Localization in random media and its effect on the homogenized behavior of materials

La localisation dans les milieux aléatoires et ses effets sur la réponse homogénéisée des matériaux

François Willot (1, 2, 3)

1 CMM - Centre de Morphologie Mathématique
2 CDM - Centre des Matériaux
3 SU - Sorbonne Université

The present manuscript is submitted in partial fulfillment of my application to the degree of ``Habilitation à diriger des recherches'' at Sorbonne University. Its main contribution is a study in theoretical mechanics devoted to homogenization problems in the context of degenerate (non-strictly convex) local response of one of the phases, which can serve as idealized models for porous or rigidly-reinforced materials exhibiting perfectly-plastic behavior. In these situations plastic flow preferentially concentrates along shear bands; as a result the material effective response is governed by those regions within the material where the field localizes. A form of localization also occurs in linear problems governed by asymptotically hyperbolic partial derivative equations, where the strain field is found to develop banding patterns. The solutions are relevant to strongly-anisotropic elastic or thermoelastic media. Such a linear problem is studied in the present work in the context of a random microstructure, specifically a polycrystal containing cracks. A related topic is whether incipient localization may develop as a result of the microstructure itself. This question is investigated in the case of a Stokes flow occurring in a porous medium around a set of obstacles exhibiting unusual spatial distribution, characterized by long-range correlations. The different model problems are addressed by means of various techniques. Limit analysis bounds are combined with integral geometry to provide insight on the material behavior. Comparisons with rigorous bounds and estimates of homogenization theories, and full-field ``Fourier-based'' numerical results, allow us to interpret and assess the response of certain random microstructures. Although a complete solution is out-of-reach, the interplay between field localization and microstructure is elucidated in specific cases.

Mots clés: Homogenization; Localization; Perfect plasticity; Nonlinear media

Mots clés: Homogénéisation; Localisation; Plasticité parfaite; Milieux non-linéaires

Domaines: Matériaux

François Willot. Localization in random media and its effect on the homogenized behavior of materials. Materials. Université Paris Sorbonne, 2019. ⟨tel-02412623⟩