Fitting Model to Compressive Strength of Composite Laminate After Impact

doi:10.16183/j.cnki.jsjtu.2020.360

Abstract

Abstract:

Carbon fiber-reinforced epoxy composites are widely used in the primary structure of aircraft, the compressive strength of which after impact is an important part in the evaluation of damage tolerance. At present, it mainly relies on a large number of tests to obtain compressive strength after impact in the engineering project. Therefore, it is necessary to develop a simple mathematical model to describe the compressive strength law after impact. A novel mathematical model for fitting compressive strength data of composite laminate after impact was proposed. Using the mathematical model and the initial model parameters, the compressive strength data after impact at different impact energy could be converted into some equivalent undamaged compressive strength data. Then, these equivalent undamaged compressive strength data were normally fitted using the maximum-likelihood estimate (MLE) method to obtain the standard deviation of normal distribution. The above steps was repeated until the minimum estimator of standard deviation was obtained. Hence, the best estimators of parameters for the mathematical model were determined. In order to further demonstrate the applicability of the mathematical model, post-impact compressive strength tests including different thicknesses, layup proportion, and material types were conducted, and the experimental data were fitted with the model. The results indicate that the mathematical model has a good applicability to the compressive strength test data after impact including different thicknesses, layup proportions, and material types.

Key words: composite, laminate, impact damage, compressive strength, fitting model, maximum-likelihood estimate (MLE)

CLC Number:

TB332

GUAN Qingyu, XIA Pinqi, ZHENG Xiaoling, WU Guanghui. Fitting Model to Compressive Strength of Composite Laminate After Impact[J]. Journal of Shanghai Jiao Tong University, 2021, 55(11): 1459-1466.

Figures/Tables 14

Fig.1

Tab.1

Fig.2

Fig.3

Fig.4

Fig.5

Fig.6

Tab.2

Damage area of M21/IMA composite on layup L0

E/J	$A - d$ /mm²	C_V/%
无损伤	—	—
20	390	6.4
30	697	11
40	936	8.1
50	1203	12

Tab.2

Tab.3

Compressive strength data after impact of M21/IMA composite on layup L0

$A - d$ / mm²	ε_CAI×10⁶
$A - d$ / mm²	1	2	3	4	5	6
1	11754	10804	11895	12522	11489	10047
390	4655	5013	5179	4443	4571	4701
697	3730	3697	3996	4019	3442	4057
936	3451	3536	3573	3674	3743	3144
1203	3268	3212	3165	2981	3285	3493

Tab.3

Fig.7

Fig.8

Tab.4

Model parameters

材料类型	铺层代码	$α^$	$β^$	$μ^$	$σ^$	B
M21/IMA	L1	0.0158	0.3581	11862	899	0.8305
	L2	0.0186	0.3870	10653	814	0.8271
X850/IM+	L0	0.0235	0.3716	9450	758	0.8159

