Mechanical Behavior of a FGM Capacitive Micro-Beam Subjected to a Heat Source

Document Type: Research Paper

Authors

Mechanical Engineering Department, Urmia University

Abstract

This paper presents mechanical behavior of a functionally graded (FG) cantilever micro-beam subjected to a nonlinear electrostatic pressure and thermal moment considering effects of material length scale parameters. Material properties through the beam thickness direction are graded. The top surface of the micro-beam is made of pure metal and the bottom surface from a mixture of metal and ceramic. The material properties through the thickness direction follow the volume fraction of the constitutive materials in exponential function form. The governing nonlinear thermo-electro-mechanical differential equation based on Euler-Bernoulli beam theory assumptions is derived using modified couple stress theory (MCST) and is solved using the Galerkin based weighted residual method. The effects of the electrostatic pressure and temperature changes on the deflection and stability of the FGM micro-beam, having various ceramic constituent percents, are studied. The obtained results are compared with the results predicted by classic theory (CT) and for some cases are verified with those reported in the literature.

Keywords


[1] Suresh S., Mortensen A., 1998, Fundamentals of Functionally Graded Materials, IQM communications, London.

[2] Kapuria S., Bhattacharyya M., Kumar A.N., 2008, Bending and free vibration response of layered functionally graded beam: A theorical model and its experimental validation, Composite Structures 82: 390-402.

[3] Khalili S.M.R., Jafari A., Eftekhari S.A., 2010, A mixture Ritz-DQ method for vibration of functionally graded beams carrying moving loads, Composite Structures 92: 2497-2511.

[4] Sadeghian H., RezaZadeh G., M.Osterberg P., 2007, Application of the generalized differential quadrature method of the study of pull-in phenomenon of MEMS switches, Journal of Micromechanics 16(6): 1334-1340.

[5] Sankar B.V., 2001, An elasticity solution for functionally graded beams, Composite Science Technology 61: 689-696.

[6] Sankar B.V., Taeng J.T., 2002, Thermal stresses in functionally graded beams, AIAA Journal 40:1228-1232.

[7] Venkataraman S., Sankar B.V., 2003, Elasticity solution for stresses in a sandwich beam with functionally graded core, AIAA Journal 41: 2501-2505.

[8] Massalas C. V., Kalpakidis V. K., 1983, Coupled thermoelastic vibration of a simply supported beam, Journal of Sound Vibration 88: 425-429.

[9] Chakraborty A., Gopalakrishnan S., Reddy J.N., 2003, A new beam finite element for the analysis of functionally graded materials, International Journal of Mechanical Science 45: 519-539 .

[10] Alibeigloo A. 2010, Thermoelasticity analysis of functionally graded beam with integrated surface piezoelectric layers, Journal of Composite Structures 92: 1535-1543.

[11] Babaei M.H., Abbasi M., Eslami M.R., 2008, Coupled thermoelasticity of functionally graded beams, Journal of Thermal Stresses 31: 680–697.

[12] Pamidighantam S., Puers R., Baert K., A C Tilmans H., 2002, Pull-in voltage analysis of electrostatically actuated beam structures with fixed–fixed and fixed–free end conditions, Journal of Micromechanics and Microengineering 12: 458-464.

[13] Ramezani A., Alasty A., Akbari A., 2007, Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular surface forces, Journal of Solids and Structures 44: 4925-4941.

[14] Hasanyan D.J., Batra R.C., Harutyunyan S., 2008, Pull-in instabilities in functionally graded micro-thermoelectromechanical systems, Journal of Thermal Stress 31: 1006-1021.

[15] Rezazadeh M. Pashapour F. Abdolkarimzadeh 2011, Mechanical behavior of bi-layer cantilever micro-beam subjected to electrostatical force, mechanical shock and thermal moment, International Journal of Applied Mechanics 3(3): 543-561.

[16] Rezazadeh G., Keyvani A., Jafarmadar S., 2012, On a MEMS based dynamic remote temperature sensor using transverse vibration of bi-layer micro-cantilever, Journal of Measurement 45 (3): 580-589.

[17] Mohammadi-alasti B., Rezazadeh G., Borghei A., Minaei S., Habibifar R., 2011, On the mechanical behavior of functionally graded micro-beam subjected to a thermal moment and nonlinear electrostatic pressure, Composite and Structures 93: 1516-1525.

[18] Kong S., Zhou S., Nie Z., Wang K., 2008, The size-dependent natural frequency of Bernoulli-Euler micro-beams, International Journal of Engineering Science 46(5): 427-437.

[19] Nix W.D., 1989. Mechanical properties of thin films. Metallurgical and Materials Transactions 20A(11): 2217-2245.

[20] Fleck N.A., Muller G.M., Ashby, M.F., Hutchinson, J.W., 1994, Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia 42(2): 475–487.

[21] Poole W.J., Ashby M.F., Fleck N.A., 1996, Micro-hardness of annealed and work-hardened copper polycrystals, Scripta Materialia 34 (4): 559-564.

[22] Yang F., Chong A.C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39 (10): 2731-2743.

[23] Abbasnejad B., Rezazadeh G., Shabani R., Stability analysis of a capacitive FGM micro-beam using modified couple stress theory, Acta Mechanica Solid and Sinica, Accepted paper.

[24] Asghari M., Ahmadian M.T., Kahrobaiyan M.H., Rahaeifard M., 2010, On the size-dependant behavior of functionally graded micro-beams, Materials and Design 31: 2324-2329.

[25] Martin H. Sadd, 2009, Elasticity, Theory, Applications, and Numerics, Second edition, Academic Press.

[26] Saeedi Vahdat A., Rezazadeh G., 2011, Effects of axial and residual stresses on thermoelastic damping in capacitive micro-beam resonators, Journal of the the Franklin Institute 348 :622-639.

[27] Toupin R.A., 1962, Elastic materials with couple stresses, Archive for Rational Mechanics and Analysis 11: 385-414.

[28] Mindlin R.D., Tiersten H.F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11: 415-448.

[29] Koiter W.T., 1964, Couple stresses in the theory of elasticity, I and II, Philosophical Transactions of the Royal Society of London B 67: 17-44.

[30] Mindlin R.D., 1964, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis 16: 51-78.

[31] Mindlin R. D., 1965, Stress functions for a Cosserat continuum, International Journal of Solids and Structures 1: 265-271.

[32] Eringen A.C., 1968, Theory of micro-polar elasticity, in: Fracture 1, edited by H. Lei-bowitz, Academic Press:621-729.

[33] Sadeghian H., Goosen H., Bossche A., Thijsse B., van Keulen F., 2011, On the size-dependent elasticity of silicon nano-cantilevers: Impact of defects, Journal of Physics D: Applied Physics 44(7): 2001-2007.