Imp1
This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA
Overview
Download & View Imp1 as PDF for free.
More details
- Words: 3,216
- Pages: 6
Composite Structures 69 (2005) 89–94 www.elsevier.com/locate/compstruct
Optimization of two-component composite armor against ballistic impact G. Ben-Dor, A. Dubinsky, T. Elperin
*
The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel Available online 19 June 2004
Abstract Using Florence’s model we determined an optimal design of a two-component ceramic-faced lightweight armor against normal ballistic impact. The solution is found in a closed form that allowed us to determine the thicknesses of the plates in the optimal armor as functions of the specified areal density of the armor, parameters determining the material properties of the armor’s components and characteristics of the impactor. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Projectile; Ballistic limit; Ceramics; Armor; Optimization
1. Introduction Florence’s model [1] yields a relatively simple expression for the ballistic limit velocity (BLV), and it is actually the only model suitable for an analytic optimization of two-component ceramic armors. Florence proposed to use his model for determining an armor with the minimum areal density and gave an example of a numerical calculation for a ceramic/aluminum shield [1]. Similar calculations for ceramic/GFRP shield were performed by Hetherington and Rajagopalan [2]. Later, Hetherington [3] considered the problem of determining the structure of two-component armor with a given areal density that provides the maximum BLV. He suggested an approximate expression for the optimum value of the front plate to back plate thicknesses ratio. Wang and Lu [4] investigated a similar problem when the total thickness of the armor is specified rather than the areal density. The problem of designing an armor with the minimum areal density for a given BLV was investigated by Ben-Dor et al. [5]. It was shown that the solution of the optimization problem can be presented in terms of the dimensionless variables whereby all the characteristics of the impactor and the armor are ex-
*
Corresponding author. Tel.: +972-8-647-7078; fax: +972-8-6472813. E-mail address: [email protected] (T. Elperin). 0263-8223/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2004.05.014
pressed as functions of two independent dimensionless parameters. The latter solution allows to find the solution for the optimization problem for an arbitrary twocomponent composite armor in a closed analytical form. In this study we comprehensively investigated a problem considered in [3] for an arbitrary two-component armor and found the solution in a closed analytical form.
2. Formulation of the problem Consider a normal impact by a rigid projectile on a two-layer composite armor consisting of a ceramic front plate and a ductile back plate. In this study we employ the following model: ae2 r2 b2 z½ ðc1 b1 þ c2 b2 Þz þ m ; 0:91m2 2 z ¼ pð R þ 2b1 Þ ;
v2bl ¼
ð1Þ
where vbl is the ballistic limit velocity, m is a projectile’s mass, R is a projectile’s radius, bi are the thicknesses of the plates, ri are the ultimate tensile strengths, e2 is the breaking strain, ci are the densities of the materials of the plates, subscripts 1 and 2 denote a ceramic plate and a back plate, respectively. For a ¼ 1 Eq. (1) recovers the model suggested by Florence [1] as it was re-worked in [2]. More recently in [5] this model was generalized slightly by introducing a coefficient a that can be
90
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94
