Interpretation of the J-Integral(Interaction between Mechanics of Materials and Mathematics)

In this paper, we present an interpretation that the J-integral proposed as a parameter of a crack by J. R. Rice is the shape derivative of a singular point. First, we introduce the definition of the original J-integral and the results of an investigation into its physical meaning. Then, we introduce proposals for the extension of the J-integral that were published based on the physical meanings. We also introduce that K. Ohtsuka had proposed the generalized Jintegral that extended the J-integral to three dimensions in mathematics at that time. After that, Ohtsuka realized that the Hadamard formula for the derivative of a functional with respect to domain variation can be derived from the generalized J-integral, and came into contact with the author who had proposed a shape optimization method. We show that the J-integral is the shape derivative of a singular point, which we learned from this exchange. Finally, we point out some points to note regarding the integrals used in the extended J-integral.