Perforation Characteristics of Three-Layer Steel Plates Subjected to Impact with Different Shapes and Velocities of Reactive Fragments

Abstract

In this paper, the AUTODYN/Smoothed Particle Hydrodynamics (SPH) method was used to study the impact of reactive fragments on three-layer equidistant steel plates. The perforation characteristics of equidistant three-layer steel plates were investigated along with the parameters of combustion energy release from reactive fragments under varied impact velocities and shape conditions. The modification of the steel plates’ perforation diameter was investigated using the dimensional analysis approach. The shock wave pressure and chemical reaction characteristics were examined using the shock wave theory. The results show that within the examined impact velocity range, the perforation diameter initially increased and then decreased as the impact velocity of the reactive fragment rose. In addition, the perforation diameter was approximately 1.5–3 times the diameter of the reactive fragment. As the impact speed increased, the active reaction generated by the reactive fragments became more sufficient. The energy released contributed to the impact’s pressure rise; in addition, the temperature of the steel plate was raised in part by the reactive fragment impact, making the steel plate more prone to melting. The results of this investigation provide important support for a detailed understanding of the rules governing the failure of steel plates under the impact of reactive fragments as well as the combustion of reactive fragments under impact.

1. Introduction

The impact problem is extensively researched in the disciplines of craters, armor and anti-armor design, and spaceship safety protection. The Whipple protection construction has clear benefits for blocking space debris [1,2]. This structure’s first layer encourages projectile fragmentation, which has an impact in the form of debris on the second layer or buffer structure and successfully prevents damage from kinetic energy impact. However, to determine how to withstand the effects of kinetic energy, consider this structure. The thin plates used in space debris protection research are mainly composed of lightweight structural materials, and their response is different from that of high-strength thin-plate materials such as steel [3,4,5,6]. Moreover, the speed of space debris is usually greater than 7 km/s, which exceeds the impact velocity possessed by projectiles under normal circumstances [7,8]. The current research focuses on hot topics with velocities ranging from 1.5 to 4.5 km/s, and most of them are spherical and cylindrical projectiles [9]. As a result, the research findings in the area of protecting against space debris no longer adhere to the current standards for projectile damage to steel plates. The failure characteristics of steel plates under general high-speed impact must therefore be studied.

Steel is the most often used structural material in engineering, and there is currently a considerable demand for studies on its dynamic behavior under impact situations [10,11,12]. Currently, little is known about how strikes, particularly those involving reactive fragment impacts, affect such equidistant steel plate constructions. Contrary to other inert materials, reactive fragments are structural materials that show properties that facilitate ignition during impact [13,14,15]. It will be interesting to observe what kind of perforation the three layers of uniformly spaced steel plate will undergo when reactive fragments impact them [16,17]. The perforation diameter of steel plates under impact conditions serves as a reference criterion for quickly estimating the damage characteristics of steel plates. As a result, the perforation diameter of steel plates under impact conditions has drawn the interest of numerous academics, making it a disputed and significant topic in modern research [18,19,20]. For the diameter of impact perforation, researchers have used semi-empirical and empirical formulations [21,22,23]. Researchers have created empirical or semi-empirical models to depict the variables that affect the perforation diameter brought on by the impact of inert bullets on steel plates [24,25,26]. However, there are few studies on semi-empirical formulations for the perforation diameter under the impact conditions of reactive fragments. In this case, it is crucial to develop a quick empirical formula for estimating the perforation diameter.

The Lagrange grid method is prone to mesh failure when simulating large deformations under high-speed impacts; when simulating large deformations under hypervelocity impacts, Euler grids are prone to the problem of empty grids [27]. Therefore, these two grid methods have certain limitations. The SPH method [28,29,30] is a meshless numerical simulation computation approach that can prevent the Lagrange mesh method’s mesh distortion failure in the case of significant deformation. In addition, the SPH method [31,32,33] can avoid the problems of empty grids that arise in simulating hypervelocity impacts. In the case of a hypervelocity impact, it is better suited for modeling the performance of the material response. As a result, the SPH approach can more accurately mimic the behavior of materials under hypervelocity impacts.

