Thông tin tài liệu
B.2 Mixed Invariants 313 B.2 Mixed Invariants The four mixed invariants of two tensors can be written as: 11 22 33ij ji tr a b a b a b a b ab (B.11) 2222 11 22 33 ij jk ki tr a a b a b a b a b ab (B.12) 2222 11 22 33ij jk ki tr a b b a b a b a b ab (B.13) 22 22 22 22 11 22 33ij jk kl li tr aa b b ab ab ab ab (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 ij df da kl a ki lj GG kl a c 1 3 ki lj ij kl GGGG 1 I ij G 2 I 1ji ij aIG 3 I 12 j kki ji ij aa aI IG 2 J 1 1 3 ji ji ij aa I c G 3 J 2jk ki ij aa J cc G tr a ij G 2 tr a 2 j i a 3 tr a 3 jk ki aa tr ab ji b 2 tr ab 2 j kki ab 2 tr ba jk ki bb 22 tr ab 2 jk kl li abb 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 analyti- cal mechanics, between strain energy and complementary energy in elasticity theory, between the various potentials that occur in thermodynamics, and be- tween 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 ex- tremum principles. Sewell (1987) presents a comprehensive account of the the- ory from this viewpoint with particular emphasis on singular points. These transformations have also been widely employed in rate formulations of elas- tic/plastic materials to transfer between stress-rate and deformation-rate poten- tials, e. g. Hill (1959, 1978, 1987); Sewell (1987). These applications are rather different from those used in this book. We review therefore those basic proper- ties of the transformation that are needed in the main text. C.2 Geometrical Representation in ( n + 1)-dimensional Space A function () i Z Xx , 1in ! , defines a surface * in 1n -dimensional , i Z x 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 i Xx are denoted by i y : i i X y x w w (C.1) 316 Appendix C Legendre Transformations Tangent hyperplane Z x i * Q(-Y, 0 i ) P(X, x i ) -Y X Figure C.1. Representation of * in ( n + 1)-dimensional space so that the normal to * in the 1n -dimensional space is 1, i y . If the tan- gent hyperplane at the point P , i Xx on * cuts the Z axis at Q ,0 i Y , the vector , i XYx lies in the tangent hyperplane (Figure C.1), and hence is or- thogonal to the normal to * at P. Forming the scalar product of these two vec- tors therefore leads to iiii Xx Y y xy (C.2) The function i Z Yy defines the family of enveloping tangent hyper- planes and hence is the required dual description of the surface *. The form of this function can be found by eliminating the n variables i x from the 1n equations in (C.1) and (C.2). This can be achieved locally, provided that (C.1) can be inverted and solved for the i x 's, i. e. provided the Hessian matrix 2 i jij y X xxx w w www , is non-singular. Points at which the determinant of the Hessian matrix vanishes are singularities of the transformation (Sewell, 1987). Differen- tiating (C.2) at a non-singular point with respect to i y gives jj ji ji i i xx XY yx xy y y ww ww ww w w (C.3) C.3 Geometrical Representation in n-dimensional Space 317 which, by virtue of (C.1) reduces to i i Y x y w w (C.4) Relations (C.1)–(C.3) define the Legendre transformation. This transforma- tion is self-dual because, if the function i Z Yy is used to define a surface c * “pointwise” in , i Z y space, then i Z Xx describes the same surface c * “planewise” because 1, i x 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 () i Xx is a quadratic form, 1 2 () iijij Xx Axx , where ij A is a non-singular, symmetrical matrix. Hence, the dual variables are iijj i X y Ax x w w , so that 1 iijj xAy , and the Legendre dual is also a quadratic form: 111 11 22 i ii i ij ij ij ij ij ij Yy xy Xx Ayy Ayy Ayy (C.5) The transformation in general is succinctly written i i X y x w w (C.1)bis iiii Xx Y y xy (C.2)bis i i Y x y w w (C.4)bis 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 con- siderations. 