Bending and buckling of thin strain gradient elastic beams

Lazopoulos, K.A.; Lazopoulos, Α.Κ.

doi:10.1016/j.euromechsol.2010.04.001

Cited by 147 publications

(73 citation statements)

References 16 publications

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“…Hence for the sake of simplicity, similar governing equation and boundary conditions are considered here. Before Lazopoulos and Lazopoulos (2010), governing equation for Euler-Bernoulli beam had been developed by Papargyri-Beskou et al (2003). But, one higher stress component which was missed in their paper has been considered by Lazopoulos and Lazopoulos (2010).…”

Section: Strain Gradient Formulations Of Nano-beams (Formulation Ii)mentioning

confidence: 99%

“…Before Lazopoulos and Lazopoulos (2010), governing equation for Euler-Bernoulli beam had been developed by Papargyri-Beskou et al (2003). But, one higher stress component which was missed in their paper has been considered by Lazopoulos and Lazopoulos (2010). The governing equations and boundary conditions without considering surface effect based on Lazopoulos and Lazopoulos (2010) work are expressed as follows:…”

Section: Strain Gradient Formulations Of Nano-beams (Formulation Ii)mentioning

confidence: 99%

“…The strain gradient Euler-Bernoulli beam formulation has been derived by Lazopoulos and Lazopoulos (2010). Hence for the sake of simplicity, similar governing equation and boundary conditions are considered here.…”

Section: Strain Gradient Formulations Of Nano-beams (Formulation Ii)mentioning

confidence: 99%

“…While, in another paper (Peddieson et al, 2003) it has been stated that the governing equations of each beam segment which is not acted upon by a distributed load has the same local or classical governing equation. Also, in a paper by Lazopoulos and Lazopoulos (2010) there is a contradiction on boundary conditions which deviates the physical interpretation. For instance, based on their results, when the dimensions of a simply supported nano-beam are large enough, the displacement field does not converge to the classical response of the beam.…”

Section: Introductionmentioning

confidence: 99%

“…In recent years, many works have been published on the analysis of an Euler-Bernoulli microbeam based on the simplified strain gradient elasticity (Lazopoulos and Lazopoulos, 2010;Rajabi and Ramezani, 2011;Lazopoulos, 2013), the modified strain gradient elasticity (Akgöz and Civalek, 2011) and modified couple stress theories (Park and Gao, 2006;Akgöz and Civalek, 2011). Furthermore, analysis of the Timoshenko beam model using the modified couple stress theory (Ma et al, 2008;Gao, 2014) and the modified strain gradient elasticity (Wang et al, 2010; Latin American Journal of Solids and Structures 12 (2015) 2208-2230 al., 2012; Kahrobaiyan et al, 2014) has been investigated and the Reddy-Levinson beam theory based on modified couple stress theory (Ma et al, 2010) has been studied.…”

Section: Introductionmentioning

confidence: 99%

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Determination of the Appropriate Gradient Elasticity Theory for Bending Analysis of Nano-beams by Considering Boundary Conditions Effect

Shokrieh

Zibaei

2015

Lat. Am. j. solids struct.

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In the present paper, a critique study on some models available in the literature for bending analysis of nano-beams using the gradient elasticity theory is accomplished. In nonlocal elasticity models of nano-beams, the size effect has not been properly considered in governing equations and boundary conditions. It means that in these models, because of replacing of the size effect with the inertia gradient effect, the size dependency has been ignored in bending analysis of nano-beams. Therefore, as the beam dimensions increase in comparison to its material length scale parameter, the obtained solution based on the gradient elasticity theory (either in the nonlocal elasticity theory or the strain gradient elasticity theory) should converge to the classical elasticity solution. Hence, satisfying of boundary conditions is a crucial point. In this paper, governing equations and boundary conditions are presented based on two gradient elasticity theories (i.e., nonlocal elasticity and strain gradient elasticity theories). Also, boundary conditions in strain gradient elasticity theory are modified based on a dimensional analysis approach. The results indicate that the strain gradient elasticity theory captures the size effect more sensitive in comparison with the nonlocal elasticity theory in bending analysis. In addition, modified boundary conditions in strain gradient elasticity theory can lead to converge the classical solution at large scales. To prove that the boundary conditions of nano-beam have the direct effect on mechanical behavior of structure, the sizedependent Young modulus of carbon nanotube (CNT) is investigated and the results show that the prediction of strain gradient elasticity theory with modified boundary conditions is in a good agreement with experimental results.

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Section: Strain Gradient Formulations Of Nano-beams (Formulation Ii)mentioning

confidence: 99%

Section: Strain Gradient Formulations Of Nano-beams (Formulation Ii)mentioning

confidence: 99%

Section: Strain Gradient Formulations Of Nano-beams (Formulation Ii)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Determination of the Appropriate Gradient Elasticity Theory for Bending Analysis of Nano-beams by Considering Boundary Conditions Effect

Shokrieh

Zibaei

2015

Lat. Am. j. solids struct.

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Bibliography

2012

Carbon Nanotubes and Nanosensors

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Discontinuous Galerkin FEMs for a gradient beam in static carbon nanotube applications

Eptaimeros

Koutsoumaris

2021

Math Methods in App Sciences

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The engineering applications of the innovative materials, such as carbon nanotubes (CNTs) and graphene sheets, constitute a developing branch of the modern science. CNTs, in particular, exhibit extraordinarily mechanical properties capable of making them a reliable material for the above applications. The simulation of the mechanical response of a CNT is effectively conducted by a generalized beam model. This work investigates the response of a CNT, subjected to a static loading, by means of a gradient beam model. Given that the existence of a boundary layer is demonstrated by the analytical solutions of the aforementioned problem, the interior penalty discontinuous Galerkin finite element methods (IPDGFEMs) are crucial to its solution. The hp‐version IPDGFEMs are developed in this study for the solution of a static gradient elastic beam in bending, derived from two different equilibrium formulations, for the first time. An a priori error analysis is also performed for the above method, and numerical simulations are then carried out. A comparison is finally drawn between the exact deflection and those of the IPDGFEM and the conforming C2‐continuous finite element method (C2CFEM) for a number of discretizations and each length scale that is investigated. The deduced results highlight the suitability, the efficiency, and the accuracy of the investigated models over the already existing ones, and they have considerable significance for the engineering design in the range of micro and nanodimensions.

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Bending and buckling of thin strain gradient elastic beams

Cited by 147 publications

References 16 publications

Determination of the Appropriate Gradient Elasticity Theory for Bending Analysis of Nano-beams by Considering Boundary Conditions Effect

Determination of the Appropriate Gradient Elasticity Theory for Bending Analysis of Nano-beams by Considering Boundary Conditions Effect

Bibliography

Discontinuous Galerkin FEMs for a gradient beam in static carbon nanotube applications

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