The response of steel plates to impact damage depends on a variety of parameters, such as the impact velocity, steel plate types, shape of the reactive fragments, thickness, distance between the steel plates, and other factors [5,11,34]. AUTODYN has a distinct advantage in simulating hypervelocity impacts [35,36]. This paper focuses on the numerical simulation study of equal mass and volume of cylindrical and spherical reactive fragments impacting Q345S (Q345 steel) or 304SS (304 stainless steel) plates with different velocities by using the numerical simulation AUTODYN. This study, which is highly innovative, used the quantitative analysis method to thoroughly analyze the impact of speed, kinetic energy, and chemical energy effects on the perforation. Additionally, this will offer techniques and suggestions for achieving a more precise fitting of perforation diameter formulas in subsequent studies. Impact experiments validated the numerical simulation results in this paper and have a certain rationality. The fitting formula for perforation diameter in this article has significant reference value for the study of steel plate damage characteristics and offers specific data assistance for the design of penetration experiments.

2. Numerical Simulation Methods

2.1. Numerical Simulation Models and Parameters

2.1.1. Numerical Simulation Models

Figure 1 depicts the model schematic diagram of spherical or cylindrical reactive fragments impacting equidistant three-layer Q345S or 304SS plates. According to

$$\frac{4}{3}\pi r^3=\pi r^{2}h$$

(1)

equal volume and equal diameter designs were employed to ensure that the two shapes of reactive fragments had the same volume and diameter. As a result, the reactive fragments used in this simulation were spherical reactive fragments with a diameter of 6 mm or cylindrical reactive fragments with a diameter of 6 mm and a length of 4 mm. It was more compelling to investigate the effects of the two shapes of reactive fragments on the size of the perforation or other damage features of the steel plate in this instance because (in accordance with Formula (1)) they had the same mass, volume, and diameter. Additionally, the kinetic energy could be directly represented by the impact velocity. Since both the cylindrical and spherical reactive fragments were formed of the same substance, their densities were equivalent. For the target plate arrangement, 2.5 mm thick Q345S and 304SS plates with similar spacing (40 mm) were chosen. Using the AUTODYN-3D/SPH approach, the numerical simulation analysis of the impact process was conducted.

2.1.2. Numerical Simulation Parameters

In order to numerically simulate the damage process and damage characteristics of the reactive fragment impacting steel plate, AUTODYN-3D is used in this paper. Following convergence testing, the boundary condition was chosen to be the transmit non-reflective boundary condition, and the smooth particle length was chosen to be 0.04 mm. Both reactive fragments and steel plates used the J-C failure model as a criterion for material failure.

Both the Q345 steel and 304 SS plates were studied using the Zerilli–Armstrong (Z–A) strength model [37]. In contrast to 304SS, which is a face-centered cubic (FCC) metal, Q345S is a body-centered cubic (BCC) metal. Different versions of the Z–A strength model exist for both face- and body-centered cubic metals [38], and they are more useful for describing the dynamic yield strength properties of metals with various crystal structures. Part of the model parameters came from the ANSYS18.2/AUTODYN software’s built-in parameters for the same type of material, such as the constant class parameters of the material model, and part of them came from the basic physical properties of the material, such as the static yield strength, shear modulus, density, melting point, and other parameters.

For FCC, the Z–A strength model is stated as

$$Y=Y_0+C_2\sqrt{\epsilon }\exp \left[-C_{3}T+C_{4}T\log \dot{\epsilon }\right]$$

(2)

For BCC, the Z–A strength model is stated as

$$Y=Y_0+C_1\exp \left[-C_{3}T+C_{4}T\log \dot{\epsilon }\right]+C_5{\epsilon }^n$$

(3)

where $\epsilon$ is the equivalent plastic strain, $\dot{\epsilon }$ is the equivalent plastic strain rate, and T is temperature. $Y_0,C_1,C_2,C_3,C_4,C_5$ and n are material constants. The Z-A model parameters of Q345S plate are shown in

Table 1. Q345S Z–A strength model parameters.

Y0 (Mbar)	C1	C3	C4	C5	n	G (Mbar)
3.45 × 10−3	0.01033	0.00698	4.15 × 10−4	0.00266	0.289	0.8

. The Z-A model parameters of 304SS plate are shown in

Table 2. 304SS Z–A strength model parameters.

Y0 (Mbar)	C2	C3	C4	G (Mbar)
2.05 × 10−3	0.0089	0.0028	1.15 × 10−4	0.85

.