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 i x , the relation ,0 ii i ii xy Xx xy CI{ (C.6) defines a family of hyperplanes in n-dimensional i y space. These hyperplanes envelope a surface in this space, the equation of which is obtained by eliminat- ing the i x between (C.4) and 0 i ii X y xx wI w ww (C.7) 318 Appendix C Legendre Transformations On comparison with (C.1) and (C.2), it follows that the equation of this sur- face is i Yy C , so that the hyperplanes defined by (C.4) envelope the level surfaces of the dual function Y. Dually, the hyperplanes defined by , ii i ii xy Yy xy C\{ (C.8) envelope the level surfaces of i Xx in i x space. These level surfaces are, of course, the “cross sections” of the 1n -dimensional surfaces i Z Xx and i Z Yy discussed above. C.4 Homogeneous Functions Of particular importance in applications in continuum mechanics are cases where the function i Z Xx is homogeneous of degree p in the i x 's, so that ii Xx pXxO O for any scalar O. From Euler's theorem for such functions, it follows that ii ii i X p Xx x xy x w w (C.9) so that from (C.2), iiii i Y qY y x y y y w w (C.10) where 11 1 pq , so that the Legendre dual i Yy is necessarily homogeneous of degree 1 p q p . In the example above, 2 p , 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 ij E H and the complementary energy ij C V are both quadratic functions of their argument and satisfy the fundamental relation, ij ij ij ij ECHV VH (C.11) Another case of particular importance in rate-independent plasticity theory occurs when X is homogeneous and of degree one, so that iii Xx xy , in which case the dual function i Yy is identically zero from (C.2). There is a simple geometric interpretation of this far-reaching result. Since ii Xx XxO O , the 1n -dimensional surface i Z Xx is a hypercone with its vertex at the origin. Hence, all tangent hyperplanes meet the Z axis at 0 Z , so that 0 i Yy for all i y . 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 treat- ment (see Appendix D). C.5 Partial Legendre Transformations Now suppose that the functions depend on two families of variables, , ii XxD say, where i x and i D are n- and m-dimensional vectors, respectively. We can perform the Legendre transformation with respect to the i x variables as above and obtain the dual function , ii YyD . The variables i D play a passive role in this transformation and are treated as constant parameters. Hence, the three basic equations are now ,, ii ii ii Xx Yy xyD D (C.12) i i X y x w w and i i Y x y w w (C.13) If the derivatives of X with respect to the passive variables i D are denoted by i E , then it follows from (C.12) that i ii XYww E wD wD (C.14) It is also possible, in general, to perform a Legendre transformation on , ii XxD with respect to the i D variables and construct a second dual function , ii VxE with the properties, ,, ii ii ii Xx VxD E DE (C.15) where i i Xw E wD , i i Vw D wE (C.16) and furthermore: i ii XV y xx ww ww (C.17) since now the i x 's are the passive variables. This process can be continued. A Legendre transformation of , ii YyD with respect to the i D variables produces a fourth function , ii WyE . The same function is obtained by transforming , ii VxE with respect to the i x 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 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 vT , the Gibbs free energy , g pT , 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 i x , so that ,, ii ii Xx X xOD OD , the value of / ii y Xx w w is unaffected by the transformation ii xxoO , and so the mapping from ii x y o is 1fo . Furthermore, since , ii i i xy X x D (C.18) the dual function , ii YyD is identically zero, as already noted above, and so 0 ii ii YY dY dy d y ww D wwD (C.19) But also from (C.13), ii ii i i ii XX x dy y dx dx d x ww D wwD (C.20) i i i i ii X x X y xX wD w E w w D , ),( i i i i ii Y y Y x yY wD w E w w D , ),( i i i i ii W x W x yW wE w D w w E , ),( i i i i ii V x V y xV wE w D w w E , ),( ii yxYX ii WY ED WV xy ii ii XV ED XYWV 0 Figure C.2. Chain of four partial Legendre transformations C.7 Legendre Transformations of Functionals 321 which by virtue of (C.1) reduces to 0 ii i i X xdy d w D wD (C.21) Hence, by comparing (C.19) with (C.21), it follows that i i Y x y w O w and ii XYww O wD wD (C.22) where O is an undetermined scalar, reflecting the non-unique nature of this sin- gular 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 sur- faces 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 trans- formations. These considerations are very important for questions of unique- ness, 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 as- pects 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, > @ ˆ ˆˆˆ , Xx Xx w d 8 KKKK ³ (C.23) where Y is the domain of K and ˆ X is a continuously differentiable function of a functional variable ˆ x . 