Ballistic impact: The status of analytical and numerical modeling

Int. J. Impact Engng Vol. 7, No. 1, pp. 9-35, 1988 Printed in Great Britain

0734-743X/88 $3.00+0.00 © 1988 Pergamon Press plc

BALLISTIC I M P A C T : T H E STATUS OF ANALYTICAL AND NUMERICAL MODELING CHARLES E. ANDERSON, JR a n d SOL R. BODNER* Southwest Research Institute, San Antonio, TX 78284, U.S.A.

(Received 21 October 1987; and in revised form 10 December 1987)

Summary--Simulationof ballistic impact generally fails into one of two categories: analytical models which assume certain dominant physical phenomena and numerical simulations, where the general continuum conservation equations are integrated in time at all points in a spatial mesh to obtain the spatial time history of stresses, strains, velocities, etc., in the projectile and target. Analytical formulations model the mechanical processes over the full field of influence which leads to approximate conservation equations for the entire region in an impact event. These formulations have met with considerable success so long as the assumptions inherent in the model are applicable. Numerical simulations (hydrocodes) compute the wave (shock) interactions important at early times. Mechanical effects, e.g. plugging, erosion, etc., require realistic constitutive modeling which includes failure criteria and failure propagation. This paper gives a broad overview of both analytical and hydrocode modeling and examines the technical issues, uncertainties and potential difficulties for advancement in predictive capability.

INTRODUCTION

Ballistic perforation is an exceedingly complex mechanical process that has been examined for the past 200 years. At the present time, there are three reasonably distinct directions for these investigations: derivation of empirical formulas based on extensive testing, development of relatively 'simple' models of the perforation process and applying the relevant equations of motion and material behavior, and full numerical solutions based on solving all the governing equations over a spatial grid at successive time increments. Because of the computer resources required and the expense in performing a large variety of parametric studies, there has been considerable interest in the intermediate or engineering modeling approach. A number of review articles on ballistic perforation have been published in the past few years. Some of these contain detailed descriptions and give appropriate reference to the various engineering models and numerical techniques that have been proposed up to the time of the surveys. Activity in the field of developing new models of the perforation process is now strong; a number of investigations are currently in progress, and new results are appearing continually in the literature. This fluid situation makes it difficult to present a definitive assessment of the current stage of the subject. As a consequence, all that can be attempted here is to call attention to some of the more readily available survey articles that have been published and to outline general procedures that are being used by various investigators. What will emerge, however, is that a comparison of problem areas facing advancement in engineering modeling approaches has much in common with advances in numerical simulations of penetration mechanics. * Current address: Technion--Israel Institute of Technology, Haifa, Israel. 9

10

('. E. ANDERS¢)N.JR and S. R. BODNI~R PUBLISHED

SURVEYS

Engineering modeling of ballistic perjoration An authoritative and thorough review of the open literature on ballistic perforation was prepared by Backman and Goldsmith [1]. It contains 278 reference citations dating from papers of the 1800s to the time of its own publication. It includes descriptions of the different physical mechanisms involved in the penetration and perforation process. A number of engineering models that have been proposed are outlined, and an indication, where possible, of their range of applicability is given. This article is probably the most complete and readily available single source of information on ballistic modeling up to 1978. A very useful set of papers is to be found in a special issue of the International Journal of Engineering Science [-2] that was also published in 1978. Two papers in that issue are particularly concerned with engineering modeling in the ordinary ballistic range, see Refs [3,4]. The reference lists in these papers complement that in Ref. [1]. Two books have been published in the past few years which include extensive discussions of the ballistic perforation problem. One is the volume entitled The Physics of Deformation and Flow [5]. The other is the book Impact Dynamics written by Zukas et al. [6]. Descriptions of a number of engineering models are presented in both books. Billington and Tate give a detailed discussion of some of the earlier models for perforation of thin ductile plates, particularly those of Bethe and Taylor. There is also a clear description of the energy method of Recht and Ipson [7] for moderately thick plates that experience plugging. A section of their book also is devoted to penetration due to very high velocity impact. This problem is currently of major interest. Regarding the book Impact Dynamics, the chapter by Zukas entitled "Penetration and Perforation of Solids" (pp. 155-214) is a widely encompassing review of experimental techniques and analytical modeling of ballistic perforation. The reference list at the end of that chapter is a useful guide to the literature and includes reports and papers in conference proceedings that are not generally known. In summary, the four references described above serve as the major source of information on the ballistic perforation problem at the present time. They contain fairly complete descriptions of the various engineering models that have been proposed. Some of the basic features and open questions on these models will be discussed in following sections.

Numerical simulation of ballistic penetration~perforation Hydrocodes are large computer programs that can be used to simulate numerically highly dynamic events, particularly those which include shocks, by approximating a continuum in a pointwise (finite difference) or piecewise (finite element) manner, and then solving the conservation equations coupled with material descriptions. Hydrocode enhancements have evolved as a result of the need of researchers to improve predictive capability. These enhancements have been a combination of numerical techniques and the inclusion of physical models. Over the last 20-25 years, a great deal of effort has gone into providing the models and algorithms to simulate the response of real materials under high rates of loading. An equation of state is necessary to account for the resistance to hydrostatic compression, the increase in internal energy due to thermodynamically non-reversible processes (e.g. shock loading, plastic work) and phase transitions (both solid-solid as well as melting and vaporization). A constitutive model is required to account for strength effects. Typically, a von Mises yield criterion is used to compute the onset of plastic flow. Strain hardening, strain rate effects, thermal softening, etc., can be accounted for by modifying the yield stress (reviewed by Anderson [8]). Often, though, the data necessary for calibrating the coefficients of the constitutive model are not known for the high strain rates typical of a penetration event so that an elastic-perfectly plastic constitutive model is commonly used for numerical simulations. For ballistic impact problems involving ductile materials, elastic effects are generally unimportant and a rigid-viscoplastic material model would seem to be an appropriate characterization. Although hydrodynamic computer codes have been used to simulate impact problems since 1958 [9], few results have been published in the open literature. Most of the problems

