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Large amplitude free vibration analysis of thermally post-buckled composite doubly curved panel embedded with SMA fibers

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Abstract

Thermal post-buckled vibration of laminated composite doubly curved panel embedded with shape memory alloy (SMA) fiber is investigated and presented in this article. The geometry matrix and the nonlinear stiffness matrices are derived using Green–Lagrange type nonlinear kinematics in the framework of higher order shear deformation theory. In addition to that, material nonlinearity in shape memory alloy due to thermal load is incorporated by the marching technique. The developed mathematical model is discretized using a nonlinear finite element model and the sets of nonlinear governing equations are obtained using Hamilton’s principle. The equations are solved using the direct iterative method. The effect of nonlinearity both in geometric and material have been studied using the developed model and compared with those published literature. Effect of various geometric parameters such as thickness ratio, amplitude ratio, lamination scheme, support condition, prestrains of SMA, and volume fractions of SMA on the nonlinear free vibration behavior of thermally post-buckled composite flat/curved panel been studied in detail and reported.

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Authors and Affiliations

  1. Department of Mechanical Engineering, NIT Rourkela, Rourkela, 769008, Odisha, India

    Subrata Kumar Panda

  2. Department of Aerospace Engineering, IIT Kharagpur, Kharagpur, 721302, West Bengal, India

    B. N. Singh

Authors
  1. Subrata Kumar Panda
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  2. B. N. Singh
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Correspondence to Subrata Kumar Panda.

Appendix

Appendix

The individual nonlinear strain terms in Eq. (4) are given below:

