An isogeometric approach to static and transient analysis of fluid-infiltrated porous metal foam piezoelectric nanoplates with flexoelectric effects and variable nonlocal parameters

Abstract

In this work, a novel refined higher-order shear deformation plate theory is integrated with nonlocal elasticity theory for analyzing the free vibration, bending, and transient behaviors of fluid-infiltrated porous metal foam piezoelectric nanoplates resting on Pasternak elastic foundation with flexoelectric effects. Isogeometric analysis (IGA) and the Navier solution are applied to the problem. The innovation in the present study is that the influence of the in-plane variation of the nonlocal parameter on the free and forced vibration of the piezoelectric nanoplates is investigated for the first time. The nonlocal parameter and material characteristics are assumed to be material-dependent and vary gradually over the thickness of structures. Based on Hamilton’s principle, equations of motion are built, then the IGA approach combined with the Navier solution is used to analyze the static and dynamic response of the nanoplate. Lastly, we investigate the effects of the porosity coefficients, flexoelectric parameters, elastic stiffness, thickness, and variation of the nonlocal parameters on the mechanical behaviors of the rectangular and elliptical piezoelectric nanoplates.

Graphical abstract

Keywords

isogeometric analysis / fluid-infiltrated porous / piezoelectric nanoplates / static and transient analysis / flexoelectric effect / variable nonlocal parameters

References

[1] Nguyen T D, Mao S, Yeh Y W, Purohit P K, McAlpine M C. Nanoscale flexoelectricity. Advanced Materials, 2013, 25(7): 946–974

[2] Wang B, Gu Y, Zhang S, Chen L Q. Flexoelectricity in solids: Progress, challenges, and perspectives. Progress in Materials Science, 2019, 106: 100570

[3] Vasquez-Sancho F, Abdollahi A, Damjanovic D, Catalan G. Flexoelectricity in Bones. Advanced Materials, 2018, 30(21): 1801413

[4] Tagantsev A K. Piezoelectricity and flexoelectricity in crystalline dielectrics. Physical Review B: Condensed Matter, 1986, 34(8): 5883–5889

[5] Yudin P V, Tagantsev A K. Fundamentals of flexoelectricity in solids. Nanotechnology, 2013, 24(43): 432001

[6] Bagheri R, Tadi B Y. On the size-dependent nonlinear dynamics of viscoelastic/flexoelectric nanobeams. Journal of Vibration and Control, 2021, 27(17−18): 2018–2033

[7] Ray M C. Benchmark solutions for the material length scale effect in flexoelectric nanobeam using a couple stress theory. Applied Mathematical Modelling, 2022, 108: 189–204

[8] Tho N C, Thanh N T, Tho T D, van Minh P, Hoa L K. Modelling of the flexoelectric effect on rotating nanobeams with geometrical imperfection. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2021, 43(11): 510

[9] Yu P, Leng W, Peng L, Suo Y, Guo J. The bending and vibration responses of functionally graded piezoelectric nanobeams with dynamic flexoelectric effect. Results in Physics, 2021, 28: 104624

[10] Ansari R, Faraji O M, Nesarhosseini S, Rouhi H. Flexoelectricity effect on the size-dependent bending of piezoelectric nanobeams resting on elastic foundation. Applied Physics. A, Materials Science & Processing, 2021, 127(7): 518

[11] BaroudiSNajarF. Modeling and vibration analysis of a nonlinear piezoelectric flexoelectric nanobeam. In: Proceedings of the International Conference Design and Modeling of Mechanical Systems. Cham: Springer International Publishing, 2021: 448–455

[12] Baroudi S, Najar F, Jemai A. Static and dynamic analytical coupled field analysis of piezoelectric flexoelectric nanobeams: A strain gradient theory approach. International Journal of Solids and Structures, 2018, 135: 110–124

[13] Chu L, Dui G, Zheng Y. Thermally induced nonlinear dynamic analysis of temperature-dependent functionally graded flexoelectric nanobeams based on nonlocal simplified strain gradient elasticity theory. European Journal of Mechanics-A/Solids, 2020, 82: 103999

