9780198506256

OUP CORRECTED PROOF FINALS, 26/5/2010, SPi

1

Introduction

This book covers fundamental aspects of the mechanics of solids and, more specically,of thin elastic solids. Its style is voluntarily informal: the methods of analysis are quitediverse and could have been put into a rigorous formalism only at the expense of theclarity of exposition. Keeping in mind that this branch of science aims at describing realworld phenomena, we have put emphasis on the derivation of explicit solutions in specicgeometries. Our hope is that a detailed analysis of these special cases will help the readerto develop a real intuition of the mechanical phenomena without being hindered by theformalism. Therefore, this book does not claim to be a comprehensive treatise on elasticityof rods, plates and shells. Instead, it focuses on some recent developments to which theauthors have contributed. The problems that are investigated have been selected in partbecause they allow various important mathematical techniques to be introduced. Thesetechniques, which include for instance boundary layer and bifurcation theories, have a rangeof applications that extends well beyond the theory of elasticity.

This book was designed to be as self-sucient as possible and no prior knowledge ofelasticity theory is required. It is intended for third year undergraduate and graduatestudents, for students in engineering schools, as well as for researchers from mechanicalengineering and from other elds such as applied mathematics and numerical analysis,aerospace engineering and biophysics. A basic understanding of Euclidean geometry and ofcalculus with real variables is assumed. The reader does not need to be familiar with other,less classical tools: the basics of calculus of variation are given in a short Appendix A; asimple but illustrative example of a boundary layer is treated in Appendix B; the geometry ofplanar and space curves is summarized in Section 1.3 below, together with other geometricalnotions; the geometry of surfaces is introduced when needed, in Part III of this book.Complex analysis is not required either.

When writing this book, both authors were employees of the French national institute forscientic research (CNRS), whose support is gratefully acknowledged. Many of the ideas andresults presented here are the result of discussions and collaborations with colleagues whoare credited in the relevant chapters. We shall express here our gratitude to Etienne Guyon,John Hutchinson, Roberto Toro, Sebastien Neukirch and Florence Bertails-Descoubes fortheir remarks and corrections on the manuscript.

1.1 Outline

Chapter 2 derives the basic equations of elasticity, and is a necessary starting point for thereader not yet familiar with continuum mechanics and with linear elasticity. What followsis organized into three mostly independent parts, each one being concerned with a family


2 Introduction

of thin elastic bodies, namely elastic rods, elastic plates and elastic shells. In each of theseparts, the rst chapter derives the governing equations (Kirchho equations for elastic rods,membrane and plate equations, etc.) and should be read rst, while the remaining chaptersfocus on specic examples and can be skipped or read in any particular order.

We shall now outline in more details the structure of this book. Chapter 2 recalls theprinciples of the so-called Hookean elasticity in three dimensional solids, as formulated ina fully explicit form by Cauchy and Poisson. It serves as a starting point to build theoriesfor one or two-dimensional elastic bodies in the following parts.

Chapter 3 is the rst chapter of Part I, and contains a derivation of the equations forstatic elastic rods. The kinematical notions of bending and twist are rst introduced. Theequations of equilibrium are then derived, leading to the so-called Kirchho equations, whichdescribe three-dimensional deformations of a rod. In Chapter 4 we look at an applicationof this theory for representing human hair. The mechanics of a single hair strand can bereduced, under reasonable assumptions, to a system with two dimensionless parameters,describing the eects of gravity and of permanent curling. In Chapter 5 we present anotherapplication of the theory of elastic rods. This is related to the spontaneous formation ofripples along the edge of a thin sheet of material when it is stretched. In a rst attemptto understand these ripples, we analyse a simple rod model that describes what happensat the edge of such a sheet. This question will be reconsidered later on in Chapter 10 in amore geometrical spirit, using the full equations for plates.

Part II is devoted to the mechanics of elastic plates. We call a plate any piece of elasticmaterial that has the shape of sheet (it has a small thickness in only one direction of space)and is planar in the absence of stress. In Chapter 6, we derive the equations of equilibrium forplates, the so-called Fopplvon Karman equations (F.von K. equations as a shorthand).These equations have an intricate mathematical structurethe source of in-plane stressis the Gauss curvature, a quadratic function of the out-of-plane deviation. A classicalapplication of the F.von K. equations is the analysis of the buckling of plates undertransverse loading. We shall present various situations where this buckling phenomenoncan be analysed, in connection with experiments. In Chapter 7, we consider buckling nearthreshold at length. Buckling at a nite distance from the threshold in loading spaceis studied in Chapter 8. Unlike mathematically related problems of instabilities in uidmechanics, it makes sense to look at the buckling instability far above threshold. Foldsand d-cones are a rst type of solutions of the F.von K. equations that can emergeat large loads, and are the building blocks for the polydedral shape of crumpled paper.They are studied in Chapter 9. In Chapter 10, we derive a dierent kind of solution,relevant to the buckling of an elastic plate under large applied edge loads, which isself-similar.

Part III is devoted to the elasticity of thin shells. By denition a shell is a piece of thinelastic material that is curved at rest. The main contribution to the elastic energy of a shellcomes from stretching eects. As a result, the mechanical behaviour of shells is related toa non-trivial question of geometry, namely whether a given surface can or cannot deformwithout stretching, that is without changing the lengths of curves drawn along it. Thisgeometrical problem is considered in some detail in Chapter 11. Geometry is only one pieceof the puzzle. A sphere, for instance, is found to be geometrically rigid but this does not tellhow a spherical shell deforms when subjected to stress. Thanks to the existence of a smallparameter, the ratio of thickness to radius of curvature, a consistent theory describes thedeformations of shells at the dominant order. The so-called membrane theory is derived in


Notations and conventions 3

Chapter 12 for shells of revolution. We also show how it can be extended to take bendingeects into accountthe membrane equations in the general case, that is without symmetry,are given in Appendix 12 for the sake of completeness.

