## 2 X-ray Micro-Tomography and Digital Volume Correlation
### Digital Image and Volume Correlation  
Digital Image Correlation (DIC) is a technique for measuring displacements on surfaces under different load conditions [33]. If grey scale conservation is assumed, the difference between taken pictures at different instants depends only on the displacement field:  
\[f(\bar{x})=g(\bar{x}+\bar{u}(\bar{x})) \tag{1}\]  
with \(f(\bar{x})\) and \(g(\bar{x})\), are respectively the reference and deformed states seen as scalars (grey levels) and \(\bar{u}\) is the displacement vector for each position \(\bar{x}\). Finding the best \(\bar{u}\) field is achieved by minimizing the optical flow equation (Equation (1)) over the region of interest.  
The digital volume correlation (DVC) approach corresponds to the extension of the two-dimensional DIC method in three dimensions [34]. In this study, DVC was carried out with YaDICs software, developed in Laboratoire de Mecanique de Lille [35]. The platform was based on C++, and optimized to process 3D volumes in a reduced time. To identify displacements, several parameters have to be defined: a metric, a sampling, an interpolator, a transformation, an optimizer and finally a regularization method (Figure 1). More details can be found in [35].  
Figure 1: Different steps of the digital volume correlation (DVC) process with YaDICs software [36].  
Displacements can be searched using a decomposition on a discrete basis:  
\[\bar{u}(\bar{x})=\sum a_{n}\bar{\psi}_{n}(\bar{x}) \tag{2}\]  
with \(a_{n}\), the sought degrees of freedom and \(\bar{\psi}_{n}\), the shape functions. In this study, finite elements method shape functions were used [37], providing a continuous displacements field on the whole studied volume. A concern is to present the same formalism for numerical simulations so that interpolation errors can be reduced for material parameters identification for example [38].  
A multi-scale resolution strategy was adopted for the DVC. This pyramid scheme reduced the problem size and thus avoided some local minimum traps. In the present case, six scales (or resolutions) are employed; the coarsest one was the'scale 5' with one'macro' voxel, which was averaged over \(2^{5}\times 2^{5}\times 2^{5}\) voxels while the full resolution image corresponds to the'scale 0' (Figure 2).  
In this study, DIC procedure was achieved with \(8\times 8\times 8\) pixels windows ensuring a good spatial resolution thanks to the good natural speckle pattern of the studied material (claystone) in X-ray micro-tomography. This size of correlation window thus helped to have a fine description of localization phenomena in the material while having a moderate uncertainty, i.e., less than 0.1 voxel. Finally, the YaDICs software enabled a regularization with a median filter to limit uncertainties. Considering a set of pixels including an aberrant value, the median filter sorted the values in ascending order to determine the median value. It allowed the outlier to be replaced by a consensus value for neighbouring values [39]. The median filter thus reduces measurement uncertainties in images correlation while maintaining discontinuities [40].  
The accuracy of DVC method was generally evaluated by using the uncertainty of displacement field, which was calculated in terms of the standard deviation of the displacement between two pictures. One was at the reference position and the other was translated by 20 microns [41; 42]. For this study, we calculated the standard deviation for the three components of the displacement vector for the element size varying from 4 to 64 voxels. It was found that the element size with 8 voxels provided a good compromise between the uncertainty and contrast of displacement field. For instance, the uncertainty was below 0.1 voxels.  
Figure 2: Pyramidal process employed in YaDICs.