Ballistic Impact: Recent Advances in Analytical Modeling of Plate Penetration Dynamics A Review

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1 Anatoly Dubinsky's Home page: Gabi Ben-Dor Anatoly Dubinsky Tov Elperin The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel Ballistic Impact: Recent Advances in Analytical Modeling of Plate Penetration Dynamics A Review This review covers studies dealing with simplified analytical models for ballistic penetration of an impactor into different solid media, namely, metals, soil, concrete, and composites at high speeds, but not at hypervelocities. The overview covers mainly papers that were published in the last decade, but not analyzed in previous reviews on impact dynamics. Both mathematical models and their engineering applications are considered. The review covers 280 citations. DOI: / Introduction Models classified as empirical, semi- or quasi-empirical, engineering, simplified, analytical, semi-analytical and approximate in the mechanics of high-speed nonhypervelocity penetration constitute the subject of this review. These models comprise algebraic relations and/or ordinary differential equations, and calculations based on such models do not require large computer resources. Analytical models are useful from the viewpoint of their direct applicability, since the qualitative laws determined by the use of these models can be considered as a basis for further theoretical and experimental investigations. Most review publications on ballistic impact Table also survey analytical models. The current review surveys, to a large extent, developments subsequent to the publications of Corbett et al. 13, Abrate 14, Kasano 17, and Goldsmith 18, which appeared between 1996 and Earlier studies are also included for the following reasons: i they have not been covered in other reviews; ii they have been analyzed in a different context; or iii for purposes of completeness and continuity of the analysis. 2 Some Common Approaches to Approximate Modeling of Penetration and Perforation 2.1 Localized Interaction Approach. Many engineering models for penetration modeling belong to the category of the so-called localized interaction models LIM 21,22, in which the integral effect of the interaction between the host medium and a moving projectile is described as the superposition of the independent local interactions of the projectile s surface elements with the medium. Each local interaction is determined both by the local geometric and kinematic parameters of the surface element primarily, by the angle between the velocity vector and the local normal vector to the projectile surface and by some global parameters that take into account the integral characteristics of the medium e.g., hardness, density, etc.. The following description is typical of LIM: df = n a ;u,v n 0 + a ;u,v 0 ds if 0 u 1 n a ;1,v n 0 ds if u =1 0 if u 0 1 where see Fig. 1 Transmitted by Assoc. Editor N. Jones. 0 = v 0 + u n 0 / 1 u 2, u = v 0 n 0 = cos 2 df is the force acting at the surface element ds of the projectile that is in contact with the host medium, n 0 and 0 are the inner normal and tangent unit vectors at a given location on the projectile s surface, respectively, v 0 is a unit vector of the velocity at the projectile v, is the angle between the vector n 0 and the vector v 0, the functions n and determine the model of the projectile-medium interaction, and a is a vector with components a 0,a 1,... that characterize, mainly, the properties of the host medium. In the most frequently used types of LIM, it is assumed that or =0 = k n where k is the coefficient of friction between the impactor and the shield. Ben-Dor et al. 23,24 proposed a unified model describing the normal penetration of a rigid striker into a non-thin shield. This formalism for a shield with a finite thickness SFT is illustrated in Figs. 2 and 3, which also show the notations used. The coordinate h, the current depth of the penetration, is defined as the distance between the nose of the impactor and the front surface of the shield. The cylindrical coordinates x,, pertain to the impactor, and the equation = x,, where is some function, determines the shape of the impactor. The part of the lateral surface of the impactor between the cross sections x= 1 and x= 2 0 h L+b interacts with the SFT, where see Fig. 3 1 h = 0 if 0 h b h b if b h b + L 5 2 h = h if 0 h L L if h L b is the thickness of the shield and L is the impactor s nose length. The total force F is determined by integrating the local force over the impactor-shield contact surface determined for some h as 0 2, 1 h x 2 h. The drag force D=F v 0 is a function of a,h,v, i.e., it does not depend on time in the explicit form. Let us consider a sharp impactor and, for simplicity, assume that it has no flat bluntness. Then Eqs. 1, 2, and 5 imply 3 4 Applied Mechanics Reviews Copyright 2005 by ASME NOVEMBER 2005, Vol. 58 / 355

2 Table 1 Basic review publications on ballistic impact Reference Year Material Type of publication Kennedy Concrete Survey Backman and Goldsmith Metal, soil Survey Jonas and Zukas Metal, soil Survey Zukas Metal Book Brown Metal, soil, concrete Survey Anderson and Bodner Metal, soil Survey Heuzé Geological materials Report Recht Metal Book Zukas and Walters Metal, soil, concrete Book Abrate Composites Survey Abrate Composites Survey Dancygier and Yankelevsky Concrete Article Corbett et al Metal, soil, concrete Survey Abrate Composites Book Teland Concrete Report Børvik et al Metal Report Kasano Fiber composites Survey Goldsmith Metal, soil Survey Cheeseman and Bogetti Fiber composites Survey Phoenix and Porwal Fiber composites Article where 2 h D a ;h,v = 1 h dx a ;u x,,v u 0 x, d 0 a ;u,v = u n a ;u,v + 1 u 2 a ;u,v u x, = x u 0 x,, u 0 x, = 2 x The equation of motion of an impactor with mass m may be given as mv dv + D a ;h,v =0 dh 9 where the velocity of the impactor v is considered as a function of h. This model is also valid for a semi-infinite shield SIS. Inthe latter case, the impactor-shield contact surface is determined as 0 2, 0 x 2 h. Let v= v imp,h be the solution of Eq. 9 with the initial condition v 0 =v imp. The ballistic limit velocity BLV, v bl, is usually considered as a characteristic of the perforation process for a SFT, and it is defined as the initial velocity of the impactor required to emerge from the shield with zero velocity. Thus, v bl is determined from the equation v bl,l+b =0. In the case of a SIS, the depth of penetration DOP, H, for the known impact velocity v imp is determined from the equation v imp,h =0. Thus, the general characteristics of the penetration, the BLV and the DOP, can be obtained by solving a first-order ordinary differential equation. Equation 6 shows that the assumption that the lateral surface of the impactor during its motion is completely immersed in the host medium, i.e., =0, 2 = L 10 implies that D does not depend on h. Thus, an equation of motion of the impactor does not contain h in explicit form. It is thus a separable equation, which simplifies the solution. However, Li et al. 25, using experimental data, confirmed the need for taking into account the incomplete immersion of the impactor in the shield at the initial stage of penetration, where the length of the impactor and a penetration depth are of the same order. Ben-Dor et al. 26 extended LIMs to the case of nonhomogeneous shields in which the properties of the material vary, depending on the depth of the shield, Fig. 2, i.e., a =a. In particular, a step function dependence of a for layered shields, including shields with air gaps 27, was analyzed. The LIMs can be readily modified for impactors with plane bluntness 21,28,29. It is very attractive to apply the localized interaction approach for investigating problems of impact dynamics, since it allows one to describe relatively easily the projectile-medium interaction taking into account the impactor s shape and to simulate the motion of an impactor in a shield. Indeed, many of the known models can be described in the framework of the localized interaction approach. To the best of our knowledge, the first model describing bulletbarrier interaction during the penetration of a projectile through a plate was suggested by Nishiwaki 30. This model can be described by means of Eqs. 1, 2, and 4 with n = a 0 + a 2 u 2 v 2 11 where a 0 is the static contact pressure and a 2 = is the density of the material of the shield. Using his experimental results for conical bullets and aluminum shields, Nishiwaki 30 drew the conclusion that a 0 is proportional to the thickness of the perforated plate. He developed a relationship between the impact velocity and the residual velocity of a cone-shaped impactor perforating a SFT, taking into consideration the change in the impactorplate contact surface during penetration. It is interesting to note that the above-described model is based on the same assumptions as the Newton s LIM 31 widely used in aerodynamics. To determine the effect of the hypersonic gas flow over the surface of the projectile, the projectile-medium interaction is modeled as the outcome of nonelastic impacts of the host medium s particles on the projectile surface. On the basis of their experimental investigations, Vitman and Stepanov 32 proposed the model described by Eqs. 3 and 11, where a 2 = and a 0 is the dynamic hardness of the material of the host medium. Golubev and Medvedkin 33 modified the model 32 to take into account the effect of viscous resistance at the initial stage of penetration. Various semiempirical models that differ from one another mainly in the choice of functions n and were collected and analyzed by Recht 8 ; this list may be supplemented by adding the model suggested by Landgzov and Sarkisyan 34. A model that is based solely on the dependence between the drag force D and the velocity of the impactor v, and does not consider the influence of the impactor s shape on its resistance or does not take it into account through some, usually empirical, 356 / Vol. 58, NOVEMBER 2005 Transactions of the ASME