The symmetry of revolution applies for the examples we consider next, the elastic torusand the elastic ball. The elastic torus is the simplest shell geometry that is partly elliptic(on its outer part) and partly hyperbolic (on its inner part): its analysis, carried out inChapter 13, provides insights into the mechanics of shells of mixed kind (i.e. shells thatare neither fully hyperbolic nor fully elliptic), a class of shells for which very few generalresults are available. As we shall see, the mechanical stiness of a torus is very dierentfrom that of more classical shells, an eect that we shall ultimately explain with the help ofdierential geometry. The calculation of the elastic response of a torus requires an elaborateboundary layer analysis. In the last chapter, Chapter 14, we focus on another simple shellgeometry, namely a spherical elastic ball, and consider a question with an almost everydayrelevance: how does a ball, such as a tennis or ping pong ball, atten when hit, and howlong does it take for it to bounce? This turns out to be a rather subtle question, becausethere are two regimes: at small forces, the sphere attens at the contact with the pushingplane, while at larger forces it forms an inverted cap and contacts the pushing plane along acircular ridge.

1.2 Notations and conventions

We have tried to keep mathematical notations as simple and consistent as possible. Probablymost of them are familiar to the reader.

1.2.1 Vectors

Real numbers, vectors and vector spacesThe set of real numbers is denoted R. A real x R is a positive, null or negative number, < x < + ( means belongs to but she shall rarely use this and others Bourbakistnotations).

A vector a (which may be seen equivalently as a point here) in the three-dimensionalEuclidean space is given by its coordinates (ax, ay, az) in an orthonormal basis (ex, ey, ez).Using a particular basis, vectors can be identied with their coordinates, that is a collectionof three real numbers (ax, ay, az). Therefore, the Euclidean three-dimensional space, whichis the set of all vectors, is denoted R3. Vectors are denoted by boldface symbols such a R3.

We shall also use the notation (a1, a2, a3), or simply (ai) (i = 1, 2, 3) for vectorcoordinates:

ax = a1, ay = a2, az = a3.

The notation rst introduced, (ax, ay, az), is more readable and is preferred for explicitcalculations. The second notation (ai) allows implicit summation (see below), which yieldscompact notations.

Similarly, the orthonormal basis whose vectors dene the x, y and z axes will be denoted(ex, ey, ez) or (e1, e2, e3).

The current point in physical space is denoted r and its coordinates are written (x, y, z),that is x = rx = r1, y = ry = r2, z = . We introduce yet another notation for the current


4 Introduction

point r = (x, y, z) in the form (x1, x2, x3), or simply (xi). This will be useful for implicitsummation:

x1 = x, x2 = y, x3 = z.

The presence of an index after the symbol x changes its meaning: xi is a generic coordinatewhile x stands for the rst coordinate. This does not introduce any ambiguity.

Operations on vectorsThe scalar product of two vectors a and a is denoted a a:

a b = ax bx + ay by + az bz. (1.1)The condition that the basis (ei)i=1,2,3 is orthonormal takes the form:

ei ej = ij for i, j = 1, 2, 3, (1.2)where we have introduced Kroneckers symbol ij with integer indices, whose value is 1 ifindices are equal and 0 otherwise.

When the vectors onto which the scalar product operates are themselves given by longexpressions, we use an alternative notation for the scalar product which is easier to read,namely a|b. The two notations are equivalent when applied to vectors:

a|b = a b. (1.3)The notation with angular brackets is used in a more general context, for instance when wedene the scalar product f |g of two functions f and g.

The norm of a vector is denoted like the absolute value of real numbers:

|a| = a a. (1.4)The cross product of two vectors a and b is a new vector c written as:

c = a b (1.5)Let ai, bi and ci be the coordinates of these three vectors (i = 1, 2, 3). The cross productoperation is dened by:

c1 = a2 b3 a3 b2, c2 = a3 b1 a1 b3, c3 = a1 b2 a2 b1. (1.6)A double cross product can be expanded using the following identity, valid for any vectors

a, b and c:

a (b c) = (a c)b (a b) c. (1.7)Note that on the right-hand side, every factor in parenthesis is given by a scalar productoperation, yielding a real number, which is multiplied by a third vector. This iden-tity can be checked by an explicit calculation, using the denition above for the crossproduct. It can also be established without calculation: this identity coincides with thedenition of the cross product when b = ey and c = ez. The equality for all b and cfollows by bilinearity and antisymmetry of both sides of the equation with respect to band c.



Another useful identity for the cross product is the Jacobi identity:1

a (b c) + b (c a) + c (a b) = 0, (1.8)valid for any vectors a, b and c. Again, this identity can be established by an explicit buttedious calculation, or simply by noting that it is true for a = ex, b = ey and c = ez, andby generalizing for all a, b and c by bilinearity and antisymmetry of both sides with respectto all pairs of vectors.

