A novel approach to construct a thickness-independent bridging law for large scale bridging in type I delamination of composites
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摘要: 大范围纤维桥接是复合材料分层扩展中的一种重要的增韧机制，通常用R曲线和桥接牵引-分离定律表征。然而R曲线和桥接牵引-分离定律都与结构厚度有关，需用不同厚度试件分别进行实验测量。近期的研究指出：桥接现象的厚度相关性实质上是由于不同厚度结构弯曲刚度不同所造成的；将裂纹张开的转角引入，构建的桥接牵引-分离转角关系与厚度无关。但是，现有的构造方法需要试件预埋光纤测量应变，试验过程复杂。本文通过弹性约束梁模型解析推导双悬臂梁预制裂纹尖端张开位移和转角，并结合J-积分法构建桥接牵引-分离转角关系，通过实验验证该关系是厚度无关的。进一步基于这种厚度无关的桥接律，提出了一种根据单一厚度试验数据逆推其他任意厚度R曲线与桥接牵引-分离定律的方法。通过不同厚度的碳纤维/环氧树脂、芳纶纤维/环氧树脂复合材料层合板双悬臂梁试验，证明逆推出来的R曲线和桥接牵引-分离关系曲线与用试验直接测量的曲线吻合很好。本文方法基于解析模型，仅需测量载荷-位移曲线即可，避免了裂纹长度的测量以及不同厚度的多次试验，简化了试验流程，为复合材料结构性能表征提供了有力工具。
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关键词:
- 复合材料结构I型分层 /
- DCB试验 /
- 纤维桥接 /
- R曲线 /
- J-积分法
Abstract: Large-scale fiber bridging plays a crucial role in enhancing the toughness of composite structures during delamination, typically analyzed through the R-curve and the bridging traction-separation law. These metrics, however, are influenced by the specimen's thickness, necessitating separate experimental evaluations for each thickness variation. Recent research indicates that this thickness-dependency originates from the varying bending stiffness across structures of different thicknesses. By incorporating the crack opening angle, a thickness-independent traction-separation and angle relationship was developed. Existing methods for establishing this relationship require embedding optical fibers to gauge the internal strain, complicating the experimental procedure. This study introduced an elastic restraint beam model to get analytical solution of the double cantilever beam (DCB) test, based on which the pre-crack tip opening displacement and deflection angle were derived. These equations, combined with the J-integral method, formed a thickness-independent bridging traction-separation and angle relationship. Experimental validation confirmed the thickness-independence of the derived bridging law. Furthermore, an iterative method was proposed to inversely calculate the R-curves and bridging traction-separation laws for any thickness based on data from a single thickness DCB experiment. Experiments on carbon fiber/epoxy resin and aramid fiber/epoxy resin composite laminates of varying thicknesses demonstrated the accuracy of the inversely deduced curves compared to direct measurements. The present method is based on an analytical model and requires only the measurement of load-displacement curves, avoiding the need to measure crack length and conduct multiple experiments with different thicknesses. This simplifies the experimental process and provides a powerful tool for characterizing the performance of composite material structures.-
Key words:
- type I delamination of composite structures /
- DCB test /
- fiber bridging /
- R curve /
- J-integral method
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图 1 J-积分法的积分路径
Figure 1. Integral path of J-integral method
$ \delta _{\text{n}}^ * $—Opening displacement of pre-crack tip; $ \delta _{\text{t}}^ * $—Sliding displacement of pre-crack tip; $ {\varGamma _{{\text{loc}}}} $—Integral path locally round the bridging zone; $ {\varGamma _{{\text{tip}}}} $—Integral path of crack tip; $ \sigma \left( \delta \right) $—Relationship between bridging stress and crack opening displacement
图 2 双悬臂梁(DCB)试件的几何形状
Figure 2. Geometric shape of double cantilever beam (DCB) specimen
a0—Initial crack length; a—Crack length; b—Width of DCB specimen; 2h—Thickness of DCB specimen
图 3 弹性梁模型及变形示意图
Figure 3. Elastic beam model and its deformation
$ P $—Load of loading point; $ \varDelta $—Displacement of loading point; $ {k_{\text{e}}} $—Spring restraint stiffness; $ a $—Crack length; $ w $—Beam deflection; $ \xi $—Opening displacement of crack tip in elastic supported beam model; $ \overline \delta $—Opening displacement of crack tip in elastic restraint beam model; $ {K_{\text{e}}} $—Spring stiffness; $ {K_{\text{r}}} $—Torsion spring stiffness; $ \psi $—Corner of beam section
图 4 (a)实际裂纹拓展；(b)等效裂纹拓展
Figure 4. (a) Actual crack growth; (b) Equivalent crack growth
Δa—Extended crack displacement
图 5 不同厚度(2h)碳/环氧树脂复合材料DCB试验的桥接律：(a) $ \sigma {\text{-}} \delta _{\text{n}}^ * $曲线；(b)$ \sigma {\text{-}} \delta _{\text{n}}^ * \theta _{\text{n}}^ * $曲线
Figure 5. Bridging law for DCB test of carbon/epoxy composites with different thicknesses (2h): (a) $ \sigma {\text{-}} \delta _{\text{n}}^ * $ curves; (b) $ \sigma{\text{-}} \delta _{\text{n}}^ * \theta _{\text{n}}^ * $ curves
$\theta _{\text{n}}^ * $—Pre-crack opening corner
图 6 $ \sigma {\text{-}}\delta _{\text{n}}^ * \theta _{\text{n}}^ * $曲线构建方法流程图
Figure 6. Flow chart of $ \sigma {\text{-}}\delta _{\text{n}}^ * \theta _{\text{n}}^ * $ curve construction
图 7 逆推法流程图
Figure 7. Flow chart of inverse method
The tilde over the symbol represents the iterative estimate value.
