Appendix a Functions, Functionals and Their Derivatives A.1 Functions and Functionals

Appendix A Functions, Functionals and their Derivatives

A.1 Functions and Functionals

The usage of variables and functions in this book will be familiar to most readers, but some of the techniques used for analysis of functionals may be unfamiliar. As a preamble to the description of functionals, first we describe the terminology for the simpler cases. The following is not intended as a rigorous or comprehensive introduction but simply a clarification of the terminology used here. A variable is a symbol used to represent an unspecified value of a set. The value of the variable is one particular member of the set and the range is the set itself. For example, the variable x might represent a real number. In this case, the range is the set of real numbers from f to f, and in a particular instance, the value of x may be the number 3.81. A function of one variable associates a value from one set (the range of the function) with each value from another set (the domain of the function). For example, the function yx 3 2 expresses y as a function of the variable x. If the domain of the function (i. e. the range of its argument x) is the set of real numbers, then the resulting range of the function y will be the set of non-negative real numbers. The variable y is called the dependent variable or value of the function, and x is the independent variable or the argument of the function. If the specific form of a function is not given, then it is written in the form y fx , or y yx . The concept is simply extended to functions of several arguments, e. g. y yx 123,, x x . The arguments may be vectorial or tensorial rather than scalar, as may be the dependent variable. A functional is loosely defined as a “function of a function”. It is a function in which one or more of the arguments is itself a function. The range of the functional may be either another function or just a variable. A functional is distin- guished here from a function by use of square brackets > @ . Thus, for instance, zzy > @ , where y yx . 306 Appendix A Functions, Functionals and their Derivatives

It is sometimes necessary to make a careful distinction between the function f itself and its value at a particular value of x, which we shall denote f x . In this regard, we follow the more usual convention, but note that in some subjects, it is common to use f x for the function and f for its value.

A.2 Some Special Functions

Throughout this book, we need to make much use of some special functions, principally related to absolute values and certain singularities. We denote the absolute value by xxx ,0t; xxx ,0 . We use the following notation for the Macaulay bracket: xxx ,0t; xx 0, 0 . We also need the derivatives of the above functions. They may be loosely defined as follows, although more formal definitions are provided by the subdifferential, which is introduced in Appendix D on convex analysis. For the time being, we define a modified signum function as S1,0 xx , S1,0 xx !, and S x as undefined for x 0 , but within the range d1S01 d. Note that this does not correspond to the conventional signum function, usually denoted by “sgn” or “sg”, and which is usually defined by sgn 0 0 . 1 We define the Heaviside step function as HS1 xx , so that 2 H0,0 xx , H1,0 xx !, and H x is undefined for x 0 , but within the range 0H01dd . We define the Dirac impulse function G x through the relationship b ³G x x00 fxdxfx , axdd0 b. It can be proven straightforwardly that a Hc xx G , where f c x denotes the differential of f x . The quantity that plays the same role as the absolute value for a symmetrical second-order tensor is xxkl kl , and we define the derivative of this with respect to xij as a generalised tensorial signum function, which we write xij Sij x ij . Like the signum function of a scalar, this quantity is unde- xxkl kl fined when xxkl kl 0 (which necessarily occurs only when xij 0 ), but we require S0S01ij ij d . It is emphasised again that some of the above definitions can be set in a much more satisfactory formalism by the use of convex analysis; see Appendix D. A.3 Derivatives and Differentials 307

A.3 Derivatives and Differentials

The derivative of a function is the instantaneous rate of change of the function df with respect to one of its arguments. The definition of the derivative { fxc dx of the function f x with respect to x should be familiar: f xxfxG fxc lim (A.1) Gox 0 Gx The above is of course the fundamental definition from which many familiar expressions for derivatives of particular functions are obtained. When the function has more than one argument, partial derivatives are defined by obtaining the derivative with respect to one argument whilst consider- ing the others as constants. Thus if f fxy , the partial derivative wf { fxyx , with respect to x is defined as: wx f xxyfxyG,, fxyx ,lim (A.2) Gox 0 Gx In the main text, the differential of an integral is required. If bx Fx ³ f xtdt, , then application of the basic definitions results in ax

bx wf db da Fxc dt fxb ,, fxa (A.3) ³ wxdxdx ax The differential of a function f x is defined by df fc x dx , where dx is an independent variable. (Note the formal distinction, often ignored, between a differential and a derivative). The total differential of a function of more than one argument, for example f xy, , is defined in the following way: wwf f df dx dy (A.4) wwxy wf where each of the terms of the form dx is a partial differential. wx The concept of the differential of a function can be extended to that of a functional by using either the Gateaux or Frechet differential, and these are devel- oped as follows. 308 Appendix A Functions, Functionals and their Derivatives

In the classical calculus of variations, the variation of a functional f >u@ is defined in terms of a variation Gu of its argument function: f >uufuHG@ > @ GG{fu>@,lim u (A.5) Ho0 H where H is a scalar. A more precise statement defining Gf is based on a choice of norm in the space of f:

fu> HG u@ fu> @ limGGfu>@ , u 0 (A.6) Ho0 H

For sufficiently well-behaved functionals, Gf will be a linear functional of its argument Gu, so that GDGf >uu,,@ DGG fuu> @ , for all scalars D, and GGGf >uu,,,12 u@ GGGG fuu> 1@ fuu> 2@ , for all Gu1 and Gu2 . In this case, the functional Gf may be presented as the operation of a linear operator f c>u@ on the function Gu, which we write in the following way:

GGf >uu,,@ fuc> @ G u fu > u@ , G u (A.7)

Note that the above expression does not represent simply the inner product of f c>u@ and Gu, although in certain simple cases, it does take this form. The linear operator f c>u@ above is known as the Gateaux derivative of the functional f. An alternative basic definition for the generalised derivative of f (the Frechet derivative) requires that f c be that linear operator satisfying

fu> G u@ fu> @ fc> u@, G u lim 0 (A.8) Gou 0 Gu where the norms are defined in some appropriate way. It can be shown that the Gateaux and Frechet definitions are equivalent when the linear operator f c is continuous in the function u, and so we shall simply refer to f c in this book as the Frechet derivative. It is not essential to retain the variational notation in the definition of the Frechet derivative; therefore, a variation Gu can be replaced by any fixed v. Furthermore, when this variation is replaced by the differential du, the resulting functional df> u,, du@ fc> u@ du will be referred to as the Frechet differential. Frechet derivatives are used in this book to define Legendre transformations of functionals (see Appendix C), and Frechet differentials are used in deriving the incremental response of material behaviour. A.4 Selected Results 309

A.4 Selected Results

A.4.1 Frechet Derivatives of Integrals

Although the Frechet derivative is defined for general functionals, here we are interested principally in functionals of the form, f >ufuwdˆˆˆ@ ³ ˆ KKKK, (A.9) 8 where Y is the domain of K and fˆ is a continuously differentiable function of the variable uˆ , which is in turn a function of K. Here and in the main text, we adopt the convention that any quantity which is a function of K is denoted by a ‘hat’ notation, e. g. wˆ . Then, according to definitions (A.7) and (A.8), one can show that the Frechet differential of the functional (A.9) is given by

wKfuˆ ˆ, df>@ uˆˆ,, du fc >@ u ˆ du ˆ du ˆ KKK w ˆ d (A.10) ³ wuˆ 8 Now consider a more general case in which there are several independent ˆ ˆ variables: f >uuˆˆ11!!NN@ ³ fu ˆ KKKKK u ˆ , w ˆ d, where f is a continu- 8 ously differentiable function of functional variables uˆi , iN 1! . When the variable uˆ in (A.7) and (A.8) is identified as the full N-dimensional space of functions uˆi K , then the Frechet differential is

N ˆ §·wKfu ˆˆ1! uN , df>@ uˆˆ,, du fc >@ u ˆ du ˆ ¨¸ du î KKK w ˆ d (A.11) ³¦¨¸wu 8 i 1©¹i For the definition of partial Legendre transformations, the variable uˆ in definition (A.7) and (A.8) is identified as an n-dimensional subspace of the full N- dimensional space of functions uî K . In this case, the corresponding Frechet derivative is given by the following operator f c , which is linear in any integrable functions vî K :

n ˆ §·wKfu ˆˆ1! uN , fuvc>@ˆˆ, ¨¸ v ˆi KKK w ˆ d (A.12) ³¦¨¸wuˆ 8 i 1©¹i 310 Appendix A Functions, Functionals and their Derivatives

