# Mass Loss and Penetration Depth of Hypervelocity Projectiles

Authors: Tianhan Xu; Yanyu Qiu; Jie Li; Yihao Cheng; Wei Chen
Journal: Shock and Vibration
 (2021)

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## Abstract

Numerous investigations recorded a dramatic decrease in penetration depth at hypervelocity. The decrease is due to the severe mass loss of the projectile. The mechanics of mass loss are complicated, so the entirely theoretical model is still absent. In this paper, we derive a semitheoretical formula for mass loss by solving the equation of motion of the projectile during penetration. The result shows that the decrease in projectile mass at hypervelocity accords with the power law. Furthermore, we obtain a continuous formula for depth of penetration in the entire velocity domain. The theoretical results agree well with experimental data. The formula for the critical velocity of depth decreasing is also obtained as a by-product of the solving procedure.


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## Body

## 1. Introduction

With the rapid development of hypervelocity air vehicles, the hitting speed of hypervelocity weapons has reached Mach 5–15 (Mach 1 = 340.3 m/s). There will be intense interactions on the interface between the projectile and the target at such high velocities. Melting, blunting, and mass loss of the projectile will occur. These phenomena weaken the projectile and diminish the terminal depth of penetration (DOP).Numerous experiments have confirmed that projectile mass decreases dramatically under hypervelocity penetration. Forrestal and Piekutowski [1] used steel projectiles to penetrate aluminum targets and recorded the vanishing of the projectile at the impact velocity of 3075 m/s with the help of radiographs. Qian et al. [2] conducted penetration experiments with tungsten alloy projectiles and concrete targets. The projectile was entirely eroded at the impact velocity of 3360 m/s. Li et al. [3] penetrated granite targets with alloy steel projectiles. The percentage of mass loss reached 88% when the impact velocity was 2378 m/s, and the velocity range of severe mass loss corresponded to that of DOP decreasing.Projectile mass loss plays an essential role during the penetration and dominates the DOP in a specific range of velocity. Researchers have conducted numerous investigations on the mechanisms of mass loss. Silling and Forrestal [4] found a linear relationship between mass loss and the initial kinetic energy of projectiles. Klepaczko and Hughes [5] analyzed the wear at high sliding speeds and defined universal parameters to measure the wear intensity during penetration. Jones et al. [6] studied the work and heat during penetration and calculated the mass loss caused by surface melting. It is clear that many physical processes participate in the penetration process, leading to projectile mass loss. He et al. [7] combined several models and provided a formula with seven variables to estimate the mass loss.These investigations were conducted on the condition that the impact velocity is below the transition velocity of semihydrodynamics, which is about 1000–1300 m/s. In this region, the DOP increases with impact velocity, though there is mass loss during penetration, and the projectiles maintain their primary forms after penetration. However, at hypervelocities, the dramatic decrease in DOP along with intense abrasion of projectiles will occur. It is evident that more complex physical processes occur during hypervelocity penetration, resulting in considerable difficulties in establishing accurate models to describe the process of mass loss. For engineering applications, a solution is to develop a semitheoretical model with unknown coefficients and to determine the coefficients by fitting experimental data.In this paper, we analyze the penetration process of a cylindrical long rod projectile into a semi-infinite rock target and establish an equation of motion of the projectile. By solving the equation, we derive a formula for intense mass loss with one unknown coefficient. We further obtain the equations for DOP from low velocities to hypervelocities. In-depth analyses of the coefficients are carried out, and their physical meanings are clarified.


## 2. Dynamic Behaviors of Projectile and Target during Penetration

Figure1 depicts the change of DOP with the impact velocity [3] (h is the DOP, L is the total length of the projectile, and υ0 is the impact velocity). As shown in the figure, when the impact velocity is low, the DOP increases until the velocity reaches a critical value and then decreases dramatically with the velocity. After reaching the minimum, the DOP increases again with the velocity increasing. The change of the DOP is closely related to the states of projectile and target. Therefore, it is necessary to understand the behaviors of projectile and target during penetration under different impact velocities.Figure 1
The curve of DOP vs. impact velocities.


### 2.1. Dynamic Behaviors of the Projectile

With increasing the impact velocity, the interactions between projectile and target become more intense, and the behaviors of the projectile can be divided into four states as follows [3]:(a)
Rigid State. When the impact velocity is relatively low, the deformation and mass loss of the projectile are negligible, and the projectile can be regarded as a rigid body.(b)
Deformation State. When the pressure on the interface between projectile and target reaches the dynamic yield stress of the projectile, the projectile nose turns to the plastic state. Melting and mass loss also occur within a thin layer on the surface of the projectile nose. These factors blunt the projectile nose and slightly weaken the penetration efficiency of the projectile.(c)
Erosion State. With increasing the impact velocity, the interactions on the interface between projectile and target become more intense. The scope of yielding, melting, and mass loss of the projectile expands, leading to the failure of the structure of the projectile. In this state, the terminal DOP decreases with the impact velocity increasing.(d)
Hydrodynamic State. When the impact velocity further increases, the projectile as a whole enters the plastic flow state and can be regarded as a steady-state fluid. The projectile mass is entirely lost at the end of the penetration.