图 8 碳/环氧树脂复合材料DCB试验的GI与裂纹拓展长度的关系对比
Figure 8. Comparison of the relationship between GI and extended crack displacement of carbon/epoxy composites in DCB test
图 9 碳/环氧树脂复合材料逆推方法结果与试验结果对比图：(a) I型断裂韧性与预制裂纹尖端张开位移的关系；(b)桥接应力与预制裂纹张开位移的关系；(c) I型断裂韧性与裂纹拓展长度的关系；(d)载荷-位移曲线
Figure 9. Comparison between the results of the inverse method and the experimental results of carbon/epoxy composites: (a) Relationship between mode I fracture toughness and pre-crack opening displacement; (b) Relationship between bridging stress and pre-crack opening displacement; (c) Relationship between mode I fracture toughness and extended crack displacement; (d) Load-displacement curves
图 11 芳纶/环氧树脂复合材料DCB试验的载荷-位移曲线
Figure 11. Load-displacement curves of aramid/epoxy composites in DCB tests
图 12 芳纶/环氧树脂复合材料DCB试验的裂纹拓展长度-加载点位移曲线
Figure 12. Extended crack displacement-loading point displacement curves of aramid/epoxy composites in DCB test
DIC—Digital image correlation
图 13 芳纶/环氧树脂复合材料DCB试验的I型断裂韧性R曲线
Figure 13. R curves of mode I fracture toughness of aramid/epoxy composites in DCB test
图 14 不同厚度芳纶/环氧树脂复合材料DCB试验的桥接律：(a) $ \sigma {\text{-}} \delta _{\text{n}}^ * $曲线；(b) $ \sigma {\text{-}} \delta _{\text{n}}^ * \theta _{\text{n}}^ * $曲线
Figure 14. Bridging law for DCB test of aramid/epoxy composites with different thicknesses: (a) $ \sigma {\text{-}} \delta _{\text{n}}^ * $ curves; (b) $ \sigma {\text{-}} \delta _{\text{n}}^ * \theta _{\text{n}}^ * $ curves
图 15 芳纶/环氧树脂复合材料逆推法结果与试验结果对比图：(a) I型断裂韧性与预制裂纹尖端张开位移的关系；(b)桥接应力与预制裂纹张开位移的关系；(c) I型断裂韧性与裂纹拓展长度的关系；(d)载荷-位移曲线
Figure 15. Comparison between the results of the inverse method and the experimental results of aramid/epoxy composites: (a) Relationship between mode I fracture toughness and pre-crack opening displacement; (b) Relationship between bridging stress and pre-crack opening displacement; (c) Relationship between mode I fracture toughness and extended crack displacement; (d) Load-displacement curves
表 1 碳/环氧树脂复合材料DCB试验$ {G_{\text{I}}} {\text{-}}\delta _{\text{n}}^ * $曲线的拟合参数
Table 1. Fitting parameters for $ {G_{\text{I}}} {\text{-}} \delta _{\text{n}}^ * $ curves of carbon/epoxy composites in DCB test
Thickness/mm $ {B_1} $ $ {B_2} $ $ {B_3} $ $ {B_4} $ 6 0.4529 2.392 1.364 0.3562 10 0.6238 4.389 1.782 0.5032 14 0.8462 9.091 2.532 0.6392 Notes: GI—Mode I fracture toughness; $ {B_1} $-$ {B_4} $—Fitting parameters. 
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表 2 碳/环氧树脂复合材料DCB试件的几何尺寸
Table 2. Geometric dimensions of carbon/epoxy composite DCB specimens
Test piece number Length/mm Width/mm Thickness/mm 01 340 10 6 02 340 10 10 03 340 10 14
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表 3 芳纶/环氧树脂复合材料参数
Table 3. Aramid/epoxy composite material parameters
$ {E_{\text{f}}} $/$ {\text{GPa}} $ $ {E_{22}} $/$ {\text{MPa}} $ $ {G_{12}} $/$ {\text{MPa}} $ 70.01 6042 5102 Notes：$ {E_{\text{f}}} $—Flexural modulu in direction 1; $ {E_{22}} $—Elastic modulu in direction 2; $ {G_{12}} $—Shear modulu in direction 12.
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表 4 芳纶/环氧树脂复合材料DCB试验$ {G_{\text{I}}} {\text{-}} \delta _{\text{n}}^ * $曲线的拟合参数
Table 4. Fitting parameters for $ {G_{\text{I}}} {\text{-}} \delta _{\text{n}}^ * $ curves of aramid/epoxy composites in DCB test
Thickness/mm $ {B_1} $ $ {B_2} $ $ {B_3} $ $ {B_4} $ 2 0.1501 0.2002 0.07974 0.0483 6 1.3230 0.1978 − 0.9344 0.1933
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