Title

Nonlinear free vibration analysis of micro-beams resting on viscoelastic foundation based on the modified couple stress theory

Journal title

Archive of Mechanical Engineering

Yearbook

2017

Volume

vol. 64

Issue

No 2

Affiliation

Jam, Jafar Eskandari : Composite Materials and Technology Cente, Malek Ashtar University of Technology, Tehran, Iran ; Noorabadi, Milad : Composite Materials and Technology Cente, Malek Ashtar University of Technology, Tehran, Iran ; Namdaran, Nader : Composite Materials and Technology Cente, Malek Ashtar University of Technology, Tehran, Iran

Authors

Jam, Jafar Eskandari ; Noorabadi, Milad ; Namdaran, Nader

Keywords

nonlinear free vibration ; size dependent ; modified couple stress theory ; Euler-Bernoulli beam model ; Galerkin method

Divisions of PAS

Nauki Techniczne

Coverage

239-256

Publisher

Polish Academy of Sciences, Committee on Machine Building

Bibliography

[1] W. Faris, E. Abdel-Rahman, and A. Nayfeh. Mechanical behavior of an electrostatically actuated micropump. In 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, Colorado, 22-25 April 2002. doi: 10.2514/6.2002-1303.
[2] X.M. Zhang, F.S. Chau, C. Quan, Y.L. Lam, and A.Q. Liu. A study of the static characteristics of a torsional micromirror. Sensors and Actuators A: Physical, 90(1):73–81, 2001. doi: 10.1016/S0924-4247(01)00453-8.
[3] X. Zhao, E. M. Abdel-Rahman, and A.H. Nayfeh. A reduced-order model for electrically actuated microplates. Journal of Micromechanics and Microengineering, 14(7):900–906, 2004. doi: 10.1088/0960-1317/14/7/009.
[4] H.A.C. Tilmans and R. Legtenberg. Electrostatically driven vacuum-encapsulated polysilicon resonators: Part II. Theory and performance. Sensors and Actuators A: Physical, 45(1):67–84, 1994. doi: 10.1016/0924-4247(94)00813-2.
[5] N.A. Fleck, G.M. Muller, M.F. Ashby, and J.W. Hutchinson. Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia, 42(2):475–487, 1994. doi: 10.1016/0956-7151(94)90502-9.
[6] J.S. Stölken and A.G. Evans. A microbend test method for measuring the plasticity length scale. Acta Materialia, 46(14):5109–5115, 1998. doi: 10.1016/S1359-6454(98)00153-0.
[7] A.C.l. Eringen. Nonlocal polar elastic continua. International Journal of Engineering Science, 10(1):1–16, 1972. doi: 10.1016/0020-7225(72)90070-5.
[8] D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, and P. Tong. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51(8):1477–1508, 2003. doi: 10.1016/S0022-5096(03)00053-X.
[9] R.A. Toupin. Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis, 11(1):385–414, 1962. doi: 10.1007/BF00253945.
[10] F. Yang, A.C.M. Chong, D.C.C. Lam, and P. Tong. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10):2731–2743, 2002. doi: 10.1016/S0020-7683(02)00152-X.
[11] J.N. Reddy. Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. International Journal of Engineering Science, 48(11):1507–1518, 2010. doi: 10.1016/j.ijengsci.2010.09.020.
[12] J.N. Reddy. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45(2-8):288–307, 2007. doi: 10.1016/j.ijengsci.2007.04.004.
[13] E. Taati, M. Molaei, and J.N. Reddy. Size-dependent generalized thermoelasticity model for Timoshenko micro-beams based on strain gradient and non-Fourier heat conduction theories. Composite Structures, 116:595–611, 2014. doi: 10.1016/j.compstruct.2014.05.040.
[14] H.M. Sedighi, A. Koochi, and M. Abadyan. Modeling the size dependent static and dynamic pull-in instability of cantilever nanoactuator based on strain gradient theory. International Journal of Applied Mechanics, 06(05):1450055, 2014. doi: 10.1142/S1758825114500550.
[15] M. Molaei, M.T. Ahmadian, and E. Taati. Effect of thermal wave propagation on thermoelastic behavior of functionally graded materials in a slab symmetrically surface heated using analytical modeling. Composites Part B: Engineering, 60:413–422, 2014. doi: 10.1016/j.compositesb.2013.12.070.
[16] M. Molaei Najafabadi, E. Taati, and H. Basirat Tabrizi. Optimization of functionally graded materials in the slab symmetrically surface heated using transient analytical solution. Journal of Thermal Stresses, 37(2):137–159, 2014. doi: 10.1080/01495739.2013.839617.
