A review on the application of modified continuum models in modeling and simulation of nanostructures

Wang, K. F.; Wang, B. L.; Kitamura, Takayuki

doi:10.1007/s10409-015-0508-4

Cited by 91 publications

(31 citation statements)

References 192 publications

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“…It should be mentioned that for micro-and nano-structures such as nanotubes and graphene sheets, in particular, the so-called non-local continuum theory has raised attention among reseachers of the field as well (see [20,12] for selective but still extensive lists of references). However, a concern about the non-local continuum theory -discovered particularly in the context of non-local beam models -is the fact that a big portion of the literature applies the so-called differential form of Eringen's constitutive equation [46] which is very recently shown to be nonequivalent to its integral form in the context of Euler-Bernoulli beam problem [20] (see [47] as well).…”

Section: Introductionmentioning

confidence: 99%

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Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models

Niiranen¹,

Balobanov

Kiendl

et al. 2017

Mathematics and Mechanics of Solids

103

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As the first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler-Bernoulli beam bending models following a simplified version of Mindlin's strain gradient elasticity theory of form II. For both models, this leads to sixth-order boundary value problems with new types of boundary conditions which are given additional attributes singly and doubly; referring to a physically relevant distinguishment between free and prescribed curvature, respectively. Second, the variational formulations are analyzed with rigorous mathematical tools: existence and uniqueness of weak solutions are established by proving continuity and ellipticity of the associated symmetric bilinear forms. This guarantees optimal convergence for conforming Galerkin discretization methods. Third, the variational analysis is extended to cover two other generalized beam models: another modification of the strain gradient elasticity theory and a * Corresponding author: jarkko.niiranen@aalto.fi

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Section: Introductionmentioning

confidence: 99%

“…In general, this means the necessity of multi-scale mechanics and implies a need for reliable and efficient multi-scale analysis tools, both analytical and computational methods alongside with experimental tests [4,10,5,11,12].…”

Section: Introductionmentioning

confidence: 99%

Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models

Niiranen¹,

Balobanov

Kiendl

et al. 2017

Mathematics and Mechanics of Solids

103

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“…For this purpose, various types of sizedependent continuum theory which can interpret size effect efficiently have been proposed to investigate the mechanical behavior of structures at both micro and nano scales [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Wang et al [20] reviewed recent advancements in the applications of such modified continuum mechanics model.…”

Section: Introductionmentioning

confidence: 99%

Surface free energy effects on the postbuckling behavior of cylindrical shear deformable nanoshells under combined axial and radial compressions

2016

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In the traditional continuum mechanics, the effects of surface free energy are generally ignored. However, this cannot be the case for nanostructures because of their high surface to volume ratio; surface energy plays an important role in the mechanical responses. In the present study, the nonlinear buckling and postbuckling characteristics of cylindrical nanoshells subjected to combined axial and radial compressions are investigated in the presence of surface energy effects. To this end, Gurtin-Murdoch elasticity theory is implemented into the classical firstorder shear deformation shell theory to develop an efficient size-dependent shell model incorporating surface free energy effects. Subsequently, a boundary layer theory is employed including surface effects in conjunction with the nonlinear prebuckling deformations, the large postbuckling deflections and the initial geometric imperfection. Finally, a solution methodology based on a two-stepped singular perturbation technique is utilized to obtain the size-dependent critical buckling loads and equilibrium postbuckling paths corresponding to the both axial dominated and radial dominated loading cases. It is observed that for the both axial dominated and radial dominated loading cases, surface free energy effects cause to increase the both critical buckling load and critical end-shortening of shear deformable nanoshell made of silicon.

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“…The incorporation of surface elasticity into continuum-based models of deformation has become widespread in the recent literature primarily due to the desire to accommodate size effects in elastic solids (see, for example, [29,31] and the references contained therein). The starting point in the development of such models is in selecting an appropriate theory of surface elasticity.…”

Section: Introductionmentioning

confidence: 99%

Existence and integral representation of solutions for plane deformations of a micropolar elastic solid with surface elasticity

Gharahi

Schiavone

2020

Z Angew Math Mech

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We consider a linear theory of elastic boundary reinforcement of a micropolar elastic solid subjected to plane-strain deformations. The reinforcement consists of a thin micropolar elastic coating bonded to part of the boundary of the solid. The elastic properties of the coating incorporate both classical and micropolar bending, extension and twisting effects. Interior and exterior mixed boundary problems are formulated and analyzed using the boundary integral equation method. The boundary value problems are reduced to systems of singular integro-differential equations to which Noether-type theorems are shown to apply. We consider also the corresponding boundary value problems based on an alternative lower-order shell model of the reinforcement. Finally, existence and uniqueness results are presented for the corresponding interior and exterior boundary value problems in the appropriate classical function spaces. K E Y W O R D Sexistence of solution, micropolar boundary reinforcement, micropolar surface effects, plane micropolar elasticity, singular integro-differential equations

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A review on the application of modified continuum models in modeling and simulation of nanostructures

Cited by 91 publications

References 192 publications

Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models

Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models

Surface free energy effects on the postbuckling behavior of cylindrical shear deformable nanoshells under combined axial and radial compressions

Existence and integral representation of solutions for plane deformations of a micropolar elastic solid with surface elasticity

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