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Hygro-Thermo-Mechanical Vibration Behavior of Viscoelastic Nanosheets Resting on Visco-Pasternak Medium Taking into Account Flexoelectric and Actual Surface Effects

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Abstract

For the first time, flexoelectric and actual surface effects are applied to analyze the free oscillation of viscoelastic nanosheets resting on visco-Pasternak medium in hygro-temperature environment. The material characteristics of nanosheets are viscoelastic based on the Kelvin–Voigt model, and the visco-Pasternak medium has two layers, in which the sliding layer and the other layer are described as a spring system interspersed with a damping system. The general equation of motion of the nanosheets is established by applying Kirchhoff plate theory along with Hamilton's principle and the nonlocal strain gradient hypothesis. To solve the free vibration equations of nanosheets with various boundary conditions, the Navier and Galerkin approaches are employed. The presence of the viscoelastic component causes the nanosheets' inherent frequency to oscillate in the general situation with both imaginary and real components. Finally, the impact of coefficients on vibration of viscoelastic nanosheets is discussed in the numerical examples. This is the work that calculates the flexoelectric, surface effects, and viscoelastic-based nanosheet structures under varied loads, which serves as a precondition for their manufacture and use in reality. The present work is general because it combines the theories of nonlocal strain gradient, flexoelectricity, and actual surface effects for viscoelastic nanoplate. Currently, no research has been performed on this issue, and this study makes clear on understanding the underlying physics of electromechanical coupling at the nanoscale.

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References

  1. Eringen, A.C.: Theory of micropolar plates. Zeitschrift für Angewandte Mathematik und Physik ZAMP. 18, 12–30 (1967)

    Article  Google Scholar 

  2. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)

    Article  MathSciNet  Google Scholar 

  3. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    Article  Google Scholar 

  4. Gurtin, M.E.; Ian, M.A.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MathSciNet  Google Scholar 

  5. Lam, D.C.C.; Yang, F.; Chong, A.C.M.; Wang, J.; Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)

    Article  Google Scholar 

  6. Abdelrahman, A.A.; Esen, I.; Daikh, A.A.; Eltaher, M.A.: Dynamic analysis of FG nanobeam reinforced by carbon nanotubes and resting on elastic foundation under moving load. Mech. Based Des. Struct. Mach. 51, 5383–5406 (2021)

    Article  Google Scholar 

  7. Pham, Q. H.; Tran, V. K.; Nguyen, P. C.: Nonlocal strain gradient finite element procedure for hygro-thermal vibration analysis of bidirectional functionally graded porous nanobeams. Waves Random Complex Media, 1–32 (2023)

  8. Ansari, R.; Faraji Oskouie, M.; Nesarhosseini, S.; Rouhi, H.: Flexoelectricity effect on the size-dependent bending of piezoelectric nanobeams resting on elastic foundation. Appl. Phys. A., 127 (2021).

  9. 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. Front. Struct. Civil Eng., pp 1–22 (2023)

  10. Mahmoudpour, E.; Hosseini-Hashemi, S.H.; Faghidian, S.A.: Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model. Appl. Math. Model. 57, 302–315 (2018)

    Article  MathSciNet  Google Scholar 

  11. Fang, J.; Zheng, S.; Xiao, J.; Zhang, X.: Vibration and thermal buckling analysis of rotating nonlocal functionally graded nanobeams in thermal environment. Aerosp. Sci. Technol. 106, 106146 (2020)

    Article  Google Scholar 

  12. Karimi, M.; Mirdamadi, H.R.; Shahidi, A.R.: Shear vibration and buckling of double-layer orthotropic nanoplates based on RPT resting on elastic foundations by DQM including surface effects. Microsyst. Technol. 23, 765–797 (2015)

    Article  Google Scholar 

  13. Thi, T. H. N.; Tran, V. K.; Pham, Q. H.: Flexoelectric and size-dependent effects on hygro-thermal vibration of variable thickness fluid-infiltrated porous metal foam nanoplates. Heliyon (2024)

  14. Mechab, B.; Mechab, I.; Benaissa, S.; Ameri, M.; Serier, B.: Probabilistic analysis of effect of the porosities in functionally graded material nanoplate resting on Winkler-Pasternak elastic foundations. Appl. Math. Model. 40, 738–749 (2016). https://doi.org/10.1016/j.apm.2015.09.093

    Article  MathSciNet  Google Scholar 

  15. Salehipour, H.; Nahvi, H.; Shahidi, A.; Mirdamadi, H.R.: 3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory. Appl. Math. Model. 47, 174–188 (2017)

    Article  MathSciNet  Google Scholar 

  16. 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. Compos. Struct. 318, 117067 (2023)

    Article  Google Scholar 

  17. Zenkour, A. M.; Radwan, A. F.: Nonlocal mixed variational formula for orthotropic nanoplates resting on elastic foundations. Euro. Phys. J. Plus., 135 (2020).

