Analytical engineering models for predicting high speed penetration of hard projectiles into concrete shields: A review

Abstract

We describe virtually all known analytical models for predicting protective properties of concrete shields against normal high-speed impact by rigid projectiles. Presented formulas can be directly used in practical calculations. Particular emphasis is given to widely used one-stage and two-stage models which are systemized in a hierarchical classification system using a unified approach. One-stage models employ the same formula along the whole trajectory of a projectile for calculating a force exerted on a penetrating projectile by a shield. In the case of two-stage models, a resistance force at the first stage of penetration is a linear function of the instantaneous depth of penetration while at the second stage of penetration normal stresses at every location on projectile-shield contact surface are polynomial function of normal velocity component. Conditions of continuity of the resistance force and velocity of a projectile are invoked in the transition point between these two sub-models. A wide variety of models can be devised by using different sub-models at each stage of penetration.

Keywords

Concrete shield impact penetration analytical models

Introduction

In a broad class of approximate engineering models we can distinguish between two major sub-classes: empirical (semi-empirical, phenomenological) models and analytical models.

The terms “empirical model” is used for relations between impact velocity and depth of penetration (DOP) for a semi-infinite shield and ballistic limit velocity (BLV) and thickness of a finite thickness shield which have been obtained by statistical analysis of the experimental results and are not based on the physical laws (excluding, probably, approaches based on dimensional analysis and similitude).

Analytical models can be physically substantiated although usually their justification requires a large number of assumptions. We consider mainly relatively simple engineering models which are characterized by the following features: either they determine the relations between “integral characteristics” of penetration in the explicit form (in algebraic form or including quadratures) or they describe local interaction between a shield and a projectile in the points of projectile–shield contact surface that yields such relations. The latter approach provides an opportunity to describe also a process of penetration.

Most of the analytical engineering models are based completely or partially on cavity expansion approximation whereby determining forces acting on penetrating striker is reduced to calculating stresses required for expansion of cavity in the material of the shield. Important beneficial feature of this approach is that it involves determining parameters of penetrator–shield interaction model which depend upon the mechanical properties of the shield. Comprehensive description of cavity expansion approach with relevant references can be found in the monograph by Ben-Dor et al. (2006a).

Engineering models for calculating penetration into concrete shields are described in the dedicated surveys by Adeli and Amin (1985), Ben-Dor et al. (2005), Brown (1986), Corbett et al. (1996), Daudeville and Malécot (2011), Kennedy (1976), Li et al. (2005), Murthy et al. (2010), Rahman et al. (2010), Teland (1998), Williams (1994), Zaidi et al. (2010), monographs by Bangash (2009), Bangash and Bangash (2006), Ben-Dor et al. (2006a), Bulson (1997), Latif el al. (2012b), Szuladziński (2010), and also in the report published by the United States Department of Energy, DOE (2006). These surveys include mainly empirical models while much less emphasis (if at all) is given to a few well known analytical models.

The main goal of the present study is to fill this gap. We describe and classify virtually all known analytical models. Particular attention has been given to two-stage models which are widely used for describing penetration into concrete shields. Presented formulas can be directly used for practical calculations of penetration.

Semi-infinite shields

Systematization of models

Two-stage models are widely often used to describe penetration into semi-infinite concrete shields whereby the first stage (cratering) is realized for $0 \leq h \leq h_{0}$ while the second stage (tunneling) comprises subsequent motion of the impactor ( $h > h_{0}$ ), where h₀ depends on the length of the projectile nose.

In order to systematize models describing penetration into semi-finite concrete shields, it is convenient to denote them as $p q$ models where p and q are associated with sub-models used at the first stage of penetration and at the second stage of penetration, correspondingly. In the existing models, it is assumed that resistance force at the first stage of penetration is a function of the instantaneous distance between leading edge of projectile and front surface of a shield, h. Hereafter the following notations are used: $p = 1$ if the resistance force is proportional to h and $p = 2$ if the dependence between the resistance force and h is linear with non-zero constant term. At the second stage, it is assumed that the resistance force is, generally, a quadratic function of the instantaneous velocity of projectile. In the case when all three terms in this quadratic polynomial are present, $q = 3$ , and $q = 2$ if a linear term is missing; if the resistance force is independent of impactor velocity then $q = 1$ .

The main developments in the field of two-stage models are summarized in the following (more detailed description of these models and their analysis are given in corresponding Section). Originally, Forrestal et al. (1994) suggested a 1&2 model that includes the “resistance parameter” determined from experiments while Frew et al. (1998) suggested formula for calculating this parameter as a function of unconfined compressive strength of the concrete. Later, Forrestal et al. (2003) returned to their original approach for determining this parameter.