$$\begin{aligned} &\bigl( \varepsilon _{1}^{4} \bigr) = \bigl[ ( u_{,\xi _{1}} )^{2} + ( v_{,\xi _{1}} )^{2} + ( w_{,\xi _{1}} )^{2} \bigr] \\ &\bigl( \varepsilon _{2}^{4} \bigr) = \bigl[ ( u_{,\xi _{2}} )^{2} + ( v_{,\xi _{2}} )^{2} + ( w_{,\xi _{2}} )^{2} \bigr] \\ & \bigl( \varepsilon _{6}^{4} \bigr) = 2 [ u_{,\xi _{1}}u_{,\xi _{2}} + v_{,\xi _{1}}v_{,\xi _{2}} + w_{,\xi _{1}}w_{,\xi _{2}} ] \\ & \bigl( \varepsilon _{5}^{4} \bigr) = 2 [ \phi _{1}u_{,\xi _{1}} + \phi _{2}v_{,\xi _{1}} ] \\ & \bigl( \varepsilon _{4}^{4} \bigr) = 2 [ \phi _{1}u_{,\xi _{2}} + \phi _{2}v_{,\xi _{2}} ] \\ & \bigl( k_{1}^{5} \bigr) = 2 \biggl[ \phi _{1,\xi _{1}}u_{,\xi _{1}} + \phi _{2,\xi _{1}}v_{,\xi _{1}} - \frac{\phi _{1}}{R_{1}}w_{,\xi _{1}} \biggr] \\ &\bigl( k_{2}^{5} \bigr) = 2 \biggl[ \phi _{1,\xi _{2}}u_{,\xi _{2}} + \phi _{2,\xi _{2}}v_{,\xi _{2}} - \frac{\phi _{2}}{R_{2}}w_{,\xi _{2}} \biggr] \\ & \begin{aligned}[t] \bigl( k_{6}^{5} \bigr) &= 2 \biggl[ \phi _{1,\xi _{2}}u_{,\xi _{1}} + \phi _{1,\xi _{1}}u_{,\xi _{2}} + 2\phi _{2,\xi _{1}}v_{,\xi _{2}} + \phi _{2,\xi _{2}}v_{,\xi _{1}}\\ &\quad - \frac{\phi _{2}}{R_{2}}w_{,\xi _{1}} - \frac{\phi _{1}}{R_{1}}w_{,\xi _{2}} \biggr] \end{aligned} \\ &\bigl( k_{5}^{5} \bigr) = 2 [ \phi _{1}\phi _{1,\xi _{1}} + 2\psi _{1}u_{,\xi _{1}} + \phi _{2}\phi _{2,\xi _{1}} + 2\psi _{2}v_{,\xi _{1}} ] \\ & \bigl( k_{4}^{5} \bigr) = 2 [ \phi _{1}\phi _{1,\xi _{2}} + 2\psi _{1}u_{,\xi _{2}} + \phi _{2}\phi _{2,\xi _{2}} + 2\psi _{2}v_{,\xi _{2}} ] \\ & \begin{aligned}[t] \bigl( k_{1}^{6} \bigr) &= \biggl[ \phi _{1,\xi _{1}}^{2} + \phi _{2,\xi _{1}}^{2} + 2\psi _{1,\xi _{1}}u_{,\xi _{1}} + 2 \psi _{2,\xi _{1}}v_{,\xi _{1}} \\ &\quad + \frac{\phi _{1}^{2}}{R_{1}^{2}} - \frac{\psi _{1}}{R_{1}}w_{,\xi _{1}} \biggr] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{2}^{6} \bigr) &= \biggl[ \phi _{1,\xi _{2}}^{2} + \phi _{2,\xi _{2}}^{2} + 2\psi _{1,\xi _{2}}u_{,\xi _{2}} + 2\psi _{2,\xi _{2}}v_{,\xi _{2}} \\ &\quad + \frac{\phi _{2}^{2}}{R_{2}^{2}} - 2\frac{\psi _{2}}{R_{2}}w_{,\xi _{2}} \biggr] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{6}^{6} \bigr) &= 2 \biggl[ \phi _{1,\xi _{1}}\phi _{1,\xi _{2}} + \phi _{2,\xi _{1}}\phi _{2,\xi _{2}} + \psi _{1,\xi _{1}}u_{,\xi _{2}} \\ &\quad + \psi _{1,\xi _{2}}u_{,\xi _{1}} + \psi _{2,\xi _{1}}v_{,\xi _{2}} + \psi _{2,\xi _{2}}v_{,\xi _{1}}\\ &\quad - \frac{\psi _{1}}{R_{1}}w_{,\xi _{2}} - \frac{\psi _{2}}{R_{2}}w_{,\xi _{1}} + \frac{\phi _{1}}{R_{1}}\frac{\phi _{2}}{R_{2}} \biggr] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{5}^{6} \bigr) &= 2 [ \phi _{1}\psi _{1,\xi _{1}} + \phi _{2}\psi _{2,\xi _{1}} + 2\psi _{1} \phi _{1,\xi _{1}} + 2\psi _{2}\phi _{2,\xi _{1}} \\ &\quad + 3\theta _{1}u_{,\xi _{1}} + 3\theta _{2}v_{,\xi _{1}} ] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{4}^{6} \bigr) &= 2 [ \phi _{1}\psi _{1,\xi _{2}} + \phi _{2}\psi _{2,\xi _{2}} + 2\psi _{1} \phi _{1,\xi _{2}} + 2\psi _{2}\phi _{2,\xi _{2}} \\ &\quad + 3\theta _{1}u_{,\xi _{2}} + 3\theta _{2}v_{,\xi _{2}} ] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{1}^{7} \bigr)& = 2 \biggl[ u_{,\xi _{1}}\theta _{1,\xi _{1}} + v_{,\xi _{1}}\theta _{2,\xi _{1}} + \phi _{1,\xi _{1}} \psi _{1,\xi _{1}} \\ &\quad + \phi _{2,\xi _{1}}\psi _{2,\xi _{1}} - \frac{\theta _{1}}{R_{1}}w_{,\xi _{1}} + \frac{\phi _{1}}{R_{1}}\frac{\psi _{1}}{R_{1}} \biggr] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{2}^{7} \bigr) &= 2 \biggl[ u_{,\xi _{2}}\theta _{1,\xi _{2}} + v_{,\xi _{2}}\theta _{2,\xi _{2}} + \phi _{1,\xi _{2}} \psi _{1,\xi _{2}} \\ &\quad + \phi _{2,\xi _{2}}\psi _{2,\xi _{2}} - \frac{\theta _{2}}{R_{2}}w_{,\xi _{2}} + \frac{\phi _{2}}{R_{2}}\frac{\psi _{2}}{R_{2}} \biggr] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{6}^{7} \bigr) &= 2\biggl[ u_{,\xi _{1}}\theta _{1,\xi _{2}} + u_{,\xi _{2}}\theta _{1,\xi _{1}} + v_{,\xi _{1}}\theta _{2,\xi _{2}} + v_{,\xi _{2}}\theta _{2,\xi _{1}} \\ &\quad + \phi _{1,\xi _{1}}\psi _{1,\xi _{2}} + \phi _{1,\xi _{2}}\psi _{1,\xi _{1}} + \phi _{2,\xi _{1}}\psi _{2,\xi _{2}} \\ &\quad + \phi _{2,\xi _{2}}\psi _{2,\xi _{1}} - \frac{\theta _{1}}{R_{1}}w_{,\xi _{2}} - \frac{\theta _{2}}{R_{2}}w_{,\xi _{1}} + \frac{\psi _{1}}{R_{1}}\frac{\phi _{2}}{R_{2}}\\ &\quad + \frac{\psi _{2}}{R_{2}}\frac{\phi _{1}}{R_{1}} \biggr] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{5}^{7} \bigr) &= 2 [ \phi _{1}\theta _{1,\xi _{1}} + \phi _{2}\theta _{2,\xi _{1}} + 2\psi _{1}\psi _{1,\xi _{1}} + 2\psi _{2}\psi _{2,\xi _{1}} \\ &\quad + 3 \theta _{1}\phi _{1,\xi _{1}} + 3\theta _{2}\phi _{2,\xi _{1}} ] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{4}^{7} \bigr) &= 2 [ \phi _{1}\theta _{1,\xi _{2}} + \phi _{2}\theta _{2,\xi _{2}} + 2\psi _{1}\psi _{1,\xi _{2}} + 2\psi _{2}\psi _{2,\xi _{2}} \\ &\quad + 3 \theta _{1}\phi _{1,\xi _{2}} + 3\theta _{2}\phi _{2,\xi _{2}} ] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{1}^{8} \bigr)& = \biggl[ \psi _{1,\xi _{1}}^{2} + \psi _{2,\xi _{1}}^{2} + 2\phi _{1,\xi _{1}}\theta _{1,\xi _{1}} + 2\phi _{2,\xi _{1}}\theta _{2,\xi _{1}} \\ &\quad + \frac{\psi _{1}^{2}}{R_{1}^{2}} + 2\frac{\phi _{1}}{R_{1}}\frac{\theta _{1}}{R_{1}} \biggr] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{2}^{8} \bigr) &= \biggl[ \psi _{1,\xi _{2}}^{2} + \psi _{2,\xi _{2}}^{2} + 2\phi _{1,\xi _{2}}\theta _{1,\xi _{2}} + 2\phi _{2,\xi _{2}}\theta _{2,\xi _{2}} \\ &\quad + \frac{\psi _{2}^{2}}{R_{2}^{2}} + 2\frac{\phi _{2}}{R_{2}}\frac{\theta _{2}}{R_{2}} \biggr] \end{aligned} \\ & \begin{aligned}[t] \bigl( k_{6}^{8} \bigr) &= \biggl[ \psi _{1,\xi _{1}}\psi _{1,\xi _{2}} + \psi _{2,\xi _{1}}\psi _{2,\xi _{2}} + 2\theta _{1,\xi _{1}}\phi _{1,\xi _{2}} \\ &\quad + 2\theta _{2,\xi _{1}}\phi _{2,\xi _{2}} + 2\theta _{1,\xi _{2}}\phi _{1,\xi _{1}} + 2\theta _{2,\xi _{2}}\phi _{2,\xi _{1}} \\ &\quad + 2\frac{\psi _{1}}{R_{1}}\frac{\psi _{2}}{R_{2}} + 2\frac{\phi _{1}}{R_{1}} \frac{\theta _{2}}{R_{2}} + 2\frac{\phi _{2}}{R_{2}}\frac{\theta _{1}}{R_{1}} \biggr] \end{aligned} \\ & \bigl( k_{5}^{8} \bigr) = 2 [ 2\psi _{1}\theta _{1,\xi _{1}} + 2\psi _{2}\theta _{2,\xi _{1}} + 3\theta _{1}\psi _{1,\xi _{1}} + 3\theta _{2}\psi _{2,\xi _{1}} ] \\ & \bigl( k_{4}^{8} \bigr) = 2 [ 2\psi _{1} \theta _{1,\xi _{2}} + 2\psi _{2}\theta _{2,\xi _{2}} + 3\theta _{1}\psi _{1,\xi _{2}} + 3\theta _{2}\psi _{2,\xi _{2}} ] \\ &\bigl( k_{1}^{9} \bigr) = 2 \biggl[ \psi _{1,\xi _{1}} \theta _{1,\xi _{1}} + \psi _{2,\xi _{1}}\theta _{2,\xi _{1}} + \frac{\psi _{1}}{R_{1}}\frac{\theta _{1}}{R_{1}} \biggr] \\ & \bigl( k_{2}^{9} \bigr) = 2 \biggl[ \psi _{1,\xi _{2}}\theta _{1,\xi _{2}} + \psi _{2,\xi _{2}}\theta _{2,\xi _{2}} + \frac{\psi _{2}}{R_{2}}\frac{\theta _{2}}{R_{2}} \biggr] \\ & \begin{aligned}[t] \bigl( k_{6}^{9} \bigr) &= 2 \biggl[ \psi _{1,\xi _{1}} \theta _{1,\xi _{2}} + \psi _{2,\xi _{1}}\theta _{2,\xi _{2}} + \theta _{1,\xi _{1}}\psi _{1,\xi _{2}}\\ &\quad + \theta _{2,\xi _{1}}\psi _{2,\xi _{2}} + \frac{\psi _{2}}{R_{2}}\frac{\theta _{1}}{R_{1}} + \frac{\psi _{1}}{R_{1}}\frac{\theta _{2}}{R_{2}} \biggr] \end{aligned} \\ &\bigl( k_{5}^{9} \bigr) = 2 [ 3\theta _{1}\theta _{1,\xi _{1}} + 3\theta _{2}\theta _{2,\xi _{1}} ] \\ & \bigl( k_{4}^{9} \bigr) = [ 2 \times 3 \times \theta _{1} \times \theta _{1,\xi _{2}} + 2 \times 3 \times \theta _{2} \times \theta _{2,\xi _{2}} ] \\ &\bigl( k_{1}^{10} \bigr) = \biggl[ \theta _{1,\xi _{1}}^{2} + \theta _{2,\xi _{1}}^{2} + \frac{\theta _{1}^{2}}{R_{1}^{2}} \biggr] \\ &\bigl( k_{2}^{10} \bigr) = \biggl[ \theta _{1,\xi _{2}}^{2} + \theta _{2,\xi _{2}}^{2} + \frac{\theta _{2}^{2}}{R_{2}^{2}} \biggr] \\ & \bigl( k_{6}^{10} \bigr) = 2 \biggl[ \theta _{1,\xi _{1}}\theta _{1,\xi _{2}} + \theta _{2,\xi _{1}} \theta _{2,\xi _{2}} + \frac{\theta _{1}}{R_{1}}\frac{\theta _{2}}{R_{2}} \biggr] \end{aligned}$$
(A.1)

