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 va- lue 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 num- bers, 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 func- tional 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 de- fined as follows, although more formal definitions are provided by the subdif- ferential, 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 de- fined 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 func- tional 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 de- fined 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 func- tional 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 varia- tion Gu can be replaced by any fixed v. Furthermore, when this variation is re- placed 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 ˆi KKK w ˆ d (A.11) ³¦¨¸wu 8 i 1©¹i For the definition of partial Legendre transformations, the variable uˆ in defi- nition (A.7) and (A.8) is identified as an n-dimensional subspace of the full N- dimensional space of functions uˆi K .