Study of Volumetric Flow Rate of a Micropump Using a Non-classical Elasticity Theory


1 Electrical Engineering Department, Urmia University, Urmia, Iran

2 Mechanical Engineering Department, Tabriz University, Tabriz, Iran

3 Mechanical Engineering Department, Urmia University, Urmia, Iran


The purpose of this research is to study the mechanical behavior of a micropump with clamped circular diaphragm which is the main component of drug delivery systems. In this paper, the non-linear governing equations of the circular microplate using Kirchhoff thin plate theory was been extracted based on the modified couple stress (MCST) and classical (CT) theories. Then, the non-linear equation of static deflection is solved using Step-by-Step Linearization Method (SSLM) in order to escape the nonlinearity of the differential equation and Galerkin-based reduced-order model is applied to investigate the dynamic motion of the microplate. Afterwards, static and dynamic stabilities of the micropump have been studied based on both MCST and CT, then compared. Also, volumetric flow rate of the micropump was been delved based on both theories and in entire research, presence of the length scale parameter in modified couple stress theory brings this opportunity to study the size effect on the mechanical behavior of the micropump.


1.     Rashvand, K., Rezazadeh, G. and Madinei, H., "Effect of length-scale parameter on pull-in voltage and natural frequency of a micro-plate", International Journal of Engineering,  Vol. 27, No. 3, (2014), 375-384.

2.     Liu, J., Martin, D.T., Kadirvel, K., Nishida, T., Cattafesta, L., Sheplak, M. and Mann, B.P., "Nonlinear model and system identification of a capacitive dual-backplate mems microphone", Journal of Sound and Vibration,  Vol. 309, No. 1-2, (2008), 276-292.

3.     Saeedivahdat, A., Abdolkarimzadeh, F., Feyzi, A., Rezazadeh, G. and Tarverdilo, S., "Effect of thermal stresses on stability and frequency response of a capacitive microphone", Microelectronics Journal,  Vol. 41, No. 12, (2010), 865-873.

4.     Zhou, Z., Wang, Z. and Lin, L., "Microsystems and nanotechnology, Springer,  (2012).

5.     Bhat, K., Nayak, M., Kumar, V., Thomas, L., Manish, S., Thyagarajan, V., Gaurav, S., Bhat, N. and Pratap, R., Design, development, fabrication, packaging, and testing of mems pressure sensors for aerospace applications, in Micro and smart devices and systems. 2014, Springer.3-17.

6.     Lemarquand, G., Ravaud, R., Shahosseini, I., Lemarquand, V., Moulin, J. and Lefeuvre, E., "Mems electrodynamic loudspeakers for mobile phones", Applied Acoustics,  Vol. 73, No. 4, (2012), 379-385.

7.     Ferrari, M., "Biomems and biomedical nanotechnology: Volume ii: Micro/nano technologies for genomics and proteomics, Springer Science & Business Media,  Vol. 2,  (2007).

8.     Ashraf, M.W., Tayyaba, S. and Afzulpurkar, N., "Micro electromechanical systems (mems) based microfluidic devices for biomedical applications", International journal of molecular sciences,  Vol. 12, No. 6, (2011), 3648-3704.

9.     Nisar, A., Afzulpurkar, N., Mahaisavariya, B. and Tuantranont, A., "Mems-based micropumps in drug delivery and biomedical applications", Sensors and Actuators B: Chemical,  Vol. 130, No. 2, (2008), 917-942.

10.   Sen, M., Wajerski, D. and Gad-el-Hak, M., "A novel pump for mems applications", Journal of Fluids Engineering, Transactions of the ASME,  Vol. 118, No. 3, (1996), 624-627.

11.   Judy, J.W., Tamagawa, T. and Polla, D.L., "Surface-machined micromechanical membrane pump", in Micro Electro Mechanical Systems, 1991, MEMS'91, Proceedings. An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots. IEEE, IEEE., (1991), 182-186.

12.   Zengerle, R., Richter, A. and Sandmaier, H., "A micro membrane pump with electrostatic actuation", in Micro Electro Mechanical Systems, 1992, MEMS'92, Proceedings. An Investigation of Micro Structures, Sensors, Actuators, Machines and Robot. IEEE, IEEE., (1992), 19-24.

13.   Machauf, A., Nemirovsky, Y. and Dinnar, U., "A membrane micropump electrostatically actuated across the working fluid", Journal of Micromechanics and Microengineering,  Vol. 15, No. 12, (2005), 2309.

14.   Liu, W.-y., "Research on electrostatic micropump pull-in phenomena based on reduced order model", in Intelligent Computation Technology and Automation (ICICTA), 2010 International Conference on, IEEE. Vol. 2, (2010), 1154-1157.

15.   Li, L., Zhu, R., Zhou, Z. and Ren, J., "Modeling of a micropump membrane with electrostatic actuator", in Advanced Computer Control (ICACC), 2010 2nd International Conference on, IEEE. Vol. 5, (2010), 630-632.

