Maxwell homogenization scheme for piezoelectric composites with arbitrarily-oriented spheroidal inhomogeneities

Abstract

In this work, the effective electro-elastic properties of piezoelectric composites are computed using the Maxwell homogenization method (MHM). The composites are made by several families of spheroidal inhomogeneities embedded in a homogeneous infinite medium (matrix). Each family of spheroidal inhomogeneities is made of the same material, and all the inhomogeneities have identical size and shape and are randomly oriented. The inhomogeneities and matrix materials exhibit piezoelectric transversely isotropic symmetry. It is shown that the shape of the “effective inclusion” substantially affects the effective piezoelectric properties. A new and simple form to calculate the aspect ratio of effective inclusion is presented. The effect on the overall piezoelectric properties due to the orientation of the inhomogeneities and different families of piezoelectric inhomogeneities is discussed. The MHM approach is applied in two examples, material with inhomogeneities having scatter orientation and composites with two different families of spheroidal inhomogeneities.

Appendices

Appendix A. ${\mathbf {T}}$, ${\mathbf {U}}$, ${\mathbf {t}}$ tensor bases representation

The electro-elastic properties of a piezoelectric transversely isotropic material can be represented in the form

$$\begin{aligned} \begin{aligned} {\mathbf {C}}(\phi ,\theta )&= C_{11} {\mathbf {T}}^{1}(\phi ,\theta ) + C_{12} ({\mathbf {T}}^{2}(\phi ,\theta ) - {\mathbf {T}}^{1}(\phi ,\theta ))\\&\quad + C_{13} ({\mathbf {T}}^{3}(\phi ,\theta ) + {\mathbf {T}}^{4}(\phi ,\theta )) + 4C_{44} {\mathbf {T}}^{5}(\phi ,\theta ) + C_{33} {\mathbf {T}}^{6}(\phi ,\theta ), \\ \mathbf {e(\phi ,\theta )}&= e_{31} {\mathbf {U}}^{1}(\phi ,\theta ) + e_{15} {\mathbf {U}}^{2}(\phi ,\theta ) + e_{33} {\mathbf {U}}^{3}(\phi ,\theta ), \\ {\varvec{\epsilon }} (\phi ,\theta )&= \epsilon _{33} {\mathbf {t}}^{1}(\phi ,\theta ) + \epsilon _{11} {\mathbf {t}}^{2}(\phi ,\theta ) \end{aligned} \end{aligned}$$

(A.1)

where $C_{11}$, $C_{12}$, $C_{13}$, $C_{33}$, $C_{44}$ are five independent elastic moduli of the transversely isotropic medium, $e_{31}$, $e_{15}$, $e_{33}$ are three piezoelectric constants, and $\epsilon _{33}$, $\epsilon _{11}$ are two permittivities. The quantities ${\mathbf {T}}^{i}(\phi ,\theta )$, ${\mathbf {U}}^{i}(\phi ,\theta )$, ${\mathbf {t}}^{i}(\phi ,\theta )$ are the elements of the tensor basis defined in Ref. [38] as follows:

$$\begin{aligned} \begin{aligned} T_{ijkl}^{1}(\phi ,\theta )&= \frac{1}{2}(\theta _{ik}\theta _{lj} + \theta _{il}\theta _{kj}), \qquad T_{ijkl}^{2}(\phi ,\theta ) = \theta _{ij}\theta _{kl}, \\ T_{ijkl}^{3}(\phi ,\theta )&= \theta _{ij}m_{k}m_{l}, \qquad T_{ijkl}^{4}(\phi ,\theta ) = m_{i}m_{j}\theta _{kl}, \\ T_{ijkl}^{5}(\phi ,\theta )&= \frac{1}{4}\left( \theta _{ik}m_{l}m_{j}+ \theta _{il}m_{k}m_{j}+\theta _{jk}m_{l}m_{i} + \theta _{jl} m_{k}m_{i}\right) , \\ T_{ijkl}^{6}(\phi ,\theta )&= m_{i}m_{j}m_{k}m_{l}, \\ U_{ijk}^{1}(\phi ,\theta )&= \theta _{ij}m_{k}, \qquad U_{ijk}^{2}(\phi ,\theta ) = m_{i}\theta _{jk}+m_{j}\theta _{ik}, \qquad U_{ijk}^{3}(\phi ,\theta ) = m_{i}m_{j}m_{k}, \\ t_{ij}^{1}(\phi ,\theta )&= m_{i}m_{j}, \qquad t_{ij}^{2}(\phi ,\theta ) = \theta _{ij}, \\ \theta _{ij}(\phi ,\theta )&= \delta _{ij}-m_{i}(\phi ,\theta )m_{j}(\phi ,\theta ) \end{aligned} \end{aligned}$$

