 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; 1000C, 2000C, 3000C,  4000C and 5000C.   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 elastic-viscoplastic solid, with Young’s modulus E, and a flow potential given by

$g\left({\sigma }_{e}\right)={\stackrel{˙}{\epsilon }}_{0}^{}{\left(\frac{{\sigma }_{e}}{Y}\right)}^{m}$

where Y, ${\stackrel{˙}{\epsilon }}_{0}$ and m are material properties.  The specimen is stress free at time t=0¸ and is then stretched at a constant (total) strain rate $\stackrel{˙}{\eta }$.

3.8.2.1.            Show that the equation governing the axial stress in the specimen can be expressed in dimensionless form as $\frac{d\stackrel{˜}{\sigma }}{d\stackrel{˜}{t}}+{\stackrel{˜}{\sigma }}^{m}=1$, where $\stackrel{˜}{\sigma }=\left(\sigma /Y\right){\left({\stackrel{˙}{\epsilon }}_{0}/\stackrel{˙}{\eta }\right)}^{1/m}$ and $\stackrel{˜}{t}=\left(E\stackrel{˙}{\eta }t/Y\right){\left({\stackrel{˙}{\epsilon }}_{0}/\stackrel{˙}{\eta }\right)}^{1/m}$ are dimensionless measures of stress and time.

3.8.2.2.            Hence, deduce that the normalized stress $\stackrel{˜}{\sigma }$ is a function only of the material parameter m and the normalized strain $\stackrel{˜}{\epsilon }=\epsilon \left(E/Y\right){\left({\stackrel{˙}{\epsilon }}_{0}/\stackrel{˙}{\eta }\right)}^{1/m}$.

3.8.2.3.            Show that during steady state creep $\stackrel{˜}{\sigma }=1$.

3.8.2.4.            Obtain an analytical solution relating $\stackrel{˜}{\sigma }$ to $\stackrel{˜}{\epsilon }$ for m=1.

3.8.2.5.            Obtain an analytical solution relating $\stackrel{˜}{\sigma }$ to $\stackrel{˜}{\epsilon }$ for very large m (note that, in this limit the material behaves like an elastic-perfectly plastic, rate independent solid, with yield stress Y).

3.8.2.6.            By integrating the governing equation for $\stackrel{˜}{\sigma }$ numerically, plot graphs relating $\stackrel{˜}{\sigma }$ to $\stackrel{˜}{\epsilon }$ 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 4000C, when deformed at a strain rate of 10-3s-1 3.8.3.      The figure shows a thin polycrystalline Al film on a substrate.  The film can be idealized as an elastic-viscoplastic solid in which the steady-state uniaxial strain rate, stress temperature relation given by $\stackrel{˙}{\epsilon }={\stackrel{˙}{\epsilon }}_{0}\mathrm{exp}\left(-Q/kT\right){\left(\sigma /Y\right)}^{m}$, where ${\stackrel{˙}{\epsilon }}_{0}$, 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 $T={T}_{0}\left(1+\beta t\right)$ .   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 first-order differential equation for the stress numerically, e.g. using the ODE solvers in MATLAB. 3.8.4.      A cylindrical, thin-walled 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 elastic-power-law viscoplastic solid with Young’s modulus E, Poisson’s ratio $\nu$, and a flow potential given by

$g\left({\sigma }_{e}\right)={\stackrel{˙}{\epsilon }}_{0}{\left(\frac{{\sigma }_{e}}{Y}\right)}^{m}$

where ${\sigma }_{e}$ is the Von-Mises eequivalent stress. Recall that the stresses in a thin-walled pressurized tube are related to the internal pressure by ${\sigma }_{zz}=pR/\left(2t\right)$, ${\sigma }_{\theta \theta }=pR/t$. Calculate the steady-state 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.