Asymptotically-Accurate Nonlinear Hyperelastic Shell Constitutive Model Using Variational Asymptotic Method

Abstract

The focus of this work is on the development of asymptotically-accurate nonlinear hyperelastic constitutive model for thin shell structures using Variational Asymptotic Method (VAM). In this work, these structures are analyzed for both geometric and material nonlinearities. The geometric nonlinearity is handled by allowing finite deformations and generalized warping functions through Green strain, while the material nonlinearity is incorporated through strain energy density function of hyperelastic material model. Using the inherent small parameters (moderate strains, very small thickness-to-wavelength ratio and very small thickness-to-initial radius of curvature) for the application of VAM, the process begins with three-dimensional nonlinear hyperelasticity and it weakly decouples the analysis into a one-dimensional through-the-thickness nonlinear analysis and a two-dimensional nonlinear shell analysis. Through-the-thickness analysis is analytical work, providing 3-D warping functions and two-dimensional nonlinear constitutive relation for Nonlinear Finite Element Analysis of shells. Current theory and code are demonstrated through standard test cases and validated with literature.

Appendix

$$\begin{aligned}&\mathscr {L}^1_{11} =\mathscr {L}^2_{22} =\frac{12 h \mu \left( 4 \mu ^3+4 \mu ^2 \lambda -3 \mu \lambda ^2-\lambda ^3\right) }{(\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^2_{11} =\mathscr {L}^1_{22} =\frac{2 h \mu \left( 32 \mu ^3+44 \mu ^2 \lambda -4 \mu \lambda ^2-3 \lambda ^3\right) }{(\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^1_{33} =\mathscr {L}^2_{33} =\mathscr {L}^3_{23} =\mathscr {L}^3_{13} =-\frac{h \mu (3 \lambda +2 \mu )}{2 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^1_{44} =\mathscr {L}^2_{55} =\mathscr {L}^4_{14} =\mathscr {L}^5_{25} =-\frac{h^3 \mu \left( 5 \lambda ^3+15 \mu \lambda ^2-8 \mu ^2 \lambda -12 \mu ^3\right) }{3 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^2_{44} =\mathscr {L}^1_{55} =\mathscr {L}^5_{15} =\mathscr {L}^4_{24} =-\frac{h^3 \mu \left( 7 \lambda ^3+16 \mu \lambda ^2-36 \mu ^2 \lambda -32 \mu ^3\right) }{6 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^1_{66} =\mathscr {L}^2_{66} =\mathscr {L}^6_{16} =\mathscr {L}^6_{26} =-\frac{h^3 \mu (7 \lambda +2 \mu )}{24 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^1_{77} =\mathscr {L}^1_{88} =\mathscr {L}^7_{17} =\mathscr {L}^8_{28} =\frac{2 h \mu (\lambda +\mu )}{3 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^2_{77} =\mathscr {L}^2_{88} =\mathscr {L}^8_{18} =\mathscr {L}^7_{27} =\frac{h \lambda \mu }{3 \lambda +6 \mu } \nonumber \\&\mathscr {L}^5_{14} =\mathscr {L}^4_{15} =\mathscr {L}^5_{24} =\mathscr {L}^4_{25} =\frac{h^3 \mu \left( -5 \lambda ^3-8 \mu \lambda ^2+44 \mu ^2 \lambda +32 \mu ^3\right) }{6 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^6_{34} =\mathscr {L}^6_{35} =\mathscr {L}^4_{36} =\mathscr {L}^5_{36} =\mathscr {L}^3_{46} =\mathscr {L}^1_{56} =-\frac{h^3 \mu (3 \lambda +2 \mu )}{24 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^8_{37} =\mathscr {L}^7_{38} =\mathscr {L}^3_{78} =\frac{{\mathscr {C}}_{88}}{5} \nonumber \\&\mathscr {L}^1_{45} =\mathscr {L}^2_{45} =\frac{h^3 \mu \left( -5 \lambda ^3-8 \mu \lambda ^2+44 \mu ^2 \lambda +32 \mu ^3\right) }{6 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {Q}^{11}_{11} =\mathscr {Q}^{22}_{22} =\frac{6 h \mu \left( 5 \lambda ^5+25 \mu \lambda ^4-72 \mu ^2 \lambda ^3-632 \mu ^3 \lambda ^2-688 \mu ^4 \lambda -240 \mu ^5\right) }{(\lambda +2 \mu )^5}, \nonumber \\&\mathscr {Q}^{12}_{11} =\mathscr {Q}^{12}_{22} =2\mathscr {Q}^{11}_{12} =\mathscr {Q}^{22}_{12} =\frac{24 h \mu \left( \lambda ^5+2 \mu \lambda ^4-54 \mu ^2 \lambda ^3-304 \mu ^3 \lambda ^2-296 \mu ^4 \lambda -96 \mu ^5\right) }{(\lambda +2 \mu )^5}, \nonumber \\&\mathscr {Q}^{77}_{77} =\mathscr {Q}^{77}_{88} =3 \mathscr {Q}^{88}_{77} =3 \mathscr {Q}^{88}_{88} =\frac{3 \mathscr {Q}^{78}_{78}}{2} =\frac{\mathscr {C}_{11}}{4}, \nonumber \\&\mathscr {Q}^{22}_{11} =\mathscr {Q}^{11}_{22} =\frac{\mathscr {Q}^{12}_{12}}{2} =\frac{h \mu \left( 9 \lambda ^5+8 \mu \lambda ^4-632 \mu ^2 \lambda ^3-3456 \mu ^3 \lambda ^2-3248 \mu ^4 \lambda -1024 \mu ^5\right) }{(\lambda +2 \mu )^5}, \nonumber \\&\mathscr {Q}^{33}_{11} =\mathscr {Q}^{33}_{22} =\mathscr {Q}^{11}_{33} =\mathscr {Q}^{22}_{33} =\frac{\mathscr {Q}^{13}_{13}}{2} =\frac{\mathscr {Q}^{23}_{23}}{2} =\frac{3h \mu \left( \lambda ^3+3 \mu \lambda ^2-4 \mu ^2 \lambda -4 \mu ^3\right) }{(\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{44}_{11} =\mathscr {Q}^{55}_{22} =\mathscr {Q}^{11}_{44} =\mathscr {Q}^{22}_{55} =2 \mathscr {Q}^{14}_{14} =\frac{\mathscr {Q}^{25}_{25}}{2} \nonumber \\&\quad =\frac{h^3 \mu \left( 33 \lambda ^5+161 \mu \lambda ^4-244 \mu ^2 \lambda ^3-2352 \mu ^3 \lambda ^2-2528 \mu ^4 \lambda -848 \mu ^5\right) }{6 (\lambda +2 \mu )^5}, \nonumber \\&\mathscr {Q}^{45}_{11} =\mathscr {Q}^{45}_{22} =\mathscr {Q}^{15}_{14} =\mathscr {Q}^{14}_{15} =\mathscr {Q}^{25}_{24} =\mathscr {Q}^{24}_{25} =2\mathscr {Q}^{11}_{45} =\frac{\mathscr {Q}^{22}_{45}}{2} \nonumber \\&\quad =\frac{h^3 \mu \left( 5 \lambda ^5+14 \mu \lambda ^4-144 \mu ^2 \lambda ^3-728 \mu ^3 \lambda ^2-656 \mu ^4 \lambda -192 \mu ^5\right) }{(\lambda +2 \mu )^5}, \nonumber \end{aligned}$$

