Probabilistic residual strength model for composite materials considering stress levels
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摘要: 针对当前大多数复合材料剩余强度模型通用化程度低、试验成本高的问题，本文提出了一个计及应力水平效应且独立于应力水平的剩余强度概率模型。首先，给出归一化强度储备的定义，并根据归一化强度储备推导出确定性剩余强度模型。然后，将一个疲劳寿命概率模型耦合进确定性剩余强度模型，进而衍生出一个新的剩余强度概率模型。最后，利用文献中的恒幅与变幅剩余强度试验数据对所提出的剩余强度概率模型的准确性和适用性进行验证。结果表明：几乎所有的恒幅试验数据点都分布在预测曲线的95%置信上限与5%置信下限之间，且50%可靠度的预测曲线对试验数据具有高拟合优度值：0.94、0.84及0.97。所提出的模型在充分考虑了复合材料剩余强度统计特征的前提下，仅用一组模型参数即可准确描述多个应力水平下的强度退化。在变幅工况下，所提出模型在升序与降序变幅加载中的预测值与试验值的相对误差均低于6%。Abstract: To address the problems of low generalization and high testing costs of most current residual strength models for composites, a probabilistic residual strength model that accounts for the effect of stress level and is independent of stress level was proposed. Firstly, the normalized strength reserve was defined and a deterministic residual strength model was derived based on the normalized strength reserve. Then, a fatigue life probability model was coupled into the deterministic residual strength model, and then a new residual strength probability model was derived. Finally, the accuracy and applicability of the proposed probabilistic residual strength model was verified using constant-amplitude and variable-amplitude residual strength experimental data from the open literatures. The results show that almost all the constant amplitude experimental data points are distributed between the upper 95% confidence limit and lower 5% confidence limit of the prediction curves, and the prediction curves with 50% reliability have high goodness-of-fit values for the experimental data: 0.94, 0.84 and 0.97. The proposed model accurately describes strength degradation at multiple stress levels using only one set of model parameters, with sufficient consideration of the statistical characteristics of the residual strength of the composite. The relative error between the predicted values of the proposed model and experimental values for both ascending and descending variable-amplitude loading is less than 6%.
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Key words:
- composite /
- strength reserve /
- residual strength /
- probability /
- fatigue life /
- constant and variable amplitude /
- stress level
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图 1 归一化强度储备的概念：(a) 剩余强度与强度储备；(b) 归一化剩余强度与归一化强度储备
Figure 1. Concept of normalized strength reserve: (a) Residual strength and strength reserve; (b) Normalized residual strength and normalized strength reserve
σ0—Tensile static strength; σc—Critical strength, numerically equal to the peak stress; N—Fatigue life
图 2 两个连续载荷块的加载示意图
Figure 2. Loading diagram of two continuous load blocks
σmax,i—Peak stress of the ith load block; σmax,i+1—Peak stress of the (i+1)th load block; ni—Cycle counts of the ith load block; ni+1—Cycle counts of the (i+1)th load block
图 3 变幅加载下的剩余强度退化路径：(a) 低-高 (L-H)；(b) 高-低 (H-L)
Figure 3. Residual strength degradation paths under variable amplitude loading: (a) Low-high (L-H); (b) High-low (H-L)
mi—Number of stress cycles in the first load block; mi+1,i—Equivalent number of cycles of the first load block mi in the second load block
图 6 模型参数拟合与VARTM E-玻璃/乙烯基酯的剩余强度预测曲线：(a) 归一化强度储备拟合曲线与试验数据[22]；(b) 置信区间为5%~95%的剩余强度预测区间与试验数据[22]；(c) 失效概率为50%的剩余强度预测曲线与试验数据[22]
