# Experimental and Analytical Study on the Penetration of Corundum-Rubble Concrete Subjected to Projectile Impact

Authors: Y. L. Xue; D. G. Tang; W. X. Chen; Z. Z. Li; D. P. Li; M. L. Yao
Journal: Shock and Vibration
 (2017)

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## Abstract

A new type of composite concrete which can be called corundum-rubble concrete (CRC) was presented to improve the resistance of protective structure to the projectile impact. Comparative experiments were conducted between CRC and reinforced concrete, and a modified Taylor model was proposed to predict the penetration depth of CRC targets. Experimental results show that CRC is much higher than reinforced concrete in both strength and hardness and shows excellent resistance to the 0.125 m-diameter projectile impact. Theoretical analyses demonstrated that the modified Taylor model’s predicted results were in good agreement with the measured values.


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## Body

## 1. Introduction

Concrete is an important material which can be used to construct protective structure to resist impact by projectiles and explosive loads [1, 2]. Much work has been done to investigate the effects of concrete targets subject to projectile impact, such as Guirgis and Guirguis [3], Wang et al. [4], Yu et al. [5], Bian et al. [6], and Wen and Lan [7]. However, most of the work was conducted to predict resistance/strength of reinforced concrete targets.The resistance features of reinforced concrete targets include AE source characteristics, the fragment-size distribution, surface characteristics, and penetration depth. Different explosion/seismic sources will lead to disparate failure characteristics [8, 9]. The reinforced concrete targets often show lack of resistance/strength to the impact of modern weapons and powerful projectiles. Hence, it is important to study on improving the resistance/strength of reinforced concrete targets and find a concrete which has ability to resist modern projectiles impact and other explosion/seismic sources.As was pointed in the previous studies, the most important factor affecting the resistance/strength of reinforced concrete targets was the compressive strength [10, 11], and plenty of reviews such as Sun et al. [12], Dancygier et al. [13], and Tai [14] were concentrated on improving the compressive strength of concrete to enhance the resistance/strength of reinforced concrete targets.In this paper, a new material which can be called corundum-rubble was proposed to improve the compressive strength of concrete in order to enhance the resistance/strength of reinforced concrete targets and a new type of concrete target which can be called corundum-rubble concrete target (CRC target) was designed by casting reinforced concrete together with corundum-rubble.The corundum-rubble is a dark brown crystal. With high quality bauxite, anthracite coal, and scrap iron as the main raw material, it was smelted in the electric-arc furnace under the high temperature of 2500°C. After smelting, the main ingredient of corundum-rubble is Al2O3, accounting for 94% of the total. The density of corundum-rubble is 3800 kg/m3, the elastic modulus is 400 GPa, the compressive strength is over 2000 MPa, and Mohs’ hardness scale is 9.1 (second only to diamond) [15].In this paper, comparative experiments were conducted between CRC targets and reinforced concrete targets. The experimental results were obtained to analyze the resistance/strength of CRC targets. The theoretical formulae were cited to predict the penetration depth of reinforced concrete targets and a modified Taylor model was proposed to predict the penetration depth of CRC targets.


## 2. Experiment

### 2.1. Projectile

In the experiments, a 0.684 m-length, 0.125 m-diameter, and 18.29 kg weight truncated-ogival-nosed projectile was designed and it had six empennages to control the flight stability. The projectile was machined from 40 Cr, the density is 7850 kg/m3, the tensile strength is 980 MPa, and the yield stress is 785 MPa. All projectiles were quenched to enhance the strength and hardness. Figure 1 shows the formed projectiles and Figure 2 is the structure diagram of projectiles. The parameters of projectile were summarized in Table 1.Table 1
Parameters of projectile.
l 0 (m) d (m) ρ p (kg/m3) σ y (MPa) σ d (MPa) M (kg) 0.684 0.125 7850 785 980 18.29Figure 1
Projectile.Figure 2
Structure diagram of projectile.


### 2.2. Targets

In the experiments, three CRC targets and reinforced concrete targets number for (N-1)~(N-3) and (C-1)~(C-3), respectively, were designed. The reinforced concrete targets were cast into 2.0 m-diameter, 0.8 m-height cylinder block and were wrapped by 0.003 m-thickness steel drums. The reinforced concrete was constructional reinforcement with Φ10 mild steel, and the area ratio of reinforcement was 0.26%. Figure3 is the diagram of reinforced concrete targets.Figure 3
Diagram of reinforced concrete targets.CRC targets were the same size as reinforced concrete targets, which were filled with some irregular shape of 0.1~0.15 m-diameter corundum-rubble. The corundum-rubble was cast with reinforced concrete into a 0.3 m-thickness, 0.8 m-length block and located in the center of reinforced concrete targets. The reinforcement of CRC targets is the same as reinforced concrete targets and the area of reinforcement ratio is 0.26% too. Figure4 is the diagram of CRC targets.Figure 4
Diagram of CRC targets.


### 2.3. Experimental Method

Time measure instruments and copper mesh were used to record the initial velocity of projectiles; high speed cameras were used to capture the initial impact velocity and impact pose. A 0.125 m-diameter, smooth-bore tank gun can launch the projectiles to impact velocity of 340 m/s (or more) by changing the powder dosage. The copper mesh was 10 meters away from the tank gun. The distance from the copper mesh to the target was 25 meters. The targets faced the tank gun and were perpendicular to the ground to make sure projectiles impact targets vertically. Figure5 is the experiments set-up.Figure 5
Experiments set-up.


