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Chapter 3
Constitutive
Models  Relations between Stress and Strain
The
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.
The
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.
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 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);
This
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 gradient;
The
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.
If
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.
These
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
constitutive equations
You
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
of thermodynamics.
(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 laws:
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
To
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
where
is the change in Kirchhoff stress, and is an increment in strain resulting from an
infinitesimal change in displacement .
This
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 constitutive equations.
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A
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 ,
while ;
for the latter ,
while .
In
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.
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