A mathematical model of a composite material randomly reinforced such as TexSol ( mixture of sand and wire) is proposed. For this study, a variational asymptotic analysis is performed in order to obtain a homogeneous deterministic structure taken into account the mechanical behavior of this material. The modeling strategy is to cut (in direction x3) a cube of TexSol into thin plates of thickness h(ε) depending on a very small parameter ε << 1. When h(ε) is small enough, we assume that each plate contains vertical fibers. Our 3D initial problem is decomposed into n 2D layer type models giving 2-dimensional formulation after passage to the limit. The resulting model is deterministic. Then, using this result on each plate, we obtain a discrete energy (according to x3), which is the sum of n 2-dimensional homogeneous and deterministic energies. We reconstruct a 3D structure by means of a variational integration respect to x3, i.e. taking a variational limit on n. This obtained energy limit, homogeneous and deterministic, is thus proposed as a TexSol model. Finally, the model is validated by a numerical study.
TEXSOL-LIKE RANDOMLY REINFORCED MATERIALS : STOCHASTIC VARIATIONAL MODELING BY HOMOGENIZATION