Some coupled terms in (A.1):

$$\begin{aligned} &u_{,\xi _{1}} = \frac{\partial \bar{u}}{\partial \xi _{1}} + \frac{\bar{w}}{R_{1}}, \qquad u_{,\xi _{2}} = \frac{\partial \bar{u}}{\partial \xi _{2}}, \qquad v_{,\xi _{1}} = \frac{\partial \bar{v}}{\partial \xi _{1}}\\ & v_{,\xi _{2}} = \frac{\partial \bar{v}}{\partial \xi _{2}} + \frac{\bar{w}}{R_{2}}, \qquad w_{,\xi _{1}} = \frac{\partial \bar{w}}{\partial \xi _{1}} - \frac{\bar{u}}{R_{1}}\\ & w_{,\xi _{2}} = \frac{\partial \bar{w}}{\partial \xi _{2}} - \frac{\bar{v}}{R_{2}},\qquad \phi _{1,\xi _{1}} = \frac{\partial \phi _{1}}{\partial \xi _{1}} \\ & \phi _{1,\xi _{2}} = \frac{\partial \phi _{1}}{\partial \xi _{2}},\qquad \phi _{2,\xi _{1}} = \frac{\partial \phi _{2}}{\partial \xi _{1}},\qquad \phi _{2,\xi _{2}} = \frac{\partial \phi _{2}}{\partial \xi _{2}}\\ & \psi _{1,\xi _{1}} = \frac{\partial \psi _{1}}{\partial \xi _{1}},\qquad \psi _{1,\xi _{2}} = \frac{\partial \psi _{1}}{\partial \xi _{2}},\qquad \psi _{2,\xi _{1}} = \frac{\partial \psi _{2}}{\partial \xi _{1}}\\ & \psi _{2,\xi _{2}} = \frac{\partial \psi _{2}}{\partial \xi _{2}},\qquad \theta _{1,\xi _{1}} = \frac{\partial \theta _{1}}{\partial \xi _{1}},\qquad \theta _{1,\xi _{2}} = \frac{\partial \theta _{1}}{\partial \xi _{2}}\\ & \theta _{2,\xi _{1}} = \frac{\partial \theta _{2}}{\partial \xi _{1}},\qquad \theta _{2,\xi _{2}} = \frac{\partial \theta _{2}}{\partial \xi _{2}} \end{aligned}$$

Linear and nonlinear thickness coordinate matrices as appeared in Eq. (4)