The shock equations [39] of state are stated as

$$P=P_H+\Gamma \rho \left(e-e_H\right)$$

(4)

$$P_H=\frac{{\rho }_0{c}_0\mu (1+\mu )}{{\left[1-\left(s-1\right)\mu \right]}^2}$$

(5)

$$e_H=\frac{1}{2}\frac{P_H}{P_0}\left(\frac{\mu }{1+\mu }\right)$$

(6)

$$\mu =\frac{\rho }{{\rho }_0}-1$$

(7)

where P is the hydrostatic pressure, ${\rho }_0$ is the initial density, C is the sound speed, e is the energy, and Γ and S are the material constants in the equation. Parameters of steel for shock equation of state are shown in

Table 3. Parameters of steel for shock equation of state.

Γ	S	C (km/s)
1.86	1.29	3.96

.

The ignition growth reaction rate equation, the shock equation of state when unreacted, and the JWL equation of state when reacted were integrated with the Steinberg–Guinan (S–G) strength model for the reactive fragments. S–G strength model parameters of reactive fragment are shown in

Table 4. S–G strength model parameters of reactive fragment.

G0 (Mbar)	Y0 (Mbar)	Tm (K)	G′p	G′T (Mbar/K)	β	n	Y′p
0.81	0.00236	1900	1.793	−0.04	0.9	0.1	0.0191

. According to the S–G strength model [40]:

$$G=G_0\left[1+\frac{G_P^{\prime }}{G_0}\frac{P}{{\left(V_0/V\right)}^{1/3}}+\frac{G_T^{\prime }}{G_0}\left(T-300\right)\right]$$

(8)

$$Y=Y_0\left(1+\frac{Y_P^{\prime }}{Y_0}\frac{P}{{\left(V_0/V\right)}^{1/3}}+\frac{G_T^{\prime }}{G_0}\left(T-300\right)\right){\left(1+\beta \epsilon \right)}^n$$

(9)

The JWL equation of state is stated as

$$p=A{e}^{-r{R}_{1}v}+B{e}^{-r{R}_{2}v}+\frac{w{c}_{v}T}{v}$$

(10)

Ignition growth model parameters of reactive fragment are shown in

Table 5. Ignition growth model parameters.

I (/μs)	a	b	c	d	e	g
400	0.0396	0.566	0.333	0.667	0.222	0.667
x	y	z	Figmax	FG1max	FG2min	G1
3.6	2.9	2.6	0.6	0.8	0.82	150
G2	A (Mbar)	B (Mbar)	R1	R2	W
500	0.106802	0.03513	0.49	0.27	0.02

. The ignition growth model [41,42] with three parts (ignition, growth, and the completion reaction rate) is stated as

$$\frac{\partial F}{\partial t}=I{\left(1-F\right)}^b{\left(\frac{\rho }{{\rho }_0}-1-a\right)}^x+G_1{\left(1-F\right)}^c{F}^d{P}^y+G_2{\left(1-F\right)}^e{F}^g{P}^z$$

(11)

The material failure is influenced by the temperature softening and strain rate strengthening in situations of high temperature, pressure, and strain rate. A J-C failure model that takes stress triaxiality, temperature, and strain rate effects into account was proposed by Johnson et al. [43]. The damage parameters are defined by the model, which uses the cumulative damage criteria:

$$D={\displaystyle \sum \frac{\Delta {\epsilon }_{eq}}{{\epsilon }_f}}$$

(12)

where $\Delta {\epsilon }_{eq}$ is the equivalent plastic strain increment for a calculated cycle, and ${\epsilon }_f$ is equivalent fracture strain [12,44,45]. The failure model parameters of Q345S, 304SS, and reactive fragment are shown in

Table 6. Failure model parameters.

