On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth

Abstract

We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models on multilattices which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.

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

“…More specifically, in these papers a version of the Cauchy-Born rule is established by considering a box containing a portion of a crystal and showing that, under the condition that the atoms in a boundary layer (whose width depends on the maximal interatomic interaction length) follow a given affine deformation, the global minimizer of the energy is given by the homogeneous deformation in which all atoms follow that affine deformation. In [BS13] we showed that these results can be combined with abstract results on integral representation to give a link in terms of Γ-convergence and, in particular, convergence of global minimizers of the atomistic energy to the continuum energy with Cauchy-Born energy density (for small strains) as the interatomic distances tend to zero. A corresponding discrete-to-continuum convergence result in which simultaneously the strain becomes infinitesimally small had been obtained by the second author in [Sch09] resulting in a continuum energy functional with the linearized Cauchy-Born energy density.…”

From Statistical Polymer Physics to Nonlinear Elasticity

“…Under the above Hypotheses the passage from discrete Hamiltonians to continuum energies is wellunderstood (e.g. by Γ-convergence in [3,4,7,8,10]; see also [9] for results on local minimizers). In this paper we are interested in the asymptotic behavior of the free energy when we prescribe boundary conditions.…”

Stochastic Homogenization of Nonconvex Discrete Energies with Degenerate Growth

“…• A ε , A ε g (A), A # (kY ) denote function spaces of piecewise affine functions (subordinate to εL or L, respectively), see (8), (12) and (19).…”

“…(6), V (ω; ·, ·) : E ×R n → [0, ∞) denotes a random interaction potential, and ω stands for a configuration sampled from a stationary and ergodic law. In [2], a general Γ-convergence result for functionals of the form (1) is proven under the assumption that the interaction potential V (ω; ·, ·) is non-degenerate, i.e., satisfies a uniform p-growth condition (see also [1,19] for the non-random case). The key point and novelty of the present paper is to consider random potentials with degenerate growth: We suppose that λ(ω; e)( 1 c |r| p − c) ≤ V (ω; e, r) ≤ c(λ(ω; e)|r| p + 1), where λ(ω; ·) : E → [0, ∞) denotes a (stationary and ergodic) random weight function that satisfies certain moment conditions, cf.…”