Semi-analytical solutions for functionally graded cubic quasicrystal laminates with mixed boundary conditions

Abstract

Tremendous attention of researchers has been attracted by the unusual properties of quasicrystals. In this paper, the static solution of functionally gradient multilayered cubic quasicrystal plates on an elastic foundation with mixed boundary conditions is presented based on the linear elastic theory of quasicrystals. The quasicrystal material properties are assumed to have an exponent-law variation along the thickness direction. The elastic foundation is taken as the Winkler–Pasternak model, which is utilized to simulate the interaction between the plate and the elastic medium. The multilayered quasicrystal structures with two opposite edges simply supported and simply/clamped/free supported boundary conditions at other edges are considered. The semi-analytical method, which makes use of the state-space method in the z-direction, the one-dimensional differential quadrature method in the x-direction, and the series solution in the y-direction, is adopted to convert the system of governing partial differential equations into the ordinary one. From the propagator matrix, the static solution can be derived by imposing the boundary conditions on the top and bottom surfaces of the multilayered plates. Finally, typical numerical examples are presented to verify the effectiveness of this method and illustrate the influence of different boundary conditions, stacking sequence, foundation parameters, and functionally gradient exponential factors on the phonon and phason variables.

Appendix A

1.1 Some parameters

$$\begin{aligned} & a_{1} = C_{44}^{0} K_{44}^{0} - R_{3}^{0} R_{3}^{0} ,\quad a_{2} = C_{44}^{0} ,\quad a_{3} = R_{3}^{0} ,\quad a_{4} = K_{44}^{0} ,\quad a_{5} = \frac{{C_{11}^{0} }}{{C_{11}^{0} K_{11}^{0} - R_{1}^{0} R_{1}^{0} }},\quad a_{6} = C_{11}^{0} K_{11}^{0} - R_{1}^{0} R_{1}^{0} , \\ & a_{7} = C_{11}^{0} R_{2}^{0} R_{2}^{0} + C_{12}^{0} C_{12}^{0} K_{11}^{0} - 2C_{12}^{0} R_{1}^{0} R_{2}^{0} ,\quad a_{8} = - C_{11}^{0} + \frac{{a_{7} }}{{a_{6} }},\quad a_{9} = - C_{12}^{0} - C_{44}^{0} + \frac{{a_{7} }}{{a_{6} }}, \\ & a_{10} = C_{11}^{0} K_{12}^{0} R_{2}^{0} - C_{12}^{0} R_{1}^{0} K_{12}^{0} + C_{12}^{0} R_{2}^{0} K_{11}^{0} - R_{1}^{0} R_{2}^{0} R_{2}^{0} ,\quad a_{11} = \frac{{a_{10} }}{{a_{6} }} - R_{1}^{0} ,a_{12} = \frac{{a_{10} }}{{a_{6} }} - R_{2}^{0} - R_{3}^{0} , \\ & a_{13} = - C_{12}^{0} K_{11}^{0} + R_{1}^{0} R_{2}^{0} ,a_{14} = - C_{11}^{0} R_{2}^{0} + C_{12}^{0} R_{1}^{0} ,\quad a_{15} = C_{11}^{0} K_{12}^{0} K_{12}^{0} - 2K_{12}^{0} R_{1}^{0} R_{2}^{0} + K_{11}^{0} R_{2}^{0} R_{2}^{0} , \\ & a_{16} = - K_{11}^{0} + \frac{{a_{15} }}{{a_{6} }},\quad a_{17} = - K_{12}^{0} - K_{44}^{0} + \frac{{a_{15} }}{{a_{6} }},\quad a_{18} = - K_{11}^{0} R_{2}^{0} + K_{12}^{0} R_{1}^{0} ,\quad a_{19} = - C_{11}^{0} K_{12}^{0} + R_{1}^{0} R_{2}^{0} , \\ & a_{20} = \frac{{K_{11}^{0} }}{{a_{6} }},\quad a_{21} = - \frac{{R_{1}^{0} }}{{a_{6} }},\quad b_{1} = a_{8} \frac{{\partial^{2} }}{{\partial x^{2} }} - a_{2} \frac{{\partial^{2} }}{{\partial y^{2} }},\quad b_{2} = a_{16} \frac{{\partial^{2} }}{{\partial x^{2} }} - a_{4} \frac{{\partial^{2} }}{{\partial y^{2} }}. \\ \end{aligned}$$

(A.1)

