A unified model for dwell and penetration during long rod impact on thick ceramic targets

International Journal of Impact Engineering 131 (2019) 304–316

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International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng

A unified model for dwell and penetration during long rod impact on thick ceramic targets

T

⁎

Salil Bavdekara, Ghatu Subhasha, , Sikhanda Satapathyb a b

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA Soldier Protection Sciences Branch, Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Brittle materials Armor ceramics Penetration Dwell Interface defeat

In this manuscript, we present an improved model for long rod penetration into thick ceramic targets by using a modified version of the Walker–Anderson model along with our dynamic expanding cavity model. This model is simpler to use and captures additional physics such as the dwell-penetration transition phenomena in ceramics as compared to the Walker–Anderson model. The target response is assumed to occur in a hemispherical region containing nested comminuted, cracked and elastic regions of deformation. A dynamic expanding cavity model, recently developed by the authors, along with an exponential pressure-shear response of brittle materials in the form of the extended Mohr-Coulomb model, is used to capture the stress fields in these regions. Incorporating this constitutive model reduces the requirement on (often expensive) experimental data as it uses a single set of parameters to predict the response of many brittle ceramics, which is especially beneficial in understanding and exploring the behavior of new materials. The predictions of the model are in good agreement with experimental results and is also used to quantify the effect of amorphization on boron carbide.

1. Introduction Analytical models for the assessment of impact and penetration resistance are needed as they can provide an economical way to guide armor design and development. These models can be used to estimate the ballistic performance of materials by quantifying the benefits of their superior mechanical properties [1]. While advancements in computational power have reduced the time and monetary expenses of simulations, computational models are yet to capture many features observed in the ballistic impact process. The results from these computational models are also dependent on the appropriate choice of constitutive models, whose parameters need recalibration for different impact conditions and boundary conditions used. Hence, the computational models, and the material models used therein, need to be calibrated/corroborated with some other approaches to validate their accuracy, and may not be mature enough to obviate the need for complimentary analytical and experimental methods. In general, the simulations may be corroborated with experimental data for the commonly used ceramics (viz. Al2O3, AlN, SiC and B4C) [2–10]; however, this experimental data is not available for newer ceramics (B6O, BAM, etc.) that are currently being considered or developed for armor applications or ceramics that may be considered in the future. Obtaining the relevant experimental data requires ballistic experiments that are ⁎

prohibitively expensive and time consuming. It also requires that the processing conditions for these newer materials be optimized so as to obtain test samples of suitable size and quality (microstructure and density), which in itself is an expensive and time-consuming venture. In the absence of such experimental data, analytical models provide another avenue to assess the performance potential of a ceramic, through parameters such as target resistance and depth of penetration, and corroborate computational simulations without much investment. This knowledge is helpful to determine whether it is worth pursuing a particular ceramic for armor applications, before a large amount of time and money is dedicated towards its development. It also provides important insight into the material parameter space for developing novel ceramic materials. Hard structural ceramics resist penetration by eroding and decelerating the projectile nose. During low velocity impact on a hard ceramic target, the projectile dwells and is fully eroded at the surface of the target, with little or no penetration, with the projectile fragments flowing radially outward. This phenomenon is termed ‘interface defeat’ [11–13]. At higher velocities, dwell is negligible and the projectile penetrates the target immediately upon impact. However, there exists an intermediate range of impact velocities where the projectile dwells and erodes at the surface, but starts to penetrate after a certain dwell time. This is termed as ‘transitional velocity’.

Corresponding author. E-mail address: subhash@ufl.edu (G. Subhash).

https://doi.org/10.1016/j.ijimpeng.2019.05.014 Received 21 November 2018; Received in revised form 8 May 2019; Accepted 13 May 2019 Available online 15 May 2019 0734-743X/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Impact Engineering 131 (2019) 304–316

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Consequently, the total depth of penetration, P, normalized by the initial length of the projectile, L, is given by

ρp

P /L =

ρt

(3)

However, the hydrodynamic theory does not account for the deceleration and erosion of the projectile or the strengths of the target and projectile. Tate [22,23] and Alekseevski [24] independently presented an equation based on modified hydrodynamic theory as

Rt +

Beyond dwell, the penetration process can be divided into four phases (illustrated in Fig. 1): transient, primary, secondary and recovery [14,15]. The transient phase (or shock phase) of penetration, characterized by high pressure and high penetration velocity, occurs at the start of the penetration process and only lasts for a few microseconds. Shock and wave effects dominate the mechanics of this phase, and these waves momentarily release the geometric confinement of the target. The duration of this phase is dependent on geometry and material parameters (e.g., the density of the target) and not the impact velocity [16]. Upon impact, a shock wave emanates from the impact site and quickly spreads into the bulk material. The projectile then penetrates the target and the penetration velocity (and hence pressure) rapidly drops, denoting the start of the primary (or steady-state) phase of penetration. The duration of this phase, which accounts for a majority of the penetration process, scales with the aspect ratio of the projectile and the impact velocity. The penetration velocity and pressure remain essentially constant during this phase and the rod is often completely eroded by the end of this phase. Any residual penetration occurs during the ‘secondary phase’ and is due to the remaining kinetic energy of the fragments of the target material and projectile debris [17–19]. This phase persists until the pressure in the target just ahead of the cavity is too low to overcome the material's strength. The final phase of penetration is the elastic recovery phase of the target, during which the crater may contract. This phase is very small in duration and magnitude and its effects are often negligible. Hydrodynamic theory was one of the first approaches proposed to analytically model the penetration of long projectiles into thick targets [20,21]. The theory invokes Bernoulli's theorem at the projectile-target interface to relate the velocities of impact (V) and penetration (U) to the densities of the target (ρt) and projectile (ρp) as

v˙ = −

l˙ = −(v − u)

ρt / ρp

(5)

(6)

