Study on V-Notch Stress Singularity in Functionally Graded Bi-Material Medium-Thickness Sector Plates With the DQM
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摘要: 对于双材料功能梯度中厚板切口尖端问题提出了一个分析应力奇性指数的实用方法：微分求积法(DQM). 首先从柱坐标系下平衡方程出发，基于切口尖端位移场的级数渐近展开假设，推导出了关于双材料功能梯度中厚板切口尖端奇性指数的常微分方程组(ODEs)特征值问题，并将切口的径向边界条件表达为奇性指数和特征角函数的组合. 然后基于DQM理论，将ODEs的特征值问题转化为标准型广义代数方程组特征值问题，求解可一次性地计算出相应边界条件下双材料功能梯度中厚板切口尖端处应力奇性指数. 首先，通过算例验证了该文DQM计算功能梯度中厚板切口尖端处应力奇性指数的结果是有效的. 然后, 用DQM计算了功能梯度与纯金属/纯陶瓷混合板应力奇性指数，结果发现随着切口夹角的改变，纯陶瓷板与纯金属板与功能梯度板的混合，分别会对应力奇性指数有不同的影响效果.Abstract: A practical method for analyzing stress singularity indexes was proposed for the problem of notch tips in bi-material functionally graded medium-thickness plates: the differential quadrature method (DQM). Firstly, according to the equilibrium equations in the cylindrical coordinate system and based on the series asymptotic expansion assumption about the displacement field at the tip of the notch, the eigenvalue problem of the ordinary differential equations (ODEs) with respect to the singularity exponent at the notch tip of functionally graded bi-material medium-thickness plates was derived, and the radial boundary condition of the notch was expressed as a combination of the singularity exponent and the characteristic angular function. Then, under the DQM theory, the eigenvalue problem of ODEs was transformed into a standard-type eigenvalue problem of generalized algebraic equations, and the stress singularity index at the notch tip under the corresponding boundary condition was calculated for only one time, and was verified to be valid by a numerical example. Finally, the stress singularity index of functionally graded and pure metal/pure ceramic hybrid plates was calculated with the DQM. The research indicates that, the mixing of pure ceramic plates and pure metal plates with functional gradient plates has different effects on the stress singularity index, respectively, with the change of the notch angle.
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图 1 陶瓷体积分数随厚度的变化曲线(幂律分布)
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 1. The ceramic volume fraction as a function of the thickness (power-law distribution)
图 2 含V形切口的功能梯度扇形板
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 2. Functional gradient sector plates with a V-notch
图 3 功能梯度双材料中厚度板切口平面图
Figure 3. The plan view of the incision in the functional gradient bi-material plate of a medium thickness
图 4 不同边界条件下，切口应力奇性指数λ1与离散单元数N的关系
Figure 4. The variations of singularity index λ1 of notch stresses with number of discrete cells N under different boundary conditions
图 5 功能梯度双材料中厚度板切口
Figure 5. The notch in the functional gradient bi-material plate of a medium thickness
图 6 功能梯度与纯金属/纯陶瓷混合板在简支-简支条件下的切口应力奇性指数
Figure 6. Singularity indexes of notch stresses of hybrid plates of functional gradient materials and pure metals/ceramics under simply-simply supported conditions
图 7 功能梯度与纯金属/纯陶瓷混合板在夹支-夹支条件下的切口应力奇性指数
Figure 7. Singularity indexes of notch stresses of functional gradient material and pure metal/ ceramic hybrid plates under clamped-clamped supported boundary conditions
表 1 自由-自由边界条件下的切口应力奇性指数
Table 1. Singularity indexes of notch stresses under free-free supported boundary conditions
