The effect of accuracy of the length scale parameter on natural frequencies of porous rectangular microplate

Document Type : Research Article

Authors

Department of Mechanical Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Abstract

The length-scale parameter as a primary factor has an important role in the approximation of natural frequencies of micro-structures. Applying the exact length scale parameters which are recently determined by researchers, natural frequencies of porous microstructures are determined. Since the assumption of constant length-scale parameter leads to deviation of natural frequencies from their exact value, this research applies a length scale parameter which is a function of plate thickness and material. To model the porous structure of the microplate, various porous models including evenly porosity mode, unevenly symmetric porosity model, and unevenly asymmetric porosity model are employed. The microplate is assumed to be thin, and classical plate theory is utilized to approximate the displacement field of the microplate. The modified couple stress theory is used to capture the microstructural behavior of the microplate. The trial functions which satisfy the boundary conditions are taken as the polynomial form. Evaluating the energy values of the system, the Rayleigh-Ritz method is employed to solve the governing equations of the system. The results obtained in the present work are validated with data given in the literature search. A parameter study is performed to study the effects of various parameters on the natural frequency of the microplate.

Keywords

Main Subjects


[1] A.C. Eringen, Nonlocal polar elastic continua, International journal of engineering science, 10(1) (1972) 1-16.
[2] K. Khorshidi, A. Fallah, A. Siahpush, Free vibrations analaysis of functionally graded composite rectangular na-noplate based on nonlocal exponential shear deformation theory in thermal environment,  (2017).
[3] K. Khorshidi, A. Siahpush, A. Fallah, Electro-Mechanical free vibrations analysis of composite rectangular piezoelectric nanoplate using modified shear deformation theories, Journal of Science and Technology of Composites, 4(2) (2017) 151-160.
[4] 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, 51(8) (2003) 1477-1508.
[5] R.A. Toupin, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, 11(1) (1962) 385-414.
[6] R. Mindlin, H. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and analysis, 11(1) (1962) 415-448.
[7] F. Yang, A. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39(10) (2002) 2731-2743.
[8] S. Park, X. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, 16(11) (2006) 2355.
[9] M.A. Khorshidi, The material length scale parameter used in couple stress theories is not a material constant, International Journal of Engineering Science, 133 (2018) 15-25.
[10] Z. Li, Y. He, J. Lei, S. Guo, D. Liu, L. Wang, A standard experimental method for determining the material length scale based on modified couple stress theory, International Journal of Mechanical Sciences, 141 (2018) 198-205.
[11] M. Kahrobaiyan, M. Asghari, M. Ahmadian, Strain gradient beam element, Finite Elements in Analysis and Design, 68 (2013) 63-75.
[12] C. Liebold, W.H. Müller, Comparison of gradient elasticity models for the bending of micromaterials, Computational Materials Science, 116 (2016) 52-61.
[13] L. He, J. Lou, E. Zhang, Y. Wang, Y. Bai, A size-dependent four variable refined plate model for functionally graded microplates based on modified couple stress theory, Composite Structures, 130 (2015) 107-115.
[14] J. Reddy, J. Kim, A nonlinear modified couple stress-based third-order theory of functionally graded plates, Composite Structures, 94(3) (2012) 1128-1143.
[15] H.-T. Thai, S.-E. Kim, A size-dependent functionally graded Reddy plate model based on a modified couple stress theory, Composites Part B: Engineering, 45(1) (2013) 1636-1645.
[16] M. Şimşek, J. Reddy, Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory, International Journal of Engineering Science, 64 (2013) 37-53.
[17] A. Bakhsheshy, K. Khorshidi, Free vibration of functionally graded rectangular nanoplates in thermal environment based on the modified couple stress theory, Modares Mechanical Engineering, 14(15) (2015) 323-330.
[18] K. Khorshidi, A. Fallah, Free vibration analysis of size-dependent, functionally graded, rectangular nano/micro-plates based on modified nonlinear couple stress shear deformation plate theories, Mechanics of Advanced Composite Structures, 4(2) (2017) 127-137.
