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.
where E is the relaxation modulus. Hence
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
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.