Material Name	D1	D2	D3	D4	D5
Q345S	0.123	0.236	−2.43	0.058	0.8856
304SS	0.47	1.31	−2.36	0.012	0.8856
Reactive fragment	1.355	1.833	−1.93	0.015	1.868

.

$${\epsilon }_f=\left[D_1+D_2\exp \left(D_3{\sigma }^{*}\right)\right]\left(1+D_4\ln \frac{{\dot{\epsilon }}_{eq}}{{\dot{\epsilon }}_0}\right)\left(1+D_5{T}^{*}\right)$$

(13)

$$T^{*}=\frac{T-T_R}{T_m-T_R}$$

(14)

In the formula above, D₁, D₂, D₃, D₄, and D₅ are the material parameters; ${\sigma }^{*}$ is the stress triaxiality; ${\dot{\epsilon }}_{eq}$ is the actual strain rate; ${\dot{\epsilon }}_0$ is the reference strain rate; $T^{*}$ is the dimensionless temperature; T is the instantaneous temperature; $T_R$ is the reference temperature; and $T_m$ is the melting temperature.

2.2. Numerical Validation

In Figure 2, the schematic diagram is displayed. The impact experiment used a two-stage light gas gun to fire reactive fragments in order to validate the results of the numerical simulation. The reactive fragments used in this experiment were cylindrical reactive fragments with a diameter of 6 mm and a length of 4 mm. The reactive fragment impact velocity was 2.38 km/s.

It can be observed in Figure 3 that there is a high degree of agreement between the simulation results and the experimental results. It was confirmed that the numerical simulation’s smooth particle length choice and model parameters were suitable. This shows that the simulation model parameters can be applied to additional simulation studies and that some reference can be made to the simulation outcomes.

The experimental results for the spherical and cylindrical reactive fragments impacting three layers of Q345S or 304SS plates are presented in Figure 4, Figure 5 and Figure 6. The results showed that at a velocity of 2.4 km/s, the spherical reactive fragments had the ability to penetrate two layers of the Q345S plate and leave impact marks on the third layer of the steel plate (Figure 4). The cylindrical reactive fragments had the ability to penetrate two layers of the Q345S plate to the limit at an impact velocity of 2.38 km/s (Figure 5). The cylindrical reactive fragments had the ability to penetrate a single layer of the 304SS plate at an impact velocity of 2.31 km/s and leave pits or small holes on the second layer of the 304SS plate (Figure 6).

Table 7. Comparison of numerical simulation results and experimental results.

Experiment Number	Fragment Shape	Steel Plate Material	V (km/s)	Total Area of First-Layer Steel Plate Perforation (mm2)		Error %
				Simulation	Experiment
1	Spherical	Q345S	2.40	127	107	18.6
2	Cylindrical	Q345S	2.38	122	126	−3.2
3	Cylindrical	304SS	2.31	153	155	−1.3

presents a comparison and analysis of the experimental and numerical simulation findings for the Q345S or 304SS steel plates impacted with spherical and cylindrical reactive fragments at similar impact velocities. There was a good agreement between the experiment and the numerical simulation results. As a result, this paper makes use of the numerical simulation results for further investigation.

2.3. Simulated Working Conditions

In this study, using experimentally verified model parameters, we conducted numerical simulation research on the characteristics of three-layer equidistant 304 or Q345 steel plates under reactive fragment impact conditions. The simulation’s SPH smooth particle length was 0.04. The respective target plate material and structure were three layers of Q345S plate or three layers of 304SS steel plate, both of which had a thickness of 2.5 mm, and the impact velocity of the cylindrical or spherical reactive fragment was 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, or 4.5 km/s.

3. Results and Discussion

3.1. Simulation Results of Three-Layer Steel Plates under Different Impact Conditions

The modeling results’ range of impact speeds is depicted in Figure 7. As the impact velocity rose, the cylindrical reactive fragment’s penetration capacity continued to rise. When the spherical reactive fragments impacted the 304 steel plate or Q345 steel plate, an interesting phenomenon emerged, and the analysis led us to conclude that at a low impact velocity, the spherical reactive fragments had a length of 6 mm in the impact direction, while the cylindrical reactive fragments had a length of 4 mm, and the spherical shape was better than the cylindrical reactive fragments in the length of the impact direction. Approximate rigid body penetration forms were mainly dominant in this impact velocity range. The metal plate was semi-fluid or nearly fluid at the time of impact when the impact velocity exceeded a certain value due to the thermal softening effect that dominated, and its shear modulus slowly decreased. However, when impacting at hypervelocity, the cylindrical reactive fragments were more likely to reflect the benefits of their penetration under high impact. The effects of the two reactive fragments roughly converged when the impact velocity continued rising.

