Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems

Abstract

Abstract. We study continuum and atomistic models for the elastodynamics of crystalline solids at zero temperature. We establish sharp criterion for the regime of validity of the nonlinear elastic wave equations derived from the well-known Cauchy-Born rule.

Trends and challenges in the mechanics of complex materials: a view

“…And this is a result of statistical mechanics. Moreover, since elastically deformed states are in general just local minimizers of the energy, we can show that the Cauchy-Born rule is always valid for elastically deformed crystals, provided appropriate choices of the unit cell characterizing lattice periodicity [20,21]. The idea underlying the Cauchy-Born rule can be generalized to multiple lattices [22] or to an expression in terms of velocities.…”

The Effect of Crystal Symmetries on the Locality of Screw Dislocation Cores

“…One then expects that the dominant error contribution is the Cauchy-Born anti-discretisation error. The results of [3,6,15] suggest that the resultant corrector should decay as O(|x| −3 ), however exploiting crystal symmetries reveals that the Cauchy-Born error is of higher order than expected and one even obtains a corrector decay of O(|x| −4 ).…”

Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory

“…Based on the observation that elastic deformations in general are merely local energy minimizers, E and Ming have pioneered a different approach. In [EM07] they show that, under suitable stability assumptions, solutions of the equations of continuum elasticity on the flat torus with smooth body forces are asymptotically approximated by corresponding atomistic equilibrium configurations. Recently these results have been generalized to a large class of interatomic potentials under remarkably mild regularity assumptions on the body forces for problems on the whole space by Ortner and Theil, cf.…”

“…With these preparations we state and prove our main result Theorem 5.1 in Section 5. Similarly as in [EM07,OT13] our goal is to find an atomistic solution in the vicinity of a continuous solution of the associated Cauchy-Born system by observing that the latter is an approximate solution of the discrete system, where now we also have to account for the additional boundary data. To this end, we begin by formulating a quantitative version of the implicit function theorem with a small parameter.…”