A Principles of Hyperplasticity part 12 potx

The principal motivation for adopting the convex analytical approach is that it allows us to deal more rigorously with the relationships between the tion function and the yield function.

262 12 Behaviour of Porous Continua

x large strains,

x fluid flow in porous media,

x heat flow in porous media,

x viscous effects,

x inertial effects

Apart from this generalization, the proposed framework places less stringent restrictions on the class of derived constitutive models in terms of the require-ments of the Second Law of Thermodynamics Within the framework of Chap-ter 4, the fact that the mechanical dissipation had to be non-negative resulted in

a condition that is more stringent than the Second Law Within the present framework, it is the total dissipation (including dissipation due to heat and fluid fluxes) that has to be non-negative, which is equivalent to the Second Law

As in the standard hyperplastic approach, the entire constitutive behaviour is completely defined by specification of two scalar potential functions However,

in the generalised framework, these functions also include properties related to different phases of the media and their interaction The fluid and heat conduc-tion laws are also built into these potentials, completing description of the con-stitutive behaviour of complex media

presen-In general, we follow the terminology employed by Han and Reddy (1999) in their book in which they make much use of convex analysis for conventional plasticity theory We acknowledge, too, that the French school of plasticity has made much use of this approach for many years

The principal motivation for adopting the convex analytical approach is that

it allows us to deal more rigorously with the relationships between the tion function and the yield function It will be recalled that so far we have treated this relationship as a special degenerate case of the Legendre transform In con-vex analysis, the Legendre transform is generalised to the Legendre-Fenchel transform (or Fenchel dual), and this allows more thorough treatment of the degenerate case The alternative cases of elastic or elastic-plastic behaviour are also treated simply by convex analysis

dissipa-The applicability of convex analysis to plasticity becomes so apparent that it seems highly likely that this will become the standard paradigm for expressing plasticity theory Many of the concepts that have been given special names by plasticity theorists have parallels in the much more widely applied field of con-vex analysis The advantages of expressing plasticity in this way are therefore twofold Firstly, there is the extra rigour that is achieved; secondly, numerous

264 13 Convex Analysis and Hyperplasticity

standard mathematical results can be employed, some of which give useful, new

insights into plasticity theory

Further advantages come in the treatment of constraints that arise in (a)

ex-treme cases such as incompressible elasticity or (b) dilation constraints in

plas-ticity These are treated by using indicator functions, which are one of the most

simple and powerful concepts in convex analysis Indicators can also be used to

express unilateral constraints, which arise, for instance, in materials that are

able to sustain compression but not tension

In hyperplasticity, the indeterminacy of the form of the yield function can be

resolved by the use of a canonical yield function which is closely related to the

gauge function of convex analysis

A brief introduction to the concepts of convex analysis and the terminology

used here is given in Appendix D, and familiarity with the material there is

as-sumed in the following sections It is strongly recommended that any reader

unfamiliar with convex analysis should study Appendix D in detail before

pro-ceeding further with this chapter Given that the notation of convex analysis is

not entirely standardised, even a reader familiar with convex analysis may find it

useful to study Appendix D, where our notation and terminology are defined

When potentials are not differentiable in the conventional sense, convex

analy-sis serves as the framework for expressing constitutive behaviour, subject only

to the limitation that the potentials must be convex This does not prove too

restrictive for our purposes A complete exposition of hyperplasticity in convex

analysis terminology would be lengthy, but suffice it to say (at least for simple

examples) that each occurrence of a differential becomes a subdifferential Thus

instead of V w wH we write f Vw H Thus the equations defining the “g f

formulation”, in which gV D and ,  dV D D are specified, may be expressed , , 

in which each of the variables may be a scalar or tensor, and the internal

vari-ables (and generalised stresses) may be a single variable, multiple varivari-ables, or

an infinite number of variables

13.3 Examples from Elasticity 265

13.3 Examples from Elasticity

Before going on to examine more complex problems in plasticity, it is useful to

gain some familiarity with the techniques of convex analysis by looking at some

problems in elasticity

As an example of the way convex analysis can be used to express constraints,

consider some simple variants on elasticity Linear elasticity (in one dimension)

is given by either of the expressions,

22

E

or

22

g E

V

Using derivations based on the subdifferential (which in this case includes

simply the derivative, because both the above are smooth strictly convex

func-tions) Vw H hence f V H , or E Hw  g V hence  H V E

Now consider a rigid material, which can be considered as the limit Eo f

The resulting f can be written in terms of the indicator function:

