Constitutively informed multi-body interactions for lattice particle models

Abstract

Discrete particle models have been widely used as an alternative to classical continuum mechanics to describe the mechanical behavior of solids, fluids and granular matter. Lattice particle models (LPM) are a subset of discrete particle models which treat the body as a set of interacting point masses with fixed sets of neighbors. Heretofore, the types of interaction forces considered in LPMs have been very limited; consisting mainly of linear pairwise interactions or empirically motivated generalizations thereof. Thus LPMs have been able to describe a very limited range of macroscopic constitutive behavior and the properties have generally been strongly dependent on the discretization.We present a new paradigm for incorporating multibody interactions in LPMs using a mesoscale analog of the Cauchy–Born rule which relates the deformation of the continuum to that of the underlying atomic lattice. We consider a Delaunay triangulation of an irregular lattice of particles comprising the body in its reference configuration. We calculate linear transformations to map the particle positions of each triangle in the current configuration to its reference configuration. Invoking the analogy of the Cauchy–Born rule, we identify the linear transformations with the continuum deformation gradient. Using this we express the continuum constitutive free energy of the body, which is a function of the deformation gradient, in terms of the current positions of the particles. The neighbor interaction forces on each of the particles are taken to be the gradients of this energy. The equations of motion for each particle are then solved numerically to obtain the particle trajectories and the overall deformation of the body. This model is validated using comparisons with classical results of stress distribution in isotropic and anisotropic material plates with circular holes. We then show how this model is able to describe formation of fine microstructure in two different materials: crack branching in brittle materials and twinning in phase transforming materials.