Nowadays, piezoelectric materials are used as smart nanostructures in many engineering applications. Polyvinylidene fluoride (PVDF) is one of the piezoelectric polymeric materials which has fascinated the attention of the scientific community due to its remarkable properties. In the present study, dynamic stability and the parametric resonance of a Mindlin PVDF nanoplate lying on an elastic medium under electromechanical loadings and a moving nanoparticle are investigated. First, Hamilton's principle is used to obtain the partial equations governing the transverse oscillations of the PVDF nanoplate. By utilizing the theory of nonlocal piezoelasticity, the small-scale effects are applied to the equations. Navier’s approach is employed to procure the solution for the simply supported nanoplate. Then, the boundaries of instability in the plane of parameters are obtained by applying the energy-rate method to the governing time-varying ODEs. The results demonstrate that the increase in the nonlocal parameter and reduction of the nanoparticle movement path decrease the dynamic stability of the PVDF nanoplate. Also, the nanoplate made of PVDF materials is dynamically more stable than the nanoplate made of PZT-4 piezoceramic materials. Moreover, the piezoelectric voltage can be used to control the parametric resonance conditions of nanostructures. The results of this study have the potential to be utilized in the accurate design of nanoscale piezoelectric mass sensors in nanoelectromechanical systems.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Vives AA (2008) Piezoelectric transducers and applications. Springer, Berlin
Chen P, Chen S, Guo W, Gao F (2018) The interface behavior of a thin piezoelectric film bonded to a graded substrate. Mech Mater 127:26–38
Khajekhabaz M, Eftekhari SA, Hashemian M, Toghraie D (2020) Optimal vibration control of multi-layer micro-beams actuated by piezoelectric layer based on modified couple stress and surface stress elasticity theories. Phys A Stat Mech Appl 546:123998
Egorov VI, Maksimova OG, Koibuchi H, Jug G, Bernard C, Chenal JM, Lame O, Diguet G, Sebald G, Cavaillé JY (2019) Finsler geometry modeling of reverse piezoelectric effect in PVDF. J Phys Conf Ser 1391:12014
Mishra S, Unnikrishnan L, Nayak SK, Mohanty S (2019) Advances in piezoelectric polymer composites for energy harvesting applications: a systematic review. Macromol Mater Eng 304:1800463
Sappati KK, Bhadra S (2018) Piezoelectric polymer and paper substrates: a review. Sensors 18:3605
Malikan M, Nguyen VB (2018) Buckling analysis of piezo-magnetoelectric nanoplates in hygrothermal environment based on a novel one variable plate theory combining with higher-order nonlocal strain gradient theory. Phys E Low-Dimens Syst Nanostruct 102:8–28
Arefi M, Kiani M, Rabczuk T (2019) Application of nonlocal strain gradient theory to size dependent bending analysis of a sandwich porous nanoplate integrated with piezomagnetic face-sheets. Compos Part B Eng 168:320–333
Pradhan SC, Kumar A (2011) Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Compos Struct 93:774–779
Pirmoradian M, Torkan E, Toghraie D (2020) Study on size-dependent vibration and stability of DWCNTs subjected to moving nanoparticles and embedded on two-parameter foundations. Mech Mater 142:103279
Ansari R, Ashrafi MA, Pourashraf T, Sahmani S (2015) Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory. Acta Astronaut 109:42–51
Ansari R, Norouzzadeh A, Gholami R, Shojaei MF, Darabi MA (2016) Geometrically nonlinear free vibration and instability of fluid-conveying nanoscale pipes including surface stress effects. Microfluid Nanofluidics 20:28
Togun N, Bağdatli SM (2016) Size dependent nonlinear vibration of the tensioned nanobeam based on the modified couple stress theory. Compos Part B Eng 97:255–262
Şimşek M, Aydın M, Yurtcu HH, Reddy JN (2015) Size-dependent vibration of a microplate under the action of a moving load based on the modified couple stress theory. Acta Mech 226:3807–3822
Zhao H-S, Zhang Y, Lie S-T (2018) Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects. Acta Mech Sin 34:676–688
Pirmoradian M, Torkan E, Zali H, Hashemian M, Toghraie D (2020) Statistical and parametric instability analysis for delivery of nanoparticles through embedded DWCNT. Phys A Stat Mech Appl 554:123911
Norouzzadeh A, Ansari R (2018) Isogeometric vibration analysis of functionally graded nanoplates with the consideration of nonlocal and surface effects. Thin-Walled Struct 127:354–372
Bensattalah T, Hamidi A, Bouakkaz K, Zidour M, Daouadji TH (2020) Critical buckling load of triple-walled carbon nanotube based on nonlocal elasticity theory. J Nano Res 62:108–119
Ansari R, Norouzzadeh A (2016) Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates: an isogeometric analysis. Phys E Low-Dimens Syst Nanostruct 84:84–97
Ansari R, Torabi J, Norouzzadeh A (2018) Bending analysis of embedded nanoplates based on the integral formulation of Eringen’s nonlocal theory using the finite element method. Phys B Condens Matter 534:90–97
Orooji Y, Derakhshandeh MR, Ghasali E, Alizadeh M, Asl MS, Ebadzadeh T (2019) Effects of ZrB2 reinforcement on microstructure and mechanical properties of a spark plasma sintered mullite-CNT composite. Ceram Int 45(13):16015–16021
Karimi-Maleh H, Alizadeh M, Orooji Y, Karimi F, Baghayeri M, Rouhi J, Tajik S, Beitollahi H, Agarwal S, Gupta VK, Rajendran S, Rostamnia S, Fu L, Saberi-Movahed F, Malekmohammadi S (2021) Guanine-based DNA biosensor amplified with Pt/SWCNTs nanocomposite as analytical tool for nanomolar determination of daunorubicin as an anticancer drug: a docking/experimental investigation. Ind Eng Chem Res 60(2):816–823
Karimi-Maleh H, Karimi F, Orooji Y, Mansouri G, Razmjou A, Aygun A, Sen F (2020) A new nickel-based co-crystal complex electrocatalyst amplified by NiO dope Pt nanostructure hybrid; a highly sensitive approach for determination of cysteamine in the presence of serotonin. Sci Rep 10(1):11699
Yan Z, Jiang LY (2012) Vibration and buckling analysis of a piezoelectric nanoplate considering surface effects and in-plane constraints. Proc R Soc A Math Phys Eng Sci 468:3458–3475
Amir, S., Khorasani, M. & BabaAkbar-Zarei, H. Buckling analysis of nanocomposite sandwich plates with piezoelectric face sheets based on flexoelectricity and first-order shear deformation theory. J. Sandw. Struct. Mater. 1099636218795385 (2018).
