About effect of layering on ballistic properties of metal shields against sharp-nosed rigid projectiles

Engineering Fracture Mechanics 102 (2013) 358–361

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Technical Note

About effect of layering on ballistic properties of metal shields against sharp-nosed rigid projectiles G. Ben-Dor, A. Dubinsky, T. Elperin ⇑ Pearlstone Center for Aeronautical Engineering, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

a r t i c l e

i n f o

Article history: Received 4 February 2013 Accepted 7 February 2013

Keywords: Metals Ductile fracture Impact Layered shield Ballistic limit velocity

a b s t r a c t On the basis of semi-empirical model (Rosenberg Z, Dekel E. Revisiting the perforation of ductile plates by sharp-nosed rigid projectiles. Int J Solids Struct 2010; 47(22–23): 3022–3033), we present a rigorous mathematical proof of the conjecture that layering does not improve ballistic properties of ductile shields against high-speed sharp nosed impactors. Within a relatively narrow range of normalized thicknesses of a monolithic shield and of layers (ratios of thicknesses to diameter of impactor), layering does not effect ballistic properties of the shield. In other cases ballistic performance of monolithic shield is better than ballistic performance of any layered shield having the same total thickness. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The term layering implies replacing a monolithic plate by several plates in contact having the same total thickness. Recent survey [1] that also includes a review of theoretical, experimental and numerical investigations of the effect of layering on ballistic properties of metal shields against rigid projectiles, has clearly demonstrated a widespread interest in this subject. Studies [2–5] that used energy approach have a direct bearing on this brief note. It was found that properties of function f that determines the dependence between ballistic limit velocity of a shield and its thickness, Vbl = f(T), play an important role in evaluating efficiency of layering. Study [3] revealed properties of function f which imply either efficiency or inefficiency of layering for metal shields. However the derived in [3] criteria do not exhaust all types of behavior of the function f, and in particular, they cannot be extended to the suggested in [4] approximation. Conducted in [5] analysis of several shields consisting of equal width plates demonstrated that described in [4] model predicts inefficiency of layering. In this note using the same model we derive a rigorous analytical solution of this problem in the general case of unequal width plates. 2. Mathematical model and statement of the problem Consider high speed normal penetration of a rigid sharp-nosed projectile into monolithic shield having thickness Tsum and into layered shield consisting of N metal plates in contact manufactured from the same material and having the same total thickness. The thicknesses of the plates in the layered shield are T(j), j = 1, 2, ..., N. Energy based approach yields the following criterion [3]: if d > 0 then layering is ineffective, if d < 0 then layering is effective, and if d = 0 then ballistic characteristics of monolithic and layered shields are the same. Here

⇑ Corresponding author. Tel.: +972 8 6477078; fax: +972 8 6479394. E-mail address: [email protected] (T. Elperin). 0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.02.004

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359

Nomenclature d G m N Si T Tmin Tmax Tsum T(j) u1, u3 Vbl V bl w1 Y d

u Note

diameter of cylindrical part of impactor function in Eq. (2) mass of impactor number of plates in layered shield variables in Eqs. (8) and (11) thickness of shield lower bound for independent variable in function G(T) upper bound for independent variable in function G(T) total thickness of layers in shield thickness of jth plate variables in Eqs. (10) and (13) ballistic limit velocity of shield reference velocity, Eq. (6) variable in Eq. (10) flow stress of shield material parameter determined in Eq. (1) function, Eq. (3) Bar over variables indicates dimensionless parameters whereby d is a characteristic length and V bl is characteristic velocity.

d ¼ GðT sum Þ 

N X GðT ðjÞ Þ;

T sum ¼

N X T ðjÞ

j¼1

ð1Þ

j¼1

and function G denotes a squared ballistic limit velocity of the shield, V 2bl , as a function of its thickness, T:

V 2bl ¼ GðTÞ:

ð2Þ

Hereafter bar over a variable indicates dimensionless parameter. In the following we use some mathematical properties of function GðTÞ which have been proved in [3] and which are used in this Note. Property 1. If GðTÞ is a increasing, convex downward, twice differentiable non-negative function which is defined in the interval T min 6 T 6 T max ð0 < T min < T max are given) and satisfies the inequality

uðT min Þ P 0; uðTÞ ¼ TG0 ðTÞ  GðTÞ P 0;

ð3Þ

then d > 0 for

T min 6 T ðjÞ 6 T max ;

T min 6 T sum 6 T max ;

j ¼ 1; 2; . . . ; N

ð4Þ

and, consequently, monolithic shield is superior over any layered shield. If we replace the constraint that GðTÞ is a convex downward function by the constraint that GðTÞ is a convex upward function and the ‘‘greater than or equal to’’ sign, P, by the ‘‘less than or equal to’’ sign, 6, in Eq. (3), then d < 0, i.e. any layered shield is superior over monolithic shield [3]. Property 2. If GðTÞ / T and Eq. (4) is satisfied then d = 0 and, consequently, layering does not affect ballistic properties of the shield. Our study is based on the model [4] that can be described using the following function G:

8 > < ð1=3 þ 2TÞT V 2bl ¼ GðTÞ ¼ T > : ð1 þ 0:4 ln TÞT

if

0 < T < 1=3

if

1=3 6 T 6 1

if

T>1

ð5Þ

where

V bl ¼

V bl ; V bl

T¼

T ; d

rffiffiffiffiffiffiffiffiffi pYd ; m

V bl ¼ d

d is diameter of projectile, m is projectile mass, Y is flow stress of the material of the shield. Mathematically, our goal is to determine the sign of d for different combinations of T ðjÞ .

