Discrete to Scale-Dependent Continua for Complex Materials: A Generalized Voigt Approach Using the Virtual Power Equivalence

Abstract

The mechanical behaviour 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 behaviour of such materials, taking into account the size, the orientation and the disposition of the heterogeneities. This often calls for the need of non-classical continuum descriptions, which can be obtained through multiscale approaches aimed at deducing properties and relations by bridging information at different levels of material descriptions.

Current researches in solid state physics as well as in mechanics of materials show that energy-equivalent continua obtained by defining direct links with lattice systems, as widely investigated by the corpuscular-continuous approaches of nineteenth century, are still among the most promising approaches in material science. The aim is here to point out the suitability of adopting discrete to scale-dependent continuous models, based on a generalization of the so-called Cauchy–Born (Voigt) rule used in crystal elasticity and in classical molecular theory of elasticity, in order to identify continua with additional degrees of freedom (micromorphic, multifield, etc.) which are essentially non-local models with internal length and dispersive properties. It is shown that, within the general framework of the principle of virtual powers, the correspondence map relating the finite number of degrees of freedom of discrete models to the continuum kinematical fields provides a guidance on the choice of the most appropriate continuum approximation for heterogeneous media. Some applications of the mentioned approach to ceramic matrix composites and masonry-like materials are discussed.