|x − x|)γkl (x ) R kl V

+ γ (|x − x|)γlk (x )] dv , P = − χ E (|x − x|)El (x ) dv , R k V χ B (|x − x|)Bl (x ) dv . R Mk = − V

(14.1.34)

334


For isotropic media, the dynamic parts of the constitutive equations read t dt {λv (|x − x|, t − t )arr (x , t )δkl D tkl = −∞

V

+ [µv (|x − x|, t − t ) + κv (|x − x|, t − t )] × akl (x , t ) + µv (|x − x|, t − t )alk (x , t )} dv(x ), t

m = dt [αv (|x − x|, t − t )brr (x , t )δkl D kl V

−∞

+ βv (|x − x|, t − t )bkl (x , t ) + γv (|x − x|, t − t )blk (x , t ) + α bT (|x − x|, t − t )lkm T,m (x , t ) ˙ + α bE (|x − x|, t − t )lkm E˙m (x , t ) + σ bE (|x − x|, t − t )lkm Em (x , t )] dv(x ), t 1

dt [kkl (|x − x|, t − t )T,k (x , t ) qk = T0 V −∞

+ α bT (|x − x|, t − t )lmk blm (x , t ) ˙ + k T E (|x − x|, t − t )E˙k (x , t )

+ σ T E (|x − x|, t − t )Ek (x , t )] dv(x ), t ˙

dt [k T E (|x − x|, t − t )T,k (x , t ) D Pk = − V

−∞

+ χ(|x − x|, t − t )E˙k (x , t ) ˙

+ α bE (|x − x|, t − t )lik bli (x , t ) ˙

+ χ EE (|x − x|, t − t )Ek (x , t )] dv(x ), t

dt γ (|x − x|, t − t )B˙ k (x , t ) dv(x ), DM = − V −∞ t

dt [σ bE (|x − x|, t − t )lik bli (x , t ) Jk = −∞ TE

+σ

V

˙ (|x − x|, t − t )T,k (x , t ) + χ EE (|x − x|, t − t )E˙k (x , t )

+ σ (|x − x|, t − t )Ek (x , t )] dv(x ).

(14.1.35)

An observation of these equations reveals several “cross effects”: (a) A thermal gradient T,k , an electric field E , and a E˙ may produce a couple stress. Conversely, the deformation-rate tensors bkl , E , and E˙ contribute to the heat qk . (b) A temperature gradient T,k a deformation-rate tensor bkl , and E˙k may induce polarization. Conversely, T,k , bkl , and E˙k may produce current. The cross effects arising from the interactions of bkl and E˙k appear not to have been noticed before.


335

14.2 Field Equations The field equations of memory-dependent, nonlocal micropolar E-M thermoelastic solids consist of the union of the balance laws and the constitutive equations. Balance Laws Conservation of Mass ∇ ·v =0 ρ˙ + ρ∇ [ρ(v − u)] · n = 0

in V − σ, on σ.

(14.2.1a) (14.2.1b)

Conservation of Microinertia Djkl + (kpr jlp + lpr jkp )νr = 0 Dt [jkl (v − u)] · n = 0

in V − σ,

(14.2.2a)

on σ.

(14.2.2b)

Balance of Momentum tkl,k + ρ(fl − v˙l ) + FlE = 0 E l [tkl + tkl + uk Gl − ρvl (vk − uk )]nk = F

in V − σ,

(14.2.3a)

on σ.

(14.2.3b)

in V − σ,

(14.2.4a)

on σ.

(14.2.4b)

Balance of Moment of Momentum mkl,k + lmn tmn + ρ(ll − j ν˙ l ) + LE l =0 l [mkl − ρjpl νp (vk − uk )]nk = L Balance of Energy ((14.1.3)) ˙ + θ˙ η + θ η) −ρ( ˙ + tkl akl + mkl blk + qk,k − Pk E˙k − Mk B˙ k + Jk Ek + ρh = 0 in V − σ, E [(tkl + tkl + uk Gl )vl + mkl νl + qk − Sk − (ρ + ρθη + E · P 1 on σ. + 2 ρv · v + 21 ρjij νi νj + 21 E 2 + 21 B 2 )(vk − uk )]nk = H (14.2.5) E E The expressions of FlE and LE l are given by (4.2.5) and (4.2.6) (C = L ). The E , the momentum G, and the Poynting vector S are given by E-M stress tensor tkl k l , L l , and H are the surface load residuals on the (4.2.12), (4.2.13), and (4.2.16). F body to bring it to natural state (see the discussion presented in Section 2.1). In linear theory (14.2.1) and (14.2.2) are discarded.

336


The Maxwell Equations (see Section 4.1) ∇ · D − qe = 0 n · [D] = we ∇ ·B=0 n · [B] = 0 1 ∂B +∇ ×E = 0 c ∂t 1 n × [E + u × B] = 0 c 1 ∂D 1 −∇ ×H + J = 0 c ∂t c ∨

n × [H − u × D] = n × [ H ]

in V − σ, on σ, in V − σ, on σ,

(14.2.6a) (14.2.6b) (14.2.7a) (14.2.7b)

in V − σ,

(14.2.8a)

on σ,

(14.2.8b)

in V − σ,

(14.2.9a)

on σ.

(14.2.9b)

By taking the divergence of (14.2.9a) and using (14.2.6a) we obtain the equation of the conservation of charge ∂qe + ∇ · J = 0. ∂t

(14.2.10)

With the constitutive equations obtained in Section 4.1, these equations are closed.

Problems 14.1 Following a similar procedure as in Section 8.4, obtain a model for the dielectric and conduction tensors for the micropolar media. 14.2 Obtain an equation for the dielectric displacement vector D replacing (8.4.21) for the micropolar media. 14.3 Obtain the constitutive equations for the Kelvin–Voigt-type memory-dependent nonlocal E-M micropolar elastic solids. 14.4 In a linear chain, between two consecutive atoms A (with mass M), there are two consecutive equidistant atoms a (with mass m). The spring constant between two consecutive A is K, and between two consecutive a and a and A is k. Determine the dissipation relations for the harmonic waves. 14.5 Construct a quasi-continuum micropolar model based on the model of Problem 14.4. 14.6 Obtain the dispersion relations for the isotropic nonlocal micropolar model in the absence of E-M fields. 14.7 Obtain the dispersion relations for the nonlocal rigid model subject to E-M fields.

15 Nonlocal Continuum Theory of Liquid Crystals

15.0 Scope In this chapter we present a brief account of the liquid crystals. Section 15.1 contains a description of various types of liquid crystals. The balance laws of liquid crystals are the same as those of micropolar fluids (Section 15.2). The constitutive equations of nonlocal electromagnetic (E-M) liquid crystls are developed in Section 15.3. These equations differ from those of micropolar fluids in that they involve an additional variable microinertia tensor in their constitution, as displayed through the invariants listed in Section 15.4. The static and dynamic members of the constitutive equations are obtained in Section 15.4. Field equations are given in Section 15.5. With these, then, the equations developed are ready for the solution of problems in nonlocal liquid crystal theory.

15.1

Description of Liquid Crystals

There exist a large number of organic compounds that exhibit simultaneously solid and liquid behavior. For example, they display birefringence reminiscent of the crystalline phase, at the same time they flow like fluids. These substances are called liquid crystals. Liquid crystals are constituted of rod-like, disk-like, or arbitrary-shaped molecules in a fluent environment. Upon motion, the molecules take new orientations that makes the fluid anisotropic. There are many different types of liquid crystals. Liquid crystals, that possess rod-like molecules, are divided into two classes: nematic and smectic. In nematic liquid crystals, the mass centers of the molecules are distributed randomly in three dimensions. In smectics, they are arranged in equidistant planes (Figure 15.1.1).

338


Figure 15.1.1. Nematic and smectic liquid crystals.

Figure 15.1.2. Cholesteric phase.

