Dynamic Concentrations and Potentials of Embedded Eccentrically Coated Magneto-Electro-Elastic Fiber Subjected to Anti-Plane Shear Waves

Abstract

This paper examines the problem of the fully coupled magneto-electro-elastic (MEE) scattering of SH-waves incident upon a heterogeneous MEE scatterer which is embedded in an unbounded medium. The scatterer consists of a circular core and a circular encapsulator with eccentricity. All three regions: the core, encapsulator, and the surrounding matrix have distinct MEE properties and fully coupled constitutive relations. The generated coupled MEE fields coexist simultaneously in all these regions without resort to any simplifying assumptions. The precise description of the multifunctionality involves the solution of three fully coupled partial differential equations in three different regions. The associated Green’s function equations involve 9 independent components of Green’s functions. The behaviors of the regions are described by the generalized constitutive equations suitable for transversely isotropic MEE properties. Conventionally, wave function approach has been used to study the elastodynamic fields associated with the purely elastic axisymmetric problems; such a treatment encounters serious difficulties in the presence of eccentricity. As a rigorous analytical remedy the dynamic magneto-electro-mechanical equivalent inclusion method (DMEMEIM) will be developed in this work. To this end, the notions of eigenstress, eigenbody-force, eigenelectric, and eigenmagnetic fields will be introduced. As it will be shown, the employment of these notions in conjunction with the eigenfunction space of the pertinent coupled field equations provides a meticulous mathematical framework for the treatment of the proposed problem. The exact analytical formulation for the fully coupled total MEE scattering cross-section is derived. The ramifications of the MEE couplings as well as the wavenumber on the induced scattered fields are considered. As it will be seen, the magnetic field has a substantial effect on the total scattering cross-section. The interfacial stresses are remarkably affected not only by the eccentricity, but also by the magnetic parameters. Moreover, the dynamic electric displacement concentration factor (DEDCF), the dynamic stress concentration factor (DSCF), the electric potential, and the magnetic potential will be examined for different wavenumbers.

Appendices

Appendix A

For a given field point, $\boldsymbol{\Theta}$ within any of the specified regions shown in Fig. 17, the functions $\mathscr{F}_{n}^{(j)}{(r)}, \mathscr{G}_{n}^{(j)}{(r)}, \mathscr{H}(r)$, and $\mathscr{K}(r)$ pertinent to Eqs. (41)-(43) in terms of the unknown functions, $\imath _{n}^{(j)}{(r)}$ and $\jmath _{n}^{(j)}{(r)} (n=0,1,2,\ldots ; j=1,2,3,4,5,6,7)$ are obtained after some manipulations

$$\begin{aligned} \frac{4 \textbf{i} \overset{\Omega}{\rho}\omega ^{2}}{k^{2}} \mathscr{F}_{n}^{(j)}{(r)} =&\pi H_{n}^{(1)}{(kr)}\int _{0}^{r} \imath _{n}^{(j)}{(r^{\prime})}J_{n}{(kr^{\prime})}r^{\prime }dr^{ \prime} \\ +&\pi J_{n}{(kr)}\int _{r}^{R_{0}}\imath _{n}^{(j)}{(r^{\prime})}H_{n}^{(1)}{(kr^{ \prime})}r^{\prime }dr^{\prime} \\ +&\alpha _{n} J_{n}{(kr)}\int _{0}^{\pi}\int _{R_{0}}^{R_{(\theta ^{ \prime})}}F_{n}^{(j)}{(r^{\prime},\theta ^{\prime})}H_{n}^{(1)}{(kr^{ \prime})}r^{\prime }dr^{\prime }d\theta ^{\prime}, \hbox{$ \boldsymbol{\Theta}\in \Omega $,} \end{aligned}$$

