A micromorphic model for the multiple scale failure of heterogeneous materials

The multi-scale micromorphic theory developed in our previous paper [Vernerey, F.J., Liu, W.K., Moran, B., 2007. Multi-scale micromorphic theory for hierarchical materials. J. Mech. Phys. Solids, doi:10.1016/j.jmps.2007.04.008] is used to predict the failure of heterogeneous materials illustrated by a high strength steel alloy possessing two populations of hard particles distributed at two distinct length scales in an alloy matrix. To account for the effect and size of microstructural features during fracture, additional kinematic variables are added, giving rise to the couple stresses associated with each population of particles. The various stress and strain measures must satisfy a set of coupled multi-scale governing equations derived from the principle of virtual power. A three-scale constitutive model is then developed to represent the failure of the alloy from nucleation, growth and coalescence of voids from each population of particles. For this, three distinct yield functions, each corresponding to a different scale, are introduced. Cell model simulations using finite elements are performed to determine the constitutive relations based on the key microstructural features. Two-dimensional failure analyses are then presented in tension and in shear, and show good agreement with direct numerical simulation of the microstructure.