6. Constitutive Laws
equations listed in the preceding sections apply to any solid or fluid that
deforms under the action of external forces.
They do not, by themselves, have a unique solution, however, because the
deformations are not related in any way to internal forces.
governing equations are completed by constitutive
laws that provide the missing connection.
Unlike deformation measures; kinetics; and conservation laws, a
constitutive law cannot be calculated or predicted from first principles,
except for a few very special cases such as small deformations of crystalline
materials, where elastic properties can be estimated using ab-initio techniques
that approximate quantum mechanical level atomic scale interactions in some
6.1 Basic assumptions local homogeneity of internal forces and
deformation; the principle of local action
Before discussing specific constitutive models, it is
helpful to review the basic assumptions that we take for granted in developing
stress-strain laws. They are listed
A very small sample that is extracted from the
solid has uniform properties;
When the solid is deformed, initially straight
lines in the solid are deformed into smooth curves (with continuous slope);
means that very short line segments (much shorter than the radius of curvature
of the curves) are just stretched and rotated by the deformation. Consequently, the deformation of a
sufficiently small volume element can be characterized by the deformation
at a point in the solid depends only on the change in shape of a vanishingly
small volume element surrounding the point.
It must therefore be a function of the deformation gradient or a strain
measure that is derived from it.
we accept the preceding assumptions, it means that we can measure the
relationship between stress and strain by doing an experiment that induces a uniform strain in a suitable sample of
the material. According to our
assumptions, the stress should also be uniform, and can be calculated from the
forces acting on the specimen.
are clearly approximations. Materials
are not really uniform at small scales, whether you choose to look at the
atomic scale, or the microstructural scale.
However, these features are usually much smaller than the size of the solid
part or component, and the material can be regarded as statistically uniform, in the sense that if you cut two specimens
with similar size out of the material they will behave in the same way. A continuum model then describes the average stress and deformation in a
region of the material that is larger than microstructural features, but small
compared with the dimensions of the part.
6.2 Restrictions on constitutive
may be called upon to develop a stress-strain law for a new material at some
point of your career. If so, it is essential to make sure that the
stress-strain law satisfies two conditions:
(i) It must obey the laws of
(ii) It must satisfy the
condition of objectivity, or material frame indifference.
In addition, it is a good idea to ensure that the
material satisfies the Drucker
stability criterion discussed
in more detail below. Of course, your
proposed law must conform to experimental measurements, and if possible should
be based on some understanding of the physical processes that govern the
response of the solid. But only the
first two conditions apply to all solids.
Thermodynamic restrictions: Constitutive laws usually start by expressing the
specific internal or free energy, specific entropy, and heat flux of a material
in terms of the temperature, parameters characterizing shape changes, and any
internal state variables (such as yield stress) that characterize the material
state. These have the general form
For a solid with mass
density (in the deformed configuration) experiencing a stretch rate and Cauchy stress ,
the first and second laws of thermodynamics then require that
for all possible processes.
In practice, only the
second law leads to any restrictions on the constitutive equations. We assume that the first law can always be
satisfied by some appropriate heat flux q.
will see how to use the second law in devising structures for constitutive
equations through specific examples in the next few sections. In practice, a simplified version of the
second law is used nearly always used, however, and we summarize this form
the majority of problems in solid and fluid mechanics, we do not attempt to model
deformation and heat conduction simultaneously (there are exceptions of course,
such as high strain rate deformation; thermoelasticity, and so on). In practice, we can often assume that the
solid is in equilibrium with a surrounding heat bath with constant temperature,
and heat flow through a solid is sufficiently rapid to ensure that the
temperature remains approximately uniform.
Under these conditions the second law reduces to
this states that the rate of work done by stresses must always equal or exceed
the rate of change of free energy of the solid.
Frame indifference: The
principle of frame indifference (which is still the subject of some debate)
states that constitutive equations must be invariant to a change of observer
(all observers, regardless of their reference frame, must describe the same
physical process). The transformation
of various field quantities under changes of observer was discussed in the
preceding section. To reiterate,
spatial coordinates map as
under a change of observer.
In a constitutive model, observers in the two frames will use some temperature or deformation measures
that we will denote by a vague symbol (they could be scalar, vector, or tensor
valued variables). These measures are
always related by a appropriate transformation relations under a change of
observer for example, deformation gradients, Lagrange
strains, and the Left stretch tensor transform according to .
The constitutive model will predict
some physical quantity of interest in terms of these variables. We will denote the predicted quantity in the
two frames by a similar vague variable ,
which could be a scalar, vector or tensor valued quantity. For example, the constitutive model could
predict the free energy ,
the heat flux vector or Cauchy or Material stress . Again, we know how these physical quantities
transform under a change of observer . The functions relating these variables to the kinematic variables must
behave correctly under a change of observer in general
a specific example, a constitutive relation specifying the Cauchy stress in
terms of deformation gradient would have to transform according to
restricts the possible form of the function . You can use a similar process to crank out
restrictions for any kind of constitutive relation.
is still some debate as to whether frame indifference should apply under a
transformation between frames in relative rotation. Usually, it is assumed that constitutive
equations must hold even in a non-inertial frame. But some fluid mechanics models of turbulent
flow, for example, attempt to describe macroscopic behavior of a volume element
of fluid that has internal angular momentum.
Constitutive laws of this kind can only hold in an inertial frame, and
so invariance need only apply during a transformation from one inertial frame
to another. These have the form
dependence is gone).
Stability: For most practical applications, the
constitutive equation must satisfy a condition known as the Drucker stability
criterion, which can be expressed as follows.
Consider a deformable solid, subjected to boundary tractions ,
which induce some displacement field . Suppose that the tractions are increased to ,
resulting in an additional displacement .
The material is said to be stable in the sense of Drucker if the work done by
the tractions through the displacements is positive or zero for all :
You can show that this
condition is satisfied as long as the stress-strain relation obeys
is the change in Kirchhoff stress, and is an increment in strain resulting from an
infinitesimal change in displacement .
is not a thermodynamic law (the work done by the change in tractions is not a physically meaningful quantity), and
there is nothing to say that real materials have to satisfy Drucker stability.
In fact, many materials show clear signs that they are not stable in the sense of Drucker. However, if you try to solve a boundary
value problem for a material that violates the Drucker stability criterion, you
are likely to run into trouble. The
problem will probably not have a unique solution, and in addition you are
likely to find that smooth curves on the undeformed solid develop kinks (and may not even be continuous)
after the solid is deformed. This kind
of deformation violates one of the fundamental assumptions underlying continuum
simple example of a stress-strain curve for material that is not stable in the
sense of Drucker is shown in the picture.
The stability criterion is violated wherever the stress decreases with strain in tension, or increases with strain in compression.
For the former we see that ,
for the latter ,