Response of Thermoelastic Beam due to Thermal Source in Modified Couple Stress Theory
Department of Mathematics, Kurukshetra University Kurukshetra Kurukshetra, India
E-mail: rajneesh_kuk@rediffmail.com
Received:
Received: 08 January 2016; revised: 14 April 2016; accepted: 26 April 2016; published online: 31 May 2016
DOI: 10.12921/cmst.2016.22.02.004
Abstract:
The present investigation deals with the problem of thermoelastic beam in the modified couple stress theory due to thermal source. The governing equations of motion for the modified couple stress theory and heat conduction equation for coupled thermoelasticity are investigated to model the vibrations in a homogeneous isotropic thin beam in a closed form by applying the Euler Bernoulli beam theory. The Laplace transform technique is used to solve the problem. The lateral deflection, thermal moment, axial stress average due to normal heat flux in the beam are derived and computed numerically. The resulting quantities are depicted graphically for a specific model. A particular case is also introduced.
Key words:
classical coupled theory, modified couple stress theory, thermoelastic beam
References:
[1] E. Cosserat, F. Cosserat, Theory of deformable bodies. Her- mann et Fils, Paris, 1909.
[2] R.A. Toupin, Elasticmaterialswithcouple-stresses,Arch.for Ratio. Mech. Analy. 11, 385-414 (1962).
[3] R.D. Mindlin, H.F. Tiersten, Effects of couple-stresses in linear elasticity, Arch. for Ratio. Mech. and Analy. 11, 415-448 (1962).
[4] F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couplestress based strain gradient theory for elasticity, Int. J. Solids Struct. 39. 2731-43. (2002).
[5] S.K. Park, X.L. Ago, Bernoulli-Euler beam model based on a modified couple stress theory, J. of Micromech. and Micro Engg., 16 2355 (2006).
[6] M. Simsek, J.N. Reddy,Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory, Int. J. of Engg. Sci. 64, 37-53 (2013).
[7] M. Mohammad-Abadi, A.R. Daneshmehr, Size dependent buckling analysis of micro beams based on modified couple stress theory with high order theories and general boundary conditions, Int. J. of Engng. Sci 74, 1-14 (2014).
[8] H. Larijani, A.H. Shahdadi, A new shear deformation model with modified couple stress theory for microplates, Acta Mech. 226 2773-2788 (2015).
[9] Y.T. Beni, F. Mehralian, H. Razavi, Free vibration analysis of size-dependent shear deformable functionally graded cylin- drical shell on the basis of modified couple stress theory, Composite Structures 120, 65-78 (2015).
[10] A.M. Dehrouyeh-Semnani M. Dehrouyeh, M. Torabi- Kafshgari, M. Nikkhah-Bahrami, A damped sandwich beam model based on symmetric-deviatoric couple stress theory, Int. J. of Engng. Sci. 92, 83-94 (2015).
[11] Y.Sun,D.Fang,M.Saka,A.K.Soh,Laser-inducedvibrations of micro-beams under different boundary conditions, Int. J. of Solids and Structures 45, 1993-2013 (2008).
[12] Y.Li,C.J.Cheng,Anonlinearmodelofthermoelasticbeams with voids, with applications J. of mech. of materials and Structures 5(5), 805-820 (2010).
[13] J.N. Sharma, Thermoelastic damping and frequency shift in Micro/Nano-Scale anisotropic beams, J. of Thermal Stresses 34, 650-666 (2011).
[14] J. Zang, Y. Fu, Pull-in analysis of electrically actuated vis- coelastic microbeams based on a modified couple stress the- ory, Meccanica 47, 1649-1658 (2012).
[15] G.Rezazadeh,A.S.Vahdat,S.Tayefeh-Rezaei,C.Cetinkaya,
Thermoelastic damping in a micro-beam resonator using mod- ified couple stress theory, Acta Mechanica 223(6), 1137-1152 (2012).
[16] X.GuoX,Y.B.Yi,S.Pourkamali,Afiniteelementanalysisof thermoelastic damping in vented MEMS beam resonators, Int. J. of Mech. Sci. 4: 73-82 (2013).
[17] A.E. Abouelregal, A.M. Zenkour, Effect of phase lags on thermoelastic functionally graded microbeams subjected to ramp-type heating, Iranian Journal of Science and Technology: Transactions of Mechanical Engineering 38(M2), 321-335 (2014).
[18] J.N.Sharma,M.Kaur,Transversevibrationsinthermoelastic- diffusive thin micro-beam resonators, J. of Thermal Stresses 37, 1265-1285 (2014).
[19] A.M.Zenkour,A.E.Abouelregal,ThermoelasticVibrationof an Axially Moving Microbeam Subjected to Sinusoidal Pulse Heating, Int. J. Str. Stab. Dyn. 15(6), 1-15 (2015).
[20] W. Nowacki, Dynamical problems of thermo diffusion in solids, Engg. Frac. Mech. 8, 261-266 (1976).
[21] S.S. Rao, Vibrations of continuous systems. John Wiley & Sons, New York 2007.
[22] I.H. EI-Sirafy, M.A. Abdou, E. Awad, Generalized lagging response of thermoelastic beams, Mathematical Problems in Engineering Article ID 780679, 1-13 (2014).
[23] G.Honig,U.Hirdes,Amethodforthenumericalinversionof the Laplace transform, J. Comput. Appl. Math. 10, 113-132 (1984).