[17] L.L. Ke and Y.S. Wang. Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Composite Structures, 93(2):342–350, 2011. doi: 10.1016/j.compstruct.2010.09.008.
[18] L.L. Ke, Y.S. Wang, J. Yang, and S. Kitipornchai. Nonlinear free vibration of size-dependent functionally graded microbeams. International Journal of Engineering Science, 50(1):256–267, 2012. doi: 10.1016/j.ijengsci.2010.12.008.
[19] S.K. Park and X.L. Gao. Bernoulli–Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering, 16(11):2355, 2006. http://stacks.iop.org/0960-1317/16/i=11/a=015.
[20] S. Kong, S. Zhou, Z. Nie, and K. Wang. The size-dependent natural frequency of Bernoulli–Euler micro-beams. International Journal of Engineering Science, 46(5):427–437, 2008. doi: 10.1016/j.ijengsci.2007.10.002.
[21] E. Taati, M. Nikfar, and M.T. Ahmadian. Formulation for static behavior of the viscoelastic Euler-Bernoulli micro-beam based on the modified couple stress theory. In ASME 2012 International Mechanical Engineering Congress and Exposition; Vol. 9: Micro- and Nano-Systems Engineering and Packaging, Parts A and B, pages 129–135, Houston, Texas, USA, 9-15 November 2012. doi: 10.1115/IMECE2012-86591.
[22] H.M. Ma, X.L. Gao, and J.N. Reddy. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids, 56(12):3379–3391, 2008. doi: 10.1016/j.jmps.2008.09.007.
[23] M. Asghari and E. Taati. A size-dependent model for functionally graded micro-plates for mechanical analyses. Journal of Vibration and Control, 19(11):1614–1632, 2013. doi: 10.1177/1077546312442563.
[24] J.N. Reddy and J. Kim. A nonlinear modified couple stress-based third-order theory of functionally graded plates. Composite Structures, 94(3):1128–1143, 2012. doi: 10.1016/j.compstruct.2011.10.006.
[25] E. Taati, M. Molaei Najafabadi, and H. Basirat Tabrizi. Size-dependent generalized thermoelasticity model for Timoshenko microbeams. Acta Mechanica, 225(7):1823–1842, 2014. doi: 10.1007/s00707-013-1027-7.
[26] H.T. Thai and D.H. Choi. Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory. Composite Structures, 95:142–153, 2013. doi: 10.1016/j.compstruct.2012.08.023.
[27] E. Taati. Analytical solutions for the size dependent buckling and postbuckling behavior of functionally graded micro-plates. International Journal of Engineering Science, 100:45–60, 2016. doi: 10.1016/j.ijengsci.2015.11.007.
[28] M.A. Eltaher, A.E. Alshorbagy, and F.F. Mahmoud. Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Applied Mathematical Modelling, 37(7):4787–4797, 2013. doi: 10.1016/j.apm.2012.10.016.
[29] B. Akgöz and Ö. Civalek. Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity. Composite Structures, 134:294–301, 2015. doi: 10.1016/j.compstruct.2015.08.095.
[30] N. Togun and S.M. Bağdatlı. Nonlinear vibration of a nanobeam on a Pasternak elastic foundation based on non-local Euler-Bernoulli beam theory. Mathematical and Computational Applications, 21(1):3, 2016.
[31] B. Akgöz and Ö. Civalek. A novel microstructure-dependent shear deformable beam model. International Journal of Mechanical Sciences, 99:10–20, 2015. doi: 10.1016/j.ijmecsci.2015.05.003.
[32] B. Akgöz and Ö. Civalek. A new trigonometric beam model for buckling of strain gradient microbeams. International Journal of Mechanical Sciences, 81:88–94, 2014. doi: 10.1016/j.ijmecsci.2014.02.013.
[33] N. Shafiei, M. Kazemi, and M. Ghadiri. Nonlinear vibration of axially functionally graded tapered microbeams. International Journal of Engineering Science, 102:12–26, 2016. doi: 10.1016/j.ijengsci.2016.02.007.
[34] R. Ansari, V. Mohammadi, M.F. Shojaei, R. Gholami, and H. Rouhi. Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory. European Journal of Mechanics – A/Solids, 45:143–152, 2014. doi: 10.1016/j.euromechsol.2013.11.002.
[35] Yong-Gang Wang, Wen-Hui Lin, and Ning Liu. Nonlinear free vibration of a microscale beam based on modified couple stress theory. Physica E: Low-dimensional Systems and Nanostructures, 47:80–85, 2013. doi: 10.1016/j.physe.2012.10.020.