This deceleration and erosion of the projectile results in the loss of its kinetic energy. For years, the Tate-Alekseevski model was the standard [2–6,26–32] for the penetration of long eroding projectiles into thick targets, and is still used in analyses today. However, it is not without its drawbacks. The value of Rt in the model cannot be calculated a priori, and was generally determined by iteratively varying (Rt − Yp ) until there is agreement between the total depth of penetration predicted by the model and experimental data [25,33]. As this is not very satisfying, numerous theoretical arguments, often in the form of quasi-static and dynamic expanding cavity models (ECMs), have been proposed to theoretically calculate Rt [18,28,34–40], but most of them also rely on some experimental data to calibrate the model. The ECMs assume a spherical or cylindrical expanding cavity encased by nested 3D axisymmetric regions of elastic and inelastic fields within an undeformed continuum. In the dynamic ECMs [35,36,39], the cavity and deformation regions expand at a constant velocity. Hence, these ECMs are restricted to the steady-state phase of penetration and do not capture the transient phases at the beginning and end of the penetration process. As discussed previously, Rt is velocity-dependent and since portions of the penetration are nonsteady, it must follow that Rt is time-dependent, i.e., not constant during penetration [30]. While the ECMs [35–40] successfully capture the 3D stress state in the target and provide a good measurement of Rt during the steadystate phase of penetration, they are unable to capture the time-dependent Rt during the non-steady phases. Additionally, they do not provide any information about the projectile and hence, cannot offer a measure of the total penetration depth and time. Modified hydrodynamic models, on the other hand, are able to provide this information but are restricted to 1D analysis, along the centerline of penetration. Hence, a combination of an expanding cavity model and modified hydrodynamic

(1)

V 1+

σp ρp l

where l is the instantaneous length of the projectile. The rate of erosion of the projectile is the difference between the nose and tail velocities given by

where, U is the velocity of the projectile/target interface (i.e., penetration velocity) and V is the impact velocity of projectile. These velocities are assumed to remain constant for the entire duration of penetration. Eq. (1) can be solved to obtain the relation between these two velocities as [20,21]

U=

(4)

where u and v are the instantaneous (and time-dependent) nose and tail velocities, respectively, of the projectile during the penetration process; Yp is the flow stress of the projectile and Rt is defined as the target's resistance to penetration in a one-dimensional formulation. Rt is not simply the flow stress of the target material but is velocity-dependent [25,26] and also accounts for the geometric effects of the penetration process [27]. Penetration parameters such as the instantaneous rate of penetration (u) and instantaneous depth of penetration (p) depend on the difference (Rt − Yp ) and not their absolute values. Tate [18,28] later modified Eq. (4) by setting Yp = 1.7σp , where σp is the dynamic compressive yield strength of the projectile. Deceleration of the projectile occurs due to elastic waves reflected off the back end of the rod, and is given by

Fig. 1. Schematic representation of dwell and the four phases of penetration (defined by Christman and Gehring [15]) presented as the pressure at the projectile/target interface.

1 1 ρ U 2 = ρp (V − U )2 2 t 2

1 2 1 ρ u = ρp (v − u)2 + Yp 2 t 2

(2) 305

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elastic regions. The continuity of the axial velocity profile at the projectile/target interface [27,44] is used to evaluate the extent of the plastic region in the projectile, s, and is given by

model may help provide a better picture of the penetration process. However, none of the models discussed so far are capable of modeling all four phases of the penetration process, depicted in Fig. 1, and calculating a time-dependent Rt. Consequently, the value of Rt obtained from these models cannot be considered to be a unique material constant. Improvements to the hydrodynamic models [41–43] were made by modifying the Bernoulli's equation approach from a momentum balance consideration along the centerline of penetration [27]. Using this approach, Walker and Anderson [44] proposed a time-dependent model for long-rod penetration into a semi-infinite target. In the following section, we examine the assumptions and features of this model and its capability in simulating the penetration of long rod projectiles into ceramic targets. We identify few limitations of this model and propose modifications for further improvement. The effectiveness of these improvements is illustrated with comparison to experimental results available in literature.

s=

v˙ = −

zi

zp

∂uz dz + ρt ∂t

∞

∫ zi

∂uz 1 1 dz + ρp (u2 − v 2) − ρt u2 − 2 ∂t 2 2

∞

∫ zi

σ0 + β Σ, Σ < Σ¯ σ=⎧ ⎨ Σ ≥ Σ¯ ⎩ σ¯ ,

∂σxz dz = 0 ∂x

where uz is the axial component of the particle velocity and zi and zp are the instantaneous locations of the projectile nose and tail, respectively. To solve Eq. (7), a velocity profile along the centerline in the projectile and target, and the evolution of shear stress (σxz) in the target must be specified. The velocity profiles were obtained through numerical simulations [27] and used to evaluate the first two terms of Eq. (7). It is assumed that the velocity gradients exist only in the plastic regions in the projectile and target. The results of the model are most sensitive to the last term in Eq. (7) and hence, the selection of the shear stress behavior is a vital part in using the model. When the model was first proposed [44], this shear stress gradient was derived assuming von Mises plastic flow in the target. An expression for the shear stress gradient was obtained in terms of velocity field gradients, by formulating the velocity flow field as the curl of a vector potential. Through these assumptions, Eq. (7) can be explicitly integrated to give

{

}

δ−1 1 ∂ v−u 2Ru ⎞ + ρt δ˙ + ρp s 2 ⎛ δ+1 2 ∂t ⎝ s ⎠ (δ + 1)2 1 1 7 = ρp (v − u)2 − ρt u2 − Yt ln δ (8) 2 2 3

where R is the radius of the impact cavity and δ and s are the extents of the plastic regions in the target and projectile, respectively.1 The radius of the impact cavity (R) is determined by the empirical relation [45]

R = Rp (1 + 0.287 v0 + 0.148 v02)

(11)

