Discrete Lattice Modeling of Wave Propagation in Materials with Heterogeneous Microstructures

Abstract

Many natural and human-made material systems (e.g., bone, shale, and cement-based composites) exhibit heterogeneous microstructures. Lattice models have reemerged to simulate such material systems because of their inherent simplicity while offering tremendous capabilities. However, current lattice models suffer from several deficiencies; for example, some lattice models cannot span the necessary range of Poisson’s ratio, some others are not isotropic (e.g., the standard square lattice), and others are not practical for complex domains (e.g., equilateral triangular lattice and hexagonal lattice). Thus, there is a need for a simple lattice that can handle Poisson’s ratio without any limitations, capture all the possible deformation modes of an isotropic elastic material, have minimal degrees-of-freedom, and provide positive definite stiffness and mass matrices. In this paper, we develop such a lattice model. Our approach hinges on equating the Lagrangians of the continuous (continuum) and discrete (lattice) systems, defining the strains consistently in terms of displacements in the lattice, adding a local interaction term to span the entire invariant space, and using an energy preserving time-stepping scheme. Using a Bloch wave analysis, we show that the lattice is, in fact, isotropic. Also, the lattice model does not suffer from volumetric locking, which is not the case with low-order finite elements. We verify the accuracy of the model using analytical solutions on benchmark problems. Finally, we demonstrate the application of the model on a practical example by performing propagation analysis in cement paste microstructure acquired from scanning electron microscopy (SEM).