This article describes the mathematics for building the foundations of the theory of the Generalized J-integral (GJ-integral) in various variational problems defined on D that gives the shape derivative on singular points such as boundaries, cracks, joints of different boundary conditions and interfaces, which is applicable in fracture mechanics and the shape gradient in shape sensitivity analysis. When the solution u is regular inside $\omega \cap D$ for an open set ω, the GJ-integral $J_{\omega}(u)$ is made so that $J_{\omega}(u)=0$ , and is divided into the sum of Path(Surface)-integral $P_{\omega}(u)$ and Region-integral $R_{\omega}(u)$ . If for all u, $R_{\omega}(u)$ is the bounded linear functional with respect to the vector fields derived from shape perturbation, then $R_{\omega}(u)$ becomes definable at any ω. Using $J_{\omega}(u)=0$ , the result of erasing smooth regions of u from $R_{D}(u)$ is the shape sensitivity of singular points expressed by the GJ-integral made in smooth regions.
Making is done from energy density functions in 2nd-order partial differential equations/systems (PDE/SYS) including nonlinearities and 4th-order PDE using Kirchhoff as an example, where $R_{D}(u)$ is associated with the Fréchet derivative of an energy functional. The material Fréchet derivative of the minimizer obtained using the adjoint variable v is expressed by the variation $\delta _{u} R_{D}(u)[v]$ of $R_{D}(u)$ with respect to u and is a generalization of Hadamard’s variational formula, and the shape sensitivity of eigenvalues are linked to the GJ-integral $\delta _{u} R_{D}^{E}$ in eigenvalue problems.
The theory can be numerically calculated and is useful for visualization of shape sensitivities and diagnosis of stress concentrations.
