Optimal flux densities for linear mappings and the multiscale geometry of structured deformations
The authors study expressions for interfacial terms in the relaxed energy associated to structured deformations of continua. Structured deformations were defined in [\textit{G. Del Piero} and \textit{D. R. Owen}, Arch. Ration. Mech. Anal. 124, No. 2, 99--155 (1993; Zbl 0795.73005)] for the study of deformations at different length scales particularly in models of plasticity and fracture. For a continuum \(U\), a structured deformation consists of a macroscopic deformation \(g\), a microscopic deformation \(G\), and a rectifiable surface in \(U\) indicating the site of disarrangement. The least energy needed to deform \(U\) by \(g\) to affect a deformation by \(G\) on small pieces is realized as the infimum of limit inferiors of energies defined on sequences \(u_n\) with \(u_n \to g\) and \(\nabla u_n \to G\). Representation formulae in [\textit{R. Choksi} and \textit{I. Fonseca}, Arch. Ration. Mech. Anal. 138, No. 1, 37--103 (1997; Zbl 0891.73078)] show that it suffices to compute the infimum of a related energy, which for \(x_0 \in U\) involves maps on the unit cube having the average gradient \(G(x_0)\) and the boundary value \(\nabla g(x_0)\). The authors study a particular form of this energy measuring switches, separations and interpenetration along the disarrangement site. The computation amounts to the evaluation of \[ \text{inf}_V \frac{1}{|V|} \int_{\partial V} |M x \cdot \nu |d\mathcal{H}^{n-1}(x) \] for \(M\) in a dense set of matrices. Here the infimum is taken over sets \(V \subset U\) with finite perimeter. The authors show that the infimum equals \(|\text{tr} M|\). Since \(|\text{tr} M|\) is a lower bound for the integral, equality comes by modifying \(V\) along the flow \(x \mapsto Mx\) to create regions in \(\partial V\) where the integrand is minimized. This provides an interesting relation between minimal flux of a linear vector field \(x \mapsto Mx\) and the minimal amount of switching, separating and interpenetrating needed to form the structured deformation \((i, I+M)\).
scientific article; zbMATH DE number 6504072