Finally, the mixed product of three vectors a, b, c is dened as the scalar (a b) c. By anexplicit calculation in coordinates, this scalar can be shown to be equal to the determinantof the matrix with the coordinates of a, b and c presented in rows in some orthonormalbasis (ex, ey, ez):

(a b) c = det

ax bx cxay by cyaz bz cz

. (1.9)

As a result, the mixed product is invariant by direct permutation of the vectors:

(a b) c = (b c) a = (c a) b. (1.10)These identities can also be shown geometrically, as the mixed product can be interpretedas the signed volume of the parallelepipedic 3D domain with edges a, b and c, positive ifthe vectors are in direct orientation and negative otherwise.

1.2.2 Tensors

Up to Part III (which is devoted to the elasticity of shells, i.e. to surfaces with an arbitraryshape at rest), we shall use tensors in a very naive manner: we make use only of orthonormalframes (making it unnecessary to distinguish covariant and contravariant indices) andintroduce tensors merely as a convenient notation (Einstein summation) for linear algebra(matrix multiplication, etc.). A more intrinsic approach is followed in Part III and furtherdetails are given in due course.

The symbol i has already been used above in the expression of a generic vector coordinatexi. More generally, we shall use Latin letters for indices of coordinates in the three-dimensional space. For instance, (xi) means (x1, x2, x3) unless a specic value of i wasproposed in the context.

In contrast, Greek indices only run over the rst two dimensions: (a) stands for the two-dimensional vector (a1, a2). This convention will be useful to distinguish between in-planevectors and arbitrary vectors in plates and shells elasticity.

The range of indices can therefore be determined unambiguously from their read-ing. This allows us to introduce the Einstein summation convention. According to thisconvention:

A sum sign is implicit for any index repeated twice in the same side of an equal sign. If an index appears once on either side of an equal sign (dangling index), the equalityis true for all values of the indexunless a particular value of this index has beenexplicitly chosen.

1 This equality denes what is called a Lie algebra when a, b and c denote innitesimal rotations.


6 Introduction

Similarly when there is no equal sign, indices that appear only once (and whose valueis unspecied) are to be understood as generic indices.

For instance, ai (with no repeated indices) collectively denotes the coordinates (a1, a2, a3) =(ax, ay, az) of a vector a. In contrast, the notation ai bi has a repeated index: a sum isimplicit, making this expression a shorthand for

3i=1

ai bi,

which is the scalar product a b.Another example is ai = bij cj , which stands for:

a1 =3

j=1

b1j cj , a2 =3

j=1

b2j cj , a3 =3

j=1

b3j cj .

Note that the index i appears once on either side of the equal sign, and so the equality musthold for all values of i: it is in fact an equality between vectors with index i. In contrast, jis repeated (in the right-hand side) and so must be summed. The above notation ai = bij cjmeans that the vector a is obtained by multiplying the matrix B with entries (bij) by thevector c.

The harmonic operator f acting on a scalar function f(x, y, z) is dened as

f =2f

x2+

2f

y2+

2f

z2. (1.11)

Using Einstein summation, this can be put into the condensed form:

f =2f

xi2 .

Here, the index i appears to be repeated when the denominator xi2 = xi xi is expanded,and so a sum is implicit. Because Greek indices scan only the rst two directions, theoperator 2f/x2 stands for the two-dimensional harmonic operator, namely 2f/x2 +2f/y2.

Both repeated and non-repeated indices are dumb: the actual symbol used is unimportantand can be changed as long as consistency is preserved. For instance, ai = bij cj can berewritten ap = bpq cq. This is because the original equality is valid for all i and involves animplicit sum over j in the right-hand side.

The following operation, called the double contraction,2 generalizes the dot product withtwo tensors with two indices:

aij bij .

Starting from two tensors with two indices, the result of the double contraction is a scalar.The density of elastic energy, for instance, involves the double contraction of the straintensor by the stress tensor, as we shall see. The following property will be useful later on:

2 More accurately, this is the double contraction of a tensor with the transpose of the other, as the doublecontraction is usually dened as a : b = aij bji.



the double contraction of a symmetric tensor by a skew-symmetric tensor is zero:

if aij = aji and bij = bji, then akl bkl = 0. (1.12)

The proof comes by noticing that terms in the double contraction with k = l are zero sinceany diagonal component bkl is zero for an skew-symmetric tensor, while for k = l, the termsin the implicit sum with indices (k, l) and with the same indices permuted cancel eachother since alk blk = akl (bkl) = (akl bkl) (no implicit summation here). It follows fromthis result that if aij is a symmetric tensor and bij and cij are two tensors with the samesymmetric part, that is (bij + bji)/2 = (cij + cji/2), then bij and cij yield the same resultby double contraction with aij : aij bij = aij cij . Indeed, when bij and cij have the samesymmetric part, their dierence is3 a skew-symmetric tensor, and equation (1.12) applies.

1.2.3 Mechanical quantities

The current point in the reference conguration is written r = (x, y, z). Upon deformation,the associated material point moves to r = (x, y, z) (the point in the actual conguration).

The displacement eld will be denoted as u. Its components (ux, uy, uz) are also denoted(u, v, w) and, by denition, are equal to (x x), (y y) and (z z). In our analysisof plates, we take (x, y) as the direction parallel to the centre plane in the referenceconguration and assume small displacements (this is the framework of Fopplvon Karmanequations). As a result, the tangent plane to the plate remains everywhere close to the(x, y) plane upon deformation, and (u, v) are the in-plane components of the displacement.The transverse component, also called the deection, is w. The coordinates along the plate,(x, y), are also written (x), with = 1, 2.

