Article Analytical engineering models for predicting high speed penetration of hard projectiles into concrete shields: A review G Ben-Dor, A Dubinsky and T Elperin 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 pene- tration (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 International Journal of Damage Mechanics 2015, Vol. 24(1) 76–94 ! The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1056789514521647 ijd.sagepub.com Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel Corresponding author: T Elperin, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P. O. Box 653, Beer-Sheva 84105, Israel. Email: [email protected]at BEN GURION UNIV NEGEV on December 8, 2014 ijd.sagepub.com Downloaded from
Transcript
Article
Analytical engineering models forpredicting high speed penetrationof hard projectiles into concreteshields: A review
G Ben-Dor, A Dubinsky and T Elperin
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.
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 pene-tration (DOP) for a semi-infinite shield and ballistic limit velocity (BLV) and thickness of a finitethickness shield which have been obtained by statistical analysis of the experimental results and are
International Journal of Damage
Mechanics
2015, Vol. 24(1) 76–94
! The Author(s) 2014
Reprints and permissions:
sagepub.co.uk/journalsPermissions.nav
DOI: 10.1177/1056789514521647
ijd.sagepub.com
Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the
Negev, Beer-Sheva, Israel
Corresponding author:
T Elperin, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P. O. Box 653, Beer-Sheva 84105, Israel.
not based on the physical laws (excluding, probably, approaches based on dimensional analysis andsimilitude).
Analytical models can be physically substantiated although usually their justification requires alarge number of assumptions. We consider mainly relatively simple engineering models which arecharacterized by the following features: either they determine the relations between ‘‘integral char-acteristics’’ of penetration in the explicit form (in algebraic form or including quadratures) or theydescribe local interaction between a shield and a projectile in the points of projectile–shield contactsurface that yields such relations. The latter approach provides an opportunity to describe also aprocess of penetration.
Most of the analytical engineering models are based completely or partially on cavity expansionapproximation whereby determining forces acting on penetrating striker is reduced to calculatingstresses required for expansion of cavity in the material of the shield. Important beneficial feature ofthis approach is that it involves determining parameters of penetrator–shield interaction model whichdepend upon the mechanical properties of the shield. Comprehensive description of cavity expansionapproach 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 dedi-cated surveys by Adeli and Amin (1985), Ben-Dor et al. (2005), Brown (1986), Corbett et al. (1996),Daudeville and Malecot (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), Bangashand Bangash (2006), Ben-Dor et al. (2006a), Bulson (1997), Latif el al. (2012b), Szuladzinski (2010),and also in the report published by the United States Department of Energy, DOE (2006). Thesesurveys include mainly empirical models while much less emphasis (if at all) is given to a few wellknown analytical models.
The main goal of the present study is to fill this gap. We describe and classify virtually all knownanalytical models. Particular attention has been given to two-stage models which are widely used fordescribing penetration into concrete shields. Presented formulas can be directly used for practicalcalculations of penetration.
Semi-infinite shields
Systematization of models
Two-stage models are widely often used to describe penetration into semi-infinite concrete shieldswhereby the first stage (cratering) is realized for 0 � h � h0 while the second stage (tunneling)comprises subsequent motion of the impactor (h4 h0), where h0 depends on the length of theprojectile nose.
In order to systematize models describing penetration into semi-finite concrete shields, it is con-venient to denote them as p & q models where p and q are associated with sub-models used at thefirst stage of penetration and at the second stage of penetration, correspondingly. In the existingmodels, it is assumed that resistance force at the first stage of penetration is a function of theinstantaneous distance between leading edge of projectile and front surface of a shield, h.Hereafter the following notations are used: p ¼ 1 if the resistance force is proportional to h andp ¼ 2 if the dependence between the resistance force and h is linear with non-zero constant term. Atthe second stage, it is assumed that the resistance force is, generally, a quadratic function of theinstantaneous velocity of projectile. In the case when all three terms in this quadratic polynomial arepresent, q ¼ 3 , and q ¼ 2 if a linear term is missing; if the resistance force is independent of impactorvelocity then q ¼ 1.
