Unification of local and nonlocal models within a stable integral formulation for analysis of defects

Abstract

Employing the discrete Cauchy-Born rule with the principle of virtual work done, a generalized model is formulated with the hope to unify local and nonlocal continuum frameworks. Despite the conventional mapping of the microscopic bond from the undeformed configuration, the consistent derivation requires a transformation on the Average Deviation of Lattice (ADL) vector in the region of influence. The new conversion proffers flexibility to the framework for the analysis of nonuniform distribution of particles in the field. We also found a compact mapping matrix which converts surface-based forces (stresses) to the nonlocal body-based forces. The transformation matrix allows reconstructing continuum models at a lower length scale in a discrete setting. To see the credibility of the model, fracture evolution in SCB specimen made of Polymethyl methacrylate (PMMA) is simulated, and the results are compared with experiments which admit an acceptable agreement.