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Buckling analysis and dynamic response of FGM sandwich cylindrical panels in thermal environments using nonlocal strain gradient theory

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Abstract

The present paper provides an analysis to obtain the critical buckling load and vibration frequencies of the sandwich cylindrical panel with functionally graded (FG) face sheets and FG porous core resting on an elastic foundation, subjected to mechanical load and in thermal environments. The panel is formulated within the framework of the nonlocal strain gradient theory for shell model and classical shell theory. Based on Hamilton’s principle and Galerkin’s method, the effects of nonlocal and strain gradient parameters, materials and geometrical characteristics, porosity, temperature and elastic foundation on buckling load, fundamental frequencies, and dynamic response of the panel are considered.

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Acknowledgements

This research is funded by the Project number QG.21.25 of VNU Hanoi “Nonlinear stability of laminated smart composite plates and shells”. The authors are grateful for this support.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering and Mechatronics, Phenikaa University, Ha Dong, Hanoi, 12116, Vietnam

    Do Quang Chan & Tran Quoc Quan

  2. PHENIKAA Research and Technology Institute (PRATI), A&A Green Phoenix Group JSC, No. 167 Hoang Ngan, Trung Hoa, Cau Giay, Hanoi, 11313, Vietnam

    Do Quang Chan

  3. Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Ha Dong, Hanoi, 12116, Vietnam

    Tran Quoc Quan

  4. University of Transport Technology, 54 Trieu Khuc, Thanh Xuan, Hanoi, Vietnam

    Bui Gia Phi

  5. Thainguyen University of Technology, Thainguyen, Vietnam

    Dang Van Hieu

  6. Departemnt of Engineering and Technology in Constructions and Transportation, VNU Hanoi - University of Engineering and Technology, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam

    Nguyen Dinh Duc

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  1. Do Quang Chan
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Contributions

NDD: Conceptualization, Funding acquisition, Methodology, Supervision, Writing—review and editing; Corresponding author. DQC: Investigation, Methodology, Validation, Writing—original draft. TQQ: Investigation, Methodology. BGP: Investigation, Methodology, Software, Validation. DVH: Investigation, Methodology, Software, Validation.

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Correspondence to Nguyen Dinh Duc.

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Appendices

Appendix 1

1.1 FG porous type 1

$$E_{1} = E_{c} \left( {h - h_{{{\text{core}}}} } \right) + \frac{{E_{mc} \left( {h - h_{{{\text{core}}}} } \right)}}{k + 1} + \frac{{E_{m} h_{{{\text{core}}}} {\mkern 1mu} \left( {\pi - 2{\mkern 1mu} e_{0} } \right)}}{\pi }$$
(52)
$$E_{2} = 0$$
$$\begin{gathered} E_{3} = {\mkern 1mu} \frac{{E_{c} \left( {h^{3} - h_{{{\text{core}}}}^{3} } \right)}}{12} + \frac{{E_{mc} \left( {2{\mkern 1mu} h^{3} + 2{\mkern 1mu} kh^{2} h_{{{\text{core}}}} + k\left( {k + 1} \right)hh_{{{\text{core}}}}^{2} - \left( {k\left( {k + 2} \right) + 1} \right)h_{{{\text{core}}}}^{3} } \right)}}{{4k\left( {k + 2} \right)\left( {k + 3} \right) + 1}} \hfill \\ \,\qquad\qquad + {\mkern 1mu} \frac{{E_{m} h_{{{\text{core}}}}^{3} \left( {\pi^{3} - 6{\mkern 1mu} e_{0} \pi^{2} + 48{\mkern 1mu} e_{0} } \right)}}{{12\pi^{3} }} \hfill \\ \end{gathered}$$

1.2 FG porous type 2

$$E_{1} = E_{c} \left( {h - h_{{{\text{core}}}} } \right) + \frac{{E_{mc} \left( {h - h_{{{\text{core}}}} } \right)}}{k + 1} + \frac{{E_{m} h_{{{\text{core}}}} {\mkern 1mu} \left( {\pi - 2{\mkern 1mu} e_{0} } \right)}}{\pi }$$
$$E_{2} = - \frac{{E_{m} h_{{{\text{core}}}}^{2} e_{0} \left( {\pi - 4} \right)}}{{\pi^{2} }}$$
(53)
$$\begin{gathered} E_{3} = {\mkern 1mu} \frac{{E_{c} \left( {h^{3} - h_{{{\text{core}}}}^{3} } \right)}}{12} + {\mkern 1mu} \frac{{E_{mc} \left( {2{\mkern 1mu} h^{3} + 2{\mkern 1mu} kh^{2} h_{{{\text{core}}}} + k\left( {k + 1} \right)hh_{{{\text{core}}}}^{2} - \left( {k\left( {k + 2} \right) + 1} \right)h_{{{\text{core}}}}^{3} } \right)}}{{4k\left( {k + 2} \right)\left( {k + 3} \right) + 1}} \hfill \\ \qquad\qquad+ {\mkern 1mu} \frac{{E_{m} h_{{{\text{core}}}}^{3} \left( {\pi^{3} - 6{\mkern 1mu} e_{0} \pi^{2} + 192{\mkern 1mu} e_{0} - 48{\mkern 1mu} \pi {\mkern 1mu} e_{0} } \right)}}{{12\pi^{3} }} \hfill \\ \end{gathered}$$

