含切口的压电准晶组合结构界面断裂分析的辛-等几何耦合方法

杨震霆; 王雅静; 聂雪阳; 徐新生; 周震寰

doi:10.21656/1000-0887.440247

含切口的压电准晶组合结构界面断裂分析的辛-等几何耦合方法

doi: 10.21656/1000-0887.440247

大连理工大学工程力学系工业装备结构分析优化与CAE软件全国重点实验室，辽宁大连 116024

基金项目:

辽宁省自然科学基金 2023-MS-118

详细信息

作者简介:
杨震霆(1994—)，男，博士生(E-mail: zhentingyang@outlook.com)

通讯作者:
周震寰(1983—)，男，教授，博士(通讯作者. E-mail: zhouzh@dlut.edu.cn)

中图分类号: O346
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- 被引次数: 0
出版历程
- 收稿日期: 2023-08-17
- 修回日期: 2023-10-12
- 刊出日期: 2024-02-01

Symplectic Isogeometric Analysis Coupling Method for Interfacial Fracture of Piezoelectric Quasicrystal Composites With Notches

State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

摘要

摘要: 发展了一种适用于含有切口的压电准晶/压电晶体/弹性体三材料组合结构界面断裂问题的高精度的半数值半解析方法.首先，通过引入Hamilton体系建立了三材料组合结构的Hamilton对偶方程，将原问题在传统Lagrange体系下的高阶偏微分控制方程转化为低阶常微分方程组.其次，通过分离变量法求解问题对应的辛本征值和本征解，将各物理场变量利用辛级数展开形式表示.最后，将辛级数与等几何分析方法相结合，获得了辛-等几何耦合列式，直接求得切口尖端附近奇异物理场及其强度因子的解析表达式.
- 准晶体 /
- 压电材料 /
- 等几何分析 /
- Hamilton体系 /
- V形切口 /
- 界面断裂
Abstract: A high-precision semi numerical and semi analytical method for interfacial fracture problem of piezoelectric quasicrystals (PQCs)/piezoelectric crystals (PZCs)/elastic material composites with notches was developed. Firstly, the Hamiltonian system was introduced and the Hamiltonian dual equations for the 3-material composite were formulated. The higher order partial differential governing equations were transformed into a set of ordinary differential equations. Secondly, the symplectic eigenvalues and eigensolutions were obtained through separation of variables. The physical quantities were expressed with the expansion of symplectic series. Finally, a symplectic isogeometric analysis (IGA) coupling equation was derived through combination of the symplectic series and the IGA. The analytical expressions of the physical quantities near the notch tip and the intensity factors were derived.
- quasicrystal /
- piezoelectric material /
- isogeometric analysis /
- Hamiltonian system /
- V notch /
- interfacial fracture

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图 1 含切口的三材料组合结构

Figure 1. The 3-material composite with a notch

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图 2 奇异区和非奇异区

Figure 2. The singular region and the non-singular region

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图 3 含内部裂纹的压电准晶体

Figure 3. The square PQC with an internal crack

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图 4 应力强度因子随模型尺寸的变化

Figure 4. The variations of the stress intensity factor vs. the size of the model

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图 5 圆形压电晶体/环氧树脂双材料结构

Figure 5. The circular PZC/epoxy bi-material

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图 6 材料参数对应力、电位移强度因子的影响

Figure 6. The effects of the material constants on the stress and electric displacement intensity factors

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图 7 含边切口的方形压电准晶体/压电晶体/环氧树脂模型

Figure 7. The square PQC/PZC/epoxy composite models

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表 1 压电准晶体/压电晶体/弹性体材料参数

Table 1. The material constants of PQC/PZC/elastic materials

	C₄₄/GPa	R₃/GPa	K₂/GPa	e₁₅/(C/m²)	d₁₅/(C/m²)	λ₁₁/(C²/(N·m²))
PQC	50	1.2	0.18	-0.138	-0.16	8.26×10^-11
PZC	25.6	-	-	12.7	-	6.46
epoxy	1.76	-	-	-	-	-

