# A Dynamic Damage Constitutive Model of Rock-like Materials Based on Elastic Tensile Strain
## 1 Introduction  
The dynamic response process of damage and fracture in rock-like materials under blast loading is highly complex, and the study of the dynamic constitutive model of these materials, including rock and concrete, represents a significant research focus. Numerous experimental studies have demonstrated that rock-like materials exhibit significant tensile britteness, the tensile-compressive strength difference effect, the damage effect, the confining pressure effect, and the strain rate effect. To accurately characterize the dynamic mechanical properties of rock-like materials, various classical dynamic damage constitutive models have been proposed in the industry. These can be broadly categorized into two groups: one being the compressive-shear plastic damage model, exemplified by the Holmquist-Johnson-Cook (HJC) model [1], the Riedel-Hiermaier-Thoma (RHT) model [2], and the Karagozian & Case (K&C) model [3], primarily employed in applications such as impact penetration; the other is the tensile statistical damage model, represented by the Grady-Kipp (GK) model [4], the Taylor-Chen-Kuszmaul (TCK) model [5], and the Kuszmaul (KUS) model [6], mainly utilized in fields like engineering blasting. In recent years, several scholars have conducted extensive research and made significant improvements regarding the applicability of classical models in specific engineering calculations. [PERSON] et al. [7, 8] contended that the HJC model demonstrates superior performance in shear-compression damage description, computational efficiency, and convenience compared to the RHT model and the K&C model, and refined its parameter determination method. [PERSON] et al. [9] conducted an analysis and concluded that the HJC model offers amore precise depiction of damage and maximum deflection in comparison to the experimental results. [PERSON] [10] integrated the TCK model for tensile damage with the RHT model for shear damage to formulate a comprehensive damage model applicable to concrete impact or penetration scenarios; the calculated results indicate that the model is capable of predicting crater formation, perforation, and post-perforation scabbing on the inner face of the tunnel and the back face of the target during penetration into concrete or reinforced concrete materials, and the residual velocity of the penetrator aligns well with experimental data. [PERSON] et al. [11] developed a novel damage model, utilizing the TCK model and the HJC model, to characterize the explosive damage effects in concrete; the numerical simulation results depict the failure of reinforced concrete and the formation of a blasting crater, specifically highlighting the compression of the central blasting cavity and the spalling due to free surface tension, which aligns with both experimental and theoretical findings. [PERSON] et al. [12] refined the treatment of the compression-tension interface and introduced the TCK-HJC composite constitutive model to characterize the penetration and perforation behavior of thin plates; the calculated results indicate that the dispersion of failed concrete fragments under two distinct penetration angles closely aligns with the experimental findings. [PERSON] et al. [13] compared five typical blast damage models and selected the KUS tensile model with superior computational result while also incorporating the Rheological-Dynamical Analogy (RDA) compression damage model to develop a novel combined tensile-compression damage model for characterizing rock blast damage, the numerical simulation results demonstrate the model's capability to accurately depict the blasting-induced damage in rock. [PERSON] et al. [14] and [PERSON] et al. [15] both enhanced the equivalent tensile fracture damage model proposed by [PERSON] et al. [16] and developed their constitutive models for tensile-compressive damage. The former's model correlates compressive damage with compressive plastic work and the history of tensile damage, while the latter's model still associates compressive damage with equivalent tensile strain.