16.   Zhao, Y.-P., Wang, L. and Yu, T., "Mechanics of adhesion in mems—a review", Journal of Adhesion Science and Technology,  Vol. 17, No. 4, (2003), 519-546.

17.   Nabian, A., Rezazadeh, G., Haddad-Derafshi, M. and Tahmasebi, A., "Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and non-uniform electrostatic pressure", Microsystem Technologies,  Vol. 14, No. 2, (2008), 235-240.

18.   Zhang, Y. and Zhao, Y.-p., "Numerical and analytical study on the pull-in instability of micro-structure under electrostatic loading", Sensors and Actuators A: Physical,  Vol. 127, No. 2, (2006), 366-380.

19.   Mobki, H., Sadeghia, M. and Rezazadehb, G., "Design of direct exponential observers for fault detection of nonlinear mems tunable capacitor", IJE Transactions A: Basics,  Vol. 28, No. 4, (2015), 634-641.

20.   Dowlati, S., Rezazadeh, G., Afrang, S., Sheykhlou, M. and Pasandi, A.M., "An accurate study on capacitive microphone with circular diaphragm using a higher order elasticity theory", Latin American Journal of Solids and Structures,  Vol. 13, No. 4, (2016), 590-609.

21.   Khanchehgardan, A., SHAH, M.A.A., Rezazadeh, G. and Shabani, R., "Thermo-elastic damping in nano-beam resonators based on nonlocal theory",  Vol. 26, No. 12, (2013), 1505-1514. 

22.   SHAH, M.A.A., Khanchehgardan, A., Rezazadeh, G. and Shabani, R., "Mechanical response of a piezoelectrically sandwiched nano-beam based on the nonlocal theory",  Vol. 26, No. 12, (2013), 1515-1524.

23.   Loh, O.Y. and Espinosa, H.D., "Nanoelectromechanical contact switches", Nature Nanotechnology,  Vol. 7, No. 5, (2012), 283.

24.   Sadeghian, H., Yang, C.-K., Goosen, J.F., Bossche, A., Staufer, U., French, P.J. and van Keulen, F., "Effects of size and defects on the elasticity of silicon nanocantilevers", Journal of Micromechanics and Microengineering,  Vol. 20, No. 6, (2010) doi: 10.1088/0960-1317/20/6/064012.

25.   Abazari, A.M., Safavi, S.M., Rezazadeh, G. and Villanueva, L.G., "Modelling the size effects on the mechanical properties of micro/nano structures", Sensors,  Vol. 15, No. 11, (2015), 28543-28562.

26.   Tsiatas, G.C., "A new kirchhoff plate model based on a modified couple stress theory", International Journal of Solids and Structures,  Vol. 46, No. 13, (2009), 2757-2764.

27.   Lazopoulos, K., "On bending of strain gradient elastic micro-plates", Mechanics Research Communications,  Vol. 36, No. 7, (2009), 777-783.

28.   Eringen, A.C., "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", Journal of Applied Physics,  Vol. 54, No. 9, (1983), 4703-4710.

29.   Reddy, J., "Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates", International Journal of Engineering Science,  Vol. 48, No. 11, (2010), 1507-1518.

30.   Toupin, R.A., "Elastic materials with couple-stresses", Archive for Rational Mechanics and Analysis,  Vol. 11, No. 1, (1962), 385-414.

31.   Cosserat, E. and Cosserat, F., "Théorie des corps déformables",  Vol., No., (1909).

32.   Mindlin, R. and Tiersten, H., "Effects of couple-stresses in linear elasticity", Archive for Rational Mechanics and Analysis,  Vol. 11, No. 1, (1962), 415-448.

33.   Yang, F., Chong, A., Lam, D.C.C. and Tong, P., "Couple stress based strain gradient theory for elasticity", International Journal of Solids and Structures,  Vol. 39, No. 10, (2002), 2731-2743.

34.   Jomehzadeh, E., Noori, H. and Saidi, A., "The size-dependent vibration analysis of micro-plates based on a modified couple stress theory", Physica E: Low-dimensional Systems and Nanostructures,  Vol. 43, No. 4, (2011), 877-883.

35.   Timoshenko, S.P. and Woinowsky-Krieger, S., "Theory of plates and shells, McGraw-Hill,  (1959).

36.   Leech, C., "Beam theories: A variational approach", International Journal of Mechanical Engineering Education.,  Vol. 5, (1977), 81-87.

37.   Vogl, G.W. and Nayfeh, A.H., "A reduced-order model for electrically actuated clamped circular plates", Journal of Micromechanics and Microengineering,  Vol. 15, No. 4, (2005), 684.

38.   Sharafkhani, N., Rezazadeh, G. and Shabani, R., "Study of mechanical behavior of circular fgm micro-plates under nonlinear electrostatic and mechanical shock loadings", Acta Mechanica,  Vol. 223, No. 3, (2012), 579-591.

39.   Nayfeh, A.H., Younis, M.I. and Abdel-Rahman, E.M., "Dynamic pull-in phenomenon in mems resonators", Nonlinear Dynamics,  Vol. 48, No. 1-2, (2007), 153-163.