A Second-Deformation-Gradient Theory of Plasticity

A. Acharya and T.G. Shawki

Department of Theoretical and A pp lied Mechanics University of Illinois at Urbana-Charnpaign

Urbana, Illinois 61801

October 28, 1993

Abstract

Second deformation gradient dependent constitutive postu­

lates for plastic materials are introduced. The Clausius-Duhem

Inequality and the Principle of Material Frame Indifference are

used to deduce restrictions on the constitutive asssumptions. These

general considerations are utilized to lay down the constitutive as­

sumption of a special class of plastic solids, namely elasto-plastic

solids. Interesting features of this special case, such as a mate­

rial length scale in the constitutive description of elasto-plastic

solids and a possible explanation of the Taylor-Quinney effect in

rate independent thermoplasticity, are illustrated. It is expected

that the introduction of a higher gradient of deformation, as op­

posed to those of internal variables, will ease the experimental

determination of specific constitutive statements.

2

1 Introduction

A significant portion of the focus of current research in continuum plasticity

has been directed towards the analysis of classical rate independent plastic­

ity models in the strain softening regime. It is well known that there are

serious drawbacks associated with simulating strain softening with classical

plasticity models. References [1], [2) and [3), in the context of finite strains

and quasi-static deformations, illustrate the essential nature of the problem;

i.e. loss of uniqueness of the solutions to the equations of equilibrium due

to material softening. In the case of dynamic motions, softening induces loss

of hyperbolicity and results in an ill-posed initial-value problem, a fact that

has been well illustrated in [4), [5).

The ill-posedness of the mathematical model primarily manifests itself

through the prediction of narrow bands of localized deformation which tend

to vanishing thickness in numerical simulations and bifurcation of solutions

in analytic quasi-statics. A common feature of the aforementioned analyses

is their inability to predict the observed characteristic width of shear bands.

These difficulties, combined with numerous observations of shear localization

in slow motions of weakly rate sensitive materials, have motivated the intro­

duction of higher order gradients of the effective plastic strain [6), [7), [8),

[9), [10) or the notion of average strain [11) in the constitutive description of

rate-independent materials in the softening regime. Gradients of deforma­

tion, whose origins in a continuum theory date back to [12), have the desired

effect of introducing a length scale to the rate independent, elastic-plastic

problem. Modeling softening materials as a polar-oriented media ( Cosserat

3

continuua) also results in the introduction of an internal length scale, an ap­

proach used in [13] and [14]. From a different viewpoint based on dislocation

theory, a higher order strain gradient-couple stress theory of plasticity has

been introduced in [15].

In summary, theories dealing with modeling rate independent softening

plasticity can be broadly classified into three groups: ( 1) theories with cou­

ple stress, (2) theories that involve ordinary stress as a function of a strain

measure which includes higher order gradients of the total deformation, with

no higher order stresses postulated and (3) theories that include gradients of

some internal variables ( e.g. effective plastic strain) in the evolution laws of

the internal variables.

Theories belonging to category (2) are incompatible with thermodynamics

in the sense of [16], [17] when formulated as a theory with internal variables.

Theories of category (1) have the complication of involving higher order stress

while those belonging to category (3), despite being the simplest in concept

as well as being consistent with thermodynamics, involve a significant prob­

lem in the implementation of the consistency condition of rate independent

plasticity which is illustrated by the following considerations.

As a slight modification on classical rate independent elasto-plasticity,

let J be the yield function expressed as a function of the stress T, effective

plastic strain r and its referential gradient VT. Thus

cp = J(T, r, Vr).

Since T can be expressed as a function of the deformation gradient F and

4

the plastic strain P, the yield function can be written as

cf>= J(F, P, r, vT).

Suppose now that a material point is at yield; 1.e. cf> = 0. The typical

algorithm, in the absence of the gradient VT in the function J, decides from

the the rate of total deformation F, with the internal variables not evolving,

whether ~ > O; i.e. if (3J/3F).F > 01 . F is generated by solving the

equation of balance of linear momentum in a dynamic case, given a stress

field that satisfies cf> ::; 0 at every material point, while in the quasi-static

case, (3J/3F).~F::; 0 for points where cf>= 0 is assumed, the equation of

equilibrium is used to generate an approximate ~F (the 'elastic predictor'

calculations) and the assumption ( 3J/ 3F).~F::; 0 checked, a contradiction

of which results in a plastic process(~r > 0) being invoked in which the

stresses as well as the internal variables are evolved according to ~cf> = 0.

Hence, in the gradient independent case, given F (~F), the effective plastic

strain rate I' ( ~r) is determined by

r = 0 for all points at which aJ .

/4 = 0 and - · F < 0 '+' 3F -

or cf>< 0 aJ .

I'= , -aff.F ,

[!t-f(F, P) + ~t] for points at which

aJ . cf>= 0 and BF.F > 0, 2

1where A.B represents the scalar product of two tensors of the same order, say 'n',

which is given by A; 1 ___ ;nB; 1 .•• in in terms of components with respect to a rectangular

Cartesian coordinate system in the underlying Euclidean point space.

