بررسی اثر گرما بر فرکانس نانو صفحه دایره ای با ساختار درجه بندی شده بر اساس نظریه تنش کوپل اصلاح شده

نوع مقاله : مقاله پژوهشی

نویسندگان

1 استادیار، گروه مهندسی مکانیک، دانشگاه بین المللی امام خمینی (ره)، قزوین، ایران

2 دانشجو، گروه مهندسی مکانیک، دانشگاه بین المللی امام خمینی (ره)، قزوین، ایران

چکیده

در این پژوهش اثر گرما بر فرکانس­ طبیعی حاصل از ارتعاش آزاد یک نانو صفحه­ی دایره­ای متقارن نازک مورد بررسی قرار می­گیرد. توزیع گرما به صورت یکنواخت در نظر گرفته شده و خواص مواد در راستای ضخامت صفحه تغییر می­کند. از نظریه تنش کوپل اصلاح شده به منظور اعمال اثر اندازه استفاده شده و معادلات حاکم برحرکت و شرایط مرزی با استفاده از اصل هامیلتون و بر اساس نظریه کلاسیک صفحه  به دست آمده و به صورت تحلیلی با استفاده از توابع بسل برای شرط مرزی کاملاً مهار شده و تکیه گاه ساده حل می­شوند. نتایج حاصله نشان می­دهد که افزایش تغییرات دما، نسبت ضخامت به پارامتر اثر اندازه، توان توزیع ماده و نسبت شعاع به ضخامت منجر به کاهش فرکانس طبیعی سیستم در هردو شرط مرزی می­شود. اثر افزایش نسبت ضخامت به پارامتر اثر اندازه بر روی فرکانس طبیعی صفحه در شرایط مرزی کاملاً مهار شده بیشتر از شرایط مرزی تکیه گاه ساده است ولی در هر دو شرط مرزی در نسبت های بالاتر این اثر کاهش می­یابد. اثر افزایش تغییرات دما بر روی فرکانس­های طبیعی در شرایط مرزی تکیه گاه ساده نسبت به شرایط مرزی کاملاً مهار شده بیشتر است به گونه­ای که تغییرات دماهای پایین تری منجر به رفتار غیر خطی صفحه می­شود.