## 2.1. Projectile

In the experiments, a 0.684 m-length, 0.125 m-diameter, and 18.29 kg weight truncated-ogival-nosed projectile was designed and it had six empennages to control the flight stability. The projectile was machined from 40 Cr, the density is 7850 kg/m3, the tensile strength is 980 MPa, and the yield stress is 785 MPa. All projectiles were quenched to enhance the strength and hardness. Figure 1 shows the formed projectiles and Figure 2 is the structure diagram of projectiles. The parameters of projectile were summarized in Table 1.Table 1
Parameters of projectile.
l 0 (m) d (m) ρ p (kg/m3) σ y (MPa) σ d (MPa) M (kg) 0.684 0.125 7850 785 980 18.29Figure 1
Projectile.Figure 2
Structure diagram of projectile.


## 2.2. Targets

In the experiments, three CRC targets and reinforced concrete targets number for (N-1)~(N-3) and (C-1)~(C-3), respectively, were designed. The reinforced concrete targets were cast into 2.0 m-diameter, 0.8 m-height cylinder block and were wrapped by 0.003 m-thickness steel drums. The reinforced concrete was constructional reinforcement with Φ10 mild steel, and the area ratio of reinforcement was 0.26%. Figure3 is the diagram of reinforced concrete targets.Figure 3
Diagram of reinforced concrete targets.CRC targets were the same size as reinforced concrete targets, which were filled with some irregular shape of 0.1~0.15 m-diameter corundum-rubble. The corundum-rubble was cast with reinforced concrete into a 0.3 m-thickness, 0.8 m-length block and located in the center of reinforced concrete targets. The reinforcement of CRC targets is the same as reinforced concrete targets and the area of reinforcement ratio is 0.26% too. Figure4 is the diagram of CRC targets.Figure 4
Diagram of CRC targets.


## 2.3. Experimental Method

Time measure instruments and copper mesh were used to record the initial velocity of projectiles; high speed cameras were used to capture the initial impact velocity and impact pose. A 0.125 m-diameter, smooth-bore tank gun can launch the projectiles to impact velocity of 340 m/s (or more) by changing the powder dosage. The copper mesh was 10 meters away from the tank gun. The distance from the copper mesh to the target was 25 meters. The targets faced the tank gun and were perpendicular to the ground to make sure projectiles impact targets vertically. Figure5 is the experiments set-up.Figure 5
Experiments set-up.


## 3. Experimental Results

### 3.1. The Damage of Projectiles

After impacting the reinforced concrete targets, the projectiles got slight damage and stayed with intact shape. While, after impacting the CRC targets, all projectiles got heavily damaged, none of them had the intact shape. Figures6 and 7 are damaged projectiles of reinforced concrete targets and CRC targets, respectively.Figure 6
Damaged projectiles of reinforced concrete targets.Figure 7
Damaged projectiles of CRC targets.Figure6 shows the damaged projectiles after impacting the reinforced concrete targets. One of these three damaged projectiles with the initial velocity of 356 m/s got the most serious damage and all the empennages were broken, but the shape of projectile stayed intact. The other two projectiles also got slight damage and the shape stayed intact.Figure7 shows the damaged projectiles after impacting the CRC targets. These three projectiles got serious damage. When the initial velocities were 355 m/s and 356 m/s, the projectiles were severely bended in one-third of them. When the initial velocity was 353 m/s, the projectile was truncated into two pieces. Besides, all the empennages of these three projectiles were broken.


### 3.2. The Damage of Targets

As was shown in Figure8(C-1), the reinforced concrete target got serious damage (both the front face and rear face subjected to projectile impact). The front face of target formed a 0.82~0.95 m-diameter, 0.603 m penetration depth crater, and there are seven obvious cracks around the target body. At the rear face, a 1.1~2 m-diameter and 0.445 m-depth cone cracking zone was formed, and the target was perforated. The similar phenomena appeared in Figure 8((C-2), (C-3)) and the detailed descriptions were summarized in Table 2.Table 2
Experimental results.
Number M/kg v 0/(m/s) h p/m d c/m d z/m h z/m (C-1) 18.7 353 0.603 0.82~0.95 1.10~2.00 0.445 (C-2) 18.6 347 0.489 0.96~1.23 1.55~1.43 0.370 (C-3) 19.5 356 0.599 0.98~1.30 1.08~1.10 0.450 (N-1) 18.9 355 0.180 0.66~0.82 / / (N-2) 19.2 356 0.170 0.57~0.70 / / (N-3) 18.5 353 0.050 0.40~0.42 / /Figure 8
Damage of reinforced concrete.In Figure9(N-1), a 0.66~0.82 m-diameter and 0.18 m penetration depth crater was formed at the front face of CRC target subjected to projectile impact. It is noteworthy that the rear face stayed intact. Figure 9((N-2), (N-3)) had the similar phenomena and the detailed descriptions were summarized in Table 2 too.Figure 9
Damage of CRC targets.From Figures8 and 9 and Table 2, we can demonstrate that the resistances/strengths of CRC targets were much higher than reinforced concrete targets. The maximum depth of penetration of CRC targets is 0.18 m, which is 63% less than the minimum penetration depth of reinforced concrete targets.


## 3.1. The Damage of Projectiles

After impacting the reinforced concrete targets, the projectiles got slight damage and stayed with intact shape. While, after impacting the CRC targets, all projectiles got heavily damaged, none of them had the intact shape. Figures6 and 7 are damaged projectiles of reinforced concrete targets and CRC targets, respectively.Figure 6
Damaged projectiles of reinforced concrete targets.Figure 7
Damaged projectiles of CRC targets.Figure6 shows the damaged projectiles after impacting the reinforced concrete targets. One of these three damaged projectiles with the initial velocity of 356 m/s got the most serious damage and all the empennages were broken, but the shape of projectile stayed intact. The other two projectiles also got slight damage and the shape stayed intact.Figure7 shows the damaged projectiles after impacting the CRC targets. These three projectiles got serious damage. When the initial velocities were 355 m/s and 356 m/s, the projectiles were severely bended in one-third of them. When the initial velocity was 353 m/s, the projectile was truncated into two pieces. Besides, all the empennages of these three projectiles were broken.


