Mesoscale Equivalent Numerical Study of Ultra-High Performance Concrete Subjected to Projectile Impact

Abstract

Numerical investigations on the performance of ultra-high performance concrete (UHPC) subjected to projectile impacts have attracted extensive attention, but there are still deficiencies in the accuracy and computational efficiency of related simulation methods. To make up for these deficiencies, a mesoscale equivalent model for UHPC is developed to simulate the response of UHPC under projectile impacts. In this model, an equivalent treatment is conducted on steel fibers to reduce their quantity under the premise that the interfacial shearing force between the fibers and the matrix remains equal. Based on the mesoscale equivalent model, numerical simulations of uniaxial compressive tests and projectile penetration tests on UHPC specimens are performed in LS-DYNA, and the numerical results are compared with the corresponding experimental results to verify the developed model. It is found that the mesoscale equivalent model could accurately reproduce the failure mode and stress-strain curve of UHPC specimens when the amplification factor of steel fibers is lower than 5. When the amplification factor is 5, the computational efficiency of the numerical models for penetration tests is significantly improved, and the maximum relative error between the numerical results of the crater diameter and penetration depth and experimental results is 11.7%. The successful application of the mesoscale equivalent model provides a more precise and in-depth perspective in simulating the response of UHPC with steel fibers subjected to projectile impact. Then, the influence of projectile striking velocities, UHPC compressive strengths, and volume percentages of steel fibers on the depth of penetration (DOP) are further numerically assessed. Based on the simulated data, modifications of the Young equation for predicting the DOP are conducted, and the maximum relative error of the modified equation is 13.9%. This demonstrates that the modified Young equation can accurately predict the DOP of UHPC subjected to projectile impacts.

1. Introduction

With the enhancement of the penetration capability of earth-penetrating weapons and the frequent occurrence of local wars, studies on the resistance of new building materials under projectile impacts have aroused wide attention from engineers and researchers. Ultra-high performance concrete (UHPC) is a kind of cement-based composite material with high strength, high toughness, excellent ductility, and good energy absorption capacity, which has a promising application on protective structures that may be subjected to projectile impacts [1,2]. Experimental investigations on UHPC structures against projectile impacts have been carried out over the past few decades [3,4,5,6,7]. However, it is expensive and time-consuming to conduct penetration experiments, which leads to the fact that the current experimental investigations are mainly aimed at small-caliber bullets or reduced-scale projectiles. Meanwhile, it is usually difficult to obtain the expected mechanical data in penetration tests due to the ultra-high strain rate and deceleration, which hinders the in-depth understanding of the penetration mechanism of UHPC.

Recently, efforts have been put into the numerical investigation of the dynamic response of UHPC subjected to projectile impacts, hoping to supplement the deficiencies of experimental investigations and even replace the costly penetration experiments. Prakash et al. [8] carried out numerical investigations on high-velocity projectiles penetrating steel fiber reinforced concrete (SFRC) panels with a volume percentage of steel fibers ranging from 0% to 10% using a modified RHT model. A design chart for determining the optimal panel thickness under different fiber contents and projectile kinetic energies was compiled. Blasone et al. [9] investigated the mechanical behavior of the ultra-high performance fiber-reinforced concrete (UHPFRC) under an armor-piercing projectile using the coupled plasticity-damage model DFHcoh-KST. The damage mechanisms are well simulated by the numerical tool, and it was found that the softening behavior provided by fibers had a significant influence on the damage pattern of UHPFRC targets under projectile impacts. Wan et al. [10] calibrated a set of HJC model parameters for UHPC with steel fiber according to the material test data and then simulated experiments of two different kinds of bullets penetrating UHPC targets using the modified HJC model. The numerical results showed that the modified material model could accurately simulate the depth of penetration (DOP), but it underestimated the damage, especially the tensile damage, inside UHPC targets. Liu et al. [11] presented a numerical study evaluating the performance of UHPC (90–190 MPa) under ogive-nosed projectile impacts with striking velocities from 300 m/s to 1000 m/s, where the effects of the compressive strength, projectile striking velocity, and projectile CRH on the DOP and cratering damage of targets are discussed by applying a calibrated and validated K&C material model. An empirical equation for the DOP prediction was proposed. In this work, the numerical results of projectile impacts lack sufficient experimental comparisons and verifications, and it was found that the empirical equation overestimates the penetration resistance of the UHPC. Zhou et al. [12] established a novel dynamic constitutive model for UHPC based on the KCC model, which was applied to numerically predict the resistance and damage pattern of UHPC subjected to projectile impacts and achieved good accuracy. Liu et al. [13] numerically explored the effects of steel wire mesh on the penetration resistance of reactive power concrete (RPC). An empirical equation was proposed to predict the DOP of steel wire mesh-reinforced RPC targets subjected to high-velocity projectile penetrations.

Generally, UHPC is treated as a homogeneous material when conducting simulations on UHPC targets subjected to projectile impacts, which has shown some shortcomings in reproducing and explaining the dynamic response. In fact, UHPC should be regarded as a two-phase composite material consisting of matrixes and reinforcing fibers. The addition of fibers is an important contributor to the high performance of UHPC [14]. Hence, treating UHPC as a homogeneous material is no longer appropriate, and a three-dimensional model that models the matrix and fibers discretely should be established. Zhang et al. [15], Liang et al. [16], and Peng et al. [17] built a mesoscale model to simulate the mechanical properties and failure characteristics of UHPC under static and dynamic loadings. The numerical results are in excellent agreement with corresponding experimental results, and the mesoscale model provides a mesoscopic perspective to analyze the responses of UHPC. Smith et al. [18] carried out simulations of UHPC slab penetration and perforation experiments using a mesoscale discrete particle model. It was found that the mesoscale model perfectly reproduces cratering damage, spalling damage, and the crack-bridging effect in UHPC slabs. Although the mesoscale model performs better than the homogeneous model in the penetration simulation, the computational cost of the mesoscale model is too high to be used in some large-scale penetration simulations. The main reason of the high cost of the mesoscale model is that the number of established fiber elements is too large. Therefore, some equivalent treatment must be performed on fibers to reduce their quantity in the penetration model to improve computational efficiency.