Whereas these models were developed for the ogive-shaped projectiles, Ben-Dor et al. (2003) and Chen and Li (2002) used generalized formulas for arbitrary bodies of revolution, including projectiles with flat bluntness when calculating the resistance force at the second stage of penetration. Forrestal and Tzou (1997) proposed instead of a $1 2$ model, a $1 3$ model which allows, by appropriate choice of the coefficients, to take into account compressibility and/or cracking of shield material. Ben-Dor et al. (2006b) suggested a similar generalization of this model.

However, these generalizations were proposed only for a sub-model describing the second stage of penetration, while a one-term sub-model, that assumes sharp impactors, was used at the first stage. In order to overcome this shortcoming Lixin et al. (2000) and Teland and Sjøl (2004) suggested $2 2$ models taking into account flat bluntness of impactors at both stages of penetration. Finally, Ben-Dor et al. (2009) suggested a generalized $2 3$ model which is a combination and refinement of the $1 3$ model by Forrestal and Tzou (1997) and $2 2$ model by Teland and Sjøl (2004). Since this model is the most general $p q$ models, it is convenient to use this model as a basis in describing the class of such models and interpret other models as particular cases of this general model.

The bibliography on one-stage models is less abundant, and they are analyzed in corresponding section. Hereafter these models are denoted as $0 q$ models where the parameter q has the same meaning as for the two-term models.

On the basis of the model proposed by Forrestal et al. (1994), Frew et al. (1998) and Sjøl and Teland (2001, 2003) suggested a three-term model (see also Gebbeken et al., 2009).

Some models for semi-infinite shields and shields having a finite thickness were suggested by Seifoori and Liaghat (2011) and Shiqiao et al. (2004, 2006, 2009).

Hereafter description of models is based on the “from general-to-specific” concept. Reference to a particular publication does not imply that authors of this publication formulated the model exactly in the same form as presented in this review. Very often in the original study the model is presented in a simplified form (e.g. for ogival nose projectiles or neglecting friction) while generalization of the model is straightforward.

Two-stage models

General 2&3 model

The general 2&3 model is characterized by the following expression for the instantaneous resistance force, D (Ben-Dor et al., 2009):

D = {D (1) (h) if 0 \leq h \leq h_{0} D (2) (v) if h \geq h_{0}

(1)

where the superscript indicates stage number,

D (1) (h) = c h + D_{*}, D_{*} = π r 2 σ_{n} (v)

(2)

D (2) (v) = A_{2} v 2 + A_{1} v + A_{0}

(3)

A_{i} = π a_{i} [r 2 + 2 \int_{0}^{L} [\frac{Φ_{x}}{\sqrt{Φ_{x}^{2} + 1}}) i Φ (Φ_{x} + μ_{fr}) d x], i = 0, 1, 2

(4)

c is constant that is determined below. At the second stage of penetration the model is described as

σ_{n} (v_{n}) = a_{2} v_{n}^{2} + a_{1} v_{n} + a_{0}, σ_{t} (v_{n}) = μ_{fr} σ_{n} (v_{n})

(5)

where

σ_{n}

and

σ_{t}

are normal stress and the tangential stress, respectively; a_i are parameters of the model, and other notations are shown in Figure 1.

Figure 1.

Notations.

Consequently, at the second stage of penetration, a three-term model is used. At the first stage, linear dependence between the resistance force $D (1) (h)$ and the instantaneous penetration depth is assumed, whereby $D (1) (0) = D_{*}$ is equal to the resistance of projectile flat bluntness determined from the model at the second stage of penetration, as proposed Teland and Sjøl (2004).

Solutions of the second Newton’s law equation taking into account the initial condition $v (0) = v_{imp}$ and the conditions of continuity of the resistance force and projectile velocity at $h = h_{0}$ which have been proposed previously by Forrestal et al. (1994) for the 1&2 model yield the following expression for the DOP:

H = h_{0} + m ψ (v_{0}), ψ (z) = \int_{0}^{z} \frac{ζ d ζ}{A_{2} ζ 2 + A_{1} ζ + A_{0}}

(6)

where

v_{0} = v_{0} (v_{imp}) = \frac{\sqrt{A_{1}^{2} + 4 (A_{2} + λ) (λ v_{imp}^{2} - A_{0} - D_{*})} - A_{1}}{2 (A_{2} + λ)}

(7)

c = c (v_{imp}) = \frac{m (v_{imp}^{2} - v_{0}^{2}) - 2 D_{*} h_{0}}{h_{0}^{2}}, λ = m / h_{0}

(8)

$v_{imp}$ is impact velocity, m is mass of projectile and $v_{0} = v (h_{0})$ ,

v_{imp} \geq \sqrt{(A_{0} + D_{*}) / λ}

(9)

and integral in equation (6) can be found analytically:

ψ (z) = {\frac{1}{2 A_{2}} [ln (\frac{χ}{A_{0}}) - A_{1} ω] if Δ \neq 0 \frac{1}{A_{2}} [ln (\frac{v + ɛ}{ɛ}) - \frac{v}{v + ɛ}] if Δ = 0

(10)

where

ω = {\frac{1}{\sqrt{- Δ}} ln [\frac{(η - \sqrt{- Δ}) (A_{1} + \sqrt{- Δ})}{(η + \sqrt{- Δ}) (A_{1} - \sqrt{- Δ})}] if Δ < 0 \frac{2}{\sqrt{Δ}} [tan - 1 (\frac{η}{\sqrt{Δ}}) - tan - 1 (\frac{A_{1}}{\sqrt{Δ}})] if Δ > 0

(11)

χ = A_{2} v 2 + A_{1} v + A_{0}, η = 2 A_{2} v + A_{1}, ɛ = \sqrt{A_{0} / A_{2}}, Δ = 4 A_{0} A_{2} - A_{1}^{2}

(12)

In the special case of 2&3 model proposed by Ben-Dor et al. (2009), $h_{0} = 4 R$ and coefficients a_i ( $i = 0, 1, 2$ ) are adopted from Forrestal and Tzou (1997) (see equation (36)).

1&2, 1&3 and 2&2 models are considered below as particular cases of the general 2&3 model.

1&2 models: General case

The general 2&3 model is reduced to a 1&2 model by setting $a_{1} = 0$ in equation (5) and $D_{*} = 0$ . In this case, equation (4) implies that $A_{1} = 0$ and equations (7), (8), (9) and (6) can be rewritten as follows:

H = h_{0} + \frac{m}{2 A_{2}} ln (1 + \frac{A_{2}}{A_{0}} v_{0}^{2})

(13)

v_{0} = v_{0} (v_{imp}) = \sqrt{\frac{{mv}_{imp}^{2} - A_{0} h_{0}}{m + A_{2} h_{0}}}, v_{imp} \geq \sqrt{\frac{A_{0} h_{0}}{m}}

(14)

c = c (v_{imp}) = \frac{m (v_{imp}^{2} - v_{0}^{2})}{h_{0}^{2}} = \frac{m (A_{2} v_{imp}^{2} + A_{0})}{h_{0} (m + A_{2} h_{0})}

(15)

Equation (15) implies that $c > 0$ and, consequently, the model does not violate the physics of penetration.

1&2 models: Sandia particular models

The above model recovers the model proposed by Forrestal et al. (1994) for non-truncated ogive-nosed impactors if

μ_{fr} = 0, h_{0} = 4 R, a_{2} = ρ_{sh}, a_{0} = sf'_{c}

(16)

A_{0} = π a_{0} R 2, A_{2} = π a_{2} R 2 Ψ, Ψ = \frac{8 K_{CRH} - 1}{24 K_{CRH}^{2}}, K_{CRH} = \frac{ρ_{og}}{2 R}

(17)

where

K_{CRH}

is the caliber radius head of ogive-nosed projectile,

ρ_{og}

is radius of the arc of the ogive,

f'_{c}

is the unconfined compressive strength of the shield material and s is parameter determined from experimental data. Taking into account equations (16) and (17), equations (14) and (13) can be rewritten as follows (Forrestal et al., 1994):

v_{0} = \sqrt{\frac{{mv}_{imp}^{2} - 4 π R 3 sf'_{c}}{m + 4 π R 3 ρ_{sh} Ψ}}, v_{imp} \geq 2 R \sqrt{\frac{π Rsf'_{c}}{m}}

(18)

H = 4 R + \frac{m}{2 π ρ_{sh} R 2 Ψ} ln (1 + \frac{ρ_{sh} Ψ}{sf'_{c}} v_{0}^{2})

(19)

These equations allow us to determine s as a function of H and $v_{imp}$ (Forrestal et al., 1994):

s = \frac{ρ_{sh} v_{imp}^{2} Ψ / f'_{c}}{(1 + 4 π ρ_{sh} R 3 Ψ / m) exp [2 π ρ_{sh} R 2 Ψ (H - 4 R) / m] - 1}

(20)

The latter equation is suggested for calculating values of s using experimental data for H . It is recommended to use the average value of s for predicting the DOP.

