Constitutive Models Relations between Stress and Strain
equations listed in Chapter 2 are universal they apply to all deformable solids. They can’t be solved, however, unless the
deformation measure can be related to the internal forces.
constitutive model for a material is
a set of equations relating stress to strain (and possibly strain history,
strain rate, and other field quantities).
Unlike the governing equations in the previous chapter, these equations
cannot generally be calculated using fundamental physical laws (although people
are trying to do this). Instead,
constitutive models are fit to experimental measurements.
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 below.
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
stress 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.
3.1 General requirements for
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 obeys the laws
(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.
Thermodynamic restrictions: The laws of thermodynamics impose two restrictions on stress-strain
1. The first law requires that the work done by stresses
must either be stored as recoverable internal energy in the solid, or be
dissipated as heat (or a combination of both).
2. The second law requires that if a sample of the
material is subjected to a cycle of deformation that starts and ends with an
identical strain and internal energy (at constant temperature, or without heat
exchange with the surroundings) the total work done must be positive or zero.
Objectivity: Strictly speaking, the term “objectivity” or
“material frame independence” is the condition that the tensor-valued functions
that relate stress to deformation measure must transform correctly under a
change of basis and change of origin for the coordinate system. A detailed mathematical derivation of the
consequences of objectivity will not be given here (see for example Malvern
“Introduction to the Mechanics of a Continuous Medium,” Prentice Hall). However, you can check whether a constitutive
law is objective using the following simple (analytical) test:
1. Load the solid (quasi-statically) by subjecting its
boundary to prescribed forces, to induce a Cauchy stress in the solid.
2. Subject both the solid, and the loads acting on the
solid, to a quasi-static rigid rotation, characterized by a rotation tensor .
3. The constitutive law must predict that after the
rotation, the stress components change to new values given by
see this, note that since the loads rotate together with the solid, the
components of traction acting normal and tangent to any interior material plane
in the solid must remain constant. With
this in mind:
1. Suppose that, just before the rotation is applied, denotes the traction acting on an interior
material plane with normal .
2. The traction vector and the normal to the interior
plane rotate with the solid, and therefore have components and after rotation.
3. By definition, . From (2), we see therefore that
4. Multiply both sides of this equation by and recall that to see that
5. Comparing this with (1), we conclude that . Finally, multiplying both sides of this
equation by we conclude that ,
giving the required result.
Drucker 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 ,
the following chapter, we outline constitutive laws that were developed to
approximate the behavior of a wide range of materials, including
polycrystalline metals and non-metals; elastomers; polymers; biological tissue;
soils; and metal single crystals. A few
additional material models, which account for material failure, are also
described in Chapter 9.