نوع مقاله: مقاله پژوهشی

نویسندگان

1 استادیار، مهندسی مکانیک، دانشگاه اراک، اراک، ایران

2 کارشناس ارشد، مهندسی مکانیک، دانشگاه اراک، اراک، ایران

چکیده

هدف این مقاله مطالعه ارتعاش آزاد نانو ورق مستطیلی مرکب پیزوالکتریک تحت بار الکترومکانیکی شامل نیروی محوری و ولتاژ خارجی بر اساس تئوری‌های تغییر شکل برشی اصلاح شده نمایی و مثلثاتی به همراه تئوری الاستیسیته غیرمحلی و شرایط مرزی چهار طرف تکیه‌گاه ساده است. در تئوری غیرمحلی تنش در هر نقطه تابعی از کرنش در تمامی نقاط محیط می‌باشد. این تئوری اهمیت بسزایی در ساختارهای با ابعاد میکرو و نانو دارد. درتئوری تغییرشکل برشی اصلاح شده نمایی و تئوری تغییر شکل برشی اصلاح شده مثلثاتی، توابع نمایی و مثلثاتی در راستای ضخامت، جهت در نظر گرفتن تاثیرات تغییر شکل برشی عرضی و اینرسی دورانی بکار رفته است. از تئوری الاستیسیته غیرمحلی جهت در نظر گرفتن پارامتر مقیاس طول کوچک نانو ورق مستطیلی مرکب پیزوالکتریک استفاده شده است. نانو ورق مرکب پیزوالکتریک مورد مطالعه در این تحلیل ورق مستطیلی ساخته شده از ماده مرکب پیزوالکتریک PZT 4 متشکل از ترکیبات بلوری سرب، زیرکونیم و تیتانیم جهت دست‌یابی به خواص فلز- سرامیک و همچنین خاصیت پیزوالکتریک می‌باشد. با بکارگیری اصل همیلتون معادلات دیفرانسیل حاکم بر ارتعاش نانو ورق مستطیلی مرکب پیزوالکتریک به‌دست آمده و برای به‌دست آوردن پاسخ فرکانسی از حل دقیق ناویر استفاده شده است. در نهایت تأثیر پارامتر مقیاس نانو، نیروهای محوری، ولتاژ خارجی و نسبت‌های هندسی روی فرکانس‌‌های طبیعی بی‌‌بعد شش مد ارتعاشی اول نانو ورق مستطیلی مرکب پیزوالکتریک مورد بحث و بررسی قرار گرفته است.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Electro-Mechanical free vibrations analysis of composite rectangular piezoelectric nanoplate using modified shear deformation theories

نویسندگان [English]

  • Korosh Khorshidi 1
  • Ali Siahpush 2
  • Abolfazl Fallah 2

1 1- Department of Mechanical Engineering, University of Arak, Arak, Iran

2 1- Department of Mechanical Engineering, University of Arak, Arak, Iran

چکیده [English]

The aim of this paper is to study the free vibration of composite rectangular piezoelectric nanoplate subjected to an electro-mechanical loading includes a biaxial force and an external voltage based on exponential shear deformation theory and trigonometric shear deformation theory in conjunction with the nonlocal elasticity theory under the simply supported boundary condition. The nonlocal theory states that stress at a point is a function of strains at all points in the continuum. The nonlocal elasticity theory becomes significant for small length scale in micro and nanostructures. In exponential shear deformation theory andtrigonometric shear deformation theory, exponential and trigonometric functions are used in terms of thickness coordinate to include the effect of transverse shear deformation and rotary inertia. Nonlocal elasticity theory is employed to investigate effect of small scale on natural frequency of composite rectangular piezoelectric nanoplate. It is assumed that the composite rectangular piezoelectric nanoplate made of PZT 4 composite piezoelectric material includes crystal compounds of Pb, Zr and Ti to achieve metal-ceramic and piezoelectric properties. The governingdifferential equations of the vibration of the composite rectangular piezoelectric nanoplate are derived by using the Hamilton’s principle, which are then solved by using the Navier method to obtain the natural frequencies of the composite rectangular piezoelectric nanoplate. The detailed parametric study is conducted to discuss the influences of the nonlocal parameter, biaxial force external electric voltage and geometrical ratios on the first six nondimensional frequencies of the composite rectangular piezoelectric nanoplate

کلیدواژه‌ها [English]

  • Composite piezoelectric nanoplate
  • Free vibration
  • Nonlocal
  • Exponential and Trigonometric Shear Deformation

[1] Jalili, N., “Piezoelectric-Based Vibration Control from Macro to Micro/Nano Scale Systems”, SpringeScience Business Media, LLC 2010, ISBN 978-1-4419-0069-2, 2009.

[2] Wang, Z.L., “ZnO Nanowire and Nanobelt Platform for Nanotechnology”, Materials Science and Engineering, Vol. 64, No. 3–4, pp. 33–71, 2009.