The strain tensor is written ij and the stress tensor ij . Volumic mass is denoted and external forces per unit mass are written g. Linear elastic materials are consideredthroughout this book: Youngs modulus is denoted by E and Poissons ratio by ; Lamecoecients are written according to the standard notation and .

In our analysis of elastic rods, the centre line has curvilinear coordinate s. The materialframe is denoted (d1,d2,d3) where d1 and d2 are along the principal direction of inertiawithin the cross-section and d3 is tangent to the centre line. Curvatures and twist aredenoted and . Flexural and torsional stinesses are denoted EI and J respectively.The internal force and moment are denoted N and M respectively. Other notations areintroduced in Chapter 3.

The geometry of surfaces is presented in Part III of this book. The principal curvaturesare denoted 1 and 2 (there could be some ambiguity with the two curvatures of a Cosseratrod but we deal with rods and shells in dierent parts), the principal radii of curvatures areR1 and R2, the mean curvature is H, and the Gaussian curvature is K.

The thickness of a plate or a shell is written as h, a length that is by assumptionmuch smaller than any other typical length scale in the problem. The bending stiness isdenoted D.

Elastic energies are denoted as E... with various indices, such as Eel (total elastic energy),Eb (bending energy), Es (stretching energy), etc.

3 This can be shown by noticing that any tensor aij with two indices can be explicitly decomposed as thesum of a symmetric and a skew-symmetric tensor, which read (aij + aji)/2 and (aij aji)/2 respectively.


8 Introduction

1.2.4 Dimensional analysis, scaled variables

Let us consider a point-like mass m that is attached to a linear spring with an additionalapplied force F (t). Its equation of motion is:

k x(t) + mx(t) = F (t),

where k is the spring stiness, t the time, and the dots denote time derivation. We denotetypical values with an asterisk: F for the force, t for the time, etc. These quantities canbe chosen arbitrarily (also some choices are more helpful than others, see below). Thesetypical quantities are used to introduce rescaled, dimensionless quantities, marked with abar, such as

t =t

t, F =

F

F .

A similar denition is used for other quantities.By convention, rescaled quantities are implicitly considered to be functions of rescaled

parameters and so x is a function of t = t/t, unlike x, which is a function of t. As a result,derivatives acting on rescaled quantities are taken with respect to rescaled variable. Since

x(t) =x(t t)

x

we get that the derivative of a rescaled quantity rescales dierently from the non-derivedquantity:

x(t) =dxdt

=t

xdxdt

=t

xx(t) =

x(t)x/t

.

This shows that the natural scale for x(t) is x/t. A rule of thumb is that if a scales likea and b scales like b, the derivative da/db scales like a/b. The only trick is not to forgetscaling factors, such as 1/t in the previous example, associated with derivatives markedwith a prime or dot shorthand.

As written above, scaling factors such as t and F can be chosen arbitrarily andindependently of each other, but the trick is to choose them in such a way that the rescaledproblem has as few dimensionless coecients as possible in the end. In our example ofthe forced linear spring, the typical intensity of the force F provides a natural typicaldisplacement in the form

x =F

k.

A typical time t can be proposed by balancing the scales associated with the rst twoterms in the equation of motion: k x = mx/(t)2. By this equation, we are in fact tryingto impose the condition that the coecients in front of the rst two terms become identicalin the rescaled problem. This yields a natural choice for the time scale

t =

m

k.

Up to a factor 2 this is the period of the natural oscillations of the springmass system.Using these rescalings, the dimensionless problem becomes:

x(t) + x(t) = F (t).


Mathematical background 9

The obvious advantage of the rescaled formulation is that it depends on fewer parameters.Here, the mass and the spring stiness have been eectively set to one. The idea is to getrid of as many parameters as possible before actually solving equations. In several placesin this book, the dimensional analysis absorbs all the parameters of the problem, makingit possible to derive a universal solution by solving once and for all an equation with xednumerical constants. Another important asset of introducing dimensionless equations is thatit allows one to directly infer the dominant physical eect out of many possible eects:4 thesubdominant eects have a small coecient in their mathematical representation once therescaling has been made, although the coecients of the leading terms remain of order one.

1.2.5 Miscellaneous notations

To allow compact notation, commas in indices denote partial derivatives. For instance,

f,x stands forf

x.

The gradient of a scalar eld f(x, y, z) can then be written as the vector with coordinates(f,xi), or simply (f,i). Using implicit summation over the index i, the harmonic operatorintroduced earlier can be written as f = f,x2 + f,y2 + f,z2 = f,xi2 = f,ii. Implicit summa-tion over index i applies here.

In the Fopplvon Karman theory of elastic plates, we use the dierential operator [f, g]acting on two smooth scalar elds f(x, y) and g(x, y) dened along the plane (x, y). Thisoperator is a combination of products of second derivatives, and is known as the MongeAmpe`re operator:

[f, g] =2f

x22g

y2+

2g

x22f

y2 2

2f

xy

2g

xy. (1.13)

In other places, we shall have to perform an expansion of various functions in powers ofa small parameter. Let f() be such a function of , small. The Taylor expansion of f(.) iswritten as

f() = f [0] + f [1] + . . . .

Finally we shall use the shorthand ODE for Ordinary Dierential Equations, PDE forPartial Dierential Equations, l.h.s. and r.h.s. for the left-hand and right-hand side (ofequations), F.von K. as a shorthand for Fopplvon Karman, b.c. for boundary conditions,as well as the classical 2D and 3D for two-dimensional and three-dimensional.

