# A Macro-Micro Damage Model for Rock under Compression Loading
## 6 Conclusions  
In this work, a new calculation model for the mode-I stress intensity factor \(K_{\text{I}}\) at wing crack tip was proposed and a corresponding damage constitutive model was also  
Figure 14: Effects of fracture toughness \(K_{\text{IC}}\) on stress-strain relationship.  
Figure 15: Effect of initial damage \(D_{0}\) on stress-strain relationship.  
established. Based on theoretical and experimental studies, the following conclusions can be drawn:  
1. Through studying the cracking and propagation laws of compressive-shear rock cracks under remote stress field, a new calculation model for the mode-I stress intensity factor \(K_{\rm I}\) at wing crack tip was proposed considering the interaction of wing cracks and the additional stress caused by rock bridge damage. The results showed that compared with the classical model proposed by [PERSON] and [PERSON] [2] (i.e., Equation (1)), the proposed calculation model can accurately predict the cracking angle of wing crack. Compared with S model (i.e., Equation (4)), when the equivalent crack propagation length is long, the proposed criterion has a higher accuracy than the Equation (4). Compared with LK model (i.e., Equation (5)), when the equivalent crack propagation length is short (\(l/a<1\)), the prediction accuracy of this model is higher than that of Equation (5). In conclusion, the proposed calculation model can simulate the whole range of variation of wing crack length from being extremely short to very long.
2. Based on the modified calculation model for the mode-I stress intensity factor \(K_{\rm I}\), a homogenization method is used to establish the macro-micro damage model for rock materials by combining the relationship between microcrack growth and macroscopic strain. By comparing the test curves with the theoretical curves, the results show that the proposed damage constitutive model can both simulate stress-strain response in the failure process of the sandstone samples, and identify the peak strength of rock well. Furthermore, comparing experimental and theoretical relations between crack initiation stress and confining pressure, model parameters (i.e., \(\mu\), \(a\), and \(\beta\)) hardly measured by test are solved. The proposed macro-micro damage model based on micromechanics can continuously describe the relationship between stress-strain curve for rock materials, which provides a simple and easy method to study the effect of microcrack growth on the progressive failure process of rock under compression loading.
3. Effects of initial microcrack size, microcrack friction coefficient, fracture toughness, initial damage and parameter \(m\) on the stress-strain relationship are studied. It was found that with the increase in the value \(m\), the peak compressive strength of rock remains unchanged, and the corresponding peak strain decreases gradually. Parameter \(m\) can be used to reflect the brittleness and ductility of rock materials to a certain extent. The peak compressive strength of rock increases with the increase of microcrack friction coefficient and fracture toughness. However, the peak strength of rock decreases with the increase of initial microcrack size and initial damage.  
Conceptualization, Z.P. and Y.Z.; methodology, Z.P.; software, Z.P. and S.C.; validation, Z.P. and X.C.; formal analysis, Z.P.; investigation, Z.P.; resources, Z.P. and Y.Z.; data curation, Z.P.; writing--original draft preparation, Z.P.; writing--review and editing, Z.P. and Y.Z.; visualization, Z.P.; supervision, Z.P. and Y.Z.; project administration, Y.Z.; funding acquisition, Y.Z.; All authors have read and agreed to the published version of the manuscript.  
This study was financially funded by \"The National Natural Science Foundation of China (No. 41772308)\". We would like to express our sincere appreciations to the anonymous reviewers for many valuable suggestions.  
Not applicable.  
Data Availability Statement: Data are available upon request.  
The authors declare no conflict of interest.