# Research and Application of Foam Filling Material in Soft Rock Roadways
## 3 Analysis of Roof Falls Mechanism in Soft Rock Roadways  
### Basic Species of Plastic Zone of Roadway-Surrounding Rock  
The section of the Songxian Shanjinyan Gold Mine roadway under study is a straight-wall semi-circular arch, as shown in Figure 4. Working according to relevant research results, the basic elastoplastic constitutive equation based on strain space, which was derived by [PERSON] et al. (2020), is adopted. This is shown in Equation (1):  
\[dq=3G\left[1-\frac{2 Gq}{2 Gq+\frac{\partial k}{\partial\varepsilon_{s}^{p}}} \right]d\varepsilon_{s} \tag{1}\]  
where \(q\) is the deviatoric stress and \(k\) is the hardening parameter of the shear yield surface. In this paper, we take the function of plastic shear strain, \(k=k(\varepsilon_{s}^{p})\).  
The morphological evolution law of the surrounding rock stress field and plastic zone in circular roadways has universal applicability, and it is still applicable to non-circular roadways. Therefore, the semi-circular arch section of the straight wall can be simplified into a circular section for subsequent analysis.  
Figure 4: Section shape of roadway. (**a**) Songxian Shanjin roadway field photo. (**b**) Straight wall semi-circular arch.  
Figure 3: Filling of wooden crib.  
Under the condition of a non-uniform stress field and a given rock lithology, the species of plastic zones in roadway-surrounding rocks are mainly divided into three shapes: circular, oval, and butterfly. According to the shape characteristics of the plastic zone boundaries, we can define them as follows:  
1. Circular plastic zone: under the condition of bidirectional isobaric stress, the boundary of the plastic zone appears circular.
2. Oval plastic zone: The oval plastic zone under non-isobaric conditions is illustrated in Figure 5. In this case, the maximum radius of the plastic zone boundary is on the vertical axis, while the minimum radius is on the horizontal axis.
3. Butterfly plastic zone: The butterfly plastic zone under non-isobaric conditions is shown in Figure 6, where the minimum radius of the plastic zone boundary is located on the horizontal axis, while the maximum radius is distributed across four quadrants. These boundary outlines take on a butterfly-like shape, with dips along the axes and prominent features in the quadrants.  
Because of the complex nature of rock strata, in practice, there are far more than just these three basic forms.  
This form can be obtained by rotating, superimposing, and combining these three basic forms. Among the three basic forms of the plastic zone, the butterfly plastic zone is the most common.  
The butterfly plastic zone is regarded as a more advanced extension form, evolving from the circular and oval plastic zone. When butterfly plastic zone failure occurs in the roadway-surrounding rock, significant deformation and failure of the surrounding rock  
Figure 5: Schematic diagram of boundary meaning of oval plastic zone.  
Figure 6: Schematic diagram of boundary meaning of butterfly plastic zone.  
often follow, bringing new problems in ensuring the stability of the roadway-surrounding rock. However, after an in-depth study of the butterfly plastic zone and reaching an understanding of its expansion behavior, the range distribution law of the butterfly plastic zone can be clarified, providing a critical positioning standard for the stability analysis of roadway-surrounding rock. This helps to better address the challenges associated with the stability of roadway-surrounding rock.