Buckling of Sandwich Tube with Foam Core Under Combined Loading

Abstract

In the framework of a general stability theory for three-dimensional bodies the buckling analysis is carried out for the nonlinearly elastic three-layer cylindrical tube subjected to axial compression under internal or external pressure. It is assumed that the middle layer (core) of the tube is made of metal or polymer foam, and to describe its behavior the model of micropolar continuum is used. Such approach allows to study in detail the influence of foam microstructure on the deformation stability, which is especially important when the macroscopic dimensions of the tube are comparable with the average size of the foam cells. The inner and outer layers (coatings) of the tube are assumed to be made of the classic non-polar materials. Applying linearization the neutral equilibrium equations have been derived, which describe the perturbed state of the cylindrical sandwich tube. By solving these equations numerically for some specific materials, the critical curves and corresponding buckling modes have been found and the stability regions have been constructed in the planes of loading parameters (relative axial compression and internal or external pressure). Using the obtained results, the influence of coatings properties, as well as the overall size of the tube, on the loss of stability has been analyzed.

Appendix: Derivation of Neutral Equilibrium Equations

With respect to the representations (3), (4), (6) and (22), the expressions for the linearized stretch tensors ${{{\mathbf {\mathsf{{Y}}}}}}^\bullet$, ${{{\mathbf {\mathsf{{U}}}}}}^\bullet _{-}$ and ${{{\mathbf {\mathsf{{U}}}}}}^\bullet _{+}$, and wryness tensor ${{{\mathbf {\mathsf{{L}}}}}}^\bullet$ have the form:

$$\begin{aligned} {{{{\mathbf {\mathsf{{Y}}}}}}}^\bullet&=\left( {\frac{\partial v_\varPhi }{\partial r}-{f}^{\prime }\omega _Z } \right) {\vec {e}}_r \otimes {\vec {e}}_\varphi + \frac{1}{r}\left( {\frac{\partial v_R }{\partial \varphi }-v_\varPhi +f \omega _Z } \right) {\vec {e}}_\varphi \otimes {\vec {e}}_r \nonumber \\&\quad +\left( {\frac{\partial v_Z }{\partial r}+{f}^{\prime } \omega _\varPhi } \right) {\vec {e}}_r \otimes {\vec {e}}_z + \left( {\frac{\partial v_R }{\partial z}-\alpha \omega _\varPhi } \right) {\vec {e}}_z \otimes {\vec {e}}_r \nonumber \\&\quad +\frac{1}{r}\left( {\frac{\partial v_Z }{\partial \varphi }-f \omega _R } \right) {\vec {e}}_\varphi \otimes {\vec {e}}_z + \left( {\frac{\partial v_\varPhi }{\partial z}+\alpha \omega _R } \right) {\vec {e}}_z \otimes {\vec {e}}_\varphi \nonumber \\&\quad +\frac{\partial v_R }{\partial r}{\vec {e}}_r \otimes {\vec {e}}_r +\frac{1}{r}\left( {\frac{\partial v_\varPhi }{\partial \varphi }+v_R } \right) {\vec {e}}_\varphi \otimes {\vec {e}}_\varphi +\frac{\partial v_Z }{\partial z}{\vec {e}}_z \otimes {\vec {e}}_z, \end{aligned}$$

(25)

$$\begin{aligned} {{{{\mathbf {\mathsf{{U}}}}}}}_\pm ^\bullet&=\frac{\partial v_R^\pm }{\partial r}{\vec {e}}_r \otimes {\vec {e}}_r +\frac{1}{r}\left( {\frac{\partial v_\varPhi ^\pm }{\partial \varphi }+v_R^\pm } \right) {\vec {e}}_\varphi \otimes {\vec {e}}_\varphi +\frac{\partial v_Z^\pm }{\partial z}{\vec {e}}_z \otimes {\vec {e}}_z \nonumber \\&\quad +\frac{1}{rf_\pm ^\prime +f_\pm }\left( {f_\pm ^\prime \left( {\frac{\partial v_R^\pm }{\partial \varphi }-v_\varPhi ^\pm } \right) +f_\pm \frac{\partial v_\varPhi ^\pm }{\partial r}} \right) \left( {{\vec {e}}_r \otimes {\vec {e}}_\varphi +{\vec {e}}_\varphi \otimes {\vec {e}}_r } \right) \nonumber \\&\quad +\frac{1}{f_\pm ^\prime +\alpha }\left( {f_\pm ^\prime \frac{\partial v_R^\pm }{\partial z}+\alpha \frac{\partial v_Z^\pm }{\partial r}} \right) \left( {{\vec {e}}_r \otimes {\vec {e}}_z +{\vec {e}}_z \otimes {\vec {e}}_r } \right) \nonumber \\&\quad +\frac{1}{f_\pm +\alpha r}\left( {f_\pm \frac{\partial v_\varPhi ^\pm }{\partial z}+\alpha \frac{\partial v_Z^\pm }{\partial \varphi }} \right) \left( {{\vec {e}}_z \otimes {\vec {e}}_\varphi +{\vec {e}}_\varphi \otimes {\vec {e}}_z } \right) , \end{aligned}$$