A.4.2 Frechet Derivatives of Integrals Containing Differential Terms

Now consider the case where the integral contains differential terms. It is b duˆ straightforward to show that the Frechet differential of fu>@ˆ dK with ³ dK a b dvˆ b respect to the function uˆ is given by fuvc>@ˆˆ, dK >@ v ˆ . Similarly, the ³ dK a a b du2 ˆ Frechet differential of fuˆ dK with respect to the function uˆ is given by >@ ³ 2 a dK b b dv2 ˆˆªº dv fuvc>@ˆˆ, dK . ³ 2 «»dK a dK ¬¼a It is straightforward to show that the Frechet differential of b 2 1 §·duˆ fu>@ˆ ¨¸ dK with respect to the function uˆ is given by ³ 2 dK a ©¹ b duˆˆ dv fuvc>@ˆˆ, dK. Integrating by parts, we obtain ³ ddKK a b b ªºduˆˆ d2 u fuvc>@ˆˆ, v ˆK vd ˆ (A.13) «»dK ³ 2 ¬¼a a dK The latter form proves convenient in some applications. b 2 1 §·du2 ˆ Similarly, the Frechet differential of f >@udˆ ¨¸K with respect to the ³ 2¨¸2 a ©¹dK b dudv22ˆˆ function uˆ is given by fuvc ˆˆ, dK. Integrating by parts twice, we >@ ³ 22 a ddKK obtain

bbb ªºªºdudv234ˆˆ du ˆ du ˆ f c>@uvˆˆ, «»«»K v ˆvd ˆ (A.14) 234dK ³ ¬¼¬¼«»«»dddKKKaaa where again this latter form again proves convenient in some applications.

Appendix B

Tensors

B.1 Tensor Definitions and Identities

A second-order Cartesian tensor a is also written aij or in matrix form for

ªºaaa11 12 13 three dimensions as «»aaa. We use the summation convention over «»21 22 23 ¬¼«»aaa31 32 33 3 a repeated index; thus, for instance, aaaakk 11 22 33 and aaij jk ¦ aa ij jk . j 1 2 Alternatively, we may write aaij jk a . The unit tensor, or Kronecker G is defined by Gij 1, ij ; Gij 0, ij z. A tensor is symmetric if it is equal to its transpose aaij ji and is antisymmetric, or skew-symmetric, if aaij ji . The 1 inverse of a tensor or matrix is defined by aaij jk G ik , and a tensor or matrix is orthogonal if its inverse is equal to its transpose aaij jk G ik .

The tensor has principal values a1 , a2 , a3 , which are the eigenvalues of the matrix, and are the solutions of the cubic equation,

32 aaIaII123 0 (B.1) where I1 , I2 , I3 are the invariants of the tensor, which are

Ia1123 ii tr a aaa (B.2)

112 Iaaaa traa2 tr 2 22 ij ji ii jj (B.3) aa12 aa 23 aa 31 312 Appendix B Tensors

1 Iaaaaaaaaa 23 3 6 ij jk ki ij ji kk ii jj kk (B.4) 1 3 2traaaaa32 3tr tr tr det aaa 6 123

Note that some authors define I2 with the opposite sign, but we prefer the notation used here; otherwise, J2 (see below), which plays a major role in the analysis of shear behaviour, is always negative. The traces of the powers may alternatively be chosen as defining the three invariants,

tr a aaaaIii 1231 (B.5)

22222 tr a aaaaaij ji 123 2 II 21 (B.6)

3 333 3 tr a aaij jk a ki a123 a a 3 I 3 3 II 211 I (B.7) The deviator of a tensor is defined as follows: 1 aac a G (B.8) ij ij3 kk ij so that I1cc tr a 0 . The second and third invariants of the deviator are also often required and may be written in a variety of forms: 111 Iccc J aa aa aa 22226ij ji ij ji ii jj 11§·2 1 traa22 tr II 23¨¸ 21 3 ©¹ (B.9) 1 aaaaaaaaa222 3 123122331 1 2 22 aa aa aa 6 12 23 31

11§· 2 I33cccc J aaij jk a ki ¨¸ aa ij jk a ki aaa ij ji kk aaa ii jj kk 33©¹ 9 11§·32 23 12 3 ¨¸traaaa tr tr tr IIII3211 33©¹ 9 327 1 222aaa aaa aaa (B.10) 27 123 231 312 1 2 aaa333 27^ 123 222 312 aa12 a 3 aa 23 a 1 aa 31 a 2 aaa 123 ` B.2 Mixed Invariants 313

B.2 Mixed Invariants

The four mixed invariants of two tensors can be written as:

tr ab aij b ji a11 b a 22 b a 33 b (B.11)

2222 tr ab aij a jk b ki a11 b a 22 b a 33 b (B.12)

2222 tr ab aij b jk b ki a11 b a 22 b a 33 b (B.13)

22 22 22 22 tr ab aaij jk b kl b li ab11 ab 22 ab 33 (B.14) where the forms expressed in terms of the principal values only apply if the principal axes coincide for the two tensors. Thus for two tensors, there are 10 invariants, three for each tensor alone and four mixed invariants.

B.2.1 Differentials of Invariants of Tensors

Since the various potentials used in this book are most often written in terms of invariants and then are differentiated to obtain the constitutive behaviour, it is convenient to note the differentials of tensors and their invariants given in Table B.1. 314 Appendix B Tensors

Table B.1. Differentials of functions of tensors and their invariants

f df daij

akl GGki lj ac 1 kl GGGG ki lj3 ij kl

I1 Gij

I2 aIjiG1 ij

I3 aajkkiG aI ji12 I ij J 1 2 aac G I ji ji3 1 ij

J3 aaccjk kiG J2 ij

tr a Gij tr a2 2aji tr a3 3aajk ki

tr ab bji tr ab2 2abjkki tr ba2 bbjk ki tr ab22 2abbjk kl li

Appendix C

Legendre Transformations

C.1 Introduction

The Legendre transformation is one of the most useful in applied mathematics, although its role is not always explicitly recognised. Well-known examples in- clude the relation between the Lagrangian and Hamiltonian functions in analytical mechanics, between strain energy and complementary energy in elasticity theory, between the various potentials that occur in thermodynamics, and between the physical and hodograph planes occurring in the theories of the flow of compressible fluids and perfectly plastic solids. The Legendre transformation plays a central role in the general theory of complementary variational and extremum principles. Sewell (1987) presents a comprehensive account of the theory from this viewpoint with particular emphasis on singular points. These transformations have also been widely employed in rate formulations of elastic/plastic materials to transfer between stress-rate and deformation-rate potentials, e. g. Hill (1959, 1978, 1987); Sewell (1987). These applications are rather different from those used in this book. We review therefore those basic properties of the transformation that are needed in the main text.