(A.2)

where $m_{i}(\phi ,\varphi ) = (\cos \theta \sin \phi ,\sin \theta \sin \phi ,\cos \phi )$ is the unit vector along the symmetry axis of the material and is defined by Eq. (3.9).

The average electro-elastic properties of a composite in which the inhomogeneities have arbitrary orientations are given by integrating over the different orientations,

$$\begin{aligned} \begin{aligned} {\mathbf {C}}^{p}&= \int _{0}^{\pi /2}\mathrm {d}\phi \, P_{p}(\phi )\sin \phi \left\{ \int _{0}^{2\pi }\mathrm {d}\theta \Big [C_{11}{\mathbf {T}}^{1}(\phi ,\theta ) + C_{12} \left( {\mathbf {T}}^{2}(\phi ,\theta ) - {\mathbf {T}}^{1}(\phi ,\theta )\right) \right. \\&\quad \left. +\, C_{13} ({\mathbf {T}}^{3}(\phi ,\theta ) + {\mathbf {T}}^{4}(\phi ,\theta )) + 4C_{44} {\mathbf {T}}^{5}(\phi ,\theta ) + C_{66} {\mathbf {T}}^{6}(\phi ,\theta )\Big ]\right\} , \\ {\mathbf {e}}^{p}&= \int _{0}^{\pi /2}\mathrm {d}\phi \, P_{p}(\phi )\sin \phi \int _{0}^{2\pi }\mathrm {d}\theta \Big [e_{31}{\mathbf {U}}^{1}(\phi ,\theta ) + e_{15} {\mathbf {U}}^{2}(\phi ,\theta ) + e_{33} {\mathbf {U}}^{3}(\phi ,\theta )\Big ], \\ {\varvec{\epsilon }} ^{p}&= \int _{0}^{\pi /2}\mathrm {d}\phi \, P_{p}(\phi )\sin \phi \int _{0}^{2\pi }\mathrm {d}\theta \Big [\epsilon _{33} {\mathbf {t}}^{1}(\phi ,\theta ) + \epsilon _{11} {\mathbf {t}}^{2}(\phi ,\theta )\Big ], \end{aligned} \end{aligned}$$

(A.3)

after taking into account Eqs. (A.1) and (3.10). The functions $P_{\lambda }(\phi )$ and $P_{\sigma }(\phi )$ described in Eqs. (3.6) and (3.7), respectively, are represented by the function $P_{p}(\phi )$. The components of ${\mathbf {C}}^{p}$, ${\mathbf {e}}^{p}$, and ${\varvec{\epsilon }}^{p}$ are given by

