Dynamic Analysis of Laminated Composite Plate Integrated with a Piezoelectric Actuator Using Four-Variable Refined Plate Theory

Abstract

This paper presents an analytical solution based on four-variable refined plate theory for dynamic analysis of cross-ply composite laminates integrated with piezoelectric fiber-reinforced composite actuator attached to the upper surface. This theory considers parabolic transverse shear strains through the plate thickness. Equations of motion for simply supported rectangular plates are derived using Hamilton’s principle, and Navier’s method is employed to solve the equations. An unconditionally stable implicit Newmark scheme is used for temporal discretization. The accuracy of the present analytical solution is proved by comparing the results with those obtained by the finite element analysis. The effects of different parameters such as thickness-to-side ratio, aspect ratios and types of mechanical and electrical loading on the deflections and stresses are investigated.

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References

  1. Abad F, Rouzegar J (2017) An exact spectral element method for free vibration analysis of FG plate integrated with piezoelectric layers. Compos Struct 180:696–708

    Article  Google Scholar 

  2. Arefi M, Zenkour AM (2017a) Effect of thermo-magneto-electro-mechanical fields on the bending behaviors of a three-layered nanoplate based on sinusoidal shear-deformation plate theory. J Sandw Struct Mater. https://doi.org/10.1177/1099636217697497

    Article  Google Scholar 

  3. Arefi M, Zenkour AM (2017b) Influence of magneto-electric environments on size-dependent bending results of three-layer piezomagnetic curved nanobeam based on sinusoidal shear deformation theory. J Sandw Struct Mater. https://doi.org/10.1177/1099636217723186

    Article  MATH  Google Scholar 

  4. Arefi M, Zenkour AM (2017c) Nonlocal electro-thermo-mechanical analysis of a sandwich nanoplate containing a Kelvin–Voigt viscoelastic nanoplate and two piezoelectric layers. Acta Mech 228(2):475–493

    MathSciNet  MATH  Article  Google Scholar 

  5. Arefi M, Zenkour AM (2017d) Size-dependent free vibration and dynamic analyses of piezo-electro-magnetic sandwich nanoplates resting on viscoelastic foundation. Physica B Condens Matter 521:188–197

    Article  Google Scholar 

  6. Arefi M, Zenkour AM (2017e) Thermo-electro-magneto-mechanical bending behavior of size-dependent sandwich piezomagnetic nanoplates. Mech Res Commun 84:27–42

    Article  Google Scholar 

  7. Arefi M, Zenkour AM (2017f) Thermo-electro-mechanical bending behavior of sandwich nanoplate integrated with piezoelectric face-sheets based on trigonometric plate theory. Compos Struct 162:108–122

    Article  Google Scholar 

  8. Arefi M, Zenkour AM (2018) Free vibration analysis of a three-layered microbeam based on strain gradient theory and three-unknown shear and normal deformation theory. Steel Compos Struct 26(4):421–437

    Google Scholar 

  9. Arefi M, Bidgoli EMR, Dimitri R, Bacciocchi M, Tornabene F (2018a) Application of sinusoidal shear deformation theory and physical neutral surface to analysis of functionally graded piezoelectric plate. Compos B Eng 151:35–50

    Article  Google Scholar 

  10. Arefi M, Zamani MH, Kiani M (2018b) Size-dependent free vibration analysis of three-layered exponentially graded nanoplate with piezomagnetic face-sheets resting on Pasternak’s foundation. J Intell Mater Syst Struct 29(5):774–786

    Article  Google Scholar 

  11. Behjat B, Salehi M, Armin A, Sadighi M, Abbasi M (2011) Static and dynamic analysis of functionally graded piezoelectric plates under mechanical and electrical loading. Sci Iran 18:986–994

    Article  Google Scholar 

  12. Farsangi MA, Saidi AR (2012) Levy type solution for free vibration analysis of functionally graded rectangular plates with piezoelectric layers. Smart Mater Struct 21:094017

    Article  Google Scholar 

  13. Huang XL, Shen HS (2005) Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators. Int J Mech Sci 47:187–208

    MATH  Article  Google Scholar 

  14. Jiang HJ, Liang LH, Ma L, Guo J, Dai HL, Wang XG (2017) An analytical solution of three-dimensional steady thermodynamic analysis for a piezoelectric laminated plate using refined plate theory. Compos Struct 162:194–209

    Article  Google Scholar 

  15. Kapuria S, Dube GP, Dumir PC, Sengupta S (1997) Levy-type piezothermoelastic solution for hybrid plate by using first-order shear deformation theory. Compos B Eng 28:535–546

    Article  Google Scholar 

  16. Kumar A, Chakraborty D (2009) Effective properties of thermo-electro-mechanically coupled piezoelectric fiber reinforced composites. Mater Des 30:1216–1222

    Article  Google Scholar 

  17. Mallik N, Ray MC (2003) Effective coefficients of piezoelectric fiber reinforced composites. AIAA J 41:704–710

    Article  Google Scholar 

  18. Mallik N, Ray MC (2004) Exact solutions for the analysis of piezoelectric fiber reinforced composites as distributed actuators for smart composite plates. Int J Mech Mater Des 1:347–364

