# Research of Method for Solving Relaxation Modulus Based on Three-Point Bending Creep Test

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## Abstract

**:**

## 1. Introduction

## 2. Interconversion of Creep Compliance and Relaxation Modulus

#### 2.1. Interconversion Equation

_{i}is the modulus of the i-th Maxwell element associated to the spring stiffness, n is the number of spring-dashpot elements accounted for and t is the time.

#### 2.2. Derivation of Relaxation Modulus

_{i}, τ

_{i}, etc., as:

_{i}(i = 1 to 6) is the elastic modulus in GMM, τ

_{i}(i = 1 to 6) is the relaxation time in GMM, ${\eta}_{1}^{\prime}$ and ${\eta}_{2}^{\prime}$ are the viscosity coefficients in the Burgers model, ${E}_{1}^{\prime}$ and ${E}_{2}^{\prime}$ are the elastic moduli in the Burgers model. Actually, this is a transcendental equation. Although the commands Solve and NSolve in Mathematica can be applied to solve exponential and trigonometric equations in a limited way, they are not designed to solve complicated transcendental equations. Fortunately, Equation (6) can be solved by Mathematica using Taylor’s formula.

#### 2.2.1. Taylor’s Formula

_{0}[30]. In other words, this type of function, with a simple form and widely applicable conditions, is very convenient to deal with some qualitative problems. Due to Equation (6), n-order derivative exists at 0, Taylor’s formula with a surplus item Peano is used to expand at 0, namely, the Maclaurin formula. In our study, there are 12 unknown parameters of the generalized Maxwell model, so the equation is expanded to 12 polynomials about t and a Peano remainder, and the required accuracy of the calculations can be acquired. We have:

_{i}(i = 1 to 6) is the elastic modulus in GMM, τ

_{i}(i = 1 to 6) is the relaxation time in GMM, ${\eta}_{1}^{\prime}$ and ${\eta}_{2}^{\prime}$ are the viscosity coefficients in the Burgers model, ${E}_{1}^{\prime}$ and ${E}_{2}^{\prime}$ are the elastic moduli in the Burgers model and the ellipsis “∙∙∙” represents a total of 11 items.

#### 2.2.2. Solutions of Transcendental Equation

_{0}, x

_{min}, x

_{max}}, Max Iterations → 200]. Here, the equation solved is lhs = = rhs, 200 iterations will be carried out in the interval [x

_{min}, x

_{max}], and the roots will be found near x

_{0}. In order to solve the unknowns of E

_{i}and τ

_{i}in Equation (7), the bending creep tests were carried out.

## 3. Creep Tests and Calculations of Relaxation Moduli

#### 3.1. Three-Point Bending Creep Tests

#### 3.1.1. Material Properties

#### 3.1.2. Sample Preparations

#### 3.1.3. Bending Creep Tests Procedure

_{0}is calculated as:

_{0}is the concentrated force applied at the middle of the specimen (in N), b is the width of the specimen section (in m), h is the height of the cross section of the specimen (in m) and L is the span of the specimen (in m).

#### 3.1.4. Test Results and Calculation of Creep Compliance

#### 3.2. Determination of Model Parameters

#### 3.3. Calculation of Relaxation Moduli

## 4. Verification of Calculated Results and Uniaxial Compression Tests

#### 4.1. Uniaxial Compression Relaxation Tests

#### 4.1.1. Determination of Constant Levels of Input Strains for Relaxation Tests

#### 4.1.2. Uniaxial Compression Relaxation Tests Procedure

#### 4.1.3. Determination Parameters of GMM Model

#### 4.1.4. Construction of Master Curves for Relaxation Modulus of Asphalt Mixture

_{1}and C

_{2}are material parameters and T

_{0}is the reference temperature, in this study, T

_{0}= 0 °C.