(52)

$$\begin{aligned} \frac{4 \textbf{i} \overset {\Gamma}{\rho}\omega ^{2}}{k^{2}} \mathscr{F}_{n}^{(j)}{(r)} =&\pi H_{n}^{(1)}{(kr)}\int _{0}^{R_{0}} \imath _{n}^{(j)}{(r^{\prime})}J_{n}{(kr^{\prime})}r^{\prime }dr^{ \prime} \\ +&\pi H_{n}^{(1)}{(kr)}\int _{R_{0}}^{r}\jmath _{n}^{(j)}{(r^{\prime})}J_{n}{(kr^{ \prime})}r^{\prime }dr^{\prime} \\ +&\alpha _{n} J_{n}{(kr)}\int _{0}^{\pi}\int _{r}^{R_{(\theta ^{ \prime})}}F_{n}^{(j)}{(r^{\prime},\theta ^{\prime})}H_{n}^{(1)}{(kr^{ \prime})}r^{\prime }dr^{\prime }d\theta ^{\prime}, \hbox{$ \boldsymbol{\Theta}\in \Gamma _{0}$,} \end{aligned}$$

(53)

$$\begin{aligned} \frac{4 \textbf{i} \overset{\circ}{\rho}\omega ^{2}}{k^{2}}\mathscr{F}_{n}^{(j)}{(r)} =& \pi H_{n}^{(1)}{(kr)}\int _{0}^{R_{0}}\imath _{n}^{(j)}{(r^{\prime})}J_{n}{(kr^{ \prime})}r^{\prime }dr^{\prime} \\ +&\pi H_{n}^{(1)}{(kr)}\int _{R_{0}}^{r}\jmath _{n}^{(j)}{(r^{\prime})}J_{n}{(kr^{ \prime})}r^{\prime }dr^{\prime} \\ +&\alpha _{n} J_{n}{(kr)}\int _{0}^{\pi}\int _{r}^{R_{(\theta ^{ \prime})}}F_{n}^{(j)}{(r^{\prime},\theta ^{\prime})} H_{n}^{(1)}{(kr^{ \prime})}r^{\prime }dr^{\prime }d\theta ^{\prime} \\ -&\alpha _{n} J_{n}{(kr)}\int _{0}^{\theta _{0}(r)}\int _{r}^{R_{( \theta ^{\prime})}}F_{n}^{(j)}{(r^{\prime},\theta ^{\prime})} H_{n}^{(1)}{(kr^{ \prime})}r^{\prime }dr^{\prime }d\theta ^{\prime} \\ +&\alpha _{n} H_{n}^{(1)}{(kr)}\int _{0}^{\theta _{0}(r)}\int _{r}^{R_{( \theta ^{\prime})}}F_{n}^{(j)}{(r^{\prime},\theta ^{\prime})} J_{n}(kr^{ \prime})r^{\prime }dr^{\prime }d\theta ^{\prime}, \hbox{$ \boldsymbol{\Theta}\in \Gamma _{1}$,} \end{aligned}$$

(54)

in which $\alpha _{0}=1$ and $\alpha _{n}=2$ for $n=1,2,3,\ldots$

$$\begin{aligned} 4\mathscr{G}_{n}^{(j)}{(r)} =&\frac{r^{-n}}{n}\int _{0}^{r}(r^{\prime})^{n} \imath _{n}^{(j)}{(r^{\prime})}r^{\prime }dr^{\prime}+ \frac{r^{n}}{n}\int _{r}^{R_{0}}(r^{\prime})^{-n} \imath _{n}^{(j)}{(r^{ \prime})}r^{\prime }dr^{\prime} \\ +&\frac{2}{\pi}\frac{r^{n}}{n}\int _{0}^{\pi}\int _{R_{0}}^{R( \theta ^{\prime})}(r^{\prime})^{-n} F_{n}^{(j)}{(r^{\prime},\theta ^{ \prime})}r^{\prime }dr^{\prime }d\theta ^{\prime},\hbox{$ \boldsymbol{\Theta}\in \Omega $,} \end{aligned}$$

(55)

$$\begin{aligned} 4\mathscr{G}_{n}^{(j)}{(r)} =&\frac{r^{-n}}{n}\int _{0}^{R_{0}}(r^{ \prime})^{n} \imath _{n}^{(j)}{(r^{\prime})}r^{\prime }dr^{\prime}+ \frac{r^{-n}}{n}\int _{R_{0}}^{r}(r^{\prime})^{n} \jmath _{n}^{(j)}{(r^{ \prime})}r^{\prime }dr^{\prime} \\ +&\frac{2}{\pi}\frac{r^{n}}{n}\int _{0}^{\pi}\int _{r}^{R(\theta ^{ \prime})}(r^{\prime})^{-n} F_{n}^{(j)}{(r^{\prime},\theta ^{\prime})}r^{ \prime }dr^{\prime }d\theta ^{\prime},\hbox{$\boldsymbol{\Theta}\in \Gamma _{0}$,} \end{aligned}$$