[24] R.S. Daliwal, A. Singh, Dynamicalcoupledthermoelasticity. Hindustan Publishers, Delhi, 1980.
The present investigation deals with the problem of thermoelastic beam in the modified couple stress theory due to thermal source. The governing equations of motion for the modified couple stress theory and heat conduction equation for coupled thermoelasticity are investigated to model the vibrations in a homogeneous isotropic thin beam in a closed form by applying the Euler Bernoulli beam theory. The Laplace transform technique is used to solve the problem. The lateral deflection, thermal moment, axial stress average due to normal heat flux in the beam are derived and computed numerically. The resulting quantities are depicted graphically for a specific model. A particular case is also introduced.
Key words:
classical coupled theory, modified couple stress theory, thermoelastic beam
References:
[1] E. Cosserat, F. Cosserat, Theory of deformable bodies. Her- mann et Fils, Paris, 1909.
[2] R.A. Toupin, Elasticmaterialswithcouple-stresses,Arch.for Ratio. Mech. Analy. 11, 385-414 (1962).
[3] R.D. Mindlin, H.F. Tiersten, Effects of couple-stresses in linear elasticity, Arch. for Ratio. Mech. and Analy. 11, 415-448 (1962).
[4] F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couplestress based strain gradient theory for elasticity, Int. J. Solids Struct. 39. 2731-43. (2002).
[5] S.K. Park, X.L. Ago, Bernoulli-Euler beam model based on a modified couple stress theory, J. of Micromech. and Micro Engg., 16 2355 (2006).
[6] M. Simsek, J.N. Reddy,Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory, Int. J. of Engg. Sci. 64, 37-53 (2013).
[7] M. Mohammad-Abadi, A.R. Daneshmehr, Size dependent buckling analysis of micro beams based on modified couple stress theory with high order theories and general boundary conditions, Int. J. of Engng. Sci 74, 1-14 (2014).
[8] H. Larijani, A.H. Shahdadi, A new shear deformation model with modified couple stress theory for microplates, Acta Mech. 226 2773-2788 (2015).
[9] Y.T. Beni, F. Mehralian, H. Razavi, Free vibration analysis of size-dependent shear deformable functionally graded cylin- drical shell on the basis of modified couple stress theory, Composite Structures 120, 65-78 (2015).
[10] A.M. Dehrouyeh-Semnani M. Dehrouyeh, M. Torabi- Kafshgari, M. Nikkhah-Bahrami, A damped sandwich beam model based on symmetric-deviatoric couple stress theory, Int. J. of Engng. Sci. 92, 83-94 (2015).
[11] Y.Sun,D.Fang,M.Saka,A.K.Soh,Laser-inducedvibrations of micro-beams under different boundary conditions, Int. J. of Solids and Structures 45, 1993-2013 (2008).
[12] Y.Li,C.J.Cheng,Anonlinearmodelofthermoelasticbeams with voids, with applications J. of mech. of materials and Structures 5(5), 805-820 (2010).
[13] J.N. Sharma, Thermoelastic damping and frequency shift in Micro/Nano-Scale anisotropic beams, J. of Thermal Stresses 34, 650-666 (2011).
[14] J. Zang, Y. Fu, Pull-in analysis of electrically actuated vis- coelastic microbeams based on a modified couple stress the- ory, Meccanica 47, 1649-1658 (2012).
[15] G.Rezazadeh,A.S.Vahdat,S.Tayefeh-Rezaei,C.Cetinkaya,
Thermoelastic damping in a micro-beam resonator using mod- ified couple stress theory, Acta Mechanica 223(6), 1137-1152 (2012).
[16] X.GuoX,Y.B.Yi,S.Pourkamali,Afiniteelementanalysisof thermoelastic damping in vented MEMS beam resonators, Int. J. of Mech. Sci. 4: 73-82 (2013).
[17] A.E. Abouelregal, A.M. Zenkour, Effect of phase lags on thermoelastic functionally graded microbeams subjected to ramp-type heating, Iranian Journal of Science and Technology: Transactions of Mechanical Engineering 38(M2), 321-335 (2014).
[18] J.N.Sharma,M.Kaur,Transversevibrationsinthermoelastic- diffusive thin micro-beam resonators, J. of Thermal Stresses 37, 1265-1285 (2014).
[19] A.M.Zenkour,A.E.Abouelregal,ThermoelasticVibrationof an Axially Moving Microbeam Subjected to Sinusoidal Pulse Heating, Int. J. Str. Stab. Dyn. 15(6), 1-15 (2015).
[20] W. Nowacki, Dynamical problems of thermo diffusion in solids, Engg. Frac. Mech. 8, 261-266 (1976).
[21] S.S. Rao, Vibrations of continuous systems. John Wiley & Sons, New York 2007.
[22] I.H. EI-Sirafy, M.A. Abdou, E. Awad, Generalized lagging response of thermoelastic beams, Mathematical Problems in Engineering Article ID 780679, 1-13 (2014).
[23] G.Honig,U.Hirdes,Amethodforthenumericalinversionof the Laplace transform, J. Comput. Appl. Math. 10, 113-132 (1984).
[24] R.S. Daliwal, A. Singh, Dynamicalcoupledthermoelasticity. Hindustan Publishers, Delhi, 1980.