3 Fig. 2 Normal impact: coordinates and notations Here see Fig. 4, the impactor s surface that interacts with the shield is divided into subareas using the planes x=x i, where 1 h = x 1 x 2 x i x i+1 x N = 2 h, and A x i Fig. 1 coefficients, may be classified as a degenerate LIM DLIM. Classic DLIMs postulate a polynomial dependence of D v 2. Heimdahl and Schulz 35 studied the motion of an impactor for an arbitrary function D v. A number of workers have proposed power-law dependences for different media Ben-Dor et al. 23 showed how the power-law version of a DLIM implies a LIM. The localized interaction approach allows one to extend models developed for conical impactors to impactors with more complex shapes Let us consider the normal penetration of a conical impactor into a medium and assume that functions n a ;,v and a ;,v in the relationship df = n a ;,v n 0 + a ;,v 0 ds, 0 /2 12 are known, where is the half angle of the apex of the cone. Equation 12 is similar to Eq. 1 and determines the effect of the host medium on the surface of the conical impactor. Then, functions determining the LIM that is suitable for sharp impactors with other shapes are determined as a ;u,v = a ;sin 1 u,v, = n, 13 The method proposed by Recht 8 can be considered as a discrete version of the localized interaction theory LIT for sharp bodies of revolution. Let us assume, for simplicity, that Eq. 3 is valid. Then Eqs. 6 8 imply 2 h D a ;h,v =2 1 h n a ;u,v x dx 14 where u=u x = x / x 2 +1=sin and = x is the angle between the tangent to the generator and the axis of the impactor. The integral in Eq. 14 can be approximated by a sum D a ;h,v 2 i i i Definition of LIM x i x i+1 n a ;u,v x dx x i+1 d 2 n a ; i dx dx,v x i n a ; i,v A x i 15 is the cross-sectional area of the impactor by a plane x=x i, A x i =A x i+1 A x i, i = x i. In both of the above-described methods, the force at the location of the interaction between the projectile and the host medium is assumed to be equal to the force at the surface of the tangent cone at this location, when the projectile velocity and the host medium are the same in both cases. Different versions of such an approach are known in aerodynamics as methods of tangent cones MTC 21, Cavity Expansion Approximations. The spherical cavity expansion approximation SCEA in a quasi-static version is widely used for constructing impactor-shield interaction models for SIS. In these models, expansion of a spherically symmetrical cavity from a zero initial radius at a constant velocity is considered by means of some continuum mechanics model of the material. Let the solution of this problem be represented in the form p = a ; * 16 where p is the stress at the boundary of the cavity, * is the radius of the hole, and is some function. Then, it is assumed that the normal stress at the surface of the impactor moving in the same medium is given by the following formula: n = a ;uv 17 i.e., it is assumed that the normal effect of the host medium on the impactor s surface at some location with instantaneous normal velocity v n =v cos =uv is equal to the stress at the boundary of the cavity that expands with constant velocity v n. Clearly, a quasistatic SCEA implies a LIM with n = a ;uv in Eq. 1. Another widely used approach is known as the cylindrical cavity expansion approximation CCEA model, method, etc.. Sometimes other names are used, e.g., the method of plane sections 43,44 and the disks model 45. The CCEA has been applied to modeling penetration into SISs and perforation of SFTs. In this approach, normal penetration of a slender body of revolution is usually considered, and it is assumed that particles of the material of the shield move in radial direction during penetration by the impactor. The shield is divided into infinitely thin layers, and, in each layer, cavity expansion caused by the moving impactor is modeled. This facilitates the determination of the stress at the boundary of the hole, the force acting at the penetrator in each layer and, respectively, at each location on the impactor s lateral surface. The common technique for applying the CCEA to penetration mechanics may be described as follows. The solution of a dynamics problem of hole expansion with time is usually represented for each layer as Applied Mechanics Reviews NOVEMBER 2005, Vol. 58 / 357

4 Fig. 4 Recht s approach 8 Fig. 3 x-h Domain of impactor-shield interaction in coordinates p = a ; *, *, * 18 Let = x be the equation of the surface of the impactor a body of revolution, then for the layer with the coordinate see Fig. 2 we can write 44 * = h, * = x ḣ, * = x ḣ 2 + x ḧ, x = h 19 The expression for the normal stress at the surface of the impactor, n = p, is obtained by substituting *, *, and * from Eq. 19 into Eq. 18. Since n is a function of x,ḣ=v and ḧ=vdv/dh, the equation of motion of an impactor in the case of CCEA that is determined by Eq. 18, is similar in structure to Eq. 9. In the case of a quasi-static model when Eq. 16 instead of Eq. 18 is assumed to be valid the cylindrical cavity expansion model yields n = a ; v = a ; uv 1 u 2 0 a ;u,v 20 i.e., the CCEA is reduced to a LIM with n = 0. The study of Bishop et al. 46 pioneered the application of cavity expansion models in penetration mechanics. They obtained solutions describing the quasi-static expansion of cylindrical and spherical cavities in an infinite medium from zero initial radius and used these solutions to determine the forces acting at a conical impactor. A survey of the state of the art up to the late 1950s concerning the dynamic expansion of cavities in solids was prepared by Hopkins 47. Useful information on this topic is summarized in the monograph of Yu 48, which consists of two parts, namely, Fundamental Solutions and Geotechnical Applications. Cavity-expansion models applied to penetration mechanics have been described and analyzed by Teland 49 and Satapathy 50. Recent studies directly associated with the application of cavity-expansion methods in modeling ballistic impact are surveyed below. The most intensive research in this field has been conducted at Moscow State University and Sandia Research Laboratories, and, as was noted by Isbell et al. 51, some of the results obtained are similar. 2.3 Lambert-Jonas Approximation. The following formula was proposed by Lambert and Jonas 52 and Lambert 53 for the reduction of ballistic impact data: n vˆ res = a 0 vˆ imp 1 1/n, vˆ imp = v imp v bl, vˆ res = v res v bl, vˆ imp 1 21 where v imp, v res and v bl are the impact velocity, the residual velocity and the BLV, respectively, and a 0 and n and, actually, v bl are the coefficients of approximation in Eq. 21. Many of the empirical and semi-empirical models can be represented in the form of Eq. 21, particularly, models based on energy and/or momentum conservation 4, Mileiko and Sarkisyan 36 and Mileiko et al. 37 see also 59 demonstrated that a solution of the equation of motion of the impactor yields Eq. 21 with a 0 =1, when a power-law dependence between the impactor s drag force and its velocity is valid. Nixdorff showed that under certain assumptions the theory of Awerbuch 64 and Awerbuch and Bodner 65 implies Eq. 21. Ben-Dor et al. 66 compared the accuracy of Eq. 21 with an arbitrary exponent n and that with n=2. Since at present Eq. 21 is used as one of the preferred methods to reduce experimental data, it is useful to understand the cause of the efficiency of this correlation. Ben-Dor et al. 67,68 addressed this problem by constructing a wide class of physically realistic models of penetration, which imply Eq. 21. They found that the following model satisfies the latter requirement: dd = G 1 x,h v 2 +G 2 x,h v 2 n dx, where dd is the differential of the drag force acting at the impactor s surface element between cross sections x and x+dx at a depth h see notations and coordinates in Fig. 2, G 1 and G 2 are non-negative functions determining the model, and the loss or accumulation of mass and change of the shape of the impactor during penetration can be taken into account in this model. Nennstiel 69 combined the Lambert-Jonas equation and the demarre formula for BLV 4 and introduced additional fitting parameters into the model. These parameters, determined by the method of nonlinear regression of experimental data, enables the model to be adapted for special pistol and revolver bullets. Grabarek 70 and Anderson et al. 71 considered Eq. 21 as only one of the possible correlations between vˆ res and vˆ imp, and used a different unified relationship: vˆ res = a 2 z 2 + a 1 z + a 0 z 0.5 / z +1, z = vˆ imp 1, z where the approximation coefficients a i i=0,1,2 are determined from regression analysis of experimental data. 3 Metal, Geological and Concrete Shields 3.1 Modeling of Penetration. Investigations that were performed at Moscow State University in the 1950s and the beginning of the 1960s in the field of soil dynamics were summarized in the monograph of Rakhmatulin et al. 44, in which penetration modeling occupies an important place. The authors described dynamic solutions for the expansion of cylindrical and spherical cavities in soil. The problem of an impactor s penetration into a SIS soil was solved by applying the developed cylindrical cavity expansion models. Solutions were found for the DOP, which took into account incomplete immersion of the impactor in the shield at the initial stage of penetration. The results obtained during the next decade were summarized 358 / Vol. 58, NOVEMBER 2005 Transactions of the ASME

