transverse loads on a yawed projectile

PERGAMON

INTERNATIONAL JOURNAL OF

IMPACT ENGINEERING

International Journal of Impact Engineering 23 (1999) 77-86 www.elsevier.com/locate/ijimpeng

TRANSVERSE LOADS ON A YAWED PROJECTILE

STEPHAN J. RLESS. SIKHAND. SATL\PATHY,

and MICHAEL J. NORMANDIA

Institute for Advanced Technology, The University of Texas at Austin 4030-2 W. Braker Lane, Austin TX 78759

Summary-Yaw has been known to greatly influence the penetration performance of long-rod projectiles. Experiments have shown that even small angles of yaw can significantly degrade performance. We show that a critical feature of a yawed impact is the transverse load on the penetrator. Transverse loads tend to decrease the misalignment of rod axis and velocity vector. We use classical cavity expansion theory to quantify the impact transients and determine the magnitude of the transverse load. Then, a steady-state slot-cutting model is used to calculate the shape and orientation of a projectile that exits a finite plate. We find that this is contrary to the findings of some previous studies considered. The strength of the projectile may be ignored compared to the inertial loads even at the relatively high impact velocities. The theory agrees well with reverse impact experimental data on finite plates. 0 1999 Elsevier Science Ltd. All rights reserved.

INTRODUCTION

The mechanics of yawed long-rod projectiles at hypervelocity has been receiving a lot of attention recently due to their importance for several contemporary terminal ballistic systems [ 11. Solutions exist for cases where the penetrator stays rigid [2] or elastic [3]. There have been a number of experimental studies [4, 51 and numerical evaluations [6, 7, 81 of yaw effects for problems where erosion is present. Due to the complexity of these problems there have been only a few attempts at analytical modeling of this important phenomena [9, lo]. Discussions of yaw mechanics have attributed penetration degradation variously to rigid-body rotation of the projectile [ll], strike of the tail on the crater lip [5, lo], collapse of the projectile into a trough [ 121, reduction in effective length [ 131, and altered penetration direction [5, 141. While all of these effects may play a role, for very high velocity impact of tungsten rods onto steel targets, the paramount importance of lateral loads that redirect the projectile velocity vector has been recently identified [9]. In this study, we have attempted to quantify the lateral load, and combine it with models for steady-state slot cutting and initial transients to describe yawed penetration of finite- plate targets.

Fig. 1 defines the geometrical quantities associated with the yaw problem. Yaw angle, y is defined as follows:

y = sin-’ [

sgn(V*I,)g], (1)

where V is the velocity vector and L is the length vector pointing from tail to tip of the projectile. The angle of obliquity is defined by,

p = sin-‘[ sgn(+ixV) y]. (2)

0734-743X/99/$ - see front matter 0 1999 Elsevier Science Ltd. All rights reserved. PII: SO734-743X(99)00064-0

78 S.J. Bless et al. /International Journal of Impact Engineering 23 (1999) 77-86

where ri is unit normal into the target surface on the impact side. As we shall discuss later, it is essential to differentiate between the path of a point on the rod and the instantaneous shape of the rod itself. Thus, 8 defines the angle that the instantaneous velocity vector at a point makes with the X-axis. This angle defines the tangent to the particle path. The angle Q at a point is the angle between the tangent to the rod and X-axis, and hence defines the shape of the rod. R is the radius of curvature of the particle path. All the calculations are made in stationary plate coordinates, where the relative velocity between the projectile and the target is assigned to the projectile. We also break up the penetrator velocity V into components parallel (VP) and transverse (V,) to the local rod axis.

Surface

Fig. 1. Geometry preimpact and during penetration.

The surface interaction uoint is defined as the noint on the rod ftamet face) which is in contact with the target (rod) for thi first time (points 0 and P in Fig. 1). ’ u ’

Velocity of the interaction point along the rod axis: & = V cos p

co@ - Y) Velocity of the interaction point normal to the rod axis: v” = Vsin y

Velocity of the interaction point along the plate surface: I$, = V ‘ln ’ co4P - r>

STAGES OF INTERACTION

Impact of a yawed and eroding rod projectile is a complex event that may consist of distinct phases. The phases may overlap in time. Figure 2 illustrates the phases. They are:

(3)

(4)

(5)

three

primarv nenetration. In this stage a penetration cavity is induced in the target by the penetrator’s mrm.llel (axial) velocitv comsonent. The uenetrator elements that strike the crater floor are unaffected by lateral penetration.

