Applied Solid Mechanics

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A P P L I E D S O L I D M E C H A N I C S

Much of the world around us, both natural and man-made, is built from and held

together by solid materials. Understanding how they behave is the task of solid

mechanics, which can in turn be applied to a wide range of areas from earthquake

mechanics and the construction industry to biomechanics. The variety of materials

(such as metals, rocks, glasses, sand, flesh and bone) and their properties (such as

porosity, viscosity, elasticity, plasticity) are reflected by the concepts and techniques

needed to understand them, which are a rich mixture of mathematics, physics, ex-

periment and intuition. These are all brought to bear in this distinctive book, which

is based on years of experience in research and teaching. Theory is related to

practical applications, where surprising phenomena occur and where innovative

mathematical methods are needed to understand features such as fracture. Starting

from the very simplest situations, based on elementary observations in engineer-

ing and physics, models of increasing sophistication are derived and applied. The

emphasis is on problem solving and on building an intuitive understanding, rather

than on a technical presentation of theoretical topics. The text is complemented by

over 100 carefully chosen exercises, and the minimal prerequisites make it an ideal

companion for mathematics students taking advanced courses, for those undertak-

ing research in the area or for those working in other disciplines in which solid

mechanics plays a crucial role.


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Cambridge Texts in Applied Mathematics

Editorial Board

Mark Ablowitz, University of Colorado, Boulder

S. Davis, Northwestern UniversityE. J. Hinch, University of Cambridge

Arieh Iserles, University of Cambridge

John Ockendon, University of Oxford

Peter Olver, University of Minnesota


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A P P L I E D S O L I D M E C H A N I C S

P E T E R H O W E L LUniversity of Oxford

G R E G O R Y K O Z Y R E F FFonds de la Recherche ScientifiqueFNRS

and Universit e Libre de Bruxelles

J O H N O C K E N D O N

University of Oxford


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CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So Paulo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-85489-4

ISBN-13 978-0-521-67109-5

ISBN-13 978-0-511-50639-0

P. D. Howell, G. Kozyreff and J. R. Ockendon 2009

2008

Information on this title: www.cambrid e.or /9780521854894

This publication is in copyright. Subject to statutory exception and to theprovision of relevant collective licensing agreements, no reproduction of any partmay take place without the written permission of Cambridge University Press.

Cambridge University Press has no responsibility for the persistence or accuracyof urls for external or third-party internet websites referred to in this publication,and does not guarantee that any content on such websites is, or will remain,accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

a erback

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Contents

List of illustrations page viiiPrologue xiii

Modelling solids 1

1.1 Introduction 1

1.2 Hookes law 2

1.3 Lagrangian and Eulerian coordinates 3

1.4 Strain 4

1.5 Stress 71.6 Conservation of momentum 10

1.7 Linear elasticity 11

1.8 The incompressibility approximation 13

1.9 Energy 14

1.10 Boundary conditions and well-posedness 16

1.11 Coordinate systems 19

Exercises 24

Linear elastostatics 28

2.1 Introduction 28

2.2 Linear displacements 29

2.3 Antiplane strain 37

2.4 Torsion 39

2.5 Multiply-connected domains 42

2.6 Plane strain 47

2.7 Compatibility 682.8 Generalised stress functions 70

2.9 Singular solutions in elastostatics 82

2.10 Concluding remark 93

Exercises 93

v


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vi Contents

Linear elastodynamics 103

3.1 Introduction 103

3.2 Normal modes and plane waves 104

3.3 Dynamic stress functions 121

3.4 Waves in cylinders and spheres 1243.5 Initial-value problems 132

3.6 Moving singularities 138

3.7 Concluding remarks 143

Exercises 143

Approximate theories 150


4.2 Longitudinal displacement of a bar 1514.3 Transverse displacements of a string 152

4.4 Transverse displacements of a beam 153

4.5 Linear rod theory 158

4.6 Linear plate theory 162

4.7 Von Karman plate theory 172

4.8 Weakly curved shell theory 177

4.9 Nonlinear beam theory 187

4.10 Nonlinear rod theory 1954.11 Geometrically nonlinear wave propagation 198


Exercises 205

Nonlinear elasticity 215


5.2 Stress and strain revisited 216

5.3 The constitutive relation 221

5.4 Examples 233


Exercises 239

Asymptotic analysis 245


6.2 Antiplane strain in a thin plate 246

6.3 The linear plate equation 248

6.4 Boundary conditions and Saint-Venants principle 2536.5 The von Karman plate equations 261

6.6 The EulerBernoulli plate equations 267

6.7 The linear rod equations 273

6.8 Linear shell theory 278


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Contents vii


Exercises 283

Fracture and contact 287


7.2 Static brittle fracture 2887.3 Contact 309


Exercises 321

Plasticity 328


8.2 Models for granular material 330

8.3 Dislocation theory 3378.4 Perfect plasticity theory for metals 344

8.5 Kinematics 358

8.6 Conservation of momentum 360

8.7 Conservation of energy 360

8.8 The flow rule 362

8.9 Simultaneous elasticity and plasticity 364

8.10 Examples 365

8.11 Concluding remarks 370Exercises 372

More general theories 378


9.2 Viscoelasticity 379

9.3 Thermoelasticity 388

9.4 Composite materials and homogenisation 391

9.5 Poroelasticity 408

9.6 Anisotropy 413


Exercises 417

Epilogue 426

Appendix Orthogonal curvilinear coordinates 428

References 440

Index 442


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Illustrations

1.1 A reference tetrahedron. page 81.2 The forces acting on a small two-dimensional element. 91.3 A small pill-box-shaped region at the boundary between two elastic

solids. 181.4 Forces acting on a polar element of solid. 221.5 A system of masses connected by springs. 252.1 A unit cube undergoing (a) uniform expansion, (b) one-dimensional

shear, (c) uniaxial stretching. 302.2 A uniform bar being stretched under a tensile force. 32

2.3 A paper model with negative Poissons ratio. 332.4 A strained plate. 342.5 A bar in a state of antiplane strain. 382.6 A twisted bar. 392.7 A uniform tubular torsion bar. 432.8 The cross-section of (a) a circular cylindrical tube; (b) a cut tube. 442.9 The unit normal and tangent to the boundary of a plane region. 492.10 A plane annulus being inflated by an internal pressure. 532.11 A plane rectangular region subject to tangential tractions on its faces. 572.12 The tractions applied to the edge of a semi-infinite strip. 592.13 The surface displacement of a half-space and corresponding surface

pressure. 652.14 A family of functions (x) that approach a delta-function as 0. 832.15 Contours of the maximum shear stress created by a point force acting

at the origin. 852.16 Four point forces. 913.1 Plots of the first three Bessel functions. 108

3.2 A P-wave reflecting from a rigid boundary. 1163.3 A layered elastic medium. 1173.4 Dispersion relation for symmetric and antisymmetric Love waves. 1203.5 Illustration of flexural waves. 1283.6 The one-dimensional fundamental solution. 1343.7 The two-dimensional fundamental solution. 135

viii


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List of illustrations ix

3.8 The cone x2 + y2 = c2t2 tangent to the plane k1x + k2y = t. 1383.9 The two-sheeted characteristic cone for the Navier equation. 1383.10 The response of a string to a point force moving at speed V. 1393.11 Wave-fronts generated by a moving force on an elastic membrane. 1413.12 P-wave- and S-wave-fronts generated by a point force moving at speed

V in plane strain. 1423.13 Group velocity versus wave-number for symmetric and antisymmetric

Love waves. 1464.1 The forces acting on a small length of a uniform bar. 1514.2 The forces acting on a small length of an elastic string. 1534.3 The forces and moments acting on a small segment of an elastic beam. 1544.4 The end of a beam under clamped, simply supported and free conditions.1554.5 The first three buckling modes of a clamped elastic beam. 1574.6 The internal force components in a thin elastic rod. 1594.7 Cross-section through a rod showing the bending moment components. 1594.8 Examples of cross-sections in the (y, z)-plane and their bending

stiffnesses. 1614.9 The forces acting on a small section of an elastic plate. 1634.10 The bending moments acting on a section of an elastic plate. 1644.11 The displacement of a simply supported rectangular plate sagging

under gravity. 1694.12 (a) A cylinder, (b) a cone, (c) another developable surface, (d) a

hyperboloid. 1754.13 Typical surface shapes with (a) zero, (b) negative and (c) positiveGauss 179

4.14 Deformations of a cylindrical shell. 1844.15 Deformations of an anticlastic shell. 1854.16 Deformations of a synclastic shell. 1864.17 A beam (a) before and (b) after bending; (c) a close-up of the

displacement field. 1874.18 (a) The forces and moments acting on a small segment of a beam.