5 in the monograph of Sagomonyan 72, in which the method of normal sections, a generalization of the method of plane sections cylindrical cavity expansion model for modeling penetration by nonslender impactors, was proposed. This method is based on the assumption that the particles of the host medium move along the direction normal to the impactor s surface after the contact. Penetration by a nonslender sharp cone-shaped impactor was studied by means of the proposed approach. Approximate solutions were obtained for some other problems, i.e., normal penetration by a sphere, oblique penetration by a cone, and ricochet from a soil surface. The results of subsequent investigations were summarized in another monograph of Sagomonyan 73, which considered the five classes of problems described below: 1. Normal impact of a slender rigid projectile (a body of revolution). A dynamic CCEA was used to model perforation of a metal plate and a brittle plate. A technique for obtaining an analytical solution for the BLV was outlined, and some calculations for cone-nosed impactors were performed. The quasi-static solution of Bagdoev and Vantsyan 74 for penetration of a slender striker into an anisotropic shield was discussed. 2. Perforation by normal impact of impactors with a cylindrical nose. Impactors and shields with different combinations of properties of the materials were considered strainhardening or rigid shield; rigid or strain-hardening impactor; elastic or elastic-plastic impactor, etc.. 3. Perforation by normal impact of truncated cones and by non-slender projectiles with a conical nose see also 75. Here, penetration of an impactor with plane bluntness into an elastic-plastic shield with a plug formation was investigated. The penetration phenomenon was considered as two simultaneous processes, namely, expansion of the cavity in the shield and motion of the plug. A model was developed to describe the expansion of a cylindrical cavity inside an elastic-plastic medium starting from nonzero radius. Solutions associated with modeling the impact by a cylindrical striker on a plate taking into account different modes of behavior of the material of the plate were used for the modeling of plugging. The version of the method of normal sections was used for modeling the penetration of nonslender impactors into a SFT. 4. Oblique impact by a rigid impactor on a plastic or elasticplastic plate see also 76. Here, motion of the impactor and simultaneous deformation of the plate were modeled by means of both analytical and numerical approaches. 5. Penetration of rigid impactors into soils. A method to increase the DOP by means of jet momentum was also studied. Studies in all these directions were continued in later years see Extensive studies of cavity expansion approximations CEA in penetration dynamics were performed at the Sandia Research Laboratories by Forrestal and his colleagues. They proposed a large variety of spherical and cylindrical cavity-expansion models for materials with different mechanical properties. The distinct features of their approach are described below. Simple quasi-static two-term or three-term models for engineering applications were developed by using theoretical studies of cavity expansion problems, numerical simulations, and experimental investigations. The friction coefficient was often introduced into a model to account for the tangent component of the impactor-shield interaction force. A limited number of impactor nose shapes cone, sphere, and ogive were considered, comprehensively, although the approach is applicable to a wide class of bodies of revolution. A simplification of Eq. 10 was used when normal penetration was modeled. Two-term SCEAs a ; * = a 0 + a 2 *2 23 were developed for concrete shields 80,81, shields manufactured from elastic-plastic materials 82,83, strain-hardening materials 84 88, and soils 89. Three-term SCEAs a ; * = a 0 + a 1 * + a 2 *2 24 were proposed for concrete shields 90 and metal shields taking into account strain hardening, compressibility, and strain-rate effects 91. The coefficients a i in Eqs. 23 and 24 depend, generally, on the mechanical properties of the material of the shield and are determined experimentally or from numerical simulations. For instance, the following model 91 can be described in greater detail: a 0 = â 0 Y, a 1 = â 1 Y, a2 = â 2 25 where Y is the quasi-static yield strength and is the density of the nondeformed shield material. Fitting coefficients â i are presented by Warren and Forrestal 91 for models with incompressible or compressible materials with or without strain rate effects four versions of the model. All these studies considered normal penetration into the SISs. The models described by Eqs. 24 and 25, taking into account the simplification of Eq. 10, allow one to obtain an expression for the DOP in a closed form. Development of the approaches based on CEA accounts for some additional features of penetration. In the studies of Littlefield et al. 92, Partom 93, and Teland and Sjøl 94, cavity expansion models were modified to account for the finite size of the shield in the direction normal to the direction of penetration. Warren and Poormon 95 generalized the model 91 to take into account the influence of the free surface of a SIS on the motion of the impactor after oblique impact. Generally, the approach may be described as follows. Spherical cavity expansion is considered under the condition that the radial stress equals zero at = +, where a sphere with a radius + represents the free surface. Then, the solution of the spherical cavity expansion problem is represented in the form p= a ; *, +, *. Clearly, an uncertainty in determining * and + emerges in the calculation of the normal stress at the surface of the impactor by means of Eq. 17. Warren and Poormon 95 calculated the distances * and + along the direction normal to the penetrator surface at a given location, namely, + was the distance between an axis of the impactor a body of revolution and the free surface, and * was the distance between an axis of the impactor and its surface. Longcope et al. 96 used a similar approach to account for the free-surface effect in modeling the penetration of an impactor into a geological shield, but they calculated * and + in a different manner. A spherical cavity expansion technique was also developed by Macek and Duffey 97 to take into account near-surface effects and layering, in the case of normal and oblique penetration into soil. Cylindrical cavity expansion models were developed for geological shields , metal shields 87, and concrete shields 110. Some two-term CCEAs and SCEAs were summarized and compared by Forrestal et al. 111 using experimental data for conical-, spherical, and ogive-nosed projectiles. The study of Brown et al. 112 describes practical applications of some cavity expansion models developed in Sandia National Laboratories. Analysis of various CEAs using experimental data and benchmark calculations has been undertaken by a number of researchers Forrestal and Longcope 117, Satapathy and Bless 118,119, Kartuzov et al. 120,121, Satapathy 122 and Mastilovic and Krajcinovic 123,124 also applied cavity expansion analysis to brittle materials. Aptukov 125,126 and Aptukov et al. 127, taking into account the influence of the free surface, spherical layering of the medium and temperature effects, ob- Applied Mechanics Reviews NOVEMBER 2005, Vol. 58 / 359

6 tained the solution for the one-dimensional problem of the expansion of a spherical cavity into a compressible elastic-plastic medium. Some engineering approximations of dynamic spherical cavity expansion solutions in an elastic-plastic medium applied to penetration problems were obtained. Kravchenko et al. 128 used a combined approach, including the method of plane sections and cylindrical spherical cavity expansion models in modeling the penetration by a solid body of revolution into soils and rocks. Bashurov et al. 129 used a three-term SCEA given by Eq. 24 in modeling penetration into concrete, metal, ice, and geological media. Ben-Dor et al. 130 obtained analytical formulas for the BLV and the DOP by applying a dynamic CCEA and Eq. 5. Yarin et al. 131 studied the penetration of a rigid projectile into an elastic-plastic shield with finite thickness. An ovoid of Rankine was considered as an impactor because it implied a reasonably simple velocity field that exactly satisfied the continuity equation and the condition of impenetrability of the projectile. Some simplifications enabled the force applied at the projectile to be calculated analytically. An equation of projectile s motion could generally be solved numerically. Simple analytical formulas for the DOP or the residual velocity were derived by including some additional assumptions. It was shown that the suggested procedure could be approximately generalized to a projectile with a tip of arbitrary shape. Roisman et al. 132 developed an analytical model for oblique penetration by a rigid projectile into an elastic-plastic shield. Yossifon et al. 133 analyzed the main difference between the approach of Yarin et al. 131 and that of Roisman et al The first solution satisfied the balance of linear momentum pointwisely in the shield region, but it satisfied the free-surface boundary conditions only in an integral sense. The second solution satisfied the balance of linear momentum only along a finite number of instantaneous streamlines, but it satisfied the boundary conditions exactly at each intersection of the streamlines with the boundary. The first solution was valid only for normal penetration, whereas the second solution could be used for oblique penetration. The predictions using these two theoretical approaches for the case of normal impact were compared to numerical simulations. It was concluded that both analytical methods yielded reliable results for the penetration depth, the BLV and the residual velocity of the projectile. Yossifon et al. 134 generalized the single-layer models 131,133 to penetration into multilayered shields by rigid projectiles and undertook a comprehensive consideration of two-layered shields. They noted that the computational time required for the proposed analytical model was only a few minutes. Using the energy-balance approach, Srivathsa and Ramakrishnan 135,136 derived a ballistic performance index to estimate and compare the ballistic quality of metal materials. This index is a function of the commonly determined mechanical properties of the material and the striking velocity of the projectile. Srivathsa and Ramakrishnan 137 represented these indexes as maps. Forrestal and Hanchak 138 extended the rigid-plastic beam analysis 139,140 to predict