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# Elastic-Plastic Fracture Mechanics. Lecture 2

These notes belong to a course on fracture mechanics

Lecture 1 described the Begley-Landes experiment, and the blunting of a crack due to large deformation. Lecture 2 is motivated by the following considerations.

When a rubber containing a crack is loaded, before the crack extends, strain everywhere in the rubber can be large. By contrast, when a metal containing a crack is loaded, before the crack extends, strain in the metal is typically small, except near the tip of the blunted crack. Consequently, to analyze deformation in the metal, at a distance a few times the crack opening displacement away from the crack tip, we can use the field theory of infinitesimal deformation.

When a metal strains far beyond the elastic limit, the stress-strain relation is often fit to a power law.

When a metal undergoes homogeneous deformation subject to proportional loading, the stress-strain behavior of plastic deformation is indistinguishable from that of elastic deformation.

The above idealizations are used to formulate a field theory of infinitesimal deformation and power-law material model. Within this theory, a large class of boundary-value problems has solutions of a remarkably simple form. This result, the Ilyushin theorem, generalizes the linearity in the linear elastic theory into a scaling relation for the power-law material.

The power-law material and the Ilyushin theorem are described in the present lecture. We then apply these ideas to elastic-plastic fracture mechanics. In particular, we will describe the HRR field.

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## Comments

## Large Scale Yielding Solution

I would like to interject that an exact large scale yielding solution has been recently published in the

Journal of Elasticity,99, 117-130 (2010). The initial geometry of this problem is that of an elliptical hole in an infinite plate. The material is perfectly plastic subject to plane stress loading conditions with tensile tractions at infinity. This exact solution should aid modelers of the near crack tip strain field for large deformations.