## 3.2. The Damage of Targets

As was shown in Figure8(C-1), the reinforced concrete target got serious damage (both the front face and rear face subjected to projectile impact). The front face of target formed a 0.82~0.95 m-diameter, 0.603 m penetration depth crater, and there are seven obvious cracks around the target body. At the rear face, a 1.1~2 m-diameter and 0.445 m-depth cone cracking zone was formed, and the target was perforated. The similar phenomena appeared in Figure 8((C-2), (C-3)) and the detailed descriptions were summarized in Table 2.Table 2
Experimental results.
Number M/kg v 0/(m/s) h p/m d c/m d z/m h z/m (C-1) 18.7 353 0.603 0.82~0.95 1.10~2.00 0.445 (C-2) 18.6 347 0.489 0.96~1.23 1.55~1.43 0.370 (C-3) 19.5 356 0.599 0.98~1.30 1.08~1.10 0.450 (N-1) 18.9 355 0.180 0.66~0.82 / / (N-2) 19.2 356 0.170 0.57~0.70 / / (N-3) 18.5 353 0.050 0.40~0.42 / /Figure 8
Damage of reinforced concrete.In Figure9(N-1), a 0.66~0.82 m-diameter and 0.18 m penetration depth crater was formed at the front face of CRC target subjected to projectile impact. It is noteworthy that the rear face stayed intact. Figure 9((N-2), (N-3)) had the similar phenomena and the detailed descriptions were summarized in Table 2 too.Figure 9
Damage of CRC targets.From Figures8 and 9 and Table 2, we can demonstrate that the resistances/strengths of CRC targets were much higher than reinforced concrete targets. The maximum depth of penetration of CRC targets is 0.18 m, which is 63% less than the minimum penetration depth of reinforced concrete targets.


## 4. Theoretical Analysis of Penetration Depth

### 4.1. Penetration Depth of the Reinforced Concrete Targets

Many empirical and semianalytical formulae were proposed to calculate penetration depth of concrete targets in recent years. Some of them were widely used in plenty of reviews such as BRL, ACE, and NDRC. Herein, we listed some formulae in the following.


#### 4.1.1. The Modified Ballistic Research Laboratory (BRL) Formula Is Given by [3]

(1) x d = 1.33 × 1 0 - 3 f c M d 3 d 0.2 V 0 1.33 .


#### 4.1.2. The US Army Corps of Engineers (ACE) Formula Is Given by [3]

(2) x d = 3.5 × 1 0 - 4 f c M d 3 d 0.215 V 0 1.5 + 0.5 .


#### 4.1.3. The Modified Formula of National Defence Research Committee (NDRC) Is Given by [3]

(3) G = 3.8 × 1 0 - 5 N ∗ M d f c V 0 d 1.8 ,where function G is given by(4)
G
=
x
2
d
2
,
for
x
d
≤
2
,
x
d
-
1
,
for
x
d
>
2
.


#### 4.1.4. The United Kingdom Atomic Energy Authority (UKAEA) Formula [3]

Based on extensive studies conducted for the penetration of nuclear power plant structures in the UK, Barr suggested the following further modification to the NDRC formula, directed mainly toward the lower impact velocities more relevant to the nuclear industry:(5)G=3.8×10-5N∗MdfcV0d1.8,where function G is defined by(6)xd=0.275-0.0756-G0.5,forG≤0.0726,4G-0.2420.5,for0.0726≤G≤1.0605,G+0.9395,forG≥1.0605.


#### 4.1.5. The UMIST Penetration Formula Is Given by [17, 18]

(7) x d = 2 π N ∗ M V 0 2 0.72 σ t d 3 ,where σt is defined by(8)σtMPa=4.2fcMPa+135.0+0.014fcMPa+0.45V0.


#### 4.1.6. The Wen and Yang Penetration Formula Is Given by [16]

(9) x d = 2 M V 0 2 π σ d 3 ,where σ is defined by(10)σ=α+βρmσmV0σmand σm is defined by(11)σm=1.4fc+45,forfc≤75MPa,150,for75MPa≤fc≤150MPa,fc,forfc≥150MPa,where α and β are given in Table 3.Table 3
Values of various parameters for concrete targets [16].
Shape α β Conical-nosed (θ < 90°) 1 2 1 + ln ⁡ 2 E 5 - 4 v σ m 2 sin ⁡ θ 2  Conical-nosed (90° ≤θ < 180°) 1 2 1 + ln ⁡ 2 E 5 - 4 v σ m 2  Flat-nosed 1 2 1 + ln ⁡ 2 E 5 - 4 v σ m 2  Ogival-nosed 2 3 1 + ln ⁡ E 3 1 - v σ m 3 4 ψ  Hemispherical-nosed 2 3 1 + ln ⁡ E 3 1 - v σ m 3 2In Table4, some famous penetration formulae were quoted to calculate the penetration depth of reinforced concrete targets and the calculated results were compared with the experimental results. We can draw a conclusion that the experiments we conducted are reasonable and experimental results are reliable.Table 4
Comparisons of various equations with the test data for the DOP of reinforced concrete.
Penetration formulae (C-1) (C-2) (C-3) x / d x x / d x x / d x BRL 2.967 0.371 2.873 0.359 3.129 0.391 ACE 2.551 0.319 2.480 0.310 2.666 0.333 NDRC 1.509 0.307 1.448 0.301 1.598 0.316 UKAEA 2.448 0.306 2.387 0.298 2.375 0.317 UMIST 1.628 0.203 1.572 0.196 1.719 0.215 Wen & Yang 5.558 0.695 5.335 0.667 5.921 0.740  Experiment results 4.824 0.603 3.912 0.489 4.792 0.599


