# Characteristics of Moduli Decay for the Asphalt Mixture under Different Loading Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Materials Preparation and Test Method

#### 2.1. Materials

#### 2.2. Specimens Preparation

#### 2.3. Test Method

## 3. Test Result and Analysis

#### 3.1. The Establishment of the Decay Model for Moduli

_{f}is fatigue life, t is stress ratio, σ is stress level. K, and n are the material parameters of asphalt mixtures.

_{f}the specimen, and E

_{0}is the initial value of modulus.

_{f}is the fatigue life, N is the loading cycles, and α and γ are the material parameters related to the stress.

#### 3.2. The Initial Values of Moduli at Different Fatigue Stress Levels

_{0}was defined as the average one of the 10 moduli, which were near the cycle ratio N/N

_{f}= 0.01. Because the specimen of the indirect tensile test is in the state of two-dimensional stress state, the pattern of double moduli decay of the indirect tensile test was considered. The average initial moduli values of four parallel specimens under different stress state and different stress levels are summarized in Table 8:

#### 3.3. The Critical Value of Moduli at Different Fatigue Stress Levels

#### 3.4. Analysis of the Fitting Results of Fatigue Tests under Different Stress Levels

_{0}for tensile moduli, as measured by direct tensile test and indirect tensile tests, was fitted with the cycle ratio by Equation (6). The fitting results of tensile moduli from the direct tensile and the indirect tensile fatigue tests in different stress levels are shown in Figure 5 and Figure 6, respectively.

## 4. Summary and Conclusions

- (1)
- Under different loading conditions, the S-N fatigue curves for the direct tensile, indirect tensile and uniaxial compression are different. Therefore, it is difficult to compare the fatigue resistance of the same asphalt mixtures in the different stress conditions, according to the traditional method.
- (2)
- There are three stages in the moduli decay curves under different stress levels and conditions, which are migration, stabilization, and destruction. The decay patterns are nonlinear.
- (3)
- The decay patterns of the direct tensile moduli and the tensile moduli, as measured by the indirect tensile test, are similar. At the same time, the decay patterns of the uniaxial compression and the compression moduli, as measured by indirect tensile test, are similar. Nevertheless, the decay patterns of the tensile and the compressive moduli are obviously different.
- (4)
- There are significant differences in the tensile and compression characteristics of asphalt mixtures. For the flexural fatigue test, the decay rate of the tensile modulus at the bottom of the specimen is greater than that of the compression modulus at the top of the specimen under the same cyclic ratio condition. So, it exhibits tensile stress failure characteristics during the flexural fatigue test of the asphalt mixture.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Direct tensile test specimen; (

**b**) Uniaxial compression specimen; and, (

**c**) Indirect tensile specimen.

**Figure 3.**(

**a**) Load waveform of fatigue test; (

**b**) Direct tensile test; (

**c**) Uniaxial compression test; and, (

**d**) Indirect tensile test.

**Figure 4.**(

**a**) Fatigue curves characterized by stress ratio (

**b**) Fatigue curves characterized by stress level.

**Figure 7.**The decay pattern of tensile moduli measured by the indirect tensile test and direct tensile test.

Test Projects | Test Standard: JTG F40-2004 (China) [21] | |||
---|---|---|---|---|

Technical Requirements | Test Results | Test Methods | ||

Penetration (25 °C, 100 g, 5 s) (0.1 mm) | 30~60 | 53.9 | T 0604-2000 | |

Penetration index PI | ≥0 | 0.533 (r = 0.997) | T 0604-2000 | |

Ductility (5 cm/min, 5 °C) (cm) | ≥20 | 36.3 | T 0605-1993 | |

Softening point (Ring ball) (°C) | ≥60 | 72.5 | T 0606-2000 | |

Flash point (°C) | ≥230 | 265 | T 0611-1993 | |

Solubility (%) | ≥99 | 99.9 | T 0607-1993 | |

135 °C kinematic viscosity (Pa.s) | ≤3 | 2.37 | T 0620-2000 | |

Rolling Thin Film Oven Test (RTFOT) 163 °C 85 min | Mass loss (%) | ≤±1.0 | 0.23 | T 0609-1993 |

Residual penetration ratio (25 °C) (%) | ≥65 | 75.2 | T 0604-2000 | |

Residual ductility (5 °C) (cm) | ≥15 | 23.4 | T 0605-1993 |

Test Item | Technical Requirements [22] | Test Results | Test Methods |
---|---|---|---|

