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

Abstract

Many modern theories for "complex" materials rely on the construction of specific potentials for inner actions by resorting to mechanistic/corpuscular descriptions of the matter at different length scales. Techniques allowing the dialogue among different material scales on the basis of energetic equivalence criteria play a central role in mechanics of materials. These multiscale techniques can be seen as originated from the molecular theories of the nineteenth century and in particular by the suggestions of Voigt and Poincaré, to deal with the paradoxes coming from the search of the exact number of elastic constants in linear elasticity. In this paper, we examine both Voigt and Poincaré's mechanistic-energetic approaches to the standard linear theory of elasticity in contrast with the mechanistic molecular model by Navier, Cauchy and Poisson. We also try to provide some conceptual guidelines among these approaches and the contemporary theories of complex material behaviour.

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

“…This rule simply imposes homogeneous deformations to the lattice system. It is worth noting that the presence of additional degrees of freedom allows to include internal lengths in the model without resorting to non-homogeneous mappings and non-local descriptions [24]. Within the framework of rigid body kinematics, the strain measures of the discrete system can be defined as the relative displacements and rotations between blocks.…”

“…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.…”

“…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. Within this framework, the possibility to provide accurate descriptions of the microstructure is closely related to refinements of the standard lattice models, which can be obtained resorting to non-homogeneous deformations or to non-central-force schemes [24].…”

A multiscale description of particle composites: From lattice microstructures to micropolar continua

“…For the chosen module with elastic coefficient of Table 1, the derived constitutive coefficients of the matrix in (23) are strictly diagonal and, as a consequence, no Poisson's effects are accounted for. However, more general assumptions are possible [3], also by resorting to multiple interactions [57]. Adopting this three steps procedure, that moves from a physically-based discrete description at the micro-level on the tracks of [40], we can rationally derive the multifield continua parameters, focusing on the micropolar constants that are generally difficult to obtain resorting to standard experimental results.…”

Nineteenth century molecular models with a glance at modern discrete—continuum theories

“…Even if they finally led back to classical continuuous formulations by introducing internal constraints: Voigt imposing the same uniform rotation to the particles and Poincaré considering only pair-interaction terms. However, even if both Voigt and Poincaré, removing the local character of the central-force scheme, offered a good solution to the controversy about the elastic constants, the mechanistic-molecular approach was abandoned in favour of the energetic-continuum approach by Green, and their works have been neglected for long time [12][13][14][15]. Now these ideas found a renewed interest with particular reference to the problem of constitutive modelling of composite materials.…”