$$\begin{aligned}&\mathscr {Q}^{55}_{11} =\mathscr {Q}^{44}_{22} =\mathscr {Q}^{22}_{44} =\mathscr {Q}^{11}_{55} =\frac{\mathscr {Q}^{15}_{15}}{2} =\frac{\mathscr {Q}^{24}_{24}}{2} \nonumber \\&\quad =\frac{h^3 \mu \left( 33 \lambda ^5+100 \mu \lambda ^4-880 \mu ^2 \lambda ^3-4560 \mu ^3 \lambda ^2-4240 \mu ^4 \lambda -1280 \mu ^5\right) }{12 (\lambda +2 \mu )^5}, \nonumber \\&\mathscr {Q}^{66}_{11} =\mathscr {Q}^{66}_{22} =\mathscr {Q}^{11}_{66} =\mathscr {Q}^{22}_{66} =\mathscr {Q}^{16}_{16} =2 \mathscr {Q}^{26}_{26} =\frac{h^3 \mu \left( 9 \lambda ^3+21 \mu \lambda ^2-52 \mu ^2 \lambda -44 \mu ^3\right) }{12 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{77}_{11} =\mathscr {Q}^{88}_{22} =\mathscr {Q}^{11}_{77} =\mathscr {Q}^{11}_{88} =\frac{\mathscr {Q}^{17}_{17}}{2} =\frac{\mathscr {Q}^{28}_{28}}{2} =\frac{h \mu \left( 4 \mu ^3-4 \mu ^2 \lambda -19 \mu \lambda ^2-5 \lambda ^3\right) }{3 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{88}_{11} =\mathscr {Q}^{77}_{22} =\mathscr {Q}^{22}_{77} =\mathscr {Q}^{22}_{88} =\mathscr {Q}^{18}_{18} =\frac{\mathscr {Q}^{27}_{27}}{2} =\frac{h \mu \left( 32 \mu ^3+44 \mu ^2 \lambda -4 \mu \lambda ^2-3 \lambda ^3\right) }{6 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{12}_{33} =\mathscr {Q}^{33}_{12} =2\mathscr {Q}^{23}_{13} =\mathscr {Q}^{13}_{23} =\frac{h \mu \left( 9 \lambda ^3+22 \mu \lambda ^2-68 \mu ^2 \lambda -56 \mu ^3\right) }{2 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{44}_{33} =\mathscr {Q}^{55}_{33} =\mathscr {Q}^{33}_{44} =\mathscr {Q}^{33}_{55} =\mathscr {Q}^{34}_{34} =\mathscr {Q}^{35}_{35} =\frac{h^3 (\lambda -\mu ) \mu \left( 5 \lambda ^2+20 \mu \lambda +12 \mu ^2\right) }{12 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{45}_{33} =\mathscr {Q}^{35}_{34} =\mathscr {Q}^{34}_{35} =2\mathscr {Q}^{33}_{45} =\frac{h^3 \mu \left( 13 \lambda ^3+30 \mu \lambda ^2-68 \mu ^2 \lambda -56 \mu ^3\right) }{24 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{66}_{33} =\mathscr {Q}^{33}_{66} =2 \mathscr {Q}^{36}_{36} =\frac{h^3 \mu (13 \lambda +6 \mu )}{96 (\lambda +2 \mu )}, \nonumber \\&\mathscr {Q}^{77}_{33} =\mathscr {Q}^{88}_{33} =\mathscr {Q}^{33}_{77} =\mathscr {Q}^{33}_{88} =\frac{\mathscr {Q}^{37}_{37}}{2} =2 \mathscr {Q}^{38}_{38} =-\frac{h \mu (5 \lambda +6 \mu )}{24 (\lambda +2 \mu )}, \nonumber \\&\mathscr {Q}^{12}_{44} =\mathscr {Q}^{12}_{55} =2\mathscr {Q}^{44}_{12} =\mathscr {Q}^{55}_{12} =\mathscr {Q}^{24}_{14} =\mathscr {Q}^{25}_{15} =\mathscr {Q}^{14}_{24} =\mathscr {Q}^{15}_{25} \nonumber \\&\quad =\frac{h^3 \mu \left( 19 \lambda ^5+64 \mu \lambda ^4-436 \mu ^2 \lambda ^3-2368 \mu ^3 \lambda ^2-2272 \mu ^4 \lambda -704 \mu ^5\right) }{3 (\lambda +2 \mu )^5}, \nonumber \\&\mathscr {Q}^{44}_{44} =\mathscr {Q}^{55}_{55} =\frac{h^5 \mu \left( 89 \lambda ^5+325 \mu \lambda ^4-1616 \mu ^2 \lambda ^3-9160 \mu ^3 \lambda ^2-9744 \mu ^4 \lambda -3312 \mu ^5\right) }{120 (\lambda +2 \mu )^5}, \nonumber \\&\mathscr {Q}^{45}_{44} =\mathscr {Q}^{45}_{55} =2\mathscr {Q}^{44}_{45} =\frac{\mathscr {Q}^{55}_{45}}{2} \nonumber \\&\quad =\frac{h^5 \mu \left( 21 \lambda ^5+29 \mu \lambda ^4-880 \mu ^2 \lambda ^3-3868 \mu ^3 \lambda ^2-3552 \mu ^4 \lambda -1056 \mu ^5\right) }{30 (\lambda +2 \mu )^5}, \nonumber \\&\mathscr {Q}^{66}_{44} =\mathscr {Q}^{66}_{55} =\mathscr {Q}^{44}_{66} =\mathscr {Q}^{55}_{66} =\frac{\mathscr {Q}^{46}_{46}}{2} =\frac{\mathscr {Q}^{56}_{56}}{2} =\frac{h^5 \mu \left( 47 \lambda ^3+79 \mu \lambda ^2-360 \mu ^2 \lambda -300 \mu ^3\right) }{720 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{77}_{44} =\mathscr {Q}^{88}_{55} =\mathscr {Q}^{44}_{77} =\mathscr {Q}^{44}_{88} =\mathscr {Q}^{47}_{47} =2 \mathscr {Q}^{58}_{58} =\frac{h^3 \mu \left( 196 \mu ^3-570 \mu ^2 \lambda -1578 \mu \lambda ^2-433\lambda ^3\right) }{756 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{88}_{44} =\mathscr {Q}^{77}_{55} =\mathscr {Q}^{55}_{77} =\mathscr {Q}^{55}_{88} =\frac{\mathscr {Q}^{48}_{48}}{2} =\frac{\mathscr {Q}^{57}_{57}}{2} =\frac{h ^3 \mu \left( 48 \mu ^3+66 \mu ^2 \lambda -11 \mu \lambda ^2-7 \lambda ^3\right) }{36 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{12}_{66} =2\mathscr {Q}^{66}_{12} =\mathscr {Q}^{16}_{16} =\mathscr {Q}^{16}_{26} =\frac{h^3 \mu \left( 29 \lambda ^3+54 \mu \lambda ^2-244 \mu ^2 \lambda -184 \mu ^3\right) }{24 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{45}_{66} =2 \mathscr {Q}^{66}_{45} =\frac{\mathscr {Q}^{56}_{46}}{2} =\mathscr {Q}^{46}_{56} =\frac{h^5 \mu \left( 137 \lambda ^3+190 \mu \lambda ^2-1332 \mu ^2 \lambda -888 \mu ^3\right) }{1440 (\lambda +2 \mu )^3}, \nonumber \end{aligned}$$

