Optimization of two-component composite armor against ballistic impact

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1 Composite Structures 69 (005) Optimization of two-component composite armor against ballistic impact G. Ben-Dor,. Dubinsky, T. Elperin * The Pearlstone Center for eronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion Universityof the Negev, P.O. Box 653, 8405 Beer-Sheva, Israel vailable online 9 June 004 bstract Using Florence s model we determined an imal 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 imal 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 004 Elsevier Ltd. ll rights reserved. Keywords: Projectile; Ballistic limit; Ceramics; rmor; Optimization. Introduction * Corresponding author. Tel.: ; fax: address: elperin@menix.bgu.ac.il (T. Elperin). Florence s model [] yields a relatively simple expression for the ballistic limit velocity (BLV), and it is actually the only model suitable for an analytic imization 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 []. Similar calculations for ceramic/gfrp shield were performed by Hetherington and Rajagopalan []. 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 imum 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 imization problem can be presented in terms of the dimensionless variables by all the characteristics of the impactor and the armor are expressed as functions of two independent dimensionless parameters. The latter solution allows to find the solution for the imization 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.. 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: v bl ¼ ae r b z½ðc b þ c b Þz þ mš ; 0:9m z ¼ pðr þb Þ ; ðþ v bl is the ballistic limit velocity, m is a projectile s mass, R is a projectile s radius, b i are the thicknesses of the plates, r i are the ultimate tensile strengths, e is the breaking strain, c i are the densities of the materials of the plates, subscripts and denote a ceramic plate and a back plate, respectively. For a ¼ Eq. () recovers the model suggested by Florence [] as it was re-worked in []. More recently in [5] this model was generalized slightly by introducing a coefficient a that can be /$ - see front matter Ó 004 Elsevier Ltd. ll rights reserved. doi:/j.compstruct

2 90 G. Ben-Dor et al. / Composite Structures 69 (005) 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, b ; b, that provide the maximum ballistic limit velocity v bl for a given areal density of the armor ¼ c b þ c b : ðþ Introduce (for details see [5]) the dimensionless variables b, b, c, c, w, using the following formulas: b i ¼ b i R; c i ¼ m pr c i; i ¼ ; ; rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ae r v bl ¼ w ; ¼ m 0:9c pr : ð3þ Using Eq. (3), the Eqs. () and () can be rewritten as follows: h w ¼ c b z c b i þ c b z þ ; ð4þ ¼ c b þ c b ; ð5þ z ¼ z pr ¼ðþ b Þ : ð6þ Substituting c b from Eq. (5) and z from Eq. (6) into Eq. (4) we obtain w ¼ wð; b; xþ; ð7þ wð; b; xþ ¼ð xþðbx þ Þ½ðbx þ Þ þ Š; ð8þ x ¼ ¼ þ ; b ¼ c ; ¼ c b ; ¼ c b : ð9þ Thus the problem is reduced to finding x, 0 6 x 6 ; ð0þ which provides the minimum w considered as a function of x. The solution of this problem depends only on two parameters, and c.if x ¼ u 0 ð; c Þ; ðþ provides the minimum w (hereafter a superscript denotes the imal parameters), then the principal dimensionless parameters associated with the imal solution (the thickness of the ceramic plate, b, the areal densities of the plates, and, and their ratio, n, the ballistic limit velocity, w ) are also some functions of and c b ¼ c u 0 ð; c Þ¼u ð; c Þ; ðþ ¼ u 0 ð; c Þ; ð3þ ¼ c b ¼ ½ u 0 ð; c ÞŠ ¼ u ð; c Þ; ð4þ n ¼ s w ¼ u 0 ð; c Þ ¼ u 3ð; c Þ; ¼ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w ; ; u c 0 ð; c Þ ¼ u 4 ð; c Þ: ð5þ ð6þ In order to elucidate the analysis based on the dimensionless variables we will refer ( it is possible) to the special kind of the armor that we will call a basic armor (B). s a B 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 inch projectile reported in [3]. For B a transition from the dimensionless to the dimensional parameters, i.e., for the areal density (kg/m ), the widths of the plates and the ballistic limit velocity v bl (m/s) is performed as follows: ¼ 370, b i ¼ 6:35 b i, v bl ¼ 33w (c ¼ 0:060 corresponds to c ¼ 3499 kg/m 3 ). 3. Investigation of the function w(; b; x) Let us calculate the derivative w x ð; b; xþ ¼ ow ¼ðbx þ Þf ð; b; xþ; ox ð7þ f ð; b; xþ ¼c 3 x 3 þ c x þ c x þ c 0 ; ð8þ c 0 ¼ ð4b Þþb ; c ¼ b½ð8b 7Þ 3Š ð9þ c ¼ b ð4b Þ; c 3 ¼ 5b 3 : Hereafter, the parameters and b are not listed as arguments of the corresponding functions. Consider a behavior of the function wð; b; xþ at the interval determined by Eq. (0). 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 þ > 0; wðþ ¼0; ð0þ ðþ f ð0þ ¼c 0 ¼ þ gð; c c Þ; ðþ f ðþ ¼ ½b 3 þ 3b þð3þþbþþš < 0; ð3þ 4ð þ Þ gð; c Þ¼ c : ð4þ þ The curve gð; c Þ¼0 divides the domain P 0, c P 0 into two sub-domains determined by the conditions g < 0 and g > 0, correspondently (see Fig. ). Consider now these two cases in more details taking into account that the third degree polynomial f ðxþ can have or 3 real roots.

