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 $v=v(y_2,t)e_1$ and Kirchoff stress fields $\tau =q(y_2,t)(e_1\otimes e_2+e_2\otimes e_1)+({\tau }_{11}{e}_1\otimes e_1+{\tau }_{22}{e}_2\otimes e_2+{\tau }_{33}{e}_3\otimes e_3)$ 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 $v(x_2)$, $\tau$ 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 $W^p=W$?