Crushed stone value (%) | ≤28 | 15.6 | T 0316-2005 |

Los Angeles weared stone value (%) | ≤30 | 19.7 | T 0317-2005 |

Apparent relative density (g/cm^{3}) | ≥2.5 | 2.6 | T 0321-2005 |

Water absorption (%) | ≤2.0 | 1.4 | T 0304-2005 |

Content of flat and elongated particles in coarse aggregate (%) | ≤15 | 8 | T 0312-2005 |

<0.075 mm particle content (Washing methods) (%) | ≤1 | 0.2 | T 0310-2000 |

Asphalt adhesion/grade | ≥4 | 6 | T 0616-1993 |

Firmness (%) | ≤12 | 5 | T 0314-2000 |

Content of soft stone (%) | ≤5 | 2.5 | T 0320-2000 |

Asphalt Aggregate Ratio (%) | Bulk Specific Gravity (g·cm^{−3}) | Volume of Air Voids VV (%) | Voids Filled with Asphalt VFA (%) | Voids in Mineral Aggregate VMA (%) | Marshall Stability (kN) | Flow Value (0.1 mm) |
---|---|---|---|---|---|---|

5.2% | 2.44 | 4.5 | 67.2 | 16.1 | 12.7 | 27.9 |

Specification Requirement [23] | / | 3~6 | 55~70 | >12.5 | >8 | 15~40 |

Factor Type | Factor Name | Factor Level Number | Levels of Factor |
---|---|---|---|

External factors | Stress ratio | 1 | 0.4 |

Test temperature (°C) | 1 | 15 | |

Load frequency(Hz) | 1 | 10 | |

Internal factors | Fatigue test method | 3 | Direct tensile test, Indirect tensile test, Uniaxial compression test. |

Asphalt types | 1 | 70 | |

Asphalt content (%) | 1 | 5.2% | |

Mixing gradation (AC) | 1 | AC-13 | |

Volume of air voids (%) | 1 | 4.5% |

Stress Level (MPa) | Maximum Load Value (KN) | Stress Ratio | Average Fatigue Life (Times) | CV% |
---|---|---|---|---|

0.25 | 0.625 | 0.223 | 398,879 | 0.176 |

0.5 | 1.25 | 0.446 | 41,129 | 0.203 |

1 | 2.5 | 0.892 | 3686 | 0.188 |

1.5 | 3.75 | 1.338 | 993 | 0.176 |

Stress Level (MPa) | Maximum Load Value (KN) | Stress Ratio | Average Fatigue Life (Times) | CV% |
---|---|---|---|---|

2 | 15.7 | 0.178 | 797,470 | 0.098 |

2.5 | 19.625 | 0.223 | 198,977 | 0.074 |

3 | 23.55 | 0.268 | 91,067 | 0.086 |

3.5 | 27.475 | 0.312 | 26,734 | 0.105 |

Stress Level (MPa) | Maximum Load Value (KN) | Stress Ratio | Average Fatigue Life (Times) | CV% |
---|---|---|---|---|

0.25 | 2.386 | 0.11 | 1,283,789 | 0.021 |

0.5 | 4.772 | 0.22 | 50,062 | 0.166 |

1 | 9.544 | 0.439 | 2236 | 0.079 |

1.5 | 14.315 | 0.659 | 435 | 0.167 |

Stress Level/MPa | 0.25 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | |
---|---|---|---|---|---|---|---|---|---|

Tensile moduli | E_{0} (MPa) | 1384 | 1686 | 2121 | 2407 | / | / | / | / |

CV (%) | 0.054 | 0.055 | 0.058 | 0.045 | / | / | / | / | |

Compression moduli | E_{0} (MPa) | / | / | / | / | 8852 | 10,102 | 11,167 | 11,943 |

CV (%) | / | / | / | / | 0.021 | 0.024 | 0.023 | 0.055 | |

Indirect tensile moduli | E_{0} (MPa) | 3560 | 4027 | 4585 | 5057 | / | / | / | / |

CV (%) | 0.011 | 0.019 | 0.014 | 0.019 | / | / | / | / | |

Compression moduli measured by indirect tensile test | E_{0} (MPa) | 7926 | 8835 | 9975 | 10,760 | / | / | / | / |

