Problems for Chapter 3
Constitutive Models: Relations between Stress and Strain
3.8. Viscoplasticity
3.8.1. Suppose that a uniaxial tensile specimen with length made from Aluminum can be characterized by a viscoplastic constitutive law with properties listed in Section 3.8.4. Plot a graph showing the strain rate of the specimen as a function of stress. Use log scales for both axes, with a stress range between 5 and 60 MPa, and show data for room temperature; 100^{0}C, 200^{0}C, 300^{0}C, 400^{0}C and 500^{0}C. Would you trust the predictions of the constitutive equation outside this range of temperature and stress? Give reasons for your answer.
3.8.2. A uniaxial tensile specimen can be idealized as an elasticviscoplastic solid, with Young’s modulus E, and a flow potential given by
$g({\sigma}_{e})={\dot{\epsilon}}_{0}^{}{\left(\frac{{\sigma}_{e}}{Y}\right)}^{m}$
where Y, ${\dot{\epsilon}}_{0}$ and m are material properties. The specimen is stress free at time t=0¸ and is then stretched at a constant (total) strain rate $\dot{\eta}$.
3.8.2.1. Show that the equation governing the axial stress in the specimen can be expressed in dimensionless form as $\frac{d\tilde{\sigma}}{d\tilde{t}}+{\tilde{\sigma}}^{m}=1$, where $\tilde{\sigma}=(\sigma /Y){({\dot{\epsilon}}_{0}/\dot{\eta})}^{1/m}$ and $\tilde{t}=\left(E\dot{\eta}t/Y\right){\left({\dot{\epsilon}}_{0}/\dot{\eta}\right)}^{1/m}$ are dimensionless measures of stress and time.
3.8.2.2. Hence, deduce that the normalized stress $\tilde{\sigma}$ is a function only of the material parameter m and the normalized strain $\tilde{\epsilon}=\epsilon (E/Y){({\dot{\epsilon}}_{0}/\dot{\eta})}^{1/m}$.
3.8.2.3. Show that during steady state creep $\tilde{\sigma}=1$.
3.8.2.4. Obtain an analytical solution relating $\tilde{\sigma}$ to $\tilde{\epsilon}$ for m=1.
3.8.2.5. Obtain an analytical solution relating $\tilde{\sigma}$ to $\tilde{\epsilon}$ for very large m (note that, in this limit the material behaves like an elasticperfectly plastic, rate independent solid, with yield stress Y).
3.8.2.6. By integrating the governing equation for $\tilde{\sigma}$ numerically, plot graphs relating $\tilde{\sigma}$ to $\tilde{\epsilon}$ for a few values of m between m=1 and m=100.
3.8.2.7.
Estimate the
time, and strain, required for a tensile specimen of Aluminum to reach steady
state creep at a temperature of 400^{0}C, when deformed at a strain
rate of

3.8.3. The figure shows a thin polycrystalline Al film on a substrate. The film can be idealized as an elasticviscoplastic solid in which the steadystate uniaxial strain rate, stress temperature relation given by $\dot{\epsilon}={\dot{\epsilon}}_{0}\mathrm{exp}(Q/kT){\left(\sigma /Y\right)}^{m}$, where ${\dot{\epsilon}}_{0}$, Q and Y are material constants, and k is the Boltzmann constant. Suppose that the film is stress free at some temperature ${T}_{0}$ time t=0. Its temperature is then raised at steady rate $T={T}_{0}(1+\beta t)$ . Using the material properties listed in Section 3.8.4, calculate and plot the variation of stress in the film with time, for various values of ${T}_{0}$ and $\beta $ . You will need to integrate the firstorder differential equation for the stress numerically, e.g. using the ODE solvers in MATLAB.

3.8.4. A cylindrical, thinwalled pressure vessel with initial radius R, length L and wall thickness t<<R is subjected to internal pressure p. The vessel is made from an elasticpowerlaw viscoplastic solid with Young’s modulus E, Poisson’s ratio $\nu $, and a flow potential given by
$g({\sigma}_{e})={\dot{\epsilon}}_{0}{\left(\frac{{\sigma}_{e}}{Y}\right)}^{m}$
where ${\sigma}_{e}$ is the VonMises eequivalent stress. Recall that the stresses in a thinwalled pressurized tube are related to the internal pressure by ${\sigma}_{zz}=pR/(2t)$, ${\sigma}_{\theta \theta}=pR/t$. Calculate the steadystate strain rate in the vessel, as a function of pressure and relevant geometric and material properties. Hence, calculate an expression for the rate of change of the vessel’s length, radius and wall thickness as a function of time.