Appendix: Scalar Formulation of the Linearized Boundary Value Problem
With respect to the representations (2), (3), (15), the expressions for the linearized stretch tensor \({{\mathbf {\mathsf{{Y}}}}}^\bullet \) and wryness tensor \({{\mathbf {\mathsf{{L}}}}}^\bullet \) have the form:
$$\begin{aligned} {{{\mathbf {\mathsf{{Y}}}}}}^\bullet&=\displaystyle \left( {\frac{\partial v_\varPhi }{\partial r}-{f}^{\prime }\omega _Z } \right) {\mathbf {e}}_r \otimes {\mathbf {e}}_\varphi + \frac{1}{r}\left( {\frac{\partial v_R }{\partial \varphi }-v_\varPhi +f \omega _Z } \right) {\mathbf {e}}_\varphi \otimes {\mathbf {e}}_r \nonumber \\&\quad +\displaystyle \left( {\frac{\partial v_Z }{\partial r}+{f}^{\prime } \omega _\varPhi } \right) {\mathbf {e}}_r \otimes {\mathbf {e}}_z + \left( {\frac{\partial v_R }{\partial z}-\alpha \omega _\varPhi } \right) {\mathbf {e}}_z \otimes {\mathbf {e}}_r \\&\quad +\displaystyle \frac{1}{r}\left( {\frac{\partial v_Z }{\partial \varphi }-f \omega _R } \right) {\mathbf {e}}_\varphi \otimes {\mathbf {e}}_z + \left( {\frac{\partial v_\varPhi }{\partial z}+\alpha \omega _R } \right) {\mathbf {e}}_z \otimes {\mathbf {e}}_\varphi \nonumber \\&\quad +\displaystyle \frac{\partial v_R }{\partial r}{\mathbf {e}}_r \otimes {\mathbf {e}}_r +\frac{1}{r}\left( {\frac{\partial v_\varPhi }{\partial \varphi }+v_R } \right) {\mathbf {e}}_\varphi \otimes {\mathbf {e}}_\varphi +\frac{\partial v_Z }{\partial z}{\mathbf {e}}_z \otimes {\mathbf {e}}_z\nonumber \end{aligned}$$
(18)
$$\begin{aligned} {{{\mathbf {\mathsf{{L}}}}}}^\bullet&=\displaystyle \frac{\partial \omega _R }{\partial r}{\mathbf {e}}_r \otimes {\mathbf {e}}_r + \frac{1}{r}\left( {\frac{\partial \omega _\varPhi }{\partial \varphi }+\omega _R } \right) {\mathbf {e}}_\varphi \otimes {\mathbf {e}}_\varphi +\frac{\partial \omega _Z }{\partial z}{\mathbf {e}}_z \otimes {\mathbf {e}}_z \nonumber \\&\quad +\displaystyle \frac{\partial \omega _\varPhi }{\partial r}{\mathbf {e}}_r \otimes {\mathbf {e}}_\varphi +\frac{1}{r}\left( {\frac{\partial \omega _R }{\partial \varphi }-\omega _\varPhi } \right) {\mathbf {e}}_\varphi \otimes {\mathbf {e}}_r + \frac{\partial \omega _Z }{\partial r}{\mathbf {e}}_r \otimes {\mathbf {e}}_z \\&\quad +\displaystyle \frac{\partial \omega _R }{\partial z}{\mathbf {e}}_z \otimes {\mathbf {e}}_r +\frac{1}{r}\frac{\partial \omega _Z }{\partial \varphi }{\mathbf {e}}_\varphi \otimes {\mathbf {e}}_z +\frac{\partial \omega _\varPhi }{\partial z}{\mathbf {e}}_z \otimes {\mathbf {e}}_\varphi \nonumber \end{aligned}$$
(19)
According to relations (2), (4), (11), (12), (15), (18) (19), the components of the linearized Piola-type stress tensor \({{\mathbf {\mathsf{{D}}}}}^\bullet \) and couple stress tensor \({{\mathbf {\mathsf{{G}}}}}^\bullet \) are written as follows:
$$\begin{aligned} {\mathbf {e}}_r \cdot {{{\mathbf {\mathsf{{D}}}}}}^\bullet \cdot {\mathbf {e}}_R&= \left( {\lambda +\chi } \right) \frac{\partial v_R }{\partial r}\,\,+\frac{\lambda }{r}\left( {\frac{\partial v_\varPhi }{\partial \varphi }+v_R } \right) +\lambda \frac{\partial v_Z }{\partial z} \nonumber \\[-0.5ex] \!\!\!\!\!\! {\mathbf {e}}_r \cdot {{{\mathbf {\mathsf{{D}}}}}}^\bullet \cdot {\mathbf {e}}_\varPhi&= \left( {\mu +\kappa } \right) \frac{\partial v_\varPhi }{\partial r}+\frac{\mu }{r}\!\left( {\frac{\partial v_R }{\partial \varphi }-v_\varPhi }\!\right) \!+\left[ \mu \left( {{f}^{\prime }+\frac{f}{r}} \right) +\lambda s-\chi \right] \omega _Z \nonumber \\[-0.5ex] {\mathbf {e}}_r \cdot {{{\mathbf {\mathsf{{D}}}}}}^\bullet \cdot {\mathbf {e}}_Z&= \left( {\mu +\kappa } \right) \frac{\partial v_Z }{\partial r}+\mu \frac{\partial v_R }{\partial z}-\left[ \mu \left( {{f}^{\prime }+\alpha } \right) +\lambda s-\chi \right] \omega _\varPhi \nonumber \\[-0.5ex] {\mathbf {e}}_\varphi \cdot {{{\mathbf {\mathsf{{D}}}}}}^\bullet \cdot {\mathbf {e}}_R&= \frac{\mu +\kappa }{r}\!\left( {\frac{\partial v_R }{\partial \varphi }-v_\varPhi } \right) +\mu \frac{\partial v_\varPhi }{\partial r}-\left[ \mu \left( {{f}^{\prime }+\frac{f}{r}} \right) +\lambda s-\chi \right] \omega _Z \nonumber \\[-0.5ex] {\mathbf {e}}_\varphi \cdot {{{\mathbf {\mathsf{{D}}}}}}^\bullet \cdot {\mathbf {e}}_\varPhi&= \lambda \frac{\partial v_R }{\partial r}\,\,+\frac{\lambda +\chi }{r}\left( {\frac{\partial v_\varPhi }{\partial \varphi }+v_R } \right) +\lambda \frac{\partial v_Z }{\partial z} \\[-0.5ex] {\mathbf {e}}_\varphi \cdot {{{\mathbf {\mathsf{{D}}}}}}^\bullet \cdot {\mathbf {e}}_Z&= \frac{\mu +\kappa }{r}\frac{\partial v_Z }{\partial \varphi }+\mu \frac{\partial v_\varPhi }{\partial z}+\left[ \mu \left( {\frac{f}{r} +\alpha } \right) +\lambda s-\chi \right] \omega _R \nonumber \\[-0.5ex] {\mathbf {e}}_z \cdot {{{\mathbf {\mathsf{{D}}}}}}^\bullet \cdot {\mathbf {e}}_R&= \left( {\mu +\kappa } \right) \frac{\partial v_R }{\partial z}+\mu \frac{\partial v_Z }{\partial r}+\left[ \mu \left( {{f}^{\prime }+\alpha } \right) +\lambda s-\chi \right] \omega _\varPhi \nonumber \\[-0.5ex] {\mathbf {e}}_z \cdot {{{\mathbf {\mathsf{{D}}}}}}^\bullet \cdot {\mathbf {e}}_\varPhi&= \left( {\mu +\kappa } \right) \frac{\partial v_\varPhi }{\partial z}+\frac{\mu }{r}\frac{\partial v_Z }{\partial \varphi }-\left[ \mu \left( {\frac{f}{r} +\alpha } \right) +\lambda s-\chi \right] \omega _R \nonumber \\[-0.5ex] {\mathbf {e}}_z \cdot {{{\mathbf {\mathsf{{D}}}}}}^\bullet \cdot {\mathbf {e}}_Z&= \lambda \frac{\partial v_R }{\partial r}\,\,+\frac{\lambda }{r}\left( {\frac{\partial v_\varPhi }{\partial \varphi }+v_R } \right) +\left( {\lambda +\chi } \right) \frac{\partial v_Z }{\partial z} \nonumber \end{aligned}$$