[14] Zhao X, Zheng S, Li Z. Effects of porosity and flexoelectricity on static bending and free vibration of AFG piezoelectric nanobeams. Thin-Walled Structures, 2020, 151: 106754

[15] Naskar S, Shingare K B, Mondal S, Mukhopadhyay T. Flexoelectricity and surface effects on coupled electromechanical responses of graphene reinforced functionally graded nanocomposites: A unified size-dependent semi-analytical framework. Mechanical Systems and Signal Processing, 2022, 169: 108757

[16] Lan M, Yang W, Liang X, Hu S, Shen S. Vibration modes of flexoelectric circular plate. Acta Mechanica Sinica, 2022, 38(12): 422063

[17] Duc D H, van Thom D, Cong P H, Van Minh P, Nguyen N X. Vibration and static buckling behavior of variable thickness flexoelectric nanoplates. Mechanics Based Design of Structures and Machines, 2023, 51(12): 7102–7130

[18] Ji X. Nonlinear electromechanical analysis of axisymmetric thin circular plate based on flexoelectric theory. Scientific Reports, 2021, 11(1): 21762

[19] van Minh P, van Ke T. A comprehensive study on mechanical responses of non-uniform thickness piezoelectric nanoplates taking into account the flexoelectric effect. Arabian Journal for Science and Engineering, 2023, 48(9): 11457–11482

[20] Thai L M, Luat D T, Phung V B, van Minh P, van Thom D. Finite element modeling of mechanical behaviors of piezoelectric nanoplates with flexoelectric effects. Archive of Applied Mechanics, 2022, 92(1): 163–182

[21] Qu Y L, Guo Z W, Zhang G Y, Gao X L, Jin F. A new model for circular cylindrical Kirchhoff–Love shells incorporating microstructure and flexoelectric effects. Journal of Applied Mechanics, 2022, 89(12): 121010

[22] Zhang J, Fan M, Tzou H. Flexoelectric vibration control of parabolic shells. Journal of Intelligent Material Systems and Structures, 2023, 34(8): 909–927

[23] Fattaheian D S, Tadi B Y. Size-dependent continuum-based model of a truncated flexoelectric/flexomagnetic functionally graded conical nano/microshells. Applied Physics. A, Materials Science & Processing, 2022, 128(4): 320

[24] Fan M, Yu P, Xiao Z. An artificial neural network model for multi-flexoelectric actuation of plates. International Journal of Smart and Nano Materials, 2022, 13(4): 1–23

[25] Zhuang X, Guo H, Alajlan N, Zhu H, Rabczuk T. Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning. European Journal of Mechanics. A, Solids, 2021, 87: 104225

[26] Guo H, Zheng H, Zhuang X. Numerical manifold method for vibration analysis of Kirchhoff’s plates of arbitrary geometry. Applied Mathematical Modelling, 2019, 66: 695–727

[27] Guo H, Zheng H. The linear analysis of thin shell problems using the numerical manifold method. Thin-walled Structures, 2018, 124: 366–383

[28] Nguyen V P, Anitescu C, Bordas S P A, Rabczuk T. Isogeometric analysis: An overview and computer implementation aspects. Mathematics and Computers in Simulation, 2015, 117: 89–116

[29] Jüttler B, Langer U, Mantzaflaris A, Moore S E, Zulehner W. Geometry + simulation modules: Implementing isogeometric analysis. Proceedings in Applied Mathematics and Mechanics, 2014, 14(1): 961–962

[30] Marussig B, Hughes T J R. A review of trimming in isogeometric analysis: Challenges, data exchange and simulation aspects. Archives of Computational Methods in Engineering, 2018, 25(4): 1059–1127

[31] Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39−41): 4135–4195

[32] Hughes T J R, Reali A, Sangalli G. Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS. Computer Methods in Applied Mechanics and Engineering, 2008, 197(49−50): 4104–4124

[33] Ha S H, Choi K K, Cho S. Numerical method for shape optimization using T-spline based isogeometric method. Structural and Multidisciplinary Optimization, 2010, 42(3): 417–428

[34] Vu-Bac N, Duong T X, Lahmer T, Zhuang X, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Computer Methods in Applied Mechanics and Engineering, 2018, 331: 427–455

