The Cauchy relations in a molecular theory of elasticity

Stakgold, Ivar

doi:10.1090/qam/36650

Cited by 81 publications

(58 citation statements)

References 2 publications

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“…This result that an amorphous solid with a random network of cohesive bonds satisfies the Cauchy relation seems to be a generalization of Stakgold's theorem [86] to an amorphous solid. A material particle in an amorphous solid satisfies centrosymmetry in a statistical sense, and the cohesive force law corresponds to a two-body potential.…”

Section: Virtual Internal Bond Modelmentioning

confidence: 56%

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Klein

Foulk

Chen

et al. 2001

Theoretical and Applied Fracture Mechanics

151

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Simulation of generalized fracture and fragmentation remains an ongoing challenge in computational fracture mechanics. There are difficulties associated not only with the formulation of physically-based models of material failure, but also with the numerical methods required to treat geometries that change in time. The issue of fracture criteria is addressed in this work through a cohesive view of material, meaning that a finite material strength and work to fracture are included in the material description. In this study, we present both surface and bulk cohesive formulations for modeling brittle fracture, detailing the derivation of the formulations, fitting relations, and providing a critical assessment of their capabilities in numerical simulations of fracture. Due to their inherent adaptivity and robustness under severe deformation, meshfree methods are especially well-suited to modeling fracture behavior. We describe the application of meshfree methods to both bulk and surface approaches to cohesive modeling. We present numerical examples highlighting the capabilities and shortcomings of the methods in order to identify which approaches are best-suited to modeling different types of fracture phenomena.

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Section: Virtual Internal Bond Modelmentioning

confidence: 56%

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Klein

Foulk

Chen

et al. 2001

Theoretical and Applied Fracture Mechanics

151

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“…There are two equivalent approaches for extracting the elastic moduli from the atomistic description, 31,32 namely the method of the homogeneous deformations, [33][34][35] also called the Cauchy-Born hypothesis, 36 and the method of the long waves within lattice-dynamical theories. 31 For examples of the application of these classical methods to the in-plane response of graphene, see Refs.…”

Section: Finite Crystal Elasticity For Curved Monolayersmentioning

confidence: 99%

“…Finite crystal elasticity has been used, for instance, to obtain elastic moduli and study the stability of strained crystals. [33][34][35]39,40 Recently, these ideas have been cast in a computational framework to solve general boundary value problems with complex geometries or loading, in what is called the quasicontinuum method. 41 This method can handle defects and fracture, by adaptively refining the continuum description down to the atomic level where required.…”

Section: Finite Crystal Elasticity For Curved Monolayersmentioning

confidence: 99%

“…Consequently, as the crystalline solid deforms, lattice vectors undergo a linear transformation. 32,33,46 This approach is often abstracted through the Cauchy-Born rule: 36,40 aϭFA, ͑1͒…”

Section: Finite Crystal Elasticity For Curved Monolayersmentioning

confidence: 99%

“…These relative shifts are called inner displacements. [33][34][35]42,46 The optical modes are the analog of the inner displacements in lattice dynamical theories. 32 Let denote the inner displacements field, which following Ref.…”

Section: Constitutive Law For Graphenementioning

confidence: 99%

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Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule

2004

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A finite deformation continuum theory is derived from interatomic potentials for the analysis of the mechanics of carbon nanotubes. This nonlinear elastic theory is based on an extension of the Cauchy-Born rule called the exponential Cauchy-Born rule. The continuum object replacing the graphene sheet is a surface without thickness. The method systematically addresses both the characterization of the small strain elasticity of nanotubes and the simulation at large strains. Elastic moduli are explicitly expressed in terms of the functional form of the interatomic potential. The expression for the flexural stiffness of graphene sheets, which cannot be obtained from standard crystal elasticity, is derived. We also show that simulations with the continuum model combined with the finite element method agree very well with zero temperature atomistic calculations involving severe deformations.Peer ReviewedPostprint (published version

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The Existence of Interfacial Couples in Infinitesimal Elasticity

Rajagopal

1960

Annalen der Physik

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SummaryThe existence of interfacial couples in general crystalline media is established and an extension of the Cauchy stress principle is formulated in this paper. A strict method of treating the homogeneous deformation of an infinite lattice is also presented. The relevance of interfacial couples in the theory of elasticity is discussed together with a critical survey of the recent work in this field.

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The Cauchy relations in a molecular theory of elasticity

Cited by 81 publications

References 2 publications

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule

The Existence of Interfacial Couples in Infinitesimal Elasticity

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