A Principles of Hyperplasticity Part 4 ppt

Collins and Houlsby also discuss the fact that a yield surface in stress space can be derived by elimination of generalised stresses from Equations 4.11.. An alternative is that O could

potential (the yield surface for the case of associated flow) with respect to stresses

ij

V Collins and Houlsby also discuss the fact that a yield surface in stress space

can be derived by elimination of generalised stresses from Equations (4.11) They

also demonstrate that non-associated flow (in the sense of conventional plasticity

theory) can be derived within this framework and is intimately linked to

stress-dependence of the dissipation function This issue is addressed in Section 4.10

4.4.3 Convexity

Since d F D tij ij 0, it follows that the condition on y is e 0

e ij ij

y

w

wF (because 0

O t ) This has a straightforward geometric interpretation and is simply the

condition that the surface y e contains the origin in generalised stress space 0

and satisfies certain convexity conditions It does not require, however, that the

yield surface should be strictly convex either in generalised stress space or in

stress space

4.4.4 Uniqueness of the Yield Function

There are also relationships for each of the passive variables x ij of the form:

e e

y d

ww

O

where x stands for any of H V Dij, ij, ij,s, or T These relationships demonstrate

that there is a close relationship between the functional forms of d and e y e

Note that, because of the nature of the singular transformation, the functional

form of y is not uniquely determined In particular, the dimension of e y is e

not determined However, the product Oy e must have the dimension

(stress) (strain rate)u If, for instance, O is chosen to have the dimension of

strain rate (i e the same dimension as D ), then it follows that ij y must be e

a homogeneous first-order function in stress Note, however, that quantities

with the dimension of stress might include the stresses V , generalised stresses ij

ij

F , and material properties with the dimension of stress An alternative is that

O could be chosen with the dimensions of (stress) (strain rate)u , in which case

the yield function must be dimensionless We place here no particular

requirement on the form of the yield function In Chapter 13, in which we

express hyperplasticity in a convex analytical framework, we will find that it is

possible to select a preferred form for the yield function, and we shall call this

the canonical yield function

4.6 A Complete Formulation 59

4.5 Transformations from Internal Variable

to Generalised Stress

For each of the functions e (u, f, h or g), a further transformation is possible,

changing the independent variable from D to ij F in the form ij e  F D e ij ij

Correspondingly, the relevant passive variable in d or e y is changed from e D ij

to F After the transformation, note the results ij ij

4.6 A Complete Formulation

Adopting the approach described above, the constitutive behaviour is entirely defined by the specification of two potentials The first is an energy potential, and the second either a dissipation function or the yield surface There are

a total of 16 different possibilities, however, for the choice of the potentials, representing all permutations of the following possibilities:

x choice of u, f, h or g or for the energy function

x dissipation function d or yield surface e y e

x transformation between D and ij F for the energy function ij

The possibilities are illustrated in Table 4.1 In principle any of the 16 formulations could be used to provide a complete specification of the constitutive behaviour of a material In each case, two potentials are specified Technically, it would be possible to specify the energy potential from one of the

16 boxes and the dissipation or yield function from another, but presumably such a mixed form would be adopted only in rather special circumstances The choice of formulation will depend on the application in hand For instance, the four forms of the energy potential in classical thermodynamics are adopted in

different cases (e g isothermal problems, adiabatic problems, etc.)

On differentiating the energy function and dissipation or yield functions with respect to the appropriate variables, the relationships in Table 4.2 are obtained Once the chosen two scalar functions have been specified, the entire constitutive behaviour can be derived from the differentials in the appropriate box in Table 4.2, together with the condition F F ij ij

