## Modeling of random heteregeneous materials: from microscale to macroscale

If the length-scales are well separated, homogenization theory can provide a robust theoretical framework for heterogeneous materials. In this context, the macroscopic properties can be retrieved from the solution to an auxiliary problem, formulated over the representative volume element (with appropriate boundary conditions). In the present work, we focus on the homog- enization of heterogeneous materials which are described at the finest scale by two different materials models (both depending on a specific characteristic length) while the homogeneous medium behaves as a classical Cauchy medium in both cases.

In the first part, the random microstructure of a Cauchy medium is considered. Solving the auxiliary problem on multiple realizations can be very costly due to constitutive phases exhibiting not well-separated characteristic length scales and/or high mechanical contrasts. In order to circumvent these limitations, our study is based on a mesoscopic description of the material, combined with information theory. In the mesostructure, defined by a filtering framework, the fine-scale features are smoothed out.

The second part is dedicated to gradient materials which induce microscopic size-effect due to the existence of microscopic material internal length(s). The random microstructure is described by a newly introduced stress-gradient model. Despite being conceptually similar, we show that the stress-gradient and strain-gradient models define two different classes of materials. Next, simple approaches such as mean-field homogenization techniques are proposed to better understand the assumptions underlying the stress-gradient model. The obtained semi-analytical results allow us to explore the influence on the homogenized properties of the model parameters and constitute a first step toward full-field simulations.

**Keywords:** Homogenization, Probabilistic modeling, Micromechanics, Gradient materials.

### Publications:

Tran, V.P., Guilleminot, J., Brisard, S., Sab, K., (2016), Stochastic modeling of mesoscopic elasticity random field, *Mechanics of Materials*, 93, 1-12, hal-01226337

Tran, V.P., Guilleminot, J., Brisard, S., Sab, K., (2015), Homogénéisation numérique à l’aide de modèles prior de la raideur mésoscopique : identification et validation, *22ème Congrès Français de Mécanique*, Lyon, France, hal-01194370