A coupled discrete/continuous method for computing lattices. Application to a masonry-like structure

Abstract

This paper presents a coupled discrete/continuous method for computing lattices and its application to a masonry-like structure. This method was proposed and validated in the case of a one dimensional (1D) railway track example presented in Hammoud et al. (2010). We study here a 2D model which consists of a regular lattice of square rigid grains interacting by their elastic interfaces in order to prove the feasibility and the robustness of our coupled method and highlight its advantages. Two models have been developed, a discrete one and a continuous one. In the discrete model, the grains which form the lattice are considered as rigid bodies connected by elastic interfaces (elastic thin joints). In other words, the lattice is seen as a “skeleton” in which the interactions between the rigid grains are represented by forces and moments which depend on their relative displacements and rotations. The continuous model is based on the homogenization of the discrete model ( Cecchi and Sab, 2009). Considering the case of singularities within the lattice (a crack for example), we develop a coupled model which uses the discrete model in singular zones (zones where the discrete model cannot be homogenized), and the continuous model elsewhere. A new criterion of coupling is developed and applied at the interface between the discrete and the continuum zones. It verifies the convergence of the coupled solution to the discrete one and limits the size of the discrete zone. A good agreement between the full discrete model and the coupled one is obtained. By using the coupled model, an important reduction in the number of degrees of freedom and in the computation time compared to that needed for the discrete approach, is observed.