/Ilr. J .. \icH:·Unf'IJr
M ..-dumic-,.
Vol. 8. pp. 261-~77, Pergamon Prn.~ 1913. Printed in Great Britain
A UNIFIED THEORY OF THERMOVISCOPLASTICITY
OF CRYSTALLINE SOLIDS
D. R.
Department
of Engineering
AblitnIet-A
unified Iheory
that a therrnodynamies
for
mechanics of materials with
tion behaviour of crystalline
plasticity arc examined.
BHANDARI
Mechanics.
and J. T.
University
ODEN
of Alabama.
Huntsville.
Alabama.
U.S.A.
of thermoviscoplasticity
of crystalline solids is presented. In parlicular
'viscoplastic'
materials can be accommodated
within the framework
memory. The basic physical concepts are derived from the consideration
solids. Relationships
of Ihe present approach 10 several of Ihe cxisling
it is shown
of modem
of disloca[heories of
I. INTRODUCTION
TIiL'; paper describes a rather general theory of thcrmoviscoplasticity
of crystalline solids,
along with the necessary thermodynamic considerations that must underline it. The theory
itself involves generalizations
of ideas proposed early in the development of modern
continuum thermodynamics,
principally in the linear theories of irreversible thermodynamics of materials with memory of Biot [1. 2] and Ziegler [3. 4]. In their theory,
dissipation was included in the governing functional as a quadratic form in the rates-ofchange of certain internal state variables. Their development was based on Onsager's work.
with its characteristic symmetries, and is often referred to as the "classica]"' thermodynamics
of irreversible processes. Similar concepts were used for special thermodynamic descriptions of certain elastic, viscoelastic and plastic materials by Drucker [5]. Dillon [6],
Vakulenko [7). Kluitenberg [8,9]. and Kestin [10]. among others. Extensions of the
internal state variable (hidden variable) approach to the thermodynamics
of non-linear
viscoelastic materials have been explored extensively by Schapery and by Valanis in a
number of papers (e.g. [11-14]).
The notion of internal or hidden variables was, at first. not easily reconciled within the
framework of modern continuum mechanics. and a number of alternate approaches to
the development of a thermodynamic
theory of materials with memory were initiated.
Perhaps the most prominent among these was the work on dissipative media by Coleman
[15J and Coleman and M izel [16]. which extended the earlier isothermal theory of Coleman and Noll [17]. ]n Coleman's thermodynamic
theory of simple materials [15J. it is
shown that certain materials can be characterized by only two constitutive functionals, one
describing the free energy, which is independent of temperature gradient. and the other
describing the heat flux. A key feature of Coleman's theory is that the stress. entropy and
the internal dissipation are determined as Frechet differentials of free energy functional.
More recently. the thermodynamics
of non-linear materials with internal state variables
has been studied by Coleman and Gurtin [18]. In their approach. the collection of constitutive equations describes what is generally referred to as a material of the evolution type.
Here a separate constitutive equation is given for the rate of change of hidden variable
261
262
D. R.
BHA.NDARI
and J. T.
ODFN
which is often referred to as an equation of evolution. Coleman and Gurtin point out that
their approach [18] to continuum thermodynamics
is but one of the several approaches
including those based on constitutive equations of differential type (e.g. Colcman and Mizel
[19]. Schapery [20]. Perzyna and Olszak [21] and others).
All these approaches are generally regarded as independent of one another. However.
attempts have been made by Coleman and Gurtin [18] and by Lubliner [22] to unify
them. For example. if it is assumed that the solutions of the evolution equations are stable.
then the stability postulate in the theory of Coleman and Gurtin [18J gives most of the
qualitative properties of Coleman's theory of materials with memory [15]. It is shown by
Lubliner [22J that under specific but fairly broad conditions, the principle of fading memory
of Coleman [15] is obeyed by non-linear evolutionary materials. Furthermore, the generalized stress relations derived by Coleman and Gurtin [18J are implicit in the work of
Coleman [15] and the stronger results of Coleman are valid under equivalent constitutive
hypothesis.
General theories of thermoviscoelasticity
derived from, say. Coleman's thermodynamics
of simple materials [15]. are generally regarded as adequate for describing non-linear
behavior in most polymers and even in certain metals if no dislocations take place (rather.
if the dislocation density is small). Since such theories generally assume that the response
of the material is governed by some mcasure of the gradient of the motion (c.g. F. Cor i').
the absolute tempcrature 0, and the temperature gradient g, it seems logical that state
variables (independent constitutive variables) must be introduced in the 'plasticity' phenomena such as yielding. strain hardening. etc., are to be encompassed by the constitutive
cquations. Some macroscopic measure of the influence of dislocations immediately arises
as a likely candidate for such additional measures.
Coleman-Gurtin
type thermodynamics
for the study of elastoplastic materials has
been used by Kratochvil and Dillon [23. 24]. Tseng [25]. and Hahn [26]. These investigators have employed certain basic concepts from the theory of dislocations in crystalline
solids to interpret various internal state variables. Arguments have been made that dislocations. their arrangements and their interactions in crystalline solids play the role of
internal state variables. The present study follows a pattern similar to that of Kratochvil
and Dillon [23] in that we use Kroner's arguments from dislocation theory to justify
the inclusion of "'hidden state variables"' which manifests itself in the form of second-order
tcnsors Alil. Howevcr. we carry the study a bit deeper by also investigating the relationship
of this theory to others existing or recently proposed in the literaturc. Perzyna and Wonjo
[27] also developed a thermodynamic
theory of viscoplasticity by introducing a secondorder tensor A called the inelastic strain tensor. which played the role of a hidden state
variable. Recently Oden and Bhandari [28] prcsented a theory of thermoplastic materials
with memory based on an extension of Coleman's thermodynamics
of simple materials
[15]. Among features of their work were that their theory does not make use of the idea
of yield surfaces and it can be reduced to either Coleman's theory [15] or the GreenNaghdi theory of plasticity [29] as special cases. In the present work it is shown that the
functional theory of Oden and Bhandari [28] can be obtaincd from the evolutionary
theory prcscnted herein by introducing certain plausible assumptions concerning properties
of the equations of evolution.
