# Power-Law Analytical Form of Interconversion

The power-law analytic form of interconversion is used to predict the relaxation modulus from measured creep-compliance data and as input for characterizing asphalt mixtures in ABAQUS. It is evident that creep and stress relaxation phe­nomena are caused by the same linear viscoelastic properties. For linear visco­elastic material, this interconversion can be done by applying Laplace transform

e(s) = sD(s)a(s) (4)

where s is the Laplace transform variable, e and r are strain and stress, respectively

r(s) = sE(s)e(s)

TABLE 2—Sigmoidal parameters and the WLF constants.

 b1, MPa b2, MPa a1 a2 c1 c2 R-square 3.70 • 10-02 6.98 • 1003 2.097 -0.908 1520 11750 0.976

where E is the relaxation modulus. Hence

1

s2D(s)

According to Eq 6, the relaxation modulus can be calculated from the creep compliance. Hence, it is easier to express the creep compliance by the power – law function

D(t)=Atn (7)

where t is time and D(t) is creep compliance (1/GPa).

In this report, the experimental creep data are limited to the linear part of the sigmoidal function with maximum slope. Hence, instead of using a sigmoi­dal function, the data is approximated by a power function for easier intercon­version of the creep data to the relaxation modulus. The power-law equation fitted to the creep-compliance data for an arbitrary reference temperature of 5 °C is expressed by Eq 7 as shown in Fig. 5(a) with, A = 7.71 • 10~02 1/GPa and n = 6.88 • 10~01. Transforming Eq 7 into the Laplace domain and substituting into Eq 6 leads to the following Eq 8 after back-transforming the equation into the time domain. The result is shown in Fig. 5(b)

where A and n are constants and Г(п + 1) is the gamma function.