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Vibration analysis of polymer composite plates reinforced with graphene platelets resting on two-parameter viscoelastic foundation

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Abstract

In this paper, the vibration behavior of composite plate reinforced with graphene platelets (GPLs) resting on viscoelastic foundation in thermal environment based on higher-order shear deformation theory is examined. Halpin–Tsai model is utilized to determine the material properties of composites plate reinforced with GPL. In the present study, four patterns of GPLs distribution in plate layers are considered. To obtain the Euler–Lagrange equations of composites plate, Hamilton’s principle is employed and Navier’s method is utilized for analyzing and solving the problem. The results of this study have been verified by checking them with the previous works. The effects of various parameters such as geometry effect, GPL weight fraction, temperature changes and viscoelastic foundation on vibrational reaction of structure are analyzed.

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Notes

  1. Graphene platelet-reinforced composite.

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Authors and Affiliations

  1. Department of Mechanics, Faculty of Engineering, Imam Khomeini International University, Qazvin, 3414916818, Iran

    Saeedeh Qaderi & Farzad Ebrahimi

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  1. Saeedeh Qaderi
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  2. Farzad Ebrahimi
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Appendix

Appendix

  • 1.

    $$ {\text{K}}_{11} = - m^{2} A_{11} - n^{2} \left( {A_{44} } \right) $$

  • 2.

    $$ K_{12} = - mnA_{12} - mnA_{44} $$

  • 3.

    $$ K_{13} = c_{1} m^{3} \left( {D_{11} } \right) + c_{1} n^{2} m\left( {D_{12} } \right) + 2c_{1} mn^{2} \left( {D_{44} } \right) $$

  • 4.

    $$ K_{14} = - m^{2} \left( {B_{11} } \right) + c_{1} m^{2} \left( {D_{11} } \right) + c_{1} n^{2} \left( {D_{44} } \right) - n^{2} B_{44} $$

  • 5.

    $$ K_{15} = mnc_{1} D_{12} - mnB_{44} + mnc_{1} D_{44} - mnB_{12} $$

  • 6.

    $$ K_{21} = K_{12} $$

  • 7.

    $$ K_{22} = - n^{2} A_{22} - m^{2} A_{44} $$

  • 8.

    $$ K_{23} = c_{1} n^{3} D_{22} + mn^{2} c_{1} D_{12} + 2m^{2} nc_{1} D_{44} $$

  • 9.

    $$ K_{24} = - mnB_{12} + c_{1} mnD_{12} - mnB_{44} c_{1} mnD_{44} $$

  • 10.

    $$ K_{25} = - n^{2} B_{22} + c_{1} n^{2} D_{22} - m^{2} B_{44} + C_{1} m^{2} B_{44} $$

  • 11.

    $$ K_{31} = K_{13} $$

  • 12.

    $$ K_{32} = K_{23} $$

  • 13.

    $$ K_{33} = - m^{4} c_{1}^{2} G_{12} + n^{2} m^{2} \left( { - 2c_{1}^{2} G_{12} - 4c_{1}^{2} G_{44} } \right) - n^{4} c_{1}^{2} G_{22} - m^{2} \left( {A_{55} - 6c_{1} C_{55} + 9c_{1}^{2} E_{55} } \right) - n^{2} \left( {A_{66} - 6c_{1} C_{66} + 9c_{1}^{2} E_{66} } \right) - K_{w} + K_{p} \left( { - n^{2} - m^{2} } \right) + N_{T} \left( {m^{2} + n^{2} } \right) $$

  • 14.

    $$ K_{34} = c_{1} m^{3} E_{11} - c_{1}^{2} m^{3} G_{11} + mn^{2} \left( {c_{1} E_{12} - c_{1}^{2} G_{12} + 2c_{1} E_{44} - 2c_{1}^{2} G_{44} } \right) - 3c_{1} mA_{55} C_{55} + 3c_{1} m\left( {C_{55} - 3c_{1} E_{55} } \right) $$

  • 15.