$$\begin{aligned}&\mathscr {Q}^{66}_{66} =\frac{h^5 (17 \lambda -26 \mu ) \mu }{640 (\lambda +2 \mu )}, ~\mathscr {Q}^{77}_{66} =\mathscr {Q}^{88}_{66} =\mathscr {Q}^{66}_{77} =\mathscr {Q}^{66}_{88} =\frac{\mathscr {Q}^{67}_{67}}{2} =\frac{\mathscr {Q}^{68}_{68}}{2} =\frac{-h^3 \mu (47 \lambda +58 \mu )}{864 (\lambda +2 \mu )}, \nonumber \\&\mathscr {Q}^{12}_{77} =\mathscr {Q}^{12}_{88} =2\mathscr {Q}^{77}_{12} =\frac{\mathscr {Q}^{88}_{12}}{2} =\mathscr {Q}^{27}_{17} =\mathscr {Q}^{28}_{18} =\mathscr {Q}^{17}_{27} =\mathscr {Q}^{18}_{28} =\frac{4 h \mu (\lambda +4 \mu ) \left( 2 \mu ^2+2 \mu \lambda -\lambda ^2\right) }{3 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{45}_{77} =\mathscr {Q}^{45}_{88} =2\mathscr {Q}^{77}_{45} =\frac{\mathscr {Q}^{88}_{45}}{2} =\mathscr {Q}^{57}_{47} =\mathscr {Q}^{58}_{48} =\mathscr {Q}^{47}_{57} =\mathscr {Q}^{48}_{58} \nonumber \\&\quad =\frac{h^3 \mu \left( 1344 \mu ^3+1524 \mu ^2 \lambda -632 \mu \lambda ^2-277 \lambda ^3\right) }{504 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{45}_{12} =\mathscr {Q}^{25}_{14} =\mathscr {Q}^{25}_{15} =\mathscr {Q}^{15}_{24} =\mathscr {Q}^{14}_{25} =\mathscr {Q}^{12}_{45} \nonumber \\&\quad =\frac{h^3 \mu \left( 25 \lambda ^5+56 \mu \lambda ^4-872 \mu ^2 \lambda ^3-4192 \mu ^3 \lambda ^2-3632 \mu ^4 \lambda -1024 \mu ^5\right) }{6 (\lambda +2 \mu )^5}, \nonumber \\&\mathscr {Q}^{46}_{13} =\mathscr {Q}^{36}_{14} =\mathscr {Q}^{34}_{16} =\mathscr {Q}^{56}_{23} =\mathscr {Q}^{36}_{25} =\mathscr {Q}^{34}_{26} =\mathscr {Q}^{16}_{34} =\mathscr {Q}^{26}_{35} =\mathscr {Q}^{14}_{36} =\mathscr {Q}^{25}_{36} \nonumber \\&\quad =\mathscr {Q}^{13}_{46} =\mathscr {Q}^{23}_{56} =\frac{h^3 \mu \left( 9 \lambda ^3+26 \mu \lambda ^2-20 \mu ^2 \lambda -24 \mu ^3\right) }{12 (\lambda +2 \mu )^3}, \nonumber \\&\mathscr {Q}^{56}_{13} =\mathscr {Q}^{36}_{15} =\mathscr {Q}^{46}_{23} =\mathscr {Q}^{36}_{24} =\mathscr {Q}^{34}_{26} =\mathscr {Q}^{26}_{34} =\mathscr {Q}^{16}_{35} =\mathscr {Q}^{24}_{36} =\mathscr {Q}^{15}_{36} =\mathscr {Q}^{23}_{46} \nonumber \\&\quad =\mathscr {Q}^{13}_{56} =\frac{h^3 \mu \left( 15 \lambda ^3+38 \mu \lambda ^2-60 \mu ^2 \lambda -56 \mu ^3\right) }{24 (\lambda +2 \mu )^3} \end{aligned}$$

$$\begin{aligned}&{\mathscr {C}}_{11}= k_{11}^{2} \frac{4 h^3 \mu (\lambda +\mu )^2 (2 \lambda +3 \mu )}{3 (\lambda +2 \mu )^3} + k_{22}^{2} \frac{h^3 \lambda ^2 \mu (2 \lambda +3 \mu )}{3 (\lambda +2 \mu )^3} + k_{12}^{2} \frac{h^3 \lambda ^2 \mu }{12 (\lambda +2 \mu )^2} + k_{21}^{2} \frac{h^3 \mu (\lambda +\mu )^2}{3 (\lambda +2 \mu )^2} \nonumber \\&\qquad \quad +k_{12}k_{21} \frac{h^3 \mu \left( 6 \lambda ^2+9 \lambda \mu +4 \mu ^2\right) }{3 (\lambda +2 \mu )^2} + k_{11}k_{22} \frac{2 h^3 \lambda ^2 \mu (\lambda +\mu )}{3 (\lambda +2 \mu )^3} \nonumber \\&{\mathscr {C}}_{22}= k_{11}^{2} \frac{h^3 \lambda ^2 \mu (2 \lambda +3 \mu )}{3 (\lambda +2 \mu )^3} + k_{22}^{2} \frac{4 h^3 \mu (\lambda +\mu )^2 (2 \lambda +3 \mu )}{3 (\lambda +2 \mu )^3} + k_{12}^{2} \frac{h^3 \mu (\lambda +\mu )^2}{3 (\lambda +2 \mu )^2} + k_{21}^{2} \frac{h^3 \lambda ^2 \mu }{12 (\lambda +2 \mu )^2} \end{aligned}$$

