Grid-based volume integration for elasticity: Traction boundary integral equation

•For nonlinear elasticity, the traction boundary integral equation has a volume term.•This volume integral is evaluated using just a regular cell grid covering the domain.•For fracture analysis, the volume integration can simply ignores the fracture surface.•Test calculations have demonstrated the correctness of the implementation. A volume integral algorithm for the non-homogeneous 3D elasticity traction boundary integral equation is presented. The body force volume integral is exactly split into a relatively simple boundary integral, together with a remainder volume integral that can be evaluated using a regular grid of cuboid cells covering the problem domain. Of particular importance for (inelastic) fracture analysis is that the volume integral over the regular grid is computed without explicit knowledge of the domain boundary, including the fracture surface. A Galerkin approximation is employed, and the numerical implementation is validated by solving body force elasticity problems with known solutions.