一维六方准晶非周期平面内中心开口裂纹的平面热弹性问题

赵雪芬; 卢绍楠; 马园园; 张保文

doi:10.21656/1000-0887.440302

一维六方准晶非周期平面内中心开口裂纹的平面热弹性问题

doi: 10.21656/1000-0887.440302

赵雪芬^{1, 2, ,},
卢绍楠³,
马园园³,
张保文³

1.
宁夏大学信息工程学院, 银川 750021
2.
宁夏大学新华学院, 银川 750021
3.
宁夏大学数学统计学院, 银川 750021

基金项目:

国家自然科学基金 12062021

国家自然科学基金 62062057

国家自然科学基金 12262033

国家自然科学基金 12062022

详细信息

通讯作者:
赵雪芬(1983—), 女, 副教授, 博士, 硕士生导师(通讯作者. E-mail: snownfen@163.com)

中图分类号: O343.8
计量
- 文章访问数: 70
- HTML全文浏览量: 23
- PDF下载量: 21
- 被引次数: 0
出版历程
- 收稿日期: 2023-10-10
- 修回日期: 2023-11-14
- 刊出日期: 2024-03-01

The Plane Thermoelastic Problem of a Central Opening Crack in the 1D Hexagonal Quasicrystal Non-Periodic Plane

1.
School of Information Engineering, Ningxia University, Yinchuan 750021, P.R.China
2.
Xinhua College of Ningxia University, Yinchuan 750021, P.R.China
3.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, P.R.China

摘要

摘要: 考虑裂纹内部介质的热传导率, 研究了一维六方准晶非周期平面内含中心开口裂纹的平面热弹性问题. 利用Fourier积分变换技术, 得到了热应力、裂纹尖端处的热应力强度因子和应变能密度因子的封闭解. 数值结果讨论了裂纹内部介质的热传导率、外载荷及声子场-相位子场耦合系数对热应力强度因子和应变能密度因子的影响. 结果表明, 声子场-相位子场耦合系数对裂纹扩展影响较大. 当声子场载荷较小或热流密度较大时, 裂纹不易扩展，热流密度在裂纹尖端处会出现集中热效应. 随着裂纹内部介质热传导率的增大, 热流密度逐渐增加而热应力强度因子逐渐减小. 该文所得结果为准晶热力学性质的实际应用提供了理论依据, 进而可用于优化准晶元器件的设计和制备.
- 一维六方准晶 /
- 中心开口裂纹 /
- Fourier积分变换 /
- 热应力强度因子 /
- 应变能密度因子
Abstract: Considering the thermal conductivity of the medium inside the crack, the plane thermoelastic problem of the 1D hexagonal quasicrystal with a central open crack in an aperiodic plane, was studied. With the Fourier integral transformation technology, the closed form solutions of thermal stresses, thermal stress intensity factors and strain energy density factors were obtained. Numerical examples were used to analyze the effects of the thermal conductivity, the external load, and the phonon field-phason field coupling coefficient on the thermal stress intensity factor and the strain energy density factor at the crack tip. The results indicate that, the heat flux density gradually increases but the thermal stress intensity factor gradually decreases with the thermal conductivity. The phonon field-phason field coupling coefficient has a significant impact on the crack propagation. When the phonon field load is relatively small or the heat flux density is relatively high, the crack is not easy to propagate. The heat flux density exhibits a concentration effect at the crack tip. The work provides a theoretical basis for the application of thermodynamic properties of quasicrystals, and the optimization of design and preparation of quasicrystal components.
- 1D hexagonal quasicrystal /
- central opening crack /
- Fourier integral transformation /
- thermal stress intensity factor /
- strain energy density factor

HTML全文

图 1 均匀的热流密度q₀和外载荷σ₀, h₀作用下的中心开口裂纹

Figure 1. A central opening crack under uniform heat flux density q₀ and stress σ₀, h₀

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图 2 q_c/q₀, K_σ/K_σ^m随λ_c/λ_z和σ₀的变化趋势

Figure 2. Variations of q_c/q₀, K_σ/K_σ^m with λ_c/λ_z and σ₀

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图 3 q_c/q₀, K_σ/K_σ^m和K_H/K_H^m随λ_c/λ_z的变化趋势