$$\begin{aligned}&\qquad \quad +k_{12}k_{21} \frac{h^3 \mu \left( 6 \lambda ^2+9 \lambda \mu +4 \mu ^2\right) }{3 (\lambda +2 \mu )^2} + k_{11}k_{22} \frac{2 h^3 \lambda ^2 \mu (\lambda +\mu )}{3 (\lambda +2 \mu )^3} \nonumber \\&{\mathscr {C}}_{33}= (k_{11}^{2} + k_{22}^{2}) \frac{5 h^3 \mu }{48} + (k_{12}^{2}+ k_{21}^{2}) \frac{h^3 \mu (\lambda +\mu )}{12 (\lambda +2 \mu )} +k_{12}k_{21} \frac{h^3 \mu (3 \lambda +4 \mu )}{12 (\lambda +2 \mu )} + k_{11}k_{22} \frac{h^3 \mu }{24} \end{aligned}$$

$$\begin{aligned}&{\mathscr {C}}_{44}= k_{11}^{2} \frac{h^5 \mu \left( 10 \lambda ^3+38 \lambda ^2 \mu +49 \lambda \mu ^2+21 \mu ^3\right) }{45 (\lambda +2 \mu )^3} + k_{12}^{2} \frac{h^5 \lambda ^2 \mu }{720 (\lambda +2 \mu )^2} + k_{21}^{2} \frac{h^5 \mu \left( 7 \lambda ^2+22 \lambda \mu +18 \mu ^2\right) }{360 (\lambda +2 \mu )^2} \nonumber \\&\qquad \quad +k_{12}k_{21} \frac{h^5 \mu \left( 17 \lambda ^2+40 \lambda \mu +24 \mu ^2\right) }{120 (\lambda +2 \mu )^2} + k_{11}k_{22} \frac{h^5 \lambda \mu \left( 3 \lambda ^2-16 \lambda \mu -24 \mu ^2\right) }{180 (\lambda +2 \mu )^3} + k_{22}^{2} \frac{h^5 \lambda ^2 \mu (2 \lambda -\mu )}{180 (\lambda +2 \mu )^3} \end{aligned}$$

$$\begin{aligned}&{\mathscr {C}}_{55}= k_{22}^{2} \frac{h^5 \mu \left( 10 \lambda ^3+38 \lambda ^2 \mu +49 \lambda \mu ^2+21 \mu ^3\right) }{45 (\lambda +2 \mu )^3} + k_{21}^{2} \frac{h^5 \lambda ^2 \mu }{720 (\lambda +2 \mu )^2} + k_{12}^{2} \frac{h^5 \mu \left( 7 \lambda ^2+22 \lambda \mu +18 \mu ^2\right) }{360 (\lambda +2 \mu )^2} \nonumber \\&\qquad \quad +k_{12}k_{21} \frac{h^5 \mu \left( 17 \lambda ^2+40 \lambda \mu +24 \mu ^2\right) }{120 (\lambda +2 \mu )^2} + k_{11}k_{22} \frac{h^5 \lambda \mu \left( 3 \lambda ^2-16 \lambda \mu -24 \mu ^2\right) }{180 (\lambda +2 \mu )^3} + k_{11}^{2} \frac{h^5 \lambda ^2 \mu (2 \lambda -\mu )}{180 (\lambda +2 \mu )^3} \end{aligned}$$

$$\begin{aligned}&{\mathscr {C}}_{66}= (k_{11}^{2}+k_{22}^{2}) \frac{h^5 \mu }{64} + (k_{12}^{2}+k_{21}^{2}) \frac{h^5 \mu (4 \lambda +\mu )}{240 (\lambda +2 \mu )} + k_{12}k_{21} \frac{h^5 \mu (11 \lambda +8 \mu )}{240 (\lambda +2 \mu )} + k_{11}k_{22} \frac{h^5 \mu }{160} \nonumber \\&{\mathscr {C}}_{77}= k_{11}^{2} \frac{25 h^3 \mu }{216} - k_{12}^{2} \frac{5 h^3 \mu }{1512} + k_{12} k_{21} \frac{5 h^3 \mu }{42} \nonumber \\&{\mathscr {C}}_{88}= k_{22}^{2} \frac{25 h^3 \mu }{216} - k_{21}^{2} \frac{5 h^3 \mu }{1512} + k_{12} k_{21} \frac{5 h^3 \mu }{42} \end{aligned}$$

$$\begin{aligned}&{\mathscr {C}}_{12}= (k_{11}^{2}+k_{22}^{2}) \frac{2 h^3 \lambda \mu \left( 2 \lambda ^2+5 \lambda \mu +3 \mu ^2\right) }{3 (\lambda +2 \mu )^3} + (k_{12}^{2}+k_{21}^{2}) \frac{h^3 \lambda \mu (\lambda +\mu )}{6 (\lambda +2 \mu )^2} \nonumber \\&\qquad \quad + k_{12}k_{21} \frac{h^3 \mu \left( 21 \lambda ^2+24 \lambda \mu +4 \mu ^2\right) }{12 (\lambda +2 \mu )^2} + k_{11}k_{22} \frac{h^3 \lambda \mu \left( 5 \lambda ^2+8 \lambda \mu +4 \mu ^2\right) }{6 (\lambda +2 \mu )^3} \end{aligned}$$

$$\begin{aligned}&{\mathscr {C}}_{13}= (k_{11}k_{21}+k_{22}k_{21}) \frac{h^3 \mu (3 \lambda +2 \mu ) (2 \lambda +3 \mu )}{12 (\lambda +2 \mu )^2} + k_{11}k_{12} \frac{h^3 \mu (3 \lambda +2 \mu ) (5 \lambda +8 \mu )}{24 (\lambda +2 \mu )^2} \nonumber \\&\qquad \quad + k_{22}k_{12} \frac{h^3 \mu (3 \lambda +2 \mu ) (3 \lambda +4 \mu )}{24 (\lambda +2 \mu )^2} \nonumber \\&{\mathscr {C}}_{14}= -k_{11} \frac{h^3 \mu \left( 3 \lambda ^2+7 \lambda \mu +4 \mu ^2\right) }{3 (\lambda +2 \mu )^2} -k_{22} \frac{h^3 \lambda ^2 \mu }{6 (\lambda +2 \mu )^2} \nonumber \\&{\mathscr {C}}_{15}= -k_{11} \frac{h^3 \lambda \mu (\lambda +\mu )}{3 (\lambda +2 \mu )^2} - k_{22} \frac{h^3 \lambda \mu (3 \lambda +4 \mu )}{6 (\lambda +2 \mu )^2} \end{aligned}$$

