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Geometrically nonlinear free vibration of CNTs reinforced sandwich conoidal shell in thermal environment

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Abstract

Geometrically nonlinear free vibration behavior of functionally graded sandwich conoidal shell under thermal environment is investigated by finite element method considering Green–Lagrange strains and higher-order shear deformation theory. The analysis is carried out using quadratic eight-noded isoparametric elements. The governing equation of motion is derived employing Lagrange’s equation, and eigenvalue solutions are computed employing a direct iterative procedure. After validation of the formulation, a parametric study of the shell is carried out varying the volume fraction of the reinforcing carbon nanotubes, length-to-thickness ratio, core-to-face sheet thickness ratio, ratio of small height to greater height, amplitude factor and temperature with clamped and simply supported boundary conditions. At the end, shifting trend of mode shapes of the shell with respect to amplitude factor is presented.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Government College of Engineering, Kalahandi, Odisha, 766003, India

    Mrutyunjay Rout

  2. Department of Civil Engineering, Haldia Institute of Technology, Haldia, West Bengal, India

    Sasank Shekhar Hota

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  1. Mrutyunjay Rout
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Appendix

Appendix

The linear mid-plane strains

$$\begin{gathered} \varepsilon_{{{\text{xl}}}}^{0} = \overline{u}_{0,x} ,\;\varepsilon_{{{\text{yl}}}}^{0} = \overline{v}_{0,y} ,\;\gamma_{{{\text{xyl}}}}^{0} = u_{0,y} + v_{0,x} + 2w_{0} /R_{{{\text{xy}}}} ,\;\kappa_{{{\text{xl}}}}^{1} = \theta_{x,x} ,\;\kappa_{{{\text{yl}}}}^{1} = \theta_{y,y} ,\; \hfill \\ \kappa_{{{\text{xyl}}}}^{1} = \theta_{x,y} + \theta_{y,x} ,\;\kappa_{{{\text{xzl}}}}^{1} = - \theta_{x} /R_{x} ,\;\kappa_{{{\text{yzl}}}}^{1} = - \theta_{y} /R_{y} ,\;\kappa_{{{\text{xl}}}}^{3} = \lambda_{x,x} ,\;\kappa_{{{\text{yl}}}}^{3} = \lambda_{y,y} , \hfill \\ \;\kappa_{{{\text{xyl}}}}^{3} = \lambda_{x,y} + \lambda_{y,x} ,\;\kappa_{{{\text{xzl}}}}^{3} = - \lambda_{x} /R{}_{x},\;\kappa_{{{\text{yzl}}}}^{3} = - \lambda_{y} /R{}_{y},\;\kappa_{{{\text{xzl}}}}^{2} = 3\lambda_{x} ,\;\kappa_{{{\text{yzl}}}}^{2} = 3\lambda_{y} , \hfill \\ \gamma_{{{\text{xzl}}}}^{0} = \theta_{x} + w_{0,x} - u_{0} /R_{x} ,\;\gamma_{{{\text{yzl}}}}^{0} = \theta_{y} + w_{0,y} - v_{0} /R_{y} , \hfill \\ \end{gathered}$$