Table 4.1 The 16 possible formulations

4.6 A Complete Formulation 61 Table 4.2 Results from differentiation of energy and dissipation functions

u

w

V wH

u s

w

T w

u ij ij

d

w

F wD

ij ij

f

w

V wH

f

wT

f ij ij

d

w

F wD

ij ij

h

w

H wV

h s

w

T w

h ij ij

d

w

F wD

ij ij

g

w

H wV

g

wT

g ij ij

d

w

F wD

ij

F

ij ij

u

w

V wH

u s

w

T w

ij ij

u

w

D wF

u ij ij

d

w

F wD

ij ij

f

w

V wH

f

wT

ij ij

f

w

D wF

f ij ij

d

w

F wD

ij ij

h

w

H wV

h s

w

T w

ij ij

h

w

D wF

h ij ij

d

w

F wD

ij ij

g

w

H wV

g

wT

ij ij

g

w

D wF

g ij ij

d

w

F wD

u

w

V wH

u s

w

T w

u ij ij

y

w

D OwF



ij ij

f

w

V wH

f

wT

f ij ij

y

w

D OwF



ij ij

h

w

H wV

h s

w

T w

h ij ij

y

w

D OwF



ij ij

g

w

H wV

g

wT

g ij ij

y

w

D OwF



ij

F

ij ij

u

w

V wH

u s

w

T w

ij ij

u

w

D wF

u ij ij

y

w

D OwF



ij ij

f

w

V wH

f

wT

ij ij

f

w

D wF

f ij ij

y

w

D OwF



ij ij

h

w

H wV

h s

w

T w

ij ij

h

w

D wF

h ij ij

y

w

D OwF



ij ij

g

w

H wV

g

wT

ij ij

g

w

D wF

g ij ij

y

w

D OwF



4.7 Incremental Response

In the numerical analysis of problems involving non-linear materials, the

incremental form of the constitutive relationship is usually required This, for

instance, often forms a central part of a finite element analysis Therefore, one of

the most important criteria that needs to be applied to the formulation of any

model is that the incremental form of the constitutive relationship should be

derived solely by applying standard procedures, without the need to introduce

either ad hoc procedures or additional assumptions Within classical plasticity

theory, more or less standardized procedures are adopted to derive incremental

response [see for example Zienciewicz (1977)], although the mathematical

treatment of the hardening behaviour tends to vary considerably

Differentiation of the energy expressions in Table 4.2 leads straightforwardly

to the results in Table 4.3 where the (symmetrical) matrix > @ucc is defined as

and the matrices > @ucc , > @f cc , ª¬fccº¼, > @hcc , ª º¬ ¼hcc , > @gcc and > @gcc are similarly

defined with appropriate permutation of the energy functions and independent

variables

These incremental relationships are true for both dissipation and yield

function formulations However, in general, the explicit stress-strain response

can be obtained only for those formulations based on the yield functions and

4.7 Incremental Response 63

e

y type For each of these forms the incremental relationships can be written

(noting that F Fij  ) in the following form: ij

where substitutions for e, a , ij b , x, and z are to be taken from the appropriate ij

column of Table 4.4 Equation (4.15) is used together with the flow rule:

e ij ij

y

w

D OwF

The multiplier O is obtained by substituting the above equations in the

consistency condition, which is obtained by differentiating the yield function:

Together with the orthogonality condition in its incremental form F Fij ij, this

can be used to derive

ij ij

Finally, this can be simplified to

ijkl ij ij

z z

mn

y A C

B

w 

e ez ez

mn

y A C

B

w 

and E stands for either D or b

4.7 Incremental Response 65

The first two rows of the matrix in Equation (4.23) describe the incremental

relationships among the stresses, strains, temperature, and entropy The third

and fourth rows are the evolution equations for the generalised stress and the

internal variable The final row allows evaluation of the plastic multiplier O for

the increment The forms of the relationships, after the appropriate substitution

of variables, are given in Table 4.5

The above solution applies only when plastic deformation occurs, i e D zij 0,

and O ! If the above solution results in 0 O  , then it implies that elastic 0

unloading has occurred In this case, the consistency equation no longer applies

but is simply replaced by the condition O For this case, it is straightforward 0

to show that the above relations are replaced by

kl kl

ij ij

ijkl ij ij

ijkl ij ij

kl

kl ij

ijkl ij

ij ijkl ij

kl

kl ij

ijkl ij

ij ijkl ij

The choice of formulation is determined by the application in hand, and to

a certain extent by personal preferences The u and h formulations are

particularly convenient for problems where changes in entropy are determined

(e g adiabatic problems), whilst the f and g formulations are appropriate for

those with prescribed temperature (e g isothermal problems) The u and f

formulations correspond to strain-space based plasticity models and are

particularly applicable when the strains are specified Conversely the h and g

formulations correspond to the more commonly used stress-space plasticity

approaches and are particularly convenient for problems with prescribed

stresses

However, by appropriate numerical manipulation, it is possible to use any of

the formulations for any application For instance, the g formulation leads

directly to the compliance matrix This can be straightforwardly inverted to give

the stiffness matrix

4.8 Isothermal and Adiabatic Conditions

Isothermal conditions can be imposed straightforwardly by the condition T  0

They are most conveniently examined using either the Helmholtz free energy or

Gibbs free energy forms of the equations Thus the isothermal elastic-plastic

stiffness matrix is D ijkl fHH and the isothermal compliance matrix is D ijkl gVV (both

