一维六方压电准晶中唇形孔口次生四条裂纹的反平面问题

王程颜; 刘官厅

doi:10.21656/1000-0887.440346

一维六方压电准晶中唇形孔口次生四条裂纹的反平面问题

doi: 10.21656/1000-0887.440346

王程颜^1,,
刘官厅^{1, 2, ,}

1.
内蒙古师范大学数学科学学院，呼和浩特 010022
2.
内蒙古自治区应用数学中心，呼和浩特 010022

基金项目:

国家自然科学基金 12162027

内蒙古自治区高等学校科学技术研究自然科学重点项目 NJZZ22574

详细信息

作者简介:
王程颜(1999—)，男，硕士生(E-mail: 2641734345@qq.com)

通讯作者:
刘官厅(1966—)，男，教授，博士(通讯作者. E-mail: guantingliu@imnu.edu.cn)

中图分类号: O346.1
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- 文章访问数: 278
- HTML全文浏览量: 134
- PDF下载量: 19
- 被引次数: 0
出版历程
- 收稿日期: 2023-12-05
- 修回日期: 2024-01-09
- 刊出日期: 2024-07-01

The Antiplane Problem of a Lip-Shaped Orifice With 4 Edge Cracks in 1D Hexagonal Piezoelectric Quasicrystal

WANG Chengyan^1
,,
LIU Guanting^{1, 2
, ,}

1.
College of Mathematics Science, Inner Mongolia Normal University, Hohhot 010022, P.R.China
2.
Inner Mongolia Center of Applied Mathematics, Hohhot 010022, P.R.China

摘要

摘要: 通过构造共形映射，利用Stroh型公式，研究了一维六方压电准晶体中唇形孔口次生四条裂纹的反平面问题，并对裂纹尖端处的应力强度因子及能量释放率进行了解析求解. 在数值算例中，分析了缺陷的几何参数和外部载荷对应力强度因子及能量释放率的影响规律. 结果表明：唇形孔口左右任意一侧裂纹长度或孔口长度的增长，对左右两侧裂纹的扩展有促进作用；上下两侧裂纹长度的增长，对左右两侧裂纹的扩展无明显的影响；唇形孔口的高度越高，对左右两侧裂纹扩展的抑制作用越显著；外部机械载荷和电载荷增大，对裂纹的扩展有促进作用. 一些特殊缺陷可由该缺陷的相关参数退化得来，如唇形孔口次生两条裂纹、唇型裂纹、Griffith裂纹等.
- 共形映射 /
- 孔边裂纹 /
- Stroh型公式 /
- 应力强度因子
Abstract: Through construction of the conformal mapping and with Stroh's formula, the antiplane problem of 4 secondary cracks at the lip-shaped orifice of 1D hexagonal piezoelectric quasicrystal, was studied. The effects of geometrical parameters and external loads on stress intensity factors and energy release rates were analyzed with numerical examples. The results show that, the crack length growth on either corner side of the orifice or the orifice length increase can promote the crack propagation. The crack length growth on the upper and lower sides has no obvious effect on the crack propagation on the left and right sides. The higher the orifice height is, the more significant the inhibition effect on the crack growth on both corner sides will be; the increase of the external mechanical load and electric load can promote the crack propagation. Some special defects can be derived from the degradation of the relevant parameters of the defects, such as the secondary 2 cracks in the orifice, the lip-shaped crack, and the Griffith crack, etc.
- conformal mapping /
- orifice edge crack /
- Stroh's formula /
- stress intensity factor

HTML全文

图 1 一维六方压电准晶中唇形孔口次生四条裂纹的缺陷模型

Figure 1. A defect model for a lip-shaped orifice with 4 edge cracks in the 1D hexagonal piezoelectric quasicrystal material

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图 2 K随a的变化曲线

Figure 2. The curves of K with ratio a

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图 3 K随L₁/a的变化曲线

Figure 3. The curves of K with ratio L₁/a

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图 4 K随h/a的变化曲线

Figure 4. The curves of K with ratio h/a

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图 5 K随h/a的变化曲线(L₁=L₂=L₃=0 m)

Figure 5. The curves of K with ratio h/a(L₁=L₂=L₃=0 m)

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图 6 L₃对K的影响曲线

Figure 6. The influence curves of L₃ on K

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图 7 不同L₁/a条件下，G/G_cr随a的变化关系

Figure 7. The change curves of G/G_cr with a under different L₁/a conditions

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图 8 在不同D₂^∞条件下，G/G_cr随σ₃₂^∞的变化曲线

Figure 8. The curves of G/G_cr with respect to σ₃₂^∞ for different D₂^∞ values

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图 9 当D₂^∞=0 C/m²，G/G_cr随σ₃₂^∞的变化关系

Figure 9. For D₂^∞=0 C/m², the curves of G/G_cr with σ₃₂^∞

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图 10 不同L₂条件下，G/G_cr随H₃₂^∞的变化曲线

Figure 10. The change curves of G/G_cr with H₃₂^∞ under different L₂ conditions

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图 11 不同L₂条件下，G/G_cr随R₃的变化关系

Figure 11. The change curves of G/G_crwith R₃ under different L₂ conditions

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图 12 G/G_cr随K₂的变化曲线

Figure 12. The change curves of G/G_cr with K₂

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图 13 G/G_cr随L₁/a的变化曲线

Figure 13. The change curves of G/G_cr with L₁/a

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图 14 G/G_cr随h/a的变化曲线

Figure 14. The change curves of G/G_cr with h/a

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图 15 L₃对G/G_cr的影响规律

Figure 15. The change curves of G/G_cr with L₃

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A1 唇形孔口次生四条裂纹的外部区域到ζ平面单位圆的内部的共形映射

A1. Conformal transformation from the exterior of the lip-shaped orifice with 4 cracks to the exterior region of the unit circle

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表 1 一维六方压电准晶材料参数^[11]

Table 1. Material parameters of the 1D hexagonal piezoelectric quasicrystal^[11]

C₄₄/GPa	R₃/GPa	K₂/GPa	e₁₅/(C·m^－2)	d₁₅/(C·m^－2)	λ₁₁/(C²·N^－1·m^－2)
50	1.2	0.3	－0.138	－0.160	8.26×10^－11

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