## 6 Conclusions  
A nonlinear rheological model of rock materials was proposed in this study. This model combined the Komamura-Huang-Bingham with the Drucker-Prager (DP) yield criterion to improve existing rheological models. A UMAT subroutine was implemented using ABAQUS software for numerical simulation of the nonlinear rheological behavior of bolt support and surrounding rock. The model was validated by comparing the model predictions with the displacements monitored on site. The influences of the length of the bolts and the degree of the squeezing problems on mechanical responses of the soft-rock tunnels were investigated. The following conclusions were drawn:  
1. When the bolt length exceeds 9 m, the elongation sensitivity to increasing the bolt length is very low. Therefore, keeping the bolt length as long as possible is not advised. Economically, there is an optimal value for the length of the bolt. When the bolt exceeds the threshold, the reinforcement effect of the bolt on the rock mass is insignificant. Such a threshold is approximately equal to the diameter of the investigated tunnel, as in the case of the lateral pressure coefficient being equal to 0.7. As for cases of different lateral pressure coefficients, the optimal length of the bolts must be scrutinized. 2. In case of few and very severe squeezing problems, corresponding to the Classes A and D, the mechanical response of the tunnels, if only bolts were used for primary support, is insensitive to the selection of rock bolt length. The disturbance effect of the bolt on the rock mass is insignificant. Even with the increase of time, the elongation of the same rock bolt length hardly changes. Even as time increases, the surrounding rock softens, and the strength decreases. As a consequence, a shorter bolt will gradually lose its function, and the anchoring end to the tunnel wall will loosen. Therefore, the anchoring capacity of the bolt will gradually decrease. 3. The size of the plastic zone is dependent on the in situ stress conditions and the time duration after the tunnel excavation, and, however, independent of the length of the bolts. The problem of the large deformation of a soft rock tunnel is challenging. The subsequent research will be on the following two aspects:  
Figure 13: Sensitivity analysis of the elongation of the bolts with respect to the lateral pressure coefficient \(\lambda\) when \(\beta=0.4\) and \(\xi=0.9\). 1. Notably, the development of constitutive models of rock has been a hot topic for several decades. Each model has advantages and disadvantages. The associated flow rule used in this paper may lead to excessive dilation. In order to improve this situation, carrying out true triaxial experiments of rocks may provide a way to obtain the yield and the potential surface, resulting in a non-associated flow rule. 2. It is emphasized that a small strain continuum model was used to model large deformations in this paper. The comparison of the displacement obtained from the simulation and the monitoring on site indicates that the error remains acceptable. However, using a finite strain continuum is a more logical choice for the future work. Conceptualization, J.-L.Z. ; Data curation, Y.J. ; Funding acquisition, J.-L.Z. ; Investigation, Y.J., N.L. and H.-H.J. ; Methodology, N.L. ; Software, N.L. and H.-H.J. ; Supervision, J.-L.Z. ; Validation, Y.J. ; Writing--original draft, Y.J. ; Writing--review & editing, M.-L.Z. and J.-L.Z. All authors have read and agreed to the published version of the manuscript. This research was funded by Yunnan Hydropower Survey, Design and Research Institute (Grant No. 02302530844) and the Shanghai Rising-Star Program (Grant No. 22 OB1405000). The APC was funded by Yunnan Hydropower Survey, Design and Research Institute. Not applicable. The data presented in this study are available on request from the corresponding author. The authors gratefully acknowledge the support of the team lead by [PERSON] who is a member of Chinese Academy of Science.