Applied Solid Mechanics () - Peter Howell

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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 increasengineer-ing 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 workundertak-ing 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 University

E. 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 L

University of Oxford

G R E G O R Y K O Z Y R E F F Fonds de la Recherche Scientifique—FNRS

and Universit´e Libre de Bruxelles J O H N O C K E N D O N

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CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press

The 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

2008

Information on this title: www.cambridge.org/9780521854894

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

Cambridge University Press has no responsibility for the persistence or accuracy of 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

paperback eBook (EBL) hardback

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Illustrations

1.1 A reference tetrahedron. page 8

1.2 The forces acting on a small two-dimensional element. 9

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

solids. 18

1.4 Forces acting on a polar element of solid. 22

1.5 A system of masses connected by springs. 25

2.1 A unit cube undergoing (a) uniform expansion, (b) one-dimensional

shear, (c) uniaxial stretching. 30

2.2 A uniform bar being stretched under a tensile force. 32

2.3 A paper model with negative Poisson’s ratio. 33

2.4 A strained plate. 34

2.5 A bar in a state of antiplane strain. 38

2.6 A twisted bar. 39

2.7 A uniform tubular torsion bar. 43

2.8 The cross-section of (a) a circular cylindrical tube; (b) a cut tube. 44 2.9 The unit normal and tangent to the boundary of a plane region. 49 2.10 A plane annulus being inﬂated by an internal pressure. 53 2.11 A plane rectangular region subject to tangential tractions on its faces. 57 2.12 The tractions applied to the edge of a semi-inﬁnite strip. 59 2.13 The surface displacement of a half-space and corresponding surface

pressure. 65

2.14 A family of functions δε(x) that approach a delta-function as ε→ 0. 83

2.15 Contours of the maximum shear stress created by a point force acting

at the origin. 85

2.16 Four point forces. 91

3.1 Plots of the ﬁrst three Bessel functions. 108

3.2 A P -wave reﬂecting from a rigid boundary. 116

3.3 A layered elastic medium. 117

3.4 Dispersion relation for symmetric and antisymmetric Love waves. 120

3.5 Illustration of ﬂexural waves. 128

3.6 The one-dimensional fundamental solution. 134

3.7 The two-dimensional fundamental solution. 135

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List of illustrations ix 3.8 The cone x2+ y2= c2t2 tangent to the plane k1x + k2y = ωt. 138

3.9 The two-sheeted characteristic cone for the Navier equation. 138 3.10 The response of a string to a point force moving at speed V . 139 3.11 Wave-fronts generated by a moving force on an elastic membrane. 141 3.12 P -wave- and S -wave-fronts generated by a point force moving at speed

V in plane strain. 142

3.13 Group velocity versus wave-number for symmetric and antisymmetric

Love waves. 146

4.1 The forces acting on a small length of a uniform bar. 151 4.2 The forces acting on a small length of an elastic string. 153 4.3 The forces and moments acting on a small segment of an elastic beam. 154 4.4 The end of a beam under clamped, simply supported and free conditions.155 4.5 The ﬁrst three buckling modes of a clamped elastic beam. 157 4.6 The internal force components in a thin elastic rod. 159 4.7 Cross-section through a rod showing the bending moment components. 159 4.8 Examples of cross-sections in the (y, z)-plane and their bending

stiﬀnesses. 161

4.9 The forces acting on a small section of an elastic plate. 163 4.10 The bending moments acting on a section of an elastic plate. 164 4.11 The displacement of a simply supported rectangular plate sagging

under gravity. 169

4.12 (a) A cylinder, (b) a cone, (c) another developable surface, (d) a

hyperboloid. 175

4.13 Typical surface shapes with (a) zero, (b) negative and (c) positive

Gauss 179

4.14 Deformations of a cylindrical shell. 184

4.15 Deformations of an anticlastic shell. 185

4.16 Deformations of a synclastic shell. 186

4.17 A beam (a) before and (b) after bending; (c) a close-up of the

displacement ﬁeld. 187

4.18 (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. 188 4.19 (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 a

buckling beam versus the force parameter. (b) Corresponding response

of the weakly nonlinear solution. 193

4.21 (a) Pitchfork bifurcation diagram of leading-order amplitude versus forcing parameter. (b) The corresponding diagram when asymmetry is

introduced. 195

4.22 A system of pendulums attached to a twisting rubber band. 199 4.23 A kink propagating along a series of pendulums attached to a rod. 200 4.24 Travelling wave solution of the nonlinear beam equations. 201

4.25 A beam clamped near the edge of a table. 206

4.26 A beam supported at two points. 207

ian curvature.