[3] Park, K.I. Xu, S. Liu, Y. Hwang, G.T. Kang, S.J. Kang, S.J.L. Wang, Z.L. and Lee K.J., “Piezoelectric BaTiO3 Thin Film Nanogenerator on Plastic Substrates”, Nano Letters, Vol.10, No 12, pp. 4939–4943, 2010.

[4] Galan, U. Lin, Y.R. Ehlert, G.J. and Sodano, H.A., “Effect of ZnO Nanowire Morphology on the Interfacial Strength of Nanowire Coated Carbon Fibers”, Composites Science and Technology, Vol. 71, No. 7, pp. 946–954, 2011.

[5] Lam, D.C.C. Yang, F. Chong, A.C.M. Wang, J. and Tong, P., “Experiments and Theory in Strain Gradient Elasticity”, Journal of the Mechanics and Physics of Solids, Vol. 51, No. 8, pp. 1477–1508, 2003.

[6] Yang, F. Chong, A.C.M. Lam, D.C.C. and Tong, P., “Couple Stress Based Strain Gradient Theory for Elasticity”, International Journal of Solids and Structures, Vol. 39, No. 10, pp. 2731–2743, 2002.

[7] Eringen, A.C., “Nonlocal Polar Elastic Continua”, International Journal of Engineering Science, Vol. No. 1, pp. 1–16, 1972.

[8] Eringen, A.C., “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves”, Journal of Applied Physics, Vol. 54, pp. 4703–4710, 1983.

[9] Eringen, A.C., “Nonlocal Continuum Field Theories”, Springer, NewYork, ISBN 978-0-387-22643-9, 2002.

[10] Li, Y.S. Feng, W.J. and Cai, Z.Y., “Bending and Free Vibration of Functionally Graded Piezoelectric Beam Based on Modified Strain Gradient Theory”, Composite Structures, Vol. 115, pp. 41–50, 2014.

[11] Alzahrani, E.O. Zenkour, A.M. and Sobhy, M., “Small Scale Effect on Hygro-Thermo-Mechanical Bending of Nanoplates Embedded in an Elastic Medium”, Composite Structures, Vol. 105, pp. 163–172, 2013.

[12] Golmakani, M.E. and Rezatalab, J., “Nonlinear Bending Analysis of Orthotropic Nanoscale Plates in an Elastic Matrix Based on Nonlocal Continuum Mechanics”, Composite Structures, Vol.111, pp. 85–97, 2014.

[13] Şimşek, M. and Yurtcu, H.H., “Analytical Solutions for Bending and Buckling of Functionally Graded Nanobeams Based on the Nonlocal Timoshenko Beam Theory”, Composite Structures, Vol. 97, pp. 378–386, 2013.

[14] Hosseini-Hashemi, S. Zare M. and Nazemnezhad, R., “An Exact Analytical Approach for Free Vibration of Mindlin Rectangular Nanoplates via Nonlocal Elasticity”, Composite Structures, Vol. 100, pp. 290–299, 2013.

[15] Sobhy, M., “Generalized Two-variable Plate Theory for Multi Layered Grapheme Sheets with Arbitrary Boundary Conditions”, Acta Mechanica, Vol. 225, No. 9, pp. 2521–2538, 2014.

[16] Chen, A.L. Wang, Y.S. Ke, L.L. Guo, Y.F. and Wang, Z.D., “Wave Propagation in Nanoscaled Periodic Layered Structures”, Journal of Computational and Theoretical Nanoscience, Vol. 10, pp. 2427–2437, 2013.

[17] Ke, L.L. and Wang, Y.S., “Thermoelectric-Mechanical Vibration of Piezoelectric NanobeamsBased on the Nonlocal Theory”, Smart Materials and Structures, Vol. 21, No. 2, 025018, 2012.

[18] Ke, L.L. Wang, Y.S. and Wang, Z.D., “Nonlinear Vibration of the Piezoelectric Nanobeams Based on the Nonlocal Theory”, Composite Structures,Vol. 94, No. 6, pp. 2038–2047, 2012.

[19] Liu, C. Ke, L.L. Wang, Y.S. Yang, J. and Kitipornchai, S., “Buckling and Post Buckling of Size-Dependent Piezoelectric Timoshenko Nanobeams Subject to Thermo-Electro-Mechanical Loadings”, International Journal of Structural Stability and Dynamics, Vol. 14, No. 03, 1350067, 2014.

[20] Arani, A.G. Roudbari, M.A. and Amir, S., “Nonlocal Vibration of SWBNNT Embedded in Bundle of CNTs Under a Moving Nanoparticle”, Physica B: Condensed Matter, Vol. 407, No. 17, pp. 3646–3653, 2012.

[21] Arani, A.G. Atabakhshian, V. Loghman, A. Shajari, A.R. and Amir S., “Nonlinear Vibration of Embedded SWBNNTs Based on Nonlocal Timoshenko Beam Theory Using DQ Method”, Physica B: Condensed Matter, Vol. 407, No. 13, pp. 2549–2555, 2012.

[22] Arani, A.G. Amir, S. Shajari, A.R. and Mozdianfard, M.R., “Electro-Thermo-Mechanical Buckling of DWBNNTs Embedded in Bundle of CNTs Using Nonlocal Piezoelasticity Cylindrical Shell Theory”, Composites Part B: Engineering, Vol. 43, No. 2, pp. 195–203, 2012.