It is hoped that the loose notation in denoting a function and its evaluation in similar

fashion will not cause confusion.

5

--------------------------------------------------

while in the gradient dependent case, the specification of f is

r = 0 for all points where aJ . aJ _o

</> = 0 and aF.F + ovT_Vf:; 0

or 4> < 0

aJ . . [aJ aJ] aJ _o

aF.F + r aP.f(F, P) + ar + avr .vr = 0 for points where (1)

where VI' is replaced by ~(Vr) as required.

In a recent article [18], the boundary conditions associated with a partial

differential equation for r, arising out of considerations similar in concept

to those that gave rise to ( 1), have been specified by invoking a variational

principle in the context of small strains. The loading-unloading conditions,

however, have not been specified, and it is claimed that the lack of a pri­

ori knowledge of the plastically loading zone presents no greater problem in

numerical simulations of plasticity problems involving such models over con­

ventional (gradient independent) models. It is to be noted that in the context

of numerical simulations, the plastically loading zone in a gradient indepen­

dent theory is completely determined by the balance of linear momentum

calculations, while in a gradient dependent theory, this step does not possess

such a simple algorithmic feature. Even when the distinction between the

elastic/plastic loading zones can be made, simulation of gradient dependent

models would require solving an extra boundary-value problem for r. 2? = I'f(F, P)

6

In this paper, we begin by examining the thermodynamics of gradient

dependent plasticity, where the second deformation gradient is introduced as

a constitutive dependency. Both viscoplasticity and rate independent plas­

ticity are considered, in the isothermal and non-isothermal settings. The aim

is to determine whether the introduction of a higher gradient of the deforma­

tion, in the absence of higher order stresses, is permissible in the conventional

theories of plasticity, keeping in mind that the presence of higher gradients

without higher order stresses is ruled out in generalized elasticity [19]. Our

motivation for introducing a higher gradient of the total deformation is that

some information regarding a material length scale ought to be present in

the constitutive description of plastic materials, as is evident from the finite

width of shear bands observed in materials that are thought to be rate­

independent and thermally insensitive, and that localization of deformation,

as the name suggests, primarily manifests itself through the lo<;:alization of

the 'observable' total deformation.

After deducing the thermodynamic implications of gradient dependent

(in the aforementioned sense) models, we introduce a particular example of

a practical, simple and easily implementable model at finite strains, of a

rate independent thermally sensitive elasto--plastic material. For this class of

materials, the constitutive response is completely defined through the specifi­

cation of the specific free energy function, the yield function and the heat flux

response. , This happens as a consequence of compatibility with thermody- ·

namics [16] and a dissipation postulate in the spirit of infinitesimal plasticity.

This example can be easily adapted to model the response of the three other

types of materials considered.

7

2 Thermodynamics

For the sake of completeness, the notions of a thermodynamic process, admis­

sibility of a thermodynamic process and compatibility with thermodynamics

will now be defined.

A set of ten appropriately valued functions of material points 'X' and

time 't' given by

1) X

2) T

3) b

4) r

5) p

6) ' 7) q

8) ,,,

9) 0

10) c

or

motion

stress (symmetric)

specific body force

specific extrinsic heat supply due to radiation

plastic strain (second order tensor valued)

hardening parameter (scalar valued)

heat flux

specific entropy

absolute temperature and

specific internal energy

specific free energy (2)

is called a thermodynamic process if the laws of balance of linear momentum,

balance of angular momentum and balance of energy given by,

divT + pb = pa (a -acceleration; p - density)

8

pc T.L - pr - divq (L - spatial velocity gradient)

respectively, are satisfied at all times 't'.

A thermodynamic process is said to be admissible if it is generated through

the use of a constitutive assumption.

A constitutive asumption is said to be compatible with thermodynamics

if the local form of the Clausius-Duhem inequality given by,

T.L - pi+ p0iJ - q.g/0 > 0 (g - spatial gradient of 0)

or equivalently,

T.L - pJ - pr,0 - q.g/0 2: 0

is satisfied by all admissible processes generated by it.

Having covered the preliminaries, we now consider the thermodynamics

of gradient dependent plastic materials (in the sense of this paper). The

principle of equipresence is not strictly adopted in the 'g' dependence of the

constitutive functions since it can be shown by arguments in [16), [20) and

[17) and the sort of reasoning to be presented subsequently ( rate independent

case) that such a dependence drops out of the stress, free-energy /internal

energy, temperature/entropy response while the evolution equations for the

internal variables and the heat flux remain unrestricted. Such a dependence

is not considered in the evolution equations for the internal variables as

our primary aim is to investigate the validity of the inclusion of the second

gradient of the deformation as a constitutive dependence.