### 2.2. Dynamic Behaviors of the Target

With the increase in the impact velocity, the pressure on the target side of the interface also increases. The stress-strain state of the target in the near-penetration region can be seen as the one-dimensional strain state, and the induced stress waves are one-dimensional plane waves [8]. Metal targets turn to plastic flow at the elastic limit. Concrete and rock targets, however, turn to a hardening state at the elastic limit, and the hardening is caused due to internal friction [9]. The target with internal friction can be modeled by a solid that consists of small spheres with cohesion and friction. The “spheres” here refer to the groups of crystals with strong mutual bonds in real rocks. Since the target is in the state of one-dimensional strain under strong impact, we can consider a model solid in a cylinder with rigid walls, where compression occurs only along the cylinder axis. With the increase in compression, the solid undergoes three states in sequence:(a)
Elastic State. When the axial compression is small, friction is great, and cohesion is not broken. The deformation of the target obeys Hooke’s law.(b)
Internal Friction State. When the axial compression reaches the elastic limit, cohesion is broken, and spheres slide on each other. Friction in this state is not negligible, and, therefore, penetration resistance of the projectile contains the term of internal friction.(c)
Hydrodynamic State. In this state, friction is negligible compared with great axial compression. The target can be considered as plastic fluid. The hydrodynamic term predominates in penetration resistance.As we can see from the above analysis, both the projectile and the target undergo complex solid and fluid dynamic processes at hypervelocities. The terminal DOP is the result of the combination of various factors, which are not only involved in complex physical processes, but also affected by the randomness caused by the material inhomogeneity. It is difficult to establish an accurate model to predict the DOP for this condition, and it is unnecessary to do so for engineering applications. The appropriate approach is to analyze the factor that dominates the DOP and to establish a semitheoretical model.


## 2.1. Dynamic Behaviors of the Projectile

With increasing the impact velocity, the interactions between projectile and target become more intense, and the behaviors of the projectile can be divided into four states as follows [3]:(a)
Rigid State. When the impact velocity is relatively low, the deformation and mass loss of the projectile are negligible, and the projectile can be regarded as a rigid body.(b)
Deformation State. When the pressure on the interface between projectile and target reaches the dynamic yield stress of the projectile, the projectile nose turns to the plastic state. Melting and mass loss also occur within a thin layer on the surface of the projectile nose. These factors blunt the projectile nose and slightly weaken the penetration efficiency of the projectile.(c)
Erosion State. With increasing the impact velocity, the interactions on the interface between projectile and target become more intense. The scope of yielding, melting, and mass loss of the projectile expands, leading to the failure of the structure of the projectile. In this state, the terminal DOP decreases with the impact velocity increasing.(d)
Hydrodynamic State. When the impact velocity further increases, the projectile as a whole enters the plastic flow state and can be regarded as a steady-state fluid. The projectile mass is entirely lost at the end of the penetration.


## 2.2. Dynamic Behaviors of the Target

With the increase in the impact velocity, the pressure on the target side of the interface also increases. The stress-strain state of the target in the near-penetration region can be seen as the one-dimensional strain state, and the induced stress waves are one-dimensional plane waves [8]. Metal targets turn to plastic flow at the elastic limit. Concrete and rock targets, however, turn to a hardening state at the elastic limit, and the hardening is caused due to internal friction [9]. The target with internal friction can be modeled by a solid that consists of small spheres with cohesion and friction. The “spheres” here refer to the groups of crystals with strong mutual bonds in real rocks. Since the target is in the state of one-dimensional strain under strong impact, we can consider a model solid in a cylinder with rigid walls, where compression occurs only along the cylinder axis. With the increase in compression, the solid undergoes three states in sequence:(a)
Elastic State. When the axial compression is small, friction is great, and cohesion is not broken. The deformation of the target obeys Hooke’s law.(b)
Internal Friction State. When the axial compression reaches the elastic limit, cohesion is broken, and spheres slide on each other. Friction in this state is not negligible, and, therefore, penetration resistance of the projectile contains the term of internal friction.(c)
Hydrodynamic State. In this state, friction is negligible compared with great axial compression. The target can be considered as plastic fluid. The hydrodynamic term predominates in penetration resistance.As we can see from the above analysis, both the projectile and the target undergo complex solid and fluid dynamic processes at hypervelocities. The terminal DOP is the result of the combination of various factors, which are not only involved in complex physical processes, but also affected by the randomness caused by the material inhomogeneity. It is difficult to establish an accurate model to predict the DOP for this condition, and it is unnecessary to do so for engineering applications. The appropriate approach is to analyze the factor that dominates the DOP and to establish a semitheoretical model.


## 3. Mechanisms of the Decrease in DOP

### 3.1. Influence of Projectile Yielding

Numerous experiments showed that when impact velocity reaches a critical value, the DOP does not increase with the velocity but decreases instead [1–3, 10, 11]. It is generally acknowledged that metal turns to plastic flow at the elastic limit. In this connection, the decrease in DOP may be due to the projectile yielding. If so, we can conveniently obtain the critical value of the impact velocity and the maximum DOP before decreasing.We assume that the elastic limit of the projectile is larger than that of the target, and thus, the target turns to plastic before the projectile. When the projectile is still elastic, the pressure on the interface at the instant of impact is(1)p=ρpcpυL=ρtDtυt,where ρp and ρt are, respectively, densities of the projectile and the target, Dt is the shock wave velocity of the target, cp is the elastic longitudinal wave velocity of the projectile, and υL and υt are, respectively, particle velocities of the projectile and the target on the interface.The shock wave velocityDt is usually written as(2)Dt=a1+a2υt,where a1 and a2 are coefficients related to materials and can be determined by fitting experimental data.Shemyakin [9] investigated the behavior of rock under dynamic loadings and concluded that the wave velocity in the vicinity of the loading source is approximately equal to elastic wave velocity, i.e. Dt≈ct, where ct is the elastic longitudinal wave velocity of the target.Assuming that the initial particle velocity of the target equals the impact velocity, we can give the condition where the projectile yields, i.e.,(3)p=ρtDtυy≈ρtctυy=σyp,where υy is the minimum impact velocity at which the projectile yields and σyp is the dynamic compression strength of the projectile.Whiffin [12] conducted Taylor impact tests with mild steel projectiles of different sizes and confirmed that the dynamic compression strength of projectiles is constant (see Figure 2).Figure 2
Dynamic compression strength of projectiles at different impact velocities.Besides, Whiffin [12] obtained the relationships between the static and dynamic compression strength of steel and alloy, i.e.,(4)σyσs=8.84−2.42lgσs,for steel and(5)σyσs=6.32−1.89lgσs,for aluminum alloy, where σs (MPa) and σy (MPa) are static and dynamic compression strengths, respectively.We calculateυy with different experimental data, and the results are listed in Table 1, where σsp is the static compression strength of the projectile, and υm is the critical velocity of decrease in DOP.Table 1
Parameters of projectiles and targets in different penetration tests.
Data sourcesσsp (MPa)σyp (MPa)ρt (kg/m3)ct (m/s)υy (m/s)υm (m/s)υm/υyForrestal and Piekutowski [1]11151634270050001218927.412451680270050001249327.5149017302700500012810868.5Li et al. [3]195219522670420017416009.2Qian et al. [2]731139622004000159260016.4As shown in Table1, velocities at which projectiles yield are much smaller than those at which DOPs decrease, indicating that projectiles have yielded before DOP decreasing. It is also worth noting that the critical velocities are nonlinearly related to the strength of the projectile and target. The analysis above confirms that the decrease in DOP is due to the combination of different factors.


