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A Refined Nonlocal Strain Gradient Theory for Assessing Scaling-Dependent Vibration Behavior of Microbeams

Authors: Xiaobai Li, Li Li, Yujin Hu, Weiming Deng, Zhe Ding

Abstract:

A size-dependent Euler–Bernoulli beam model, which accounts for nonlocal stress field, strain gradient field and higher order inertia force field, is derived based on the nonlocal strain gradient theory considering velocity gradient effect. The governing equations and boundary conditions are derived both in dimensional and dimensionless form by employed the Hamilton principle. The analytical solutions based on different continuum theories are compared. The effect of higher order inertia terms is extremely significant in high frequency range. It is found that there exists an asymptotic frequency for the proposed beam model, while for the nonlocal strain gradient theory the solutions diverge. The effect of strain gradient field in thickness direction is significant in low frequencies domain and it cannot be neglected when the material strain length scale parameter is considerable with beam thickness. The influence of each of three size effect parameters on the natural frequencies are investigated. The natural frequencies increase with the increasing material strain gradient length scale parameter or decreasing velocity gradient length scale parameter and nonlocal parameter.

Keywords: Euler-Bernoulli Beams, free vibration, higher order inertia, nonlocal strain gradient theory, velocity gradient.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1129209

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References:


[1] L. Li, Y. Hu, and L. Ling, “Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory,” Composite Structures, vol. 133, pp. 1079–1092, 2015.
[2] L. Li, Y. Hu, and L. Ling, “Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory,” Physica E: Low-dimensional Systems and Nanostructures, vol. 75, pp. 118–124, 2016.
[3] L. Li and Y. Hu, “Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory,” Computational Materials Science, vol. 112, pp. 282–288, 2016.
[4] P. Khodabakhshi and J. Reddy, “A unified beam theory with strain gradient effect and the von k´arm´an nonlinearity,” ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 2016.
[5] L. Li and Y. Hu, “Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material,” International Journal of Engineering Science, vol. 107, pp. 77–97, 2016.
[6] F. Ebrahimi and M. R. Barati, “Magneto-electro-elastic buckling analysis of nonlocal curved nanobeams,” The European Physical Journal Plus, vol. 131, no. 9, p. 346, 2016.
[7] L. Li and Y. Hu, “Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory,” International Journal of Engineering Science, vol. 97, pp. 84–94, 2015.
[8] F. Ebrahimi and M. R. Barati, “On nonlocal characteristics of curved inhomogeneous euler–bernoulli nanobeams under different temperature distributions,” Applied Physics A, vol. 122, no. 10, p. 880, 2016.
[9] F. Ebrahimi and M. R. Barati, “A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved fg nanobeams,” Composite Structures, vol. 159, pp. 174–182, 2017.
[10] S. Guo, Y. He, D. Liu, J. Lei, L. Shen, and Z. Li, “Torsional vibration of carbon nanotube with axial velocity and velocity gradient effect,” International Journal of Mechanical Sciences, 2016.
[11] D. Zhang, Y. Lei, and Z. Shen, “Vibration analysis of horn-shaped single-walled carbon nanotubes embedded in viscoelastic medium under a longitudinal magnetic field,” International Journal of Mechanical Sciences, vol. 118, pp. 219–230, 2016.
[12] L. Li, Y. Hu, and X. Li, “Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory,” International Journal of Mechanical Sciences, vol. 115, pp. 135–144, 2016.
[13] L. Li, X. Li, and Y. Hu, “Free vibration analysis of nonlocal strain gradient beams made of functionally graded material,” International Journal of Engineering Science, vol. 102, pp. 77–92, 2016.
[14] A. C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,” Journal of Applied Physics, vol. 54, no. 9, pp. 4703–4710, 1983.
[15] E. C. Aifantis, “On the role of gradients in the localization of deformation and fracture,” International Journal of Engineering Science, vol. 30, no. 10, pp. 1279–1299, 1992.
[16] R. D. Mindlin, “Micro-structure in linear elasticity,” Archive for Rational Mechanics and Analysis, vol. 16, no. 1, pp. 51–78, 1964.
[17] C. W. Lim, G. Zhang, and J. N. Reddy, “A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation,” Journal of the Mechanics and Physics of Solids, vol. 78, pp. 298–313, 2015.
