# Improved Block Element Method for Simulating Rock Failure
## 3 Constitutive Model Considering Coupled Normal and Shear Stress Behavior  
Although other improved BEMs [20; 21; 22; 23] under various new displacement modes have improved the calculation accuracy by introducing complex displacement modes, their complex theoretical derivation has significantly increased the difficulty of engineering applications. In contrast, the rigid BEM has always maintained an irreplaceable position in engineering practice due to its simple theory and efficient implementation. This method simplifies the layer elements' stress into two basic components: normal and shear stresses. This simplification not only greatly reduces the complexity of the calculation but also effectively characterizes the key mechanical behavior between blocks. Due to its good computational efficiency, this simplification establishes the classical Mohr-Coulomb-based model as the preferred constitutive model in the BEM. However, in previous BEMs, when applying the Mohr-Coulomb criterion, rock and soil materials were typically simplified into ideal elastic-plastic or sliding friction models, neglecting the critical mechanical behavior of softening after material failure. To better characterize the progressive damage and softening behavior of materials, this work develops an enhanced constitutive model based on the traction-separation law [30]. Drawing inspiration from [PERSON]'s work [31] and simplifying its initial assumptions, the model employs quadratic functions to define the initial and failure damage surfaces and establishes a damage evolution equation that accounts for normal-tangential stress coupling through geometric relationships. This model overcomes the limitations of conventional brittle-fracture assumptions in the traditional BEM and incorporates a progressive damage mechanism that accurately reflects the nonlinear softening behavior of materials. Consequently, it enables a realistic simulation of the entire damage process from initial cracking to complete failure. It is assumed that the layer element enters the nonlinear stage when the stress satisfies the following condition:  
\[\left(\frac{\langle\sigma_{n}\rangle}{f_{t}}\right)^{2}+\left(\frac{\sigma_{s} }{f_{s}}\right)^{2}=1 \tag{12}\]  
In this expression, \(f_{t}\) denotes the tensile strength in the normal direction, and \(f_{s}\) denotes the shear strength in the tangential direction. The symbol \(\langle\ \rangle\) denotes the Macaulay bracket. \[\langle x\rangle= \begin{cases}x&x\geq 0\\ 0&x<0\end{cases}  \tag{13}\]  
The damage evolution criterion is used to characterize the degradation process of material strength after the onset of damage, eventually leading to complete failure. Once a layer element satisfies the damage initiation condition, it enters the damage evolution phase. The evolution of damage index \(\omega\) is typically described using either a bilinear or exponential degradation model. Among these, the bilinear damage model is widely used in rock material simulations thanks to its simple parameterization and computational efficiency [30,31]. Accordingly, this study adopts the bilinear damage model illustrated in Figure 3 to characterize the degradation process, assuming the material has a constitutive relationship similar to Equation (12) at both the point of the initial damage and at the point of complete failure.