# Uneven Stiffness Coal Seam: A New Structural Factor Prone to Coal Burst Based on Stiffness Theory
## 2 Details in Stiffness Theory  
In 1970, [PERSON] put forth the conditions for specimen unstable failure through energy analysis [11]. It is mathematically given as:  
\[k_{\mathrm{p}}/|\lambda_{\mathrm{b}}|<1 \tag{1}\]  
where \(k_{\mathrm{p}}\) is the stiffness of the testing machine, \(\lambda_{\mathrm{b}}\) is the average slope of the load-displacement curve of the specimen during the post-peak strain softening stage, and \(|\lambda_{\mathrm{b}}|\) is referred to as the post-peak stiffness. The instability failure of the specimen on the flexible testing machine bears resemblance to coal burst. In both scenarios, pressure is exerted by the pressure provider, resulting in the failure of the pressure bearer while the pressure provider remains relatively intact. Both the specimen and coal mass serve as pressure bearers, with the relevant variables denoted by subscript b. Similarly, the testing machine and surrounding rock (the local mine composed of roof, coal, and floor strata around the coal mass) act as pressure providers, with the relevant variables indicated by subscript p. Therefore, Equation (1), initially depicting the necessary condition for specimen instability failure, can also be considered a prerequisite for coal burst, it can be referred to as the stiffness criterion, where \(|\lambda_{\mathrm{b}}|\) denotes the post-peak stiffness of the coal mass that undergo failure after the peak, and \(k_{\mathrm{p}}\) represents the local mine stiffness (LMS), which is the equivalent stiffness of the surrounding rock (pressure provider). According to the definition of LMS and the calculating method proposed by [PERSON] [11; 23; 24], The LMS of a coal pillar can be calculated with Equation (2). \[k_{\mathrm{p}}=\frac{\Delta F_{\mathrm{p}}}{\Delta h}=\frac{f_{1}}{d_{1}-d_{2}} \tag{2}\]  
where \(f_{1}\) is load suffered by the pillar, \(d_{1}\) and \(d_{2}\) are the distances between the roof and the floor before and after coal mass is mined, respectively. as shown in Figure 1.  
By analyzing the stiffness theory, the following three details that were easily overlooked in previous research can be identified:  
1. The stiffness criterion is a post-peak instability condition. So when discussing the stiffness criterion, it is already assumed that the coal mass satisfies the failure criterion. Therefore, the necessary conditions for coal burst according to the stiffness theory contain both the failure criterion and the stiffness criterion. The former determines whether the coal undergoes failure and enters the post-peak stage, providing the  
Figure 1: LMS calculation using numerical modeling [24]. basis for the latter. The latter determines whether the system instability occurs and is crucial in determining the burst intensity of failure. 2. The boundary between the specimen and testing machine is clear-cut, whereas determining the exact boundary between the coal mass and surrounding rock prior to the peak is diffcult. The determination of \(\mid\lambda_{\text{b}}\mid\) laterally becomes more complex due to this factor. In this case, it might be wiser to initially evaluate the relative risk of coal bursts by utilizing Equation (3). \[r_{\text{CB}}=r_{\text{F}}\times r_{\text{I}}\] (3) where \(r_{\text{CB}}\) is the probability of rock burst occurrence, known as the risk of coal burst; \(r_{\text{F}}\) is the probability of coal mass failure, known as the risk of failure; \(r_{\text{I}}\) is the probability that the system meeting the stiffness criterion and experiencing post-peak instability, known as the risk of instability. Higher values of dynamic and static loads, as well as lower coal strength and confining pressure, increase the likelihood of meeting the failure criterion, resulting in a higher \(r_{\text{F}}\). While, lower values of \(k_{\text{P}}\) or higher values of \(\mid\lambda_{\text{b}}\mid\) make it easier to satisfy the stiffness criterion and post-peak instability, leading a higher \(r_{\text{I}}\). 3. Stiffness is a structural parameter rather than a material parameter.