A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams

doi:10.1080/15376494.2017.1410903

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A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams

Journal Article•DOI•

A nonlocal higher-order curved beam finite model including thickness stretching effect for bending analysis of curved nanobeams

T. Merzouki, M. Ganapathi¹, Olivier Polit²•Institutions (2)

VIT University¹, Paris West University Nanterre La Défense²

03 Apr 2019-Mechanics of Advanced Materials and Structures (Taylor & Francis)-Vol. 26, Iss: 7, pp 614-630

TL;DR: In this article, a finite element approach is developed for the static analysis of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a hig...

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Abstract: In the present work, a finite element approach is developed for the static analysis of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a hig...

...read moreread less

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Journal Article•DOI•

A review of size-dependent elasticity for nanostructures

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Mohammad Hosseini¹, Amin Hadi², Ahmad Malekshahi¹, Mohammad Shishesaz¹•Institutions (2)

Shahid Chamran University of Ahvaz¹, University of Tehran²

01 Jun 2018-Applied and Computational Mechanics

TL;DR: This review includes the last researches on bending, buckling, and vibration of nano-plates, nano-beams, nanorods, and nanotubes which were investigated by non-local elasticity theory and nonlocal strain gradient theory.

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Abstract: Nanotechnology is one of the pillars of human life in the future. This technology is growing fast and many scientists work in this field. The behavior of materials in nano size varies with that in macro dimension. Therefore scientists have presented various theories for examining the behavior of materials in nano-scale. Accordingly, mechanical behavior of nano-plates, nanotubes nano-beams and nano-rodes are being investigated by Non-classical elasticity theories. This review includes the last researches on bending, buckling, and vibration of nano-plates, nano-beams, nanorods, and nanotubes which were investigated by non-local elasticity theory and nonlocal strain gradient theory. Great scholars have written valuable reviews in the field of nanomechanics. Therefore, given a large number of researches and the prevention of repetition, the articles in the past year are reviewed.

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41 citations

Cites methods from "A nonlocal higher-order curved beam..."

...[51] developed a finite element approach for the static analysis of curved nanobeams using nonlocal elasticity beam theory based on Eringen formulation coupled with a higher-order shear deformation accounting for through-thickness stretching....
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Journal Article•DOI•

A review of mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled carbon nanotubes using Eringen’s nonlocal elasticity theory

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Chih Ping Wu¹, Jung Jen Yu¹•Institutions (1)

National Cheng Kung University¹

01 Sep 2019-Archive of Applied Mechanics

TL;DR: In this article, a review of various nonlocal beam and shell theories incorporating Eringen's nonlocal elasticity theory and the application of strong-and weak-form-based formulations to the current issue is presented.

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Abstract: This article is intended to present an overview of various mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled (SW-, DW-, and MW-) carbon nanotubes (CNTs) with combinations of simply supported, free, and clamped edge conditions embedded or non-embedded in an elastic medium, including bending, free vibration, buckling, coupled thermo-elastic and hygro-thermo-elastic, dynamic instability, wave propagation, geometric nonlinear bending, and large amplitude vibration analyses. This review introduces the development of various nonlocal beam and shell theories incorporating Eringen’s nonlocal elasticity theory and the application of strong- and weak-form-based formulations to the current issue. Based on the principle of virtual displacements and Reissner’s mixed variational theorem, the corresponding strong- and weak-form formulations of the local Timoshenko beam theory are reformulated for the free vibration analysis of rectangular nanobeams and SW-, DW-, and MW-CNTs, and presented for illustrative purposes. A comparative study of the results obtained using assorted nonlocal beam and shell theories in combination with the analytical and numerical methods is carried out.

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32 citations

Journal Article•DOI•

Recent Developments in the Dynamic Stability of Elastic Structures

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Ida Mascolo¹•Institutions (1)

University of Salerno¹

11 Oct 2019-Frontiers in Applied Mathematics and Statistics

TL;DR: This paper provides a survey of selected topics of current research interest and aims to collate the key recent developments and international trends, as well as describe any possible future challenges.

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Abstract: Dynamic instability in the mechanics of elastic structures is a fascinating topic, with many issues still unsettled. Accordingly, there is a wealth of literature examining the problems from different perspectives (analytical, numerical, experimental etc.), and coverings a wide variety of topics (bifurcations, chaos, strange attractors, imperfection sensitivity, tailor-ability, parametric resonance, conservative or non-conservative systems, linear or nonlinear systems, fluid-solid interaction, follower forces etc.). This paper provides a survey of selected topics of current research interest. It aims to collate the key recent developments and international trends, as well as describe any possible future challenges. A paradigmatic example of Ziegler's paradox on the destabilizing effect of small damping is also included.