determined using the available experimental data in order to increase the accuracy of the predictions. The objective of the present study is to find the thicknesses of the plates, b1 ; b2 , that provide the maximum ballistic limit velocity vbl for a given areal density of the armor A ¼ c 1 b1 þ c 2 b2 :
ð2Þ
Introduce (for details see [5]) the dimensionless vari , A using the following formulas: ables b1 , b2 , c1 , c2 , w m bi ¼ bi R; ci ¼ 3 ci ; i ¼ 1; 2; pR rffiffiffiffiffiffiffiffiffiffiffiffiffi ae2 r2 m ; A ¼ 2 A: ð3Þ vbl ¼ w pR 0:91c2 Using Eq. (3), the Eqs. (1) and (2) can be rewritten as follows: h i 2 ¼ c2 w b2z c1 b1 þ c2 b2 z þ 1 ; ð4Þ A ¼ c1 b1 þ c2 b2 ;
opt
Aopt 1 2 1 ¼ u3 ðA; c1 Þ; opt opt ¼ c1 Þ u A ðA; A1 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2A ¼ Aw A; ; u0 ðA; c1 Þ ¼ u4 ðA; c1 Þ: c1
nopt ¼
opt w
A2
¼
ð5Þ
wx ðA; b; xÞ ¼
2 ¼ AwðA; b; xÞ; w
where
ð7Þ 2
wðA; b; xÞ ¼ ð1 xÞðbx þ 1Þ½Aðbx þ 1Þ þ 1; 2A b¼ ; c1
ð8Þ
A1 ¼ c1 b1 ; A2 ¼ c2 b2 :
Thus the problem is reduced to finding x, ð10Þ
0 6 x 6 1;
which provides the minimum w considered as a function of x. The solution of this problem depends only on two parameters, A and c1 . If ¼ u0 ðA; c1 Þ;
ow ¼ ðbx þ 1Þf ðA; b; xÞ; ox
A u ðA; c1 Þ ¼ u1 ðA; c1 Þ; bopt 1 ¼ c1 0 opt
ð12Þ
A1 ¼ Au0 ðA; c1 Þ;
ð13Þ
opt A2
ð14Þ
c2 bopt 2
c0 ¼ Að4b 1Þ þ 2b 1;
¼ A½1 u0 ðA; c1 Þ ¼ u2 ðA; c1 Þ;
c2 ¼ Ab ð4b 11Þ;
c1 ¼ b½Að8b 7Þ 3
c3 ¼ 5Ab3 :
ð17Þ
ð18Þ ð19Þ
Hereafter, the parameters A and b are not listed as arguments of the corresponding functions. Consider a behavior of the function wðA; b; xÞ at the interval determined by Eq. (10). To this end let us calculate the values of the functions w (x) and f ðxÞ at the end points of this interval wð0Þ ¼ A þ 1 > 0;
ð20Þ
wð1Þ ¼ 0;
ð21Þ
ð11Þ
provides the minimum w (hereafter a superscript opt denotes the optimal parameters), then the principal dimensionless parameters associated with the optimal solution (the thickness of the ceramic plate, bopt 1 , the opt opt areal densities of the plates, A1 and A2 , and their opt ) are also some ratio, nopt , the ballistic limit velocity, w functions of A and c1
¼
Let us calculate the derivative
2
ð9Þ
x
3. Investigation of the function w(A; b; x)
f ðA; b; xÞ ¼ c3 x3 þ c2 x2 þ c1 x þ c0 ;
where
opt
ð16Þ
In order to elucidate the analysis based on the dimensionless variables we will refer (where it is possible) to the special kind of the armor that we will call a ‘‘basic armor’’ (BA). As a BA we selected the ceramic/ GFRP armor, and used the experimental data on perforation of the armors with different thicknesses of the plates by a 0.50 inch projectile reported in [3]. For BA a transition from the dimensionless to the dimensional parameters, i.e., for the areal density A (kg/m2 ), the widths of the plates and the ballistic limit velocity vbl (m/s) is performed as follows: A ¼ 370A, bi ¼ 6:35 bi , vbl ¼ 133 w (c1 ¼ 0:060 corresponds to c1 ¼ 3499 kg/m3 ).
where z 2 b1 Þ : ð6Þ z ¼ 2 ¼ ð1 þ 2 pR Substituting c2 b2 from Eq. (5) and z from Eq. (6) into Eq. (4) we obtain
A1 A1 ; x¼ ¼ A A1 þ A2
ð15Þ
f ð0Þ ¼ c0 ¼
Aþ1 gðA; c1 Þ; c1
f ð1Þ ¼ ½Ab3 þ 3Ab2 þ ð3A þ 1Þb þ A þ 1 < 0;
ð22Þ ð23Þ
where gðA; c1 Þ ¼
4Að2A þ 1Þ c1 : Aþ1
ð24Þ
The curve gðA; c1 Þ ¼ 0 divides the domain A P 0, c1 P 0 into two sub-domains determined by the conditions g < 0 and g > 0, correspondently (see Fig. 1). Consider now these two cases in more details taking into account that the third degree polynomial f ðxÞ can have 1 or 3 real roots.
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94 0.3
91
0.18 A > A
0.16
A = A
0.14 0 .0 1 4 8
0.12
f( A , β,0 )< 0
0.2
g( A ,
γ1 ) =
0
A < A
0.10
w
γ1
*
A = 0 .0 2 *
*
0 .0 1
0.08 0.06 0.04
0.1
f( A , β,0 )> 0
γ 1 = 0 .0 6
0.02 0.00 0.0
0 .2
0.4
(a)
0 .6
0.8
1.0
x = A1 / A
0.0 0.00
0.02
0.04
0.06
0.08
4.5
A
A = 0 .2 0
4.0
Fig. 1. The sign of AðA; b; 0Þ depending on the parameters A and c1 .
