## 2 Materials and Methods  
### Sensor Setup and Integration into Materials Testing Setup  
The aim was to design an optical strain measurement system replacing mechanical extensometers with a strain measurement rate above 1 kHz and with low latency. It should be compatible to standard material testing equipment like an Instron 8862 LCF testing machine and an Instron 8800 PID controller. The latter one requires an analogue 0-10 V elongation signal \(\Delta I(t)\) with 5 V representing zero elongation with base length \(\mathbf{l_{0}}\), i.e., the initial distance of the templates on the samples:  
\[\Delta l(t)=\left|\overset{\rightarrow}{r_{2}}(t)-\overset{\rightarrow}{r_{1 }}(t)\right|-l_{0}\text{ with }l_{0}=\left|\overset{\rightarrow}{r_{2}}(0)- \overset{\rightarrow}{r_{1}}(0)\right|. \tag{1}\]  
In this equation, \(\overset{\rightarrow}{r_{1}}(t)=\overset{\rightarrow}{r_{1}}(\mathbf{0})+ \overset{\rightarrow}{d_{1}}(t)\) and \(\overset{\rightarrow}{r_{2}}(t)=\overset{\rightarrow}{r_{2}}(\mathbf{0})+ \overset{\rightarrow}{d_{2}}(t)\) denote the xy-position vectors of the templates \(\mathbf{T_{1}}\) and \(\mathbf{T_{2}}\) at time \(t\) within the focus plane of the lens, i.e., on the specimen surface. These positions \(\overset{\rightarrow}{r_{k}}(t)\) are displaced from the initial positions \(\overset{\rightarrow}{r_{k}}(\mathbf{0})\) by the displacement vectors \(\overset{\rightarrow}{d_{k}}(t)\). So \(\Delta I(t)\) is positive (or above 5 V) when the sample is elongated and negative (or below 5 V) in the case of a compressed sample. This vector-based definition differs from the one-dimensional one of strain in DIN ISO 9513  
\[\varepsilon(t)=\Delta l(t)/l_{0}\text{ with }\Delta l(t)=\left|\overset{ \rightarrow}{r_{2}}(t)-\overset{\rightarrow}{r_{1}}(t)\right| \tag{2}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\varepsilon\) & **Mechanical Strain** & \(\mathbf{n}\) & **Correlation Size** \\ \hline \(\varepsilon_{o}\) & Strain amplitude & \(n_{T}\) & Template subset size \\ \(\dot{\varepsilon}\) & Strain rate & \(n_{S}\) & Search subset size \\ \(\sigma\) & Mechanical stress & \(p(u,v)\) & Polynomial fitting \\ \(\sigma_{row}(i)\) & Standard deviation of & & the correlation peak \\  & row \(i\) & \(r\) & Subset position in \\ \(I(i,j)\) & Image intensity & & focal plane \\ \(C(u,v)\) & Correlation amplitude & \(S\) & Search subset \\ \(d\) & Displacement & \(T\) & Template subset \\ \(l_{0}\) & Extensometer base & \(u\), \(v\) & Correlation image \\  & length & & coordinates \\ \(\Delta l\) & Elongation & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notification table. The main advantage of the elongation-based formulation \(\Delta l(t)\) is that it can be calibrated like a mechanical extensometer using a digital dial gauge (Sylvac S_Dial Work Nano 2.5 mm [27]). As shown in Figure 1 the sample surface is recorded by a high-resolution telecentric lens with a 1:1 magnification (VS-THVI-110/S) on a fast CameraLink camera (Basler ACE 2040-25 gm). The working distance is 110 mm and the resulting field-of-view \(11.2\times 1.4\) mm\({}^{2}\). The telecentric lens is required in 2D-DIC systems to avoid any changes in magnification caused by movements of the sample surface along the optical axis [28]. A blue LED illumination with 2.5 W electrical power is coupled coaxially into the lens enabling reproducible image acquisition on cylindrical surfaces with an exposure time below 0.5 milliseconds. For high temperature measurements, a filter is integrated into the telecentric lens to block the blackbody radiation from the hot specimen surface. The image data is transferred to a PC using a CameraLink frame grabber (National Instruments PCIe-1433) with a speed of 680 MB/s. In order to achieve a frame rate of 1.2 kHz, the original image size of 2040 \(\times\) 2040 pixel was reduced to 2040 \(\times\) 256 pixel.