Simulation of buckling and delamination propagation of composite laminates with fiber bridging
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摘要: 纤维增强复合材料层合板由于层间力学性能弱，容易出现分层损伤。分层的扩展往往伴随着纤维桥联效应，纤维桥联能显著增大层合板尤其是多向层合板分层扩展的阻力。考虑纤维桥联效应的三线性内聚力模型能表征分层扩展实验中断裂韧性的“R曲线”特征，比传统的双线性模型能更为准确地描述复合材料的分层扩展行为。本论文基于三线性内聚力模型，对含圆形分层复合材料层合板的轴向压缩进行数值模拟，探讨纤维桥联效应对分层扩展及后屈曲行为的影响规律。研究结果发现，纤维桥联对层合板的屈曲载荷影响较小；混合屈曲模式下，三线性模型预测的上下子板相对法向位移明显低于双线性模型；相同分层深度下，三线性模型预测的层合板后屈曲更早转变为整体屈曲模式。随着分层深度的增加，层合板的屈曲模式由局部屈曲逐步过渡为混合屈曲和整体屈曲；当分层深度较浅时，I型分层扩展占主导；随着分层深度的增加，I型分层逐渐减弱，而II型和III型分层扩展则显著增强；当分层接近板中面时，I型分层停止扩展，以II型及III型分层为主。Abstract: Fiber reinforced composite laminates are susceptible to delamination damage due to relatively weak inter-laminar mechanical properties. The growth of delamination is always accompanied by fiber bridging, which can significantly increase the resistance of delamination propagation, especially in the case of multidirectional lami-nates. Compared with the traditional bilinear model, a trilinear cohesive zone model with fiber bridging effects can describe the “R curve” characteristic of the fracture toughness shown in the delamination test, and consequently better characterize the delamination propagation behavior of composite laminates. In order to evaluate the effects of fiber bridging on the behaviors of delamination growth and post-buckling in composite laminates, a trilinear cohesive zone model was built to investigate the compressive behavior of composite laminates with a circular delamination. The results demonstrate that, fiber bridging has little effect on the buckling load of the laminates. The relative deflection between the upper and lower sub-laminate predicted by the trilinear model is much smaller than that predicted by the bilinear model under mixed buckling mode. The buckling mode predicted by the trilinear model transits to global buckling earlier than that predicted by the bilinear model at identical delamination depth. The post-buckling modes change from local buckling mode to mix mode buckling and finally global buckling with the increase of delamination depth. For shallow delamination, mode I delamination is prominent. With the increase of delamination depth, the mode I delamination gradually disappears, while the mode II and III delamination propagation increases significantly. For delamination close to the mid-plane of the laminate, the delamination propagation is dominated by mode II and mode III, without mode I growth.
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图 1 内聚力单元
Figure 1. Cohesive element
δn—Relative displacement in the normal direction; δs, δt—Relative displacement in the tangential direction
图 2 双线性内聚力模型
Figure 2. Bilinear cohesive zone model
σi, δi (i = n, s, t)—Traction and relative displacement in the normal and tangential directions of the cohesive element; K— Stiffness of the cohesive element; K0—Initial stiffness of the cohesive element; d—Damage coefficient; σ0, δ0—Traction and relative displacement at the initial moment of element injury; δf—Relative displacement at element failure
图 3 传统三线性内聚力模型
Figure 3. Traditional trilinear cohesive zone model
σb, δb—Traction force and relative displacement at matrix failure; G1—Fracture toughness of the matrix; G2—Fracture toughness of bridged fibers; GC—Overall fracture toughness; m—Ratio of matrix fracture toughness to overall fracture toughness; n—Ratio of bridged fibers fracture toughness to overall fracture toughness
图 5 含圆形分层的层合板示意图：(a) 几何模型构成；(b)边界条件
