SiC/AZ91D复合材料中孔隙缺陷对裂纹萌生和扩展行为的影响

李步炜; 尧军平; 陈国鑫; 李怡然; 梁超群

doi:10.13801/j.cnki.fhclxb.20230711.002

SiC/AZ91D复合材料中孔隙缺陷对裂纹萌生和扩展行为的影响

doi: 10.13801/j.cnki.fhclxb.20230711.002

南昌航空大学　航空制造工程学院，南昌 330000

基金项目: 国家自然科学基金(52065046；51661024)；江西省科技重点研发计划(20202BBEL53024)；研究生创新专项(2030009101050)

详细信息

通讯作者:
尧军平，博士，教授，硕士生导师，研究方向为金属基复合材料　E-mail: yyyjpsz@126.com

中图分类号: TB331
计量
- 文章访问数: 414
- HTML全文浏览量: 150
- PDF下载量: 21
- 被引次数: 0
出版历程
- 收稿日期: 2023-05-24
- 修回日期: 2023-06-28
- 录用日期: 2023-07-01
- 网络出版日期: 2023-07-11
- 刊出日期: 2024-03-01

Effect of porosity defects on crack initiation and propagation behavior in SiC/AZ91D composites

School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang 330000, China

Funds: National Natural Science Foundation of China (52065046; 51661024); Jiangxi Provincial Science and Technology Key Research and Development Program (20202BBEL53024); Postgraduate Innovation Special Fund Project (2030009101050)

摘要

摘要: 采用内聚力模型及有限元分析方法，在含真实形貌SiC颗粒增强AZ91D镁基复合材料中引入孔隙缺陷，分析不同孔隙率及孔隙形状在单轴拉伸过程中对SiC/AZ91D复合材料力学行为的影响。结果表明：孔隙长径比为1时，孔隙率为0%、0.5%、1.0%、1.5%的复合材料的抗拉强度分别为351.214 MPa、339.452 MPa、325.735 MPa、306.791 MPa，抗拉强度随孔隙率的增加逐渐降低，复合材料中裂纹萌生和裂纹扩展时间均随孔隙率增加而提前。孔隙长径比越大，其尖端部位应力集中越严重，复合材料抗拉强度也越低。无孔隙缺陷的SiC/AZ91D复合材料裂纹萌生扩展机制是颗粒与基体交界处萌生微裂纹，微裂纹相互连接形成主裂纹绕开颗粒进行扩展致使材料断裂，含孔隙的SiC/AZ91D复合材料裂纹萌生扩展机制为微裂纹在孔隙周围萌生，与颗粒和基体交界处产生的微裂纹相互连接，汇集成主裂纹绕开颗粒扩展使材料断裂。
- SiC/AZ91D复合材料 /
- 裂纹萌生 /
- 裂纹扩展 /
- 有限元分析 /
- 孔隙率 /
- 内聚力模型
Abstract: Using the finite element analysis method, this study introduced porosity defects into SiC/AZ91D magnesium matrix composites with realistic SiC particle morphology, and analyzed the influence of different porosity rates and shapes on the mechanical behavior of SiC/AZ91D composites during uniaxial tensile process. The results show that when the aspect ratio of the pore length to width is 1, the tensile strengths of the composites with void contents of 0%, 0.5%, 1.0%, and 1.5% are 351.214 MPa, 339.452 MPa, 325.735 MPa and 306.791 MPa, respectively. The tensile strength gradually decreases with the increase of porosity rate, and the initiation and propagation time of cracks in the composite material advances with the increase of porosity rate. As the aspect ratio of the pore length to width increases, the stress concentration at the tip of the pore becomes more severe, resulting in lower tensile strength of the composite material. The crack initiation and propagation mechanism in the SiC/AZ91D composite material without porosity defects involves the initiation of microcracks at the particle-matrix interface, followed by their interconnection to form a main crack, which propagates around the particles leading to material fracture. In the case of SiC/AZ91D composites with porosity, microcracks initiate around the pores and interconnect with microcracks generates at the particle-matrix interface, ultimately converging into a main crack that propagates around the particles, causing material fracture.
- SiC/AZ91D composites /
- crack initiation /
- crack extension /
- finite element analysis /
- porosity /
- cohesive model

