M- and $$\hbox {M}_{\text {int}}$$ M int -integrals for cracks normal to the interface of anisotropic bimaterials

A contour integral termed $$\hbox {M}_{\mathrm{int}}$$ is presented for describing the fracture behavior of cracks passing through or terminating normally at a bimaterial interface. The $$\hbox {M}_{\mathrm{int}}$$ -integral is defined by performing the conventional M-integral along a contour enclosing the cracks, with the coordinate system properly originated for measure of the integration points. The presented formulation is considered to be feasible for problems with generally anisotropic elastic materials. Physically, the energy parameter $$\hbox {M}_{\mathrm{int}}$$ is shown to be equivalent to twice the released potential energy required for creation of the cracks. Such relation is exactly valid when the crack length is small compared with the size of the specimen, and approximately satisfied for problems containing relatively large cracks. Also, due to path-independence, the $$\hbox {M}_{\mathrm{int}}$$ -integral can be performed along an arbitrary outer contour, which is chosen to be far from the crack tips. With this property, the complicated singular stress field in the near-tip areas can be avoided so that a complicated finite element model around the crack tips is not required in the calculation.