Extending the Lattice Boltzmann Method to Non-linear Solid Mechanics

TL;DR

This paper develops an extended Lattice Boltzmann Method for nonlinear solid mechanics, enabling simulation of large deformations and dynamic loading with validated benchmarks and complex wave propagation modeling.

Contribution

It introduces a novel Lattice Boltzmann approach incorporating nonlinear constitutive laws via a forcing term for solid mechanics applications.

Findings

Successfully simulates large deformations under dynamic loads.

Validates method with benchmark tests for tension and shear.

Demonstrates capability to model bending wave propagation.

Abstract

This work outlines a Lattice Boltzmann Method (LBM) for geometrically and constitutively nonlinear solid mechanics to simulate large deformations under dynamic loading conditions. The method utilizes the moment chain approach, where the non-linear constitutive law is incorporated via a forcing term. Stress and deformation measures are expressed in the reference configuration. Finite difference schemes are employed for gradient and divergence computations, and Neumann- and Dirichlet-type boundary conditions are introduced. Numerical studies are performed to assess the proposed method and illustrate its capabilities. Benchmark tests for weakly dynamic uniaxial tension and simple shear across a range of Poisson's ratios demonstrate the feasibility of the scheme and serve as validation of the implementation. Furthermore, a dynamic test case involving the propagation of bending waves in a cantilever beam highlights the potential of the method to model complex dynamic phenomena.