$$\begin{aligned} \begin{aligned} C_{11}^{(p)}&= g_{11}(p) C_{11} + g_{12}(p) C_{13} + \frac{3}{8}g_{6}(p) C_{33} + 2g_{12}(p) C_{44}, \\ C_{12}^{(p)}&= \frac{1}{8}g_{6}(p) C_{11} + (1-2g_{1}(p)) C_{12} + g_{10}(p) C_{13} + \frac{1}{8}g_{6}(p) C_{33} -\frac{1}{2}g_{6}(p) C_{44}, \\ C_{13}^{(p)}&= g_{4}(p) C_{11} + g_{1}(p) C_{12} + g_{5}(p) C_{13} + g_{4}(p) C_{33} - 4g_{4}(p) C_{44}, \\ C_{33}^{(p)}&= g_{6}(p) C_{11} + 4g_{4}(p) C_{13} + g_{7}(p) C_{33} + 8g_{4}(p) C_{44}, \\ C_{44}^{(p)}&= g_{8}(p) C_{11} - \frac{1}{2}g_{1}(p) C_{12} - 2g_{4}(p) C_{13} + g_{4}(p) C_{33} + g_{9}(p) C_{44}, \\ e_{31}^{(p)}&= (g_{2}(p) + g_{3}(p)) e_{31} - 2g_{2}(p) e_{15} + g_{2}(p) e_{33}, \\ e_{15}^{(p)}&= - g_{2}(p) e_{31} + g_{3}(p) e_{15} + g_{2}(p) e_{33}, \\ e_{33}^{(p)}&= 2g_{2}(p) e_{31} + 4g_{2}(p) e_{15} + g_{3}(p) e_{33}, \\ \epsilon _{11}^{(p)}&= (1 - g_{1}(p)) \epsilon _{11} + g_{1}(p) \epsilon _{33}, \\ \epsilon _{33}^{(p)}&= 2g_{1}(p) \epsilon _{11} + (1 - 2g_{1}(p)) \epsilon _{33}. \end{aligned} \end{aligned}$$

(A.4)

When $p = \lambda$, the functions $g_{i}(\lambda )$ are defined as

$$\begin{aligned} \begin{aligned} g_{1}(\lambda )&= \frac{3}{9+\lambda ^{2}}-\lambda e^{-\frac{\pi }{2}\lambda }\frac{3+\lambda ^{2}}{6(9+\lambda ^{2})}, \\ g_{2}(\lambda )&= \frac{e^{-\frac{\pi }{2}\lambda }\bigg (40+ 24e^{\frac{\pi }{2}\lambda }(1+\lambda ^{2})+\lambda (64+\lambda ( 44+\lambda (20+\lambda (4+\lambda ))))\bigg )}{8(4+\lambda ^{2})(16+\lambda ^{2})}, \\ g_{3}(\lambda )&= \frac{e^{-\frac{\pi }{2}\lambda }\bigg (24+ 4e^{\frac{\pi }{2}\lambda }(1+\lambda ^{2})(10+\lambda ^{2})+ \lambda (64+\lambda (24+20\lambda +\lambda ^{3}))\bigg )}{4(4+\lambda ^{2})(16+\lambda ^{2})}, \\ g_{4}(\lambda )&= \frac{3(5+\lambda ^{2})}{(9+\lambda ^{2})(25+\lambda ^{2})}+\lambda e^{-\frac{\pi }{2}\lambda }\frac{12+(1+\lambda ^{2}) (18+\lambda ^{2})}{15(9+\lambda ^{2})(25+\lambda ^{2})}, \\ g_{5}(\lambda )&= \frac{96+(1+\lambda ^{2})(24+\lambda ^{2})}{(9+\lambda ^{2})(25+\lambda ^{2})}+\lambda e^{-\frac{\pi }{2}\lambda }\frac{192+(1+\lambda ^{2}) (63+\lambda ^{2})}{30(9+\lambda ^{2})(25+\lambda ^{2})}, \\ g_{6}(\lambda )&= \frac{1800}{15(9+\lambda ^{2})(25+\lambda ^{2})} -\lambda e^{-\frac{\pi }{2}\lambda }\frac{435+178\lambda ^{2}+ 7\lambda ^{4}}{15(9+\lambda ^{2})(25+\lambda ^{2})}, \\ g_{7}(\lambda )&= \frac{24+(1+\lambda ^{2})(21+\lambda ^{2})}{(9+\lambda ^{2})(25+\lambda ^{2})}+\lambda e^{-\frac{\pi }{2}\lambda }\frac{72+(1+\lambda ^{2}) (33+\lambda ^{2})}{5(9+\lambda ^{2})(25+\lambda ^{2})}, \\ g_{8}(\lambda )&= \frac{3(35+3\lambda ^{2})}{2(9+\lambda ^{2})(25+\lambda ^{2})}-\lambda e^{-\frac{\pi }{2}\lambda }\frac{192+(1+\lambda ^{2}) (63+\lambda ^{2})}{60(9+\lambda ^{2})(25+\lambda ^{2})}, \\ g_{9}(\lambda )&= \frac{10+\lambda ^{2}}{25+\lambda ^{2}}-\lambda e^{-\frac{\pi }{2}\lambda }\frac{-5+\lambda ^{2}}{10(25+\lambda ^{2})}, \\ g_{10}(\lambda )&= \frac{6(20+\lambda ^{2})}{(9+\lambda ^{2})(25+\lambda ^{2})}-\lambda e^{-\frac{\pi }{2}\lambda }\frac{1065+382\lambda ^{2}+ 13\lambda ^{4}}{60(9+\lambda ^{2})(25+\lambda ^{2})}, \\ g_{11}(\lambda )&= \frac{93+(1+\lambda ^{2})(27+\lambda ^{2})}{(9+\lambda ^{2})(25+\lambda ^{2})}+\lambda e^{-\frac{\pi }{2}\lambda }\frac{1695+586\lambda ^{2}+ 19\lambda ^{4}}{120(9+\lambda ^{2})(25+\lambda ^{2})}, \\ g_{12}(\lambda )&= \frac{6(10+\lambda ^{2})}{(9+\lambda ^{2})(25+\lambda ^{2})}+\lambda e^{-\frac{\pi }{2}\lambda }\frac{-195-26\lambda ^{2}+ \lambda ^{4}}{60(9+\lambda ^{2})(25+\lambda ^{2})}. \end{aligned} \end{aligned}$$