(b) The sign convention for the forces at the ends of the beam. 1884.19 (a) Final angle of a diving board versus applied force parameter.(b) Deflection of a diving board for various values of the force parameter.191

4.20 (a) Response diagram of the amplitude of the linearised solution for abuckling beam versus the force parameter. (b) Corresponding responseof the weakly nonlinear solution. 193

4.21 (a) Pitchfork bifurcation diagram of leading-order amplitude versusforcing parameter. (b) The corresponding diagram when asymmetry isintroduced. 195

4.22 A system of pendulums attached to a twisting rubber band. 1994.23 A kink propagating along a series of pendulums attached to a rod. 2004.24 Travelling wave solution of the nonlinear beam equations. 2014.25 A beam clamped near the edge of a table. 2064.26 A beam supported at two points. 207

iancurvature.

4.27 The first three buckling modes of a vertically clamped beam. 213


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x List of illustrations

5.1 The deformation of a small scalene cylinder. 2185.2 Typical forcestrain graphs for uniaxial tests on various materials. 2335.3 A square membrane subject to an isotropic tensile force. 2345.4 Response diagrams for a biaxially-loaded incompressible sheet of

MooneyRivlin material. 2355.5 Scaled pressure inside a balloon as a function of the stretch for various

values of the MooneyRivlin parameter. 2365.6 Gas pressure inside a cavity as a function of inflation coefficient for

various values of the MooneyRivlin parameter. 2386.1 The edge of a plate subject to tractions. 2546.2 The geometry of a deformed two-dimensional plate. 2697.1 Definition sketch of a thin crack. 2887.2 Definition sketch for contact between two solids. 288

7.3 (a) A Mode III crack. (b) A cross-section in the (x, y)-plane. 2907.4 Definition sketch for the function

z2 c2 . 292

7.5 Displacement field for a Mode III crack. 2937.6 (a) A planar Mode II crack. (b) The regularised problem of a thin

elliptical crack. 2977.7 Contour plot of the maximum shear stress around a Mode II crack. 3017.8 The displacement of a Mode II crack under increasing shear stress. 3037.9 A Mode I crack. 3047.10 Contour plot of the maximum shear stress around a Mode I crack. 306

7.11 The displacement of a Mode I crack under increasing normal stress. 3077.12 Solution for the contact between a string and a level surface. 3107.13 Three candidate solutions for a contact problem. 3117.14 The contact between a beam and a horizontal surface under a uniform

pressure. 3147.15 Contact between a rigid body and an elastic half-space. 3177.16 The penetration of a quadratic punch into an elastic half-space. 3197.17 A flexible ruler flattened against a table. 3267.18 A wave travelling along a rope on the ground. 3268.1 A typical stressstrain relationship for a plastic material. 3298.2 The stressstrain relationship for a perfectly plastic material. 3308.3 The forces acting on a particle at the surface of a granular material. 3318.4 The normal force and frictional force acting on a surface element inside

a granular material. 3328.5 The Mohr circle. 3338.6 The triaxial stress factor versus angle of friction. 3368.7 An antiplane cut-and-weld operation. 339

8.8 The displacement field in an edge dislocation. 3408.9 An edge dislocation in a square crystal lattice. 3418.10 A moving edge dislocation. 3428.11 The normalised torque versus twist applied to an elastic-plastic

cylindrical bar. 3478.12 The normalised torque versus twist applied to an elastic-plastic

cylindrical bar, showing the recovery phase. 348


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List of illustrations xi

8.13 The free-boundary problem for an elastic-perfectly plastic torsion bar. 3498.14 Residual shear stress in a gun barrel versus radial distance for different

values of the maximum internal pressurisation. 3528.15 The Tresca yield surface. 3558.16 The von Mises yield surface. 3568.17 The Coulomb yield surface. 3588.18 Luders bands in a thin sheet of metal. 3698.19 The Mohr surface for three-dimensional granular flow. 3738.20 The normalised torque versus twist applied to an elastic-plastic

cylindrical bar undergoing a loading cycle. 3759.1 (a) A spring; (b) a dashpot; (c) a spring and dashpot connected in

parallel; (d) a spring and dashpot connected in series. 3809.2 (a) Applied tension as a function of time. (b) Resultant displacement

of a linear elastic spring. (c) Resultant displacement of a linear dashpot.3819.3 Displacement of a Voigt element due to the applied tension shown in

Figure 9.2(a). 3829.4 Displacement of a Maxwell element due to the applied tension shown

in Figure 9.2(a). 3839.5 (a) The variation of Youngs modulus with position in a bar. (b) The

corresponding longitudinal displacement. 3929.6 (a) The variation of Youngs modulus with position in a bar. (b) The

corresponding longitudinal displacement. 3959.7 A periodic microstructured shear modulus. 3969.8 A symmetric, piecewise constant shear modulus distribution. 4009.9 Some modulus distributions that are antisymmetric about the diago-

nals of a square. 4029.10 Dimensionless wavenumber versus the Youngs modulus non-uniformity

parameter. 4079.11 The one-dimensional squeezing of a sponge. 4119.12 Dimensionless stress applied to a sponge versus dimensionless time for

different values of the Peclet number. 4129.13 A Jeffreys viscoelastic element. 418

9.14 A system of masses connected by springs and dashpots in parallel. 4189.15 A system of masses connected by springs and dashpots in series. 4199.16 Dimensionless wavenumber versus Youngs modulus contrast for a

piecewise uniform bar. 424A1.1 A small reference box. 432A1.2 Cylindrical polar coordinates. 437A1.3 Spherical polar coordinates. 438


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Prologue

Although solid mechanics is a vitally important branch of applied mechan-ics, it is often less popular, at least among students, than its close relative,

fluid mechanics. Several reasons can be advanced for this disparity, such as

the prevalence of tensors in models for solids or the especial difficulty of han-

dling nonlinearity. Perhaps the most daunting prospect for the student is the

multitude of different behaviours that can occur and cause elementary theo-

ries of elasticity to become irrelevant in practice. Examples include fracture,

buckling and plasticity, and these pose intellectual challenges in solid me-

chanics that are every bit as fascinating as concepts like flight, shock waves

and turbulence in fluid dynamics. Our principal objective in this book is to

demonstrate this fact to undergraduate and beginning graduate students.

We aim to give the subject as wide an accessibility as possible to math-

ematically-minded students and to emphasise the interesting mathematical

issues that it raises. We do this by relating the theory to practical applica-

tions where surprising phenomena occur and where innovative mathematical

methods are needed.Our layout is essentially pragmatic. Although more advanced texts in solid

mechanics often begin with quite general theories founded on basic mechan-

ical and thermodynamic principles, we start from the very simplest models,

based on elementary observations in engineering and physics, and build our

way towards models that are the basis for current applied research in solid

mechanics. Hence, we begin by deriving the basic Navier equations of linear

elasticity, before illustrating the mathematical techniques that allow these

equations to be solved in many different practically relevant situations, bothstatic and dynamic. We then proceed to describe some approximate theories

for the elastic deformation of thin solids, namely bars, strings, beams, rods,

plates and shells. We soon discover that many everyday phenomena, such as

the buckling of a beam under a compressive load, cannot be fully described

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xiv Prologue

using linear theories. We therefore give a brief exposition of the general the-

ory of nonlinear elasticity, and then show how formal asymptotic methods

allow simplified linear and weakly nonlinear models to be systematically de-

duced. Although we regard such asymptotic techniques as invaluable to any

applied mathematician, these last two topics may both be omitted on a firstreading without loss of continuity. We go on to present simple models for

fracture and contact, comparing and contrasting these apparently similar

phenomena. Next, we show how plasticity theory can be used to describe

situations where a solid yields under a sufficiently high stress. Finally, we

show how elasticity theory may be generalised to include further physical ef-

fects, such as thermal stresses, viscoelasticity and porosity. These combined

fields of solid mechanics are increasingly finding applications in industrial

and medical processes, and pose ever more elaborate modelling questions.Despite the breadth of the models and relevant techniques that will emerge

in this book, we will usually try to present the theoretical developments

ab initio. Nonetheless, the book is very far from being self-contained. Any

student who aspires to becoming a solid mechanics specialist will have to

delve further into the literature, and we will provide references to help with

this.

We assume only that the reader has a reasonable familiarity with the

calculus of several variables. Fluency with the more advanced techniques

required for Chapters 6 and 7, in particular, will readily be acquired by

a student who works through the exercises in the early Chapters, espe-

cially those cited in the text. Indeed, we firmly believe that solid mechanics

provides a wonderful arena in which to build an understanding of such im-

portant mathematical areas as linear algebra, partial differential equations,

complex variable theory, differential geometry and the calculus of variations.

Our hope is that, having read this book, a student should be able to confrontany practical problem that may be encountered in everyday solid mechanics

with at least some idea of the basic mathematical modelling that will be

required.

During the writing of this book, we received a great deal of help and inspi-

ration as a result of discussions with David Allwright, Jon Chapman, Sam

Howison, L. Mahadevan, Roman Novokshanov and Domingo Salazar, as well

as many other colleagues and students too numerous to thank individually.

We would like to express our particular gratitude to Gareth Jones, HilaryOckendon and Tom Witelski who gave invaluable advice on draft Chapters.

We are also indebted to David Tranah and his colleagues at Cambridge

University Press for helping to make this book a reality.