### 3.2. Dominant Mechanism of the Decrease in DOP

Existing investigations confirmed that blunting and mass loss of projectiles weaken the penetration efficiency of projectiles [4, 7, 13–17], provided that impact velocities are less than the critical velocities. In this subsection, we analyze the influences of blunting and mass loss at hypervelocities, aiming to figure out the dominant mechanism of the decrease in DOP.Figure3 depicts the variation of caliber-radius-head (CRH) and DOP with the impact velocity, where fc is the axial compression strength of targets. It is evident that the CRH value of the posttest projectiles decreases with the impact velocity and approaches a limit of 0.5, which represents a hemispherical projectile nose. Chen et al. [19] indicated that the limit of CRH defines a theoretical limit of projectile mass loss. As shown in Figure 3, DOP keeps increasing with the velocity after CRH reaches its limit. We can thus infer that the blunting of the projectile nose contributes little to the decrease in DOP.Figure 3
Curves of CRH and DOP vs. impact velocity. (a) Data from reference [18]. (b) Data from reference [10].
(a)(b)Figure4 shows the variation of the ratio of posttest projectile mass with the impact velocity. As shown in the figure, with increasing the impact velocity, the residual projectile mass decreases slowly at first and then drops rapidly at a critical value of the velocity. In Figure 4(a), the ratio of residual mass decreases from 90% to 20% when the impact velocity increases from 1600 m/s to 1800 m/s. In Figure 4(b), the ratio of residual mass decreases from 42% to nearly 0 when the impact velocity increases from 2860 m/s to 3080 m/s. It is worth noting that the velocity region in which DOP decreases is consistent with that in which the residual mass decreases dramatically (the region is stressed by a hatched pattern in the figure). This consistency indicates that the decrease in DOP is due to the severe mass loss of the projectile.Figure 4
Curves of DOP and the ratio of residual mass vs. impact velocity. (a) Data from reference [20]. (b) Data from reference [2].
(a)(b)Figure5 shows six posttest radiographs of targets and projectiles from Reference [1], and the corresponding DOPs are depicted in Figure 6 by points A-F, respectively. The radiographs with υ0=781m/s and υ0=932m/s show slight bending and blunting of the projectiles. Points A and B in Figure 6 indicate DOP increasing in this velocity region. When the velocity increases to 1037 m/s and 1193 m/s, large bending and shortening of the projectiles occur, leading to a dramatic decrease in DOP, as shown by points C and D in Figure 6. The surface of the projectile is extremely rough at υ0=1193m/s, which implies severe abrasion and mass loss of the projectile. At the velocities of 1802 m/s and 3075 m/s, abrasion and mass loss are more intense, the projectile bodies are broken, and the DOP increases again with the velocity increasing.Figure 5
Posttest radiographs of targets and projectiles. (a) 781 m/s. (b) 932 m/s. (c) 1037 m/s. (d) 1193 m/s. (e) 1802 m/s. (f) 3075 m/s.
(a)(b)(c)(d)(e)(f)Figure 6
The curve of DOP vs. the impact velocity.Recalling the dynamic behaviors of the projectile discussed in Subsection2.1, we can now analyze the mechanics of DOP in different velocity ranges. In the rigid and deformation states, although yielding, blunting, and slight mass loss of the projectile have occurred, DOP still increases with the velocity. This is because the increase in the impact velocity compensates for the decrease in DOP caused by mass loss and blunting of the projectile. In this region, the projectile can be seen as a rigid body, since the DOP increases linearly with the velocity. With increasing the impact velocity, the scopes of abrasion, yielding, and melting expand. At a critical velocity, the projectile loses the capacity to maintain its primary form, and severe mass loss occurs. In this state, i.e., abrasion state, the increase in the impact velocity can no longer compensate for the decrease in DOP caused by mass loss; thus, DOP decreases dramatically with the velocity increasing. With a further increase in the velocity, the projectile turns to the hydrodynamic state and can be regarded as a steady-state flow. DOP increases again with the velocity increasing, because the residual mass is so small that the effect of mass loss is little in this region.


