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Theoretical and Applied Fracture Mechanics 33 (2000) 185±190

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Optimization of two component ceramic armor for a given impact velocity G. Ben-Dor a, A. Dubinsky a, T. Elperin a,*, N. Frage b a

b

Department of Mechanical Engineering, The Pearlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel Department of Materials Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

Abstract Using FlorenceÕs model a problem of two-component ceramic-faced lightweight armors design against ballistic impact is solved. Approximate analytical formulas are derived for areal density and thicknesses of the plates of the optimal armor as functions of parameters determining the properties of the materials of the armor components, crosssection and mass of an impactor, and of the expected impact velocity. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction Ballistic perforation of multi-layered plates has been a subject of intensive research during recent years since non monolithic con®gurations are considered feasible for designing shields or elements of the shields. Simpli®ed analytical models were derived and used for the analysis and optimization of the shields consisting of the layers manufactured from di€erent materials, e.g., ductile multi-layered shields [1±7], aluminum/Lexan combinations [8], ceramic-faced armors [9±19]. Since the model [10] provides a relatively simple expression for the ballistic limit velocity, it is widely used for the investigation of the ballistic properties of two-component ceramic armors [11±13,15,16]. The model [10] was used to determine the armor with the minimum areal density with an example for the numerical calculation of a ceramic/aluminum

shield [10]. Similar calculations for ceramic/GFRP (glass®bre reinforced plastic) shield can be found also in [11]. The problem of designing an armor with the maximum ballistic limit velocity was investigated analytically in the case when the areal density is given in [14] and in the case when the total thickness of the armor is given in [17]. In this study the problem of designing an armor with the minimum areal density which was formulated in [10], is completely investigated for an arbitrary two-component armor, and the approximate analytical solution is found. 2. Model description 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, the following model is employed: ae2 r2 h2 z‰…q1 h1 ‡ q2 h2 †z ‡ mŠ ; 0:91m2 z ˆ p…R ‡ 2h1 †2 ; v2 ˆ

*

Corresponding author. Fax: +972-7-6472813990. E-mail address: [email protected] (T. Elperin).

0167-8442/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 0 0 ) 0 0 0 1 3 - 6

…1†

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G. Ben-Dor et al. / Theoretical and Applied Fracture Mechanics 33 (2000) 185±190

where v is the ballistic limit velocity, m the projectile's mass, R the projectile's radius, h1 , h2 are the plate's thicknesses, r the ultimate tensile strength, e is the breaking strain, q is the density, subscripts 1 and 2 refer to a ceramic plate and a back plate, respectively. For a ˆ 1 Eq. (1) describes the model suggested in [10] as it is re-worked in [11]. We generalized slightly this model introducing a coecient a which can be determined using the available experimental data in order to increase the accuracy of the predictions. Expressions for the areal density A of the armor reads A ˆ q 1 h 1 ‡ q2 h 2 :

…2†

The objective of the present study is to ®nd the thicknesses of the plates h1 , h2 which provide the minimum areal density of the armor A for a given ballistic limit velocity v . Introduce the following dimensionless variables hi pR3 qi ; i ˆ 1; 2; hi ˆ ; qi ˆ R s m 0:91q2 pR2 A  ˆ v : w ; A ˆ ae2 r2 m

…3†

Then Eqs. (1) and (2) can be rewritten, respectively, as follows: h  i  2 ˆ q2 h2z q1 h1 ‡ q2 h2 z ‡ 1 ; …4† w A ˆ q1 h1 ‡ q2 h2 ; where  2 z z ˆ 2 ˆ 1 ‡ 2h1 : pR

…5†

…6†

Eq. (4) is a quadratic equation with respect to q2 h2 Solving this equation and substituting the obtained expression for q2 h2 into Eq. (5) yields r  2  2 q1 h1z ‡ 1 ‡ 4w   q1 h1z ÿ 1 ‡    ˆ : A h1 ; q1 ; w 2z …7† Thus the problem is reduced to ®nding a positive  The value h2 can h1 that provides the minimum A. be then found from Eq. (5).

It is important to emphasize that the dimensionless areal density A is a function of one variable h1 and depends on only two parameters, q1  If and w.  q1 ; w† hopt 1 ˆ u1 …

…8†

 then the dimensionless provides the minimum A, minimum areal density Aopt and the optimal ratio of the areal density of the second plate to the areal density of the ®rst plate are also functions of q1  and w: h i  q1 ; w  ˆ uA …  q1 ; w†; q1 ; w†; …9† Aopt ˆ A u1 … Aopt q hopt q2 hopt 2 ˆ 2 2opt ˆ ÿ1 opt q 1 h1 q1 h1 q1 hopt 1 ˆ

 uA … q1 ; w†  ÿ 1 ˆ u0 … q1 ; w†  q1 ; w† q1 u1 …

…10†

and opt ÿ q1 hopt q2 hopt 2 ˆ A 1  ÿ q1 u1 …  ˆ uA … q1 ; w† q1 ; w†  q1 ; w†: ˆ u2 …