Ballistic impact: analytical and numerical modeling

11

analyzed are documented either in government reports or presented at specialized conferences such as the International Ballistic Symposia. In the last 10 or so years, there have been several review articles which discuss and compare features of various one-, two- and three-dimensional hydrocodes [3, 6, 10-13,]. Anderson ['8-] has written a review article on the bases and use of hydrocodes, and Johnson and Anderson [14] discuss application of hydrocodes to hypervelocity impact. To the authors' knowledge, penetration mechanics problems addressed by hydrocode analyses which exist in the open (peer reviewed) literature consist only of the papers by Bjork [15], Riney and Halda [16], Wilkins and Guinan [-17], Mescall and Papirno [18,], Bertholf et al. [19-], Misey [-20-1,Johnson [-21], the special issue on Penetration Mechanics edited by Eringen [2] and the special issue of the International Journal of Impact Engineering [22]. The articles by Wilkins [23] and Jonas and Zukas [3] are concerned with armor-type applications; Byers [24-] examines earth penetration; and Sedgwick et al. [25] discuss both hypervelocity impact and impact at ordnance velocities. The papers in the Anderson et al. volume [22] deal exclusively with hypervelocity impact. As already mentioned, many more papers exist in proceedings of technical meetings as well as in Government reports. A number of papers concerning shock propagation and material response modeling have been published which have used hydrocode simulations. Additionally, a number of papers exist in the literature where impact problems are used as examples, but the emphasis is on the algorithm development, such as those coauthored by Gordon Johnson for the EPIC family of codes, and by John Hallquist for the DYNA family of codes (see [8] for references). The above references are reasonably comprehensive with regard to the use of hydrocodes for impact problems. Objectives This paper will concentrate on giving an overview of progress in ballistic modeling. First, the discussion will focus on engineering models for penetration and will highlight essential features of these models which have become quite sophisticated in recent years. Then, a review of some of the major advances in hydrocode modeling of penetration problems will be given. Next, a discussion of current problems of interest will show where further enhancement is required both for the engineering models as well as for numerical (hydrocode) simulations. ENGINEERING MODELS Basic models: thin plates, rigid projectiles The early models of the ballistic perforation process concentrated on single deformation mechanisms and rigid projectiles. Since these models assume no stress or deformation gradient through the target thickness, they are essentially applicable to relatively thin target plates (compared to the projectile diameter). Prominent among these are the models of Bethe and Taylor who calculated the plastic work expended in perforation of ductile plates. In particular, they computed the work done in expanding the hole to its final radius; the difference in the two models lies mainly in the stress distribution about the hole. Taylor also examined the work done in dishing (see pp. 527-531 of [5-]). Thomson [26] extended Taylor's analysis to account for the kinetic energy imparted to the plate material. A number of other investigators have taken the same general approach and have examined in detail the plastic work and the kinetic energy of target material associated with the perforation process for various possible deformation configurations. Elastic-plastic cavity expansion as the primary dissipative mechanism has been recently investigated by Forrestal and colleagues [-27] with applications to geological and soft metal targets. The particular case of petalling of the target plate has received renewed attention in recent years, and an extensive study for this deformation mode has been reported by Landkof and Goldsmith [28]. For thin plates, the nose shape of the projectile, the impact velocity range and the ductility of the target material are important factors on the geometry of the perforation mode. Because of the variability in the geometry, some experiments are generally required to

12

C, E, ANDERSON, JR and S. R. BODNER

determine the applicable perforation mode for a given situation. Single mechanism analytical models corresponding to the observed failure can then be used to calculate the velocity loss in perforation which is usually the main parameter of interest. For corresponding perforation mechanisms, these simple models can generally supply reasonable predictions of the velocity drop for cases of rigid projectiles and relatively thin plates.

Basic models: moderately thick plates, rigid projectiles For moderately thick target plates, i.e. those having thicknesses up to several projectile diameters, there is generally more than one physical mechanism effective during the penetration process. In these cases, plugging, i.e. ejection of a cylindrical or conical mass of target material, usually occurs at some later stage of the process. If the target plug m a s s m t b is known, then the residual velocity Vr can be determined by a combined energy and momentum balance analysis developed by Recht and Ipson [7]: M

2

(1)

where M is the projectile mass and Vo is the initial impact velocity. This is a useful formulation, but depends on knowing, in addition to the plug mass, the ballistic limit VL which is defined as the minimum impact velocity for perforation, or the maximum impact velocity for which the residual velocity is zero. Alternative definitions for VL can lead to different results in actual experimental work (which also require further statistical definitions), but they reduce to the same value in analytical formulations. In principle, both the plug mass and the ballistic limit can be obtained experimentally, but the objective of ballistic modeling is to obtain formulations that are as self-contained as possible with minimum recourse to empirical quantities. A second step in generating a model for moderately thick plates was provided by the onedimensional, three-stage perforation model of Awerbuch and Bodner [29, 30]. This model considers three interconnected perforation stages with an initial stage characterized by compressive and inertial resisting forces, Fc and F~, followed by a second stage which adds a shearing force, Fs, and a third stage of plug ejection, Fig. 1. The equation of motion of the perforation process in the direction of motion is: d

dt (my) = - (F i q- F c + Fs).

(2a)

Assuming, in a one-dimensional context, that a plastic zone of target material in front of the projectile moves with the same velocity as the projectile, then the equation of motion becomes: dv

prAy2 + (m° + PtAz)v dzz = - (Fi + Fc + F~),

(2b)

where m0 is the projectile mass, Pt is the target density, z is the distance from the target impact surface to the front of the moving target material and A is the cross-sectional area of the penetrator. The compressive force (Fc) is required to bring the target material to the inelastic condition; the inertial force (Fi) is that required to bring the target material from rest to the projectile velocity; and the shear force (Fs) resists relative movement between material forward of the projectile and the bulk of the target. All three resisting forces are operative during stage 2. The compressive force decreases to zero at the end of stage 2, and only the shear force is operative in stage 3. The Fc term is proportional to the target strength and the F i term is proportional to v 2. The shear force appears as a constant term if the shear strength is taken as a constant, but if strain rate effects are important, the shearing force also includes a term proportional to the velocity. Equation (2b) is therefore in the form of a generalized Poncelet equation.

13

Ballistic impact: analytical and numerical modeling

,- Added Target ~-Z-mZ Mass r~. I--'lillll|l

2Rp

i

(a) First Stage

~r

: Io2Fz

(b) Second Stage

Ar

~rrZ

I

Dq

Iq -,....~

(c) End of Second Stage

~r

(d) Third Stage

FIG. 1. Awerbuch-Bodner model.

A plastic zone moving with the projectile velocity is assumed to exist forward of the projectile which leads to the additional inertial terms in the equation of motion. These are intended to account for both longitudinal (ptAgv dv/dz) and radial flow effects (ptAv 2) within a one-dimensional formulation. An interpretation of the term prAy 2 in equation (2b) is that it is due to the convective transfer of target material in the radial direction from the projectile path (which cannot be modeled explicitly in a one-dimensional framework and enters as part of the variable projectile mass effect). In other one-dimensional analyses, the constraint to radial flow in the target is modeled by increasing the uniaxial stress material strength by some constant factor, e.g. Ref. [31], while the actual material strength is used for Fc in the Awerbuch-Bodner model. Predictions based on this model have shown reasonable agreement with tests, but require some empirical factors, in particular the plug length, due to the inherent limitations of a one-dimensional model. An extensive discussion of the Awerbuch-Bodner perforation model was published by Nixdorff E32, 33] who showed that the predicted residual velocity is a special case of a general form proposed by Lampert and Jonas [34]. This general form is also described by Zukas E6], and is consistent with the energy balance procedure of Recht and Ipson E7]that is, Nixdorff showed that integration of the Awerbuch-Bodner force equation leads to the energy solution of Recht-Ipson. A method to couple the Awerbuch-Bodner model with overall structural deformation was developed by Marom and Bodner [35] using a redefinition of the ballistic limit and a generalization of the energy balance equations of Recht and Ipson E7]. Other treatments of the coupling of ballistic perforation with overall structural deformation are described in the review article by Backman and Goldsmith [1]. More recent papers on the topic are by Corran et al. E36] and Shadbolt et al. [37]. An essential difficulty of the formulation of Awerbuch and Bodner and similar analyses in the literature is that they are one-dimensional treatments of what is basically a two-

14

c.E. ANDERSON.JR and S. R. BODNER

dimensional mechanism. That is, the deformations and stresses vary in the radial as well as in the longitudinal (projectile path) direction. A two-dimensional analysis is inherently more suitable for the ballistic perforation problem, but is an order of magnitude more complicated. A discussion of a two-dimensional ballistic model is given in a later section of this article. However, because of the relative simplicity and success of the one-dimensional models, they have been used as the basis for further investigations involving deformable projectiles.

