The couple stress analysis of Timoshenko micro-beams based on new considerations
Document Type : Research Paper
Authors
Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran.
Abstract
In this paper, a new approach is proposed for the couple stress analysis of micro-beams. As the main assumption, power series expansions are assumed for the axial displacement. The lateral and transverse displacements are adopted according to the classical beam theories. It is demonstrated that this consideration imposes a decisive constraint of skew-symmetry on the couple-stress tensor. So, in the case of micro-beams, there is no need for referring to the main arguments in modified couple stress theory (M-CST). This approach also allows for revising the conventional boundary conditions in couple stress analysis of micro-beams. For the special case of Timoshenko micro-beams, the axial displacement is approximated by a first-order polynomial and a new set of boundary conditions similar to the classical model is developed. Benchmark problems are then considered for demonstrating the advantages of the proposed model.
Keywords
[1] A. Hadjesfandiari, G. Dargush, Couple stress theory for solids, International Journal of Solids and Structures, Vol. 48, No. 18, pp. 2496-2510, 2011.
[2] W. Koiter, Couple stresses in the theory of elasticity I,II, Proceedings of the Koninklijke Nederlandse Akademie Van Wetenschappen, Series BB Vol. 67, pp. 17-44, 1964.
[3] S. Park, X. Gao, Variational formulation of a modified couple stress theory and its application to a simple shear problem, Zeitschrift für angewandte Mathematik und Physik, Vol. 59, No. 5, pp. 904-917, 2008.
[4] R. Mindlin, H. Tiersten, Effects of couple stresses in linear elasticity, Archive for Rational Mechanics and Analysis, Vol. 11, pp. 415-447, 1962.
[5] R. Toupin, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, Vol. 11, pp. 385-414, 1962.
[6] A. Eringen, Theory of micropolar plates, Zeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 18, No. 1, pp. 12-30, 1967.
[7] S. Ramezani, R. Naghdabadi, S. Sohrabpour, Analysis of micropolar elastic beams, European Journal of Mechanics-A/Solids, Vol. 28, No. 2, pp. 202-208, 2009.
[8] H. Ma, X. Gao, J. Reddy, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, Vol. 56, No. 12, pp. 3379-3391, 2008.
[9] D. Steigmann, Equilibrium of prestressed networks, IMA journal of applied mathematics, Vol. 48, No. 2, pp. 195-215, 1992.
[10] D. Lam, F. Yang, A. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, Vol. 51, No. 8, pp. 1477-1508, 2003.
[11] E. Alavi, M. Sadighi, M. Pazhooh, J. Ganghoffer, Development of size-dependent consistent couple stress theory of Timoshenko beams, Applied Mathematical Modelling, Vol. 79, pp. 685-712, 2020.
[12] M. Asghari, M. Kahrobaiyan, M. Rahaeifard, M. Ahmadian, Investigation of the size effects in Timoshenko beams based on the couple stress theory, Archive of Applied Mechanics, Vol. 81, No. 7, pp. 863-874, 2011.
[13] S. Park, X. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, Vol. 16, No. 11, pp. 2355, 2006.
[14] T. Kaneko, On Timoshenko's correction for shear in vibrating beams, Journal of Physics D: Applied Physics, Vol. 8, No. 16, pp. 1927, 1975.
[15] M. Kahrobaiyan, M. Asghari, M. Ahmadian, A Timoshenko beam element based on the modified couple stress theory, International Journal of Mechanical Sciences, Vol. 79, pp. 75-83, 2014.
[16] M. Mohammadi, A. Farajpour, A. Rastgoo, Coriolis effects on the thermo-mechanical vibration analysis of the rotating multilayer piezoelectric nanobeam, Acta Mechanica, Vol. 234, No. 2, pp. 751-774, 2023/02/01, 2023.
[17] M. Mohammadi, A. Farajpour, A. Moradi, M. Hosseini, Vibration analysis of the rotating multilayer piezoelectric Timoshenko nanobeam, Engineering Analysis with Boundary Elements, Vol. 145, pp. 117-131, 2022.
[18] M. Mohammadi, A. Rastgoo, Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium, Mechanics of Advanced Materials and Structures, Vol. 27, No. 20, pp. 1709-1730, 2020.
[19] M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, A. Rastgoo, Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads, European Journal of Mechanics-A/Solids, Vol. 77, pp. 103793, 2019.
[20] M. Mohammadi, A. Rastgoo, Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core, Structural Engineering and Mechanics, An Int'l Journal, Vol. 69, No. 2, pp. 131-143, 2019.
[21] A. Farajpour, A. Rastgoo, M. Mohammadi, Vibration, buckling and smart control of microtubules using piezoelectric nanoshells under electric voltage in thermal environment, Physica B: Condensed Matter, Vol. 509, pp. 100-114, 2017.
[22] A. Farajpour, M. H. Yazdi, A. Rastgoo, M. Loghmani, M. Mohammadi, Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates, Composite Structures, Vol. 140, pp. 323-336, 2016.
[23] A. Farajpour, M. H. Yazdi, A. Rastgoo, M. Mohammadi, A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment, Acta Mechanica, Vol. 227, pp. 1849-1867, 2016.
[24] M. Mohammadi, M. Safarabadi, A. Rastgoo, A. Farajpour, Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment, Acta Mechanica, Vol. 227, pp. 2207-2232, 2016.
[25] M. Goodarzi, M. Mohammadi, M. Khooran, F. Saadi, Thermo-mechanical vibration analysis of FG circular and annular nanoplate based on the visco-pasternak foundation, Journal of Solid Mechanics, Vol. 8, No. 4, pp. 788-805, 2016.
[26] M. Safarabadi, M. Mohammadi, A. Farajpour, M. Goodarzi, Effect of surface energy on the vibration analysis of rotating nanobeam, 2015.
[27] M. Mohammadi, A. Farajpour, A. Moradi, M. Ghayour, Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites Part B: Engineering, Vol. 56, pp. 629-637, 2014.
[28] S. Asemi, A. Farajpour, M. Mohammadi, Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory, Composite Structures, Vol. 116, pp. 703-712, 2014.
[29] M. Mohammadi, M. Ghayour, A. Farajpour, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites Part B: Engineering, Vol. 45, No. 1, pp. 32-42, 2013.
[30] M. Mohammadi, M. Goodarzi, M. Ghayour, A. Farajpour, Influence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory, Composites Part B: Engineering, Vol. 51, pp. 121-129, 2013.
[31] W. Xia, L. Wang, L. Yin, Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration, International Journal of Engineering Science, Vol. 48, No. 12, pp. 2044-2053, 2010.
)
Volume 54, Issue 1
March 2023Pages 19-35
- Receive Date: 26 November 2022
- Revise Date: 25 December 2022
- Accept Date: 29 December 2022