Free Vibration Analysis of Sandwich Micro Beam with Piezoelectric Based on Modified Couple Stress Theory and Surface Effects

الموضوعات : فصلنامه شبیه سازی و تحلیل تکنولوژی های نوین در مهندسی مکانیک

Mohammad khajekhabaz ¹ (Khomeinishahr branch, Islamic Azad Uni., Esfahan, Iran.)
Ali Eftekhari ² (Khomeinishahr branch, Islamic Azad Uni., Esfahan, Iran.)
Mohammad Hashemian ³ (Islamic Azad University, Khomeinishahr Branch)

تاريخ الإرسال : 27 الأحد , ربيع الثاني, 1439 تاريخ التأكيد : 27 الإثنين , رمضان, 1439 تاريخ الإصدار : 27 الأربعاء , شوال, 1439

الکلمات المفتاحية: Free vibration analysis, Micro-beam, Modified couple stress elasticity theory, Gurtin- Murdoch surface theory,

ملخص المقالة :

In this paper, the free vibration analysis of sandwich micro beam with piezoelectric layers based on the modified couple stress and surface elasticity theories are investigated. The Hamilton’s principle is employed to derive the sandwich micro beam with piezoelectric based on modified couple stress theory incorporating with Gurtin- Murdoch surface theory. The generalized differential quadrature method is used to discretize the partial differential equation into the ordinary differential equation. The effect of various parameters such as thickness to material length scale parameter ratio, the surface residual stress, Young's modulus of surface layer, surface mass density and surface piezoelectric constant are investigated by comparing the results obtained using the modified couple and classical theories. Numerical results of this problem evaluate the effect of micro length scale parameters on natural frequency. The results show that surface parameter effects are significant when the model is small, but can be neglected with increasing model size.

المصادر:

[1] N. Fleck, G. Muller, M. Ashby, J. Hutchinson, “Strain gradient plasticity: theory and experiment,” Acta Metallurgica et Materialia, vol. 42, 1994, pp. 475-487.

[2] J. Stölken, A. Evans, “A microbend test method for measuring the plasticity length scale,” Acta Materialia, vol. 46, 1998, pp. 5109-5115.

[3] A.C. Eringen, D. Edelen, “On nonlocal elasticity,” International Journal of Engineering Science, vol.  10, 1972, pp. 233-248.

[4] U.B. Ejike, “The plane circular crack problem in the linearized couple-stress theory,” International Journal of Engineering Science, vol. 7, 1969, pp. 947-961.

[5] F. Yang, A. Chong, D.C.C. Lam, P. Tong, “Couple stress based strain gradient theory for elasticity,” International Journal of Solids and Structures, vol. 39, 2002, pp. 2731-2743.

[6] D.C. 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, 2003, pp. 1477-1508.

[7] S. Park, X. Gao, “Bernoulli–Euler beam model based on a modified couple stress theory,” Journal of Micromechanics and Microengineering, vol. 16, 2006, pp. 2355.

[8] S. Kong, S. Zhou, Z. Nie, K. Wang, “The size-dependent natural frequency of Bernoulli–Euler micro-beams,” International Journal of Engineering Science, vol. 46, 2008, pp. 427-437.

[9] R.A. Alashti, A. Abolghasemi, “A size-dependent Bernoulli-Euler beam formulation based on a new model of couple stress theory,” International Journal of Engineering TRANSACTIONS C, vol. 27, 2014, pp. 951-960.

[10] H. Farokhi, M.H. Ghayesh, M. Amabili, “Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory,” International Journal of Engineering Science, vol. 68, 2013, pp. 11-23.

[11] B. Akgöz, Ö. Civalek, “Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams,” International Journal of Engineering Science, vol. 49, 2011, pp. 1268-1280.

[12] S. Kong, S. Zhou, Z. Nie, K. Wang, “Static and dynamic analysis of micro beams based on strain gradient elasticity theory,” International Journal of Engineering Science, vol. 47, 2009, pp. 487-498.

[13] X. Liang, S. Hu, S. Shen, “A new Bernoulli–Euler beam model based on a simplified strain gradient elasticity theory and its applications,” Composite Structures, vol. 111, 2014, pp. 317-323.

