Problems for Chapter 3
Constitutive Models: Relations between
Stress and Strain
3.5. Hyperelasticity
3.5.1. Derive the stress-strain relations for an
incompressible, Neo-Hookean material subjected to
3.5.1.1.
Uniaxial tension
3.5.1.2.
Equibiaxial
tension
3.5.1.3.
Pure shear
Derive expressions for the Cauchy stress, the Nominal
stress, and the Material stress tensors (the solutions for nominal stress are
listed in the table in Section 3.5.6).
You should use the following procedure: (i) assume that the specimen
experiences the length changes listed in 3.5.6; (ii) use the formulas in
Section 3.5.5 to compute the Cauchy stress, leaving the hydrostatic part of the
stress p as an unknown; (iii)
Determine the hydrostatic stress from the boundary conditions (e.g. for
uniaxial tensile parallel to you know ;
for equibiaxial tension or pure shear in the plane you know that )
3.5.2.
Repeat problem
3.5.1 for an incompressible Mooney-Rivlin material.
3.5.3.
Repeat problem
3.5.1 for an incompressible Arruda-Boyce material
3.5.4.
Repeat problem
3.5.1 for an incompressible Ogden
material.
3.5.5. Using the results listed in the table in Section
3.5.6, and the material properties listed in Section 3.5.7, plot graphs showing
the nominal stress as a function of stretch ratio for each of (a) a Neo-Hookean material; (b) a
Mooney-Rivlin material; (c) the Arruda-Boyce material and (c) the Ogden
material when subjected to uniaxial tension, biaxial tension, and pure shear
(for the latter case, plot the largest tensile stress ).
3.5.6.
A foam specimen
is idealized as an Ogden-Storakers foam with strain energy density
where
and are material properties. Calculate:
3.5.6.1.
The Cauchy stress
in a specimen subjected to a pure volume change with principal stretches
3.5.6.2.
The Cauchy stress
in a specimen subjected to volume preserving uniaxial extension
3.5.6.3.
The Cauchy stress
in a specimen subjected to uniaxial tension, as a function of the tensile
stretch ratio .
(To solve this problem you will need to assume that the solid is subjected to
principal stretches parallel to ,
and stretches parallel to and . You will need to determine from the condition that in a uniaxial tensile test.
3.5.7.
Suppose that a
hyperelastic solid is characterized by a strain energy density where
are invariants of the Left Cauchy-Green deformation
tensor . Suppose
that the solid is subjected to an infinitesimal strain, so that B can be approximated as ,
where is a symmetric infinitesimal strain
tensor. Linearize the constitutive
equations for ,
and show that the relationship between Cauchy stress and infinitesimal strain is equivalent to the isotropic linear elastic
constitutive equation. Give formulas
for the bulk modulus and shear modulus for the equivalent solid in terms of the
derivatives of .
3.5.8.
The constitutive
law for a hyperelastic solid is derived from a strain energy potential ,
where
are the invariants of the Left Cauchy-Green
deformation tensor .
3.5.8.1.
Calculate the
Cauchy stress induced in the solid when it is subjected to a rigid rotation,
followed by an arbitrary homogeneous deformation. Hence, demonstrate that the constitutive law
is isotropic.
3.5.8.2.
Apply the simple
check described in Section 3.1 to test whether the constitutive law is
objective.
3.5.9. The strain energy density of a hyperelastic solid is
sometimes specified as a function of the right
Cauchy-Green deformation tensor ,
instead of as described in Section 3.5. (This procedure must be used if the material
is anisotropic, for example)
3.5.9.1.
Suppose that the
strain energy density has the general form . Derive formulas for the Material stress,
Nominal stress and Cauchy stress in the solid as functions of ,
and
3.5.9.2.
Apply the simple
check described in Section 3.1 to demonstrate that the resulting stress-strain
relation is objective.
3.5.9.3.
Calculate the
Cauchy stress induced in the solid when it is subjected to a rigid rotation,
followed by an arbitrary homogeneous deformation. Hence, demonstrate that the constitutive law
is not, in general, isotropic.
3.5.9.4.
Suppose that the
constitutive law is simplified further by writing the strain energy density as
a function of the invariants of C, i.e. ,
where
Derive
expressions relating the Cauchy stress components to .
3.5.9.5.
Demonstrate that
the simplified constitutive law described in 3.5.9.4 characterizes an isotropic
solid.