The Anti-Plane Problem of Regular n-Polygon Holes With Radial Edge Cracks in 1D Hexagonal Piezoelectric Quasicrystals
-
摘要: 利用复变函数方法和Schwarz-Christoffel(SC)变换, 构造了保形映射函数, 研究了一维六方压电准晶中正n边形孔边裂纹的反平面问题.首先由一维六方压电准晶反平面问题的本构方程、平衡方程和几何方程推导得出其控制方程.在电不可导通边界条件下, 应用Cauchy积分公式, 得出任意正n边形孔边裂纹尖端附近应力强度因子和电位移强度因子的解析解, 并针对n=3,5,6时, 给出数值算例, 可以看出这些特殊情形可退化为已有的结果.研究结果表明:等效场强度因子K的值随着孔边长a和裂纹长度L/a的增加而增大; 孔洞的尺寸对等效场强度因子K的影响特别显著, 易导致破坏.该文所给结果对计算等效强度因子具有一般性, 适用于任意正n边形孔边裂纹的求解问题, 从而为工程力学、材料的制备和应用等提供了良好的理论依据.
-
关键词:
- 一维六方压电准晶 /
- Schwarz-Christoffel变换 /
- 正n边形孔口 /
- 裂纹 /
- 等效强度因子
Abstract: The heat conduction is a common problem in engineering practice. Compared with those of isotropic materials, the heat conduction problem of anisotropic materials is more complicated, so it is of great significance to accurately predict the internal temperature distribution. The numerical manifold method (NMM) was developed to solve typical continuous and discontinuous heat conduction problems in anisotropic materials. According to the governing differential equation, boundary conditions and variational principles, the NMM discrete equations for such problems were derived. Several representative examples involving continuous and discontinuous situations were analyzed with the uniform mathematical cover independent of all physical boundaries. The results prove the feasibility and accuracy of the method and indicate that the NMM can simulate the heat conduction problem of anisotropic materials well. Besides, the influences of the material properties and crack configurations on the temperatures were also investigated. -
[1] VALDOVINOS J. Pediatric mechanical circulatory support applications for frequency-leveraged piezoelectric hydraulic pumps[D]. PhD Thesis. Los Angeles: University of California, 2014. [2] WANG Y J, GAO C F. The mode Ⅲ cracks originating from the edge of a circular hole in a piezoelectric solid[J]. International Journal of Solids and Structures,2008,45(16): 4590-4599. [3] WANG Y J, GAO C F, SONG H. The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material[J]. Mechanics Research Communications,2015,65: 17-23. [4] GHERROUS M, FERDJANI H. Analysis of a Griffith crack at the interface of two piezoelectric materials under anti-plane loading[J]. Continuum Mechanics and Thermodynamics,2016,28(6): 1683-1704. [5] SHECHTMAN D G, BLECH I A, GRATIAS D, et al. Metallic phase with long-range orientational order and no translational symmetry[J]. Physical Review Letters,1984,53(20): 1951-1953. [6] LI L H, FAN T Y. Exact solutions of two semi-infinite collinear cracks in a strip of one dimensional hexagonal quasicrystal[J]. Applied Mathematics and Computation,2008,196(1): 1-5. [7] FAN T Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications [M]. Berlin: Springer, 2011. [8] HU C Z, WANG R H, DING D H, et al. Piezoelectric effects in quasicrystals[J]. Physical Review B,1997,56(5): 2463-2468. [9] RAO K R M, RAO P H, CHAITANYA B S K. Piezoelectricity in quasicrystals: a group-theoretical study[J]. Pramana: Journal of Physics,2007,68(3): 481-487. [10] ALTAY G, DKMECI 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] 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. [12] 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. [13] GUO J H, PAN E. Three-phase cylinder model of 1D hexagonal piezoelectric quasicrystal composites[J]. Journal of Applied Mechanics,2016,〖STHZ〗 83(8): 081007. [14] YU J, GUO J H, PAN E, et al. General solutions of plane problem in one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics[J]. Applied Mathematics and Mechanics(English Edition),2015,36(6): 793-814. [15] FAN T Y, LI X F, SUN Y F. A moving screw dislocation in a one-dimensional hexagonal quasicrystals[J]. Acta Physica Sinica (Overseas Edition),1999,8(4): 288-295. [16] LI X, HUO H S, SHI P P. Analytic solutions of two collinear fast propagating cracks in a symmetrical strip of one-dimensional hexagonal piezoelectric quasicrystals[J]. Chinese Journal of Solid Mechanics,2014,35(2): 1-7. [17] TUPHOLME