M- and M_c-integrals for multicracked problems in three dimensions

Aproblem-invariantM_c-integral is proposed as an energy parameter for describing the degradation of structural integrity caused by irreversible evolution of multiple cracks in three-dimensional (3D) elastic solids. The physical meaning for 3DM_c, which is related to the surface energy corresponding to creation of the cracks, does not hold in the same manner as that for two-dimensional (2D)M_c and needs to be properly reformulated. Also, the 3D integration is shown to be surface-independent in a modified sense. With this property, by choosing a closed surface remote from the crack fronts, the 3D M_c can then be accurately evaluated with finite-element (FE) solutions even when the near-front areas are not simulated with very fine grids.