SHEAR BAND OBSERVATIONS AND DERIVATIONS OF REQUIREMENTS FOR A SHEAR BAND MODEL

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SHEAR BAND OBSERVATIONS AND DERIVATIONS OF REQUIREMENTS FOR A SHEAR BAND MODEL

L. Seaman, D. Curran, D. Erlich, T. Cooper, O. Dullum

To cite this version:

L. Seaman, D. Curran, D. Erlich, T. Cooper, O. Dullum. SHEAR BAND OBSERVATIONS AND DERIVATIONS OF REQUIREMENTS FOR A SHEAR BAND MODEL. Journal de Physique Col- loques, 1985, 46 (C5), pp.C5-273-C5-282. �10.1051/jphyscol:1985535�. �jpa-00224766�

JOURNAL DE PHYSIQUE

Colloque C5, supplCrnent au nos, Tome 46, aoQt 1985 page C5-273

SHEAR BAND OBSERVATIONS AND DERIVATIONS OF REQUIREMENTS FOR A SHEAR BAND MODEL

L. Seaman, D.R. Curran, D.C. Erlich, T. Cooper and 0. ~ullum*

SRI I n t e m a t i o n a Z , 333 Ravenswood Avenue, Menlo Park, C a l i f o r n i a 94025, U.S.A.

R6sum4 - On a analys6 les observations de cisaillement adiabatique dans des essais de fragmentation de cylindres et des essais d'impact et de pgnbtra- tion de projectiles, afin d'obtenir des indications pour le d6veloppement d'un modPle thgorique. Les essais de fragmentation de cylindres ont conduit B un modSle microm6canique des processus de germination et de croissance pour des bandes de cisaillement microscopiques. Le modsle d6velopp6 sur la base des observations dans les cylindres est capable de prgdire la limite balistique pour la pbngtration d'un projectile dans une cible d'acier. I1 reprgsente correctement l'ampleur et la nature de l'endommagement dans la cible pour un domaine de vitesses d'impact. Par contre, l'endommagement du projectile par bandes de cisaillement n'est pas bien repr6sent6 par ce mo- dPle. I1 a 6tG n6cessaire de modifier la relation d4formation-contrainte en y incorporant l'adoucissement thermique, et de retarder le d6but de la ger- mination des bandes dans le modsle afin de ~ouvoir reproduire la position et l'orientation des bandes dans le projectile.

Abstract - Observations of shear banding in fragmenting cylinders and in dynamic punch tests were analyzed for guidance in development of a theoret- ical model. The fragmenting cylinder tests led to a micro- mechanical model containing nucleation and growth processes for micro shear bands. The model constructed from the cylinder observations was able to predict the critical impact velocity for punching through a steel plate and to correctly repre- sent the extent and nature of the damage in the plate for a range of punch velocities. Shear band damage to the punching rod was not well described by this model. Instead, it was necessary to expand the stress-strain relation to include thermal softening and to delay the onset of band nucleation in the model to approximate the band locations and orientations in the rod.

I - INTRODUCTION

We are studying shear banding through several types of experiments, development of a micromechanical model based on the experiments, and simulations of the experi- ments. First, we studied contained fragmenting cylinder (CFC) experiments: these tests are useful because they produce relatively uniform strain and stress through- out the test cylinder at strain rates of interest (about 10~). From these data we fully characterized the micro processes of shear band development. Then we used the model based on these processes to predict rod penetration into hard steel plates, emphasizing the behavior of the plate material. This comparison was surprisingly satisfactory, suggesting the validity of the model and confirming that shear banding is the primary damage mechanism in the plate punching process. Finally, we simu- lated Taylor tests /l/ (impact of a rod of test material onto a hard anvil). In these tests the shear banding processes were not represented well by the model.

* Current address, Norwegian Defence Research Establishment, Division for Weapon and Equipment, P.O. Box 25 - N

-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985535

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From these comparisons we drew conclusions about the nature of shear banding and about the means for constructing a model to represent the phenomena under a range of experimental conditions.