$$\begin{aligned} [ H ]_{L} &= \left [ \begin{array}{@{}cccccccccccccccccccc@{}} 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 &\zeta ^{2} & 0 & 0 & 0 & 0 & \zeta ^{3} & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0& 0 & \zeta ^{2} & 0 & 0 & 0 & 0 & \zeta ^{3} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0& 0 & 0 & \zeta ^{2} & 0 & 0 & 0 & 0 & \zeta ^{3} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 & \zeta ^{2} & 0 & 0 & 0 & 0 & \zeta ^{3} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 & \zeta ^{2} & 0 & 0 & 0 & 0 & \zeta ^{3} \end{array} \right . \\ {[H ]}_{\mathrm{NL}} &= \left [ \begin{array}{@{}cccccccccccccccc@{}} 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 & \zeta ^{2} & 0 & 0 & 0 & 0 & \zeta ^{3} \\ 0 & 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 & \zeta ^{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 & \zeta ^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 & \zeta ^{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 & \zeta ^{2} & 0 \end{array} \right . \\ &\quad \left . \begin{array}{@{}ccccccccccccccccc@{}} 0 & 0 & 0 & 0 & \zeta ^{4} & 0 & 0 & 0 & 0 & \zeta ^{5} & 0 & 0 & 0 & 0 & \zeta ^{6} & 0 & 0 \\ \zeta ^{3} & 0 & 0 & 0 & 0 & \zeta ^{4} & 0 & 0 & 0 & 0 & \zeta ^{5} & 0 & 0 & 0 & 0 & \zeta ^{6} & 0 \\ 0 & \zeta ^{3} & 0 & 0 & 0 & 0 & \zeta ^{4} & 0 & 0 & 0 & 0 & \zeta ^{5} & 0 & 0 & 0 & 0 & \zeta ^{6} \\ 0 & 0 & \zeta ^{3} & 0 & 0 & 0 & 0 & \zeta ^{4} & 0 & 0 & 0 & 0 & \zeta ^{5} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \zeta ^{3} & 0 & 0 & 0 & 0 & \zeta ^{4} & 0 & 0 & 0 & 0 & \zeta ^{5} & 0 & 0 & 0 \end{array} \right ] \end{aligned}$$
(A.2)

Individual terms of the matrix [A] as appeared in Eq. (23)