(12)

where σ¯ is the maximal strength of the ceramic and Σ¯ = (σ¯ − σ0)/ β . At low pressures (Σ < Σ¯ ), the flow strength is linearly dependent on the confinement pressure, Σ, by a zero pressure strength (σ0) and a slope β. At higher pressures (Σ ≥ Σ¯ ), the strength is pressure-independent and constant. The above model captures the penetration behavior of ceramics reasonably well; however, the Drucker–Prager model parameters (σ0, σ¯, β ) were selected by adjusting them to fit the model to experimental penetration data [46–48]. This limits the predictive capability of the model to well-established materials for which this data is readily available, and not for newly developed exploratory materials or for materials where these parameters are unknown. Further, the bilinear Drucker–Prager model is unable to capture the non-linear behavior shown by multiple ceramics at pressures close to and above HEL [50–52]. The bilinear nature of the model also requires the solution of additional simultaneous equations to determine the transition point between the constant strength and pressure-dependent strength regions, which further complicates the evaluation of the model. Hence, an alternative constitutive model is desired. It has been demonstrated [51] that the extended Mohr–Coulomb model, developed by Shafiq and Subhash [52], can capture the nonlinear pressure-dependent shear strength of intact and comminuted ceramics over a wide range of pressures using a single curve. This model, also referred to as the SSEMC model, uses an exponential function to capture the normalized strength of a brittle material, as shown in Fig. 2. Additionally, the constants used in this constitutive model are universal and applicable to most brittle materials, which reduces the requirement on experimental data. The model was later expanded to be applicable to damaged and comminuted ceramics as well [36,51]. Knowledge of the material's HEL is the only material property required to utilize the model. These features are desirable for use in an analytical penetration model as a single set of universal constants applicable to numerous ceramics allows the model to be used for new materials in the absence of experimental data. The cylindrical expanding cavity model used in the Walker–Anderson model [44,46–48] consists of plastic and elastic regions in the target. However, in recent years, expanding cavity models for ceramics include comminuted, cracked and elastic regions, as shown in Fig. 3, to more accurately model the response of a ceramic to impact and penetration [35–38,53–55]. The SS-EMC model has recently been incorporated into a dynamic expanding cavity model (hereafter referred to as d-ECM) which contains these comminuted, cracked and elastic regions [36]. This d-ECM can be used to calculate both, the extent of the comminuted and cracked regions, as well as the shear

(7)

ρp v˙ (L − s ) + u˙ ρp s + ρt R

σp v−u s˙ ⎡1 + + ⎤ ρp (L − s ) ⎣ c c⎦

where c = E / ρp , the elastic bar wave speed in the projectile. Eqs. (6), (8) and (11) completely define the model for metal targets [44]. The model was later extended to ceramic targets [46–48] using the bilinear Drucker–Prager (DP) constitutive model to characterize the material's flow strength as a function of confinement pressure, Σ, as [49]

The Walker–Anderson (WA) model, initially developed for penetration into metal targets [44], successfully captured the steady and non-steady phases of penetration and was shown to reduce to the Tate–Alekseevski model under certain simplified conditions. The model is based on the integration of the Eulerian momentum conservation equation in Cartesian co-ordinates which, along the axis of penetration (z), simplifies to

∫

(10)

Similar to the Tate–Alekseevski model, it is assumed that the projectile decelerates due to elastic waves reflecting off the tail surface. However, at the nose, the elastic waves are assumed to reflect off the plastic region and not the projectile/target interface. The deceleration of the tail of the projectile as a result of these waves is given by

2. The Walker–Anderson model

ρp

R v−u 1 ⎞ ⎛ ⎞ ⎛1 − 2 ⎝ u ⎠⎝ δ2 ⎠

(9)

where v0 is the impact velocity (in km/s) and Rp is the radius of the projectile. The extent of the plastic region in the target (δ) is obtained through a cylindrical expanding cavity model with nested plastic and 1 The extent of the plastic region in Ref. [44] is denoted as α. However, δ is used here to remain consistent with the notation used in the d-ECMs [35,36] and the following sections of this manuscript.

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the spherical d-ECM, based on the SS-EMC model [36,52], to calculate the stresses in the target. It will be shown that this formulation greatly reduces the complexity of the model. The new model is consistent with experimental data and also includes a formulation for dwell/interface defeat, to capture the erosion and radial flow of the projectile on the surface of the target, before the onset of penetration. 3. Unified model for penetration and dwell 3.1. Penetration In this section, a modified momentum balance model is derived using a spherical coordinate system. The projectile and target are considered to be axisymmetric, with the axis of penetration along the raxis (θ = ϕ = 0 ) in the target (see Fig. 3). The axial velocity is denoted as ur. The location of the nose and tail of the projectile are denoted by ri(t) and rp(t), respectively and their velocities are

Fig. 2. Extended Mohr–Coulomb (SS-EMC) model for intact and failed strength of brittle materials along with the universal constants [36,52]. The strength and pressure are normalized (denoted by *) using the respective HEL values.

ur (ri ) = u

(13a)

ur (rp) = v

(13b)

The origin lies at the interface and hence, ri (0) = 0 and rp (0) = −L . The target is assumed to be semi-infinite. The r-component of the momentum conservation equation in spherical coordinates is

uθ2 + uϕ2 uϕ ∂ur u ∂u ∂u ⎡ ∂u ρ ⎢ r + ur r + θ r + − r ∂θ r sin θ ∂ϕ r ∂r ⎣ ∂t 1 ∂ 2 1 ∂ (σrθ sin θ) 1 ∂σrθ = 2 (r σrr ) + + r ∂r r sin θ r sin θ ∂ϕ ∂θ

⎤ ⎥ ⎦ −

σθθ + σϕϕ r

(14)

Due to the spherical symmetry, along the axis of penetration (θ = ϕ = 0 ), uθ = uϕ = 0 and σθ = σϕ . Hence, Eq. (14) simplifies to

ρ

2(σr − σθ ) ∂ur ∂u ∂σ + ρur r = r + ∂t ∂r ∂r r

(15)

Eq. (15) is then integrated along the axis of penetration over the projectile and target to give

Fig. 3. Schematic of deformation regions in the projectile and target during penetration, along with the coordinate notations.