I

lateral nenetration. In this stage, the projectile comes in contact with the side of the cavity. This usually happens first on the target surface. At the velocities considered here the projectile cuts a slot in the cavity wall. The lateral penetration is due to the transverse component of the penetrator velocity. If there is no or negligible open cavity from primary penetration, the rod also penetrates into the target from its leading face, which penetration is driven by the local parallel velocity component.

terminal penetration. In the final stage, the rod that has exited the slot strikes the bottom or sides of the cavity where it may cause additional penetration. The exiting rod has an altered velocity vector and may be broken. This phase is important for thick targets and is most likely absent for relatively thin plates.

S.J. Bless et al. /International Journal of Impact Engineering 23 (1999) 77-86 19

B (a) primary penetration

(b) lateral penetration

(c) terminal penetration (may or may not occur- depending on impact conditions)

Fig. 2. Stages in yawed rod penetration: (a) primary penetration. (b) lateral penetration.

(c) terminal penetration

INITIAL TRANSIENTS

Lateral penetration begins after a delay associated with the time required for the rod to contact the crater wall. This “cavitation delay” is over when:

Vi,0 a(t)

where a is the cavity radius (at the surface).

(6)

If the crater size is large enough, the entire rod can pass through without interacting with the side-wall, and the effect of initial yaw is minimal. Bjerke et al. [5] presented a formula to calculate the maximum yaw (critical yaw) for this condition. Bjerke et al. assumed that the crater grows


rapidly enough that the final size is attained almost instantaneously. Recently, the time-dependence of crater growth was reconsidered by Lee and Bless [15] but their treatment does not include the initial impact crater.

The growth rate of the initial cavity can be estimated by assuming spherical cavitation. The differential equation that describes the dynamics of a spherical cavity in an incompressible elastic- plastic medium is [ 161:

ati+Gd’+${l+ln(g)}=E (7)

where a denotes the cavity radius, P is the initial pressure, p is the density, Y and E are the flow stress and Young’s modulus respectively. The initial conditions for this problem have to be chosen carefully. The first is that the initial cavity radius is equal to the rod radius. The initial condition on the velocity is intriguing. As in any initial-boundary value problems in solid mechanics, both the velocity and the pressure cannot be specified simultaneously at the cavity wall. We examined the condition that a = V at t = 0. We found, as shown in Fig. 3, that agreement with Bjerke et al’s empirical relations for final cavity diameter for normal impacts was rather good. Figure 4 shows the crater diameter as a function of time that is implied by this solution, compared to the interaction point motion (Eqn. 5) for some cases of interest that are discussed in the next section. We note that for oblique targets, the crater is not centered on the rod axis, and this will affect the time for onset of lateral penetration predicted by Fig. 4.

Fig. 3. Final impact crater size at different velocities.

Impact Velocity, m/s

3’

2.75

2.5

2.25

2

1.75

1.5

1.25

Shot-265: Interaction point

Shot-259: Interaction point

1 x 1K6 Tim,sec

2 x lo-” 3 x 1O-6 4x 1tl-6

Fig. 4. Trajectories of interaction point and impact crater on target surface. Embedment starts where the trajectories cross.


There is an additional transient associated with lateral penetration: radial embedment. At the velocities considered here, embedment is analogous to indentation [ 171. It is driven by the geometrical interference of the rod and target. In such a situation, penetration due to the frontal portion of the rod, which is already inside the crater at the instant of initial lateral contact, is not affected. The rod starts to embed at a speed V,, given by Eqn. (4). The lateral load is the transverse component of the load due to the cavity expansion pressure acting on the partially embedded cross-section [9]. The embedment factor, f, which is the ratio of current load, F, to the maximum load F,, is given by:

f=’ max = min{l,~cos-l[l-~siny]}. (8)

The minimum function is used to denote the fact that after full embedment the embedment factor is unity.

SLOT-CUTTING DYNAMICS

From this point, we restrict our analysis to consider finite plates that are perforated. For this problem, once embedment is complete, the slot cutting becomes steady state, and the process is self similar (unless the penetrator fractures). The terminal phase (c) of yawed rod penetration is considered to be absent.

To calculate the slot evolution in the target, it is necessar’ to distinguish between the instantaneous rod shape and the path followed by a point on the ro;l (particle path, as shown in Fig. 1).

The target-penetrator interaction is assumed to be due to: i) frontal resistance to penetration along the rod axis, and ii) lateral resistance to slot-cutting. We assume that Alekseevski-Tate theory [ 181 applies in the axial direction, by virtue of which erosion and deceleration in the axial direction can be calculated. The lateral resistance is given by cavity expansion pressure, which acts in the normal direction to the rod.