9

3 Thermally sensitive, rate dependent plas­

tic solid

Our constitutive assumption is,

<p J(F, VF, P,0, 1)

T T(F, VF, P, 0, 1)

'f/ ~(F, VF, P, 0, ,)

1P '1/;(F, VF, P, 0, 1)

q q( F, VF, P, 0, , , g)

p - H(cp)II(F, VF, P, 0, 1) 3

1 H(cp) S(F, VF, P,0,1). (3)

To derive necessary conditions for compatibility with thermodynamics,

we assume that

T.L- p,J;- pr,i)- q.g/0?: 0 \/ admissible processes (4)

which, given the form (3), implies that

{ 0 for•< 0

3 H(·) 1 for · 2: 0

10

(5)

Choose a state characterized by F*, VF*, P*, 0*, 1* and any value g* from

the domains of the functions in (:3) 3 cp < 0. Since an arbitrary choice of a

motion, a temperature field and referential distributions of plastic strain and

the hardening parameter generates an admissible process ( due to .the liberty

in choosing b and r in the definition of an admissible process), we construct

the abovementioned entities such that, at an arbitrarily fixed time 't' and

at the material point whose constitutive assumption is being considered, the

values of F, VF, P, 0, 1 and g are given by F*, VF*, P*, 0*, 1* and g*, respec­

tively, with arbitrary rates F, V~F, and iJ. This can be done by considering

truncated Taylor series type expansions for x and 0 in the referential vari­

ables and time. (These argume~1ts are familiar; see e. g. [17].) Since (5) has

to hold for this process, it follows that

yp-T

aJ 8VF

and - q.g/0 > 0

aJ 80

V(F*,VF*,P*,0*,1*) 3 J(F*,VF*,P*,0*,1*) < 0. (6)

For a choice of state F*, VF*, P*, 0*, 1* 3 cp 2: 0, the fact that an admissible

process may be created with uniform 0, along with the reasoning given above

yields

yp-T

aJ 8VF

11

> 0

> 0

aJ 80

V(F*,VF*,P*,0*,,y*) 3 ef>(F*,VF*,P*,0*,,y*) 2: 0. (7)

It is easy to see that if (6) and (7) hold then (5) is satisfied. Thus, (6) and

(7) are equivalent to compatibility with thermodynamics of (3). This implies

that for the class of materials characterized by the constitutive response

(3), the stress and the entropy are independent of VF at the current time

( although a dependence on its prior history is not ruled out), but it is possible

for the evolution equations for the plastic strain and the hardening parameter

to exhibit such a dependence.

12

4 The thermally sensitive, rate independent,

plastic solid

An admissible process for a solid of the present class is defined by the addition

of an extra field, over material points and time and denoted by </> ( the yield

parameter), to the set given in (2), with the added require~ent that, along

with balance of linear momentum, angular momentum and energy, the yield

criteria, namely,

< 0

and </> ~ 0 whenever </> = 0 (8)

are satisfied for all material points of the body at any given time. The

additional requirement reflects the physical notion that no material point of

a rate independent plastic body can violate the yield condition.

We now define the typical solid in this class by the constitutive assump-

tion,

</>

T

r7

VJ

q

e p

r

J( F, v' F, P, 0,,)

T(F, v' F, P, 0,,)

~(F, v' F, P, 0, ,)

1fa ( F, v' F, P, 0, , )

q(F, v' F, P,0,,,g)

~IP,-r constanti loading function

H( </>)H(()>.(F, v' F, P, 0, ,, F, VF, 0)IT(F, v' F, P, 0, ,) 4

H(</>)H(() >.(F, v' F, P, 0, ,, F, VF, 0)S(F, v' F, P, 0,,), (9)

13

with the additional structure that

• A) For fixed values of the first four arguments, ¢( F, VF, P, 0, "() :S 0

can be solved for "/ > 0.

• B) The evolution equations for P and "/ are such that (Sh is satisfied

for arbitrary rates F', VF, 0, i.e. , II and 3 are specified and the scalar

valued function .A is determined from (8)2. This procedure ensures

rate-independence of the constitutive assumption.

• C) The stress function T and the entropy function fJ are continuous on

the set of points in ( F, VF, P, 0, "() space defined by ¢( F, VF, P, 0, "() :S

0.

• D) The specific free energy function -0 is continuously differentiable on

the set ~f points defined by ¢( F, VF, P, 0, "() :S 0.

and

• E) The set of points defined by ¢( F, VF, P, 0, "Y) = 0 are accumula­

tion points of the set defined by ef;(F, VF,P,0,7) < 0. Under certain

smoothness assumptions on efJ this assumption can be shown to be a

proven consequence.

It is also to be understood that the domains of the functions T, fJ, ,(/J, II and

3 are the set defined by <p :S 0 and derivatives of ,(/J with respect to its

arguments, are defined on the set <p = 0 by continuous extension of the

{ o for· S 0 4H(·) I

for•> 0

14

said derivatives on the set cp < 0. Of course, the tacit assumption that the

set cp < 0 defines a.n open set in ( F, VF, P, 0, 1 ) space is involved in this

agreement.

