Problems for Chapter 3
Constitutive Models: Relations between
Stress and Strain
3.10. Large Strain
Viscoelasticity
3.10.1. A cylindrical specimen is made from a material that
can be idealized using the finite-strain viscoelasticity model described in
Section 3.10. The specimen may be
approximated as incompressible.
3.10.1.1.
Let L denote the length of the deformed
specimen, and denote the initial length of the
specimen. Write down the deformation
gradient in the specimen in terms of
3.10.1.2.
Let denote the decomposition of stretch in to
elastic and plastic parts. Write down
the elastic and plastic parts of the deformation gradient in terms of and find expressions for the elastic and
plastic parts of the stretch rate in terms of
3.10.1.3.
Assume that the
material can be idealized using Arruda-Boyce potentials
Obtain an expression for the stress in the specimen in
terms of ,
using only the first two term in the expansion for simplicity. Your answer should include an indeterminate
hydrostatic part.
3.10.1.4.
Calculate the
deviatoric stress measure
in
terms of ,
and hence find an expression for in terms of
3.10.1.5.
Suppose that the
specimen is subjected to a harmonic cycle of nominal strain such that . Use the results of 3.10.1.2 and 3.10.1.4 to
obtain a nonlinear differential equation for
3.10.1.6.
Use the material
data given in Section 3.10.5 to calculate (numerically) the variation of Cauchy
stress in the solid with time induced by cyclic straining. Plot the results as a curve of Cauchy stress
as a function of true strain. Obtain
results for various values of and frequency .