The mesoscale equivalent treatment on fibers in UHPC has rarely been studied before. In the present study, an efficient mesoscale equivalent model based on the bond-slip constitutive between steel fibers and matrixes is first developed for simulating the dynamic response of UHPC under projectile impacts and is verified by the uniaxial tests and penetration tests. Then, the validated numerical model is applied to numerically investigate the effects of the projectile striking velocity, UHPC compression strength, and volume percentage of steel fibers on the DOP of UHPC targets subjected to projectile impacts. Moreover, a modified Young equation for predicting the DOP of UHPC subjected to the projectile penetration is fitted in terms of the simulated data.

2. Penetration Experiments

2.1. UHPC Targets

In the present tests, two cuboid UHPC targets are cast for the projectile impact under different striking velocities, as shown in Figure 1. The plain sizes of the two targets are both 2.0 m × 2.0 m, while the thicknesses are 0.9 m and 1.5 m, respectively. Each target is wrapped by 1 cm thick steel plates except for the front and back to weaken the lateral boundary effects. Straight copper-coated steel fibers with 0.2 mm diameter and 13 mm long, as shown in Figure 2, are incorporated into the matrix at a volume content (V_f) of 2%. The yield strength of the steel fiber is 2100 MPa, and the density is 7830 kg/m³.

Quasi-static uniaxial compression tests are carried out on prismatic specimens with dimensions of 100 mm × 100 mm × 300 mm using an electro-hydraulic servo testing machine with a capacity of 300 tons to determine the uniaxial compressive strength of UHPC used for targets and corresponding matrixes without steel fibers, as shown in Figure 3. A total of six specimens are tested, three of which are made of UHPC with steel fibers, and the other three are made of UHPC without steel fibers (which can be named as matrix), complying with the test procedure in Chinese Standard GB/T 50081-2019 [19]. Before the compression tests, all prismatic specimens are steam cured using an automatic control system [20]. The experimental uniaxial compressive strengths of UHPC specimens with steel fibers are 144.1 MPa, 152.3 MPa, and 148.6 MPa, respectively, and the results of matrix specimens are 139.8 MPa, 134.5 MPa, and 145.3 MPa, respectively. In the subsequent discussion, the uniaxial compressive strength of UHPC and matrix are taken as the average values of the corresponding three specimens, i.e., 148.3 MPa and 139.9 MPa, respectively. At the same time, the density of UHPC specimens with steel fibers and matrix specimens is also measured, and the average values are 2430 kg/m³ and 2237 kg/m³, respectively.

2.2. Projectiles

The projectile is a type of tangent ogive-nose projectile with a caliber of 8 cm (d = 8 cm), and detailed dimensions are shown in Figure 4. The projectiles are composed of the outer shell and inner filling, and each weighs 11.9 kg. The outer shell is made of high-strength steel; the yield strength is 1350 MPa. In order to facilitate the launch of the projectiles, the centering ring and sabot are attached to each projectile.

2.3. Experimental Setup

Figure 5 shows the test setup of the penetration tests. A 100 mm smoothbore gun is used to launch the projectiles. The targets are placed vertically on the testing bed. Some thick normal concrete blocks are placed tightly on the back of the targets to restrain the rigid body displacement in the thickness direction of the targets. The smoothbore gun barrel is adjusted to be perpendicular to the front of the targets to ensure that the projectiles can strike in the center of targets vertically. During the flight of projectiles, the centering rings and sabots would automatically separate from the projectiles, so the impact mass (m) is 11.9 kg. Two sets of high-speed photography equipment are used to measure the actual striking velocity of the projectile (V₀) and to capture the penetration process.

2.4. Experimental Results

Figure 6 shows a typical moment that the projectile contacts the target in the penetration process. Detailed experimental data, including the DOP and crater diameter (d_c), are listed in

Table 1. Penetration tests data.

Target	Plain Size (m)	Thickness (m)	fc (MPa)	Vf	m (kg)	V0 (m/s)	DOP (cm)	dc (cm)
U-1	2.0 × 2.0	0.9	148.3	2%	11.9	410	42.1	51.3
U-2	2.0 × 2.0	1.5				664	78.6	78.2

.

The frontal damages of targets are shown in Figure 7. When the projectile hits the target, concrete around the contact location is ejected out and forms a circular fragment cloud. The projectile with a speed of 410 m/s bounced off the target, while the one at a striking velocity of 664 m/s was stuck in the target. The projectile impact formed an obvious funnel-shaped crater on the impact surface first and then a tunnel deep into the targets. An evident spalling phenomenon could be observed at the edge of the crater. Moreover, the concrete around the tunnel is crushed into powders by ultra-high compressive stress.

As listed in Table 1, the diameter of the crater increases with the growth of the striking velocity of the projectile. In the present penetration tests, The DOP included both the depth of the crater and the depth of the tunnel. The DOP of the projectiles at striking velocities of 410 m/s and 664 m/s is 42.1 cm and 78.6 cm, respectively, which also shows an increasing tendency.

