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 ${\stackrel{˙}{\lambda }}_{e},{\stackrel{˙}{\lambda }}_{p}$

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

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

${U}_{T}={\mu }_{T}\left\{\frac{1}{2}\left({\overline{I}}_{1}^{}-3\right)+\frac{1}{20{\beta }_{T}^{2}}\left({\overline{I}}_{1}^{2}-9\right)+\frac{11}{1050{\beta }_{T}^{4}}\left({\overline{I}}_{1}^{3}-27\right)+...\right\}$

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

${{\tau }^{\prime }}_{ij}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\left[\frac{1}{{J}_{e}^{2/3}}\left(\frac{\partial {U}_{T}}{\partial {\overline{I}}_{1}^{e}}+{\overline{I}}_{1}^{e}\frac{\partial {U}_{T}}{\partial {\overline{I}}_{2}^{e}}\right){B}_{ij}^{e}-\frac{{\overline{I}}_{1}^{e}}{3}\frac{\partial {U}_{T}}{\partial {\overline{I}}_{1}^{e}}{\delta }_{ij}-\frac{1}{{J}_{e}^{4/3}}\frac{\partial {U}_{T}}{\partial {\overline{I}}_{2}^{e}}{B}_{ik}^{e}{B}_{kj}^{e}\right]$

in terms of ${\lambda }_{e}$, and hence find an expression for ${\stackrel{˙}{\lambda }}_{p}$ in terms of ${\lambda }_{e}$

3.10.1.5.        Suppose that the specimen is subjected to a harmonic cycle of nominal strain such that $L=\alpha {L}_{0}\mathrm{sin}\omega t$.  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 $\omega$.