Several modifications of the model by Forrestal et al. (1994) are proposed. Frew et al. (1998) suggested using the following relationship for s:

s = 82.6 (106 / f'_{c}) 0.544

(21)

instead of processing of experimental data. Li and Chen (2003) proposed another approximation:

s = 72.0 \times 103 / \sqrt{f'_{c}}

(22)

Li and Chen (2002, 2003) used formula $h_{0} = 1.414 R + L$ instead of equation (16). Forrestal and Tzou (1997) suggested, among others, the incompressible, elastic-plastic two-term model that yields formulas $a_{0} = 5.18 f'_{c}, a_{2} = 3.88 ρ_{sh}$ instead of expressions in equation (16).

Forrestal et al. (2003), Frew et al. (2000) (for limestone shield) and Frew et al. (2006) returned to the approach suggested by Forrestal et al. (1994) and preferred interpreting the resistance parameter $sf' = a_{0}$ as a measure of shield resistance which can be determined from experiments using equation (20). Frew et al. (2000) showed that this parameter depends on projectile shank diameter, and proposed equation that describes dependence of the resistance parameter on diameter. Frew et al. (2006) studied the effect of shield diameter on deceleration and penetration depth of 3.0 caliber-radius-head ogive-nosed projectiles, and found negligible changes in the penetration depth and only small decrease of deceleration as shield diameter was reduced.

Teland and Sjøl (2000) studied the boundary effects in penetration, namely, effects associated with finite size of a shield in the direction normal to the direction of penetration, and introduced a correcting parameter into the model of Forrestal et al. (1994) that was supposed to take into account these effects. Although they did not succeed to derive a formula for the correction coefficient, they found that boundary effects can be neglected in most cases if the ratio of shield diameter to projectile diameter is larger than 15.

Apart from the analysis conducted by the authors of this review, some additional evaluations of Sandia 1&2 models based on the experimental data were performed by Forrestal et al. (1996), Frew et al. (1998), Tham (2006) and Vahedi et al. (2008).

Gomez and Shukla (2001) extended the 1&2 model of Forrestal et al. (1994)(, 1996) and Frew et al. (1998) to multiple impacts introducing an empirical coefficient that is a function of the number of impacts. On the basis of the same 1&2 model, Choudhury et al. (2002), Siddiqui et al. (2002) and Siddiqui et al. (2009) derived expressions for the DOP for a buried shield and applied sensitivity analysis to study the influence of various random variables on projectile reliability and shield safety.

1&2 models: Case of small impact velocities

If $v_{imp} < \sqrt{A_{0} h_{0} / m}$ then the radicand the expression for v₀ in equation (14) is negative, and the model in its original form cannot be used. In this case, projectile is stopped at the first stage of penetration and the second stage does not occur. Formula for the DOP can be obtained from the equation of projectile motion at the first stage of penetration:

H = \sqrt{\frac{m}{c}} v_{imp} = \sqrt{\frac{h_{0} (m + A_{2} h_{0})}{A_{2} v_{imp}^{2} + A_{0}}} v_{imp}

(23)

assuming that equation (15) for parameter c remains valid. This model (

h_{0} = 4 R

a_{2} = ρ_{sh}

) is proposed by Li and Chen (2003), and it can be considered as generalization of the model of Forrestal et al. (1994) and Frew et al. (1998).

1&2 models: Other versions

Li and Chen (2003) (see also Li and Chen 2002) found that the generalized model of Forrestal et al. (1994) and Frew et al. (1998) allows to determine the dimensionless DOP as a function of two dimensionless parameters.

In a more general case of 1&2 model, it is convenient to introduce similar dimensionless parameters as follows:

N = \frac{π m}{4 d A_{2}}, I = \frac{π {mv}_{imp}^{2}}{4 d A_{0}}, k = \frac{h_{0}}{d} = \frac{h_{0}}{2 R}

(24)

Then equations (13) and (23) recover the same relationships:

\frac{H}{d} = {\sqrt{\frac{1 + 0.25 k π / N}{1 + I / N} \frac{4 k}{π} I} if \frac{H}{d} \leq k, or I \leq 0.25 k π k + \frac{2 N}{π} ln (\frac{1 + I / N}{1 + 0.25 k π / N}) if \frac{H}{d} > k, or I > 0.25 k π

(25)

which have been obtained by Li and Chen (2003) when parameters of the model are selected using the model of Forrestal et al. (1994) and Frew et al. (1998):

N = \frac{m}{8 ρ_{sh} d} / \int_{0}^{L} \frac{Φ Φ_{x}^{3}}{Φ_{x}^{2} + 1} d x, I = \frac{{mv}_{imp}^{2}}{82.6 f'_{c} (106 / f'_{c}) 0.544 d 3}

(26)

In the limiting case, $N > > 1, I / N < < 1$ , Li and Chen (2003) suggested the following simplified formulas:

\frac{H}{d} = {\sqrt{\frac{4 k}{π} I} if \frac{H}{d} \leq k \frac{k}{2} + \frac{2 I}{π} if \frac{H}{d} > k

(27)

Apart from the analysis performed by Li and Chen (2003), comparison of this model with results of experiments was conducted also by Shiu et al. (2008). Latif et al. (2012a, 2012b) slightly improved the model by Li and Chen (2003), by introducing, in particular, additional frictional factor for ogive-nose projectiles.