$$\begin{aligned} &{[ A ]}_{1\_1} = u_{,\xi _{1}},\qquad [ A ]_{1\_3} = v_{,\xi _{1}},\qquad [ A ]_{1\_5} = w_{,\xi _{1}},\qquad { [ A ]}_{2\_2} = u_{,\xi _{2}},\qquad [ A ]_{2\_4} = v_{,\xi _{2}},\qquad [ A ]_{2\_6} = w_{,\xi _{2}} \\ & {[ A ]}_{3\_1} = u_{,\xi _{2}},\qquad [ A ]_{3\_2} = u_{,\xi _{1}},\qquad [ A ]_{3\_3} = v_{,\xi _{2}},\qquad [ A ]_{3\_4} = v_{,\xi _{1}},\qquad [ A ]_{3\_5} = w_{,\xi _{2}},\qquad [ A ]_{3\_6} = w_{,\xi _{1}} \\ &[ A ]_{4\_1} = \phi _{1},\qquad [ A ]_{4\_3} = \phi _{2},\qquad [ A ]_{4\_22} = u_{,\xi _{1}},\qquad [ A ]_{4\_23} = v_{,\xi _{1}},\qquad [ A ]_{5\_2} = \phi _{1},\qquad [ A ]_{5\_4} = \phi _{2} \\ & [ A ]_{5\_22} = u_{,\xi _{2}}, \qquad [ A ]_{5\_23} = v_{,\xi _{2}},\qquad [ A ]_{6\_1} = \phi _{1,\xi _{1}},\qquad [ A ]_{6\_3} = \phi _{2,\xi _{1}},\qquad [ A ]_{6\_5} = - \frac{\phi _{1}}{R_{1}} \\ & [ A ]_{6\_7} = u_{,\xi _{1}} ,\qquad [ A ]_{6\_9} = v_{,\xi _{1}},\qquad [ A ]_{6\_22} = - \frac{w_{,\xi _{1}}}{R_{1}}, \qquad [ A ]_{7\_2} = \phi _{1,\xi _{2}}, \qquad [ A ]_{7\_4} = \phi _{2,\xi _{2}} \\ & [ A ]_{7\_6} = - \frac{\phi _{2}}{R_{2}},\qquad [ A ]_{7\_8} = u_{,\xi _{2}} ,\qquad [ A ]_{7\_10} = v_{,\xi _{2}},\qquad [ A ]_{7\_23} = - \frac{w_{\xi _{2}}}{R_{2}},\qquad [ A ]_{8\_1} = \phi _{1,\xi _{2}} \\ &[ A ]_{8\_2} = \phi _{1,\xi _{1}},\qquad [ A ]_{8\_3} = \phi _{2,\xi _{2}}, \qquad [ A ]_{8\_4} = \phi _{2,\xi _{1}}, \qquad [ A ]_{8\_5} = - \frac{\phi _{2}}{R_{2}}, \qquad [ A ]_{8\_6} = - \frac{\phi _{1}}{R_{1}} \\ & [ A ]_{8\_7} = u_{,\xi _{2}},\qquad [ A ]_{8\_8} = u_{,\xi _{1}},\qquad [ A ]_{8\_9} = v_{,\xi _{2}},\qquad [ A ]_{8\_10} = v_{,\xi _{1}}, \qquad [ A ]_{8\_22} = - \frac{w_{\xi _{2}}}{R_{1}} \\ & [ A ]_{8\_23} = - \frac{w_{,\xi _{1}}}{R_{2}}, \qquad [ A ]_{9\_1} = 2\psi _{1}, \qquad [ A ]_{9\_3} = 2\psi _{2},\qquad [ A ]_{9\_7} = \phi _{1},\qquad [ A ]_{9\_9} = \phi _{2} \\ & [ A ]_{9\_22} = \phi _{1,\xi _{1}},\qquad [ A ]_{9\_23} = \phi _{2,\xi _{1}},\qquad [ A ]_{9\_24} = 2u_{,\xi _{1}},\qquad [ A ]_{9\_25} = 2v_{,\xi _{1}}, \qquad [ A ]_{10\_2} = 2\psi _{1} \\ & [ A ]_{10\_4} = 2\psi _{2},\qquad [ A ]_{10\_8} = \phi _{1},\qquad [ A ]_{10\_10} = \phi _{2},\qquad [ A ]_{10\_22} = \phi _{1,\xi _{2}}, \qquad [ A ]_{10\_23} = \phi _{2,\xi _{2}} \\ & [ A ]_{10\_24} = 2u_{,\xi _{2}},\qquad [ A ]_{10\_25} = 2v_{,\xi _{2}},\qquad [ A ]_{11\_1} = \psi _{1,\xi _{1}},\qquad [ A ]_{11\_3} = \psi _{2,\xi _{1}},\qquad [ A ]_{11\_5} = - \frac{\psi _{1}}{R_{1}} \\ & [ A ]_{11\_7} = \phi _{1,\xi _{1}}, \qquad [ A ]_{11\_9} = \phi _{2,\xi _{1}},\qquad [ A ]_{11\_11} = u_{,\xi _{1}},\qquad [ A ]_{11\_13} = v_{,\xi _{1}},\qquad [ A ]_{11\_22} = \frac{\phi _{1}}{R_{1}^{2}} \\ & [ A ]_{11\_24} = - \frac{w_{,\xi _{1}}}{R_{1}},\qquad [ A ]_{12\_2} = \psi _{1,\xi _{2}},\qquad [ A ]_{12\_4} = \psi _{2,\xi _{2}},\qquad [ A ]_{12\_6} = - \frac{\psi _{2}}{R_{2}},\qquad [ A ]_{12\_8} = \phi _{1,\xi _{2}} \\ & [ A ]_{12\_10} = \phi _{2,\xi _{2}},\qquad [ A ]_{12\_12} = u_{,\xi _{2}},\qquad [ A ]_{12\_14} = v_{,\xi _{2}},\qquad [ A ]_{12\_23} = \frac{\phi _{2}}{R_{2}^{2}},\qquad [ A ]_{12\_25} = - \frac{w_{,\xi _{2}}}{R_{2}} \\ & [ A ]_{13\_1} = \psi _{1,\xi _{2}},\qquad [ A ]_{13\_2} = \psi _{1,\xi _{1}},\qquad [ A ]_{13\_3} = \psi _{2,\xi _{2}},\qquad [ A ]_{13\_4} = \psi _{2,\xi _{1}},\qquad [ A ]_{13\_5} = - \frac{\psi _{2}}{R_{2}} \\ & [ A ]_{13\_6} = - \frac{\psi _{1}}{R_{1}},\qquad [ A ]_{13\_7} = \phi _{1,\xi _{2}},\qquad [ A ]_{13\_8} = \phi _{1,\xi _{1}},\qquad [ A ]_{13\_9} = \phi _{2,\xi _{2}},\qquad [ A ]_{13\_10} = \phi _{2,\xi _{1}} \\ & [ A ]_{13\_11} = u_{,\xi _{2}},\qquad [ A ]_{13\_12} = u_{,\xi _{1}},\qquad [ A ]_{13\_13} = v_{,\xi _{2}},\qquad [ A ]_{13\_14} = v_{,\xi _{1}}, \qquad [ A ]_{13\_22} = \frac{1}{R_{1}}\frac{\phi _{2}}{R_{2}} \\ & [ A ]_{13\_23} = \frac{1}{R_{2}}\frac{\phi _{1}}{R_{1}},\qquad [ A ]_{13\_24} = - \frac{w_{,\xi _{2}}}{R_{1}},\qquad [ A ]_{13\_25} = - \frac{w_{,\xi _{1}}}{R_{2}},\qquad [ A ]_{14\_1} = 3\theta _{1},\qquad [ A ]_{14\_3} = 3\theta _{2} \\ & [ A ]_{14\_7} = 2\psi _{1},\qquad [ A ]_{14\_9} = 2\psi _{2},\qquad [ A ]_{14\_11} = \phi _{1},\qquad [ A ]_{14\_13} = \phi _{2}, \qquad [ A ]_{14\_22} = \psi _{1,\xi _{1}} \\ & [ A ]_{14\_23} = \psi _{2,\xi _{1}}, \qquad [ A ]_{14\_24} = 2\phi _{1,\xi _{1}},\qquad [ A ]_{14\_25} = 2\phi _{2,\xi _{1}},\qquad [ A ]_{14\_26} = 3u_{,\xi _{1}}, \qquad [ A ]_{14\_27} = 3v_{,\xi _{1}} \\ & [ A ]_{15\_2} = 3\theta _{1},\qquad [ A ]_{15\_4} = 3\theta _{2}, \qquad [ A ]_{15\_8} = 2\psi _{1},\qquad [ A ]_{15\_10} = 2\psi _{2}, \qquad [ A ]_{15\_12} = \phi _{1} \\ & [ A ]_{15\_14} = \phi _{2},\qquad [ A ]_{15\_22} = \psi _{1,\xi _{2}},\qquad [ A ]_{15\_23} = \psi _{2,\xi _{2}}, \qquad [ A ]_{15\_24} = 2\phi _{1,\xi _{2}}, \qquad [ A ]_{15\_25} = 2\phi _{2,\xi _{2}} \\ & [ A ]_{15\_26} = 3u_{,\xi _{2}},\qquad [ A ]_{15\_27} = 3v_{,\xi _{2}}, \qquad [ A ]_{16\_1} = \theta _{1,\xi _{1}},\qquad [ A ]_{16\_3} = \theta _{2,\xi _{1}}, \qquad [ A ]_{16\_5} = - \frac{\theta _{1}}{R_{1}} \\ &[ A ]_{16\_7} = \psi _{1,\xi _{1}},\qquad [ A ]_{16\_9} = \psi _{2,\xi _{1}},\qquad [ A ]_{16\_11} = \phi _{1,\xi _{1}},\qquad [ A ]_{16\_13} = \phi _{2,\xi _{1}},\qquad [ A ]_{16\_15} = u_{,\xi _{1}} \\ & [ A ]_{16\_17} = v_{,\xi _{1}},\qquad [ A ]_{16\_22} = \frac{\psi _{1}}{R_{1}^{2}},\qquad [ A ]_{16\_24} = \frac{\phi _{1}}{R_{1}^{2}}, \qquad [ A ]_{16\_26} = - \frac{w_{,\xi _{1}}}{R_{1}}, \qquad [ A ]_{17\_2} = \theta _{1,\xi _{2}} \\ &[ A ]_{17\_4} = \theta _{2,\xi _{2}},\qquad [ A ]_{17\_6} = - \frac{\theta _{2}}{R_{2}},\qquad [ A ]_{17\_8} = \psi _{1,\xi _{2}},\qquad [ A ]_{17\_10} = \psi _{2,\xi _{2}}, \qquad [ A ]_{17\_12} = \phi _{1,\xi _{2}} \\ & [ A ]_{17\_14} = \phi _{2,\xi _{2}},\qquad [ A ]_{17\_16} = u_{,\xi _{2}},\qquad [ A ]_{17\_18} = v_{,\xi _{2}},\qquad [ A ]_{17\_23} = \frac{\psi _{2}}{R_{2}^{2}},\qquad [ A ]_{17\_25} = \frac{\phi _{2}}{R_{2}^{2}} \\ & [ A ]_{17\_27} = - \frac{w_{,\xi _{2}}}{R_{2}},\qquad [ A ]_{18\_1} = \theta _{1,\xi _{2}},\qquad [ A ]_{18\_2} = \theta _{1,\xi _{1}},\qquad [ A ]_{18\_3} = \theta _{2,\xi _{2}},\qquad [ A ]_{18\_4} = \theta _{2,\xi _{1}} \\ & [ A ]_{18\_5} = - \frac{\theta _{2}}{R_{2}},\qquad [ A ]_{18\_6} = - \frac{\theta _{1}}{R_{1}},\qquad [ A ]_{18\_7} = \psi _{1,\xi _{2}},\qquad [ A ]_{18\_8} = \psi _{1,\xi _{1}}, \qquad [ A ]_{18\_9} = \psi _{2,\xi _{2}} \\ & [ A ]_{18\_10} = \psi _{2,\xi _{1}},\qquad [ A ]_{18\_11} = \phi _{1,\xi _{2}},\qquad [ A ]_{18\_12} = \phi _{1,\xi _{1}},\qquad [ A ]_{18\_13} = \phi _{2,\xi _{2}}, \qquad [ A ]_{18\_14} = \phi _{2,\xi _{1}} \\ & [ A ]_{18\_15} = u_{,\xi _{2}},\qquad [ A ]_{18\_16} = u_{,\xi _{1}},\qquad [ A ]_{18\_17} = v_{,\xi _{2}},\qquad [ A ]_{18\_18} = v_{,\xi _{1}},\qquad [ A ]_{18\_22} = \frac{1}{R_{1}}\frac{\psi _{2}}{R_{2}} \\ & [ A ]_{18\_23} = \frac{1}{R_{2}}\frac{\psi _{1}}{R_{1}},\qquad [ A ]_{18\_24} = \frac{1}{R_{1}}\frac{\phi _{2}}{R_{2}},\qquad [ A ]_{18\_25} = \frac{1}{R_{2}}\frac{\phi _{1}}{R_{1}},\qquad [ A ]_{18\_26} = - \frac{w_{,\xi _{2}}}{R_{1}} \\ & [ A ]_{18\_27} = - \frac{w_{,\xi _{1}}}{R_{2}},\qquad [ A ]_{19\_7} = 3\theta _{1},\qquad [ A ]_{19\_9} = 3\theta _{2},\qquad [ A ]_{19\_11} = 2\psi _{1},\qquad [ A ]_{19\_13} = 2\psi _{2} \\ & [ A ]_{19\_15} = \phi _{1},\qquad [ A ]_{19\_17} = \phi _{2},\qquad [ A ]_{19\_22} = \theta _{1,\xi _{1}},\qquad [ A ]_{19\_23} = \theta _{2,\xi _{1}}, \qquad [ A ]_{19\_24} = 2\psi _{1,\xi _{1}} \\ & [ A ]_{19\_25} = 2\psi _{2,\xi _{1}}, \qquad [ A ]_{19\_26} = 3\phi _{1,\xi _{1}}, \qquad [ A ]_{19\_27} = 3\phi _{2,\xi _{1}},\qquad [ A ]_{20\_8} = 3\theta _{1}, \qquad [ A ]_{20\_10} = 3\theta _{2} \\ & [ A ]_{20\_12} = 2\psi _{1}, \qquad [ A ]_{20\_14} = 2\psi _{2},\qquad [ A ]_{20\_16} = \phi _{1},\qquad [ A ]_{20\_18} = \phi _{2}, \qquad [ A ]_{20\_22} = \theta _{1,\xi _{2}} \\ & [ A ]_{20\_23} = \theta _{2,\xi _{2}}, \qquad [ A ]_{20\_25} = 2\psi _{2,\xi _{2}}, \qquad [ A ]_{20\_26} = 3\phi _{1,\xi _{2}}, \qquad [ A ]_{20\_27} = 3\phi _{2,\xi _{2}},\qquad [ A ]_{21\_7} = \theta _{1,\xi _{1}} \\ & [ A ]_{21\_9} = \theta _{2,\xi _{1}},\qquad [ A ]_{21\_11} = \psi _{1,\xi _{1}},\qquad [ A ]_{21\_13} = \psi _{2,\xi _{1}}, \qquad [ A ]_{21\_15} = \phi _{1,\xi _{1}},\qquad [ A ]_{21\_17} = \phi _{2,\xi _{1}} \\ & [ A ]_{21\_22} = \frac{\theta _{1}}{R_{1}^{2}},\qquad [ A ]_{21\_24} = \frac{\psi _{1}}{R_{1}^{2}},\qquad [ A ]_{21\_26} = \frac{\phi _{1}}{R_{1}^{2}},\qquad [ A ]_{22\_8} = \theta _{1,\xi _{2}},\qquad [ A ]_{22\_10} = \theta _{2,\xi _{2}} \\ & [ A ]_{22\_12} = \psi _{1,\xi _{2}},\qquad [ A ]_{22\_14} = \psi _{2,\xi _{2}}, \qquad [ A ]_{22\_16} = \phi _{1,\xi _{2}},\qquad [ A ]_{22\_18} = \phi _{2,\xi _{2}}, \qquad [ A ]_{22\_23} = \frac{\theta _{2}}{R_{2}^{2}} \\ & [ A ]_{22\_25} = \frac{\psi _{2}}{R_{2}^{2}},\qquad [ A ]_{22\_27} = \frac{\phi _{2}}{R_{2}^{2}}, \qquad [ A ]_{23\_7} = \theta _{1,\xi _{2}},\qquad [ A ]_{23\_8} = \theta _{1,\xi _{1}},\qquad [ A ]_{23\_9} = \theta _{2,\xi _{2}} \\ & [ A ]_{23\_10} = \theta _{2,\xi _{1}},\qquad [ A ]_{23\_11} = \psi _{1,\xi _{2}}, \qquad [ A ]_{23\_12} = \psi _{1,\xi _{1}},\qquad [ A ]_{23\_13} = \psi _{2,\xi _{2}},\qquad [ A ]_{23\_13} = \psi _{2,\xi _{2}} \\ & [ A ]_{23\_14} = \psi _{2,\xi _{1}},\qquad [ A ]_{23\_15} = \phi _{1,\xi _{2}},\qquad [ A ]_{23\_16} = \phi _{1,\xi _{1}},\qquad [ A ]_{23\_17} = \phi _{2,\xi _{2}},\qquad [ A ]_{23\_18} = \phi _{2,\xi _{1}} \\ & [ A ]_{23\_22} = \frac{1}{R_{1}}\frac{\theta _{2}}{R_{2}},\qquad [ A ]_{23\_23} = \frac{1}{R_{2}}\frac{\theta _{1}}{R_{1}}, \qquad [ A ]_{23\_24} = \frac{1}{R_{1}}\frac{\psi _{2}}{R_{2}},\qquad [ A ]_{23\_25} = \frac{1}{R_{2}}\frac{\psi _{1}}{R_{1}} \\ &[ A ]_{23\_26} = \frac{1}{R_{1}}\frac{\phi _{2}}{R_{2}},\qquad [ A ]_{23\_27} = \frac{1}{R_{2}}\frac{\phi _{1}}{R_{1}}, \qquad [ A ]_{24\_11} = 3\theta _{1}, \qquad [ A ]_{24\_13} = 3\theta _{2},\qquad [ A ]_{24\_15} = 2\psi _{1} \\ & [ A ]_{24\_17} = 2\psi _{2},\qquad [ A ]_{24\_24} = 2\theta _{1,\xi _{1}},\qquad [ A ]_{24\_25} = 2\theta _{2,\xi _{1}},\qquad [ A ]_{24\_26} = 3\psi _{1,\xi _{1}},\qquad [ A ]_{24\_27} = 3\psi _{2,\xi _{1}} \\ & [ A ]_{25\_12} = 3\theta _{1},\qquad [ A ]_{25\_14} = 3\theta _{2}, \qquad [ A ]_{25\_16} = 2\psi _{1}, \qquad [ A ]_{25\_18} = 2\psi _{2},\qquad [ A ]_{25\_24} = 2\theta _{1,\xi _{2}} \\ & [ A ]_{25\_25} = 2\theta _{2,\xi _{2}}, \qquad [ A ]_{25\_26} = 3\psi _{1,\xi _{2}}, \qquad [ A ]_{25\_27} = 3\psi _{2,\xi _{2}}, \qquad [ A ]_{26\_11} = \theta _{1,\xi _{1}},\qquad [ A ]_{26\_13} = \theta _{2,\xi _{1}} \\ & [ A ]_{26\_15} = \psi _{1,\xi _{1}},\qquad [ A ]_{26\_17} = \psi _{2,\xi _{1}}, \qquad [ A ]_{26\_24} = \frac{\theta _{1}}{R_{1}^{2}}, \qquad [ A ]_{26\_26} = \frac{\psi _{1}}{R_{1}^{2}}, \qquad [ A ]_{27\_12} = \theta _{1,\xi _{2}} \\ & [ A ]_{27\_14} = \theta _{2,\xi _{2}},\qquad [ A ]_{27\_16} = \psi _{1,\xi _{2}}, \qquad [ A ]_{27\_18} = \psi _{2,\xi _{2}}, \qquad [ A ]_{27\_25} = \frac{\theta _{2}}{R_{2}^{2}}, \qquad [ A ]_{27\_27} = \frac{\psi _{2}}{R_{2}^{2}} \\ & [ A ]_{28\_11} = \theta _{1,\xi _{2}}, \qquad [ A ]_{28\_12} = \theta _{1,\xi _{1}}, \qquad [ A ]_{28\_13} = \theta _{2,\xi _{2}},\qquad [ A ]_{28\_14} = \theta _{2,\xi _{1}},\qquad [ A ]_{28\_15} = \psi _{1,\xi _{2}} \\ & [ A ]_{28\_16} = \psi _{1,\xi _{1}},\qquad [ A ]_{28\_17} = \psi _{2,\xi _{2}},\qquad [ A ]_{28\_18} = \psi _{2,\xi _{1}},\qquad [ A ]_{28\_24} = \frac{1}{R_{1}}\frac{\theta _{2}}{R_{2}},\qquad [ A ]_{28\_25} = \frac{1}{R_{2}}\frac{\theta _{1}}{R_{1}} \\ & [ A ]_{28\_26} = \frac{1}{R_{1}}\frac{\psi _{2}}{R_{2}},\qquad [ A ]_{28\_27} = \frac{1}{R_{2}}\frac{\psi _{1}}{R_{1}}, \qquad [ A ]_{29\_15} = 3\theta _{1},\qquad [ A ]_{29\_17} = 3\theta _{2}, \qquad [ A ]_{29\_26} = 3\theta _{1,\xi _{1}} \\ & [ A ]_{29\_27} = 3\theta _{2,\xi _{1}}, \qquad [ A ]_{30\_16} = 3\theta _{1},\qquad [ A ]_{30\_18} = 3\theta _{2},\qquad [ A ]_{30\_26} = 3\theta _{1,\xi _{2}},\qquad [ A ]_{30\_27} = 3\theta _{2,\xi _{2}} \\ & [ A ]_{31\_15} = \theta _{1,\xi _{1}}, \qquad {[ A ]}_{31\_17} = \theta _{2,\xi _{1}},\qquad [ A ]_{31\_26} = \frac{\theta _{1}}{R_{1}^{2}},\qquad [ A ]_{32\_16} = \theta _{1,\xi _{2}},\qquad [ A ]_{32\_18} = \theta _{2,\xi _{2}} \\ &[ A ]_{32\_27} = \frac{\theta _{2}}{R_{2}^{2}}, \qquad [ A ]_{33\_15} = \theta _{1,\xi _{2}},\qquad [ A ]_{33\_16} = \theta _{1,\xi _{1}},\qquad [ A ]_{33\_17} = \theta _{2,\xi _{2}},\qquad [ A ]_{33\_18} = \theta _{2,\xi _{1}} \\ & [ A ]_{33\_26} = \frac{1}{R_{1}} \frac{\theta _{2}}{R_{2}}, \qquad {[ A ]}_{33\_27} = \frac{1}{R_{2}}\frac{\theta _{1}}{R_{1}} \end{aligned}$$
(A.3)

