## 1 Introduction  
The most significant failure characteristic of concrete materials is the generation and expansion of cracks, accompanied by the deformation and localization of damaged areas [1]. Therefore, the crack evolution process, deformation development, and damage localization are key research hotspots. Various methods, such as scanning electron microscopy (SEM) [2], optical microscopy (OM) [3], digital image correlation (DIC) [4], and acoustic emission (AE) [5], have been used to investigate the generation and propagation of cracks in concrete. However, these traditional methods can only be used to analyze the surface failure, and it is very difficult to investigate and visualize the interior failure mode of concrete.  
In recent years, X-ray computed tomography (X-ray CT) has been widely used to analyze the failure of cement-based materials. X-ray CT is superior to traditional methods for observing crack development in materials. It has been used to not only observe the generation and propagation of internal cracks, but also to detect crack shape [6, 7]. With the development of CT and loading equipment, the research of in situ tests is gradually attracting scholars' attention. Real-time observation through in situ scanning is of great significance for the application of cement-based materials in practical engineering. The fracture behavior of concrete under uniaxial compressive loading has been analyzed by some scholars through X-ray CT, including the qualitative analysis of crack morphology and distribution and the quantitative analysis of crack volume and width [8; 9]. The variation in the CT image gray value [10] or building numerical models based on CT images has been used to analyze the damage evolution of concrete [11; 12]. However, due to the errors in the selection of average gray values or in building models, the accuracy of the characterization of internal damage to materials still needs to be improved.  
The deformation of material is a significant indication of damage during force application. It is difficult to measure strain with a strain gauge during an in situ loading test due to the limitations of the CT loading device. In addition, the traditional strain measurement method can only obtain the strain in a specific area, not the full-field strain in the material. Digital volume correlation (DVC) is an ideal tool to solve this problem. DVC is an extension of DIC, and their working principles are the same. DVC combined with X-ray CT can measure the internal displacement field, and the three-dimensional distribution of the strain can be obtained through a calculation [13; 14]. The unique advantages of this method have attracted the attention of many scholars. Some of them used X-ray CT combined with DVC to characterize the fracture evolution of concrete, and we compared the results with the finite element simulation (established based on CT images) [15; 16]. The results show that the pattern of fracture evolution obtained by the two methods is consistent. Studies on concrete materials mainly focus on the optimization of the DVC calculation accuracy or the description of strain cloud diagrams [17; 18; 19; 20]. The quantitative analysis of the strain and a further analysis of material internal damage based on the strain remain to be studied.  
In this paper, X-ray CT combined with the DVC method was used to characterize the localized damage of cement mortar under uniaxial compression. X-ray CT was used to obtain 2D and 3D crack distribution and the width of the cracks so as to analyze the crack evolution of the specimen under different loads. DVC was employed to obtain the strain field under different stress states and analyze the strain distribution inside the specimen. The gray value and damage index of CT images were used to characterize the damage degree of cement mortar. In order to analyze the effect of localization damage on specimen failure, the localization factor was used to characterize damage to the high-amplitude strain region in the 3D scale.