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The Lattice Boltzmann Method (LBM), e.g. in [ 1] and [2 ], can be interpretedas an alternative method for the numerical solution of partial differentialequations. Consequently, although the LBM is usually applied to solve fluidflows, the above interpretation of the LBM as a general numerical tool, allowsthe LBM to be extended to solid mechanics as well. In this spirit, the LBM hasbeen studied in recent years. First publications [3], [4] presented an LBMscheme for the numerical solution of the dynamic behavior of a linear elasticsolid under simplified deformation assumptions. For so-called anti-plane sheardeformation, the only non-zero displacement component is governed by atwo-dimensional wave equation. In this work, an existing LBM for thetwo-dimensional wave equation is extended to more general plane strainproblems. The proposed algorithm reduces the plane strain problem to thesolution of two separate wave equations for the volume dilatation and thenon-zero component of the rotation vector, respectively. A particular focus ison the implementation of types of boundary conditions that are commonlyencountered in engineering practice for solids: Dirichlet and Neumann boundaryconditions. Last, several numerical experiments are conducted that highlightthe performance of the new LBM in comparison to the Finite Element Method.

A Lattice Boltzmann Method for Elastic Solids Under Plane Strain Deformation