Static analysis of tapered nanowires based on nonlocal Euler-Bernoulli beam theory via differential quadrature method

Janghorban, Maziar

doi:10.1590/s1679-78252012000200007

Cited by 12 publications

(5 citation statements)

References 21 publications

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“…By applying the relation (18) at " S ¼ 1 and then using the condition (39), the unknown constant C can be obtained as…”

Section: Fixed-free Nano-beamsmentioning

confidence: 99%

“…where z is the rotation at the in°ection point. By enforcing (52) together with (18), the constant C is obtained, for this particular case, as…”

Section: Fixed-rollered Nano-beamsmentioning

confidence: 99%

“…By using the relations (15), (18), (53) along with the natural boundary condition " mð1Þ ¼ " m 0 , it leads to "…”

Section: Fixed-rollered Nano-beamsmentioning

confidence: 99%

“…However, most of existing studies were found limited to certain nano-scale in°uences. For instance, the classical beam theory enhanced by Eringen nonlocal constitutive law was proposed to predict the buckling loads, post-buckling shapes, bending and vibration responses of nano-rods/tubes/ribbons, 31,34,44,48 nano-wires, 18 nano-elements and nano-beams. 5,6,11,22,30,41,43,45 Results from those studies indicated that nonlocal parameters play a crucial role on both value and size-dependent behavior of predicted solutions when the characteristic length of elements falls within the nano-scale range.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear Analysis for Bending, Buckling and Post-buckling of Nano-Beams with Nonlocal and Surface Energy Effects

Nguyen

Reddy

Rungamornrat

et al. 2019

Int. J. Str. Stab. Dyn.

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The modeling and analysis for mechanical response of nano-scale beams undergoing large displacements and rotations are presented. The beam element is modeled as a composite consisting of the bulk material and the surface material layer. Both Eringen nonlocal elasticity theory and Gurtin–Murdoch surface elasticity theory are adopted to formulate the moment–curvature relationship of the beam. In the formulation, the pre-existing residual stress within the bulk material, induced by the residual surface tension in the material layer, is also taken into account. The resulting moment-curvature relationship is then utilized together with Euler–Bernoulli beam theory and the elliptic integral technique to establish a set of exact algebraic equations governing the displacements and rotations at the ends of the beam. The linearized version of those equations is also established and used in the derivation of a closed-form solution of the buckling load of nano-beams under various end conditions. A discretization-free solution procedure based mainly upon Newton iterative scheme and a selected numerical quadrature is developed to solve a system of fully coupled nonlinear equations. It is demonstrated that the proposed technique yields highly accurate results comparable to the benchmark analytical solutions. In addition, the nonlocal and surface energy effects play a significant role on the predicted buckling load, post-buckling and bending responses of the nano-beam. In particular, the presence of those effects remarkably alters the overall stiffness of the beam and predicted solutions exhibit strong size-dependence when the characteristic length of the beam is comparable to the intrinsic length scale of the material surface.

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“…By applying the relation (18) at " S ¼ 1 and then using the condition (39), the unknown constant C can be obtained as…”

Section: Fixed-free Nano-beamsmentioning

confidence: 99%

“…where z is the rotation at the in°ection point. By enforcing (52) together with (18), the constant C is obtained, for this particular case, as…”

Section: Fixed-rollered Nano-beamsmentioning

confidence: 99%

“…By using the relations (15), (18), (53) along with the natural boundary condition " mð1Þ ¼ " m 0 , it leads to "…”

Section: Fixed-rollered Nano-beamsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear Analysis for Bending, Buckling and Post-buckling of Nano-Beams with Nonlocal and Surface Energy Effects

Nguyen

Reddy

Rungamornrat

et al. 2019

Int. J. Str. Stab. Dyn.

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show abstract

“…The nonlocal elasticity theory was initiated by Eringen (1972Eringen ( , 1983Eringen and Wegner, 2003) and Eringen and Edelen (1972). In recent decades, many researchers have used the nonlocal elasticity theory to analyze mechanical behaviors of micro and nano-beams in static loading conditions (Peddieson et al, 2003;Reddy, 2007;Wang and Liew, 2007;Kiani, 2010aKiani, , 2010bJanghorban, 2012).…”

Section: Introductionmentioning

confidence: 99%

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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Analysis of axially functionally graded nano-tapered Timoshenko beams by element-based Bernstein pseudospectral collocation (EBBPC)

Rajasekaran

2017

Engineering with Computers

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Static analysis of tapered nanowires based on nonlocal Euler-Bernoulli beam theory via differential quadrature method

Cited by 12 publications

References 21 publications

Nonlinear Analysis for Bending, Buckling and Post-buckling of Nano-Beams with Nonlocal and Surface Energy Effects

Nonlinear Analysis for Bending, Buckling and Post-buckling of Nano-Beams with Nonlocal and Surface Energy Effects

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

Analysis of axially functionally graded nano-tapered Timoshenko beams by element-based Bernstein pseudospectral collocation (EBBPC)

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