ARTICLE
Free vibration of a hyper-elastic microbeam using a new “augmented Biderman model”
 
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Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran
Online publish date: 2019-07-15
Publish date: 2019-07-15
Submission date: 2018-04-08
Acceptance date: 2019-04-03
 
Journal of Theoretical and Applied Mechanics 2019;57(3):739–750
KEYWORDS
ABSTRACT
A new augmented Biderman model inspired by the modified couple stress theory has been introduced to investigate the size effect in addition to nonlinear material effects. Then, this model is used to investigate free vibration of a hyper-elastic microbeam. Classical Biderman strain energy does not include the effect of small size in hyper-elastic materials. In order to consider the effect of small size, terms inspired by the modified couple stress theory are added to the classical Biderman strain energy function. In order to provide the possibility of calculating these terms, a relation between the material constants in the hyper-elastic Biderman model and the linear elastic constants is obtained. The equations of motion of the microbeam is obtained based on the extended Hamilton principle, and then is solved using Galerkin discretization and perturbation methods. The effect of thickness to length scale ratio on the normalized frequency is studied for different modes. It is shown that when thickness gets larger in comparison with the length scale parameter, the normalized frequency tends to classical Biderman results. The results obtained are validated by results of the Runge-Kutta numerical method and indicate an excellent agreement. Mode shapes of the microbeam based on the classical and the augmented models are depicted, where the augmented model anticipates stiffer behavior for hyperelastic microbeams.
 
REFERENCES (25)
1.
Abbasi M., Mohammadi A.K., 2014, Study of the sensitivity and resonant frequency of the flexural modes of an atomic force microscopy microcantilever modeled by strain gradient elasticity theory, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 228, 1299-1310.
 
2.
Akgöz B., Civalek Ö., 2013, A size-dependent shear deformation beam model based on the strain gradient elasticity theory, International Journal of Engineering Science, 70, 1-14.
 
3.
Aranda-Ruiz J., Loya J., Fernández-Sáez J., 2012, Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory, Composite Structures, 94, 2990-3001.
 
4.
Carpi F., De Rossi D., Kornbluh R., Pelrine R.E., Sommer-Larsen P., 2011, Dielectric Elastomers as Electromechanical Transducers: Fundamentals, Materials, Devices, Models and Applications of an Emerging Electroactive Polymer Technology, Elsevier.
 
5.
Chakravarty U.K., 2014, On the resonance frequencies of a membrane of a dielectric elastomer, Mechanics Research Communications, 55, 72-76.
 
6.
Danaee Barforooshi S., Karami Mohammadi A., 2016, Study neo-Hookean and Yeoh hyper-elastic models in dielectric elastomer-based micro-beam resonators, Latin American Journal of Solids and Structures, 13, 1823-1837.
 
7.
Dubois P., Rosset S., Niklaus M., Dadras M., Shea H., 2008, Voltage control of the resonance frequency of dielectric electroactive polymer (DEAP) membranes, Journal of Microelectromechanical Systems, 17, 1072-1081.
 
8.
Eringen A.C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science, 10, 1-16.
 
9.
Feng C., Jiang L., Lau W.M., 2011, Dynamic characteristics of a dielectric elastomer-based microbeam resonator with small vibration amplitude, Journal of Micromechanics and Microengineering, 21, 16-23.
 
10.
Feng C., Yu L., Zhang W., 2014, Dynamic analysis of a dielectric elastomer-based microbeam resonator with large vibration amplitude, International Journal of Nonlinear Mechanics, 65, 63-68.
 
11.
Kahrobaiyan M.H., Asghari M., Ahmadian M.T., 2014, A Timoshenko beam model based on the modified couple stress theory, International Journal of Mechanical Sciences, 79, 75-83.
 
12.
Karami Mohammadi A., Danaee Barforooshi S., 2017, Nonlinear forced vibration analysis of dielectric elastomer based microbeam with considering Yeoh hyper-elastic model, Latin American Journal of Solids and Structures, 14, 643-656.
 
13.
Ke L.L., Wang Y.S., Yang J., Kitipornchai S., 2012, Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory, Journal of Sound and Vibration, 331, 94-106.
 
14.
Löwe C., Zhang X., Kovacs G., 2005, Dielectric elastomers in actuator technology, Advanced Engineering Materials, 7, 361-367.
 
15.
Marckmann G., Verron E., 2006, Comparison of hyperelastic models for rubber-like materials, Rubber Chemistry and Technology, 79, 835-858.
 
16.
Martins P.A.L., Natal Jorge R.M.A., Ferreira J.M.A., 2006, A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues, Strain, 42, 135-147.
 
17.
Mindlin R.D., Tiersten H.F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11, 415-448.
 
18.
Mockensturm E.M., Goulbourne N., 2006, Dynamic response of dielectric elastomers, International Journal of Non-Linear Mechanics, 41, 388-395.
 
19.
Ogden R.W., Roxburgh D.G., 1993, The effect of pre-stress on the vibration and stability of elastic plates, International Journal of Engineering Science, 31, 1611-1639.
 
20.
Perline R.E., Kornbluh R.D., Stanford P.Q., Oh S., Eckerle J., Full R.J., Rosenthal M.A., Meijer K., 2002, Dielectric elastomer artificial muscle actuator: toward biomimetic motion, Proceeding of SPIE Electroactive Polymer Actuators and Devices, 126-137.
 
21.
Soares R.M., Gonçalves P.B., 2012, Nonlinear vibrations and instabilities of a stretched hyperelastic annular membrane, International Journal of Solids Structures, 49, 514-526.
 
22.
Verron E., Khayat R.E., Derdouri A., Peseux B., 1999, Dynamic inflation of hyperelastic spherical membrane, Journal of Rheology, 43, 1083-1097.
 
23.
Wang Y.G., Lin W.H., Liu N., 2015, Nonlinear bending and post-buckling of extensible microscale beams based on modified couple stress theory, Applied Mathematical Modeling, 39, 117-127.
 
24.
Yang F., Chong A., Lam D., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39, 2731-2743.
 
25.
Zhang G., Gaspar J., Chu V., Conde J.P., Electrostatically actuated polymer microresonators, Applied Physics Letters, 87, 1-3.
 
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