the BLV of a plate with a rectangular crosssection against a flat-nosed nondeforming projectile with a rectangular cross section. Dinovitzer et al. 141 developed an analytical model to predict the number of armor debris fragments produced in a ballistic penetration into a single or multiplelayered metal shield. Gupta and Madhu 142 and Madhu et al. 143 used their experimental data to derive empirical formulas for calculating the residual velocity velocity drop of an impactor. Liaghat and Malekzadeh 144 modified the model of Dikshit and Sundararajan 145 for perforation into thick plates by introducing an analytical equation instead of an empirical relationship for determining the size of the plastic zone. Chen and Li 146 developed a penetration model for perforation of a thick plate by a rigid projectile with various nose shapes. The model takes into account two perforation mechanisms, namely, hole expansion for sharp-nosed projectiles and plug formation for blunt-nosed impactors. Explicit formulas or algebraic equations were obtained to predict the BLV and the residual velocity of the impactor. The proposed model of Chen and Li 147 included submodels of shear plugging, bending and membrane deformations, and hole expansion for a range of plate thicknesses against blunt impactors and was used to obtain analytical formulas for the BLV and the residual velocity. Wu and Batra 148 proposed a four-stage model of perforation of a thick strain-hardening plate by hemisphericalnosed rigid cylindrical rods. The first and the second stages corresponded to the cases in which 0 h R and R h 2R, respectively, where h is the penetration depth and R is the penetrator s shank radius. During the third stage, the tunnel continued to grow, and it was assumed that a bulge was not formed at the rear surface of the shield. Then in the fourth stage, a plug developed ahead of the projectile and ejected out of the shield when the bulge radius reached a limiting value. The fourth stage may not necessarily occur for a relatively thick plate or for low-impact velocity, and the second and/or the third stages may not necessarily occur for relatively thin shields. Holt et al. 149 described a simple model of plugging based on the equations of momentum and energy conservation and the assumption of constant shear stress acting at the interface between the plug and the shield. This model included parameters determined from experiments. Chen and Davies 150 obtained a relationship between energy absorption during plugging and the initial impact velocity and investigated the behavior of energy absorption versus impact velocity. Grigoryan 151 suggested a simple model describing the penetration of a rigid impactor into soft soil and derived the solution to the problem in an explicit form. He noted that his investigation was performed in 1969, although it was published much later in Foster et al. 152 proposed a simple combined analytical model for penetration into geological shields. The force component of the model included the model given by Eq. 11 with the equation for the tangent force being assumed to be a linear function of the normal force. The mass loss from a projectile due to surface melting was estimated using thermodynamic considerations and calculating the work performed by tangential forces during penetration. Børvik et al. 16 analyzed the best-known phenomenological models for normal impact on steel plates by blunt cylindrical projectiles some generalizations of the results of this analysis can be found in 153 and compared the results obtained using these models to those obtained from experiments. In the study of Børvik et al. 153, the models suggested by Wen and Jones 154, Bai and Johnson 155, and Ravid and Bodner 156 were analyzed using experimental results and benchmark calculations for a wide range of shield thicknesses. For the reader s convenience, a short description of these models is presented in the Appendix. Wierzbicki 157 developed a new analytical model for perforation of thin plates struck normally by conical-nose projectiles accompanied by petaling. A closed-form solution was derived for the total energy absorbed by the system, the BLV, the number of petals, and the final deformed shape of the plate as a function of plate flow stress, thickness, and parameters of the external loading. Gupta et al. 158 suggested analytical and empirical relations for determining the BLV and the residual velocity based on theoretical considerations and experiments performed for thin plates and ogive-nosed projectiles. Atkins et al. 159 investigated the formation of multiple necks and cracks around perforations in ductile materials and obtained expressions for the number of plane-strain radial necks formed by conical and round-ended projectiles penetrating into plane targets. Empirical formulas describing the impact on concrete shields were analyzed in the studies listed in Table 1 and were also discussed by a number of researchers 113, Dancygier 164 modified the existing empirical perforation formulas to include the reinforcement ratio as a variable. Luk and Forrestal 80,81 developed a model to estimate the DOPs and the forces acting at the surface of ogive- and spherical- 360 / Vol. 58, NOVEMBER 2005 Transactions of the ASME

7 nose projectiles penetrating into a concrete SIS. The model was based on a quasi-static SCEA. Forrestal et al. 110 proposed a cylindrical cavity expansion model and an iterative procedure for determining its parameters. Forrestal et al. 165 suggested the following two-stage penetration model. In the first stage of the penetration, the resistance force was given by the notations are shown in Fig. 2 D = ch, 0 h R 26 where c is a constant and =4. In the second stage h R, it was assumed that the spherical cavity expansion model 89 described by Eq. 23 is valid, with a 0 =sf c, a 2 =, where f c is the unconfined compressive strength and dimensionless empirical constant s can be calculated from experimental data. The constant c is determined using the condition of continuity of the resistance force at h= R. This model yielded an explicit solution for the DOP. More recently, Forrestal et al. 166 and Frew et al. 167 showed that s can be considered as a function of f c and plotted the corresponding curve. Frew et al. 167 suggested the approximation s=82.6 f c f c in MPa, but Forrestal et al. 168 preferred to interpret a 0 as the measure of the shield resistance. Although Forrestal and Luk 89 developed their model for ogiveshaped impactors, the model may be easily generalized to arbitrary bodies of revolution and projectiles with plane bluntness. The corresponding formulas for cone, truncated-ogive and segmental-spherical noses were derived by Chen and Li 29. Lixin et al. 169 introduced an empirical constant to take into account the truncation effect of the ogive-nose projectile. In some studies 169,170 it was proposed that in Eq. 26 should be considered as an empirical constant. Li and Tong 171 generalized the model, taking into account plug formation. A modification of this model has also been proposed by Teland and Sjøl 172. Forrestal and Tzou 90 compared different spherical cavity expansion penetration models for concrete shields. An elasticcracked model based on the SCEA was developed by Xu et al Yankelevsky 160 suggested a two-stage model for concrete slab penetration in which the disks model 45 was used. Gomez and Shukla 170 extended the model of Forrestal and Luk 89 to multiple impacts. To this end, an empirical coefficient that is a function of the number of impacts was introduced into the formula derived by Forrestal and Luk 89. On the basis of the same model, Choudhury et al. 174 and Siddiqui et al. 175 derived expressions for the DOP in a buried shield and applied sensitivity analysis to study the influence of various random variables on projectile reliability and shield safety. Chen and Li 29 and Li and Chen 162,176 performed a dimensional analysis of analytical perforation models for concrete, metal, and soil and concluded that two dimensionless parameters would suffice to describe the DOP with reasonable accuracy. Me-Bar 177 proposed a method for scaling the phenomena of ballistic penetration into concrete shields. The authors suggested to separate the energy absorbed by the shield during penetration into the energy expended for surface effects and the energy expended for volume effects. Then using energy balance they derived expressions that account for the irregularity in scaling. Frew et al. 178 and Forrestal and Hanchak 179 proposed that the model of Forrestal and Luk 89 could be applied to a limestone shield with a 2 = and a 0 = R/R 0, where 0 =607 MPa, 1 =86 MPa, R 0 = m. The modeling of penetration into a shield with a predrilled cavity has also attracted the attention of researchers in the field. To determine the DOP, Murphy 180 and Folsom 181 modified models developed for homogeneous shields. A better substantiated approach takes into account the influence of the predrilled cavity on the contact surface between an impactor and a shield during penetration. The area and shape of this contact surface affect a drag force. Teland 182,183 suggested an SCEA-based model and noted that a similar approach had been developed independently by Szendrei 184. Yankelevsky 185 developed a model based on his own version of CCEA. Clearly, a LIM can be easily generalized to this situation. Young 186 suggested the following formulas for the DOP of an impactor penetrating into soil, rock, or concrete: H = a 3a 4 m/s ln v imp if v imp a 3 a 4 m/s v imp 30.5 if v imp 61 where for soil, and 27 = 0.27m0.4 if 2 m 27 1 if m = 0.46m0.15 if 5 m 182 1ifm for rock and concrete, where S 0 is a characteristic cross-sectional area of the impactor and the coefficients a 3 and a 4 depend on the impactor s nose shape and on the properties of the material of the shield, respectively a method for calculating a 3 and a 4 was also suggested by Young 186. All the parameters in Eqs are in SI units. Similar formulas were also developed by Young 186 for ice and frozen soil. 3.2 Shape Optimization