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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