(26)

$$\begin{aligned} {{{{\mathbf {\mathsf{{L}}}}}}}^\bullet&=\frac{\partial \omega _R }{\partial r}{\vec {e}}_r \otimes {\vec {e}}_r + \frac{1}{r}\left( {\frac{\partial \omega _\varPhi }{\partial \varphi }+\omega _R } \right) {\vec {e}}_\varphi \otimes {\vec {e}}_\varphi +\frac{\partial \omega _Z }{\partial z}{\vec {e}}_z \otimes {\vec {e}}_z \nonumber \\&\quad +\frac{\partial \omega _\varPhi }{\partial r}{\vec {e}}_r \otimes {\vec {e}}_\varphi +\frac{1}{r}\left( {\frac{\partial \omega _R }{\partial \varphi }-\omega _\varPhi } \right) {\vec {e}}_\varphi \otimes {\vec {e}}_r + \frac{\partial \omega _Z }{\partial r}{\vec {e}}_r \otimes {\vec {e}}_z \nonumber \\&\quad + \frac{\partial \omega _R }{\partial z}{\vec {e}}_z \otimes {\vec {e}}_r +\frac{1}{r}\frac{\partial \omega _Z }{\partial \varphi }{\vec {e}}_\varphi \otimes {\vec {e}}_z +\frac{\partial \omega _\varPhi }{\partial z}{\vec {e}}_z \otimes {\vec {e}}_\varphi . \end{aligned}$$

(27)

According to the relations (3), (5), (6), (15)–(17), (22), (25)–(27), the linearized Piola-type stress tensor ${{{\mathbf {\mathsf{{D}}}}}}^\bullet$ and couple stress tensor ${{{\mathbf {\mathsf{{G}}}}}}^\bullet$, and the linearized Piola stress tensors ${{{\mathbf {\mathsf{{D}}}}}}_{-}^\bullet$ and ${{{\mathbf {\mathsf{{D}}}}}}_{+}^\bullet$ are written as follows:

$$\begin{aligned} {{{{\mathbf {\mathsf{{D}}}}}}}^\bullet&=\left[ \left( {\zeta +\tau } \right) \frac{\partial v_R }{\partial r}\,\,+\frac{\lambda }{r}\left( {\frac{\partial v_\varPhi }{\partial \varphi }+v_R } \right) +\lambda \frac{\partial v_Z }{\partial z} \right] {\vec {e}}_r \otimes {\vec {e}}_R \nonumber \\&\quad +\left[ \tau \frac{\partial v_\varPhi }{\partial r}+\frac{\mu }{r}\!\left( {\frac{\partial v_R }{\partial \varphi }-v_\varPhi }\!\right) \!+B_3 \omega _Z \right] {\vec {e}}_r \otimes {\vec {e}}_\varPhi \nonumber \\&\quad +\left[ \tau \frac{\partial v_Z }{\partial r}+\mu \frac{\partial v_R }{\partial z}-B_2\omega _\varPhi \right] {\vec {e}}_r \otimes {\vec {e}}_Z +\left[ \tau \frac{\partial v_R }{\partial z}+\mu \frac{\partial v_Z }{\partial r}+B_2\omega _\varPhi \right] {\vec {e}}_z \otimes {\vec {e}}_R \nonumber \\&\quad +\left[ \frac{\tau }{r}\!\left( {\frac{\partial v_R }{\partial \varphi }-v_\varPhi } \right) +\mu \frac{\partial v_\varPhi }{\partial r}-B_3 \omega _Z \right] {\vec {e}}_\varphi \otimes {\vec {e}}_R \nonumber \\&\quad +\left[ \lambda \frac{\partial v_R }{\partial r}\,\,+\frac{\zeta +\tau }{r}\left( {\frac{\partial v_\varPhi }{\partial \varphi }+v_R } \right) +\lambda \frac{\partial v_Z }{\partial z} \right] {\vec {e}}_\varphi \otimes {\vec {e}}_\varPhi \nonumber \\&\quad +\left[ \frac{\tau }{r}\frac{\partial v_Z }{\partial \varphi }+\mu \frac{\partial v_\varPhi }{\partial z}+B_1 \omega _R \right] {\vec {e}}_\varphi \otimes {\vec {e}}_Z + \left[ \tau \frac{\partial v_\varPhi }{\partial z}+\frac{\mu }{r}\frac{\partial v_Z }{\partial \varphi }-B_1\omega _R \right] {\vec {e}}_z \otimes {\vec {e}}_\varPhi \nonumber \\&\quad +\left[ \lambda \frac{\partial v_R }{\partial r}\,\,+\frac{\lambda }{r}\left( {\frac{\partial v_\varPhi }{\partial \varphi }+v_R } \right) +\left( {\zeta +\tau } \right) \frac{\partial v_Z }{\partial z} \right] {\vec {e}}_z \otimes {\vec {e}}_Z, \end{aligned}$$

(28)

$$\begin{aligned} {{{{\mathbf {\mathsf{{G}}}}}}}^\bullet&=\left[ (\gamma +\gamma _2)\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} \right] {\vec {e}}_r \otimes {\vec {e}}_R \nonumber \\&\quad +\left[ \gamma _2 \frac{\partial \omega _\varPhi }{\partial r}+\frac{\gamma _3 }{r}\left( {\frac{\partial \omega _R }{\partial \varphi }-\omega _\varPhi } \right) \right] {\vec {e}}_r \otimes {\vec {e}}_\varPhi \nonumber \\&\quad +\left[ \gamma _2 \frac{\partial \omega _Z }{\partial r}+\gamma _3 \frac{\partial \omega _R }{\partial z} \right] {\vec {e}}_r \otimes {\vec {e}}_Z +\left[ \gamma _2 \frac{\partial \omega _R }{\partial z}+\gamma _3 \frac{\partial \omega _Z }{\partial r} \right] {\vec {e}}_z \otimes {\vec {e}}_R \nonumber \\&\quad +\left[ \frac{\gamma _2 }{r}\left( {\frac{\partial \omega _R }{\partial \varphi }-\omega _\varPhi } \right) +\gamma _3 \frac{\partial \omega _\varPhi }{\partial r} \right] {\vec {e}}_\varphi \otimes {\vec {e}}_R \nonumber \\&\quad +\left[ \gamma _1 \frac{\partial \omega _R }{\partial r}\,\,+\frac{\gamma +\gamma _2 }{r}\left( {\frac{\partial \omega _\varPhi }{\partial \varphi }+\omega _R } \right) +\gamma _1 \frac{\partial \omega _Z }{\partial z} \right] {\vec {e}}_\varphi \otimes {\vec {e}}_\varPhi \nonumber \\&\quad +\left[ \frac{\gamma _2 }{r}\frac{\partial \omega _Z }{\partial \varphi }+\gamma _3 \frac{\partial \omega _\varPhi }{\partial z} \right] {\vec {e}}_\varphi \otimes {\vec {e}}_Z +\left[ \gamma _2 \frac{\partial \omega _\varPhi }{\partial z}+\frac{\gamma _3 }{r}\frac{\partial \omega _Z }{\partial \varphi } \right] {\vec {e}}_z \otimes {\vec {e}}_\varPhi \nonumber \\&\quad +\left[ \gamma _1 \frac{\partial \omega _R }{\partial r}\,\,+\frac{\gamma _1 }{r}\left( {\frac{\partial \omega _\varPhi }{\partial \varphi }+\omega _R } \right) +(\gamma +\gamma _2 )\frac{\partial \omega _Z }{\partial z} \right] {\vec {e}}_z \otimes {\vec {e}}_Z, \end{aligned}$$

