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 $L_0$ denote the initial length of the specimen.  Write down the deformation gradient in the specimen in terms of $\lambda =L/L_0$

3.10.1.2.        Let $\lambda ={\lambda }_e{\lambda }_p$ 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 ${\lambda }_e,{\lambda }_p$ and find expressions for the elastic and plastic parts of the stretch rate in terms of ${\dot{\lambda }}_e,{\dot{\lambda }}_p$

3.10.1.3.        Assume that the material can be idealized using Arruda-Boyce potentials

$U_{\infty }={\mu }_{\infty }\left\{\frac{1}{2}({\overline{I}}_1^{}-3)+\frac{1}{20{\beta }_{\infty }^2}({\overline{I}}_1^2-9)+\frac{11}{1050{\beta }_{\infty }^4}({\overline{I}}_1^3-27)+\mathrm{...}\right\}+\frac{K}{2}{\left(J-1\right)}^2$

$U_T={\mu }_T\left\{\frac{1}{2}({\overline{I}}_1^{}-3)+\frac{1}{20{\beta }_T^2}({\overline{I}}_1^2-9)+\frac{11}{1050{\beta }_T^4}({\overline{I}}_1^3-27)+\mathrm{...}\right\}$

Obtain an expression for the stress in the specimen in terms of ${\lambda }_e,{\lambda }_p$, 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

${{\tau }^{\prime }}_{ij}=2\left[\frac{1}{J_e^{2/3}}\left(\frac{\partial U_T}{\partial {\overline{I}}_1^e}+{\overline{I}}_1^e\frac{\partial U_T}{\partial {\overline{I}}_2^e}\right)B_{ij}^e-\frac{{\overline{I}}_1^e}{3}\frac{\partial U_T}{\partial {\overline{I}}_1^e}{\delta }_{ij}-\frac{1}{J_e^{4/3}}\frac{\partial U_T}{\partial {\overline{I}}_2^e}{B}_{ik}^e{B}_{kj}^e\right]$

in terms of ${\lambda }_e$, and hence find an expression for ${\dot{\lambda }}_p$ in terms of ${\lambda }_e$

3.10.1.5.        Suppose that the specimen is subjected to a harmonic cycle of nominal strain such that $L=\alpha L_0\sin \omega t$.  Use the results of 3.10.1.2 and 3.10.1.4 to obtain a nonlinear differential equation for ${\lambda }_e$

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 $\alpha$ and frequency $\omega$.