In the present paper what might be called a Coleman-Gurtin
type thermodynamics
of
non-linear materials with internal state variables is dcveloped and is applied to a combincd
treatment of rheologic and plastic phenomena. A theory of viscoplasticity of crystalline
A uniJled Theory of Ihermol'iscoplasTicily o( crysTalline solid.,
263
matcrials is constructed which is unified. in that a thcrmodynamics
for "viscoplastic"
materials is accommodatcd
within thc framcwork of modern continuum mechanics of
materials with memory. The basic physical concepts are dcrived from the consideration
of dislocation behaviour of crystalline solids, and emphasis is placed on what appears to be
a logical identification of internal state variables. Finally. the relationships of this approach
to sevcral of the existing theories of plasticity are examined.
2. 50\1E
PHYSICAL ASPECTS FRO"
DISLOCATION
THEORY
Before stating specifically the constitutive equations for our thermoplastically
simple
materials, it is necessary to choose a sct of state variables suitable for describing plastic
phenomena. In the following, we briefly summarize elcments of the theory of dislocations
which are useful in choosing appropriate state quanti tics for the class of materials studied
here. For a detailed discussion of dislocation behaviour in crystalline solids. see. for example.
the articles of Bilby [30]. Taylor [31]. Kondo [32]. Mura [33]. and books ofCottrel [34].
Read [35]. Kroner [36]. and Gilman [37].
It is well established in crystal physics that "plasticity"' phenomena such as yielding,
creep and work-hardening, etc .. are the fundamental mechanical properties exhibited by
most solids with crystalline structurc. The basic feature of a crystalline solid is tbe regularity
and periodicity of crystal lattice structure. A dislocation is a line discontinuity in the
atomic lattice: it represents a defect in the regularity or the ordered state of an otherwise
perfect lattice. One important result of the microscopic theories of plasticity of crystallinc
solids is that among the crystal lattice defects such as impurity atoms, vacancies. grainboundaries, etc .. dislocations play the most important role: it can be argued that their
motion and generation in crystals account for nearly all the plasticity phenomena. Furthermore. the plastic (or irrecoverable) deformation in crystalline solids is a now process. the
basic now mechanism being a slipping of crystals caused by the motion of dislocations.
Burger's vector B [34] is frequcntly cmployed to charm;terize the magnitude and direction
of such slip movements in crystals: the magnitude of the vector indicates the amount of
slip occurring and the direction indicates the direction of relative movcment undergonc
by two originally contiguous points.
Advances in solid state physics and metal physics (e.g. see [34-36]) have shown that
any dislocation can be constructed from edge and screw segments and any motion resolved
into components in the slip plane (glide) and normal to the slip plane (climb). The gliding
of dislocations causes layers of crystals to slip over one another producing the now process
during plastic deformation. The amount of slip in one plane is always equal to a multiple
of the Burgers vector. so that the crystal lattice pattern after slip always retains its regularity. Thus. one significant feature of plastic deformation in a crystalline solid is that it
changes the shape of crystals without destroying its crystallinity. The shape change due
to slip is generally referred to as plastic distortion.
Another important aspect in dislocation theory is the multiplication and intersection
of dislocations. The density and distribution of dislocation lines in a crystal usually increase
during plastic deformation by dislocation multiplication processes such as Frank-Reed
sources and multiple cross glides. The increase in dislocation density raises the internal
energy of the crystal and thereby facilitatcs the plastic now. but at the same timc dcvelops
resistance (drag) forccs to further dislocation motion. This interaction of dislocations in a
crystal may account for work-hardening
phenomena. It appears that the complex nature
264
D. R.
BHASDARJ
and 1. 1'. OnEN
of plastic phenomena in crystallinc solids is basically different from that of non-crystalline
solids.
From the above discussion, it seems obvious to assume that for microscopic theories
of such materials, crystal defects (dislocations), their exact arrangements and motion are
the important parameters in describing plastic phenomena. Then the gross behaviour
of these parameters must constitutc the macroscopic plasticity phenomena, the main
intcrest of our study. To complctely specify these dctails on a macroscopic level would
naturally require an infinite numbcr of these statc quantities. However. Kroner [38, 39]
has shown that due to the randomness of dislocation distributions,
it is generally not
necessary to use an infinite set of such measures to construct a reasonable continuum
thcory of plasticity. Kroner points out that dislocation arrangements can be describcd in a
macroscopic way by taking the mean values and the first. second. etc .. moments of the
dislocation distribution. In fact, the order of moments may be determined by cxperimental
investigations. and experimental cvidence seems to support the notion that only a small
number of these is sufficient to adequately describe dislocation arrangements. We follow
the suggestion of Kroner. and introducc a /inite number of quantities Alii (i = I. 2, .... N)
for describing the dislocation arrangements. In gcncral, the quantitics Ali) will bc tcnsors
of differcnt rank. since these consist of various orders of moments reprcsenting various
levels of approximation.
However, in this formulation. to be consistent with our other
basic variables. we shall consider the Ali) to be second order tensors. For example, in the
macroscopic theories, the plastic strain, denoted ". may be interpreted as a limit of the
average of the local geometrical changes in a volumc element.
These physical observations and results from dislocation theory guide us in the present
work to postulate that the inelastic strain" and a Iinite set of dislocation arrangement
tensors Ali) constitute the internal (structural) statc variables suitable for describing the
plastic behaviour in crystalline solids.
3. KNEMA TICS
Most of the usual kinematical relations are assumed to hold. We consider a material
body ~. the elements of which are material particles X. We wish to trace the motion of
the body relative to a reference configuration Co in three-dimensional
space E3-i.e .. at
some reference timc r = 0, the particles X are in onc-to-one correspondence with spatial
point (places) x. When convenient, we shall associate with each x a triple Xi of rectangular
coordinates which give the location of x relative to a fixed spatial frame of referencc in Co:
X = (Xl. X2• X3) denote labels (material coordinates) of a particle X at x in Co which
instantaneously coincide with xj at r = O.
