Problems for Chapter 3
Constitutive Models: Relations between
Stress and Strain
3.7. Small Strain
Plasticity
3.7.1. The stress state induced by stretching a large plate
containing a cylindrical hole of radius a
at the origin is given by
Here,
is the stress in the plate far from the hole.
(Stress components not listed are all zero)
3.7.1.1.
Plot contours of
von-Mises equivalent stress (normalized by ) as a function of and ,
for a material with Hence identify the point in the solid that
first reaches yield.
3.7.1.2.
Assume that the
material has a yield stress Y .Calculate
the critical value of that will just cause the plate to reach yield.
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3.7.2. The stress state (expressed in cylindrical-polar
coordinates) in a thin disk with mass density that spins with angular velocity can be shown to be
Assume that the disk is made from an elastic-plastic
material with yield stress Y and .
3.7.2.1.
Find a formula
for the critical angular velocity that will cause the disk to yield, assuming
Von-Mises yield criterion. Where is the
critical point in the disk where plastic flow first starts?
3.7.2.2.
Find a formula
for the critical angular velocity that will cause the disk to yield, using the
Tresca yield criterion. Where is the
critical point in the disk where plastic flow first starts?
3.7.2.3.
Using parameters
representative of steel, estimate how much kinetic energy can be stored in a
disk with a 0.5m radius and 0.1m thickness.
3.7.2.4.
Recommend the
best choice of material for the flywheel in a flywheel energy storage
system.
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3.7.3. An isotropic, elastic-perfectly plastic thin film with
Young’s Modulus ,
Poisson’s ratio , yield stress in uniaxial tension Y
and thermal expansion coefficient is bonded to a stiff substrate. It is stress free at some initial temperature
and then heated. The substrate prevents the film from stretching in its own
plane, so that ,
while the surface is traction free, so that the film deforms in a state of
plane stress. Calculate the critical temperature change that will cause the film to yield, using (a)
the Von Mises yield criterion and (b) the Tresca yield criterion.
3.7.4. Assume that the thin film
described in the preceding problem shows so little strain hardening behavior
that it can be idealized as an elastic-perfectly plastic solid, with uniaxial
tensile yield stress Y. Suppose the film is stress free at some
initial temperature, and then heated to a temperature ,
where is the yield temperature calculated in the
preceding problem, and .
3.7.4.1.
Find the stress in the film at this
temperature.
3.7.4.2.
The film is then cooled back to its original
temperature. Find the stress in the film
after cooling.
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3.7.5. Suppose that the thin film described in the preceding
problem is made from an elastic, isotropically hardening plastic material with
a Mises yield surface, and yield stress-v-plastic strain as shown in the
figure. The film is initially stress
free, and then heated to a temperature ,
where is the yield temperature calculated in problem
1, and .
3.7.5.1.
Find a formula
for the stress in the film at this temperature.
3.7.5.2.
The film is then
cooled back to its original temperature.
Find the stress in the film after cooling.
3.7.5.3.
The film is cooled further by a temperature change . Calculate the critical value of that will cause the film to reach yield again.
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3.7.6. Suppose that the thin film described in the preceding
problem is made from an elastic, linear
kinematically hardening plastic
material with a Mises yield surface, and yield stress-v-plastic strain as shown
in the figure. The stress is initially
stress free, and then heated to a temperature ,
where is the yield temperature calculated in problem
1, and .
3.7.6.1.
Find a formula
for the stress in the film at this temperature.
3.7.6.2.
The film is then
cooled back to its original temperature.
Find the stress in the film after cooling.
3.7.6.3.
The film is cooled further by a temperature change . Calculate the critical value of that will cause the film to reach yield again.
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3.7.7. A thin-walled tube of mean radius a and wall
thickness t<<a is subjected to an axial load P which
exceeds the initial yield load by 10% (i.e. ). The
axial load is then removed, and a torque Q is applied to the tube. You may assume that the axial load induces a
uniaxial stress while the torque induces a shear stress . Find the magnitude of Q to cause
further plastic flow, assuming that the solid is
3.7.7.1.
an isotropically
hardening solid with a Mises yield surface
3.7.7.2.
a linear
kinematically hardening solid with a Mises yield surface
Express
your answer in terms of and appropriate geometrical terms, and assume
infinitesimal deformation.
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3.7.8. 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 isotropic
elastic-plastic solid with Young’s modulus E,
Poisson’s ratio ,
and its yield stress varies with accumulated plastic strain as . Recall that the stresses in a thin-walled
pressurized tube are related to the internal pressure by ,
3.7.8.1.
Calculate the
critical value of internal pressure required to initiate yield in the solid
3.7.8.2.
Find a formula
for the strain increment resulting from an increment in pressure
3.7.8.3.
Suppose that the
pressure is increased 10% above the initial yield value. Find a formula for the change in radius,
length and wall thickness of the vessel.
Assume small strains.
3.7.9. Write a simple program that will compute the history
of stress resulting from an arbitrary history of strain applied to an
isotropic, elastic-plastic von-Mises solid.
Assume that the yield stress is related to the accumulated effective
strain by ,
where ,
and n
are material constants. Check your code
by using it to compute the stress resulting from a volume preserving uniaxial
strain ,
and compare the predictions of your code with the analytical solution. Try one other cycle of strain of your
choice.
3.7.10. In a classic paper, Taylor, G. I., and Quinney, I., 1932, “The Plastic Distortion of Metals,” Philos. Trans. R. Soc.
London, Ser. A, 230, pp. 323362
described a series of experiments designed to investigate the plastic
deformation of various ductile metals. Among other things, they compared their
experimental measurements with the predictions of the von-Mises and Tresca
yield criteria and their associated flow rules.