[14] K. Lazopoulos, A. Lazopoulos, “Bending and buckling of thin strain gradient elastic beams,” European Journal of Mechanics-A/Solids, vol. 29, 2010, pp. 837-843.

[15] M.A. Trindade, A. Benjeddou, “Refined sandwich model for the vibration of beams with embedded shear piezoelectric actuators and sensors,” Computers & Structures, vol. 86, 2008, pp. 859-869.

[16] M.-R. Ghazavi, G. Rezazadeh, S. Azizi, “Pure parametric excitation of a micro cantilever beam actuated by piezoelectric layers,” Applied Mathematical Modelling, vol. 34, 2010, pp. 4196-4207.

[17] S. Sahmani, M. Bahrami, “Size-dependent dynamic stability analysis of microbeams actuated by piezoelectric voltage based on strain gradient elasticity theory,” Journal of Mechanical Science and Technology, vol. 29, 2015, pp. 325.

[18] A.G. Arani, M. Abdollahian, R. Kolahchi, “Nonlinear vibration of a nanobeam elastically bonded with a piezoelectric nanobeam via strain gradient theory,” International Journal of Mechanical Sciences, vol. 100, 2015, pp. 32-40.

[19] M.E. Gurtin, A.I. Murdoch, “A continuum theory of elastic material surfaces,” Archive for rational mechanics and analysis, vol. 57, 1975, pp. 291-323.

[20] Y. Fu, J. Zhang, “Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies,” Applied Mathematical Modelling, vol. 35, 2011, pp. 941-951.

[21] M. Mohammadimehr, M.M. Mohammadi Najafabadi, H. Nasiri, B. Rousta Navi, “Surface stress effects on the free vibration and bending analysis of the nonlocal single-layer graphene sheet embedded in an elastic medium using energy method,” Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, vol. 230, 2016, pp. 148-160.

[22] H.M. Sedighi, A. Koochi, M. Abadyan, “Modeling the size dependent static and dynamic pull-in instability of cantilever nanoactuator based on strain gradient theory,” International Journal of Applied Mechanics, vol. 6, 2014, pp. 1450055.

[23] M. Mohammadimehr, H. Mohammadi Hooyeh, H. Afshari, M.R. Salarkia, “Free vibration analysis of double-bonded isotropic piezoelectric Timoshenko microbeam based on strain gradient and surface stress elasticity theories under initial stress using differential quadrature method,” Mechanics of Advanced Materials and Structures, vol.24, 2017, pp. 287-303.

[24] M. Keivani, A. Koochi, A. Kanani, M.R. Mardaneh, H.M. Sedighi, M. Abadyan, “Using strain gradient elasticity in conjunction with Gurtin–Murdoch theory for modeling the coupled effects of surface and size phenomena on the instability of narrow nano-switch,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 231, 2017, pp. 3277-3288.

[25] S.A. Mirkalantari, M. Hashemian, S.A. Eftekhari, D. Toghraie, “Pull-in instability analysis of rectangular nanoplate based on strain gradient theory considering surface stress effects,” Physica B: Condensed Matter, vol. 519, 2017, pp. 1-14.

[26] M. Keivani, A. Koochi, H.M. Sedighi, A. Abadian, M. Abadyan, “A Nonlinear Model for Incorporating the Coupled Effects of Surface Energy and Microstructure on the Electromechanical Stability of NEMS,” Arabian Journal for Science and Engineering, vol. 41, 2016, pp. 4397-4410.

[27] S. Foroutan, A. Haghshenas, M. Hashemian, S.A. Eftekhari, D. Toghraie, “Spatial buckling analysis of current-carrying nanowires in the presence of a longitudinal magnetic field accounting for both surface and nonlocal effects,” Physica E: Low-dimensional Systems and Nanostructures, vol. 97, 2018, pp. 191-205.

[28] K.-M. Hu, W.-M. Zhang, Z.-K. Peng, G. Meng, “Transverse vibrations of mixed-mode cracked nanobeams with surface effect,”  Journal of Vibration and Acoustics, vol. 138, 2016, pp. 011020.