## 2 Methods and Materials
### Theoretical Background for Data Processing
#### 2.1.1 Continuum Mechanics and Digital Image Correlation Measurements  
The digital image correlation technique combines image recording by optical cameras and software tracking procedures for accurate measurements of changes occurring in images due to deformation. Full-field displacement and strain maps are obtained by comparing surface images acquired along the deformation path. The algorithm recognises unique surface features and allocates their coordinates to the image pixels. It then tracks the relative displacement of these features in a set of sequentially obtained images and compares it to the first reference image, which often represents the undeformed state. The tracking is based on the iterative optimisation of a deformation vector to achieve a high correlation between the pairs of images. Thus, a stochastic, high contrast pattern on the surface throughout the entire test is required for the DIC method to find unique correlations. This can be obtained by painting the surface of the specimen with a speckle pattern. However, in many cases, the natural surface morphology inherited from the specimen preparation, e.g., grinding, is sufficient for a successful analysis [59]. The DIC analysis returns a field of displacements \(\mathbf{u}\) expressed in coordinates corresponding to either reference \(\mathbf{X}\) or deformed configuration \(\mathbf{x}\). Displacements are used to find the deformation gradient \(\mathbf{F}\), which is a second-order tensor defined as  
\[\mathbf{F}=\frac{\partial\mathbf{x}}{\partial\mathbf{X}}=\mathbf{I}+\frac{ \partial\mathbf{u}}{\partial\mathbf{X}} \tag{1}\]  
containing information about both the stretch and rotation of the body. Here \(\mathbf{I}\) is the identity matrix, and \(\partial\) denotes the partial differential. \(\mathbf{F}\) is commonly used in calculations of virtually any type of strain defined in continuum mechanics.  
For elastic energy (or power) density calculations, stresses and strains must be energetically conjugate. For the Cauchy stress tensor \(\sigma\), its energetic conjugate is the deformation rate tensor \(\mathbf{D}\), which, in turn, is calculated from the velocity gradient \(\mathbf{L}\) as  
\[\mathbf{D}=\frac{1}{2}(\mathbf{L}\cdot\mathbf{L}^{T}) \tag{2}\]  
The velocity gradient is defined as the spatial derivative of the velocity \(\mathbf{v}\) with respect to coordinates in the deformed configuration \(\mathbf{x}\)  
\[\mathbf{L}=\frac{\partial\mathbf{v}}{\partial\mathbf{x}} \tag{3}\]  
Tracking the velocity of a given point in the deformed configuration must account for that the point moves with strain or time. To simplify calculations, the velocity gradient \(\mathbf{L}\) can be obtained from the deformation gradient \(\mathbf{F}\) expressed in the reference configuration according to Equation (4). The data given in the reference configuration make it easy to track a point of interest at large deformations and displacements.  
\[\mathbf{L}=\dot{\mathbf{F}}\cdot\mathbf{F}^{-1} \tag{4}\]  
When the Cauchy stress tensor and the deformation rate tensor are known, the mechanical power density can be calculated as  
\[p_{M}=\boldsymbol{\sigma}:\mathbf{D} \tag{5}\]