(A.5)

For $p = \sigma$, the functions $g_{i}(\sigma )$ are defined as follows:

$$\begin{aligned} g_{1}(\sigma )= & {} \frac{\sigma \coth \sigma - 1}{\sigma ^{2}}, \nonumber \\ g_{2}(\sigma )= & {} \text {csch}\sigma \frac{\sigma ^{2} + 2(3 + \sigma ^{2})\cosh \sigma - 6\sigma \sinh \sigma - 6}{2\sigma ^{3}},\nonumber \\ g_{3}(\sigma )= & {} \frac{6\sigma + \sigma ^{3} -3(2 + \sigma ^{2}) \coth \sigma + 6\text {csch}\sigma }{\sigma ^{3}},\nonumber \\ g_{4}(\sigma )= & {} \frac{-12 - 5\sigma ^{2} - \sigma (12 + \sigma ^{2}) \coth \sigma }{\sigma ^{4}},\nonumber \\ g_{5}(\sigma )= & {} (8 + \sigma ^{2})\frac{3 + \sigma ^{2} - 3\sigma \coth \sigma }{\sigma ^{4}},\nonumber \\ g_{6}(\sigma )= & {} \frac{24 + 8\sigma ^{2} - 24\sigma \coth \sigma }{\sigma ^{4}},\\ g_{7}(\sigma )= & {} \frac{24 + 12\sigma ^{2} + \sigma ^{4} - 4\sigma (6 + \sigma ^{2})\coth \sigma }{\sigma ^{4}},\nonumber \\ g_{8}(\sigma )= & {} \frac{-24 - 11\sigma ^{2} + 3\sigma (8 + \sigma ^{2}) \coth \sigma }{2\sigma ^{4}},\nonumber \\ g_{9}(\sigma )= & {} \frac{48 + 21\sigma ^{2} + \sigma ^{4} - \sigma (48 + 5\sigma ^{2})\coth \sigma }{\sigma ^{4}},\nonumber \\ g_{10}(\sigma )= & {} \frac{-6 - 4\sigma ^{2} + 2\sigma (3 + \sigma ^{2}) \coth \sigma }{\sigma ^{4}},\nonumber \\ g_{11}(\sigma )= & {} \frac{9 + 5\sigma ^{2} + \sigma ^{4} - \sigma (9 + 2\sigma ^{2})\coth \sigma }{\sigma ^{4}},\nonumber \\ g_{12}(\sigma )= & {} 2\frac{-9 - 4\sigma ^{2} + \sigma (9 + \sigma ^{2}) \coth \sigma }{\sigma ^{4}}.\nonumber \end{aligned}$$