  18. Ansari, R.; Gholami, R.: Surface effect on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates with various edge supports. Acta Astronaut. 118, 72–89 (2016)

    Article  Google Scholar 

  19. Thi, T. T. T.; Tran, V. K.; Pham, Q. H.: Static and dynamic analyses of multi-directional functionally graded porous nanoplates with variable nonlocal parameter using MITC3+ element. J. Vibrat. Eng. Technolo., pp 1–25 (2023)

  20. Nami, M.R.; Janghorban, M.; Damadam, M.: Thermal buckling analysis of functionally graded rectangular nanoplates based on nonlocal third-order shear deformation theory. Aerosp. Sci. Technol. 41, 7–15 (2015)

    Article  Google Scholar 

  21. Sahmani, S.; Fattahi, A.M.; Ahmed, N.A.: Radial postbuckling of nanoscaled shells embedded in elastic foundations based on Ru’s surface stress elasticity theory. Mech. Based Des. Struct. Mach. 47, 787–806 (2019)

    Article  Google Scholar 

  22. Duong, K. D.; Mai, D. N.; Minh, P. V.; Ke, T. V.: An isogeometric approach to free vibration analysis of bi-directional functionally graded porous doubly-curved shallow microshells with variable length-scale parameters. Front. Struct. Civil Eng., 1–24 (2024)

  23. Zeighampour, H.; Tadi Beni, Y.; Botshekanan, D.M.: Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory. Thin-Walled Struct 122, 378–386 (2018)

    Article  Google Scholar 

  24. Pham, Q.H.; Nguyen, T.A.; Do, N.T.; Tran, V.K.; Nguyen, M.N.: Static and vibration analyses of functionally graded porous shell structures by using an averaged edge/node-based smoothed MITC3 element. Comput. Math. Appl. 153, 56–70 (2024)

    Article  MathSciNet  Google Scholar 

  25. Zhang, F.; Bai, C.; Wang, J.: Study on dynamic stability of magneto-electro-thermo-elastic cylindrical nanoshells resting on Winkler–Pasternak elastic foundations using nonlocal strain gradient theory. J. Braz. Soc. Mech. Sci. Eng. 45 (2022).

  26. Pouresmaeeli, S.; Ghavanloo, E.; Fazelzadeh, S.A.: Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium. Compos. Struct. 96, 405–410 (2013)

    Article  Google Scholar 

  27. Jalaei, M.H.; Arani, A.G.: Analytical solution for static and dynamic analysis of magnetically affected viscoelastic orthotropic double-layered graphene sheets resting on viscoelastic foundation. Physica B 530, 222–235 (2018)

    Article  Google Scholar 

  28. Karličić, D.; Kozić, P.; Pavlović, R.: Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium. Compos. Struct. 115, 89–99 (2014)

    Article  Google Scholar 

  29. Sobhy, M.; Zenkour, A.M.: Nonlocal thermal and mechanical buckling of nonlinear orthotropic viscoelastic nanoplates embedded in a Visco-Pasternak medium. Int. J. Appl. Mech. 10, 1850086 (2018)

    Article  Google Scholar 

  30. Zhang, Z.; Yan, Z.; Jiang, L.: Flexoelectric effect on the electroelastic responses and vibrational behaviors of a piezoelectric nanoplate. J. Appl. Phys. 116 (2014).

  31. Liang, X.; Hu, S.; Shen, S.: Effects of surface and flexoelectricity on a piezoelectric nanobeam. Smart Mater. Struct. 23, 035020 (2014)

    Article  Google Scholar 

  32. Joueid, N.; Zghal, S.; Chrigui, M.; Dammak, F.: Thermoelastic buckling analysis of plates and shells of temperature and porosity dependent functionally graded materials. Mech. Time-Dependent Mater. (2023). https://doi.org/10.1007/s11043-023-09644-6

  33. Zghal, S.; Dammak, F.: Vibrational behavior of beams made of functionally graded materials by using a mixed formulation. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 234, 3650–3666 (2020)

    Article  Google Scholar 

  34. Zghal, S.; Joueid, N.; Tornabene, F.; Dimitri, R.; Chrigui, M.; Dammak, F.: Time-dependent deflection responses of FG porous structures subjected to different external pulse loads. J. Vibrat. Eng. Technol. 12, 857–876 (2024)

    Article  Google Scholar 

  35. Zghal, S.; Trabelsi, S.; Dammak, F.: Post-buckling behavior of functionally graded and carbon-nanotubes based structures with different mechanical loadings. Mech. Based Des. Struct. Mach. 50, 2997–3039 (2022)

    Article  Google Scholar 

  36. Xiang, S.; Lee, K.Y.; Li, X.F.: Elasticity solution of functionally graded beams with consideration of the flexoelectric effect. J. Phys. D Appl. Phys. 53, 105301 (2020)

    Article  Google Scholar 

  37. 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. Mech. Syst. Signal Process. 169, 108757 (2022)

    Article  Google Scholar 

  38. Yang, W.; Liang, X.; Shen, S.: Electromechanical responses of piezoelectric nanoplates with flexoelectricity. Acta Mech. 226, 3097–3110 (2015)