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5.1 The deformation of a small scalene cylinder. 218 5.2 Typical force–strain graphs for uniaxial tests on various materials. 233 5.3 A square membrane subject to an isotropic tensile force. 234 5.4 Response diagrams for a biaxially-loaded incompressible sheet of

Mooney–Rivlin material. 235

5.5 Scaled pressure inside a balloon as a function of the stretch for various

values of the Mooney–Rivlin parameter. 236

5.6 Gas pressure inside a cavity as a function of inﬂation coeﬃcient for

various values of the Mooney–Rivlin parameter. 238

6.1 The edge of a plate subject to tractions. 254

6.2 The geometry of a deformed two-dimensional plate. 269

7.1 Deﬁnition sketch of a thin crack. 288

7.2 Deﬁnition sketch for contact between two solids. 288

7.3 (a) A Mode III crack. (b) A cross-section in the (x, y)-plane. 290

7.4 Deﬁnition sketch for the function√z2_{− c}2_. ₂₉₂

7.5 Displacement ﬁeld for a Mode III crack. 293

7.6 (a) A planar Mode II crack. (b) The regularised problem of a thin

elliptical crack. 297

7.7 Contour plot of the maximum shear stress around a Mode II crack. 301 7.8 The displacement of a Mode II crack under increasing shear stress. 303

7.9 A Mode I crack. 304

7.10 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. 307 7.12 Solution for the contact between a string and a level surface. 310

7.13 Three candidate solutions for a contact problem. 311

7.14 The contact between a beam and a horizontal surface under a uniform

pressure. 314

7.15 Contact between a rigid body and an elastic half-space. 317 7.16 The penetration of a quadratic punch into an elastic half-space. 319

7.17 A ﬂexible ruler ﬂattened against a table. 326

7.18 A wave travelling along a rope on the ground. 326

8.1 A typical stress–strain relationship for a plastic material. 329 8.2 The stress–strain relationship for a perfectly plastic material. 330 8.3 The forces acting on a particle at the surface of a granular material. 331 8.4 The normal force and frictional force acting on a surface element inside

a granular material. 332

8.5 The Mohr circle. 333

8.6 The triaxial stress factor versus angle of friction. 336

8.7 An antiplane cut-and-weld operation. 339

8.8 The displacement ﬁeld in an edge dislocation. 340

8.9 An edge dislocation in a square crystal lattice. 341

8.10 A moving edge dislocation. 342

8.11 The normalised torque versus twist applied to an elastic-plastic

cylindrical bar. 347

8.12 The normalised torque versus twist applied to an elastic-plastic

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List of illustrations xi 8.13 The free-boundary problem for an elastic-perfectly plastic torsion bar. 349 8.14 Residual shear stress in a gun barrel versus radial distance for diﬀerent

values of the maximum internal pressurisation. 352

8.15 The Tresca yield surface. 355

8.16 The von Mises yield surface. 356

8.17 The Coulomb yield surface. 358

8.18 L¨uders bands in a thin sheet of metal. 369

8.19 The Mohr surface for three-dimensional granular ﬂow. 373 8.20 The normalised torque versus twist applied to an elastic-plastic

cylindrical bar undergoing a loading cycle. 375

9.1 (a) A spring; (b) a dashpot; (c) a spring and dashpot connected in parallel; (d) a spring and dashpot connected in series. 380 9.2 (a) Applied tension as a function of time. (b) Resultant displacement

of a linear elastic spring. (c) Resultant displacement of a linear dashpot.381 9.3 Displacement of a Voigt element due to the applied tension shown in