HTML全文

图 1 SiC颗粒的SEM图像及其建模

Figure 1. SEM images and modeling of SiC particles

下载: 全尺寸图片幻灯片

图 2 不同孔隙率的SiC/AZ91D镁基复合材料模型：(a) VC=0 (理想状况)；(b) VC=0.5%；(c) VC=1.0%；(d) VC=1.5%

Figure 2. Modeling diagram of SiC/AZ91D magnesium-based composite materials with different void contents: (a) VC=0 (Ideal condition); (b) VC=0.5%; (c) VC=1.0%; (d) VC=1.5%

下载: 全尺寸图片幻灯片

图 3 SiC/AZ91D镁基复合材料模型及网格划分和载荷施加：(a) SiC/AZ91D模型；(b) 网格划分；(c) 载荷施加

Figure 3. Modeling, meshing and load application of SiC/AZ91D magnesium-based composite materials: (a) SiC/AZ91D model; (b) Meshing; (c) Load application

下载: 全尺寸图片幻灯片

图 4 不同孔隙形状的SiC/AZ91D镁基复合材料模型：(a) 长径比r=1；(b) r=2；(c) r=4

Figure 4. Model of SiC/AZ91D magnesium-based composite materials with different pore shapes: (a) Aspect ratio of the pore length to width r=1; (b) r=2; (c) r=4

下载: 全尺寸图片幻灯片

图 5 内聚力单元受力变形模型

Figure 5. Force-deformation model of cohesive element

下载: 全尺寸图片幻灯片

图 6 双线性内聚力模型

Figure 6. Bilinear cohesive zone model

$ {\delta }_{\mathrm{m}}^{\text{max}} $—Maximum value of the effective displacement; $ {\delta }_{\mathrm{m}}^{\mathrm{f}} $—Effective displacement at complete failure; $ {\delta }_{\mathrm{m}}^{0} $—Effective displacement at the initiation of damage; $ {\tau }_{\mathrm{m}}^{0} $—Maximum separation stress; K—Elasticity coefficient or spring constant; D—Damage amount

下载: 全尺寸图片幻灯片

图 7 拉伸过程中含不同孔隙率SiC/AZ91D镁基复合材料应力-应变曲线

Figure 7. Stress-strain curves of SiC/AZ91D magnesium-based composite materials with different void content during tensile process

下载: 全尺寸图片幻灯片

图 8 拉伸过程中不同孔隙率SiC/AZ91D复合材料屈服应力值、抗拉强度和伸长率

Figure 8. Yield stress, tensile strength and elongation of SiC/AZ91D composite materials with different void contents during the tensile process

下载: 全尺寸图片幻灯片

图 9 孔隙周围A、B、C方向示意图

Figure 9. Schematic diagram of the A, B, and C directions around the void

S—Quivalent stress (MPa)

下载: 全尺寸图片幻灯片

图 10 孔隙周围基体A、B、C 方向应力随施载时间变化曲线

Figure 10. Stress variation curves with respect to loading time in the A, B, and C directions of the matrix around the void

下载: 全尺寸图片幻灯片

图 11 含不同孔隙率 SiC/AZ91D 复合材料裂纹长度随时间变化曲线

Figure 11. Crack length variation curves with respect to time in SiC/AZ91D composite materials with different void contents

下载: 全尺寸图片幻灯片

图 12 不同孔隙率的SiC/AZ91D复合材料裂纹扩展路径：(a) VC=0%；(b) VC=0.5%；(c) VC=1.0%；(d) VC=1.5%

Figure 12. Crack propagation paths in SiC/AZ91D composite materials with different void contents: (a) VC=0%; (b) VC=0.5%; (c) VC=1.0%; (d) VC=1.5%

下载: 全尺寸图片幻灯片

图 13 含不同孔隙形状的SiC/AZ91D复合材料的拉伸应力-应变曲线

Figure 13. Stress-strain curves during tension of SiC/AZ91D composite materials with different void shapes

下载: 全尺寸图片幻灯片

图 14 拉伸过程中含不同孔隙形状SiC/AZ91D 复合材料抗拉强度和伸长率

Figure 14. Tensile strength and elongation of SiC/AZ91D composite materials with different void shapes during the tensile process

下载: 全尺寸图片幻灯片

图 15 含不同形状孔隙的复合材料同一区域的应力场：(a) r=1；(b) r=2；(c) r=4

Figure 15. Stress field in the same region of composite materials with different void shapes: (a) r=1; (b) r=2; (c) r=4