When the rod-like elments of nematics are organized into adjacent planes, slightly rotated forming a helical structure, this is called cholesteric or chiral nematic (Figure 15.1.2). Cholesterics exhibit birefringence and optical activity. There exist other types of liquid crystals. The polymeric type, whose molecules consist of rigid bars attached to each other by flexible chains or by a single long flexible fiber with attached bars as side chains (Figure 15.1.3). Polymeric liquid crystals can be in nematic, chiral nematic, or smectic orders. There exists, still, a different class of liquid crystals with biologically important molecules, e.g., phospholipids, soap is also in this category. These are called amphiphilic molecules, which are either water seeking or water repelling (Figure 15.1.4). For a detailed description of these substances, we refer the reader to the literature. Many books have been written, here are a few of them, Collins [1990], Gray and Winsor [1974], Wright [1995], de Jeu [1980], de Gennes and Prost [1993], Chandrasekhar [1992], and Eringen [2001].

15.1 Description of Liquid Crystals

339

Figure 15.1.3. Liquid crystalline polymers: (a) main chain; (b) side chain.

Figure 15.1.4. Amphiphilic molecules: (a) soap molecules; (b) micelle; (c) vesicle; (d) phospholipid molecule.

340


Here we are interested in giving an account of the nonlocal theory of liquid crystals which was proposed by Eringen [1981]. Although the method of approach presented here can easily be extended to the microstretch type (polymeric liquid crystals having stretch degree) cf. Eringen [2001], and more complicated micromorphic (fully polymeric) cases, we shall not discuss these cases because of space limitations.

15.2 Balance Laws The balance laws of the liquid crystals are the same as those given in Section 14.2, equations (14.2.1a) to (14.2.9b). These equations are valid for all classes of liquid crystals. The main difference of liquid crystal theory is in the constitutive equations.

15.3 Constitutive Equations of Nonlocal Electromagnetic Liquid Crystals For the development of the constitutive equations, the Clausius–Duhem (C–D) inequality given by (14.1.4) is fundamental 1 ˙ + ηη) −ρ( ˙ + tkl akl + mkl blk + qk θ,k − Pk E˙k − Mk B˙ k + Jk Ek dv ≥ 0. θ V (15.3.1) The state of liquid crystals, subject to E-M fields, is determined by the characterization of the response functions Z = {, tkl , mkl , qk , Pk , Mk , Jk },

(15.3.2)

as functionals of the independent variables ˙ jkl , γkl , akl , bkl , θ,k , Ek , Bk }. Y = {ρ, θ, θ,

(15.3.3)

Here we note that the independent variable set includes the extra variables jkl and γkl . Of these, the microinertia tensor jkl is responsible for the characterization of the anisotropy which changes with the motion, as indicated by the microinertia balance law (14.2.2a). The microinertia tensor was introduced by Eringen [1964]. He showed that jkl is the same thing as the so-called order parameter, except that in Eringen’s theory of liquid crystals [1978a] jkl changes with motion. For a discussion, see Eringen [2001]. The wryness tensor γkl characterizes the solid properties of the liquid crystals. It represents the twist elasticity. We decompose the response functions Z into static and dynamic parts η = R η + D η,

tkl = R tkl + D tkl , mkl = R mkl + D mkl , qk = D qk , Pk = R Pk + D Pk , Mk = R Mk + D Mk , Jk = D Jk . (15.3.4)

15.3 Nonlocal E-M Liquid Crystals

341

where qk and Jk possess no static parts. For ψ, we propose a constitutive equation in the form of a symmetric linear functional ρψ =

V

F [ρ , R Y ; ρ, R Y; |x − x|] dv ,

(15.3.5)

= {θ, jkl , γkl , Ek , Bk }.

(15.3.6)

where R Y = R Y(x , t) and R Y(x, t)

Function F is symmetric, i.e., F [ρ , R Y ; ρ, R Y; |x − x|] = F [ρ, R Y; ρ , R Y ; |x − x |].

(15.3.7)

The C–D inequality (15.3.1) will show that ∂F D

··· + |x − x| dv dv ≥ 0. ∂|x − x| Dt V V Since D|x − x|/Dt can be varied independently throughout V, we conclude that ∂F dv = 0. (15.3.8)

− x| ∂|x V This places the conditions on F on the surface of the body, moreover it must be valid for all x. Thus, we are forced to take ∂F /∂|x − x| = 0 as obtained for fluids in Section 3.3. The independence of F from |x − x| indicates that ψ will loose nonlocality, since F can be integrated intrinsically to eliminate R Y from the argument of F . This implies that, in fact, we have a constitutive equation in the local form ψ = (ρ, θ, jkl , γkl , Ek , Bk ). (15.3.9) The time-rate of is given by ∂ ∂ ∂ ∂ ∇ ·v+ θ˙ − γrl akl + blk ∂ρ ∂θ ∂γrk ∂γlk ∂ ˙ ∂ ˙ Ek + Bk + Rkl νkl , + ∂Ek ∂Bk

˙ = −ρ

where Rkl =

∂F ∂F ∂F ∂F jrl + jrl + γlr + γrl . ∂jkr ∂jrk ∂γkr ∂γrk

(15.3.10)

(15.3.11)

In calculating ρ˙ and Djkl /Dt, we used (14.2.1a) and (14.2.2a). For γ˙kl we substituted (13.2.21). The axiom of objectivity requires that ψ will depend on jkl and γkl through their invariants. It is simple to show that Rkl = Rlk . Since νkl = −νlk the last term in (15.3.10) vanishes.

342


Substituting (15.3.10) into the C–D inequality, we will have ∂ ∂ 2 ∂ ˙ γrl akl θ + R tkl + ρ δkl + ρ −ρ R η + ∂θ ∂ρ ∂γrk V ∂ ˙ ∂ ˙ ∂ Ek − R Mk + ρ Bk blk − R Pk + ρ + R mkl − ρ ∂Ek ∂Bk ∂γlk 1 − ρ D ηθ˙ + D tkl akl + D mkl blk + qk θ,k θ

− D Pk E˙k − D Mk B˙ k + Jk Ek dv ≥ 0.

(15.3.12)

We are free to select the static constitutive equations as

Rη R tkl

=−

∂ , ∂θ

= −πδkl − ρ

∂ γrl , ∂γrk

∂ , ∂γlk ∂ , R Pk = −ρ ∂Ek ∂ . R Mk = −ρ ∂Bk R mkl

π ≡ ρ2

∂ , ∂ρ

=ρ

(15.3.13)

Here, π is the thermodynamic pressure. The second term in R tkl indicates that the liquid crystals are distorted like solids, since the wryness tensor γkl represents the twist strain. With (15.3.13), the inequality (15.3.12) is reduced to

−ρ D ηθ˙ + D tkl akl + D mkl blk +

1 qk θ,k − D Pk E˙k − D Mk B˙ k + Jk Ek − Kˆ ≥ 0, θ (15.3.14)

where V

Kˆ dv = 0.

(15.3.15)

15.4 Constitutive Equations of Nematic Liquid Crystals

343

The C–D inequality (15.3.14) has the same form as (2.3.1). Consequently, from (2.3.3) we have the constitutive equations for the dynamic parts −ρ D η = D tkl

=

D mkl

=

qk /θ = −D Pk = −D Mk = Jk =

∂ + U η, ∂ θ˙ ∂ + Uklt , ∂akl ∂ + U m, ∂blk ∂ q + Uk , ∂θ,k ∂ p + Uk , ∂ E˙k ∂ + UkM , ∂ B˙ k ∂ + UkJ , ∂Ek

(15.3.16)

where is the dissipation potential and U η , . . . , UkJ are the constitutive residuals which do not contribute to the dissipation of energy, i.e., q U η θ˙ + Uklt akl + Uklm blk + Uk θ,k + UkP E˙k + UkM B˙ k + UkJ Ek = 0. (15.3.17)

The constitutive response functions D η, D tkl , . . . , Jk vanish with the independent variables θ˙ , akl , . . . , E˙k . For the linear theory, following the Onsager postulate, the constitutive residuals are ignored. With (15.3.13) and (15.3.16), the constitutive equations are completely expressed in terms of a function ψ and a functional .