They used the apparatus shown in the figure. Thin walled cylindrical tubes were first
subjected to an axial stress .
The stress was sufficient to extend the tubes plastically. The axial stress was then reduced to a
magnitde ,
with ,
and a progressively increasing torque was applied to the tube so as to induce a
shear stress in the solid. The twist, extension and
internal volume of the tube were recorded as the torque was applied. In this problem you will compare their
experimental results with the predictions of plasticity theory. Assume that the
material is made from an isotropically hardening rigid plastic solid, with a Von Mises yield surface, and yield
stress-v-plastic strain given by .
3.7.10.1.
One set of
experimental results is illustrated in the figure to the right. The figure shows the ratio required to initiate yield in the tube during
torsional loading as a function of m. Show that theory predicts that
3.7.10.2.
Compute the
magnitudes of the principal stresses at the point of yielding under combined axial
and torsional loads in terms of and m.
3.7.10.3.
Suppose that, for
a given axial stress ,
the shear stress is first brought to the critical value
required to initiate yield in the solid, and is then increased by an
infinitesimal increment . Find expressions for the resulting plastic
strain increments ,
in terms of m, ,
h and .
3.7.10.4.
Hence, deduce
expressions for the magnitudes of the principal strains increments resulting from the stress increment .
3.7.10.5.
Using the results
of 11.2 and 11.6, calculate the so-called “Lode parameters,” defined as
and
show that the theory predicts
3.7.11. The Taylor/Quinney experiments show that the
constitutive equations for an isotropically hardening Von-Mises solid predict
behavior that matches reasonably well with experimental observations, but there
is a clear systematic error between theory and experiment. In this problem, you will compare the predictions
of a linear kinematic hardening
law with experiment. Assume that the solid has a yield function
and hardening law given by
3.7.11.1.
Assume that
during the initial tensile test, the axial stress in the specimens reached a magnitude ,
where is the initial tensile yield stress of the
solid and is a scalar multiplier. Assume that the axial stress was then reduced
to and a progressively increasing shear stress
was applied to the solid. Show that the
critical value of at which plastic deformation begins is
given by
Plot against m for various values of .
3.7.11.2.
Suppose that, for
a given axial stress ,
the shear stress is first brought to the critical value required to initiate
yield in the solid, and is then increased by an infinitesimal increment . Find expressions for the resulting plastic
strain increments, in terms of m, ,
c and .
3.7.11.3.
Hence, deduce the
magnitudes of the principal strains in the specimen .
3.7.11.4.
Compute the
magnitudes of the principal stresses at the point of yielding under combined axial
and torsional loads in terms of m,
,
and .
3.7.11.5.
Finally, find
expressions for Lode’s parameters
3.7.11.6.
Plot versus for various values of ,
and compare your predictions with Taylor and Quinney’s measurements.
3.7.12.
An elastic-
nonlinear kinematic hardening solid has Young’s modulus ,
Poisson’s ratio ,
a Von-Mises yield surface
where Y is
the initial yield stress of the solid, and a hardening law given by
where and are material properties. In the undeformed solid, .
Calculate the formulas relating the total strain increment to the state of stress ,
the state variables and the increment in stress applied to the solid
3.7.13. Consider a rigid
nonlinear kinematic hardening solid, with yield surface and hardening law
described in the preceding problem.
3.7.13.1.
Show that the constitutive law implies that
3.7.13.2.
Show that under uniaxial loading with ,
3.7.13.3.
Suppose the material is subjected to a monotonically increasing uniaxial
tensile stress . Show that the uniaxial stress-strain curve
has the form (it is simplest to calculate as a function of the strain and then use the
yield criterion to find the stress)
3.7.14. Suppose that a solid contains a large number of
randomly oriented slip planes, so that it begins to yield when the resolved
shear stress on any plane in the
solid reaches a critical magnitude k.
3.7.14.1.
Suppose that the
material is subjected to principal stresses . Find a formula for the maximum resolved shear
stress in the solid, and by means of appropriate sketches, identify the planes
that will begin to slip.
3.7.14.2.
Draw the yield
locus for this material.
3.7.15. Consider a rate independent plastic material with
yield criterion . Assume that (i) the constitutive law for the
material has an associated flow rule, so
that the plastic strain increment is related to the yield criterion by ;
and (ii) the yield surface is convex, so that
for all stress
states and satisfying and and . Show that the material obeys the principle of
maximum plastic resistance.
3.7.16. The yield strength of a
frictional material (such as sand) depends on hydrostatic pressure. A simple model of yield and plastic flow in
such a material is proposed as follows:
Yield criterion
Flow rule
Where is a material constant (some measure of the
friction between the sand grains).
3.7.16.1.
Sketch the yield surface for this material in principal stress space
(note that the material looks like a Mises solid whose yield stress increases
with hydrostatic pressure. You will need
to sketch the full 3D surface, not just the projection that is used for
pressure independent surfaces)
3.7.16.2.
Sketch a vector indicating the direction of plastic flow for some point
on the yield surface drawn in part (3.8.3.1)
3.7.16.3.
By finding a counter-example, demonstrate that this material does not
satisfy the principle of maximum plastic resistance
(you can do this
graphically, or by finding two specific stress states that violate the
condition)
3.7.16.4.
Demonstrate that the material is not stable in the sense of Drucker i.e. find a cycle of loading for which the
work done by the traction increment through the displacement increment is
non-zero.
3.7.16.5.
What modification would be required to the constitutive law to make it
satisfy the principle of maximum plastic resistance and Drucker stability? How does the physical response of the stable
material differ from the original model (think about compaction under combined
shear and pressure).