[35] Bazilevs Y, Calo V M, Zhang Y, Hughes T J R. Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Computational Mechanics, 2006, 38(4−5): 310–322

[36] Auricchio F, Da Veiga L B, Hughes T J R, Reali A, Sangalli G. Isogeometric collocation methods. Mathematical Models and Methods in Applied Sciences, 2010, 20(11): 2075–2107

[37] Vu-Bac N, Duong T X, Lahmer T, Areias P, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis of thermal expansion induced morphing of thin shells. Computer Methods in Applied Mechanics and Engineering, 2019, 350: 480–510

[38] Schmidt R, Kiendl J, Bletzinger K U, Wüchner R. Realization of an integrated structural design process: Analysis-suitable geometric modelling and isogeometric analysis. Computing and Visualization in Science, 2010, 13(7): 315–330

[39] BazilevsJ. Isogeometric Analysis of Turbulence and Fluid-Structure Interaction. Austin: the University of Texas at Austin, 2006

[40] Wang Y, Wang Z, Xia Z, Poh L H. Structural design optimization using isogeometric analysis: A comprehensive review. Computer Modeling in Engineering & Sciences, 2018, 117(3): 455–507

[41] Vu-Bac N, Rabczuk T, Park H S, Fu X, Zhuang X. A NURBS-based inverse analysis of swelling induced morphing of thin stimuli-responsive polymer gels. Computer Methods in Applied Mechanics and Engineering, 2022, 397: 115049

[42] BontinckZCornoJdeGersem HKurzSPelsASchöpsSWolfFdeFalco CDölzJVázquezRRömerU. Recent advances of isogeometric analysis in computational electromagnetics. arXiv preprint arXiv:1709.06004, 2017

[43] de Lorenzis L, Wriggers P, Hughes T J R. Isogeometric contact: A review. GAMM Mitteilungen, 2014, 37(1): 85–123

[44] Hamdia K M, Ghasemi H, Zhuang X, Rabczuk T. Multilevel monte carlo method for topology optimization of flexoelectric composites with uncertain material properties. Engineering Analysis with Boundary Elements, 2022, 134: 412–418

[45] Ghasemi H, Park H S, Alajlan N, Rabczuk T. A computational framework for design and optimization of flexoelectric materials. International Journal of Computational Methods, 2020, 17(01): 1850097

[46] Ghasemi H, Park H S, Zhuang X, Rabczuk T. Three-dimensional isogeometric analysis of flexoelectricity with MATLAB implementation. Computers, Materials & Continua, 2020, 65(2): 1157–1179

[47] HamdiaK MGhasemiHZhuangXAlajlanNRabczukT. Computational machine learning representation for the flexoelectricity effect in truncated pyramid structures. Computers, Materials & Continua, 2019: 79–87

[48] Gao J, Xiao M, Zhang Y, Gao L. A comprehensive review of isogeometric topology optimization: Methods, applications and prospects. Chinese Journal of Mechanical Engineering, 2020, 33(1): 1–14

[49] Hung P T, van Phung P, Chien H. A refined isogeometric plate analysis of porous metal foam microplates using modified strain gradient theory. Composite Structures, 2022, 289(1): 115467

[50] van Phung P, Ferreira A J M, Nguyen-Xuan H, Thai C H. Scale-dependent nonlocal strain gradient isogeometric analysis of metal foam nanoscale plates with various porosity distributions. Composite Structures, 2021, 268: 113949

[51] Barati M R. Nonlocal-strain gradient forced vibration analysis of metal foam nanoplates with uniform and graded porosities. Advances in Nano Research, 2017, 5(4): 393–414

[52] Cemal Eringen A. Nonlocal polar elastic continua. International Journal of Engineering Science, 1972, 10(1): 1–16

[53] Pham Q H, Nguyen P C, Tran V K, Nguyen-Thoi T. Finite element analysis for functionally graded porous nano-plates resting on elastic foundation. Steel and Composite Structures, 2021, 41(2): 149–166

[54] Biot M A. Theory of buckling of a porous slab and its thermoelastic analogy. Journal of Applied Mechanics, 1964, 31(2): 194–198

