Multi-Value Microstructural Descriptors for Complex Materials: Analysis of Ground States

Mechanics of Material Mutations

“…• The weakening of the continuity assumption along the lines indicated in Šilhavý (1991, 2005). Difficulties arise when we considerν (·) as a multivalued function over M, with values determined modulo a permutation, as proposed for a refined description of material complexity in Focardi et al (2014), where the conditions for the existence of the ground states for the relevant elastic energy are determined in this case.…”

“…-The appropriate interpretations in this setting, even extensions when this is the case, of concepts in calculus of variations, such as the notion of quasiconvexity, which can allow us to determine the existence of minimizers (ground states) for an energy depending on that type of maps and their gradients, besides the standard deformation, are necessary. Answers to these problems are given in Focardi et al (2014). A key ingredient for them is the completeness of M. Affirming that M is a complete manifold means that the notion of a geodetic curve is available for it and any pair of elements of M can be connected by a geodetic path.…”

“…In general, there is no reason to presume a priori a hierarchy between the elements of the microstructural family for they are identical with one another. This point of view, presented first in Focardi, Mariano, and Spadaro (2014), implies a number of analytical problems: -Although we can give meaning to the notion of differentiability for a manifold-r-valued map, there is no representation such that each component mapping x −→ ν α (x) is differentiable, taking into account the quotient with respect to the permutations. -Even in the case in which M is a smooth manifold, the set of r-valued maps over M defined above is not a smooth manifold anymore, and it has to be treated as a metric space only.…”

Equilibrium configurations of mixture thin films undergoing large strains

“…If we want to consider such a view in the case of mixture thin films, we must account for the mixing of the mixture's components, and it is exactly what I propose here. • From the technical side . This work rests heavily on a previous paper by M. Focardi, E. Spadaro, and myself on the general model‐building framework of the mechanics of complex materials . Some definitions are quoted essentially verbatim from that work.…”

“…First, I introduce here a special class of weak diffeomorphisms coupled with the layer‐descriptor‐map and prove that is it closed. Such a space does not appear in Focardi et al; precisely, in that reference, a kinematic link between the deformation map and the descriptor of the microstructure's morphology does not appear. Then, I specify a condition (H2‐c2 below), which is necessary also in Focardi et al, where it is just implicit.…”