The non-linear mid-plane strain terms are

$$\varepsilon_{{{\text{xnl}}}}^{0} = \left( {\overline{u}_{0,x}^{2} + \overline{v}_{0,x}^{2} + \overline{w}_{0,x}^{2} } \right)$$
$$\varepsilon_{{{\text{ynl}}}}^{0} = \left( {\overline{u}_{0,y}^{2} + \overline{v}_{0,y}^{2} + \overline{w}_{0,y}^{2} } \right)$$
$$\gamma_{{{\text{xynl}}}}^{0} = 2\left( {\overline{u}_{0,x} \overline{u}_{0,y} + \overline{v}_{0,x} \overline{v}_{0,y} + \overline{w}_{0,x} \overline{w}_{0,y} } \right)$$
$$\kappa_{{{\text{xnl}}}}^{1} = 2\left( {\overline{u}_{0,x} \theta_{x,x} + \overline{v}_{0,x} \theta_{y,x} - \overline{w}_{0,x} \theta_{x} /R_{x} } \right)$$
$$\kappa_{{{\text{ynl}}}}^{1} = 2\left( {\overline{u}_{0,y} \theta_{x,y} + \overline{v}_{0,y} \theta_{y,y} - \overline{w}_{0,y} \theta_{y} /R_{y} } \right)$$
$$\kappa_{{{\text{xynl}}}}^{1} = 2\left( {\overline{u}_{0,x} \theta_{x,y} + \overline{u}_{0,y} \theta_{x,x} + \overline{v}_{0,x} \theta_{y,y} + \overline{v}_{0,y} \theta_{y,x} - \overline{w}_{0,x} \theta_{y} /R_{y} - \overline{w}_{0,y} \theta_{x} /R_{x} } \right)$$
$$\kappa_{{{\text{xznl}}}}^{1} = 2\left( {\theta_{x,x} \theta_{x} + \theta_{y,x} \theta_{y} } \right)$$
$$\kappa_{{{\text{yznl}}}}^{1} = 2\left( {\theta_{x,y} \theta_{x} + \theta_{y,y} \theta_{y} } \right)$$
$$\kappa_{{{\text{xnl}}}}^{3} = 2\left( {\overline{u}_{0,x} \lambda_{x,x} + \overline{v}_{0,x} \lambda_{y,x} - \overline{w}_{0,x} \lambda_{x} /R_{x} } \right)$$
$$\kappa_{{{\text{ynl}}}}^{3} = 2\left( {\overline{u}_{0,y} \lambda_{x,y} \lambda_{y,y} - \overline{w}_{0,y} \lambda_{y} /R_{y} } \right)$$
$$\kappa_{{{\text{xynl}}}}^{3} = 2\left( {\overline{u}_{0,x} \lambda_{x,y} + \overline{u}_{0,y} \lambda_{x,x} + \overline{v}_{0,x} \lambda_{y,y} + \overline{v}_{0,y} \lambda_{y,x} - \overline{w}_{0,x} \lambda_{y} /R_{y} - \overline{w}_{0,y} \lambda_{x} /R_{x} } \right)$$
$$\kappa_{{{\text{xznl}}}}^{3} = 2\left( {\theta_{x} \lambda_{x,x} + \theta_{x,x} 3\lambda_{x} + \theta_{y,x} 3\lambda_{y} + \theta_{y} \lambda_{y,x} } \right)$$
$$\kappa_{{{\text{yznl}}}}^{3} = 2\left( {\theta_{x,y} 3\lambda_{x} + \theta_{x} \lambda_{x,y} + \theta_{y} \lambda_{y,y} + \theta_{y,y} 3\lambda_{y} } \right)$$
$$\kappa_{{{\text{xnl}}}}^{2} = \left( {\theta_{x,x}^{2} + \theta_{y,x}^{2} + \theta_{x}^{2} /R_{x}^{2} } \right)$$
$$\kappa_{{{\text{ynl}}}}^{2} = \left( {\theta_{x,y}^{2} + \theta_{y,y}^{2} + \theta_{y}^{2} /R_{y}^{2} } \right)$$
$$\kappa_{{{\text{xynl}}}}^{2} = 2\left( {\theta_{x,x} \theta_{x,y} + \theta_{y,x} \theta_{y,y} + \theta_{x} \theta_{y} /R_{x} R_{y} } \right)$$
$$\kappa_{{{\text{xznl}}}}^{2} = 2\left( {\overline{u}_{0,x} 3\lambda_{x} + \overline{v}_{0,x} 3\lambda_{y} } \right)$$
$$\kappa_{{{\text{yznl}}}}^{2} = 2\left( {\overline{u}_{0,y} 3\lambda_{x} + \overline{v}_{0,y} 3\lambda_{y} } \right)$$
$$\gamma_{{{\text{xznl}}}}^{0} = 2\left( {\overline{u}_{0,x} \theta_{x} + \overline{v}_{0,x} \theta_{y} } \right)$$
$$\gamma_{{{\text{yznl}}}}^{0} = 2\left( {\overline{u}_{0,y} \theta_{x} + \overline{v}_{0,y} \theta_{y} } \right)$$
$$\kappa_{{{\text{xnl}}}}^{4} = 2\left( {\theta_{x,x} \lambda_{x,x} + \theta_{y,x} \lambda_{y,x} + \theta_{x} \lambda_{x} /R_{x}^{2} } \right)$$
$$\kappa_{{{\text{ynl}}}}^{4} = 2\left( {\theta_{x,y} \lambda_{x,y} + \theta_{y,y} \lambda_{y,y} + \theta_{y} \lambda_{y} /R_{y}^{2} } \right)$$
$$\kappa_{{{\text{xynl}}}}^{4} = 2\left( {\theta_{x,x} \lambda_{x,y} + \theta_{x,y} \lambda_{x,x} + \theta_{y,x} \lambda_{y,y} + \theta_{y,y} \lambda_{y,x} + \theta_{x} \lambda_{y} /R_{x} R_{x} + \theta_{y} \lambda_{x} /R_{x} R_{y} } \right)$$
$$\kappa_{{{\text{xznl}}}}^{5} = 2\left( {\lambda_{x,x} 3\lambda_{x} + \lambda_{y,x} 3\lambda_{y} } \right)$$
$$\kappa_{{{\text{yznl}}}}^{5} = 2\left( {\lambda_{x,y} 3\lambda_{x} + \lambda_{y,y} 3\lambda_{y} } \right)$$
$$\kappa_{{{\text{xnl}}}}^{6} = \left( {\lambda_{x,x}^{2} + \lambda_{y,x}^{2} + \lambda_{x}^{2} /R_{x}^{2} } \right)$$
$$\kappa_{{{\text{ynl}}}}^{6} = \left( {\lambda_{x,y}^{2} + \lambda_{y,y}^{2} + \lambda_{y}^{2} /R_{y}^{2} } \right)$$
$$\kappa_{{{\text{xynl}}}}^{6} = 2\left( {\lambda_{x,x} \lambda_{x,y} + \lambda_{y,x} \lambda_{y,y} + \lambda_{x} \lambda_{y} /R_{x} R_{y} } \right)$$

