A Dynamic Damage Constitutive Model of Rock-like Materials Based on Elastic Tensile Strain

Abstract

To accurately characterize the damage of rock-like materials under simultaneous or alternating tensile and compressive loading, a dynamic damage constitutive model for rock-like materials based on elastic tensile strain is developed by integrating the classical compressive plastic damage model and the tensile elastic damage model. The model is based on the Holmquist–Johnson–Cook (HJC) and Kuszmaul (KUS) models, categorizing the element stress state into tensile and compressive states through positive and negative elastic volumetric strain. It utilizes elastic tensile strain to enhance the calculation method for tensile cracks, determining the tensile strength of the principal direction based on the contribution rate of tensile principal stress for uniaxial/multiaxial loading. Additionally, it establishes a maximum elastic tensile strain rate function to rectify the model’s effect on the tensile strain rate. Through the LS-DYNA subroutine development, the model proficiently delineates the distribution of ring-shaped cracks on the frontal side and strip-shaped cracks on the rear side of the reinforced concrete slab subjected to impact loading. Numerical simulations demonstrate that the model provides more accurate damage prediction results for stress conditions involving simultaneous or alternating compression and tension, offering valuable insights for damage analysis in engineering blasting or impact penetration.

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 brittleness, 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. Xiong 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. Du et al. [9] conducted an analysis and concluded that the HJC model offers a more precise depiction of damage and maximum deflection in comparison to the experimental results. Jin [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. Wu 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. Qiang 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. Hu 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. Chen et al. [14] and Xie et al. [15] both enhanced the equivalent tensile fracture damage model proposed by Yang 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. These studies demonstrate that directly integrating a compressive damage model and a tensile damage model into a new framework is a scientifically sound, efficient, and practical approach based on classical models.

Building on the aforementioned concepts, this study endeavors to integrate the widely acknowledged HJC model and KUS model to formulate a composite damage constitutive model known as the HJC-KUS (HK) composite damage model. This new model combines the attributes of the HJC model for characterizing distortion/volumetric plastic damage under shear and compression, along with those of the KUS model for describing microcrack extension damage under volumetric tensile deformation. In the course of this investigation, several typical issues are identified in the direct composite model (HK₀), including the influence of compression plastic history on tensile strength, excessive suppression of tensile damage under volumetric tensile conditions, and discrepancies between uniaxial/multiaxial tensile strength and experimental observations. Consequently, targeted improvements are implemented in this study to obtain the HK model, and the HK model is integrated into LS-DYNA for further validation.

2. The Construction of the HK Model

2.1. Damage Evolution

2.1.1. Evolution of Compressive Damage

Holmquist et al. [1] proposed that under strong compressive loading, the accumulation of volumetric plastic strain and equivalent (shear) plastic strain leads to material strength degradation or failure. The compressive damage D_c is defined as in Equation (1).

$$D_c=\sum {\displaystyle \frac{\Delta {\epsilon }^p+\Delta u^p}{{\epsilon }_f^p+u_f^p}}$$

(1)

$${\epsilon }_f^p+u_f^p=D_1{(p^{*}+T^{*})}^{D_2}$$

(2)

In these equations, $\Delta {\epsilon }^p$ and $\Delta u^p$ represent the increments in equivalent plastic strain and volumetric plastic strain at the current time step, D₁ and D₂ are material parameters, $p^{*}=p/f_c$ is the normalized pressure (p denotes hydrostatic pressure, f_c represents uniaxial compressive strength), and $T^{*}=T/f_c$ is the normalized maximum tensile hydrostatic pressure (T denotes the maximum tensile hydrostatic pressure).

2.1.2. Evolution of Tensile Damage

Kuszmaul et al. [6] proposed that when a material is subjected to volumetric tensile loading, internal cracks will be activated, and the relationship between the tensile damage D_t, the Poisson’s ratio $\nu$, crack density C_d and its evolution rate can be expressed as in Equations (3)–(5).

$$D_t={\displaystyle \frac{16}{9}}\times {\displaystyle \frac{1-{\overline{\nu }}^2}{1-2\overline{\nu }}}{C}_d$$

(3)

$$\overline{\nu }=\nu (1-{\displaystyle \frac{16}{9}}{C}_d)$$

(4)

$${\dot{C}}_d={\displaystyle \frac{5}{2}}km{\left({\displaystyle \frac{K_{IC}}{{\rho }_{0}V{\dot{\mu }}_{\max }}}\right)}^2{\mu }^{m-1}\dot{\mu }(1-D_t)$$

(5)

In these equations, k and m represent material parameters, μ denotes the volumetric strain acting in tension (non-negative), and ${\dot{\mu }}_{\max }$ is the maximum volumetric strain rate. The symbol ${\rho }_0$ represents density, V stands for sound velocity, and K_IC signifies the first-type fracture toughness. The crack density C_d represents the ratio of the volume of the cracked region to the total volume, and it increases with more severe damage.

