立方准晶压电材料的半空间问题

李光芳; 刘昉昉; 于静; 李联和

doi:10.21656/1000-0887.430221

立方准晶压电材料的半空间问题

doi: 10.21656/1000-0887.430221

李光芳^{1, 2,},
刘昉昉^{2, 3},
于静²,
李联和^{2, 3, ,}

1.
内蒙古农业大学理学院，呼和浩特 010018
2.
内蒙古自治区应用数学中心，呼和浩特 010022
3.
内蒙古师范大学数学科学学院，呼和浩特 010022

基金项目:

国家自然科学基金项目 11962026

国家自然科学基金项目 12002175

国家自然科学基金项目 12162027

国家自然科学基金项目 62161045

内蒙古自然科学基金项目 2020MS-01018

内蒙古自然科学基金项目 2021MS01013

内蒙古自然科学基金项目 2022ZD05

内蒙古自然科学基金项目 2023QN01007

内蒙古自治区高等学校科学技术研究项目 NJZY22519

详细信息

作者简介:
李光芳(1989—)，女，讲师，硕士(E-mail: liguangfang@126.com)

通讯作者:
李联和(1978—)，男，教授，博士，博士生导师(通讯作者. E-mail: nmglilianhe@163.com)

中图分类号: O29
计量
- 文章访问数: 341
- HTML全文浏览量: 152
- PDF下载量: 69
- 被引次数: 0
出版历程
- 收稿日期: 2022-07-04
- 修回日期: 2022-10-24
- 刊出日期: 2023-07-01

The Half Space Problem of Cubic Quasicrystal Piezoelectric Materials

LI Guangfang^{1, 2
,},
LIU Fangfang^{2, 3},
YU Jing²,
LI Lianhe^{2, 3
, ,}

1.
College of Science, Inner Mongolia Agricultural University, Hohhot 010018, P.R.China
2.
Center for Applied Mathematics Inner Mongolia, Hohhot 010022, P.R.China
3.
College of Mathematics Science, Inner Mongolia Normal University, Hohhot 010022, P.R.China

摘要

摘要: 考虑了立方准晶压电材料的半空间问题. 给出了反平面机械载荷和面内电载荷作用下立方准晶压电材料弹性问题的控制方程，结合半无限区域表面边界条件，利用算子理论和复变函数方法获得了立方准晶压电材料半空间问题一般解的表达式. 基于一般解得到了集中线力作用下，半空间问题的声子场和相位子场的位移、应力以及电位移的解析表达式.
- 立方准晶压电材料 /
- 复变函数法 /
- 半空间 /
- 线性力
Abstract: The half space problem of cubic quasicrystal piezoelectric materials was considered. The governing equations of elasticity for cubic quasicrystal piezoelectric materials under anti-plane deformation and in-plane electric field were given. Combined with the surface boundary conditions in the semi-infinite region, a general solution was obtained by means of the operator theory and the complex function method. Then the analytical expressions of the displacements and stresses of the phonon field and the phason field, and the electric displacements of the half space problem under concentrated linear surface forces, were derived.
- cubic quasicrystal piezoelectric material /
- complex function method /
- half space /
- linear force

HTML全文

图 1 力电载荷作用下的半无限区域

Figure 1. The semi-infinite region under electromechanical load

下载: 全尺寸图片幻灯片

图 2 无量纲声子场位移$ \bar{u}_z$等势图

Figure 2. The contour of dimensionless phonon field displacement $ \bar{u}_z$

下载: 全尺寸图片幻灯片

图 3 无量纲相位子场位移$ \bar{w}_z$等势图

Figure 3. The contour of dimensionless phason field displacement $ \bar{w}_z$

下载: 全尺寸图片幻灯片

图 4 无量纲电势Φ等势图

Figure 4. The contour of dimensionless potential Φ

下载: 全尺寸图片幻灯片

参考文献(24)

