Problems for Chapter 3
Constitutive Models: Relations between
Stress and Strain
3.9. Viscoelasticity
3.9.1. The figure shows a thin film of material that is
deformed plastically during a pressure-shear plate impact experiment. The goal of this problem is to derive the
equations governing the velocity and stress fields in the specimen. Assume that:
- The film deforms in
simple shear, and that the velocity and Kirchoff stress fields are independent of
- The material has mass
density and isotropic elastic response, with
shear modulus and Poisson’s ratio
- The film can be idealized
as a finite strain viscoplastic solid with power-law Mises flow potential,
as described in Section 3.9. Assume
that the plastic spin is zero.
3.9.1.1.
Calculate the
velocity gradient tensor L, the
stretch rate tensor D and spin
tensor W for the deformation,
expressing your answer as components in the basis shown in the figure
3.9.1.2.
Find an
expression for the plastic stretch rate, in terms of the stress and material
properties
3.9.1.3.
Use the elastic
stress rate-stretch rate relation to obtain an expression for the time
derivative of the shear stress q and
the stress components in terms of ,
and appropriate material properties
3.9.1.4.
Write down the
linear momentum balance equation in terms of and .
3.9.1.5.
How would the
governing equations change if ?