Analysis of Love-type acoustic wave in a functionally graded piezomagnetic plate sandwiched between elastic layers

Abstract

The objective of the present research article is to explore the behavior of phase velocity of surface acoustic Love-type wave in a layer of functionally graded piezomagnetic material (FGPM) sandwiched between a thin layer of heterogeneous fiber-reinforced material and heterogeneous elastic substrate. The fiber-reinforced layer is considered exponentially graded and under initial stress (compressive/ tensile). The interfaces between the FGPM layer and elastic layer/substrate are mechanically imperfect. The material properties of the piezomagnetic layer and elastic substrate are assumed to be varying in quadratic way along with the depth of the structure. Mechanical displacement components for each layer and magnetic potential for the piezomagnetic layer are obtained by solving linear mechanical/coupled magneto-mechanical field equations. Using suitable boundary conditions, transcendental dispersion relations are obtained for magnetically open and short cases. A numerical example is provided for the layers and substrate. The effects of heterogeneity of two elastic layers, gradient factor of piezomagnetic layer, initial stress, reinforcement direction, and mechanical imperfections on the fundamental mode phase velocity of Love-type wave are illustrated graphically. The present investigation may find practical application in the design of piezomagnetic sensors and transducers.

Appendix

1.1 For magnetically open case:

$H_{11} = [\mu '_T\alpha _1 + a^2 _1(\mu '_L - \mu '_T)\alpha _1 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _1(h_1 + h_2)},\quad$ $H_{12} = [\mu '_T\alpha _2 + a^2 _1(\mu '_L - \mu '_T)\alpha _2 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _2(h_1 + h_2)},\quad$ $H_{23} = \frac{h^0_{15}}{\mu ^0_{11}}\cos {k \lambda _1 h_1},\quad$ $H_{24} = -\frac{h^0_{15}}{\mu ^0_{11}}\sin {k \lambda _1 h_1},\quad$ $H_{25} = e^{-kh_1},\quad$ $H_{26} = e^{kh_1},\quad$ $H_{31} = e^{-\gamma ' h_1}[\mu '_T\alpha _1 + a^2 _1(\mu '_L - \mu '_T)\alpha _1 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _1 h_1},\quad$ $H_{32} = e^{-\gamma ' h_1}[\mu '_T\alpha _2 + a^2 _1(\mu '_L - \mu '_T)\alpha _2 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _2 h_1},\quad$ $H_{33} = -\bar{c_{44}}[-b\cos {k\lambda _1 h_1} + (1-bh_1)k\lambda _1\sin {k\lambda _1 h_1}],\quad$ $H_{34} = -\bar{c_{44}}[b\sin {k\lambda _1 h_1} + (1-bh_1)k\lambda _1\cos {k\lambda _1 h_1}],\quad$ $H_{35} = h^0_{15}[-be^{-kh_1} + (1-bh_1)ke^{-kh_1}],\quad$ $H_{36} = h^0_{15}[-be^{kh_1} - (1-bh_1)ke^{kh_1}],\quad$ $H_{41} = e^{-\gamma ' h_1}[\mu '_T\alpha _1 + a^2 _1(\mu '_L - \mu '_T)\alpha _1 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _1 h_1}+k_m e^{-\alpha _1 h_1},\quad$ $H_{42} = e^{-\gamma ' h_1}[\mu '_T\alpha _2 + a^2 _1(\mu '_L - \mu '_T)\alpha _2 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _2 h_1}+k_me^{-\alpha _2 h_1},\quad$ $H_{43} = -k_m\frac{1}{1 - bh_1}\cos {k\lambda _1 h_1},\quad$ $H_{44} = k_m\frac{1}{1 - bh_1}\sin {k\lambda _1 h_1},\quad$ $H_{53} = -k_n,\quad$ $H_{57} = c^s_{44}[-a-k\xi ]+k_n,\quad$ $H_{63} = -\bar{c_{44}}b,\quad$ $H_{64} = \bar{c_{44}}k\lambda _1 ,\quad$ $H_{65} = h^0 _{15}[k-b],\quad$ $H_{66} = - h^0 _{15}[k+b],\quad$ $H_{67} = c^s_{44}[-a-k\xi ],\quad$ $H_{75} = 1,\quad$ $H_{76} = 1,\quad$ $H_{73} = \frac{h^0 _{15}}{\mu ^0_{11}}.$   

1.2 For magnetically short case:

$G_{11} = [\mu '_T\alpha _1 + a^2 _1(\mu '_L - \mu '_T)\alpha _1 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _1(h_1 + h_2)},\quad$ $G_{12} = [\mu '_T\alpha _2 + a^2 _1(\mu '_L - \mu '_T)\alpha _2 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _2(h_1 + h_2)}$,   $G_{25} = [b-k(1-bh_1)]e^{-kh_1}$,   $G_{26} = [b+k(1-bh_1)]e^{kh_1}$,   $G_{31} = e^{-\gamma ' h_1}[\mu '_T\alpha _1 + a^2 _1(\mu '_L - \mu '_T)\alpha _1 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _1 h_1}$,   $G_{32} = e^{-\gamma ' h_1}[\mu '_T\alpha _2 + a^2 _1(\mu '_L - \mu '_T)\alpha _2 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _2 h_1}$,   $G_{33} = -\bar{c_{44}}[-b\cos {k\lambda _1 h_1} + (1-bh_1)k\lambda _1\sin {k\lambda _1 h_1}]$,   $G_{34} = -\bar{c_{44}}[b\sin {k\lambda _1 h_1} + (1-bh_1)k\lambda _1\cos {k\lambda _1 h_1}]$,   $G_{35} = h^0_{15}[-be^{-kh_1} + (1-bh_1)ke^{-kh_1}]$,   $G_{36} = h^0_{15}[-be^{kh_1} - (1-bh_1)ke^{kh_1}]$,   $H_{41} = e^{-\gamma ' h_1}[\mu '_T\alpha _1 + a^2 _1(\mu '_L - \mu '_T)\alpha _1 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _1 h_1}+k_m e^{-\alpha _1 h_1},\quad$ $H_{42} = e^{-\gamma ' h_1}[\mu '_T\alpha _2 + a^2 _1(\mu '_L - \mu '_T)\alpha _2 + a_1a_2ik((\mu '_L - \mu '_T)]e^{-\alpha _2 h_1}+k_me^{-\alpha _2 h_1},\quad$ $H_{43} = -k_m\frac{1}{1 - bh_1}\cos {k\lambda _1 h_1},\quad$ $H_{44} = k_m\frac{1}{1 - bh_1}\sin {k\lambda _1 h_1},\quad$ $H_{53} = -k_n,\quad$ $H_{57} = c^s_{44}[-a-k\xi ]+k_n,\quad$ $G_{63} = -\bar{c_{44}}b$,   $G_{64} = \bar{c_{44}}k\lambda _1$,   $G_{65} = h^0 _{15}[k-b]$,   $G_{66} = - h^0 _{15}[k+b]$,   $G_{67} = c^s_{44}[-a-k\xi ]$,   $G_{75} = -b+k$,   $G_{76} = -b-k$.