Skip to main content
SpringerLink
Log in

Size-dependent generalized thermoelasticity model for Timoshenko microbeams

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A size-dependent, explicit formulation for coupled thermoelasticity addressing a Timoshenko microbeam is derived in this study. This novel model combines modified couple stresses and non-Fourier heat conduction to capture size effects in the microscale. To this purpose, a length-scale parameter as square root of the ratio of curvature modulus to shear modulus and a thermal relaxation time as the phase lag of heat flux vector are considered for predicting the thermomechanical behavior in a microscale device accurately. Governing equations and boundary conditions of motion are obtained simultaneously through variational formulation based on Hamilton’s principle. As for case study, the model is utilized for simply supported microbeams subjected to a constant impulsive force per unit length. A comparison of the results with those obtained by the classical elasticity and Fourier heat conduction theories is carried out. Findings indicate that simultaneous considering the length-scale parameter and thermal relaxation time has strong influence on the thermoelastic behavior of microbeams. In dynamic thermoelastic analysis of the microbeam, while the non-Fourier heat conduction model is employed, the modified couple stress theory predicts larger deflection compared with the classical theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pei J., Tian F., Thundat T.: Glucose biosensor based on the microcantilever. Anal. Chem. 76, 292–297 (2004)

    Article  Google Scholar 

  2. McMahan, L.E., Castleman, B.W.: Characterization of vibrating beam sensors during shock and vibration. In: Record—IEEE PLANS, Position Location and Navigation Symposium, pp. 102–110, Monterey, CA, USA (2004)

  3. Lun F.Y., Zhang P., Gao F.B., Jia H.G.: Design and fabrication of micro-optomechanical vibration sensor. Microfabr. Technol. 120, 61–64 (2006)

    Google Scholar 

  4. Yun W., Peilong D., Zhenying X., Hua Y., Jiangping W., Jingjing W.: A constitutive model for thin sheet metal in micro-forming considering first order size effects. Mater. Des. 31, 1010–1014 (2010)

    Article  Google Scholar 

  5. Fleck N.A., Muller G.M., Ashby M.F., Hutchinson J.W.: Strain gradient plasticity theory and experiment. Acta Metall. Mater. 42, 475–487 (1994)

    Article  Google Scholar 

  6. Stolken J.S., Evans A.G.: Microbend test method for measuring the plasticity length scale. Acta Mater. 46, 5109–5115 (1998)

    Article  Google Scholar 

  7. Lam D.C.C., Chong A.C.M.: Indentation model and strain gradient plasticity law for glassy polymers. J. Mater. Res. 14, 3784–3788 (1999)

    Article  Google Scholar 

  8. Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)

    Article  MATH  Google Scholar 

  9. McFarland A.W., Colton J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15, 1060–1067 (2005)

    Article  Google Scholar 

  10. Lakes, R.: Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Mühlhaus, H.B. (Ed.) Continuum Models for Materials with Microstructure, pp. 1–25. Wiley, Chichester (1995)

  11. Chen C.P., Lakes R.S.: Dynamic wave dispersion and loss properties of conventional and negative Poisson’s ratio polymeric cellular materials. Cell. Polym. 8, 343–369 (1989)

    Google Scholar 

  12. Mindlin R.D., Tiersten H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mindlin R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  14. Koiter W.T.: Couple-stresses in the theory of elasticity, I and II. Proc. K Ned Akad Wet B. 67, 17–44 (1964)

    MATH  Google Scholar 

  15. Yang F., Chong A.C.M., Lam D.C.C., Tong P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)

    Article  MATH  Google Scholar 

  16. Maurer M.J, Thompson H.A.: Non-Fourier effects at high heat flux. J. Heat Transf. 95, 284–286 (1973)

    Article  Google Scholar 

  17. Babaei M.H, Chen Z.T.: Hyperbolic heat conduction in a functionally graded hollow sphere. Int. J. Thermophys. 29, 1457–1469 (2008)

    Article  Google Scholar 

  18. Papargyri-Beskou S., Polyzos D., Beskos D.E.: Dynamic analysis of gradient elastic flexural beams. Struct. Eng. Mech. 15, 705–716 (2003)

    Article  Google Scholar 

  19. Papargyri-Beskou S., Beskos D.E.: Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates. Arch. Appl. Mech. 78, 625–635 (2008)

    Article  MATH  Google Scholar 

  20. Papargyri-Beskou S., Polyzos D., Beskos D.E.: Wave dispersion in gradient elastic solids and structures: a unified treatment. Int. J. Solids Struct. 46, 3751–3759 (2009)

    Article  MATH  Google Scholar 

  21. Papargyri-Beskou S., Giannakopoulos A.E., Beskos D.E.: Variational analysis of gradient elastic flexural plates under static loading. Int. J. Solids Struct. 47, 2755–2766 (2010)