The motion of 8d relative to Co is given by the relation
x
= I(x,n
(3.1)
where I describes a mapping which carries the particle X onto its place x in E3 at time I.
Effectively, I(X. I) is a one-parameter family of mappings of Co onto current configurations
C, c E3. As is customary. we denote as the deformation gradient at time I with respect to
matcrial particles X thc tensor
F(x. t) = VI(X,
I)
(3.2)
A IIl/i(/ed /Ileory of /hermoviscoplas/ici/y
265
o( cry.~/(Jlline solids
and wc assume thaI dct F > 0 for cvery X and I. WC also introducc the Grecn--Saint Venant
strain tcnsor)' and the Cauchy-Green
deformation tensor C by thc relalions
y
=
1fC -
l)
and
C
= FTF
where I is the unil tensor and FT denotes the transpose
Consider a configuration C i' 0 :::; t :::; I. intermediate
the place of X at time l
y
(3.3a, b)
of F.
between Co and C,. and define as
(3.4)
= X(x'i)
Formally, y = X(ic- I(X). i) where k(X) = X defines the place of particle X in C" thus C;
(or, for that matter. Co and C.) need not be a configuration actually occupied hy.!Jf during
its motion. The deformation gradient at C; is then
F
For fixcd
t. wc assume
= VX(x'i).
that (3.4) is invertible so that we can write X
x = X(X(y). i) = i(y, t)
(3.5)
= X-Ilyll,:/.
Then
(3.6)
and
13.7)
wherc F is given by 13.5) and i = V),1.
Introducing (3.7) into (3.3), we see that
(3.8)
which can be written in the form
(3.9)
wherc
(3.IOa, b)
and in which the dependence on X and t is understood.
We shall refer to y as the total strain tensor. In the absence of a more appropriate term.
we follow the classical terminology and refer to " as the inleastic slrai/l tensor even though
we rccognize that at this point" is a purely kinematical quantity and that y - " may embody
strains which are permanent in the usual sense of the term. The tensor ~ = )' - " shall be
referred to as the difference slrai/llensor.
In most crystalline solids, plastic deformation (i.e.. yielding in thc sense of permanent
deformation) is attributed to a flow process of crystalline lattice defects normally described
in terms of devclopmcnt and propagation
or dislocations. In such situations we shall
interpret the homogeneous deformation 1(t) of (3.1) of the body PA as consisting of homogcneous lattice distortion and homogeneous shape distortion produced by homogeneous
motion of dislocations. The latticc distortion is restorable, and on rcstoration thc latticc
distortion disappears completcly (except locally at dislocation Jines) and the body fJB
D. R.
266
occupics a differcnt configuration
BHANDARI
and J. T.
ODIN
C;. Then in view of (3.7) wc may write
I
= ii
(3.11 )
where
i is thc
homogeneous latticc distortion
homogeneous plastic distortion due to homogeneous motion of dislocation.
That is. the total deformation I is the composition of two deformations i and i. where i is
that part of the deformation associated with plastic dislocations rather than lattice distortions and. therefore. not rccovcrable according to our hypothesis. We associate with this part
of the deformation a strain tcnsor
i is the
(3.12)
where
t=
VI.
~. THER\lODY""\lIC
PROCESSES
For the purpose of establishing notation and some rcsults for future reference. we rcview
briefly here certain notations. now fairly standard. on thermodynamic
processes. We shall
assume that couple stresses and body couples are absent in the body !!4 and that there is
no diffusion of mass in tM. A thcrmodynamic proccss of tM can then be described by a set
of nine functions {I. a, h. cp, q. h, s. e. (Xlii} of the particle X and time t. The function I(X. t)
defines the motion of fJd, a(X t) is the second Piola-Kirchhoff
stress tensor (cf. [40]. pp.
124]. h(X t) is the body force vector per unit mass, cp(X t) dcnotes the free energy per unit
mass, q(X t} the heat flux vector, h(X t) the heat supply per unit mass per unit time, S(X. t)
the entropy per unit mass, O(X, t) the absolute temperature, and a(i)(X, I) (i = 1,2 .... , n)
are internal state variables. This set of nine functions defined for all X in fJI and for all
time t is called a thermodynamic
process in fJI if and only if it is compatible with the laws
of balance of linear momentum and conservation of energy (cf. [40]. pp 295). Under
appropriate smoothness assumptions, the local forms of thcse Jaws are
Div (Fa)
+
ph
=
(4.1 )
pii
and
tf(ayT)
-
p(cp
+ sO + sO) +
Div q
where p is the mass density in the reference configuration
+
ph
=
0
Co and the superimposed
(4.2)
dots
indicate time rates.
To specify a thermodynamic process it suffices to prescribe the seven functions {X, a. qJ.
q. s. e. 2(i)}. the remaining two functions band h are then dctermined from (4.1) and (4.2)
A thermodynamic process in fM, compatible with the constitutive equations at each point
X of fJI and all time t is called an admissible process (cf. [40]. pp. 365).
TilE CLAUSIUS-DUllEr •• INEQUALITY
If qlO is regarded to be an "entropy flux" due to the heat flow and hiD to be the entropy
supply due to the radiation (say), then the specific rate r of production of entropy is given by
A unified theory of {hermol'iscopla~{ici{y
pl"
The Clausius-Duhem
negative:
=
267
of cry.Hnlline solid.,
. [ph0 + Dlv. (q/O) ] .
(4.3)
ps -
inequality asserts that the rate-of-production
of entropy is non-
r ~ o.
