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Propagation characteristics of Love waves in a layered piezomagnetic structure

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Abstract

Propagation characteristics of Love waves in a layered structure composed of a piezomagnetic layer attached to an elastic substrate exposed to a magnetic field and compressive stress are investigated theoretically in this article. The effective elastic, piezomagnetic and magnetic permeability constants of a piezomagnetic material can affect by a magnetic field and compressive stress. The effects of magnetic field and compressive stress on the phase velocity, group velocity, mode shape, and magnetic potential of the Love wave are discussed in detail. It is found that the number of modes increases as the intensity of magnetic field increases while this tendency is reverse when applying compressive stress. As the intensity of magnetic field increases, the group velocity decreases but the magnitude of surface displacement of a piezomagnetic layer increases. The findings presented in this article are useful for improving the performance of surface acoustic wave (SAW) devices.

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Acknowledgements

The National Natural Science Foundation of China (Nos. 11962015, 11872194 and 11862012) and the Natural Science Foundation of Shandong Province (No. ZR2020KA006) are gratefully acknowledged for their financial support.

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Authors and Affiliations

  1. Key Laboratory of Mechanics On Disaster and Environment in Western China, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Fangming Lei & Yuanwen Gao

  2. School of Science, Lanzhou University of Technology, Lanzhou, 730000, Gansu, People’s Republic of China

    Fangming Lei, Zhengxi Chen & Chunlong Gu

  3. School of Architecture and Art, Weifang University of Science and Technology, Weifang, 262700, Shandong, People’s Republic of China

    Liansheng Ma

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  1. Fangming Lei
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  2. Zhengxi Chen
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  3. Chunlong Gu
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  4. Liansheng Ma
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  5. Yuanwen Gao
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Correspondence to Fangming Lei or Yuanwen Gao.

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Appendices

Appendix 1

The matrix forms of Eq. (1)

The matrix forms of Eq. (1) are described as follows (using the engineering shear strain, i.e., γxy = 2εxy; \(G = \frac{E}{{2\left( {1 + \nu } \right)}}\)):