To derive necessary conditions for the compatibility of (9) with thermo­

dynamics, we again assume that (4) holds. This, along with (9) implies,

T a~ . a~ ~ a~ . (T p- - P f}F).F - Pav F.v F + (-pry - P 80 )0-

H(<P)fl(()>. [P ;;.n + P :~s] -q.g/0 2 o V admissible processes. (10)

By the definition of an admissible process and assumption A) above,

we see that, for the given constitutive assumption, an arbitrary choice of a

motion, a temperature field and a referential distribution of plastic strain

determines at least one referential distribution of the hardening parameter,

all of which in turn determine an admissible process. Then, in a similar way

as in the previous section, we arbitrarily fix a material point 'X' and time

't', choose F*, VF*, P*, 0* and g* and construct a motion for the body such

that at the point 'X', whose constitutive description is being considered,

and time 't', the values of F, VF, P, 0 and g are given by F*, VF*, P*, 0*

and g*, respectively, with arbitrary rates P, V~F, and iJ. The corresponding

admissible process is then created by using a distribution of ,* consistent

with cp ~ 0 over the whole body, and the motion, temperature and plastic

strain distribution just created. The point' X', then, is either at yield ( cp = 0)

or below ( cp < 0). Since the process can be repeated for each point of the

15

body, it follows from (9) and (10) that if a material point is below yield,

TF-T

'f/

8J p8F

0

8J 80

and - q.g/0 > 0

V(F*, v'F*,P*,0*,,*) 3 efy(F*, v'F*,P*,0*, 1*) < 0 (11)

has to hold. However, for a choice of state F*, v' F*, P*, 0*, 1* 3 <f> = 0,

the rates P, VF, iJ cannot be varied arbitrarily while all other terms in (10)

remain constant and therefore the usual arguments in the derivation of the

stress and entropy relations cannot be made. An appeal to assumptions C),

D), E), along with the knowledge that 1) a linear function on a finite dimen­

sional vector space is continuous, 2) the composition of inversion and trans­

position of an invertibe tensor is continuous, and 3) tensor multiplication of

two tensor valued continuous functions ( on the same domain) is continuous,

allows us to make a continuous extension argument, which shows that even

for a state at yield,

TF-T 8~ p8F

8~ 0

8'1F

'f/ 8~ 80

- [ 01i &1,_l -H(~)>.. p8P.IT+pa,::::.. > 0

and- q.g/0 > 0

16

V(F*,VF*,P*,0*,,*) 3 J(F*,VF*,P*,0*,,*)=0, (12)

where the last assertion is due to the rate independence of the evolution

equations for the internal variables and a choice of an admissible process

in which the point 'X' is at yield with F = 0 and 0 = 0. The results

(11) and (12) are the implications of compatibility with thermodynamics of

the constitutive assumption (9). Again, it is easy to <;:heck that these are

also sufficient conditions for compatibility with thermodynamics of (9). It is

observed that the nature of the possible dependencies of the rate independent

material is formally the same as that of the rate dependent material, with

obvious modifications in the forms of the evolution equations for the internal

variables.

17

5 The thermally insensitive, rate dependent

plastic solid

For thermally insensitive constitutive assumptions, it is convenient to work

with the specific internal energy and temperature as dependent variables

with the specific entropy as an independent variable. We also assume that

the material is a non-conductor. Thus our constitutive equation for the

generic material of this class is taken to be,

rp ¢( F, \7 F, P, , , TJ)

T T(F,\1F,P, 1 ,r7)

0 iJ ( F, v' F, P, 1 , r7) 00 constant

E i ( F, v' F, P, 1 , r7)

q q ( F, v' F, P, 1 , TJ, g) 0

P H ( rp) IT ( F, v' F, P, , , 77)

' H( rp) S(F, v' F, P, ,, r7). (13)

The necessary conditions for compatibility with thermodynamics arise out of

assummg

T.L - pi - p017 2: 0 V admissible processes,

which implies, for the constitutive assumption (13),

18

Using similar arguments as in Section 3, we have

E 0Or7 + E(F, v' F, P, ,)

TF-T OE poF

ot OE = 0 -- --ov'F ov'F

V( F*, v' F*, P*, 'l, r() 3 ¢( F*, v' F*, P*, 1 *, ry*) < 0. (14)

and

E 0017 + E(F, v' F, P, ,)

TF-T oE PoF

oi oE 0 -- --

ov'F ov'F

[ OE OE~] - Po P .n + Po,::::. > 0

V(F*, v' F*, P*, ,*, r]*) 3 ¢(F*, v'F*,P*, 1*,ry*) 2_ 0. (15)

It is easy to check the sufficiency of (14) and (15) for compatibility with

thermodynamics of (13). Thus, we have that the stress is independent of the

entropy and the second gradient of the deformation ( at the current time),

but the evolution equations remain unrestricted.

19

6 The thermally insensitive, rate indepen­

dent plastic solid

The constitutive assumption for the typical solid in this class is taken to be,

<p ¢( F, VF, P, 1 , 'f/)

T T(F, VF, P,,, ry)

0 0(F, VF, P, 1 , ry) 00 constant

6 €(F, VF, P, ,, ry)

q q(F, VF, P,,,g) - 0

e ¢IP,.,, constant; loading function

p - H(q;) iI(t) >..(F, VF, P;,, 'f/, F', VF)IT(F, VF, P, ,, ry)

1 H(</>) fl(t) >..(F, VF, P, 1, ry, F, VF)S(F, VF, P,,, ry). (16)

We work with the same notion of an admissible process and assumptions

regarding our constitutive assumption as in Section 4, with obvious modifi­

cations for the change in the thermal dependent and independent variables.