15.4 Constitutive Equations of Nematic Liquid Crystals A. Static Constitutive Equations Nematic liquid crystals possess centers of symmetry, i.e., the two ends of rodlike particles are indistinguishable. According to the axiom of objectivity, the free energy ψ must be invariant under the full group of orthogonal transformations of the spatial frame of reference. The complete set of invariants of jkl , γkl , Ek , and Bk can be read from the tables available (Eringen [1980, p. 533], Spencer [1971, p. 293]). Among these invariants tr j = JKK = const. can be excluded from this list. Moreover, we would like to consider only the invariants that contain jkl in the

344


first degree, since |jkl | is a small quantity. However, this is important directionally I1 = tr γ s ,

I2 = tr(jγγ s ),

I5 = tr(γγ 2A ), I8 = E · E ,

I6 = tr(jγγ 2A ), I9 = E · j · E , I12 = tr(γγ D E ),

I11 = B · j · B,

I3 = tr(γγ 2s ),

I4 = tr(jγγ 2s ),

I7 = tr(jγγ s γ A ), I10 = B · B, I13 = tr(jγγ D E ),

(15.4.1)

where γ S ≡ 21 (γγ + γ T ),

γ A = 21 (γγ − γ T ),

γDk = ij k γij .

(15.4.2)

The free energy is then expressed to a second-degree θ, by ρψ = = 0 (ρ) − ρη0 θ − 21 ρC0 θ 2 + 21 [B1 I12 + (B2 + B3 )I3 + 2B4 I1 I2 + 2(B5 + B6 )I4 + (B2 − B3 )I5 + 2(B5 − B6 )I6 + 4B7 I7 + E1E · E + E2E · j · E + C1 B · B + C2 B · j · B + G1 I12 + G2 I13 ],

(15.4.3)

where η0 , C0 , and Bi (i = 1, 2, . . . , 7), E1 , E2 , C1 , C2 , G1 , and G2 are material moduli. In general, they may also depend on ρ, θ, and the invariants of jkl . However, here they are considered constants. 0 is a function of ρ. The free energy (14.4.3) may be expressed as ρψ = = T + U.

(15.4.4)

Here, T is the thermal energy and U is the strain energy T = 0 (ρ) − ρη0 θ − ρC0 θ 2 , U = 21 C0 θ 2 + 21 Bij kl γij γkl + 21 Ekl Ek El + 21 Ckl Bk Bl + Gij k γij Ek ,

(15.4.5)

where Bij kl = B1 δij δkl + B2 δil δj k + B3 δik δj l + B4 (jkl δij + jij δkl ) + B5 (jil δj k + jj k δil ) + B6 (jj l δik + jik δj l ) + B7 (jj l δik − jik δj l ), Eij = Ej i = E1 δij + E2 jij , Cij = Cj i = C1 δij + C2 jij , Gij k = G1 ij k + G2 lij jlk .

(15.4.6)

15.4 Constitutive Equations of Nematic Liquid Crystals

345

Upon substituting (15.4.4) into (5.3.13) we obtain the static part of the constitutive equations:1 ∂ = η0 + C0 θ, ∂θ R tkl = −πδkl − R mkj γj l , Rη

R Pk

=−

= −Ekl El − Gij k γij ,

∂ , ∂ρ R mkl = Bij lk γij , π = ρ2

R Mk

= −Ckl Bl .

(15.4.7)

We observe that the stress tensor is nonlinear in the wryness tensor γkl . Since the couple stress R mkl also depends on γkl , we see that the twisting strains are responsible for the solid properties of liquid crystals. Moreover, these are anisotropic in character, since Bij kl depends on jkl . The anisotropy changes with the motion, and this change occurs according to equation (14.2.2a) of the microinertia conservation. We also note that polarization and magnetization are also anisotropic at each instant of motion with an evolving anisotropy with the motion. In equation (15.4.7) of R P also occurs a “cross effect" showing that the wryness tensor gives rise to polarization. This effect was recognized by Meyer [1969] in a director theory of liquid crystals, based on the Osean–Frank director theory. Sometimes, it is called the Flexoelectric effect (de Gennes and Prost [1993]). However, here, this effect is anisotropic with evolving anisotropy, where as in the director theory they are constants. For other discussions regarding the differences between the micropolar theory and the director theory the reader is referred to Eringen [2001].

B. Dynamic Constitutive Equations The dynamic portions of the constitutive equation require the composition of the ˙ the dielectric and dissipation functional . We ignore the independent variable θ, magnetic losses, so that the independent variables jkl , akl , bkl , θ,k , Ek , and Bk form the functional , with these variables being linear jkl . Here, we propose a linear functional in the sense of Friedman and Katz (Section 3.2):

2 = [αij kl (aij akl + aij akl ) + βij kl (bij bkl + bij bkl ) V

+ dij k (bij T,k + bij T,k ) + kkl (T,k T,l + T,k T,l )

(Bk Bl + Bk Bl ) + σkl (Ek El + Ek El ) + ckl

+ fkl (Ek T,l + Ek T,l ) + gij k (bij Ek + bij Ek )] dv ,

(15.4.8)

where a prime denotes the dependence on x , e.g.,

akl = akl (x , t),

αij kl = αij kl (|x − x|, j (x ); j(x)),

(15.4.9)

1 The micropolar theory of liquid crystals first appeared in Eringen [1978a]. Subsequent papers, including E-M interaction, are Eringen [1979a], [1979b], [1993a], [1994], [1997a], [1997b], [2001]. For the nonlocal theory see Eringen [1981].

346


and the material tensors are given by αij kl = α1 δij δkl + α2 δil δj k + α3 δik δj l + 21 α4 [(jkl + jkl )δij + (jij + jij )δkl ] + 21 α5 [(jil + jil )δj k + (jj k + jj k )δil ]

+ 21 α6 [(jj l + jj l )δik + (jik + jik )δj l ]

+ 21 α7 [(jj l + jj l )δik − (jik + jik )δj l ],

σij = σ1 δij + 21 σ2 (jij + jij ), gij k = g1 ij k + 21 g2 lij (jlk + jlk ).

(15.4.10)

The tensor βij kl is similar to αij kl , with αi replaced by βi . The material tensor dij k

, and f

is in the form gij k , with gi replaced by di . The material tensor kij , cij ij

have the same composition as σij , with σi replaced by ki , ci , and fi , respectively. All material moduli αi , βi , di , ki , σi , ci , fi , and gi are functions of |x − x|, e.g., αi = αi (|x − x|). Substituting , given by (15.4.8) and (15.3.16) and ignoring the constitutive residuals, we obtain the dynamic parts of the constitutive equations t = αij kl (|x − x|; j + j)aij (x , t) dv(x ), D kl V [βij lk (|x − x| ; j + j)bij (x , t) + dlkj (|x − x|; j + j)T,j (x , t) D mkl = V

+ glkj (|x − x|; j + j)Ej (x , t)] dv(x ), 1 qk = [kkl (|x − x|; j + j)T,l (x , t) + dij k (|x − x|; j + j)bij (x , t) T0 V + flk (|x − x|; j + j)El (x , t)] dv(x ), Jk = [σkl (|x − x|; j + j)El (x , t) + fkl (|x − x|; j + j)T,l (x , t) V

+ gij k (|x − x|; j + j)bij (x , t)] dv(x ).

(15.4.11)

The dynamic constitutive equations possess nonlocality. Here we see the appearance of jkl and jkl , to a first degree, in the arguments of the material tensors αij kl , βij kl , . . . , gij k . This brings nonlinearity into the dynamic constitutive equations. These moduli may also depend on ρ and T . However, further linearization is possible by taking jkl = jkl = j0kl = const. tensor and ρ = ρ0 = const. T = T0 = const. in the material moduli. In this case, then the material moduli will depend on |x − x| only, but the evolution of the anisotropy with motion is eliminated. From the dynamical constitutive equations (15.4.11), we notice several “cross physical phenomena”: (i) They are all anisotropic, with the anisotropy evolving with the motion. (ii) The temperature gradient and electric field cause couple stress.


347

(iii) Heat is generated by the electric field (Peltier effect) and the deformation rate tensor bij . (iv) Current is generated by the thermal gradient (Seebeck effect) and bij . All of these effects are nonlocal. This means that even distant electric fields create heat, and distant thermal gradients cause a flow of current. An instantaneous degree of ordering (jkl ) changes anisotropy except in the linear case. It can be shown that, in some special cases, from these constitutive equations, we can derive director-type constitutive equations. For these, and other aspects of ˙ and the theory, we refer the reader to Eringen [2001]. Upon substituting for using (15.4.7) the energy equation (14.2.5a) reduces to −ρθ R η˙ + D tkl akl + D mkl blk + qk,k + Jk Ek + ρh = 0.