Figure 9.2(a). 382

9.4 Displacement of a Maxwell element due to the applied tension shown

in Figure 9.2(a). 383

9.5 (a) The variation of Young’s modulus with position in a bar. (b) The

corresponding longitudinal displacement. 392

9.6 (a) The variation of Young’s modulus with position in a bar. (b) The

corresponding longitudinal displacement. 395

9.7 A periodic microstructured shear modulus. 396

9.8 A symmetric, piecewise constant shear modulus distribution. 400 9.9 Some modulus distributions that are antisymmetric about the

diago-nals of a square. 402

9.10 Dimensionless wavenumber versus the Young’s modulus non-uniformity

parameter. 407

9.11 The one-dimensional squeezing of a sponge. 411

9.12 Dimensionless stress applied to a sponge versus dimensionless time for

diﬀerent values of the P´eclet number. 412

9.13 A Jeﬀreys viscoelastic element. 418

9.14 A system of masses connected by springs and dashpots in parallel. 418 9.15 A system of masses connected by springs and dashpots in series. 419 9.16 Dimensionless wavenumber versus Young’s modulus contrast for a

piecewise uniform bar. 424

A1.1 A small reference box. 432

A1.2 Cylindrical polar coordinates. 437

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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, both static 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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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 first reading 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 confront any 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, Hilary Ockendon 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 Hooke’s law, and use mathematical ideas to combine this evidence with the 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.

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By basing our theory on Hooke’s 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 system eventually 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 even more dramatic changes can take place as the load increases, the most strik-ing phenomenon bestrik-ing 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 Hooke’s law

Robert Hooke (1678) wrote

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

Hooke’s 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. Hooke’s 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)

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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 twice as 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 coordinates

Suppose 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 displacement u(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 particle’s initial position and its current position.

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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 ∂xi ∂Xj . (1.3.2)

The physical signiﬁcance of J is that it measures the change in a small volume compared with its initial volume:

dx1dx2dx3 = J dX1dX2dX3, or dx = J dX (1.3.3) as shorthand. The positivity of J 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 ρ(X, t), by M (t) = V (t) ρ dx = V (0) ρJ dX. (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) ρJ dX = M (t) = M (0) = V (0) ρ0dX, (1.3.5)

where ρ0 is the density in the rest state. Since V is arbitrary, we deduce that

ρJ = ρ0. (1.3.6)

Hence, we can calculate the density at any time t in terms of ρ0 and the displacement ﬁeld. The initial density ρ0 is usually taken as constant, but (1.3.6) also applies if ρ0 = ρ0(X).

1.4 Strain

To generalise the concept of strain introduced in Section 1.2, we consider the deformation of a small line segment joining two neighbouring particles with initial positions X and X + δX. At some later time, the solid deforms such that the particles are displaced to X + u(X, t) and X + δX + u(X + δX, t) respectively. Thus we can use Taylor’s theorem to show that the line element

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1.4 Strain 5

δX that joins the two particles is transformed to

δx = δX + u(X + δX, t)− u(X, t) = δX + (δX · ∇)u(X, t) + · · · , (1.4.1) where (δX· ∇) = δX1 ∂ ∂X1 + δX2 ∂ ∂X2 + δX3 ∂ ∂X3 . (1.4.2)

Let L =|δX| and = |δx| 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 =|δX + (δX · ∇)u(X, t)|2. (1.4.3) Although we will try in subsequent chapters to minimise the use of suﬃces, it is helpful at this stage to introduce components so that

X = (Xi) = (X1, X2, X3)Tand similarly for u. Then (1.4.3) may be written in the form 2− L2 = 2 3 i,j=1 EijδXiδXj, (1.4.4) where Eij = 1 2 ∂ui ∂Xj + ∂uj ∂Xi + 3 k=1 ∂uk ∂Xi ∂uk ∂Xj . (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+ 2EijδXiδXj, where Eij = 1 2 ∂ui ∂Xj + ∂uj ∂Xi + ∂uk ∂Xi ∂uk ∂Xj . (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 δXTE δX. (1.4.7)

It is clear from (1.4.4) that the stretch is measured by the quantitiesEij; in particular, the stretch is zero for all line elements if and only ifEij ≡ 0. It is thus natural to identifyEij 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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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 = P X, u = P u. (1.4.8)

Since P is orthogonal, (1.4.8) may be inverted to give X = PTX. Alterna-tively, using suﬃx notation, we have

Xβ = pjβXj, ui = piαuα. (1.4.9) The strain in the new coordinate system is denoted by

Eij = 1 2 ∂u_i ∂X_j + ∂u_j ∂X_i + ∂u_k ∂X_i ∂u_k ∂X_j , (1.4.10)