(4.4)
This implies that (4.3) can be written in the form
.
p8s - ph -
.
q
DIV
+ B1 q . 9
where 9 = grad 0. Now for each thermodynamic
(4.2) enables us to write
1
(1*
where we call the quantity
(1*
+ (jq.g
~
0
process, the energy-balance
=
equation
(4.6)
~ 0
the internal dissipation. Clearly, our
(1*
(4.5)
rr((1)·T) - p(¢
(1*
+ sO).
is defincd by
(4.7)
The inequality (4.6) is then called the general dissipation inequality.
5. CONSTITUTIVE
EQUATIONS
In the dcvclopment of constitutive equations for a non-linear theory of viscoplasticity,
we shall assume that a simple crystalline material at point X is characterizcd by four
response functions {p, fr, q and .~.which determine the value of cp, (1. q and S when the GreenSaint Venant strain y, the absolute temperature O. the temperature gradient 9 and structural
(or internal) state variables lX(i) are known at point X and time t. Specifically, we consider
a material which is characterized by the following system of constitutive equations:
cp
= (PlY, 8. g. aY))
(S.la)
(1
=
(5.1b)
q
= ti(y.
S
= Sly. O. g, IXlil).
In addition, the internal state variables
relationships of the type
n(y. O. g.
IXU)
lX(i))
(], g. IXlil)
(5.1c)
(5.1d)
are assumed to be given by a set of functional
(5.le)
The influence of the histories of y, (] and possibly even 9 on the current responsc can often
be introduced through equations of the type in (5.1e): equations (5.1e) are sometimes
refcrrcd to as equatio/ls of <TOllitio/l, since they describc the evolution of the internal state
of the material over time. We shall further assume that the constitutive equations (5.1)
satisfy the principle of material frame-indifference as postulated by Truesdell and Toupin
[40].
D. R.
268
and J. T.
BHANDARI
ODEN
So far, the constitutivc assumptions (5.1) arc csscntially of the typc studied by Colcman
and Gurtin [18]. The rcmarkable feature of this approach is that these equations apply to
almost all materials irrespective of their constitution.
In fact, as discussed carlier, the
constitutive properties of the material depend on !X(i) which characterize the internal
state of the body. In crystalline solids the behavior of dislocations, their distribution and their
interactions. play the role of internal state variables. In accordance with our previous
discussion of section 2 and motivated by the physical results from dislocation theory of
crystalline solids we now postulate that the internal state variables !XH) consist of a second
order tensor 'I called the 'plastic' (or inelasticl strain and a set A(i) of dislocation arrangemcnt
tensors, so that
(5.2)
Then in view of (5.le) and (5.2) our plastic evoilltionary
eqllations
are written in the form
;, = ~(y, O. 'I.Alii)
A
Ii)
(5.3a)
= A HI(y, 0, 'I, A H)).
(5.3b)
Constitutive assumptions (5.3) are the immediate consequences
of the basic physical
results of dislocation theory: that is the plastic flow in crystalline solids is a dissipative and
ti!11e-dcpendent process determined by the dynamical motion of dislocations. It is assumed
that the values of'l and AH) at time t are uniquely determined by the solution of(5.3) subject
to the initial conditions (say) '1(0) = 0 and A(i)(O) = A~).
We now require that for every admissible thermodynamic
process in 14, the response
functions appearing in (5.1aH5.1d) and (5.3) must be such that the postulatc (4.6) of positive
entropy production is satisfied at each point X of ~ and for any time t. This is equivalent
to the inequality
• .
(r(ayT)
As a consequence
of(5.la)
1
A
+ su) + 0 q
- p(cp
• g ~ O.
(5.4)
and (5.2) we can rewritc (5.4) as follows:
(r[(a -pil,.q,)YT]
- p{tr[(a,q,);,T]
- p(S
+ 0eq,)8
+ tr[(o)illjlIATCil]}
- POgq, . 9
+~q.g
~ 0
(5.5)
where o.,q,. oeq,. iJ,q" etc .. denote the partial differentiation of q, with respect [0 y. O. and 'I,
respectively. We now follow the arguments similar to those of. say. Coleman and Gurtin
[18]. i.e. we observe that by fixing y, 0, g. q and A(l) at time ( we also fix;' and A(i) (as a
result of(5.3)) but 0 and are left arbitrary. Thus for the inequality (5.5) to hold independent
of the signs of 8 and
their coefficients must vanish. Consequently. we obtain
y,
y,
g,
9
o q,(.)
9
0'=
S
The Clausius-Duhem
=
= 0
(5.6a)
po,.ljl(.)
(5.6b)
-8eljl(.).
(5.6c)
inequality (5.5) reduces to
A unified theory of rhamolJiscoplasriciry
(!( cryHallint'
solidI
269
which is called the qeneral dissipation inequality.
The general dissipation inequality (5.6d) implies that when 9 = 0, the illtemul dissipation
inequality
(5.7)
a* by
holds. Now if we define the internal dissipation
e. 't.A (il)
a* = U(y.
(5.8)
we can write the intcrnal dissipation
inequality (5.7) in the form
u*(y, e. 't, AliI)
~ O.
Further, for any fixed value of a* we can arbitrarily
(5.6d) by varying g. Hence, it follows that
q.g
~
(5.9)
vary the last term of the incquality
(5.10)
0
which is heal conduction inequality.
Summarizing. from the results of(5.6), (5.7) and (5.10) we have the following consequences:
(i) The response functions
namely
(p. iT and
S
are independent
cp
=
(p(y. O. fl. A(il)
(5.lla)
(f
=
&(y. O. fl. A(i))
(5.1 Lb)
s = S(},. 0, 't. Ali))
(ii) ip determines stress
(f
through
(f
of the temperature
gradient g:
(5.llc)
the relation
= pD7(p()'. 0,
(iii) cP determines entropy S through
fl. A fiI)
(5.12)
the relation
S = - De(p(y. O. '1. AliI)
(5.L3)
(iv) (p. r" A(il and q obey the general dissipation inequality (5.6d). The internal dissipation and heat conduction inequalities hold and are given by (5.9) and (5.10).