…11†

The above property of the model, i.e., the existence of only two independent dimensionless parameters, allows us to investigate the problem completely and for a general case, namely, for arbitrary combination of materials of the plates. The results for a given materials can be determined using the solution obtained in dimensionless form:  q1 ; w†; hopt 1 ˆ Ru1 … m  u … q ; w†; hopt 2 ˆ pR2 q2 2 1 m  q1 ; w†; Aopt ˆ 2 uA … pR q2

…12†

where superscript opt indicates the corresponding  are determined by optimal parameters, q and w Eq. (3). 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 which will be referred to as a ``basic armor'' (BA). To this end, select the ceramic/GFRP armor, and use the experimental data in [11] for perforation of the

G. Ben-Dor et al. / Theoretical and Applied Fracture Mechanics 33 (2000) 185±190

armors with di€erent thicknesses of the plates by a 0.50 inch projectile. For BA a change from the dimensionless to the dimensional parameters can 2 be performed for areal density A…kg=m †, widths of the plates hi (mm) and the ballistic limit velocity  hi ˆ 6:35hi ; v ˆ 133w  v (m/s) follows: A ˆ 370A; … q1 ˆ 0:060 corresponds to q1 ˆ 3499 kg=m3 ). The approach suggested in [11±13] is based on the assumption that the momentum transferred from projectile to the armor is independent on the impact velocity, while the model in [10] is used for calculating the ballistic limit velocity. This approach implies the following correlation between the impact velocity vimp , the residual velocity vres and the ballistic limit velocity v vimp vres ÿ ˆ 1: v v

3. Optimal armor Now we will analyze the dependence on A on h1 .  ˆ …q ÿ 1†=2 where Eqs. (6) and (7) imply that A…0† p  2 ‡ 1 > 0: …14† q ˆ 4w  is independent on h1 When h1 ! 1 The value A…0†  h1 † attains the asymptotic value function A…   A ˆ q1 h1 . The derivative dA A0 ˆ  dh1

  p  1z ‡ 1 q1z z ÿ 4 ÿ 16w  2 q h 1 2 q r   ˆ 1 ‡ p ‡ 2 2 z z p 2 2 z q1 h1z ‡ 1 ‡ 4w …15†

…13†

If the experimental data on vimp and vres are available, Eq. (13) can be used for ®nding the best value of the parameter a in Eq. (1) and for determining the corresponding ballistic limit velocities. Certainly, it is assumed that a is constant for a given combination of the materials of the plates. For the data presented in [11], it is found that a ˆ 0:90. Comparison of Eq. (13) with the experimental data is showed in Fig. 1 where solid circles correspond to the experimental vimp and vres [11] together with the calculating v using Eq. (1).

187

is too involved for the complete analytical analysis. Consider its behavior for h1 ˆ 0 2 q1 ÿ 4 ÿ 16w p 2 1 ‡ 4w 2 ÿ4q ‡ … q1 ‡ 4†q ‡ q1 : ˆ q

2A0 …0† ˆ q1 ‡ 4 ‡

…16†

The positive solution of the equation q1 ‡ 4†q ‡ q1 ˆ 0 reads ÿ4q2 ‡ …  q q1 ‡ 16 : …17† q0 ˆ 0:125 q1 ‡ 4 ‡ q21 ‡ 24 Thus A0 …0† 6 0

if q P q0

…18†

if q 6 q0 :

…19†

and A0 …0† P 0

Fig. 1. Comparison of theoretical predictions with experimental data from [11] combined with the model given by Eq. (1).

The condition given by Eq. (19) is satis®ed only for very small ballistic limit velocities (e.g., for BA for v < 17 m=s). Therefore, consider the case when a condition expressed by Eq. (18) is satis®ed.  h1 † is The typical behavior of function A…  showed in Fig. 2 for q1 ˆ 0:06 and di€erent w where solid circles are plotted using experiments  h1 † provides the [11]. The minimum of function A… solution of the problem. However, the variation is quite small in the neighborhood of the minimum (see also [11]). This means that the thickness of the

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Fig. 2. Areal density of armor vs. thickness of ceramic plate; solid circles ± experimental data from [11] combined with model given by Eq. (1).

ceramic plate may be changed in the vicinity of the optimal value without considerable loss in areal density. The solution of the optimization problem in a graphical form is showed in Figs. 3 and 4. Useful conclusions about the properties of optimal armor can be made using Fig. 5 where the ratio of the

Fig. 4. Optimal thickness of ceramic plate vs. ballistic limit velocity.

areal densities of the plates in the optimal armor A2 =A1 is plotted. Inspection of this ®gure shows  the ratio A2 =A1 is that even for a relatively small w, close to a constant value 0.30 (for BA it corre P 4 the ratio sponds to v ˆ 400 m=s). Thus, for w of the thicknesses of the plates in the optimal armor is inversely proportional to the ratio of their densities: hopt q1 2 : opt  0:3 q2 h1

Fig. 3. Areal density of optimal armor vs. ballistic limit velocity.

