Modified couple stress model for thermoelastic microbeams due to temperature pulse heating

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt


In this research, vibration frequency analysis of a microbeam under a temperature pulse is investigated. In view of the modified couple stress theory and generalized Lord-Shulman (LS) hyperbolic heat conduction model with a single relaxation time, the thermoelastic coupled equations for clamped microbeams have been determined. The analytical terminologies for temperature, deflection, axial displacement, dilatation, flexure moment, couple stress, and axial stress in the microbeam have been acquired utilizing Laplace transform technique. Furthermore, examinations have been displayed in graphs to figure the effect of particular boundaries, for example, the couple stress and pulse of temperature on every one of the thoughts about factors. The couple stress parameter significantly affects all the field distributions. The higher temperature pulses show many disagreements between the results of the present couple stress model and the classical LS one. Alternate estimations of thermal relaxation time have been utilized to the curves anticipated by two unique theories of thermoelasticity that gotten as exceptional instances of the current LS model. Numerical inferences explain that evaluation of deflection anticipated by brand new theory is lower than that of classical LS one.


[1]           A. M. Zenkour, A. E. Abouelregal, Effect of harmonically varying heat on FG nanobeams in the context of a nonlocal two-temperature thermoelasticity theory, European Journal of Computational Mechanics, Vol. 23, No. 1-2, pp. 1-14, 2014.
[2]           E. Carrera, A. Abouelregal, I. Abbas, A. Zenkour, Vibrational analysis for an axially moving microbeam with two temperatures, Journal of Thermal Stresses, Vol. 38, No. 6, pp. 569-590, 2015.
[3]           A. E. Abouelregal, A. M. Zenkour, Thermoelastic problem of an axially moving microbeam subjected to an external transverse excitation, Journal of Theoretical and Applied Mechanics, Vol. 53, No. 1, pp. 167-178, 2015.
[4]           A. Abouelregal, A. 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, Vol. 38, No. M2, pp. 321, 2014.
[5]           W. Duan, C. M. Wang, Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory, Nanotechnology, Vol. 18, No. 38, pp. 385704, 2007.
[6]           Q. Wang, K. Liew, Application of nonlocal continuum mechanics to static analysis of micro-and nano-structures, Physics Letters A, Vol. 363, No. 3, pp. 236-242, 2007.
[7]           G. Rezazadeh, F. Khatami, A. Tahmasebi, Investigation of the torsion and bending effects on static stability of electrostatic torsional micromirrors, Microsystem Technologies, Vol. 13, No. 7, pp. 715-722, 2007.
[8]           J.-Y. Chen, Y.-C. Hsu, S.-S. Lee, T. Mukherjee, G. K. Fedder, Modeling and simulation of a condenser microphone, Sensors and Actuators A: Physical, Vol. 145, pp. 224-230, 2008.
[9]           A. C. Chong, D. C. Lam, Strain gradient plasticity effect in indentation hardness of polymers, Journal of Materials Research, Vol. 14, No. 10, pp. 4103-4110, 1999.
[10]         A. W. McFarland, J. S. Colton, Role of material microstructure in plate stiffness with relevance to microcantilever sensors, Journal of Micromechanics and Microengineering, Vol. 15, No. 5, pp. 1060, 2005.
[11]         J. Reddy, Microstructure-dependent couple stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids, Vol. 59, No. 11, pp. 2382-2399, 2011.
[12]         F. Yang, A. Chong, D. C. C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International journal of solids and structures, Vol. 39, No. 10, pp. 2731-2743, 2002.
[13]         M. Malikan, Buckling analysis of a micro composite plate with nano coating based on the modified couple stress theory, Journal of Applied and Computational Mechanics, Vol. 4, No. 1, pp. 1-15, 2018.
[14]         M. Malikan, Electro-thermal buckling of elastically supported double-layered piezoelectric nanoplates affected by an external electric voltage, Multidiscipline Modeling in Materials and Structures, 2019.
[15]         B. Akgöz, Ö. Civalek, A size-dependent shear deformation beam model based on the strain gradient elasticity theory, International Journal of Engineering Science, Vol. 70, pp. 1-14, 2013.
[16]         B. Akgöz, Ö. Civalek, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science, Vol. 49, No. 11, pp. 1268-1280, 2011.
[17]         M. Malikan, V. B. Nguyen, R. Dimitri, F. Tornabene, Dynamic modeling of non-cylindrical curved viscoelastic single-walled carbon nanotubes based on the second gradient theory, Materials Research Express, Vol. 6, No. 7, pp. 075041, 2019.