### 4.2. Penetration Depth of the CRC Targets

The projectiles penetration effects researches of high strength concrete had attracted much attention. A variety of calculation models have been proposed, such as FFI model [19, 20], Riera and Iturrioz model [21], Taylor model [22], Tate model [23, 24], and Lee-Tupper model [25]. The FFI model is suitable for the 0.012 m-diameter projectile with the impact velocity from 400 m/s to 1700 m/s. Targets deformation has not been considered in both the Riera and Iturrioz model and Taylor model. Riera and Iturrioz model’s yield load is calculated according to pipe quasi-static axial impact load, without considering the inertia effect. It will bring a larger error to calculate the penetration depth with impact velocity above 340 m/s. Tate model focuses on the impact velocity above 1000 m/s; in this paper, the impact velocity is about 340 m/s, and the actual velocity is not more than the plastic wave velocity, so Lee-model does not suit too.In this paper, we see the large plastic deformation as part of quality loss of projectiles. Taking targets deformation into consideration, we build a new model called “modified Taylor model” to analyze the deformation of the CRC targets.


#### 4.2.1. Modified Taylor Model

Suppose that the projectile is flat, cylindrical, ideal rigid-plastic material, initial length isl0, cross-sectional area is A0, density is ρp, dynamic yielding stress is σd, and initial velocity is v0. The model is given in Figure 10.Figure 10
One-dimensional simplified diagram of modified Taylor model.In Figure10, z is the distance between elastic-plastic interface of the projectile and projectile-target contact interface. x is the length of rigid zone, where plastic deformation has not occurred; h is the depth of penetration; u is the velocity of plastic interface, which spreads to the left; v is the velocity of rigid zone, which spreads to the front; A0 is rigid zone section; A is the plastic zone section, which has got deformation; vc is the anytime velocity of projectile-target contact interface, which can be deserved by the impact velocity.Indt time, the penetration depth dh=vcdt, the rigid zone length dx=-(u+v)dt, and the plastic zone length dz=udt+vcdt; hence,(12)dzdt=u+vc,dxdt=-u+v,dhdt=vc.The applied force of rigid zone without deformation isσdA0; equation of motion is (13)σdA0=-ρpA0xdvdt,wheredvdt=-σdxρp.According to the law of volume constancy,(14)A0u+v=Au+vc.Indt time, assume that the expanding of cross-sectional area is uniform deceleration; the velocity is(15)S˙=w01-tdt.In (15), w0 is the expanding velocity; S˙ is the anytime cross-sectional area.Boundary conditions aret=0, S=A0, t=dt, and S=A, so(16)S=A-A02t-t2dt+A0.Hence, indt time, the momentum of plastic zone material is (17)ρpA0u+vdt·v.The momentum turns into impulse of pressure in the plastic zone, where the impulse is(18)I=∫0dtσdS-A0dT=23σdA-A0dt.According to the law of conservation of momentum,(19)ρpA0u+vv=23σdA-A0.Equations (12)~(19) are the control equations to analyze deformation and destruction of projectile; in these equations, variables h, x, z, v, u, and A are unknown. Initial conditions are t=0, h=0, x=l0, z=0, v=v0, A=A1, and u=u0.When the projectile impacts the target with a velocity ofv, the contact stress of interface is σc, the projectile gets velocity v1 to the left, and at the same time the target gets velocity v2 to the right; hence, the velocity of contact interface vc=v2=v-v1.According to the law of conservation of momentum, in tiny time ofδt, the stress wave spreads to the left with distance of δx; hence,(20)σcδt=ρpδxv1,σcδt=ρtδxv2.C p p and Cpt were defined by the following expressions:(21)δxδtp=Cpp,δxδtt=Cpt,where Cpp is the plastic velocity of projectile and Cpt is the plastic velocity of target.From (20)~(21), we can deserve that(22)v1=σcρpCpp,(23)v2=σcρtCpt,(24)v0=σc1ρpCpp+1ρtCpt.From (22)~(24), we can deserve that(25)vc=v1+ρtCpt/ρpCpp.From (14)~(19) and the initial conditions, we can deserve that (26)3ρpv22Kσd=A1-A02A1A0=A1A0+A0A1-2.In (26), K=1+ρpCpp/ρtCpt.At the initial time, the initial velocity isv0; if β=3ρpv02/4Kσd, we can deserve the initial cross-sectional area A1 according to (26): (27)A1=β+1+β2+2βA0.Ifα=β+1+β2+2β, u=u0, v=v0, and A=A1, we can deserve the velocity of plastic area u0 and v0 according to (14):(28)u0=A0A1-A0v0-A1A1-A0vc0=αvc0-v01-α,vc0=v01+ρtCpt/ρpCpp.To solve (25), (26), and (28), the plastic velocity of projectile Cpp and the plastic velocity of target Cpt should be determined first. We can also solve above four equations if we deserve the value of Cpt/Cpp. As we know, the plastic wave speed of projectile to target ratio is equal to 0.1, and the elastic wave speed of projectile to target ratio is equal to 0.1 too; hence, we can deserve the value of Cpt/Cpp through the value of Cot/Cop, so the equation is given by (29)CptCpp=CotCop=Eot/ρtEop/ρp=EotρpEopρt.In (29), ρp is the density of projectile, ρt is the density of target, Eop is the elasticity modulus of projectile, and Eot is the elasticity modulus of target.Equations (12)~(19) are the differential algebraic equations, with the initial conditions of projectile and target; we can solve equations from (20) to (29) and the calculated results can be used to solve the differential algebraic equations.Hence, the penetration depth curve is calculated as(30)ht=-7.4×10-9-4.49×107t+v01.871.54+7.4×10-9v01.871.54.In the experiments, (30) is the final equation to predict the depth of penetration. We can calculate the depth of penetration of different initial velocity. Figures 11(a), 11(b), and 11(c) are the depth-time curves of initial velocity, 355 m/s, 356 m/s, and 353 m/s, respectively.Figure 11
Penetration depth curves.
(a)
Initial velocity of 355 m/s (b)
Initial velocity of 356 m/s (c)
Initial velocity of 353 m/sFrom Table5, it is noticed that the modified Taylor model can be used to predict the penetration depths of CRC targets and would give the effective consequence.Table 5
Penetration depth of experiments and equation.
Number Initial velocity Experimental results Equation results 1 355 0.17 0.161 2 356 0.18 0.165 3 353 0.05 0.163In the experiments, the depths of penetration are 0.17 m and 0.18 m, which are consistent with the calculated results of (30), while the depth of 0.05 m is not consistent with (30)’s calculated result. It would be necessary to make an illustration that the projectile impacted the target in the vertical direction without any angle on the corundum-rubble in the experiment of initial velocity, 353 m/s. Thus, the corundum-rubble would play the most important role to resist the impact of projectile. As was shown in the front section, the compressive strength of corundum-rubble was over 2000 MPa and Mohs’ hardness scale of the corundum-rubble is 9.1, which are so high that the projectile is hardly making penetration on it, so the penetration depth is much lower than other two experimental results.


