Extending the Lattice Boltzmann Method to Non-linear Solid Mechanics

2025-02-02

AI Summary

This work addresses the computational challenge of simulating large deformations, dynamic loading, and complex constitutive behavior in strongly nonlinear solid mechanics. We present the first systematic extension of the lattice Boltzmann method (LBM) to solids exhibiting simultaneous geometric and constitutive nonlinearity. A novel LBM formulation is developed within a moment-based framework, where all stress and deformation measures are defined on the reference configuration, and a nonlinear constitutive force term is incorporated. Geometric nonlinearity, material nonlinearity, and dynamic boundary effects are consistently modeled using finite-difference approximations for gradients/divergences and a hybrid Neumann–Dirichlet boundary treatment. Numerical validation demonstrates high accuracy across a wide Poisson’s ratio range for canonical uniaxial tension and simple shear benchmarks, and successfully captures bending wave propagation in a cantilever beam. The method overcomes the traditional LBM limitation to fluid dynamics or infinitesimal-deformation solids, exhibiting superior accuracy and robustness.

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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 nonlinear 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.

Problem

• Lattice Boltzmann Method
• Complex Solid Mechanics
• Large Deformation

Innovation

• Lattice Boltzmann Method (LBM)
• Moment Lattice Technique
• Finite Difference Method