# A Dynamic Damage Constitutive Model of Rock-like Materials Based on Elastic Tensile Strain
## 2 The Construction of the HK Model
### Damage Evolution
#### 2.1.3 Issues and Resolutions  
Referring to the constitutive coupling method [12], the zero value of elastic volumetric strain is utilized as the classification criterion for compressive and tensile states, enabling direct coupling of the compressive damage evolution of the HJC model with the tensile damage evolution of the KUS model. This results in the formulation of a damage constitutive model (HK\({}_{0}\) model) expressed by Equations (1)-(5). Nevertheless, there are numerous issues associated with directly employing this model for numerical calculations. To enhance its applicability, this paper addresses the following problems to develop the final HK model. 1. The impact of accumulated compressive plasticity. In the context of explosion and layered fracture, compressive loading precedes tensile loading. If the material undergoes plastic deformation under compression during this process, a discrepancy may arise between the zero value of hydrostatic pressure and the zero value of volumetric strain when transitioning from compression to tension:  
\[p=0\rightarrow\mu^{e}=0,\;\mu^{p}<0,\;\mu=\mu^{e}+\mu^{p}<0 \tag{6}\]  
In this equation, \(\mu^{e}\) and \(\mu^{p}\) represent the elastic and plastic strains of the material, respectively. Subsequently, the strains are increased as follows:  
\[\mu^{e}=-\mu^{p}>0\rightarrow\mu=\mu^{e}+\mu^{p}=0\to p<0 \tag{7}\]  
During the loading process from \(p\) = 0 and \(\mu\) < 0 to \(p\) < 0 and \(\mu\) = 0, the tensile stress will continue to increase based on elastic calculations without accounting for damage. The damage evolution relationship is unable to cover all stress states, ultimately resulting in a significantly higher calculated tensile strength than the actual/input strength. Since the sign of hydrostatic pressure is consistent with the sign of the elastic body strain in the elastic state, this study exclusively employs the elastic component of the volumetric strain to assess tensile damage, aiming to avoid the impact of compressive plastic loading history on both tensile damage and strength. Conservative predictions of tensile damage. Owing to the evident tensile brittle characteristics of rock-like materials, their tensile strength is approximately one order of magnitude lower than their compressive strength. Under compression with \(p>0\), the material may still undergo tensile failure, as observed in Brazilian split testing, where an approximate relationship is as shown in Equation (8) [17]. \[\big{|}\sigma_{\text{ymin}}\big{|}=3|\sigma_{\text{xmax}}| \tag{8}\]  
In this equation, \(\sigma_{\text{ymin}}\) represents the minimum compressive stress in the direction of the load, and \(\sigma_{\text{xmax}}\) represents the maximum tensile stress in the perpendicular direction of the load. Currently, the hydrostatic pressure of the adjacent unit near the specimen center is positive, and the volumetric strain of the adjacent unit near the specimen center is negative. According to Equation (3), it exclusively computes crack density and tensile damage when \(\mu>0\). Despite Equation (3) indicating that no tensile damage will occur at the center of the specimen under Brazilian split loading, in reality, tensile failure typically occurs at this location. Therefore, Equation (3) yields an excessively conservative prediction for tensile damage. [PERSON] et al. [16] posited that the equivalent tensile strain is the primary cause of material damage, and utilizing this measure to calculate damage could broaden the conditions under which material damage occurs. The equivalent tensile strain \(\theta\) is defined as depicted in Equation (9), as described by the authors. \[\theta=\sum_{i=1}^{3}\left(\varepsilon_{i}+|\varepsilon_{i}|\right)/2 \tag{9}\]  
In this equation, \(\varepsilon_{i}\) denotes the strain in the three principal directions. While directly substituting the equivalent tensile strain \(\theta\) for the volumetric strain \(\mu\) in Equation (2) can broaden the conditions for calculating tensile damage, it results in excessively lenient criteria.