(56)

$$\begin{aligned} 4\mathscr{G}_{n}^{(j)}{(r)} =&\frac{r^{-n}}{n}\int _{0}^{R_{0}}(r^{ \prime})^{n} \imath _{n}^{(j)}{(r^{\prime})}r^{\prime }dr^{\prime}+ \frac{r^{-n}}{n}\int _{R_{0}}^{r}(r^{\prime})^{n} \jmath _{n}^{(j)}{(r^{ \prime})}r^{\prime }dr^{\prime} \\ +&\frac{2}{\pi}\frac{r^{n}}{n}\int _{0}^{\pi}\int _{r}^{R(\theta ^{ \prime})}(r^{\prime})^{-n} F_{n}^{(j)}{(r^{\prime},\theta ^{\prime})}r^{ \prime }dr^{\prime }d\theta ^{\prime} \\ -&\frac{2}{\pi}\frac{r^{-n}}{n}\int _{0}^{\theta _{0}(r)}\int _{r}^{R( \theta ^{\prime})}(r^{\prime})^{n} F_{n}^{(j)}{(r^{\prime},\theta ^{ \prime})}r^{\prime }dr^{\prime }d\theta ^{\prime} \\ -&\frac{2}{\pi}\frac{r^{n}}{n}\int _{0}^{\theta _{0}(r)}\int _{r}^{R( \theta ^{\prime})}(r^{\prime})^{-n} F_{n}^{(j)}{(r^{\prime},\theta ^{ \prime})}r^{\prime }dr^{\prime }d\theta ^{\prime},\hbox{$ \boldsymbol{\Theta}\in \Gamma _{1}$,} \end{aligned}$$

(57)

$$ \mathscr{H}(r)=\frac{1}{r}\int _{0}^{r} \frac{\overset{\Omega}{\mu}}{\overset{\Omega}{a^{*}}}\left ( \overset{\Omega}{\kappa}\ \imath _{0}^{(4)}{(r^{\prime})}+ \overset{\Omega}{d}\ \imath _{0}^{(6)}{(r^{\prime})}\right )r^{\prime }dr^{ \prime },\hbox{$\boldsymbol{\Theta}\in \Omega $,} $$

(58)

$$\begin{aligned} \mathscr{H}(r) =&\frac{1}{r}\int _{0}^{R_{0}} \frac{\overset{\Gamma}{\mu}}{\overset{\Gamma}{a^{*}}}\left ( \overset{\Gamma}{\kappa}\ \imath _{0}^{(4)}{(r^{\prime})}+ \overset{\Gamma}{d}\ \imath _{0}^{(6)}{(r^{\prime})}\right )r^{ \prime }dr^{\prime} \\ +&\frac{1}{r}\int _{R_{0}}^{r} \frac{\overset{\Gamma}{\mu}}{\overset{\Gamma}{a^{*}}}\left ( \overset{\Gamma}{\kappa}\ \jmath _{0}^{(4)}{(r^{\prime})}+ \overset{\Gamma}{d}\ \jmath _{0}^{(6)}{(r^{\prime})}\right )r^{ \prime }dr^{\prime },\hbox{$\boldsymbol{\Theta}\in \Gamma _{0}$,} \end{aligned}$$

(59)

$$\begin{aligned} \mathscr{H}(r) =&\frac{1}{r}\int _{0}^{R_{0}} \frac{\overset{\circ}{\mu}}{\overset{\circ}{a^{*}}}\left ( \overset{\circ}{\kappa}\ \imath _{0}^{(4)}{(r^{\prime})}+ \overset{\circ}{d}\ \imath _{0}^{(6)}{(r^{\prime})}\right )r^{\prime }dr^{ \prime} \\ +&\frac{1}{r}\int _{R_{0}}^{r} \frac{\overset{\circ}{\mu}}{\overset{\circ}{a^{*}}}\left ( \overset{\circ}{\kappa}\ \jmath _{0}^{(4)}{(r^{\prime})}+ \overset{\circ}{d}\ \jmath _{0}^{(6)}{(r^{\prime})}\right )r^{\prime }dr^{ \prime} \\ -&\frac{1}{r\pi}\int _{0}^{\theta _{0}(r)}\int _{r}^{R(\theta ^{ \prime})}\frac{\overset{\circ}{\mu}}{\overset{\circ}{a^{*}}}\left ( \overset{\circ}{\kappa}\ F_{0}^{(4)}{(r^{\prime},\theta ^{\prime})}+ \overset{\circ}{d}\ F_{0}^{(6)}{(r^{\prime},\theta ^{\prime})}\right )r^{ \prime }dr^{\prime }d\theta ^{\prime },\hbox{$\boldsymbol{\Theta}\in \Gamma _{1}$,} \end{aligned}$$