C.2 Geometrical Representation in (n + 1)-dimensional Space

A function Z Xx()i , in 1! , defines a surface * in n 1 -dimensional Z,xi space. However, the same surface can be regarded as the envelope of tangent hyperplanes. One way of describing the Legendre transformation is that it allows one to construct the functional representation that describes Z in terms of these tangent hyperplanes. This relationship is a well-known duality in ge- ometry. The gradients of the function Xx i are denoted by yi : wX (C.1) yi wxi 316 Appendix C Legendre Transformations

Z *

Tangent hyperplane

P(X, xi)

Q(-Y, 0i) -Y

x i Figure C.1. Representation of * in (n + 1)-dimensional space

so that the normal to * in the n 1 -dimensional space is 1, yi . If the tangent hyperplane at the point P Xx, i on * cuts the Z axis at Q Y,0i , the vector XYx , i lies in the tangent hyperplane (Figure C.1), and hence is orthogonal to the normal to * at P. Forming the scalar product of these two vectors therefore leads to (C.2) Xx iiii Yy xy

The function Z Yy i defines the family of enveloping tangent hyperplanes and hence is the required dual description of the surface *. The form of this function can be found by eliminating the n variables xi from the n 1 equations in (C.1) and (C.2). This can be achieved locally, provided that (C.1) can be inverted and solved for the xi 's, i. e. provided the Hessian matrix wy w2X i , is non-singular. Points at which the determinant of the Hessian wwwxxxjij matrix vanishes are singularities of the transformation (Sewell, 1987). Differen- tiating (C.2) at a non-singular point with respect to yi gives wwXYwwxxjj yxji (C.3) wwxyji w y i w y i C.3 Geometrical Representation in n-dimensional Space 317 which, by virtue of (C.1) reduces to wY (C.4) xi wyi Relations (C.1)–(C.3) define the Legendre transformation. This transformation is self-dual because, if the function Z Yy i is used to define a surface *c “pointwise” in Z, yi space, then Z Xx i describes the same surface *c “planewise” because 1, xi define the normal to *c and X is the intercept of the tangent plane with the Z axis from (C.2). The transformation is not in general straightforward to perform analytically. An exception is when Xx() is a quadratic form, Xx() 1 Axx , where A is i iijij 2 ij a non-singular, symmetrical matrix. Hence, the dual variables are wX 1 yiijj Ax , so that xAyiijj , and the Legendre dual is also a quadratic wxi form: Yy xyXx Ayy11111 Ayy Ayy (C.5) i ii i ijij 22 ijij ijij The transformation in general is succinctly written wX (C.1)bis yi wxi (C.2)bis Xx iiii Yy xy wY (C.4)bis xi wyi The choice of the sign of the dual function is somewhat arbitrary, and Y is sometimes written instead of Y. The choice is usually governed by physical considerations.

C.3 Geometrical Representation in n-dimensional Space

An alternative geometrical visualisation in n-dimensional space is also valuable in gaining understanding of formal results. For fixed C but variable xi , the relation (C.6) I{ xyii,0 Xx i xyC ii defines a family of hyperplanes in n-dimensional yi space. These hyperplanes envelope a surface in this space, the equation of which is obtained by eliminating the xi between (C.4) and wI wX (C.7) yi 0 wwxxii 318 Appendix C Legendre Transformations

On comparison with (C.1) and (C.2), it follows that the equation of this surface is Yy i C, so that the hyperplanes defined by (C.4) envelope the level surfaces of the dual function Y. Dually, the hyperplanes defined by (C.8) \{ xyii, Yy i xyC ii envelope the level surfaces of Xx i in xi space. These level surfaces are, of course, the “cross sections” of the n 1 -dimensional surfaces Z Xx i and Z Yy i discussed above.

C.4 Homogeneous Functions

Of particular importance in applications in continuum mechanics are cases where the function Z Xx i is homogeneous of degree p in the xi 's, so that Xx Oii O pXx for any scalar O. From Euler's theorem for such functions, it follows that

wX (C.9) pXx ii x xy ii wxi so that from (C.2),

wY (C.10) qY yiiii x y y wyi 11 where 1 , so that the Legendre dual Yy is necessarily homogeneous pq i p of degree q . p 1 In the example above, p 2 , so that X and Y are both homogeneous of degree two. A familiar example of this situation is in linear elasticity where the elastic strain energy E Hij and the complementary energy C Vij are both quadratic functions of their argument and satisfy the fundamental relation, (C.11) EC HVij ij VH ij ij Another case of particular importance in rate-independent plasticity theory occurs when X is homogeneous and of degree one, so that Xx iii xy, in which case the dual function Yy i is identically zero from (C.2). There is a simple geometric interpretation of this far-reaching result. Since Xx Oii O Xx , the n 1 -dimensional surface Z Xx i is a hypercone with its vertex at the origin. Hence, all tangent hyperplanes meet the Z axis at Z 0 , so that Yy i 0 for all yi . This special case is pursued further later, C.5 Partial Legendre Transformations 319 and the terminology of convex analysis will prove particular useful in its treatment (see Appendix D).

C.5 Partial Legendre Transformations

Now suppose that the functions depend on two families of variables, Xx ii,D say, where xi and Di are n- and m-dimensional vectors, respectively. We can perform the Legendre transformation with respect to the xi variables as above and obtain the dual function Yy ii,D . The variables Di play a passive role in this transformation and are treated as constant parameters. Hence, the three basic equations are now (C.12) Xx ii,,D Yy ii D xy ii

wX and wY (C.13) yi xi wxi wyi

If the derivatives of X with respect to the passive variables Di are denoted by Ei , then it follows from (C.12) that

wwXY (C.14) Ei wDii wD It is also possible, in general, to perform a Legendre transformation on Xx ii,D with respect to the Di variables and construct a second dual function Vx ii,E with the properties,

Xx ii,,D Vx ii E DE ii (C.15) where

wX , wV (C.16) Ei Di wDi wEi and furthermore:

wwXV (C.17) yi wwxxii since now the xi 's are the passive variables. This process can be continued. A Legendre transformation of Yy ii,D with respect to the Di variables produces a fourth function Wy ii,E . The same function is obtained by transforming Vx ii,E with respect to the xi variables. A closed chain of transformation is hence produced as shown in Figure C.2, where the basic differential relations are summarised. The best known example 320 Appendix C Legendre Transformations

X (xi ,Di ) wX wX yi , Ei wxi wDi

X Y xi yi V X DiEi

Y (yi ,Di ) V (xi ,Ei ) wY wY XYWV 0 wV wV xi , Ei yi , Di wyi wDi wxi wEi

Y W DiEi WV xyii

W (yi ,Ei ) wW wW xi , Di wx wE i i Figure C.2. Chain of four partial Legendre transformations of such a closed chain of transformations is in classical thermodynamics, where the four functions are the internal energy usv , , the Helmholtz free energy f T,v , the Gibbs free energy g T, p , and the enthalpy hsp , , where T , s, v, and p are the temperature, entropy, specific volume, and pressure respectively, e. g. Callen (1960). Other examples are given by Sewell (1987).

C.6 The Singular Transformation

When X is homogeneous of order one in xi , so that ODXx ii,, XOD x ii, the value of yii wXx/ w is unaffected by the transformation xxiioO , and so the mapping from xiio y is fo1 . Furthermore, since (C.18) xyii X x i,D i the dual function Yy ii,D is identically zero, as already noted above, and so wwYY (C.19) dY dyiiD d 0 wwDyii But also from (C.13),

wwXX (C.20) xii dy y ii dx dx i D d i wwDxii C.7 Legendre Transformations of Functionals 321 which by virtue of (C.1) reduces to

wX (C.21) xdyiiD d i 0 wDi Hence, by comparing (C.19) with (C.21), it follows that

wY and wwXY (C.22) xi O O wyi wDii wD where O is an undetermined scalar, reflecting the non-unique nature of this singular transformation. The above development is classical in the sense that all the functions are as- sumed to be sufficiently smooth for all derivatives to exist. In practice, the surfaces encountered in plasticity theory, on occasion, contain flats, edges, and corners. Such surfaces and the functions defining them can be included in the general theory using some of the concepts of convex analysis. In particular, the commonly defined derivative is replaced by the concept of a “subdifferential”, and the simple Legendre transformation is generalised to the “Legendre-Fenchel transformation” or “Fenchel dual”. For simplicity of presentation, we have so far used the classical notation, and convex analysis is introduced in Appendix D. Treatments of the mechanics of elastic/plastic materials that use convex analysis notation may be found in Maugin (1992), Reddy and Martin (1994), and notably Han and Reddy (1999). Because our main concern here is to exhibit the overall structure of the theory as it affects the developments of constitutive laws, we have not highlighted the behaviour of any convexity properties of the various functions under the transformations. These considerations are very important for questions of uniqueness, stability, and the proof of extremum principles, which are beyond the scope of this book, but are fruitful areas for future research. Some of these aspects of Legendre transformations are considered at length in the book by Sewell (1987).