Thus. the complete set of constitutive equations for thermoviscoplastic
materials takes
the form
(5.14a)
270
D. R.
(1
=
BHANDARI
o-(y, 0,
s = Sly.
and J. T.
ODES
fl, A IiI) = (le/p( ,)
0, fl. Ali))
= -
(5.14b)
0e4J(.)
(5.14c)
q = q(y, 0, g,,,. Ali))
(5.14d)
;, = q(y, e, fl. Ali))
(5.15a)
and
.4(i) =
6. RELATIONSHIP
Ali)(y,
O. fl, A(i)).
WITH
OTHER
(5.15b)
EXISTING
THEORIES
Several thermodynamic
theories of elastic-plastic materials can be obtaincd as special
cases of (5.14) and (5.15) by imposing further restrictions on the constitutive equations
(5.15). For example. onc important class of materials results from excluding all those
state quantitics from the set AliI which arc responsible for viscous effects in crystalline
solids. In other words, wc neglect those quantitics which describe grain boundary sliding,
internal slipping of grains. twinning etc. In agreement with Kroiier's conclusions, we also
assume that a single state variable A (say the dislocation loop density tensor) is sufficient
to includc effects like Bauschinger effect. Then the constitutive equations (5.15a) are
replaced by a quasi-linear transformation
ofliu into Aij:
(6.1 )
Relation (6.1) may be referred to as a dislocation production law. and thc fourth order
tensor QjjU as the dislocation production tensor [38]. The implication of (6.1) is that it
simply restricts the occurrence of dislocation densities to the process in which plastic
deformations occur.
Constitutive equations (5.14) with (6.1) are not sufficient to formulatc a determinate
problem in the sense of classical theory of elastoplasticity. This is due to the fact that when
in (6.1) ;, = 0, which also means A = 0, the response may be reversible, and the theory
reduces to one for thermoelastic materials. Thcrcfore, in order to formulate a determinate
problem (that is, to determine a "plastic" stress-strain relation). one needs an additional
postulate of yielding. This postulate can be considered as a consequence of physical
assumptions of defining a limit point on the strain path before irreversible displacements
take place (i.e. when;' :F 0). A convenient (but not essential) way of expressing this threshold
character of the response function q is to make thc assumption of the existence of yield
surface. Since the postulate of yielding and its consequences are well known, we shall not
elaborate on these here.
It has been shown by Owen [41,42] that a good theory of plasticity can be constructed
without introducing eXplicitly the notion of yield functions. In his theory, Owen introduced
the concept of an "elastic-range"
which seems to be equivalent to using the yield function
to express the threshold character of inelastic deformations described by Kratochvil and
Dillon [23].
Green and N aghdi's theory oj plast icit y [29].
The continuum theory of plasticity developed
by Green and Naghdi characterizes
the
A unified theory of thermori.fcop/asticity
(~f crystalline
solids
271
rate-independent
plastic behavior of crystalline solids. The charactcristic of their theory
is that the governing plastic conslitulive cquations arc homogeneous in time of the first
degree in the "state variables": whercas thc plastic evolutionary equations (15.5a. b) in the
present paper include time dependent plastic behavior and hence belongs to viscoplasticity
theory. Although. from a theoretical point of view. these two models arc different. a rateindependent theory similar to Green and Naghdi can be constructed from the present
formulation if the plastic evolutionary equations (15.5) are made homogeneous in time of
the first degree. Once this is done. the collection of constitutive equations togethcr. with
the usual assumption of the existence of a yield surface. leads to results similar to those of
Green and Naghdi [29].
nellY'S theury (!f" elastic-plastic crystallille solids [25].
Tseng. in his constitutive theory of clastic-plastic crystallinc solids [25] assumed that a
scalar quantity ct, called the internal structural density, is to be added to the constitutive
variables of thermo-elasticity.
In Tseng's work, the constitutive equations for thermoelasticity are functions of FI'». 0, g and :x; whereas the plastic evolutionary equations for
e(or pr) andi are assumed to be functions of (/, Ft', () and ct. It appears the forms (15.5a. b)
arc more convenient for making a thermodynamic analysis.
Theory of thermoplastic materials with mel/lOr)' [28].
We shall now show that the results of Section 5 in thc present theory are nearly identical
to those presented by Oden and Bhandari [28] in their earlier work on thermoplastic
materials with memory. Following the arguments of Lubliner [22]. we observe that under
suitable assumptions the present value of A(iI(e) from the evolutionary equation (5.15b)
can be determined in terms of the 'past histories' ofy. 0 and". Then the solution of(5.15b)
can be written in the form of a functional
~=r
Am(t)
= )7t(i) {['(r)}
r= .-
(6.2)
00
where for the sake of conciseness we have used the notion r = (y. O. "" and the dependence
of A(i) and r on X is understood. Thcn r'(r) denotes the restrictions of nr) to r < t. Functions ne) (t > - (0) permitting uniquc, continuous solutions Alil(l) of (5.15b) which satisfy
Am( - 'l.:') = 0 arc considered admissible. In fact. for Alil(tl to be continuous. A'Ii) nt),
Ali» and hence r(t) need not be continuous in t (see [22J).
Substituting (6.2) into (5.14) and (5.15a), and making use of the more familiar notation
adopted in [28]. we obtain the functional forms
<rj
cp =
where
cP[r(c).
r~(s)= ru - s)[o < S < 00]
Jlili){ F(r)}] =
1'=._
and r(t}
a)
(/ = ~[~(S):f(I)]
5=0
= fI
5=0
[r~(s): r(cn
(6.3a)
.1=0
Similarly
= P(O).
= pey <I> [r~: r]
'"
(6.3b)
5=0
~
S
00
<I>
~
[r~(s):
ru)] = -a,,$
s=1l
[r~: r]
(6.3c)
D. R.
272
BHANDARI
00
f2
=
q
[r~(s):
and J. T.