γ/(°) λ1 λ2 λ3 present ref. [26] present ref. [26] present ref. [26] 360 0.499 996 0.500 000 0.499 996 0.500 000 0.499 999 0.500 000 330 0.501 438 0.501 453 0.545 454 0.545 454 0.598 155 0.598 191 300 0.512 722 0.512 221 0.599 999 0.600 000 0.730 264 0.730 900 270 0.544 465 0.544 484 0.666 666 0.666 666 0.908 261 0.908 529 250 0.586 284 0.586 279 0.719 999 0.720 000 1.000 001 1.000 000 240 0.615 787 0.615 731 0.749 999 9 0.750 000 - - 210 0.751 991 0.751 974 0.857 142 0.857 142 - - 200 0.818 636 0.818 696 0.899 999 0.900 000 - - 190 0.900 098 0.900 044 0.947 368 0.947 368 - - 180 - - - - - - 
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表 2 给定不同边界条件和不同离散单元数N时的切口应力奇性指数
Table 2. Singularity indices of notch stresses given different boundary conditions and different numbers of discrete cells N
boundary condition λ1 λ2 λ3 N=12 N=40[26] N=12 N=40[26] N=12 N=40[26] free-free supported 0.512 722 0.512 722 0.599 999 0.599 999 0.730 264 0.730 264 clamped-clamped supported 0.555 367 0.555 367 0.599 999 0.599 999 0.652 638 0.652 638 simply-simply supported 0.400 000 0.400 000 0.600 000 0.600 000 0.800 000 0.800 000
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表 3 FGM板的材料属性
Table 3. Material properties of the FGM plate
material property metal ceramic Ti Al SiC Al2O3 Young’s modulus/GPa 110 70 290 380 Poisson’s ratio 0.3 0.3 0.3 0.3
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表 4 双材料功能梯度板在自由-自由边界条件下的切口应力奇性指数
Table 4. Singularity indexes of notch stresses of the bi-material functional gradient plate under free-free supported boundary conditions
γ/(°) λ1 λ2 λ3 bi-material mono-material[26] bi-material mono-material[26] bi-material mono-material[26] 360 0.499 980 0.500 000 0.499 980 0.500 000 0.499 999 0.500 000 330 0.507 192 0.501 453 0.545 454 0.545 454 0.592 299 0.598 191 300 0.515 571 0.512 221 0.599 999 0.600 000 0.726 888 0.730 900 270 0.546 979 0.544 484 0.666 666 0.666 666 0.903 624 0.908 529 250 0.588 040 0.586 279 0.719 999 0.720 000 1.000 000 1.000 000 240 0.616 945 0.615 731 0.749 999 0.750 000 0.999 900 - 210 0.750 629 0.751 974 0.857 142 0.857 142 0.999 972 - 200 0.815 914 0.818 696 0.899 999 0.900 000 0.999 972 - 190 0.895 010 0.900 044 0.947 368 0.947 368 - - 180 0.992 171 - 0.999 804 - - -
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表 5 双材料功能梯度板在夹支-夹支和简支-简支边界条件下的切口应力奇性指数
Table 5. Singularity indexes of notch stresses of the bi-material functional gradient plate under clamped-clamped supported/simply-simply supported boundary conditions
boundary condition γ/(°) λ1 λ2 λ3 bi-material mono-material[26] bi-material mono-material[26] bi-material mono-material[26] clamped- clamped supported 360 0.499 999 0.499 999 0.499 999 0.499 999 0.499 999 0.499 999 330 0.538 790 0.523 515 0.545 454 0.545 454 0.554 112 0.569 327 300 0.560 687 0.555 368 0.599 999 0.600 000 0.647 693 0.652 638 270 0.606 790 0.604 045 0.666 666 0.666 666 0.742 978 0.744 462 240 0.680 941 0.681 239 0.749 999 0.750 000 0.838 111 0.834 887 210 0.795 638 0.803 999 0.857 142 0.857 142 0.931 255 0.917 938 200 0.844 277 0.859 324 0.899 999 0.900 000 0.965 408 0.944 752 190 0.896 430 0.924 049 0.947 368 0.947 368 - 0.971 899 180 0.947 900 - - - - - simply-simply supported 360 0.499 999 0.500 000 0.500 000 0.500 000 0.500 007 0.500 000 330 0.456 182 0.454 545 0.545 454 0.545 454 0.635 669 0.636 368 300 0.405 273 0.400 000 0.599 999 0.600 000 0.798 681 0.800 000 270 0.295 285 0.333 333 0.371 629 0.333 333 0.666 666 0.666 666 240 0.243 594 0.250 000 0.506 631 0.500 000 0.749 999 0.750 000 210 0.140 400 0.142 857 0.716 795 0.714 286 0.857 142 0.857 142 200 0.098 415 0.099 999 0.801 697 0.800 000 0.899 999 0.900 000 190 0.051 848 0.052 633 0.895 510 0.894 737 0.947 368 0.947 369 180 0.999 993 - 0.999 999 - - -
