On the Optimal Prediction of the Stress Field Associated with Discrete Element Models

This work presents an optimized and convergent regularization procedure for the computation of the stress field exhibited by particle systems subject to self-equilibrated short-range interactions. A regularized definition of the stress field associated with arbitrary force networks is given, and its convergence behavior in the continuum limit is demonstrated analytically, for the first time in the literature. The analyzed systems of forces describe pair interactions between lumped masses in ‘atomistic’ models of 2D elastic bodies and 3D membrane shells based on non-conforming finite element methods. We derive such force networks from polyhedral stress functions defined over arbitrary triangulations of 2D domains. The stress function associated with an unstructured force network is projected onto a structured triangulation, producing a new force network with ordered structure. The latter is employed to formulate a ‘microscopic’ definition of the Cauchy stress of the system in the continuum limit. The convergence order of such a stress measure to its continuum limit is given, as the mesh size approaches zero. Benchmark examples illustrate the application of the proposed regularization procedure to the prediction of the stress field exhibited by a variety of 2D and 3D membrane networks.