Problems for Chapter 3
Constitutive Models: Relations between
Stress and Strain
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
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
where Y, and m
are material properties. The specimen is
stress free at time t=0¸ and is then
stretched at a constant (total) strain rate .
Show that the
equation governing the axial stress in the specimen can be expressed in
dimensionless form as ,
where and are dimensionless measures of stress and time.
that the normalized stress is a function only of the material parameter m and the normalized strain .
Show that during
steady state creep .
analytical solution relating to for m=1.
analytical solution relating to for very large m (note that, in this limit the material behaves like an
elastic-perfectly plastic, rate independent solid, with yield stress Y).
the governing equation for numerically, plot graphs relating to for a few values of m between m=1 and
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 ,
Q and Y are material constants, and k
is the Boltzmann constant. Suppose that
the film is stress free at some temperature time t=0. Its temperature is then raised at steady rate
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 and . 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 ,
and a flow potential given by
where is the Von-Mises eequivalent stress. Recall
that the stresses in a thin-walled pressurized tube are related to the internal
pressure by ,
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.