# Electroelastic Coupled-Wave Scattering and Dynamic Stress Concentration of Triangular Defect Piezoceramics

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^{2}

^{3}

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## Abstract

**:**

_{0.85}Ca

_{0.15})(Zr

_{0.1}Ti

_{0.9})O

_{3}and set different temperatures, Ce doping amount, and incident wave number for numerical simulation. The numerical results show that the incident wave number, piezoelectric properties, and the shape parameters and deflection angle of the triangular defect have a great influence on the dynamic stress around the defect, and some meaningful laws are summarized through analysis.

## 1. Introduction

_{0.85}Ca

_{0.15})(Zr

_{0.1}T

_{i0.9})O

_{3}(one of BCZTs) as an example, the DSCF results under different Ce doping amount, temperature, and shape parameters were calculated and analyzed (see Section 6). The effects of incident wavenumber, piezomagnetic properties, and piezoelectric parameters on the DSCF around the defect are also discussed (see Section 7).

## 2. The Basic Equations of Two-Dimensional Anti-plane Problems in Cylindrical Coordinates

## 3. Solutions of Incident, Scattered, and Total Electroelastic Waves in Piezoelectric Materials with Triangular Defects

## 4. Boundary Conditions and Determination of Mode Coefficients

## 5. Dynamic Stress Concentration Factor

## 6. Numerical Examples Simulation and Discussion

_{0.85}Ca

_{0.15})(Zr

_{0.1}Ti

_{0.9})O

_{3}of BCZT is used as a specific calculation example, and its initial related material constants are obtained by conventional tests on the samples as follows

_{0.05}ZT as an example to explore the effect of $\alpha $ on stress concentration.

## 7. Conclusions

_{0.85}Ca

_{0.15})(Zr

_{0.1}Ti

_{0.9})O

_{3}) under the influence of electroelastic waves is calculated, and the following conclusions are finally drawn.

- Supported by the currently available data, Ce doping and elevated temperature (between 1350–1500 °C) will increase the piezoelectric constant of the (Ba
_{0.85}Ca_{0.15})(Zr_{0.1}Ti_{0.9})O_{3}. - As ka increases, the stress concentration around the defect fluctuates more violently, and the increase of the piezoelectric coefficient ${e}_{15}$ will make the stress concentration more serious. In addition, the stress concentration of the triangular defects is obvious at the three vertices.
- When ka changes from 0 to 5, the stress concentration trend around the defect is similar to that of ka = 0.1 and ka = 0.5. The only thing that changes greatly is the maximum value of stress concentration (stress concentration coefficients on the three vertices). With the increase of ka, the maximum value of the stress concentration factor has a general trend of first increasing and then decreasing. The stress concentration factor peaks in the range of ka = 4~5.
- Under different deflection angles, the stress concentration phenomenon is still concentrated at the vertex of the triangle. However, as the deflection angle $\alpha $ increases, the magnitude of the stress concentration factor becomes smaller.
- The maximum value of the stress concentration factor decreases as the absolute value of the deflection angle increases. The variation trend of the maximum stress concentration factor with $\alpha $ can be considered to be almost symmetric at about x = 0 and take a maximum value at x = 0.
- The difference in temperature and the amount of Ce doping is essentially the difference in piezoelectric constant. Within the data range of this study, the better the piezoelectric performance, the more intense the piezoelectric effect and the greater the magnitude of the stress concentration factor.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the setting of the coordinate system and the incidence of SH waves (The inserted figures (unpublished) show triangular-like microcracks observed by scanning electron microscopy for BCZT and other ferroelectric ceramics.).

**Figure 3.**Polar plot of stress concentration of BCZT at different incident wavenumbers and temperatures.

**Figure 4.**Polar plot of stress concentration of BCCe

_{0.05}ZT at different incident wavenumbers and temperatures.

**Figure 6.**Stress concentration graph under different $\alpha $${(\mathrm{BCCe}}_{0.05}\mathrm{ZT}:ka=0.5)$.

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**MDPI and ACS Style**

Lin, J.; Ji, H.; Zhou, C.; Fan, J.; Han, X.; Bao, J.; Gong, Y.; Ni, J.; Zhou, W.
Electroelastic Coupled-Wave Scattering and Dynamic Stress Concentration of Triangular Defect Piezoceramics. *Actuators* **2022**, *11*, 106.
https://doi.org/10.3390/act11040106

**AMA Style**

Lin J, Ji H, Zhou C, Fan J, Han X, Bao J, Gong Y, Ni J, Zhou W.
Electroelastic Coupled-Wave Scattering and Dynamic Stress Concentration of Triangular Defect Piezoceramics. *Actuators*. 2022; 11(4):106.
https://doi.org/10.3390/act11040106

**Chicago/Turabian Style**

Lin, Jiang, Huawei Ji, Chuanping Zhou, Jiawei Fan, Xiao Han, Junqi Bao, Yongping Gong, Jing Ni, and Weihua Zhou.
2022. "Electroelastic Coupled-Wave Scattering and Dynamic Stress Concentration of Triangular Defect Piezoceramics" *Actuators* 11, no. 4: 106.
https://doi.org/10.3390/act11040106