下载: 全尺寸图片幻灯片

图 16 含不同孔隙形状 SiC/AZ91D 复合材料裂纹长度随时间变化曲线

Figure 16. Crack length variation curves with respect to time in SiC/AZ91D composite materials with different void shapes

下载: 全尺寸图片幻灯片

图 17 含不同形状孔隙的SiC/AZ91D复合材料的裂纹扩展路径：(a) r=1；(b) r=2；(c) r=4

Figure 17. Crack propagation paths in SiC/AZ91D composite materials with different void shapes: (a) r=1; (b) r=2; (c) r=4

下载: 全尺寸图片幻灯片

图 18 拉伸片试样

Figure 18. Tensile sheet sample

下载: 全尺寸图片幻灯片

图 19 实验拉伸片试样尺寸

Figure 19. Experimental tensile sheet sample size

R—Circle radius

下载: 全尺寸图片幻灯片

图 20 实验使用的WDW-E100D微机控制电子万能试验机

Figure 20. WDW-E100D microcomputer-controlled electronic universal testing machine used in the experiment

下载: 全尺寸图片幻灯片

图 21 含不同孔隙率的SiC/AZ91D复合材料仿真与实验拉伸应力-应变曲线对比

Figure 21. Comparison of simulated and experimental stress-strain curves during tensile testing of SiC/AZ91D composite materials with different void contents

下载: 全尺寸图片幻灯片

图 22 含不同孔隙率SiC/AZ91D复合材料拉伸应力-应变误差带图：(a) VC=0.5%；(b) VC=1.0%；(c) VC=1.5%

Figure 22. Tensile stress-strain error band diagram of SiC/AZ91D composite materials with different void contents: (a) VC=0.5%; (b) VC=1.0%; (c) VC=1.5%

下载: 全尺寸图片幻灯片

表 1 排水法测量的SiC/AZ91D复合材料孔隙率(VC)

Table 1. Void content (VC) measurement for SiC/AZ91D composite material using drainage method

Sample serial number	VC/%
1	1.56
2	0.84
3	0.85
4	1.62
5	0.83
6	0.41
7	1.65
8	0.94
9	0.48
10	1.18

下载: 导出CSV

表 2 AZ91D镁合金和SiC颗粒的基本参数

Table 2. Basic parameters of AZ91D magnesium alloy and SiC particles

Material	$ \rho $/(kg·m⁻³)	E/GPa	$ \mu $	σ_b/MPa
AZ91D	1800	45	0.33	164
SiC	3215	450	0.17	2000
Notes: $ \rho $—Material density; E—Modulus of elasticity; $ \mu $—Poisson's ratio; σ_b—Tensile strength.

下载: 导出CSV

表 3 SiC/AZ91D颗粒-界面的本构模型参数

Table 3. Constitutive model parameters of SiC/AZ91D particle-interface

$ {t}_{\mathrm{n}}/\mathrm{M}\mathrm{P}\mathrm{a} $	$ {t}_{\mathrm{t}}/\mathrm{M}\mathrm{P}\mathrm{a} $	$ {\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}/\mathrm{m}\mathrm{m} $	$ {\delta }_{\mathrm{f}}/\mathrm{m}\mathrm{m} $
400	400	0.00015	0.00005
Notes: $ {t}_{\mathrm{n}} $—Interface normal nominal stress; $ {t}_{\mathrm{t}} $—Interfacial tangential nominal stress; $ {\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}} $—Destruction displacement; $ {\delta }_{\mathrm{f}} $—Material complete failure separation.

下载: 导出CSV

表 4 AZ91D镁合金的Johnson-Cook (J-C)本构参数

Table 4. Johnson-Cook (J-C) constitutive model parameters for AZ9ID magnesium alloy

A/MPa	B/MPa	n	C	${u}_{{\rm{f}}}^{\mathrm{p}\mathrm{l} }/{\rm{mm} }$
164	600	0.283	0.021	0.00015
Notes: A—Yield strength of AZ91D matrix under static load; B—Hardening constant; n—Hardening exponent; C—Strain rate constant; $ {u}_{\mathrm{f}}^{\mathrm{p}\mathrm{l}} $—Failure displacement.