(29)

$$\begin{aligned} {{{{\mathbf {\mathsf{{D}}}}}}}_\pm ^\bullet&=\left[ \left( {\zeta _\pm +\mu _\pm } \right) \frac{\partial v_R^\pm }{\partial r}\,\,+\frac{\lambda _\pm }{r}\left( {\frac{\partial v_\varPhi ^\pm }{\partial \varphi }+v_R^\pm } \right) +\lambda _\pm \frac{\partial v_Z^\pm }{\partial z} \right] {\vec {e}}_r \otimes {\vec {e}}_R \nonumber \\&\quad +\left[ \left( {\mu _\pm +B^{\pm }_3} \right) \,\frac{\partial v_\varPhi ^\pm }{\partial r}+\frac{\mu _\pm - B^{\pm }_3}{r}\left( {\frac{\partial v_R^\pm }{\partial \varphi }-v_\varPhi ^\pm } \right) \right] {\vec {e}}_r \otimes {\vec {e}}_\varPhi \nonumber \\&\quad +\left[ \left( {\mu _\pm +B^{\pm }_2} \right) \frac{\partial v_Z^\pm }{\partial r}+\left( {\mu _\pm -B^{\pm }_2} \right) \frac{\partial v_R^\pm }{\partial z} \right] {\vec {e}}_r \otimes {\vec {e}}_Z \nonumber \\&\quad +\left[ \frac{\mu _\pm + B^{\pm }_3}{r}\left( {\frac{\partial v_R^\pm }{\partial \varphi }-v_\varPhi ^\pm } \right) +\left( {\mu _\pm -B^{\pm }_3} \right) \frac{\partial v_\varPhi ^\pm }{\partial r} \right] {\vec {e}}_\varphi \otimes {\vec {e}}_R \nonumber \\&\quad +\left[ \lambda _\pm \frac{\partial v_R^\pm }{\partial r}\,\,+\frac{\zeta _\pm +\mu _\pm }{r}\left( {\frac{\partial v_\varPhi ^\pm }{\partial \varphi }+v_R^\pm } \right) +\lambda _\pm \frac{\partial v_Z^\pm }{\partial z} \right] {\vec {e}}_\varphi \otimes {\vec {e}}_\varPhi \nonumber \\&\quad +\left[ \frac{\mu _\pm + B^{\pm }_1}{r}\frac{\partial v_Z^\pm }{\partial \varphi }+\left( {\mu _\pm -B^{\pm }_1} \right) \frac{\partial v_\varPhi ^\pm }{\partial z} \right] {\vec {e}}_\varphi \otimes {\vec {e}}_Z \nonumber \\&\quad +\left[ \left( {\mu _\pm +B^{\pm }_2} \right) \frac{\partial v_R^\pm }{\partial z}+\left( {\mu _\pm -B^{\pm }_2} \right) \frac{\partial v_Z^\pm }{\partial r} \right] {\vec {e}}_z \otimes {\vec {e}}_R \nonumber \\&\quad +\left[ \left( {\mu _\pm +B^{\pm }_1} \right) \frac{\partial v_\varPhi ^\pm }{\partial z}+\frac{\mu _\pm - B^{\pm }_1}{r}\frac{\partial v_Z^\pm }{\partial \varphi } \right] {\vec {e}}_z \otimes {\vec {e}}_\varPhi \nonumber \\&\quad +\left[ \lambda _\pm \frac{\partial v_R^\pm }{\partial r}\,\,+\frac{\lambda _\pm }{r}\left( {\frac{\partial v_\varPhi ^\pm }{\partial \varphi }+v_R^\pm } \right) +\left( {\zeta _\pm +\mu _\pm } \right) \frac{\partial v_Z^\pm }{\partial z} \right] {\vec {e}}_z \otimes {\vec {e}}_Z. \end{aligned}$$

(30)