[19] A. Bakhsheshy, H. Mahbadi, The effect of multidimensional temperature distribution on the vibrational characteristics of a size-dependent thick bi-directional functionally graded microplate, Noise & Vibration Worldwide,  (2019) 0957456519883265.
[20] A. Bakhsheshy, H. Mahbadi, The effect of fluid surface waves on free vibration of functionally graded microplates in interaction with bounded fluid, Ocean Engineering, 194 (2019) 106646.
[21] K. Al-Basyouni, A. Tounsi, S. Mahmoud, Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position, Composite Structures, 125 (2015) 621-630.
[22] R. Aghazadeh, E. Cigeroglu, S. Dag, Static and free vibration analyses of small-scale functionally graded beams possessing a variable length scale parameter using different beam theories, European Journal of Mechanics-A/Solids, 46 (2014) 1-11.
[23] K. Magnucki, P. Stasiewicz, Elastic buckling of a porous beam, Journal of Theoretical and Applied Mechanics, 42(4) (2004) 859-868.
[24] E. Farzaneh Joubaneh, A. Mojahedin, A. Khorshidvand, M. Jabbari, Thermal buckling analysis of porous circular plate with piezoelectric sensor-actuator layers under uniform thermal load, Journal of Sandwich Structures & Materials, 17(1) (2015) 3-25.
[25] M. Jabbari, A. Mojahedin, A. Khorshidvand, M. Eslami, Buckling analysis of a functionally graded thin circular plate made of saturated porous materials, Journal of Engineering Mechanics, 140(2) (2013) 287-295.
[26] P. Stasiewicz, K. Magnucki, Elastic buckling of a Pours beam, Theoretical and Applied Mechanics, 140 (2008) 287-298.
[27] M. Rezaei, A. Mojahedin, M.R. Eslami, Mechanical Buckling of FG Saturated Porous Rectangular Plate under Temperature Field, Iranian Journal of Mechanical Engineering Transactions of the ISME, 17(1) (2016) 61-78.
[28] A.R.Y.N. Yadegari Naeini, A. Ghasemi, Analysis of Bending and Buckling of Circular Porous Plate Using First-Order Shear Deformation Theory, Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering, 6(4) (2014) 55-62.
[29] Ş.D. Akbaş, Vibration and static analysis of functionally graded porous plates, Journal of Applied and Computational Mechanics, 3(3) (2017) 199-207.
[30] P. Leclaire, K.V. Horoshenkov, M.J. Swift, D.C. Hothersall, THE VIBRATIONAL RESPONSE OF A CLAMPED RECTANGULAR POROUS PLATE, Journal of Sound and Vibration, 247(1) (2001) 19-31.
[31] P. Leclaire, K. Horoshenkov, A. Cummings, Transverse vibrations of a thin rectangular porous plate saturated by a fluid, Journal of Sound and Vibration, 247(1) (2001) 1-18.
[32] E. Arshid, A. Khorshidvand, Flexural Vibrations Analysis of Saturated Porous Circular Plates Using Differential Quadrature Method, Modares Mechanical Engineering, 19(1) (2017) 78-100.
[33] A.K. Vashishth, V. Gupta, Vibrations of porous piezoelectric ceramic plates, Journal of Sound and Vibration, 325(4) (2009) 781-797.
[34] G. Altintas, Natural vibration behaviors of heterogeneous porous materials in micro scale, Journal of Vibration and Control, 20(13) (2014) 1999-2005.
[35] C. Della, D.W. Shu, Vibration of porous beams with embedded piezoelectric sensors and actuators,  (2015).
[36] I. Mechab, B. Mechab, S. Benaissa, B. Serier, B.B. Bouiadjra, Free vibration analysis of FGM nanoplate with porosities resting on Winkler Pasternak elastic foundations based on two-variable refined plate theories, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(8) (2016) 2193-2211.
[37] N. Shafiei, S.S. Mirjavadi, B. MohaselAfshari, S. Rabby, M. Kazemi, Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams, Computer Methods in Applied Mechanics and Engineering, 322 (2017) 615-632.
[38] M. Şimşek, M. Aydın, Size-dependent forced vibration of an imperfect functionally graded (FG) microplate with porosities subjected to a moving load using the modified couple stress theory, Composite Structures, 160 (2017) 408-421.
[39] J. Zhao, Q. Wang, X. Deng, K. Choe, R. Zhong, C. Shuai, Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions, Composites Part B: Engineering, 168 (2019) 106-120.
[40] K. Khorshidi, A. Bakhsheshy, Free Vibration analysis of Functionally Graded Rectangular plates in contact with bounded fluid, Modares Mechanical Engineering, 14(8) (2014) 165-173.
[41] L. Yin, Q. Qian, L. Wang, W. Xia, Vibration analysis of microscale plates based on modified couple stress theory, Acta Mechanica Solida Sinica, 23(5) (2010) 386-393.