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Chapter 3
Constitutive
Models  Relations between Stress and Strain
3.4 Generalized
Hooke’s law elastic materials subjected to small
stretches but large rotations
Recall that the
stress-strain law for an anisotropic, linear elastic material (Sect 3.1) has
the form
where is stress (any stress measure you like), is the infinitesimal strain, and is the tensor of elastic moduli.
This stress-strain relation can only be used if the material is
subjected to small deformations, and small rotations. This is partly because the infinitesimal
strain for a finite rotation so the law predicts that a nonzero stress is
required to rotate a solid.
There are some situations where a solid is subjected to small shape
changes, but large rotations. For
example, the picture shows a long slender beam bent into a circle by moments
applied to its ends. The strains in
the beam are of order ,
where h is the thickness of the
beam and R is its curvature. The ends of the beam have rotated through
a full 90 degrees, however. The
linear elastic constitutive equations would not predict the correct stress in
the beam.
It is easy to
fix this problem: provided we choose a sensible (nonlinear) strain measure,
together with the appropriate work-conjugate stress measure, we can still use
a linear stress-strain relation. To
make this precise, suppose that a solid is subjected to a displacement field . Define
 The deformation
gradient and its jacobian
The Lagrange strain
The Eulerian strain
The rotation tensor (see 2.1.13 for the best
way to compute R in practice)
The Cauchy (“true”) stress ,
defined so that
The Material stress (work conjugate to
Lagrange strain)
The Material stress-Lagrange strain
relation can be expressed as
where is the tensor of elastic moduli for the material with orientation in the
undeformed configuration. This is
identical to the stress-strain relation for a linear elastic solid, except
that the stress measure has been replaced by Material stress, and the strain
measure has been replaced by Lagrange strain. You can therefore use all the
matrix representations and tables of data given in Section 3.1 to apply the
constitutive equation. The Cauchy (“true”) stress can be computed from the
material stress as
The Cauchy stress-Eulerian strain
relation: Alternatively, the
stress-strain relation can be expressed in terms of stress and deformation
measures that characterize the deformed solid, as
where is the tensor of elastic moduli for the material with orientation of the deformed
configuration. This tensor is
related to by
For the special
case of an isotropic material with
Young’s modulus E and Poisson’s
ratio
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