Here the following notations are used:

$$\begin{aligned} \zeta =\lambda +\mu , \quad \zeta _\pm =\lambda _\pm +\mu _\pm , \quad \tau =\mu +\kappa , \quad \gamma =\gamma _1 +\gamma _3, \quad \xi =n^2+r^2\beta ^2+1, \end{aligned}$$

$$\begin{aligned} B_1= & {} \mu \left( {\frac{f}{r} +\alpha } \right) +\lambda s-\chi , \qquad B_1^\pm =\mu _\pm +\frac{\left( \lambda _\pm s_\pm - 2 \mu _\pm \right) r}{f_\pm +\alpha r},\\ B_2= & {} \mu \left( {{f}^{\prime } +\alpha } \right) +\lambda s-\chi , \qquad \;\; B_2^\pm =\mu _\pm +\frac{ \lambda _\pm s_\pm - 2 \mu _\pm }{f^\prime _\pm +\alpha }, \\ B_3= & {} \mu \left( {{f}^{\prime }+\frac{f}{r}} \right) +\lambda s-\chi , \qquad B_3^\pm =\mu _\pm +\frac{\left( \lambda _\pm s_\pm - 2 \mu _\pm \right) r}{r f^\prime _\pm + f_\pm }. \end{aligned}$$

By taking into account the expressions (4), (9), (22)–(24), (28)–(30), we derive a neutral equilibrium equations (14):

$$\begin{aligned}&\left( \zeta +\tau \right) V^{\prime \prime }_{R} +\frac{\zeta +\tau }{r} V^{\prime }_{R} -\frac{\zeta +\tau \xi }{r^{2} } V_{R} +\frac{n\zeta }{r} V^{\prime }_{\varPhi } -\frac{n\left( \zeta +2\tau \right) }{r^{2} } V_{\varPhi } \nonumber \\&\quad +\beta \zeta V^{\prime }_{Z} +\beta B_{2} \varOmega _{\varPhi } -\frac{nB_{3} }{r} \varOmega _{Z} = 0, \nonumber \\&\tau V^{\prime \prime }_{\varPhi } -\frac{n\zeta }{r} V^{\prime }_{R} -\frac{n\left( \zeta +2\tau \right) }{r^{2} } V_{R} +\frac{\tau }{r} V^{\prime }_{\varPhi } -\frac{\zeta n^{2} +\tau \xi }{r^{2} } V_{\varPhi } -\frac{n\beta \zeta }{r} V_{Z} \nonumber \\&\quad -\beta B_{1} \varOmega _{R} +B^{\prime }_{3} \varOmega _{Z} +B_{3} \varOmega ^{\prime }_{Z} =0, \nonumber \\&\tau V^{\prime \prime }_{Z} -\beta \zeta V^{\prime }_{R} -\frac{\beta \zeta }{r} V_{R} -\frac{n\beta \zeta }{r} V_{\varPhi } +\frac{\tau }{r} V^{\prime }_{Z} -\left( \zeta \beta ^{2} +\frac{\xi -1}{r^{2} } \tau \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, \nonumber \\&\left( \gamma +\gamma _{2} \right) \left( \varOmega ^{\prime \prime }_{R} +\frac{\varOmega ^{\prime }_{R} }{r} \right) -\left[ \frac{\gamma +\gamma _{2} \xi }{r^{2} } -\left( \frac{f}{r} +\alpha \right) B_{1} \right] \varOmega _{R} -\beta B_{1} V_{\varPhi } \nonumber \\&\quad +\frac{nB_{1} }{r} V_{Z} +\frac{n\left( \gamma +2\gamma _{2} \right) }{r^{2} } \varOmega _{\varPhi } -\frac{n\gamma }{r} \varOmega ^{\prime }_{\varPhi } -\beta \gamma \varOmega ^{\prime }_{Z} =0, \nonumber \\&\gamma _{2} \varOmega ^{\prime \prime }_{\varPhi } +\frac{\gamma _{2} }{r} \varOmega ^{\prime }_{\varPhi } -\left[ \frac{\gamma n^{2} +\gamma _{2} \xi }{r^{2} } -\left( f^{\prime } +\alpha \right) B_{2} \right] \varOmega _{\varPhi } +B_{2} V^{\prime }_{Z} +\beta B_{2} V_{R} \nonumber \\&\quad +\frac{n\gamma }{r} \varOmega ^{\prime }_{R} +\frac{n\left( \gamma +2\gamma _{2} \right) }{r^{2} } \varOmega _{R} -\frac{n\beta \gamma }{r} \varOmega _{Z} =0, \end{aligned}$$