The yield function, although a function in ( F, VF, P, 1 , ry) space, is under­

stood to be independent of ry. Then, using almost exactly similar arguments

as in Se~tion 4, we derive the necessary and sufficient conditions for the

compatibility with thermodynamics of (16) to be,

rp-T

8€ 8VF

0077 + g(F, VF, P, ,) ag

Pap . ag 8VF

20

0

and

V(F*, VF*,P*,,*,"l*) 3 J(F*, VF*,P*,,*,'l/*) < 0

t 00 'f/ + i(F, VF, P,,)

TF-T

at avF

- [ ai ai ] -H(e)A Pap.rr+p 812

0

V(F*, VF*, P*,,*, "I*) 3 J(F*, VF*, P* ,,*,"I*)= 0.

Thus, the nature of the constitutive dependence on VF for this class of

materials remains formally the same as for the rate dependent thermally

insensitive material only when the yield domain is endowed with additional

structure.

21

7 Consequences of Material frame Indiffer-

ence

In order to have a properly invariant constitutive theory for any mate­

rial response, the consequences of Material frame indifference5 and Material

symmetry, apart from the thermodynamic restrictions due to the Clausius­

Duhem inequality, have to be examined. In what follows, we do not assume

that the material enjoys any preferred symmetry with respect to any configu­

ration. Also, the restrictions posed by frame indifference on the constitutive

equations of only the thermally sensitive, rate independent solid are deduced.

The other cases are straightforward specializations of the results in this case.

Motivated by the physical interpretation of the internal variable P as a

strain, we endow it with the indifference properties of the total strain tensor

E = l /2( C - I), where I is the second order identity tensor. It is also

postulated that I remains invariant under a change of frame. With this

understanding, the Principle of Material frame indifference requires that

J(F(t), P(t),,(t), 0(t))

Q(t)T(F(t), P(t), ,(t), 0(t))QT(t)

r}(F(t), P(t), ,(t), 0(t))

Q(t)q(F(t), 'v F(t), P(t), 1(t), 0(t),g(t))

J(F(t), 'v F(t), P(t), ,(t), 0(t))

J( Q(t)F(t), P(t), ,(t), 0(t)) 6

T( Q(t)F(t), P(t), ,(t), 0(t))

r}( Q(t)F(t), P(t), 1(t), 0(t))

q( Q(t)F(t), Q(t)'v F(t), P(t), 7

,(t), 0(t), Q(t)g(t))

¢( Q( t)F( t), Q( t) 'v F( t), P( t),

5 An exposition of the fundamental issue can be found in [23].

22

,(t), O(t))

VQ( ·) that are orthogonal tensor valued functions of time on the translation

space ½ of the three dimensional Euclidean point space E3 , which is our

physical setting.

In order to derive necessary conditions for the frame indifference of the

response functions ,J;, T, fJ, q and efJ we make the choice Q(t) = RT(t), where

R(t) is defined by the unique decomposition for P(t) = R(t)U(t), R(t) is an

orthogonal second order tensor and U(t) a symmetric posjtive definite secopd

order tensor. Thus we have,

~(F(t), P(t), ,(t), 0(t)) ,(j;(U(t), P(t), ,(t), 0(t))

'ifJ(C(t), P(t), ,(t), 0(t))

T(F(t), P(t), ,(t), O(t)) - R(t)T(U(t), P(t), ,(t), O(t))RT(t)

r7(F(t), P(t), ,(t), O(t))

ij(F(t), V F(t), P(t), ,(t), O(t), g(t))

F(t)T( C(t), P(t), ,(t), O(t))FT(t)

fJ(U(t), P(t), ,(t), O(t))

- ij(C(t), P(t), ,(t), O(t))

R(t)ij(U(t), RT(t)V F(t), P(t),

1(t), O(t), RT(t)g(t))

F(t)q(C(t), FT(t)V F(t), P(t),

1(t), 0(t), FT(t)g(t))

6 B ( •) is the history of the quantity B upto time 't'. 71n Cartesian components with respect to an orthonormal basis { ei} in the three di-

mensional underlying vector space Va, for A a second order tensor and B a third order

tensor, (AB),jk = A,mBmjk•

23

ct(F(t), V F(t), P(t), 1(t), 0(t)) J(U(t), Rr(t)V F(t), P(t), 1(t), 0(t))

J( C(t), FT(t)V F(t), P(t), 1(t), 0(t) ),

(17)

where functions denoted by overhead 'v' are particular functions of their ar­

guments, determined by the the structure of the corresponding 'A ' function,

the polar decomposition of F and the relation C(t) = FT(t)F(t) = U(t)U(t).

It can be verified that if the response functions ,J, T, ft, q and J have either of

the two forms given in (17) for each of them, then material frame indifference

is satisfied. Thus these forms are equivalent to demanding frame indifference

for the 1/J, T, r7, q and q> response for this class of material.