(60)

$$ \mathscr{K}(r)=\frac{1}{r}\int _{0}^{r} \frac{\overset{\Omega}{\kappa}}{\overset{\Omega}{a^{*}}}\left (- \overset{\Omega}{d}\ \imath _{0}^{(4)}{(r^{\prime})}+ \overset{\Omega}{\mu}\ \imath _{0}^{(6)}{(r^{\prime})}\right )r^{ \prime }dr^{\prime }, \hbox{$\boldsymbol{\Theta}\in \Omega $,} $$

(61)

$$\begin{aligned} \mathscr{K}(r) =&\frac{1}{r}\int _{0}^{R_{0}} \frac{\overset{\Gamma}{\kappa}}{\overset{\Gamma}{a^{*}}}\left (- \overset{\Gamma}{d}\ \imath _{0}^{(4)}{(r^{\prime})}+ \overset{\Gamma}{\mu}\ \imath _{0}^{(6)}{(r^{\prime})}\right )r^{ \prime }dr^{\prime} \\ +&\frac{1}{r}\int _{R_{0}}^{r} \frac{\overset{\Gamma}{\kappa}}{\overset{\Gamma}{a^{*}}}\left (- \overset{\Gamma}{d}\ \jmath _{0}^{(4)}{(r^{\prime})}+ \overset{\Gamma}{\mu}\ \jmath _{0}^{(6)}{(r^{\prime})}\right )r^{ \prime }dr^{\prime }, \hbox{$\boldsymbol{\Theta}\in \Gamma _{0}$,} \end{aligned}$$

(62)

$$\begin{aligned} \mathscr{K}(r) =&\frac{1}{r}\int _{0}^{R_{0}} \frac{\overset{\circ}{\kappa}}{\overset{\circ}{a^{*}}}\left (- \overset{\circ}{d}\ \imath _{0}^{(4)}{(r^{\prime})}+ \overset{\circ}{\mu}\ \imath _{0}^{(6)}{(r^{\prime})}\right )r^{ \prime }dr^{\prime} \\ +&\frac{1}{r}\int _{R_{0}}^{r} \frac{\overset{\circ}{\kappa}}{\overset{\circ}{a^{*}}}\left (- \overset{\circ}{d}\ \jmath _{0}^{(4)}{(r^{\prime})}+ \overset{\circ}{\mu}\ \jmath _{0}^{(6)}{(r^{\prime})}\right )r^{ \prime }dr^{\prime} \\ -&\frac{1}{r\pi}\int _{0}^{\theta _{0}(r)}\int _{r}^{R(\theta ^{ \prime})}\left (-\overset{\circ}{d}\ F_{0}^{(4)}{(r^{\prime},\theta ^{ \prime})}+\overset{\circ}{\mu}\ F_{0}^{(6)}{(r^{\prime},\theta ^{ \prime})}\right )r^{\prime }dr^{\prime }d\theta ^{\prime }, \hbox{$ \boldsymbol{\Theta}\in \Gamma _{1}$,} \end{aligned}$$

(63)

where

$$ F_{n}^{(j)}(r^{\prime},\theta ^{\prime})=\cos (n\theta ^{\prime}) \sum _{m=0}^{\infty}\jmath _{m}^{(j)}(r^{\prime})\cos (m\theta ^{ \prime}), j=1,3,4,6, $$

(64)

$$ F_{n}^{(j)}(r^{\prime},\theta ^{\prime})=\sin (n\theta ^{\prime}) \sum _{m=1}^{\infty}\jmath _{m}^{(j)}(r^{\prime})\sin (m\theta ^{ \prime}), j=2,5,7. $$