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1

Modelling solids

1.1 Introduction

In everyday life we regularly encounter physical phenomena that apparently

vary continuously in space and time. Examples are the bending of a paper

clip, the flow of water or the propagation of sound or light waves. Such phe-

nomena can be described mathematically, to lowest order, by a continuum

model, and this book will be concerned with that class of continuum models

that describes solids. Hence, at least to begin with, we will avoid all consid-eration of the atomistic structure of solids, even though these ideas lead

to great practical insight and also to some beautiful mathematics. When

we refer to a solid particle, we will be thinking of a very small region of

matter but one whose dimension is nonetheless much greater than an atomic

spacing.

For our purposes, the diagnostic feature of a solid is the way in which it

responds to an applied system of forces and moments. There is no hard-and-fast rule about this but, for most of this book, we will say that a continuum is

a solid when the response consists of displacements distributed through the

material. In other words, the material starts at some reference state, from

which it is displaced by a distance that depends on the applied forces. This is

in contrast with a fluid, which has no special rest state and responds to forces

via a velocity distribution. Our modelling philosophy is straightforward. We

take the most fundamental pieces of experimental evidence, for example

Hookes law, and use mathematical ideas to combine this evidence withthe basic laws of mechanics to construct a model that describes the elastic

deformation of a continuous solid. Following this simple approach, we will

find that we can construct solid mechanics theories for phenomena as diverse

as earthquakes, ultrasonic testing and the buckling of railway tracks.

1


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2 Modelling solids

By basing our theory on Hookes law, the simplest model of elasticity,

for small enough forces and displacements, we will first be led to a system

of differential equations that is both linear, and therefore mathematically

tractable, and reversible for time-dependent problems. By this we mean

that, when forces and moments are applied and then removed, the systemeventually returns to its original state without any significant energy being

lost, i.e. the system is not dissipative.

Reversibility may apply even when the forces and displacements are so

large that the problem ceases to be linear; a rubber band, for example,

can undergo large displacements and still return to its initial state. How-

ever, nonlinear elasticity encompasses some striking new behaviours not

predicted by linear theory, including the possibility of multiple steady states

and buckling. For many materials, experimental evidence reveals that evenmore dramatic changes can take place as the load increases, the most strik-

ing phenomenon being that of fracture under extreme stress. On the other

hand, as can be seen by simply bending a metal paper clip, irreversibility

can readily occur and this is associated with plastic flow that is significantly

dissipative. In this situation, the solid takes on some of the attributes of a

fluid, but the model for its flow is quite different from that for, say, water.

Practical solid mechanics encompasses not only all the phenomena men-

tioned above but also the effects of elasticity when combined with heat

transfer (leading to thermoelasticity) and with genuine fluid effects, in cases

where the material flows even in the absence of large applied forces (leading

to viscoelasticity) or when the material is porous (leading to poroelastic-

ity). We will defer consideration of all these combined fields until the final

chapter.

1.2 Hookes lawRobert Hooke (1678) wrote

it is . . . evident that the rule or law of nature in every springing body is that theforce or power thereof to restore itself to its natural position is always proportionateto the distance or space it is removed therefrom, whether it be by rarefaction, orseparation of its parts the one from the other, or by condensation, or crowding ofthose parts nearer together.

Hookes observation is exemplified by a simple high-school physics experi-

ment in which a tensile force T is applied to a spring whose natural length

is L. Hookes law states that the resulting extension of the spring is propor-

tional to T: if the new length of the spring is , then

T = k( L), (1.2.1)

where the constant of proportionality k is called the spring constant.


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1.3 Lagrangian and Eulerian coordinates 3

Hooke devised his law while designing clock springs, but noted that it

appears to apply to all springy bodies whatsoever, whether metal, wood,

stones, baked earths, hair, horns, silk, bones, sinews, glass and the like. In

practice, it is commonly observed that k scales with 1/L; that is, everything

else being equal, a sample that is initially twice as long will stretch twiceas far under the same force. It is therefore sensible to write (1.2.1) in the

form

T = k L

L, (1.2.2)

where k is the elastic modulus of the spring, which will be defined more

rigorously in Chapter 2. The dimensionless quantity ( L)/L, measuring

the extension relative to the initial length, is called the strain.Equation (1.2.2) is the simplest example of the all-important constitutive

law relating the force to displacement. As shown in Exercise 1.3, it is possible

to construct a one-dimensional continuum model for an elastic solid from

this law, but, to generalise it to a three-dimensional continuum, we first need

to generalise the concepts of strain and tension.

1.3 Lagrangian and Eulerian coordinatesSuppose that a three-dimensional solid starts, at time t = 0, in its rest

state, or reference state, in which no macroscopic forces exist in the solid

or on its boundary. Under the action of any subsequently applied forces

and moments, the solid will be deformed such that, at some later time t, a

particle in the solid whose initial position was the point X is displaced

to the point x (X, t). This is a Lagrangian description of the continuum: if

the independent variable X is held fixed as t increases, then x(X, t) labels

a material particle. In the alternative Eulerian approach, we consider the

material point which currently occupies position x at time t, and label its

initial position by X(x, t). In short, the Eulerian coordinate x is fixed in

space, while the Lagrangian coordinate X is fixed in the material.

The displacementu(X, t) is defined in the obvious way to be the difference

between the current and initial positions of a particle, that is

u(X, t) = x(X, t) X. (1.3.1)

Many basic problems in solid mechanics amount to determining the dis-

placement field u corresponding to a given system of applied forces.

The mathematical consequence of our statement that the solid is a con-

tinuum is that there must be a smooth one-to-one relationship between X

and x, i.e. between any particles initial position and its current position.


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4 Modelling solids

This will be the case provided the Jacobian of the transformation from X

to x is bounded away from zero:

0 < J < , where J = det xiXj

. (1.3.2)The physical significance of J is that it measures the change in a small

volume compared with its initial volume:

dx1dx2dx3 = JdX1dX2dX3, or dx = JdX (1.3.3)

as shorthand. The positivity ofJ means that we exclude the possibility that

the solid turns itself inside-out.

We can use (1.3.3) to derive a kinematic equation representing conserva-tion of mass. Consider a moving volume V(t) that is always bounded by

the same solid particles. Its mass at time t is given, in terms of the density

(X, t), by

M(t) =

V(t)

dx =

V(0)

JdX. (1.3.4)

Since V(t) designates a fixed set of material points, M(t) must be a constant,namely its initial value M(0):

V(0)JdX= M(t) = M(0) =

V(0)

0 dX, (1.3.5)

where 0 is the density in the rest state. Since V is arbitrary, we deduce that

J = 0. (1.3.6)

Hence, we can calculate the density at any time t in terms of 0 and the

displacement field. The initial density 0 is usually taken as constant, but

(1.3.6) also applies if 0 = 0(X).

1.4 Strain

To generalise the concept of strain introduced in Section 1.2, we consider thedeformation of a small line segment joining two neighbouring particles with

initial positions X and X+ X. At some later time, the solid deforms such

that the particles are displaced to X+u(X, t) and X+ X+u(X+ X, t)

respectively. Thus we can use Taylors theorem to show that the line element


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1.4 Strain 5

X that joins the two particles is transformed to

x = X+u(X+ X, t)u(X, t) = X+ (X)u(X, t) + , (1.4.1)

where

(X ) = X1 X1

+ X2 X2

+ X3 X3

. (1.4.2)

Let L = |X| and = |x| denote the initial and current lengths respectively

of the line segment; the difference L is known as the stretch. Then, to

lowest order in L,

2 = |X+ (X )u(X, t)|2. (1.4.3)

Although we will try in subsequent chapters to minimise the use ofsuffices, it is helpful at this stage to introduce components so that

X= (Xi) = (X1, X2, X3)T and similarly for u. Then (1.4.3) may be written

in the form

2 L2 = 23

i,j =1

Eij XiXj , (1.4.4)

where

Eij =1

2

uiXj

+ujXi

+3

k=1

ukXi

ukXj

. (1.4.5)

By way of introduction to some notation that will be useful later, we point

out that (1.4.4) may be written in at least two alternative ways. First, we

may invoke the summation convention, in which one automatically sums over

any repeated suffix. This avoids the annoyance of having to write explicit

summation, so (1.4.4) is simply

2 = L2 + 2Eij XiXj , where Eij =1

2

uiXj

+ujXi

+ukXi

ukXj

.

(1.4.6)

Second, we note that 2 L2 is a quadratic form on the symmetric matrix

E whose components are (Eij ):

2 L2 = 2 XTEX. (1.4.7)

It is clear from (1.4.4) that the stretch is measured by the quantities Eij ;

in particular, the stretch is zero for all line elements if and only ifEij 0. It

is thus natural to identify Eij with the strain. Now let us ask: what happens

when we perform the same calculation in a coordinate system rotated by an


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6 Modelling solids

orthogonal matrix P = (pij )? Intuitively, we might expect the strain to be

invariant under such a rotation, and we can verify that this is so as follows.