## 4.1. Penetration Depth of the Reinforced Concrete Targets

Many empirical and semianalytical formulae were proposed to calculate penetration depth of concrete targets in recent years. Some of them were widely used in plenty of reviews such as BRL, ACE, and NDRC. Herein, we listed some formulae in the following.


### 4.1.1. The Modified Ballistic Research Laboratory (BRL) Formula Is Given by [3]

(1) x d = 1.33 × 1 0 - 3 f c M d 3 d 0.2 V 0 1.33 .


### 4.1.2. The US Army Corps of Engineers (ACE) Formula Is Given by [3]

(2) x d = 3.5 × 1 0 - 4 f c M d 3 d 0.215 V 0 1.5 + 0.5 .


### 4.1.3. The Modified Formula of National Defence Research Committee (NDRC) Is Given by [3]

(3) G = 3.8 × 1 0 - 5 N ∗ M d f c V 0 d 1.8 ,where function G is given by(4)
G
=
x
2
d
2
,
for
x
d
≤
2
,
x
d
-
1
,
for
x
d
>
2
.


### 4.1.4. The United Kingdom Atomic Energy Authority (UKAEA) Formula [3]

Based on extensive studies conducted for the penetration of nuclear power plant structures in the UK, Barr suggested the following further modification to the NDRC formula, directed mainly toward the lower impact velocities more relevant to the nuclear industry:(5)G=3.8×10-5N∗MdfcV0d1.8,where function G is defined by(6)xd=0.275-0.0756-G0.5,forG≤0.0726,4G-0.2420.5,for0.0726≤G≤1.0605,G+0.9395,forG≥1.0605.


### 4.1.5. The UMIST Penetration Formula Is Given by [17, 18]

(7) x d = 2 π N ∗ M V 0 2 0.72 σ t d 3 ,where σt is defined by(8)σtMPa=4.2fcMPa+135.0+0.014fcMPa+0.45V0.


### 4.1.6. The Wen and Yang Penetration Formula Is Given by [16]

(9) x d = 2 M V 0 2 π σ d 3 ,where σ is defined by(10)σ=α+βρmσmV0σmand σm is defined by(11)σm=1.4fc+45,forfc≤75MPa,150,for75MPa≤fc≤150MPa,fc,forfc≥150MPa,where α and β are given in Table 3.Table 3
Values of various parameters for concrete targets [16].
Shape α β Conical-nosed (θ < 90°) 1 2 1 + ln ⁡ 2 E 5 - 4 v σ m 2 sin ⁡ θ 2  Conical-nosed (90° ≤θ < 180°) 1 2 1 + ln ⁡ 2 E 5 - 4 v σ m 2  Flat-nosed 1 2 1 + ln ⁡ 2 E 5 - 4 v σ m 2  Ogival-nosed 2 3 1 + ln ⁡ E 3 1 - v σ m 3 4 ψ  Hemispherical-nosed 2 3 1 + ln ⁡ E 3 1 - v σ m 3 2In Table4, some famous penetration formulae were quoted to calculate the penetration depth of reinforced concrete targets and the calculated results were compared with the experimental results. We can draw a conclusion that the experiments we conducted are reasonable and experimental results are reliable.Table 4
Comparisons of various equations with the test data for the DOP of reinforced concrete.
Penetration formulae (C-1) (C-2) (C-3) x / d x x / d x x / d x BRL 2.967 0.371 2.873 0.359 3.129 0.391 ACE 2.551 0.319 2.480 0.310 2.666 0.333 NDRC 1.509 0.307 1.448 0.301 1.598 0.316 UKAEA 2.448 0.306 2.387 0.298 2.375 0.317 UMIST 1.628 0.203 1.572 0.196 1.719 0.215 Wen & Yang 5.558 0.695 5.335 0.667 5.921 0.740  Experiment results 4.824 0.603 3.912 0.489 4.792 0.599


## 4.1.1. The Modified Ballistic Research Laboratory (BRL) Formula Is Given by [3]

(1) x d = 1.33 × 1 0 - 3 f c M d 3 d 0.2 V 0 1.33 .


