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Problems
for Chapter 9
Modeling
Material Failure
9.2. Stress and
Strain Based Fracture and Fatigue Criteria
9.2.1.
A flat specimen
of glass with fracture strength  ,
Young’s modulus E and Poisson’s
ratio  is indented by a hard metal sphere with
radius R, Young’s modulus  and Poisson’s ratio . Using solutions for contact stress fields
given in Chapter 5, calculate a formula for the load P that will cause the glass to fracture, in terms of geometric
and material parameters. You can
assume that the critical stress occurs on the surface of the glass.
9.2.2.
The figure
shows a fiber reinforced composite laminate.Â
(i)
When loaded in uniaxial tension parallel to the fibers, it fails at a stress
of 500MPa.
(ii)
When loaded in uniaxial tension transverse to the fibers, it fails at a
stress of 250 MPa.Â
(iii) When loaded at 45 degrees to the fibers, it
fails at a stress of 223.6 MPa
The laminate is then loaded in uniaxial tension at
30 degrees to the fibers. Calculate
the expected failure stress under this loading, assuming that the material
can be characterized using the Tsai-Hill failure criterion.
9.2.3.
A number of
cylindrical specimens of a brittle material with a 1cm radius and length 4cm
are tested in uniaxial tension.  It is
found 60% of the specimens withstand a 150MPa stress without failure; while
30% withstand a 170 MPa stress without failure.
9.2.3.1.
Calculate
values for the Weibull parameters  and m
for the specimens
9.2.3.2.
Suppose that a
second set of specimens is made from the same material, with length 8cm and
radius 1cm.  Calculate the stress
level that will cause 50% of these
specimens to fail.
9.2.4.
A beam with length L, and rectangular cross-section  is made from a brittle material with Young’s
modulus E, Poisson’s ratio ,
and the failure probability distribution of a volume  is characterized by Weibull parameters  and .
9.2.4.1.
 Suppose that the beam is loaded in uniaxial
tension parallel to its length.Â
Calculate the stress level  corresponding to 63% failure, in terms of
geometric and material parameters.
9.2.4.2.
Suppose that
the beam is loaded in 3 point bending.Â
Let  denote the maximum value of stress in the
beam (predicted by beam theory). Find
an expression for the stress distribution in the beam in terms of
9.2.4.3.
Hence, find an
expression for the value of  that corresponds to 63% probability of
failure in the beam.  Calculate the
ratio .
9.2.5.
A glass shelf
with length L and rectangular
cross-section  is used to display cakes in a bakery. As a
result, it subjected to a daily cycle of load (which may be approximated as a
uniform pressure acting on it surface) of the form  where  is the time the store has been open, and  is the total time the store is open each
day. As received, the shelf has a
tensile strength ,
and the glass can be characterized by static fatigue parameters  and .  Find an expression for the life of the
shelf, in terms of relevant parameters.
9.2.6.
A cylindrical
concrete column with radius R,
cross-sectional radius R, and
length L is subjected to a
monotonically increasing compressive axial load P. Assume that the
material can be idealized using the constitutive law given in Section 9.2.4,
with the compressive yield stress-v-plastic strain of the form
where  and m
are material properties. Assume small
strains, and a homogeneous state of stress and strain in the column. Neglect elastic deformation, for
simplicity.
9.2.6.1.
Calculate the
relationship between the axial stress  and strain ,
in terms of the plastic properties c, Â and m
9.2.6.2.
Calculate the
volume change of the column, in terms of ,
c, Â and m
9.2.6.3.
Suppose that
the sides of the column are subjected to a uniform traction q.Â
Repeat the calculations in parts 9.2.6.1 and 9.2.6.2.
9.2.7.
Suppose that
the column described in the previous problem is encased in a steel tube, with
(small) wall thickness t. The steel can be idealized as a rigid
perfectly plastic material with yield stress . Calculate the relationship between the
axial stress  and strain ,
in terms of geometric and material properties.Â
9.2.8.
Extend the
viscoplastic finite element program described in Section 8.5 to model the
behavior of a porous plastic material with constitutive equations given in
Section 9.2.5. This will involve the
following steps:
9.2.8.1.
Develop a
procedure to calculate the stress ,
the void volume fraction ,
the effective strain measures  at the end of the time increment, given
their values at the start of the increment and given an increment in plastic
strain . You should use a fully implicit update, as
discussed in Section 8.5.  The
simplest approach is to set up, and solve, three simultaneous nonlinear
equations for (  ) using Newton-Raphson iteration, and
subsequently compute the stress distribution.
9.2.8.2.
Calculate the
tangent stiffness  for the material
9.2.8.3.
Implement the
new constitutive equations in the viscoplastic finite element program
9.2.8.4.
Test your code
by simulating the behavior of a uniaxial tensile specimen subjected to
monotonic loading.
9.2.9.
A specimen of
steel has a yield stress of 500MPa.Â
Under cyclic loading at a stress amplitude of 200 MPa it is found to
fail after  cycles, while at a stress amplitude of
100MPa it fails after  cycles.Â
This material is to be used to fabricate a plate, with thickness h, containing circular holes with
radius a<<h.  The plate will be subjected to constant
amplitude cyclic uniaxial stress far from the holes, and must have a life of
at least  cycles. Â
What is the maximum stress amplitude that the plate can withstand?
9.2.10. A spherical pressure vessel with internal radius a and external radius b=1.5a is repeatedly pressurized from zero internal pressure to a
maximum value . The sphere has yield stress Y, and its fatigue behavior of the
vessel (under fully reversed uniaxial tension) can be characterized by
Basquin’s law .
9.2.10.1.
Find an
expression for the fatigue life of the vessel in terms of ,
and relevant geometric and material properties. Assume that the effects of mean stress can
be approximated using Goodman’s rule.Â
Assume that
9.2.10.2.
Suppose that
the vessel is first pressurized to its collapse load and then unloaded, so as
to induce a distribution of residual stress in the cylinder.  It is subsequently subjected to a cyclic
pressure with magnitude . Calculate the mean stress and stress
amplitude as a function of position in the vessel wall, and hence deduce an
expression for its fatigue life.
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9.2.11. The figure shows a solder joint on a printed circuit
board. The printed circuit board can
be idealized as a pinned-pinned beam with thickness h, length L, Young’s
modulus E and mass density . The board vibrates in its fundamental mode
with a frequency  and mode shape . The yield stress of solder is so low it can
be neglected, and it is firmly bonded to the printed circuit board. As a result, it is subjected to a cyclic
plastic strain equal to the strain at the surface of the beam. The fatigue life of solder can be
characterized by a Coffin-Manson law .  Find an expression for the time to failure
of the solder joint, in terms of relevant geometric and material parameters.
9.2.12. A specimen of steel is tested under cyclic
loading. It is found to have a fatigue
threshold ,
and fails after  cycles when tested at a stress amplitude .   Suppose that, in service, the material
spends 80% of its life subjected to stress amplitudes ,
10% of its life at ,
and the remainder at .  Calculate the life of the component during
service (assume that the mean stress  during both testing and service).
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