Damage identification method for fiber-reinforced composite laminates based on element-level damage indicators
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摘要: 本文针对复合材料层合板结构提出了一种能够综合反映其承载能力缺失的单元级损伤指标−层合单元损伤指标，该指标既可以反映层合板面内外方向刚度的损伤情况，又具有参数数量较少、较易识别的优点。为保证所提损伤指标的合理性，本文利用数学和力学算子对单元级损伤指标与材料级损伤指标进行了等效，并比较了不同损伤指标在表征损伤程度之间的差异性。并提出了基于单元级损伤参数的复合材料结构损伤识别流程，即首先利用单元应变能差值指标对损伤单元进行筛选，然后利用优化方法对候选单元的损伤程度进行辨识。本文所提方法通过数值算例和一个试验进行了验证，分析了单元级损伤参数各个元素之间的相关性，并验证了基于单元级损伤参数的复合材料层合板结构损伤识别流程。本文的研究成果补充了现有复合材料结构健康监测理论。Abstract: An element-level damage index, damage index of laminated element, was proposed for composite laminate structures that can comprehensively reflect the degradation of load-bearing capacity, which can reflect the damage of stiffness in both internal and external directions and has the advantages of fewer parameters and easier identification. In order to ensure the rationality of the proposed damage index, mathematical and mechanical operators was used to equate the element-level damage index with the material-level damage index, and the differences between different damage indexes in characterizing the degree of damage were compared. The damage identification process of composite structures based on element-level damage parameters was also proposed, i.e., the damage elements were firstly screened by using modal strain energy change ratio, and then the damage degree of the candidate elements was identified by using optimization methods. The proposed method was validated by numerical examples and an experiment work, and the correlation between the elements of the element-level damage parameters was analyzed and the damage identification process of the composite laminate structure based on the element-level damage parameters was validated. The results of this paper complement the existing theory of health monitoring of composite structures.
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图 1 3类损伤指标所需损伤变量个数及特点
Figure 1. Number of damage variables required for the three types of damage indicators and their characteristics
w—Traditional damage variable; ΔAn—Normallized area; ΔAw—Damaged area; Mi—Vector containing material-level damage indicators;f(d1, d2, d3)i—Function of material-level damage indicators
图 2 材料级和单元级损伤指标等效过程示意图
Figure 2. Schematic diagram of equivalent process for material-level and element-level damage indicators
${\boldsymbol{\varPsi }} $—Stiffness matrix of symmetric laminates; A—Tensile stiffness matrix; D—Bending stiffness matrix; Q1—Stiffness matrix of the 1st layer; Q1 d—Stiffness matrix of the 1st layer of the damaged structure; Qk—Stiffness matrix of the kth layer; Qk d—Stiffness matrix of the kth layer of the damaged structure; $E_1^{dk}, E_{\rm{2}}^{dk}, v_{12}^{dk}, v_{{\rm{21}}}^{dk}, G_{{\rm{12}}}^{dk}$—Longitudinal, transverse elastic moduli, Poisson's ratio in both in-face directions, and shear modulus of the kth layer of the damaged structure; $E_1^{k}, E_{\rm{2}}^{k}, v_{12}^{k}, v_{{\rm{21}}}^{k}, G_{{\rm{12}}}^{k}$—Longitudinal, transverse elastic moduli, Poisson's ratio in both in-face directions, and shear modulus of the kth layer of the undamaged structure; $d_{11}^k, d_{22}^k, d_{{\rm{12}}}^k$—Material-level damage parameters in fiber direction, matrix direction and shear direction for the kth layer; ${\boldsymbol{\varPsi }}\left( \theta \right) $—Stiffness matrix expressed by element-level damage parameters; $\Delta {\boldsymbol{\varPsi }}_\theta ^{} $—Residual stiffness matrix expressed by element-level damage parameters; ${\boldsymbol{\varPsi }}\left( {d_{ij}^{}} \right) $—Stiffness matrix expressed by material-level damage parameters; $\Delta{\boldsymbol{ \varPsi}} _d^{} $—Residual stiffness matrix expressed by material-level damage parameters; $\theta _A^p, \theta _D^p$—In-plane and out-of-plane element-level damage indicators under the pth failure mode; $g_q^{} $—The qth kind of operator; $\left\| \cdot \right\|_2 $—The 2-norm of matrix; h—Thickness of laminates; ${z_k} $—Midplane coordinates of the kth layer
