Bending of Threaded and Unthreaded Tungsten 

Rods Under Oblique Impact 

by Martin N. Raftenberg and Todd W. Bjerke 

ARL-TR-3578 August 2005 

Approved for public release; distribution is unlimited.
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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5069 

ARL-TR-3578 August 2005 

Bending of Threaded and Unthreaded Tungsten 

Rods Under Oblique Impact 

by Martin N. Raftenberg and Todd W. Bjerke 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited.
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5a. CONTRACT NUMBER 

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4. TITLE AND SUBTITLE 

 Bending of Threaded and Unthreaded Tungsten Rods Under Oblique Impact

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 1L162618AH80 

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6. AUTHOR(S) 

 Martin N. Raftenberg and Todd W. Bjerke (both of ARL) 

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U.S. Army Research Laboratory 

Weapons and Materials Research Directorate 

Aberdeen Proving Ground, MD 21005-5069

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

We employed ballistic experiments and finite element (FE) analysis to explore the extent to which, and the mechanism by 

which, the threading on a tungsten heavy alloy rod increases its susceptibility to break up under conditions of oblique impact. 

In the laboratory, small-scale rods with and without threads were launched into thin rolled homogeneous armor plates at 

1300 m/s and 60 degrees’ obliquity. Fracture near the nose of the rod occurred more consistently with the threaded rods. LSDYNA1

 was used to analyze plane strain elastic impact of threaded and unthreaded rods striking a rigid surface at 10 m/s and 

60 degrees’ obliquity. These calculations revealed a greater degree of bending in a threaded rod than in an unthreaded rod of 

the same diameter. The maximum principal stress, which has been used as a criterion for brittle tensile fracture, also attained 

larger values in the threaded than in the unthreaded rod. With the aid of static FE analyses of cantilevered, point-loaded, 

threaded and unthreaded rods, the results of which could be compared with an analytical result from plate theory, we 

associated the increased bending with a decrease in effective rod thickness. 

1

LS-DYNA, which is not an acronym, is a trademark of Livermore Software Technology Corp. 

15. SUBJECT TERMS

 bending; KE rod; oblique impact; threading; tungsten heavy alloy 

16. SECURITY CLASSIFICATION OF:

19a. NAME OF RESPONSIBLE PERSON

 Martin N. Raftenberg 

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iii

Contents 

List of Figures iv

List of Tables v

1. Introduction 1

2. Experimental Observations 1

3. Plane-Strain Modeling of Oblique Impact 7

3.1 Problem Definition..........................................................................................................7 

3.2 Meshes.............................................................................................................................9 

3.3 WHA Properties ..............................................................................................................9 

3.4 Results and Discussion..................................................................................................11 

3.4.1 Contour Plots of Txx Throughout Rods 7-NT and 7-T.......................................11 

3.4.2 History Plots of Txx in Rods 7-NT and 7-T at X = 34.74 mm............................12 

3.4.3 Profiles of Txx, Tyy, Tzz, and Txy in Rods 7-NT and 7-T at X = 34.74 mm 

 and t = 32 μs.............................................................................................................18 

3.4.4 Contour Plots of Txx in Rod 7-T in the Vicinity of Threads 1 Through 8..........20 

3.4.5 History Plots of Txx in Rod 7-T at X = 36.04 mm..............................................20 

3.4.6 Profiles of Principal Stresses in Rods 7-NT and 7-T at X = 35.74 mm 

 and t = 32 μs.............................................................................................................23 

4. Conclusions 27

4.1 Summary of Results ......................................................................................................27 

4.2 Assessment ....................................................................................................................28 

5. References 29

Appendix A. Flexural Wave Analysis 31

Appendix B. Static Bending of Rods 7-NT and 7-T 35

List of Symbols 41 

Distribution List 42 
iv

List of Figures 

Figure 1. Geometry of rod 6A-T.....................................................................................................2 

Figure 2. Geometry of rod 7-T........................................................................................................2 

Figure 3. Sketch of the experiments; a tungsten rod strikes an RHA plate at 1300 m/s and 

60 degrees’ obliquity..................................................................................................................3 

Figure 4. Shadowgraphs of rod 6A-T at three times following perforation of an RHA plate; 

this pattern of separation of a single large nose fragment occurred in all five experiments 

with 6A-T...................................................................................................................................4 

Figure 5. Shadowgraphs of rod 7-T at three times following perforation of an RHA plate; 

this pattern of separation of a single nose fragment was repeated in four of five 

experiments with 7-T. ................................................................................................................5 

Figure 6. Shadowgraphs of rod 7-T at three times following perforation of an RHA plate 

from the single experiment in which 7-T fragmented at multiple locations..............................6 

Figure 7. A post-perforation shadowgraph of rod 7-NT from each of four experiments; the 

results show greater experiment-to-experiment variability than with 6A-T and 7-T, and in 

one experiment, the rod remained intact....................................................................................7 

Figure 8. Sketch of the plane-strain problems studied in section 3: a tungsten rod strikes a 

rigid target at 10 m/s and 60 degrees’ obliquity. .......................................................................8 

Figure 9. The front portion of rod 7-T, with material and spatial coordinate systems 

indicated (dimensions are in millimeters)..................................................................................8 

Figure 10. FE mesh for rod 7-T, (a) overview, (b) front, (c) back, (d) thread..............................10 

Figure 11. FE mesh for rod 7-NT, (a) overview, (b) front, (c) back, (d) “top”. ..........................11 

Figure 12. Txx in rod 7-NT at specific times in the range of 5 to 50 μs after impact....................13 

Figure 13. Txx in rod 7-NT at specific times in the range of 55 to 100 μs after impact................14 

Figure 14. Txx in rod 7-T at specific times in the range of 5 to 50 μs after impact.......................15 

Figure 15. Txx in rod 7-T at specific times in the range of 55 to 100 μs after impact...................16 

Figure 16. Txx in rod 7-NT along X = 34.74 mm as a function of time after impact. (Note the 

peaks at t ≅ 32 μs.)..................................................................................................................17 

Figure 17. Txx in rod 7-T along X = 34.74 mm as a function of time after impact. (Note the 

peaks at t ≅ 32 μs.)..................................................................................................................17 