γ = 0 .0 6 1
3.5 3.0
Assume that
*
2.5
w
gðA; c1 Þ < 0:
A > A
ð25Þ
0 .1 6
2.0 1.5
Since f ðxÞ ! þ1 when x ! 1 and f ð0Þ < 0, the equation f ðxÞ ¼ 0 has a root at the interval 1 < x < 0 and, consequently, 0 or 2 roots at the interval 0 < x < 1. Let us assume now that
0 .1 2 0 .1 0
1.0
0 .0 7
0.5 0.0 0.0
gðA; c1 Þ > 0:
ð26Þ
Taking into account Eq. (23) one can conclude that the equation f ðxÞ ¼ 0 has 1 or 3 roots at the interval 0 < x < 1. Since wx ð0Þ > 0, then there exists an arbitrary small 1 such that wð1Þ > wð0Þ > wð1Þ and, consequently, a maximum wðxÞ is attained not at the end points of the interval [0,1] but at some interior point where wx ðxÞ ¼ f ðxÞ ¼ 0. Numerical simulation shows that equation f ðxÞ ¼ 0 does not have roots at the interval [0,1] if Eq. (25) is valid, and it has one root in the opposite case. Thus, the is attained at the point x ¼ 0 (the first case) maximum w and at the point where f ðxÞ ¼ 0 (the second case). The ðxÞ is shown in Fig. 2(a) and behavior of the function w (b), whereas Fig. 2(a) illustrates the transition from the
case 1 (A < A ) to the case 2 (A > A ), where A ¼ A is the solution of the equation gðA; c1 Þ ¼ 0. Let us consider now the case determined by Eq. (25). Formally, wðxÞ attains its maximum value for b1 ¼ 0 when the employed physical model is not valid. Let us show that the case determined by Eq. (25) is of no practical significance. The inequality given by Eq. (25) can be solved for positive A A < Hðc1 Þ;
ð27Þ
0.2
(b)
0.4
0.6
0.8
1.0
x = A1 / A
ðxÞ depending on A. Fig. 2. (a,b) Different versions of the behavior w
where Hðc1 Þ ¼ ð1=16Þ½c1 4 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c21 þ 24c1 þ 16:
ð28Þ
On the other hand, decreasing wðxÞ implies the inequality 6w ð0Þ ¼ w
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AWð0Þ ¼ AðA þ 1Þ:
ð29Þ
Combining these inequalities we obtain < hðc1 Þ; w where hðc1 Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hðc1 Þ½Hðc1 Þ þ 1:
ð30Þ
ð31Þ
Let us now estimate c1 . Substituting the mass of the cylindrical impactor in term of its density cimp , length L and radius of the base R, m ¼ pR2 L;
ð32Þ
into the second equation in Eq. (3) for c1 we obtain c1 ¼
R c1 ; L cimp
ð33Þ
92
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94
i.e., indeed c1 is much less than 1. Then Taylors series expansion for small c1 yields
7
γ1
6
2
Hðc1 Þ ¼ 0:25c1 ð0:25c1 Þ þ Oðc31 Þ;
ð34Þ
pffiffiffiffi 5=2 hðc1 Þ ¼ 0:5 c1 þ Oðc1 Þ:
ð35Þ
0 .0 4 0 .0 5 0 .0 6
5
Clearly, inequality given by Eq. (30) taking into account that does not Eq. (35) is valid only for very small w correspond to a ballistic impact conditions. Thus, e.g., for ‘‘basic armor’’ c1 ¼ 0:06 and Eq. (30) implies that vbl < 17 m/s. Note that equation A 0:25c1 ¼ 0 is a good approximation of the curve shown in Fig. 1.
0 .0 7 0 .0 8
4
0 .0 9 opt
b1
0 .1 0
3 2 1 0 0.05
0.10
0.15
0.20
0.25
0.30
0.35
A
4. Optimal armor
Fig. 4. Optimal thickness of ceramic plate vs. given areal density of the armor.
In a case when Eq. (26) is valid, the equation f ðxÞ ¼ 0 has one real root that is the location of the maximum of a function wðxÞ. This root can be determined using Cardano’s formulae [6] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi 3 3 opt x ¼ D þ 0:5q ð1=3Þc2 =c3 ; ð36Þ D 0:5q
1.8 1.6
γ
1
0 .1 0
1.4
0 .0 9
1.2
where
0 .0 8 0 .0 7
3
2
D ¼ ðp=3Þ þ ðq=2Þ ;
ð37Þ
ξ
opt
1.0
0 .0 6 0 .0 5
0.8
p¼
3c1 c3 c22 ; 3c23
q¼
2c32 c1 c2 c0 þ 27c33 3c23 c3
ð38Þ
0 .0 4
0.6 0.4
The solution of the optimization problem in a graphical form is shown in Figs. 3–6. Fig. 4. shows that the dependence bopt 1 vs. A is close to a linear function for every c1 . Even Eq. (36) represents the solution of the considered problem in closed form. The family of the curves plotted in Fig. 4 can be more simply approximated with the average accuracy of 3% in the range 0:04 6 c1 6 0:1, 0:05 6 A 6 0:35 as follows:
0.2 0.05
0.10
0.15
0.20
0.25
0.30
0.35
A
Fig. 5. Optimal ratio of the areal density of the back plate, A2 , to the areal density of the ceramic plate, A1 , vs. given areal density of the armor.