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 (HK₀ 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.

• 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:

$$p=0\to {\mu }^e=0,{\mu }^p<0,\mu ={\mu }^e+{\mu }^p<0$$

(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:

$${\mu }^e=-{\mu }^p>0\to \mu ={\mu }^e+{\mu }^p=0\to p<0$$

(7)

During the loading process from p = 0 and μ < 0 to p < 0 and μ = 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.

2.

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].

$$\left|{\sigma }_{y\min }\right|=3\left|{\sigma }_{x\max }\right|$$

(8)

In this equation, ${\sigma }_{y\min }$ represents the minimum compressive stress in the direction of the load, and ${\sigma }_{x\max }$ 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 μ > 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.

Yang 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 θ is defined as depicted in Equation (9), as described by the authors.

$$\theta ={\displaystyle \sum _{i=1}^3({\epsilon }_i+\left|{\epsilon }_i\right|)/2}$$

(9)

In this equation, ${\epsilon }_i$ denotes the strain in the three principal directions. While directly substituting the equivalent tensile strain θ for the volumetric strain μ in Equation (2) can broaden the conditions for calculating tensile damage, it results in excessively lenient criteria. In the elastic state, the strain in the principal direction comprises both the strain induced by tensile stress and that caused by the Poisson’s effect as shown in Equation (10).

$${\epsilon }_i^e={\displaystyle \frac{1}{E}}{\sigma }_i+{\displaystyle \frac{\nu }{E}}({\sigma }_i-{\sigma }_{kk}),i=1,2,3$$

(10)

In this equation, ${\epsilon }_i^e$ denotes the elastic strain in the three principal directions, ${\sigma }_i$ represents the stress in these directions, and E stands for Young’s modulus. In an elastic state, the Poisson 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).

$$\begin{array}{l}{\epsilon }_1={\epsilon }_2=-\nu {\epsilon }_3>{\epsilon }_3\\ \mu =(1-2\nu ){\epsilon }_3<0\\ \theta =2\nu {\epsilon }_3>0\end{array}$$

(11)

In this scenario, substituting the equivalent tensile strain θ for the volumetric strain μ 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 ${\varphi }^e$ is defined as in Equation (12).

$${\varphi }^e={\displaystyle \sum _{i=1}^3\left({\sigma }_i+\left|{\sigma }_i\right|\right)}/2E$$

(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).

$${\dot{C}}_d={\displaystyle \frac{5}{2}}km{\left({\displaystyle \frac{K_{IC}}{{\rho }_{0}V{\dot{\varphi }}_{\max }^e}}\right)}^2{\left({\varphi }^e\right)}^{m-1}{\dot{\varphi }}^e(1-D_t)$$

(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).

$${\lambda }_i={\displaystyle \frac{{\sigma }_i+\left|{\sigma }_i\right|}{{\displaystyle \sum _{i=1}^3\left({\sigma }_i+\left|{\sigma }_i\right|\right)}}},i=1,2,3$$

(14)

$${\dot{C}}_{di}={\dot{C}}_d\ast {\lambda }_i^m,i=1,2,3$$

(15)

By combining Equations (3) and (4), the tensile damage D_ti 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:

$$D_t=\max \left(D_{t1},D_{t2},D_{t3}\right)$$

(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. To accurately characterize the strain rate effect of materials, an auxiliary strain rate ${\dot{\varphi }}_0$ is introduced to establish the lower limit of the strain rate parameter ${\dot{\varphi }}_{\max }^e$, thus achieving a segmented depiction of the tensile strain rate effect. ${\dot{\varphi }}_{\max }^e$ is shown in Equation (17).

$${\dot{\varphi }}_{\max }^e=\left\{\begin{array}{l}\begin{array}{lll}{\dot{\varphi }}_{\max }^e\hfill & & \hfill {\dot{\varphi }}_{\max }^e\ge {\dot{\varphi }}_0\end{array}\hspace{0.33em}\\ \begin{array}{lll}{\dot{\varphi }}_0\hfill & & \hfill {\dot{\varphi }}_{\max }^e<{\dot{\varphi }}_0\end{array}\end{array}\right.$$

(17)

Therefore, the damage calculation expressions of the HK model are as follows:

$$\left\{\begin{array}{l}{D}_{ti}={\displaystyle \frac{16}{9}}\times {\displaystyle \frac{1-{\overline{\nu }}^2}{1-2\overline{\nu }}}{C}_{di}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\varphi }^e>0\hspace{0.33em}\\ D_t=\max \left(D_{t1},D_{t2},D_{t3}\right)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\\ D_c=\sum {\displaystyle \frac{\Delta {\epsilon }^p+\Delta u^p}{{\epsilon }_f^p+u_f^p}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mu }^e\le 0\end{array}\right.$$

(18)

The damage variable D of the material is defined as the maximum of D_t and D_c, taking into consideration the most adverse effects of damage.