Figure 18. Profiles of Txx, Tyy, Tzz, and Txy in rod 7-NT along X = 34.74 mm and at t = 32 μs. ..18 

Figure 19. Profiles of Txx, Tyy, Tzz, and Txy in rod 7-T along X = 34.74 mm and at t = 32 μs. .....19 

Figure 20. Profiles of Txx in rods 7-NT and 7-T along X = 34.74 mm and at t = 32 μs. ..............20 

Figure 21. Txx in rod 7-T at 20 and 30 μs after impact. ................................................................21 

Figure 22. Txx in rod 7-T at 40 and 50 μs after impact. ................................................................22 
v

Figure 23. Txx in rod 7-T along X = 36.04 mm as a function of time after impact. (Note the 

generally small values of Txx at Y = ± 5.89 mm, corresponding to the thread’s interior.).......23 

Figure 24. Impact experiment described in Bjerke and Edmanson (2004) and subsequent 

SEM of fracture surface...........................................................................................................24 

Figure 25. Profiles of principal Cauchy stresses in rod 7-NT along X = 34.74 mm and at t = 

32 μs.........................................................................................................................................25 

Figure 26. Profiles of principal Cauchy stresses in rod 7-T along X = 34.74 mm and at t = 32 

μs..............................................................................................................................................26 

Figure 27. Profiles of maximum principal Cauchy stress in rods 7-NT and 7-T along X = 

34.74 mm and at t = 32 μs. ......................................................................................................26 

Figure 28. Profiles of maximum Cauchy shear stress in rods 7-NT and 7-T along X = 34.74 

mm and at t = 32 μs. ................................................................................................................27 

Figure A-1. An element of bent rod at time t................................................................................31 

Figure B-1. The quasi-static, plane-strain plate problem studied in section B.1: (a) the plate 

is cantilevered at one end and point loaded at the other, (b) free-body diagram, (c) 

bending moment diagram. .......................................................................................................36 

Figure B-2. The mid-plane deflection field computed for rod 7-NT with LS-DYNA closely 

agrees with the analytical solution for a plate with 12-mm thickness. ....................................37 

Figure B-3. The mid-plane deflection field computed for rod 7-T with LS-DYNA is 

bounded by the analytical solution for plates with 12-mm and 10.72-mm thickness and is 

closer to the latter.....................................................................................................................37 

Figure B-4. The quasi-static plane strain problems studied in section B.2: rods 7-NT, 7-T, 

and 7-TE are each cantilevered at one end and point loaded at the other...............................38 

Figure B-5. The profiles of rods 7-NT, 7-T, and 7-TE.................................................................39 

List of Tables 

Table 1. Properties of WHA in its linearly elastic representation. .................................................9 

Table 2. Peak values of Txx at X = 34.74 mm and t = 32 μs. ........................................................19 

Table 3. Maximum values of T1 at X = 34.74 mm and t = 32 μs.................................................27 

Table B-1. Effective thicknesses of the three plates.....................................................................39 
vi

INTENTIONALLY LEFT BLANK
1

1. Introduction 

Kinetic energy penetrators are generally threaded in order to accommodate a sabot. We 

determined to explore the extent to which, and the mechanisms by which, such threading 

increases a rod’s susceptibility to breakup under conditions of oblique impact. 

Section 2 describes experiments in which small-scale rods composed of a 93.1% tungsten-4.7% 

nickel-2.2% cobalt (93.1W-4.7Ni-2.2Co) tungsten heavy alloy (WHA), with and without threads, 

were launched with a gas gun into 9.5-mm-thick rolled homogeneous armor (RHA) targets at 60 

degrees’ obliquity and a speed of 1300 m/s. Differences in breakup patterns of threaded and 

unthreaded rods were noted. 

Section 3 describes finite element (FE) simulations of the elastic impact on a rigid surface at a 

speed of 10 m/s and 60 degrees’ obliquity of a threaded and an unthreaded geometry used in the 

experiments. The goal was to investigate differences between the two rods in terms of patterns 

of elastic stress waves and of structural bending. These dynamic calculations were performed in 

plane strain with the LS-DYNA1 code (Livermore Software Technology Corp., 2003) in explicit 

mode. In order to further explore differences in bending characteristics, we also performed static 

plane-strain analyses of each rod cantilevered at one end and point loaded at the other. These 

static calculations employed LS-DYNA in the implicit mode. 

Our general finding is that the threads, by decreasing the effective thickness of the rod, lead to a 

greater degree of bending under oblique impact. Axial bending stresses therefore attain larger 

magnitudes in the threaded rod. The threads also lead to a locally more triaxial state of stress 

and thus to a greater departure from a state of pure bending. Furthermore, the maximum 

principal stress upon which a brittle tensile failure criterion can be based attains larger values in 

the threaded rod, particularly in the vicinity of the threads, than are attained anywhere in the 

unthreaded rod. 

Our results are summarized and an assessment is provided in section 4. 

2. Experimental Observations 

WHA rods with the threaded geometries (see figures 1 and 2) were fired into 9.5-mm- (3/8-inch) 

thick RHA plates at 1300 m/s and 60 degrees’ obliquity (figure 3). (Obliquity is defined as the 

angle between the projectile’s trajectory and the normal to the target.) The length-to-diameter

(L/D) ratio was 16.2 for rod 6A-T and 16.7 for rod 7-T. The rods used in the experiments were 

 1LS-DYNA, which is not an acronym, is a trademark of Livermore Software Technology Corp.
2

composed of a tungsten-nickel-cobalt alloy with the composition 93.1W-4.7Ni-2.2Co. All were 

machined from a single lot of stock material obtained from Aerojet Ordnance Tennessee, Inc. 

Figure 1. Geometry of rod 6A-T. 

Figure 2. Geometry of rod 7-T. 
3

Figure 3. Sketch of the experiments; a tungsten rod strikes an RHA plate at 

1300 m/s and 60 degrees’ obliquity. 

Five experiments were performed with the smaller rod, 6A-T. Figure 4 presents shadowgraphs 

at three post-perforation times from one of these experiments. The nose of the rod, extending 

from the front to a location between the third and fourth threads, has separated from the 

remainder as a single large fragment. A similar pattern was observed in the four other 

experiments with rod 6A-T. 