Tab.4

Fig.9

Fig.10

References 27

[1]	杜善义. 先进复合材料与航空航天[J]. 复合材料学报, 2007, 24(1):1-12.
	DU Shanyi. Advanced composite materials and aerospace engineering[J]. Acta Materiae Compositae Sinica, 2007, 24(1):1-12.
[2]	杜善义, 关志东. 我国大型客机先进复合材料技术应对策略思考[J]. 复合材料学报, 2008, 25(1):1-10.
	DU Shanyi, GUAN Zhidong. Strategic considerations for development of advanced composite technology for large commercial aircraft in China[J]. Acta Materiae Compositae Sinica, 2008, 25(1):1-10.
[3]	LIU H B, FALZON B G, TAN W. Experimental and numerical studies on the impact response of damage-tolerant hybrid unidirectional/woven carbon-fibre reinforced composite laminates[J]. Composites Part B: Engineering, 2018, 136:101-118. doi: 10.1016/j.compositesb.2017.10.016 URL
[4]	林智育, 许希武. 含冲击损伤复合材料加筋层板压缩剩余强度[J]. 航空学报, 2009, 30(1):56-61.
	LIN Zhiyu, XU Xiwu. Residual compressive strength of stiffened composite laminates with impact damage[J]. Acta Aeronautica et Astronautica Sinica, 2009, 30(1):56-61.
[5]	AHN S S, HONG S W, KOO J M, et al. Evaluation of compressive residual strength in composite material under impact damage[J]. Transactions of the Korean Society of Mechanical Engineers A, 2013, 37(4):503-509. doi: 10.3795/KSME-A.2013.37.4.503 URL
[6]	ANDREW J J, SRINIVASAN S M, AROCKIARAJAN A, et al. Parameters influencing the impact response of fiber-reinforced polymer matrix composite materials: A critical review[J]. Composite Structures, 2019, 224:111007. doi: 10.1016/j.compstruct.2019.111007 URL
[7]	CMH-17 Committee. Composite materials handbook. Vol 3: Polymer matrix composites materials usage, design, and analysis[M]. Detroit: SAE International, 2012: 2-33.
[8]	杨宇, 孙侠生, 杨胜春, 等. 含冲击损伤复合材料层压板压缩破坏机制试验研究[J]. 复合材料学报, 2012, 29(3):197-202.
	YANG Yu, SUN Xiasheng, YANG Shengchun, et al. Experimental study on compressive failure mechanism of low-velocity-impact-damaged composite laminates[J]. Acta Materiae Compositae Sinica, 2012, 29(3):197-202.
[9]	童谷生, 孙良新, 刘英卫, 等. 复合材料层压板低能量冲击后剩余抗压强度的工程估算[J]. 机械工程材料, 2004, 28(3):19-21.
	TONG Gusheng, SUN Liangxin, LIU Yingwei, et al. A simplified method of evaluating compression after impact strength for composite laminate[J]. Materials for Mechanical Engineering, 2004, 28(3):19-21.
[10]	燕瑛, 曾东. 复合材料层板低速冲击剩余强度的研究[J]. 航空学报, 2003, 24(2):137-139.
	YAN Ying, ZENG Dong. Study on the post-impact compressive strength of composite lamin ates[J]. Acta Aeronautica et Astronautica Sinica, 2003, 24(2):137-139.
[11]	PAPANICOLAOU G C, STAVROPOULOS C D. New approach for residual compressive strength prediction of impacted CFRP laminates[J]. Composites, 1995, 26(7):517-523. doi: 10.1016/0010-4361(95)96809-K URL
[12]	ATAS A, MOHAMED G F, SOUTIS C. Modelling delamination onset and growth in pin loaded composite laminates[J]. Composites Science and Technology, 2012, 72(10):1096-1101. doi: 10.1016/j.compscitech.2011.07.005 URL
[13]	张子龙, 程小全, 益小苏. 复合材料冲击损伤及冲击后压缩强度的等效实验方法[J]. 实验力学, 2001, 16(3):313-319.
	ZHANG Zilong, CHENG Xiaoquan, YI Xiaosu. An effective test method for characterization of impact damage and gaining the compression properties after impact of composite laminates[J]. Journal of Experimental Mechanics, 2001, 16(3):313-319.
[14]	DOST E F, ILCEWICZ L B, AVERY W B, et al. Effects of stacking sequence on impact damage resistance and residual strength for quasi-isotropic laminates[J]. Composite Materials: Fatigue and Fracture, 1991, 3:476-500.
[15]	DOST E F, ILCEWICZ L B, GOSSE J H. Sublaminate stability based modeling of impact damaged composite laminates[C]∥Proceeding of 3rd Technology Conference of American Society for Composites. Seattle, Washington, USA: Technomic Publishing Co., Ltd., 1988: 354-363.
[16]	崔海坡, 温卫东, 崔海涛. 层合复合材料板的低速冲击损伤及剩余压缩强度研究[J]. 机械科学与技术, 2006, 25(9):1013-1017.
	CUI Haipo, WEN Weidong, CUI Haitao. Research on low velocity impact damage and residual compressive strength of laminated composites[J]. Mechanical Science and Technology, 2006, 25(9):1013-1017.
[17]	HORTON R E, WHITEHEAD R S. Damage tolerance of composites[R]. Washington, USA: Federal Aviation Administration, 1998.
[18]	HOSUR M V, MURTHY C R L, RAMURTHY T S. Compression after impact testing of carbon fiber reinforced plastic laminates[J]. Journal of Composites Technology and Research, 1999, 21(2):51-64. doi: 10.1520/CTR10947J URL
[19]	黄骁, 王进, 韩涛, 等. 复合材料层板冲击后压缩强度经验预测公式[J]. 复合材料学报, 2018, 35(5):1158-1165.
	HUANG Xiao, WANG Jin, HAN Tao, et al. An empirical prediction formula for compressive strength of composite laminates after impact[J]. Acta Materiae Compositae Sinica, 2018, 35(5):1158-1165.
[20]	管清宇, 严文军, 吴光辉, 等. 碳纤维/环氧树脂复合材料层压板冲击凹坑的回弹特性[J]. 复合材料学报, 2020, 37(2):284-292.
	GUAN Qingyu, YAN Wenjun, WU Guanghui, et al. Impact dent relaxation characteristic of carbon fiber/epoxy resin composite laminate[J]. Acta Mater-iae Compositae Sinica, 2020, 37(2):284-292.
[21]	LEVENBERG K. A method for the solution of certain non-linear problems in least squares[J]. Quarterly of Applied Mathematics, 1944, 2(2):164-168. doi: 10.1090/qam/1944-02-02 URL
[22]	MARQUARDT D W. An algorithm for least-squares estimation of nonlinear parameters[J]. Journal of the Society for Industrial and Applied Mathematics, 1963, 11(2):431-441. doi: 10.1137/0111030 URL
[23]	CMH-17 Committee. Composite materials handbook. Vol 1: Polymer matrix composites guidelines for characterization of structural materials[M]. Detroit: SAE International, 2012: 8-38.
[24]	ASTM Committee. Standard test method for measuring the damage resistance of a fiber-reinforced polymer matrix composite to a drop-weight impact event: ASTM D7136/D7136M—15[S]. West Conshohocken, USA: ASTM International, 2015.
[25]	ASTM Committee. Standard test method for compressive residual strength properties of damaged polymer matrix composite plates: ASTM D7137/D7137M—12[S]. West Conshohocken, USA: ASTM International, 2012.
[26]	ASTM Committee. Standard test method for compressive residual strength properties of damaged polymer matrix composite plates: ASTM D6641/D6641M—14[S]. West Conshohocken, USA: ASTM International, 2014.
[27]	盛骤, 谢式千, 潘承毅. 概率论与数理统计[M]. 第4版. 北京: 高等教育出版社, 2008, 213-216.
	SHENG Zhou, XIE Shiqian, PAN Chengyi. Probability theory and mathematical statistics[M]. 4th ed. Beijing: Higher Education Press, 2008: 213-216.

材料类型	铺层代码	铺层顺序	铺层比例(0/±45/90)	名义厚度/mm	E/J
M21/IMA	L0	[45/0/-45/90]_2S	25/50/25	2.992	无损伤、20、30、 40、50
	L1	[45/0/-45/90]_3S	25/50/25	4.448
	L2	[45/0/-45/90/45/0/-45/0/90/0]_S	40/40/20	3.740
X850/IM+	L0	[45/0/-45/90]_2S	25/50/25	2.992