# A Dynamic Damage Constitutive Model of Rock-like Materials Based on Elastic Tensile Strain
## 2 The Construction of the HK Model
### Comprehensive Loading Numerical Calculation  
[PERSON] et al. studied the damage results of reinforced concrete slabs through explosion experiments [20]. In the experiment, the dimensions of the reinforced concrete slab were 1.25 m \(\times\) 1.25 m \(\times\) 0.05 m, with a single layer of bi-directional \(\varphi\)6@75 mm reinforcement. The slab was constrained by clamp plates on both sides. The explosive was located directly above the center of the slab, 50 cm away from the slab, and the explosive charge was 640 g. Based on the experimental conditions, a numerical simulation was conducted to analyze the impact of an explosion on a reinforced concrete slab. The complete model was utilized for the numerical computation, with fully constrained vertical and rotational displacements on both sides of the slab and a joint connection between the steel bar and the concrete. A Lagrange mesh with a grid size of 0.5 cm was applied for the reinforced concrete slab, while a Euler mesh with a grid size of 1 cm was used for explosives and air. Additionally, material interaction was implemented using the structured-arbitrary Lagrange Euler method. The material parameters of the concrete are detailed in Table 2.  
Figure 2: Results of damage in split specimen test: (**a**) damage cloud map of the HK model; (**b**) damage cloud map of the HK\({}_{0}\) model; (**c**) typical experimental damage result. compressive damage model and KUS tensile damage model. Specific improvements have been made to the model to enhance its applicability, including:  
1. Replacing volumetric strain with elastic tensile strain to enhance the method for calculating tensile crack density, thus overcoming the influence of plastic accumulation on tensile stress calculation and expanding the range of damage assessment under tensile loading,
2. Calculating the crack density on the principal direction based on the contribution rate of the principal stress to unify the tensile strength under uniaxial and multiaxial loading conditions,
3. Introducing a maximum elastic tensile strain rate function to make the description of the material strain rate effect more accurate. Based on this, the HK model is imported into the finite element software and verified through calculations for both simple and combined scenarios. Some advantages are demonstrated by the computational results. The HK model overcame some inherent limitations of creating the HK\({}_{0}\) model by directly combining the HJC and KUS models, thus avoiding overly conservative estimates of tensile damage, which helps reduce safety risks related to the design. The HK model has significantly higher accuracy than the HJC, RHT, and K&C models in predicting damage in complex loading conditions involving tensile and compressive loads. It is suggested that the HK model proposed in this study holds practical utility for predicting damage in engineering applications such as blasting or impact penetration. However, the HK model established in this paper only utilized elastic tensile strain for calculating tensile damage. Therefore, the model is unable to depict the shear failure process under a tensile state. Subsequent enhancements will be implemented to refine the calculation method for shear damage under a tensile state, thereby improving the model's capacity to characterize damage in rock-like materials. Concurrently, verification experiments on various rock-like materials will be conducted to further validate the rationality of the model. Conceptualization, X.Z. and Y.X. ; methodology, X.Z., Y.Z., W.W. and F.Z. ; validation, X.Z., Y.X. and L.W. ; resources, X.Z. and Y.Z. ; data curation, X.Z. and L.W. ; writing--original draft preparation, X.Z. ; writing--review and editing, X.Z., Y.X., W.W. and F.Z. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The original contributions presented in this study are included in this article; further inquiries can be directed to the corresponding authors.