INTRODUCTION

TO THE MECHANICS

OF A CONTINUOUS MEDIUM

Lawrence E. MalvernProfessor of Mechanics

College oIEngineenng

Mtchtgan State University

Prentice-Hall, Inc.

l.nglcv: ood CIr(fI. SCII Jersey

e 1969 byPrentice-Hall, Inc.

Englewood Cliffs, N J

All rights reserved No part of this bookmay be reproduced in any form or by any meanswithout permission in wrrtmg from the publisher.

Current printing (last digit):

10 9 8 7

13-487603-2Library of Congress Catalog Card Number 69-13712

Printed In the Unrted States of America

Preface

This book offers a unified presentation of the concepts and general princi­ples common to all branches of solid and fluid mechanics, designed to appealto the intuition and understanding of advanced undergraduate or first-yearpostgraduate students in engineering or engineering science.

The book arose from the need to provide a general preparation in contin­uum mechanics for students who WIll pursue further work in specialized fieldssuch as viscous fluids, elasticity, viscoelasticity, and plasncity, Originally thebook was introduced for reasons of pedagogical economy-to present the com­mon foundations of these specialized subjects in a unified manner and also toprovide some introduction to each subject for students who will not takecourses in all of these areas. This approach develops the foundations morecarefully than the traditional separate courses where there is a tendency tohurry on to the applications, and moreover provides a background for lateradvanced study in modem nonlinear continuum mechanics,

The first fivechapters devoted to general concepts and principles applicableto all continuous media are followed by a chapter on constitutive equations, theequations defining particular media. The chapter on constitutive theory beginsWith sections on the specific constitutive equations of linear viscosity, linearizedelasticity, linear viscoelasucity, and plasticity, and concludes with two sectionson modem constitutive theory. There are also a chapter on fluid mechanics andone on linearized elasticity to serve as examples of how the general principlesof the first five chapters are combined with a constituuve equation to formu­late a complete theory. Two appendices on curvilinear teosor componentsfollow, which may be omitted altogether or postponed until after the mainexposition is completed.

Although the book grew out of lecture notes for a one-quarter course forfirst-year graduate students taught by the author and several colleagues duringthe past 12 years, It contains enough material for a two-semester course and iswritten at a level suitable for advanced undergraduate students. The only

v

vi Preface

prerequisites are the basic mathematics and mechanics equivalent to that usu­ally taught in the first two or three years of an undergraduate engineeringprogram. Chapter 2 reviews vectors and matrices and introduces what tensormethods are needed. Part of this material may be postponed until needed, butit is collected in Chap. 2 for reference.

The last 15 to 20 years have seen a great expansion of research and publi­cation in modern continuum mechanics. The most notable developments havebeen jn the theory of constitutive equations, especially in the formulation ofvery general principles restricting the possible forms that constitutive equationscan take. These new theoretical developments are especially addressed to theformulation of nonlinear constitutive equations, which are only briefly touchedupon in this book. But the new developments have also pointed up the limita­tions of some of the widely used linear theories. This does not mean that anyof the older linear theories must be discarded, but the new developments pro­vide some guidance to the conditions under which the older theories can beused and the conditions where they are subject to significant error. The lasttwo sections of Chap: 6 survey modern constitutive theory and provide refer­ences to original papers and to more extended treatments of the modern theorythan that given in this, introductory text.

The book is a carefully graduated approach to the subject in both contentand-style. The earlier part of the book is written with a great deal of illustrativedetail in the development of the basic concepts of stress and deformation andthe mathematical formulation used to represent the concepts. Symbolic formsof the equations, 'using dyadic notation, are supplemented by expanded Carte­sian component forms, matrix forms, and indicial forms of the same equationsto give the student abundant opportunity to master the notations. There arealso many simple exercises involving interpretation of the general ideas in con­crete examples. In Chaps. 4 and 5 there is a gradual transition to more relianceon compact notations and a gradual increase in the demands on the reader'sability to comprehend general statements.

Until the end of Sec. 4.2, each topic considered is treated fairly completelyand (except for the brief section on stress resultants in plate theory) onlyconcepts that will be used repeatedly in the following sections are introduced.Then there begin to appear concepts and formulations whose full implementa­tion is beyond the scope of the book. These include, for example, the relativedescription of motion, mentioned in Sec. 4.3 and also in some later sections,and the finite rotation and stretch tensors of Sec. 4.6, which are important insome of the modern developments referred to in the last two sections of Chap.6. The aim in presenting this material is to heighten the reader's awarenessthat the subject of continuum mechanics is in a state of rapid development,and to encourage his reading of the current literature. The chapters on fluidsand on elasticity also refer to published methods and results in addition tothose actually presented.