of Impactors. At the early stages of investigations of problems of an impactor s shape optimization, indirect criteria were used. Kucher 187 optimized the penetrator s shape using as a criterion the dynamic work from Thomson s theory 188 for thin plates. Nixdorff 189 compared the efficiency of conical, different power-law, and ogival heads and found that there are indeed shapes that are superior to Kucher s optimum head, which was determined by solving the corresponding variational problem. Ben-Dor et al. 28 explained this paradox with reference to the correct solution 190 see also 191 of the mathematically similar variational problem in hypersonic aerodynamics, namely, determining a thin head with minimum drag by means of the Newton-Busemann model for projectilemedium interaction. Gendugov et al. 192, Bunimovich and Yakunina , Ostapenko and Yakunina 196, and Ostapenko 197 determined the shapes of three-dimensional 3D bodies with the minimum shape factor that is equivalent to the minimum resistance during the motion of an impactor inside a dense medium with constant velocity. Using the previously developed disks model 45, Yankelevsky 198 optimized the shape of a projectile penetrating into soil by minimizing the instantaneous resistance force. The optimal shape was found to be determined by a single parameter depending on the velocity and deceleration of the impactor and the properties of the medium. As direct criteria for optimization, the maximum DOP for a given impact velocity in the case of a SIS and the BLV for a SFT were used. Yankelevsky and Gluck 199, using the disks model 45, obtained formulas for the penetration depth of an ogiveshaped projectile into soil and analyzed the influence of the impactor s shape parameters and the characteristics of the shield material on the criterion. Bondarchuk et al. 200 used a simple LIM for shape optimization of 3D impactors penetrating into SISs soil and metal. Numerical calculations and experiments showed that 3D impactors offer advantages over bodies of revolution when the DOP is considered as the criterion of optimization. Additional calculations associated with determining efficient 3D penetrators are to be found in the monographs 201,202. Ostapenko et al. 203 found numerically the optimum cross section of a 3D conical impactor with the maximum DOP for the class of models with n =a 2 uv 2 +a 1 uv +a 0 and given by Eq. 4. Ostapenko and Yakunina 204 used this criterion in their analytical investi- Applied Mechanics Reviews NOVEMBER 2005, Vol. 58 / 361

8 gation of a variational problem based on the model given by Eqs. 4 and 11 ; they considered slender bodies with self-similar cross sections. Jones et al. 205 investigated the problem of impactor body of revolution optimization using the model given by Eqs. 3, 10, and 11 and the shape factor as the optimization criterion. Jones and Rule 206 showed later that the criterion of the maximum DOP implies the same variational problem. Jones et al. 205 used an approximate perturbation method and a numerical simulation for solving the Euler-Lagrange equation. Jones and Rule 206 and Rule and Jones 207 investigated analytically and numerically the problem of maximizing the DOP using the model given by Eqs. 4, 10, and 11, i.e., taking friction into account. It was found that blunt-nosed impactors have the most favorable geometry. In 206 it was observed that numerical simulations showed that for some values of the parameters the predicted impactor tip is as close to blunt-ended as possible. Ben- Dor et al. 28 noted that the formulation of impactor shape optimization problems must allow for the existence of the plane bluntness with the unknown size. They drew attention to the analytical solution of the mathematical problem investigated by Jones et al. 205, which was obtained earlier in aerodynamics the optimal projectile with a plane bluntness 191,208. If friction is taken into account, then the optimal impactors also have a plane bluntness 209. The shape of the impactor that attains the maximum DOP for concrete or limestone SIS for a given impact velocity was determined by Ben-Dor et al. 210 using the two-stage model 165. They found that the optimum shape is close to a blunt cone and that it is independent of the properties of the material of the shield. The optimum shapes among spherical-conic, sharp-conic, and truncated-ogive impactors were also found. A new formulation of the problem of the nose optimization of an impactor against a SFT when the impactor can have a plane bluntness with a hitherto unknown radius that causes a plug formation was suggested by Ben-Dor et al Ben-Dor et al. 28 investigated the properties of the following model used for shape optimization of impactors: n = a 1,... ;u 1 v + a 0 0 v, =0 30 where 0 and 1 are some functions. They described a procedure for averaging the drag force for 3D impactors with plane bluntness that generalizes and interprets the widely used replacement of Eq. 5 by Eq. 10. The properties established for the simplified model are described below. Determining the shape of the impactor with the minimum BLV for a SFT is equivalent to determining the shape of the impactor with the maximum DOP for a SIS; in these determinations the optimal shape of the impactor must be selected from shapes with a given cross-sectional area of the base. The two problems can be reduced to the minimization of the same functional; this functional and the optimal shape of the impactor are independent of the choice of functions 0 and 1.In addition, this optimal impactor has minimum resistance in the hypothetical situation in which the impactor is completely immersed in a homogeneous shield and moves with constant velocity. If in Eq. 30 =a 1 u, where u is some function, then the shape of the optimal impactor does not depend on the properties of the material of the shield parameters a 0 and a 1. Bunimovich and Dubinsky 21 showed that the LIM model given by Eqs. 1 and 2 implies the condition u= v 0 n 0 =u for the projectile with the minimum drag among 3D projectiles with a given cross-sectional area of the base flying at a zero angle of attack, where u is a constant determined by the model adopted for projectile-medium interaction. Yakunina 212,213 proposed a procedure, using the above-mentioned condition, for constructing projectiles with the minimum drag from conical and plane elements. Later, she used DOP as a criterion of optimization and the following model to extend this theory to 3D impactors 214 : n = a 2 uv 2 + a 0, = a 4 uv 2 + a 3 31 Using a LIM with n =a 2 u v 2 +a 0 and Eq. 3, Ben-Dor et al. 26,215 studied the problem of impactor shape optimization for nonhomogenous layered shields They considered 3D sharp conical impactors with a given form of the longitudinal contour, length, and volume. It was found that an impactor having the minimum drag during its hypothetical motion with constant velocity in a homogenous shield penetrates to the maximum depth in a SIS and has the minimum BLV when it penetrates into a SFT, regardless of the distribution of the properties of the material of the shield along its depth and of the number of the layers. By analogy with the hypersonic flow over flying projectiles, Ben-Dor et al. 215 predicted that the optimal impactors have a star-shaped cross section. It was shown 215 that if the ballistic performance of one of the two impactors is better in penetrating a reference homogenous or nonhomogenous shield, then the same property remains valid for any other shield if both impactors have the same longitudinal contours, are manufactured from the same material, and have the same lengths of the nose and of the cylindrical part. Aptukov and Pozdeev 216 considered the minimax problem for determining the shape of an impactor that penetrates to the maximum depth under the most unfavorable distribution of the mechanical properties along the depth of SIS with a given areal density. The model given by Eqs. 3 and 11 was used with a linear relationship between the parameters a 0 and a 2. On the basis of the LIM given by Eqs. 1 and 2, Ben-Dor et al. 24,217 determined the area rules for penetrating impactors that facilitate the prediction of the influence of a small variation of the impactor s shape on its BLV. It was found that the BLV of a 3D impactor whose shape is close to the shape of any reference impactor body of revolution or body with polygonal cross sections is determined, mainly, by the distribution of the crosssectional area along the longitudinal axis. If this area distribution is the same for a 3D impactor and the reference impactor, then the difference in their BLVs is of the order of 2, where is the order of the difference in their shapes. 3.3 Modeling, Analysis and Optimization of Multilayered Plates. Marom and Bodner 218 conducted a combined analytical and experimental comparative study of monolithic, layered, and spaced thin aluminum shields. They found that the ballistic resistance of a monolithic shield is higher than that of a multilayered shield in contact with the same areal density and lower than the ballistic resistance of a spaced shield. The study of Radin and Goldsmith 219 was also based on semi-empirical models and experimental investigations. They found a monolithic aluminum shield to be superior to a layered shield with the same total thickness for both conical-nose and blunt projectiles, whereas spaced shields were less effective. Nixdorff compared the ballistic performance of a monolithic metal shield to that of a shield having the same total thickness, but consisting of several plates in contact, where the two shields were manufactured from the same material. Using the theory developed by Awerbuch and Bodner 64,65, Nixdorff showed that division of a homogeneous shield into several layers implies a reduction of the BLV of the shield. Woodward and Cimpoeru 220 developed a simple semiempirical model that considers the perforation of laminates as a two-stage process of indentation on the impact side of the shield and either shear or dishing failure on the exit side, depending on the shield configuration. Experimental data for laminated aluminum alloy shields perforated by plane-ended or conical penetrators were used in this investigation. Young 186, using semiempirical formulas for soil, rock, and concrete homogeneous shields, developed a technique for calculating the penetration into layered shields. Aptukov et al. 221 and Aptukov 222, using Pontrjagin s maximum principle, determined the optimum distribution of the mechanical characteristics of a nonhomogeneous plate. The areal density of the shield along the trajectory of the impactor s motion, up to its stop, was used as a criterion and cylindrical and conenosed impactors were considered. The model determined by Eqs. 362 / Vol. 58, NOVEMBER 2005 Transactions of the ASME