1.3 FG porous type 3

$$E_{1} = E_{c} \left( {h - h_{{{\text{core}}}} } \right) + \frac{{E_{mc} \left( {h - h_{{{\text{core}}}} } \right)}}{k + 1} + E_{m} h_{{{\text{core}}}} {\mkern 1mu} \left( {1 - 2\lambda_{0} e_{0} } \right)$$
$$E_{2} = 0$$
$$\begin{gathered} E_{3} = {\mkern 1mu} \frac{{E_{c} \left( {h^{3} - h_{{{\text{core}}}}^{3} } \right)}}{12} + {\mkern 1mu} \frac{{E_{mc} \left( {2{\mkern 1mu} h^{3} + 2{\mkern 1mu} kh^{2} h_{{{\text{core}}}} + k\left( {k + 1} \right)hh_{{{\text{core}}}}^{2} - \left( {k\left( {k + 2} \right) + 1} \right)h_{{{\text{core}}}}^{3} } \right)}}{{4k\left( {k + 2} \right)\left( {k + 3} \right) + 1}} \hfill \\ \qquad\qquad+ {\mkern 1mu} \frac{{E_{m} h_{{{\text{core}}}}^{3} \left( {1 - {\mkern 1mu} e_{0} \lambda_{0} } \right)}}{{12\pi^{3} }} \hfill \\ \end{gathered}$$
(54)