(A.6)

Taking the limit of parallel orientations $p\rightarrow \infty$, the components of $\mathbf{C }^{(p)}$, $\mathbf{e }^{(p)}$, ${\varvec{\epsilon }}^{(p)}$ have the following form:

$$\begin{aligned} \begin{aligned} \lim \limits _{p\rightarrow \infty } C_{11}^{(p)}&= C_{11}, \qquad \lim \limits _{p\rightarrow \infty } C_{12}^{(p)} = C_{12}, \qquad \lim \limits _{p\rightarrow \infty } C_{13}^{(p)} = C_{13}, \qquad \lim \limits _{p\rightarrow \infty } C_{33}^{(p)} = C_{33}, \\ \lim \limits _{p\rightarrow \infty } C_{44}^{(p)}&= C_{44}, \qquad \lim \limits _{p\rightarrow \infty } e_{31}^{(p)} = e_{31}, \qquad \lim \limits _{p\rightarrow \infty } e_{15}^{(p)} = e_{15}, \qquad \lim \limits _{p\rightarrow \infty } e_{33}^{(p)} = e_{33}, \\ \lim \limits _{p\rightarrow \infty } \epsilon _{11}^{(p)}&= \epsilon _{11}, \qquad \lim \limits _{p\rightarrow \infty } \epsilon _{33}^{(p)} = \epsilon _{33}, \end{aligned} \end{aligned}$$

(A.7)

after substitution of $g_{i}(\lambda )$ or $g_{i}(\sigma )$ into Eq. (A.4). The limits in Eq. (A.7) are in agreement with the representation of a transversely isotropic medium $\mathbf{C }(0,\theta )$, $\mathbf{e }(0,\theta )$, ${\varvec{\epsilon }}(0,\theta )$ in the tensor basis given by Eq. (A.2).

Appendix B. Components of the tensors $\bar{{\mathbf {S}}}^{(r)}_{x}$, $\bar{{\mathbf {M}}}^{(r)}_{x}$, $\bar{{\mathbf {S}}}^{(r)}_{,x}$, and $\bar{{\mathbf {M}}}^{(r)}_{,x}$.

The tensor ${\mathbf {S}}^{(r)}_{x}$ is given by Eq. (2.5). Due to the $\varphi$ and u dependence in Eq. (2.5), the only non-vanishing components of $S^{(r)}_{x(klij)}$ are $S^{(r)}_{x(1111)} = S^{(r)}_{x(2222)}$, $S^{(r)}_{x(1122)} = S^{(r)}_{x(2211)}$, $S^{(r)}_{x(3333)}$, $S^{(r)}_{x(1133)} = S^{(r)}_{x(2233)} = S^{(r)}_{x(3311)} = S^{(r)}_{x(3322)}$, $S^{(r)}_{x(2323)} = S^{(r)}_{x(2332)} = S^{(r)}_{x(3223)} = S^{(r)}_{x(3232)} = S^{(r)}_{x(1313)} =S^{(r)}_{x(1331)} = S^{(r)}_{x(3113)} = S^{(r)}_{x(3131)}$ and $S^{(r)}_{x(1212)} = S^{(r)}_{x(1221)} = S^{(r)}_{x(2112)} = S^{(r)}_{x(2121)}$. Then, using the Voigt two-index notation,

$$\begin{aligned} S^{(r)}_{x(11)}&= \int _{-1}^{1}\mathrm {d}u\, \frac{1-u^{2}}{4\pi |{\bar{\Gamma }}_{li}|} \Big (\Lambda _{c}(\Lambda _{a}+3\Gamma _{b})- \Lambda ^{2}_{ac}\Big ), \end{aligned}$$