Individual terms of the matrix [G]

$$\begin{aligned} &[ G ]_{1\_1} = \frac{\partial}{\partial \xi _{1}},\qquad [ G ]_{1\_3} = \frac{1}{R_{1}},\qquad [ G ]_{2\_1} = \frac{\partial}{\partial \xi _{2}},\qquad [ G ]_{3\_2} = \frac{\partial}{\partial \xi _{1}}, \qquad [ G ]_{4\_2} = \frac{\partial}{\partial \xi _{2}},\qquad [ G ]_{4\_3} = \frac{1}{R_{2}} \\ & [ G ]_{5\_1} = - \frac{1}{R_{1}},\qquad [ G ]_{5\_3} = \frac{\partial}{\partial \xi _{1}},\qquad [ G ]_{6\_2} = - \frac{1}{R_{2}},\qquad [ G ]_{6\_3} = \frac{\partial}{\partial \xi _{2}},\qquad [ G ]_{7\_4} = \frac{\partial}{\partial \xi _{1}} \\ & [ G ]_{8\_4} = \frac{\partial}{\partial \xi _{2}}, \qquad [ G ]_{9\_5} = \frac{\partial}{\partial \xi _{1}},\qquad [ G ]_{10\_5} = \frac{\partial}{\partial \xi _{2}},\qquad [ G ]_{11\_6} = \frac{\partial}{\partial \xi _{1}},\qquad [ G ]_{12\_6} = \frac{\partial}{\partial \xi _{2}} \\ & [ G ]_{13\_7} = \frac{\partial}{\partial \xi _{1}},\qquad [ G ]_{14\_7} = \frac{\partial}{\partial \xi _{2}},\qquad [ G ]_{15\_8} = \frac{\partial}{\partial \xi _{1}},\qquad [ G ]_{16\_8} = \frac{\partial}{\partial \xi _{2}},\qquad [ G ]_{18\_9} = \frac{\partial}{\partial \xi _{2}} \\ & [ G ]_{19\_1} = 1, \qquad [ G ]_{20\_2} = 1,\qquad [ G ]_{21\_3} = 1,\qquad [ G ]_{22\_4} = 1,\qquad [ G ]_{23\_5} = 1,\qquad [ G ]_{24\_6} = 1 \\ & [ G ]_{25\_7} = 1,\qquad [ G ]_{26\_8} = 1,\qquad [ G ]_{27\_9} = 1 \end{aligned}$$
(A.4)

