# Research on Seismic Wave Quality Factor of Marble Jointed Rock Mass under SHPB Impact
## 5 Numerical Simulation of Damage and Fracture of Rock Mass under Explosive Dynamic Loading
### Selection of Numerical Models for Materials and Explosives
#### 5.2.4 The Failure Model  
(1) Failure model of specimen material  
When the strength of the local stress field of material under the action of external force reaches or exceeds its strength limit, the failure phenomenon will occur. In order to better simulate the dynamic fracture morphology of rock in a physical test, a reasonable failure model should be selected in numerical simulation. In this paper, the modified maximum principal stress criterion is used to describe the fracture failure behavior of rock under explosive dynamic load. In general, the description of the maximum principal stress failure criterion for the material state can also be simplified as: when the principal stress \(\sigma_{1}\) of A certain element reaches the dynamic tensile strength \(\sigma_{T}\) or the maximum shear stress \(\tau_{max}\) reaches the dynamic shear strength \(\tau_{c}\) of the material, cracks begin to form and are in a state of damage. When the crack runs through the whole unit, it has completely failed. In this case, \(\sigma_{1}\) = 0 of the unit is described by Equation (20). \[\sigma_{T}\leq\sigma_{T}\text{ or }\tau_{max}\leq\tau_{c} \tag{20}\]  
In combination with the crack tip stress intensity factor and its critical stress criterion Equation in linear elastic fracture mechanics, the relationship between the critical stress of model element and the crack propagation length within the element can be obtained, as shown in Equations (21) and (22). \[K=\alpha\cdot\sqrt{\pi a}=K_{IC}^{d} \tag{21}\]  
\[\sigma_{1}=\frac{K_{IC}^{d}}{\alpha\cdot\sqrt{\pi a}}=\frac{\sqrt{E_{d}G_{c}} }{\alpha\cdot\sqrt{\pi a}} \tag{22}\]  
where \(K_{IC}^{d}\) is the material dynamic fracture toughness, \(E_{d}\) is the material dynamic elastic modulus, \(a\) is the dynamic crack propagation length, \(G_{c}\) is the energy required per unit thickness model crack dynamic fracture length, and \(\alpha\) is the shape coefficient related to crack size and location. To describe the gradual failure of the element, this paper adopts the maximum principal stress linear tensile fracture softening damage model, and the relationship between the maximum principal stress and strain of the element is shown in Figure 11.  
In the fracture softening model, the dynamic fracture strain after complete failure of model element is \(\varepsilon_{\mu}\), the energy required for crack propagation per unit length on specimen with unit thickness is \(G_{C}\), and the dynamic fracture strain \(\varepsilon_{\mu}\) is expressed as Equation (23):  
\[\varepsilon_{\mu}=\frac{2G_{C}}{\sigma_{T}L} \tag{23}\]  
where \(\sigma_{T}\) is the dynamic tensile strength of the material, and \(L\) is the characteristic size in the direction of maximum principal stress. In addition, the damage variable \(Dam\) of the model element is the ratio of dynamic fracture strain \(\varepsilon_{f}\) to complete fracture strain \(\varepsilon_{\mu}\). \[Dam=\frac{\varepsilon_{f}}{\varepsilon_{\mu}}=\frac{\varepsilon_{f}\sigma_{T} L}{2G_{C}} \tag{24}\]  
In the Equation (24), dynamic fracture strain \(\varepsilon_{f}\) can be obtained by backward Euler method. Therefore, when the model element is damaged from the beginning to complete failure, the maximum principal stress \(\sigma_{1}\) that it can withstand can be expressed as follows:  
\[\sigma_{1}=\sigma_{T}(1-Dam) \tag{25}\]  
According to Equation (25), it can be seen that the whole fracture damage process of model element changes with \(Dam=(\,0\sim 1)\). (2) Failure model of explosive  
As with the strength model for explosives, the failure model for explosives is None. Damage and Fracture Analysis of Rock Mass with Different Quality Factors under Explosive Dynamic Load  
Based on the above theoretical analysis, nonlinear dynamic finite difference software AUTODYN-Code was used to analyze the damage and breakage of different quality factors under dynamic explosion loading.