Preface

The sections on the constitutive equations of viscoelasticity and plasticityare introduced by accounts of the observed responses of real materials in orderto motivate and also to point up the limitations of the idealized representationsthat follow. The second section on plasticity includes work-hardening. a partof the theory not in a satisfactory state, but so important in engineering appli­cations that it was believed essential to mention and point out some of theshortcomings of the available formulations.

A one-quarter course might well include most of the first five chapters, onlypart of Chap. 6, and either Chap. 7 on fluids or Chap. 8 on elasticity. Section3.6 on stress resultants in plates and those parts of Sees. 5.3 and 5.4 treatingcouple stress can be omitted without destroying the continuity, as also can Sees.6.5 and 6.6 on plasticity. Section 4.6 can be given only minor emphasis, oromitted altogether if the last two sections of Chap. 6 are not to be covered.The second appendix, presenting only physical components in orthogonal cur­vilinear coordinates might be included if time permits; although not needed inthe text, it is useful for applications.

A two-term course could include the first appendix on general curvilineartensor components, useful as a preparation for reading some of the modern lit­erature. There is sufficient textual material in the book for a full year course,but it should probably be supplemented with some challenging applicationsproblems. Most of the exercises in the text are teaching devices to illuminatethe theory, rather than applications.

The book is a textbook, designed for classroom teaching or self-study, nota treatise reporting new scientific results. Obviously the author is indebted tohundreds of investigators over a period of more than two centuries as well asto earlier books in the field or in its specialized branches. Some of these inves­tigators and authors are named in the text, but the bibliography at the end ofthe book includes only the twentieth-century writings cited. Extensive bibli­ographies may be found in the two Encyclopedia of Physics treatises; "TheClassical Field Theories," by C. Truesdell and R. A. Toupin, Vol. III.' 1, pp.226-793 (1960), and "The Non-Linear Field Theories of Mechanics." by C.Truesdell and W. Noll, Vol. lUj3 (1965), published by Springer-Verlag,Berlin. These two valuable comprehensive treatises are among the referencesfor collateral reading cited at the end of the introduction. Many of thehistorical allusions in the text are based on these two sources.

The author is indebted to several colleagues at Michigan State Universitywho have used preliminary versions of the book in their classes. These includeDr. C. A. Tatro (now at the Lawrence Radiation Laboratory. Livermore,California)and Professors M. A. Medick. R. W. Little, and K. N. Subramanian.Professors John Foss and Merle Potter read the first version of the materialon fluid mechanics. Encouragement and helpful criticism have been providedby these colleagues and also by the dozens of students who have taken thecourse.

Preface

The author is also indebted to Michigan State University for sabbaticalleave during 1966-67 to work 00 the book and to Prentice-Hall, Iae., for theircooperation and assistance in preparing the final text and illustrations.

Finally~ thanks arc due to the author's wife for inspiration, encouragementand forbearance.

LAWRENCB E. MALVBR.N

WI Lan~ing~ Michigan

Contents

1. Introduction

1.1 The Continuous Medium

2. Vectors and Tensors

2.1 Introduction 72.2 Vectors; Vector Addition; Vector and Scalar Components;

Indicia! Notation; Finite Rotations not Vectors 102.3 Scalar Product and Vector Product 172.4 Change of Orthonormal Basis (Rotation of Axes); Tensors

as Linear Vector Functions; Rectangular Cartesian TensorComponents; Dyadics; Tensor Properties; Review of Ele­mentary Matrix Concepts 25

2.5 Vector and Tensor Calculus; Differentiation; Gradient, Di­vergence and Curl 48

3. Stress

3.1 Body Forces and Surface Forces 643.2 Traction or Stress Vector; Stress Components 693.3 Principal Axes of Stress and Principal Stresses; Invariants;

Spherical and Deviatoric Stress Tensors 853.4 Mohr's Circles 953.5 Plane Stress; Mohr's Circle 1023.6 Stress Resultants in the Simplified Theory of Bending ofThin

Plates 112

4. Strain and Deformation

4.1 Small Strain and Rotation in Two Dimensions 1204.2 Small Strain and Rotation in Three Dimensions 129

ix

1

7

64

120

" Contents

4.3 Kinematics of a Continuous Medium; Material Derivatives138

4.4 Rate-of-Deformation Tensor (Stretching); Spin Tensor (Vor-ticity); Natural Strain Increment 145

4.5 Finite Strain and Deformation; Eulerian and LagrangianFormulations; Geometric Measures of Strain; Relative De-formation Gradient 154