C.7 Legendre Transformations of Functionals

C.7.1 Integral Functional of a Single Function

Consider a functional, Xx> ˆˆˆ@ ³ Xxˆ KKKK, w d (C.23) 8 where Y is the domain of K and Xˆ is a continuously differentiable function of a functional variable xˆ . 322 Appendix C Legendre Transformations

wKKXxˆ ˆ , If yˆ K , then the Legendre transform of the function Xˆ is wKxˆ

Yyˆˆ ˆˆˆˆ KK,, x K y K Xx KK (C.24) It follows from the standard properties of the transform that wKKYyˆ ˆ , xˆ K . wKyˆ The functional defined by

Yy> ˆˆˆ@ ³³ Yyˆ KKKK, w d x ˆˆˆˆ KKKK y w d Xx> @ (C.25) 88 may then be considered the Legendre transform of the original functional, and using definitions of Appendix A, it can be confirmed that this definition satisfies the appropriate differential conditions.

C.7.2 Integral Functional of Multiple Functions

A case of interest in the present work is a Legendre transform of a functional of the form,

Xxu> ˆˆ,,,@ ³ Xxˆ ˆ KKKKK u ˆ w ˆ d (C.26) 8 where Xˆ is a continuously differentiable function of the variables xˆ K and uˆ K . wKKKXxˆ ˆˆ ,, u Denoting yˆ K , the Legendre transform of the function wKxˆ Xˆ with respect to the variable xˆ K is defined as

Yyuˆˆ ˆˆ,,K x ˆ K yˆ K Xx ˆ K , u ˆ K , K (C.27) From the standard properties of the transform, it follows that ˆ wKKKYy ˆˆ ,, u xˆ K (C.28) wKyˆ ˆˆ wYy ˆˆ K,, u KK w Xx ˆˆ K ,, u KK (C.29) wKuuˆˆ wK Then, the Legendre transformation of functional (C.26) in function xˆ , where function uˆ is a passive variable, is given by the functional, YyuYyu> ˆˆ,,,@ ³³ˆ ˆ KKKKK ˆ wd ˆ xywdXxu ˆ KKKK ˆ ˆ > ˆˆ ,@ (C.30) 88 C.7 Legendre Transformations of Functionals 323 and using definitions of Appendix A, it can be confirmed that this definition satisfies the appropriate differential conditions. When xˆ is not a function but a variable, denoted x, all above equations are valid, except that Equation (C.30) may be rewritten as Yyu> ,,,ˆˆˆ@ ³ Yyuˆ KKKK w d xyXxu> , ˆ@ (C.31) 8 where wKKXxuˆ ,, y wdˆ KK (C.32) ³ wx 8 When the function Xˆ is a continuously differentiable function of the function xˆ (or variable x) and any finite number N of functions uˆi , the same Equa- tions (C.27)–(C.30) are still valid, except that Equation (C.29) unfolds into N equations: wKwKYyuˆˆ ˆˆ,,!! u ˆ Xxu ˆˆ ,,, yˆ u ˆ 11NN ,1iN ! (C.33) wwuuˆˆii

C.7.3 The Singular Transformation

An important case in rate-independent plasticity theory occurs when functional Xxuˆ ˆˆ,,K in (C.26) is homogeneous of degree one in, say, xˆ K : Xxˆˆ OKˆˆ ,, u KK O Xx ˆˆ K ,, u KK (C.34) From Euler’s theorem, it follows that wKKKXxˆ ˆˆ ,, u Xxˆ ˆˆ KKK,, u x ˆ K yˆˆ KK x (C.35) wKxˆ Then the Legendre transformation of the function Xxˆ ˆˆ KKK,, u with respect to uˆ K , when other variables and functions are passive, is defined by Equation (C.27), so that after substitution of (C.35), we obtain Yyˆˆ ˆˆ KKK,, u x ˆˆ KKKKK{ y Xx ˆˆ ,,0 u (C.36) The properties of this transformation are wKKKYyˆ ˆˆ ,, x xˆ K OKˆ (C.37) wKyˆ wXxˆˆ ˆˆ K,, u KK w Y yˆˆ K ,, u KK Oˆ K (C.38) wKuuˆˆ wK 324 Appendix C Legendre Transformations where OKˆ is an undetermined scalar, reflecting the non-unique nature of this singular transformation. Then the Legendre transformation of functional (C.26) in function xˆ , when function uˆ is a passive variable, is given by the functional, Yyu>@ˆˆ,,, ³³ Yyˆ ˆ KKKKK u ˆ w ˆ d x ˆ KKKK{ yˆ w ˆ d Xxu>@ ˆˆ ,0 (C.39) 88 and using definitions of Appendix A, it can be confirmed that this definition satisfies the appropriate differential conditions.

Appendix D

Convex Analysis

D.1 Introduction

The terminology of convex analysis allows a number of the issues relating to hyperplastic materials to be expressed succinctly. In particular, through the definition of the subdifferential, it allows rigorous treatment of functions with singularities of various sorts. These arise, for instance, in the treatment of the yield function. A brief summary of some basic concepts of convex analysis is given here. The terminology is based chiefly on that of Han and Reddy (1999). A more detailed introduction to the subject is given by Rockafellar (1970). No attempt is made to provide rigorous, comprehensive definitions here. For a fuller treatment, reference should be made to the above texts. Although it is currently used by only a minority of those studying plasticity, it seems likely that in time convex analysis will become the standard paradigm for expressing plasticity theory.

D.2 Some Terminology of Sets

We use brackets ^ ` to indicate a set, so that ^0,1, 3.5` is simply a set containing the numbers 0, 1 and 3.5. A closed set containing a range of numbers is denoted by > , @ , thus >ab, @ ^ x add x b` , where the meaning of the contents of the final bracket is “x, such that axbdd”. We use to denote the null (empty) set. In the following, C is a subset in a normed vector space V (in simple terms a space in which a measure of distance is defined), usually with the dimension of Rn (with n finite), but possibly infinite dimensional. The notation , is used for an inner product, or more generally the action of a linear operator on a function. The space Vc is the space dual to V under the inner product xx*, , so 326 Appendix D Convex Analysis that xV and xV* c . More generally, Vc is termed the topological dual space of V (the space of linear functionals on V). The operation of summation of two sets, illustrated in Figure D.1a, is defined by (D.1) CC12 ^ xxxCxC 121122 , ` The operation of scalar multiplication of a set, illustrated in Figure D.1b is defined by OCxxC ^O ` (D.2) It is also convenient to define the operation of multiplication of a set C by a set S of scalars: SC ^O x O S, x C` (D.3) The definitions of the interior and boundary of a set are intuitively simple concepts, but their formal definitions depend first on the definition of distance. In Rn , the Euclidian distance is defined as 12 dxy ,, x y x yx y (D.4) and we define the open ball of radius r and centered at xo as (D.5) Bx oo,, r ^ x dxx r` The interior of C is then defined as intCx ^ H!H 0, BxC , ` (D.6)

(a) (b) Figure D.1. (a) Summation of sets; (b) scalar multiple of a set D.3 Convex Sets and Functions 327

This means that there exists some H (possibly very small) so that a ball of radius H is entirely contained in C. The closure of C is defined as the intersection of all sets obtained by adding a ball of non-zero radius to C: clCCB ^ HH! 0, 0,` (D.7) Finally, the boundary of C is that part of the closure of C that is not interior:

bdyCC cl \ int C (D.8)

D.3 Convex Sets and Functions

A set C is convex if and only if 1OxyC O , xy, C, 01O (D.9) where, for instance, xy, C means “for all x and y belonging to C”. Simple examples of convex and non-convex sets in two-dimensional space are given in Figure D.2. A function f whose domain is a convex subset C of V and whose range is real or rf is convex if and only if

fxyfxfy 11O O d O O , xy, C, 01O (D.10) This is illustrated for a function of a single variable in Figure D.3. Convexity requires that NPd NQ for all N between X and Y. This property has to be true for all pairs of X,Y within the domain of the function. A function is strictly convex if d can be replaced by < in (D.10) for all x z y . The effective domain of a function is defined as the part of the domain for which the function is not; thus, dom fx ^ x V fx f` .