where

$$\begin{aligned} \overline{u}_{0,x} &= u_{0,x} + w_{0} /R_{x} ,\;\overline{v}_{0,x} = v_{0,x} + w_{0} /R_{xy} ,\;\overline{w}_{0,x} = w_{0,x} - u_{0} /R_{x} ,\;\overline{u}_{0,y} = u_{0,y} \\ &\quad+ w_{0} /R_{xy} ,\;\overline{v}_{0,y} = v_{0,y} + w_{0} /R_{y} ,\;\overline{w}_{0,y} = w_{0,y} - v_{0} /R_{y} , \\ \end{aligned}$$

.

The linear thickness coordinate matrix is given by

$$\begin{gathered} \left[ {T_{l} } \right] = \left[ {\begin{array}{lllllllllllllllll} 1 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} }\hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill\\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} } \hfill & 0 \hfill & {z^{2}} \hfill & 0 \hfill & 1 \hfill & 0 \hfill\\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} } \hfill & 0 \hfill & {z^{2} } \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right] \hfill \\ \quad \quad \quad \quad \quad \quad \quad \ \\ \end{gathered}$$

The nonlinear thickness coordinate matrix is given by

$$\begin{gathered} \left[ {T_{{nl}} } \right] = \left[ {\begin{array}{llllllllllllllllllllllllllll} 1 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{2} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{6} } \hfill & 0 \hfill & 0 \hfill\\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{2} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{6} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{2} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{4} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{6} } \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{2} } \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{5} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & z \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{3} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{2} } \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {z^{5} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right] \end{gathered}$$

