# Coupled Elastic-Plastic Damage Modeling of Rock Based on Irreversible Thermodynamics
## 6 Conclusions  
This paper combines plastic increment theory and irreversible thermodynamics to construct an elastoplastic damage coupled constitutive equation of layered rock, implemented numerically through the UMAT development in ABAQUS. The code for the elastoplastic damage coupled model has also been provided. By comparing indoor experimental data from La Biche shale in Canada and Eastern Yunnan-Western Chongqing shale in China, the model's validity is verified, and the impact of model parameters on elastoplastic and damage behavior is analyzed. The main conclusions are as follows:  
1. According to irreversible thermodynamics, the damage evolution equation is defined as a function related to damage and damage-driving force. Plastic body strain increments and equivalent plastic strain increments are used to build the plastic thermodynamic potential. Additionally, the effects of confining pressure and layered structure on shale mechanical behavior are considered. Furthermore, damage, plasticity, and elastoplastic damage are coupled corrected using the regression mapping algorithm and integrated into ABAQUS 2022 through the UMAT subroutine development. 2. Simulation analysis of La Biche shale's triaxial experiments shows that the model can reflect the characteristics of increased peak stress with confining pressure and decreased post-peak stress and simulate the volumetric expansion of shale under stress and the reduction in volumetric expansion with increased confining pressure. Further simulation analysis of the experimental data from Eastern Yunnan-Western Chongqing shale demonstrates that the constitutive model can effectively describe the increase in shale strength under high confining pressure, the macroscopic mechanical characteristics under different bedding plane angles, and the material softening behavior caused by the evolution of internal microcracks after the yield point. 3. Through parameter sensitivity analysis, it is found that the plastic hardening rate \(b\) determines the plastic hardening. The larger \(b\) is, the more evident the plastic hardening and the greater the peak strength. The damage evolution rate \(B_{\omega}\) affects the post-peak strain. The smaller \(B_{\omega}\) is, the faster the post-peak stress decreases, and the greater the strain under the same stress condition. The weighting coefficient \(r\) reflects damage behavior. The larger \(r\) is, the slower the damage grows, and the smaller the  
Figure 15: Sensitivity analysis of \(a\): (**a**) The relationship between axial strain and deviatoric stress; (**b**) The relationship between axial strain and damage variable \(D\). stress peak and residual strength are. The damage softening \(a\) indicates the degree of post-peak stress decrease. The larger \(a\) is, the faster the stress decreases, and the lower the peak strength and residual strength are. The following supporting information can be downloaded at [[https://www.mdpi.com/article/10.3390/app142310923/sl](https://www.mdpi.com/article/10.3390/app142310923/sl)]([https://www.mdpi.com/article/10.3390/app142310923/sl](https://www.mdpi.com/article/10.3390/app142310923/sl)). Table S1: Parameter Definition for the Elastic-Plastic Damage Coupled Model; Code S1: UMAT Program for the Elastic-Plastic Damage Coupled Model. Conceptualization, X.J. and K.Q. ; methodology, X.J. and Y.D. ; software, Y.D. and R.H.; validation, X.J. and Y.D. ; formal analysis, Y.D. and K.Q. ; investigation, C.F. and K.Q. ; resources, Y.D. and J.W. ; data curation, X.J. and C.F. ; writing--original draft preparation, X.J. and Y.D. ; writing--review and editing, X.J. and K.Q. ; visualization, Y.D. and C.F. ; supervision, J.W. ; project administration, X.J. and J.W. ; funding acquisition, J.W. All authors have read and agreed to the published version of the manuscript. This research was funded by the Science and Technology Innovation Project of the Chinese Institute of Coal Science, grant number 2021-KXYJ-001. Not applicable. The original contributions presented in this study are included in the article.