20
0.80
γ
0.75
1
0 .0 4
0.70
15
0.65
γ
x
0 .0 6 1
0 .0 7
0 .0 4
0.60 opt
0 .0 5
0 .0 5
0.55
wopt
0 .0 8
10
0 .0 9
0 .0 6
0 .1 0
0 .0 7
0.50
0 .0 8
5
0 .0 9
0.45
0 .1 0
0.40 0.35 0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 0.05
0.10
0.15
0.20
0.25
0.30
0.35
A
Fig. 3. Values x corresponding to the optimal armor vs. given areal density of the armor.
Fig. 6. Maximum ballistic limit velocity vs. given areal density of the armor.
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94
c1 Þ bopt 1 ¼ u1 ðA;
93
5.0
pffiffiffiffi ¼ ð588:5c1 407:2 c1 þ 78:2ÞA 0:25:
opt
4.5
ð39Þ
b1
4.0
After substitution u0 ðA; c1 Þ ¼
3.5
ð40Þ b
Eqs. (13)–(16) can also be rewritten using the function u1 . It was noted in literature [2,7] that variation of the BLV is quite small in the neighborhood of the maximum, i.e., the thicknesses of the plates may be changed in the vicinity of the optimal values for a given areal density of the armor without considerable loss in the BLV. The typical results that support this conclusion are showed in Figs. 7–9 for c1 ¼ 0:06. Let us consider the on some parameter, g (for dependence of the BLV and w
opt
ξ
4.0 γ = 0 .0 6 1
3.0 2.5
ξ
inf
2.0
ε=
sup
5%
3%
1%
1.5 1.0 0.5 0.05
0.10
0.15
0.20
0.25
0.30
0.9
0.8
0.7
x
inf sup
0.6
ε=
0.5
2.5 2.0
1
1.5
ε=
5%
3%
1%
1.0 0.5 0.05
0.10
0.15
0.20
0.25
0.30
0.35
A
Fig. 9. The boundaries of the segment [binf ; bsup ] for the thickness of , inside the segment the ceramic b1 that implies BLV, w opt ], vs. given areal density, A. [ð1 eÞwopt ; w
3%
5. Concluding remarks A closed-form solution for the two-component armor optimization problem is found among the designs with a given areal density when the ballistic limit velocity (BLV) of the impactor is a target function. In addition to the exact solution, the simplified approximate solution is proposed as well. The behavior of the BLV near the optimum is investigated. It is shown that the thicknesses of the plates can be changed in a quite broad range in the neighborhood of the optimal design of the armor without decline in its defense properties.
5%
References
0.4
0.3 0.05
1%
b
inf 1 sup
0.35
Fig. 7. The boundaries of the segment [ninf ; nsup ] for the parameter inside the segment [ð1 eÞwopt ; w opt ],vs. n ¼ A2 =A1 that implies BLV, w given areal density, A.
x
= 0 .0 6
instance, the thickness of the ceramic plate), that varies within the limits specified in the formulation of the problem. The lower, ginf , and the upper, gsup , values of in the interval the interval for g that imply variation of w opt for several given values of e are shown ½ð1 eÞ wopt ; w in Figs. 7–9 for g ¼ n ¼ A2 =A1 ¼ A2 =A1 (Fig. 7), g ¼ x ¼ A1 =A ¼ A1 =A (Fig. 8) and g ¼ b1 (Fig. 9). The results showed in Figs. 7–9 support the above property of the optimal solution. Thus, there exists a broad range of possible designs for the optimal lightweight two-component armor among the designs with almost identical ballistic performance.
4.5
ξ
1
3.0
c1 u1 ðA; c1 Þ: A
3.5
γ
γ 1 = 0.06
0.10
0.15
x
0.20
0.25
0.30
opt
0.35
Fig. 8. The boundaries of the segment [xinf ; xsup ] for the parameter inside the segment [ð1 eÞwopt ; w opt ], x ¼ A2 =A1 that implies BLV, w vs. given areal density, A.
[1] Florence AL. Interaction of projectiles and composite armor. Part 2. AMMRC-CR- 69-15, Stanford Res Inst, Menlo Park, California. 1969. [2] Hetherington JG, Rajagopalan BP. An investigation into the energy absorbed during ballistic perforation of composite armors. Int J Impact Eng 1991;11(1):33–40. [3] Hetherington JG. Optimization of two component composite armours. Int J Impact Eng 1993;12(3):409–14.
94
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94
[4] Wang B, Lu G. On the optimisation of two-component plates against ballistic impact. J Mater Process Technol 1996;57(1–2):141–5. [5] Ben-Dor G, Dubinsky A, Elperin T, Frage N. Optimization of two component ceramic armor for a given impact velocity. Theor Appl Fract Mech 2000;33(3):185–90.
[6] Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New-York: McGraw-Hill Book Company; 1968. [7] Lee M, Yoo YH. Analysis of ceramic/metal armour systems. Int J Impact Eng 2001;25(9):819–29.
