Applications of Mathematics
Kumbakonam R. Rajagopal
On implicit constitutive theories
Applications of Mathematics, Vol. 48 (2003), No. 4, 279–319
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48 (2003) APPLICATIONS OF MATHEMATICS No. 4, 279–319
ON IMPLICIT CONSTITUTIVE THEORIES
, College Station
(Received December 17, 2002)
Abstract. In classical constitutive models such as the Navier-Stokes fluid model, and the
Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical
quantities. Models for viscoelastic and inelastic responses on the other hand are usually
implicit relationships between the stress and the kinematical quantities. Another class of
problems wherein it would be natural to develop implicit constitutive theories, though sel-
dom resorted to, are models for bodies that are constrained. In general, for such materials
the material moduli that characterize the extra stress could depend on the constraint reac-
tion. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction
associated with the constraint of incompressibility. In the linear case, this would be the
pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of
bodies described by an implicit constitutive relation for the specific Helmholtz potential
that depends on both the stress and strain, and which does not dissipate in any admissible
process. The stress in such a material is not derivable from a potential, i.e., the body is not
hyperelastic (Green elastic).
Keywords: constitutive relations, constraint, Lagrange multiplier, Helmholtz potential,
rate of dissipation, elasticity, inelasticity, viscoelasticity
MSC 2000 : 76A02, 76A05, 76A10, 74D10, 74A20, 74C99
1. Introduction
I discuss here a general framework for describing the thermomechanics of continu-
ous media. Most of the theories that are in place are specializations of the framework
which is being proposed. The new framework however allows for a much more gen-
eral structure to study the response of bodies when subject to external as well as
internal stimuli and allows one to bring under one unifying theme a much richer and
wider class of material response. In fact, the framework provides an embarrassment
of riches that can and has to be kept in check by appealing to sound physics.
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It is customary to start by defining what is meant by a body and then delineat-
ing the set of states that are accessible to the bodies that belong to the Material
Universe of interest (of course some of the bodies might be defined by a subset of
the state parameters that have been identified, i.e., the state space associated with
them is a subspace or submanifold of the state space of choice). In fact, identifying
functions which will be associated with (defined on) our body of interest (i.e., the
state functions) and their co-domains defines the class of bodies of interest (i.e., we
are interested in those abstract sets B on which these state functions are defined).
However, the notion of a fixed abstract set B with which a certain set of properties
can be associated (i.e., the assumption that certain functions can be defined on B×
)
is quite untenable as a general framework for defining a body. For instance, if one is
interested in growth or adaptation of biological matter due to external and internal
stimuli, then the only set that qualifies to be designated as the body is the set
that is present at current time, the body everchanging with time. Thus, the only
tenable descriptor of a body is a (differentiable) manifold from an eulerian point
of view: the set of particles that exist at a certain instant of time having a certain
measure and topological structure. This set, frozen in time, compared with other sets
similarly frozen at other instants of time, allow us to make measurements concerning
members of subsets which have a commonality that can be mathematically identified.
Moreover, no body or for that matter system is ever a closed system which receives no
stimulus from that which is external to it. All systems are open systems. However, in
some of them, we can neglect the external stimuli and treat them as closed systems.
The only system, if any, that can be viewed as a closed system, is the total universe
of objects that have nothing external to them, but one can immediately see the
dangerous path down which the existence of such a set leads towards as one is then
postulating a set of all bodies that qualifies to be a body. Trying to put such a
general mathematical structure into place is not only beyond my ability, but were
I able to do so, would make what follows so complicated that it would defeat the
purpose of the article, namely a discussion of the physics of deformable continua.
Thus, I will dispense with the attempt to infuse such rigor and discuss issues from
an intuitive standpoint.
The classical definition of a body that endows a set with a measure theoretic
structure is itself a partial specification of what we mean by the state space of a body
for it endows an abstract set (of “particles”) with the property of mass. Associating
other properties with the abstract set of particles is not any the different. Thus, the
state space is nothing more than the space defined by the properties associated with
an abstract set (of particles). With the same set of particles we can associate different
properties with the passage of time. At different times, with the same abstract set of
particles we thus can associate different response capabilities. The paths traversed
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in the state space are called processes. Not all points in the state space can be
connected by processes, for a given set of particles, i.e., a particular body cannot be
subject to every possible process. Thus, with the same set of particles that resembles
a “solid” body at one instant of time, we might associate a body that is a “fluid” or
a gas at a later instant of time. At this stage we of course cannot make precise what
we mean by a solid, fluid or a gas. These comments are merely meant to motivate
the main ideas.