No erosion is envisioned in the lateral direction. We have experimentally verified this assumption for yaw angles 5 10” and speeds to 2.6 km/s for tungsten penetrating armor steel [8]. Erosion can occur during this process, however.

From Alekseevski-Tate theory, we obtain the following equations for motion in the rod’s axial direction:

;P~(“~ -u)’ + Y = fp,u2 + R,

(9)

where p, is the projectile density, p, is the target density, u is the penetration velocity of the rod (assumed to be in the parallel direction), Y is the projectile strength, and R, is the target resistance.

Equations (9) provide very useful insight regarding the penetration direction during the primary penetration stage. These equations imply that the velocity of the projectile/target interface is the vector sum of u and V,.

In the transverse direction, the cavity expansion speed is V,. Using self-similarity transformations, Forrestal [19] has solved the field equations for dynamic cylindrical cavity expansion problem with an elastic-plastic constitutive law. His solution is in the form of a set of non-linear equations, evaluation of which yields the pressure required to open a cylindrical cavity at a speed V,. We have numerically solved these equations for RHA and have plotted the same in Fig. 5. A quadratic curve tit formula of the following form represents the solution reasonably well:


o,=o,+bp,V,’ (10)

where ‘3, is the quasi-static cylincl+l cavity expansion pressure. For RHA steel, with a flow stress of 1.2 GPa, c~” is 3.3 GPa and b is equal 2.2.

2 L -- equations

- - --Quadratic Curve-fit 1

0 + ~__-_____~+_ ___ -_+------~- +~----~ -+ ~~~

0 100 200 300 400 500 600

Cavity Expansion Velocity, mlsec

Fig. 5. Dynamic cylindrical cavity expansion solution.

We use this cavity expansion pressure as an approximation to the lateral stress present at the projectile-target interface. Considering an element of penetrator length AL, we express Newton’s 2nd law for the element as follows:

(11)

The first term in the square bracket of the RHS represents the acceleration of the point in the target in contact with the rod element. The second term in the square bracket of the RHS is the normal component of the acceleration of the rod element with respect to the target point that is required to maintain the rod in a curvilinear motion.

The angle, 8, is given by,

tan(O) = V,/V,.

Differentiating Eqn. (12) with respect to time we obtain:

(12)

(13)

From geometry, V, and V, are related to V, and V, through:

V, = V,cosql+~sin@

’ V, = V,sin@-V,cos$ I

Inserting Eqn. (14) into Eqn. (13) we obtain:

(14)

(1%

where we have used V, = Vcos(+B) and V,= Vsin($-6).

S.J. Bless et al. /International Journal of Impact Engineering 23 (I 999) 77-86

Solving Eqns. (11) and (15) along with the relation R = V/b, we obtain:

83

v~cos(q4) I’, =

+[*I( O” +~‘v’)-&sin2($-f3)

1+cos2(~-0)

and

At any given time step, the angle, $ and its derivative are given by,

(16)

(17)

Finally, to conserve mass, incompressibility in the rod material dictates that axial derivative of V, be zero. Thus V, is taken to be a constant for the whole rod at any given instance.

The solutions to equations (9), (16), and (17) comprise our transverse load model, which is called SLOTPEN. These equations form a set of ODES which can be time integrated along with Eqn. (18) and the initial conditions to obtain a time-dependent evolution of the rod shape. It should be noted that when the normal component of the particle velocity, V, becomes zero, 8 becomes equal to $. When V, is zero the cavity stress should be zero since the rod is not

interacting with the target in the lateral direction. Thus from Eqn. 16, v, is also zero. In our formulation of the lateral interaction, we have assumed that any penetrator deformation

occurs only at the load location. This is consistent with observations [20] and can be theoretically justified as follows: The rod strength is clearly negligible and a plastic hinge develops when the moment at any section is enough to yield the entire penetrator cross-section. From elementary beam theory, for a cantilever beam of circular cross section, this should happen at a distance,

x n Y T- -- -= 120,. D

(19)

For the tungsten-steel interaction this value is about 0.6. Thus the cavity stress causes local plastic flow and there is no effective elastic resistance of the rod to bending within the cavity. A similar argument applies to shear loads. The section yields completely under shear at,

x x2 _=_-

D 20~ (20)

when 7 is the shear strength of the penetrator. This ratio is about 0.3 for our materials. These arguments imply that only axial stress can be transmitted along the axis of the rod; the off-axis components should cause local plastic deformation.

