  The Lattice Boltzmann Method (LBM), e.g. in [ 1] and [2 ], can be interpreted
as an alternative method for the numerical solution of partial differential
equations. Consequently, although the LBM is usually applied to solve fluid
flows, the above interpretation of the LBM as a general numerical tool, allows
the LBM to be extended to solid mechanics as well. In this spirit, the LBM has
been studied in recent years. First publications [3], [4] presented an LBM
scheme for the numerical solution of the dynamic behavior of a linear elastic
solid under simplified deformation assumptions. For so-called anti-plane shear
deformation, the only non-zero displacement component is governed by a
two-dimensional wave equation. In this work, an existing LBM for the
two-dimensional wave equation is extended to more general plane strain
problems. The proposed algorithm reduces the plane strain problem to the
solution of two separate wave equations for the volume dilatation and the
non-zero component of the rotation vector, respectively. A particular focus is
on the implementation of types of boundary conditions that are commonly
encountered in engineering practice for solids: Dirichlet and Neumann boundary
conditions. Last, several numerical experiments are conducted that highlight
the performance of the new LBM in comparison to the Finite Element Method.