ð6Þ

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G. Ben-Dor et al. / Engineering Fracture Mechanics 102 (2013) 358–361

3. Investigation of the problem 3.1. Application of the results obtained in [3] If the total thickness of the shield and the thicknesses of the layers lie inside the same sub-interval for GðTÞ the results from [3] can be directly applied in the analysis. Let T min ¼ 0; T max ¼ 1=3. Then conditions in Property 1 are satisfied and d > 0, i.e., monolithic shield is superior over any layered shield. If T min ¼ 1=3; T max ¼ 1 then in this situation Property 2 can be used: d = 0, i.e., layering does not affect ballistic properties of the shield. Let T min ¼ 1; T max ¼ 1. Then

G0 ðTÞ ¼ 1:4 þ 0:4 ln T > 0;

GðTÞ ¼ ð1 þ 0:4 ln TÞT > 0; 00

G ðTÞ ¼ 0:4=T > 0;

ð7Þ

uðT min Þ ¼ 0:4 > 0

and according to Property 1 d > 0, i.e., monolithic shield is superior over any layered shield. 3.2. General case when 1=3 6 T sum 6 1 In this case, the thickness of monolithic shield lies in second sub-interval (see Eq. (4)) while the thicknesses of the layers can lie in the first and in the second sub-intervals:

1=3 6 T sum 6 1;

T sum ¼ S1 þ S2 ;

X

S1 ¼

T ðjÞ ;

X

S2 ¼

0
T ðjÞ :

ð8Þ

1=36T ðjÞ 61:0

Then

d ¼ ðS1 þ S2 Þ  ðu1 þ S1 Þ ¼ 2w1 ;

ð9Þ

where

u1 ¼

X

ð1=3 þ 2T ðjÞ ÞT ðjÞ ;

X

w1 ¼

0
ð1=3  T ðjÞ ÞT ðjÞ :

ð10Þ

1=36T ðjÞ 61:0

Since w1 > 0, then d > 0, i.e., monolithic shield is superior over layered shields. 3.3. General case when T sum > 1 In this case, the thickness of monolithic shield lies in third sub-interval (see Eq. (5)) while the thicknesses of the layers can lie in the first, in the second and in the third sub-intervals:

T sum > 1:0;

T sum ¼ S1 þ S2 þ S3 ;

S3 ¼

X

T ðjÞ :

ð11Þ

T ðjÞ >1:0

Then

d ¼ ½1 þ 0:4 lnðS1 þ S2 þ S3 ÞðS1 þ S2 þ S3 Þ  ½u1 þ S2 þ ¼ 2w1 þ 0:4½ðS1 þ S2 þ S3 Þ lnðS1 þ S2 þ S3 Þ 

X

X

ð1 þ 0:4 ln T ðjÞ Þ T ðjÞ 

T ðjÞ >1:0

T ðjÞ ln T ðjÞ :

ð12Þ

T ðjÞ >1:0

Since u1 P 0; S1 P 0; S2 P 0, Eq. (12) implies that

d P 0:4u3 ;

u3 ¼ S3 ln S3 

X

T ðjÞ ln T ðjÞ :

ð13Þ

T ðjÞ >1:0

In order to prove that u3 > 0 let us consider a model with

GðTÞ ¼ T ln T;

T min ¼ 1;

T max ¼ 1:

ð14Þ

Hence

G0 ðTÞ ¼ 1 þ ln T > 0;

G00 ðTÞ ¼ 1=T > 0;

uðT min Þ ¼ 1 > 0

and according to Property 1 u3 > 0, i.e., monolithic shield is superior over layered shields.

ð15Þ

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4. Concluding remarks On the basis of semi-empirical model [4] we presented a rigorous mathematical proof of the conjecture that layering does not improve ballistic properties of ductile shields against high-speed sharp nosed impactor. When the dimensionless thickness (ratio of thickness to diameter of impactor) of monolithic shield and the thicknesses of the layers lie in the range from 1/ 3 to 1, layering does not affect ballistic properties of the shield. In other cases, monolithic shield is superior over any layered shield having the same total thickness. These theoretical predictions are confirmed by comparing with experimental or numerical results in [3]. References [1] Ben-Dor G, Dubinsky A, Elperin T. Investigation and optimization of protective properties of metal multi-layered shields: a review. Int J Protect Struct 2012;3(3):275–91. [2] Ben-Dor G, Dubinsky A, Elperin T. Ballistic properties of multilayered concrete shields. Nucl Engng Des 2009;239(10):1789–94. [3] Ben-Dor G, Dubinsky A, Elperin T. Effect of layering on ballistic properties of metallic shields against sharp-nosed rigid projectiles. Engng Fract Mech 2010;77(14):2791–9. [4] Rosenberg Z, Dekel E. Revisiting the perforation of ductile plates by sharp-nosed rigid projectiles. Int J Solids Struct 2010;47(22–23):3022–33. [5] Rosenberg Z, Dekel E. Terminal ballistics. Springer; 2012.