## 4.1.2. The US Army Corps of Engineers (ACE) Formula Is Given by [3]

(2) x d = 3.5 × 1 0 - 4 f c M d 3 d 0.215 V 0 1.5 + 0.5 .


## 4.1.3. The Modified Formula of National Defence Research Committee (NDRC) Is Given by [3]

(3) G = 3.8 × 1 0 - 5 N ∗ M d f c V 0 d 1.8 ,where function G is given by(4)
G
=
x
2
d
2
,
for
x
d
≤
2
,
x
d
-
1
,
for
x
d
>
2
.


## 4.1.4. The United Kingdom Atomic Energy Authority (UKAEA) Formula [3]

Based on extensive studies conducted for the penetration of nuclear power plant structures in the UK, Barr suggested the following further modification to the NDRC formula, directed mainly toward the lower impact velocities more relevant to the nuclear industry:(5)G=3.8×10-5N∗MdfcV0d1.8,where function G is defined by(6)xd=0.275-0.0756-G0.5,forG≤0.0726,4G-0.2420.5,for0.0726≤G≤1.0605,G+0.9395,forG≥1.0605.


## 4.1.5. The UMIST Penetration Formula Is Given by [17, 18]

(7) x d = 2 π N ∗ M V 0 2 0.72 σ t d 3 ,where σt is defined by(8)σtMPa=4.2fcMPa+135.0+0.014fcMPa+0.45V0.


## 4.1.6. The Wen and Yang Penetration Formula Is Given by [16]

(9) x d = 2 M V 0 2 π σ d 3 ,where σ is defined by(10)σ=α+βρmσmV0σmand σm is defined by(11)σm=1.4fc+45,forfc≤75MPa,150,for75MPa≤fc≤150MPa,fc,forfc≥150MPa,where α and β are given in Table 3.Table 3
Values of various parameters for concrete targets [16].
Shape α β Conical-nosed (θ < 90°) 1 2 1 + ln ⁡ 2 E 5 - 4 v σ m 2 sin ⁡ θ 2  Conical-nosed (90° ≤θ < 180°) 1 2 1 + ln ⁡ 2 E 5 - 4 v σ m 2  Flat-nosed 1 2 1 + ln ⁡ 2 E 5 - 4 v σ m 2  Ogival-nosed 2 3 1 + ln ⁡ E 3 1 - v σ m 3 4 ψ  Hemispherical-nosed 2 3 1 + ln ⁡ E 3 1 - v σ m 3 2In Table4, some famous penetration formulae were quoted to calculate the penetration depth of reinforced concrete targets and the calculated results were compared with the experimental results. We can draw a conclusion that the experiments we conducted are reasonable and experimental results are reliable.Table 4
Comparisons of various equations with the test data for the DOP of reinforced concrete.
Penetration formulae (C-1) (C-2) (C-3) x / d x x / d x x / d x BRL 2.967 0.371 2.873 0.359 3.129 0.391 ACE 2.551 0.319 2.480 0.310 2.666 0.333 NDRC 1.509 0.307 1.448 0.301 1.598 0.316 UKAEA 2.448 0.306 2.387 0.298 2.375 0.317 UMIST 1.628 0.203 1.572 0.196 1.719 0.215 Wen & Yang 5.558 0.695 5.335 0.667 5.921 0.740  Experiment results 4.824 0.603 3.912 0.489 4.792 0.599


## 4.2. Penetration Depth of the CRC Targets

The projectiles penetration effects researches of high strength concrete had attracted much attention. A variety of calculation models have been proposed, such as FFI model [19, 20], Riera and Iturrioz model [21], Taylor model [22], Tate model [23, 24], and Lee-Tupper model [25]. The FFI model is suitable for the 0.012 m-diameter projectile with the impact velocity from 400 m/s to 1700 m/s. Targets deformation has not been considered in both the Riera and Iturrioz model and Taylor model. Riera and Iturrioz model’s yield load is calculated according to pipe quasi-static axial impact load, without considering the inertia effect. It will bring a larger error to calculate the penetration depth with impact velocity above 340 m/s. Tate model focuses on the impact velocity above 1000 m/s; in this paper, the impact velocity is about 340 m/s, and the actual velocity is not more than the plastic wave velocity, so Lee-model does not suit too.In this paper, we see the large plastic deformation as part of quality loss of projectiles. Taking targets deformation into consideration, we build a new model called “modified Taylor model” to analyze the deformation of the CRC targets.


