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Nonlinear frequencies of porous functionally graded piezo-elasto-magneto plates with non-uniform thickness: a hybrid FEM-ANN predictive approach

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Abstract

Predicting coupled frequency response of piezo-elasto-magnetic (PEM) structures is crucial for accurately designing sensors/actuators, energy harvesters and other smart structures. The numerous coupling parameters involved make the numerical analysis through the conventional finite element methods (FEM) cumbersome and time-consuming, particularly for non-uniformly shaped/variable thickness structures. Hence, in this article, a hybrid approach integrating the computational benefits of FEM and artificial neural network (ANN) models has been proposed to predict the coupled nonlinear frequency response (NLFR) of porous functionally graded PEM (PFG-PEM) plates with non-uniform geometries. Through this approach, the computational efforts are substantially reduced retaining appreciable accuracy. A FEM model based on Hamilton’s principle, von Karman’s nonlinearity and higher-order shear deformation theory (HSDT) was initially developed for non-uniform PFG-PEM plates. The large datasets collected from the nonlinear FEM simulation are used to train an ANN model that can accurately predict the NLFR of non-uniform PFG-PEM plates for out-of-range input data sets. The plates in the current study have non-uniform thicknesses varying bi-linearly, linearly, and exponentially. The different variants of PFG-PEM composites and porosity patterns are evaluated, whose material property varies across the thickness according to a power law distribution. In addition, two forms of electromagnetic boundary conditions, such as open and closed circuits, are enforced on the plate, and its NLFR is assessed. Further, several numerical examples are presented to understand the interdependency of several material and geometrical parameters on the overall NLFR of PFG-PEM plates. This predictive tool can be readily used for further optimisation of smart structural design, significantly reducing the time consumed.

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The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.

Abbreviations

u, v, and w :

Displacement components in x-, y- and z-direction

u 0, v 0, and w 0 :

Mid-plane displacement components

a, b :

Plate’s length and width

h(x, y):

Variable thickness

h ab :

Maximum thickness of the plate varying in y- and x-directions

h b :

Maximum thickness of the plate varying in the y-direction

h a :

Initial thickness of the plate

P fg :

Effective material properties of PFG-PEM composites

P t :

The material property of PFG-PEM composites at the top-most layer

P b :

The material property of PFG-PEM composites at the bottom-most layer

V por :

Porosity distribution patterns

k :

Grading index

p :

Porosity volume

z :

Distance of the point under evaluation from the mid-surface

T p, T k :

Potential and kinetic energy

θ x and θ y :

Rotations of normal in xz-plane and yz-plane

κ x, κ y :

Higher-order rotational terms

β, χ :

Taper ratio in y- and x-direction

\(\left\{{\varepsilon }_{b}\right\}, \left\{{\varepsilon }_{s}\right\}\) :

Bending and shear strain components

\(\left\{{\varepsilon }_{b\_L}\right\}, \left\{{\varepsilon }_{b\_\mathrm{NL}}\right\}\) :

Linear and nonlinear bending strain components

\({\Omega }^{\mathrm{fg}}\) :

The volume of the FG layer

\({\omega }{\prime}\) :

Nonlinear frequency ratio

\({\omega }_{\mathrm{nl}}\) :

Nonlinear frequency

\({\omega }_{l}\) :

Linear frequency

ρ :

Density

[C]:

Elastic stiffness coefficient matrix

[B tb], [B rb], [B ts], [B rs]:

Strain–displacement matrices

[e], [q], [m], [η] and [µ]:

Piezoelectric, magnetostrictive, magnetoelectric, dielectric and permeability matrices

{σ}, {D} and {B}:

Stress, electric displacement and magnetic flux density vectors

{E}, {H}:

Electric and magnetic field intensity

References

  1. Zhai, J., Xing, Z., Dong, S., Li, J., Viehland, D.: Magnetoelectric laminate composites: an overview. J. Am. Ceram. Soc. 91(2), 351–358 (2008). https://doi.org/10.1111/j.1551-2916.2008.02259.x

    Article  CAS  Google Scholar 

  2. Duc, N.D., Dat, N.D., Anh, V.T.T., Giang, V.D., Thinh, P.N.: Effects of the magneto-electro-elastic layer on the CNTRC cylindrical shell. Arch. Appl. Mech. 93(3), 997–1021 (2023)

    ADS  Google Scholar 

  3. Arshid, E., Soleimani-Javid, Z., Amir, S., Duc, N.D.: Higher-order hygro-magneto-electro-thermomechanical analysis of FG-GNPs-reinforced composite cylindrical shells embedded in PEM layers. Aerosp. Sci. Technol. 126, 107573 (2022)