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The main developments in the field of two-stage models are summarized in the following (moredetailed 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 par-ameter 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) andChen and Li (2002) used generalized formulas for arbitrary bodies of revolution, including project-iles 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, byappropriate choice of the coefficients, to take into account compressibility and/or cracking of shieldmaterial. 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 ofpenetration, while a one-term sub-model, that assumes sharp impactors, was used at the first stage. Inorder to overcome this shortcoming Lixin et al. (2000) and Teland and Sjøl (2004) suggested 2 & 2models 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 the1 & 3 model by Forrestal and Tzou (1997) and 2 & 2 model by Teland and Sjøl (2004). Since thismodel is the most general p & qmodels, it is convenient to use this model as a basis in describing theclass 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 correspondingsection. Hereafter these models are denoted as 0 & q models where the parameter q has the samemeaning as for the two-term models.
On the basis of the model proposed by Forrestal et al. (1994), Frew et al. (1998) and Sjøl andTeland (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 bySeifoori 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 aparticular publication does not imply that authors of this publication formulated the model exactlyin the same form as presented in this review. Very often in the original study the model is presentedin a simplified form (e.g. for ogival nose projectiles or neglecting friction) while generalization of themodel is straightforward.
Two-stage models
General 2&3 model. The general 2&3 model is characterized by the following expression for theinstantaneous resistance force, D (Ben-Dor et al., 2009):
D ¼Dð1ÞðhÞ if 0 � h � h0
Dð2ÞðvÞ if h � h0
(ð1Þ
where the superscript indicates stage number,
Dð1ÞðhÞ ¼ c hþD�, D� ¼ �r2�nðvÞ ð2Þ
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where �n and �t are normal stress and the tangential stress, respectively; ai are parameters of themodel, and other notations are shown in Figure 1.
Consequently, at the second stage of penetration, a three-term model is used. At the first stage,linear dependence between the resistance force Dð1ÞðhÞ and the instantaneous penetration depth isassumed, whereby Dð1Þð0Þ ¼ D� is equal to the resistance of projectile flat bluntness determined fromthe 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 conditionvð0Þ ¼ vimp and the conditions of continuity of the resistance force and projectile velocity ath ¼ h0 which have been proposed previously by Forrestal et al. (1994) for the 1&2 model yieldthe following expression for the DOP:
H ¼ h0 þm ðv0Þ, ðzÞ ¼
Z z
0
�d�
A2�2 þ A1� þ A0ð6Þ
Figure 1. Notations.
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and integral in equation (6) can be found analytically:
ðzÞ ¼
1
2A2ln
�
A0
� �� A1!
� �if � 6¼ 0
1
A2ln
vþ "
"
� ��
v
vþ "
� �if � ¼ 0
8>>><>>>:
ð10Þ
where
! ¼
1ffiffiffiffiffiffiffiffi��p ln
ð��ffiffiffiffiffiffiffiffi��p
ÞðA1 þffiffiffiffiffiffiffiffi��p
Þ
ð�þffiffiffiffiffiffiffiffi��p
ÞðA1 �ffiffiffiffiffiffiffiffi��p
Þ
� �if �5 0
2ffiffiffiffi�p tan�1
�ffiffiffiffi�p
� �� tan�1
A1ffiffiffiffi�p
� �� �if �4 0
8>>><>>>:
ð11Þ
� ¼ A2v2 þ A1vþ A0, � ¼ 2A2vþ A1, " ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiA0=A2
p, � ¼ 4A0A2 � A2
1 ð12Þ
In the special case of 2&3 model proposed by Ben-Dor et al. (2009), h0 ¼ 4R and coefficients ai(i ¼ 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 a1 ¼ 0 inequation (5) and D� ¼ 0 . In this case, equation (4) implies that A1 ¼ 0 and equations (7), (8), (9) and(6) can be rewritten as follows:
Equation (15) implies that c4 0 and, consequently, the model does not violate the physics ofpenetration.
