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Analysis for the thermoelastic bending of a functionally graded material cylindrical shell

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Abstract

An analytic study for thermoelastic bending of a functionally graded material (FGM) cylindrical shell subjected to a uniform transverse mechanical load and non-uniform thermal loads is presented. Based on the classical linear shell theory, the equations with the radial deflection and horizontal displacement are derived out. An arbitrary material property of the FGM cylindrical shell is assumed to vary through the thickness of the cylindrical shell, and exact solution of the problem is obtained by using an analytic method. For the FGM cylindrical shell with fixed and simply supported boundary conditions, the effects of mechanical load, thermal load and the power law exponent on the deformation of the FGM cylindrical shell are analyzed and discussed.

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Acknowledgements

The authors wish to thank reviewers for their valuable comments and the research is supported by the National Natural Science Foundation of China (11372105), New Century Excellent Talents Program in University (NCET-13-0184), Key Laboratory of Manufacture and Test Techniques for Automobile Parts, Ministry of Education (2012), Hunan Provincial Natural Science Foundation for Creative Research Groups of China (12JJ7001), and the central colleges of basic scientific research and operational costs (funded by the Hunan University).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, China

    Hong-Liang Dai & Ting Dai

  2. Key Laboratory of Manufacture and Test Techniques for Automobile Parts, Ministry of Education, Chongqing University of Technology, Chongqing, 400054, China

    Hong-Liang Dai

  3. Department of Engineering Mechanics, College of Mechanical & Vehicle Engineering, Hunan University, Changsha, 410082, China

    Hong-Liang Dai & Ting Dai

Authors
  1. Hong-Liang Dai
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  2. Ting Dai
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Correspondence to Hong-Liang Dai.