$$\begin{aligned}&{\mathscr {C}}_{16}={\mathscr {C}}_{26}= - (k_{12}+k_{21}) \frac{h^3 \mu (3 \lambda +2 \mu )}{12 (\lambda +2 \mu )} \nonumber \\&{\mathscr {C}}_{23}= (k_{11}k_{12}+k_{22}k_{12}) \frac{h^3 \mu (3 \lambda +2 \mu ) (2 \lambda +3 \mu )}{12 (\lambda +2 \mu )^2} + k_{11}k_{21} \frac{h^3 \mu (3 \lambda +2 \mu ) (3 \lambda +4 \mu )}{24 (\lambda +2 \mu )^2} \nonumber \\&\qquad \quad +k_{22}k_{21} \frac{h^3 \mu (3 \lambda +2 \mu ) (5 \lambda +8 \mu )}{24 (\lambda +2 \mu )^2} \nonumber \\&{\mathscr {C}}_{24}= - k_{11} \frac{h^3 \lambda \mu (3 \lambda +4 \mu )}{6 (\lambda +2 \mu )^2} - k_{22} \frac{h^3 \lambda \mu (\lambda +\mu )}{3 (\lambda +2 \mu )^2} \end{aligned}$$

$$\begin{aligned}&{\mathscr {C}}_{25}= - k_{11} \frac{h^3 \lambda ^2 \mu }{6 (\lambda +2 \mu )^2} - k_{22} \frac{h^3 \mu \left( 3 \lambda ^2+7 \lambda \mu +4 \mu ^2\right) }{3 (\lambda +2 \mu )^2} \nonumber \\&{\mathscr {C}}_{34}={\mathscr {C}}_{35}= - (k_{12}+k_{21}) \frac{h^3 \mu (\lambda +\mu )}{6 (\lambda +2 \mu )} \nonumber \\&{\mathscr {C}}_{36}= - (k_{11}+k_{22}) \frac{h^3 \mu }{12} \nonumber \\&{\mathscr {C}}_{45}= (k_{11}^2+k_{22}^2) \frac{h^5 \lambda \mu \left( 21 \lambda ^2+46 \lambda \mu +28 \mu ^2\right) }{360 (\lambda +2 \mu )^3} + (k_{12}^2+k_{21}^2) \frac{h^5 \lambda \mu (3 \lambda +4 \mu )}{720 (\lambda +2 \mu )^2} \nonumber \\&\qquad \quad + k_{12}k_{21} \frac{h^5 \mu \left( 25 \lambda ^2+44 \lambda \mu +12 \mu ^2\right) }{240 (\lambda +2 \mu )^2} + k_{11}k_{22} \frac{h^5 \mu \left( 21 \lambda ^3+16 \lambda ^2 \mu -36 \lambda \mu ^2-48 \mu ^3\right) }{360 (\lambda +2 \mu )^3} \end{aligned}$$

$$\begin{aligned}&{\mathscr {C}}_{46}= k_{11}k_{12} \frac{h^5 \mu \left( 51 \lambda ^2+106 \lambda \mu +48 \mu ^2\right) }{720 (\lambda +2 \mu )^2} + k_{11}k_{21} \frac{h^5 \mu \left( 89 \lambda ^2+168 \lambda \mu +60 \mu ^2\right) }{1440 (\lambda +2 \mu )^2} \nonumber \\&\qquad \quad + k_{22}k_{12} \frac{h^5 \mu \left( 25 \lambda ^2+48 \lambda \mu +36 \mu ^2\right) }{720 (\lambda +2 \mu )^2} + k_{22}k_{21} \frac{h^5 \mu \left( 63 \lambda ^2+140 \lambda \mu +108 \mu ^2\right) }{1440 (\lambda +2 \mu )^2} \nonumber \\&{\mathscr {C}}_{56}= k_{11}k_{12} \frac{h^5 \mu \left( 63 \lambda ^2+140 \lambda \mu +108 \mu ^2\right) }{1440 (\lambda +2 \mu )^2} + k_{11}k_{21} \frac{h^5 \mu \left( 25 \lambda ^2+48 \lambda \mu +36 \mu ^2\right) }{720 (\lambda +2 \mu )^2} \nonumber \\&\qquad \quad + k_{22}k_{12} \frac{h^5 \mu \left( 89 \lambda ^2+168 \lambda \mu +60 \mu ^2\right) }{1440 (\lambda +2 \mu )^2} + k_{22}k_{21} \frac{h^5 \mu \left( 51 \lambda ^2+106 \lambda \mu +48 \mu ^2\right) }{720 (\lambda +2 \mu )^2} \nonumber \\&{\mathscr {C}}_{78}= (k_{11}k_{12}+k_{22}k_{21}) \frac{5 h^3 \mu }{84} + (k_{11}k_{21}+k_{22}k_{12}) \frac{85 h^3 \mu }{1512} \end{aligned}$$

(16)

