Generalized continua for discontinuous complex materials. A Voigt–like approach using the principle of virtual works.

The mechanical behaviour of complex materials, characterised 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 to describe the macroscopic behaviour of such materials, taking into account the size, orientation and disposition of the micro heterogeneities. Attention will be focused on multiscale approaches which aim to deduce properties and relations at a given macro-scale by bridging information at proper underlying micro-level via energy equivalence criteria. Focus will be on physically-based corpuscular-continuous models originated by the molecular models developed in 19th century to give an explanation per causas of elasticity. In particular, the ‘mechanisticenergetic’ approach by Voigt and Poincaré will be examined. When dealing with the paradoxes coming from the search of the exact number of elastic constants in linear elasticity, they respectively introduced moment and multi-body interactions models which allow to by-pass the experimental discrepancies related to the so-called central–force scheme, originally adopted by Navier, Cauchy and Poisson [1–3]. Current researches in solid state physics as well as in mechanics of materials show that energyequivalent continua obtained by defining direct links with lattice systems are still among the most promising approaches in material science. Aim of this presentation is pointing out the suitability of adopting discrete-continuous Voigt-like approaches, based on generalization of the so-called Cauchy–Born rule used in crystal elasticity and classical molecular theory of elasticity, for identifying continua with additional degrees of freedom (micromorphic, multifield, etc.). These generalized continua are essentially non-local models with internal length and dispersive properties. It will be shown that, within the general framework of the principle of virtual work, the assumed generalized correspondence map relating the finite number of degrees of freedom of discrete models to the continuum kinematic fields implies the selection of the continuum equivalent to the defined discrete medium; thus providing a guidance for non–standard continuous approximation of heterogeneous media. The circumstances in which, not very differently than in the past, empirical inadequacies still calls for the need of removing the local character of the classical hypothesis of lattice mechanics (central–forces or homogeneous deformations) will be also discussed. Some applications of such approaches will be finally shown with reference to masonry–like material as micropolar [4], second gradient and classical continua.