2.2. Constitutive Relationship

2.2.1. Compressive Constitutive Relationship

The HJC model is used to describe the elastic–plastic response of materials under compression, and the distortion behavior of the material is represented by an equivalent strength function, as shown in Equation (19).

$${\sigma }_y^{*}=[A(1-D_c)+B{p}^{*n}](1+C\ln {\dot{\epsilon }}^{*})$$

(19)

In this equation, ${\sigma }_y^{*}$ represents the normalized equivalent intensity, and ${\dot{\epsilon }}^{*}$ denotes the dimensionless equivalent strain rate, while A, B, and n are parameters defining the intensity limit surface, and C stands for the strain rate coefficient. When the equivalent stress surpasses the equivalent strength, the material undergoes plastic deformation, and the plastic strain is determined using the corresponding flow law.

The volumetric deformation behavior of materials is characterized by a three-stage state equation, as shown in Equation (20).

$$p=\left\{\begin{array}{l}K{\mu }^e\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}elastic stage\\ p_{crush}+K_{tran}(\mu -{\mu }_{crush})\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}transitional stage\\ K_1\overline{\mu }+K_2{\overline{\mu }}^2+K_3{\overline{\mu }}^3\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}compaction stage\end{array}\right.$$

(20)

In this equation, $p_{crush}$ and ${\mu }_{crush}$ represent the pressure and volumetric strain at which the cavity initiates collapse, K denotes the bulk modulus, K_tran represents the bulk modulus of the transitional stage, while K₁, K₂, and K₃ are material constants, and $\overline{\mu }$ is the modified volumetric strain.

2.2.2. Tensile Constitutive Relationship

According to the KUS model, the tensile state is described by a linear elastic damage constitutive equation, as shown in Equation (21), which causes a decrease in modulus due to tensile damage.

$$\begin{array}{l}p=K\left(1-D\right){\mu }^e\\ S_{ij}=2G\left(1-D\right)e_{ij}\end{array}$$

(21)

In this equation, K and G are the initial bulk modulus and shear modulus, respectively, and $e_{ij}$ is the deviatoric strain, while $S_{ij}$ is the deviatoric stress.

Equations (18)–(21) constitute the theoretical basis of the HK model.

3. The HK Model Testing

The HK model theory is implemented in the commercial software LS-DYNA to conduct simple and comprehensive loading numerical calculations aiming to validate its rationality.

3.1. Material Parameter Determination

The parameters for the compression stage can be obtained by referring to the parameter fitting method of the HJC model [7]. The parameters for the stretching stage include K_IC, k, and m. K_IC can be derived from experimental data. The parameter m for granite is commonly assigned as 7 [5,13], whereas for concrete, it is typically set at 6 [10,11,12]. The parameter k is fitted based on the uniaxial tensile strength of the material. By employing the aforementioned approach, the material parameters for the HK model of high-strength steel fiber concrete are derived and presented in

Table 1. Parameters of high-strength steel fiber concrete [7,10,11,12].

G/GPa	fC/MPa	T/MPa	C	A	B	n	Smax
17.3	131.2	11.4	0.007	0.54	1.53	0.82	10
${\epsilon }_f^{\mathrm{m}\mathrm{i}\mathrm{n}}$	D1	D2	PC/MPa	UC	PL/MPa	UL	K1/GPa
0.02	0.04	1.0	92.4	0.004	800	0.1	85
K2/GPa	K3/GPa	KIC/(MN·cm−3/2)	k/(cm−3)	m
−171	208	2.747 × 10−4	3.0 × 1014	6

.

3.2. Simple Loading Numerical Calculations

3.2.1. Numerical Calculation of a Single-Element Model

The numerical calculation of a single-element model is hypothetical. A single-element model is established, and numerical simulations are performed using the HK and HK₀ models for pre-compression followed by tensile loading, uniaxial and triaxial equal loading, as well as tensile loading at different strain rates. The material parameters are detailed in Table 1, with corresponding results presented in Figure 1. In Figure 1a, the initiation of damage occurs earlier in the HK model as compared to the HK₀ model. The hydrostatic tensile strength calculated by the HK model aligns with the input value of parameter T, while that computed by the HK₀ model significantly exceeds the specified strength. In Figure 1b, the uniaxial and triaxial tensile strengths determined by the HK model are identical, whereas substantial disparities exist in those calculated by the HK₀ model. In Figure 1c, under loading conditions with a strain rate greater than the reference, both material strengths computed by the HK and HK₀ models exhibit rapid escalation with an increasing strain rate. Conversely, under loading conditions with a strain rate lower than the reference, while material strength remains constant for the HK model, it declines sharply for the HK₀ model.