The same set of particles, let us say a piece of steel, would respond like what is
referred to as a linearized elastic solid when subject to sufficiently small deformations;
behave in an inelastic manner when subject to sufficiently large deformations; if the
conditions are right it could twin or undergo solid to solid phase transition; melt
or for that matter exist in a gaseous state. It could thus happen that we need to
associate different parts of the state space with the same set of particles, the body.
It is important to recognize that for a specific set of particles, say in its “solid-like”
form, not all the “states” that go to define the state space are necessary to describe its
response in the sense that the body’s response is very well approximated by a certain
subset of states. This does not mean that the other functions that go to define the
state-space are unimportant, it is just that they do not play a significant role in
describing the body’s behavior, given the state that it is in, for a certain class of
processes of interest (one could think of the case of space, time and velocity as being
state variables, and at sufficiently high velocities our constitutive theories would have
to be different). Thus, for a body that is formed by the liquid that oozes from the
bark of a rubber tree, under certain conditions, we need to know only the extent
of its deformation to describe its state of stress reasonably well. (Both the stress
and the deformation gradient are state functions. Constitutive relations are merely
relations between the values of these state functions. Constitutive relations define
subsets in the state-space.) This does not mean that how fast the body deforms or
other such information is unnecessary. It merely means that for a class of processes
that one is interested in, the response seems to be independent of how fast the body
deforms. Thus, it might be more accurate to state that, as long as the body is
deformed “sufficiently” slowly, the stress in the material is captured quite accurately
by the deformation (which also has to be within certain limits). Thus, we do not
need to know exactly how slowly we deform, provided it is below a certain value. Of
course similar caveats would apply for the rate of heating, etc. We cannot exhaust
the specification of all the quantities that need to meet certain restrictions in order
that the stress response depend only on the state of the deformation. Nonetheless,
we do find it most useful to discuss this ideal case without detailing all the caveats,
and this idealization explains the response of materials such as rubber reasonably
well for a large class of processes.
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How a body responds may depend on how the body got to the state (i.e., its past
history), but on the other hand it need not. In fact, theories which are currently
available that describe the response of a body in terms of the history of certain states,
can be recast within a framework where such a historical description is unnecessary;
an explicit relationship between the stress and the history of the strain, say, can be
described by an implicit theory in which the history of the strain is replaced by an
equivalent differential equation and an initial condition.
Constitutive theories and constraints can be thought of as defining subsets in
the state space. In fact, one could argue for a more general framework where even
“balance laws” define appropriate subsets of the state space; for instance the balance
of mass might define a subdomain in the state space with the possibility of radioactive
decay outside this subdomain. That is, there could be processes in the state space
wherein the balance of mass is violated. Such a process cannot lie within the subset
of the state space wherein the balance of mass is always met. I shall not try to create
such a grand framework. Here, I shall limit myself and rest content with a state-
space wherein processes are required to meet the usual balance laws of continuum
mechanics.
While discussing the state-space and process classes, it might be appropriate to
discuss invariance requirements on constitutive quantities and balance laws. In dif-
ferent domains of the state-space and the process classes we might require differ-
ent invariance requirements. For instance, we might require Galilean invariance for
a certain class of processes. In classical Newtonian Mechanics, we require such an
invariance for our balance laws. The requirement of frame-indifference for the con-
stitutive relations which is routinely enforced for continua is somewhat questionable
as they are ultimately substituted into the balance laws wherein we merely require
Galilean invariance. For other classes of processes, we might require more general
balance laws, say Maxwell’s equations and invariance under Lorentz transformations.
The requirement of Galilean invariance could be viewed as the invariance that this
class of processes satisfies under certain conditions, i.e., Galilean invariance could be
viewed as being an appropriate reduction of Lorentz invariance under the conditions
under consideration. While both Galilean and Lorentz invariance seem to be appro-
priate restrictions in that they are consistent with one another, frame-indifference
that allows for rotations that are time dependent between frames is not consistent
with Lorentz invariance. It has been shown in [37] that frame-indifference cannot
be required amongst all observers and that the rotation of the earth is sufficient to
create problems as time goes on. In fact, it is easy to see that if Q(t) is not zero
velocities can exceed the speed of light for sufficiently large |x|.
It is also worth noting that to obtain the constitutive models for the stress in
elasticity, one does not need to require frame-indifference, Galilean invariance would
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