下载: 导出CSV

表 5 SiC颗粒本构模型及失效参数

Table 5. Constitutive model and failure parameters of SiC particles

Parameter	Value
G/GPa	193
A₀	0.96
B₀	0.35
C₀	0.009
M	1
N	0.65
T₁/MPa	750
S_FMAX/MPa	1300
L_HE/MPa	11700
P_HEL/MPa	5130
D₁	0.48
D₂	0.48
K₁	220000
K₂	361000
K₃	0
Notes: G—Shear modulus; A₀—Strength parameter before damage; B₀—Strength parameter when damage occurs; C₀—Strain rate constant; M—Pressure index when damage occurs; N—Pressure index when no damage occurs; T₁—Cut-off pressure; S_FMAX—Maximum fracture strength; L_HE—Hugoniot elastic limit; P_HEL—Hugoniot elastic limit pressure; D₁, D₂—Fracture constant; K₁, K₂, K₃—Material parameter.

下载: 导出CSV

参考文献(34)

[1]	褚兵强. 低能量冲击对含孔隙CFRP层合板力学性能的影响研究[D]. 哈尔滨: 哈尔滨工业大学, 2011. CHU Bingqiang. The effects of low-energy impact on the mechanical properties of carbon fiber reinforced polymer laminates with voids[D]. Harbin: Harbin Institute of Technology, 2011(in Chinese).
[2]	李钊. 碳纤维复合材料孔隙率超声检测与评价技术研究[D]. 杭州: 浙江大学, 2014. LI Zhao. Research on ultrasonic detection and evaluation technique for porosity of carbon fiber composites[D]. Hangzhou: Zhejiang University, 2014(in Chinese).
[3]	李亚星, 李洪双, 陈普会. 考虑孔隙随机性的SiC/SiC复合材料力学性能参数概率分析[J/OL]. 应用力学学报: 1-12[2024-01-08]. LI Yaxing, LI Hongshuang, CHEN Puhui. Probability analysis of mechanical properties parameters of SiC/SiC composites considering pore randomness[J/OL]. Chinese Journal of Applied Mechanics: 1-12[2024-01-08](in Chinese).
[4]	HONG Y, WANG W J, LIU J Q, et al. Effect of porosity and interface structures on thermal and mechanical properties of SiCp/6061Al composites with high volume fraction of SiC[J]. Transactions of Nonferrous Metals Society of China,2019,29(5):941-949. doi: 10.1016/S1003-6326(19)65003-X
[5]	GAO M Q, GUO E Y, CHEN Z N, et al. Revealing the role of micropore defects in tensile deformation of a B₄Cp/Al composite using an actual three-dimensional model[J]. Journal of Materials Research and Technology,2023,22:3146-3155. doi: 10.1016/j.jmrt.2022.12.145
[6]	CHAMBERS A R, EARL J S, SQUIRES C A, et al. The effect of voids on the flexural fatigue performance of unidirectional carbon fibre composites developed for wind turbine applications[J]. International Journal of Fatigue,2006,28(10):1389-1398. doi: 10.1016/j.ijfatigue.2006.02.033
[7]	KOSMANN N, KARSTEN J M, SCHUETT M, et al. Determining the effect of voids in GFRP on the damage behaviour under compression loading using acoustic emission[J]. Composites Part B: Engineering,2015,70:184-188.
[8]	张阿樱. 孔隙率对碳纤维/环氧树脂层压板力学性能的影响[J]. 中国机械工程, 2010, 21(24):3014-3018. ZHANG A'ying. Experimental characterization of porosity and mechanical properties of carbon/epoxy composite laminates[J]. China Mechanical Engineering,2010,21(24):3014-3018(in Chinese).
[9]	张阿樱, 张东兴. 不同孔隙率 CFRP 层合板冲击后力学性能试验表征[J]. 材料科学与工艺, 2014, 46(3): 54-60. ZHANG A'ying, ZHANG Dongxing. Research on the static mechanical properties of CFRP laminates with different void contents[J]. Materials Science and Technology, 2014, 46(3): 54-60(in Chinese).