3. Mesoscale Equivalent Model for UHPC

The mesoscale model with explicit modeling of fibers is better at simulating and explaining the response of UHPC under all kinds of loadings. However, building fibers according to actual size will lead to excessive fibers for typical FE models. For example, for the 2.0 m × 2.0 m × 1.5 m cuboid target U-2 used in the above penetration tests, the number of steel fibers would be more than 70 million in a quarter model. Therefore, to reduce the computational cost of simulation, a mesoscale equivalent model is first developed in this section for UHPC with steel fibers.

3.1. Generation of Fibers

In a UHPC specimen, fibers are uniformly and randomly distributed. The generation of random straight round fibers inside a certain specimen volume V can be as follows. First, the number of fibers is calculated by N = 4V_fV/(πd_f²L_f) according to the fiber diameter d_f, length L_f and volume content V_f. Random points with the number of N are uniformly generated in the specimen space and assigned to be the initial point of each fiber, labeled as (x_1i, y_1i, z_1i). Then, for each fiber, the initial direction is defined by two random spatial angles φ_i and θ_i in the spherical coordinate system with the initial point as the origin. The ending point of a fiber, labeled as (x_2i, y_2i, z_2i), is determined according to the initial point, random angles, and L_f, as shown in Equation (1).

$$\left\{\begin{array}{l}{x}_{2i}=x_{1i}+L_f\sin {\theta }_i\cos {\phi }_i\\ y_{2i}=y_{1i}+L_f\sin {\theta }_i\sin {\phi }_i\\ z_{2i}=z_{1i}+L_f\cos {\theta }_i\end{array}\right.$$

(1)

With the initial point and ending point determined, any fiber in the given specimen space can be identified. Equation (1) is cycled N times until all fibers are generated. All generated points and fibers will be written into node keyword file and element keyword file, respectively, which will facilitate subsequent model establishment.

3.2. Bond-Slip Constitutive Model

For building the mesoscale equivalent model for UHPC with steel fibers, the failure mode and interfacial behavior between steel fibers and matrixes must be determined. Deng et al. [21] pointed out that the fiber pullout is the most common failure mode when UHPC is subjected to loading due to the weak fiber-matrix interface and high tensile strength of the steel fiber. Hence, the interfacial behavior between steel fibers and matrixes can be regarded as a kind of bond-slip behavior. Su et al. [22] carried out single steel fiber pullout tests and put forward a bond-slip constitutive model for UHPC with steel fibers. In this model, the interfacial shear stress τ is simplified to be constant at a given relative slip S. For different values of the relative slip, the interfacial shear stress is determined by Equation (2), where τ increases linearly with S in the bonding phase and declines exponentially with S in the debonding phase.

$$\tau =\left\{\begin{array}{l}{G}_{s}S S\le S_{max}\\ G_s{S}_{max}\times e^{-EXP\times D} S>S_{max} \end{array}\right.$$

(2)

where G_s is the interfacial shear modulus, EXP is the exponent in the debonding phase, and D is the accumulated plastic displacement in all the integral time steps.

The pivotal constitutive parameters for UHPC (150 MPa) are determined according to the pullout load-slip curve, where G_s is 2393 MPa, S_max is 0.00125, and EXP is 0.2. Then, simulations on the static split tension test and SHPB test are performed, where the numerical results agree well with experimental results, and more details can be seen in Reference [22]. This bond-slip constitutive model is proposed, and the determined parameters are validated by static and dynamic tests and are adopted in the present study.

3.3. Equivalent Treatment on Steel Fibers

Since the basic reason for the ultra-high computational cost of the original mesoscale model is the huge number of steel fibers, the key to establishing a mesoscale equivalent model is to reduce the number through the equivalent treatment of steel fibers. Steel fibers play a bridging role in UHPC, which limits the deformation and crack development in the matrix. This kind of bridging effect is reflected by the interaction between steel fibers and matrixes; the essence is the transmission of the interfacial shear force. Therefore, the bridging effect can be equivalent as long as the interfacial shear force can be guaranteed to be unchanged when the equivalent treatment on steel fibers is performed [23,24]. The specific implementation procedure of the equivalent treatment on steel fibers is as follows.

The first step is to amplify steel fibers geometrically under the condition that the aspect ratio of fibers remains unchanged and replace the original fibers with amplified fibers.

$$d_{af}=n{d}_f, L_{af}=n{L}_f$$

(3)

where $d_{af}$ and $L_{af}$ are the diameter and length of the amplified fiber, respectively, and n is the geometric amplification factor. According to the premise that the volume percentage of steel fibers V_f remains unchanged, the number of fibers N_af after equivalent treatment is determined as:

$$N_{af}=\frac{V_{f}V}{\pi {(d_{af})}^2{L}_{af}}=\frac{V_{f}V}{\pi {(n{d}_f)}^2(n{L}_f)}=\frac{1}{n^3}N$$

(4)

At the same time, due to the fibers being uniformly and randomly distributed, the number of fibers in any direction is the same, where the number is labeled as $N^{\kappa }$ for original fibers and $N_{af}^{\kappa }$ for amplified fibers $N_{af}^{\kappa }=N^{\kappa }/n^3$.