$$ \begin{gathered} \left\{ \begin{gathered} \varepsilon _{x} \hfill \\ \varepsilon _{y} \hfill \\ \varepsilon _{z} \hfill \\ \gamma _{{yz}} \hfill \\ \gamma _{{zx}} \hfill \\ \gamma _{{xy}} \hfill \\ \end{gathered} \right\} = \left[ {\begin{array}{*{20}c} {{1 \mathord{\left/ {\vphantom {1 E}} \right. \kern-\nulldelimiterspace} E}} &\quad {{{ - \nu } \mathord{\left/ {\vphantom {{ - \nu } E}} \right. \kern-\nulldelimiterspace} E}} &\quad {{{ - \nu } \mathord{\left/ {\vphantom {{ - \nu } E}} \right. \kern-\nulldelimiterspace} E}} &\quad 0 &\quad 0 &\quad 0 \\ {{{ - \nu } \mathord{\left/ {\vphantom {{ - \nu } E}} \right. \kern-\nulldelimiterspace} E}} &\quad {{1 \mathord{\left/ {\vphantom {1 E}} \right. \kern-\nulldelimiterspace} E}} &\quad {{{ - \nu } \mathord{\left/ {\vphantom {{ - \nu } E}} \right. \kern-\nulldelimiterspace} E}} &\quad 0 &\quad 0 &\quad 0 \\ {{{ - \nu } \mathord{\left/ {\vphantom {{ - \nu } E}} \right. \kern-\nulldelimiterspace} E}} &\quad {{{ - \nu } \mathord{\left/ {\vphantom {{ - \nu } E}} \right. \kern-\nulldelimiterspace} E}} &\quad {{1 \mathord{\left/ {\vphantom {1 E}} \right. \kern-\nulldelimiterspace} E}} &\quad 0 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad {{1 \mathord{\left/ {\vphantom {1 G}} \right. \kern-\nulldelimiterspace} G}} &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad {{1 \mathord{\left/ {\vphantom {1 G}} \right. \kern-\nulldelimiterspace} G}} &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad {{1 \mathord{\left/ {\vphantom {1 G}} \right. \kern-\nulldelimiterspace} G}} \\ \end{array} } \right]\left\{ \begin{gathered} \sigma _{{\text{x}}} \hfill \\ \sigma _{{\text{y}}} \hfill \\ \sigma _{{\text{z}}} \hfill \\ \tau _{{{\text{yz}}}} \hfill \\ \tau _{{{\text{zx}}}} \hfill \\ \tau _{{{\text{xy}}}} \hfill \\ \end{gathered} \right\} \hfill \\ \quad \quad \; + \frac{{\lambda _{s} }}{{M_{s}^{2} }}\left[ {\begin{array}{*{20}c} {1 - {{\tilde{\sigma }_{{\text{x}}} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{{\text{x}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{{\text{x}}} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{{\text{x}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{{\text{x}}} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{{\text{x}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad 0 &\quad 0 &\quad 0 \\ { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{{\text{y}}} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{{\text{y}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad {1 - {{\tilde{\sigma }_{{\text{y}}} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{{\text{y}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{{\text{y}}} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{{\text{y}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad 0 &\quad 0 &\quad 0 \\ { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{{\text{z}}} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{{\text{z}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{{\text{z}}} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{{\text{z}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad {1 - {{\tilde{\sigma }_{{\text{z}}} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{{\text{z}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad 0 &\quad 0 &\quad 0 \\ { - {{2\tilde{\tau }_{{{\text{yz}}}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{{\text{yz}}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad { - {{2\tilde{\tau }_{{{\text{yz}}}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{{\text{yz}}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad { - {{2\tilde{\tau }_{{{\text{yz}}}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{{\text{yz}}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad 3 &\quad 0 &\quad 0 \\ { - {{2\tilde{\tau }_{{{\text{zx}}}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{{\text{zx}}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad { - {{2\tilde{\tau }_{{{\text{zx}}}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{{\text{zx}}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad { - {{2\tilde{\tau }_{{{\text{zx}}}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{{\text{zx}}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad 0 &\quad 3 &\quad 0 \\ { - {{2\tilde{\tau }_{{{\text{xy}}}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{{\text{xy}}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad { - {{2\tilde{\tau }_{{{\text{xy}}}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{{\text{xy}}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} & { - {{2\tilde{\tau }_{{{\text{xy}}}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{{\text{xy}}}} } {\sigma _{{\text{s}}} }}} \right. \kern-\nulldelimiterspace} {\sigma _{{\text{s}}} }}} &\quad 0 &\quad 0 &\quad 3 \\ \end{array} } \right]\left\{ \begin{gathered} M_{{\text{x}}}^{2} \hfill \\ M_{{\text{y}}}^{2} \hfill \\ M_{{\text{z}}}^{2} \hfill \\ M_{{\text{y}}} M_{{\text{z}}} \hfill \\ M_{{\text{z}}} M_{{\text{x}}} \hfill \\ M_{{\text{x}}} M_{{\text{y}}} \hfill \\ \end{gathered} \right\}, \hfill \\ \end{gathered}$$
(A1)
$$ \begin{gathered} \left\{ \begin{gathered} H_{{\text{x}}} \hfill \\ H_{{\text{y}}} \hfill \\ H_{{\text{z}}} \hfill \\ \end{gathered} \right\} = \frac{1}{{{\text{k}}M}}f^{ - 1} \left( {\frac{M}{{M_{{\text{s}}} }}} \right)\left[ {\begin{array}{*{20}c} 1 &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad 0 \\ 0 &\quad 0 &\quad 1 \\ \end{array} } \right]\left\{ \begin{gathered} M_{{\text{x}}} \hfill \\ M_{{\text{y}}} \hfill \\ M_{{\text{z}}} \hfill \\ \end{gathered} \right\} \hfill \\ \quad\quad\quad \quad \;\, - \frac{{\lambda_{{\text{s}}} }}{{\mu_{0} M_{{\text{s}}}^{2} }}\left[ {\begin{array}{*{20}c} {2\tilde{\sigma }_{{\text{x}}} - \frac{{I_{\sigma }^{2} - 3II_{\sigma } }}{{\sigma_{{\text{s}}} }}} &\quad {2\tilde{\tau }_{{{\text{xy}}}} } &\quad {2\tilde{\tau }_{{{\text{zx}}}} } \\ {2\tilde{\tau }_{{{\text{xy}}}} } &\quad {2\tilde{\sigma }_{{\text{y}}} - \frac{{I_{\sigma }^{2} - 3II_{\sigma } }}{{\sigma_{{\text{s}}} }}} &\quad {2\tilde{\tau }_{{{\text{yz}}}} } \\ {2\tilde{\tau }_{{{\text{zx}}}} } &\quad {2\tilde{\tau }_{{{\text{yz}}}} } &\quad {2\tilde{\sigma }_{{\text{z}}} - \frac{{I_{\sigma }^{2} - 3II_{\sigma } }}{{\sigma_{{\text{s}}} }}} \\ \end{array} } \right]\left\{ \begin{gathered} M_{{\text{x}}} \hfill \\ M_{{\text{y}}} \hfill \\ M_{{\text{z}}} \hfill \\ \end{gathered} \right\}. \hfill \\ \end{gathered} $$
(A2)