9 3 and 11 and a linear relationship between the parameters a 0 and a 2 was used. Aptukov et al. 223 solved the discrete problem of optimization of a layered plate when the shield consisted of several layers of material and the material itself could be chosen from a given set of materials. The cylindrical cavity expansion model suggested by Sagomonyan 224 was used in this study. All these investigations are summarized in the monograph of Aptukov et al Ben-Dor et al. 225 considered multilayered shields consisting of several plates in contact with no interaction between plates. It was assumed that the model given by the equation n = a i 2 u v 2 + a i 0, =0 32 was valid for the ith plate in the initial order of the layers in the shield. The effect of the order of the plates on the BLV of the shield was studied, and the results can be summarized as follows. If a i 2 are the same for all the plates in the shield, the shield is perforated by 3D nonconical impactor, and a i 0 a i+1 0 for any two adjacent plates, then the BLV of the shield can be increased by interchanging the plates. The maximum BLV of the shield is achieved when the plates are arranged in the order of increasing values a i 0 ; the minimum BLV is attained when the plates are arranged in the armor in the inverse order. In the general case in which a 0 and a 2 are different for different plates, the maximum BLV for a two-layered shield against a 3D conical impactor is attained when the plates are arranged according to increasing magnitudes of the parameter, i = a i i 0 /a 2 33 In additional assumption that the plates are perforated sequentially one would expect that this is approximately valid if the length of the impactor is much less than the thickness of each plate facilitated the extension of the latter conclusion to shields consisting of more than two plates. Ben-Dor et al. 27, studied analytically the influence of air gaps between the plates on the BLV of a multi-layered shield. Using the general LIM given by Eqs. 1 and 2, they 27 found that the ballistic performance of the shield against 3D conical-nosed impactors is independent of the widths of the air gaps and of the sequence of plates in the shield and that it is determined only by the total thickness of the plates if the plates are manufactured from the same material. The influence of air gaps on the BLV of a shield that consists of two plates manufactured from different materials was studied by Ben-Dor et al. 226 using the model described by Eq. 32. It was found that if , then the BLV decreases increases with increasing the air gap thickness from zero to the length of the impactor and becomes constant with further increases of the air gap thickness. If 1 = 2 then the BLV does not depend on the thickness of the air gap. Using the CCE model 73, Ben-Dor et al. 227 studied the effect of air gaps on the ballistic performance of a spaced shield comprising plates manufactured from the same material, which was penetrated by a conical-nosed impactor. It was found that the BLV of the shield increased with the increase of the widths of air gaps; this effect was very small for slender impactors and could be intensified by increasing the half angle of the cone apex. The case of large air gaps and plates manufactured from different materials was also considered by Ben-Dor et al A number of other ballistic properties of multilayered shields perforated by a 3D conical impactor were established by Ben-Dor et al. 229,230. In the first case to be considered, the armor consisted of adjacent plates manufactured from one of two possible materials and the total thickness of the plates manufactured from each material was fixed 229. It was found that the maximum BLV was attained for a two-layered shield without interchanging the plates manufactured from different materials, i.e., the front plate in the optimum shield must be the plate manufactured from the material with the smaller value of the parameter. The second case considered the use of a number of materials with different properties for manufacturing the plates in a multilayered shield 230. For a shield with a given areal density mass per unit surface area and thickness, the goal was to determine the structure i.e., the order and the thicknesses of the plates of the different materials that would provide the maximum BLV of the shield against a normal impact by a 3D conical impactor. It was found that the shield having the maximum BLV must contain one plate manufactured from the material with the maximum. The shield with the minimum BLV will contain one plate manufactured from the material with the minimum. The magnitudes of the BLV of different shields with a given areal density and thickness will vary between these two limiting solutions. 4 Composite Shields Florence 231 developed an analytical model for a twocomponent composite armor consisting of a ceramic front plate and a ductile back plate impacted normally by a rigid projectile at ballistic velocity. This model, as reworked by Hetherington and Rajagopalan 56, yields the following expression for the BLV, v bl : v 2 bl = 2 2 b 2 z 1 b b 2 z + m 0.91m 2, z = R +2b where b i are the plate s thicknesses, i are the ultimate tensile strengths, 2 is the breaking strain, i are the densities of the materials of the plates, and subscripts 1 and 2 refer to a ceramic plate and a back plate, respectively. The impactor was modeled as a short cylindrical rod that strikes the ceramic plate. The ceramic plate then breaks progressively into a cone of fractured material. The impact energy is transferred to the back plate, which is deformed like a uniform membrane. The simplifying assumptions that enabled an analytical expression to be obtained for the BLV were elucidated by Hetherington and Rajagopalan 56 as follows: i the diameter of the circular area at the back plate over which the momentum is distributed is equal to the base diameter of the fracture conoid in the ceramic facing, and the angle of the conoid is chosen to be equal 63 deg; ii the deformation history of the back plate is modeled by the motion of a membrane clamped around the perimeter of the base of the fractured conoid, and the initial conditions for the membrane s motion are determined by the condition imposed by the projectile s impact and by the conservation of momentum within the projectile-shield system; iii failure occurs when the maximum tensile strain in the membrane attains the ultimate breaking strain of the back plate. Florence s model was used for solving problems associated with armor optimization. Some numerical results obtained with this approach were presented by Hetherington and Rajagopalan 56 and by Florence 231. Later, Hetherington 232 investigated analytically the problem of determining the structure of twocomponent armor with a given areal density that provides the maximum BLV. He suggested an approximate expression for the optimum value of the ratio of the front plate width to the back plate width. Wang and Lu 233 investigated a similar problem in which the total thickness of the armor rather than the areal density was a given characteristic. The problem of designing twocomponent armor with the minimum areal density for a given BLV was comprehensively investigated by Ben-Dor et al It was shown that the solution of the optimization problem could be presented in terms of the dimensionless variables, where all the characteristics of the impactor and the armor were expressed as a function of two independent dimensionless parameters. The latter condition enables the determination of a solution of the optimization problem for an arbitrary two-component composite armor in an analytical form. Approximate formulas were derived for the areal density and thicknesses of the plates in the optimal armor as functions of the parameters determining the properties of the ma- Applied Mechanics Reviews NOVEMBER 2005, Vol. 58 / 363

