Generalized mathematical homogenization of the lattice discrete particle model

Concrete, ceramics, fiber and particle reinforced composites, as well as porous media, are materials widely used in industry and engineering. All these materials are heterogeneous at a certain scale and some of their specific macroscopic behaviors during damage can be traced back to their micro structural behavior. In order to obtain realistic results in the numerical simulations of heterogeneous materials, one needs either to perform computationally intensive fine scale simulations or to adopt a multi-scale technique that is able to reduce the computational cost of the analysis while it retains enough accuracy on the quantities of interest. In this paper the mathematical homogenization approach is used to upscale the Lattice Discrete Particle Model (LDPM), that have been successfully formulated to simulate concrete at the scale of the major heterogeneities. The Lattice Discrete Particle Model (LDPM) simulates concrete at the meso-scale considered to be the length scale of coarse particle pieces. Contrarily to continuum-based approaches, in discrete models like LDPM, the displacement and rotation fields are only defined in a finite number of points representing the center of coarse aggregate particles. The mechanical interaction between adjacent particles is governed by meso-scale constitutive equations. In this work LDPM is homogenized through the classical mathematical homogenization by employing first order asymptotic expansions for displacements and rotations. Numerical simulations are carried out to analyze the behavior of the resulting homogenized macroscopic constitutive equation.