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Hw4.pdf
Homework 4
Fracture, Fatigue
1
A thin metal plate (half-height = 1, half width = 0.2 and thickness = 0.01 all in meters)
is loaded in uniaxial tension parallel to its height that produces a stress σ = 107 P a at its
border. The plate has an elliptical hole (major semi-axis a and minor semi-axis b = 0.01) at
its center with the major axis perpendicular to the loading direction. Assume isotropic linear
elastic behavior with E = 1011 P a, ν = 0.3. Develop a finite element model of the loaded
plate and use it to determine the stress concentration factor due to the hole as a function of
the ratio a/b. Considere values 1 ≤ a/b ≤ 10. Compare your results with standard against
those presented in a standard reference such as Roark’s.
2
A large steel sheet has a central crack of length 2a = 0.04 m and undergoes catastrophic
fracture at a fracture stress σF = 480 M P a. A second large sheet of the same steel has a
central crack of length 2a = 0.1 m. Use the fracture stress equation in the form
C
σF = √
a
where C is a constant and calculate its fracture stress.
3
A forensic examination of a fatigue specimen showed that an initial crack of size = 0.003 m,
grew to a final size of 0.008 m by fast fracture during testing. Use the Paris equation in the
1
form
m
da
= C(∆K)m = C(∆σ)m (πa) 2
dN
with m = 3, to obtain, by integration an expression relating the initial and final crack
sizes with the stress amplitude and the number of cycles to failure. Then, use the resulting
expression to determine the increase in fatigue life that can be expected in the specimen if
the initial crack size is reduced to 0.001 m (without changes in the final crack size). Give
the increased life as the ratio of the number of cycles to failure for the specimen with the
smaller initial crack to that for the original specimen.
2
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