Genesis of the multiscale approach for materials with microstructure

Abstract

This paper presents an overview of the origin of multiscale approaches in mechanics. While the pioneer molecular models of linear elastic bodies by Navier, Cauchy and Poisson were contradicted by experiments, the phenomenological energetic approach by Green still seems suitable for simple materials only. Voigt's molecular model, here reinterpreted in the light of contemporary mechanics, reconciled the two approaches providing a conceptual guideline for developing constitutive models based on a direct link between continuum and discrete solid mechanics. Such a theoretical background proves to be especially suitable for new complex materials. An example referred to masonry-like materials is given.

Voigt and Poincaré’s mechanistic–energetic approaches to linear elasticity and suggestions for multiscale modelling

“…His results in terms of number of elastic constants ('multi-constant' theory [5,22,38]) were confirmed by experiments.…”

“…In the former case, the equivalent continuum derived belongs to the family of microcontinua such as: micropolar continua equivalent to lattices with dislocations, used to describe plasticity [1], or to masonry-like materials [18,28,36,37]; higher order continua [2,15,33]; multifield continua equivalent to fibre-composite microcracked materials [31,39,40]. It has been shown that, applying the same conceptual framework, Voigt's coarse-graining process leads to simple continua, which in turn are micropolar continua with constrained degrees of freedom [38]. The latter is the case of continua originated by multi-body potential descriptions, which has been widely explored in the field of solid matter physics in order to calculate chemical-physical properties important for the design of high performance complex materials [10,25,26].…”

“…In the present paper, we will sketch Voigt's refined molecular approach [5,38] and Poincaré's energetic approach to linear elasticity. The original texts are re-interpreted and re-written in terms of modern tensor algebra in order to understand how these approaches can be still valid and predictive in the theories of new, non-standard materials.…”

Block masonry as equivalent micropolar continua: the role of relative rotations

“…Alternatively, the identification of the stress-strain relationships of the multifield continuum can be based on the definition of a direct energy equivalence with discrete descriptions of the matter [11,12]. Such energy-based lattice-continuum approaches have their origin in the non-standard molecular models of elasticity of the eighteenth and nineteenth century [22][23][24] and, also at present, are among the most promising approaches in materials science [25,26]. The basic idea of these approaches is the assumption that there is an implicit mapping from the set of degrees of freedom of the discrete model to the displacement vector fields, like the so-called Cauchy-Born rule used in crystal elasticity.…”

“…In the paper [12], the same result has been alternatively obtained by equating the mean lattice strain energy to the continuum strain energy density. It has already been recognized that the identified micropolar continuum reduces to an anisotropic classical continuum when the block rotations are constrained to equal the local rigid rotation and when the interaction couples between blocks are null (Voigt's constraints [22]). …”