The element of matrix [A] is

$$\begin{gathered} A(1,1) = \overline{u}_{0,x} ,\,A(1,3) = \overline{v}_{0,x} ,\;A(1,5) = \overline{w}_{0,x} ,\;A(2,2) = \overline{u}_{0,y} ,\;A(2,4) = \overline{v}_{0,y} ,\;A(2,6) = \overline{w}_{0,y} ,\;A(3,1) = \overline{u}_{0,y} , \hfill \\ A(3,2) = \overline{u}_{0,x} ,\;A(3,3) = \overline{v}_{0,y} ,\;A(3,4) = \overline{v}_{0,x} ,\;A(3,5) = \overline{w}_{0,y} ,\;A(3,6) = \overline{w}_{0,x} ,\;A(4,1) = \theta_{x,x} ,\;A(4,1) = \theta_{y,x} , \hfill \\ A(4,5) = - \theta_{x} /R{}_{x},\;A(4,7) = \overline{u}_{0,x} ,\;A(4,9) = \overline{v}_{0,x} ,\;A(4,15) = - \overline{w}_{0,x} /R_{x} ,\;A(5,2) = \theta_{y,x} ,\;A(5,4) = \theta_{y,y} , \hfill \\ \end{gathered}$$
$$\begin{gathered} A(5,6) = - \theta_{y} /R_{y} ,\;A(5,8) = \overline{u}_{0,y} ,\;A(5,10) = \overline{v}_{0,y} ,\;A(5,16) = - \overline{w}_{0,y} /R_{y} ,\;A(6,1) = \theta_{x,y} ,\;A(6,2) = \theta_{x,x} , \hfill \\ A(6,3) = \theta_{y,y} ,\;A(6,4) = \theta_{y,x} ,\;A(6,5) = - \theta_{y} /R_{y} ,\;A(6,6) = - \theta_{x} /R{}_{x},\;A(6,7) = \overline{u}_{0,y} ,\;A(6,8) = \overline{u}_{0,x} , \hfill \\ A(6,9) = \overline{v}_{0,y} ,\;A(6,10) = \overline{v}_{0,x} ,\;A(6,15) = - \overline{w}_{0,y} /R_{x} ,\;A(6,16) = - \overline{w}_{0,x} /R_{y} ,\;A(7,7) = \theta_{x} ,\;A(7,9) = \theta_{y} , \hfill \\ \end{gathered}$$
$$\begin{gathered} A(7,15) = \theta_{x,x} ,\;A(7,16) = \theta_{y,x} ,\;A(8,8) = \theta_{x} ,\;A(8,10) = \theta_{y} ,\;A(8,15) = \theta_{x,y} ,\;A(8,16) = \theta_{y,y} ,\;A(9,1) = \lambda_{x,x} , \hfill \\ A(9,3) = \lambda_{y,x} ,\;A(9,5) = - \lambda_{x} /R_{x} ,\;A(9,11) = \overline{u}_{0,x} ,\;A(9,13) = \overline{v}_{0,x} ,\;A(9,17) = - \overline{w}_{0,x} /R_{x} ,\;A(10,2) = \lambda_{x,y} , \hfill \\ A(10,4) = \lambda_{y,y} ,\;A(10,6) = - \lambda_{y} /R_{y} ,\;A(10,12) = \overline{u}_{0,y} ,\;A(10,14) = \overline{v}_{0,y} ,\;A(10,18) = - \overline{w}_{0,y} /R_{y} , \hfill \\ A(11,1) = \lambda_{x,y} ,\;A(11,2) = \lambda_{x,x} ,\;A(11,3) = \lambda_{y,y} ,\,A(11,4) = \lambda_{y,x} ,\;A(11,5) = - \lambda_{y} /R_{y} ,\;A(11,6) = - \lambda_{x} /R_{x} , \hfill \\ A(11,11) = \overline{u}_{0,y} ,\;A(11,12) = \overline{u}_{0,x} ,\;A(11,13) = \overline{v}_{0,y} ,\;A(11,14) = \overline{v}_{0,x} ,\;A(11,17) = - \overline{w}_{0,y} /R_{x} , \hfill \\ A(11,18) = - \overline{w}_{0,x} /R_{y} ,\;A(12,7) = 3\lambda_{x} ,\;A(12,9) = 3\lambda_{y} ,\;A(12,11) = \theta_{x} ,\;A(12,13) = \theta_{y} ,\;A(12,15) = \lambda_{x,x} , \hfill \\ \end{gathered}$$