(15.4.12)

This is the equation of heat conduction.

15.5 Field Equations Field equations consist of the union of the balance laws and the constitutive equations. Two different approaches are available.

A. Field Equations Using γ and j In this case, in addition to the balance laws, we also need to employ compatibility conditions. Thus, we have the mechanical balance laws: ∇ · v = 0, ρ˙ + ρ∇ Djkl + (kpr jlp + lpr jkp )νr = 0, Dt Dvl (R tkl + D tkl ),k + ρ fl − + FlE = 0, Dt (R mkl + D mkl ),k + lij (R tij + D tij ) D + ρ ll − (jlk νk ) + LE l = 0, Dt ρθ

∂ 2 ∂π ∂ R mkl θ˙ − θ ∇ ·v−θ (γlr akr − blk ) ∂θ 2 ∂θ ∂θ + D tkl akl + D mkl blk + qk,k + Jk Ek + ρh = 0.

(15.5.1) (15.5.2) (15.5.3)

(15.5.4)

(15.5.5)

Compatibility Conditions kpq (2γlp,q + lmn γnp γmq ) = 0,

(15.5.6)

348


The Maxwell Equations ∇ · D − qe ∇ ·B 1 ∂B +∇ ×E c ∂t 1 ∂D 1 −∇ ×H + J c ∂t c

= 0, = 0,

(15.5.7) (15.5.8)

= 0,

(15.5.9)

= 0.

(15.5.10)

Constitutive Equations They are given by (15.4.7) and (15.4.11). The union of balance laws, compatibility conditions, and the constitutive equations give the field equations. These are integro-partial differential equations for the determination of the fields ρ, vk , jkl , γkl , νk , T , Ek , Bk , and Jk .

B. Field Equations Using Variables n and φ According to (13.1.19), χkK is expressed in terms of an angle φ(x, t) and a unit vector n. Using this expression, we may express γkl , νk , and jkl in terms of φ and n: γkl = nk φ,l + sin φnk,l + (1 − cos φ)kmn nm nn,l , ˙ k + sin φ n˙ k + (1 − cos φ)kmn nm n˙ n . νk = φn 1 2 3 jkl = jlk = J(kl) cos2 φ + J(kl) sin2 φ + J(kl) sin φ cos φ 5 6 4 + J(kl) cos φ + J(kl) sin φ + J(kl) ,

(15.5.11) (15.5.12) (15.5.13)

where Jklα are given by Jkl1 = Jij (δki δlj − 2nk ni δlj + nk nl ni nj ), Jkl2 = Jij kip lj r np nr , Jkl3 = 2Jij (kip np nl nj − kip np δlj ), Jkl4 = 2Jij (nk ni δlj − nk nl ni nj ), Jkl5 = −2Jij kip np nl nj , Jkl6 = Jij ni nj nk nl , Jij = JKL δiK δj L .

(15.5.14)

In this case, then, all field equations are expressible in terms of the independent fields ρ, φ, nk , vk , T , EK , BK . Since now there is no need for jkl , γkl , and νk (as they are expressed in terms of three variables nk and φ), the total number of field equations is reduced, discarding (15.5.2) from the list of equations and replacing γkl , νk , and jkl by (15.5.11) to (15.5.13).


349

The linearized forms of (15.5.11) to (15.5.13) are given by γkl = φk,l , νk = φ˙ k , jkl = Jkl − (Jkp lpm + Jlp kpm )φm .

(15.5.15)

These are adequate to replace the nonlinear quantities (15.5.11) to (15.5.13), in the field equations and in the constitutive equations. These results are obtained through the linearization process discussed in Section 13.1, where we set χ kK = (δkl − klm φm )δlK ,

(15.5.16)

i.e., φ = φn,

n · n = 1,

(15.5.17)

so that vector φ , which denotes a rotation vector with three components, having three degrees of freedom, as against the Oseen–Frank (director theory), which has only two degrees of freedom, namely a unit vector n only.

Problems 15.1 Carrying out all the intermediate missing steps, show how you can obtain, from expression (15.3.9) and the C–D inequality, the static constitutive equations (15.3.13). 15.2 Obtain the expressions (15.5.11) and (15.5.12). 15.3 Obtain the expressions (15.5.13) and (15.5.14). 15.4 For two-dimensional deformation, the field equations of micropolar fluids are greatly simplified if the rotations of the nematic particles take place about an axis perpendicular to a reference plane. Obtain the field equations for the static case. 15.5 In the x3 = const. plane, rod-like elements of nematic liquid crystals are oriented in the x2 -direction. A constant magnetic field is acting parallel to the x1 -axis. Determine the critical distance at which the nematic rods are rotated to become parallel to the x1 -axis. Ignore any dyamic effects. 15.6 The constitutive equations of liquid crystals are nonlinear since they contain directional effects arising from the dependence on jkl and the stress tensor involves the products mγγ . Construct a completely linear theory.


Appendix A

Here we present a brief account of the derivation of the Riemann–Christoffel curvature tensor, and the Laplacian of the stress tensor in curvilinear coordinates. For the background necessary we refer the readers, who are not familiar with tensor analysis, to Eringen [1980, Appendix C], Eringen [1971], or any books on tensor calculus. In the curvilinear coordinates x k , the metric tensor is denoted gkl (x) so that the square of the element of arclength is given by ds 2 = gkl dx k dx l .

(A.1)

The inverse g kl of gkl is the solution of g ki gil = δ kl .

(A.2)

By means of gkl and g kl we can raise and lower the indices of any tensor to obtain the covariant, mixed, and contravariant components, e.g., Akl = g ki Ail ,

Akl = g kj Al j ,

Akl = gj l Akj ,

Akl = gik Ail .

(A.3)

In orthogonal curvilinear coordinates, gkl and g kl possess only three components each, and they are related to each other by gk k =

1 , gkk

gkl = 0,

k = l,

(A.4)

352

Appendix A

where the underbars denote the suspension of the summation on repeated indices. If zk denotes the Cartesian coordinates with base vectors ik , and x k the curvilinear coordinates with base vectors gk (x), the partial derivative of a vector u(z) in rectangular coordinates read ∂u ∂(uk ik ) ∂uk = = ik . ∂zl ∂zl ∂zl However, in the curvilinear coordinates x k , we have ∂u ∂(uk gk ) ∂uk ∂gk = = gk + uk , ∂x l ∂xl ∂x l ∂xl but

∂gk ∂ = l ∂x l ∂x

∂zn in ∂x k

=

∂ 2 zn in . ∂x k ∂x l

Upon replacing in = (∂x k /∂zn )gk , we obtain ∂gk i = g, l kl i ∂x i where is the Christoffel symbol of the second kind defined by kl

i kl

=

∂ 2 zn ∂x i . ∂x k ∂x l ∂zn

(A.5)

(A.6)

The Christoffel symbol of the second kind is related to the first kind [ i j, k ] by r k [ i j, k ] = gkr , = g kr [ i j, r ]. (A.7) ij ij Using (A.6) we can express both symbols in terms of a metric tensor ∂gj k ∂gij ∂gik [ i j, k ] = 21 . + − ∂x j ∂x i ∂x k

(A.8)

Using (A.5) we can express the partial derivative of a vector in curvilinear coordinates by ∂u ∂ = i (uk gk ) = ∂x i ∂x

∂uk k + uj gk ≡ uk;i gk , ij ∂x i

where uk;i , defined by

uk;i

=

uk,i

k + ij

uj ,

(A.9)

Appendix A

353

is called the covariant derivative of a contravariant vector. The covariant derivative of a covariant vector is similarly obtained j uk;i = uk,i − uj . (A.10) ki Covariant derivatives are tensors and can be covariant differentiated further. Thus, for example, covariant derivatives of a second-order tensor are given by treating each index as a vector index, e.g., k l jl kl kl A + Akj , A ;i ≡ A ,i + ij ij j k j k k k A l;i ≡ A l,i − Aj + A l, li ij j j Akl;i ≡ Akl,i − Aj l − Akj . (A.11) ki li It is simple to verify that the covariant derivatives of metric tensors vanish, i.e., gkl;i = g kl;i = 0.