4.6 Rotation and Stretch Tensors 1724.7 Compatibility Conditions; Determination of Displacements

When Strains are Known 183

s. General Principles

5.1 Introduction; Integral Transformations; Flux 1975.2 Conservation of Mass; The Continuity Equation 2055.3 Momentum Principles; Equations of Motion and Equilib­

rium; Couple Stresses 2135.4 Energy Balance; First Law of Thermodynamics; Energy

Equation 2265.5 Principle of Virtual Displacements 2375.6 Entropy and the Second Law of Thermodynamics; the

Clausius-Dubem Inequality 2485.7 The Caloric Equation of State; Gibbs Relation; Thermody­

namic Tensions; Thermodynamic Potentials; DissipationFunction 260

6. Constitutive Equations

6.1 Introduction; Idea I Materials 2736.2 Classical Elasticity; Generalized Hooke's Law; Isotropy;

Hyperelasticity; The Strain Energy Function or Elastic Po­tential Function; Elastic Symmetry; Thermal Stresses 278

6.3 Fluids; Ideal Frictionless Fluid; Linearly Viscous (New-tonian) Fluid; Stokes Condition of Vanishing Bulk Vis­cosity; Laminar and Turbulent Flow 295

6.4 Linear Viscoelastic Response 3066.5 Plasticity I. Plastic Behavior of Metals; Examples of Theo­

ries Neglecting Work-Hardening: Levy-Mises PerfectlyPlastic; Prandtl-Reuss Elastic. Perfeetly Plastic; and Visco­plastic Materials 327

6.6 Plasticity II. More Advanced Theories; Yield Conditions;Plastic-Potential Theory; Hardening Assumptions; OlderTotal-Strain Theory (Deformation Theory) 346

6.7 Theories of Constitutive Equations 1: Principle of Equi­presence; Fundamental Postulates of a Purely MechanicalTheory; Principle of Material Frame-IndilTerence 378

197

273

Contents

6.8 Theories of Constitutive Equations II: Material SymmetryRestrictions on Constitutive Equations of Simple Materials;Isotropy 406

7. Fluid Mechanics

7.1 Field Equations of Newtonian Fluid: Navier-Stokes Equa­tions; Example: Parallel Plane Flow of Incompressible FluidBetween Flat Plates 423

7.2 Perfect Fluid: Euler Equation; Kelvin's Theorem; BernoulliEquation; Irrotational Flow; Velocity Potential; AcousticWaves; Gas Dynamics 434

7.3 Potential Flow of Incompressible Perfect Fluid 4487.4 Similarity of Flow Fields in Experimental Model Analysis;

Characteristic Numbers; Dimensional Analysis 4627.5 Limiting Cases: Creeping-Flow Equation and Boundary­

Layer Equations for Plane Flow of Incompressible ViscousFluid 475

8. Linearized Theory of Elasticity

8.1 Field Equations 4978.2 Plane Elasticity in Rectangular Coordinates 5058.3 CYlindrical Coordinate Components; Plane Elasticity in

Polar Coordinates 5258.4 Three-Dimensional Elasticity; Solution for Displacements;

Vector and Scalar Potentials; Wave Equations; GalerkinVector; Papkovich-Neuber Potentials; Examples, IncludingBoussinesq Problem 548

Appendix I. Tensors

I. I Introduction; Vector-Space Axioms; Linear Independence;Basis; Contravariant Components of a Vector; EuclideanVector Space; Dual Base Vectors; Covariant Componentsof a Vector 569

1.2 Change of Basis; Unit-Tensor Components 5761.3 Dyads and Dyadics: Dyadics as Second-Order Tensors;

Determinant Expansions; Vector (cross) Products 5881.4 Curvilinear Coordinates; Contravariant and Covariant

Components Relative to the Natural Basis; The MetricTensor 596

I. 5 Physical Components of Vectors and Tensors 606I. 6 Tensor Calculus; Covariant Derivative and Absolute De-

xi

423

497

569

xii Contents

rivative of a Tensor Field; Christoffel Symbols; Gradient,Divergence. and Curl; Laplacian 614