For the evolution equations for the internal variables, it is easier to con­

sider the frame indifference of a quantity, say A, given by the constitutive

equation

A= ~(P(t), 1(t),F(·), VF(·),0(·)),

with the added stipulation that A remain invariant under a change of frame.

It can now be deduced that the frame indifference of A is equivalent to

demanding

8'(P(t), 1(t), F(·), V F(·),0(·)) 'Zs(P(t), 1(t), U(·), RT(-)V F(·), 0(·))

CJ'( P(t), 1(t), C( · ), FT(-)V F( · ), 0( ·) ),

where, again, CJ' is a particular function of its arguments, determined through

the structure of ~ and the relationships between F, R, U and C.

Noticing the dependencies of the 'v' set of functions, and realizing that

the pair C and F is a more convenient set of quantities in practical imple-

24

mentations of theory over R and U,. we find that a material response defined

by

'ljJ(t) ;/;(C(t), P(t), ,y(t), 0(t))

T(t) F(t)T( C(t), P(t), ,y(t), 0(t) )FT(t)

ry(t) ij(C(t), P(t), ,(t), 0(t))

q(t) F(t)q(C(t), FT(t)'V F(t), P(t),

,y(t), 0(t), FT(t)g(t))

</J(t) J(C(t), FT(t)'V F(t), P(t), 1(t), 0(t))

A(t) 8'( C(t), FT(t)V F(t), P(t), ,y(t), C(t), FTV F(t), 0(t)), (18)

where the ' ~ ' set of functions are arbitrary upto ,the smoothness require­

ments of the fields they represent, satisfies the Principle of material frame

indifference. In the next section, we utilize this result for laying down the

constitutive equations for a special class of plastic solids that we shall bein~

terested in. Since 7 and P remain invariant under a change of frame, so do i' and P and the deliberations for the indifference requirements of the response

function for A apply to the response functions for i' and P '. Thus, any ap­

propriately valued function, say S, with the dependencies on the arguments

of § such that

S(C(t), FT(t)V F(t), P(t), 1(t), aC(t), aFTV F(t), a0(t))

aS(C(t), FT(t)V F(t), P(t), ,y(t), C(t), FTV F(t), 0(t)) Va E ~

could be a frame indifferent response function for i' and P.

25

8 A special case; thermally sensitive, rate

independent elasto-plastic solids

Oul3 point of view regarding the modeling of plasticity is that every choice

of a motion determines a value of strain at a point at a given time, and

the history of the motion and temperature field upto that time decides the

plastic strain (internal state variables) at the point through a constitutive

statement. Given such a constitutive statement, the material response is

completely specified. An elastic strain may be defined in certain cases from .

E and P if the physics so demands - however, the notion is not essential to

plasticity.

In this section9 we consider a special solid belonging to the general class of

Section 4, which fits the intuitive notion of elastO:-plastic solids, as outlined

in [22). · Assuming P to be a symmetric second order tensor, we define a

quantity, the elastic strain, as

Ee= E-P,

and require that

• 1) it be possible to determine Ee, at a material point, from a knowledge

of the stress, temperature and the deformation gradient at that point.

and

8This appears to be the understanding in [22]. 9 Points of similarity in the considerations of this section and [21] exist. However, the

divergence in concept and basic assumptions prevented us from making a close comparison.

26

• 2) That the constitutive equation for the stress be of such a form that,

when the stress at a point vanishes and the temperature is brought to

some base value, the value of Ee is the zero second order tensor. It is

clear that this requirement implies that at an unloaded state (in terms

of stress and temperature), the total strain is equal to the plastic strain.

That this requirement can be realized by at least one idealized material

in the general class of Section 4 will be illustrated in the following

considerations.

The specific free energy is taken to be of the form,

'Ip

( 8U)T aEe

The heat flux is assigned to be

U(Ee,0) + H(,,0)

8H 8100 =/=. O

q= FGFTg,

where G is a negative semi-definite second order tensor. The choice of q is

governed by simplicity and an effort in moving closer to fulfilling the sufficient

conditions for compatibility with thermodynamics and frame indifference.

From the thermodynamic considerations of Section 4 the stress has the form,

(19)

and the entropy,

27

Since p = p0 /det(F) and det(F) = (det(C)) 1!2 , p0 being the density in the

reference configuration, it is clear that the specific free energy, stress, heat

flux and the entropy are frame indifferent.

To demonstrate the earlier claim of providing an example of an idealized

elasto-plastic material, in the sense of this section, consider a material for

which,

:~e = CEe + (0- 00 )M,

where C is a fourth order tensor, symmetric in its first two indices when

expressed in terms of components with respect to an orthonormal basis in

½ with the the relation A = CB being invertible ( A and B second order

tensors), 00 is the ambient temperature and M is a symmetric second order

tensor. Clearly, this material satisfies the requirements I) and 2) laid down

before for an idealized elasto-plastic material. In order to demonstrate that

the identity E = Ee+ P (often referred to as a 'decomposition' of E) does

not imply an uncoupling of elastic and plastic phenomena, consider

C = 2( Ee + P) + I.