(A.12)

Consequently, we raise and lower the indices of (A.11) by using metric tensors, e.g., kj kj Akl;i = A ;i gj l , A ;i = Akl;i g j l . (A.13) Covariant derivatives of higher-order tensors follow the same rules displayed in the compositions of (A.11).

I. Riemann–Christoffel Curvature Tensor From calculus we know that the order of mixed partial derivatives is not important, i.e., ∂ 2φ ∂ 2φ = j i. i j ∂x ∂x ∂x ∂x The question then arises: Under what conditions does the second-order covariant partial derivative commute? For example: When can we write Ak;lm = Ak;ml ? The answer is found by forming both sides of this equation and subtracting one from the other. We have r Ak;l = Ak,l − Ar , kl

and Ak;lm

r r = (Ak;l ),m − Ar;l − Ak;r , km lm

354

Appendix A

or

r r r Ar − Ar,m − Ar,l k l ,m kl km r s r r s + As − Ak,r + As . km rl lm lm kr

Ak;lm = Akl,m −

(A.14)

Interchanging the indices l and m and subtracting we obtain Ak;lm − Ak;ml = R rklm Ar , where R rklm

=

r km

,l

r − kl

s + km ,m

r sl

(A.15)

s − kl

r . sm

(A.16)

This fourth-order tensor is called the Riemann–Christoffel curvature tensor. Clearly, R rklm is independent of the vector Ar . It is formed in terms of the metric tensor only. Hence, we have proved: Theorem. Cross covariant derivatives of any vector commute if and only if the Riemann–Christoffel tensor vanishes identically. By lowering the index r, we obtain a fourth-order tensor Rklmn = gkr R rlmn ,

(A.17)

which is known as the curvature tensor. In three dimensions the nonvanishing components of Rklmn are six: R1212 , R1313 , R2323 , R1213 , R2123 , and R3132 . In two dimensions the only nonvanishing component is R1212 . These constitute the compatibility conditions when the metric tensor is ckl or CKL . The Riemann–Christoffel curvature tensor was used by Einstein to develop his theory of general relativity.

II. Laplacian of a Tensor We wish to calculate the Laplacian of a second-order symmetric tensor. To this end, we need a second covariant derivative, e.g., k l Akl;ij = Akl,i + Arl + Akr ir ir ,j k n l nl rl nr + A ,i + A + A jn ir ir l k n kn rn kr + A ,i + A + A jn ir ir n k l kl rl kr − A ,n + . (A.18) A + A ij nr nr

Appendix A

355

The Laplacian of a contravariant tensor Akl is given by ∇ 2 Akl = Akl;ij g ij .

(A.19)

Consequently, in orthogonal curvilinear coordinates, we have 1 k l 2 kl ∇ A = Akl,i + Arl + Akr ir ir gii ,i i n l k Arl + Anr + Anl,i + ir ir in l k n + Akn,i + Arn + Akr in ir ir n k l − Akl,n + Arl + Akr . ii nr nr

(A.20)

(k) (l) :

We must now replace Akl by its physical components, given by A (k) √ (l) / gk k gl l .

Akl = A

(A.21) (k) (l)

With this then, (A.20) gives the physical components of the Laplacian (∇ 2 A) in the orthogonal curvilinear coordinates 1 A(k)(l) √ (k) 2 (∇ A) (l) = gk k gl l √ gii gk k gl,l n r i

+

k ir

(r) A (l)

(k) A (r)

+

l ir

(k) A (r)

,i

1 k √ √ grr gl l gk k grr ,i gii i n (n) (r) (n) A (l) A (l) A (r) n l × + + √ √ √ ir ir gnn gl l grr gl l gnn grr ,i (k) (r) A (n) A (n) 1 l k + + √ √ ir gii i n gk k gnn grr gnn

+

,i

n i r √gk k grr (k) (r) A (l) A (l) 1 n k + − √ √ nr gii i i gk k gl l grr gl l (k) A (r) l , + √ nr gk k grr +

(A.22)

where the summation is suspended on the indices k and l. For the indices i, n, and r the conventional summation applies.

356

Appendix A

In orthogonal curvilinear coordinates we have ds 2 = g11 (dx 1 )2 + g22 (dx 2 )2 + g33 (dx 3 )2 , 1 gk k = , gk k 1 ∂gk k ∂ √ l k = = l (ln gk k ), , kk kl 2gl l ∂x l ∂x ∂ √ k k = k (ln gk k ), = 0, k = l = m. kk lm ∂x

(A.23)

For example, in the cylindrical coordinates (r, θ, z), we have 1 g11 = g 11 = g33 = g 33 = 1, g22 = 22 = r 2 , g 1 2 2 1 k = = , = −r all other = 0. 12 21 22 lm r

(A.24)

Using these in (A.22), we calculate the Laplacian of the stress tensor t kl in cylindrical coordinates 4 ∂trθ 2 − 2 (trr − tθ θ ), r 2 ∂θ r 4 ∂trθ 2 = ∇ 2 tθ θ + 2 + 2 (trr − tθ θ ), r ∂θ r 4 2 ∂ = ∇ 2 trθ − 2 trθ + 2 (trr − tθ θ ), r r ∂θ 1 2 ∂tθ z 2 = ∇ trz − 2 trz − 2 , r r ∂θ 1 2 ∂trz = ∇ 2 tθ z − 2 tθ z + 2 r r ∂θ = ∇ 2 tzz ,

(∇ 2 t)rr = ∇ 2 trr − (∇ 2 t)θ θ (∇ 2 t)rθ (∇ 2 t)rz (∇ 2 t)θ z (∇ 2 t)zz where

(A.25)

1 ∂ 2f ∂ 2f 1 ∂f ∂ 2f + + + . (A.26) ∂r 2 r ∂r r 2 ∂θ 2 ∂z2 These equations were given by Povstenko [1995] without their derivations. ∇2f =

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Index

Acceleration field, 262 Admissibility, Axiom of, 37 Admissible thermoelastic state, 91 Airy’s stress function, 126 Alfven velocity, 221 Alfven waves, 220–221 Ampère’s law, 50 Amphiphilic molecules, 338, 339 Angular momentum, balance of, 56 Anomalous skin effect, 209 Antiplane strain, 127–128 Approximate models in nonlocal linear elasticity, 98–105 Atomic models, matching dispersion curves with, 98–100 Attenuating neighborhood hypothesis, 34–35, 178, 197, 233 Axial vector, 6 Balance of angular momentum, 56 of energy, 15–24, 56 of moment of momentum, 22, 274 of momentum, 22, 56, 274

of momentum moments, 22, 274 Balance laws electromagnetic, 49–53 of liquid crystals, 340 local, 22 mechanical, 56–58, 347 of microcontinuum mechanics, 272 of micromorphic continua, 271–274 of micropolar continua, 275–276 of microstretch continua, 274–275 nonlocal, see Nonlocal balance laws of nonlocal linear electromagnetic theory, 194 for thermomechanics, 18–21 Body loads, 15–16 Bold brackets, 17 Born–Kármán lattice model, 98–99 Bravais lattice, 79 Burger’s vector, 106