    Google Scholar 

  4. Quan, T.Q., Van Quyen, N., Duc, N.D.: An analytical approach for nonlinear thermo-electro-elastic forced vibration of piezoelectric penta–Graphene plates. Eur. J. Mech. A/Solids 85, 104095 (2021)

    ADS  MathSciNet  Google Scholar 

  5. Zheng, Y.F., Liu, L.C., Qu, D.Y., Chen, C.P.: Nonlinear postbuckling analysis of magneto-electro-thermo-elastic laminated microbeams based on modified couple stress theory. Appl. Math. Model. 118, 89–106 (2023)

    MathSciNet  Google Scholar 

  6. Singh, S.K., Singh, I.V.: Extended isogeometric analysis for fracture in functionally graded magneto-electro-elastic material. Eng. Fract. Mech. 247, 107640 (2021)

    Google Scholar 

  7. Nguyen, D.T.D., Javidan, F., Attar, M., Natarajan, S., Yang, Z., Ooi, E.H., Song, C., Ooi, E.T.: Fracture analysis of cracked magneto-electro-elastic functionally graded materials using scaled boundary finite element method. Theoret. Appl. Fract. Mech. 118, 103228 (2022)

    Google Scholar 

  8. Sh, E.L., Kattimani, S., Trung, N.T.: Frequency response analysis of edge-cracked magneto-electro-elastic functionally graded plates using extended finite element method. Theoret. Appl. Fract. Mech. 120, 103417 (2022)

    Google Scholar 

  9. Zhou, L., Li, M., Tian, W., Liu, P.: Coupled multi-physical cell-based smoothed finite element method for static analysis of functionally grade magneto-electro-elastic structures at uniform temperature. Compos. Struct. 226, 111238 (2019)

    Google Scholar 

  10. Zhou, L., Qu, F., Ren, S., Mahesh, V.: An inhomogeneous stabilized node-based smoothed radial point interpolation method for the multi-physics coupling responses of functionally graded magneto-electro-elastic structures. Eng. Anal. Bound. Elem. 151, 406–422 (2023)

    MathSciNet  Google Scholar 

  11. Zhang, P., Qi, C., Fang, H., Sun, X.: A semi-analytical approach for the flexural analysis of in-plane functionally graded magneto-electro-elastic plates. Compos. Struct. 250, 112590 (2020)

    Google Scholar 

  12. Tang, Y., Li, C.L., Yang, T.: Application of the generalized differential quadrature method to study vibration and dynamic stability of tri-directional functionally graded beam under magneto-electro-elastic fields. Eng. Anal. Bound. Elem. 146, 808–823 (2023)

    Google Scholar 

  13. Jiang, W.W., Gao, X.W., Liu, H.Y.: Multi-physics zonal Galerkin free element method for static and dynamic responses of functionally graded magneto-electro-elastic structures. Compos. Struct. 321, 117217 (2023)

    Google Scholar 

  14. Zhang, S.Q., Zhao, Y.F., Wang, X., Chen, M., Schmidt, R.: Static and dynamic analysis of functionally graded magneto-electro-elastic plates and shells. Compos. Struct. 281, 114950 (2022)

    CAS  Google Scholar 

  15. Zhao, Y.F., Zhang, S.Q., Wang, X., Ma, S.Y., Zhao, G.Z., Kang, Z.: Nonlinear analysis of carbon nanotube reinforced functionally graded plates with magneto-electro-elastic multiphase matrix. Compos. Struct. 297, 115969 (2022)

    Google Scholar 

  16. Mahesh, V.: Nonlinear damping of auxetic sandwich plates with functionally graded magneto-electro-elastic facings under multiphysics loads and electromagnetic circuits. Compos. Struct. 290, 115523 (2022)

    Google Scholar 

  17. Mahesh, V., Harursampath, D.: Nonlinear vibration of functionally graded magneto-electro-elastic higher order plates reinforced by CNTs using FEM. Eng. Comput. 38(2), 1029–1051 (2022)

    Google Scholar 

  18. Mahesh, V.: Porosity effect on the energy harvesting behaviour of functionally graded magneto-electro-elastic/fibre-reinforced composite beam. Eur. Phys. J. Plus 137(1), 48 (2021)

    Google Scholar 

  19. Mahesh, V., Harursampath, D.: Large deflection analysis of functionally graded magneto-electro-elastic porous flat panels. Eng. Comput. 38(Suppl 2), 1615–1634 (2022)