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表 2 情况1时切口奇异性指数随角度Δθ的变化

Table 2. The variations of the singularity orders vs. Δθ in case 1

	Δθ
	50°	60°	70°	80°	90°
μ₁-1	-0.209 8	-0.243 8	-0.275 9	-0.306 0	-0.334 5
μ₂-1	-0.013 2	-0.018 8	-0.028 0	-0.046 2	-0.083 7

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表 3 情况2时切口奇异性指数随角度Δθ的变化

Table 3. The variations of the singularity orders vs. Δθ in case 2

	Δθ
	50°	60°	70°	80°	90°
μ₁-1	-0.476 8	-0.481 0	-0.484 2	-0.486 7	-0.488 7
μ₂-1	-0.240 0	-0.273 6	-0.304 1	-0.332 5	-0.358 0

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表 4 情况3时切口奇异性指数随角度Δθ的变化

Table 4. The variations of the singularity orders vs. Δθ in case 3

	Δθ
	50°	60°	70°	80°	90°
μ₁-1	-0.838 9	-0.852 8	-0.863 6	-0.872 4	-0.879 7

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表 5 界面1强度系数(情况1)

Table 5. The intensity coefficients at interface 1 (case 1)

	a
	1	2	3	4	5
K₁^σ	0.949 0	1.042 3	1.058 4	1.049 7	1.023 7
K₁^D	0.011 0	0.012 0	0.012 2	0.012 1	0.011 8
K₁^H	0	0	0	0	0
K₂^σ	-0.021 3	-0.021 5	-0.021 6	-0.021 6	-0.021 6
K₂^D	0.010 9	0.011 0	0.011 0	0.011 0	0.011 0
K₂^H	0	0	0	0	0

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表 6 界面2强度系数(情况1)

Table 6. The intensity coefficients at interface 2 (case 1)

	a
	1	2	3	4	5
K₁^σ	0.051 2	0.056 3	0.057 1	0.056 7	0.055 3
K₁^D	0	0	0	0	0
K₂^σ	0.000 9	0.000 9	0.000 9	0.000 9	0.000 9
K₂^D	0	0	0	0	0

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表 7 界面1强度系数(情况2)

Table 7. The intensity coefficients at interface 1 (case 2)

	a
	1	2	3	4	5
K₁^σ	-0.014 5	-0.014 4	-0.014 0	-0.012 8	-0.009 9
K₁^D	0.007 9	0.007 8	0.007 6	0.007 0	0.005 4
K₁^H	0	0	0	0	0
K₂^σ	0.932 9	1.028 9	1.044 1	1.031 7	0.994 9
K₂^D	0.011 7	0.012 9	0.013 1	0.013 0	0.012 5
K₂^H	0	0	0	0	0

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表 8 界面2强度系数(情况2)

Table 8. The intensity coefficients at interface 2 (case 2)

	a
	1	2	3	4	5
K₁^σ	0.000 4	0.000 4	0.000 4	0.000 4	0.000 3
K₁^D	0	0	0	0	0
K₁^H	0	0	0	0	0
K₂^σ	0.048 9	0.053 9	0.054 7	0.054 1	0.052 1
K₂^D	0	0	0	0	0
K₂^H	0	0	0	0	0

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表 9 界面1强度系数(情况3)

Table 9. The intensity coefficients at interface 1 (case 3)

	a
	1	2	3	4	5
K₁^σ	0.058 0	0.075 1	0.076 0	0.073 6	0.073 6
K₁^D	0	0	0	0	0
K₁^H	0	0	0	0	0

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表 10 界面2强度系数(情况3)

Table 10. The intensity coefficients at interface 2 (case 3)

	a
	1	2	3	4	5
K₁^σ	0.058 3	0.075 4	0.076 4	0.074 0	0.074 0
K₁^D	0	0	0	0	0

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