366

Index

Cartesian base vectors, 2 Cartesian coordinates, 1 Cauchy data, 84 Cauchy deformation tensor, 5 Cauchy–Green deformation tensors, right and left, 6 Causality, Axiom of, 32–33 Channel flow in nonlocal fluid dynamics, 181–184 Characteristic determinant, 7 Characteristic lengths, viii internal, 77 Chiral nematics, 338 Cholesterics, 338 Christoffel symbol, 352 Classical field theories, v Classical stress field, 144 Cohesive distance, 22 Cohesive shear stress, 141 Cohesive zone, 34 Compatibility conditions, 12–13 in nonlocal microcontinua, 260–261 Concentrated force, nonlocal elastic half-plane under, 166–171 Conservation of magnetic flux, 50 of mass, 22, 56, 265, 271 of microinertia, 265, 271 of microstretch inertia, 274 Constitutive axioms, 31–38 Constitutive-dependent variables, 33 Constitutive equations, vii, 31–47 for electromagnetic solid media, 199–200 of isotropic solids, 85–86 linear, 73–77 of memory-dependent nonlocal electromagnetic elastic solids, 62–66 of memory-dependent nonlocal electromagnetic thermoviscous fluids, 66–70

of memory-dependent nonlocal micromorphic elastic solids, 278–285 of memory-dependent nonlocal micropolar elastic solids, 293–296 of memory-dependent nonlocal micropolar electromagnetic elastic solids, 325–334 of memory-dependent nonlocal microstretch elastic solids, 286–293 of memory-dependent nonlocal thermoelastic solids, 38–43 of memory-dependent nonlocal thermofluids, 43–47 of nematic liquid crystals, 343– 347 of nonlocal electromagnetic liquid crystals, 340–343 of nonlocal fluid dynamics, 178–179 of nonlocal linear thermo-micropolar elasticity, 302 of nonlocal micropolar elasticity, 296–297 of rigid media, 242–243 Constitutive residual, 27 Continuity equation for incompressible fluids, 185–186 Continuity requirements, 88 Continuous distribution of dislocations, 123–132 Continuous media, theory of, 2 Continuum field theories, nonlocal, see Nonlocal continuum field theories Continuum mechanics, v Convolution, 86 nonlocal micropolar elasticity formulation by means of, 301 Convolution product, 93 Cosserat elasticity, 257n

Index

Couple, 53–57 at discontinuity surface, 55 Couple stress tensor, 269, 270 Covariant derivatives, 353 Cracks, interaction of dislocations with, 144–153 Debye continuum, 81 Debye frequency, 99 Debye screening, 204 Defects interaction between dislocations and, 153–161 interaction energy between, 156– 160 kinematical tensor of, 154 Deformation gradients, 2 Deformation-rate tensors, 9, 25, 263– 264 Deformation tensor, 258 Determinism, Axiom of, 33 Dielectric tensor, 200 models for, 200–204 Difference histories, 41, 282–283 Dilatation center, 159 Dipoles, force, 153 Director theories, vii Disclination, 161 straight wedge, 161–163 Discontinuity surface couple at, 55 electromagnetic force at, 55 power at, 55 small, 155 Dislocation density, true, 123 Dislocation free zone, 108, 111 Dislocations, 153 continuous distribution of, 123– 132 interaction of with cracks, 144– 153 with defects, 153–161 stress fields for special distributions of, 128–132

367

Dispersion curves, matching, with atomic models, 98– 100 Displacement potentials in nonlocal microcontinua, 320–321 Displacement vector, 73 Dissipation, postulate of, 42 Dissipation function density, 195 Dissipation inequality, 40 Dissipation potential, 26–29, 41, 46, 280 E-M, see Electromagnetic entries Eddy currents, 209 Edge dislocation, 112–116, see also Dislocations along line segment, 128 Eigenvalues, 7 Elastic distortion, 123 Elastic poles, 155 Elastic-solid dielectric tensor model, 201–203 Elasticity, nonlocal, ix Electromagnetic (E-M) balance laws, 49–53 Electromagnetic fields, absence of, 242 Electromagnetic force, 53–57 at discontinuity surface, 55 Electromagnetic solid media, constitutive equations for, 199–200 Electromagnetic theory, v nonlocal, 49–58 Energy, 23 Energy balance, 15–24, 56 principle of, 267 Energy balance law, v, 15–24 Equipresence, Axiom of, 33 Eulerian strain tensor, 6 Exponential-integral function, 162 Extrinsic body loads, 15–16 Fading Memory, Axiom of, 36–37 Fading-memory hypothesis, 36, 233

368

Index

Faraday’s law, 50 Field equations of memory-dependent nonlocal electromagnetic elastic solids, 243–245, 252 of memory-dependent nonlocal electromagnetic thermoviscous fluids, 234–235 of memory-dependent nonlocal micropolar electromagnetic elastic solids, 335–336 of nematic liquid crystals, 347– 349 of nonlocal fluid dynamics, 179– 181 of nonlocal linear elasticity, 82– 87 of nonlocal micropolar elasticity, 297–300 Finger deformation tensor, 5 Finite microrotation tensor, 257 Force dipoles, 153 Fracture criterion, 129 Free charge density, 54 Function space, 38 Fundamental solution, 165–166 Fundamental solutions in nonlocal microcontinua, 323–324 Galerkin representation, 171 generalization of, 322 Gauss’ law, 50 Gradient theories, vii Green deformation tensor, 5 Green function of linear differential operator, 103– 105 nonlocal, 126 Green–Gauss theorem, 17 Griffith crack, 132 hoop stress near tip of, 136, 137 nonlocal stress field at, 132–137 Gyrotropic media, 210–212 Hall current, 28

Heat input, 23 Helmholtz free energy, 25 Hookean stress, 100 Hookean stress components, 135 Hooke’s law, 81 Hoop stress near tip of Griffith crack, 136, 137 Hydrodynamic lubrication problem, 184 Ideal fluids, 47 Incompatibility tensor, 124 Incompressible fluids, 181 continuity equation for, 185–186 Inertia in nonlocal microcontinua, 264–266 Influence function, 34 Interaction energy between defects, 156–160 Interatomic attractions, viii Internal characteristic length, 77 Internal energy density, 16 Inverse microdeformation tensor, 256 Inverse motion, 2 Isotropic media, material moduli for, 227 Isotropic micropolar media, 295–296 Isotropic microstretch media, 291– 293 Isotropic solids, 75–77 constitutive equations of, 85–86 material moduli for, 285 Jump conditions, 23, 194 Jump discontinuities, 55 Kelvin problem, 165 Kelvin–Voigt model, 227–228 Kernel function, 76–77 Kinematical tensor of defects, 154 Kinematically admissible states, 91, 311 Kinematics, 254–261 Kinetic energy, in nonlocal microcontinua, 264–266

Index

Kinetic energy density per unit mass, 266 LA (longitudinal acoustic branch), 319 Lagrange’s equation, 79 Lagrangian strain tensor, 6 Laplacian of tensors, 354–356 Lattice dynamical foundations of linear elasticity, 78–82 Left stretch tensor, 6 Limited nonlocality, vii Line crack subject to shear, 138–143 Line distribution, 127 Linear chains, 100–102 Linear constitutive equations, 73–77 of memory-dependent nonlocal electromagnetic elastic solids, 237–243 of memory-dependent nonlocal electromagnetic thermoviscous fluids, 231–234 of memory-dependent nonlocal microstretch elastic solids, 287–291 of memory-dependent nonlocal thermoelastic solids, 223– 228 of micromorphic elastic solids, 281–285 of nonlocal linear electromagnetic theory, 195–198 Linear differential operator, Green function of, 103–105 Linear function space, 91 Liquid crystals balance laws of, 340 description of, 337–340 nematic, see Nematic liquid crystals nonlocal continuum theory of, 337–349 polymeric, 338 LO (longitudinal optic branch), 319 Local balance laws, 22 Local media, 242

369

with memory, 242 Local theory of superconductivity, 218–219 Localization, 21 London depth, 212 London equation first, 212 second, 213 London gauge, 213 Longitudinal acoustic branch (LA), 319 Longitudinal optic branch (LO), 319 Lubricant film flow on rotating disk, 189–192 Lubrication, in microscopic channels, 184–189 Lubrication problem, hydrodynamic, 184 Macromotion, 254 Magnetic flux, conservation of, 50 Magnetic vector potential, 219 Magnetization vector, 54 Magnetoelectric effect, 239 nonlocal, 197 Magnetohydrodynamic (MHD) waves, 220–221 Mass conservation of, 22, 56, 265, 271 in nonlocal microcontinua, 264–266 Mass density, 43 Mass density residuals, 18 Material derivative, 8, 262 Material frame-indifferent quantities, 10 Material Invariance, Axiom of, 33–34 Material moduli, 75 for isotropic media, 227 for isotropic solids, 285 Material particles, 254–255 Material points of body, vii Material stability, 82 Material tensors, 38