## 3.1. Influence of Projectile Yielding

Numerous experiments showed that when impact velocity reaches a critical value, the DOP does not increase with the velocity but decreases instead [1–3, 10, 11]. It is generally acknowledged that metal turns to plastic flow at the elastic limit. In this connection, the decrease in DOP may be due to the projectile yielding. If so, we can conveniently obtain the critical value of the impact velocity and the maximum DOP before decreasing.We assume that the elastic limit of the projectile is larger than that of the target, and thus, the target turns to plastic before the projectile. When the projectile is still elastic, the pressure on the interface at the instant of impact is(1)p=ρpcpυL=ρtDtυt,where ρp and ρt are, respectively, densities of the projectile and the target, Dt is the shock wave velocity of the target, cp is the elastic longitudinal wave velocity of the projectile, and υL and υt are, respectively, particle velocities of the projectile and the target on the interface.The shock wave velocityDt is usually written as(2)Dt=a1+a2υt,where a1 and a2 are coefficients related to materials and can be determined by fitting experimental data.Shemyakin [9] investigated the behavior of rock under dynamic loadings and concluded that the wave velocity in the vicinity of the loading source is approximately equal to elastic wave velocity, i.e. Dt≈ct, where ct is the elastic longitudinal wave velocity of the target.Assuming that the initial particle velocity of the target equals the impact velocity, we can give the condition where the projectile yields, i.e.,(3)p=ρtDtυy≈ρtctυy=σyp,where υy is the minimum impact velocity at which the projectile yields and σyp is the dynamic compression strength of the projectile.Whiffin [12] conducted Taylor impact tests with mild steel projectiles of different sizes and confirmed that the dynamic compression strength of projectiles is constant (see Figure 2).Figure 2
Dynamic compression strength of projectiles at different impact velocities.Besides, Whiffin [12] obtained the relationships between the static and dynamic compression strength of steel and alloy, i.e.,(4)σyσs=8.84−2.42lgσs,for steel and(5)σyσs=6.32−1.89lgσs,for aluminum alloy, where σs (MPa) and σy (MPa) are static and dynamic compression strengths, respectively.We calculateυy with different experimental data, and the results are listed in Table 1, where σsp is the static compression strength of the projectile, and υm is the critical velocity of decrease in DOP.Table 1
Parameters of projectiles and targets in different penetration tests.
Data sourcesσsp (MPa)σyp (MPa)ρt (kg/m3)ct (m/s)υy (m/s)υm (m/s)υm/υyForrestal and Piekutowski [1]11151634270050001218927.412451680270050001249327.5149017302700500012810868.5Li et al. [3]195219522670420017416009.2Qian et al. [2]731139622004000159260016.4As shown in Table1, velocities at which projectiles yield are much smaller than those at which DOPs decrease, indicating that projectiles have yielded before DOP decreasing. It is also worth noting that the critical velocities are nonlinearly related to the strength of the projectile and target. The analysis above confirms that the decrease in DOP is due to the combination of different factors.


## 3.2. Dominant Mechanism of the Decrease in DOP

Existing investigations confirmed that blunting and mass loss of projectiles weaken the penetration efficiency of projectiles [4, 7, 13–17], provided that impact velocities are less than the critical velocities. In this subsection, we analyze the influences of blunting and mass loss at hypervelocities, aiming to figure out the dominant mechanism of the decrease in DOP.Figure3 depicts the variation of caliber-radius-head (CRH) and DOP with the impact velocity, where fc is the axial compression strength of targets. It is evident that the CRH value of the posttest projectiles decreases with the impact velocity and approaches a limit of 0.5, which represents a hemispherical projectile nose. Chen et al. [19] indicated that the limit of CRH defines a theoretical limit of projectile mass loss. As shown in Figure 3, DOP keeps increasing with the velocity after CRH reaches its limit. We can thus infer that the blunting of the projectile nose contributes little to the decrease in DOP.Figure 3
Curves of CRH and DOP vs. impact velocity. (a) Data from reference [18]. (b) Data from reference [10].
(a)(b)Figure4 shows the variation of the ratio of posttest projectile mass with the impact velocity. As shown in the figure, with increasing the impact velocity, the residual projectile mass decreases slowly at first and then drops rapidly at a critical value of the velocity. In Figure 4(a), the ratio of residual mass decreases from 90% to 20% when the impact velocity increases from 1600 m/s to 1800 m/s. In Figure 4(b), the ratio of residual mass decreases from 42% to nearly 0 when the impact velocity increases from 2860 m/s to 3080 m/s. It is worth noting that the velocity region in which DOP decreases is consistent with that in which the residual mass decreases dramatically (the region is stressed by a hatched pattern in the figure). This consistency indicates that the decrease in DOP is due to the severe mass loss of the projectile.Figure 4
Curves of DOP and the ratio of residual mass vs. impact velocity. (a) Data from reference [20]. (b) Data from reference [2].
(a)(b)Figure5 shows six posttest radiographs of targets and projectiles from Reference [1], and the corresponding DOPs are depicted in Figure 6 by points A-F, respectively. The radiographs with υ0=781m/s and υ0=932m/s show slight bending and blunting of the projectiles. Points A and B in Figure 6 indicate DOP increasing in this velocity region. When the velocity increases to 1037 m/s and 1193 m/s, large bending and shortening of the projectiles occur, leading to a dramatic decrease in DOP, as shown by points C and D in Figure 6. The surface of the projectile is extremely rough at υ0=1193m/s, which implies severe abrasion and mass loss of the projectile. At the velocities of 1802 m/s and 3075 m/s, abrasion and mass loss are more intense, the projectile bodies are broken, and the DOP increases again with the velocity increasing.Figure 5
Posttest radiographs of targets and projectiles. (a) 781 m/s. (b) 932 m/s. (c) 1037 m/s. (d) 1193 m/s. (e) 1802 m/s. (f) 3075 m/s.
(a)(b)(c)(d)(e)(f)Figure 6
The curve of DOP vs. the impact velocity.Recalling the dynamic behaviors of the projectile discussed in Subsection2.1, we can now analyze the mechanics of DOP in different velocity ranges. In the rigid and deformation states, although yielding, blunting, and slight mass loss of the projectile have occurred, DOP still increases with the velocity. This is because the increase in the impact velocity compensates for the decrease in DOP caused by mass loss and blunting of the projectile. In this region, the projectile can be seen as a rigid body, since the DOP increases linearly with the velocity. With increasing the impact velocity, the scopes of abrasion, yielding, and melting expand. At a critical velocity, the projectile loses the capacity to maintain its primary form, and severe mass loss occurs. In this state, i.e., abrasion state, the increase in the impact velocity can no longer compensate for the decrease in DOP caused by mass loss; thus, DOP decreases dramatically with the velocity increasing. With a further increase in the velocity, the projectile turns to the hydrodynamic state and can be regarded as a steady-state flow. DOP increases again with the velocity increasing, because the residual mass is so small that the effect of mass loss is little in this region.