Five ballistic experiments were also performed with the larger rod, 7-T. Figure 5 presents 

shadowgraphs at three post-perforation times from one of these experiments. The pattern is quite 

similar to that noted in figure 4, namely, the nose of the rod has separated as a single large 

fragment, this time at a location between the seventh and eighth threads. Three of the four other 

experiments involving rod 7-T also displayed the pattern in figure 5 of separation of the nose 

region as a single large fragment. Shadowgraphs from the remaining experiment with 7-T are 

shown in figure 6. Here, fragmentation is seen to have occurred at several locations. We 

speculate that some pre-impact yaw may have been present in the experiment of figure 6. 

The post-perforation shadowgraphs in figure 7 show rod 7-NT, identical in geometry to 7-T but 

without threads, in four different experiments. These small-scale prototype rods were launched 

without a sabot by means of a pusher plate, which made testing of the 7-NT possible. Results 

with 7-NT displayed greater experiment-to-experiment variability than did results with the two 

threaded geometries. Two of four experiments (figures 7a and 7b) displayed the pattern familiar 

from experiments with the threaded rods, namely, separation of a single fragment from the nose 

region. In one of the four experiments (figure 7c), the rod fragmented into two pieces near its 

center. In the remaining experiment with 7-NT (figure 7d), the rod remained intact. On the basis 

of this fourth experiment, we draw the conclusion that the presence of threading increases the 

susceptibility of tungsten rods to fracture under conditions of oblique impact. 

30°

9.5 mm 

RHA 

Penetrator 

1300 m/s 
4

Figure 4. Shadowgraphs of rod 6A-T at three times following perforation of an 

RHA plate; this pattern of separation of a single large nose fragment 

occurred in all five experiments with 6A-T. 
5

Figure 5. Shadowgraphs of rod 7-T at three times following perforation 

of an RHA plate; this pattern of separation of a single nose 

fragment was repeated in four of five experiments with 7-T. 
6

Figure 6. Shadowgraphs of rod 7-T at three times following perforation of an RHA 

plate from the single experiment in which 7-T fragmented at multiple 

locations. 
7

 (a) (b) 

 (c) (d) 

Figure 7. A post-perforation shadowgraph of rod 7-NT from each of four experiments; the results show greater 

experiment-to-experiment variability than with 6A-T and 7-T, and in one experiment, the rod remained intact. 

3. Plane-Strain Modeling of Oblique Impact 

3.1 Problem Definition 

The goal of our finite element modeling was to explore sources of the increased fracture 

susceptibility associated with threading. We decided to pursue an elastic plane-strain analysis of 

rod 7 in figure 2, both threaded (7-T) and unthreaded (7-NT). The possibility for inherently 

three-dimensional (3-D) and/or inelastic sources for increased fracture susceptibility can be 

explored in more costly, future analyses outside the scope of this report. In the present report, 

we are pursuing the hypothesis that increased bending in the threaded rod, compared to the 

unthreaded rod, accounts, at least in part, for its increased tendency to fragment under oblique 

impact. This phenomenon can be isolated and studied in its simplest form in this twodimensional (2-D) elastic model. 

Figure 8 shows the problem that we analyzed. This problem is simpler yet relevant to that of 

figure 3 which we studied experimentally. The RHA target has been replaced by a rigid surface, 

still inclined at 30 degrees to the rod’s initial trajectory. A state of plane strain is assumed for 

both the rod and plate. The rod’s impact speed has been reduced to 10 m/s in order to keep the 

results linear and elastic. 
8

Figure 8. Sketch of the plane-strain problems studied in section 3: 

a tungsten rod strikes a rigid target at 10 m/s and 60 degrees’ 

obliquity. 

Figure 9 shows the front portion of rod 7-T and indicates the material and spatial reference 

frames and Cartesian coordinate systems employed in our analyses. Our material coordinates, 

(X,Y,Z), specify a material point’s location at time t = 0. The origin, X = Y = 0, coincides with 

the central point on the front edge of the rod. Coordinate X measures distance along the rod, and 

Y measures distance above and below the mid-plane of the rod. At t = 0, the x, y, and z axes of 

our fixed spatial coordinate system coincide with the X, Y, and Z axes, respectively. Motion and 

deformation are assumed to be confined to the x-y plane. 

The LS-DYNA finite element code was applied in its explicit mode to all calculations in 

section 3. The rigid surface inclined at 30 degrees to the x-axis was specified with the use of 

the feature *RIGIDWALL_GEOMETRIC_FLAT. 

Figure 9. The front portion of rod 7-T, with material and spatial coordinate systems indicated (dimensions are in 

millimeters). 

30o 

Rigid 

Surface Penetrator 

10 m/s 

X, x 

Y, y

0

X=34.74 Y=-6.0 X=0

Y=6.0

X=36.04
9

3.2 Meshes 

We used AutoCAD LT2 (Autodesk2

 Corporation, 2005) to create geometry files in “dxf” format 

for rods 7-NT and 7-T. In the latter case, the 2-D cross-sectional geometry shown in figure 2, 

including the thread details, was accurately represented. 

These “dxf” files were then imported into HyperMesh3

 (Altair3 Engineering, Inc., 2005) to create 

the meshes shown in figures 10 and 11 for rods 7-T and 7-NT, respectively. Our restriction to 

2-D plane strain allowed for the small element size of about 0.2 mm to be used throughout the 

mesh. The mesh for 7-T contained 71,038 nodes and 69,898 elements. Almost all these 

elements were four-node quadrilaterals. A small number of three-node triangles was employed 

in the vicinity of each thread. The mesh for 7-NT consisted of 72,689 nodes and 71,398 

elements; here, all the elements were four-node quadrilaterals. The meshes were exported from 

HyperMesh in LS-DYNA format. 

In LS-DYNA, *SECTION_SHELL with ELFORM=13 was used to specify plane strain. 

Reduced integration (one Gaussian quadrature point per element) was employed. 