The evaluation steps of the geometric strain vector \(\{ \bar{\varepsilon} _{G} \}\) and the material property matrix [D G ] as appeared in Eq. (21) are derived as follows:

$$\begin{aligned} \{ \bar{\varepsilon} _{G} \} = \frac{1}{2}\left \{ \begin{array}{@{}l@{}} {[ ( \bar{u}_{,\xi _{1}} )^{2} + ( \bar{v}_{,\xi _{1}} )^{2} + ( \bar{w}_{,\xi _{1}} )^{2} ]} \\ {[ ( \bar{u}_{,\xi _{2}} )^{2} + ( \bar{v}_{,\xi _{2}} )^{2} + ( \bar{w}_{,\xi _{2}} )^{2} ]} \\ 2 [ ( \bar{u}_{,\xi _{1}} ) ( \bar{u}_{,\xi _{2}} ) + ( \bar{v}_{,\xi _{1}} ) ( \bar{v}_{,\xi _{2}} ) \\ \quad + ( \bar{w}_{,\xi _{1}} ) ( \bar{w}_{,\xi _{2}} ) ] \end{array} \right \} \quad \mbox{or,}\quad \{ \bar{\varepsilon} _{G} \} = [ H_{ G } ] \{ \varepsilon _{G} \} = [ H_{ G } ] [ A _{ G } ] \{ \beta _{ G } \} \end{aligned}$$

The values of [A G ] and {β G } are

$$ {[ A _{ G } ]} = \left \{ \begin{array}{@{}l@{}} {[ ( u_{,\xi _{1}} ) + ( v_{,\xi _{1}} ) + ( w_{,\xi _{1}} ) ]} \\ {[ ( u_{,\xi _{2}} ) + ( v_{,\xi _{2}} ) + ( w_{,\xi _{2}} ) ]} \\ {}[ ( u_{,\xi _{1}} ) ( u_{,\xi _{2}} ) + ( v_{,\xi _{1}} ) ( v_{,\xi _{2}} ) \\ \quad + ( w_{,\xi _{1}} ) ( w_{,\xi _{2}} ) ] \end{array} \right \} \quad \mbox{and}\quad \{ \beta _{ G } \} = \left [ \begin{array}{@{}c@{}} u_{,\xi _{1}} \\ u_{,\xi _{2}} \\ v_{,\xi _{1}} \\ v_{,\xi _{2}} \\ w_{,\xi _{1}} \\ w_{,\xi _{2}} \end{array} \right ] $$
(A.5)

The values of material property matrix are obtained by the following procedure:

$$\begin{aligned} [ D_{G} ] = \sum_{k = 1}^{N} \int_{\zeta _{k - 1}}^{\zeta _{k}} [ H_{ G } ]^{\mathrm{T}} \{ \bar{S} \}^{k} [ H_{ G } ] \quad \mbox{where}\ \{ \bar{S} \}^{k} = \left [ \begin{array}{@{}cccccc@{}} N_{\Delta t1} & N_{\Delta t12} & 0 & 0 & 0 & 0 \\ N_{\Delta t12} & N_{\Delta t2} & 0 & 0 & 0 & 0 \\ 0 & 0 & N_{\Delta t1} & N_{\Delta t12} & 0 & 0 \\ 0 & 0 & N_{\Delta t12} & N_{\Delta t2} & 0 & 0 \\ 0 & 0 & 0 & 0 & N_{\Delta t1} & N_{\Delta t12} \\ 0 & 0 & 0 & 0 & N_{\Delta t12} & N_{\Delta t2} \end{array} \right ] \end{aligned}$$
(A.6)

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Panda, S.K., Singh, B.N. Large amplitude free vibration analysis of thermally post-buckled composite doubly curved panel embedded with SMA fibers. Nonlinear Dyn 74, 395–418 (2013). https://doi.org/10.1007/s11071-013-0978-5

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