Cellular automata model for elastic solid material

TL;DR

This paper introduces a new two-dimensional cellular automaton model for elastic solid materials that accurately captures wave phenomena and deformation coupling, aligning with classical elasticity equations.

Contribution

The paper presents a novel CA model with coefficient matrices that enable realistic elastic behavior and Poisson ratio effects, advancing solid dynamics simulation.

Findings

Model recovers Navier equations in the continuum limit.

Captures wave phenomena related to Poisson ratio effects.

Mathematically described as a conservative system.

Abstract

The Cellular Automaton (CA) modeling and simulation of solid dynamics is a long-standing difficult problem. In this paper we present a new two-dimensional CA model for solid dynamics. In this model the solid body is represented by a set of white and black particles alternatively positioned in the $x$- and $y$- directions. The force acting on each particle is represented by the linear summation of relative displacements of the nearest-neighboring particles. The key technique in this new model is the construction of eight coefficient matrices. Theoretical and numerical analyses show that the present model can be mathematically described by a conservative system. So, it works for elastic material. In the continuum limit the CA model recovers the well-known Navier equations. The coefficient matrices are related to the shear module and Poisson ratio of the material body. Compared with previous CA model for solid body, this model realizes the natural coupling of deformations in the $x$- and $y$- directions. Consequently, the wave phenomena related to the Poisson ratio effects are successfully recovered. This work advances significantly the CA modeling and simulation in the field of computational solid dynamics.