$$\begin{gathered} A(16,8) = \theta_{x,x} ,\;A(16,9) = \theta_{y,y} ,\;A(16,10) = \theta_{y,x} ,\;A(16,15) = \theta_{y} /(R{}_{x}R{}_{y}),\;A(16,16) = \theta_{x} /(R{}_{x}R{}_{y}), \hfill \\ A(17,1) = 3\lambda_{x} ,\;A(17,3) = 3\lambda_{y} ,\;A(17,17) = 3\overline{u}_{0,x} ,\;A(17,18) = 3\overline{v}_{0,x} ,\;A(18,2) = 3\lambda_{x} ,\;A(18,4) = 3\lambda_{y} , \hfill \\ A(18,17) = 3\overline{u}_{0,y} ,\;A(18,18) = 3\overline{v}_{0,y} ,\;A(19,1) = \theta_{x} ,\;A(19,3) = \theta_{y} ,\;A(19,15) = \overline{u}_{0,x} ,\;A(19,16) = \overline{v}_{0,x} , \hfill \\ \end{gathered}$$
$$\begin{gathered} A(20,2) = \theta_{x} ,\;A(20,4) = \theta_{y} ,\;A(20,15) = \overline{u}_{0,y} ,\;A(20,16) = \overline{v}_{0,y} ,\;A(21,7) = \lambda_{x,x} ,\;A(21,9) = \lambda_{y,x} ,\; \hfill \\ A(21,11) = \theta_{x,x} ,\;A(21,13) = \theta_{y,x} ,\;A(21,15) = \lambda_{x} /R_{x}^{2} ,\;A(21,17) = \theta_{x} /R_{x}^{2} ,\;A(22,8) = \lambda_{x,y} , \hfill \\ A(22,10) = \lambda_{y,y} ,\;A(22,12) = \theta_{x,y} ,\;A(22,14) = \theta_{y,y} ,\;A(22,16) = \lambda_{y} /R_{y}^{2} ,\;A(22,18) = \theta_{y} /R_{y}^{2} , \hfill \\ \end{gathered}$$
$$\begin{gathered} A(23,7) = \lambda_{x,y} ,\;A(23,8) = \lambda_{x,x} ,\;A(23,9) = \lambda_{y,y} ,\;A(23,10) = \lambda_{y,x} ,\;A(23,11) = \theta_{x,y} ,\;A(23,12) = \theta_{x,x} , \hfill \\ A(23,13) = \theta_{y,y} ,\;A(23,14) = \theta_{y,x} ,\;A(23,15) = \lambda_{y} /(R_{x} R_{y} ),\;A(23,16) = \lambda_{x} /(R_{x} R_{y} ), \hfill \\ A(23,17) = \theta_{y} /(R_{x} R_{y} ),\;A(23,18) = \theta_{x} /(R_{x} R_{y} ),\;A(24,10) = 3\lambda_{x} ,\;A(24,13) = 3\lambda_{y} ,\;A(24,17) = 3\lambda_{x,x} , \hfill \\ \end{gathered}$$
$$\begin{gathered} A(24,18) = 3\lambda_{y,x} ,\;A(25,12) = 3\lambda_{x} ,\;A(25,14) = 3\lambda_{y} ,\;A(25,17) = 3\lambda_{x,y} ,\;A(25,18) = 3\lambda_{y,y} , \hfill \\ A(26,11) = \lambda_{x,x} ,\;A(26,13) = \lambda_{y,x} c,\;A(26,17) = \lambda_{x} /R_{x}^{2} ,\;A(27,12) = \lambda_{x,y} ,\;A(27,14) = \lambda_{y,y} , \hfill \\ A(27,18) = \lambda_{y} /R_{y}^{2} ,\;A(28,11) = \lambda_{x,y} ,\;A(28,12) = \lambda_{x,x} ,\;A(28,13) = \lambda_{y,y} ,\;A(28,14) = \lambda_{y,x} , \hfill \\ A(28,17) = \lambda_{y} /(R_{x} R_{y} ),\;A(28,18) = \lambda_{x} /(R_{x} R_{y} ) \hfill \\ \end{gathered}$$