[18] Y. Zhang, C. Wang, and N. Challamel, “Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model,” Journal of Engineering Mechanics, vol. 136, no. 5, pp. 562–574, 2009.
[19] J. Zang, B. Fang, Y.-W. Zhang, T.-Z. Yang, and D.-H. Li, “Longitudinal wave propagation in a piezoelectric nanoplate considering surface effects and nonlocal elasticity theory,” Physica E, vol. 63, pp. 147–150, 2014.
[20] B. Wang, Z. Deng, H. Ouyang, and J. Zhou, “Wave propagation analysis in nonlinear curved single-walled carbon nanotubes based on nonlocal elasticity theory,” Physica E, vol. 66, pp. 283–292, 2015.
[21] M. Z. Nejad, A. Hadi, and A. Rastgoo, “Buckling analysis of arbitrary two-directional functionally graded euler–bernoulli nano-beams based on nonlocal elasticity theory,” International Journal of Engineering Science, vol. 103, pp. 1–10, 2016. (Online). Available: http://www.sciencedirect.com/science/article/pii/S0020722516300180
[22] A. Daneshmehr, A. Rajabpoor, and A. Hadi, “Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories,” International Journal of Engineering Science, vol. 95, pp. 23–35, 2015.
[23] F. Ebrahimi and M. R. Barati, “A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams,” Arabian Journal Forence & Engineering, vol. 41, no. 5, pp. 1679–1690, 2015.
[24] N. Challamel and C. Wang, “The small length scale effect for a non-local cantilever beam: a paradox solved,” Nanotechnology, vol. 19, no. 34, p. 345703, 2008.
[25] J. Fern´andez-S´aez, R. Zaera, J. Loya, and J. Reddy, “Bending of euler–bernoulli beams using eringens integral formulation: a paradox resolved,” International Journal of Engineering Science, vol. 99, pp. 107–116, 2016.
[26] N. Challamel, Z. Zhang, C. Wang, J. Reddy, Q. Wang, T. Michelitsch, and B. Collet, “On nonconservativeness of eringens nonlocal elasticity in beam mechanics: correction from a discrete-based approach,” Archive of Applied Mechanics, vol. 84, no. 9-11, pp. 1275–1292, 2014.
[27] E. Benvenuti and A. Simone, “One-dimensional nonlocal and gradient elasticity: closed-form solution and size effect,” Mechanics Research Communications, vol. 48, pp. 46–51, 2013.
[28] F. Yang, A. Chong, D. Lam, and P. Tong, “Couple stress based strain gradient theory for elasticity,” International Journal of Solids and Structures, vol. 39, no. 10, pp. 2731–2743, 2002.
[29] N. Fleck, G. Muller, M. Ashby, and J. Hutchinson, “Strain gradient plasticity: theory and experiment,” Acta Metallurgica et Materialia, vol. 42, no. 2, pp. 475–487, 1994.
[30] N. Fleck and J. Hutchinson, “A phenomenological theory for strain gradient effects in plasticity,” Journal of the Mechanics and Physics of Solids, vol. 41, no. 12, pp. 1825 – 1857, 1993. (Online). Available: http://www.sciencedirect.com/science/article/pii/002250969390072N
[31] C. Polizzotto, “A gradient elasticity theory for second-grade materials and higher order inertia,” International Journal of Solids and Structures, vol. 49, no. 15, pp. 2121–2137, 2012.
[32] C. Polizzotto, “A second strain gradient elasticity theory with second velocity gradient inertia–part i: Constitutive equations and quasi-static behavior,” International Journal of Solids and Structures, vol. 50, no. 24, pp. 3749–3765, 2013.
[33] C. Polizzotto, “A second strain gradient elasticity theory with second velocity gradient inertia–part ii: Dynamic behavior,” International Journal of Solids and Structures, vol. 50, no. 24, pp. 3766–3777, 2013.
[34] H. Askes and E. C. Aifantis, “Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results,” International Journal of Solids and Structures, vol. 48, no. 13, pp. 1962–1990, 2011.
[35] H. Askes and E. C. Aifantis, “Gradient elasticity and flexural wave dispersion in carbon nanotubes,” Physical Review B, vol. 80, no. 19, p. 195412, 2009.
[36] S. Papargyri-Beskou, D. Polyzos, and D. Beskos, “Wave dispersion in gradient elastic solids and structures: a unified treatment,” International Journal of Solids and Structures, vol. 46, no. 21, pp. 3751–3759, 2009.
[37] D. Polyzos and D. Fotiadis, “Derivation of mindlins first and second strain gradient elastic theory via simple lattice and continuum models,” International Journal of Solids and Structures, vol. 49, no. 3, pp. 470–480, 2012.
[38] S. N. Iliopoulos, D. G. Aggelis, and D. Polyzos, “Wave dispersion in fresh and hardened concrete through the prism of gradient elasticity,” International Journal of Solids and Structures, vol. 78, pp. 149–159, 2016.