1&2 models: Sandia particular models. The above model recovers the model proposed byForrestal et al. (1994) for non-truncated ogive-nosed impactors if
�fr ¼ 0, h0 ¼ 4R, a2 ¼ sh, a0 ¼ sf 0c ð16Þ
A0 ¼ �a0R2, A2 ¼ �a2R
2�, � ¼8KCRH � 1
24K2CRH
, KCRH ¼og2R
ð17Þ
where KCRH is the caliber radius head of ogive-nosed projectile, og is radius of the arc of the ogive,f 0c is the unconfined compressive strength of the shield material and s is parameter determined fromexperimental data. Taking into account equations (16) and (17), equations (14) and (13) can berewritten as follows (Forrestal et al., 1994):
These equations allow us to determine s as a function of H and vimp (Forrestal et al., 1994):
s ¼shv
2imp�=f 0c
ð1þ 4�shR3�=mÞexp½2�shR2�ðH� 4RÞ=m� � 1ð20Þ
The latter equation is suggested for calculating values of s using experimental data for H . It isrecommended 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 ¼ 82:6ð106=f 0cÞ0:544
ð21Þ
instead of processing of experimental data. Li and Chen (2003) proposed another approximation:
s ¼ 72:0� 103=ffiffiffiffiffif 0c
pð22Þ
Li and Chen (2002, 2003) used formula h0 ¼ 1:414Rþ L instead of equation (16). Forrestal andTzou (1997) suggested, among others, the incompressible, elastic-plastic two-term model that yieldsformulas a0 ¼ 5:18f 0c, a2 ¼ 3:88sh instead of expressions in equation (16).
Forrestal et al. (2003), Frew et al. (2000) (for limestone shield) and Frew et al. (2006) returned tothe approach suggested by Forrestal et al. (1994) and preferred interpreting the resistance parametersf 0 ¼ a0 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, andproposed equation that describes dependence of the resistance parameter on diameter. Frew et al.
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(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 onlysmall decrease of deceleration as shield diameter was reduced.
Teland and Sjøl (2000) studied the boundary effects in penetration, namely, effects associated withfinite size of a shield in the direction normal to the direction of penetration, and introduced acorrecting parameter into the model of Forrestal et al. (1994) that was supposed to take into accountthese effects. Although they did not succeed to derive a formula for the correction coefficient, theyfound that boundary effects can be neglected in most cases if the ratio of shield diameter to projectilediameter is larger than 15.
Apart from the analysis conducted by the authors of this review, some additional evaluations ofSandia 1&2 models based on the experimental data were performed by Forrestal et al. (1996), Frewet 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 Frewet al. (1998) to multiple impacts introducing an empirical coefficient that is a function of the numberof impacts. On the basis of the same 1&2 model, Choudhury et al. (2002), Siddiqui et al. (2002) andSiddiqui et al. (2009) derived expressions for the DOP for a buried shield and applied sensitivityanalysis to study the influence of various random variables on projectile reliability and shield safety.
1&2 models: Case of small impact velocities. If vimp 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA0h0=m
pthen 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. Formulafor the DOP can be obtained from the equation of projectile motion at the first stage of penetration:
assuming that equation (15) for parameter c remains valid. This model (h0 ¼ 4R , a2 ¼ sh) isproposed by Li and Chen (2003), and it can be considered as generalization of the model ofForrestal 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 thegeneralized model of Forrestal et al. (1994) and Frew et al. (1998) allows to determine the dimen-sionless DOP as a function of two dimensionless parameters.