    Article  MathSciNet  Google Scholar 

  39. Yue, Y.M.; Xu, K.Y.; Chen, T.: A micro scale Timoshenko beam model for piezoelectricity with flexoelectricity and surface effects. Compos. Struct. 136, 278–286 (2016)

    Article  Google Scholar 

  40. Li, Y.S.; Pan, E.: Bending of a sinusoidal piezoelectric nanoplate with surface effect. Compos. Struct. 136, 45–55 (2016)

    Article  MathSciNet  Google Scholar 

  41. Ebrahimi, F.; Barati, M.R.: Dynamic modeling of embedded nanoplate systems incorporating flexoelectricity and surface effects. Microsyst. Technol. 25, 175–187 (2018)

    Article  Google Scholar 

  42. Assadi, A.: Size dependent forced vibration of nanoplates with consideration of surface effects. Appl. Math. Model. 37, 3575–3588 (2013)

    Article  MathSciNet  Google Scholar 

  43. Yan, Z.; Jiang, L.Y.: Surface effects on the vibration and buckling of piezoelectric nanoplates. EPL (Europhysics Letters). 99, 27007 (2012)

    Article  Google Scholar 

  44. Reddy, J.N.; Lim, C.W.; Zhang, G.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)

    Article  MathSciNet  Google Scholar 

  45. 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. Eng. Comp. 39, 993–1018 (2022)

    Article  Google Scholar 

  46. Quoc-Hoa Pham, P-C.N.; Van-Ke, T.; Nguyen-Thoi, T.: Finite element analysis for functionally graded porous nano-plates resting on elastic foundation. Steel Compos. Struct. 41, 149–166 (2021).

  47. Zhang, Z.; Jiang, L.: Size effects on electromechanical coupling fields of a bending piezoelectric nanoplate due to surface effects and flexoelectricity. J. Appl. Phys. 116 (2014).

  48. 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 Stud. Thermal Eng.. 40, 102478 (2022)

    Article  Google Scholar 

  49. 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. Compos. Struct. 264, 113737 (2021)

    Article  Google Scholar 

  50. Ebrahimi, F.; Barati, M.R.: Wave propagation analysis of quasi-3D FG nanobeams in thermal environment based on nonlocal strain gradient theory. Appl. Phys. 122(9), 843 (2016). https://doi.org/10.1007/s00339-016-0368-1

    Article  Google Scholar 

  51. Ebrahimi, F.; Barati, M.R.: A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment. Appl. Phys. A 122(9), 1–14 (2016). https://doi.org/10.1007/s00339-016-0322-2

    Article  Google Scholar 

  52. Zenkour, A.M.; Sobhy, M.: Thermal buckling of various types of FGM sandwich plates. Compos. Struct. 93(1), 93–102 (2010). https://doi.org/10.1016/j.compstruct.2010.06.012

    Article  Google Scholar 

  53. Zenkour, A.M.; Sobhy, M.: Thermal buckling of functionally graded plates resting on elastic foundations using the trigonometric theory. J. Therm. Stress. 34(11), 1119–1138 (2011). https://doi.org/10.1080/01495739.2011.606017

    Article  Google Scholar 

  54. Sobhy, M.: Thermomechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium. Physica E 56, 400–409 (2014)

    Article  Google Scholar 

  55. Shen, L.; Shen, H.-S.; Zhang, C.-L.: Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Comput. Mater. Sci. 48, 680–685 (2010)

    Article  Google Scholar 

  56. Van Minh, P.; Van Ke, T.: A comprehensive study on mechanical responses of non-uniform thickness piezoelectric nanoplates taking into account the flexoelectric effect. Arab. J. Sci. Eng. 48, 11457–11482 (2022)

    Article  Google Scholar 

  57. Babu, B.; Patel, B.P.: A new computationally efficient finite element formulation for nanoplates using second-order strain gradient Kirchhoff’s plate theory. Compos. B Eng. 168, 302–311 (2019)

    Article  Google Scholar 

  58. 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. Sandwich Struct. Mater. 22, 2186–2209 (2018)

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Engineering and Technology, Nguyen Tat Thanh University, Ho Chi Minh City, Vietnam

    Quoc Hoa Pham

  2. Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi, Vietnam

    Van Ke Tran

  3. Advanced Structural Engineering Laboratory, Department of Structural Engineering, Faculty of Civil Engineering, Ho Chi Minh City Open University, Ho Chi Minh City, Vietnam

    Phu-Cuong Nguyen

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  1. Quoc Hoa Pham
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  2. Van Ke Tran
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  3. Phu-Cuong Nguyen
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Correspondence to Phu-Cuong Nguyen.

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Pham, Q.H., Tran, V.K. & Nguyen, PC. Hygro-Thermo-Mechanical Vibration Behavior of Viscoelastic Nanosheets Resting on Visco-Pasternak Medium Taking into Account Flexoelectric and Actual Surface Effects. Arab J Sci Eng (2024). https://doi.org/10.1007/s13369-024-09017-2

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