(65)

Fig. 17

The geometric details of the integration bounds for the calculation of $I_{\Omega \cup \Gamma}$

Appendix B

The unknown functions, $\mathscr{P}(r)$ and $\mathscr{Q}(r)$ and the unknown coefficients, $\mathscr{S}_{n}, \mathscr{N}_{n}, \mathscr{M}_{n}, \mathscr{L}_{n}^{(4)}, \mathscr{L}_{n}^{(5)}, \mathscr{L}_{n}^{(6)}$, and $\mathscr{L}_{n}^{(7)} (n=0,1,2,\ldots )$ appearing in relations (46)-(48) are given as

$$\begin{aligned} \mathscr{P}(r) =& \frac{1}{r}\bigg[\int _{0}^{R_{0}}( \frac{\overset{D}{\mu}}{\overset{D}{a^{*}}})(\overset{D}{\kappa}\ \imath _{0}^{(4)}(r^{\prime})+\overset{D}{d}\ \imath _{0}^{(6)}(r^{ \prime}))r^{\prime }dr^{\prime} \\ +&\frac{1}{\pi}\int _{0}^{\pi}\int _{R_{0}}^{R(\theta ^{\prime})}( \frac{\overset{D}{\mu}}{\overset{D}{a^{*}}})(\overset{D}{\kappa}\ F_{0}^{(4)}(r^{ \prime},\theta ^{\prime})+\overset{D}{d}\ F_{0}^{(6)}(r^{\prime}, \theta ^{\prime}))r^{\prime }dr^{\prime }d\theta ^{\prime}\bigg], \end{aligned}$$

(66)

$$\begin{aligned} \mathscr{Q}(r) =& \frac{1}{r}\bigg[\int _{0}^{R_{0}}( \frac{\overset{D}{\kappa}}{\overset{D}{a^{*}}})(-\overset{D}{d}\ \imath _{0}^{(4)}(r^{\prime})+\overset{D}{\mu}\ \imath _{0}^{(6)}(r^{ \prime}))r^{\prime }dr^{\prime} \\ +&\frac{1}{\pi}\int _{0}^{\pi}\int _{R_{0}}^{R(\theta ^{\prime})}( \frac{\overset{D}{\kappa}}{\overset{D}{a^{*}}})(-\overset{D}{d}\ F_{0}^{(4)}(r^{ \prime},\theta ^{\prime})+\overset{D}{\mu}\ F_{0}^{(6)}(r^{\prime}, \theta ^{\prime}))r^{\prime }dr^{\prime }d\theta ^{\prime}\bigg], \end{aligned}$$

(67)

$$\begin{aligned} \mathscr{L}_{n}^{(j)} = \frac{1}{2}\int _{0}^{R_{0}}(r^{\prime})^{n} \imath _{n}^{(j)}(r^{\prime})r^{\prime }dr^{\prime }+\frac{2}{\pi} \int _{0}^{\pi}\int _{R_{0}}^{R(\theta ^{\prime})}(r^{\prime})^{n} F_{n}^{(j)}(r^{ \prime},\theta ^{\prime})r^{\prime }dr^{\prime }d\theta ^{\prime}, \\ j=4,5,6,7; n=0,1,2,\ldots , \end{aligned}$$

(68)

$$\begin{aligned} \mathscr{S}_{0} =& 2\Pi _{0}^{(3)}+k\left [\Pi _{1}^{(1)}+\Pi _{1}^{(2)}+ \overset{D}{\kappa}\frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}}\Pi _{1}^{(4)}+ \overset{D}{\kappa} \frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}}\Pi _{1}^{(5)}+ \overset{D}{\mu}\frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}}\Pi _{1}^{(6)}+ \overset{D}{\mu} \frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}}\Pi _{1}^{(7)} \right ], \end{aligned}$$

