Variational Principles of Continuum Mechanics. II. Applications

Interaction of Mechanics and Mathematics For further volumes: www.springer.com/series/5395 Victor L. Berdichevsky Va...

Author: Victor Berdichevsky

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Interaction of Mechanics and Mathematics

For further volumes: www.springer.com/series/5395

Victor L. Berdichevsky

Variational Principles of Continuum Mechanics II. Applications

With 33 Figures

123

IMM Advisory Board D. Colton (USA) . R. Knops (UK) . G. DelPiero (Italy) . Z. Mroz (Poland) . M. Slemrod (USA) . S. Seelecke (USA) . L. Truskinovsky (France) IMM is promoted under the auspices of ISIMM (International Society for the Interaction of Mechanics and Mathematics).

Author V.L. Berdichevsky Professor of Mechanics Department of Mechanical Engineering Wayne State University Detroit, MI 48202 USA [email protected]

ISSN 1860-6245 e-ISSN 1860-6253 ISBN 978-3-540-88468-2 e-ISBN 978-3-540-88469-9 DOI 10.1007/978-3-540-88469-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2008942378 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Contents - II. Applications

Part III Some Applications of Variational Methods to Development of Continuum Mechanics Models 14 Theory of Elastic Plates and Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 14.1 Preliminaries from Geometry of Surfaces . . . . . . . . . . . . . . . . . . . . . . 590 14.2 Classical Shell Theory: Phenomenological Approach . . . . . . . . . . . . 598 14.3 Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 14.4 Derivation of Classical Shell Theory from Three-Dimensional Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 14.5 Short Wave Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 14.6 Refined Shell Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 14.7 Theory of Anisotropic Heterogeneous Shells . . . . . . . . . . . . . . . . . . . 665 14.8 Laminated Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 14.9 Sandwich Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 14.10 Nonlinear Theory of Hard-Skin Plates and Shells . . . . . . . . . . . . . . . 701 15 Elastic Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 15.1 Phenomenological Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 15.2 Variational Problem for Energy Density . . . . . . . . . . . . . . . . . . . . . . . 725 15.3 Asymptotic Analysis of the Energy Functional of Three-Dimensional Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 16 Some Stochastic Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 16.1 Stochastic Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 16.2 Stochastic Quadratic Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 16.3 Extreme Values of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 16.4 Probability Distribution of Energy: Gaussian Excitation . . . . . . . . . 768 16.5 Probability Distribution of Energy: Small Excitations . . . . . . . . . . . 771 16.6 Probability Distribution of Energy: Large Excitations . . . . . . . . . . . . 789 16.7 Probability Distribution of Linear Functionals of Minimizers . . . . . 797 16.8 Variational Principle for Probability Densities . . . . . . . . . . . . . . . . . . 801

v

vi

Contents - II. Applications

17 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 17.1 The Problem of Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 17.2 Homogenization of Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . 818 17.3 Some Non-asymptotic Features of Homogenization Problem . . . . . 833 17.4 Homogenization of Random Structures . . . . . . . . . . . . . . . . . . . . . . . 840 17.5 Homogenization in One-Dimensional Problems . . . . . . . . . . . . . . . . 849 17.6 A One-Dimensional Nonlinear Homogenization Problem: Spring Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856 17.7 Two-Dimensional Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862 17.8 Two-Dimensional Incompressible Elastic Composites . . . . . . . . . . . 875 17.9 Some Three-Dimensional Homogenization Problems . . . . . . . . . . . . 883 17.10 Estimates of Effective Characteristics of Random Cell Structures in Terms of that for Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . 892 18 Homogenization of Random Structures: a Closer View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899 18.1 More on Kozlov’s Cell Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899 18.2 Variational Principle for Probability Densities . . . . . . . . . . . . . . . . . . 918 18.3 Equations for Probability Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 922 18.4 Approximations of Probability Densities . . . . . . . . . . . . . . . . . . . . . . 927 18.5 The Choice of Probabilistic Measure . . . . . . . . . . . . . . . . . . . . . . . . . 931 18.6 Entropy of Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934 18.7 Temperature of Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939 18.8 Entropy of an Elastic Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943 19 Some Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961 19.1 Shallow Water Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961 19.2 Models of Heterogeneous Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . 966 19.3 A Granular Material Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976 19.4 A Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011

Contents - I. Fundamentals

Part I Fundamentals 1 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Prehistory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Mopertuis Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Euler’s Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Lagrange Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Jacobi Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Hamilton Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Hamiltonian Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Physical Meaning of the Least Action Principle . . . . . . . . . . . . . . . .

3 3 11 15 20 26 26 32 36

2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Thermodynamic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Entropy and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Gibbs Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Nonequilibrium Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Secondary Thermodynamics and Higher Order Thermodynamics . .

45 45 47 51 59 59 60 64

3 Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1 Continuum Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Basic Laws of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.3 Classical Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.4 Thermodynamic Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4 Principle of Least Action in Continuum Mechanics . . . . . . . . . . . . . . . . 117 4.1 Variation of Integral Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.2 Variations of Kinematic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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4.3 4.4 4.5 4.6

Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Variational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Models with High Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Tensor Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5 Direct Methods of Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.2 Quadratic Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.3 Existence of the Minimizing Element . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.4 Uniqueness of the Minimizing Element . . . . . . . . . . . . . . . . . . . . . . . 168 5.5 Upper and Lower Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.6 Dual Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.7 Legendre and Young-Fenchel Transformations . . . . . . . . . . . . . . . . . 181 5.8 Examples of Dual Variational Principles . . . . . . . . . . . . . . . . . . . . . . . 201 5.9 Hashin-Strikman Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . 216 5.10 Variational Problems with Constraints . . . . . . . . . . . . . . . . . . . . . . . . 224 5.11 Variational-Asymptotic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 5.12 Variational Problems and Functional Integrals . . . . . . . . . . . . . . . . . . 270 5.13 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Part II Variational Features of Classical Continuum Models 6 Statics of a Geometrically Linear Elastic Body . . . . . . . . . . . . . . . . . . . . 285 6.1 Gibbs Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 6.2 Boundedness from Below . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.3 Complementary Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.4 Reissner Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.5 Physically Linear Elastic Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.6 Castigliano Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 6.7 Hashin-Strikman Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . 306 6.8 Internal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 6.9 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 6.10 Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 6.11 Continuously Distributed Dislocations . . . . . . . . . . . . . . . . . . . . . . . . 328 7 Statics of a Geometrically Nonlinear Elastic Body . . . . . . . . . . . . . . . . . 341 7.1 Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 7.2 Gibbs Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 7.3 Dual Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 7.4 Phase Equilibrium of Elastic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . 369 8 Dynamics of Elastic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 8.1 Least Action vs Stationary Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375


8.2 8.3 8.4

ix

Nonlinear Eigenvibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Linear Vibrations: The Rayleigh Principle . . . . . . . . . . . . . . . . . . . . . 379 The Principle of Least Action in Eulerian Coordinates . . . . . . . . . . . 381

9 Ideal Incompressible Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 9.1 Least Action Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 9.2 General Features of Solutions of Momentum Equations . . . . . . . . . . 392 9.3 Variational Principles in Eulerian Coordinates . . . . . . . . . . . . . . . . . . 396 9.4 Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 9.5 Variational Features of Kinetic Energy in Vortex Flows . . . . . . . . . . 408 9.6 Dynamics of Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 9.7 Quasi-Two-Dimensional and Two-Dimensional Vortex Flows . . . . . 427 9.8 Dynamics of Vortex Filaments in Unbounded Space . . . . . . . . . . . . . 433 9.9 Vortex Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 9.10 Symmetry of the Action Functional and the Integrals of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 9.11 Variational Principles for Open Flows . . . . . . . . . . . . . . . . . . . . . . . . . 453 10 Ideal Compressible Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 10.1 Variational Principles in Lagrangian Coordinates . . . . . . . . . . . . . . . 455 10.2 General Features of Dynamics of Compressible Fluid . . . . . . . . . . . 457 10.3 Variational Principles in Eulerian Coordinates . . . . . . . . . . . . . . . . . . 461 10.4 Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 10.5 Incompressible Fluid as a Limit Case of Compressible Fluid . . . . . . 470 11 Steady Motion of Ideal Fluid and Elastic Body . . . . . . . . . . . . . . . . . . . . 473 11.1 The Kinematics of Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 11.2 Steady Motion with Impenetrable Boundaries . . . . . . . . . . . . . . . . . . 475 11.3 Open Steady Flows of Ideal Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 11.4 Two-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 11.5 Variational Principles on the Set of Equivortical Flows . . . . . . . . . . . 484 11.6 Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 11.7 Regularization of Functionals in Unbounded Domains . . . . . . . . . . . 493 12 Principle of Least Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 12.1 Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 12.2 Creeping Motion of Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 12.3 Ideal Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 12.4 Fluctuations and Variations in Steady Non-Equilibrium Processes . 505 13 Motion of Rigid Bodies in Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 13.1 Motion of a Rigid Body in Creeping Flow of Viscous Fluid . . . . . . 509 13.2 Motion of a Body in Ideal Incompressible Fluid . . . . . . . . . . . . . . . . 514 13.3 Motion of a Body in a Viscous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 521

x


Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 A. Holonomic Variational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 B. On Variational Formulation of Arbitrary Systems of Equations . . . . 538 C. A Variational Principle for Probability Density . . . . . . . . . . . . . . . . . 543 D. Lagrange Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 E. Microdynamics Yielding Classical Thermodynamics . . . . . . . . . . . . 553 Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

Chapter 14

Theory of Elastic Plates and Shells

˚ in three-dimensional space and, at each point on the surface, Consider the surface erect a segment of length h directed along the normal to the surface; the centers ˚ The segments cover some three-dimensional region, V˚ of the segments are on . (Fig. 14.1). If h is much smaller than the minimum curvature radius of the surface ˚ R, and the characteristic size of the surface , ˚ L, , h ≪ 1, R

h ≪ 1, L

then an elastic body occupying the region V˚ in its undeformed state is called an ˚ is a plane, i.e. R = ∞, then there is only one small parameter, h/L. elastic shell. If One can expect that the deformation of the elastic shells can be approximately described by functions which depend only on the two surface coordinates and time. The problem of constructing the shell theory consists of the proper choice for these functions, derivation of the governing equations for these functions and establishing the link between the two-dimensional characteristics and the three-dimensional stress state. These issues are addressed in this chapter. The last three sections of the chapter are concerned with the theory of laminated plates and shells, and, in

Fig. 14.1 Notation for shells

V.L. Berdichevsky, Variational Principles of Continuum Mechanics, Interaction of Mechanics and Mathematics, DOI 10.1007/978-3-540-88469-9 1, C Springer-Verlag Berlin Heidelberg 2009

589

590

14


particular, hard-skin plates and shells. In case of a hard skin an additional small parameter, the ratio of elastic moduli of the core and the skin, comes into play and changes the leading asymptotics. We begin with introduction of necessary facts from theory of surfaces.

14.1 Preliminaries from Geometry of Surfaces The surface tensors. Consider in three-dimensional space a two-dimensional surface , defined by the parametric equations x i = r i (ξ α ) ,

(14.1)

where ξ α are the surface parameters and the small Greek indices α, β, γ , . . . run values 1, 2. The parametric equations (14.1) contain more information than just the definition of the surface because they distinguish the individual points on the surface marked by the parameters ξ α . The specific choice of the parameters on the surface is not essential, and it appears to be necessary to consider the invariance of all relationships ′ with respect to the transformation group of the surface coordinates, ξ α → ξ α , ′ ′ ξ α = ξ α ξβ .

(14.2)

The vectors and tensors with respect to this group are called the surface vectors and tensors, and the corresponding tensor indices the surface indices. The tangent vectors and the metric tensor. The derivatives, rαi = r i /ξ α , are the components of two vectors in the observer’s frame, r1i and r2i . The three-dimensional vectors, r1i and r2i , are tangential to the surface . At the same time, for each fixed index i, they form a surface vector with respect to index α. The observer’s metrics, allowing one to measure distances in three-dimensional space, induces the surface intrinsic metrics on , which determines the distances between the points of the surface: the squared distance, ds 2 , between the points ξ α and ξ α + dξ α , j ds 2 = gi j r i (ξ α + dξ α ) − r i (ξ α ) r j (ξ α + dξ α ) − r j (ξ α ) = gi j rαi dξ α rβ dξ β ,

can be written as

ds 2 = aαβ dξ α dξ β , where the tensor j

aαβ = gi j rαi rβ , is called the surface metric tensor or the first quadratic form of the surface.

(14.3)

14.1

Preliminaries from Geometry of Surfaces

591

The contravariant components of the surface metric tensor, a αβ , are introduced as the solutions of the system of linear equations a αβ aγβ = δγα . According to (3.20), 1 a , (14.4) a aαβ % % where a is the determinant of the matrix %aαβ %. Using the space and the surface metrics, we can juggle the space and the surface indices; for example, for rαi we have a αβ =

riα ≡ gi j rαj ,

riα ≡ a αβ riβ .

(14.5)

The Levi-Civita tensor. The two-dimensional Levi-Civita tensor is defined as εαβ =

√ aeαβ

with eαβ being the two-dimensional Levi-Civita symbol (e11 = e22 = 0, e12 = −e21 = 1). By definition, eαβ = eαβ . One can check that 1 ′ ′ εαβ ≡ a αα a ββ εα′ β ′ = √ eαβ . a Note the identities εαβ εγβ = δγα ,

eαβ eγβ = δγα .

(14.6)

The normal vector. Consider a vector with the components 1 1 j n i = √ εijkr1 r2k = εαβ εijk rαj rβk . 2 a

(14.7)

Here εijk is the three-dimensional Levi-Civita tensor (see Sect. 3.1). The vector, n i , is orthogonal to the surface, since, due to (14.7), n i rαi = 0.

(14.8)

Let us show that the vector n i has the unit length g i j n i n j = 1. From (14.7) and (3.19),

(14.9)

592

14


1 ij g εikl r1k r2l εjmn r1m r2n = a 1 1 2 a11 a22 − a12 = 1. = (gkm gln − gkn glm ) r1k r2l r1m r2n = a a gi j ni n j =

So, n i are the components of the unit vector normal to the surface indeed. It follows from (14.7) and (3.19) that n i εijk = εαβ rαj rβk .

(14.10)

εi jk rαj rβk = n i εαβ .

(14.11)

Note also the relation

j

It can be obtained from the following reasoning. The scalar product of εi jk rα rβk with j r1i and r2i is zero. Hence, for each fixed α and β, εi jk rα rβk is proportional to n i , and one can write εi jk rαj rβk = cαβ n i .

(14.12)

Tensor cαβ must be antisymmetric. Therefore, cαβ = cεαβ . Contracting (14.12) with n i εαβ and using (14.7), we obtain the value of c : c = 1. The area element. The area element, dω, of the surface, , is the area of the infinitesimally small parallelogram with the sides, r1i dξ 1 and r2i dξ 2 . It is equal to j the length of the vector product of these two εijkr1 dξ 1r2k dξ 2 . The vector √ vectors, product, according to (14.7), is equal to n i adξ 1 dξ 2 . Since the normal vector, n i , has the unit length, dω =

√ adξ 1 dξ 2 .

(14.13)

The surface in the initial state. Let the position of the surface change with time and be given by the functions x i = r i (ξ α , t). The position of the surface at the ˚ and all the other quantities in the initial state will initial instant, t0 , is denoted by , ◦ be furnished with the symbol . In particular, r i (ξ α , t0 ) ≡ r˚ i (ξ α ) , % % a˚ = det %a˚ αβ % ,

a˚ αβ

˚r i j , a˚ αβ = gi j r˚αi r˚β , ξ α 1 a˚ 1 j = , n˚ i = √ εijk r˚1 r˚2k , a˚ a˚ αβ a˚ r˚αi ≡

r˚iα = gi j r˚αi ,

r˚iα = a˚ αβ r˚iβ .

(14.14)

The decomposition of Kronecker’s delta. The following identity holds: rαi r αj + n i n j = δ ij .

(14.15)

14.1


593

In order to check that, it is sufficient to project (14.15) onto the tangent vectors and j the normal vector (contract (14.15) with rα and n j ) and inspect that the resulting 1 equations are identities. The decomposition of Kronecker’s delta is used in constructing the projections onto the tangent plane and the normal direction. For example, the vector with components T i can be represented by the sum of a vector tangent to the plane and a normal vector, T i = T i δ ij = T j rαi r αj + T j n j n i = T α rαi + T n i , where T α = T i riα is a surface vector, T α rαi is the vector tangent to the surface (i.e. T α rαi n i = 0), and T = T j n j is a scalar. Similarly, the tensor of the second order can be written as

j β j T i j = T kl δki δl = T kl rαi rkα + n i n k rβ rl + n j n l = j

= T αβ rαi rβ + T1α rαi n i + T2α rαj n i + T n i n j ,

(14.16)

β

where T αβ = T kl rkα rl is the surface tensor, T1α = T kl rkα n l , T2α = T kl n k rlα are the surface vectors which coincide in the case of a symmetric tensor T i j , and T = T i j n i n j is a scalar. The first term of the sum (14.16) “lies in the tangent plane” to the surface in the sense that it is orthogonal to the normal vector with respect to both indices, the second term is orthogonal to the normal vector with respect to index i, the third term – with respect to index j, and the fourth term is “orthogonal to the tangent plane” (its contraction with the tangent vectors is equal to zero). The decomposition of Kronecker’s delta is also used in the decomposition of the gradient along the tangent and the normal directions, j = δi = rαj riα + n j n i = riα α + n i . i j j x x x ξ n

(14.17)

1

The decomposition of the Kronecker’s delta (14.15) uses the space metrics. Actually, such decomposition does not need metrics and can be done without the use of the metric properties. Indeed, consider on the surface a vector field, r3i , which is not tangent to at any point. This means that the determinant, r, of the matrix with the components r1i , r2i and r3i is not zero. Let us define three vector fields gi1 , gi2 , gi3 by the system of linear equations rαi g αj + r3i g 3j = δ ij .

(14.18)

Since r = 0, the solution of the system of equations (14.18) exists, and it is unique. Equation (14.18) is the sought decomposition of the Kronecker’s delta. Note that vector gi3 is normal to in the sense that gi3 rαi = 0.

Indeed,

gi3 rαi =

1 r r r3i

rαi =

1 j eijk r1 r2k rαi ≡ 0. r

594

14


a Here /n ≡ n j /x j is the derivative along the normal vector, i while /ξ ≡ j j rα /x is the derivative along the surface: for any function ϕ x considered on the surface, ϕ x i = ϕ r i (ξ α ) ,

i x r i (ξ α ) ϕ r i (ξ α ) = = α ϕ r i (ξ α ) . i α i x ξ r ξ

ϕ rαi

The decomposition of the Kronecker’s delta in terms of the initial state, r˚αi r˚ αj + n˚ i n˚ j = δ ij , yields similar relations. Two covariant derivatives. In the same coordinates system, ξ α , we have two metric tensors, aαβ and a˚ αβ . We introduce two Christoffel’s symbols: for the surface metrics, aαβ , γ

αβ = and the surface metrics, a˚ αβ ,

1 γδ a aαδ,β + aβδ,α − aαβ,δ , 2

1 γ ˚ αβ = a˚ γ δ a˚ αδ,β + a˚ βδ,α − a˚ αβ,δ . 2

(14.19)

(14.20)

The comma before a Greek index, α, denotes partial derivative with respect to ξ α . The corresponding covariant derivatives are denoted by the bar and the semicolon in indices. For example, for a surface vector, T α , T|βα =

T α + αλβ T λ , ξ β

T;βα =

T α + ˚ αλβ T λ , ξ β

for a surface tensor, Tαβ , Tαβ − λαγ Tλβ − λβγ Tαλ , ξ γ Tαβ − ˚ λαγ Tλβ − ˚ λβγ Tαλ , = ξ γ

Tαβ|γ = Tαβ;γ

and for a surface scalars, like r i (ξ α ) or n i (ξ α ) , i i i rαi = r,α = r|α = r;α ,

n i,α = n i|α = n i;α .

One can show by direct inspection, using (14.19), that the covariant derivatives of aαβ vanish: aαβ|γ = 0,

αβ

a|γ = 0,

14.1


595

and, similarly, αβ a˚ ;γ = 0.

a˚ αβ;γ = 0, Besides,2

αβ

εαβ|γ = 0,

ε|γ = 0.

(14.21)

Note the relation √ 1 a β αβ = √ a ξ α

(14.22)

which follows from (14.19) similarly to (4.80). The second quadratic form of the surface. Denote by ei the basic vectors of Cartesian coordinates in three-dimensional space. Consider the increment of the tangent vectors tα = rαi ei when the point ξ α is shifted along the surface for dξ α : i dtα = rα,β ei dξ β .

(14.23)

i The coefficients, rα,β , are symmetric with respect to α, β as the second partial derivatives of the functions r i (ξ α ) ,

i = rα,β

2 r i (ξ ) rαi = α β. β ξ ξ ξ

Let us breakdown the vectors ek into their tangent and normal components by means of (14.15): γ ek = e j δki = tγ rk + n k ei n i . (14.24)

Substituting (14.24) into (14.23), we obtain

Here

γ dtα = αβ tγ dξ β + bαβ dξ β n i ei . γ

γ

k αβ = rα,β rk , 2

k nk . bαβ = rα,β

(14.25)

(14.26)

Indeed, differentiating the identity, εαβ εαβ = 2, we have αβ

εαβ εαβ|γ + εαβ ε|γ = 0. αβ

Since the covariant derivatives of the surface metric tensor are zero, εαβ εαβ|γ = εαβ ε|γ . That yields εαβ εαβ|γ = 0. The tensor, εαβ|γ , is antisymmetric over α, β. Therefore, the only non-zero components have the indices, α = 1, β = 2 and α = 2, β = 1. Hence, εαβ|γ = 0.

596

14


γ

The object, αβ , introduced by (14.26)1 coincides with that of (14.19). Indeed, from (14.19) and (14.3) we have

⎛ i ⎞ i rβi riδ r r rα riβ 1 iδ γ α ⎠= αβ = a γ δ ⎝ + − β α 2 ξ ξ ξ δ =

1 γδ i i i i i i a r,αβ riδ + riα r,βδ . + r,βα riδ + riβ r,δα − r,αδ riβ − riα r,βδ 2

In the brackets the first and the third terms are equal while the second and last terms cancel out as well as the fourth and fifth terms, and we arrive at (14.26)1 . The object bαβ (14.26)2 can be written in terms of the covariant derivatives of the tangent vectors, i i − λαβ rλi = rα,β rα|β

i i − ˚ λαβ rλi . = rα,β or rα;β

(14.27)

Since, according to (14.8) and (14.27), i i i n i = rα;β n i = rα,β ni , rα|β

we have i i i bαβ = rα,β n i = rα|β n i = rα;β ni .

(14.28)

Formula (14.28) shows that bαβ form the components of a surface tensor. They are called the components of the second quadratic form of the surface. Tensor bαβ is i i i = r,αβ = r,βα . Using (14.8), the definition of bαβ (14.26)2 symmetric, because rα,β can also be written as bαβ = −rαi n i,β .

(14.29)

The derivatives of the tangent and normal vectors can be expressed in terms of i bαβ . To obtain such relations, we note that the three-dimensional vectors, rα|β , are directed along the normal vector to the surface: contracting (14.27) with the tangent vectors, riγ , i i riγ rα|β = riγ rα,β − aγ λ λαβ ,

(14.30)

and using (14.26)1 , we see that the right hand side of (14.30) is zero. The magnitudes i of vectors, rα|β , are determined by (14.28). Hence, for the derivatives of the tangent vectors, we obtain i i i = rβ|α = r|αβ = bαβ n i . rα|β

(14.31)

To find the derivatives of the normal vector, we note the relation n i n i,α = 0,

(14.32)

14.1


597

which is obtained by differentiation of the equation, n i n i = 1, with respect to the surface coordinates. Equation (14.32) means that the three-dimensional vectors, n i,α , are tangent to the surface, and hence can be presented as a sum of tangent vectors. The coefficients of the sum can be obtained by projecting n i,α on rβi . From (14.29) these coefficients are −bαβ . Finally, n i,α = −bαβ rβi .

(14.33)

Equation (14.33) can also be taken as the initial definition of the second quadratic form. It shows that bαβ are the measures of the rate of the normal vector when the point is moving over the surface. The key role of the two quadratic forms of the surface, aαβ and bαβ , in the surface geometry is explained by the following statement: each surface is determined uniquely (up to a rigid motion) by its quadratic forms, aαβ and bαβ . Curvatures. Consider the eigenvectors of the second quadratic form, i.e. the vectors t α which are solutions of a system of linear equations, bαβ t β = κaαβ t β .

(14.34)

In a generic case, there are two eigenvectors, t1α and t2α , and two corresponding eigenvalues, κ1 and κ2 . The vectors t1α and t2α determine the directions of principal curvature. The corresponding eigenvalues are called the principle curvatures, and their inverse, R1 = 1/κ1 and R2 = 1/κ2 , radii of curvature. The lines tangent to the eigenvectors are called curvature lines. There is a special coordinate system the coordinate lines of which are the curvature lines. The coordinates of this system are called principal coordinates. The two invariants of the second quadratic form, 1 1 H = bαα = 2 2

1 1 + R1 R2

% % det %bαβ % 1 % %= and K = , R1 R2 det %a αβ %

are called the mean and the Gaussian curvature, respectively.

Compatibility conditions. The tensors, aαβ and bαβ , are not independent because their six components are expressed in terms of three functions, r i (ξ α ) . Therefore, there must be some compatibility relations linking these tensors. These relations are the Codazzi equations: bαβ|γ − bαγ |β = 0,

(14.35)

Rσ αβγ = bσβ bαγ − bσ γ bαβ ,

(14.36)

and the Gauss equations:

where Rσ αβγ is the curvature tensor of the surface: aσ γ aαγ aαβ aσβ 1 + σ γ − α γ − σ β . Rσ αβγ = 2 ξ α ξ β ξ ξ ξ ξ ξ ξ

598

14


Fig. 14.2 Notation to the divergence theorem

The compatibility equations (14.35) and (14.36) are the equations containing only aαβ , bαβ and their derivatives. Due to antisymmetry of (14.35) with respect to β, γ and (14.36) with respect to σ, α and β, γ , there are two independent equations (14.35) and one independent equation (14.36). So, there are three constraints for the six components of the two quadratic forms of the surface. →

Divergence theorem. Let v be a vector field tangent to the surface . Denote its α surface components by v α . For the surface divergence, v|α the following divergence theorem holds: α v|α dω = v α να ds, (14.37)

→

→

where ν is the unit tangent vector to which is normal to the tangent vector τ of the curve (Fig. 14.2), να are its surface components, s the arc length along . The analytical origin of (14.37) is the formula following from (14.22) α v|α

√ v α 1 av α α β = α + αβ v = √ , ξ a ξ α

(14.38)

which yields α v|α dω

=

√ α av dξ 1 dξ 2 . ξ α

Therefore, the covariant formula (14.37) is equivalent to the usual statement for an integral of divergence over a two-dimensional region.

14.2 Classical Shell Theory: Phenomenological Approach It is natural to model the position of a thin elastic shell by a surface. Then the key kinematic characteristics of the elastic shell are the functions x i = r i (ξ α , t),

14.2

Classical Shell Theory: Phenomenological Approach

599

defining the position of the surface at the instant t. To obtain the dynamical equations of the shell theory we have to construct the action functional. Elastic energy must depend on the functions r i (ξ α , t) in a very special way because energy does not feel rigid motion. Therefore, we forewarn the construction of the action functional by the description of the surface deformation measures. Strain measures. As was mentioned, any surface, , up to its rigid motion, is determined by the first and second quadratic forms of the surface, aαβ and bαβ . Their ˚ are denoted by a˚ αβ and b˚ αβ . Recall that values in the initial state, , k n˚ k , b˚ αβ = r˚α;β

n˚ i,α = −b˚ αβ r˚βi .

(14.39)

The juggling of indices of b˚ αβ and other tensors in the initial state is done by means of the metric tensor of the initial state, a˚ αβ . Juggling of indices in the deformed state, if not otherwise stated, is done by means of the current metric tensor, aαβ . The tensors Aαβ =

1 aαβ − a˚ αβ and Bαβ = bαβ − b˚ αβ 2

(14.40)

characterize the surface deformation and can serve as the strain measures of the surface. The tensor Aαβ is a measure of elongations of the surface; if Aαβ = 0, the distances between any two points of the surface measured along the surface do not change. The tensor Bαβ is a measure of bending. To appreciate better the role of Bαβ as a bending measure, note that a plane can be deformed in a cylindrical surface without change of the lengths of any line on the surface. For such deformation, Aαβ = 0. The only indicator of the deformation occurred is the tensor Bαβ . Deformations for which Aαβ = 0 are called pure bending. Juggling the indices in Aαβ and Bαβ and other strain measures, encountered further, is done by means of the metric tensor of the initial state, a˚ αβ . Energy. It is natural to assume that kinetic and free energies of the shell possess the surface densities, i.e. they can be written as the surface integrals K=

K d ω, ˚

˚

F=

d ω, ˚

˚

˚ d ω˚ being an area element of . In classical shell theory kinetic and free energies are considered as functionals of the position vector of the surface, r i (ξ α , t) . One assumes that K =

1 i ρr ¯ ri,t , 2 ,t

600

14


ρ¯ being the surface mass density, while is a function of the strain and bending measures: = Aαβ , Bαβ .

Variational principle. First let no external forces act on the shell. Then the action functional for an elastic shell is I r i (ξ α , t) =

t1 t0 ˚

1 i ρr ¯ ri,t − Aαβ , Bαβ d ωdt. ˚ 2 ,t

(14.41)

The true motion of the shell is a stationary point of the functional (14.41) on a set of all functions r i (ξ α , t) with given initial and final values, r i (ξ α , t0 ) = r˚ i (ξ α ) ,

1

r i (ξ α , t1 ) = r i (ξ α ) ,

and, possibly, some boundary values. Consider a typical setting of the kinematic boundary constraints. Kinematic boundary conditions. If the dependence of on Aαβ and Bαβ is not degenerated, one can show that free energy, F, “feels” (see Sect. 5.5) the change of the values of r i and n i at the boundary. Therefore, the kinematic boundary conditions ˚ include an assigning of r i and n i on a part of the boundary, ˚ u , of the surface : i , r i = r(b)

n i = n i(b)

on ˚ u .

(14.42)

The index u in ˚ u emphasizes that on ˚ u the displacements of the shell, u i ≡ r i (ξ α , t) − r i (ξ α , t0 ), are known. Among the six boundary conditions (14.42), only four conditions are independent. Indeed, let σ be a parameter on the curve ˚ u , and ξ α = ξ α (σ ) are the parametric equations of ˚ u . Then equation (14.42)1 can be written as i r i (t, ξ α (σ )) = r(b) (t, σ ) .

It determines a space curve, u , given by the parametric equation i x i = r(b) (t, σ ) .

The normal vector to the surface, n i , must be orthogonal to u , and therefore the prescribed boundary values of the normal vector, n i(b) (t, σ ) , must obey the two equations

14.2


n i(b)

i r(b) (t, σ )

σ

601

n i(b) n i(b) = 1,

= 0,

which leave only one independent constraint (14.42)2 . Variations of characteristics of the surface. To derive the governing equations of the shell theory we first need to find the variations of the geometrical characteristics of the deformed surface. Let functions r i (ξ α , t) acquire infinitesimally small increments, δr i . Varying (14.3), we get3 δaαβ = riα δr i ,β + (α ↔ β) .

(14.43)

It is taken into account that δrαi = δr i ,α due to the permutability of the operators δ and /ξ α . If the projections of the vector, δr i , on the normal and the tangent vectors are used, δr i = δr α rαi + δr n i , then (14.43) can be written as4 δaαβ = riα δr γ rγi + δr n i ,β + (α ↔ β) = = riα (δr γ )|β rγi + δr γ riα rγi |β + δr,β riα n i + δrriα n i,β + (α ↔ β) .

(14.44)

γ

In the brackets, the first term is equal to aαγ δr|β due to (14.3); since the covariant derivatives of the metric tensor are zeros, it can also be written equal to δrα|β where γ

δrα = aαγ δr γ = aαγ ri δr i = riα δr i . The second and the third terms are zero in accordance with (14.31) and (14.8). The last term is equal to −bαβ δr due to (14.33). Finally, δaαβ = δrα|β + δrβ|α − 2bαβ δr.

3

(14.45)

Recall that the notation (α ↔ β) means the previous term in the equation with the indices α and β replaced by β and α, respectively. 4 Since δr γ r i γ ,β , for each i, is a surface scalar, it can be written in various ways: γ i δr rγ ,β = (δr γ ),β rγi + δr γ rγi ,β = γ

γ

= δr|β rγi + δr γ rγi |β = δr;β rγi + δr γ rγi ;β .

We use the second option, the covariant form of this equation, for the deformed state.

602

14


The variation of the normal vector can be found by varying the equations n i rαi = 0,

n i n i = 1.

We have rαi δn i = −n i δrαi ,

n i δn i = 0.

(14.46)

Equations (14.46) define the projections of the vector δn i on the tangent vectors and the normal vector. The projection on the normal vector is zero. Consequently, δn i = −riα n k δr k ,α .

(14.47)

i Let us find δbαβ . Since in the Cartesian observer’s frame, bαβ ≡ n i r,αβ , we have

i δbαβ = n i δr i ,αβ + r,αβ δn i .

Taking into account (14.47) and (14.26)1 , we get γ

i i δbαβ = n i δrα,β − rα,β ri n k δrγk

γ i i i = n i δrα,β − αβ δrγi = n i δrα|β = n i δr|αβ .

(14.48)

From (14.48) and (14.43), the variations of the two deformation measures of the surface are5 i δ Aαβ = ri(α δr,β) ,

δ Bαβ =

(14.49)

i . n i δr|αβ

The system of equations. Everything is prepared now to proceed to the derivation of the governing equations of shell dynamics. In accordance with (14.49), the variation of the free energy F is δF =

˚

i d ω. ˚ riα δr i,β + n i δr|αβ Aαβ Bαβ

(14.50)

Since we have to integrate by parts, and the second term contains the covariant derivatives in the deformed state, it is convenient to transform the integral (14.50) to ˚ and the integral over the deformed surface . We note that the area elements of are linked by a factor θ : d ω˚ = θ dω, 5

θ=

√ √ ˚ a a/

Recall that the parenthesis in indices mean symmetrization, i.e. 1 i i i + riβ δr,α . ri(α δr,β) = riα δr,β 2

(14.51)

14.2


603

This factor may be interpreted as the ratio of the surface mass densities in the deformed and undeformed states. Introducing the notations S αβ = θ

(A, B) , Aαβ

M αβ = −θ

( A, B) , Bαβ

(14.52)

we have δF =

i S αβ rαi δr i,β −M αβ n i δr|αβ dω.

(14.53)

The tensors S αβ and M αβ are symmetric. The tensor S αβ “works” on the surface strains, while M αβ “works” on the surface bending; therefore, they are called the stress resultants and the stress moments, respectively. For the variation of kinetic and free energies we have6

δ

t1

Kdt =

t1

δr i d ωdt ˚ =− ρr ¯ i,t t

=−

ρr ¯ i,tt δr i d ωdt ˚ =

t0 ˚

t0 ˚

t0

t1

t1

ρθr ¯ i,tt δr i dωdt,

t0

δ

t1

Fdt =

t1

t0

t0

=

i i dωdt = S αβ rαi δr,β − M αβ n i δr|αβ

t1 t0

−

t1

S αβ rβi + M αβ n i |β δri,α dωdt − i M αβ n i νβ δr|α dsdt

t0

=−

t1 t0

+

t1 t0

6

S αβ rβi + M αβ n i |β δri dωdt + |α

i S αβ rβi + M αβ n i |β να δri − M αβ n i νβ δr|α dsdt.

Here we use a covariant form of the divergence theorem (14.37) for a surface .

604

14


i Taking first the variations δr i and δr,α equal to zero at the boundary, we obtain from the condition, δ I = 0, Euler equations,

ρθ ¯

2r i (t, ξ α ) αβ i αβ i

= S rβ + M n |β . |α t 2

(14.54)

Equations (14.54), along with the constitutive equations for S αβ and M αβ (14.52) and the kinematical relations (14.3), (14.7), (14.29) and (14.40), form a closed system of equations for three functions, r i (t, ξ α ). Boundary conditions. In the case of non-zero variations on the following equality is to be satisfied at any instant: ?

@

S αβ rαi + M αβ n i |α δri − M αβ n i δri,α νβ ds = 0.

(14.55)

i . They The integral (14.55) contains the variations δr i and their derivatives δr,α i are not independent: the derivative of δr along the contour is determined completely by the values of δr i on this contour. To obtain the boundary conditions from (14.55), first we have to rewrite (14.55) in the form containing only independent variations. To this end we break down the two-dimensional Kronecker’s delta, δαβ , in terms of the tangent vector, τ α , and the normal vector, ν α ,

δαβ = τα τ β + να ν β , and make the corresponding decomposition of the derivatives, i i β i β i β δr,α = δr,β δα = δr,β τ τα + δr,β ν να

(14.56)

If ξ β = ξ β (s) are the parametric equations of the contour , s being the arc length of , then τβ =

dξ β (s) , ds

and the first term of (14.56) can be written as i β δr,β τ τα = τα

dδr i . ds

The second term in (14.56) contains the normal derivative of δr i which we denote by dδr i /dν : dδr i i β ≡ δr,β ν . dν

14.2


605

The integral (14.55) takes the form

dδr i dδr i − M αβ να n i νβ ds = 0. S αβ rαi + M αβ n i |α δri − M αβ τα n i ds dν

Obviously, δr i and n i dδr i /dν are independent. Integrating the second term by parts we obtain the equation containing only independent variations:

S αβ rαi

αβ i

+ M n

|α

dδr i d αβ i αβ νβ + M τα νβ n i δr − M να νβ n i ds = 0. ds dν

(14.57)

Variations δr i and n i dδr i /dν are zero at ˚ u due to (14.42). On the other part of the boundary, ˚ f = ˚ − ˚ u , the arbitrariness of δri and n i dδr i /dν, yields the boundary conditions on ˚ f :

d αβ S αβ rαi + M αβ n i |α νβ + M τα νβ n i = 0, ds

M αβ νa νβ = 0.

(14.58)

These boundary conditions are quite unusual: in contrast to other problems we have dealt with in continuum mechanics, they contain the derivatives of normal vector to the boundary, and, thus, curvatures of the boundary. This is caused by the presence of the second derivatives of position vector in energy density and occurs for both plates and shells. The issue of the proper boundary conditions for plates was a long-standing problem in the nineteenth century. It was solved by Kirchhoff. He was the first to apply the energy method to the derivation of the equations and the boundary conditions of plate theory and make the transformation from (14.55), (14.56) and (14.57) in case of plates. Interestingly, it took more than half a century to understand how to get Kirchhoff’s boundary conditions from the differential equations of linear elasticity. External forces. Let the external forces now be non-zero. Denote by Q i and Ri the forces working on the variations of the positions of the points of and , respectively, and by M the “generalized force” working on the rotations of the fibers normal to . Then the variation of the action functional (14.41) must be equated to the negative work of the external forces on the shell displacements, ⎤ ⎡ t1 i dδr ds ⎦ dt. Ri δr i + Mn i − ⎣ Q i δr i dω + dν t0

(14.59)

This yields the corresponding contributions to the momentum equations ρθ ¯

2r i (t, ξ α ) αβ i αβ i

= S rα + M n |β + Q i , t 2

(14.60)

606

14


and the boundary conditions

d αβ M τα n β n i = R i , S αβ rαi + M αβ n i |α νβ + ds M αβ να νβ = M.

(14.61)

The governing system of equations projected to the tangent plane and the normal vector. It is often convenient to write down the “intrinsic” system of equations projected to the tangent directions and the normal vector. According to (14.33), αβ S αβ rαi + M αβ n i |α = S αβ − bλα M λβ rαi + M |α n i .

It is convenient to introduce a non-symmetric tensor, T αβ , as T αβ = S αβ − bλα M λβ .

(14.62)

Projecting equations (14.60) on the tangent vectors and the normal vector, we have λβ

αβ

2r i , t 2 2r i + Q = ρθ ¯ ni 2 . t

¯ iα T|β − bλα M|β + Q α = ρθr αβ

M|αβ + bαβ T αβ

(14.63)

Here the projections of the external forces are denoted by Q α = riα Q i ,

Q = ni Qi ,

R α = R i riα ,

R = Ri ni .

The projections of the boundary conditions are T αβ νβ − M γβ τγ νβ τ λ bλα = R α , d αβ αβ νβ M |α + M τα νβ = R, ds M αβ νa νβ = M.

(14.64)

If the function is known, (14.63) and (14.64) augmented by the constitutive equations (14.52), and the initial conditions, form a closed system of equations of shell dynamics. Physically linear theory. As in elasticity theory, by physically linear one means a simplification of general theory based on smallness of strains. The strains in shell theory are characterized by two dimensionless parameters: 1/2 ε A = max Aαβ Aαβ

and

1/2 . ε B = h max Bαβ B αβ

(14.65)

14.2


607

In physically linear shell theory one neglects the contributions on the order of ε A and ε B with respect to unity. This considerably simplifies the system of equations. First of all, one can replace the covariant differentiation over by the covariant ˚ Indeed, there is a simply verified identity: differentiation over .

γ

γ γ αβ − ˚ αβ = a γ σ Aασ ;β + Aβσ ;α − Aαβ;σ .

(14.66)

γ

Therefore, αβ − ˚ αβ ∼ ε A /l, where l is the characteristic length of the stress state γ γ on ,7 and replacement of αβ by ˚ αβ corresponds to neglecting the terms on the order of ε A in comparison with unity. Second, θ can be replaced by unity. Third, bαβ can be replaced by b˚ αβ when they enter in the products like bλα M λβ or T αβ bαβ . Finally, the governing equations take the form 2 i

r λβ − b˚ λα M ;β + Q α = ρr ¯ iα 2 , t 2 i r αβ ¯ i 2, M ;αβ + b˚ αβ T αβ + Q = ρn t T αβ = S αβ − b˚ λα M λβ . T

αβ ;β

(14.67)

In the boundary conditions vectors τ α and ν α may be replaced by τ˚ α and ν˚ α , ˚ s˚ being respectively.8 Indeed, let ξ α = ξ α (˚s ) be the equation for the contour , ˚ Then τ˚ α = dξ α /d s˚ . Since the contour ˚ does not move over the arc lengths on . particles, the vector τ α is proportional to dξ α /d s˚ . The proportionality coefficient, ds/d s˚ , differs from unity by a small term on the order of ε A . Consequently, τ α = τ˚ a + O (ε A ). The vectors ν α and ν˚ a are defined by the equations aαβ ν α ν β = 1, aαβ ν α τ β = 0, and a˚ αβ ν˚ α = 1, a˚ αβ ν˚ α τ˚ β = 0. Therefore, ν α = ν˚ α + 0 (ε A ). So the boundary conditions are T αβ ν˚ β − M γβ τ˚γ ν˚ β τ˚ λ b˚ λα = R α , d αβ M τ˚α ν˚ β = R, ν˚ β M αβ ;α − d s˚ M αβ ν˚ α ν˚ β = M.

(14.68)

In physically linear theory energy is a quadratic form of Aαβ and Bαβ . Accordingly, S αβ and M αβ are linear functions of Aαβ and Bαβ . We consider these relations further. Geometrically linear theory. Another case where considerable simplifications are possible is the case of small displacements, u i = r i − r˚ i . Then, setting δr i = u i in (14.49), we get 7

About the characteristic length (see Sect. 5.11); a complete definition will be given below. Note that the three-dimensional vectors rαi τ α and rαi ν α may differ considerably from r˚αi τ˚ α and r˚αi ν˚ α because rαi − r˚αi may be not small even for small strains. 8

608

14


i Aαβ = r˚(α u i,β) ,

Bαβ =

(14.69)

n˚ i u i;αβ .

As in three-dimensional elasticity, even for small displacements, theory can be, in principle, physically nonlinear, i.e. the dependence of S αβ and M αβ on the strain measures is nonlinear. Equations (14.63) and (14.64) can be simplified due to smallness of displacements: the acceleration terms riα 2 r i /t 2 and n i 2r i /t 2 can be replaced by 2 u α /t 2 and 2 u/t 2 , respectively, with u α and u being the projection of the displacements to the tangent plane and the normal vector of the initial state, u α ≡ r˚iα u i , u ≡ n˚ i u i , ˚ and covariant derivatives over may be replaced by covariant derivatives over . We obtain the governing equations 2 u α λβ αβ = T ;β − b˚ λα M ;β + Q α , t 2 2 u α αβ ρ¯ 2 = M ;αβ − b˚ αβ T αβ + Q, t T αβ = S αβ − b˚ λα M λβ , ρ¯

(14.70)

and the boundary conditions T αβ ν˚ β − M γβ τ˚γ ν˚ β τ˚ λ b˚ λα = R α , d αβ M τ˚α ν˚ β = R, ν˚ β M αβ ;α − dξ M αβ ν˚ α ν˚ β = M.

(14.71)

In linear shell theory equations (14.70) and (14.71) are closed by the linear relations between S αβ , M αβ and Aαβ , Bαβ and the linear relations between the deformation measures and displacements (14.69). Various bending measures. Instead of the bending measure Bαβ one can use another bending measure ραβ which is a function of Bαβ and Aαβ as long as the couple (Aαβ , Bαβ ) is in one-to-one correspondence with the couple ( Aαβ , ραβ ). If the energy density is given as a function of Aαβ and Bαβ , it can be computed in terms of Aαβ and ραβ , and we get another form of the theory. In linear shell theory we have an additional opportunity to drop small terms of order ε A and ε B in comparison to unity and, thus, get a different set of equations which still have the same accuracy. For example, let be a quadratic function, which we write in a symbolic form: 1 = A2 + h 2 B 2 , μh with μ being the characteristic value of the shear modulus. If we take λ ραβ = Bαβ − b˚ (α Aλβ) ,

14.2


609

then in the expression for , 2 1 1 2 2 = A +h ρ+ A , μh R we may drop the terms, h 2 /R 2 A2 and h 2 ρ A/R, which are of the order ε2A (h/R)2 and ε B ε A h/R, respectively, and thus smaller than the leading terms of the order ε2A and ε2B . The equations of the theory with the energy density 1 = A2 + h 2 ρ 2 , μh differ from the original ones but have the same accuracy. Nowadays, the simplification of the equations achieved in this way are not important: computers will not feel much difference between the various versions of the theory. However, half a century ago when most results of the shell theory were obtained analytically, the simplifications meant a lot. Here we summarize the various versions of the classical shell theory related to the different choices of the bending measure. α Consider a surface dξα . Its positions in the initial and the deformed ielement i i α states are r˚α dξ and r˚α + u ,α dξ . The out-of-tangent-plane rotation of the vector r˚αi dξ α is characterized by the “angles of rotation”: n˚ i

1 0 i r˚α + u i,α − r˚αi = n˚ i u i,α .

For example, for a plate, n˚ i is a constant vector, and n˚ i u i,α are the derivatives of the normal displacement, u = n˚ i u i : n˚ i u i,α = u ,α . The derivative u ,1 is the angle of rotation of the fiber directed along the coordinate line, ξ 1 , in the plane passing through dξ 1 and the normal vector. Bending of the surface is characterized by the derivatives of the angles, n˚ i u i,α . Let us single out the derivatives of n˚ i u i,α in the expression for Bαβ . Since Bαβ = n˚ i u i;αβ ≡ n˚ i u i;(αβ) , the symmetrization over α, β of the relation γ

n˚ i u i;αβ = (n˚ i u i,α );β + r˚γi u i,α b˚ β yields the formula γ

Bαβ = (n˚ i u i,(α );β) + r˚γi u i,(α b˚ β) .

(14.72)

The first term in (14.72), the derivatives of the rotation angles, characterizes bending. Let us simplify the second term as much as possible. The gradient of displacements can be presented as a sum of its symmetric and antisymmetric parts,9 9 In this chapter, from now on we will denote the Levi-Civita tensor by e , since the notation ε γα αβ will be used for the tangential components of the strain tensor.

610

14

r˚γi u i,α = Aγ α + φeγ α ,

φ=


1 αβ i 1 1 e r˚α u i,β = eαβ u α;β = √ u 1,2 − u 2,1 . 2 2 2 a˚

The parameter φ has the meaning of the angle of rotation of the surface elements around the normal vector. Substituting this expression into (14.72) we obtain λ Bαβ = ραβ + b˚ (α Aλβ) ,

(14.73)

γ ραβ = (n˚ i u i,(α );β) + φeγ (α b˚ β) .

(14.74)

where ραβ denotes the tensor

One can use ραβ as a bending measure instead of Bαβ . Note that it is not possible to “simplify” the bending measure even further by discarding the last term in (14.74), because in this case ραβ will not be invariant with respect to the motion of the shell as a rigid body (discarding the terms containing Aαβ apparently does not affect the feature of the bending measure to be zero for rigid motions). The bending measure ραβ was introduced by Koiter [150] and Sanders [267]. In papers by A. Goldenweiser, V. Novozhilov, and L. Balabuch other bending measures, κ1 , κ2 , and τ , were used. These measures, κ1 , κ2 and τ , can be defined by ˚ the following condition: in the principal coordinates of the surface , κ1 = −ρ11 , κ2 = −ρ22 , τ = −B12 . ¯¯

¯¯

¯¯

(14.75)

In writing the physical components of a tensor, the corresponding indices are underlined, in particular, ρ11 = ¯¯

ρ11 ρ22 B12 , ρ22 = , B12 = √ . ¯¯ ¯¯ a˚ 11 a˚ 22 a˚ 11 a˚ 22

To put the relations of such a theory into the tensor from, we introduce the tensor καβ , the physical components of which coincide with the bending measures in the principal coordinates: κ11 = κ1 , κ22 = κ2 , κ12 = τ. ¯¯

¯¯

¯¯

˚ which do not have To find καβ in any coordinate system, consider first the surfaces umbilical points, i.e. points at which the curvature radii coincide. Let us introduce the tensor γ b˜ αβ = b˚ (α eγβ) .

If the coordinate axes coincide with the curvature lines, then this tensor has only one non-zero component:

14.2


611

1 1√ 1 , a˚ b˜ 12 = − − 2 R1 R2 ˚ The quantity where R1 and R2 are the principal curvature radii on the surface . 1 b˜ = b˜ αβ b˜ αβ is equal to zero only at the umbilical points and does not vanish on 2 the surface considered. Let us define the tensor dαβ as b˜ αβ dαβ = √ . b˜ In the principal coordinates, the components of this tensor have the following values: d11 = d22 = 0, d12 = d21 = 1 if R1 > R2 , d12 = d21 = −1 if R1 < R2 . Here, the dependence of dαβ , as well as of the Livi-Civita tensor, on the choice of ˚ becomes apparent. The fourth order the orientation of the coordinate system on tensor dαβ d γ δ does not depend on the orientation of the coordinate system. In each system of coordinates, the physical components of the tensor dαβ are bounded above √ by the number 2, since dαβ d αβ = 2. At the umbilical points, an uncertainty of the type 0/0 appears in the definition of dαβ . Therefore, if the surface has umbilical points, the definition of dαβ at these points needs to be supplemented in any way that would satisfy the condition dαβ d αβ = 2. Consider the tensor καβ = −ραβ −

1 ˚ H dαβ d γ δ Aγ δ , 2

(14.76)

˚ Since, according to (14.73), in the curvature ˚ = 1/2b˚ αα being mean curvature of . H lines, ˚ A 12 , B12 = ρ12 + H ¯¯

¯¯

¯¯

d11 = d22 = 0, ¯¯

¯¯

d12 d 1¯2¯ = 1, ¯¯

the physical components of the tensor καβ coincide with κ1 , κ2 , τ . Thus the tensor (14.76) is the bending measure sought. Bending measure ραβ . In what follows we will use as the bending measure the tensor ραβ . For finite displacements, ραβ may be defined by (14.73). Using this bending measure simplifies the transition from the geometrically nonlinear theory to the most elegant form of the linear theory suggested by Koiter and Sanders. In physically linear theory, the energy density, , is a quadratic function of Aαβ and ραβ , while θ (14.51) can be set equal to unity. Denote the derivatives of with respect to Aαβ and ραβ by

612

14

n αβ =


( A, ρ) ( A, ρ) , m αβ = . Aαβ ραβ

(14.77)

By t αβ we denote the sum t αβ = n αβ + b˚ γ[α m γβ] . The tensors, T αβ and M αβ , introduced earlier by (14.52) and (14.62), are expressed in terms of n αβ and m αβ as T αβ = t αβ ,

M αβ = −m αβ .

(14.78)

The second equation (14.78) follows from (14.73) and (14.52). To obtain the first one, we note that, according to (14.52), (14.62) and (14.73),

T αβ = n αβ + b˚ γ[α m γβ] + bγα − b˚ γα m γβ . The last term here is in the order of ε B (ε B + ε A ) and can be omitted in a physically linear theory. The dynamical equations of the physically linear shell theory become 2 i

r λβ + b˚ λα m ;β + Q α = ρr ¯ iα 2 , t 2 i r αβ ¯ i 2. −m ;αβ + b˚ αβ n αβ + Q = ρn t t

αβ

;β

(14.79)

Linear shell theory. In linear shell theory, we can replace the projections of acceleration on the deformed surface, , by the projections to the undeformed surface. Thus, t

αβ ;β

−m

+ b˚ λα m

αβ ;αβ

λβ ;β

2 u α , t 2 2 u + Q = ρ¯ 2 . t

+ Q α = ρ¯

+ b˚ αβ n αβ

(14.80)

These equations along with the boundary conditions

ν˚ β m αβ ;α

t αβ ν˚ β + m αβ τ˚α ν˚ β τ˚ λ b˚ λα = R α , d αβ m ν˚ β τ˚α = −R, m αβ ν˚ α ν˚ β = −M, + d s˚

(14.81)

the constitutive equations (14.77) and geometrical relations (14.74) and (14.69) form a closed system of equations.

14.2


613

Computation of the physical characteristics of elastic shells. To complete the construction of the shell theory, we have to formulate the rule for calculation of the shell energy density in terms of the three-dimensional characteristics of the elastic material, to establish the link between the three-dimensional stress state and the two-dimensional characteristics of deformation and give the relationships to find the generalized forces entering the governing equations, Q i , Ri , M in terms of the actual surface forces acting on the shell. We consider the last two tasks further in this chapter while here we give a recipe for computation of energy density in the leading approximation. This recipe is justified in further sections where the sufficient conditions for its applicability are also given. α 3 ξ 3 directed Consider a Lagrangian coordinate 3 system, ξ , ξ , with the coordinate 3 ˚ along the normal vector to , ξ ≤ h/2; on the middle surface ξ = 0. Index 3 is usually omitted, so ξ 3 ≡ ξ. Denote by εαβ , εα3 and ε33 the strain tensor components. Energy density per unit volume, F, is a known function of strains: F = F εαβ , εα3 , ε33 .

The material of the shell can be inhomogeneous; therefore the material characteristics, and, hence, F, may depend on coordinates, but we do not emphasize this in our notation. The material may be physically nonlinear, so energy density is not necessarily a quadratic function of strains. Computation of the shell energy density, , consists of the two steps. First, we compute the function F|| εαβ = min F εαβ , εα3 , ε33 . εα3 ,ε33

(14.82)

Then, we assume that the tangential strains are the linear functions of the transverse coordinates εαβ = Aαβ − ξραβ

(14.83)

and find the shell energy density by integration of F|| Aαβ − ξραβ over the transverse coordinate:

Aαβ , ραβ =

h/2

−h/2

F|| Aαβ − ξραβ dξ.

(14.84)

The condition (14.82) means that, in the first approximation, the stress components, σ α3 =

F F and σ 33 = , εα3 ε33

614

14


are zeros. The tangential components of stresses are given by the formula σ

αβ

F|| . = εαβ εαβ =Aαβ −ξραβ

(14.85)

Actually, the stress components σ α3 and σ 33 are not zeros, but they are much smaller than σ αβ in typical cases and may be neglected. Note that there are some exceptions, like, e.g., sandwich plates with a weak core, or shells under a concentrated load. The assumption (14.83) follows from vanishing of εα3 in the first approximation, as will be seen further in Sect. 14.4. An immediate consequence of (14.84) is the physical meaning of tensors n αβ , αβ m and M αβ , which enter into the governing equations. Differentiating (14.84) with respect to Aαβ and ραβ we have h/2 h/2 F|| n = σ αβ dξ, dξ = = Aαβ −h/2 −h/2 εαβ εαβ =Aαβ −ξραβ h/2 h/2 F|| αβ m = =− ξ dξ = − σ αβ ξ dξ, ραβ −h/2 εαβ εαβ =Aαβ −ξραβ −h/2 h/2 M αβ = −m αβ = σ αβ ξ dξ. (14.86) αβ

−h/2

These equations justify the terms of the resultants and the bending moments for the tensors n αβ and M αβ . Let us apply this rule to computation of energy density of homogeneous isotropic shell. Isotropic homogeneous shell. For an isotropic body, 1 2 F εαβ , εα3 , ε33 = λ g˚ i j εi j + μg˚ i j g˚ kl εik ε jl , 2 where g˚ i j are the contravariant components of the metric tensor in the coordinate system ξ α , ξ 3 . This coordinate system is curvilinear, and therefore g˚ i j are some functions of coordinates. These functions are explicitly computed further (see (14.125)). The analysis of (14.125) show that, in the first approximation for thin shells, g˚ αβ = a˚ αβ , g˚ α3 = 0, g˚ 33 = 1. Thus F=

2 1 αβ 2 λ a˚ εαβ + ε33 + μ a˚ αγ a˚ βδ εαβ εγ δ + 2a˚ αβ εα3 εβ3 + ε33 . 2

14.2


615

This function can be identically rewritten as @ ? 2 F = μ σ a˚ αβ εαβ + a˚ αγ a˚ βδ εαβ εγ δ + 2μa˚ αβ εα3 εβ3 +

2 ν 1 λ = . (14.87) + (λ + 2μ) ε33 + σ a˚ αβ εαβ , σ = 2 λ + 2μ 1 − ν

Minimum with respect to ε33 is achieved at ε33 = −σ a˚ βδ εαβ . This means that the coefficient σ plays the role of a “two-dimensional Poisson’s coefficient”: for a given extension of the shell, the contraction, ε33 , of the normal fibers10 occurs due to vanishing of σ 33 . Minimum with respect to εα3 is achieved for εα3 = 0. That means that the fibers normal to the middle surface remain normal to the deformed midsurface in the course of deformation. So, minimization of F with respect to εα3 and ε33 gives @ ? 2 F|| εαβ = μ σ a˚ αβ εαβ + a˚ αγ a˚ βδ εαβ εγ δ .

(14.88)

From (14.84) and (14.88),

αβ

αβ

μh 3 α 2 2 σ ρ α + ραβ ρ . = μh σ Aαα + Aαβ A + 12

(14.89)

The stress resultants and the bending moments are related to the strain measures by the constitutive equations

αβ n αβ = 2μh σ Aλλ a˚ +Aαβ ,

m αβ =

μh 3 λ αβ αβ σρλ a˚ +ρ . 6

(14.90)

The energy expression (14.89) and the constitutive equations (14.90) were suggested by Koiter [150] and Sanders [267]. Let us denote the extension energy by (A): αβ

2 (A) = μh σ Aαα + Aαβ A .

(14.91)

1 Then, the bending energy is equal to (hρ) and the energy of the shell, according 12 to the Koiter-Sanders theory, is = (A) +

10

1 (hρ) . 12

(14.92)

Usual Poisson’s coefficient characterizes the contraction of the fibers normal to the axis of a beam.

616

14


Historically, the first energetic linear shell theory was the Novozhilov-Balabuch theory [233], with the energy density αβ

αβ

μh 3 α 2 2 + σ κα + καβ κ , = μh σ Aαα + Aαβ A 12

(14.93)

where καβ is the bending measure (14.76). According to (14.77), in the NovozhilovBalabuch theory, the constitutive equations are:11

μh 3 αβ ˚ d αβ d γ δ κγ δ H n αβ = 2μh σ Aλλ a˚ +Aαβ − 12 μh 3 λ αβ αβ

σ κ λ a˚ +κ . M αβ = −m αβ = 6

(14.94)

The energy densities (14.89) and (14.93) differ by the terms of the form μh 3 b˚ AB. These terms are small compared to those included in the expressions (14.89) and (14.93) because μh 3 b˚ AB ≤ μh

1 2 A + h2 B2 , R

(14.95)

and the interaction terms between extension and bending are on the order of h/R compared with energy of extension and energy of bending. Similarly, the expression αβ

αβ

μh 3 α 2 2 σ B α + Bαβ B + = μh σ Aαα + Aαβ A 12

(14.96)

is of the same order of accuracy as (14.89) and (14.93). The shell theories12 with the energy densities (14.89), (14.93) and (14.96) will be called classical. The classical shell theories, as will be seen later, describe correctly the shell stress state. This, however, is not always true for displacements: there are problems in which the classical theory can yield 100% error for displacements. To obtain the displacements correctly one needs to use a refined shell theory. The mechanism of the accuracy loss of classical shell theories is described further in Sect. 14.6. Isotropic inhomogeneous shells. If the elastic moduli of the material of the shell change along the normal, like in sandwich shells, then in (14.88) μ and σ are some functions of ξ. Applying (14.84) we find

11

If we rewrite the constitutive equations (14.94) as equations for T1 = T11 T2 = T22 , S = S12 , M1 = M11 , M2 = M22 , H = M12 , then they will become the equations derived in [115].

12

We follow the tradition and refer to the shell models as shell theories.

14.2


Aαβ , ραβ =

h/2

−h/2

617

? @ 2 μ σ Aαα − ξραα + a˚ αγ a˚ βδ Aαβ − ξραβ Aγ δ − ξργ δ dξ =

= ext Aαβ + int Aαβ , ραβ + bend ραβ

where extension energy, ext , bending energy, bend , and extension-bending interaction energy, int , are

2 ext Aαβ = h μσ Aαα + μ Aαβ Aαβ , int Aαβ , ραβ = −h 2 μσ ξ Aαα ραα + 2 μξ Aαβ ρ αβ ,

: ; 2 : ; bend ραβ = h μσ ξ 2 ραα + μξ 2 ραβ ρ αβ .

Here by · we denoted averaging over the thickness: for any function, ϕ (ξ ), 1

ϕ = h

h/2

ϕdξ =

−h/2

1/2

ϕdζ,

ζ ≡ ξ/ h.

(14.97)

−1/2

An interesting feature of this case is the interaction between extension and bending: extension of the shell causes non-zero bending moments, m αβ , and, inversely, bending of the shell causes non-zero resultants, n αβ . If the material properties are symmetric with respect to the middle surface, then μσ ξ = μξ = 0, and int = 0. Thus, there is no interaction between extension and bending for shells with material symmetry. On other versions of classical physically linear and geometrically nonlinear shell theory. The energy density, , is a complex nonlinear function of the derivatives of the particle trajectories of the shell, r i (ξ α , t). Since this function is an approximate one, it is natural to simplify it within the framework of the accuracy with which it was obtained. i , and this nonlinearity The strain measures, Aαβ , are quadratic with respect to r,α is unlikely to be further simplified in the general case. Therefore, the energy density, , is at least a polynomial of the fourth order with respect to rαi . The components of the second quadratic form, and, consequently, the bending measure, Bαβ , depend on the derivatives of r i in quite a complex way: i bαβ = n i rα;β ,

1 j n i = √ eijk r1 r2k . a

(14.98)

One simplification here is straightforward: a can be substituted by a˚ in the expression for n i (it involves omitting the terms on the order of ε A ). Then bαβ becomes a cubic form with respect to the derivatives of r i ,

618

14


1 j i , bαβ = √ eijkr1 r2k r;αβ a˚

(14.99)

while the bending energy becomes a form of the sixth order. Let us show that within the accuracy of the classical theory, the bending energy can be taken as a form of the third order. Using the decomposition of the Kronecker’s delta (14.15), we can write the product bαβ bγ δ as i j j j i i i bαβ bγ δ = n i rα;β n j rγ ;δ = δi j − riσ r jσ rα;β rγ ;δ = rα;β riγ ;δ − riσ rα;β r jσ rγ ;δ .

Similarly to the derivation of (14.26)1 from (14.19), one can obtain from (14.19) that i = σαβ − ˚ σαβ . riσ rα;β

Then, according to (14.66), i =O riσ rα;β

ε

A

l

,

l being the characteristic length of the stress state. Omitting this term we make an error in the energy density on the order of the h 3 (ε A /l)2 which is acceptable in classical shell theory. Therefore, we can put i riγ ;δ bαβ bγ δ = rα;β

and the bending energy density of isotropic homogeneous shell becomes bend =

μh 3 iα β β σ r ;α ri ;β − 2σ bαα b˚ β + r iα;β riα;β − 2b˚ αβ bαβ . 12

(14.100)

In (14.100) an additive constant depending on b˚ αβ is dropped. The tensor bαβ is cubic with respect to the derivatives of r i ; consequently, the energy density (14.100) is a cubic form with respect to the derivatives of r i . For plates b˚ αβ = 0, and the bending energy becomes quadratic with respect to the second derivatives of r i : bend =

μh 3 iα β σ r ;α ri ;β + r iα;β riα;β . 12

(14.101)

Since we dropped the terms containing the derivatives of Aαβ , the boundary conditions for shell theory with bending energy (14.100) or (14.101) differ from that of classical shell theories. Now the energy functional “feels” the boundary values of r i i and r,α . An artificial boundary layer of the thickness of h may appear in the theory;

14.2


619

however, the two-dimensional nonlinear theory becomes overall simpler than the theory with the energy (14.96). All the relations of this section remain valid if the shell thickness, h, depends on h ,1 ≪ coordinates, but the derivatives of h with respect to coordinates are small: 1, h ,2 ≪ 1.

Computation of the stress components σ α3 and σ 33 . If σ αβ are determined from the shell theory by formula (14.85) then σ α3 and σ 33 can be found in statics from the equilibrium equations of three-dimensional elasticity. For example, for plates we have σ α3 σ αβ =− β , ξ ξ σ α3 σ 33 =− α , ξ ξ

σ α3 ξ =h/2 = P+α ,

−σ α3 ξ =−h/2 = P−α ,

σ 33 ξ =h/2 = P+ ,

(14.102)

σ 33 ξ =−h/2 = −P− ,

(14.103)

σ αβ dξ.

(14.104)

where P±α , P± are the surface forces at the faces of plate, ξ = ±h/2. Equations (14.102) are ordinary differential equations of the first order. These equations are overdetermined because we have for them the boundary conditions at both ends, ξ = h/2 and ξ = −h/2. The compatibility condition is obtained by the integration of (14.102) over ξ : P+α + P−α = −

ξ β

h/2 −h/2

The compatibility condition for (14.103) is obtained in the following way: we integrate (14.103) over ξ to get P+ + P− = −

ξ α

h/2

σ α3 dξ,

(14.105)

−h/2

and then express the integral of σ α3 in terms of σ αβ . This can be done by multiplying (14.102) by ξ and integrating over ξ : h/2

h α σ α3 P+ − P−α − dξ = ξ ξ 2 −h/2

h/2

σ dξ = − β ξ −h/2 α3

h/2

σ αβ ξ dξ. (14.106)

−h/2

Plugging (14.106) in (14.105), we find a constraint for σ αβ : P+ + P− = −

ξ α

h α P+ − P−α + β 2 ξ

Finally, the compatibility conditions are

h/2

−h/2

σ αβ ξ dξ .

(14.107)

620

14


n αβ + P+α + P−α = 0, ξ β h α 2 m αβ P − P−α + P+ + P− = 0. + α ξ α ξ β ξ 2 +

(14.108)

Comparing the compatibility conditions with the equilibrium equatations for plates (14.80) (t αβ = n αβ , b˚ αβ and the acceleration terms are zero), we see that the compatibility conditions coincide with the equilibrium equations for an appropriate choice of Q, Q α , and, thus, are satisfied automatically. So from (14.102) and (14.103) we can find σ α3 and σ 33 : σ α3 = −P−α − σ

33

= −P− − α ξ

ξ −h/2

ξ β −P−α

ξ

σ αβ dξ, −h/2

− β ξ

(14.109)

η

σ −h/2

αβ

dξ dη.

The computation of σ α3 and σ 33 provides a useful check for the validity of the classical shell theory for a particular problem: the magnitudes of σ α3 and σ 33 , computed from (14.109), must be much smaller than σ αβ . Otherwise, the classical shell theory does not describe the stress state of the shell correctly.

14.3 Plates The equations of classical shell theory simplify considerably in the case of plates when b˚ αβ = 0. In discussion of the plate theory we assume that the material of the plate is isotropic and homogeneous, and, hence, the energy density is given by formula (14.89) or, equivalently, (14.91) and (14.92). Linear theory. For plates, the coordinates ξ α may be chosen Cartesian, and coinciding with Eulerian coordinates, x α ; thus a˚ αβ = a˚ αβ = δαβ , while the covariant derivatives coincide with the partial derivatives. The position vector is r α = x α + uα ,

r 3 = u.

In linear theory, according to (14.69) and (14.74), Aαβ = u (α,β) ,

ραβ = u ,αβ .

The energy density splits into the sum of the extension energy ext Aαβ deα pending only on the longitudinal displacements, u α (t, x ) , and the bending energy, bend ραβ , depending only on the normal displacements, u (t, x α ):

14.3

Plates

621

2 ext Aαβ = μh σ u α,α + u (α,β) u α,β ,

μh 3 σ (u)2 + u ,αβ u ,αβ . bend ραβ = 12

Accordingly, the action functional splits into the sum of functionals depending on u α and u only: I = Iext + Ibend , t1 1 α α ˚ ρu ¯ α,t u ,t − ext Aαβ + Q α u d ωdt, Iext = 2 Ibend =

t2 ˚ t1

t2 ˚

(14.110)

1 2 ρu ¯ − bend ραβ + Qu d ωdt. ˚ 2 ,t

˚ Varying the functionals Here we assume that u α , u and du/dν are prescribed at . (14.110) we find the equations for longitudinal vibrations, ρu ¯ α,tt = 2μσ hu γ,γ ,α + 2μhu (α,β) ,β + Q α ,

and for bending vibrations,

ρu ¯ ,tt + D2 u = Q, where the bending rigidity, D, is D = 2μ (1 + σ )

h3 . 12

Nonlinear theory. In general, for finite gradients of the plate displacements the plate becomes “a shell” in the deformed state, and no simplifications of the equations are possible except setting b˚ αβ = 0. There is an important special case, however, when the gradients, u α,β , are much smaller than unity. Then in the expression for strains,

@ 1 i 1 ? γ δγ α + u γ ,α δ γ + u ,β + u ,α u ,β − δαβ = r,α ri,β − a˚ αβ == 2 2 1 1 γ = u (α,β) + u ,α u ,β + u ,α u γ ,β , 2 2

Aαβ =

we can drop the last term in comparison with first one; we obtain 1 Aαβ = u (α,β) + u ,α u ,β . 2

622

14


To write down the bending measures, we note that n 3 = r1,1r2,2 − r1,2r2,1 = 1 + u 1,1 1 + u 2,2 − u 1,2 u 2,1 ≈ 1, β

β

3 3 n γ = eγ 3β r,1 r,2 + eγβ3r,1r,2 = −eγβ u ,1 rβ,2 + eγβ rβ,1 u ,2 ≈ −eγβ u ,1 δβ2 − u ,2 δβ1 = −u ,γ .

Hence, from (14.99),

γ

γ

ραβ = bαβ = u ,αβ + n γ u ,αβ = u ,αβ − u ,γ u ,αβ .

(14.111)

Assuming that u ,αβ and u ,α are of the same order we may neglect the last term in γ (14.111) as containing the small factors u ,αβ . Finally, ραβ = u ,αβ . The energy density becomes 1 = ext u (α,β) + u α , u ,β + bend hu ,αβ . 2 In statics, the true displacements provide minimum for the functional

˚

1 α ext u (α,β) + u ,α u ,β + bend hu ,αβ − Q α u − Qu d ω˚ 2

(14.112)

for the prescribed values of u α , u and u/ν at the boundary. In general, the functional (14.112) cannot be simplified further. There is, however, a wide class of problems in which bending energy can be neglected. Such problems can be characterized implicitly by the following condition: the minimization problem, 1 ext u (α,β) + u ,α u ,β − Q α u α − Qu d ω˚ → 2

˚

min

u α ,u: u α| =u α(b) ,u | =u (b)

, (14.113)

has a unique minimizer. Then the bending energy in (14.112) generates just a small correction, since for thin plates the terms of the bending energy, like h 3 u ,αβ . u ,αβ , 2 are much smaller than the term in the extension energy h u ,α u ,α . The plate theory with the energy functional (14.113) is called von Karman’s theory. The equations of this theory are

14.3

Plates

623

n αβ

αβ ,β

αβ + Q α = 0, n u ,β ,α + Q = 0, 1 1 = 2μ σ u γ,γ + u ,γ u ,γ δαβ + u (α,β) + u ,α u ,β . 2 2 n

Tension vs compression. Von Karman’s theory is, perhaps, the simplest one, which shows a clear distinction between tension and compression in thin elastic bodies. To see that distinction, consider a one-dimensional case, when u and u α depend only on one coordinate, x ≡ x 1 , and u 2 ≡ 0. Further we denote u 1 by v. Suppose that at the ends, x = 0 and x = L , the plate is attached to hinges, u = 0 at x = 0, L; the hinge at x = 0 is immobile: v = 0 at x = 0, and there is a longitudinal external force, R, acting on the hinge, x = L . Then the energy functional (14.113) with the added work of the edge force becomes (after dividing by the plate length in x 2 -direction) I (u, v) =

L 0

μh(1 + σ )A2 d x − Rv(L),

A ≡ v ,x +

1 2 u ,x . 2

To reveal the non-convex nature of this functional, it is convenient to make a change of variables, v → A. For a given u(x), there is a one-to-one correspondence between A and v,x . Function v,x is arbitrary, and so is A ( the boundary condition, v = 0, selects v for a given v,x , and does not affect A). Since v(L) =

L

v,x d x =

0

L 0

1 2 A− d x, u ,x 2

the energy functional takes the form I (u, A) =

0

L

μh(1 + σ )A2 d x − R

L

0

A−

1 2 d x. u ,x 2

The stationary point over A is unique and corresponds to the minimum of the energy functional, achieved at ˇ = A

R . 2μh(1 + σ )

After minimization over A, the energy functional becomes a functional of u(x) only: I (u) =

0

L

R 2 R2 L u ,x d x − . 2 2μh(1 + σ )

(14.114)

We have to minimize this functional over all u(x) such that u(0) = u(L) = 0. The functional (14.114) behaves drastically differently for R > 0 (tension) and R < 0 (compression). If the plate is under tension, the energy functional is convex, and its only minimizer is u(x) ≡ 0. If the plate is under compression, the energy functional is not bounded below, and the variational problem is not well-posed. The reason is

624

14


that we neglected the bending energy. If we keep the bending energy, the energy functional is I (u, A) =

L 0

2 R 2 μh 3 (1 + σ ) R2 L dx − u ,x + u ,x x . 2 12 2μh(1 + σ )

This functional is convex if R is not large enough. Indeed, the function u ,x is an arbitrary function, satisfying, due to the boundary conditions for u(x), the constraint

L

u ,x d x = 0.

0

Hence, from the Wirtinger inequality (5.24) in the case of the constraint (5.17), (2π )

2

L 2

0

(u ,x ) d x ≤ L

2

L

(u ,x x )2 d x.

0

The constant (2π )2 is the best constant in the Wirtinger inequality; therefore, the equality sign in this inequality is achievable. For R < 0, decreasing the functional by means of the Wirtinger inequality, we have I (u, A) ≥

0

L

|R| L 2 μh 3 (1 + σ ) − 12 2(2π )2

2 u ,x x d x −

R2 L . 2μh(1 + σ )

We see that the energy remains positive, when |R| L 2 μh 3 (1 + σ ) − ≥ 0. 12 2(2π )2 This occurs if the compression force, R, does not exceed the critical value, Rcr =

2μh 3 (1 + σ )π 2 . 3L 2

For such R, u(x) ≡ 0. If |R| = Rcr , then the functional is still bounded below, but the minimum is attainable on a “ray” of non-zero functions: these are the functions for which the equality sign is achieved in the Wirtinger inequality. If |R| > Rcr , then the functional is not bounded below. A complete analysis of the case, |R| > Rcr , requires keeping all the terms of the geometrically nonlinear theory. At the stationary points in this case u(x) = 0, and one says that the straight shape, u = 0, is unstable. Membrane theory of plates. If the plate is stretched so much that the strains, u (α,β) , and the stress resultants, n αβ , can be assumed practically constant, n αβ = n¯ αβ = 2 const, while u ,α ≪ 1, and u ,α can be neglected in comparison with unity, then

14.3

Plates

625

the functional (14.113) can be simplified further. Keeping only the leading terms containing u ,α we have

˚

1 αβ n¯ u ,α u ,β − Qu d ω. ˚ 2

(14.115)

Plate with energy functional (14.115) is called also a membrane. The governing equation of the membrane theory is αβ n¯ u ,β ,α + Q = 0.

The variational problem for the functional (14.115) is well-posed if the tensor n¯ αβ is positive definite, i.e. the plate is stretched in both directions. Otherwise the functional is not bounded from below, and the membrane theory does not make sense. Membrane theory of shells. By membrane shell theory one means the shell theory where the bending energy is neglected. In statics we obtain the variational problem

˚

0

1 ext Aαβ − Q α u α − Qu d ω˚ →

min

u α ,u: u α|˚ =u α(b) ,u |˚ =u (b)

.

(14.116)

In linear approximation, Aα,β = u (α,β) − ubαβ . The applicability of membrane theory depends on whether or not energy has a kernel (i.e. energy vanishes for non-zero fields u α , u) for the prescribed boundary conditions. This relates to the existence of non-zero solutions of the equations u (α,β) − u b˚ αβ = 0 u α|˚ = 0. A non-zero solution of these equations is called infinitesimally small bending of ˚ If an infinitesimally small bending exists, the extension energy has the surface . a kernel, and the bending energy which was neglected in (14.116) can, in fact, not be small on the kernel and should be kept. If the shell does not admit the infinitesimally small bending, the bending energy is much smaller than the extension energy (it contains a small factor, thickness to the power two) and can be dropped. The membrane theory is sensible in this case. Later in this chapter we will justify the formulas of classical shell theory by variational-asymptotic method of Sect. 5.11. In addition, we establish the relations allowing one to calculate the three-dimensional stress state if a two-dimensional problem of classical shell theory is solved, give the link between the actual surface forces acting on the shell, and generalized forces Q i , Ri , and M, and consider refined shell theories which take into account small corrections of classical theory.

626

14


This is followed by the discussion of hard-skin plates and shells, for which the classical theory does not work.

14.4 Derivation of Classical Shell Theory from Three-Dimensional Elasticity Variational principle of three-dimensional elasticity. Let us introduce in V˚ a curvilinear coordinate system, ξ α , ξ , according to the equations x i = r˚ i (ξ α ) + ξ n˚ i (ξ α ) .

(14.117)

˚ |ξ | ≤ h/2; the The coordinates ξ α , ξ vary in a cylinder of the height, h: ξ α ∈ , ˚ The coordinate region run by ξ α , as well as the middle surface, are denoted by . ξ has a special role, and is, therefore, not numbered; in the equations where all the Lagrangian coordinates play a similar role, the coordinate ξ acquires the number 3: ξ ≡ ξ 3 . Along with ξ , we will use the scaled coordinate ζ, ζ = ξ/ h, |ζ | ≤ 1/2. The motion of an elastic body is described by the functions x i (ξ α , ξ, t). We will assume that at a part of the boundary, ˚ u × [−h/2, h/2] , the functions x i (ξ α , ξ, t) are given: i x i (ξ α , ξ, t) = x(b) at ˚ u × [−h/2, h/2] .

(14.118)

We assume also that some dead surface forces, Pi , act on the faces of the shell, ˚ ± , defined by the equations ξ = ±h/2, and on the part of the shell’s edge S˚ = ˚ p × [−h/2, h/2] (˚ p = ˚ − ˚ u ). Then, the true motion of the elastic body is the stationary point of the functional ⎡ ⎤ t1 1 i ⎢ ⎥ vi v − F d V˚ + Pi x i d ω˚ ± + Pi x i d σ˚ ⎦ dt ⎣ ρ0 2 t0

V˚

˚±

(14.119)

S˚

on the set of functions x i (ξ α , ξ, t), taking the assigned values on ˚ u × [−h/2, h/2] and at the initial and the final instants. In (14.119), d ω˚ ± are the area elements of ˚ ±. 3D problem vs 2D problem. Denote the characteristic lengths of the stress state along the coordinates ξ 1 , ξ 2 by l1 and l2 respectively; they depend on the point of the shell. The character of the stress state depends significantly on the relative magnitude of the parameters h, l1 , and l2 . The shell can be split into three parts, V1 , V2 and V3 : in V1 , the ratio of h to one of the characteristic lengths, l1 , l2 , is much smaller than unity and to the other on the order of or greater than unity (say, h/l1 ≪ 1, h/l2 ≈ 1); in V2 , the ratio of h to each of the characteristic lengths, l1 , l2 , is much smaller than

14.4

Derivation of Classical Shell Theory from Three-Dimensional Elasticity

627

unity; in V3 , the parameters h, l1 , and l2 are on the same order of magnitude. The regions, V1 , V2 and V3 , are some cylinders. In region, V1 , it is possible to replace the three-dimensional problem by a one-dimensional problem (a beam). In region, V2 , the three-dimensional problem, as we will see, can be approximated by a twodimensional one. In V3 , the problem remains three-dimensional. Actual location of regions V1 , V2 and V3 in the shell depends on the external forces. We assume that the external forces are such that only the regions V1 and V2 are presented, V˚ = V1 + V2 , i.e. a complete “dimension reduction” is possible. Usually, the region V1 is adjacent to the edge of the shell. Taking the presence of such region into account is essential only in the refined shell theories to achieve the accuracy required. Moreover, we identify V2 with V˚ , because the theory of the edge beam is not consider here. Though that narrows considerably the class of problems, asymptotics of which can be treated consistently, an extension to a wider class of problems is achieved by just adding the edge beam energy to the energy functional which we derive. Small parameters and the classification of the two-dimensional shell theories. In the problem under consideration, there are two small parameters: the geometric parameter h/R and the stress state parameter h/l, where l = min {l1 , l2 }. We will introduce one more small parameter by the assumption that the strain amplitude 1 ε = max εab εab 2 V˚

is much smaller than unity. Moreover, we will limit the consideration to low frequency shell vibrations, i.e. the vibrations the characteristic time scale of which, τ, is linked to l as l τ= , c

(14.120)

√ where c is the characteristic speed of shear waves, for isotropic body c = μ/ρ. We will construct the two-dimensional shell theories by means of the formal ˚ is fixed (and, limit procedure h → 0. It will be assumed that the middle surface, , consequently, so is the characteristic curvature radius, R). The forces may change with h, accordingly l, τ and ε can change as well. The characteristic time τ is linked to l by (14.120). Therefore, there are three independent small parameters: h/l, h/R and ε. The existence of three small parameters brings about a great variety of asymptotics. The number of possible cases can be limited by the following reasoning. For metals, ε ∼ 10−5 ÷ 10−3 , and it does not make sense to retain the terms on the order of ε in comparison with unity. For thin shells (h/R ≤ 10−2 ), the terms on the order of h/R should be also neglected. For the shells of medium thickness (h/R ∼ 10−1 ÷2·10−1 ), taking into account the corrections on the order of h/R may be of interest. Computations suggest that for smooth but sufficiently fast changing stress states, two-dimensional theories are sometimes satisfactory up to h/l ∼ 1/2, and it is appropriate to construct a two-dimensional theory where the corrections on

628

14


the order of h/l and (h/l)2 are taken into account. Due to this, one can consider the following hierarchy of the approximation theories:

Classical theory Refined main theory Theory taking into account geometrical corrections Theory taking into account transverse shear

2

2

2

2

2

1+ 1+

h l h l

h + Rh + hl2 + lR + Rh 2 +ε + . . . , 2 h h2 h2 + R + l2 + lR + Rh 2 +ε + . . . ,

1+

h l

+

h R

+

h2 l2

h + lR + Rh 2 +ε + . . . ,

1+

h l

+

h R

+

h2 l2

+

h2 lR

2

+ Rh 2 +ε + . . . .

Here, the order of the kept and the omitted terms is shown. The name of the last theory is due to the fact that taking into account the transverse shear leads to the corrections on the order of h 2 /l 2 . Separation of the refinements of the classical theory into three separate theories is methodologically convenient; however, it should be noted that such separation is somewhat conventional. The sequence in which the refined theories are presented assumes that l ≪ R and that ε ≪ h/R. If, for example, l ∼ R, then the next approximation after the classical theory is given by the theory which takes into account the geometrical correction, and the theory which takes into account the transverse shear is asymptotically inconsistent, since it keeps the corrections on √ the order of h 2 /l 2 and omits the corrections on the order of h 2 /R 2 . If l ∼ h R, then the first correction of the classical theory is given by the main refined theory, while the next one is the theory which takes into account the transverse shear, since h 2 /l 2 ∼ h/R. In the main refined theory, it is assumed that ε ≪ h/l; in the theory which takes into account the geometrical correction and the theory which takes into account transverse shear it is assumed that ε ≪ h/l + h/R. For ε ∼ h/R or ε ∼ h/l, all the above-mentioned refined theories are asymptotically inconsistent. There is another important cause of inconsistencies of the refined theories: the forces at the shell edge are usually not known precisely; at best, we know only their resultants and moments. The self-equilibrated part of the edge forces produces the corrections to the stress state inside the shell on the order of h/l. Therefore, strictly speaking, only the classical shell theory makes sense. There are exceptions though: closed shells (they do not have an edge) and the shells with a free edge (the edge load is known precisely). We consider refined theories, keeping in mind these exceptions and the experimental evidence that in many problems the stresses caused by a self-equilibrated load at the shell edge are small. Geometrical relations. To write down the energy density, we need to know the components of the metric tensor of the Lagrangian coordinate system in the initial state. Differentiating (14.117) with respect to ξ α , and using the relations (14.39), we get i x˚ ,α =μ ˚ βα r˚βi , x˚ ,ξi = n˚ i

(14.121)

14.4


629

where μ ˚ βα ≡ δαβ − ξ b˚ αβ .

(14.122)

From (14.121), we find the components of the metric tensor i i g˚ αβ = x˚ ,α x˚ iβ = a˚ αβ − 2ξ b˚ αβ + ξ 2 c˚ αβ , g˚ α3 = 0, g˚ 33 = 1,

where c˚ αβ are the components of the third quadratic form of the middle surface: c˚ αβ = b˚ αλ b˚ λβ . The tensor, c˚ αβ , can be expressed in terms of a˚ αβ and b˚ αβ as % % % ˚ = 1 b˚ αα . ˚ b˚ αβ , K˚ = det % ˚ H c˚ αβ = − K˚ a˚ αβ + 2 H %b˚ αβ % /a, 2

(14.123)

In the principal coordinate % % system of the tensor a˚ αβ it is easy to calculate the determinant, g˚ = det %g˚ αβ %: κ≡

√ ˚ ξ + K˚ ξ 2 . g˚ / a˚ = 1 − 2 H

(14.124)

Equation (14.124) is invariant with respect to coordinate transformations; therefore, it is valid for any coordinates, ξ α . It follows from (14.124) that d V˚ = κd ωdξ, ˚ ˚ where d ω˚ is an area element on : d ω˚ =

√ ˚ 1 dξ 2 , adξ

˚ + and ˚ −, while on ˚ h + 1 K˚ h 2 d ω, ˚ d ω˚ + = 1 − H 4

˚ h + 1 K˚ h 2 d ω. d ω˚ − = 1 + H ˚ 4

The contravariant components of the metric tensor are given by the formulas g˚ αβ =

2

1 ˚ ξ a˚ αβ + 2ξ 1 − 2 H ˚ ξ b˚ αβ + ξ 2 c˚ αβ , g˚ α3 = 0, g˚ 33 = 1. 1 − 2 H κ2 (14.125)

Equations (14.125) can be obtained, for example, in the following way. Let us denote the inverse tensor of μ ˚ βα = δαβ − ξ b˚ αβ by λ˚ βα . Tensor λ˚ βα is the solution of the equations,

630

14


γ

μ ˚ βα λ˚ β = δαγ . We seek the solution of these equations in the form λ˚ αβ = Aδβα + B b˚ βα . Using (14.122), we get the linear system of equations for A and B A + ξ K˚ B = 1,

˚ ξ B = 0, −ξ A + 1 − 2 H

which yields, A=

˚ξ 1 − 2H , κ

B=

ξ . κ

Thus,

1 ˚ ξ δαβ + ξ b˚ αβ . 1 − 2H λ˚ αβ = κ

(14.126)

The components of the matrix ξ˚ia , which is inverse to the matrix x˚ ai , are ξ˚iα = λ˚ αβ r˚iβ ,

ξ˚i3 = n˚ i ;

(14.127)

(it is sufficient to check that the components of the matrix (14.127) satisfy the equalities, x˚ bi ξ˚ia = δab ). To complete the derivation of (14.125), it remains to note that g˚ ab = ξ˚ia ξ˚ bi . Free energy. Consider first a shell made of an isotropic elastic material: 2 2F = λ g˚ ab εab + 2μg˚ ab g˚ cd εab εbd = 2 2 . = λ g˚ αβ εαβ +ε33 + 2μg˚ αβ g˚ γ δ εαγ εβδ + 4μg˚ αβ εα3 εβ3 + 2με33

(14.128)

The dependence of g˚ αβ on the transverse coordinate is defined by (14.125). The components of the strain tensor are expressed in terms of the particle trajectories according to the general formulas of the geometrically nonlinear theory as i xi,β − g˚ αβ , 2εαβ = x,α

i 2εα3 = x,α xi,ξ ,

2ε33 = x,ξi xi,ξ − 1.

(14.129)

Of all the derivatives of the functions x i (ξ α , ξ, t) with respect to the coordinates ξ , ξ α , and t, the derivative x,ξi has the “largest” value because, among the corresponding scales h, l, and l/c, the scale h is the smallest one. Accordingly, the strains εα3 and ε33 are formally the largest. Therefore, it is convenient to transform the expression for energy (14.128) to the form where the leading role of the strain

14.4


631

tensor components εα3 and ε33 is taken into account. We will do this in the following way. We introduce the “longitudinal” energy, F , as the result of the minimization of the energy density with respect to the leading strain tensor components, εα3 and ε33 , F εαβ = min F εαβ , εα3 , ε33 . εα3 ,ε33

(14.130)

The longitudinal energy does not depend on εα3 and ε33 and coincides with F when the stress tensor components σ α3 = ρF/εα3 , and σ 33 = ρF/ε33 are equal to zero. We introduce also the “transverse” energy as the part of energy that vanishes due to minimization over εα3 and ε33 : F⊥ = F − F .

(14.131)

It depends on all strain components. From (14.128), (14.130) and (14.131) @ ? 2 F = μ σ g˚ αβ εαβ + g˚ αγ g˚ βδ εαβ εγ δ

F⊥ =

2 1 (λ + 2μ) ε33 +σ g˚ αβ εαβ + 2μg˚ αβ εα3 εβ3 . 2

(14.132)

The constant σ has been already introduced in (14.87). External forces. The condition of the boundedness of deformations (ε < 1) for h → 0 puts some constraints on the external forces. First of all, it is clear that the work of the external forces in the action functional I (14.119) must be of the same 2 ˚ order of magnitude as energy, i.e. με h . Since x i (ξ α , ξ, t) are finite for h → 0, in a generic case, Pi = O (h). Now we tend ε to zero. The external forces are proportional to the stresses, therefore, Pi = O (μεh). The estimate must contain a small dimensionless parameter, and h should be replaced either by h/l, h/R or the sum h/l + h/R. Hence, as a main assumption, we have to accept that the external forces on the shell faces satisfy the condition h h . + Pi = O με l R

(14.133)

Set M0 . The shell theory will be constructed by means of the variational-asymptotic method. First, we set the external forces on the shell faces equal to zero and identify the leading terms in the functional. Since the functions x i (ξ α , ξ, t) have the smallest characteristic length along the coordinate ξ , the leading term of the functional is 1 2

t1 t0 V˚

2 (λ + 2μ) x,ξi xi,ξ d V˚ dt.

(14.134)

632

14


The functional (14.134) does not hold the boundary conditions with respect to ξ α and t (see Sect. 5.5); its minimum is equal to zero. It is reached on the functions not depending on ξ : x i = r i (ξ α , t). The functions r i (ξ α , t) are arbitrary and make up the set M0 of the general scheme of the variational-asymptotic method. Further, we look for the stationary points of the form x i = r i (ξ α , t) + x ′i (ξ α , ξ, t) , where x ′i are much smaller than r i . Without loss of generality, we can put the constraints :

; x ′i (ξ α , ξ, t) = 0.

(14.135)

The angle brackets, · , the average over the thickness, were defined by (14.97). Had the functions x ′i not satisfied (14.135), the consistency with (14.135) could have been attained by the substitution r i → r i + x ′i , x ′i → x ′i − x ′i . Let us extract the leading terms containing x ′i . Due to (14.135), the interaction 1 i terms between xt′i and r,ti and the interaction terms between Aαβ = r riβ − a˚ αβ 2 α ′i and x,α in the longitudinal energy will be much smaller than the interaction terms between Aαβ and xξ′i in the transverse energy. In the first approximation, we keep in 1 ′ the transverse energy a˚ αβ instead of g˚ αβ , Aαβ instead of εαβ , and rαi xi,ξ instead of 2 εα3 . To determine x ′i , we obtain the functional 1 2

t1

2 ′ ′ j ′

rβ x j,ξ d V˚ dt. (14.136) (λ + 2μ) ε33 + σ a˚ αβ Aαβ + μa˚ αβ rαi xi,ξ t0 V˚

′ ′ Here ε33 = 12 xξ′i xi,ξ −1 . The minimum of the functional (14.136) is equal to zero and reached at the functions x ′i = n i ξ ϕ,

1 2 ϕ − 1 = −σ a˚ αβ Aαβ . 2

(14.137)

Therefore, the next term of the asymptotic expansion is completely determined by the previous one, and the set N coincides with the set M0 in the general scheme of the variational-asymptotic method and comprises functions r i (ξ α , t). The left hand side of (14.137)2 is small and it has the order of ε A . Therefore, ϕ = 1 + O (ε A ). In the leading approximation, x ′i = ξ n i .