ODE:-
nt}. g(t)]
(6.3d)
no].
(6.3e)
.<=0
00
;, = K[r~(s):
.<=0
We note that the stress a and entropy
S are derivable
from the free energy functional
00
cD [:]
which is independent
of 9 as implied by (6.3b) and (6.3c).
.<=0
The general dissipation
inequality
in this case is:
00
- p{tr[(o~
<I>
1
,7)
[:]);'1"] + bf
.=0
Ctl
cD [-Ir~]} + 0 f2 [:].
5=0
(6.4)
9 ~ 0
.<=0
<I)
where b,.
<D [ - : I] denotes
a frcchet
differential
linear in arguments
to the right of
.=0
vertical stroke:
The internal dissipation
and heat conduction
inequalities
:T.I
a*
<D [:]);'1']
= -O'-l{tr[(o~
are now given by
rS)
+ bf
,=0
<D [-It~]
~0
(6.6)
.=0
and
<Xl
fl
[~.r,9]·9
~ O.
(6.7)
s=o
The constitutive
given in [28].
equations (63) together with the rcsults (6.4)--(6.7) are essentially of the type
Valanis's 'emiochronic' theory of viscoplasticity [43]
Valanis, in his recent work [43. 44]. has also used the concepts of hidden variables in
devcloping a functional "cndochronic"
theory of viscoplasticity. His developmcnt is based
on Onsager's relation. a move which, to some. has proved to be controversial (see, Truesdell
and Toupin [45]). furthermore.
Valanis's work appears to be restricted to infinitesimal
deformations
and isothermal processes. We note that under appropriate
additional
assumptions Valanis's work can be obtained as a special case of the general formulation
given by (6.3). To prove that this is so, consider the special case in which the free energy
functional (6.3a) has the following form
f f ..
,
f)(P =
cp
o
+ -1
2
oy (I") dr' dt"
A')k I (t - t' t - t") ~oy.. (r')..-M
'or'
at"
o
I
+
II
o
I
0
I
.~
B'J It 0
t', t -
?y
t") ~
at'
al
at"
(t') ---!l (t") dt' dl"
A ll/l(fied Iheor.v of thermol'iscoplasticity
ff
"
ff
I
+~
2
o
-
t' r -
at'
0
I
I
ff
2f f
o
0
,
I
1
D'J(t - t', t -
dt' dr"
ot'
at"
..
0'/, . 00
(t'l01"
at"
ae
F(r - t', t -
t") at' (t')
(t")dt'
dt"
((")
dt' dt"
ae
ar<n
(6.8)
dt' dt".
0
kernels. With the aid of (6.3b) and (6.3c). we obtain
I
fAiikl(t
o
00
E'J(t - t', t - t") ~
,
=
a",.
t")~(t')--
Here Aijk/(.). BijUO etc. are material
from (6.8) the stress and entropy:
(J'ij
(t') atlu (t")
01"
r") ~
.
I
o
o
Cijk/(r
0
+
+
273
I
I
+
of crystalline solid,
aYkl (r') d(
- t')
+
at'
fBijk'(t
a,/
- t')
u (t') dt'
at'
0
"f
I
+
ae (t') dt'
D'J(r - t')-
(6.9)
at'
o
f
t
I
- pS =
Dij(t - r') ~
+
(t') dl'
at'
o
£ii(t -
n~
(t') dt'
at'
0
I
+
r
•
f
F(I - t')
ao
a1 (t')
dt'.
(6.10)
o
For the sake of illustration. assume that the constitutive
given in terms of the histories of Y and (] only: e.g.
I
,.., =
I'J
equation
for tiii is
I
f e-.
'kl
'J
ay
t') ~
(t -
k/
at'
dt'
+
o
f
of) (t')
-'j
£' (t - t') -
~
ot'
dt'
.
(6.11 )
0
Moreover. to obtain an explicit form of the constitutive
consider only isotropic materials for which
A
(evolution)
ijkl
Bijkl
=
A0 t
=
BO {) I).. bkl
.I:
uil'kl
equation
'.1:
.1:')
+ A 1(c\k(Jil
+ ut/5jk
+ B I (S 'kc).Jl + {)'le> J''k)
.
I
J
for stress n. we further
[). R.
274
+ C1(c5jkc5}1 +
Cijkl = CObl'kl
E'i =
and J. 1'. ODfr-:
BIIA:-;DARI
c5i/()}k)
E°c5..
I)
Vij = D°c5..
etc.
I}
(6.12)
Then with the aid of (6.11) and (6.12), we rewrite (6.9) in the form
.. f
t
a
l
}
f
t
=
2JI(1 -
°
oy" (I') dl'
I') ~
at'
.
+ ().
I}
(t') dt'
°
t
. I' ( ')at'o£l ( ')
+ (j..
n-°Yor'kk
K(t -
I. I -
I
-
I
dt '
(6.13)
I)
where the material kernels JI( .), and K(.) and i.(.) are now given by
2JI(t)
=
2,4 1(1)
+ 4B'(r)*C1(t)
K(t) = AO(I)
+ 2B1(1)*('0(1)
=
+ 2B1(I)*Eo(1)
i.(1)
0°(1)
(6.14)
and the symbol * in (6.14) denotes the convolution operator.
Finally, we obtain Valanis's "cndochronic" theory of visco plasticity (i.e. a theory in which
stress, among other properties, is a functional of strain history. defined with respect to an
intrinsic time scale, the lattcr being the property of the material at hand) by introducing a
time scale z which is independent of t, the external time measured by clock. but which is
intrinsically related to the dcformation and temperature. To illustrate this. we introduce in
the manner of Pipkin and Rivlin [46] a non-negativc monotone increasing timc invariant
parameter
,
z( r) =
J
+ (())2]t
[trW);)
(6.15)
dr'
°
where superposed
dot indicates dilTerentiation
.
d)',
Y = dr' (r I: B
with respect to time r'. i.e.