In the case when the second stage of penetration is not realized (the basic model by Forrestal et al., 1994 and Frew et al., 1998), similar dimensionless variables were suggested and accuracy of the dimensionless formula for the DOP was studied by Sjøl et al. (2002), Sjøl and Teland (2000) and Teland and Sjøl (2000) (see also Teland and Sjøl, 2004).

Note that using the following dimensionless variables,

ω_{1} = \frac{m}{A_{2} h_{0}}, ω_{2} = \frac{{mv}_{imp}^{2}}{A_{0} h_{0}}

(28)

allows us obtaining a more simple formula for the DOP:

\frac{H}{h_{0}} = {\sqrt{\frac{ω_{2} (ω_{1} + 1)}{ω_{1} + ω_{2}}} if 0 < ω_{2} \leq 1 1 + ω_{1} ln (\frac{ω_{1} + ω_{2}}{ω_{1} + 1}) if ω_{2} > 1

(29)

1&1 model

For relatively small impact velocities, Forrestal et al. (2003) used the 1&1 model

D (1) (h) = ch, D (2) (v) = A_{0}

(30)

Substituting $A_{2} = 0$ into equations (15) and (14) we arrive at the simplified expressions for c and v₀:

c = A_{0} / h_{0}, v_{0} = \sqrt{({mv}_{imp}^{2} - A_{0} h_{0}) / m}

(31)

Formula for the DOP is obtained by passing to the limit $A_{2} \to 0$ in equation (13) and taking into account equation (14). Applying the L’Hospital’s rule we find that

H = 0.5 h_{0} + \frac{{mv}_{imp}^{2}}{2 A_{0}}

(32)

1&3 models

Models of this type can be obtained from the general 2&3 model setting $D_{*} = 0$ . Similar to the 2&3 model, this model can be applied for describing penetration by projectiles without flat bluntness.

For this model, equations (8) and (7) can be rewritten as follows:

c = c (v_{imp}) = \frac{m (v_{imp}^{2} - v_{0}^{2})}{h_{0}^{2}}

(33)

v_{0} = v_{0} (v_{imp}) = \frac{\sqrt{A_{1}^{2} + 4 (A_{2} + λ) (λ v_{imp}^{2} - A_{0})} - A_{1}}{2 (A_{2} + λ)}

(34)

and equation (6) and inequality in equation (14) are satisfied. Clearly, the model is mechanistically sound only if

c > 0

, or

v_{0} < v_{imp}

. Taking into account equation (34), the latter inequality reduces to the following constraint:

A_{2} v_{imp}^{2} + A_{1} v_{imp} + A_{0} > 0

(35)

Forrestal and Tzou (1997) suggested the 1&3 model with

a_{i} = f'_{c} (\frac{ρ_{sh}}{f'_{c}}) i / 2 {\tilde{a}}_{i}, i = 0, 1, 2

(36)

where a_i are coefficients in equation (5),

{\tilde{a}}_{i}

are the known dimensionless coefficients which determine the model of the shield material (see Table 1). Forrestal and Tzou (1997) assumed that

μ_{fr} = 0

and

h_{0} = 4 R

, and considered non-truncated ogive-nosed projectiles for which equation (17) must be used for calculation of

A_{0}

while

A_{1} = π a_{2} R 2 Ψ_{1}, Ψ_{1} = 2 K_{CRH} (1 - 2 K_{CRH}) \times sin - 1 (\frac{\sqrt{4 K_{CRH} - 1}}{2 K_{CRH}}) + \frac{\sqrt{4 K_{CRH} - 1}}{6 K_{CRH}} (12 K_{CRH}^{2} - 4 K_{CRH} + 1)

(37)

Table 1.

Coefficients for different models of shield material (Forrestal and Tzou, 1997).