The vectors X and u are transformed to X and u in the new coordinate

system, where

X = PX, u = Pu. (1.4.8)

Since P is orthogonal, (1.4.8) may be inverted to give X= PTX. Alterna-

tively, using suffix notation, we have

X = pjX

j , ui = pi u. (1.4.9)

The strain in the new coordinate system is denoted by

Eij =1

2

uiXj

+ujXi

+ukXi

ukXj

, (1.4.10)

which may be manipulated using the chain rule, as shown in Exercise 1.4,

to give

Eij = pipjE. (1.4.11)

In matrix notation, (1.4.11) takes the form

E = PEPT, (1.4.12)

so the 3 3 symmetric array (Eij ) transforms exactly like a matrix repre-

senting a linear transformation of the vector space R3. Arrays that obey the

transformation law (1.4.11) are called second-rank Cartesian tensors, and

E= (Eij ) is therefore called the strain tensor.

Almost as important as the fact that E is a tensor is the fact that itcan vanish without u vanishing. More precisely, if we consider a rigid-body

translation and rotation

u = c + (Q I)X, (1.4.13)

where I is the identity matrix while the vector c and orthogonal matrixQ are constant, then E is identically zero. This result follows directly from

substituting (1.4.13) into (1.4.6) and using the fact that QQT = I, andconfirms our intuition that a rigid-body motion induces no deformation.

The word tensor as used here is effectively synonymous with matrix, but it is easy togeneralise (1.4.11) to a tensor with any number of indices. A vector, for example, is a tensorwith just one index.


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1.5 Stress 7

1.5 Stress

In the absence of any volumetric (e.g. gravitational or electromagnetic) ef-

fects, a force can only be transmitted to a solid by being applied to its

boundary. It is, therefore, natural to consider the force per unit area or

stress applied at that boundary. To do so, we now analyse an infinitesimalsurface element, whose area and unit normal are da and n respectively. If it

is contained within a stressed medium, then the material on (say) the side

into which n points will exert a force df on the element. (By Newtons third

law, the material on the other side will also exert a force equal to df.) In

the expectation that the force should be proportional to the area da, we

write

df = da, (1.5.1)

where is called the traction or stress acting on the element.

Perhaps the most familiar example is that of an inviscid fluid, in which

the stress is related to the pressure p by

= pn. (1.5.2)

This expression implies that (i) the stress acts only in a direction normal

to the surface element, (ii) the magnitude of the stress (i.e. p) is indepen-dent of the direction ofn. In an elastic solid, neither of these simplifying

assumptions holds; we must allow for stress which acts in both tangential

and normal directions and whose magnitude depends on the orientation of

the surface element.

First consider a surface element whose normal points in the x1-direction,

and denote the stress acting on such an element by 1 = (11 , 21, 31)T. By

doing the same for elements with normals in the x2- and x3-directions, we

generate three vectors j (j = 1, 2, 3), each representing the stress acting

on an element normal to the xj -direction. In total, therefore, we obtain nine

scalars ij (i, j = 1, 2, 3), where ij is the i-component ofj , that is

j = ijei, (1.5.3)

where ei is the unit vector in the xi-direction.

The scalars ij may be used to determine the stress on an arbitrary surface

element by considering the tetrahedron shown in Figure 1.1. Here ai denotesthe area of the face orthogonal to the xi-axis. The fourth face has area

a =

a21 + a22 + a

23; in fact if this face has unit normal n as shown, with

components (ni), then it is an elementary exercise in trigonometry to show

that ai = ani.


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8 Modelling solids

a2

a3

x1

x3

x2

n

a1

Fig. 1.1 A reference tetrahedron; ai is the area of the face orthogonal to the xi -axis.

The outward normal to the face with area a1 is in the negative x1-direction

and the force on this face is thus a11. Similar expressions hold for the

faces with areas a2 and a3. Hence, if the stress on the fourth face is denoted

by , then the total force on the tetrahedron is

f = a ajj . (1.5.4)

When we substitute for aj and j , we find that the components of f aregiven by

fi = a (i ij nj ) . (1.5.5)

Now we shrink the tetrahedron to zero volume. Since the area a scales

with 2, where is a typical edge length, while the volume is proportional

to 3, if we apply Newtons second law and insist that the acceleration be

finite, we see that f/a must tend to zero as 0.

Hence we deduce an

Readers of a sensitive disposition may be slightly perturbed by our glibly letting the dimensionalvariable tend to zero: if is reduced indefinitely then we will eventually reach an atomic scaleon which the solid can no longer be treated as a continuum. We reassure such readers that(1.5.6) can be more rigorously justified provided the macroscopic dimensions of the solid arelarge compared to any atomistic length-scale.


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1.5 Stress 9

G11

11

22

21

22

12

12

21

x2

x1

x2

x1

Fig. 1.2 The forces acting on a small two-dimensional element.

expression for :

i = ij nj , or = n. (1.5.6)

This important result enables us to find the stress on any surface element

in terms of the nine quantities (ij ) = .

Now let us follow Section 1.4 and examine what happens to ij when we

rotate the axes by an orthogonal matrix P. In the new frame, (1.5.6) will

become

= n (1.5.7)

where, since and n are vectors, they transform according to

= P, n = Pn. (1.5.8)

It follows that n = (P PT)n and so, since n is arbitrary,

= P PT, or ij = pipj. (1.5.9)

Thus ij , like Eij , is a second-rank tensor, called the Cauchy stress tensor.

We can make one further observation about ij by considering the angular

momentum of the small two-dimensional solid element shown in Figure 1.2.The net anticlockwise moment acting about the centre of mass G is (per

unit length in the x3-direction)

2 (21x2)x1

2 2 (12 x1)

x22

,


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10 Modelling solids

where 21 and 12 are evaluated at G to lowest order. By letting the rectangle

shrink to zero (see again the footnote on page 8), and insisting that the

angular acceleration be finite, we deduce that 12 = 21 . This argument can

be generalised to three dimensions (see Exercise 1.5) and it shows that

ij ji (1.5.10)

for all i and j, i.e. that ij , like Eij , is a symmetric tensor.

1.6 Conservation of momentum

Now we derive the basic governing equation of solid mechanics by apply-

ing Newtons second law to a material volume V(t) that moves with the

deforming solid:

d

dt

V(t)

uit

dx =

V(t)

gi dx +

V(t)

ij nj da. (1.6.1)

The terms in (1.6.1) represent successively the rate of change of momentum

of the material in V(t), the force due to an external body force g, such as

gravity, and the traction exerted on the boundary of V, whose unit normal

is n, by the material around it. We differentiate under the integral (using

the fact that dx = 0 dX is independent of t) and apply the divergencetheorem to the final term to obtain

V(t)

2uit2

dx =

V(t)

gi dx +

V(t)

ijxj

dx. (1.6.2)

Assuming each integrand is continuous, and using the fact that V(t) is ar-

bitrary, we arrive at Cauchys momentum equation:

2u

it2 = g

i +

ijxj

. (1.6.3)

This may alternatively be written in vector form by adopting the following

notation for the divergence of a tensor: we define the ith component of

to be

( )i =jixj

. (1.6.4)

Since is symmetric, we may thus write Cauchys equation as

2u

t2= g + . (1.6.5)

This equation applies to any continuous medium for which a displacement

u and stress tensor can be defined. The distinction between solid, fluid


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1.7 Linear elasticity 11

or some other continuum comes when we impose an empirical constitutive

relation between and u.

For solids, (1.6.5) already confronts us with a distinctive fundamental

difficulty. The most obvious generalisation of Hookes law is to suppose that

a linear relationship exists between the stress and the strain E. But wenow recall that E was defined in Section 1.4 in terms of the Lagrangian

variables X; indeed, the time derivative in (1.6.5) is taken in a Lagrangian

frame, with Xfixed. On the other hand, the stress tensor has been defined

relative to Eulerian coordinates and is differentiated in (1.6.5) with respect

to the Eulerian variable x. It is not immediately clear, therefore, how the

stress and strain, which are defined in different frames of reference, may be

self-consistently related. We will postpone the full resolution of this difficulty

until Chapter 5 and, for the present, restrict our attention to linear elasticityin which, as we shall see, the two frames are essentially identical.

1.7 Linear elasticity

The theory of linear elasticity follows from the assumption that the dis-

placement u is small relative to any other length-scale. This assumption

allows the theory developed thus far to be simplified in several ways. First,it means that ui/Xj is small for all i and j. Second, we note from (1.3.1)

that x and X are equal to lowest order in u. Hence, if we only consider

leading-order terms, there is no need to distinguish between the Eulerian

and Lagrangian variables: we can simply replace X by x and ui/Xj by

ui/xj throughout. A corollary is that the Jacobian J is approximately

equal to one, so (1.3.6) tells us that the density is fixed, to leading order,

at its initial value 0. Finally, we can use the smallness of ui/xj to neglect

the quadratic term in (1.4.5) and hence obtain the linearised strain tensor

Eij eij =1

2

uixj

+ujxi

. (1.7.1)

Much of this book will be concerned with this approximation. Therefore,

and with a slight abuse of notation, we will write E= (eij ).