Appendix 2

The effective material constants

The effective material constants of the piezomagnetic material can be expressed as follows:

$$ \begin{gathered} c_{{{\text{ijkl}}}} \left( {{\varvec{H}},{\varvec{\sigma}}} \right) = {\text{S}}_{{_{{{\text{ijkl}}}} }}^{ - 1} \left( {{\varvec{H}},{\varvec{\sigma}}} \right), \hfill \\ q_{{{\text{mij}}}} \left( {{\varvec{H}},{\varvec{\sigma}}} \right) = c_{{{\text{ijkl}}}} \left( {{\varvec{H}},{\varvec{\sigma}}} \right)\overline{q}_{{{\text{mkl}}}} \left( {{\varvec{H}},{\varvec{\sigma}}} \right), \hfill \\ \mu_{{{\text{nm}}}} \left( {{\varvec{H}},{\varvec{\sigma}}} \right) = \overline{\mu }_{{{\text{nm}}}} \left( {{\varvec{H}},{\varvec{\sigma}}} \right) - q_{{{\text{nkl}}}} \left( {{\varvec{H}},{\varvec{\sigma}}} \right){\text{S}}_{{{\text{ijkl}}}} \left( {{\varvec{H}},{\varvec{\sigma}}} \right)q_{{{\text{mij}}}} \left( {{\varvec{H}},{\varvec{\sigma}}} \right), \hfill \\ \end{gathered} $$
(B1)