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20 citations

Journal Article•DOI•

Vibration analysis of nonlocal beams using higher-order theory and comparison with classical models

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Aleksander Czekanski¹, V. V. Zozulya¹•Institutions (1)

York University¹

18 Jun 2021-Mechanics of Advanced Materials and Structures

TL;DR: In this article, a higher-order model for plane rods and beams based on the linear theory of nonlocal elasticity was developed, and the one-dimensional higher order theory is based on two-dimensional equatio...

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Abstract: New higher-order models are developed for plane rods and beams based on the linear theory of nonlocal elasticity. The one-dimensional higher-order theory is based on two-dimensional equatio...

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12 citations

Journal Article•DOI•

Free vibration analysis of Euler–Bernoulli curved beams using two-phase nonlocal integral models:

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Pei Zhang¹, Hai Qing¹•Institutions (1)

Nanjing University of Aeronautics and Astronautics¹

28 May 2021-Journal of Vibration and Control

TL;DR: Eringen's nonlocal elastic model has been widely applied to address the size-dependent response of micro-/nanostructures, which is observed in experimental tests and molecular dynamics simulation as discussed by the authors.

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Abstract: Eringen’s nonlocal elastic model has been widely applied to address the size-dependent response of micro-/nanostructures, which is observed in experimental tests and molecular dynamics simulation. ...

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12 citations

References

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Journal Article•DOI•

On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves

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A. Cemal Eringen

01 Sep 1983-Journal of Applied Physics

TL;DR: In this article, the integropartial differential equations of the linear theory of nonlocal elasticity are reduced to singular partial differential equations for a special class of physically admissible kernels.

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Abstract: Integropartial differential equations of the linear theory of nonlocal elasticity are reduced to singular partial differential equations for a special class of physically admissible kernels. Solutions are obtained for the screw dislocation and surface waves. Experimental observations and atomic lattice dynamics appear to support the theoretical results very nicely.

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3,929 citations

"A nonlocal higher-order curved beam..." refers background in this paper

...Conclusions The bending analysis of curved nanobeams is carried out using finite element approach developed based on Eringen’s constitutive equations for nonlocal analysis in conjunction with a higher-order shear deformation theory accounting for throughthickness stretching effect....
[...]
...[6] A. C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,” J. Appl....
[...]
...[4] A. C. Eringen, “Nonlocal polar elastic continua,” Int....
[...]
...Nonlocal elasticity theory has been introduced by Eringen [4–6] and Eringen and Edelen [5]....
[...]
...[5] A. C. Eringen andD....
[...]

Journal Article•DOI•

Indentation size effects in crystalline materials: A law for strain gradient plasticity

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William D. Nix¹, Huajian Gao¹•Institutions (1)

Stanford University¹

01 Mar 1998-Journal of The Mechanics and Physics of Solids

TL;DR: In this article, the indentation size effect for crystalline materials can be accurately modeled using the concept of geometrically necessary dislocations, which leads to the following characteristic form for the depth dependence of the hardness: H H 0 1+ h ∗ h where H is the hardness for a given depth of indentation, h, H 0 is a characteristic length that depends on the shape of the indenter, the shear modulus and H 0.

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Abstract: We show that the indentation size effect for crystalline materials can be accurately modeled using the concept of geometrically necessary dislocations. The model leads to the following characteristic form for the depth dependence of the hardness: H H 0 1+ h ∗ h where H is the hardness for a given depth of indentation, h, H0 is the hardness in the limit of infinite depth and h ∗ is a characteristic length that depends on the shape of the indenter, the shear modulus and H0. Indentation experiments on annealed (111) copper single crystals and cold worked polycrystalline copper show that this relation is well-obeyed. We also show that this relation describes the indentation size effect observed for single crystals of silver. We use this model to derive the following law for strain gradient plasticity: ( σ σ 0 ) 2 = 1 + l χ , where σ is the effective flow stress in the presence of a gradient, σ0 is the flow stress in the absence of a gradient, χ is the effective strain gradient and l a characteristic material length scale, which is, in turn, related to the flow stress of the material in the absence of a strain gradient, l ≈ b( μ σ 0 ) 2 . For materials characterized by the power law σ 0 = σ ref e 1 n , the above law can be recast in a form with a strain-independent material length scale l. ( builtσ σ ref ) 2 = e 2 n + l χ l = b( μ σ ref ) 2 = l ( σ 0 σ ref ) 2 . This law resembles the phenomenological law developed by Fleck and Hutchinson, with their phenomenological length scale interpreted in terms of measurable material parametersbl].