For a single original steel fiber embedded in the matrix, as shown in Figure 8, an increment of the axial force in an infinitesimal segment of fiber is:

$$dP={\tau }_0\cdot \pi d_f\cdot dl$$

(5)

where τ₀ is the actual interfacial shear stress, the interfacial shear force, i.e., the axial force of the fiber, can be computed as:

$$F={\displaystyle {\int }_0^{L_e}dP=}\pi d_f{\displaystyle {\int }_0^{L_e}{\tau }_{0}dl}$$

(6)

Based on the equivalence of interfacial shear stress distribution [22], Equation (6) is simplified to:

$$F= \tau \cdot \pi d_f\cdot L_e$$

(7)

Similarly, the interfacial shearing force of a single amplified fiber is:

$$F_{af}= \tau \cdot \pi (n{d}_f)\cdot n{L}_e=n^{2}F$$

(8)

Then the resultant interfacial shearing force of the original fibers in a direction can be computed as:

$$F^{\kappa }= N^{\kappa }F=N^{\kappa }\tau \cdot \pi d_f\cdot L_e$$

(9)

and the resultant force of amplified fibers in the same direction is:

$$F_{af}^{\kappa ’}=N_{af}^{\kappa }\tau \cdot \pi (n{d}_f)\cdot n{L}_e=\frac{1}{n}{F}^{\kappa }$$

(10)

However, as Equation (10) shows, $F_{af}^{\kappa ’}$ is not equal $F^{\kappa }$. Therefore, the second step is to modify the interfacial shear modulus G_s to be $G_{saf}=n{G}_s$ and then the final resultant interfacial shearing force of amplified fibers is:

$$F_{af}^{\kappa }=N_{af}^{\kappa }(n\tau )\cdot \pi (n{d}_f)\cdot n{L}_e=F^{\kappa }$$

(11)

Equation (11) indicates that the interfacial shearing force of equivalent steel fibers and original steel fibers are ensured to be equal, which means the bridging effect of steel fibers is unchanged and then demonstrates that the equivalent treatment on steel fibers is theoretically reasonable. Ultimately, the mesoscale equivalent model for UHPC with steel fibers is developed based on the equivalent treatment of fibers.

4. Verification on the Mesoscale Equivalent Model

To further verify the developed model, numerical simulations on both uniaxial compression tests and penetration tests presented in Section 1 are conducted in LS-DYNA employing the mesoscale equivalent model. The numerical results are compared with corresponding experimental results, which show that the developed mesoscale equivalent model could well reproduce the behavior of UHPC under static and dynamic loadings.

4.1. Model Verification with Uniaxial Compression Tests

4.1.1. Numerical Model

The typical mesoscale finite element model for prismatic specimens under uniaxial compression is shown in Figure 9. The specimen is placed on a fixed loading plate, and the load is applied to the specimen by imposing displacement on the moving loading plate. An automatic surface-to-surface contact algorithm considering the friction effect is adopted to simulate the contact behavior between the loading plate and the specimen. The solid element SOLID164 is used for modeling the matrix of the specimen and loading plates. The steel fibers are modeled by beam element BEAM161. The keyword *CONSTRAINED_BEAM_IN_SOLID (CBIS) [25] in LS-DYNA is adopted to simulate the bond-slip behavior between steel fibers and matrixes.

For different amplification factors n, the dimensions and number of steel fibers in the mesoscale equivalent model are different, and the effect of the amplification factor on the numerical needs to be investigated. In this study, numerical simulations on uniaxial compression with different amplification factors n (n = 1, 3, 4, 5, 6, 7, n = 1 means no equivalence treatment on steel fibers) are performed. Figure 10 shows the distribution of steel fibers in a specimen with different n, where the number of steel fiber elements decreases fast with the increasing n. The critical parameters of the CBIS algorithm for the mesoscale equivalent model with different n are listed in

Table 2. Critical CBIS algorithm parameters under different n.

d	l	GS	Smax	EXP
0.2·n mm	13·n mm	2393·n MPa	1.25 × 10−3	0.2

.

Without adding steel fibers, the UHPC matrix could be regarded as high-strength concrete, modeled by the RHT model [26] in the present study. The critical RHT failure surface parameters A_fail and N_fail are determined by fitting the triaxial compression test data of 140 MPa HSC in Reference [27], as shown in Figure 11, where A_fail and N_fail are 1.78 and 0.35, respectively. The other critical parameters are self-defined according to Refs. [10,28,29]. Due to the stiffness of loading plates in practice being much larger than that of the specimen, the loading plates are modeled as a rigid body with the material model *MAT_RIGID. The isotropic and kinematic hardening material model *MAT_PLASTIC_KINEMATIC (*MAT_003) is chosen to build the steel fibers. Parameters of material models used in the uniaxial compression simulations are listed in

Table 3. Parameters of material models for the UHPC matrix, loading plate, and steel fibers (units: cm-g-μs).

Material	Material Model	Input Parameter	Value
UHPC matrix	*MAT_RHT	Shear modulus G	0.19
		Mass density ρ0	2.273
		Compressive strength fc	0.0014
		Failure surface constant Afail	1.78
		Failure surface constant Nfail	0.35
		Residual Surface constant B	0.8
		Residual Surface constant m	0.3
		Damage constant D1	0.045
		Damage constant D2	1.0
		Minimum strain at fracture EFMIN	0.011
Loading plate	*MAT_RIGID	Mass density RO	7.83
		Young’s modulus E	2.0
		Poisson’s ratio PR	0.25
Steel fibers	*MAT_003	Mass density RO	7.83
		Young’s modulus E	2.0
		Poisson’s ratio PR	0.30
		Yield strength SIGY	0.021
		Tangent modulus ETAN	0.0021
		Failure strain FS	0.2

.

