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Experimental investigation on size-dependent higher-mode vibration of cantilever microbeams

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Abstract

A new perspective is presented to study size-dependent elasticity experimentally within micron scale utilizing dynamic approach. The size-dependent vibration of cantilever microbeams made of nickel is studied for the first three transverse modes. The normalized natural frequency of the first mode manifests strong size effect as reported. Remarkably, the normalized natural frequencies of the second and third mode also increase to 1.9 times as the thickness of microbeams decreases from 15 to 2.1 μm. Similarly the normalized bending rigidity increases to about 3.5 times. It is the first time that elastic size effect is observed in vibration of higher modes. Moreover, the size-dependent vibration of the first mode is interpreted in light of the modified couple stress theory, and the theoretical prediction also fit the experimental results of high modes very well. Hence it is confirmed that modified couple stress theory is valid for vibration of higher modes too.

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References

  • Akgöz B, Civalek Ö (2011) Buckling Analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories. J Comput Theor Nanosci 8:1821–1827

    Article  MATH  Google Scholar 

  • Akgöz B, Civalek Ö (2012) Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory. Arch Appl Mech 82:423–443

    Article  MATH  Google Scholar 

  • Akgöz B, Civalek Ö (2013) A size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int J Eng Sci 70:1–14

    Article  MathSciNet  MATH  Google Scholar 

  • Andrew WM, Jonathan SC (2005) Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15:1060

    Article  Google Scholar 

  • Ansari R, Gholami R (2016) Nonlocal free vibration in the pre- and post-buckled states of magneto-electro-thermo elastic rectangular nanoplates with various edge conditions. Smart Mater Struct 25:095033

    Article  Google Scholar 

  • Al-Basyouni KS, Tounsi A, Mahmoud SR (2015) Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position. Compos Struct 125:621–630

    Article  Google Scholar 

  • Ansari R, Gholami R, Faghih Shojaei M, Mohammadi V, Darabi MA (2015a) Size-dependent nonlinear bending and postbuckling of functionally graded Mindlin rectangular microplates considering the physical neutral plane position. Compos Struct 127:87–98

    Article  Google Scholar 

  • Ansari R, Gholami R, Faghih Shojaei M, Mohammadi V, Sahmani S (2015b) Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. Eur J Mech A Solid 49:251–267

    Article  MathSciNet  MATH  Google Scholar 

  • Ansari R, Gholami R, Shahabodini A (2016a) Size-dependent geometrically nonlinear forced vibration analysis of functionally graded first-order shear deformable microplates. J Mech 32:539–554

    Article  Google Scholar 

  • Ansari R, Shojaei MF, Gholami R (2016b) Size-dependent nonlinear mechanical behavior of third-order shear deformable functionally graded microbeams using the variational differential quadrature method. Compos Struct 136:669–683

    Article  Google Scholar 

  • Asghari M, Kahrobaiyan MH, Ahmadian MT (2010) A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int J Eng Sci 48:1749–1761

    Article  MathSciNet  MATH  Google Scholar 

  • Baltacıoglu AK, Akgöz B, Civalek Ö (2010) Nonlinear static response of laminated composite plates by discrete singular convolution method. Compos Struct 93:153–161

    Article  Google Scholar 

  • Banerjee JR, Cheung CW, Morishima R, Perera M, Njuguna J (2007) Free vibration of a three-layered sandwich beam using the dynamic stiffness method and experiment. Int J Solids Struct 44:7543–7563

    Article  MATH  Google Scholar 

  • Chen WJ, Li XP (2013) Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory. Arch Appl Mech 83:431–444

    Article  MATH  Google Scholar 

  • Chen D, Yang J, Kitipornchai S (2015) Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos Struct 133:54–61

    Article  Google Scholar 

  • Chong ACM, Lam DCC (2011) Strain gradient plasticity effect in indentation hardness of polymers. J Mater Res 14:4103–4110

    Article  Google Scholar 

  • Civalek Ö, Demir Ç (2011) Buckling and bending analyses of cantilever carbon nanotubes using the Euler–Bernoulli beam theory based on non-local continuum model. Asian J Civil Eng 12:651–661

    Google Scholar 

  • Chong ACM, Yang F, Lam DCC, Tong P (2011) Torsion and bending of micron-scaled structures. J Mater Res 16:1052–1058

    Article  Google Scholar 

  • Dai HL, Wang L (2017) Size-dependent pull-in voltage and nonlinear dynamics of electrically actuated microcantilever-based MEMS: a full nonlinear analysis. Commun Nonlinear Sci Numer Simul 46:116–125

    Article  MathSciNet  Google Scholar 

  • Dai HL, Wang YK, Wang L (2015) Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory. Int J Eng Sci 94:103–112