∞

ri

ρp

stress distribution in the target. Further, the velocity profile in the target, assumed by Walker and Anderson [44], is scaled in a way such that the particle velocity drops to zero at the end of the plastic region (r = δR ), whereas, for spherical symmetry, it should drop as 1/r2 throughout the target [56]. The scaling also implies that, at θ = π/2 , the target material flows back at a constant speed, uθ = u/(δ 2 − 1) , throughout the plastic region, which can range from 3 to 10 times the cavity radius [48]. This behavior seems somewhat nonphysical. The Walker–Anderson model is derived using the momentum conservation equation in Cartesian co-ordinates. The extent of the plastic zone is calculated using a cylindrical expanding cavity model and the shear stress gradients in Eq. (7) are calculated assuming a spherical flow field. The inconsistency in co-ordinate systems due to the implementation of a cylindrical cavity expansion solution in a Cartesian momentum conservation equation for a region having a spherical flow field seems unreasonable. Using a spherical expanding cavity model with spherical conservation equations is preferable for consistency and simplicity in analytical calculations. Lastly, and most importantly, the Walker–Anderson model does not include a solution for dwell and interface defeat. Hence, it cannot be used to model the impact response of ceramics at lower velocities, where the pressure at the projectile/target interface is not high enough to cause penetration immediately after impact. In the next section, a new momentum balance model for the penetration of ceramics by long-rod projectiles is proposed by integrating the momentum conservation equation using (i) a spherical coordinate system, (ii) a more suitable function for the spherical flow field, and (ii)

∫ ∂∂utr dr + ρt ∫ ∂∂utr dr + 12 ρp ur2 rp

∞ = σr +2 rp

ri

ri rp

+

∞ 1 ρ ur2 ri 2 t

∞

∫ (σr −r σθ) dr

(16)

rp

Assuming a negligible change in density for both materials [36,38,44,57,58] allows for the density terms to be pulled out of the integrals. The projectile tail is a stress-free surface, hence, σr (rp) = 0 ; and for a semi-infinite target, σr (∞) = 0 . Further, σr − σθ = 2τ . Inserting these relations and the definitions in Eqs. (13), along with the assumption that the shear stress in the long-rod projectile is negligible results in

ρp

ri

∞

∞

rp

ri

ri

∫ ∂∂utr dr + ρt ∫ ∂∂utr dr + 12 ρp (u2 − v2) = 12 ρt u2 + 4 ∫ rτ dr

(17)

It is interesting to note that Eq. (17) has a similar form to Eq. (7), as along the axis of penetration, r and z are equivalent. The major difference is in the last term, as the solution to Eq. (7) requires the evaluation of the shear stress gradient while the solution to Eq. (17) only requires the calculation of the shear stress. This is a key difference as the models are most sensitive to this term and, as will be discussed later, this term in Eq. (17) is easier to evaluate. Walker and Anderson [44] showed that, for the projectile, the solution to first term in Eq. (17) can be given by ri

2

∫ ∂∂utr dr = v˙ (l − s) + us˙ + (v − u) u + ∂∂t ⎛⎝ v −s u ⎞⎠ s2 rp

307

(18)

International Journal of Impact Engineering 131 (2019) 304–316

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(σθ) exceeds its tensile strength and is comminuted (pulverized) when the radial stress (σr) exceeds its compressive strength. Eq. (15) is integrated in each region, along with the respective constitutive equations and boundary conditions, to derive the field solutions2 for the radial and hoop stresses at a given penetration velocity, u. The constitutive behavior of the comminuted ceramic is described using the SSEMC model [52]. The field solutions for σr(r) and σθ(r) are then used to evaluate τ(r) and the final integral in Eq. (17). As these field solutions are lengthy and do not have an explicit analytical formulation in the comminuted region, an algebraic expression for the value of the integral is not provided. Instead, the integral is numerically evaluated within the d-ECM at each time step, and its value is denoted by St. It is important to note that Eq. (15) forms the basis of both, the new momentum balance model as well as the d-ECM, which adds to the consistency in the model's formulation. The continuity of the velocity gradient at the projectile/target interface [27,44] is used to evaluate the extent of the plastic region, s, in the projectile. Over this length, the axial velocity in the projectile reduces linearly from the tail velocity, v, to the nose velocity, u. The slope of this velocity profile in the projectile can be equated to the derivative of Eq. (22a) or Eq. (26) at the interface (r = R and θ = 0 ) to give

The solution for the second term in Eq. (17) over the target requires a velocity profile to be specified. Using numerical simulations to guide them, Walker and Anderson [44] determined the velocity profile using spherical plastic flow fields, which can be obtained from the curl of a vector potential

A = f (r )sin θe^ϕ

(19)

where f(r) is a suitable function that represents the flow field. And the radial and hoop velocities are given by

ur (r , θ) =

2f (r ) cos θ r

uθ (r , θ) = −

1 d(rf (r )) sin θ 2 dr

(20a) (20b)

As discussed in Section 2, the assumed function for f(r) in Walker and Anderson's model results in an unsatisfactory behavior. Models and simulations predict only a radial flow and no backward flow during penetration [59,60], with the velocity scaling with 1/r2 throughout the target [56]. To obtain these features, we choose the following function

f (r ) =

uR2 2r

(21)

u−v 2u =− s R

which, when substituted into Eqs. (20), results in

R 2 ur = u cos θ ⎛ ⎞ ⎝r ⎠

(22a)

uθ = 0

(22b)

or

s=

The same velocity field can be derived from the radial (r) component of the mass conservation equation in spherical coordinates, given by

∂ρ 1 ∂ (ρr 2ur ) 1 ∂ (ρuθ sin θ) 1 ∂ (ρuϕ) =0 + + + 2 r r sin θ r sin θ ∂ϕ ∂θ ∂r ∂t

(23)

(24)

ri

∫ ∂∂utr dr = v˙ (l − s) + us˙ ⎛⎝1 +

(25)

rp

(31)

(26)

1 s ˙ ⎛1 + ⎞ + (v − u) u ⎤ + ρt u2 + ρt uR ˙ + ρp (u2 − v 2) ρp ⎡v˙ (l − s ) + us 2 R⎠ ⎝ ⎣ ⎦ 1 2 − ρt u − 4St = 0 (32) 2

∞

ri

s ⎞ + (v − u) u R⎠

All the terms in the model have now been evaluated and can be combined. Substituting the solutions for the integrals given in Eqs. (31) and (27), along with the numerical value of St obtained from the dECM, into Eq. (17) gives

Substituting the relation for ur (θ = 0) , given by Eqs. (22a) and (26), into the second integral term in Eq. (17) yields

˙ ∫ ∂∂utr dr = u2 + uR

(30)

Now, Eq. (18) can be simplified to

where D1 is the constant of integration. Applying the boundary condition that the particle velocity at the cavity surface, ur (r = R) = u cos θ , results in

R 2 ur = u cos θ ⎛ ⎞ ⎝r ⎠

(29)