1.2 State equations for SSSS

$$\begin{aligned} \frac{{{\text{d}}\tilde{u}_{{xr}} }}{{{\text{d}}z}} & = \frac{{a_{4} }}{{a_{1} }}\tilde{\sigma }_{{xzr}} - \frac{{a_{3} }}{{a_{1} }}\tilde{H}_{{xzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{u}_{{yr}} }}{{{\text{d}}z}} & = \frac{{a_{4} }}{{a_{1} }}\tilde{\sigma }_{{yzr}} {\text{ + }}\frac{{a_{4} }}{{a_{1} }}\tilde{H}_{{yzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)q\tilde{u}_{{zr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{{xr}} }}{{{\text{d}}z}} & = - \frac{{a_{3} }}{{a_{1} }}\tilde{\sigma }_{{xzr}} + \frac{{a_{2} }}{{a_{1} }}\tilde{H}_{{xzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{{yr}} }}{{{\text{d}}z}} & = - \frac{{a_{3} }}{{a_{1} }}\tilde{\sigma }_{{yzr}} + \frac{{a_{2} }}{{a_{1} }}\tilde{H}_{{yzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)q\tilde{w}_{{zr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{zzr}} }}{{{\text{d}}z}} & = - \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{\sigma }_{{xzr}} } + q\tilde{\sigma }_{{yzr}} - a_{2} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{u}_{{zk}} } - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{w}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{H}_{{zzr}} }}{{{\text{d}}z}} & = - \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{H}_{{xzr}} } + q\tilde{H}_{{yzr}} - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{u}_{{zk}} } - a_{4} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{w}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{xzr}} }}{{{\text{d}}z}} & = a_{8} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{xk}} } + a_{2} q^{2} \tilde{u}_{{xr}} - \frac{{a_{7} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{u}_{{xk}} } - a_{9} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{yk}} } + a_{{11}} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{xk}} } + a_{3} q^{2} \tilde{w}_{{xr}} \\ & \quad - \frac{{a_{{10}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{w}_{{xk}} } - a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{yk}} } + \frac{{a_{{13}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{\sigma }_{{zzk}} } + \frac{{a_{{14}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{H}_{{zzk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{yzr}} }}{{{\text{d}}z}} & = a_{9} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - a_{2} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{yk}} } - a_{8} q^{2} \tilde{u}_{{yr}} + a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } \\ & \quad - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{yk}} } - a_{{11}} q^{2} \tilde{w}_{{yr}} + \frac{{a_{{13}} }}{{a_{6} }}q\tilde{\sigma }_{{zzr}} + \frac{{a_{{14}} }}{{a_{6} }}q\tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{H}_{{xzr}} }}{{{\text{d}}z}} & = a_{{11}} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{xk}} } + a_{3} q^{2} \tilde{u}_{{xr}} - \frac{{a_{{10}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{u}_{{xk}} } - a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{yk}} } + a_{{16}} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{xk}} } + a_{4} q^{2} \tilde{w}_{{xr}} \\ & \quad - \frac{{a_{{15}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{w}_{{xk}} } - a_{{17}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{yk}} } + \frac{{a_{{18}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{\sigma }_{{zzk}} } + \frac{{a_{{19}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{H}_{{zzk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{H}_{{yzr}} }}{{{\text{d}}z}} & = a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{yk}} } - a_{{11}} q^{2} \tilde{u}_{{yr}} + a_{{17}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } - a_{4} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{yk}} } \\ & \quad - a_{{16}} q^{2} \tilde{w}_{{yr}} + \frac{{a_{{18}} }}{{a_{6} }}q\tilde{\sigma }_{{zzr}} + \frac{{a_{{19}} }}{{a_{6} }}q\tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{u}_{{zr}} }}{{{\text{d}}z}} & = \frac{{a_{{13}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - \frac{{a_{{13}} }}{{a_{6} }}q\tilde{u}_{{yr}} + \frac{{a_{{18}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } - \frac{{a_{{18}} }}{{a_{6} }}q\tilde{w}_{{yr}} + a_{{20}} \tilde{\sigma }_{{zzr}} + a_{{21}} \tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{{zr}} }}{{{\text{d}}z}} & = \frac{{a_{{14}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - \frac{{a_{{14}} }}{{a_{6} }}q\tilde{u}_{{yr}} + \frac{{a_{{19}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } - \frac{{a_{{19}} }}{{a_{6} }}q\tilde{w}_{{yr}} + a_{{21}} \tilde{\sigma }_{{zzr}} + a_{5} \tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right). \\ \end{aligned}$$

(A.2)

