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

Theil, Florian

doi:10.1051/m2an/2010106

Cited by 17 publications

(11 citation statements)

References 22 publications

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“…In the finite domain case, one would need to account for surface relaxation effects, which we have entirely avoided here by choosing an anti-plane model (however, see Remark 3.4. The phenomenon of surface relaxation in discrete problems seems a difficult one, and to the authors' knowledge, has yet to be addressed systematically in the Applied Analysis literature, but for some results in this direction, see [27]. A possible way forward would be to impose an additional stability condition on the boundary, similar to our condition (STAB), which could then be investigated separately.…”

Section: Possible Extensionsmentioning

confidence: 99%

Analysis of Stable Screw Dislocation Configurations in an Antiplane Lattice Model

Hudson¹,

Ortner²

2015

SIAM J. Math. Anal.

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We consider a variational anti-plane lattice model and demonstrate that at zero temperature, there exist locally stable states containing screw dislocations, given conditions on the distance between the dislocations and on the distance between dislocations and the boundary of the crystal. In proving our results, we introduce approximate solutions which are taken from the theory of dislocations in linear elasticity, and use the inverse function theorem to show that local minimisers lie near them. This gives credence to the commonly held intuition that linear elasticity is essentially correct up to a few spacings from the dislocation core.Date: December 6, 2018. 2000 Mathematics Subject Classification. 74G25, 74G65, 70C20, 49J45, 74M25, 74E15. STABLE SCREW DISLOCATION CONFIGURATIONS 2 here, and using a different set of analytical techniques. In particular, our analysis employs discrete regularity results which enable us to provide quantitative estimates on the equilibrium configurations, while previous results only provide estimates on the energies.1.1. Outline. The setting for our results is similar to that described in [18]: our starting point is the energy difference functionalwhere Ω ⊂ Λ is a subset of a Bravais lattice, B Ω is a set of pairs of interacting (lines of) atoms, Dy b is a finite difference, and ψ is a 1-periodic potential.We call a deformation y a locally stable equilibrium if u = 0 minimises E Ω (y + u; y) among all perturbations u which have finite energy, and are sufficiently small in the energy norm. The key assumption upon which we base our analysis is the existence of a local equilibrium in the homogeneous infinite lattice containing a dislocation which satisfies a condition which we term strong stabilitythis notion is made precise in §3.2.Under this key assumption, our main result is Theorem 3.3. This states that, given a number of positive and negative screw dislocations, there exist locally stable equilibria containing these dislocations in a given domain as long as the core positions satisfy a minimum separation criterion from each other and from the boundary of the domain. Furthermore, these configurations may only be globally stable if there is one dislocation in an infinite lattice.The proof of Theorem 3.3 is divided into two cases, that in which Ω = Λ, and that in which Ω is a finite convex lattice polygon: these are proved in §5 and §6 respectively. Preliminaries2.1. The lattice. Underlying the results presented in this paper is the structure of the triangular lattice Λ := a 1 +a 2 3 + [a 1 , a 2 ] · Z 2 , where a 1 = (1, 0) T and a 2 = 1 2 ,

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Section: Possible Extensionsmentioning

confidence: 99%

Analysis of Stable Screw Dislocation Configurations in an Antiplane Lattice Model

Hudson¹,

Ortner²

2015

SIAM J. Math. Anal.

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“…Since E a and E scb are Fréchet differentiable in suitable function spaces it should be possible, using nonlinear analysis techniques such as the inverse function theorem, to extend the results from the linearized model problem to the fully nonlinear problem, provided that the stiffness parameter α is sufficiently large. Techniques of this kind have been used, for example, in [37]. …”

Section: This Immediately Implies (216)mentioning

confidence: 99%

“…Although our analysis is elementary, it makes three novel contributions: (1) we show that the "correct" approximation parameter is the stiffness of the interaction potential (however, Theil [37] uses similar ideas for an analysis of surface relaxation); and (2) we show that the mean strain (which is an important quantity of interest) has a much lower relative error than the strain field. of the SCB method at moderate additional computational cost.…”

Section: Introductionmentioning

confidence: 99%

“…The main difference in our case is the discrete setting which does not give us the opportunity to let the mesh-size tend to zero. For a mathematical analysis of thin atomistic structures, surface energies and surface relaxation we refer to [3,31,32,37] and references therein. Our work also draws inspiration from [8,9] where a similar linearised model problem is used to analyze the accuracy of atomistic-to-continuum coupling methods.…”

Section: Introductionmentioning

confidence: 99%

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An analysis of the boundary layer in the 1D surface Cauchy–Born model

et al. 2012

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Abstract. The surface Cauchy-Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is O(1) in the mesh size; however, we are able to identify an alternative "approximation parameter" -the stiffness of the interaction potential -with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.Mathematics Subject Classification. 70C20, 70-08, 65N12, 65N30.

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“…A precise and rigorous mathematical treatment of such surface relaxation effects is currently still out of reach (but cf. [The11]). In order to allow for as many atomistic boundary conditions (and body forces) as possible, we consider general convergence rates ε γ in our main theorem, Theorem 5.4, and only restrict γ as much as necessary.…”

Section: Introductionmentioning

confidence: 99%

Connecting Atomistic and Continuous Models of Elastodynamics

Braun

2017

Arch Rational Mech Anal

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We prove long-time existence of solutions for the equations of atomistic elastodynamics on a bounded domain with time-dependent boundary values as well as their convergence to a solution of continuum nonlinear elastodynamics as the interatomic distances tend to zero. Here, the continuum energy density is given by the CauchyBorn rule. The models considered allow for general finite range interactions. To control the stability of large deformations we also prove a new atomistic Gårding inequality.

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

Cited by 17 publications

References 22 publications

Analysis of Stable Screw Dislocation Configurations in an Antiplane Lattice Model

Analysis of Stable Screw Dislocation Configurations in an Antiplane Lattice Model

An analysis of the boundary layer in the 1D surface Cauchy–Born model

Connecting Atomistic and Continuous Models of Elastodynamics

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