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

Eij = piαpjβEαβ. (1.4.11) In matrix notation, (1.4.11) takes the form

E _{= P}_EPT_, _(1.4.12)

so the 3× 3 symmetric array (Eij) transforms exactly like a matrix repre-senting a linear transformation of the vector spaceR3. 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 it can 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 matrix Q 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, and conﬁrms 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 to generalise (1.4.11) to a tensor with any number of indices. A vector, for example, is a tensor with 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 inﬁnitesimal surface 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 Newton’s 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 = σ da, (1.5.1)

where σ 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

σ =−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 of n. 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 τ1= (τ11, τ21, τ31)T. By doing the same for elements with normals in the x2- and x3-directions, we generate three vectors τj (j = 1, 2, 3), each representing the stress acting on an element normal to the xj-direction. In total, therefore, we obtain nine scalars τij (i, j = 1, 2, 3), where τij is the i-component of τj, that is

τj = τijei, (1.5.3)

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

The scalars τij may be used to determine the stress on an arbitrary surface element by considering the tetrahedron shown in Figure 1.1. Here ai denotes the area of the face orthogonal to the xi-axis. The fourth face has area a = a2₁+ a2₂+ a2₃; 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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a

2

a

3

x

1

x

3

x

2

n

a

1

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

The outward normal to the face with area a1is in the negative x1-direction and the force on this face is thus −a1τ1. Similar expressions hold for the faces with areas a2 and a3. Hence, if the stress on the fourth face is denoted by σ, then the total force on the tetrahedron is

f = aσ− ajτj. (1.5.4) When we substitute for aj and τj, we ﬁnd that the components of f are given by

fi= a (σi− τijnj) . (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 Newton’s 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 dimensional variable tend to zero: if is reduced indeﬁnitely then we will eventually reach an atomic scale on 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 are large compared to any atomistic length-scale.

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1.5 Stress 9 G τ11 τ11 τ22 τ21 τ22 τ12 τ12 τ21 x2 x1 δx2 δx1

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

expression for σ:

σi= τijnj, or σ = τ n. (1.5.6) This important result enables us to ﬁnd the stress on any surface element in terms of the nine quantities (τij) = τ .

Now let us follow Section 1.4 and examine what happens to τij when we rotate the axes by an orthogonal matrix P . In the new frame, (1.5.6) will become

σ = τn (1.5.7)

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

σ = P σ, n = P n. (1.5.8)

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

τ = P τ PT, or τ_ij = piαpjβταβ. (1.5.9) Thus τij, likeEij, is a second-rank tensor, called the Cauchy stress tensor.

We can make one further observation about τij 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 (τ21δx2) δx1

2 − 2 (τ12δx1) δx2

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where τ21 and τ12are 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 τ12 = τ21. This argument can be generalised to three dimensions (see Exercise 1.5) and it shows that

τij ≡ τji (1.5.10)

for all i and j, i.e. that τij, likeEij, is a symmetric tensor.

1.6 Conservation of momentum

Now we derive the basic governing equation of solid mechanics by apply-ing Newton’s second law to a material volume V (t) that moves with the deforming solid: d dt V (t) ∂ui ∂t ρ dx = V (t) giρ dx + ∂V (t) τijnjda. (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 ρ dx = ρ0dX is independent of t) and apply the divergence theorem to the ﬁnal term to obtain

V (t) ∂2ui ∂t2 ρ dx = V (t) giρ dx + V (t) ∂τij ∂xj dx. (1.6.2)

Assuming each integrand is continuous, and using the fact that V (t) is ar-bitrary, we arrive at Cauchy’s momentum equation:

ρ∂ 2_u i ∂t2 = ρgi+ ∂τij ∂xj . (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 ∇ · τ to be

(∇ · τ)_i= ∂τji ∂xj

. (1.6.4)

Since τ is symmetric, we may thus write Cauchy’s equation as ρ∂

2_u

∂t2 = ρg +∇ · τ. (1.6.5)

This equation applies to any continuous medium for which a displacement

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1.7 Linear elasticity 11

or some other continuum comes when we impose an empirical constitutive relation between τ and u.

For solids, (1.6.5) already confronts us with a distinctive fundamental difficulty. The most obvious generalisation of Hooke’s law is to suppose that a linear relationship exists between the stress τ and the strain E. But we now 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 X fixed. On the other hand, the stress tensor τ 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 elasticity in 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 ρ is ﬁxed, to leading order, at its initial value ρ0. 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 ∂ui ∂xj +∂uj ∂xi . (1.7.1)

Much of this book will be concerned with this approximation. Therefore, and with a slight abuse of notation, we will writeE = (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 close to 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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but rather for displacements of the form

u = c + ω× x, (1.7.2)

where c and ω are constant (see Exercise 1.6).