1.3 State equations for CSCS

$$\begin{aligned} \frac{{{\text{d}}\tilde{u}_{{xr}} }}{{{\text{d}}z}} & = \frac{{a_{4} }}{{a_{1} }}\tilde{\sigma }_{{xzr}} - \frac{{a_{3} }}{{a_{1} }}\tilde{H}_{{xzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{u}_{{yr}} }}{{{\text{d}}z}} & = \frac{{a_{4} }}{{a_{1} }}\tilde{\sigma }_{{yzr}} {\text{ + }}\frac{{a_{4} }}{{a_{1} }}\tilde{H}_{{yzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)q\tilde{u}_{{zr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{{xr}} }}{{{\text{d}}z}} & = - \frac{{a_{3} }}{{a_{1} }}\tilde{\sigma }_{{xzr}} + \frac{{a_{2} }}{{a_{1} }}\tilde{H}_{{xzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{{yr}} }}{{{\text{d}}z}} & = - \frac{{a_{3} }}{{a_{1} }}\tilde{\sigma }_{{yzr}} + \frac{{a_{2} }}{{a_{1} }}\tilde{H}_{{yzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)q\tilde{w}_{{zr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{zzr}} }}{{{\text{d}}z}} & = - \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{\sigma }_{{xzr}} } + q\tilde{\sigma }_{{yzr}} - a_{2} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{u}_{{zk}} } - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{w}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{H}_{{zzr}} }}{{{\text{d}}z}} & = - \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{H}_{{xzr}} } + q\tilde{H}_{{yzr}} - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{u}_{{zk}} } - a_{4} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{w}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{xzr}} }}{{{\text{d}}z}} & = a_{8} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{xk}} } + a_{2} q^{2} \tilde{u}_{{xr}} - \frac{{a_{7} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{u}_{{xk}} } - a_{9} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{yk}} } + a_{{11}} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{xk}} } + a_{3} q^{2} \tilde{w}_{{xr}} \\ & \quad - \frac{{a_{{10}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{w}_{{xk}} } - a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{yk}} } + \frac{{a_{{13}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{\sigma }_{{zzk}} } + \frac{{a_{{14}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{H}_{{zzk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{yzr}} }}{{{\text{d}}z}} & = a_{9} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - a_{2} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{yk}} } - a_{8} q^{2} \tilde{u}_{{yr}} + a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } \\ & \quad - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{yk}} } - a_{{11}} q^{2} \tilde{w}_{{yr}} + \frac{{a_{{13}} }}{{a_{6} }}q\tilde{\sigma }_{{zzr}} + \frac{{a_{{14}} }}{{a_{6} }}q\tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{H}_{{xzr}} }}{{{\text{d}}z}} & = a_{{11}} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{xk}} } + a_{3} q^{2} \tilde{u}_{{xr}} - \frac{{a_{{10}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{u}_{{xk}} } - a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{yk}} } + a_{{16}} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{xk}} } + a_{4} q^{2} \tilde{w}_{{xr}} \\ & \quad - \frac{{a_{{15}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {f_{{rk}}^{{}} \tilde{w}_{{xk}} } - a_{{17}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{yk}} } + \frac{{a_{{18}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{\sigma }_{{zzk}} } + \frac{{a_{{19}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{H}_{{zzk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{H}_{{yzr}} }}{{{\text{d}}z}} & = a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{yk}} } - a_{{11}} q^{2} \tilde{u}_{{yr}} + a_{{17}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } - a_{4} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{yk}} } \\ & \quad - a_{{16}} q^{2} \tilde{w}_{{yr}} + \frac{{a_{{18}} }}{{a_{6} }}q\tilde{\sigma }_{{zzr}} + \frac{{a_{{19}} }}{{a_{6} }}q\tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{u}_{{zr}} }}{{{\text{d}}z}} & = \frac{{a_{{13}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - \frac{{a_{{13}} }}{{a_{6} }}q\tilde{u}_{{yr}} + \frac{{a_{{18}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } - \frac{{a_{{18}} }}{{a_{6} }}q\tilde{w}_{{yr}} + a_{{20}} \tilde{\sigma }_{{zzr}} + a_{{21}} \tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{{zr}} }}{{{\text{d}}z}} & = \frac{{a_{{14}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - \frac{{a_{{14}} }}{{a_{6} }}q\tilde{u}_{{yr}} + \frac{{a_{{19}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } - \frac{{a_{{19}} }}{{a_{6} }}q\tilde{w}_{{yr}} + a_{{21}} \tilde{\sigma }_{{zzr}} + a_{5} \tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right). \\ \end{aligned}$$

(A.3)

