# Determination of Nonlinear Creep Parameters for Hereditary Materials
## 5 Application of Proposed Algorithm and Methods
### Asphalt Concrete
At present, it has been accepted in road science and practice that mechanical properties of asphalt concrete depend on the value and duration of the applied load and temperature, and this dependence can be described in a sufficiently accurate way with the use of linear viscoelasticity theory [39, 40, 41, 42]. The essential nonlinearity of asphalt concrete strain determined above shows the unacceptability of linear viscoelasticity theory and necessitates the use of nonlinear viscoelasticity theory. Figure 8: Isochronous creep curves of the asphalt concrete. Figure 7: Creep curves of the asphalt concrete at various stresses. Graphs of experimental rheological parameter at various stresses, constructed according to the calculation results under formula (17) with the use of data of Table 4, are shown in Figure 9.  
It is clearly seen that experimental rheological parameters of the asphalt concrete at various stresses do not merge into one curve; most of them diverge considerably from each other, which also shows the essential nonlinearity of the asphalt concrete strain at the considered temperatures and stresses. Creep curves of the asphalt concrete at minimum stress, equal in our case to 0.041 MPa, have been approximated by Equation (5) with the use of [PERSON]'s kernel and the following values have been obtained for creep parameters: \(\alpha=0.3\), \(\epsilon_{0}=0.058\), \(\delta=0.0138\). Experimental and calculated creep curves of the asphalt concrete at the stress are represented in Figure 10 for comparison, which shows clearly that experimental creep curve of the asphalt concrete has been satisfactorily approximated by the equation of linear viscoelasticity with the use of [PERSON]'s kernel. Figure 10: Experimental and calculated creep curves of the asphalt concrete at stress 0.041 MPa. Figure 9: Graphs of experimental rheological parameter of the asphalt concrete at various stresses. Then the values of instantaneous strain \(\varepsilon_{0}^{m}\) are required to calculate under formula (20) at other stresses. For this purpose, we select one value of load duration, for example, 570 s. Under the following expression, we obtain the value of calculated rheological parameter:  
\[k_{m}(t) = 1+\ \frac{\delta}{1-\alpha}\ t^{(1-\alpha)}. \tag{32}\]  
According to the expression (32), we find that \(k_{m}\) (570) = 2.6745 for the selected load duration and at known values of creep parameters \(\alpha\) and \(\delta\). We have values of creep strain, equal to 0.1759%, 0.2349%, 0.3632%, and 0.5741%, respectively from Table 4 at \(t\) = 570 for stresses 0.074 MPa, 0.111 MPa, 0.148 MPa, and 0.183 MPa. According to formula (20) we find the following values of instantaneous strain of the asphalt concrete equal to 0.0658%, 0.0878%, 0.1358%, and 0.2147% at the abovementioned stresses. Now the calculated values of creep strain for the asphalt concrete at various stresses, except for minimum ones, can be determined under formula (21). It is found out that calculated values of creep strain in initial time moments at maximum stress equal to 0.183 MPa do not sufficiently converge with experimental ones. Therefore, we redetermine the value of instantaneous strain at this stress under formula (22), having included into the calculation the values of experimental creep strain and model rheological parameter at t = 90,150, 270, and 570 s. The new value of instantaneous strain at stress 0.183 MPa has been obtained, which is equal to 0.2037. Calculated and experimental creep curves of the asphalt concrete at all stresses in comparison are represented in Figure 11. As it is seen, the calculated curves converge with sufficient accuracy with experimental ones at all stresses.