(20)
$$\begin{aligned} {\mathbf {e}}_r \cdot {{{\mathbf {\mathsf{{G}}}}}}^\bullet \cdot {\mathbf {e}}_R&= (\gamma _1 +\gamma _2 +\gamma _3)\frac{\partial \omega _R }{\partial r}\,\,+\frac{\gamma _1 }{r}\left( {\frac{\partial \omega _\varPhi }{\partial \varphi }+\omega _R } \right) +\gamma _1 \frac{\partial \omega _Z }{\partial z} \nonumber \\[-0.5ex] {\mathbf {e}}_r \cdot {{{\mathbf {\mathsf{{G}}}}}}^\bullet \cdot {\mathbf {e}}_\varPhi&= \gamma _2 \frac{\partial \omega _\varPhi }{\partial r}+\frac{\gamma _3 }{r}\left( {\frac{\partial \omega _R }{\partial \varphi }-\omega _\varPhi } \right) \nonumber \\[-0.5ex] {\mathbf {e}}_r \cdot {{{\mathbf {\mathsf{{G}}}}}}^\bullet \cdot {\mathbf {e}}_Z&= \gamma _2 \frac{\partial \omega _Z }{\partial r}+\gamma _3 \frac{\partial \omega _R }{\partial z} \nonumber \\[-0.5ex] {\mathbf {e}}_\varphi \cdot {{{\mathbf {\mathsf{{G}}}}}}^\bullet \cdot {\mathbf {e}}_R&= \frac{\gamma _2 }{r}\left( {\frac{\partial \omega _R }{\partial \varphi }-\omega _\varPhi } \right) +\gamma _3 \frac{\partial \omega _\varPhi }{\partial r} \nonumber \\[-0.5ex] {\mathbf {e}}_\varphi \cdot {{{\mathbf {\mathsf{{G}}}}}}^\bullet \cdot {\mathbf {e}}_\varPhi&= \gamma _1 \frac{\partial \omega _R }{\partial r}\,\,+\frac{\gamma _1 +\gamma _2 +\gamma _3}{r}\left( {\frac{\partial \omega _\varPhi }{\partial \varphi }+\omega _R } \right) +\gamma _1 \frac{\partial \omega _Z }{\partial z} \\[-0.5ex] {\mathbf {e}}_\varphi \cdot {{{\mathbf {\mathsf{{G}}}}}}^\bullet \cdot {\mathbf {e}}_Z&= \frac{\gamma _2 }{r}\frac{\partial \omega _Z }{\partial \varphi }+\gamma _3 \frac{\partial \omega _\varPhi }{\partial z} \nonumber \\[-0.5ex] {\mathbf {e}}_z \cdot {{{\mathbf {\mathsf{{G}}}}}}^\bullet \cdot {\mathbf {e}}_R&= \gamma _2 \frac{\partial \omega _R }{\partial z}+\gamma _3 \frac{\partial \omega _Z }{\partial r} \nonumber \\[-0.5ex] {\mathbf {e}}_z \cdot {{{\mathbf {\mathsf{{G}}}}}}^\bullet \cdot {\mathbf {e}}_\varPhi&= \gamma _2 \frac{\partial \omega _\varPhi }{\partial z}+\frac{\gamma _3 }{r}\frac{\partial \omega _Z }{\partial \varphi } \nonumber \\[-0.5ex] {\mathbf {e}}_z \cdot {{{\mathbf {\mathsf{{G}}}}}}^\bullet \cdot {\mathbf {e}}_Z&= \gamma _1 \frac{\partial \omega _R }{\partial r}\,\,+\frac{\gamma _1 }{r}\left( {\frac{\partial \omega _\varPhi }{\partial \varphi }+\omega _R } \right) +(\gamma _1 +\gamma _2 +\gamma _3)\frac{\partial \omega _Z }{\partial z} \nonumber \end{aligned}$$
(21)
By taking into account the expressions (3), (6), (15), (16), (20), (21), we derive a scalar form of the linearized equilibrium equations (10)
