### Effect of Temperature Dependency on Thermoelastic Behavior of Rotating Variable Thickness FGM Cantilever Beam

Document Type : Research Paper

Authors

Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

10.22034/jsm.2019.666768

Abstract

Thermoelastic behavior of temperature-dependent (TD) and independent (TID) functionally graded variable thickness cantilever beam subjected to mechanical and thermal loadings is studied based on shear deformation theory using a semi-analytical method. Loading is composed of a transverse distributed force, a longitudinal distributed temperature field due to steady-state heat conduction from root to the tip surface of the beam and an inertia body force due to rotation. A successive relaxation (SR) method for solving temperature-dependent steady-state heat conduction equation is employed to obtain the accurate temperature field. The beam is made of functionally graded material (FGM) in which the mechanical and thermal properties are variable in longitudinal direction based on the volume fraction of constituent. Using first-order shear deformation theory, linear strain–displacement relations and Generalized Hooke’s law, a system of second order differential equation is obtained. Using division method, differential equations are solved for every division. As a result, longitudinal displacement, transverse displacement, and consequently longitudinal stress, shear stress and effective stress are investigated. The results are presented for temperature dependent and independent properties. It has been found that the temperature dependency of the material has a significant effect on temperature distribution, displacements and stresses. This model can be used for thermoelastic analysis of simple turbine blades.

