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Chapter 3
Constitutive Laws - Relations between Stress and Strain
3.3 Hypoelasticity  elastic materials with a nonlinear
stress-strain relation under small deformation
Hypoelasticity
is used to model materials that exhibit nonlinear, but reversible, stress
strain behavior even at small strains.
Its most common application is in the so-called `deformation theory of
plasticity,’ which is a crude approximation of the behavior of metals loaded
beyond the elastic limit.
A
hypoelastic material has the following properties
 The solid has a preferred shape
The specimen deforms reversibly: if you remove the loads, the solid returns
to its original shape.
The strain in the specimen depends only on
the stress applied to it it doesn’t depend on the rate of loading, or
the history of loading.
The
stress is a nonlinear function of strain, even when the strains are
small, as shown in the picture above.
Because the strains are small, this is true whatever stress measure we
adopt (Cauchy stress or nominal stress), and is true whatever strain measure
we adopt (Lagrange strain or infinitesimal strain).
We will assume here that the material is isotropic (i.e. the response of a
material is independent of its orientation with respect to the loading
direction). In principle, it would be
possible to develop anisotropic hypoelastic models, but this is rarely done.
The stress strain law is constructed as follows:
Strains and rotations are assumed to be
small. Consequently, deformation is
characterized using the infinitesimal strain tensor defined in Section 2.1.7. In addition, all stress measures are taken
to be approximately equal. We can use
the Cauchy stress as the stress measure.
When
we develop constitutive equations for nonlinear elastic materials, it is
usually best to find an equation for the strain
energy density of the material as a function of the strain, instead of
trying to write down stress-strain laws directly. This has several advantages: (i) we can
work with a scalar function; and (ii) the existence of a strain energy
density guarantees that deformations of the material are perfectly
reversible.
If the material is isotropic, the strain
energy density can only be a function strain measures that do not depend on
the direction of loading with respect to the material. One can show that this means that the
strain energy can only be a function of invariants of the strain tensor that is to say, combinations of strain
components that have the same value in any basis (see Appendix B). The strain tensor always has three
independent invariants: these could be the three principal strains, for
example. In practice it is usually
more convenient to use the three fundamental scalar invariants:
Here, is a measure of the volume change associated
with the strain; is a measure of the shearing caused by the
strain, and I can’t think of a good physical interpretation for . Fortunately, it doesn’t often appear in
constitutive equations.
Strain energy density:
In principle, the strain energy density
could be any sensible function . In most practical applications, nonlinear
behavior is only observed when the material is subjected to shear deformation
(characterized by ); while stress varies linearly with volume
changes (characterized by ).
This behavior can be characterized by a strain energy density
where are material properties (see below for a
physical interpretation).
Stress-strain behavior
For this strain energy density function, the stress follows as
The strain can also be calculated in terms of stress
where is the second invariant of the stress
tensor.
To interpret these results, note that
If the solid is subjected to uniaxial tension, (with
stress and all other stress components zero); the nonzero
strain components are
If the solid is subjected to hydrostatic stress
(with and all other stress components zero) the
nonzero strain components are
If the solid is subjected to pure shear stress
(with and all other stress components zero) the
nonzero strains are
Thus, the solid responds linearly to
pressure loading, with a bulk modulus K. The relationship between shear stress and
shear strain is a power law, with exponent n.
This is just an example of a hypoelastic stress-strain law many other forms could be used.
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