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
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 .
3.8.2.1.
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.
3.8.2.2.
Hence, deduce
that the normalized stress is a function only of the material parameter m and the normalized strain .
3.8.2.3.
Show that during
steady state creep .
3.8.2.4.
Obtain an
analytical solution relating to for m=1.
3.8.2.5.
Obtain an
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).
3.8.2.6.
By integrating
the governing equation for numerically, plot graphs relating to 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
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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 ,
where ,
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.
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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.