Models Jor moderately thick plates, deformable projectiles In most cases of ballistic perforation, there is deformation and flattening of the projectile due to the high resisting forces. This change of geometry generally causes further increase in those forces, and the interaction between projectile deformation and penetration is an important aspect in the development of analytical models. In Awerbuch and Bodner [28], for example, the cross-sectional area of the projectile is taken to be an empirical function of the longitudinal co-ordinate. It is usually sufficient in practical use of that model to consider a mean cavity diameter based on averaging the entry and exit holes. For a given projectile and target plate, it is generally possible to extrapolate limited test results for the cavity size and shear zone extent over a wider range of impact velocities and target thicknesses. Dehn [38, 39] also accounts for mushrooming in a phenomenological manner. Furthermore, he treats finite thickness effects in a similar phenomenological manner [38], where an average parameter, interpreted as the target thickness at the moment of failure, is used analogous to an average presented area (i.e. 'mushrooming'). However, these empirical or semi-empirical approaches are limiting, and it is desirable to formulate ballistic models that are as deterministic as possible. Most efforts to determine projectile deformation analytically are based on Taylor's 'mushrooming' theory of a deformable rod normally striking a rigid surface. The theory utilizes plastic wave theory and the plastic incompressibility condition to obtain the residual deformed shape of the projectile. A detailed description of Taylor's theory and of subsequent analytical and experimental work on the problem appears in the book by W. Johnson [40], pp. 229-249. Wilkins and Guinan [17] combine careful experimental work with numerical simulation (hydrocode analysis) of Taylor anvil tests. Analytically, Taylor's theory was extended by Lee and Tupper [41] and more recently by Recht [4]. Recht applied his results to predict the deformed shape of realistic projectiles subsequent to impact, but he did not couple the theory with a ballistic perforation model. Recht's work is particularly useful in giving insight into the very early features of ballistic impact which are usually ignored in most ballistic modeling work. There are some recent and ongoing investigations that attempt to couple the projectile mushrooming effect with a one-dimensional perforation model. Woodward [42] treats the problem of a deformable projectile striking a thick target. The analysis is based on considering projectile and target to be one-dimensional impacting rods with the radial constraint effect on the target compensated by an enhanced flow stress value. Agreement of the model with various test results appears to be reasonable. Moderately thick target plates are of major practical interest, and rear surface effects have an important role in the overall perforation process. Some investigations have generalized the Awerbuch-Bodner model to obtain an analytical model in which all the important response parameters, including the cavity size and the shear zone length, are determined as part of the formulation. Liss et al. [43] examined the plugging mechanism in some detail for the case of a rigid striker, and also considered local structural deformation of the target plate. Wenxue and coworkers [44] studied both the mushrooming effect as well as plugging. The main drawback of these investigations is that they are still limited by the essentially onedimensional treatment of the problem since many of the effects cannot be properly represented in a one-dimensional formulation.

Ballistic impact:analyticaland numericalmodeling

15

Modeling for high velocity impact

Higher impact velocities, i.e. greater than about 1 km s- 1, tend to generate hydrodynamic effects which include erosion of both projectile and target material. A one-dimensional model due to Alekseevskii [45] and to Tate [46, 47] has become the standard reference in the high velocity regime. Their equation takes the form of a modified Bernoulli equation: 1

1

ptu 2 + T = ~ pp(V - u) 2 + Y,

(3)

where T is the strength of the target (usually on the order of three times the uniaxial ultimate tensile stress); Y is the strength of the projectile (taken as the uniaxial ultimate tensile stress); Pt and pp are the target and projectile densities, respectively; v is the speed of the rear of the projectile; and u is the penetration speed. Figure 2 depicts a representation of the hydrodynamic-like flow. In effect, the force acting on the projectile to decelerate it is given by Y, so the equation of motion is: (4)

ppl ~tt = -- Y'

where I is the current rod length. The length of the rod changes since v and u are different. Jones et al. [48] have modified Tate's work to incorporate mushroom-type deformation at the impact end of the rod, and deceleration at the opposite, rigid end. The latter effect leads to increased penetration over the standard Tate theory as a consequence of decreasing the erosion rate. However, front-end mushrooming usually more than offsets this gain so that the penetration depth is reduced. The 'modified Bernoulli' model serves as a useful guide to the penetration process, but has a number of limitations as discussed by Wright [49, 50] and others. Among other restrictions, the strength-modified hydrodynamic models apply to thick (mathematically semi-infinite) target plates and do not account for the influence of the rear surface of the plate. In addition, the projectile and target strength terms are treated as constant values which cannot be determined within the context of those analyses. A direction of current ballistic modeling research is to attempt to couple the modeling work done for intermediate thickness plates in the standard impact velocity range with the

Target

U

~

I I

V--U

I I I

Pp

Ih

T

I

I

I

FIG. 2. Hydrodynamicpenetration.

16

C. F~.ANDERSON,JR and S. R. BODNER

hydrodynamic effects at the higher velocities. An interesting 'unified' treatment of the ballistic perforation process has been developed by Dehn [39] based on a one-dimensional equation for the resistive force as a function of velocity. Using this equation, the equation of motion can be developed to include projectile mass erosion. The resulting solutions provide good guidelines on the penetration characteristics of a wide class of projectiles including long rod penetrators and shaped charge jets.

Two-dimensional modeling of ballistic perforation A two-dimensional ballistic model for intermediate thickness target plates struck by blunt-nosed, rigid projectiles was published by Ravid and Bodner [51]. The basic procedure of that model is somewhat analogous to that used for problems of punch indentation in rigid-viscoplastic media, namely, determination of the flow field consistent with the geometric constraints thereby obtaining the required indentation force. In the Ravid-Bodner model, inertial forces of displaced target material are considered as well as a succession of deformation modes due to the finite thickness of the target plate. A variational theorem for dynamic plastic deformations is used to determine the longitudinal and radial extent of the plastic flow zone in the initial stage. In all, a five-stage sequence of deformation modes is used in the analysis which includes bulging of target material at the rear surface and plugging as the final exit stage, Fig. 3. Overall structural deformations and projectile deformability are not included in this analysis. The motivation for the two-dimensional treatment with its added computational complications was to develop a model with a minimum of physical ambiguities and empirical factors which was capable of further generalization. In the two-dimensional analysis, the position and motion of the target material are completely determined at all times during the perforation process and empirical information is not required. The assumptions required on the plastic zone formation and velocity in the one-dimensional treatment are therefore not needed. In addition, the target material flow stress, modified for strain rate sensitivity, is used in a direct manner since the effect of the radial constraint on target deformation is part of the formulation. For the two-dimensional penetration model, various failure modes are considered to be potentially operative upon the onset of the second deformation stage. These include plugging during stages 2 and 3 of a truncated conical segment caused by relatively brittle failure in shear (a strain limit) or by adiabatic shear band formation. Reasonably ductile target plates have an alternate deformation mechanism during stages 2 and 3, failing by a limiting strain criterion in stage 4. There can be variations in the detailed geometry of the cylindrical plug which forms during this type of failure [52]. In principle, the failure mechanism should be automatically determined by the associated computational procedure. However, if ductile shear plugging is the mode of failure, some simplification can be achieved by choosing the detailed geometry of the plug from the variety of possible cases. Predictions of the residual velocity of projectiles for different types and thicknesses of steel alloy target plates obtained from the three-stage Awerbuch-Bodner and the five-stage Ravid-Bodner model are given in Fig. 4. Also plotted are data from ballistic experiments. The projectiles were armor-piercing rounds where only the (non-deforming) hard core of the rounds was considered for the calculations. The impact velocities of the projectiles were approximately 845 + 10 m s- a for all tests. It is seen that the self-contained five-stage model is in excellent agreement with the ballistic data and gives considerably better agreement than the partially empirical three-stage model. A few subtle points are worth mentioning concerning Fig. 4. Two different exit velocities are computed for the 9 mm target, representing different plugging mode geometries. In one case, the mode is a cylindrical plug with an ogival nose, while in the other case, the mode is a full cylindrical plug; a combination of the two failure modes was observed in the experiment. Also, various steel alloys were used for the target plates in the examples of Fig. 4. The large difference between the exit velocity of the 8.0 and 8.1 mm plates is due to the much higher yield stress of the 8.1mm plate. Further discussion concerning the predictions and experimental results is given in Ravid and Bodner [51].

