Scattering of shear horizontal waves by the piezomagnetic material cylinder embedded in an elastic medium

Abstract

Scattering behaviors of shear horizontal (SH) waves by the piezomagnetic cylinder embedded in an infinite elastic medium are investigated. The effective material constants of the piezomagnetic cylinder are changed with the magnetic field and compressive stress. By applying the method of wave function expansion, the governing differential equations of the scattered SH waves are solved and the mechanical displacement, dynamic stress, magnetic potential, and magnetic induction strength affected by the magnetic field and compressive stress are obtained. It is found that the mechanical displacement increases as the magnetic field strength increases while this tendency is reversed when applying compressive stress. The circumferential dynamic stress, magnetic potential, and magnetic induction strength increase as the magnetic field strength increases, respectively. But, the variation of the radial dynamic stress with the magnetic field is complex. The findings presented in this article are useful for designing magneto-elastic acoustic wave devices with high performance.

Appendices

Appendix A: The matrix forms of Eq. (1)

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

$$\begin{aligned} \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 _{x} \hfill \\ \sigma _{y} \hfill \\ \sigma _{z} \hfill \\ \tau _{{yz}} \hfill \\ \tau _{{zx}} \hfill \\ \tau _{{xy}} \hfill \\ \end{gathered} \right\} \\ & \quad + \frac{{\lambda _{s} }}{{M_{s}^{2} }}\left[ {\begin{array}{*{20}c} {1 - {{\tilde{\sigma }_{x} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{x} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{x} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{x} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{x} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{x} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad 0 & \quad 0 & \quad 0 \\ { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{y} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{y} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad {1 - {{\tilde{\sigma }_{y} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{y} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{y} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{y} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad 0 & \quad 0 & \quad 0 \\ { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{z} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{z} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {{\tilde{\sigma }_{z} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{z} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad {1 - {{\tilde{\sigma }_{z} } \mathord{\left/ {\vphantom {{\tilde{\sigma }_{z} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad 0 & \quad 0 & \quad 0 \\ { - {{2\tilde{\tau }_{{yz}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{yz}} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {{2\tilde{\tau }_{{yz}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{yz}} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {{2\tilde{\tau }_{{yz}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{yz}} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad 3 & \quad 0 & \quad 0 \\ { - {{2\tilde{\tau }_{{zx}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{zx}} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {{2\tilde{\tau }_{{zx}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{zx}} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {{2\tilde{\tau }_{{zx}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{zx}} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad 0 & \quad 3 & \quad 0 \\ { - {{2\tilde{\tau }_{{xy}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{xy}} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {{2\tilde{\tau }_{{xy}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{xy}} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad { - {{2\tilde{\tau }_{{xy}} } \mathord{\left/ {\vphantom {{2\tilde{\tau }_{{xy}} } {\sigma _{s} }}} \right. \kern-\nulldelimiterspace} {\sigma _{s} }}} & \quad 0 & \quad 0 & \quad 3 \\ \end{array} } \right]\left\{ \begin{gathered} M_{x}^{2} \hfill \\ M_{y}^{2} \hfill \\ M_{z}^{2} \hfill \\ M_{y} M_{z} \hfill \\ M_{z} M_{x} \hfill \\ M_{x} M_{y} \hfill \\ \end{gathered} \right\}, \hfill \\ \end{aligned}$$

(A1)

$$\begin{aligned} \left\{ \begin{gathered} H_{x} \hfill \\ H_{y} \hfill \\ H_{z} \hfill \\ \end{gathered} \right\} &= \frac{1}{{\overline{k}M}}f^{ - 1} \left( {\frac{M}{{M_{s} }}} \right)\left[ {\begin{array}{*{20}c} 1 & \quad 0 & \quad 0 \\ 0 & \quad 1 & \quad 0 \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]\left\{ \begin{gathered} M_{x} \hfill \\ M_{y} \hfill \\ M_{z} \hfill \\ \end{gathered} \right\}\\ & \quad - \frac{{\lambda_{s} }}{{\mu_{0} M_{s}^{2} }}\left[ {\begin{array}{*{20}c} {2\tilde{\sigma }_{x} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} & \quad {2\tilde{\tau }_{xy} } & \quad {2\tilde{\tau }_{zx} } \\ {2\tilde{\tau }_{xy} } & \quad {2\tilde{\sigma }_{y} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} & \quad {2\tilde{\tau }_{yz} } \\ {2\tilde{\tau }_{zx} } & \quad {2\tilde{\tau }_{yz} } & \quad {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \\ \end{array} } \right]\left\{ \begin{gathered} M_{x} \hfill \\ M_{y} \hfill \\ M_{z} \hfill \\ \end{gathered} \right\}. \hfill \\ \end{aligned}$$

(A2)

Appendix B: The effective material constants

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

$$\begin{gathered} c_{ijgl} \left( {{\varvec{H}},{\varvec{\sigma}}} \right) = S_{{_{ijgl} }}^{ - 1} \left( {{\varvec{H}},{\varvec{\sigma}}} \right), \hfill \\ q_{mij} \left( {{\varvec{H}},{\varvec{\sigma}}} \right) = c_{ijgl} \left( {{\varvec{H}},{\varvec{\sigma}}} \right)\overline{q}_{mgl} \left( {{\varvec{H}},{\varvec{\sigma}}} \right), \hfill \\ \mu_{nm} \left( {{\varvec{H}},{\varvec{\sigma}}} \right) = \overline{\mu }_{nm} \left( {{\varvec{H}},{\varvec{\sigma}}} \right) - q_{ngl} \left( {{\varvec{H}},{\varvec{\sigma}}} \right)S_{ijgl} \left( {{\varvec{H}},{\varvec{\sigma}}} \right)q_{mij} \left( {{\varvec{H}},{\varvec{\sigma}}} \right), \hfill \\ \end{gathered}$$

