The mechanically based non-local elasticity: an overview of main results and future challenges

Paola, Mario Di; Failla, Giuseppe; Pirrotta, Antonina; Sofi, Alba; Zingales, Massimiliano

doi:10.1098/rsta.2012.0433

Cited by 77 publications

(36 citation statements)

References 45 publications

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“…(35) for local thermal energy transfer q x; t ð Þ involves non-monotonic increments of the entropy function s ¼ sðuÞ of CIT introduced in previous section as observed in previous papers [39,40]. Previous consideration leads to assess the thermodynamical consistency of the model in terms of the extended irreversible thermodynamics (EIT) as it is shown in the following section.…”

Section: A Non-equilibrium Model Of Fractional-order Thermal Energy Tmentioning

confidence: 99%

Fractional-order theory of heat transport in rigid bodies

Zingales¹

2014

Communications in Nonlinear Science and Numerical Simulation

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a b s t r a c tThe non-local model of heat transfer, used to describe the deviations of the temperature field from the well-known prediction of Fourier/Cattaneo models experienced in complex media, is framed in the context of fractional-order calculus. It has been assumed (Borino et al., 2011 [53], Mongioví and Zingales, 2013 [54]) that thermal energy transport is due to two phenomena: (i) A short-range heat flux ruled by a local transport equation; (ii) A long-range thermal energy transfer proportional to a distance-decaying function, to the relative temperature and to the product of the interacting masses. The distance-decaying function is assumed in the functional class of the power-law decay of the distance yielding a novel temperature equation in terms of a-order Marchaud fractional-order derivative ð0 6 a 6 1Þ. Thermodynamical consistency of the model is provided in the context of Clausius-Plank inequality. The effects induced by the boundary conditions on the temperature field are investigated for diffusive as well as ballistic local heat flux. Deviations of the temperature field from the linear distributions in the neighborhood of the thermostated zones of small-scale conductors are qualitatively predicted by the used fractional-order heat transport model, as shown by means of molecular dynamics simulations.

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Section: A Non-equilibrium Model Of Fractional-order Thermal Energy Tmentioning

confidence: 99%

Fractional-order theory of heat transport in rigid bodies

Zingales¹

2014

Communications in Nonlinear Science and Numerical Simulation

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“…Along the interface x 3 = h, to within an exponentially small error, erfc (−h/a) = 2 and, consequently, 12) and the non-local stresses s αβ tend to their local analogues σ αβ in (2.9).…”

Section: ) Is O(a/h) It Is Less Than O(a/hmentioning

confidence: 99%

“…The fundamental concepts underpinning contemporary non-local continuum models were developed in a series of wellknown papers by Kroner [4], Eringen [5], Eringen & Edelen [6]; see also [7] and references therein. The state of art has been presented by a number of authors throughout the area's scientific development [8][9][10][11][12][13]. The latter paper addresses Piola's important contribution, not widely known for a long time to a broad international audience, see also references to Piola's original papers in Dell'Isola et al [13].…”

Section: Introductionmentioning

confidence: 99%

Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

Chebakov¹,

Kaplunov²,

Rogerson³

2016

Proc. R. Soc. A.

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The dynamic response of a homogeneous halfspace, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the 'local' problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of nonlocal elastic phenomena on the Rayleigh wave speed is investigated.

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“…First time the fractional derivatives with respect to space coordinates have been applied in the elasticity theory of nonlocal continuum by Gubenko [29,30] in 1957. Recently, the fractional-order derivatives is used to describe continuum with power-law type of non-locality (for example, see [26, [31][32][33][34][35][36][37][38][39][40][41][42][43]). Fractional models of integral non-local elasticity are considered in papers [44][45][46][47][48][49][50], where the microscopic models of fractional integral elasticity are also described.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional lattice models with long-range interactions of Grünwald–Letnikov type for fractional generalization of gradient elasticity

Tarasov

2015

Meccanica

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Models of three-dimensional lattices with long-range interactions of Grünwald-Letnikov type for fractional gradient elasticity of non-local continuum are suggested. The lattice long-range interactions are described by fractional-order difference operators. Continuous limit of suggested three-dimensional lattice equations gives continuum differential equations with the Grünwald-Letnikov derivatives of non-integer orders. The proposed lattice models give a new microstructural basis for elasticity of materials with power-law type of non-locality. Moreover these lattice models allow us to have a unified microscopic description for fractional and usual (non-fractional) gradient elasticity continuum.

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The mechanically based non-local elasticity: an overview of main results and future challenges

Cited by 77 publications

References 45 publications

Fractional-order theory of heat transport in rigid bodies

Fractional-order theory of heat transport in rigid bodies

Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

Three-dimensional lattice models with long-range interactions of Grünwald–Letnikov type for fractional generalization of gradient elasticity

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