Optimization of two-component composite armor against ballistic impact G. Ben-Dor, A. Dubinsky, T. Elperin
*
The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel Available online 19 June 2004
Abstract Using Florence’s model we determined an optimal design of a two-component ceramic-faced lightweight armor against normal ballistic impact. The solution is found in a closed form that allowed us to determine the thicknesses of the plates in the optimal armor as functions of the specified areal density of the armor, parameters determining the material properties of the armor’s components and characteristics of the impactor. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Projectile; Ballistic limit; Ceramics; Armor; Optimization
1. Introduction Florence’s model [1] yields a relatively simple expression for the ballistic limit velocity (BLV), and it is actually the only model suitable for an analytic optimization of two-component ceramic armors. Florence proposed to use his model for determining an armor with the minimum areal density and gave an example of a numerical calculation for a ceramic/aluminum shield [1]. Similar calculations for ceramic/GFRP shield were performed by Hetherington and Rajagopalan [2]. Later, Hetherington [3] considered the problem of determining the structure of two-component armor with a given areal density that provides the maximum BLV. He suggested an approximate expression for the optimum value of the front plate to back plate thicknesses ratio. Wang and Lu [4] investigated a similar problem when the total thickness of the armor is specified rather than the areal density. The problem of designing an armor with the minimum areal density for a given BLV was investigated by Ben-Dor et al. [5]. It was shown that the solution of the optimization problem can be presented in terms of the dimensionless variables whereby all the characteristics of the impactor and the armor are ex-
*
Corresponding author. Tel.: +972-8-647-7078; fax: +972-8-6472813. E-mail address: [email protected] (T. Elperin). 0263-8223/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2004.05.014
pressed as functions of two independent dimensionless parameters. The latter solution allows to find the solution for the optimization problem for an arbitrary twocomponent composite armor in a closed analytical form. In this study we comprehensively investigated a problem considered in [3] for an arbitrary two-component armor and found the solution in a closed analytical form.
2. Formulation of the problem Consider a normal impact by a rigid projectile on a two-layer composite armor consisting of a ceramic front plate and a ductile back plate. In this study we employ the following model: ae2 r2 b2 z½ ðc1 b1 þ c2 b2 Þz þ m ; 0:91m2 2 z ¼ pð R þ 2b1 Þ ;
v2bl ¼
ð1Þ
where vbl is the ballistic limit velocity, m is a projectile’s mass, R is a projectile’s radius, bi are the thicknesses of the plates, ri are the ultimate tensile strengths, e2 is the breaking strain, ci are the densities of the materials of the plates, subscripts 1 and 2 denote a ceramic plate and a back plate, respectively. For a ¼ 1 Eq. (1) recovers the model suggested by Florence [1] as it was re-worked in [2]. More recently in [5] this model was generalized slightly by introducing a coefficient a that can be
90
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94
determined using the available experimental data in order to increase the accuracy of the predictions. The objective of the present study is to find the thicknesses of the plates, b1 ; b2 , that provide the maximum ballistic limit velocity vbl for a given areal density of the armor A ¼ c 1 b1 þ c 2 b2 :
