## 2 Methods and Materials
### Correction for Lens Distortion  
Like the majority of microscopic lenses, the lens used for FCG tests distorts the image. The Ncorr DIC algorithm has a rather simple way for accounting for the error in displacements caused by lens distortions, provided the lens distortions are known. To quantify and eliminate the distortions, the method proposed by [PERSON] et al. [117] was employed. The linear least-squares algorithm estimates the distortion coefficient from the distorted displacements obtained from the rigid body, in-plane translation DIC analysis. Ncorr utilises this coefficient to correct the distorted displacement fields. Corrected displacements are, however, still in distorted coordinates. This affects the strain calculations, as the Ncorr algorithm assumes an equal and constant spacing between points, which is not the case with non-negligible lens distortions. To express the displacements in undistorted coordinates, first the valid undistorted coordinates within the analysed region were found. The mask containing the valid pixel locations was undistorted, creating a new mask consisting of undistorted coordinates. To find the displacements in the undistorted coordinates, the equivalent distorted coordinates were calculated with the distortion model. Eventually, the displacement data could be interpolated at these locations, resulting in displacements in the equivalent undistorted coordinates. Results for raw and corrected displacements are exemplified and compared in Figure 9: owing to these implemented measures, the improvement of displacement measurements after lens distortion correction is remarkable, which, in turn, results in the enhanced accuracy of strain field estimates. Deformation Gradient and Strain Tensors  
Once the coordinates and displacements have been corrected, the strains can be seamlessly calculated using the deformation gradients calculated by Ncorr. As a result, the displacement vector \(\mathbf{u}\) is obtained in the reference and deformed configurations, \(\mathbf{X}\) and \(\mathbf{x}\), respectively, as well as the derivatives of the displacement vector with respect to coordinates in both configurations. Figure 9: Comparison of raw and corrected displacements. The global displacement field between the undeformed and cyclically deformed material is denoted as \(\mathbf{u}_{cu}\). Images obtained during cyclic test were processed one by one with respect to the \"cyclic reference\" image, which was acquired immediately after the cyclic images. The data for this sequential cyclic analysis will be denoted by the subscript index \(c\) (\(\mathbf{u}_{c}\), \(\mathbf{x}_{c}\), and \(\mathbf{X}_{c}\)), and data from the analysis with undeformed body will have the subscript index \(u\) (\(\mathbf{u}_{u}\), \(\mathbf{x}_{u}\), and \(\mathbf{X}_{u}\)). Since \(\mathbf{u}=\mathbf{x}-\mathbf{X}\), and \(\mathbf{X}_{c}=\mathbf{x}_{u}\), we can express \(\mathbf{u}_{u}\) as  
\[\mathbf{u}_{u}=\mathbf{X}_{c}-\mathbf{X}_{u} \tag{17}\]  
The goal is to find the deformation gradient as if the cyclic analysis would be performed with respect to the undeformed state, \(\frac{\partial\mathbf{u}_{cu}}{\partial\mathbf{X}_{u}}\). After differentiating Equation (17) with respect to \(\mathbf{X}_{c}\), and rearranging it, we obtain  
\[\frac{\partial\mathbf{X}_{c}}{\partial\mathbf{X}_{u}}=\left(\mathbf{I}-\frac {\partial\mathbf{u}_{u}}{\partial\mathbf{X}_{c}}\right)^{-1} \tag{18}\]  
Since \(\frac{\partial\mathbf{u}_{cu}}{\partial\mathbf{X}_{c}}=\frac{\partial \mathbf{u}_{c}}{\partial\mathbf{X}_{c}}+\frac{\partial\mathbf{u}_{u}}{ \partial\mathbf{X}_{c}}\), and both terms on the right hand side are known from the DIC analysis, it can be combined with Equation (18).