    Article  Google Scholar 

  19. Mindlin RD (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18:31–38

    MATH  Google Scholar 

  20. Phung-Van P, Nguyen-Thoi T, Le-Dinh T, Nguyen-Xuan H (2013) Static and free vibration analyses and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cell-based smoothed discrete shear gap method (CS-FEM-DSG3). Smart Mater Struct 22(9):095026

    Article  Google Scholar 

  21. Phung-Van P, Abdel-Wahab M, Liew KM, Bordas SPA, Nguyen-Xuan H (2015a) Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory. Compos Struct 123:137–149

    Article  Google Scholar 

  22. Phung-Van P, De Lorenzis L, Thai CH, Abdel-Wahab M, Nguyen-Xuan H (2015b) Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements. Comput Mater Sci 96:495–505

    Article  Google Scholar 

  23. Phung-Van P, Nguyen LB, Tran LV, Dinh TD, Thai CH, Bordas SPA, Abdel-Wahab M, Nguyen-Xuan H (2015c) An efficient computational approach for control of nonlinear transient responses of smart piezoelectric composite plates. Int J NonLinear Mech 76:190–202

    Article  Google Scholar 

  24. Phung-Van P, Lieu QX, Nguyen-Xuan H, Wahab MA (2017a) Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Compos Struct 166:120–135

    Article  Google Scholar 

  25. Phung-Van P, Tran LV, Ferreira AJM, Nguyen-Xuan H, Abdel-Wahab M (2017b) Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads. Nonlinear Dyn 87(2):879–894

    MATH  Article  Google Scholar 

  26. Phung-Van P, Thanh CL, Nguyen-Xuan H, Abdel-Wahab M (2018) Nonlinear transient isogeometric analysis of FG-CNTRC nanoplates in thermal environments. Compos Struct 201:882–892

    Article  Google Scholar 

  27. Ray MC, Mallik N (2004) Finite element analysis of smart structures containing piezoelectric fiber-reinforced composite actuator. AIAA J 42:1398–1405

    Article  Google Scholar 

  28. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12:69–77

    MathSciNet  MATH  Google Scholar 

  29. Rouzegar J, Abad F (2015a) Analysis of cross-ply laminates with piezoelectric fiber-reinforced composite actuators using four-variable refined plate theory. J Theor Appl Mech 53:439–452

    Article  Google Scholar 

  30. Rouzegar J, Abad F (2015b) Free vibration analysis of FG plate with piezoelectric layers using four- variable refined plate theory. Thin Wall Struct 89:76–83

    Article  Google Scholar 

  31. Rouzegar J, Abbasi A (2017) A refined finite element method for bending of smart functionally graded plates. Thin Wall Struct 120:386–396

    Article  Google Scholar 

  32. Rouzegar J, Abbasi A (2018) A refined finite element method for bending analysis of laminated plates integrated with piezoelectric fiber-reinforced composite actuators. Acta Mech Sin 34:689–705

    MathSciNet  MATH  Article  Google Scholar 

  33. Rouzegar J, Sharifpoor RA (2015) A Finite Element Formulation for bending analysis of isotropic and orthotropic plates based on two-variable refined plate theory. Sci Iran Trans B Mech Eng 22:196–207

    Google Scholar 

  34. Rouzegar J, Sharifpoor RA (2016) Finite element formulations for free vibration analysis of isotropic and orthotropic plates using two-variable refined plate theory. Sci Iran Trans B Mech Eng 23:1787–1799

    Google Scholar 

  35. Rouzegar J, Sharifpoor RA (2017) Finite element formulations for buckling analysis of isotropic and orthotropic plates using two-variable refined plate theory. Iran J Sci Technol Trans Mech Eng 41:177–187

    Article  Google Scholar 

  36. Shahraeeni M, Shakeri R, Hasheminejad SM (2015) An analytical solution for free and forced vibration of a piezoelectric laminated plate coupled with an acoustic enclosure. Comput Math Appl 69:1329–1341

    MathSciNet  Article  Google Scholar 

  37. Shimpi RP (2002) Refined plate theory and its variants. AIAA J 40:137–146

    Article  Google Scholar 

  38. Shimpi RP, Patel HG (2006) A two variable refined plate theory for orthotropic plate analysis. Int J Solids Struct 43:6783–6799

    MATH  Article  Google Scholar 

  39. Shivakumar J, Ray MC (2008) Nonlinear analysis of smart cross-ply composite plates integrated with a distributed piezoelectric fiber reinforced composite actuator. Mech Adv Mater Struct 15(1):40–52

    Article  Google Scholar 

  40. Shiyekar SM, Kant T (2011) Higher order shear deformation effects on analysis of laminates with piezoelectric fibre reinforced composite actuators. Compos Struct 93:3252–3261

    Article  Google Scholar 

  41. Thai HT, Choi DH (2012) A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation. Compos B Eng 43:2335–2347

    Article  Google Scholar 

  42. Thanh CL, Phung-Van P, Thai CH, Nguyen-Xuan H, Wahab MA (2018) Isogeometric analysis of functionally graded carbon nanotube reinforced composite nanoplates using modified couple stress theory. Compos Struct 184:633–649

    Article  Google Scholar 

  43. Xia XK, Shen HS (2009) Nonlinear vibration and dynamic response of FGM plates with piezoelectric fiber reinforced composite actuators. Compos Struct 90:254–262

    Article  Google Scholar 

Download references

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Correspondence to Jafar Rouzegar.

