The Influence of Uncertainty in Crack or Slip Plane Orientations on Cracking or Localization Probabilities

Realistic mesh-independent simulations of localization in a macroscopically homogenous material must incorporate (explicitly or statistically) length and orientation scales that reflect microscale inhomogeneities. Weibull statistics (or similar approaches) have long been used to allow for uncertainty in the size and spatial locations of microcracks, but little work has been done to determine the role of uncertainty in flaw orientation within the Weibull framework. Mohr-Coulomb (MC) failure theory initiates damage when Mohr’s circle first touches the failure line, but this presupposes that a crack exists in that orientation. If not, the stress state can be intensified beyond the failure line. The failure probability is then proportional to the solid angle of unit normals that map above the failure line. By allowing for variability in flaw size and orientation, realistic failure probabilities may be generated that change the discrete MC failure threshold into a fuzzy boundary. Though MC theory is used as an example, the basic ideas apply to any classical discrete threshold failure theory, or to crystal plasticity (in which the “flaw orientation” is re-interpreted as a slip plane).