Figure D.2. Non-convex and convex functions 328 Appendix D Convex Analysis

z = f(x) z

x + (1-O)y = Ox + (1-O)y

x X N Y (1-O) O

Figure D.3. Graph of a convex function of one variable

D.4 Subdifferentials and Subgradients

The concept of the subdifferential of a convex function is a generalisation of the concept of differentiation. It allows the process of differentiation to be extended to convex functions that are not smooth (i. e. continuous and differentiable in the conventional sense to any required degree). If V is a vector space and Vc is its dual under the inner product , , then xV* c is said to be a subgradient of the function f x , xV , if and only if fy t fx x*, y x , y . The subdifferential, denoted by wf x , is the subset of Vc consisting of all vectors x * satisfying the definition of the subgradient: wfx ^ x**,, Vc f y fx t x y x y` (D.11) For a function of one variable, the subdifferential is the set of the slopes of li- nes passing through a point on the graph of the function, but lying entirely on or below the graph. The concept is illustrated in Figure D.4. The concept of the subdifferential allows us to define “derivatives” of non- differentiable functions. For example, the subdifferential of wx is the signum function, which we now define as a set-valued function: ^ 1,` x 0 ° S1,1,0 xwx w ®>@ x (D.12) ° ¯ ^`!1,x 0 D.5 Functions Defined for Convex Sets 329

w = f(x)

Figure D.4. Subgradients of a function at a non-smooth point

Thus at a point x, wf x may be a set consisting of a single number equal to wwfx, or a set of numbers, or (in the case of a non-convex function) may be empty.

D.5 Functions Defined for Convex Sets

The indicator function of a set C is a convex function defined by 0, xC IxC ® (D.13) ¯f, xC so that the indicator function is simply zero for any x that is a member of the set and f elsewhere. Although this appears at first sight to be a rather curious function, it proves to have many applications. In particular, it plays an important role in plasticity in that it is closely related to the yield function. The normal cone NxC of a convex set C is the set-valued function defined by (D.14) NxC ^ xV**,0,c x y dx y C` 330 Appendix D Convex Analysis

It is straightforward to show that NxC ^0` if xCint (the point is in the interior of the set), that NxC can be identified geometrically with the cone of normals to C at x if xCbdy (the point is on the boundary of the set), and further that NxC is empty if xC (the point is outside the set). Furthermore, the subdifferential of an indicator function of any convex set is the normal cone of that set: wIxCC Nx . Another important function defined for a convex set is the gauge function or Minkowski function, defined for a set C as (D.15) JC xxC inf^PtP 0 ` where inf ^x` denotes the infimum, or lowest value of a set. In other words, JC x is the smallest positive factor by which the set can be scaled, and x is a member of the scaled set. The meaning is most easily under- stood for sets that contain the origin (which proves to be the case for all sets of interest in hyperplasticity). In the following, we shall therefore assume that C is convex and contains the origin. It is straightforward to see in this case that JC x 1 for any point on the boundary of the set, is less than unity for a point inside the set, and is greater than unity for a point outside the set. At the origin, JC 00 . In the context of (hyper)plasticity, it is immediately obvious that the gauge may be related to the conventional yield function. If the set C is the set of (generalised) stresses F that are accessible for any given state of the internal variables (the elastic region), then the yield function is a function conventionally taken as zero at the boundary of this set (the yield surface), negative within, and positive without. One possible expression for the yield function would therefore be y F JC F1 . Other functions could of course be chosen as the yield function, but this is perhaps the most rational choice; so we follow Han and Reddy (1999) in calling this the canonical yield function. To emphasise the case where the yield surface is written in this way, we shall give it the special notation y F JC F1 . The gauge function is always homogeneous of order one in its argument x, so that JCC Oxx OJ . (In the language of convex analysis, such functions are simply referred to as positively homogeneous.) The canonical yield function is therefore conveniently written in the form of a positively homogeneous function of the (generalised) stresses, minus unity. It is also clear that the gauge of the set of accessible generalised stresses contains exactly the same information as the yield function, canonical or otherwise, and there may be benefits from specify- ing the gauge rather than the yield function. It is useful to note that at the boundary of C, the normal cone can also be written NxCC w Ix OwJ C x,0 dOdf. This proves to be a convenient form that allows the normal cone to be expressed in terms of the subdifferential of the gauge function and therefore (in hyperplasticity), of the canonical yield function. D.6 Legendre-Fenchel Transformation 331

It is straightforward to see that the definition (D.15) can be inverted. Given a positively homogeneous function f x , one can define a set C, such that f x is the gauge function of C: Cxfx ^ d1` (D.16) which has the property that JC xfx . It is worth noting, too, that the indicator function of a set containing the origin can always be expressed in the following form, and this proves useful in the application of convex analysis to (hyper)plasticity: (D.17) IxCC I>f,0@ J x1

The function Ix>f,0@ is simply zero for all non-positive values of x and f for positive values.

D.6 Legendre-Fenchel Transformation

The Legendre-Fenchel transformation (often simply called the Fenchel dual, or conjugate function) is a generalisation of the concept of the Legendre transformation. If f x is a convex function defined for all xV , its Legendre-Fenchel transformation is f ** x , where xV* c is defined by fx**sup*, ^ xxfx ` (D.18) xV where sup means the supremum, or highest value for any xV . xV It is straightforward to show the Fenchel dual is a generalisation of the Leg- endre transform. We use the notation that if xfx*w and f ** x is the Fenchel dual of f x , then xfxw ** . Some useful Fenchel duals, together with their subdifferentials are given in Table D.1. The dual of the sum of two functions involves the process of infimal convolution: f gx inf ^ fx y g y ` (D.19) yV Table D.1 also includes some special functions that are defined in Section D.10. 332 Appendix D Convex Analysis

Table D.1. Some Fenchel duals and their subdifferentials f x wf x f ** x wf ** x

Ix^`0 Nx^`0 0 ^0`

Ix>f,0@ Nx>f,0@ Ix>0,f@ * Nx>0,f@ *

Ix>1,1@ Nx>1,1@ x * S* x

Ix>0,1@ Nx>0,1@ x * H* x

pos x H x Ix>f,0@ *1 Nx>f,0@ *1

1 ^0` Ix^`0 * Nx^`0 *

x ^1` Ix^`1 * Nx^`1 * x2 2 ^x` x *22 ^x *` nn1 ½11 n n1 §·x * °°§·x * xn , n 1 nx n 1 ®¾ ! ^ ` ¨¸ ¨¸n ©¹n ¯¿°°©¹ expx ^expx` xxx*log ** ^logx *` f xgx wwf xgx f * g ** x 1 f ax afw ax f ** xa wfxa** a

D.7 The Support Function

The support function is also defined for a convex set. For a convex set C in V, if xV* c , then the support function is defined by (D.20) VC xxxxC*sup,* ^ ` Note that although C is a set of values of the variable x, the argument of the support function is the variable x * conjugate to x. It can be shown that the support function is the Fenchel dual of the indicator function. The support function is always homogeneous of order one in x * , i. e. it is positively homogeneous. It follows that any homogeneous order-one function defines a set in dual space. In hyperplasticity, one can observe that the dissipation function is homogeneous and order one in the internal variable rates. It can thus be interpreted as a support function, and the set it defines in the dual space of generalised stresses is the set of accessible generalised stress states (the elastic region). The Fenchel dual of the dissipation function is the indicator function for this set of accessible states, which is zero throughout the set. We can identify this indicator function D.7 The Support Function 333 with the Legendre transform wy O of the dissipation function introduced in Chapter 4. Equation (D.20) can be inverted to obtain the set C from the support function. If f ** x is a homogeneous first-order function in x * , then the set defined by solving the system of inequalities, Cxxxfx ^ ,*d * *, x *` (D.21) satisfies the condition that VC xfx*** . Application of (D.21), with f ** x as the dissipation function, allows the set of accessible (generalised) stresses to be derived from the dissipation function in a systematic manner; hence the elastic region can be derived from the dissipation function. The subdifferential of the support function defines a set called the maximal responsive map (see Han and Reddy, 1999, although we depart from their notation here):

PCC xx** wV (D.22) The normal cone and the maximal responsive map are inverse in the sense that (D.23) xxxNxPCC ** It also follows [see Lemma 4.2 of Han and Reddy (1999)] that C is simply related to the support function by the subdifferential at the origin, i. e. the maximal responsive map at the origin. Thus

C PCC 00 wV (D.24) Both the gauge and support functions are positively homogeneous. Defining the domain of the support function Sx dom VC * , it can be shown (see Han and Reddy, 1999) that xx*, (D.25) JC x sup 0*zxSVC x * D and JC x is called the polar of VC x * , written JCC V . The process is D symmetrical so that VCC J and, defining the domain of the gauge function Gx dom JC , xx*, (D.26) VC x *sup 0zxGJC x Further, we have the following inequality: (D.27) VJtCC xxxxxSxG**,,*, 334 Appendix D Convex Analysis

and the equality holds for xxwVC * : (D.28) VJCC xxxxxxxS**,,*,* wV C

Application of (D.25), together with y F JC F1 , allows the canonical yield function to be determined directly from the dissipation function.