I1 - NATURE OF THE SRI SHEAR BANDING MODEL

According to our observations in fragmenting cylinders, shear bands resemble penny- or half-penny shaped cracks: they are not planes as is usually assumed in theoretical studies. Figure 1 shows the most common orientation for shear bands in fragmenting cylinders and indicates their resemblance to shear cracks. They have the geometry of macroscopic dislocations. Bands occur with densities of 103 to 105/cm3. The planes of damage observed by others and named "shear bands" actually represent the coalescence of many of these micro shear bands. From the CFC tests, we conclude that the micro shear bands begin to nucleate at a critical strain level and that they appear in a range of sizes. With continuing strain, they grow by enlarging their radii. With more strain, they may coalesce with other bands to form fragments. The model contains these micro processes for the bands.

P - Length in Direction Perpendicular to Slip Motion d - Depth Along 45'

Slip Plane B - Shear Displacement

Along Slip Plane Z - Axial Direction

r - Radial Direction B - Circumferential

Direction

FIGURE 1 DOMINANT ORIENTATION FOR SHEAR BANDS IN FRAGMENTING CYLINDER EXPERIMENTS

The model (termed SHEAR4 /2,3/), which was developed to represent shear banding in fragmenting cylinders, is a multiple-plane plasticity model in which all plastic flow, shear banding, and tensile opening occurs on the selected planes. The chosen set of nine planes (seven for two-dimensional problems) is shown in Fig. 2. On each of the planes, the shear banding is represented by a density of shear bands with a range of sizes, not by individual shear bands. Nucleation of bands can occur in the model after the shear strain on a plane exceeds a threshold given by:

where E: is the plastic shear strain on the ith orientation plane and E: is the strain rate. The increase in the total number of shear bands is given by the nucleation rate:

where Ni is the number of bands on the ith plane. Similarly, the growth or enlargement of the radii R of the bands is given by

Here CG is a growth coefficient determined from the fragmenting cylinder observations. The extent of coalescence or fragmentation is determined from the dimensionless quantity D:

D = nyx N R ~ ( 4 )

FIGURE 2 RELATIVE LOCATIONS OF THE COORDINATE DIRECTIONS AND INITIAL ORIENTATION OF THE SHEARING PLANES

Full fragmentation occurs when D reaches 1. Shear strain in the material causes a shear offset between the faces of the bands, as indicated in Figure 1. When this offset reaches a critical level B = b / ~ , where b is a constant, growth of the band occurs. The ratio b was also found from the cylinder experiments. The model also provides for tensile opening, either in an elastic manner, following Sneddon and Lowengrub /4/, or by plastic deformation, using results of He and Hutchinson /5/.

Stress-strain relations in the model provide for isotropic behavior in the undamaged state and for the development of anisotropy caused by the plastic strain and damage.

111 - CALIBRATION OF THE MODEL WITH FRAGMENTING ROUND DATA

The SHEAR4 model has been developed both qualitatively and quantitatively from observations of CFC experiments. Here we introduce the experimental procedure, the measurements of shear bands, and the method of determining the micro processes from these measurements.

A schematic of the CFC experiment is shown in Fig. 3. In this experiment the test material is in the form of a cylinder, which is filled with a low density explosive

(low density so that the driving pressures are 5 to 20 GPa). Around the test cylin- der is a tube of PMMA or other plastic. These two cylinders are encased in a heavy steel tube. When the explosive is detonated, the explosive pressure expands the test cylinder into the plastic tube, thus introducing a large amount of shear strain into the test material at a high strain rate (104/s). This rapid expansion is halted by the large resistance of the outer steel tube. In the usual experiment, the test material is expanded to the point where there are many shear bands in the tube, but the tube is in one piece or a few pieces: the stage of full fragmentation has been avoided. By varying the density of the explosive (and thus the peak pres- sure) and the thickness of the plastic tube (and the amount of expansion of the test tube), we can study a range of damage from zero to fragmentation with the CFC test.

