Molecular Approaches for Multifield Continua: origins and current developments

Abstract

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. Attention is centred on multiscale approaches which aim to deduce properties and relations at a given macroscale by bridging information at proper underlying microlevel via energy equivalence criteria. Focus is on physically–based corpuscular–continuous models originated by the molecular models developed in the 19th century to give an explanation per causas of elasticity. In particular, the ‘mechanistic–energetistic’ approach by Voigt and Poincaré who, when dealing with the paradoxes deriving from the search of the exact number of elastic constants in linear elasticity, respectively introduced molecular models with moment and multi–body interactions is examined. Thus overcoming the experimental discrepancies related to the so–called central–force scheme, originally adopted by Navier, Cauchy and Poisson.

Current research in solid state physics as well as in mechanics of materials shows that energy equivalent continua obtained by defining direct links with lattice systems are still among the most promising approaches in material science. This study aims at emphasizing the suitability of adopting discrete–continuous approaches, based on a generalization of the so–called Cauchy–Born rule used in crystal elasticity and in the classical molecular theory of elasticity, to identify continua with additional degrees of freedom (micromorphic, multifield, etc.), which are essentially ‘non–local’ models with internal length and dispersive properties. By lacking in internal length 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 microheterogeneities. Within the general framework of the principle of virtual power, it is described as the selection of a correspondence map, relating the finite number of degrees of freedom of discrete models to the kinematical fields of equivalent continua, provides a guidance for non-standard continuous approximations of heterogeneous media by–passing the intrinsic limits of scale separation of classical continua formulations. The circumstances in which, not very differently than in the past, empirical inadequacies still call for the need of removal of the local character of the classical hypothesis of lattice mechanics (central-forces or homogeneous deformations) are also discussed. A sample application of discrete–continuum homogenization approach leading to multifield description is finally shown with reference to microcracked composite materials, which can be representative of fiber–reinforced composites, ceramic matrix composites or porous metal–ceramic composites, as well as concrete and masonry–like materials.