NanoMechanics: Elasticity in Nano-Objects

Merchan, Lina; Szoszkiewicz, Robert

doi:10.1007/978-3-540-36807-6_12

Cited by 3 publications

(3 citation statements)

References 154 publications

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“…The proposed lattice model can also be easily generalized for the case of the high-order gradient elasticity and the correspondent fractional extension by using the next terms of fractional Taylor series. The suggested lattice models with long-range interactions can be important to describe the nonlocal elasticity of materials at microscale and ISRN Condensed Matter Physics nanoscales [34][35][36], where the interatomic and intermolecular interactions are prevalent in determining the properties of these materials.…”

Section: Resultsmentioning

confidence: 99%

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Fractional Gradient Elasticity from Spatial Dispersion Law

Tarasov

2014

ISRN Condensed Matter Physics

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Nonlocal elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak nonlocality) and the integral nonlocal models (strong nonlocality). This paper focuses on the fractional generalization of gradient elasticity that allows us to describe a weak nonlocality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We demonstrate how the continuum limit transforms the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in Riesz's form. A weak nonlocality of power-law type in the nonlocal elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The continuum equations with derivatives of noninteger orders, which are obtained from the lattice model, can be considered as a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity.

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Section: Resultsmentioning

confidence: 99%

“…If we use the fractional Taylor's formula (see Appendix A) we get a finite number of terms. For example, Taylor's series in the Odibat-Shawagfeh form that contains the Caputo fractional derivative has two terms for (34). Using…”

Section: Fractional Taylor Series Approachmentioning

confidence: 99%

Fractional Gradient Elasticity from Spatial Dispersion Law

Tarasov

2014

ISRN Condensed Matter Physics

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“…We also assume that the suggested lattice model can be generalized to get discrete (lattice) models for dislocations in the gradient elasticity continuum [32,33,34,35,36,37,38,39], and then it will be possible to extend them to the fractional non-local case. The suggested lattice models with long-range interactions, which are suggested for the gradient elasticity continuum, can be important to describe the non-local elasticity of materials at micro and nano scales [42,43,44,45], where the interatomic and intermolecular interactions are prevalent in determining the properties of these materials.…”

Section: Resultsmentioning

confidence: 99%

General lattice model of gradient elasticity

Tarasov

2014

Mod. Phys. Lett. B

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New lattice model for the gradient elasticity is suggested. This lattice model gives a microstructural basis for second-order strain-gradient elasticity of continuum that is described by the linear elastic constitutive relation with the negative sign in front of the gradient. Moreover the suggested lattice model allows us to have a unified description of gradient models with positive and negative signs of the strain gradient terms. Possible generalizations of this model for the high-order gradient elasticity and three-dimensional case are also suggested.

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Determination of mechanical properties of geomaterials based on nano-indentation tests and fraction order models

Журавков

Romanova

2016

J Min Sci

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NanoMechanics: Elasticity in Nano-Objects

Cited by 3 publications

References 154 publications

Fractional Gradient Elasticity from Spatial Dispersion Law

Fractional Gradient Elasticity from Spatial Dispersion Law

General lattice model of gradient elasticity

Determination of mechanical properties of geomaterials based on nano-indentation tests and fraction order models

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