17

Ballistic impact: analytical and numerical modeling R

STAGE I Dynamic Plastic Penetrotion

,r" -;

Final

H'~I~/1 i R

STAGE IgeFormations... t.~~_~J~ /Final II

STAGE tTt

.-

Vle t ~ ProjSTAGE ectiExi

~"T~'% R

,FinExi al t

I~ Plug FIG. 3. Ravid-Bodner model.

Using the same general approach as that in the Ravid-Bodner analysis, Westine [53] has carried out a simplified treatment of the perforation process by assuming that the plastic velocity flow field is hemispherical and depends only upon the radial distance from the penetrator nose which is also assumed hemispherical. There are fewer stages in his work and two failure models are treated in a one-dimensional manner: petalling and strain to plugging. In addition, procedures for including projectile erosion effects for high velocity impact conditions have been modeled in a quasi one-dimensional manner assuming that the flow fields of the eroding penetrator and deforming target do not interact. The target and penetrator are coupled through a hydrodynamic approximation similar to equation (3). Current work is allowing for a variety of penetrator nose shapes; it has been found that the simplified flow field originally assumed leads to physically unacceptable solutions, and a description similar to that of Ravid and Bodner is being adopted in further work.

18

C E. ANDERSON, JR and S. R. BODNER

•

Awerbach-Bodner (3 Stage)

•

Ravid-Bodner (5 Stage)

I

Experiment

700

E

600

-'r"

g > 500

t

400

300

t

I

I

I

I

I

6

7

8

9

10

11

12

Plate Thickness (mm)

FIG. 4. Comparison of Awerbuch-Bodner and Ravid-Bodner models to experimental data.

A combined experimental and computational study by Rand et al. [54] tends to support the formation of a plastic flow field during penetration. A polycarbonate projectile was fired into a semi-infinite polycarbonate target at impact velocities greater than 1.5 km s -1. A permanent 'bubble' was observable in the target in all the experiments. Since polycarbonate is a birefringent material, its index of refraction is sensitive to the stress state. Analysis of hydrocode calculations led to the interpretation that the bubble is a visualization of the extent of the plastic zone field. In the paper by Ravid et al. [55], the details of the initial stage of very high velocity impact (i.e. up to 5 km s- 1 for most metals) are examined. This two-dimensional analysis considers the effects of shock waves, their subsequent reduction due to rarefaction waves, and ensuing plastic deformation of the components leading to mushrooming of the projectile and the development of a target cavity wider than the projectile. Figure 5 depicts the assumed shock and the plastic flow field slightly after impact. This initial impact stage terminates when relief waves 'catch' the initial shock along the centerline of the projectile. Further work [56] has modified the previous work to account for the case of relatively soft projectiles that could experience deceleration during this initial stage. Subsequent penetration stages would be required to consider hydrodynamic and projectile erosion effects in addition to the rear surface effects of bulging, plug formation and breakout. Development of such an all-inclusive model of the ballistic perforation process is one of the objectives of the analytical modeling procedure. O'Donoghue et al. [57] have taken one of the two examples given in Ravid et al. [55], and used a hydrocode to solve the complete set of field equations, using the same material parameters as the analytical example. Preliminary examination of the computational results shows that excellent agreement exists between almost all the numerical and analytical results. The largest discrepancy is in the extent of the crater diameter which appears to be due to the assumption in the analytical model (to keep the algebra tractable) that the interface between the projectile and target remains 'straight' (Fig. 5).

Ballistic impact: analyticaland numerical modeling

°n°leeci Tit

19

R0

ShOCk

H

front

o

Shocked

region

c

.

FIG. 5. Shock phase and plastic deformation. The idealized penetration of a rigid projectile into a semi-infinite deformable target, assuming that the rate of penetration and all flow fields are steady as seen from the nose of the penetrator and that no shear stress is transmitted across the target-penetrator interface, has been examined by Batra and Wright [58], and Batra [59, 60]. The initial study [58] assumed the target was rigid-perfectly plastic, and the projectile had a hemispherical nose. The other two studies have used different constitutive models for the target (viscoplastic and thermoviscoplastic, respectively), and also examined the influence of the nose shape. Recent work has examined the steady state deformation of a perfectly plastic rod striking a rigid target crater [61], which quantifies the amount of projectile material that has turned and is flowing in the opposite direction as a function of the ratio of specific kinetic energy to penetrator uniaxial tensile flow stress. The purpose of these studies has been to examine the kinematics and associated stress fields which could then be used to devise and/or check simpler engineering theories. For example, Batra and Wright [58] found that target material adjacent to the sides of the penetrator extrudes rearward in a uniform block that is separated from the bulk of the stationary target by a narrow region with a sharp velocity gradient. This is a partial and independent confirmation of the velocity field assumed by Ravid and Bodner [51] in their work.

Engineerin9 models--closure The point could be raised whether developing more complicated models of ballistic perforation is worthwhile since it may appear to bring the procedure closer to that of the full numerical solution. Aside from the benefit of having a better physical understanding of the phenomenon, there is still a large gap in the computational effort required between the two approaches. Computational solutions based on both Bodner's and Westine's models can be run on a microcomputer in a matter of minutes, while a full numerical solution can take hours of computational time on a large computer. For parametric studies as well as for a large class of engineering problems, the analytical modeling approach to the ballistic perforation problem is a very useful tool. COMPUTATIONAL

MODELING

The first numerical simulations of impact and penetration investigated hypervelocity impact. Early formulations did not include strength effects; thus, metals were treated as a fluid, with no viscosity, and the expression 'hydrodynamic computer codes' was used to refer