…20†

Fig. 5. Optimal ratio of the areal densities of the plates vs. ballistic limit velocity.

G. Ben-Dor et al. / Theoretical and Applied Fracture Mechanics 33 (2000) 185±190

The families of curves plotted in Figs. 3 and 5 can be approximated with the average accuracy  6 10 as of 3% in the range 0:04 6 q1 6 0:1, 1 6 w follows:  ˆ …0:04 ‡ 1:12  0:425 ; q1 ; w† q1 †w Aopt ˆ uA …

…21†

q2 hopt 2  ˆ 0:29 ‡ …0:1 ‡ q1 †w  ÿ1:47 : ˆ u0 … q1 ; w† q1 hopt 1

…22†

and hopt can be expressed Using Eqs. (9)±(12) hopt 1 2 through functions uA and u0 :  ˆ q1 ; w† hopt 1 ˆ u1 … ˆ

 u … q ; w† h A 1 i  ‡1 q1 ; w† q1 u0 …

 1:895 …0:04 ‡ 1:12 q 1 †w  ;  1:47 ‡ 0:1 q1 q1 ‡ 1:29w

 0 …  1 u … q ; w†u q1 ; w† i:  ˆ Ah 1 u … q ; w† hopt 2 ˆ q2 2 1  ‡1 q ; w† q u … 2

0

…23†

…24†

1

Taking into account Eq. (12) the expressions for the solution given by Eqs. (21)±(24) provides characteristics of the optimal armor in term of the parameters determining the properties of the materials of the armor components, its cross-section area, mass of the impactor and the expected impact velocity.

4. Concluding remarks It is showed that the solution of the optimization problem for a two-layer armor can be presented in terms of dimensionless variables whereby all the characteristics of the impactor and the armor are expressed as a function of two independent dimensionless parameters. The latter allows to present the solution of the optimization problem for an arbitrary two-component composite armor in analytical form which is convenient for practical applications. The validity of the result obtained using the above simpli®ed model must be investigated experimentally for di€erent materials applied for armor component. Clearly, experimental proce-

189

dure must be adjusted for testing the optimality of the obtained solution. Investigations of Functionally Graded Composites (FGC) have received considerable attention during recent years (see, e.g., [20]). The results of our research in this ®eld [21, 22] show that analysis of two-component materials can be useful for optimization of FGC. Acknowledgements We are grateful to Prof. A.L. Florence (International Stanford Research Institute) and Prof. R. Zaera (Carlos III University, Madrid) for providing us with their research reports and reprints of their papers. This study was partially supported by the Israel Ministry of Science (Grant No. 8450-1-98). References [1] V.N. Aptukov, A.V. Murzakaev, A.V. Fonarev, Applied Theory of Penetration [Prikladnaja Teorija Pronikanija], Moscow, Nauka, 1992 (in Russian). [2] G. Ben-Dor, A. Dubinsky, T. Elperin, Optimal 3D impactors penetrating into layered targets, Theoret. and Appl. Fract. Mech. 27 (3) (1997) 161±166. [3] G. Ben-Dor, A. Dubinsky, T. Elperin, Some ballistic properties of non-homogeneous shields, Composites A 30 (6) (1999) 733±736. [4] G. Ben-Dor, A. Dubinsky, T. Elperin, On the order of plates providing the maximum ballistic limit velocity of a layered armor, Int. J. Impact Eng. 22 (1999) 741±755. [5] G. Ben-Dor, A. Dubinsky, T. Elperin, E€ect of air gap and of order of plates on ballistic resistance of twolayered armor, Theoret. Appl. Fract. Mech. 31 (1999) 233±241. [6] G. Ben-Dor, A. Dubinsky, T. Elperin, Optimization of layered shield with a given areal density, Int. J. Fract. 91 (1998) L9±L14. [7] G. Ben-Dor, A. Dubinsky, T. Elperin, The optimum arrangement of the plates in a multi-layered shields, Int. J. Solids and Struct. 37 (2000) 687±696. [8] J. Radin, W. Goldsmith, Normal projectile penetration and perforation of layered targets, Int. J. Impact Eng. 7 (1988) 229±259. [9] I.S. Chocron Benloulo, V. Sanchez-Galvez, A new analytical model to simulate impact onto ceramic/composite armors, Int. J. Impact Eng. 21 (6) (1998) 461±471. [10] A.L. Florence, Interaction of projectiles and composite armor. Part 2. Stanford Research Institute, Menlo Park, AMMRC-CR-69-15, August 1969.

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