4.1.2. Results and Discussion

The numerical failure modes of specimens under different amplification factors n are shown in Figure 12. For the cases of n = 1, 3, 4, 5, the specimens exhibit ductile shearing failure modes, where a main oblique crack develops from the end of the specimen to the middle in the direction of about 45 degrees, which is the same as the typical experimental failure mode shown in Figure 12g. However, for the case of n = 6 and n = 7, the failure mode is quite different from the other cases and the experimental result, where the damage occurs firstly in the middle of the specimen and is approximately concentrated in the middle, which displays as brittle splitting failure mode to a certain content. Shearing failure is a type of ductile failure, while splitting failure belongs to brittle failure. Although the volume percentage of steel fibers is the same, the failure mode tends to change from ductile failure to brittle failure with the increase of the amplification factor. The difference between failure modes under different amplification factors indicates that steel fibers greatly influence the ductility of UHPC.

Figure 13 shows the uniaxial compressive curves with different amplification factors n and the average full stress-strain curve obtained from uniaxial compression tests. There is little difference in the ascending branches of stress-strain curves obtained from the numerical simulations under different amplification factors, and all the ascending branches are in good agreement with the test data. As for the descending branches of the stress-strain curves, the numerical results of n = 1, 3, 4, and 5 are in good accord with the test data. Nevertheless, for the cases of n = 6 and n = 7, the numerical descending branches drop more sharply than the other cases and have much lower residual strength, which presents a great difference from the experimental result. If taking the area under the full stress-strain curve of uniaxial compression as an index to characterize the toughness of specimens, Figure 13 shows that the toughness decreases with the increase of the amplification factor. The variation of the toughness index caused by the change in amplification factor shows that steel fiber influences the toughness of UHPC significantly.

Steel fibers improve the ductility and toughness of UHPC by the bridging effect in the matrix. According to the above comparison and discussion, the bridging effect gradually weakens with the increase in the amplification factor. In the present study, the mesoscale equivalent model can accurately simulate the uniaxial compression behavior of UHPC when the amplification factor is lower than 5. However, with further increases in the amplification factor, the mesoscale equivalent model loses its validity. This phenomenon may be because the equivalence of the resultant interfacial shearing force of steel fibers, as shown in Equation (11), is destroyed due to the excessive amplification in the fiber dimension. The equivalent treatment on fibers is based on the assumption that there are enough original fibers in any direction to synthesize a certain number of amplified fibers. When the amplification factor is too large, such as n = 6 or n = 7, the number of original fibers in some directions may not be enough to synthesize sufficient amplified fibers, leading to the uneven distribution of amplified fibers and causing Equation (11) to fail. Therefore, to ensure the accuracy of the simulation, it is recommended that the amplification factor should not be greater than 5.

4.2. Model Verification with Penetration Tests

Taking the target U-2 as an example, the estimated computation time of penetration simulations under n = 1–5 is listed in

Table 4. The estimated computation time of penetration simulations on target U-2 under different amplification factors.

n	1	3	4	5
computation time	1400 h 16 min	51 h 54 min	23 h 18 min	11 h 35 min

. The computation time falls quickly as the amplification factor increases. Consequently, the amplification factor adopted for penetration simulations is determined to be 5 for balancing the accuracy and computation time of numerical calculation.

4.2.1. Numerical Model

Since no obvious yaw phenomenon is observed in the penetration process, quarter finite element models based on the equivalence method are established, as shown in Figure 14. The keyword *CONSTRAINED_GLOBAL is used to simulate the symmetric boundary condition by defining two global boundary constraint planes. As for the outer surfaces of targets, the fixed boundary condition is imposed according to the experimental setup. Matrix elements (SOLID164) within five times the projectile diameter are refined, and the minimum matrix element size is 0.8 cm. There are 352,512 steel fiber elements (BEAM161) uniformly distributed in the target with 90 cm thickness and 587,571 steel fiber elements in the target with 150 cm thickness. The outer shell and inner filling of the projectile are modeled by the solid element SOLID164 and are connected by joint nodes. The eroding surface-to-surface contact algorithm is used to simulate the contact behavior between the projectile and matrix.

In the process of penetrating, the outer shell of the projectile withstands high temperatures, which will soften the material. Thus, the Johnson–Cook model [30,31] (*MAT_015) applied for metals subjected to large strains, high strain rates, and high temperatures, coupled with *EOS_GRUNEISEN, is adopted to build the outer shell. Material model *MAT_003 is used to build the inner filling of the projectile. The material models and parameters for the projectile are listed in

Table 5. Parameters of material models for projectile [3] (units: cm-g-μs).

Material	Material Model	Input Parameter	Value
Outer shell of projectile	*MAT_015	Shear modulus	0.84
		Mass density	7.83
		Poisson’s ratio	0.25
		a/b/n/c/m	0.01350/0.00477/0.18/0.012/1.0
		Failure stress	−2
		D1	0.15
		D2	0.72
		D3	1.66
		C/S1/γ/A	0.4596/1.357/1.71/0.43
Inner filling of projectile	*MAT_003	Mass density RO	1.63
		Young’s modulus E	0.1
		Poisson’s ratio PR	0.45
		Yield strength SIGY	0.0006
		Tangent modulus ETAN	0.001
		Failure strain FS	3.0

.