1.4 State equations for CSSS

$$\begin{aligned} \frac{{{\text{d}}\tilde{u}_{{xr}} }}{{{\text{d}}z}} & = \frac{{a_{4} }}{{a_{1} }}\tilde{\sigma }_{{xzr}} - \frac{{a_{3} }}{{a_{1} }}\tilde{H}_{{xzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{zk}} } \left( {2 \le r \le N} \right), \\ \frac{{{\text{d}}\tilde{u}_{{yr}} }}{{{\text{d}}z}} & = \frac{{a_{4} }}{{a_{1} }}\tilde{\sigma }_{{yzr}} {\text{ + }}\frac{{a_{4} }}{{a_{1} }}\tilde{H}_{{yzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)q\tilde{u}_{{zr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{{xr}} }}{{{\text{d}}z}} & = - \frac{{a_{3} }}{{a_{1} }}\tilde{\sigma }_{{xzr}} + \frac{{a_{2} }}{{a_{1} }}\tilde{H}_{{xzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{zk}} } \left( {2 \le r \le N} \right), \\ \frac{{{\text{d}}\tilde{w}_{{yr}} }}{{{\text{d}}z}} & = - \frac{{a_{3} }}{{a_{1} }}\tilde{\sigma }_{{yzr}} + \frac{{a_{2} }}{{a_{1} }}\tilde{H}_{{yzr}} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)q\tilde{w}_{{zr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{zzr}} }}{{{\text{d}}z}} & = - \sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{\sigma }_{{xzr}} } + q\tilde{\sigma }_{{yzr}} - a_{2} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{1rk}}^{{}} \tilde{u}_{{zk}} } - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{1rk}}^{{}} \tilde{w}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{H}_{{zzr}} }}{{{\text{d}}z}} & = - \sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{H}_{{xzr}} } + q\tilde{H}_{{yzr}} - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{1rk}}^{{}} \tilde{u}_{{zk}} } - a_{4} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{1rk}}^{{}} \tilde{w}_{{zk}} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{xzr}} }}{{{\text{d}}z}} & = a_{8} \sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(2)}} \tilde{u}_{{xk}} } + a_{2} q^{2} \tilde{u}_{{xr}} - \frac{{a_{7} }}{{a_{6} }}\sum\limits_{{k = 2}}^{N} {f_{{1rk}} \tilde{u}_{{xk}} } - a_{8} \sum\limits_{{k = 2}}^{N} {f_{{Nrk}} \tilde{u}_{{xk}} } - a_{9} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{yk}} } \\ & \quad + a_{{11}} \sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(2)}} \tilde{w}_{{xk}} } + a_{3} q^{2} \tilde{w}_{{xr}} - \frac{{a_{{10}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{N} {f_{{1rk}} \tilde{w}_{{xk}} } - a_{{11}} \sum\limits_{{k = 2}}^{N} {f_{{Nrk}} \tilde{w}_{{xk}} } - a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{yk}} } \\ & \quad + \frac{{a_{{13}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{\sigma }_{{zzk}} } + \frac{{a_{{14}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{H}_{{zzk}} } \left( {2 \le r \le N} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{yzr}} }}{{{\text{d}}z}} & = a_{9} q\sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - a_{2} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{yk}} } - a_{8} q^{2} \tilde{u}_{{yr}} + a_{{12}} q\sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } \\ & \quad - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{yk}} } - a_{{11}} q^{2} \tilde{w}_{{yr}} + \frac{{a_{{13}} }}{{a_{6} }}q\tilde{\sigma }_{{zzr}} + \frac{{a_{{14}} }}{{a_{6} }}q\tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{H}_{{xzr}} }}{{{\text{d}}z}} & = a_{{11}} \sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(2)}} \tilde{u}_{{xk}} } + a_{3} q^{2} \tilde{u}_{{xr}} - \frac{{a_{{10}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{N} {f_{{1rk}} \tilde{u}_{{xk}} } - a_{{11}} \sum\limits_{{k = 2}}^{N} {f_{{Nrk}} \tilde{u}_{{xk}} } - a_{{12}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{u}_{{yk}} } + a_{{16}} \sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(2)}} \tilde{w}_{{xk}} } \\ & \quad + a_{4} q^{2} \tilde{w}_{{xr}} - \frac{{a_{{15}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{N} {f_{{1rk}} \tilde{w}_{{xk}} } - a_{{16}} \sum\limits_{{k = 2}}^{N} {f_{{Nrk}} \tilde{w}_{{xk}} } - a_{{17}} q\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{w}_{{yk}} } + \frac{{a_{{18}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{\sigma }_{{zzk}} } \\ & \quad + \frac{{a_{{19}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(1)}} \tilde{H}_{{zzk}} } \left( {2 \le r \le N} \right), \\ \frac{{{\text{d}}\tilde{H}_{{yzr}} }}{{{\text{d}}z}} & = a_{{12}} q\sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - a_{3} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{u}_{{yk}} } - a_{{11}} q^{2} \tilde{u}_{{yr}} + a_{{17}} q\sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } - a_{4} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{rk}}^{{(2)}} \tilde{w}_{{yk}} } \\ & \quad - a_{{16}} q^{2} \tilde{w}_{{yr}} + \frac{{a_{{18}} }}{{a_{6} }}q\tilde{\sigma }_{{zzr}} + \frac{{a_{{19}} }}{{a_{6} }}q\tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{u}_{{zr}} }}{{{\text{d}}z}} & = \frac{{a_{{13}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - \frac{{a_{{13}} }}{{a_{6} }}q\tilde{u}_{{yr}} + \frac{{a_{{18}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } - \frac{{a_{{18}} }}{{a_{6} }}q\tilde{w}_{{yr}} + a_{{20}} \tilde{\sigma }_{{zzr}} + a_{{21}} \tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{{zr}} }}{{{\text{d}}z}} & = \frac{{a_{{14}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{u}_{{xk}} } - \frac{{a_{{14}} }}{{a_{6} }}q\tilde{u}_{{yr}} + \frac{{a_{{19}} }}{{a_{6} }}\sum\limits_{{k = 2}}^{N} {X_{{rk}}^{{(1)}} \tilde{w}_{{xk}} } - \frac{{a_{{19}} }}{{a_{6} }}q\tilde{w}_{{yr}} + a_{{21}} \tilde{\sigma }_{{zzr}} + a_{5} \tilde{H}_{{zzr}} \left( {2 \le r \le N - 1} \right), \\ \end{aligned}$$

(A.4)

with $f_{1rk} = X_{r1}^{(1)} X_{1k}^{(1)} , \, f_{Nrk} = X_{rN}^{(1)} X_{Nr}^{(1)} , \, f_{rk} = f_{1rk} + f_{Nrk} .$