The first layer of steel plate developed a specified range of debris cloud damage regions following the production of an ellipsoidal debris cloud on the second layer. These areas were around 3 to 4 cm in diameter and exhibited an increasing trend as the impact velocity rose. In this area, it was mainly due to the first layer of metal debris hitting the second layer of the steel-plate-formed crater and reactive debris from the reaction products impacting the crater. When the impact velocity exceeded a certain value, the second layer of steel formed debris in the air after unloading the impact of the liquid material, forming a crater impact.

3.2. Perforation Diameter Analysis

3.2.1. Impact Process on Velocity Vector of Reactive Fragment and MIS.STRESS of Steel Plate

(1)

Impact process on velocity vector of reactive fragment

As depicted in Figure 8, the SPH particle velocity vectors of the reactive fragment were all in the direction of impact prior to the onset of impact perforation. An SPH particle velocity vector with radial expansion was gradually formed along the steel plate during the initial stage of impact perforation. In the lateral corner area of the reactive fragment during impact perforation, an SPH particle velocity vector with the opposite impact direction was formed. These SPH particle velocity components caused the steel plate to generate a backsplash fragment cloud and an ellipsoidal forward-fragment cloud. The reactive fragment then underwent shock-induced chemical reaction processes to release energy throughout the impact process. As a result, a variety of physical characteristics of the steel plate and reactive fragments, as well as the impact process, influenced the specific perforation diameter of the steel plate. It was determined that the diameter of the reactive fragments was invariably less than the perforation diameter of the steel plate.

(2)

Impact process on MIS.STRESS of steel plate

As shown in Figure 9, a cylindrical reactive fragment projectile collided with a first layer of steel plate when it was moving at a speed of 1.5 km/s. In the impact process, the MIS.STRESS was the primary reason for plastic deformation in the area immediately surrounding the perforation of the steel plate. Stress spread throughout the perforation of the steel plate in a radial direction. The steel plate material experienced material fracture failure, perforation, and fracture in the direction of impact when the MIS.STRESS was greater than the dynamic tensile strength of the material. The reflection of the shock compression wave off the back surface of the steel plate and the generation of a tensile sparse wave in the impact direction reduced the dynamic yield strength of the steel plate. Additionally, the temperature rise and weakening caused by the impact encouraged the steel plate’s fracture.

During the impact process, the energetic fragments broke apart, forming a velocity vector that spread outward and even formed a reverse velocity vector, which resulted in the perforation of the steel plate being larger than the diameter of the energetic fragments. In addition, by continuously forming, the MIS.STRESS wave spreads toward the surrounding area.

3.2.2. Impact of Velocity on Perforation Diameter

According to the simulation results, at impact velocities between 1.5 and 4.5 km/s, the first layer of the steel plate would probably be perforated in a circular pattern that was 1.6–3 times the diameter of the reactive fragment (see the analysis in Figure 10). The perforation diameter of the first layer of steel plate normally increased with increasing impact velocity, reaching a high value at a specific impact velocity and then decreasing, according to an analysis of the cylindrical reactive fragment impact. With a focus on the diameter of the perforation in the first layer of steel plates, the equation was fitted using the quantitative analysis approach.

3.2.3. Quantitative Analysis Method for Perforation Diameter

Based on the principle of quantitative analysis, the size of the perforation is related to

$$d=f(d_p,l_p,c_p,{\rho }_p,E_k,Q_s,Y_p,V,T_t,c_t,{\rho }_t,Y_t)$$

(15)

According to the basic principle of the analysis of the quantities and π theorem, (as shown in

Table 8. Physical units and dimensions.

Object	Physical Quantities	Symbols	Units	Dimensions
Reactive fragment	Diameter	dp	mm	L
	Cylindrical length	lp	mm	L
	Sound velocity	cp	km/s	LT−1
	Density	ρp	g/cm3	ML−3
	Kinetic energy	Ek	kJ	L2MT−2
	Chemical energy released	Qs	kJ	L2MT−2
	Yield strength	Yp	GPa	L−1MT−2
	Impact velocity	V	km/s	LT−1
Steel plate	Thickness	Tt	mm	L
	Sound velocity	ct	km/s	LT−1
	Density	ρ	g/cm3	ML−3
	Yield strength	Yt	GPa	L−1MT−2

