# 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
In the elastic state, the strain in the principal direction comprises both the strain induced by tensile stress and that caused by the [PERSON]'s effect as shown in Equation (10). \[\varepsilon_{i}^{e}=\frac{1}{E}\sigma_{i}+\frac{\
u}{E}(\sigma_{i}-\sigma_{kk} ),i=1,2,3 \tag{10}\]  
In this equation, \(\varepsilon_{i}^{e}\) denotes the elastic strain in the three principal directions, \(\sigma_{i}\) represents the stress in these directions, and \(E\) stands for [PERSON]'s modulus. In an elastic state, the [PERSON] effect causes tensile and compressive strains in one principal direction to produce strains in the other principal directions. This causes Equation (9) to include an unnecessary Poisson effect that influences in its calculated value, as observed in the case of the uniaxial compression condition where the following relationships exist in Equation (11). \[ \begin{array}{l}\varepsilon_{1}=\varepsilon_{2}=-\
u\varepsilon_{3}> \varepsilon_{3}\\ \mu=(1-2\
u)\varepsilon_{3}<0\\ \theta=2\
u\varepsilon_{3}>0\end{array}  \tag{11}\]  
In this scenario, substituting the equivalent tensile strain \(\theta\) for the volumetric strain \(\mu\) in Equation (5) will result in the calculation of tensile cracking, contradicting the expectation that the material should experience compressive damage rather than tensile fracture under uniaxial compression. Utilizing equivalent tensile strain for the calculation of tensile crack density may lead to an overestimation in damage assessment. This study posits that the elastic strain resulting from tensile stress in the principal direction of the strain is responsible for causing tensile damage in materials, rather than the additional strain induced by the Poisson effect. In accordance with the method of equivalent tensile strain, the elastic tensile strain \(\phi^{e}\) is defined as in Equation (12). \[\phi^{e}=\sum\limits_{i=1}^{3}(\sigma_{i}+|\sigma_{i}|)/2E \tag{12}\]  
Ultimately, the substitution of elastic tensile strain for volumetric strain is employed to enhance the crack density calculation method, thereby expanding the scope of tensile damage assessment. The crack density evolution rate function is shown in Equation (13). \[\dot{C}_{d}=\frac{5}{2}km\left(\frac{K_{IC}}{\rho_{0}\
u\dot{\phi}_{\max}^{e}} \right)^{2}(\phi^{e})^{m-1}\dot{\phi}^{e}(1-D_{t}) \tag{13}\]  
3. Prediction of anomalies in multi-axial tensile strength. A substantial body of experimental evidence suggests that the tensile strength of rock-like materials remains consistent under uniaxial, biaxial, and triaxial loading conditions [18]. However, most existing models fail to account for this behavior; for instance, the HJC model predicts a tensile strength with \(f_{t}/3=2f_{tt}/3=f_{ttt}\), while the KUS model also exhibits similar discrepancies in predicting tensile strength under uniaxial/multiaxial loading conditions. It is imperative to consider the evolution of tensile damage in different principal directions to enhance the characterization of tensile strength within the model. Therefore, the principal stress crack density contribution factor \(\lambda_{i}\) and the increment in crack density in the principal direction \(\dot{C}_{di}\) are introduced, as shown in Equations (14) and (15). \[\lambda_{i}=\frac{\sigma_{i}+|\sigma_{i}|}{\sum\limits_{i=1}^{3}(\sigma_{i}+| \sigma_{i}|)},i=1,2,3 \tag{14}\]  
\[\dot{C}_{di}=\dot{C}_{d}*\lambda_{i}^{m},i=1,2,3 \tag{15}\]  
By combining Equations (3) and (4), the tensile damage \(D_{li}\) in various principal directions can be determined. The tensile damage accumulates separately in the three principal directions, and the maximum value is taken as the final tensile damage:  
\[D_{t}=\max(D_{t1},D_{t2},D_{t3}) \tag{16}\]  
4. The impact of tensile strain rate. Experimental results indicate a strain rate dependency in the tensile strength of rock-like materials [19], with a weaker effect at low strain rates and a more pronounced impact at high strain rates.