## 4. Theoretical Model of Mass Loss

To analyze the motion of projectiles in the erosion stage, we study the process of a projectile penetrating vertically into a semi-infinite target. As stated in Subsection3.2, the shape of the projectile nose has changed to a hemisphere when severe mass loss happens. Therefore, the original nose shape has little effect on the DOP in this stage. In this connection, we assume that a cylindrical projectile impacts the target orthogonally with the initial velocity υ0, taking the initial static target as the reference frame. During the penetration, the instant velocity of the projectile tail is υp and the instant particle velocity of the target on the interface is υt, as shown in Figure 7.Figure 7
Schematic diagram of the projectile penetrating the target.The particle velocity of the projectile nose relative to the projectile tail is(6)υL=υp−υt.On the interface of projectile and target, considering that shock wave velocities approximately equal to elastic wave velocities, we have [21](7)ρpcpυL=ρtctυt.Combining equations (6) and (7) yields(8)υL=υp1+ρpcp/ρtct,υt=υp1+ρtct/ρpcp.The resistance force against penetration can be expressed as [20](9)σc=αs+βsυt,where αs and βs are coefficients, and σc is resistance force per unit area on the cross-section of the projectile. At medium and high velocities, we can prove that βsυt≫αs; thus, αs is negligible.When the initial impact velocity,υ0, exceeds the critical velocity, υm, the projectile enters the erosion state, where the equation of motion of the projectile is(10)−σcA0=ρpA0ldυpdt,where A0 is the cross-section area of the projectile, and l is the instant length of the entire residual projectile during the penetration. In fact, the particle velocity υL equals the rate of projectile shortening, i.e., dl/dt=−υL.Equation (10) can thus be written as(11)βsυtA0dll=ρpA0υLdυp.Substituting equation (8) into equation (11) and integrating both sides yield(12)l=CexpρpβsZ2Z1υp,where C is the undetermined coefficient, Z1=1+ρpcp/ρtct, and Z2=1+ρtct/ρpcp.Taking into account the initial condition thatυp|l=L=υ0, we have(13)lL=expρpβsZ2Z1υp−υ0.It follows from equation (13) that the instant mass of the projectile, m, is(14)mm0=expρpβsZ2Z1υp−υ0,provided that l/L=m/m0, where m0 is the initial mass of the projectile.Equation (14) can be recast as(15)m=m0expαυp−υ0υm,where α=ρp/βsZ2/Z1υm.The projectile turns to the rigid state whenυp decreases to υm, and thus, the residual mass of the projectile is(16)mres=m0expα1−υ0υm.It follows from equations (15) and (16) that α is the rate of mass loss when υ0>υm. In the above solution procedure, α is determined by the penetration resistance and strengths of projectile and target. As we have known, however, there are at least six factors that have effects on mass loss. Therefore, α can be regarded as an unknown coefficient that is determined by fitting experimental data.


## 5. Calculation of DOP

### 5.1. DOP of Rigid Penetration

Whenυ0≤υm, the mass loss has little effect on DOP, and the projectile can be seen as a rigid body. For a rigid projectile, υt=υp, and thus, the equation of motion of the projectile can be expressed as(17)−σcA0=m0dυpdt=m0dυtdt=m0d2h′dt2,where h′ is the instant DOP during the penetration process.Substituting equation (9) into equation (17) yields the formula for DOP of rigid projectile:(18)h=−m0βsA0∫υ001−αsαs+βsυ0dυp=m0βsA0υ0−αsβsln1+βsαsυ0.The coefficientsαs and βs are determined as follows [20]:(19)αs=2τs,βs=ρtctλdN2,λd=1−1.12nε0χn/4,χ=r0Δ,Δ=Kcτs2,ε0=τsG,N2=12ψ2−4ψ+16ψ4ψ−1−2ψ2ψ−1π2−θ0,θ0=arc sin1−0.5ψ,where n = 1.6–1.8, ε0 is the limit of the shear strain of the target, τs is the shear strength of the target, G is the shear modulus of the target, r0 is the projectile radius, Kc is the fracture toughness of the projectile, and ψ is the CRH value.For rock targets,βsυ0≫αs when υ0>500m/s, and thus, αs is negligible [3, 20]. Then, equation (18) can be simplified to(20)h=m0A0βsυ0,0<υ0≤υm.


### 5.2. DOP of Erosion Penetration

When the impact velocity,υ0, exceeds the critical velocity, υm, the projectile turns to the erosion state, where projectile mass and DOP decrease dramatically. Because the instant velocity, υp, keeps decreasing during the penetration process, the DOP consists of two parts, i.e., the erosion part when υp>υm and the rigid part when υp≤υm.Forυp>υm, the equation of motion of the projectile can be expressed as(21)mdυpdt=−βsυtπr02=−βsπr02dhdt.Substituting equation (15) for the projectile mass in equation (21) yields(22)m0expαυp−υ0υmdυp=−βsπr02dh.The velocity decreases fromυ0 to υm while DOP increases from 0 to hs1; thus, the DOP of the erosion stage can be obtained by integrating equation (21), i.e.,(23)hs1=m0υmβsπr02α1−expα1−υ0υm.Whenυp decreases to υm, the residual projectile keeps penetrating as a rigid body. This part is equivalent to the rigid penetration with the initial impact velocity υm and the projectile mass mres. The DOP of this part can be obtained by substituting equation (16) for the projectile mass in equation (19), i.e.,(24)hs2=m0υmπr02βsexpα1−υ0υm.The terminal DOP is(25)hs=hs1+hs2=λcm0υ0βsπr02,υ0>υm,λc=υmυ01α1+α−1expα1−υ0υm,where λc can be termed as “decrease coefficient.” hs has the same form as equation (20). When υ0>υm, λc<1, implying that DOP is less than that in the rigid state at the same impact velocity. In this connection, we can examine the validity of equation (16) by comparing theoretical and experimental results of DOP.


