Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours

Romano, Giovanni; Luciano, Raimondo; Barretta, Raffaele; Diaco, Marina

doi:10.1007/s00161-018-0631-0

Cited by 81 publications

(34 citation statements)

References 30 publications

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“…The size effect of doubly-curved nanoshells starts considering the strain-driven gradient model by Eringen [13] to include possible nonlocal long-range interactions, which is also discussed for beam applications in References [50][51][52][53]. Thus, the stress-strain relations for both thin and thick isotropic nanoshells, accounting for small effects, are expressed by the equation below…”

Section: Governing Equations Of Doubly-curved Nanoshellsmentioning

confidence: 99%

Nonlocal Elasticity Response of Doubly-Curved Nanoshells

Dindarloo

Dimitri

et al. 2020

Symmetry

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In this paper, we focus on the bending behavior of isotropic doubly-curved nanoshells based on a high-order shear deformation theory, whose shape functions are selected as an accurate combination of exponential and trigonometric functions instead of the classical polynomial functions. The small-scale effect of the nanostructure is modeled according to the differential law consequent, but is not equivalent to the strain-driven nonlocal integral theory of elasticity equipped with Helmholtz’s averaging kernel. The governing equations of the problem are obtained from the Hamilton’s principle, whereas the Navier’s series are proposed for a closed form solution of the structural problem involving simply-supported nanostructures. The work provides a unified framework for the bending study of both thin and thick symmetric doubly-curved shallow and deep nanoshells, while investigating spherical and cylindrical panels subjected to a point or a sinusoidal loading condition. The effect of several parameters, such as the nonlocal parameter, as well as the mechanical and geometrical properties, is investigated on the bending deflection of isotropic doubly-curved shallow and deep nanoshells. The numerical results from our investigation could be considered as valid benchmarks in the literature for possible further analyses of doubly-curved applications in nanotechnology.

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Section: Governing Equations Of Doubly-curved Nanoshellsmentioning

confidence: 99%

Nonlocal Elasticity Response of Doubly-Curved Nanoshells

Dindarloo

Dimitri

et al. 2020

Symmetry

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“…These issues have been addressed in [25] for inflected beams, by showing that proper CBCs have to be imposed to close the relevant integral problem. Well-posed nonlocal integral elasticity models have been formulated by generalized functions in [26] and numerically solved by iterative techniques in [2].…”

Section: Structural Problems Of Technical Interest Are Instead Definementioning

confidence: 99%

“…Let us preliminarily recall the definition of integral convolution of a scalar field s with a averaging kernel (12) with x and points of the interval and the scalar kernel fulfilling positivity, parity, symmetry, normalization and limit impulsivity properties [25,26].…”

Section: Bishop Rod In Nonlocal Integral Elasticitymentioning

confidence: 99%

A consistent variational formulation of Bishop nonlocal rods

Barretta

Faghidian

Sciarra

2019

Continuum Mech. Thermodyn.

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Thick rods are employed in Nanotechnology to build modern electro mechanical systems. Design and optimization of such structures can be carried out by nonlocal continuum mechanics which is computationally convenient when compared to atomistic strategies. Bishop's kinematics is able to describe small-scale thick rods if a proper mathematical model of nonlocal elasticity is formulated to capture size effects. In all papers on the matter, nonlocal contributions are evaluated by replacing Eringen's integral convolution with the consequent (but not equivalent) differential equation governed by Helmholtz's differential Both hardening and softening structural responses are predictable with a suitable tuning of the parameters.

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“…Indeed, the constitutive boundary conditions associated with Eringen's integral convolution in nonlocal structural problems of engineering interest are in contrast with equilibrium requirements, and thus, the strain-driven law can lead to ill-posed elastic formulations [36].…”

Section: Introductionmentioning

confidence: 99%

Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions

et al. 2019

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Nonlocal strain gradient continuum mechanics is a methodology widely employed in literature to assess size effects in nanostructures. Notwithstanding this, improper higher-order boundary conditions (HOBC) are prescribed to close the corresponding elastostatic problems.In the present study, it is proven that HOBC have to be replaced with univocally determined boundary conditions of constitutive type, established by a consistent variational formulation.The treatment, developed in the framework of torsion of elastic beams, provides an effective approach to evaluate scale phenomena in smaller and smaller devices of engineering interest.Both elastostatic torsional responses and torsional free vibrations of nano-beams are investigated by applying a simple analytical method. It is also underlined that the nonlocal strain gradient model, if equipped with the inappropriate HOBC, can lead to torsional structural responses which unacceptably do not exhibit nonlocality. The presented variational strategy is instead able to characterize significantly peculiar softening and stiffening behaviours of structures involved in modern Nano-Electro-Mechanical-Systems (NEMS). KeywordsTorsion; nano-beams; nonlocal strain gradient model; strain gradient elasticity; integral elasticity; size effects; analytical modelling; NEMS. d J J dx J J dx J

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Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours

Cited by 81 publications

References 30 publications

Nonlocal Elasticity Response of Doubly-Curved Nanoshells

Nonlocal Elasticity Response of Doubly-Curved Nanoshells

A consistent variational formulation of Bishop nonlocal rods

Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions

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