20

C, L ANDERSON~JR and S. R. BODNER

to these computer programs. Johnson and Anderson [14] give a short history on the evolution and development of hydrocodes, particularly with respect to impact problems. It has been found that in simulating numerically a penetration event, as well as for many other dynamic scenarios, a methodology must be adopted where the 'continuum' is modified to permit fracture, stress relaxation, the introduction of new free surfaces and other manifestations of material failure. Anderson [8] reviews failure modeling using hydrocodes and, in particular, discusses failure modeling as applicable to spallation and in the Lagrangian modeling of deep penetration. The models most used to predict material failure in hydrocode simulations require some critical stress, strain, plastic work or other instantaneous criteria, since some appropriate critical level can be estimated from known material properties. However, with damage initiation, a time-dependent description of failure growth is required (with a threshold below which no damage occurs); these models predict increasing damage rates as the threshold is exceeded. If the threshold is exceeded by a sufficiently large quantity, the damage rate saturates and the model effectively can be replaced by a single criterion, sudden failure model. To date, computational modeling of failure has essentially been limited to only a single failure mode. The primary reason for this limitation is that modeling fracture computationally requires a model for material failure as well as a method for representing failure and failure propagation in the calculation grid. The latter requirement has probably been the dominant difficulty in numerical simulation, though this should not be interpreted as underestimating the difficulties in understanding the mechanics of the failure process. That failure is very important to consider is demonstrated by the work of Misey et al. [62]. They used both Eulerian (HELP) and Lagrangian (EPIC-2) hydrocodes to simulate the normal impact of a long steel rod into rolled homogeneous armor at 1000 m s- t. Hauver [63,64] developed a procedure for measuring the strains in a long rod. Strains were computed at three positions along the rod in the numerical simulations and compared against the experimental data. The computational results from both codes showed good agreement with the experimental data during the elastic phase of the deformation. However, with the onset of plastic deformation, agreement was dependent on the material failure criterion incorporated in the code. Lack of a failure condition produced excessively large strains 20 mm from the impact end of the rod (47 Yostrain compared to the experimental data of 16 Yostrain) at 20/is, Fig. 6. Using the HELP code, fairly good agreement is obtained with a simple failure-in-tension model until approximately 18 #s, after which the calculated strain deviates sharply from the experimental data. EPIC-2 was used for a third calculation where failure was allowed when the equivalent plastic strain exceeded 100~o; relatively good agreement with the experimental data over the complete 20/~s is observed. The authors 50

/ EPIC-2 Strains without Failure

i

40

/

EPIC-2 Strains with Failure

ra,tOata

HELP Strains w i t h Failure =~ 30 v

/

/ f

/J

.=_ 20

/

10

I 0

4

8

12

16

20

Time (~s)

FIG. 6. Compressive strain time record at 20 mm gage position (0-20 ,us) (after Misey et al. [62]).

Ballisticimpact: analyticaland numericalmodeling

21

report that similar results were obtained for gage locations 40 and 60 mm from the front of the penetration. These results are significant in view of the fact that the strains are compared 20 mm from the projectile-target interface, indicating that the results are dependent on material failure which occurs at or close to the interface. Large deformations and material failure are generally prevalent in a penetration mechanics problem. Simulating the fracture and separation of material in a continuum code has been a particular challenge. Several innovative numerical techniques have been implemented over the last several years to simulate material failure. First, however, it is instructive to review the types of failure that can occur during the penetration and perforation process. Failure modes

Material failure can be divided into two main categories, namely that attributed directly to wave propagation, i.e. transient effects, and that caused by continuous loading. The engineering models discussed in the previous sections generally account only for the second category. Hydrocodes, on the other hand, simulate both the wave propagation and the continuous mechanical effects. Failure mechanisms can be further subdivided with three types usually evident in a penetration event of moderately thick targets: spall, plugging and erosion. At impact, compressive waves propagate into the projectile and target. Rarefaction waves are generated in the presence of free surfaces to satisfy the condition of zero stress at a free surface. These rarefaction waves can cause the material to go into tension and, if the tensile stress is of sufficient amplitude and duration, the material will fracture. The timeframe for the wave propagation effects in chunky type projectiles is on the order of the projectile length or the target plate thickness divided by the sound speed. For typical fragment sizes, wave propagation effects are completed in the first 10/~s or so. Tensile failure can be modeled fairly well with hydrocodes. Accumulative damage models such as the Tuler-Butcher model [65] and the Cochran-Banner model [66] permit damage to accumulate as a function of applied stress and duration. The parameters necessary to 'calibrate' the models are not too difficult to obtain with controlled flyer plate experiments. However, if the interest is in modeling damage near conditions of damage initiation, then zoning of the problem must be sufficiently fine in order to provide good stress resolution within the grid. For example, Bertholf et al. [19] found that the numerical simulation did not predict spall if the finite difference zoning were too coarse. Better spatial resolution, however, did predict spallation in agreement with experiments. The difficulties in providing good stress resolution should not be minimized, particularly for multidimensional problems. Doubling the grid resolution increases the computational run time by a factor of 8 for two-dimensional calculations and a factor of 16 in three-dimensional calculations. Failure models used in hydrocodes range from a simple 'clamp' on the tensile stress, usually called Pmin, to accumulative damage modeling. Historically, failure has been difficult to implement because of the lack of algorithm development for tracking failure propagation on the computational grid. Thus, many problems are simulated using only a Pmin criterion. A more realistic damage criterion sets all tensile and shear stresses to zero within a computational zone when a maximum stress in tension is reached; however, the zone can support hydrostatic compression. Bertholf and Kipp [67], developed a more sophisticated biaxial failure model where the accumulated plastic work for each zone was monitored. If the plastic work exceeded a critical value for the material, and if the principal stress state was such that tensile yielding was occurring, then the material failed perpendicular to the direction of the maximum normal stress. The partially failed material could no longer support shear stress in the plane of the failure or tensile stress normal to the failure. If further loading resulted in failure normal to this direction, then the material was assumed to have failed completely and could only support hydrostatic compression. Micromechanical based models for the nucleation and growth of voids have been developed by SRI International researchers [68]. Physically based criteria prescribe the

22

C, E. ANDERS()N.JR and S. R. B~,)DNER

initiation of ductile, brittle or adiabatic shear failure. The propagation of failure in the form of the nucleation and growth of voids, cracks, or shear bands then is prescribed by experimentally determined rate equations. Sedgwick et al. [25] used the nucleation and growth model to compute spall; and Erlich et al. [69,70] used the model to compute fragment size distributions for fragmenting explosively loaded cylinders and shear banding for armor penetration by a projectile. The difficulty in these micromechanical models is experimentally obtaining the micromechanical dynamic fracture parameters required by the models. As already discussed, failure can result either by tensile stresses induced by rarefaction waves during wave propagation or from the continuous loading of the projectile on the target. What has emerged in recent years for metallic targets is the use of either immediate damage criteria (e.g. a maximum stress in tension) or damage accumulation models to describe spallation, and a simple strain to failure criterion for failure during continuous loading. The strain to failure criterion will be discussed in the paragraphs below. The failure mechanism generally treated with Eulerian codes is tensile failure. If a cell goes into tension beyond some prescribed fracture stress, then a void is introduced into the cell to relax the stress back to the limiting tensile value. This void is treated as 'material', and the cell void volume is advected similar to other materials. Interior cracks and spall can be modeled in this manner. In principle, the fracture criteria can also be written in terms of the maximum shear stress as opposed to a maximum tensile stress with a void being introduced for stress relaxation. Accumulative damage modeling, however, is intrinsically more difficult since a damage number must also be advective. The Eulerian code H E L P has been used to simulate plugging failure using a maximum plastic work value as a failure criterion. After failure, a slip surface is permitted to propagate through the target in the direction of the maximum shear stress [25]. Considerable algorithm enhancement of the computer code was required to permit tracking of the damage. To date, the most sophisticated material constitutive modeling has been done with Lagrangian codes, since Lagrangian co-ordinates track the material point history within a computational zone. (The one exception has been H E L P ; in H E L P , tracer particles are used to define material and free surfaces, and are introduced to track damage.) In Lagrangian codes, stress can be computed directly as a function of strain, strain rate, internal energy and damage, e.g. Ref. [71]. Plugging and erosion have been modeled using algorithms for dynamically redefining slide surfaces; these will be described below. Plugging tends to be the principal mode of target failure, while erosion is an important projectile failure mode at high ordnance velocities. Numerical modeling of these failure mechanisms can ultimately be traced back to the modeling of material response. A discussion of plugging and erosion modeling can highlight certain success with predictive modeling, and also indicates some uncertainties in current understanding. This discussion of plugging and erosion will focus on the response of metals.