=
dO,
dr' (1." )
and z represents the arc length of the path in the ten-dimensional
temperaturc. Then introducing the time scale z into (6.13), we obtain
.
aii = 2
f
o
space of strain and
.-
JI(Z -
-7') ~oz'
(z') dz'
°.
+ c5i} f
(6.16)
+ c5 Ij
f
K("" ~
-'I OYkk
oz' (z') dz'
-
°
,t(z -
z') ~O (z') dz'.
oz'
(6.17)
A limped theory of thermoriscop/asticily of crystalline solids
275
We observe that hy introducing the "reduced"' time the form of the constitutivc
does not alter and (6.17) is esscntially the one givcn by Valanis [43].
equations
Prandt/-RellSS relations
As a final example, we show that the Prandtl-Reuss
relations of classical plasticity can
be obtained from (6.17) as a special case. For isothermal proccsscs and inlinitcsimal
strains. reduces to
a;j = 2
f
µ(z -
(z') dz'
z') ~
o
-
= 3
au
f
d"
(6.18)
z') ~;~ (z') dz'
K(z -
o
a;j
where
and l;j are the deviatoric stress and strain tensors and
mean stress and the dilatation. By selecting
t au
=
0'0
and
j'u
are the
(6.19)
we see that
= 2µo
a;j
f-
(!t(:. :')
di'iiz')
(6.20)
o
which. when differentiated
yields
•
(X
d)'..
1
.
2/10
I)
•
= -dza ..+ -da ...
I)
2µ0
I)
(6.21)
Then using the relation
(6.22a)
we can write
1
= 2µ0
a;j
and
,
dll.·
')
(x.
= dIll')' = -dza
...
2µ0
We recognize these results as the familiar Prandt-Reuss
(6.22b)
I)
equations
of classical plasticity.
Acknoll'ledgeml'nt- The support of this work by the U.S. Air Force omcc of Scientific Rcscarch under Contrac[
F44620·69·C·O 124 is gralefully acknowledged.
276
D. R. BHANDARIand J. T. ODm
REFERENCES
[1] M. A. BlOT. Theory <lnd stress-str<lin relations in <In isotropic viscoelasticity and n:laxalion phenomenon,
J. IIppl. Phy". 25. 13g5-1391 (1954).
[2] M. A. BlOT, V<lriation<ll principles in irreversible thermodynamics
with applications
10 viscoelasticity.
Php. Ret·. 97. (6). 1463-1469 (1955).
[3] H. ZIWLER. Thermodynamik
und Rhcologische Problclllc. Ing. Arch. 25. 58-70 (1957).
[4] H. ZtEGLER, An <llIempl to generalize Onsager"s principle. and its significancc for rheological problems.
Z. A M. P. 9.749-763 (1958).
[5] D. C. DRuCK!:R. Proceedings. IllIemational Symposium on Second Ordl'r Effects in Ela.l"tici/y. Plasticity. and
Fluid Dynamics. Haifa. April 21-29. 1962. Pergamon Press. Oxford. p. 331 (1964).
(6] O. W. DILLON. J. Mech. Ph)'s. SO/itlf II, 21 (1963).
[7] A. A. VAKULENKO.Dokl. Akad. Nauk. SSR. 126.736 (1959).
[8] G. A. KLuIT!:NnERG. On rheology and thermodynamics
of irreversible pro£jjllCs. Phpica. 28. 1173 (1962).
[9] G. A. KLUITENRERG.On the lhermodynamics
ofviseosity and plasticily. Plflf5ica. 29. 633 () 963).
[10] J. KEsnN. On Ihc application
of the principles of thermodynamics
to straincd solid materials, IUTAM
Sympo.fillm onlrrel'ersible Aspects of Cominullm Mechanics. Vienna. June 22-25 (1966).
[II] R. A. SCHAP!:RY.Application of thermodynamics
to themlOmeehanical.
fracture. and birefringent pheno·
menon in viscoelastic media, J. appl. Phys. 35. 1415-1465 (1964).
[12] R. A. Schapery. On the characterization
of nonlinear viscoel<lstic 1lI<11eriaJs.l'olyma £ngng Sl'i. 9. 295-310
(1969).
[131 R. A. SClIAPr;RY. Procee(!ings. IUTAM S)'mpo.,ium. East Kilbridge. June 25-28 (1968).
/141 K. c. VALANIS. Symposium on Mec/umiCtlI 8ehavior of Materials Uncll'r Dynllmic Ltltulf. San Antonio.
Texas. September 6-8 (1967).
[15] B. D. COLEMAN.Thermodynamics
of materials wilh memory. Arc/IS. ration Mec/l. Analysis 17. 1-46 (1964).
/161 B. D. COLEMANand V. J. MIZEl. A general theory of dissipation in materials with memory. Arch,. ration
Alech. Analysis 27.255-274 (1967).
[17] B. D. COLEMAN<lnd W. NOLI., The thermodynamics
of elastic materials with heat conduction and viscosity.
Arc/IS. ration Mech. Analpis 13. 167-178 (1963).
[18] R. D. COtEMAN and M. E. GURTIN. Thcrmodynamics
with internal statc variables. 1. d/{'III. Phys. 47.597613 (1967).
[19] B. D. COLEMANand V. J. VIZEl. Existencc of caloric cquations of state in thermodynamics.
J. chem. Phys.
40.116-1125
(1964).
[20] R. A. SCHAPERY, A theory of nonlinear thermoviscoelasticity
based on irreversible thermodynamics.
Procee(iings 5/h U.S. Natiollal COllgress of Applied Mechallics. A.S.,H.E .. pp. 511-530 (1966).
[21] P. PERZ\~A and W. OLSZAK. IUTAM Symposium 011 Irrerersible Aspect.\" of COlltinuum Mechallics. Vienna.
1966. pp. 279-291. Springcr Verlag (1968).
[22] L. LUBLINFR. On fading memory·m<llerials
of evolution typc. Acw Mech. 8. 75-81 (1969).