[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): 1951-1954. doi: 10.1103/PhysRevLett.53.1951
[2]	DING D, YANG W, HU C, et al. Linear elasticity theory of quasicrystals and defects in quasicrystals[J]. Materials Science Forum, 1994, 150/151: 345-354. doi: 10.4028/www.scientific.net/MSF.150-151.345
[3]	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 Application[M]. Beijing: Beijing Institute of Technology Press, 1999. (in Chinese)
[4]	郭俊宏, 刘官厅. 一维六方准晶中带双裂纹的椭圆空口问题的解析解[J]. 应用数学和力学, 2008, 29(4): 439-446. http://www.applmathmech.cn/article/id/1059 GUO Junhong, LIU Guanting. Analytic solutions of problem about an elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals[J]. Applied Mathematics and Mechanics, 2008, 29(4): 439-446. (in Chinese) http://www.applmathmech.cn/article/id/1059
[5]	LI X F, XIE Y L, FAN T Y. Elasticity and dislocations in quasicrystals with 18-fold symmetry[J]. Physics Letters A, 2014, 377(39): 2810-2814.
[6]	肖万伸, 张春雨, 邹伟生. 一维六方准晶复合材料界面层中螺型位错分析[J]. 材料科学与工程学报, 2014, 32(2): 215-218. https://www.cnki.com.cn/Article/CJFDTOTAL-CLKX201402012.htm XIAO Wanshen, ZHANG Chunyu, ZOU Weisheng. Elastic analysis of a screw dislocation in an interfacial layer in 1D hexagonal quasicrystal composites[J]. Journal of Materials Science & Engineering, 2014, 32(2): 215-218. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-CLKX201402012.htm
[7]	WANG X, SCHIAVONE P. Elastic field near the tip of an anticrack in a decagonal quasicrystalline material[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(3): 401-408. doi: 10.1007/s10483-020-2582-8
[8]	ZHANG Z G, DING S H, LI X. Two kinds of contact problems for two-dimensional hexagonal quasicrystals[J]. Mechanics Research Communications, 2021, 113: 103683. doi: 10.1016/j.mechrescom.2021.103683
[9]	DING D H, QIN Y L, WANG R H, et al. Generalization of Eshelby's method to the anisotropic elasticity theory of dislocations in quasicrystals[J]. Acta Physica Sinica, 1995, 4(11): 816-824.
[10]	ALTAY G, DÖMECI M C. On the fundamental equations of piezoelasticity of quasicrystal media[J]. International Journal of Solids and Structures, 2012, 49(23/24): 3255-3262.
[11]	ZHANG L L, ZHANG Y M, GAO Y. General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect[J]. Physics Letters A, 2014, 378(37): 2768-2776. doi: 10.1016/j.physleta.2014.07.027
[12]	LI X Y, LI P D, WU T H, et al. Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect[J]. Physics Letters A, 2014, 378(10): 826-834. doi: 10.1016/j.physleta.2014.01.016
[13]	FAN C Y, LI Y, XU G T, et al. Fundamental solutions and analysis of three-dimensional cracks in one-dimensional hexagonal piezoelectric quasicrystals[J]. Mechanics Research Communications, 2016, 74: 39-44. doi: 10.1016/j.mechrescom.2016.03.009
[14]	白巧梅, 丁生虎. 一维六方准晶压电中正六边形孔边裂纹的反平面问题[J]. 应用数学和力学, 2019, 40(10): 1071-1080. doi: 10.21656/1000-0887.390362 BAI Qiaomei, DING Shenghu. An anti-plane problem of cracks at edges of regular hexagonal holes in 1D hexagonal piezoelectric quasicrystals[J]. Applied Mathematics and Mechanics, 2019, 40(10): 1071-1080. (in Chinese) doi: 10.21656/1000-0887.390362
[15]	LI Y, QIN Q H, ZHAO M H. Analysis of 3D planar crack problems in one-dimensional hexagonal piezoelectric quasicrystals with thermal effect, part Ⅰ: theoretical formulations[J]. International Journal of Solids and Structures, 2020, 188/189: 269-281. doi: 10.1016/j.ijsolstr.2019.10.019
[16]	刘兴伟, 李星, 汪文帅. 一维六方压电准晶中正n边形孔边裂纹的反平面问题[J]. 应用数学和力学, 2020, 41(7): 713-724. https://www.cnki.com.cn/Article/CJFDTOTAL-YYSX202007002.htm LIU Xingwei, LI Xing, WANG Wenshuai. The anti-plane problem of regular n-polygon holes with radial edge cracks in 1D hexagonal piezoelectric quasicrystals[J]. Applied Mathematics and Mechanics, 2020, 41(7): 713-724. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YYSX202007002.htm
[17]	CHENG J X, LIU B J, CAO X L, et al. Applications of the Trefftz method to the anti-plane fracture of 1D hexagonal piezoelectric quasicrystals[J]. Engineering Analysis With Boundary Elements, 2021, 131: 194-205. doi: 10.1016/j.enganabound.2021.06.025
[18]	周旺民, 宋玉海. 立方准晶材料中的运动螺型位错[J]. 应用数学和力学, 2005, 26(12): 1459-1462. http://www.applmathmech.cn/article/id/636 ZHOU Wangmin, SONG Yuhai. Moving screw dislocation in cubic quasicrystal[J]. Applied Mathematics and Mechanics, 2005, 26(12): 1459-1462. (in Chinese) http://www.applmathmech.cn/article/id/636
[19]	GAO Y, ZHANG L L. Plane problems of cubic quasicrystal media with an elliptic hole or a crack[J]. Physics Letters A, 2011, 375(28): 2775-2781.
[20]	LI L H, LIU G T. Stroh formalism for icosahedral quasicrystal and its application[J]. Physics Letters A, 2012, 376(8/9): 987-990.
[21]	LONG F, LI X F. Thermal stresses of a cubic quasicrystal circular disc[J]. Mechanics Research Communications, 2022, 124: 103913.
[22]	王仁卉, 胡承正, 桂嘉年. 准晶物理学[M]. 北京: 科学出版社, 2004. WANG Renhui, HU Chengzheng, GUI Jianian. Quasicrystal Physics[M]. Beijing: Science Press, 2004. (in Chinese)
[23]	CHIANG C R. Mode-Ⅲ crack problems in a cubic piezoelectric medium[J]. Acta Mechanica, 2013, 224: 2203-2217.
[24]	JOHNSON K L. Contact Mechanics[M]. Cambridge: Cambridge University Press, 1985.