[23] Arani, A.G. Abdollahian, M. Kolahchi, R. and Rahmati,A.H., “Electro-Thermo-Torsional Buckling of an Embedded Armchair DWBNNT Using Nonlocal Shear Deformable Shell Model”, Composites Part B: Engineering, Vol. 51, pp. 291–299, 2013.

[24] Liu, C. Ke, L.L. Wang, Y.S. Yang, J. and Kitipornchai, S., “Thermo-Electro-Mechanical Vibration of Piezoelectric Nanoplates Based on the Nonlocal Theory”, Composite Structures, Vol. 106, pp. 167–174, 2013.

[25] Ke, L.L. Liu, C. and Wang, Y.S., “Free Vibration of Nonlocal Piezoelectric Nanoplates under Various Boundary Conditions”, Physica E: Low-dimensional Systems and Nanostructures, Vol. 66, pp. 93–106, 2015.

[26] Sayyad, A.S. and Ghugal, Y.M., “Bending and Free Vibration Analysis of Thick Isotropic Plates by Using Exponential Shear Deformation Theory”, Applied and Computational Mechanics, Vol. 6, No. 1, pp. 65–82, 2012.

[27] Ghugal, Y.M. and Sayyad, A.S., “Stress Analysis of Thick Laminated Plates Using Trigonometric Shear Deformation Theory”, International Journal of Applied Mechanics Vol. 5, No. 1, 1350003, 2013.

[28] Kharde, S.B. Mahale, A.K. Bhosale, K.C. and Thorat S.R., “Flexural Vibration of Thick Isotropic Plates by Using Exponential Shear Deformation Theory”, International Journal of Emerging Technology and Advanced Engineering, Vol. 3, No. 1, pp. 369-374, 2013.

[29] Mantari, J.L. Oktem, A.S. and Soares, C.G., “A New Trigonometric Shear Deformation Theory for Isotropic, Laminated Composite and Sandwich Plates”, International Journal of Solids and Structures, Vol. 49, No. 1, pp. 43–53, 2012.

[30] Tounsi, A. Houari, M.S.A. Benyoucef, S. and Bedia, A.A., “A Refined Trigonometric Shear Deformation Theory for Thermoelastic Bending of Functionally Graded Sandwich Plates”, Aerospace Science and Technology, Vol. 24, No. 1, pp. 209–220, 2013.

[31] Rango, R.F. Nallim, L.G. and Oller, S., “Formulation of Enriched Macro Elements Using Trigonometric Shear Deformation Theory for Free Vibration Analysis of Symmetric Laminated Composite Plate Assemblies”, Composite Structures, Vol. 119, pp. 38–49, 2015.

[32] Khorshidi, K. and Fallah, A., “Buckling Analysis of Functionally Graded Rectangular Nano-plateBased on Nonlocal Exponential Shear Deformation Theory“, International Journal of Mechanical Sciences, Vol. 113, pp. 94-104, 2016.

[33] Khorshidi, K. Asgari, T. and Fallah, A., “Free Vibrations Analysis of Functionally Graded Rectangular Nanoplates Based on Nonlocal Exponential Shear Deformation Theory“. Mechanics of Advanced Composite Structures, Vol. 2, No. 2, pp. 79-93,‎2015.

[34] Quek, S.T. and Wang, Q., “On Dispersion Relations in Piezoelectric Coupled-Plate Structures”, Smart Materials and Structures, Vol. 9, No. 6, pp. 859–867, 2000.

[35] Wang, Q. and Quek, ST., “Flexural Vibration Analysis of Sandwich Beam Coupled withPiezoelectric Actuator”, Smart Materials and Structures, Vol. 9, No 1, pp. 103–109, 2000.

[36] Wang, Q. Quek, S.T. Sun, T.C. and Liu, X., “Analysis of Piezoelectric Coupled CircularPlate”, Smart Materials and Structures, Vol. 10, No. 2, pp. 229–239, 2001.

[37] Wang, Q., “Axisymmetric Wave Propagation in a Cylinder Coated with a Piezoelectric Layer”, International Journal of Solids and Structures,Vol. 39, No. 11, pp. 3023–3037, 2002.

[38] Zhao, M. Qian, C. Lee, S.W.R. Tong, P. Suemasu, H. and Zhang, T.Y., “Electro-Elastic Analysis of Piezoelectric Laminated Plates”, Advanced Composite Materials, Vol.16, No. 1, pp. 63–81, 2007.

[39] Pietrzakowski, M., “Piezoelectric Control of Composite Plate Vibration: Effect of Electric PotentialDistribution”, Computers & Structures, Vol.86, No. 9, pp. 948–954, 2008.

[40] Bodaghi,M. and Shakeri,M., “An analytical Approach for Free Vibration and Transient Response of Functionally Graded Piezoelectric Cylindrical Panels Subjected to Impulsive Loads”, Composite Structures, Vol. 94, No. 5, pp. 1721–1735, 2012.