Lattice-Spring Analogy for Isotropic Elasticity

Abstract

This study introduces an innovative Isotropic Elastic Lattice Spring Model (IELSM) that addresses the fundamental limitation of classical lattice spring models: the constraint of fixed Poisson's ratio. By amending the total strain energy within the Lattice Spring Model (LSM), IELSM provides a self-consistent formulation for simulating isotropic elastic materials with arbitrary Poisson's ratios. The model's core innovation lies in augmenting classical axial spring frameworks with additional volumetric constraints, establishing a direct and exact mapping between IELSM's parameters and macroscopic elastic constants. This enables simulation across the full admissible Poisson's ratio: -1 < {\nu} < 1 under plane stress and -1 < {\nu}< 0.5 under plane strain conditions. Eigenvalue analysis indicates that the IELSM has better numerical stability compared to the standard bilinear quadrilateral element and the constant strain triangular element. The characteristic of the numerical implementation lies in directly decomposing the additional volumetric constraints into an equivalent combination of standard mechanical components (axial, shear and rotational springs), laying the foundation for the realization of fracture simulation based on discrete methods. Comprehensive validation through uniaxial tension, pure shear, stress concentration around a circular hole, and stress singularity analyses for central and crucifix-shaped cracks demonstrates IELSM's exceptional accuracy, convergence and computational robustness. The model exhibits excellent performance in stress intensity factor calculations at crack tips, validating its effectiveness for singular stress field analysis. This work bridges the gap between LSM and continuum mechanics, establishing an analog framework that maintains theoretical consistency while offering computational accuracy for the solution of elastic boundary value problems.