    Google Scholar 

  20. Mahesh, V.: Porosity effect on the nonlinear deflection of functionally graded magneto-electro-elastic smart shells under combined loading. Mech. Adv. Mater. Struct. 29(19), 2707–2725 (2022)

    Google Scholar 

  21. Nguyen-Thoi, T., Ly, K.D., Truong, T.T., Nguyen, S.N., Mahesh, V.: Analysis and optimal control of smart damping for porous functionally graded magneto-electro-elastic plate using smoothed FEM and metaheuristic algorithm. Eng. Struct. 259, 114062 (2022)

    Google Scholar 

  22. Ly, K.D., Nguyen-Thoi, T., Truong, T.T., Nguyen, S.N.: Multi-objective optimization of the active constrained layer damping for smart damping treatment in magneto-electro-elastic plate structures. Int. J. Mech. Mater. Des. 18(3), 633–663 (2022)

    CAS  Google Scholar 

  23. Vinyas, M.: On frequency response of porous functionally graded magneto-electro-elastic circular and annular plates with different electro-magnetic conditions using HSDT. Compos. Struct. 240, 112044 (2020)

    Google Scholar 

  24. Esen, I., Özmen, R.: Thermal vibration and buckling of magneto-electro-elastic functionally graded porous nanoplates using nonlocal strain gradient elasticity. Compos. Struct. 296, 115878 (2022)

    Google Scholar 

  25. Esayas, L.S., Kattimani, S.: Effect of porosity on active damping of geometrically nonlinear vibrations of a functionally graded magneto-electro-elastic plate. Def. Technol. 18(6), 891–906 (2022)

    Google Scholar 

  26. Hung, P.T., Thai, C.H., Phung-Van, P.: A C0-HSDT free vibration of magneto-electro-elastic functionally graded porous plates using a moving Kriging meshfree method. Aerosp. Sci. Technol. 137, 108266 (2023)

    Google Scholar 

  27. Saffari, P.R., Sirimontree, S., Thongchom, C., Jearsiripongkul, T., Saffari, P.R., Keawsawasvong, S.: Effect of uniform and nonuniform temperature distributions on sound transmission loss of double-walled porous functionally graded magneto-electro-elastic sandwich plates with subsonic external flow. Int. J. Thermofluids 17, 100311 (2023)

    Google Scholar 

  28. Li, Z., Wang, Q., Qin, B., Zhong, R., Yu, H.: Vibration and acoustic radiation of magneto-electro-thermo-elastic functionally graded porous plates in the multi-physics fields. Int. J. Mech. Sci. 185, 105850 (2020)

    Google Scholar 

  29. Ebrahimi, F., Jafari, A., Barati, M.R.: Vibration analysis of magneto-electro-elastic heterogeneous porous material plates resting on elastic foundations. Thin-Walled Struct. 119, 33–46 (2017)

    Google Scholar 

  30. Liu, H., Liu, H., Yang, J.: Vibration of FG magneto-electro-viscoelastic porous nanobeams on visco-Pasternak foundation. Compos. B Eng. 155, 244–256 (2018)

    Google Scholar 

  31. Kiran, M.C., Kattimani, S.C., Vinyas, M.: Porosity influence on structural behaviour of skew functionally graded magneto-electro-elastic plate. Compos. Struct. 191, 36–77 (2018)

    Google Scholar 

  32. Kumar, V., Singh, S.J., Saran, V.H., Harsha, S.P.: Vibration characteristics of porous FGM plate with variable thickness resting on Pasternak’s foundation. Eur. J. Mech. A/Solids 85, 104124 (2021)

    ADS  MathSciNet  Google Scholar 

  33. Minh, P.P., Duc, N.D.: The effect of cracks on the stability of the functionally graded plates with variable-thickness using HSDT and phase-field theory. Compos. B Eng. 175, 107086 (2019)

    Google Scholar 

  34. Xu, Y., Zhou, D.: Three-dimensional elasticity solution of functionally graded rectangular plates with variable thickness. Compos. Struct. 91(1), 56–65 (2009)

    Google Scholar 

  35. Aksogan, O., Sofiyev, A.H.: Dynamic buckling of a cylindrical shell with variable thickness subject to a time-dependent external pressure varying as a power function of time. J. Sound Vib. 254(4), 693–702 (2002)

    ADS  Google Scholar 

  36. Sofiyev, A.H., Aksogan, O.: Buckling of a conical thin shell with variable thickness under a dynamic loading. J. Sound Vib. 270(4–5), 903–915 (2004)

