From molecular theories to non-local continuum descriptions: revisiting 19th century ideas for current multiscale modelling

Molecular theories of elasticity were originally conceived to derive constitutive relations for continuum models by attributing the origin of elastic behaviour to interactions among discrete material points. Classical centralforce models by Cauchy, Navier and Poisson led to local continuum descriptions that exhibited intrinsic limitations, such as an insufficient number of elastic constants and inconsistencies with experimental data. Later refinements by Voigt and Poincaré — based on force and couple interactions between rigid molecules or multibody angular potentials, while preserving a common homogenisation scheme — addressed these issues by introducing internal lengths, thus providing a mechanical basis for scale-dependent (implicitly non-local) continuum descriptions. These enriched molecular models preserved a common homogenisation scheme and, when supplemented with internal constraints, were able to overcome the limitations related to the insufficient number of elastic constants. This paper revisits and reinterprets these foundational contributions, showing how they provide a mechanical framework for linking discrete microstructures to effective scale-dependent continuum models, with particular relevance to continua with additional degrees of freedom (multifield continua). The paper is further complemented by a paradigmatic example illustrating a discrete-to-scale-dependent model for composite materials and the resulting dispersive wave propagation.