1.5 State equations for CSFS

$$\begin{aligned} \frac{{{\text{d}}\tilde{u}_{xr} }}{{{\text{d}}z}} & = \frac{{a_{4} }}{{a_{1} }}\tilde{\sigma }_{xzr} - \frac{{a_{3} }}{{a_{1} }}\tilde{H}_{xzr} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)\sum\limits_{k = 2}^{N} {X_{rk}^{(1)} \tilde{u}_{zk} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{u}_{yr} }}{{{\text{d}}z}} & = \frac{{a_{4} }}{{a_{1} }}\tilde{\sigma }_{yzr} { + }\frac{{a_{4} }}{{a_{1} }}\tilde{H}_{yzr} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)q\tilde{u}_{zr} \left( {2 \le r \le N} \right), \\ \frac{{{\text{d}}\tilde{w}_{xr} }}{{{\text{d}}z}} & = - \frac{{a_{3} }}{{a_{1} }}\tilde{\sigma }_{xzr} + \frac{{a_{2} }}{{a_{1} }}\tilde{H}_{xzr} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)\sum\limits_{k = 2}^{N} {X_{rk}^{(1)} \tilde{w}_{zk} } \left( {2 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{yr} }}{{{\text{d}}z}} & = - \frac{{a_{3} }}{{a_{1} }}\tilde{\sigma }_{yzr} + \frac{{a_{2} }}{{a_{1} }}\tilde{H}_{yzr} + \left( {\frac{{a_{3}^{2} - a_{2} a_{4} }}{{a_{1} }}} \right)q\tilde{w}_{zr} \left( {2 \le r \le N} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{zzr} }}{{{\text{d}}z}} & = - \sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{\sigma }_{xzr} } + q{\mathbf{E}}_{1} \tilde{\sigma }_{yzr} - a_{2} \sum\limits_{k = 2}^{N} {f_{1rk}^{{}} \tilde{u}_{zk} } - a_{3} \sum\limits_{k = 2}^{N} {f_{1rk}^{{}} \tilde{w}_{zk} } \left( {1 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{H}_{zzr} }}{{{\text{d}}z}} & = - \sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{H}_{xzr} } + q{\mathbf{E}}_{1} \tilde{H}_{yzr} - a_{3} \sum\limits_{k = 2}^{N} {f_{1rk}^{{}} \tilde{u}_{zk} } - a_{4} \sum\limits_{k = 2}^{N} {f_{1rk}^{{}} \tilde{w}_{zk} } \left( {1 \le r \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{xzr} }}{{{\text{d}}z}} & = a_{8} \sum\limits_{k = 2}^{N - 1} {X_{rk}^{(2)} \tilde{u}_{xk} } + a_{2} q^{2} \tilde{u}_{xr} - \frac{{a_{7} }}{{a_{6} }}\sum\limits_{k = 2}^{N - 1} {f_{1rk} \tilde{u}_{xk} } - \left( {\frac{{a_{13} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{4} }}{{a_{6} c_{1} }}} \right)\sum\limits_{k = 2}^{N - 1} {f_{Nrk} \tilde{u}_{xk} } - a_{9} q\sum\limits_{k = 2}^{N} {X_{rk}^{(1)} \tilde{u}_{yk} } \\ & \quad + \left( {\frac{{a_{13} c_{3} }}{{a_{6} }} + \frac{{a_{14} c_{5} }}{{a_{6} }}} \right)q{\mathbf{E}}_{6} \tilde{u}_{yr} - \frac{{a_{8} }}{q}\sum\limits_{k = 2}^{N} {F_{Nrk} \tilde{u}_{yk} } + \frac{{a_{7} }}{{a_{6} q}}\sum\limits_{k = 2}^{N} {\overline{F}_{Nrk} \tilde{u}_{yk} } + \left( {\frac{{a_{13} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{4} }}{{a_{6} c_{1} }}} \right)\frac{1}{q}\sum\limits_{k = 2}^{N} {\overline{\overline{F}}_{Nrk} \tilde{u}_{yk} } \\ & \quad + a_{11} \sum\limits_{k = 2}^{N - 1} {X_{rk}^{(2)} \tilde{w}_{xk} } + a_{3} q^{2} \tilde{w}_{xr} - \frac{{a_{10} }}{{a_{6} }}\sum\limits_{k = 2}^{N - 1} {f_{1rk} \tilde{w}_{xk} } - \left( {\frac{{a_{13} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{6} }}{{a_{6} c_{1} }}} \right)\sum\limits_{k = 2}^{N - 1} {f_{Nrk} \tilde{w}_{xk} } - a_{12} q\sum\limits_{k = 2}^{N} {X_{rk}^{(1)} \tilde{w}_{yk} } \\ & \quad + \left( {\frac{{a_{13} c_{5} }}{{a_{6} }} + \frac{{a_{14} c_{7} }}{{a_{6} }}} \right)q{\mathbf{E}}_{6} \tilde{w}_{yr} - \frac{{a_{11} }}{q}\sum\limits_{k = 2}^{N} {F_{Nrk} \tilde{w}_{yk} } + \frac{{a_{10} }}{{a_{6} q}}\sum\limits_{k = 2}^{N} {\overline{F}_{Nrk} \tilde{w}_{yk} } + \left( {\frac{{a_{13} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{6} }}{{a_{6} c_{1} }}} \right)\frac{1}{q}\sum\limits_{k = 2}^{N} {\overline{\overline{F}}_{Nrk} \tilde{w}_{yk} } \\ & \quad + \frac{{a_{13} }}{{a_{6} }}\sum\limits_{k = 2}^{N} {X_{rk}^{(1)} \tilde{\sigma }_{zzk} } + \frac{{a_{14} }}{{a_{6} }}\sum\limits_{k = 2}^{N} {X_{rk}^{(1)} \tilde{H}_{zzk} } \left( {2 \le r \le N - 1} \right), \\ \end{aligned}$$