Figure 6. Model parameter fitting and residual strength prediction curves for VARTM E-glass/vinyl ester: (a) Normalized strength reserve fitting curve vs. experimental data[22]; (b) Residual strength prediction band with confidence interval of 5%-95% vs. experimental data[22]; (c) Residual strength prediction curves with 50% failure probability vs. experimental data[22]
μ, ν—Model parameters for the normalized strength reserve model
图 7 二级应力水平变幅加载的T300/934石墨/环氧树脂剩余强度预测：(a) 高-低；(b) 低-高
Figure 7. Residual strength prediction of T300/934 graphite/epoxy under variable amplitude loading containing 2 stress levels: (a) H-L; (b) L-H
图 8 三级应力水平变幅加载的T300/934石墨/环氧树脂剩余强度预测：(a) 高-低；(b) 低-高
Figure 8. Residual strength prediction of T300/934 graphite/epoxy under variable amplitude loading containing 3 stress levels: (a) H-L; (b) L-H
图 9 22级应力水平加载的VARTM E-玻璃/乙烯基酯剩余强度预测曲线与试验数据[22]：(a) 50%失效概率的剩余强度预测；(b) 5%失效概率的剩余强度预测
Figure 9. Residual strength prediction curves of VARTM E-glass/vinyl ester for 22 stress level loading vs. experimental data[22]: (a) Residual strength prediction for 50% probability of failure; (b) Residual strength prediction for 5% probability of failure
表 1 铺层顺序为[0/+45/90/−45/0]s的VARTM E-玻璃/乙烯基酯的静拉伸强度[22]
Table 1. Static tensile strength of VARTM E-glass/vinyl ester with a lay-up sequence of [0/+45/90/−45/0]s[22]
Test number σ0/MPa Test number σ0/MPa 1 338 11 343 2 342 12 333 3 350 13 333 4 333 14 327 5 321 15 323 6 328 16 338 7 326 17 331 8 353 18 332 9 341 19 341 10 332 20 321 
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表 2 铺层顺序为[0/45/90/−452/90/45/0]2的T300/934石墨/环氧树脂的静拉伸强度[26]
Table 2. Static tensile strength of T300/934 graphite/epoxy with a lay-up sequence of [0/45/90/−452/90/45/0]2[26]
Test number σ0/MPa Test number σ0/MPa 1 427.49 14 481.96 2 444.04 15 481.96 3 445.42 16 486.79 4 445.42 17 486.79 5 449.55 18 491.61 6 455.07 19 492.30 7 455.07 20 495.06 8 466.10 21 496.44 9 468.86 22 500.58 10 477.82 23 503.33 11 477.82 24 511.61 12 477.80 25 519.88 13 480.58
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表 3 T300/934石墨/环氧树脂的非线性累积损伤与线性累积损伤的对比[7]
Table 3. Non-linear cumulative damage of T300/934 graphite/epoxy in comparison to linear cumulative damage[7]
σ1 σ2 σ3 n1 n2 n3 Ref. [7] Eq.(14) Eq.(15) H-L 345 290 — 20000 40000 — 0.76 0.63 0.45 L-H 290 345 — 40000 20000 — 0.67 0.63 0.50 H-L 400 345 290 4000 8000 20000 0.86 0.62 0.48 L-H 290 345 400 20000 8000 4000 0.70 0.62 0.46 Notes: σ1—Peak stress of the first loading; σ2—Peak stress of the second loading; σ3—Peak stress of the third loading; n1—Number of cycles for the first loading; n2—Number of cycles for the second loading; n3—Number of cycles for the third loading.
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表 4 VARTM E-玻璃/乙烯基酯的变幅测试载荷谱[22]
Table 4. Load spectra for variable amplitude testing of VARTM E-glass/vinyl ester[22]
Load cycle Stress/MPa Load cycle Stress/MPa 337535 74 394 157 182664 81 213 150 98852 88 115 164 53496 95 62 170 28950 102 34 177 15667 109 18 183 8478 116 10 190 4588 123 5 196 2483 130 3 203 1344 144 2 209 727 137 1 216
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表 5 两个算例(VARTM E-玻璃/乙烯基酯和T300/934石墨/环氧树脂)涉及的模型参数
Table 5. Model parameters involved in the two cases (VARTM E-glass/vinyl ester and T300/934 graphite/epoxy)
$ \alpha $ $\beta $ $\gamma $ $\delta $ $\mu $ $\nu $ Case 1 0.0544 0.1730 490 20 0.2 0.25 Case 2 0.1909 0.1932 340.6 20.99 0.1 0.13 Notes: α, β—Model parameters of the S-N curve; γ—Scale parameter of the Weibull distribution of static strength; δ—Shape parameter of the Weibull distribution of static strength.
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