    Article  MathSciNet  MATH  Google Scholar 

  • Dehrouyeh-Semnani AM, Dehrouyeh M, Torabi-Kafshgari M, Nikkhah-Bahrami M (2015) An investigation into size-dependent vibration damping characteristics of functionally graded viscoelastically damped sandwich microbeams. Int J Eng Sci 96:68–85

    Article  MathSciNet  MATH  Google Scholar 

  • Fang J, Gu J, Wang H (2018) Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. Int J Mech Sci 136:188–199

    Article  Google Scholar 

  • Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42:475–487

    Article  Google Scholar 

  • Ghayesh MH (2018) Dynamics of functionally graded viscoelastic microbeams. Int J Eng Sci 124:115–131

    Article  MathSciNet  MATH  Google Scholar 

  • Ghayesh MH, Farokhi H (2018) Bending and vibration analyses of coupled axially functionally graded tapered beams. Nonlinear Dyn 91:17–28

    Article  Google Scholar 

  • Gholami R, Ansari R (2016) A most general strain gradient plate formulation for size-dependent geometrically nonlinear free vibration analysis of functionally graded shear deformable rectangular microplates. Nonlinear Dyn 84:2403–2422

    Article  MathSciNet  Google Scholar 

  • Gholami R, Ansari R (2017) A unified nonlocal nonlinear higher-order shear deformable plate model for postbuckling analysis of piezoelectric-piezomagnetic rectangular nanoplates with various edge supports. Compos Struct 166:202–218

    Article  Google Scholar 

  • Gholami R, Ansari R (2018) Grain size and nanoscale effects on the nonlinear pull-in instability and vibrations of electrostatic actuators made of nanocrystalline material. Mater Res Express 5:015012

    Article  Google Scholar 

  • Gholami R, Ansari R, Rouhi H (2015) Studying the effects of small scale and Casimir force on the non-linear pull-in instability and vibrations of FGM microswitches under electrostatic actuation. Int J Non-Linear Mech 77:193–207

    Article  Google Scholar 

  • Jaworski JW, Dowell EH (2008) Free vibration of a cantilevered beam with multiple steps: comparison of several theoretical methods with experiment. J Sound Vib 312:713–725

    Article  Google Scholar 

  • Kahrobaiyan M, Asghari M, Ahmadian M (2013) Strain gradient beam element. Finite Elem Anal Des 68:63–75

    Article  MathSciNet  MATH  Google Scholar 

  • Karamanlı A, Vo TP (2018) Size dependent bending analysis of two directional functionally graded microbeams via a quasi-3D theory and finite element method. Compos Part B-Eng 144:171–183

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Kong S, Zhou S, Nie Z, Wang K (2009) Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int J Eng Sci 47:487–498

    Article  MathSciNet  MATH  Google Scholar 

  • Krysko AV, Awrejcewicz J, Pavlov SP, Zhigalov MV, Krysko VA (2017) Chaotic dynamics of the size-dependent non-linear micro-beam model. Commun Nonlinear Sci Numer Simul 50:16–28

    Article  MathSciNet  Google Scholar 

  • Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508

    Article  MATH  Google Scholar 

  • Lei J, He Y, Zhang B, Gan Z, Zeng P (2013) Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory. Int J Eng Sci 72:36–52

    Article  MATH  Google Scholar 

  • Lei J, He Y, Zhang B, Liu D, Shen L, Guo S (2015) A size-dependent FG micro-plate model incorporating higher-order shear and normal deformation effects based on a modified couple stress theory. Int J Mech Sci 104:8–23

    Article  Google Scholar 

  • Lei J, He Y, Guo S, Li Z, Liu D (2016) Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity. Aip Adv 6:105202

    Article  Google Scholar 

  • Li L, Hu Y (2015) Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int J Eng Sci 97:84–94

    Article  MathSciNet  MATH  Google Scholar 

  • Li Z, He Y, Lei J, Guo S, Liu D, Wang L (2018) A standard experimental method for determining the material length scale based on modified couple stress theory. Int J Mech Sci 141:198–205

    Article  Google Scholar 

  • Liebold C, Müller WH (2016) Comparison of gradient elasticity models for the bending of micromaterials. Comput Mater Sci 116:52–61

    Article  Google Scholar 

  • Liu D, He Y, Tang X, Ding H, Hu P, Cao P (2012) Size effects in the torsion of microscale copper wires: experiment and analysis. Scripta Mater 66:406–409

    Article  Google Scholar 

  • Mercan K, Civalek Ö (2016) DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Compos Struct 143:300–309

    Article  Google Scholar 

  • Mercan K, Civalek Ö (2017) Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ. Compos Part B-Eng 114:34–45

    Article  Google Scholar 

  • Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech An 16:51–78

    Article  MathSciNet  MATH  Google Scholar 

  • Mohammad-Abadi M, Daneshmehr AR (2014) Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions. Int J Eng Sci 74:1–14