) the basic physical quantities are L, M, and T. The basic physical quantities $d_p,{\rho }_t,Y_t$ are taken as the basic physical quantities.

$$\frac{d}{d_p}=f\left(\frac{l_p}{d_p},\frac{c_p}{{\rho }_t^{-\frac{1}{2}}{Y}_t^{\frac{1}{2}}},\frac{{\rho }_p}{{\rho }_t},\frac{E_k}{d_p^3{Y}_t},\frac{Q_s}{d_t^3{Y}_t},\frac{Y_p}{Y_t},\frac{V}{{\rho }_t^{-\frac{1}{2}}{Y}_t^{\frac{1}{2}}},\frac{T_t}{d_p},\frac{c_t}{{\rho }_t^{-\frac{1}{2}}{Y}_t^{\frac{1}{2}}}\right)$$

(16)

A hypothesis was proposed based on the impact process of the reactive fragments while assuming that 80% of the chemical reaction energy is used to increase the hydrostatic pressure generated by the impact process and that the degree of chemical reaction is exponentially correlated with the impact velocity. Then, the chemical reaction energy released could be considered as

$$Q_s=0.8\times Q\times (1-\exp (-V))$$

(17)

In the case of the same reactive fragments impacting the same steel plate at different velocities:

$$(d_p,l_p,c_p,{\rho }_p,Y_p,T_t,c_t,{\rho }_t,Y_t)=const$$

(18)

So, the functional relationship equation is simplified as

$$\frac{d}{d_p}=g\left(\frac{E_k}{d_p^3{Y}_t},\frac{V}{{\rho }_t^{-\frac{1}{2}}{Y}_t^{\frac{1}{2}}},\frac{Q_s}{d_p^3{Y}_t}\right)$$

(19)

It is thought that the chemical energy Q_s, kinetic energy E_k, and impact velocity V all work together to determine the steel plate’s perforation diameter. Based on the results of the simulation, the following assumptions were made regarding the above function expressions:

$$d=d_p\times \left(A{\left(\frac{Q_s}{d_p^3{Y}_t}\right)}^B+\mathrm{C}{\left(\frac{V}{{\rho }_t^{-\frac{1}{2}}{Y}_t^{\frac{1}{2}}}\right)}^D+E{\left(\frac{E_K}{d_p^3{Y}_t}\right)}^F\right)$$

(20)

The coefficients A, B, C, D, E, and F were produced as shown in

Table 9. Coefficients of fitting parameters.

Shape of Reactive Fragments	Material of Steel Plate	A	B	C	D	E	F
Cylindrical	Q345	10.6052	0.7148	−0.1040	0.7951	−0.7334	−0.1631
Cylindrical	304SS	19.1378	3.4247	0.9822	−0.4941	0.1122	−0.5479
Spherical	Q345	6.3527	0.8689	−0.0003	2.3834	0.9713	0.5260
Spherical	304SS	9.5401	1.4532	−0.2560	0.5421	3.4162	2.0337

when the perforation diameters in the table data were fitted by using regression analysis in MATLAB for various reactive fragments. The equation for the steel plate’s perforated diameter based on the fit is displayed below.

$$d_1=d_p\times \left(10.6052{\left(\frac{Q_s}{d_p^3{Y}_t}\right)}^{0.7148}-0.1040{\left(\frac{V}{{\rho }_t^{-\frac{1}{2}}{Y}_t^{\frac{1}{2}}}\right)}^{0.7951}-0.7334{\left(\frac{E_K}{d_p^3{Y}_t}\right)}^{-0.1631}\right)$$

(21)

$$d_2=d_p\times \left(19.1378{\left(\frac{Q_s}{d_p^3{Y}_t}\right)}^{3.4247}+0.9822{\left(\frac{V}{{\rho }_t^{-\frac{1}{2}}{Y}_t^{\frac{1}{2}}}\right)}^{-0.4941}+0.1122{\left(\frac{E_K}{d_p^3{Y}_t}\right)}^{-0.5479}\right)$$

(22)

$$d_3=d_p\times \left(6.3527{\left(\frac{Q_s}{d_p^3{Y}_t}\right)}^{0.8689}-0.0003{\left(\frac{V}{{\rho }_t^{-\frac{1}{2}}{Y}_t^{\frac{1}{2}}}\right)}^{2.3834}+0.9713{\left(\frac{E_K}{d_p^3{Y}_t}\right)}^{0.5260}\right)$$