(69)

$$\begin{aligned} \mathscr{S}_{1}= 2\Pi _{1}^{(3)}+ k&\left [\Pi _{2}^{(1)}+\Pi _{2}^{(2)}+ \overset{D}{\kappa}\frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}}\Pi _{2}^{(4)}+ \overset{D}{\kappa} \frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}}\Pi _{2}^{(5)} +\overset{D}{\mu}\frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}}\Pi _{2}^{(6)}+ \overset{D}{\mu}\frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}}\Pi _{2}^{(7)} \right ] \\ -2 k&\left [\Pi _{0}^{(1)}+\overset{D}{\kappa} \frac{\overset{D}{b^{*}}}{\overset{D}{a*}}\Pi _{0}^{(4)}+ \overset{D}{\mu}\frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}} \Pi _{0}^{(6)} \right ], \end{aligned}$$

(70)

$$\begin{aligned} \mathscr{S}_{n} = 2\Pi _{n}^{(3)}+k\left [\Pi _{n+1}^{(1)}+\Pi _{n+1}^{(2)}+ \overset{D}{\kappa}\frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}}\Pi _{n+1}^{(4)}+ \overset{D}{\kappa} \frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}}\Pi _{n+1}^{(5)} +\overset{D}{\mu}\frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}}\Pi _{n+1}^{(6)}+ \overset{D}{\mu}\frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}}\Pi _{n+1}^{(7)} \right ] \\ - k\left [\Pi _{n-1}^{(1)}-\Pi _{n-1}^{(2)} +\overset{D}{\kappa} \frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}}\Pi _{n-1}^{(4)}- \overset{D}{\kappa}\frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}}\Pi _{n-1}^{(5)} +\overset{D}{\mu}\frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}}\Pi _{n-1}^{(6)} -\overset{D}{\mu}\frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}}\Pi _{n-1}^{(7)} \right ] , \\ n = 2,3,\ldots ,\qquad \qquad \qquad \end{aligned}$$

(71)

$$\begin{aligned} \mathscr{N}_{0} =& \frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}} \Bigg\{ 2\Pi _{0}^{(3)}+k\bigg[\Pi _{1}^{(1)}+\Pi _{1}^{(2)} - \overset{D}{\kappa}\frac{\overset{D}{q}-\overset{D}{\mu}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{1}^{(4)}-\overset{D}{\kappa}\frac{\overset{D}{q}-\overset{D}{\mu}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{1}^{(5)} \\ +& \overset{D}{\mu}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{1}^{(6)} +\overset{D}{\mu}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{1}^{(7)}\bigg]\Bigg\} , \end{aligned}$$

(72)

$$\begin{aligned} \mathscr{N}_{1} =& \frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}} \Bigg\{ 2\Pi _{0}^{(3)}+k\bigg[\Pi _{1}^{(1)}+\Pi _{1}^{(2)} - \overset{D}{\kappa}\frac{\overset{D}{q}-\overset{D}{\mu}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{1}^{(4)}-\overset{D}{\kappa}\frac{\overset{D}{q}-\overset{D}{\mu}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{1}^{(5)} \\ +& \overset{D}{\mu}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{1}^{(6)} +\overset{D}{\mu}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{1}^{(7)}\bigg] \\ -&2k\bigg[\Pi _{0}^{(1)}-\overset{D}{\kappa}\frac{\overset{D}{q}-\overset{D}{\mu}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{0}^{(4)}+ \overset{D}{\mu}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{1}^{(6)}\bigg]\Bigg\} , \end{aligned}$$

(73)

$$\begin{aligned} \mathscr{N}_{n} =& \frac{\overset{D}{b^{*}}}{\overset{D}{a^{*}}} \Bigg\{ 2\Pi _{n}^{(3)}+k\bigg[\Pi _{n+1}^{(1)}+\Pi _{n+1}^{(2)}- \overset{D}{\kappa}\frac{\overset{D}{q}-\overset{D}{\mu}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{n+1}^{(4)}-\overset{D}{\kappa}\frac{\overset{D}{q}-\overset{D}{\mu}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{n+1}^{(5)} \\ +&\overset{D}{\mu}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{n+1}^{(6)}+ \overset{D}{\mu}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{n+1}^{(7)}\bigg]-k\bigg[\Pi _{n-1}^{(1)}-\Pi _{n-1}^{(2)} \\ -&\overset{D}{\kappa}\frac{\overset{D}{q}-\overset{D}{\mu}(\overset{D}{\hat{\eta}} -\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{n-1}^{(4)} + \overset{D}{\kappa}\frac{\overset{D}{q}-\overset{D}{\mu}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{n-1}^{(5)}+ \overset{D}{\mu}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{n-1}^{(6)} \\ -&\overset{D}{\mu}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{b^{*}}} \Pi _{n-1}^{(7)}\bigg]\Bigg\} , \qquad n=2,3,\ldots , \end{aligned}$$