(14.138)

14.4


633

In what follows, r i (ξ α , t) will denote the average of the functions, x i (ξ α , ξ, t) , with respect to the normal coordinate: r i (ξ α , t) = x i (ξ α , ξ, t), while by we will mean the surface defined by the equations x i = r i (ξ α , t). Let us make the following change of the required functions: x i (ξ α , ξ, t) → i y (ξ α , ξ, t): x i (ξ α , ξ, t) = r i (ξ α , t) + ξ n i (ξ α , t) + hy i (ξ α , ξ, t) .

(14.139)

The previous consideration can be viewed as a motivation for the change of the required functions (14.139). According to the definition of r i , the functions, y i , satisfy the constraints

y i = 0.

(14.140)

The formula (14.139) establishes -a one-to-one correspondence between all the func. tions, x i (ξ α , ξ, t) , and the pairs, r i (ξ α , t) , y i (ξ α , ξ, t) , where y i satisfy the conditions (14.140). Following the general scheme of the variational asymptotic method we fix r i (ξ α , t) and seek the dependence of y i on r i from the variational principle. Characteristic lengths. Let yα ≡ rαi yi and y ≡ n i yi be the projections of y i on the tangent and normal vectors, ζ the scaled normal coordinate, ζ = ξ/ h, and α ≡ max yα,ζ , ≡ max y,ζ . V˚

V˚

Due to the inequality (5.18), |yα | ≤ α , |y| ≤ , and if α = = 0, then yα = y = 0. ˚ The best constants l1 , l2 in the inequalities,13 Consider a point on the surface . Aαβ,γ ≤ ε A , h Bαβ,γ ≤ ε B , v i ≤ c (ε A + ε B ) , v i ≤ c (ε A + ε B ) , ,α lγ lγ l cα c α , y,t ≤ , (14.141) , max y,α ≤ , yα,t ≤ max yα,β ≤ ζ ζ lβ lα l l 13

Recall that we denoted by c the characteristic speed of sound in elastic material.

634

14


where ε A and ε B are defined by (14.65), will be called the characteristic lengths of the stress state along the middle surface. ˚ is defined as the best constant The characteristic curvature radius of the surface in the inequalities

1 1 αβ 2 ≤ , b˚ αβ b˚ R

21 1 ≤ 2. b˚ αβ;γ b˚ αβ;γ R

Some estimates. We need several estimates which follow from the definition of the characteristic length. Denote by m the numbers which depend neither on h nor on the stress state. The letter m may have different values in different inequalities or even in different terms of the same inequality. It is obvious that a˚ αβ ≤ m, αβ a˚ ≤ m, and, since ε A < m, also aαβ < m, a αβ < m. Moreover, ˚ γ ≤ m/d, αβ

˚ (the maximum distance between two points of ). ˚ where d is the diameter of Similarly, m ˚ bαβ ≤ , R

m ˚ bα;γ ≤ 2 . R

According to the definition of ε A and ε B , we have Aαβ ≤ mε A , Bαβ ≤ mε B , h bαβ ≤ m γ

h h + ε B , h bαβ ≤ m + εB . R R (14.142)

For the Christoffel’s symbols αβ on , it follows from (14.66) that εA γ γ αβ − ˚ αβ ≤ m , l

m εA γ +m . αβ ≤ d l

(14.143)

In what follows, l will denote the smallest of two numbers, d and l. Then, the estimate for the Christoffel’s symbols is m γ αβ ≤ . l

(14.144)

The following estimate also holds: h 2 Cαβ ≡

h h2 h2 λ cαβ −˚cαβ = h 2 b˚ (α Bλβ) + O ε2B + ε A ε B + 2 ε2A . 2 R R

(14.145)

Here, cαβ = a μν baμ bβν are the components of the third quadratic form of the surface . The relation (14.145) follows from the identity

1 1 λ cαβ −˚cαβ = b˚ (α Bλβ) + Bαλ Bβλ + (a μν − a˚ μν ) b˚ αμ + Bαμ b˚ βν + Bβν 2 2

14.4


635

and the estimate |a μν − a˚ μν | ≤ mε A . Let α < 1, < 1. Then, for sufficiently small h/l, h/R, ε, α ≤ mε,

≤ mε,

ε A ≤ mε,

ε B ≤ mε.

(14.146)

To prove the estimates (14.146) we first write down the expressions for the strain tensor components in terms of the functions r i and y i : 1 μ i i yi,β , εαβ = Aαβ − h Bαβ ζ + h 2 Cαβ ζ 2 + hr(α yi,β) − h 2 b(α rμi yi,β)ζ + h 2 y,α 2 i 2εα3 = rαi yi,ζ − hbαμ rμi yi,ζ + hn i yi,α + hy,α yi,ζ , (14.147) 1 ε33 = n i yi,ζ + y,ζi yi,ζ . 2 Let us rewrite the definition of ε33 as 1 1 y,ζ 1 + y,ζ = ε33 − a αβ yα,ζ y,ζ . 2 2 Since max y,ζ = < 1, 1 + y,ζ /2 ≥ 1/2. Therefore, ζ

y,ζ ≤ 2 |ε33 | + a αβ α β ≤ 2 |ε33 | + m 2 + 2 . 2 1

(14.148)

The definition of εα3 can be written as

yα,ζ = 2εα3 + hbαμ ζ yμ,ζ − hy,α − hbαμ yμ − y,α + bαλ yλ y,ζ − ha μλ yμ|α − bμα y yλ,ζ .

(14.149)

1 + 2 . Using According to (14.141) and (14.144), yμ|α = yμ,α − λμα yλ ≤ m 2 also (14.141), (14.142), (14.143) and (14.144), from (14.149) we get h h h + ε B (1 + 2 ) + + ε B (1 + 2 ) + R l R h h h h 2 + + ε B (1 + 2 ) + m . +m (1 + 2 ) ≤ 2 |εα3 | + m l l R l

yα,ζ ≤ 2 |εα3 | + m

From this, using (14.148), we have 1 + 2 ≤ 2 (|ε13 | + |ε23 |) + m

h h h + + ε B (1 + 2 ) + m |ε33 | . l R l

Consequently, for sufficiently small values of h/l + h/R + ε B , 1 + 2 ≤ mε. It then follows from (14.148) that ≤ mε.

636

14


Setting ζ = 0 in the equation for εαβ (14.147), we get the following estimate of Aαβ :

Aαβ = εαβ − h y(α|β) − bαβ y − 1 h 2 y,α + bλ yλ y,β + bμ yμ α β 2 1 − h 2 a μν yμ|α − bμα y yν|β − bνβ y ≤ 2 ζ =0 2 h h h h + εB + m + + ε B (1 + 2 ) ≤ mε + m (1 + 2 ) + m l R R R 2 h h + ε B ≤ mε . +m (1 + 2 ) + l R Therefore, the third relation (14.146) holds. The fourth relation (14.146) can be derived analogously. In order to do that, Bαβ should be determined from the equation which is obtained by multiplying the equation for εαβ (14.147) by ζ and integrating over ζ . In what follows, it is assumed that the conditions α < 1 and < 1, for which the estimates (14.146) have been proved, are satisfied. Calculation of yi . Let us find y i . The action functional contains many terms after the substitution (14.139) is made. In order to find the leading terms and the secondary ones we have to know, first of all, the order of yα and y. To determine the orders of yα and y, consider an approximate expression for strains: εαβ = Aαβ − h Bαβ ζ,

2εα3 = yα,ζ + hy,α ,

ε33 = y,ζ .

(14.150)

For such strains, the functions yα and y enter only in the transverse energy. Besides, we ignore for the moment the kinetic energy and the work of external forces. In the first approximation, we can set κ = 1, g˚ αβ = a˚ αβ . The terms of the action functional containing yα and y are

h

t1

t0 ˚

J=

F⊥ d ωdζ ˚ =h

t1

J d ω, ˚

(14.151)

t0 ˚

2 1 (λ + 2μ) y,ζ + σ Aαα − σ h Bαα ζ + μa˚ αβ yα,ζ + hy,α (α → β) . 2 (14.152)

The functional (14.151) does not keep the boundary conditions, and the problem is reduced to minimizing J with the constraints yα = y = 0. The minimum of J is equal to zero and is reached at the functions

14.4


yα =

1 1 1 1 β β − σ h 2 Bβ,α ζ ζ 2 − , σ h Aβ,α ζ 2 − 2 12 6 4 1 1 . y = −σ Aαα ζ + σ h Bαα ζ 2 − 2 12

637

(14.153)

Consequently, y ∼ ε A + εB ,

yα ∼

h (ε A + ε B ) . l

(14.154)

Now, let us check whether the use of the approximate expressions for strains (14.150) and omitting of the kinetic energy and the work of external forces was legitimate. It is seen from (14.147) that, due to the estimates (14.154), all dropped terms in the expressions for εα3 and ε33 make small corrections to the terms kept in (14.150). In the expression for εαβ , the term h 2 Cαβ ζ 2 may be omitted due to the estimate (14.145). The terms in kinetic energy, containing y, are small in comparison to those i yi,β) = hy(α|β) − hbαβ y have the smallest in J . Among the terms containing y, hr(α order of magnitude. Consider the interaction terms brought about by this term in i the expression for energy (the quadratic term h 2 y,α yi,β , according to the definition 2 of the characteristic lengths (14.141), is small compared to the terms yα,ζ and 2 y,ζ in J ): ; : h g˚ αγ g βδ Aαβ − h Bαβ hyδ|γ .

Note that, according to (14.125), for small h/R, 2 h αβ αβ αβ ˚ g˚ = a˚ + 2ξ b + O , R2 2 1 1 √ h 1 ˚ αβ 1 , bαβ ≡ b˚ αβ − H a˚ . g αβ ≡ κ g˚ αβ = a˚ αβ + 2ζ bαβ + O 2 R 2

(14.155)

: ; Due to the constraint yα = 0, we have Aαβ yδ|γ = 0. The leading interaction 1 αγ βδ 2 term between yα and Aαβ is of the form h a˚ b Aαβ yδ|γ , and therefore is small

compared to the interaction term : between ; yα,ζ and y,ζ in J . The leading interaction term between yα and Bαβ , h 2 Bαβ yδ|γ , is of the same order of magnitude as the interaction term between yα,ζ and y,α in J . Therefore it should be kept for the correct calculation of yα . The quadratic term in kinetic energy, ρ0 h 2 y,ti yi,t , may be ignored as small com pared to the terms μ y,ζα yα,ζ + y,ζ2 in J . According to (14.124), (14.140) and : ; (14.141), the interaction term ρ0 h r,ti yi,t κ in kinetic energy is on the order of (ε A + ε B ) ( + 1 + 2 ) and is considerably smaller than the terms in J ; therefore, it should also be dropped.

638

14


Let us project the external force at the shell faces on the normal and tangent directions to the surface, : P i = Pn i + P α rαi . According to (14.133), the work, P y, will contribute a term on the order of h h μ ε2 and is not significant for the determination of y in the first approx+ l R imation. The work, P α yα , is one the order of με2 (h/l)2 and makes a contribution on the order of μy,ζα yα,ζ . Therefore it has to be taken into account. Retaining the above-mentioned term does not change the order of yα and y and the equation for y (14.153). So the estimates (14.154) hold. From these estimates and the formulae (14.147), it follows that εαβ ≤ m (ε A + ε B ) , |εα3 | ≤ m h (ε A + ε B ) , |ε33 | ≤ m (ε A + ε B ) . l

Therefore, ε ≤ m (ε A + ε B ) and, as can be seen from (14.146), the measures of smallness of deformations ε A + ε B and ε are asymptotically equivalent. Classical shell theory. In calculation of energy with the accuracy of the classical shell theory, due to (14.153) and the estimates (14.154), 1 1 , y = −σ Aαα ζ + σ h Bαα ζ 2 − 2 12

yα = 0.

Moreover, it can also be set g˚ αβ = a˚ αβ and κ = 1. Hence, in the first approximation, the transverse component of the elastic energy is equal to zero, and total energy coincides with the longitudinal energy calculated for the deformations εαβ = Aαβ − h Bαβ ζ,

(14.156)

or εαβ = Aαβ − hραβ ζ, or εαβ = Aαβ − hκαβ ζ. After dropping the interaction terms between the strain and bending, we get one of the expressions for given in Sect. 14.2. The surface kinetic energy density is K =

1 ρ0 hr,ti ri,t . 2

14.4


639

In the expression for the work of the external forces on the shell faces, we can keep only the terms related to work on r i + ξ n i ; therefore,

˚±

h i i {Pi } r + [Pi ] n d ω, ˚ Pi x d ω˚ ± = 2 i

(14.157)

˚

where {Pi } = Pi |ξ = h + Pi |ξ =− h , [Pi ] = Pi |ξ = h − Pi |ξ =− h . 2 2 2 2 Varying the functional (14.157) one should employ the formula for the variations of the normal vector (14.47). Comparing the variation of the functional (14.157) with (14.59), we find the generalized force, Q i , in terms of the surface forces: Q i = {Pi } +

h 0 1 jα Pj r ni 2

.

(14.158)

;α

The displacements, in case they are given at the shell edge, are assumed to be i consistent with the asymptotic expansion. This means that the boundary values, x(b) , as well as the particle trajectories (14.139), can be presented as i i x(b) = r(b) (s) + n i(b) (s) ξ + h O (ε) .

In the first approximation, this leads to the boundary conditions (14.42). If the region V1 is adjacent to the shell boundary and has the width h, then the ˚ 2 energy of the body in this region is on the order of μh 2 ε and can be ignored in comparison with the free energy of the shell. Therefore, the “edge beam” problem does not appear in the classical approximation. The leading terms of the work of external forces at the shell edge are determined by the work on r i + ξ n i . Therefore, the generalized edge forces of the shell theories, Ri and M, are related to the surface forces by the formulae Ri = h P i +

d (h Pk ξ τ κ n i ) , ds

M = h Pi ξ ν i .

(14.159)

The three-dimensional strain field is constructed from the two-dimensional characteristics according to the equations εαβ = Aαβ − h Bαβ ζ . The components εα3 , which characterize the angle between the normal to and the normal fiber, are much smaller and have the order of εh/l. The strain component, ε33 , is defined by the equation, σ 33 = 0. Note that the equalities i i rα,t = v,α ,

k n i,t = −r iα n k v,α

(14.160)

640

14


and the estimates (14.141) yield the relations ρ0 vi,t rαi = ρ0 vα,t + O ρ0 c2 ε2 /l ,

ρ0 vi,t n i = ρ0 v,t + O ρ0 c2 ε2 /l , (14.161)

where vα = rαi vi , v = n i vi . Therefore, the right-hand sides of the dynamic equations ¯ ,t . (14.67) can also be written as ρv ¯ α,t and ρv This completes the justification of the classical geometrically nonlinear and physically linear shell theory.

14.5 Short Wave Extrapolation Before proceeding to the derivation of the refined shell theories an important notion of short wave extrapolation has to be introduced. The stress states for which h/l ≪ 1 are called the long wave states. The shell theory is derived in the region of long waves with h/l ≪ 1 and is a theory of long wave stress states. However, for a complete theory, it is not sufficient to have the equations written only for a long wave range, because the formulation of the boundary value problems of mathematical physics assumes that the differential equation are defined for waves of all wave lengths. Moreover, the type of equation (hyperbolic, elliptic, parabolic), and, consequently, the character of the natural boundary and initial conditions, is defined by the behavior of the differential operator for short waves: the type of differential equation is determined by the highest derivatives, and it is precisely the case of short waves when the highest derivatives make the leading contribution. Therefore, besides the derivation of the equations in the long wave range, the construction of a shell theory contains another logically independent step – the extrapolation of the equations to the short waves, i.e. to the stress states for which the equations derived are certainly not valid. Suppose that we obtain some approximate system of equations in a long wave range from asymptotical reasoning. All terms, which are small within the approximation considered, are dropped. We can apply the equations obtained both for long and short waves. We call such an extrapolation trivial. The nontrivial extrapolation supposes adding some terms which are small in the long wave range, but significant for the short waves. Consider an example. Let us take two differential equations: 2 2 u 2 u − a = 0, t 2 x 2

2 4 2 u 2 u 2 2 u − a + a h = 0. t 2 x 2 x 4

For long waves, these equations coincide in the first approximation; since a2h2

h 2 2 2 u 4 u ∼ a , x 4 l 2 x 2

(14.162)

14.5

Short Wave Extrapolation

641

the term a 2 h 2 4 u/x 4 in the second equation can be dropped. For short waves, the equations are different: in the second equation the term a 2 2 u/x 2 is small compared to a 2 h 2 4 u/x 4 , and for short waves the second equation becomes the equation for the lateral beam vibrations: 4 2 u 2 2 u + a h = 0. t 2 x 4

The second equation (14.162) can be considered as one of the possible ways of nontrivial short wave extrapolation of the first equation (14.162). The two equations (14.162) are significantly different: the first equation requires two boundary condition, the second requires four. Equations of the shell theory are derived for long wave stress states. It is clear that various nontrivial extrapolations are possible for the same system of equations of shell theory. This should not be perplexing, because short wave stress states do not admit two-dimensional description, and only a qualitative approximation or, maybe, a satisfactory description of the integral characteristics, may be expected. Above, in constructing the classical shell theory, it was implicitly assumed that trivial extrapolation is carried out. Luckily, the trivial extrapolation yields a sensible system of equations.14 In constructing the refined theories, the situation is more complex due to the possibility of equivalent in accuracy, but different in form, systems of equations, so that trivial extrapolation is acceptable for some of them, but not for the others. As an example, consider linear equations describing lateral plate vibrations. In the first approximation, the normal displacement component, u, obeys the equation 2 u + a 2 h 2 2 u = 0, t 2

a2 =

μ (λ + μ) . 3 (λ + 2μ) ρ

(14.163)

The refined equation for u is [24, 25] 2 2 u 2 2 2 2 u + a h u − αh = 0, t 2 t 2

α=

1 2

9 2μ . − 10 3 (λ + 2μ)

(14.164)

The last term in (14.164) is on the order of h 2 /l 2 compared to the previous one for long waves, since, according to (14.163), in the first approximation, 2 u ∼ a 2 h 2 2 u, t 2 and, therefore,

14

This is not always the case for the first approximation. For example, trivial extrapolation of the equations for high frequency long wave shell vibrations brings elliptic equations (see [29, 34, 173]).

642

14

αh 2


2 u h2 2 2 2 4 2 3 ∼ αh a u ∼ αh a u. t 2 l2

The term αh 2 2 u/t 2 is small; hence, without loss of accuracy, we can replace 2 u/t 2 in this term by its value in the first approximation from (14.163), −a 2 h 2 2 u. We arrive at the equation 2 u + a 2 h 2 2 u + αa 2 h 4 3 u = 0. t 2

(14.165)

Consider now the equation 2 4 2 u 2 2 2 2 u 2 u + a h u − αh + βh = 0. t 2 t 2 t 4

(14.166)

For long waves, (14.164) and (14.166) differ in the term βh 2 4 u/t 4 , which is on the order of h 4 /l 4 a 4 h 2 2 u (since 4 u/t 4 ∼ a 2 h 2 2 2 u/t 2 ∼ a 4 h 4 4 u). Consequently, for long waves, the three equations (14.164), (14.165) and (14.166), which refine the equation of the first approximation (14.163), have the same accuracy. However, for short waves, the differences are significant: Equations (14.164), (14.165) and (14.166) require different numbers of boundary and initial conditions; equation (14.166), unlike equations (14.164) and (14.165), is hyperbolic for a proper range of parameter, β. Which extrapolation of (14.164), (14.165) and (14.166) should be preferred? Many computations of test problems show that the hyperbolic extrapolations on the whole better describe the stress state than the non-hyperbolic ones. The reason is that hyperbolic extrapolations can better capture the features of the dispersion curves in the short wave range. This turns out to be significant for the correct prediction of the behavior of the integral characteristics. In constructing the refined shell theories by the variational-asymptotic method, the short wave extrapolations are made in the following way. First, the elastic and kinetic energies are found from the asymptotic analysis of the action functional of the three-dimensional elastic theory. Then, by means of a few operations (change of the required function, integration by parts, etc.), which do not violate the asymptotic accuracy, the expressions for the elastic and kinetic energies are put into a form convenient for extrapolations. Such a form supposes positive definiteness and sufficient simplicity for all wavelengths. Postulating these expressions for the energies, the system of equations and boundary conditions are found from the least action principle.

14.6 Refined Shell Theories We begin with the formulation of the results of the asymptotical analysis, then the derivation will follow.

14.6

Refined Shell Theories

643

Theory taking into account the geometrical correction. Constructing more accurate approximations of the stress state, one has to take into account that the refined equations depend on small changes in the choice of the required functions. For example, either the position vector of the middle surface, x i (ξ α , 0, t) , or the average position vector of the transverse fibers, r i (ξ α , t) = x i (ξ α , ξ, t), or the position vector of the shell face + , x i (ξ α , h/2, t), or − , x i (ξ α , −h/2, t), etc., can be taken as the basic kinematic characteristics of the shell deformation. Within the accuracy of the classical shell theory, it is not essential to know which one of these characteristics is used: the equations are the same. However, the equations of refined shell theories are different. It is natural to choose the required functions in such a way as to make the energy expression, and, consequently, the equations, as simple as possible. Further, the functions r¯ i (ξ α , t) = r i (ξ α , t) −

σ h2 i μ n ρμ 60

(14.167)

are chosen as the required functions. The reason for such a choice will become clear later. According to (14.167), the tangent components of r i (ξ α , t) and r¯ i (ξ α , t) coincide, while the normal components are different. It will be seen from the further derivation that the normal component n i r¯ i (ξ α , t) has the meaning of the function n i x i (ξ α , ξ, t), averaged with the weight, ξ 2 − h 2 /4 : n i x i (ξ α , ξ, t) ξ 2 − n i r¯ i (ξ α , t) = 2 ξ 2 − h4

h2 4

.

We denote the extension and the bending measures, constructed for the functions r¯ i , by γαβ and ραβ , respectively: γαβ =

1 i r¯,α r¯i,α −a˚ αβ , 2

λ γλβ) . ραβ = n¯ i r¯ i ,αβ − b˚ αβ − b˚ (α

(14.168)

Here, n¯ i are the components of the unit vector normal to the surface x i = r¯ i (ξ α , t). Using the formulae for variations (14.49), it is easy to see that γαβ = Aαβ + O

h ε , R

and the tensors ραβ , defined by (14.168) and (14.73), differ by the terms of the 2 negligible order h 2 ε/l ; this is why we use for both tensors the same notation. The energy density in the theory taking into account geometrical correction is the sum of the energy density of classical theory, cl (γ , ρ) , and the interaction energy between bending and extension, int (γ , ρ) :

644

14


= cl (γ , ρ) + int (γ , ρ) ,

μh 3 2 2 σ ραα + ραβ ρ αβ , cl = μh σ γαα + γαβ γ αβ + 12 3 μh 3 int (γ , ρ) = − ρ αβ b′λa γβλ +σρ αβ b˚ αβ γλλ + σρλλ b˚ αβ γαβ 3 5

6 ˚ γμμ . σ −1 ρλλ H (14.169) +σ 5 ˚ b′λ = b˚ λ − Here, bα′λ is the deviator of the second quadratic form of the surface : α α λ ˚ H δα . Additional terms appear in the work of external forces, Asurf , on the shell faces: Asurf =

˚

=

˚ [Pi ] r¯ i + 1 [Pi ] h n¯ i − d ω, {Pi } − h H ˚ 2

σh σh {P} ρλλ , [P] γλλ − 2 10

P ≡ Pi n i .

(14.170)

The equations of motion in the theory taking into account geometrical corrections are the same as in the classical theory and have the form (14.67). There are changes in the expression for the external forces: 0 1 ˚ [Pi ] + 1 h P α n¯ i . Q i = {Pi } − h H ;α 2 The changes in the constitutive equations are as follows. In the refined theories, the constitutive equations describe the appearance of the bending moment caused by the extension of the shell and the reciprocal effect of the appearance of the stress resultants caused by the bending of the shell: ( + ) = 2μh σ γλλ a˚ αβ + γ αβ − γαβ 1 3 (αλ ′ β) 3 λ ˚ αβ 6 μν αβ ˚ λ ˚ − μh ρ b λ + σ a˚ bμν ρ + σ −1 H ρλ + σρλ b 3 5 5 1 + σ h [P] a˚ αβ , 2

( + ) 1 αβ − m αβ = = μh 3 σρλλ a˚ αβ + ρ ραβ 6 1 3 (αλ ′ β) 3 ˚ μν 6 λ ˚ αβ αβ λ ˚ − μh γ b λ + σ a˚ b γμν + σ −1 H γλ +σ γλ b 3 5 5 1 (14.171) + σ h 2 {P} a˚ αβ . 10 n αβ =

14.6


645

Generally speaking, the boundary conditions also require refinement (see [30]). However, for many problems they are not significant computationally, and it is possible to use the natural boundary conditions, i.e. the boundary conditions of the classical theory. Accuracy loss in classical shell theory. The theory taking into account the geometrical correction in some problems becomes the theory of the first approximation. This is for the following reason. It is known (see [143, 151, 152, 220, 280]) that the classical shell theory correctly predicts the stress state (at least in the energy norm) and, consequently, the tangential strains, εαβ . As has been shown in the previous section, in the first approximation, εαβ = γαβ − ζ hραβ . If γαβ ∼ hραβ , both terms in the expression for εαβ are on the same order of magnitude and, in the classical theory, γαβ and ραβ , and, consequently, the displacements are predicted correctly. However, if γαβ ≫ ραβ , then the terms containing ραβ move to the category of the correcting terms, and, in order to find the actual values of ραβ , it is necessary to know the corrections on the order of h/R. In the classical theory, these corrections are dropped. Therefore, the classical theory cannot guarantee that the values of ραβ , and, consequently, of the displacements, are predicted correctly. The class of problems for which the theory taking into account geometrical correction becomes the theory of the leading approximation is characterized by the following conditions: (1) the shell is non-shallow (otherwise, the interaction energy int is small); (2) the shell is highly stretched (γαβ ≫ hραβ ); (3) the shell is not “well clamped” (it admits infinitesimally small bending). The meaning of the last condition has pure energetic origin. If, due to the boundary conditions, the shell does not have infinitesimally small bending (in the linear theory), then ραβ are completely defined by γαβ , and the interaction energy is small compared to cl due to the inequality (14.95). If an infinitesimally small bending exists, then ραβ acquire additional degrees of freedom, which are functionally independent of γαβ . In order to determine these degrees of freedom, it is necessary to minimize the energy with γαβ held fixed. In this case, the interaction energy makes a contribution of the same order of magnitude as the bending energy. The simple examples illustrating the loss of accuracy of classical shell theory were constructed in [62, 211]. The theory taking into account the transverse shear. This theory has two new required functions, ϕ¯ α (transverse shears). The transverse shears, ϕ¯ α , relate to the particle trajectories by the formulae

ϕ¯ α = ϕα −

σ h2 μ ρ , 60 μ,α

ϕα ≡

rαi xi (ξ α , t) ξ . h 2 /12

(14.172)

The formula for the extension measure does not change, while the bending measure is now the tensor: i λ ραβ = n¯ i r¯,αβ − b˚ αβ − b˚ (α γλβ) − ϕ¯ (α;β) .

(14.173)

646

14


One can check that the tensor (14.173) differs from the tensor (14.73) by the 2 2 terms which go beyond the accuracy of the theory (terms of the order o h ε/l ). The energy density, , is = 1 (γ , ρ) +

5 μh ϕ¯ a ϕ¯ a , 12

(14.174)

where 1 (γ , ρ) is the function (14.169). Now we can explain why (14.167) and (14.172) were chosen as the required functions. It is precisely for these functions that the theory taking into account transverse shear has an especially simple form, and the complication of the expression for which accompanies an increase in accuracy consists only in adding the transverse shear energy in (14.174) and modifying the bending measure (14.173). The work of the forces on the shell faces is

˚ [Pi ] r¯ i + 1 [Pi ] h n¯ i − d ω, {Pi } − h H ˚ Asurf = 2 ˚

σh 1 0 α1 1 1 β β α 2 {P} {P } = + h P ;α ρβ − σh [P] + h ;α γβ − 2 6 10 12 5 0 α1 P h ϕ¯ a , P α = r¯ iα Pi . − (14.175) 12

Taking the variation of the energy functional results in the system of equilibrium equations t qα

;α

αβ ;β

1 0 β1 ˚α ˚ δβα , h P bβ + 2 H (14.176) 2 1 0 1 ˚ [P] , + {P} + h P α ;α = h H t αβ = n αβ + b˚ σ[α m σβ] , 2 αβ q α + m ;β = Q,

− q β bβα + {P α } =

+ t αβ bαβ

and the constitutive equations 0 1 1 1 αβ σ h 2 {P μ };μ a˚ αβ , m αβ = m 1 − σ h 3 P μ ;μ a˚ αβ , (14.177) 12 120 5 5 0 α1 α α q = μh ϕ¯ − h P , 6 12 αβ

n αβ = n 1 +

αβ

αβ

where n 1 and m 1 are given by (14.171) An extension to dynamics is considered here only in the case of the low frequency vibrations in the linear approximation. Let the displacements of the shell, u i = r¯ i − r˚ i , be infinitesimally small. We introduce the new required functions instead of

14.6


647

ϕ¯ α , the rotation angles ψα = ϕ¯ α − n˚ k u k,α . The extension and the bending measures are expressed in terms of the required functions as i γαβ = r˚(α u i,β) ,

1 1 ραβ = −ψ(α;β) + u [λ,α] b˚ βλ + u [λ,β] b˚ αλ . 2 2

(14.178)

Here, 2u [λ,α] = u λ,α − u α,λ , u α = r˚αi u i . It is shown further in this section that, within the accuracy of the theory taking into account transverse shear, the kinetic energy density is 2 1 1 2 2 λ 2 1 i 2 α . 1 − σ h ψ,t ψα,t + σ h u ,λt K = ρh u ,t u i,t + 2 12 5 12

(14.179)

Therefore, the variation of the energy functional yields the equations 1 ρh 3 σ 2 △u α ,tt = 0, 12 q α;α + t αβ b˚ αβ − ρhu ,tt = 0, 2 1 αβ 3 α q + m ;β + ρh 1 − σ ψ α ,tt = 0, 12 5 5 t αβ = n αβ + b˚ σ[α m σβ] , qα = μh ψα + n k u k,α , 6 t

αβ ;β

− q β bβα − ρhu α,tt +

(14.180)

where n αβ and m αβ are linked to the extensional and bending measures by the formulae (14.171), and △ is the surface Laplace’s operator. For simplicity, it is assumed in (14.180) that the external forces Pi on the shell faces are equal to zero. Taking into account Pi in (14.180) is simple: they enter these equations in the same manner as the static equations (14.176) and (14.177). The asymptotically exact boundary conditions are not known for the theory taking into account transverse shear; however, many numerical simulations show that employing the natural boundary conditions results in sufficiently good accuracy. Now we proceed to the derivation of the relations summarized above. Surface energy density in refined shell theories. We shall construct the energy 2 expression retaining the terms on the order of h/l, h/R, and h 2 /l , and omitting 2 2 2 the terms on the order of ε, h /l R, h /R and of the higher orders. To do that, it is necessary to calculate the functions yα which, as has been established, are on the order of hε/l. The determination of yα involves retaining the terms on the order of hε (1 + 2 ) /l in the expression for the elastic energy. The terms of such order of magnitude are present in longitudinal as well as in transverse energies. In order to simplify the calculations by not taking into account the terms in the longitudinal energy, instead of the substitution (14.139), we make the substitution x i (ξ α , ξ, t) = r i (ξ α , t) + ϕ i (ξ α , t) ξ + hz i (ξ α , ξ, t) .

(14.181)

648

14


The functions r i and ϕ i are assumed to be independent prescribed functions for which we will construct further the two-dimensional problem. For given functions r i and ϕ i we seek the functions, z i . The presence of three additional independent required functions, ϕ i , allows us to put six constraints on the “remainder,” z i :

z i = 0,

z i ξ = 0.

(14.182)

The functions r i and ϕ i have a simple meaning: they are the average position vector and the average “rotation” of the transverse fibers: r i = x i (ξ α , ξ, t),

ϕi =

x i (ξ α , ξ, t) , ξ . h 2 /12

Equation (14.181) establishes correspondence between all functions x i - i ia one-to-one . α i (ξ , ξ, t) and all triples r , ϕ , z , where z i satisfies the constraints (14.182). From the previous analysis, it follows that the transverse shear ϕα = rαi ϕi has the order εh/l, while the transverse elongation ϕ − 1 ≡ ϕi n i − 1 is in the order of ε. Assuming that r i and ϕ i are known, we accept that ϕα ∼ hε/l, and seek for the functions, z i . The constraints (14.182) cause the interaction terms between z i and r i , ϕ i in the longitudinal energy to be negligibly small. In constructing the approximate theories of higher accuracy, we need to estimate the higher derivatives of the required functions. Therefore, the inequalities containing higher derivatives are introduced into the definition of the characteristic lengths. Generally, adding inequalities decreases the value of characteristic lengths. In the case of the substitution (14.181), it is enough to add the estimates for the derivatives of the new degrees of freedom, ϕα and ϕ. In what follows, the characteristic length, l, is the best constant in the inequalities Aαβ,γ ≤ ε A , Bαβ,γ ≤ ε B , l l

|ϕα | ≤

h (ε A + ε B ) , l

i r ≤ c (ε A + ε B ) , ,t

(14.183) i c (ε A + ε B ) i c (ε A + ε B ) r ≤ r ≤ |ϕ − 1| ≤ ε A + ε B , , , ,αt ,αβt l l2 i c (ε A + ε B ) ε A + εB ϕ ≤ ϕ,α ≤ ϕα,β ≤ h (ε A + ε B ) , , , ,t l2 l l i c¯ ¯ ¯ z ≤ max z ,γ ≤ , . max z α,γ ≤ , ,t ζ ζ l l l Here, z α = rαi z i , z = n i z i , ¯ = max z 1,ζ + z 2,ζ + z ,ζ . Since ζ

z i = y i − r iα ϕα ζ − (ϕ − 1) n i ζ,

the quantity ¯ is on the order of ε. In terms of r i , ϕ i , and z i , the strain tensor components are

14.6


649

1 (ϕ) (ϕ) 2 i i i εαβ = Aαβ − h Bαβ ζ + h 2 Cαβ ζ + hr(α z i,β) + hϕ(α z i,β) ζ + h 2 z ,α z i,β , 2 1 i i 2εα3 = z α,ζ + ϕα + hϕ i z i,α + h ϕ i ϕi ,α ζ + hϕ,α z i,ζ ζ + hz ,α z i,ζ , 2 1 i 1 i ϕi ϕ − 1 + z ,ζ z i,ζ . (14.184) ε33 = z ,ζ + 2 2

Here,

(ϕ) i = −r(α ϕi,β) − b˚ αβ , Bαβ

Note that

(ϕ) = Cαβ

1 i ϕ,α ϕi,β − c˚ αβ . 2

h h (ϕ) ε , + = Bαβ − ϕ(α|β) + (ϕ − 1) bαβ = Bαβ + O Bαβ l R h2 (ϕ) ˚ λ (ϕ) h 2 Cαβ = h 2 Bλ(α bβ) + O ε2A + ε2B + (ε A + ε B ) . Rl From (14.181) follow the estimates i i k ¯ 1+ 1 , z = n n z k,α + r iβ r k z k,α ≤ m β ,α l R i ϕ − n i = n i n k ϕk,α + r iβ r k ϕk,α + r iβ bβα ≤ m (ε A + ε B ) 1 + 1 , ,α ,α β l R i 1 1 ¯ h 2 ϕ,α ε. − n i,α z i,β ≤ m + l R

These estimates show that the last two terms in the formula for εaβ , from which 1 1 2 + the terms h 2 n i,α z i,β are deducted, are on the order of ε2 , and can thus l R i z i,β) and h 2 n i(,α z i,β) ζ , be dropped. Due to the constraints (14.182), the terms, r(α 2 h h make a contribution to the longitudinal energy on the order of ε A + εB R2 R h h ¯ . Since, in the transverse energy, there are interaction terms between + l R h ¯ z i and r,i ϕ i on the order of ε, the interaction terms in the longitudinal enl ergy can be dropped. Analogously, due to (14.182) and the estimates (4.2.21a), the : ; : ; : ; h2 terms, h ρ0r,ti z i,t κ , h 2 ρ0 ϕ,ti z i,t κ and h 2 ρ0 z ,ti z i,t κ , are of the order ρ0 c2 ε2 , lR 2 3 2 2h 2 2 h ρ0 c ε 2 and ρ0 c ε 2 , respectively. Therefore, in determining the first approxil R l mation of z i , these terms enter only in the expression for transverse energy. For the components of the deformation tensor εαβ , the following approximate expression can be used:

650

14


(ϕ) (ϕ) 2 εαβ = Aαβ − h Bαβ ζ + h 2 Cαβ ζ .

(14.185)

In the transverse energy, a simplification is also possible by droppingthe terms i h h i i ¯ + hϕ,α z i,ζ ζ and hz ,α z i,ζ ζ in the formula for εα3 (since hϕ,α z i,ζ ζ ≤ m l R h ¯ i z i,ζ ζ ≤ m (h/l+h/R) 2 ), and by the changes of (ε A +ε B ) + m , and hz ,α R i 2 i i ϕ i −1 by 2 (ϕ− 1), and ϕ z i,α by z ,α , (since ϕi ϕ − 2 (ϕ − 1) ≤ m (ε A +ε B ) and ϕ ϕ i z i,α − n i z i,α ¯ Besides, the approximate expressions g˚ αβ = a˚ αβ and ≤ 1l (ε A +ε B )2 + m hl ε). αβ αβ g˚ = a˚ can be used for the metric tensor components, since the values for z i are sought in the first approximation. Taking into account the work of the external forces on z i , to determine z i , we get the functional h (J∠ (z α , z) + J⊥ (z)) d ω, ˚ (14.186) ˚

1

1 J∠ (z α , z) = 2

2

− 12

μa˚ αβ z α,ζ + ϕα + hϕ,α ζ + hz ,α (α → β) dζ − {P α z α } ,

1

1 J⊥ (z) = 2

2

− 12

2 (λ + 2μ) z ,ζ + ϕ − 1+σ Aαα −σ h B αα ζ dζ − {P z} .

˚ The functional (14.186) does not feel the kinematic boundary conditions on ; therefore, determination of z α and z is reduced to minimizing the sum J∠ + J⊥ at ˚ with respect to z α and z, which are considered to be each point of the surface arbitrary functions of ζ , satisfying the constraints

α

z α = 0,

z α ζ = 0,

z = 0,

zζ = 0.

Coordinates ξ play the role of parameters in this variational problem. In order to calculate z α , it is necessary to know the function z in the first approximation. The interaction terms between z and z α in J∠ are small compared to interaction terms between z and Aαβ , h Bαβ . Therefore, the first approximation for z is found by minimizing the functional J⊥ with respect to z. Let us make the change of variables, z → z¯ : 1 σ + z¯ . (14.187) z = h Bαα ζ 2 − 2 12 Then J⊥ = −

2 1 σ {P}h Bαα + J¯⊥ , J¯⊥ =

(λ + 2μ) z¯ ,ζ + ϕ¯ − {P z¯ } 12 12

14.6


651

where ϕ¯ = ϕ − 1 + σ Aαα . The function z¯ satisfies the conditions ¯z = 0, ¯z ζ = 0. Introducing the Lagrange multipliers, ρ and π , for the minimizing element we get the following problem: d (λ + 2μ) z¯ ,ζ + ϕ¯ = ρ + π ζ, ¯z = 0, ¯z ζ = 0, dζ (λ + 2μ) z¯ ,ζ + ϕ¯ ζ = 1 = P+ , (λ + 2μ) z¯ ,ζ + ϕ¯ ζ =− 1 = −P− . (14.188) 2

2

1 The indices ± mark the values at ζ = ± . Thus,15 2 1 1 + ρζ + a, (λ + 2μ) z¯ ,ζ + ϕ¯ = π ζ 2 − 2 4

a=

1 [P], 2

ρ = {P}.

(14.189) Multiplying (14.189) by ζ 2 − 1/4, integrating over ζ and taking into account that 2 2 : ζ −2 1/4 ;= 1/30 and, due to the condition ¯z ζ = 0, the equality holds z¯ ,ζ ζ − 1/4 = 0, we get

1 π = −10 (λ + 2μ) ϕ¯ − [P] . 2

From (14.176), we find z¯ : 1 5 1 3 3 5 ϕ¯ + [P] ζ ζ 2 − + {P} ζ 2 − . z¯ = − ζ ζ 2 − 3 20 2(λ + 2μ) 3 20 12 (14.190) Since, according to (14.188),

2 ; : d {P z¯ } = (λ + 2μ) − (λ + 2μ)ϕ¯ z¯ ,ζ + ϕ¯ , z¯ z¯ ,ζ + ϕ¯ = (λ + 2μ) z¯ ,ζ + ϕ¯ dζ we have 2 1 1 + ϕ¯ a − π = J¯⊥ = − (λ + 2μ) z¯ ,ζ + ϕ¯ 2 12 < 2 = 1 1 1 1 2 =− π ζ − + ρζ + a + ϕ¯ a − π , 2 (λ + 2μ) 2 4 12 and, finally

15

Recall the definitions of the symbols [·] and {·}, due to which [P] = P+ − P− ,

{P} = P+ + P− .

652

14

min J¯⊥ =


5 1 ¯ (λ + 2μ) ϕ¯ 2 + ϕ[P]. 12 12

(14.191)

The terms quadratic with respect to the components of the external surface forces are omitted, because, as is easy to check, their variation contributes the terms on the order of (h/l)2 ε2 in comparison with unity. In the linear theory, the terms depending only on the external forces are not varied, and play the role of additive constants in the energy functional. Suppose that ϕ¯ = O (hε/l); we shall check further that this is really the case. Then, according to (14.190) and the condition Pi = 0 (με (h/l + h/R)), we see that z¯ = O (hε/l), and, in the first approximation, z is σ 1 z = h Bαα ζ 2 − . (14.192) 2 12 Substituting the expression (14.192) into the formula for the functional J∠ , we find that z α is the minimizer of the functional 1

J∠ =

1 2

2

− 21

1 σ β μa˚ αβ z α,ζ + ϕα + hϕ,α ζ + h 2 Bβ,α ζ 2 − (α → β) dζ 2 12

(14.193) on the set of functions satisfying the constraints z α = z α ζ = 0. Let us make the change of the required functions: z α → z¯ α , σ 2 β 1 3 1 2 2 − h Bβ,α ζ ζ − + z¯ α . z α = − hϕ,α ζ − 2 12 6 20

(14.194)

In terms of z¯ α , the functional J∠ takes the form J∠ =

σ 1 β {P a }hϕ,α + [P α ]h 2 Bβ,α + J¯∠ , 12 120 1

1 J¯∠ = 2

2

− 21

ϕ¯ α ≡ ϕα −

μa˚ αβ z¯ α,ζ + ϕ¯ α (α → β)dζ − {P α z¯ α }, σ 2 β h Bβ,α . 60

(14.195)

Calculation of z¯ α is reduced to minimizing the functional J∠ over all functions z¯ α satisfying the conditions ¯z α = ¯z α ζ = 0. This problem is completely analogous to the problem of minimizing the functional J¯⊥ . Therefore, its solution is given by (14.190), where z¯ is replaced by z¯ α , λ + 2μ by μ, and P and Pα :

14.6


653

- . 1 5 0 β1 5 1 3 3 + + Pβ ζ 2 − , z¯ α = − ϕζ P ζ ζ2 − ¯ ζ2 − a˚ αβ 3 20 2μ 3 20 12 (14.196) and 5 1 min J¯∠ = μϕ¯ α ϕ¯ α + [P α ]ϕ¯ α . 12 12

(14.197)

From (14.195) and (14.197), we have min J∠ =

1 1 σ 5 β μϕ¯ α ϕ¯ α + [P α ]ϕ¯ α + {P α }hϕ,α + [P α ]h 2 Bβ,α . 12 2 12 120

(14.198)

The terms in the expression (14.198) are on the order of μ (h/l)2 ε2 . The term, 5 μϕ¯ α ϕ¯ α , has the meaning of the surface energy of the transverse shear; the ∠ = 12 other terms are related to the work of external forces. In order to complete the construction of energy, it remains to calculate the longitudinal surface energy density = h U κ. 1

1

1

Let us introduce the tensor, εαβ = g αγ εγβ , where g αγ are defined by (14.155). For 1

F κ as a function of ε αβ one can write

2 1α 1β 1α F κ = μ σ εα + εβ εα . 1

Expand ε αβ over the first Legendre’s polynomials. From (14.155) and (14.185) we have 1

εαβ =

Aαβ −

1 h 2 1 αγ 1 h2 . b Bγ δ + Cβα +h Bβ(ϕ)α − 2bαγ Aβγ ζ +h 2 (. . .) ζ 2 − 6 12 12 1

Move the upper index of ε αβ down by means of metrics a˚ αβ . Then

2 αβ 1 αγ βδ 1 1 F|| κ = μ σ a˚ ε αβ + a˚ a˚ ε αβ εγ δ . 1

Tensor ε αβ is not symmetric; however, its antisymmetric part gives the contribution in F|| κ on the order of ε2 h 2 /R 2 and can be neglected. For computation of || = 1 1 : ; 1 F|| κ , one has to plug in F|| the tensor ε (αβ) = Aαβ − h B αβ ζ where 1

Aαβ = Aαβ −

h 2 1 λ (ϕ) h 2 (ϕ) b(α B β)λ + Cαβ , 6 12

1

1

(ϕ) B αβ = Bαβ − 2bλ(α Aβ)λ .

654

14


For the surface density of the longitudinal energy we obtain the relations of the same form as in classical shell theory: 1 1 1 = A + h B . 12

(14.199) 1

The difference is in the expressions for the extension and bending measures, Aαβ 1

1

and B αβ . Within the framework of the accuracy considered, in the formula for Aαβ (ϕ) (ϕ) we may replace Bαβ and Cαβ by Bαβ and Cαβ , respectively. Besides, extracting in (ϕ) the terms depending on ϕα and ϕ and using (14.145), we have Bαβ 1

Aαβ = Aαβ −

h 2 ′λ b B , 12 (α β)λ

1

′λ B αβ = Bαβ − ϕ(α;β) + (ϕ − 1) bαβ − 2b(α Aβ)λ .

Let us write Euler equation for ϕ. In the two-dimensional theory, energy is a sum of the functions (14.199) and the functions (14.198) and (14.191), multiplied by h. Energy depends on the derivatives of ϕ linearly; therefore, Euler equation becomes a linear algebraic equation: 1 h 5 (λ − 2μ) ϕ − 1+σ Aαα + [P] − {P α };α + h −1 m αβ bαβ = 0. 6 12 12

(14.200)

From this equation, in the first approximation,

ϕ = 1 − σ Aαα .

(14.201)

For zero external forces, the error of (14.201) is O (hε/R), and, consequently, within the accuracy of the theory taking into account the transverse shear, the term 5 in energy, (λ − 2μ) h ϕ¯ 2 , can be neglected. 6 For non-zero external forces, the error of (14.201) is O ((h/R+h/l)ε). Then ϕ¯ 2 may be on the order of h 2 ε2 /l 2 , and, at first glance, it should be retained in the energy. However, after additionthe workof external forces on ϕ, ¯ the sum turns out to be at least on the order of O h 2 ε2 /Rl , and can therefore be dropped. Let us take as the bending measure the tensor λ ραβ = Bαβ − ϕ(α;β) − b˚ (α Aλβ) .

(14.202)

If the transverse shear ϕα is neglected, then the tensor (14.202) becomes the Koiter1

1

Sanders bending measure. The tensors Aαβ and B αβ , in terms of Aαβ and ραβ take the form 1

Aαβ = Aαβ −

h 2 ′λ b ρβ)λ , 12 (α

1

′λ B αβ = ραβ − b(α Aβ)λ − σ Aλλ b˚ αβ .

14.6


655

Here we used (14.201). The energy expression is still too cumbersome. Especially “unpleasant” is the β gradient of Bβ , which enters the energy of the transverse shear through the quantity σ 2 β h Bβ,α . Since ϕα are independent required functions, it is natural ϕ¯ α = ϕα − 60 β to get rid of the gradient of Bβ by means of a change of the required functions. β It should be made in such a way that the gradient of Bβ would not appear in the expression for ραβ , which also contain ϕα . It is easy to check that the formula for ραβ will not change if, along with the transition form ϕα to the new shear angles ϕ¯ α the requires functions r i are also changed by the functions r¯i = ri − n i

σ 2 β h Bβ . 60

(14.203)

β

Indeed, −n i σ h 2 Bβ /60 is a small correction on the order of hε to ri ; therefore, the corresponding change of the second quadratic form of the surface can be found by means of the formula (14.49). It can be checked by direct inspection that the following equation holds for ραβ : λ i ραβ = n¯ i r¯;αβ γλβ) , − b˚ αβ − ϕ¯ (α;β) − b˚ (α

where γλβ denotes the tensor Aλβ constructed by the functions r¯i : γλβ =

1 i r¯,α r¯i,β − a˚ αβ , 2

while the unit vector, n¯ i , is defined by the relations

n¯ i n¯ i = 1.

i n¯ i r¯,α = 0, 1

1

The tensors Aαβ and B αβ become 1

Aαβ = γαβ −

h 2 ′λ σ b(α ρλβ) − h 2 ρμμ b˚ αβ , 12 60

1

′λ B αβ = ραβ − b(α γλβ) − σ γλλ b˚ αβ . 1

1

After the substitution of the expressions of Aαβ and B αβ in terms of γαβ and ραβ , and dropping the terms on the order of h 2 /R 2 in comparison to the kept ones, the surface energy density gets the final form (14.174). The work of external forces is given by (14.175). Kinetic energy. Let us move on to calculation of the kinetic energy. According to the formulae for the variations of Bαβ (14.49) and the normal vector (14.47), we have i Bαβ,t = n i v|αβ ,

k n i,t = −r iα n k v,α .

(14.204)

656

14


From (14.183), (14.160), (14.194), (14.196), (14.192) and (14.204), the estimates follow i rα,t

cε ∼ , l

n i,t

cε ∼ , l

α h Bα,t =O

ϕ,ti

=

n i,t

hcε , l2

i

ϕ,tα rαi

+ ϕ,t n + =O hcε . z ,ti = O l

cε2 l

,

Within the accepted accuracy, for the kinetic energy density we have 1 1 h2 i i i K = ρ0 h x,t xi,t = ρ0 h vi v + ϕi,t ϕ,t = 2 2 12 2 1 σ 2 h 2 α 2 h k = ρ0 h vi v i + a˚ αβ n k v,α . Aα,t − ϕα,t (α → β) + 2 12 12 The formula for kinetic energy in terms of v i and ϕ,ti is simple enough. However, after the transition to the variables r¯ i and ϕ¯ α for which the elastic energy takes a simple form, the formula for kinetic energy complicates: K =

1 σ h2 ρ0 h v¯ i v¯ i + v¯ i n¯ i n¯ k ¯v k 2 30 k σ 2 h 2 α 2 h2 , γα,t − ϕ¯ α,t (α → β) + + a˚ αβ n¯ k v¯ ,α 12 12

(14.205)

where is the surface Laplace’s operator, and it is used that, with the accepted accuracy, according to (14.204), i

v¯ i ≡ r¯ ,t = v i − n i

σ 2h2 n¯ k ¯v k , 60

α

α Aα,t = γα,t .

(14.206)

The kinetic energy expression (14.205) is not suitable for extrapolation, since it is linear with respect to the higher (second) derivatives of velocity and, for short waves, the kinetic energy is not positively definite. In order to transform the kinetic energy expression into a form convenient for extrapolation, let us integrate the second term in (14.205) by parts. It becomes −ρ0 σ h 3 a˚ αβ v¯ i,α n¯ i n¯ j v¯ j,β /60 (derivatives of the normal vector produce the terms of the negligible order). Adding small terms α involving cϕ¯ α,t and replacing γαα by v¯ ;α (which is admissible within the accepted accuracy), we get K =

k 2σ 1 σ 2 h 2 α 2 h2 1− . a˚ αβ n¯ k v¯ ,α v¯ ;α − ϕ¯ α,t (α → β) + ρ0 h v¯ i v¯ i + 2 12 5 12 (14.207)

14.6


657

The terms which were transferred to the boundary in integration by parts are ignored, since the asymptotically accurate boundary conditions are not discussed here. Linearization in the expression (14.207) yields the relation (14.179). Tensors nαβ and mαβ and the integral characteristics of the stress state. To figure out the physical meaning of the tensors n αβ and m αβ we now establish the relationships between n αβ , m αβ , and q α , introduced above, and the integral characteristics of the stress state. We restrain our consideration by statics of isotropic shells. Denote by σ ab the contravariant components of the Cauchy stress tensor in the Lagrangian coordinate system ξ α , ξ ≡ ξ 3 , and by T αβ , M αβ and N α the integrals T

αβ

=

h/2

κμαλ σ βλ dξ,

M

αβ

=

h/2

κμαλ σ βλ ξ dξ,

α

N =

κσ α3 dξ.

−h/2

−h/2

−h/2

h/2

(14.208) The tensors, T αβ and M αβ , are not symmetric. We are going to show that within the accepted accuracy of the two-dimensional shell theories, t αβ = T αβ , n αβ = T αβ + b˚ λ[α M λβ] ,

m αβ = −M (αβ) ,

q α = −m

αβ

;β

= N α.

(14.209)

Consider the equilibrium equations of an elastic shell as a three-dimensional body ρ0 i ab ∇˚ a = 0. (14.210) xb σ ρ On the faces of the shell, the following conditions are satisfied: j j σ ab xai xb n +j h = P+i σ ab xai xb n −j = Pi . h ξ= 2

ξ =− 2

(14.211)

Here, n i± are the components of the normal vector on the shell faces in the deformed state. Equation (14.210) can also be written as ξ a

ρ0 g˚ xbi σ ab = 0. ρ

(14.212)

Within the accepted accuracy, ρ0 /ρ can be replaced by unity and x i (ξ α , ξ ) by the expression r i (ξ α ) + ξ n i (ξ α ), because σ ab = O (με), and, according to (14.139) and (14.154), x i (ξ α , ξ ) = r i (ξ α ) + ξ n i (ξ α ) + h O (ε). Then, in (14.211), (14.212), we can set i xαi = μβα rβi , μβα ≡ δαβ − ξ bαβ , x,3 = n i , n i+ = n i , n i− = −n i ,

(14.213)

658

14


and (14.211) and (14.212) become β3 α i 33 i

λβ α i β3 i

˚ ˚ g σ g σ g˚ σ μβ rα + g˚ σ n = 0, (14.214) + μ r + n λ α ξ β ξ h σ λ3 ξ =± h δλα ± bλα rαi ± σ 33 n i = P±i . (14.215) 2 h 2 ξ =± 2

Using the relation (14.38) written for the undeformed state, √ β √ β ˚ ˚ ;β , av = av ξ β

setting

(14.216)

√ g˚ = κ a˚ in (14.214) and introducing the notation τ αβ = κμαλ σ βλ , τ α = κσ α3 , τ = κσ 33 ,

(14.217)

we can write (14.214) as τ

αβ ;β

α β μ τ = 0, ξ β + τ αβ bαβ + τ = 0. ξ

− τ β bβα +

(14.218)

β

(14.219)

τ

;β

Note that the tensor, τ αβ , is non-symmetric. The stress resultants, T αβ , the bending moments, M αβ , and the shear force, N α , (14.208) are expressed in terms of τ αβ and τ α as

T αβ =

h

h

h

2

2

2

τ αβ dξ,

M αβ =

− h2

τ αβ ξ dξ,

− h2

Na =

τ α dξ.

(14.220)

− h2

The tensors T αβ and M αβ are non-symmetric. Integrating over ξ (14.218) and (14.219), and (14.218) multiplied by ξ , results in the system of “averaged” equations of equilibrium: αβ

α ;β − bβ N + N α ;α + T αβ bαβ

T

M

αβ

;β

0 α β3 1 μβ κσ = 0, 0 33 1 + κσ = 0, @ ? α α β3 = 0. − N + ξ κμβ σ

Now, using the boundary conditions (14.215), we finally get

14.6


659

T

αβ ;β

− bβα N β + {κ P α } = 0,

N α ;α + T αβ bαβ + {κ P} = 0, h 0 α1 αβ κ P = 0. M ;β − N α + 2

(14.221)

Juggling the indices may be done by means of both the metric a˚ αβ and the metric aαβ , since aαβ − a˚ αβ = O (ε). From the definition of the tensors T αβ and M αβ (14.220) and the symmetry of the stress tensor σ αβ there follows another equation, the so-called sixth equilibrium equation:

T αβ + bσα M σβ eαβ = 0.

(14.222)

Let us emphasize that in the derivation of (14.221), only the quantities on the order of ε were discarded, while (14.222) is exact.16 If the terms on the order of h 2 /l R and h 2 /R 2 are also dropped, then κ|ξ =± h2 in (14.221) can be set equal to ˚ . The equilibrium equations become κ± = 1 ∓ h H T

αβ ;β

0 1 ˚ P α = 0, − bβα N β + {P α } − h H

˚ [P] = 0, N α ;α + T αβ bαβ + {P} − h H h 0 α1 h ˚ αβ P − h H {P α } = 0. M ;β − N α + 2 2

(14.223)

Now, let us calculate the tensors n αβ and m αβ in the linear theory in terms of T and M αβ . For simplicity, only the case when the material is isotropic and the tangential components of the external forces on the shell faces, P±α , are equal to zero will be considered. According to the definition of the stress tensor αβ

σ αβ =

Thus,

F F F⊥ F = + = + λ ε33 +σ g˚ μν εμν g˚ αβ , εαβ εαβ εαβ εαβ F = (λ + 2μ) ε33 +σ g˚ μν εμν . σ 33 = ε33 F = σ αβ − σ σ 33 g˚ αβ . εαβ

16

(14.224)

One can also obtain the exact averaged equilibrium equations, which do not contain approximations, in the coordinate system linked to the deformed state. However, such equations are not of much value because this coordinate system is not known.

660

14


According to the formula for z (14.187) and equation (14.190), in the first approximation 1 1 1 − ζ2 + σ 33 = 5 (λ + 2μ) ϕ¯ 5ζ 2 − [P] + {P}ζ. 4 2 4 Function ϕ¯ is given in the first approximation by (14.200). Hence, 3 1 + {P}ζ. σ 33 = − m αβ bαβ 1 − 4ζ 2 + 3 [P] ζ 2 − 2h 12

We will need a more exact equation, which takes into account corrections on the order of h/l. Without pausing for the derivation, we give the result for σ 33 : 3 αβ m bαβ 1 − 4ζ 2 (14.225) 2h 1 4 1 3 + n αβ bαβ 1 − 4ζ 2 ζ + ζ 1 − ζ 2 {κ P} + [κ P] , 2 2 3 2

σ 33 = −

and for relation between εμν and γμν , ρμν :

h2 λ ρh 2 λ ˚ λ γλν) −σ γ λλ b˚ μν ζ + εμν = γμν + b˚ (μ ρλλν) − ρλ bμν − h ρμν +b(μ 12 60 σ h2 ˚ 1 1 λ − + hz (μ;ν) . bμν ραα ζ 2 − (14.226) +h 2 b(μ ρλν) ζ 2 − 12 2 12 The quantities z α are functions of ϕ¯ α only. From the definition of n αβ and m αβ we have

n αβ

h2 h F κ + F εμν κ = dξ + = = γαβ εμν γαβ γαβ − h2

h

=

2

− h2

, κ σ μν − σ σ 33 a˚ μν δμ(α δμβ) − b˚ μ(α b˚ νβ) ξ + σ b˚ μν a˚ αβ ξ dξ + γαβ

h2 h F κ + F εμν = dξ + = = κ ραβ εμν ραβ ραβ

m αβ

− h2

h

=

2

− h2

h 2 (α β) σ h 2 μν 33 μν αβ (α β) ˚ δ μ δμ − bμν a˚ − ξ δμ δν dξ + κ σ − σ σ a˚ . 12 60 ρ αβ (14.227)

14.6


661

1 In (14.227), the terms which contain the factor ζ 2 − are omitted because, after 12 integration over ξ , they become ignorably small. Now, using the formula (14.225) and the expressions for T αβ and M αβ (14.220) which, within the considered accuracy, can be written as h

h

2

2

κσ αβ dξ = T αβ + b˚ σα M βσ ,

− h2

κσ αβ ξ dξ = M αβ +

h 2 ˚ α βσ b T , (14.228) 12 σ

− h2

we get the second and the third equalities (14.209). Note, that the relation (14.222) follow from the first equation (14.228). An analogy (14.222) for the second equation (14.228) is

M αβ +

h 2 ˚ α βσ eαβ = 0. bσ T 12

(14.229)

This equation serves to determine the anti-symmetric part of M αβ , M [αβ] , which does not enter into the equations of two-dimensional shell theory: M [αβ] = −

h 2 [α β]σ b T . 12 σ

Note that the second equation (14.209) can also be written as n αβ = T (αβ) , while the non-symmetric tensor, t αβ , in terms of which the equilibrium equations were written, coincides with T αβ . Besides, according to the third equation (14.220), αβ the vector q α = −m ;β coincides with the shear force, N α . The rest of this section is concerned with the comparison of the obtained equations with the equations derived by other means by A.L. Goldenveizer and E. Reissner. This comparison contains some technicalities which are useful for better understanding the refined theories. Theory taking into account geometrical correction and Goldenveizer’s iteration theory. Let us compare the derived relations with the relations obtained by A.L. Goldenveizer by means of the asymptotic analysis of linear static equations of the three-dimensional elasticity. The following notation was used in [115] to characterize the forces and the shell geometry: G 1 = M11 , S12 = T12 ,

G 2 = −M22 , S21 = T21 ,

H12 = M12 , m = [P],

H21 = M21 , T1 = T11 , T2 = T22 , 1 1 Z ′ = {P}, = −b˚ 11 , = −b˚ 22 . R1 R2

662

14


One can check that the equilibrium equations (14.80) of the theory taking into account geometrical correction completely coincide with the equilibrium equations in [115]. In order to compare the constitutive equations, (14.171) should be written in terms of the displacements of the middle surface, which are used as the required functions in [115]. All the quantities related to the middle surfaced are furnished with the letter m. From the formulae (14.95), (14.139) and (14.153), it follows that (m) ri

= r¯i −

σ h2 n i ρμμ , 40

(m) αβ

γαβ = γ

−

σ h2 μ ˚ ρ bαβ , 40 μ

(m)

ραβ = ρ αβ . (m) αβ

Recalculating the constitutive equations (14.171) in terms of γ we get

αβ

(m) σ γ λλ a˚ αβ

(m)αβ

(14.230) (m)

and ρ αβ ,

− +γ = 2μh 3 μh 3 (αλ β) ˚ a˚ αβ + 3 σρ λλ b˚ αβ + σ h [P]a˚ αβ . − σ − 1 ρλλ H ρ bλ +σ b˚ μν ρ μν a˚ αβ +σ 3 2 4 2 n

(m)

(m)

Within the considered accuracy, the dependence of m αβ on γ αβ and ρ αβ does not differ from the dependence of m αβ on γαβ , ραβ (14.171). For the moments, A.L. Goldenveizer obtained the constitutive equations (see (27.13.10) in [115]) 1 1 ν Eh 3 1 ε1 −σ κ 1 + νκ 2 + − + (ε1 +ε2 ) 12 (1 − nu) R2 R1 R1 R2 2 h − σ {P}, 12 ω ω Eh 3 Eh 3 , H21 = − . (14.231) τ− τ− =− 12 (1 + ν) 2R2 12 (1+ν) 2R1

G1 = −

H12

Here, the extensional and bending measures are defined by (m)

κ 1 = −ρ11 , (m) 11 ,

ε1 = γ

κ 2 = −ρ22 , (m) 22 ,

ε2 = γ

˚ (m) τ = − B 12 = −ρ12 − H γ 12 , (m) 12 ,

ω = 2γ

and E = 2μ (1 + ν) is the Young modulus. The equations for the mixed components of the bending moments tensor coincide in (14.171) and (14.231) (in order to check that this is true, the equality (14.229) should be used). The diagonal components of the bending moments are different in (14.171) and (14.231). However, it turns out that in the long wave range they differ

14.6


663

by terms of an ignorably small magnitude. Indeed, from the second equilibrium equation (14.223) and the expression for {P}, in the first approximation (for P α = 0), we have {P} = −T αβ b˚ αβ − M αβ ;αβ .

(14.232) αβ

Substituting (14.232) into (14.231) and discarding the term h 2 M ;αβ , which is small compared to G 1 , we transform Goldenveizer’s equations into form (14.171). The equations for the extensional resultants coincide completely. The constitutive equations (14.171) have some advantages over Goldenveizer’s constitutive equations. They are more exact and retain their form in the theory taking αβ into account transverse shear (dropping the term h 2 M ;αβ , which is necessary for transforming (14.171) into the form of Goldenveizer’s equations, but is not allowed in the theory taking shear into account). Besides, the extension and bending measures, constructed in terms of the middle surface displacements are not the parameters for which the resultant and moment tensors are potential. Indeed, according to (14.171), (14.223) and (14.230), δ ( + ) = n αβ δγαβ + m αβ δραβ = T (αβ) δγαβ − M (αβ) δραβ = σ h 2 μν ˚ (m) (m) T bμν a˚ αβ δ ρ αβ . = T (αβ) δ γ αβ − M (αβ) + 40 (m)

Therefore, if the parameters γ

(m)

αβ

measures, then T (αβ) and M (αβ)

and ρ αβ are used as the extension and the bending σ h 2 μν ˚ T bμν a˚ αβ , not T (αβ) and M (αβ) , will have + 40

the elastic potential. In the nonlinear case, the equations taking into account geometrical correction were derived by Niordson [231] from some hypotheses refining the Kirchhoff-Love hypotheses. Niordson retained all the corrections on the order of h/R and h 2 /R 2 . It should be noted that it does not make sense to retain the corrections on the order of h 2 /R 2 and not take into account the transverse shear, which yields, in the best case scenario, l ∼ R, the corrections of the same order, h 2 /R 2 , but for l ≪ R brings more considerable corrections (excluding some “exotic” cases, where the transverse shear is small). If all terms on the order of h 2 /R 2 are dropped, and the expression for energy obtained by Niordson is recalculated in terms of Aαβ and Bαβ (Niordson used the components of the middle surface position vector as the required functions, (m)

(m)

and A αβ and B αβ as the strain measures), then one obtains required expression (14.169). Asymptotic theory taking into account transverse shear and Reissner theory. Reissner derived the shell theory from the Castigliano principle by means of several a priori hypotheses regarding the dependence of stresses on the transverse coordinate, and obtained the energy expression in terms of the stress resultants and moments. To compare the Reissner theory with that obtained above, the energy as

664

14


a function of n αβ , m αβ and q α should be found. For simplicity, assume the external forces on the shell faces to be equal to zero. Moreover, small terms on the order of h 2 /R 2 are everywhere omitted. The calculation is significantly simplified by means of the following reasoning. Let be a quadratic form with respect to the variables x i : (x) =

1 1 ij a xi x j + εbi j xi x j , 2 2

(14.233)

where ε is a small parameter. We need to find as a function y i = /xi , retaining only the terms on the order of ε. Let us show that is (y) =

1 (−1) i j 1 i j (−1) (−1) k l a y y − εb aik a jl y y 2 ij 2

(14.234)

where ai(−1) is the inverse tensor of the tensor a i j . j Indeed, y i are defined by the equations y i = a i j x j + εbi j x j .

(14.235)

j In the first approximation, xi = ai(−1) j y . Taking into account the corrections on the order of ε, from (14.235) we have

j y −εb jk akl−1 y l . xi = ai(−1) j

(14.236)

1 i y xi and (14.236). 2 The relation (14.234) shows that calculation of in terms of n αβ , m αβ , and q a is reduced to calculation of extension energy, (γ ), bending energy, (hρ) , and interaction energy, int (γ , ρ) , in terms of n αβ and m αβ , when the equations of state of the classical theory are used for n αβ and m αβ ,

Equation (14.234) follows from the equality =

n αβ = 2μh σ γλλ a˚ αβ + γ αβ ,

m αβ =

μh 3 λ αβ αβ σρ λ a˚ + ρ , 6

while the transverse shear is expressed in terms of shear force, q α , from the equation qα = As a result, we get

5 μh ϕ¯ α . 6

14.7

Theory of Anisotropic Heterogeneous Shells

=

665

α 2 1 αβ σ 3 n n αβ − + 2 m αβ m αβ n 4 4 (1 + 2σ ) α h α 2 σ 2σ 2 ′ m λ b˚ αβ n αβ m α + n αβ bαλ − 2 m λβ − − h (1 + 2σ ) 5 (1 + 2σ ) λ σ ˚ m μμ n νν + 3 qα q α . (14.237) H + 1 + 2σ 5

1 μh

The expression (14.237) can be rewritten using the identity 2

′ n αβ bαλ m λβ

1 2 λ ˚ αβ μ ν ˚ − m λ bαβ n + H m μ n ν = m λλ b˚ αβ n αβ + b˚ αβ m αβ n λλ . 5 5

(14.238)

It can be checked by breaking down all the tensors into their spherical and deviator parts and using the identity ′α bα′β n ′σ β m σ = 0,

which is valid for any three deviator tensors in two-dimensional space. If we use as the primary physical characteristics the Young modulus, E = 2μ (1 + ν), and Poisson’s coefficient, ν, then, taking into account that σ/ (1 + 2σ ) = ν/ (1 + ν), we get =

1 + ν αβ ν α 2 6 (1 + ν) αβ 6ν 2 nα + m m αβ − 2 m αα + n n αβ − 2 2 2 h h @ 3 ν ′ λ qα q α . (14.239) +2n αβ bαλ m β + m λλ b˚ αβ n αβ + ν b˚ αβ m αβ n λλ + 5 5μh 1 Eh

It is easy to check that, in the curvature lines, this equation becomes the expression obtained by Reissner.17

14.7 Theory of Anisotropic Heterogeneous Shells The derivation of the equations of the anisotropic heterogeneous shell theory repeats the derivation of the isotropic shell theory almost verbatim; thus, it will be presented in less detail than in the previous section. Elastic moduli. The energy density of the anisotropic body is a quadratic form with respect to the strain tensor,

17

Note that in Reissner’s theory, only the equations for the integral characteristics are exact, while the distributions over the thickness are not.

666

14


2F =E abcd εab εcd = E αβγ δ εαβ εγ δ + E 3333 ε33 ε33 + 4E α3β3 εα3 εβ3 + + 2E αβ33 εαβ ε33 + 4E αβγ 3 εαβ εγ 3 + 4E α333 εα3 ε33 .

Calculating the longitudinal elastic energy defined by the formula (14.82) we obtain F =

1 αβγ δ εαβ εγ δ . E 2

(14.240)

Here, 1 αβ33 γ δ33 E E − Hμν G αβμ G γ δν , (14.241) E 1 G αβμ = E αβμ3 − E αβ33 , E = E 3333 , E and Hμν is the inverse tensor of the tensor E μ3ν3 − E μ333 E ν333 /E. As is easy to check, the transverse elastic energy is a positive quadratic form with respect to αβγ δ

E

= E αβγ δ −

γα = 2εα3 + E αμν εμν and γ = ε33 + E αβ εαβ , where E αμν = Hαβ G μνβ ,

E αβ =

E αβ33 − E μαβ E μ333 E

.

(14.242)

The transverse elastic energy is 2F⊥ = E α3β3 γα γβ + 2E α333 γα γ + Eγ 2 .

(14.243)

The relation, F = F + F⊥ , and the formulae (14.240) and (14.243) may be checked by substituting the expressions for the “two-dimensional” elastic moduli E , E α33 , E γ 333 , E, E a μν and E αβ into (14.240) and (14.243) in terms of the elastic moduli of the three-dimensional body (14.241) and (14.242) and opening the parenthesis. Two-dimensional elastic moduli satisfy the symmetry conditions αβγ δ

βαγ δ

αβδγ

γ δαβ

, E α3β3 = E β3α3 , E αβ = E βα , E αμν = E ανμ , (14.244) and the constraints caused by positiveness of elastic energy, E

= E

= E

= E

14.7

Theory of Anisotropic Heterogeneous Shells αβγ δ

E

εαβ ε γ δ ≥ mεαβ εαβ ,

667

m = const > 0,

E α3β3 γα γβ + 2E α333 γα γ + Eγ 2 ≥ m γα γ α + γ 2 ,

(14.245)

m = const > 0. (14.246)

It is natural to take the two-dimensional elastic moduli as the independent components of the elastic moduli tensor. According to (14.244), there are 21 elastic αβγ δ constants; 12 of them – E , E α3β3 , E α333 , and E – have the dimension of shear modulus and have to satisfy the conditions (14.245) and (14.246), while the other 9, E αβ , and E αμν , the two-dimensional Poisson coefficients, are dimensionless and may take on arbitrary values. Dependence of elastic moduli on transverse coordinate. The Lagrangian coordinate system is curvilinear in the initial state; therefore the components of the elastic moduli tensor depend on the coordinates even in the case of homogeneous shells. We have to separate the dependence of the elastic moduli on coordinates due to geometrical reasons and the dependence of the elastic moduli on coordinates due to the heterogeneity of the shell. In the undeformed state, the basis vectors of the Lagrangian coordinate system, ˚ by the equations e˚ α , relates to the tangent vectors aα of the surface e˚ α = μ ˚ βα a˚ β , where μ ˚ βα are defined by the formulae (14.122), the vector e˚ 3 does not depend on ξ ˚ coincides with the unit normal vector to the surface . ˚ Due and, on the surface, , αβγ δ to that, the two-dimensional tensors of the elastic moduli E e˚ α e˚ β e˚ γ e˚ δ , . . . may be expanded over the vectors a˚ α , and αβγ δ

E

αβγ δ

e˚ α e˚ β e˚ γ e˚ δ = E

μ ˚ σα μ ˚ νβ μ ˚ ωγ μ ˚ θδ a˚ σ a˚ ν a˚ ω a˚ θ , . . .

We introduce the notation αβγ δ

C σ νωθ = E

μ ˚ σα μ ˚ νβ μ ˚ ωγ μ ˚ θδ , Cσ ν = E αβ μ ˚ σα μ ˚ νβ , Cσων = E γαβ μ ˚ σα μ ˚ νβ λ˚ γω ,

˚ σα μ ˚ ωβ , C σ = E α333 μ ˚ σα . G σ ω = E α3β3 μ

(14.247)

Here, λ˚ ωγ is the tensor (14.126). The two-dimensional tensors (14.247) have the following meaning. If a (nonholonomic) “physical” system of coordinates is chosen on the surface, then the values of the components of the two-dimensional tensors (14.247) and the twodimensional scalar E coincide with the physical components of the elastic moduli tensor in this system of coordinates. Therefore, the formulae (14.247) perform the sought separation of the dependence of elastic moduli on curvilinearity of the coordinate system and heterogeneity of the shell: the heterogeneity is described by the dependence of the tensors (14.247) on ξ α and ζ = ξ/ h, while all the geometric factors are included in μ ˚ βα and λ˚ βα . Knowing the elastic properties of the shell material means knowing the tensors (14.247).

668

14 αβγ δ

Resolving (14.247) with respect to E αβγ δ

E ||


, . . . ., we have

= C σ νωθ λ˚ ασ λ˚ βν λ˚ γω λ˚ δθ , E αβ = Cσ ν λ˚ ασ λ˚ βν ,

(14.248)

˚ ωγ , E α3β3 = G σ ω λ˚ ασ λ˚ βω , E α333 = C σ λ˚ ασ . E γαβ = Cσων λ˚ ασ λ˚ βν μ

The formulae (14.248), (14.122) and (14.126) define the dependence of the elastic moduli on the transverse coordinate. The small parameter h/R enters these relations through μ ˚ βα and λ˚ βα . In what follows we need only the correction of the order h/R in αβγ δ comparison to unity. Tensor E enters the energy density with the coefficient κ, αβγ δ therefore we make the expansion for κ E . Since αβγ δ

κ E

= C σ νωθ κ 1/4 λ˚ ασ κ 1/4 λ˚ βν κ 1/4 λ˚ γω κ 1/4 λ˚ δθ ,

1 1 1 ˚ α 1 κ 4 λ˚ ασ = δσα + ξ bσα + . . . , bσα ≡ b˚ σα − H δ , 2 σ

we have αβγ δ κ E

=C

αβγ δ

α

(ξ , ζ ) + ξ C 1

′αβγ δ

2 h ¯ 2 , (ξ , ζ ) + O μ R α

1

1

(14.249)

1

C ′αβγ δ = C αβγ σ bσδ + C αβσ δ bσγ + C ασ γ δ bσβ + C σβγ δ bσα . Here and in the following estimates, μ ¯ is the maximum eigenvalue of the elastic moduli tensor, i.e. the best constant in the inequality E abcd εab εcd ≤ με ¯ ab εab . Using that λ˚ ασ = δσα +ξ b˚ σα + O h 2 /R 2 , we obtain

2 h E = C + ξC + O μ ¯ 2 , C ′αβ ≡ Cασ b˚ σα + Cσ β b˚ σα , R 2 h ασ ˚ β αβ ˚ σ αβ αβ ′αβ ˚α E γ = Cγ + ξ Cγ + O μ ¯ 2 , Cγ′αβ ≡ Cσβ γ bσ + Cγ bσ − Cσ bγ , R h h α3β3 αβ α333 α , E . (14.250) E =G +O μ ¯ =C +O μ ¯ R R αβ

αβ

′αβ

Major cases of elastic symmetry. Planes of elastic symmetry parallel to the middle surface. If the elastic properties are invariant with respect to the reflections across the planes parallel to the middle surface, then the “two-dimensional” elastic moduli with an odd number of indices are all equal to zero,

14.7


Cαβ γ = 0,

669

C α = 0.

(14.251)

Transversal isotropy. If the elastic properties are also invariant with respect to rotation around the vector normal to the middle surface, the “two-dimensional” elastic modulus tensors of the second order are spherical: G αβ = G a˚ αβ ,

Cαβ = σ a˚ αβ ,

(14.252)

while the tensor C αβγ δ contains the two parameters, λ′ and μ, C αβγ δ = λ′ a˚ αβ a˚ γ δ + μ a˚ αγ a˚ βδ + a˚ βγ a˚ αδ .

(14.253)

The elastic properties of the transversely isotropic body are characterized by five parameters, λ′ , μ, G, E, and σ . The parameters, μ, G, E, and λ′ + μ are to be positive, while the coefficient σ is arbitrary. Isotropy. In this case, there are two independent parameters: λ and μ, and λ′ = 2μσ , λ , E = λ + 2μ, G = μ. The parameter σ for isotropic material is not σ = λ + 2μ arbitrary because it is linked to λ and μ. In the general case of anisotropy, the tensor G αβ may be called the shear modulus tensor, while the tensors Cαβ and Cαβ γ have the meaning of the “two-dimensional” Poisson coefficients. Homogeneous shells. For such shells, the tensors C, Cαβ , G αβ , Cαβ γ , C α and E do not depend on ζ . Functions yα and y. Let us first consider in the framework of statics a generic case, μν where the shell is heterogeneous and Ca are non-zero. As for isotropic shell, we make the substitution of the required functions (14.139), and seek yα , y. For εα3 and ε33 we accept the approximate expressions (14.150) and disregard the work of external forces. For εα3 we take the formula 2εα3 = yα,ζ . The required functions yα and y minimize the functional F⊥ with the constraints yα = y = 0. Apparently, the minimum of the functional is equal to zero and is reached at the functions for which γα = γ = 0. Thus, in order to define yα , y, we have a system of ordinary differential equations: Aβγ − h Bβγ ζ = 0, yα,ζ + Cβγ α y,ζ + Cαβ Aαβ − h Bαβ ζ = 0,

yα = 0,

y = 0.

(14.254)

It can be easily checked that the Lagrange multipliers for the constraints yα =

y = 0 are equal to zero, and we do not include them in (14.254). According to (14.254), yα , and y are on the order of ε. The analysis of the terms dropped in the expression for the strain tensor components (14.147) shows that all of them are smaller than the retained ones. The contribution of the work of external forces is also ignorably small. Therefore, equations (14.254) give yα and y in the first approximation.

670

14


The functions yα describe the tangential displacements of the fibers perpendicular to the middle surface. Equations (14.254) show that for the shells with Cαβ γ = 0 (in particular, these are the shells having the planes of elastic symmetry parallel to the middle surface), yα = 0. Therefore, in the first approximation, a fiber normal to ˚ transforms into a straight segment perpendicular to the deformed middle surface, . As a heuristic hypothesis, this property was used by Kirchhoff to derive his plate theory, and subsequently by Love in constructing the shell theories. For bodies with non-zero tensor Cαβ γ , the transverse fibers deviate from the positions normal to the deformed middle surface for an angle on the order of ε (for isotropic shells, this angle is much smaller, on the order of hε/l), while the dependence of Cαβ γ on ζ causes a fiber to be bent even in the first approximation. Note that according to the second equation (14.254), the transverse contraction of the fibers occurs for isotropic as well as anisotropic shells (in the former case, Cαβ = σ a˚ αβ ). The physical interpretation of vanishing the transverse elastic energy is simple: from the constitutive equations σ α3 =

F F⊥ F F⊥ =2 and σ 33 = = εα3 γa ε33 γ

we see that the equation F⊥ = 0 is equivalent to vanishing of the stress tensor components, σ α3 and σ 33 . Shell energy in the classical approximation. Let us first calculate the elastic energy of the anisotropic heterogenous shell neglecting the interaction terms between bending and extension on the order of μhε ¯ A ε B h/R. In the expression for the components of the strain tensor εαβ , the terms containing yα and y may be omitted, as well as the term h 2 Cαβ ζ 2 . Therefore, the surface energy density is ; 1 : αβγ δ α h C (ξ , ξ ) Aαβ − hραβ ζ Aγ δ − hργ δ ζ 2 = int (A) + bend (hρ) + int (A, hρ) .

= h F κ =

Here, the extension energy, ext (A), the bending energy, bend (hρ), and the interaction energy, int (A, hρ), are 1 h C αβγ δ Aαβ Aγ δ , 2 1 bend (hρ) = h 3 C αβγ δ ζ 2 ραβ ργ δ , 2 int (A, hρ) = −h 2 C αβγ δ ζ Aαβ ργ δ . ext (A) =

(14.255)

If the tensor C αβγ δ ζ is not equal to zero, then, for heterogeneous shells, the interaction energy is on the same order as the extension and bending energies, and may not be omitted. The interaction energy describes the appearance of the bending moments caused by extension, and the reciprocal effect of the appearance of the

14.7


671

stress resultants caused by bending. These effects are significant in the description of the deformation of heterogeneous shells. Note, for example, a shell consisting of two homogeneous shells with different elastic properties: αβγ δ

(ξ α ) for ζ > 0,

αβγ δ

(ξ α ) for ζ < 0.

C αβγ δ = C+

C αβγ δ = C−

Then

C αβγ δ ζ =

1 αβγ δ αβγ δ C+ − C− . 8

If the shell is homogeneous in the lateral direction, then C αβγ δ ζ = 0 and 1 = hC αβγ δ 2

Aαβ Aγ δ

h2 + ραβ ργ δ . 12

(14.256)

So, the constitutive equations are n αβ = hC αβγ δ Aγ δ ,

m αβ =

h 3 αβγ δ C ργ δ . 12

γ

Note that for Cαβ = 0, the error of the expressions (14.255) and (14.256) is larger than for shells with elastic symmetry planes parallel to the middle surface. It is on the order of μhε2 h/l because yα ∼ ε. Shell energy taking into account geometrical corrections and transverse shear. Refined shell theories will be considered for the case of elastic symmetry with respect to the planes parallel to the middle surface. Thus (14.251) holds while γα = 2ε3α . In the first approximation, the following formula may be used for F⊥: 2 2F⊥ = G αβ 2εα3 2ε3 + E ε33 + Cαβ εαβ .

(14.257)

Moreover, we assume that the dependence of the tensor C αβγ δ on the transverse coordinate is of quite special type: C αβγ δ = C¯ αβγ δ (ξ α ) f (ζ ) ,

f (ζ ) ≥ const > 0.

(14.258)

This assumption limits the possible types of heterogeneity. For example, for isotropic shells it means that only the change in the shear modulus is allowed along the thickness, while the Poisson coefficient is constant. If the elastic moduli are not of the type (14.258), then the investigation is more complicated because it is not possible to reduce the determination of the functions z i in (14.181) to minimization of the transverse energy. The function f (ζ ) is normalized by the condition f = 1.

672

14


In order to simplify the following formulae, we will assume that the tangential components of the external forces are equal to zero on the shell faces. Let us make the change of required functions (14.181). Instead of the constraints (14.182), the functions z i will be restricted by the constraints

f (ζ )z i = 0,

ζ f (ζ )z i = 0.

(14.259)

The relation between the functions r i , ϕ i and the position vector of the body in the deformed state is determined from the equations r i + h ζ f ϕ i = f x i ,

ζ f r i + h ζ 2 f ϕ i = ζ f x i .

(14.260)

If ζ f = 0, then r i can be interpreted as the components of the average position vector of the transverse fiber in the deformed state with the weight, f , r i = f x i , while ϕ i has the meaning of the average rotation with the weight, f : hϕ i =

ζ f x i .

ζ 2 f

The characteristic lengths are defined as before by the formulae (14.141). Due to the special structure of the elastic moduli tensor (14.258) and to the coni z i,β) = hz (α;β) − hbαβ z and hn i,α z i,β ζ in (14.184) straints (14.259), the terms hr(α make a negligible contribution to the longitudinal energy. Therefore, an approximate expression (14.185) may be used for εαβ and, consequently, the calculation of z in the first approximation is reduced to the minimization problem for the functional 1

1 J⊥ = 2

2

− 12

2 E(ζ ) z ,ζ + ϕ − 1+Cαβ (ζ ) Aαβ −Cαβ (ζ ) h B αβ ζ dζ − {P z} (14.261)

on the set of functions z, satisfying the conditions

f (ζ ) z = 0,

ζ f (ζ ) z = 0

(14.262)

while determination of z α is reduced to the minimization problem for the functional 1

J∠ =

1 2

2

− 21

G αβ (ζ ) z α,ζ + ϕ α +hϕ,α ζ +hz ,α (α → β)dζ

(14.263)

14.7


673

on the set of functions, z α , satisfying the conditions

f (ζ ) z α = 0,

ζ f (ζ ) z α = 0.

(14.264)

These problems must be solved at each surface point ξ α . The function z which enters in (14.263) is the minimizing function of the functional (14.261). We begin with consideration of the variational problem (14.261) and (14.262). αβ Let us introduce the functions R αβ and R1 as the continuous solutions of the ordinary differential equations d R αβ = −Cαβ , dζ

αβ

d R1 = ζ Cαβ , dζ

(14.265)

and make the change of the required function, z → z¯ : αβ

¯ + z¯ . z = − (ϕ − 1) ζ + R αβ Aαβ + R1 h Bαβ + ϕζ Here, the following notation is used: αβ

ϕ¯ = ϕ − 1 −

ζ f R1

ζ f R αβ Aαβ − h Bαβ .

ζ 2 f

ζ 2 f

(14.266)

αβ

The additive constants of the functions R αβ and R1 are selected by the conditions

ζ 2 f f R αβ = ζ f ζ f R αβ ,

αβ

αβ

ζ 2 f f R1 = ζ f ζ f R1 .

(14.267)

Then the constraints (14.259) for z become

f z¯ = 0,

ζ f z¯ = 0.

(14.268)

The functional J⊥ depends on z¯ as D

C . 1 αβ αβ J⊥ = J¯⊥ − P R αβ Aαβ − P R1 h Bαβ +

ζ f R αβ Aαβ + ζ f R1 h Bαβ , 2 1 2 2 1 J¯⊥ = (14.269) E z¯ ,ζ + ϕ¯ dζ − {P z¯ }. 2 − 21

For the same reasons as in the isotropic case, the functional J¯⊥ is small and can be neglected. However, since we will further encounter a problem analogous to, but more cumbersome than, minimization of J¯⊥ , we will give here the solution of this simpler problem for further reference.

674

14


Let us drop the work of the external forces in (14.269). Then 1

1 J¯⊥ = 2

2

2 E z¯ ,ζ + ϕ¯ dζ.

− 21

(14.270)

Consider the dual problem 1

min J¯⊥ = max J¯⊥∗ ,

J¯⊥∗ =

2

p (ζ ) ϕ¯ −

− 21

1 2 p dζ, 2E

(14.271)

where the maximum is sought over all p (ζ ) and the numbers a and a1 satisfying the equations dp = a1 f + aζ f, dζ

1 = 0. p ± 2

(14.272)

The numbers a and a1 are the Lagrange multipliers for the constraints (14.268). It follows from (14.272) that a and a1 are linked: a1 + a ζ f = 0. Denote by g and g1 the functions defined by the Cauchy problem dg = ζ f, dζ

1 = 0, g − 2

dg1 = f, dζ

1 = 0. g1 − 2

(14.273)

Then the general solution of (14.272) can be written as p = a (g − ζ f g1 ) .

(14.274)

Substituting (14.274) into (14.271) and minimizing with respect to the parameter a we obtain min J¯⊥ =

1¯ 2 E ϕ¯ , 2

g − ζ f g1 2 . E¯ = 1 2 (g − ζ f g1 ) E

(14.275)

Now, we proceed to solve the variational problem (14.263) and (14.264). In (14.263), the function z in the first approximation is z + (ϕ − 1)ζ = R μν Aμν + R1μν h Bμν . The formulae (14.263) and (14.266) suggest the substitution z α → z¯ α :

(14.276)

14.7


675

? @ μν z α = −h Q μν Aμν −h 2 z 1 Bμν +Aζ − ζ f A + z¯ α , ,α

where A does not depend on ζ, while Q μν and Q μν 1 are the solutions of the ordinary differential equations d Q μν = R μν , dζ

d Q 1 μν = R1μν . dζ

(14.277)

The additive constants in the solutions of (14.277) and A should be chosen in such a way as to make the functions z¯ α satisfy the constraints

f z¯ a = 0,

ζ f z¯ a = 0.

In order to do so, we set

f Q μν = 0,

h

ζ f Q μν Aμν + ζ f Q 1 μν h B μν , q q = ζ 2 f − ζ f 2 . (14.278)

μν

A=

f Q 1 = 0,

The functional J∠ depends on z¯ α as 1

1 J∠ = 2

2

− 12

G αβ z¯ α,ζ + ϕ¯ α (α → β) dζ.

(14.279)

Here, the notation ϕ¯ α = ϕα + A,α

(14.280)

is introduced. The minimization problem for the functional (14.279) is solved in the same way as the minimization problem the functional (14.270). The solution is 1 1 ¯ αβ ∠ ≡ min J∠ = G ϕ¯ α ϕ¯ β , h 2

¯ αβ = g − ζ f g1 2 H (−1)αβ . (14.281) G

2 Here, H (−1)αβ is the inverse tensor of the tensor G (−1) αβ (g − ζ f g1 ) . Let us make the substitution of the required functions ri → r¯i , ϕα → ϕ¯ α , ri = r¯i − n i A and, as in the previous section, denote by γαβ and ραβ the tensors

γαβ =

1 i i λ r¯ r¯i,β − a˚ αβ , ραβ = n¯ i r¯,αβ − b˚ αβ − ϕ¯ (α;β) − b˚ (α γλβ) . 2 ,α

676

14


In terms of the new variables, the transverse energy takes the form of the Reissner shear energy. To calculate the longitudinal energy, we write the tensors, Aαβ and (ϕ) Bαβ , in terms of r¯i , ϕ¯ α : : μν ; Aαβ = γαβ + hq −1 b˚ αβ ζ f Q μν γμν + ζ f Q 1 hρμν , μν

ζ f R 1

ζ f R μν (ϕ) λ ˚ ˚ Bαβ = ραβ + b(α γλ) + bαβ γμν + hρ μν .

ζ 2 f

ζ 2 f Substituting these expression into the strains, we obtain (ϕ) λ εαβ = Aαβ − h Bαβ ζ + h 2 b˚ (α ρλβ) ζ 2 ,

; : and for the longitudinal energy = h C¯ αβγ δ f + ξ C ′αβγ δ εαβ εγ δ we get : ; 2 = h Aαβγ δ γαβ γσ δ + h 3 ζ 2 f B αβγ δ ραβ ργ δ + 2h 2 D αβγ δ γαβ ργ δ .