1.3 Mathematical background

1.3.1 Rigid-body rotations, innitesimal rotations

Let a be a three-dimensional vector, a = |a| its norm, and b be an another arbitraryvector. When a = 0, let u = a/a be the unit vector parallel to a, and b and b be the

4 See for instance our analyses of the hair in Chapter 4, and the regularization of the torus solution bynon-linear versus bending eects based on the parameter in Chapter 13.


10 Introduction

x

n

t

y

r

s

C

Fig. 1.1 Geometry of a planar curve. Curvature is associated with rotation (s) of the frame(t,n) when one follows the curve.

longitudinal and perpendicular components of b with respect to the axis spanned by u,respectively:

b = (u b)u, b = b b.When a = 0, we set by convention b = b and b = 0.

Let us call rotation with vector a, denoted Ra, the rotation with angle a = |a| in radians,whose axis is spanned by u and passes through the origin of the coordinates (the rotationwith vector a = 0 is by denition the identity mapping). The eect of a rotation Ra on avector b is expressed by the following linear mapping:

Ra(b) = b + b cos a + (u b) sin a.Upon this rotation, the component b of b along the axis of rotation remains unchangedwhile its normal component rotates by an angle a in the plane perpendicular to u, spannedby b and (u b).

In the case of an innitesimal rotation represented by a vector a whose norm is small,a 1, the above formula becomes, at linear order with respect to a:

Ra(b) b + (u b) a + . . .since b + b = b. Therefore, the change of b due to an innitesimal rotation with vectora reads

b = Ra(b) b a b,using a = (a)u. Now, a b = 0 by construction and so an innitesimal rotation ischaracterized by

b a b. (1.14)Consider the case where b is the position vector of a material point on a solid undergoinga rigid-body rotation with rotation vector . Then, b = v t is the change of its positionduring a time interval t, and the angle of rotation reads a = t. The equation aboveyields the well-known form of the velocity eld v in a rigid body in rotation with angularvelocity , namely v = b.



1.3.2 Geometry of planar curves, curvature

We shall introduce here the geometry of curves and in particular the notion of curvature,which lies at the heart of the theory of rods. Let C be a smooth planar curve in the (x, y)plane, see Fig. 1.1. Let s be the curvilinear coordinate along this curve. We denote byr(s) its arc length parameterisation: r(s) is the current point along the curve in its actualconguration.5 Since s is an arc length parameterisation, the tangent vector to the curve

t(s) =drds

(1.15)

is unitary (its norm is one). Being a vector contained in the (x, y) plane, it can be writtenin the form:

t(s) = cos (s) ex + sin (s) ey. (1.16a)

Let us introduce the unitary normal vector to the curve (there is a unique normal vectorbecause the curve is drawn on a plane):

n(s) = sin (s) ex + cos (s) ey. (1.16b)This vector is normal to the tangent, hence to the curve, and is such that (t(s),n(s)) isoriented in the trigonometric direction.

Assuming that the curve C is C2-smooth (the parameterisation r(s) twice dierentiablewith continuous second derivatives), it is possible to choose a continuously dierentiabledetermination of the angle (s). By derivation of equations (1.16a) and (1.16b), we get:

t(s) = (s)n(s) and n(s) = (s) t(s), (1.17)where primes denote derivation with respect to s. Introducing the vector

(s) = (s) ez, (1.18)

where z is the direction perpendicular to the plane, equations (1.17) can be put in the form:

t(s) = (s) t(s) and n(s) = (s) n(s). (1.19)Now consider a solid body in rotation with instantaneous rotation vector (t). Let r be

the actual position in space of an arbitrary point of this solid, with the origin r = 0 lying onthe axis of rotation. The velocity eld due to this rotation is r = r. Comparison withequation (1.19) yields a simple interpretation of the vector : along the curve (that is whens varies) the frame (t(s),n(s)) rotates according to the rotation vector (s). As the frameis always contained in the (x, y) plane, the rotation vector has indeed to be perpendicularto this plane, that is along e3. The rotation velocity around the direction e3 is (s) fromequation (1.18)note that this is actually not a rotation velocity but instead a rate ofrotation per unit curvilinear length. This is exactly the common sense notion of curvature:a curve is highly curved when the orientation of the tangent changes rapidly when onefollows the curve. Therefore, we shall call curvature the quantity (s) = (s).

5 According to our general conventions, the current point along the curve in its actual congurationshould be denoted by r(s) and not by r(s). However, in the geometry of curves (and in the elasticityof rods), the current point in the reference conguration is directly parameterised by s and the referenceconguration is of little use. Moreover, the prime notation in r(s) could be misinterpreted as a derivationwith respect to s. Therefore, in the particular case of rods, r shall refer to the actual conguration.


12 Introduction

x

yz

t

n gk

b

Fig. 1.2 A three-dimensional curve and the associated SerretFrenet frame. Rotation of the

SerretFrenet frame, as described by the Darboux vector, involves a combination of a rotation

about the tangent (torsion) and the binormal (curvature).

In the case of a circle of radius R, the parametric equation (x = R cos, y = R sin)yields s = R (curvilinear coordinate) and (s) = + /2 = s/R + /2, hence a constantcurvature (s) = 1/R. Note that by following the circle in the opposite direction, we wouldhave obtained (s) = 1/R.

