Ground states in complex bodies

Abstract

Abstract.A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance of configurational actions follows. After describing a list of possible applications of the general results collected here, a concrete discussion of the existence of ground states in thermodynamically stable quasicrystals is presented at the end.

Global Existence for Rate-Independent Gradient Plasticity at Finite Strain

“…in Ω which means that there is no local self-interpenetration. Following [CiN87] (see also [MaM07] for a similar approach using currents), we may define the "non-self-interpenetration" version of the space Y of admissible deformations via…”

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

“…Such a system is what we consider here to be the material microstructure. We take Q ≥ 2, for the case Q = 1 corresponds to the original format of the mechanics of complex materials (for the pertinent existence theorem for the minimizers of the energy in this case see [41], while for the existence theorems in the case of simple bodies undergoing finite strains see [4,27].…”

“…When ν is single-M-valued, a general expression of the elastic energy of complex materials and the existence of relevant minimizers have been analyzed in [41]. 6 Here we consider an energy with less general form.…”

Semicontinuity theorem in the micropolar elasticity

“…The first existence theorems for geometrically exact Cosserat and micromorphic models, based on convexity arguments are also given by Neff in [17] (micromorphic elasticity is more general theory than micropolar elasticity). Also, for generalized continua with microstructure the existence theorem is given in [11] where convexity in the derivative of the variable which describes microstructure is demanded (in the micropolar case that would mean convexity in ∇R). In our work we extend these developments in the micropolar case to more general constitutive behavior.…”