10 terials of the armor components, the crosssection and mass of the impactor, and the expected impact velocity. Lee and Yoo 235 conducted a combined numerical and experimental study that supported the results for the armor s optimization based on the model suggested by Florence 231. Hetherington and Lemieux 236 and Sadanandan and Hetherington 237 generalized Florence s model to the case of an oblique impact. Ben-Dor et al. 234 used fitting coefficient in Eq. 34 that is determined from experimental data in order to improve the accuracy of the model. Woodward 238 developed two simple models for the perforation of ceramic composite armor by a flat-ended cylinder, one for the perforation of a shield with a thin back plate and the other for a shield with a thick back plate. Analytical models for the perforation of a ceramic/metal armor with thin back plate have been proposed by a number of researchers Zaera et al. 242 developed an engineering model taking into account the effect of the adhesive layer used to bond ceramic tiles to the back metallic plate on the ballistic behavior of ceramic/metal armors. Fellows and Barton 243 suggested a model that predicts penetration of projectiles into ceramic-faced semi-infinite armor. Zhang et al. 244 proposed a three-stage dynamic model of ballistic impact by a cylindrical projectile against ceramic fiber-reinforced plastic FRP composites. Du and Zhao 245 proposed a model that facilitates the prediction of the BLV of shield manufactured from a ceramic/aluminum alloy. James 246 suggested an empirical formula to estimate the optimum ratio between the thickness of the ceramic plate and the thickness of the back metallic plate in a two-component armor for an oblique impact. Ben-Dor et al. 247 developed a procedure to optimize a two-component armor on the basis of the experimental data. In the model suggested by Wen 248,249 see also 250, it was assumed that the deformations are localized and that the pressure at the surface of the impactor immerged into the shield can be broken down into two components: the cohesive quasi-static resistive pressure caused by elastic-plastic deformations of the armor material and the dynamic resistance associated with the kinetic energy of the impactor. It was assumed that the normal stress at the surface of the projectile penetrating into a FRP laminate could be represented as n = c e + e v imp 35 where e is the quasi-static linear elastic limit of the material of the armor, c=1, and is a constant depending on the shape of the impactor. The values of the parameter were assumed to be =2 sin for a conical-nosed impactor, where is the half angle of the apex of the cone, =2 for a cylindrical flat-faced impactor, and =1.5R/Rˆ for an ogival-nosed impactor with ogive generator radius Rˆ. Analytical equations were obtained for the DOP SIS and the BLV SFT struck at normal incidence by impactors with conical, truncated, plane, ogival and hemispherical nose shapes. Using the MTC, Ben-Dor et al. 40 generalized the model suggested by Wen 248,249 to impactors with an arbitrary shape, n = e +2 e uv imp, studied the ballistic properties of impactors that have the shape of a body of revolution against SFT and derived formulas for the BLV. The shapes of the impactors with the minimum and the maximum BLV were found analytically, i.e., the interval of possible values of the BLV for impactors with different shapes of the nose was determined. It was shown that the optimum impactor with the minimum BLV has plane bluntness and its BLV is very close to that for the optimum blunt cone impactor, i.e., a quasi-optimum impactor shape can be selected among the blunt cones. A comparison of impactors with different shapes showed that the optimum impactor has a significant advantage over the sharp-cone and ogive impactors. The shape of the impactor that attains a given DOP with the minimum impact velocity against semi-infinite FRP laminates was found by means of a numerical procedure for solving the corresponding nonclassical variational problem 41. It was shown that the optimum shape depends on the DOP, which is assumed to be larger than the length of the nose of the impactor. For a relatively small DOP, the optimal impactor is a sharp awl-shaped body. With an increase of the DOP, the optimal nose geometry of the impactor changes and becomes close to a blunt cone. It was demonstrated analytically that a plane-nosed cylinder requires the maximum impact velocity so as to attain a given DOP. The performances of the impactors with different head shapes penetrating into SISs was studied. Penetration of monolithic semi-infinite and finite FRP laminates struck by a 3D projectile was studied by Ben-Dor et al. 42, and formulas for calculating the characteristics of penetration and perforation were derived. It was shown that some typical problems of optimization of the shape of the impactor for finite shields and SISs can be reduced to the same variational problem. The study of Ben-Dor et al. 42 predicted the advantage of 3D conical impactors over conical impactors having the shape of a body of revolution. Wen 251 generalized his model given by Eq. 35 to metal, concrete, and soil shields, as well as to eroding penetrators. He proposed that parameter c, which represents the constraining effect of the shield material, be determined either experimentally or by cavity expansion analysis. The parameter e was assigned the meaning of static yield stress for metallic shields and of shear strength for concrete or soil shields. To adapt the model to eroding projectiles, it was assumed that the nose of the projectile could be approximated by a semi-sphere with effective radius R = R, where is an empirical coefficient that equals to 1.6 for steel armor the value 1.55 was given in 252. It was also proposed that the ejection of the shield material from its distal side due to surface effects such as spalling or plugging be taken into account by introducing the effective thickness b =b R, where 0 is an empirical constant. Wen also discussed the suggested models in an additional paper 253. The accuracy of the models 248,249 for various composites has also been analyzed in 254,255. Wang and Chou 256 developed a semi-empirical model to describe the resistance force at each penetration stage by a cylindrical-conical impactor into a glass-fiber-reinforced plastic GRP plate by taking into account the incomplete immersion of the impactor in the armor. The model facilitates the estimation of the energy absorption and the residual velocity of the projectile. Empirical parameters in the model must be determined experimentally. Gellert et al. 257 conducted ballistic tests for cylinders with two diameters and three nose shapes against GRP composite plates of various thicknesses. Reduction of the test data showed that the kinetic energy of the impactor at the BLV can be represented as a two-component piecewise linear function of the shield thickness. A simple model was suggested in order to explain this dependence. Based on the energy balance and taking into account the energy absorbed by the shield and transferred to spallation, Czarnecki 258 developed three semi-empirical procedures to estimate the BLV of composite laminates. For fiber composites, Lee et al. 259 and Song and Egglestone 260 proposed an empirical relationship between the BLV and the areal density, v bl =c 0 A c 1, where c 0 and c 1 are material-dependent constants whose values are given for Spectra-1000, Spectra-900, Kelvar-29, and S-2 glass. Morye et al. 261 developed a simple model for calculating energy absorption by polymer composites during ballistic impact. The energy loss of the projectile was subdivided into three terms, namely, energy absorbed in tensile failure of the composite, energy converted into elastic deformation of the composite, and energy converted into the kinetic energy of the moving portion of the composite. These contributions were combined in the model to determine the BLV of the composite. The required input parameters for the model were determined by physical characterization and high-speed photography. The experimentally measured values of the BLV were found to vary within the range predicted by the model for three examined composite 364 / Vol. 58, NOVEMBER 2005 Transactions of the ASME

11 systems. It was found that the dominant energy absorption mechanism was associated with the kinetic energy of the moving portion of the composites. Kasano 262 studied analytically and experimentally the impact perforation of carbon fiber composite laminates struck by a steel-ball projectile. Two analytical relationships between the impact velocity and the residual velocity identical in form and similar to that determined in 54 were obtained from the conservation laws for energy and momentum. The unknown coefficients in this unified relationship must be determined from impact tests. Prosser 263 studied the perforation of nylon panels by rods with a truncated wedge-shaped leading surface of the nose. He used the model given by Eq. 21 with n=2, a 0 =1, which was based on the assumption of independence of the projectile s energy loss during perforation of the shield on the striking velocity. A linear approximation of the square of the BLV versus the number of layers in the panel was proposed. Parga-Landa and Hernandez-Olivares 264 developed an analytical model for predicting the impact behavior of soft armors against a rigid projectile. They made the following basic assumptions: separation distance between the adjacent armor layers is constant; the projectile moves between the adjacent layers with constant acceleration; the shield material properties remain constant during penetration; there is no friction between the projectile and the armor plate; fiber behavior is linear elastic; and every layer contributes to decelerating the projectile until it reaches its fracture strain. Cunniff 265, taking into account mass, shape, impact velocity, impact obliquity of the striker, and the areal density of the shield, developed a parametric model that facilitates the prediction of the BLV and the residual velocity of a projectile penetrating into textilebased body armor. The model was based on multidimensional nonlinear regression analysis of extensive test data for Kevlar-29. This model was also used to estimate the armor s areal density requirements for protection against anti-personnel mines. This model, describing the flight of a mine fragment in air, assumed that the drag force is a piecewise linear function of the velocity of the fragment. Vinson and Zukas 266 and Vinson and Walker 267 modeled the ballistic impact on textile fabric and fiberreinforced composite shields by a blunt-nosed projectile by analyzing the conical shell formed on impact. Simple computational procedures were proposed to determine the residual velocity of the impactor. Additional analysis of the conical shell approach was given by Focht and Vinson 268. Chocron-Benloulo et al. 269 developed an analytical model to describe impact on textile armor based on the results by Royance et al. 270 and Smith et al. 271 and proposed a failure criterion. Chocron-Benloulo et al. 272 complemented the model with a delamination equation. Billon and Robinson 273 developed an analytical model for ballistic impact on single-layered and multilayered fabric armor. The model employed the assumptions that the rate of change of the projectile s kinetic energy remained constant with increasing penetration depth this value depends on the properties of the material of the layer, and