### 4.2.1. Modified Taylor Model

Suppose that the projectile is flat, cylindrical, ideal rigid-plastic material, initial length isl0, cross-sectional area is A0, density is ρp, dynamic yielding stress is σd, and initial velocity is v0. The model is given in Figure 10.Figure 10
One-dimensional simplified diagram of modified Taylor model.In Figure10, z is the distance between elastic-plastic interface of the projectile and projectile-target contact interface. x is the length of rigid zone, where plastic deformation has not occurred; h is the depth of penetration; u is the velocity of plastic interface, which spreads to the left; v is the velocity of rigid zone, which spreads to the front; A0 is rigid zone section; A is the plastic zone section, which has got deformation; vc is the anytime velocity of projectile-target contact interface, which can be deserved by the impact velocity.Indt time, the penetration depth dh=vcdt, the rigid zone length dx=-(u+v)dt, and the plastic zone length dz=udt+vcdt; hence,(12)dzdt=u+vc,dxdt=-u+v,dhdt=vc.The applied force of rigid zone without deformation isσdA0; equation of motion is (13)σdA0=-ρpA0xdvdt,wheredvdt=-σdxρp.According to the law of volume constancy,(14)A0u+v=Au+vc.Indt time, assume that the expanding of cross-sectional area is uniform deceleration; the velocity is(15)S˙=w01-tdt.In (15), w0 is the expanding velocity; S˙ is the anytime cross-sectional area.Boundary conditions aret=0, S=A0, t=dt, and S=A, so(16)S=A-A02t-t2dt+A0.Hence, indt time, the momentum of plastic zone material is (17)ρpA0u+vdt·v.The momentum turns into impulse of pressure in the plastic zone, where the impulse is(18)I=∫0dtσdS-A0dT=23σdA-A0dt.According to the law of conservation of momentum,(19)ρpA0u+vv=23σdA-A0.Equations (12)~(19) are the control equations to analyze deformation and destruction of projectile; in these equations, variables h, x, z, v, u, and A are unknown. Initial conditions are t=0, h=0, x=l0, z=0, v=v0, A=A1, and u=u0.When the projectile impacts the target with a velocity ofv, the contact stress of interface is σc, the projectile gets velocity v1 to the left, and at the same time the target gets velocity v2 to the right; hence, the velocity of contact interface vc=v2=v-v1.According to the law of conservation of momentum, in tiny time ofδt, the stress wave spreads to the left with distance of δx; hence,(20)σcδt=ρpδxv1,σcδt=ρtδxv2.C p p and Cpt were defined by the following expressions:(21)δxδtp=Cpp,δxδtt=Cpt,where Cpp is the plastic velocity of projectile and Cpt is the plastic velocity of target.From (20)~(21), we can deserve that(22)v1=σcρpCpp,(23)v2=σcρtCpt,(24)v0=σc1ρpCpp+1ρtCpt.From (22)~(24), we can deserve that(25)vc=v1+ρtCpt/ρpCpp.From (14)~(19) and the initial conditions, we can deserve that (26)3ρpv22Kσd=A1-A02A1A0=A1A0+A0A1-2.In (26), K=1+ρpCpp/ρtCpt.At the initial time, the initial velocity isv0; if β=3ρpv02/4Kσd, we can deserve the initial cross-sectional area A1 according to (26): (27)A1=β+1+β2+2βA0.Ifα=β+1+β2+2β, u=u0, v=v0, and A=A1, we can deserve the velocity of plastic area u0 and v0 according to (14):(28)u0=A0A1-A0v0-A1A1-A0vc0=αvc0-v01-α,vc0=v01+ρtCpt/ρpCpp.To solve (25), (26), and (28), the plastic velocity of projectile Cpp and the plastic velocity of target Cpt should be determined first. We can also solve above four equations if we deserve the value of Cpt/Cpp. As we know, the plastic wave speed of projectile to target ratio is equal to 0.1, and the elastic wave speed of projectile to target ratio is equal to 0.1 too; hence, we can deserve the value of Cpt/Cpp through the value of Cot/Cop, so the equation is given by (29)CptCpp=CotCop=Eot/ρtEop/ρp=EotρpEopρt.In (29), ρp is the density of projectile, ρt is the density of target, Eop is the elasticity modulus of projectile, and Eot is the elasticity modulus of target.Equations (12)~(19) are the differential algebraic equations, with the initial conditions of projectile and target; we can solve equations from (20) to (29) and the calculated results can be used to solve the differential algebraic equations.Hence, the penetration depth curve is calculated as(30)ht=-7.4×10-9-4.49×107t+v01.871.54+7.4×10-9v01.871.54.In the experiments, (30) is the final equation to predict the depth of penetration. We can calculate the depth of penetration of different initial velocity. Figures 11(a), 11(b), and 11(c) are the depth-time curves of initial velocity, 355 m/s, 356 m/s, and 353 m/s, respectively.Figure 11
Penetration depth curves.
(a)
Initial velocity of 355 m/s (b)
Initial velocity of 356 m/s (c)
Initial velocity of 353 m/sFrom Table5, it is noticed that the modified Taylor model can be used to predict the penetration depths of CRC targets and would give the effective consequence.Table 5
Penetration depth of experiments and equation.
Number Initial velocity Experimental results Equation results 1 355 0.17 0.161 2 356 0.18 0.165 3 353 0.05 0.163In the experiments, the depths of penetration are 0.17 m and 0.18 m, which are consistent with the calculated results of (30), while the depth of 0.05 m is not consistent with (30)’s calculated result. It would be necessary to make an illustration that the projectile impacted the target in the vertical direction without any angle on the corundum-rubble in the experiment of initial velocity, 353 m/s. Thus, the corundum-rubble would play the most important role to resist the impact of projectile. As was shown in the front section, the compressive strength of corundum-rubble was over 2000 MPa and Mohs’ hardness scale of the corundum-rubble is 9.1, which are so high that the projectile is hardly making penetration on it, so the penetration depth is much lower than other two experimental results.