3.3 WHA Properties 

The rods used in the experiments were composed of a tungsten-nickel-cobalt alloy with the 

composition 93.1W-4.7Ni-2.2Co. The properties assigned in the finite element analyses are 

listed in table 1. The value for density ρ was reported in Johnson (1993) for a tungsten-nickelFE alloy with composition 90W-7Ni-3Fe. In our low-velocity impact calculations, the WHA 

was assumed to behave as a homogeneous, isotropic, linearly elastic continuum. The values 

used for its elastic Young’s modulus E and Poisson’s ratio ν are also listed in table 1. In LSDYNA, Material *MAT_ORTHOTROPIC_ELASTIC was applied to the WHA in the 

degenerate isotropic case. 

Table 1. Properties of WHA in its linearly elastic representation. 

ρ (kg/m3) E (GPa) ν 

16980 327.4 0.3193 

 2

AutoCAD LT and Autodesk are registered trademarks of Autodesk Corporation.

3HyperMesh and Altair are registered trademarks of Altair Engineering, Inc.
10

(a) 

(b) 

(c) 

(d) 

Figure 10. FE mesh for rod 7-T, (a) overview, (b) front, (c) back, (d) thread. 
11

(a) 

(b) 

(c) 

(d) 

Figure 11. FE mesh for rod 7-NT, (a) overview, (b) front, (c) back, (d) “top”. 

3.4 Results and Discussion 

3.4.1 Contour Plots of Txx Throughout Rods 7-NT and 7-T 

Figures 12 and 13 show contours of the Cauchy stress component Txx at 5-μs intervals throughout the entire rod 7-NT. Figures 14 and 15 do the same for rod 7-T. Time is measured from the 

instant of initial impact on the rigid surface. Txx is resolved in terms of the fixed x-axis direction, 
12

but since little rigid-body rotation occurs during the 100 μs covered in these figures, Txx can be 

closely identified with the rod’s axial stress. “Red” indicates axial tension and “blue” indicates 

axial compression. Note that the scale is very sensitive, with a full range of ±10 MPa. 

Figures 12 and 14 show an initial “blue” axial compression wave that traverses the 200-mm 

length of the rod in about 43 μs. This traversal time corresponds very closely to speed 

(1 ν )ρ 2 E − or 4634 m/s with properties in table 1. This wave speed is derived with the 

assumption, in addition to plane strain, that Tyy and Txy are approximately zero throughout the 

rod. The motivation for this assumption is that Tyy and Txy are zero on the bounding free surfaces 

at the maximum and minimum Y values and the rod is thin (see Kolsky, 1963, pp. 79-83). 

In the wake of the axial compression wave, figures 12 and 14 show a slower traveling region of 

alternating tension and compression on either side of the rod’s mid-surface. This is a characteristic bending pattern, in which the rod is axially extended on one side of its neutral surface and 

axially compressed on the other. The slower wave is therefore a flexural wave. In appendix A, 

we derive a condition on the flexural wavelength in order for the flexural wave to move more 

slowly than the axial compression wave. At later times, in figures 13 and 15, the entire rod is in a 

standing-wave pattern of flexure. 

3.4.2 History Plots of Txx in Rods 7-NT and 7-T at X = 34.74 mm 

Figure 16 presents a family of plots of Txx at X = 34.74 mm as functions of time for rod 7-NT. 

Figure 17 presents the same for rod 7-NT. The strip of elements with centroid location X = 

34.74 mm is shown in “green” in figures 9, 10b, and 11b. In rod 7-T, this strip intersects the two 

free surfaces at the approximate center of the trough between threads 3 and 4. Each plot in the 

family corresponds to a particular value of Y. The FE solution has been sampled at a Y-interval 

of every three elements, or about 0.6 mm. In each figure, the curves at Y = ±0.31 mm pertain to 

elements closest to the rod’s mid-surface. In figure 16, Y = ±5.89 mm pertains to elements 

having one edge on rod 7-NT’s free surface at Y = ±6.00 mm. In figure 17, Y = ±5.26 mm 

pertains to elements having one edge on rod 7-T’s free surface at Y = ±5.36 mm. 

In both figures, every plot attains a local and perhaps a global extremum at about 32 μs after 

initial impact. These early-time extrema are of particular interest because our results for impact 

on a rigid surface (figure 8) are of greatest relevance to the RHA perforation problem of figure 3 

at early times. The remaining analysis therefore focuses on this particular time. 
13

Figure 12. Txx in rod 7-NT at specific times in the range of 5 to 50 μs after impact. 

Stress contours (Pa) 

20 μs 

5 μs 

10 μs 

15 μs 

25 μs 

30 μs 

35 μs 

40 μs 

45 μs 

50 μs 
14

Figure 13. Txx in rod 7-NT at specific times in the range of 55 to 100 μs after impact. 

70 μs 

55 μs 

60 μs 

65 μs 

75 μs 

80 μs 

85 μs 

90 μs 

Stress contours (Pa) 

95 μs 

100 μs 
15

Figure 14. Txx in rod 7-T at specific times in the range of 5 to 50 μs after impact. 

20 μs 

5 μs 

10 μs 

15 μs 

25 μs 

30 μs 

35 μs 

40 μs 

Stress contours (Pa) 

45 μs 

50 μs 
16

Figure 15. Txx in rod 7-T at specific times in the range of 55 to 100 μs after impact. 

70 μs 

55 μs 

60 μs 

65 μs 

75 μs 

80 μs 

85 μs 

90 μs 

Stress contours (Pa) 

95 μs 

100 μs 
17

Figure 16. Txx in rod 7-NT along X = 34.74 mm as a function of time after impact. (Note the peaks at t ≅ 32 μs.) 

Figure 17. Txx in rod 7-T along X = 34.74 mm as a function of time after impact. (Note the peaks at t ≅ 32 μs.) 
18

3.4.3 Profiles of Txx, Tyy, Tzz, and Txy in Rods 7-NT and 7-T at X = 34.74 mm and t = 32 μs 

Plots of Txx, Tyy, Tzz, and Txy as functions of Y (“Y-profiles”) for X = 34.74 mm and t = 32 μs are 

given in figure 18 for rod 7-NT. In appendix A, we describe a state of pure bending, in which 

the plate is bent into locally circular arcs and Txx increases linearly with distance Y from the 

neutral surface (see equation A-2). The neutral surface can be identified with the mid-plane in 

the case of our symmetric geometries. Figure 18 shows that a state of pure bending closely 

approximates our solution for rod 7-NT. Note the small negative offset from zero in Txx at Y = 0, 

which is indicative of a small superimposed axial compression. 