(B. 1)

$$\begin{aligned} S^{(r)}_{x(12)}&= \int _{-1}^{1}\mathrm {d}u\, \frac{1-u^{2}}{4\pi |{\bar{\Gamma }}_{li}|} \Big (\Lambda ^{2}_{ac}- \Lambda _{ab}\Lambda _{c}\Big ), \end{aligned}$$

(B. 2)

$$\begin{aligned} S^{(r)}_{x(13)}&= \int _{-1}^{1}\mathrm {d}u\, \frac{u\sqrt{1-u^{2}}}{4\pi |{\bar{\Gamma }}_{li}|} \Lambda _{ac}\Big (\Lambda _{ab}-\Lambda _{a}- 3\Gamma _{b}\Big ), \end{aligned}$$

(B. 3)

$$\begin{aligned} S^{(r)}_{x(33)}&= \int _{-1}^{1}\mathrm {d}u\, \frac{u^{2}}{4\pi |{\bar{\Gamma }}_{li}|} \Big (\Lambda ^{2}_{a}-\Lambda ^{2}_{ab}+ 6\Lambda _{a}\Gamma _{b}+\Gamma ^{2}_{b}\Big ), \end{aligned}$$

(B. 4)

$$\begin{aligned} S^{(r)}_{x(44)}&= \int _{-1}^{1}\mathrm {d}u\, \frac{1-u^{2}}{8\pi |{\bar{\Gamma }}_{li}|} \Big (\Lambda ^{2}_{a}-\Lambda ^{2}_{ab}+ 6\Lambda _{a}\Gamma _{b}+\Gamma ^{2}_{b}\Big ) \nonumber \\&\quad + \int _{-1}^{1}\mathrm {d}u\, \frac{u\sqrt{1-u^{2}}}{2\pi |{\bar{\Gamma }}_{li}|} \Lambda _{ac}\Big (\Lambda _{ab}-\Lambda _{a}- 3\Gamma ^{0}_{b}\Big ) \nonumber \\&\quad +\int _{-1}^{1}\mathrm {d}u\, \frac{u^{2}}{\pi |{\bar{\Gamma }}_{li}|} \Big (\Lambda _{c}(\Lambda _{a}+\Gamma _{b})- \Lambda ^{2}_{ac}\Big ), \end{aligned}$$

(B. 5)

$$\begin{aligned} S^{(r)}_{x(66)}&= \int _{-1}^{1}\mathrm {d}u\, \frac{1-u^{2}}{8\pi |{\bar{\Gamma }}_{li}|} \Big (\Lambda _{c}(\Lambda _{a}+3\Gamma _{b}+ \Lambda _{ab})-2\Lambda ^{2}_{ac}\Big ). \end{aligned}$$

(B. 6)

Here, the determinant $|{\bar{\Gamma }}_{li}|$ of ${\bar{\Gamma }}_{li}$ is given by

$$\begin{aligned} |{\bar{\Gamma }}_{li}| = -\frac{\Gamma _{b}}{\epsilon }\Big (2\Gamma _{ac}\gamma _{a}\gamma _{c}- \Gamma _{a}\gamma ^{2}_{c}+ \epsilon \Gamma ^{2}_{ac}-\Gamma _{c} \Big (\gamma ^{2}_{a}+\epsilon \Gamma _{a}\Big )\Big ) \end{aligned}$$

(B. 7)

where $\Lambda _{a}({\varvec{\zeta }})$, $\Lambda _{ab}({\varvec{\zeta }})$, $\Lambda _{ac}({\varvec{\zeta }})$, $\Lambda _{c}({\varvec{\zeta }})$, $\Gamma _{a}({\varvec{\zeta }})$, $\Gamma _{b}({\varvec{\zeta }})$, $\Gamma _{c}({\varvec{\zeta }})$, $\Gamma _{ab}({\varvec{\zeta }})$, $\Gamma _{ac}({\varvec{\zeta }})$, $\gamma _{a}({\varvec{\zeta }})$, $\gamma _{c}({\varvec{\zeta }})$, and $\epsilon ({\varvec{\zeta }})$ are given by

