On the characteristics of shear acoustic waves propagating in an imperfectly bonded functionally graded piezoelectric layer over a piezoelectric cylinder

Abstract

A theoretical approach is taken into consideration to investigate the propagation behaviour of shear acoustic waves in a piezoelectric cylindrical layered structure composed of a piezoelectric material cylinder imperfectly bonded to a concentric functionally graded piezoelectric material (FGPM) cylindrical layer of finite width. The functional gradient in the FGPM cylindrical layer is considered to vary continuously along the radial direction (function of radial coordinate), and the imperfection of the interface of the cylindrical structure is taken into account which may practically exist due to some mechanical and/or electrical damage. By means of mathematical transformation, the governing electromechanical coupled field differential equations are reduced to Bessel’s equations. An analytical treatment has been employed to determine the dispersion relations of propagating shear acoustic waves for both electrically short and electrically open conditions, which are further validated by reducing the obtained results to the pre-established standard results and classical Love wave equation as a special case of the problem. The effects of functional gradient parameter, radii ratio, wave number, order of Bessel’s function appearing in the dispersion relations and mechanical/electrical imperfection parameters associated with the imperfect bonding of a piezoelectric material cylinder and FGPM layer on the phase velocity of shear acoustic waves have been reported through numerical simulation and graphical demonstration. For the sake of numerical computation, the data of PZT-5H for the FGPM cylindrical layer and AlN for a piezoelectric material cylinder have been considered.

Appendices

Appendix A

$$\begin{aligned} M_{11}= & {} N_{11} =\bar{{c}}_{44} \left( {\frac{\omega r_2 }{\beta _1 }{J}'_p \left( {\frac{\omega r_2 }{\beta _1 }} \right) -\frac{\ell }{2}J_p \left( {\frac{\omega r_2 }{\beta _1 }} \right) } \right) r_2^{-\left( {\frac{\ell }{2}+1} \right) } , \\ M_{12}= & {} N_{12} =\bar{{c}}_{44} \left( {\frac{\omega r_2 }{\beta _1 }{Y}'_p \left( {\frac{\omega r_2 }{\beta _1 }} \right) -\frac{\ell }{2}Y_p \left( {\frac{\omega r_2 }{\beta _1 }} \right) } \right) r_2^{-\left( {\frac{\ell }{2}+1} \right) } , \\ M_{13}= & {} N_{13} =-r_2^{-p-\frac{\ell }{2}-1} e_{15}^{\left( F \right) } \left( {\frac{\ell }{2}+p} \right) , \qquad M_{14} =N_{14} =r_2^{p-\frac{\ell }{2}-1} e_{15}^{\left( F \right) } \left( {p-\frac{\ell }{2}} \right) , \\ M_{15}= & {} N_{15} =0,\qquad M_{16} =N_{16} =0, \qquad N_{17} =0, \\ M_{21}= & {} \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_p \left( {\frac{\omega r_2 }{\beta _1 }} \right) r_2^{-\frac{\ell }{2}} , \qquad M_{22} =\frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_p \left( {\frac{\omega r_2 }{\beta _1 }} \right) r_2^{-\frac{\ell }{2}} ,\qquad M_{23} =r_2^{-p-\frac{\ell }{2}} , \\ M_{24}= & {} r_2^{p-\frac{\ell }{2}} , \qquad M_{25} =0, \qquad M_{26} =0, \\ N_{21}= & {} \frac{e_{15}^{\left( f \right) } }{\varepsilon _{11}^{\left( f \right) } }r_2^{-\frac{\ell }{2}} J_p \left( {\frac{\omega r_2 }{\beta _1 }} \right) , \qquad N_{22} =\frac{e_{15}^{\left( f \right) } }{\varepsilon _{11}^{\left( f \right) } }r_2^{-\frac{\ell }{2}} Y_p \left( {\frac{\omega r_2 }{\beta _1 }} \right) , \qquad