Constructing continuum-like measures based on a nonlocal lattice particle model: Deformation gradient, strain and stress tensors

Abstract

Explicit expressions for continuum-like measures of deformation gradient, strain and stress tensors were constructed using the techniques of weighted least squares and energy conjugate. Although presented based on a nonlocal lattice particle model, the formulations of these measures are general in nature and can be applied to other numerical models. Deformation gradient was formulated based on the deformation states of neighboring material points. Within the small strain limit, the infinitesimal strain tensor was derived from normal strains of bonds connecting material point of interest with all its neighbors. The second Piola–Kirchhoff stress tensor was constructed using the energy conjugate concept with respect to Green–Lagrangian strain tensor. The first Piola–Kirchhoff and Cauchy stress tensors were also formulated based on the derived second Piola–Kirchhoff stress tensor. For problems of homogeneous and inhomogeneous deformations, all three constructed measures yield very good predictions comparing to local continuum mechanics solutions. However, the second Piola–Kirchhoff formulation predicts less accurate results in surface regions due to incomplete neighbor list of material points with these regions.