Following the CFC experiment, the cylinder is sectioned to reveal the shear bands on the inner surface (as in Fig. 4a). The shear bands appear as axial lines or cracks on the inner surface; actually, they are generally semicircular planes of separation within the material, as indicated in Fig. 1. The center of the semi- circle is usually on the inner surface of the cylinder. The rest of the shear band extends into the thickness of the tube at 45 degrees to the radius of the tube. The process of counting the shear bands is begun by photographing the bands on the inner surface of the tube. The photographs are divided into sections of similar levels of damage (sections A through F in Fig. 4a). Then the sizes of the bands are measured in each section. These counts are arranged into the cumulative size distributions

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Reduced Density

FIGURE 3 SCHEMATIC OF CONTAINED FRAGMENTING CYLINDER EXPERIMENTS FOR STUDYING SHEAR BAND KINETICS

Shear Bands

A 8 C D

C

(a) Cross Section of

a

Cylinder with Damage z k

5

RADIUS - cm (b) Surface Data

r a 2i'r

W a B \

a

RADIUS

-

(C) Volume Data

M A-7893-98 FIGURE 4 STEPS IN OBTAINING CUMULATIVE SHEAR BAND

DISTRIBUTIONS FROM CONTAINED FRAGMENTING CYLINDER DATA

shown in Fig. 4b. In this schematic representation, the maximum damage occurred in section D, near the center of the tube, and very little damage occurred in the end sections A and F; this distribution of damage is customary. Hence a range of damage is observed in each experiment. Next we use a statistical transformation due to Scheil / 6 , 7 / to obtain the volumetric distributions shown in Fig. 4c. These last distributions represent a shear band density and hence a state of the material.

Following the counting process, we simulate the CFC experiment and obtain the plas- tic strain history in each of the selected sections A through F in the tube. As one might expect, the maximum strain occurs in the middle of the tube and lesser amounts of strain occur near the ends. Hence there is a correlation between the amounts of strain and the numbers and sizes of shear bands. The increase in the numbers of bands is termed nucleation, and increase in the sizes is called growth. By plotting the numbers versus the plastic strain, we determined the factors in the nucleation equation (Eq. 2). A similar plot leads to the parameters in the growth equation (Eq. 3). For determining the nucleation and growth processes, we perform several (3 to 5) CFC tests with a range of conditions in an attempt to obtain at least one test with a very low level of damage and another with partial fragmentation. Then, when

the shear band parameters are determined, they represent fairly well the entire range of damage. Thus, from CFC tests in a material, we can characterize our shear band model.

A sample result obtained from a simulation of a CFC experiment using the SHEAR4 model is shown in Fig. 5. This cross section shows the configuration and strain

levels 100 ps after detonation. By this time, the expansion has ceased. The plastic strain contours shown are for plastic strain only in the plane that lies 45 degrees between the radial and circumferential directions and along the axis of the tube. The largest strain is clearly near the inner surface of the tube and near the midlength of the tube. Simulations such as this can generally "predict" the final position of the inner surface of the cylinder with high precision and can provide the numbers and sizes of bands along the cylinder within about 50%.

xis ^of Symmetry

JA-314522-77

FIGURE 5 PLASTIC STRAIN CONTOURS FOR THE PRIMARY DAMAGE MODE FROM ASlMULATlON OF A CONTAINED FRAGMENTING CYLINDER TEST OF 4340 STEEL OF RC 40 USING THE SHEAR4 MODEL: 1 0 0 ~ s AFTER DETONATION

From CFC experiments in several steels and in depleted uranium, and simulations of these experiments, we have concluded that the SHEAR4 model can represent shear banding in fragmenting cylinders, not with great precision, but satisfactorily for many purposes.

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IV - SIMULATIONS OF DYNAMICALLY PUNCHED PLATES

With the shear banding model based on CFC tests, we attempted to predict the process of punching of rods into hard steel plates. These calculations are purely predic- tive because the shear band model has been fully characterized from the CFC data.

Our use of the model was based on the assumption that (1) the primary failure mechanism of the plates (and rods) is shear banding, and (2) the shear banding in CFC experiments is the same as that occurring in the punch tests.

First, let us review some experimental information about the punching process. At the limiting velocity at which perforation of a tough steel plate just occurs, the rod severs a plug from the plate, leaving a fairly clean hole with a diameter like that of the rod. At lower velocities, the plate may be indented, but no cracking or shear banding appears. At velocities well above the limiting velocity, a hole much larger in diameter than the rod is made in the target, and the plug (and rod) are fragmented during the event.