To further verify the locality of rod deformation, we utilized DYNA-3D [21]. A lateral triangular pressure pulse of a width of 3.3 ~.ts was set to travel at 2.37 km/s along the axis of a tungsten long rod. The length and diameter of the rod were 88 mm and 2.6 mm, respectively. The height of the pressure pulse was 4.6 GPa. The constitutive response of tungsten was assumed to be elastic-perfectly plastic with a flow stress of 2 GPa. Figure 6 shows the total velocity of a node in the middle of the rod with time. The bar wave speed for tungsten is about 4.5 km/s. Thus the elastic wave should arrive at this location at 9.8 I&, and the load pulse should arrive at 18.5 l.ts.


m I s

600, ! ! ! !

._._____________________:_._____________________

I 400 _.______---.--___________~,_._____________~_-_____

.______---._________---- 1 _.________-______-___

0-L

fi _ ( _____________------_-- -___-----________-____ / _______.._____________-.

._______________-_____ ._._._.._.....______~_,-_~------~~~~~----~~~~ ________._______________ i i

ore00 eke-OS k-05

Time, s

Fig. 6. DYNA-3D simulation of total velocity at the center of the projectile subjected to a traversing load pulse.

From Fig. 6 it appears that a small amplitude deformation arrives at this point corresponding to the axial elastic wave speed. The bulk of the deformation however accompanies the traveling pulse. There is no appreciable deformation ahead of this pulse.

COMPARISON WITH EXPERIMENTS

Bless et al. [4] have reported reverse ballistic testing with yawed penetrators. We have chosen two cases, one each with a positive and a negative yaw angle. Table 1 summarizes the experimental data.

Table 1. Experimental Data (Bless et al.)

Shot Plate Obliquity Yaw Impact Rod Rod Erosion Erosion No. Thickness Velocity Diameter Length (Exp.) (computed)

(mm) (km/s) (mm) (mm) (mm) (mm)

259 3.13 70.4 -9.61 2.39 2.19 78.96 11 9.4

265 4.5 53 6.9 2.31 2.58 88.3 6 5.52

In Figs. 7 through Fig. 10 we have plotted the shape of the rod predicted by our SLOTPEN model along with the radiographs taken during the experiments at different time. The model includes erosion, cavitation delay, embedment transient, and steady-state slot cutting. The data show the rod emerging from the plate with a bent nose and a rotated or deflected shank. The nose bend is due to the embedment transients; it experiences less deflection than the shank.

The embedment speeds calculated from Eqn. (4) are 285 m/s for shot 265 and 400 m/s for shot 259. The velocity of the interaction point along the target surface are 567 m/s and 821 m/s for shot 265 and shot 259 respectively. This entails a cavitation delay of about 2 ms to 3 ms on the target surface (see Fig. 4).

The agreement for shot 265 is excellent. Both the steady-state slope and the transient parts are matched reasonably well. For shot 259 we are able to obtain good agreement with data as far as the steady-state slope is concerned. But the transient part is not matched well with the calculated embedment speed. However, we obtained much better agreement when we set the embedment speed to 100 m/s. We believe that this discrepancy may be due to a more complex form of the embedment transient when the yaw and obliquity have opposite signs. In this case, the initial cavity profile is shaped so that first contact with the crater wall occurs on the exit side of the crater,


rather than on the entrance side as was assumed in Fig. 4. It is possible then that the cavity has a component of growth due to the penetration and that the appropriate embedment speed is a relative velocity between the embedment and the cavity growth.

Fig. 7. Shot 265: t = 20 us. The solid lines are the model prediction for rod axial position. The white lines are enhanced outlines of the image visible on the radiograph. The dark lines show the rod axis and plate position. (In stationary plate coordinates, the rod travels from left to right.)

Fig. 8. Shot 265: t = 35 us.

Fig. 9. Shot 265: t = 55 us.

Fig. 10. Shot 259: time = 55 us. Dark lines are model predictions: A) embedment speed, V,, calculated from Eqn. 4 = 400 m/s; B) adjusted embedment speed, V,,= 100 m/s. Outlined images are from radiographs, including, in the last image, the initial undisturbed image of the rod.