The matrix [G] is defined as

$$\left[ G \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} {N_{i,x} } & 0 & {N_{i} /R_{x} } & 0 & 0 & 0 & 0 \\ {N_{i,y} } & 0 & {N_{i} /R_{xy} } & 0 & 0 & 0 & 0 \\ 0 & {N_{i,x} } & {N_{i} /R_{xy} } & 0 & 0 & 0 & 0 \\ 0 & {N_{i,y} } & {N_{i} /R_{y} } & 0 & 0 & 0 & 0 \\ { - N_{i} /R_{x} } & 0 & {N_{i,x} } & 0 & 0 & 0 & 0 \\ 0 & { - N_{i} /R_{y} } & {N_{i,y} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {N_{i,x} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {N_{i,y} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {N_{i,x} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {N_{i,y} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {N_{i,x} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {N_{i,y} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {N_{i,x} } \\ 0 & 0 & 0 & 0 & 0 & 0 & {N_{i,y} } \\ 0 & 0 & 0 & {N_{i} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {N_{i} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {N_{i} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {N_{i} } \\ \end{array} } \right]$$

The matrix \(\left[ Z \right]\) is defined as

$$[Z] = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & z & 0 & {z^{3} } & 0 \\ 0 & 1 & 0 & 0 & z & 0 & {z^{3} } \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ \end{array} } \right]$$

The matrix \(\left[ S \right]\) is defined as

$$\left[ S \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} {N_{x} } & {N_{xy} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {N_{xy} } & {N_{y} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {N_{x} } & {N_{xy} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {N_{xy} } & {N_{y} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {N_{x} } & {N_{xy} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {N_{xy} } & {N_{y} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{N_{x} h^{2} }}{12}} & {\frac{{N_{xy} h^{2} }}{12}} & 0 & 0 & {\frac{{N_{x} h^{4} }}{80}} & {\frac{{N_{xy} h^{4} }}{80}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{N_{xy} h^{2} }}{12}} & {\frac{{N_{y} h^{2} }}{12}} & 0 & 0 & {\frac{{N_{xy} h^{4} }}{80}} & {\frac{{N_{y} h^{4} }}{80}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{N_{x} h^{2} }}{12}} & {\frac{{N_{xy} h^{2} }}{12}} & 0 & 0 & {\frac{{N_{x} h^{4} }}{80}} & {\frac{{N_{xy} h^{4} }}{80}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{N_{xy} h^{2} }}{12}} & {\frac{{N_{y} h^{2} }}{12}} & 0 & 0 & {\frac{{N_{xy} h^{4} }}{80}} & {\frac{{N_{y} h^{4} }}{80}} \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{N_{x} h^{4} }}{80}} & {\frac{{N_{xy} h^{4} }}{80}} & 0 & 0 & {\frac{{N_{x} h^{6} }}{448}} & {\frac{{N_{xy} h^{6} }}{448}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{N_{xy} h^{4} }}{80}} & {\frac{{N_{y} h^{4} }}{80}} & 0 & 0 & {\frac{{N_{xy} h^{6} }}{448}} & {\frac{{N_{y} h^{6} }}{448}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{N_{x} h^{4} }}{80}} & {\frac{{N_{xy} h^{4} }}{80}} & 0 & 0 & {\frac{{N_{x} h^{6} }}{448}} & {\frac{{N_{xy} h^{6} }}{448}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{N_{xy} h^{4} }}{80}} & {\frac{{N_{y} h^{4} }}{80}} & 0 & 0 & {\frac{{N_{xy} h^{6} }}{448}} & {\frac{{N_{y} h^{6} }}{448}} \\ \end{array} } \right]$$

where \(N_{x}\), \(N_{y}\) and \(N_{xy}\) are the thermal in-plane stress resultants.

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Rout, M., Hota, S.S. Geometrically nonlinear free vibration of CNTs reinforced sandwich conoidal shell in thermal environment. Acta Mech 234, 2677–2694 (2023). https://doi.org/10.1007/s00707-023-03508-3

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