N_{23} =r_2^{-\left( {\frac{\ell }{2}+p} \right) } , \qquad N_{24} =r_2^{-\left( {\frac{\ell }{2}-p} \right) } ,\\ N_{25}= & {} 0, \qquad N_{26} =0, \qquad N_{27} =-r_2^{-n} ,\\ N_{31}= & {} 0, \qquad N_{32} =0, \qquad N_{33} =r_2^{-\left( {\ell /{2+p+1}} \right) } \varepsilon _{11}^{\left( F \right) } \left( {\frac{\ell }{2}-p} \right) ,\\ N_{34}= & {} r_2^{-\left( {\ell /{2-p+1}} \right) } \varepsilon _{11}^{\left( F \right) } \left( {\frac{\ell }{2}+p} \right) , \qquad N_{35} =0, \qquad N_{36} =0,\qquad N_{37} =\varepsilon _{11}^{\left( 0 \right) } \nu r_2^{-n-1} ,\\ M_{31}= & {} N_{41} =\left[ {\alpha _1 r_1 J_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) -\left( {\frac{r_1 }{r_2 }} \right) ^{\ell }\bar{{c}}_{44} \left( {\frac{\omega r_1 }{\beta _1 }{J}'_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) -J_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) \frac{\ell }{2}} \right) } \right] r_1^{-\frac{\ell }{2}-1} ,\\ M_{32}= & {} N_{42} =\left[ {\alpha _1 r_1 Y_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) -\left( {\frac{r_1 }{r_2 }} \right) ^{\ell }\bar{{c}}_{44} \left( {\frac{\omega r_1 }{\beta _1 }{Y}'_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) -Y_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) \frac{\ell }{2}} \right) } \right] r_1^{-\frac{\ell }{2}-1} ,\\ M_{33}= & {} N_{43} =e_{15}^{\left( F \right) } r_1^{-p-\frac{\ell }{2}-1} \left( {\frac{\ell }{2}+p} \right) , \qquad M_{34} =N_{44} =e_{15}^{\left( F \right) } r_1^{p-\frac{\ell }{2}-1} \left( {p-\frac{\ell }{2}} \right) ,\\ M_{35}= & {} N_{45} =-\,\alpha _1 J_n \left( {\frac{\omega r_1 }{\beta _2 }} \right) , \qquad M_{36} =N_{46} =0,\quad N_{37} =0,\\ M_{41}= & {} N_{51} =\alpha _2 \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }r_1^{-\ell /2} J_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \qquad M_{42} =N_{52} =\alpha _2 \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }r_1^{-\ell /2} Y_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \\ M_{43}= & {} N_{53} =r_1^{-\left( {\ell /{2+p+1}} \right) } \left( {\alpha _2 r_1 -\varepsilon _{11}^{\left( F \right) } \left( {\frac{\ell }{2}+p} \right) } \right) ,\qquad M_{44} =N_{54} =r_1^{-\left( {\ell /{2-p+1}} \right) } \left( {\alpha _2 r_1 -\varepsilon _{11}^{\left( F \right) } \left( {\frac{\ell }{2}-p} \right) } \right) , \\ M_{45}= & {} N_{55} =-\,\alpha _2 \frac{e_{15}^{\left( P \right) } }{\varepsilon _{11}^{\left( P \right) } }J_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \qquad M_{46} =N_{56} =-\,\alpha _2 r_1^{-n} , \qquad N_{57} =0, \\ M_{51}= & {} N_{61} =0, \qquad M_{52} =N_{62} =0, \qquad M_{53} =N_{63} =r_1^{-\left( {\ell /{2+p+1}} \right) } \varepsilon _{11}^{\left( f \right) } \left( {\frac{\ell }{2}-p} \right) , \\ M_{54}= & {} N_{64} =r_1^{-\left( {\ell /{2-p+1}} \right) } \varepsilon _{11}^{\left( f \right) } \left( {\frac{\ell }{2}+p} \right) , \qquad M_{55} =N_{65} =0, \qquad M_{56} =N_{66} =-n\varepsilon _{11}^{\left( p \right) } r_1^{-n-1} ,\qquad N_{67} =0 \\ M_{61}= & {} N_{71} =\bar{{c}}_{44} \left( {\frac{\omega r_1 }{\beta _1 }{J}'_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) -\frac{\ell }{2}J_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) } \right) r_1^{-\left( {\frac{\ell }{2}+1} \right) } , \\ M_{62}= & {} N_{72} =\bar{{c}}_{44} \left( {\frac{\omega r_1 }{\beta _1 }{Y}'_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) -\frac{\ell }{2}J_p \left( {\frac{\omega r_1 }{\beta _1 }} \right) } \right) r_1^{-\left( {\frac{\ell }{2}+1} \right) } , \\ M_{63}= & {} N_{73} =-e_{15}^{\left( F \right) } r_1^{-\left( {\ell /2+p+1} \right) } \left( {\frac{\ell }{2}+p} \right) , \qquad M_{64} =N_{74} =e_{15}^{\left( F \right) } r_1^{-\left( {\ell /2-p+1} \right) } \left( {p-\frac{\ell }{2}} \right) , \\ M_{65}= & {} N_{65} =-\bar{{\bar{{c}}}}_{44} \frac{\omega }{\beta _2 }{J}'_n \left( {\frac{\omega r_1 }{\beta _2 }} \right) , \qquad M_{66} =N_{76} =ne_{15}^{\left( P \right) } r_1^{-\left( {n+1} \right) } ,\qquad c_{77} =0. \end{aligned}$$

Appendix B

$$\begin{aligned} {\overline{M}} _{11}= & {} {\overline{N}} _{11} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{J}'_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad {\overline{M}} _{12} ={\overline{N}} _{12} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{Y}'_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\\ {\overline{M}} _{13}= & {} {\overline{N}} _{13} =-r_2^{-n-1} e_{15}^{\left( F \right) } n, \quad {\overline{M}} _{14} ={\overline{N}} _{14} =r_2^{n-1} e_{15}^{\left( F \right) } n, \quad {\overline{M}} _{15} ={\overline{N}} _{15} =0, \qquad {\overline{M}} _{16} ={\overline{N}} _{16} =0, \qquad {\overline{N}} _{17} =0,\\ {\overline{M}} _{21}= & {} \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) , \qquad {\overline{M}} _{22} =\frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) , \qquad {\overline{M}} _{23} =r_2^{-n} , \\ {\overline{M}} _{24}= & {} r_2^n , \qquad {\overline{M}} _{25} =0, \qquad {\overline{M}} _{26} =0, \\ {\overline{N}} _{21}= & {} \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad {\overline{N}} _{22} =\frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad {\overline{N}} _{23} =r_2^{-n} , \qquad {\overline{N}} _{24} =r_2^n , \\ {\overline{N}} _{25}= & {} 0, \qquad {\overline{N}} _{26} =0, \qquad {\overline{N}} _{27} =-r_2^{-n} , \\ {\overline{N}} _{31}= & {} 0, \qquad {\overline{N}} _{32} =0, \qquad {\overline{N}} _{33} =-nr_2^{-\left( {n+1} \right) } \varepsilon _{11}^{\left( F \right) } , \\ {\overline{N}} _{34}= & {} r_2^{n+1} \varepsilon _{11}^{\left( F \right) } n, \qquad {\overline{N}} _{35} =0, \qquad {\overline{N}} _{36} =0,\qquad {\overline{N}} _{37} =\varepsilon _{11}^{\left( 0 \right) } nr_2^{-n-1} , \\ {\overline{M}} _{31}= & {} {\overline{N}} _{41} =\left[ {\alpha _1 r_1 J_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) -\bar{{c}}_{44} \frac{\omega }{\beta _1 }{J}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) } \right] , \\ {\overline{M}} _{32}= & {} {\overline{N}} _{42} =\left[ {\alpha _1 r_1 Y_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) -\bar{{c}}_{44} \frac{\omega }{\beta _1 }{Y}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) } \right] , \\ {\overline{M}} _{33}= & {} {\overline{N}} _{43} =e_{15}^{\left( F \right) } r_1^{-n-1} n, \qquad {\overline{M}} _{34} ={\overline{N}} _{44} =-e_{15}^{\left( F \right) } r_1^{n-1} n, \\ {\overline{M}} _{35}= & {} {\overline{N}} _{45} =-\,\alpha _1 J_n \left( {\frac{\omega r_1 }{\beta _2 }} \right) , \qquad {\overline{M}} _{36} ={\overline{N}} _{46} =0,\qquad {\overline{N}} _{37} =0, \\ {\overline{M}} _{41}= & {} {\overline{N}} _{51} =\alpha _2 \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) ,\qquad {\overline{M}} _{42} ={\overline{N}} _{52} =\alpha _2 \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \\ {\overline{M}} _{43}= & {} {\overline{N}} _{53} =r_1^{-\left( {n+1} \right) } \left( {\alpha _2 r_1 -\varepsilon _{11}^{\left( F \right) } n} \right) ,\qquad {\overline{M}} _{44} ={\overline{N}} _{54} =r_1^{\left( {n+1} \right) } \left( {\alpha _2 r_1 +\varepsilon _{11}^{\left( F \right) } n} \right) , \\ {\overline{M}} _{45}= & {} {\overline{N}} _{55} =-\,\alpha _2 \frac{e_{15}^{\left( P \right) } }{\varepsilon _{11}^{\left( P \right) } }J_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) ,\qquad {\overline{M}} _{46} ={\overline{N}} _{56} =-\,\alpha _2 r_1^{-n} ,\qquad {\overline{N}} _{57} =0, \\ {\overline{M}} _{51}= & {} {\overline{N}} _{61} =0, \qquad {\overline{M}} _{52} ={\overline{N}} _{62} =0, \qquad {\overline{M}} _{53} ={\overline{N}} _{63} =-nr_1^{-\left( {n+1} \right) } \varepsilon _{11}^{\left( F \right) },\quad \\ {\overline{M}} _{54}= & {} {\overline{N}} _{64} =nr_1^{\left( {n+1} \right) } \varepsilon _{11}^{\left( F \right) } , \qquad {\overline{M}} _{55} =\overline{N} _{65} =0, \qquad {\overline{M}} _{56} ={\overline{N}} _{66} =-n\varepsilon _{11}^{\left( P \right) } r_1^{-n-1} ,\qquad {\overline{N}} _{67} =0, \\ {\overline{M}} _{61}= & {} {\overline{N}} _{71} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{J}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) ,\qquad {\overline{M}} _{62} ={\overline{N}} _{72} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{Y}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) ,\\ {\overline{M}} _{63}= & {} {\overline{N}} _{73} =-e_{15}^{\left( F \right) } r_1^{-\left( {n+1} \right) } n,\qquad {\overline{M}} _{64} ={\overline{N}} _{74} =e_{15}^{\left( F \right) } r_1^{\left( {n-1} \right) } n, \\ {\overline{M}} _{65}= & {} {\overline{N}} _{65} =-\bar{{\bar{{c}}}}_{44} \frac{\omega }{\beta _2 }{J}'_n \left( {\frac{\omega r_1 }{\beta _2 }} \right) , \qquad {\overline{M}} _{66} ={\overline{N}} _{76} =ne_{15}^{\left( P \right) } r_1^{-\left( {n+1} \right) } , \qquad {\overline{N}} _{77} =0. \\ \overline{{\overline{M}} } _{11}= & {} \overline{{\overline{N}} } _{11} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{J}'_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad \overline{{\overline{M}} } _{12} =\overline{{\overline{N}} } _{12} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{Y}'_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\\ \overline{{\overline{M}} } _{13}= & {} \overline{{\overline{N}} } _{13} =-r_2^{-n-1} e_{15}^{\left( F \right) } n, \qquad \overline{{\overline{M}} } _{14} =\overline{{\overline{N}} } _{14} =r_2^{n-1} e_{15}^{\left( F \right) } n, \quad \overline{{\overline{M}} } _{15} =\overline{{\overline{N}} } _{15} =0, \quad \overline{{\overline{M}} } _{16} =\overline{{\overline{N}} } _{16} =0, \quad \overline{{\overline{N}} } _{17} =0,\\ \overline{{\overline{M}} } _{21}= & {} \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) , \qquad \overline{{\overline{M}} } _{22} =\frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad \overline{{\overline{M}} } _{23} =r_2^{-n} , \\ \overline{{\overline{M}} } _{24}= & {} r_2^n , \qquad \overline{{\overline{M}} } _{25} =0,\qquad \overline{{\overline{M}} } _{26} =0, \\ \overline{{\overline{N}} } _{21}= & {} \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad \overline{{\overline{N}} } _{22} =\frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad \overline{{\overline{N}} } _{23} =r_2^{-n} , \overline{{\overline{N}} } _{24} =r_2^n , \\ \overline{{\overline{N}} } _{25}= & {} 0, \qquad \overline{{\overline{N}} } _{26} =0, \qquad \overline{{\overline{N}} } _{27} =-r_2^{-n} , \\ \overline{{\overline{N}} } _{31}= & {} 0, \qquad \overline{{\overline{N}} } _{32} =0, \qquad \overline{{\overline{N}} } _{33} =-nr_2^{-\left( {n+1} \right) } \varepsilon _{11}^{\left( F \right) } , \\ \overline{{\overline{N}} } _{34}= & {} r_2^{n+1} \varepsilon _{11}^{\left( F \right) } n, \qquad \overline{{\overline{N}} } _{35} =0,\qquad \overline{{\overline{N}} } _{36} =0,\qquad \overline{{\overline{N}} } _{37} =\varepsilon _{11}^{\left( 0 \right) } nr_2^{-n-1} , \\ \overline{{\overline{M}} } _{31}= & {} \overline{{\overline{N}} } _{41} =\left[ {\alpha _1 r_1 J_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) -\bar{{c}}_{44} \frac{\omega }{\beta _1 }{J}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) } \right] , \\ \overline{{\overline{M}} } _{32}= & {} \overline{{\overline{N}} } _{42} =\left[ {\alpha _1 r_1 Y_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) -\bar{{c}}_{44} \frac{\omega }{\beta _1 }{Y}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) } \right] , \\ \overline{{\overline{M}} } _{33}= & {} \overline{{\overline{N}} } _{43} =e_{15}^{\left( F \right) } r_1^{-n-1} n, \qquad \overline{{\overline{M}} } _{34} =\overline{{\overline{N}} } _{44} =-e_{15}^{\left( F \right) } r_1^{n-1} n, \\ \overline{{\overline{M}} } _{35}= & {} \overline{{\overline{N}} } _{45} =-\,\alpha _1 J_n \left( {\frac{\omega r_1 }{\beta _2 }} \right) ,\qquad \overline{{\overline{M}} } _{36} =\overline{{\overline{N}} } _{46} =0,\qquad \overline{{\overline{N}} } _{37} =0, \\ \overline{{\overline{M}} } _{41}= & {} \overline{{\overline{N}} } _{51} =\alpha _2 \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \qquad \overline{{\overline{M}} } _{42} =\overline{{\overline{N}} } _{52} =\alpha _2 \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \\ \overline{{\overline{M}} } _{43}= & {} \overline{{\overline{N}} } _{53} =r_1^{-\left( {n+1} \right) } \left( {\alpha _2 r_1 -\varepsilon _{11}^{\left( F \right) } n} \right) ,\qquad \overline{{\overline{M}} } _{44} =\overline{{\overline{N}} } _{54} =r_1^{\left( {n+1} \right) } \left( {\alpha _2 r_1 +\varepsilon _{11}^{\left( F \right) } n} \right) , \\ \overline{{\overline{M}} } _{45}= & {} \overline{{\overline{N}} } _{55} =0,\qquad \overline{{\overline{M}} } _{46} =\overline{{\overline{N}} } _{56} =-\,\alpha _2 r_1^{-n} ,\qquad \overline{{\overline{N}} } _{57} =0, \\ \overline{{\overline{M}} } _{51}= & {} \overline{{\overline{N}} } _{61} =0,\qquad \overline{{\overline{M}} } _{52} =\overline{{\overline{N}} } _{62} =0,\qquad \overline{{\overline{M}} } _{53} =\overline{{\overline{N}} } _{63} =-nr_1^{-\left( {n+1} \right) } \varepsilon _{11}^{\left( F \right) } , \\ \overline{{\overline{M}} } _{54}= & {} \overline{{\overline{N}} } _{64} =nr_1^{\left( {n+1} \right) } \varepsilon _{11}^{\left( F \right) } ,\qquad \overline{{\overline{M}} } _{55} =\overline{{\overline{N}} } _{65} =0,\qquad \overline{{\overline{M}} } _{56} =\overline{{\overline{N}} } _{66} =-n\varepsilon _{11}^{\left( P \right) } r_1^{-n-1} ,\qquad \overline{{\overline{N}} } _{67} =0, \\ \overline{{\overline{M}} } _{61}= & {} \overline{{\overline{N}} } _{71} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{J}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \qquad \overline{{\overline{M}} } _{62} =\overline{{\overline{N}} } _{72} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{Y}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) ,\\ \overline{{\overline{M}} } _{63}= & {} \overline{{\overline{N}} } _{73} =-e_{15}^{\left( F \right) } r_1^{-\left( {n+1} \right) } n, \qquad \overline{{\overline{M}} } _{64} =\overline{{\overline{N}} } _{74} =e_{15}^{\left( F \right) } r_1^{\left( {n-1} \right) } n, \\ \overline{{\overline{M}} } _{65}= & {} \overline{{\overline{N}} } _{65} =-c_{44}^{\left( P \right) } \frac{\omega }{\beta _2 }{J}'_n \left( {\frac{\omega r_1 }{\beta _2 }} \right) ,\qquad \overline{{\overline{M}} } _{66} =\overline{{\overline{N}} } _{76} =0, \qquad \overline{{\overline{N}} } _{77} =0. \\ \overline{\overline{{\overline{M}} } } _{11}= & {} \overline{\overline{{\overline{N}} } } _{11} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{J}'_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad \overline{\overline{{\overline{M}} } } _{12} =\overline{\overline{{\overline{N}} } } _{12} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{Y}'_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\\ \overline{\overline{{\overline{M}} } } _{13}= & {} \overline{\overline{{\overline{N}} } } _{13} =-r_2^{-n-1} e_{15}^{\left( F \right) } n,\quad \overline{\overline{{\overline{M}} } } _{14} =\overline{\overline{{\overline{N}} } } _{14} =r_2^{n-1} e_{15}^{\left( F \right) } n,\quad \overline{\overline{{\overline{M}} } } _{15} =\overline{\overline{{\overline{N}} } } _{15} =0, \quad \overline{\overline{{\overline{M}} } } _{16} =\overline{\overline{{\overline{N}} } } _{16} =0,\quad \overline{\overline{{\overline{N}} } } _{17} =0,\\ \overline{\overline{{\overline{M}} } } _{21}= & {} \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) , \qquad \overline{\overline{{\overline{M}} } } _{22} =\frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) , \qquad \overline{\overline{{\overline{M}} } } _{23} =r_2^{-n} , \\ \overline{\overline{{\overline{M}} } } _{24}= & {} r_2^n ,\qquad \overline{\overline{{\overline{M}} } } _{25} =0, \qquad \overline{\overline{{\overline{M}} } } _{26} =0, \\ \overline{\overline{{\overline{N}} } } _{21}= & {} \frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad \overline{\overline{{\overline{N}} } } _{22} =\frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_n \left( {\frac{\omega r_2 }{\beta _1 }} \right) ,\qquad \overline{\overline{{\overline{N}} } } _{23} =r_2^{-n} , \qquad \overline{\overline{{\overline{N}} } } _{24} =r_2^n , \\ \overline{\overline{{\overline{N}} } } _{25}= & {} 0, \qquad \overline{\overline{{\overline{N}} } } _{26} =0, \qquad \overline{\overline{{\overline{N}} } } _{27} =-r_2^{-n} , \\ \overline{\overline{{\overline{N}} } } _{31}= & {} 0, \qquad \overline{\overline{{\overline{N}} } } _{32} =0, \qquad \overline{\overline{{\overline{N}} } } _{33} =-nr_2^{-\left( {n+1} \right) } \varepsilon _{11}^{\left( F \right) } , \\ \overline{\overline{{\overline{N}} } } _{34}= & {} r_2^{n+1} \varepsilon _{11}^{\left( F \right) } n,\qquad \overline{\overline{{\overline{N}} } } _{35} =0, \qquad \overline{\overline{{\overline{N}} } } _{36} =0,\qquad \overline{\overline{{\overline{N}} } } _{37} =\varepsilon _{11}^{\left( 0 \right) } nr_2^{-n-1} , \\ \overline{\overline{{\overline{M}} } } _{31}= & {} \overline{\overline{{\overline{N}} } } _{41} =r_1 J_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \qquad \overline{\overline{{\overline{M}} } } _{32} =\overline{\overline{{\overline{N}} } } _{42} =r_1 