$$\begin{aligned} \begin{aligned} \Lambda _{a}({\varvec{\zeta }})&= \Gamma _{a}({\varvec{\zeta }})+ \frac{\gamma _{a}^{2}({\varvec{\zeta }})}{\epsilon ({\varvec{\zeta }})},&\Lambda _{ab}({\varvec{\zeta }})&= \Gamma _{ab}({\varvec{\zeta }}) +\frac{\gamma _{a}^{2}({\varvec{\zeta }})}{\epsilon ({\varvec{\zeta }})}, \\ \Lambda _{ac}({\varvec{\zeta }})&= \Gamma _{ac}({\varvec{\zeta }})+ \frac{\gamma _{a}({\varvec{\zeta }}) \gamma _{c}({\varvec{\zeta }})}{\epsilon ({\varvec{\zeta }})},&\Lambda _{c}({\varvec{\zeta }})&= \Gamma _{c}({\varvec{\zeta }})+ \frac{\gamma _{c}^{2}({\varvec{\zeta }})}{\epsilon ({\varvec{\zeta }})}, \\ \Gamma _{a}({\varvec{\zeta }})&= \frac{1}{a^{2}_{r}}\left( C^{0}_{11}(1-u^{2})+ \frac{C^{0}_{44} u^{2}}{\delta ^{2}_{r}}\right) ,&\Gamma _{ab}({\varvec{\zeta }})&= \frac{C^{0}_{11}+C^{0}_{12}}{2 a^{2}_{r}}(1-u^{2}), \\ \Gamma _{b}({\varvec{\zeta }})&= \frac{1}{a^{2}_{r}}\left( C^{0}_{66}(1-u^{2})+ \frac{C^{0}_{44} u^{2}}{\delta ^{2}_{r}}\right) ,&\Gamma _{ac}({\varvec{\zeta }})&= \frac{C^{0}_{13}+C^{0}_{44}}{a^{2}_{r}\delta _{r}} u\sqrt{1-u^{2}}, \\ \Gamma _{c}({\varvec{\zeta }})&= \frac{1}{a^{2}_{r}}\left( C^{0}_{44}(1-u^{2})+ \frac{C^{0}_{33} u^{2}}{\delta ^{2}_{r}}\right) ,&\gamma _{a}({\varvec{\zeta }})&= \frac{e^{0}_{31}+e^{0}_{15}}{a^{2}_{r}\delta _{r}} u\sqrt{1-u^{2}}, \\ \gamma _{c}({\varvec{\zeta }})&= \frac{1}{a^{2}_{r}}\left( e^{0}_{15}(1-u^{2})+ \frac{e^{0}_{33} u^{2}}{\delta ^{2}_{r}}\right) ,&\epsilon ({\varvec{\zeta }})&= \frac{1}{a^{2}_{r}}\left( \epsilon ^{0}_{11}(1-u^{2})+ \frac{\epsilon ^{0}_{33} u^{2}}{\delta ^{2}_{r}}\right) . \end{aligned} \end{aligned}$$

(B. 8)

The tensor ${\mathbf {S}}^{(r)}_{,x}$ is obtained from Eq. (2.6). The only nonzero components of $S^{(r)}_{,x(k4ij)}$ are $S^{(r)}_{,x(1413)} = S^{(r)}_{,x(2423)} = S^{(r)}_{,x(1431)} = S^{(r)}_{,x(2432)}$, $S^{(r)}_{,x(3411)} = S^{(r)}_{x(3422)}$, and $S^{(r)}_{,x(3433)}$, due to the $\varphi$ and u dependence in Eq. (2.6). Then, in Voigt’s notation,

$$\begin{aligned} S^{(r)}_{,x(31)}&= \int _{-1}^{1}\mathrm {d}u\, \frac{u\sqrt{1-u^{2}}}{4\pi \epsilon |{\bar{\Gamma }}_{li}|} \bigg (\Lambda _{a}-\Lambda _{ab}+3\Gamma _{b}\bigg ) \bigg (\Lambda _{c}\gamma _{a} - \Lambda _{ac}\gamma _{c}\bigg ), \end{aligned}$$