### 5.3. DOP of Hydrodynamic Penetration

In the hydrodynamic state, the behaviors of projectile and target can be described as steady-state fluids, and the modified Bernoulli equation applies, i.e., [3](26)12ρpυp−υt2=H+κ2ρtυt2,where H is the dynamic hardness of the target, and κ is the coefficient describing the state of the target. κ=1 stands for the ideal liquid state of the target. The relationship between κ and the impact velocity is determined by Boltzmann’s function [3]:(27)κ=κ0,M∗≤1.5,2κ0−1+expM∗′1+expM∗′,M∗>1.5,where M∗′=M∗−1.5/0.5, M∗=υ0/c, c=2H/ρt, κ0=1+ν/31−ν, and ν is Poisson’s ratio of the target.The DOP in the hydrodynamic state can be derived from equation (26), i.e.,(28)h=Lλλ−θλθ−κ,where λ=ρp/ρt and θ=κ+1−κ/λ2/M∗2.We can see thatθ⟶κ as M∗, i.e., υ0, increases. Besides, it follows from equation (27) that κ⟶1 as υ0 increases. Equation (28) can thus be simplified into(29)hL=λ.Equation (29) is the DOP of ideal jet flow. The projectile length L is equivalent to the length of the jet flow.In equation (28), the whole projectile turns to the hydrodynamic state. As shown in Figures 5(a) and 5(b), however, the projectile, though a large proportion of which is lost, still remains at the end of hypervelocity penetration. In this connection, there is no clear boundary between the erosion state and the hydrodynamic state. To obtain the continuous calculation formula for DOP, we assume that the erosion part of the projectile keeps penetrating in the form of fluid.The equivalent jet length of the erosion part is(30)L′=Lm0−mresm0=L1−expα1−υ0υm.Substituting equation (30) into equation (28) yields the DOP of the jet flow, i.e.,(31)hf=Lλλ−θλθ−κ1−expα1−υ0υm.Recalling equation (25), the total DOP when υ0>υm is(32)h=hs+hf.Equation (32) is valid for conditions, where impact velocities are greater than the critical velocity. The only unknown coefficient α can be determined by fitting experimental data. When the impact velocity is relatively low, hs dominates the terminal DOP; when the impact velocity is high, however, hf predominates.


### 5.4. Analysis of the Penetration Formulas

If we rewrite the initial projectile mass as(33)m0=ϑρpπr02L,where ϑ is the coefficient related to the geometry of the projectile, then the formulas of hs1, hs2, and hf can be expressed in normalized forms that are independent of projectile size, i.e.,(34)hs1L=ϑλ2υmctλdN2α1−expα1−υ0υm,(35)hs2L=ϑλ2υmctλdN2expα1−υ0υm,(36)hfL=λ21−θ/λθ−κ/λ1−expα1−υ0υm.It follows from equation (16) that(37)mresm0=expα1−υ0υm,m0−mresm0=1−expα1−υ0υm.The left-hand sides of equation (37) represent the proportions of residual mass and erosion mass, respectively. Equations (33)–(35) can be recast as(38)hs1L=ϑλ2υmctλdN2m0−mresm0ϕ,(39)hs2L=ϑλ2υmctλdN2mresm0,(40)hfL=λ21−θ/λθ−κ/λm0−mresm0,where ϕ=1/α. We can see that equations (38) and (39) are similar in form. Considering that equation (39) represents the DOP of a rigid projectile with the mass of mres and the initial velocity of υm, while equation (38) represents the DOP of a projectile, whose mass is entirely lost just at the end of penetration, we can explain the physical essence of the coefficient ϕ: the value of ϕ equals the ratio of DOP of an entirely worn projectile to DOP of a rigid projectile with the same mass and impact velocity.We also notice that the bracket part in equations (38) and (39) equals the DOP of the projectile with the mass of m0 at the critical velocity and that the product of the first two terms in equation (40) equals the DOP of jet flow with the length of L. Therefore, the physical significance of each term in equations (37)-(39) is clear.In general, one needs numerous experimental results to determineα and υm by data fitting. However, we find that, in theory, one set of data is sufficient to determine all coefficients.It follows from equation (37) that(41)α=υmυm−υ0lnmresm0.Substituting equation (41) into equation (32) yields the formula for the critical velocity:(42)υm=h−hfβsπr02+υ0m0−mres/lnmres/m0mres+m0−mres/lnmres/m0.All parameters in equation (42) are either known or determined by one experiment at hypervelocity. In practice, one usually needs to conduct numerous experiments in a wide velocity range to determine υm. With the help of equation (42), however, one can estimate υm by several experiments at arbitrary hypervelocities.


## 5.1. DOP of Rigid Penetration

Whenυ0≤υm, the mass loss has little effect on DOP, and the projectile can be seen as a rigid body. For a rigid projectile, υt=υp, and thus, the equation of motion of the projectile can be expressed as(17)−σcA0=m0dυpdt=m0dυtdt=m0d2h′dt2,where h′ is the instant DOP during the penetration process.Substituting equation (9) into equation (17) yields the formula for DOP of rigid projectile:(18)h=−m0βsA0∫υ001−αsαs+βsυ0dυp=m0βsA0υ0−αsβsln1+βsαsυ0.The coefficientsαs and βs are determined as follows [20]:(19)αs=2τs,βs=ρtctλdN2,λd=1−1.12nε0χn/4,χ=r0Δ,Δ=Kcτs2,ε0=τsG,N2=12ψ2−4ψ+16ψ4ψ−1−2ψ2ψ−1π2−θ0,θ0=arc sin1−0.5ψ,where n = 1.6–1.8, ε0 is the limit of the shear strain of the target, τs is the shear strength of the target, G is the shear modulus of the target, r0 is the projectile radius, Kc is the fracture toughness of the projectile, and ψ is the CRH value.For rock targets,βsυ0≫αs when υ0>500m/s, and thus, αs is negligible [3, 20]. Then, equation (18) can be simplified to(20)h=m0A0βsυ0,0<υ0≤υm.