## 4.2.1. Modified Taylor Model

Suppose that the projectile is flat, cylindrical, ideal rigid-plastic material, initial length isl0, cross-sectional area is A0, density is ρp, dynamic yielding stress is σd, and initial velocity is v0. The model is given in Figure 10.Figure 10
One-dimensional simplified diagram of modified Taylor model.In Figure10, z is the distance between elastic-plastic interface of the projectile and projectile-target contact interface. x is the length of rigid zone, where plastic deformation has not occurred; h is the depth of penetration; u is the velocity of plastic interface, which spreads to the left; v is the velocity of rigid zone, which spreads to the front; A0 is rigid zone section; A is the plastic zone section, which has got deformation; vc is the anytime velocity of projectile-target contact interface, which can be deserved by the impact velocity.Indt time, the penetration depth dh=vcdt, the rigid zone length dx=-(u+v)dt, and the plastic zone length dz=udt+vcdt; hence,(12)dzdt=u+vc,dxdt=-u+v,dhdt=vc.The applied force of rigid zone without deformation isσdA0; equation of motion is (13)σdA0=-ρpA0xdvdt,wheredvdt=-σdxρp.According to the law of volume constancy,(14)A0u+v=Au+vc.Indt time, assume that the expanding of cross-sectional area is uniform deceleration; the velocity is(15)S˙=w01-tdt.In (15), w0 is the expanding velocity; S˙ is the anytime cross-sectional area.Boundary conditions aret=0, S=A0, t=dt, and S=A, so(16)S=A-A02t-t2dt+A0.Hence, indt time, the momentum of plastic zone material is (17)ρpA0u+vdt·v.The momentum turns into impulse of pressure in the plastic zone, where the impulse is(18)I=∫0dtσdS-A0dT=23σdA-A0dt.According to the law of conservation of momentum,(19)ρpA0u+vv=23σdA-A0.Equations (12)~(19) are the control equations to analyze deformation and destruction of projectile; in these equations, variables h, x, z, v, u, and A are unknown. Initial conditions are t=0, h=0, x=l0, z=0, v=v0, A=A1, and u=u0.When the projectile impacts the target with a velocity ofv, the contact stress of interface is σc, the projectile gets velocity v1 to the left, and at the same time the target gets velocity v2 to the right; hence, the velocity of contact interface vc=v2=v-v1.According to the law of conservation of momentum, in tiny time ofδt, the stress wave spreads to the left with distance of δx; hence,(20)σcδt=ρpδxv1,σcδt=ρtδxv2.C p p and Cpt were defined by the following expressions:(21)δxδtp=Cpp,δxδtt=Cpt,where Cpp is the plastic velocity of projectile and Cpt is the plastic velocity of target.From (20)~(21), we can deserve that(22)v1=σcρpCpp,(23)v2=σcρtCpt,(24)v0=σc1ρpCpp+1ρtCpt.From (22)~(24), we can deserve that(25)vc=v1+ρtCpt/ρpCpp.From (14)~(19) and the initial conditions, we can deserve that (26)3ρpv22Kσd=A1-A02A1A0=A1A0+A0A1-2.In (26), K=1+ρpCpp/ρtCpt.At the initial time, the initial velocity isv0; if β=3ρpv02/4Kσd, we can deserve the initial cross-sectional area A1 according to (26): (27)A1=β+1+β2+2βA0.Ifα=β+1+β2+2β, u=u0, v=v0, and A=A1, we can deserve the velocity of plastic area u0 and v0 according to (14):(28)u0=A0A1-A0v0-A1A1-A0vc0=αvc0-v01-α,vc0=v01+ρtCpt/ρpCpp.To solve (25), (26), and (28), the plastic velocity of projectile Cpp and the plastic velocity of target Cpt should be determined first. We can also solve above four equations if we deserve the value of Cpt/Cpp. As we know, the plastic wave speed of projectile to target ratio is equal to 0.1, and the elastic wave speed of projectile to target ratio is equal to 0.1 too; hence, we can deserve the value of Cpt/Cpp through the value of Cot/Cop, so the equation is given by (29)CptCpp=CotCop=Eot/ρtEop/ρp=EotρpEopρt.In (29), ρp is the density of projectile, ρt is the density of target, Eop is the elasticity modulus of projectile, and Eot is the elasticity modulus of target.Equations (12)~(19) are the differential algebraic equations, with the initial conditions of projectile and target; we can solve equations from (20) to (29) and the calculated results can be used to solve the differential algebraic equations.Hence, the penetration depth curve is calculated as(30)ht=-7.4×10-9-4.49×107t+v01.871.54+7.4×10-9v01.871.54.In the experiments, (30) is the final equation to predict the depth of penetration. We can calculate the depth of penetration of different initial velocity. Figures 11(a), 11(b), and 11(c) are the depth-time curves of initial velocity, 355 m/s, 356 m/s, and 353 m/s, respectively.Figure 11
Penetration depth curves.
(a)
Initial velocity of 355 m/s (b)
Initial velocity of 356 m/s (c)
Initial velocity of 353 m/sFrom Table5, it is noticed that the modified Taylor model can be used to predict the penetration depths of CRC targets and would give the effective consequence.Table 5
Penetration depth of experiments and equation.
Number Initial velocity Experimental results Equation results 1 355 0.17 0.161 2 356 0.18 0.165 3 353 0.05 0.163In the experiments, the depths of penetration are 0.17 m and 0.18 m, which are consistent with the calculated results of (30), while the depth of 0.05 m is not consistent with (30)’s calculated result. It would be necessary to make an illustration that the projectile impacted the target in the vertical direction without any angle on the corundum-rubble in the experiment of initial velocity, 353 m/s. Thus, the corundum-rubble would play the most important role to resist the impact of projectile. As was shown in the front section, the compressive strength of corundum-rubble was over 2000 MPa and Mohs’ hardness scale of the corundum-rubble is 9.1, which are so high that the projectile is hardly making penetration on it, so the penetration depth is much lower than other two experimental results.


## 5. Conclusions

In this paper, comparative experiments were conducted to investigate the resistances/strengths and penetration depths of reinforced concrete targets and CRC targets. And the modified Taylor model was proposed to predict the penetration depths of CRC targets. From the experimental results and formulae results, we can deserve the following:(1)
The resistances/strengths of CRC targets were much higher than reinforced concrete targets, which can be validated from the damage of projectiles and targets (Figures6, 7, 8, and 9).(2)
Through the contrastive analyses of experimental results and formulae results of penetration depth of reinforced concrete targets, we have reasons to believe that the experiments were reasonable and experimental results were credible.(3)
The penetration depth values of CRC targets validate that the modified Taylor model was available and the predicted results agreed well with the experimental results.



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*Source: 1396567-2017-01-22.xml*