Thermal post-buckling analysis of graded sandwich curved structures under variable thermal loadings

Abstract

In the present research, finite element solutions of thermal post-buckling load-bearing strength of functionally graded (FG) sandwich shell structures are reported by adopting a higher-order shear deformation type kinematics. For the numerical calculation, nine nodes are considered for each element. A specialized MATLAB code is developed incorporating the present mathematical model to evaluate the numerical buckling temperature. The Green–Lagrange nonlinear strain is adopted for the formulation of the sandwich structure. The eigenvalue equation of the FG sandwich structure is solved to predict the post-buckling temperature values of the structure. Moreover, three kinds of temperature distributions across the panel thickness are assumed, viz., uniform, linear and nonlinear. In addition, the properties are described using the power law distributions. The numerical solutions are first validated and, subsequently, the impact of alterations of structural parameters, viz., the curvature ratios, core–face thickness ratios, support conditions and power law index (n_Z) including the amplitude ratio on the thermal post-buckling response of FG sandwich curved panels have been studied in details. The investigation reveals different interesting outcomes, which may help for future references for the analysis and design of the graded sandwich structure.

Appendices

Appendix A

$\left\{ \sigma \right\} = \left\{ {\begin{array}{*{20}c} {\sigma_{x} } & {\sigma_{y} } & {\sigma_{z} } & {\tau_{yz} } & {\tau_{xz} } & {\tau_{xy} } \\ \end{array} } \right\}^{T} ,$

$\left\{ \varepsilon \right\} = \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } & {\varepsilon_{y} } & {\varepsilon_{z} } & {\gamma_{yz} } & {\gamma_{xz} } & {\gamma_{xy} } \\ \end{array} } \right\}^{T} ,$

$\left\{ \alpha \right\} = \left\{ {\begin{array}{*{20}c} {\alpha_{x} } & {\alpha_{y} } & 0 & 0 & 0 & {2\alpha_{xy} }\\ \end{array} } \right\}^{T} .$

Appendix B

$u_{0} = \sum\limits_{i = 1}^{9} {N_{i} u_{0i} }, v_{0} = \sum\limits_{i = 1}^{9} {N_{i} v_{0i} }, w_{0} = \sum\limits_{i = 1}^{9} {N_{i} w_{0i} }, \phi_{x} = \sum\limits_{i = 1}^{9} {N_{i} \phi_{xi} }, \phi_{y} = \sum\limits_{i = 1}^{9} {N_{i} \phi_{yi} }, u_{0}^{*} = \sum\limits_{i = 1}^{9} {N_{i} u_{0i}^{*} },v_{0}^{*} = \sum\limits_{i = 1}^{9} {N_{i} v_{0i}^{*} }, \phi_{x}^{*} = \sum\limits_{i = 1}^{9} {N_{i} \phi_{xi}^{*} }, \phi_{y}^{*} = \sum\limits_{i = 1}^{9} {N_{i} \phi_{yi}^{*} }.$

Appendix C

$\left[ {D_{1} } \right] = \int\limits_{ - h/2}^{ + h/2} {\left[ {T^{l} } \right]^{T} \left[ Q \right]\left[ {T^{l} } \right]^{T} {\text{d}}z}$, $\left[ {D_{2} } \right] = \int\limits_{ - h/2}^{ + h/2} {\left[ {T^{l} } \right]^{T} \left[ Q \right]\left[ {T^{nl} } \right]^{T} {\text{d}}z}$, $\left[ {D_{3} } \right] = \int\limits_{ - h/2}^{ + h/2} {\left[ {T^{nl} } \right]^{T} \left[ Q \right]\left[ {T^{l} } \right]^{T} {\text{d}}z}$, $\left[ {D_{4} } \right] = \int\limits_{ - h/2}^{ + h/2} {\left[ {T^{nl} } \right]^{T} \left[ Q \right]\left[ {T^{nl} } \right]^{T} {\text{d}}z}$.