We simulated an impact of a 4340 steel rod (R 4 0 ) into a rolled homogeneous armor (RHA) steel plate at 750 m/s, approximately tge limiting velocity. Figure 6 shows that full fragmentation occurred in the plate in cells that formed a line through the plate. There was little collateral damage in the target. Hence the plug will be forced out, the rod will continue through, but there will be little additional damage after the time shown in the figure; this result agrees well with our experi- mental data at this velocity.

(b) 12.3 ps After Impact

JA-314532-99A

FIGURE 6 SIMULATION OF A BLUNT 4340 STEEL ROD PENETRATING A RHA PLATE AT THE BALLISTIC LIMIT (750 mls)

Shaded cells were fully shear-banded in at least one orientation; cells with a cross contain some bands.

Many models have been able to predict a known limiting velocity. However, usually they also show penetration for a great range of impact velocities. So we tried a similar impact at 500 m/s. A dent was made in the plate, but no cells in the target were fragmented; this result also agrees with our observations.

Next we attempted to simulate an impact at 1550 m/s, over twice the limiting veloc- ity for this configuration. Figure 7 shows a large zone of damage ahead of the rod and no defined shear band that will release a plug. Again, this behavior is like that observed in highly overmatched cases: the rod is eroded, but passes through, accompanied by a shower of fragments from both the rod and the plate.

l

I

Fully Shear Banded

-

3

Fully Shear

0.50 0.0 -0.50 -1.00 -1.50 -2.00 -2.50 AXIAL POSITION (cm)

JA-314532-102A

FIGURE 7 SIMULATION OF A BLUNT 4340 STEEL ROD PENETRATING A RHA PLATE AT 1550 m/s (AT 28 us)

Thus it appears that, with parameters determined from CFC tests, SHEAR4 is able to predict the limiting velocity for penetration, extent of collateral damage, and the changing trend in the damage distribution in the plate. Hence many of the phenomena in the model must be accurate.

V - TAYLOR TEST SIMULATIONS

The Taylor test, the impact of a blunt rod onto a hard anvil, is very similar to the penetrating impact discussed above, so we expected to make satisfactory simula- tions with the SHEAR4 model. Here, however, the emphasis is on the rod instead of on the plate. Figure 8 shows views of a rod after impacting a quartz plate. The rod has been shear banded into a separate cone in the center, an outer ring, plus the remainder of the rod, which is now sharpened. We should expect to find these results in simulations of the Taylor test.

Figure 9 shows the results of simulations of a Taylor test of 4340 steel at RC 40 launched at 457 m/s. The contours show the plastic strains in the orientation that can lead to a cone at the front of the rod. The contour lines show that the most important shear strain levels are in the center at the impact plane, rather than in the regions that actually shear band in the tests.

Next, we considered the hypothesis that the stress-strain relations should include thermal softening and that the threshold criterion for shear banding should be adjusted. Figure 10 shows a yield curve for the material, indicating an initial work-hardening region, followed by a peak of the curve where thermal softening becomes important. But we now assume that, although this peak is important for softening effects, it is not sufficient for shear banding. Shear banding is assumed to occur at a strain somewhat farther along. Clifton and co- workers /8/ recently found analytically that the peak point of the curve may not cause instability, but some continuing strain is required.

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FIGURE 8 THREE VIEWS OF ASTEEL CYLINDER AFTER IMPACTING A QUARTZITE TARGET AT 140 mls

FIGURE 9 CONTOURS OF PLASTIC STRAIN I N THE DIRECTION TO FORM A CONE AT THE HEAD OF THE PROJECTILE: SHEAR4 SIMULA- TION OF A TAYLOR TEST WITH 4340 STEEL

FIGURE 10 PROPOSED STRESS-STRAIN PATH SHOWING THERMAL SOFTENING AND THE NUCLEATION THRESHOLD FOR SHEAR BANDING

With this expanded criterion for shear banding, we simulated the Taylor test again, this time with a thermal softening process, but no shear banding. Some results are shown in Fig. 11. Note that the computational grid is now severely distorted. Here we see the development of a region of intense shear that is forming a cone adjacent

to the impact plane. Now the largest strains are along the cone-forming plane and not along the axis or in the other shear band orientations (not shown) as before.