Figure 18. Profiles of Txx, Tyy, Tzz, and Txy in rod 7-NT along X = 34.74 mm and at t = 32 μs. 

Y-profiles of Txx, Tyy, Tzz, and Txy for X = 34.74 mm and t = 32 μs are shown in figure 19 for rod 

7-T. Figure 19 shows that rod 7-T was also approximately in a state of pure bending within its 

central core region bounded by Y ≅ ±4.5 mm. Outside this core region, in the vicinity of the 

threads, two observations can be made for rod 7-T, based on figure 19. First, Txx increases 

steeply with further increases in Y, and second, a more triaxial state of stress develops as Tyy, Tzz, 

and Txy all increase substantially in absolute value (but remain smaller in absolute value than 

Txx). 
19

Figure 19. Profiles of Txx, Tyy, Tzz, and Txy in rod 7-T along X = 34.74 mm and at t = 32 μs. 

Y-profiles of Txx at X = 34.74 mm and t = 32 ms for rods 7-NT and 7-T are compared in 

figure 20. The 7-NT and 7-T results intersect at Txx = −16 MPa, offset from zero because of the 

superimposed axial compression. Note that with the exception of the immediate vicinity of this 

single intersection point, Txx is greater in rod 7-T than in 7-NT. In rod 7-T, Txx attains its 

maximum of 688 MPa at Y = 5.26 mm and its minimum of −766 MPa at Y = −5.26 mm. In rod 

7-NT, Txx attains its maximum of 463 MPa at Y = 5.89 mm and its minimum of −517 MPa at 

Y = −5.89 mm. These observations are collected in table 2. 

Table 2. Peak values of Txx at X = 34.74 mm and t = 32 μs. 

rod 7-NT rod 7-T 

max 

(MPa) 

at Y

(mm) 

min 

(MPa) 

at Y

(mm) 

max 

(MPa) 

at Y

(mm) 

min 

(MPa) 

at Y

(mm) 

463 5.89 −517 −5.89 688 5.26 −766 −5.26 
20

Figure 20. Profiles of Txx in rods 7-NT and 7-T along X = 34.74 mm and at t = 32 μs. 

3.4.4 Contour Plots of Txx in Rod 7-Tin the Vicinity of Threads 1 Through 8 

Figures 21 and 22 show Txx contours in rod 7-T in the vicinity of the first eight threads. Note 

the reduced contour sensitivity; in figures 21 and 22, the full range is ±200 MPa, whereas in 

figures 12 through 14, it was ±10 MPa. 

The contours in figures 21 and 22 clearly show the threads’ interiors to be at a lower level of 

axial stress than in the central core of the rod. This is not surprising since the stress-free 

boundary conditions require Txx to be zero on all three faces of each thread. 

3.4.5 History Plots of Txx in Rod 7-T at X = 36.04 mm 

In rod 7-T, the location X = 36.04 mm corresponds to the approximate center of the fourth thread 

(figure 9). Figure 23 contains a family of plots of Txx at this X location in rod 7-T as functions of 

time. The FE solution has again been sampled at a Y-interval of every three elements, or about 

0.6 mm. 
21

Figure 21. Txx in rod 7-T at 20 and 30 μs after impact. 

Stress contours (Pa) 

20 μs 

30 μs 
22

Figure 22. Txx in rod 7-T at 40 and 50 μs after impact. 

Stress contours (Pa) 

40 μs 

50 μs 
23

Figure 23. Txx in rod 7-T along X = 36.04 mm as a function of time after impact. (Note the generally small values 

of Txx at Y = ± 5.89 mm, corresponding to the thread’s interior.) 

Note that throughout the time range shown, the largest values for Txx occur at Y = 4.64 mm, 

just outside the edge of the central core region. As Y is increased beyond this value, Txx

decreases. Thus, figure 23 supports the observation made in section 3.4.4 that the threads in rod 

7-T are largely unloaded relative to the central core of the rod. 

In appendix B, plane-strain static analyses of rods 7-NT and 7-T, cantilevered at the back end 

and point loaded at the front, are performed with LS-DYNA in implicit mode. The results for 

mid-plane deflection are compared with an analytical solution from plate theory. These 

comparisons further support the picture that the threads of rod 7-T are largely outside the axial 

load path and decrease the effective thickness of the rod, thereby leading to a larger degree of 

bending. 

3.4.6 Profiles of Principal Stresses in Rods 7-NT and 7-T at X = 35.74 mm and t = 32 μs 

Bjerke and Edmanson (2004) describe a set of experiments in which a beam composed of 

93W-5.6Ni-1.4Fe and with dimensions 127×9×9 mm was impacted at the center of its length by 

a cylindrical projectile of the same material. The projectile had a 44.5-mm length and an 8.9-mm 

diameter, and the impact speed was 50 m/s (figure 24). The beam split into two pieces in each 

experiment. Scanning electron microscope (SEM) fractography was performed on the main 

fracture surfaces. Figure 24 shows an SEM of a fracture surface near its intersection with what 
24

had been the back face of the beam. Signs of planar cleavage are apparent. This suggests the 

use of a failure criterion based on maximum principal stress. 

Figure 24. Impact experiment described in Bjerke and Edmanson 

(2004) and subsequent SEM of fracture surface. 

