Statistical characterization of meso-scale uniaxial compressive strength in brittle materials with randomly occurring flaws

Increasingly fine spatial resolution in numerical models of brittle materials promises to improve prediction and characterization of dynamic failure in these materials. However, as the resolution of these numerical models begins to approach the material micro-scale, the associated discretization requires a definitive connection to the microstructure. In many cases a numerical model (e.g., a finite element mesh) that explicitly resolves each flaw within the material is not feasible for macro-scale analyses. As an alternative, each element can be treated as a meso-scale continuum with constitutive properties that reflect the characteristics of the underlying microstructure. Small scale elements will exhibit random variations in the constitutive properties as a result of the random variations in the number and types of flaws and the flaw sizes contained within each element. The present paper proposes a technique for assigning probability distributions to these element properties, which can be thought of as the meso-scale constitutive properties. In particular, the strain-rate dependent compressive uniaxial strength of a ceramic is modeled using a two-dimensional analytical model developed by Paliwal and Ramesh (2008). The effect on the probability distribution of meso-scale (or element-level) strength from flaw density, flaw size distribution, flaw clustering, and strain rate are studied. Higher strain rates, more flaw clustering, and decreasing element size all contribute to greater scatter in uniaxial compressive strength. Variations in flaw size increase the scatter in the strength more for low strain rate loadings and less clustered microstructures. The results provide interesting comparisons to the classical assumption of a two-parameter Weibull-distributed strength, showing that a three-parameter Weibull distribution and even a lognormal distribution fit better with the simulated strength data.