∂ v−u 2 ∂u ⎛ ⎞= ∂t ⎝ s ⎠ R ∂t

Integrating Eq. (24) yields

D ur = 21 r

R v−u ⎞ ⎛ 2⎝ u ⎠

This definition for s is slightly different from the one presented in Eq. (10). However, for structural ceramics, δ ranges from 3 to 10 [48]. Hence, 0.9 ≤ (1 − 1/ δ 2) ≤ 0.99, and Eqs. (10) and (29) are relatively equivalent. Eq. (28) can be partially differentiated with respect to t to obtain

which, due to spherical symmetry and the assumption that the change in density is negligible [36,38,44,57,58], reduces to

ρt ∂ 2 (r ur ) = 0 r 2 ∂r

(28)

(27)

which can be further simplified to

The new function f(r) not only yields a more realistic velocity profile, but also greatly simplifies the evaluation of the second integral, as compared to that in the Walker–Anderson model (Eq. (16) in Ref. [44]). To solve for the last term in Eq. (7), Walker and Anderson [44] calculated the shear stress gradient in terms of a plastic flow field using the Drucker–Prager constitutive model. In our case, the last term in Eq. (17) is solved using a spherical dynamic expanding cavity model (dECM), which has been recently developed by the authors [36] and proven to better capture the response of semi-infinite brittle targets to penetration. This d-ECM postulates the formation of nested comminuted, cracked and elastic deformation regions under the cavity (see Fig. 3). The brittle material is assumed to crack when the hoop stress

s 1 1 ˙ ⎛1 + ⎞ ⎤ + ρp (v − u)2 = ρt u2 + ρt˙ uR + 4St − ρp ⎡v˙ (l − s ) + us R 2 2 ⎝ ⎠ ⎣ ⎦ (33) Eqs. (33), (11) and (6) completely define the model. If we take the limit s → 0, this model reduces to the Tate–Alekseevski model given in Eq. (4). Hence, the first term in Eq. (33) may be considered to be equivalent to Yp and the final term, 4St, equivalent to Rt. Hence, 2 A detailed derivation of the d-ECM and its field solutions for stresses are given in Refs. [35,36].

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s ˙ ⎛1 + ⎞ ⎤ + ρt uR ˙ Rt − Yp = 4St + ρp ⎡v˙ (l − s ) + us R ⎠⎦ ⎝ ⎣ 1 1 = ρp (v − u)2 − ρt u2 2 2

where the superscript (*) denotes normalized quantities with their corresponding HEL components, i.e. τ * = τ / τHEL and Σ* = Σ/ΣHEL . Σ is the hydrostatic pressure in the target due to the projectile load as well as any external confinement. a = 1.15, b = −1.06, k = 1.78, a1 = 1.11 and k1 = 0.38 are universal model constants, applicable to most brittle materials [36,51,52]. One important exception is B4C, for which a1 = 0.46 , leading it to have a lower strength for the failed material. This lower value is speculated to be due to a unique deleterious mechanism activated at high pressures, known as amorphization [36,62,63], in this ceramic. From Eqs. (36) and (37), it is evident that the maximum shear stress in the target is dependent on the impact velocity, V. If this velocity is sufficiently high such that τ0 > τi, there is no dwell. At lower velocities, when τ0 < τi, the projectile is unable to penetrate and dwells at the surface of the target. Damage (Δ) is assumed to accumulate in the ceramic target directly beneath the impact site based on the overstress principle [64]. According to this principle, the rate of damage accumulation is proportional to the overstress in the target, and given by

(34)

The model is solved by adopting the following procedure: For the first time step, the boundary conditions are trivial: l = L , the initial projectile length; v = V , the impact velocity; and s = 0 . The only boundary condition that is not specified is the initial penetration velocity (u = u 0) , which can be calculated from the Rankine–Hugoniot shock conditions [44]. However, it was observed that, for a projectile with a high aspect ratio, the model is not sensitive to this initial value and hence, the hydrodynamic penetration velocity, given by Eq. (2) may be used as a simpler alternative. For each successive time step, u(t), v(t) and l(t) are known quantities obtained from the previous step. Hence, for the given u(t), the d-ECM is used to compute St. Next, Eq. (29) is employed to calculate s, which is subsequently used to obtain v˙ from Eq. (11). Then, u˙ and l˙ are obtained using Eqs. (33) and (6), respectively. These derivatives are now used to calculate the respective parameters for the successive time step; e.g., v (t + dt ) = v (t ) + v˙ (t )dt . The process is iterated until u = 0 or l = 0 , which marks the end of the penetration process. Finally, the depth of penetration at any time, t, can be calculated as

Δ˙ = A (τ0 − τf )

where A is a constant damage increment parameter, determined by fitting the model to experimental data. The best fit was achieved using A = 15 × 103 (Pa.s)−1. Δ = 0 for the intact material and Δ = 1 for the completely failed material. As damage accumulates, the shear strength of the target can be represented by

t

p (t ) =

∫ udt 0

(35)

τt* (Δ) = τi* − Δ(τi* − τ f*)

It is important to note that the new model, presented in Eq. (33), is much less complicated than the original Walker–Anderson model [44], presented in Eq. (8). Further, the extent of the damaged region, δ, does not appear in any of the current model equations. These reductions in complexity do not arise from any additional assumptions or simplifications of the mechanics of the penetration process. They are a result of employing (i) a spherical coordinate system which eliminates the need for a stress gradient, (ii) a simpler formulation for the velocity profile in the target, and (iii) the use of the d-ECM to directly evaluate the stresses in the target. It will be shown in the next section that the new model is able to capture the experimentally observed penetration behavior for a number of structural ceramics.

