## 3 Experiment and Results  
### Internal Strain Investigation of Red Sandstone Exposed to Compression  
A cylindrical specimen of red sandstone of 25 mm in diameter and 50 mm in length is scanned under different compression loadings in situ. The scan voltage is 120 kV, and the current is 200 \(\upmu\)A. The experimental process is divided into 8 steps as shown in the stress-strain curve in Figure 3a. In each step, the specimen is scanned when the force is stable. In step 1, we take two consecutive scans of the specimen in identical settings and without moving (except the tomographic rotation) or deforming the specimen and designate them as Scans 1 and 2, respectively. Those two scans are used to evaluate artifacts and image noise influences in Section 4.2. In step 7, the loading value decreases due to the development of micro cracks, but the specimen still has loading capacity till the loading reaches the peak value of 12.08 MPa. In step 8, the broken specimen is scanned. In Figure 3, three sectional images at \(y\) = 6.20 mm, 12.50 mm and 18.80 mm of the specimen in step 6 and step 8 are shown, respectively. From the gray images shown Figure 3b-d, it is difficult to ascertain the damage. With reference to Figure 3e-h, the coalescence of tensile cracks mainly results in the failure of the specimen. At the lower left and right corners of the middle sectional image shown in Figure 3f, there are also some cracks due to shear stress. We select the volumetric image of step 1 as the reference image, and volumetric images of other steps as the deformed images, respectively. The size of the volumetric image has 560 \(\times\) 560 \(\times\) 801 voxels and the dimension of a unit voxel is 45 \(\upmu\)m \(\times\) 45 \(\upmu\)m \(\times\) 45 \(\upmu\)m. By using the MS-DVSP algorithm, displacement fields are calculated. The final subset size is 32 \(\times\) 32 \(\times\) 32 voxels and the shift of the subset is 10 voxels. The calculated region of each section is the rectangular area marked with the  
Figure 2: Schematic of the X-ray microtomography system and the in situ loading setup. dashed lines in Figure 3b-d. The \(u\), \(v\) and \(w\) displacement fields of all sectional images of step 6 are plotted in Figure 4. As expected, the specimen mainly bore compression deformation along the \(z\) axis, and maximum displacements exist in the top region. Illustrated in Figure 4a,d,g, displacement fields of sectional images are divided into the negative and positive displacement regions, which make the specimen tensile along \(x\) axis. The main cracks shown in Figure 3e-h can be predicated by the zero displacement interface. This tensile deformation can also be investigated along \(y\) axis. In Figure 4b,e,h, the displacement of the sectional image at \(y\) = 6.20 mm is negative, while the displacement of most area of the sectional image at \(y\) = 18.80 mm is over zeros, especially in the lower region. These displacement distribution characteristics manifest that the expansion deformation occurs in circumference. Figure 4: _Cont._  
Figure 3: Stress–strain curve and reconstructed sectional images along \(y\) = 6.20 mm, 12.5 mm and 18.80 mm: (**a**) Stress–strain curve and three orthogonal sectional images of step 6; (**b–d**) sectional images of step 6, respectively; (**e–g**) sectional images of step 8, respectively. Strain estimation can be derived from computed displacement fields by using numerical computation methods. During this procedure, the noise in computed displacement fields will result in decreasing the reliability of strain estimation. It is necessary to smoothing the calculated displacement dataset before strain estimation [44]. Here, we first apply multivariate kernel smoothing regression [45] to smooth the displacement fields. After that, a point-wise least-squares (PLS) [46] approach is used to compute the internal strain fields. The size of the strain calculation cubic element is \(33\times 33\times 33\) voxels, and the distance between the center points of the neighbor element is 10 voxels.