(31)

$$\begin{aligned}&\gamma _{2} \varOmega ^{\prime \prime }_{Z} -\left[ \gamma \beta ^{2} +\frac{\xi -1}{r^{2} } \gamma _{2} -B_{3} \left( f^{\prime } +\frac{f}{r} \right) \right] \varOmega _{Z} -B_{3} \left( \frac{n}{r} V_{R} +V^{\prime }_{\varPhi } +\frac{V_{\varPhi } }{r} \right) \nonumber \\&\quad +\frac{\gamma _{2} }{r} \varOmega ^{\prime }_{Z} +\beta \gamma \left( \varOmega ^{\prime }_{R} +\frac{\varOmega _{R} }{r} -\frac{n}{r} \varOmega _{\varPhi } \right) =0, \nonumber \\&\left( \zeta _{\pm } +\mu _{\pm } \right) \left[ \left( V_{R}^{\pm } \right) ^{{^{\prime \prime }} } +\frac{\left( V_{R}^{\pm } \right) ^{\prime } }{r} \right] -\frac{1}{r^{2} } \left( \zeta _{\pm } +\xi \mu _{\pm } +\beta ^{2} r^{2} B_{2}^{\pm } +n^{2} B_{3}^{\pm } \right) V_{R}^{\pm } \nonumber \\&\quad +\frac{n}{r} \left( \zeta _{\pm } -B_{3}^{\pm } \right) \left( V_{\varPhi }^{\pm } \right) ^{{^{\prime }} } -\frac{n}{r^{2} } \left( \zeta _{\pm } +2\mu _{\pm } +B_{3}^{\pm } \right) V_{\varPhi }^{\pm } +\beta \left( \zeta _{\pm } -B_{2}^{\pm } \right) \left( V_{Z}^{\pm } \right) ^{{^{\prime }} } =0, \nonumber \\&\left( \mu _{\pm } +B_{3}^{\pm } \right) \left[ \left( V_{\varPhi }^{\pm } \right) ^{{^{\prime \prime }} } +\frac{\left( V_{\varPhi }^{\pm } \right) ^{{^{\prime }} } }{r} \right] -\frac{1}{r^{2} } \left( n^{2}\zeta _{\pm } +\xi \mu _{\pm } +B_{3}^{\pm } +\beta ^{2} r^{2} B_{1}^{\pm } \right) V_{\varPhi }^{\pm } \nonumber \\&\quad -\frac{n}{r} \left( \zeta _{\pm } -B_{3}^{\pm } \right) \left( V_{R}^{\pm } \right) ^{{^{\prime }} } -\frac{n}{r^{2} } \left( \zeta _{\pm } +2\mu _{\pm } +B_{3}^{\pm } \right) V_{R}^{\pm } -\frac{n\beta }{r} \left( \zeta _{\pm } -B_{1}^{\pm } \right) V_{Z}^{\pm } =0, \nonumber \\&\left( \mu _{\pm } +B_{2}^{\pm } \right) \left( V_{Z}^{\pm } \right) ^{{^{\prime \prime }} } + \left( \frac{\mu _{\pm }}{r} +\frac{1}{f^{\prime }_{\pm } +\alpha } \left[ B_{1}^{\pm } \frac{f_{\pm } +\alpha r}{r^2} -B_{2}^{\pm } f^{\prime \prime }_{\pm } \right] \right) \left( V_{Z}^{\pm } \right) ^{{^{\prime }} } \nonumber \\&\quad -\beta \left( \zeta _{\pm } -B_{2}^{\pm } \right) \left( V_{R}^{\pm } \right) ^{{^{\prime }} } -\beta \left( \frac{\zeta _{\pm }}{r} -\frac{1}{f^{\prime }_{\pm } +\alpha } \left[ B_{1}^{\pm } \frac{f_{\pm } +\alpha r}{r^2} -B_{2}^{\pm } f^{\prime \prime }_{\pm } \right] \right) V_{R}^{\pm } \nonumber \\&\quad -\frac{n\beta }{r} \left( \zeta _{\pm } -B_{1}^{\pm } \right) V_{\varPhi }^{\pm } -\left( \left[ \zeta _{\pm } +\mu _{\pm } \right] \beta ^{2} +\frac{n^{2} }{r^{2} } \left[ \mu _{\pm } +B_{1}^{\pm } \right] \right) V_{Z}^{\pm } =0. \nonumber \end{aligned}$$