Figure 3. Variations of q_c/q₀, K_σ/K_σ^m and K_H/K_H^m with λ_c/λ_z

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图 4 q_c/q₀, K_σ/K_σ^m和K_H/K_H^m随σ₀的变化趋势

Figure 4. Variations of q_c/q₀, K_σ/K_σ^m and K_H/K_H^m with σ₀

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图 5 K_σ/K_σ^m随q₀变化

Figure 5. Variations of K_σ/K_σ^m with q₀

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图 6 K_H/K_H^m随q₀变化

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 6. Variations of K_H/K_H^m with q₀

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图 7 q₀不同时, S/S_m随σ₀变化

Figure 7. Variations of S/S_m with σ₀ for different q₀ values

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图 8 z不同时, q_z/q₀随x变化

Figure 8. Variations of q_z/q₀ with x for different z values

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图 9 q_z/q₀在xOz面上的变化趋势

Figure 9. Variations of q_z/q₀ on plane xOz

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表 1 一维六方压电准晶体弹性常数^[15]

Table 1. 1D hexagonal piezoelectric quasicrystal elastic constants^[15]

C₁₁/GPa	C₁₂/GPa	C₁₃/GPa	C₃₃/GPa	C₄₄/GPa	K₁/GPa	K₂/GPa
150	55	45	90	50	0.084	0.036

R₁/GPa	R₂/GPa	R₃/GPa	β₁/(MPa/K)	β₃/(MPa/K)	λ_x/(W/(m·K))	λ_z/(W/(m·K))
-1.68	1.2	1.2	1.798	1.383	12.4	12.4

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参考文献(26)