    $$ K_{35} = m^{2} n\left( { - c_{1}^{2} G_{12} + 2c_{1} E_{44} - 2c_{1}^{2} G_{44} + c_{1} E_{12} } \right) - n^{3} c_{1}^{2} G_{22} - 3c_{1} nA_{66} C_{66} + 3c_{1} n\left( {C_{66} - 3c_{1} E_{66} } \right) + c_{1} n^{3} E_{22} $$

  • 16.

    $$ K_{41} = K_{14} $$

  • 17.

    $$ K_{42} = K_{24} $$

  • 18.

    $$ K_{43} = K_{34} $$

  • 19.

    $$ K_{44} = - m^{2} \left( {C_{11} - 2c_{1} E_{11} + c_{1}^{2} G_{11} } \right) - n^{2} (C_{44} - 2c_{1} E_{44} + c_{1}^{2} G_{44} + 3c_{1} \left( {C_{55} - 3c_{1} E_{55} } \right) - 3c_{1} A_{55} C_{55} $$

  • 20.

    $$ K_{45} = - mn(c_{1} E_{12} + C_{44} - c_{1} E_{44} + c_{1}^{2} G_{12} - c_{1} E_{44} + c_{1}^{2} G_{44} + C_{12} - c_{1} E_{12} $$

  • 21.

    $$ K_{51} = K_{15} $$

  • 22.

    $$ K_{52} = K_{25} $$

  • 23.

    $$ K_{53} = K_{35} $$

  • 24.

    $$ K_{54} = K_{45} $$

  • 25.

    $$ K_{55} = - n^{2} \left( {C_{22} - 2c_{1} E_{22} + c_{1}^{2} G_{22} } \right) - m^{2} \left( {C_{44} - 2c_{1} E_{44} c_{1}^{2} G_{44} } \right) + 3c_{1} C_{66} - A_{66} + 3c_{1} \left( {C_{66} - 3c_{1} E_{66} } \right) $$

  • 26.

    $$ M_{11} = - I_{0} $$

  • 27.

    $$ M_{12} = M_{21} = 0 $$

  • 28.

    $$ M_{13} = c_{1} mI_{3} $$

    .

  • 29.

    $$ M_{14} = - I_{1} + c_{1} I_{3} $$

  • 30.

    $$ M_{15} = M_{51} = 0 $$

  • 31.

    $$ M_{22} = - I_{0} $$

  • 32.

    $$ M_{23} = M_{32} = c_{1} nI_{3} $$

  • 33.

    $$ M_{24} = M_{42} = 0 $$

  • 34.

    $$ M_{25} = - I_{1} + c_{1} I_{3} $$

  • 35.

    $$ M_{31} = c_{1} mI_{3} $$

  • 36.

    $$ M_{32} = M_{23} $$

  • 37.

    $$ M_{33} = - I_{0} - 2c_{1}^{2} I_{6} \left( {m^{2} + n^{2} } \right) $$

  • 38.

    $$ M_{34} = M_{43} = c_{1} mI_{4} - mc_{1}^{2} I_{6} $$

  • 39.

    $$ M_{35} = c_{1} nI_{4} - nc_{1}^{2} I_{6} $$

  • 40.

    $$ M_{44} = M_{55} = - I_{2} + 2c_{1} I_{4} - c_{1}^{2} I_{6} $$

  • 41.

    $$ C_{11} = C_{12} = C_{13} = C_{14} = C_{15} = C_{21} = C_{22} = C_{23} = C_{24} = C_{25} = C_{31} = C_{32} = C_{34} = C_{35} = C_{41} = C_{42} = C_{43} = C_{44} = C_{45} = C_{51} = C_{52} = C_{53} = C_{54} = C_{55} = 0 $$

  • 42.

    $$ C_{33} = C_{d} $$

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Qaderi, S., Ebrahimi, F. Vibration analysis of polymer composite plates reinforced with graphene platelets resting on two-parameter viscoelastic foundation. Engineering with Computers 38, 419–435 (2022). https://doi.org/10.1007/s00366-020-01066-z

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