Assuming the validity of (1.7.1), we can now generalise Hooke’s law by postulating a linear relationship between the stress and strain tensors. We assume that τ 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

τij = Cijkek. (1.7.3)

The symmetry of τij 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

Cijkpiipjjpkkp ≡ Cijk (1.7.4) for all orthogonal matrices P = (pij). It can be shown (see, for example, Ockendon & Ockendon, 1995, pp. 7–9) that this is suﬃcient to reduce the speciﬁcation of Cijk to just two scalar quantities λ and µ, such that

Cijk = λδijδk+ 2µδikδj, (1.7.5) where δij is the usual Kronecker delta, which represents the identity matrix; consequently,

τij = λ (ekk) δij+ 2µeij. (1.7.6) This relation can also be inverted to give the strain corresponding to a given stress, that is eij = 1 2µ τij − λ(τkk) (3λ + 2µ)δij . (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 λ and µ are known as the Lam´e constants, and µ is called the shear modulus.As we shall see in Chapter 2, λ and µ measure a material’s 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 λ (GPa) µ (GPa) Cartilage 3× 10−5 9× 10−5 Rubber 0.04 0.003 Polystyrene 2.3 1.2 Granite 10 30 Glass 28 28 Copper 86 37 Steel 100 78 Diamond 270 400

Table 1.1 Typical values of the Lam´e constants λ and µ for some everyday materials (1 GPa = 109N m−2= 104atmospheres; 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 Lam´e equation,

ρ∂ 2_u

∂t2 = ρg + (λ + µ) grad div u + µ∇

2_u. _(1.7.8)

Recall that ρ does not vary to leading order, so (1.7.8) comprises three equations for the three components of u. It may alternatively be written in component form ρ∂ 2_u i ∂t2 = ρgi+ (λ + µ) ∂2uj ∂xi∂xj + µ∂ 2_u i ∂x2 j , (1.7.9)

where the ﬁnal ∂x2_j is treated as a repeated suﬃx, or

ρ∂ 2_u

∂t2 = ρg + (λ + 2µ) grad div u− µ curl curl u, (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 approximation

There 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 Lam´e constant λ is much larger than the shear modulus µ.

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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 λ

µ = 1

ε, (1.8.1)

where ε is a small parameter. From (1.7.8), we expect that, in the limit ε→ 0, div u will be of order ε. Hence, if we deﬁne a scalar function p such that

pε =− µ

ε div u, (1.8.2)

then pε will approach a ﬁnite limit p as ε→ 0.

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

ρ∂ 2_u

∂t2 = ρg− ∇p + µ∇

2_u, _(1.8.3a)

along with the limit of (1.8.2), that is

div u = 0. (1.8.3b)

The condition (1.8.3b) means that each material volume is conserved during the 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 ρ∂ 2_u i ∂t2 ∂ui ∂t dx = V ρgi ∂ui ∂t dx + V ∂τij ∂xj ∂ui ∂t dx. (1.9.1) The ﬁnal term may be rearranged, using the divergence theorem, to

V ∂τij ∂xj ∂ui ∂t dx = ∂V ∂ui ∂t τijnjda− V τij ∂eij ∂t 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 2ρ ∂u_∂t 2 dx + V W dx = V ρgi ∂ui ∂t dx + ∂V ∂ui ∂t τijnjda, (1.9.3) where W is a scalar function of the strain components that is chosen to satisfy

∂W ∂eij

= τij. (1.9.4)

With τij given by (1.7.6), we can integrate (1.9.4) to determineW up to an arbitrary constant as W = 1 2τijeij = 1 2λ (ekk) 2 + µ (eijeij). (1.9.5) Here the summation convention is invoked such that (ekk)2 is the square of the trace of E, while (eijeij) 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 the terms 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 µ, λ > 0, we can easily see from (1.9.5) thatW 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 µ, (λ + 2µ/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 =

D

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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-posedness

Suppose 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 of u and ∂u/∂n to be prescribed on ∂D. In many physical problems, we specify either the displacement u or the traction τ 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

∂τij ∂xj

+ ρgi = 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 uiτijnjda = D eijτijdx = 2 D W dx, (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, rather than the displacement, is speciﬁed:

τ n = σ(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 that

∂D τijnjda + D ρgidx = 0 (1.10.5)

and hence that

∂D

σ da +

D

ρg 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 acting on D may also be obtained by taking the cross product of x with (1.10.2) before integrating, to give

∂D x×σ da + D ρx×g 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 σ 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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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 bonded together 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

τ(1)n = τ(2)n, (1.10.8)

where τ(1) and τ(2) are the values of τ 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 τ(1)_{n and τ}(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

∂ 2_w ∂t2 = T

∂2w

∂x2 (1.10.9)

which describes small transverse waves on a string with tension T and line density (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 19

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 are orthogonal 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 {τxx, τxy, . . .} rather than {τ11, τ12, . . .}, and similarly for the strain components. The linear constitutive relation (1.7.6) gives

τxx = (λ + 2µ)exx+ λeyy+ λezz, τxy = 2µexy, τyy = λexx+ (λ + 2µ)eyy+ λezz, τxz = 2µexz,

τzz = λexx+ λeyy+ (λ + 2µ)ezz, τyz = 2µeyz, (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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and the three components of Cauchy’s momentum equation are ρ∂ 2_u ∂t2 = ρgx+ ∂τxx ∂x + ∂τxy ∂y + ∂τxz ∂z , ρ∂ 2_v ∂t2 = ρgy+ ∂τxy ∂x + ∂τyy ∂y + ∂τyz ∂z , ρ∂ 2_w ∂t2 = ρgz+ ∂τxz ∂x + ∂τyz ∂y + ∂τzz ∂z , (1.11.3)

where the body force is g = (gx, gy, gz)T. In terms of the displacements, the Navier equation reads (assuming that λ and µ are constant)

ρ∂ 2_u ∂t2 = ρgx+ (λ + µ) ∂ ∂x(∇ · u) + µ∇ 2_u, ρ∂ 2_v ∂t2 = ρgy+ (λ + µ) ∂ ∂y(∇ · u) + µ∇ 2_v, ρ∂ 2_w ∂t2 = ρgz+ (λ + µ) ∂ ∂z (∇ · u) + µ∇ 2_w. _(1.11.4)

1.11.2 Cylindrical polar coordinates

We deﬁne cylindrical polar coordinates (r, θ, z) in the usual way and de-note the displacements in the r-, θ- and z-directions by ur, uθ and uz re-spectively. The stress components are denoted by τij where now i and j are equal to either r, θ or z and, as in Section 1.5, τij 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

τrr = (λ + 2µ)err+ λeθθ+ λezz, τrθ = 2µerθ, τθθ = λerr+ (λ + 2µ)eθθ+ λezz, τrz = 2µerz,

τzz = λerr+ λeθθ+ (λ + 2µ)ezz, τθz = 2µeθz, (1.11.5) where the strain components are now given by

err = ∂ur ∂r , 2erθ= 1 r ∂ur ∂θ + ∂uθ ∂r − uθ r , eθθ = 1 r ∂uθ ∂θ + ur , 2erz = ∂ur ∂z + ∂uz ∂r , ezz = ∂uz ∂z , 2eθz = ∂uθ ∂z + 1 r ∂uz ∂θ . (1.11.6)

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The three components of Cauchy’s momentum equation (1.6.3) read

ρ∂ 2_u r ∂t2 = ρgr+ 1 r ∂ ∂r(rτrr) + 1 r ∂τrθ ∂θ + ∂τrz ∂z − τθθ r , ρ∂ 2_u θ ∂t2 = ρgθ+ 1 r ∂ ∂r(rτrθ) + 1 r ∂τθθ ∂θ + ∂τθz ∂z + τrθ r , ρ∂ 2_u z ∂t2 = ρgz+ 1 r ∂ ∂r(rτrz) + 1 r ∂τθz ∂θ + ∂τzz ∂z , (1.11.7)

where the body force is g = grer + gθeθ + gzez. Written out in terms of displacements, these become