Number of the model	Characteristic of the model	${\tilde{a}}_{0}$	${\tilde{a}}_{1}$	${\tilde{a}}_{2}$
1	Incompressible, elastic–plastic	5.18	0.0	3.88
2	Incompressible, elastic-cracked-plastic	4.05	1.36	3.51
3	Compressible, elastic–plastic	4.50	0.75	1.29
4	Compressible, elastic-cracked-plastic	3.45	1.60	1.12

2&2 models: Version 1

Consider the 2&2 model as a particular case of the general 2&3 model for which $a_{1} = 0$ in equation (5). Then $A_{1} = 0$ , and equations (8) and (7) can be rewritten as follows:

v_{0} = \sqrt{\frac{η}{m + h_{0} A_{2}}}, η = (m - h_{0} B_{2}) v_{imp}^{2} - h_{0} (A_{0} + B_{0})

(38)

c = \frac{({mA}_{2} - {mB}_{2} - 2 h_{0} A_{2} B_{2}) v_{imp}^{2} + {mA}_{0} - {mB}_{0} - 2 h_{0} B_{0} A_{2}}{h_{0} (m + h_{0} A_{2})}

(39)

η \geq 0, B_{i} = π r 2 a_{i}, i = 0, 2

(40)

The DOP in this case is determined by equation (13).

Ben-Dor et al. (2010) took notice to the fact that this model allows positive and negative values of the parameter c . Analysis shows that $c < 0$ for relatively large values of the parameter $\bar{r} = r / R$ .

Teland and Sjøl (2004) suggested that $h_{0} = L$ and described their model for two classes of projectile shapes. In the case of truncated ogive-nosed impactors,

A_{2} = π a_{2} R 2 Ψ, Ψ = \frac{(1 - \bar{r}) 2 [3 \bar{r} 2 + (2 \bar{r} + 1) (8 K_{CRH} - 1)]}{24 K_{CRH}^{2}}

(41)

We are not aware about the experiments which allow evaluating accuracy of the linear approximation of the function $D (1) (h)$ . However, there exist indirect arguments in favor of the linear model, and these arguments are commonly accepted for substantiating the empirical models. First of all, Teland and Sjøl (2004) validated the model by comparing results of calculations based on this model with experimental data on the DOP. Secondly, a linear model is a simple and natural generalization of the model suggested by Forrestal et al. (1994). This generalization does not require introducing additional empirical parameters and employs the same method for determining the value of only one coefficient. Finally, introducing this model preserves the hierarchy of the models whereby in a particular case of a sharp impactor, the model of Teland and Sjøl (2004) recovers the model of Forrestal et al. (1994) (with another choice of h₀).

If $η \geq 0$ then the impactor penetrates at the depth $h = h_{0}$ , and the above conditions of continuity at $h = h_{0}$ have a real physical meaning. A different situation arises when $η < 0$ , and the models suggested by Forrestal et al. (1994) and Teland and Sjøl (2004) cannot be directly applied in this case. Li and Chen (2003) solved a similar problem for $1 2$ model by using the same expression for c. Ben-Dor et al. (2010) employed this approach for the 2&2 model. For $η < 0$ , the sub-model is applicable if

γ_{1}^{2} + γ_{0} \geq 0, γ_{0} = {mv}_{imp}^{2} / c, γ_{1} = (B_{2} v_{imp}^{2} + B_{0}) / c

(42)

and the DOP is obtained using the following formula:

H = {\sqrt{γ_{1}^{2} + γ_{0}} - γ_{1} if c > 0 - \sqrt{γ_{1}^{2} + γ_{0}} - γ_{1} if c \leq 0

(43)

2&2 models: Version 2

Lixin et al. (2000) suggested a model for truncated ogive-nosed impactors. They assumed that the resistance force is determined by equation (1) where

D (1) (h) = CC' (h + L_{\times})

(44)

D (2) (v) = C' D_{\times}^{(2)} (v), D_{\times}^{(2)} (v) = A_{2} v 2 + A_{0}

(45)

C' = 1 + κ_{1} (r / R) 2, h_{0} = κ_{0} R

(46)

C is constant that is determined below while

κ_{0}

(

3 \leq κ_{0} \leq 5

) and

κ_{1}

are parameters which are determined from experiments. The resistance force of the hypothetical sharp ogive-nosed impactor having the length

L + L_{\times}

(see Figure 2),

D_{\times}^{(2)}

, is calculated on the basis of two-term model using equations (16), (17) and (21) for determining parameters A₀ and A₂ .

Figure 2.

Shape of projectile: illustration to the model by Lixin et al. (2000).

