Molecular Approaches for Multifield Continua: origins and current developments

Dynamical properties of a composite microcracked bar based on a generalized continuum formulation

“…The two lattices are linked together by nonlinear elastic bonds. For the sake of simplicity, the fibers are considered 'frozen' in such a way that the microrotation and the work-conjugated microcouple are null [14].…”

“…Dynamical properties of a composite microcracked bar The constitutive equations of the multifield continuum are directly derived from a proper micromechanical description via energy equivalence [31,32]. Following the homogenization formulation presented in [33], to refer to for all details, and under the above-mentioned assumption of 'frozen' fibers, the linearized stress-strain relationships in the one-dimensional framework read [14]:…”

“…However, by lacking in material internal length parameters, the simple (Grade 1) classical continuum does not always seem appropriate for representing the macroscopic behavior of these materials, taking into account the size, the orientation, and the disposition of the heterogeneities. This often calls for the need of nonclassical (generalized) continuum descriptions [8][9][10][11][12] that can be considered as 'explicitly' or 'implicitly' nonlocal models due to the presence of internal length parameters and dispersion properties in wave propagation [9,13,14]. In this framework, strain gradient, or non-local models of explicit type [15], have been presented [16][17][18], as well as models referable to micromorphic continua, in particular, the micropolar continua, that can be considered non-local models of implicit type [19][20][21][22][23][24][25][26][27].…”

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.…”

“…In this way, discrete-to-scale dependent nonlocal continua can be naturally derived using proper correspondence maps between discrete and continuum kinematic descriptors. Examples of such approaches are found in the works [12,15,16], where multifield continua with additional degrees of freedom [10] are derived from a lattice system made of rigid particles and distributed voids/defects (pores, microcracks, etc.) and then adopted for the description of ceramic matrix (polycrystals with interfaces) and fibre-reinforced composites.…”