Let T1, T2, and T3 denote the principal stresses, listed in order of increasing magnitude. In plane 

strain, T2 is Tzz throughout the rod, and the Mohr’s Circle relationships (Fung, 1965, pp. 69-76) 

can be used to evaluate T1 and T3, as well as the maximum shear stress, Tshear. We have 

2

xy

2

xx yy xx yy

1 2 2

T

T T T T

T + 













 − +

+ = (1) 

WHA 

WHA 
25

T2 = Tzz (2) 

2

xy

2

xx yy xx yy

3 2 2

T

T T T T

T + 













 − − + = (3) 

2

xy

2

xx yy

shear 2

T

T T

T + 













 − = (4) 

The resulting Y-profiles of the three principal stresses at X = 34.74 mm and t = 32 μs are 

presented for rods 7-NT and 7-T in figures 25 and 26, respectively. Profiles of the maximum 

principal stress, T1, and the maximum shear stress, Tshear, are compared from rods 7-NT and 7-T 

in figures 27 and 28. Both T1 and Tshear exhibit larger magnitudes in the vicinity of the threads 

in rod 7-T than occur anywhere in rod 7-NT. The maximum values computed for T1 in rods 

7-NT and 7-T at X = 34.74 mm and t = 32 μs are compared in table 3. The maximum observed 

in 7-T is seen to be 49% larger than that in 7-NT. This suggests a greater likelihood of brittle 

fracture in rod 7-T than in rod 7-NT even if the fracture threshold stress were the same in the two 

rods. The greater likelihood of 7-T to fracture would be accentuated if the fracture threshold 

stress were reduced in the vicinity of the threads by the introduction of flaws during the 

machining process. 

Figure 25. Profiles of principal Cauchy stresses in rod 7-NT along X = 34.74 mm 

and at t = 32 μs. 
26

Figure 26. Profiles of principal Cauchy stresses in rod 7-T along X = 34.74 mm and 

at t = 32 μs. 

Figure 27. Profiles of maximum principal Cauchy stress in rods 7-NT and 7-T along 

X = 34.74 mm and at t = 32 μs. 
27

Figure 28. Profiles of maximum Cauchy shear stress in rods 7-NT and 7-T along 

X = 34.74 mm and at t = 32 μs. 

Table 3. Maximum values of T1 at X = 34.74 mm and 

t = 32 μs. 

rod 7-NT rod 7-T 

max 

(MPa) 

at Y

(mm) 

max 

(MPa) 

at Y

(mm) 

463 5.89 690 5.26 

4. Conclusions 

4.1 Summary of Results 

In section 2, we presented data from experiments in which small-scale rods composed of 93.1W4.7Ni-2.2Co WHA were launched into a 9.5-mm-thick RHA plate at 60 degrees’ obliquity and a 

speed of 1300 m/s. Two rod geometries were employed: rod 6A with a 10.2-mm diameter and 

an L/D ratio of 16.2, and rod 7 with a 12-mm diameter and an L/D of 16.7. In ten experiments 

with threaded rods, five with 6A and five with 7, fracture consistently occurred near the front of 

the rod. In the four experiments with unthreaded rods, all of rod 7, the results were more varied. 

In one experiment, fracture did not occur. In another experiment, fracture occurred near the 
28

rod’s center. In two other experiments, fracture occurred near the front, as with the threaded 

rods. 

In section 3, we presented FE simulations of the elastic impact of rod 7, both threaded (7-T) 

and unthreaded (7-NT), upon a rigid surface at a speed of 10 m/s and 60 degrees’ obliquity. 

The LS-DYNA code was used in explicit mode. The impact was found to generate both axial 

compression waves and flexural waves, with the latter predominant. We focused our postprocessing at the distance from the nose of the rod corresponding to the trough between the 

third and fourth threads in 7-T. We found the axial stress along this cross section to peak at 

32 μs after initial impact, so we further focused our post-processing on this particular time. 

At this cross section and time, rod 7-NT was found to be approximately in a state of pure 

bending with small superimposed axial compression. The solution in rod 7-T was found to 

consist of a central core, extending from the center to about 0.8 mm from the base of the threads, 

in which a state of nearly pure bending plus small axial compression was observed. Outside this 

central core, the threaded rod displayed a more triaxial state of stress. 

Within its central core region, the threaded rod displayed a greater amplitude of bending than did 

the unthreaded rod at the same distance from the centerline. In appendix B, the source of this 

enhanced bending in the threaded rod was explored by means of a static analysis of the rods, 

threaded and unthreaded, point loaded at one end and cantilevered at the other. These static FE 

analyses were performed with LS-DYNA in implicit mode. The FE solutions were compared 

with an analytical result from plate theory. The threads were found to have reduced the effective 

thickness of the rod and to have decreased the rod’s flexural rigidity. 

Outside the central core region of rod 7-T and thus in the vicinity of the threads, axial stress 

attained values substantially larger than any observed in 7-NT. Furthermore, a Mohr’s circle 

analysis revealed that the maximum principal stress and the maximum shear stress also attained 

larger values near the threads of rod 7-T than anywhere in 7-NT. A fracture criterion based on 

maximum principal stress would therefore predict that the threads cause enhanced susceptibility 

to fracture under oblique impact. 

4.2 Assessment 

The FE simulations in this report involved 10-m/s impact upon a rigid surface. The results 

suggested possible sources for the enhanced tendency of the threaded rod to fracture. The 

threads reduce the rod’s effective thickness and thus its flexural rigidity. The resulting enhanced 

bending leads to larger axial stresses, which peak near the base of the threads. In addition, the 

threads resulted in greater triaxiality, particularly in the vicinity of the threads. 

Additional possible sources for the enhanced tendency to fracture may be revealed in a 3-D 

perforation analysis at 1300 m/s. Such an analysis would include out-of-plane modes of 

deformation, plasticity properties of the WHA, and interactions between the threads and the wall 

of the penetration channel developed in the RHA. 
29

5. References 

Altair Engineering, Inc., http://www.altair.com/software/hw_hm.htm, 2005. 

Autodesk Corporation, http://usa.autodesk.com/adsk/servlet, 2005. 

Bjerke, T. W.; Edmanson, W. R. High Rate Deformation and Failure of Tungsten-Sintered 

Metals. Proceedings of the 2004 Society for Experimental Mechanics Annual Conference 

and Exhibition, Costa Mesa, CA, paper 90, 2004. 

Fung, Y. C. Foundations of Solid Mechanics, Prentice-Hall: Englewood Cliffs, NJ, pp. 69-76, 

1965. 

Johnson, G. R. Material Characterization for Warhead Computations, Tactical Missile 

Warheads, editor: J. Carleone, Progress in Astronautics and Aeronautics, vol. 55, 1993. 

Kolsky, H. Stress Waves in Solids, Dover: New York, 1963. 

Livermore Software Technology Corporation. LS-DYNA Keyword User’s Manual. Version 970, 

Livermore, CA, April 2003. 