So far, we have derived a model for penetration into a thick ceramic target. However, penetration only occurs when the pressure at the interface between the projectile and the target (termed as stagnation pressure) is sufficiently high to overcome the strength of the target material. If this condition is not met, the projectile dwells on the surface of the target. During dwell, the approximate relation for the maximum normal pressure at the projectile-target interface is given by [10]

4. Results and discussion In this section, we compare the results of the new momentum balance model to experimental data available in literature. This data is obtained from penetration experiments of long projectiles, typically slender tungsten rods with L/ D = 20 , on AD995 Alumina [2], AlN [3], SiC [4] and B4C [5]. The material properties used in the model for the tungsten and gold projectiles as well as the four structural ceramics are listed in Table 1. First, high velocity impacts, where there is no dwell, are considered. The model predictions for the positions of the nose and tail of a tungsten projectile impacting an AlN target at 2.98 km/s are plotted as a function of time in Fig. 4. Both quantities are normalized by the projectile's initial length (L). The hydrodynamic solution (Eqs. (1)–(3)) and results from experimental data [3] are also shown. The hydrodynamic approximation is fairly reasonable for the tail position; however, it greatly over-predicts the nose position, and consequently, the depth of penetration. The momentum balance model, however, captures the experimental data well. Penetration ceases when the rod is completely

(36)

1 ρ v2 2 p

and Kp is the bulk modulus of the projectile. The where, qp = maximum shear stress in the target due to this normal pressure is given by [61]

τ0 =

p0 π

(37)

For consistency, the shear strength of the target material is assumed to be pressure-dependent and is described by the SS-EMC model (Fig. 2). The strength of the intact material (τi) is given by [52]

τi* = a + be−kΣ*

(38)

and the strength of the failed material (τf) is given by [36]

τ f* = a1 (1 − e−k1 Σ*)

(41)

where the superscript (*) denotes that each quantity is normalized by τHEL. Hence, with increasing damage, the shear strength of the material, τt, decreases from τi to τf during dwell. To solve the new model with dwell, τ0 is evaluated at the instant of impact (v = V ) from Eqs. (36) and (37). If τ0 > τi, there is no dwell and the model is solved as described in the previous section. If τ0 < τi, the deceleration and erosion rate of the projectile are calculated using Eqs. (5) and (6), respectively. These quantities are used in the next step, to obtain the instantaneous projectile velocity, v, and projectile length, l, and maximum shear stress, τ0. The damage, Δ, accumulated in each time step is obtained from Eq. (40) and the resulting strength of the damaged target, τt, is evaluated using Eq. (41). If this strength drops below τ0, the projectile begins to penetrate the target. If τ0 < τf or if the rod is completely eroded before enough damage has accumulated for the stress developed in the target to overcome its strength, there is no penetration and the projectile is completely defeated at the interface. The highest impact velocity for which this condition (i.e., interface defeat) is satisfied is termed as the critical velocity, Vcr.

3.2. Dwell

qp σp ⎞ ⎛ + 3.27 ⎟ p0 = qp ⎜1 + 0.5 Kp qp ⎝ ⎠

(40)

(39) 309

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Table 1 Material properties for the projectiles and selected structural ceramics.

Tungsten Gold AD995 AlN SiC B4C

ρ (kg/m3)

E (GPa)

K (GPa)

HEL (GPa)

PHEL (GPa)

τHEL (GPa)

References

19,300 19,300 3890 3226 3215 2510

482 202 373 315 449 462

313 179 230 201 227 233

– – 6.7 9.0 11.7 19.0

– – 3.97 5.0 5.9 8.71

– – 2.72 3.0 6.5 7.72

[2–5,73] [66,74] [2,75] [3,76] [4,77,78] [5,79]

Fig. 5. Normalized kinetic energy of a tungsten projectile (L/ D = 20 ) penetrating an AlN target. Impact velocity = 2.98 km/s. Most of the kinetic energy is lost due to the erosion of the projectile.

Fig. 4. Normalized nose and tail positions of a tungsten projectile (L/ D = 20 ) penetrating an AlN target compared with experimental data [3]. Impact velocity = 2.98 km/s. The nose position also corresponds to the instantaneous depth of penetration (p). P is the total depth of penetration.

eroded. The total depth of penetration is denoted as P. The kinetic energy (KE) of the projectile is given by

KE =

1 ρ Ap lv 2 2 p

(42)

Fig. 6. Nose (penetration) and tail velocities for a tungsten projectile (L/ D = 20 ) penetrating an AlN target. Impact velocity = 2.98 km/s. U is the steady-state penetration velocity.

πRp2

is the cross-sectional area of the projectile. During where Ap = dwell and penetration, the rod undergoes deceleration as well as erosion, thereby losing its kinetic energy. Differentiating Eq. (42) with respect to time gives

1 dKE = ρp Ap lvv˙ + ρp Ap v 2l˙ 2 dt

given by either side of Eq. (33). This interface pressure and the quantity in Eq. (34) that is equivalent to (Rt − Yp) in the Tate–Alekseevski model are plotted in Fig. 7 for two impact velocities. As expected, in both cases, the pressure profile is identical to the nose velocity profile shown in Fig. 6. (Rt − Yp) intially has a very low value during the intial shock phase, indicating that the target material provides very little resistance during this phase, resulting in a high interface pressure and rate of penetration. This values rapidly increases and stays fairly constant in the steady-state phase, especially at higher velocities. The interface pressure and (Rt − Yp) both drop near the end of the penetration process, when the projectile length is reduced to a few diameters and the velocity has significantly subsided. Interestingly, this drop is much sharper at higher velocites, as seen in Fig. 7. This implies that as the impact velocity increases, the penetration process approaches closer to the hydrodynamic solution where the pressure is constant during the entire penetration process and drops immediately to zero at the end. This behavior can also be inferred from the total depth of penetration (P/L) data, plotted in Fig. 8, for the four structural ceramics. For each material, the total depth of penetration approaches the hydrodynamic limit as the impact velocity increases. There is good agreement between the model and the experimentally observed depth of penetration for AD995 Alumina, AlN and B4C. In the case of SiC, the model tends to underpredict the penetration. However, the general trend is still the same as the experimental data. This underprediction is possibly due to the use of universal constants of the SS-EMC model. The model constants for the comminuted material are obtained from experimental

(43)

The first term on the right hand side in Eq. (43) represents the loss in KE due to deceleration and the second term represents the loss in KE due to erosion. This decay of kinetic energy during penetration is illustrated in Fig. 5, in which the kinetic energy of the projectile is normalized by its 1 kinetic energy at impact (KE0 = 2 ρp Ap LV 2 ). A majority of the kinetic energy is lost due to projectile erosion, with projectile deceleration accounting for a very minor portion. A similar observation was made by Anderson and Walker [31] for the loss in kinetic energy of a long projectile during the dwell phase. The nose and tail velocities for the same impact case as above are plotted in Fig. 6. The tail velocity remains mostly constant for a majority of the penetration process. This behavior is expected as deceleration of the projectile (v˙ ) has a minimal impact on its loss of kinetic energy, as seen in Fig. 5. The nose velocity immediately after impact is higher than the hydrodynamic solution. However, it rapidly drops to a steady-state value and remains at that value for most of the penetration process, until the projectile is almost completely eroded. Once the projectile is eroded to the size of a few diameters, the nose and tail velocities drop rapidly and penetration ceases. The steady-state penetration velocity, U, is defined as the median of the nose velocity, u(t), over the penetration process. The pressure at the projectile/target interface during penetration is 310