[23] J. KRATOCHVil and O. W. DillON. Thermodynamics
of clastic-plastic materials as a theory with internal
state variables. J. appl. Phys. 40.3207-3218
(1969).
[241 J. Kratochvil and O. W. Dillon. Thcrmodynamics
of crystalline c1astie·viseo-plastie
materials. 1. appl. "hys.
41. 1470-1479 (1970).
[25] W. S. Tseng. A constitutive theory for e1astic·plastic crystalline simple solids. Ph.D. Di.f.'l.'rllllioll. University
of California. Berkeley (1971).
[26] H. T. HAHN. A disloc<ltion theory of plasticity. Tech. Rep. No.3. Dept. of Engng Mech .. Penn. Slale Univ ..
Pennsylvania (197 I).
[27] P. PFRZYNA and W. WONJO. Thermodynamics
of a rate sensitive plastic material. Arch. Mech. Stos. 20.
501-510 (1968).
[28] J. T. ODE~ ;lI1d D. R. BHANDARI. A theory of thermo'plaslic
materials with mcmory. J. Ellgllg Mech. Dip.
ASC£ (To be published).
[291 A. E. GRlfN and P. M. NAGHDI. A general theory for e1astic·elastic continuum. Arch.,. ratioll Alech. Allalysis
t8. 251-281 (1965).
[30] B. A. BILBY.Geometry and continuum mechanics. IUTA M Symposium 011 Mec/ulllic.l" of Genera /iced COlltillua
(Ed E. KRONER) Springcr Verlag (1968).
[31] J. W. TAHOR. Dislocation dynamics and dynamic yielding. J. appl. Ph)".I".36. (10).3146-3150
(1965).
[32] K. KONDO. On Ihe two main curren Is of the geometrical theory of imperfect continua. IUTAM Symposium
Oil Mechallic.' ofGellera/i=ed
Cal/lillua pp. 200-213 (Ed E. KR(lNFR) Springer Verlag (l968).
[33] T. MURA, Conlinuous distribution
of dislocations and the mathcmallcaJ
theory of plasticity. Phys. State
Sol. 10.447-453 (1965).
[341 A. H. COTTR!'t. Dislocati01l.f alld Plastic' Floll' i/l Crystals. Oxford University Prcss. London (1952),
[351 W. T. RFAl>. f)i.,IOCl/tio,,",;/I Crystals, McGr;tw·llili Hook Company. New York (1953).
(361 E. KR(;NI'R. KOlltilllllllll.\· 71ll'o,.h· tier Vl'rset=llllgc"1Ulld Eigen·Spal/llllgell. Springer Verl;tg. Berlin (1958).
[37] J. J. GilMAN. Micromedlllllic.l" (~rFlolI" ill Solid.,. MeGraw·Hili
Book Company (1969).
A lin (fled Ilreory of Ilrermol"i.<copla.<licilr of cryslallinc solid<
277
[381 E. KRljSER. DisloC;llion: A new concept in the continuulll theory of plasticity. J. lIIolh. PhI'.<. "2. 27-37
(1962).
(391 E. KRO~ER. !low the internal state ofa plastically deformed body is to be described in a continuum theory.
Proceedin~.< oIllre fimrllr IllIemotional Congress on Rheology. 1963 (Ed. 1'. H. LEE). Int('rscien('e. New York
(1965).
[401 C. A. TRul'sDRL and W. Noll.. The Non·lilwar Field Theories of Mechanics. Holltlhuck der I'hy.<ik (Ed.
FlUGGE) III/3. Springer Verlag. Berlin (1965).
[41] D. R. OWE~. Thermodynamics
of materials with elastic range. Arc/Is. ralion Alec},. Analr.<i.<31, 91-112 (1968).
(421 D. R. OWEN. A lll('ehanical th('ory of materials with elastic range. Arc/IS. ralion Alech. Anoly.<i.,·37. 85-110
(1970).
[431 K. c. VAlANIS. A theory of viscoplasticily
without a yield surface·1. Arch. Alech. Slo.<.. 23. (4). 517-533
(1971 ).
[441 K. c. V AlANIS. A theory of viscoplasticity
without a yield surface·II, An·h. Alec},. 510.<.. 23. (4). 535-551
(1971).
[451 c. A. TRUF_~Dfl.l .!Od R. A. TOt.ll'IN. The Classical Field Theories. Ilalldhllch der I'hy.<ik, III/I. Springer
Verlag (1960).
[46] A. C. PIPKIN and R. S. RIVlIN. Mechanics of rate independcnt materials. Teclrnical Reporl No. 95. Division
of Applied Mathematies.
Brown University (1964).
(Recei"ed
12 Seplelllber
1972)
Resume-On
presenle une lheorie unifiee de la thermoviscoplaslicite
des solides cristallins. En partieulier nous
pouvons montrer que la thermodynamique
des materiaux "viscoplastiques"
peut etre adaptee dans Ie cadre de
la mecanique des milieux continus moderne des materiaux
memo ire. Lcs concepts physiques de base sont
dCduits de la consideration du comportement
des dislocations dans les solides cristallins. On regarde Ics rclalions
entre notre approche actuelle et plusieurs lheories exislantes de la plastidte.
a
Zusammcnfassung-Einc
einheitJiche Theorie der Thermoviskoplastizitat
kristalliner
Fc,tkorpcr
wird dar·
gcstcllt 1m besondcren war cs uns miiglich zu zeigcn, dass cine Thermodynamik
"viskoplaslischer"
slorrc in den
Rahmen moderner Kontinuumsmechanik
der Storre mil Gcdiiclllnis cingepasst werden kann. Die physikalischc
Grundkonzepte
werden durch die Betrachtung
des Verhaltens von Versetzungen
in kristallinen Fcslkorpern
hergeleitet. Zusammenhange
unserer Darstellung mit einigen bestehenden Plastilitiitstheorien
werden untersucht.
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