## 1 Introduction
[25] have collected and analyzed acoustic emission (AE) signals (the AE activity, amplitude, duration, and energy) in a compression experiment and found the crackling noise distribution of the avalanche criticality. [PERSON] et al. [26] and [PERSON] et al. [27] have investigated the crack development patterns of rocks under compression stress, including the fracture process zone (FPZ) and the relevant fracture parameters, by measuring the surface strain fields of rock materials. [PERSON] et al. [28] have investigated the strain feature of single-flaw rock samples with the particle imaging velocimetry (PIV) technique and found that the displacement direction lines significantly changed in the region with flaws. [PERSON] et al. [29] have explained the initiation mechanism of secondary cracks by analyzing the surface strain field, AE event, and distribution during the uniaxial compression tests of rock-like specimens with an internal open-type flaw. [PERSON] et al. [30] have observed the AE and charge signal of coal compression tests and found that the richness of the signal increases with the deterioration of coal. [PERSON] et al. [31] described the strain dispersion of complex fractured rock masses by the standard deviations of the strain field and quantitatively identified the crack initiation stress (CIS). [PERSON] et al. [32] have built an elastic-plastic rheological model to calculate the specific crack orientation resulting in different types of crack propagation based on the variability of deformation localization near pre-existing shear cracks. [PERSON] et al. [33] have built a micromechanics-based model, taking the interaction among sliding cracks into account, for analyzing the damage and deformation localization of brittle rock. Furthermore, the strain field of rock-like material has been statistically analyzed to describe the damage accumulation and some precursor signals of the final collapse. [PERSON] et al. [34] explored that the variation coefficients of the vertical linear strain and the maximum shear strain might be the failure precursors of the coal specimens under uniaxial compression. [PERSON] et al. [35] have found that the differential rate of the strain field-axial strain could be used to recognize the precursor before the failure of the fractured rock mass. Through analyzing the absolute standard deviation (ASD) and absolute variation coefficient (AVC) of the strain field obtained from laboratory tests and numerical simulation, [PERSON] et al. [36] found that the variation laws of the ASD and AVC of the shear strain field were highly correlated with the rock fracture evolution and obtained the precursory indices of full-field strain information. Some researchers developed a damage variable based on the standard deviation of the apparent equivalent strain fields to evaluate the damage degree of rock samples [20; 26; 37]. [PERSON] et al. [38; 39] have scanned the full-field view of axial and circumferential strains on the sandstone specimen surfaces by the distributed fiber optic strain sensing (DFOSS) technology and detected the initialization of microcrack nucleation. This paper has used two-dimensional (2D) discrete element models to simulate the whole damage process of the intact and fractured rock based on the compression tests of the rock-like material samples. The volumetric strain rate field was obtained by 200 measurement circles covered on each BPM. The statistical data of the strain rate fields, including the mean, variance, simple mean deviation, and correlation coefficient, were analyzed in relation to the characteristics of the damage evolution and failure precursor among the pre-peak stage of compression tests. Moreover, the results of the intact BPM were compared with the results of the BPM with 45\({}^{\circ}\) flaw to find the influence of pre-existing flaws on those characteristics. Based on the results of the 2D models, the evolution characteristics of localized deformation have been quantitatively analyzed.