## 5.2. DOP of Erosion Penetration

When the impact velocity,υ0, exceeds the critical velocity, υm, the projectile turns to the erosion state, where projectile mass and DOP decrease dramatically. Because the instant velocity, υp, keeps decreasing during the penetration process, the DOP consists of two parts, i.e., the erosion part when υp>υm and the rigid part when υp≤υm.Forυp>υm, the equation of motion of the projectile can be expressed as(21)mdυpdt=−βsυtπr02=−βsπr02dhdt.Substituting equation (15) for the projectile mass in equation (21) yields(22)m0expαυp−υ0υmdυp=−βsπr02dh.The velocity decreases fromυ0 to υm while DOP increases from 0 to hs1; thus, the DOP of the erosion stage can be obtained by integrating equation (21), i.e.,(23)hs1=m0υmβsπr02α1−expα1−υ0υm.Whenυp decreases to υm, the residual projectile keeps penetrating as a rigid body. This part is equivalent to the rigid penetration with the initial impact velocity υm and the projectile mass mres. The DOP of this part can be obtained by substituting equation (16) for the projectile mass in equation (19), i.e.,(24)hs2=m0υmπr02βsexpα1−υ0υm.The terminal DOP is(25)hs=hs1+hs2=λcm0υ0βsπr02,υ0>υm,λc=υmυ01α1+α−1expα1−υ0υm,where λc can be termed as “decrease coefficient.” hs has the same form as equation (20). When υ0>υm, λc<1, implying that DOP is less than that in the rigid state at the same impact velocity. In this connection, we can examine the validity of equation (16) by comparing theoretical and experimental results of DOP.


## 5.3. DOP of Hydrodynamic Penetration

In the hydrodynamic state, the behaviors of projectile and target can be described as steady-state fluids, and the modified Bernoulli equation applies, i.e., [3](26)12ρpυp−υt2=H+κ2ρtυt2,where H is the dynamic hardness of the target, and κ is the coefficient describing the state of the target. κ=1 stands for the ideal liquid state of the target. The relationship between κ and the impact velocity is determined by Boltzmann’s function [3]:(27)κ=κ0,M∗≤1.5,2κ0−1+expM∗′1+expM∗′,M∗>1.5,where M∗′=M∗−1.5/0.5, M∗=υ0/c, c=2H/ρt, κ0=1+ν/31−ν, and ν is Poisson’s ratio of the target.The DOP in the hydrodynamic state can be derived from equation (26), i.e.,(28)h=Lλλ−θλθ−κ,where λ=ρp/ρt and θ=κ+1−κ/λ2/M∗2.We can see thatθ⟶κ as M∗, i.e., υ0, increases. Besides, it follows from equation (27) that κ⟶1 as υ0 increases. Equation (28) can thus be simplified into(29)hL=λ.Equation (29) is the DOP of ideal jet flow. The projectile length L is equivalent to the length of the jet flow.In equation (28), the whole projectile turns to the hydrodynamic state. As shown in Figures 5(a) and 5(b), however, the projectile, though a large proportion of which is lost, still remains at the end of hypervelocity penetration. In this connection, there is no clear boundary between the erosion state and the hydrodynamic state. To obtain the continuous calculation formula for DOP, we assume that the erosion part of the projectile keeps penetrating in the form of fluid.The equivalent jet length of the erosion part is(30)L′=Lm0−mresm0=L1−expα1−υ0υm.Substituting equation (30) into equation (28) yields the DOP of the jet flow, i.e.,(31)hf=Lλλ−θλθ−κ1−expα1−υ0υm.Recalling equation (25), the total DOP when υ0>υm is(32)h=hs+hf.Equation (32) is valid for conditions, where impact velocities are greater than the critical velocity. The only unknown coefficient α can be determined by fitting experimental data. When the impact velocity is relatively low, hs dominates the terminal DOP; when the impact velocity is high, however, hf predominates.


## 5.4. Analysis of the Penetration Formulas

If we rewrite the initial projectile mass as(33)m0=ϑρpπr02L,where ϑ is the coefficient related to the geometry of the projectile, then the formulas of hs1, hs2, and hf can be expressed in normalized forms that are independent of projectile size, i.e.,(34)hs1L=ϑλ2υmctλdN2α1−expα1−υ0υm,(35)hs2L=ϑλ2υmctλdN2expα1−υ0υm,(36)hfL=λ21−θ/λθ−κ/λ1−expα1−υ0υm.It follows from equation (16) that(37)mresm0=expα1−υ0υm,m0−mresm0=1−expα1−υ0υm.The left-hand sides of equation (37) represent the proportions of residual mass and erosion mass, respectively. Equations (33)–(35) can be recast as(38)hs1L=ϑλ2υmctλdN2m0−mresm0ϕ,(39)hs2L=ϑλ2υmctλdN2mresm0,(40)hfL=λ21−θ/λθ−κ/λm0−mresm0,where ϕ=1/α. We can see that equations (38) and (39) are similar in form. Considering that equation (39) represents the DOP of a rigid projectile with the mass of mres and the initial velocity of υm, while equation (38) represents the DOP of a projectile, whose mass is entirely lost just at the end of penetration, we can explain the physical essence of the coefficient ϕ: the value of ϕ equals the ratio of DOP of an entirely worn projectile to DOP of a rigid projectile with the same mass and impact velocity.We also notice that the bracket part in equations (38) and (39) equals the DOP of the projectile with the mass of m0 at the critical velocity and that the product of the first two terms in equation (40) equals the DOP of jet flow with the length of L. Therefore, the physical significance of each term in equations (37)-(39) is clear.In general, one needs numerous experimental results to determineα and υm by data fitting. However, we find that, in theory, one set of data is sufficient to determine all coefficients.It follows from equation (37) that(41)α=υmυm−υ0lnmresm0.Substituting equation (41) into equation (32) yields the formula for the critical velocity:(42)υm=h−hfβsπr02+υ0m0−mres/lnmres/m0mres+m0−mres/lnmres/m0.All parameters in equation (42) are either known or determined by one experiment at hypervelocity. In practice, one usually needs to conduct numerous experiments in a wide velocity range to determine υm. With the help of equation (42), however, one can estimate υm by several experiments at arbitrary hypervelocities.


