Internal Length Gradient (ILG) Material Mechanics Across Scales and Disciplines

Aifantis, Elias C.

doi:10.1016/bs.aams.2016.08.001

Cited by 107 publications

(62 citation statements)

References 230 publications

(259 reference statements)

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“…These will generalize some results already obtained for fractal elasticity [58]. In [59] an operator split method (the Ru-Aifantis theorem) has been used to obtain solutions to gradient elasticity in terms of solutions of corresponding problems in classical elasticity [8,9,60]. A generalization of the Ru-Aifantis operator split method can be used to solve boundary value problems for fractal gradient elasticity by using the solution for fractal non-gradient elasticity given in [58].…”

Section: Discussionsupporting

confidence: 53%

On fractional and fractal formulations of gradient linear and nonlinear elasticity

Aifantis

2019

Acta Mech

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In this paper we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of threedimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel we consider non-linear field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relationship that we elaborate on, can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally we discuss fractal materials described by continuum models with non-integer dimensional spaces. Using the recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

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Section: Discussionsupporting

confidence: 53%

On fractional and fractal formulations of gradient linear and nonlinear elasticity

Aifantis

2019

Acta Mech

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“…All these problems were successfully and efficiently addressed by incorporating internal lengths in the standard constitutive equations of elasticity and diffusion, as scalar multipliers of newly introduced Laplacian terms of the persistent constitutive variables to account for weakly nonlocal effects. The resulting internal length gradient (ILG) framework and its applications to various areas of material mechanics are reviewed in a recent article by the author [1], where extensive bibliography can also be found. In the same article a brief account of fractional and fractal generalization of the ILG framework is given.…”

Section: Introductionmentioning

confidence: 99%

Fractional generalizations of gradient mechanics

Aifantis¹

2019

Applications in Physics, Part A

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“…In previous publications by Aifantis et al [2,4,14,27,28], it was shown that the GradEla model can conveniently capture the occurrence of elastic size effects in small one-dimensional objects and specimens with holes commonly used for advanced technology applications. More recently, its extension to describe size effects due to thermomechanical, chemomechanical, and electromechanical couplings has been reviewed in [7]. The purpose of this section is to expand this discussion for an emerging technological problem of internal stress development and capacity fade in Lithium-ion (Li-ion) rechargeable battery electrodes by elaborating on the classical elasticity treatment first used for this problem in the pioneering work of [8] (see also related chapters and references in [9]).…”

Section: Static and Fatigue Assessment Of Notched Barsmentioning

confidence: 99%

“…See also the analyses in [15,17,19], as well as a more recent discussion in [6,7]. Since the presence of cracks reduces drastically the convergence rate of standard finite element methodologies, very fine meshes are required in the neighbourhood of the stress riser to ensure that convergence of the solution is reached, with significant increasing in computational cost.…”

Section: Convergence In the Presence Of Singularities And Recommendatmentioning

confidence: 99%

Gradient-enriched finite element methodology for axisymmetric problems

2017

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Due to the axisymmetric nature of many engineering problems, bi-dimensional axisymmetric finite elements play an important role in the numerical analysis of structures, as well as advanced technology micro/ nano-components and devices (nano-tubes, nano-wires, micro-/nano-pillars, micro-electrodes). In this paper, a straightforward C 0 -continuous gradient-enriched finite element methodology is proposed for the solution of axisymmetric geometries, including both axisymmetric and non-axisymmetric loads. Considerations about the best integration rules and an exhaustive convergence study are also provided along with guidances on optimal element size. Moreover, by applying the present methodology to cylindrical bars characterised by a circumferential sharp crack, the ability of the present methodology to remove singularities from the stress field has been shown under axial, bending, and torsional loading conditions. Some preliminary results, obtained by applying the proposed methodology to notched cylindrical bars, are also presented, highlighting the accuracy of the methodology in the static and fatigue assessment of notched components, for both brittle and ductile materials. Finally, the proposed methodology has been applied to model the unit cell of the anode of Li-ion batteries showing the ability of the methodology to account for size effects.

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Internal Length Gradient (ILG) Material Mechanics Across Scales and Disciplines

Cited by 107 publications

References 230 publications

On fractional and fractal formulations of gradient linear and nonlinear elasticity

On fractional and fractal formulations of gradient linear and nonlinear elasticity

Fractional generalizations of gradient mechanics

Gradient-enriched finite element methodology for axisymmetric problems

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