基于可解释性机器学习算法的碳纤维增强复合材料剩余刚度预测方法

doi:10.7527/S1000-6893.2025.32249

Abstract

Abstract:

The conventional standard regression models often struggle to accurately predict the residual stiffness of Carbon Fiber Reinforced Polymers （CFRP）， while data-driven approaches typically lack interpretability. To address these challenges， we introduce a novel method that integrates Back Propagation Neural Network （BPNN） with Symbolic Regression （SR）. A stiffness degradation dataset is constructed using static and fatigue test data from T800. Key features， including stress level， normalized life， and strength， are selected through methods such as Pearson Correlation Coefficient （PCC）， Max-Relevance and Min-Redundancy （mRMR）， and SHAP analysis. SR is employed to uncover clear physical principles， while BPNN effectively captures complex relationships among multiple parameters. The results indicate that SR significantly outperforms traditional models in predicting the combined effects of stress level and normalized life. Additionally， BPNN demonstrates greater accuracy in predicting the interactions among stress level， normalized life and strength， maintaining low prediction errors across varying conditions. This integrated framework successfully merges physical interpretability with the capacity to model intricate relationships， offering a valuable tool for precise and transparent fatigue damage assessment in composite materials.

Key words: carbon fiber reinforced polymer, fatigue test, residual stiffness, symbolic regression, BP neural network

CLC Number:

V258

Yubin LU, Xiaohua NIE, Zhen WU. A residual stiffness prediction approach for carbon fiber reinforced composite materials based on interpretable machine learning algorithms[J]. Acta Aeronautica et Astronautica Sinica, 2025, 46(21): 532249.

Figures/Tables 22

Table 1

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Fig.6

Fig.7

Table 2

Fig.8

Table 3

Feature and parameter setting for SR

特征/参数	数值
输入	n/N， S， X
输出	E（n）/E₀
运算符	$+, -, ×, ÷,^, e, l n$
约束	$^: (1,5), e : 5, l n : 5$
嵌套约束	$l n : {l n : 0}, e : {e : 0,^: 0, l n : 0},^{e : 0, l n : 0}$
迭代次数	500
复杂度	20，30
损失函数	MSE
选择标准	Best

Table 3

Table 4

Equations obtained by SR for 20 maxsize

复杂度	MSE	SCORE	E（n）/E₀
1	0.026 5	1.594 0	0.910 9
4	0.020 2	0.898 5 0.212 7	$l n n N$
5	0.019 6	0.287 2	$0.235 9 n N 0.315 4$
6	0.019 4	0.092 1	$0.100 8 / e x p n N + 0.851 6$
7	0.017 0	0.131 7	$X n N + 1.005 - 0.133 3$
8	0.013 1	0.261 7	$0.220 6 S 1.064 n N$
9	0.012 0	0.085 2	$0.250 3 S 0.310 1 0.936 1 n N$
10	0.009 1	0.276 7	$0.223 8 l n S n N / 0.015 2 - 0.111 9 0.015 8$
12	0.008 9	0.008 4	$0.294 3 S 0.278 4 n N / 0.006 9 + 0.257 5 0.014 81$
13	0.008 6	0.041 1	$0.294 3 S - n N 0.278 1 n N / 0.006 9 + 0.212 3 0.014 06$
15	0.008 4	0.030 5	$0.294 3 S - n N 0.278 1 n N / 0.006 9 + 0.212 3 0.014 06$
19	0.008 3	0.001 2	$0.294 3 S - 2 n N 2 0.278 1 n N / 0.006 9 + 0.212 3 0.014 06$
20	0.007 9	0.076 5	$l n S 0.209 n N + 0.006 0 0.018 2$

Table 4

Table 5

Fig.9

Fig.10

Fig.11

Fig.12

Table 6

Equations obtained by SR for 20 to 30 Maxsize

复杂度	MSE	SCORE	方程
22	0.015 2	0.023 7	$y = X + S - 0.03 n N + 1.02 - 1.38 S$
25	0.013 3	0.017 4	$y = 0.62 + n N X + S - 0.03 - 1.36 S$
27	0.012 9	0.003 9	$y = n N X + S + 2.22 - 0.03 + 0.98 - 1.47 S$
30	0.015 1	0.002 2	$y = n N S - 0.03 + X + 1.01 - 1.38 S$