In a more general case of 1&2 model, it is convenient to introduce similar dimensionless param-eters as follows:
N ¼�m
4dA2, I ¼
�mv2imp
4dA0, k ¼
h0d¼
h02R
ð24Þ
Then equations (13) and (23) recover the same relationships:
which have been obtained by Li and Chen (2003) when parameters of the model are selected usingthe model of Forrestal et al. (1994) and Frew et al. (1998):
N ¼m
8shd
Z L
0
��3x
�2x þ 1
dx, I ¼mv2imp
82:6f 0cð106=f 0cÞ
0:544d3ð26Þ
In the limiting case, N44 1, I=N55 1 , Li and Chen (2003) suggested the following simplifiedformulas:
H
d¼
ffiffiffiffiffiffiffiffi4k
�I
rif
H
d� k
k
2þ2I
�if
H
d4 k
8>><>>: ð27Þ
Apart from the analysis performed by Li and Chen (2003), comparison of this model with results ofexperiments was conducted also by Shiu et al. (2008). Latif et al. (2012a, 2012b) slightly improvedthe 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 Forrestalet al., 1994 and Frew et al., 1998), similar dimensionless variables were suggested and accuracy of thedimensionless formula for the DOP was studied by Sjøl et al. (2002), Sjøl and Teland (2000) andTeland and Sjøl (2000) (see also Teland and Sjøl, 2004).
Note that using the following dimensionless variables,
!1 ¼m
A2h0, !2 ¼
mv2imp
A0h0ð28Þ
allows us obtaining a more simple formula for the DOP:
Formula for the DOP is obtained by passing to the limit A2! 0 in equation (13) and taking intoaccount equation (14). Applying the L’Hospital’s rule we find that
H ¼ 0:5h0 þmv2imp
2A0ð32Þ
1&3 models. Models of this type can be obtained from the general 2&3 model setting D� ¼ 0 .Similar to the 2&3 model, this model can be applied for describing penetration by projectiles withoutflat bluntness.
For this model, equations (8) and (7) can be rewritten as follows:
and equation (6) and inequality in equation (14) are satisfied. Clearly, the model is mechanisticallysound only if c4 0 , or v0 5 vimp. Taking into account equation (34), the latter inequality reduces tothe following constraint:
A2v2imp þ A1vimp þ A0 4 0 ð35Þ
Forrestal and Tzou (1997) suggested the 1&3 model with
ai ¼ f 0cshf 0c
� �i=2
~ai, i ¼ 0, 1, 2 ð36Þ
where ai are coefficients in equation (5), ~ai are the known dimensionless coefficients which determinethe model of the shield material (see Table 1). Forrestal and Tzou (1997) assumed that �fr ¼ 0 andh0 ¼ 4R , and considered non-truncated ogive-nosed projectiles for which equation (17) must beused for calculation of A0 while
2&2 models: Version 1. Consider the 2&2 model as a particular case of the general 2&3 model forwhich a1 ¼ 0 in equation (5). Then A1 ¼ 0 , and equations (8) and (7) can be rewritten as follows:
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 c5 0 for relatively large values of the parameter �r ¼ r=R .Teland and Sjøl (2004) suggested that h0 ¼ L and described their model for two classes of pro-
jectile shapes. In the case of truncated ogive-nosed impactors,
A2 ¼ �a2R2�, � ¼
ð1� �rÞ2½3�r2 þ ð2�rþ 1Þð8KCRH � 1Þ�
24K2CRH
ð41Þ
We are not aware about the experiments which allow evaluating accuracy of the linear approxi-mation of the function Dð1ÞðhÞ. 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 modelwith experimental data on the DOP. Secondly, a linear model is a simple and natural generalizationof the model suggested by Forrestal et al. (1994). This generalization does not require introducingadditional empirical parameters and employs the same method for determining the value of only onecoefficient. Finally, introducing this model preserves the hierarchy of the models whereby in aparticular case of a sharp impactor, the model of Teland and Sjøl (2004) recovers the model ofForrestal et al. (1994) (with another choice of h0).
