Abstract: Based on the solid mechanics of the discrete form and its variational principles pro-posed by Niu^[1,2], this paper puts forward four kinds of the boundary integral-variational theorems of an aroitrary element. In the course of the fracture analysis,they can be used to compute the energy release rate along the normal direction of the crack boundary.When there is the hole in the solid, and there are the given surface forces on the hole boundary or there are not any given surface forces on the hole boundary, they can be used to compute the variation of the energy along the normal direction of the hole boundary. In the course of the discrete analysis, they can be used to establish the discrete equations,so that the values of the unknown functions are solved. At the same time, from this paper we know that the J-integral proposed by Rice^[3] represents an integral to be independent of a path imperfectly.