CV (%) | 0.015 | 0.018 | 0.018 | 0.014 | / | / | / | / | |

Tensile moduli measured by indirect tensile test | E_{0} (MPa) | 2547 | 2935 | 3464 | 3858 | / | / | / | / |

CV (%) | 0.024 | 0.021 | 0.027 | 0.012 | / | / | / | / |

Stress Level (MPa) | 0.25 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | |
---|---|---|---|---|---|---|---|---|---|

Tensile moduli | E_{min} (MPa) | 609 | 716 | 869 | 961 | / | / | / | / |

CV (%) | 0.049 | 0.057 | 0.052 | 0.051 | / | / | / | / | |

Compression moduli | E_{min} (MPa) | / | / | / | / | 7480 | 8086 | 8528 | 8787 |

CV (%) | / | / | / | / | 0.021 | 0.022 | 0.009 | 0.010 | |

Indirect tensile moduli | E_{min} (MPa) | 895 | 958 | 1040 | 1106 | / | / | / | / |

CV (%) | 0.021 | 0.019 | 0.053 | 0.042 | / | / | / | / | |

Compression moduli measured by indirect tensile test | E_{min} (MPa) | 5073 | 5593 | 6184 | 6563 | / | / | / | / |

CV (%) | 0.027 | 0.025 | 0.018 | 0.018 | / | / | / | / | |

Tensile moduli measured by indirect tensile test | E_{min} (MPa) | 560 | 616 | 693 | 733 | / | / | / | / |

CV (%) | 0.036 | 0.024 | 0.036 | 0.021 | / | / | / | / |

Stress Level (MPa) | Parameters | 0.25 | 0.5 | 1 | 1.5 |
---|---|---|---|---|---|

Tensile moduli | m | 0.222 | 0.265 | 0.337 | 0.413 |

n | 0.229 | 0.235 | 0.248 | 0.2636 | |

Tensile moduli measured by indirect tensile test | m | 0.638 | 0.743 | 0.964 | 1.255 |

n | 0.503 | 0.586 | 0.732 | 0.844 |

**Table 11.**Inflection points and slopes of decay curves of tensile moduli under different test conditions.

Test Type | Inflection Point | Tangency Point | Tangent Slope |

Direct tensile | 0.381 | (0.381, 0.727) | −0.416 |

Tensile moduli measured by indirect tensile test | 0.114 | (0.114, 0.908) | −0.791 |

Stress Level (MPa) | Parameters | 0.25 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 |
---|---|---|---|---|---|---|---|---|---|

Compression moduli | m | / | / | / | / | 0.087 | 0.106 | 0.144 | 0.319 |

n | / | / | / | / | 0.068 | 0.084 | 0.087 | 0.097 | |

Compression moduli measured by indirect tensile test | m | 1.108 | 1.246 | 1.472 | 1.938 | / | / | / | / |

n | 0.191 | 0.203 | 0.215 | 0.223 | / | / | / | / |

**Table 13.**Fitting parameters of fatigue moduli decay curve in different tests method under 1 MPa stress level.

Test Type | m | n |
---|---|---|

Compression moduli | 0.086 | 0.053 |

Compression moduli measured by indirect tensile test | 1.472 | 0.215 |

**Table 14.**Inflection points and slopes of decay curves of compression moduli under different test conditions.

Test Type | Inflection Point | Tangency Point | Tangent Slope |
---|---|---|---|

Uniaxial compression | 0.370 | (0.370, 0.877) | −0.120 |

Compression moduli measured by indirect tensile test | - | (0.370, 0.945) | −0.243 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lv, S.; Fan, X.; Xia, C.; Zheng, J.; Chen, D.; You, L.
Characteristics of Moduli Decay for the Asphalt Mixture under Different Loading Conditions. *Appl. Sci.* **2018**, *8*, 840.
https://doi.org/10.3390/app8050840

**AMA Style**

Lv S, Fan X, Xia C, Zheng J, Chen D, You L.
Characteristics of Moduli Decay for the Asphalt Mixture under Different Loading Conditions. *Applied Sciences*. 2018; 8(5):840.
https://doi.org/10.3390/app8050840

**Chicago/Turabian Style**

Lv, Songtao, Xiyan Fan, Chengdong Xia, Jianlong Zheng, Dong Chen, and Lingyun You.
2018. "Characteristics of Moduli Decay for the Asphalt Mixture under Different Loading Conditions" *Applied Sciences* 8, no. 5: 840.
https://doi.org/10.3390/app8050840