#### 4.2. Verification of Calculated Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 9.**Comparisons of relaxation test results and calculation results at 15 °C. (

**a**) 0.3 mm; (

**b**) 0.9 mm; (

**c**) 1.5 mm; (

**d**) 3.0 mm; (

**e**) 6.0 mm.

Model Parameters | 0 °C | 10 °C | 15 °C |
---|---|---|---|

${E}_{1}^{\prime}$ | 3771.931 | 1893.139 | 1105.924 |

${E}_{2}^{\prime}$ | 5996.615 | 1272.372 | 705.477 |

${\eta}_{1}^{\prime}$ | 2.4 × 10^{7} | 4.2 × 10^{6} | 2.0 × 10^{6} |

${\eta}_{2}^{\prime}$ | 5 × 10^{6} | 1.2 × 10^{6} | 4.6 × 10^{5} |

R^{2} | 0.998 | 0.999 | 0.999 |

Model Parameters | 15 °C | 0 °C |
---|---|---|

E_{1}/MPa | 2032 | 431 |

E_{2}/MPa | 2415.65 | 675.65 |

E_{3}/MPa | 3742.5 | 1452.5 |

E_{4}/MPa | 7464 | 431 |

E_{5}/MPa | 5645 | 3713 |

E_{6}/MPa | 2032 | 431 |

τ_{1}/s | 0.01 | 0.0015 |

τ_{2}/s | 1.54 | 0.00019 |

τ_{3}/s | 0.011 | 0.00012 |

τ_{4}/s | 0.00018 | 0.00089 |

τ_{5}/s | 0.0000007 | 0.0000003 |

τ_{6}/s | 0.0000005 | 0.0000006 |

Model Parameters | 0.3 mm | 0.9 mm | 1.5 mm | 3.0 mm | 6.0 mm |
---|---|---|---|---|---|

E_{1}/MPa | 1 | 1.3 | 1200 | 12.21 | 65.442 |

E_{2}/MPa | 1.6 | 1.33 | 25.56 | 1 | 114.98 |

E_{3}/MPa | 159 | 106.4 | 182.06 | 95.30 | 54.45 |

E_{4}/MPa | 341 | 181.3 | 94.33 | 567.09 | 31.396 |

E_{5}/MPa | 341.6 | 181.3 | 89.3 | 567309 | 31.396 |

E_{6}/MPa | 1 | 1.3 | 145.1 | 1 | 114.93 |

τ_{1}/s | 1 | 1 | 1 | 1 | 1 |

τ_{2}/s | 1 | 1 | 1 | 18.84 | 3.16 |

τ_{3}/s | 12.248 | 7.63 | 11.43 | 18.84 | 19.5 |

τ_{4}/s | 1 | 1 | 1 | 1 | 286.09 |

τ_{5}/s | 1 | 1 | 1 | 1 | 296.14 |

τ_{6}/s | 1 | 1 | 1 | 18.84 | 3.16 |

Temperature/°C | Temperature Fluctuation $\mathbf{\Delta}\mathit{T}/\xb0\mathbf{C}$ | Shift Factors lgα_{T} |
---|---|---|

15 | 15 | 1.6636 |

−5 | 5 | −0.5959 |

−15 | 15 | −1.8569 |

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**MDPI and ACS Style**

Sun, Y.; Gu, Z.; Wang, J.; Yuan, X.
Research of Method for Solving Relaxation Modulus Based on Three-Point Bending Creep Test. *Materials* **2019**, *12*, 2021.
https://doi.org/10.3390/ma12122021

**AMA Style**

Sun Y, Gu Z, Wang J, Yuan X.
Research of Method for Solving Relaxation Modulus Based on Three-Point Bending Creep Test. *Materials*. 2019; 12(12):2021.
https://doi.org/10.3390/ma12122021

**Chicago/Turabian Style**

Sun, Yazhen, Zhangyi Gu, Jinchang Wang, and Xuezhong Yuan.
2019. "Research of Method for Solving Relaxation Modulus Based on Three-Point Bending Creep Test" *Materials* 12, no. 12: 2021.
https://doi.org/10.3390/ma12122021