(23)

$$d_4=d_p\times \left(9.5401{\left(\frac{Q_s}{d_p^3{Y}_t}\right)}^{1.4532}-0.2560{\left(\frac{V}{{\rho }_t^{-\frac{1}{2}}{Y}_t^{\frac{1}{2}}}\right)}^{0.5421}+3.4162{\left(\frac{E_K}{d_p^3{Y}_t}\right)}^{2.0337}\right)$$

(24)

where the fit formulas for the perforation diameter of the cylindrical reactive fragment impacting the Q345S plate, the cylindrical reactive fragment impacting the 304SS plate, the spherical reactive fragment impacting the Q345S plate, and the spherical reactive fragment impacting the 304SS plate are d₁, d₂, d₃, and d₄. However, the aforementioned technique can be utilized to estimate the perforation diameter with reasonable accuracy and can be used as a semi-empirical formula to fit the perforation diameter for various operating situations. This will have profound significance in the estimation of the experimental perforation diameter and experimental design. The following is a MATLAB plot of the equation mentioned above.

As shown in Figure 11, the steel plate’s perforation diameter changed when the reactive fragment impacted it, first increasing and then decreasing as the impact velocity rose. The perforation diameter was approximately 1.5–3 times the diameter of the reactive fragment. The steel plate’s perforation diameter was enhanced by the combined action of tensile and shear waves during the impact process.

As a result of the compressive shock wave’s reflection as a tensile sparse wave on the surface of the steel plate, steel plates close to the impact surface shattered, resulting in a debris cloud. The direction of impact resulted in the creation of tensile stress. The shear stress produced by the movement of shear waves caused the steel plate surrounding the perforated zone to deform or undergo internal microscopic modifications. When the impact velocity was low in this range, the perforation diameter of the steel plate increased as the impact velocity increased because the reactive fragments still exhibited some structural strength. When the impact velocity was higher and the shock wave pressure was significantly greater than the material strength, the material exhibited a near-fluid state, and its shear modulus eventually went to zero, causing the perforation diameter to decrease.

3.3. Theoretical Analysis of Shock Wave Pressure on Steel Plates at Different Impact Velocities

Based on the one-dimensional shock wave theory, characteristics of the reactive fragment impacts on the steel plate, such as the particle velocity and shock wave velocity at the point of contact, were theoretically estimated. The details are displayed below. The prerequisites for interface continuity are as follows:

$$v=u_{p1}+u_{p2}$$

(25)

$$P_{rea}={\rho }_{01}\left(c_1+s_1{u}_{p1}\right)u_{p1}$$

(26)

$$P_{stel}={\rho }_{02}\left(c_2+s_2{u}_{p2}\right)u_{p2}$$

(27)

$$P_{che}=1/2{\rho }_0{U}^2(\gamma +1)+Q_S{\rho }_0(\gamma -1)$$

(28)

where P_rea is the reactive fragment’s pressure caused by the impact, P_stel is the steel plate’s pressure generated by the impact, and P_che is the pressure generated by the reactive fragment’s chemical reaction at different impact velocities. The assumed formula for the energy released by chemical reactions is (17). In the equation above, v is the impact velocity; u_p1 and u_p2 are the particle velocities in the reactive fragment and the steel plate, respectively; and u_s1 and u_s2 are the shock wave velocities in the reactive fragment and the steel plate, respectively. The reactive fragment’s zero pressure volume sound velocity was 3.96 mm/μs and the S₁ was 1.29 according to one-dimensional shock wave theory. It had an 8.13 g/cm³ density. Assuming that the steel plate’s zero pressure volume sound velocity was 4.569 mm/μs and the S₂ was 1.49, it had a 7.83 g/cm³ density.

As shown in Figure 12, the curve of the relationship between the pressure and particle velocity of the reactive fragment impacting the 304SS steel plate obtained via the impedance matching method is shown. V₁–V₇ are the results at impact velocities of 1.5–4.5 km/s, respectively, where each intersection represents the equilibrium pressure reached by P_rea, P_stel, and P_che in different chemical reaction states at different impact velocities.

