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Beyond the classical Cauchy-Born rule

Andrea Braides, Andrea Causin, Margherita Solci, Lev Truskinovsky

TL;DR

This work investigates discrete, non-convex lattice energies with long-range interactions and boundary constraints, showing that geometric frustration can invalidate the classical Cauchy-Born rule. It develops a Generalized Cauchy-Born framework via the $\widehat{Q}_{\mathbf m} f$ relaxation and the $Q_{\mathbf m} f$ transform, together with a phase-function $\theta(z)$ and locking states, to distinguish parameter regimes where local (cell-based) reductions suffice from those requiring mesoscopic, laminated descriptions. Two archetypal kernel classes are analyzed in depth: concentrated kernels (finite-range interactions) and exponential kernels (rapidly decaying long-range interactions), revealing explicit structures of minimizers, energy partitions into phases, and the dependence on scale parameters. The results connect discrete variational problems to continuum homogenization via Gamma-convergence, highlighting how nonlocality and discreteness profoundly shape the homogenized energy and the validity of the generalized CB rule. The framework yields explicit formulas for relaxation in key prototypical energies (truncated quadratic and bi-well potentials) and demonstrates how locking states enable manageable, periodic descriptions of otherwise complex microstructures, with implications for lattice fracture, phase transitions, and material design.

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.

Beyond the classical Cauchy-Born rule

TL;DR

This work investigates discrete, non-convex lattice energies with long-range interactions and boundary constraints, showing that geometric frustration can invalidate the classical Cauchy-Born rule. It develops a Generalized Cauchy-Born framework via the relaxation and the transform, together with a phase-function and locking states, to distinguish parameter regimes where local (cell-based) reductions suffice from those requiring mesoscopic, laminated descriptions. Two archetypal kernel classes are analyzed in depth: concentrated kernels (finite-range interactions) and exponential kernels (rapidly decaying long-range interactions), revealing explicit structures of minimizers, energy partitions into phases, and the dependence on scale parameters. The results connect discrete variational problems to continuum homogenization via Gamma-convergence, highlighting how nonlocality and discreteness profoundly shape the homogenized energy and the validity of the generalized CB rule. The framework yields explicit formulas for relaxation in key prototypical energies (truncated quadratic and bi-well potentials) and demonstrates how locking states enable manageable, periodic descriptions of otherwise complex microstructures, with implications for lattice fracture, phase transitions, and material design.

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.
Paper Structure (47 sections, 43 theorems, 427 equations, 32 figures)

This paper contains 47 sections, 43 theorems, 427 equations, 32 figures.

Table of Contents

  1. Introduction
  2. Nonlocal discrete problems and their relaxation
  3. Nearest-neighbour interaction and quadratic penalization
  4. General properties of $\widehat{Q}_{\bf m}f(z)$
  5. Lower bound: optimization on nearest-neighbour clusters
  6. Upper bound: optimization over periodic patterns
  7. The $\bf m$-transform of $f$
  8. Interpolation by parameterized kernels
  9. Description of minimizers by a phase function
  10. Phase-constrained relaxation and related properties
  11. Phase function and locking states
  12. Phase-constrained analysis for decoupled interactions
  13. Convexification as an envelope of phase-constrained problems
  14. Case (a): the non-convexity set is a bounded interval.
  15. Case (b): the non-convexity set is a half line.
  16. ...and 32 more sections

Key Result

Proposition 2.1

Let $f\colon\mathbb R\to\mathbb R$. Then, the convex envelope of $f$ is

Figures (32)

  • Figure 1: representation of exponential and concentrated kernels.
  • Figure 2: the function $Q_{\bf m}f$ in Remark \ref{['corNN2']} with $f(z)=(1-z^2)^2$ for different values of $m_1<1$.
  • Figure 3: Graph of $Q_{\bf 0}f^t(\theta,z)$ for different values of $\theta$.
  • Figure 4: Graph of the phase function $\theta(z)$ for the function $f^t$ in Example \ref{['dwm0']}.
  • Figure 5: Graph of $Q_{\bf 0}f^{t}(\theta,z)$ with $\theta$ fixed and increasing values of ${t}$.
  • ...and 27 more figures

Theorems & Definitions (156)

  • Proposition 2.1: a characterization of the convex envelope
  • Remark 2.2: additivity
  • Definition 2.3: relaxation with kernel $\bf m$
  • Remark 2.4: nearest-neighbour interactions
  • Remark 2.5: $\widehat{Q}_{\bf m}f$ as a $\Gamma$-limit
  • Remark 2.6
  • Proposition 2.7: convexity of $\widehat{Q}_{\bf m}f$
  • Remark 2.8: alternative statements of boundary conditions
  • Remark 2.9: estimates by decomposition for $\widehat{Q}_{\bf m}f$
  • Lemma 2.10: minimization of the quadratic part
  • ...and 146 more