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SH-wave propagation in a piezoelectric layer over a heterogeneous dry sandy half-space

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Abstract

The present paper investigates the behavior of SH-wave propagation in a heterogeneous dry sandy half-space bonded by a piezoelectric layer abutting the vacuum. The vacuum is assumed as a layer of air. Solutions for mechanical displacement and electrical potential functions are obtained by solving the coupled field equations of the piezoelectric layer with the help of the separation of variables technique. The rigidity and density of the half-space are assumed to vary exponentially with depth. Suitable boundary conditions are applied to obtain the dispersion equation of the SH-wave for electrically open and short cases. Some special cases of the problem are extracted, and the results obtained match the classical Love wave equation, which validates the authenticity of the considered problem. The effect of physical parameters such as piezoelectric, dielectric constant, inhomogeneity, and sandy parameters on the phase velocity of SH-wave is investigated through numerical calculations and presented graphically. Also, a comparative study has been done to analyze the effect of parameters by considering two piezoelectric materials, PZT-4 and \(\text {BaTiO}_3\). It is observed that the phase velocity increases with the increase of the dielectric constant, inhomogeneity, and sandy parameters, while simultaneously decreasing with the rise of the piezoelectric constant for both piezoelectric materials. This study can be applied to many scientific and engineering disciplines using sensors, actuators, capacitors, and the design of various acoustic surface wave devices.

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Abbreviations

\((u_1^{(u)},\,v_1^{(u)},\,w_1^{(u)})\) :

Displacement components of the piezoelectric layer

\((u_2^{(l)},\,v_2^{(l)},\,w_2^{(l)})\) :

Displacement components of the half-space

\(\sigma _{pq}^{(u)}\), \(S_{rt}^{(u)}\) :

Stress and strain tensors of the layer

\(\psi _1^{(u)}\) :

Electrical potential function of the layer

\(E_r^{(u)}\), \(D_q^{(u)}\) :

Electrical potential field and electrical displacement of the layer

\(c_{pqrt}^{(u)}\), \(e_{qrt}^{(u)}\), \(\varepsilon _{qr}^{(u)}\) :

Elastic, piezoelectric, and dielectric coefficients of the layer

\(\rho _1^{(u)}\), \(\rho _2^{(l)}\) :

Densities of the layer and half-space

\({\nabla ^2}\) :

Laplacian operator in two dimensional

\(c_{44}^{(u)}\), \(\varepsilon _{11}^{(u)}\), \(e_{15}^{(u)}\) :

Elastic, dielectric and piezoelectric constants of the layer

\(\phi ^0\), \(\varepsilon ^0\), \(D_q^{0}\) :

Electrical potential function, dielectric constant, and electric displacement in the vacuum

\({\eta }\), \(\rho _2^{(l)}\), \({\mu _2^{(l)}}\) :

Sandy parameter, density, and rigidity of the half-space

\({\mu _2}\), \({\rho _2}\) :

The initial values of \({\mu _2^{(l)}}\) and \(\rho _2^{(l)}\) at \(x_3=0\)

\(\alpha \) :

The inhomogeneity parameter of the half-space has dimensions that are the inverse of length.

k, c :

Wave number and common wave velocity

H :

Thickness of the layer

\(c_0\) :

Bulk shear-wave velocity of the layer

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Acknowledgements

The authors are indebted to the Indian Institute of Technology (ISM) for providing all the research facilities.

Funding

The author expresses their gratitude to the Government of India’s University Grants Commission (UGC) for awarding Mr. MOHD SADAB the UGC-JRF award Ref. No. JUN20C05768.

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

    Mohd Sadab & Santimoy Kundu

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Correspondence to Mohd Sadab.

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Sadab, M., Kundu, S. SH-wave propagation in a piezoelectric layer over a heterogeneous dry sandy half-space. Acta Mech 234, 5841–5854 (2023). https://doi.org/10.1007/s00707-023-03708-x

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