Table 6

Fig.13

Fig.14

Fig.15

Fig.16

References 25

[1]	KUMAR C H， SWAMY R P. Fatigue life prediction of glass fiber reinforced epoxy composites using artificial neural networks［J］. Composites Communications， 2021， 26： 100812.
[2]	马辉东，连喆，杨鑫源，等. 变频加载纤维增强复合材料剩余强度预测［J］. 航空学报， 2025， 46（8）： 231068.
	MA H D， LIAN Z， YANG X Y， et al. Residual strength prediction for fiber-reinforced composites under variable frequency loading［J］. Acta Aeronautica et Astronautica Sinica， 2025， 46（8）： 231068 （in Chinese）.
[3]	DEGRIECK AND J， VAN PAEPEGEM W. Fatigue damage modeling of fibre-reinforced composite materials： Review［J］. Applied Mechanics Reviews， 2001， 54（4）： 279-300.
[4]	HWANG W， HAN K S. Fatigue of composites： Fatigue modulus concept and life prediction［J］. Journal of Composite Materials， 1986， 20（2）： 154-165.
[5]	YANG J N， LEE L J， SHEU D Y. Modulus reduction and fatigue damage of matrix dominated composite laminates［J］. Composite Structures， 1992， 21（2）： 91-100.
[6]	WU F Q， YAO W X. A fatigue damage model of composite materials［J］. International Journal of Fatigue， 2010， 32（1）： 134-138.
[7]	SUZUKI T， MAHFUZ H， TAKANASHI M. A new stiffness degradation model for fatigue life prediction of GFRPs under random loading［J］. International Journal of Fatigue， 2019， 119： 220-228.
[8]	闫海，邓忠民. 基于深度学习的短纤维增强聚氨酯复合材料性能预测［J］. 复合材料学报， 2019， 36（6）： 1413-1420.
	YAN H， DENG Z M. Prediction of properties of short fiber reinforced urethane polymer composites based on deep learning［J］. Acta Materiae Compositae Sinica， 2019， 36（6）： 1413-1420 （in Chinese）.
[9]	DING X X， HOU X N， XIA M， et al. Predictions of macroscopic mechanical properties and microscopic cracks of unidirectional fibre-reinforced polymer composites using deep neural network （DNN）［J］. Composite Structures， 2022， 302： 116248.
[10]	SHIROLKAR N， PATWARDHAN P， RAHMAN A， et al. Investigating the efficacy of machine learning tools in modeling the continuous stabilization and carbonization process and predicting carbon fiber properties［J］. Carbon， 2021， 174： 605-616.
[11]	MENTGES N， DASHTBOZORG B， MIRKHALAF S M. A micromechanics-based artificial neural networks model for elastic properties of short fiber composites［J］. Composites Part B： Engineering， 2021， 213： 108736.
[12]	ZHOU J， WU Z， LIU Z L， et al. A novel normalized fatigue progressive damage model for complete stress levels based on artificial neural network［J］. International Journal of Fatigue， 2024， 187： 108447.
[13]	ZHENG T， GUO L C， WANG Z X， et al. A reliable progressive fatigue damage model for life prediction of composite laminates incorporating an adaptive cyclic jump algorithm［J］. Composites Science and Technology， 2022， 227： 109587.
[14]	ASTM. Standard test method for tensile properties of polymer matrix composite materials：［S］. West Conshohocken： ASTM， 2014.
[15]	ASTM. Standard test method for determining the compressive properties of polymer matrix composite laminates using a Combined Loading Compression （CLC） test fixture：［S］. West Conshohocken： ASTM， 2009.
[16]	ASTM. Standard test method for in-plane shear response of polymer matrix composite materials by tensile test of a ±45° laminate：［S］. West Conshohocken： ASTM， 2013.
[17]	ASTM. Standard test method for tension-tension fatigue of polymer matrix composite materials：［S］. West Conshohocken： ASTM， 2019.
[18]	LUNDBERG S M， LEE S I. A unified approach to interpreting model predictions［C］∥Proceedings of the 31st International Conference on Neural Information Processing Systems. New York： ACM， 2017： 4768-4777.
[19]	BI Y， XIANG D X， GE Z Y， et al. An interpretable prediction model for identifying N⁷-methylguanosine sites based on XGBoost and SHAP［J］. Molecular Therapy Nucleic Acids， 2020， 22： 362-372.
[20]	YANG C， CHEN M Y， YUAN Q. The application of XGBoost and SHAP to examining the factors in freight truck-related crashes： An exploratory analysis［J］. Accident Analysis & Prevention， 2021， 158： 106153.
[21]	UDRESCU S M， TEGMARK M. AI Feynman： A physics-inspired method for symbolic regression［J］. Science Advances， 2020， 6（16）： eaay2631.
[22]	马菲，张琼，赖培军，等. 基于BP神经网络的试飞训练安全性量化模型［J］. 航空学报， 2024， 45（5）： 529957.
	MA F， ZHANG Q， LAI P J， et al. BP neural network-based quantitative classification model for safety in experimental flight training［J］. Acta Aeronautica et Astronautica Sinica， 2024， 45（5）： 529957 （in Chinese）.
[23]	梁益铭，李广宁，徐敏. 基于机器学习的智能控制数值虚拟飞行方法［J］. 航空学报， 2023， 44（17）： 128098.
	LIANG Y M， LI G N， XU M. Method for numerical virtual flight with intelligent control based on machine learning［J］. Acta Aeronautica et Astronautica Sinica， 2023， 44（17）： 128098 （in Chinese）.
[24]	HAN H Y， LI W D， ANTONOV S， et al. Mapping the creep life of nickel-based SX superalloys in a large compositional space by a two-model linkage machine learning method［J］. Computational Materials Science， 2022， 205： 111229.
[25]	ZHANG Y D， ZHENG T， LIU G， et al. Predicting the fatigue life of T800 carbon fiber composite structural component based on fatigue experiments of unidirectional laminates［J］. International Journal of Fatigue， 2025， 190： 108622.

静力	疲劳	应力水平S/%	应力比R	数量
D0-T	D0-T-T	80，75，70，65	0.1	11
D0-C	D0-C-C	75，70，65	10.0	7
D45-T	D45-T-T	60，55，50，45	0.1	10

模型	MSE/10^-3	MAPE/%	R²
传统模型	5.048	1.78	0.531 8
SR模型（复杂度为20）	1.578	0.91	0.855 7

A residual stiffness prediction approach for carbon fiber reinforced composite materials based on interpretable machine learning algorithms

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参数	数值
输入	｛S，n/N，X，R｝
输出	E（n）/E₀
最大迭代次数	500
学习率	0.1
决策树的最大深度	5

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