370

Index

Maxwell equations, 50, 336 Mechanical balance laws, 56–58, 347 Mechanical variables, independent, 32 Media with absorption dielectric tensor model, 203 gyrotropic, 210–212 isotropic, see Isotropic media isotropic micropolar, 295–296 isotropic microstretch, 291–293 local, see Local media Meissner experiments, 212 Memory Axiom of, 35–36 local media with, 242 nonlocal elastic solids without, 42–43 nonlocal electromagnetic fluids without, 251–252 nonlocal electromagnetic solids without, 65–66 nonlocal media without, 242 nonlocal thermoviscous fluids without, 69–70 thermoviscous fluids without, 47 Memory-dependence, vii Memory-dependent nonlocal electromagnetic elastic solids, 237– 245, 247–252 constitutive equations of, 62–66, 247–252 field equations of, 243–245, 252 linear constitutive equations of, 237–243 Memory-dependent nonlocal electromagnetic thermoviscous fluids, 231–235 constitutive equations of, 66–70 field equations of, 234–235 linear constitutive equations of, 231–234 Memory-dependent nonlocal micromorphic elastic solids, con-

stitutive equations of, 278– 285 Memory-dependent nonlocal micropolar elastic solids, constitutive equations of, 293– 296 Memory-dependent nonlocal micropolar electromagnetic elastic solids, 325–336 constitutive equations of, 325– 334 field equations of, 335–336 Memory-dependent nonlocal microstretch elastic solids constitutive equations of, 286–293 linear constitutive equations of, 287–291 Memory-dependent nonlocal Peltier effect, 251 Memory-dependent nonlocal Seebeck effect, 251 Memory-dependent nonlocal thermoelastic solids, 223–229 boundary-initial value problems of, 228–229 constitutive equations of, 38–43 linear constitutive equations of, 223–228 Memory-dependent nonlocal thermofluids, constitutive equations of, 43–47 Memory functionals, 29 MHD (magnetohydrodynamic) waves, 220–221 Microcontinuum mechanics, balance law of, 272 Microdeformation tensor, 256, 258 Microelements, 201 Microgyration tensor, 262 Microinertia, conservation of, 265, 271 Microinertia tensors, 265 Micromorphic continua, 255–256

Index

balance laws of, 271–274 strain measures of, 258–259 Micromorphic elastic solids, linear constitutive equations of, 281– 285 Micromotion, 254 Micropolar continua, 257 balance laws of, 275–276 strain measures of, 260 Micropolar media, isotropic, 295–296 Micropolar moduli, nonlocal, 311– 316 Microscopic channels, lubrication in, 184–189 Microstress moments, 267 Microstress tensor, 268 Microstretch continua, 257 balance laws of, 274–275 strain measures of, 259–260 Microstretch inertia, conservation of, 274 Microstretch media, isotropic, 291– 293 Microstretch rotary inertia, 266 Microstretch scaler inertia, 266 Mixed boundary-initial value problem, general, 235 Moment of momentum, balance of, 22, 274 Momentum, balance of, 22, 56, 274 Momentum moments, balance of, 22, 274 Motion, 1–5 inverse, 2 Navier equations, 100 Neighborhood, Axiom of, 34–35 Nematic liquid crystals, 337–338 constitutive equations of, 343– 347 field equations of, 347–349 Nondimensional shear stress, 121– 122 Nonlocal balance laws, reduction of, 23–24

371

Nonlocal continuum field theories, v defined, vii lattice dynamical foundations of, 78–82 literature on, ix–x Nonlocal continuum theory of liquid crystals, 337–349 Nonlocal elastic half-plane under concentrated force, 166–171 Nonlocal elastic half-space, rigid stamp on, 171–175 Nonlocal elastic solids, 38 without memory, 42–43 Nonlocal elasticity, ix Nonlocal electromagnetic fluids without memory, 251–252 Nonlocal electromagnetic liquid crystals, constitutive equations of, 340–343 Nonlocal electromagnetic solids, without memory, 65–66 Nonlocal electromagnetic theory, 49– 58 Nonlocal fluid dynamics, 177–192 channel flow in, 181–184 constitutive equations of, 178– 179 field equations of, 179–181 Nonlocal Green function, 126 Nonlocal hexagonal elastic solids, screw dislocation in, 116– 123 Nonlocal linear elasticity, 71–175 approximate models in, 98–105 field equations of, 82–87 uniqueness theorem of, 87–91 Nonlocal linear electromagnetic theory, 193–221 balance laws of, 194 linear constitutive equations of, 195–198 point charge in, 204 Nonlocal linear thermo-micropolar elasticity, 301–305

372

Index

constitutive equations of, 302 uniqueness theorem for, 305 Nonlocal magnetoelectric effect, 197 Nonlocal media with no memory, 242 Nonlocal microcontinua, 253–324 compatibility conditions in, 260– 261 displacement potentials in, 320– 321 fundamental solutions in, 323– 324 inertia in, 264–266 kinetic energy in, 264–266 mass in, 264–266 propagation of plane waves in, 316–320 reciprocal theorem for, 306–308 variational principles for, 308– 311 Nonlocal micropolar elasticity, 296– 301 boundary conditions, 299 constitutive equations of, 296– 297 field equations of, 297–300 formulation by means of convolution, 301 initial conditions, 299–300 Nonlocal micropolar moduli, 311– 316 Nonlocal Peltier effect, 197 memory-dependent, 251 Nonlocal pyroelectricity, 197 Nonlocal residuals, 271 nature of, 21–22 Nonlocal Seebeck effect, 197 memory-dependent, 251 Nonlocal solid media with absorption dielectric tensor model, 204 Nonlocal stress field at Griffith crack, 132–137

Nonlocal theory of superconductivity, 214–220 Nonlocal thermoviscous fluids without memory, 69–70 Nonlocality, v concept of, viii limited, vii Nonsimple materials of gradient type, 34 Normed distance, 36 Objective tensors, 10–12 Objectivity, 11 Axiom of, 33 Onsager postulate, 28, 224 Onsager reciprocity relations, 27 Optical activity, 211 Optical waves, 205–206 Oscillator model, 201 Peach–Koehler formula, 126 Peltier effect, nonlocal, see Nonlocal Peltier effect Perfect body, 24 Perfect continuum, 24 Permutation symbols, 3 Phospholipids, 338 Piezoelectric effect, 239 Piezoelectricity, Voigt’s, 238 Piezomagnetic effect, 239 Piola deformation tensor, 5 Pippard’s theory of superconductivity, 213, 219–220 Plane strain, 126–127 Plane waves, propagation of, in nonlocal microcontinua, 316– 320 Point charge in nonlocal linear electromagnetic theory, 204 Point defects, 153–156, see also Defects Poiseuille flow profile, 182 Polaritons, 206–209 Polarization vector, 54 Poles, elastic, 155

Index

Polymeric liquid crystals, 338 Power, 53–57 at discontinuity surface, 55 Power and energy theorem, 92–93 Poynting vectors, 54 Pyroelectricity, nonlocal, 197 Pyromagnetic effect, 197 Quasi-continuum, defined, 155–156 Quasi-continuum representation, 79 Reciprocal theorem, 93–94 for nonlocal microcontinua, 306– 308 References, 357–364 Response functionals, vii Response objects, vii Riemann–Christoffel curvature tensor, derivation of, 351–354 Right stretch tensor, 6 Rigid body motions, restrictions for, 37–38 Rigid body susceptibility, 206 Rigid media, constitutive equations of, 242–243 Rigid stamp on nonlocal elastic halfspace, 171–175 Rotating disk, lubricant film flow on, 189–192 Screw dislocation(s), 106–112, see also Dislocations along line segments, 129 distribution of, 109–110 in half-plane, 110–111 in nonlocal hexagonal elastic solids, 116–123 uniform distribution of, along circle, 130–132 Second law of thermodynamics, 24– 26, 57–58, 276–278 Second sound, 28 Seebeck effect, nonlocal, see Nonlocal Seebeck effect Shannon’s theorem, 80

