Deterministic and stochastic analyses of fracture processes in a brittle microstructure system

Variations in constituent properties, phase morphology, and phase distribution cause deformation and failure at the microstructural level to be inherently stochastic. This paper focuses on the stochasticity of fracture processes that arises as a result of measurement uncertainties in the properties of the constituents in the heterogeneous microstructures of an Al₂O₃/TiB₂ ceramic composite system. Basic postulate here is that for a microstructure local material properties vary around their macroscopically measured value with the macroscopically measured value being the mean of the variation. A micromechanical cohesive finite element framework with explicit resolution of arbitrary fracture patterns and arbitrary microstructural morphologies is used in the analyses carried out in this paper. The randomness in the constituent properties at any given point in the microstructure is specified relative to the local mean values of the corresponding properties. A deterministic analysis and a stochastic analysis are carried out simultaneously. The combination of determinism and stochasticity is achieved by integrating a perturbation analysis of the influence of stochastic property variations around their mean values and a deterministic analysis for the microstructure with the mean values of the constituent properties. Calculations are carried out for actual and idealized microstructures of the Al₂O₃/TiB₂ material system. Calculations focus on analyzing the fracture response variation with varying levels of variation of material properties for a particular microstructural morphology as well as on analyzing the variations in fracture response with variations in microstructural morphology. It is observed that microstructural morphology is intricately linked to the variations in fracture response when material properties have stochastic origin. A microstructure less prone to fracture shows higher variations in fracture response when compared to the one which offers least resistance to the crack propagation. In addition, for a particular microstructural morphology, the levels of variations in the crack surface area generated and the variations in the energy release rate are of the same order as the levels of variations in constituent properties. The observations support the conclusion that a material designer needs to make conservative estimates for a material’s performance if its microstructural construction imparts uncertainty to local material properties.