Dynamic relocation of sliding interfaces Sliding interfaces are used in numerical simulations when large relative motions are expected at material boundaries, such as the interface of an explosive and metal case. They are also usefully employed where large shears or fractures develop. In their initial formulation, sliding interfaces were specified at the beginning of a problem and remained unchanged. Methodologies have now been developed to permit the dynamic relocation of the sliding surface [72-78]. First, a suitable material criterion establishes failure of a computational zone. Upon failure, the element can no longer support stress or pressure (total failure) and all forces acting on the element are set to zero while only the mass is retained. The sliding surface is then redefined to follow the failure of the element. A failure strain criterion that has been found useful is setting the equivalent (von Mises) strain ~ equal

Ballistic impact: analytical and numerical modeling

23

to some prescribed 'ultimate' strain, where: e =-3-

-

+

-

+

-

(5)

1/5

and the e~s are the principal strains.

Plugging Ringers 1-79,80] has studied plugging using the finite-element code EPIC-2 and has compared computational results against experiments performed by Woodward and coworkers [81,82]. Impact by steel cylinders into Ti 125 (99 ~o pure titanium) and Ti 318 (a titanium alloy--90 ~o Ti, 6 ~o A1, 4 ~o V) resulted in plugging failures in both cases. Figure 7 depicts numerical results for the impact of the steel projectile on the two different titanium targets (6.35 mm thick) at 10/~s with all failure criteria suppressed. The impact velocities were 350 m s- t against the Ti 125 target and 450 m s- 1 against the Ti 318 target; these impact velocities are slightly above their respective ballistic limits. Properties for the two titanium targets are given in Table 1. A comparison of the two figures indicates that Ti 318 is stronger than Ti 125 due to its higher strength; however, experimental results show that both targets are defeated at these respective impact velocities. The reason for the large differences in response is that the failure mechanism for Ti 318 is different from that of Ti 125, and occurs at an earlier stage in the perforation process. The failure for the Ti 125 target is considered to be the direct result of strain limitation, while the failure ofTi 318 is considered to be the result of formation of adiabatic shear bands.

.32

-Iv'-u"-,P

',.v" ~.~,

= 350 m/s

V

o x

~ ~

~,

go~i N

.32

-

x

~

-.16

~o

~

N

k

-.16

X -32

-.32

N I

0

I

I

I

.16

I

.32

R AXIS

(m x

I

1

•48

!

I

.64

0

I0")

I

.t6

I

I

.32

R AXIS

I

.48

I

1

.64

(m x 10-')

Steel Cylinder versus Ti 318 Target

Steel Cylinder versus Ti 125 Target

FIG. 7. Impact by a blunt steel projectile into titanium targets (from Ringers [79]). TABLE 1. TITANIUM TARGET PROPERTIES (AFTER RINGERS [74, 79]) Ti 125

Material Density Yield Ultimate tru eu Young's modulus Ballistic limit (6.35 mm target)

99~o Pure Ti 4.51 x 103 kg m - 3 522.5 M P a 600 M P a 0.45-0.5 1.158 x 105 M P a 300 m s - 1

Ti 318 90~o Ti, 6 ~ AI, 4 % V 4,43 x 103 kg m - 3 1029 M P a 1209 M P a 0.2 1,158 x l0 s M P a 440 m s - 1

24

C.E. ANDERSON, JR and S. R. BODNER

The computational algorithm should 'mimic' the physics of the failure process. A failure initiation condition can require relations between the stress components in addition to a limiting parameter value (e.g. equivalent strain) in order to determine the onset of failure. Other physically motivated criteria are used for determining the direction of failure propagation [79, 80]. Ringers [79] used maximum equivalent strain values between 0.5 and 1.0 as possible failure criteria to model the plugging of Ti 125 target, Figs 8 and 9. The smallest critical strain (0.5) predicted target perforation at 350 m s 1, in agreement with experiment; whereas higher critical strains required higher impact velocities. The smaller critical strain is consistent with the static ultimate strain for Ti 125 (Table 1). For the higher impact velocity of 500 m s 1, the computational results were not sensitive to the value of the critical strain since the penetrator overmatched the target. Figure 10 shows a time sequence of the penetration process and the formation of the plug. A critical strain criterion for perforation of the Ti 318 produced results that were not consistent with the test results. Cross-sectioning of target material and microscopically examining the failed target showed that the Ti 318 target formed adiabatic shear bands. Wulf [83] found that Ti 318 has a considerably lower work-hardening rate, a much higher yield strength and a higher thermal softening rate than Ti 125; these properties are conducive to the formation of adiabatic shear bands. In the computational model, the temperature rise due to plastic work is first computed for each computational element [79]. Based on arguments by Recht [84] when thermal softening overcomes the rate of strain hardening, i.e. 141

'

I

12 I--

'

I

Vs

'

1

500 m/s

L Original

i

"

0

20

30

10

'

I

'

I

'

I

'

I

'

V s = 350 m/s

--

t "

_i,o

0

A ~

jl

40

50

60

70

80

Time (its) FIG. 8. Plugging---depth of penetration vs time (from Ringers [79]). 500

I

X

'

I

r

I

r

I

'

I

'

I

'

I

'

Vs = 500 m/s

400

\

E

-o 300 o o ¢L in

'~ 200 o

2 100 f

erforation

D.

~=1 0

10

20

30

40 50 60 70 Time (its) FIG. 9. Plugging--projectile speed vs time (from Ringers [79]).

80

Ballistic impact: analytical and numerical modeling

E~

25

m

O X

E ~ -.12-

10/1.s

-.16I

l

•04

I

!

.08

-.{

-.Q

-.1

-.I

-.1

-.I

I

.12

0

.04

.08

.12

0

.04

.08

.12

R AXIS (m x 10-')

V s = 500 m / s

PERFORATION @ 3 5 . 6 ~ s

~" = 1 VR = 9 3 7 m / s FIG. 10. Computational modeling of plugging failure (from Ringers [-79]).