(14.282)

The “effective” elastic moduli are given by the formulae A

αβγ δ

: ; ; : ζ f R γ δ ζ f ζ f Qγ δ h ˜ αβγ δ αβμν

ζ f + C¯ + =C + C h b˚ μν + 2 q

ζ 2 f

: ; ; : ζ f Q αβ ζ f R αβ ζ f γ δμν ˚ ¯ − h bμν +C , q

ζ 2 f ¯ αβγ δ

h ζ 3 f ˜ αβγ δ C B αβγ δ = C¯ αβγ δ + 2 ζ 2 f

ζ f R1 γ δ ζ f Q 1 γ δ ζ f αβμν ˚ ¯ − + [(α, β) ←→ (γ , δ)] , + C h bμν

ζ 2 f

ζ 2 f q

D

αβγ δ

ζ f Q αβ ζ f h ˜ αβγ δ 2 αβ γ δμν ˚ αβγ δ ¯ ¯

ζ f + C h bμν ζ f R − = −C

ζ f − C + 2 q

ζ f Q 1 γ δ ζ f R γ δ ζ f +C¯ αβμν h b˚ μν − . (14.283) q

ζ 2 f

Here, the notation is introduced: C˜ αβγ δ = C¯ αβγ σ bσ′δ + C¯ αβσ δ bσ′γ + C¯ ασ γ δ bσ′β + C¯ σβγ δ bσ′α . The surface energy density is the sum = + ∠ ,

(14.284)

14.7


677

where is given by the formula (14.282), and ∠ is given by the formula (14.281). The work of external forces is of the form (14.175), where is the function . . = −h P R αβ γαβ −h P R1 αβ hραβ +h {P} q −1 ζ f Q μν γ μν + ζ f Q 1 μν hρ μν . (14.285)

Homogeneous shells. In the particular case of homogeneous shells, f = 1, αβ

Q1

1 1 αβ ζ2 − Cαβ = −Q αβ , R αβ = −ζ Cαβ , R1 = 2 12 1 1 Cαβ , C¯ αβγ δ = C αβγ δ ; = ζ ζ2 − 6 4

therefore, the longitudinal surface energy density is = (γ ) +

1 (hρ) + int (γ , hρ) , 12

where 1 αβγ δ hC γαβ γσ δ, 2 2 αβμν ˚ μνγ δ ˚ αβ γδ γαβ ργ δ . bμν C + C bμν C + 2C 5 (14.286)

(γ ) = h 3 ˜ αβγ δ C int (γ , hρ) = − 24

Using the formulae (14.273) and (14.281) to calculate the transverse energy density, we get ∠ =

5h αβ G ϕ¯ a ϕ¯ β . 12

(14.287)

The function becomes =

h2 h [P]Cαβ γαβ − {P}Cαβ ραβ . 2 10

(14.288)

If the expressions for tensors C αβγ δ and Cαβ for isotropic shells are substituted into the formulae (14.286), (14.287) and (14.288), these formulae are transformed into the corresponding ones for homogeneous shell obtained in the previous section. Note that these formulae remain valid if the shell is inhomogeneous over the middle surface. Low frequency vibrations of heterogeneous shell. Classical approximation. Let the estimates for time derivatives (14.183) hold. Then, for heterogeneous shells,

678

14


2 2h 2 2 2 i 2h = O ρ0 c ε , h ρ0 ϕ,t z i,t κ = O ρ0 c 2 ε , l l 2 h h 2 ρ0r,ti z i,t = O ρ0 c2 2 ε2 , c ≡ μ/ρ. (14.289) l

h ρ0r,ti z i,t κ

Here, the generic case of ρ0 ≡ const · f (ζ ) is considered; μ denotes the least eigenvalue of the elastic moduli tensor. If ρ0 = const · f (ζ ) then, due to the constraints (14.259), stronger estimates than (14.289) are valid. We do not pause to consider this special case. It can be seen from (14.289) that the terms in the kinetic energy containing the functions z i affect the calculation of z i in the first approximation; however, it does not change the order of z i . Dropping the terms on the order of h/l and h/R, we obtain equations (14.67), where ρ¯ should be replaced by ρ0 , while T αβ and M αβ are determined from the constitutive equations for the surface energy (14.255). The error of the equations of the leading approximation is h/l + h/R. In dynamics, better approximations are impossible without changing the model significantly, because of the above-mentioned effect of the influence of the inertial terms on the calculation of z i . Low frequency vibrations of homogeneous shells. A refined dynamic theory similar to the theory taking into account transverse shear can be developed for the shells homogeneous in the thickness direction. For homogeneous shells, the estimates made for inertial terms in the previous section hold. Therefore, the inertial terms are not significant in calculation of z i in the first approximation. Consequently, the surface energy will be of the form (14.286) and (14.287). The derivation of the kinetic energy expression is analogous to that for isotropic shells. The only difference is that for anisotropic shells, the substitution ri → r¯i is defined by the equality ri = r¯i − n i A, and in the homogeneous case, according to (14.278), h2 A = − Cαβ Bαβ . Therefore, 60 K =

2 h 2 αβ 2 h2 1 κ a˚ αβ − Cαβ n¯ κ v¯ ,α . C vα;β ρh v¯ i v¯ i + − ϕ¯ α (α → β) + 2 12 5 12

(14.290)

In the isotropic case, the tensor 2 a˚ αβ − Cαβ (14.291) 5 is reduced to the spherical tensor 1 − 2σ5 a˚ αβ , and, since σ ≤ 1, it is positive. For anisotropic shells the tensor (14.291) may not be positive. In this case, the formula (14.292) results in a non-hyperbolic, and, apparently, unsatisfactory extrapolation. However, it correctly describes long wave vibrations.

14.8

Laminated Plates

679

Shells with small shear rigidity. Above, it was assumed that the elastic moduli do not depend on small parameters. However, in applications, shells with an anomalously small shear modulus tensor G αβ are of interest. Let us analyze to which extent the relations obtained are applicable in this case. Our derivation was based on determination of the functions z i by minimization of the transverse energy, and on neglection the interaction terms between z i and Aαβ , Bαβ . Let C αβγ δ ∼ μ, G αβ ∼ G. Then, the interaction terms between z α and h2 ¯ and in the Aαβ , Bαβ in the longitudinal energy are on the order of μ (ε A +ε B ) , lR h transverse energy are on the order of G ¯ (ε A +ε B ). The leading terms with respect l to z α are on the order of G ¯ 2 in the transverse energy and on the order of μ (h/l)2 ¯ 2 in the longitudinal energy. Therefore, all the conclusions still hold for the shells for which G h ≪ , R μ

2 h G ≪ . l μ

(14.292)

If the conditions (14.292) are not satisfied, the problem becomes extremely complicated; moreover, if the second constraint (14.292) does not hold, the two-dimensional description of the deformed state becomes impossible.

14.8 Laminated Plates We have seen that the classical two-dimensional plate theory can be used for any inhomogeneous plates, including the laminated plates (i.e. plates with a piece-wise constant dependence of the elastic moduli of the normal coordinate), if the thickness of the plate, h, is much smaller than the characteristic length of the stress state, l, h ≪ 1, l

(14.293)

and there are no other small parameters, in particular, the elastic moduli of different layers are of the same order. In various industrial applications, however, laminated plates are used with a high ratio of elastic moduli of laminae, which could reach several orders of magnitude. Especially beneficial are the sandwich plates (Fig. 14.3) with a high ratio of the shear moduli of the skin and the core, μs and μc , and a very thin skin thickness, h s , on the order of a few per cent of the thickness of the core, h c . For such plates the two new small parameters come into play, μc /μs and h s / h c . Apparently, the classical two-dimensional plate theory may not work due to the presence of additional small parameters, and a question arises as to what is the leading asymptotic two-dimensional plate theory. We develop such a theory for linear sandwich plate in the next section, and for nonlinear sandwich plates and shells in Sect. 14.10. In

680

14


Fig. 14.3 Notation for sandwich plates

this section we prepare a starting point for the asymptotic analysis, a linear twodimensional plate theory that employs the presence of only one small parameter, h/l, and allows for any relative magnitude of the elastic moduli of the layers. Inside each layer, the elastic moduli are assumed to be constants of the same order. The number of the layers can be any. Before proceeding to the derivation of this theory we outline the major peculiarities of the hard-skin sandwich plates. Hard-skin sandwich plates. We will call the sandwich plates, for which the parameter, μc h c /μs h s , is small, the hard-skin plates. The behavior of such plates depends crucially on the interplay between the two small parameters, μc h c /μs h s and18 h/l. It is convenient to introduce a shear stiffness parameter α by the relation μc h c = μs h s

α h , l

(14.294)

or, alternatively,

μc h c α = ln μs h s

E E μs h s h l = ln . ln ln l μc h c h

(14.295)

Here h s could be of the order of h c (thick skin) or much smaller than h c (thin skin). For hard-skin plates, α is a positive number. In bending problems there are three qualitatively and quantitatively different situations: 0 < α < 2,

α = 2,

α > 2.

(14.296)

In the case of “not very hard skin”, 0 < α < 2, the plate can be described in the leading approximation by the classical two-dimensional plate theory, though various simplifications appear due to the presence of additional small parameters. The case of “very hard skin”, α > 2, is quite different: the plate behaves as a membrane (even in the absence of the extension forces). In both cases, α < 2 and α > 2, the transverse shear is determined uniquely in terms of the displacements of the mid-plane. For “not very hard skin” (α < 2), transverse shear follows the Kirchhoff

18

For symmetric sandwich plates h = h c + 2h s .

14.8

Laminated Plates

681

hypothesis of classical plate theory: the normal fibers rotate to stay normal to the deformed mid-plane. For “very hard skin” (α > 2), the skin prevents the normal fibers from rotating, and the normal fibers remain normal to the initial mid-plate. The intermediate case, α = 2, is exceptional: the transverse shears cannot be found explicitly and remain additional kinematic parameters to be found from a boundary value problem. This is shown schematically in Fig. 14.4. We focus on static problems. However, the results are easily extended to lowfrequency vibrations, i.e. the vibrations, the characteristic frequency of which, ω, satisfies the condition ω∼

> 1 μc ρc , l

(14.297)

where ρc is mass density of the core. In the low frequency case, the two-dimensional dynamical equations are obtained from the static equations by adding the leading inertia terms to the equations of statics. If (14.297) does not hold, then a twodimensional theory must include high-frequency effects.

Fig. 14.4 Deformation of normal fibers for differerent plate stiffness parameter α

682

14


The shear stiffness α depends on the physical characteristics, the geometry of the sandwich, and on the force (through the characteristic length of the stress state, l). Therefore, the same plate can be described by classical theory for some force, but for a force with a shorter characteristic length may need a modified theory. The shear stiffness α decreases as l increases. Therefore, for sufficiently large l, α becomes smaller than 2, and one can apply classical plate theory. The membrane regime appears for sufficiently short characteristic length (but still l ≫ h). The smaller parameter h/l, the more pronounced is the difference between the two regimes and the sharpness of the transition at α = 2. We derive a two-dimensional theory, which is valid for any value of α and a universal two-dimensional theory which is the leading approximation for all values of elastic moduli, μs ∼ μc and μc ≪ μs . It turns out that in extension problems, the leading approximation is always given by classical plate theory. The deviations from classical plate theory become noticeable only in higher order terms. The leading correction is related to the core thickening. To characterize these corrections, we introduce parameter β by the relation μc h c = μs h s

√

hs hc l

β

,

and show that the core thickening is determined explicitly for β < 4 and for β > 4, while for β = 4 one has to solve a boundary value problem to find the core thickening. Asymptotic analysis of an elastic layer with the prescribed displacements of the faces. We begin the derivation with a solution of an auxiliary static problem: find the leading terms of the asymptotics for a homogeneous elastic layer, occupying the region x α ∈ , −h/2 x h/2, when the tangent displacements, wα , and the normal displacement, w, are prescribed at x = ±h/2 : wα |x=±h/2 = wα± x β

w|x=±h/2 = w ± x β .

(14.298)

We seek the asymptotics away from the plate edges. The boundary conditions at the edges do not affect the asymptotics. According to the Saint-Venant principle for the prescribed displacements at the boundary, any edge load exponentially decays away from the edge at the distances on the order of h, and does not influence the interior stress state. The characteristic length, l, of functions wα± , w ± is assumed to be much bigger than h. For material, which possesses a plane of elastic symmetry parallel to the midplane, the problem splits into a superposition of the bending problem and the extension problem (see subsection on evenness and oddness of minimizers in Sect. 5.13). This is symbolically shown in Fig. 14.5. In the bending problem, w and wα are even and odd functions of x, respectively; in the extension problem, w is an odd function of x, while wα are even functions.

14.8

Laminated Plates

683

Fig. 14.5 Split of an arbitrary problem with the prescribed displacements at the faces into the sum of the bending problem and the extension problem

The true displacements minimize the free energy functional

εαβ ≡

1 2

wα wβ + α β x x

h/2 −h/2

,

F εαβ , εα3 , ε33 dd x, εα3 ≡

1 2

wα w + α x x

(14.299)

,

ε33 ≡

w , x (14.300)

on the set of displacements extracted by the constraints (14.298). Let first the material be isotropic. For an isotropic material the free energy density is F=

2 1 α 2 λ εα + ε33 + μ εαβ εαβ + 2εα3 εα3 + ε33 . 2

(14.301)

Following the general recipe (see (14.130) and (14.131)), we split F into the sum of two functions F and F⊥ : F = F + F⊥

2 F εαβ ≡ min F εαβ , εα3 , ε33 = μ σ εαα + εαβ εαβ ,

εα ,ε

F⊥ ≡ F − F = 2μεα3 εα3 +

2 1 (λ + 2μ) ε33 + σ εαα . 2

(14.302) λ σ = , λ + 2μ

Consider first the bending problem. In the bending problem the displacements satisfy the constraints at the faces: α

w |x=±h/2 = u(x ), Here we introduced the notation u≡

h wα x=±h/2 = ±ψα (x β ) . 2

w+ + w− , 2

ψα ≡

wα+ − wα− . h

(14.303)

(14.304)

Denote the order of displacements at the faces by w. ¯ Without loss of generality we may assume that all components of displacements, w± and wα± , have the same

684

14


order (if they do not, than the corresponding smaller contributions will drop out from the leading approximation we are to find). The strains have the orders εαβ ∼

w ¯ , l

εα3 ∼

w ¯ , h

ε33 ∼

w ¯ . h

Therefore, all the leading terms are in F⊥ , and, keeping only the leading terms we have F⊥ =

μ wα w α λ + 2μ + 2 x x 2

w x

2

.

(14.305)

Minimization of the leading term of energy,

h/2 −h/2

F⊥ dd x,

on the set of all displacements w, wα , which satisfy the boundary conditions (14.303) yields w = u,

(14.306)

wα = ψα x.

At the next step of the variational-asymptotic method we seek the minimizer in the form w = u + w′ ,

wα = ψα x + wα′

(14.307)

where w ′ ≪ u ∼ w, ¯ wα′ ≪ ψα x ∼ w. ¯ We plug (14.307) in (14.302) and keep only the leading terms, containing w ′ and wα′ , and the leading interaction terms between w ′ , wα′ and u, ψα . The leading terms of wα′ in F ,

w ′α 2 ′ ′(α,β) μ σ + w(α,β) w , x α

′ w(α,β)

1 ≡ 2

are much smaller than the leading terms of wα′ in F⊥ , 1 wα′ w ′α μ . 2 x x The interaction terms between wα′ and ψα in F ψα w ′β 2μ σ α x β + xψ(α,β) w ′(α,β) x x

wβ′ wα′ + x α x α

14.8

Laminated Plates

685

are much smaller than the interaction term in F⊥ , u w ′ μ α ψα + α . x x The interaction term in F⊥ , w ′ w ′α , x x α is much smaller19 than w ′ ψ α x . x x α Retaining only the leading terms in w′ , wα′ and the leading interaction terms we arrive at the minimization problem of the functional,

h/2 −h/2

′α w ′α w u u + ψα + α + ψα + x x x xα ! ′ 2 ψ α λ + 2μ w + σx α d xd, + 2 x x

μ 2

on the set of all functions w ′ , wα′ , which vanish at the faces of the plate. The minimizer, obviously, is 2 ψ α h 1 − x2 wα′ = 0, w′ = σ , (14.308) 2 4 x α while the minimum of the functional is the shear energy h/2 μ ψα + u ,α (ψ α + u ,α ) dd x. −h/2 2

(14.309)

One can check that further terms of the asymptotic expansion give a smaller contribution to energy. So, finally, in the leading approximation, 1 w=u+ σ 2 =

19

h/2 −h/2

Fd x =

h2 α − x 2 ψ,α , 4

wα = ψα x,

(14.310)

μh μh 3 α 2 ψα + u ,α (ψ α + u ,α ) . σ ψ,α + ψ(α,β) ψ (α,β) + 12 2 (14.311)

The reader interested in more detailed explanation can find it in the paper where this derivation was given; the paper is cited in Bibliographic Comments.

686

14


Similarly, in the extension problem one obtains in the leading approximation 1 wα = u α + 2

w = ψ x,

=

h2 − x2 4

ψ , x α

uα ≡

wα+ + wα− , 2

ψ≡

w+ − w− , h (14.312)

h/2

λ + 2μ 2 2 Fd x = μh σ u α,α + u (α,β) u (α,β) + h ψ + σ u α,α . 2 −h/2 (14.313)

Equations (14.310)–(14.313) determine how energy of an elastic layer depends on the displacements, prescribed at the faces, in the leading approximation. For anisotropic materials, energy of a layer can be obtained in the same way. For materials possessing a plane of elastic symmetry that is parallel to the faces, free energy density has the form F=

1 αβγ δ 1 2 εαβ εγ δ + Eε33 + 2G αβ εα3 εβ3 + E αβ εαβ ε33 . E 2 2

(14.314)

The calculation of F and F⊥ yields the relations (see (14.240) and (14.241)): F = F + F⊥ ,

F = min F = εα ,ε

1 αβγ δ E εαβ εγ δ , 2

αβγ δ

E

= E αβγ δ −

1 αβ γ δ E E , E

(14.315)

2 1 1 F⊥ = F − F = 2G αβ εα3 εβ3 + E ε33 + E αβ εαβ . 2 E In the bending problems, in the leading approximation w=u+

=

1 αβ E ψα,β 2E

h/2 −h/2

Fd x =

h2 − x2 , 4

wα = ψα x,

(14.316)

h 3 αβγ δ h E ψα,β ψγ ,δ + G αβ ψα + u ,α ψβ + u ,β . 24 2

In the extension problem, in the leading approximation w = ψx

wα = u α +

1 2

h2 − x2 4

ψ x α

(14.317)

14.8

Laminated Plates

=

687

2 1 αβ h αβγ δ 1 Fd x = E u α,β u γ ,δ + E ψ + E u α,β . 2 2 E −h/2 h/2

A theory of laminated plates. The relations obtained in the previous subsection easily allow us to construct the governing two-dimensional equations of laminated plates, which contain all leading effects, but may include smaller effects as well.

± ± Denote by Fk w(k) , w(k)α the elastic energy of the kth layer with the prescribed + + − − displacements on the upper face, w(k) , w(k)α , and the lower face, w(k) , w(k)α . For isotropic materials, according to (14.311) and (14.313),

μ h3 2 k k (α,β) ± ± α + ψ(k)α,β ψ(k) σk ψ(k),α Fk w(k) , w(k)α = 12

α μk h k ψ(k)α + u (k),α ψ(k) + + u ,α (k) 2

2 (α,β) + μk h k σk u α(k),α + u (k)α,β u (k) 2 λk + 2μk d, h k ψ(k) + σk u α(k),α + 2

(14.318)

where σk = λk /(λk +2μk ), h k is the thickness of the kth layer, and λk , μk its Lame’s constants. u (k) = ψ(k)α =

− + + w(k) w(k)

2 + − w(k)α − w(k)α hk

u (k)α =

,

− + + w(k)α w(k)α

2

,

ψ(k) =

− + − w(k) w(k)

hk

,

(14.319)

,

Then, the total energy of the laminated plate, containing N layers, is N

± ± ± ± ± ± F w(1) , w w = F , ...w , w , ...w(N k (k)α . (k) (N )α (1)α )

(14.320)

k=1

To find a deformed state of the plate subject to the external forces acting on the top ± ± ± ± 20 , ...w(N face, one has to minimize over w(1) ) , w(1)α , ...w(N )α the functional I =F

± ± , ...w(N w(1) ),

± ± , ...w(N w(1)α )α

−

+ + d. Pw(1) + P α w(1)α

(14.321)

± ± The functions w(1) , ..., w(N )α must satisfy the conditions of continuity of displacements on the interfaces:

20

We number the layers starting from the top layer.

688

14 − + w(k) = w(k+1) ,

− + w(k)α = w(k+1)α ,


k = 1, ...

(14.322)

If the material is anisotropic, then one has to replace (14.318) by the sum of bending and extension energies (14.316) and (14.317). The energy functional contains small parameters, h k , and can be simplified further. The result, however, depends on the relative orders of elastic moduli in different layers. We consider in the next section only the case of symmetric hard-skin sandwich plates. The functional (14.321) can be a starting point for the asymptotic analysis in all other cases as well.

14.9 Sandwich Plates First we formulate the results of the asymptotic analysis. This will be followed by the derivation. Governing equations for isotropic plates. In the linear elasticity theory, the solution of any21 three-dimensional problem for a plate can be presented as a superposition of two solutions, one with even function of x, w (x α , x) , and odd functions of α β α x, w x , x and another one with odd function of x, w (x , x) , and even functions α β of x, w x , x . The first is called a bending problem, the second an extension problem. Symbolically, this is shown in Fig. 14.6. As was mentioned, the leading approximation in the extension problem is given by the classical plate theory. Therefore, here we summarize the results only for the bending problems. Consider first the isotropic plates. The elastic properties are characterized by the following dependence of Lame’s constants, λ and μ, on the normal coordinate. μ(x) =

6 |x| < h c /2 μc , μs h c /2 < |x| < h c /2 + h s

λ(x) =

6

λc λs

|x| < h c /2 . h c /2 < |x| < h c /2 + h s

Indices c and s mark the values in the core and in the skin. In case of thin skin (h s ≪ h c ), in all asymptotic estimates, h c can be replaced by h = h c + 2h s .

Fig. 14.6 Split of the load into the sum of loads causing bending and extension

21

It is assumed that elastic properties are symmetric with respect to the planes that are parallel to the mid-plane, and elastic moduli are even functions of x, i.e. the sandwich is symmetric.

14.9

Sandwich Plates

689

The plate is loaded at the top face by a surface force with components P, P α . The external tangent force at the faces, P α , is assumed to be on the order of the normal force, P, or smaller: P α = O(P).

(14.323)

The edge conditions are not essential in the derivation of the leading approximation. First let the skin be thin, i.e. one can neglect corrections on the order of h s / h c near unity. As was mentioned, there are three different cases. Case α < 2. In this case the displacement distribution over the normal coordinate is ⎧ 2

⎨ 1 σc h c − x 2 u 2 4

w = u− 1 2 ⎩ σs h s − x ′2 u 2

4

⎫ |x| h c /2 ⎬ , , ′ x h s /2 ⎭

wα = −u ,α x,

|x| h/2. (14.324)

Here x ′ is a local normal coordinate in the skin, x′ = x −

1 (h c + h s ) , 2

and a two-dimensional Laplace’s operator. The displacement field has the only field variable, u(t, x α ). It satisfies the Kirchhoff equation, μs h s h 2c (1 + σs ) 2 u = P.

(14.325)

This is the Euler equation of the functional

μs h s h 2c σs u 2 + u ,αβ u ,αβ − Pu d. 2

(14.326)

Specifying the geometrical constraints at the boundary and varying the functional (14.326) one gets the boundary conditions as well. Note that the bending rigidity in the functional (14.326), μs h s h 2c , coincides with the usual expression,

h/2

h3 2 μx d x = μc c + μs 12 3 −h/2 2

hc + hs 2

3

−

hc 2

3

,

≈

μs h s h 2c , 2

when μc h c ≪ μs h s andh s ≪ h c . Emphasize that the quadratic terms in (14.324) (and further in 14.337) must be taken into account in spite of the fact that they are much smaller than the previous leading term: these small quadratic terms in the displacement distribution give a finite contribution to energy.

690

14


For low-frequency vibrations, the only difference in the governing equation is the additional leading inertia term, μs h s h 2c (1 + σs ) 2 u + ρ¯

2 u = P, t 2

where ρ¯ is the mass density per unit area. The transition from statics to lowfrequency dynamics in all other cases considered further is similar and, thus, will not be mentioned again. Case α > 2. In this case, in the core w = u,

: ; wα = − u ,α + ωe ˚ αβ x β x.

(14.327)

Here · is the averaging over the plate

· =

1 ||

·d,

d,

(14.328)

I = xα x α .

(14.329)

|| =

and ω˚ a constant22 , : ; ω˚ = − u ,α xβ eαβ /I,

So, the in-plane displacements, wα , :are ;linear functions of the normal coordinate, x. They depend on three constants, u ,α and ω. ˚ It is assumed in (14.327) that the boundary conditions of the three-dimensional elasticity problem at the plate edge do not contain kinematic constraints on the displacements of the core. Otherwise, wα = 0.

(14.330)

Equation (14.330) also holds, if the core displacements are not constrained at the edge, but the plate is:clamped at the edge in normal direction (i.e. u = 0 at the edge). ; Then the constants, u ,α: and ω, ; ˚ are equal to zero due to the divergence theorem. For zero constants, u ,α and ω, ˚ the normal to the mid-plane in the undeformed state maintains its direction in space.: If ;the plate is clamped only at a part of the boundary (e.g., as a cantilever), then u ,α and ω˚ can be non-zero. In the upper skin, x > 0, the displacements are w = u,

hc : ; − x ′ u ,α . ˚ αβ x β wα = − u ,α − ωe 2

(14.331)

There are additional terms that provide continuity of the displacements (14.327) and (14.331) at the interface, but they give a small contribution to energy. The governing equation for u(x α ) is the membrane equation with bending 22

To simplify the formulas, the coordinates x α are chosen in such a way that x α = 0.

14.9

Sandwich Plates

691

μc h c u −

μs h 3s (1 + σs ) 2 u + P = 0. 12

(14.332)

Note the difference between the bending rigidities in (14.332) and (14.325). The first two terms of (14.332) have the orders μc h c

u¯ μs h 3s u¯ and , 2 l 10l 4

where u¯ is the order of displacements. The ratio of the second term to the first is on the order of μs h 3s /10μc h c l 2 . If the skin is so thin that

hs l

2

≪ 10

μc h c , μs h s

(14.333)

then the second term in (14.332) can be dropped and (14.332) transforms to the membrane equation, μc h c u + P = 0.

(14.334)

In many cases the condition (14.333) holds. For example, for h s / h c = 1/10, h c /l = 1/10,(14.333) becomes 10−6 ≪ μc /μs , which is usually true. However, since our task is to cover all cases when the the transition from the three-dimensional to a two-dimensional theory is possible, there is no reason to ignore the second term in (14.332). For a localized load, this term describes the boundary layer. The width of the boundary layer l ∗ = μs h 3s /10μc h c , must be much bigger than h in order for our two-dimensional theory to be sensible. Therefore,

μs 10μc

hs h

3

≫ 1.

(14.335)

In principle, there could be materials for which (14.333) and (14.335) hold, but they do not seem to be in industrial use. Equation (14.332) is the Euler equation of the functional

μc h c : ; μs h 3s σs u 2 + u ,αβ u ,αβ + u ,α − u ,α − ωe ˚ αβ x β 6 2 ,α 1 αγ ,α ˚ × u − u − ωe xγ − Pu d. (14.336)

: ; Note that the constants, u ,α and ω, ˚ though not appearing in the Euler equation, enter the boundary conditions, which are obtained by varying the functional (14.336). Case α = 2. The displacement distribution over the thickness is

692

14

w=u+

⎧ ⎨

h 2c 1 α σ − x 2 ψ,α 2 c 4

2 ⎩ − 1 σs h s − x ′2 u 2 4


|x| h c /2 , ′ x h s /2

⎧ xψα ⎪ ⎪ 2

⎨ 1 1 h ′2 h c α + h ψ + σ − x ψ , c α s wα = 2 2 42

2 ,α ⎪ ⎪ h h 1 1 ′2 c ⎩ − h c ψα − σs −x ψα , 2 2 4 2 ,α

|x| h c /2 ′ x h s /2, ′ x h s /2,

(14.337)

x >0 . x 2, where ψα is a linear function of in-plane coordinates, for α = 2, u and ψα are linked by a system of differential equations:

1 β 2 μs h s h c (1 + 2σs ) ψ,β + ψα + μc h c ψα + u ,α = 0 ,α 2 μc h c (ψ α + u ,α ),α + P = 0.

(14.338)

(14.339)

The boundary conditions can be obtained by varying the functional

μh α μs h s h 2 α 2 c c ,α (αβ) ψα + u ,α (ψ + u ) − Pu d. σs ψ,α + ψ(αβ) ψ + 2 2

Universal approximation for hard-skin plates with thin skin. Any asymptotic theory assumes some limit procedure. For example, the equations of classical plate theory may be thought of as equations for displacements of the mid-plane, which are obtained in the limit h → 0. In this limit procedure, one should fix the dependence of forces on the space coordinates and reduce the magnitude of forces in such a way that the displacements remain finite. In the case of hard-skin plates, one should simultaneously change elastic moduli and the ratio h s / h c to keep the shear parameter α constant. Though such asymptotic procedure helps to recognize the major elastic effects involved, in practical problems all the characteristics of the plate are given and are not a subject of a limit procedure. Therefore, it is desirable to have a two-dimensional theory, which is applicable for any α. We call such theory universal. It is easy to see that all three cases are the approximations of a two-dimensional theory with the energy functional I =

μs h s h 2c α 2 μs h 3s σs u 2 + u ,αβ u ,αβ + σs ψ,α + ψ(α,β) ψ (α,β) 6 2 α μc h c ψα + u ,α (ψ + u ,α ) − Pu d. + (14.340) 2

14.9

Sandwich Plates

693

For example, to obtain the relations of the case α > 2, we note that for α > 2 the leading term of the functional (14.340), which contains ψα , is

μs h s h 2c α 2 σs ψ,α + ψ(α,β) ψ α,β d. 2

Its minimum is zero. It is achieved at the function, ψα , that are solutions of the equations ψα ψβ + α = 0. x β x

(14.341)

The general solution of (14.341) is ψα = ψ˚ α − ωe ˚ αβ x β ,

(14.342)

where ψ˚ α , ω˚ are some constants. Considering the functional (14.340) on the fields (14.342) we obtain the functional

μc h c μs h 3s σs u 2 + u α,β u ,αβ + u ,α + ψ˚ α − ωe ˚ αβ x β 6 2

@ (14.343) × u ,α + ψ˚ α − ωe ˚ αβ xγ − Pu d.

The functional (14.343) must be minimized over the constants ψ˚ α and ω. ˚ The corresponding equations for ψ˚ α and ω˚ are

u ,α + ψ˚ α − ωe ˚ αβ x β d = 0

(14.344)

u ,α + ψ˚ α − ωe ˚ αβ x β eαγ xγ d = 0.

: ; Hence, ψ˚ α = − u ,α , ω˚ is given by (14.329), and (14.343) transforms to (14.336). A universal theory of sandwich plates. The equations of the previous subsection hold for hard-skin thin plates. In practice, there are plates with intermediate values of parameters, when the skin is hard but “not very hard” or thin, but not “very thin”. Therefore, it is desirable to have a two-dimensional theory which works for all values of parameters: μs could be on the order of μc , h s on the order of h c , but the cases μc ≪ μs and h s ≪ h c should also be included. The energy functional of such theory, derived further, is

694

I =

14


μs h s 2 μs h 3s α σs u 2 + u α,β u ,αβ + σs h c ψ,α − h s u 6 2

μc h 3c α 2 σc ψ,α + ψα,β ψ α,β + h c ψ(α,β) − h s u ,αβ h c ψ (α,β) − h s u ,αβ + 12 α μc h c ,α ψα + u ,α (ψ + u ) − Pu d. (14.345) + 2

In special cases considered, it transforms to the functionals of the previous subsection. In case of the hard-skin thick plate the fourth term can be omitted. In the case considered by classical theory of plates, μs ∼ μc , the functional can be simplified using the presence of the small parameter, h/l. The leading term of the functional is the shear energy,

μc h c ψα + u ,α (ψ α + u ,α ) d. 2

Minimum of the shear energy is zero and achieved for (14.346)

ψα = −u ,α . The functional (14.345) on the field (14.346) is ?

@ A (u)2 + Bu ,αβ u ,αβ − Pu d

(14.347)

where A=

B=

μs h 3s μs h s (h c + h s )2 μc h 3c σs + σs + σc = 6 2 12 μs h 3s μs h s (h c + h s )2 μc h 2c + + = 6 2 12

h/2

μ (x) σ (x) x 2 d x −h/2

(14.348)

h/2

μ (x) x 2 d x.

−h/2

The functional (14.347) with the coefficients (14.348) is the energy functional of the classical plate theory. The particular value of the shear factor, μc h c , in (14.345) is not essential for the theory to be correct in the case μs ∼ μc . However, this value is important to embed the case of the hard-skin plates. Now we proceed to the derivation. Bending of isotropic hard-skin plates. In bending problems, w is an even function of x, while wα are odd functions. Therefore, the number of arguments of the energy functional is reduced, and, as such, one can take the kinematic characteristics of the skin, u s , ψαs , u sα , ψ s , and the kinematic characteristics of the bending problem for

14.9

Sandwich Plates

695

the core, u c , ψαc . In what follows we drop index of the core and write u ≡ uc,

ψα ≡ ψαc ,

while for the skin kinematic parameters we use capital letters: u s = U,

ψαs = α ,

u sα = Uα ,

ψ s = .

Since for the skin, w+ + w− , 2 w + + wα− Uα = α , 2 U =

w+ − w− , hs w + − wα− α = α , hs =

the skin surface displacements are expressed in terms of U, , α as w+ = U +

hs , 2

w− = U −

hs , 2

wα+ = Uα +

hs α , 2

wα− = Uα −

hs α . 2

For the core, according to (14.310), w + = u,

wα+ =

hc ψα . 2

(14.349)

The continuity of displacements put constraints on the admissible functions: U−

hs = u, 2

Uα −

hs hc α = ψα . 2 2

(14.350)

The energy functional to be analyzed is I =2

μh μs h 3s α 2 s s σs ,α + (α,β) (α,β) + α + U,α (α + U ,α ) 12 2

(λ + 2μ ) h s s s α 2 α 2 + σs U,α + μs h s σs U,α d + U(α,β) U (α,β) + 2

μh μc h 3c α 2 c c σc ψ,α + ψ(α,β) ψ (α,β) + ψα + u ,α (ψ α + u ,α ) d + 12 2 h hs s d. (14.351) P U + + P α U α + α − 2 2

The functional depends on nine functions, u, ψα , U, α , Uα , , which are linked by three constraints (14.350). We can choose six functions u, ψα , , α as independent kinematic variables. Then from (14.350)

696

14

U =u+

hs , 2

Uα =


1 (h c ψα + h s α ) . 2

(14.352)

The work of external force becomes

P (u + h s ) + P

α

1 h c ψα + h s α 2

d.

The functional does not contain derivatives of , and minimization over is an algebraic problem. To shorten the way to the final result, we fix u, ψα and seek , α minimizing the functional. Let us assume that can be neglected in the first equation (14.352), i.e. U = u,

(14.353)

and that Uα ∼ hu ,α ,

α ∼ u ,α .

(14.354)

Besides, let us put P α = 0. After determining , α from this simplified problem, we will check that the resulting equations are consistent with our assumptions. Indeed, minimization over yields ˇ = −σs U α . ,α

(14.355)

In minimization over α , the terms with derivatives of α can be dropped as small in comparison with the terms in α + u ,α (α + u ,α ) . Hence, the minimizing ˇ α , are functions, ˇ α = −u ,α .

(14.356)

Plugging (14.353)–(14.356) in the energy functional we obtain 1 I =2 μs h s

1 2 h 2s α σs u 2 + u ,αβ u ,αβ + σs h c ψ,α − h s u 4 12 d + h c ψ(α,β) − h s u ,αβ h c ψ (α,β) − h s u ,αβ 2

α 2 hc μc h c σs ψ,α + + ψ(α,β) ψ (α,β) μs h s 12 1 1 Pud. + ψα + u ,α (ψ α + u ,α ) d − 2 μs h s (14.357)

14.9

Sandwich Plates

697

The first integral is the energy of the skin, the second the energy of the core, and the last the work of external force (all are referred to μs h s ). In the skin energy, the first term is the bending energy of the skin, and the second the extension energy. Let us minimize the functional (14.357) over ψ α . If α < 2 then the leading term containing ψα is the core shear energy, ψα + u ,α (ψ α + u ,α ) . Minimum is achieved at ψα = −u ,α .

(14.358)

In the extension energy of the skin, h c ψ(α,β) − h s u ,αβ can be replaced by −h c u ,αβ . The bending energy of the skin and the bending energy of the core can be neglected compared with extension energy of the skin. Finally, we arrive at the functional of classical plate theory, in which small terms are dropped: I =

μs h s h 2c 2 ,αβ σs u + u ,αβ u − Pu d. 2

The energy of the plate in this case is just the extension energy of the skin. If α > 2, then the leading terms of (14.357) containing ψα are in the extension energy of the skin: 1 I =2 μs h s

The minimizer is

2 1 α σs h c ψ,α − h s u 4 1 + h c ψ(α,β) − h s u ,αβ h c ψ (α,β) − h s u ,αβ d.

ψα =

hs u ,α + ψ˜ α , hc

(14.359)

(14.360)

where ψ˜ α is any solution of the equations ψ˜ α,β + ψ˜ β,α = 0.

(14.361)

The general solution of (14.361) is ψ˜ α = ψ˚ α + ωe ˚ αβ x β .

(14.362)

It contains three arbitrary constants, ψ˚ α and ω. ˚ The minimum of the functional (14.359) is zero. Function ψα enters in shear energy in the sum with u ,α . Therefore, for thin skin plates, the first term in (14.360) can be dropped, and the leading terms of energy functional becomes

698

14

I =


μs h 3s σs u 2 + u ,αβ u ,αβ 6

μc h c γ ,α α αγ ˚ ˚ u ,α + ψα + ωe ˚ αγ x u + ψ + ωe ˚ xγ − Pu d. + 2 (14.363)

The energy consists of the shear energy of the core and the bending energy of the skin. Functional (14.363) should be minimized over the normal displacements, u, ˚ These constants are determined from (14.344). and the constants, ψ˚ α and ω. ˚ energy of the plate is identical to membrane energy with the For zero ψ˚ α , ω, extensional force μc h c . If, additionally, the skin is so thin that μs h 3s ≪ 1, μc h c l 2

(14.364)

then the functional simplifies to

I =

μc h c u ,α u ,α − Pu d. 2

For thick skin plates, the term μc h c μs h s

h 2c α 2 σs ψ,α + ψ(α,β) ψ α,β d 12

considered on the field (14.360), becomes μc h c μs h s

h 2s σs u 2 + u ,αβ u ,αβ d. 12

For any positive α it is much smaller than the first term of (14.357). Finally, for the thick skin plate, I =

μc h c μs h 3s hs γ 2 ,αβ ˚ 1+ u ,α + ψα + ωe σs u + u ,αβ u ˚ αγ x + 6 2 hc hs (14.365) u ,β + ψ˚ β + +ωe × 1+ ˚ βγ xγ − Pu d. hc

The values of the constants, ψ˚ α , ω, ˚ change accordingly: hs u ,α , ψ˚ α = − 1 + hc

; > hc : u ,α xβ eαβ I. ω˚ = − 1 + hc

If α = 2, then the terms containing ψα in the core shear energy and the skin extension energy are of the same order. To find ψα one has to solve a boundary

14.9

Sandwich Plates

699

value problem. Clearly, ψα is going to be on the order of u ,α . Thus, the bending energy of the core and the skin can be dropped, the difference, h c ψ(α,β) − h s u ,αβ , in the extension energy of the skin can be replaced for thin skin by h c ψ(α,β) , and the energy functional for thin skin plates simplifies to I =

μh α μs h s h 2c α 2 c c α,β ,α σs ψ,α + ψα,β ψ + ψα + u ,α (ψ + u ) − Pu d. 2 2

To combine all three cases for thin skin plate, one keeps in the functional (14.357) the skin bending energy, the core shear energy, drop the core bending energy and neglect in the skin extension energy h s u ,αβ . So, we get the functional (14.340). For a thick skin plate one has to use the functional (14.364). To check the validity of the simplifications made we note that the relations (14.354) follow from the second formula (14.352) and from (14.356). Thus, from (14.355), ∼ hu, and the neglection of in the first relation (14.350) was indeed possible. Besides, the terms in the work of external forces containing P α are small in comparison with P, if the assumption (14.323) holds. This justifies the simplifications made. Bending of anisotropic hard-skin plates. For anisotropic plates a variety of possible asymptotics becomes much richer. We consider here only the cases when the elastic moduli within each layer are of the same order. The derivation repeats that of the previous subsection and yields the following energy functionals for thin skin plates: for α < 2

h s h 2c αβγ δ E s u ,αβ u ,γ δ − Pu d 2

(14.366)

for α > 2

h 3s αβγ δ hc E s u ,αβ u ,γ δ + G αβ d u u − Pu ,α ,β 6 2 c

(14.367)

for α = 2

h s h 2c αβγ δ hc E s ψ,αβ ψ,γ δ + G αβ ψ + u ψ + u − Pu d (14.368) α ,α β ,β 2 2 c

Combining the results, we get for any α the energy functional

h 3s αβγ δ h s h 2c αβγ δ E u ,αβ u ,γ δ + E s ψα,β ψγ ,δ 6 2 hc ψ ψ − Pu d. + G αβ + u + u α ,α β ,β 2 c

(14.369)

700

14


For thick skin plate, the only essential simplification from the general functional of the previous section is the possibility to find explicitly23 and α : 1 α = − E αβ U,α , E

α = 0.

(14.370)

These relations were used in the derivation of (14.366)–(14.368). So, the energy functional of anisotropic thick skin plate is

h 3s αβγ δ h s αβγ δ E s u α,β u γ ,δ + E s h c ψα,β − h s u ,αβ h c ψγ ,δ − h s u ,αβ 6 2

h c αβ h 3c αβγ δ + E c ψα,β ψγ ,δ + G c ψα + u ,α ψβ + u ,β − Pu d. 12 2 Extension of the isotropic hard-skin sandwich plates. In the extension problem, wα are even functions of x, w an odd function of x. We characterize the kinematics of the sandwich plates by the core parameters, u α and ψ, and the skin parameters, Uα , , U, and α . Energy is the sum of energy of the core and energy of the skin: F=

λ + 2μ 2 2 c c d h c ψ + σc u α,α μc h c σc u α,α + u (α,β) u α,β + 2

μh μs h 3s α 2 s s +2 α + U,α (α + U ,α ) σs ,α + α,β α,β + 12 2

λ + 2μ s s α 2 α 2 . h s + σs U,α + U(α,β) U (α,β) + +μs h s σs U,α 2 (14.371)

The continuity of the displacements on the interfaces allows us to express Uα and U in terms of other parameters: Uα = u α +

hs α , 2

U=

1 (h c ψ + h s ) . 2

(14.372)

Functions u α , ψ, α and may be considered independent. Let us assume that the terms h s α and h s in (14.372) are small, thus setting Uα = u α ,

23

U=

1 h c ψ. 2

(14.373)

We use for anisotropic plates the same notation for kinematic parameters as for isotropic ones.

14.10

Nonlinear Theory of Hard-Skin Plates and Shells

701

We put also P = 0. Keeping in the energy functional only the leading terms in and α and leading interaction terms we obtain α = U,α = −

hc ,α , 2

α = −σs U,α = −σs u α,α .

(14.374)

The energy functional becomes 2 2 λc + 2μc μc h c 1 σc u α,α + u (α,β) u (α,β) + d I = ψ + σc u α,α μs h s μs h s 2μc 2 2 2 hs hc σc ψ 2 + ψ,αβ ψ ,α,β + σs u α,α + u (α,β) u (α,β) d +2 12 · 4 1 − P α u α d. (14.375) μs h s 2 If β < 4, then the leading term in ψ is the core “thickening” energy ψ + σc u α,α . Minimization of the leading term yields ψ = −σc u α,α ,

(14.376)

and, neglecting the core extension energy, which is small, we arrive at the classical plate theory: I =

?

@ 2 μs h s σc u α,α + u (α,β) u (α,β) − P α u α d.

(14.377)

If β > 4, then the leading terms in ψ are in the skin energy. Minimization over ψ yields ψ = 0. The core extension energy is smaller than that of the skin. So, in the leading approximation we again arrive at the functional (14.377). If β = 4, then the terms with ψ in the core and the skin energy are of the same order, and, to find ψ, one has to solve a boundary value problem. However, the corresponding corrections to energy functional are of negligible order. Therefore, the leading approximation is still given by the classical theory. The result holds for anisotropic plates as long as all eigenvalues of the elastic moduli tensor are of the same order.

14.10 Nonlinear Theory of Hard-Skin Plates and Shells By a hard-skin thin shell we mean a sandwich-type shell, for which the characteristic elastic modulus of the skin, μs , is much bigger than that of the core, μc , while the

702

14


thickness of the skin, h s , is much smaller the core thickness, h c : μc h c ≪ 1, μs h s

hs ≪ 1. hc

(14.378)

The parameters (14.378) are additional to the two key small parameters of any twodimensional shell theory, the ratio of the plate thickness, h, to the characteristic length of the stress state along the shell, l, and the ratio of h to the characteristic curvature radius of the shell in the unloaded state, R: h ≪ 1, l

h ≪ 1. R

(14.379)

The parameters h/l and h/R must be small in that order the three-dimensional stress state can be described by a two-dimensional theory. The additional parameters (14.378), as we have shown for plates, can change the leading approximation of an asymptotic theory. The same, of course, occurs for shells, and we derive in this section the corresponding leading asymptotic two-dimensional theory. We assume that material is physically linear, and the magnitude of strains, ε, is much smaller than any of the small parameters (14.378) and (14.379). Displacements of the shell, however, can be large. We also accept that R = ∞. The nonlinear equations for hard-skin plates can be obtained by tending R to infinity in the relations obtained for shells. To simplify the estimates, we assume that R ∼ l when R = ∞. As for plates, we introduce the shear stiffness parameter α by the relation μc h c = μs h s

α h . l

(14.380)

In the case of shells, there are two distinctive situations: α < 2,

and α 2.

For α < 2 the classical shell theory remains valid. For α 2 the classical shell theory does not work: the transverse shear enters the leading approximation. In contrast to plates, the transverse shear is not determined explicitly for shells, if α > 2, and should be found from a boundary value problem. First we give the formulation of the leading approximation of the two-dimensional theory for isotropic shell. Nonlinear two-dimensional theory of hard-skin shells. We consider a generic case of shell deformations, when bendings and extensions are of the same order and have the order of strains, ε : Aαβ ∼ ε,

h Bαβ ∼ ε.

14.10


703

It is shown further that position vectors of the material points of the core have the form 1 h2 x i (ξ α , ξ ) = r i (ξ α ) + ξ n i (ξ α ) (1 + ϕ (ξ α )) + rβi ϕ β (ξ α ) ξ + n i σc Bαα ξ 2 − c . 2 4 (14.381)

formula that (14.381) contains six two-dimensional fields, characterizing the deformation of the shell: r i (ξ α ) ,

ϕ β (ξ α )

and ϕ (ξ α ) .

Functions ϕ α and ϕ describe the transverse shear and the core thickening, respectively. Energy of the shell in the leading approximation is the sum of energy of the skin and a functional F r i (ξ α ) , F=

2 h 2 2 ˚ + F r i (ξ α ) . 2μs h s σs Aαα + Aαβ Aαβ + s σs Bαα + Bαβ B αβ d ˚ 12 (14.382)

The functional F r i (ξ α ) is determined from the solution of the following minimization problem: F r i (ξ α ) = min

ϕ,ϕα

2

μs h s h 2c σs B˜ αα + ϕ b˚ αα + B˜ αβ + ϕ b˚ αβ B˜ αβ + ϕ b˚ αβ ˚ 2 λc + 2μc μc h c α α 2 ˚ ϕα ϕ + h c ϕ + σc A α d . (14.383) + 2 2

Here B˜ αβ = Bαβ − ϕ(α;β) .

(14.384)

Minimization over ϕ can be done explicitly; however, it is better to keep the general form (14.383) since it clarifies the origin of different terms. In minimization over ϕα , one should take into account the kinematic constraints, if such are set at the shell edge. In general equations (14.382) and (14.383) form the final result and do not admit further simplifications. In some particular the simplifications and even an icases, α can be done. We describe some explicit computation of the functional F r (ξ ) such cases. First of all, functional F r i (ξ α ) can be computed explicitly if α < 2. In this case, the two last terms in (14.383) are the leading terms of the functional. Minimization of these two terms yields ϕα = 0,

ϕ = −σc Aαα .

(14.385)

704

14


Functional F considered on the fields (14.385) becomes F r i (ξ α ) =

˚

μs h s h 2c α 2 ˚ σs Bα + Bαβ B αβ d . 2

(14.386)

Energy at equations (14.382) and (14.386) corresponds to classical nonlinear shell theory. If α > 2, then an explicit computation of F is usually not possible. Remarkably, this can be done for plates, if in a nonlinear problem the bending measure can be expressed in terms of normal displacement, u, as in linear theory, Bαβ = u ,αβ .

(14.387)

Indeed, for plates b˚ αβ = 0, and ϕ enters only the last term in (14.383); therefore ϕ = −σc Aαα .

(14.388)

Then F r i (ξ α ) = min ϕα

˚

μs h s h 2c α α 2 σs Bα − ϕ;α + 2 αβ μc h c α (α;β) ˚ ϕα ϕ d . + + Bαβ − ϕ(α;β) B − ϕ 2 (14.389)

Note that for plates ϕ(α;β) = ϕ(α,β) . Assume that there are no kinematic constraints for ϕα at the edge. For α > 2 the last term in (14.389) is much smaller than the term μs h s h 2c ϕ(α;β) ϕ (α;β) and can be ignored in calculation of ϕα . We arrive at the variational problem: min ϕα

˚

μs h s h 2c α 2 σs u − ϕ,α . (14.390) + u ,αβ − ϕ(α,β) u ,αβ − ϕ (α,β) 2

The minimizer of the variational problem (14.390) can be found explicitly: ϕα = u ,α + ϕ˚ α − ωe ˚ αβ x β .

(14.391)

Here ϕ˚ α and ω˚ are arbitrary constants. The functional (14.389) on the fields (14.391) takes the form μc h c ˚ u ,α + ϕ˚ α − ωe ˚ αβ x β u ,α + ϕ˚ α − ωe ˚ αγ xγ d . ˚ 2 This integral must be minimized over parameters ϕ˚ α and ω. ˚ That links the values of the constants with the normal displacements:

14.10


: ;> ω˚ = eαβ u ,α ξβ I,

: ; ϕ˚ α = − u ,α ,

705

I = ξ α ξα ,

where · means average value over the plate, 1 ˚ ˚ ˚

· = = d , ·d , ˚ ˚ ˚

and the coordinate system ξ α is chosen in such a way that ξ α = 0. Finally, : ; : ; μc h c ˚ F= u ,α − u ,α − ωe ˚ αβ ξ β u ,α − u ,β − ωe ˚ αγ ξγ d . ˚ 2

Emphasize that the extension measure is still nonlinear, and the theory remains nonlinear. In particular, to study buckling, one can assume the normal displacements to be small; thus (14.387) holds, while in the extension measure, Aαβ , one should keep the quadratic term in u ,α , 1 Aαβ = u (α,β) + u ,α u ,β , 2 2 because the tangent displacements, u α , could be on the order of u ,α . Simplifications in (14.389) are also possible in a linear shell theory. As we have seen in Sect.14.2, without loss of accuracy, the bending measure, Bαβ , can be replaced by another bending measure, ραβ : γ ραβ = n˚ i u i(,α ;β) + φeγ (α b˚ β) ,

where u α are the tangent components of displacement vector, and φ the rotation of the mid-surface around the normal vector, φ=

1 αβ e u α;β , 2

u α = r˚αi u i .

Replacing Bαβ in (14.383) by ραβ and changing variables, ϕα → ϕ˜ α , ϕα = n˚ i u i,α + ϕ˜ α , we obtain the following variational problem for F F = min ϕ,ϕα

˚

2

μs h s h 2c γ α σs −ϕ˜ ;α + ϕ b˚ αα + −ϕ˜ (α;β) + φeγ (α b˚ β) + ϕ b˚ αα 2

μ h c c ϕ˜ α + n˚ i u i,α ϕ˜ α + n˚ i u i,α −ϕ˜ (α;β) + φeγ (α b˚ γβ) + ϕ b˚ αβ + 2 λc + 2μc α 2 ˚ d . (14.392) h c ϕ + σc Aα + 2

706

14


For α > 2, the last two terms in (14.392) can be ignored in computation of ϕ˜ α and ϕ. Minimization over ϕ yields ϕ = ϕ˜ α;β b˚ αβ

F b˚ γ δ b˚ γ δ .

In general, minimization over ϕ˜ α is a non-elementary problem. However, for γ example, for spherical shells, since eγ (α b˚ β) = 0, one gets ϕ˜ (α;β) = 0,

ϕ = 0.

Interestingly, the spherical geometry of hard skins prevents the core from thickening (ϕ = 0). Let the edge conditions allow only zero values of ϕ˜ α . Then F=

˚

2 μc h c λc + 2μc ˚ d . h c σc Aαα n˚ i u i,α n˚ j u j,α + 2 2

(14.393)

For hard-skin shells, the last term in (14.393) is small in comparison with the first term of (14.382) and can dropped. Finally, for spherical shells, F r i (ξ α ) =

˚

μc h c ˚ n˚ i u i,α n˚ j u j,α d . 2

In general, for non-spherical shells and, moreover, in non-linear shell theory for α 2, the variational problem (14.383) does not admit explicit solution. Equations (14.382) and (14.383) correspond to a universal theory, which holds for any α. The equations and boundary conditions are obtained by equating the variation of energy to the work of external forces. We give here only a system of equations, assuming that there is only normal force acting on the shell face. We introduce the notations, 1 0 S αβ = 4μs h s Aγγ a˚ αβ + Aαβ + (λc + 2μc ) h c ϕ + σc Aαα M αβ = −

@ ?

1 μs h 3s 0 σs Bγγ a˚ αβ + B αβ − μs h s h 2c σs B˜ γγ + ϕ b˚ γγ a˚ αβ + B˜ αβ + ϕ b˚ αβ . 3

Then the governing equations are

S αβ rβi + M αβ n i ;β − Pn i = 0 ;α

(14.394)

@

?

β μs h s h 2c σs B˜ αα + ϕ b˚ αα b˚ β + B˜ αβ + ϕ b˚ αβ b˚ αβ + (λc + 2μc ) h c ϕ + σ Aαα = 0

14.10


707

?

@ μc h c ϕα + μs h s h 2c σs B˜ γγ + ϕ b˚ γγ + B˜ αβ + ϕ b˚ αβ = 0. ;β

For low-frequency vibrations, the only change would be the inertial term in the right hand side of the first equation (14.394), ρ ¯ 2 x i /t 2 . Now we proceed to the derivation of (14.382) and (14.383). Elastic shells with the prescribed values of displacements at the faces. In this subsection we give a solution of an auxiliary problem: find the leading approximation for energy of an elastic shell when the displacements are prescribed at the faces of the shell and the characteristic length of the prescribed displacements, l, is much bigger than the shell thickness h. First of all, it is clear that the magnitude of the prescribed displacements cannot be arbitrary, otherwise our condition, that the strain magnitude, ε, is small can be violated. To find the corresponding constraints for the prescribed displacements, we present the position vector of a material point, x i (ξ α , ξ ) , in the form x i (ξ α , ξ ) = r i (ξ α ) + ξ n i (ξ α ) + hy i (ξ α , ξ ) .

(14.395)

If we define r i (ξ α ) in terms of x i (ξ α , ξ ) as 1 r (ξ ) = h i

α

h/2

x i (ξ α , ξ ) dξ,

(14.396)

−h/2

then equation (14.395) can be viewed as the definition of functions y i (ξ α , ξ ) . It was shown in Sect. 14.4 that y i = O(ε). So, the position vectors of the two faces, ξ = ±h/2, should have the form x+i (ξ α ) ≡ x i (ξ α , h/2) = r i (ξ α ) + x−i (ξ α ) ≡ x i (ξ α , −h/2) = r i (ξ α ) −

h i α n (ξ ) + hy+i (ξ α ) , 2

h i α n (ξ ) + hy−i (ξ α ) , 2

(14.397)

where r i (ξ α ) , y+i (ξ α ) and y−i (ξ α ) are some given functions, and y−i , y+i are on the order of ε. Let now the positions of the face particles, x+i (ξ α ) and x−i (ξ α ) be given. We define r i (ξ α ) as r i (ξ α ) =

1 i α x+ (ξ ) + x−i (ξ α ) , 2

(14.398)

and define y+i and y−i by equations (14.397). We assume that the prescribed functions, x+i and x−i , are such that y+i and y−i are on the order of ε. From (14.397) and (14.398),

708

14


y+i (ξ α ) = −y−i (ξ α ) .

(14.399)

Equations -(14.397) and (14.398) establish a one-to-one correspondence between all . . functions x+i (ξ α ) , x−i (ξ α ) and all functions r i (ξ α ) , y+i (ξ α ) , y−i (ξ α ) obeying (14.399). We denote y+i (ξ α ) by ϕ i (ξ α ) /2 and finally write the boundary conditions as x+i (ξ α ) = r i (ξ α ) +

h h i α n (ξ ) + ϕ i (ξ α ) , 2 2

(14.400)

x−i (ξ α ) = r i (ξ α ) −

h h i α n (ξ ) − ϕ i (ξ α ) . 2 2

(14.401)

We aim to find energy of the shell in the limit h/l → 0 when the positions of the face points are given by (14.400) and (14.401). The strains, εαβ , εα3 , ε33 are linked to the particle positions by the formulas, εαβ =

1 i x xi,β − g˚ αβ , 2 ,α

i εα3 = x,α xi,ξ ,

ε33 = x,ξi xi,ξ − 1.

(14.402)

The total free energy of the shell is F x i (ξ α , ξ ) =

˚

h/2 −h/2

˚ F εαβ , εα3 , ε33 d dξ.

(14.403)

In case of isotropic material, F = F + F⊥ ,

2 F εαβ = min F εαβ , εα3 , ε33 = μ σ g˚ αβ εαβ + g˚ αβ g˚ γ δ εαγ εβδ εα3 ,ε33

F⊥ ≡ F − F = 2μg˚ αβ εα3 εβ3 +

2 1 (λ + 2μ) ξ33 + σ g˚ αβ εαβ . 2

The contravariant components of the metric tensor in initial state, g˚ αβ , can be set equal to a˚ αβ in the leading approximation. We seek the minimum value of the functional F x i (ξ α , ξ ) on the set of all functions x i (ξ α , ξ ) satisfying the constraints (14.400, 14.401). This variational problem corresponds to the case of the free edge of the shell. However, any other edge boundary condition would introduce only small corrections to the stress state located at a vicinity of the edge of the size h. The variational problem contains a small parameter h/l, and we will study it by the variational-asymptotic method.

14.10


709

Let us make a substitution of the required functions, x i (ξ α , ξ ) → y i (ξ α , ξ ) , defined by (14.395). In (14.395) r i (ξ α ) are assumed to be taken from the boundary conditions (14.400, 14.401). Therefore, (14.396) does not necessarily hold. Plugging (14.395) into (14.402) we obtain the strains, 1 μ i i εαβ = Aαβ − h Bαβ ζ + h 2 Cαβ ζ 2 + hr(α yi,β) − h 2 b(α rμi yi,β)ζ + h 2 y,α yi,β , 2 (14.404) i 2εα3 = rαi yi,ζ − hbαμ rμi yi,ζ + hn i yi,α + hy,α yi,ζ , 1 ε33 = n i yi,ζ + y,ζi yi,ζ . 2

Here Cαβ =

1 γ bα bγβ − b˚ αγ bγβ . 2

(14.405)

To determine the orders of y i , we take first the approximate formulas for strains: εαβ = Aαβ − h Bαβ ,

2εα3 = yα,ζ ,

ε33 = yα,ζ .

(14.406)

Here yα and y are the projections of y i on the tangent plane to and on the normal to : yα ≡ rαi yi ,

y ≡ n i yi .

The leading terms of energy functional which contain yα , y are h

˚

1 2

− 21

λ + 2μ μ αβ α α 2 ˚ y,ζ + σ Aα − σ h Bα ζ + a˚ yα,ζ yβ,ζ d dζ. 2 2

(14.407)

Here Aαα = a˚ αβ Aαβ , Bαα = a˚ αβ Bαβ . Minimum of the functional (14.407) must be sought on the set of functions yα , y satisfying the boundary conditions 1 yα ζ = 1 = ϕα , 2 2

1 yα ζ =− 1 = − ϕα , 2 2

where ϕα ≡ rαi ϕi , ϕ ≡ r i ϕi . The minimizer is yα = ϕα ζ,

1 y ζ = 1 = ϕ, 2 2

1 y ζ =− 1 = − ϕ, 2 2 (14.408)

1 1 2 α . y = ϕζ + σ h Bα ζ − 2 4

For the minimum value of energy we have

(14.409)

710

14

h

˚


2 μ λ + 2μ ˚ ϕ + σ Aαα + a˚ αβ ϕα ϕβ d . 2 2

(14.410)

We see that yα = O(ε), y = O(ε). Since (see 14.145) λ Bλβ) + O(ε2 ), h 2 Cαβ = h 2 b˚ (α

the leading terms in (14.404) are indeed given by (14.406). Finally, the shell energy is F r i , ϕα , ϕ =

μh 3

2 2 μh σ Aαα + Aαβ Aαβ + σ Bαα + Bαβ B αβ + ˚ 12 +

2 μh αβ λ + 2μ ˚ ϕ + σ Aαα + a˚ ϕα ϕβ d . 2 2

(14.411)

Skin energy. In Sect. 14.9 we have shown that for hard-skin plates the skin energy coincides with the two-dimensional energy of the plate, the displacements of which are equal to the displacements of the interface. Here we skip the corresponding derivation for shells and accept that the skin energy is equal to the two-dimensional energy of the shell, the position vector of which is (14.400) for upper skin and (14.401) for lower skin. Denote the corresponding extension and bending measures + − − by A+ αβ , Bαβ and Aαβ , Bαβ . For isotropic shells the two-dimensional energy of the two skin layers is

μ h3 2 s s + +αβ +α 2 +αβ + ˚ σ μs h s σs A+α d B + + B B + A A s α α αβ αβ ˚ 12 (14.412) +(+ → −). The symbol (+ → −) denotes the previous expression in which the index + is replaced by the index −. + To find A+ αβ , Bαβ in terms of Aαβ , Bαβ , ϕ, ϕα we plug (14.400) in (14.40). Writing (14.400) in the form x+i (ξ α ) = r i (ξ α ) +

hc i hc n (1 + ϕ) + rβi ϕ β , 2 2

and using the relations (14.31) and (14.33), n i,α = −bαβ rβi ,

i rβ|α = bβα n i

(14.413)

14.10


711

we have24 x+i hc i hc β hc β β i δ b ϕ n ϕ,α + bαβ ϕ β . − = r + α β α |α + α ξ 2 2 2

(14.414)

Note that the covariant derivative over the deformed surface, , can be replaced in ˚ this introduces errors (14.414) by the covariant derivative over initial surface, ; 2 of the negligible error ε (see Sect. 14.2). The unit normal vector to the deformed interface surface, n i+ , can be found from the following reasoning. We present n i+ in the form n i+ = cα rαi + cn i with some undetermined parameters, cα and c. These parameters can be found from the relations n i+ n +i = 1,

n +i

x+i = 0. ξ α

(14.415)

Equations (14.415) yield a system of linear equations for cα , c: aαβ cα cβ + c2 = 1, hc hc β hc +c ϕ,α + bαβ ϕ β = 0. cβ aαβ δαβ − bαβ + ϕ;α 2 2 2

(14.416)

We solve these equations taking into account the first correction to the leading terms. In the leading approximation, c = 1, cα = 0. Therefore, the first correction is cβ = −a βα and

hc ϕ,α + bαγ ϕ γ , 2

n i+ = n i − rβi a βα

hc ϕ,α + bαγ ϕ γ . 2

(14.417)

In the undeformed state, i.e., for ϕ = ϕ β = 0 and r i , n i replaced by r˚ i , n˚ i , the position vector of the interface is x˚ +i = r˚ i (ξ α ) +

24

hc i α n˚ (ξ ) . 2

After computing derivative we neglect ϕ, which is O (ǫ) , is comparison with unity.

712

14


Accordingly, the tangent vectors to the interface surface in the undeformed state are x˚ +i hc ˚ β β i ˚ δ . b − = r α β ξ α 2 α

(14.418)

˚ Both tangent vectors are linear combinations of r˚1i and r˚2i , and, thus, tangent to . ˚ and equal to n˚ i . Therefore the normal vector to the interface is also normal to From (14.413), (14.414), (14.417), (14.418) and (14.40) we find the extension measure of the upper interface: x˚ +i x˚ +i 1 x+i x+i − 2 ξ α ξ β ξ α ξ β 2 1 hc hc γ hc σ hc hc = δβσ − δβσ + ϕ;β + aγ σ δαγ − bαγ + ϕ;α 2 2 2 2 2 2 hc ˚ γ hc ˚ σ γ γ σ σ δβ − δβ . × ϕ,α + bαγ ϕ ϕ,β + bβσ ϕ −a˚ γ σ δα − bα 2 2

A+ αβ =

Keeping only the leading terms in Aαβ , Bαβ , ϕ, ϕα and dropping all the terms of order ε2 we get hc 1 ˜ bαβ ϕ, A+ αβ = Aαβ − h c Bαβ − 2 2

(14.419)

where we used the notation of (14.384) + To find Bαβ we also need the projection of the second derivatives of the position i vector x+ to the normal vector n i+ . Since the projections of n i+ to rαi are small, we have to keep in the projections of the second derivatives to n i the leading terms and the first correction, while in projection to rαi only the leading terms. We have

x+i ξ α

hc γ ϕ,α + bαγ ϕ ;β = n bαβ + 2

hc i γ γ γ σ . + bβ ϕ,α + bασ ϕ − rγ bα − ϕ;β ;β 2 i

β

+ are Therefore, the leading terms of Bαβ

+ = Bαβ + Bαβ

@ h c ? γ . ϕ,α + bαγ ϕ γ ;β + bαβ ϕ;α 2

(14.420)

− − − Computation of A− αβ and Bαβ is similar. We obtain for Aαβ and Bαβ equations (14.419) and (14.420) in which n i , ϕ α and ϕ are replaced by −n i , −ϕ α , −ϕ :

14.10


713

1 ˜ 1 ˚ A− αβ = Aαβ + h c Bαβ + h c bαβ ϕ 2 2 − Bαβ = Bαβ −

@ h c ? γ ϕ,α + bαγ ϕ γ ;β + bγβ ϕ;α . 2

(14.421)

Plugging (14.419)–(14.421) into (14.412) we find the energy of the skin layers: Fskin

? 2 σs Aαα + Aαβ Aαβ = 2μs h s ˚

2

h 2c α α αβ αβ ˜ ˚ ˜ ˚ ˜ ˚ σs Bα + ϕ bα + Bαβ + ϕ bαβ B + ϕ b + 4 2 2 h2 h 2 + s σs Bαα + Bαβ B αβ + c σs ϕ ,α + bγα ϕ γ ;α 12 4 @ @ ? ;β h 2c ? γ ˚ ϕ,α + bαγ ϕ γ ;β + bγβ ϕ;α d . ϕ ,α + bγα ϕ γ + bγβ ϕ γ ;α + 4 (14.422)

Asymptotic analysis of the shell energy. The energy of the shell is a sum of energy (14.411), where one should put index c at λ, μ, σ and h, indicating that this is energy of the core, and energy (14.422). Since μs h s ≫ μc h c , the first term in (14.411) can be neglected in comparison with the first term in (14.422). The second term in (14.411) can be neglected in comparison with the second term in (14.422). If we drop the last two terms in (14.422), we arrive at the functional (14.382) and (14.383). It is seen from the minimization of this functional over ϕ that μs h s h c ˜ μs h s h c ϕ ∼ Aαβ + Bαβ 1+ 2 μc R μc R or

2−α 2−α h h 1+ Bαβ · l. ϕ ∼ Aαβ + l l

That means that either ϕ ∼ Aαβ or ϕ ∼ Bαβ l or ϕ ∼ Aαβ + l Bαβ . The transverse shear, “in the worst case scenario”, is on the order of l Bαβ . On such functions, the last two terms in (14.422) are smaller than the terms retained and can be neglected indeed. Anisotropic hard-skin shells. We give here the formula for energy of a hard-skin shell under conditions that material has a plane of elastic symmetry, which is parallel to the mid-surface, and within each layer the elastic moduli are of the same order. The asymptotic analysis that is similar to the one for the isotropic case yields the following formula for energy:

714

14

F=

˚

F r i (ξ α ) = min ϕ,ϕα

αβγ δ E s

˚


h 2s ˚ + F r i (ξ α ) Aαβ Aγ δ + Bαβ Bγ δ d 12

1 αβγ δ E s h s h 2c B˜ αβ + ϕ b˚ αβ B˜ γ γ δ + ϕ b˚ γ δ + 2

(14.423)

2 ! 1 αβ 1 αβ 1 ˚ + G c h c ϕα ϕβ + Ec h c ϕ + E Aαβ d . 2 2 Ec c The elastic constants here are defined by (14.241). For isotropic material this functional transforms in the functional (14.382) and (14.383).

Chapter 15

Elastic Beams

15.1 Phenomenological Approach ˚ in three-dimensional space and a region, V˚ , Beams. Consider a space curve, , ˚ at every point, S is orthogonal which is formed by motion of a flat figure, S, along ; ˚ to (Fig. 15.1). Denote the diameter of S (the maximum distance between two points of S) by h, the length of ˚ by L 0 and the minimum curvature-torsion radius of ˚ by R. If h ≪ 1, R

h ≪ 1, L0

then an elastic body occupying in its undeformed state the region, V˚ , is called an elastic beam. Let the beam be deformed by some external forces; either the surface forces or the displacements are given at the ends of the beam. It seems plausible that the threedimensional elastic problem can be approximated by a one-dimensional problem,

Fig. 15.1 Notation in beam theory


715

716

15

Elastic Beams

which contains only the functions of the longitudinal coordinate, ξ , along ˚ (we discuss here only the statics of beams). We will first consider a heuristic beam theory, which was founded by Kirchhoff and Clebsch, and then discuss the variational problem that must be solved to find energy of the beam in one-dimensional theory. This will be followed by the systematic derivation of the one-dimensional beam theory from the three-dimensional elasticity theory using the variational-asymptotic method. Kinematics of beams. The beam will be modeled by a space curve, , at every point of which an orthogonal vector triad is attached; one of the triad vectors is tangent to the curve. For a given position of the curve, the triad is defined up to rotation around the tangent vector. The corresponding degree of freedom serves to describe the relative rotation of the transverse cross-sections of the beam. So, the beam has four functional degrees of freedom. Let x i = r i (ξ ) be the parametric equations of the curve in some Cartesian coordinate system x i , τ i (ξ ) the unit tangent vector to , and ταi (ξ ), α = 1, 2, the two other vectors of the triad; as in the previous chapter, Greek indices1 run through values 1, 2. The position ˚ Its parametric equations are of the curve in the undeformed state is denoted by . x i = r˚ i (ξ ), ˚ The parameter ξ is τ˚ i (ξ ) and τ˚αi (ξ ) being the vectors of the orthonormal triad at . identified with the arc length in the undeformed state; thus τ˚ i =

d r˚ i . dξ

The parameter ξ changes on the segment 0 ≤ ξ ≤ L 0 . To introduce the strain measures for the beam, we first need to define the beam curvatures. The vectors, τ˚ i and τ˚αi , as the vectors of an orthogonal triad, satisfy the conditions τ˚i τ˚ i = 1,

τ˚αi τ˚iβ = δαβ ,

τ˚i τ˚αi = 0.

(15.1)

Let us project the vectors d τ˚ i /dξ and d τ˚αi /dξ on to the vectors τ˚ i , τ˚αi . Taking the derivative of the first equation (15.1) with respect to ξ ,

1

Small Greek indices number the two vectors of the triad that are orthogonal to the tangent vector. They are also used for the projections on these vectors. Greek indices can be put in either upper or lower position in correspondence with the general rule of summation over repeating upper and lower indices; quantities with upper and lower indices coincide; in particular ταi = τ iα , ταi = τiα .

15.1

Phenomenological Approach

717

τ˚i

d τ˚ i = 0, dξ

we see that the vector d τ˚ i /dξ is orthogonal to the vector τ˚ i and, consequently, has nonzero projections only on the vectors τ˚αi . Denote these projections by −ω˚ α : d τ˚ i = −ω˚ α τ˚αi . dξ

(15.2)

Analogously, the vector d τ˚αi /dξ is orthogonal to the vector τ˚αi and, therefore has ·β nonzero projections on the vectors,2 τ˚ i and eα· τ˚βi . The projections of d τ˚αi /dξ on τ˚ i are equal to ω˚ α , as follows from the third equation (15.1) differentiated with respect to ξ : d τ˚ i d τ˚ i i τ˚α + τ˚ i α = 0 dξ dξ

or

d τ˚ i i τ˚ = ω˚ α . dξ α

Therefore, d τ˚αi ·β i = ω˚ α τ˚ i + ωe ˚ α· τ˚β . dξ

(15.3)

The quantities, ω˚ 1 and ω˚ 2 , will be called the curvatures, and ω˚ the torsion of the beam. In order to avoid confusion, we note that often ω˚ 1 and ω˚ 2 denote the projections of d τ˚ i /dξ onto −τ˚2i and τ˚1i , correspondingly. The notation used in the text allows us to simplify the tensor form of the basic relations. Curvatures and torsion, ω˚ α and ω, ˚ determine the triad uniquely by the system of ordinary differential equations (15.2), (15.3), if τ˚ i and τ˚αi are given at one point ˚ of . ˚ there is an arbitrariness in the choice of the triad: the vectors, τ˚ i , For a given , α can be rotated around the tangent vector. Accordingly, the curvatures change. If the vectors τ˚αi are chosen in such a way that ω˚ 2 is equal to zero, then the vector τ˚1i is ˚ called the principal normal vector, τ˚2i the binormal vector, −ω˚ 1 the curvature of , i ˚ and ω˚ the torsion of . In beam theory, it is convenient to link the vectors τ˚α to the geometric or the physical properties of the cross-section of the beam, for example by directing τ˚αi along the symmetry axes of the cross-section (in case when such axes exist). If, after the vectors τ˚αi are linked to the properties of the cross-section, it turns out that ω˚ = 0, the beam is called naturally twisted or pre-twisted. For a deformed state, the formulae analogous to (15.2) and (15.3) are dτ i = −ωα τ iα , ds 2

·β

dτ iα ·β i = ωα τ i + ωeα· τβ , ds

(15.4)

As before, eαβ = eα· = eαβ are the two-dimensional Levi-Civita symbols: e11 = e22 = 0, e12 = −e21 = 1.

718

15

Elastic Beams

where s is the arc length along . The elongation of the axis of the beam is described by the function s = s(ξ ). The function

ds 2 1 1 dr i dri γ = −1 −1 = 2 dξ 2 dξ dξ will be used as the elongation measure. Since ds = 1 + 2γ , dξ the formulae (15.4) can be conveniently written in terms of the derivatives of τ i and ταi with respect to ξ as dτ i = − 1 + 2γ ωα ταi , dξ

dταi .β i = 1 + 2γ ωα τ i + 1 + 2γ ωeα. τβ . dξ

(15.5)

Comparison of (15.2), (15.3) and (15.5) shows that the curvature and torsion measures may be introduced in the following way: α =

1 + 2γ ωα (s(ξ )) − ω˚ α (ξ ),

=

1 + 2γ ω (s(ξ )) −ω(ξ ˚ ).

(15.6)

˚ ), and the vectors If γ (ξ ), α (ξ ), (ξ ), the curvature and torsion, ω˚ α (ξ ) and ω(ξ τ i , and ταi at one of the points of the curve are known, then τ i (ξ ) and ταi (ξ ) can be uniquely computed from the system of ordinary differential equations (15.5). The curve is found then from the differential equations, dr i = 1 + 2γ (ξ )τ i (ξ ). dξ The deformed and undeformed positions of the beam (and the corresponding vector triad) coincide if, and only if, the strain measures γ , α and are identically equal to zero. The curvature and torsion measures may be expressed in terms of the derivatives of the triad vectors with respect to ξ from (15.5) and (15.6): α = τi

dτi dταi − ω˚ α = −ταi − ω˚ α , dξ dξ

=

1 αβ dταi e τiβ − ω. ˚ 2 dξ

(15.7)

In (15.7) the vector τ i is defined by the functions r i (ξ ) according to the equations 1 dr i , τi = √ 1 + 2γ dξ

γ =

1 2

dr i dri −1 . dξ dξ

(15.8)

15.1


719

Variational principle. Since the strain measures, γ , α and , completely determine the deformed state of the beam, it is natural to assume that the elastic energy density per unit length of the initial state is a function of γ , α and : = (γ , α , ) . The external forces do work on the variations of r i and ταi . Denote the corresponding “generalized forces” by Q i and Q iα , and, for simplicity, assume that the ends of the i i beam are clamped: r i (ξ ) and ταi (ξ ) take the fixed values r0i , r1i , τα0 , τα1 , respectively. The forces are supposed to be “dead forces”, i.e. they do not depend on the deformed state of the beam but may depend on ξ. The deformed state of the beam is a stationary point of the functional L 0

(γ , α , ) dξ −

L 0 0

0

Q i r i + Q iα ταi dξ

(15.9)

on the set of all admissible positions of the curve and all orientations of the vectors of the triad, which are subject to the constraints τi ταi = 0,

ταi τiβ = δαβ

(15.10)

in which τ i is the vector (15.8). Variation of the strain measures. Six functions ταi satisfy the five constraints (15.10). Therefore, there is only one independent variation among six variations, → δταi . It corresponds to the infinitesimally small rotations of the vectors τ α around → the vector τ . The infinitesimally small angle of rotation will be denoted by δϕ : δϕ =

1 αβ i e τβ δτiα . 2

So, there are four independent variations, δr i and δϕ, and, correspondingly, four equilibrium equations. From (15.8), the variation of the tangent vector is δτ i =

dδr k dδr i − τ i τk . ds ds

(15.11)

It follows from (15.11) that the variation of the tangent vector is orthogonal to the tangent vector: τi δτ i = 0; the same conclusion, of course, is obtained by varying the equation τi τ i = 1. → → Denote the projections of the vector δ τ onto the vectors τ α by δθ α : τiα δτ i ≡ δθ α . According to (15.11),

720

15

δθα =τiα

Elastic Beams

dδr i . ds

Varying (15.10) and (15.8) we obtain the following expressions for the variations of the vectors τ i and ταi : δταi = −δθα τ i + τ iβ eαβ δϕ.

δτ i = ταi δθ α ,

From (15.7) and (15.12), we find the variations of the strain measures: dδθα δα = 1 + 2γ − + eαβ ωβ δϕ + ωeαβ δθ β , ds dδϕ αβ dδr i +e ωβ δθα , δγ = (1 + 2γ ) τ i . δ = 1 + 2γ ds ds

(15.12)

(15.13)

For the clamped beam ends, δr i = δτ i = δταi = 0

at

ξ = 0, L 0 .

According to (15.12), at the ends of the beam the variations δτ i and δϕ vanish: δτ i = δθα = δϕ = 0 Governing equations of beam theory. by T, M α , and M : T =

1 + 2γ

, γ

at

ξ = 0, L 0 .

(15.14)

Denote the derivatives of energy density

Mα =

, α

M=

.

(15.15)

They have the meaning of tension, bending moments and torque, respectively. M 1 is → → the bending moment in the plane τ , τ 1 , and M 2 is the bending moment in the plane → → τ , τ 2. For the variation of the energy of the beam, we find using (15.13):

δ

L 0 0

dξ =

L 0

T dδr i + (1 + 2γ ) τi ds 1 + 2γ 0 dδθα +M α 1 + 2γ − + eαβ ωβ δϕ + ωeαβ δθ β + ds dδϕ + eαβ ωβ δθα dξ. +M 1 + 2γ ds √

In order to integrate by parts, it is convenient to change the integration variable, ξ , to s. Since

15.1


721

ds = we have

δ

L 0

dξ =

0

L

1 + 2γ dξ,

dδθα dδr i +M α − + eαβ ωβ δϕ + ωeαβ δθ β + ds ds 0 dδϕ αβ +M + e ωβ δθα ds. ds T τi

Here L is the beam length in the deformed state. After integration by parts of the second and the third terms and use of the end conditions (15.14) we obtain

δ

L 0

dξ =

0

L

T τi

dδr i + δθ α ds

0

d Mα dM +eαβ Mωβ − M β ω + δϕ − +eαβ M α ωβ ds. ds ds

Equating variation of energy to the work of the external forces and using the formula for variation δθα (15.12), after additional integration by parts we arrive at the governing system of equations of the beam theory: Qi d d Mα T τi + +eαβ Mωβ − M β ω + Rα τ iα + √ = 0, (15.16) ds ds 1 + 2γ dM − eαβ M α ωβ + Q = 0. (15.17) ds The bending and the twisting moments of the external forces R α and R are linked to the “generalized forces” Q iα from (15.9) by the equalities Qα τ i , Rα = √ i 1 + 2γ

.β α i eα. Q i τβ Q=√ . 1 + 2γ

To form a closed system of equations, (15.16) and (15.17) must be supplemented by the constitutive equations (15.15) and the kinematical formulae (15.7). Equation (15.16) admits a reduction of order. Indeed, let q i be such that dq i = Qi . dξ Then T τi +

d Mα +eαβ Mωβ − M β ω + Rα τ iα + q i = 0. ds

(15.18)

722

15

Elastic Beams

If Q i ≡ 0, then q i are some constants that should be found from the boundary conditions. Projecting (15.18) on the triad, we obtain the usual form of the equilibrium equations (15.16): T + qi τ i = 0,

d Mα +eαβ Mωβ − M β ω + Rα + qi ταi = 0. ds

(15.19)

Physically linear theory. The beam model is determined by prescribing the energy density, (γ , α ,). If the amplitude of the deformations is small, then for many materials may be taken as a quadratic form with respect to γ , α and : ¯ 2 + Aαβ α β + C2 + 2γ (Aα α +A) + 2B α α . 2 = Eγ

(15.20)

The deformation measures, γ , α and , are kinematically independent; therefore, all the terms in the expression (15.20) are significant. The constitutive equations (15.15) become ¯ + Aα α + A, T = Eγ

M α = Aαβ β + Aα γ + B α , M = C + Aγ + B α α .

(15.21)

√ Here we used the fact that in physically linear theory γ ≪ 1, and 1 + 2γ may be replaced by unity. The coefficient E¯ has the meaning of the effective Young modulus, Aαβ the bending rigidity, C the torsion rigidity, and the coefficients Aα , A and B α characterize the interaction effects between bending, torsion and extension. Physically and geometrically linear theory. Denote by u i the displacements of the beam axis, u i = r i − r˚ i . Suppose that the derivatives of the displacements du i /dξ are small in comparison to unity. Let also the increments of the triad vectors 1 ταi − τ˚αi be small. Then the rotation angle of the cross-section eαβ ταi − τ˚αi τ˚iβ 2 is also small. If the derivatives of the displacements and ϕ have a certain order of smallness , then γ , α , ∼ . In linear theory one neglects all terms on the order of in comparison with unity. T , Mα and M are on the order of . Therefore in the equilibrium equations one can replace ωα by ω˚ α and ω by ω, ˚ and differentiation over s by differentiation over ξ : T = −qi τ˚ i , d Mα +eαβ M ω˚ β − M β ω˚ = −Rα − qi τ˚αi , dξ dM − eαβ M α ω˚ β = −Q. dξ

(15.22)

15.1


723

The most simplifications occur in the kinematic relations linking γ , α , and with displacements and ϕ. Setting δr i = u i , δθ = ϕ in the formulae for variations (15.13), we obtain the linearized expressions for α , and γ : du i d ·β i du i τ˚iα + eαβ ω˚ β ϕ + ωe , α = − ˚ α· τ˚β dξ dξ dξ du i du i dϕ + eαβ ω˚ β τ˚αi , γ = τ˚i . = dξ dξ dξ

(15.23)

The system of equations of the linear beam theory comprises the constitutive equations (15.21), the equilibrium equations (15.22), and the kinematical relations (15.23). Physically nonlinear effects. As will be seen, usually the coefficients Aα , A and B α are equal to zero, and the leading interaction terms are, in fact, cubic. Although these terms are small, they describe effects which are missing in the classical theory (15.20). For example, consider the cubic term, Bγ 2 , in energy ¯ 2 + Aαβ α β + C2 + 2Bγ 2 . 2 = Eγ It generates an additional term B2 in constitutive equation for the tension T , and additional term, 2Bγ , in torque: ¯ + B2 , T = Eγ

M = C + 2Bγ .

The additional terms describe the well-known experimental fact that any twist is accompanied by some tension, while the simultaneous twist and elongation of the beam cause an additional torque, 2Bγ . The latter effect becomes pronounced only for finite strains. The asymmetry of the term Bγ 2 with respect to γ and results in the asymmetry of the interaction effect it describes: if an elongation, γ , is inflicted upon an initially non-twisted beam ( = 0), then the torque remains equal to zero, while there is always nonzero tension, caused by the twist, B2 . The Kirchhoff-Clebsch theory. Let energy be quadratic with respect to the deformation measures. The coefficients of the quadratic form (15.20) depend on the elastic moduli and the geometry of the cross-section of the beam. If E is the characteristic value of the elastic moduli, then from the dimension reasoning, E¯ ∼ Eh 2 , Aαβ ∼ Eh 4 , C ∼ Eh 4 , Aα ∼ Eh 3 , B α ∼ Eh 4 . The energy terms have different orders of the cross-section size, h, and one can try to simplify the theory using the smallness of the parameter h. This can be done by means of the variational-asymptotic method. To describe the idea, we set the external forces, Q i and Q iα , to be equal to zero and assume that the beam is deformed by prescribing its displacements at the beam ends. The first step is to minimize the formally leading term of the energy functional

724

15

L 0

1¯ 2 Eγ dξ. 2

Elastic Beams

(15.24)

0

Assume that among the admissible deformed curves, , there are curves of the ˚ This is obviously true for length equal to the length of the undeformed curve, . curvilinear beams if the displacements of its ends are sufficiently small, and is not true for straight beams subject to elongation. With the assumption made, the minimum of the functional (15.24) is equal to zero and reached at the functions, r¯ i (ξ ), for which 2γ =

d r¯ i d r¯i − 1 = 0. dξ dξ

(15.25)

Therefore, the set M0 of the general scheme of the variational-asymptotic method comprises the functions, r¯ i (ξ ), taking on the assigned values at the ends of the beam and satisfying the incompressibility condition (15.25). Fixing r¯ i , we seek for the next term in the expansion r i = r¯ i + r ′i where r ′i is much less than r¯ i . One can show (we do not pause to do this) that, in many problems, r ′i are uniquely defined by r¯ i (i.e., in terms of the variational-asymptotic scheme, M0 = N ) and make small contributions to energy. This leads to the Kirchhoff-Clebsch beam theory: the deformed state of the beam is the stationary point of the energy functional,

0

L

1 αβ (A α β + C2 + 2Bγ 2 )ds, 2

(15.26)

on the set of functions r¯ i (ξ ) and ταi , satisfying the kinematic conditions at the ends of the beam and the incompressibility-inextensibility condition (15.25). Initially straight incompressible beams. There is a case when Kirchhoff-Clebsch theory admits a considerable simplification. Let the energy density be the sum of the bending energy and the twist energy, 2 = Aαβ α β + C2 , the bending rigidity tensor be spherical, Aαβ = Aδ αβ , and the beam is straight in the undeformed state (ω˚ α = 0). We are going to show that the energy density can be written just as a quadratic form with respect to second derivatives of r i : 2 = A

d 2 r i d 2 ri + C2 . dξ 2 dξ 2

15.2

Variational Problem for Energy Density

725

Therefore, the nonlinearity of the theory remains only in the incompressibility constraint (15.25) and the expression for ω in terms of ταi (15.7). Indeed, using the decomposition of the Kroneker delta, j

δ i = τi τ j + τiα ταj , the first formula (15.7), and the equations τi =

dri , dξ

τi

dτ i = 0, dξ

we can write Aαβ α β = Aταi

dτ dτ dτ i jα dτ j d 2 r i d 2 ri i j i τ = A δi j − τ τ j =A 2 dξ dξ dξ dξ dξ dξ 2

as claimed.

15.2 Variational Problem for Energy Density Construction of energy density, , in terms of the elastic characteristics and the geometry of the beam cross-section is not as elementary as for shells where the computation of energy density is reduced to solution of some algebraic problem and integration. It turns out that for beams the energy density is the minimum value in some variational problem for a functional defined on functions of the cross-sectional coordinates. In this section we discuss the formulation of this variational problem and some of its consequences. The derivation of the variational problem from threedimensional elasticity by the variational-asymptotic method is given in the next section. We consider only the first approximation, where the small contributions on the order of h/R, h/l and ε (l being the characteristic length of the stress state along the beam, and ε the strain amplitude) are ignored. Locally, a curved beam with a large curvature-torsion radius R (R ≫ h) can be viewed as a cylinder with constant cross-section S. Denote the coordinates in the cross-section by ξ α ; the third coordinate, ξ 3 , is directed along the cylinder axis. The free energy density of the material, F, is a function of strains: F = F εαβ , εα3 , ε33 . We assume that the material is physically linear, and F is a quadratic form of εαβ , εα3 , and ε33 . We define three functions: the longitudinal energy: F|| (ε33 ) = min F εαβ , εα3 , ε33 , εαβ ,εα3

(15.27)

726

15

Elastic Beams

the shear energy: F∠ (εα3 , ε33 ) = min F εαβ , εα3 , ε33 − F|| (ε33 ) , εαβ

(15.28)

and the transverse energy: F⊥ = F − F|| − F∠ .

(15.29)

Accordingly, the energy density is the sum F = F|| + F∠ + F⊥ .

(15.30)

The direct computation given in the next section yields the expressions: 1 2 Eε , 2 33 1 F∠ = G αβ (2εα3 − Cα ε33 ) (α → β) , 2 1 σ 2εσ 3 (α, β → γ , δ) . F⊥ = C αβγ δ εαβ + Cαβ ε33 + Cαβ 2 F|| =

(15.31)

The coefficients appearing in (15.31) can be taken as primary elastic characteristics of the material. For inhomogeneous beams, they can be functions of the crosssectional coordinates, ξ α . Their explicit relations to the conventional elastic moduli are given in the next section. To comprehend better the meaning of such energy splitting, we note that the longitudinal and shear energies do not depend on the cross-sectional in-plane strains, εαβ . Therefore, in-plane stresses are σ αβ =

F⊥ F = = C αβγ δ εγ δ + Cγ δ ε33 + Cγσ δ 2εσ 3 . εαβ εαβ

Similarly3 , σ α3 =

F F⊥ F∠ = + = σ γ δ Cγαδ + G αβ 2εβ3 − Cβ ε33 . (2εα3 ) (2εα3 ) (2εα3 )

These relations show that the transversal energy, F⊥ , depends only on the stress components, σ αβ , and F⊥ is equal to zero if and only if σ αβ = 0. If σ αβ = 0, then the shear energy, F∠ , depends only on σ α3 , and F∠ = 0 if and only if σ α3 = 0. The 3

See the note on the differentiation of a scalar function over components of symmetric tensors in Sect. 3.3.

15.2


727

longitudinal energy, F|| , is the energy which the body has in case when σ αβ = 0 and σ α3 = 0. Therefore, the coefficient E has the meaning of the Young modulus of anisotropic elastic body. Let u α ξ β and u ξ β be some functions of cross-sectional coordinates. Consider the functional

∠ (u) =

1 2

S

(u α , u) = ∠ (u) + ⊥ (u α , u) , G αβ u ,α + eσ α ξ σ + Cα (γ + σ ξ σ ) (α → β) dξ 1 dξ 2

1 ⊥ (u α , u) = 2

S

C αβγ δ u (α,β) + Cαβ (γ + σ ξ σ )

λ u ,λ + eσ λ ξ σ (α, β → γ , δ) dξ 1 dξ 2 . +Cαβ

Comma in indices denotes differentiation over cross-sectional coordinates. The functional (u α , u) depends on the parameters γ , α and . Denote its minimum value by (γ , α , ) : (γ , α , ) = min . u α ,u

(15.32)

Now we can formulate the key formula of the one-dimensional beam theory: the energy density of the beam is (γ , α , ) =

1 2

E (γ + σ ξ σ )2 dξ 1 dξ 2 + (γ , α , ) .

(15.33)

S

The derivation of this formula is given further while here we consider some of its consequences. Functional (u α , u) is invariant with respect to transformations u → u + c,

u α → u α + cα + κeαβ ξ β ,

cα , c, κ being constants. In order to select the unique minimizer (we will see in the next section its physical meaning) we set the constraints

u = 0,

u α = 0,

: ; u α,β eαβ = 0.

(15.34)

Dual variational problem. In general, finding the minimum of the functional is equivalent to solving the Neuman-type boundary value problem for a system of three elliptic equations of the second order with the variable coefficients. The transition to the dual variational problem allows one to replace the system of three equation of the second order by a system of two equations, one of the second order with respect to u and one of the fourth order with respect to the stress function χ .

728

15

Elastic Beams

Let the function u be fixed. Then, determining of u α is reduced to minimization of ⊥ with respect to u α . Let us write ⊥ as ⊥ = max σ αβ

C ? @ σ αβ u (α,β) + C αβ (γ + σ ξ σ )+C λαβ (u ,λ + eσ λ ξ σ ) S

(15.35)

1 (−1) αβ γ δ D 1 2 dξ dξ , − Cαβγ δσ σ 2 (−1) αβγ δ where Cαβγ , and maximum is sought over δ is the inverse tensor of the tensor C αβ βα all symmetric tensor fields σ = σ . Then

min ⊥ = min max {·} , ua

uα

(15.36)

σ αβ

where {·} is the integral in the right-hand side of (15.35). Changing the order of maximization and minimization in (15.36) (which is possible when the assumptions of the general scheme of Sect. 5.8 for integral functionals are satisfied) and calculating min {·}, we find that it is equal to uα

∗⊥ (σ αβ , u) =

S

−

-

0 1 σ αβ Cαβ (γ + σ ξ σ ) + C λαβ (u ,λ +eσ λ ξ σ )

1 (−1) αβ γ δ D 1 2 C dξ dξ , σ σ 2 αβγ δ

(15.37)

if σ

αβ ,β

=0

in S, σ αβ νβ = 0

on S,

(15.38)

and −∞ if (15.38) does not hold. Therefore, ⊥ = max ∗⊥ (σ αβ , u), min α u

σ αβ

(15.39)

where maximum is sought over all σ αβ satisfying the constraints (15.38). Each solution of (15.38) may be written as (see Sect. 6.6) σ αβ = eαμ eβ χ,μν , where χ is some function in S satisfying the boundary constraints χ,α = const on S.

(15.40)

For a given stress state, function χ is determined up to an arbitrary linear function. One can choose this function in such a way that for a simply connected region,

15.2


729

χ,α = 0 on S, while for multiple-connected region, χ,α = 0 at one of the components of the boundary of S. So 0 1 min = min(∠ + ⊥ ) = min max ∠ (u) + ∗⊥ (χ , u) , u α ,u

u α ,u

u

χ

(15.41)

where the functional ∗⊥ (χ , u) is obtained by substituting the expressions for σ αβ in terms of χ into (15.37): ∗⊥ (χ , u)

=

S

−

-

0 σ 1 eαμ eβν χ,μν Cαβ (γ + σ ξ σ ) + C αβ λ (u ,λ + eσ λ ξ )

D 1 (−1) αμ βν γ λ δκ Cαβγ δ e e e e χ,μν χ,λκ dξ 1 dξ 2 . 2

The Euler equations of the minimax problem (15.41) are the two equations of the second and the fourth orders for u and χ . As a rule, the variational problem under consideration may be solved only by numerical methods. However, there is a number of important particular cases, where the problem admits considerable simplifications or the exact solutions. The most important simplifications are related to the homogeneity properties, the existence of the plane of elastic symmetry perpendicular to the beam central line and of the central symmetry of elastic characteristics and the geometry of the cross-section. We begin from consideration of homogeneous beams. Homogeneous beams. A beam is called homogeneous if C αβγ δ , G αβ , C αβ ,Cαβ λ and Cα do not depend on ξ α . Let us show that for general anisotropy and arbitrary geometry of the crosssection, the minimum value of the functional can be found analytically up to a multiplicative constant; to compute this constant, one has to solve some variational problem. To obtain this result, we change the required function, u → v:

: α β; ξ ξ 1 + v. u = −Cα ξ α γ − Cα β ξ α ξ β − |S| 2 By · we denote further the integral over cross-section

· =

S

by |S| the area of the cross-section. The functional ∠ becomes

·dξ 1 dξ 2 ,

(15.42)

730

15

∠ =

Elastic Beams

1 αβ ˜ σ α ξ σ (α → β), v ,α + e

G 2

˜ denotes the parameter: where ˜ = − 1 eμν Cμ ν . 2 Substitution of (15.42) into the functional ⊥ results in the relation ⊥ =

1 2

S

? 1 C αβγ δ u (α,β) + D αβ +Dαβλ ξ λ + Cαβ λ v,λ [α, β → γ , δ]dξ 1 dξ 2 .

