Bending and buckling of nonlocal strain gradient elastic beams

Xu, Xiaojian; Wang, Xuancang; Zheng, Min; Ma, Zheng

doi:10.1016/j.compstruct.2016.10.038

Cited by 136 publications

(32 citation statements)

References 98 publications

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“…The conditions alluded in table 1 can satisfy the geometrical and force boundary conditions given in table 2. The stress resultants in boundaries can be written in the nonlocal strain gradient forms as force boundary conditions [57]. There are also other references in which the nonlocal boundary conditions are presented [58][59][60].…”

Section: Rayleigh-ritz Solution Processmentioning

confidence: 99%

Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

Malikan

Eremeyev

2020

Mater. Res. Express

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This research predicts theoretically post-critical axial buckling behavior of truncated conical carbon nanotubes (CCNTs) with several boundary conditions by assuming a nonlinear Winkler matrix. The post-buckling of CCNTs has been studied based on the Euler-Bernoulli beam model, Hamilton's principle, Lagrangian strains, and nonlocal strain gradient theory. Both stiffness-hardening and stiffness-softening properties of the nanostructure are considered by exerting the second stressgradient and second strain-gradient in the stress and strain fields. Besides small-scale influences, the surface effect is also taken into consideration. The effect of the Winkler foundation is nonlinearly taken into account based on the Taylor expansion. A new admissible function is used in the Rayleigh-Ritz solution technique applicable for buckling and post-buckling of nanotubes and nanobeams. Numerical results and related discussions are compared and reported with those obtained by the literature. The significant results proved that the surface effect and the nonlinear term of the substrate affect the CCNT considerably.

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Section: Rayleigh-ritz Solution Processmentioning

confidence: 99%

Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

Malikan

Eremeyev

2020

Mater. Res. Express

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“…Non-standard essential boundary conditions, i.e. w 00 0 ð Þ ¼ w 00 1 ð Þ ¼ 0, have been selected as in [39]. The integral constitutive equations read…”

Section: Governing Equation Standard and Nonstandard Boundary Conditmentioning

confidence: 99%

“…The new formulation allows to analyse stiffer (or more flexible) structures than those corresponding to the classical case, appropriately selecting the two scale-dependent material parameters included in the model. The theory has become very popular, and nowadays it is being currently used by a large number of authors, see [36][37][38][39][40][41][42][43][44][45] just to mention a few. When it is applied to bounded domains, the fulfilment of both standard and non-standard boundary conditions, together with the constitutive boundary conditions are required.…”

Section: Introductionmentioning

confidence: 99%

“…When it is applied to bounded domains, the fulfilment of both standard and non-standard boundary conditions, together with the constitutive boundary conditions are required. In some of the quoted papers the solution only satisfies the standard and the non-standard boundary conditions, dismissing the constitutive ones [36,37,38,39,42,45], meanwhile in other works only satisfies the standard and the constitutive boundary conditions, dismissing the non-standard ones [40,41,43,44]. In this respect Zaera et al [46] showed the inconsistency of the NSGT when applied to finite solids.…”

Section: Introductionmentioning

confidence: 99%

“…In order to stress the aforementioned inconsistency, in the present work we point out the consequences of dismissing certain mandatory boundary conditions, either non-standard or constitutive, to derive an apparently well-posed problem from a factually overconstrained problem. In particular, we analyse the bending behaviour of a simply supported Bernoulli-Euler beam subjected to an uniformly distributed static load, which was considered in different works [39,44,46]. Through this straightforward example we show that: (i) if constitutive boundary conditions are omitted, the equilibrium fields are incompatible with the constitutive equations of the material, as it was pointed out by Barretta and de Sciarra [40]; (ii) if non-standard boundary conditions are omitted, the energy conservation principle is infringed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Non-standard and constitutive boundary conditions in nonlocal strain gradient elasticity

2020

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Zaera et al. (Int J Eng Sci 138:65-81, 2019) recently showed that the nonlocal strain gradient theory (NSGT) is not consistent when it is applied to finite solids, since all boundary conditions associated to the corresponding problems cannot be simultaneously satisfied. Given the large number of works using the NSGT being currently published in the field of generalized continuum mechanics, it is pertinent to evince the shortcomings of the application of this theory. Some authors solved the problem omitting the constitutive boundary conditions. In the current paper we show that, in this case, the equilibrium fields are not compatible with the constitutive equation of the material. Other authors solved it omitting the nonstandard boundary conditions. Here we show that, in this case, the solution does not fulfil conservation of energy. In conclusion, the inconsistency of the NSGT is corroborated, and its application must be prevented in the analysis of the mechanical behaviour of nanostructures.

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Two‐phase local/nonlocal gradient mechanics of elastic torsion

Faghidian

2020

Math Methods in App Sciences

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The higher order two‐phase local/nonlocal elasticity model and the higher order strain gradient theory are unified via an abstract variational scheme. The higher order constitutive integral convolution is established in a consistent variational framework governed by ad hoc functional space of test fields. Equivalent differential constitutive law equipped with nonclassical boundary conditions of constitutive type is determined. The proposed higher order elasticity theory provides as special cases a range of well‐known size‐dependent elasticity models such as nonlocal, two‐phase local/nonlocal, strain gradient, modified nonlocal strain gradient, and nonlocal strain‐driven gradient models. Evidences of well‐posedness of the introduced higher order two‐phase local/nonlocal gradient problems are elucidated by rigorous examination of the elastostatic torsional response of structural schemes of applicative interest in nano‐mechanics. The exact analytical solution of the torsion problem of elastic nano‐beams is derived, graphically demonstrated, and compared with analogous outcomes in the literature. The conceived higher order elasticity theory can efficiently characterize advanced nano‐materials and structural elements of modern nano‐systems.

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Bending and buckling of nonlocal strain gradient elastic beams

Cited by 136 publications

References 98 publications

Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

Post-critical buckling of truncated conical carbon nanotubes considering surface effects embedding in a nonlinear Winkler substrate using the Rayleigh-Ritz method

Non-standard and constitutive boundary conditions in nonlocal strain gradient elasticity

Two‐phase local/nonlocal gradient mechanics of elastic torsion

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