[10]	RICOTTA M, QUARESIMIN M, TALREJA R. Mode I strain energy release rate in composite laminates in the presence of voids[J]. Composites Science and Technology,2008,68(13):2616-2623. doi: 10.1016/j.compscitech.2008.04.028
[11]	SUO Y Y, WANG B, JIA P R, et al. The effect of fabrication defects on the mechanical behaviors of metal matrix composites[J]. Materialstoday Communications,2022,25:101663. doi: 10.1016/j.mtcomm.2020.101663
[12]	乔海燕, 任学冬, 史亦韦, 等. GH4169高温合金涡轮盘表面径轴向裂纹的渗透检测可行性[J]. 航空材料学报, 2016, 36(6):92-96. doi: 10.11868/j.issn.1005-5053.2016.6.015 QIAO Haiyan, REN Xuedong, SHI Yiwei, et al. Feasibility of penetrant detection of radial and axial cracks on the surface of GH4169 superalloy turbine disk[J]. Journal of Aeronautical Materials,2016,36(6):92-96(in Chinese). doi: 10.11868/j.issn.1005-5053.2016.6.015
[13]	李灼华. 射线检测在复合材料无损检测中的应用[J]. 科技传播, 2014, 6(3): 179, 175. LI Zhuohua. Application of radiographic testing in nondestructive testing of composite materials[J]. Science and Technology Communication, 2014, 6(3): 179, 175(in Chinese).
[14]	陈文玮. 超声检测技术在金属材料焊接成型中的应用[J]. 山西冶金, 2022, 45(3):264-265. doi: 10.16525/j.cnki.cn14-1167/tf.2022.03.108 CHEN Wenwei. Application of ultrasonic testing technology in welding and forming of metal materials[J]. Shanxi Metallurgy,2022,45(3):264-265(in Chinese). doi: 10.16525/j.cnki.cn14-1167/tf.2022.03.108
[15]	孟祥姝, 李武胜. 单向复合材料孔隙率测试技术与分布规律研究[J]. 高科技纤维与应用, 2021, 46(4):52-55. doi: 10.3969/j.issn.1007-9815.2021.04.008 MENG Xiangshu, LI Wusheng. Research on porosity testing technology and distribution law of unidirectional composite materials[J]. High-tech Fiber and Application,2021,46(4):52-55(in Chinese). doi: 10.3969/j.issn.1007-9815.2021.04.008
[16]	凌振宝, 邹得宝, 张堃, 等. 岩矿石孔隙度测量方法[J]. 吉林大学学报(地球科学版), 2011, 41(3):921-924. doi: 10.13278/j.cnki.jjuese.2011.03.032 LING Zhenbao, ZOU Debao, ZHANG Kun, et al. Measuring method of rock ore porosity[J]. Journal of Jilin University (Earth Science Edition),2011,41(3):921-924(in Chinese). doi: 10.13278/j.cnki.jjuese.2011.03.032
[17]	耿昆. SiCp/Al复合材料基于微细观的有限元建模拟实[D]. 上海: 上海交通大学, 2017. GENG Kun. Finite element modeling and simulation of SiCp/Al composites based on micro view[D]. Shanghai: Shanghai Jiao Tong University, 2017(in Chinese).
[18]	邱鑫. 挤压铸造SiCp/AZ91镁基复合材料的显微结构与性能[D]. 哈尔滨: 哈尔滨工业大学, 2006. QIU Xin. Microstructure and properties of squeeze cast SiCp/AZ91 magnesium matrix composites[D]. Harbin: Harbin Institute of Technology, 2006(in Chinese).
[19]	王涛. 高体积分数SiCp/Al复合材料高速铣削基础研究[D]. 北京: 北京理工大学, 2015. WANG Tao. Fundamental study on high speed milling of high volume fraction SiCp/Al composite[D]. Beijing: Beijing Institute of Technoloy, 2015(in Chinese).
[20]	TANG J M, DENG Q C. Experiment and simulation of SiC particle-reinforced aluminum matrix composites fabricated by friction stir processing[J]. The International Journal of Advanced Manufacturing Technology,2022,122(2):895-910. doi: 10.1007/s00170-022-09967-7
[21]	张磊刚. SiCp/Mg 多级纳米复合材料的压缩断裂特性研究[D]. 成都: 西南交通大学, 2018. ZHANG Leigang. Research on compressive fracture properties of SiCp/Mg hierarchical nanocomposites[D]. Chengdu: Southwest Jiaotong University, 2018(in Chinese).