Appendix

Appendix

$$\begin{aligned} &{\begin{array}{l} A_{10} = \varTheta_{10} + \varTheta_{12}, \qquad B_{10} = \varTheta_{11} + \varTheta_{13}, \\ \varGamma_{1}^{*} = \varGamma_{1} + \frac{1}{a}\varGamma_{2},\qquad A_{20} = \varTheta_{10},\\ B_{20} = \varTheta_{11},\qquad A_{30} = - a\varTheta_{12} - \varTheta_{13},\\ B_{30} = - a\varTheta_{13} - \varTheta_{14},\qquad \varGamma_{2}^{*} = - \varGamma_{2} - \frac{1}{a}\varGamma_{3}\end{array}} \end{aligned}$$
(A.1)
$$\begin{aligned} &{\varTheta_{10} = \int_{ - h/2}^{h/2} \frac{E_{2}}{1 - v^{2}} \biggl(1 + (E_{r} - 1) \biggl(\frac{h - 2z}{2h} \biggr)^{n} \biggr)\mathrm{d}z} \\ &{\varTheta_{11} = \int_{ - h/2}^{h/2} \frac{E_{2}}{1 - v^{2}} \biggl(1 + (E_{r} - 1) \biggl(\frac{h - 2z}{2h} \biggr)^{n} \biggr)z\mathrm{d}z} \\ &{\varTheta_{12} = \frac{1}{a}\int_{ - h/2}^{h/2} \frac{E_{2}}{1 - v^{2}} \biggl(1 + (E_{r} - 1) \biggl(\frac{h - 2z}{2h} \biggr)^{n} \biggr)} \\ &{\hphantom{\varTheta_{12} =} {}\times z\mathrm{d}z} \\ &{\varTheta_{13} = \frac{1}{a}\int_{ - h/2}^{h/2} \frac{E_{2}}{1 - v^{2}} \biggl(1 + (E_{r} - 1) \biggl(\frac{h - 2z}{2h} \biggr)^{n} \biggr)} \\ &{\hphantom{\varTheta_{13} =} {}\times z^{2}\mathrm{d}z} \\ &{\varTheta_{14} = \frac{1}{a}\int_{ - h/2}^{h/2} \frac{E_{2}}{1 - v^{2}} \biggl(1 + (E_{r} - 1) \biggl(\frac{h - 2z}{2h} \biggr)^{n} \biggr)} \\ &{\hphantom{\varTheta_{14} =} {}\times z^{3}\mathrm{d}z} \end{aligned}$$
(A.2)
$$\begin{aligned} &{\varGamma_{1} = \int_{ - h/2}^{h/2} \frac{E(z)}{1 - v}\alpha (z)T(z)\mathrm{d}z = \varDelta _{10} + \varDelta _{11} + \varDelta _{12}} \\ &{\hphantom{\varGamma_{1} =} {} + \varDelta _{13} + \varDelta _{14} + \varDelta _{15}} \\ &{\varGamma_{2} = \int_{ - h/2}^{h/2} \frac{E(z)}{1 - v}\alpha (z)T(z)z\mathrm{d}z = \varDelta _{20} + \varDelta _{21} + \varDelta _{22}} \\ &{\hphantom{\varGamma_{2} =} {} + \varDelta _{23} + \varDelta _{24} + \varDelta _{25}} \\ &{\varGamma_{3} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}\alpha_{r}z^{2}\mathrm{d}z = \varDelta _{30} + \varDelta _{31} + \varDelta _{32}} \\ &{\hphantom{\varGamma_{3} =} {} + \varDelta _{33} + \varDelta _{34} + \varDelta _{35}} \end{aligned}$$
(A.3)
$$\begin{aligned} &{\varDelta _{10} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}\alpha_{r}\mathrm{d}z,} \\ &{\varDelta _{11} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(1 - 2 \alpha_{r} + \alpha_{r}E_{r}) \biggl( \frac{h - 2x}{2h}\biggr)^{n}\mathrm{d}x} \\ &{\varDelta _{12} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(1 - \alpha_{r}) (E_{r} - 1) \biggl( \frac{h - 2z}{2h}\biggr)^{2n}\mathrm{d}z} \\ &{\varDelta _{13} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(T_{r} - 1)\alpha_{r}\frac{\varPsi_{a}}{\varPsi_{b}} \mathrm{d}z} \\ &{\varDelta _{14} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(T_{r} - 1) (1 - 2\alpha_{r}} \\ &{\hphantom{\varDelta _{14} =} {} + \alpha_{r}E_{r}) \biggl(\frac{h - 2z}{2h} \biggr)^{n}\frac{\varPsi_{a}}{\varPsi_{b}}\mathrm{d}z} \\ &{\varDelta _{15} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(T_{r} - 1) (1 - \alpha_{r})} \\ &{\hphantom{\varDelta _{15} =} {}\times (E_{r} - 1) \biggl(\frac{h - 2z}{2h}\biggr)^{2n} \frac{\varPsi_{a}}{\varPsi_{b}}\mathrm{d}z} \\ &{\varDelta _{20} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}\alpha_{r}z\mathrm{d}z} \\ &{\varDelta _{21} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(1 - 2\alpha_{r}} \\ &{\hphantom{\varDelta _{21} =} {} + \alpha_{r}E_{r}) \biggl(\frac{h - 2z}{2h}\biggr)^{n}z\mathrm{d}z} \\ &{\varDelta _{22} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(1 - \alpha_{r})} \\ &{\hphantom{\varDelta _{22} =} {}\times (E_{r} - 1) \biggl( \frac{h - 2x}{2h}\biggr)^{2n}x\mathrm{d}x} \\ &{\varDelta _{23} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(T_{r} - 1)\alpha_{r}\frac{\varPsi_{a}}{\varPsi_{b}}z \mathrm{d}z} \end{aligned}$$
(A.4)
$$\begin{aligned} &{\varDelta _{24} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(T_{r} - 1) (1 - 2\alpha_{r} + \alpha_{r}E_{r})} \\ &{\hphantom{\varDelta _{24} =} {}\times \biggl(\frac{h - 2z}{2h} \biggr)^{n}\frac{\varPsi_{a}}{\varPsi_{b}}z\mathrm{d}z} \\ &{\varDelta _{25} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(T_{r} - 1) (1 - \alpha_{r}) (E_{r} - 1)} \\ &{\hphantom{\varDelta _{25} =} {}\times \biggl(\frac{h - 2z}{2h}\biggr)^{2n} \frac{\varPsi_{a}}{\varPsi_{b}}z\mathrm{d}z} \\ &{\varDelta _{31} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(1 - 2\alpha_{r}} \\ &{\hphantom{\varDelta _{31} =} {} + \alpha_{r}E_{r}) \biggl(\frac{h - 2x}{2h}\biggr)^{n}z^{2}\mathrm{d}z} \\ &{\varDelta _{32} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(1 - \alpha_{r})} \\ &{\hphantom{\varDelta _{32} =} {}\times (E_{r} - 1) \biggl( \frac{h - 2x}{2h}\biggr)^{2n}z^{2}\mathrm{d}z} \\ &{\varDelta _{33} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(T_{r} - 1)\alpha_{r}\frac{\varPsi_{a}}{\varPsi_{b}}z^{2} \mathrm{d}z} \\ &{\varDelta _{34} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(T_{r} - 1) (1 - 2\alpha_{r} + \alpha_{r}E_{r})} \\ &{\hphantom{\varDelta _{34} =} {}\times \biggl(\frac{h - 2z}{2h} \biggr)^{n}\frac{\varPsi_{a}}{\varPsi_{b}}z^{2}\mathrm{d}z} \\ &{\varDelta _{35} = \int_{ - h/2}^{h/2} \frac{E_{2}\alpha_{1}T_{1}}{1 - v}(T_{r} - 1) (1 - \alpha_{r}) (E_{r} - 1)} \\ &{\hphantom{\varDelta _{35} =} {}\times \biggl(\frac{h - 2x}{2h}\biggr)^{2n} \frac{\varPsi_{a}}{\varPsi_{b}}z^{2}\mathrm{d}z} \\ &{\alpha_{r} = \frac{\alpha_{2}}{\alpha_{1}},\qquad E_{r} = \frac{E_{1}}{E_{2}},\qquad K_{r} = \frac{K_{1}}{K_{2}}} \end{aligned}$$
(A.5)
$$\begin{aligned} &{\begin{array}{l} \varPsi_{a} = \displaystyle\int_{ - h/2}^{z} \frac{1}{1 + (K_{r} - 1)(\frac{h - 2z}{2h})^{n}}\mathrm{d}z,\\ \varPsi_{b} = \int _{ - h/2}^{h/2} \frac{1}{1 + (K_{r} - 1)(\frac{h - 2z}{2h})^{n}}\mathrm{d}z \end{array}} \end{aligned}$$
(A.6)

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Dai, HL., Dai, T. Analysis for the thermoelastic bending of a functionally graded material cylindrical shell. Meccanica 49, 1069–1081 (2014). https://doi.org/10.1007/s11012-013-9853-1

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