Propagation of SH Wave in a Rotating Functionally Graded Magneto-Electro-Elastic Structure with Imperfect Interface

Abstract

Purpose

The SH waves in a rotating functionally graded magneto-electro-elastic system comprising an elastic substrate and functionally graded magneto-electro-elastic layer varying linearly are studied analytically. The layer and substrate are assumed that they are not perfectly connected at the interface in terms of mechanically, electrically, and magnetically.

Methods

Dispersion relations have been obtained for four cases of electrically open and magnetically open, electrically short and magnetically short, electrically open and magnetically short, and electrically short and magnetically open situations with imperfect interfaces. It has been derived utilizing the Bessel function of zero-order for layer and variable separable method for elastic substrate.

Results

Based on the numerical results, the properties of SH waves through the proposed framework and the conditions depending on various physical and geometrical parameters have been examined. Special cases are taken into account for different combinations of imperfections in the interface. The study examines the simultaneous simulated results of several physical parameters, including inhomogeneity, thickness, rotation, phase velocity, and imperfect SH wave interface distribution in the structure under consideration, which were created using Mathematica 7. Graphical representations of the cut-off frequency curves for the first three modes have also been included. The examined model could be helpful for the development of surface acoustic wave (SAW) devices.

Conclusion

The phase velocity curves of the rotating SH wave are impacted by distinct parameters such as inhomogeneity parameter, thickness, rotation parameter, and imperfect interfaces.

