Predicting the Effect of the Loading Rate on the Fracture Toughness of Hydraulic Asphalt Concrete Based on the Weibull Distribution
Abstract
:1. Introduction
2. Test Materials and Methods
2.1. Raw Materials and Mixing Ratios
2.2. Specimen Preparation and Test Program
2.3. Fracture Index Calculation
Stress Intensity Factor
3. Analysis of Test Results
3.1. Optimal Span
3.2. Rationalization Verification
4. Minimum Number of Specimens for Semicircular Bending Test KIC
4.1. Preliminary Analysis of KIC Data
4.2. The Minimum Number of KIC Specimens Is Determined
- When m = 7, and : unsatisfied.
- When m = 8, and : critical.
- When m = 9, and : satisfied.
5. Conclusions and Discussion
- The loading rate has a significant effect on the critical stress intensity factor of hydraulic asphalt concrete, and the KIC dispersion is large at each loading rate, with the dispersion of the data at a loading rate of 10 mm/min differing by a factor of nearly 10 in the variance value compared to 0.2 mm/min. Consideration of the average fracture toughness values obtained from a limited number (3–5) of repetitive tests does not necessarily provide a suitable index for indicating actual damage behavior.
- The two-parameter Weibull model fits the distribution of the fracture strength of hydraulic asphalt concrete well and with high accuracy, thus allowing a better evaluation of the actual fracture behavior of the material; by assuming a simple displacement factor, distribution probability curves can be predicted for any desired loading rate based on the parameters of the referenced Weibull model.
- On the basis of existing tests, the KIC of hydraulic asphalt concrete was found to obey a lognormal distribution using statistical methods, and the minimum number of specimens for the SCB test was estimated to be 9 when γ = 0.95 and under the condition that the relative deviation does not exceed 5%.
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Notation
KIC | critical stress intensity factor |
P | critical load |
r | radius |
t | thickness |
a | length of the cutout |
YI(0.8) | Type I fracture stress intensity factor |
Pf | probability of failure |
K0 | normalization factor |
LR | loading rate |
S.F | shift factor |
m | size of deviation |
mean value | |
s | standard deviation |
μ | expected value |
m | sample size |
γ | confidence level |
α | significance level |
δ | relative deviation |
δ* | corrected relative deviation |
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Test Item | Units | SL 501-2010 | Specimen Test Results |
---|---|---|---|
Needle penetration (25 °C,100 g, 5 s) | 0.1 mm | 60~80 | 66.5 |
Ductility (5 cm/min, 10 °C) | cm | ≥20 | 55 |
Ductility (5 cm/min, 15 °C) | cm | - | >150 |
Softening point (Globe method) | °C | ≥45 | 53.0 |
Flash point | °C | ≥260 | 320 |
Mass change after heating in a film oven | % | - | −0.06 |
Residual needle entry ratio (25 °C) | % | ≥61 | 83.3 |
Residual ductility (5 cm/min, 10 °C) | cm | ≥6 | 15.1 |
Raw Material | Grain Size/mm | Pitch | ||||||
---|---|---|---|---|---|---|---|---|
19–16 | 16–13.2 | 13.2–9.5 | 9.5–4.75 | 4.75–2.36 | 2.36–0.075 | <0.075 | ||
Ratio% | 6.6 | 6.8 | 10.5 | 18.2 | 15.9 | 30.0 | 12.0 | 7.5 |
Fracture Indicator | Loading Rate (mm/min) | KIC Mean Value (Mpa·m0.5) | KIC Maximum | KIC Minimum | KIC Spread | Median KIC | Variance Results |
---|---|---|---|---|---|---|---|
KIC | 0.2 | 0.1945 | 0.2578 | 0.1149 | 0.1429 | 0.1934 | 0.0010 |
1 | 0.3533 | 0.4309 | 0.2745 | 0.1564 | 0.3562 | 0.0017 | |
5 | 0.6929 | 0.8232 | 0.5442 | 0.2790 | 0.7038 | 0.0070 | |
10 | 0.8479 | 1.0271 | 0.6540 | 0.3731 | 0.8584 | 0.0102 |
Loading Rate | K0 | m | K Mean | R2 |
---|---|---|---|---|
(mm/min) | (MPa·m0.5) | (MPa·m0.5) | ||
0.2 | 0.205 | 9.004 | 0.1945 | 0.9613 |
1 | 0.371 | 9.324 | 0.3533 | 0.9847 |
5 | 0.731 | 8.509 | 0.6929 | 0.9754 |
10 | 0.892 | 8.886 | 0.8479 | 0.9870 |
Model Parameters | LR = 0.05 mm/min | LR = 3 mm/min | ||||
---|---|---|---|---|---|---|
Test Data | Predicted Values | Error | Test Data | Predicted Values | Error | |
Average KIC | 0.121 | 0.116 | 4.13% | 0.553 | 0.561 | 1.45% |
K0 | 0.129 | 0.123 | 4.65% | 0.574 | 0.589 | 2.61% |
i | KIC/MPa·m0.5 | lgKIC | pi | µi |
---|---|---|---|---|
1 | 0.6540 | −0.1844 | 0.952 | 1.6646 |
2 | 0.6637 | −0.1780 | 0.905 | 1.3106 |
3 | 0.7113 | −0.1480 | 0.857 | 1.0669 |
4 | 0.7271 | −0.1384 | 0.810 | 0.8779 |
5 | 0.7633 | −0.1173 | 0.762 | 0.7128 |
6 | 0.7897 | −0.1025 | 0.714 | 0.5651 |
7 | 0.8193 | −0.0865 | 0.667 | 0.4316 |
8 | 0.8371 | −0.0772 | 0.619 | 0.3029 |
9 | 0.8481 | −0.0715 | 0.571 | 0.1789 |
10 | 0.8539 | −0.0686 | 0.524 | 0.0602 |
11 | 0.8629 | −0.0640 | 0.476 | −0.0602 |
12 | 0.8716 | −0.0597 | 0.429 | −0.1789 |
13 | 0.8848 | −0.0531 | 0.381 | −0.3029 |
14 | 0.9056 | −0.0430 | 0.333 | −0.4316 |
15 | 0.9331 | −0.0301 | 0.286 | −0.5651 |
16 | 0.9350 | −0.0292 | 0.238 | −0.7128 |
17 | 0.9424 | −0.0257 | 0.190 | −0.8779 |
18 | 0.9551 | −0.0199 | 0.143 | −1.0669 |
19 | 0.9723 | −0.0122 | 0.095 | −1.3106 |
20 | 1.0271 | 0.0116 | 0.048 | −1.6646 |
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He, J.; Ding, X.; Yang, W.; Yang, H.; Liu, L. Predicting the Effect of the Loading Rate on the Fracture Toughness of Hydraulic Asphalt Concrete Based on the Weibull Distribution. Materials 2024, 17, 803. https://doi.org/10.3390/ma17040803
He J, Ding X, Yang W, Yang H, Liu L. Predicting the Effect of the Loading Rate on the Fracture Toughness of Hydraulic Asphalt Concrete Based on the Weibull Distribution. Materials. 2024; 17(4):803. https://doi.org/10.3390/ma17040803
Chicago/Turabian StyleHe, Jianxin, Xinyu Ding, Wu Yang, Haihua Yang, and Liang Liu. 2024. "Predicting the Effect of the Loading Rate on the Fracture Toughness of Hydraulic Asphalt Concrete Based on the Weibull Distribution" Materials 17, no. 4: 803. https://doi.org/10.3390/ma17040803