The current Taylor test results should be considered exploratory and not definitive because the yield curve used was selected arbitrarily and not for accuracy in repre- senting 4340 steel.

FIGURE 11 RESULTS OF ASHEARISIMULATION OF A TAYLOR TEST.

USING THERMAL SOFTENING: CONTOUR LINES ARE FOR THE CONE-FORMING ORIENTATION OF SHEAR BANDS

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From these comparisons, we conclude that some thermal softening process must be reflected in the yield curve to obtain the correct plastic strain distributions in the rod. The adjustments needed to handle phenomena in impacting rods suggest that the stress-strain relations are more important for rods. In the CFC experi- ment, the shear banding processes are greatly overdriven by the explosive, so the stress-strain relations have less influence. In the target ahead of the rod, the stress-strain relations are important in representing the anisotropy necessary to correctly orient the plane of separation; however, the material is well confined by the surrounding target. In the rod, however, there are free surfaces nearby, permitting large amounts of slip to occur. These differences in the three types of tests appear to be fruitful areas for pursuing our understanding of the shear banding processes.

V1 - CONCLUSIONS

The multiple-plane, micromechanical model SHEAR4 can represent CFC experiments in considerable detail, including the final deformed shape of the cylinder, numbers and sizes of shear bands, and the distribution of the bands throughout the length of the cylinder.

With the model based on CFC data, surprisingly accurate predictions can be made for rod impacts onto hard steel plates. The predictions include the limiting velocity for penetration, collateral damage in the plate, and the nature of the damage in the plate. At the present level of development, the model is not accurate for the damage in the rod. Preliminary calculations indicate that inclusion of thermal softening and a modification in the nucleation threshold would improve the accuracy of the simulations of the rod.

We further conclude that shear banding is the dominant failure mechanism in all three types of experiments, although the current results suggest that the shear banding process in the rod may be more dependent on the details of thermal softening.

AKNOWLEDGEMENT

This research was supported partially by the U.S. Army Research Office and partially by SRI International.

REFERENCES

/l/ Taylor, G. I., Proc. Roy. Soc. London A= (1948) 289-299.

/2/ Seaman, L., Curran, D. R., and Shockey, D. A., Scaling of Shear Band Fracture Processes, Proceedings of the U.S. Army Sagamore Conference on Solid Mechanics, July 1982, published as Material Behavior Under High Stress and Ultrahigh Loading Rates, John Mescal1 and Volker Weiss, Eds. (Plenum Press, New York, 1983).

/3/ Seaman, L., and Dein, J. L., Representing Shear Band Damage at High Strain Rates, Proceedings of IUTAM Symposium on Nonlinear Deformation Waves, in Tallinn, Estonia, August 1982, U. Nigul and J. Engelbrecht, Eds. (Springer-Verlag, Berlin and Heidelberg, 1983).

/ h / Sneddon, I., and Lowengrub, M., Crack Problems in the Classical Theory of Elasticity, (John Viley & Sons, Inc., New York, 1969).

/5/ He, M. Y., and Hutchinson, J. W., The Penny-Shaped Crack and the Plane Strain Crack in an Infinite Body of Power-Law Material, J. Appl. Mech., Trans. ASME 48,

(1981) 830-840.

/ 6 / Scheil, E., Die Berechnung der Anzahl und Grossenverteilung kugel-formiger

Krigtalle in undurchsichtigen Koerpern mit Hilfe durch einen ebenen Schnitt erhaltenen Schnittkreise, 2. Anorg. Allgem. Chem. 201 (1931) 259.

/ 7 / Scheil, E., Statistiche Gefugeuntersuchungen I, 2. Metallk. 27, (1935) 199.

/8/ Clifton, R. J., Duffy, J., Hartley, K. A., and Shawki, T. G., Scripta Met.18, (1984) 443.