Remembering (1.4.13), we note that it is possible to approximate E by

(1.7.1) even when u is not small compared with X, just as long as u is closeto a rigid-body translation and rotation. This situation is called geometric

nonlinearity and we will encounter it frequently in Chapters 4 and 6. It

occurs because Eij is identically zero for rigid-body motions of the solid

given by (1.4.13); however eij does not vanish for such rigid-body motions,


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12 Modelling solids

but rather for displacements of the form

u = c + x, (1.7.2)

where c and are constant (see Exercise 1.6).

Assuming the validity of (1.7.1), we can now generalise Hookes law bypostulating a linear relationship between the stress and strain tensors. We

assume that is zero when E is; in other words the stress is zero in the

reference state. This is not the case for pre-stressed materials, and we will

consider some of the implications of so-called residual stress in Chapter 8.

Even with this assumption, we apparently are led to the problem of defining

81 material parameters Cijk (i,j,k, = 1, 2, 3) such that

ij = Cijkek . (1.7.3)

The symmetry of ij and eij only enables us to reduce the number of

unknowns to 36. This can be reduced to a more manageable number by

assuming that the solid is isotropic, by which we mean that it behaves the

same way in all directions. This implies that Cijk must satisfy

Cijkpiipjjpkk p Cijk (1.7.4)

for all orthogonal matrices P = (pij ). It can be shown (see, for example,Ockendon & Ockendon, 1995, pp. 79) that this is sufficient to reduce the

specification of Cijk to just two scalar quantities and , such that

Cijk = ij k + 2ik j, (1.7.5)

where ij is the usual Kronecker delta, which represents the identity matrix;

consequently,

ij = (ekk ) ij + 2eij . (1.7.6)This relation can also be inverted to give the strain corresponding to a given

stress, that is

eij =1

2

ij

(kk )

(3 + 2)ij

. (1.7.7)

In Chapter 9 we will consider solids, such as wood or fibre-reinforced mate-

rials, that are not isotropic, and for which (1.7.6) must be generalised.

The material parameters and are known as the Lame constants, and is called the shear modulus.As we shall see in Chapter 2, and measure a

materials ability to resist elastic deformation. They have the units of pres-

sure; typical values for a few familiar solid materials are given in Table 1.1.

It will be observed that these values may be very large for relatively hard


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1.8 The incompressibility approximation 13

(GPa) (GPa)

Cartilage 3 105 9 105

Rubber 0.04 0.003Polystyrene 2.3 1.2Granite 10 30

Glass 28 28Copper 86 37Steel 100 78Diamond 270 400

Table 1.1 Typical values of the Lame constants and for some everyday

materials (1 GPa = 109 N m2 = 104 atmospheres; a typical car tyre

pressure is two atmospheres).

materials, the significance being that tractions much less than these values

will result in small deformations, so that linear elasticity is valid.

Now we substitute our linear constitutive relation (1.7.6) into the momen-

tum equation (1.6.3) and replace X with x to obtain the Navier equation,

also known as the Lame equation,

2u

t2 = g + ( + )graddivu +

2

u. (1.7.8)Recall that does not vary to leading order, so (1.7.8) comprises three

equations for the three components ofu. It may alternatively be written in

component form

2uit2

= gi + ( + )2uj

xixj+

2uix2j

, (1.7.9)

where the final x

2

j is treated as a repeated suffix, or

2u

t2= g + ( + 2)graddivu curl curlu, (1.7.10)

where we have used the well-known vector identity

del squared equals grad div minus curl curl. (1.7.11)

1.8 The incompressibility approximationThere is an interesting and important class of materials that, although elas-

tic, are virtually incompressible, so they may be sheared elastically but are

highly resistant to tension or compression. In linear elasticity, this amounts

to saying that the Lame constant is much larger than the shear modulus .


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14 Modelling solids

The values given in Table 1.1 show that rubber has this property, as do many

biomaterials such as muscle.

If a material is almost incompressible, we can set

=

1

, (1.8.1)

where is a small parameter. From (1.7.8), we expect that, in the limit

0, divu will be of order . Hence, if we define a scalar function p such

that

p =

divu, (1.8.2)

then p will approach a finite limit p as 0.

When we now substitute (1.8.1) and (1.8.2) into the Navier equation(1.7.8) and let 0, we obtain

2u

t2= g p + 2u, (1.8.3a)

along with the limit of (1.8.2), that is

divu = 0. (1.8.3b)

The condition (1.8.3b) means that each material volume is conserved duringthe deformation, and it imposes an extra constraint on the Navier equation.

The extra unknown p, representing the isotropic pressure in the medium,

gives us the extra freedom we need to satisfy this constraint.

1.9 Energy

We can obtain an energy equation from (1.6.3) by taking the dot product

with u/t and integrating over an arbitrary volume V:V

2uit2

uit

dx =

V

giuit

dx +

V

ijxj

uit

dx. (1.9.1)

The final term may be rearranged, using the divergence theorem, toV

ijxj

uit

dx =

V

uit

ij nj da

V

ijeijt

dx. (1.9.2)


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1.9 Energy 15

Hence (1.9.1) may be written in the form

d

dt

V

1

2

u

t

2

dx +

VWdx

=

V

giuit

dx +

V

uit

ij nj da, (1.9.3)

where W is a scalar function of the strain components that is chosen to

satisfy

W

eij= ij . (1.9.4)

With ij given by (1.7.6), we can integrate (1.9.4) to determine W up to anarbitrary constant as

W=1

2ij eij =

1

2 (ekk )

2 + (eij eij ). (1.9.5)

Here the summation convention is invoked such that (ekk )2 is the square of

the trace of E, while (eij eij ) is the sum of the squares of the components

ofE.

The first term in braces in (1.9.3) is the net kinetic energy in V, while theterms on the right-hand side represent the rate of working of the external

body force g and the tractions on V respectively. Hence, in the absence of

other energy sources resulting from, say, chemical or thermal effects, we can

interpret equation (1.9.3) as a statement of conservation of energy. The dif-

ference between the rate of working and the rate of change of kinetic energy

is the rate at which elastic energy is stored in the material as it deforms; W

is therefore called the strain energy density. This is analogous to the energy

stored in a stretched spring (see Exercise 1.1) and, at a fundamental scale,

is a manifestation of the energy stored in the bonds between the atoms. If

, > 0, we can easily see from (1.9.5) that W is a non-negative function

of the strain components, whose unique global minimum is attained when

eij = 0. In fact, Exercise 1.7 demonstrates that it is only necessary to have

, ( + 2/3) > 0.

The net conservation of energy implied by (1.9.3) reflects the fact that the

Navier equation is not dissipative. Furthermore, even without the constitu-tive relation (1.7.6), the steady Navier equation is a necessary condition for

the net gravitational and strain energy in an elastic body D, namely

U =

DW g udx, (1.9.6)


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16 Modelling solids

to be minimised, as shown in Exercise 1.8. However, the situation changes

when thermal effects are important, as we will see in Chapter 9.

1.10 Boundary conditions and well-posednessSuppose that we wish to solve (1.7.8) for u(x, t) when t is positive and x

lies in some prescribed domain D. We now ask: what sort of boundary

conditions may be imposed on D to obtain a well-posed mathematical

problem, in other words, one for which a solution u exists, is unique and

depends continuously on the boundary data? For boundary-value problems

in linear elasticity, it is generally far easier to discuss questions of uniqueness

than it is to prove existence. Hence in this section we will focus only on

establishing uniqueness.

In elastostatic problems, in which the left-hand side of (1.7.8) is zero,

the Navier system is, roughly speaking, a generalisation of a scalar elliptic

equation. By analogy, it seems appropriate for either u or three linearly

independent scalar combinations ofu and u/n to be prescribed on D.

In many physical problems, we specify either the displacement u or the

traction n everywhere on the boundary, and we will now examine each of

these in turn.First consider a solid body D on whose boundary the displacement is

prescribed, that is

u = ub(x) on D. (1.10.1)

Inside D, u satisfies the steady Navier equation

ijxj

+ gi = 0, (1.10.2)

and we will now show that, if a solution u of (1.7.6), (1.10.2) with the

boundary condition (1.10.1) exists, then it is unique.

Suppose that two solutions u(1) and u(2) exist and let u = u(1)u(2) . Thus

u satisfies the homogeneous problem, with ub = g = 0. Now, by multiplying

(1.10.2) by ui, integrating over D and using the divergence theorem, we

obtain

D

uiij nj da =

Deij ij dx = 2

DWdx, (1.10.3)

where W is given by (1.9.5). The left-hand side of (1.10.3) is zero by the

boundary conditions, while the integrand W on the right-hand side is non-

negative and must, therefore, be zero. It follows that the strain tensor eij


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1.10 Boundary conditions and well-posedness 17

is identically zero in D, and the displacement can therefore only be a rigid-

body motion (i.e. a uniform translation and rotation; see Exercise 1.6). Since

u is zero on D, we deduce that it must be zero everywhere and, hence, that

u(1) u(2) .