(15.43)

Here Dαβ = Cαβ −C αβ λ Cλ γ ,

σ σ Dαβλ = Cαβ λ − Cαβ C(σ λ) + Cαβ eλσ .

Formula (15.43) suggests the substitution u α → vα : 1 u α = −Dαβ ξ β − aαβγ 2

:

ξβξγ ξβξγ − |S|

;

+ vα .

(15.44)

where aαβγ are some constants yet to be defined. The constants aαβγ are symmetric with respect to the indices β, γ . We set aαβγ to be a solution of the linear system of equations a(αβ)γ ≡

1 (aαβγ + aβαγ ) = Dαβγ . 2

(15.45)

It can be checked by direct inspection that for any tensor aαβγ of the third order, which is symmetric with respect to the last two indices, the identity holds: aαβγ = a(αβ)γ + a(αγ )β − a(βγ )α .

(15.46)

Therefore, the solution of the system of equations (15.45) is aαβγ = D(αβ)γ + D(αγ )β − D(βγ )α .

(15.47)

After the changes of variables, the functional , due to (15.45), becomes =

1 2

S

0

˜ σ α ξ σ (α → β) G αβ v ,α + e

@ +C αβγ δ v (α,β) + C αβ λ v,λ (α, β → γ , δ) dξ 1 dξ 2

(15.48)

15.2


731

˜ therefore The minimizing element of the functional is proportional to ; =

1 ˜2 C , 2

where C is the minimum value of the functional αβ G v¯ ,α + eσ α ξ σ (α → β)+ C = min v¯ ,¯vα

(15.49)

S

+C αβγ δ v¯ (α,β) + C αβ λ v¯ ,λ (α, β → γ , δ) dξ 1 dξ 2 .

˜ v¯ α = vα /. ˜ The constant C is called the torsional rigidity of the Here, v¯ = v/, beam. So, for a homogeneous beam, does not depend on γ , while α and enter ˜ into through the combination : 2 1 1 μν = C − e Cμ ν . 2 2

(15.50)

Elliptic cross-section. Let us find the torsional rigidity for a beam with an elliptic cross-section, bαβ ξ α ξ β ≤ 1, where bαβ is a positive symmetric tensor.4 Let us find the functions, v¯ α and v¯ , from the system of equations G αβ v¯ ,β + eσβ ξ σ = aeαλ bλμ ξ μ ,

v¯ (α,β) + C λαβ v¯ ,λ = 0,

(15.51)

where a is a constant yet to be defined. From the first equation (15.51), we have

(−1) αλ e bλμ − eμβ ξ μ . v¯ ,β = aG αβ

(15.52)

These are two equations for one function, v¯ . We choose a to make these equations compatible: the derivative 2 v¯ /ξ 1 ξ 2 found from (15.52) for β = 1 and the derivative 2 v¯ /ξ 2 ξ 1 found from (15.52) for β = 2 must be equal, or

Hence,

μβ αλ = 0. eμβ v¯ ,β ,μ = aG (−1) αβ e bλμ − eμβ e a=

2 αλ μβ G (−1) αβ e e bλμ

.

(15.53)

From (15.52), we find that 4

If the coordinates ξ 1 , ξ 2 are directed along the axes of the ellipse, it will not result in simplifications, because there is an arbitary anisotropy of the elastic properties. Due to this, all relations are written in tensor form.

732

15

v¯ =

: ;F

1 (−1) αλ aG βα e bλμ − eμβ ξ μ ξ β − ξ μ ξ β |S| . 2

Elastic Beams

(15.54)

Substituting (15.54) into the second equation (15.51) results in

v¯ (α,β) + Cσλλ aG αβ (−1) eσ ν bνμ − eμλ ξ μ = 0.

(15.55)

The solution of (15.55) is

v¯ α =

1 a¯ αβγ ξ β ξ γ , 2

(15.56)

where the constant tensor, a¯ αβγ , is symmetric with respect to the indices β, γ and satisfies the system of linear equations

λ σν a¯ (αβ)γ = −Cαβ aG (−1) . e b − e νγ γ λ σλ This system is analogous to the system of equations (15.45), and its solution is given by the formula (15.47). It is easy to check that the functions, v¯ and v¯ α , satisfying equations (15.51), also satisfy the Euler equations for the functional (15.49) and the corresponding natural boundary conditions. Substituting them into (15.49) we get αλ βσ μν C = a 2 G (−1) αβ e e bλμ bσ ν I ,

where I μν are the moments of the cross-section, I μν = ξ μ ξ γ dξ 1 dξ 2 . In the principal coordinates of the ellipse with the semi-axes, b1 , b2 , b11 = b22 =

1 1 1 , b12 = 0, I11 = π b13 b2 , I22 = π b23 b1 , and 2 4 4 b2 C=

π b13 b23 2 G (−1) 11 b1

+

2 G (−1) 22 b2

=

4 −1 G (−1) 11 (I22 )

(−1) (I11 )−1 + G 22

.

1 , b12

(15.57)

Note that we did not assume that the tensors G αβ and I αβ are coaxial. If they are, (−1) −1 −1 then G (−1) and 11 = (G 11 ) , G 22 = (G 22 ) C=

π G 11 G 22 b13 b23 . G 22 b12 + G 11 b22

In the case of a transversally-isotropic material, G αβ = Gδ αβ , and

15.2


C=

733

π Gb13 b23 4G 4G = (−1)α , = 2 2 −1 −1 (I ) + (I ) b1 + b2 11 22 Iα

(−1) is the inverse tensor of I αβ . where Iαβ

Estimate of the torsion rigidity of homogeneous anisotropic beam with an arbitrary cross-section. Let us show that for arbitrary geometry of the beam cross-section the following estimate for the torsional rigidity is true: C≤

4 (−1) G αβ (−1) eαμ eβν Iμν

.

(15.58)

The inequality (15.58) is a generalization to the anisotropic case of the Nikolai’s inequality, C≤

4G I11 I22 , I11 + I22

or, in tensor notation, C≤

4G Iα(−1)α

.

Let us use the functions (15.54) and (15.56) as the trial functions in the variational principle for torsional rigidity, with bαβ being some constants. This gives the following upper estimate of the torsional rigidity: C≤

¯ μν bμλ bνσ I λσ 4G . ¯ μν bμν 2 G

(15.59)

αμ βν ¯ μν = G (−1) Here G αβ e e . Let us minimize the right-hand side of (15.59) with respect to bαβ . Since the right-hand side of (15.59) is not affected by the transformation bαβ → λbαβ , this ¯ μν bμλ bνσ I λσ , with problem is equivalent to minimization of the quadratic form 4G μν ¯ bμν = 1. It is easy to see that the minimum value of the quadratic the constraint G (−1) ¯ λσ (−1) −1 form is reached at the tensor bμν = Iμν (G Iλσ ) . This proves the assertion made. The inequality (15.58) allows one to establish some extremal features of the torsional rigidity. Consider the torsional rigidities of the beams with different crosssections, and, possibly, with different elastic moduli, but such that the contraction αμ βν (−1) is the same. From the comparison of the relations (15.57) and G (−1) αβ e e Iμν (15.58), it is seen that on this set of beams, the beams with an elliptic cross-section have the maximum torsional rigidity. If, in addition, the tensor G αβ is spherical (and αμ βν (−1) = G −1 Iα(−1)α , and, strengthening the = G (−1) δαβ ), then G (−1) G (−1) αβ e e Iμν αβ estimate (15.58) by means of the inequality

734

15

1 (−1)μ

Iμ

≤

Elastic Beams

1 μ I , 4 μ

we get C ≤ G Iμμ . This inequality becomes an equality for circular beams; consequently, of all the cross-sections with the same polar moment Iμμ , the circle has the maximum torsional rigidity. Heterogeneous beams. We limit the consideration of heterogeneous beams with the case when the beam has a plane of elastic symmetry perpendicular to the central line. The “two-dimensional” elastic moduli tensors with an odd number of indices γ are equal to zero (Cαβ = 0, Cα = 0) and the minimization problem for the functional splits into two independent problems: the minimization problem for the functional 1 αβ ∠ (u) = G u ,α +eσ α ξ σ (α → β) dξ 1 dξ 2 , 2 S

and the minimization problem for the functional ⊥ (u α ) =

1 2

C αβγ δ u (α,β) + C αβ (γ + σ ξ σ ) (α, β → γ , δ) dξ 1 dξ 2 . S

(15.60)

The first problem is, in essence, the well-known Saint-Venant torsion problem (see [223]). The minimizing function of the functional ∠ (u) is proportional to ; therefore, min ∠ (u) = u

1 C2 , 2

where the torsion rigidity, C, is the minimum value of the functional C = min u

G αβ (u ,α + eσ α ξ σ )(α → β) dξ 1 dξ 2 .

S

The second problem corresponds to a problem of two-dimensional elasticity. Since the minimizing functions depend on γ and α linearly, the minimum value of ⊥ is a quadratic form with respect to γ and α : ⊥ = min ⊥ = uα

1 αβ E ⊥ γ 2 + 2E ⊥α γ α + E ⊥ α β . 2

15.2


735

Fig. 15.2 Examples of central-symmetric cross-sections which do not possess two axes of symmetry

So, for the beams with an elastic symmetry plane perpendicular to the central line, the function is a sum of the torsion energy, C2 /2, and a contribution to the extension and bending energies ⊥ ; function does not contain the interaction terms between torsion and extension, γ ·, and between torsion and bending, ·α . More substantial conclusions can be made if the cross-section and the elastic properties of the beam also have the central symmetry. The cross-section is called central-symmetric if, for every point with the coordinates ξ α , it contains a point with the coordinates −ξ α . Note that the cross-section may be central-symmetric and not have two symmetry axes (two such cross-sections are shown in Fig. 15.2). For functions defined in a central-symmetric regions, the notion of evenness may be introduced: a function is even if it has the same values at the points ξ α and -ξ α . By definition, the elastic characteristics of the beam are central-symmetric if C, G αβ and C αβ are even functions of ξ α . Let us write u α as a sum of odd and even functions (they are denoted by one and two primes, respectively), u α = u ′α + u ′′α . Then the functional ⊥ becomes a sum of two functionals, one of which depends only on u ′α and the other only on u ′′α : ⊥ = ′⊥ + ′′⊥ , 1 ′⊥ = C αβγ δ u ′(α,β) + Cαβ γ (α, β → γ , δ) dξ 1 dξ 2 , 2 S 1 ′′ C αβγ δ u ′′(α,β) + Cαβ σ ξ σ (α, β → γ , δ) dξ 1 dξ 2 . ⊥ = 2 S

Functionals ′⊥ and ′′⊥ can be minimized independently. Minimum of ′⊥ is proportional to γ 2 , minimum of ′′⊥ is a quadratic form of α . So αβ

2⊥ = E ⊥ γ 2 + E ⊥ α β , and energy does not contain the interaction terms between γ , α and . As is easy to see, the similar conclusion can be made if C αβγ δ are even functions of ξ α , G αβ are arbitrary while C αβ are either odd or even functions of ξ α .

736

15

Elastic Beams

Heterogeneous beams with constant Poisson coefficients. It turns out that, for the beams with constant Poisson coefficients, the following remarkable feature holds: contains only the torsion energy 1 C2 2

=

and

⊥ = 0.

(15.61)

To prove that, we use the fact that any value of the functional gives an upper bound of . We take 1 u α = −Cαβ γ ξ β − aαβγ ξ β ξ γ . 2

(15.62)

The constants aαβγ are chosen as the solution of the linear system of equations a(αβ)γ = Cαβ γ .

(15.63)

For these functions, ⊥ = 0. Consequently, ⊥ =min ⊥ = 0. uα

The minimizing functions u α have a universal form for an arbitrary cross-section and an arbitrary dependence of the elastic moduli C αβγ δ on the coordinates; this form is found from (15.62), (15.63) and (15.46): 1 u α = −Cαβ γ ξ − Cα(β γ ) − Cβγ α ξ β ξ γ . 2 β

(15.64)

Criterion of vanishing of ⊥ . The quadratic form, ⊥ , is identically equal to zero not only for beams with constant Poisson coefficients, but also for some beams with variable Poisson coefficients. Let us prove the following assertion. Let the region S be divided into two parts, S1 and S2 , by a smooth line L. In both sub-regions, the Poisson coefficients, Cαβ , are continuous, but they may be discontinuous on the line L. Then, in order for ⊥ to be equal to zero, it is necessary and sufficient that there is a function c(ξ α ) such that in the regions of continuity of the Poisson coefficients, Cαβ = c,αβ

(15.65)

[c,α ] = const.

(15.66)

while on the discontinuity line,5

5 Recall that the symbol [A] denotes the difference between the values of A on the two sides of the discontinuity line.

15.2


737

Sufficiency. Let us set u α = −c,α γ − c,α σ ξ σ + cα + ωeασ ξ σ + aα ,

(15.67)

where ω, aα are constants which may have different values in S1 and S2 . Then, inside S1 and S2 , u (α,β) + Cαβ γ + Cαβ σ ξ σ = 0.

(15.68)

Let us choose the constants ω and aα in such a way that u α be continuous on L, − rα γ − rα σ ξ σ + [c]α + [ω]eασ ξ σ + [aα ] = 0,

(15.69)

where rα denote the constants equal to [c,α ]. From the kinematic compatibility conditions dξ α d[c] = rα , dσ dσ where σ is a parameter on L, it follows that [c] = rα ξ α + r,

r = const.

(15.70)

The constant terms in (15.69) cancel out if we set [aα ] = rα γ + r α .

(15.71)

The terms linear with respect to ξ σ have the form (rσ α − rα σ + [ω]eασ ) ξ σ . Therefore, setting [ω] = eσ α rσ α ,

(15.72)

we completely satisfy (15.69). The formulae (15.71) and (15.72) give the differences between the constants ω and aα in the regions S1 and S2 . If one sets the constraints on u α (15.34), then these constants are completely defined. The functions u α are admissible and the functional ⊥ is equal to zero at these functions. Consequently, ⊥ = 0. Necessity. Let ⊥ = 0. Then since the quadratic form F⊥ is non-degenerate, the equalities (15.68) hold. The parameters γ and α are independent; therefore, as follows from (15.68), the compatibility conditions for the tensors Cαβ and Cαβ ξ σ should be satisfied in S1 and S2 to guarantee that Cαβ and Cαβ ξ σ may be presented as a symmetric part of the gradient of a vector field. For an arbitrary tensor εαβ , which

738

15

Elastic Beams

admits such representation, εαβ = u (α,β) , the necessary and sufficient compatibility conditions are μ

μ

λ − εα,βμ − εβ,αμ = 0, εαβ + ελ,αβ

(15.73)

where is the two-dimensional Laplace operator. Substitution of Cαβ and Cαβ ξ σ into (15.73) results in the system of equations μ

μ

λ − Cα,βμ − Cβ,αμ = 0, Cαβ + Cλ,αβ ν ν ν ν δλα − Cαλ,β − Cβλ,α = 0. 2Cαβ,λ + Cν,α − Cα,ν δλβ + Cν,β − Cβ,ν

(15.74) (15.75)

Contracting the equalities (15.75) with respect to α, β, we get ν ν = Cλ,ν . Cν,λ

(15.76)

Equation (15.75) can be simplified using (15.76)6 2Cαβ,λ = Cαλ,β + Cβλ,α .

(15.77)

Let us write these equations substituting indices, β → λ, λ → β: 2Cαλ,β = Cαβ,λ + Cλβ,α ,

(15.78)

and plug Cαλ,β in (15.77) from (15.78). Due to symmetry Cαβ , we obtain Cβα,λ = Cβλ,α . Hence, there exists a vector Cα such that Cαβ = Cα,β . The symmetry of Cαβ yields that vector Cα is potential, Cα = c,α , and, therefore, (15.65) hold. Equations (15.74) and (15.75) are satisfied for Cαβ = c,αβ . The equalities (15.67) follow from the relations (15.65) and (15.68): the displacement, u α , is the sum of the general solution of the homogeneous equations, rigid motions (the last two terms of (15.67)) and a solution of the inhomogeneous 6

If there were no coefficient 2 on the left-hand side of (15.77), these equations can be interpreted as the conditions that the Kristoffel’s symbols of the two-dimensional metric, Cαβ , vanish; that is possible only for Cαβ = const.

15.2


739

equations (15.68) (the first three terms in (15.67)). It remains to show that the condition of the continuity of u α results in the constraints (15.66). Indeed, since the parameters γ and α are arbitrary, [ω] = θ γ + θ α α , [aα ] = rα γ + rαλ λ where θ , θ α , rα and rαβ are constants. The equations [u α ] = 0, are reduced to the system of equations 0 1 c,α = rα + θ eασ ξ σ ,

[c]δαβ = (rα + θ eασ ξ σ ) ξ β − θ β eασ ξ σ − rαβ .

(15.79)

Let us set α = 1, β = 2 and α = 2, β = 1 in the second equations (15.79). Then, on the line of discontinuity, r1 ξ 2 + θ (ξ 2 )2 − θ 2 ξ 2 − r12 = 0,

r2 ξ 1 − θ (ξ 1 )2 + θ 1 ξ 1 − r21 = 0.

At least one of these relations has to be an identity with respect to the coordinates ξ 1 , ξ 2 ; otherwise, the first one defines the line ξ 2 = const, while the second one the line ξ 1 = const. Let the first relation be identically satisfied. Then θ = 0, and it follows from the first equation (15.79) that [c,α ] = const. The same result is obtained if the second relation is identically satisfied. For [c,α ] = const, the continuity of u α may be achieved by the choice of the constants ω and aα in the regions S1 and S2 , as has already been checked. It is obvious that analogous conclusions also hold for several lines of discontinuity. Note several consequences of the criterion obtained. 1. In order for ⊥ ≡ 0 for a heterogeneous beam with discontinuous Poisson coefficients, it is necessary that the conditions 1 0 Cαβ τ β = 0

(15.80)

be satisfied on the discontinuity line, where τ β is the tangent vector to the discontinuity line. The relations (15.80) are obtained by differentiation of (15.66) along the discontinuity line. 2. For a transversely isotropic body, ⊥ ≡ 0 if and only if the Poisson coefficient ν is constant. Indeed, if Cαβ = νδαβ , then (15.65) becomes c,12 = 0, c,11 = c,22 and has a unique solution, ν = c,11 = c,22 = const; there is no curve for which the conditions (15.80) can be satisfied for [ν] = 0. 3. In order for ⊥ ≡ 0 for a heterogeneous beam with piece-wise constant Poisson coefficients, it is necessary that % % det %[C αβ ]% = 0

740

15

Elastic Beams

and the discontinuity lines are straight and perpendicular and the principal axis of the tensor [Cαβ ] for which the corresponding eigenvalue does not equal to zero. This assertion is an elementary consequence of (15.80). Summary. Now we outline the most important results obtained. According to (15.115) and (15.87), for homogeneous anisotropic beams, 1 = 2

E|S|γ 2 + E I

αβ

2 1 μν α β + C − e Cμ ν , 2

(15.81)

where E is the longitudinal Young modulus, I αβ are the moments of the crosssection, torsional rigidity, C, is the minimum value of the functional (15.49), and Cμ the “Poisson coefficient vector”. If the beam has a plane of elastic symmetry perpendicular to the central line, then Cα = 0 and the expression (15.81) becomes the classical one:

1 αβ = E|S|γ 2 + E I α β + C2 . (15.82) 2 Otherwise, there is the interaction term between torsion and bending, −Ceμν Cμ ν , in the expression for energy, while the effective bending rigidities are 1 E I αβ + Ceμα eνβ Cμ Cν . 4 Note that the interaction of torsion and bending and the increase in the effective bending rigidity occur only for bending in the plane perpendicular to the vector Cμ . For heterogeneous beams with an elastic symmetry plane perpendicular to the central line, the function , along with the torsion energy, contains contributions to the extension and bending energies, and the formula for becomes

=

⎡⎛

⎞

⎛

⎞

1 ⎣⎝ Edξ 1 dξ 2 +E ⊥ ⎠ γ 2 + 2 ⎝ Eξ α dξ 1 dξ 2 +E ⊥a ⎠ α γ + 2 S S ⎛ ⎤ ⎞ 1 +⎝ Eξ α ξ β dξ dξ 2 +E ⊥ αβ ⎠ α β + C2 ⎦ .

S

αβ

Additional rigidities E ⊥ , E ⊥a , E ⊥ are found from the solution of the minimization problem for the functional ⊥ . Emphasize that all these results pertain to geometrically non-linear theory. Nonlinearity enters through the dependence of deformation measures, γ , α and , on r i (ξ ) and ταi (ξ ).

15.3

Asymptotic Analysis of the Energy Functional of Three-Dimensional Elasticity

741

15.3 Asymptotic Analysis of the Energy Functional of Three-Dimensional Elasticity Geometry of undeformed state. We will use the curvilinear system of coordinates, ξ α , ξ, in the region V˚ , introduced by the equations x i = x˚ i (ξ α , ξ ) = r˚ i (ξ ) + τ˚αi ξ α .

(15.83)

The points ξ α are the points of the beam cross-section S which could be, in general, a multi-connected region. It is assumed that the point with the coordinates ξ α = 0 coincides with the center of gravity of S, i.e. ξ α = 0. The projections on the axis ξ are marked by index 3, which will sometimes be omitted (in particular, ξ 3 ≡ ξ ). The coordinates ξ α , ξ are assumed to be Lagrangian. To find the components of the metric tensor in the Lagrangian coordinate system, we first find the derivatives of functions (15.83). Using (15.3), we have i x˚ ,α = τ˚αi ,

·α β i x˚ ,ξi = 1 + ω˚ β ξ β τ˚ i + ωe ˚ β· ξ τ˚α .

As usual, the comma before the Greek indices denotes derivative with respect to ξ α , and the comma before ξ denotes the derivative with respect to ξ . Thus, 2 g˚ αβ = δαβ , g˚ α3 = ωe ˚ αξ α, ˚ βα ξ β , g˚ 33 = 1 + ω˚ β ξ β + ωξ % % 2 g˚ ≡ det %g˚ αβ % = 1 + ω˚ β ξ β .

(15.84)

To determine the contravariant components of the metric tensor we have to find the solutions of the linear system of equations g˚ αβ g˚ βγ + g˚ α3 g˚ 3γ = δγα , g˚ 3β g˚ βγ + g˚ 33 g˚ 3γ = 0, g˚ 3β g˚ β3 + g˚ 33 g˚ 33 = 1. (15.85) From the second equation (15.85) and (15.84), we have g˚ 3β = −g˚ 33 g˚ 3β . Substituting this into the third equation (15.85) and using (15.84) for g˚ 3β , we obtain γ·

ωe ˚ ·σ ξ σ 1 3γ g˚ 33 = 2 , g˚ = 2 . 1 + ω˚ β ξ β 1 + ω˚ β ξ β

(15.86)

From the first equation (15.85), and from the equations (15.84) and (15.86), we get the expression for the other components of the metric tensor: β·

g˚ αβ = δ αβ +

α· ωe ˚ ·γ e·δ ξ γ ξ δ

(1 + ω˚ σ ξ σ )2

.

(15.87)

We will assume that ω˚ α and ω˚ are smooth functions of ξ . The best constant, R, in the inequalities

742

15

|ω˚ α | ≤

1 , R

d ω˚ α 1 dξ ≤ R 2 ,

|ω| ˚ ≤

1 , R

Elastic Beams

d ω˚ ≤ 1 , dξ R2

(15.88)

is called the characteristic radius of curvature-torsion.

Three-dimensional energy functional. We are going to consider the problem of equilibrium of an elastic beam under the actions of dead surface and body forces assuming that the beam ends are clamped: i x i (ξ α , 0) = r0i + τα0 ξ α,

i x i (ξ α , L 0 ) = r1i + τα1 ξ α,

(15.89)

i i where τα0 τiβ0 = δαβ , τα1 τiβ1 = δαβ . The equilibrium positions of the beam are the stationary points of the functional

I =

V˚

Fd V˚ −

V˚

gi x i (ξ α , ξ ) d V˚ −

f i x i (ξ α , ξ ) d A,

(15.90)

V˚

on the set of all continuously differentiable functions, x i (ξ α , ξ ) , satisfying the end conditions (15.89). The energy density per unit volume, F, is assumed to be a positive quadratic form of the strains: 2F = E abcd εab εcd = E αβγ δ εαβ εγ δ +E 3333 ε33 ε33 + 4E α3β3 εα3 εβ3 + +2E αβ33 εαβ ε33 + 4E αβγ 3 εαβ εγ 3 + 4E α333 εα3 ε33 .

(15.91)

We expect that the functional (15.90) tends to a “one-dimensional” functional as h → 0. The dependence of the external body and surface forces, gi and f i , and the elastic moduli tensor on h is to be specified; we will discuss these dependences later. Let us make the change of variable ξ α → ζ α : ξ α = hζ α . After this change, the region of values of ζ α does not depend on h, and h enters the functional explicitly. The region of values of ζ α is also denoted by S. Change of required functions. We may begin in the same way as we have done in construction of the shell theory by looking for the set N of the general scheme of the variational-asymptotic method. Then in the first step we would find that the functions x i (ξ α , ξ ) do not depend on ξ α : x i = r i (ξ ); in the second step that x ′i (ξ α , ξ ) are linear functions of ξ α : x ′i = ταi (ξ )ξ α , where the vectors τ1i and τ2i are orthogonal to each other and orthogonal to the vector dr i /dξ , and, hence, have an additional degree of freedom, rotation around the tangent vector of . In the next step, the functions x ′′i are completely determined by r i and ταi and, therefore, the set N consists of the functions r i (ξ ) and ταi (ξ ). We omit these considerations, having “guessed” the set N and move on to the appropriate change of the required functions. Let us introduce the functions

15.3


r i (ξ ) =

x i (ξ α , ξ ) . |S|

743

(15.92)

The functions r i have the meaning of the components of the average position vector of the cross-section in the deformed state. The curve with the equation x i = r i (ξ ) is called the deformed center line of the beam. The parameter ξ is not the arc length of the deformed center line. The elongation of the center line is characterized by the axial strain γ introduced above. At the curve we introduce two unit vectors ταi which are orthogonal to each other and are also orthogonal to the vector dr i 1 , τi = √ 1 + 2γ dξ and then make the following change of the required functions, x i (ξ α , ξ ) → y i (ξ α , ξ ): x i (ξ α , ξ ) = r i (ξ ) + ταi (ξ )ξ α + hy i (ξ α , ξ ) .

(15.93)

Due to the definition of r i (15.92), the functions y i (ξ α , ξ ) satisfy the constraints

y i = 0.

(15.94)

The extra degree of freedom, which appears in prescribing ταi , allows us to put an additional constraint on y i . For definiteness, as such we take the equality

yα|β eαβ = 0,

(ya ≡ ταi yi ).

(15.95)

The vertical line before the Greek indices denotes a derivative with respect to ζ α . The equality (15.95) means that the “average” rotation of the cross-section is described by rotation of the vectors ταi . Equation (15.93) establishes -a one-to-one correspondence between all functions . x i (ξ α , ξ ) and all sets of triplets r i (ξ ), ταi (ξ ), y i (ξ α , ξ ) , satisfying the constraints (15.10), (15.8), (15.94) and (15.95). Indeed, for given r i , ταi and y i (ξ α , ξ ) one can find x i (ξ α , ξ ). Let us show the opposite: for given x i (ξ α , ξ ), it is possible to find uniquely r i , ταi and y i which satisfy the constraints (15.10), (15.8), (15.94) and (15.95). First, we determine r i by (15.92). Then we take some vectors τμ′i which satisfy the equalities (15.10). These vectors may be rotated, τμ′i → ταi , ταi = τμ′i sαμ , with sαμ being the components of some two-dimensional orthogonal matrix performing the rotation for angle σ . We define the angle σ by the equation μ

i i ′

x,α τiβ eαβ = x,α τiμ sβ eαβ = 0.

(15.96)

Equation (15.96) is of the form A cos σ + B sin σ = 0, where A and B are expressed ′ i in terms of x,α τiμ , and, consequently, is always solvable. The solution defines the

744

15

Elastic Beams

vectors ταi uniquely. Finally, we introduce y i by the equation hy i = x i (ξ α , ξ ) − r i (ξ ) − ταi (ξ )ξ α . Then, the relation (15.91) holds, and the functions y i satisfy the constraints (15.94) and (15.95). At the beam ends, the functions r i and ταi should be chosen according to the boundary conditions (15.89): τ i (0) = r0i ,

r i (L 0 ) = r1i ,

i ταi (0) = τα0 ,

i ταi (L 0 ) = τα1 .

(15.97)

Also, y i (ξ α , 0) = y i (ξ α ,L 0 ) = 0.

(15.98)

Let us define the bending-twist amplitude, ε , and the central line elongation amplitude εγ as ε = h max α α + 2 , ξ

εγ = max |γ |. ξ

The number ε = ε + εγ characterizes the magnitude of three-dimensional strains. The characteristic length of the deformed state l is introduced as the best constant in the system of inequalities dωα ε dξ ≤ l ,

dω ε ≤ , dξ l

dγ εγ ≤ , dξ l

|y,ξi | ≤

1 max l V˚

j

|α

y|α y j . (15.99)

The characteristic length l is a function of ξ . Suppose that the strain state of the beam is such that l can be on the order of h only at the end regions of the beam, while away from the ends of the beam h ≪ 1. l The size of the end regions is, by our assumption, on the order of h. Regarding the external forces, we accept the conditions h fi = O μ ε , l

ε

gi = O μ , l

(15.100)

and, for simplicity, limit our consideration to the body forces which are constant over the cross-section. In these estimates, μ is the characteristic value of the components of the elastic moduli tensor. Let us show that longitudinal, transverse and shear energies defined by (15.27), (15.28) and (15.29) are

15.3


1 1 αβ 2 , F∠ = G ∠ (2εα3 + E α ε33 ) (α → β), E ε33 2 2 1 αβγ δ F⊥ = E εαβ +E αβ ε33 + E σαβ 2εσ 3 (α, β → γ , δ), 2

745

F =

(15.101)

αβ

σ and find the relationship between the coefficients E , G ∠ , E α , E αβ , E αβ and the components of the elastic modulus tensor. The first, fourth and fifth terms in (15.91) contain εαβ . Extracting a complete square, let us write these terms as

E αβγ δ εαβ +E αβ ε33 + E σαβ 2εσ 3 (α, β → γ , δ) − E αβγ δ E αβ E γ δ ε33 ε33 − β

α α −4E μνλσ E μν E λσ εα3 εβ3 − 4E μνλσ E μν E λσ εα3 ε33

(15.102)

(coefficient 2 in front of εσ 3 is introduced in order to simplify further relations). σ The coefficients E αβ and E αβ are the solutions of the system of linear equations E αβγ δ E γ δ = E αβ33 ,

σ

E αβγ δ E γ δ = E αβσ 3 .

(−1) (−1) γ δμν = δα(μ δβν) ), the Denoting the inverse tensor to E αβγ δ by E αβγ δ , (i.e., E αβγ δ E solution of this system can be written as (−1) αβ33 E γ δ = E αβγ , δE

(−1) αβσ 3 . E γσ δ = E αβγ δE

From (15.91) and (15.102), for F we have αβ 2F = E αβγ δ εαβ +E αβ ε33 + E σαβ 2εσ 3 (α, β → γ , δ) + 4G ∠ εα3 εβ3

a εα3 ε33 + E 3333 − E αβγ δ E αβ E γ δ ε33 ε33 . (15.103) +4 E α333 − E μνλσ E μν E λσ Here, the notation is introduced: αβ

β

α G ∠ = E α3β3 − E μνλσ E μν E λσ .

The second and third terms of (15.103) contain the strain components εα3 . Extracting a complete square in these terms, we write them as follows: αβ

αβ

G ∠ (2εα3 +E α ε33 ) (α → β) − G ∠ E α E β ε33 ε33 , where E α is the solution of the system of linear equations αβ

α G ∠ E β = E α333 − E μνλσ E μν E λσ .

This solution is

746

15

Elastic Beams

α , E β = αβ E α333 − E μνλσ E μν E λσ αβ

where αβ is the inverse tensor of G ∠ . Substituting the expression for the second and the third terms into (15.103), for F we finally get 2F = E αβγ δ εαβ +E αβ ε33 + E σαβ 2εσ 3 (α, β → γ , δ) αβ +4G ∠

(15.104)

(2εα3 +E α ε33 ) (α → β) + E ε33 ε33 ,

where αβ

E = E 3333 − E αβγ δ E αβ E γ δ −G ∠ E α E β . Let us make a non-degenerated change of variables εαβ , εα3 , ε33 → γαβ , γα , ε33 in the quadratic form (15.104): σ γαβ = εαβ + E αβ ε33 + E αβ 2εσ 3 ,

γα = 2εα3 + E a ε33 .

Since γαβ , γα and ε33 are independent, and there are no interaction terms between γαβ and γα , γαβ and ε33 , γα and ε33 in (15.104), the quadratic form (15.104) is positive, if the inequalities E αβγ δ γαβ γσ δ ≥ Eγαβ γ αβ ,

αβ

G ∠ γa γβ ≥ Gγα γ α ,

E > 0

(15.105)

hold where E and G are some positive constants. For fixed ε33 the minimum of (15.104) with respect to εαβ and εα3 is reached for γαβ = 0, γα = 0 and is equal to 12 E (ε33 )2 . Therefore, the first relation (15.101) holds. The two other relations (15.101) are checked similarly. The intermediate equalities which came up in derivation of the formula (15.104) relate the coefficients of the quadratic forms (15.91) and (15.104). Two-dimensional elastic moduli tensors. The two-dimensional tensors γ αβ E αβγ δ , G ∠ , E αβ and E αβ are subject to the symmetry conditions αβ

βα

E αβγ δ = E βαγ δ = E αβδγ = E γ δαβ , G ∠ = G ∠ , γ γ E αβ =E βα , E αβ = E βα . It is natural to consider these tensors, along with the vector E α and the scalar E , as the independent components of the elastic modulus tensor. We call them two-dimensional elastic modulus tensors. αβ The components of the tensors E αβγ δ , G ∠ and E have the dimensionality of γ the Young modulus, while the components of the tensors E αβ , E αβ and E α are dimensionless.

15.3


747

The positiveness of elastic energy puts constraints (15.105) on the tensors E αβγ δ , γ and E . The dimensionless tensors E αβ , E αβ and E α may take on any values. The elastic moduli are functions of ξ α and ξ , and have the form E = E (ζ α , ξ, h/R). The dependence on the parameter h/R is caused by curvilinearity of the Lagrangian coordinate system. For the limit values of the elastic moduli as h/R → 0, we introduce the following notation: in the limit, h/R → 0, αβ G∠

αβ

E = E (ζ σ , ξ ) , G ∠ = G αβ (ζ σ , ξ ) , E αβ = Cαβ (ζ σ , ξ ) , E α = Cα (ζ σ , ξ ) ,

E αβγ δ = C αβγ δ (ζ σ , ξ ) , γ γ E αβ = Cαβ (ζ σ , ξ ) .

As has already been mentioned, if the elastic characteristics are symmetric with respect to the plane perpendicular to the central line of the beam, the twodimensional tensors with an odd number of indices are equal to zero: Cα = 0,

γ

Cαβ = 0.

If, moreover, the elastic characteristics are invariant with respect to rotation in the cross-sectional plane (i.e., if the body is transversely isotropic), then the tensors C αβγ δ , C αβ and G αβ have the special form C αβγ δ = λδ αβ δ γ δ + μ δ αγ δ βδ + δ αδ δ γβ , G αβ = Gδ αβ , C αβ = νδ αβ .

The elastic properties of a transversely isotropic body are characterized by five parameters, E, G, λ, μ and ν; the parameters, E, G, μ and λ + μ are positive, while ν is arbitrary. For an isotropic body, E = 2μ (1 + ν) is the Young modulus, G = μ the shear modulus, and ν = λ/2 (λ + μ) the Poisson coefficient. In the anisotropic case, G αβ and C αβ have the meaning of the shear modulus tensor and the transverse Poisson coefficient tensor, respectively. The dimensionless λ can also be interpreted as some “Poisson coefficients”. tensors Cα and Cαβ Asymptotic analysis of the energy functional. Let us write the strain tensor components in terms of r i , ταi and y i . Taking the derivative of (15.93), we have i i , = ταi + y|α x,α

x,ξi =

1 0 1 + 2γ (1 + ωσ ξ σ ) τ i + ωeσ·β· ξ σ τβi + hy,si ,

where the comma before the index s denotes a derivative with respect to s. Projecting y i to the vector triad, y i = y˜ τ i + y λ τ iλ , and using the relation y,si = y˜ ,s + y λ ωλ τ i + y,sλ − ωλ y˜ + ωy σ eσ·λ· τ iλ ,

we obtain

εαβ = y(α,β) +

1 1 y˜ ,α y˜ ,β + yαi yλ,β , 2 2

(15.106)

748

15

Elastic Beams

0 2εα3 = (1 + ωσ ξ σ )y|α + ωeσ α ξ σ + 1 + 2γ hyα,s − hωα y˜ + hωeσ α y σ + 1 +ωeσ·β· ξ σ yβ|α + h y˜ ,s + ωλ yλ y|α + h y,sλ − ωλ y˜ + y σ ωeσ·λ· yλ|α ,

1 + 2γ − 1 ω˚ σ ξ σ + ω˚ α β ξ α ξ β + ε33 = γ + 1 + 2γ σ ξ σ + ? 1 1 +ωξ ˚ α ξα + α β ξ α ξ β + 2 ξα ξ α + (1 + 2γ ) h(1 + ωσ ξ σ )( y˜ ,s + y λ ωλ )+ 2 2 1 σ ·λ σ +hωξ eσ · (yλ,s − ωλ y˜ + y eσ λ ω) + h 2 ( y˜ ,s + y λ ωλ )2 + 2 1 2 λ λ σ ·λ + h y,s − ω y˜ + y eσ · ω yλ,s − ωλ y˜ + y μ eμλ ω , y ≡ 1 + 2γ y˜ . 2

Holding r i (ξ ) and ταi (ξ ), let us seek the functions y i (ζ α , ξ ) in the leading approximation. First, we take the approximate expressions for strains: 2εα3 = y|α + heβα ζ β ,

εαβ = y(α|β) ,

ε33 = γ + hβ ζ β .

(15.107)

Then, the derivatives of yα and y with respect to ξ do not enter into the functional, the functional does not keep the end conditions (15.98) for yα , y, and the problem of finding yα and y is reduced to minimizing for each ξ the functional − h2

( f α yα + f y) dϕ,

S

1 1 : αβ 0 G (yα , y) = y|α + heσ α ζ σ +Cα (γ + hσ ζ σ ) [α → β]+ 2 ? +C αβγ δ y(α|β) + C αβ (γ +hσ ζ σ ) 1 ; λ y|λ + heσ λ ζ σ [α, β → γ , δ] , + Cαβ

(15.108)

with the constraints

yα = 0,

y = 0,

:

; yα|β eαβ = 0.

(15.109)

Here, dϕ is the arc length element on S divided by h, f α ≡ f i ταi , and f = f i τ i . The work of external body forces is dropped, since due to the first two constraints (15.109) and the assumption (15.100), it is on the order of μh 2 ε2 /l R and is small compared to the interaction terms between yα , y and hσ , h and γ , which are taken into account in (15.108). The functional is the integral over the cross-section of the sum of the transverse and shear energies. Consider first the minimization problem for the functional , i.e. the minimization problem for the functional (15.108) with zero external forces on S. The functional is strictly convex, bounded from below, quadratic functional. Its minimizing element yα , y linearly depends on hα , h and γ ; therefore,

15.3


yˇ α ∼ ε,

yˇ ∼ ε.

749

(15.110)

The minimum value, , of the functional is a quadratic form with respect to hα , h and γ . For nonzero external forces, yα and y can be written as yα = yˇ α + z α ,

y = yˇ + z.

(15.111)

The functions, z α , z, satisfy the constraints

z α = 0,

z α|β eαβ = 0.

z = 0,

(15.112)

Substituting (15.111) into the functional (15.108) results in the expression − R α α − R − N γ + G αβ z |α z |β +

αβγ δ λ 2 C z (α|β) + C αβ z |λ (α, β → γ , δ) − h ( f α z α + f z)dϕ,

(15.113)

S

where R α , R and N are coefficients at α , and γ , respectively, in the work of external forces on the displacements yˇ α , yˇ : α

R α + R + N γ = h

2

( f α yˇ α + f yˇ )dϕ.

(15.114)

S

They can be determined as soon as the dependence of yˇ α and yˇ on α , and γ is found. In derivation of (15.113), it is used that, due to the Euler equations for yˇ α , yˇ , there are no interaction terms between yˇ α , yˇ and z α , z. The part of the functional (15.113) which is quadratic with respect to z α and z, is equal to zero on the fields z α = cα + eαβ ζ β κ, z = c, cα , c, κ = const. These fields are excluded by the conditions (15.113), and it can be proved that the functional (15.113) is bounded from below. The minimizing element of the functional (15.113), i zˇ α , zˇ , linearly

depends on the external forces and, according to the condition f = O μεh/l , the values of the last two terms in (15.113) are ignorably small. Due to the same condition, R α α + R + N γ is on the order of h/l and can, therefore, be dropped in the first approximation. Note, however, that, as the additional analysis shows, for a cross-section with central symmetry, all other corrections are smaller, and the main refinement of the classical beam theory is in keeping this linear form. The terms omitted in the expression for strains (15.101) are on the order of h/R, h/l, ε and εγ , as can be seen from (15.106) and the estimates (15.110). Therefore, the solution of the minimization problem for the functional does indeed give the leading approximation of yα and y.

750

15

Elastic Beams

Dropping the small terms in the work of external forces, we get generalized forces Q i and Q iα introduced in the heuristic beam theory: Qi = h

f i dϕ + |S|Fi ,

Q iα = h 2

S

f i ζ α dϕ.

S

The energy density per unit length of the beam = F + F⊥ + F∠ , where only the terms on the order of μh 2 ε2 are retained, is

=

1

E (γ + hσ ζ σ )2 + (γ , α ,) . 2

(15.115)

The first term in (15.115) (the average value of the longitudinal energy) characterizes the extension and bending energies related to the elongation of the longitudinal beam fibers; the second term (the average value of transverse energy and shear energy) includes the twist energy and the additional contribution of the extension and bending energies caused by the deformation of the beam in the transverse directions. The minimization problem for the functional is a variational problem for a quadratic functional to be minimized with respect to three functions, yα and y, of the cross-sectional coordinates. To construct the “classical” beam theory, one has to find only the minimum value, (γ , α , ), of this functional as a function of four parameters, γ , α and . To describe the stresses inside the beam, one has to find the minimizer of as well.

Chapter 16

Some Stochastic Variational Problems

Stochastic variational problems appear in many branches of physics and mechanics. This chapter gives a brief overview of the features caused by the variational nature of the problems.

16.1 Stochastic Variational Problems In many problems of mechanics and physics, the functionals being minimized depend on parameters which can be considered as random variables. Such parameters might be the elastic moduli and heat conductivity coefficients of inhomogeneous bodies, eigen-strain of bodies, the shape of a body, etc. In this section we discuss the set-up of variational problems for these cases. A field a is called a random field defined in a region V ⊂ Rn if a is a function of two variables – a point x of region V and an event, an element ω of some set, : a = a (x, ω). The set, , is endowed with a probability measure, i.e. for each subset of , A, one can compute the probability of the event ω ∈ A. If A is an infinitesimally small vicinity of some point ω then this probability is denoted by μ(dω). In physical applications one can always assume that there exists a probability density, a non-negative function p(ω), such that, μ(dω) = p(ω)dω, and

μ(dω) =

p(ω)dω = 1.

Probability of the event ω ∈ A is, by definition,

A

μ(dω) =

p(ω)dω. A


751

752

16


Mathematical expectation of a random quantity ϕ(ω) is the integral of ϕ(ω) over : ϕ(ω)μ(dω). Mϕ =

We use for mathematical expectation the symbol M. In mathematics, a standard notation for mathematical expectation is E (expectation), but we retain it for energy. A functional I depending on two variables u and ω, where u is an element of a set M and ω is an element of , will be called a stochastic functional on M. Consider the stochastic functional of the form u κ (x) d V, (16.1) I (u,ω) = L a κ (x, ω) , u κ (x) , x i V

where a κ (x, ω) are some random fields (in what follows, the index κ will be ˇ for every ω. The minomitted). Let I (u,ω) have only one minimizing element, u, imizing element depends on ω and, therefore, is a random field: uˇ = uˇ (x,ω). The minimum value of the functional Iˇ is a random number. The question arises: what ˇ e.g. its averaged value, are the probability characteristics of u, (16.2) v (x) = M uˇ = uˇ (x, ω) μ (dω),

variance,

1

σ 2 (x) = M (uˇ (x, ω) − v (x)) (uˇ (x, ω) − v (x)) ,

(16.3)

the correlation function, b (x, y) = M (uˇ (x, ω) · uˇ (y, ω)) ,

(16.4)

etc., and also the probability characteristics of the minimum value of the functional. We can also be interested in determining more delicate characteristics of the random ˇ ω), like the one-point probability density, field u(x, ˇ ω)), f 1 (x, u) = Mδ(u − u(x, the two-point probability density, ˇ ω))δ u ′ − u(x ˇ ′ , ω) , f 2 (x, u; x ′ , u ′ ) = Mδ(u − u(x, 1

In (16.3) and (16.4) the indices are omitted. Actually, σ and b have two indices, and, for example, the right hand side of (16.3) can be written more elaborately as ′

′

M(uˇ κ (x, ω) − v κ (x))(uˇ κ (x, ω) − v κ (x)); variance is the square root of this tensor.

16.1

Stochastic Variational Problems

753

etc. For all these, we will construct some variational problems. Let us denote the minimum value of I (u, ω) by Iˇω , thus emphasizing that it is a random number. Suppose that the functional I (u, ω) is bounded from below on M by a non-random constant. Then, adding a constant to I (u, ω), we get a positive functional. Therefore, without loss of generality we can set I (u, ω) > 0 and Iˇω > 0. We begin with establishing a simple but quite fundamental fact: the order of the minimization and the probabilistic average can be changed: min

u(x,ω)

I (u (x, ω) , ω) μ (dω) =

min I (u (x, ω) , ω) μ (dω) =M Iˇω .

u(x,ω)

(16.5)

Consider first a set consisting of just two elements, ω1 and ω2 . The probabilities of the events ω1 and ω2 are p1 and p2 , respectively ( p1 + p2 = 1). Then 0 I (u (x, ω) , ω) μ (dω) = min min I (u (x, ω1 ) , ω1 ) p 1 u(x,ω1 ),u(x,ω2 )

u(x,ω)

1 +I (u (x, ω2 ) , ω2 ) p 2 .

Since u (x, ω1 ) and u (x, ω2 ) are independent, the minimum of the sum of two functionals is equal to the sum of minima: p1 min I (u (x, ω1 ) , ω1 ) + p2 min I (u (x, ω2 ) , ω2 ) . u(x,ω1 )

u(x,ω2 )

This can be written as

min I (u (x, ω) , ω) μ (dω) ,

u(x,ω)

as claimed. The same reasoning works for any finite set . As was mentioned on several occasions, finite-dimensional approximations should be sufficient to model the reality. Further, admitting infinite sets, , as the limits of finite sets, we assume that and the probabilistic measure are such that (16.5) holds true. Similarly one can show that the order of the probabilistic averaging and the maximization can be changed: I (u (x, ω) , ω) μ (dω) = max I (u (x, ω) , ω) μ (dω) . max u(x,ω)

u(x,ω)

Moreover, the same is true for minimax variational problems: min max

u(x,ω) v(x,ω)

I (u (x, ω) , v (x, ω) , ω) μ (dω) =

min max I (u (x, ω) , v (x, ω) , ω) μ(dω) .

u(x,ω) v(x,ω)

(16.6)

754

16


To justify (16.6) we check the validity of (16.6) for a set consisting of two elements, ω1 and ω2 . We have min

max

0

I (u (x, ω1 ) , v (x, ω1 ) , ω1 ) p1 1 + I (u (x, ω2 ) , v (x, ω2 ) , ω2 ) p2 = p1 max I (u (x, ω1 ) , v (x, ω1 ) , ω1 ) = min u(x,ω1 ),u(x,ω2 ) v(x,ω1 ) + p2 max I (u (x, ω2 ) , v (x, ω2 ) , ω2 ) =

u(x,ω1 ),u(x,ω2 ) v(x,ω1 ),v(x,ω2 )

v(x,ω2 )

= p1 min max I (u (x, ω1 ) , v (x, ω1 ) , ω1 ) + u(x,ω1 ) v(x,ω1 )

+ p2 min max I (u (x, ω1 ) , v (x, ω2 ) , ω2 ) = u(x,ω2 ) v(x,ω2 ) = min max I (u (x, ω) , v (x, ω) , ω) μ (dω) . u(x,ω) v(x,ω)

The same is true for any finite set and, by our assumption, to the infinite sets we are dealing with. Further we focus mostly on minimization and maximization problems. Formulas that are analogous to (16.5) can be written for all the moments of the random variable Iˇω : n M Iˇω =

n min I (u (x, ω) , ω) μ (dω) = min I n (u (x, ω) , ω) μ (dω). u(x,ω)

u(x,ω)

(16.7)

Denote by J (v, ω) the dual functional: max J (v, ω) = Iˇω .

v(x,ω)

Then M Iˇω =

max J (v(x, ω), ω) μ (dω)

v(x,ω)

= max

v(x,ω)

J (v (x, ω) , ω) μ (dω).

(16.8)

The variational problems (16.7) and (16.8) yield, in particular, the bounds for the random number Iˇω . The variational principle (16.5) can be used to obtain the equations for probabilistic characteristics of the required random field, u (x, ω). Suppose, for example, that we need to find the equations for the averaged field, v (x), the dispersion,

16.1

Stochastic Variational Problems

755

σ (x), and the correlation function, b (x, y) . Let us apply the cross-section principle (see Sect. 5.5) and seek the minimum in (16.5) with the constraints (16.2), (16.3) and (16.4). The corresponding minimum value, G, is a functional of v (x), σ (x), b (x, y). The true values vˇ (x), σˇ (x), bˇ (x, y) provide the minimum value for the functional G (v, σ, b). Therefore, the construction of equations for v (x), σ (x) and b (x, y) is reduced to solving the variational problem (16.5) with the additional constraints (16.2), (16.3), (16.4) and (16.5). The constraints should include all the characteristic of the random field taken into consideration. In the actual construction of the functional G, the direct methods of calculus of variations and the approximate methods, like the Rayleigh-Ritz method, may be employed. The most crude approximation is obtained if we set in (16.5): u (x, ω) = v (x) .

(16.9)

Then G (v) =

I (v, ω) μ (dω).

A refined approximation takes into account the variance u (x, ω) = v (x) + σ (x) ϕ (ω) ,

(16.10)

where ϕ (ω) is a prescribed function with zero mean and unit variance. An additional minimization can be performed over all functions ϕ. The above-mentioned yields an important conclusion: in stochastic variational problems, the equations for the probabilistic characteristics of the required field always possess a variational structure, i.e. they are the Euler equations of some functionals. The fields (16.9) and (16.10) are the admissible fields of the stochastic variational problem. Therefore, computing I (u, ω) μ (dω) on these fields we obtain upper

estimates of M Iˇω . For example, for the field (16.9) we get an important inequality: M Iˇω ≤ min G (v) . v

(16.11)

The lower estimates of M Iˇω are derived similarly using the dual variational principle. The estimates of Iˇω result in some estimates of its distribution function F (ξ ):2 F (ξ ) = Mθ ξ − Iˇω . 2

Recall that the notation θ(ξ ) is used for the step function: θ(ξ ) = 1 for ξ 0, θ(ξ ) = 0 for ξ < 0.

756

16


Indeed, suppose that we have obtained some upper estimate of Iˇω by the RayleighRitz method (by narrowing the set of admissible functions and calculating the corresponding minimum value of the functional, Iˇω+ ): Iˇω ≤ Iˇω+ . The set of elements ω for which Iˇω ≤ ξ is wider than the set of elements ω for which Iˇω+ ≤ ξ : if ω is such that Iˇω+ ξ, then also Iˇω ξ ; if for some ω Iˇω ξ, then the inequality, Iˇω+ ξ, may not hold. Thus, we get for F (ξ ) an estimate, F (ξ ) ≥ F + (ξ ) , where F + (ξ ) is the distribution function of Iˇω+ . Similarly, the upper estimates of F (ξ ) are obtained.

16.2 Stochastic Quadratic Functionals In this and the next four sections we focus mostly on the stochastic variational problems for quadratic functionals, I (u, ω) =

1 (A (ω) u, u) − (l (ω) , u) → min . u 2

(16.12)

We assume that the set of admissible elements, u, form a linear space; the quadratic part of the functional, 21 (A (ω) u, u), is strictly positive on this space, i.e. (A (ω) u, u) > 0 if u = 0, and the functional is bounded below for each ω. ˇ are linear: For quadratic functionals, the equations for the minimizer, u, A(ω)uˇ = l(ω).

(16.13)

If the operator A does not depend on the event ω the problem is called weakly stochastic, otherwise the variational problem is called strongly stochastic. Many physical theories provide examples of such types of problems. The analytical results can be obtained mostly for weakly stochastic problems. For such problems, uˇ = A−1l(ω),

(16.14)

ˇ and for where A−1 is the inverse operator. For the averaged minimizer, u¯ = M u, ¯ we find from (16.13) the equations the fluctuations, uˇ ′ = uˇ − u, ′

¯ u¯ = A−1l, −1 ′

uˇ = A l (ω),

l¯ ≡ Ml(ω),

¯ l (ω) ≡ l(ω) − l. ′

(16.15) (16.16)

Obviously, u¯ and uˇ ′ are the minimizers in the following variational problems:

16.2

Stochastic Quadratic Functionals

757

1 ¯ u → min, (Au, u) − l, u 2 ′ 1 (Au, u) − l (ω) , u → min . u 2

(16.17) (16.18)

ˇ is not equal to zero if the linThe minimizer of the variational problem, u, ear functional, (l (ω) , u) , is non-zero. Therefore, we call the linear functional, (l (ω) , u) , the excitation. As is usual in the probabilistic approach, the probabilistic modeling is especially effective if one can identify in the phenomenon to be modeled the statistically independent (or slightly dependent) events.3 Analytical investigation can be advanced considerably if there are many statistically independent events involved. Further we consider three cases of excitations with a large number of statistically independent events: small excitations, large excitations and Gaussian excitations. Small excitations. This is the case when (l, u) is a sum of small independent excitations. More precisely, there is a large number, N , of independent identically distributed random variables, r1 , . . . , r N , the event, ω, is the vector, ω = (r1 , . . . , r N ), and there is a given random linear functional, (l0 (r ), u). Then the linear functional of the variational problem, (l(ω), u) , is defined as an “empirical average” of N values of (l0 (r ), u): (l(ω), u) =

N 1 (l0 (ra ), u) . N a=1

(16.19)

According to (16.14), the minimizer of the variational problem can be written as

N N 1 1 −1 ′ −1 (16.20) l0 (ra ) = A−1l¯ + A l0 (ra ), uˇ = A N a=1 N a=1 l¯ = l¯0 ≡ Ml0 (r ),

l0′ (r ) ≡ l0 (r ) − l¯0 .

/N The sum of statistically independent variables, a=1 A−1 l0′ (ra ), is on the order of √ N ; indeed, to estimate the magnitude of this sum, we can find its variance: M

N a=1

A−1l0′ (ra )

2

=

N a=1

M(A−1l0′ (ra ))2 = N M(A−1l0′ (r ))2 .

Here we used the statistical independence of l0′ (ra ) for different a. The number 2 M A−1l0′ (r ) is a finite number, which does not depend on N . Therefore, the /N /N squared sum√ a=1 A−1l0′ (ra ) is of the order of N , the sum a=1 A−1 l0′ (ra ) √ is of the order of N , and the fluctuations of the minimizer uˇ are of the order of 1/ N . 3

Recall that two random variables, ϕ(ω) and ψ(ω), are statistically independent if Pr ob [ϕ(ω) = x, ψ(ω) = y] = Pr ob [ϕ(ω) = x] Pr ob [ψ(ω) = y] .

758

16


There is a small probability of finite deviations of uˇ from u¯ though. As we will see, the study of such deviations are of interest in some applications. The corresponding problems are called the problems of large deviations. Note for further references that, according to (16.15), (16.16), (16.17) and ¯ (16.18), the averaged minimizer, u, ¯ u¯ = M uˇ = A−1 l,

(16.21)

and the minimizer’s fluctuations, u ′ , N 1 −1 ′ A l0 (ra ), u = uˇ − u¯ = N a=1 ′

are the solutions of the variational problems 1 (Au, u) − l¯0 , u → min, u 2 N ′ 1 1 l (ra ), u → min . (Au, u) − u 2 N a=1 0

(16.22) (16.23)

Most results obtained for the functional (16.19) can be extended to the functionals that are the sums of different statistically independent linear functionals N 1 (l, u) = (la (ra ), u) . N a=1

(16.24)

Large excitations. If the average value of each member of the sums (16.19) or (16.24) is zero, then u¯√= 0, and it becomes sensible to scale the fluctuations, u ′ , multiplying√ them by N . This is equivalent to consideration of the excitations, which are N times bigger. In the case of the sum of equal random functionals we obtain the functional N 1 (l, u) = √ (l0 (ra ), u) , N a=1

M (l0 , u) = 0.

(16.25)

Energy of a large excitation is finite. Gaussian excitation. The excitation is called Gaussian if, for any real u and any complex number z, ¯

1 2

Me z[(l(ω),u)−(l,u )] = e 2 z

(Bu,u)

,

(16.26)

where (Bu, u) is a non-negative quadratic functional. Gaussian excitations appear, for example, as the limit case for the functionals of the form (16.25) as N → ∞.

16.2

Stochastic Quadratic Functionals

759

Equation (16.26) is assumed to hold not only for real u, but for complex-valued u as well. Formula (16.26) can be viewed as a compact way to provide the information on all moments of the random number, (l (ω) , u): expanding both sides of (16.26) in Taylor’s series with respect to z, ¯

∞ k 1 k ¯ u ] , z [(l (ω) , u) − l, k! k=0 k ∞ 1 2k 1 z (Bu, u) , = k! 2 k=0

e z[(l(ω),u)−(l,u )] = 1 2

e2z

(Bu,u)

and equating the coefficients at the same powers of z in the right and the left hand sides of (16.26), we obtain ¯ u ] = 0, M[(l (ω) , u) − l, 2 ¯ u ] = (Bu, u), M[(l (ω) , u) − l, 3 ¯ u ] = 0, M[(l (ω) , u) − l, 4 ¯ u ] = 3[(Bu, u)]2 , M[(l (ω) , u) − l,

(16.27) etc.

The second equation (16.27) determines the meaning of the quadratic functional ¯ u . (Bu, u): it is the squared variance of the linear functional (l (ω) , u) − l, We conclude this section by introducing for further reference a notion which is closely related to Gaussian excitations: Gaussian random field. Gaussian fields. A random field, a(x, ω), is called Gaussian if formula (16.26) holds for the functional, a(x, ω)u(x)d V. (l (ω) , u) = V

Due to the arbitrariness of u, formula (16.26) contains all the information on the ¯ “local” characteristics of a(x, ω). Indeed, denoting by a(x) the average value of the ¯ field a(x, ω) at a point x : a(x) = Ma(x, ω), we have from (16.27): ′ ¯ ¯ ′ ))]u(x)u(x ′ )dVdV ′ . (Bu, u) = M[(a(x, ω) − a(x))(a(x , ω) − a(x V

V

The function in the integrand ′ ¯ ¯ ′ ))] , ω) − a(x B(x, x ′ ) = M[(a(x, ω) − a(x))(a(x

is called the correlation function of the field a(x, ω). So, the functional (Bu, u) is known if the correlation function of the field a(x, ω) is given: B(x, x ′ )u(x)u(x ′ )dVdV′ . (16.28) (Bu, u) = V

V

760

16


Let us show that (16.26) allows us to find probability densities of the field a(x, ω). First we choose u(x) in (16.26) as δ-function: u(x) = δ(x − x1 ). Then, 1 2

¯ 1 )] Me z[a(x1 ,ω)−a(x = e2z

B(x1 ,x1 )

(16.29)

.

To compute the probability density, f (a, x1 ), of the random field a(x, ω) at the point x1 , f (a, x1 ) = Mδ(a − a(x1 , ω)), we use a representation of the δ-function:4 1 δ (E) = 2πi

i∞

e E z dz.

(16.30)

−i∞

We have

1 f (a, x1 ) = Mδ(a − a(x1 , ω)) = M 2πi

i∞

e z(a−a(x1 ,ω)) dz

−i∞

=

1 2πi

i∞

e za Me−za(x1 ,ω) dz

i∞

¯ 1 )] ¯ 1 )] e z[a−a(x Me−z[a(x1 ,ω)−a(x dz.

−i∞

1 = 2πi

−i∞

Plugging here (16.29), f (a, x1 ) =

1 2πi

i∞

1 2

¯ 1 )] 2 z e z[a−a(x e

B(x1 ,x1 )

dz,

−i∞

and using (5.310), we find f (a, x1 ) = √

1 1 ¯ 1 )]2 [a−a(x − . e 2B(x1 ,x1 ) 2π B(x1 , x1 )

4

The integral in (16.30) does not converge absolutely, and its meaning has to be explained. Formula (16.30) can be understood in the following way: for any function with a compact support, ϕ(E), +∞ iα +∞ 1 δ (E) ϕ(E)d E = lim ϕ(E)e E z dzdE = ϕ(0). α→∞ 2πi −∞ −iα −∞

16.3

Extreme Values of Energy

761

We see that the one-point probability density of the random field at a point x1 is Gaussian with the variance B(x1 , x1 ). In the same way one can determine the joint probability density of the values of the field a(x, ω) at the points x1 , . . . , xn : f (a1 , . . . , an ) = Mδ(a1 − a(x1 , ω)) . . . δ(an − a(xn , ω)) = i∞ i∞ 1 ¯ 1 ))+...+z n (a1 −a(x ¯ n )) e z (a1 −a(x ... −i∞

−i∞

dz n dz 1 ... . 2πi 2πi 0 We set in (16.28) u(x) to be a sum of δ-functions: u(x) = i z 1 δ(x − x1 ) + . . . + z n δ(x − xn )], z k being pure imaginary. Then, Me−[z

1

¯ 1 ))+...+z n (a(xn ,ω)−a(x ¯ n ))] (a(x1 ,ω)−a(x

(Bu, u) = −B jk z j z k ,

B jk = B(x j , xk ),

and, using (16.26) with z = i we find f (a1 , . . . , an ) =

i∞

...

−i∞

i∞

ez

1

¯ 1 ))+...+z n (a1 −a(x ¯ n )) (a1 −a(x

1

e 2 B jk z

−i∞

j k

z

dz n dz 1 ... . 2πi 2πi

Applying (5.320), we obtain 1 1 (−1) jk ¯ j ))(ak −a(x ¯ k )) (a j −a(x f (a1 , . . . , an ) = √ , e− 2 B n (2π ) B % % where B (−1) jk is the inverse tensor to B jk , and B = det % B jk % . So the joint probability density of the values of the field, a(x, ω), at any finite set of points is also Gaussian. The random Gaussian field, a(x, ω), is called delta-correlated if for any u(x), Me

¯ V V [a(x,ω)−a(x)]u(x)d

1

= e2

V

B(x)u 2 (x)d V

.

This corresponds to the case when the correlation function B(x, x ′ ) is the δ-function B(x, x ′ ) = B(x)δ(x − x ′ ).

16.3 Extreme Values of Energy Variational problems for extreme values. It is often important to know the range in which the minimum value of the functional Iˇω can change. We denote the extreme values of Iˇω by Iˇ − and Iˇ + : Iˇ − ≤ Iˇω ≤ Iˇ + .

762

16


To find the extreme values one can use the possibility to change the order of computation of minima (or maxima) of any functional of two variables u and ω; in particularly, for I (u, ω) and its dual functional J (v, ω), we have min min I (u, ω) = min min I (u, ω), u

ω

u

ω

max max J (v, ω) = max max J (v, ω). ω

v

v

ω

Denoting by I − (u) and J + (v) the functionals I − (u) = min I (u, ω),

J + (v) = max J (v, ω),

ω

ω

we obtain for the extreme values Iˇ − and Iˇ + the variational problems Iˇ − = min I − (u),

Iˇ + = max J + (v).

u

v

(16.31)

If, as for quadratic functionals (16.12), the minimum value is negative, and can be identified with negative energy of the system, −E(ω), E(ω) = − min I (u, ω),

(16.32)

u

then energy changes between the extreme values E min and E max : E min = − Iˇ + ,

E max = − Iˇ − .

Relations (16.31) provide the variational problems for the extreme values of energy: E min = − max J + (v), v

E max = − min I − (u). u

(16.33)

In the weakly stochastic case, the functionals J + (v) and I − (u) can be easily found. For example, for I − (u) we have I − (u) = min I (u, ω) = ω

1 1 (Au, u) + min[− (l (ω) , u)] = (Au, u)−max (l (ω) , u) . ω ω 2 2 (16.34)

The functional maxω (l (ω) , u) can be a nonlinear functional of u. Therefore, even in this simplest case, the computation of the extreme values is a non-elementary problem. It is essential that the computation of the minimum value of energy involves the dual variational problem: if we attempt to get the value of E min using the original variational problem (16.32),

E min = min − min I (u, ω) = − max min I (u, ω), ω

u

ω

u

16.3


763

then we arrive at a minimax problem for which it is not always possible to change the order of minimization and maximization. Sometimes it is beneficial to reformulate the variational problems for the extreme values of energy in terms of functional integrals. To do that we need to employ Laplace’s asymptotics of integrals which is briefly outlined in the next subsection. Laplace’s asymptotics. Consider an integral which depends on a parameter λ in the following way: I (λ) =

f (x)eλS(x) d V, V

where V is a bounded region of n-dimensional space, f (x) and S(x) are some smooth functions. We wish to find the asymptotics of this integral as λ → +∞. Laplace suggested that the leading terms of the asymptotics of I (λ) are the same as that of the integral over the vicinities of the points where the function S(x) has the maximum value. Then the asymptotics can be easily found. Indeed, let S(x) achieve its maximum value only at one point,% xˆ ; this point is%an internal point of V, and the matrix of the second derivatives, %2 S(xˆ )/x i x j % is non-degenerated, i.e. its determinant, , is non-zero. We can write I (λ) = f (xˆ )eλS(xˆ )

V

f (x) −λ[S(xˆ )−S(x)] e d V. f (xˆ )

In a small vicinity of the point xˆ we can replace S(xˆ ) − S(x) by the non-degenerated quadratic form 1 S(xˆ ) − S(x) ≈ − Si j (x i − xˆ i )(x j − xˆ j ), 2

(16.35)

where Si j = 2 S(xˆ )/x i x j . Note that the quadratic form (16.35) is positive because xˆ is the point of maximum of S(x). In a small vicinity of xˆ we can replace f (x)/ f (xˆ ) by unity thus obtaining I (λ) ≈ f (xˆ )e

λS(xˆ )

1

i

i

e−λ[− 2 Si j (x −xˆ )(x

j

−xˆ j )]

d V.

(16.36)

small vicinity of xˆ

0 1 Since λ → +∞, the function exp −λ[− 12 Si j (x i − xˆ i )(x j − xˆ j )] decays very fast away from xˆ . We do not pause to justify that the expansion of the integration region from a small vicinity of xˆ to the entire space, R,causes only exponentially small corrections in (16.36). Thus we can write I (λ) ≈ f (xˆ )eλS(xˆ )

1

i

i

e−λ[− 2 Si j (x −xˆ )(x R

j

−xˆ j )]

d V.

764

16


The integral here, according to the Gauss formula (5.295), is equal to Finally, the leading term of the asymptotics is

I (λ) ≈

(2π )n f (xˆ )eλS(xˆ ) . λn ||

(2π )n /λn ||.

(16.37)

As a more elaborate derivation shows, the error of the formula (16.37) is on the order of (eλS(xˆ ) /λn/2 )/λ. If S(x) achieves its maximum at several internal points, one should sum the contributions (16.37) of all points. One can check that in the cases of the point of maximum lying on the boundary and/or degeneration of the quadratic form − 21 Si j (x i − xˆ i )(x j − xˆ j ) the asymptotics remains qualitatively the same, I (λ) ≈ prefactor(λ)eλS(xˆ ) ,

(16.38)

with the prefactor being a decaying power function of λ. The prefactor is a constant independent on λ if S(x) has maximum value on a set with non-zero volume. By Laplace’s asymptotics we mean further the asymptotics of the form (16.38) where the prefactor changes slower than the exponential function of λ: 1 ln prefactor(λ) → 0 λ

as λ → ∞.

Changing in the previous consideration S(x) by −S(x) we obtain the asymptotics

V

f (x)e−λS(x) d V ≈ prefactor(λ)e−λS(xˇ ) ,

(16.39)

where xˇ is the point of minimum of S(x). In applications to the variational problems, we also need to know the asymptotics of integrals of the form (16.39) for complex values of λ. In this case we denote the parameter by z, I (z) =

f (x)e zS(x) d V, V

and consider the asymptotics of I (z) as |z| → ∞. Note first of all that I (z) is an analytical function of z at any finite point z if the integral, as we accept, converges absolutely:

V

| f (x)| eRe zS(x) d V < ∞.

The point z = ∞ can, however, be the singular point of I (z). Usually, the singularity is essential, i.e. the asymptotics of I (z) along different paths, z → ∞, are different.

16.3


765

It turns out that Laplace’s asymptotics,

I (z) ≈

1 (2π )n zS(xˆ ) , 1+O f (xˆ )e z n || z

(16.40)

holds true for all paths z → ∞ such that |Arg z| ≤ π/2 − ε for some small ε. For other paths, this asymptotics does not hold. This is seen from studying the asymptotics when z → ∞ along the imaginary axis, z = i y, |y| → ∞. It turns out that in this case the leading contribution to the asymptotics is provided not only by the point of maximum of S(x), but by all stationary points of S(x), particularly by all points of local maxima and minima. This asymptotics is called the stationary phase asymptotics; we do not dwell on it here since it will not be used further. Functional integrals and extreme values of energy. Let us assume that the set of events is a bounded region in a finite dimensional space. As we have discussed, due to the approximate nature of all the ingredients of physical theories, such an assumption does not constrain the generality of our consideration. Then, as follows from (16.39), the minimum value of energy can be presented as 1 ln λ→+∞ −λ

E min = lim

e−λE(ω) μ(dω).

(16.41)

Similarly, for the maximum value of energy we have E max

1 ln = lim λ→+∞ λ

eλE(ω) μ(dω).

(16.42)

If Laplace’s asymptotics holds for the functional integral,

e−λI (u,ω) Du ≈ prefactor(λ)e−λ minu I (u,ω) = prefactor(λ)eλE(ω) ,

then formula (16.42) is consistent with the formula 1 ln λ→+∞ λ

E max = lim

e−λI (u,ω) Duμ(dω).

(16.43)

One can also express this fact in the form

e−λI (u,ω) Duμ(dω) ≈ prefactor(λ)e−λ minu,ω I (u,ω) = prefactor(λ)eλEmax .

Similarly, if the Laplace asymptotics holds for the functional integrals,

eλJ (v,ω) Dv ≈ prefactor(λ)eλ maxv J (v,ω) ,

766

16


then, since maxv,ω J (v, ω) = maxω [−E(ω)] = −E min , one can present the minimum value of energy in terms of a functional integral: 1 ln λ→+∞ −λ

E min = lim

eλJ (v,ω) Dvμ(dω).

(16.44)

In the case of weakly stochastic quadratic functionals formula (16.43) takes the form 1 1 E max = lim ln e−λ 2 (Au,u) eλ(l(ω),u) μ(dω) Du. λ→+∞ λ Applying to the integral over ω Laplace’s asymptotics, we have 1 ln λ→+∞ λ

E max = lim

1 ln λ→+∞ λ

1

e−λ 2 (Au,u)+λ maxω (l(ω),u) Du = lim

e−λI

−

(u)

Du. (16.45)

Here we also used (16.34). If one can use Laplace’s asymptotics for the functional integral of exp[−λI − (u)], then formula (16.45) coincides with the second relation (16.33). Similar functional relations can be written for the minimum energy, if one uses the dual functional. Example 1. Vortex gas.

I (u, r1 , . . . , r N ) =

Consider a functional 1 2

V

N γa u(ra ). u ,α u ,α + ε2 u,αβ u ,αβ d V −

(16.46)

a=1

Here V is a bounded two-dimensional region, Greek indices run through values 1,2, function u(x α ) vanishes on the boundary of the region V, u(x α ) = 0

on V,

(16.47)

comma in indices denotes derivative: u ,α = u/x α , r1 , . . . , r N are the points of V chosen randomly and independently, ε a small parameter. The minimum value of the functional (16.46) has the meaning of (negative) kinetic energy of a flow of ideal incompressible fluid caused by the vortices with intensities, γ1 , . . . , γ N , placed at the points r1 , . . . , r N ; ε plays the role of the size of the vortex core (see Sect. 9.7). The same minimization problem appears in dislocation theory: the minimizer is the stress function for the internal stresses caused by screw dislocations located at the points, r1 , . . . , r N (see Sect. 6.10). Let us find the maximum value of energy. We have

16.3


E max = − min min ra

u∈(16.47)

1 2

767

V

N γa u(ra ) u ,α u ,α + ε2 u,αβ u ,αβ d V − a=1

!

! N 1 ,α 2 ,αβ = − min min γa u(ra ) u ,α u + ε u,αβ u dV − u∈(16.47) ra 2 V a=1 !

N 1 ,α 2 ,αβ = − min γa u(ra ) u ,α u + ε u,αβ u d V + min − ra u∈(16.47) 2 V a=1 ! N 1 ,α 2 ,αβ = − min γa u(ra ) . u ,α u + ε u,αβ u d V − max ra u∈(16.47) 2 V a=1

If all γa are positive, then max ra

N a=1

γa u(ra ) = γ max u(r ), r

(16.48)

where γ is the total vorticity: γ =

N

γa .

a=1

If there are vortices of different signs, then max ra

N a=1

γa u(ra ) = γ + max u(r ) + γ − min u(r ) = max (γ + u(r + ) + γ − u(r − )), + − r

r

r ,r

(16.49)

where γ + ≥ 0 and γ − ≤ 0 are the total intensities of positive and negative vortices. Consider first the case of vortices of the same sign. To find the maximum value of energy we have to solve the variational problem: 1 u ,α u ,α + ε2 u,αβ u ,αβ d V − max γ u(r ) → min . r u∈(16.47) 2 V We can change the order of minu∈(16.47) and maxr :

1 u ,α u ,α + ε2 u,αβ u ,αβ d V − max γ u(r ) = r u∈(16.47) 2 V 1 u ,α u ,α + ε2 u,αβ u ,αβ d V + min(−γ u(r )) = min r u∈(16.47) 2 V 1 u ,α u ,α + ε2 u,αβ u ,αβ d V − γ u(r ) = min min r u∈(16.47) 2 V = min(−E 0 (r )) = − max E 0 (r ), min

r

r

768

16


where E 0 (r ) is the energy of one vortex of the intensity γ located at the point r. Hence, E max = max E 0 (r ). r

For particular regions, the function E 0 (r ) can be easily found. In the case of vortices of two signs, similarly, E 0 (r + , r − ), E max = max + − r ,r

where E 0 (r + , r − ) is the energy of a pair of vortices located at r + and r − and having the intensities, γ + and γ − . Computation of the minimum value of energy of point vortices is a more difficult problem, and we do not dwell on it here.

16.4 Probability Distribution of Energy: Gaussian Excitation Probability distribution of energy is important to know in various problems of mechanics, in particular in statistical mechanics of point vortices and mechanics of composite materials. First we consider this issue for the case of Gaussian excitation. To find the probability density function of the random number, E = − min I ,

f (E) = Mδ E + min I (u, ω) ,

(16.50)

u

we plug in (16.50) the presentation of δ-function (16.30),

f (E) = M

1 2πi

i∞

e

E z+z min I (u) u

dz =

1 2πi

−i∞

i∞

e E z Me

z min I (u) u

dz,

(16.51)

−i∞

and use (5.325): 1 f (E) = 2πi

i∞

−i∞

=

1 2πi

i∞

⎡

eEz ⎣M

i∞

e

√ − 12 (Au,u)+i z(l,u)

−i∞

1

e E z− 2 (Au,u) Mei

−i∞

For Gaussian excitation, according to (16.26),

√

z(l,u)

D A udz.

⎤

D A u ⎦ dz (16.52)

16.4

Probability Distribution of Energy: Gaussian Excitation

1 f (E) = 2πi

i∞

e E z− 2 (Au,u)+i

i∞

e E z e− 2 ((A+z B)u,u)+i

1

769

√ ¯ z(l,u) − 21 z(Bu,u)

e

D A udz

−i∞

1 = 2πi

1

√ ¯ z(l,u)

D A udz.

(16.53)

−i∞

The functional integral in (16.53) can be computed. To this end, consider some finite-dimensional truncation of the quadratic and linear functionals, and denote them by (Am u, u) , ((A + z B)m u, u) and (l¯m , v) . Then5

√ ¯ 1 e− 2 ((A+z B)m u,u)+i z(lm ,u) z m

m

det Am d u = e

¯ ¯ −z 21 ((A+z B)−1 m lm ,lm )

det Am , det(A + z B)m

where (A + z B)−1 m is the inverse matrix for the matrix ( A + z B)m . The ratio det(A + z B)m / det Am is a function of z and m; denote the logarithm of this ratio by m (z). We assume that the limits exist: det(A + z B)m = (z), lim m (z) = lim ln m→∞ m→∞ det Am ¯ ¯ lim (A + z B)−1 m l m , l m = (z). m→∞

Then

1

e− 2 ((A+z B)u,u)+i

√ ¯ z(l,u)

1

1

D A u = e− 2 z(z)− 2 (z)

and, finally, the probability density function is given by the integral 1 f (E) = 2πi

i∞

1

1

e E z− 2 z(z)− 2 (z) dz.

(16.54)

−i∞

Example 2. Consider a variational problem

a 0

1 du 2 − g(x, ω)u(x) d x → 2 dx

min

u(x): u(0)=u(a)=0

,

(16.55)

where g(x, ω) is a δ-correlated Gaussian random field with zero mean:

5

Recall that, according to notation of Sect. 5.12, the factor (2π )−m/2 is included in the volume element, d m u.

770

16

Me

a 0

1

g(x,ω)u(x)d x

= e2

a 0

Some Stochastic Variational Problems Bu 2 (x)d x

(16.56)

.

Here B is a constant. A possible physical interpretation of the minimizer is the lateral deflection of a beam clamped at the two ends; the deflection is caused by a random normal force. The minimizer can be found explicitly: it is the solution of the boundary value problem d 2u = −g(x, ω), dx2

u(0) = u(a) = 0.

(16.57)

The solution is du(x, ω) =− dx

x

g(x, ω)d x + c,

0

and the constant c is determined from the boundary conditions c=

1 a

a

dx

x′

g(x ′ , ω)d x ′ .

0

0

A typical behavior of the solution is a random curve, y = u(x, ω), passing through the beam ends, (x = 0, y = 0) and (x = a, y = 0). The average deviation from the underformed position, the straight line y = 0, is equal to zero because l¯ = 0. The probability density function of energy can be computed using the explicit solution. It is easier, however, to follow the above-described approach. Since l¯ = 0, we have (z) = 0. To find (z), consider quadratic functionals (Au, u) =

a

0

du dx

2

d x and ((A + z B)u, u) =

0

a

du dx

2

+ zBu2 d x

on the set of functions vanishing at the points x = 0, a. We extrapolate u(x), defining it on the segment [−a, a] as an odd function, and expand the extrapolated function in Fourier series on [−a, a]: u(x) =

∞

u k sin

k=1

π kx . a

Then 1 (Au, u) = 2

a

−a

∞ πk k=1

∞ π2 2 2 k uk , = 2a k=1

π kx u k cos a a

2

2 a ∞ π kx 1 πk cos2 dx = uk dx 2 k=1 a a −a

16.5

Probability Distribution of Energy: Small Excitations

771

2 ∞ ∞ π2 2 2 π kx 1 a ((A + z B)u, u) = u k sin k uk + z B dx 2a k=1 2 −a k=1 a =

∞ ∞ π 2 2 2 az B 2 k uk + u . 2a k=1 2 k=1 k

Hence,

(z) = lim ln m→∞

m G

k=1

1 (π 2 k 2 2a m G

k=1

+ a 2 z B)

π 2k2 2a

m H a2 z B 1+ 2 2 . = ln lim m→∞ π k k=1

Function (z) can be expressed in terms of elementary functions due to the relation ∞ H sinh z z2 1+ 2 2 . = z π k k=1 We have √ sinh(a z B) (z) = ln . √ a zB

(16.58)

From (16.54) and (16.58) we obtain the formula for probability density function of energy, 1 f (E) = 2πi

i∞

e

Ez

−i∞

√ a zB dz, √ sinh(a z B)

(16.59)

or, in a dimensionless form, after the change z → z/a 2 B, 1 f (E) = 2πia 2 B

i∞

−i∞

e

E a2 B

z

√ sinh

z √ dz. z

16.5 Probability Distribution of Energy: Small Excitations Probability distribution. In this section we find the probability distribution of energy in weakly stochastic variational problems in the case of small excitation, I (u, r ) =

N 1 1 (Au, u) − (l0 (ra ), u) → min, u 2 N a=1

772

16


where (l0 (r ), u) is a given linear functional and r1 , . . . , r N are identically distributed statistically independent variables. One may expect that, as in the central limit theorem, the linear part of the functional converges as N → ∞ to the averaged functional ¯ u) = M (l0 (r ), u) . (l,

(16.60)

¯ u): Denote by E¯ energy, corresponding to (l, 1 ¯ ¯ (Au, u) − (l, u) . − E = min u 2

(16.61)

The minimizer of this variational problem is the averaged minimizer u¯ (16.21). The ¯ A typical magnitude is small in the energy fluctuates around the average value, E. −1/2 order of N . We will seek the probability of large fluctuations, when E differs from E¯ for some finite number; such probability should be exponentially small. We are going to show that the computation of the probability density can be reduced to the solution of the following deterministic variational problem. Consider a functional of real functions, u, and two real parameters, E and z: z S(E, z, u) = E z + (Au, u) + ln Me−z(l0 ,u) . 2

(16.62)

For a given E, this functional has many stationary points. We choose the point at which the value of the functional is maximum. This value is denoted further by S(E), while the value of z at this stationary point is denoted by β. In physical problems, S(E) and β usually have the meaning of entropy and inverse temperature. We use these terms for S(E) and β in what follows, and call S(E, z, u) the entropy functional. To emphasize that the functional I and the probability density of energy f (E) depend on N , we further write I N and f N (E), respectively. Note that the entropy functional (16.62) and entropy S(E) do not depend on N . We are going to motivate the asymptotic relation: for large N , f N (E) = const e N S(E) .

(16.63)

Before proceeding to the derivation of this relation, we briefly describe some of its features which will be further considered in detail. First, it turns out that entropy is negative: S(E) ≤ 0 for all E. Second, entropy is a concave function defined on the segment [E min , E max ]; its qualitative sketch is shown in Fig. 16.1. Third, entropy ¯ which is the most probable value reaches the maximum value at some point E = E, of energy. In the vicinity of this point, 1 d 2 S ¯ 2. (E − E) S(E) = 2 d E 2 E= E¯

16.5


773

Fig. 16.1 A qualitative dependence of entropy on energy in case of small excitations with Ml0 = 0

The second derivative is negative; denote it by −1/σ 2 . Hence, in vicinity of E¯ the probability density of energy is Gaussian, f (E) ∼ e

−

¯ 2 (E− E) √ 2(σ/ N )2

,

√ a typical deviation of energy from the most with a small variance, σ/ N . Therefore, √ probable value is on the order of 1/ N indeed. In the limit N → ∞ the probability ¯ The derivative density converges to δ-function with the support at the point E = E. d S(E)/d E coincides with β at the stationary point in accordance with the rule of differentiation of the stationary value with respect to parameter (Sect. 5.13): β=

d S(E) . dE

This formula motivates the term inverse temperature for β. As is seen from Fig. 16.1, temperature, T = 1/β, can be positive and negative: it is positive for E < E¯ and ¯ Accordingly, one says that the states with negative temperature negative for E > E. are hotter than that with positive temperature: the energies in negative temperature states are bigger. The transition from positive to negative temperatures occurs by passing the point T = ∞; therefore it is often more convenient to operate with coldness, β, for which this transition occurs at the point β = 0. This point is the point of the most probable value of energy. Now we proceed to a more detailed discussion. Properties of the entropy functional. Euler equations of the entropy functional. Varying the entropy functional, we obtain the equations for its stationary points, z × and u × : 1 × × Me−z (l0 ,u )l0 , Me−z × (l0 ,u × ) × × 1 Me−z (l0 ,u ) (l0 , u × ) E + (Au × , u × ) − = 0. 2 Me−z × (l0 ,u × ) Au × =

(16.64) (16.65)

774

16


As follows from (16.64), (Au × , u × ) =

1 Me−z × (l0 ,u × )

Me−z

×

(l0 ,u × )

(l0 , u × ).

Therefore, (16.65) is equivalent to the equation 1 (Au × , u × ) = E, 2

(16.66)

which shows that E has the meaning of energy of the stationary point. At a stationary point, the entropy functional has the value 2E z × + Me−z

×

(l0 ,u × )

.

(16.67)

¯ For β = 0, (16.64) becomes If β = 0, then E = E. ¯ Au × = Ml0 = l, ¯ ¯ and E = E. i.e. u × coincides with the averaged minimizer, u, Entropy functional S(E, z, u) is convex over u for positive z. Denote by J (u) the functional J (u) = ln Me−z(l0 ,u) . It is convex:6 √ u1 + u2 1 1 = ln Me− 2 z(l0 ,u 1 )− 2 z(l0 ,u 2 ) ≤ ln Me−z(l0 ,u 1 ) Me−z(l0 ,u 2 ) J 2 1 = (J (u 1 ) + J (u 2 )) . 2 Therefore, the entropy functional (16.62) is convex over u for each fixed positive z. Entropy functional is concave over u for small negative z. Let us expand functional ln Me−z(l0 ,u) over z and keep only the terms of the first and the second order: 1 2 2 −z(l0 ,u) (16.68) = ln M 1 − z(l0 , u) + z (l0 , u) ln Me 2 ¯ u) + 1 z 2 [M(l0 , u)2 − (l, ¯ u) + 1 z 2 M(l0 , u)2 = −z(l, ¯ u)2 ]. = ln 1 − z(l, 2 2 The coefficient at z 2 is positive:

6

Here we used the inequality M[ f g] ≤

M f 2 Mg 2 ,

which is similar to the Cauchy inequality and is derived similarly (see a footnote in Sect. 5.1).

16.5


775

¯ u)2 = [M(l0 , u)]2 = [M[1 · (l0 , u)]]2 < M12 M(l0 , u)2 = M(l0 , u)2 . (l, The entropy functional becomes z 1 ¯ u)2 ] − z(l, ¯ u). S(E, z, u) = E z + (Au, u) + z 2 [M(l0 , u)2 − (l, 2 2 The third term for small z is negligible compared to the second one, and the functional behavior is determined by the second term. Hence, for small negative z the entropy functional is concave. Change of variables. It is convenient to make a change of variable in the entropy functional, u → v = u/z: S(E, z, v) = E z +

1 (Av, v) + ln Me−(l0 ,v) . 2z

(16.69)

Then the last term becomes independent on z. To indicate which variables are used we employ the notations S(E, z, u) and S(E, z, v) for functionals (16.62) and (16.69), respectively. The stationary points of the functionals S(E, z, u) and S(E, z, v) have, obviously, the same stationary values of z, while the stationary functions u × and v × are linked as v× =

u× . z×

(16.70)

The stationary points z × , v × obey the equations 1 1 −(l0 ,v × ) Av × = l0 , × ) Me × −(l ,v 0 z Me

1 (Av × , v × ) = E. 2z ×2

(16.71)

Entropy functional S(E, z, v) is convex for positive z. As we have shown, the functional ln Me−(l0 ,u) is convex. Therefore S(E, z, v) is convex over v for each fixed positive z. Besides, for each fixed v, S(E, z, v) is a convex function of z for positive z because E and (Av, v) are positive. Let us show that S(E, z, v) is also convex as a functional of two variables z and v. Since the first and the third terms in (16.69) are convex, we have to prove that 1 1 v1 + v2 v1 + v2 1 1 ≤ , A , , v , v + (Av (Av ) ) 1 1 2 2 1 2 2 2 z1 z2 (z + z 2 ) 2 1 or, multiplying by z 1 + z 2 , z2 z1 (A(v1 + v2 ), (v1 + v2 )) ≤ 1 + (Av1 , v1 ) + 1 + (Av2 , v2 ) . z1 z2 Denote the ratio z 2 /z 1 by t. Then the inequality we need to prove takes the form 2 (Av1 , v2 ) ≤ t (Av1 , v1 ) + It can be rewritten as

1 (Av2 , v2 ) . t

776

16


t 2 (Av1 , v1 ) − 2t (Av1 , v2 ) + (Av2 , v2 ) = (A(tv1 − v2 ), (tv1 − v2 )) ≥ 0, and, therefore, holds true. We assume that the functionals ( Av, v) and Me−(l0 ,v) are such that S(E, z, v) is strictly convex and, thus, there can be only one stationary point for positive z. If a stationary point for positive z exists, then the stationary value at this point is bigger then any other stationary value. Let z 1× , v1× and z 2× , v2× be two stationary points of the entropy functional. We are going to show that for any two stationary points, (16.72) S(E, z 1× , v1× ) − S(E, z 2× , v2× ) ≥ 2E(z 1× − z 1× ).

In fact, the inequality is strict under non-constraining conditions, e.g. for strictly convex functional ln Me−(l0 ,v) . This inequality has important consequences. If z 1× ≥ 0, then the right hand side of (16.72) is zero, and the stationary value at z 1× , v1× is not smaller than any other stationary value. Hence, if a stationary point with positive z × exists, then entropy S(E) is the stationary value of the entropy functional at this point. For other stationary points with negative z × we obtain, setting in (16.72) z 1× < 0, and z 2× > 0, an estimate of the stationary values from below: S(E, z 1× , v1× ) ≥ S(E) − 4E z 1× .

If in (16.72) both z 1× and z 2× are negative, then (16.72) means that the sum S E, z × , v × + 4E z × exceeds any other stationary value. So let z 1× , v1× and z 2× , v2× be solutions of the equations 1 1 × −(l0 ,v1× ) l0 , × Me × Av1 = −(l ,v ) 0 1 z1 Me 1 1 × −(l0 ,v2× ) l0 , × Me × Av2 = −(l ,v ) z2 Me 0 2

1 2z 1×2 1 2z 2×2

(Av1× , v1× ) = E,

(16.73)

(Av2× , v2× ) = E,

(16.74)

and we wish to estimate from below the difference S(E, z 1× , v1× ) − S(E, z 2× , v2× ). × Since S(E, z × , v × ) = 2E z × + J (v × ), where J (v) = ln Me−(l0 ,v ) , S(E, z 1× , v1× ) − S(E, z 2× , v2× ) = 2E(z 1× − z 2× ) + J (v1× ) − J (v2× ).

(16.75)

For the difference J (v1× ) − J (v2× ) we can write J (v1× ) − J (v2× ) =

0

1

d J (tv1× + (1 − t)v2× ) dt. dt

The function tv1× + (1 − t)v2× belongs to the line of the functional space passing through the points v1× and v2× . Functional J (v) is convex, and so is a function of t, J (tv1× + (1 − t)v2× ). Hence, the second derivative, d 2 J (tv1× + (1 − t)v2× )/dt 2 , is

16.5


777

non-negative, and the first derivative is an increasing function of t, in particular d J (tv1× + (1 − t)v2× ) d J (tv1× + (1 − t)v2× ) ≥ . dt dt t=0

This results in inequality: J (v1× )

−

J (v2× )

≥

1 0

d J (tv1× + (1 − t)v2× ) d J (tv1× + (1 − t)v2× ) dt = dt dt t=0 t=0 ×

=−

Me−(l0 ,v2 ) (l0 , v1× − v2× ) ×

Me−(l0 ,v2 )

.

(16.76)

Using the first equation (16.74), we put the right hand side of (16.76) in the final form: J (v1× ) − J (v2× ) ≥ −

1 (Av2× , v1× − v2× ). z 2×

(16.77)

Substituting (16.77) in (16.75) and using the second equations (16.73) and (16.74) we have 1 (Av2× , v1× − v2× ) z 2× 1 1 1 = 2E(z 1× − z 2× ) + × (Av2× , v2× ) − × (Av2× , v1× ) = 2E z 1× − × (Av2× , v1× ) z2 z2 z 2 1 1 ≥ 2E z 1× − × α(Av1× , v1× ) + (Av2× , v2× ) α 2 z 2 1 1 2 2 = 2E z 1× − × α2 z 1× E + 2 z 2× E = 2E z 1× − 2E z 1× . α 2 z 2 S(E, z 1× , v1× ) − S(E, z 2× , v2× ) ≥ 2E(z 1× − z 2× ) −

Here we chose the optimum value of an arbitrary positive constant α : α = z 2× /z 1× . So, we arrive at (16.72). Entropy is negative. We are going to show that for any solution of (16.71), z × , v × , the stationary value of the entropy functional, S(E, z × , v × ), is negative or zero: S(E, z × , v × ) = Eβ +

1 × (Av × , v × ) + ln Me−(l0 ,v ) ≤ 0. 2β

Therefore, S(E) ≤ 0.