## 6. Comparison with Experimental Results

Li et al. [3] conducted penetration experiments in a wide range of velocities. The experiments were divided into two stages. In the first stage, projectiles with a diameter of 10.8 mm were ejected towards targets with impact velocities between 1200 and 1810 m/s. In the second stage, projectiles with a diameter of 7.2 mm were ejected in the velocity range of 1830–4200 m/s. Projectiles of the two stages were ogive-nose steel rods with a CRH value of 3.0 and a length-diameter ratio of 5. Targets were made of granite and were of the same size in two stages. The photos of posttest projectiles are shown in Figure 8, and the experimental and theoretical DOPs are shown in Figure 9(a) Comparing the photos of projectiles and the curves of DOPs, it is clear that DOPs decrease when dramatic mass loss of projectiles occurs.Figure 8
Posttest projectiles at different impact velocities. (a)υ0=1196m/s. (b) υ0=1426m/s. (c) υ0=1430m/s. (d) υ0=1600m/s. (e) υ0=1654m/s. (f) υ0=1752m/s. (g) υ0=1789m/s. (h) υ0=1808m/s.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 9
Theoretical and experimental DOPs of penetration experiments. (a)υm=1600m/s and α=35. (b) υm=2600m/s and α=4.3. (c) υm=1500m/s and α=7. (d) υm=967m/s and α=10.
(a)(b)(c)(d)Qian et al. [2] conducted penetration experiments with tungsten alloy projectiles and concrete targets. The impact velocities range from 1820 to 3660 m/s. The projectiles were flat-nose rods, whose diameter was 3.45 mm, and length was 10.5 mm. The experimental and theoretical DOPs are shown in Figure 9(b).As for metal targets, 6061-T6511 aluminum targets were used by Piekutowski et al. [22]. VAR 4340, 3-CRH-nosed, steel rods were launched at velocities between 550 and 3000 m/s. The total length of projectiles was 71.1 mm, and the diameter was 7.11 mm. The results for experimental and theoretical DOPs are shown in Figure 9(c).Forrestal and Piekutowski [1] conducted experiments with the same materials as in Piekutowski et al. [22]. However, the nose of projectiles was spherical rather than ogival. Figure 9(d) shows the experimental and theoretical DOPs of spherical-nose projectiles.In general, we can see that theoretical results are in good agreement with experimental data. Whenυ0>υm, the calculation formula describes well the transition of the projectile from the erosion state to the hydrodynamic state. Comparing Figure 9(c) with Figure 9(d), we can infer that the nose shape of projectiles has effects on the value of υm. Blunter nose means larger resistance force acting on projectiles, which naturally results in more dramatic interactions and, consequently, smaller υm for DOP decreasing. The calculated DOPs are kind of smaller than experimental DOPs in Figure 9(d), when the velocity is in the hydrodynamic region. This may be due to the decrease in density of the metal target under high temperature induced by severe interaction.We can also validate equation (42) against experimental data in Figure 4(a). The initial projectile mass is 32.45 g, and the results are listed in Table 2. The mean value of calculated υm is 1885.5 m/s, which is quite close to the experimental result, 1600 m/s.Table 2
Calculatedυm with different experimental results.
υ0 (m/s)DOP (cm)mres (g)υm (m/s)17528.7410.21899.317898.319.32028.418089.3410.31889.3206710.556.41636.6216510.475.21712.5235610.985.11986.3237810.153.82046.3


## 7. Conclusions

A decrease in DOP as impact velocity increases has been observed in numerous penetration experiments. The decrease in DOP occurs at a critical velocity and, in general, is accompanied by a dramatic erosion of projectile mass, as well as the yielding and blunting of projectiles. To investigate the mechanism of DOP decreasing, we analyzed the relationship between the critical velocity and various factors.We found that, before reaching the critical velocity, the yielding of projectiles has occurred, and the nose shape of projectiles has come to constant geometry, i.e., sphericity. However, the velocity of dramatic mass loss is nearly equal to the critical velocity of DOP decreasing. Thus, we can infer that the decrease in DOP is the result of the dramatic mass loss of projectiles.The projectile mass loss is the result of multiple factors, including yielding, melting, and wearing of the projectile. When the impact velocity is relatively low, the mass loss is too slight to reverse the DOP, and the DOP increases as the impact velocity increases. At a critical velocity, however, the severe interaction between the projectile and target violates the stability of the projectile body, leading to dramatic mass loss of the projectile and, consequently, the decrease in DOP with impact velocity increasing.We established a semitheoretical model for mass loss of the projectile and obtained a power law of mass decreasing. Then, a continuous formula for DOP at hypervelocity was derived considering the mass loss of the projectile. We compared the theoretical DOP with experimental results of steel projectiles penetrating different targets, including rock, concrete, and metal. The calculated DOP agrees with experimental results and describes well the transition of the projectile from the erosion state to the hydrodynamic state.At last, the formula for the critical velocityυm was obtained as a by-product of the solving procedure for DOP. One can predict the range of υm with several experimental results at hypervelocities.



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*Source: 7320711-2021-07-23.xml*