330
8 Thermoelastic Vibrations of Timoshenko Microbeams
42. Massalas, C.V., Kalpakidis, V.K.: Coupled thermoelastic vibrations of a simply supported
beam. J. Sound Vib. 88(3), 425–429 (1983)
43. Shieh, R.C.: Thermoelastic vibration and damping for circular Timoshenko beams. J. Appl.
Mech. 42(2), 405–410 (1975)
44. Parayila, D.V., Kulkarnia, S.S., Pawaskara, D.N.: Analytical and numerical solutions for thick
beams with thermoelastic damping. Int. J. Mech. Sci. 94–95, 10–19 (2015)
45. Prevost, J.H., Tao, D.: Finite element analysis of dynamic coupled thermoelasticity problems
with relaxation times. J. Appl. Mech. 50(4), 817–822 (1983)
46. Prabhakar, S., Vengallatore, S.: Theory of thermoelastic damping in micromechanical resonators with two-dimensional heat conduction. J. Microelectromech. Sys. 17(2), 494–502
(2008)
47. Silver, M.J., Peterson, L.D., Erwin, R.S.: Predictive elastothermodynamic damping in finite
element models using a perturbation formulation. AIAA 43(12), 2646–2653 (2005)
48. Sun, Y., Fang, D., Soh, A.K.: Thermoelastic damping in micro-beam resonators. Int. J. Sol.
Struct. 43, 3213–3229 (2006)
49. Lepage, S.: Stochastic finite element method for the modeling of thermoelastic damping
in micro-resonators, Ph.D. thesis, Vibrations et Identification des Structures, Département
d’Aérospatiale et Méchanique. Université de Liége, Liége (2006)
50. Guo, X., Yi, Y.B., Pourkamali, S.: A finite element analysis of thermoelastic damping in
vented MEMS beam resonators. Int. J. Mech. Sci. 74, 73–82 (2013)
51. De, S.K., Aluru, N.R.: Theory of thermoelastic damping in electrostatically actuated
microstructures. Phys. Rev. B 74, 14305 (2006)
52. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory
and experiments. Acta Metall. Mater. 42, 475–487 (1994)
53. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain
gradient elasticity. J. Mech. Phys. Soli 51, 1477–1508 (2003)
54. Liu, H.K., Pan, C.H., Liu, P.P.: Dimension effect on mechanical behavior of silicon microcantilever beams. Measurement 41, 885–895 (2008)
55. Tsiatas, G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J.
Sol. Struct. 46(13), 2757–2764 (2009)
56. Wang, F.C., Yang, F.Q., Zhao, Y.P.: Size effect on the coalescence-induced self-propelled
droplet. Appl. Phys. Lett. 98(5), 053112 (2011)
57. Brezny, R., Green, D.J.: Characterization of edge effects in cellular materials. J. Mater. Sci.
25, 4571–4578 (1990)
58. Onck, P.R., Andrews, E.W., Gibson, L.J.: Size effects in ductile cellular solids. Part I: modeling. Int. J. Mech. Sci. 43(3), 681–699 (2001)
59. Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Hermann et Fils, Paris (1909)
60. Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient
elasticity theory. Euro. J. Mech. A/Sol. 29, 591–599 (2010)
61. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Rat. Mech.
Anal. 11, 415–448 (1962)
62. Yang, F., Chong, M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for
elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)
63. Park, S.K., Gao, X.L.: Bernoulli-Euler beam model based on a modified couple stress theory.
Micromech. Microeng. 16(11), 2355–2359 (2006)
64. 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)
65. Kong, S., Zhou, S., Nie, Z., Wang, K.: Static and dynamic analysis of micro beams based on
strain gradient elasticity theory. Int. J. Eng. Sci. 47, 487–498 (2009)
66. Kahrobaiyan, M.H., Asghari, M., Rahaeifard, M., Ahmadian, M.T.: Investigation of the sizedependent dynamic characteristics of atomic force microscope microcantilevers based on the
modified couple stress theory. Int. J. Eng. Sci. 48(12), 1985–1994 (2010)
67. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H., Rahaeifard, M.: On the size-dependent
behavior of functionally graded micro-beams. Mater. Des. 31, 2324–2329 (2010)
8 Thermoelastic Vibrations of Timoshenko Microbeams
42. Massalas, C.V., Kalpakidis, V.K.: Coupled thermoelastic vibrations of a simply supported
beam. J. Sound Vib. 88(3), 425–429 (1983)
