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The mechanical behavior of complex materials, characterized at finer scales by the presence of heterogeneities of significant size and texture, strongly depends on their microstructural features. By lacking in material internal scale parameters, the classical continuum does not always seem appropriate for describing the macroscopic behavior taking into account the size, orientation and disposition of the micro-heterogeneities. This calls for the need of non-classical continuum theories that can be constitutively characterized through multiscale approaches, which allow us to deduce material properties and relations at different levels of description by bridging information at proper underlying sub-levels.
In the framework of a multiscale modeling, aimed at deriving suitable homogeneous anisotropic continua for these materials, the non-local character of the description is crucial for avoiding physical inadequacies and theoretical computational problems. In particular, in problems in which a characteristic internal (material) length, l, is comparable to a macroscopic (structural) length, L. Among non-local theories, it is useful to distinguish between ‘explicit’ or ‘strong’ and ‘implicit’ or ‘weak’ non-locality, where implicit non-locality concerns continua with extra degrees of freedom [10], such as the multifield continua here adopted. In this presentation, particular attention will be devoted to show the effectiveness of continua with local rigid and/or deformable microstructure (micropolar/ micromorphic or affine microstructure) [3, 4], here adopted for describing fiber reinforced/particle composites, such as: ceramic/metal/polymer matrix composites, i.e. polycrystals with interfaces (grain boundaries or thin/thick interfaces); short fiber-reinforced composites; masonry-like materials (brick/block masonry, roman concrete, rocks).
A two-step multiscale (three levels) procedure is developed for deriving the constitutive equations for special kinds of these materials. As an example, at the microscopic level the material is described as a lattice system, at the mesoscopic level as a two-phase micropolar continuum and at the macroscopic level as a homogeneous micropolar continuum. For the transition from the discrete micro-level to the two-phases meso-level (i) a coarse graining procedure based on a generalized Cauchy-Voigt correspondence maps and energy equivalence is adopted [5, 6, 11]. For the meso-macro level transition (ii), a statistical homogenization procedure is developed, basing on the solution of Boundary Value Problems (BVP), posed on Statistically Representative Elements (SVE), with Boundary Conditions (BC) derived from a generalized macrohomogeneous condition of Hill’s type [7, 8]. This procedure provides hierarchies of bonds and aims at estimating the size of the Representative Volume Element (RVE) to adopt for performing homogenization. In this framework, a new criterion of convergence introduced allow us to estimate the elastic, classical and micropolar, constitutive moduli for particular classes of particle composites. Finally, some applications of the mentioned approach to periodic as well as to random particle composite materials, by varying the material contrast between matrix and inclusions, will be reported and discussed.
Keywords: Composite materials, Continua with Microstructure, Homogenization,

Composite materials, Continua with Microstructure, Homogenization