1. Introduction

2. Experimental work process

2.1 Materials

2.2 In-Direct Tensile (IDT) mixture creep test

3. Relaxation modulus computation approach

4. Relaxation modulus computation results

5. Summary and conclusion

## 1. Introduction

The low temperature cracking is one of the serious distresses in asphalt pavement (Moon, 2010). Thermal stress initiates in restricted asphalt pavement layer when temperature drops below 0°C. Then transverse thermal cracking happens when increased internal thermal stress exceeds the endurable temperature limit in asphalt pavement layer (Marasteanu et al., 2009). The major issue of thermal cracking is: water and moisture can be infiltrated easily into asphalt pavement layer and finally rutting, fatigue cracking and pot hole issue can occur which can be a serious issue in highway management (Moon, 2010; Moon, 2012; Moon et al., 2014).

It is well known that low temperature creep testing is required to compute thermal stress of given asphalt materials (e.g. asphalt binder and asphalt mixture: Marasteanu et al., 2009; Moon, 2010; Moon; 2012; Moon et al., 2014). Because of this reason, various types of Simple Performance Test (SPT) were developed and introduced in many pavement agencies in U.S. with application of viscoelastic theory (AASHTO, 2012; Marasteanu et al., 2009; Moon, 2010; Moon, 2012; Moon et al., 2014; Cannone Falchetto et al., 2014). Bending Beam Rheometer (BBR) is used for measuring low temperature creep properties for asphalt binder based on current AASHTO specification (AASHTO, 2012). In case of asphalt mixture, In-Direct Tensile (IDT) test (AASHTO, 2003) is widely used for measuring low temperature creep properties. Generally, creep compliance: D(t) and corresponding creep stiffness: S(t) results are generated using Euler-Bernoulli beam theory (Findley, 1976; Ferry 1980) in creep testing. Secondly, relaxation modulus: E(t), is computed by applying several inter-conversion techniques then thermal stress: σ(T°), is computed by using various mathematical (and/or numerical) approaches with viscoelastic theory (Marasteanu et al., 2009; Moon, 2010; Moon, 2012; Moon et al., 2014). However, only few pavement research agencies can convert experimental creep compliance: D(t), into relaxation modulus: E(t), due to its complex computing approaches. The major objective of this paper is to introduce an inter-conversion approach using Power-law function with Laplace transformation theory (Park and Kim, 1999). By using this computation approach, it is expected that more pavement research and managing agencies can easily generate relaxation modulus of given asphalt materials (binder and mixture) from creep testing.

## 2. Experimental work process

2.1 Materials

Two different asphalt mixtures were prepared by means of Superpave Gyratory Compactor (SGC) with two different types of asphalt binder (AASHTO, 2010). Table 1 presents brief mixture design information of prepared asphalt mixtures and corresponding binders in this paper. All the asphalt mixtures were designed and prepared in Korea Expressway Corporation Pavement Research Division (KECPRD) based on the current asphalt pavement material specification in Korea (MOLIT, 2015). Then all the prepared materials were sent to ISBS (Germany) for further experimental works.

Table 1. Prepared asphalt materials

Mixture number | Mixture design information | Binder grade | Other materials |

1 |
- WC-1, NMAS=13 mm(Wearing Course, type-1) - Passing sieve (%)13 mm: 100%, 10 mm: 86%, 5 mm: 54%, 2.5 mm: 43%, 0.6 mm: 20%, 0.075 mm: 7% |
Performance-Grade PG 64-28 (Un modified binder) |
None (Aggregate grade 1) |

2 |
- WC-1, NMAS=13 mm - Passing sieve (%)13 mm: 100%, 10 mm: 84%, 5 mm: 56%, 2.5 mm: 43%, 0.6 mm: 18%, 0.075 mm: 7% |
Performance-Grade PG 58-34 (Unmodified binder) |
RAP (Reclaimed Asphalt Pavement), 25% |

2.2 In-Direct Tensile (IDT) mixture creep test

Low temperature creep properties (e.g. creep compliance and creep stiffness) of given asphalt mixtures were measured by means of In-Direct Tensile (IDT) testing equipment (Zhang et al. 1997; AASHTO, 2003) located in ISBS (Germany). In IDT test, cylinder shaped specimen contains dimension of 150 mm^{3} (diameter) 403 mm (thickness) is prepared from SGC (AASHTO, 2010). Secondly, total four Linear Variable Differential Transducers (LVDTs) are used to estimate the vertical and horizontal creep displacements for seconds with application of constant load equal to 7.0~8.5 kN. Fig. 1 and Table 2 show experimental work set up and schematic information of IDT test performed, respectively.

Table 2. Information of IDT (In-Direct Tensile) mixture creep test

Mixture Number | Testing temp. [°C] | Number of specimens | Applied load [kN] |

1 | -12, -24 | 3 (-6°C), 3 (-18 °C) | 7.0~8.5 |

2 | -6, -18 | 3 (-18°C), 3 (-24 °C) | 7.0~8.5 |

Two different temperature conditions were considered in this paper and at each temperature condition, three specimens were prepared. Therefore, total six specimens were used for IDT test per mixture set in this paper.