D.8 Further Results in Convex Analysis

Whilst the above are the most important results needed in Chapter 13, it is worth noting some further relationships between convex sets and functions. The polar Cq of a convex set C is defined as (D.29) Cxq ^ **1 VC x d ` in other words, it is the set for which the support function of C is the gauge. The polar f q of a non-negative convex function f, which is zero at the origin, is defined by fxq *inf0 ^Pt xx ,*1 dP fx , x` (D.30) and it can be shown that Equations (D.25) and (D.26) can be derived from this more general relationship. Finally, the indicator and the gauge of a convex set are said to be obverse to each other, where the obverse g of a function f is defined by gx inf^O!Od 0 f x 1` (D.31) and the operation f Oxf OO 1 x is called right scalar multiplication [note fx that f 0 xI ^`0 x if fx zf and f 0 xfx if f].

D.9 Summary of Results for Plasticity Theory

In summary, we have the following concepts from convex analysis which are of relevance in plasticity theory: x A convex set C in V. x The indicator function IxC of the set. x The gauge function JC x of the set. x The support function VC x * , which is the Fenchel dual of the indicator, and is also the polar of the gauge function. x The normal cone NxCC w Ix which is a set in Vc which is the subdifferential of the indicator function. D.9 Summary of Results for Plasticity Theory 335 x The maximal responsive map, which is the subdifferential of the support function PCC xx** wV . x The set C is the subdifferential of the support function at the origin C PCC 00 wV . The relationships among these quantities are illustrated in Figure D.5 for the simple one-dimensional set Cab >, @ . The roles of these concepts in hyperplasticity are explored in more detail in Chapter 11, but Table D.2 gives the correspondences among some concepts in conventional plasticity theory and in the convex analytical approach.

Table D.2 Correspondences between conventional plasticity theory and the convex analytical approach

Conventional plasticity Convex analytical approach to (hyper)plasticity theory Elastic region A convex set in (generalised) stress space. Yield surface The indicator function or (for some purposes) the gauge function, or equivalently the canonical yield function. Plastic potential and flow Gauge function (or equivalently the canonical yield func- rule tion) and the normal cone. Plastic work Support function (equal to the dissipation function). NB: For models in which energy can be stored through plastic straining, this is not equal to plastic work. (No equivalents in con- The indicator of the elastic region and the support func- ventional theory) tion (dissipation) are Fenchel duals. The gauge function of the elastic region and the support function (dissipation) are polars.

336 Appendix D Convex Analysis

C >@ a,b Cq >@1 a ,1 b

JC x ICq x *

IC x VC x *

x* wIC x NC x x wVC x * PC x *

Figure D.5. Relationships among functions of a convex set in one dimension

D.10 Some Special Functions

We have already introduced the signum function (D.12), which we treat as a set- valued function. Closely related is the Heaviside step function: ^0,` x 0 1 ° HS10,1,0 xx ^` ®>@ x (D.32) 2 ° ¯ ^`!1,x 0 D.10 Some Special Functions 337

It is also useful to define a closely related set-valued function: ,0x ° H,1,0 xx ®>@f (D.33) ° ¯ ^`!1,x 0

H x , which can also be written as Nx>0,f@ , is useful because it is the subdifferential of the positive values of x, defined as f,0x pos x ® (D.34) ¯ xx,0t Note that careful distinction is needed among pos x , the absolute value x , and the Macaulay bracket x . Note that all of these functions are convex. References

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Index

A cohesionless soils 209 cohesive material 270, 340 adiabatic 43–46, 49, 66, 67, 79, 256 compatibility 8–11, 278, 280, 282 advanced plasticity models 118 complementary energy 48, 77, 164, 168, anisotropic elasticity 172 170, 172, 318 anisotropy 120, 163, 164, 169, 172, 210, compliance matrix 22, 23, 49, 66, 171 340 conjugate variables 57 associated flow rule 18, 21, 28, 29, 32, conservation of energy 37 58, 69, 74, 83, 86, 96, 100, 108–114, consistency condition 20–25, 63, 91, 97, 118, 130, 155, 176–179, 186, 209, 210, 109, 111, 115, 124, 145, 179, 197, 239 261, 342 constitutive behaviour 10, 21, 42, 59, 69, 74, 75, 88, 91–93, 96–98, 102, 124, B 125, 148, 149, 155, 180, 190, 195, 197, back stress 27, 68, 89, 97, 108, 123, 130, 254, 258, 262, 266, 302, 313 149, 153, 179, 180, 184, 207, 209 constitutive models 1–4, 8, 11, 19, 62, back stress function 144 65, 74, 79, 115, 117, 156, 182, 210, backbone curve 192, 232, 233 220, 255, 256, 259, 262, 273, 304, 340– bar structure 279 343 bending moment 299, 300 constraints 1, 71, 73, 84, 205, 264–266, bending stiffness 298 270, 278–282, 302 body force 9, 245, 246 continuous field of yield surfaces 119, bounding surface plasicity 105–110, 151, 155, 340 118 continuous hyperplastic model 142, bulk modulus 14, 44, 79, 162, 190 177, 191 continuous hyperplasticity 133, 146, C 155, 203, 210, 224, 300, 341 continuous material memory 119 canonical yield function 17, 58, 268– continuum mechanics 6, 8–11, 48, 243, 271, 275, 330, 334, 335 253, 260, 318, 342 Cauchy small strain tensor 7 contraction 18, 87 Cauchy stress 8, 11, 286 convective derivative 9, 242 classical thermodynamics 35, 36, 47, convex analysis 4, 17, 217, 263–266, 59, 256, 320 271, 275, 304, 306, 321, 325, 330, 331, classical thermodynamics of fluids 40, 334 43, 48 convex function 265, 328–331, 334 Clausius-Duhem inequality 38, 161 convex sets 327, 334 346 Index convexity 58, 321 E coupled materials 32, 176 effective angle of friction 259 creep 211, 237, 238 effective stresses 159–162, 187 creep rupture 238 elastic material 13, 15, 133, 288, 339 critical state 28, 87, 186, 187, 191, 195, elastic strains 16, 19, 20, 176 203, 204, 209, 210, 340 elasticity 11–16, 20, 77–80, 111, 162– cross-coupling 112, 180, 184 164, 209, 265, 277, 286, 288, 339, 340 elastic-viscoplastic model 216 D elliptical yield surfaces 178, 186 damage mechanics 274 end bearing 293–296 damage parameter 274, 277 endochronic theory 117, 118, 343 Darcy’s law for fluid flow 254 energy functional 134, 193, 230 decoupled materials 103 energy function 42, 45, 47, 49, 68, 80, deformation gradient tensor 6 120, 137, 190, 215, 256, 298 degenerate transform 57, 263 entropy 36, 38, 40–46, 55, 65–70, 74, 75, density 9, 47, 172, 242–244, 254, 286 80, 248–251, 302, 320 deviatoric stress 78, 84, 94, 99, 129, 154, entropy flux 54, 249, 301 159, 182, 200, 201 equation of state 36, 45, 46 differential 10, 11, 20, 47, 57, 59, 68, 74, equations of motion 248, 254 88, 120, 138, 150, 181, 193, 216, 217, equilibrium 8–11, 32, 36, 41, 51, 248, 225–227, 232, 233, 247, 250, 255, 264, 278, 280, 295, 296 295, 296, 299, 300, 306–310, 313, 322– Euclidian distance 326 324, 328 Euler’s theorem 56, 136, 212, 323 dilation 29, 73, 86, 205, 259, 261, 270, Euler-Almansi tensor 7 271, 340 Eulerian formulation 7, 242, 250 Dirac impulse function 306 evolution equations 19, 50, 53, 65, 142, displacement 6–11, 31, 114, 179, 257, 230 290–297 extensive quantities 40, 47, 242, 251 displacement gradient tensor 6 extremum principles 2, 303, 321 dissipation 3, 5, 30, 37, 40, 41, 50, 53– 62, 66–75, 82–103, 121–131, 136, 139, F 143–145, 149, 153, 161, 176, 178, 183, Fenchel dual 265, 266, 275, 304, 321, 188–196, 204–217, 221, 223, 229, 241, 331–335 249, 251, 256, 257, 262, 268–271, 275, fibre-reinforced material 288 278, 284, 285, 289, 293, 294, 297, 298, finite element 2, 62, 112, 138, 172, 230, 301–304, 332–335, 339 343 dissipation functional 136, 138, 143, First Law 36, 38, 246 144, 152, 178, 230 flexible pile 294, 298 dissipative coupling 176 flow potential 213, 215, 218, 219, 222, dissipative generalised stress 3, 54, 56, 224, 230, 233, 270, 277, 303 75, 91, 94, 96, 99, 125, 129, 152, 178, flow potential functional 230, 234 213, 261 flow rule 17, 22, 24, 29, 57, 63, 83–86, dissipative generalised stress function 89, 94, 99, 101, 103, 111, 122, 143– 136, 143, 148, 191 146, 179, 210, 261, 335 dissipative materials 48, 54, 274, 304, fluid 40, 242–248, 253–262, 315 342 fluxes 241–243, 250–254, 262 Drucker’s stability postulate 4 force potential 213, 215, 218, 219, 223, Drucker-Prager model 87, 261 225, 251–256, 259, 261, 268, 275, 277, dry density 243, 250 303 dummy subscripts 21 force potential functional 229, 230 Index 347