^ `0

which has the Fenchel dual  g 0

The subdifferential of f gives VN^ `0 H , which gives V f f for > , @ H , 0

and is otherwise empty, so that there is zero strain for any finite stress

Con-versely, non-zero strain is impossible The subdifferential of  (just consisting g

of the derivative) gives H directly, irrespective of the stress In a comparable 0

way, the limit E o , i e an infinitely flexible material, is obtained from either 0

0

f or  g I^ `0 V

The above considerations become of more practical application as one moves

to two and three dimensions For instance, triaxial linear elasticity is given by

32

G

266 13 Convex Analysis and Hyperplasticity

or

26

q g G

without the need to introduce a separate constraint Note that whenever it is

required to constrain a variable x to a zero value, where x is one of the

argu-ments of an energy function, one simply adds the indicator function ^ `I0 x In

the dual form, the Fenchel dual does not depend on the variable conjugate to x

The above results can of course very simply be extended to full continuum

models

Unilateral constraints can also be treated using convex analysis A

one-dimensional material with zero stiffness in tension (i e a “cracking” material)

can obtained from

22

where we recall that the Macaulay bracket is defined such that x if x xt 0

and x if 0 x (where we use a tensile positive convention) Such a model 0

might, for instance, be the starting point for modelling masonry materials,

con-crete, or soft rocks

Another case, rigid in tension and with zero stiffness in compression (in other

words, the “light inextensible string” found in many elementary textbooks) is

In each of the above cases, elementary application of the subdifferential

for-mulae gives the required constitutive behaviour, effectively applying the

“con-straints” (unilateral or bilateral) as required Table 13.1 gives the forms of both f

and g required to specify a number of different types of “elastic” materials,

to-gether with the derived stresses and strains The table illustrates how the convex

analytical framework can be used to express concisely the behaviour of materials

with “corners” in the response, e g at the tension to compression transition

13.3 Examples from Elasticity 267

268 13 Convex Analysis and Hyperplasticity

The dissipation function (which is in this case the same as the force potential)

d d D  z D is a first-order function of D, and the conjugate generalised

stress is defined by Fw D , which is the generalisation of d F w wD d

The set X (capital F) of accessible stress states can be found by identifying the

dissipation function as the support function of a convex set of F; hence applying

Equation (D.21) from Appendix D,

Note that here the notation , is used for an inner product, or more generally

the action of a linear operator on a function

The indicator and gauge functions of X can be determined in the usual way

Note that the indicator is the dual of the support function, so it is the flow

where D w w F N& F , which is the generalisation of D w w wF It is useful

at this stage to obtain the gauge function,

Furthermore, we define the canonical yield function (in the usual sense

adopted in hyperplasticity) as y F J& F  Then, applying Equation 1

where O t [see Lemma 4.5 of Han and Reddy (1999)] The above is the equiva-0

lent of the usual D Ow wF y Clearly, w F plays the role of y y w wF, and O has

its usual meaning In particular, O for a point within the yield surface (inte-0

rior of X) and takes any value in the range >0,f for a point on the yield sur-@

face (boundary of X)

13.4 The Yield Surface Revisited 269

It can be seen, however, that the assumption made in developments in

Chap-ter 4 that, because D w w wF Ow wFy with O an arbitrary multiplier, one could

deduce that w O was slightly too simplistic a step y

Now we are in a position to address the process of obtaining either a yield

surface from a dissipation function or vice versa If we start with d z d D ,

then we apply (13.7) to find the set of admissible states X, and then use (D.15),

together with the definition of the canonical yield function:

so that y F can in principle be determined directly from d D This is an

important result Then, D Ow F y

Conversely, if we first specify the yield surface y F in the normal way, then

X is easily obtained from & F^ y F d0`, and the dissipation function is then

the support function of this set:

d D V & D  F D F& F D y F d (13.18)

so that d D can in principle be determined directly from y F This too is an

important result, although it is more obvious than the transformation from

dissipation to yield

It is not essential for (13.18), but there is a clear preference for expressing the

yield surface in canonical form such that J& F y F  is a homogeneous 1

first-order function of F, so that it can be interpreted as the gauge function of

the set X Note that the yield function is not itself positively homogeneous, but it

is, however, expressible as a positively homogeneous function minus unity If it

is chosen this way, then y is dimensionless, so that O has the dimension of

stress times strain rate

If y is expressed in canonical form, then the dissipation function can be

ex-pressed directly as the polar,

The results are summarised as follows:

Option 1: start from the specified dissipation function d z d D :

d

Fw D (13.20)