[55] PhamQ HTranT TTranV KNguyenP CNguyen-ThoiTZenkourA M. Bending and hygro-thermo-mechanical vibration analysis of a functionally graded porous sandwich nanoshell resting on elastic foundation. Mechanics of Advanced Materials and Structures, 2022, 29(27): 5885–5905

[56] TranV KPhamQ HNguyen-ThoiT. A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations. Engineering with Computers, 2022: 1–26

[57] Wang B, Li X F. Flexoelectric effects on the natural frequencies for free vibration of piezoelectric nanoplates. Journal of Applied Physics, 2021, 129(3): 034102

[58] Zhang Z, Yan Z, Jiang L. Flexoelectric effect on the electroelastic responses and vibrational behaviors of a piezoelectric nanoplate. Journal of Applied Physics, 2014, 116(1): 014307

[59] Shu L, Wei X, Pang T, Yao X, Wang C. Symmetry of flexoelectric coefficients in crystalline medium. Journal of Applied Physics, 2011, 110(10): 104106

[60] Pham Q H, Tran V K, Nguyen P C. Hygro-thermal vibration of bidirectional functionally graded porous curved beams on variable elastic foundation using generalized finite element method. Case Studies in Thermal Engineering, 2022, 40: 102478

[61] PhamQ HNhanH TTranV KZenkourA M. Hygro-thermo-mechanical vibration analysis of functionally graded porous curved nanobeams resting on elastic foundations. Waves in Random and Complex Media, 2023, 16(2): 256978379

[62] Dung N T, van Ke T, Huyen T T H, van Minh P. Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory. Comptes Rendus. Mécanique, 2022, 350(G1): 121–141

[63] Thai L M, Luat D T, van Ke T, van Phung M. Finite-element modeling for static bending analysis of rotating two-layer FGM beams with shear connectors resting on imperfect elastic foundations. Journal of Aerospace Engineering, 2023, 36(3): 04023013

[64] PhamQ HTranV KNguyenP C. Nonlocal strain gradient finite element procedure for hygro-thermal vibration analysis of bidirectional functionally graded porous nanobeams. Waves in Random and Complex Media, 2023, 1–32

[65] Pham Q H, Malekzadeh P, Tran V K, Nguyen-Thoi T. Free vibration analysis of functionally graded porous curved nanobeams on elastic foundation in hygro-thermo-magnetic environment. Frontiers of Structural and Civil Engineering, 2023, 17(4): 1–22

[66] Sobhy M. A comprehensive study on FGM nanoplates embedded in an elastic medium. Composite Structures, 2015, 134: 966–980

[67] Pham Q H, Tran V K, Tran T T, Zenkour A M. Nonlocal higher-order finite element modeling for vibration analysis of viscoelastic orthotropic nanoplates resting on variable viscoelastic foundation. Composite Structures, 2023, 318: 117067

[68] Tran T T, Tran V K, Pham Q H, Zenkour A M. Extended four-unknown higher-order shear deformation nonlocal theory for bending, buckling and free vibration of functionally graded porous nanoshell resting on elastic foundation. Composite Structures, 2021, 264: 113737

[69] Pham Q H, Nguyen P C, Tran V K, Nguyen-Thoi T. Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium. Defence Technology, 2022, 18(8): 1311–1329

[70] Pham Q H, Nguyen P C, Tran V K, Lieu Q X, Tran T T. Modified nonlocal couple stress isogeometric approach for bending and free vibration analysis of functionally graded nanoplates. Engineering with Computers, 2023, 39(1): 993–1018

[71] Pham Q H, Tran V K, Tran T T. Vibration characteristics of sandwich plates with an auxetic honeycomb core and laminated three-phase skin layers under blast load. Defence Technology, 2022, 24: 148–163

[72] Newmark N M. A method of computation for structural dynamic. Journal of the Engineering Mechanics Division, 1959, 85(3): 67–94

[73] Ebrahimi F, Habibi S. Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate. Steel and Composite Structures, 2016, 20(1): 205–225

[74] Yang W, Liang X, Shen S. Electromechanical responses of piezoelectric nanoplates with flexoelectricity. Acta Mechanica, 2015, 226(9): 3097–3110

[75] Reddy J N. Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering, 2000, 47(1−3): 663–684

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