Now we attempt the same calculation when the surface traction, ratherthan the displacement, is specified:

n = (x) on D. (1.10.4)

Like the Neumann problem for a scalar elliptic partial differential equation

(Ockendon et al., 2003, p. 154), the Navier equation only admits solutions

satisfying (1.10.4) if so-called solvability conditions are satisfied. If we inte-

grate (1.10.2) over D and use the divergence theorem, we find thatD

ij nj da +

D

gi dx = 0 (1.10.5)

and hence that D

da +

D

g dx = 0. (1.10.6)

This represents a net balance between the forces, namely surface traction

and gravity, acting on D. An analogous balance between the moments actingon D may also be obtained by taking the cross product of x with (1.10.2)

before integrating, to giveD

x da +

D

xg dx = 0, (1.10.7)

as shown in Exercise 1.9. As well as representing physical balances on the

system, (1.10.6) and (1.10.7) may be interpreted as instances of the Fredholm

Alternative (see Ockendon et al., 2003, p. 43).Now suppose the solvability conditions (1.10.6) and (1.10.7) are satisfied

and that two solutions u(1) and u(2) of (1.10.2) and the boundary condition

(1.10.4) exist. As before, the difference u = u(1) u(2) satisfies the homo-

geneous version of the problem, with g and set to zero. By an argument

analogous to that presented above, we deduce that the strain tensor eij must

be identically zero. However, since u is now not specified on D, we can only

infer from this that the displacement is a rigid-body motion, as shown in

Exercise 1.6. Thus the solution of (1.10.2) subject to the applied traction(1.10.4) is determined only up to the addition of an arbitrary translation

and rotation.

As well as the boundary conditions (1.10.1) and (1.10.4), there are gener-

alisations in which the traction is specified on some parts of the boundary


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18 Modelling solids

solid 1

solid 2

n

Fig. 1.3 A small pill-box-shaped region at the boundary between two elastic solids.

and the displacement on others, for example in contact problems and in frac-

ture, as described in Chapter 7. Another common generalisation of (1.10.1)

and (1.10.4) occurs when two solids with different elastic moduli are bondedtogether across a common boundary D, as shown in Figure 1.3. Then the

displacement vectors are the same on either side of D and, by balancing

the stresses on the small pill-box-shaped region shown in Figure 1.3, we see

that

(1)n = (2)n, (1.10.8)

where (1) and (2) are the values of on either side of the boundary. Thus

there are six continuity conditions across such a boundary.

On the other hand, if two unbonded solids are in smooth contact, only

the normal displacement is continuous across D. However, this loss of in-

formation is compensated by the fact that the four tangential components

of (1)n and (2)n are zero and the normal components of these tractions

are continuous. Frictional contact between rough unbonded surfaces poses

serious modelling challenges, as we will see in Chapter 7.For elastodynamic problems, we may anticipate that (1.7.8) admits wave-

like solutions. It may, therefore, be viewed as a generalisation of a scalar

wave equation, such as the familiar equation

2w

t2= T

2w

x2(1.10.9)

which describes small transverse waves on a string with tension T and linedensity (see Section 4.3). We will examine elastic waves in more detail

in Chapter 3 but, in the meantime, we expect to prescribe Cauchy initial

conditions for u and u/t at t = 0, as well as elliptic boundary conditions

such as (1.10.1) or (1.10.4).


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1.11 Coordinate systems

In the next two chapters, we will construct some elementary solutions of the

Navier equation (1.7.8). In doing so, it is often useful to employ coordinate

systems particularly chosen to fit the geometry of the problem being consid-

ered. A detailed derivation of the Navier equation in an arbitrary orthogonal

coordinate system may be found in the Appendix. Here we state the main

results that will be useful in subsequent chapters for the three most popular

coordinate systems, namely Cartesian, cylindrical polar and spherical polar

coordinates.

All three of these coordinate systems are orthogonal; in other words the

tangent vectors obtained by varying each coordinate in turn are mutually

perpendicular. This means that the coordinate axes at any fixed point areorthogonal and may thus be obtained by a rotation of the usual Cartesian

axes. Under the assumptions of isotropic linear elasticity, the Cartesian stress

and strain components are related by (1.7.6), which is invariant under any

such rotation. Hence the constitutive relation (1.7.6) applies literally to any

orthogonal coordinate system.

1.11.1 Cartesian coordinates

First we write out in full the results derived thus far using the usual Carte-

sian coordinates (x,y,z). To avoid the use of suffices, we will denote the

displacement components by u = (u, v, w)T. It is also conventional to la-

bel the stress components by {xx, xy , . . .} rather than {11 , 12 , . . .}, and

similarly for the strain components. The linear constitutive relation (1.7.6)

gives

xx = ( + 2)exx + eyy + ezz , xy = 2exy ,

yy = exx + ( + 2)eyy + ezz , xz = 2exz ,

zz = exx + eyy + ( + 2)ezz , yz = 2eyz , (1.11.1)

where

exx =

u

x , 2exy =

u

y +

v

x ,

eyy =v

y, 2eyz =

v

z+

w

x,

ezz =w

z, 2exz =

u

z+

w

x, (1.11.2)


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20 Modelling solids

and the three components of Cauchys momentum equation are

2u

t2= gx +

xxx

+xy

y+

xzz

,

2

vt2

= gy + xyx

+ yyy

+ yzz

,

2w

t2= gz +

xzx

+yzy

+zzz

, (1.11.3)

where the body force is g = (gx, gy , gz )T. In terms of the displacements, the

Navier equation reads (assuming that and are constant)

2

ut2 = gx + ( + ) x ( u) +

2u,

2v

t2= gy + ( + )

y( u) + 2v,

2w

t2= gz + ( + )

z( u) + 2w. (1.11.4)

1.11.2 Cylindrical polar coordinatesWe define cylindrical polar coordinates (r,,z) in the usual way and de-

note the displacements in the r-, - and z-directions by ur, u and uz re-

spectively. The stress components are denoted by ij where now i and j

are equal to either r, or z and, as in Section 1.5, ij is defined to be the

i-component of stress on a surface element whose normal points in the j-

direction. As noted above, the constitutive relation (1.7.6) applies directly

to this coordinate system, so that

rr = ( + 2)err + e + ezz , r = 2er ,

= err + ( + 2)e + ezz , rz = 2erz ,

zz = err + e + ( + 2)ezz , z = 2ez , (1.11.5)

where the strain components are now given by

err =

ur

r , 2er =

1

r

ur

+

u

r

u

r ,

e =1

r

u

+ ur

, 2erz =

urz

+uzr

,

ezz =uzz

, 2ez =uz

+1

r

uz

. (1.11.6)


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The three components of Cauchys momentum equation (1.6.3) read

2urt2

= gr +1

r

r(rrr ) +

1

r

r

+rzz

r

,

2

ut2 = g + 1r r (r

r ) + 1r

+ z

z + rr ,

2uzt2

= gz +1

r

r(rrz ) +

1

r

z

+zzz

, (1.11.7)

where the body force is g = grer + ge + gzez . Written out in terms of

displacements, these become

2ur

t2

= gr + ( + )

r

( u) + 2ur urr

2

2

r2

u

,

2ut2

= g +( + )

r

( u) +

2u

ur2

+2

r2ur

,

2uzt2

= gz + ( + )

z( u) + 2uz , (1.11.8)

where

u =

1

r

r (rur) +

1

r

u

+

uz

z ,

2ui =1

r

r

r

uir

+

1

r22ui2

+2uiz2

(1.11.9)

are the divergence ofu and the Laplacian of ui respectively, expressed in

cylindrical polars.

Detailed derivations of (1.11.6) and (1.11.7) are given in the Appendix.

Notice the undifferentiated terms proportional to 1/r which are not present

in the corresponding Cartesian expressions (1.11.2) and (1.11.3). The originof these terms may be understood in two dimensions (r, ) by considering

the equilibrium of a small polar element as illustrated in Figure 1.4, in which

= (r + r,) = + r

r+ ,

= (r, + ) = +

+ , (1.11.10)

when we expand using Taylors theorem. Summing the resultant forces inthe r- and -directions to zero results in

rr (r + r) rr r r sin + r r cos r r = 0,

r cos r + r r sin + r (r + r) r r = 0. (1.11.11)


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22 Modelling solids

r

rrr

e

r

rr r

r r

er

Fig. 1.4 Forces acting on a polar element of solid.

Now letting ,r 0 and using (1.11.10), we obtain

rrr +

1

r

r +

rr

r = 0,

rr +1

r

+

2rr = 0, (1.11.12)

which are the components of the two-dimensional steady Navier equation in

plane polar coordinates with no body force; cf (1.11.7). The stress component

is the so-called hoop stress in the -direction that results from inflating

an elastic object radially; we will see an explicit example of hoop stress in

Section 2.6.

1.11.3 Spherical polar coordinates

The spherical polar coordinates (r,,) are defined in the usual way, such

that the position vector of any point is given by

r(r,,) =

r sin cos r sin sin

r cos

. (1.11.13)

Again, we can apply the constitutive relation (1.7.6) literally, to obtainrr = ( + 2)err + e + e , r = 2er ,

= err + ( + 2)e + e , r = 2er ,

= err + e + ( + 2)e , = 2e . (1.11.14)


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The linearised strain components are now given by

err =urr

, 2er =1

r

ur

+ur

ur

,

e = 1r

u

+ ur

, 2er = 1r sin

ur

+ ur

ur

,

e =1

r sin

u

+urr

+u cot

r, 2e =

1

r sin

u

+1

r

u

u cot

r.