from Table 4.5) For elastic conditions, these reduce to

Adiabatic conditions are slightly more complex In reversible

thermodynamics, the adiabatic condition (no heat flow across boundaries) is

associated with isentropic conditions, but in the presence of dissipation, the

adiabatic condition becomes T  T  F D s d s ij ij 0 Adiabatic conditions are

most conveniently expressed using the internal energy or the enthalpy forms of

the equations Multiplying the fourth line of the appropriate matrix equations is

Table 4.5 by F and substituting the adiabatic condition ij T F Ds ij ij gives

4.9 Plastic Strains 67

which can simply be rearranged to solve for s in terms of either the stress or

strain increment Substituting in the first line of the appropriate matrix equation

in Table 4.5 gives the adiabatic stiffness or compliance behaviour as

Similar substitutions for the entropy increment are necessary in the second to

fifth lines of the equations to solve for the other incremental quantities

Note that for the elastic case, adiabatic and isentropic conditions are

iden-tical, and the stiffness and compliance matrices are simply

So far, no particular interpretation has been placed on the internal variable D ij

By a suitable choice of D , Collins and Houlsby (1997) showed that it is ij

normally possible to write the Gibbs free energy so that the only term that

involves both V and ij D is linear in ij D : ij

1 ij 2 ij 3 ij ij

Furthermore, if g is also linear in the stresses, then Collins and Houlsby (1997) 3

showed that no elastic-plastic coupling occurs In this case, it is always possible

(again by suitable choice of D ) to choose ij g3 V D For this case, it follows ij ij

The interpretation of the above is that D plays exactly the same role as the ij

conventionally defined plastic strain Hij p It is convenient to define elastic strain

H H  D w wV Furthermore, the generalised stress simply differs

from the stress by the term wg2 wD , and it is convenient to introduce the ij

“back stress” defined as U V  F wij ij ij g2 wD Note that ij e e

U U D For this case, the development of the incremental response

equations can be considerably simplified by noting that the differential

ij kl ij kl ij kl

By using the back stress and the elastic strain, further Legendre

transformations are possible that can lead to certain simpler forms of the other

energy functions, but this topic is not further pursued here

4.10 Yield Surface in Stress Space

Consider the case where a material is specified by choosing the Gibbs free energy

 ij, ij

g g V D and the yield function y g y gV D Fij, ij, ij 0 Note that

because the yield function is the Legendre transform of the dissipation function,

either can be used to specify the material

Noting that F F w wD , we can express the generalised stress as ij ij g ij

a function of the true stress and internal variable F Fij ijV Dij, ij Substituting

this in the expression for the yield surface, we obtain

4.11 Conversions Between Potentials 69

We observe that the plastic strain increments are in the direction wy g wFij

They will be “associated” in the conventional sense, i e normal to the yield

surface in true stress space if they are in the direction y* wV Clearly, this is ijonly the case if wy g wV ij 0, that is, if the yield function is independent of the stresses (or, exceptionally, if wy g wVij is always parallel to wy g wFij) From (4.13), we observe that wy g wV ij 0 only if wd g wV ij 0, so that associated flow only occurs if the dissipation is independent of the true stress Conversely,

if the dissipation depends on the stress, it is an inevitable consequence of our approach that flow should be non-associated in the conventional sense Frictional materials involve dissipation which depends on the stresses, and so

we conclude that frictional materials will always involve non-associated flow This observation is entirely consistent with experimental observations on granular materials

4.11 Conversions Between Potentials

In the formulation described here, much emphasis has been placed on the concept that, once two scalar functions are known, then the entire constitutive behaviour of the material is determined Emphasis has also been placed on the fact that there are many possible combinations of functions that can be used, and that these are interrelated through a series of Legendre transformations Different functions may be required for different applications For instance,

a hypothesis about the constitutive behaviour of a material might best be expressed as an assumption about the form of the dissipation function, whereas the incremental response is most conveniently derived from the yield function The ability to transfer between the various functions is therefore vitally important

4.11.1 Entropy and Temperature

The simplest transformations are those between u and h, in terms of entropy, and f and g, in terms of temperature Take the example of the u to f

transformation The equation T T H D ij, ij,s w wu s has to be solved for

the inversion may need considerable ingenuity, or may not even be expressible

in conventional mathematical functions All other transformations between entropy and temperature are possible, subject to analogous conditions

Table 4. 1 The 16 possible formulations