In IDT test, the creep compliance: D(t) is calculated as (Zhang et al., 1997):

$$\left\{\begin{array}{c}\mathrm D(t)=\frac1{S(t)}=\frac{H_m^{(t)}\cdot d\cdot t}{P\cdot GL}\cdot C_{compliance}\;\;\;\\C_{compliance}=0.6354\times\left(\frac xy\right)^{-1}-0.332\\\frac xy=\frac{\varepsilon_{x\;in\;500\;seconds}}{\varepsilon_{y\;in\;500\;seconds}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\end{array}\right.$$ | (1) |

Where D(*t*) = Creep compliance, 1/MPa;

*S*(*t*) = Creep stiffness, MPa (=1/D(t));

${H}_{m}^{\left(t\right)}$= Measured horizontal deflection at time t, mm;

*d*= Diameter of specimen (=150 mm, prepared from SGC);

*t*= Thickness of specimen (=40 mm, prepared from SGC);

*GL*= Gauge Length (=38 mm);

*P*= Constant applied load, kN(=7.0~8.5);

*m*_{x }= strain in horizontal direction, mm/mm;

*m*_{y} = strain in vertical direction, mm/mm;

In IDT creep test, the m-value can easily be computed as seen in Equation (2).

$$m(t)=\begin{vmatrix}\frac{d\log S(t)}{d\log(t)}\end{vmatrix}$$ | (2) |

## 3. Relaxation modulus computation approach

To compute thermal stress of given asphalt mixture, relaxation modulus: E(t), needs to be computed from experimentally measured (and/or computed) creep compliance: D(t) (and/or creep stiffness: S(t)) results by means of various types of inter-conversion approaches. Rather than applying complicated Hopkins and Hamming’s algorithm (1967), approximated simple inter-conversion approach named Power-law function transformation algorithm was applied (Park and Kim, 1999).

It is well known that D(t) and E(t) can be expressed well with Power-law function based on several experimental experiences as can be seen in Equation (3) (Ferry, 1980; Park and Kim, 1999; Moon et al., 2019):

$$D(t)\cong D_1\cdot t^n\;\;,\;\;E(t)\cong E_1\cdot\frac1{t^n}=E_1\cdot t^{-n}$$ | (3) |

Moreover, D(t) and E(t) are inter-related based on convolution integral as:

$$\begin{array}{l}\int\limits_0^t=E(t-\tau)\cdot D(t)d\tau=\int\limits_0^tE(t-\tau)\cdot\frac1{S(t)}d\tau=t\\\int\limits_0^t=E(t)\cdot D(t-\tau)d\tau=\int\limits_0^tE(t)\cdot\frac{\displaystyle1}{\displaystyle S(t-\tau)}d\tau=t\end{array}$$ | (4) |

In this paper, a relationship between D(t) and E(t) was set based on empirical experience as:

$$E(t)\cdot D(t)\cong1$$ | (5) |

With application of Laplace transformation, Equation (4) can be changed as:

$$L\left(\int\limits_0^tE(t-\tau)\cdot D(t)d\tau\right)=L(t)\rightarrow\overline E(S)\cdot\overline D(S)=\frac1{s^2}$$ | (6) |

By using Equations (1~6) and Gamma function with Euler’s reflective formula, Equations (4) to (5) can be finally re-expressed as (Park and Kim, 1999; Moon et al., 2019):

$$\begin{array}{l}\left\{\begin{array}{l}E(t)\cdot D(t)=\frac{\sin(n\cdot\pi)}{n\cdot\pi}\\n={\left|\frac{\mathrm{dlog}D(t)}{\log\tau}\right|}_{\tau=t}or\;n={\left|\frac{\mathrm{dlog}E(t)}{\log\tau}\right|}_{\tau=t}\end{array}\right.\\\end{array}$$ | (7) |

Finally, E(t) can be estimated directly from experimental D(t) by applying Equations (1) to (7) in IDT test.

## 4. Relaxation modulus computation results

Values of relaxation modulus: E(t), were computed based on Equations (1~7) with experimental IDT results. The results of numerical E(t) (i.e. computed modulus at constant strain condition) were compared with experimental S(t) (i.e. computed stiffness at constant loading condition) in this study. All the computed results are presented in Figs. 2 to 5.

Based on the computation results from Figs. 2 to 5, it can be said that in all cases lower and similar data generation trends were observed in E(t) computation process with application of Power-law transformation approach. Generally, testing and measuring Relaxation modulus: E(t), tends to be relatively complex compared to S(t) (and/or D(t)) measurement. Therefore, reliable inter-conversion technique is needed to compute E(t). Even though Power-law approach is approximated mathematical transformation not an exact analogical transformation approach, it can be said that this conversion approach can successfully be applied for computing (and/or estimating) E(t) of given asphalt mixture including RAP mixture. However, due to limited testing conditions performed in this paper more extensive experimental works and analyses are needed as future research.

## 5. Summary and conclusion

In this paper, feasibility of applying simple Power-law transformation for computing relaxation modulus of given asphalt mixture was evaluated. As an experimental work, IDT creep testing was considered. Two different asphalt mixtures including RAP addition was considered during material preparation process. Even though Power-law transformation approach is an approximated method (not exact analogical method), this method was proven to provide reliable relaxation modulus data prediction ability. More extensive experimental and mathematical works are needed to further verify findings in this paper.