Three-Dimensional Mixed-Mode Fracture Analysis in Finite Structures Using a Generalized Domain Integral: Crack Front Energy Partition and Thickness Effects

This paper presents a three-dimensional generalization of the M-integral, formulated as an interaction integral based on a bilinear strain energy density, for the mixed-mode decoupling of crack front energies in finite structural components. The proposed Mθ3D integral combines real and virtual mechanical fields within a local spherical reference frame, enabling the separate evaluation of mode I (opening), mode II (in-plane shear) and mode III (out-of-plane shear) energy release rates along arbitrary crack front lines. The theoretical framework, derived from Noether’s theorem and the virtual work principle, is implemented in the Cast3M finite element code using a toroidal integration domain with a local theta weighting function. Numerical validations are conducted on the Mixed-Mode Crack Growth (MMCG) specimen, a geometry representative of structural components subjected to combined tension and shear. Three key findings are demonstrated: (i) practical domain independence is achieved for all three fracture modes; (ii) the three-dimensional approach converges to the plane-stress solution for thin specimens and reveals significant deviations from plane-strain assumptions; (iii) even under nominally mode I + II loading, a non-negligible mode III component emerges due to Poisson-induced out-of-plane effects, with magnitude increasing at free surfaces and for thicker geometries. These results indicate that finite-thickness and out-of-plane effects can significantly affect the partition of fracture energy between modes. For the MMCG configuration investigated here, the three-dimensional formulation shows the limitations of two-dimensional assumptions and provides an energetic basis for the analysis of mixed-mode fracture in finite-thickness components.