$$\begin{aligned}&\mathscr {L}^1_{11} =\frac{12 h \mu \left( 4 \mu ^3+4 \mu ^2 \lambda -3 \mu \lambda ^2-\lambda ^3\right) }{(\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^4_{11} =k_{11}\frac{h^3 \mu \left( 23 \lambda ^4+93 \lambda ^3 \mu -12 \lambda ^2 \mu ^2-192 \lambda \mu ^3-104 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad +k_{22} \frac{h^3 \lambda \mu \left( 5 \lambda ^3+7 \lambda ^2 \mu -66 \lambda \mu ^2-48 \mu ^3\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{5}_{11}=\mathscr {L}^{1}_{15} =k_{11}\frac{h^3 \mu \left( 9 \lambda ^4+21 \lambda ^3 \mu -98 \lambda ^2 \mu ^2-168 \lambda \mu ^3-64 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad +k_{22}\frac{h^3 \mu \left( 19 \lambda ^4+46 \lambda ^3 \mu -192 \lambda ^2 \mu ^2-336 \lambda \mu ^3-128 \mu ^4\right) }{6 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{6}_{11}=\mathscr {L}^{6}_{22} =(k_{12}+k_{21})\frac{h^3 \mu \left( 17 \lambda ^3+44 \lambda ^2 \mu -56 \lambda \mu ^2-56 \mu ^3\right) }{12 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{4}_{22}=\mathscr {L}^{2}_{24} = k_{11}\frac{h^3 \mu \left( 19 \lambda ^4+46 \lambda ^3 \mu -192 \lambda ^2 \mu ^2-336 \lambda \mu ^3-128 \mu ^4\right) }{6 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad +k_{22}\frac{h^3 \mu \left( 9 \lambda ^4+21 \lambda ^3 \mu -98 \lambda ^2 \mu ^2-168 \lambda \mu ^3-64 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^5_{22} =k_{11} \frac{h^3 \lambda \mu \left( 5 \lambda ^3+7 \lambda ^2 \mu -66 \lambda \mu ^2-48 \mu ^3\right) }{3 (\lambda +2 \mu )^4} +k_{22}\frac{h^3 \mu \left( 23 \lambda ^4+93 \lambda ^3 \mu -12 \lambda ^2 \mu ^2-192 \lambda \mu ^3-104 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^4_{33}= k_{11}\frac{h^3 \mu \left( 5 \lambda ^2+10 \lambda \mu +4 \mu ^2\right) }{12 (\lambda +2 \mu )^2} + k_{22}\frac{h^3 \mu \left( 5 \lambda ^2+8 \lambda \mu +4 \mu ^2\right) }{24 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^5_{33}= k_{11}\frac{h^3 \mu \left( 5 \lambda ^2+8 \lambda \mu +4 \mu ^2\right) }{24 (\lambda +2 \mu )^2} +k_{22}\frac{h^3 \mu \left( 5 \lambda ^2+10 \lambda \mu +4 \mu ^2\right) }{12 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{6}_{33}=\mathscr {L}^{3}_{36}= (k_{12}+k_{21})\frac{h^3 \mu (11 \lambda +6 \mu )}{48 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^{4}_{44}= k_{11}\frac{h^5 \mu \left( 51 \lambda ^4+181 \lambda ^3 \mu -180 \lambda ^2 \mu ^2-760 \lambda \mu ^3-408 \mu ^4\right) }{60 (\lambda +2\mu )^4} + k_{22}\frac{h^5 \lambda \mu \left( 4 \lambda ^3-23 \lambda ^2 \mu -186 \lambda \mu ^2-128 \mu ^3\right) }{60 (\lambda +2 \mu )^4} \nonumber \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{5}_{44}= k_{11}\frac{h^5 \mu \left( 62 \lambda ^4+111 \lambda ^3 \mu -918 \lambda ^2 \mu ^2-1808 \lambda \mu ^3-768 \mu ^4\right) }{180 (\lambda +2\mu )^4} \nonumber \\&\qquad \quad +k_{22}\frac{h^5 \mu \left( 25 \lambda ^4+18 \lambda ^3 \mu -496 \lambda ^2 \mu ^2-784 \lambda \mu ^3-320 \mu ^4\right) }{120 (\lambda +2\mu )^4} \nonumber \\&\mathscr {L}^{6}_{44}= \mathscr {L}^{6}_{55}= (k_{12}+k_{21}) \frac{h^5 \mu \left( 47 \lambda ^3+110 \lambda ^2 \mu -214 \lambda \mu ^2-204 \mu ^3\right) }{360 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{4}_{55}= k_{11}\frac{h^5 \mu \left( 25 \lambda ^4+18 \mu \lambda ^3-496 \mu ^2 \lambda ^2-784 \mu ^3 \lambda -320 \mu ^4\right) }{120 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad + k_{22}\frac{h^5 \mu \left( 62 \lambda ^4+111 \mu \lambda ^3-918 \mu ^2 \lambda ^2-1808 \mu ^3 \lambda -768 \mu ^4\right) }{180 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{5}_{55}= k_{11}\frac{h^5 \lambda \mu \left( 4 \lambda ^3-23 \mu \lambda ^2-186 \mu ^2 \lambda -128 \mu ^3\right) }{60(\lambda +2 \mu )^4} + k_{22}\frac{h^5 \mu \left( 51 \lambda ^4+181 \mu \lambda ^3-180 \mu ^2 \lambda ^2-760 \mu ^3 \lambda -408 \mu ^4\right) }{60 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{4}_{66}= k_{11}\frac{h^5 \mu \left( 11 \lambda ^2+13 \mu \lambda -3 \mu ^2\right) }{180 (\lambda +2\mu )^2} + k_{22}\frac{h^5 \mu \left( 43 \lambda ^2+44 \mu \lambda +36 \mu ^2\right) }{1440 (\lambda +2 \mu )^2} \nonumber \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{5}_{66}= k_{11}\frac{h^5 \mu \left( 43 \lambda ^2+44 \mu \lambda +36 \mu ^2\right) }{1440 (\lambda +2 \mu )^2} + k_{22}\frac{h^5 \mu \left( 11 \lambda ^2+13 \mu \lambda -3 \mu ^2\right) }{180 (\lambda +2\mu )^2} \nonumber \\&\mathscr {L}^{6}_{66}= (k_{12}+k_{21}) \frac{h^5 (3 \lambda -2 \mu ) \mu }{64 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^{4}_{77}= -k_{11} \frac{h^3 \mu \left( 943 \lambda ^2+2614 \mu \lambda +1624 \mu ^2\right) }{1512 (\lambda +2 \mu )^2} -k_{22} \frac{h^3 \lambda ^2 \mu }{36 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{5}_{77}= -k_{11} \frac{h^3 \lambda \mu (137 \lambda +246 \mu )}{504 (\lambda +2 \mu )^2} -k_{22} \frac{h^3 \lambda \mu (2 \lambda +3 \mu )}{18 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{6}_{77}= -k_{12} \frac{h^3 \mu (52 \lambda +41 \mu )}{756 (\lambda +2 \mu )} -k_{21} \frac{h^3 \mu (25 \lambda +36 \mu )}{168 (\lambda +2 \mu )} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{4}_{88}= -k_{11} \frac{h^3 \lambda \mu (2 \lambda +3 \mu )}{18 (\lambda +2 \mu )^2} -k_{22} \frac{h^3 \lambda \mu (137 \lambda +246 \mu )}{504 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{5}_{88}= -k_{11} \frac{h^3 \lambda ^2 \mu }{36 (\lambda +2 \mu )^2} -k_{22} \frac{h^3 \mu \left( 943 \lambda ^2+2614 \mu \lambda +1624 \mu ^2\right) }{1512 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{6}_{88}= -k_{12} \frac{h^3 \mu (25 \lambda +36 \mu )}{168 (\lambda +2 \mu )} -k_{21} \frac{h^3 \mu (52 \lambda +41 \mu )}{756 