Timoshenko, S.; Woinowsky-Krieger, S. Theory of Plates and Shells, Second Edition, McGrawHill: New York, 1959. 
30

INTENTIONALLY LEFT BLANK
31

Appendix A. Flexural Wave Analysis 

Kolsky (1963) on pages 48-54 discusses “flexural vibrations of rods.” His “rods” have a finite zwidth, so his analysis must be modified slightly to fit our situation of plates with effectively 

infinite z-width (plane strain). In figure A-1, the element of rod between x and x+∆x is bent into 

a circular arc with local radius of curvature R(x,t) at time t. Vy(x,t) is the resultant shear force per 

unit z-width on face x at time t. Mz(x,t) is the resultant bending moment per unit z-width on face 

x at time t. Mid-surface deflection δy(x,t) is defined in terms of the y-displacement field by 

(x,t) u (x,0,t) δ y ≡ y (A-1) 

Figure A-1. An element of bent rod at time t. 

The circular arc assumption implies that throughout the rod, 

2

2 ( , )

( , ) ( , , )

x

x t

y R x t

y e x y t xx ∂

∂ = − ≅ − δ (A-2) 

in which y is distance above the rod’s mid-surface. 

If we introduce the assumption of plane strain and the approximation Tyy = 0, based on the 

thinness of the rod, we have 

2

2

2 2

( , )

1 ( , , ) 1 ( , , )

x

E y x t

e x y t

E T x y t x

xx xx ∂

∂

⋅ − = − − = δ

ν ν

 (A-3) 
32

The resulting bending moment on face x per unit z-width is 

2

2

2

3 2

2

( , )

12(1 ) ( , ) ( , , )x

E h x t M x t yT x y t dy x h

h z xx ∂

∂

− = − = − ∫−

δ

ν

 (A-4) 

Conservation of y-momentum applied to the rod element in figure A-1 requires that 

x

V

t

h y y

∂

∂ = ∂

∂

2

2

δ

ρ (A-5) 

If we neglect rotary inertia, a reasonable assumption when the wavelength is substantially larger 

than the rod thickness (Kolsky 1963, pp. 52-53), then conservation of z-angular momentum 

requires that 

x

M V z

y ∂

∂ = − (A-6) 

so that 

4

4

2

2

2

2

12 (1 ) x

E h

t

y y

∂

∂

− = ∂

∂ δ

ν

δ

 (A-7) 

Consider a traveling wave solution of the form 

(x,t) sin[k(x c t)] δ y = − f (A-8) 

Here, cf is the flexural wave speed, and k is the wave number related to the wavelength λ by 

λ

2π k = (A-9) 

Substitution into equation A-7 yields the characteristic equation 

4

2

2

2 2

12 (1 )

k E h k c f −ν =

or 

0 0 6 3

3

c

h

c hk c f = = ⋅ λ

π (A-10) 

c0 is the axial compression wave speed given by 

(1 ν )ρ

0 2 − = E

c (A-11) 

0 c c f < whenever 

1

3

<

λ

π h

or 
33

h 1.814 h

3

> = π λ (A-12) 

For rod 7-NT, h = 12 mm and 0 c c f < whenever λ > 21.77 mm. 
34

INTENTIONALLY LEFT BLANK
35

Appendix B. Static Bending of Rods 7-NT and 7-T 

B.1 Analytical Solution 

In appendix A, we developed equations of motion for the dynamic bending of a plate in plane 

strain. We now consider the special case of quasi-static bending, so that inertial effects are 

negligible. Then δy, deflection of the plate’s mid-surface, and Mz, the resultant bending moment 

per unit z-width, can both be treated as functions of x only. Equation A-4 then becomes 

D

M x

dx

d y z ( )

2

2

= − δ (B-1) 

D is the plate’s flexural rigidity, a measure of its resistance to bending, and is given by 

12(1 )

2

3

−ν = E h D (B-2) 

E and ν are Young’s modulus and Poisson’s ratio, respectively.4 

Consider the specific boundary conditions shown in figure B-1a. The plate is point loaded at 

x = 0 and cantilevered at x = L. The applied load per unit z-width is F. The cantilevered 

boundary conditions are 

δ y (L) = 0 (B-3) 

(L) = 0

dx

dδ y (B-4) 

The rod’s free-body diagram and bending moment diagram are shown in figure B-1b and B-1c. 

The resultant bending moment on the cross section at location x is 

M z (x) = xF ; 0 < x < L (B-5) 

The substitution of equation B-5 into B-1 and the application of boundary conditions B-3 and 

B-4 yields the deflection solution 















 



 



  +



 



  ⋅ −  





⋅ − = −

3 3 2

y L

x

2

1

L

x

2

3 1

h

L

E

4(1 )F (x) ν δ (B-6) 

 4

Equation B-1 is derived in chapter 1 of Timoshenko and Woinowsky-Krieger (1959) as a special case of plate 

theory, in which the plate’s bent configuration involves only one non-zero radius of curvature.
36

(a) 

(b) 

(c) 

Figure B-1. The quasi-static, plane-strain plate problem studied in section B.1: (a) the plate is cantilevered at one 

end and point loaded at the other, (b) free-body diagram, (c) bending moment diagram. 

Equation B-6 is evaluated in figures B-2 and B-3. Material constants E and ν are assigned the 

values for WHA in table 1. F is set to 100 kN/m; L is set to 200 mm, the length of rods 7-NT 

and 7-T. Two different values are assigned to thickness h. In figures B-2 and B-3, we consider 

h = 12 mm, which is the diameter of the stock of rod 7, as seen in figure 2; 12 mm is therefore 

the thickness of the entire rod 7-NT and the thickness of rod 7-T based on the peaks of the 

threads. The other h value considered in figure B-3 is 10.72 mm. Since each thread has a 

0.64-mm dimension (figure 2), 10.72 mm is the thickness of 7-T based on the troughs between 

the threads. 

F

F

LF 

x

x = L 

0

Mz(x)

LF

x h

F 

y 

cantilevered 

L
37

Figure B-3 shows that an 11% decrease in h produces a 40% increase in δy at each x location. As 

a consequence of the 3 h dependence of D, small changes in h lead to large decreases in bending 

resistance. 

