Interactions in a Multi-scale Representation of Sparse Media: From Mechanics to Thermodynamics

Homogenization of Complex Lattices for Metamaterials: Open Problems and Conjectures

“…The associated affine deformation gradient G, itself a function of x and t, is a solution to the equation Ġ = BG , with initial datum that can be selected equal to the second-rank identity tensor; the superposed dot indicates total time derivative. The resulting equations are such that the scheme is, in general, multi-field (compare [48] with [49,50] for different settings where these ideas apply). In the presence of internal constraints, the picture simplifies because the resulting balances collapse, reducing the equations (also depending on whether we consider fluctuations c := w * − v − By).…”

“…These equations (see references [49,50] for an analysis of their origin) are naturally complemented by initial conditions…”

“…In this case, although the prototypical discrete structure is taken as a simple, statistically periodical (i.e., the distribution of masses is independent of the specific space window) discrete one, the resulting representation becomes always multi-field: the gross deformation describes the motion of the window center of mass, and a tensor field represents the around-mass-center motion of mass points into the window. Pertinent interactions emerge, as shown in the example of Figure 2, and they have to be appropriately balanced [47,49,50].…”