(1.11.15)

Cauchys equation of motion leads to the three equations

2

urt2

= gr + 1r2

(r2

rr )r

+ 1r sin

(sin r )

+1

r sin

r

+

r,

2ut2

= g +1

r2(r2r )

r+

1

r sin

(sin )

+1

r sin

+r cot

r,

2ut2

= g +1

r2(r2r )

r+

1

r sin

(sin )

+1

r sin

+r + cot

r, (1.11.16)

where the body force is g = grer +ge +ge. Again, (1.11.15) and (1.11.16)

may be derived using the general approach given in the Appendix or more

directly by analysing a small polar element. In terms of displacements, the

Navier equation reads

2urt2

= gr + ( + )

r( u)

+

2ur

2urr2

2

r2 sin

(u sin )

2

r2 sin

u

,

2ut2

= g +( + )

r

( u)

+

2u +

2

r2ur

u

r2 sin2

2cos

r2 sin2

u

,


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24 Modelling solids

2ut2

= g +( + )

r sin

( u)

+

2u +

2

r2 sin

ur

+2cos

r2 sin2

u

u

r2 sin2

, (1.11.17)

where

u =1

r2

r

r2ur

+

1

r sin

(sin u) +

1

r sin

u

,

2ui =1

r2

r

r2

uir

+

1

r2 sin

sin

ui

+

1

r2 sin2

2ui2

.

(1.11.18)

Exercises

1.1 A light spring of natural length L and spring constant k hangs freely

with a mass m attached to one end and the other end fixed. Show

that the length of the spring satisfies the differential equation

md2

dt2+ k( L) mg = 0.

Deduce that

1

2m

d

dt

2+

1

2k( L)2 + mg(L ) = const.

and interpret this result in terms of energy.

1.2 A string, stretched to a tension Talong the x-axis, undergoes small

transverse displacements such that its position at time t is given

by the graph z = w(x, t). Given that w satisfies the wave equation

(1.10.9), where is the mass per unit length of the string, show that,if x = a and x = b are any two points along the string,

d

dt

1

2

ba

w

t

2 dx +

ba

1

2

w

x

2Tdx

=

T

w

x

w

t

ba

.

Interpret this result in terms of conservation of energy.

1.3 A system of masses m along the x-axis at positions Xn = nL

(n = 0, 1, 2, . . .) are linked by springs satisfying Hookes law (1.2.1),as shown in Figure 1.5. If each mass is displaced by a distance un(t),

show that the tension Tn joining Xn to Xn+1 satisfies

Tn Tn1 = md2undt2

, where Tn = k (un+1 un).


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Exercises 25

m mm

Tn1 Tn1 Tn Tn

unun1 un+1

Fig. 1.5 A system of masses connected by springs along the x-axis.

Deduce that

md2undt2

= k (un+1 2un + un1)

and show that this is a spatial discretisation of the partial differential

equation

2ut2

= c2 2u

X2,

where X = nL, un(t) = u(X, t) and c2 = kL2/m.

[It is not so easy to use discrete element models in more than

one dimension; for example, it can be shown that it is impossible to

retrieve the Navier equation as the limit of a lattice of masses joined

by springs aligned along three orthogonal axes.]

1.4 If x, x, u and u are related by (1.4.9), use the chain rule to showthat

uixj

= pipjux

.

Hence establish equation (1.4.11) to show that Eij transforms as a

tensor under a rotation of the coordinate axes.

[Hint: note that, since P is orthogonal, pklpkm lm .]

1.5 Consider a volume V(t) that is fixed in a deforming solid body. Showthat conservation of angular momentum for V(t) leads to the equa-

tion

d

dt

V(t)

xu

t dx =

V(t)

xg dx +

V(t)

x(n) da.

From this and Cauchys momentum equation (1.6.3), deduce that

ij ji.

1.6 Show that the linearised strain eij , given by (1.7.1), is identically zeroif and only ifu = c + x, where c and are spatially-uniform

vectors. Show that this approximates the rigid-body motion (1.4.13)

when P is close to I. Ifu is of this form, and known to be zero at

three non-collinear points, deduce that c = = 0.


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26 Modelling solids

1.7 By writing the linearised strain tensor eij as the sum of a zero-

trace contribution eij (1/3)ij ekk and a purely diagonal contribu-

tion (1/3)ij ekk , show that W can be rewritten as

W=

2 +

3

(ekk )2

+

eij 1

3 ij ekk

eij 1

3 ij ekk

.

Deduce that, for W to have a single global minimum at eij = 0, it is

sufficient for and +2/3 to be positive. By considering particular

values of eij , show that it is also necessary.

1.8 Suppose a solid body occupies the region D and the displacement u

is prescribed on D. Let

U =

D W g udx,

where W is the strain energy density. Show that, if ui is changed by

a small virtual displacement i, then the corresponding leading-order

change in U is

U =

D

ixj

ij gii

dx.

[Hint: use the fact thatij is symmetric.] Use the divergence theoremto show that

U =

D

iij nj da

D

ijxj

+ gi

i dx,

and deduce that the minimisation of U with respect to all displace-

ments satisfying the given boundary condition leads to the steady

Navier equation. Deduce also that, if no boundary condition is im-

posed, the natural boundary condition is the vanishing of the tractionij nj on D.

Show also that the minimisation of U subject to the constraint

divu 0 leads to the steady incompressibleNavier equation (1.8.3a),

where p is a Lagrange multiplier.

1.9 An elastic body D at rest is subject to a traction n = (x) on its

boundary D. By taking the cross product ofx with (1.10.2) before

integrating over D, derive (1.10.7) and deduce that the net moment

acting on D must be zero.1.10 Suppose that u satisfies the steady Navier equation (1.10.2) in a

region D and the mixed boundary condition (x)u+(x)n = f(x)

on D. Show that, if a solution exists, it is unique provided > 0

and 0.


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Exercises 27

[If and take different signs, then there is no guarantee of

uniqueness. This is analogous to the difficulty associated with the

Robin boundary condition for scalar elliptic partial differential equa-

tions (Ockendon et al., 2003, p. 154).]

1.11 (a) Show that, in plane polar coordinates (r, ), the basis vectorssatisfy

derd

= e ,ded

= er .

(b) Consider a small line segment joining two particles whose po-

lar coordinates are (r, ) and (r + r, + ). Show that the

vector joining the two particles is given to leading order by

X rer + re .

(c) If a two-dimensional displacement field is imposed, with

u = ur (r, )er () + u(r, )e(), show that the line element

X is displaced to

x = X+

urr

r +ur

u

er

+

ur

r + u

+ ur e .

(d) Hence show that

|x|2 = |X|2 +(r,r)

err erer e

r

r

where, to leading order in the displacements,

err = urr , 2er = 1r u

r + u

r u

r , e

= 1r

u + ur

.

[These are the elements of the linearised strain tensor in plane

polar coordinates, as in (1.11.6).]


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2

Linear elastostatics

2.1 Introduction

This chapter concerns steady state problems in linear elasticity. This topic

may appear to be the simplest in the whole of solid mechanics, but we will

find that it offers many interesting mathematical challenges. Moreover, the

material presented in this chapter will provide crucial underpinning to the

more general theories of later chapters.

We will begin by listing some very simple explicit solutions which give

valuable intuition concerning the role of the elastic moduli introduced in

Chapter 1. Our first application of practical importance is elastic torsion,

which concerns the twisting of an elastic bar. This leads to a class of exact

solutions of the Navier equation in terms of solutions of Laplaces equation

in two dimensions. However, as distinct from the use of Laplaces equation

in, say, hydrodynamics or electromagnetism, the dependent variable is the

displacement, which has a direct physical interpretation, rather than a po-

tential, which does not. This means we have to be especially careful to ensurethat the solution is single-valued in situations involving multiply-connected

bars.

These remarks remain important when we move on to another class of

two-dimensional problems called plane strain problems. These have even

more general practical relevance but involve the biharmonic equation. This

equation, which will be seen to be ubiquitous in linear elastostatics, poses

significant extra difficulties as compared to Laplaces equation. In particular,

we will find that it is much more difficult to construct explicit solutions using,for example, the method of separation of variables.

An interesting technique to emerge from both these classes of problems is

the use ofstress potentials, which are the elastic analogues of electrostatic or

gravitational potentials, say, or the stream function in hydrodynamics. We

28


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will find that a large class of elastostatic problems with some symmetry, for

example two-dimensional or axisymmetric, can be described using a single

scalar potential that satisfies the biharmonic equation.

Fully three-dimensional problems are mostly too difficult to be suitable for

this chapter. Nonetheless, we will be able to provide a conceptual frameworkwithin which to represent the solution by generalising the idea of Greens

functions for scalar ordinary and partial differential equations. The necessary

Greens tensor describes the response of an elastic body to a localised point

force applied at some arbitrary position in the body. This idea opens up one

of the most distinctive and fascinating aspects of linear elasticity: because

of the intricacy of (1.7.8), many different kinds of singular solutions can

be constructed using stress functions and Greens tensors, each being the

response to a different kind of localised forcing, and the catalogue of thesedifferent responses is a very helpful toolkit for thinking about solid mechanics

more generally.