ð2Þ
Introduce (for details see [5]) the dimensionless vari , A using the following formulas: ables b1 , b2 , c1 , c2 , w m bi ¼ bi R; ci ¼ 3 ci ; i ¼ 1; 2; pR rffiffiffiffiffiffiffiffiffiffiffiffiffi ae2 r2 m ; A ¼ 2 A: ð3Þ vbl ¼ w pR 0:91c2 Using Eq. (3), the Eqs. (1) and (2) can be rewritten as follows: h i 2 ¼ c2 w b2z c1 b1 þ c2 b2 z þ 1 ; ð4Þ A ¼ c1 b1 þ c2 b2 ;
opt
Aopt 1 2 1 ¼ u3 ðA; c1 Þ; opt opt ¼ c1 Þ u A ðA; A1 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2A ¼ Aw A; ; u0 ðA; c1 Þ ¼ u4 ðA; c1 Þ: c1
nopt ¼
opt w
A2
¼
ð5Þ
wx ðA; b; xÞ ¼
2 ¼ AwðA; b; xÞ; w
where
ð7Þ 2
wðA; b; xÞ ¼ ð1 xÞðbx þ 1Þ½Aðbx þ 1Þ þ 1; 2A b¼ ; c1
ð8Þ
A1 ¼ c1 b1 ; A2 ¼ c2 b2 :
Thus the problem is reduced to finding x, ð10Þ
0 6 x 6 1;
which provides the minimum w considered as a function of x. The solution of this problem depends only on two parameters, A and c1 . If ¼ u0 ðA; c1 Þ;
ow ¼ ðbx þ 1Þf ðA; b; xÞ; ox
A u ðA; c1 Þ ¼ u1 ðA; c1 Þ; bopt 1 ¼ c1 0 opt
ð12Þ
A1 ¼ Au0 ðA; c1 Þ;
ð13Þ
opt A2
ð14Þ
c2 bopt 2
c0 ¼ Að4b 1Þ þ 2b 1;
¼ A½1 u0 ðA; c1 Þ ¼ u2 ðA; c1 Þ;
c2 ¼ Ab ð4b 11Þ;
c1 ¼ b½Að8b 7Þ 3
c3 ¼ 5Ab3 :
ð17Þ
ð18Þ ð19Þ
Hereafter, the parameters A and b are not listed as arguments of the corresponding functions. Consider a behavior of the function wðA; b; xÞ at the interval determined by Eq. (10). To this end let us calculate the values of the functions w (x) and f ðxÞ at the end points of this interval wð0Þ ¼ A þ 1 > 0;
ð20Þ
wð1Þ ¼ 0;
ð21Þ
ð11Þ
provides the minimum w (hereafter a superscript opt denotes the optimal parameters), then the principal dimensionless parameters associated with the optimal solution (the thickness of the ceramic plate, bopt 1 , the opt opt areal densities of the plates, A1 and A2 , and their opt ) are also some ratio, nopt , the ballistic limit velocity, w functions of A and c1
¼
Let us calculate the derivative
2
ð9Þ
x
3. Investigation of the function w(A; b; x)
f ðA; b; xÞ ¼ c3 x3 þ c2 x2 þ c1 x þ c0 ;
where
opt
ð16Þ
In order to elucidate the analysis based on the dimensionless variables we will refer (where it is possible) to the special kind of the armor that we will call a ‘‘basic armor’’ (BA). As a BA we selected the ceramic/ GFRP armor, and used the experimental data on perforation of the armors with different thicknesses of the plates by a 0.50 inch projectile reported in [3]. For BA a transition from the dimensionless to the dimensional parameters, i.e., for the areal density A (kg/m2 ), the widths of the plates and the ballistic limit velocity vbl (m/s) is performed as follows: A ¼ 370A, bi ¼ 6:35 bi , vbl ¼ 133 w (c1 ¼ 0:060 corresponds to c1 ¼ 3499 kg/m3 ).
where z 2 b1 Þ : ð6Þ z ¼ 2 ¼ ð1 þ 2 pR Substituting c2 b2 from Eq. (5) and z from Eq. (6) into Eq. (4) we obtain
A1 A1 ; x¼ ¼ A A1 þ A2
ð15Þ
f ð0Þ ¼ c0 ¼
Aþ1 gðA; c1 Þ; c1
f ð1Þ ¼ ½Ab3 þ 3Ab2 þ ð3A þ 1Þb þ A þ 1 < 0;
ð22Þ ð23Þ
where gðA; c1 Þ ¼
4Að2A þ 1Þ c1 : Aþ1
ð24Þ
The curve gðA; c1 Þ ¼ 0 divides the domain A P 0, c1 P 0 into two sub-domains determined by the conditions g < 0 and g > 0, correspondently (see Fig. 1). Consider now these two cases in more details taking into account that the third degree polynomial f ðxÞ can have 1 or 3 real roots.
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94 0.3
91
0.18 A > A
0.16
A = A
0.14 0 .0 1 4 8
0.12
f( A , β,0 )< 0
0.2
g( A ,
γ1 ) =
0
A < A
0.10
w
γ1
*
A = 0 .0 2 *
*
0 .0 1
0.08 0.06 0.04
0.1
f( A , β,0 )> 0
γ 1 = 0 .0 6
0.02 0.00 0.0
0 .2
0.4
(a)
0 .6
0.8
1.0
x = A1 / A
0.0 0.00
0.02
0.04
0.06
0.08
4.5
A
A = 0 .2 0
4.0
Fig. 1. The sign of AðA; b; 0Þ depending on the parameters A and c1 .