500

"~ =k 300

E 200

/-

100% Plastic W o r k

Converted to Heat ,o0

-

Converted to Heat - / 0 0

I 5

] 10

~ 15

~ I ~ 20

~ 25

30

Projectile Speed (m/s)

FIG. 11. Projectile speed vs time (from Ringers [80]).

c~tr/Oe= 0, it is assumed that material instability occurs and slip is permitted. All other criteria, including the direction of failure propagation, are determined exactly as for the critical strain failure model. Ringers examined the penetration process for impact velocities of 247 and 455 m s- 1. An uncertainty in the calculations was the percentage of plastic work converted to heat; values of 85 and 100 ~o were used with only a small difference observed in the projectile speed vs time, Fig. 11. The pattern of elements reaching the failure criteria and initiating or furthering the failure propagation is depicted in Fig. 12. In conclusion, plugging has been simulated by two different failure conditions, strain to failure and formation of adiabatic shear bands. Upon failure, computational elements split and the sliding interface is relocated to follow the failure. The propagation of splitting elements simulates the fracture process. In both failure modes, the failure criteria are based on physically reasonable assumptions which tend to mimic the actual failure process. Some uncertainties still exist, but overall, the results are in good agreement with experimental data. Recent work has examined shear band failure using a micromechanical approach. Shockey et al. [85] have developed a computational shear band model which computes the number and sizes of shear bands as a function of applied loading history. The kinetics of the shear band process were deduced and are described mathematically by correlating computed strain and strain rate histories with experimentally measured shear band size distributions. This shear band model was incorporated into a version of HEMP with a

26

C.E. ANDERSON, J r and S. R. BODNEr

V~~ 455mls cry{PENETRATOR" 2290MPo Z L

[ ~ ~ R ~

1

~

~

J

EDGEOF PROJECTILE AT IMPACT / FRONTAL g TAROETSURFACE

0.711~s 1.01 ps 1.01 /~s

~u ~ 1 . 2 6 / ~ s INDICATES ~ L37ps AN ELEMENT TOTALLY FAILED ~ 1.54ps

2.429.s ~ ~ ~ ~ ' -

3.25~s

1007. Plastic Nork FIG. 12. Pattern of elements reaching failurecriteriaand initiatingor furtheringfailurepropagation (from Ringers [80]).

damage variable as part of the constitutive relation. The example of impact of a 6.35 mm diameter rod of 4340 steel into a 6.35 mm thick plate of rolled homogeneous armor (RHA) at 750ms -~ is shown in Fig. 13. Lightly shaded regions indicated cells with some shear banding; the darkly shaded regions indicate computational cells where the shear banding damage has saturated, i.e. complete damage. The front of the projectile has failed by 7.7 ps, although the target directly beneath the projectile has sustained little damage. By 12.3 #s, the line of damaged cells in the target, near the periphery of the projectile, indicates that perforation of the target by plugging is imminent. The simulation was stopped after 12.3/~s because of excessive cell distortion. The authors report [85]: The computation of penetration by plugging was accomplished naturally by allowing cells subjected to high shear strains to fail in shear. No slide lines or other predetermined, artificially introduced surfaces of weakness were involved. To our knowledge, this calculation is the first to compute penetration by plugging using only material failure properties and without the use of artificial surface weakness.

It is instructive to show a comparison of the previous example at approximately 12.3/~s with and without the shear damage model activated, Fig. 14. The upper profile of Fig. 14 (no shear failure allowed) shows more compression of the target plate, less shear damage in the penetrator and less jetting at the projectile-target interface than the profile where shear damage was explicitly modeled. The deformation in the lower profile is more localized, concentrated in the vicinity of the zones near the periphery of the projectile and where it interacts with the target. Large shears and the onset of plugging are much more prevalent in the lower profile. The large zone distortion requires that the grid be rezoned for the computation to continue. Thus, the failure methodology does not have to use sliding interfaces, but it results in excessive grid distortion terminating the penetration problem.

27

Ballistic impact: analytical and numerical modeling

4340 Steel

RHA

(a) 7.7 /as after Impact

HA

(b) 9.9/zs after Impact

IA

(¢) 12.3 /as after Impact

FIG. 13. Simulation of ballistic impact using C-HEMP and SHEAR3 (from Shockeyet al. [85]).

This can be contrasted with the work of Ringers where the use of sliding interfaces permits complete perforation without rezoning, but with the attendant question of the approximations inherent in using sliding interfaces. Nevertheless, a sliding interface is a closer approximation to observations of actual shear bands, and in principle, failure by fracture on shear banding represents in the limit the creation of two surfaces. The shear zone widths of Fig. 14 are large compared to observed shear zones in failure. B o t h methodologies require a sufficient number of computational elements to provide adequate resolution of the strains and stresses for failure predictions to be accurate. Erosion

Erosion is characterized by total material failure. From a numerical viewpoint, total failure occurs when the equivalent strain exceeds a user prescribed erosion strain, ~e. Erosion strains of 1.0, 1.5 and 2.0 have been used; Stecher and Johnson [75] found that the results were relatively insensitive to the value chosen. Stecher and Johnson make a distinction between an erosion strain and a failure strain. Brittle materials may fracture before they

28

C, E, ANDERSON, JR and S. R. BODNER Upper Profile - -

Lower Profile - -

Simulation No. 4 (with EP model), 12.5 #s after Impact

Simulation No. 3 (with SHEAR3 model), 12.3 #s after Impact

FIG. 14. Effect of the shear banding model SHEAR3 on the ballistic impact simulation (from Shockey et al. [85]).

erode (gf < ~e); when ~fis exceeded, shear or tensile stresses are not allowed, but the material can still support hydrostatic compression. However, if the strain reaches ~e, then the material fails completely and is no longer capable of supporting hydrostatic pressure. Once an element has 'completely' failed, the motion of the mass is unrestrained by the adjacent material. Only when the mass encounters a sliding surface, where conservation of momentum is imposed, can the velocity of the failed element change. Results for the impact of long rod penetrators made of three different materials on a semiinfinite steel target are shown in Fig. 15 [76]. The numerical results are contrasted against one-dimensional hydrodynamic (no strength) theory. The Lagrangian (EPIC) calculations also are compared to an Eulerian (DORF) computation for the case of the copper penetrator. In general, the Lagrangian calculations indicated slightly deeper craters having somewhat smaller crater diameters than the Eulerian calculations. Crater dimensions are obtained by disregarding totally failed material. Since computational results have not been compared with matching experimental data, it is not clear which computation is closer to the actual physical circumstance. Kimsey and Zukas [75] have compared their results using an erosion algorithm against experimental data for both semi-infinite and finite targets and have obtained good agreement. The normalized penetration depth and normalized crater diameter vs striking velocity are compared against data by Hohler and Stilp [85] in Figs 16 and 17. It is clear that the use of dynamic material properties obtained from high strain rate data leads to better predictions than static material values, particularly at the lowest impact velocity where strength effects tend to dominate the impact problem more than at higher velocities.

Ballistic impact: analytical and numerical modeling 3.0

I

I

I

I ~ J Tungsten -~_ ~ , ~

• V = 5000 m/s I L = 1.35 cm V D = 0.30cm

2.5 --

Proje.ile ~

1°

/Copper ~ _

_

--

29

_

~

eAiumlnum

~ 1.0-a.

ions f ~ - f

fT"

•••

Eulerian Computation

I

1-D

I

Approximations

o.o •

I

I

I

I

I

I

0

2

4

5

8

10

12

14

Time, (Its)

FIG. 15. Long rod penetrations into semi-infinite target vs time (from Stecher and Johnson [76]).

