A Principles of Hyperplasticity part 14 doc

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 const

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

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

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 Z X x( )i , i ! , defines a surface * in 1 n n1-dimensional

Z x space However, the same surface can be regarded as the envelope of , i

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 X x are denoted by i y : i

i i

X y x

w

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

so that the normal to * in the n1-dimensional space is 1,y i If the

tan-gent hyperplane at the point P X x, i on * cuts the Z axis at Q Y,0i, the

vector X Y x , i 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

i i i i

The function Z Y y i 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 x from the i n1

equations in (C.1) and (C.2) This can be achieved locally, provided that (C.1)

can be inverted and solved for the x 's, i e provided the Hessian matrix i

, 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 y gives i

C.3 Geometrical Representation in n-dimensional Space 317

which, by virtue of (C.1) reduces to

i i

Y x y

w

Relations (C.1)–(C.3) define the Legendre transformation This

transforma-tion is self-dual because, if the functransforma-tion Z Y y i is used to define a surface

c

* “pointwise” in Z y, i space, then Z X x i describes the same surface c*

“planewise” because 1,x i define the normal to * and X is the intercept of c

the tangent plane with the Z axis from (C.2)

The transformation is not in general straightforward to perform analytically

An exception is when X x is a quadratic form, ( )i X x( )i 12A x x ij i j, where A is ij

a non-singular, symmetrical matrix Hence, the dual variables are

X y x

Y x y

ww (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 x , the relation i

x y i, i X x i x y i i C 0

defines a family of hyperplanes in n-dimensional y space These hyperplanes i

envelope a surface in this space, the equation of which is obtained by

eliminat-ing the x between (C.4) and i

0

i

X y

wI w



On comparison with (C.1) and (C.2), it follows that the equation of this

sur-face is Y y i C, so that the hyperplanes defined by (C.4) envelope the level

surfaces of the dual function Y Dually, the hyperplanes defined by

x y i, i Y y i x y i i C

envelope the level surfaces of X x in i x space These level surfaces are, of i

course, the “cross sections” of the n1-dimensional surfaces ZX x i and

i

Z Y y discussed above

C.4 Homogeneous Functions

Of particular importance in applications in continuum mechanics are cases

where the function Z X x i is homogeneous of degree p in the x 's, so that i

In the example above, p , so that X and Y are both homogeneous of degree 2

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,

ij ij ij ij

Another case of particular importance in rate-independent plasticity theory

occurs when X is homogeneous and of degree one, so that X x i x y i i, in

which case the dual function Y y is identically zero from (C.2) There is i

a simple geometric interpretation of this far-reaching result Since

X Ox OX x , the n1-dimensional surface Z X x i is a hypercone

with its vertex at the origin Hence, all tangent hyperplanes meet the Z axis at

0

Z , so that Y y i for all 0 y This special case is pursued further later, i

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, X x i,D i

say, where x and i D are n- and m-dimensional vectors, respectively We can i

perform the Legendre transformation with respect to the x variables as above i

and obtain the dual function Y y i,D The variables i D play a passive role in i

this transformation and are treated as constant parameters Hence, the three

basic equations are now

 i, i i, i i i

i i

X y x

ww

and i

i

Y x y

ww (C.13)

If the derivatives of X with respect to the passive variables D are denoted by i

since now the x 's are the passive variables i

This process can be continued A Legendre transformation of Y y i,D with i

respect to the D variables produces a fourth function i W y i,E The same i

function is obtained by transforming V x i,E with respect to the i x variables i

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

of such a closed chain of transformations is in classical thermodynamics, where

the four functions are the internal energy u s v , the Helmholtz free energy  , 

 , 

f Tv , the Gibbs free energy gT,p, and the enthalpy h s p , 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 x , so that i OX x i,D i XOx i,D , the i

value of y i wX/w is unaffected by the transformation x i x io O , and so the x i

mapping from x ioy i is f o Furthermore, since 1

i i X x

X y

x X

wD

wEww

D,

),(

i i

W x

W x

y W

wE

w

Dw

w



E,

),(

i

i i i

i i V x

V y

x V

wE

wDw

w



E,

),(

i

i y x Y

X

i i W

i i X

X Y  W V 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

Y x y

w O

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

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 ˆ ,ˆ ,ˆ

Then, the Legendre transformation of functional (C.26) in function ˆx , where

function ˆu is a passive variable, is given by the functional,

Y y u>ˆ ˆ, @ Y yˆˆ ,uˆ , wˆ d xˆ yˆ wˆ d X x u>ˆ ˆ, @

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 ˆx is not a function but a variable, denoted x, all above equations are

valid, except that Equation (C.30) may be rewritten as

When the function Xˆ is a continuously differentiable function of the

func-tion ˆx (or variable x) and any finite number N of functions ˆ u , the same Equa- i

tions (C.27)–(C.30) are still valid, except that Equation (C.29) unfolds into N

C.7.3 The Singular Transformation

An important case in rate-independent plasticity theory occurs when functional

Then the Legendre transformation of the function X xˆ ˆ K ,uˆ K K with re-, 

spect 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

where O K is an undetermined scalar, reflecting the non-unique nature of this ˆ

singular transformation

Then the Legendre transformation of functional (C.26) in function ˆx , when

function ˆu is a passive variable, is given by the functional,

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 ticity theory

plas-D.2 Some Terminology of Sets

We use brackets ^ ` to indicate a set, so that ^0, 1, 3.5 is simply a set contain-`

ing the numbers 0, 1 and 3.5 A closed set containing a range of numbers is noted by > @, , thus >a b, @ ^x ad dx b`, where the meaning of the contents of

de-the final bracket is “x, such that a x bd d ” 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

n

R (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

func-tion The space V c is the space dual to V under the inner product x x , so *,

that x V and *x Vc More generally, V c 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

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 R , the Euclidian distance is defined as n

D.3 Convex Sets and Functions 327

This means that there exists some H (possibly very small) so that a ball of

ra-dius 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:

D.3 Convex Sets and Functions

A set C is convex if and only if

1 Ox O  , y C x y C,  , 0  O  (D.9) 1where, for instance, x y 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

f  O x Oy d  O f x  Of y , x y C,  , 0  O  (D.10) 1

This is illustrated for a function of a single variable in Figure D.3 Convexity

requires that NP NQd 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

con-vex if d can be replaced by < in (D.10) for all x yz

The effective domain of a function is defined as the part of the domain for

which the function is not; thus, dom f x ^x V f x  f `

Figure D.2 Non-convex and convex functions

z

Q P

Y N

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 V c is

its dual under the inner product , , then x*Vc is said to be a subgradient of

the function f x , x V , if and only if f y f x t x*,y x  , y

The subdifferential, denoted by wf x , is the subset of V c consisting of all

vectors x satisfying the definition of the subgradient: *

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 w x is the signum

function, which we now define as a set-valued function:

Figure C.2 Chain of four partial Legendre transformations

C.7... 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... Geometrical Representation in n-dimensional Space

An alternative geometrical visualisation in n-dimensional space is also valuable

in gaining understanding of formal