[1]	SHECHTMAN D, BLECH I, GRATIAS D, et al. Metallic phase with long-range orientational order and no translational symmetry[J]. Physical Review Letters, 1984, 53(20): 1951-1953. doi: 10.1103/PhysRevLett.53.1951
[2]	范天佑. 准晶数学弹性理论及应用[M]. 北京: 北京理工大学出版社, 1999. FAN Tianyou. Mathematical Theory of Elasticity of Quasicrystals and Its Applications[M]. Beijing: Beijing Institute of Technology Press, 1999. (in Chinese)
[3]	ZHANG Z, URBAN K. Transmission electron microscope observations of dislocations and stacking faults in a decagonal Al-Cu-Co alloy[J]. Philosophical Magazine Letters, 1989, 60(3): 97-102. doi: 10.1080/09500838908206442
[4]	LI P D, LI X Y, KANG G Z. Crack tip plasticity of a half-infinite Dugdale crack embedded in an infinite space of one-dimensional hexagonal quasicrystal[J]. Mechanics Research Communications, 2015, 70: 72-78. doi: 10.1016/j.mechrescom.2015.09.007
[5]	YU J, GUO J H. Analytical solution for a 1D hexagonal quasicrystal strip with two collinear mode-Ⅲ cracks perpendicular to the strip boundaries[J]. Crystals, 2023, 13(4): 661. doi: 10.3390/cryst13040661
[6]	卢绍楠, 赵雪芬, 马园园. 一维六方压电准晶双材料界面共线裂纹问题[J]. 应用数学和力学, 2023, 44(7): 809-824. doi: 10.21656/1000-0887.430111 LU Shaonan, ZHAO Xuefen, MA Yuanyuan. Problem of interface collinear cracks between one dimensional hexagonal piezoelectric quasicrystal bimaterials[J]. Applied Mathematics and Mechanics, 2023, 44(7): 809-824. (in Chinese) doi: 10.21656/1000-0887.430111
[7]	GRUSHKO B, HOLLAND-MORITZ D, HOLLAND-MORITZ D, et al. Transition between periodic and quasiperiodic structures in Al-Ni-Co[J]. Journal Alloys and Compounds, 1998, 280(1): 215-230.
[8]	HIRAGA K, OHSUNA T, SUN W, et al. The structural characteristics of Al-Co-Ni decagonal quasicrystals and crystalline approximants[J]. Journal Alloys and Compounds, 2002, 342(1): 110-114.
[9]	YAVAS B, LI M X, LEONARD H R, et al. Identifying experimental parameters for in situ TEM heating experiments on metastable microstructures: application to a quasicrystal-reinforced Al alloy[J]. Microscopy and Microanalysis, 2002, 28(S1): 1840-1842.
[10]	DANG H Y, QI D P, ZHAO M H, et al. The thermally induced interfacial behavior of a thin two-dimensional decagonal quasicrystal film[J]. International Journal of Fracture, 2023, 224(1/2): 1-14.
[11]	NGUYEN H V, DO N B, NGUYEN T H O, et al. Synthesis and magnetic properties of Al-Cu-Fe quasicrystals prepared by mechanical alloying and heat treatment[J]. Journal of Materials Research, 2023, 38(3): 644-653. doi: 10.1557/s43578-022-00846-1
[12]	SLADEK J, SLADEK V, REPKA M, et al. Gradient theory of thermoelasticity for interface crack problems with a quasicrystal layer[J]. International Journal of Solids and Structures, 2023, 264: 112097. doi: 10.1016/j.ijsolstr.2022.112097
[13]	FAN C Y, YUAN Y P, PAN Y B, et al. Analysis of cracks in one-dimensional hexagonal quasicrystals with the heat effect[J]. International Journal of Solids and Structures, 2017, 120: 146-156. doi: 10.1016/j.ijsolstr.2017.04.036
[14]	LI P D, LI X Y, KANG G Z. Axisymmetric thermo-elastic field in an infinite one-dimensional hexagonal quasi-crystal space containing a penny-shaped crack under anti-symmetric uniform heat fluxes[J]. Engineering Fracture Mechanics, 2018, 190: 74-92. doi: 10.1016/j.engfracmech.2017.12.001
[15]	ZHANG X, FAN C Y, LU C S, et al. Three-dimensional thermal fracture analysis of a one-dimensional hexagonal quasicrystal coating with interface cracks[J]. Engineering Fracture Mechanics, 2023, 277: 1-20.
[16]	GUO J H, YU J, XING Y M, et al. Thermoelastic analysis of a two-dimensional decagonal quasicrystal with a conductive elliptic hole[J]. Acta Mechanica, 2016, 227(9): 2595-2607. doi: 10.1007/s00707-016-1657-7
[17]	DING D H, YANG W G, HU C Z. Generalized elasticity theory of quasicrystals[J]. Physical Review B, 1993, 48: 7003-7010. doi: 10.1103/PhysRevB.48.7003
[18]	LI L H, LIU G T. Icosahedral quasicrystals solids with an elliptic hole under uniform heat flow[J]. Chinese Physics B, 2014, 23(5): 056101. doi: 10.1088/1674-1056/23/5/056101
[19]	ZHONG X C, LONG X Y, ZHANG L H. An extended thermal-medium crack model[J]. Applied Mathematical Modelling, 2018, 56: 202-216. doi: 10.1016/j.apm.2017.11.016
[20]	EDE A J. An Introduction to Heat Transfer: Principles and Calculations[M]. New York: Pergamon Press, 1967.
[21]	范天佑. 断裂理论基础[M]. 北京: 科学出版社, 2006. FAN Tianyou. The Theoretical Basis of Fracture[M]. Beijing: Science Press, 2006. (in Chinese)
[22]	FABRIKANT V I. Computation of infinite integrals involving three Bessel functions by introduction of new formalism[J]. ZAMM Journal of Applied Mathematics and Mechanics, 2003, 83(6): 363-374. doi: 10.1002/zamm.200310059
[23]	FAN T Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications[M]. Beijing: Science Press, 2017.
[24]	ZHONG X C, WU B, ZHANG K S. Thermally conducting collinear cracks engulfed by thermomechanical field in a material with orthotropy[J]. Theoretical and Applied Fracture Mechanics, 2013, 65: 61-68. doi: 10.1016/j.tafmec.2013.05.009
[25]	ZHONG X C, WU B. Thermoelastic analysis for an opening crack in an orthotropic material[J]. International Journal of Fracture, 2012, 173(1): 49-55. doi: 10.1007/s10704-011-9665-z
[26]	BADALIANCE R. Application of strain energy density factor to fatigue crack growth analysis[J]. Engineering Fracture Mechanics, 1980, 13(3): 657-666. doi: 10.1016/0013-7944(80)90094-6