2.0

'

I

'

I

'

I

'

Hohler & Stilp (1977)

1.5

r'l Zk

Static Data L Numerical Dynamic Data I Simulations

Q

a

O

.m 1.o o O. N

~O 0.5

z

L

I

i

1

0.2 0.4 0.6 Normalized Striking Velocity, v/c

I

o.8

FIG. 16. Penetration depth vs striking velocity (from Kimsey and Zukas [75]).

The penetration and perforation of a steel long rod (L/D = 10) into a 2.54 cm armor steel plate is shown in Fig. 18 [75]. The comparisons of the computational results against experiment for two tests are given in Table 2. Again, good agreement is obtained between the calculations and the test. Even with the success of modeling failure by erosion, there still are questions which need to be answered. Long rod penetrations by aluminum, copper, and tungsten into a semi-infinite steel targets are shown in Fig. 19 [76]. Total penetration depth is clearly a function of projectile density. However, the final impact craters all show an indentation on the axis, which is more pronounced as the penetrator density increases. The target eroding strains were set to ~o = 3.0 when the projectile was completely eroded. This higher value of ~ tends to reduce the centerline problem--the justification for increasing ~e is that the erosion process no longer exists after the projectile has eroded away, and that it helps to relieve the

30

C.E. ANDERSON,JR and S. R. B
i

I

'

T

'

I

H o h l e r & Stilp (1977) i-I

c

4.0

A

1

Static Data

Dynamic Data

Numerical Simulations

-

~5

~ 3.0-~3

~ 2.0--

1.o

~

I 0.2 Normalized

J

I

i

0.4

I 0.6

Striking Velocity, v/c

FIG. 17. Crater diameter vs striking velocity (from Kimsey and Zukas [-75]).

FIG. 18. Eroding penetrator and finite target, impact velocity 1103 m s 1 (from Kimsey and Zukas [75]).

centerline 'jet'. However, more work is necessary to understand this portion of the numerical simulation. It is worth noting that the criterion for erosion is much simpler than for plugging. For plugging, not only are there criteria for failure, but the direction of failure propagation must also be found. For erosion, however, the sliding surface is redefined simply by total element failure. As stated earlier, it is easier to characterize damage when damage levels are well above a threshold criterion.

Ballistic impact: analytical and numerical modeling

31

TABLE 2. COMPARISON Ol~ CALCULATZONSAND I~XI'ERIMENTS(FROM KlSEY AND ZUKAS [75]) D = 1.0 cm Residual velocity (m s - l)

10, M = 65 g,

Residual mass (g)

(m s - 1)

Calculated

Measured

Calculated

Measured

1219 1103

925 709

910 690

34.5 32.1

39.1" 32.7

*Estimated from radiograph.

ALUMINUM PROJECTILE

Fm :

:

!.:

COPPER PROJECTILE

-?

I

t = 1 2 IJS

HOTE~ -

L/D =

PRESSURE CONTOUR8 ~ 4 O W N AT INTERVALS OF 10 GPA

- EROOING STRAta : 1.5 FOR ALL C A S E 8

FIG. 19. Penetration into a semi-infinite target (courtesy of G. Johnson).

32

( . E. ANDERSON, JR and S. R. BODNER COMPARISONS

BETWEEN

ENGINEERING MODELING MODELING

AND COMPUTATIONAL

The preceding discussions have highlighted a number of features for both the engineering models and numerical (hydrocode) simulations. Both approaches have matured over the years and have become more versatile in their applications. However, a fundamental difference between the engineering modeling and numerical simulation lies in the extensions of the methodologies to two dimensions. Numerically, extending the conservation equations from one dimension to two or more dimensions is straightforward. The components of vector and tensorial quantities must now be treated and the programming logic must be written to accommodate the extra dimension(s). The biggest limitation is the demand on computer resources as discussed by Johnson and Anderson [14]. The challenge with extending the engineering modeling approach is understanding the physical picture of the dynamic processes during penetration and formulating a mathematical description of this mechanics. Thus, the advances in the engineering modeling have been in the formulation of the mechanics of dynamic plasticity during the interaction of the projectile and the target. Initial work considered only a rigid projectile at the lower ordnance velocities. Shock effects associated with high velocity impact, in combination with dynamic plasticity, have now been formulated in a two-dimensional framework [55]. This initial impact stage now needs to be rigorously coupled with the subsequent stages of penetration of the Ravid Bodner model. Projectile and target erosion and associated hydrodynamic effects must also be explicitly modeled. A seven-stage model for the complete very high velocity penetration and perforation process is envisaged. Except for the more complex two-dimensional modeling, engineering models have been limited to a primary mode of failure. Westine [53], in his Bodner-type model, has a phenomenological criterion for differentiating between petalling or plugging. The Ravid-Bodner model has a number of possible failure modes: the location and the type of failure are deduced from physically based criteria and the governing one is essentially automatically determined by the associated computational program. In numerical simulations, the work to date has anticipated a priori the primary mode of failure, e.g. spallation, plugging or erosion. A current effort is to combine plugging and erosion within the same computational framework [87], but it is not clear what difficulties are going to be encountered when or if the code tries to switch between failure modes, or if the slideline algorithm for erosion will accommodate and be compatible with the algorithm for plugging. We iterate a comment made earlier, namely, that it is desirable to formulate models that are as deterministic as possible, i.e. independent of empirical factors. What are the problems facing the ballistic modeler? In the one-dimensional engineering models, impact obliquity is accounted for in the energy formulation of Recht and Ipson [7]. At low angles of obliquity, i.e. near normal impact, obliquity factors have done an adequate job in accounting for obliquity effects by increasing the line-of-sight thickness of the target, e.g. Awerbuch and Bodner [88]. At high angles of obliquity, predictive results are not so good. The rewriting of the two-dimensional formulation into a quasi three-dimensional or truly three-dimensional model will be a major undertaking. Things do not fare much better with numerical simulations because of the demanding requirements on computer resources for a threedimensional problem. Johnson and Anderson [ 14] demonstrate the 'exponential' growth on memory and CPU time as a problem is analyzed in one, two, and then three dimensions. Unfortunately, many problems are three-dimensional, and a common recourse is ballistic testing. Another common area that needs improvement is an understanding and incorporation of physically based failure models. An important application of ballistic modeling is in the analysis of armor systems. Ceramics have come to play an increasingly important role in armor protection, yet the dynamic response of ceramics is just starting to be understood. In general, the ability to be accurate in predictions of either the engineering or numerical models requires proper inclusion of the dynamic response of materials to high rates of strain, failure initiation and failure propagation.

Ballistic impact: analytical and numerical modeling

33

CLOSURE

A very brief overview of ballistic penetration engineering models has been given. An examination of failure modeling within the context of numerical simulations has also been discussed. Differences and similarities between the two approaches to modeling ballistic penetration and perforation have been presented. Finally, difficulties in extending the predictive capability and range of applicability for both types of models were briefly discussed. Though differences and similarities exist between the two approaches, both have the objective of predicting penetration and target response from first principles without the aid of empiricism. Major advances in formulating the mechanics in analytical modeling, and the development of failure algorithms in computational modeling, have led to significant advancement in both modeling approaches. In general, as the science of ballistic modeling has advanced, it has provided considerable insight to the overall understanding of penetration mechanics.

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