Higher-order strain/higher-order stress gradient models derived from a discrete microstructure, with application to fracture

Higher-order gradient models are derived from a discrete particle structure. A general set of constitutive equations is found in which zeroth and higher-order stress terms are related to zeroth and higher-order strain terms. As special cases, a model with only higher-order strain terms is considered as well as a model with both higher-order stress and higher-order strain terms. The model without higher-order stress is found to be unstable. The model with higher-order stress, on the other hand, is stable. The model with a higher-order stress term and a higher-order strain term can successfully be used to model softening phenomena.