Since Chas a unique tensor square root U which is related to the deformation

gradient by the relation F = RU, R being the rotation tensor, F can be

expressed as

F = RPsym(ll2)(2(Ee + P) + I),

where Psym112 (·) is the tensor square root function of a symmetric positive

definite second order tensor. Substituting this expression for the deformation

gradient into equation (19) it can be seen that the elastic strain Ee is indeed

28

coupled to the plastic strain P through the stress constitutive equation in

this special example. We now take a look at the dissipation, apart from the

heat conduction, given by

which for this special case becomes

Defining O'. = -pa,J;;a, and DP= p-T p p-1 in analogy with D = F-T j;;p-I,

D the symmetric part of the spatial velocity gradient L, Dtoc takes the form

D1oc = T.DP + 0'.1.

In keeping with the spirit of infinitesimal plasticity [24], it is now postulated . ~ .

that, for fixed F, 0, VF, P, ,, C, VF and 0,

• 1) There exists an elastic domain in (T, a) space whose boundary points

(the yield surface cp = J(T, a) = 0) are dissipative states.

• 2) For any material point, the responses DP and 1 are completely de­

termined from the knowledge of T and a at that point.

• 3) For a point (T*, a*) on the yield surface, the values of DP(T*, a*) and

,y(T*, a*) are such that the value of D10 c is a maximum (not necessarily

absolute) over all posssible dissipative states.

10The terminology is in the spirit of [23].

29

These assumptions translate to the mathematical statement that

L(T, a)= T.DP(T*, a*)+ cry(T*, a*) - )\(p(T, a)

is a maximum at (T*, a*), where A is a Lagrange multiplier. The last sentence

implies

and the fact that the surface <p = 0 in (T, a) space is convex. Following the

consequences of the dissipation postulate, we now define the yield surface by

the function

ef>(T, a)= IITl'2 - p ~~ - r(IIV Fib 0), 11

where r( I IV Fl '3, 0) ~ 0 for all values of the arguments. It can easily be seen

that ¢(T, a)= 0 is a convex surface in (T, a) space.

To show that the <p response satisfies frame indifference, it is sufficient

to show that IIV Fib is a function of the arguments in (185), since IITllz =

( C. ~9J;e C JJ;e )112 . This is indeed the case, as is shown in the following demon­

stration.

IIRTV Fll3

11rr1 Frv Fll3

f(C, FTV F, P, 1 , 0),

11 I I · I In is the norm of an nth order tensor argument, defined in terms of the nth order

scalar product as IIAlln = (A.A) 112

;30

where J( ·) is a particular function of its arguments. The evolution equations,

with the assumed yield function, have the form

DP H(cp)H(t) .\T/IITll2

, H(c/>)H(() .\,

where ,\ is determined from the condition cp = 0 whenever cp = 0 and

( = ~IP,~( constant > 0. It is to be noted that, in the context of numeri­

cal implementation, the 'elastic predictor' calculations completely determine

the value of ( even with the gradient dependence. To verify the fact that the

P and ,y responses for this model are frame indifferent, it is convenient to

express the evolution equations in the form

p

and note that ,\ and ( are functions of the arguments of the reduced frame

indifferent yield function J (185 ) and the rates E, IIVFlb and 0. Since E,

IIV Fll3 and 0 remain invariant under a change of frame, so do their rates, and

hence the constitutive equations for the evolution of the internal variables

are indifferent.

Due to the specific choice of the yield function, the local dissipation in

the- model, apart from heat conduction, is given by

- [ 8H] D1oc H(cp)H(() ,\ IITll2 - pa, H(cp)H(() .\r(IIV Fib, 0).

31

It is evident at this point that the choice of the heat flux together with the

form of the local dissipation for this model ensures satisfaction of (124,5 ). It

is also clear that the dissipation along with all other dependent constitutive

quantities are dependent on an internal material length scale associated with

the non-dimensionalization of VF. An interesting observation is made when

the temperature evolution equation is analyzed at a material point loading

plastically for this model . In general, balance of energy yields,

. . p0iJ = T.D - p'lj; - p170 - divq - r.

Clearly, for this special ( elasto-plastic) material,

. . T.L - p'lj; - pr70

Denoting Ts= -p0[)2 H/8081 , and keeping in mind that Ts ~ 0 ( p0 > 0 and

-82H/8081 = 8r7/8, ~ 0) 12 , the temperature equation takes the form

{ f)2 ( U + H) } · _ . 82 U · e • •

- p0 a02 0 - ,[T - Ts]+ p0 80f)Ee E - divq - r' (20)

and it is interesting to observe that the term 1Ts on the right side of (20)

accounts for the dissipation of energy due to strain hardening that is not dis­

sipated as heat, and may be interpreted as the cause for the Taylor-Quinney

effect [2.5].

12where the intuitive notion that the entropy has to be nondecreasing with increasing

strain hardening has been used.

32

9 Conclusion

The thermodynamical restrictions on the constitutive assumptions of second

gradient of deformation dependent plastic solids have been fully analyzed.