(B1)

where

$$S_{11} = \frac{1}{E} - \frac{{\lambda_{s} M_{z}^{2} }}{{\sigma_{s} M_{s}^{2} }} + \frac{{\overline{k}\lambda_{s}^{2} \frac{{M_{z}^{2} }}{{M_{s}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{x} }}{{\sigma_{s} }}} \right)^{2} }}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B2)

$$S_{12} = - \frac{v}{E} + \frac{{\lambda_{s} M_{z}^{2} }}{{2\sigma_{s} M_{s}^{2} }} + \frac{{\overline{k}\lambda_{s}^{2} \frac{{M_{z}^{2} }}{{M_{s}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{x} }}{{\sigma_{s} }}} \right)\left( {1 + \frac{{2\tilde{\sigma }_{y} }}{{\sigma_{s} }}} \right)}}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B3)

$$S_{13} = - \frac{v}{E} + \frac{{\lambda_{s} M_{z}^{2} }}{{2\sigma_{s} M_{s}^{2} }} - \frac{{2\overline{k}\lambda_{s}^{2} \frac{{M_{z}^{2} }}{{M_{s}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{x} }}{{\sigma_{s} }}} \right)\left( {1 - \frac{{\tilde{\sigma }_{z} }}{{\sigma_{s} }}} \right)}}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B4)

$$S_{22} = \frac{1}{E} - \frac{{\lambda_{s} M_{z}^{2} }}{{\sigma_{s} M_{s}^{2} }} + \frac{{\overline{k}\lambda_{s}^{2} \frac{{M_{z}^{2} }}{{M_{s}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{y} }}{{\sigma_{s} }}} \right)^{2} }}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B5)

$$S_{23} = - \frac{v}{E} + \frac{{\lambda_{s} M_{z}^{2} }}{{2\sigma_{s} M_{s}^{2} }} - \frac{{2\overline{k}\lambda_{s}^{2} \frac{{M_{z}^{2} }}{{M_{s}^{2} }}\left( {1 + \frac{{2\tilde{\sigma }_{y} }}{{\sigma_{s} }}} \right)\left( {1 - \frac{{\tilde{\sigma }_{z} }}{{\sigma_{s} }}} \right)}}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B6)

$$S_{33} = \frac{1}{E} - \frac{{\lambda_{s} M_{z}^{2} }}{{\sigma_{s} M_{s}^{2} }} + \frac{{4\overline{k}\lambda_{s}^{2} \frac{{M_{z}^{2} }}{{M_{s}^{2} }}\left( {1 - \frac{{\tilde{\sigma }_{z} }}{{\sigma_{s} }}} \right)^{2} }}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B7)

$$S_{44} = \frac{1}{G} - \frac{{3\lambda_{s} M_{z}^{2} }}{{\sigma_{s} M_{s}^{2} }} + \frac{{9\overline{k}\lambda_{s}^{2} \frac{{M_{z}^{2} }}{{M_{s}^{2} }}}}{{3\mu_{0} M_{s} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{y} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B8)

$$S_{55} = \frac{1}{G} - \frac{{3\lambda_{s} M_{z}^{2} }}{{\sigma_{s} M_{s}^{2} }} + \frac{{9\overline{k}\lambda_{s}^{2} \frac{{M_{z}^{2} }}{{M_{s}^{2} }}}}{{3\mu_{0} M_{s} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{x} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B9)

$$S_{66} = \frac{1}{G} - \frac{{3\lambda_{s} M_{z}^{2} }}{{\sigma_{s} M_{s}^{2} }},$$

(B10)

$$\overline{q}_{31} = - \frac{{\overline{k}\mu_{0} \lambda_{s} M_{z} \left( {1 + \frac{{2\tilde{\sigma }_{x} }}{{\sigma_{s} }}} \right)}}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B11)

$$\overline{q}_{32} = - \frac{{\overline{k}\mu_{0} \lambda_{s} M_{z} \left( {1 + \frac{{2\tilde{\sigma }_{y} }}{{\sigma_{s} }}} \right)}}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B12)

$$\overline{q}_{33} = \frac{{2\overline{k}\mu_{0} \lambda_{s} M_{z} \left( {1 - \frac{{\tilde{\sigma }_{z} }}{{\sigma_{s} }}} \right)}}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B13)

$$\overline{q}_{24} = \frac{{3\overline{k}\mu_{0} \lambda_{s} M_{z} }}{{3\mu_{0} M_{s} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{y} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B14)

$$\overline{q}_{15} = \frac{{3\overline{k}\mu_{0} \lambda_{s} M_{z} }}{{3\mu_{0} M_{s} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{x} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B15)

$$\overline{\mu }_{11} = \mu_{0} + \frac{{\overline{k}\mu_{0}^{2} M_{s}^{2} }}{{3\mu_{0} M_{s} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{x} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B16)

$$\overline{\mu }_{22} = \mu_{0} + \frac{{\overline{k}\mu_{0}^{2} M_{s}^{2} }}{{3\mu_{0} M_{s} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{y} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B17)

$$\overline{\mu }_{33} = \mu_{0} + \frac{{\overline{k}\mu_{0}^{2} M_{s}^{2} }}{{\frac{{\mu_{0} M_{s} }}{{\frac{1}{{M_{3}^{2} }} - \csc {\text{h}}^{2} \left( {M_{3} } \right)}} - \overline{k}\lambda_{s} \left( {2\tilde{\sigma }_{z} - \frac{{{\text{I}}_{\sigma }^{2} - 3{\text{II}}_{\sigma } }}{{\sigma_{s} }}} \right)}},$$

(B18)

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