Research on the high-speed impact of projectiles on reinforced concrete equivalent targets

Due to the prolonged production cycle of reinforced concrete target plates and the need for a series of preparations and tests before use, it is essential to study the equivalent relationship between reinforced concrete target plates and steel target plates. Numerical simulations of the process of projectiles vertically penetrating reinforced concrete target plates were conducted using LS-DYNA software. Models for both reinforced concrete and steel targets were established, and the equivalent thickness conversion relationship between the two was obtained. Simulation results indicated an inverse proportional relationship between projectile residual kinetic energy and target thickness. Curve fitting using Origin software was performed to obtain the conversion relationship between the thicknesses of reinforced concrete target plates and steel targets. The simulation results were validated, showing an error in concrete target plate thickness conversion of less than 10%. The research results provide theoretical support and data references for the equivalent study of reinforced concrete target plates and steel target plates.


Introduction
Concrete material reinforced with steel bars is widely used in both military and civilian applications, including command centers, bunkers, bridges, and buildings.Concrete, as a composite material, is typically composed of a mixture of cement, mortar, aggregate (crushed stone), and water.It exhibits high compressive strength but lower tensile strength.However, with the addition of an appropriate amount of steel reinforcement, it can enhance the overall structural load-bearing capacity and ductility.
The concrete target plate often requires a lengthy curing period during the manufacturing phase, during which the concrete strength gradually increases [1] .The strength of concrete is influenced by various factors such as materials, mix proportions, temperature, humidity, and manual compaction [2] .For reinforced concrete target plates, the reinforcement ratio and the selection of steel bars also significantly affect the concrete strength.Testing to meet experimental requirements typically takes a considerable amount of time.Therefore, establishing an equivalent target for reinforced concrete is necessary to replace actual reinforced concrete targets in ballistic resistance studies.Considering factors such as usage versatility, cost, and material strength, 45# steel is chosen as the equivalent target material for relevant experimental research, replacing reinforced concrete.This approach aims to achieve savings in labor, time, and economic costs while ensuring research progress.
Numerical simulation studies on high-speed penetration of medium-caliber projectiles into reinforced concrete target plates of varying thickness were conducted using LS-DYNA dynamics software.A finite element model for the penetration of projectiles into reinforced concrete target plates was established, and the reliability of the model was verified.The study involved analyzing the damage characteristics of reinforced concrete targets during the penetration process and determining the equivalent relationship between target plates of different thicknesses and steel targets.

Target equivalent principle
The target plate equivalence principle is mainly divided into the following four types, namely, the material strength equivalence principle, the ultimate penetration speed equivalence principle, the remaining penetration depth equivalence principle, and the target body absorption kinetic energy equivalence principle [3] .The principle of equivalent kinetic energy absorbed by the target, that is, the kinetic energy loss of the projectile during the process of penetrating the target plate is an equivalent index and measurement standard [4] .This principle can reflect the value of the remaining kinetic energy after the projectile penetrates the target plate.The remaining kinetic energy can be considered a criterion to measure whether the target body is equivalent, the equivalence principle of target absorption kinetic energy is selected as the equivalence principle in this article.Expressed as: In the equation: p m is the mass of the projectile; 0 v is the initial velocity of the projectile; r v is the remaining velocity of the projectile.
This paper uses LS-DYNA to carry out numerical simulation research on the vertical penetration of projectiles into reinforced concrete target plates and equivalent steel targets.The principle of equivalent target body absorption kinetic energy is selected to conduct equivalent target body research, that is, projectiles with the same material properties and the same speed penetrate vertically into steel targets with different thicknesses s h and reinforced concrete target plates with different thicknesses c h .During the numerical simulation calculation process, assuming that the impact of projectile abrasion and deformation is not considered, the kinetic energy lost by the projectile during the penetration process can be calculated by the above formula when the projectile mass, projectile initial velocity, and projectile residual velocity are known.get.When the residual velocity of the projectile is the same, the steel target with thickness s h and the reinforced concrete target plate with thickness c h are considered equivalent targets.

Numerical simulation of projectile penetration into reinforced concrete target plates
Based on the target energy absorption equivalent principle, dynamic simulations of high-speed projectile penetration into reinforced concrete target plates were conducted using LS-DYNA software.The Lagrange dynamic algorithm was employed, taking into account material failure and erosion contact.The projectile, steel reinforcement, and steel target utilized the JOHNSON-COOK constitutive model, while concrete was modeled using the KCC constitutive model [5] .Material failure was defined using the *MAT_ADD_EROSION keyword, and erosion contact was defined using the *CONTACT_ERODING _SURFACE_TO_SURFACE keyword.

Three-dimensional model and material parameter setting
The projectile and target plate have symmetry, in order to save calculation time, 1/4 model is used for simulation calculation.The oval solid projectile is selected to simulate the projectile, with a diameter of 90 mm and a length of 250 mm.Modeling and meshing are shown in Figure 1.The cross section of the concrete target 1/4 model and steel target 1/4 model is 400 mm × 400 mm, the steel bars are cylinders with a diameter of 6 mm, and the interval between steel bars is 80 mm.The modeling is carried out in a common node way, and the interval between each layer of steel mesh is 50 mm, so as to ensure that the reinforcement ratio of reinforced concrete target is constant at 1.5%.The modeling and mesh division are shown in Figure 2, global symmetry constraints are carried out, and boundary constraints are set outside the target.The simulation time step is set to 0.69 s.The projectile material is 30CrMnSiA steel, the concrete material is C40 concrete, the reinforcement material is Q235 steel, and the steel target material is 45 # steel.The material properties are shown in Table 1 and Table 2. [6][7] Table 1 3.2×10-4Under the numerical simulation condition, the projectile penetrates the reinforced concrete target with different thicknesses and a reinforcement ratio of 1.5% at the velocity of 1000 m/s, and the residual velocity after the projectile penetrates the target is obtained.In the process of numerical simulation, it is found that there is no failure grid, that is, the mass of the projectile does not change during penetration.In this paper, the thickness of a reinforced concrete target is equivalent to that of a steel target when the residual velocity of the projectile is basically the same and the mass of the projectile is constant.Then the projectile penetrates steel targets with different thicknesses at the velocity of 1000 m/s, and the residual velocity of the projectile is obtained.The specific working conditions of projectile penetrating reinforced concrete targets are shown in Table 3.