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^{4}_{12}= -k_{11} \frac{h^3 \mu \left( 13 \lambda ^4+38 \mu \lambda ^3-98 \mu ^2 \lambda ^2-220 \mu ^3 \lambda -96 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad -k_{22} \frac{h^3 \mu \left( 13 \lambda ^4+24 \mu \lambda ^3-168 \mu ^2 \lambda ^2-216 \mu ^3 \lambda -64 \mu ^4\right) }{6 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{5}_{12}= -k_{11} \frac{h^3 \mu \left( 13 \lambda ^4+24 \mu \lambda ^3-168 \mu ^2 \lambda ^2-216 \mu ^3 \lambda -64 \mu ^4\right) }{6 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad -k_{22} \frac{h^3 \mu \left( 13 \lambda ^4+38 \mu \lambda ^3-98 \mu ^2 \lambda ^2-220 \mu ^3 \lambda -96 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{6}_{12}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 29 \lambda ^3+66 \mu \lambda ^2-140 \mu ^2 \lambda -120 \mu ^3\right) }{24 (\lambda +2\mu )^3} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{4}_{13}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 19 \lambda ^3+56 \mu \lambda ^2-36 \mu ^2 \lambda -48 \mu ^3\right) }{24 (\lambda +2\mu )^3} \nonumber \\&\mathscr {L}^{5}_{13}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 8 \lambda ^3+21 \mu \lambda ^2-28 \mu ^2 \lambda -28 \mu ^3\right) }{12 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{6}_{13}= k_{11} \frac{h^3 \mu \left( 11 \lambda ^2+22 \mu \lambda +8 \mu ^2\right) }{24 (\lambda +2 \mu )^2} + k_{22} \frac{h^3 \mu \left( 4 \lambda ^2+7 \mu \lambda +2 \mu ^2\right) }{12 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{1}_{14}= k_{11} \frac{h^3 \mu \left( 23 \lambda ^4+93 \mu \lambda ^3-12 \mu ^2 \lambda ^2-192 \mu ^3 \lambda -104 \mu ^4\right) }{3 (\lambda +2 \mu )^4} + k_{22} \frac{h^3 \lambda \mu \left( 5 \lambda ^3+7 \mu \lambda ^2-66 \mu ^2 \lambda -48 \mu ^3\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{2}_{14}= k_{11} \frac{h^3 \mu \left( 13 \lambda ^4+38 \mu \lambda ^3-98 \mu ^2 \lambda ^2-220 \mu ^3 \lambda -96 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad + k_{22} \frac{h^3 \mu \left( 13 \lambda ^4+24 \mu \lambda ^3-168 \mu ^2 \lambda ^2-216 \mu ^3 \lambda -64 \mu ^4\right) }{6 (\lambda +2 \mu )^4} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{3}_{14}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 19 \lambda ^3+56 \mu \lambda ^2-36 \mu ^2 \lambda -48 \mu ^3\right) }{24 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{2}_{15}= k_{11} \frac{h^3 \mu \left( 13 \lambda ^4+24 \mu \lambda ^3-168 \mu ^2 \lambda ^2-216 \mu ^3 \lambda -64 \mu ^4\right) }{6 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad + K_{22} \frac{h^3 \mu \left( 13 \lambda ^4+38 \mu \lambda ^3-98 \mu ^2 \lambda ^2-220 \mu ^3 \lambda -96 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{3}_{15}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 8 \lambda ^3+21 \mu \lambda ^2-28 \mu ^2 \lambda -28 \mu ^3\right) }{12 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{1}_{16}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 17 \lambda ^3+44 \mu \lambda ^2-56 \mu ^2 \lambda -56 \mu ^3\right) }{12 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{2}_{16}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 29 \lambda ^3+66 \mu \lambda ^2-140 \mu ^2 \lambda -120 \mu ^3\right) }{24 (\lambda +2 \mu )^3} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{3}_{16}= k_{11} \frac{h^3 \mu \left( 11 \lambda ^2+22 \mu \lambda +8 \mu ^2\right) }{24 (\lambda +2 \mu )^2} +k_{22} \frac{h^3 \mu \left( 4 \lambda ^2+7 \mu \lambda +2 \mu ^2\right) }{12 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{4}_{23}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 8 \lambda ^3+21 \mu \lambda ^2-28 \mu ^2 \lambda -28 \mu ^3\right) }{12 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{5}_{23}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 19 \lambda ^3+56 \mu \lambda ^2-36 \mu ^2 \lambda -48 \mu ^3\right) }{24 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{6}_{23}= k_{11} \frac{h^3 \mu \left( 4 \lambda ^2+7 \mu \lambda +2 \mu ^2\right) }{12 (\lambda +2 \mu )^2} + k_{22} \frac{h^3 \mu \left( 11 \lambda ^2+22 \mu \lambda +8 \mu ^2\right) }{24 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{1}_{24}= k_{11} \frac{h^3 \mu \left( 13 \lambda ^4+38 \mu \lambda ^3-98 \mu ^2 \lambda ^2-220 \mu ^3 \lambda -96 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad + k_{22} \frac{h^3 \mu \left( 13 \lambda ^4+24 \mu \lambda ^3-168 \mu ^2 \lambda ^2-216 \mu ^3 \lambda -64 \mu ^4\right) }{6 (\lambda +2 \mu )^4} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{3}_{24}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 8 \lambda ^3+21 \mu \lambda ^2-28 \mu ^2 \lambda -28 \mu ^3\right) }{12 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{1}_{25}= k_{11} \frac{h^3 \mu \left( 13 \lambda ^4+24 \mu \lambda ^3-168 \mu ^2 \lambda ^2-216 \mu ^3 \lambda -64 \mu ^4\right) }{6 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad + k_{22} \frac{h^3 \mu \left( 13 \lambda ^4+38 \mu \lambda ^3-98 \mu ^2 \lambda ^2-220 \mu ^3 \lambda -96 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{2}_{25}= k_{11} \frac{h^3 \lambda \mu \left( 5 \lambda ^3+7 \mu \lambda ^2-66 \mu ^2 \lambda -48 \mu ^3\right) }{3 (\lambda +2 \mu )^4} + k_{22} \frac{h^3 \mu \left( 23 \lambda ^4+93 \mu \lambda ^3-12 \mu ^2 \lambda ^2-192 \mu ^3 \lambda -104 \mu ^4\right) }{3 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{3}_{25}=\mathscr {L}^{1}_{34}=\mathscr {L}^{2}_{35}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 19 \lambda ^3+56 \mu \lambda ^2-36 \mu ^2 \lambda -48 \mu ^3\right) }{24 (\lambda +2\mu )^3} \nonumber \\&\mathscr {L}^{1}_{26}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 