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[1] GARG A, CHALAK H D. A review on analysis of laminated composite and sandwich structures underhygrothermal conditions[J]. Thin-Walled Structures, 2019, 142: 205-226. doi: 10.1016/j.tws.2019.05.005 [2] SHIOTA I, MIYAMOTO Y. Functionally Graded Materials 1996[C]//Proceedings of the 4th International Symposium on Functionally Graded Materials. Tsukuba, Japan: Elsevier Science, 1996. [3] SHEN H S. Functionally Graded Materials: Nonlinear Analysis of Plates and Shells[M]. Boca Raton: CRC Press, 2009. [4] 王素粉. 基于ANSYS的功能梯度材料静态断裂力学分析[J]. 机械工程与自动化, 2020(1): 87-88.WANG Sufen. Static fracture mechanics analysis of functionally graded materials based on ANSYS[J]. Mechanical Engineering & Automation, 2020(1): 87-88. (in Chinese) [5] ARSLAN K, GUNES R. Penetration mechanics of ceramic/metal functionally graded plates under ballistic impact: an experimental perspective[J]. Composite Structures, 2024, 331: 117897. doi: 10.1016/j.compstruct.2024.117897 [6] PU N, ZHANG Y F, MA W T. Fracture mechanics analysis of functionally graded materials by an efficient and accurate meshless method[J]. Theoretical and Applied Fracture Mechanics, 2024, 130: 104313. doi: 10.1016/j.tafmec.2024.104313 [7] 平学成, 谢基龙, 陈梦成, 等. 各向异性两相材料尖劈奇性场的非协调元分析[J]. 力学学报, 2005, 37(1): 24-31.PING Xuecheng, XIE Jilong, CHEN Mengcheng, et al. A non-confirming finite element analysis of singular fields in prismatic anisotropic bimaterial wedges[J]. Acta Mechanica Sinica, 2005, 37(1): 24-31. (in Chinese) [8] CHEN W H, XU Y H, CHEN G L, et al. Experimental investigation of compressive behaviors of functionally graded composites material based on engineering cementitious composites and normal concrete[J]. Case Studies in Construction Materials, 2024, 20: e02751. doi: 10.1016/j.cscm.2023.e02751 [9] LIU T, ZHENG H Y, ZHANG W, et al. Nonlinear forced vibrations of functionally graded three-phase composite cylindrical shell subjected to aerodynamic forces, external excitations andhygrothermal environment[J]. Thin-Walled Structures, 2024, 195: 111511. doi: 10.1016/j.tws.2023.111511 [10] 高正阳, 石先杰. 功能梯度石墨烯增强复合材料锥壳动力学建模及特性分析[J/OL]. 振动工程学报, 2024(2024-07-15)[2024-11-14]. https://link.cnki.net/urlid/32.1349.TB.20240711.1517.002 .GAO Zhengyang, SHI Xianjie. Dynamic modeling and characteristic analysis of FG-GPLRC conical shell[J/OL]. Journal of Vibration Engineering, 2024(2024-07-15)[2024-11-14].https://link.cnki.net/urlid/32.1349.TB.20240711.1517.002 . (in Chinese)[11] 王强. 含V形切口结构强度与疲劳寿命研究[D]. 合肥: 合肥工业大学, 2013.WANG Qiang. Research on the strength and fatigue life of the V-notched structures[D]. Hefei: Hefei University of Technology, 2013. (in Chinese) [12] ZUCCHINI A, HUI C Y, ZEHNDER A T. Crack tip stress fields for thin, cracked plates in bending, shear and twisting: a comparison of plate theory and three-dimensional elasticity theory solutions[J]. International Journal of Fracture, 2000, 104: 387-407. doi: 10.1023/A:1007699314793 [13] KOTOUSOV A. Effect of plate thickness on stress state at sharp notches and the strength paradox of thick plates[J]. International Journal of Solids and Structures, 2010, 47(14/15): 1916-1923. [14] LI J. Singularity analysis of near-tip fields for notches formed from several anisotropic plates under bending[J]. International Journal of Solids and Structures, 2002, 39(23): 5767-5785. doi: 