3 G. Ben-Dor et al. / Composite Structures 69 (005) =0.0 > * = * 0. f(,β,0 )< 0 g(, γ ) = < * w γ 0. f(,β,0 )> γ = 0.0 ssume that gð; c Þ < 0: ð5þ Since f ðxþ!þ when x! and f ð0þ < 0, the equation f ðxþ ¼0 has a root at the interval < x < 0 and, consequently, 0 or roots at the interval 0 < x <. Let us assume now that gð; c Þ > 0: ð6þ Taking into account Eq. (3) one can conclude that the equation f ðxþ ¼0 has or 3 roots at the interval 0 < x <. Since w x ð0þ > 0, then there exists an arbitrary small such that wðþ > wð0þ > wðþ and, consequently, a maximum wðxþ is attained not at the end points of the interval [0,] but at some interior point w x ðxþ ¼f ðxþ ¼0. Numerical simulation shows that equation f ðxþ ¼0 does not have roots at the interval [0,] if Eq. (5) is valid, and it has one root in the opposite case. Thus, the maximum w is attained at the point x ¼ 0 (the first case) and at the point f ðxþ ¼0 (the second case). The behavior of the function wðxþ is shown in Fig. (a) and (b), as Fig. (a) illustrates the transition from the case ( < ) to the case ( > ), ¼ is the solution of the equation gð; c Þ¼0. Let us consider now the case determined by Eq. (5). Formally, wðxþ attains its maximum value for b ¼ 0 when the employed physical model is not valid. Let us show that the case determined by Eq. (5) is of no practical significance. The inequality given by Eq. (5) can be solved for positive < Hðc Þ; Fig.. The sign of ð; b; 0Þ depending on the parameters and c. ð7þ (a) w (b) Hðc Þ¼ð=6Þ½c 4 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c þ 4c þ 6Š: ð8þ On the other hand, decreasing wðxþ implies the inequality w 6 wð0þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi Wð0Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð þ Þ: ð9þ Combining these inequalities we obtain w < hðc Þ; ð30þ p hðc Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hðc Þ½Hðc ÞþŠ: ð3þ Let us now estimate c. Substituting the mass of the cylindrical impactor in term of its density c imp, length L and radius of the base R, m ¼ pr L; ð3þ into the second equation in Eq. (3) for c we obtain c ¼ R L c c imp ; γ = > * x = / = x = / Fig.. (a,b) Different versions of the behavior wðxþ depending on. ð33þ