This suggests the following denitions, valid for an arbitrary curve. The rate of rotationof the frame (t,n) is called the signed curvature:

(s) = (s). (1.20)

Its inverse is called the signed radius of curvature:

R(s) =1

(s)=

1(s)

. (1.21)

In general, this quantity is not constant along the curve. It can be shown that this R(s) is alsothe radius of the circle that best approximates the curve near a particular point r(s). Thesigns of both the curvature and the radius of curvature depend on the chosen orientation inthe plane, as well as on the orientation of the curve set by the parameterisation s. Unsignedcurvature and radius of curvature are dened as their absolute values, and are geometricquantities (they do not depend on choices of orientation).

1.3.3 Geometry of curves in 3D space, SerretFrenet frame

The notion of curvature for a planar curve can be extended to the case of a space curveby means of the SerretFrenet frame. Let again C be a space curve given by its arc lengthparameterisation r(s), and t(s) = r(s) its tangent. We shall rst remark that:

t(s) t(s) = 12d (t(s) t(s))

ds=

12d

(|t(s)|2)ds

= 0, (1.22)

since t(s) is a unitary vector: |t(s)|2 = 1 for all s. Therefore, the derivative of any unitaryvector (in fact, of any vector of constant norm) with respect to some parameter yields avector that is perpendicular to it. Whenever t(s) = 0, let us introduce the positive quantityk(s) = |t(s)| and n(s) be the unitary vector parallel to it: n(s) = t(s)/|t(s)| = t(s)/k(s).This vector n is the normal vector. By construction, n(s) is perpendicular to t(s), and sothe new vector b(s) = t(s) n(s) denes an orthonormal frame (t(s),n(s),b(s)), called the



n

S

S

Fig. 1.3 Notations for the divergence theorem in two dimensions.

SerretFrenet frame. The vector b is the binormal vector . Let us assume further that n(s)is dierentiable at s and let us introduce g(s) = n(s) b(s). The functions k(s) and g(s)are called the curvature and the torsion respectively. Note that k(s) and g(s) are dierent,although related, to the curvatures and twist introduced in the context of a Cosserat curve,introduced in Chapter 3.

The derivatives of the vectors of the SerretFrenet frame can be expressed in terms ofthe curvature and torsion. Indeed, by denition of k(s), t(s) = k(s)n(s). By decompositionin the orthonormal SerretFrenet frame, n = (n t) t + (n b)b, using n n = 0 by thesame argument as in equation (1.22). Now, by a similar argument again, we nd:

n t = d(n t)ds

n t = n t = k, (1.23)

since n and t are perpendicular for all s. This yields n(s) = k(s) t(s) + g(s)b. Similarly,one can show that b(s) = g(s)n(s). Introducing the so-called Darboux vector :

(s) = g(s) t(s) + k(s)b(s), (1.24)

one can put all these equations into the compact form:

t(s) = (s) t(s), n(s) = (s) n(s), b(s) = (s) b(s). (1.25)By these relations, (s) can be interpreted as the local rate of rotation of the SerretFrenetframe.6 Note that, by construction, (s) n(s) = 0: the SerretFrenet does not rotate aboutn(s), as indicated by the cross in Fig. 1.2.

The SerretFrenet frame arises naturally when one deals with mathematical curves, butit is not well suited to the mechanics of rods. One of the problems associated with it is thatit is not always well dened, even when the curve is smooth: this happens for a straightline, for instance, for which t(s) vanishes everywhere and so n(s) is ill-dened. Even worse,the SerretFrenet might not be continuous even though the curve is C (case of a planarcurve with an inexion point), and k(s) and/or g(s) may not go to zero even though theunderlying curve converges uniformly to a straight line (the case of a helix with constantpitch and innitesimal radius).

In the case of a planar curve, the SerretFrenet frame is either (t2D,n2D, ez) if > 0or (t2D,n2D,ez) if < 0, where t2D, n2D refer to the quantities introduced in ouranalysis of planar curves above. Note that if = 0, the SerretFrenet frame is undened.By identication, one shows that k(s) = |(s)| and g(s) = 0 for a planar curve. In fact, the

6 The existence of the Darboux vector dened by equation (1.25) follows from the fact that the SerretFrenet frame, being orthonormal, denes a rigid-body rotation in the Euclidean space with parameter s.By identifying the parameter s with time, the Darboux vector is just the instantaneous rotation velocity ofthe rigid body attached to the frame.


14 Introduction

converse is true, and g(s) = 0 for all s implies that the curve is planar (provided the curveis smooth enough for the SerretFrenet frame to exist).

1.3.4 Divergence theorem

We recall here the divergence theorem, which will be used later to derive the equationsof mechanical equilibrium (2.51). Let V be a bounded region in three-dimensional space7

with boundary S = V . Let a(r) be a continuous, dierentiable vector eld dened over V .The divergence theorem relates the volume integral of the divergence of a over V with thesurface integral of a over S:

V

(div a)(r) d3r =

S

a(r) n(r) dA, (1.26)

where d3r = dxdy dz is the element of volume, n is the outward pointing normal to S denedalong S, dA is the element of area, and div a denotes the dierential operator acting on avector eld a:

div a = ai,i =aixi

=axx

+ayy

+azz

. (1.27)

The right-hand side of equation (1.26) is called the ux8 of vector a across surface S.Note that the divergence theorem is a multi-dimensional generalization of the fundamental

formula of calculus: qp

f (x) dx = f(q) f(p) (1.28)

in which p < q denes a real interval V = [p, q] with boundaries p and q. The outwardspointing normals at p and q are just +ex and ex respectively, making the divergencetheorem (1.26) an extension of equation (1.28). Equation (1.28) expresses the fact that theincrease of f from x to x + dx is given to rst order by f (x) dx. A very similar reasoningshows that the ux of a across the boundary of the innitesimal cube dened by (x, y, z) [x, x + dx] [y, y + dy] [z, z + dz] is equal to rst order to (div a) dxdy dz, hence the formof the divergence theorem (1.26).

