Multimodal frequency and phase velocity spectrum of shear wave in microstructural flexomagnetic plate loaded with complex fluid

Magnetoelastic (ME) effect-based sensors have been innovatively engineered for diverse applications, ranging from ensuring food safety to facilitating immunoassays and structural health monitoring. These cutting-edge devices leverage piezomagnetic/flexomagnetic plates to propagate waves, with the top surface of the plate frequently encountering contact with liquids. This article delves into the intricacies of analytically modeling the flexomagnetic effect, focusing on strain gradients within a finite-thickness plate. Additionally, it investigates the dispersion of shear horizontal (SH) waves within the same flexomagnetic plate when immersed in a Newtonian liquid region. The study meticulously considers the influences of plate thickness, flexomagnetic coefficient, and the viscosity of the liquid, particularly for a Cobalt-ferrite plate. The obtained dispersion equations are used to plot phase velocity spectra and frequency spectra. The implications of this research extend to optimizing surface acoustic wave sensors (SAW) designed for underwater applications, bio-sensing, and chemo-sensing. The comprehensive understanding of the dispersion characteristics of SH waves in piezomagnetic plates with flexomagnetic effect equips researchers and engineers to enhance the performance and efficiency of SAW devices across a spectrum of under-liquid scenarios.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

No funding was received to assist with the preparation of this manuscript.

Authors and Affiliations

1. Department of Mathematics and Computing, Indian Institute of Technology (ISM) Dhanbad, Dhanbad, Jharkhand, 826004, India

   Mahargha Biswas & Sanjeev A. Sahu

Corresponding author

Correspondence to Sanjeev A. Sahu.

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.

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Informed consent was obtained from all individual participants included in the study.

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Appendix

1.1 For magnetically short case

\(Q_{11}=-\eta \lambda ^l \exp {-\lambda ^l}h_1,\) \(Q_{12} = \left( sc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_1 e^{-m_1 h_1}\), \(Q_{13} = -\left( sc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_1 e^{m_1 h_1}\), \(Q_{14} = \left( tc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_2 e^{m_2 h_1},\) \(Q_{15} = -\left( tc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_2 e^{m_2 h_1}.\)

\(Q_{21}=\exp {\lambda h_1},\) \(Q_{22}=i c k_1 s\exp {-m_1 h_1},\) \(Q_{23}=i c k_1 s\exp {m_1 h_1},\) \(Q_{24}=i c k_1 t\exp {-m_2 h_1},\) \(Q_{25}=i c k_1 t\exp {-m_2 h_1}.\)

\(Q_{32}=\left( -\mu ^p_{11}+sh^p _{15}+ik_1s\left( h_{41}+\frac{h_{52}}{2}\right) \right) m_1 e^{-m_1 h_1}\), \(Q_{33}=-\left( -\mu ^p_{11}+sh^p _{15}+ik_1s\left( h_{41}+\frac{h_{52}}{2}\right) \right) m_1 e^{m_1 h_1}\), \(Q_{34}=\left( -\mu ^p_{11}+th^p _{15}+ik_1t\left( h_{41}+\frac{h_{52}}{2}\right) \right) m_2 e^{-m_2 h_1}\), \(Q_{35}=-\left( -\mu ^p_{11}+th^p _{15}+ik_1t\left( h_{41}+\frac{h_{52}}{2}\right) \right) m_2 e^{m_2 h_1},\)

\(Q_{42} = \left( sc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_1\), \(Q_{43} = -\left( sc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_1\), \(Q_{44} = \left( tc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_2\) \(Q_{45} = -\left( tc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_2,\)

\(Q_{32}=\left( -\mu ^p_{11}+sh^p _{15}+ik_1s\left( h_{41}+\frac{h_{52}}{2}\right) \right) m_1 \), \(Q_{33}=-\left( -\mu ^p_{11}+sh^p _{15}+ik_1s\left( h_{41}+\frac{h_{52}}{2}\right) \right) m_1 \), \(Q_{34}=\left( -\mu ^p_{11}+th^p _{15}+ik_1t\left( h_{41}+\frac{h_{52}}{2}\right) \right) m_2 \), \(Q_{35}=-\left( -\mu ^p_{11}+th^p _{15}+ik_1t\left( h_{41}+\frac{h_{52}}{2}\right) \right) m_2,\)

1.2 For magnetically open case

\(T_{11}=-\eta \lambda ^l \exp {-\lambda ^l}h_1,\) \(T_{12} = \left( sc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_1 e^{-m_1 h_1}\), \(T_{13} = -\left( sc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_1 e^{m_1 h_1}\), \(T_{14} = \left( tc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_2 e^{m_2 h_1},\) \(T_{15} = -\left( tc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_2 e^{m_2 h_1}.\)

\(T_{21}=\exp {\lambda h_1},\) \(T_{22}=i c k_1 s\exp {-m_1 h_1},\) \(T_{23}=i c k_1 s\exp {m_1 h_1},\) \(T_{24}=i c k_1 t\exp {-m_2 h_1},\) \(T_{25}=i c k_1 t\exp {-m_2 h_1}.\)

\(T_{32}= e^{-m_1 h_1}\), \(T_{33}= e^{m_1 h_1}\), \(T_{34}= e^{-m_2 h_1}\), \(T_{35}= e^{m_2 h_1},\)

\(T_{42} = \left( sc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_1\), \(T_{43} = -\left( sc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_1\), \(T_{44} = \left( tc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_2\) \(T_{45} = -\left( tc^p_{44}+h^p _{15}-ik_1\frac{h_{41}}{2}\right) m_2,\) \(T_{52}=1\) \(T_{53}=1\), \(T_{54}=1\), \(T_{55}=1\),

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