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Fig. 7. The pressure at the projectile/target interface and the equivalent Tate–Alekseevski (Rt − Yp) value, as defined in Eq. (34), for a tungsten projectile (L/ D = 20 ) penetrating an AlN target. Note that this value is essentially constant during the steady-state phase of penetration.

followed by penetration, and penetration without dwell for long gold projectiles (L/ D = 70 ) impacting SiC targets. Dwell time is defined as the duration from the time of impact to the time when the pressure at the interface is just high enough to overcome the damaged strength of the material. At low velocities, this time isn't perfectly defined. For example, for an impact velocity of 800 m/s, the dwell ceases at ∼20 μs. However, there is negligible penetration after that (P/L ≈ 0.05). Hence, the projectile can be considered to be defeated at the surface of the target and this case may be classified as "sustained dwell". For higher velocites, e.g., V = 950 m/s and V = 1200 m/s, the model predicts a dwell time of ∼12 μs and ∼2.5 μs respectively. Following dwell, the target accumulates enough damage and penetration continues into the

data on a variety of brittle materials and some deviation is expected when used for specific materials [36]. However, if a value more suitable to the SiC data (k1 = 0.22 ) is used, the model fits the data better, as shown in Fig. 8(c). Fig. 9 displays the steady-state penetration velocity, U, calculated in the model versus experimental measurements. There is excellent agreement between the model and experiments for AD995 Alumina, AlN and B4C. The model underpredicts the penetration velocity for SiC; however, the discrepency is quite small and the overall trend is captured well. Once again, using k1 = 0.22 for this material provides much better agreement with experimental data. Fig. 10 illustrates the various cases of interface defeat, dwell

Fig. 8. Comparison of experimental data and model results for normalized depth of penetration as a function of impact velocity for a long tungsten projectile (L/ D = 20 ) impacting different stuctural ceramics. 311

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Fig. 9. Comparison of experimental data and model results for steady-state penetration velocity as a function of impact velocity for a long tungsten projectile (L/ D = 20 ) impacting different structural ceramics.

fidelity of the data is not high. A plot of this experimental data along with the model predictions for dwell time for impact velocities up to 1600 m/s is presented in Fig. 11(a). No dwell is predicted above 1250 m/s. The depth of penetration, normalized by the projectile length, is plotted in Fig. 11(b) and the penetration velocity is plotted in Fig. 11(c). Note that there is little penetration (P/L < 0.05) up to ∼800 m/s and the corresponsing penetration velocities are also negligible. Past this, the depth of penetration significantly increases with impact velocity, upto ∼1250 m/s. Past this, no dwell is predicted and

target until the projectile is completely eroded. Thus, the depth of penetration increases with increasing impact velocity. At V = 1600 m/s, penetration begins immediately after impact and there is no dwell. These results agree reasonably well with the experimental data [65,66] on bare SiC targets, in which they observed sustained dwell with no penetration at impact velocities under 776 m/s, ∼10 μs of dwell at 958 m/s, ∼2.6 μs of dwell at 1212 m/s and no dwell at impact velocities over 1550 m/s. However, it is cautioned that the data is very limited, i.e., one experiment at each impact velocity, and hence the

Fig. 10. Normalized nose (dashed lines) and tail (solid lines) positions of a long gold projectile (L/ D = 70 ) after impact on a bare SiC target illustrating dwell followed by penetration (V ≤ 1200 m/s) and penetration with negligible dwell (V = 1600 m/s). 312

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Fig. 11. Comparison of experimental data and model predictions for (a) dwell time, (b) normalized depth of penetration, and (c) penetration velocity as a function of impact velocity for a long gold projectile (L/ D = 70 ) impacting a bare SiC target.

ceramic targets. What is remarkable about the model is that very few material parameters are required to solve the model. With only the density and two elastic constants for the projectile and target, along with the HEL components of the target, the model can be used to predict the impact response of brittle materials. This is possible due to the utilization of the dynamic expanding cavity model (d-ECM) to evaluate

penetration begins immediately after impact. The model is able to capture the depth of penetration into the ceramic target reasonably. While the model follows a similar trend to the experimental data for penetration velocity, it does result in an under-prediction at impact velocities greater than 1.1 km/s. Similar to the penetration results for SiC (Fig. 9), this is likely due to the use of universal constants of the SSEMC model. It can be seen that using the constant for SiC results in better agreement at these higher velocities. While the model and experimental results reported here are for an unconfined target with no cover plate or buffer, it should be noted that most penetration experiments are carried out with these additions. The inclusion of external confinement into the model for dwell is trivial, the pressure, Σ, in Eqs. (38)–(39) is increased by an amount corresponding to the external confinement pressure, thereby increasing the shear strength of the target and leading to an increase in the critical velocity, Vcr [67]. The inclusion of a metal buffer, however, is more tedious. While the buffer greatly prolongs dwell by attenuating the initial shock wave and reducing the pressure on the target [64–66], it does not significantly affect the penetration phase. Hence, the absence of cover plates in the proposed model should not affect its applicability to the ranking and selection of materials (existing and new) for armor. The effect of the cover plate can be thought to be similar to the effect of a pulse shaper (or metal cushion) used in a split Hopkinson pressure bar (SHPB) for the dynamic testing of ceramics. Nemat-Nasser et al. [68] provided an analytical solution for the response of this pulse shaper during a SHPB test. While this solution is for the uniaxial stress state encountered in the experiment, it can be modified to be applicable to the appropriate stress-state encountered by the cover plate used in ballistic experiments. From Figs. 4–11, it can be seen that the new model is capable of capturing dwell and penetration phases of long projectiles into thick

