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 $v=v\left({y}_{2},t\right){e}_{1}$ and Kirchoff stress fields $\tau =q\left({y}_{2},t\right)\left({e}_{1}\otimes {e}_{2}+{e}_{2}\otimes {e}_{1}\right)+\left({\tau }_{11}{e}_{1}\otimes {e}_{1}+{\tau }_{22}{e}_{2}\otimes {e}_{2}+{\tau }_{33}{e}_{3}\otimes {e}_{3}\right)$ are independent of ${x}_{1}$
• The material has mass density $\rho$ and isotropic elastic response, with shear modulus $\mu$ and Poisson’s ratio $\nu$
• 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 ${e}_{1},{e}_{2}$ 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 $\stackrel{\nabla }{{\tau }_{ij}}={C}_{ijkl}{D}_{kl}^{e}$  to obtain an expression for the time derivative of the shear stress q and the stress components ${\tau }_{11},{\tau }_{22},{\tau }_{33}$ in terms of $v\left({x}_{2}\right)$, $\tau$ and appropriate material properties

3.9.1.4.            Write down the linear momentum balance equation in terms of $\tau$ and $v$.

3.9.1.5.            How would the governing equations change if ${W}^{p}=W$?