When no chemical reactions occurred during the impact process, the state was represented by P₁′, while P₁ represented the condition in which all of the chemical reactions present in the active material were released. Due to the presence of metal substances that release energy through chemical reactions during impact, the impedance matching method generally does not involve the energy release portion of chemical reactions. Therefore, this article is based on Mayers’ treatment method for shock wave-induced NI-Sb chemical reactions [46]. Chemical reaction factors were added during the impact process, and the degree of chemical reaction was a function of the impact velocity.

Consequently, a number of assumptions were possible. The first was the superposition of the pressure and detonation waves produced by the impact, as seen in the pink state 1 in Figure 12. Second, as illustrated in black state 2 in Figure 12, the product was still solid after the impact despite the release of chemical reaction energy. Third, as indicated by the blue state 3 in Figure 12, the result of the impact was a mixed state of solid and gas that was accompanied by the release of chemical reaction energy but did not result in a detonation wave. Fourth, the product remained in a solid form following the impact, and no chemical reaction released energy, as shown in the red state 4 in Figure 12. The pressure particle velocity relationships of various states of reactive fragments were established by applying the impedance matching approach. The pressure value produced by the impact was determined purely by the formula. It was the junction point of the shock wave pressure and particle velocity relationship curves of the 304 SS steel plate at various impact velocities. P₁ is the pressure value at which the detonation wave pressure is superimposed on the shock wave pressure produced by the impact (for instance, at an impact velocity of 4 km/s (V6)), and P₁′ is the pressure value at which no chemical reaction releases energy. The resulting pressure value may fall between pressures P₁ and P₁′ since the situation may really be between two states.

3.4. P_CJ Analysis of Impact Process

Based on the assumption of the impact velocity and energy released by chemical reactions, the energy released by reactive fragments triggered by impact at different impact velocities could be obtained. By analyzing the simulation results, the rationality of the theoretical analysis of the shock wave pressure calculation was verified. The impedance matching method provided an important reference for predicting the impact pressure. The shock wave pressure and particle relationships were obtained at different impact velocities through the impedance matching method. If chemical reactions were added, their effects would be affected. The curve shifted upward, and the pressure generated by the chemical reaction occurred [47].

For the convenience of the numerical simulation calculation, the average value of P_CJ was taken as 2.45 GPa, the average energy per unit volume was taken as 0.056 Terg/cm³, and the detonation wave velocity was 0.123 cm/μs. Based on the above results, it was found that the reactive fragments induced chemical reactions under impact conditions, causing gas expansion that was similar to combustion to work externally, thereby releasing energy and generating pressure enhancement. This was consistent with the phenomenon observed in the impact experiment. According to Figure 13, the intersection of the P-up (red lines) and Rayleigh tangents (blue lines) was the bursting wave pressure. This process did not produce the bursting process but instead produced a slow gas expansion to release the chemical energy process, but the bursting wave theory and the existence of similarity to this process (that is, the enhancement effect on the impact process) would produce a certain pressure, so this paper adopted the bursting wave theory to describe the process, which has a certain degree of reasonableness with a certain value of reference.

4. Conclusions

Using the SPH numerical simulation approach, a study of reactive fragments striking Q345S or 304SS plates at 1.5–4.5 km/s was carried out in this research. The results yielded the following findings in particular:

(1)

The results of the numerical simulation and the experimental results were in good agreement, demonstrating the accuracy of the SPH simulation method in simulating the properties of steel plates under impact. The Z–A strength model and parameters chosen in this study can better reflect the physical change process of Q345S or 304SS plates under the impact of reactive fragments.

(2)

The damage failure characteristics of the steel plate were significantly influenced by the shape of the reactive fragments, the impact velocity, and the type of steel plate. The quick prediction of the perforation diameter of a 2.5 mm thick steel plate affected by reactive fragments at varied velocities was made possible by the fitting equation of the perforation diameter of the first layer of the steel plate based on the method of quantitative analysis.

(3)

The impedance matching approach was used to examine the pressure variation trend in the steel plates and their behavior under various velocity impact situations. The pressure-added value produced by the impact-induced chemical reaction of reactive fragments rose as the impact velocity rose.

According to the research findings presented in this article, further investigation is required to examine the effects of large-sized spherical and cylindrical reactive fragments striking 304SS or Q345S plates at varying velocities.