4.2.2. Results and Discussion

Figure 15 shows the numerical results of the penetration process for U-1, where the striking velocity is 410 m/s and the target thickness is 90 cm. The “History Variable #4” refers to the damage parameter D in the RHT model, where D = 0 means no damage and D = 1 means fracture of the material. It can be seen that severe damage mainly occurs around the ballistic trajectory, and the region of damage distribution expands as penetration depth increases.

Frontal damage contours of targets U-1 and U-2 are shown in Figure 16. As the striking velocity of the projectile increases, the impact surface of the target tends to be subjected to more severe local damage. The numerical results of the crater diameter on targets U-1 and U-2 are 53.8 cm and 77.3 cm, respectively. Comparing the numerical and experimental results, the maximum relative error in the crater diameter is just 5.9%, showing perfect consistency. In addition to the crater, the projectile impact also forms cracks on the impact surface. The numerical results of cracks show that there are four main cracks on the impact face of target U-1 and eight main cracks on the impact face of target U-2, and these cracks develop radially and orthogonally. The width, number, and distribution range of cracks expand with increased striking velocity. For target U-1, the number and distribution of the simulated cracks are in good agreement with the experimental results. While for target U-2, the difference between the numerical results of the cracks and the experimental results is a little large, which may be caused by the slight yaw of the projectile penetrating into U-2 in the penetration tests.

The time-displacement curves and time-velocity curves of the projectiles are shown in Figure 17. The numerical values of DOPs at striking velocities of 410 m/s and 664 m/s are 40.4 cm and 87.1 cm, respectively. The numerical results and experimental results of DOPs are compared in Figure 18, where the relative errors are −3.8% and 11.7%, respectively. The reason for the slightly larger error in the case of 664 m/s may be that the projectile has a slight yaw, as shown in Figure 7b. Although the yaw is not obvious, it could increase the resistance of the projectile penetration into the target and then reduce the DOP. The maximum relative error of the DOP is lower than 15%, which indicates that the numerical method could reasonably predict the DOP when projectiles penetrate UHPC targets. Figure 17 shows that the velocities of the three projectiles gradually decrease from positive values to negative values with the increase of the DOP. The negative values of the velocities mean that the projectiles bounce in the opposite direction, which is consistent with the experimental phenomenon. The rebound velocities of projectiles at striking velocities of 410 m/s and 664 m/s are 16.7 m/s and 22.6 m/s, respectively.

Comparing the frontal damage and the DOP between experiments and numerical simulations indicates that the mesoscale equivalent model can reasonably simulate the dynamic performance of UHPC under projectile penetration when the amplification factor is 5.

5. Numerical Investigation on DOP

The DOP is the most important and noteworthy index, which reflects the resistance of concrete defensive structure under the projectile impact, and it is greatly influenced by the projectile’s striking velocity and the concrete’s compression strength [5]. As for UHPC, steel fibers play an important role in improving tensile strength and toughness, which makes it attractive to study the influence of steel fibers on the DOP. Therefore, based on the validated mesoscale equivalent model, 62 penetration scenarios considering different striking velocities, compression strength, and volume percentage of steel fibers are simulated to investigate the specific effect of these parameters on the DOP. All the simulated results of the DOP are listed in

Table 6. Numerical results of the DOP.

Simulation No.	V0 (m/s)	fc (MPa)	Vf	Simulated DOP (cm)
1	340	100	2%	36.4
2	400			43.8
3	450			54.0
4	500			63.8
5	550			75.2
6	600			85.7
7	650			98.0
8	700			115.2
9	750			131.3
10	800			151.6
11	340	120	2%	33.2
12	400			41.4
13	450			46.3
14	500			56.3
15	550			65.4
16	600			75.9
17	650			88.9
18	700			103.1
19	750			119.2
20	800			139.9
21	340	140	2%	31.3
22	450			44.5
23	500			53.4
24	550			59.2
25	600			71.7
26	650			82.6
27	700			95.0
28	750			110.0
29	800			131.6
30	400		1%	44.3
31			1.5%	40.9
32			2%	38.8
33			2.5%	37.6
34			3%	36.8
35			3.5%	36.4
36			4%	36.2
37	340	160	2%	30.0
38	400			35.9
39	450			41.0
40	500			48.1
41	550			56.1
42	600			67.1
43	650			81.1
44	700			89.9
45	750			107.4
46	800			126.7
47	340	180	2%	28.7
48	450			39.0
49	500			45.8
50	550			53.2
51	600			64.2
52	650			75.8
53	700			89.1
54	750			103.9
55	800			123.2
56	400		1%	38.7
57			1.5%	36.8
58			2%	35.8
59			2.5%	34.5
60			3%	34.3
61			3.5%	34.0
62			4%	33.8

.

5.1. Effect of Striking Velocity

Figure 18 shows the DOPs of UHPC targets with 2% steel fiber volume content at striking velocities of 340 m/s, 400m/s, 450 m/s, 500m/s, 550 m/s, 600 m/s, 650m/s, 700 m/s, 750 m/s, and 800 m/s. The numerical results manifest that for the UHPC targets of 100 MPa, 120 MPa, 140 MPa, 160 MPa, and 180 MPa, the DOP increases with the increasing striking velocity. Taking the scenario of the 140 MPa UHPC target as an example, the DOPs at striking velocities of 400 m/s, 500 m/s, 600 m/s, 700 m/s, and 800 m/s are 38.8 cm, 53.4 cm, 71.7 cm, 95.0 cm, and 131.6 cm, respectively. The increments between adjacent striking velocities are 14.6 cm, 18.3 cm, 24.3 cm, and 39.0 cm, respectively, and present an exponentially growing trend. For UHPC targets with other compression strengths, the growing trend is similar. The above findings show an exponential relationship between the DOP and striking velocity for UHPC.

