# From Complex Modulus E* to Creep Compliance D(t): Experimental and Modeling Study
## 1 Introduction
During the test, vertical and horizontal deformations are measured and the creep compliance represents the time-dependent horizontal deformation by the applied stress, which can be calculated by the given Equation (3):  
\[D(t)=\frac{\Delta X_{tm,t}\times D_{avg}\times b_{avg}}{P_{avg}\times GL}\times C _{compl} \tag{3}\]  
\[C_{compl}=0.6354\times\left(\frac{X}{Y}\right)^{-1}-0.332 \tag{4}\]  
where: \begin{tabular}{l l} D(t): & creep compliance at time t (1/kPa). \\ GL: & gauge length in meters (mm). \\ D\({}_{\text{avg}}\): & average diameter of all specimens (mm). \\ b\({}_{\text{avg}}\): & average thickness of all specimens (mm). \\ Pavg: & average creep load (kN). \\ \(\Delta\)X\({}_{\text{tm,t}}\): & trimmed mean of the normalized horizontal deformations of all specimen faces at time t (mm/mm). \\ C\({}_{\text{cmpl}}\): & correction factor. \\ \(\hat{\psi}\): & absolute value of the ratio of the normalized, trimmed mean of the horizontal deformations (i.e., \(\Delta\)X\({}_{\text{tm,t}}\)) to the normalized, trimmed mean of the vertical deformations (i.e, \(\Delta\)Y\({}_{\text{tm,t}}\)) at a time \\  & corresponding to \(\frac{1}{2}\) the total creep test time for all specimen faces. \\ \end{tabular}   
The National Cooperative Highway Research Program (NCHRP) in its report 530 presents a laboratory test program to evaluate and compare measurement of creep compliance using IDT configuration, uniaxial compression, and uniaxial tension tests [11; 12]. The objective was to review the test procedure described in AASHTO T322-07. The results show that the behavior is very anisotropic. The compliance measured with the IDT test was lower than the compliance determined in uniaxial compression, which in turn was lower than the uniaxial tension compliance. The NCHRP chose to validate the test procedure described in AASHTO T322-07 because the compliance measured in perpendicular plane in relation to the compaction is generally higher than that determined parallel to compaction. However, we note that thermal cracks at low temperatures are generally perpendicular to the direction of compaction of the pavement. Lowering road temperature creates contraction stress; the contraction is blocked in the longitudinal direction of the road which leads to transverse cracking [4; 13]. The IDT configuration allows determining parallel creep compliance contrary to the direct tensile and direct compression configurations. Additionally, we note that compaction procedure with gyratory compactor is not very representative of field compaction. The viscoelastic theory offers the possibility to get creep compliance results directly from complex modulus results. [PERSON] et al. conducted a study at binder scale for the prediction of BBR results (creep compliance test) from shear complex modulus results (G*) [14]. Complex modulus tests were performed by Metravib (tensile/compression configuration) then the results were converted to shear complex modulus using [PERSON]'s ratio (\(|\)E*\(|\) = 2 (1 + \(\
u\)) \(|\)G*\(|\)). incompressibility was supposed, so that \(\
u\) = 0.5 and the approximation \(|\)E*\(|\) = 3 \(|\)G*\(|\)was used [4; 14]. To change from the frequency domain to the time domain, [PERSON] et al. used four approximate equations namely [PERSON] equation (1958), [PERSON] equation (1982), Ninomiya and Ferry equation (1959) and Schwarzl and Struik equation (1968) [14; 15; 16]. Creep compliance results obtained from BBR data were compared with the converted results from the approximate equations (prediction). The predicted results describe the experimental data well. Any linear viscoelastic body can be modelled by the generalized Kelvin-Voigt model (GKV) (Figure 1a) [17]. This model is an assembly of simple rheological elements like springs and dashpots, the complex modulus and the creep compliance are given by Equations (5) and (6).