Solution of the second Newton’s law equation taking into account initial condition $v (0) = v_{imp}$ and conditions of continuity of the resistance force and impactor velocity at $h = h_{0}$ implies the following expression for the DOP:

H = h_{0} + \frac{m}{2 A_{2} C'} ln (1 + \frac{A_{2}}{A_{0}} v_{0}^{2})

(47)

where

v_{0} = \sqrt{\frac{{mv}_{imp}^{2} (h_{0} + L_{\times}) - A_{0} C' h_{0} (h_{0} + 2 L_{\times})}{m (h_{0} + L_{\times}) + A_{2} C' h_{0} (h_{0} + 2 L_{\times})}}

(48)

C = \frac{A_{2} v_{0}^{2} + A_{0}}{h_{0} + L_{\times}} > 0

(49)

v_{imp} \geq \frac{A_{0} C' h_{0} (h_{0} + 2 L_{\times})}{m (h_{0} + L_{\times})}

(50)

One-stage models

Luk and Forrestal (1987) proposed spherical cavity expansion model for the material described with a locked hydrostat and constant shear strength. This model is a 0&2 model that is determined by equation (5) with $a_{1} = 0$ and $μ_{fr} = 0$ where

a_{0} = (2 / 3) Y (1 - ln η_{*})

(51)

a_{2} = \frac{\frac{3 Y}{E} + η_{*} (1 - \frac{3 Y}{2 E}) 2 + \frac{3 η_{*}^{2 / 3} - η_{*} (4 - η_{*})}{2 (1 - η_{*})}}{[(1 + \frac{Y}{2 E}) 3 + η_{*} - 1] 2 / 3} ρ_{sh}

(52)

Y is the yield stress, E is Young’s modulus, $ρ_{sh}$ is the initial density of shield material and $η_{*}$ is the locked volumetric strain.

DOP can be calculated using equation (13) where $h_{0} = 0$ , A₀ and A₂ are given by equation (4). In the case of ogive-nose projectiles that was considered by Luk and Forrestal (1987), equation (4) must be replaced by equation (17).

Luk and Forrestal (1989) modified the expression for the DOP in Luk and Forrestal (1987) by taking into account friction, and showed that a proper choice of friction coefficient improves agreement between experimental results and theoretical predictions. However, taking into account a current level of understanding of the mechanism of interaction of projectiles with concrete shields, this friction coefficient can be considered only as a fudge factor for improving predictions of the model.

Xu et al. (1997) suggested 0&2 elastic-cracked spherical cavity expansion model. Chen and Li (2002) did not consider the first stage of penetration (cratering) and applied $0 2$ models with parameters which are determined either by the model of Forrestal et al. (1994) or the model of Luk and Forrestal (1987). In this case, equation (13) with $h_{0} = 0$ allows us to calculate the DOP. Using the dimensionless variables defined by equation (24) or (26) expression for the DOP can be written as follows:

\frac{H}{d} = \frac{2 N}{π} ln (1 + \frac{I}{N}) = \frac{2 I}{π} ϕ (ζ), ϕ (ζ) = \frac{ln (1 + ζ)}{ζ}, ζ = \frac{I}{N}

(53)

Function $ϕ (ζ)$ for $0 < ζ < 1$ can be expanded into converging power series with alternating signs:

ϕ (ζ) = \sum_{n = 1}^{\infty} (- 1) n + 1 \frac{ζ n - 1}{n}

(54)

If $I < < N$ ( $ζ < < 1$ ) the following approximate formula for the DOP can be obtained by keeping only the first term in this series (Chen and Li, 2002):

\frac{H}{d} \approx \frac{2}{π} I = 0.637 I

(55)

Rosenberg and Dekel (2010) proposed a 0&1 model that is based on the assumption that projectile moves inside a shield with constant deceleration, a. Since $D = ma$ , under this assumption solution of the equation of motion, $H = 0.5 v_{imp}^{2} / a$ , can be written as

H = 0.5 ρ_{imp} L_{eff} v_{imp}^{2} / R_{sh}

(56)

where

R_{sh}

is the resisting stress of a shield,

ρ_{imp}

is density of projectile,

L_{eff}

is effective length of projectile and

R_{sh} = ρ_{imp} L_{eff} a, L_{eff} = m / (π R 2 ρ_{imp})

(57)

Equation (56) is recommended for calculating the DOP, where $L_{eff}$ is found from equation (57), while the following formula is suggested for determining $R_{sh}$ (in GPa) as a function of $f'_{c}$ (in MPa):

R_{sh} = 0.22 ln (f'_{c}) - 0.285

(58)

It is assumed that $v_{imp} < v_{c}$ , where critical velocity $v_{c}$ is defined as $v_{c} = \sqrt{R_{t} / (K_{shape}^{(0)} ρ_{sh})}$ and $K_{shape}^{(0)}$ is a projectile nose shape factor (for instance, $K_{shape}^{(0)} = 0.5$ for a hemispherical nose).

Wen (2002) suggested the model in which normal stress on projectile surface that is in contact with concrete shield is a linear function of the impact velocity.