5.2. Effect of Compression Strength

As shown in Figure 19, the DOP decreases with increasing compression strengths of UHPC targets when the striking velocity is fixed. Comparing the numerical results of UHPC targets under projectile penetration with the striking velocity at 550 m/s, where DOPs of 100 MPa, 120 MPa, 140 MPa,160 MPa, and 180 MPa targets are 75.2 cm, 65.4 cm, 59.2 cm, 56.1 cm, and 53.2 cm, respectively, it can be observed that the increments between the adjacent compression strength are 9.8 cm, 6.2 cm, 3.1 cm, and 2.9 cm, respectively, showing a decreasing trend. Hence, the enhancement of compression strength contributes to improving the resistance of the UHPC target against projectile penetration. However, it should also be noted that the improvement effect diminishes with increasing compression strengths.

5.3. Effect of Steel Fiber

Axial force contours of steel fibers for target U-2 in four different moments, along with internal damages and cracks in the matrix, are listed in

Table 7. Development of internal damages and steel fibers axial force in target U-2.

Time	Internal Damages of Target U-2		The Axial Force of Steel Fibers
t = 630 μs
t = 1350 μs
t = 2070 μs
t = 2700 μs

. From t = 630 μs to t = 2700 μs, the axial force in steel fibers develops synchronously with the expansion of internal damages and cracks. Since damage is accumulated by the plastic strain, the damage parameter can reflect the deformation of matrix elements. This deformation will cause a relative slip between the matrix and steel fibers embedded in the matrix, resulting in the generation of axial force in the steel fibers. The axial force will increase until the maximum relative slip S_max is reached and then decline until total debonding happens. Therefore, taking the regions around the ballistic trajectory at t = 630 μs and regions where the cracks develop at t = 2070 μs as examples, the axial force of steel fibers in the regions with damage is larger than that in the undamaged regions. The above findings show that the steel fibers can exert a bridging effect by limiting the deformation of matrix elements and the development of cracks, improving the resistance of UHPC targets under projectile penetration.

Figure 19 shows the DOPs of 140 MPa and 180 MPa UHPC targets with volume percentages of steel fibers of 1.0%, 1.5%, 2.0%, 2.5%, 3.0%, 3.5%, and 4.0%, where the striking velocity is 400 m/s. No matter the UHPC target of 140 MPa or 180 MPa, the DOP decreases with the increasing volume percentage of steel fibers. DOPs into 140 MPa UHPC targets with 1.0% and 4.0% V_f is 44.3 cm and 36.2 cm, respectively, where the 3% increase in the volume percentage of steel fibers results in an 18.3% decrease in the DOP. For the 180 MPa UHPC targets, the corresponding DOPs are 38.7 cm and 33.8 cm, respectively, and the decrease in the DOP is 12.7%. It can be seen from Figure 19 that when V_f increases from 3% to 4%, the reduction in the DOP is not obvious, which demonstrates that there is a limit to reducing the DOP by adding more steel fibers.

5.4. Modification of Young Equations for DOP Prediction

Based on an extensive experimental database of projectile impact, Sandia National Laboratories proposed the empirical Young penetration equations to predict the DOP into natural earth materials and concrete. With the expansion of the experimental database, Young equations have been constantly updated, and the latest version for concrete is as follows (in SI units):

$$DOP=0.00000153 N K_{\mathrm{e}}{K}_h{(t_c{T}_c)}^{-0.06}(11-P){({35/f}_c)}^{0.3}{(m /A)}^{0.7}(V_0-30.5), V_0\ge 61 m/s$$

(12)

$$N=0.18{(CRH-0.25)}^{0.5}+0.56, \mathrm{for} \mathrm{tangent} \mathrm{ogive} \mathrm{nose} \mathrm{shapes}$$

(13)

$$K_{\mathrm{e}}={(F/ W_1)}^{0.3}$$

(14)

$$K_h= 0.46 {(m)}^{0.15}, \mathrm{when} m < 182 \mathrm{kg}; \mathrm{else}, K_h=1.0$$

(15)

where N is the nose performance coefficient of the projectile, K_e is the correction coefficient for edge effects in concrete targets, and K_h is the correction coefficient for the lightweight projectile. More details about the three coefficients are presented in Reference [32]. t_c is the cure time of concrete, T_c is the thickness of the target, P is the volumetric percentage of rebars in concrete targets, f_c is the compression strength of concrete, m is the mass of the projectile, A is the cross-sectional area of the projectile, V₀ is the striking velocity of the projectile.

The Young equations comprehensively consider the parameters that affect the DOP, which makes them applicable in many penetration scenarios. For normal concrete and high-strength concrete with compression strength under 100 MPa, the accuracy of the Young equations is well validated. However, according to the above parametric analysis, the functional relationship between the DOP and the striking velocity of the projectile for UHPC is different from the linear relationship described in the original Young equations. At the same time, the original Young equations do not consider the influence of steel fibers on the penetration depth. Consequently, the original Young equations are not applicable to predicting the DOP into UHPC with ultra-high strength and steel fibers and should be modified. The parametric analysis shows that the functional relationships between the DOP and striking velocity V₀, compression strength f_c, and volume percentage of steel fibers V_f can be described by an exponential function, power function, and quadratic polynomial, respectively. Hence, the modified Young equation for UHPC without rebar is expressed as:

$$DOP=\alpha ·N{K}_e{K}_h·{(t_c{T}_c)}^{-0.06}(f_c^{\beta })(k_0+k_1{V}_f+k_2{V}_f^2){(m/A)}^{0.7}(e^{\gamma V_0})$$

(16)

where N, K_e, and K_h are the same as the original equations, and α, β, k₀, k₁, k_2, and γ are the undetermined coefficients.

