Abstract
This paper investigates the bandgap properties of a piezoelectric periodic nano-beam considering size and surface effects using a modified couple stress theory. The nano-beam is made of some finite periodic arrays of piezoelectric (PZT-5H) and epoxy segments. The Bloch theorem for periodic materials together with the transfer matrix method are employed for analyzing the problem. The band structure analyzed by the current model incorporates both the material length scale parameter and surface effects in the bulk and surface layer of the beam, respectively. The main objective of this study is to investigate the effects of different parameters such as external electrical loading, length scale parameter, surface effects and geometrical properties of the nano-beam on the width of bandgaps and the starting frequencies. It is found that when the external electrical field is increased, the surface effects on the bandgaps are increased. Also, for high values of length to height ratio, ignoring the surface effects reduces the number of bandgaps. The results of the current study may be helpful in designing piezoelectric periodic nano-sensing devices.
Similar content being viewed by others
References
Baas, A.F.D.: Nanostructured Metamaterials: Exchange Between Experts in Electromagnetics and Material Science. EUR-OP, Belgium (2010)
Claeys, C.C., Vergote, K., Sas, P., Desmet, W.: On the potential of tuned resonators to obtain low-frequency vibrational stop bands in periodic panels. Sound Vib. (2013). https://doi.org/10.1016/j.jsv.2012.09.047
Sigalas, M., Economou, E.: Elastic and acoustic wave band structure. Sound Vib. (1992). https://doi.org/10.1016/0022-460x(92)90059-7
Yu, D., Liu, Y., Zhao, H., Wang, G., Qiu, J.: Flexural vibration band gaps in Euler–Bernoulli beams with locally resonant structures with two degrees of freedom. Phys. Rev. B (2006). https://doi.org/10.1103/PhysRevB.73.064301
Wen, J.: Theory and experimental investigaion of flexural wave propagation in thin rectangular plate with periodic structure. Chin. J. Mech. Eng. (2005). https://doi.org/10.3901/cjme.2005.03.385
Yan, Z., Jing, H., Linhua, J.: Flexural vibration band gaps characteristics in phononic crystal Euler beams on two-parameter foundation. Adv. Mech. Eng. (2013). https://doi.org/10.1155/2013/935258
Xiang, H.J., Shi, Z.: Vibration attenuation in periodic composite Timoshenko beams on Pasternak foundation. Struct. Eng. Mech. (2011). https://doi.org/10.12989/sem.2011.40.3.373
Casalotti, A., El-Borgi, S., Lacarbonara, W.: Metamaterial beam with embedded nonlinear vibration absorbers. Int. J. Non Linear Mech. (2018). https://doi.org/10.1016/j.ijnonlinmec.2017.10.002
Beli, D., Arruda, J.R.F., Ruzzene, M.: Wave propagation in elastic metamaterial beams and plates with interconnected resonators. Int. J. Solids Struct. (2018). https://doi.org/10.1016/j.ijsolstr.2018.01.027
Ebrahimi, F., Daman, M.: Nonlocal thermo-electro-mechanical vibration analysis of smart curved FG piezoelectric Timoshenko nanobeam. Smart Struct. Syst. (2017). https://doi.org/10.12989/sss.2017.20.3.351
Ebrahimi, F., Dehghan, M., Seyfi, A.: Eringen’s nonlocal elasticity theory for wave propagation analysis of magneto-electro-elastic nanotubes. Adv. Nano Res. (2019). https://doi.org/10.12989/anr.2019.7.1.001
Ghorbanpour, A.A., Pourjamshidian, M., Arefi, M.: Application of nonlocal elasticity theory on the wave propagation of flexoelectric functionally graded (FG) timoshenko nano-beams considering surface effects and residual surface stress. Smart Struct. Syst. (2019). https://doi.org/10.12989/sss.2019.23.2.141
Kaghazian, A., Hajnayeb, A., Foruzande, H.: Free vibration analysis of a Piezoelectric nanobeam using nonlocal elasticity theory. Struct. Eng. Mech. (2017). https://doi.org/10.12989/sem.2017.61.5.617
Ansari, R., Ashrafi, M.A., Hosseinzadeh, S.: Vibration characteristics of piezoelectric microbeams based on the modified couple stress theory. Shock Vib. (2014). https://doi.org/10.1155/2014/598292
Ebrahimi, F., Safarpour, H.: Vibration analysis of inhomogeneous nonlocal beams via a modified couple stress theory incorporating surface effects. Wind Struct. (2018). https://doi.org/10.12989/was.2018.27.6.431
Ehyaei, J., Akbarizadeh, M.R.: Vibration analysis of micro composite thin beam based on modified couple stress. Struct. Eng. Mech. (2017). https://doi.org/10.12989/sem.2017.64.4.403
Kocatürk, T., Akbaş, Ş.D.: Wave propagation in a microbeam based on the modified couple stress theory. Struct. Eng. Mech. (2013). https://doi.org/10.12989/sem.2013.46.3.417
Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequency of Bernoulli-Euler micro-beams. Eng. Sci. (2008). https://doi.org/10.1016/j.ijengsci.2007.10.002
Gurtin, M.E., Murdoch, A.L.: A continuum theory of elastic material surfaces. Ration. Mech. Anal. (1975). https://doi.org/10.1007/BF00261375
Gao, X.L.: A new Timoshenko beam model incorporating microstructure and surface energy effects. Acta Mech. (2015). https://doi.org/10.1007/s00707-014-1189-y
Gao, X.L., Mahmoud, F.F.: A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects. Z. Angew. Math. Phys. (2014). https://doi.org/10.1007/s00033-013-0343-z
Juntarasaid, C., Pulngern, T., Chucheepsakul, S.: Bending and buckling of nanowires including the effects of surface stress and nonlocal elasticity. Physica E Low Dimens. Syst. Nanostruct. (2012). https://doi.org/10.1016/j.physe.2012.08.005
Yan, Z., Jiang, L.: Surface effects on the electromechanical coupling and bending behaviours of piezoelectric nanowires. Phys. D Appl. Phys. (2011). https://doi.org/10.1088/0022-3727/44/7/075404
Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. (1994). https://doi.org/10.1016/0956-7151(94)90502-9
Zbib, H.M., Aifantis, E.C.: Size effects and length scales in gradient plasticity and dislocation dynamics. Scr. Mater (2003). https://doi.org/10.1016/S1359-6462(02)00342-1
Sab, K., Legoll, F., Forest, S.: Stress gradient elasticity theory: existence and uniqueness of solution. J. Elast. (2016). https://doi.org/10.1007/s10659-015-9554-1
Polizzotto, C.: A note on the higher order strain and stress tensors within deformation gradient elasticity theories: physical interpretations and comparisons. Int. J. Solids Struct. (2016). https://doi.org/10.1016/j.ijsolstr.2016.04.001
Ebrahimi, F., Zokaee, F., Mahesh, V.: Analysis of the size-dependent wave propagation of a single lamellae based on the nonlocal strain gradient theory. Biomater. Biomed. Eng. (2019). https://doi.org/10.12989/bme.2019.4.1.045
Ebrahimi, F., Dabbagh, A.: Wave dispersion characteristics of nonlocal strain gradient double layered graphene sheets in hygro-thermal environments. Struct. Eng. Mech. (2018). https://doi.org/10.12989/sem.2018.65.6.645
Ghorbanpour, A.A., Pourjamshidian, M., Arefi, M.: Influence of electro-magneto-thermal environment on the wave propagation analysis of sandwich nano-beam based on nonlocal strain gradient theory and shear deformation theories. Smart Struct. Syst. (2017). https://doi.org/10.12989/sss.2017.20.3.329
Narendar, S., Gopalakrishnan, S.: Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes. Comput. Mater. Sci. (2009). https://doi.org/10.1016/j.commatsci.2009.09.021
Wang, L.: Wave propagation of fluid-conveying single-walled carbon nanotubes via gradient elasticity theory. Comput. Mater. Sci. (2010). https://doi.org/10.1016/j.commatsci.2010.06.019
Yang, Y., Zhang, L., Lim, C.W.: Wave propagation in double-walled carbon nanotubes on a novel analytically nonlocal Timoshenko-beam model. J. Sound Vib. (2011). https://doi.org/10.1016/j.jsv.2010.10.028
Narendar, S., Gupta, S.S., Gopalakrishnan, S.: Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler–Bernoulli beam theory. Appl. Math. Model. (2012). https://doi.org/10.1016/j.apm.2011.11.073
Aydogdu, M.: Longitudinal wave propagation in multiwalled carbon nanotubes. Compos. Struct. 107, 578–584 (2014). https://doi.org/10.1016/j.compstruct.2013.08.031
Liu, Z., Rumpler, R., Feng, L.: Broadband locally resonant metamaterial sandwich plate for improved noise insulation in the coincidence region. Compos. Struct. (2018). https://doi.org/10.1016/j.compstruct.2018.05.033
Miranda, E.J.P., Nobrega, E.D., Ferreira, A.H.R., Dos Santos, J.M.C.: Flexural wave band gaps in a multi-resonator elastic metamaterial plate using Kirchhoff–Love theory. Mech. Syst. Signal Process. (2019). https://doi.org/10.1016/j.ymssp.2018.06.059
Sheng, M., Guo, Zh, Qin, Q., He, Y.: Vibration characteristics of a sandwich plate with viscoelastic periodic cores. Compos. Struct. (2018). https://doi.org/10.1016/j.compstruct.2018.07.110
Zouari, S., Brocail, J., Génevaux, J.M.: Flexural wave band gaps in metamaterial plates: a numerical and experimental study from infinite to finite models. J. Sound Vib. (2018). https://doi.org/10.1016/j.jsv.2018.07.030
Ebrahimi, F., Barati, M.: Thermo-mechanical vibration analysis of nonlocal flexoelectric/piezoelectric beams incorporating surface effects. Struct. Eng. Mech. (2018). https://doi.org/10.12989/sem.2018.65.4.435
