A stable extended/generalized finite element method with Lagrange multipliers and explicit damage update for distributed cracking in cohesive materials

A flexible, general and stable mixed formulation is developed to model distributed cracking in cohesive grain-based materials in the framework of the extended/generalized finite element method. The displacement field is discretized on each grain separately, and the continuity of the displacement and traction fields across the interfaces between grains is enforced by Lagrange multipliers. The design of the discrete Lagrange multiplier space is detailed for bilinear quadrangular elements with the potential presence of multiple interfaces/discontinuities within an element. We give numerical evidence that the designed Lagrange multiplier space is stable and provide examples demonstrating the robustness of the method. Relying on the stable discretization, a cohesive zone formulation equipped with a damage constitutive formulation expressed in terms of the traction is used to model propagation of multiple cracks at the interfaces between grains. The damage formulation makes use of an explicit solution procedure, couples the normal and tangential failure modes, accounts for different tension and compression behaviours and takes into account a compression-dependent fracture energy in mixed mode. The framework is applied to complex 2D problems inspired by indirect tension tests of heterogeneous rock-like materials.