Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Meng, Licheng; Lai, Hairong; Xie, Zhijun; Zou, Dajun; Guo, Zhen; He, Xingui; Gao, Cun‐Fa

doi:10.1007/s10483-018-2395-9

Cited by 18 publications

(4 citation statements)

References 35 publications

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“…The correspondent boundary conditions are given by (11a), (12b) and the pair of additional conditions (18). In this case, we arrive at γ 1 = 2 and γ 2 = 3 for the left and for the right boundary layer, respectively.…”

Section: Beam With Clamped and Simply Supported Endsmentioning

confidence: 92%

“…Let both beam ends be simply supported (S-S conditions), as given by the boundary conditions (12) rewritten in differential form (16a,16b), together with the additional constraints (18). Substituting the expansions (21,24,25) into these conditions, we determine the strength of either boundary layer γ 1 = γ 2 =…”

Section: Beam With Simply Supported Endsmentioning

confidence: 99%

“…This model is immune from the inconsistencies of the PNLM and it has been adopted to solve the problem of static bending [33] and buckling [36] of Euler-Bernoulli (E-B) beams. Static axial deformation of a beam is considered in [26,37], while semi-analytical solutions for the combined action of axial and flexural static loadings is given in [18]. Axial and flexural free vibrations of beams have also been considered in [19] and in [9].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory

Mikhasev

Nobili

2020

International Journal of Solids and Structures

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Section: Beam With Clamped and Simply Supported Endsmentioning

confidence: 92%

Section: Beam With Simply Supported Endsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory

Mikhasev

Nobili

2020

International Journal of Solids and Structures

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“…With the help of Laplace transform, Zhang et al [37] and Zhang and Qing [38] utilized the two-phase local/nonlocal integral model directly to study the static bending of curved Euler-Bernoulli and Timoshenko beams. Meng et al [39] deduced the semi-analytic solution for static bending Euler-Bernoulli beam with the axial force. A consistent softening effect was obtained while using the two-phase strain-driven local/nonlocal integral model.…”

Section: Introductionmentioning

confidence: 99%

On the consistency of two-phase local/nonlocal piezoelectric integral model

Ren

Qing

2021

Appl. Math. Mech.-Engl. Ed.

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In this paper, we propose general strain- and stress-driven two-phase local/nonlocal piezoelectric integral models, which can distinguish the difference of nonlocal effects on the elastic and piezoelectric behaviors of nanostructures. The nonlocal piezoelectric model is transformed from integral to an equivalent differential form with four constitutive boundary conditions due to the difficulty in solving intergro-differential equations directly. The nonlocal piezoelectric integral models are used to model the static bending of the Euler-Bernoulli piezoelectric beam on the assumption that the nonlocal elastic and piezoelectric parameters are coincident with each other. The governing differential equations as well as constitutive and standard boundary conditions are deduced. It is found that purely strain- and stress-driven nonlocal piezoelectric integral models are ill-posed, because the total number of differential orders for governing equations is less than that of boundary conditions. Meanwhile, the traditional nonlocal piezoelectric differential model would lead to inconsistent bending response for Euler-Bernoulli piezoelectric beam under different boundary and loading conditions. Several nominal variables are introduced to normalize the governing equations and boundary conditions, and the general differential quadrature method (GDQM) is used to obtain the numerical solutions. The results from current models are validated against results in the literature. It is clearly established that a consistent softening and toughening effects can be obtained for static bending of the Euler-Bernoulli beam based on the general strain- and stress-driven local/nonlocal piezoelectric integral models, respectively.

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Exact and asymptotic bending analysis of microbeams under different boundary conditions using stress‐derived nonlocal integral model

2019

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Eringen's nonlocal differential elasticity is widely applied to model nano‐ and micro‐structures, although previous studies have shown some inconsistences, e.g., the nonlocal parameter maybe has no effect on the deflection of Euler‐Bernoulli and Timoshenko beams subjected to some kinds of boundary and loading conditions. In this paper, the static bending analysis of Euler‐Bernoulli and Timoshenko beams is performed with the application of stress‐driven nonlocal integral model. The Fredholm type integral constitutive equations of the first kind are transformed to Volterra integral equations of the first kind through simply adjusting the limit of integrals, and the general solutions to the deflection as well as bending moment and so on are derived and obtained explicitly through solving the integro‐differential governing equations with the Laplace transformation. Exact solutions are derived explicitly for different loading and boundary conditions, especially for the paradoxical beam problem while using nonlocal differential model. The analytical and asymptotic expressions of the beam deflections obtained in this paper are validated against to those existing analytical and numerical results in literature. It is clearly established that, with the application of the stress‐driven nonlocal integral model, a consistent toughening effect can be obtained for both Euler‐Bernoulli and Timoshenko beams.

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Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Cited by 18 publications

References 35 publications

On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory

On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory

On the consistency of two-phase local/nonlocal piezoelectric integral model

Exact and asymptotic bending analysis of microbeams under different boundary conditions using stress‐derived nonlocal integral model

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