Appendix

Appendix

Components of matrices K, M and F in Eq. (26) are defined as

$$f_{1} = - \left( {\frac{m\pi }{a}} \right)\mu_{1} \quad \,f_{2} = 0$$
$$f_{3} = - q_{mn} + \left( {\frac{m\pi }{a}} \right)^{2} \mu_{2} \quad f_{4} = - q_{mn} + \left( {\frac{m\pi }{a}} \right)^{2} \mu_{3}$$
$$m_{11} = m_{22} = - I_{0} \quad m_{12} = m_{21} = 0$$
$$m_{13} = m_{31} = \left( {\frac{m\pi }{a}} \right)I\quad m_{14} = m_{41} = \left( {\frac{m\pi }{a}} \right)J_{1}$$
$$m_{23} = m_{32} = \left( {\frac{n\pi }{b}} \right)I_{1} \quad m_{24} = m_{42} = \left( {\frac{n\pi }{b}} \right)J_{1}$$
$$m_{33} = - I_{0} - \left( {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right)I_{2} \quad m_{34} = m_{43} = - I_{0} - \left( {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right)J_{2}$$
$$m_{44} = - I_{0} - \left( {\left( {\frac{m\pi }{a}} \right)^{2} + \left( {\frac{n\pi }{b}} \right)^{2} } \right)K_{2}$$
$$k_{11} = - \left( {\frac{m\pi }{a}} \right)^{2} A_{11} - \left( {\frac{n\pi }{b}} \right)^{2} A_{66}$$
$$k_{12} = k_{21} = - \left( {\frac{m\pi }{a}} \right)\left( {\frac{n\pi }{b}} \right)\left( {A_{12} + A_{66} } \right)$$
$$k_{13} = k_{31} = \left( {\frac{m\pi }{a}} \right)^{3} B_{11} + \left( {\frac{m\pi }{a}} \right)\left( {\frac{n\pi }{b}} \right)^{2} \left( {B_{12} + 2B_{66} } \right)$$
$$k_{14} = k_{41} = \left( {\frac{m\pi }{a}} \right)^{3} C_{11} + \left( {\frac{m\pi }{a}} \right)\left( {\frac{n\pi }{b}} \right)^{2} \left( {C_{12} + 2C_{66} } \right)$$
$$k_{22} = - \left( {\frac{n\pi }{b}} \right)^{2} A_{22} - \left( {\frac{m\pi }{a}} \right)^{2} A_{66}$$
$$k_{23} = k_{32} = \left( {\frac{n\pi }{b}} \right)^{3} B_{22} + \left( {\frac{m\pi }{a}} \right)^{2} \left( {\frac{n\pi }{b}} \right)\left( {B_{12} + 2B_{66} } \right)$$
$$k_{24} = k_{42} = \left( {\frac{n\pi }{b}} \right)^{3} C_{22} + \left( {\frac{m\pi }{a}} \right)^{2} \left( {\frac{n\pi }{b}} \right)\left( {C_{12} + 2C_{66} } \right)$$
$$k_{33} = - \left( {\frac{m\pi }{a}} \right)^{4} E_{11} - \left( {\frac{m\pi }{a}} \right)^{2} \left( {\frac{n\pi }{b}} \right)^{2} \left( {2E_{12} + 4E_{66} } \right) - \left( {\frac{n\pi }{b}} \right)^{4} E_{22}$$
$$k_{34} = k_{43} = - \left( {\frac{m\pi }{a}} \right)^{4} F_{11} - \left( {\frac{m\pi }{a}} \right)^{2} \left( {\frac{n\pi }{b}} \right)^{2} \left( {2F_{12} + 4F_{66} } \right) - \left( {\frac{n\pi }{b}} \right)^{4} F_{22}$$
$$k_{44} = - \left( {\frac{m\pi }{a}} \right)^{4} H_{11} - \left( {\frac{m\pi }{a}} \right)^{2} \left( {\frac{n\pi }{b}} \right)^{2} \left( {2H_{12} + 4H_{66} } \right) - \left( {\frac{n\pi }{b}} \right)^{4} H_{22} - \left( {\frac{m\pi }{a}} \right)^{2} P_{55} - \left( {\frac{n\pi }{b}} \right)^{2} P_{44}$$

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Rouzegar, J., Koohpeima, R. & Abad, F. Dynamic Analysis of Laminated Composite Plate Integrated with a Piezoelectric Actuator Using Four-Variable Refined Plate Theory. Iran J Sci Technol Trans Mech Eng 44, 557–570 (2020). https://doi.org/10.1007/s40997-019-00284-1

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Keywords

  • Forced vibration
  • Piezoelectric
  • Actuator
  • Newmark scheme
  • Refined plate theory
  • Smart structure