270 13 Convex Analysis and Hyperplasticity

Note that if y is not expressed in canonical form, it cannot be readily

con-verted to the gauge, and so the dissipation function cannot simply be obtained

as the polar of the gauge

The function w (the flow potential) is the indicator of the set of admissible

generalised stress states

If y F is in canonical form such that JX F y F 1 is homogeneous of

order one, then applying Option 2 to obtain d and then applying Option 1 to

obtain y will return the original function If this condition is not satisfied, then

applying this procedure will give a different functional form of the yield function

(the canonical form), but specifying the same yield surface Thus if the yield

surface is not originally defined in canonical form, it can be converted to

ca-nonical form by first applying Equation (13.18) and then (13.17) (although in

specific instances there may often be more straightforward ways of achieving the

same objective)

We first consider how indicator functions can be used to introduce dilation

constraints A plastically incompressible cohesive material in triaxial space, with

cohesive strength (maximum allowable shear stress) c, can be defined by

q

y c

F

 (13.27)

13.5 Examples from Plasticity 271

Alternatively both the plastic strain components are introduced, but the

volumetric component is constrained to zero This approach proves more

fruit-ful for further development In the past, this has been achieved by imposing

a separate constraint, but now we do so by introducing an indicator function

into the dissipation:

The yield function is unchanged for this case, and is again given by (13.23)

This model is readily altered to frictional, non-dilative plasticity by changing the

dissipation to

^ `0

Note that we have introduced a Macaulay bracket on p which we did not use

before, but strictly it is necessary to ensure that the dissipation cannot be

nega-tive The corresponding canonical yield function is

The virtue of introducing the plastic volumetric strain is seen in that the

model can now be further modified to include dilation by changing d to

This can be compared with the yield locus y F q Mp B F used in the p 0

earlier example in Chapter 5

The above are some simple examples of the way in which expressions using

convex analysis terminology can provide a succinct description of plasticity

models for geotechnical materials They may provide the starting point for using

this approach in more sophisticated modelling

Chapter 14

Further Topics in Hyperplasticity

14.1 Introduction

In the previous chapters, we developed a technique for describing the behaviour

of materials based on the use of two scalar functions We placed great emphasis

on the fact that, once these functions (or functionals) are known, then the entire response of the material can be determined As we concentrated principally on the behaviour of rate-independent materials, we termed this approach “hyper-plasticity”, although in Chapter 11, we showed that rate-dependent materials, too, can be described by this method

In this chapter, we extend the hyperplasticity approach to modelling lems in a number of different areas The sections of this chapter are not con-nected, but represent different developments, each of which takes the hyperplas-ticity approach as the point of departure They are intended to illustrate the fact that, once an engineer is familiar with this approach to constitutive modelling, it can provide a powerful technique capable of wide generality In these extensions

prob-of the applications prob-of hyperplasticity, the two most important features to be borne in mind are (a) the emphasis on using two scalar functions to define ma-terial behaviour (different choices of functions are available through the use of

Legendre transforms), and (b) no ad hoc assumptions are required; once the

functions are specified, the entire material behaviour follows

Two of the applications considered below (in Sections 14.3 and 14.5) involve non-dissipative systems, so that strictly we should call them applications of hyperelasticity, not hyperplasticity, but we use them here simply to demonstrate the continuity of our approach

274 14 Further Topics in Hyperplasticity

So far, the only rate-independent dissipative materials we have considered are

those in which the internal parameter plays the role of the plastic strain In this

section, we investigate some different forms of the free energy and dissipation

expressions and discover that for these forms the internal variable plays the role

of the damage parameter in conventional isotropic damage models This

devel-opment demonstrates the versatility and generality of the hyperplastic approach

Consider a Helmholtz free energy,

where D is a “damage parameter” such that 0d D d The factor 1 1 D is effec-

tively applied as a reduction factor on the elastic stiffness E The symbol D or ] is

often used in the literature of “continuum damage mechanics” (CDM) for the

damage parameter, but here we use D to emphasise that this is simply

inter-preted as an internal variable and is treated in the same way in the formulation

as before Note, however, that although the role of D is again associated with

irreversible behaviour, it no longer has the physical interpretation of “plastic

F 

The first equation reveals that the usual value of the stress V H for an elas-E

tic model is simply reduced by the factor 1 D When  D (no damage), the 0

model is simply elastic, and as D increases, the stress reduces by comparison

with the undamaged case

It follows, too, that

coupled) plasticity models It can, however, be interpreted as a type of coupled

plasticity in which the reduction of stiffness is so extreme that, on a return to

zero stress, there is no irreversible strain