图 3 层合板面内载荷系数退化曲线
Figure 3. In-plane loading coefficient degradation curves of laminates
图 4 层合板拉弯组合工况系数退化曲线
Figure 4. Combined tension-bending condition coefficient degradation curves of laminates
图 7 损伤工况1损伤定位指标
Figure 7. Damage condition 1 damage positioning index
NMSECR—Normalized modal strain energy change ratio
图 9 识别参数平均值(MV)随噪声水平变化曲线
Figure 9. Curves of mean value (MV) of identification parameterswith noise level
A—In-plane damage; D—Out-plane damage
图 10 识别参数标准差(std)随噪声水平变化曲线
Figure 10. Curve of standard deviation (std) of identification parameters with noise level
图 12 模态测试设备示意图
Figure 12. Schematic diagram of the modal test equipment
DAE—Data acquisition equipment
图 15 未损伤和损伤悬臂梁的模态信息
Figure 15. Modal information of undamaged and damaged cantilever beams
图 19 计算等效刚度的工况
Figure 19. Working conditions for calculating equivalent stiffness
My—Moment of force coupling around the y axis
图 20 等效刚度计算结果和位移云图
Figure 20. Equivalent stiffness calculation results and displacement nephogram
Kxz u —Rotational stiffness around the y direction in the undamaged state; Kxz d—Rotational stiffness around the y direction in the damaged state
表 1 复合材料层合板材料属性
Table 1. Material parameters of composite laminate
E1/GPa E2/GPa G12/GPa ν12 Xt/GPa Xc/GPa Yt/GPa Yc/GPa S12/GPa 140 10 5 0.3 1500 1200 50 250 70 Notes: E1 and E2—Longitudinal and transverse elastic moduli; G12—In-plane shear modulus; v12—Poisson's ratio; Xt and Xc—Longitudinal tensile and compressive strengths; Yt and Yc—Transverse tensile and compressive strengths; S12—In-plane shear strength. 
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表 2 仿真工况
Table 2. Simulation conditions
Number Description of working conditions and load vectors [Nx; Ny; Nxy] [Mx; My; Mxy] 1 Tensile [100; 100; 0] [0; 0; 0] 2 Tensile and bending [100; 100; 0] [100; 100; 0] Notes: [Nx; Ny; Nxy]—Combined internal force per element width of laminate;[Mx; My; Mxy]—Combined internal moment per element width of laminate.
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表 3 层合板两种载荷的失效路径
Table 3. Failure paths of two loads of laminates
No. Failure paths 1 1MT, 3MT, 5MT, 7MT→2MT, 6MT→4MT→1FT,
2FT, 3FT, 4FT, 5FT, 6FT, 7FT2 6MT, 7MT→1FC, 7FT→5MT→1MC→3MT→2MC→4MT,
5FT, 6FT→3FC→1MT, 4FTNotes: MT and MC—Tensile and compression of matrix; FT and FC—Tensile and compression of fiber.
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表 4 损伤工况设置
Table 4. Damage condition settings
Damage condition Damage
elementExtent of damage θA θD 1 2 0.3 0.25 2 2 0.3 0.25 5 0.2 0.25
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表 5 无噪声工况识别结果
Table 5. Identification results without noise condition
Damage conditions Damaged element θA θD Identified value Error Identified value Error Single damage 2 0.3 0 0.25 0 24 0 — 0 — Multi-damage 2 0.3 0 0.25 0 5 0.2 0 0.25 0 23 0 — 0 —
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表 6 含噪声工况识别结果
Table 6. Identification results with noise conditions
Damage conditions Damaged element θA θD Identified value Error Identified value Error Single damage 2 0.2993 0.20% 0.2513 −0.52% 24 0 — 0 — Multi-damage 2 0.3002 0.07% 0.2491 −0.36% 5 0.1902 −4.90% 0.2401 −3.96% 23 0.0001 — 0 —
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表 7 悬臂梁修正后的固有频率
Table 7. Corrected modal frequency of cantilevered beam
Modal order Simulation values/Hz Test values/Hz Relative error/% 1 6.47 6.34 2.21 2 38.72 38.54 0.49 3 96.01 96.57 −0.53 4 188.76 190.91 −1.13 5 336.17 339.77 −1.06
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表 8 悬臂梁修正后的弹性参数
Table 8. Elastic parameters of cantilever beam after model updating
E11/GPa E22/GPa G12/GPa v12 ρ/(kg·m−3) 119.57 8.79 5.70 0.30 1794.67 Note: ρ—Density.
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