322 Appendix C Legendre Transformations If ˆ ˆ , ˆ ˆ Xx y x wKK K wK , then the Legendre transform of the function ˆ X is ˆˆ ˆˆˆˆ ,,Yy x y Xx KK K K KK (C.24) It follows from the standard properties of the transform that ˆ ˆ , ˆ ˆ Yy x y wKK K wK . The functional defined by > @ > @ ˆ ˆˆˆ ˆˆˆ ˆ ,Yy Yy w d x y w d Xx 88 KKKK KKKK ³³ (C.25) 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, > @ ˆ ˆˆ ˆ ˆ ˆ ,,, Xxu Xx u w d 8 KKKKK ³ (C.26) where ˆ X is a continuously differentiable function of the variables ˆ x K and ˆ u K . Denoting ˆ ˆˆ ,, ˆ ˆ Xx u y x wKKK K wK , the Legendre transform of the function ˆ X with respect to the variable ˆ x K is defined as ˆˆ ˆˆ ˆ ˆ ˆ ˆ ,, , ,Yyu x y Xx uK K K K K K (C.27) From the standard properties of the transform, it follows that ˆ ˆˆ ,, ˆ ˆ Yy u x y wKKK K wK (C.28) ˆˆ ˆˆ ˆˆ ,, ,, ˆˆ Yy u Xx u uu w K KK w K KK wK wK (C.29) Then, the Legendre transformation of functional (C.26) in function ˆ x , where function ˆ u is a passive variable, is given by the functional, > @ > @ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ,,, ,Yyu Yy u wd xywdXxu 88 KKKKK KKKK ³³ (C.30) [...]... 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 theory Convex analytical approach to (hyper)plasticity Elastic region A convex set in (generalised)... generally the action of a linear operator on a function The space V is the space dual to V under the inner product x*, x , so 326 Appendix D Convex Analysis that x V and x* V More generally, V 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 C1 C2 (D.1) x1 x2 x1 C1 , x2 C2 The operation of. .. (C.39) 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... x 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 g x and the operation f that f 0 x I 0 x x if f x inf f 0 1 f x (D.31) 1 x is called right scalar multiplication [note and f 0 x f x if f x ] D.9 Summary of. .. at a point x, f x may be a set consisting of a single number equal to f x , 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 IC x 0, x C ,x C (D.13) so that the indicator function is simply zero for any x that is a member of the set and elsewhere Although this appears at... 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 1 C The gauge function is always homogeneous of order one in its argument x, so that C x C x (In the language of convex analysis, such functions are simply referred to as positively... that at the boundary of C, the normal cone can also be written NC x This proves to be a convenient IC x C x ,0 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. .. 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): C x* C (D.22) x* The normal cone and the maximal responsive map are inverse in the sense that x C x* (D.23) x* N C x It also follows [see Lemma 4.2 of Han and Reddy (1999)] that C is simply related to the support... Transformations of Functionals 323 and using definitions of Appendix A, it can be confirmed that this definition satisfies the appropriate differential conditions ˆ When x is not a function but a variable, denoted x, all above equations are valid, except that Equation (C.30) may be rewritten as ˆ ˆ Y y, u ˆ Y y, u d ˆ xy X x, u (C.31) ˆ w ˆ , w d (C.32) where ˆ X x, u y , x ˆ When the function X is a. .. defined as d x, y x y x y, x y 12 (D.4) and we define the open ball of radius r and centered at xo as B xo , r x d x, xo (D.5) r The interior of C is then defined as int C x 0, B x, (D.6) C x2 C C x1 (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 (possibly very small) so that a ball of radius is entirely contained . i x variables as above and obtain the dual function , ii YyD . The variables i D play a passive role in this transformation and are treated as constant parameters. Hence, the three basic. 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. 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 i x , the relation
Ngày đăng: 10/08/2014, 11:21
Xem thêm: A Principles of Hyperplasticity part 14 doc