The divergence theorem can be extended to the case of a tensorial eld bij(r). By xingthe rst index i in this tensor, we dene the vector eld a(i)(r) given by the correspondingrow in bij , such that (a(i))j = bij . Applying the divergence theorem to a(i) yields:

V

bij(r)xj

d3r =

S

bij nj dA. (1.29)

In this equation, the integrand in the left-hand side is the divergence of a tensor of rank two,this divergence being itself a vector (index i is dangling). In the right-hand side, (bij nj) isthe vector obtained by contracting the tensor bij with the unit outwards normal nj . Notethat consistency requires that the same index of bij (here, the second one, labelled j) is

7 The divergence theorem can be readily adapted to an arbitrary number of dimensions.8 As might be familiar to the reader, the divergence theorem can be used to write conservation laws

in continuum mechanics: for instance, when a = v is the Eulerian velocity of an incompressible uid, theright-hand side of (1.26), which is the rate of variation of the mass contained in V , has to vanish. Thisholds for any volume V , and so the integrand in the left-hand side has to vanish too, yielding the classicalcondition of incompressibility div v = 0.



used for dierentiation in the divergence of the left-hand side and for contraction with thenormal in the right-hand side.

1.3.5 Constrained minimization

In elasticity, equilibrium congurations of a body minimize the elastic energy. Consider thesimple case where the conguration of a body depends on only two discrete parameters,say X and Y . Let E(X,Y ) be the total energy to be minimized. It is well-known that anecessary condition for (X,Y ) to be a minimum is the stationarity condition E = 0, whereE is the gradient of E with respect to X and Y . In practice, this condition is enforced bycomputing the rst-order variation E for arbitrary changes of X and Y (denoted X andY ) and imposing that E vanishes for all X and Y . For instance, E(X,Y ) = X2 + Y 2 +XY yields the condition E = (2X + Y ) X + (2Y + X) Y = 0, hence 2X + Y = 0 and2Y + X = 0. This linear system has the unique solution (X,Y ) = (0, 0), which is indeed theminimum of E .

Let us now consider the more interesting case where the parameters X and Y areconnected by a constraint of the form g(X,Y ) = 0, and so cannot be varied independently.For instance, imagine (X,Y ) is a parameterisation of a surface with elevation z = E(X,Y )along which a point mass is constrained to move. We consider the case where the massis further constrained to move, say on the cylinder with equation g(X,Y ) = (X 1)2 +(Y + 1)2 1 = 0. The problem is now to nd the minimum of E(X,Y ) among allowedcongurations, such that g(X,Y ) = 0. For the particular function g chosen here, there isan obvious parameterisation of the allowed congurations (X,Y ), namely X() = 1 + cos and Y () = 1 + sin(). The stationarity condition takes the form:

0 =dE(X(), Y ())

d= X ()

EX

+ Y ()EY

= t E . (1.30)This equation has a very simple geometric interpretation: the gradient of E has to beorthogonal to the tangent t to the curve of allowed congurations. By denition, g is zeroalong this curve, and so equation (1.30) must also be satised when E is replaced by g. Thisshows that both E and g are perpendicular to the tangent t, and so are aligned: thereshould exist some scalar9 such that

(E + g) = 0.This form is more convenient than (1.30), as it can be used directly from the implicitequation g(X,Y ) = 0 and does not require an explicit parameterisation.

In practice, the stationarity condition of a constrained minimization problem is obtainedby adding10 to the variation of E a new term consisting of a yet undetermined scalarcoecient , multiplied by the function g expressing the constraint in an implicit form.

9 The form of the new function to be minimized, E + g, is reminiscent of the denition of thethermodynamic potentials. Indeed, a system that has a xed volume and cannot exchange heat has aninternal energy U(S, V ) that is minimum at equilibrium. When the condition of xed volume is replaced bya condition of xed pressure, it is the enthalpy, H(S, P ) = U + P V , that is minimum at equilibrium. Todeal with the constraint of a xed pressure, a new term, P V , has been incorporated in the potential to beminimized. This new term is minus the product of the constraint, P , times a Lagrange multiplier, (V ).

10 By moving along the curve with equation g = 0, the function g does not change and so g = 0. Thenew term added in the variation is therefore formally zero. Its only purpose is to introduce an additionalparameter that can be tuned, making it possible for the constraint to be satised.


16 Introduction

-1 0 1 2 3-3

-2

-1

0

1

x

y

A

B

Fig. 1.4 Example of constrained minimization: nding the extrema of E(X, Y ) = X2 + Y 2 + X Y(contours shown in background) along the circle with equation g(X, Y ) = 0 with g(X, Y ) =

(X 1)2 + (Y + 1)2 1.

The coecient , called the Lagrange multiplier, is determined by requiring that the pointobtained in this way does indeed satisfy the constraint g = 0. With our example functions,

(E + g) = (2X + Y + 2 (X 1)) X + (2Y + x + 2 (Y + 1)) Y .This leads to the equations 2X + Y + 2 (X 1) = 0 and 2Y + X + 2 (Y + 1). Thesolutions of these equations are

X() =2

1 + 2, Y () =

21 + 2

.

They all lie on the line X = Y , which is a bisector line of the coordinate axes. As explainedabove, is found by imposing that (X(), Y ()) satises the equation g(X(), Y ()) = 0.This yields = (12)/2. The two corresponding solutions A and B are shown inFig. 1.4: A is a minimum of E along the circle while B is a maximum.

