A multi-scale formulation for polycrystalline materials accounting for cohesive micro-cracks: Homogenisation of the traction-separation law

Abstract

This contribution investigates the modelling of failure in polycrystalline materials due to the nucleation and propagation of cohesive cracks at the micro-scale. First, a two-scale formulation that enables the analysis of failure, even when the evolution and propagation of cohesive micro-cracks induce material instability, is proposed. The development of this macro-continuous/micro-discontinuous formulation is based on the Method of Multi-Scale Virtual Power, from which dynamic equilibrium and homogenisation equations are derived, ensuring a variationally consistent scale transition. The respective minimal kinematical constraint is retrieved and enforced with the Lagrange multiplier method. The formulation is also extended for periodic boundary conditions. The discretisation of the resulting equilibrium equations, with the finite element method and the generalised-α method, is described in detail.Second, a fracture-based computational homogenisation procedure is developed to obtain homogenised traction-separation laws and fracture properties from microstructural volume elements. It is derived from a crack-based Hill-Mandel principle and employs a novel energetic-based damage variable, within the Park-Paulino-Roesler cohesive model, proposed to define the crack domain and compute the crack homogenised quantities. A strategy for accurately computing the homogenised unit normal vector of the equivalent macroscopic crack is also suggested.Numerical examples demonstrate the formulation’s ability to analyse fracture problems, including a study on the impact of different microscopic boundary conditions. This work also reveals the convergence of the traction-separation law with increasing microstructure model size when using the fracture-based computational homogenisation. Furthermore, it investigates how advanced deformation mechanisms, like slip plasticity and martensitic transformation, affect intergranular crack propagation mechanisms.