43. Shieh, R.C.: Thermoelastic vibration and damping for circular Timoshenko beams. J. Appl.
Mech. 42(2), 405–410 (1975)
44. Parayila, D.V., Kulkarnia, S.S., Pawaskara, D.N.: Analytical and numerical solutions for thick
beams with thermoelastic damping. Int. J. Mech. Sci. 94–95, 10–19 (2015)
45. Prevost, J.H., Tao, D.: Finite element analysis of dynamic coupled thermoelasticity problems
with relaxation times. J. Appl. Mech. 50(4), 817–822 (1983)
46. Prabhakar, S., Vengallatore, S.: Theory of thermoelastic damping in micromechanical resonators with two-dimensional heat conduction. J. Microelectromech. Sys. 17(2), 494–502
(2008)
47. Silver, M.J., Peterson, L.D., Erwin, R.S.: Predictive elastothermodynamic damping in finite
element models using a perturbation formulation. AIAA 43(12), 2646–2653 (2005)
48. Sun, Y., Fang, D., Soh, A.K.: Thermoelastic damping in micro-beam resonators. Int. J. Sol.
Struct. 43, 3213–3229 (2006)
49. Lepage, S.: Stochastic finite element method for the modeling of thermoelastic damping
in micro-resonators, Ph.D. thesis, Vibrations et Identification des Structures, Département
d’Aérospatiale et Méchanique. Université de Liége, Liége (2006)
50. Guo, X., Yi, Y.B., Pourkamali, S.: A finite element analysis of thermoelastic damping in
vented MEMS beam resonators. Int. J. Mech. Sci. 74, 73–82 (2013)
51. De, S.K., Aluru, N.R.: Theory of thermoelastic damping in electrostatically actuated
microstructures. Phys. Rev. B 74, 14305 (2006)
52. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory
and experiments. Acta Metall. Mater. 42, 475–487 (1994)
53. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain
gradient elasticity. J. Mech. Phys. Soli 51, 1477–1508 (2003)
54. Liu, H.K., Pan, C.H., Liu, P.P.: Dimension effect on mechanical behavior of silicon microcantilever beams. Measurement 41, 885–895 (2008)
55. Tsiatas, G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J.
Sol. Struct. 46(13), 2757–2764 (2009)
56. Wang, F.C., Yang, F.Q., Zhao, Y.P.: Size effect on the coalescence-induced self-propelled
droplet. Appl. Phys. Lett. 98(5), 053112 (2011)
57. Brezny, R., Green, D.J.: Characterization of edge effects in cellular materials. J. Mater. Sci.
25, 4571–4578 (1990)
58. Onck, P.R., Andrews, E.W., Gibson, L.J.: Size effects in ductile cellular solids. Part I: modeling. Int. J. Mech. Sci. 43(3), 681–699 (2001)
59. Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Hermann et Fils, Paris (1909)
60. Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient
elasticity theory. Euro. J. Mech. A/Sol. 29, 591–599 (2010)
61. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Rat. Mech.
Anal. 11, 415–448 (1962)
62. Yang, F., Chong, M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for
elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)
63. Park, S.K., Gao, X.L.: Bernoulli-Euler beam model based on a modified couple stress theory.
Micromech. Microeng. 16(11), 2355–2359 (2006)
64. 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)
65. Kong, S., Zhou, S., Nie, Z., Wang, K.: Static and dynamic analysis of micro beams based on
strain gradient elasticity theory. Int. J. Eng. Sci. 47, 487–498 (2009)
66. Kahrobaiyan, M.H., Asghari, M., Rahaeifard, M., Ahmadian, M.T.: Investigation of the sizedependent dynamic characteristics of atomic force microscope microcantilevers based on the
modified couple stress theory. Int. J. Eng. Sci. 48(12), 1985–1994 (2010)
67. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H., Rahaeifard, M.: On the size-dependent
behavior of functionally graded micro-beams. Mater. Des. 31, 2324–2329 (2010)