Figure B-2. The mid-plane deflection field computed for rod 7-NT with LS-DYNA 

closely agrees with the analytical solution for a plate with 12-mm thickness. 

Figure B-3. The mid-plane deflection field computed for rod 7-T with LS-DYNA is 

bounded by the analytical solution for plates with 12-mm and 10.72-mm 

thickness and is closer to the latter. 
38

B.2 FE Solutions for Rods 7-NT and 7-T 

Figure B-4 shows rod 7-T point loaded at its front end and cantilevered at its back end. We used 

LS-DYNA in implicit mode to perform a quasi-static analysis. Force per unit width, F, was 

quasi-statically raised to 100 kN/m. This amplitude was chosen so that strains nowhere 

exceeded 0.005, thus ensuring that the small strain and small displacement approximations were 

applicable. The meshes in figures 10 and 11 were again used for the two rods. Plane strain 

(ELFORM=13 on *SECTION_SHELL) was again imposed, but in these quasi-static analyses, 

the two-dimensional elements were fully integrated with four Gaussian quadrature points per 

element. Material *MAT_ORTHOTROPIC_ELASTIC was again applied in the degenerate 

isotropic case. The values for ρ, E, and ν in table 1 were specified. 

Figure B-4. The quasi-static plane strain problems studied in section B.2: rods 7-NT, 7-T, and 

7-TE are each cantilevered at one end and point loaded at the other. 

Figure B-2 shows the FE results for δy(x), the y-displacement field of the mid-surface, sampled at 

5-mm intervals of x in rod 7-NT. This FE solution is compared with the analytical solution from 

plate theory, equation B-6, evaluated with h = 12 mm. The FE solution for rod 7-NT agrees 

closely with this plate theory solution. 

The FE results for δy(x) in rod 7-T are added to figure B-3. The FE solution is bounded by the 

plate theory solutions for h =12.00 and 10.72 mm, and is closer to that for h = 10.72 mm. In this 

sense, the threads have decreased the effective h of rod 7 with regard to bending. 

Rod 7-TE, shown in figure B-5, is created from rod 7-T by the extension of the threaded region 

into the front and rear of the plate. Threads span 96.8% of the length of 7-TE, as compared with 

78.8% of 7-T. LS-DYNA results for 7-TE, point loaded at one end and cantilevered at the other, 

are added to figure B-3. As expected, δy(x) results from 7-TE lie between those from 7-T and the 

analytical solution for h = 10.72 mm. 

A plate’s effective thickness, heff, can be defined from the analytical solution in equation B-6 as 

( ) 1 3

2

eff (0)

4 1















 − = E y

F h L

δ

ν (B-7) 

F

x y 

cantilevered 
39

7-NT 

7-T 

7-TE 

Figure B-5. The profiles of rods 7-NT, 7-T, and 7-TE. 

Here, δy(0) is the midplane deflection at x = 0. Results for the three plates are compared in 

table B-1. The effective thickness of rod7-TE is 0.19 mm larger than the thickness of rod 7 

based on the trough between the threads. 

Table B-1. Effective thicknesses of the three plates 

plate δy(0) (mm) heff (mm) 

rod 7-NT 5.087 11.99 

rod 7-T 6.460 11.08 

rod 7-TE 6.750 10.91 
40

INTENTIONALLY LEFT BLANK
41

List of Symbols 

D flexural rigidity 

E Young’s modulus 

F applied force per unit thickness 

L rod length 

Mz resultant bending moment per unit z-width 

Tshear maximum Cauchy shear stress 

Txx, Tyy, Tzz, 

Txy, Txz, Tyz components of the Cauchy stress tensor 

T1, T2, T3 principal values of the Cauchy stress tensor 

Vy resultant shear force per unit z-width 

X, Y, Z material coordinates 

c0 speed of axial compression wave 

cf speed of flexural wave 

exx, eyy, ezz, 

exy, exz, eyz components of the infinitesimal strain tensor 

h plate thickness 

heff plate’s effective thickness 

k wave number 

t time 

ux, uy, uz displacement components 

x, y, z spatial (laboratory) coordinates 

δy y-displacement of the plate’s mid-surface 

λ wavelength 

ν Poisson’s ratio 

ρ material density 

}

}
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 BLDG 309 

 25 DIRECTOR 

 US ARMY RSCH LAB 

 ATTN AMSRD ARL WM TD 

 T BJERKE (5 CYS) 

 M RAFTENBERG (15 CYS) 

 M GREENFIELD S BILYK 

 E RAPACKI D CASEM 

 J CLAYTON 

 BLDG 4600 

 5 DIRECTOR 

 US ARMY RSCH LAB 

 ATTN AMSRD ARL WM TC 

 K KIMSEY L MAGNESS 

 D SCHEFFLER S SCHRAML 

 W WALTERS 

 BLDG 309 

 1 DIRECTOR 

 US ARMY RSCH LAB 

 ATTN AMSRD ARL WM TE B LOVE 

 BLDG 309 
 45

NO. OF 

COPIES ORGANIZATION

 1 P CHANTERET 

 INSTITUT SAINT-LOUIS 

 MULHOUSE 

 FRANCE 

 2 RAFAEL BALLISTICS CTR 

 M MAYSELESS ZVI ROSENBERG 

 PO BOX 2250 

 HAIFA 31021 

 ISRAEL 

 1 UNIV OF WATERLOO 

 MECHANICAL ENGNRG 

 M J WORSWICK 

 200 UNIVERSITY AVE 

 WEST WATERLOO 

 ONTARIO N2L3G1 

 CANADA 

 1 ARL ERO 

 S SAMPATH 

 AERO MECH ENG 

 223 MARYLEBONE RD 

 LONDON NW1 5TH 

 UNITED KINGDOM 

 1 AMC SCI & TECH CTR 

 EUROPE 

 T J MULKERN 

 POSTFACH 81 

 55247 MAINZ KASTEL 

 GERMANY 

 2 FRAUNHOFER-INSTITUT 

 ERNST-MACH-INSTITUT 

 V HOHLER E STRASSBURGER 

 ECKERSTRASSE 4 

 79104 FREIBURG 

 GERMANY