Linear elastic chain with a hyper-pre-stress

Charlotte, Miguel; Truskinovsky, Lev

doi:10.1016/s0022-5096(01)00054-0

Cited by 53 publications

(40 citation statements)

References 22 publications

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“…The -limit result uses the separation-ofscale arguments of Chambolle [7] and Braides, Dal Maso & Garroni [4], and combines them with the analysis of internal boundary layers. It serves to emphasize that whereas other scalings proposed in the past (Charlotte & Truskinovsky [8], Braides & Cicalese [3]) separate the effects of bulk and surface contribution, the scaling employed here considers both effects with the same order.…”

Section: δ N ) = I φ(δ I )mentioning

confidence: 99%

Effective Cohesive Behavior of Layers of Interatomic Planes

Braides

Lew

Ortiz

2006

Arch. Rational Mech. Anal.

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A simple model of cleavage in brittle crystals consists of a layer of material containing N atomic planes separating in accordance with an interplanar potential under the action of an opening displacement δ prescribed on the boundary of the layer. The problem addressed in this work concerns the characterization of the constrained minima of the energy E N of the layer as a function of δ as N becomes large. These minima determine the effective or macroscopic cohesive law of the crystal. The main results presented in this communication are: (i) the computation of the limit E 0 of E N as N → ∞; (ii) the characterization of the minimum values of E 0 as a function of the macroscopic opening displacement; (iii) a proof of uniform convergence of the minima of E N for the case of nearest-neighbor interactions; and (iv) a proof of uniform convergence of the derivatives of E N , or tractions, in the same case. The scaling on which the present -convergence analysis is based has the effect of separating the bulk and surface contributions to the energy. It differs crucially from other scalings employed in the past in that it renders both contributions of the same order.

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Section: δ N ) = I φ(δ I )mentioning

confidence: 99%

Effective Cohesive Behavior of Layers of Interatomic Planes

Braides

Lew

Ortiz

2006

Arch. Rational Mech. Anal.

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“…For results on the continuum manifestation of some further interesting atomistic effects like finite-range interactions, the reader is referred to [25,9].…”

Section: Continuum Limits Of Atomistic Energiesmentioning

confidence: 99%

“…The case of infinity-range interactions is completely unclear. For examples of atomistic models with finite-range interactions and their relation to continuum theories, see [25,9].…”

Section: Remarks On Extensions To Two and Three Dimensionsmentioning

confidence: 99%

Gradient Flows as a Selection Procedure for Equilibria of Nonconvex Energies

Ortner¹

2006

SIAM J. Math. Anal.

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Abstract. For atomistic material models, global minimization gives the wrong qualitative behavior; a theory of equilibrium solutions needs to be defined in different terms. In this paper, a concept based on gradient flow evolutions, to describe local minimization for simple atomistic models based on the Lennard-Jones potential, is presented. As an application of this technique, it is shown that an atomistic gradient flow evolution converges to a gradient flow of a continuum energy as the spacing between the atoms tends to zero. In addition, the convergence of the resulting equilibria is investigated in the case of elastic deformation and a simple damaged state.

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“…They have been derived previously for various, mostly one-dimensional models, cf. [3,6,9,18]. Depending on the details of the model, surfaces can be dominated by one of several effects…”

Section: Previous Work and Conceptsmentioning

confidence: 99%

“…In the context of one-dimensional models this effect has received significant attention in the literature, e.g. [4,9].…”

Section: Previous Work and Conceptsmentioning

confidence: 99%

Surface energies in a two-dimensional mass-spring model for crystals

Theil¹

2011

ESAIM: M2AN

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Abstract.We study an atomistic pair potential-energy E (n) (y) that describes the elastic behavior of two-dimensional crystals with n atoms where y ∈ R 2×n characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E (n) admits an asymptotic expansion involving fractional powers of n:The bulk energy density E bulk is given by an explicit expression involving the interaction potentials. The surface energy E surface can be expressed as a surface integral where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous, but nowhere differentiable function of the surface normal.Mathematics Subject Classification. 74Q05.

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Linear elastic chain with a hyper-pre-stress

Cited by 53 publications

References 22 publications

Effective Cohesive Behavior of Layers of Interatomic Planes

Effective Cohesive Behavior of Layers of Interatomic Planes

Gradient Flows as a Selection Procedure for Equilibria of Nonconvex Energies

Surface energies in a two-dimensional mass-spring model for crystals

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