According to the numerical results of the DOP listed in Table 6 and the equation form expressed as Equation (13), multivariate nonlinear fitting is performed in MATLAB to determine the pending parameters. The values of α, β, k₀, k₁, k₂, and γ are determined to be 0.00344, −0.43, 292.8, −5048, 77946, and 0.003, respectively. In the end, the modified Young equation is as follows:

$$DOP=0.00344N{K}_e{K}_h{(t_c{T}_c)}^{-0.06}(f_c^{-0.43}) (292.8-5048V_f+77946V_f^2) {(m/A)}^{0.7} (e^{0.003V_0})$$

(17)

where f_c is in the unit of “MPa”, V_f is a unitless percentage, m is in the unit of “kg”, A is in the unit of “m²”, V₀ is in the unit of “m/s”, and eventually the DOP is in the unit of “cm”. The correlation index R² of the fitted function is 0.9977, indicating that the modified equation for the DOP prediction has high goodness of fit.

The DOPs calculated by the modified Young equation and the original Young equation are compared with the experimental data presented in this study and References [3,5,33], as listed in

Table 8. Comparison of the DOPs between results of equation calculation and experimental results for different UHPC targets.

Specimen	fc (MPa)	Vf (%)	m (kg)	d (cm)	V0 (m/s)	DOP in Tests (cm)	DOP of Modified Equation (cm)	Relative Error	DOP of Original Equation (cm)	Relative Error
U-1	148	2	11.9	8	410	42.1	39.2	−6.9%	52.7	25.2%
U-2	148	2	11.9	8	664	78.6	84.0	6.8%	92.7	17.9%
UHPC-SF-1 [3]	140	3	0.329	2.53	553	12.9	12.9	−0.2%	16.4	26.9%
UHPC-SF-2 [3]	140	3	0.329	2.53	683	16.6	18.7	12.7%	20.4	23.1%
UHPC-SF-3 [3]	140	3	0.329	2.53	808	20.8	23.7	13.9%	25.8	24.0%
A-5-1 [5]	114	3	0.341	2.53	510	13.4	12.7	−4.9%	16.9	26.1%
1:3 [33]	153	1.5	6.3	7.5	622	51.0	52.2	2.4%	59.7	17.1%

. It can be seen that the original Young equation greatly underestimates the penetration resistance of UHPC, and the maximum relative error is 26.9%. However, the relative error between the calculated DOPs by the modified Young equation and experimental data is within 14%, which indicates that the modified Young equation proposed in the present study can accurately predict the DOP of UHPC targets subjected to projectile impact.

6. Conclusions

In this work, a mesoscale equivalent model is first developed to numerically investigate the dynamic response of UHPC subjected to projectile impacts in a more refined and efficient way. Experiments on UHPC subjected to uniaxial compression and projectile impacts are conducted and are used to validate the developed model. Relying on the mesoscale equivalent model, the influence of projectile striking velocities, UHPC compression strengths, and volume percentages of steel fibers on the DOP is numerically investigated, and a modified Young equation for predicting the DOP is proposed. The main conclusions can be drawn as follows:

(1)

The equivalent treatment on steel fibers is to amplify the size of the fibers and the interfacial shearing modulus between fibers and the matrix by n times synchronously. The interfacial shearing force is analytically proven to be equal to that before the equivalent treatment conducted on steel fibers, demonstrating that the equivalent treatment on steel fibers is theoretically reasonable. The mesoscale equivalent model for UHPC with steel fibers is successfully developed based on the equivalent treatment on fibers. When the amplification factor of steel fiber is lower than 5, the proposed model can accurately simulate the uniaxial compression behavior of UHPC specimens. However, when the amplification factor is greater than 5, the model cannot well characterize the ductility and toughness of UHPC.

(2)

When the amplification factor of steel fibers is lower than 5, the mesoscale equivalent model can accurately reproduce the failure mode and stress-strain curve of the UHPC specimens under the uniaxial compression. The computational cost of the numerical simulations of penetration experiments decreases rapidly with the increase of the amplification factor. With an amplification factor of 5, the maximum relative error between the numerical results of the cater diameter and penetration depth and experimental results is 11.7%, indicating that the mesoscale equivalent model has high accuracy.

(3)

The mesoscale equivalent model provides a more refined and in-depth perspective into numerically investigating the response of UHPC subjected to projectile impacts. The numerical investigation of the DOP shows that the DOP increases exponentially with the increase of the projectile striking velocity. The decreasing relationships between the DOP and the compression strength and volume percentage of steel fibers can be described by the power function and quadratic polynomial, respectively. Steel fibers exert a bridging effect by limiting the deformation of matrix elements to improve the penetration resistance of UHPC, but there is a limit to reducing the DOP by adding more steel fibers.

(4)

Based on the simulated data of the DOP, a modified Young equation is proposed for predicting the DOP of UHPC targets subjected to projectile impacts. The maximum relative error between the modified equation and experimental data is 13.9%, showing the proposed equation has high accuracy.

Future studies will focus on establishing a mesoscale equivalent model for UHPC with fibers in different shapes and materials or numerically investigating the dynamic response of UHPC under blast loadings based on the mesoscale equivalent model.