Arani AG, Kolahchi R, Vossough H (2012) Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory. Phys B Condens Matter 407:4458–4465
Liu C, Ke L-L, Yang J, Kitipornchai S, Wang Y-S (2018) Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory. Mech Adv Mater Struct 25:1252–1264
Arani AG, Kolahchi R, Afshar HG (2015) Dynamic analysis of embedded PVDF nanoplate subjected to a moving nanoparticle on an arbitrary elliptical path. J Braz Soc Mech Sci Eng 37:973–986
Fard KM, Kavanroodi MK, Malek-Mohammadi H, Pourmoayed AR (2020) Buckling and vibration analysis of a double-layer Graphene sheet coupled with a piezoelectric nanoplate. Continuum, New York 7:9
Liu C, Ke L-L, Wang Y-S, Yang J (2015) Nonlinear vibration of nonlocal piezoelectric nanoplates. Int J Struct Stab Dyn 15:1540013
Li C, Liu JJ, Cheng M, Fan XL (2017) Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces. Compos Part B Eng 116:153–169
Ebrahimi F, Barati MR (2017) Damping vibration analysis of smart piezoelectric polymeric nanoplates on viscoelastic substrate based on nonlocal strain gradient theory. Smart Mater Struct 26:65018
Kolahchi R, Hosseini H, Esmailpour M (2016) Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocal-piezoelasticity theories. Compos Struct 157:174–186
Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248
Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710
Oveissi S, Toghraie D, Eftekhari SA (2016) Longitudinal vibration and stability analysis of carbon nanotubes conveying viscous fluid. Physica E 83:275–283
Norouzzadeh A, Ansari R (2017) Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity. Phys E Low-Dimens Syst Nanostruct 88:194–200
Fernández-Sáez J, Zaera R, Loya JA, Reddy JN (2016) Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci 99:107–116
Torkan E, Pirmoradian M, Hashemian M (2019) Instability inspection of parametric vibrating rectangular Mindlin plates lying on Winkler foundations under periodic loading of moving masses. Acta Mech Sin 35:242–263
Pirmoradian M, Torkan E, Karimpour H (2018) Parametric resonance analysis of rectangular plates subjected to moving inertial loads via IHB method. Int J Mech Sci 142:191–215
Pirmoradian M, Torkan E, Abdali N, Hashemian M, Toghraie D (2019) Thermo-mechanical stability of single-layered graphene sheets embedded in an elastic medium under action of a moving nanoparticle. Mech Mater 141:103248
Torkan E, Pirmoradian M, Hashemian M (2019) Dynamic instability analysis of moderately thick rectangular plates influenced by an orbiting mass based on the first-order shear deformation theory. Modares Mech Eng 19:2203–2213
Torkan E, Pirmoradian M, Hashemian M (2018) On the parametric and external resonances of rectangular plates on an elastic foundation traversed by sequential masses. Arch Appl Mech 88:1411–1428
Torkan E, Pirmoradian M (2019) Efficient higher-order shear deformation theories for instability analysis of plates carrying a mass moving on an elliptical path. J Solid Mech. https://doi.org/10.22034/JSM.2019.668763
Jazar GN (2004) Stability chart of parametric vibrating systems using energy-rate method. Int J Non Linear Mech 39:1319–1331
Jazar RN, Mahinfalah M, Mahmoudian N, Rastgaar MA (2008) Energy-rate method and stability chart of parametric vibrating systems. J Brazilian Soc Mech Sci Eng 30:182–188
Shen Z-B, Tang H-L, Li D-K, Tang G-J (2012) Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal Kirchhoff plate theory. Comput Mater Sci 61:200–205
Pradhan SC, Phadikar JK (2009) Nonlocal elasticity theory for vibration of nanoplates. J Sound Vib 325:206–223
Cheng Z-Q, Lim CW, Kitipornchai S (2000) Three-dimensional asymptotic approach to inhomogeneous and laminated piezoelectric plates. Int J Solids Struct 37:3153–3175
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Technical Editor: Aurelio Araujo.
Dimensionless parameters are defined as follows:
The components of vectors and matrices appearing in Eq. (48) are as follows:
About this article
Cite this article
Pirmoradian, M., Torkan, E., Hashemian, M. et al. Out-of-plane dynamic instability of nonlocal shear deformable nanoplates made of polyvinylidene fluoride materials subjected to electromechanical forces. J Braz. Soc. Mech. Sci. Eng. 43, 145 (2021). https://doi.org/10.1007/s40430-021-02846-4
- Dynamic stability
- Parametric resonance
- Polyvinylidene fluoride material
- Nonlocal piezoelasticity theory
- Mindlin plate theory