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The J integral
For a crack in an elastic body subject to a load, the elastic energy stored in the body is a function of two independent variables: the displacement of the load, and the area of the crack. The energy release rate is defined by the partial derivative of the elastic energy of the body with respect to the area of the crack.
This definition of the energy release rate assumes that the body is elastic, but invokes no field theory. Indeed, the energy release rate can be determined experimentally by measuring the load-displacement curves of identically loaded bodies with different areas of the cracks. No field need be measured.
Many materials, however, can be modeled with a field theory of elasticity. When a material is modeled by such a field theory, the energy release rate can be represented in terms of field variables by an integral, the J integral.
This lecture describes the J integral, along with examples of calculation. Uses of the J integral are often better appreciated in the context of individual applications, which we will describe in later lectures.
The J integral can be developed for both linear and nonlinear elastic theories. The nonlinear elastic theory will be used in class, and the linear elastic theory will be used in a homework problem.
These notes belong to a course on fracture mechanics