γ = 0 .0 6 1
3.5 3.0
Assume that
*
2.5
w
gðA; c1 Þ < 0:
A > A
ð25Þ
0 .1 6
2.0 1.5
Since f ðxÞ ! þ1 when x ! 1 and f ð0Þ < 0, the equation f ðxÞ ¼ 0 has a root at the interval 1 < x < 0 and, consequently, 0 or 2 roots at the interval 0 < x < 1. Let us assume now that
0 .1 2 0 .1 0
1.0
0 .0 7
0.5 0.0 0.0
gðA; c1 Þ > 0:
ð26Þ
Taking into account Eq. (23) one can conclude that the equation f ðxÞ ¼ 0 has 1 or 3 roots at the interval 0 < x < 1. Since wx ð0Þ > 0, then there exists an arbitrary small 1 such that wð1Þ > wð0Þ > wð1Þ and, consequently, a maximum wðxÞ is attained not at the end points of the interval [0,1] but at some interior point where wx ðxÞ ¼ f ðxÞ ¼ 0. Numerical simulation shows that equation f ðxÞ ¼ 0 does not have roots at the interval [0,1] if Eq. (25) is valid, and it has one root in the opposite case. Thus, the is attained at the point x ¼ 0 (the first case) maximum w and at the point where f ðxÞ ¼ 0 (the second case). The ðxÞ is shown in Fig. 2(a) and behavior of the function w (b), whereas Fig. 2(a) illustrates the transition from the
case 1 (A < A ) to the case 2 (A > A ), where A ¼ A is the solution of the equation gðA; c1 Þ ¼ 0. Let us consider now the case determined by Eq. (25). Formally, wðxÞ attains its maximum value for b1 ¼ 0 when the employed physical model is not valid. Let us show that the case determined by Eq. (25) is of no practical significance. The inequality given by Eq. (25) can be solved for positive A A < Hðc1 Þ;
ð27Þ
0.2
(b)
0.4
0.6
0.8
1.0
x = A1 / A
ðxÞ depending on A. Fig. 2. (a,b) Different versions of the behavior w
where Hðc1 Þ ¼ ð1=16Þ½c1 4 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c21 þ 24c1 þ 16:
ð28Þ
On the other hand, decreasing wðxÞ implies the inequality 6w ð0Þ ¼ w
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AWð0Þ ¼ AðA þ 1Þ:
ð29Þ
Combining these inequalities we obtain < hðc1 Þ; w where hðc1 Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hðc1 Þ½Hðc1 Þ þ 1:
ð30Þ
ð31Þ
Let us now estimate c1 . Substituting the mass of the cylindrical impactor in term of its density cimp , length L and radius of the base R, m ¼ pR2 L;
ð32Þ
into the second equation in Eq. (3) for c1 we obtain c1 ¼
R c1 ; L cimp
ð33Þ
92
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94
i.e., indeed c1 is much less than 1. Then Taylors series expansion for small c1 yields
7
γ1
6
2
Hðc1 Þ ¼ 0:25c1 ð0:25c1 Þ þ Oðc31 Þ;
ð34Þ
pffiffiffiffi 5=2 hðc1 Þ ¼ 0:5 c1 þ Oðc1 Þ:
ð35Þ
0 .0 4 0 .0 5 0 .0 6
5
Clearly, inequality given by Eq. (30) taking into account that does not Eq. (35) is valid only for very small w correspond to a ballistic impact conditions. Thus, e.g., for ‘‘basic armor’’ c1 ¼ 0:06 and Eq. (30) implies that vbl < 17 m/s. Note that equation A 0:25c1 ¼ 0 is a good approximation of the curve shown in Fig. 1.
0 .0 7 0 .0 8
4
0 .0 9 opt
b1
0 .1 0
3 2 1 0 0.05
0.10
0.15
0.20
0.25
0.30
0.35
A
4. Optimal armor
Fig. 4. Optimal thickness of ceramic plate vs. given areal density of the armor.
In a case when Eq. (26) is valid, the equation f ðxÞ ¼ 0 has one real root that is the location of the maximum of a function wðxÞ. This root can be determined using Cardano’s formulae [6] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi 3 3 opt x ¼ D þ 0:5q ð1=3Þc2 =c3 ; ð36Þ D 0:5q
1.8 1.6
γ
1
0 .1 0
1.4
0 .0 9
1.2
where
0 .0 8 0 .0 7
3
2
D ¼ ðp=3Þ þ ðq=2Þ ;
ð37Þ
ξ
opt
1.0
0 .0 6 0 .0 5
0.8
p¼
3c1 c3 c22 ; 3c23
q¼
2c32 c1 c2 c0 þ 27c33 3c23 c3
ð38Þ
0 .0 4
0.6 0.4
The solution of the optimization problem in a graphical form is shown in Figs. 3–6. Fig. 4. shows that the dependence bopt 1 vs. A is close to a linear function for every c1 . Even Eq. (36) represents the solution of the considered problem in closed form. The family of the curves plotted in Fig. 4 can be more simply approximated with the average accuracy of 3% in the range 0:04 6 c1 6 0:1, 0:05 6 A 6 0:35 as follows:
0.2 0.05
0.10
0.15
0.20
0.25
0.30
0.35
A
Fig. 5. Optimal ratio of the areal density of the back plate, A2 , to the areal density of the ceramic plate, A1 , vs. given areal density of the armor.
