Reflection and transmission of SH waves at the interface of a V-notch and a piezoelectric/piezomagnetic half-space

• Published in: Journal of engineering mathematics Vol. 148; no. 1
• Main Authors: Zhang, Xi-meng, Qi, Hui
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.10.2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Bessel functions; Boundaries; Computational Mathematics and Numerical Analysis; Exact solutions; Finite element method; Fredholm equations; Half spaces; Magnetic fields; Mathematical analysis; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Orthogonal functions; Piezoelectricity; Scattering; SH waves; Standing waves; Stress concentration; Superposition (mathematics); Theoretical and Applied Mechanics; Wave reflection; Expansion method; Piezoelectric/piezomagnetic half-space; Graf theorem; V-notch; SH waves; Concentration factors; Fractional Bessel functions
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-024-10392-w

This paper investigates the dynamic behavior of a V-notch with non-trivial boundaries in a piezoelectric/piezomagnetic half-space. We start by considering a SH wave impinging on the piezoelectric/piezomagnetic half-space. Upon employing the superposition principle, an expression for the scattering wave is derived, which meets the required conditions at the boundary of the half-space. Subsequently, we provide the analytic expression for the standing wave, formulated to meet the stress-free assumptions and electric/magnetic insulation at the boundaries of the V-notch. This is done using an expansion in fractional Bessel functions and the Graf theorem. Finally, a method based on Green’ functions is employed to divide the half-space along the vertical interface, where in-plane electric and magnetic fields and out-of-plane forces are exerted. This leads to the formulation of integral Fredholm equations, which are solved using an expansion into orthogonal functions and an effective truncation technique. Our results describe the scattering effect on the concentration factors of the dynamic stress, and of electric and magnetic fields in relevant conditions. The analytic solutions are validated using finite element method, and results confirm the accuracy of our findings.