CONCLUSIONS AND DISCUSSION

We presented a theory for yawed penetration into oblique plate targets. The theory includes models for the cavitation delay, embedment transient phase, and the steady-state slot-cutting phase. The theory has been incorporated in a mathematical model called SLOTPEN. The agreement between the theory and data for the rod exit shapes indicates that the lateral load calculated from cylindrical cavity expansion theory is accurate. This is an important conclusion since it can lead to more efficient design procedures for yaw-tolerant projectiles. This agreement also confirms the assumption that the material strength is much less important than inertia in affecting the lateral response for the cases considered. Both the data and the theory show that the lateral deformation is, to first order, a local phenomena and does not propagate or affect the upstream portions of the, rod significantly. The good agreement for the erosion for both cases indicates that the travel path in the target plate which determines the erosion amount is correctly approximated from the theory. The qualitative success of predicting nose deflection shows that it is the embedment transient which causes the rotation of the projectile components after perforation.

Acknowledgment-This work was supported by the U.S. Army Research Laboratory (ARL) under contract DAAA21-93-C-0101.

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REFERENCES

W. Goldsmith, “Non ideal projectile impacts on targets,” Int. J. Impact Engng., 22 (2-3), 95-395 (1999). A. J. Piekutowski, M. J. Forrestal and K. L. Poorman, T. L. Warten, “Perforation of aluminum plates with ogivenose steel rod at normal and oblique impacts,” Int. J. Impact Engng.. 18 (8). 877 (1996). S. Satapathy, A. Bedford and S. Bless, “Behavior of a yawed projectile penetrating a thin plate,” to appear in ht. J. Impact Engng. S. Bless, M. Normandia, J. Campos and V. Chart, “Experimental study of yawed impacts on thin plates using a reverse impact technique,” in preparation (1998). T. W. Bjerke, G. F. Silsby, D. R. Scheffler and R. M. Mudd, “Yawed long-rod armor penetration,” ht. J. Impact Engng., 12, 281-292 (1992). G. R. Johnson and W. H. Cook, “Lagrangian EPIC code computations for oblique, yawed-rod impacts onto thin-plate and spaced-plate targets at various velocities,” Inr. J. Impact Engng., 14,373-383 (1993). G. C. Bessette and D. L. Littlefield, “Analysis of transverse loading in long-rod penetrators by oblique plates,” Proc. 1997 Conf. Shock Compression of Condensed Matter, Amherst, MA (1997). C. E. Anderson Jr., S. J. Bless, T. R. Sharon, S. Satapathy and M. J. Normandia, “Investigation of yawed impact into a finite target,” Proc. 1997 Conf. Shock Compression of Condensed Matter, Amherst, MA (1997). S. Bless and S. Satapathy, “Penetration of thick targets by yawed long rods,” Proc. 1997 Conference Shock Compression of Condensed Matter, Amherst, MA (1997). M. Lee and S. J. Bless, “A discreet impact model for effect of yaw angle on penetration by rod projectiles,” 1997 Conf. Shock Compression of Condensed Matter, Amherst, MA (1997).

Y. I. Bukharev and V. I. Zhukov, “Model of the penetration of a metal barrier by a rod projectile with an angle of attack,” Fizika Goreniya Ivzryva, 31, 104-109 (1995). S. Bless, J. Barber, R. Bertke and H. Swift, “Penetration mechanics of yawed rods,” Inr. J. Engng. Sci., 16, 829 (1978). D. Yaziv, Z. Rosenberg and J. P. Riegel, “Penetration of yawed long rod penetrators,” Proc. 12th International Ballistics Symposium, San Antonio, TX (1990). M. Nonnandia, “Model for yawed penetrator impacting stationary, finite oblique targets,” 1998 Hypervelocity Impact Symp., Classified Session, Huntsville, AL (1998). M. Lee and S. Bless, “Cavity models for solid and hollow projectiles,” to appear Int. J. Impact Engng. H. G. Hopkins, “Dynamic expansion of spherical cavities in metal,” Progress in Solid Mechanics, 1, chapter III (edited by I. N. Sneddon and R. Hill), North-Holland Publishing Co., Amsterdam, NY (1960). S. Bless, S. Satapathy and M. Normandia, “Projectile damage in yawed impacts,” Proc. ICES98, Atlanta, GA, (1998). A. Tate, “Further results in the theory of long rod penetration,” J. Mech. Phys. Solids, 17, 141 (1969). M. J. Forrestal, “Penetration into dry porous rock,” ht. J. Solids Structures, 22 (12) 1485-1500 (1986). S. Bless and J. Barber, “Bending waves in yawed rod impacts,” J. OfBallistics, 2,281-298 (1978). S. Satapathy and S. Bless, “Response of long rods to moving lateral pressure pulse: numerical evaluation in DYNA-3D and Autodyn,” in preparation (1998).

transverse loads on a yawed projectile

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