The restrictions due to material frame indifference on these constitutive

models have been laid down. The thermodynamic considerations hold for

any theory of plasticity with internal variables where plasticity is character­

ized by a tensor valued internal variable and a scalar internal variable. In

particular, the conclusions presented hold for the usual gradient independent

plasticity theories. The frame indifference considerations hold for theories

in which the indifference requirement for the tensor valued internal variable

is that it remain invariant under a change of frame. When the indifference

requirement for the tensor internal variable is different from what has been

considered here, but is spelled out explicitly, the method used in this paper

can be easily adapted to deduce its consequences.

In variance with generalized elasticity in the sense of Gurtin [19], a theory

with internal variables is seen to be able to exhibit a higher gradient ofdefor­

mation dependence, albeit in a restricted fashion and only through certain

functions, in the absence of multi polar stresses; it is also found that this hap­

pens without the introduction of additional kinematic entities ( e. g. Cosserat

continuua) in the geometric description of classical continuua. Such a depen­

dence, in our opinion, is of importance, since it implies the introduction of

a material length scale through the constitutive theory of plastic materials

at no extra cost in terms of complexity and change in the classical notions

about plasticity. It is also to be noted that the length scale enters through

33

an 'observable' quantity, in contrast with theories that introduce gradients of

the internal variables which necessarily involves difficulties with the notion

of local ( restricted to a material point) plastic loading, thus keeping open the

avenue of experimental measurement of such effects in constitutive response

with relative ease. As it appears to us, the current suggestion also seems to

obviate the need for extra boundary conditions in posing the initial boundary

value problem of the classical balance laws.

Of interest is also the derivation of the thermodynamic stress and entropy

relations for the rate independent material, when viewed in the context of a

theory with internal variables. In this regard, the first use of Coleman and

Noll's procedure [16) for plasticity theory seems to he that in (p. 266)(22).

However, it is not clear in their work how, having derived the aforementioned

relations for points below yield, it is valid to use the relations for a point at

yield, since it is not possible to find a process in which, in their terms, e' and T, can be varied arbitrarily at such states, while all other terms in the

Clausius-Duhem inequality remain constant. In this regard, the remark in

[26), [27),(28) that the classical stress and entropy relations do not follow

from 'Coleman's method' requires a judgement caUin acceptance. Clearly,

the relations do not follow by the usual arguments; however, it has to be

kept in mind that Coleman and Gurtin's [17] deliberations did not include a·

yield type phenomenon. As is shown in this paper, if the assumptions (not

totally unplausible) on the nature of the yield domain are granted, then the

relations do follow. It is also to be kept in mind that the 'Coleman and Noll'

procedure typically requires conditions of equivalence to be established for

compatibility with thermodynamics of constitutive assumptions. Lubliner's

34

[26), [27],[28] alternative yields sufficient conditions for compatibility with

thermodynamics. To see this, using Lubliner's language and notation from

[26], it is demanded that the Clausius-Planck inequality hold for a point at

yield for all admissible processes. This implies that (equation after (6), p.

127 [26]) 8¢ . . .

(I: - BA ).A+ o-(A, q, g(A, q)) < n.A >~ 0, (21)

If the further assumption, suggested by Lubliner, is now made "that an elastic

process - one in which n.A :::; 0 by hypothesis - is quasi-reversible, that is,

in such a process the Clausius-Planck inequality holds as an equality", then

(21) implies that

0

whenever n.A :::; 0.

In particular,

0

whenever n.A= 0,

which in turn implies,

where f3 is a scalar multiplier. Thus, the stress and entropy relations do not

follow as necessary conditions for compatibility with thermodynamics despite

the added assumption. Of course, the necessary conditions are important in

this case, since they tell us which conditions cannot be violated in order to

remain compatible with thermodynamics.

35

In this paper is also illustrated the constitutive assumptions of a spe­

cial class of gradient dependent rate independent plastic solids called elasto­

plastic solids. The definition of such a material is laid down, and it is shown

with the aid of simple arguments that the coupling of elastic and plastic

deformation occurs in this model through the stress response. It is perhaps

. important to note that the mistaken notion, that a theory involving the use

of the identity E = Ee + P excludes the possibilty of exhibiting the afore­

mentioned coupling, was one of the root causes for the introduction of the

so-called 'multiplicative decomposition' [29], [30]. It is emphasized that the

content of this paper, apart from this special example, is general in terms of

applicability. It is found that the plastic flow, as it arises naturally in this

model, is non isochoric - a fact that finds justification in [31 ]. Apart from

illustrating the dependence of the constitutively dependent variables and the

dissipation in the model on an internal length scale, the results of this section

also provide an argument which may be taken to be an explanation of the

Taylor-Quinney effect.

Further work will involve numerical implementation of the aforementioned

models. It is hoped that the contents of this paper will find use in the analysis

of important problems such as deformation localization in plastic solids.

Acknowledgements

The first author would like to express his deep gratitude to Prof. Don Carlson

without whose training, guidance and· constant encouragement, this paper

would not have come to existence. A conversation with Prof. David Owen,

36

which helped in the understanding of thermodynamics of plasticity, is also

hereby acknowledged.

37

---- -----------------------

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41

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