A research into bi-Helmholtz type of nonlocal elasticity and a direct approach to Eringen’s nonlocal integral model in a finite body

Koutsoumaris, C. Chr.; Eptaimeros, K.G.

doi:10.1007/s00707-018-2180-9

Cited by 29 publications

(3 citation statements)

References 67 publications

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“…It is noted that, in the current study, the constitutive law for the nonlocal theory will be assumed as a two‐phase material (local and nonlocal phases) [6] to resolve the issue of the ill‐posedness of the problem [55–57]. In other words, the nonlocal constitutive equation will be considered as follows:

\begin{equation}{{\bm{t}}^{ve}} \left( {{{\bf x}},t} \right) = {\zeta _1}\ {{\bm{\sigma }}^{ve}}\left( {{{\bf x}},t} \right) + {\zeta _2}\int_V \alpha \left( {\left| {{{\bf x^{\prime}}} - {{\bf x}}} \right|,\ \tau } \right){{\bm{\sigma }}^{ve}}\left( {{{\bf x^{\prime}}},t} \right)dV\left( {{{\bf x^{\prime}}}} \right),\quad \forall {{\bf x}} \in V\end{equation}

…”

Section: Constitutive Equationsmentioning

confidence: 99%

“…It is noted that, in the current study, the constitutive law for the nonlocal theory will be assumed as a two-phase material (local and nonlocal phases) [6] to resolve the issue of the ill-posedness of the problem [55][56][57]. In other words, the nonlocal constitutive equation will be considered as follows: In which, 𝜁 1 and 𝜁 2 are the positive constants referring to, respectively, the local and nonlocal fractions of the so-called two-phase material.…”

Section: Constitutive Equationsmentioning

confidence: 99%

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Nonlinear post‐buckling analysis of viscoelastic nano‐scaled beams by nonlocal integral finite element method

Naghinejad

Ovesy

2022

Z Angew Math Mech

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The viscoelastic buckling and nonlinear post‐buckling behavior of nano‐scaled beams are analyzed using the nonlocal integral elasticity theory. Eringen's nonlocal theory is one of the well‐known and popular size‐dependent theories, which has been used by several researchers to study the mechanical behavior of, mostly, the elastic nanostructures. A finite element method is developed using Hamilton's principle based on the two‐phase nonlocal integral theory and taking into account the buckling related terms and viscoelastic effects. The corresponding formulations are derived by implementing the Euler–Bernoulli beam theory and Kelvin–Voigt viscoelastic model. Furthermore, for analyzing the post‐buckling and viscoelastic buckling behavior, the nonlinear strains, and initial displacement (imperfection) have been considered. By employing the variational relations, the governing equations are obtained and then solved numerically using the finite difference and Newton–Raphson methods. Because of the finite element nature of the current method, various boundary conditions can be properly implemented. The results of the current study are compared with those available in the literature and the effects of nonlocal parameter, viscoelastic parameter, axial compressive load and boundary conditions on the viscoelastic buckling, and postbuckling behavior have been investigated.

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\begin{equation}{{\bm{t}}^{ve}} \left( {{{\bf x}},t} \right) = {\zeta _1}\ {{\bm{\sigma }}^{ve}}\left( {{{\bf x}},t} \right) + {\zeta _2}\int_V \alpha \left( {\left| {{{\bf x^{\prime}}} - {{\bf x}}} \right|,\ \tau } \right){{\bm{\sigma }}^{ve}}\left( {{{\bf x^{\prime}}},t} \right)dV\left( {{{\bf x^{\prime}}}} \right),\quad \forall {{\bf x}} \in V\end{equation}

…”

Section: Constitutive Equationsmentioning

confidence: 99%

Section: Constitutive Equationsmentioning

confidence: 99%

Nonlinear post‐buckling analysis of viscoelastic nano‐scaled beams by nonlocal integral finite element method

Naghinejad

Ovesy

2022

Z Angew Math Mech

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“…Fakher et al (2018) studied the vibration of a mass nanosensor, consisting of a cantilevered carbon nanotube with a size-dependent nonlocal elastic foundation, by the finite element model of integral nonlocal elasticity. Also, Koutsoumaris and Eptaimeros (2018) explored the strain of the beams under bending via bi-Helmholtz operators and its kernel function in the integral nonlocal elasticity.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method

Fakher

Hosseini-Hashemi

2020

Journal of Vibration and Control

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It has been proved that using pure nonlocal elasticity, especially in differential form, leads to inconsistent and unreliable results. Therefore, to obviate these weaknesses, Eringen’s two-phase local/nonlocal elasticity has been recently used by researchers to consider the nonlocal size dependency of nanostructures. Given this, for the first time, the size-dependent nonlinear free vibration of nanobeams is investigated in this article within the framework of two-phase elasticity by using the Galerkin method. Contrary to differential nonlocal elasticity, the size dependency of the axial tension force, due to von Karman nonlinearity, is considered, and its effect on the nonlinear vibration is examined. The correct procedure of using the Galerkin method for studying the nonlinear vibration of two-phase nanobeams is introduced. It is shown that, although it is possible to extract the linear mode shapes of two-phase nanobeams using its equal differential equation, integral form of two-phase elasticity should be considered in the Galerkin method. Furthermore, it is observed that in two-phase elasticity, applying classic mode shapes in the Galerkin method leads to significant errors, especially in higher nonlocal parameters and lower values of local phase fraction and amplitude ratios.

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On the geometrically nonlinear vibration of a piezo‐flexomagnetic nanotube

Malikan

Eremeyev

2020

Math Methods in App Sciences

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In order to describe the behavior of thin elements used in micro‐electro‐mechanical systems (MEMS) and nano‐electro‐mechanical systems (NEMS), it is essential to study a nonlinear free vibration of nanotubes under complicated external fields such as magnetic environment. In this regard, the magnetic force applied to the conductive nanotube with piezo‐flexomagnetic elastic wall is considered. By the inclusion of Euler‐Bernoulli beam and using Hamilton's principle, the equations governing the system are extracted. More importantly, a principal effect existed in a nonlinear behavior such as axial inertia is thoroughly analyzed which is not commonly investigated. We then consider the effects of nanoscale size using the nonlocal strain gradient theory (NSGT). Hereafter, the frequencies are solved as semianalytical solutions on the basis of Rayleigh‐Ritz method. The piezo‐flexomagnetic nanotube (PF‐NT) is calculated with different boundary conditions. In order to validate, the results attained from the present solution have been compared with those available in the open literature. We realized that the nonlinear frequency analysis is so significant when a nanotube has fewer degrees of freedom at both ends and its length is long.

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A research into bi-Helmholtz type of nonlocal elasticity and a direct approach to Eringen’s nonlocal integral model in a finite body

Cited by 29 publications

References 67 publications

Nonlinear post‐buckling analysis of viscoelastic nano‐scaled beams by nonlocal integral finite element method

Nonlinear post‐buckling analysis of viscoelastic nano‐scaled beams by nonlocal integral finite element method

Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galerkin method

On the geometrically nonlinear vibration of a piezo‐flexomagnetic nanotube

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