## 3 Data Analysis of Similarity Calculation Method for Infield and Outfield Environmental Tests
### Similarity Calculation Method of Infield and Outfield Tests  
The test data of the infield and outfield low-temperature steady-state response show high elastic similarity of the infield and outfield specimens.  
In order to make the derived curve keep the curve characteristics of the infield curve, the method of scaling translation, i.e., linear transformation, is selected to transform the curve when deriving the outfield-derived curve based on the infield curve. During the linear transformation, the new curve will not change the function relationship with the original curve or the function properties of the original curve.  
When similar transformation is made to the curve of the infield test,  
\[T_{1}=kT+dT^{\prime} \tag{10}\]  
where \(T\) is the temperature of the infield test, \(T_{1}\) is the temperature of outfield transformation from the infield test, \(k\) is the scaling ratio of the curve, and \(dT^{\prime}\) is the translation parameter.  
To obtain the derived outfield curve, the following procedures were conducted. First, select a low-temperature (\(-25\)\({}^{\circ}\)C) outfield data sampling point A(\(T_{a},L_{a}\)) and a high-temperature (\(-5\)\({}^{\circ}\)C) outfield data sampling point B(\(T_{b},L_{b}\)). Then, transform the infield fitting curve in the way of translation and scaling transformation to make the transformed curve pass through the sampling points. Suppose that the temperature dependence of the tensile value of the infield is  
\[L=F(T) \tag{11}\]  
For A and B,  
\[L_{a}=F\big{(}T_{a}^{\prime}\big{)} \tag{12}\]  
\[L_{b}=F\big{(}T_{b}^{\prime}\big{)} \tag{13}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fitting Type** & **A Springback** & **B Springback** & **A Elongation** & **B Elongation** \\ \hline Polynomial fitting & 0.947 & 0.9746 & 0.92481 & 0.93142 \\ \hline Exponential fitting & 0.9378 & 0.9803 & 0.91336 & 0.9297 \\ \hline Fourier fitting & 0.94579 & 0.97416 & 0.92499 & 0.93074 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Goodness of fit of infield and outfield similarity transformation \(R^{2}\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fitting Type** & **A Springback** & **B Springback** & **A Elongation** & **B Elongation** \\ \hline Polynomial fitting & 6.5365 & 1.273 & 3.927 & 0.57 \\ \hline Exponential fitting & 6.4853 & 1.3771 & 3.7818 & 0.4719 \\ \hline Fourier fitting & 6.5585 & 0.89 & 3.906 & 0.04976 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Fitting parameter of infield and outfield similarity transformation \(dT\).  
When \(L_{a}\) and \(L_{b}\) are known, \(T_{a}^{\prime}\) and \(T_{b}^{\prime}\) of the infield temperature in the corresponding range can be obtained, which are the infield temperature when the length of the test piece is \(L_{a}\) and \(L_{b}\) in the infield test, as shown in Figure 9.  
Substituting Equations (12) and (13) into Equation (10) yields  
\[T_{a}=kT_{a}^{\prime}+dT^{\prime} \tag{14}\]  
\[T_{b}=kT_{b}^{\prime}+dT^{\prime} \tag{15}\]  
The scaling ratio \(k\) and the translation parameter \(dT^{\prime}\) are obtained by simultaneous solution.  
By changing the temperature of the infield curve according to Equation (10), the derived curve of the low-temperature elastic properties of the rubber material in the outfield can be obtained, which leads to the similarity calculation method of the infield and outfield low-temperature test based on the translation and scaling transformation of the similar curve.