The present method can be extended to an arbitrary number c of scalar constraintsg1(r) = 0, . . . , gc(r) = 0 and when r is a vector of dimension n. Then, one has to introducec Lagrange multipliers (1, . . . , c) and cancel the rst-order variation of:

r(E(r) + 1 g1(r) + + c gc(r)) = 0, (1.31)leading to solutions in the form r(1, . . . , c). As earlier, the values of the Lagrangemultipliers are found by solving c equations:

gi(r(1, . . . , c)) = 0 for 1 i c. (1.32)


2

Three-dimensional elasticity

2.1 Introduction

In this chapter, we consider solid materials with volumic extension and derive the funda-mental equations of the theory of elasticity in three dimensions. These equations will bespecialized in the following chapters to the case of thin elastic bodies such as rods andplates.

We shall be concerned with two very dierent but not unrelated approximations through-out this book, which we introduce straight away: the small strain and the small displacementapproximation. The small strain approximation, on the one hand, allows one to considerthat the elastic response of a given material is linear (this contrasts with the case of nite, orlarge strain, where solids deform non-linearly and may even break). The small displacementapproximation, on the other hand, is concerned with rotations of the material: volumeelements may undergo rigid-body rotations, which are irrelevant for computing the elasticdeformations. Keeping track of these rotations requires one to introduce nonlinearity intothe equations. When the material is only slightly perturbed from rest, all such rotationsremain innitesimal. This brings many simplications into the equations and denes whatis called the approximation of small displacement.

Therefore, it appears that there are two sources of nonlinearity in the theory of elasticity:one comes from the elastic constitutive law and the other from geometry. The rst one isassociated with the intrinsic response of a material, that is how a volume element deformsunder a given stress. It is well illustrated by stretching a piece of rubber: it becomes stierand stier and eventually breaks beyond a critical (and largesomething that is ratherspecic to rubber) stretching. The other source of nonlinearity will be discussed in detailsin this book, and is tightly connected with the word Geometry in its title. These twokinds of non-linear contributions are directly related to the two families of approximationsintroduced above. When the strain remains small, elastic constitutive laws can be linearized;when displacements are small, geometrical nonlinearity is negligible. If only the strain issmall, geometrical nonlinearity may still play a role.

Our approach in this book is to focus on geometrical nonlinearity and on importantassociated physical eects such as buckling. It turns out that geometrical nonlinearity isessential for predicting the elastic behaviour of thin bodies such as rods, plates and shells.Therefore, we do not limit our analysis to small displacements (we shall write that weconsider nite displacement). However, to avoid adding too many diculties to an alreadynon-trivial topic, we always assume small strain and consider linear constitutive laws; thisis the framework of Hookean elasticity. The equations of elasticity are given at the end ofthis chapter for small strain and nite displacementto avoid a too steep presentation,


18 Three-dimensional elasticity

we make an intermediate step and start by assuming small strain and small displacement,and generalize to nite displacement.

Classically the equations of elasticity in dimension three are introduced in two ways.The rst is to derive the equations of equilibrium by balancing the forces and torquesacting on a small element of the elastic material. This approach follows closely our physicalintuition of forces and deformations. However it is sometimes more convenient, especiallywhen considering thin elastic structures, to use a more abstract energy approach: insteadof balancing forces and torques, one derives the equilibrium equations for an elastic solid byimposing the condition that its conguration minimizes the elastic energy, a function of thedisplacements that is computed explicitly. In practice this energy approach is often simplerthan the force balance approach because the requirements coming from the symmetries orfrom constraints such as inextensibility (see the theory of rods in Chapter 3) can be takeninto account more easily. Another advantage is that it displays more clearly the variousphysical eects at play: it is easier to identify the bending and stretching energies thanthe corresponding contributions to forces and torques. This ultimately makes clearer theanalysis of the order of magnitude of these dierent physical eects. Such an analysis isused extensively throughout the following chapters. To illustrate that the minimization ofenergy gives the same result as the balance of forces we use both methods successively, inSections 2.3.4 and 2.5.8.

In the following sections, 2.2 and 2.3, we introduce the fundamental notions of the theoryof elasticity: the strain, which provides a measure of the deformation, and the stress, whichaccounts for the interior forces within a solid. The next step, addressed in Section 2.4, relatesthese quantities, by means of a so-called constitutive relation.

2.2 Strain

The theory of elasticity deals with solids that, in the presence of mechanical stress,depart from their natural conguration. Unlike uids, which can ow, the equilibriumconguration of an elastic solid is imprinted in matter, and recovered as soon as the stressis released. The strain provides a geometrical characterization of deformation: it measuresby how much the solid departs from its natural conguration.

2.2.1 Transformation

The deformation of a solid is mathematically expressed1 by a vector eld r(r) giving thenal position in the physical space of any material point in the solid initially located atr = (x, y, z), see Fig. 2.1. By nal, we mean the state of the solid after deformation,and by initial before deformation. Making reference to time can be misleading andso we shall follow a classical terminology and call actual conguration the deformedconguration (the nal one, which we are observing) and reference conguration theinitial one.

As implied by its name, dening the reference conguration is purely a matter ofconvention. It is often chosen to be a natural (that is, stress-free) conguration of the

1 We recall that, in the context of three-dimensional elasticity, primes denote quantities in the deformedconguration (actual conguration), as opposed to the reference conguration.

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