If � � 0 then the impactor penetrates at the depth h ¼ h0, and the above conditions of continuityat h ¼ h0 have a real physical meaning. A different situation arises when �5 0, and the modelssuggested by Forrestal et al. (1994) and Teland and Sjøl (2004) cannot be directly applied inthis case. Li and Chen (2003) solved a similar problem for 1 & 2 model by using the same expressionfor c. Ben-Dor et al. (2010) employed this approach for the 2&2 model. For �5 0, the sub-model isapplicable if
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
Dð1ÞðhÞ ¼ CC0ð hþ L�Þ ð44Þ
Dð2ÞðvÞ ¼ C0Dð2Þ� ðvÞ, Dð2Þ� ðvÞ ¼ A2v2 þ A0 ð45Þ
C0 ¼ 1þ �1ðr=RÞ2, h0 ¼ �0R ð46Þ
C is constant that is determined below while �0 (3 � �0 � 5) and �1 are parameters which aredetermined from experiments. The resistance force of the hypothetical sharp ogive-nosed impactorhaving the length Lþ L� (see Figure 2), Dð2Þ� , is calculated on the basis of two-term model usingequations (16), (17) and (21) for determining parameters A0 and A2 .
Solution of the second Newton’s law equation taking into account initial condition vð0Þ ¼ vimp
and conditions of continuity of the resistance force and impactor velocity at h ¼ h0 implies thefollowing expression for the DOP:
Luk and Forrestal (1987) proposed spherical cavity expansion model for the material described witha locked hydrostat and constant shear strength. This model is a 0&2 model that is determined byequation (5) with a1 ¼ 0 and �fr ¼ 0 where
a0 ¼ ð2=3ÞYð1� ln ��Þ ð51Þ
a2 ¼
3Y
Eþ �� 1�
3Y
2E
� �2
þ3�2=3� � ��ð4� ��Þ
2ð1� ��Þ
1þY
2E
� �3
þ�� � 1
" #2=3sh ð52Þ
Y is the yield stress, E is Young’s modulus, sh is the initial density of shield material and �� is thelocked volumetric strain.
DOP can be calculated using equation (13) where h0 ¼ 0 , A0 and A2 are given by equation (4). Inthe case of ogive-nose projectiles that was considered by Luk and Forrestal (1987), equation (4) mustbe replaced by equation (17).
Luk and Forrestal (1989) modified the expression for the DOP in Luk and Forrestal (1987) bytaking into account friction, and showed that a proper choice of friction coefficient improves agree-ment between experimental results and theoretical predictions. However, taking into account acurrent 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 themodel.
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 withparameters which are determined either by the model of Forrestal et al. (1994) or the model of Lukand Forrestal (1987). In this case, equation (13) with h0 ¼ 0 allows us to calculate the DOP. Usingthe dimensionless variables defined by equation (24) or (26) expression for the DOP can be written asfollows:
H
d¼
2N
�ln 1þ
I
N
� �¼
2I
�’ð�Þ, ’ð�Þ ¼
lnð1þ �Þ
�, � ¼
I
Nð53Þ
Function ’ð�Þ for 05 �5 1 can be expanded into converging power series with alternating signs:
’ð�Þ ¼X1
n¼1ð�1Þnþ1
�n�1
nð54Þ
If I55N (�55 1) the following approximate formula for the DOP can be obtained by keepingonly the first term in this series (Chen and Li, 2002):
H
d�
2
�I ¼ 0:637I ð55Þ
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Rosenberg and Dekel (2010) proposed a 0&1 model that is based on the assumption that pro-jectile moves inside a shield with constant deceleration, a. Since D ¼ ma, under this assumptionsolution of the equation of motion, H ¼ 0:5v2imp=a, can be written as
H ¼ 0:5impLeffv2imp=Rsh ð56Þ
where Rsh is the resisting stress of a shield, imp is density of projectile, Leff is effective length ofprojectile and
Rsh ¼ impLeffa, Leff ¼ m=ð�R2impÞ ð57Þ
Equation (56) is recommended for calculating the DOP, where Leff is found from equation (57),while the following formula is suggested for determining Rsh(in GPa) as a function of f 0c(in MPa):
Rsh ¼ 0:22 lnð f 0cÞ � 0:285 ð58Þ
It is assumed that vimp 5 vc, where critical velocity vc is defined as vc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRt=ðK
ð0ÞshapeshÞ
qand K
ð0Þshape is
a projectile nose shape factor (for instance, Kð0Þshape ¼ 0:5 for a hemispherical nose).