Date

2017

Type

Artykuły / Articles

Identifier

DOI: 10.1515/meceng-2017-0015 ; ISSN 0004-0738, e-ISSN 2300-1895

Source

Archive of Mechanical Engineering; 2017; vol. 64; No 2; 239-256

References

Kong (2008), The size - dependent natural frequency of Bernoulli Euler micro - beams, International Journal of Engineering Science, 20, 427, doi.org/10.1016/j.ijengsci.2007.10.002 ; Reddy (2010), Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, International Journal of Engineering Science, 11, 1507, doi.org/10.1016/j.ijengsci.2010.09.020 ; Sedighi (2014), Modeling the size dependent static and dynamic pull - in instability of cantilever nanoactuator based on strain gradient theory, International Journal of Applied Mechanics, 14, 1450055, doi.org/10.1142/S1758825114500550 ; Thai (2013), Size - dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory, Composite Structures, 26, 142, doi.org/10.1016/j.compstruct.2012.08.023 ; Akgöz (2015), A novel microstructure - dependent shear deformable beam model, International Journal of Mechanical Sciences, 31, 10, doi.org/10.1016/j.ijmecsci.2015.05.003 ; Park (2006), Bernoulli Euler beam model based on a modified couple stress theory http stacks iop org, Journal of Micromechanics and Microengineering, 19, 2355. ; Taati (2014), Size - dependent generalized thermoelasticity model for Timoshenko micro - beams based on strain gradient and non - Fourier heat conduction theories, Composite Structures, 13, 116, doi.org/10.1016/j.compstruct.2014.05.040 ; Zhang (2001), A study of the static characteristics of a torsional micromirror A :, Sensors and Actuators Physical, 2, 73, doi.org/10.1016/S0924-4247(01)00453-8 ; Ke (2012), Nonlinear free vibration of size - dependent functionally graded microbeams, International Journal of Engineering Science, 18, 256, doi.org/10.1016/j.ijengsci.2010.12.008 ; Toupin (1962), Elastic materials with couple - stresses for and, Archive Rational Mechanics Analysis, 9, 385, doi.org/10.1007/BF00253945 ; Reddy (2012), A nonlinear modified couple stress - based third - order theory of functionally graded plates, Composite Structures, 24, 1128, doi.org/10.1016/j.compstruct.2011.10.006 ; Yong (2013), Gang Wen Hui Nonlinear free vibration of a microscale beam based on modified couple stress theory : Low - dimensional Systems and Nanostructures, Physica E, 35, 80, doi.org/10.1016/j.physe.2012.10.020 ; Fleck (1994), Strain gradient plasticity : theory and experiment, Acta Metallurgica et Materialia, 5, 475, doi.org/10.1016/0956-7151(94)90502-9 ; Taati (2016), Analytical solutions for the size dependent buckling and postbuckling behavior of functionally graded micro - plates ijengsci, International Journal of Engineering Science, 27, 45. ; Ma (2008), A microstructure - dependent Timoshenko beam model based on a modified couple stress theory the and of, Journal of Mechanics Physics Solids, 22, 3379, doi.org/10.1016/j.jmps.2008.09.007 ; Yang (2002), Couple stress based strain gradient theory for elasticity and, International Journal of Solids Structures, 10, 2731, doi.org/10.1016/S0020-7683(02)00152-X ; Akgöz (2014), A new trigonometric beam model for buckling of strain gradient microbeams, International Journal of Mechanical Sciences, 32, 81, doi.org/10.1016/j.ijmecsci.2014.02.013 ; Tilmans (1994), Electrostatically driven vacuum - encapsulated polysilicon resonators : Part II Theory and performance A :, Sensors and Actuators Physical, 4, 67, doi.org/10.1016/0924-4247(94)00813-2 ; Zhao (2004), A reduced - order model for electrically actuated microplates, Journal of Micromechanics and Microengineering, 3, 900, doi.org/10.1088/0960-1317/14/7/009 ; Ansari (2014), Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory of Mechanics A /, European Journal Solids, 34, 143, doi.org/10.1016/j.euromechsol.2013.11.002 ; Ke (2011), Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory, Composite Structures, 17, 342, doi.org/10.1016/j.compstruct.2010.09.008 ; Molaei Najafabadi (2014), Tabrizi Optimization of functionally graded materials in the slab symmetrically surface heated using transient analytical solution, Journal of Thermal Stresses, 16, 137, doi.org/10.1080/01495739.2013.839617 ; Reddy (2007), Nonlocal theories for bending , buckling and vibration of beams, International Journal of Engineering Science, 12, 288, doi.org/10.1016/j.ijengsci.2007.04.004 ; Eringen (1972), Nonlocal polar elastic continua, International Journal of Engineering Science, 10, 1, doi.org/10.1016/0020-7225(72)90070-5 ; Asghari (2013), A size - dependent model for functionally graded micro - plates for mechanical analyses, Journal of Vibration and Control, 23, 1614, doi.org/10.1177/1077546312442563 ; Lam (2003), Experiments and theory in strain gradient elasticity the and of, Journal of Mechanics Physics Solids, 8, 1477, doi.org/10.1016/S0022-5096(03)00053-X ; Shafiei (2016), Nonlinear vibration of axially functionally graded tapered microbeams, International Journal of Engineering Science, 33, 12, doi.org/10.1016/j.ijengsci.2016.02.007