Beyond the classical Cauchy-Born rule

TL;DR

This paper investigates the complex interplay of non-convexity, non-locality, and discreteness in variational problems with long-range interactions, revealing conditions under which a generalized Cauchy-Born rule applies despite geometric frustration.

Contribution

It introduces a new class of problems with geometrical frustration that obey the Generalized Cauchy-Born rule in certain parameter regimes, and develops a general approach for mixed behavior cases.

Findings

Identifies conditions where GCB rule applies despite frustration

Shows existence of global solutions describing continuum limits

Develops a framework for problems with mixed GCB behavior

Abstract

Physically motivated variational problems involving non-convex energies are often formulated in a discrete setting and contain boundary conditions. The long-range interactions in such problems, combined with constraints imposed by lattice discreteness, can give rise to the phenomenon of geometric frustration even in a one-dimensional setting. While non-convexity entails the formation of microstructures, incompatibility between interactions operating at different scales can produce nontrivial mixing effects which are exacerbated in the case of incommensuration between the optimal microstructures and the scale of the underlying lattice. Unraveling the intricacies of the underlying interplay between non-convexity, non-locality and discreteness, represents the main goal of this study. While in general one cannot expect that ground states in such problems possess global properties, such as periodicity, in some cases the appropriately defined global solutions exist, and are sufficient to describe the corresponding continuum (homogenized) limits. We interpret those cases as complying with a Generalized Cauchy-Born (GCB) rule, and present a new class of problems with geometrical frustration which comply with GCB rule in one range of (loading) parameters while being strictly outside this class in a complementary range. A general approach to problems with such mixed behavior is developed.