∞

St = ∫ (τ / r ) dr . The constitutive behavior of the comminuted material ri

in the d-ECM is described using SS-EMC model, and the universal nature of the constants in this constitutive model eliminates the need for experimental data of the failed material, which is often unavailable and expensive to obtain for brittle materials. A sensitivity analysis using the same d-ECM was performed in a previous study [36], in which the authors perturbed relevant material parameters independently using a multiplicative factor (1 + κ ) and the resulting mean value of the target resistance, Rt, over a range of penetration velocities was compared to the original mean value of the target resistance with no parameter perturbation, Rt (κ = 0) . The analysis identified that the parameters that most significantly affect the penetration response of a ceramic are the comminuted material parameters, a1 and k1, in the SS-EMC model; a perturbation of 20% in each of the above parameters resulted in ∼18% and ∼14% change in Rt, respectively. Thus, the properties of the comminuted ceramic are most relevant to its penetration resistance. The target's compressive strength and elastic modulus have a much smaller effect, with a 20% perturbation in these parameters only producing ∼10% and ∼4% change in Rt, respectively. The effect of the target's tensile strength was found to be negligible as even doubling or halving this parameter resulted in <2% change in Rt. In the model, the material's tensile strength defines the extent of the cracked region in the target and its compressive strength defines the extent of the comminuted region. However, the 313

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change called amorphization under high pressure loads [62,63,71,72] despite being one of the hardest materials with high HEL value (see Table 1). It is speculated that this phenomenon is the reason that the shear strength of B4C is much lower than expected, given its high HEL [36,70], as seen in Fig. 13(a). It is theorized that if an amorphizationresistant B4C is developed, its performance would be significantly enhanced. The model presented here can provide quantitative estimates of benefits and help assess if efforts being pursued are worthwhile. If the linear fit in Fig. 13(a) is extrapolated (represented by the dotted line) to the HEL corresponding to B4C, its expected shear strength is expected to be 8.8 GPa (represented by the orange triangle), resulting in a1 = 1.14 , which is close to the universal value of that constant. If this value of a1 is used in the model, the theoretical ballistic performance of an amorphization-resistant B4C can be quantified. The normalized depth of penetration, calculated in this manner, is plotted in Fig. 13(b). It can be seen that the projectile penetrates to a much lower depth into an amorphization-resistant B4C target. This provides credence to the claim that the loss in shear strength due to amorphization is a major factor in limiting the ballistic capability of B4C and that mitigating this deleterious phenomenon would tremendously improve its performance in armor systems. Hence, the proposed analytical model allows the user to consider ‘what-if’ like these scenarios in a quick and efficient manner at a significantly reduced cost. One of the limitations of using the d-ECM to evaluate St is that it assumes that Δ = 1 throughout the comminuted region. During penetration, the shear stress, τ = 0 at r = ∞ and increases as one moves towards the interface. Therefore, at some point, the damage threshold is met and from this point onwards, Δ increases until the material is fully damaged. However, τ has the highest value immediately below the interface and decreases rapidly as r increases. Consequently, the integral of (τ/r) is most influenced by the response of the material closest to the interface. In fact, a majority of the integral value is evaluated in a small region ranging out to ∼2 times the cavity radius. Hence, applying the damage parameter of the material at the surface to the entire comminuted region for the calculation of St is a reasonable assumption for this model. A more thorough solution with Δ varying throughout the comminuted region is excesvively tediuos for an analytical model and more suited for a computational study.

∞

Fig. 12. An example demonstrating the evaluation of St = ∫ (τ / r ) dr in an AlN ri

target, subjected to penetration velocity of 1.27 km/s. (a) The value of the integrand (τ/r) decreases rapidly as r increases. The abscissa r is normalized with the cavity radius R and the ordinate (τ/r) is normalized such that the area under ∞

the curve is unity. (b) A majority of the integral, St = ∫ (τ / r ) dr , is evaluated in ri

the region closest to the cavity surface.

stresses in the elastic and cracked regions are much lower than the stresses closer to the projectile/target interface, i.e., in the comminuted region. The shear stress, τ, has the highest value at the interface and decreases rapidly as r increases. Consequently, the response of the comminuted material closest to the cavity surface is more influential in ∞

the calculation of St =

∫ (τ /r ) dr , as demonstrated in Fig. 12. Thereri

fore, while the material's tensile strength does impact the size of the cracked region in the target material, it is ultimately insignificant in resisting the penetration of the projectile. These findings are consistent with other studies that conclude that the properties of the comminuted material have a greater influence on the penetration response than the properties of its intact form [38,58,69,70] while the tensile strength has a negligible influence [1]. Hence, two variants of the same material; having similar properties such as density, elastic moduli and HEL; would not be expected to have different impact responses. However, if they have different microstructures, they may have different HELs and shear strength. This HEL-dependent shear strength is captured by the normalized universal formulation for shear strength used in the SS-EMC constitutive model. The predictive capability of the new analytical model developed here can be demonstrated by applying it to a unique phenomenon that is observed in boron carbide, which undergoes a deleterious structural

5. Conclusions Identifying a few limitations of the Walker–Anderson model, a new momentum balance model for the penetration of ceramics by long projectiles is developed. The spherical formulations used throughout the model decrease the complexity of the solution and leads to consistency between the co-ordinate systems, velocity field and dynamic

Fig. 13. (a) The relationship between the maximum shear strength of comminuted ceramics and their HEL. The dotted line is the extrapolation of the linear fit to the HEL of B4C. (b) Comparison of momentum balance model results for normalized depth of penetration as a function of impact velocity for a long tungsten projectile (L/D = 20) impacting an amorphized and a theoretical amorphization-resistant B4C ceramic. 314

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expanding cavity solution used in the model. The expanding cavity solution utilizes the SS-EMC model to evaluate the stresses in the target. The universal nature of the constants in this constitutive model allows it to be used on a wide variety of brittle materials, without the need for additional experimental data on both, intact and damaged materials, extending the applicability of the penetration model to materials for which relevant data may not be readily available. The penetration model agrees well with experimental data and also includes a formulation for dwell/interface defeat, a feature that is absent in most penetration models. The predictive capability of the model was demonstrated by considering a hypothetical amorphization-resistant B4C and quantifying the superior penetration resistance compared to a conventional material that is prone to amorphization.

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