2.2 Linear displacements

We will begin by neglecting the body force g so the steady Navier equation

reduces to

= ( + )graddivu + 2u = 0. (2.2.1)This vector partial differential equation for u is the starting point for all

we will say in this chapter. As discussed in Section 1.10, it needs to be

supplemented with suitable boundary conditions, which will vary depending

on the situation being modelled.

If the displacement u is a linear function of position x, then the strain

tensor Eis spatially uniform. It follows that the stress tensor is also uniformand, therefore, trivially satisfies (2.2.1). Such solutions provide considerableinsight into the predictions of (2.2.1) and also give a feel for the significance

of the parameters and . To avoid suffices, we will write u = (u, v, w)T

and x = (x,y,z)T.

2.2.1 Isotropic expansion

As a first example, suppose

u =

3x, (2.2.2)

where is a constant scalar, which must be small for linear elasticity to be

valid. When > 0, this corresponds to a uniform isotropic expansion of the


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30 Linear elastostatics

(a)(c)(b)

1 + /3

1 + /3

1 + /3

1

11y

x

z

1

1

1 +

Fig. 2.1 A unit cube undergoing (a) uniform expansion (2.2.2), (b) one-dimensionalshear (2.2.6), (c) uniaxial stretching (2.2.8).

medium so that, as illustrated in Figure 2.1(a), a unit cube is transformedto a cube with sides of length 1 + /3. (Of course, if is negative, the

displacement is an isotropic contraction.) Since is small, the relative change

in volume is thus 1 +

3

3 1 . (2.2.3)The strain and stress tensors corresponding to this displacement field are

given by

eij =

3ij and ij =

+

2

3

ij . (2.2.4)

This is a so-called hydrostatic situation, in which the stress is characterised

by a scalar isotropic pressure p, and ij = pij . The pressure is related tothe relative volume change by p = K, where

K = +2

3 (2.2.5)

measures the resistance to expansion or compression and is called the bulk

modulus or modulus of compression; from Exercise 1.7, we know that K is

positive.


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2.2.2 Simple shear

As our next example, suppose

u = u

vw = y

00 , (2.2.6)where is again a constant scalar. This corresponds to a simple shear of

the solid in the x-direction, as illustrated in Figure 2.1(b). The strain and

stress tensors are now given by

E= 2

0 1 0

1 0 0

0 0 0

, =

0 1 0

1 0 0

0 0 0

. (2.2.7)

Note that does not affect the stress, so the response to shear is accounted

for entirely by , which is therefore called the shear modulus.

2.2.3 Uniaxial stretching

Our next example is uniaxial stretching in which, as shown in Figure 2.1(c),

the solid is stretched by a factor in (say) the x-direction. We suppose,for reasons that will emerge shortly, that the solid simultaneously shrinks

by a factor in the other two directions. The corresponding displacement,

strain and stress are

u =

xyz

, E= 1 0 00 0

0 0

, (2.2.8)

=

(1 2) + 2 0 00 (1 2) 2 00 0 (1 2) 2

. (2.2.9)This simple solution may be used to describe a uniform elastic bar that is

stretched in the x-direction under a tensile force T, as shown in Figure 2.2.

Notice that, since the bar is assumed not to vary in the x-direction, the

outward normal n to the lateral boundary always lies in the (y, z)-plane.

If the curved surface of the bar is stress-free, then the resulting boundarycondition n = 0 may be satisfied identically by ensuring that yy = zz = 0,

which occurs if

=

2 ( + ). (2.2.10)


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n = 0

z

y

x AT T

Fig. 2.2 A uniform bar being stretched under a tensile force T.

Hence the bar, while stretching by a factor in the x-direction, must shrink

by a factor in the two transverse directions; if happened to be neg-ative, this would correspond to an expansion. The ratio between lateral

contraction and longitudinal extension is called Poissons ratio.

With given by (2.2.10), the stress tensor has just one non-zero element,

namely

xx = E, (2.2.11)

where

E = (3 + 2) +

(2.2.12)

is called Youngs modulus. If the cross-section of the bar has area A, then

the tensile force T applied to the bar is related to the stress by

T = Axx = AE; (2.2.13)

thus AE is the elastic modulus k referred to in (1.2.2). By measuring T, the

corresponding extensional strain and transverse contraction , one may

thus infer the values of E and for a particular solid from a bar-stretching

experiment like that illustrated in Figure 2.2. The Lame constants may then

be evaluated using

=E

(1 + )(1 2) , =E

2(1 + ). (2.2.14)

We note that the constitutive relation (1.7.7) can be written in terms of E

and as

Eeij = (1 + )ij kk ij . (2.2.15)While E is a positive constant with the dimensions of pressure, is dimen-

sionless and constrained on physical grounds to lie in the range

1 < < 1/2. The lower bound for comes from the condition +2/3 > 0


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(a)

(b)

Fig. 2.3 A paper model with negative Poissons ratio. Each line segment is a stripof paper viewed end-on.

required for convexity of the strain energy, as shown in Exercise 1.7. The up-

per bound 1/2 is approached as , in other words as the materialbecomes incompressible, as discussed in Section 1.8.

For most solids, is positive, but it is possible (see Figure 2.3) to construct

simple hinged paper models with negative Poissons ratio; try extending acrumpled piece of paper! So-called auxetic materials, in which < 0, have

been developed whose microscopic structure mimics such paper models so

they also display negative values of and they expand in all directions when

pulled in only one (see Lakes, 1987).

2.2.4 Biaxial strain

As a final illustrative example, we generalise Section 2.2.3 and consider an

elastic plate strained in the (x, y)-plane as illustrated in Figure 2.4. Suppose

the plate experiences a linear in-plane distortion while shrinking by a factor

in the z-direction, so the displacement is given by

u = uvw = ax + bycx + dyz , (2.2.16)

and, as in Section 2.2.3, the stress and strain tensors are both constant. Here

we choose to satisfy the condition zz = 0 on the traction-free upper and


51/468


y

x

z

h

Tyx

Txx

Txx

Tyy

TxyTyx

Tyy

Txy

Fig. 2.4 A plate being strained under tensions Txx , Ty y and shear forces Txy , Ty x .

lower surfaces of the plate, so that

=

+ 2

(a + d) =

1

(a + d), (2.2.17)

where again denotes Poissons ratio. With this choice, and with E again

denoting the Youngs modulus, the only non-zero stress components are

xx =

E(a + d)

1 2 , xy =E(b + c)

2(1 + ) , yy =

E(a + d)

1 2 . (2.2.18)

We denote the net in-plane tensions and shear stresses applied to the plate

by Tij = hij , as illustrated in Figure 2.4. We can use (2.2.18) to relate these

to the in-plane strain components by

Txx =Eh

1 2 (exx + eyy ), (2.2.19a)

Txy = Tyx = Eh1 +

exy , (2.2.19b)

Tyy =Eh

1 2 (exx + eyy ). (2.2.19c)

These will provide useful evidence when constructing more general models

for the deformation of plates in Chapter 4.

If no force is applied in the y-direction, that is Txy = Tyy = 0, then (2.2.19)

reproduces the results of uniaxial stretching, with d =

a and Txx = Eha.

On the other hand, it is possible for the displacement to be purely in the

(x, z)-plane, with

b = c = d = 0, yy =Ea

1 2 , xx =Ea

1 2 . (2.2.20)


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Thus a transverse stress yy must be applied to prevent the plate from

contracting in the y-direction when we stretch it in the x-direction. Notice

also that the effective elastic modulus E/(1 2) is larger than E whenever is non-zero, which shows that purely two-dimensional stretching is always

more strenuous than uniaxial stretching.

2.2.5 General linear displacements

The simple linear examples considered above shed useful light on more gen-

eral solutions of the Navier equation. Indeed, any displacement field, when

expanded in a Taylor series about some point x0, is linear to leading order:

u(x) = u(x0) + u (x0)T(x x0) + . (2.2.21)Here u is the displacement gradient matrix, whose entries are conven-

tionally defined to be u = (uj /xi), and may be written as the sum of

symmetric and skew-symmetric parts:

ujxi

=1

2

uixj

+ujxi

+

1

2

ujxi

uixj

. (2.2.22)

We recognise the symmetric part ofu as the linear strain tensorE

, while

the skew-symmetric part may be written in the form

1

2

uuT =

0 3 23 0 12 1 0

, (2.2.23)so that, with = (1, 2, 3)

T, (2.2.21) becomes

u(x) = u0

+

(xx

0) +

E(x

x

0) +

. (2.2.24)

The terms on the right-hand side represent a small rigid-body translation

and rotation, followed by a linear deformation characterised by the matrix E.Since Eis real and symmetric, it has real eigenvalues, say {1, 2, 3}, and

orthogonal eigenvectors. These eigenvalues are referred to as the principal

strains, and the directions defined by the eigenvectors as the principal di-

rections. If we use an ortho

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