interaction between layers was neglected. This simplified model was used to maximize the BLV of a multilayered fabric armor, and it was found that the BLV reached its maximum value when all the fabric layers in the assembly were manufactured from the material with the highest single-layer BLV. Gu 274 suggested an analytical model, based on the energy conservation law, to calculate the decrease of the kinetic energy and the residual velocity of a projectile penetrating into a multilayered plane plain-woven fabric shield. Walker 275,276 proposed simple models that imply the following expression for the BLV of the fabric: v bl = 1+ c f f 1/ 2 z, = AS 0 m z = / where A is the areal density of the shield, c f is the sound speed of the fiber and f is the failure strain of the fabric break, For the model 276, and 1 are constants and z =1, whereas for the model 275, =4.5, 0.5, z =1/ z 2 2z +3. The BLV of fabrics with resin was calculated as 276 : vˆ bl = v bl 37 where is the mass fraction of the resin in the system and v bl is given by Eq. 36. Additional analysis of the assumptions made by Walker 275,276 was performed by Orphal et al. 277 and Walker 278. Phoenix and Porwal 20 developed an analytical membrane model for the response of fibrous materials against ballistic impact by a blunt-nosed projectile. They noted that many studies in this field were performed in former USSR during 1940s and 1950s and are unknown in the West, and they tried to fill this gap. Hoo Fatt and Park 279 found an analytical solution for the BLV of a honeycomb plate subjected to normal impact by blunt and spherical projectiles, which is based on the three-stages perforation model. 5 Concluding Remarks Despite the capabilities of numerical simulations, the simplified methods retain their importance in ballistic impact dynamics. In situations in which the two types of method compete, the choice is determined by the available resources and the requirements for accuracy and reliability of the results. Very often, there is no alternative to the use of the simplified approaches because the lack of fundamental physical conceptions about the phenomena precludes the use of more exact models. Approximate models are also important because, by allowing us to determine general governing laws in analytical form, they can stimulate new engineering solutions and indicate directions for further theoretical and experimental investigations. Since this review is dedicated to analytical models, the authors set out to classify as far as was feasible the models according to their mathematical description Sec. 2 and to distinguish between the three main classes of model that are used in various modifications for different penetration conditions. The comprehensive analysis of a localized interaction approach in the review allowed us to determine its close connection to cavity-expansion approximations and other methods used in highspeed penetration mechanics HSPM. Since many models employed in HSPM and developed from different physical assumptions happen to be LIM, the review may be expected to stimulate interest in studying LIMs and applying this approach to solving practical problems. The LIT has been successfully used in aerodynamics since the 1960s 21,22,280. In this review, we have demonstrated that some approaches and results in aerodynamics may be applied in HSPM, e.g., MTC and results for shape optimization of projectiles. However, although in aerodynamics the main field of application of the LIT is the calculation and analysis of aerodynamic characteristics of projectiles during their specified motion, in HSMP there are no laws of motion for impactors, and important characteristics, such as BLV and DOP, are determined by solving equations of impactor motion. Thus, the direct similarity between HSPM and LIT applies only for determining forces acting on impactor. Since applied problems in aerodynamics and HSPM are different, it may be concluded that developing a LIT for HSPM is a separate, independent problem. Analysis in the review of publications in this field showed that the number of studies using and developing the LIT concept is growing, although the authors of these publications do not necessarily associate their investigations with LIT. Applied Mechanics Reviews NOVEMBER 2005, Vol. 58 / 365

12 An important problem related to the use of approximate models is the analysis of their accuracy. The majority of studies comprise comparisons between results obtained with suggested or approximate models and experimental results or benchmark calculations. Since the results of such comparisons can be easily predicted, we did not discuss the corresponding sections of those studies in this review. However, although this is formally beyond the scope of our survey, we did mention a number of studies that analyzed models suggested by other authors because we believe that such investigations should be encouraged. Analysis of the publications covered in this review showed that the number of studies dedicated to simplified models in HSPM and their use for investigating applied problems is quite large. There is no reasons to doubt that this tendency will continue within the coming years. axis of cylindrical coordinates associated with the impactor Fig. 2 * radius of a hole in CEA 0 tangent unit vector at a given location at the projectile s surface Fig. 1 1 h, 2 h functions describing part of the impactor s lateral surface that interacts with a shield Figs. 2 and 3 n normal stress at the surface of the impactor n, normal and tangent stress at the surface of the conical impactor, respectively angle between vector n 0 and vector v 0 Fig. 1, 0 functions determining a model in CEA Nomenclature Unless otherwise stated in the text the following notations apply: BLV ballistic limit velocity CCEA cylindrical cavity expansion approximation/approach CEA cavity expansion approximation/approach DLIM degenerate localized interaction model DOP depth of penetration FRP fiber-reinforced plastic HSPM high speed penetration mechanics LIM localized interaction model LIT localized interaction theory MTC method of tangent cones SCEA spherical cavity expansion approximation/approach SFT shield with a finite thickness SIS semi-infinite shield D drag force acting at the impactor H DOP L impactor s nose length R impactor s shank radius S 0 characteristic cross sectional area of impactor a vector with components a 0,a 1,... a 0,a 1,... parameters of models that characterize, mainly, properties of a shield may have different meanings in different models b thickness of the shield Fig. 2 h current DOP, the coordinate Fig. 2 k coefficient of friction between the impactor s surface and a shield m mass of the impactor n 0 inner normal unit vector at a given location at the projectile s surface Fig. 1 p stress at the boundary of a cavity in CEA u =cos v velocity of the projectile v n normal component of the velocity of the projectile v 0 unit vector of the velocity of the projectile at a given location at its surface Fig. 1 v bl BLV v imp impact velocity x axis of cylindrical coordinates associated with the impactor Fig. 2 x, function determining a shape of the impactor n, functions defining the projectile-shield LIM apex half angle of a tangent cone material density of the shield axis of cylindrical coordinates associated with the impactor Fig. 2 coordinate associated with a shield Fig. 2 References 1 Kennedy, R. 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17 Gabi Ben-Dor is a Professor in the Department of Mechanical Engineering and the Director of the Institutes for Applied Research of Ben-Gurion University of the Negev (Beer-Sheva, Israel). His parents survived the Holocaust and immigrated to Israel, where he was born in He completed a B.Sc. degree (1976) at Ben-Gurion University and received a Ph.D. (1978) from the Institute for Aerospace Studies (UTIAS) of the University of Toronto in Ontario, Canada, being awarded the G. N. Paterson Award for outstanding performance in his graduate studies. He has been with the Department of Mechanical Engineering ever since. From 1987 to 1991 he served as the Head of the Department and from 1994 to 2000 he served as the Dean of the Faculty of Engineering Sciences. He has conducted extensive research in Fluid Mechanics and Mechanics of Solids and has been widely published over the course of his academic career, including a monograph entitled Shock Wave Reflection Phenomena, about 200 papers in scientific journals, and 60 chapters in collective volumes. Over 290 papers based on his research have been presented at scientific conferences and symposia. He is a co-editor of a recently published three-volume Handbook of Shock Waves. In 2000, he became the first incumbent of the Dr. Morton and Toby Mower Professorial Chair in Shock Wave Studies. In 2002, he was awarded the Professor Hanin Award for his research on Shock Wave Reflection. Anatoly Dubinsky is a Senior Researcher in the Department of Mechanical Engineering at Ben-Gurion University of the Negev (Beer-Sheva, Israel). He received his M.A. (1971) and Ph.D. (1982) degrees in Mathematics and Mechanics from Moscow Lomonosov State University (Russia). He has authored about 150 papers and three books on Impact Dynamics, Aeromechanics, and Operations Research. Tov Elperin is a Professor in the Department of Mechanical Engineering and Head of the Laboratory of Turbulent Multiphase Flows at Ben-Gurion University of the Negev (Beer-Sheva, Israel). He completed his M. Sc. (1971) at Minsk State University (the former USSR) in Theoretical Physics and received his Ph.D. (1984) in Nuclear Engineering from Ben-Gurion University of the Negev. He has conducted extensive research in Fluid Mechanics, Mechanics of Solids, Heat and Mass Transfer, and Applied Physics. He is the author of over 200 papers in scientific journals and co-editor of the recently published three-volume Handbook of Shock Waves. He holds 13 patents, and over 200 papers based on his research have been presented at scientific conferences and symposia. Applied Mechanics Reviews NOVEMBER 2005, Vol. 58 / 371