Elastic-Plastic Fracture Mechanics. Lecture 1

These notes belong to a course on fracture mechanics

Decouple elastic deformation of the body and inelastic process of separation. Up to this point we have been dealing with the following situation. When a load causes a crack to extend in a body, a large part of the body is elastic, and the inelastic process of separation occurs in a zone around the front of the crack. Inelastic process of separation includes, for example, breaking of atomic bonds, growth of voids, and hysteresis in deformation.

So long as the inelastic zone is much smaller than the body, we mentally decouple the elastic deformation of the body and the inelastic process of separation. The elastic deformation of the body is used to define the driving force for the extension of the crack, the energy release rate. The inelastic process of separation is left to experimental measurement. We have visited several landmarks:

• For glass, Griffith (1921) modeled the body by the linear elastic theory, and represented the process of separation by surface energy.
• For a metal, Irwin and Orowan (1950s) modeled the body by the linear elastic theory, and lumped the process of separation into a single parameter, the fracture energy, to be measured experimentally.
• For rubber, Rivlin and Thomas (1953) modeled the body by the nonlinear elastic theory, and lumped the process of separation into a single parameter, the fracture energy, to be measured experimentally.

The small-scale yielding condition is difficult to apply to ductile metals. When a load causes a crack to extend in a ductile metal, the size of the plastic zone often exceeds 1 cm. To satisfy the small-scale yielding condition would require a specimen of the size of a file cabinet. While such specimens have indeed been used, they are often impractical. Furthermore, even if the fracture energy is measured for such a ductile metal, the small-scale yielding condition limits the utility of the fracture energy to large structures containing large cracks.

Elastic-Plastic Fracture Mechanics in a nutshell. The phenomenon of plastic deformation is readily demonstrated by simple experiments, such as bending a paper clip. But the theory of plasticity is rather intricate, which you will have to learn properly in a separate course. Fortunately, in formulating the elastic-plastic fracture mechanics, we need only a very simple and small part of the theory of plasticity:

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

For a crack extending in a body, one can separately consider inelasticity of two types:

• Messy inelasticity. Growing voids. Breaking bonds. Hysteretic deformation. Lump everything that you don’t want to deal with into the fracture process zone.
• Tidy inelasticity. The kind of inelasticity whose stress-strain behavior is indistinguishable from nonlinear elasticity. Model the body as a fictitious nonlinear elastic body.

So long as the zone of messy inelasticity (i.e., the fracture process zone) is much smaller than the specimen, one can extend the approach of Griffith to the fictitious nonlinear elastic solids. This idea was initially developed in several theoretical works. The J integral was developed for the fictitious nonlinear elastic solids (Rice, 1968). The crack-tip field, the HRR field, was obtained for elastic-plastic solids (Hutchinson, 1968; Rice and Rosengren, 1968). These theoretical works inspired the first experimental demonstration of elastic-plastic fracture mechanics for ductile metals (Begley and Landes, 1972).