کلیدواژه‌ها

موضوعات


[1]          Koizumi M., FGM activities in Japan, Compos. Part B Eng., Vol. 28, No. 1–2, pp. 1–4, 1997.
[2]          Ansari R., Gholami R., Shojaei M. F., Mohammadi V., and S. Sahmani, “Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory, Eur. J. Mech., Vol. 49, pp. 251–267, 2015.
[3]          Cheng Z.-Q. and Batra R. C., Three-dimensional thermoelastic deformations of a functionally graded elliptic plate, Compos. Part B Eng., Vol. 31, No. 2, pp. 97–106, 2000.
[4]          Hosseini Hashemi S., Es'haghi M. and Karimi M., Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers, Int. J. Mech. Sci., Vol. 52, No. 3, pp. 410–428, 2010.
[5]          Hosseini-Hashemi S., Taher H. R. D. and Akhavan H., Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations, Compos. Struct., Vol. 92, No. 7, pp. 1734–1743, 2010.
[6]          Kiani Y. and Eslami M. R., Thermal Postbuckling of Imperfect Circular Functionally Graded Material Plates: Examination of Voigt, Mori--Tanaka, and Self-Consistent Schemes, J. Press. Vessel Technol., Vol. 137, No. 2, p. 21201, 2015.
[7]          Y. Kiani and M. R. Eslami, “An exact solution for thermal buckling of annular FGM plates on an elastic medium,” Compos. Part B Eng., vol. 45, no. 1, pp. 101–110, 2013.
[8]          Kim J. and Reddy J. N., Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory, Compos. Struct., Vol. 103, pp. 86–98, 2013.
[9]          Li S.-R., Wang X. and Batra R. C., Correspondence Relations Between Deflection, Buckling Load, and Frequencies of Thin Functionally Graded Material Plates and Those of Corresponding Homogeneous Plates, J. Appl. Mech., Vol. 82, No. 11, p. 111006, 2015.
[10]        Reddy J. N. and Berry J., Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress, Compos. Struct., Vol. 94, No. 12, pp. 3664–3668, 2012.
[11]        Asgari M. and Akhlaghi M., Thermo-mechanical analysis of 2D-FGM thick hollow cylinder using graded finite elements,” Adv. Struct. Eng., Vol. 14, No. 6, pp. 1059–1073, 2011.
[12]        Asgari M., Material distribution optimization of 2D heterogeneous cylinder under thermo-mechanical loading, Struct. Eng. Mech., Vol. 53, No. 4, pp. 703–723, 2015.
[13] محمد مهدی نجفی زاده و شهروز یوسف زاده،  کمانش حرارتی ورق مستطیل شکل FGM بر اساس تئوری مرتبه­ی اول تغییر شکل برشی صفحات، مهندسی مکانیک ایران، سال ششم، شماره اول، صفحات 75-100، سال 1383.
[14]        Toupin R. A., Elastic materials with couple-stresses, Arch. Ration. Mech. Anal., Vol. 11, No. 1, pp. 385–414, 1962.
[15]        Koiter W. T., Couple-stresses in the theory of elasticity, I & II, 1969.
[16]        Mindlin R. D. and Tiersten H. F., Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal., Vol. 11, No. 1, pp. 415–448, 1962.
[17]        Mohammad Abadi M. and  Daneshmehr a. R., An investigation of modified couple stress theory in buckling analysis of micro composite laminated Euler-Bernoulli and Timoshenko beams,  Int. J. Eng. Sci., Vol. 75, pp. 40–53, 2014.
[18]        Yang F., Chong A. C. M., Lam D. C. C. and Tong P., Couple stress based strain gradient theory for elasticity, Int. J. Solids Struct., Vol. 39, No. 10, pp. 2731–2743, 2002.
[19]        Ma H. M., Gao X. L. and Reddy J. N., A microstructure-dependent Timoshenko beam model based on a modified couple stress theory,” J. Mech. Phys. Solids, vol. 56, no. 12, pp. 3379–3391, 2008.
[20]        Asghari M., Ahmadian M. T., Kahrobaiyan M. H. and Rahaeifard M., On the size-dependent behavior of functionally graded micro-beams, Mater. Des., Vol. 31, No. 5, pp. 2324–2329, 2010.
[21]        Asghari M., Rahaeifard M., Kahrobaiyan M. H. and Ahmadian M. T., The modified couple stress functionally graded Timoshenko beam formulation,” Mater. Des., Vol. 32, No. 3, pp. 1435–1443, 2011.
[22]        Salamat-talab M., Nateghi A. and Torabi J., Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory, Int. J. Mech. Sci., Vol. 57, No. 1, pp. 63–73, 2012.
[23]        Ansari R., Shojaei M. F., Mohammadi V., Gholami R., and Darabi M. A., Nonlinear vibrations of functionally graded Mindlin microplates based on the modified couple stress theory, Compos. Struct., Vol. 114, pp. 124–134, 2014.