[18]         M. Malikan, R. Dimitri, F. Tornabene, Transient response of oscillated carbon nanotubes with an internal and external damping, Composites Part B: Engineering, Vol. 158, pp. 198-205, 2019.
[19]         G.-L. She, F.-G. Yuan, Y.-R. Ren, H.-B. Liu, W.-S. Xiao, Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory, Composite Structures, Vol. 203, pp. 614-623, 2018.
[20]         A. C. Eringen, D. Edelen, On nonlocal elasticity, International journal of engineering science, Vol. 10, No. 3, pp. 233-248, 1972.
[21]         A. C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of applied physics, Vol. 54, No. 9, pp. 4703-4710, 1983.
[22]         M. Malikan, V. B. Nguyen, A novel one-variable first-order shear deformation theory for biaxial buckling of a size-dependent plate based on Eringen’s nonlocal differential law, World Journal of Engineering, 2018.
[23]         M. Malikan, On the buckling response of axially pressurized nanotubes based on a novel nonlocal beam theory, Journal of Applied and Computational Mechanics, Vol. 5, No. 1, pp. 103-112, 2019.
[24]         R. Ansari, J. Torabi, Nonlocal vibration analysis of circular double-layered graphene sheets resting on an elastic foundation subjected to thermal loading, Acta Mechanica Sinica, Vol. 32, No. 5, pp. 841-853, 2016.
[25]         M. Ahmad Pour, M. Golmakani, M. Malikan, Thermal buckling analysis of circular bilayer graphene sheets resting on an elastic matrix based on nonlocal continuum mechanics, Journal of Applied and Computational Mechanics, Vol. 7, No. 4, pp. 1862-1877, 2021.
[26]         R. Mindlin, H. Tiersten, Effects of couple-stresses in linear elasticity, Columbia Univ New York,  pp. 1962.
[27]         R. Toupin, Elastic materials with couple-stresses, Archive for rational mechanics and analysis, Vol. 11, No. 1, pp. 385-414, 1962.
[28]         K. Rajneesh, Response of thermoelastic beam due to thermal source in modified couple stress theory, CMST, Vol. 22, No. 2, pp. 95-101, 2016.
[29]         S. Park, X. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, Vol. 16, No. 11, pp. 2355, 2006.
[30]         A. Abouelregal, Response of thermoelastic microbeams to a periodic external transverse excitation based on MCS theory, Microsystem Technologies, Vol. 24, No. 4, pp. 1925-1933, 2018.
[31]         A. Zenkour, Refined two-temperature multi-phase-lags theory for thermomechanical response of microbeams using the modified couple stress analysis, Acta Mechanica, Vol. 229, No. 9, pp. 3671-3692, 2018.
[32]         M. M. Benhamed, A. Abouelregal, Influence of temperature pulse on a nickel microbeams under couple stress theory, Journal of Applied and Computational Mechanics, Vol. 6, No. 4, pp. 777-787, 2020.
[33]         A. M. Zenkour, Modified couple stress theory for micro-machined beam resonators with linearly varying thickness and various boundary conditions, Archive of Mechanical Engineering, Vol. 65, No. 1, 2018.
[34]         M. Shishesaz, M. Hosseini, K. Naderan Tahan, A. Hadi, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica, Vol. 228, No. 12, pp. 4141-4168, 2017.
[35]         A. Hadi, M. Z. Nejad, M. Hosseini, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, Vol. 128, pp. 12-23, 2018.
[36]         A. Hadi, M. Z. Nejad, A. Rastgoo, M. Hosseini, Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory, Steel and Composite Structures, An International Journal, Vol. 26, No. 6, pp. 663-672, 2018.
[37]         M. Hosseini, A. Hadi, A. Malekshahi, M. Shishesaz, A review of size-dependent elasticity for nanostructures, Journal of Computational Applied Mechanics, Vol. 49, No. 1, pp. 197-211, 2018.
[38]         M. M. Khoram, M. Hosseini, A. Hadi, M. Shishehsaz, Bending analysis of bidirectional FGM Timoshenko nanobeam subjected to mechanical and magnetic forces and resting on Winkler–Pasternak foundation, International Journal of Applied Mechanics, Vol. 12, No. 08, pp. 2050093, 2020.
[39]         H. W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, Vol. 15, No. 5, pp. 299-309, 1967.
[40]         A. R. Hadjesfandiari, G. F. Dargush, Couple stress theory for solids, International Journal of Solids and Structures, Vol. 48, No. 18, pp. 2496-2510, 2011.
[41]         G. Hoing, A method for the numerical inversion of the Laplace transform, J. Comp. Appl. Math., Vol. 10, pp. 113-132, 1984.
[42]         D. Y. Tzou, Experimental support for the lagging behavior in heat propagation, Journal of thermophysics and heat transfer, Vol. 9, No. 4, pp. 686-693, 1995.
Volume 53, Issue 1
March 2022
Pages 83-101
  • Receive Date: 19 November 2021
  • Revise Date: 17 March 2022
  • Accept Date: 17 March 2022