(16.78)

778

16


Indeed, according to the second equation (16.71): S(E, z × , v × ) =

1 1 × × (Av × , v × ) + ln Me−(l0 ,v ) = Me−(l0 ,v ) (l0 , v × ) z× Me−(l0 ,v× ) × + ln Me−(l0 ,v ) .

We can rewrite this formula as d J (tv × ) S(E, z , v ) = J (v ) − . dt t=1 ×

On the other hand,

×

×

J (v ) − J (0) =

×

0

1

d J (tv × ) dt. dt

(16.79)

(16.80)

Function J (tv × ) is a convex function of t, therefore d 2 J (tv × ) ≥ 0, dt 2 and d J (tv × )/dt is an increasing function of t. Hence, d J (tv × ) d J (tv × ) ≤ . dt dt t=1

We can increase the right hand side of (16.80) by replacing d J (tv × )/dt with a bigger number, d J (tv × )/dt t=1 . Then d J (tv × ) J (v × ) − J (0) ≤ . (16.81) dt t=1

Since J (0) = 0, (16.78) follows from (16.79) and (16.81). ¯ The coldness, β = d S(E)/d E, Entropy reaches the maximum value at E = E. is equal to zero at the point of maximum of entropy. To find the stationary points, which correspond to β = 0, we cannot set β = 0 in (16.71) due to singularity at β = 0. Therefore, we use the system of equations (16.64) and (16.66). Setting β = 0 in (16.64) we obtain for the stationary point the equation ¯ Au × = l,

i.e. u × coincide with the minimizer of the variational problem (16.61), and, as was mentioned, has the meaning of the averaged minimizer of the original variational ¯ problem, A−1l. Entropy functional S(E, z, u) is equal to zero for z = 0, and therefore entropy, ¯ Let us show that entropy S(E) is concave in the vicinity of the S(E), is zero at E= E.

16.5


779

¯ To this end we expand the entropy functional S(E, z, u) in the vicinity point E= E. ¯ keeping only the terms up to the second order: of the point z = 0, u = u, ¯ u) ¯ + 1 z 2 M(l0 , u) ¯ 2 − (l, ¯ 2 . S(E, z, u¯ + u ′ ) = (E − E)z 2

The linear terms over u ′ cancel out because u¯ is a stationary point at z = 0. At the stationary point over z the entropy functional is equal to S(E) = −

¯ 2 (E − E) 1 , ¯ u) 2 M(l0 , u) ¯ 2 − (l, ¯ 2

and S(E) is indeed concave. Asa by-product of this consideration, we found the ¯ u) ¯ 2 − (l, ¯ 2 /N . variance of energy: it is equal to M(l0 , u)

Derivation of (16.62) and (16.63) for positive temperatures. We split the derivation of formulas (16.62) and (16.63) into two parts considering first the case of positive temperatures. Let, for a chosen value of E, the functional S(E, z, v) have a stationary point, (β, v × ), over real z and real v, and β > 0. It was shown that this point is the point of minimum, and therefore we write vˇ instead of v × . Consider (16.51): 1 f N (E) = 2πi

i∞

e E z Me z minu I N (u) dz.

−i∞

Using (5.328), we have i∞

1 f N (E) = 2πi

1

e E z e 2z (Av,v)−(l,v) D 1 A vdz z

−i∞

1 = 2πi

i∞

e E z+ 2z (Av,v) Me− N

i∞

N 1 1 e E z+ 2z (Av,v) Me− N (l0 (r ),v) D 1 A vdz.

−i∞

1 = 2πi

−i∞

1

1

/N

a=1 (l0 (ra ),v)

D 1 A vdz z

z

(16.82)

N 1 ¯ by its limit value for N → ∞, e−(l,v) , then the If we replace Me− N (l0 (r ),v) integrals in (16.82) are computed explicitly and we obtain ¯ as N → ∞. f N (E) → δ(E − E)

780

16


This is an indication that E deviates slightly from E¯ for most events. To obtain a better approximation, we scale the variables z, v in (16.82), z → N z, v → N v, to put (16.82) in the form N f N (E) = 2πi

i∞

e N S(E,z,v) D Nz A vdz.

(16.83)

−i∞

In this functional integral, the “volume element” D Nz A v is such that i∞

N

e 2z (Av,v) D Nz A v = 1.

−i∞

We seek the limit behavior of this integral as N → ∞. We move the integration over imaginary functions v and imaginary z to the integration over the “line,” z = β + i y, v = vˇ + iv ′ . For small y and v ′ consider the functional S(y, v ′ ) = S(E, β + y, vˇ + v ′ ) − S(E, β, vˇ ). In this functional we keep the terms up to the second order. The functional is quadratic over y and v ′ : it does not contain the linear terms because (β, vˇ ) is a stationary point. The quadratic functional S(y, v ′ ) is positive as (β, vˇ ) is the point of minimum. Therefore, the quadratic functional S(i y, iv ′ ) is real and negative. Thus, Re S(E, β + i y, vˇ + iv ′ ) − S(E, β, vˇ ) ≥ 0 for small y and v ′ , i.e. Re S(E, β + i y, vˇ + iv ′ ) has a local maximum on the “line,” z = β + i y, v = vˇ + iv ′ at y = v ′ = 0. In fact, Re S(E, β + i y, vˇ + iv ′ ) has a global maximum at y = v ′ = 0. Indeed, 1 (A(ˇv + iv ′ ), vˇ + iv ′ ) Re S(E, β + i y, vˇ + iv ′ ) = Re E(β + i y) + 2(β + i y) @ ′ + ln Me−(l0 ,ˇv+iv ) , and S(E, β, vˇ ) = Eβ +

1 (Avˇ , vˇ ) + ln Me−(l0 ,ˇv) . 2β

Therefore, Re S(E, β + i y, vˇ + iv ′ ) − S(E, β, vˇ ) = 1 0 β (Avˇ , vˇ ) − (Av ′ , v ′ ) + 2y(Avˇ , v ′ ) 1 −(l0 ,ˇv+iv′ ) Me − (A v ˇ , v ˇ ) + ln = 2(β 2 + y 2 ) 2β − ln Me−(l0 ,ˇv) . (16.84)

16.5


781

˜ by the following relation: Let us introduce an auxiliary probabilistic measure, M, for any random variable, ϕ(r ), e−(l0 (r ),ˇv) ˜ Mϕ(r )=M ϕ(r ). Me−(l0 ,ˇv) It differs from the original measure by a factor in probability density: e−(l0 (r ),ˇv) . Me−(l0 ,ˇv) The auxiliary measure is probabilistic, i.e. ˜ M[1] = 1. We can rewrite (16.84) as ′ Re S(E, β+i y, v+iv ˇ )−S(E, β, vˇ ) = −

(A(βv ′ − y vˇ ), βv ′ − y vˇ ) ˜ −i(l0 ,v′ ) Me +ln . 2β(β 2 + y 2 )

Since the quadratic functional ( A(βv ′ − y vˇ ), βv ′ − y vˇ ) is positive, and ˜ −i(l0 ,v′ ) ˜ = 0, ˜ e−i(l0 ,v′ ) = ln M1 ln Me ≤ ln M

Re S(E, β + i y, vˇ + iv ′ ) has maximum for y = v ′ = 0, and, at least for a finite-dimensional truncation, the asymptotics (16.62) and (16.63) hold. If functionals are “good enough,” they also hold for an infinite-dimensional case. The term “good enough” still needs to be spelled out; some first results in this direction are cited in Bibliographical Comments. What we described here is, in fact, the steepest descent method in functional spaces. Derivation of (16.62) and (16.63) for negative temperatures. If, for a chosen value of E, β < 0, then the above reasoning does not work because the relation we have started with, (16.82), holds only for Re z ≥ 0, and we cannot move the line of integration over z from imaginary axis into the right half-plane. To proceed in this case we use formula (5.329), e

z min I (u,ω) u

=

∞

1

e z[ 2 (Au,u)−(l(ω),u)] D−z A u,

(16.85)

−∞

in which integration is conducted over real u and, thus, holding for Re z < 0. To avoid the issue of possibility to move the line of integration over z, we consider not the probability density function of energy but probability distribution: FN (E) = M[− min I N (u, ω) ≤ E] = u

E

−∞

f N (E ′ )d E ′ .

782

16


Then probability density function is obtained by differentiation: f N (E) =

d FN (E) . dE

Probability distribution can be written in terms of the step function, θ (E) =

I if E 0 , if E < 0

6

1 0

as FN (E) = Mθ [E + min I N (u, ω)].

(16.86)

u

The step function has the presentation7 1 θ (E) = 2πi

a+i∞

eEz

dz , z

a > 0,

a−i∞

where the integral is taken over any line in the right half-plane [a − i∞, a + i∞], a > 0. The line of integration can be moved to the left half-plane. When passing the singularity z = 0, the residue of the integrand is added, and we have for the integration line in the left half-plane:

θ (E) = 1 +

1 2πi

a+i∞

eEz

dz , z

a < 0.

a−i∞

We set here a = β/N and plug in (16.86):

7

The integral does not converge absolutely, and we rectify its meaning as θ(E) = lim

b→∞

1 2πi

a+ib

eEz

dz . z

a−ib

Note that in such a presentation θ(0) = 1/2. Indeed, lim

b→∞

1 2πi

a+ib

a−ib

1 1 dz a + ib b 1 = lim ln = lim 2Arc tan = . b→∞ 2πi z a − ib b→∞ 2π a 2

We use this presentation in combination with integrals over the probabilistic space, . Therefore, we obtain the probability distribution correctly if the measure of the event, E + min I N (u, ω) = 0, u

is zero. If it is not zero, then the corresponding correction must be made.

16.5


1 FN (E) = 1 + M 2πi

β/N +i∞

e

783

E z+z min I N (u,ω) dz u

z

(16.87)

.

β/N −i∞

Now using (16.85) we obtain 1 FN (E) = 1 + M 2πi

β/N +i∞ ∞

1

e E z e z[ 2 (Au,u)−(l(ω),u)] D−z A u

dz z

β/N −i∞ −∞

= 1+

1 2πi

β/N +i∞ ∞

β/N −i∞ −∞

N dz 1 z e E z+z 2 (Au,u) Me− N (l0 (r ),u) D−z A u . z

Scaling z , z → N z, we finally have 1 FN (E) = 1 + 2πi

β+i∞ ∞

e N S(E,z,u) D−N z A u

dz , z

β−i∞ −∞

where S(E, z, u) is the entropy functional. We consider the asymptotics as N → ∞. At the stationary point, β, u × , the entropy functional has minimum over z and maximum over u. We assume that the following inequality holds for all u ′ = 0 : ′

˜ −β(l0 ,u ) + β(Au × , u ′ ) < ln Me

|β| (Au ′ , u ′ ), 2

(16.88)

˜ is the auxiliary probabilistic measure where M ×

−β(l0 (r ),u ) ˜ =Me . M Me−β(l0 ,u × )

Note that the Euler equation (16.64) in terms of the auxiliary measure is written simply as ˜ 0. Au × = Ml

(16.89)

˜ 0 , u ′ ), and the left hand side of (16.88) for small u ′ is Therefore, ( Au × , u ′ ) = M(l ′ quadratic over u : 1 2˜ ′ 2 −β(l0 ,u ′ ) × ′ ′ ˜ ˜ ln Me + β(Au , u ) ≈ ln 1 − β M(l0 , u ) + β M(l0 , u ) + β(Au × , u ′ ) 2 0 1 @ 1 2? ˜ ˜ 0, u′) 2 . ≈ β M(l0 , u ′ )2 − M(l 2

784

16


Therefore, inequality (5.11.28) yields that |β| should not exceed the squared minimum eigenvalue λ: λ2 = min ′ u

(Au ′ , u ′ ) 1 . 0 ˜ 0, u′) 2 ˜ 0 , u ′ )2 − M(l M(l

It follows from (16.88) that Re S(E, β + i y, u × + u ′ ) has maximum over y, u ′ : Re S(E, β + i y, u × + u ′ ) =

β + iy × ′ (A(u × + u ′ ), u × + u ′ ) + ln Me−(β+i y)(l0 ,u +u ) 2 β = Eβ + (Au × , u × ) + 2(Au × , u ′ ) + (Au ′ , u ′ ) 2 × ′ × ′ + ln Me−β(l0 ,u ) e−β(l0 ,u ) e−i y(l0 ,u +u ) β ˜ −β(l0 ,u ′ ) −i y(l0 ,u × +u ′ ) = S(E, β, u × ) + β(Au × , u ′ ) + (Au ′ , u ′ ) + ln Me e 2 ˜ −β(l0 ,u ′ ) + β(Au × , u ′ ) − |β| (Au ′ , u ′ ) > S(E, β, u × ). ≥ S(E, β, u × ) + ln Me 2

= Re E(β + i y) +

Hence, at least for a finite-dimensional truncation, the asymptotics follows from the Laplace reasoning. The dependence of the volume element, D−N z A u, on N does not affect the asymptotics because ∞

1

e N z 2 (Au,u) D−N z A u = 1.

−∞

So, FN (E) ∼ 1 − ce N S(E) ,

(16.90)

with a positive constant c (the sign minus appears because z in the denominator in (16.87) should be set equal to β, which is negative). Differentiating (16.90) and taking into account that d S(E)/d E is negative, we obtain the required asymptotics (16.63). Meaning of stationary points of entropy functional. Let the event r be a point of some multi-dimensional region and have a probability density function f (r ). We are going to show that the probability of the event { r1 belongs to a small vicinity of a point, r ∗ , of volume r } under the condition that energy takes a prescribed value, E, is ∗

×

e−β(l0 (r ),u ) Probability = f (r ∗ )r, Me−β(l0 ,u × )

(16.91)

16.5


785

˜ An immediate consequence of i.e. the corresponding probability measure is M. that is the interpretation of the solution of the Euler equation for stationary points (16.89): u × is the averaged minimizer of the original variational problem under the condition that energy takes a given value, E. To prove (16.91) we note that for any function, ϕ(r1 , . . . , r N ) its averaged value is Mϕ =

ϕ(r1 , . . . , r N ) f (r1 ) . . . f (r N )dr1 . . . dr N .

The conditional average under condition that E(r1 , . . . , r N ) = E is ϕ(r1 , . . . , r N )δ(E − E(r1 , . . . , r N )) f (r1 ) . . . f (r N )dr1 . . . dr N . Mϕ = δ(E − E(r1 , . . . , r N )) f (r1 ) . . . f (r N )dr1 . . . dr N

The denominator here is f N (E). Taking ϕ = ϕ(r1 ) and noting that for any such ϕ, Mϕ =

ϕ(r1 ) ˜f (r1 | E)dr1 ,

we have ˜f (r1 | E) =

δ(E − E(r1 , r2 . . . , r N )) f (r2 ) . . . f (r N )dr2 . . . dr N f (r1 ). f N (E)

Let temperature be positive. To compute the numerator, we present it in the form 1 2πi

i∞

−i∞

N = 2πi

N −1 1 1 1 e E z+ 2z (Av,v)− N (l0 (r1 ),v) Me− N (l0 (r ),v) D 1 A vdz = z

i∞

1

e N [E z+ 2z (Av,v)+

N −1 N

ln Me−(l0 (r ),v) ] −(l0 (r1 ),v)

e

D Nz A vdz.

−i∞

Its asymptotics is × × e−(l0 (r1 ),v ) f N (E) = e−β (l0 (r1 ),u ) f N (E),

˜ The justification for negative and therefore the statement made holds, and M = M. temperatures is similar. In the same way one can show that the joint probability density function, ˜f (r1 , r2 | E), of the event (r1 , r2 ) under the condition that energy takes a prescribed value E is ˜f (r1 , r2 | E) = ˜f (r1 | E) ˜f (r2 | E) 1 + o 1 , N

786

16


i.e., in the leading approximation, r1 and r2 are independent random variables distributed with the probability density, ˜f (r | E). Prefactor. There is an exact relation f N (E) = R e N S(E) where the prefactor R is ′ × Nβ ˜ e− 2 (A−1 l ′ ,l ′ ) δ 1 (A−1l ′ , l ′ ) + (l , v ) . R=M 2 β ˜ ). This relation can be used for the calculation of the Here l ′ (r ) = l(r ) − Ml(r prefactor. The proof can be found in [48]. A generalization. All previous results are extended to the case when the linear functional is a sum of different random functionals: (l(ω), u) =

N 1 (la (ra ), u) . N a=1

In particular, the entropy functional becomes N 1 z ln Me−z(la ,u) , S(E, z, u) = E z + (Au,u) + 2 N a=1

(16.92)

while the probability density f a (r | E) to find ra at a point r when the value of energy E is given, is ×

e−β(la (r ),u ) f (r ). f a (r | E) = Me−β(la ,u × )

(16.93)

Example 3. Vortex gas. Consider the functional (16.46). / N We assume that the vortex intensity is small while the total vorticity, γ = a=1 γa , is finite. So we set γa = σa /N , σa ∼ γ . The functional takes the form 1 I (u, r1 , . . . , r N ) = 2

V

N 1 σa u(ra ). u ,α u ,α + ε2 u,αβ u ,αβ d V − N a=1

Its minimum value is the Hamiltonian H (r1 , . . . , r N ) of a dynamical system of N vortices (see Sect. 9.7). As we have seen in Chap. 2, a key role in statistical mechanics of Hamiltonian systems is played by the phase volume

16.5


(E) =

... V

V

787

θ (E − H (r1 , . . . , r N ))dr1 . . . dr N .

(16.94)

The phase volume can be interpreted in probabilistic terms: dividing (16.94) by the area, |V | , of the flow region to the power N , (E) = |V | N

... V

V

θ (E − H (r1 , . . . , r N ))

dr N dr1 ... , |V | |V |

we see that (E)/ |V | N is the probability distribution of energy if vortex positions r1 , . . . , r N are considered as identically distributed independent random variables with a constant probability density, 1/ |V | . Therefore, the relations of this section form the basis of statistical mechanics of point vortices. Let us specify these relations for the case of vortex gas. The entropy functional of a vortex gas is z S(E, z, u) = E z + 2

V

N dr 1 ,α 2 ,αβ . ln e−zσa u(r ) u ,α u + ε u,αβ u dV + |V | N a=1 V

The term with high derivatives ε2 u,αβ u ,αβ does not affect the smooth stationary points of this functional and can be dropped. Then the equations for the stationary points are N 1 σa e−βσa u(r ) u = − , N a=1 V e−βσa u(r ) dr 1 u ,α u ,α d V = E. 2 V

u|V = 0,

(16.95) (16.96)

As follows from (16.93), the probability of finding the ath vortex at a point r is f a (r | E) =

e−βσa u(r ) . −βσa u(r ) dr V e

(16.97)

Therefore, the right hand side / Nof (16.95) can be interpreted as the average value of (negative) instant vorticity, a=1 γa δ(x − ra ). Thus, the solution of (16.95) is the averaged stream function, and its derivatives determine the averaged velocity field of the fluid. In several cases, (16.95) can be solved analytically. Consider one example. Let V be a circular domain of radius R, and all vortices have the same intensity, σ/N . One can check by direct inspection that the solution is u=

r2 βσ 2 2 1− 2 , ln 1 + βσ 8π R

788

16


Fig. 16.2 Dependence of entropy on dimensionless energy for gas of vortices of equal intensity in a circle

where r is the distance from the current point to the circle center. Energy E is linked to β by the relation 8π 1 βσ 2 1− . E= ln 1 + β βσ 2 8π

(16.98)

Plugging these formulas into the entropy functional, we find entropy as a function of β : βσ 2 16π . ln 1 + S(β) = 2 − 1 + βσ 2 8π

(16.99)

Formulas (16.98) and (16.99) define parametrically the dependence of entropy on energy. It is shown in Fig. 16.2 for dimensionless energy E ∗ = 8π E/σ 2 . The dependence of dimensionless inverse temperature, β ∗ = βσ 2 /8π, on dimensionless

Fig. 16.3 Dependence of dimensionless inverse temperature on dimensionless energy for gas of vortices of equal intensities in a circle

16.6

Probability Distribution of Energy: Large Excitations

789

energy is shown in Fig. 16.3. It is seen that the range of negative β is bounded: β ∗ ≥ −1.

16.6 Probability Distribution of Energy: Large Excitations If E¯ decreases and goes to zero, the branch of positive temperatures disappears. This happens when the linear functional (l0 (r ), u) has zero mean value. Indeed, let us write down energy in terms of the linear functional explicitly. Since uˇ = A−1l, we have E=

1 −1 1 1 ˇ u) ˇ = (l, A−1l) = A l0 (ra ), l0 (rb ) . (Au, 2 2 2 2N a,b

(16.100)

Hence, the averaged energy is8

N 1 1 −1 ME= M A l0 (ra ), l0 (rb ) = M A−1 l0 (ra ), l0 (ra ) 2 2 2N 2N a,b a=1 ⎞ 1 ¯ l) ¯ . N M(A−1l0 , l0 ) + N (N − 1)(A−1l, A−1 l0 (ra ), l0 (rb ) ⎠ = + 2 2N a=b

(16.101)

8

Note a useful version of (16.101): if (l, u) =

N (lk , u), k=1

where lk are statistically independent and have zero mean value, Mlk = 0,

Mlk lm = 0

for k = m,

then N

1 ˇ u) ˇ = M E = M (Au, M Ek , 2 k=1

1 −E k = min[ (Au, u) − (lk , u)]. u 2 It is proved by the direct inspection: since uˇ = A−1

N

lk ,

k=1

we have N N N 1 1 −1 1 −1 ˇ u) ˇ = M( M (Au, lk , A lm ) = (A lk , lk ). 2 2 2 k=1 m=1 k=1

790

16


Fig. 16.4 Dependence of entropy on energy for small excitations when Ml0 = 0

If l¯ = 0, the last term in (16.101) dominates and, in the limit N → ∞, averaged energy M E coincides with E¯ : 1 ¯ l). ¯ M E = E¯ = (A−1l, 2 If l¯ = 0, then the last term in (16.101) vanishes, and ME =

1 M(A−1l0 , l0 ). 2N

(16.102)

We see that averaged energy, as it should be, is never zero, but becomes small in the order of N −1 and tends to zero as N tends to infinity. This yields the disappearance of the range of positive temperatures in the limit N → ∞. A typical graph of the dependence of entropy on energy in this case is shown in Fig. 16.4. Negativeness of temperature when l¯ = 0 can be seen from the Euler equation of the entropy functional (16.64), (16.66): u × = 0 is a solution of this equation if l¯ = 0; for positive z, the entropy functional is convex and u × = 0 is its only stationary point; however, u × = 0 does not satisfy to (16.66). Therefore, the stationary point must have negative temperature. The range of positive temperatures is quite important in some applications. In order to study this range, we have to rescale energy, multiplying it, as follows from (16.102), by N . Therefore, we have to study the minimization problem for the functional ! N N 1 1 N (Au, u) − (l0 (ra ), u). (l0 (ra ), u) = (Au, u) − N 2 N a=1 2 a=1

16.6


791

√ After the change of variable, u → u/ N , the quadratic part of the functional becomes independent of N , while the linear functional takes the form of the functional of large excitations (16.25). To emphasize in further relations that they are valid when Ml0 = 0, we write l0′ instead of l0 , which is equal to l0′ in this case. Probability density of energy of large excitations can also be obtained explicitly. This is done similarly to the case of large excitations. Let us introduce a positive operator B by the formula (Bu, u) = M(l0′ (r ), u)2 , and consider the eigenvalue problem, (16.103)

Aϕ = μBϕ.

Let the eigenfunctions of this problem form a basis in the functional space. Quadratic forms (Au, u) and (Bu, u) are diagonal in this basis: (Au, u) =

∞

λk u 2k

(Bu, u) =

,

k=1

∞

bk u 2k .

k=1

We define an analytic function of complex variable z: (z) = ln

∞ H i=1

zbk 1+ λk

.

(16.104)

Here it is assumed that 0 ≤ b1 /λ1 ≤ b2 /λ2 ≤ · · · and ∞ bk < ∞. λ k=1 k

Denote by ˆf N (z) Laplace’s transformation of probability density of energy: 1 f N (E) = 2πi

i∞

e E z ˆf N (z) dz.

−i∞

Then 1 lim ˆf N (z) = e− 2 (z) ,

N →∞

where (z) is the function (16.104). The derivation proceeds as follows:

(16.105)

792

16


i∞

1 f N (E) = Mδ E + min I N (u, ω) = u 2πi

=

1 2πi

i∞

eEz M

−i∞

1 = 2πi

i∞

e E z Me

z min I N (u,ω) u

dz =

−i∞

1

e− 2 (Au,u)+i

√

z(l(ω),u)

1

e E z e− 2 (Au,u)+N ln Me

i

D A u dz =

√z

′ N (l0 (r ),u)

D A u dz.

−i∞

Changing the function N ln M ei

√z

N

(l0′ (r ),u)

by its limit value as N → ∞, z z − M(l0′ (r ), u)2 = − (Bu,u) , 2 2 we find 1 f (E) = lim f N (E) = N →∞ 2πi

i∞

e

Ez

lim

m→∞

−i∞

det Am dz, det(A + z B)m

where Am and (A + z B)m are m-dimensional truncations of the operators A and A + z B. This formula yields (16.105) if all changes of the order of limit procedures made in this derivation are legitimate. Note a straightforward generalization: if the linear functional contains a deterministic part I (u, ω) =

N 1 ¯ u) − √1 (Au,u) − (l, (l0′ (ra ), u), 2 N a=1

then the probability density function of minimum value becomes 1 f (E) = 2πi

i∞

1

1

−1 ¯

e E z− 2 (z)− 2 ((A+z B)

¯ l,l)

dz.

−i∞

Example 4. Energy distribution in neutral vortex gas. Point vortices are thrown randomly and statistically independently into a square cell, C, and their positions are periodically duplicated over the plane. Point vortices generate some fluid flow. The kinetic energy of the flow per cell depends on the vortex positions and, thus,

16.6


793

is random. We wish to find the probability density of energy. As was mentioned, knowing this function one can determine the phase volume and other thermodynamic characteristics of the vortex gas. The true stream function of the flow is the minimizer of the functional (16.46) on the set of periodic functions. The functional (16.46) is not bounded below for arbitrary set of vortex intensities/γa : shift of N γa = 0. u(x) for a constant can pull the value of the functional to −∞, if a=1 Therefore, for correctness of the problem it is necessary to set the vortex gas to be neutral: N a=1

γa = 0.

One can show that this condition is also sufficient for the boundedness of the functional from below. We accept for simplicity that there is an equal number √ of positive and negative point vortices of the same intensity γ . By setting γ = σ/ N we make this problem a problem of large excitations. Denote by r1+ , . . . , r N+ and r1− , . . . , r N− the positions of positive and negative vortices, respectively. Then r1 is a pair (r1+ , r1− ) and (l0 (r1 ), u) = (l0 (r1+ , r1− ), u) = σ (u(r1+ ) − u(r1− )).

(16.106)

To use the functional integral presentation we need to have a non-degenerated functional, i.e. a functional without a kernel. The only degeneration in the problem under consideration is caused by the shifts of u(x) for a constant. To eliminate the kernel we impose an additional constraint on admissible functions,

u = 0,

(16.107)

where · denotes the space averaging over the cell: for any ϕ,

ϕ ≡

1 L2

ϕ d 2 x,

C

L being the size of the cell. The probability density of energy can be found explicitly. Indeed, for the linear functional l0 (16.106), (Bu,u) = σ 2 M(u(r + ) − u(r − ))2 = 2σ 2 u 2 . Here we used the fact that computing of mathematical expectation is equivalent to averaging over the cell, M(·) = ·, and we also took into account the condition (16.107). The eigenvalue problem (16.103) now reads − ϕ + ε2 2 ϕ = λ

2σ 2 ϕ. L2

(16.108)

794

16


It is convenient to make this equation dimensionless by introducing dimensionless coordinates and a dimensionless parameter ǫ¯ according to x¯ =

x , L

ǫ¯ =

ε . L

Then (16.108) takes the form ¯ + ǫ¯ 2 ¯ 2 ϕ = λ2σ 2 ϕ. − ϕ

(16.109)

Here ¯ is Laplace’s operator in dimensionless x¯ -coordinates. The periodic solutions of (16.109) are ϕ = aei2πk·x¯ ,

k ∈ Z′2 .

(16.110)

We denote by Z′2 the 2D square lattice with unit spacing, in which the point k = (0, 0) is excluded. Substituting (16.110) into (16.108) we obtain |k|2 + ǫ¯ 2 4π 2 |k|4 = λk

σ2 . 2π 2

Each point of the lattice Z′2 corresponds to one real eigenfunction. We may set this correspondence, for example, in the following way. Consider a point (k1 , k2 ) in the first quadrant, k1 > 0, k2 > 0 and the associate points, (k1 , −k2 ), (−k1 , −k2 ), (−k1 , k2 ), in three other quadrants that give the same eigenvalue, λk . We set ϕ1 = a1 sin 2π (k1 x¯ 1 + k2 x¯ 2 ) ϕ2 = a2 cos 2π (k1 x¯ 1 − k2 x¯ 2 ) ϕ3 = a3 cos 2π (k1 x¯ 1 + k2 x¯ 2 ) ϕ4 = a4 sin 2π (k1 x¯ 1 − k2 x¯ 2 )

for k = (k1 , k2 ),

for k = (k1 , −k2 ), for k = (−k1 , −k2 ),

for k = (−k1 , k2 ).

For k1 = 0, k2 = 0 we put ϕ1 = a1 sin 2π k2 x¯ 2 for k2 > 0 and ϕ2 = a2 cos 2π k2 x¯ 2 for k2 < 0. Similarly, we put ϕ1 = a1 sin 2π k1 x¯ 1 for k1 > 0, k2 = 0 and ϕ2 = a2 cos 2π k1 x¯ 1 for k1 < 0, k2 = 0. The coefficients ai of the eigenfunctions are chosen from the normalization condition ϕi2 = 1. Introducing the quantities ǫ = 2π ǫ¯ ,

e0 =

γ2 , 2π 2

λ¯ k = λk e0 ,

we get for λ¯ k a simple equation: λ¯ k = |k|2 + ǫ 2 |k|4 .

16.6


795

The function (z) and the probability density of the energy f (E) depend on the parameter ǫ. To emphasize this in our notation we attach the label ǫ to each of these functions and write ǫ (z) and f ǫ (E), respectively. Then f ǫ (E) =

1 2πi

i∞

ˆ

e E z−1/2ǫ (z) dz.

(16.111)

−i∞

For the function ǫ (z) we have ǫ (z) =

k∈Z′2

ln 1 +

z |k|2 + ǫ 2 |k|4

.

(16.112)

One can show (see the details in [174]) that, if ǫ → 0, the function ǫ (z) in (16.111) can be replaced by the function ε (z) ∼ φ0 (z) + E 1 z,

(16.113)

where φ0 (z) does not depend on ǫ: z z ln 1 + 2 − 2 , φ0 (z) = |k| |k| ′

(16.114)

k∈Z2

and E 1 corresponds to the self-energy of vortices: E1 = 4

1 1 =4 . 2 + ǫ 2 |k|4 ¯k |k| 4 λ ′ k

(16.115)

k∈Z2

The self-energy depends on the value of the parameter ǫ, and tends to infinity as ǫ → 0. More precisely, i∞

e

E z−1/2ǫ (z)

dz −

−i∞

i∞

e E z−1/2φ0 (z)−E1 z dz → 0

as ǫ → 0

(16.116)

−i∞

if the interaction energy, E ′ = E − E 1 , is kept fixed. In other words, the probability density function of the interaction energy f ǫ (E ′ ) has the limit as ǫ → 0, and this limit is equal to 1 f (E ) = 2πi ′

i∞

−i∞

e

E ′ z−1/2φ0 (z)

1 dz = 2π

∞

−∞

′

ei E y−1/2φ0 (i y) dy.

(16.117)

796

16


Fig. 16.5 Probability density function of vortex interaction energy obtained analytically (solid line) and by numerical experiments (dots)

The computation of the function φ0 (z) can be conducted in the following way. Since the series (16.114) converges slowly, one can calculate φ0 (z) by summing up the terms with k inside a circle of radius ̺ (|k| < ̺) and approximating the rest of the series by a double integral: z z z z ln 1 + 2 − 2 d 2 k. ln 1 + 2 − 2 ≈ |k| |k| |k| |k| |k|>̺ |k|>̺

This integral can be found exactly. It is equal to

π

∞ ̺2

? z z@ ̺2 ln 1 + − dt = π z + ̺2 ln . + z t t z + ̺2

The convergence is fast, and for ̺ > 3 one obtains practically the limit function, φ0 (z). The graph of f (E ′ ) calculated according to (16.117) is shown in Fig. 16.5. To obtain f (E) we have to shift this function to the right by E 1 . Note that the mean value of E ′ is zero (this is a consequence of the periodic boundary condition). In Fig. 16.5 we show function f (E ′ ); the dots are the results of numerical statistical experiments by Campbell and O’Neil [78], who casted 160 point vortices randomly and independently into a square box with periodic boundary conditions, evaluated the energy of each configuration, and computed probability density. Further details regarding this example can be found in [174].

16.7

Probability Distribution of Linear Functionals of Minimizers

797

16.7 Probability Distribution of Linear Functionals of Minimizers In applications, it often appears necessary to know the probability distribution of some functionals of minimizers. In case of Gaussian excitation and small and large excitations considered above, the probability distribution can be found analytically. Gaussian excitation. Let uˇ be the minimizer of the variational problem 1 (Au,u) − (l (ω) , u) → min, u 2

I (u, ω) =

where the linear functional, (l (ω) , u) , is Gaussian, i.e. for any z and u, ¯

1 2

Me z[(l(ω),u)−(l,u )] = e 2 z

(Bu,u)

(16.118)

.

We wish to find the probability density function of a deterministic linear functional, ˇ = (L , u): ˇ L(u) ˇ f (X ) = Mδ(X − (L , u)). ˇ = (L , A−1 l(ω)) = (A−1 L , l(ω)), it is convenient to Since uˇ = A−1l(ω) and (L , u) introduce an element, u 0 , which is the solution of the equation (16.119)

Au 0 = L , i.e. u 0 = A−1 L . This element is the minimizer in the variational problem 1 (Au,u) − (L , u) → min . u 2 Since ˇ = (l(ω), u 0 ), (L , u)

(16.120)

we have for the probability density function, 1 f (X ) = M 2πi

i∞

e

ˇ X z−z(L ,u)

1 dz = M 2πi

−i∞

=

1 2πi

i∞

−i∞

i∞

−i∞ ¯

′

e(X −(l,u 0 ))z Me−z(l (ω),u 0 ) dz,

e X z−z(l(ω),u 0 ) dz

798

16


¯ According to (16.118), where l ′ (ω) = l(ω) − l. 1 2

′

Me−z(l (ω),u 0 ) = e 2 z

(Bu 0 ,u 0 )

,

and therefore, 1 f (X ) == 2πi

i∞

¯

1 2

e(X −(l,u 0 ))z+ 2 z

(Bu 0 ,u 0 )

−i∞

¯ ))2 (X −(l,u 0 1 − dz = √ e 2(Bu0 ,u0 ) , 2π (Bu 0 , u 0 )

ˇ is i.e. it is Gaussian. The variance of the random variable, (L , u),

√ (Bu 0 , u 0 ).

Example 5. Consider a variational problem (16.55)

a 0

1 du 2 − g(x, ω)u(x) d x → 2 dx

min

u(x): u(0)=u(a)=0

(16.121)

,

with g(x, ω) being a δ-correlated Gaussian random field with zero mean, Me

a 0

g(x,ω)u(x)d x

1

= e2

a

Bu 2 (x)d x

0

(16.122)

B = const.

,

The minimizer is du(x, ω) =− dx

x 0

g(x, ω)d x +

1 a

a

dx 0

x′

g(x ′ , ω)d x ′ .

0

Let us find the probability density function of uˇ at some point x0 . The local value of the minimizer is selected by the linear functional (L , u) = u(x0 ) =

0

a

δ(x − x0 )u(x)d x.

The function u 0 is the minimizer in the deterministic variational problem

0

a

1 du 2 − δ(x − x0 )u(x) d x → 2 dx

min

u(x): u(0)=u(a)=0

Thus, u 0 , is the solution of the boundary value problem d 2u0 = −δ(x − x0 ), dx2 The solution is

u 0 (0) = u 0 (a) = 0.

.

16.7

Probability Distribution of Linear Functionals of Minimizers

u 0 (x) =

a−x0 x a x0 (a − a

799

x ≤ x0 x ≥ x0 .

x)

(16.123)

Therefore, (Bu 0 , u 0 ) = B

a

u 20 (x)d x =

0

B (x0 )2 (a − x0 )2 . 3a

ˇ 0 ) has a Gaussian distribution with zero mean and the variance So, u(x √ B/3ax0 (a − x0 ) . The variance vanishes at the clamped ends and grows quadratically away from the ends. It reaches the maximum value at the center of the segment. ˇ Then the linear Consider now the probability distribution of the derivatives of u. functional (L , u) should be taken as du(x0 ) = (L , u) = dx

a 0

δ(x − x0 )

du(x) d x. dx

The function u 0 is the minimizer in the deterministic variational problem

0

a

du(x) 1 du 2 − δ(x − x0 ) dx → 2 dx dx

min

u(x): u(0)=u(a)=0

.

Hence, u 0 , is the solution of the boundary value problem d 2u0 = δ ′ (x − x0 ), dx2

u 0 (0) = u 0 (a) = 0,

where δ ′ (x) is the derivative of δ-function. The solution is the negative derivative of function (16.123):

u 0 (x) =

0 − a−x a

x0 a

x ≤ x0 x ≥ x0 .

Computing the variance (Bu 0 , u 0 ) = B

0

a

u 20 (x)d x =

B x0 (a − x0 ) , a

we find that the derivative of the minimizer √ at each point is a Gaussian random variable with zero mean and the variance Bx0 (a − x0 ) /a. Small excitations. Now let (l(ω), u) have the form

800

16

(l(ω), u) =


N 1 (la (ra ), u). N a=1

(16.124)

ˇ converges to a We are going to show that, as N → ∞, a linear functional (L , u) deterministic number, lim

N →∞

N 1 ¯ (la , u 0 ), N a=1

l¯a ≡ Mla (ra ),

(16.125)

where u 0 is the solution of (16.119). Indeed, uˇ = A

−1

N N 1 −1 1 la (ra ) = A la (ra ), N a=1 N a=1

and ˇ = (L , u)

N N 1 1 (L , A−1 la (ra )) = (la (ra ), u 0 ). N a=1 N a=1

(16.126)

ˇ is a sum of independent random variables. As N → ∞, such a We see that (L , u) sum converges to the arithmetical average of mathematical expectations (16.125). ˇ from its average value, The deviation of (L , u)

M

! N N 1 1 ¯ (la (ra ), u 0 ) = (la , u 0 ), N a=1 N a=1

is characterized by the variance N 1 ¯ ˇ − σ = M (L , u) (la , u 0 ) N a=1 2

!2

=M

N 1 ′ (l (ra ), u 0 ) N a=1 a

!2

.

Due to the independence of ra ,

M

N 1 ′ (l (ra ), u 0 ) N a=1 a

!2

=

N 1 2 σ , N 2 a=1 a

σa2 ≡ M(la′ (ra ), u 0 )2 .

Here we assume that the variance of each member is finite. If the series N 1 2 σ N a=1 a

(16.127)

16.8

Variational Principle for Probability Densities

801

ˇ tends to zero, and (L , u) ˇ tends to converges as N → ∞, then the variance of (L , u) its averaged value ( of course, this also happens under a weaker condition: the sum (16.127) is o(N )). ˇ under the condition that energy If we seek probability density function of (L , u) ˇ still converges to its averaged value; however takes a prescribed value; then (L , u) ˜ defined in Sect. 16.5. by M one should mean the probabilistic measure, M, Large excitations. For large excitations, N 1 ˇ = (l(ω), u 0 ) = √ (la (ra ), u 0 ). (L , u) N a=1

Here the averaged value of each member of the sum is zero: M(la (ra ), u) = 0. Then, if the sum N 1 (la (ra ), u 0 ) M √ N a=1

!2

=

N 1 M(la (ra ), u 0 )2 N a=1

ˇ is Gaussian random variable with zero mean and has a limit, σ 2 , as N → ∞, (L , u) the variance σ.

16.8 Variational Principle for Probability Densities In this Section we return to consideration of variational problems for general stochastic integral functionals: I (u, ω) =

V

L (x, a(x, ω), u,i ) d V → min . u

(16.128)

For simplicity, we assume that I is a functional of only one function, u(x), and do not impose boundary constraints on u(x); besides, a(x, ω) is also a scalar field. The generalizations will be made at the end of the section. Due to the applications to be considered in Chap. 18, the region V is assumed to be a region in three-dimensional space, though, in fact, all further relations have the same form for other dimensions. The variational problem (16.128) involves a random field, a(x, ω), which characterizes the physical properties of the continuum. A usual way to describe a random field is to prescribe its probability densities: one point probability density, f (x, a) = Mδ(a − a(x, ω)),

802

16


two-point probability density, f (x, a; x ′ , a ′ ) = Mδ(a − a(x, ω))δ(a ′ − a(x ′ , ω)), etc. The minimizer in the variational problem (16.128) is also a random field. It is desirable to get the solution of the variational problem also in terms of the probability densities of the minimizer. To this end, we have to reformulate the variational problem in terms of probability densities. Such a reformulation is the subject of this section. Some further applications to studying continua with random microstructures are given in Chap. 18. We begin with a brief excerpt from the theory of random fields. Probability densities of random fields. A random field, a(x, ω), is uniquely defined by an infinite set of n-point probability densities: f n (x1 , a1 ; . . . ; xn , an ) = Mδ(a1 − a(x1 , ω)) . . . δ(an − a(xn , ω)),

n = 1, 2, . . . (16.129)

If the arguments of the probability densities are explicitly mentioned, then there is no need in index n, and we write for the n-point probability density, f (x1 , a1 ; . . . ; xn , an ) . On the other hand, if the actual dependence on the arguments is not essential, we use for it the symbol f n . As follows from the definition (16.129) probability densities obey the compatibility conditions: (a) f (x1 , a1 ; . . . ; xn−1 , an−1 ) = f (x1 , a1 ; . . . ; xn−1 , an−1 ; xn , an ) dan (16.130)

(b) f n is symmetric with respect to the transposition of any couple of arguments, xi , ai and x j , a j (c) f (x1 , a1 ; . . . ; xn , an ) 0, n = 1, 2, . . . (d) f (x, a)da = 1 Kolmogorov’s theorem9 states that for any set of functions, f n (x1 , a1 ; . . . ; xn , an ) , n = 1, 2, . . . , satisfying the constraints (16.130), a random field, a(x, ω), exists for which these functions are the probability densities. According to (16.130a), the n-point probability density, f n , determines uniquely the probability densities f 1 , f 2 , . . . , f n−1 . Sometimes, one-point probability density can determine higher order functions. Though such a case is exceptional, we discuss it here because this situation will be encountered further. Let f 1 be a δ-function: f (x, a) = δ(a − a ∗ ). 9

See, e.g., [82].

16.8


803

According to (16.130a),

f (x1 , a1 ; x2 , a2 )da2 = δ(a1 − a ∗ ).

(16.131)

The right hand side of (16.131) is equal to zero for a1 = a ∗ . The integral of a nonnegative function, f 2 , can be equal to zero only if f 2 = 0. Hence, f (x1 , a1 ; x2 , a2 ) = 0 for a1 = a ∗ . By symmetry with respect to the transposition of the arguments (16.130)b, f (x1 , a1 ; x2 , a2 ) = 0 if a2 = a ∗ . Thus, the only function which satisfies (16.131) is f (x1 , a1 ; x2 , a2 ) = δ(a1 − a ∗ )δ(a2 − a ∗ ). Continuing this reasoning, one obtains for every n f (x1 , a1 ; . . . ; xn , an ) = δ(a1 − a ∗ ) . . . δ(an − a ∗ ). The set of random fields, a(x, ω) and u i (x, ω), is described by the joint probability densities f x1 , a1 , u i1 ; . . . ; xn , an , u in = = Mδ (a1 − a(x1 , ω))δ u i1 − u i (x1 , ω ) . . . δ (an − a(xn , ω))δ u in − u i (xn , ω) .

By δ(u i ) we mean the product of three δ-functions:

δ(u i ) = δ(u 1 )δ(u 2 )δ(u 3 ). Probability densities of the random field, a(x, ω), are obtained by integration of the joint probability densities over u i -variables. For examples, for one-point probability density, f (x, a) =

f (x, a, u i )d 3 u.

Here d 3 u ≡ du 1 du 2 du 3 . Similarly, f (x, u i ) =

f (x, a, u i )da.

The first point of the reformulation. We have seen in Sect. 16.1 that the minimization of the functional I (u, ω) over u(x) for each ω is equivalent to the minimization of the functional M I (u(x, ω), ω) over the random fields u(x, ω). The mathematical expectation, M I , of the functional (16.128) can be written in terms of one-point probability density, f (x, a, u i ):

804

16

MI =


L (x, a, u i ) f (x, a, u i )dxdad3 u.

(16.132)

Thus, the minimization of the M I over the random fields u(x, ω) is reduced to minimization of the functional (16.132) over the one-point probability density f (x, a, u i ). This function must obey the constraints that follow from the compatibility conditions: (a)

f (x, a, u i ) = f n−1 =

f x, a, u i ; x (2) , a (2) , u i(2) da (2) d 3 u (2) , . . . ,

(16.133)

f n dan d 3 u (n) , . . .

(b) f n is symmetric with respect to the transposition of any two triples of arguments, separated by semicolons (c)

f n 0,

n = 1, 2, . . .

Besides, the random field a(x, ω) is given. Thus, the n-point probability

density functions of a(x, ω), ˚f n , are known, and f x1 , a1 , u i(1) ; . . . ; xn , an , u i(n) must satisfy the constraints

f x1 , a1 , u i(1) ; . . . ; xn , an , u i(n) d 3 u (1) . . . d 3 u (n) = ˚f (x1 , a1 ; . . . ; xn , an ) .

(16.134)

In particular, for the one-point probability density we have

f (x, a, u i ) d 3 u = ˚f (x, a) .

(16.135)

Note that the prescribed function, ˚f (x, a) , must obey the condition (16.130d)

˚f (x, a) da = 1.

Therefore, the corresponding condition for f (x, a, u i ) ,

f (x, a, u i ) dad3 u = 1,

is satisfied automatically due to (16.135). In all these formulas, u i (x, ω) is an arbitrary random vector field. However, in the original variational problem, the vector field u i is special: it is potential. So we

16.8


805

have to formulate the potentiality constraint in terms of the probability densities. We will call this constraint the gradient compatibility. Gradient compatibility. The condition of potentiality of random vector fields can be given several different forms. We begin with the one which seems most appropriate for numerical simulations. Consider a deterministic divergence-free smooth vector field, σ i (x), with the vanishing normal component on the boundary of V : σ i (x) = 0 in V, x i

σ i (x)n i = 0 on V.

(16.136)

If u i (x, ω) is a smooth potential field, u i (x, ω) =

u(x, ω) , x i

then the integral of u i σ i vanishes: u i (x, ω)σ i (x)d V = 0,

(16.137)

V

as follows from integration by parts. The inverse statement is also true: if (16.137) holds for any vector field satisfying to (16.136), then the smooth vector field u i (x, ω) is potential. Indeed, the general solution of (16.136) is σ i = eijk j ψk in V. Functions ψk (x) are arbitrary functions of x which satisfy the boundary condition σ i n i = eijk n i j ψk = εαβ rαi rβk j ψk = εαβ α ψβ = 0. Here x i = r i (ξ α ) are the parametric equations of V, rαi ≡ r i /ξ α , ψα = rαi ψi the tangent components of ψi , and the formula (14.10) was used. Hence, at the boundary, ψα is a two-dimensional potential vector. Since one can add to ψk any potential vector without changing σ i (x), one can choose this potential vector in such a way as to make ψα equal to zero. The integral of σ i u i after integration by parts becomes

V

σ i ui d V =

V

eijk j ψk u i d V =

V

eijk n j ψk u i d A −

V

ψk eijk j u i d V = 0. (16.138)

The boundary term vanishes because ψα = 0 at V . Since the volume integral is zero for arbitrary ψk in V , curl of u i must be zero: eijk j u i = 0 in V. Hence, u i is a potential vector field. The integral (16.137) is zero for (almost) every realization if

806

16

M

i

u i (x, ω)σ (x)d V V


2

= 0.

(16.139)

This equation can be written in terms of the two-point probability density: M

V

2 u i (x, ω)σ i (x)d Vx =M u j (x ′ , ω)σ j (x ′ )d Vx ′ = u i (x, ω)σ i (x)d V V V = σ i (x)σ j (x ′ )Mu i (x, ω)u j (x ′ , ω)d Vx d Vx ′ = V V σ i (x)σ j (x ′ )u i u ′j f (x, u i ; x ′ , u i′ )d 3 ud 3 u ′ d Vx d Vx ′ = 0. = V

V

So a random field u i (x, ω) is potential if its two-point probability density satisfies the gradient constraint (16.140) σ i (x)σ j (x ′ )u i u ′j f (x, u i ; x ′ , u i′ )d 3 ud ′3 u ′ d Vx d Vx ′ = 0. V

V

This equation must hold for any divergence-free vector field σ i (x) with zero normal component on the boundary. Conversely, if there is a set of n-point probability densities, f x1 , u i(1) , . . . ,

xn , u i(n) , which obey the compatibility conditions, while f 2 satisfies (16.140), the random field u i (x, ω) is potential. Indeed, according to Kolmogorov’s theorem, since f n obey the compatibility conditions, there exists a random field, u i (x, ω), the probability density functions of which are f n . Therefore, (16.140) can be rewritten as (16.139). Hence, (almost) every realization of the random field is potential. There is another form of the gradient compatibility. Consider the correlation function of a random field, u(x, ω): B(x, x ′ ) = Mu(x, ω)u(x ′ , ω).

(16.141)

Differentiating (16.141) with respect to x i and x ′ j , we obtain 2 B(x, x ′ ) u(x, ω) u(x ′ , ω) =M . i ′ j x x x i x ′ j Hence, if a random field u i (x, ω) is potential, then there exists such a function B(x, x ′ ) that Mu i (x, ω)u j (x ′ , ω) =

u i u ′j f (x, u i ; x ′ , u i′ )d 3 ud 3 u ′ =

2 B(x, x ′ ) . x i x ′ j

(16.142)

On the other hand, if f 2 is the two-point probability density from a compatible set of probability distribution functions, and there is a function B(x, x ′ ) such that (16.142)

16.8


807

holds, then the random field u i (x, ω) is potential. Indeed, plugging (16.142) into (16.140) we see that (16.140) is true. Thus u i (x, ω) is potential. Detailed gradient compatibility. If the Lagrangian L = L (x, a, u, u i ) also depends on u, then the condition that the vector field u i is potential is not enough. We need a criterion that a random field u i (x, ω) is the gradient of a certain random field u(x, ω). Such a criterion can be formulated in terms of the two-point ′ ′ ′ probability density, f (x, u, u i ; x , u , u i ). Let f n = f x (1) , u (1) , u i(1) ; . . . ; x (n) , u (n) ,

u i(n) be the joint n-point probability density of random fields u and u i , and the set of probability densities (n =1, 2,. . . ) satisfies the compatibility conditions: for all n, the function f n is symmetric with respect to the transposition of any pair of arguments, separated by semicolons, besides, f n 0, f n−1 =

f n du (n) d 3 u (n) ,

(16.143)

f 1 dud 3 u = 1.

(16.144)

Then the following statement holds: the set of probability densities of the fields u and u i is the set of probability densities of a random field u(x, ω) and the field of its derivatives u ,i (x, ω) if f n satisfy the compatibility conditions (16.143) and (16.144), and f 2 = f x, u, u i ; x ′ , u ′ , u i′ is a solution of the equation 2 x i x ′ j

f 2 d 3 ud 3 u ′ + ′ j (16.145) f 2 u i d 3 ud 3 u ′ x u 2 f 2 u ′j d 3 ud 3 u ′ + f 2 u i u ′j d 3 ud 3 u ′ = 0. + i ′ x u uu ′

Here, d 3 u ′ is a volume element in the space of variables u i′ . It is assumed that all the integrals and derivatives in (16.145) exist. If, in addition, f 2 → 0 as u → ∞, u ′ → ∞ so fast that the functions of u and u ′ , uu

′

′ 3

3 ′

u i f 2 du d ud u and u

u i u j f 2 d 3 ud 3 u ′ ,

tend to zero as |u| → ∞ or u ′ → ∞, then the conditions mentioned are also sufficient. To prove the necessity, consider a function, α x, u; x ′ , u ′ , which vanishes for xǫV or x ′ ǫV and for sufficiently large |u| or u ′ . Let u(x, ω) be a realization of a differentiable the function β(x, x ′ ) of two arguments: random field. We construct β(x, x ′ ) = α x, u(x, ω); x ′ , u(x ′ , ω) . Since α is zero for x, x ′ ǫV , then

808

16


2 β(x, x ′ ) 3 3 ′ d xd x = 0. x i x ′ j

Writing this equation in terms of function α, we have

2 α 2 α u(x, ω) 2 α u(x ′ , ω) + + x i x ′ j ux ′ j x i u ′ x i x ′ j 2 α u(x, ω) u(x ′ , ω) d 3 xd 3 x ′ = 0. + uu ′ x i x ′ j

(16.146)

Taking the mathematical expectation of the left hand side, we have

2 α 2 α 2 α 2 α ′ + + u + u u i u ′j i j x i x ′ j ux ′ j u ′ x i uu ′ ′

f 2 dudu ′ d 3 ud 3 u ′ d 3 xd 3 x ′ = 0. (16.147)

′

Since α(x, u; x , u ) is arbitrary, (16.145) follows from (16.147) after integration by parts. The necessity of the compatibility conditions (16.143) and (16.144) follows from the definition of probability densities. Assume now that we have a set of probability densities f 1 , f 2 . . . that satisfy the compatibility conditions (16.143) and (16.144). Then, by Kolmogorov’s theorem, there are random fields, u(x, ω) and u i (x, ω), whose distribution functions form this family. We have to show that, with probability one, by virtue of (16.145), u i (x, ω) are the derivatives of u(x, ω). For this, it suffices to show that 2 M u(x, ω) − u(x ′ , ω) − u i (x, ω)d x i = 0

(16.148)

where is a contour joining the points x and x ′ . Indeed, opening the brackets in (16.148) and taking the mathematical expectation, we obtain B(x, x) + B(x ′ , x ′ ) − 2B(x, x ′ ) + 2

Bi j (z, z ′ )dz i dz j −

(16.149)

u i′ u f x, u, u i ; z, u ′ , u i′ − f x ′ , u, u i ; z, u ′ , u i′ dudu ′ d 3 ud 3 u ′ dz i = 0.

Here ′

′

B(x, x ) = Mu(x, ω)u(x , ω) = Bi j (x, x ′ ) = Mu i (x, ω)u j (x ′ , ω) =

u u ′ f x, u; x ′ , u ′ dudu′ ,

u i u ′j f x, u i ; x ′ , u i′ d 3 ud 3 u ′ .

Let us multiply (16.145) by uu ′ and integrate the result with respect to uu ′ . We get the equation

16.8


809

2 B(x, x ′ ) (16.150) u ′ u i′ f x, u, u i ; x ′ , u ′ , u i′ dud3 udu′ d 3 u ′ − ′j x i x ′ j x − i uu ′j f x, u, u i ; x ′ , u ′ , u i′ dud3 udu′ d 3 u ′ + Bi j (x, x ′ ) = 0. x Integration of this equation over the contour with respect to x i and x ′ j , yields (16.149). Thus, ( 16.148) follows from (16.145), and for (almost) every realization of the random field, ′

u(x, ω) − u(x , ω) −

u i (z, ω)dz i = 0.

Consequently, u i (x, ω) are the derivatives of u(x, ω), as claimed. Note that the one-point probability density satisfies the equation f (x, u) + x i u

u i f (x, u, u i )d 3 u = 0.

(16.151)

It is obtained, similarly to (16.145), from the equation

V

α(x, u(x, ω))d V = 0, x i

(16.152)

which holds for any function α(x, u) that vanishes for xǫV. Taking mathematical expectation of (16.152), V

α(x, u) f (x, u)dudV + x i

V

α(x, u) u i f (x, u, u i )dVdud3 u = 0, u

integrating by parts and using the arbitrariness of α, we arrive at (16.151). Completion of reformulation. So, the functional to be minimized is a linear functional of the one-point probability density, I( f ) =

L(x, a, u i ) f (x, a, u i )dad3 udV.

(16.153)

V

The function f (x, a, u i ) must be such that it can be included in a set of n-point probability density functions, f n , satisfying the compatibility conditions (16.133). Besides, the constraints (16.134) and (16.140) must be also obeyed. We arrive at ˇ ω) are Variational principle. The true probability densities of the minimizer u(x, determined by the minimization of the linear functional (16.153) on the set selected by the linear constraints (16.133), (16.134) and (16.140).

810

16


It is interesting that the functional to be minimized and the constraints are linear while the original problem can be highly non-linear. Approximations of probability densities. Usually the information on the random fields of physical characteristics is quite poor. Typically one knows the onepoint probability density ˚f (x, a) and, sometimes, the two-point probability density ˚f (x, a; x ′ , a ′ ). Additional information might be available which is not given in terms of probability densities. For example, the material can be a polycrystal, i.e. a continuum formed by a large number of grains that are connected without gaps and overlappings. To be used, such information needs to be expressed in terms of probability densities. In any case, the field a(x, ω) is never completely known. That yields an additional arbitrariness in the setting of the variational problem. Here we consider the situation when one knows only the one-point probability density of the physical characteristics ˚f (x, a). Obviously, the minimum value of I ( f ) in this case is the minimum possible value of I (u, ω), when the information on the higher order probability densities is not available. So, we have to minimize the functional (16.154) I ( f ) = L(x, a, u i ) f (x, a, u i )dad3 udV over all positive functions f (x, a, u i ) with a prescribed integral over u i ,

f (x, a, u i )d 3 u = ˚f (x, a).

(16.155)

f (x, a, u i ; x ′ , a ′ , u i′ ) = f (x ′ , a ′ , u i′ ; x, a, u i ) 0

(16.156)

f (x, a, u i ) 0, Besides, there is a function

such that

f (x, a, u i ; x ′ , a ′ , u i′ )da ′ d 3 u ′ = f (x, a, u i ),

(16.157)

and for any divergence-free vector field σ i (x) with zero normal component at the boundary,

u i u ′j f (x, a, u i ; x ′ , a ′ , u i′ )σ i (x)σ j (x ′ )dxdad3 udx′ da ′ d 3 u ′ = 0.

(16.158)

Now we have to add one more constraint. Consider a functional of f 2 and vector field q i (x): N ( f, q) =

u i u ′j f (x, a, u i ; x ′ , a ′ , u i′ )q i (x)q j (x ′ )dxdad3 udx′ da ′ d 3 u ′ . (16.159)

16.8


811

In fact, it does not depend on the probability characteristics of the field, a(x, ω), because

f (x, a, u i ; x ′ , a ′ , u i′ )dada′ = f (x, u i ; x ′ , u i′ ),

but the form (16.159) will be more convenient. Functional, N ( f, q) can be written in terms of the correlation function, Bi j (x, x ′ ): N ( f, q) =

Bi j (x, x ′ )q i (x)q j (x ′ )dVdV′ .

If f (x, u i ; x ′ , u i′ ) is a two-point probability density of a random field, u i (x, ω), then N ( f, q) = M

u i (x, ω)q i (x)d V V

2.

.

(16.160)

Hence,10 N ( f, q) 0 for any q i (x).

(16.161)

Such a condition is a constraint on admissible f 2 . It did not appear previously, because the assumed existence of an infinite set of probability densities warrants the existence of the corresponding random field. Then (16.160) and, thus, (16.161) hold automatically. Now all probability densities with n > 2 are dropped from the consideration. Therefore, in principle, for a given f 2 , there may be no random field u i (x, ω) for which f 2 is its two-point probability density. Then the transition from (16.159) to (16.160) becomes impossible. Remarkably, the constraint (16.161) turns out to be sufficient for the existence of u i (x, ω) : if f 2 satisfies (16.161), then there is a random field u i (x, ω) whose two-point probability density is f 2 .11 So we have to minimize the functional I ( f ) on the set of probability densities selected by the constraints (16.155), (16.156), (16.157), (16.158) and (16.161). Out of these constraints the most “unpleasant” is (16.157): it says, in particularly, that the integral of f 2 over a ′ and u i′ does not depend on x ′ . 10

i Condition (16.161) is usually written / iin an other form by choosing q (x) to be concentrated at some points x1 , . . . , xk : q i (x) = s α(s) δ(x − xs ). Then, for any choice of x1 , . . . , xk , and for i , any numbers, α(s) j i Bi j (xs , xr )α(s) α(r ) 0. s,r

11

In fact this statement was proven for homogeneous random fields in [319]; however homogeneity does not seem to be essential.

812

16


One can obtain an explicit solution if L and ˚f do not depend on the space point: L = L(a, u i ),

˚f = ˚f (a).

Indeed, let us drop all the constraints except (16.155). Then we have to find the minimum value of the linear functional I ( f ) = L(a, u i ) f (a, u i )dad3 u (16.162) on the set of all non-negative functions, f (a, u i ), which obey the constraint

f (a, u i )d 3 u = ˚f (a).

(16.163)

Obviously, to get the minimum value, we have to minimize the integral

L(a, u i ) f (a, u i )d 3 u

for each a. The minimum value is achieved when f (a, u i ) is concentrated at the point u i where L(a, u i ) has the minimum value. Denote this point by ϕi (a). So f (a, u i ) = prefactor δ(u i − ϕi (a)). The value of the prefactor is found from (16.155). Finally, f (a, u i ) = ˚f (a)δ(u i − ϕi (a)).

(16.164)

The corresponding minimum value of the functional is Iˇ =

min L(a, u i ) ui

˚f (a)da.

(16.165)

Since this value is obtained by dropping some constraints and, accordingly, by an expansion of the set of admissible functions, it gives a low estimate of the true minimum value. Let us show that, in fact, it coincides with the true minimum value. To this end, it suffices to demonstrate that function (16.164) satisfies all the constraints. Then, taking (16.164) as an admissible function, we get an upper estimate, and, since for the probability density (16.164) the value of the functional is (16.165), the lower and upper estimates coincide and, thus, give the exact solution. As we have seen earlier in this section, the δ-functional dependence of the one-point probability density on its argument yields a δ-functional dependence of the two-point probability density. Thus f (a, x, u i ; a ′ , x ′ , u i′ ) = prefactor δ(u i − ϕi (a))δ(u i′ − ϕi (a ′ ))

16.8


813

where the prefactor does not depend on u i and u i′ . From (16.134), written for n = 2, we can find the prefactor f (a, x, u i ; a ′ , x ′ , u i′ ) = ˚f (a, x; a ′ , x ′ )δ(u i − ϕi (a))δ(u i′ − ϕi (a ′ )).

(16.166)

Here, however, ˚f (a, x; a ′ , x ′ ) remains arbitrary, because the two-point information is not provided. Choosing ˚f (a, x; a ′ , x ′ ), we can satisfy the gradient compatibility. Indeed, let us take the gradient compatibility condition in the form (16.142). We have

ϕi (a)ϕ j (a ′ ) ˚f (x, a; x ′ , a ′ )dada′ =

2 B(x, x ′ ) . x i x ′ j

(16.167)

Equation (16.167) can be satisfied, for example, for the function ˚f (x, a; x ′ , a ′ ) = ˚f (a) ˚f (a ′ ). Then the left hand side of (16.167) is a constant tensor ϕ¯ i ϕ¯ j (ϕ¯ i = Mϕi ), and one can set B(x, x ′ ) = ϕ¯ i x i ϕ¯ j x ′ j + const. The condition (16.161) is also satisfied:

′

i

j

′

′

Bi j (x, x )q (x)q (x )dVdV =

i

ϕ¯ i q (x)d V V

2

0.

Hence, the one-point probability density is indeed an admissible function, and the solution obtained is exact. In general, in the absence of complete information on the random structure, the one-point probability density remains undetermined. However, there is a need to have it at least approximately. Formula (16.164) motivates the following approximation. According to this formula, the probability density is concentrated at the minimum of L(a, u i ). Clearly, the true probability density is not concentrated and may be expected to be some smooth function. Various functions can be used for smoothing the δ-function in (16.164). Perhaps the simplest candidate is the function f (a, u i ) =

1 ˚f (a)e−β L(a,u i ) Z (β, α)

(16.168)

where Z (β, α) =

e−β L(a,u i ) d 3 u

(16.169)

814

16


and β > 0. The parameter β can be found if, for example, one knows the value of the “effective Lagrangian”: L eff = M

1 |V |

LdV. V

The parameter β is determined from the equation12

1 ˚f (a)L(a, u i )e−β L(a,u i ) d 3 ud a = L eff . Z (β, α)

(16.170)

Equation (16.170) can also be written as ˚f (a) ln Z (β, α) da = L eff . β

(16.171)

If β is large, then function (16.168) is concentrated near the points u i = ϕi (a) as is suggested by the approximate solution. Some generalizations. The extension of the previous results to the case of many material characteristics and many required fields is straightforward: at a space point, the probability densities become functions of the material characteristics values, a 1 , . . . , a m , and the required functions values, u i(1) , . . . , u i(n) . Accordingly, the integration of the probability densities is conducted in the space of these variables. One point, however, is worth mentioning: strain compatibility. If Lagrangian depends on strains, 1 i u j + j u i , (16.172) εi j = 2 then the gradient compatibility is modified in the following way. Let σ i j (x) be an arbitrary symmetric tensor obeying the equilibrium equations σ i j (x) = 0 in V, x j

σ i j n j = 0 on V,

σ i j = σ ji .

(16.173)

If a symmetric random tensor, εi j (x, ω), is expressed in terms of a random vector field, u i (x, ω), by (16.172), then

V

εi j (x, ω)σ i j (x)d V = 0.

12

(16.174)

The admissible values of L eff cannot be arbitrary. As follows from (16.170), L eff are constrained from below by the condition 1 ˚f (a)L min (a)eβ L(a,u i ) d 3 u da = L min (a) ˚f (a)da, L eff Z (β, α) where L min (a) is the minimum over u i of the Lagrangian, L(a, u i ).

16.8


815

Thus, M

ij

εi j (x, ω)σ (x)d V V

2

= 0,

(16.175)

or, for any σ i j (x), satisfying (16.173),

′ σ i j (x)σ kl (x ′ )d 6 εd 6 ε′ dVdV′ = 0. f (x, ε; x ′ , ε′ )εi j εkl

(16.176)

Besides, for any symmetric tensor field q i j (x),

′ ij q (x)q kl (x ′ )d 6 εd 6 ε′ dVdV′ 0. f (x, ε; x ′ , ε′ )εi j εkl

(16.177)

Conditions (16.176) and (16.177) are sufficient for the existence of a random field u i (x, ω) such that (16.172) holds. Indeed, due to (16.177) a random field εi j (x, ω) exists, whose two-point probability density is f (x, ε; x ′ , ε′ ). Therefore, (16.176) can be rewritten in the form (16.175). Hence, for (almost) all ω and any admissible σ i j , (16.174) holds. This yields (16.172). Elimination of gradient compatibility condition. One can avoid the constraint of gradient compatibility if the variational problem is put in Hashin-Strikman form described in Sect. 5.9. The functional does not depend on the derivatives of the required functions pi and the gradient compatibility constraint does not appear. As was mentioned in Sect. 5.9, the transformation of the variational problems to Hashin-Strikman form is not always possible. In such cases, the gradient compatibility condition remains among other constraints of the variational problem.

Chapter 17

Homogenization

Homogenization is a procedure that allows one to replace a precise description of highly inhomogeneous media by an averaged one, an equivalent homogeneous media. The homogenization problem has a long and rich history which began in the first quarter of the nineteenth century. Relatively recently it was recognized that the homogenization problem has an asymptotic nature. This brought a complete understanding of the static behavior of periodic structures and a considerable insight for that of random structures. In this chapter the homogenization problems are considered in cases when they admit a variational formulation.

17.1 The Problem of Homogenization Most materials which we attempt to model by a homogeneous continua are, in fact, inhomogeneous on small scales. A typical example is shown in Fig. 17.1. This is a titanium alloy. It contains the particles of the average size of about 10 μm (about one-fifth of the thickness of human hair). The alloy can be modeled by a continuum. This continuum is inhomogeneous: the properties of the particles and the matrix are different. On the other hand, any piece of the alloy of a macroscopic size, say, more than 1 mm, behaves as a homogeneous media. The macroscopic behavior of the alloy is completely determined by the properties of the particles and the matrix. The question arises: How to compute the macroproperties if the microproperties are known? This is a typical question of homogenization theory. The answer was obtained for media with periodic microstructure; it will be discussed further. For random structures the mathematical setting of the problem is more or less understood while the particular results are quite far from being comprehensive. We will discuss the situation for random structures as well. There is a more difficult issue than establishing a link between micro and macroproperties: characterization of local fields. We will touch upon this issue in the next chapter. Our two major examples are the homogenization in a scalar problem, when the quadratic part of the functional depends on only one function, u(x), 1 ij u u dV a (x) i x u j V 2 V.L. Berdichevsky, Variational Principles of Continuum Mechanics, Interaction of Mechanics and Mathematics, DOI 10.1007/978-3-540-88469-9 4, C Springer-Verlag Berlin Heidelberg 2009

817

818

17

Homogenization

Fig. 17.1 Titanium alloy with 10 m average size of the particles

and in the linear elasticity problem with the quadratic part of the functional of the form

V

1 ijkl C (x)εi j εkl d V, 2

εi j =

1 2

u j u i + i j x u

.

The key results are true, in fact, for any integral functional with a convex Lagrangian, and will be formulated in this general case.

17.2 Homogenization of Periodic Structures Homogenization in one-dimensional case. We start with a one-dimensional example. Consider the elongation of an elastic beam with the Young modulus a and the displacement u(x). The beam is inhomogeneous: a = a(x). Let the left end of the beam be clamped, u(0) = 0, and a force P is applied at the right end. The true displacement minimizes the functional

I (u) =

1 2

L 0

a (x)

du dx

2

d x − Pu (L) .

(17.1)

17.2

Homogenization of Periodic Structures

819

Fig. 17.2 Notation for a beam with periodic microstructure

The minimizing function is the solution of the boundary value problem: d du du a (x) = 0, u (0) = 0, a = P. dx dx d x x=L

(17.2)

This problem can be solved analytically: du (x) = P, a (x) dx

u (x) = P

x

dx . a (x)

(17.3)

0

Let the beam have a periodic microstructure like that shown in Fig. 17.2: the sequence of alternating short beams with two different Young’s moduli. The length of the periodic cell, ε, is assumed to be much smaller than the length of the beam, L. We may introduce the average strain, , as the total relative elongation of the beam:

=

u (L) , L

and define the effective Young’s modulus, aeff , by the relation P = aeff .

(17.4)

According to the exact solution, the constant, aeff , is ⎛

aeff = ⎝

1 L

L 0

⎞−1

dx ⎠ a (x)

.

(17.5)

It is instructive to compute also the minimum value of the functional (17.1). Using Clapeyron’s theorem (5.47) we have 1 1 Iˇ = − Pu (L) = − P 2 2 2

L 0

1 P2 dx =− L. a(x) 2 aeff

(17.6)

820

17

Homogenization

This number, Iˇ , (17.6) can also be obtained as the minimum value of the “homogenized” functional of a homogeneous elastic beam with an effective Young’s modulus, aeff , and a homogenized displacement v(x): Iˇ = min I¯ (v (x)) , v(x): v(0)=0

1 I¯ (v) = 2

L

aeff

0

dv dx

2

d x − Pv (L) .

(17.7)

This can be checked by direct inspection using the fact that the minimizing function in the variational problems (17.7) is v (x) =

P x. aeff

(17.8)

The coincidence of the minimum values in the variational problems (17.1) and (17.7) does not occur by chance. To reveal the underlying cause let us look more closely at the character of the minimizing function (17.3). Young’s modulus a(x), and, thus, the compliance a −1 (x), are the periodic functions with the period ε. The integral, x

dx , a (x)

0

computed at the points xk = kε, k = 1, 2, 3, . . . , is equal to kε

dx =k a (x)

ε

dx . a (x)

0

0

Obviously, due to periodicity of a(x), ε 0

1 dx =ε a (x) ε

ε 0

dx 1 =ε a (x) L

L

dx ε = . a (x) aeff

(17.9)

0

Here we assumed that there is an integer number of cell in the beam (i.e. L/ε is an integer). If there is an incomplete cell and L/ε is not an integer, then (17.9) contains a small correction of the order ε/L which can be neglected in the first approximation. So

17.2


u (xk ) = P

821

kε

P dx = kε . a (x) aeff

(17.10)

0

The derivative du/d x is positive. Therefore, between the points xk and xk+1 the displacement grows monotonically increasing from the value kε P/aeff to (k + 1) ε P/aeff . At the points kε, the values of u(x) and v(x) coincide. The qualitative graphs of u(x) and v(x) are shown in Fig. 17.3. We see that u(x) is a fast oscillating function which does not deviate much from a slow changing function v(x): u (x) = v (x) + u ′ (x) .

(17.11)

The correction, u ′ (x), is equal to ′

u (x) =

x

1 dx −x . a (x) aeff

0

It is a periodic function with the period ε. If we decrease ε, v(x) does not change while u ′ decreases: it is of the order of ε. In the first approximation, u(x) ≈ v(x) for ε → 0. This is not true for the derivatives: 1 dv du ′ P du 1 . (17.12) − = + = +P dx dx dx aeff a (x) aeff We see that the derivatives of u ′ (x) are of the same order as the derivatives of v(x). The same can be observed from Fig. 17.3.

Fig. 17.3 A qualitative dependence of the true and the averaged displacements, u(x) and v(x), on the coordinate

822

17

Homogenization

Now we have all the ammunition necessary to obtain the “averaged” functional (17.7) directly from the original one. The functional (17.1) can be written as xk+1 2 1 du a (x) d x − Pu(L). I (u) = 2 d x k

(17.13)

xk

Let us search for the minimizer of this functional of the form (17.11) where v(x) has constant derivative in each cell while u ′ (x) is a periodic function. Then, on each cell, we obtain the same variational problem, the cell problem: 1 min u ′ (0)=u ′ (ε) 2

ε 0

du ′ 2 d x, a (x) vx + dx

vx ≡

dv . dx

(17.14)

For vx = const, the minimizer obeys the boundary value problem du ′ d = 0, a (x) vx + dx dx

u ′ (0) = u ′ (ε) .

(17.15)

The differential equation yields du ′ a (x) vx + = λ = const. dx The value of the constant, λ, is determined from the periodicity condition ε

du ′ dx = dx

ε 0

0

λ − vx d x = 0, a (x)

i.e. λ = aeff vx . Therefore, 1 min u ′ (0)=u ′ (ε)=0 2

ε 0

dv du ′ + a (x) dx dx

2

1 d x = εaeff 2

dv dx

2

.

(17.16)

Plugging (17.16) into (17.13) and replacing the sum by an integral over x and, in the linear term, u (L) by v (L) (they differ by a small amount on the order of ε), we arrive at the functional (17.7).

17.2


823

It turns out that this procedure to find the averaged functional may be used for many inhomogeneous media with periodic microstructure as discussed in the next subsection. Homogenization of periodic structures: a heuristic approach. Consider a threedimensional grid built on the vectors τ(1) , τ(2) , τ(3) with the coordinates τ(ip) ; the nodes of the grid are-the points. ε k 1 τ(1) , k 2 τ(2) , k 3 τ(3) , k 1 , k 2 , k 3 being integers. For each node k = k 1 , k 2 , k 3 , we build the cell Ck , a parallelepiped with the center at the point k p τ( p) and the sides ετ(1) , ετ(2) , ετ(3) (Fig. 17.4a). The length of at least one of vectors, τ1 , τ2 , τ3 , is set equal to unity. Therefore, the factor ε determines the size of the cells. Consider an integral functional, V

L a, u, u ,i d V,

where the integrand depends on the field variable u(x) and its first derivatives, and on the physical characteristics of the material; denote these characteristics by a(x). For periodic structures, a(x) are periodic functions of x: a x + εk p τ( p) = a (x) .

It is convenient to introduce the scaled coordinates y i = x i /ε in which the cells have a finite size; y-coordinates change in the cell C (Fig. 17.4b). We consider a sequence of periodic structures for which ε → 0 while the physical characteristics as functions of y-coordinates, a = a (y), do not change. To prescribe the physical properties one has to specify the functions a(y). These functions a (y) may be piece-wise continuous; in particular, they may have different values

Fig. 17.4 Geometry of a periodic structure

824

17

Homogenization

at the opposite sides of the cell C. An important case is the piece-wise constant functions, a(y), when a(y) take the value a2 in some subregion, B, of C (inclusion) and the value a1 in C − B (matrix) (Fig. 17.4b). In terms of the observer’s coordinate x, a=a

x

ε

.

In the x-space, the continuum with characteristics a(x/ε) corresponds to a microinhomogeneous periodic structure, Fig. 17.4a, with the characteristic size of inhomogeneity ε. We are going to study the minimization problem: I (u) =

V

x

, u, u ,i d V − fudA → min . L a ε

(17.17)

V

We expect that for small ε, i.e. for ε being much smaller than the size of the region V and the characteristic length on which the external force f changes, the solution of this problem behaves in the same way as that of the one-dimensional problem considered in the previous subsection. So we assume that for ε → 0 the asymptotics of the minimizing element has the form u (x, ε) = v (x) + εψ (y, x) + terms of higher order.

(17.18)

Here the first term of the expansion does not depend on ε, the second is on the order of ε, and the further terms are not needed in construction of the leading approximation. Function ψ is a periodic function of the fast variables y; it may depend on the slow variables x as well. The function ψ(y, x) can be neglected in computation of u(x, ε), but it gives a finite contribution in the computation of the derivatives of u(x, ε) : v (x) ψ (y, x) u(x, ε) = + . i x x i y i

(17.19)

Here we dropped the small term εψ (y, x) /x i . So to find the first approximation for u(x, ε) and their derivatives we must find v(x) and ψ (y, x). Let us plug (17.18) in (17.17). We obtain I (u) = I (v, ψ) =

k C k

ψ f (v + εψ) d A. L a (y) , v + εψ, v,i + i d V − y V

(17.20)

The term εψ is small compared with v and can be omitted in both volume and surface integrals. Functions v and v,i do not depend on ε and, for ε → 0, they are

17.2


825

practically constant on each cell. Therefore, on each cell one has to minimize with respect to ψ the functional

C

ψ L a (y) , v, v,i + i d V. y

The minimum value of this functional is a function of v and vi . We introduce a function of parameters v and vi , L¯ v, v,i =

1 min periodic ψ(y) |C|

C

ψ L a (y) , v, vi + i d V, y

(17.21)

where |C| is the cell volume, while v and vi are considered as some constants.1 After minimization over ψ in (17.20) each member of the sum becomes equal to L¯ (v, vi ) |C| , and the sum transforms into the integral I¯ (v) =

V

L¯ v, v,i d V −

fvdA.

(17.22)

V

It corresponds to a homogeneous medium, since the explicit dependence on the space coordinates disappeared. To find the function v(x), we have to minimize the averaged functional (17.22) with respect to all v(x):

L¯ (v, vi ) d V −

fvdA → min . v(x)

(17.23)

V

V

So the solution of the original variational problem is reduced to the solution of the cell problem (17.22) which gives the effective Lagrangian, L¯ (v, vi ) , and to the solution of the homogenized variational problem (17.23). The same reasonings apply if there are a number of the field variables, u κ , κ = 1, . . . , m. The cell problem takes the form L¯ v κ , v,iκ =

1 min periodic ψ(y) |C|

C

ψ κ κ κ d V. L a (y) , v , vi + y i

In linear problems, when the Lagrangian is a quadratic functions of the arguments, the cell problem is also linear and v κ , viκ enter linearly into the Euler equations of the variational problem. Therefore, ψˇ κ depend linearly on v κ and viκ and 1

For derivatives of v we write vi when they play the role of constant parameters and v,i when it is essential that they are derivatives of v.

826

17

Homogenization

the averaged Lagrangian is also quadratic, rendering the linear averaged equations with some constant effective coefficients. The situation with homogenization of nonlinear equations is much more complex: after averaging one can get the averaged equations which may look quite different from the original ones. Before proceeding to the derivation of the above statements by the variationalasymptotic method we consider a dual formulation of the cell problem. i the variables that are dual to u κ Dual cell problem. Denote by pκ ,i , and by ∗ κ i L (a, u , pκ ) the Young-Fenchel transformation of the Lagrangian

0 i κ 1 i ) = max pκ u i − L(a, u κ , u κ L ∗ (a, u κ , pκ ,i ) . κ ui

Assume that L ∗ (a, u κ , u iκ ) is a convex function of u iκ . Then

0 i κ 1 i u i − L ∗ (a, u κ , pκ ) , L(a, u κ , u iκ ) = max pκ i pκ

and the cell variational problem can be rewritten as a minimax problem: L¯ v κ , viκ =

1 min max i periodic ψ κ pκ |C|

i pκ

(y)

C

viκ

ψ κ + y i

∗

κ

− L (a (y) , v ,

piκ )

d V.

Changing the order of minimum and maximum, we obtain L¯ v κ , viκ = max

min

i periodic ψ κ pκ

+

C

i pκ

⎡ 1 ⎣ i pκ (y) viκ − L ∗ (a, v κ , piκ ) d V |C| C ⎤

ψ κ dV ⎦ . y i

Minimization with respect to the periodic functions ψ κ narrows the set of all adi missible pκ to the functions obeying the constraints i pκ = 0, y i

0

i pκ

1

(k)

n i = 0,

k = 1, 2, 3,

(17.24)

where [ϕ](k) means the difference of the values of function ϕ at the two opposite sides of the cell; the couples of opposite sides of the cell are marked by index k. So L¯ v κ , viκ =

;1 0 κ: i ; : ∗ i max . a (y) , v κ , pκ

vi pκ − L i ∈ 17.24 pκ

(17.25)

17.2


827

It is convenient this result in another form by introducing an interme: ito; rewrite i = p¯ κ , diate constraint, pκ L¯ v κ , viκ = max i p¯ κ

;1 0 κ: i ; : ∗ i max vi pκ − L a(y), v κ , pκ i ∈ 17.24 pκ ( )

pκi = p¯ κi

⎡

⎢ i κ = max ⎢ ⎣ p¯ κ vi − i p¯ κ

⎤

;⎥ : ∗ i ⎥. L a(y), v κ , pκ min ⎦ i ∈ 17.24 pκ ( ) i i

pκ = p¯ κ

(17.26)

i the Young-Fenchel transformation of the averaged Let us denote by v κ , p¯ κ Lagrangian, 1 0 i κ i p¯ κ vi − L¯ v κ , viκ = max v κ , p¯ κ κ vi

1 0 i κ L¯ v κ , viκ = max p¯ κ vi − v κ , p¯ iκ . i p¯ κ

(17.27)

Comparing (17.26) and (17.27) we obtain the following relation for , i = v κ , p¯ κ

min i pκ ∈(17.24)

pκi = p¯ κi

0:

;1 i L ∗ a (y) , v κ , pκ .

(17.28)

¯ by Briefly, one can write the rule of computation of the averaged Lagrangian, L, means of the dual variational problem as a succession of two operations, : ; = min L ∗ ,

L¯ = ∗ .

Homogenization of periodic structures: variational-asymptotic method. The previous derivation was based on the ansatz (17.18). It is desirable to obtain it by the asymptotic analysis of the functional instead of making an a priori hypothesis, moreover that such ansatz may not be correct in some variational problems as we will see from the further examples. So let us derive (17.18) using the variationalasymptotic method. We have to study the functional depending on small parameter ε: I (u) =

V

u x , u, i d V − fudA. L a ε x

(17.29)

V

The material characteristics, a( xε ), change fast in space: they change periodically on a distance which is much smaller than the characteristic size of the volume V .

828

17

Homogenization

Therefore, we suspect that the minimizer also changes periodically in space. In addition to the periodic change, the minimizer may change slowly from cell to cell. So, we assume that

x , x, ε , (17.30) u=u ε where u(y, x, ε) is a periodic smooth function of y. For the derivatives of the functions (17.27) we have u 1 u = + i u, x i ε y i

(17.31)

where i u is the partial derivative of u(y, x, ε) when y are kept constants: i u =

u (y, x, ε) . x i

In the first approximation, u 1 u = , x i ε y i and the functional I (y) can be written as

u 1 x I (u) = d V − fudA = , u, i L a ε y ε V V 1 u d V − fudA. = L a (y) , u, ε y i k

(17.32)

V

Ck

Assume that L(a, u, u i ) → ∞ as u i → ∞ . Then for ε → 0, the integrand L tends to ∞ and the last term in (17.32) can be neglected. One can keep in the sum in (17.32) only the cells which lie inside the region V : the cells which enter in V only partially give a contribution in (17.32) which is small compared with the kept terms. Then minimization of the functional 1 u dV (17.33) L a (y) , u, ε y i k Ck

is reduced to the minimization on each cell the integral 1 u dV L a (y) , u, ε y i

(17.34)

C

over all periodic functions u(y). The minimum of the functional (17.34) is achieved on constant functions. This occurs, for example, for the Lagrangian

17.2


829

1 ij u u . a (y) i 2 x x j

(17.35)

1 ij u u a (y) i + b (x) u 2 . 2 x x j

(17.36)

L= Another example is L= In this case,

1 u u 1 u = 2 ai j i L a (y) , u, + b (x) u 2 ; ε y i 2ε y y j

(17.37)

the second term in (17.37) is negligible compared with the first one and, after we drop it, we obtain the same results as for the Lagrangian (17.35). So the first term in the asymptotic expansion of the minimizing element, u(y, x, ε), does not depend on the fast variables and, thus, is a function of x only. Denote the first term of the asymptotic expansion by v(x). So u(y, x, ε) = v (x, ε) + u ′ (y, x, ε)

(17.38)

where u ′ is small compared with v(x). The order of magnitude of u ′ is not known yet. Denote by < · > the volume average over the cell,

ψ (y) ≡

1 |C|

ψ (y) d V.

C

Without loss of generality we can put : ′ ; u (y, x, ε) = 0.

Indeed, if < u ′ > not zero,: we ; ; can redefine v(x) making the change of vari: were ables, v1 = v + u ′ , u ′1 = u ′ − u ′ . Then u = v1 (x, ε) + u ′1 (y, x, ε) ,

: ′; u 1 = 0,

as claimed. Let us plug (17.38) into the functional. Dropping the term u ′ /x i which is small compared with ε−1 u ′ /y i and, similarly, u ′ in the sum v + u ′ , we obtain I v, u ′ =

V

1 u ′ v d V − fvdA. L a (y) , v, i + x ε y i

(17.39)

V

The functional (17.39) can be minimized successively, searching first the minimum with respect to u ′ for some fixed function v(x, ε), and then the minimum with

830

17

Homogenization

respect to v(x, ε). For ε → 0, v and v/x i are practically constant within each cell. Therefore, on each cell we have to minimize the functional 1 u ′ dV L a (y) , v, v,i + ε y i

C

(17.40)

over all periodic functions u ′ . Here v and v,i must be considered as some constant. Assume that v and v,i do not depend on ε (we will check the consistency of such an assumption at the end of the derivation). Then we make a change of the variable, u ′ → ψ , u ′ = εψ , and obtain the following variational problem:

C

ψ L a (y) , v, v,i + i d V → min . periodic ψ y

(17.41)

¯ It is a function of the parameters, Denote the minimum value of the integral by L. v and vi : L¯ (v, vi ) =

min

periodic ψ

1 |C|

C

ψ L a (y) , v, vi + i d V. y

(17.42)

The functional (17.39) takes the form I¯ (v) =

V

L¯ v, v,i d V −

fvdA.

(17.43)

V

It does not depend on ε, and so does its minimizer, vˇ . If the variational problems for the functionals (17.42) (17.43) are well-posed, then our derivation is self and consistent. Function L¯ v, v,i has the meaning of the effective Lagrangian, or the Lagrangian of the homogenized medium. ˇ depends on v and vi . The depenThe minimizer of the cell problem (17.42), ψ, dence of ψˇ on the slow variables x appears due to the dependence of v and v,i on x. So, the first terms of the asymptotic expansion of the minimizing element for the micro-inhomogeneous periodic media are u (x, ε) = v (x) + εψ (y, x) .

(17.44)

This is a multi-dimensional generalization of the one-dimensional result which we began with. According to (17.44) the function u(x, ε) tends to the solution of the homogenized problem, v(x), as ε → 0. For the derivatives this is not true: according to (17.44),

17.2


831

u ψ = v,i + i . i x y

(17.45)

So to find the function u(x, ε) in the first approximation, one has to solve the ¯ and then solve the homogecell problem (17.42), find the effective Lagrangian, L, nized problem. The solution of the cell problem determines the distribution of the local fields in the cells. Note that the averages over the cell values of the derivatives coincide with v,i : from (17.45),

u = v,i . x i

(17.46)

The changes of the boundary conditions do not affect the averaged functional. For example, if the original variational problem is

x , u, u ,i d V → min , I (u) = L a u(x): ε u(x)|V =u (b)

V

where the boundary values u (b) do not depend on ε, then the averaged variational problem is ¯I (v) = L¯ v, v,i d V → min (17.47) v(x): v(x)|V =u (b)

V

with the same effective Lagrangian (17.42). If L is a quadratic function of derivatives u ,i , then the original problem is linear, and so is the cell problem. Therefore, the minimizer of the cell problem ψˇ depends linearly on the constants v,i , and the averaged Lagrangian is a quadratic function of v,i : 1 ij L¯ = aeff v,i v,i 2

(17.48)

ij

with some effective coefficients, aeff . The averaged problem is linear as well. Our derivation was based on a few assumptions the violation of which change the result. Here is one example. Consider the Lagrangian L=

1 1 x 2 1 u ,i u ,i + 2 a u + 2 f (y) u. 2 ε ε ε

If we seek the solution of the problem in the form u = u(y, x, ε), then, in the first approximation, L=

1 u u 1 1 + 2 a (y) u 2 + 2 f (y) u, 2ε2 y i y i ε ε

and the leading term of the asymptotic expansion depends on the fast variable, y.

832

17

Homogenization

Another example of this type is a variational problem with the values of the admissible functions prescribed in each cell. Such is the flow of viscous fluid through a periodic array of obstacles. One can model in this way the fluid motion in porous media. The first term of the approximation depends on fast variables. We outline briefly the corresponding derivation. D’Arcy law for flows in porous media. Consider a flow of viscous fluid through a region V containing a large number of periodically placed rigid obstacles. At the boundary of the region the velocity, vi , is given: vi = u i

on V,

(17.49)

while at the obstacles we set the no-slip condition vi = 0

on Bk .

(17.50)

The fluid is assumed to be incompressible: vi = 0. xi

(17.51)

The flow is so slow that the inertial forces can be neglected. Then the true velocity provides minimum to the total dissipation (see Sect. 12.2), μv(i, j) v (i, j) d V, D(v) = V −Bk

on the set of all velocity fields obeying the constraints (17.49), (17.50) and (17.51). Let vi ≡ v¯ i be slow changing functions. They obviously must obey the incompressibility condition ¯vi = 0. xi

(17.52)

Let us fix v¯ i and seek the minimum of the functional I (v) under the additional constraint vi = v¯ i . Then in the leading approximation, at each cell, a flow develops which is determined from the cell problem : ; D(¯vi ) = min μv(i, j) v (i, j) . periodic and divergence-free v v|B =0, vi =¯vi

The solution of this cell problem provides the first term of the asymptotic expansion. The minimum value of the cell problem is a quadratic function of v¯ i : D( v¯ i ) = D i j v¯ i v¯ j .

17.3

Some Non-asymptotic Features of Homogenization Problem

833

¯ In the leading approximation the functional D(v) is ¯ v) = D(¯

D i j v¯ i v¯ j d V.

(17.53)

V

Its minimization on the set of velocity fields v¯ i , subject to the incompressibility constraint (17.52) and boundary conditions v|V = u i , determines the true average velocity field. Note that the functional I¯ (¯v ) does not feel all three constraints v|V = u i and selects the one which it can keep in minimization, vi |V n i = u i n i . Denote by p the Lagrange multiplier for the incompressibility condition. Then, the velocity field minimizing the dissipation, D, obeys the equation D i j v¯ j = −

p . xi

(17.54)

One can check that p has the meaning of the average pressure. Equation (17.54) is the D’Arcy law for porous media. The derivation can be extended to random porous media following the general recipe of the next section.

17.3 Some Non-asymptotic Features of Homogenization Problem Effective coefficients. We complement the above consideration with some elementary comments which pertain to both periodic and random cases and do not use the asymptotic reasoning. We discuss, for simplicity the linear scalar case. Let, for definiteness of terminology, u(x) and a i j (x) be temperature and the heat conductivity coefficients, respectively. In a homogeneous body, the heat flux, q i , is a linear function of the temperature gradient, u ,i : −q i = a i j

u , x j

with some constant heat conductivity coefficients, a i j . In a micro-inhomogeneous body a i j are some functions of coordinates. In macro-homogeneous body that models the micro-inhomogeneous one, one can write for some “macroscopic heat flux,” q¯ i , and “macroscopic temperature gradient,” which still need to be defined, a relation ij

− q¯ i = aeff ij

u¯ , x j

(17.55)

where aeff are the effective heat conductivities. We are going to motivate the relation: the macroscopic heat flux is the volume average of the true heat flux: : ; q¯ i = q i .

834

17

Homogenization

Fig. 17.5 A scheme for experimental measurement of heat conductivity

Here · means the volume average: for any function, ϕ(x),

ϕ =

1 |V |

ϕ(x)d V. V

: ; : ; Moreover, q i depends linearly on u/x i , and the effective heat conductivities are the coefficients in this dependence: : i; u ij . − q = aeff x j

(17.56)

Indeed, consider a usual experimental setting to determine heat conductivities shown in Fig. 17.5. On the two sides of the specimen, A and B, a constant heat flux, q¯ 1 , is given, while other sides of the body are adiabatically isolated: q¯ i n i = 0 on these sides. The boundary conditions for the true heat flux, q i , can be written for this experiment as q i n i = q¯ i n i

on

V,

(17.57)

where q¯ i are the constants, q¯ 2 = q¯ 3 = 0. We make a generalization and consider a problem with arbitrary constants q¯ i in (17.57). The true temperature field in this problem is the minimizer of the functional

I (u) =

V

1 ij u u a (x) i dV + 2 x x j

q¯ i n i ud A.