20
0.80
γ
0.75
1
0 .0 4
0.70
15
0.65
γ
x
0 .0 6 1
0 .0 7
0 .0 4
0.60 opt
0 .0 5
0 .0 5
0.55
wopt
0 .0 8
10
0 .0 9
0 .0 6
0 .1 0
0 .0 7
0.50
0 .0 8
5
0 .0 9
0.45
0 .1 0
0.40 0.35 0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 0.05
0.10
0.15
0.20
0.25
0.30
0.35
A
Fig. 3. Values x corresponding to the optimal armor vs. given areal density of the armor.
Fig. 6. Maximum ballistic limit velocity vs. given areal density of the armor.
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94
c1 Þ bopt 1 ¼ u1 ðA;
93
5.0
pffiffiffiffi ¼ ð588:5c1 407:2 c1 þ 78:2ÞA 0:25:
opt
4.5
ð39Þ
b1
4.0
After substitution u0 ðA; c1 Þ ¼
3.5
ð40Þ b
Eqs. (13)–(16) can also be rewritten using the function u1 . It was noted in literature [2,7] that variation of the BLV is quite small in the neighborhood of the maximum, i.e., the thicknesses of the plates may be changed in the vicinity of the optimal values for a given areal density of the armor without considerable loss in the BLV. The typical results that support this conclusion are showed in Figs. 7–9 for c1 ¼ 0:06. Let us consider the on some parameter, g (for dependence of the BLV and w
opt
ξ
4.0 γ = 0 .0 6 1
3.0 2.5
ξ
inf
2.0
ε=
sup
5%
3%
1%
1.5 1.0 0.5 0.05
0.10
0.15
0.20
0.25
0.30
0.9
0.8
0.7
x
inf sup
0.6
ε=
0.5
2.5 2.0
1
1.5
ε=
5%
3%
1%
1.0 0.5 0.05
0.10
0.15
0.20
0.25
0.30
0.35
A
Fig. 9. The boundaries of the segment [binf ; bsup ] for the thickness of , inside the segment the ceramic b1 that implies BLV, w opt ], vs. given areal density, A. [ð1 eÞwopt ; w
3%
5. Concluding remarks A closed-form solution for the two-component armor optimization problem is found among the designs with a given areal density when the ballistic limit velocity (BLV) of the impactor is a target function. In addition to the exact solution, the simplified approximate solution is proposed as well. The behavior of the BLV near the optimum is investigated. It is shown that the thicknesses of the plates can be changed in a quite broad range in the neighborhood of the optimal design of the armor without decline in its defense properties.
5%
References
0.4
0.3 0.05
1%
b
inf 1 sup
0.35
Fig. 7. The boundaries of the segment [ninf ; nsup ] for the parameter inside the segment [ð1 eÞwopt ; w opt ],vs. n ¼ A2 =A1 that implies BLV, w given areal density, A.
x
= 0 .0 6
instance, the thickness of the ceramic plate), that varies within the limits specified in the formulation of the problem. The lower, ginf , and the upper, gsup , values of in the interval the interval for g that imply variation of w opt for several given values of e are shown ½ð1 eÞ wopt ; w in Figs. 7–9 for g ¼ n ¼ A2 =A1 ¼ A2 =A1 (Fig. 7), g ¼ x ¼ A1 =A ¼ A1 =A (Fig. 8) and g ¼ b1 (Fig. 9). The results showed in Figs. 7–9 support the above property of the optimal solution. Thus, there exists a broad range of possible designs for the optimal lightweight two-component armor among the designs with almost identical ballistic performance.
4.5
ξ
1
3.0
c1 u1 ðA; c1 Þ: A
3.5
γ
γ 1 = 0.06
0.10
0.15
x
0.20
0.25
0.30
opt
0.35
Fig. 8. The boundaries of the segment [xinf ; xsup ] for the parameter inside the segment [ð1 eÞwopt ; w opt ], x ¼ A2 =A1 that implies BLV, w vs. given areal density, A.
[1] Florence AL. Interaction of projectiles and composite armor. Part 2. AMMRC-CR- 69-15, Stanford Res Inst, Menlo Park, California. 1969. [2] Hetherington JG, Rajagopalan BP. An investigation into the energy absorbed during ballistic perforation of composite armors. Int J Impact Eng 1991;11(1):33–40. [3] Hetherington JG. Optimization of two component composite armours. Int J Impact Eng 1993;12(3):409–14.
94
G. Ben-Dor et al. / Composite Structures 69 (2005) 89–94
[4] Wang B, Lu G. On the optimisation of two-component plates against ballistic impact. J Mater Process Technol 1996;57(1–2):141–5. [5] Ben-Dor G, Dubinsky A, Elperin T, Frage N. Optimization of two component ceramic armor for a given impact velocity. Theor Appl Fract Mech 2000;33(3):185–90.
[6] Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New-York: McGraw-Hill Book Company; 1968. [7] Lee M, Yoo YH. Analysis of ceramic/metal armour systems. Int J Impact Eng 2001;25(9):819–29.
