Upscaling Crack Propagation and Random Interactions in Brittle Materials Under Dynamic Loading

Abstract

Multi-scale mechanics problems require the use of an appropriate upscaling approach to link micro-scale architecture (e.g., grain structure, defect population, crystal orientation) to macro-scale effective response (e.g., stress, elastoplastic properties, damage). For brittle materials, failure is often associated with crack growth from pre-existing flaws in the material, and with the subsequent coalescence of these cracks. For low strain rates, the simplifying assumption that failure is governed by the largest flaw may be justified. For high strain rates, however, a wider range of the pre-existing flaw population leads to crack initiation prior to failure, making the analysis significantly more complex. Previous analytical models of this high strain-rate behavior apply one of two assumptions: flaws are uniformly sized and periodically located; or, cracks initiated from flaws do not interact. Both of these assumptions are questionable given that these materials are quite heterogeneous at the micro-scale and that fragmentation of these materials exhibit clear crack interactions. Paliwal & Ramesh [1,2] recently addressed this issue by applying a self- consistent model that treats each flaw as residing within an ellipse of pristine (uncracked) material surrounded by a damaged material. In this way, crack growth from each flaw is at least affected indirectly by the presence of cracks from other flaws; however, this model does not address explicit interactions of cracks. In the current research, a probabilistic approach is used to define the proportion of cracks that intersect each other in a given load step. Once two cracks interact, then they are treated as a single larger crack. This model provides a stress-time or stress-strain history at a fixed strain rate loading. Results show that the predictions of strength from the model with explicit crack interactions are reduced from those predicted by the non-probabilistic self-consistent model.