    Article  MathSciNet  MATH  Google Scholar 

  • Park SK, Gao XL (2006) Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng 16:2355–2359

    Article  Google Scholar 

  • Raville ME, Ueng E-S, Lei M-M (1961) Natural frequencies of vibration of fixed-fixed sandwich beams. J Appl Mech 28:367–371

    Article  MathSciNet  MATH  Google Scholar 

  • Shafiei N, Kazemi M, Ghadiri M (2016) On size-dependent vibration of rotary axially functionally graded microbeam. Int J Eng Sci 101:29–44

    Article  MATH  Google Scholar 

  • Sourki R, Hoseini SAH (2016) Free vibration analysis of size-dependent cracked microbeam based on the modified couple stress theory. Appl Phys A 122:413

    Article  Google Scholar 

  • Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Metall 46:5109–5115

    Google Scholar 

  • Su Z, Jin G, Ye T (2018) Electro-mechanical vibration characteristics of functionally graded piezoelectric plates with general boundary conditions. Int J Mech Sci 138–139:42–53

    Article  Google Scholar 

  • Tang C, Alici G (2011) Evaluation of length-scale effects for mechanical behaviour of micro-and nanocantilevers: II. Experimental verification of deflection models using atomic force microscopy. J Phys D Appl Phys 44:335502

    Article  Google Scholar 

  • Tao C, Fu Y (2017) Thermal buckling and postbuckling analysis of size-dependent composite laminated microbeams based on a new modified couple stress theory. Acta Mech 228:1711–1724

    Article  MathSciNet  MATH  Google Scholar 

  • Thai H-T, Vo TP, Nguyen T-K, Lee J (2015) Size-dependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory. Compos Struct 123:337–349

    Article  Google Scholar 

  • Thai CH, Ferreira AJM, Lee J, Nguyen-Xuan H (2018) An efficient size-dependent computational approach for functionally graded isotropic and sandwich microplates based on modified couple stress theory and moving Kriging-based meshfree method. Int J Mech Sci 142–143:322–338

    Article  Google Scholar 

  • Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech An 11:385–414

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Xu YY, Ni Q (2013) Size-dependent vibration analysis of three-dimensional cylindrical microbeams based on modified couple stress theory: a unified treatment. Int J Eng Sci 68:1–10

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Liu W-B, Dai H-L (2015) Dynamics and instability of current-carrying microbeams in a longitudinal magnetic field. Physica E 66:87–92

    Article  Google Scholar 

  • Wu D, Huang L, Pan B, Wang Y, Wu S (2014) Experimental study and numerical simulation of active vibration control of a highly flexible beam using piezoelectric intelligent material. Aerosp Sci Technol 37:10–19

    Article  Google Scholar 

  • Xia W, Wang L, Yin L (2010) Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int J Eng Sci 48:2044–2053

    Article  MathSciNet  MATH  Google Scholar 

  • Yang W, He D, Chen W (2017) A size-dependent zigzag model for composite laminated micro beams based on a modified couple stress theory. Compos Struct 179:646–654

    Article  Google Scholar 

  • Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743

    Article  MATH  Google Scholar 

  • Zhang B, He Y, Liu D, Gan Z, Shen L (2013a) A non-classical Mindlin plate finite element based on a modified couple stress theory. Eur J Mech A-Solid 42:63–80

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang B, He Y, Liu D, Gan Z, Shen L (2013b) A novel size-dependent functionally graded curved mircobeam model based on the strain gradient elasticity theory. Compos Struct 106:374–392

    Article  Google Scholar 

  • Zhang B, He Y, Liu D, Gan Z, Shen L (2014) Non-classical Timoshenko beam element based on the strain gradient elasticity theory. Finite Elem Anal Des 79:22–39

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was financially supported by the NSFC (No. 11772138) and the Ministry of Science and Technology of China (No. 2018YFF01014200). The authors are also grateful for the support of the NSFC (Nos. 11802101, 11702103) and the China Postdoctoral Science Foundation (No. 2018M642819).

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Authors and Affiliations

  1. Department of Mechanics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Zhenkun Li, Yuming He, Jian Lei, Shihao Han, Song Guo & Dabiao Liu

  2. Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Wuhan, 430074, China

    Zhenkun Li, Yuming He, Jian Lei, Shihao Han, Song Guo & Dabiao Liu

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  1. Zhenkun Li
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  2. Yuming He
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  3. Jian Lei
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Correspondence to Yuming He.

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Li, Z., He, Y., Lei, J. et al. Experimental investigation on size-dependent higher-mode vibration of cantilever microbeams. Microsyst Technol 25, 3005–3015 (2019). https://doi.org/10.1007/s00542-018-4244-0

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