Voigt and Poincare's mechanistic-energetic approaches to linear elasticity and suggestions for multiscale modelling

摘要

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 Poincare, 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 Poincare'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.