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Problems
for Chapter 9
Modeling
Material Failure
9.5. Plastic Fracture
Mechanics
9.5.1.
Explain briefly
the main concepts underlying the use of the J integral as a fracture criterion in components experiencing
large-scale plastic deformation.
9.5.2.
The figure
shows the tip of a semi-infinite crack in an elastic-plastic material with a
bi-linear uniaxial stress-strain curve, as indicated in the figure. To provide some insight into the nature of
the crack tip fields, the constitutive behavior can be approximated as a
hypoelastic material, characterized by a strain energy density W such that  . Suppose that the solid is subjected to
remote mode I loading (so that the shear stresses on ).
9.5.2.1.
Construct the
full stress-strain equations for the hypoelastic material, using the approach
described in Section 3.3
9.5.2.2.
Consider a
material point that is very far from the crack tip, and so is subjected to a
very low stress. Write down the
asymptotic stress field in this region, in terms of an arbitrary constant that characterizes the magnitude of the
remote mode I loading
9.5.2.3.
Consider a
material point that is very close to the crack tip, and so is subjected to a
very large stress. Write down the
asymptotic stress field in this region, in terms of an arbitrary constant that characterizes the magnitude of the near
tip stresses.
9.5.2.4.
Using the path
independence of the J integral, find a relationship between ,
,
and the slopes of the uniaxial stress-strain curve
9.5.2.5.
Suppose that
the material fractures when the stress at a small distance ahead of the crack tip reaches a critical
magnitude . Assume that the critical distance is much
smaller than the region of high stress considered in 2.4. Calculate the critical value of that will cause the crack to grow, in terms
of relevant material parameters.
9.5.2.6.
Consider a
finite sized crack with length a in
the hypoelastic material. Assume that
the solid is subjected to a remote uniaxial stress far from the crack. Discuss qualitatively how the stress field
around the crack evolves as the remote stress is increased. Discuss the implications of this behavior
on the validity of the fracture criterion derived in 9.5.2.5.
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