    ADS  Google Scholar 

  37. Kumar, H.N., Kattimani, S.: Effect of different geometrical non-uniformities on nonlinear vibration of porous functionally graded skew plates: a finite element study. Def. Technol. 18(6), 918–936 (2022)

    Google Scholar 

  38. Ly, D.K., Truong, T.T., Nguyen-Thoi, T.: Multi-objective optimization of laminated functionally graded carbon nanotube-reinforced composite plates using deep feedforward neural networks-NSGAII algorithm. Int. J. Comput. Methods 19(03), 2150065 (2022)

    MathSciNet  Google Scholar 

  39. Nguyen, T., Ly, K.D., Nguyen-Thoi, T., Nguyen, B.P., Doan, N.P.: Prediction of axial load bearing capacity of PHC nodular pile using Bayesian regularization artificial neural network. Soils Found. 62(5), 101203 (2022)

    Google Scholar 

  40. Zhao, S., Zhang, Y., Zhang, Y., Zhang, W., Yang, J., Kitipornchai, S.: Buckling of functionally graded hydrogen-functionalized graphene reinforced beams based on machine learning-assisted micromechanics models. Eur. J. Mech. A/Solids 96, 104675 (2022)

    ADS  MathSciNet  Google Scholar 

  41. Zhao, S., Zhang, Y., Zhang, Y., Yang, J., Kitipornchai, S.: Vibrational characteristics of functionally graded graphene origami-enabled auxetic metamaterial beams based on machine learning assisted models. Aerosp. Sci. Technol. 130, 107906 (2022)

    Google Scholar 

  42. Birky, D., Ladd, J., Guardiola, I., Young, A.: Predicting the dynamic response of a structure using an artificial neural network. J. Low Freq. Noise Vib. Act. Control 41(1), 182–195 (2022)

    Google Scholar 

  43. Mahesh, V.: Artificial neural network (ANN) based investigation on the static behaviour of piezo-magneto-thermo-elastic nanocomposite sandwich plate with CNT agglomeration and porosity. Int. J. Non-Linear Mech. 153, 104406 (2023)

    ADS  Google Scholar 

  44. Mahesh, V., Mahesh, V., Ponnusami, S.A.: FEM-ANN approach to predict nonlinear pyro-coupled deflection of sandwich plates with agglomerated porous nanocomposite core and piezo-magneto-elastic facings in thermal environment. In: Mechanics of Advanced Materials and Structures, pp. 1–24 (2023). https://doi.org/10.1080/15376494.2023.2201927

  45. Mahesh, V.: Machine learning assisted nonlinear deflection analysis of agglomerated carbon nanotube core smart sandwich plate with three-phase magneto-electro-elastic skin. In: Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, p. 14644207231180459 (2023)

  46. Labusch, M., Etier, M., Lupascu, D.C., Schröder, J., Keip, M.A.: Product properties of a two-phase magneto-electric composite: synthesis and numerical modeling. Comput. Mech. 54, 71–83 (2014)

    MathSciNet  Google Scholar 

  47. Nan, C.W., Liu, G., Lin, Y., Chen, H.: Magnetic-field-induced electric polarization in multiferroic nanostructures. Phys. Rev. Lett. 94(19), 197203 (2005)

    ADS  PubMed  Google Scholar 

  48. Reddy, J.N.: An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, Cambridge (2004)

    Google Scholar 

  49. Tran, T.T., Nguyen, N.H., Do, T.V., Minh, P.V., Duc, N.D.: Bending and thermal buckling of unsymmetric functionally graded sandwich beams in high-temperature environment based on a new third-order shear deformation theory. J. Sandw. Struct. Mater. 23(3), 906–930 (2021)

    Google Scholar 

  50. Shabanpour, S., Razavi, S., Shooshtari, A.: Nonlinear vibration analysis of laminated magneto-electro-elastic rectangular plate based on third-order shear deformation theory. Iran. J. Sci. Technol. Trans. Mech. Eng. 43(1), 211–223 (2019)

    Google Scholar 

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Acknowledgements

The financial support by The Royal Society of London through Newton International Fellowship (NIF\R1\212432) is sincerely acknowledged by the author Vinyas Mahesh.

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    Vinyas Mahesh

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Mahesh, V. Nonlinear frequencies of porous functionally graded piezo-elasto-magneto plates with non-uniform thickness: a hybrid FEM-ANN predictive approach. Acta Mech 235, 633–657 (2024). https://doi.org/10.1007/s00707-023-03768-z

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