(A5)

$$\begin{aligned} \frac{{{\text{d}}\tilde{\sigma }_{yzr} }}{{{\text{d}}z}} & = a_{9} q\sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{u}_{xk} } - \left( {\frac{{a_{13} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{4} }}{{a_{6} c_{1} }}} \right)q{\mathbf{E}}_{4} \tilde{u}_{xr} - a_{2} \sum\limits_{k = 2}^{N} {X_{rk}^{(2)} \tilde{u}_{yk} } - a_{8} q^{2} \tilde{u}_{yr} + (\frac{{a_{13} c_{3} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{5} }}{{a_{6} c_{1} }})q^{2} {\mathbf{E}}_{3} \tilde{u}_{yr} \\ & \quad - a_{9} \sum\limits_{k = 2}^{N} {f_{Nrk} \tilde{u}_{yk} } + \left( {\frac{{a_{13} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{4} }}{{a_{6} c_{1} }}} \right){\mathbf{E}}_{5} \tilde{u}_{yr} + a_{12} q\sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{w}_{xk} } - \left( {\frac{{a_{13} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{6} }}{{a_{6} c_{1} }}} \right)q{\mathbf{E}}_{4} \tilde{w}_{xr} \\ & \quad - a_{3} \sum\limits_{k = 2}^{N} {X_{rk}^{(2)} \tilde{w}_{yk} } - a_{11} q^{2} \tilde{w}_{yr} + \left( {\frac{{a_{13} c_{5} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{7} }}{{a_{6} c_{1} }}} \right)q^{2} {\mathbf{E}}_{3} \tilde{w}_{yr} - a_{12} \sum\limits_{k = 2}^{N} {f_{Nrk} \tilde{w}_{yk} } + \left( {\frac{{a_{13} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{14} c_{6} }}{{a_{6} c_{1} }}} \right){\mathbf{E}}_{5} \tilde{w}_{yr} \\ & \quad + \frac{{a_{13} }}{{a_{6} }}q{\mathbf{E}}_{2} \tilde{\sigma }_{zzr} + \frac{{a_{14} }}{{a_{6} }}q{\mathbf{E}}_{2} \tilde{H}_{zzr} \left( {2 \le r \le N} \right), \\ \frac{{{\text{d}}\tilde{H}_{xzr} }}{{{\text{d}}z}} & = a_{11} \sum\limits_{k = 2}^{N - 1} {X_{rk}^{(2)} \tilde{u}_{xk} } + a_{3} q^{2} \tilde{u}_{xr} - \frac{{a_{10} }}{{a_{6} }}\sum\limits_{k = 2}^{N - 1} {f_{1rk} \tilde{u}_{xk} } - \left( {\frac{{a_{18} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{4} }}{{a_{6} c_{1} }}} \right)\sum\limits_{k = 2}^{N - 1} {f_{Nrk} \tilde{u}_{xk} } - a_{12} q\sum\limits_{k = 2}^{N} {X_{rk}^{(1)} \tilde{u}_{yk} } \\ & \quad + \left( {\frac{{a_{18} c_{3} }}{{a_{6} }} + \frac{{a_{19} c_{5} }}{{a_{6} }}} \right)q{\mathbf{E}}_{6} \tilde{u}_{yr} - \frac{{a_{11} }}{q}\sum\limits_{k = 2}^{N} {F_{Nrk} \tilde{u}_{yk} } + \frac{{a_{10} }}{{a_{6} q}}\sum\limits_{k = 2}^{N} {\overline{F}_{Nrk} \tilde{u}_{yk} } + \left( {\frac{{a_{18} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{4} }}{{a_{6} c_{1} }}} \right)\frac{1}{q}\sum\limits_{k = 2}^{N} {\overline{\overline{F}}_{Nrk} \tilde{u}_{yk} } \\ \, & \quad \, + a_{16} \sum\limits_{k = 2}^{N - 1} {X_{rk}^{(2)} \tilde{w}_{xk} } + a_{4} q^{2} \tilde{w}_{xr} - \frac{{a_{15} }}{{a_{6} }}\sum\limits_{k = 2}^{N - 1} {f_{1rk} \tilde{w}_{xk} } - \left( {\frac{{a_{18} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{6} }}{{a_{6} c_{1} }}} \right)\sum\limits_{k = 2}^{N - 1} {f_{Nrk} \tilde{w}_{xk} } - a_{17} q\sum\limits_{k = 2}^{N} {X_{rk}^{(1)} \tilde{w}_{yk} } \\ \, & \quad \, + \left( {\frac{{a_{18} c_{5} }}{{a_{6} }} + \frac{{a_{19} c_{7} }}{{a_{6} }}} \right)q{\mathbf{E}}_{6} \tilde{w}_{yr} - \frac{{a_{16} }}{q}\sum\limits_{k = 2}^{N} {F_{Nrk} \tilde{w}_{yk} } + \frac{{a_{15} }}{{a_{6} q}}\sum\limits_{k = 2}^{N} {\overline{F}_{Nrk} \tilde{w}_{yk} } + \left( {\frac{{a_{18} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{6} }}{{a_{6} c_{1} }}} \right)\frac{1}{q}\sum\limits_{k = 2}^{N} {\overline{\overline{F}}_{Nrk} \tilde{w}_{yk} } \\ \, & \quad \, + \frac{{a_{18} }}{{a_{6} }}\sum\limits_{k = 1}^{N - 1} {X_{rk}^{(1)} \tilde{\sigma }_{zzk} } + \frac{{a_{19} }}{{a_{6} }}\sum\limits_{k = 1}^{N - 1} {X_{rk}^{(1)} \tilde{H}_{zzk} } \left( {2 \le r \le N - 1} \right), \\ \end{aligned}$$

