## 8 Conclusions  
In order to account for some major stratigraphic characteristics of the layered rocks, i.e., features that have a major influence on the mechanical behavior of excavations in these deposits, the application of two modified strength criteria to a Cosserat-like medium constitutive model for bedded rocks is proposed. The major conclusions are summarized as follows:  
1. The numerical simulation examples showed that based on M-C and D-P criteria, the Cos-FEA was effective in analyzing the flexural deformation of an underground cave  
Figure 14: The effect of convergence by two criteria.  
Figure 15: Vertical displacements (\(u_{3}\)) of all tracking points using Cosserat FEM and traditional FEM.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Tracking Points**} & \multicolumn{3}{c}{**D-P**} & \multicolumn{3}{c}{**M-C**} \\ \cline{2-7}  & **\(u_{1}\)/mm** & \(u_{3}\)/mm** & \(\omega_{2}\)/mrad** & \(u_{1}\)/mm** & \(u_{3}\)/mm & \(\omega_{2}\)/mrad \\ \hline
1 (0, 15, 6) & 0.000 & \(-\)4.350 & 0.000 & 0.000 & \(-\)3.619 & 0.000 \\
2 (1, 15, 6) & \(-\)0.174 & \(-\)2.740 & \(-\)1.806 & \(-\)0.  
in an alternating rock mass. The vertical displacement (\(u_{3}\)) value of the top cavern calculated using the D-P criterion was found to be higher than that of the modified M-C method. However, horizontal displacements of the former were lower than the latter.
2. Under the same calculation capacity, the D-P criterion-based Cos-FEA was observed to have a higher convergence speed and a lower number of iterations when compared with that based on the M-C criterion.
3. Since the M-C hexagonal yield surface is not smooth but has corners, these corners of the hexagon can cause numerical difficulty in its application to the asymmetric Cosserat constitutive relationship. However, the D-P criterion can be viewed as a smooth approximation of the M-C criterion to avoid such difficulty, and the D-P criterion can be made to match the M-C criterion by adjusting the size of the cone. Meanwhile, the M-C criterion does not take into account the influence of the intermediate principal stress on strength, but the D-P criterion is able to reflect the combined effect of the three principal stresses. Thus, the D-P criterion-based Cos-FEA is observed to have a higher convergence speed and stability.  
It should be noted that the present numerical model is based on a simplified version of the proposed Cosserat expanded constitutive theory for taking into account the influence of bending and needs further development, especially for creep damage, as well as the influence of possible healing.  
N.B.: Conceptualization, Methodology, Software, Investigation, and Writing--review and editing. J.Z.: Conceptualization, Methodology, Software, Formal analysis, and Writing--original draft. Z.J. and X.J.: Methodology and Software. X.G.: Writing--review and editing and Supervision. All authors assisted in the analysis. All authors have read and agreed to the published version of the manuscript.  
The research was supported by the National Science Foundation for Post-doctoral Scientists of China (Grant No. 2019M661710), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20211101), and the Social Development Science and Technology Innovation Project of Suzhou City, Jiangsu Province, China (Grant No. SS202126).  
Not applicable.  
All data generated or analyzed in this research were included in this published article. Additionally, readers can access all data used to support conclusions of the current study from the corresponding author upon request.  
The authors declare that they have no conflicts of interest.