4 9 G. Ben-Dor et al. / Composite Structures 69 (005) i.e., indeed c is much less than. Then Taylors series expansion for small c yields Hðc Þ¼0:5c ð0:5c Þ þ Oðc 3 Þ; ð34þ 5= hðc Þ¼0:5 c þ Oðc Þ: ð35þ Clearly, inequality given by Eq. (30) taking into account Eq. (35) is valid only for very small w that does not correspond to a ballistic impact conditions. Thus, e.g., for basic armor c ¼ 0:06 and Eq. (30) implies that v bl < 7 m/s. Note that equation 0:5c ¼ 0 is a good approximation of the curve shown in Fig.. 4. Optimal armor In a case when Eq. (6) is valid, the equation f ðxþ ¼0 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 x 3 3 ¼ D 0:5q D þ 0:5q ð=3þc =c 3 ; ð36þ D ¼ðp=3Þ 3 þðq=þ ; ð37þ p ¼ 3c c 3 c ; q ¼ c3 c c þ c 0 ð38þ 3c 3 7c 3 3 3c 3 c 3 The solution of the imization problem in a graphical form is shown in Figs Fig. 4. shows that the dependence b vs. is close to a linear function for every c. 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 c 6 0:, 0: :35 as follows: b 3 γ Fig. 4. Optimal thickness of ceramic plate vs. given areal density of the armor. ξ γ Fig. 5. Optimal ratio of the areal density of the back plate,, to the areal density of the ceramic plate,, vs. given areal density of the armor γ x γ 0.0 w Fig. 3. Values x corresponding to the imal armor vs. given areal density of the armor Fig. 6. Maximum ballistic limit velocity vs. given areal density of the armor.

5 G. Ben-Dor et al. / Composite Structures 69 (005) b ¼ u ð; c Þ ¼ð588:5c 407: c þ 78:Þ 0:5: ð39þ fter substitution γ = b u 0 ð; c Þ¼ c u ð; c Þ: ð40þ Eqs. (3) (6) can also be rewritten using the function u. It was noted in literature [,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 imal 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 c ¼ 0:06. Let us consider the dependence of the BLV and w on some parameter, g (for b inf sup.0 b.5.0 ε = % Fig. 9. The boundaries of the segment [b inf ; b sup ] for the thickness of the ceramic b that implies BLV, w, inside the segment [ð eþw ; w ], vs. given areal density,. 3% 5% ξ ξ inf sup γ = ε = % 3% 5% ξ instance, the thickness of the ceramic plate), that varies within the limits specified in the formulation of the problem. The lower, g inf, and the upper, g sup, values of the interval for g that imply variation of w in the interval ½ð eþw ; w Š for several given values of e are shown in Figs. 7 9 for g ¼ n ¼ = ¼ = (Fig. 7), g ¼ x ¼ = ¼ = (Fig. 8) and g ¼ b (Fig. 9). The results showed in Figs. 7 9 support the above property of the imal solution. Thus, there exists a broad range of possible designs for the imal lightweight two-component armor among the designs with almost identical ballistic performance. Fig. 7. The boundaries of the segment [n inf ; n sup ] for the parameter n ¼ = that implies BLV, w inside the segment [ð eþw ; w ],vs. given areal density,. x inf x sup γ = ε = % Fig. 8. The boundaries of the segment [x inf ; x sup ] for the parameter x ¼ = that implies BLV, w inside the segment [ð eþw ; w ], vs. given areal density,. 3% 5% x 5. Concluding remarks closed-form solution for the two-component armor imization 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 imum is investigated. It is shown that the thicknesses of the plates can be changed in a quite broad range in the neighborhood of the imal design of the armor without decline in its defense properties. References [] Florence L. Interaction of projectiles and composite armor. Part. MMRC-CR- 69-5, Stanford Res Inst, Menlo Park, California [] Hetherington JG, Rajagopalan BP. n investigation into the energy absorbed during ballistic perforation of composite armors. Int J Impact Eng 99;(): [3] Hetherington JG. Optimization of two component composite armours. Int J Impact Eng 993;(3):409 4.

6 94 G. Ben-Dor et al. / Composite Structures 69 (005) [4] Wang B, Lu G. On the imisation of two-component plates against ballistic impact. J Mater Process Technol 996;57( ):4 5. [5] Ben-Dor G, Dubinsky, Elperin T, Frage N. Optimization of two component ceramic armor for a given impact velocity. Theor ppl Fract Mech 000;33(3): [6] Korn G, Korn TM. Mathematical Handbook for Scientists and Engineers. New-York: McGraw-Hill Book Company; 968. [7] Lee M, Yoo YH. nalysis of ceramic/metal armour systems. Int J Impact Eng 00;5(9):89 9.

Shape Optimization of Impactors Against a Finite Width Shield Using a Modified Method of Local Variations #

Mechanics Based Design of Structures and Machines, 35: 113 125, 27 Shape Optimization of Impactors Against a Finite Width Shield Using a Modified Method of Local Variations # G. Ben-Dor, A. Dubinsky, and