Keywords

[1] Yamanouti M., Koizumi M., Shiota I., 1990, Functionally gradient materials forum, Proceedings of the First International Symposium on Functionally Gradient Materials, Sendai, Japan.
[2] Kapania R.K., Raciti S., 1989, Recent advances in analysis of laminated beams and plates, Part I - Sheareffects and buckling, AIAA Journal 27(7): 923-935.
[3] Romano F., Zingone G., 1992, Deflections of beams with varying rectangular cross section, Journal of Engineering Mechanics 118(10): 2128-2134.
[4] Romano F., 1996, Deflections of Timoshenko beam with varying cross-section, International Journal of Mechanical Sciences 38(8): 1017-1035.
[5] Sankar B.V., 2001, An elasticity solution for functionally graded beams, Composites Science and Technology 61(5): 689-696.
[6] Sankar B.V., Tzeng J.T., 2002, Thermal stresses in functionally graded beams, AIAA Journal 40(6): 1228-1232.
[7] Chakraborty A., Gopalakrishnan S., Reddy J.N., 2003, A new beam finite element for the analysis of functionally graded materials, International Journal of Mechanical Sciences 45(3): 519-539.
[8] Ching H.K., Yen S.C., 2006, Transient thermoelastic deformations of 2-D functionally graded beams under nonuniformly convective heat supply, Composite Structures 73(4): 381-393.
[9] Kadoli R., Akhtar K., Ganesan N., 2008, Static analysis of functionally graded beams using higher order shear deformation theory, Applied Mathematical Modelling 32(12): 2509-2525.
[10] Li X.F., 2008, A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams, Journal of Sound and Vibration 318(4): 1210-1229.
[11] Giunta G., Belouettar S., Carrera E., 2010, Analysis of FGM beams by means of classical and advanced theories, Mechanics of Advanced Materials and Structures 17(8): 622-635.
[12] Giunta G., Crisafulli D., Belouettar S., Carrera E., 2013, A thermo-mechanical analysis of functionally graded beams via hierarchical modelling, Composite Structures 95: 676-690.
[13] Ramesh M.N.V., Mohan Rao N., 2013, Free vibration analysis of pre-twisted rotating FGM beams, International Journal of Mechanics and Materials in Design 9(4): 367-383.
[14] Zhang B., Li Y., 2016, Nonlinear vibration of rotating pre-deformed blade with thermal gradient, Nonlinear Dynamics 86(1): 459-478.
[15] Cao D., Liu B., Yao B., Zhang W., 2017, Free vibration analysis of a pre-twisted sandwich blade with thermal barrier coatings layers, Science China Technological Sciences 60(11): 1747-1761.
[16] Panigrahi B., Pohit G., 2018, Effect of cracks on nonlinear flexural vibration of rotating Timoshenko functionally graded material beam having large amplitude motion, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232(6): 930-940.
[17] Reddy J.N., 2011, Microstructure-dependent couple stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids 59(11): 2382-2399.
[18] Arbind A., Reddy J.N., Srinivasa A.R., 2014, Modified couple stress-based third-order theory for nonlinear analysis of functionally graded beams, Latin American Journal of Solids and Structures 11: 459-487.
[19] Kiani Y., Eslami M.R., 2012, Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation, Archive of Applied Mechanics 82(7): 891-905.
[20] Ma L.S., Lee D.W., 2012, Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading, European Journal of Mechanics - A/Solids 31(1): 13-20.
[21] Nguyen T.-K., Vo T.P., Thai H.-T., 2013, Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory, Composites Part B: Engineering 55: 147-157.
[22] Niknam H., Fallah A., Aghdam M.M., 2014, Nonlinear bending of functionally graded tapered beams subjected to thermal and mechanical loading, International Journal of Non-Linear Mechanics 65: 141-147.
[23] Filippi M., Carrera E., Zenkour A.M., 2015, Static analyses of FGM beams by various theories and finite elements, Composites Part B: Engineering 72: 1-9.
[24] Vo T.P., Thai H.-T., Nguyen T.-K., Inam F., Lee J., 2015, Static behaviour of functionally graded sandwich beams using a quasi-3D theory, Composites Part B: Engineering 68: 59-74.
[25] Arefi M., Zenkour A.M., 2017, Electro-magneto-elastic analysis of a three-layer curved beam, Smart Structures and Systems 19(6): 695-703.
[26] Arefi M., Zenkour A.M., 2017, Size-dependent electro-elastic analysis of a sandwich microbeam based on higher-order sinusoidal shear deformation theory and strain gradient theory, Journal of Intelligent Material Systems and Structures 29(7): 1394-1406.
[27] Ebrahimi F., Jafari A., 2018, A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities, Mechanics of Advanced Materials and Structures 25(3): 212-224.
[28] Oh S.-Y., Librescu L., Song O., 2003, Vibration of turbomachinery rotating blades made-up of functionally graded materials and operating in a high temperature field, Acta Mechanica 166(1): 69-87.
[29] Karami Khorramabadi M., 2009, Free vibration of functionally graded beams with piezoelectric layers subjected to axial load, Journal of Solid Mechanics 1(1): 22-28.
[30] Shahba A., Attarnejad R., Marvi M.T., Hajilar S., 2011, Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites Part B: Engineering 42(4): 801-808.
[31] Yas M.H., Kamarian S., Jam J.E., Pourasghar A., 2011, Optimization of functionally graded beams resting on elastic foundations, Journal of Solid Mechanics 3(4): 365-378.
[32] Huang Y., Yang L.-E., Luo Q.-Z., 2013, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Composites Part B: Engineering 45(1): 1493-1498.
[33] Mashat D.S., Carrera E., Zenkour A.M., Al Khateeb S.A., Filippi M., 2014, Free vibration of FGM layered beams by various theories and finite elements, Composites Part B: Engineering 59: 269-278.
[34] Jafari-Talookolaei R.A., 2015, Analytical solution for the free vibration characteristics of the rotating composite beams with a delamination, Aerospace Science and Technology 45: 346-358.
[35] Oh Y., Yoo H.H., 2016, Vibration analysis of rotating pretwisted tapered blades made of functionally graded materials, International Journal of Mechanical Sciences 119: 68-79.
[36] Carrera E., Pagani A., Banerjee J.R., 2016, Linearized buckling analysis of isotropic and composite beam-columns by Carrera Unified Formulation and dynamic stiffness method, Mechanics of Advanced Materials and Structures 23(9): 1092-1103.
[37] Rafiee M., Nitzsche F., Labrosse M., 2017, Dynamics, vibration and control of rotating composite beams and blades: A critical review, Thin-Walled Structures 119: 795-819.
[38] Musuva M., Mares C., 2015, The wavelet finite element method in the dynamic analysis of a functionally graded beam resting on a viscoelastic foundation subjected to a moving load, European Journal of Computational Mechanics 24(5): 171-209.
[39] Paul A., Das D., 2018, Free vibration behavior of tapered functionally graded material beam in thermal environment considering geometric non-linearity, shear deformability and temperature-dependent thermal conductivity, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications.
[40] Shankar K.A., Pandey M., 2016, Nonlinear dynamic analysis of cracked cantilever beam using reduced order model, Procedia Engineering 144: 1459-1468.
[41] Babu K.R.P., Kumar B.R., Rao K.M., 2016, Modal testing and finite element analysis of crack effects on turbine blades, Journal of Solid Mechanics 8(3): 614-624.
[42] Beglinger V., Bolleter U., Locher W.E., 1976, Effects of shear deformation, rotary inertia, and elasticity of the support on the resonance frequencies of short cantilever beams, Journal of Engineering for Power 98(1): 79-87.
[43] Reinhardt A.K., Kadambi J.R., Quinn R.D., 1995, Laser vibrometry measurements of rotating blade vibrations, Journal of Engineering for Gas Turbines and Power 117(3): 484-488.
[44] Touloukian Y.S., 1967, Thermophysical Properties of High Temperature Solid Materials: Elements, A.F.M. Laboratory, Macmillan.
[45] Kordkheili S.A.H., Naghdabadi R., 2007, Thermoelastic analysis of a functionally graded rotating disk, Composite Structures 79(4): 508-516.
[46] Cook R.D., 2001, Concepts and Applications of Finite Element Analysis, Wiley.
[47] Shen H.-S., Li S.-R., 2008, Postbuckling of sandwich plates with FGM face sheets and temperature-dependent properties, Composites Part B: Engineering 39(2): 332-344.