Fourier heat conduction law 261 homogeneous first-order function 56, Frechet derivative/differential 144, 193, 58, 70, 73, 75, 121, 188, 229, 269, 303, 195, 295, 298, 302, 308–310 333 free energy functional 134, 143, 144, homogeneous function 88, 212, 214, 196, 234, 296 269, 318, 331 friction 28, 29, 32, 37, 74, 84–86, 159, Hooke’s law 95, 100, 130, 155 205, 209, 210, 261, 271, 340 hyperbolic stress-strain law 191, 192 frictional material 28–32, 69, 90, 103, hyperelastic material 14, 15, 48 204, 210 hyperelasticity 15, 20, 253, 273 hypoelastic material 13, 15 G hypoelasticity 15, 20 hypoplasticity 117 gas constant 45 hysteretic behaviour 28, 107, 110, 111, Gateaux derivative/differential 307, 308 162, 200, 233 gauge function 264, 268, 269, 304, 330– 335 I Gauss’s divergence theorem 243 generalised fluxes 242 Il'iushin's postulate of plasticity 32 generalised forces 242, 296 image point 106–110 generalised failure criterion 207 incompressibility condition 72, 94, 98– generalised signum function 82 102, 128, 130, 152, 155 generalised stress 53–58, 65, 68, 70, 73, incompressibility constraint 79, 287 75, 82, 85–93, 96, 97, 123, 124, 130, incompressible elasticity 78, 81 135, 138, 144, 150, 192, 195, 205, 207, incremental response 48, 62, 68, 69, 74, 213, 229, 261, 268, 270, 278, 289, 303, 75, 90, 92, 96, 123, 124, 138, 145, 150, 332 215, 230, 234, 239, 303, 308 generalised stress function 135, 137, incremental strain vector 107, 108 144, 145, 194, 229 incremental stress vector 106, 107, 116 generalised tensorial signum function incremental stress-strain relationship 83, 306 2, 19–21, 64, 112, 142, 239 generalised thermodynamics 1, 3, 54, indicator function 264–266, 270, 271, 133, 155, 341 329–335 geotechnical materials 2, 28, 74, 142, inertial effects 261, 262 160, 205, 210, 222, 271, 339 initial and boundary conditions 8, 255 gravitational acceleration vector 9 initial stiffness 147, 162, 192, 234, 276 Green-Lagrange strain tensor 7 intensive quantities 40, 253 internal coordinate 134, 137, 228, 285, H 290 internal energy 36–46, 49, 54, 55, 66, hardening laws 28 78, 246–255, 279–282, 303, 320 hardening modulus 22, 24, 109, 110, internal function 103, 121, 134–137, 148 155, 179, 198, 228–234, 342 hardening parameters 18, 19 internal variables 1, 10, 33, 49, 53, 54, hardening plasticity 19, 22, 24, 342 71, 74, 84, 103, 120–125, 131–135, heat capacity 259 142, 155, 173, 198, 224, 225, 228–230, heat engine 39, 40 241, 242, 251, 264, 278, 280, 289, 301, heat flow/flux 36, 39, 41, 44, 50, 66, 74, 330 161, 245, 249, 262, 301 intrinsic time 117 heat supply 37, 40, 41, 54 invariants of the tensor 311 Heaviside step function 306, 336 irrecoverable behaviour 15 Hessian 70, 71, 316 irreversible behaviour 50, 51, 117, 274 hierarchy of models 15, 80, 102, 220 348 Index isentropic 43–46, 67 mean stress 84, 159, 162, 169, 172, 202 isothermal 14, 43, 46–49, 66, 74, 79, mechanical dissipation 50, 55, 75, 86, 102, 258, 259, 287 136, 161, 262, 302 isotropic elasticity 78, 83 mechanical power 36, 37 isotropic hardening 25–28, 92–95, 101, memory of stress reversals 120 103, 210, 341 micromechanical energy 209 isotropic thermoelasticity 49, 79 Minkowski function 330 Iwan model 125–127, 149, 150 mixed invariants 313 Modified Cam-Clay model 162 K modulus coupling 176 modulus of subgrade reaction 297 kinematic hardening 27, 28, 97–103, multiple internal variables 53, 120, 131, 112–115, 119, 121, 123, 130, 142, 147, 135, 224–228, 231 151, 155, 156, 185, 186, 196, 207, 209, multiple stress reversals 177 228, 231, 233, 342 multiple surface models 111, 118, 125, kinematic internal variable 53, 103, 142 120, 175, 225, 257 multisurface hyperplasticity 119 kinetic energy 245, 246, 255 N L nested surface models 111, 118 Lagrangian formulation 7, 242, 250 non-associated plastic flow 2, 32, 204 Lagrangian multiplier 72, 87, 206, 261, non-dilative plasticity 271 278, 280, 287 non-dissipative materials 48 large displacement theory 9 non-intersection condition 112–117 large strain analysis 5, 242 non-linear elasticity 1, 165 Laws of Thermodynamics 15, 162, 210 non-linear viscous behaviour 219 Legendre transform 4, 42, 43, 46–49, non-uniqueness 190 56, 57, 68, 69, 72, 73, 82, 88, 89, 122, normal cone 329, 330, 333–335 123, 137, 143, 144, 167, 205, 212, 213, normality 18, 31, 103, 123, 144 255, 263, 273, 309, 315–324, 331, 333 notation 5 Legendre-Fenchel transformation 82, 230, 256, 261, 321, 331 O limiting strain 182 linear elastic region 100, 119, 179, 181 one-dimensional elastoplasticity 81 linear elasticity 13, 14, 78, 265, 318 Onsager reciprocity relationships 254 linear hardening 27, 96–98, 127, 128, orthogonality condition 53, 56, 63, 226, 341 232, 296 link to conventional plasticity 102, 121 overconsolidated clays 177, 187, 342, loading history 110, 159, 172 343 loading surface 106, 107, 118 overconsolidation ratio 172, 175, 200, logarithmic stress-strain curve 180, 191 341