373

Shear, line crack subject to, 138–143 Shear-plane model, 188 Shear stress cohesive, 141 nondimensional, 121–122 Shifters, 257 Simple lattice, 79 Slowly varying fields, 98 Smectic liquid crystals, 337–338 Smooth Memory, Axiom of, 36 Smooth neighborhood hypothesis, 34 Solution of the mixed problems, 91 Somigliana-type representation, 164–165 Spin-inertia per unit mass, 266 Spin tensor, 10 Spring-dashpot dielectric tensor model, 200–201 Stokes’ theorem, 17 Straight-edge dislocation, 112–116, see also Dislocations Straight wedge disclination, 161–163 Strain energy density, 74, 281 Strain invariants, 7 Strain measures of micromorphic continua, 258– 259 of micropolar continua, 260 of microstretch continua, 259– 260 Stress, 15–29, 267 Stress field(s) classical, 144 for special distributions of dislocations, 128–132 Stress intensity factors, critical, 150 Stress moment tensor, 268 Stress-strain relations, 32 Stress tensor, 23 Stretch, 7 Superconductivity, 212–213 local theory of, 218–219 nonlocal theory of, 214–220

374

Index

Pippard’s theory of, 213, 219– 220 Surface heat, 180 Surface loads, 15–16 Surface traction residual, 180 Symmetric function, 39 TA (transverse acoustic branch), 319 Temperature change, 73 Tensors Laplacian of, 354–356 objective, 10–12 time-rate of, 8–10, 261–264 Thermodynamic equilibrium, defined, 25–26 Thermodynamic flux, 26, 41, 62, 277 Thermodynamic force, 26, 41, 62, 277 Thermodynamic pressure, 342 Thermodynamics, second law of, 24– 26, 57–58, 276–278 Thermomechanics, balance laws for, 18–21 Thermostatic equilibrium, 277 Thermostatic flux, 62 Thermostatic force, 62 Thermoviscous fluids without memory, 47

Time-rate of tensors, 8–10, 261–264 Time-symmetric terms, 196 Titchmarch’s theorem, 86 TO (transverse optic branch), 319 Transport theorems, 17 Transverse acoustic branch (TA), 319 Transverse optic branch (TO), 319 True dislocation density, 123 Twist elasticity, 340 Uniqueness theorem for nonlocal linear elasticity, 87–91 for nonlocal linear thermomicropolar elasticity, 305 Variational principles, 95–98 for nonlocal microcontinua, 308–311 Voigt’s piezoelectricity, 238 Volterra dislocation, 115 Volume defects, 153, see also Defects Volume loads, 15–16 Vorticity vector, 10 Wryness tensor, 258

Errata

375

Errata for “Microcontinuum Field Theories I: Foundations and Solids" Page

Location

Misprint

Correction

xiii

line 17

Axisymmetric XK = Xˆ k (x, t)

Antisymmetric XK = Xˆ K (x, t)

5

Eq. (1.2.4)

10

last sentence

In order

, in order

16

Eq. (1.5.23)

ckl = j ckl

ckl = j ckl

21

Eq. (1.7.6)

· · · + lmn cpm γnq

· · · + lmn cpm γnq

22

Definition 1.

f (x, ", t)

f (X, ", t)

26

Eq. (1.8.29)

27

Eq. (1.8.33)

3ν ≡ C˙ KL = 6j 2 νδKL C˙ KL = 6j 2 νδKL

ν≡ C˙KL = 6j 2 νδKL C˙KL = 2j 2 νδKL ckl = 13 δkl φ˙ l = Qln (t)

27

Eq. (1.8.33)4

27

Eq. (1.8.34)2

28

Eq. (1.8.43)

ckl = νδkl φ˙ lk =

29

Eq. (1.9.3)

Qlm (t)

43

line 16

tklm

49

Eq. (2.3.1)

53

Line after Eq. (2.4.9)

U (Y, 0)

U (Y, ω)

55

Eq. (2.4.18-a)

[Y (t − x); · · · ]

[Y (t − s); · · · ]

55

Eq. (2.4.18-b)

P [Y (t − x); · · · ]

P [Y (t − s); · · · ]

67

Eqs. (3.3.14) & (3.3.15)

3ρj

ρj

4

ρh ∂ V −σ θ

4

mklm

ρh ∂ V −σ θ dv

86

Eq. (4.1.13-a)

(∇E) · B

(∇E) · P

86

Eq. (4.1.13-b)

W E = FE · v + ρE · · ·

94

Eq. (4.4.3)

{· · · , EK (t), BK (t)}

W E = ρE · · · {· · · , E˙K (t), B˙ K (t)}

94

Eq. (4.4.5) of R Pk

Ek

EK

94

Eq. (4.4.5) of R Mk

Bk

94–96

Eqs. (4.4.6) & (4.4.13) of D Pk

EK

BK E˙K

94–96

Eqs. (4.4.6) & (4.4.13) of D Mk

BK

B˙ K

94–96

In Section 4.4) has a new definition namely:

97, 98

= − θη + ρ −1 Mk Bk − ρ0−1 Mk Bk − ρ0−1 K EK E˙k Ek Eqs. (4.4.5) & (4.5.11) of D Pk B˙ k Eqs. (4.4.5) & (4.5.11) of D Mk Bk

97, 98

376

Errata Page

Location

Misprint

Correction

108

last line

Mαβ = Mαβ

Mαβ = Mβα

136

Theorem 2

209

Eq. (5.21.27)

{U, φ, T } √ (O c − r)

{u, φ, T } √ O( c − r)

235

Eq. (5.27.2)

un (x1 , x1 , t)

un (x1 , x2 , t)

250

Eq. (6.1.1)

j = 1 + 3φ + ·

j =1+φ+·

250

Eq. (6.1.2)

K 3γk δkK

K = γk δkK

250

Eq. (6.1.3)

γk = 3φ,k

γk = φ,k

250

Eq. (6.1.3)

e = 3φ

e=φ

253

Eq. (6.1.21)

mkl = αφr,r + · · ·

mkl = αφr,r δkl + · · ·

273

Eq. (7.1.15)

T = · · · −

ρC0 T 2 2T0

289

item (e)

λTklE

T = · · · − T 0 T 2 0 λTk E

289

item (e)

λTklB

λTk B

ρC

Errata for “Microcontinuum Field Theories II: Foundations and Solids" Page

Location

Misprint

Correction

19

Line 6

Lukaszewicz

Łukaszewicz

95

Line 7

Megneto-

289

Eqs. (17.7.18)

... − p − 21 (a1 − a2 )ν1

= 0

Magneto. . . − p + 21 (a1 − a2 )ν1

= 0

290

Eq. (17.7.23)

. . . (λ2 − a32 h2 ),

. . . (λ2 − a32 h2 ) = 0,

291

Eq. (17.7.26)

1 [(σ − κ )Q + . . . f (y) = µν +κ ν ν 1 ν

1 [(−µ + κ + (σ . . . f (y) = λ +µ ν ν ν ν ν − κν )Q1 + σν Q2 )(y 2 − 1) + . . .

291

Eq. (17.7.30)

i22 = . . .

i12 = . . .

319

[6]

Lukaszewicz

Łukaszewicz

326

[95]

Lukaszewicz

Łukaszewicz

Nonlocal Continuum Field Theories

Nonlocal Continuum Field Theories

Nonlocal Continuum Field Theories

Statistical continuum theories

Gauge-Field Theories

Gauge Field Theories

Noncommutative quantum field theories

Gauge field theories

Microcontinuum Field Theories

Gauge Field Theories

Gauge field theories

Gauge field theories

Global integrability of field theories

Conceptual Developments of 20th Century Field Theories

Conformal field theories and integrable models

Conceptual developments of 20th century field theories

Integrable Quantum Field Theories and Their Applications

Effective field theories in flavour physics

Effective Field Theories in Flavour Physics

Supersymmetric Field Theories: Geometric Structures and Dualities

Integrable quantum field theories and their applications

Microcontinuum Field Theories: II Fluent Media

Effective field theories in flavour physics

The non-linear field theories of mechanics

Conformal Field Theories and Integrable Models

Microcontinuum field theories: foundations and solids

The non-linear field theories of mechanics

Mean Field Theories and Dual Variation

Gauge Field Theories: An Introduction with Applications

Continuum

Nonlocal Continuum Field Theories