# SH Wave Scattering and Dynamic Stress Concentration in Piezomagnetic Materials with Non-Circular Openings

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{4}piezoelectric cubic single crystals. These research results provide theoretical guidance for analyzing and designing magnetic and electric transducers in functional gradient materials and provide academic advice for ultrasound non-destructive evaluation. Additionally, piezoelectric–piezomagnetic materials have extensive applications in biomedical engineering, such as ultrasound imaging and treatment.

## 2. Governing Equations

_{44}, h

_{15,}and μ

_{11}represent the elastic constants, piezomagnetic stress constants, and magnetic constants, respectively. ϕ represents the magnetic potential within the medium.

## 3. Boundary Conditions and Modal Coefficients

_{0}is the vacuum permeability.

**Z**) and integrating over interval (0, 2π), we can obtain an infinite system of algebraic equations in the following form

**X**is a (4N + 2) × 1 order matrix of undetermined coefficients, we employ linear algebra theory.

**Ω**is a square matrix of order (4N + 2) × (4N + 4), constructed by ${F}_{1n}$, ${F}_{2n}$, ${I}_{1n}$, ${I}_{2n}$, ${P}_{1n}$, ${P}_{2n}$, ${Q}_{1}$, and ${Q}_{2}$, serving as the coefficient matrix. Meanwhile,

**Ψ**is a non-homogeneous coefficient matrix, which is a (4N + 2) × 1 matrix constructed by ${\epsilon}_{1}$ and ${\epsilon}_{2}$. We then use $X={\Omega}^{-1}\Psi $ to obtain the solution for the undetermined coefficients.

## 4. Dynamic Stress Concentration Coefficient around Elliptical Opening

## 5. Numerical Calculation and Discussion

_{2}O

_{4}as the material of interest, with a relative material constant of [24]

## 6. Conclusions

- The DSCC value around the elliptical opening is related to the illumination region. In the case of incident waves with the same frequency, a larger illumination region will generate a larger DSCC. Therefore, elastic waves traveling along the minor axis of the elliptical opening will cause a larger DSCC than those traveling along the major axis.
- Elastic waves in the low-frequency range are more significant to consider. In most instances, the DSCC produced by low-frequency waves is higher than that produced by high-frequency waves. Moreover, the DSCC in the illumination region is generally higher than in the shadow region.
- The DSCC changes differently at different locations in the piezoelectric material with an elliptical opening. The DSCC in specific areas can be much higher than in other places, making them more prone to damage.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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**Figure 1.**The outer domain of the non-circular boundary of the z-plane is mapped to the outer domain of the unit circle of the ζ-plane.

**Figure 2.**Comparison of distribution of DSCC in elliptical openings under different geometric parameters; incident angle φ, wave number ka and elliptical geometric parameters e.

**Figure 3.**Comparison of distribution of DSCC in elliptical openings under different wave numbers; incident angle φ, wave number ka and elliptical geometric parameters e.

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**MDPI and ACS Style**

Zhou, C.; Weng, J.; Wang, Z.; Pei, W.; Hu, N.; Zhou, R.; Gong, Y.
SH Wave Scattering and Dynamic Stress Concentration in Piezomagnetic Materials with Non-Circular Openings. *Appl. Sci.* **2023**, *13*, 6972.
https://doi.org/10.3390/app13126972

**AMA Style**

Zhou C, Weng J, Wang Z, Pei W, Hu N, Zhou R, Gong Y.
SH Wave Scattering and Dynamic Stress Concentration in Piezomagnetic Materials with Non-Circular Openings. *Applied Sciences*. 2023; 13(12):6972.
https://doi.org/10.3390/app13126972

**Chicago/Turabian Style**

Zhou, Chuanping, Jiayou Weng, Zhiwen Wang, Wanrong Pei, Ning Hu, Rougang Zhou, and Youping Gong.
2023. "SH Wave Scattering and Dynamic Stress Concentration in Piezomagnetic Materials with Non-Circular Openings" *Applied Sciences* 13, no. 12: 6972.
https://doi.org/10.3390/app13126972