The computational results indicate the following: (1) When subjected to plastic compression followed by reverse stretching, the HK model effectively accounts for the influence of accumulated plasticity on damage calculation. Additionally, it synchronously calculates tensile damage and stress without encountering situations where the evolution relationship of damage fails to cover tensile stress as observed in the HK₀ model calculations, as shown in Figure 1a. (2) In comparison to the tensile strength predicted by the HK₀ model for uniaxial/triaxial tensile tests, the tensile strength predicted by the HK model is found to be consistent, aligning with established experimental findings, as shown in Figure 1b. (3) The HK model can reasonably describe the strain rate effect of the segmented form of the material, and the tensile strength obtained under low strain rate loading is more reasonable than that obtained using the HK₀ model, as shown in Figure 1c.

3.2.2. Numerical Computation of Split Test

The numerical computation of the split test is a hypothetical calculation. The split test is a classical experimental technique used for indirectly measuring the tensile strength of rock-like materials. In this test, the units near the center of the specimen experience both tensile and compressive stresses simultaneously. Performing split calculations enables the analysis of the damage behavior of the specimen under combined tensile and compressive loading conditions. The specimen is sized to be Φ5.04 cm × 2.4 cm, and the displacement load is applied to the specimen through the opposite motion of two steel plates, with a loading strain rate of 5/s. The numerical computations are conducted using the HK and HK₀ constitutive models, with the material parameters detailed in Table 1.

The numerical results from the split loading simulations (Figure 2a,b) indicate that in the HK model, the damage distribution is primarily concentrated at the loading location and in the vertical direction, exhibiting a central crack consistent with the typical experimental failure pattern (Figure 2c). In contrast, for the HK₀ model, damage distribution is solely focused on the loading location without any damage occurring at the center of the specimen, thus failing to simulate tensile failure behavior.

3.3. Comprehensive Loading Numerical Calculation

Wang 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 × 1.25 m × 0.05 m, with a single layer of bi-directional φ6@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. Parameters of concrete [9,10,11,12].

G/GPa	fC/MPa	T/MPa	C	A	B	n	Smax
16.7	39.5	4.2	0.007	0.79	1.6	0.61	7.0
${\epsilon }_f^{\mathrm{m}\mathrm{i}\mathrm{n}}$	D1	D2	PC/MPa	UC	PL/MPa	UL	K1/GPa
0.01	0.035	1.0	23.3	0.00158	600	0.158	17.4
K2/GPa	K3/GPa	KIC/(MN·cm−3/2)	k/(cm−3)	m
38.8	29.8	2.747 × 10−4	5.55 × 1016	6

.

The numerical calculations for the front and back surfaces of the thin plate are shown in Figure 3. In the experimental findings [20], a small crack is observed at the front center of the slab, along with a visible curved crack running along the symmetry centerline and several circular cracks on the surface. At the rear of the slab, there are pull-off pits, circular cracks, and radial cracks extending outward from the center. Compared with the experimental findings, the computed results of the HK model demonstrate the formation of ring-shaped cracks, partial vertical fractures, and damage at the central position on the front surface of the slab. Additionally, damage is observed on the back surface, including a central area affected, ring-shaped cracks, and cracks extending outward from the center. The simulation effectively characterizes the damage morphology of both surfaces of the slab.

Du et al. compared the damage calculation results of the HJC model, RHT model, and K&C model based on the experiment [9]. On the front side of the slab, the damage calculation results of the HJC model indicate a lateral damage zone caused by the central curved deformation, and an approximately circular damage area at the center with a total damage value of less than 0.3, while the damage calculation results of the RHT model show lateral bending damage at the center and simulated radial damage cracks around it. Meanwhile, for the K&C model, extensive surface damage is observed. On the rear side of the plate, the simulation results of the HJC model do not exhibit any significant damage phenomena different from the other models, while the simulation results of the RHT model show cross-crack patterns and radial cracks on the surface, and the simulation results of the K&C model also show extensive surface damage. The degree of similarity of the damage distribution of the HK model to the experimental results is significantly superior to that attained by employing the HJC model, RHT model, and K&C model.

4. Conclusions

This study adopts the constitutive coupling method to establish the HK model capable of describing rock-like material damage evolution under impact loading, based on the HJC compressive damage model and KUS tensile damage model. Specific improvements have been made to the model to enhance its applicability, including:

• 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,
• 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,
• 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 HK₀ 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.