$$\begin{aligned} \frac{{{\text{d}}\tilde{H}_{yzr} }}{{{\text{d}}z}} & = a_{12} q\sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{u}_{xk} } - \left( {\frac{{a_{18} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{4} }}{{a_{6} c_{1} }}} \right)q{\mathbf{E}}_{4} \tilde{u}_{xr} - a_{3} \sum\limits_{k = 2}^{N} {X_{rk}^{(2)} \tilde{u}_{yk} } - a_{11} q^{2} \tilde{u}_{yr} + \left( {\frac{{a_{18} c_{3} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{5} }}{{a_{6} c_{1} }}} \right)q^{2} {\mathbf{E}}_{3} \tilde{u}_{yr} \\ \, & \quad - a_{12} \sum\limits_{k = 2}^{N} {f_{Nrk} \tilde{u}_{yk} } + \left( {\frac{{a_{18} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{4} }}{{a_{6} c_{1} }}} \right){\mathbf{E}}_{5} \tilde{u}_{yr} + a_{17} q\sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{w}_{xk} } - \left( {\frac{{a_{18} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{6} }}{{a_{6} c_{1} }})} \right)q{\mathbf{E}}_{4} \tilde{w}_{xr} \\ \, & \quad \, - a_{4} \sum\limits_{k = 2}^{N} {X_{rk}^{(2)} \tilde{w}_{yk} } - a_{16} q^{2} \tilde{w}_{yr} + \left( {\frac{{a_{18} c_{5} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{7} }}{{a_{6} c_{1} }}} \right)q^{2} {\mathbf{E}}_{3} \tilde{w}_{yr} - a_{17} \sum\limits_{k = 2}^{N} {f_{Nrk} \tilde{w}_{yk} } \\ \, & \quad + \left( {\frac{{a_{18} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{19} c_{6} }}{{a_{6} c_{1} }}} \right){\mathbf{E}}_{5} \tilde{w}_{yr} + \frac{{a_{18} }}{{a_{6} }}q{\mathbf{E}}_{2} \tilde{\sigma }_{zzr} + \frac{{a_{19} }}{{a_{6} }}q{\mathbf{E}}_{2} \tilde{H}_{zzr} \left( {2 \le r \le N} \right), \\ \frac{{{\text{d}}\tilde{u}_{zr} }}{{{\text{d}}z}} & = \frac{{a_{13} }}{{a_{6} }}\sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{u}_{xk} } - \left( {\frac{{a_{20} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{21} c_{4} }}{{a_{6} c_{1} }}} \right){\mathbf{E}}_{4} \tilde{u}_{xr} - \frac{{a_{13} }}{{a_{6} }}q\tilde{u}_{yr} + \left( {\frac{{a_{20} c_{3} }}{{a_{6} c_{1} }} + \frac{{a_{21} c_{5} }}{{a_{6} c_{1} }}} \right)q{\mathbf{E}}_{3} \tilde{u}_{yr} - \frac{{a_{13} }}{{a_{6} q}}\sum\limits_{k = 2}^{N} {f_{Nrk} \tilde{u}_{yk} } \\ \, & \quad + \left( {\frac{{a_{18} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{21} c_{4} }}{{a_{6} c_{1} }}} \right)\frac{1}{q}{\mathbf{E}}_{5} \tilde{u}_{yr} + \frac{{a_{18} }}{{a_{6} }}\sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{w}_{xk} } - \left( {\frac{{a_{20} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{21} c_{6} }}{{a_{6} c_{1} }}} \right){\mathbf{E}}_{4} \tilde{w}_{xr} - \frac{{a_{18} }}{{a_{6} }}q\tilde{w}_{yr} \\ \, & \quad + \left( {\frac{{a_{20} c_{5} }}{{a_{6} c_{1} }} + \frac{{a_{21} c_{7} }}{{a_{6} c_{1} }}} \right)q{\mathbf{E}}_{3} \tilde{w}_{yr} - \frac{{a_{18} }}{{a_{6} q}}\sum\limits_{k = 2}^{N} {f_{Nrk} \tilde{w}_{yk} } + \left( {\frac{{a_{20} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{21} c_{6} }}{{a_{6} c_{1} }}} \right)\frac{1}{q}{\mathbf{E}}_{5} \tilde{w}_{yr} \\ \, & \quad + a_{20} {\mathbf{E}}_{2} \tilde{\sigma }_{zzr} + a_{21} {\mathbf{E}}_{2} \tilde{H}_{zzr} \left( {2 \le r \le N} \right), \\ \frac{{{\text{d}}\tilde{w}_{zr} }}{{{\text{d}}z}} & = \frac{{a_{14} }}{{a_{6} }}\sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{u}_{xk} } - \left( {\frac{{a_{21} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{5} c_{4} }}{{a_{6} c_{1} }}} \right){\mathbf{E}}_{4} \tilde{u}_{xr} - \frac{{a_{14} }}{{a_{6} }}q\tilde{u}_{yr} + \left( {\frac{{a_{21} c_{3} }}{{a_{6} c_{1} }} + \frac{{a_{5} c_{5} }}{{a_{6} c_{1} }}} \right)q{\mathbf{E}}_{3} \tilde{u}_{yr} \\ \, & \quad - \frac{{a_{14} }}{{a_{6} q}}\sum\limits_{k = 2}^{N} {f_{Nrk} \tilde{u}_{yk} } + \left( {\frac{{a_{21} c_{2} }}{{a_{6} c_{1} }} + \frac{{a_{5} c_{4} }}{{a_{6} c_{1} }}} \right)\frac{1}{q}{\mathbf{E}}_{5} \tilde{u}_{yr} + \frac{{a_{19} }}{{a_{6} }}\sum\limits_{k = 2}^{N - 1} {X_{rk}^{(1)} \tilde{w}_{xk} } - \left( {\frac{{a_{21} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{5} c_{6} }}{{a_{6} c_{1} }}} \right){\mathbf{E}}_{4} \tilde{w}_{xr} \\ \, & \quad - \frac{{a_{19} }}{{a_{6} }}q\tilde{w}_{yr} + \left( {\frac{{a_{21} c_{5} }}{{a_{6} c_{1} }} + \frac{{a_{5} c_{7} }}{{a_{6} c_{1} }}} \right)q{\mathbf{E}}_{3} \tilde{w}_{yr} - \frac{{a_{18} }}{{a_{6} q}}\sum\limits_{k = 2}^{N} {f_{Nrk} \tilde{w}_{yk} } + \left( {\frac{{a_{21} c_{4} }}{{a_{6} c_{1} }} + \frac{{a_{5} c_{6} }}{{a_{6} c_{1} }}} \right)\frac{1}{q}{\mathbf{E}}_{5} \tilde{w}_{yr} \\ \, & \quad \, + a_{21} {\mathbf{E}}_{2} \tilde{\sigma }_{zzr} + a_{5} {\mathbf{E}}_{2} \tilde{H}_{zzr} \left( {2 \le r \le N} \right), \\ \end{aligned}$$

