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This work presents concepts and algorithms for the simulation of dynamic fractures with a Lattice Boltzmann method (LBM) for linear elastic solids. This LBM has been presented previously and solves the wave equation, which is interpreted as the governing equation for antiplane shear deformation. Besides the steady growth of a crack at a prescribed crack velocity, a fracture criterion based on stress intensity factors has been implemented. This is the first time that crack propagation with a mechanically relevant criterion is regarded in the context of LBMs. Numerical results are examined to validate the proposed method. The concepts of crack propagation introduced here are not limited to mode III cracks or the simplified deformation assumption of antiplane shear. By introducing a rather simple processing step into the existing LBM at the level of individual lattice sites, the overall performance of the LBM is maintained. Our findings underline the validity of the LBM as a numerical tool to simulate solids in general as well as dynamic fractures in particular.