A grain level model for the study of failure initiation and evolution in polycrystalline brittle materials. Part I: Theory and numerical implementation

A model is presented to analyze material microstructures subjected to quasi-static and dynamic loading. A representative volume element (RVE) composed of a set of grains is analyzed with special consideration to the size distribution, morphology, chemical phases, and presence and location of initial defects. Stochastic effects are considered in relation to grain boundary strength and toughness. Thermo-mechanical coupling is included in the model so that the evolution of stress induced microcracking, from the material fabrication stage, can be captured. Intergranular cracking is modeled by means of interface cohesive laws motivated by the physics of breaking of atomic bonds or grain boundary sliding by atomic diffusion. Several cohesive laws are presented and their advantages in numerical simulations are discussed. In particular, cohesive laws simulating grain boundary cracking and sliding, or shearing, are proposed. The equations governing the problem, as well as their computer implementation, are presented with special emphasis on selection of cohesive law parameters and time step used in the integration procedure. This feature is very important to avoid spurious effects, such as the addition of artificial flexibility in the computational cell. We illustrate this feature through simulations of alumina microstructures reported in part II of this work. A technique for quantifying microcrack density, which can be used in the formulation of continuum micromechanical models, is addressed in this analysis. The density is assessed spatially and temporally to account for damage anisotropy and evolution. Although this feature has not been fully exploited yet, with the continuous development of cheaper and more powerful parallel computers, the model is expected to be particularly relevant to those interested in developing new heterogeneous materials and their constitutive modeling. Stochastic effects and other material design variables, although difficult and expensive to obtain experimentally, will be easily assessed numerically by Monte Carlo grain level simulations. In particular, extension to three-dimensional simulations of RVEs will become feasible.