[24]        Jomehzadeh E., Noori H. R. and Saidi A. R., The size-dependent vibration analysis of micro-plates based on a modified couple stress theory, Phys. E Low-dimensional Syst. Nanostructures, Vol. 43, No. 4, pp. 877–883, 2011.
[25]        Ke L.-L., Yang J., Kitipornchai S. and Bradford M. A., Bending, buckling and vibration of size-dependent functionally graded annular microplates, Compos. Struct., Vol. 94, No. 11, pp. 3250–3257, 2012.
[26]        Nateghi A. and Salamat-talab M., Thermal effect on size dependent behavior of functionally graded microbeams based on modified couple stress theory,” Compos. Struct., vol. 96, pp. 97–110, 2013.
[27]        Wang Y.-G., Lin W.-H. and Liu N., Large amplitude free vibration of size-dependent circular microplates based on the modified couple stress theory,” Int. J. Mech. Sci., vol. 71, pp. 51–57, 2013.
[28]        Wang Y.-G., Lin W.-H. and Zhou C.-L., Nonlinear bending of size-dependent circular microplates based on the modified couple stress theory,” Arch. Appl. Mech., vol. 84, no. 3, pp. 391–400, 2014.
[29]        Dinh Duc N., Hong Cong P. and Pham D., Nonlinear vibration of thick FGM plates on elastic foundation subjected to thermal and mechanical loads using the first-order shear deformation plate theory, Cogent Eng., Vol. 2, No. 1, p. 1045222, 2015.
[30]        Van Dung D. and others, Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment, Compos. Part B Eng., Vol. 69, pp. 378–388, 2015.
[31]        Gupta A. K. andKumar L., Thermal effect on vibration of non-homogenous visco-elastic rectangular plate of linearly varying thickness, Meccanica, Vol. 43, No. 1, pp. 47–54, 2008.
[32]        Ghadiri M., Mahinzare M., Shafiei N., and Ghorbani K., On size-dependent thermal buckling and free vibration of circular FG Microplates in thermal environments,” Microsyst. Technol., no. 2015, 2017.
[33]        V Joshi P., Jain N. K. and Ramtekkar G. D., Effect of thermal environment on free vibration of cracked rectangular plate: An analytical approach, Thin-Walled Struct., Vol. 91, pp. 38–49, 2015.
[34]        Lanhe W., Thermal buckling of a simply supported moderately thick rectangular FGM plate, Compos. Struct., vol. 64, no. 2, pp. 211–218, 2004.
[35]        Ma L. S. and Wang T. J., Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings, Int. J. Solids Struct., Vol. 40, No. 13, pp. 3311–3330, 2003.
[36]        Najafizadeh M. M. and Heydari H. R., Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory, Eur. J. Mech., Vol. 23, No. 6, pp. 1085–1100, 2004.
[37]        Najafizadeh M. M. and Heydari H. R., An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression,” Int. J. Mech. Sci., Vol. 50, No. 3, pp. 603–612, 2008.
[38]        Van Tung H., Thermal and thermomechanical postbuckling of FGM sandwich plates resting on elastic foundations with tangential edge constraints and temperature dependent properties, Compos. Struct., vol. 131, pp. 1028–1039, 2015.
[39]        Najafi F., Shojaeefard M. H. and Googarchin H. S., Nonlinear dynamic response of FGM beams with Winkler--Pasternak foundation subject to noncentral low velocity impact in thermal field, Compos. Struct., Vol. 167, pp. 132–143, 2017.
[40]        Najafi F., Shojaeefard M. H. and Googarchin H. S., Nonlinear low-velocity impact response of functionally graded plate with nonlinear three-parameter elastic foundation in thermal field, Compos. Part B Eng., Vol. 107, pp. 123–140, 2016.
[41]        Najafi F., Shojaeefard M. H. and Googarchin H. S., Low-velocity impact response of functionally graded doubly curved panels with Winkler--Pasternak elastic foundation: An analytical approach, Compos. Struct., Vol. 162, pp. 351–364, 2017.
[42]        W. Leissa A., vibration of plates. Washington D.C.: NASA, 1969.
[43]        Lam D. C. C., Yang F., Chong A. C. M., Wang J. and Tong P., Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids, Vol. 51, No. 8, pp. 1477–1508, 2003.
[44]        Yalcin H. S., Arikoglu A. and Ozkol I., Free vibration analysis of circular plates by differential transformation method, Appl. Math. Comput.,Vol. 212, No. 2, pp. 377–386, 2009.