ρ∂ 2_u r ∂t2 = ρgr+ (λ + µ) ∂ ∂r(∇ · u) + µ ∇2_u r− ur r2 − 2 r2 ∂uθ ∂θ , ρ∂ 2_u θ ∂t2 = ρgθ+ (λ + µ) r ∂ ∂θ(∇ · u) + µ ∇2_u θ− uθ r2 + 2 r2 ∂ur ∂θ , ρ∂ 2_u z ∂t2 = ρgz+ (λ + µ) ∂ ∂z(∇ · u) + µ∇ 2_u z, (1.11.8) where ∇ · u = 1 r ∂ ∂r(rur) + 1 r ∂uθ ∂θ + ∂uz ∂z , ∇2_u i = 1 r ∂ ∂r r∂ui ∂r + 1 r2 ∂2ui ∂θ2 + ∂2ui ∂z2 (1.11.9)

are the divergence of u 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 origin of these terms may be understood in two dimensions (r, θ) by considering the equilibrium of a small polar element as illustrated in Figure 1.4, in which

˜ ταβ = ταβ(r + δr, θ) = ταβ+ δr ∂ταβ ∂r +· · · , ˆ ταβ = ταβ(r, θ + δθ) = ταβ + δθ ∂ταβ ∂θ +· · · , (1.11.10) when we expand using Taylor’s theorem. Summing the resultant forces in the r- and θ-directions to zero results in

˜

τrr(r + δr) δθ− τrrrδθ− ˆτθθδr sin δθ + ˆτrθδr cos δθ− τrθδr = 0, ˆ

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τθθ ˜ τrθ δθ ˆ τrθ ˜ τrr eθ ˆ τθθ δr τrr τrθ r τrθ er

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

Now letting δθ, δr→ 0 and using (1.11.10), we obtain ∂τrr ∂r + 1 r ∂τrθ ∂θ + τrr − τθθ r = 0, ∂τrθ ∂r + 1 r ∂τθθ ∂θ + 2τrθ r = 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 τθθ is the so-called hoop stress in the θ-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, θ, φ) are deﬁned in the usual way, such that the position vector of any point is given by

r(r, θ, φ) =



r sin θ cos φr sin θ sin φ r cos θ



 . (1.11.13)

Again, we can apply the constitutive relation (1.7.6) literally, to obtain τrr = (λ + 2µ)err + λeθθ+ λeφφ, τrθ = 2µerθ,

τθθ = λerr + (λ + 2µ)eθθ+ λeφφ, τrφ = 2µerφ,

(40)

The linearised strain components are now given by

err = ∂ur ∂r , 2erθ = 1 r ∂ur ∂θ + ∂uθ ∂r − uθ r , eθθ = 1 r ∂uθ ∂θ + ur , 2erφ = 1 r sin θ ∂ur ∂φ + ∂uφ ∂r − uφ r , eφφ= 1 r sin θ ∂uφ ∂φ + ur r + uθcot θ r , 2eθφ = 1 r sin θ ∂uθ ∂φ + 1 r ∂uφ ∂θ − uφcot θ r . (1.11.15) Cauchy’s equation of motion leads to the three equations

ρ∂ 2_u r ∂t2 = ρgr + 1 r2 ∂(r2τrr) ∂r + 1 r sin θ ∂(sin θτrθ) ∂θ + 1 r sin θ ∂τrφ ∂φ − τθθ+ τφφ r , ρ∂ 2_u θ ∂t2 = ρgθ+ 1 r2 ∂(r2τrθ) ∂r + 1 r sin θ ∂(sin θτθθ) ∂θ + 1 r sin θ ∂τθφ ∂φ + τrθ− cot θτφφ r , ρ∂ 2_u φ ∂t2 = ρgφ+ 1 r2 ∂(r2τrφ) ∂r + 1 r sin θ ∂(sin θτθφ) ∂θ + 1 r sin θ ∂τφφ ∂φ + τrφ+ cot θτθφ r , (1.11.16)

where the body force is g = grer+gθeθ+gφeφ. 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

ρ∂ 2_u r ∂t2 = ρgr + (λ + µ) ∂ ∂r(∇ · u) + µ ∇2_u r − 2ur r2 − 2 r2_{sin θ} ∂ ∂θ(uθsin θ)− 2 r2_{sin θ} ∂uφ ∂φ , ρ∂ 2_u θ ∂t2 = ρgθ+ (λ + µ) r ∂ ∂θ(∇ · u) + µ ∇2_u θ+ 2 r2 ∂ur ∂θ − uθ r2_sin2_θ − 2 cos θ r2_sin2_θ ∂uφ ∂φ ,

Applied Solid Mechanics () - Peter Howell

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

Contents

Illustrations

Prologue

1

Modelling solids

a

a

x

x

x

n

a