# From Complex Modulus E* to Creep Compliance D(t): Experimental and Modeling Study
## 1 Introduction  
The thermomechanical behavior of bituminous asphalt mixes is very complex. According to [PERSON] [1] four behaviors are observed depending on the strain amplitude (\(|\varepsilon|\)) and the number of cycles (N) of the applied loading. For small strain amplitudes (\(|\varepsilon|<10^{-4}\)) and a few hundred loading cycles, the behavior can be considered as linear viscoelastic (LVE). For tens of thousands of cycles, the phenomenon of fatigue manifests. Putting is created by the accumulation of irreversiblepermanent deformations caused by near-breaking amplitudes. The variation of the temperature can cause deformations of a few percent (\(|\epsilon|>>10^{-4}\)), the observed behavior is strongly nonlinear. The viscoelastic linear behavior of asphalt mixes is mainly characterized by two types of tests: quasi-static loading tests in the time domain (relaxation and creep) and cyclic loading tests in the frequency domain (complex modulus). Relaxation is the evolution of the stress in time for the application of a constant deformation, and creep is the evolution of the strain for a constant stress applied. As for the complex modulus, it is the expression of rigidity of viscoelastic materials under cyclic loading; it represents the ratio between the cyclic stress by the cyclic strain. Laplace-Carson transform is a mathematical tool allowing substituting variables, in particular to go from the time domain to the frequency domain. Because integral equations describing material LVE behavior in a time domain are difficult to apply, the Laplace-Carson transform is used to turn integral equations into algebraic, therefore simplifying calculations (Equation (1)) [1; 2]. \[g^{*}(p)=p\int_{0}^{+\infty}e^{-p\mathrm{f}}g(t)dt \tag{1}\]  
where g(t) is a time dependent function, g*(p) is function dependent on p, p is a complex variable, corresponding to time in the transform domain. If p = i\(\omega\) (\(\omega\) is the angular frequency) we have:  
\[E^{*}(\omega)\times D^{*}\left(\omega\right)=1 \tag{2}\]  
where E* is the complex modulus (relaxation in frequency domain) and D* is the complex compliance (creep compliance in frequency domain). Creep compliance (ratio between deformations in time by the maximum stress applied) is a very important input for low temperature or thermal cracking module in the Mechanistic-Empirical Pavement Design Guide (MEPDG) software [2; 3]. Too low creep may lead to cracking at low temperature and too high creep may lead to high potential of rutting at service temperatures [4]. This type of test (creep) is also a very important element for characterization of bitumen at low temperature; the PG High-Low temperature system is based on a creep test (bending beam rheometer, BBR) for the determination of the low temperature of bitumen [5; 6; 7; 8; 9]. The American Association of State Highway and Transportation Officials (AASHTO T322-07) describes a test protocol for the creep compliance [10]. The standard proposes to work in indirect tensile test (IDT) configuration on a cylindrical test specimen of 150 mm in diameter and 38 mm to 50 mm of thickness. The specimen compaction is performed by the gyratory compactor. The test is generally carried out at three different temperatures ranging from 10 \({}^{\circ}\)C to \(-30\)\({}^{\circ}\)C depending on the grade of the bitumen. A static load is applied along a diametral axis in order to have an indirect tensile stress. The creep compliance is a non-destructive test; the applied loading must always be in the linear viscoelastic domain. However, the load must be high enough to cause sufficient horizontal deformation so the noise in the data acquisition process is insignificant.