Appendix 2

$$X_{11} = - {\mkern 1mu} \frac{{\pi^{2} \left[ {R^{2} \theta_{0}^{2} \left( {L^{2} + \pi^{2} l^{2} m^{2} } \right) + n^{2} \pi^{2} l^{2} L^{2} } \right]}}{{4L^{3} R^{3} \theta_{0}^{3} }}\left( {m^{2} R^{2} \theta_{0}^{2} A_{11} + n^{2} L^{2} A_{33} } \right)$$
(55)
$$X_{12} = - {\mkern 1mu} \frac{{n\pi^{2} m\left[ {R^{2} \theta_{0}^{2} \left( {L^{2} + \pi^{2} l^{2} m^{2} } \right) + n^{2} \pi^{2} l^{2} L^{2} } \right]}}{{4R^{2} \theta_{0}^{2} L^{2} }}\left( {A_{12} + A_{33} } \right)$$
(56)
$$X_{13} = {\mkern 1mu} \frac{{\pi {\mkern 1mu} m\left[ {R^{2} \theta_{0}^{2} \left( {L^{2} + \pi^{2} l^{2} m^{2} } \right) + n^{2} \pi^{2} l^{2} L^{2} } \right]}}{{4L^{4} R^{3} \theta_{0}^{3} }}\left[ {\left( {\pi^{2} m^{2} RB_{11} - L^{2} A_{12} } \right)R\theta_{0}^{2} + n^{2} \pi^{2} L^{2} \left( {B_{12} + B_{33} } \right)} \right]$$
(57)
$$X_{21} = - {\mkern 1mu} \frac{{n\pi^{2} m\left[ {\left( {\pi^{2} l^{2} m^{2} + L^{2} } \right)R^{2} \theta_{0}^{2} + n^{2} \pi^{2} l^{2} L^{2} } \right]}}{{4L^{2} R^{2} \theta_{0}^{2} }}\left( {A_{21} + A_{33} } \right)$$
(58)
$$X_{22} = - {\mkern 1mu} \frac{{\pi^{2} \left[ {\left( {\pi^{2} l^{2} m^{2} + L^{2} } \right)R^{2} \theta_{0}^{2} + n^{2} \pi^{2} l^{2} L^{2} } \right]}}{{4L^{3} R^{3} \theta_{0}^{3} }}\left( {n^{2} L^{2} A_{22} + m^{2} R^{2} \theta_{0}^{2} A_{33} } \right)$$
(59)
$$X_{23} = {\mkern 1mu} {\mkern 1mu} \frac{{n\pi {\mkern 1mu} \left[ {\left( {\pi^{2} l^{2} m^{2} + L^{2} } \right)R^{2} \theta_{0}^{2} + n^{2} \pi^{2} l^{2} L^{2} } \right]}}{{4L^{3} R^{4} \theta_{0}^{4} }}\left[ {L^{2} \left( {n^{2} \pi^{2} B_{22} - R\theta_{0}^{2} A_{22} } \right) + \pi^{2} m^{2} R^{2} \theta_{0}^{2} \left( {B_{21} + B_{33} } \right)} \right]$$
(60)
$$X_{31} = {\mkern 1mu} \frac{{\pi {\mkern 1mu} m\left[ {\left( {L^{2} + \pi^{2} l^{2} m^{2} } \right)\theta_{0}^{2} R^{2} + n^{2} \pi^{2} l^{2} L^{2} } \right]}}{{4L^{4} R^{3} \theta_{0}^{3} }}\left[ {\left( {m^{2} R\pi^{2} B_{11} - L^{2} A_{21} } \right)R\theta_{0}^{2} + L^{2} \pi^{2} n^{2} \left( {B_{21} + 2{\mkern 1mu} B_{66} } \right)} \right]$$
(61)
$$X_{32} = {\mkern 1mu} \frac{{n\pi {\mkern 1mu} \left( {\left( {L^{2} + \pi^{2} l^{2} m^{2} } \right)\theta_{0}^{2} R^{2} + n^{2} \pi^{2} l^{2} L^{2} } \right)}}{{4L^{3} R^{4} \theta_{0}^{4} }}\left[ {L^{2} \left( {n^{2} \pi^{2} B_{22} - R\theta_{0}^{2} A_{22} } \right) + \pi^{2} m^{2} R^{2} \theta_{0}^{2} \left( {B_{12} + 2{\mkern 1mu} B_{66} } \right)} \right]$$
(62)
$$X_{33} = {\mkern 1mu} - \frac{{\left[ {\left( {L^{2} + \pi^{2} l^{2} m^{2} } \right)\theta_{0}^{2} R^{2} + n^{2} \pi^{2} l^{2} L^{2} } \right]}}{{4L^{5} R^{5} \theta_{0}^{5} }}\left[ \begin{gathered} L^{4} R^{2} \theta_{0}^{4} A_{22} + n^{2} \pi^{2} L^{4} \left( {n^{2} \pi^{2} D_{22} - 2{\mkern 1mu} R\theta_{0}^{2} B_{22} } \right) \hfill \\ - \pi^{2} L^{2} \theta_{0}^{4} m^{2} R^{3} \left( {B_{12} + B_{21} } \right) + m^{4} R^{4} \theta_{0}^{4} \pi^{4} D_{11} \hfill \\ + L^{2} \theta_{0}^{2} m^{2} n^{2} R^{2} \pi^{4} \left( {D_{12} + D_{21} + 2{\mkern 1mu} D_{66} } \right) \hfill \\ \end{gathered} \right]$$
(63)
$$X_{34} = {\mkern 1mu} \frac{{h\pi^{2} m^{2} \left[ {\mu^{2} \pi^{2} n^{2} L^{2} + \left( {\mu^{2} \pi^{2} m^{2} + L^{2} } \right)R^{2} \theta_{0}^{2} } \right]}}{{4RL^{3} \theta_{0} }}$$
(64)
$$X_{35} = {\mkern 1mu} \frac{{R^{2} L^{2} \theta_{0}^{2} + \mu^{2} \pi^{2} m^{2} R^{2} \theta_{0}^{2} + \mu^{2} \pi^{2} n^{2} L^{2} }}{{4RL\theta_{0} }}$$
(65)
$$X_{36} = {\mkern 1mu} \frac{{\pi^{2} \left( {R^{2} \theta_{0}^{2} m^{2} + n^{2} L^{2} } \right)\left( {R^{2} L^{2} \theta_{0}^{2} + \mu^{2} \pi^{2} m^{2} R^{2} \theta_{0}^{2} + \mu^{2} \pi^{2} n^{2} L^{2} } \right)}}{{4R^{3} L^{3} \theta_{0}^{3} }}$$
(66)
$$X_{37} = {\mkern 1mu} - \frac{{LR\theta_{0} \left[ { - 1 + ( - 1)^{m} } \right]\left[ { - 1 + ( - 1)^{n} } \right]}}{{nm\pi^{2} }}$$
(67)

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Chan, D.Q., Quan, T.Q., Phi, B.G. et al. Buckling analysis and dynamic response of FGM sandwich cylindrical panels in thermal environments using nonlocal strain gradient theory. Acta Mech 233, 2213–2235 (2022). https://doi.org/10.1007/s00707-022-03212-8

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