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International Journal for Multiscale Computational Engineering

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A STRAIN-DIFFERENCE BASED NONLOCAL ELASTICITY THEORY FOR SMALL-SCALE SHEAR-DEFORMABLE BEAMS WITH PARAMETRIC WARPING

Volume 18, Issue 1, 2020, pp. 83-102
DOI: 10.1615/IntJMultCompEng.2019030885
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ABSTRACT

The strain-difference based nonlocal elasticity theory devised by the authors is applied to homogeneous isotropic beams subjected to static loads. Shear deformation is taken into account and a warping parameter ω is used to fix the warping shape of the cross sections. On letting ω vary from zero to infinity, a continuous family of beam models is generated, which spans from the Euler-Bernoulli beam (ω = 0) to the Timoshenko beam (ω → ∞), and identifies itself with the Reddy beam for ω = 2. Taking as basic unknowns the axial stretching e, the Euler-Bernoulli curvature χEB, and the shear curvature η, the boundary-value problem proves to be governed by three uncoupled integral equations whose input terms contain, besides the load data, eight arbitrary constants. These equations are solved by addressing a set of eight uncoupled auxiliary integral equations independent of the boundary conditions, each of which is either a Fredholm integral equation of the second kind, or is more complex but has strong similarities with the latter type of equation. This makes it possible to express (e, χEB, η), the axial and transverse displacements (u, w), and the shear angle γ to within the mentioned constants, which is useful to enforce the eight boundary conditions. The numerical solutions for simple beam problems are reported and graphically illustrated with particular concern for size effects and for their sensitivity to shear deformation.

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CITED BY
  1. Pisano Aurora A., Fuschi Paolo, Polizzotto Castrenze, Shear Effects in Elastic Nanobeams, in Proceedings of XXIV AIMETA Conference 2019, 2020. Crossref

  2. Moghtaderi Saeed H., Faghidian S. Ali, Asghari Mohsen, Nonlinear vibrations of gradient and nonlocal elastic nano-bars, Mechanics Based Design of Structures and Machines, 2020. Crossref

  3. Faghidian S Ali, Flexure mechanics of nonlocal modified gradient nano-beams, Journal of Computational Design and Engineering, 8, 3, 2021. Crossref

  4. Faghidian S. Ali, Ghavanloo Esmaeal, Unified higher-order theory of two-phase nonlocal gradient elasticity, Meccanica, 56, 3, 2021. Crossref

  5. Pisano A. A., Fuschi P., Polizzotto C., Euler–Bernoulli elastic beam models of Eringen’s differential nonlocal type revisited within a $$\mathbf{C }^{0}-$$continuous displacement framework, Meccanica, 56, 9, 2021. Crossref

  6. Faghidian S. Ali, Contribution of nonlocal integral elasticity to modified strain gradient theory, The European Physical Journal Plus, 136, 5, 2021. Crossref

  7. Pisano Aurora Angela, Fuschi Paolo, Polizzotto Castrenze, Integral and differential approaches to Eringen's nonlocal elasticity models accounting for boundary effects with applications to beams in bending, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 101, 8, 2021. Crossref

  8. De Domenico Dario, Ricciardi Giuseppe, Askes Harm, A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type—part II: boundary-value problems in the one-dimensional case, Meccanica, 56, 3, 2021. Crossref

  9. De Domenico Dario, Ricciardi Giuseppe, Askes Harm, A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type: part I—analytical formulation and thermodynamic framework, Meccanica, 56, 3, 2021. Crossref

  10. Polizzotto Castrenze, Fuschi Paolo, Pisano Aurora Angela, Strain-Difference Based Nonlocal Elasticity Theories: Formulations and Obtained Results, in 50+ Years of AIMETA, 2022. Crossref

  11. Polizzotto Castrenze, Elishakoff Isaac, Fuschi Paolo, Shear Deformable Elastic Beam Models in Vibration and Sensitivity of Natural Frequencies to Warping Effects, in Recent Approaches in the Theory of Plates and Plate-Like Structures, 151, 2022. Crossref

  12. Fazlali Mahdad, Moghtaderi Saeed H, Faghidian S Ali, Nonlinear flexure mechanics of beams: stress gradient and nonlocal integral theory, Materials Research Express, 8, 3, 2021. Crossref

  13. Faghidian S. Ali, Mohammad-Sedighi Hamid, Dynamics of nonlocal thick nano-bars, Engineering with Computers, 38, 3, 2022. Crossref

  14. Polizzotto C., Fuschi P., Pisano A.A., A 2D warping theory for shear deformable elastic beams of axisymmetric cross section in flexure, European Journal of Mechanics - A/Solids, 96, 2022. Crossref

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