where

$$\begin{aligned} c_{1} & = C_{12}^{0} K_{12}^{0} - R_{2}^{0} R_{2}^{0} ,c_{2} = C_{11}^{0} C_{11}^{0} K_{12}^{0} - 2C_{11}^{0} R_{1}^{0} R_{2}^{0} - C_{12}^{0} (c_{1} + R_{1}^{0} R_{1}^{0} ),c_{3} = C_{11}^{0} - C_{12}^{0} , \\ c_{4} & = C_{11}^{0} R_{1}^{0} K_{12}^{0} + C_{12}^{0} R_{1}^{0} K_{11}^{0} - C_{11}^{0} R_{2}^{0} K_{11}^{0} - C_{12}^{0} R_{2}^{0} K_{12}^{0} - R_{1}^{0} R_{1}^{0} R_{2}^{0} + R_{2}^{0} R_{2}^{0} R_{2}^{0} ,c_{5} = R_{1}^{0} - R_{2}^{0} , \\ c_{6} & = C_{12}^{0} (K_{11}^{0} - K_{12}^{0} )(K_{11}^{0} + K_{12}^{0} ) - 2R_{1}^{0} R_{2}^{0} K_{11}^{0} + (R_{1}^{0} R_{1}^{0} + R_{2}^{0} R_{2}^{0} )K_{12}^{0} ,c_{7} = K_{11}^{0} - K_{12}^{0} , \\ \end{aligned}$$

(A6)

$$\begin{aligned} {\mathbf{E}}_{1} & = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & 0 \\ {{\mathbf{I}}_{(N - 2) \times (N - 2)} } & {\mathbf{0}} \\ \end{array} } \right],{\mathbf{E}}_{2} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{I}}_{(N - 2) \times (N - 2)} } \\ 0 & {\mathbf{0}} \\ \end{array} } \right],{\mathbf{E}}_{3} = \left[ {\begin{array}{*{20}c} {{\mathbf{0}}_{(N - 2) \times (N - 2)} } & {\mathbf{0}} \\ {\mathbf{0}} & 1 \\ \end{array} } \right], \\ {\mathbf{E}}_{4} & = \left[ {\begin{array}{*{20}c} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \\ {X_{N2}^{(1)} } & \cdots & {X_{N(N - 1)}^{(1)} } \\ \end{array} } \right],{\mathbf{E}}_{5} = \left[ {\begin{array}{*{20}c} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \\ {X_{NN}^{(1)} X_{N2}^{(1)} } & \cdots & {X_{NN}^{(1)} X_{NN}^{(1)} } \\ \end{array} } \right], \\ {\mathbf{E}}_{6} & = \left[ {\begin{array}{*{20}c} 0 & \cdots & 0 & {X_{N2}^{(1)} } \\ \vdots & \ddots & \vdots & \vdots \\ 0 & \cdots & 0 & {X_{N(N - 1)}^{(1)} } \\ \end{array} } \right], \\ \end{aligned}$$

(A7)

with $F_{Nrk} = X_{rN}^{(2)} X_{kN}^{(1)} ,\overline{F}_{Nrk} = f_{1rN}^{{}} X_{kN}^{(1)} ,\overline{\overline{F}}_{Nrk} = f_{NrN}^{{}} X_{kN}^{(1)} .$