(B. 9)

$$\begin{aligned} S^{(r)}_{,x(15)}&= \int _{-1}^{1}\mathrm {d}u\, \frac{u\sqrt{1-u^{2}}}{\pi \epsilon |{\bar{\Gamma }}_{li}|} \bigg (\Lambda _{a}-\Lambda _{ab}+3\Gamma _{b}\bigg ) \bigg (\Lambda _{c}\gamma _{a}- \Lambda _{ac}\gamma _{c}\bigg ) \nonumber \\&\quad +\int _{-1}^{1}\mathrm {d}u\, \frac{1-u^{2}}{4\pi \epsilon |{\bar{\Gamma }}_{li}|} \gamma _{c}\bigg (\Lambda ^{2}_{a}- \Lambda ^{2}_{ab}+6\Lambda _{a}\Gamma _{b} +\Gamma ^{2}_{b}\bigg ) \nonumber \\&\quad +\int _{-1}^{1}\mathrm {d}u\,\frac{1-u^{2}}{2\pi \epsilon |{\bar{\Gamma }}_{li}|}\Lambda _{ac}\gamma _{a} \bigg (\Lambda _{ab}-\Lambda _{a}-3\Gamma _{b}\bigg ), \end{aligned}$$

(B. 10)

$$\begin{aligned} S^{(r)}_{,x(33)}&= \int _{-1}^{1}\mathrm {d}u\, \frac{u^{2}}{4\pi \epsilon |{\bar{\Gamma }}_{li}|} \gamma _{c}\bigg (\Lambda ^{2}_{a}- \Lambda ^{2}_{ab}+6\Lambda _{a}\Gamma _{b} +\Gamma ^{2}_{b}\bigg ) \nonumber \\&\quad +\int _{-1}^{1}\mathrm {d}u\, \frac{u^{2}}{4\pi \epsilon |{\bar{\Gamma }}_{li}|} 2\Lambda _{ac}\gamma _{a}\bigg (\Lambda _{ab}- \Lambda _{a}-3\Gamma _{b}\bigg ). \end{aligned}$$

(B. 11)

The tensor ${\mathbf {M}}^{(r)}_{,x}$ is given by Eq. (2.7). Due to the $\varphi$ and u dependence in Eq. (2.7), the only nonzero components of $M^{(r)}_{,x(k44j)}$ are $M^{(r)}_{,x(1441)} = M^{(r)}_{,x(2442)}$ and $M^{(r)}_{,x(3443)}$. Then

$$\begin{aligned} M^{(r)}_{,x(11)}&= \int _{-1}^{1}\mathrm {d}u\frac{1-u^{2}}{4} \bigg (\epsilon -\frac{\Gamma _{b}}{|\Gamma _{ij}|} \Big (\gamma _{c}(2\Gamma _{ac}\gamma _{a}- \Gamma _{a}\gamma _{c})- \Gamma _{c}\gamma ^{2}_{a}\Big )\bigg )^{-1}, \end{aligned}$$

(B. 12)

$$\begin{aligned} M^{(r)}_{,x(33)}&= \int _{-1}^{1}\mathrm {d}u\frac{u^{2}}{2} \bigg (\epsilon -\frac{\Gamma _{b}}{|\Gamma _{ij}|} \Big (\gamma _{c}(2\Gamma _{ac}\gamma _{a}- \Gamma _{a}\gamma _{c})- \Gamma _{c}\gamma ^{2}_{a}\Big )\bigg )^{-1}. \end{aligned}$$

(B. 13)

Here, the determinant $|\Gamma _{li}|$ of $\Gamma _{li}$ is written as

$$\begin{aligned} |\Gamma _{li}| = -\Gamma _{b}\Big ( \Gamma ^{2}_{ac}-\Gamma _{a}\Gamma _{c}\Big ). \end{aligned}$$

(B. 14)