    Article  MATH  Google Scholar 

  22. Park S.K., Gao X.L.: Bernoulli–Euler beam model based on a modified couple stress theory. Micromech. Microeng. 16, 2355–2359 (2006)

    Article  Google Scholar 

  23. Ma H.M., Gao X.L., Reddy J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids. 56, 3379–3391 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kong S., Zhou S., Nie Z., Wang K.: The size-dependent natural frequency of Bernoulli–Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008)

    Article  MATH  Google Scholar 

  25. Cheng S.H., Feng B.: Size effect in micro-scale cantilever beam bending. Acta Mech. 219, 291–307 (2011)

    Article  Google Scholar 

  26. Ma H.M., Gao X.L., Reddy J.N.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011)

    Article  MATH  Google Scholar 

  27. Gao, X.L., Huang, J.X., Reddy, J.N.: A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech (2013). doi:10.1007/s00707-013-0880-8

  28. Landau L.D., Lifshitz E.M.: Theory of Elasticity. Pergamon Press, Oxford (1959)

    Google Scholar 

  29. Manolis G.D., Beskos D.E.: Thermally induced vibrations of beam structures. Comput. Meth. Appl. Mech. Eng. 21, 337–355 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  30. Massalas C.V., Kalpakidis V.K.: Coupled thermoelastic vibration of a simply supported beam. J. Sound Vib. 88, 425–429 (1983)

    Article  MATH  Google Scholar 

  31. Massalas C.V., Kalpakidis V.K.: Coupled thermoelastic vibration of a Timoshenko beam. Lett. Appl. Eng. Sci. 22, 459–465 (1984)

    MATH  Google Scholar 

  32. Givoli D., Rand O.: Dynamic thermoelastic coupling effects in a rod. AIAA J. 33, 776–778 (1995)

    Article  Google Scholar 

  33. Lifshitz R., Roukes M.L.: Thermoelastic damping in micro- and nanomechanical systems. Phys. Rev. B. 61, 5600–5609 (2000)

    Article  Google Scholar 

  34. Rezazadeh G., Saeedivahdat A., Pesteii S.M., Farzi B.: Study of thermoelastic damping in capacitive micro-beam resonators using hyperbolic heat conduction model. Sens. Transducers J. 108, 54–72 (2009)

    Google Scholar 

  35. Vahdat A.S., Rezazadeh G.: Effects of axial and residual stresses on thermoelastic damping in capacitive micro-beam resonator. J. Franklin Inst. 348, 622–639 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Guo F.L., Rogerson G.A.: Thermoelastic coupling effect on a micro-machined beam machined beam resonator. Mech. Res. Commun. 30, 513–518 (2003)

    Article  MATH  Google Scholar 

  37. Sun Y., Fang D., Soh A.K.: Thermoelastic damping in micro-beam resonators. Int. J. Solids Struct. 43, 3213–3229 (2006)

    Article  MATH  Google Scholar 

  38. Rezazadeh G., Vahdat A., Tayefeh-rezaei S., Cetinkaya C.: Thermoelastic damping in a micro-beam resonator using modified couple stress theory. Acta Mech. 223, 1137–1152 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. Duwel A., Gorman J., Weinstein M., Borenstein J., Ward P.: Experimental study of thermoelastic damping in MEMS. Gyros Sens. Actuator A 103, 70–75 (2003)

    Article  Google Scholar 

  40. Lopez Molina J.A., Rivera M.J., Trujillo M., Berjano E.J.: Thermal modeling for pulsed radiofrequency ablation: analytical study based on hyperbolic heat conduction. Med. Phys. 36, 1112–1119 (2009)

    Article  Google Scholar 

  41. Shih T.C., Kou H.S., Liauh C.T., Lin W.L.: The impact of thermal wave characteristics on thermal dose distribution during thermal therapy: a numerical study. Med. Phys. 32, 3029–3036 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Sharif University of Technology, Azadi Ave., Tehran, Iran

    Ehsan Taati

  2. Thermoelasticity Center of Excellence, Mechanical Engineering Department, Amirkabir University of Technology, No. 424, Hafez Ave., PO Box 15875-4413, Tehran, Iran

    Masoud Molaei Najafabadi & Hassan Basirat Tabrizi

Authors
  1. Ehsan Taati
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masoud Molaei Najafabadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hassan Basirat Tabrizi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hassan Basirat Tabrizi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Taati, E., Molaei Najafabadi, M. & Basirat Tabrizi, H. Size-dependent generalized thermoelasticity model for Timoshenko microbeams. Acta Mech 225, 1823–1842 (2014). https://doi.org/10.1007/s00707-013-1027-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-013-1027-7

Keywords

Search

Navigation