M P Macaulay brackets 92, 116, 179, 217 partial derivative/differential 307 mapping rule 106, 118 partial Legendre transformations 319 Masing rules 28, 147, 151, 185 passive variables 59, 122, 319 mass balance equations 244, 246, 254 perfect gas 35, 36, 41, 44–46 mass flux 243 perfect plasticity 18–23, 32, 81, 103, 233 material derivative 242, 246, 250 permeability coefficient 259 Maxwell’s relations 43 pile capacity 290 Index 349 pin-jointed structures 277 Q Piola-Kirchhoff stress tensor 250 quadratic functions 78, 318 plastic moduli 148 quasi-homogeneous dissipation plastic modulus function 151, 178, 192 function 254 plastic multiplier 18, 20, 23, 65, 108, 111, 207 R plastic potential 2, 17–22, 29, 32, 33, 58, 86, 111, 122, 210, 261 rate effects 211, 239 plastic strain 16–29, 32, 33, 49, 57, 67, rate process theory 221, 223, 233, 236 73, 82–86, 89, 90, 93–99, 105–114, rate-dependent materials 212, 215, 221, 117, 121–123, 126, 129, 131, 138, 144, 228, 230, 239, 273 149, 150, 154, 156, 172–177, 180, 188, rate-dependent models 224 189, 192, 207, 234, 261, 271, 274, 275, rate-independent materials 1, 3, 51, 291, 335 117, 136, 230, 303 plastic strain increments 16, 18, 69, 173 rates of the plastic strains 87 plastic strain rate tensor 88, 121 rational mechanics 49, 133, 155 plastic work 19, 24, 29, 30, 90, 209, 210, rational thermodynamics 2, 3 335 redundant structure 281 plasticity theory 1–5, 16, 18, 28, 33, 35, reservoir 38–40 57, 58, 62, 89, 107, 117, 143, 177, 263, reversibility 40, 117, 191 264, 300, 304, 321, 323, 334, 335 reversible materials 40 Poisson's ratio 162 reversible processes 41, 49, 341 polar function 269, 270, 304, 333–335 rigid pile 290, 296, 297 pore fluid 160, 161, 243–249, 253, 254, rigid-plastic materials 84 257 rubber elasticity 286, 287 pore water pressure 160 porosity 243, 256 S porous continua 241, 339 saturated granular materials 160 porous medium 241–243, 248, 253 secant shear stiffness 177, 191 potential functionals 142, 148, 151, 177, Second Law 3, 38, 54, 161, 248 210 shear modulus 14, 79, 162, 188, 290, potential functions 2, 74, 88, 89, 93, 98, 341 102, 121, 125, 128, 156, 209, 238, 241, sign convention 8 254, 258, 262, 303, 341 simple shear 16, 26, 94, 99, 102 potentials 2, 59, 74, 122, 173, 176, 213, singular transformation 58, 71, 73, 138, 217, 232, 262, 264, 302, 303, 313, 315, 230, 320, 321, 324 340 skin friction 293 power input 37, 160, 161 sliding element 97, 98, 126–128, 134 Prager’s translation rule 28, 114 slip stress 97, 98, 126, 134, 149 preconsolidation pressure 162, 172– small deformations 6–8 175, 196, 198, 235 small displacement 7, 8 pressure 36, 40–47, 74, 160–163, 166, small strain analysis 5–8, 47 172, 175, 196, 202, 204, 210, 245, 247, small strain region 179 250, 253, 258, 259, 320, 341 small strain stiffness 162 principal stretches 287 small strains 6–8, 50, 162, 183, 203, 257, prismatic beams 284 286 property 36, 38, 42, 46, 54, 57, 74, 88, soil skeleton 161, 242–250, 253–261 229, 253, 301, 331 soils 2, 28, 32, 33, 74, 107, 112, 118, 119, proportional loading 131, 155, 180–183 159, 162, 163, 172, 174, 183, 186, 191, 195, 198, 204, 221, 339–343 350 Index source of heat 37 subgradient 275, 328 specific enthalpy 42, 256 subscript notation 5, 301 specific entropy 40, 45, 248 support function 268, 269, 304, 332– specific Gibbs free energy 42, 93, 96, 98, 335 101, 102, 121, 142, 175, 177, 256 surroundings 36–38, 245 specific Gibbs free energy functional 179 T specific heat 44–46 tangent modulus 98, 127, 128, 150 specific heat at constant pressure 45 temperature 32, 36–46, 54, 55, 65, 66, specific heat at constant volume 45, 46 69, 70, 75, 79, 80, 88, 133, 221, 253, specific Helmholtz free energy 42, 175, 301, 320 256 temperature gradient 41, 55 specific internal energy 40, 74, 246, 256, Terzaghi’s principle of effective stress 301 253 specific volume 36, 40–44, 47, 188, 286, thermal conductivity coefficient 259 320 thermal dissipation 41, 50, 54, 75, 301 S-shaped curve 177 thermal expansion 44, 46, 49, 79, 80 standard material 5, 303 thermal expansion coefficient 259 state variables 35–37, 42, 49, 251 thermally activated processes 221, 233 stiffness 44, 46, 67, 109–112, 147, 151, thermodynamic closed system 35 156, 162, 163, 166, 168–174, 177, 178, thermodynamic efficiency 39 182, 183, 192, 193, 196, 198, 201, 202, thermodynamic equilibrium 36, 51 210, 234, 266, 274–277, 280, 287, 291, thermodynamic process 6, 7, 20, 21, 26, 340, 342 36–46, 50, 51, 75, 80, 81, 110, 148, stiffness matrix 20–25, 32, 48, 66, 170– 183, 211, 221, 222, 233, 242, 249, 269, 172, 178 275, 302, 319, 331, 333 strain contours 169, 170 thermodynamics 1, 2, 4, 5, 15, 18, 31, strain decomposition 20 35, 36, 40, 42, 47–51, 54, 66, 133, 137, strain energy potential 14 162, 242, 256, 304, 341 strain hardening 19, 24, 89, 93, 118, thermodynamics of fluids 40, 47 123, 144, 289 thermodynamics with internal variables strain-hardening hyperplasticity 88 1, 49 strains 7–16, 23, 28, 31, 32, 66, 71, 72, thermoelasticity 48, 79, 80, 261, 340 79, 85, 87, 117, 159, 160, 163, 164, thermomechanics of continua 47 167–170, 177, 186, 187, 191, 195, 203, Third Law 36 205, 226, 233, 243, 251, 262, 266, 277, tortuosity 245, 255, 256 287–289 total differential 307 strain-softening behaviour 31 tractions 10, 245, 247 strength parameters 148 triaxial test 159, 160, 183, 339 stress history 111, 112, 118, 200, 202 true stress space 68, 69, 86, 92, 95, 97, stress reversal 98, 114, 127, 151, 184– 100, 123, 130, 144, 179 186, 201 stress tensor 9, 47, 154, 163, 175, 301 U stress–dilatancy relation 210 stress-induced anisotropy 169, 172 unchanged system 37–39 stretch 286, 287 uncoupled materials 32 structural analysis 277, 300 undamaged Helmholtz free energy 277 structural anisotropy 172 unified soil models 191 St-Venant model 97, 98 uniqueness 58, 190, 303, 321 subdifferential 82, 264–266, 321, 328– unsaturated granular material 161, 340 335, 337 Index 351

V work conjugacy 10, 160 work hardening 19, 24, 341 velocity 9, 160, 243–245, 255, 258 virgin consolidation line 198 Y visco-hyperplastic model 233, 234 viscous materials 212 yield stress 16, 26, 211, 284 voids ratio 162 yield surface 2, 16–33, 57–59, 68–70, volumetric behaviour 29, 78, 100, 129, 83–91, 95, 96, 100, 103–131, 142–147, 153, 165–172, 182, 209, 271, 289 150–155, 177–180, 184, 188–192, 195– volumetric thermal expansion 201, 204–210, 217, 225, 231, 233, 239, coefficients 258 261, 268–270, 289, 304, 330 von Mises yield surface 16, 26, 27, 83, 95, 100, 101, 130, 155 Z Zeroth Law 36 W Ziegler’s orthogonality condition 3, 75, weighting function 143, 148–151, 177, 225, 254, 257, 261, 302 228 Ziegler’s translation rule 28, 114, 186 Winkler method 290