Wen (2002) suggested the model in which normal stress on projectile surface that is in contactwith 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. Inthis case, the DOP of ogive-nose projectile is calculated using equation (13) where
A0 ¼ �a0R2ð1� �2Þ, � ¼ d0=d, h0 ¼ 0 ð59Þ
A2 ¼�a2R
2½ð8� 24R2 � 16R3ÞKCRH � 3R4 þ 8R3 þ 6R2 � 1�
24K2CRH
ð60Þ
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 penetrationand perforation of concrete shields by ogive-nose projectiles. Shear dilatancy as well as com-pressibility of material in comminuted region are considered by introducing a dilatant–kinematicrelation. Solution of cavity expansion problem allows determining coefficients of approximation inequation (5) were and applying it directly for penetration modeling.
Sun and Yuan (2012) adapted the model suggested by Yankelevsky and Adin (1980) for the caseof penetration into concrete.
Shields having finite thickness
As a rule, analytical models describing penetration and perforation of finite thickness shields employvarious 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 perforationoccurs after plug formation, and projectile motion prior plug formation is described, as in the case ofsemi-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) usedsimilar approach to reinforced concrete shield. Ben-Dor et al. (2010) suggested a model that is the
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combination of the models proposed by Li and Tong (2003) and Teland and Sjøl (2004) and isapplicable 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, vbl:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� f 0csd
parameter s is determined by equation (21) while � 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 suggestedusing 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 shieldthickness, taking into account a possibility of plug formation, is a very hard problem that requiresmaking serious simplifying assumptions. According to the definition in the ‘‘Introduction’’ sectionsuch models cannot be classified as engineering models, and they are not considered herecomprehensively.
Concluding remarks
In this study, we presented more or less comprehensively all widely used analytical models whichwere suggested for describing high-speed penetration into concrete shields.
The performed analysis showed that most models can be considered as one-stage and two-stagemodels. One-stage models employ the same formula along the whole trajectory of a projectile forcalculating a force exerted on a penetrating projectile by a shield. In the case of two-stage models, aresistance force at the first stage of penetration is a linear function of the instantaneous DOP whileat the second stage of penetration normal stresses at every location on projectile-shield contactsurface are polynomial function of normal velocity component. Conditions of continuity of theresistance force and velocity of a projectile are invoked in the transition point between these twosub-models. A wide variety of models can be devised by using different sub-models at each stage ofpenetration.
The adopted unified approach allowed systemizing all models in a hierarchical classificationsystem (see Figure 3).
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In descriptions of the models, we usually included arguments in favor of the particular modelwhich are based on theoretical considerations, experimental results or ‘‘exact’’ calculations.Nevertheless, it must be emphasized that these arguments or comparisons with other models haveonly illustrative character.
Clearly, we realize the importance and urgency of theoretical and experimental investigations forthe purpose of comparing performance of various analytical models and refining the ranges of theirvalidity. However, we believe that this is a separate and very challenging problem that will be asubject of future research. The present survey is a necessary step in this direction, and we conceivethat it is of interest in its own right.
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Dð1Þ drag force at the first stage of penetrationDð2Þ drag force at the second stage of penetrationDð2Þ� resistance force acting on hypothetical sharp ogive-nosed projectileD� parameter, equation (2)f 0c unconfined compressive strength (Pa)h instantaneous depth of penetration, Figure 1h0 depth at which cratering stage changes to tunneling stageH depth of penetrationI parameter, equations (24) and (26)k h0=dL length of nose of projectile, Figure 1
L� see Figure 2m mass of impactorN equations (24) and (26)r radius of flat bluntness of projectile noseR radius of projectile shanks parameter of model, equations (20) to (22)T thickness of shieldv instantaneous velocity of projectilev0 velocity of projectile at h ¼ h0
vimp impact velocityx coordinate associated with projectile, Figure 1y coordinate associated with projectile, Figure 1
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