where

$$ {\text{S}}_{11} = \frac{1}{E} - \frac{{\lambda_{{\text{s}}} M_{{\text{z}}}^{2} }}{{\sigma_{{\text{s}}} M_{{\text{s}}}^{2} }} + \frac{{{\text{k}}\lambda_{{\text{s}}}^{2} \frac{{M_{{\text{s}}}^{2} }}{{M_{{\text{s}}}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{{\text{x}}} }}{{\sigma_{{\text{s}}} }}} \right)^{2} }}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B2)
$$ {\text{S}}_{12} = - \frac{v}{E} + \frac{{\lambda_{{\text{s}}} M_{z}^{2} }}{{2\sigma_{{\text{s}}} M_{{\text{s}}}^{2} }} + \frac{{{\text{k}}\lambda_{{\text{s}}}^{2} \frac{{M_{z}^{2} }}{{M_{{\text{s}}}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{{\text{x}}} }}{{\sigma_{{\text{s}}} }}} \right)\left( {1 + \frac{{2\tilde{\sigma }_{{\text{y}}} }}{{\sigma_{{\text{s}}} }}} \right)}}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B3)
$$ {\text{S}}_{13} = - \frac{v}{E} + \frac{{\lambda_{{\text{s}}} M_{{\text{z}}}^{2} }}{{2\sigma_{{\text{s}}} M_{{\text{s}}}^{2} }} - \frac{{2k\lambda_{{\text{s}}}^{2} \frac{{M_{{\text{z}}}^{2} }}{{M_{{\text{s}}}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{{\text{x}}} }}{{\sigma_{{\text{s}}} }}} \right)\left( {1 - \frac{{\tilde{\sigma }_{{\text{z}}} }}{{\sigma_{{\text{s}}} }}} \right)}}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B4)
$$ {\text{S}}_{22} = \frac{1}{E} - \frac{{\lambda_{{\text{s}}} M_{{\text{z}}}^{2} }}{{\sigma_{{\text{s}}} M_{{\text{s}}}^{2} }} + \frac{{{\text{k}}\lambda_{{\text{s}}}^{2} \frac{{M_{{\text{z}}}^{2} }}{{M_{{\text{s}}}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{{\text{y}}} }}{{\sigma_{{\text{s}}} }}} \right)^{2} }}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B5)
$$ {\text{S}}_{23} = - \frac{v}{E} + \frac{{\lambda_{{\text{s}}} M_{{\text{z}}}^{2} }}{{2\sigma_{{\text{s}}} M_{{\text{s}}}^{2} }} - \frac{{2{\text{k}}\lambda_{{\text{s}}}^{2} \frac{{M_{{\text{z}}}^{2} }}{{M_{{\text{s}}}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{{\text{y}}} }}{{\sigma_{{\text{s}}} }}} \right)\left( {1 - \frac{{\tilde{\sigma }_{{\text{z}}} }}{{\sigma_{{\text{s}}} }}} \right)}}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{s} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B6)
$$ {\text{S}}_{33} = \frac{1}{E} - \frac{{\lambda_{{\text{s}}} M_{{\text{z}}}^{2} }}{{\sigma_{{\text{s}}} M_{{\text{s}}}^{2} }} + \frac{{4{\text{k}}\lambda_{{\text{s}}}^{2} \frac{{M_{{\text{z}}}^{2} }}{{M_{{\text{s}}}^{2} }}\left( {1 - \frac{{\tilde{\sigma }_{{\text{z}}} }}{{\sigma_{{\text{s}}} }}} \right)^{2} }}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B7)
$$ {\text{S}}_{44} = \frac{1}{G} - \frac{{3\lambda_{{\text{s}}} M_{{\text{z}}}^{2} }}{{\sigma_{{\text{s}}} M_{{\text{s}}}^{2} }} + \frac{{9k\lambda_{{\text{s}}}^{2} \frac{{M_{{\text{z}}}^{2} }}{{M_{{\text{s}}}^{2} }}}}{{3\mu_{0} M_{{\text{s}}} - k\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{y}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B8)
$$ {\text{S}}_{55} = \frac{1}{G} - \frac{{3\lambda_{{\text{s}}} M_{{\text{z}}}^{2} }}{{\sigma_{{\text{s}}} M_{{\text{s}}}^{2} }} + \frac{{9{\text{k}}\lambda_{{\text{s}}}^{2} \frac{{M_{{\text{z}}}^{2} }}{{M_{{\text{s}}}^{2} }}}}{{3\mu_{0} M_{{\text{s}}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{x}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B9)
$$ {\text{S}}_{66} = \frac{1}{G} - \frac{{3\lambda_{{\text{s}}} M_{{\text{z}}}^{2} }}{{\sigma_{{\text{s}}} M_{{\text{s}}}^{2} }}, $$
(B10)
$$ \overline{q}_{31} = - \frac{{{\text{k}}\mu_{0} \lambda_{{\text{s}}} M_{{\text{z}}} \left( {1 + \frac{{2\tilde{\sigma }_{{\text{x}}} }}{{\sigma_{{\text{s}}} }}} \right)}}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B11)
$$ \overline{q}_{32} = - \frac{{{\text{k}}\mu_{0} \lambda_{{\text{s}}} M_{{\text{z}}} \left( {1 + \frac{{2\tilde{\sigma }_{{\text{y}}} }}{{\sigma_{{\text{s}}} }}} \right)}}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B12)
$$ \overline{q}_{33} = \frac{{2{\text{k}}\mu_{0} \lambda_{{\text{s}}} M_{{\text{z}}} \left( {1 - \frac{{\tilde{\sigma }_{{\text{z}}} }}{{\sigma_{{\text{s}}} }}} \right)}}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B13)
$$ \overline{q}_{24} = \frac{{3{\text{k}}\mu_{0} \lambda_{{\text{s}}} M_{{\text{z}}} }}{{3\mu_{0} M_{{\text{s}}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{y}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B14)
$$ \overline{q}_{15} = \frac{{3{\text{k}}\mu_{0} \lambda_{{\text{s}}} M_{{\text{z}}} }}{{3\mu_{0} M_{{\text{s}}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{x}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B15)
$$ \overline{\mu }_{11} = \mu_{0} + \frac{{{\text{k}}\mu_{0}^{2} M_{{\text{s}}}^{2} }}{{3\mu_{0} M_{{\text{s}}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{x}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B16)
$$ \overline{\mu }_{22} = \mu_{0} + \frac{{{\text{k}}\mu_{0}^{2} M_{{\text{s}}}^{2} }}{{3\mu_{0} M_{{\text{s}}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{y} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B17)
$$ \overline{\mu }_{33} = \mu_{0} + \frac{{{\text{k}}\mu_{0}^{2} M_{{\text{s}}}^{2} }}{{\frac{{\mu_{0} M_{{\text{s}}} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - {\text{k}}\lambda_{{\text{s}}} \left( {2\tilde{\sigma }_{{\text{z}}} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)}}, $$
(B18)

with \(M_{{\text{z}}} = M_{{\text{s}}} \left( {\cot \;hM_{3} - \frac{1}{{M_{3} }}} \right)\) and \(M_{3} = {\text{k}}H_{{\text{z}}} + \frac{{{\text{k}}\lambda_{{\text{s}}} M_{{\text{z}}} }}{{\mu_{0} M_{{\text{s}}}^{2} }}\left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{{\text{s}}} }}} \right)\).

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Lei, F., Chen, Z., Gu, C. et al. Propagation characteristics of Love waves in a layered piezomagnetic structure. Acta Mech 234, 5101–5113 (2023). https://doi.org/10.1007/s00707-023-03644-w

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