(17.58)

V

The minimizer obeys the equations q i = 0, x i and the boundary condition (17.57).

−q i = a i j (x)

u , x j

(17.59)

17.3


835

Let us show that :

Indeed, the identity holds:

V

; q i = q¯ i .

qi d V =

(17.60)

q j n j x i d A.

(17.61)

V

It is obtained by applying the divergence theorem to the right hand side of (17.61) and using the first equation (17.59):

V

q j n j xi d A =

V

(q j x i ) dV = x j

V

q j δ ij d V =

q i d V. V

Then (17.60) follows from (17.61) and the boundary condition (17.57):

V

q j n j xi d A =

V

q¯ j n j x i d A = q¯ j

V

n j x i d A = q¯ j

V

x i d V = q¯ i |V | . x j

The boundary value problem (17.59) and (17.57) depends linearly on: the con; stants q¯ i . Therefore, the minimizer depends linearly on q¯ i , and so do u/x i . Hence, we can write

: ; u = Ai j q¯ j = Ai j q j , i x

(17.62)

% % % % where Ai j are some constants. Assuming that the matrix %aieff(−1) % has a non-zero j determinant and inverting (17.62), we obtain (17.56). Formula (17.62) can be considered : ;as the definition of the effective coefficients. Note that the volume average, u/x i , can be expressed, due to the divergence theorem, in terms of the boundary values of u:

1 u = un j d A. |V | V x j

(17.63)

Therefore, they can be measured experimentally. For example, in the experiment of Fig. 17.5,

uB − uA u = , x 1 l

where u B and u A are the temperatures at the sides B and A.

836

17

Homogenization

The minimum value of the functional Iˇ is a quadratic function of q¯ i . Denote the coefficients of this quadratic function by a¯ i(−1) : j 1 q¯ i q¯ j |V | . Iˇ = − a¯ i(−1) 2 j

(17.64)

The following relation holds: the tensor a¯ i(−1) is inverse to the tensor aeff . Indeed, j i ˇ differentiating I with respect to q¯ we have ij

Iˇ ¯ j |V | . = −a¯ i(−1) j q q¯ i

(17.65)

On the other hand, according to the rule of differentiation of the minimum value of the functional (17.58) with respect to the parameters q¯ i ,2 Iˇ =− q¯ i

un i d A.

(17.66)

V

The link between a¯ i(−1) and aeff follows from (17.60), (17.65), (17.66), (17.63) j and (17.58). So, ij

1 q¯ i q¯ j |V | . Iˇ = − aieff(−1) 2 j

(17.67)

Formula (17.67) can also be written as u 1 i j u |V | . Iˇ = − aeff 2 x i x j

(17.68)

This equation can also be considered as a primary definition of the effective ij coefficients aeff . The dual problem. The variational problem that is dual to the minimization problem for the functional (17.58) is (see Sects. 5.6 and 5.8) 1 −1 ai j (x)q i (x)q j (x)d V Iˇ = max − q i (x) V 2

(17.69)

where maximum is sought over all vector fields q i (x) satisfying the constraints q i (x) = 0 in V, q i (x)n i = q¯ i n i on V. x i

2

See Sect. 5.13.

(17.70)

17.3


837

From (17.67) and (17.69), min q i ǫ(17.70)

V

1 −1 1 a (x)q i (x)q j (x)d V = aieff(−1) q¯ i q¯ j . 2 ij 2 j

(17.71)

Another definition of effective coefficients. One can give another definition of effective coefficients. Consider the minimization problem for the functional I1 (u) =

V

1 ij u u a (x) i dV 2 x x j

(17.72)

on the set of all functions u(x) linearly depending on the coordinates at the boundary u = vi x i .

(17.73)

Here vi are some constants. The minimizer depends linearly on the constants vi . Therefore, the minimum value of the functional is a quadratic form of vi , and one can define the effective coefficients as the coefficients of this quadratic form: min I1 =

1 ij a vi v j |V | . 2 eff

(17.74) ij

The further analysis of the homogenization problem given shows that aeff in (17.68) and (17.74) are practically the same. The dual problem to the minimization problem of the functional (17.72) is (see Sect. 5.8) Iˇ1 = max qi

i

V

j

q (x)n i v j x d A −

V

1 −1 i j a (x)q (x)q (x)d V , 2 ij

(17.75)

where maximum is taken over all divergence-free vector fields, q i (x) = 0. x i

(17.76)

Since, due to the divergence theorem,

i

V

j

q ni v j x d A =

V

1 i q vi d V, 2

(17.77)

(17.75) can be also written as Iˇ =

: i; 1 −1 i j |V | . q vi − ai j q q max 2 q i ǫ(17.76)

(17.78)

838

17

Homogenization

Effective elastic moduli. Similar statements hold for an elastic body. Consider the minimization problem I (u) =

V

1 ijkl C (x)εi j εkl d V − 2

V

σ¯ i j n j u i d A → min u(x)

where σ¯ i j are some constants. Then : ij; σ = σ¯ i j ,

the effective elastic moduli can be defined by the equation

besides,

: ij; ijkl σ = Ceff εkl , 1 ijkl : ; min I = − Ceff εi j εkl . 2

(17.79)

One can also introduce the effective coefficients considering the minimization problem for the functional I1 (u) =

V

1 ijkl C (x)εi j εkl d V 2

on the set of displacements u i (x) that are some linear functions of coordinates at the boundary u i = ε¯ i j x j at V,

ε¯ i j = const.

(17.80)

The minimum value is a quadratic function of ε¯ i j , min I1 =

1 ijkl C ε¯ i j ε¯ kl |V | , 2 eff

(17.81)

and the coefficients in (17.79) and (17.81) can be considered as the effective elastic moduli. They are equal to the effective coefficients in (17.79) in the asymptotic limit of the homogenization problem to be discussed further. The minimizer of I1 (u) possesses the properties : ; : ; ijkl εi j = ε¯ i j and σ i j = Ceff ε¯ i j .

(17.82)

17.3


839

The first relation (17.82) is obtained by means of the divergence theorem : ; 1 εi j = |V |

V

1 u (i n j) d V = ε¯ (ik x k n j) d A = |V | V V x k 1 = ε¯ (ik d V = ε¯ i j . |V | V x j)

(i u j) d V =

1 |V |

To justify the second formula (17.82) we write the formula for the variation of Iˇ1 : δ Iˇ1 =

σ i j j δu i d V.

(17.83)

V

Here σ i j is the true stress field, i.e. the stress field at the point of minimum of I1 (u), and δu i the variation of displacements caused by variation of ε¯ i j . If δu i were equal to zero at V, then δ Iˇ1 would be zero as well as the variation of the functional at the point of minimum on the admissible variations. The variation of the true displacements which is due to variation of ε¯ i j is not zero at the boundary: δu i = δ ε¯ i j x j at V. We put δu i = δ ′ u i + δ ε¯ i j x j , where δ ′ u i = 0 at V, and plug this in (17.83). The term with δu i′ vanishes, and we get : ; δ Iˇ1 = σ i j δ ε¯ i j |V | .

(17.84)

Varying (17.81) with respect to ε¯ i j ,

ijkl

δ Iˇ1 = Ceff ε¯ kl δ ε¯ i j |V | ,

(17.85)

equating (17.85) and (17.84), and using the arbitrariness of δ ε¯ i j , we obtain the second formula (17.82). Voigt-Hill estimate. Variational formulas (17.64) and (17.74) allow one easily to estimate the effective coefficients from above and below. Let us take the trial field u = vi x i . Plugging it in (17.72), we obtain the estimate: for any constants vi , : ; ij aeff vi v j a i j vi v j .

(17.86)

It is called the Voigt-Hill estimate after Voigt, who suggested the use of the linear fields to approximate the state of inhomogeneous bodies, and Hill, who showed that

840

17

Homogenization

such fields yield the estimate (17.86). The flux, corresponding to the linear fields −q i = a i j (x)v j , varies over the body. For elastic bodies, the Voigt-Hill estimate takes the form: for any constants, ε¯ i j , : ; ijkl Ceff ε¯ i j ε¯ kl C ijkl ε¯ i j ε¯ kl .

(17.87)

Reuss-Hill estimate. The constant flux, q i , is an admissible field in the variational problem (17.78). Plugging q i (x) = q¯ i = const in (17.78) we obtain 1 i j 1 ij ¯ ¯ . q q aeff vi v j max q¯ i vi − ai−1 2 2 j q¯ i Since 1 ij 1 eff(−1) i j i a vi v j max q¯ vi − ai j q¯ q¯ , 2 eff 2 q¯ i ¯ i q¯ j , and we arrive at the estimate: function aieff(−1) q¯ i q¯ j should not exceed ai−1 j q j for any q¯ i :

i j ¯ i q¯ j . ¯ ¯ q q ai−1 aieff(−1) j q j

(17.88)

It is called the Reuss-Hill estimate after Reuss who used a constant flux to describe the fields in microinhomogeneous bodies (then u(x) changes according to equation i u ,i = −ai−1 j q ) and Hill who obtained (17.88). The corresponding estimate for elastic bodies is: for any constants σ¯ i j , eff(−1) i j kl (−1) σ¯ i j σ¯ kl . σ¯ σ¯ Cijkl Cijkl

(17.89)

The inverse tensor of elastic moduli is defined by (6.32). It is essential that the Reuss-Hill estimate is the best possible estimate if one only knows the average value : −1i j ; a , and nothing is known about the microgeometry of the material. The equality sign corresponds to layered materials, as we will see in Sect. 17.5.

17.4 Homogenization of Random Structures Probabilistic measure. We perceive the material shown in Fig. 17.1 as a material with random structure. Which meaning do we put in this notion? Mostly, we imply that this material possesses the following two features: • If we disturb the positions of inclusions, the effective properties of the material do not change; this may be true even if the shifted positions of the inclusions go

17.4

Homogenization of Random Structures

841

far away from the original ones, and the material gains a quite different microgeometry. • Such independence of the effective properties on the microgeometry is observed only for large samples; for small samples the effective properties depend on the particular microgeometry. To be specific, consider the heat conductivity problem I (u) =

u u 1 ij a (x) i d V → min . u(x): 2 x x j u|V =u

V

(17.90)

(b)

ij The inhomogeneities are described 1 by2 the3 dependence of the heat conductivities, a (b), on the space coordinates, x = x , x , x . The boundary values of temperature u are assumed to be some linear functions of coordinates: u (b) = vi x i . The constants vi have the meaning of the macroscopic temperature gradient. The problem is linear, and the minimizing function depends linearly on vi . Thus, the minimum value of the functional is a quadratic function of vi :

1 ij Iˇ = aeff vi v j |V | . 2

(17.91)

As we discussed in Sect. 17.3, (17.91) may be viewed as the definition of the effective coefficients for volumes of any size. To model the random structure mathematically, we admit for consideration many fields a i j (x). We put a i j = a i j (x, ω) , where the parameter ω runs through some set . Each value of ω corresponds to a possible realization of the microstructure. Functions a i j (x, ω) describe an ensemble of the microstructures which, as we think, model the physical situation like the one shown in Fig. 17.1. The construction of the ensemble, i.e. the functions a i j (x, ω) is similar to the mathematical modeling in other physical situations: we design the model and check it against the experimental results. If the coincidence is satisfactory, the ensemble constructed indeed captures the physical reality. If the physics of the problem is adequately captured by its mathematical formulation (17.90), then all that the ensemble a i j (x, ω) must do is to model properly the microgeometry of the microstructures. Usually, ω-variables have the meaning of the parameters determining the microgeometry. For example, if the particles on Fig. 17.1 were spheres of equal radii, one can mean by ω the positions of the centers of the spheres. If the spheres had different radii, the radii should be included in the ω-variables. We will call the set the parameter space. The mathematical notion of random field involves, in addition to the construction of the ensemble a i j (x, ω), one more ingredient, the probabilistic measure on the set of realizations . This additional structure accounts for the frequency of a certain realization ω to be observed in experiments. Knowing the probability measure, one can compute various probabilistic characteristics, e.g., the average value of the field

842

17

Homogenization

a i j (x, ω) at a space point, Ma i j (x, ω) , correlations, Ma i j (x, ω) a i j x ′ , ω , prob ij ability density of local characteristics, Mδ a − a i j (x, ω) , etc. The questions arises: how to construct the probabilistic measure on the set ? Certainly, such a construction should reflect the information on microstructure obtained from experiments. We will address this issue further in Sect. 18.6, and first focus on the features of the homogenization problem which do not depend on the particular choice of the probabilistic measure. Homogenization as an asymptotic problem. The key parameter of the homogenization problems is the characteristic length ε on which the physical characteristics a i j (x, ω) become statistically independent. This means that for all |x1 − x2 | ≫ ε, D C ij ij Probability of the event a i j (x1 , ω) = a1 , a i j (x2 , ω) = a2 ≃ D D C C ij ij Probability of the event a i j (x1 , ω) = a1 · Probability of the event a i j (x2 , ω) = a2 .

If ε is much smaller than the size of the region V, then we observe the effective characteristics which do not depend on the microstructure. If ε becomes comparable with the size of the region V, then the effective characteristics fluctuate. The presence of a small parameter ε makes the homogenization problem an asymptotic one. The asymptotic analysis is especially simple for variational problems. The setting of homogenization in variational problems is as follows. Consider a stochastic variational problem u (17.92) I (u, ω) = L a (x, ω) , u, i d V − fudA → min . u(x) x V

V

The random field of physical characteristics, a (x, ω) , is assumed to possess quite a special structure. To describe it, we introduce an auxiliary three-dimensional space, R y , of “fast variables.” To prescribe a microstructure, we specify in this space a random field, a (y, ω). Then the actual physical characteristics in (17.92) are taken as x

,ω , (17.93) a=a ε where ε is a small parameter. The fast variables y are dimensionless, while x has the dimension of length. The stochastic variational problem (17.92) becomes depending on the small parameter ε : I (u, ω) =

V

u x

, ω , u, i d V − fudA. L a ε x

(17.94)

V

In the limit ε → 0, it may admit a homogenized description, i.e. Iˇ = min I (u) ≃ min I¯ (v) u

v

(17.95)

17.4


843

where I¯ (v) is some “homogenized” functional I¯ (v) =

L¯ v, v,i d V −

f vd A.

(17.96)

V V It turns out that this is indeed the case if the field a (y, ω) is ergodic and L a, u, u ,i obeys some physically non-restricting conditions. For ergodicity of physical characteristics, it is enough that the values of the random field a (y, ω) be statistically independent at sufficiently remote points y and y ′ . More precisely, this means that the two-point probability density, f 2 a, y; a ′ , y ′ , becomes the product of the onepoint probability densities for sufficiently remote points,

f 2 a, y; a ′ , y ′ − f 1 (a, y) f 1 a ′ , y ′ → 0 as y − y ′ → ∞.

In this case, the effective Lagrangian is determined by a variational problem which is analogous to the cell problem for periodic structures. We proceed to the construction of this problem. Kozlov’s cell problem. The cell problem for random structures is a variational problem for functionals defined on the functions of the fast y-variables. The domain of such functions is the entire R y -space, i.e. the “cell” for random structures is the three-dimensional space, R y . For any function ψ (y) one can introduce the space average · as 1

ψ (y) = lim λ→∞ |Bλ |

ψ (y) dy

Bλ

where Bλ is the ball of radius λ, and |Bλ | its volume. The averaged Lagrangian is determined from the following variational cell problem first found by S. Kozlov: L¯ (v, vi ) = min

ψ(y): ψ i =0

ψ . L a (y, ω) , v, vi + i y

(17.97)

y

In this problem, v and vi are considered as constant parameters. For ergodic fields, a (y, ω), the minimum value in (17.97) is the same for all ω (up to a set of ω of zero measure). Therefore, we may drop the argument ω in (17.97). Formula (17.97) assumes that Lagrangian L a, u, u ,i is strictly convex with respect to the field gradient u ,i . If it is not, one has to search for stationary values of the cell functional ψ L a, v, vi + i . y

844

17

Homogenization

Then the averaged Lagrangian, L¯ v, v,i , and the homogenized functional (17.96) may deliver many stationary points. Derivation of Kozlov’s cell problem. The derivation proceeds as follows. Consider a continuous function of fast and slow variables, f (x, y), for which the volume average f (x, y) exists for every x. Then, the following equality holds: lim

ε→0

V

f

x

ε

, x d V = f (x, y) d V.

(17.98)

V

˜ In order to prove this, we partition the x region V into cubes Cn with the size where is so small that the function f ε , x can be considered constant in each cube with respect to the second argument. Since, for ≫ ε,

C˜ n

f

⎛

1 , x d 3 x ≈ 3 ⎝ 3 ε λ

x

λC

⎞ xn 3 ⎠ f y + λ , x d y ≈ f (y, x)3 ,

where y = (x − xn ) /ε, xn is the center of the cube Cn and λ = /ε, the equality (17.98) holds. It is clear that the continuity condition for f (y, x) is not significant; it is only necessary that all operations be meaningful in (17.98). Let a (y) be some realization of the microstructure. We seek for a solution of the form u = u xε , x,ε where the functions u (y, x,ε) are defined for all y in the y-space. In the first approximation, 1 u(y, x,ε) u = , i x ε y i and, according to (17.98), in (17.94) the volume integral becomes 1 u(y, x,ε) L a (y) , u (y, x,ε) , d V. ε y i

(17.99)

V

It is assumed that the realization of the microstructure a (y) and the admissible fields u (y, x,ε) are such that L exists. Let L a, u, 1ε u(y,x,ε) ,ε → ∞ for ε → 0. Then, in the first approximation, the y i work of external forces can be neglected in (17.94). Finding the stationary points in the first approximation is reduced to finding the stationary points of the functional for a “cell,” 1 u(y, x,ε) . (17.100) L a (y) , u (y, x,ε) , ε y i

17.4


845

The variable x is a parameter in this functional. As in the case of periodic structures, what follows significantly depends on the structure of the stationary points of the functional (17.100). Here, the Lagrangians for which the stationary points of the functional (17.100) do not depend on fast variables will be considered. Everything said about other cases for periodic structures is true for the random structures as well. So, let the stationary points of the functional (17.100) do not depend on the fast variables in the first approximation: u = v (x). The functions v (x) make up the set M0 of the general scheme of the variational-asymptotic method. Holding v, we look for its first correction, u = v + u ′ (y, x,ε) ,

(17.101)

where u ′ is smaller than v in the asymptotic sense, i.e. u ′ → 0 for ε → 0. We fix v by the condition v = u. Then

u ′ = 0.

(17.102)

Substituting (17.101) into the functional (17.94), analogously to the periodic case, and neglecting the terms i u compared to ε−1 u(y, x,ε) /y i and u ′ compared to v, we obtain the functional 1 u ′ d V. (17.103) L a (y) , v (x) , v,i (x) + ε y i V

After the change of the required functions, u ′ → ψ, u ′ = εψ, determining of ψ is reduced to finding the stationary points of the functional ψ (y) L a (y) , v, vi + , y i

(17.104)

where v and vi are some constant parameters. The condition of u ′ being smaller than v in the asymptotic sense means that ψ (y) grows more slowly than the linear function as y → ∞, ψ (y) →0 |y|

for |y| → ∞,

(17.105)

1 where |y| = yi y i 2 . Indeed, if ψ (y) ∼ |y| for |y| → ∞, then εψ ∼ x for y → ∞, i.e. the second term of the expansion (17.101) makes the same contribution as the first one. If ψ (y) increases faster than |y| for |y| → ∞, then the contribution of the second term in the expansion (17.101) will be greater than that of the first. Therefore, the condition (17.105) has to be satisfied. Condition (17.105) yields

ψ = 0. y i

(17.106)

846

17

Homogenization

In order to justify (17.106), it is sufficient to make a transition from the volume to the surface integral in the definition of volume average and use (17.105). The stationary points of the functional (17.104) are sought on the set of functions ψ satisfying the condition (17.105) and the constraint ψ = 0, which follows from (17.102). The condition (17.105) may be replaced by (17.106). The stationary points of the cell functional (17.104) do not depend on ε, and the first two terms of the expansion of u with respect to ε are u = v + εψˇ y, v, v,i .

(17.107)

Assume that the cell functional has only one stationary point. Then the second term of the expansion is defined by v, and the set N of the general scheme of the variational-asymptotic method coincides with the set M0 . The value of the cell functional at the stationary point is L¯ v, v,i . One can check that the following terms in the expansion (17.107) make small contributions to the functional. Therefore, in the first approximation, the functional (17.92) coincides with the functional (17.96) and the true value of the field vˇ is the stationary point of the functional (17.96). In this reasoning it is not essential how many stationary points the cell functional (17.104) and the homogenized functional (17.96) have: we pick them all. However, if L is a strictly convex function of u ,i , then the cell problem is reduced to the minimization problem (17.97) with the unique minimizer. Usually, the minimizer of the cell functional ψˇ is bounded at infinity; therefore, it is sufficient to look for the minimum in (17.97) on a set of functions bounded on infinity. Moreover, for physical problems, it is natural to assume that ψ/y i and L/ψ ,i are bounded on infinity. Further we will consider the Lagrangians and the microstructures for which this property holds. One comment about the Euler equation for the variational cell problem (17.97) is now in order. Taking the variation of the functional L and integrating by parts, we get

δL = 0, δψ δψ

δL L ≡− i . δψ y ψ ,i

(17.108)

One cannot conclude from this equation that δL/δψ = 0. Equation (17.108) is satisfied, for example, if δL/δψ is an arbitrary smooth function in R y vanishing δL ˜ = 0 for any function outside the ball of some finite radius: in this case ψ δψ

˜ since the integral of ψ˜ δL over R y is finite. In order to guarantee the equality ψ, δψ δL/δψ = 0, one has to assume that δL/δψ “behaves in the same way” in all remote parts of the space. For example, if δL/δψ is a periodic function, then choosing δψ to δL is reduced to an integral over the be a periodic function as well, we see that δψ δψ δL = 0 yields the Euler equation δL/δψ = 0. The cell; therefore the equality δψ δψ same is true for almost periodic functions considered further in Sect. 18.1: if δL/δψ

17.4


847

is an almost periodic function and δψ is an arbitrary almost periodic function, then (17.108) yields δL/δψ = 0. i Dual cell problem. Let L ∗ a, v κ , pκ be the Young-Fenchel transformation of the Lagrangian with respect to field gradients: i κ i = max pκ u i − L a, v κ , u iκ . L ∗ a, v κ , pκ κ ui

Then, for convex L , i κ i , u i − L ∗ a, v κ , pκ L a, v κ , u iκ = max pκ i pκ

and the cell problem can be written as ψ κ κ ∗ κ i i ¯L v κ , viκ = min max pκ vi + − L a, v , pκ . κ i y i pκ ψκ : ψ y i

=0

Changing the order of minimum and maximum, we obtain the dual cell problem 0: i ; κ : ∗ ;1 i L¯ v κ , viκ = max pκ vi − L a (y) , v κ , pκ (y) , i pκ

(17.109)

i where maximum is sought over all functions pκ satisfying the conditions i pκ = 0, y i

i pκ are bounded on infinity.

(17.110)

As for periodic structures, the variational problem (17.109) can be written in terms of the Young-Fenchel transformation of the averaged Lagrangian i κ vi − L¯ v κ , viκ , v κ , p¯ xi = max p¯ κ i p¯ κ

i κ L¯ v κ , vix = max p¯ κ vi − v κ , p¯ iκ . κ p¯ i

(17.111)

: i ; i To this end we introduce in (17.109) an intermediate constraint, pκ = p¯ κ , to put (17.109) in the form, L¯ v κ , vix = max κ p¯ i

max i pκ ∈(17.110)

pκi = p¯ κi

0:

;1 ; x : ∗ i i vi − L a(y), v κ , pκ pκ

848

17

⎡

⎢ i x = max ⎣ pκ vi − κ p¯ i

min

i pκ ∈(17.110)

pκi = p¯ κi

Homogenization

⎤ : ∗ ;⎥ i L a(y), v κ , pκ ⎦.

(17.112)

Comparing (17.112) with (17.111), we see that the dual variational problem can be written as v κ , p¯ iκ =

min

i pκ ∈(17.110) i i pκ = p¯ κ

:

; i . L ∗ a, v κ , pκ

(17.113)

Therefore, as for periodic structures, the computation of the averaged Lagrangian for random structures is reduced to the solution of the dual cell problem and the subsequent computation of the Young-Fenchel transformation, L¯ = ∗ .

: ; = min L ∗ ,

If material characteristics a (y) are periodic, the minimizers of the cell problem are periodic functions of y. Indeed, if the function L a, u, u ,i is strictly convex with respect to u ,i , the cell problem has the only minimizer. The solutions of the periodic cell problem belong to the set of admissible functions and satisfy the Euler equations for the variational cell problem, and, consequently, provide the required minimum. Therefore, for periodic structures, the set of admissible functions in (17.97) can be reduced to the set of periodic functions. Hence, the space averaging over R y becomes the space averaging over a cell of the periodic structure, and the variational formulae (17.97) and (17.111) transform into the corresponding formulae for periodic structures. Further, we will use the general formulae (17.97) and (17.111), implying that they are true for both periodic and random structures. Some features of the cell problem. A number of assertions about the qualitative properties of microstructures can be derived from (17.97) and (17.111). The most important one is the strict convexity of L¯ with respect to vi ; it takes place for the strictly convex micro-Lagrangians L with respect to u ,i . Indeed, let ψˇ (y, v, vi ) be 1

2

1

2

the minimizer of the cell problem (17.97), and v i and v i be two values of vi , v i = v i .

1

2

Setting vi = 21 v i + v i in (17.97) and choosing an admissible field, ψ (y) =

2 1 1 ψˇ y, v, v i + ψˇ y, v, v i , we get an estimate: 2

2 1 1 ψˇ 1 ¯L v, 1 v1 i + v2 i ≤ L a (y) , v, v i + i y, v, v i + v i + 2 2 y

ψˇ 2 + i y, v, v i . y

17.5

Homogenization in One-Dimensional Problems

849

Due to convexity of L, this yields

1 1

2

¯L v, 1 v1 i + v2 i ≤ L¯ v, v i + L¯ v, v i , 2 2 as claimed. If the micro-Lagrangian L is strictly convex with respect to u and u i , the strict convexity of L¯ with respect to v and vi is proved analogously. So the composite comprising the phases with “good” properties may not have non-convex anomalies in energy. The only degenerations which may arise relate to vanishing of L¯ or L¯ ∗ ; the corresponding examples are given in the next sections. Note another consequence of the general homogenization formulae: it turns out i that the “average stresses,” p¯ κ , are potential functions of the “average strains,” viκ , and the potential is the averaged Lagrangian, :

i pκ

;

L¯ v κ , viκ L i ≡ , ≡ p¯ κ = u iκ viκ

viκ

i v κ , p¯ κ = . i p¯ κ

(17.114)

This follows directly from (17.109). There are just a few exact solutions to the cell problem. Some of them are considered in the next sections.

17.5 Homogenization in One-Dimensional Problems The homogenization problem is easily solved when the medium characteristics and the required functions depend on one independent variable: du κ (x) , L = L a, u κ , dx

y=

x . ε

Let us show that in this case the calculation of the averaged Lagrangian is reduced to an algebraic problem of calculating the Young-Fenchel transformation and to subsequent integration. The easiest way to find the averaged Lagrangian is to use the dual variational problem (17.111). Indeed, the constraints in the dual variational problem are dpκ = 0, dy

pκ = p¯ κ ;

therefore the set of admissible fields, pκ (y) , consists of one element pκ (y) = const = p¯ κ . It follows from (17.113) that is obtained by integration of L ∗ over y: : ; v κ , p¯ κ = L ∗ (a (y) , v κ , p¯ κ ) .

(17.115)

850

17

Homogenization

¯ the Young-Fenchel transformation of v κ , p¯ κ with respect to p¯ κ To find L, should be computed: κ dv κ κ ¯L v κ , dv = max p¯ κ − (v , p¯ κ ) . p¯ κ dx dx The minimizing functions of the cell problem ψˇ κ are determined by the relation connecting the minimizing field of the dual variational cell problem and ψˇ κ : d ψˇ κ L ∗ (a (y) , v κ , p¯ κ ) dv κ + = dx dy p¯ κ

(17.116)

The ordinary differential equations (17.116) should be supplemented by the conditions (see (17.102)) : κ; ψˇ = 0,

and the algebraic equations linking dv κ /d x and p¯ κ (see (17.114)) ; v κ , p¯ κ dv κ : ∗ = = L (a (y) , v κ , p¯ κ ) dx p¯ κ p¯ κ

(17.117)

Equations (17.117) can also be considered as a necessary condition for the solvability of (17.116) in the class of functions ψ κ (y) bounded at infinity (for periodic structures, in the class of periodic functions): they are obtained by applying the averaging · to (17.116). Equations (17.116) and (17.117) determine ψˇ κ y, v κ , vxκ . The functions ψˇ κ increase more slowly at infinity than the linear function y; under some additional constraints on L and a κ (y) , functions ψˇ κ are bounded at infinity. In particular, ψˇ κ are bounded at infinity and periodic if L is a continuous function of a κ , and a κ (y) are periodic. Consider some examples. Example 1. Consider a straight extendable beam on an elastic foundation: 2 du 1 1 + C (y) u 2 . L = E (y) 2 dx 2 The Young-Fenchel transformation of the function L with respect to du/d x is L ∗ (v, p) = According to (17.115),

1 1 p 2 − C (y) v 2 . 2E (y) 2

17.5


(v, p¯ ) =

851

1 1 1 p¯ 2 − C (y)v 2 . 2 E (y) 2

Calculating the Young-Fenchel transformation of the function 1 (v, p¯ ) with respect to p¯ , we find the averaged Lagrangian L: 2 dv 1 1 dv = L¯ v, + Cv 2 . 1 dx d x 2 2 E(y)

(17.118)

The “effective” beam behaves as an extendable beam on an elastic foundation with the macrocharacteristics E −1 −1 and C. Note the different averaging rules for the coefficients E and C. Let the beam have a periodic structure, and each cell consist of two sections of the same lengths with the characteristics E 1 , C1 and E 2 , C2 . If the properties differ considerably, E 1 ≪ E 2 , C1 ≪ C2 , then

E −1 −1 = 2

1 1 + E1 E2

−1

≈ 2E 1 , C =

1 1 (C1 + C2 ) ≈ C2 , 2 2

i.e. the composite as a whole acquires the compressibility of the weak phase and the foundation rigidity of the strong phase. Example 2. Consider a mixture of n homogeneous media with the Lagrangians du κ du κ , . . . , L n uκ , . L 1 uκ , dx dx The components have volume concentrations, c1 , . . . , cn . According to (17.115), (v κ , p¯ κ ) = c1 L ∗1 (v κ , p¯ κ ) + . . . + cn L ∗n (v κ , p¯ κ ) .

(17.119)

Let us take a special case of power functions for energies of each phase, an a1 du α1 L1 = , . . . , Ln = α1 d x αn

αn du . dx

The powers, α1 , . . . , αn , are numbered in the increased order. From Example 3 of Sect. 5.7, L ∗1 ( p) = where

a1 β1

β1 βn p , . . . , L ∗ ( p) = an p , n a βn an 1

1 1 1 1 + = 1, . . . , + = 1. α1 β1 αn βn

852

17

Homogenization

From (17.119) we have ( p¯ ) =

β1 p¯ + . . . + cn an a βn 1

c1 a1 β1

βn p¯ . a

(17.120)

n

In order to construct the function L¯ (dv/d x), one has to calculate the YoungFenchel transformation of the function (17.120). It cannot be expressed in terms of elementary functions. However, the asymptotics of L¯ for dv/d x → 0 and dv/d x → ∞ can easily be found. For dv/d x → 0, we also have p¯ → 0. Since the numbers β1 , . . . , βn are arranged in decreasing order, the last term in (17.120) is the leading one for p¯ → 0. Calculating its Young-Fenchel transformation, we find that L¯

dv dx

an

∼

αn

αn cnβn

αn dv dx

dv as → 0. dx

α1 dv dx

dv as → ∞. dx

(17.121)

The asymptotics of L¯ for |dv/d x| → ∞ is constructed similarly: for |dv/d x| → ∞, also | p¯ | → ∞; therefore, the first term in (17.120) becomes the leading one, and L¯

dv dx

∼

a1 α1 β

α1 c1 1

(17.122)

These seemingly paradoxical results are related to the fact that it is always the most soft elements that are excited. For large “stresses” p¯ , the most soft elements are those with small α, while for small p¯ , the most weak elements are those with n −α β

α

− β1

large α. The rigidity coefficients, an cn n and a1 c1 1 , increase significantly for small concentration, because the soft elements are to make a small contribution to the total “deformation” of the composite, dv/d x. Example 3. The composite may demonstrate the new properties, which the phases do not possess. We illustrate this feature by the following model example. Consider a composite comprised of the phases with the Lagrangians a L= α

α du , dx

where the coefficient a is the same for all phases, while α is a function of y, and α(y) > 1. Then (see Example 3 of Sect. 5.7) L ∗ ( p) = and, according to (17.115),

a β

p β , a

1 1 + = 1, α β

17.5


853

0,

b > 0, (17.124)

where (b) is the gamma-function (b) =

∞

t b−1 e−t dt.

0

The parameters of the gamma-distribution, the constants b and c, are expressed in terms of the mathematical expectation β0 and the dispersion σ of the power β as c=

β0 − 1 (β0 − 1)2 , b= σ σ

β0 = Mβ, σ = M (β − β0 )2 .

According to (17.123) and (17.124), cb | p¯ | ( p¯ ) = (b)

∞ 0

ξ b−1 −(c−ln| p¯ |)ξ a e dξ. 1+ξ

(17.125)

It is seen from (17.125) that ( p¯ ) = ∞ if the amplitude of the stress p¯ is greater than the number pmax ≡ aec . Therefore, the composite does not admit large stresses and, in this sense, behaves similar to elastic-plastic medium. Layered composites. Layered composites are those for which characteristics change only in one direction while the required functions may depend on all coordinates. Let us show that the averaged Lagrangian of the layered composites is calculated essentially in the same way as for one-dimensional structures. Denote the coordinate along the direction in which the physical characteristics change by x, the corresponding fast coordinate by y, and the other coordinates by x α , where small

854

17

Homogenization

Greek indices run the values 1, 2, . . . , n − 1. A comma before Greek index α denotes derivative with respect to x α ; a comma before index x denotes derivative with κ α κ κ respect to x: u κ ,α ≡ u /x , u ,x ≡ u /x. The solution will be sought in a class κ κ of functions of the form u = u (y, x, x α ) . κ Let L 0 a , u κ ,x be the leading (in asymptotic sense) part of the Lagrangian as κ u ,x → ∞. Assume that the variational problem u κ κ → min, L a (y) , u(y) y where · is the averaging operator over y, has only constant minimizing functions, u κ = const. Then the first term of the asymptotic expansion does not depend on the fast variable, and the solution has the form u κ = v κ (x, x α ) + εψ κ (y, x, x α ) . Calculating the averaged Lagrangian (17.97), we take into account that ψ κ depend only on one fast variable, y. Then, ψ κ κ κ κ L a , v , L¯ v κ , vxκ , vακ = min , v , v + (y) x α ψκ y

(17.126)

where the minimum is sought over all functions ψ κ (y) which are bounded at infinκ κ ity, while v κ , vxκ ≡ v,x , and vακ ≡ v,α are some parameters. Let us construct the dual variational problem. Denote the Young-Fenchel trans⊕ formation of the Lagrangian L with respect to u κ ,x by L : 1 0 κ κ κ pκ u κ L ⊕ a, u κ , pκ , u κ ,x − L a, u , u ,x , u ,α . ,α = max κ u ,x

The symbol ⊕ is introduced in order to distinguish this transformation from L ∗ , the Young-Fenchel transformation of the Lagrangian L with respect to all derivaκ tives, u κ ,x , u ,α . Since κ ⊕ κ L a, u κ , u κ a, u κ , pκ , u κ ,α ,x , u ,α = max pκ u ,x − L pκ

we obtain from (17.126) ;1 0 : L¯ v κ , vxκ , vακ = max pκ vxκ − L ⊕ a (y) , v κ , pκ , vακ . pκ (y)

(17.127)

Here, the maximum is sought over all functions pκ (y) satisfying the condition dpκ /dy = 0. Since the admissible functions pκ (y) are constant, pκ (y) = const = p¯ κ , (17.127) can be written as

17.5


;1 : 0 L¯ v κ , vxκ , vακ = max p¯ κ vxκ − L ⊕ a (y) , v κ , p¯ κ , vακ . p¯ κ

855

(17.128)

So the computation of the effective Lagrangian is reduced to the algebraic : problem ; of computation of the Young-Fenchel transformation, L ⊕ , integration, L ⊕ , and one more Young-Fenchel transformation (17.128). Example 4. Layered elastic composites. Consider an elastic layered geometrically linear composite with free energy density per unit volume F εi j , εi j = 1 u i, j + u j,i . Let the elastic characteristics change along axis 3, x 3 ≡ x. Suppose 2 p that for εi j εi j → ∞, U ≥ const εi j εi j , where p > 21 . Then the variational problem arising at the first step of the variational-asymptotic method has the solution u i = const. Consequently, the leading term of the asymptotic expansion of displacements does not depend on y, and the averaged energy is found from the variational problem 1 dψα dψ F¯ ε¯ αβ , ε¯ 3α , ε¯ 33 = min F a (y) , ε¯ αβ , + ε¯ 3α , + ε¯ 33 , ψα ,ψ 2 dy dy where ε¯ αβ = v(α,β) , 2¯ε3α = vα,3 + v3,α and ε¯ 33 = v3,3 are the macrodeformations. Denote the Young-Fenchel transformation of F with respect to ε¯ α3 and ε¯ 33 by F ⊕ , and the Young-Fenchel transformation of F¯ ε¯ αβ , ε¯ 3α , ε¯ 33 with respect to ε¯ α3 and ε¯ 33 by F¯ ⊕ : F ⊕ a (y) , εαβ , p α3 , p 33 = max 2 p α3 εα3 + p 33 ε33 − F a (y) , εαβ , ε3α , ε33 , ε3α ,ε33 F¯ ⊕ ε¯ αβ , p¯ α3 , p¯ 33 = max 2 p¯ α3 ε¯ α3 + p¯ 33 ε¯ 33 − F¯ ε¯ αβ , ε¯ 3α , ε¯ 33 . ε¯ 3α ,¯ε33

The coefficient 2 is introduced to make p α3 and p¯ α3 the components of a tensor (see a note on differentiation of scalar functions with respect to components of a symmetric tensor in Sect. 3.3). From the dual cell problem we find that : ; F¯ ⊕ ε¯ αβ , p¯ α3 , p¯ 33 = F ⊕ a κ (y) , ε¯ αβ , p¯ α3 , p¯ 33

(17.129)

0 1 F¯ ε¯ αβ , ε¯ 3α , ε¯ 33 = max 2 p¯ α3 ε¯ α3 + p¯ 33 ε¯ 33 − F¯ ⊕ ε¯ αβ , p¯ α3 , p¯ 33 .

(17.130)

where p¯ α3 and p¯ 33 are the constant parameters. After integration in (17.129), the macroenergy is reconstructed by the Young-Fenchel transformation, p¯ α3 , p¯ 33

So the calculation of macroenergy is reduced to an integration and the calculation of the Young-Fenchel transformation. Let us find the macroenergy in the particular case of a layered composite made up of linearly elastic isotropic phases,

856

17

Homogenization

2F = λ (y) (εii )2 + 2μ (y) εi j εi j ,

(17.131)

or, explicitly writing the dependence on the strain tensor components εαβ , εα3 and ε33 , 2 2F = λ(εαα )2 + 2μεαβ εαβ + 2λεαα ε33 + (λ + 2μ)ε33 + 4μεα3 εα3 .

(17.132)

Calculating F ⊕ , we get 1 2F ⊕ = (λ + 2μ)−1 ( p 33 − λεαα )2 + μ−1 pα3 p α3 − λ(εαα )2 − 2μεαβ εαβ . (17.133) 2 The expression for F¯ ⊕ follows from (17.129) and (17.133): 2 F¯ ⊕ = (λ + 2μ)−1 ( p¯ 33 )2 − 2 λ(λ + 2μ)−1 p¯ 33 ε¯ αα + λ2 (λ + 2μ)−1 (¯εαα )2 + 1 (17.134) + μ−1 p¯ α3 p¯ α3 − λ(¯εαα )2 − 2 μ¯εαβ ε¯ αβ . 2 ¯ From (17.131) and (17.135), we find F: 2 F¯ = ( λ + λ(λ + 2μ)−1 2 (λ + 2μ)−1 −1

− λ2 (λ + 2μ)−1 )(¯εαα )2 + 2 μ ε¯ αβ ε¯ αβ + 4 μ−1 −1 ε¯ α3 ε¯ α3

2 +2 λ(λ + 2μ)−1 (λ + 2μ)−1 −1 ε¯ αα ε¯ 33 + (λ + 2μ)−1 −1 ε¯ 33 .

(17.135)

The effective coefficients look quite odd; however, the “mechanism” of their construction is absolutely lucid physically: the true stresses p 33 and p α3 must be constant in the first approximation. If λ, μ = const, then the expression (17.135) transforms into (17.132), as it should.

17.6 A One-Dimensional Nonlinear Homogenization Problem: Spring Theory A spring with a large number of coils can approximately be considered as a onedimensional continuum (an equivalent beam) whose “particles” are the coils of the spring (Fig. 17.6). The stress state of the spring oscillates rapidly. The exact problem can be replaced by a “homogenized” one in which the required functions vary slowly from coil to coil. Homogenized spring behaves as a beam with some effective properties. It is called the equivalent beam. Derivation of the governing equations for the equivalent beam is a one-dimensional homogenization problem. The problem is nonlinear because the displacements of the spring can be large. Due to nonlinearity, one can expect some peculiar interactions between extension and torsion of the equivalent beam which are hard to guess from pure phenomenological reasoning. In this section

17.6

A One-Dimensional Nonlinear Homogenization Problem: Spring Theory

857

Fig. 17.6 A spring and its equivalent beam

we outline the results of asymptotic analysis of the energy functional referring the reader for details to the paper cited in Bibliographic Comments. Spring geometry in undeformed state. A spring is an elastic beam which has in the undeformed state the shape of a spiral. To describe this shape analytically, we first introduce a smooth curve ¯ 0 which will further have the meaning of the shape of the equivalent beam in the undeformed state, so that the spring in the undeformed state 0 twines around ¯ 0 . The parametric equations of ¯ 0 are written as x i = R˚ i (ζ ), where ζ is the arc length along ¯ 0 . At each point of ¯ 0 we attach an orthotriad, t˚ai (ζ ), a = 1, 2, 3, such that t˚3i = d R˚ i dζ . The parametric equations of any other curve 0 twining around ¯ 0 can be written as r˚ i (ζ ) = R˚ i (ζ ) + ρ˚ a (ζ ) t˚ai (ζ ).

(17.136)

The functions ρ˚ a (ζ ) have the meaning of projections of the “local position vector” ˚ r˚ i − R˚ i , on the orthotriad, t˚i . of , a The curve ˚ is also endowed with an orthotriad τ˚αi . Denote the projections of this orthotriad on the vectors t˚ai by αab : ·b ˚i ta . τ˚ai = α˚ a·

(17.137)

% ·b % % is orthogonal. The matrix %α˚ a· The arc length of ˚ is denoted by ξ. The coordinate ξ changes on the segment [0, L]. We assume that there is a one-to-one correspondence between ξ and ζ : ζ = ζ˚ (ξ ).

858

17

Homogenization

Functions ζ˚ (ξ ) and ρ˚ a (ζ ) are linked by the condition d r˚ i d r˚i = 1. dξ dξ

(17.138)

Formulas (17.136) and (17.137) can be written for the undeformed state of an arbitrary beam. By definition, we call an elastic beam the spring if, for some choice ·b of R˚ i (ζ ) and t˚ai (ζ ), the functions ρ a (ξ ) and αa· (ξ ) and the physical characteristics of the beam can be presented as periodic functions of a fast variable η and, possibly, a slow variable ξ , i.e. they have a form f (η, ξ ) where f is a periodic function of η with the period unity. In particular, we can write ·b ·b ρ a = ρ a (η, ξ ) and αa· = αa· (η, ξ ).

The fast variable, η, is assumed to be a function of ξ : η = η(ξ ). This function contains a small parameter. It can be introduced as the characteristic length, , of η(ξ ), i.e. by the relation, dη 1 ≡ . dξ The characteristic length, , has the meaning of the length of a coil: when η changes from 0 to 1, ξ increases for . Usually for industrial springs, = const. Further equations hold true in a general case when is a function of slow variable, i.e. the coil length changes slowly along the spring. The spring contains a large number of coils, i.e. ≪ L. We assume also that the parameters,

κ˚ ≡

d ζ˚ 1 and ω(0)a = eabc t˚,ξib t˚ic , dξ 2

change slowly along the spring. Consider an example. Let the equivalent beam in the undeformed state be straight: R˚ i = a i ξ , t˚1i = bi , t˚2i = ci , where a i , bi , ci are orthogonal constant unit vectors; the curvature-twist parameters of ¯ 0 , ωa , are zeros. We set

17.6


ρ 1 = R cos 2π η , ρ 2 = R sin 2π η , ρ 3 = 0,

859

(17.139)

η = ξ/ , ζ = κξ ˚ , κ˚ = sin α , α ∈ [−π/2, π/2] .

Here R is the radius of the spring, the coil length, α the pitch of the coil. Due to (17.138), the parameters R, and α are linked by the relation 2π R = cos α. The curve ˚ defined by (17.136) and (17.139) is the spiral. We choose τ˚1i and τ˚2i ˚ Then the components of the orthogonal matrix αab to be normal and binormal on . representing the projections of τ˚ai on t˚ai are: α1b = (− cos 2π η , − sin 2π η , 0)

(17.140)

α2b = (sin α sin 2π η , − sin α cos 2π η , cos α) α3b = (− cos α sin 2π η , cos α cos 2π η , sin α) Springs so defined are called regular cylindrical springs, and their geometry is ˚ ω˚ a , are given by two parameters, R and α. The curvature-twist parameters of , constant in this case: ω˚ 1 = 0 , ω˚ 2 = 2π −1 cos α , ω˚ 3 = 2π −1 sin α. The energy functional. Springs possess a rich set of inextensional deformations; therefore their deformations are described by the Kirchhoff-Clebsch theory (see Sect. 15.1). We assume that the spring is deformed by prescribing some kinematic conditions at the ends. Then the true deformed state is a stationary point of the functional L 0

1 αβ A α β + C2 + 2B α α ds 2

(17.141)

on the set of all inextensional deformations obeying the boundary conditions. The Greek indices run values 1, 2. The curvature-twist measures α and are defined in Sect. 15.1. To have shorter relations which involve summation over Latin indices, we identify and 3 . Deformed state. To describe the deformed state of the spring, we introduce the ¯ equivalent beam, a curve, , x i = R i (ζ ) , endow it with an orthotriad, tai (ζ ), and assume that the deformed state of the spring is described by the formula similar to (17.136):

860

17

Homogenization

r i = R i (ζ ) + ρ a tai (ζ ).

(17.142)

The parameter ζ is no longer the arc length: in deformed state, the equivalent beam is extended. The parameter ξ remains the arc length in the deformed state because the spring is inextensible. The dependence ζ (ξ ) characterizes the extension of the equivalent beam. As the measure of the extension, we use the parameter γ =

1 κ 2 −1 , 2 κ˚

κ≡

dζ , dξ

κ˚ ≡

d ζ˚ . dξ

Functions, r i and ρ a in (17.142) are considered as the functions of slow and fast variables periodic with respect to the fast variable η. The same is true for the pro·b , defined as jections of τai on the orthotriad of the equivalent beam, αa· ·b i τai = αa· tb (ζ ) .

We now have to link the characteristics of the equivalent beam with that of the spring. To this end, we define the operation of averaging over a coil, · , as integration over η : for any function, f (η, ξ ) , of the fast and slow variables η and ξ, 1

f = f (η, ξ ) dη. 0

We put on the functions ρ a the constraints : a; ρ = 0.

(17.143)

This specifies the meaning of the position vector of the equivalent beam : ; Ri = r i .

To fix tai , we note that for given r i and ταi , (and, thus, R i ) the orthotriad tai is not determined uniquely: the rotation of the vectors t1i and t2i around t3i does not change ·b : r i , ταi and R i if simultaneously one changes the matrix αa· ·γ tαi → tγi aα· , t3i → t3i

ααβ → ρα →

·γ ααγ aβ· , αα3 → αα3 ·γ ργ aα· , ρ3 → ρ3 ,

(17.144) , α3α → α3α , α33 → α33

·γ

aα· being an orthogonal two-dimensional matrix. Hence, we set an additional condition: ρ 2 |η=0 = 0 , ρ 1 |η=0 > 0.

(17.145)

17.6


861

It makes t1i coincide with r i − R i (ζ ) − ρ 3 t3i (ζ ) /ρ 1 at η = 0. The deformation of the equivalent beam is characterized by its elongation and curvature-twist measures a = κωa − κω ˚ (0)a , 1 1 ωa = eabc t,ξib tic , ω(0)a = eabc t˚,ξib t˚ic . 2 2 The deformation of the spring is characterized by the curvature spring measures a = ωa − ω˚ a , where ωa are calculated from the formulas ωa =

1 1 ·c θa + καab ωb + eabc α|ξbd αd· , 2

θa ≡

1 bd ·c eabc α|η αd· . 2

(17.146)

The vertical bar in the subscripts before the η and ξ denote partial differentiation of the functions dependent on the fast and slow variables with respect to η, ξ , while comma before ξ denotes “full derivative,” i.e. derivative with respect to ξ which take into account that η = η(ξ ). Energy of the equivalent beam. The energy density of the equivalent beam F is the minimum value in the following variational cell problem: F = min

; 1 : αβ A α β + C2 + 2B α α 2

(17.147)

where the curvature-twist measures are a =

1 b θa − θ˚a + αab + (αab − α˚ ab ) κω ˚ b(0) .

The minimum in (17.147) is taken over all periodic functions αab (η) and ρ a (η) satisfying the constraints

α3a

·c = δab , αac αb·

b 1 = κδ3a + ρa|η + eabc + κ˚ ω˚ b ρ c , : a; ρ = 0 , ρ2 |η=0 = 0 , ρ1 |η=0 > 0.

(17.148) (17.149) (17.150)

The equivalent beam characteristics, a , , κ˚ and ωb(0) in the problem (17.147), (17.148), (17.149) and (17.150) are constant parameters, while the geometrical characteristics of the spring in the initial state, θ˚a and α˚ ab , and the physical characteristics of the spring, Aαβ , C and B α , are given periodic functions of η.

862

17

Homogenization

The boundary conditions for the functions R i and tai are found by the minimizers ρˇ (η) and αˇ ab (η) , and the known boundary values of r i and τai . This variational problem can be solved analytically in the case3 of a regular cylindrical spring when the bending of the equivalent beam, α , are small, so that the interaction terms between α and ≡ 3 , γ can be neglected, the bending rigidities of the spring, Aαβ , are isotropic, Aαβ = I δ αβ , and there are no interactions between extension and bending of the spring, B α = 0. In such a case, a

2 2π 2π 2 2 + 1−κ − 1 − κ◦ 2F = I 2π 2π κ˚ 2 +C + κ− IC α + α . √ 1 2 C − 2 (C − I ) 1 − κ˚

(17.151)

The deformed position of the equivalent beam, R i , tai , is a stationary point of the functional L

Fdξ.

(17.152)

0

The energy density, F, is not a convex function of the twist, , and the elongation, κ; accordingly, the energy functional is not convex on the set of admissible positions of the equivalent beam. This is manifested in the well-known instability of springs under torque.

17.7 Two-Dimensional Structures Two-dimensional cell problems, as a rule, do not admit exact solutions. However, in a number of exceptional cases, it is possible to find the averaged Lagrangian without knowing the solutions of the cell problem just by using the general features of the original and the dual variational cell problems. Besides, these features yield some universal relationships for the effective coefficients. This and the next section are concerned with these issues. Consider a scalar problem for a two-dimensional structure. According to (17.97), ψ ¯ L(v, vα ) = min L a, v, vα + α . ψ y 3

There are solutions under more general assumptions (see [284]).

(17.153)

17.7

Two-Dimensional Structures

863

The small Greek indices α, β, γ run values 1, 2. Construction of the averaged Lagrangian for two-dimensional structures is based on the remarkable feature of the dual variational problem in the two-dimensional case. It turned out that the dual variational problem may be put into the same form as the original one. Let us first establish this fact. Dual variational problem. Consider the function (v, p¯ α ), (v, p¯ α ) = min L ∗ (a, v, p α ),

(17.154)

where minimum is sought over all bounded functions p α satisfying the conditions pα = 0, y α

p α = p¯ α .

(17.155)

The general solution of the first equation (17.155) is p α = eαβ

χ , y β

(17.156)

where eαβ is the two-dimensional Levi-Civita symbol, and χ an arbitrary function, which increases at infinity not faster than a linear function. We extract explicitly the linearly growing part of χ and denote the reminder by ϕ : χ = kα y α + ϕ,

(17.157)

kα being some constants. Plugging (17.156) and (17.157) in to (17.154), we obtain the variational problem : ; (v, p¯ α ) = min L ∗ a, v, eαβ kβ + eαβ ϕ,β . ϕ

(17.158)

Assume that the minimizer of the problem (17.158), ϕ, ˇ is bounded at infinity. Then one can take as admissible only the functions ϕ that are bounded at infinity. This allows us to find the value of the constant kβ : from the second constraint (17.155), which becomes eαβ kβ = p¯ α , we have kβ = eαβ p¯ α . So, all admissible fields can be presented as ϕ p α = eαβ kβ + β , kβ = eαβ p¯ α . y

(17.159)

Since, as follows from (14.6), eαβ kβ = p¯ α , the variational problem takes the form ϕ (v, p¯ α ) = min L ∗ a (y) , v, p¯ α + eαβ β , ϕ y

(17.160)

864

17

Homogenization

where minimum is sought over all functions ϕ bounded at infinity. Let us rewrite the dual variational problem (17.160) making a rotation of the coordinate system for an angle π/2 and a translation for a vector cα , y α → y ′α :

α. β y ′α = e.β y + cα ,

We get

′β ·α y α = eβ· y − cβ .

′ ϕ α (v, p¯ ) = min L a y y , v, p¯ + ′α . ϕ y α

∗

(17.161)

Comparing (17.161) and (17.153) we see that the dual variational problem (17.161) has the same form as the original one; the only difference is that the Lagrangian L ∗ is taken at the shifted points obtained by the rotation for the angle π/2 and some translation. Emphasize that the material characteristics a(y), which are usually the components of some tensors, are treated in the transformation y → y ′ as scalars. This result is based on the special structure of the variational cell problem: the functional is a space average over unbounded space, not over a bounded region, as for the usual integral functionals. Recall that there is a difference between the dual and the original variational problems for a finite region. For example, the dual problem to the Dirichlet problem is the Neuman problem. A way to find averaged Lagrangian. Symbolically, we can write the dual variational problem as

Besides, according to (17.111),

; : = min L ∗shifted . ∗ = L¯ = min L .

; : Therefore, if we can link L ∗shifted with L in such a way that min L ∗shifted can be expressed in terms of min L, then we get an equation for or for the effective Lagrangian L¯ = ∗ . There are several cases in which this idea yields the computation of the effective Lagrangian. Equation for the averaged Lagrangian. Consider the Lagrangians which depend ¯ α ), only on the derivatives of the required function: L = L(a, u ,α ). Then, L¯ = L(v = ( p¯ α ). Suppose that for some value of a constant, b, and for all constants, vα , the equality holds: ϕ ϕ min L ∗ a y(y ′ ) , b vα + ′α = min L a(y), vα + α . (17.162) ϕ ϕ y y

17.7


865

For example, this equality is true if, at each point, the shifted and scaled L ∗ coincides with L : ϕ ϕ L ∗ a y(y ′ ) , b vα + ′α = L a(y ′ ), vα + ′α . y y

(17.163)

Let us show that the condition (17.162) yields the following equation for the averaged Lagrangian: ¯ α ). L¯ ∗ (bvα ) = L(v

(17.164)

Indeed, from (17.161), we have ϕ (bvα ) = min L ∗ a y y ′ , bvα + ′α ϕ y ϕ˜ . = min L ∗ a y y ′ , b vα + ′α ϕ˜ y

(17.165)

Here, ϕ˜ = ϕ/b. From (17.165) and (17.162), ϕ (bvα ) = min L a(y), vα + α = L(vα ). ϕ y It remains to use the fact that (see (17.111)) ( p¯ α ) = L¯ ∗ ( p¯ α ) . Equation (17.164) admits the only solution: independently on the physical features of the phases, as soon as the condition (17.162) holds, the averaged medium must be linear and isotropic and have the effective coefficient equal to b : 1 L¯ = bvα v α . 2

(17.166)

To prove this fact, consider a slightly more general problem: let a function, f (x) , of several variables x = {x1 , . . . , xn }, obey the equation f (x) = f ∗ (bx).

(17.167)

We are going to prove that the only solution of this equation is the quadratic function f (x) =

1 bxi x i . 2

866

17

Homogenization

In particular, if f (x) = f ∗ (x) , then f (x) = 12 xi x i . Any function, f (x), obeys the inequality f (x) + f ∗ (x ∗ ) ≥ xi∗ x i (see (5.105)). We put in this inequality, xi∗ = bxi , to obtain the estimate f (x) + f ∗ (bx) ≥ bxi x i . Since f ∗ (bx) = f (x), we get a low bound for f (x): f (x) ≥

1 bxi x i . 2

On the other hand, if we take the definition of f ∗ (x ∗ ), f ∗ (x ∗ ) = max(xi∗ x i − f (x)), x

and replace f (x) by a smaller function, 12 bxi x i , we obtain an upper estimate of f ∗ (x ∗ ) : f ∗ (x ∗ ) ≤

1 −1 ∗ ∗i b xi x . 2

Setting x ∗ = bx, we get an estimate of f (x) from above: f (x) = f ∗ (bx) ≤

1 bxi x i . 2

Upper and lower estimates of f (x) coincides. Consequently, f (x) = 12 bxi x i . Note that a generalization of (17.167) by including a factor λ, f (x) = λ f ∗ (bx),

(17.168)

does not yield a new solution: the only function satisfying this equation is also quadratic: f (x) =

1 √ b λxi x i . 2

(17.169)

To see that we note that, according to (17.168), f ∗ (x) = f (x/b)/λ; therefore, 1 y f (x) = max(y · x − f ∗ (y)) = max y · x − f ( ) λ b 1 ∗ 1 = max (y · λbx − f (y)) = f (λbx). λ λ On the other hand, from (17.168), f (x) is equal to λ f ∗ (bx). Hence, changing bx by x, we have

17.7


867

λ2 f ∗ (x) = f ∗ (λx). Therefore f ∗ (x), and thus f (x), are quadratic functions, and the only solution of (17.168) is (17.169). There are many cases where condition (17.162) is satisfied. Consider some of them. Equirepresented phases. Let the composite be comprised of several homogeneous phases. The phases are called equirepresented if the averaged Lagrangian does not change after some of the phases are interchanged. For a two phase composite, the interchange of phases implies the following operation: the first phase is placed in the region occupied by the second phase and vice versa. Generally speaking, the relative areas c1 and c2 occupied by the first and the second phases do not necessarily coincide, i.e. the interchange of phases may involve adding of one phase and reducing the amount of the another one. If the relative areas of the equirepresented phases are equal, then c1 = c2 = 12 . An example is a checker-board structure. In three-phase composites, the phase interchange can occur according to the rules 1 → 2 → 3 → 1 or 1 → 2 → 1, 3 → 3, etc. Suppose that the operation of Young-Fenchel transformation in (17.162) corresponds to phase permutation, i.e. L ∗ is the Lagrangian of the composite with the rearranged phases, and after such a rearrangement the averaged Lagrangian does not change. Then the averaged Lagrangian is quadratic and given by formula (17.166). Consider some examples. Two-phase composites. Let the composite be microscopically linear and isotropic: L=

1 au ,α u ,α . 2

Then L ∗ (a, u ,α ) =

1 u ,α u ,α . 2a

Suppose that the composite is comprised of two phases, in which the coefficient a takes on the values a1 and a2 , respectively. The condition (17.162) is satisfied for √ b = a1 a2 . Indeed, for this value of b, L ∗ (a (y) , bu ,α ) =

a1 a2 u ,α u ,α . 2a (y)

This is the Lagrangian of the composite with interchanged phases. The equality (17.162) means that the phases are equirepresented. According to (17.166), such a composite is macroisotropic L=

1 αβ a¯ vα vβ , 2

¯ αβ , a¯ αβ = aδ

868

17

Homogenization

Fig. 17.7 Checker-board microstructure

Fig. 17.8 Equirepresented two-phase periodic structure

and its effective coefficient is given by the Dykhne formula a¯ =

√ a1 a2 .

(17.170)

It is difficult to represent visually randomly placed equirepresented phases; therefore, all further illustrations are given for periodic structures. A simple example of the periodic structure for which formula (17.170) is true is a checker-board structure, a cell of which is shown in Fig. 17.7. The indices 1 and 2 denote the regions occupied by the first and the second phases, respectively. An example of a two-phase composite with a more complex geometry is shown in Fig. 17.8. For each of the microstructures shown, the Dykhne formula (17.170) holds. In these examples, the relative areas of the equirepresented phases are equal; c1 = c2 = 12 . This, however, is not necessary: all that is needed is that the effective characteristics remain the same after permutation of phases. The formula (17.170) can be generalized in the following way. Let the continuum be macroscopically and microscopically isotropic and let the coefficient a be a positive random field. Let us introduce a random field σ by the equation a = a0 e σ

(17.171)

and assume that the distribution function of σ is even, while the field σ (y) is ergodic. Then substituting the coefficient a0 eσ (y) by a0 e−σ (y) , we do not change the effective characteristics. Setting b = a0 , we see that the equality (17.162) holds. Therefore a¯ = b = a0 . An interpretation of the number a0 can be obtained by applying the mathematical expectation to the equality ln a = ln a0 + σ.

17.7


869

We have a0 = e M ln a . So we obtain the Dykhne formula: a¯ = e M ln a .

(17.172)

√ If a takes on the values a1 and a2 with 1/2, then a0 =√ a1 a2 , and √the probability the quantity σ = ln a − ln a0 = ln a/ a1 a2 takes on the values ln a1 /a2 and √ − ln a1 /a2 with the probabilities 1/2, and, consequently, has an even distribution function, and (17.172) transforms to (17.170). Non-linear two-phase composites. Homogenization for non-linear structures is much more complicated. One of the reasons is that in non-linear problems the macroscopic properties are characterized not only by the tensor of the second rank a αβ , but also by tensors of higher ranks; for example, the averaged Lagrangian may contain terms like a¯ αβγ δ vα vβ vγ vδ . Geometrical considerations are not sufficient for significantly decreasing the number of effective characteristics; for example, there are tensors invariant with respect to rotations by π/2 which cannot be expressed in β γ β γ terms of the metric tensor. For example, the tensor a¯ αβγ δ = e1α e1 e1 e1δ + e2α e2 e2 e2δ , α α where e1 and e2 are unit vectors directed along the axes of a square lattice, is such a tensor; it does not change for rotation by π/2, since e1α → e2α , e2α → −e1α . Therefore it is impossible to derive, for example, the isotropic property of non-linear checkerboard structure by means of solely geometric considerations. However, if the condition (17.162) is satisfied, the composite will be macroisotropic (and even linear); this is expressed by the formula (17.166). There is a large class of microstructures for which the equality (17.162) holds true. These microstructures have the Lagrangian L which, for some value of b, satisfies the equation L ∗ (a(y(y ′ )), bu ,α ) = L(a(y ′ ), u ,α ),

·α ′β y α = eβ· (y − cβ ).

(17.173)

Consider an example. Let the composite have the microstructure shown in Fig. 17.7. The phases 1 and 2 are homogeneous and described by the Lagrangians, L 1 a1κ , u ,α and L 2 (a2κ , u ,α ). Suppose that the Lagrangians L 1 and L 2 are linked by the equations L ∗1 a1κ , bu ,α = L 2 a2κ , u ,α ,

L ∗2 a2κ , bu ,α = L 1 a1κ , u ,α .

(17.174)

Then (17.173) and, consequently, (17.162) hold, and the composite is macroscopically linear and isotropic, even though the phases may not be so, and its averaged Lagrangian is given by formula (17.166). Let us find the arbitrariness which remains in the choice of the functions L 1 and L 2 obeying (17.174). Applying the operation ∗ to the first equation, we get (L ∗1 )∗

κ α 1 κ α1 = = max p u ,α − L 1 a1 , bu ,α = max p kα − L 1 a1 , k kα b u ,α 1 = L 1 a1κ , p α = L ∗2 a2κ , p α . b 0

α

870

17

Homogenization

Setting pα = bu ,α , we see that the second equality (17.174) is equivalent to the first. Therefore the function L 1 and the parameter b may be chosen arbitrarily, while the function L 2 is then found from the first equality (17.174). Consider the case when L 1 is a power function r

L 1 = a1

(u ,α u ,α ) 2 , r

(17.175)

L 2 , according to the first equality (17.174) and example 3 of Section 5.7 should be taken as the power function as well, s

(u ,α u ,α ) 2 , L 2 = a2 s

r −1 + s −1 = 1.

(17.176)

From the example 3 of Section 5.7, L ∗1 u ∗α =

1 sa1s−1

∗ ∗α 2s uα u ,

L ∗2 u ∗α =

1 ra2r −1

∗ ∗α r2 uα u .

Equations (17.174) take the form bs a1s−1

br

= a2 ,

a2r −1

= a1 .

The first equation yields s−1

1

1

1

b = a1 s a2s = a1r a2s . The second equation gives the same value of b because r −1 + s −1 = 1. Finally, according to (17.166), the effective medium is linear and isotropic, and its effective coefficient is given by the formula obtained by S. Kozlov, 1

1

a¯ = a1r a2s .

(17.177)

An interesting feature of such continuum is that it becomes non-conducting, i.e. a¯ = 0, as long as one of the phases is non-conducting (a1 or a2 is zero). In summary, to get the averaged Lagrangian for equirepresented two-phase composites one can choose L 1 arbitrarily and then find L 2 from the first equality (17.174). Three-phase composites. One can check that for three-phase composites the only permutation that yields non-trivial result is 1 ←→ 2, 3 ←→ 3. Therefore, the Lagrangian of the first phase can be chosen arbitrary, the Lagrangian of the second phase is

17.7


871

L 2 a2κ , u ,α =L ∗1 a1κ , bu ,α , while the Lagrangian of the third phase satisfies the equation L ∗3 a3κ , bu ,α =L 3 a3κ , u ,α . Hence, the third phase must be linear and isotropic and has the characteristic a3 ¯ coinciding with the effective coefficient a,

L3 =

1 ¯ α vα . av 2

The Lagrangians L 1 and L 2 can be, for example, the power functions (17.176). Then 1

1

1

1

a3 must be equal to a1r a2s , and the effective coefficient of the composite is a1r a2s as well. Four-phase composites. The permutations of the phases in four-phase composites bringing non-trivial results are 1 ←→ 2, 3 ←→ 4. Therefore, all statements made for two-dimensional composites apply. One can choose the Lagrangians of two phases, say, phase 1 and phase 3, arbitrarily, and take the Lagrangians of phase 2 and phase 4 as their Young-Fenchel transformations. For example, for power functions we have ′

L1 =

,α r2 ′ (u ,α u ) 2a1 ′

r

′

,

L2 =

,α s2 ′ (u ,α u ) 2a2 ′

,

1 1 + ′ = 1, r′ s

L4 =

,α s2 ′′ (u ,α u ) 2a2 ′′

,

1 1 + ′′ = 1. ′′ r s

′′

′′

L3 =

,α r2 ′′ (u ,α u ) 2a1 ′′

r

s

,

s

The parameters a1′ , a2′ , a1′′ , a2′′ , r ′ and r ′′ should satisfy the constraint 1

1

1

1

(a1′ ) r ′ (a2′ ) s′ = (a ′′1 ) r ′′ (a2′′ ) s′′ ,

(17.178)

because the factor b is common in the permutations 1 ←→ 2 and 3 ←→ 4. The ¯ An common value (17.178) is equal to the effective coefficient of the composite a. example of such composite is considered further in Section 17.8 in case of elastic media. Effective characteristics as functionals. The effective characteristics of linear micro-heterogenious continua possess some remarkable properties that follow from the variational cell problems (17.153) and (17.160). To derive these properties, we note that for a linear continuum,

872

17

1 αβ a u ,α u ,β , 2 1 L¯ = a¯ αβ v,α v,β , 2

Homogenization

1 −1 α β a p p , 2 αβ 1 −1 α β p¯ p¯ , L¯ ∗ = a¯ αβ 2 L∗ =

L=

and therefore (17.153) and (17.160) can be written as ψ ψ vβ + β , (17.179) a¯ αβ vα vβ = min a αβ vα + α ψ y y ϕ ϕ (−1) αμ βν (−1) αλ βν a¯ αβ e e kλ kν = min aαβ e e kμ + μ kν + ν . (17.180) ϕ y y In two-dimensional space, the inverse tensor can be written explicitly in terms of the original tensor and Levi-Civita symbol, (−1) αμ βν aαβ e e =

a μν , A2

A≡

det a μν .

(17.181)

According to (17.181), (17.180) takes the form kα kβ a¯ αβ 2 ¯ A

a αβ = min ϕ A2

ϕ ka + α y

ϕ kβ + β . y

(17.182)

¯ = det a¯ αβ . Here, A The variational principle (17.179) defines the effective coefficients a¯ αβ as some functionals of the fields a αβ . If a¯ αβ are considered as functionals of a αβ , we write a¯ αβ = a¯ αβ (a μν ). All follows from (17.182), the functionals a¯ αβ (a μν ) satisfy the equations μν

a¯ αβ (a ) = a¯ αβ ¯ 2 (a μν ) A

a μν A2

(17.183)

.

μν Equation (17.183) suggest that the tensors a¯ αβ (a μν ) and a¯ αβ aA2 are coaxial. Moreover, the eigenvalues, a¯ 1 (a μν ) and a¯ 2 (a μν ) of the tensor a¯ αβ are linked: a¯ 2 (a μν )a¯ 1

a μν A2

a¯ 1 (a μν )a¯ 2

= 1,

a μν A2

= 1.

Calculating the determinants of the matrices on the right and the left hand side of (17.183), we obtain the relation found by Kozlov: ¯ 2 (a μν ) A ¯2 A

a μν A2

= 1.

(17.184)

17.7


873

The Dykhne formula (17.170) can be derived from Kozlov’s equation: since ¯ αβ (the continuum is macroisotropic), a μν = aδ μν (the continuum is a¯ αβ = aδ ¯ = a, ¯ A = a, and the Kozlov’s equation becomes microisotropic), we have A ¯ μν )a¯ a(aδ

1 μν δ a

= 1.

Due to semi-linearity, a¯ (δ μν /a) = a¯ (a1 a2 δ μν /a) /a1 a2 . For equirepresented phases, a¯ (a1 a2 δ μν /a) is the effective coefficient of the composite with interchanged phases and, thus, equal to a¯ (aδ μν ). Therefore, a¯ 2 (aδ μν ) = a1 a2 , √ and, consequently, a¯ = a1 a2 . Here is one more example of application of (17.184). Example 5. Let every square of a checker-board be occupied by an anisotropic square with the eigenvalues of the tensor a αβ equal to a1 and a2 . In the system of coordinates with its axes along the axes of the square, the tensor a αβ is diagonal. The probabilities of the two possible orientation of the square are equal. The symmetry group of the composite contains rotations for the right angle, π/2. According to the Hermann-German theorem [129, 111], a two-dimensional tensor of the second order, which is invariant with respect to the group containing rotation for an angle less than π , is spherical. Consequently, the tensor a¯ αβ is spherical. Since the determinant of a αβ , A2 = a1 a2 , does not change from point to point, we find from (17.184) and ¯ (17.189) the effective coefficient, a: a¯ =

√ a1 a2 .

Apparently this result admits the following generalization: for any macroscopically isotropic continuum, for which the product, A2 , of the eigenvalues of the tensor a αβ , is the same at every point, a¯ = A. In particular, that gives the effective coefficient of two-dimensional polycrystals. The functionals a¯ αβ (a μν ) contain a lot of information about the solution of the cell problem. For example, let us vary a αβ (y) by some infinitesimally small field, δa αβ , which is constant in some part of the plane with non-zero specific area c and zero outside of . Then, using the formula for derivative of the minimum value in variational problem with respect to parameter (see Sect. 5.13), one finds from (17.179) that δ a¯ αβ vα vβ= c δa μν

ψˇ ψˇ vν + ν , vμ + μ y y

(17.185)

where ψˇ is the minimizer of the cell problem, and · is the averaging operator over the region ,

874

17

1 λ→∞ |λB ∩ |

f = lim

Homogenization

f d 2 y.

λB∩

Equation (17.185) gives the average of the minimizer over any set with non-zero specific area. In particular, taking the derivative of the Dykhne formula (17.170) with respect to a1 and a2 , we find the average of the gradient of the true field over the phases, ψˇ vμ + vμ + μ y a1 ψˇ μ vμ + vμ v = vμ + μ a2 y

a2 vμ v μ = a1

ψˇ , y μ 1 ψˇ . y μ 2

(17.186)

Here, ·1 and ·2 are the averages over the first and the second phases, respectively, and it is assumed that c1 = c2 = 12 . It can be seen from the formulae (17.186) that, on average, the gradient of the field is greater for the phase with the smaller value 1 of the coefficient a, and as a decreases, it increases as a − 4 . Multiplying the first relation (17.186) by a1 and the second one by a2 , we see that the energy averages coincide for both phases: 1 ψˇ ψˇ ψˇ ψˇ 1 μ μ a1 vμ + μ a2 vμ + μ v + v + = = 2 y yμ 1 2 y yμ 2 1√ = a1 a2 vμ v μ . 2 Multiplying the relations (17.186) by a12 and a22 , correspondingly, we find the average value of the flux p α = a (v α + ψ/y α ) for the phases: α

pα p 1 =

a1 p¯ α p¯ α , a2

α

pα p 2 =

a2 p¯ α p¯ α , a1

p¯ α =

√ a1 a2 vα .

(17.187)

The formulae (17.187) show that the average flow over a phase increases with the 1 increase of the coefficient, a, of a phase as a 4 (for the fixed average flow, p¯ α ). Note two general properties of the functionals a¯ αβ (a μν ). The functionals, a¯ αβ (a μν ), are concave: for any vα , a¯

αβ

1 μν μν 1 μν μν (a 1 +a 2 ) vα vβ ≥ (a¯ αβ (a1 ) + a¯ αβ (a2 ))vα vβ . 2 2

Indeed, from (17.179)

(17.188)

17.8

Two-Dimensional Incompressible Elastic Composites

875

ψ ψ 1 μν μν 1 μν μν vν + ν ≥ (a 1 +a 2 ) vα vβ = min (a 1 +a 2 ) vμ + μ ψ 2 2 y y ψ ψ ψ ψ 1 μν μν vν + ν + min a2 vμ + μ vν + ν . min a1 vμ + μ ≥ ψ 2 ψ y y y y

a¯ αβ

Moreover, according to (17.172), the functionals, a¯ αβ (a μν ), are semi-linear, i.e. for any k > 0, a¯ αβ (ka μν ) = k a¯ αβ (a μν ).

(17.189)

17.8 Two-Dimensional Incompressible Elastic Composites The results of the previous section can be extended to two-dimensional incompressible geometrically linear elastic composites, because the dual variational cell problem is reduced to the original cell problem in that case. The cell problem. Let the energy density of the composite F(a κ , εαβ ) be a convex functions of the strain components εαβ = u (α,β) . According to (17.97), the macroenergy density F¯ is a function of the macrodeformations ε¯ αβ which can be found from the variational problem, ¯ εαβ ) = min F(a κ , ε¯ αβ + ψ(α|β) ), F(¯ ψα

ψα|β ≡

ψα . y β

(17.190)

In the first approximation, the true deformations are εαβ = ε¯ αβ + ψ(α|β) . Therefore, if the continuum is microscopically incompressible, ψ α satisfy the additional condition α = 0. ε¯ αα + ψ|α

(17.191)

Equation (17.191) holds at every point of the continuum. We assume that the continuum does not have cavities or discontinuities (the lines on which the displacements ψα may be discontinuous). Then α

ψ|α = 0,

and, applying the operation · to (17.191), we get ε¯ αα = 0, i.e. the continuum is macroscopically incompressible.

876

17

Homogenization

The constraint (17.191) becomes α ψ|α = 0.

(17.192)

Dual variational cell problem. Denote the deviators of tensor by a prime: by definition, for any tensor a αβ , a ′αβ means the tensor a ′αβ = a αβ − 12 aσσ δ αβ . Due to the micro and macro-incompressibility, ′ ε¯ αβ = ε¯ αβ ,

′ εαβ = εαβ .

The energy density F of the incompressible continuum may be considered as a ′ function of the strain deviator εαβ . Denote the Young-Fenchel transformation of F ′ ∗ with respect to εαβ by F : κ 1 0 ′αβ ′ ′ . p , ε − F a ε F ∗ a κ , p ′αβ = max (y) αβ αβ ′

(17.193)

εαβ

′

′ The maximum is sought over all εαβ with zero trace εαα = 0. Obviously, in this case ∗ F depends only onthe deviator of stresses p ′αβ . ′αβ ∗ ¯ ¯ we denote the Young-Fenchel transformation of the funcSimilarly, by F p ′ ′ . tion F¯ ε¯ αβ with respect to all trace-free tensors ε¯ αβ

It is easy to check that the dual variational cell problem (17.111) becomes : ; G p¯ ′αβ = min F ∗ a κ (y) , p ′αβ , p′αβ

(17.194)

where minimum is sought over all fields p ′αβ bounded at infinity and subject to the constraints αβ

p |α = 0,

p ′αβ = p αβ −

1 γ αβ p δ , 2 γ

:

; p ′αβ = p¯ ′αβ .

(17.195)

The general solution of the first two equations (17.195) can be written as μβ) p ′αβ = e(α· , ·μ γ

γ μβ = χ (μ|β) ,

(17.196)

where χ μ is an arbitrary vector field satisfying the incompressibility condition χ|αα = 0. Indeed, the general solution of the equation χ|αα = 0 can be written as χ α = eαβ χ|β , where χ is an arbitrary function. From (17.196), we find p ′αβ in terms of χ :

17.8

Two-Dimensional Incompressible Elastic Composites ′ ′ p11 = − p22 = γ12 =

1 χ|22 − χ|11 , 2

877

′ p12 = γ22 = −γ11 = −χ|12 ,

which coincide with the general solution of the equilibrium equations in terms of Airy functions (see (6.64)). From the boundedness of p ′αβ , it follows that χ α do not increase faster than a linear function. Separating the part of χ α , which increases at infinity linearly, we get χ α =γ¯ αβ yβ +ϕ α . We assume that ϕ α are bounded at infinity. Since γαα = 0, also γ¯αα = 0. Consequently, ϕ α|α = 0.

(17.197)

The constants γ¯ αβ , are linked to p¯ ′αβ by the equation that follows from (17.195) and (17.196): μβ) = p¯ ′αβ . e(α· ·μ γ¯

Hence, the dual variational cell problem may be written as : ∗ κ ; μβ) μβ) = min F a (y) , e(α· G e(α· ·μ γ¯ ·μ γ α ϕ

(17.198)

where γ μβ = γ¯ μβ + ϕ (μ|β) , and the minimum is sought over all fields ϕ α bounded at infinity and satisfying the incompressibility condition (17.197). Let us put the dual variational problem (17.198) into a form completely analogous to the original cell problem. In order to do so, we introduce a new coordinate system, y˜ α , which is rotated for the angle π/4 and translated for some vector cα , F√ y˜ α = sβα y β + cα , s11 = −s21 = +s12 = +s22 = 1 2.

The tensor components in the new coordinate system are marked by the symbol ∼ . The components of any deviator tensor, tαβ , in the new coordinate system are ·μ t˜αβ = −e(α· tμβ) .

Denote the derivatives of function ϕ˜ α with respect to y˜ β by ϕ˜ α||β . Since μβ) e(α· = p¯ ′αβ , ·μ γ¯

μβ) (μ|β )) e(α· = p¯ ′αβ +e(α· = p¯ ′αβ − ϕ˜ (α||β) , ·μ γ ·μ ϕ

after changing ϕ˜ α by −ϕ˜ α , the dual variational problem becomes : ∗ κ ′ ′αβ (α||β) ; F a y y , p¯ +ϕ˜ , G p¯ ′αβ = min α ϕ˜

i.e. it gets the form of the original cell problem.

(17.199)

878

17

Homogenization

Microscopically linear isotropic continua. Let the equality min F ∗ a (y( y˜ )) , b(¯εαβ + ϕ˜ (αβ) ) = min F(a κ , ε¯ αβ + ψ(α|β) ) ϕ˜ a

ψa

(17.200)

hold for some constant b. Then the macroenergy satisfies the equation F¯ ∗ (b¯εαβ ) = F(¯εαβ ). As has been shown above, this equation has the only solution ¯ εαβ ) = μ¯ F(¯ ¯ εαβ ε¯ αβ ,

μ ¯ =

b . 2

(17.201)

Therefore, any elastic composite which satisfies the condition (17.200) macroscopically behaves as a linear elastic body with the shear modulus b/2. Here is one sufficient condition for (17.200) to be satisfied. Let a macroisotropic composite comprise two phases, which are so well mixed that the phase interchange does not affect the macroscopic properties of the com′ ′ posite. Suppose that the phase energies F1 (εαβ ) and F2 (εαβ ) are such that, for some constant b, the equality ′ ′ F2 (εαβ ) = F1∗ (bεαβ )

(17.202)

holds. Then the analogous equality holds for the Young-Fenchel transformation of the second phase energy: 0 ′ ′αβ 1 F2∗ (bεαβ ) = max bεαβ p − F2 p ′αβ = p′αβ 0 ′ 1 ′ = max εαβ bp ′αβ − F1∗ bp ′αβ = F1 εαβ . bp′αβ

For the macroisotropic bodies, the functions a (y( y˜ )) in (17.200) can be replaced by a( y˜ ). Then, on the left hand side of (17.200), we have macroenergy calculated for a composite with the permutated phases. Equation (17.200) means that permutation of the phases does not change macroenergy. Therefore, the equality (17.200), and, consequently (17.201), hold. Consider some examples. Example 1. Let each phase be linear and isotropic: F1 = μ1 εαβ εαβ ,

F2 = μ2 εαβ εαβ .

√ The condition (17.202) is satisfied for b = 2 μ1 μ2 , and the shear modulus of the composite is μ ¯ =

√

μ1 μ2 .

17.8


879

Example 2. For composites with the same microstructure, but with non-linear properties s

r

F1 = μ1

(εαβ εαβ ) 2 , r

F2 = μ2

(εαβ εαβ ) 2 , s

r −1 + s −1 = 1

we find, analogously, that the composite behaves as linearly elastic and isotropic body with the shear modules 1

1

μ ¯ = μ1r μ2s . Example 3. Consider a composite with the same microstructure comprising nonlinearly elastic phases with the energies F1 = F2 =

6

μ1 εαβ εαβ +∞

if εαβ εαβ ≤ k12 if εαβ εαβ ≥ k12

μ2 εαβ εαβ

εαβ εαβ − k2 + μ2 k22 a

if εαβ εαβ ≤ k22

if εαβ εαβ ≥ k22

.

√ Suppose that the phase characteristics are related as μ1 k12 =μ2 k22 , a=2 μ1 μ2 k1 . As in Example 2, the composite behaves as isotropic and linearly elastic, with the √ shear modulus equal to μ ¯ = μ1 μ2 . Example 4. Consider a visco-plastic composite with the same geometric microstructure. The phase dissipation densities are defined by the formulae

D2 =

D1 = k1 eαβ eαβ + μ1 eαβ eαβ ,

0

μ2

eαβ eαβ − k2

2

if eαβ eαβ ≤ k22 if eαβ eαβ ≥ k22

.

Here, eαβ have the meaning of the strain rate tensor. If the characteristics of the phases are related as k1 = 4μ1 μ2 k2 , then the condition (17.202) is satisfied, and the √ composite behaves as a linearly viscous fluid with the viscosity μ ¯ = μ1 μ2 , which does not depend on the plastic moduli k1 , k2 . Invariance of microproperties for rotations by π /2. Let us rewrite (17.199) for ′ κ ∗ an elastic composite with the energy F (a (y ( y˜ ))) , εαβ . We get ′ : ; ′ F¯ ε¯ αβ = min F (a κ (y ( y˜ (z)))) , ε¯ αβ + ϕ˜ (α|||β) , ϕ˜

β where ϕ˜ α|||β ≡ ϕ/z ˜ , z β are the coordinates rotated for π/4 with respect to the coordinates, y˜ β . The coordinates z β are rotated for π/2 with respect to the original

880

17

Homogenization

coordinates y β . On the right-hand side of this equation we have the energy density of the composite with the microstructure rotated by π/2; the equation means that the macroenergy does not change for such a rotation. This property is not obvious. For example, consider a four-phase composite with a periodic microstructure shown in Fig. 17.9. Rotation for π/2 corresponds to some interchange of phases, and there are no reasons to expect that the macromoduli do not change. However, they do not change, as follows from the equality obtained. Apparently, this fact relates to the incompressibility of the elastic continuum and does not take place for compressible media.

Fig. 17.9 Elastic composite with microstructure that is not invariant with respect to π/2-rotation. For incompressible elastic media the macroproperties of such composite are invariant for π/2rotations

For continua described by one required function, the analog of rotation for π/2 is rotation by π ; there are also periodic structures for which the invariance of macroproperties with respect to a rotation by π is not obvious. Many two-dimensional periodic structures have the rotation by π/2 as a member of the symmetry group. This yields the equation of the type (17.164) guaranteeing that the body is macroscopically isotropic. Rotation by π/4 is not a symmetry element for any periodic structure. Therefore, generally speaking, there are no analogous equations for elastic composites, the elastic continuum is characterized by several moduli, and the possibility to write the dual problem in a form analogous to the original variational problem yields only some relationships between the macromoduli without admitting their exact calculation. Let us establish these relationships for linear elastic composites. Effective characteristics as functionals. We begin with constructing a general form of energy density for two-dimensional incompressible elastic anisotropic bod′ has two independent components, ies. In every system of coordinates, the tensor εαβ ′ ′ ′ ′ ε1 = ε11 and ε2 = ε12 (while ε22 = −ε11 ). Therefore, for F we can write F = b11 ε12 + 2b12 ε1 ε2 + b22 ε22 . If the coordinate system is rotated for an angle, θ , the quantities ε1 and ε2 transform as vector components for a rotation by 2θ :

17.8


881

ε˜ 1 = ε1 cos 2θ + ε2 sin 2θ, ε˜ 2 = −ε1 sin 2θ + ε2 cos 2θ, where ε˜ 1 , ε˜ 2 are the 11-component and 12-component of the tensor in the rotated coordinate system. Consequently, the material characteristics b11 , b12 and b22 transform as coordinates of a symmetric tensor for a rotation by −2θ , and there exists a system of coordinates for which this tensor is diagonal (b12 = 0). Let us introduce √ a symmetric tensor aαβ with zero trace, which has the values a11 = −a22 = 1/ 2, a12 = 0 in the principle coordinate system of the tensor bαβ . In the principal coordinate system b11 = 2 (μ + η) , b22 = 2μ. Then the function 2 ′ ′ ε′αβ + η a αβ εαβ μεαβ

coincides with energy density in this coordinate system and, since the energy density is a scalar, this is true in all coordinate systems. So for energy density of two-dimensional incompressible elastic body, without loss of generality, we can write 2 ′ ′ ε′αβ + η a αβ εαβ , F = μεαβ

(17.203)

where μ, η, and a αβ are physical characteristics of the elastic body, and a αβ = a βα , aαα = 0. The tensor a αβ is normalized by the condition a αβ aαβ = 1. So a αβ have one degree of freedom: they are defined by the angle of rotation, ϕ, of the principal axes with respect to some fixed frame. The elastic properties are characterized by three parameters, μ, η, ϕ. The energy is positive if and only if μ > 0 and μ + η >0. For an isotropic continuum, η = 0. The function F ∗ p ′αβ is F

∗

p

′αβ

1 η αβ ′ 2 ′αβ ′ . p pαβ − a pαβ = 4μ μ+η

In terms of γ αβ , the function F ∗ is written as (see (17.196)) F∗ =

1 η γαβ γ αβ + (a αβ γαβ )2 . 4(μ + η) 4μ(μ + η)

¯ The formula (17.203) can be applied to macroscopic energy density F:

(17.204)

882

17

Homogenization

′ F¯ = μ¯ ¯ εαβ ε¯ ′αβ + η( ¯ a¯ αβ ε¯ ′αβ )2 .

The effective characteristics μ, ¯ η¯ and ϕ¯ are some functionals of μ(y), η (y) and ϕ (y). The functionals μ ¯ and η¯ are semi-linear with respect to μ (y) and η (y). We use the index × to denote their values on the fields (μ + η)−1 , ημ−1 (μ + η)−1 and ϕ. It follows from (17.190), (17.198) and (17.204) that the following relations hold between the functionals μ, ¯ η¯ and ϕ: ¯ μ( ¯ μ ¯ × + η¯ × ) = 1,

η¯ × , (μ ¯ × + η¯ × ) ϕ¯ = ϕ¯ × .

η¯ μ ¯× =

(17.205)

The second relation (17.205) can be rewritten in a symmetric form by means of the first relation: η¯ μ ¯ × = η¯ × μ. ¯

(17.206)

Let us apply this relations to the analysis of elastic properties of checker-board structures. Example 5. Consider the checker-board structure shown in Fig. 17.7. Let the phases be isotropic and have the shear moduli μ1 , μ2 . Consider the functional 1 . μ1 μ2 μ ¯ ≡ μ1 μ2 μ ¯ μ ×

Due to semi-linearity of the functional μ(μ), ¯ we have μ1 μ2 μ ¯× = μ ¯

μ1 μ2 μ

.

The function μ1 μ2 /μ takes on the value μ1 in phase 2 and μ2 in phase 1 of the checker-board structures. Since the phase permutation does not affect the value of ¯ × = μ. ¯ Similarly, μ1 μ2 η¯ × = η. ¯ Substituting these values of μ ¯ × and η¯ × μ, ¯ μ1 μ2 μ into the first equality (17.205), we find the relation which connects the micro and macro elastic moduli, μ( ¯ μ ¯ + η) ¯ = μ1 μ2 .

(17.207)

If the elastic checker-board structure were macroisotropic (η¯ = 0), we find from (17.207) the effective shear modulus:

17.9

Some Three-Dimensional Homogenization Problems

μ ¯ =

√

883

μ1 μ2 .

However, there are no reasons to expect that the checker-board is macroisotropic: extension along the sides of the squares and along their diagonals may cause different energy increments. Therefore, the above reasoning allows us to find only the relationship between the macromoduli, not their values. Example 6. Consider a four-phase composite with the periodic structure shown in Fig. 17.9. The phases are characterized by the shear moduli μ1 , μ2 , μ3 , μ4 . Let the shear moduli of the phases be related by the constraint (17.208)

μ1 μ2 = μ3 μ4 .

Then the function μ1 μ2 /μ (y) takes on the values μ2 , μ1 , μ4 and μ3 in the first, second, third and fourth phases, respectively. They correspond to the permutation of the first and the second and the third and the fourth phases. This permutation may be carried out by a rotation for π/2. Therefore, the macromoduli do not feel that phase permutation. Consequently,

μ1 μ2 μ1 μ2 μ ¯ =μ ¯ μ ×

=μ, ¯

μ1 μ2 η¯

×

= η¯

×

μ1 μ2 μ

= η.

The relationship between the micro and macro elastic moduli, follows from (17.206), μ( ¯ μ ¯ + η) ¯ = μ1 μ2 =

√

μ1 μ2 μ3 μ4 .

This relationship has three free parameters, μ1 , μ2 and μ3 ; the fourth is found from the constraint (17.208) Consider one particular case. We set μ1 = μ2 = μ and tend μ3 to zero and μ4 to infinity in such a way that μ3 μ4 = μ2 . In the limit, we get a periodic structure containing cavities and rigid inclusions. The macromoduli are related to the matrix shear modulus μ by the formula μ( ¯ μ ¯ + η) ¯ = μ2 . This relation still holds if the boundaries of the inclusions are collapsed into segments, and we obtain a homogeneous body containing periodically placed “cuts” and rigid supports.

17.9 Some Three-Dimensional Homogenization Problems Ideal fluid mixture. The simplest three-dimensional homogenization problem is the computation of the macroscopic properties of a heterogeneous mixture comprising two or more ideal compressible fluids with different properties.

884

17

Homogenization

We describe motion in Lagrangian coordinates. To distinguish micro- and macrocharacteristics, we introduce fast and slow Lagrangian coordinates Y a and X a respectively. We assume that the physical characteristics (initial density ρ0 , specific heat cv , etc.) are arbitrary functions of fast variables Y a , defined in threedimensional space RY . If the fluid has two phases, each of the physical characteristics is a piece-wise constant function of Y a which takes on two values. The physical properties of the mixture are described by the internal energy density, U (Y, ρ, S). The initial positions of the particles, x˚ i , are functions of slow variables and, for simplicity, we set % i % % x˚ % % det % % X a % = 1.

The current particle positions, x i = x i (Y a , X a , t), Y a = X a /ε, depend on both the fast and the slow variables. It is natural to assume that, as ε → 0, the specific volume % i% % x % det % X a % 1 = (17.209) ρ ρ0 is bounded. If the asymptotic expansion of particle trajectories with respect to ε begins with the term that does not depend on ε, x i = x¯ i X a , t + εy i Y a , X a , t, ε ,

(17.210)

; ρ0 y i = 0,

(17.211)

and the next term, εy i , is small, while y i and their first derivatives are bounded in RY , then the specific volume is bounded. In what follows, we assume that the asymptotic formula (17.210) holds true. Without loss of generality, we can impose the constraints on y i : :

where · is the average over RY . Then x¯ i have the meaning of the particle trajectories averaged over mass: : i; ρ0 x x¯ = , ρ¯ 0 i

ρ¯ 0 = ρ0 .