I. 7 Deformation; Two-Point Tensors; Base Vectors; MetricTensors; Shifters; Total Covariant Derivative 629

I. 8 Summary of General-Tensor Curvilinear-ComponentForms of Selected Field Equations of Continuum Me­chanics 634

Appendix II. Orthogonal Curvilinear Coordinates,Physical Components of Tensors

H. 1 Coordinate Definitions; Scale Factors; Physical Compo­nents; Derivatives of Unit Base Vectors and of Dyadies641

H.2 Gradient. Divergence, and Curl in Orthogonal CurvilinearCoordinates 650

II. 3 Examples oC Field Equations of Continuum Mechanics,Using Physical Components in Orthogonal CurvilinearCoordinates 659

n.4 Summary oC Differential Formulas in Cylindrical andSpherical Coordinates 667

Bibliography. Twentieth-Century Authors Citedin the Text

Author Index

Subject Index

641

673

685

691

Sec. 4.6 Strain and Deformation 173

additive decomposition of the displacement gradient into the sum of a purestrain plus a pure rotation, since when the displacement-gradient componentsare not small compared to unity the two matrices no longer represent purestrain and pure rotation, respectively. But a multiplicative decomposition ofthe deformation gradient F = x~ into the product of two tensors, one of whichrepresents a rigid-body rotation, while the other is a symmetric positive-definitetensor is always possible. If R denotes the orthogonal rotation tensor, whichrotates the principal axes of C at X into the directions of the principal axesofD- I at x, then we will see in Eqs. (14) to (21) below that there exist two tensorsU and V satisfyingt

so that

F=R·U=V·R

dx =(R.U).dX =(V.R).dX

(4.6.1a)

with rectangular Cartesian component forms (referring XI and XI to the samereference axes and using lower-case indices for both)

such that

Xfc,p = RkqUqp = VkqR qp and

dx, = RkqUqpdXp = VkqRqpdXp

R1mRJm = s; and RkpRkq = s; or}

R·R" = 1 and R"·R = 1.

(4.6.1b)

(4.6.1 c)

U is called the right stretch tensor, and V is called the left stretch tensor.Either stretch tensor operating as U· or as V· on the set of all vectors at apoint produces length changes (stretch) in the vectors and also produces addi­tional rotation of all vectors except those in the principal directions of thestretch tensor, in addition to the rigid-body rotation R of the whole set ofvectors. Both U and V are symmetric and positive-definite.

Equations (1) show that we may consider the motion and deformation ofan infinitesimal volume element at X to consist of the successive application] of:

tThe notation here is that of Truesdell and Noll (1965), following Noll (1958), exceptthat they omit the dots. giving F = RU = VR.

tThe fact that the deformation at a point may be considered as the result of a translationfollowed by a rotation of the principal axes of strain, and stretches along the principalaxes, was apparently recognized by Thomson and Tail in 1867, but first explicitly statedby Love in 1892.

402 Constitutive Equations Chap. (I

unaffected by the change of frame. frame-indifference does place restrictionson possible nonlinear viscous constitutive equations. In fact frame-indifferenceimplies that the function reo) in Eq. (6.3.8) must be an isotropic tensor func­tion. See, for example, Truesdell and Toupin (/960), Sec. 299.

Co-rotational and convected stress rates, stress flux. Some particularconstitutive equations that have been proposed include the material time deriv­ative t of the stress T among the constitutive variables, for example the Max­well model and the standard linear solid of linear viscoelasticity (see Sec. 6.4)and the Prandtl-Reuss elastic plastic material (Sec. 6.5). Such equations donot satisfy the principle of material frame-indifference for arbitrary motions.One part of the difficulty is that the material derivative t does not transformaccording to Eq. (7) under an orthogonal change of the spatial reference frameeven though T does. That is, t is not frame-indifferent, even though T is. Weshall demonstrate that t is not frame-indifferent, and in the demonstration weshall discover a group of terms which together are frame-indifferent. This groupis called the co-rotational stress rate and denoted by or with a superposedsmall circle instead of a dot. It is equal to the material derivative of the stressas it would appear to an observer in a frame of reference attached to theparticle and rotating with it at an angular velocity equal to the instantaneousvalue of the angular velocity w of the material. We define:

Co-rotational stress rate (6.7.69)

where W is the spin tensor of Sec. 4.4.To lead to the definition of Eq, (69) we begin by differentiating the equation

given by Eq. (7), obtaining

'I'. Q 'I' QT' I r~ 'I' QT' I Q TnT'= •• * - \..l... - . • .'l.

(6.7.70)

(6.7.71)

Because of the last two terms in Eq. (71), the tensor t is not frame-indifferent.We now eliminate Qand QT' from Eq. (71) as follows. We substitute A = Q oQ"in Eq. (68b) and then multiply Eq. (68b) by Q from the right, to obtain

whence, since QToQ = 1, we obtain

Q= W· oQ - QoW with transpose

Q" = _Q".W. -\- WoQT.(6.7.72)

(Recall that Wand W· are skew, so that, for example W" = - W.) Substitut-