29 \lambda ^3+66 \mu \lambda ^2-140 \mu ^2 \lambda -120 \mu ^3\right) }{24 (\lambda +2 \mu )^3} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{2}_{26}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 17 \lambda ^3+44 \mu \lambda ^2-56 \mu ^2 \lambda -56 \mu ^3\right) }{12 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{3}_{26}= k_{11} \frac{h^3 \mu \left( 4 \lambda ^2+7 \mu \lambda +2 \mu ^2\right) }{12 (\lambda +2 \mu )^2} + k_{22} \frac{h^3 \mu \left( 11 \lambda ^2+22 \mu \lambda +8 \mu ^2\right) }{24 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{2}_{34}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 8 \lambda ^3+21 \mu \lambda ^2-28 \mu ^2 \lambda -28 \mu ^3\right) }{12 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{3}_{34}= k_{11}\frac{h^3 \mu \left( 5 \lambda ^2+10 \mu \lambda +4 \mu ^2\right) }{12 (\lambda +2 \mu )^2} + k_{22} \frac{h^3 \mu \left( 5 \lambda ^2+8 \mu \lambda +4 \mu ^2\right) }{24 (\lambda +2 \mu )^2} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{1}_{35}= (k_{12}+k_{21}) \frac{h^3 \mu \left( 8 \lambda ^3+21 \mu \lambda ^2-28 \mu ^2 \lambda -28 \mu ^3\right) }{12 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{3}_{35}= k_{11} \frac{h^3 \mu \left( 5 \lambda ^2+8 \mu \lambda +4 \mu ^2\right) }{24 (\lambda +2 \mu )^2} +k_{22} \frac{h^3 \mu \left( 5 \lambda ^2+10 \mu \lambda +4 \mu ^2\right) }{12 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{1}_{36}= k_{11} \frac{h^3 \mu \left( 11 \lambda ^2+22 \mu \lambda +8 \mu ^2\right) }{24 (\lambda +2 \mu )^2} + k_{22} \frac{h^3 \mu \left( 4 \lambda ^2+7 \mu \lambda +2 \mu ^2\right) }{12 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{2}_{36}= k_{11} \frac{h^3 \mu \left( 4 \lambda ^2+7 \mu \lambda +2 \mu ^2\right) }{12 (\lambda +2 \mu )^2} + k_{22} \frac{h^3 \mu \left( 11 \lambda ^2+22 \mu \lambda +8 \mu ^2\right) }{24 (\lambda +2 \mu )^2} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{4}_{45}= k_{11} \frac{h^5 \mu \left( 62 \lambda ^4+111 \mu \lambda ^3-918 \mu ^2 \lambda ^2-1808 \mu ^3 \lambda -768 \mu ^4\right) }{180 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad + k_{22} \frac{h^5 \mu \left( 25 \lambda ^4+18 \mu \lambda ^3-496 \mu ^2 \lambda ^2-784 \mu ^3 \lambda -320 \mu ^4\right) }{120 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{5}_{45}= k_{11} \frac{h^5 \mu \left( 25 \lambda ^4+18 \mu \lambda ^3-496 \mu ^2 \lambda ^2-784 \mu ^3 \lambda -320 \mu ^4\right) }{120 (\lambda +2 \mu )^4} \nonumber \\&\qquad \quad +k_{22} \frac{h^5 \mu \left( 62 \lambda ^4+111 \mu \lambda ^3-918 \mu ^2 \lambda ^2-1808 \mu ^3 \lambda -768 \mu ^4\right) }{180 (\lambda +2 \mu )^4} \nonumber \\&\mathscr {L}^{6}_{45}=\mathscr {L}^{5}_{46}=\mathscr {L}^{4}_{56}= (k_{12}+k_{21}) \frac{h^5 \mu \left( 143 \lambda ^3+290 \mu \lambda ^2-916 \mu ^2 \lambda -696 \mu ^3\right) }{1440 (\lambda +2 \mu )^3} \nonumber \\&\mathscr {L}^{4}_{46}=\mathscr {L}^{5}_{56}= (k_{12}+k_{21}) \frac{h^5 \mu \left( 47 \lambda ^3+110 \mu \lambda ^2-214 \mu ^2 \lambda -204 \mu ^3\right) }{360 (\lambda +2 \mu )^3} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{6}_{46}= k_{11} \frac{h^5 \mu \left( 11 \lambda ^2+13 \mu \lambda -3 \mu ^2\right) }{180 (\lambda +2 \mu )^2} +k_{22} \frac{h^5 \mu \left( 43 \lambda ^2+44 \mu \lambda +36 \mu ^2\right) }{1440 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{7}_{47}= -k_{11} \frac{h^3 \mu \left( 943 \lambda ^2+2614 \mu \lambda +1624 \mu ^2\right) }{1512 (\lambda +2 \mu )^2} -k_{22} \frac{h^3 \lambda ^2 \mu }{36 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{8}_{47}=\mathscr {L}^{7}_{48}= -k_{12} \frac{h^3 \mu (77 \lambda +72 \mu )}{336 (\lambda +2 \mu )} -k_{21} \frac{41 h^3 \mu (11 \lambda +4 \mu )}{3024 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^{8}_{48}= -k_{11} \frac{h^3 \lambda \mu (2 \lambda +3 \mu )}{18 (\lambda +2 \mu )^2} -k_{22} \frac{h^3 \lambda \mu (137 \lambda +246 \mu )}{504 (\lambda +2 \mu )^2} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{6}_{56}= k_{11} \frac{h^5 \mu \left( 43 \lambda ^2+44 \mu \lambda +36 \mu ^2\right) }{1440 (\lambda +2 \mu )^2} k_{22} \frac{h^5 \mu \left( 11 \lambda ^2+13 \mu \lambda -3 \mu ^2\right) }{180 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{7}_{57}= -k_{11} \frac{h^3 \lambda \mu (137 \lambda +246 \mu )}{504 (\lambda +2 \mu )^2} -k_{22} \frac{h^3 \lambda \mu (2 \lambda +3 \mu )}{18 (\lambda +2 \mu )^2} \nonumber \\&\mathscr {L}^{8}_{57}=\mathscr {L}^{7}_{58}= -k_{12} \frac{41 h^3 \mu (11 \lambda +4 \mu )}{3024 (\lambda +2 \mu )} -k_{21} \frac{h^3 \mu (77 \lambda +72 \mu )}{336 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^{8}_{58}= -k_{11} \frac{h^3 \lambda ^2 \mu }{36 (\lambda +2 \mu )^2} -k_{22} \frac{h^3 \mu \left( 943 \lambda ^2+2614 \mu \lambda +1624 \mu ^2\right) }{1512 (\lambda +2 \mu )^2} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{7}_{67}= -k_{12} \frac{h^3 \mu (52 \lambda +41 \mu )}{756 (\lambda +2 \mu )} -k_{21} \frac{h^3 \mu (25 \lambda +36 \mu )}{168 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^{8}_{67}=\mathscr {L}^{7}_{68}=\mathscr {L}^{6}_{78}= -(k_{11}+k_{22}) \frac{29 h^3 \mu }{432} \nonumber \\&\mathscr {L}^{8}_{68}= -k_{12} \frac{h^3 \mu (25 \lambda +36 \mu )}{168 (\lambda +2 \mu )} -k_{21} \frac{h^3 \mu (52 \lambda +41 \mu )}{756 (\lambda +2 \mu )} \nonumber \\&\mathscr {L}^{4}_{78}= -k_{12} \frac{h^3 \mu (77 \lambda +72 \mu )}{336 (\lambda +2 \mu )} -k_{21} \frac{41 h^3 \mu (11 \lambda +4 \mu )}{3024 (\lambda +2 \mu )} \end{aligned}$$

$$\begin{aligned}&\mathscr {L}^{5}_{78}= -k_{12} \frac{41 h^3 \mu (11 \lambda +4 \mu )}{3024 (\lambda +2 \mu )} -k_{21} \frac{h^3 \mu (77 \lambda +72 \mu )}{336 (\lambda +2 \mu )} \end{aligned}$$