Analysis of penetration results
In order to verify the rationality of the numerical simulation method adopted in this paper, the experimental condition of the projectile penetrating concrete target at 130 m/s in Xu's experiment is numerically simulated [8] , and the results are compared with the experimental data in the literature.From the numerical simulation results, the peak acceleration of the projectile is about 1.79 × 10 4 g, and the penetration depth is 16.9 cm.In Xu's experiment, the peak acceleration of the projectile is about 1.6 × 10 4 g, and the penetration depth is 17 cm.

ICAMIM-2023
Journal of Physics: Conference Series 2720 (2024) 012038 IOP Publishing doi:10.1088/1742-6596/2720/1/0120384 Using LS-DYNA software for dynamic simulation, the acceleration curves of projectiles with different thicknesses are obtained, as shown in Figure 2, which is basically consistent with the trend of projectile acceleration time history curve in Huang's simulation experiment [9] .And the same laminar cracking phenomenon occurs in the penetration process as Wang's experiment [10] .
When the thickness of the target is less than 45 cm, the whole mass of the target is smaller.The kinetic energy distribution in the part of the target increases with the increase of the thickness, and the kinetic energy lost by the projectile in the penetration process also increases with the increase of the thickness of the target.When the thickness of the target plate is greater than 45 cm, the compression wave generated by the contact between the projectile and the target will be transmitted to a longer distance, and the transmission time of the compression wave will also take longer.
When the warhead completely invades the target plate, the energy transmitted by the projectile to the target plate will be more widely dispersed in all positions of the target plate, resulting in the acceleration of the projectile unable to reach the maximum when the cross-sectional area of the projectile invasion is maximum.The time that the projectile is affected by resistance increases with the increase of target thickness, which is reflected in the velocity of the projectile, that is, when the projectile finally penetrates the target, the velocity is lower.In terms of energy loss, the larger the target thickness, the more the projectile energy loss.The residual velocity of the projectile is shown in Table 4.

Projectile penetration into an equivalent steel target
Compared with projectile penetrating reinforced concrete targets, reinforced concrete appears to surface craters, cracks, shedding, and spallation.These phenomena do not occur when the projectile penetrates the steel target, and a smooth round hole will be left in the center of the steel target when the projectile penetrates the target completely.Based on the residual velocity of projectile penetrating reinforced concrete target, the projectile penetrates steel targets with different thicknesses at a speed of 1000 m/s, and the residual velocities of projectile penetrating steel targets with different thicknesses are obtained.Table 4 lists the residual velocities of projectile penetrating steel targets which are roughly consistent with Table 4.The time history curve of projectile acceleration is shown in Figure 3.

Equivalent relationship between reinforced concrete target plate and steel plate
According to the equivalent thickness of the reinforced concrete target plate and steel plate at the velocity of 1000 m/s obtained by the above numerical simulation, it is considered that the target plate corresponding to the same residual velocity of the projectile in Table 4 is equivalent.
According to the discrete data of the target thickness of the two different materials, the equivalent thickness conversion curves of the two materials are established by fitting with Origin software.Taking the thickness of the reinforced concrete target plate as an independent variable and the equivalent target thickness as a dependent variable, the equivalent equation between them at the velocity of 1000 m/s is obtained after fitting.This equation is approximate to the cubic equation, and the equivalent equation is: ( 2 )

Equivalent relation verification
Based on the equivalent equation obtained by fitting the equivalent target thickness data, the conversion between 25 cm and 35 cm reinforced concrete target plate and equivalent steel target thickness is verified, and the equivalent steel target thickness of 25 cm and 35 cm reinforced concrete target plate is obtained by numerical simulation.Compared with the steel target thickness calculated by the equivalent equation, the error of the conversion result is less than 10%, which is shown in Table 5.

Conclusion
According to the principle of equivalent kinetic energy absorbed by the target, the equivalent relationship between reinforced concrete target and steel target is established.The equivalent thickness between the reinforced concrete target and steel target is obtained from the residual velocity of the projectile obtained by numerical simulation.And the thickness conversion relationship between the reinforced concrete target and steel target is obtained by fitting, which is verified again by numerical simulation.The error between calculated results and simulation results is less than 10%, which provides a reference for the study of reinforced concrete targets and equivalent targets.

Figure 1 .
Figure 1.1/4 projectile model and mesh generation.The projectile material is 30CrMnSiA steel, the concrete material is C40 concrete, the reinforcement material is Q235 steel, and the steel target material is 45 # steel.The material properties are shown in Table1 and Table 2.[6][7] Table1.Material parameter.

Figure 2 .
Figure 2. Acceleration time history curve of projectile reinforced concrete target plate.

Figure 3 .
Figure 3. Acceleration time history curve of projectile penetrating steel target.Table4.Residual velocity of projectile.

Table 4 .
Residual velocity of projectile.

Table 5 .
Verification of equivalent target thickness transformation relationship.