[22]	SU Y S, LI Z, YU Y, et al. Structural modeling and tensile simulation of graphene-reinforced metal matrix composites[J]. Science China Materials,2018,61(1):112-124. doi: 10.1007/s40843-017-9142-2
[23]	梁超群, 尧军平, 李怡然, 等. SiC/AZ91D镁基复合材料单轴压缩过程中裂纹萌生扩展机制[J]. 复合材料学报, 2023, 40(7): 4282-4293. LIANG Chaoqun, YAO Junping, LI Yiran, et al. Mechanism of crack initiation and propagation during uniaxial compression of SiC/AZ91D magnesium matrix composites[J]. Acta Materiae Compositae Sinica, 2023, 40(7): 4282-4293(in Chinese).
[24]	夏聪聪. 裂纹尖端拘束对单边缺口拉伸试样裂纹扩展阻力曲线影响研究[D]. 天津: 天津大学, 2021. XIA Congcong. Research on the effect of crack tip constraint on the crack propagation resistance curve of unilateral notched tensile specimens[D]. Tianjin: Tianjin University, 2021(in Chinese).
[25]	ZHANG J, OUYANG Q B, GUO Q, et al. 3D microstructure-based finite element modeling of deformation and fracture of SiCp/Al composites[J]. Composites Science and Technology,2016,123:1-9. doi: 10.1016/j.compscitech.2015.11.014
[26]	魏俊磊. 颗粒增强镁基复合材料力学行为的研究[D]. 太原: 太原理工大学, 2018. WEI Junlei. Study on mechanical behavior of particle reinforced magnesium matrix composites[D]. Taiyuan: Taiyuan University of Technology, 2018(in Chinese).
[27]	卢子兴. 复合材料界面的内聚力模型及其应用[J]. 固体力学学报, 2015, 36(S1):85-94. doi: 10.19636/j.cnki.cjsm42-1250/o3.2015.s1.014 LU Zixing. The cohesion model of composite material interface and its application[J]. Journal of Solid Mechanics,2015,36(S1):85-94(in Chinese). doi: 10.19636/j.cnki.cjsm42-1250/o3.2015.s1.014
[28]	翁琳. 基于微观结构的颗粒增强复合材料力学性能数值分析[D]. 上海: 上海交通大学, 2015. WENG Lin. Numerical analysis of mechanical properties of particle-reinforced composites based on microstructure[D]. Shanghai: Shanghai Jiao Tong University, 2015(in Chinese).
[29]	张炯, 屈展, 黄其青, 等. 基于内聚力模型的圆形夹杂与基体界面渐进脱粘分析[J]. 西安石油大学学报(自然科学版), 2014, 29(3):106-110, 12. ZHANG Jiong, QU Zhan, HUANG Qiqing, et al. Progressive debonding analysis of circular inclusion and matrix interface based on cohesion model[J]. Journal of Xi'an Shiyou University (Natural Science Edition),2014,29(3):106-110, 12(in Chinese).
[30]	周圣杰. B4C/2024Al复合材料变形断裂行为研究[D]. 哈尔滨: 哈尔滨工业大学, 2020. ZHOU Shengjie. Research on deformation and fracture behavior of B4C/2024Al composites[D]. Harbin: Harbin Institute of Technology, 2020(in Chinese).
[31]	周爽. 纳米增强体镁基复合材料力学性能数值模拟[D]. 沈阳: 沈阳工业大学, 2017. ZHOU Shuang. Numerical simulation of mechanical properties of nano-reinforced magnesium matrix composites[D]. Shenyang: Shenyang University of Technology, 2017(in Chinese).
[32]	JOHNSON G R, HOLMQUIST T J. An improved computational constitutive model for brittle materials[J]. AIP Conference Proceedings, 1994, 309(1): 981-984.
[33]	XIANG D H, SHI Z L, FENG H, et al. Finite element analysis of ultrasonic assisted milling of SiCp/Al composites[J]. The International Journal of Advanced Manufacturing Technology,2019,105:3477-3488.
[34]	张世强. 曲线回归的拟合优度指标的探讨[J]. 中国卫生统计, 2002, 19(1):9-11. doi: 10.3969/j.issn.1002-3674.2002.01.003 ZHANG Shiqiang. Study on the goodness-of-fit in dex of curve regression[J]. China Health Statistics,2002,19(1):9-11(in Chinese). doi: 10.3969/j.issn.1002-3674.2002.01.003