Figure 5. Diagram of laminates with a circular delamination: (a) Composition of the geometric model; (b) Boundary conditions
D—Delamination depth; L, W—Length and width of finite element model; H—Thickness of finite element model; r—Radius of the initial delamination; RP-1, RP-2—Reference points for applying the load; U1, U2, U3—Displacement in three directions
图 6 不同分层深度下两种模型预测的T700/TDE85层合板载荷-位移曲线
Figure 6. Force-displacement curves predicted by two models with different delamination depths for T700/TDE85 laminates
图 7 分层深度 D/H = 0.1的T700/TDE85层合板分层扩展及屈曲：(a) 面外位移-载荷曲线；((b), (c)) 不同载荷下两种模型的屈曲模式；((d), (e)) 不同模型的分层扩展预测
Figure 7. Delamination growth and buckling for normalized delamination depth D/H = 0.1 of T700/TDE85 laminates: (a) Out-of-plane deflection vs. loading displacement; ((b), (c)) Buckling modes of two models under different loads; ((d), (e)) Delamination propagation predicted by different model
图 11 分层深度D/H = 0.5的T700/TDE85层合板分层扩展及屈曲：(a) 面外位移-载荷曲线；(b) 两种模型的屈曲模式；((c), (d)) 不同模型的分层扩展预测
Figure 11. Delamination growth and buckling for normalized delamination depth D/H = 0.5 of T700/TDE85 laminates: (a) Out-of-plane deflection vs. loading displacement; (b) Buckling mode of two models; ((c), (d)) Delamination propagation predicted by different model
图 8 分层深度D/H=0.25的T700/TDE85层合板分层扩展及屈曲：(a) 面外位移-载荷曲线；((b), (c)) 不同载荷下两种模型的屈曲模式；((d), (e)) 不同模型的分层扩展预测
Figure 8. Delamination growth and buckling for normalized delamination depth D/H=0.25 of T700/TDE85 laminates: (a) Out-of-plane deflection vs. loading displacement; ((b), (c)) Buckling modes of two models under different loads; ((d), (e)) Delamination propagation predicted by different model
图 9 分层深度D/H=0.3的T700/TDE85层合板分层扩展及屈曲：(a) 面外位移-载荷曲线；((b), (c), (d), (e)) 不同载荷下两种模型的屈曲模式；((f), (g)) 不同模型的分层扩展预测
Figure 9. Delamination growth and buckling for normalized delamination depth D/H=0.3 of T700/TDE85 laminates: (a) Out-of-plane deflection vs. loading displacement; ((b), (c), (d), (e)) Buckling modes of two models under different loads; ((f), (g)) Delamination propagation predicted by different model
图 10 分层深度D/H = 0.35的T700/TDE85层合板分层屈曲：(a) 面外位移-载荷曲线；(b) 两种模型的屈曲模式；((c), (d)) 不同模型的分层扩展预测
Figure 10. Delamination growth and buckling for normalized delamination depth D/H = 0.35 of T700/TDE85 laminates: (a) Out-of-plane deflection vs. loading displacement; (b) Buckling mode of two models; ((c), (d)) Delamination propagation predicted by different model
表 1 T700/TDE85单向板力学性能参数[35]
Table 1. Mechanical parameters of T700/TDE85 unidirectional laminates[35]
E11/MPa E22/MPa E33/MPa G12/MPa G13/MPa G23/MPa ν12 ν13 ν23 138000 10160 10160 5860 5860 4790 0.28 0.3 0.3 Notes: Eij—Young’s modulus; Gij—Shear modulus; vij—Poisson’s ratio, where subscripts i, j denote the principal material axes. 
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K11/(N·mm−3) K22/(N·mm−3) K33/(N·mm−3) Nmax/MPa Smax/MPa Tmax/MPa GIC/(N·mm−1) GIIC/(N·mm−1) GIIIC/(N·mm−1) α 106 106 106 5 20 20 0.276 0.8 0.8 2 Notes: Kij—Interface stiffness; Nmax, Smax, Tmax—Interfacial strength in longitudinal and transverse directions, respectively; GiC—Fracture toughness; α—Power exponent factor.
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表 3 三线性内聚力模型界面参数
Table 3. Interface parameters of the trilinear cohesive zone model
K11/(N·mm−3) K22/(N·mm−3) K33/(N·mm−3) Nmax/MPa Smax/MPa Tmax/MPa GIC/(N·mm−1) GIIC/(N·mm−1) GIIIC/(N·mm−1) α σb δb 106 106 106 5.25 21 21 0.359 1.04 1.04 2 0.2 0.11 Notes: σb—Maximum bridging stress; δb—Initial displacement of bridge fiber damage.
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