## 1 Introduction  
The Cosserat continuum theory is superior to the classical continuum theory (without an internal length parameter) to describe the mechanical behaviors of materials within their micro-structure. It has become a hot topic and has increasingly caught the attention of many researchers. The Cosserat continuum model, an asymmetrical elastic theory for layered rocks with bending stiffness, was first proposed by the [PERSON] brothers [1] at the beginning of the past century. In the proposed theory, the influence of defined characteristic length of the material was considered by introducing the concepts of couple stress and bending curvature. The stress tensor and strain tensor in the Cosserat continuum theory are generally not symmetric, and the difference in the shear components of stress is equilibrated by micromoments. When the characteristic length of the medium is regarded as infinitesimal, the Cosserat continuum would degenerate as the classical continuum. [PERSON], [PERSON], [PERSON], [PERSON] and their co-workers [2; 3; 4; 5; 6; 7] have extensively developed the Cosserat theory in relation to layered materials. A new 3D Cosserat expanded constitutive model [8] is based on the macro-average significance, in which the influence of bending was taken into account. In this model, a new unit cell, namely the material point containing two alternate layers, was employed to simulate the coordination of the meso-displacement between two layers with mismatched mechanical properties. This model provides a new way for analyzing the deformation and failure of layered rock. Firstly, using the elastic model and having the Cosserat stresses, namely the macro-average ones, which have been determined, the conventional stresses of the different layers in a unit cell could be obtained in sequence. Then, the conventional stresses could be utilized through a routine way for strength and failure analysis. Figure 1 shows the schematic of the stress components of the three-dimensional Cosserat continuum on an alternating layered rock mass. For the derivation of the 3D Cosserat expanded constitutive model in relation to the layered materials, the following assumptions are made:  
1. The alternating layered medium is composed of a medium bonding different mechanical properties, and the bonding properties of the interface of every layered medium is good. There is no sliding, opening or embedding between them. 2. When subjected to deformations of stretching and bending, four side faces of the block element maintain the plane (Plane Section Assumption). 3. The material strain is very small (Small Deformation Assumption). Since the 1980s, the Cosserat theory has been applied to geotechnical engineering. [PERSON] [9], [PERSON] [10], [PERSON] [11], [PERSON] [12] and their co-workers have conducted a substantial amount of work to verify the theory, especially for bedded rock masses. The results indicated that Cosserat medium method was better and more applicable to simulate failure features for a bedded rock mass with slipping and bending deformations. Based on the classical Cosserat medium theory for jointed rocks, [PERSON] et al. established a new 2D and 3D Cosserat expanded constitutive model for bedded salt rocks [13; 14; 15; 16; 17; 18]. It is well known that there are many criteria in geomechanics [19; 20; 21; 22]. The M-C and D-P criteria are applied widely among them. The M-C criterion hexagonal yield surface is not smooth but has corners. These corners of the hexagon can cause numerical computation difficulty in its application to Cosserat elasto-plastic analysis. In order to improve convergence, many researchers have suggested using the rounded edges and  
Figure 1: Schematic of the stress components of the three-dimensional Cosserat continuum on the alternating layered rock mass. vertices for approximating the M-C criterion. Among them, one of the most extensive is the D-P strength criterion, and it can be seen as the M-C criterion used in order to avoid the difficulties of smoothing approximation.