Y_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \\ \overline{\overline{{\overline{M}} } } _{33}= & {} \overline{\overline{{\overline{N}} } } _{43} =0, \qquad \overline{\overline{{\overline{M}} } } _{34} =\overline{\overline{{\overline{N}} } } _{44} =0, \qquad \overline{\overline{{\overline{M}} } } _{35} =\overline{\overline{{\overline{N}} } } _{45} =-J_n \left( {\frac{\omega r_1 }{\beta _2 }} \right) ,\qquad \overline{\overline{{\overline{M}} } } _{36} =\overline{\overline{{\overline{N}} } } _{46} =0, \quad \overline{\overline{{\overline{N}} } } _{37} =0, \\ \overline{\overline{{\overline{M}} } } _{41}= & {} \overline{\overline{{\overline{N}} } } _{51} =\frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }J_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \qquad \overline{\overline{{\overline{M}} } } _{42} =\overline{\overline{{\overline{N}} } } _{52} =\frac{e_{15}^{\left( F \right) } }{\varepsilon _{11}^{\left( F \right) } }Y_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \\ \overline{\overline{{\overline{M}} } } _{43}= & {} \overline{\overline{{\overline{N}} } } _{53} =r_1^{-n} ,\qquad \overline{\overline{{\overline{M}} } } _{44} =\overline{\overline{{\overline{N}} } } _{54} =r_1^{\left( {n+2} \right) } ,\qquad \overline{\overline{{\overline{M}} } } _{45} =\overline{\overline{{\overline{N}} } } _{55} =0, \qquad \overline{\overline{{\overline{M}} } } _{46} =\overline{\overline{{\overline{N}} } } _{56} =-r_1^{-n} ,\qquad \overline{\overline{{\overline{N}} } } _{57} =0, \\ \overline{\overline{{\overline{M}} } } _{51}= & {} \overline{\overline{{\overline{N}} } } _{61} =0, \qquad \overline{\overline{{\overline{M}} } } _{52} =\overline{\overline{{\overline{N}} } } _{62} =0, \qquad \overline{\overline{{\overline{M}} } } _{53} =\overline{\overline{{\overline{N}} } } _{63} =-nr_1^{-\left( {n+1} \right) } \varepsilon _{11}^{\left( F \right) } , \\ \overline{\overline{{\overline{M}} } } _{54}= & {} \overline{\overline{{\overline{N}} } } _{64} =nr_1^{\left( {n+1} \right) } \varepsilon _{11}^{\left( F \right) } , \qquad \overline{\overline{{\overline{M}} } } _{55} =\overline{\overline{{\overline{N}} } } _{65} =0, \qquad \overline{\overline{{\overline{M}} } } _{56} =\overline{\overline{{\overline{N}} } } _{66} =0,\qquad \overline{\overline{{\overline{N}} } } _{67} =0, \\ \overline{\overline{{\overline{M}} } } _{61}= & {} \overline{\overline{{\overline{N}} } } _{71} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{J}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) , \qquad \overline{\overline{{\overline{M}} } } _{62} =\overline{\overline{{\overline{N}} } } _{72} =\bar{{c}}_{44} \frac{\omega }{\beta _1 }{Y}'_n \left( {\frac{\omega r_1 }{\beta _1 }} \right) ,\\ \overline{\overline{{\overline{M}} } } _{63}= & {} \overline{\overline{{\overline{N}} } } _{73} =-e_{15}^{\left( F \right) } r_1^{-\left( {n+1} \right) } n, \qquad \overline{\overline{{\overline{M}} } } _{64} =\overline{\overline{{\overline{N}} } } _{74} =e_{15}^{\left( F \right) } r_1^{\left( {n-1} \right) } n, \\ \overline{\overline{{\overline{M}} } } _{65}= & {} \overline{\overline{{\overline{N}} } } _{65} =-\mu _2 \frac{\omega }{\beta _2 }{J}'_n \left( {\frac{\omega r_1 }{\beta _2 }} \right) ,\qquad \overline{\overline{{\overline{M}} } } _{66} =\overline{\overline{{\overline{N}} } } _{76} =0, \qquad \overline{\overline{{\overline{N}} } } _{77} =0. \end{aligned}$$