# Elastoplastic Constitutive Model for Energy Dissipation and Crack Evolution in Rocks
## 7 Conclusions  
This study focuses on the construction of an elastoplastic constitutive model for rock energy dissipation and crack evolution, encompassing systematic theoretical, algorithmic, and validation research. The key findings include the following:  
1. The elastic parameters can be derived from the secant modulus of the stress-strain curve at the expansion stress and the corresponding Poisson's ratio. The yield function and potential function can be based on the Mohr-Coulomb strength criterion. The plastic shear strain, as a plastic parameter incorporating deviatoric stress, quantifies the degree of plastic strain and reflects the current stress state. These findings lay a foundational basis for the elastoplastic modeling of rock mechanical properties.
2. The dissipated energy is defined as the integral of the stress-strain curve minus the elastic strain energy, with energy dissipation quantitatively described in terms of cohesion and internal friction angle related to the stress-strain curve. The crack volumetric strain is equivalent to plastic volumetric strain, and crack evolution can be indirectly quantified using the dilatancy angle associated with increments in plastic volumetric strain. These insights provide a theoretical basis for the elastoplastic modeling of rock energy dissipation and crack evolution.
3. A corresponding theoretical elastoplastic constitutive model was established, and its three-dimensional finite difference format was derived. The incremental iterative relationships among the various mechanical parameters and their detailed computational processes within FLAC\({}^{3\text{D}}\) were clarified, thus facilitating the implementation of the numerical algorithm for the theoretical model.
4. The simulated stress-strain curves closely align with the experimental data, and the simulated dissipated energy and crack volumetric strain are consistent with the theoretical calculations. This validates that the theoretical model effectively predicts the mechanical response, energy dissipation, and crack evolution of rocks, thus providing computational support for a more accurate description and prediction of the actual failure processes of rocks.  
Conceptualization, L.C. and Z.Y.; methodology, L.C.; software, L.C.; validation, L.C.; formal analysis, L.C. and Z.Y.; investigation, L.C.; resources, L.C.; data curation, L.C.; writing--original draft preparation, L.C.; writing--review and editing, L.C.; visualization, L.C.; supervision, X.L.; project administration, L.C.; funding acquisition, Z.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 52304127).  
Not applicable.  
Data Availability Statement: The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
We would like to express our gratitude to Professor [PERSON] from Hunan University of Science and Technology and [PERSON] from Chengdu University of Technology for their invaluable support. This assistance is not covered in the author contribution or funding sections.  
The authors declare no conflicts of interest.