The functions x¯ i (X a , t) describe the macromotion. We define the mass density of macromotion, ρ, ¯ as ρ¯ =

ρ¯ %0 %. % x¯ i % det % X a %

(17.212)

17.9


885

We make the following two assumptions: (1) the processes in the continuum are so slow, that there is enough time to reach locally thermodynamic equilibrium; consequently, temperature and pressure are functions only of slow variables, and (2) true velocities deviate small from the average velocities: ε2 y,ti yi,t ≪ v¯ i v¯ i ,

v¯ i ≡ x¯ ,ti .

The problem is to find the action of the mixture t1

ρ0

t2 V˚

1 i vi v − U (Y, ρ, S) d V˚ dt 2

(17.213)

as a functional of macrocharacteristics. First, we formulate the result: in the first approximation the action (17.213) as a functional of macrocharacteristics is t1 t2 V˚

ρ¯ 0

1 i v¯ i v¯ − U¯ ρ, ¯ S¯ d V˚ dt, 2

(17.214)

where S¯ is averaged entropy,

ρ0 S S¯ = , ρ¯ 0

(17.215)

while the macroenergy U¯ as a function of macrodensity ρ¯ and macroentropy S¯ is calculated from the formula

ρ0 μ (Y a , p, T ) p ¯ ¯ ¯ (17.216) U ρ, ¯ S = max − +TS . p,T

ρ0 ρ¯ In (17.216) μ is the chemical potential (see Example 8 in Sect. 5.7), and the maximum is sought over all constants p and>T . Therefore, in order to find macroenergy, we first need to find the integral, ρ0 μ ρ0 , which depends on parameters p and T , and then make a transformation of the Young-Fenchel type. The rule for calculating the macroenergy is reminiscent of the rule for averaging in one-dimensional structures. The only differences are that the average is calculated over a three-dimensional space and there is an additional thermodynamic variable, entropy. The proof of the formulae (17.214) and (17.216) is based on the following remarkable relation: ρ¯ 0 ρ0 = . (17.217) ρ ρ¯

886

17

Homogenization

For ρ0 ≡ const, it takes the form 1 1 = , ρ ρ¯ which shows that macroscopic specific volume is equal to the average of the microscopic specific volumes. Formula (17.217) follows from the continuity equation % i % % x % ρ0 % = det % % X a % , ρ

the formulae for the determinant (3.14) and the asymptotic expansion (17.210). Indeed, taking the derivative of (17.210) with respect to X a , we have x i = x¯ ai + y|ai , X a

y|ai ≡

y i . Y a

Consequently,

: % %; j 1 i ρ0 j = det %xai % = eijk x¯ a + y|ai x¯ b + y|b x¯ ck + y|ck eabc = ρ 3! 1 j j = eijk x¯ ai x¯ b x¯ ck eabc + 3 eijk x¯ |ai y|b y|ck eabc + 3!

j

j

+3 eijk x¯ ai x¯ b yck eabc + eijk y|ai y|b y|ck eabc

.

Since x¯ ai do not depend on the fast variables, and

j y|b y|ck eabc = y j y|ck eabc |b = 0,

because y i and their derivatives are bounded in RY , we have

% i % % x¯ % ρ0 % = det % % X a % , ρ

as claimed. At first glance, (17.217) contradicts the relation ρ¯ = ρ1 c1 + ρ2 c2 ,

(17.218)

which is usually used for a two-phase mixture. However, the formula (17.218) also holds true. Indeed, using the continuity equation (17.209) we can write the definition of ρ¯ (17.212) as

17.9


887

% i% % x % = det % X a % ρ¯ 0 % %= ρ % % . ρ¯ = i % x¯ % % x¯ i % det % X det % X a% a%
R 2 0

17.10

Estimates of Effective Characteristics of Random Cell Structures

895

The admissible values of r are inside a smaller cube: |ri |

1 − R, 2

i = 1, 2, 3.

All values of r are assumed to be equiprobable. Then Ma(y, r ) is a periodic func¯ tion of y; denote it by a(y). The homogenized media is, obviously, isotropic. The estimate of the effective coefficient aeff takes the form: for any constants, vi , i

aeff vi v

min

periodic ψ(y)

ψ ψ i ¯ v + d V. a(y) vi + i y yi C

(17.244)

The dependence of a¯ on y 1 is shown in Fig. 17.11 for a two-dimensional problem, ¯ 1 , y 2 ), in the case of pores (b = −1) for R = 0.29 and R = 0, 33 (the a¯ = a(y corresponding porosities, π R 2 , are 0.27 and 0.34; a1 is set to be equal to 1).

0.10

0.15

0.08 0.10

0.06 0.04

0.05

0.02 –0.4

–0.2

0.2

0.4

–0.4

–0.2

0.2

0.4

Fig. 17.11 The dependence of a¯ on y 1 in a two-dimensional problem for pores (b = −1) and R = 0.29 (a) and R = 0.33 (b)

If one takes as a trial field in (17.244) the function ψ = 0, then one obtains an estimate ¯ aeff a(y) cell . It coincides with the Voigt-Hill estimate for the random cell structure. Hence, the exact solution of the periodic problem (17.244) provides a better estimate. Example 3. Nanocrystalline materials. The properties of polycrystals become drastically different if the grain size becomes on the order of dozens of nanometers. If such a polycrystal is obtained by a severe plastic deformation, then it appears that the actual structure of the polycrystal is similar to that shown in Fig. 17.12. The grain boundaries become thick, in the order of 10–20 nm. The material inside the grain boundaries is highly defected and, perhaps, can be considered as amorphous and isotropic. The polycrystal can be modeled as a multi-phase material comprised of a matrix and randomly oriented monocrystals embedded in the matrix. To mimic

896

17

Homogenization

Fig. 17.12 Microgeometry of nano-grain polycrystal

such microgeometry, we can consider a random cell structure with the spherical inclusions of radius R placed in the centers of the cells. The orientations of the inclusions are random and independent. In a linear scalar problem, the estimate ¯ (17.243) takes the form (17.244) with a(y) replaced by 6 ij a matrix |y| R , a¯ (y) = Ma i j |y| < R ij

ij

a matrix and a i j being the characteristics of the matrix and the inclusions, respectively.

Fig. 17.13 A realization of random chessboard structure

17.10

Estimates of Effective Characteristics of Random Cell Structures

897

Example 4. A random checker-board. Usually, periodic cell problems admit only numerical solution. However, there are a few cases when the periodic cell problem can be solved analytically as, for example, the scalar problem for a checkerboard. In such a case the estimate (17.243) can be completed without a numerical investigation. Consider an example. Let each cell have constant properties a1 or a2 . We put in each white cell of a checker-board the materials 1 or 2 with probabilities p and 1 − p, and in each black cell the same materials with probabilities p ′ and 1 − p ′ . A realization of such structure is shown in Fig. 17.13. In such a small piece, 10 × 10, the microgeometry looks completely random. The imposed black-white probabilistic structure of the board can be observed only at long distances. Averaging yields a periodic structure with the material characteristics pa1 + (1 − p)a2 and p ′ a1 + (1 − p ′ )a2 in the white and black cells, respectively. According to Dykhne formula, the estimate (17.243) takes the form aeff

( pa1 + (1 − p)a2 ) ( p ′ a1 + (1 − p ′ )a2 ).

Obviously it is generalized to an arbitrary set of materials as aeff

Ma | white Ma | black

where Ma | white and Ma | black are the probabilistic averages of the characteristics placed in the white and black cells.

Chapter 18

Homogenization of Random Structures: a Closer View

18.1 More on Kozlov’s Cell Problem As we have seen in Sect. 17.4, the averaged Lagrangian and the local fields in random structures are determined by Kozlov’s cell problem (17.97) L¯ (v, vi ) =

ψ L a (y, ω) , v, vi + i . min y ψ(y): ψ/y i =0

(18.1)

Here v, vi are constant parameters, and · means space average in the threedimensional space, R y , of fast y-variables: for any function ψ(y), 1

ψ (y) = lim λ→∞ |Bλ |

ψ (y) dy,

(18.2)

Bλ

Bλ being the ball of radius λ and |Bλ | ball’s volume. The variational problem (18.1) is quite different from the variational problems for the integral functionals considered in the other parts of the book. In a normal variational problem, the values of the integral are finite, even if the integral is taken over the entire space,

Ry

Ldy < ∞.

If such an integral was finite in Kozlov’s cell problem then, as follows from the definition of the space average (18.2), L = 0. In fact, in Kozlov’s cell problem,

Ry

Ldy = ∞.

To see that, let us take a linear scalar problem, and set ψ = 0. Then L = 21 a i j vi v j is a constant, and the integral of L over the entire space diverges. The space average

L is finite. The space average L can also be finite for the nonzero admissible V.L. Berdichevsky, Variational Principles of Continuum Mechanics, Interaction of Mechanics and Mathematics, DOI 10.1007/978-3-540-88469-9 5, C Springer-Verlag Berlin Heidelberg 2009

899

900

18


fields ψ (y) that do not decay at infinity. This suggests to use as the admissible fields the so-called almost periodic functions and the homogeneous random fields. Before turning to the discussion of Kozlov’s cell problem (18.1), we give a brief overview of these two notions. Almost periodic functions. We begin with the functions of one variable, y. The simplest functions that do not decay at infinity are periodic functions, like a cos y + b sin y. Any periodic function with the period 2π can be presented by a Fourier series, i.e. the series over sin kx and cos kx, k = 1, 2 . . . . It is convenient to deal with complex-valued functions u(x) and write the Fourier series as an expansion over eiky : u(y) =

+∞

u k eiky .

(18.3)

k=−∞

To express the coefficients of the series u(x) in terms of u(x), we note the relation : iky ; e =

6

1 k=0 0 k= m

(18.4)

where · is the space average (18.2) which, in the one-dimensional case, takes the form 1 λ→∞ 2λ

u = lim

λ

(18.5)

u(y)dy.

−λ

For periodic functions, it coincides with space average over one period:

u =

1 2π

2π

u(y)dy. 0

Multiplying (18.3) by e−iky , taking the space average and using (18.4) we find the Fourier coefficients u k of u(x): ; : u k = u(y)e−iky .

(18.6)

If u(y) is real then, denoting the complex conjugation by ∗, we can write u(y) =

+∞

k=−∞

u k eiky =

+∞

k=−∞

u −k e−iky = u ∗ (y) =

+∞

u ∗k e−iky .

k=−∞

Hence, the Fourier coefficients of a real-valued function obey the condition u −k = u ∗k .

(18.7)

18.1

More on Kozlov’s Cell Problem

901

The following fundamental statement holds: : 2; |u k |2 . |u| =

(18.8)

k

It is called Parseval’s equation. For a Fourier series, containing a finite number of members, it can be checked by direct inspection: < = : ; : 2; : ∗; iky ∗ −iky |u| = u u = uk e um e u k u ∗m ei(k−m)y = = k

=

k,m

m

k,m

u k u ∗m δkm =

k

|u k |2 .

An example of a function that may be non-periodic and non-decaying at infinity is eiαy + eiβy . The periods of the two terms are 2π/α and 2π/β, respectively ( α > 0, β > 0). If the ratio α/β is rational, i.e. there are integers p and q such that p/α = q/β, then the two terms have a common period, 2π p/α : e

iα y+k 2πα p

+e

iβ y+k 2πα p

= eiαy e2πikp + eiβy e2πikq = eiαy + eiβy

for any integer k.

If α/β is irrational, then the function is not periodic. In particular, the only point where it takes the value 2 is y = 0. In general, let us take the real numbers, k1 , k2 , . . . , kn , and some complex numbers, u 1 , . . . , u n , and form a sum: u(y) =

u s eiks y .

(18.9)

s

The function u(y) is typically non-periodic. We also admit into consideration infinite sums. The sums of the form (18.9) are called almost periodic functions. It is assumed that the series is converging. The meaning of the convergence can be different. Accordingly, there are different spaces of almost periodic functions. The set of points k1 , k2 . . . is called the spectrum of u(y). The spectrum can be found from the following statement: the space average : ; u(y)e−iλy

is zero if λ is not a point of the spectrum, and equal to the coefficient at eiky in the expansion (18.9), if λ = k.

902

18


If u(y) is real, and its spectrum contains a point k, then it also contains the point −k, and the corresponding coefficients are complex-conjugated: : ; : ;∗ u(y)e−iky = u(y)eiky .

Remarkably, Parseval’s equation (18.8) also holds for almost periodic functions, if the summation is taken over all points of the spectrum. As in the periodic case, Parseval’s equation for finite sums (18.9) can be checked by direct inspection. A periodic function of the period 2π has the spectrum k = 0, ±1, ±2, . . . . A periodic function of the period a has the spectrum k = 0, ±2π/a, ±4π/a, . . . . The spectrum of an almost periodic function is an arbitrary discrete set. For almost periodic functions of three variables (y 1 , y 2 , y 3 ) = y, the spectrum is a set of points in three-dimensional space k(s) , s = 1, 2 . . . . Almost periodic functions are sums of the form u(y) = u (s) eik(s) ·y (18.10) s

where k(s) · y is scalar product of the vectors k(s) and y. In the multi-dimensional case, formula (18.4) and Perseval’s equation, : 2 ; 2 u (s) , |u| = s

hold true. For real-valued functions u(y) along with each point k, the spectrum contains the point −k. If the point k has the number s, i.e. denoted by k(s) , it is convenient to use for the point −k, the number −s, and for the point k = 0, the number s = 0. A function u(y) is real-valued if and only if u (−s) = u ∗(s) .

(18.11)

For practical purposes, it is enough to deal with functions having a discrete spectrum. If the spectrum is dense, it would be convenient to include into consideration also the limit case, the functions with a continuum spectrum. However, this faces the following difficulty. Suppose we increase the number of points in the spectrum, making it more dense. Then it seems natural to replace the sum (18.10) by the integral u(y) ∼ u(k)eik·y dk. However, this is not possible: if the integral is converging, then it tends to zero as |y| → ∞, while we are considering the functions that do not decay at infinity. This is why it is hard to treat the functions with continuum spectrum within the framework of classical analysis. Nevertheless, a way to introduce the functions with continuum spectra exists. It is based on a radical change of the point of view: the

18.1


903

functions u(y) are assumed to be random.1 We turn now to consideration of the relations brought by this point. Homogeneous random fields. Now consider a random field (18.10), where k(s) are some deterministic vectors, while u (s) are random. Without loss of generality, we may assume that the mathematical expectation of function (18.10) is zero (by deducting, if necessary, the function Mu(y, ω)). Thus, Mu (s) = 0. To deal with ergodic functions, i.e. functions for which

u(y, ω) = Mu(y, ω) = 0, we set u (0) = u(y, ω) = 0. A crucial assumption is that any two coefficients, u (s) and u (r ) , are statistically independent if s = −r. In particular, Mu (s) u (r ) = Mu (s) Mu (r ) = 0

for s = −r.

(18.12)

Due to (18.11), this means that both real and imaginary parts of the coefficients, u ′(s) = Re u (s) and u ′′(s) = Im u (s) , are statistically independent for different s and r : Mu ′(s) u ′(r ) = 0 for s = −r, Mu ′(s) u ′′(r ) = 0 for any s, r,

Mu ′′(s) u ′′(r ) = 0

for s = −r,

besides, 2 2 M u ′(r ) = M u ′′(r ) .

Consider the correlation function of the field, u(y, ω), B(y, y ′ ) = Mu(y, ω)u(y ′ , ω). Not every function of two variables, B(y, y ′ ), can be the correlation function of some random field u(y, ω): to be a correlation function of some field, B(y, y ′ ) must satisfy some conditions. To obtain these conditions, consider a function, q(y), which is an arbitrary deterministic function vanishing for sufficiently large y. Then M

1

q(y)u(y, ω)dy

2

0.

This idea was suggested and developed by Khinchine [147]; see also [219].

904

18


Since M

q(y)u(y, ω)dy

2

=M

q(y)u(y, ω)q(y ′ )u(y ′ , ω)dydy′ ,

for any q(y) the inequality must hold

B(y, y ′ )q(y)q(y ′ )dy 3 d 3 y ′ 0.

(18.13)

This is a necessary condition which any correlation function must obey. For the random fields under consideration, the correlation function possesses an additional feature: it depends on y and y ′ through the difference τ = y − y ′ . Indeed, from (18.10) and (18.12) we have B(y, y ′ ) = M

s

u (s) eik(s) ·y

′

r

′

u (r ) eik(r ) ·y =

s

2 ′ M u (s) eik(s) ·(y−y ) .

(18.14)

The corresponding field, u(y, ω), is called a homogeneous (in a broad sense) random field. If the coefficients u (s) satisfy some additional conditions, which we do not consider, then the field is (strictly) homogeneous, i.e. for any τ,2 f n (y1 + τ, u 1 ; . . . ; yn + τ, u n ) = f n (y1 , u 1 ; . . . ; yn , u n ). In contrast to u(y), B(τ ) is expected to decay as |τ | → ∞, because the values of u(y) become independent at remote points. Therefore, the Fourier transformation B(k) of B(τ ),

B(k) =

B(τ )e−ik·τ d 3 τ,

(18.15)

could make sense. We assume that B(k) indeed exists. Then, if the spectrum becomes dense, and there are many points k(s) in small vicinities k of every point k, then (18.14) should transform to the inversion of (18.15),3 1 B(τ ) = B(k)eik·τ d 3 k. (18.16) (2π )3

2

For example, the one-point distribution function does not depend on y if, for each s, u (s) and Arg u (s) are statistically independent and Arg u (s) is homogeneously distributed.

3

See a subsection on Fourier transformation in Sect. 6.7.

18.1


905

Comparing (18.16) with (18.14) we conclude that 1 M |u s |2 . B(k)k ≈ 3 (2π )

(18.17)

k(s) ∈k

We see that the amplitude of each member of sum (18.10) decays as the number of points of the spectrum N increases. In the limit N → ∞ one can obtain a continuous (or piece-wise continuous) function B(k). In this case, one says that the random field has a continuum spectrum. An important consequence of (18.17) is that B(k) is a real and non-negative function. This feature, as was shown by Bochner in a different setting, is necessary and sufficient for the correlation function to satisfy (18.13). Kozlov’s cell problem. Consider Kozlov’s cell problem for a linear scalar case: 1 ij ψ ψ 1 ij vj + . aeff vi v j = a (y) vi + i min 2 y y j ψ: ψ/y i =0 2

(18.18)

This problem makes perfect sense if a i j (y) are almost periodic functions. It also makes sense if a i j (y) are random homogeneous functions: a i j = a i j (y, ω). In this case, the cell problem may be considered for every realization ω. Accordingly, the ˇ minimizer ψ is a random function, ψˇ = ψ(y, ω). Under some conditions on the ij coefficients a (y, ω), the minimum value in (18.18) is the same for (almost) all realizations. Roughly, the correlations of a i j (y, ω) at two points y and y ′ should ′ decay fast enough as y − y → ∞. Besides, ergodicity holds: the space average can be replaced by mathematical expectation. To avoid the discussion of these mathematical issues, we just put in what follows: 1 ij 1 ij ψ ψ vj + . aeff vi v j = M a (y, ω) vi + i min 2 y y j ψ: ψ/y i =0 2

(18.19)

Sometimes it is beneficial to present (18.19) in a different form by means of Hashin-Strikman transformation considered in Sects. 5.9 and 6.7. We add and substract in (18.19) the functional

1 ψ ψ i v + a0 vi + i 2 y yi

where a0 is a deterministic constant. We have 1 ψ ψ 1 ij i v + a vi v j = M a0 vi + i min 2 eff y yi ψ: ψ/y i =0 2 1 ij ψ ψ vj + . + a − a0 g i j vi + i 2 y y j

(18.20)

906

18


Then we present the last term in (18.20) by means of the Young-Fenchel transformation, 1 1 ij ψ ψ ψ i j i vj + = max − . p v a − a0 g i j vi + i p p + b ij i 2 y y j y i 2 pi Here bi j is the inverse tensor to a i j − a0 g i j : bi j a jk − a0 g jk = δik .

(18.21)

Changing the order of minimization and maximization4 we obtain 1 ψ ψ ψ 1 ij min aeff vi v j = M max a0 vi + i vi + + pj vj + 2 y yi y j pi (y,ω) ψ: ψ/y i =0 2 1 − bi j pi p j . (18.22) 2

The minimum over ψ is : ; 1 a0 vi v i + pi vi + J ( p) 2

where J ( p) =

min

ψ: ψ/y i =0

1 ψ ψ ψ a0 i + pj j . 2 y yi y

(18.23)

: ; Here we used the fact that a0 v i ψ/y i = 0. We arrive at the following Variational principle. The macro-Lagrangian can be found from the solution of the variational problem : j; 1 ij 1 1 i i j a vi v j = M max a0 vi v + v j p − bi j p p + J ( p) . 2 eff 2 pi (y,ω) 2

(18.24)

ˇ are Note that the maximizer in (18.24), pˆ i , and the minimizer in (18.23), ψ, linked as bi j pˆ j = vi +

ψˇ . y i

(18.25)

This follows from the minimax problem (18.22). Therefore, determining the maximizer pˆ i in (18.24), we also find the minimizer ψˇ of the original variational problem

4

The possibility of such change can be checked in the same way as in Sect. 5.8.

18.1


907

(18.18). The minimizer in the auxiliary variational problem (18.23) has no relation ˇ because p j (y) in (18.23) is an arbitrary field. to ψ, The variational principle (18.24) holds for any value of the constant a0 . Neither the maximum value nor the maximizer depends on a0 . The derivative of the maximum value with respect to a0 must vanish. Let us show that this yields the relation ; 1: 1 1 bik bkj pˆ i pˆ j + J ( pˆ ) = vi v i . 2 a0 2

(18.26)

Indeed, differentiating (18.21) with respect to a0 we determine the derivative of bi j : bi j = bkj bk j . a0

(18.27)

The derivative of an extreme value of a functional with respect to a parameter is computed in accordance with the rule formulated in Sect. 5.13. Therefore, differentiating (18.24) with respect to a0 , we get J ( pˆ ) 1 i 1 k i j vi v − bik b j pˆ pˆ + = 0. 2 2 a0

(18.28)

Similarly, J ( pˆ ) 1 ψˇ ψˇ . = a0 2 y i yi On the other hand, from Clapeyron theorem (5.46), J ( pˆ ) =

ˇ ˇ 1 ψ ψ . a0 2 y i yi

(18.29)

So, J ( pˆ ) 1 = − J ( pˆ ). a0 a0

(18.30)

Formula (18.26) follows from (18.28) and (18.30). The validity of (18.26) can also be checked by direct inspection by plugging (18.25) in (18.26) and using (18.29). If the variational principle is used to obtain estimates of the effective coefficients by computing the functional on the trial fields, then a0 becomes an arbitrary parameter that can be adjusted to get a better estimate. We have seen in Sect. 16.1 that the order in which one computes mathematical expectation and maximization can be changed. Therefore, the variational principle (18.24) can also be written in the form

908

18

1 1 ij a vi v j = a0 vi v i + max 2 eff 2 pi (y,ω)


1 v j M p j − M bi j pi p j + M J ( p) . (18.31) 2

Explicit form of J(p). The minimizer of the variational problem (18.23) is a solution of the equation y i

ψ i + p (y) = 0. a0 yi

(18.32)

Let p j (y) be some almost periodic functions p j (y) =

j

p(s) eik(s) ·y .

s

The functional (18.23) is invariant with respect to shifts of pi (y) for a constant, ; : i because ψ/y = 0. Therefore, we may replace pi (y) in (18.23) by : ; p˜ i (y) = pi (y) − pi .

Obviously

: i; p˜ = 0

and

p˜ i (y) =

(18.33)

i p(s) eik(s) ·y .

s=0

The minimizer in (18.23) is determined up to a constant, and we put additionally

ψ = 0. The spectrum of any solution of (18.32) must coincide with the spectrum of p j (y) for, if the spectrum of ψ(y) has a point k which does not belong to the spectrum of p˜ j (y), the coefficient at eik·y in the expansion of ψ(y), ψ(y) =

ψ(s) eik(s) ·y ,

s

as follows from (18.32), must be zero. From (18.32), j

ψ(s)

ik(s) j p(s) = 2 , a0 k(s)

2 k(s) = k(s) j k j , (s)

s = 0.

18.1


909

According to Clapeyron’s theorem (5.47), J ( p) =

1 j ψ . p˜ 2 y j

Therefore, = < 1 j ik(s) ·y ik(r ) ·y J ( p) = p e ψ(r ) ik(r ) j e = 2 s=0 (s) r =0 2 = < j −ik(r ) ·y 1 p(s) k(s) j 1 j ik(s) ·y ∗ ψ(r ) −ik(r ) j e p e =− = . (18.34) 2 s=0 (s) 2a0 s=0 k(s) 2 r =0

The variational problem (18.31) involves the mathematical expectation of J ( p). We have for M J ( p), M J ( p) = −

1 k(s)i k(s) j ∗j i M p(s) p(s) . 2a0 s=0 k(s) 2

(18.35)

j

i Now let the number of points in the spectrum of pi increases while M p(s) p(s) decrease in such a way, that for the points of the spectrum lying in a small vicinity, k, of a point k,

k(s) ∈k

∗j

i M p(s) p(s) ≈

1 B i j (k)k. (2π )3

(18.36)

Then the sum in (18.35) transforms into the integral 1 M J ( p) = − 2a0 (2π )3

km k j m j B (k)d 3 k. |k|2

(18.37)

It is assumed that B i j (k) decay fast enough as |k| → ∞ for the absolute convergence of the integral (18.37). The tensor B i j (k) must be non-negative, i.e. for any k and any ξ , B i j (k)ξi ξ j ≥ 0.

(18.38)

This follows from the definition of B i j (k) (18.36): i 2 1 ∗j i B i j (k)kξi ξ j ≈ M p(s) p(s) ξi ξ j = M p(s) ξi ≥ 0. 3 (2π ) k(s) ∈k

k(s) ∈k

Let B i j (τ ) be the Fourier transformations of B i j (k) :

910

18

B m j (k) =

B m j (τ )e−ik·τ d 3 τ, B m j (τ ) =


1 (2π )3

B m j (k)eik·τ d 3 k.

(18.39)

We assume that the integral

B i j (k)Bi j (k)d 3 k

is finite. Then, according to Parseval’s equation, the integral

B i j (τ )Bi j (τ )d 3 τ

is also finite. Hence, B i j (τ ) tend to zero as |τ | → ∞. Tensor B i j (τ ) has the meaning of correlations of the random field p˜ i (y, ω), : ; : ; B i j (τ ) = M p˜ i (y, ω) p˜ j (y + τ, ω) = M pi (y, ω) − pi p j (y, ω) − p j . (18.40) At large distances, p˜ i (y, ω) and p˜ i (y + τ, ω) become statistically independent, and B i j (τ ) = M p˜ i (y, ω)M p˜ i (y + τ, ω) → 0

as |τ | → ∞.

It is convenient to rewrite (18.37) in terms of B i j (τ ). We have seen in Sect. 6.7 that the Fourier transformation of 1/4π |x| is 1/ |k|2 (see (6.106)). Besides, ki k j B i j (k) is the Fourier transformation of −2 B i j (τ )/τ i τ j . Therefore, due to Parseval’s equation,

u(x)v ∗ (x)d 3 x =

1 (2π )3

u(k)v ∗ (k)d 3 k,

(18.37) can be written as 1 M J ( p) = 2a0

1 2 B i j (τ ) 3 d τ. 4π |τ | τ i τ j

(18.41)

This, of course, assumes that B i j (τ ) are sufficiently smooth. We accept additionally that B i j (τ ) are continuous and have continuous second derivatives at τ = 0. This allows us to move the derivatives from B i j (τ ) to 1/ |τ | . To perform such a transformation we note that, due to the convergence of the integral (18.41) at τ = 0, 1 M J ( p) = lim 8πa0 ε→0 In (18.42), one can integrate by parts:

|τ |ε

1 2 B i j (τ ) 3 d τ. |τ | τ i τ j

(18.42)

18.1


|τ |ε

911

1 B i j (τ ) 1 2 B i j (τ ) 3 1 ij B nj dA d τ =− ni − |τ | τ i τ j |τ | τ j |τ | ,i |τ |=ε 1 ,i j B i j (τ )d 3 τ. (18.43) + |τ |ε |τ |

Here n i are the components of the external normal to the sphere |τ | = ε. For the derivatives of 1/ |τ | we have

1 |τ |

,i

τi ni , ni ≡ , 2 |τ | |τ | 1 = − 3 gi j − 3n i n j . |τ |

=−

1 |τ |

,i j

(18.44) (18.45)

Note a relation for the average value of n i n j over the unit sphere:5 1 4π

|τ |=1

ni n j d A =

1 gi j . 3

(18.46)

An important consequence of (18.46) is that the tensor gi j −3n i n j , which appears in (18.45), has zero average over any sphere:

|τ |=const

gi j − 3n i n j d A = 0.

(18.47)

Therefore, also

|τ |=const

1 |τ |

,i j

d A = 0.

(18.48)

In the surface integral in (18.43) the first term tends to zero as ε → 0, because ij B, j are continuous at τ = 0. The limit of the second term, due to (18.44) and B i (0). So, finally, (18.46), is − 4π 3 i 1 M J ( p) = 2a0

5

gi j − 3n i n j i j 1 i 3 − Bi (0) − lim B (τ )d τ . ε→0 |τ |ε 3 4π |τ |3

(18.49)

It can be obtained in the following way: the integral of n i n j over the unit sphere is a tensor that is invariant with respect to orthogonal transformations. Therefore, it has the form const · gi j . The value of the constant can be found by computing the trace of the left hand side of (18.46), which is unity.

912

18


One can get rid of the limit in (18.49) and write just gi j − 3n i n j i j B (τ )d 3 τ, 4π |τ |3

(18.50)

if, in the vicinity of τ = 0, one sets the order of computation of integrals over spheres, |τ | = ρ = const, and over ρ : first, the integrals over the spheres are found, and then the integral over ρ is taken. Indeed, due to (18.47), the integral of the first ij ij term of the expansion B i j (τ ) = B i j (0) + B,k (0)τ k + B,kl (0)τ k τ l + . . . vanishes; the integral of the second term of the expansion is zero due to antisymmetry of τ k gi j − 3n i n j ; the integrals of all other terms are finite. Further we accept such convention on the order of integration and drop the limit in (18.49). Formula (18.49) corresponds to the following functional J ( p) (without mathematical expectation):

J ( p) = −

1 2a0

; 1: p˜ i p˜ i + 3

; 3 : ; gi j − 3n i n j : i j ˜ ˜ p (y, ω) p (y + τ, ω) d τ , p˜ i = pi − pi . 3 y 4π |τ | (18.51)

Here · y is the space average over y-variables. Note that functions of ; : τ, p˜ i (y, ω) p˜ j (y + τ, ω) y , may not decay as |τ | → ∞. For example, in the case of almost periodic functions, < = ∗j : l ; j l ik(s) ·y −ik(m) ·(y+τ ) p˜ (y, ω) p˜ (y + τ, ω) y = p(s) e p(m) e = s

=

m

y

∗j l p(s) p(s) e−ik(s) ·τ .

s

These functions, in general, do not decay as |τ | → ∞. However, since the integral over the unit sphere |n| = 1, gi j − 3n i n j e−ik(s) ·n|τ | d A, |n|=1

decays as |τ | → ∞, the integral (18.51) is converging. Explicit form of J(p) in two-dimensional problems. The derivation of the explicit form of J ( p) in two-dimensional problems requires some modifications. As before, in k-representation we obtain for M J ( p) a relation similar to (18.37): kα kβ αβ 1 B (k)d 2 k. (18.52) M J ( p) = − 2a0 (2π )2 |k|2 The small Greek indices run through values 1,2. In order to write this relation in τ −variables, we cannot just use the Parseval equation for the functions 1/ |k|2 and kα kβ B αβ (k) because the Fourier transformation of 1/ |k|2 , 1 ik·τ 2 e d k, |k|2

18.1


913

does not exist in the usual sense: the integral diverges at k = 0.6 Therefore, we write M J ( p) in the form kα kβ 1 lim B αβ (k)d 2 k. M J ( p) = − 2 2a0 (2π ) ε→0 |k|2 + ε The Fourier transformation of (|k|2 + ε)−1 , G ε (τ ) =

1 (2π )2

eik·τ d 2 k, |k|2 + ε

is meaningful. Therefore, M J ( p) =

1 lim 2a0 ε→0

G ε (τ )

2 B αβ (τ ) 2 d τ, τ α τ β

(18.53)

where B αβ (τ ) are the Fourier transformation of B αβ (k), 1 αβ B (τ ) = B αβ (k)eik·τ d 2 k = M p˜ α (y, ω) p˜ β (y + τ, ω), (2π )2 and p˜ α (y, ω) ≡ p α (y, ω) − p α . In the limit ε → 0, G ε (τ ) has the asymptotics,7 6

At k = ∞ the integral converges because, after the transformation to polar coordinates, the integral over θ, 2π ei|k|·|τ | cos θ dθ, 0

− 21

decays as |k| → ∞ as |τ |

7

.

Indeed,

1 (2π )2

eik·τ |k|2 + ε

d2k =

1 (2π )2

= 1 = (2π )2

0

ρdρ ρ 2 + ε |τ |2

2π 0

∞ 0

∞ 0

⎛ 1 ⎝

1 (2π )2 ρdρ

ρ 2 + ε |τ |2

ρdρ ρ2 + ε

2π

2π

eiρ|τ | cos θ dθ =

0

eiρ cos θ dθ =

0

1 + eiρ cos θ − 1 dθ +

∞ 1

ρdρ ρ 2 + ε |τ |2

2π 0

e

iρ cos θ

⎞

dθ ⎠ =

⎞ ⎛ 1 2π 2π ∞ iρ cos θ ρdρ ρdρ 1 ⎝ 1 1 + ε |τ |2 iρ cos θ ln = 2π + e − 1 dθ + e dθ ⎠ . (2π )2 2 ε |τ |2 ρ 2 + ε |τ |2 ρ 2 + ε |τ |2 0

0

1

0

914

18

G ε (τ ) =


1 1 1 − ln ln ε + const + 0(ε). 2π |τ | 4π

(18.54)

The integral of 2 B αβ /τ α τ β is zero because, presumably, B αβ (τ ) tend to zero at infinity fast enough. So, M J ( p) =

1 2a0

1 2 B αβ (τ ) 2 1 ln d τ. 2π |τ | τ α τ β

(18.55)

It is convenient to put M J ( p) in another form by getting rid of the derivatives of B αβ (τ ). To this end we note that, due to the convergence at τ = 0, the integral can be presented as a limit:

M J ( p) =

1 lim 2a0 ε→0

|τ |ε

1 1 2 B αβ (τ ) 2 d τ. ln 2π |τ | τ α τ β

Integrating by parts we get

M J ( p) =

⎡

1 ⎢ lim ⎣ 2a0 ε→0

|τ |=ε

+

|τ |ε

B αβ (τ )

−

1 1 1 B αβ 1 αβ ds+ ln + B n ln nβ α 2π |τ | τ α 2π τ β |τ | ⎤

2

1 2 ⎥ 1 ln d τ⎦ , τ α τ β 2π |τ |

(18.56)

where nβ =

τβ . |τ |

Since 1 nβ ln =− , |τ | τ β |τ |

2 τ α τ β

ln

1 1 = − 2 (gαβ − 2n α n β ), |τ | |τ |

Since B αβ (τ ) is vanishing at large |τ | , it is enough to consider the asymptotic behavior as ε → 0 for finite |τ | . The first term here yields the first two terms of (18.54). The other two terms are constant in the limit ε → 0.

18.1


915

and 1 2π

2π

0

n α n β dθ =

1 gαβ , 2

(18.57)

equation (18.56) can be finally written as

M J ( p) = −

⎡

1 ⎢1 α ⎣ B (0) + lim ε→0 2a0 2 α

|τ |ε

⎤

1 ⎥ (gαβ − 2n α n β )B αβ (τ )d 2 τ ⎦ . 2π |τ |2

(18.58) As in the three-dimensional case, limε→0 in (18.58) can be dropped if we accept a convention that in the vicinity of the point τ = 0 the integral is computed in polar coordinates in the following order: we take first the integral over the angle and after that the integral over the radius. Then, due to (18.57), the average value of (gαβ − 2n α n β )B αβ (τ ) over the circle vanishes in the leading approximation, and the integral over the radius becomes converging. Elasticity cell problem. The cell problem for geometrically linear elastic bodies has the form F(¯εi j ) =

ψ j 1 ψi F C(y, ω), ε¯ i j + . + 2 y j y i ψi : ψi /y j =0 min

(18.59)

Here C(y, ω) are the random elastic moduli. For the same reason, as in the scalar case, we introduce in (18.59) the mathematical expectation and write ψ j 1 ψi . F C(y, ω), ε¯ i j + + min F(¯εi j ) = M 2 y j y i ψi : ψi /y j =0

(18.60)

In the geometrically and physically linear theory, when F=

1 ijkl C εi j εkl , 2

we have 1 ijkl 1 ijkl ψk ψi ε¯ kl + . C ε¯ i j ε¯ kl = M C (y, ω) ε¯ i j + min 2 eff y j y l ψi : ψi /y j =0 2 (18.61) Here we used the symmetry of C ijkl over i, j and k, l. Now perform the transformations similar to that in Sect. 6.7:

916

18


1 ijkl 1 ijkl ψk ψk ψi ψi ε ¯ = ε ¯ + ε¯ i j + ε ¯ C C + + + kl kl ij 2 y j y l 2 0 y j y l

ψi ψk 1 ijkl C ijkl − C0 ε ¯ = ε¯ i j + + + kl 2 y j y l : ; 1 1 ijkl 1 ijkl = C0 ε¯ i j ε¯ kl + C0 ψi, j ψk,l + pi j ε¯ i j + max pi j ψi, j − Hijkl pi j p kl . 2 2 2 pi j (18.62)

Here pi j = p ji are random fields, and Hijkl (y, ω) the inverse tensor to C ijkl (y, ω) − ijkl C0 . From (18.61) and (18.62) we get the following Variational principle. The effective coefficients and the local fields of elastic random structures are determined by the variational problem : ij; 1 ijkl 1 ijkl 1 i j kl C ε¯ i j ε¯ kl = M max C0 ε¯ i j ε¯ kl + p ε¯ i j − Hijkl p p + J ( p) , (18.63) 2 eff 2 pi j (y,ω) 2 1 ijkl ij J ( p) = M min (18.64) C0 ψi, j ψk,l + p ψi, j . ψ i : ψ i /y j =0 2 The explicit form of J ( p) is found in the same way as in the scalar case. We will do it for an isotropic body, when C ijkl = λ0 g i j g kl + μ0 g ik g jl + g il g jk .

Then the minimizer in (18.64) is a solution of the equation (λ0 + μ0 )

pl j ψ j l + μ ψ + = 0. 0 yl y j y j

(18.65)

Assume first that pl j are almost periodic functions, lj p(s) eik(s) ·y . pl j = s

lj

The additive constants p(0) do not contribute to the functional (18.64) due to the : ; constraint ψ i /y j = 0. The functional J ( p) depends actually on the random field : ; l j ik(s) ·y p˜ l j (y, ω) = pl j (y, ω) − pl j = p(s) e . s=0

We seek the solution of (18.65) in the form ψj =

s

j

ψ(s) eik(s) ·y .

18.1


917

The minimizer in (18.64) is determined up to an additive constant. Thus, we put : ; i = ψ i = 0. ψ(0)

j

The spectra of ψ i and pi , as follows from (18.65), coincide. The coefficients ψ(s) are determined by the linear equations j

lj

l 2 l k(s) j ψ(s) + μ0 k(s) ψ(s) = ik(s) j p(s) . (λ0 + μ0 ) k(s)

(18.66)

The solution of these equations can be written explicitly. First, we contract (18.66) with k(s)l to obtain j

lj

2 k(s) j ψ(s) = ik(s)l k(s) j p(s) , (λ0 + 2μ0 ) k(s)

or lj

j k(s) j ψ(s)

=

ik(s)l k(s) j p(s) 2 (λ0 + 2μ0 ) k(s)

,

s = 0.

(18.67)

l Plugging this result in (18.66), we find ψ(s) :

l ψ(s)

l k k + μ λ 1 (s)m 0 0 (s) mj i δml − = k(s) j p(s) , 2 2 λ0 + 2μ0 k(s) μ0 k(s)

s = 0.

According to Clapeyron’s theorem, 1 lj ∗ 1 l j ψl =− p p ψ ik(s) j = j 2 y 2 s=0 (s) (s)

1 λ0 + μ0 k(s)m k(s)l mj ln . =− glm − k(s) j p(s) k(s)n p(s) 2 2 λ + 2μ 2μ k k 0 0 0 (s) (s) s=0

J ( p) =

Hence, M J ( p) = −

s=0

1 2 2μ0 k(s)

glm

λ0 + μ0 k(s)m k(s)l − 2 λ0 + 2μ0 k(s)

mj l n . k(s) j k(s)n M p(s) p(s)

(18.68) As before, we assume that, for a sufficiently large number of points of the spectrum,

s∈k

mj l n ≈ M p(s) p(s)

1 B mjl n (k)k, (2π )3

918

18


and the sum (18.68) transforms into the integral M J ( p) = −

k j kn 1 2μ0 k 2 (2π )3

glm −

λ0 + μ0 km kl λ0 + 2μ0 |k|2

B mjl n (k)d 3 k.

(18.69)

Functions B mjl n (k) are the Fourier transformations of B mjl n (τ ) = M p˜ m j (y, ω) p˜ l n (y + τ, ω). As in the scalar case, the convergence of the integral

B mjl n (k)Bmjl n (k)d 3 k

yields the convergence of the integral

B mjl n (τ )Bmjl n (τ )d 3 τ.

Thus, at infinity the correlations B mjl n (τ ) decay faster than 1/ |τ |3/2 . The tensor λ0 + μ0 km kl 1 g − ml λ0 + 2μ0 |k|2 μ0 |k|2 is the Fourier transformation of Green’s tensor τm τl 1 (3 − 4ν0 )gml + . G ml (τ ) = 16π μ0 (1 − ν0 ) |τ | |τ |2 This follows from (6.90) and (6.102). Hence, using Parseval’s equation, we finally write (18.69) as 1 M J ( p) = 2

G ml (τ )

2 B mjl n (τ ) 3 d τ. τ j τ n

(18.70)

18.2 Variational Principle for Probability Densities Variational problems of the previous section admit a reformulation in terms of probability densities in the same way as for usual integral functionals in Sect. 16.8. We illustrate this reformulation for the case of a scalar problem,

1 ij ψ ψ vj + → min a (y, ω) vi + i . 2 y y j ψ: ψ/y i =0

18.2


919

Let the random fields a(y, ω) and ψi (y, ω) be homogeneous and ergodic. Homogeneity yields the independence of the one-point probability density on y: f 1 = f (a, ψ). Due to ergodicity,

1 ij 1 ij a (vi + ψi ) v j + ψ j = a (vi + ψi ) v j + ψ j f (a, ψ)dad3 ψ. 2 2 (18.71) Here the notations used in Sect. 16.8 are changed slightly: since only the gradients of ψ appear in the problems of this chapter, and ψ does not enter explicitly in Lagrangian, we denote by ψ, for brevity, the set of components of the vector ψi . Accordingly, d 3 ψ means the volume element in the space of variables, ψi : d 3 ψ = dψ1 dψ2 dψ3 . By da we denote the volume element in the space of variables a i j . The meaning of da may vary depending on the character of randomness of a i j (y, ω). For example, for polycrystals a i j (y, ω) change over the space in a quite special way: there is a constant matrix, a˚ i j , and a i j at a point y is obtained from a˚ i j by an orthogonal transformation, oki : j

a i j (y, ω) = a˚ kl oki (y, ω)ol (y, ω).

(18.72)

The matrix oki (y, ω) is a random field. It can be specified by three angles of rotation, θ i (y, ω). Accordingly, in this case da = dθ 1 dθ 2 dθ 3 . The one-point probability density must be included in an infinite family of n-point probability densities, f n = f y1 , a1 , ψ1; y2 , a2 , ψ2; . . . ; yn , an , ψn ,

in such a way, that the compatibility equations hold:

f n+1 dan+1 d 3 ψn+1 = f n .

(18.73)

The n-point probability densities are symmetric with respect to transposition of the groups of arguments separated by semicolons. Besides, due to the homogeneity of the random fields a i j (y, ω) and ψi (y, ω), f (y1 + τ, a1 , ψ1 ; . . . ; yn + τ, an , ψn ) = f (y1 , a1 , ψ1 ; . . . ; yn , an , ψn ) . (18.74) The random field a i j (y, ω) is given. Thus, we know all its probability densities: f˚n = f (y1 , a1 ; . . . ; yn , an , ) . This yields the constraints

f n d 3 ψ1 . . . d 3 ψn = f˚n ,

f n 0.

(18.75)

920

18


The random field ψi (y, ω) has zero space average. Due to ergodicity, this means that Mψi = 0, i.e.

3

ψi f (ψ)d ψ = 0,

f (ψ) ≡

f (a, ψ)da.

(18.76)

The random field ψi (y, ω) is potential. The gradient compatibility condition is derived in the same way as in Sect. 16.8. It involves the two-point probability density f (y, ψ; y ′ , ψ ′ ), which we write using homogeneity of the random field ψi (y, ω) as f (ψ ′ ; τ, ψ), where τ = y ′ − y. Then the symmetry condition of the two-point probability density has the form f (ψ ′ ; τ, ψ, ) = f (ψ; −τ, ψ ′ ).

(18.77)

The gradient compatibility condition (16.140) in the case under consideration reads: for any divergence-free vector fields σ i (y), σ i = 0, y i

(18.78)

the following equation holds:

σ i (y)σ j (y ′ )ψi ψ ′j f ψ; y − y ′ , ψ ′ d 3 ψd 3 ψ ′ d 3 yd 3 y ′ = 0.

(18.79)

Functions σ i (y) can be taken as zero for sufficiently large |y| . Then the integral in (18.79) is actually an integral over a finite region. Note that the minimizer of the cell problem is expected to be statistically independent at remote space points. Therefore, f ψ; y − y ′ , ψ ′ should tend to the product of one-point densities f (ψ) and f (ψ ′ ) as y − y ′ → ∞. Thus, as y − y ′ → ∞,

ψi ψ ′j

f ψ; y − y ′ , ψ ′ d 3 ψd 3 ψ ′ →

3

ψi f (ψ)d ψ

ψi′ f (ψ ′ )d 3 ψ ′ = 0.

Minimization of the functional (18.71) over all random fields ψ(y, ω) is equivalent to minimization of the functional (18.72) over all probability densities satisfying the constraints stated. We obtain the following Variational principle. The true probability densities minimize the linear functional I( f ) =

1 ij a (vi + ψi ) v j + ψ j f (a, ψ) dad3 ψ 2

(18.80)

on the set of probability density functions selected by the linear constraints (18.73), (18.74), (18.75), (18.76) and (18.79).

18.2


921

If the Hashin-Strikman transformation of the cell problem is performed then, according to (18.31) and (18.49), the functional should be taken as I( f ) =

−

1 2a0

p¯ i ≡

1 1 v j p j − bi j pi p j f (a, p)dad3 p a0 vi v i + 2 2 1 − ( pi − p¯ i )( pi − p¯ i ) f ( p)d 3 p 6a0

(18.81)

gi j − 3n i n j i ( p − p¯ i )( p ′ j − p¯ j ) f ( p; τ, p ′ )d 3 pd 3 p ′ d 3 τ, 4π |τ |3 pi f ( p)d 3 p,

and its maximum value should be sought. The vector field pi (y, ω) is arbitrary, and no gradient compatibility is needed. The reformulation of the cell problem in terms of probability densities in other cases is done similarly. We note here only some peculiarities concerning the dual variational principles. The dual variational principle for Kozlov’s cell problem reads : ;1 0 ¯ L(v, vi ) = M max v i pi − L ∗ (a(y, ω), v, p)

(18.82)

where maximum is taken over all divergence-free random vector fields pi (y, ω), pi (y, ω) = 0. y i As we did: before, the maximization can ; : be ; split into maximization over all constants p¯ i = pi and all fields p˜ i = pi − pi , p˜ i (y, ω) = 0. y i

The constraint (18.83) can also be written as ψ p˜ i (y, ω) i = 0 for any ψ(y) and any ω. y

(18.83)

(18.84)

We can replace the latter condition “for any ω” by the condition “for almost any ω” setting: ψ 2 M p˜ i (y, ω) i = 0 for any ψ. y This can be written in terms of the correlation function B i j (τ ) = M p˜ i (y, ω) p˜ j (y + τ, ω)

(18.85)

922

18


as ψ ψ 3 3 ′ d yd y = 0 y i y ′ j

B i j (y − y ′ )

ψ ψ ′ d 3 yd 3 pd 3 y ′ d 3 p ′ = 0. p ′ j − p¯ j f p; y − y ′ , p y i y ′ j (18.86)

for any ψ,

or, for any ψ,

pi − p¯ i

We arrive at Variational principle. The macro-Lagrangian and the microfields can be found from the variational problem i ¯ L(v, vi ) = max v pi − L ∗ (a, v, p) f (a, p)dad3 p f

where maximum is sought over all probability density functions obeying the compatibility conditions and the condition (18.86). The constraint (18.86) is the condition that is dual to the gradient compatibility condition.

18.3 Equations for Probability Densities Probability densities obey an infinite chain of equations which we first derive from Euler’s equations of the variational cell problem, and then directly from the variational principle for probability densities. For definiteness, we consider a linear scalar case (18.24); the generalizations are straightforward. We will use an explicit form of the functional J ( p) (18.51). So, 1 1 1 ij aeff vi v j = M max a0 vi v i + vi pi − bi j pi p j 2 2 pi (y,ω) 2 : : ;; 1 1 − ( pi − pi ) pi − pi + 2a0 3 : i ; j : j ;; 3 gi j − 3n i n j : i d τ . p (y, ω) − p p (y + τ, ω) − p + y |τ |3 (18.87) : ; Denote by Ai the variational derivative over p˜ i = pi − pi : : j ; 3 gi j − 3n i n j j 1 1 Ai = p (y + τ, ω) − p d τ . ( pi − pi ) + a0 3 |τ |3 Obviously

:

; Ai = 0.

18.3

Equations for Probability Densities

923

Therefore :

: ; : ; : : ;; : ; ; Ai δ p˜ i = Ai δpi − δpi = Ai δpi − Ai δpi = Ai δpi ,

and Euler equation of the variational problem (18.87) is bi j p j + Ai = vi , or, without intermediate notation,

: ; gi j − 3n i n j j p (y + τ, ω) − p j d 3 τ = vi . 3 |τ | (18.88) Note an exact relation following from (18.88):

1 1 bi j p + ( pi − pi ) + 3a0 i a0 j

; : bi j p j = vi .

(18.89)

Let us multiply (18.88) by δ(a − a(y, ω))δ( p − p(y, ω)) and apply the mathematical expectation to the result. We get

1 bi j (a) p j + ( pi − p¯ i ) f (a, p) + 3a0 gi j − 3n i n j ′ j 1 + p − p¯ j f a, p; τ, a ′ , p ′ da ′ dp ′ d 3 τ = vi f (a, p). 3 a0 |τ | (18.90)

Here p¯ i =

pi f (a, p)dadp.

(18.91)

This is an integral equation for the one-point probability density. It also in volves the two-point probability density f a, p; τ, a ′ , p ′ . To obtain an equation for the two-point probability density, we multiply (18.88) by δ(a − a(y, ω))δ( p − p(y, ω))δ(a ′ − a(y ′ , ω))δ( p ′ − p(y ′ , ω)) and compute the mathematical expectation of the result. We obtain 1 bi j (a) p j + ( pi − p¯ i ) f (a, p; τ, a ′ , p ′ ) + 3a0 gi j − 3n i n j j 1 ˜ p˜ d ad ˜ p˜ d 3 τ + p˜ − p¯ j f a, p; τ, a ′ , p ′ ; τ˜ , a, 3 a0 |τ˜ | = vi f a, p; τ, a ′ , p ′ . (18.92)

924

18


This equation is also not closed: it contains the three-point probability density. One can continue this process indefinitely to derive an infinite chain of equations for probability densities. Consider now how to derive the same chain of equations from the variational principle for probability densities, 1 1 1 ij vi pi − bi j (a) pi p j f (a, p)dadp− aeff vi v j = max a0 vi v i + f 2 2 2 1 1 − ( pi − p¯ i ) pi − p¯ i f (a, p)dadp+ 2a0 3 ′j gi j − 3n i n j i ′ i j ′ ′ ′ 3 ¯ ¯ p − p p − p f (a, p; τ, a , p )dadpda dp d τ . + |τ |3 (18.93) Here probability densities must be constrained by the compatibility conditions. Accordingly, the variations of the probability densities are not arbitrary. Let us take, for example, the constraint f (a, p)dp = f˚(a). Since f˚(a) is assumed to be given, δ f (a, p) must obey the equation δ f (a, p)dp = 0. Any function that satisfies this equation can be presented in the form δ f (a, p) =

i ξ (a, p), pi

(18.94)

where ξ i (a, p) decay fast enough as p → ∞. However, not every function suffices, because other compatibility conditions must be satisfied as well. We have to design the variations of probability densities that automatically satisfy all compatibility conditions. To this end we note that all compatibility conditions are satisfied by the following presentation of probability densities: f n = Mδ (a1 − a (y1 , ω)) δ ( p1 − p (y1 , ω)) . . . δ (an − a (yn , ω)) δ ( pn − p (yn , ω)) . (18.95) Therefore, the variations of probability densities that are due to a variation of p(y, ω) and computed from (18.95) will automatically respect the compatibility conditions. Let us find the variation of the one-point probability density δ f (a, p) = δ Mδ (a − a (y, ω)) δ ( p − p (y, ω)) δ ( p − p (y, ω)) i δp (y, ω). = −Mδ (a − a (y, ω)) pi

18.3

Equations for Probability Densities

925

We consider a special variation of the random field setting δpi (y, ω) to be a deterministic function of a(y, ω) and pi (y, ω) : δpi = δpi (a(y, ω), p(y, ω)) .

(18.96)

Then δ f (a, p) = −Mδ (a − a (y, ω))

δ ( p − p (y, ω)) i δp (a(y, ω), p(y, ω)) = pi

Mδ (a − a (y, ω)) δ ( p − p (y, ω)) δpi (a(y, ω), p(y, ω)) pi = − i f (a, p)δpi (a, p). (18.97) p =−

Certainly, such δ f (a, p) has the form (18.94). In addition, we can guarantee that other compatibility conditions can be satisfied. Let us find the variation of the two-point probability density: δ f a, p; τ, a ′ , p ′

= δ Mδ (a − a (y, ω)) δ ( p − p (y, ω)) δ a ′ − a (y + τ, ω) δ p ′ − p (y + τ, ω) δ ( p − p (y, ω)) i = −Mδ (a − a (y, ω)) δp (a(y, ω), p(y, ω)) δ a ′ − a (y + τ, ω) pi × δ p ′ − p (y + τ, ω) ′ − Mδ (a − a (y, ω)) δ p − p (y + τ, ω) δ a ′ − a (y + τ, ω) δ p ′ − p (y + τ, ω) × δpi (a(y + τ, ω), p(y + τ, ω)) p ′i = − i f a, p; τ, a ′ , p ′ δpi (a, p) − ′i f a, p; τ, a ′ , p ′ δpi a ′ , p ′ . (18.98) p p

Varying the functional in (18.93) and using (18.97) and (18.98), we have f (a, p)δpi (a, p) 1 k k j vk p − bk j p p − dadp 2 pi f (a, p)δpi 1 dadp + ( pk − p¯ k ) p k − p¯ k 6a0 pi gk j − 3n k n j ′k 1 p − p¯ k p ′ j − p¯ j + 3 2a0 |τ | ′ ′ i i dadpdadpd3 τ = 0. a p) + f δp f δp , p × (a, 2 2 pi p ′i This equation yields (18.90). To derive the second equation of the chain, for each given value of τ = y ′ − y, a ′ and p ′ , we choose δpi (y, ω) to be

926

Then

18


δpi (y, ω) = ξ i (a(y, ω), p(y, ω)) δ a ′ − a(y ′ , ω) δ p ′ − p(y ′ , ω) . δ f (a, p) = −

f a, p; τ, a ′ , p ′ ξ i (a, p) . pi

Similarly, δ f a, p; τ, a ′ , p ′ will be expressed in terms of the three-point probability density, and the variation of the functional in (18.93) will yield the second equation of the chain. Other equations of the chain are obtained analogously. In conclusion, note two cases in which the integral of f 2 disappears from the first equation of the chain (18.90). Let us integrate this equation over a and p. The integral of the second term in the left hand side is zero because f a, p; τ, a ′ , p ′ dadp = f a ′ , p ′ and

f a ′ , p ′ p ′ j − p¯ j da ′ dp ′ = 0.

Therefore, (18.90) is reduced to bi j (a) p j f (a, p)dadp = vi .

Due to ergodicity, this equation coincides with (18.89). Another case is the one in which f a, p; τ, a ′ , p ′ depends on τ only through |τ | . In this case the solution of (18.90) can be obtained explicitly. Indeed, the second term in the left hand side is zero due to (18.47). Equation (18.90) takes the form 1 bi j (a) p j + (18.99) ( pi − p¯ i ) − vi f (a, p) = 0. 3a0 The only solution of (18.99) is f (a, p) = f˚(a)δ( p − ϕ(a)) where ϕi (a) is the solution of the system of equations bi j (a)ϕ j (a) +

1 (ϕi (a) − p¯ i ) = vi , 3a0

ϕi (a)f˚(a)da = p¯ i .

This system can be solved explicitly: writing the first equation in the form 1 p¯ i , ϕ j (a) = ci j (a) vi + 3a0

(18.100)

18.4

Approximations of Probability Densities

927

where ci j (a) is the inverse tensor to bi j (a) + both sides, we obtain an equation for p¯ i :

1 g , 3a0 i j

and taking the average value of

1 p¯ i , c¯ i j ≡ ci j (a)f˚(a)da. p¯ i = c¯ i j vi + 3a0 It has a solution, p¯ i = dik c¯ k j v j , where dik is the inverse tensor to g i j − finally obtain

1 ij c¯ . 3a0

Plugging this result in (18.100), we

1 ϕ i (a) = ci j (a) δ kj + d jm c¯ mk vk . 3a0

(18.101)

Note that the assumption on the dependence of f 2 on τ only through |τ | can serve only as an approximation, because the random field pi (y, ω) depends linearly on the vector v i and, thus, the dependence on τ must not be isotropic.

18.4 Approximations of Probability Densities At the moment almost nothing is known on probability densities of local fields in random structures either experimentally or theoretically. Few exceptions are concerned with some weakly stochastic problems [136, 332, 286, 69]. Most probably strongly stochastic cases can be treated only numerically. If, however, the variational problem for probability densities is simplified by dropping most of the constraints, then the problem may admit an analytical solution. First we give an example of such a case. Consider the minimization problem for the functional (18.71): I( f ) =

1 ij a (vi + ψi ) v j + ψ j f (a, ψ)dadψ. 2

(18.102)

Out of an infinite chain of constraints, let us keep only three:

f (a, ψ)dψ = f˚(a), f (a, ψ) 0, f (a, ψ)ψi dψ = 0.

(18.103) (18.104)

Then the problem can be solved exactly. We are going to show that f (a, ψ) = f˚(a)δ (ψi − ϕi (a)) ,

(18.105)

928

18


where functions ϕi (a) are determined from the system of linear equations: 1 ij a (vi + ϕi (a)) = σ i , 2

j vi = a¯ i−1 j σ ,

(18.106)

˚ ai−1 j f (a)da.

a¯ i−1 j ≡

(18.107)

To this end we note that the constraint (18.104) can be taken care of by introducing the Lagrange multiplier σ : min

max J ( f, σ ), σ 1 ij a (vi + ψi ) v j + ψ j − σ i ψi f (a, ψ)dadψ. J ( f, σ ) = 2 f ǫ(18.103),(18.104)

I( f ) =

min

f ǫ(18.103)

(18.108)

If we change the order of minimum and maximum in (18.108), then the result can only decrease (see (5.81)). So, min

f ǫ(18.103),(18.104)

I ( f ) max σi

min

f ǫ(18.103)

J ( f, σ ).

Minimum of J ( f, σ ) over f (a, ψ) that are subject to the constraints (18.103) is easy to find: for each given a, function f (a, ψ) should be concentrated at a point ψi = ϕi (a) that minimizes the quadratic form, 1 ij a (vi + ψi ) v j + ψ j − σ i ψi . 2

Functions ϕi (a) solve the system of linear equations (18.106). So, j ϕi (a) = ai−1 j v − vi ,

(18.109)

while the probability density is given by (18.105). The minimum value of J ( f, σ ) is min

f ǫ(18.103)

1 σiσ j. J ( f, σ ) = σ i vi − a¯ i−1 2 j

Maximization over σ i yields (18.107). Not surprisingly, as follows from (18.109), (18.107) means that the condition (18.104) is satisfied. Finally, min I ( f ) max σ

min

f ǫ(18.103)

J ( f, σ ) =

1 −1 −1 a¯ vi v j 2 ij

(18.110)

−1 where a¯ i−1 is the inverse tensor to a¯ i−1 j j . Function (18.105) with ϕi (a) determined by (18.109) and (18.107) satisfies the constraints (18.103) and (18.104) and, therefore, is an admissible function in the

18.4

Approximations of Probability Densities

929

original variational problem. Plugging it in the functional I ( f ) we obtain an upper estimate min

f ǫ(18.103),(18.104)

I( f )

1 −1 −1 a¯ vi v j . 2 ij

(18.111)

1 −1 −1 a¯ vi v j , 2 ij

(18.112)

As follows from (18.110) and (18.111), min

f ǫ(18.103),(18.104)

I( f ) =

as claimed. Formula (18.112) is the Reuss-Hill low bound of the effective coefficients, while (18.105) gives the corresponding probability density. As we discussed in Sect. 16.8, an immediate consequence of (18.105) is the relation for n-point probability densities, f (a, ψ; τ1 , a1 , ψ1 ; . . . ; τn , an , ψn ) = = f˚ (a; τ1 , a1 ; . . . ; τn , an ) δ (ψ − ϕ(a)) δ (ψ1 − ϕ(a1 )) . . . δ (ψn − ϕ(an )) .

(18.113)

Therefore, all the constraints for probability densities are satisfied. The only constraint that may be violated and, thus, make the solution obtained an approximate solution, is the constraint of gradient compatibility. Whether the gradient compatibility constraint is violated or not depends on the microstructure, i.e. on the functions f˚ (a; τ1 , a1 ; . . . ; τn , an ) . If the microstructure is a layered composite then, as we have seen in Sect. 17.5, the Reuss-Hill bound is achieved and the solution obtained is exact. One can check that for a layered composite the probability densities (18.113) satisfy the gradient compatibility condition. Of course, there are materials for which the solution obtained is not exact. To give an example, consider the two-point probability density characteristics of a polycrystal in scalar problems a i j (y, ω). With overwhelming probability, the difference a i j (y + τ, ω) − a i j (y, ω) tends to zero if τ → 0, i.e. the measure of points for which such difference is finite tends to zero. Hence, for the two-point probability density we have f˚(a; τ, a ′ ) → f˚(a)δ(a − a ′ )

as τ → ∞.

(18.114)

If τ → ∞, then a i j (y, ω) and a i j (y + τ, ω) become statistically independent, and f˚(a; τ, a ′ ) → f˚(a) f˚(a ′ ) as τ → 0.

(18.115)

A simplest two-point probability density which satisfies (18.114) and (18.115) is f˚(a; τ, a ′ ) = f˚(a)δ(a − a ′ )χ (τ ) + f˚(a) f˚(a ′ )(1 − χ (τ )),

(18.116)

930

18


where function χ (τ ) is such that 0 χ (τ ) 1, χ (τ ) → 1 as τ → 0, χ (τ ) → 0 as τ → ∞.

(18.117)

For an isotropic medium χ (τ ) must depend only on |τ |. To check the gradient compatibility we take it in the form (16.142): ′ 2 ρ y, y ′ , Mψi (y, ω)ψ j y , ω = y i y ′ j

or, for y ′ − y = τ and ρ = ρ (τ ):

Mψi ψ ′j = −

2 ρ (τ ) . τ i τ j

(18.118)

In the case of probability densities (18.113) and (18.116),

′ Mψi ψ j = ϕi (a)ϕ j (a ′ ) f˚ a)δ(a − a ′ χ (τ ) + f˚(a) f˚(a ′ ) (1 − χ (ω) dada′ = ϕi ϕ j χ (τ ) + ϕ¯ i ϕ¯ j (1 − χ (τ )) , ϕi ϕ j ≡ ϕi (a)ϕ j (a)f˚(a)da. (18.119) Note that ϕi depends linearly on the constants vi while the material characteristic χ (τ ) does not depend on vi . Function ρ(τ ) coincides with Mψ(y)ψ(y + τ ) and, thus, is a quadratic function of v i : ρ = α i j (τ )vi v j . For isotropic material, α i j (τ ) = αg i j + βτ i τ j with α and β being some functions of |τ | . Writing down the corresponding right hand side of (18.118) and plugging in the left hand side of the expression for Mψi ψ ′j (18.119) one can check that the condition (18.118) cannot be satisfied. The probability density (18.105) has the following meaning: the random field ψi (y, ω) is a function of the random field a i j (y, ω). In particular, if a i j (y, ω) = const in some piece of material, ψi (y, ω) are constant in this piece as well. For example, if a i j (y, ω) are the characteristics of a polycrystal and, thus, constant in each grain, ψi (y, ω) are also constant in each grain. Moreover, in all similarly oriented grains the constants ψi are the same. The probability density (18.105) violates the gradient compatibility. To avoid the necessity to satisfy the gradient compatibility, we may consider the variational problem for probability densities after the HashinStrikman transformation is made (see (18.81)): 1 1 v j p j − bi j pi p j f (a, p)dad3 p a0 vi v i + 2 2 1 − ( pi − p¯ i )( pi − p¯ i ) f ( p)d 3 p 6a0 gi j − 3n i n j i 1 − ( p − p¯ i )( p ′ j − p¯ j ) f ( p; τ, p ′ )d 3 pd 3 p ′ d 3 τ, 3 2a0 |τ | 4π i i 3 p¯ ≡ p f ( p)d p. (18.120)

I( f ) =

18.5

The Choice of Probabilistic Measure

931

We can seek an approximate maximizer of I ( f ) of the form f (a, p) = f˚ (a) δ ( pi − ϕi (a)) .

(18.121)

For such one-point probability density the two-point probability density must also be of special form: f a, p; τ, a ′ , p ′ = f˚ a; τ, a ′ δ ( pi − ϕi (a)) δ pi′ − ϕi (a ′ ) .

If the two-point probability density of the material characteristics f˚ a; τ, a ′ depends on τ only through |τ | , then the last term in (18.120) vanishes, and for the determination of f (a, p) one obtains a variational problem: 1 1 vi pi − bi j (a) pi p j − ( pi − p¯ i )( pi − p¯ i ) f (a, p) dadp f 0 2 6a0

max

with the constraint

f (a, p) dp = f˚ (a) .

The solution is the function (18.121) where ϕi (a) is the maximizer in the variational problem 1 1 vi ϕ i (a) − bi j (a)ϕ i (a)ϕ j (a) − (ϕi (a) − ϕ¯ i )(ϕ j (a) − ϕ¯ j ) f˚ (a) da. ϕi (a) 2 6a0 (18.122) Here ϕ¯ i = ϕi (a) f˚ (a) da. max

Since this is an approximate solution of the original problem, an additional optimization over a0 should be performed. The variational problem (18.122) yields a low estimate of the effective coefficients first obtained by Hashin and Strikman from another reasoning.

18.5 The Choice of Probabilistic Measure Now we turn to the discussion of a proper choice of probabilistic characteristics of random microstructures. Consider again Fig. 17.1. To simplify the matter, let us assume that each precipitate is an isotropic ellipsoid with the elastic properties different from the properties of the matrix. Then each precipitate is specified by the coordinates of its center, the orientation and the semiaxes, i.e. by nine numbers. If there are N ellipsoids in region V , then the entire structure has n = 9N degrees of

932

18


freedom; denote them by ω = {ω1 , . . . , ωn }. The set in n-dimensional space run by these parameters has a quite complex geometry because we assume additionally that the ellipsoids do not overlap. The set has a finite volume || =

dω1 . . . dωn ≡

dω.

To model the microstructure we have to specify the probabilistic measure on . This measure crucially depends on which parameters we are able to measure experimentally. Let, say, all that we know about the microstructure be the volume concentration of the precipitates ρ. The volume concentration is some function of ω1 , . . . , ωn , ρ = (ω1 , . . . , ωn ). Let the experimental values of ρ lie in the interval [ρ, ρ + △ρ]. The admissible values of ω1 , . . . , ωn are in the region ρ ≤ (ω1 , . . . , ωn ) ≤ ρ + △ρ,

ω ∈ .

All points of this region should be assigned with equal probability. For, if the probabilities were not equal for different points, we know about the microstructure more than we claimed. In essence, this is Laplace’s principle of insufficient reason. Tending △ρ to zero, we arrive at the probability measure f (ω1 , . . . , ωn ) = cδ(ρ − (ω1 , . . . , ωn )),

ω ∈ .

(18.123)

Here δ(ρ) is δ-function, c the normalizing constant, c

δ(ρ − (ω1 , . . . , ωn )) dω = 1.

(18.124)

Such probability density can be interpreted as the conditional probability density (under condition, (ω1 , . . . , ωn ) = ρ) if the unconditional probability density on the set is constant: f (ω1 , . . . , ωn ) =

1 . ||

(18.125)

We see that the probability measure is quite similar to that in classical statistical mechanics. As in classical statistical mechanics the probability density is constant, while (18.123) is equivalent to microcanonical distribution with the function (ω1 , . . . , ωn ) playing the role of Hamilton function. The physical reasonings behind the constancy of the probability densities are quite different though: in statistical mechanics the probability density is constant in the phase space due to the invariance of the probabilistic measure with respect to Hamiltonian phase flow. Another difference, which yields various deviations from the the relations of classical statistical mechanics, is that admissible ω belong to a set with quite complex geometry.

18.5

The Choice of Probabilistic Measure

933

Any additional information on microstructure changes the probabilistic measure. If we know from experiments the values, ρ1 , . . . , ρk , of k characteristics, 1 (ω1 , . . . , ωn ), . . . , k (ω1 , . . . , ωn ), then repeating the previous reasoning we obtain f (ω) = cδ(ρ1 − 1 (ω)) . . . δ(ρk − k (ω)),

ω ∈ ,

(18.126)

where c

δ(ρ1 − 1 (ω)) . . . δ(ρk − k (ω))dω = 1.

(18.127)

In addition, we may know from experiments the values of effective coefficients. This is equivalent to knowing the value of energy; denote it by H (ω). Then the measure is8 f (ω) = cδ(E − H (ω))δ(ρ1 − 1 (ω)) . . . δ(ρk − k (ω)),

ω ∈ .

(18.128)

If the effective coefficients are all that we know, we obtain the probabilistic measure which is especially close to the microcanonical distribution, f (ω) = cδ(E − H (ω1 , . . . , ωn )),

ω ∈ .

(18.129)

In summary, to model a random structure means to construct its probabilistic measure; we split all that we know about the microstructure into two categories: all that we know and all that we do not know; the parameters describing what we do not know, ω, are endowed with the constant probability. As soon as a probabilistic measure on is known, one may ask the probabilistic questions: what is the probability distribution of the effective coefficients? what is the probability distribution of the local fields? etc. The situation in modeling of random structures is quite similar to the mathematical modeling in other physical circumstances where we design the model, find the solution, and check the outcomes against the experimental results. In the case of composite materials, we construct the probabilistic measure. If the coincidence of the effective characteristics, their fluctuations, local fields, etc., with the experimental data is satisfactory, the model is adequate indeed. 8

Another measure is obtained if one uses the maximum principle for the information entropy: Sinf = − f (ω) log[ f (ω)]dω.

The major motivation for the maximum information entropy principle in statistical mechanics is that it yields the firmly established Gibbs’ distribution. In homogenization problems, Gibbs’ distribution can be used as long as it approximates well the distribution (18.129). One can expect that this is the case in the limit n → ∞. For finite n the maximum information entropy principle brings the distribution which differs from (18.129).

934

18


The above-mentioned suggests another interpretation of the probabilistic measure (18.129). Consider random cell structures discussed in Sect. 17.10. Let ωk denote the set of random parameters on which the characteristics of the kth cell depend. The parameters ω1 , . . . , ωn are random and statistically independent. Accordingly, the probability density is given by formula (18.125). Suppose, we compute the effective characteristics and find that energy of the specimen differ from that found experimentally. This means that our probabilistic model (18.125) is not adequate. We have to modify the probabilistic model. Formula (18.129) suggests such a modification. If the effective coefficients and, thus, energy, are not known, then an additional characteristic appears describing scattering of the values of energy. This is entropy of microstructure to discussion of which we proceed.

18.6 Entropy of Microstructure Entropy of microstructure. To model the macroscopic behavior of materials one has to specify the parameters describing the material, ρ1 , . . . , ρk , and the dependence of energy E on these parameters and the thermodynamic entropy S : E = E(S, ρ1 , . . . , ρk ).

(18.130)

For simplicity, we consider here macroscopically homogeneous states; otherwise (18.130) holds for the corresponding densities. As soon as the equation of state (18.130) is specified and some additional assumptions regarding the nature of the irreversible processes are made, the usual thermodynamic formalism yields a closed system of governing equations (see Sect. 3.3 and 3.4). Modeling of the behavior of random structures faces the following difficulty. Consider, for example, the material shown in Fig. 17.1. If the microstructure of the alloy does not change in the course of deformation and the applied force is small enough to cause pure elastic macroscopic deformations, then one can use linear elasticity to describe the response of the material, and E=

1 ijkl C εi j εkl |V | + E 0 (S), 2

(18.131)

where εi j and C ijkl are the components of the strain tensor and elastic moduli tensor, respectively, |V | the volume of the region V occupied by the specimen, and E 0 (S) energy of the material at zero strains. One can measure the elastic moduli experimentally. They depend on microstructure; however, since the microstructure does not change in the course of deformation, these moduli are all that we need to know to describe the material response. The situation becomes essentially different if the microstructure changes. First of all, we have to say how to characterize the changes, i.e. we need to introduce the

18.6

Entropy of Microstructure

935

parameters describing the state of the microstructure. The number of such parameters is necessarily small not only because the model must be simple enough but also due to the limited experimental data on the microstructure. In what follows, by ρ1 , . . . , ρk we mean these parameters. In the case of the microstructure of Fig. 17.1, we may use as such parameters the volume concentration of the precipitates, their average eccentricity, etc. If all that one wishes to know is the elastic macroscopic response, then one needs to know only the values of elastic moduli: they accumulate all necessary information on the microstructure. However, if one also wishes to characterize the changes in microstructure, i.e. the evolution of the parameters ρ1 , . . . , ρk , and follow the usual thermodynamic formalism, one has to specify how energy depends on ρ1 , . . . , ρk . The problem that arises is that energy of the specimen is not determined uniquely by any finite set of such parameters. This is clearly seen from the homogenization problem we have considered. Since the parameters ρ1 , . . . , ρk are all that we have at hand, we have to admit that energy E can take different values for different samples even if these samples have the same values of parameters ρ1 , . . . , ρk . Thus, energy becomes a random number. We accept that, for given ρ1 , . . . , ρk , energy has some probability density function9 f (E | ρ1 , . . . , ρk ). We define entropy of microstructure, Sm (E, ρ1 , . . . , ρk ), by the Einstein-type formula10 f (E | ρ1 , . . . , ρk ) = ceSm (E,ρ1 ,...,ρk ) . Here c is the normalizing constant: ceSm (E,ρ1 ,...,ρk ) d E = 1.

(18.132)

(18.133)

Entropy of microstructure is defined up to an additive constant due to the presence of the constant c in (18.132). There are two qualitatively different situations in modeling of random structures: entropy of microstructure, Sm (E, ρ1 , . . . , ρk ), can be a smooth function of E or it may have a sharp maximum. In the latter case, as we will see from further examples, Sm contains a large factor N , Sm (E, ρ1 , . . . , ρk ) = N Sm (E, ρ1 , . . . , ρk ),

(18.134)

while Sm (E, ρ1 , . . . , ρk ) is a smooth function of E. In the further examples, the large parameter N has the meaning of a “number of inhomogeneities” in the random structure: N = |V | /a 3 , a being the correlation radius of the microstructure. ˆ appears with overIn the case (18.134), the most probable value of energy, E, whelming probability. At this value function Sm (E, ρ1 , . . . , ρk ) has maximum over 9

For simplicity, we consider adiabatic processes, and, since thermodynamic entropy is constant, it is dropped from the set of parameters. 10

Regarding Einstein’s formula see Sect. 2.4.

936

18


ˆ is a function of E for fixed ρ1 , . . . , ρk . The most probable value of energy, E, ρ1 , . . . , ρk : ˆ 1 , . . . , ρk ). Eˆ = E(ρ

(18.135)

Equation (18.135) can be considered as the equation of state (18.130). Since energy, up to small fluctuations, is a function of ρ1 , . . . , ρk , the parameters ρ1 , . . . , ρk may be viewed as the thermodynamic parameters of the system. The case when entropy of microstructure, Sm (E, ρ1 , . . . , ρk ), does not have sharp maximum is different: energy becomes an independent parameter of state additional to the parameters ρ1 , . . . , ρk . To return to the usual framework of classical thermodynamics, we have to admit that there is an additional parameter of state which “absorbs” the arbitrariness of energy for given ρ1 , . . . , ρk , entropy of microstructure, Sm , and E = E(Sm , ρ1 , . . . , ρk ).

(18.136)

Function (18.136) can be viewed as the inversion of the function Sm (E, ρ1 , . . . , ρk ) introduced by (18.132).11 To find entropy of microstructure experimentally, one has to consider many samples, to measure for each sample the values of the parameters, E, ρ1 , . . . , ρk , determine probability density function and compute entropy of microstructure from (18.132). In the case of crystal plasticity, when one is interested in modeling of motion and nucleation of crystal defects, energy can be found experimentally by comparing the amount of heat needed to melt the sample and the corresponding defectless monocrystal. If entropy has a sharp maximum and (18.134) holds, one may seek the joint probability density function of the parameters, E, ρ1 , . . . , ρk , f (E, ρ1 , . . . , ρk ). It is given by the formula f (E, ρ1 , . . . , ρk ) = c1 eSm (E,ρ1 ,...,ρk )

(18.137)

where function Sm (E, ρ1 , . . . , ρk ) is the same as in (18.132) while the constant c1 is c1 eSm (E,ρ1 ,...,ρk ) d Edρ1 . . . dρk = 1. Obviously, (18.132) follows from (18.137) and (18.134) asymptotically as N → ∞. For macrosamples, Sm is a quantitative measure for the parameters ρ to be thermodynamic parameters: if Sm has a sharp maximum, the parameters ρ are the thermodynamic parameters, otherwise they are not. In the former case, Sm does not enter the governing thermodynamical relations. For microsamples, Sm is always an essential characteristic of the system. 11

In fact, formula (18.132) is meaningful only in the case (17.8.1.5), and for smoothly changing entropy it must be rectified; this is done further (formula (18.141)).

18.6

Entropy of Microstructure

937

Dependence of microstructure entropy on energy. Consider the case when microstructure entropy depends only on energy. Usually, the parameters of the system change in some limits, and so does energy: there are minimum and maximum values of energy, E min and E max , and E min ≤ E ≤ E max . So, entropy, Sm (E) , is defined on the segment [E min , E max ]. A typical dependence of entropy on energy is shown in ¯ Fig. 18.1. Entropy takes its maximum 1 at some value of energy, E, and tends to −∞ 0 ¯ at both ends of the segment E − , E + . The value of energy, E, is the most probable. For large N , the maximum is very sharp, and the true value of energy fluctuates in ¯ Such a behavior is typical for macrosamples. The a small vicinity of the point E. deviations are pronounced in microsamples when the number N is not large. Fig. 18.1 A typical dependence of entropy on energy

Entropy of microstructure and distribution of macroscopic parameters. Let the probability density be given by formula (18.128). As in classical thermodynamics, we introduce the “parameter volume,” (E, ρ1 , . . . , ρk ) =

dω,

ω∈, H (ω)≤E,1 (ω)≤ρ1 ,...,k (ω)≤ρk

and define the entropy of microstructure as Sm (E, ρ1 , . . . , ρk ) = ln

(E, ρ1 , . . . , ρk ) . ||

(18.138)

Entropy is dimensionless and , since ≤ ||, always negative. The constant c in (18.128) can be expressed in terms of derivatives of entropy or the parameter volume. Indeed, writing the parameter volume in terms of the step function, θ (E), ( θ (E) = 0 for E < 0, θ (E) = 1 for E ≥ 0), (E, ρ1 , . . . , ρk ) =

θ (E − H (ω))θ (ρ1 − 1 (ω)) . . . θ(ρk − k (ω))dω,

ω∈

and using that dθ (E)/d E = δ(E), we have

938

18

k+1 (E, ρ1 , . . . , ρk ) = Eρ1 . . . ρk


δ(E − H (ω))δ(ρ1 − 1 (ω)) . . . δ(ρk − k (ω))dω.

ω∈

From this relation and the normalization condition for c, c=

k+1 (E, ρ1 , . . . , ρk ) Eρ1 . . . ρk

−1

(18.139)

.

In particular, if only the value of energy is known, and the probability density is (18.129), then c=

1 , E

E ≡

d(E) , dE

(E) ≡

dω.

(18.140)

ω∈, H (ω)≤E

Let one choose samples with all possible values of ω, and seek the joint probability density function f (E, ρ1 , . . . , ρk ) to observe the values E, ρ1 , . . . , ρk of the characteristics H, 1 , . . . , k . According to (18.125), 1 f (E, ρ1 , . . . , ρk ) = ||

δ(E − H (ω))δ(ρ1 − 1 (ω)) . . . δ(ρk − k (ω))dω,

or f (E, ρ1 , . . . , ρk ) =

1 k+1 (E, ρ1 , . . . , ρk ) . || Eρ1 . . . ρk

In terms of entropy (18.138) the joint probability density is f (E, ρ1 , . . . , ρk ) =

k+1 1 eSm (E,ρ1 ,...,ρk ) . || Eρ1 . . . ρk

(18.141)

If entropy contains a large parameter, as in (18.134), then formula (18.141) is asymptotically equivalent to (18.137).12 Otherwise, (18.141) is an exact relation which replaces (18.137) if entropy is a smooth function of energy.13 It should be used to find entropy from the experimental data in that case. Since the parameter space has a finite volume || , it is worth considering the complementary parameter volume ∗ , defined as ∗ = || − , 12 13

For positive temperatures, see next section.

Note the similarity with the theory of large thermodynamic fluctuations in Hamiltonian ergodic systems with a small number of degrees of freedom [46].

18.7

Temperature of Microstructure

939

and the complementary entropy Sm∗ = log

∗ . ||

The parameter volume increases as energy grows, and so does entropy, Sm . The complementary parameter volume ∗ and the complementary entropy Sm∗ decay as energy increases.

18.7 Temperature of Microstructure In this section we consider the case when only energy is known. Then Sm (E) = ln

(E) , ||

Sm∗ (E) = ln

∗ (E) . ||

(18.142)

We define the temperature of microstructure, Tm , by the “thermodynamic relation” Sm (E) 1 = . Tm (E) E

(18.143)

Accordingly, the complementary temperature is 1 Tm∗ (E)

=

Sm∗ (E) . E

Temperature of microstructure, Tm (E), is always positive because (E) (and, thus, Sm (E)) increases as energy grows. The complementary temperature Tm∗ (E) is always negative because ∗ (E) and Sm∗ (E) are decaying functions of energy. In general, entropies introduced by formulas (18.132) and (18.138) are different: entropy (18.138) is always increasing while entropy in (18.132) may have a maximum. Therefore, we use for entropy in (18.132) the notation S˜m (E). One can define the corresponding temperature of microstructure, T˜m (E), as 1 S˜m (E) = . E T˜m (E) ˆ its derivative and, accordingly, If entropy S˜m (E) has a maximum at a point E, ˜ ˆ ˜ Tm (E), changes the sign: Tm (E) is positive for E < Eˆ and negative for E > E. We will show for a bar with random structure (see Sect. 18.8) that ˆ S˜m (E) = Sm (E) for E < E,

ˆ S˜m (E) = Sm∗ (E) for E > E.

940

18


ˆ Accordingly, T˜m (E) coincide with Tm (E) for E < Eˆ and with Tm∗ (E) for E > E. Negative temperature states correspond to larger energies and are “hotter” than positive temperature states as in other branches of physics. The physical interpretation of temperature in classical thermodynamics is based on the equipartition law. The analogous relation for microstructures is

H H = . . . = ωN = Tm . ω1 ω1 ω N

(18.144)

where · means averaging over the surface H (ω) = E. The equipartition of energy would hold if the surface H (ω) = E was the boundary of the region H (ω) ≤ E, ω ∈ . However, usually this is not the case: the set may have its own boundaries due to the geometrical constraints on the parameters ω. For example, if the parameters ω are the position vectors of the inclusions in a composite, the set looks like a cheese with many holes. Therefore, some corrections may appear in (18.144). These corrections and the physical meaning of the energy equipartition depend on the meaning of the parameters ω. Consider some examples. Let an elastic rectangular plate be weakened by a series of notches (Fig. 18.2). One side of the plate is clamped, and at the opposite side some constant displacement is given. We know the experimental value of the effective coefficient/energy. The lengths of the notches, l1 , l2 , . . . , ln , play the role of the parameters ω. Each length changes within the limits 0 ≤ ls ≤ a, s = 1, . . . , n, a being the width of the plate. If the length of any of the notches is equal to a, then the energy of the plate is zero: the plate is split into two disconnected pieces. Derivative of energy with respect to ls has the meaning of the energy release rate of the sth notch, Es . We are going to show that the average values ls Es are the same for all notches and, moreover, they are equal to the complementary temperature of the microstructure:

Fig. 18.2 A rectangular plate weakened by a series of notches

18.7

Temperature of Microstructure

941

l1 E1 = l2 E2 = . . . = ln En = Tm∗ .

(18.145)

Energy release rate of each notch is negative (since the elastic energy of the plate decays as a notch elongates); therefore, not surprisingly, the complementary temperature, which is negative, enters (18.145). To prove (18.145) consider the average, 1

l1 E1 = E

a

...

a

l1

H δ (E − H (l1 , . . . , ln )) dl1 . . . .dln . l1

0

0

It can be rewritten as 1

l1 E1 = − E

a

...

a

l1

θ (E − H (l1 , . . . , ln )) dl1 . . . .dln . l1

0

0

Integrating by parts over l1 we have a

1

l1 E1 = − E E

...

a

aθ (E − H (a, l2 . . . , ln )) dl2 . . . .dln .

0

0

Since H (a, l2 . . . , ln ) = 0,

l1 E1 =

− || ∗ ∗ =− = ∗ = Tm∗ . E E E

We obtain the same result for l2 E2 , . . . , ln En . Thus (18.145) holds true. A weak version of (18.144),

H H ω1 = . . . = ωN , ω1 ω N

(18.146)

may be valid for some structures just for symmetry reasons. Let, for example, a periodic cell consist of N boxes, B1 , . . . , B N . Each box has its own heat conductivity play ak which may take any values within the limits a− and a+ . Heat conductivities 0 1 the role of parameters ω and the set is the product of N segments, a− , a+ . Energy14 is determined by the variational problem H (a1 , . . . , a N ) =

14

min

periodic ψ

N 1 k=1

2

ak

ψ ψ vi + d V. vi + i y yi

(18.147)

Bk

In case of heat conductivity, it would be more appropriate to call it dissipation, but this is of no importance for our consideration.

942

18


Since H 1 = ak 2

ψˇ ψˇ i v + vi + i d V, y yi

Bk

H (no ψˇ being the minimizer in the variational problem (18.147), the number ak a k summation over k) has the meaning of the averaged energy of the kth box. If all boxes are congruent and can be obtained one from another by translation, then equipartition of energy (18.146) holds due to symmetry. If the boxes are not congruent, then some “generalized equipartition” takes place. To obtain it, we note that, in accordance with (18.129) and (18.140), 1 H H = a1 a1 δ (E − H (a1 , . . . , a N )) da1 . . . .da N = a1 E a1 1 θ (E − H (a1 , . . . , a N )) =− a1 da1 da2 . . . .da N . E a1

Integrating by parts, we find 1 0 1 H =− a+ θ E − H a+ , a ′ − a− θ E − H a− , a ′ da ′ + . a1 a1 E E (18.148) Here a ′ = {a2 , . . . , a N }, da ′ = da2 . . . .da N . Denote by s (E, a) the parameter volume of the sth box, s (E, a) =

θ (E − H (a1 , . . . , as−1 , a, as+1 , . . . , a N )) da1 . . . das−1 das+1 . . . da N .

(18.149) It relates to probability density function, f s (E, a) , of the parameter as for a given value of energy,

f s (E, a) = δ (a − ak ) =

1 E

δ (a − as ) δ (E − H (a1 , . . . , a N )) da1 . . . da N ,

as f s (E, a) =

1 s (E, a) . E E

(18.150)

Then, from (18.148) and (18.149),

energy of the first box +

a+ 1 (E, a+ ) − a− 1 (E, a− ) = Tm . E

(18.151)

18.8

Entropy of an Elastic Bar

943

Similar relations hold for all other boxes. Formula (18.151) determines the meaning of equipartition of energy in this particular problem: it is not the averaged energies that are the same and equal to the temperature of microstructure, but the energies corrected by a term related to the parameter volumes of the extreme values of heat conductivity. In the case of elastic bar considered further this correction is a function of temperature. Therefore, the average energy of each box, though not equal to the microstructure temperature, is a universal function of the microstructure temperature.

18.8 Entropy of an Elastic Bar In this section the notions introduced above are illustrated by an example where all calculations can be done analytically. Consider a bar made up of a sequence of homogeneous sections. The number of sections is N , their Young’s moduli are a1 , a2 , . . . , a N , the length of each section is . The Young’s moduli, a1 , a2 , . . . , a N , play the role of parameters ω. The left end of the bar is clamped, the right end is shifted for u L . The true displacement is the minimizer in the variational problem: 1 I (u) = 2

L 0

a (x)

du dx

2

dx →

min

u(x): u(0)=0, u(L)=u L

.

(18.152)

The function a (x) is piece-wise constant on the segment [0, L] (L = N ) and takes the value as when (s − 1) ≤ x ≤ s. The effective Young modulus is defined by the equation H = min I (u) = u

1 aeff ε¯ 2 L , 2

ε¯ ≡

uL . L

We consider three situations. In the first, each Young’s modulus ak can take any value in the segment, a− ≤ ak ≤ a+ ,

(18.153)

and this is the only information which we have about the microstructure. The set is a product of N segments, [a− , a+ ], and || = [a] N ,

[a] ≡ a+ − a− .

Since we do not provide more information about the microstructure, each admissible sequence, {a1 , . . . , a N } , is endowed with the same probability: if the probabilities were different, then we know more than we claimed. So, the probabilistic measure on is

944

18

f (a1 , . . . , an ) =


1 = const, [a] N

and we can determine the frequency (the probability) of measuring a certain value of energy or the effective Young’s modulus (energy is in one-to-one correspondence with the effective Young’s modulus for a given average strain ε¯ ). The second situation is that the value of energy/effective Young’s modulus is known. Thus, the moduli a1 , . . . , an have the distribution on : f (a1 , . . . , an ) =

1 δ(E − H (a1 , . . . , an )). E

In the third case, instead of the effective modulus, we have other information on the microstructure: we know the arithmetic average of the local moduli: 1 (a1 + . . . + a N ) = ρ. N

(18.154)

Thus, the probability distribution is f (a1 , . . . , an ) = const · δ(ρ −

1 (a1 + . . . + a N )). N

We begin with the first case. First Case. The “micro-problem” (18.152) has, obviously, the minimizer x

u (x) = P

dx , a (x)

0

where P is a constant determined by the boundary condition

u (L) = P

L

dx . a (x)

0

Therefore, the effective Young’s modulus is given by the formula aeff =

1 1 N

1 a1

+

1 a2

+ ... +

1 aN

.

(18.155)

The effective Young’s modulus can take any value in the segment [a− , a+ ]: a− ≤ aeff ≤ a+ .

(18.156)

18.8


945

The value aeff = a can be obtained, for example, when all a1 , . . . , a N are equal to a. Accordingly, energy changes within the limits E− ≤ E ≤ E+,

E− =

1 a− ε¯ 2 L , 2

E+ =

1 a+ ε¯ 2 L . 2

Denote the probability density function of the effective Young’s modulus by f (t) . The corresponding probability distribution of energy is f (E/ 21 ε¯ 2 L)/ 12 ε¯ 2 L . Function f (t) is finite when t ∈ [a− , a+ ] and is equal to zero outside [a− , a+ ]. It turns out that the asymptotics of f (t) for large N has the form 1 f (t) = 2 t

N N S(t) , e 2π S ′′

a− ≤ t ≤ a+ .

(18.157)

Here S (t) is the minimum value of the function x

S (t) = min S (t, x) ,

S (t, x) = ln Me a −

x

x , t

(18.158)

where M is mathematical expectation: for any function ϕ(a), 1 Mϕ ≡ [a]

a+

ϕ(a)da.

a−

We postpone the derivation of (18.157) and (18.158) to the end of this section while here we consider some of their consequences. The point xˇ where S(t, x) reaches its minimum value is determined from the equation a+ a

−a+ a−

1 xˇ /a e da a e xˇ /a da

=

1 ; t

(18.159)

it is a function of t: xˇ = xˇ (t). The second derivative 2 S(t, x)/x 2 , at the point ′′ x = xˇ (t), is denoted by S ; it is also a function of t. The dependence of the dimensionless solution of (18.159), xˇ , on t is shown in Fig. 18.3 for a− /a+ = 0.1 (in all figures the parameters which have the dimension of a, in particular xˇ and t, are referenced to a+ ). When t changes from a− to a+ , the solution xˇ runs over the real axis from +∞ to −∞. The corresponding function S(t) is presented in Fig. 18.4. To find the most probable value of the effective coefficient, i.e. the value of t for which S(t) reaches its maximum, we note that xˇ d S (t, xˇ (t)) d S(t) = = 2. dt dt t

(18.160)

946

18


Fig. 18.3 Dependence of the solution of (18.159) on t for a− /a+ = 0.1

Fig. 18.4 Dependence of entropy on the value of the effective coefficient for a− /a+ = 0.1

Hence, at the point where d S(t)/dt = 0, we have xˇ = 0. From (18.159) we see that the maximum value of the function S (t) is reached at the point

am.p. =

M

1 a

−1

(18.161)

.

