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.

(c) A.F. Bower, 2008
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