(74)

$$\begin{aligned} \mathscr{M}_{0} =& \frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}} \Bigg\{ 2\Pi _{0}^{(3)}+k\bigg[\Pi _{1}^{(1)}+\Pi _{1}^{(2)} + \overset{D}{\kappa}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{1}^{(4)} +\overset{D}{\kappa}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{1}^{(5)} \\ -& \overset{D}{\mu}\frac{\overset{D}{e}^{2}-\overset{D}{\kappa}(\hat{\eta}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{1}^{(6)} -\overset{D}{\mu}\frac{\overset{D}{e}^{2}-\overset{D}{\kappa}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{1}^{(7)}\bigg]\Bigg\} , \end{aligned}$$

(75)

$$\begin{aligned} \mathscr{M}_{1} =& \frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}} \Bigg\{ 2\Pi _{1}^{(3)}+k\bigg[\Pi _{1}^{(1)}+\Pi _{1}^{(2)} + \overset{D}{\kappa}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{1}^{(4)}+\overset{D}{\kappa}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{1}^{(5)} \\ -& \overset{D}{\mu}\frac{\overset{D}{e}^{2}-\overset{D}{\kappa}(\hat{\eta}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{1}^{(6)} -\overset{D}{\mu}\frac{\overset{D}{e}^{2}-\overset{D}{\kappa}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{1}^{(7)}\bigg] \\ -& 2k\bigg[\Pi _{0}^{(1)}+\overset{D}{\kappa}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{n-1}^{(4)}-\overset{D}{k} \frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{n-1}^{(5)} \\ -&\overset{D}{\mu}\frac{\overset{D}{e}^{2}-\overset{D}{\kappa}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{n-1}^{(6)} +\overset{D}{\mu}\frac{\overset{D}{e}^{2}-\overset{D}{\kappa}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{n-1}^{(7)}\bigg]\Bigg\} , \end{aligned}$$

(76)

$$\begin{aligned} \mathscr{M}_{n} =& \frac{\overset{D}{c^{*}}}{\overset{D}{a^{*}}} \Bigg\{ 2\Pi _{n}^{(3)}+k\bigg[\Pi _{n+1}^{(1)}+\Pi _{n+1}^{(2)} + \overset{D}{\kappa}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{n+1}^{(4)} \\ +& \overset{D}{\kappa}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{n+1}^{(5)}-\overset{D}{\mu}\frac{\overset{D}{e}^{2}-\overset{D}{\kappa}(\hat{\eta}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{n+1}^{(6)}- \overset{D}{\mu}\frac{\overset{D}{e}^{2}-\overset{D}{\kappa}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{n+1}^{(7)}\bigg] \\ -& k\bigg[\Pi _{n-1}^{(1)}-\Pi _{n-1}^{(2)}+\overset{D}{\kappa}\frac{\overset{D}{e}\ \overset{D}{q}-\overset{D}{d}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{0}^{(4)}- \overset{D}{\mu}\frac{\overset{D}{e}^{2}-\overset{D}{\kappa}(\overset{D}{\hat{\eta}}-\overset{D}{\eta})}{\overset{D}{c^{*}}} \Pi _{0}^{(6)}\bigg]\Bigg\} , \\ n&=2,3,\ldots , \end{aligned}$$

(77)

in which

$$ \frac{4\mathbf{i}\overset{D}{\rho}\omega ^{2}}{k^{2}} \Pi _{n}^{(j)}= \pi \int _{0}^{R_{0}}\imath _{n}^{(j)}(r^{\prime})J_{n}(kr^{\prime})r^{ \prime }dr^{\prime}+\alpha _{n}\int _{0}^{\pi}\int _{R_{0}}^{R( \theta ^{\prime})}F_{n}^{(j)}(r^{\prime},\theta ^{\prime})J_{n}(kr^{ \prime})r^{\prime }dr^{\prime }d\theta ^{\prime}. $$

(78)