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3,655 citations

"A nonlocal higher-order curved beam..." refers methods in this paper

...Nix and Gao [1] have studied using model based on strain gradient theory, whereas coupled stress theory CONTACT O. Polit olivier.polit@u-paris.fr LEME, UPL, Univ. Paris Nanterre, rue de Sevres, Ville d’Avray, France....
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...References [1] W. D. Nix and H. Gao, “Indentation size effects in crystalline materials: A law for strain gradient plasticity,” J. Mech....
[...]
...[3] H. M. Ma, X. L. Gao, and J. N. Reddy, “A microstructure-dependent timoshenko beam model based on a modified couple stress theory,” J. Mech....
[...]
...Nix and Gao [1] have studied using model based on strain gradient theory, whereas coupled stress theory...
[...]

Journal Article•DOI•

On nonlocal elasticity

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A. Cemal Eringen¹, A. Cemal Eringen², D.G.B. Edelen¹, D.G.B. Edelen²•Institutions (2)

Lehigh University¹, Princeton University²

01 Mar 1972-International Journal of Engineering Science

TL;DR: In this article, a theory of non-local elasticity is presented via the vehicles of global balance laws and the second law of thermodynamics via the use of a localized Clausius-Duhem inequality and a variational statement of Gibbsian global thermodynamics.

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Abstract: Via the vehicles of global balance laws and the second law of thermodynamics, a theory of nonlocal elasticity is presented. Constitutive equations are obtained for the nonlinear theory, first through the use of a localized Clausius-Duhem inequality and second through a variational statement of Gibbsian global thermodynamics.

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2,201 citations

"A nonlocal higher-order curved beam..." refers background in this paper

...Conclusions The bending analysis of curved nanobeams is carried out using finite element approach developed based on Eringen’s constitutive equations for nonlocal analysis in conjunction with a higher-order shear deformation theory accounting for throughthickness stretching effect....
[...]
...[6] A. C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,” J. Appl....
[...]
...[4] A. C. Eringen, “Nonlocal polar elastic continua,” Int....
[...]
...Nonlocal elasticity theory has been introduced by Eringen [4–6] and Eringen and Edelen [5]....
[...]
...[5] A. C. Eringen andD....
[...]

Journal Article•DOI•

Nonlocal polar elastic continua

[...]

A. Cemal Eringen¹•Institutions (1)

Princeton University¹

01 Jan 1972-International Journal of Engineering Science

TL;DR: In this article, a continuum theory of non-local polar bodies is developed for nonlinear micromorphic elastic solids, and the balance laws and jump conditions are given.

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Abstract: A continuum theory of nonlocal polar bodies is developed. Both the micromorphic and the non-polar continuum theories are incorporated. The balance laws and jump conditions are given. By use of nonlocal thermodynamics and invariance under rigid motions, constitutive equations are obtained for the nonlinear micromorphic elastic solids. The special case, nonpolar, nonlocal elastic solids, is presented.

...read moreread less

1,788 citations

Journal Article•DOI•

Nonlocal theories for bending, buckling and vibration of beams

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J. N. Reddy¹, J. N. Reddy²•Institutions (2)

National University of Singapore¹, Texas A&M University²

01 Feb 2007-International Journal of Engineering Science

TL;DR: In this article, the Euler-Bernoulli, Timoshenko, Reddy, and Levinson beam theories are reformulated using the nonlocal differential constitutive relations of Eringen.

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Abstract: Various available beam theories, including the Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories, are reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived, and variational statements in terms of the generalized displacements are presented. Analytical solutions of bending, vibration and buckling are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies. The theoretical development as well as numerical solutions presented herein should serve as references for nonlocal theories of beams, plates, and shells.

...read moreread less

1,519 citations

"A nonlocal higher-order curved beam..." refers background in this paper

...[28] R. Aghababaei and J. N. Reddy, “Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates,” J. Sound Vib., vol. 326, pp. 277–289, 2009....
[...]
...[10] J. N. Reddy, “Nonlocal nonlinear formulation for bending of classical and shear deformation theories of beams and plates,” Int....
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...The higher-order beam theory is assumed by Reddy [8], Thai [15], Ansari and Sahmani [16], Challamel [21], and Aydogdu [22]....
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...The bending analysis of nanoplates is examined by Aghababaei and Reddy [28], Phadikar and Pradhan [29], Yan et al. [30], and Nguyen, et al. [31], whereas buckling study is investigated in Refs....
[...]
...[12] C. M. C. Roque, A. J. M. Ferreira, and J. N. Reddy, “Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method,” Int....
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