Given the substitutions (23) and (24), the expressions for the linearized boundary conditions (18) take the form:

• on the inner and outer surfaces of the tube $\left( {r=r_{\pm } } \right)$

  $$\begin{aligned} \left( \zeta _{\pm }\! +\!\mu _{\pm } \right) \left( V_{R}^{\pm } \right) ^{\prime } \!+\!\frac{\lambda _{\pm }\! +\!\alpha p_{\pm } }{r_{\pm } } \left( V_{R}^{\pm }\! +\!nV_{\varPhi }^{\pm } \right) \!+\! \beta \left( \lambda _{\pm } \!+\!\frac{f_{\pm } }{r_{\pm } } p_{\pm } \right) V_{Z}^{\pm }= & {} 0, \nonumber \\ \frac{\alpha p_{\pm } +B_{3}^{\pm } -\mu _{\pm } }{r_{\pm } } \left( nV_{R}^{\pm } +V_{\varPhi }^{\pm } \right) +\left( \mu _{\pm } +B_{3}^{\pm } \right) \left( V_{\varPhi }^{\pm } \right) ^{\prime }= & {} 0,\qquad \\ \beta \left( \frac{f_{\pm } }{r_{\pm } } p_{\pm } +B_{2}^{\pm } -\mu _{\pm } \right) V_{R}^{\pm } +\left( \mu _{\pm } +B_{2}^{\pm } \right) \left( V_{\mathrm{Z}}^{\pm } \right) ^{\prime }= & {} 0, \nonumber \end{aligned}$$

  (32)

• on the interfaces between the coatings and the foam core $\left( {r=c_{\pm } } \right)$

  $$\begin{aligned}&\left( \zeta _{\pm } +\mu _{\pm } \right) \left( V_{R}^{\pm } \right) ^{\prime } +\frac{\lambda _{\pm } }{c_{\pm } } \left( V_{R}^{\pm } +nV_{\varPhi }^{\pm } \right) +\beta \lambda _{\pm } V_{Z}^{\pm } - \left( \zeta +\tau \right) V^\prime _{R} \nonumber \\&\quad -\frac{\lambda }{c_{\pm } } \left( V_{R} +nV_{\varPhi } \right) -\beta \lambda V_{Z} =0, \nonumber \\&\frac{B_{3}^{\pm } -\mu _{\pm } }{c_{\pm } } \left( nV_{R}^{\pm } +V_{\varPhi }^{\pm } \right) +\left( \mu _{\pm } +B_{3}^{\pm } \right) \left( V_{\varPhi }^{\pm } \right) ^{\prime } -\tau V^\prime _{\varPhi } \nonumber \\&\quad + \frac{\mu }{c_{\pm } } \left( nV_{R} +V_{\varPhi } \right) -B_{3} \varOmega _{Z} =0, \\&\beta \left( B_{2}^{\pm } -\mu _{\pm } \right) V_{R}^{\pm } +\left( \mu _{\pm } +B_{2}^{\pm } \right) \left( V_{\mathrm{Z}}^{\pm } \right) ^{\prime } + \mu \beta V_{R} -\tau V^\prime _{Z} +B_{2} \varOmega _{\varPhi } =0, \nonumber \\&\left( \gamma +\gamma _{2} \right) \varOmega ^\prime _{R} +\frac{\gamma _{1} }{c_{\pm } } \left( \varOmega _{R} -n\varOmega _{\varPhi } \right) -\gamma _{1} \beta \varOmega _{Z} =0, \nonumber \\&\frac{\gamma _{3} }{c_{\pm } } \left( n\varOmega _{R} -\varOmega _{\varPhi } \right) +\gamma _{2} \varOmega ^\prime _{\varPhi } =0, \qquad \gamma _{3} \beta \varOmega _{R} +\gamma _{2} \varOmega ^\prime _{Z} =0, \nonumber \\&v_{R} -v_{R}^{\pm } =0, \qquad v_{\varPhi } -v_{\varPhi }^{\pm } =0, \qquad v_{Z} -v_{Z}^{\pm } =0. \nonumber \end{aligned}$$

  (33)