$$\begin{aligned}&\left( {\lambda +\chi } \right) {V}^{\prime \prime }_R + \left( \lambda ^{\prime } +\chi ^{\prime }+\frac{\lambda +\chi }{r}\right) {V}^{\prime }_R -\frac{\lambda -\lambda ^{\prime }r+\left( {\mu +\kappa } \right) \zeta +\mu }{r^2}V_R \nonumber \\&\quad + \,\frac{n\left( {\lambda +\mu } \right) }{r}{V}^{\prime }_\varPhi - \frac{n\left( {\lambda - \lambda ^{\prime }r +3\mu +2\kappa } \right) }{r^2}V_\varPhi + \beta \left( {\lambda +\mu } \right) {V}^{\prime }_Z \nonumber \\&\quad + \,\beta \lambda ^{\prime }{V}_Z +\beta B_2 \varOmega _\varPhi -\frac{nB_3 }{r}\varOmega _Z =0 \nonumber \\[5pt]&\left( {\mu +\kappa } \right) {V}^{\prime \prime }_\varPhi -\frac{n\left( {\lambda +\mu } \right) }{r}{V}^{\prime }_R -\frac{n\left( {\lambda +\mu ^{\prime } r+ 3\mu +2\kappa } \right) }{r^2}V_R \nonumber \\&\quad + \,\left( \mu ^{\prime } +\kappa ^{\prime } + \frac{\mu +\kappa }{r}\right) {V}^{\prime }_\varPhi - \frac{\left( {\lambda +\mu } \right) n^2+\mu ^{\prime } r+\left( {\mu +\kappa } \right) \zeta }{r^2}V_\varPhi \nonumber \\&\quad - \,\frac{n\beta \left( {\lambda +\mu } \right) }{r}V_Z -\beta B_1 \varOmega _R +B_3 {\varOmega }^{\prime }_Z +B^{\;\prime }_3 {\varOmega }_Z =0 \nonumber \\[5pt]&\left( {\mu +\kappa } \right) {V}^{\prime \prime }_Z -\beta \left( {\lambda +\mu } \right) {V}^{\prime }_R -\frac{\beta \left( {\lambda +\mu ^{\prime } r + \mu } \right) }{r}V_R -\frac{n\beta \left( {\lambda +\mu } \right) }{r}V_\varPhi \nonumber \\&\quad + \,\left( \mu ^{\prime } +\kappa ^{\prime } + \frac{\mu +\kappa }{r}\right) {V}^{\prime }_Z -\left[ {\left( {\lambda +\mu } \right) \beta ^2+\frac{\zeta -1}{r^2}\left( {\mu +\kappa } \right) } \right] V_Z \nonumber \\&\quad + \,\frac{nB_1 }{r}\varOmega _R -B_2 {\varOmega }^{\prime }_\varPhi -\left( B^{\;\prime }_2 + \frac{B_2 }{r}\right) \varOmega _\varPhi =0 \\[5pt]&\gamma {\varOmega }^{\prime \prime }_R +\left( \gamma ^{\;\prime }+\frac{\gamma }{r}\right) {\varOmega }^{\prime }_R -\left[ {\frac{\gamma -\gamma ^{\;\prime }_1 r+\gamma _2 \left( {\zeta -1} \right) }{r^2}-\left( {\frac{f}{r}+\alpha } \right) B_1 } \right] \varOmega _R \nonumber \\&\quad - \,\frac{n\left( {\gamma -\gamma _2 } \right) }{r}{\varOmega }^{\prime }_\varPhi +\frac{n\left( {\gamma -\gamma ^{\;\prime }_1 r +\gamma _2 } \right) }{r^2}\varOmega _\varPhi -\beta \left( {\gamma -\gamma _2 } \right) {\varOmega }^{\prime }_Z \nonumber \\&\quad - \,\beta \gamma ^{\;\prime }_1{\varOmega }_Z-\beta B_1 V_\varPhi +\frac{nB_1 }{r}V_Z=0 \nonumber \\[5pt]&\gamma _2 {\varOmega }^{\prime \prime }_\varPhi +\frac{n\left( {\gamma -\gamma _2 } \right) }{r}{\varOmega }^{\prime }_R +\frac{n\left( {\gamma + \gamma ^{\;\prime }_3 r+ \gamma _2 } \right) }{r^2}\varOmega _R +\left( \gamma ^{\;\prime }_2 + \frac{\gamma _2 }{r}\right) {\varOmega }^{\prime }_\varPhi \nonumber \\&\quad - \,\left[ {\frac{\left( {\gamma -\gamma _2 } \right) n^2+\gamma ^{\;\prime }_3 r+\gamma _2 \zeta }{r^2}-\left( {{f}^{\prime }+\alpha } \right) B_2 } \right] \varOmega _\varPhi \nonumber \\&\quad - \,\frac{n\beta \left( {\gamma -\gamma _2 } \right) }{r}\varOmega _Z +\beta B_2 V_R +B_2 {V}^{\prime }_Z =0 \nonumber \\[5pt]&\gamma _2 {\varOmega }^{\prime \prime }_Z +\beta \left( {\gamma -\gamma _2 } \right) {\varOmega }^{\prime }_R +\frac{\beta \left( {\gamma + \gamma ^{\;\prime }_3 r -\gamma _2 } \right) }{r}\varOmega _R - \frac{n\beta \left( {\gamma -\gamma _2 } \right) }{r}\varOmega _\varPhi \nonumber \\&\quad + \,\left( \gamma ^{\;\prime }_2 + \frac{\gamma _2 }{r}\right) {\varOmega }^{\prime }_Z -\left[ {\gamma \beta ^2+\frac{n^2}{r^2}\gamma _2 -B_3 \left( {{f}^{\prime }+\frac{f}{r}} \right) } \right] \varOmega _Z \nonumber \\&\quad - \,\frac{nB_3 }{r}V_R -B_3 {V}^{\prime }_\varPhi -\frac{B_3 }{r}V_\varPhi =0 \nonumber \end{aligned}$$
(22)
Here the following notations are used:
$$\begin{aligned} \gamma =\gamma _1 +\gamma _2 +\gamma _3, \qquad \zeta =n^2+r^2\beta ^2+1 \end{aligned}$$
$$\begin{aligned} B_1&=\mu \left( {\frac{f}{r} +\alpha } \right) +\lambda s-\chi , \\ B_2&=\mu \left( {{f}^{\prime }+\alpha } \right) +\lambda s-\chi , \\ B_3&=\mu \left( {{f}^{\prime }+\frac{f}{r}} \right) +\lambda s-\chi \end{aligned}$$
The scalar representation of the linearized boundary conditions (13) on the lateral surface of the rod is written as:
$$\begin{aligned}&\left[ {\lambda (r_0) +\chi (r_0) } \right] {V}^{\prime }_R(r_0) + \frac{\lambda (r_0) +\alpha p}{r_0 }\left[ {V_R(r_0) +nV_\varPhi (r_0) } \right] \nonumber \\&\quad + \,\beta \left[ \lambda (r_0) +p \frac{f(r_0)}{r_0}\right] V_Z(r_0) =0 \nonumber \\&\left[ {\mu (r_0) +\kappa (r_0) } \right] {V}^{\prime }_\varPhi (r_0) + \frac{\alpha p-\mu (r_0) }{r_0 }\left[ {nV_R(r_0) +V_\varPhi (r_0) } \right] +B_3(r_0) \varOmega _Z(r_0) =0 \nonumber \\&\left[ {\mu (r_0) +\kappa (r_0) } \right] {V}^{\prime }_Z(r_0) +\beta \left[ p \frac{f(r_0)}{r_0}-\mu (r_0) \right] V_R(r_0)-B_2(r_0) \varOmega _\varPhi (r_0) =0 \nonumber \\&\gamma (r_0) {\varOmega }^{\prime }_R(r_0) +\frac{\gamma _1(r_0) }{r_0 }\left[ {\varOmega _R(r_0) -n \varOmega _\varPhi (r_0) } \right] -\beta \gamma _1(r_0) \varOmega _Z(r_0) =0 \nonumber \\&\frac{\gamma _3(r_0) }{r_0 }\left[ {n\varOmega _R(r_0) -\varOmega _\varPhi (r_0) } \right] +\gamma _2(r_0) {\varOmega }^{\prime }_\varPhi (r_0) =0 \nonumber \\&\beta \gamma _3(r_0) \varOmega _R(r_0) +\gamma _2(r_0) {\varOmega }^{\prime }_Z(r_0) =0 \end{aligned}$$
(23)