G E. A non-uniformly loaded anti-plane crack embedded in a half-space of a one-dimensional piezoelectric quasicrystal[J]. Meccanica,2018,53(4/5): 973-983. [18] ZHOU Y B, LI X F. Two collinear mode-Ⅲ cracks in one-dimensional hexagonal piezoelectric quasicrystal strip[J]. Engineering Fracture Mechanics,2018,189: 133-147. [19] KIRSCH G. Die theorie der elastizitt und die bedürfnisse der festigkeitslehre[J]. Zantralblatt Verlin Deutscher Ingenieure,1898,42(29): 797-807. [20] UKADGAONKER V G, AWASARE P J. A novel method of stress analysis of an infinite plate with circular hole with uniform loading at infinity[J]. Indian Journal of Science and Technology,1993,31: 539-541. [21] WANG W S, YUAN H T, LI X, et al. Stress concentration and damage factor due to central elliptical hole in functionally graded panels subjected to uniform tensile traction[J]. Materials,2019,12(3): 422. [22] UKADGAONKER V G, AWASARE P J. A novel method of stress analysis of an infinite plate with rounded corners of a rectangular hole under uniform edge loading[J]. Indian Journal of Engineering and Materials Sciences,1994,1: 17-25. [23] REZAEEPAZH J, JAFARI M. Stress concentration in metallic plates with special shaped cutout[J]. International Journal of Mechanical Sciences,2010,52(1): 96-102. [24] DAI L C, GUO W L, WANG X. Stress concentration at an elliptic hole in transversely isotropic piezoelectric solids[J]. International Journal of Solids and Structures,2006,43(6): 1818-1831. [25] 崔建斌, 姬安召, 熊贵明. 基于Schwarz-Christoffel变换的圆形隧道围岩应力分布特征研究[J]. 应用数学和力学, 2019,〖STHZ〗 40(10): 1089-1098.(CUI Jianbin, JI Anzhao, XIONG Guiming. Research on surrounding rock stress distributions for circular tunnels based on the Schwarz-Christoffel transformation[J]. Applied Mathematics and Mechanics,2019,40(10): 1089-1098.(in Chinese)) [26] YANG J, LI X, DING S H. Anti-plane analysis of a circular hole with three unequal cracks in one-dimensional hexagonal piezoelectric quasicrystals[J]. Chinese Journal of Engineering Mathematics,2016,33(2): 184-198. [27] 杨娟, 李星, 周跃亭. 一维六方压电准晶中圆孔边周期裂纹分析[J]. 振动与冲击, 2019,38(18): 62-71.(YANG Juan, LI Xing, ZHOU Yueting. Analysis of periodic cracks emanating from a circular hole in one-dimensional hexagonal piezoelectric quasicrystals[J]. Journal of Vibration and Shock,2019,38(18): 62-71.(in Chinese)) [28] YU J, GUO J H, XING Y M. Complex variable method for an anti-plane elliptical cavity of one-dimensional hexagonal piezoelectric quasicrystals[J].Chinese Journal of Aeronautics,2015,28(4): 1287-1295. [29] 樊世旺, 郭俊宏. 一维六方压电准晶三角形孔边裂纹反平面问题[J]. 应用力学学报, 2016,〖STHZ〗 33(3): 421-426.(FAN Shiwang, GUO Junhong. Anti-plane problem of an edge crack emanating from a triangle hole in one-dimensional hexagonal piezoelectric quasicrystals[J]. Chinese Journal of Applied Mechanics,2016,33(3): 421-426.(in Chinese)) [30] 白巧梅, 丁生虎. 一维六方准晶中正六边形孔边裂纹的反平面问题[J]. 应用数学和力学, 2019,〖STHZ〗 40(10): 1071-1080.(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)) [31] GUO J H, LU Z X. Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals[J]. Applied Mathematics and Computation,2011,217(22): 9397-9403. [32] MUSKHELISHVILI N I. Some Fundamental Problems of the Mathematical Theory of Elasticity [M]. Moscow: Nauka, 1966. [33] 路见可. 平面弹性复变方法[M]. 武汉: 武汉大学出版社, 2002.(LU Jianke. Plane Elastic Complex Method [M]. Wuhan: Wuhan University Press, 2002.(in Chinese)) [34] SHARMA D S. Stress distribution around polygonal holes[J]. International Journal of Mechanical Sciences,2012,〖STHZ〗 65(1): 115-124. [35] 邵阳, 郭俊宏. 一维六方准晶中正方形孔边双裂纹的反平面问题[J]. 内蒙古工业大学学报, 2014,33(2): 81-87.(SHAO Yang, GUO Junhong. Anti-plane analysis of double cracks originating from a square hole in one-dimensional hexagonal quasicrystals[J]. Journal of Inner Mongolia University of Technology,2014, 33(2): 81-87.(in Chinese)) [36] GUO J H, LIU G T. Stress analysis of the problem about a circular hole with asymmetry collinear cracks[J]. Journal of Inner Mongolia Normal University,2007,36(4): 418-422.
点击查看大图
计量
- 文章访问数: 989
- HTML全文浏览量: 163
- PDF下载量: 290
- 被引次数: 0