The number (18.161) has the meaning of the most probable value of the effective Young’s modulus. The maximum of the probability distribution (18.157) is very sharp for large N ; the bigger N , the sharper probability distribution; a comparison of the probability distributions of various N is shown in Fig. 18.5. Total entropy of the microstructure, S˜m , according to (18.157), is S˜m (E) =N S

E 1 2 ε ¯ L 2

,

(18.162)

18.8


947

Fig. 18.5 Probability density function of the effective coefficient for various numbers of inhomogeneities, N , N = 10 (upper curve), N = 100 (middle curve), N = 1000 (bottom curve). Probability density functions are normalized to have the same maximum value

where S(t) is the function (18.158). It is further shown that, for positive xˇ , the dependence of microstructure temperature on energy is determined from the relation d S˜m (E) 1 dSm (E) = . = Tm dE dE

(18.163)

Therefore, from (18.163), (18.162) and (18.160), 1 N xˇ (t) 1 xˇ (t) d(N S(t)) dt = 1 = . = 1 2 Tm dt dE ε¯ 2 L t 2 t= 1 E2 ε¯ t 2 t= 1 E2 2 2 2

ε¯ L

2

(18.164)

ε¯ L

Hence, if N , L and E tend to infinity in such a way that energy per unit length, E/L , and the segment size, = L/N , are finite, temperature of microstructure is finite. For xˇ < 0, the microstructure temperature goes to infinity in this limit as we shall see further. The complementary temperature is finite and given by (18.164) for xˇ < 0, d S˜m (E) 1 xˇ (t) 1 dSm∗ (E) = = 1 , = Tm∗ dE dE ε¯ 2 t 2 t= 1 E2 2 2

(18.165)

ε¯ L

Tm∗ tending to −∞ as N → ∞ for xˇ > 0 .

Second case. Consider now the second case when the value of energy is known. An interesting question here is: what is the distribution of local Young’s modulus, say, a1 , if one knows the value of the effective Young’s modulus, aeff . It is shown further at the end of this section that the probability density function of each Young modulus becomes exponential, β

f (a|aeff ) = ce a ,

(18.166)

948

18


where c is a normalizing coefficient while β is a function of aeff determined from the equation M a1 eβ/a 1 = . β/a Me aeff

(18.167)

This probability density function is homogeneous only when β = 0, i.e. when ¯ the effective coefficient has the value a, 1 1 1 =M ≡ a¯ a [a]

a+

da . a

a−

Third case. In the third case we seek for the probability density of the effective Young’s modulus under condition that the arithmetic average of the local moduli is known: a1 + . . . + a N f (t | ρ) = cM δ (t − aeff ) δ ρ − . (18.168) N Here the constant c is determined by the normalizing condition

f (t | ρ) dt = 1.

Its asymptotics as N → ∞ is given by the formula f (t | ρ) = const e N S(t,ρ) ,

(18.169)

where S (t, ρ) is the minimum value in the variational problem: S (t, ρ) = min S (t, ρ, x, ξ ) ,

(18.170)

x,ξ

x S (t, ρ, x, ξ ) = − + ξρ + ln t

a+

x

e a −ξ a

da . [a]

(18.171)

a−

The minimizing point {xˇ , ξˇ } of the variational problem (18.170) is determined by the system of two nonlinear equations: a+ 1

xˇ

ˇ

e a −ξ a da a

a− a+ a−

e

xˇ ˇ a −ξ a

da

1 = , t

a+

a−

xˇ

ˇ

ae a −ξ a da

a− a+

e

xˇ ˇ a −ξ a

da

= ρ.

(18.172)

18.8


949

The most probable value corresponds to the point where

Since

d S t, ρ, xˇ (t), ξˇ (t) = 0. dt xˇ d S t, ρ, xˇ (t), ξˇ (t) = 2 dt t

at this point, we have xˇ = 0. So, for xˇ = 0, t is the value of the most probable effective coefficient, am.p. . From (18.172), for a given ρ and xˇ = 0, ξˇ must be found from the equation a+

ˇ

ae−ξ a da

a− a+

(18.173)

= ρ.

e−ξˇ a da

a−

If (18.173) is solved with respect to ξˇ , and this value of ξˇ , ξˇ (ρ), is inserted in the first equation (18.172), one obtains the dependence of the most probable value of the effective coefficient, am.p. , on ρ : a+

ˇ

e−ξ (ρ)a da

a−

am.p. =

a+ 1

a−

a

.

(18.174)

e−ξˇ (ρ)a da

The relation (18.174) can be interpreted, similarly to (18.161), as an inverse mathematical expectation of a −1 , am.p. =

˜ 1 M a

−1

(18.175)

,

with an “effective” probabilistic density ˇ

˜f (a) =

e−ξ (ρ)a a+

.

e−ξˇ (ρ)a da

a−

If ρ coincides with an “average value” of the “unconstrained” case, 1 ρ= [a]

a+

a−

ada =

a− + a+ , 2

(18.176)

950

18


Fig. 18.6 Dependence of the most probable effective coefficient on the value of the parameter ρ (the coefficients a referenced to a+ , and the ratio, a− /a+ is 0.1)

then, from (18.173), ξˇ = 0 and, from (18.176), f˜ (a) = const = [a]−1 ; hence M = ˜ and the most probable effective coefficient (18.175) coincides with (18.161). For M, all other values of ρ, the most probable effective coefficient differs from (18.161). The dependence of the most probable effective coefficient on ρ is shown in Fig. 18.6. Derivation of (18.157). In derivation of the asymptotic formulas like (18.157) it is convenient to deal with a smoother probability characteristics of the probability distribution like (18.177)

g (t) = Probability [aeff ≤ t] . Obviously, f (t) =

dg (t) . dt

(18.178)

The definition (18.177) can be written as

g (t) = Mθ t −

N a1−1 + . . . + a −1 N

.

(18.179)

The step function can be presented as the following integral in the complex plane z = x + i y, 1 θ (t) = 2πi

α+i∞

et z

dz . z

(18.180)

α−i∞

The integral is taken along the line [α − i∞, α + i∞] in the right half-plane, α > 0. The integrand is an analytical function whose absolute values tend to zero along any line x = Re z = const. Therefore the line can be moved arbitrarily in the right half-plane.

18.8


951

Plugging (18.180) into (18.177) we have α+i∞

1 M g (t) = 2πi

Nz 1 t z− a−1 +...+a −1 N dz. 1 e z

(18.181)

α−i∞

Let us make a change of variables z → 1 g (t) = M 2πi

α+i∞

a1−1 +...+a −1 N z. t

Then

1 (a −1 +...+a −1 Nz N )z− t dz. e 1 z

(18.182)

α−i∞

Changing the order of computation the mathematical expectation and the integral over z we have 1 g (t) = 2πi

α+i∞

1 N S(t,z) dz, e z

z z S (t, z) = ln Me a − . t

(18.183)

α−i∞

It is more convenient to deal with another integral which converges absolutely:

G (t) =

1 2πi

α+i∞

1 N S(t,z) dz. e z2

(18.184)

α−i∞

Function g (t) is expressed in terms of G (t) as g (t) =

dG t 2 . dt N

(18.185)

To find the asymptotics of the integral (18.184) as N → ∞ we use the method of steepest descent. To this end, we note that on the real axis the function S (t, z) is real and strictly convex. Therefore, for each value of t it has the only minimum. The minimum can be achieved though at −∞ or +∞. If the point of minimum xˇ is finite, it is the solution of the equation M a1 e xˇ /a 1 = . Me xˇ /a t

(18.186)

The point of minimum is a function of t : xˇ = xˇ (t). Function S (t, xˇ (t)) is also denoted as S(t). Function S(t) is negative. Indeed, for any convex function ϕ(x), ϕ(x) − ϕ(0) =

0

x

dϕ(x) dϕ(x) dx ≤ x, dx dx

952

18


because its second derivative is positive and, thus, dϕ(x)/d x is a non-decreasing function of x. Function ϕ(x) = log Me x/a is a convex function of x (since its second derivative

2 a+ 1 x/a a+ x/a a+ 1 x/a e da e da − e da 2 a− a− a a− a

2 a+ x/a da a− e is positive15 ); therefore, for any x,

log Me x/a ≤

M a1 e x/a x. Me x/a

Plugging here instead of x the solution xˇ (t) of (18.186), we obtain S(t) ≤ 0 as claimed. The function of x a+ 1 x/a da a ae −a+ (18.187) x/a da a− e

is an increasing function of x because its derivative is positive (see the footnote). The ratio (18.187) tends to 1/a− when x → −∞ and to 1/a+ as x → +∞. Accordingly, (18.186) has a solution when a− < t < a+ . Let t be from this interval, and let xˇ be positive. We choose α in (18.184) equal to xˇ . In the vicinity of the point xˇ , S (t, z) ≃ S (t, xˇ ) +

1 d 2 S (z − xˇ )2 . 2 dz 2 xˇ

(18.188)

The second derivative, d 2 S/dz 2 , at the point xˇ is a real number , 2 M a12 e xˇ /a Me xˇ /a − M a1 e xˇ /a d2 S , = 2 dz 2 Me xˇ /a

which is a positive (see the footnote) function of t. We denote it by S ′′ . Emphasize that S ′′ = d 2 S(t)/dt 2 . On the line [xˇ − i∞, xˇ + i∞] function Re S(t, z) has, according to (18.188), a maximum at the point y = 0 : 15

Positiveness follows from the inequality (we give it in a more general setting assuming that the random number, a, has some probability density function, f (a) ) : 2 xˇ /2a 2 2 1 e 1 xˇ /a = M e xˇ /a f (a)e xˇ /2a f (a)da ≤ e f (a) da = a a a xˇ /a e e xˇ /a ≤ f (a) da e xˇ /a f (a) da = M 2 Me xˇ /a . 2 a a

18.8


xˇ y 1 xˇ Re S (t, xˇ + i y) = ln Me a +i a − ≃ S(t) − S ′′ y 2 , t 2

953

S(t) ≡ ln Me

xˇ (t) a

−

xˇ (t) . t

The values of Re S (t, xˇ + i y) at any point y is less than Re S (t, xˇ ) = S(t) : xˇ y xˇ Re S (t, xˇ + i y) = ln Me a +i a − ≤ t xˇ y xˇ xˇ xˇ a +i a ≤ ln M e − = ln Me a − = S (t) . (18.189) t t

A more instructive estimate is to use the Cauchy inequality,

fg

2

≤

f2

g2,

2 2 y 2 y y xˇ xˇ xˇ e a sin f (a) da = e a cos f (a) da + Me a +i a = a a 2 y xˇ xˇ = e 2a f (a)e 2a f (a) cos da a 2 y xˇ xˇ + e 2a f (a)e 2a f (a) sin da ≤ a y xˇ ≤ e a f (a) da ρ f (a) cos2 da a y xˇ xˇ + e a f (a) da e a f (a) sin2 da = a 2

2 xˇ xˇ a = (18.190) e f (a) da = Me a . In the Cauchy inequality, the equality sign holds only if the functions f and g are proportional. In the estimate (18.190), the functions involved are not proportional, and thus we can conclude that Re S (t, xˇ + i y) < S (t, xˇ ) for y = 0. (18.191) y xˇ Moreover, Me a +i a decays as |y|−1 , and Re S (t, xˇ + i y) becomes negative for sufficiently large |y| : a+ xˇ y y xˇ i +i e a e a f (a) da = Me a a = a −a+ 1 y xˇ = e a f (a) a 2 d ei a = y a −? @

1 xˇ y a+ 1 a+ i y d xˇ = e a f (a) a 2 ei a e a f (a) a 2 ≤ ea − a− y y a− da const c ≤ = . (18.192) |y| |y|

954

18


Split the integral (18.184) into two parts, for |y| ≤ ε and |y| ≥ ε: 1 G (t) = 2π

ε

−ε

1 1 1 ′′ 2 e N ( S(t,xˇ )− 2 S y ) dy + 2 2π (xˇ + i y)

|y|≥ε

1 e N S(t,xˇ +i y) dy. (xˇ + i y)2

(18.193) According to (18.192), for any constant δ we can choose a constant l such that Re S (t, xˇ + i y) < δ for |y| ≥ l. On the finite segment |y| ≤ l, function of y, Re S (t, xˇ + i y) , is continuous, does not exceed S (t, xˇ ) and takes the value S (t, xˇ ) only at one point, y = 0. Therefore, there is a small ε such that Re S (t, xˇ + i y) < Re S (t, xˇ + iε) for ε < |y| < l. We see that Re S (t, xˇ + i y) < Re S (t, xˇ + iε) for all |y| ≥ ε. Therefore the second integral in (18.193) can be estimated as ∞ 1 N Re S(t,xˇ +iε) 1 dy 1 N S(t,xˇ +i y) e dy ≤ e 2 . 2 2π xˇ 2 + y 2 2π ˇ x + i y) ( |y|≥ε 0

(18.194)

Note that 1 2π

ε

−ε

1 1 1 ′′ 2 e− 2 N S y dy = 2 2π (xˇ + i y)

∞

−∞

1

′′ 2

e− 2 N S y 1 dy − 2 2π (xˇ + i y)

|y|≥ε

− 1 N S ′′ y 2

e 2 dy. (xˇ + i y)2 (18.195)

In the first integral, 1 2π

∞

−∞

1

e− 2 N S

∞ 2 xˇ 2 − y 2 − 2i xˇ y xˇ − y 2 − 1 N S ′′ y 2 1 dy, dy = 2 e 2 2 + y2 2 2π ˇ xˇ 2 + y 2 x −∞

′′ 2

y

√ we make the change of variables, y → y/ N S ′′ . We obtain 1 √ 2π N S ′′

∞

−∞

xˇ 2 −

y2 N S ′′

xˇ 2 +

y2 N S ′′

2 e

− 12 y 2

dy.

As N → ∞, this integral is equal to 1 1 √ ′′ ˇ x 2π N S 2

∞

−∞

e

− 12 y 2

dy =

1 1 . 2π N S ′′ xˇ 2

(18.196)

18.8


955

The second integral in (18.195) does not exceed the number 2

∞ 0

dy

xˇ 2

+

2 y2

1

e− 2 N S

′′ 2

ε

,

which is asymptotically smaller than (18.196). Similarly, the right hand side of (18.194), which is in the order of exp [N Re S (t, xˇ + iε)] , is asymptotically smaller than (18.196) (recall that Re S (t, xˇ + iε) < S (t, xˇ )). Finally we obtain in the leading approximation 1 e N S(t) . G (t) = √ 2π N S ′′ (t)xˇ 2

(18.197)

Differentiating (18.197) and taking into account that the derivative of the prefactor gives a negligible contribution, from (18.185) and (18.160) we have g (t) = √

e N S(t)

for xˇ > 0.

2π N S ′′ xˇ (t)

(18.198)

If xˇ < 0, then the line of integration [α + i∞, α + i∞] in (18.184) must be moved in the left half-plane. The singularity of the integrand at z = 0 causes the addition of the residual at z = 0 : 1 G (t) = 2πi

α+i∞

α+i∞

1 = 2πi

α+i∞

1 N S(t,z) de N S(t,z) dz + e = z2 dz z=0 1 N S(t,z) dz + N e z2

α+i∞

1 1 , α < 0. M − a t

(18.199)

From (18.185) and (18.199), 1 g (t) = 2πi

α+i∞

1 N S(t,z) e dz + 1, α < 0. z

(18.200)

α+i∞

Applying the steepest descent method to the integral in (18.200) we find g (t) = √

e N S(t) 2π N S ′′ xˇ (t)

+ 1, xˇ < 0.

(18.201)

Probability density function f (t) = dg(t)/dt is obtained by differentiation of (18.198) and (18.201) with the use of (18.160). We arrive at the universal relation holding for both positive and negative xˇ :

956

18

f (t) = √


N e N S(t) 2π N S ′′ t 2

(18.202)

.

If t < a− or t > a+ , then f (t) = 0. This follows directly from the definition of g(t) and θ (t) or can be derived from (18.183) or (18.184). For example, if t < a− , then S(t, x) → −∞ as x → ∞; on the other hand, Re S(t, x + i y) ≤ S(t, x), and moving the line of integration by increasing α to infinity, we see that G(t) = 0 for t < a− . It is seen from the derivation that √ we consider the limit N → ∞ for a fixed nonzero number xˇ ; therefore e N S(t) / 2π N S ′′ xˇ in (18.198) and (18.201) is exponentially small due to negativeness of entropy. In the formula for the probability density (18.202) the singularity 1/xˇ disappears and (18.202) holds for all a− < t < a+ . As follows from (18.202), S˜m (E) = N S (t)|t= 1 E . 2

ε¯ 2 L

On the other hand, for the parameter volumes we have

,

(E) = log g Sm (E) = log ||

(E) = || g

E 1 2 ε¯ L 2

∗

(E) = || 1 − g

E 1 2 ε¯ L 2

.

(18.203)

Therefore, E 1 2 ε¯ L 2

=

6

for xˇ > 0

N S (t)|t= 1 E ε¯ 2 L 2 N S(t) √ e E 2π N S ′′ xˇ (t) t=

. for xˇ < 0

1 ε¯ 2 L 2

Here we neglected the logarithm of the prefactor in (18.198) as small compared to the leading term N S and used the fact that the first term in (18.201) is much smaller than unity due to negativeness of S (t) . Accordingly, ˆ S˜m (E) = Sm (E) for E < E. Similarly,

Sm∗ (E) = log

∗

(E) = log(1−g ||

E )= 1 2 ε¯ L 2

6−

N S(t) √ e 2π N S ′′ xˇ (t) t=

N S (t)|t= 1 E 2

Hence, as was claimed ˆ S˜m (E) = Sm∗ (E) for E > E.

ε¯ 2 L

E 1 ε¯ 2 L 2

for xˇ > 0 . for xˇ < 0

18.8


957

Accordingly, T˜m (E) ≡ (d S˜m (E)/d E)−1 coincides with Tm (E) for E < Eˆ and ˆ with Tm∗ (E) for E > E. Derivation of (18.166). To derive (18.166) we note that the probability density function f (a|aeff ) of Young modulus a1 is

N f (a|aeff ) = c δ (a − a1 ) δ aeff − 1 da1 . . . da N = + . . . + a1N a1 N = c δ aeff − 1 da2 . . . da N , + a12 + . . . + a1N a where c is the normalizing constant. We will compute the function F (a|aeff ) such that f (a|aeff ) =

F (a|aeff ) . aeff

Obviously, F (a|aeff ) = c

θ aeff −

1 a

+

1 a2

N + ... +

1 aN

da2 . . . da N .

Proceeding as in the previous derivation, we have z dz zaeff − a1 + a1 N+...+ 1 aN 2 F (a|aeff ) = c e da2 . . . da N 2πi z

F aeff , or, changing z → z a1 + . . . + a1N F (a|aeff ) = c =c

dz e 2πi z

1 1 a +...+ a N

dz z ea e 2πi z

(N −1) ln

z−N a 1 z

eff

1 [a]

a+

a−

da2 . . . da N

e z/a da − aN z

Applying the steepest descent method, we obtain β

F (a|aeff ) = const e a , where β is the point of minimum of the function 1 ln [a]

a+

a−

this point is determined by (18.167).

e x/a da −

x ; aeff

eff

da2 . . . da N .

958

18


Derivation of (18.169), (18.170) and (18.171). In the case when we know the value of the arithmetic average of the local moduli, we have for the probability distribution of the effective coefficient, aeff , g (t, ρ) = Probability {aeff ≤ t} = a+ a+ N a1 + . . . + a N da1 . . . da N , = c . . . θ t − −1 δ ρ − N a1 + . . . + a −1 N a−

a−

where

c

a+

a−

a+ a1 + . . . + a N da1 . . . da N = 1, ... δ ρ − N

(18.204)

a−

and f (t, ρ) =

g(t, ρ) . t

Using the presentation for the step function (18.180) and the corresponding presentation for δ-function, 1 δ (E) = 2πi

i∞

e E z dz,

(18.205)

−i∞

we obtain

g (t, ρ) = c

=c

a+

a1 +...+a N Nz dz dζ zt+ζρ− a−1 +...+a −1 −ζ N N 1 da1 . . . da N e 2πi z 2πi

a−

a+ i∞ i∞

a− −i∞ −i∞

a+

a+ i∞ i∞

dz N dζ (a −1 +...+a −1 N )z+N ζρ−N z−ζ (a1 +...+a N ) da . . . da e 1 1 N 2πi z 2πi

...

...

a− −i∞ −i∞

a−

= cN

i∞

i∞

dz dζ N S(t,ρ,z,ζ ) , e 2πi z 2πi

(18.206)

−i∞ −i∞

where S (t, ρ, z, ζ ) is the function (18.171). The integral (18.206) contains a large parameter N , and its asymptotics is computed by the steepest descent method. The constant c can be found from (18.204) by means of the presentation of the δ-function (18.205):

18.8


c

−1

=

a+

a−

=

a+

a+ a1 + . . . + a N da1 . . . da N = ... δ ρ − N a−

a+ i∞

1 zρ −z a1 +...+a N N e e dzda1 . . . da N = 2πi

i∞

⎛

...

a− −i∞

a−

=

959

N 2πi

−i∞

e N zρ ⎝

a+

a−

⎞N

e−za da ⎠ dz.

This integral can be computed exactly, but we need only its asymptotic value as N → ∞. As before, we can use the steepest descent method to obtain c−1

N = 2πi

i∞

N zρ+ln

e

a+

e−za da

a−

dz =

−i∞

1 ∗ = √ e N S1 (x ,ρ) , ′′ 2π N S where x ∗ is the minimizer of the function S1 (x, ρ) = xρ + ln

a+

a−

e−xa da.

(18.207)

Chapter 19

Some Other Applications

19.1 Shallow Water Theory Setting of the problem. Consider a potential flow of an ideal incompressible fluid over a plane. The coordinate normal to the plane is denoted by x, the Cartesian coordinates in the plane by x α , and Greek indices run through the values 1, 2. The depth of the fluid at a point x α at time t is denoted by h(x α , t) (Fig. 19.1). The fluid is under action of the gravity force with the potential gx. The pressure on the free surface of the fluid is assumed to be constant. Since pressure is defined up to an additive constant inside the fluid, it can be set equal to zero on the free surface. The fluid is either unbounded in the plane directions or contained in a vessel with the vertical walls. It seems plausible that motion can be described sufficiently well by functions only of x α and t, if the ratio of the height of the fluid to the characteristic longitudinal wave length is small. The construction of the corresponding approximate equations is the subject of shallow water theory. We derive here the equations of shallow water theory by the variational-asymptotic method. Derivation of shallow water equations. As the starting point we use Luke variational principle for the functional (9.72). According to this variational principle, the potential of the flow, ϕ(x, x α , t), and the height of the fluid, h(x α , t), are the extremals of the functional

Fig. 19.1 Notation in shallow water theory


961

962

19

I =

t1

t0


h(x α ,t)

Ld xd x 1 d x 2 dt,

0

1 2 1 L = ϕ,t + ϕ,x + ϕ,α ϕ ,α + gx. 2 2

(19.1)

Here is the region run by (x 1 , x 2 ). For further estimates, it is convenient to redefine the field variable, ϕ, by making the change ϕ → ϕ − gh 0 t, where the constant h 0 is the characteristic fluid height. The parameter h 0 may be the fluid height at the initial instant, or the fluid height at infinity, depending on the specifics of the problem. After the change, the Lagrangian L becomes 1 1 2 + ϕ,α ϕ ,α + g(x − h 0 ). L = ϕ,t + ϕ,x 2 2

(19.2)

To derive the approximate expression for L, we introduce ϕ, the characteristic magnitude of the variation of the potential, l, the characteristic length of the potential variation in the longitudinal directions, and τ, the characteristic time of the potential variation. The parameter h 0 is the characteristic length of the potential variation in the transverse direction. According to the definition of the characteristic lengths, ϕ,α ∼

ϕ , l

ϕ,x ∼

ϕ , h0

ϕ,τ ∼

ϕ . τ

The possibility to replace the three-dimensional problem by a two-dimensional one is based on the inequality h 0 ≪ l. In addition to this inequality, we assume that the characteristic √ velocity, l/τ, is on the order of the speed of propagation of small disturbances, ρh 0 , while the inertial term ϕ,t is on the order of potential energy of the field weight, g (h − h 0 ). Then, introducing the notation α = max

we have

(h − h 0 ) , h0

19.1

Shallow Water Theory

963

ϕ,t ∼ g (h − h 0 ) ∼ αgh 0 , 2 ϕ,x ∼

ϕ ∼ ϕ,t τ ∼ g (h − h 0 ) l

l2 (ϕ)2 ∼ 2 α 2 gh 0 , 2 h0 h0

ϕ,α ϕ ,α ∼

F

gh 0 ∼ α gh 0l,

(ϕ)2 ∼ α 2 gh 0 . l2

(19.3)

Holding the function h(x α , t), let us find the potential ϕ. Since l/ h 0 ≫ 1, the leading term of the functional is t1

t0

h(x α ,t)

1 2 ϕ d xd x 1 d x 2 dt. 2 ,x

(19.4)

0

The minimizer is ϕ = (x α , t),

(19.5)

i.e., in the first approximation the flow potential does not depend on the transverse coordinate. The function (x α , t) remains arbitrary at this point. The functions ϕ of the form (19.5) make up the set M0 of the general scheme of the variationalasymptotic method. As the next step, we take the potential ϕ in the form ϕ = (x α , t) + ϕ ′ (x, x α , t),

(19.6)

where the characteristic magnitude of ϕ ′ is much smaller than that of ϕ, ϕ ′ ≪ ϕ, while the characteristic lengths of and ϕ ′ in longitudinal directions are the same. Since the function is not yet defined, we can make the substitution

′

′

′

ϕ → ϕ − ϕ ,

′

→ + ϕ ,

1

ϕ ≡ h ′

h

ϕ ′ d x.

0

Then, has the meaning of the average of the three-dimensional potential, ϕ, in the transverse direction, = ϕ, while the function ϕ ′ satisfies the constraint h 0

ϕ ′ d x = 0.

(19.7)

964

19


The constraint (19.7) allows one to determine uniquely and ϕ ′ by the known potential ϕ, and there is a one-to-one correspondence between all functions ϕ and all pairs {, ϕ ′ }. Substituting (19.6) into (19.2), we get 1 ′2 1 1 ′ ′ ,α ′ L = ,t + ϕ,t′ + ϕ,x + ,α ,α + ,α ϕ,α + ϕ,α ϕ + g(x − h 0 ). 2 2 2

(19.8)

′2 . However, unlike the Lagrangian (19.2), the The leading term, containing ϕ ′ , is 21 ϕ,x Lagrangian (19.8) contains the interaction terms between ϕ ′ , and h. The interaction part of the functional,

t1

1

2

Fd x d x dt,

F=

t0

h 0

′ ′ ϕ,t + ,α ϕ,α d x,

can be transformed into a more convenient form by moving the derivatives with respect to t and x α outside the integral. Using the constraint (19.7), we get F= t

h(x α ,t)

ϕ ′ d x + ,α α x

0

h(x α ,t) 0

˙ ϕ ′ d x − ϕ ′ x=h h˙ = −ϕ ′ x=h h.

Here, the notation h˙ = h ,t + ,σ h ,σ is used. Selecting in the functional the leading term with respect to ϕ ′ and the leading interaction term, we obtain the functional t1

1

2

J d x d x dt,

t0

J=

h 0

1 ′2 ˙ ′ ϕ,x d x − hϕ . x=h 2

(19.9)

The meaning of the constraint (19.7) can be seen from the equation (19.9): with this constraint the quadratic part of the functional J does not have a kernel. The functional (19.9) is strictly convex on the set of functions, ϕ ′ , satisfying the condition (19.7). Therefore, it has a unique minimizer. Hence, the next term of the expansion is uniquely defined by the preceding one, and the set N of the general scheme of the variational asymptotic method coincides with the set M0 . The function ϕ ′ is found from the Euler equations of the functional J : ′ ϕ,x x = λ,

′ ϕ,x |x=0 = 0,

′ ˙ ϕ,x |x=h = h.

(19.10)

19.1

Shallow Water Theory

965

Here λ is the Lagrange multiplier for the constraint (19.7). The solution of (19.10) and (19.7) is given by the formula 1 h˙ h2 2 ϕ = x − . 2h 3 ′

(19.11)

According to the Clapeyron formula (5.45), at the minimizing element the functional J has the value 1 1˙ ′ |x=h = − h h˙ 2 . J = − hϕ 2 6

(19.12)

Now, consider the action functional on the flow potential (19.5). We get t1 g(h − h 0 ) 1 ,σ I = d x 1 d x 2 dt. + h ,t + ,σ 2 2

(19.13)

t0

h2

Here, the constant term g 20 is dropped. The functional (19.13) corresponds to the classical shallow water theory. Taking the variation of (19.13) with respect to , we get the continuity equation h t + (h,σ ),σ = 0, and taking the variation with respect to h, we get the Cauchy-Lagrange integral 1 ,t + ,α ,α + g(h − h 0 ) = 0. 2 To justify the classical shallow water theory we have to show that taking ϕ ′ into account results in corrections of a higher order. Indeed, substituting (19.11) into (19.8), integrating with respect to x and taking into account (19.12), we get t1

Ld x 1 d 2 dt,

t0

4 1 2 ˙ ˙ ,α g(h − h 0 )2 1 1 ˙2 ,α ,α L = h ,t + ,α − h 1 − h ,α h + h h ,α h + . 2 6 15 90 2 (19.14) Let us show that the additional terms are small. Our basic assumption was that the parameter β ≡ (h 0 /l)2 is small. We also assume that h 0 can be chosen in such a 0) small as well. way as to make the parameter α = max (h−h h0 √ From the estimates (19.3) and the condition l/τ ∼ gh 0 , we have

966

19

ht ∼

h − h0 ∼ α β gh 0 , τ

h ,σ ∼

h − h0 ∼ α β, l


,σ ∼

ϕ ∼ α gh 0 . l

Therefore, the term ,σ h ,σ can be ignored compared to h t in the expression for h˙ : h˙ ≈ h ,t . The additional terms are at least on the order of βα 2 gh 20 and are small compared to the gravitation energy, 12 g(h − h 0 )2 ∼ α 2 gh 20 . Therefore, the classical shallow water theory does indeed correspond to the first approximation of the exact theory. Of all the additional term, the leading one is hh 2,t ≈ h 0 h 2,t . Retaining the term, we get the second approximation (the Bussinesque approximation): t1 g(h − h 0 )2 1 1 h ,t + ,α ,α + − h 0 h 2,t d x 1 d x 2 dt. I = 2 2 6

(19.15)

t0

Other short wave extrapolations. Here, we present two other versions of the short wave extrapolations1 in the Bussinesque approximation: g(h − h 0 )2 1 2 1 ,α , L = h ,t + ,α − h ,t + 2 6 2 g(h − h 0 )2 1 1 . L = h ,t + ,α ,α − h˙ 2 + 2 6 2

(19.16)

The equivalence of (19.16) and (19.15) in the long wave approximation follows from the fact that (h − h 0 ) h 2,t is small compared to h 0 h 2,t .

19.2 Models of Heterogeneous Mixtures The mechanics of heterogeneous mixtures is concerned with the following problem. There are two continuous media in region V . The region occupied by one medium is connected, while the region occupied by the other medium consists of a large number of components, “particles.” The first medium is called continuous or matrix phase, and the second discrete phase. In its turn, the discrete phase may consist of particles with different physical properties. Let the models describing the continuous and the discrete phase as well as the boundary and initial conditions be given. Then the behavior of the system is completely determined. It is clear that for a sufficiently large number of particles one can only try to describe some average characteristics of motion. Construction and investigation of the equations for average characteristics is the subject of mechanics of heterogeneous mixtures. 1

For the notion of short wave extrapolation see Sect. 14.5.

19.2

Models of Heterogeneous Mixtures

967

Mechanics of heterogeneous mixtures is similar in many respects to statistical mechanics; however, it is more complex due to the greater complexity of the interactions between the particles. For example, in the mechanics of heterogeneous mixtures, the particle interactions are pair interactions only in very special cases. As a rule, interaction force between two particles depends on the motion of all other particles. The general idealization of statistical mechanics, which presents particles as material points, is also usually not applicable in mechanics of mixtures. The phenomenological approach in mechanics of heterogeneous mixtures is based on modeling of matrix and discrete phases by two continua. If the two continua did not interact, then constructing the equations describing their motion would be easy. The crux of the problem is in the description of the interaction. The interaction of continua is caused by three types of physical effects: the inertial effects, described by the kinetic energy, “elastic” and reversible heat effects, described by the internal energy, and the irreversible effects, which are described by the dissipation. Usually, the irreversible effect of the viscous friction type are the most important, and their description, as the description of other “coarse” effects, does not cause difficulties. It is much more difficult to characterize the inertial interaction. The variational method is, in fact, the only phenomenological method adequately describing the inertial interactions. Description of the inertial effects based on the variational approach is the subject of this section. Field variables. Consider an ideal compressible barotropic fluid which contains a large number of solid particles or bubbles of barotropic gas. The average radius of the particles in the case of liquid with bubbles may vary with time. The number of particles per unit volume is sufficiently small to ignore their collisions. Both the continuous and the discrete phases will be modeled by some continua. The Lagrangian coordinates of the continuous phase are denoted by ξ a , and that of the discrete phase by η p . Small Latin indices a, b, c and p, q, r run values 1, 2, 3. Since the coordinates ξ and η are transformed by different groups, different groups of indices, (a, b, c) and ( p, q, r ), are chosen for projections onto the Lagrangian axes of the first and second continua. The particle trajectories of the first continuum, x i = x1i (ξ, t), serve as the field variables of the continuous phase, and the particle trajectories of the second continuum, x i = x2i (η, t), and the average radius of the inclusions, a(η, t), serve as the field variables of the discrete phase. The indices 1 and 2 mark the quantities corresponding to the continuous and discrete phases, respectively. Denote the time derivatives when the Lagrangian coordinates ξ and η held fixed by d1 /dt and d2 /dt, correspondingly. By definition, the coordinates ξ and η are introduced in such a way that the velocities of the continuous and the discrete phases, v1i = d x1i (ξ, t)/dt and v2i = d x2i (η, t)/dt, are the velocities averaged over the volume. The average radius a is calculated over the average volume of the inclusions a3. |A| according to the formula |A| = 4π 3 The theory will also use the true density of the continuous phase, ρ1 (ξ, t), the true density of the discrete phase, ρ2 (η, t), the number of particles per unit volume, n(η, t), the volume concentration of the discrete phase,

968

19

c=


4 3 πa n, 3

(19.17)

and the volume concentration of the continuous phase, c1 = 1 − c. For a more symmetric form of the formulae, the volume concentration of the discrete phase, c, is also denoted by c2 (c1 + c2 = 1). The number of particles is assumed to be constant: % i% % x2 % % n det % % ηa % = f (η).

(19.18)

The phase transitions fluid↔particles is not considered; therefore, the particle mass 4 ρ2 πa 3 = M2 (η), 3

(19.19)

and the fluid mass % i% % x1 % % ρ1 c1 det % % ξ a % = M1 (ξ ),

(19.20)

are preserved. The right-hand sides of (19.18), (19.19) and (19.20) are given functions of the Lagrangian coordinates. Kinetic energy and internal energy. The kinetic energy per unit volume K and the internal energy per unit volume U are given by the equations 1 1 1 ′ ′ ′i 1 ′ ′ ′i 1 ′ ′ ′i ρ vi v d V ≡ ρ1 vi v d V + ρ v v d V, 2 V 2 V 2 2 i V V1 V2 1 1 1 1 ′ ′ 1 ′ ′ 1 ′ ′ ρ U dV = ρ U dV + ρ U d V. (19.21) U = V 2 V 2 1 1 V 2 2 2 K =

1 V

V

V1

V2

Here, V is the representative volume containing a large number of particles and is much smaller than the macrovolume, V ; V1 and V2 are the regions of V occupied by the continuous and respectively, the prime denoting the true phases, discrete the microfields, and U1′ ρ1′ and U2′ ρ2′ are the energy densities per unit mass of the continuous and the discrete phases,2 respectively. In accordance with (19.21), we can write K = ρ1 c1 K 1 + ρ2 c2 K 2 , 2

U = ρ1 c1 U1 + ρ2 c2 U2 .

(19.22)

Remember that by the energy density of the baratropic gas with the state equation, p = p (ρ), we mean the function U (ρ) defined by the equation p (ρ) = ρ 2 dU/dρ.

19.2


969

Here, K 1 , K 2 , U1 , U2 are the kinetic and the internal energies per unit mass of the continuous and discrete phases. The kinetic energy of the fluid phase, K 1 , is comprised of the kinetic energy of the translational motion of the fluid, 21 v12 , and the kinetic energy K¯ of the relative motion of fluid and particles: K1 =

1 2 v + K¯ . 2 1

(19.23)

It is natural to assume that K¯ is quadratic with respect to the relative particle velocity v2i − v1i and the radius rate: 2 K¯ = β (v1 − v2 )2 + γ a˙ 2 .

(19.24)

The coefficients β and γ are supposed to be known functions of c. The calculation of K¯ within the framework of ideal fluid in approximation of small particle concentration, results in the following values of the coefficients β and γ : β = 21 c, γ = 3c (see Sect. 13.3). Taking into account viscosity brings the additional terms describing memory. The form of these terms can be seen in the relation (13.56). For simplicity, in what follows, the memory is not taken into account, and for the kinetic energy equation (19.24) is used. The internal energy of the continuous phase, U1 , is assumed to be a known function of ρ1 : U1 = U1 (ρ1 ).

(19.25)

Usually, U1 can be taken coinciding with the true internal energy of the continuous phase: U1 (ρ1 ) = U1′ (ρ1 ). The kinetic energy density per unit particle mass, K 2 , is defined as K2 =

1 2 v . 2 2

(19.26)

In the absence of collisions, the internal energy of the discrete phase, U2 , is reduced to the internal energy of the particles. This means that in the case of the non-deformable solid particles, ρ2 = const, while for gas bubbles, ρ2 c2 U2 = ρ2 c2 U2′ (ρ2 ) + 4πa 2 σ n,

(19.27)

where σ is the surface tension coefficient. From (19.22), (19.23), (19.24) (19.25), (19.26) and (19.27), it follows that the Lagrangian of the system is L = ρ1 c1 L 1 + ρ2 c2 L 2 ,

˙ a, n , L 1 = L 1 v1i , v2i , a,

L 2 = L 2 v2i , a, n . (19.28)

970

19


The governing equations will be derived for the general case of the Lagrangian (19.28), without taking into account the special dependences (19.22), (19.23), (19.24), (19.25), (19.26) and (19.27). Variational principle. We assume that the motion occurs in some vessel V . The vessel does not move. The walls of the vessel are impenetrable both to the fluid and the particles. Then the fluid-particle system in the vessel V is conservative, and the true motion is a stationary point of the action functional

I =

t 2

(19.29)

Ld V dt,

t1 V

which, according to (19.28), may be written as the sum of the action functionals of the continuous and the discrete phases,

I = I1 + I2 ,

I1 =

t2

ρ1 c1 L 1 d V dt,

I2 =

t 2

ρ2 c2 L 2 d V dt.

(19.30)

t1 V

t1 V

Interaction of the continua occurs due to the dependence of L 1 on the field variables of the second continuum, v2i , a, a˙ and n. Let us derive the corresponding Euler equations. The functions x1i (ξ, t) , x2i (η, t) and a (η, t) are varied independently. The variations with ξ and η held constant are denoted by δξ and δη , respectively, while the variation with the Eulerian coordinates held constant is denoted by , as before. The variation of the particle trajectories, x1i (ξ, t) , is used only when ξ are held constant, and the variation of x2i (η, t) is used only with η held constant; therefore, the indices ξ and η are omitted, and δx1i ≡ δξ x1i , δx2i ≡ δξ x2i . Analogously, in varying v2i , a, c, n with η and v1i , c1 , . . . with ξ held constant, the indices ξ and η are omitted. They are retained only in the cases of variation with constant “alien” Lagrangian coordinates, for example δξ n or δη v1i . Let f (x, t) be a function of Eulerian coordinates. It can also be considered as a function of the Lagrangian coordinates of the continuous phase, f (x, t) = f (x1 (ξ, t), t), and of the discrete phase, f (x, t) = f (x2 (η, t), t). The variations δ ξ f , δ η f and f are related as δ ξ f = f ′ (ξ, t) − f (ξ, t) = f ′ x1′ (ξ, t) , t − f (x1 (ξ, t) , t) = f + δx1i i f, η f = f + δx2i i f, δ ξ f = δ η f + δx1i − δx2i i f. (19.31) In particular, for a, a˙ and v2i , we have

d2 δa i ˙ + δx1 − δx2i i a, dt d2 δx2i i δ ξ v i2 = δv2i + δx1i − δx2i k v2i = + δx1 − δx2i k v2i . dt

δ ξ a = δa+ δx1i − δx2i i a,

δ ξ a˙ =

19.2


971

The formulae, analogous to (19.31), hold for the time derivatives, in particular, d2 f d1 f = + v1i − v2i i f. dt dt

For the variations of the arguments of the Lagrangian we have δc = 4πa 2 nδa − ci δx2i , δc1 = −δξ c = −δc − δx1i − δx2i i c, 3ρ2 ρ1 δρ2 = − δa, ρ1 = −ρ1 i δx1i − δc1 . a c1 δn = −ni δx2i ,

Due to (19.17), (19.18), (19.19) and (19.20)

δ I1 = δ

t2 t1 V

t2

δ I2 = δ

t2

t2

ρ1 c1 L 1 d V dt =

ρ1 c1 δξ L 1 d V dt,

ρ2 c2 L 2 d V dt =

ρ2 c2 δη L 2 d V dt.

t1 V

(19.32)

t1 V

t1 V

The variations of the Lagrangians of the phases are L 1 d2 δx2k L 1 L 1 d1 δx1k L 1 −1 L 1 i δx2i + + − n + ρ δa+ cc 1 1 n ρ1 a v1k dt v2k dt L 1 d2 δa L 1 L 1 + − ρ1 i δx1i + 4πa 2 nρ1 c1−1 δa + ρ1−1 c1−1 Fi δx1i − δx2i , a˙ dt ρ1 ρ1 k L 2 L 2 L 2 d2 δx 2 −n i δx2i + δa. (19.33) δη L 2 = n a v2k dt

δξ L 1 =

Here, the notation is introduced: ρ1−1 c1−1 Fi =

L 1 L 1 L 1 L 1 L 1 i n + i a + i a˙ + ρ1 c1−1 i c. i v2k + n a a˙ ρ1 v2k

If the Lagrangian of the continuous phase has the form ˙ − U1 (ρ1 ), L 1 = K 1 (v1 , v2 , n, a, a) then the expression for Fi is simplified to Fi = ρ1 c1

K 1 K 1 k − k i v1 − pi c, x i v1

972

19


where p = ρ1

U1 ρ1

is the “true” pressure in the continuous phase. System of governing equations. After substitution of (19.33) into (19.32) and integration by parts, we obtain from stationarity of the action functional the following system of governing equations: The momentum equations for the continuous phase t J1i + k J1i v1k = k p1ik + F i , L J1i = i , p1ik = − p1 ik , p1 = c1 p v1

(19.34) (19.35)

The momentum equations for the discrete phase

J2i =

L , v2i

t J2i + k J2i v2k = k p2ik − F i , L 2 L 1 − ρ2 c2 n p2ik = − p2 δ ik , p2 = c2 p − ρ1 c1 n n n

(19.36) (19.37)

The generalized Rayleigh equation L 1 L 2 t J(a) + k J(a) v2k = −4πa 2 np + ρ1 c1 + ρ2 c2 , a a

J(a) =

L a˙

(19.38)

If the continuous phase is incompressible,

ρ1 = ρ1 (ξ ),

(19.39)

the equation p = ρ1 U1 /ρ1 should be replaced by (19.39) while the function p becomes an additional independent required function. It has the meaning of the Lagrange multiplier for the constraint (19.39). The consideration of the non-zero values of variations on the boundary of the ij ij region V ×[t0 , t1 ] shows that the tensors p1 and p2 “work” on infinitesimally small i i displacements of the two continua δx1 and δx2 and, therefore, have the meaning of the corresponding stress tensors. The vectors J1i and J2i “work” on δx1i ad δx2i in the initial and final instants and, consequently, are the momenta of the phases. Emphasize that for a mixture, the momentum does not coincide with the product of the density and the velocity. The body interaction force of two continua,3 Fi , in general, does not have the form of divergence of some tensors. 3

Sometimes the body interaction force is defined differently by the condition that the left-hand side of the momentum equation is the product of the mass density and the acceleration.

19.2


973

Let us write the relations obtained within the framework of the models with the energy (19.22), (19.23), (19.24), (19.25), (19.26) and (19.27). Continuous phase. From (19.34) and (19.35), we get: Momentum of the continuous phase (19.40) J1i = ρ1 c1 v1i + β v1i − v2i Pressure in the continuous phase

(19.41)

p1 = c1 p The body interaction force of the continuous and the discrete phases F i = ρ1 c1 i K¯ − β v1k − v2k i v1k + pi c1

(19.42)

t (ρ1 c1 ) + k ρ1 c1 v1k = 0

(19.43)

t J1i + k J1i v1k = −i p1 + F i

(19.44)

The continuity equation for the continuous phase (a consequence of (19.20))

Momentum equation for the continuous phase

Based on (19.40) (19.41), (19.42) and (19.43), the momentum equation can be written as ρ1 c1

d1 i v1 + β v1i − v2i = −c1 i p + ρ1 c1 i K¯ − ρ1 c1 β v1k − v2k i v1k (19.45) dt

Discrete phase. We have: Momentum of the discrete phase

J2i = ρ2 c2 v2i + ρ1 c11 β v1i − v2i

(19.46)

Pressure in the discrete phase

p2 = c2 p − ρ1 c1

γ a˙ 2 β (v1 − v2 )2 +c c c 2 c 2

(19.47)

The conservation of the number of particles (a consequence of (19.18)) t n + k nv2k = 0

The conservation of a particle mass (a consequence of (19.19))

(19.48)

974

19


4 4 t ρ2 πa 3 + v2k k ρ2 πa 3 = 0 3 3

(19.49)

The continuity equation for the discrete phase (a consequence of (19.48) and (19.49)) t (ρ2 c2 ) + k ρ2 c2 v2k = 0

(19.50)

t J2i + k J2i v2k = −i p2 − F i

(19.51)

p2 = c2 p − ρ1 c1 K¯ .

(19.52)

The momentum equation for the discrete phase

Consider the simplifications occurring in the momentum equations in the limit of small concentrations, when only the terms on the order of c are preserved in the equation. In this case, β and γ are linear function of c, and for pressure we have

Equation (19.51) becomes d2 v2i d2 v2i d2 v1i i + = c2 p + ρ1 c1 β − ρ2 c2 dt dt dt d2 ρ1 c1 β + v1i − v2i n + ρ1 c1 β v1k − v2k i v1k − k v1i + K¯ i ρ1 c1 . dt n

(19.53)

One can set c1 = 1 and β = c/2 in (19.53). If, moreover, the continuous phase is incompressible, the last term in (19.53) can be dropped, while for the derivative of ρ1 β/n one can use the equality n

3a˙ d2 ρ1 β = ρ1 β , dt n a

which holds for ρ1 = const and β = c/2. The result is d2 v2i d2 v2i c d2 v1i i ρ2 c2 + = −c2 p + ρ1 − dt 2 dt dt ρ1 c 3a˙ c + v1i − v2i + ρ1 v1k − v2k i v1k − k v1i . 2 a 2

(19.54)

Taking into account the compressibility of the matrix phase causes an addition of the two terms i c dρ1 + K¯ i ρ1 v1 − v2i 2 dt

to the right-hand side of (19.54).

(19.55)

19.2


975

The generalized Rayleigh equation becomes d2 n dt

3 γ4 3 3 K¯ 2σ ρ1 c1 + ρ1 c1 c πa a˙ = c pg − p − . c3 2 a a c

(19.56)

Here, pg = ρ2 dU2 /dρ2 is the gas pressure in a bubble. If β and γ are linear with respect to c, the continuous phase is incompressible and the concentrations are small, it follows from (19.56) that γ a¨ =

3c ρ1 a

pg − p −

2σ a

+

3 ¯ 2 K − γ a˙ . a

For β = c/2, γ = 3c, we obtain the well-known equation 1 a¨ = ρ1 a

2σ pg − p − a

1 + a

1 3a˙ 2 (v1 − v2 )2 − 4 2

.

(19.57)

For a compressible fluid, the term a˙

1 d ρ1 ρ1 dt

must be added to the left-hand side of (19.57). Total momentum equation. After summing up (19.44) and (19.51) and making some simple transformations using (19.52), we obtain the total momentum equation d2 v2i d1 v1i + ρ2 c2 + k ρ1 c1 β v1i − v2i v1k − v2k = −i p − ρ1 c1 K¯ . dt dt (19.58) Note that the third term on the left-hand side of (19.58) is related to the summation of the momenta and there are no reasons to include it in the total stress tensor. The total pressure is not reduced to the true pressure in the fluid phase, p, but also contains the kinetic energy of relative motion, K¯ . The total momentum equation can also be written using the phase continuity equations (19.43) and (19.50) in the divergence form, ρ1 c1

t ρ1 c1 v1i +ρ2 c2 v2i + +k ρ1 c1 v1i v1k + ρ 2 c2 v2i v2k + ρ 1 c1 β v1i − v2i v1k − v2k + p − ρ1 c1 K¯ δik = 0,

which clearly shows that the law of conservation of total momentum holds true, as it should.

976

19


19.3 A Granular Material Model In this section we demonstrate an application of the variational approach to constructing the models of granular media. Here, within the framework of the assumptions made, the variational method is, in fact, the only method of constructing the governing equations. Consider a mechanical system comprising a very large number of chaotically packed, identical and absolutely rigid spherical particles. The particles tightly pack the vessel V in such a way as to make the reduction of the total particle volume impossible by infinitesimally small displacement of the particle centers.4 Particles interact only by means of normal contact forces; the friction forces are not taken into account. The system is put in a field of external body forces; the external forces may also be applied to a part of the boundary V f of the region V . The rest of the boundary, Vu , is impenetrable to the particles. Only small deviations from the equilibrium state will be considered. It seems sensible to model the particles by some continuum occupying volume V . The displacements of continuum, u i , can be assumed infinitesimally small, so the following theory is geometrically linear. The considered medium has some special properties. First, the volume cannot be decreased by any deformation of the initial state. Second, if an infinitesimally small shear, described by the deviator of the strain tensor γi j = εi j − 31 εkk δi j , occurs in the medium, then, due to strictly geometrical reasons (particles becomes less tightly packed), the displacement causes loosening of the medium and increase of its volume; therefore the trace of the strain tensor characterizing the relative volume change is some function of γi j : εii = θ γi j .

(19.59)

The function θ γi j can be found from the statistical theory of random packing. For small γi j , we can assume that 1 k 2 ij ij , (19.60) ε θ = μγi j γ = μ εi j ε − 3 k where μ is some dimensionless coefficient depending on the volume concentration of the tight packing. Besides the volume change caused by the shear, also admissible is an independent increase in the volume due to the “dilatation” of the particles. Therefore, the admissible displacement fields should be subject to the kinematic constraint εii

4

≥ θ γi j ,

1 εi j ≡ 2

u j u i + i x j x

.

(19.61)

On the macrolevel this should be understood in the following way: for a tightly packed granular medium, the measure of the set of microdisplacements leading to a reduction of the volume is negligibly small.

19.3

A Granular Material Model

977

The internal energy density of a system of the absolutely rigid particles is obviously equal to zero, and the energy functional of the system is simply the work of external forces: ⎛

⎜ I (u) = − ⎝

ρgi u i d V +

V f

V

⎞

⎟ f i u i d A⎠ .

(19.62)

The equilibrium positions are the points of minimum of the functional (19.62) under the condition that the admissible displacement fields satisfy the constraint (19.61), and also the impenetrability condition of the part of the boundary Vu , u i n i ≤ 0 on Vu .

(19.63)

Here, n i is the outward normal unit vector on Vu . The variational problem (19.61), (19.62) and (19.63) is not correctly posed for all external forces; the boundedness from below of the functional (19.62) on the displacement fields satisfying the conditions (19.61) and (19.63) is a necessary condition for the existence of static solution. Supposing that the functional (19.62) is bounded from below, let us find the equations which are to be satisfied by the minimizing element. Introducing the Lagrange multipliers p and P for the constraints (19.61) and (19.63), respectively, we rewrite the variational problem as the minimax problem min

I (u) = 19.61 , 19.63 ⎤ ⎡ ⎥ ⎢ Pu i n i d A⎦ . fi u i d A + = min max ⎣ ρ −gi u i + p θ γi j − εii d V −

u∈

u

p≥0,P≥0

V f

V

Vu

Changing the order of calculation of minimum and maximum, we obtain the equations ij

σ, j + g i = 0 in V,

σ i j n j = f i on V f ,

σ i j = − pi j + τ i j ,

σ i j n j = −Pn i ,

τij = p

θ , γi j

p ≥ 0.

P ≥ 0 on Vu , (19.64) (19.65)

Equations (19.64) are the usual equilibrium equations and the boundary conditions of a continuum. However, the constitutive equations (19.65) are quite peculiar: the Lagrange multiplier, the pressure p, enters into the constitutive equation for the stress deviator τ i j . If, for example, θ = μγi j γ i j , then the stress deviator, as in

978

19


a linearly elastic body, is proportional to the strain deviator γ i j : τ i j = 2μpγ i j ; however, the corresponding “shear modulus” turns out to be linearly dependent on pressure. This corresponds to the physics of the phenomenon: as the outside pressure increases, the shear modulus of the dry substance has to increase, while for zero pressure, the granular media “cannot withstand” the shear.

19.4 A Turbulence Model The key problem of turbulence theory is the construction of the governing equations for the averaged characteristics of turbulent motion. Most probably there is no a universal system of equations which works for any flow, and every flow geometry requires a special treatment. Nevertheless, it is worth seeing how far one can advance in constructing the universal relations. In this section we develop the averaged equations obtained by “homogenization” of the action functional of ideal fluid. One can expect that this approach is meaningful for the flows in which molecular viscosity is negligible, i.e. for the flows away from the walls like mixing layers, jet flows, wakes, etc. Turbulence kinematics. We begin with a formalization of the notion of turbulent motion. The motion of a continuum is described by functions x (t, X ). Without loss of generality, these functions may be considered as slow changing functions of X at the initial instant t0 ; for example, one can set x (t0 , X ) = X . For a laminar motion, functions x (t, X ) remain slow changing functions of X . In a turbulent motion, x (t, X ) are fast changing functions of X : the characteristic feature of turbulence, mixing, means that two close material points move far away from each other. For a developed turbulent motion in a bounded region V after mixing each small Lagrangian volume is presented in each subregion of the region V . Therefore, it is sensible to use as the key kinematic characteristics of turbulence the characteristics of mixing. Let us partition the region V in a large number of small cubes with the size . In the process of motion, the cube with the center at the point X 0 spreads over other cubes. One can introduce the function f (t, x|X 0 ), the portion of the volume of the cube with the center at the point x, which is occupied, at the instant t by the points of the cube with the center at the point X 0 . So, f (t, x|X 0 ) 3 is the volume of the region in the cube with the center at the point x which is occupied by the fluid particles who came from the cube with the center at X 0 . If we make a sum of these volumes over all cubes, we should get the volume of the cube at the point X 0 :

f (t, x|X 0 ) d 3 x = 3 .

V

It is convenient to redefine f dividing it by 3 . In what follows only this ratio is used; we keep for it the same notation f . So

19.4

A Turbulence Model

979

f (t, x|X 0 ) d 3 x = 1.

V

Functions x (t, X ) and f (t, x|X ) have the opposite properties of smoothness for laminar and turbulent motions. For a laminar motion, functions x (t, X ) are smooth while f (t, x|X ) is singular: one can set f (t, x|X ) = δ (x − x (t, X )) . For turbulent motion, functions x (t, X ) change fast while the function f (t, x|X ) changes smoothly. The function f (t, x|X ) can be considered as the key kinematic characteristic of turbulent motion replacing the fast oscillating functions x (t, X ). At a space point, velocity of the flow v i fluctuates, and one can introduce the average velocity v¯ i and the velocity fluctuations v ′i by the equations v i = v¯ i + v ′i ,

v ′i = 0.

Averaging is denoted by bar; it can mean time average or ensemble average. One can also introduce as characteristics of turbulent motion the averaged Lagrangian coordinates ξ a : a

ξ (t, x) =

f (t, x|X ) X a d 3 x.

V

In contrast to the true Lagrangian coordinates X a , which are conserved along the particle trajectories, X a X a + v i i = 0, t x

(19.66)

the averaged Lagrangian coordinates are not conserved by the averaged flow, ξ a ξ a + v¯ i i = 0. t x The simplest equation for averaged Lagrangian coordinates modeling the turbulent diffusion is a a ξ a i ξ i j ξ (19.67) + v¯ = i D t x i x x j where D i j are diffusion coefficients. Tensor D i j is positive. It could, in principle, be non-symmetric. We assume for simplicity that it is symmetric; the changes caused by the non-symmetric diffusion are straightforward.

980

19


Equations (19.66) allow one to express velocity in terms of functions X a (t, x) (see (3.58)). Similarly, from (19.67) one can find average velocity v¯ i in terms of averaged Lagrangian coordinates ξ a (t, x) and diffusion coefficients D i j (t, x) : 2 ξ a . (19.68) x k x j % % Here x¯ ai are the components of the inverse matrix to %ξ a /x i % . Remarkably, ik j are similar to Christoffel’s symbols (compare with (4.83)). Averaged Lagrangian coordinates may be considered as coefficients in an expansion of f (t, x|X ) over X a : v¯ i = −x¯ ai

ξ a (t, x) + j D i j + D k j ik j , x i

ik j = x¯ ai

f (t, x|X ) = f 0 (t, x) + ξ a (t, x) Aab X b + . . . . Here Aab is inverse tensor to moments of inertia of the region V , Aab

X b X c d V = δac ,

V

and the origin of the Lagrangian coordinate system is assumed to be in the center of gravity of V . The diffusion equation for f which corresponds to (19.67) is f (t, x|X ) f (t, x|X ) + v¯ i = i t x i x

f (t, x|X ) D x j ij

(19.69)

which must be complimented by the initial condition f (t0 , x|X ) = δ (x − X ) .

(19.70)

So we have two kinematic descriptions of turbulent motion: a shortened one, by functions ξ a (t, x) and D i j (t, x) , and a detailed one, by functions f (t, x|X ) and D i j (t, x) . We begin with using the shortened description, and then extend it to the detailed one. Variational principle. Mixing in turbulent motion occurs mostly due to inertia. Therefore, to describe mixing it is natural to use as the starting point the Hamilton variational principle which controls inertial features of matter. We consider motion of ideal incompressible fluid in a bounded domain V , and set mass density equal to unity. According to Lin variational principle (Sect. 9.3) the true motion is an extremal of the action functional,

I =

t1 t0 V

K d V dt,

K =

1 i vi v , 2

19.4

A Turbulence Model

981

on the set of all functions X a (t, x) and v i (t, x), satisfying incompressibility condition, v i = 0, x i

(19.71)

impermeability of the boundary, V , of the region, V , and conservation of Lagrangian coordinates (19.66) which take the prescribed values at the initial and the final instants: X (t0 , x) = x,

X (t1 , x) = X 1 (x)

(19.72)

Different choices of X 1 (x) correspond to different final positions of the fluid particles and, accordingly, to different initial velocities of the fluid. If, in particular, X 1 (x) corresponds to mixing of fluid particles, then some turbulent motion develops in the region V . Our goal is to average the action functional, i.e. to present it as a functional of the smooth averaged characteristics of the fluid motion, and then to obtain the governing equations of turbulence as Euler equations of the averaged variational problem. We will make a number of phenomenological assumptions. As we will see, some relations that follow remain the same regardless of the assumptions made. We begin with the shortened kinematical description. The idea is that the constraint for Lagrangian coordinates (19.66) must be replaced by the averaged constraint (19.67) while the action functional is taken as a functional of the averaged velocity and the diffusion coefficients. In what follows, the true velocity does not appear; therefore we drop the bar and denote the average velocity by v i . The kinetic energy is the sum of the kinetic energy of averaged motion and the energy of fluctuations: K =

1 i vi v + k, 2

k=

1 ′ ′i vv 2 i

(19.73)

We assume that k is a function of the diffusion coefficients and gradient of averaged velocity. The simplest scalar which can be formed from these parameters is k = −β D i j vi, j

(19.74)

where β is a dimensionless positive number. As we will argue further, for positive β, the negative sign in (19.74) is consistent with positiveness of k. Formula (19.74) makes the variational problem closed. Governing equations. Introducing the Lagrange multipliers λa and ϕ for the constraints (19.67) and the incompressibility condition, respectively, we obtain the Lagrangian

982

19

L=

1 i vi v − β D i j vi, j − λa 2 v i +ϕ i . x

ξ a ξ a + v¯ i i − i t x x


Di j

ξ a x j

(19.75)

Varying ξ a , v i and D i j we obtain the following system of equations: λa λa i j λa + vi i + D = 0, t x x j x i j D ϕ ξ a vi = λa i + i − β ij , x x x βv(i, j) + λa,(i ξ,aj) = 0.

(19.76) (19.77) (19.78)

Consider a consequence of this system, momentum equations. Momentum equations. Let us show that momentum equations for turbulent motion have the form

τij

p τ i j dv i =− − , dt xi ρ x j

d Di j j i , = −β D k j v,k + D ki v,k + β dt p 1 dϕ = v 2 − β D i j vi, j − . ρ 2 dt

(19.79) (19.80) (19.81)

Equation (19.80) is a generalization of Boussinesque’s law linking Reynolds stresses τ i j and gradient of averaged velocity, τi j = −νT vi, j + v j,i ,

νT being the kinematic turbulent viscosity. It shows that the turbulent viscosity is anisotropic and can be expressed in terms of diffusion coefficients. Thus, the factor β has the meaning of Prandtl’s number. Equation (19.81) is a generalization of the Cauchy-Lagrange integral for turbulent motion. To derive (19.79), (19.80) and (19.81) we use formulae (4.14) and (4.15) of j Sect. 4.1. In we take tensor Pi with the negative sign, i.e. for a κtheseκ formulae κ function L u , u ,t , u ,i we put Pi = −

L κ u ,i , u κ ,t

Then, the momentum equation is

j

Pi = −

L κ j u ,i + Lδi . u κ ,j

(19.82)

19.4

A Turbulence Model

983 j

P Pi + ij = 0. t x

(19.83)

In the case of the Lagrangian (19.75), the set of functions u κ consists of functions vi , λa , ξ a , D i j and ϕ. To have the Lagrangian which depends only on the first derivatives of the required functions, we replace the term λa j D i j ξ,ia by −λa, j D i j ξ,ia +a divergence term, which can be dropped. Thus, Pi = −

L a ξ , ξ,ta ,i

j

Pi = −

L L L j vk,i − a ξ,ia − λa,i + Lδi . vk, j ξ, j λa, j

(19.84)

Plugging in (19.84) function L (19.75) we obtain Pi = λa ξ,ia ,

j

j

j

Pi = β D k j vk,i − ϕv,i + λa v j ξ,ia + λa,k D k j ξ,ia + λa,i D k j ξ,ka + Lδi .

These relations can be modified using (19.76), (19.77) and (19.78). We have j

Pi = vi − ϕ,i + β Di, j . Therefore, j P˜ vi + ij = 0, t x

j

D ϕ j j j δi + β i . P˜ i = Pi − t t

j The tensor P˜ i can be written as j D j k P˜ i = β D k j vk,i + v j vi − ϕ,i + β Di,k + β i + D k j λa,k ξ,ia + λa,i ξ,ka + t ϕ j j δi − ϕv,i = + L− t

j d Di j k = β D k j vk,i + v j vi + β − βv k Di,k + βv j Di,k − β D k j vk,i + vi,k + dt dϕ j j j k δi + v ϕ,k δi − v j ϕ,i − ϕv,i . (19.85) + L− dt

Here we used (19.77) and (19.78). The last three terms in (19.85) can be dropped because their divergence is zero. Note, that j

j

k = −β Dik v,k + term with zero divergence. −βv k Di,k + βv j Di,k

Therefore, up to terms with zero divergence, p j j j P˜ i = vi v j + δi + τi , ρ and momentum equations are (19.79), (19.80) and (19.81) as claimed.

984

19


The most interesting here is the link between the diffusion of particles and the diffusion of momentum. It turns out that this link is preserved if we keep complicating the model. Hyperbolic diffusion. In elementary (kinematic) theory of homogeneous turbulent diffusion one can derive a relationship between the Reynolds stresses τ i j = v ′i v ′ j and the diffusion coefficients D i j : τij =

1 ij D . θ

(19.86)

Here, θ is the characteristic correlation time of velocity fluctuations (v ′i at the instants t and t + τ are statistically independent if τ > θ ). Formula (19.86) has an asymptotic character: it is obtained in the limit θ → 0, f ,i → 0. So the range of its applicability is the developed turbulent flows. According to (19.86) one has to put k=

1 i D. 2θ i

(19.87)

Then an additional field variable enters the theory, the correlation time θ . The variational problem is ill-posed with respect to θ : for Dii = 0, one can vanish the energy of fluctuations by tending θ to +∞ (though we get out of the range of applicability of the relation (19.87)). The simplest possibility to make the large values of θ “energetically unfavorable” is to modify (19.69). In the diffusion equation, f q i f + vi i = i , t x x

q i n i V = 0,

(19.88)

instead of setting for the diffusion flux the law, q i = D i j f /x j , we put θ

f dq i + q i = Di j j . dt x

(19.89)

This makes the diffusion equation hyperbolic and provides a finite speed of diffusion. Without changing the hyperbolic character of diffusion, we may replace θ dq i /dt by other terms, θ

v k dq i + qk i dt x

or θ

dqi vi + qk k dt x

or θ

dqi vi − qk k dt x

.

They correspond to time derivatives of the diffusion flux in different senses. We will use the following modification of the constitutive equation (19.89):

19.4

A Turbulence Model

θ

985

v i dq i − qk k dt x

+ q i = Di j

f . x j

(19.90)

The reason for that will be explained later. The characteristic times θ in (19.90) and (19.87) could be, in general, different. Therefore, we further understand under θ the characteristic time in (19.90), while the characteristic time in (19.86) is assumed to be α times smaller than θ , i.e. (19.86) is replaced by the equation τij =

α ij D . θ

(19.91)

Accordingly, the energy of fluctuations is k=

1 i α i τi = D. 2 2θ i

For inhomogeneous flows we also add the term (19.74). Finally, k=

α i D − β D i j vi, j . 2θ i

(19.92)

Formula (19.91) explains why the second term in (19.92) has the negative sign: in shear flows with small velocity gradient and small θ , −τ i j vi, j ≥ 0, while τ i j are proportional to D i j . Hence, D i j vi, j ≤ 0. These equations complete the formulation of the variational problem. Varying the action functional we obtain the governing equations dλ + j D i j σi = 0, dt d (θ σi ) + θ σk v,ik , σi D i j n j V = 0, σi = λ,i + dt : ; : ; : ; ij vi = λ f ,i − θ σk q,ik − θ σi q i , j + ϕ,i − β D, j , : ; α δi j = σ(i f , j) − βv(i, j) , 2θ i dq α i j i . − q = σ D v i ,j 2θ 2 i dt

(19.93)

Here λ, σi and ϕ are the Lagrange multipliers for the constraints (19.88), (19.90) and (19.71). Functions f , λ, σi , q i depend on t, x and the initial point, X. The symbol · means integration over X. As in the previous subsection, the governing equations yield the momentum equations (19.79) with

986

19


d Di j α ij j i D − β D k j v,k , + D ki v,k + β θ dt p v2 α dϕ : j ; = + 2 Dkk − β D i j vi, j − − λq , j . ρ 2 2θ dt

τij =

(19.94)

We see that the transition to hyperbolic diffusion causes only the addition of the term with the correlation time in Reynolds stresses while other terms are unchanged. Now we can explain the reason for taking the constitutive equation for the diffusion flux q i in the form (19.90): for other choices, the constitutive equation for Reynolds stresses has additional contributions. Kinetic energy of fluctuations, k, as follows from (19.94) is k=

α i 1 d Di j 1 i τi = Di − β D i j vi, j + β . 2 2θ 2 dt

(19.95)

The right-hand sides of (19.95) and (19.92) differ by a divergence term. One can define k by (19.95) without changing the governing equations to keep theory selfconsistent. For shortened turbulence kinematics, one uses functions ξ a (t, x) instead of f (t, x|X ) . Therefore, (19.88) and (19.90) are replaced by the equations ξ a q ai ξ a + vi i = , t x x i ai dq ξ a v i θ − q ak k + q ai = D i j j . dt x x One can check that the constitutive equations for Reynolds stresses (19.94) do not change. Every version of the theory presented is quite complex. However, the major point is simple: the diffusion of fluid particles causes the diffusion of momentum, and the variational principle controls the link between the characteristics of the scalar diffusion and the momentum diffusion.

Bibliographic Comments

Chapter 14 Foundations of elastic plate theory were laid down by Kirchhoff [148] and later extened to elastic shells by Love [187]. Defects contained in Love’s equations were corrected by A.L. Goldenveizer in the 1940s. The improved Love’s equations did not, however, have an energy form. The first variational shell theory was obtained by Novozhilov and Balabukh (see [233]). Novozhilov-Balabukh’s equations were written in curvature lines, and it remained unclear whether they have a tensor form. In particular, Budiansky and Sanders [77] showed that Novozhilov-Balabukh’s equations cannot be written in tensor form if one uses as bending measures the tensors γδ γδ Bαβ + Q αβ Aγ δ , where Q αβ is a tensor function of a˚ αβ and b˚ αβ , which is linear with respect to b˚ αβ . The tensor form of Novozhilov-Balabukh’s equations given in the text were found from a simple reasoning: the curvature lines are uniquely defined (except some degenerated cases), and the relations written in the curvature lines can be transformed into any other coordinate system by tensor transformation. Doing γδ γδ that one finds that Q αβ depend on b˚ αβ nonlinearly (Q αβ are homogeneous functions of b˚ αβ of first order). Equations of linear shell theory the tensor form of which is quite simple and elegant were suggested by Koiter [150] and Sanders [267]. Classical shell theory is considered in [115, 233, 245]. Regarding justifications of classical shell theory see [143, 151, 167, 220, 280] and the reviews [152, 309]. There are many papers concerned with the refinements of classical shell theory. Asymptotically consistent incorporation of the geometrical corrections was done by Goldenveizer [115] and the author [31]. The difference between the refined theories of [115] and [31] is similar to the difference between Love’s equations (corrected by Goldenveizer) and Koiter-Sanders’ equations: after neglecting the refining terms, equations of [115] and [31] transform into Love’s equations and Koiter-Sanders’ equations, respectively. The influence of the interaction terms between bending and extension on the correct computation of displacements was noticed for the first time by Darevsky [83] who discovered in one problem that the use of different constitutive equations yields a difference in displacement fields on the order of 1%. This difference, however, 987

988


does not tend to zero as h/R → 0. Independently, this fact was established by Misiura and the author in [33]. The underlying energy mechanism was described in [38]. Later, the simple examples were constructed with the error in calculation of displacements on the order of 100% [67, 216]. Shown in the text the asymptotic equivalence of equations of linear statics of shells taking into account the transverse shear [30] and Reissner’s equations [254] means that the latter are asymptotically correct for integral characteristics (the opposite statements have been made in literature on this matter). Theory of anisotropic shells has been developed by Ambartsumjan [2], Naghdi [224], Widera [317], Westbrook [315] et al. In [2] the equations were obtained in curvature lines; one can check that they correspond on the energy density = 1 λ Aλβ) . This causes the differences from the (h κ), ¯ where κ¯ = −ραβ + b(α (A) + 12 equations given in the text. The derivation of the equations of elastic shell theory by the variational-asymptotic method in both linear and nonlinear cases was given by the author [30, 31]. The application of the variational-asymptotic method to developing the governing equations for high-frequency vibrations of shells and plates can be found in [29, 34, 262, 263]; this is reviewed in the monograph by K.C. Le [173]. Other applications of the variational-asymptotic method in plate and shell theory are given by D. Hodges, W. Yu and V. Volovoi [321, 322, 323, 324]. The major mechanical effects in defrmation of sandwich plates are well recognized in engineering literature [1]. An asymptotic theory of Sect. 14.8–14.10 follows the papers [59, 60]. Chapter 15 This chapter follows the papers [28, 35]. Most further developments with practical applications are due to D. Hodges, V. Soutyrine, V. Volovoi, E. Armanios, W. Yu, A. Badir and their collaborators [134, 305, 306, 307, 308]. Note also the papers [39, 45, 62, 64]. Chapter 16 Section 16.1 follows the paper [37], Sects. 16.5, 16.6 paper [48]. For real-valued integrals, the Laplace method in infinite-dimensional spaces has been developed by Ellis and Rosen [96] (see also a review paper [246]). The problems of Sect. 16.5 are similar in nature to the problems considered in the theory of large deviations by S. Varahdan [301] and M. Freidlin and A. Wentzel [104]. The technique of functional integration is discussed in [100, 113, 144, 103, 281, 282]. Proofs of some of statements made in Sects. 16.5 and 16.6 are given in [55]. The variational principle for probability densities of Sect. 16.8 is a generalization of a similar variational principle in homogenization theory [42]. More detailed study of the issues touched upon in Sect. 16.7 is given in [51]. Some other settings of stochastic variational problems were considered by R. Bellman [20]. There are also many stochastic variational problems studied within the framework of linear programming.


989

Chapter 17 A history of homogenization goes back about 200 years. Though many particular problems have been solved, until recently there was no clear understanding of how to attack the homogenization problems. Each researcher needed to develop his own tools to overcome the hurdles. A breakthrough in understanding homogenization occurred only in the 1970s, when it was recognized that homogenization of the media with periodic microstructure has an asymptotic nature and may be treated as a two-scale asymptotic expansion. This was done independently by SanchezPalencia [264, 265], Bakhvalov [9] and Babushka [8]. This expansion is a generalization to partial differential equations of the asymptotic expansions in celestial mechanics constructed by Lindstedt, Poincaré and Bogoljubov. The first proof of the statement, which may be called within the framework of the homogenization theory the homogenized equation existence theorem, was actually obtained earlier by Freidlin [102] who was motivated by pure mathematical issues. Note also the pioneering works on the existence of the homogenized equations by De Giorgi and Spagniola [86], Bensoussan, Lions and Papanicolaou [21], and De Giorgi [85], and a special homogenization problem studied by Marchenko and Khruslov [200]. A complete asymptotic analysis of the problem was given by Bakhvalov in a series of publications. Remarkably, the computation of the coefficients of the homogenized equations and the local fields can be found from the solution of some cell problem (Bakhvalov [9, 10]). In the homogenization problems possessing the variational structure, the cell problem also has a variational structure in both linear and nonlinear cases (formulas (17.21) and (17.22)). This result was obtained by the author [27], and later on rediscovered in several publications (see e.g. [199]). The next important development was an extension of the asymptotic procedure to quasi-periodic and random structures given by Kozlov [154, 155]. It turned out that the homogenization of random structures is also reduced to the solution of some cell problem. Kozlov’s cell problem was extended to nonlinear structures in [37]. The reviews can be found in Bensoussan, Lions and Papanicolaou [22], Sanchez-Palencia [266], Berdichevsky [38], Bakhvalov and Panasenko [11], Jikov, Kozlov and Oleinik [142], Milton [214], Torquato [296], Marchenko, Khruslov, Goncharenko and Shepelsky [201], Beliaev [19], and the collection of papers in memory of S. Kozlov [63]). A relevant mathematical formalization is related to the notion of G-convergence and -convergence which were advanced in many studies (see Jikov, Kozlov and Oleinik [142]). Section 17.7 contains generalizations of the results by Dykhne [93] and Kozlov [156], Sect. 17.9 a generalization of Hill’s formula [132]; these generalizations were given in [38]. The homogenization theory was applied to develop a nonlinear spring theory by Soutyrine and Berdichevsky [68]. An interesting issue of describing all possible values of effective coefficients, pioneered by Lurie and Cherkaev [191], was covered in detail by Milton [214].

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Chapter 18 A theory of almost periodic functions was created by H. Bohr and presented in his remarkable book [72]. Another space of almost periodic functions was constructed by Besicovitch [70]. Further developments are reviewed by Levitan and Jikov [177]. Regarding the theory of homogeneous random field see [219]. A variational principle for probability densities of local fields was suggested by the author [42]. Entropy and temperature of microstructure and the rule to choose the probabilistic measure were introduced in [61]. Chapter 19 Equations of motion of mixtures were developed in many papers (see a review in [230]). The system of equations in the text was obtained in [32]. Regarding further developments employing the variational approach see [106, 107, 108, 109]. The model of granular media was suggested by Vaisman and Goldstik [300], a turbulence model by the author [41].

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Index

A almost periodic functions, 900 spectrum, 902 B barotropic flow, 461 basic vectors, 137 beam theory Bernoulli-Euler, 254 energy density, 725 governing equations, 720 kinematics, 716 Kirchhoff-Clebsch, 723 linear, 722 nonlinear effects, 723 phenomenology, 715 physically linear, 722 Timoshenko, 254 Bernoulli problem, 8 Birkhoff-Khinchine theorem, 49 C Cauchy-Lagrange integral, 459 characteristic length, 253, 633 chemical potential, 193 tensor, 374 Christoffel’s symbols, 137 Clapeyron theorem, 165 Clebsch’s potentials, 395, 461 compatibility detailed gradient, 807 gradient, 805, 815 strain, 302 surface quadratic forms, 597 velocity-distortion, 70 configurational space, 20 constitutive equation, 55 geometrically nonlinear elastic body, 351 perfect gas, 56

constraint essential/inessential, 174 non-holonomic, 257 unlocking, 241 coordinates Eulerian, 67, 76 Lagrangian, 67, 76 cross-section principle, 242 D d’Alambert principle, 129 energy form, 130 D’Arcy law, 832 deformation geometrically linear, 286 derivative covariant, 138 surface, 594 Lagrangian covariant, 139 material time, 80 normal vector, 597 of inverse distortion, 83 of Lagrangian coordinates, 83 space, 79 time, 79, 140 direct tensor notations, 141 discontinuity condition, 207, 373 distortion, 69 identity for derivatives, 87 inverse, 70 inverse, identity for derivatives, 88 divergence theorem, 88 invariant form, 90 two-dimensional, 598 E effective characteristics, 871, 880, 892 effective coefficients, 833

1005

1006 eigenvibrations linear, 379 nonlinear, 377 elastic moduli, 295 bulk modulus, 296 inverse, 295 isotropic body, 295 Lame’s constants, 296 Poisson’s coefficient, 297 symmetry conditions, 295, 668 Young’s modulus, 297 energy, 191 complementary, 293 isotropic elastic body, 296 conservation, 23, 35 equivalent beam, 861 extreme values, 761 free elastic body, 286 geometrically nonlinear elastic body, 341 ideal compressible gas, 192 non-convex, 347 internal, 23 kinetic, 22 particle-wall interaction, 27 perfect gas, 191 shell theory, 599, 613 spring, 28, 859 energy surface, 47 energy-momentum tensor, 121 enthalpy, 193 entropy, 51 elastic bar, 943 gas, 55 maximum, 57 microstructure, 934 perfect gas, 57 entropy functional, 773 equation Codazzi, 597 conservation of mass, Eulerian form, 85 conservation of mass, Lagrangian form, 84 energy, 96 entropy, 98 equilibrium equation general solution, 299 Euler in curvilinear coordinates, 278 Gauss, 597 Hamilton-Jacobi, 544 internal energy rate, 98 momentum, differential form, 94

Index momentum, general features, 392, 457 momentum, in Lagrangian coordinates, 96 momentum, integral form, 93 Parseval, 310, 901 equipartition law, 49, 940 ergodic system, 47 ergodicity, 51 estimate changing the functional, 173 changing the set of admissible functions, 172 closeness of minimizer and its approximation, 176 constraint unlocking, 172 non-convex functional, 216 Reuss-Hill, 840 Voigt-Hill, 839 Euler equation, 16 excitation Gaussian, 758, 797 probability distribution of energy, 768 large, 758, 801 probability distribution of energy, 789 small, 757, 799 probability distribution of energy, 771 F formula Basset, 527 Dykhne, 873 Einstein, 59 Gauss, 270 function concave, 193 convex, 181 stress, 299 functional bilinear, 154 boundedness from below, 154 elasticity, 291 necessary condition, 160 complex-valued, 280 convex, 169 derivative of minimum value, 280 Dirichlet, 201 linear, 154 probability distribution, 797 quadratic, 154, 163 identity, 163 stochastic, 756 regularization, 493 symmetry, 446

Index functional integral Laplace’s asymptotics, 766 G generalized coordinates, 20 generalized momenta, 33 Green’s function, 219 Green’s tensor, 309 unbounded region, 310 H Hamilton function, 33, 34 Hamiltonian equations, 34 Hamiltonian structure, 42, 47 heat supply, 58, 97 homogeneous function, 23 identity, 23 homogenization, 817 layered composites, 853 mixture of elastic phases, 889 mixture of ideal fluids, 883 one-dimensional, 849 periodic structures, 818 random structures, 842 two-dimensional, 862 two-dimensional elastic composites, 875 I indices Eulerian , 76 juggling, 76 Lagrangian , 76 symmetrization, 81 inequality Cauchy, 157 Korn, 290 Poincare-Steklov-Fridrichs-Erlich, 162 Wirtinger, 158 integration functional, 277 invariant, 84 isentropic flow, 461 isovorticity group, 449 J Jacobian, 74, 84 material time derivative, 85 K Kolmogorov’s theorem, 802 Kronecker’s delta, 69 decomposition, 592

1007 L Lagrange equations, 25 Lagrange function, 25, 42 divergence terms, 279 double pendulum, 31 particle in a box, 31 pendulum, 30 vibrating suspension point, 258 spring, 29 Lagrange multipliers, 224 Laplace’s asymptotics, 763 least action principle, 127, 375 ideal compressible fluid, 455, 462 ideal incompressible fluid, 389, 398 in Eulerian coordinates, 381 minimum action, 44 nonlocal nature, 43 rigid body in ideal fluid, 514 vortex line dynamics, 422 least dissipation principle heat conduction, 497 plastic body, 503 viscous flow, 499, 500 Levi-Civita symbol identity, 72 three-dimensional, 72 Levi-Civita tensor Lagrangian components, 75 three-dimensional, 73 two-dimensional, 591 M main lemma of calculus of variations, 17, 119 mass, 23 attached, 40, 518 mass density, 85 matrix inverse , 71, 73 orthogonal, 78 orthogonal , 77 Maxwell rule, 199 metric tensor, 71 contravariant components, 71 contravariant Lagrangian components, 72 Lagrangian components, 72, 75 minimizing sequence, 162 mixing, 48 model Cattaneo, 101 dislocations, 323 continuously distributed, 328 elastic body, 101 geometrically linear, 285

1008 geometrically nonlinear, 341 Moony, 343 physically linear, 294 semi-linear, 368 Treloar, 344 entropic elasticity, 103 granular material, 976 heat conductivity, 100 heterogeneous mixtures, 966 ideal compressible fluid, 105 ideal compressible gas, 191 perfect, 191 incompressible ideal fluid, 107 internal stresses, 318 perfect gas, 106 plastic body, 109, 502 von Mises, 111 shallow water, 961 turbulent flow, 978 van der Waals gas, 194 viscous compressible fluid, 107 viscous incompressible fluid, 108 with high derivatives, 134 N Newton’s polygon rule, 249 P particle positions, 67 phase equilibrium elastic body, 369 liquids, 374 phase transition, 194 phase volume, 50, 56 Plank’s constant, 44 polar decomposition, 77 Polia-Shiffer’s theorem, 518 position current , 68 initial, 68 principle of virtual displacements, 129 problem cell, 825, 875 dual, 826, 876 elasticity, 915 random structure, 848 Dirichet, 201 Kozlov’s cell, 844, 899, 905 linear, 164 von Neuman, 161, 206 process adiabatic, 57

Index dissipative, 63 nonequilibrium, 60 R random field Gaussian, 759 homogeneous, 903 probability densities, 802 Rayleigh-Ritz method, 172 reciprocity inertial, 40 of interactions, 37 Onsager, 62 relabeling group, 446 rigid motion, 81 S Schrödinger equation, 44 set convex, 169 shell theory anisotropic heterogeneous, 665 bending measures, 607 boundary conditions, 604 energy, 613 geometric relations, 589 geometrically linear, 607 isotropic shell, 613 linear, 612, 620 low frequency vibrations, 677 membrane, 624 phenomenology, 598 physically linear, 606 small parameters, 627 strain measures, 599 von Karman, 622 short wave extrapolation, 640, 966 Snell law, 5, 6 spring theory, 856 step function, 49 strain measures, 77 tensor, 77, 78 strain rate tensor Eulerian components, 81 Lagrangian components, 80 stress tensor, 94 Piola-Kirchhoff, 95 surface geometry area element, 592 compatibility conditions, 597 covariant derivatives, 594 curvatures, 597 divergence theorem, 598

Index metric tensor, 590 normal vector, 591 derivatives, 597 second quadratic form, 595 surface tensors, 590 T temperature, 47, 50 gas, 56 torsional rigidity, 733 transformation Fourier, 240, 310 Legendre, 185 Young-Fenchel, 188 free energy of geometrically nonlinear elastic body, 365 quadratic form, 201 transverse shear, 671 V variation and fluctuations, 505 beam strain measures, 719 contravariant components of Lagrangian metrics, 123 density, 462 Eulerian of mass density, 125 Eulerian of velocity, 125 integral functional, 117 inverse distortion, 123 Jacobian, 122 Lagrangian coordinates, 125, 396, 461 mass density, 122 particle trajectories of rigid body, 123 strain tensor, 123 surface characteristics, 601 tensor, 142 scalar function, 145 velocity, 122, 399, 462 variational derivative, 25 variational equation, 128, 531 d’Alambert, 129 holonomic, 531 Sedov, 132 virtual displacements, 129 variational principle potential flows, 405 Arnold, 487 Arnold-Grinfeld, 488 Bateman, 466, 467 Bateman-Dirichlet, 491 Bateman-Kelvin, 493 beam energy density, 725, 727 bubble vibration, 527

1009 Castigliano, 298 for stress functions, 299, 300 two-dimensional problems, 302 cell problem, 916 Dirichlet, 408 dual anti-plane problem, 306 Bateman, 467 Dirichlet problem, 201 dislocation, 326 general integral functional, 212 general scheme, 178 geometrically nonlinear elastic body, 355, 361 heat conduction, 497 internal stresses, 320 kinetic energy of vortex flow, 410, 412 plastic body, 505 potential flows, 406, 469 semi-linear elastic body, 368 stress function, 321 viscous flow, 502 elastic body, 626 steady flow, 490 Fermat, 4 Gibbs, 59, 60 anti-plane problem, 305 dislocation, 325 elastic body, 285 geometrically nonlinear elastic body, 351 internal stresses, 320 Giese, 482 Giese-Kraiko, 482 Gurtin-Tonti, 542 Hamilton, 27, 215 Hashin-Strikman, 216 elastic body, 306, 317 ideal compressible fluid, 465 free surface, 469 open steady flow, 481 steady flow, 476 ideal fluid with free surface, 402 ideal incompressible fluid, 402 open steady flow, 481 steady flow, 479 Jacobi, 26, 457 Kelvin, 407, 469 kinetic energy of vortex flow, 409, 412 Kozlov’s cell problem, 906 Lagrange, 25, 549 Lin, 400, 463 Lin-Rubinov, 484

1010 Luke, 407 Migdal, 445 Mopertuis, 11 Mopertuis-Lagrange, 456 Morse-Feshbach, 541 non-equilibrium processes, 506 open flows, 453 point vortices, 433 Pontrjagin, 237 potential flows, 407 probabilities densities, 801, 809 probability densities, 920 probability density, 544 Rayleigh, 380 Reissner, 294 shell theory, 600 steady vortex flow, 485, 487 two-dimensional vortex flow, 431 vortex filament, 441, 443 vortex line dynamics, 419, 420 variational problem and functional integrals, 270 existence of minimizer, 167 extreme values, 761 minimax, 179 minimum drag body, 228 modification, 280 quadratic functional various forms, 165

Index setting, 150 stochastic, 751 uniqueness of minimizer, 168 with constraints, 224 integral constraints for derivatives, 238 variational-asymptotic method, 243 beam theory, 742 compressible flow, 472 homogenization of periodic structures, 827 homogenization of random structures, 844 shallow water, 961 shell theory, 631 vector product, 72 velocity, 70 angular , 83 volume element in Lagrangian coordinates, 84 in Eulerian coordinates, 84 in initial state, 84 vortex filament, 434 kinetic energy, 437 self-induction approximation, 443 vortex gas, 766, 786, 792 vortex line, 415 vortex sheet, 444 vorticity, 393, 408, 458 W Whitham’s method, 262

Notation

x a point in three-dimensional space, x i are its coordinates. Indices, i, j, k, l, m, run through values 1, 2, 3. In consideration of mathematical issues, x is a point in n-dimensional space, x i are its coordinates, and small Latin indices, i, j, k, l, m, run through values 1, 2, . . . , n. Usually, writing the arguments of a function, the indices are suppressed, and the notation, f (x), is used for function f (x 1 , . . . , x n ). The notation, f (x i ), is used if it desirable to emphasize that f is a function of several arguments. Summation is always conducted over repeated low and upper indices. Indices of vectors and tensors are written as low or upper indices depending on convenience and in accordance with the rule of summation over repeated low and upper index. Indices, which do not have tensor nature are put usually in parentheses; for example, the boundary values of a function, u, is denoted by u (b) . t Rn R3 R4 Xa a, b, c, d α, β, γ , δ gi j , g i j g , g ab %ab % %ai j % ai j g gˆ ˆ

time n-dimensional space three-dimensional Euclidean space four-dimensional space-time Lagrangian coordinates small Latin indices run through values 1, 2, 3 and correspond to projections on Lagrangian axes small Greek indices run through values 1, 2 and correspond to projections on a two-dimensional coordinate frame components of the metric tensor in observer’s frame components of the metric tensor in Lagrangian frame matrix with the components, ai j determinant of the matrix with the components ai j determinant of the matrix with the components gi j determinant of the matrix with the components gab this symbol marks quantities in Lagrangian coordinates in cases when an ambiguity appears without such a mark; it also marks a maximizer – the particular meaning is seen from the context. 1011

1012

Notation

◦ ei jk εi jk j δi xai X ia εab εi j ε eab ei j σ ab σij pia |x|ab αai , α ij a (−1)i j T11

this symbol marks quantities in the initial state symbol Levi-Civita tensor Levi-Civita Kronecker’s delta distortion inverse distortion components of the strain tensor in Lagrangian coordinates components of the strain tensor in Eulerian coordinates magnitude of deformation; small parameter components of the strain rate tensor in Lagrangian coordinates components of the strain rate tensor in Eulerian coordinates components of the stress tensor in Lagrangian coordinates components of the stress tensor in Eulerian coordinates components of Piola-Kirchhoff’s tensor modulus of distortion orthogonal matrices % % the components of matrix inverse to the matrix %ai j %

physical (11)-component of the tensor Ti j (the corresponding indices are underlined) parentheses a(i j) in indices mean symmetrization: a(i j) ≡ 1 a + a ij ji 2 brackets in indices mean antisymmetrization: a[i j] ≡ a[i j] 1 ai j − a ji 2 (b) u parenthesis for a single index are used to emphasize its nontensor nature; e.g., u (b) usually denotes the boundary value of u λ γλβ) being combined with the contraction, the symmetrizab(α λ tion does not act on the dummy index: b(α γλβ) ≡

1 λ λ bα γλβ + bβ γλα 2 the expression in the previous parentheses with index i (i → j) changed by j the expression in the previous parentheses with the substitu(i ↔ j) tion of indices: i → j, j → i κ multi-index; it denotes a set of indices of various physical nature field variables uκ u, x, X sometimes we drop indices and write u instead of u κ , x instead of x i , X instead of X a . U, F, K , S densities of internal energy, free energy, kinetic energy and entropy per unit mass U, F, K, S total internal energy, free energy, kinetic energy and entropy of the body ¯¯

Notation

ρ t ≡

1013

t

d dt

i ≡ ∇i a ≡ ∇a

mass density ≡ (·)t = (·),t time derivative at constant Eulerian coordinates, x time derivative at constant Lagrangian coordinates, X

x i X a

= (·),i = (·),a

∇˚ a δ δ(x) θ (x) V V |V | dV || || V f , Vu dn x = d x1 . . . d xn d3x dA I, J I Iˇ Iˆ L φ u ∈(1.1) ∗ × ≡

partial space derivative in Eulerian variables covariant space derivatives in Eulerian coordinates partial space derivatives in Lagrangian variables Lagrangian covariant space derivatives in the deformed state Lagrangian covariant space derivatives in the initial state the Jacobian of transformation from Lagrangian to Eulerian coordinates variation at constant X δ-function; if x is a point in n-dimensional space, then δ(x) is the product of n one-dimensional δ-functions the step function: θ (x) = 0 for x < 0, θ (x) = 1 for x ≥0 usually a region in three-dimensional space boundary of region V volume of region V volume element usually a surface area of the surface usually a curve length of curve parts of the boundary of elastic body in which one prescribes forces and displacements, respectively volume element in Rn volume element in R3 in Cartesian coordinates area element usually functionals usually action functional minimum value of the functional maximum value of the functional Lagrange function or Lagrangian symbol of empty set function u satisfies the constraint (1.1) usually the symbol of Young-Fenchel transformation; complex conjugation in consideration of Fourier transformation symbol of Legendre transformation this sign is usually used for definitions

1014

[ϕ] [ϕ]tt10

Notation

difference of values of function ϕ on the two sides of the discontinuity surface difference of values of function ϕ at the instant t1 and t0

Variational Principles of Continuum Mechanics. II. Applications

Variational Principles of Continuum Mechanics. I. Fundamentals

Variational Principles of Continuum Mechanics: II. Applications (Interaction of Mechanics and Mathematics)

The Variational Principles of Mechanics

Variational Principles of Continuum Mechanics: I. Fundamentals (Interaction of Mechanics and Mathematics)

Principles and Applications of Variational Methods