Calculus of Variations and Geometric Measure Theory

A. Braides - A. Causin - M. Solci - L. Truskinovsky

Beyond the classical Cauchy-Born rule

created by braidesa on 13 Sep 2023
modified on 12 Feb 2024


Published Paper

Inserted: 13 sep 2023
Last Updated: 12 feb 2024

Journal: Archive for Rational Mechanics and Analysis
Volume: 247
Number: 107
Year: 2023
Doi: 10.1007/s00205-023-01942-0


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 incommensurability 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 complimentary range. A general approach to problems with such mixed behavior is developed.