Appendix

$$\begin{aligned}{} & {} q=\sqrt{\dfrac{\rho _2(c^2k^2+\Omega ^2+2i\Omega kc)-C_{44}^{(2)}}{C_{44}^{(2)}}}.\\{} & {} a_{11}=ik\delta (1-\alpha h)\left( \overline{C_{44}^{(10)}}+\text {m}e_{15}^{(10)}+\text {n}h_{15}^{(10)}\right) J_0'(\eta _1),\\{} & {} a_{12}=-ik\delta (1-\alpha h)\left( \overline{C_{44}^{(10)}}+\text {m}e_{15}^{(10)}+\text {n}h_{15}^{(10)}\right) Y_0'(\eta _1),\\{} & {} a_{13}=ik(1-\alpha h)e_{15}^{(10)}J_0'(\gamma _1),\,a_{14}=-ik(1-\alpha h)e_{15}^{(10)}Y_0'(\gamma _1),\\ {}{} & {} a_{15}=ik(1-\alpha h)h_{15}^{(10)}J_0'(\gamma _1),\\{} & {} a_{16}=-ik(1-\alpha h)h_{15}^{(10)}Y_0'(\gamma _1),\\{} & {} a_{21}=ik\delta (1-\alpha h)\left( e_{15}^{(10)}-\text {m}\kappa _{11}^{(10)}-\text {n}\beta _{11}^{(10)}\right) J_0'(\eta _1),\\{} & {} a_{22}=ik\delta (1-\alpha h)\left( e_{15}^{(10)}-\text {m}\kappa _{11}^{(10)}-\text {n}\beta _{11}^{(10)}\right) Y_0'(\eta _1),\\ {}{} & {} a_{23}=-ik(1-\alpha h)\kappa _{11}^{(10)}J_0'(\gamma _1),\\{} & {} a_{24}=ik(1-\alpha h)\kappa _{11}^{(10)}Y_0'(\gamma _1), a_{25}=-ik(1-\alpha h)\beta _{11}^{(10)}J_0'(\gamma _1),\\ {}{} & {} a_{26}=ik(1-\alpha h)\beta _{11}^{(10)}Y_0'(\gamma _1),\\{} & {} a_{210}=k\kappa _0e^{-kh},\, a_{31}=\text {m}J_0(\eta _1),\,a_{32}=\text {m}Y_0(\eta ),\\ {}{} & {} a_{33}=J_0(\gamma _1),\,a_{34}=Y_0(\gamma _1),\,a_{310}=-e^{-kh},\\{} & {} a_{41}=ik\delta (1-\alpha h)\left( h_{15}^{(10)}-\text {m}\beta _{11}^{(10)}-\text {n}\mu _{11}^{(10)}\right) J_0'(\eta _1),\\ {}{} & {} a_{42}=-ik\delta (1-\alpha h)\left( h_{15}^{(10)}-\text {m}\beta _{11}^{(10)}-\text {n}\mu _{11}^{(10)}\right) Y_0'(\eta _1),\\{} & {} a_{43}=-ik(1-\alpha h)\beta _{11}^{(10)}J_0'(\gamma _1),\,a_{44}=ik(1-\alpha h)\beta _{11}^{(10)}Y_0'(\gamma _1),\\ {}{} & {} a_{45}=-ik(1-\alpha h)\mu _{11}^{(10)}J_0'(\gamma _1),\\{} & {} a_{46}=ik(1-\alpha h)\mu _{11}^{(10)}Y_0'(\gamma _1),\, a_{411}=k\mu _0e^{-kh}, a_{51}=\text {n}J_0(\eta _1), \\ {}{} & {} a_{52}=\text {n}Y_0(\eta _1),\,a_{55}=J_0(\gamma _1),\,a_{56}=Y_0(\gamma _1),\\{} & {} a_{511}=-e^{-kh},\,a_{61}=-I_m J_0(\eta _2),\, a_{62}=-I_m Y_0(\eta _2),\\ {}{} & {} a_{67}=I_m-q C_{44}^{(2)},\,a_{71}=-\text {m}I_e J_0(\eta _2),\, a_{72}=-\text {m}I_e Y_0(\eta _2),\\{} & {} a_{73}=-I_e J_0(\gamma _2),\, a_{74}=-I_e Y_0(\gamma _2),\,a_{78}=I_e+k\kappa _{11}^{(2)},\\ {}{} & {} a_{81}=-\text {n}I_{mg} J_0(\eta _2),\, a_{82}=-\text {n}I_{mg} Y_0(\eta _2),\\{} & {} a_{85}=-I_{mg} J_0(\gamma _2),\, a_{86}=-I_{mg}Y_0(\gamma _2),\,a_{89}=I_{mg}+k\mu _{11}^{(2)}, \\ {}{} & {} a_{91}=ik\delta \left( \overline{C_{44}^{(10)}}+\text {m}e_{15}^{(10)}+\text {n}h_{15}^{(10)}\right) J_0'(\eta _2),\\{} & {} a_{92}=-ik\delta \left( \overline{C_{44}^{(10)}}+\text {m}e_{15}^{(10)}+\text {n}h_{15}^{(10)}\right) Y_0'(\eta _2),\\ {}{} & {} a_{93}=ike_{15}^{(10)}J_0'(\gamma _2),\,a_{94}=-ike_{15}^{(10)}Y_0'(\gamma _2),\\{} & {} a_{95}=ik h_{15}^{(10)}J_0'(\gamma _2),\, a_{96}=-ikh_{15}^{(10)}Y_0'(\gamma _2), a_{98}=q C_{44}^{(2)},\\ {}{} & {} a_{101}=ik\delta \left( e_{15}^{(10)}-\text {m}\kappa _{11}^{(10)}-\text {n}\beta _{11}^{(10)}\right) J_0'(\eta _2),\\{} & {} a_{102}=ik\delta \left( e_{15}^{(10)}-\text {m}\kappa _{11}^{(10)}-\text {n}\beta _{11}^{(10)}\right) Y_0'(\eta _2),\\ {}{} & {} a_{103}=-ik\kappa _{11}^{(10)}J_0'(\gamma _2),\, a_{104}=ik\kappa _{11}^{(10)}Y_0'(\gamma _2),\\{} & {} a_{105}=-ik\beta _{11}^{(10)}J_0'(\gamma _2),\, a_{106}=ik\beta _{11}^{(10)}Y_0'(\gamma _2),\, a_{108}=-k\kappa _{11}^{(2)},\\ {}{} & {} a_{111}=ik\delta \left( h_{15}^{(10)}-\text {m}\beta _{11}^{(10)}-\text {n}\mu _{11}^{(10)}\right) J_0'(\eta _2),\\{} & {} a_{112}=-ik\delta \left( h_{15}^{(10)}-\text {m}\beta _{11}^{(10)}-\text {n}\mu _{11}^{(10)}\right) Y_0'(\eta _2),\\ {}{} & {} a_{113}=-ik\beta _{11}^{(10)}J_0'(\gamma _2),\,a_{114}=ik\beta _{11}^{(10)}Y_0'(\gamma _2),\\{} & {} a_{115}=-ik\mu _{11}^{(10)}J_0'(\gamma _2),\, a_{116}=ik\mu _{11}^{(10)}Y_0'(\gamma _2),\, a_{119}=k\mu _{11}^{(2)}. \\{} & {} \eta _1=ik\delta \left( \dfrac{1-\alpha h}{\alpha }\right) ,\,\eta _2=\dfrac{ik\delta }{\alpha },\,\gamma _1=ik \left( \dfrac{1-\alpha h}{\alpha }\right) ,\gamma _2=\dfrac{ik}{\alpha }.\\{} & {} b_{ij}=a_{ef},\forall \, i,j,f=1,2,\ldots ,9;\, e=1,3,5,6,7,8,9,10,11.\\{} & {} c_{ij}=a_{ef},\forall \, i,j,f=1,2,\ldots ,10;\, e=1,2,3,5,6,7,8,9,10. d_{ij}=a_{ef},\forall \, i,j,f=1,2,\ldots ,10;\\ {}{} & {} e=1,3,4,5,6,7,8,9,10. \end{aligned}$$