Zhang, W.M., Hu, K.M., Peng, Z.K., Meng, G.: Tunable micro and nanomechanical resonators. Sensors (2015). https://doi.org/10.3390/s151026478
Wagner, M., Graczykowski, B., Sebastian, R.J., El Sachat, A., Sledzinska, M., Alzina, F., Sotomayor, T.C.: Two-dimensional phononic crystals: disorder matters. Nano Lett. (2016). https://doi.org/10.1021/acs.nanolett.6b02305
Yan, Z., Wei, C., Zhang, C.: Band structures of elastic SH waves in nanoscale multi-layered functionally graded phononic crystals with/without nonlocal interface imperfections by using a local RBF collocation method. Acta Mech. Solida Sin. (2017). https://doi.org/10.1016/j.camss.2017.07.012
Goncalves, B.R., Karttunen, A.T., Romanoff, J.: A nonlinear couple stress model for periodic sandwich beams. Compos. Struct. (2019). https://doi.org/10.1016/j.compstruct.2019.01.034
Zhang, G.Y., Gao, X.L., Ding, S.R.: Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects. Acta Mech. (2018). https://doi.org/10.1007/s00707-018-2207-2
Zhang, G.Y., Gao, X.L., Bishop, J.E.: Band gaps for elastic wave propagation in a periodic composite beam structure incorporating microstructure and surface energy effects. Compos. Struct. (2018). https://doi.org/10.1016/j.compstruct.2017.11.040
Zhang, S., Gao, Y.: Surface effect on band structure of flexural wave propagating in magneto-elastic phononic crystal nanobeam. Phys. D Appl. Phys. (2017). https://doi.org/10.1088/1361-6463/aa8878
Aly, A.H., Nagaty, A., Mehaney, A.: Thermal properties of one-dimensional piezoelectric phononic crystal. Eur. Phys. J. B 91(10), 1–5 (2018). https://doi.org/10.1140/epjb/e2018-90297-y
Qian, D.: Bandgap properties of a piezoelectric phononic crystal nanobeam based on nonlocal theory. J. Mater. Sci. (2019). https://doi.org/10.1007/s10853-018-3124-4
Chen, A.L., Yan, D.J., Wang, Y.S., Zhang, C.: In-plane elastic wave propagation in nanoscale periodic piezoelectric/piezomagnetic laminates. Mech. Sci. (2019). https://doi.org/10.1016/j.ijmecsci.2019.02.017
Qian, D.: Bandgap properties of a piezoelectric phononic crystal nanobeam with surface effect. Appl. Phys. (2018). https://doi.org/10.1063/1.5039952
Seo, J.H.: Wide bandgap semiconductor based micro/nano devices. Micromachines (2019). https://doi.org/10.3390/mi10030213
Li, W., Meng, F., Chen, Y., Li, Y.F., Huang, X.: Topology optimization of photonic and phononic crystals and metamaterials: a review. Adv. Theory Simul. (2019). https://doi.org/10.1002/adts.201900017
Wang, Z.L., Song, J.: Piezoelectric nanogenerators based on zinc oxide nanowire arrays. Science (2006). https://doi.org/10.1126/science.1124005
Wang, G.F., Feng, X.Q.: Effect of surface stresses on the vibration and buckling of piezoelectric nanowires. Eur. Phys. Lett. (2010). https://doi.org/10.1209/0295-5075/91/56007
Yan, Z., Jiang, L.Y.: The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects. Nanotechnology (2011). https://doi.org/10.1088/0957-4484/22/24/245703
Park, S.K., Gao, X.L.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. Angew. Math. Phys. (2008). https://doi.org/10.1007/s00033-006-6073-8
Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. (1963). https://doi.org/10.1007/bf02327219
Ke, L.L., Wang, Y.S., Reddy, J.N.: Thermo-electro-mechanical vibration of size-dependent piezoelectric cylindrical nanoshells under various boundary conditions. Compos. Struct. (2014). https://doi.org/10.1016/j.compstruct.2014.05.048
Gao, X.L., Mall, S.: Variational solution for a cracked mosaic model of woven fabric composites. Int. J. Solids Struct. (2001). https://doi.org/10.1016/S0020-7683(00)00047-0
Liu, L., Hussein, M.: Wave motion in periodic flexural beams and characterization of the transition between Bragg scattering and local resonance. Appl. Mech. (2012). https://doi.org/10.1115/1.4004592
Li, Z., He, Y., Lei, J., Guo, S., Liu, D., Wang, L.: A standard experimental method for determining the material length scale based on modiÞed couple stress theory. Int. J. Mech. Sci. (2018). https://doi.org/10.1016/j.ijmecsci.2018.03.035
Huang, G.Y., Yu, S.W.: Effect of surface piezoelectricity on the electromechanical behaviour of a piezoelectric ring. Phys. Status Solidi (b) (2006). https://doi.org/10.1002/pssb.200541521
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Espo, M., Abolbashari, M.H. & Hosseini, S.M. Band structure analysis of wave propagation in piezoelectric nano-metamaterials as periodic nano-beams considering the small scale and surface effects. Acta Mech 231, 2877–2893 (2020). https://doi.org/10.1007/s00707-020-02678-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-020-02678-8