EFFECT OF STRAIN RATE ON PLASTIC FLOW AND

FAILURE IN POLYCRYSTALLINE TUNGSTEN

T. DUMMER1, J. C. LASALVIA2, G. RAVICHANDRAN3 and M. A. MEYERS4{1Department of Mechanical Engineering, Universitat Karlsruhe (TH), Karlsruhe, Germany, 2MaterialsDivision, Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, Maryland 21005, U.S.A.,3Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California, U.S.A.and 4Department of Applied Mechanics and Engineering Sciences, University of California, San Diego,

La Jolla, CA 92093, U.S.A.

(Received 27 August 1996; accepted 22 June 1998)

AbstractPolycrystalline tungsten (less than 100 p.p.m. impurities) was subjected to dierent heat treat-ments to yield dierent grain morphologies and tested at quasi-static (3 103) and dynamic (1034103/s) strain rates. Three mechanisms of deformation were identified and evaluated: slip, twinning, and inter-granular cracking. Whereas plastic flow by slip has considerable strain-rate sensitivity in tungsten (which isfound to be well represented by the Mechanical Threshold Stress constitutive equation) the cohesivestrength of the grain boundaries was found to decrease with heat treatment temperature, but was insensi-tive to strain-rate changes. Low-strain-rate deformation yielded limited damage at strains as high as 0.25,whereas high-strain-rate deformation led to catastrophic failure at strains between 0.05 and 0.10. Slip andgrain-boundary decohesion being competing deformation mechanisms, the material undergoes a ductile-to-brittle transition as the strain rate is increased from 103 to 103/s. Two failure modes are identified:debonding initiated by shear along a grain-boundary facet (similar to the wing-crack mechanism) anddebonding initiated at voids. The interactions between microcracks and twins are characterized, and thereis both evidence of fracture initiation at twins (intergranular cracks), and twin initiation at cracks (trans-granular cracks). Calculations based on existing wing-crack models enable the estimation of the grain-boundary cohesive energies. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rightsreserved.

1. INTRODUCTION

Refractory metals such as tungsten (W) and its

alloys are finding increasing structural applications

which involve impact loading [13]. The high den-

sity (r= 19.3), high strength, good ductility, andhigh melting point makes tungsten an attractive

candidate material in such applications. One such

application involves kinetic energy penetrators

which are long rods fired from smooth bore, large

caliber guns in tanks; muzzle velocities above

1000 m/s are achieved, and rolled homogeneous

armor thicknesses exceeding 1 m can be pierced.

The massive clean-up eort after Desert Storm

revealed the major problem with the past gener-

ation of kinetic energy penetrators: they were made

of depleted uranium which has the desired density

(r= 19.05) and ductility, but is pyrophoric andhighly toxic after impact. A new generation of kin-

etic energy penetrators utilizing tungsten alloys

which have an equally high density is being con-

sidered. The mechanical properties of the material

at high strain rates is of great importance in the de-

sign and analysis of such applications. Although the

principal (non-uranium) technological materials

developed for a penetrator are W-based composites,in which the continuous phase is a NiFe, NiCoor other alloy, it is important to fully understand

the strain-rate dependent response of pure tungsten.Tungsten undergoes a ductile-to-brittle transition.

The transition temperature (DBTT) increases when

the strain rate is increased, leading to fracture withlittle or no macroscopic plastic deformation belowDBTT [4]. Lassila and Gray [5] reported a ductile-to-brittle transition in W as the temperature is

reduced from 400 to 228C. Ravichandran [6] andSubhash et al. [7, 8] obtained such a transition whenthe strain rate is increased from 103 to 103/s.Chiem and Shyan [9] have carried out similar exper-iments. In addition to slip, twinning and grain-boundary decohesion can play major roles during

deformation. The objective of the present investi-gation is to study and quantify the micromechan-isms responsible for the ductile-to-brittle transitionundergone by polycrystalline tungsten, when

deformed in compression under a wide range ofstrain rates.

2. EXPERIMENTAL PROCEDURES

The as-received material consists of powder met-allurgical, hot sintered, polycrystalline tungsten. It

Acta mater. Vol. 46, No. 17, pp. 62676290, 1998# 1998 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/98 $19.00+0.00PII: S1359-6454(98)00255-9

{To whom all correspondence should be addressed.