Teland (2001) proposed a 0&2 model to predrilled shields penetrated by ogive-nose projectile. In this case, the DOP of ogive-nose projectile is calculated using equation (13) where

A_{0} = π a_{0} R 2 (1 - η 2), η = d_{0} / d, h_{0} = 0

(59)

A_{2} = \frac{π a_{2} R 2 [(8 - 24 R 2 - 16 R 3) K_{CRH} - 3 R 4 + 8 R 3 + 6 R 2 - 1]}{24 K_{CRH}^{2}}

(60)

d₀ is hole diameter, and coefficients a₀ and a₂ are determined by the model of Forrestal et al. (1994, 1996).

He et al. (2011) proposed a spherical cavity expansion penetration model to predict penetration and perforation of concrete shields by ogive-nose projectiles. Shear dilatancy as well as compressibility of material in comminuted region are considered by introducing a dilatant–kinematic relation. Solution of cavity expansion problem allows determining coefficients of approximation in equation (5) were and applying it directly for penetration modeling.

Sun and Yuan (2012) adapted the model suggested by Yankelevsky and Adin (1980) for the case of penetration into concrete.

Shields having finite thickness

As a rule, analytical models describing penetration and perforation of finite thickness shields employ various modifications and generalizations of the model suggested by Forrestal et al. (1994).

Li and Tong (2003) (see also Chen et al., 2004, Li et al., 2006, 2005) assume that perforation occurs after plug formation, and projectile motion prior plug formation is described, as in the case of semi-infinite shields, by 1&2 model of Forrestal et al. (1994), Frew et al. (1998) and Li and Chen (2003). A modification of this model was suggested by Dai et al. (2005). Chen et al. (2007, 2008) used similar approach to reinforced concrete shield. Ben-Dor et al. (2010) suggested a model that is the combination of the models proposed by Li and Tong (2003) and Teland and Sjøl (2004) and is applicable for projectiles with a flat bluntness.

Taking into account small correction described in Li et al. (2005) the model by Li and Tong (2003) in the simplified version yields the following formulas for the BLV, $v_{bl}$ :

v_{bl} = {G_{1} (T) for v_{bl} \leq W G_{2} (T) for v_{bl} > W

(61)

where

G_{1} (T) = \frac{d (w - 1)}{2 tan α} \sqrt{\frac{π kf'_{c} d}{3 sm}}, W = \frac{d}{2} \sqrt{\frac{π ksf'_{c} d}{m}}

(62)

G_{2} (T) = d \sqrt{\frac{'_{c} sd}{2 m} [\frac{T}{d} - \frac{k}{2} - \frac{\sqrt{\sqrt{3} s tan α + 1} - 1}{2 tan α}]}

(63)

w = \frac{3 s 2}{16 k 2} [\sqrt{1 + \frac{8 k}{\sqrt{3} s} (1 + \frac{2 k}{\sqrt{3} s} + \frac{2 T}{d} \tan α)} - 1] 2

(64)

parameter s is determined by equation (21) while α is the angle between the axis of the conoid (truncated cone) that describes the plug and its generatrix.

The above discussed three-phase model suggested by Sjøl and Teland (2001, 2003) for semi-infinite shields can be modified for shields having a finite thickness. He et al. (2011) also suggested using their model not only for semi-infinite shield but also for shields having a finite thickness.

It must be noted that deriving explicit formulas for calculating BLV as a function of the shield thickness, taking into account a possibility of plug formation, is a very hard problem that requires making serious simplifying assumptions. According to the definition in the “Introduction” section such models cannot be classified as engineering models, and they are not considered here comprehensively.

Concluding remarks

In this study, we presented more or less comprehensively all widely used analytical models which were suggested for describing high-speed penetration into concrete shields.

The performed analysis showed that most models can be considered as one-stage and two-stage models. One-stage models employ the same formula along the whole trajectory of a projectile for calculating a force exerted on a penetrating projectile by a shield. In the case of two-stage models, a resistance force at the first stage of penetration is a linear function of the instantaneous DOP while at the second stage of penetration normal stresses at every location on projectile-shield contact surface are polynomial function of normal velocity component. Conditions of continuity of the resistance force and velocity of a projectile are invoked in the transition point between these two sub-models. A wide variety of models can be devised by using different sub-models at each stage of penetration.

The adopted unified approach allowed systemizing all models in a hierarchical classification system (see Figure 3).

Figure 3.

Systematized references on p&q models.

In descriptions of the models, we usually included arguments in favor of the particular model which are based on theoretical considerations, experimental results or “exact” calculations. Nevertheless, it must be emphasized that these arguments or comparisons with other models have only illustrative character.

Clearly, we realize the importance and urgency of theoretical and experimental investigations for the purpose of comparing performance of various analytical models and refining the ranges of their validity. However, we believe that this is a separate and very challenging problem that will be a subject of future research. The present survey is a necessary step in this direction, and we conceive that it is of interest in its own right.

Footnotes

Appendix 1

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