10.1016/S0020-7683(02)00406-7 [15] TREIFI M, OYADIJI S O, TSANG D K L. Computations of modes Ⅰ and Ⅱ stress intensity factors of sharp notched plates under in-plane shear and bending loading by the fractal-like finite element method[J]. International Journal of Solids and Structures, 2008, 45(25/26): 6468-6484. [16] SAIDI A R, HEJRIPOUR F, JOMEHZADEH E. On the stress singularities and boundary layer in moderately thick functionally graded sectorial plates[J]. Applied Mathematical Modelling, 2010, 34(11): 3478-3492. doi: 10.1016/j.apm.2010.02.036 [17] HUANG C S. Stress singularities at angular corners in first-order shear deformation plate theory[J]. International Journal of Mechanical Sciences, 2003, 45(1): 1-20. doi: 10.1016/S0020-7403(03)00046-8 [18] HUANG C S. Corner singularities in bi-materialmindlin plates[J]. Composite Structures, 2002, 56(3): 315-327. doi: 10.1016/S0263-8223(02)00018-1 [19] BARATI E, MOHANDESI J A, ALIZADEH Y. The effect of Notch depth on J-integral and critical fracture load in plates made of functionally graded aluminum-silicone carbide composite with U-notches under bending[J]. Materials & Design, 2010, 31(10): 4686-4692. [20] CIAVARELLA M. Cancelling the effect of sharp notches or cracks with graded elastic modulus materials[J]. Journal of the Mechanics and Physics of Solids, 2024, 192: 105809. doi: 10.1016/j.jmps.2024.105809 [21] SINGH H, BHARDWAJ G, GROVER N. Modeling and static analysis of porous functionally graded and FG-sandwich plates[J]. Structures, 2024, 68: 107034. doi: 10.1016/j.istruc.2024.107034 [22] HUANG C S, CHANG M J. Corner stress singularities in an FGM thin plate[J]. International Journal of Solids and Structures, 2007, 44(9): 2802-2819. doi: 10.1016/j.ijsolstr.2006.08.024 [23] HUANG C S. Corner stress singularities in a high-order plate theory[J]. Computers & Structures, 2004, 82(20/21): 1657-1669. [24] HUANG C S, CHANG M J. Geometrically induced stress singularities of a thick FGM plate based on the third-order shear deformation theory[J]. Mechanics of Advanced Materials and Structures, 2009, 16(2): 83-97. doi: 10.1080/15376490802543634 [25] 李聪, 胡斌, 牛忠荣. 反平面塑性V形切口尖端应力和位移渐近解[J]. 应用数学和力学, 2021, 42(12): 1258-1275. doi: 10.21656/1000-0887.420045 LI Cong, HU Bin, NIU Zhongrong. Asymptotic solutions of plastic stress and displacement at V-Notch tips under anti-plane shear[J]. Applied Mathematics and Mechanics, 2021, 42(12): 1258-1275. (in Chinese) doi: 10.21656/1000-0887.420045 [26] 程长征, 丁昊, 周伟, 等. 功能梯度中厚板切口奇性分析[J]. 合肥工业大学学报(自然科学版), 2015, 38(7): 938-943.CHENG Changzheng, DING Hao, ZHOU Wei, et al. Singularity analysis of V-Notch located in functionally graded plate[J]. Journal of Hefei University of Technology (Natural Science), 2015, 38(7): 938-943. (in Chinese) [27] BELLMAN R, CASTI J. Differential quadrature and long-term integration[J]. Journal of Mathematical Analysis and Applications, 1971, 34(2): 235-238. doi: 10.1016/0022-247X(71)90110-7 [28] FARZAM A, HASSANI B. Isogeometric analysis of in-plane functionally graded porous microplates using modified couple stress theory[J]. Aerospace Science and Technology, 2019, 91: 508-524. doi: 10.1016/j.ast.2019.05.012 [29] 穆耶赛尔·艾合买提, 阿布都热西提·阿布都外力. 改进复合梯形求积公式[J]. 首都师范大学学报(自然科学版), 2016, 37(6): 1-4.MUYASSAR Ahmat, ABDURIXIT Abduweli. A corrector composite trapezoidal formula[J]. Journal of Capital Normal University (Natural Science Edition), 2016, 37(6): 1-4. 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