# Research on Material Viscoelasticity and Its Influence on Indentation Rolling Resistance
## 5 Research on Calculation of Indentation Rolling Resistance Based on Relaxation Model Parameters
### Numerical Calculation of Indentation Rolling Resistance under Different Time Ranges  
Different time ranges can denote the testing time range corresponding to the relaxation modulus and the relaxation time span of the relaxation time spectrum. These distinct relaxation time ranges also correspond to different characteristic regions of the material, such as glass state, glass transition zone, equilibrium state, and viscous flow state. In this section, the numerical characteristics of indentation rolling resistance under different time range conditions are explored using the \"spectral line reduction method\". The parameters of the working conditions were an idler radius (\(R\)) = 0.08 m; a thickness of the conveyor belt cover layer of 0.008 m; and a vertical load of 2000 N/m. Simulation analysis was conducted by varying the magnitude of the speed. For the Lorentz-type relaxation spectrum model, considering \(\tau_{0}=10^{-8}\) and \(r=1\), the number of spectral lines was 61. The calculation of indentation rolling resistance across different time ranges was conducted to determine the sensitivity of viscoelastic materials to stress over time. Consequently, four testing time intervals were selected in the logarithmic coordinate system: \([-14,\,6]\), \([-14,\,-8]\), \([-8,\,-2]\), and \([-2,\,4]\). Using the \"spectral line reduction method\", the relaxation time range in these intervals were determined as follows: \([-16,\,7]\), \([-16,\,-4]\), \([-10,\,2]\), and \([-4,\,7]\). Subsequently, the iterative equation solving method outlined in the previous section was used to determine the indentation rolling resistance. In the interval \([-14,\,6]\), the ratio of the sum of relaxation spectral line intensities to the total relaxation spectral intensity was 0.9999. Therefore, the relaxation spectrum within the interval adequately represents the relaxation spectrum of the Lorentz-type material. The calculated indentation rolling resistance within this interval can be considered as the actual indentation rolling resistance. The calculated indentation rolling resistance value in other intervals can be compared with this established actual indentation rolling resistance value. The findings showed that the calculated indentation rolling resistance in the interval \([-14,\,6]\) aligns closely with the actual indentation rolling resistance (Figure 8). Conversely, the calculated indentation rolling resistance values within the intervals \([-14,\,-8]\) and \([-2,\,4]\) were smaller than the true values, with the values in the interval \([-2,\,4]\) exhibiting the smallest values. Notably, the calculated indentation rolling resistance values within the interval \([-8,\,-2]\) were closest to the actual indentation rolling resistance values. Figure 8: The indentation rolling resistance value of the Lorentz-type relaxation spectrum. Similarly, \(\mu=-11\) and \(\sigma^{2}=7.6\) were set as the parameters for the Gaussian-type relaxation spectrum model, and the number of spectral lines was set to 61. Four testing time intervals were selected in the logarithmic coordinate system: \([-10,\,2]\), \([-10,\,-6]\), \([-6,\,-2]\), and \([-2,\,2]\). The process of solving the indentation rolling resistance was similar to that used for the Lorentz-type. The indentation rolling resistance calculated on the interval \([-10,\,2]\) was considered as the actual value. The values obtained for the other intervals were compared with the set of actual indentation rolling resistance values. The following numerical patterns were obtained by evaluating the results presented in Figures 8 and 9:  
1. Within the interval \([-14,\,-8]\), the Lorentz-type material exhibited a glassy state with low viscosity. When the roller rolled on the surface of the material, the strain lagged behind the stress briefly, resulting in a lower indentation rolling resistance than the actual value. 2. As the Lorentz-type material transitioned from the \([-8,\,-2]\) interval, it moved between the glassy and the elastic platform zone.