6267

was obtained from Sylvania GTE Products

Corporation in the form of cylindrical bars with adiameter of 25.4 mm. Tungsten powder was pressed

at a temperature of 12008C, then sintered at 26008Cfor 1 h by self-resistance heating in a hydrogen at-

mosphere. The particle size of the powder was ap-proximately 4 mm. The sintered rods of 1 inchdiameter were then rolled at 12008C to the final di-ameter of 6.35 mm. The average chemical compo-

sition of the material as determined by the

manufacturer is given in Table 1. It is consideredhigh purity tungsten, with a total impurity content

of less than 0.01%.In order to obtain dierent grain morphologies,

the as-received tungsten was subjected to threedierent heat treatments. All the material was

annealed in a vacuum furnace at 17508C for 15 h inorder to recrystallize the textured microstructureproduced by the rolling process. In order to obtain

three dierent grain sizes, portions of the tungstenrods were further heat treated at 26008C for 3.5 hand portions at 28008C for 3.5 h. The heat treat-ments were conducted in an ultra-pure dry argon

atmosphere (impurities in p.p.m.: H2O, 3; N2, 6;O2, 2; THC, 1) to prevent both oxidation and

hydrogen embrittlement.

After heat treatment, the tungsten rods [6.35 mmdiameter (D)] were cut into cylindrical specimens

for mechanical testing. The quasi-static testing wasconducted using a servo-hydraulic material testing

system and the high-strain-rate testing was doneusing a split Hopkinson (Kolsky) pressure bar sys-

tem. The specimens for quasi-static testing were cutto a length of L= 6.35 mm (L/D01). The speci-mens for dynamic split Hopkinson bar compressiontests were cut to a length of L= 3.68 mm in ac-

cordance with the recommended ratio L/D00.5 [10]. Finally, the faces of the cylinders wereground flat and parallel. The split Hopkinson bar

experiments were carried out at strain rates of 103,3103, and 4103/s. The bars in the splitHopkinson bar set-up were made of maraging steel12.5 mm in diameter. The details of split

Hopkinson pressure bar testing of metals can befound in Follansbee [10].

In order to prevent indentation of the specimen

into the bar interfaces, tungsten-carbide (WC) pla-tens were placed between each specimen-bar inter-

face. It is necessary to match the acousticimpedances (rcA) of the bar material and that ofthe platens so as to minimize spurious reflections atthe interfaces which will aect the reflected and

transmitted signals. Due to the dierence in acous-tic impedance of the bar material (steel) and platens

(WC), the diameter of the platens was dictated by

the impedance equilibrium

A1r1c1 A2r2c2 1where A1, A2, r1, r2, c1, and c2 are the cross-sec-tional areas, densities, and longitudinal bar wavespeeds, respectively, for the bar and the platen ma-

terial. The impedance matching yields a WC platendiameter of 8.4 mm for a bar diameter of 12.5 mmfor maraging steel. The use of platens to minimize

indentation into the bars and pulse distortion hasbeen analyzed in detail by Chen et al. [11].

The pulses in a split Hopkinson bar contain oscil-lations (PochammerChree) due to geometric dis-persion of wave propagation in the bars. To

minimize the high frequency components in theinput pulse, copper cushions were placed at theinterface between the incident bar and the striker

bar. This yields a stress pulse which is relatively freeof oscillations within the incident bar after impact

of the striker. By adjusting the diameter and thick-ness of the cushion, it is possible to achievereflected signals of constant amplitude. Thus a

constant strain rate during deformation can beachieved. Further details of this technique are givenby Nemat-Nasser et al. [12]. To limit the extent of

deformation of the specimen, a stopper ring wasused. The stopper ring is made of a high strength

steel alloy. The stopper ring has a much largerouter diameter than the specimen and the length issmaller than the specimen and is placed concentri-

cally over the specimen. The inner diameter is cho-sen such that no contact between the specimen and

the ring takes place during deformation. When thespecimen is initially subjected to deformation, thestopper ring does not carry any load but when the

specimen is deformed to the same length as thestopper ring, the ring also carries load. In the caseof high-strain-rate deformation, the amplitude of

the incident pulse is chosen such that the specimendoes not undergo further deformation once the ring

begins to carry load. Under quasi-static loading,with displacement control, a sudden increase inload due to the contact with stopper rings can be

used to terminate the deformation of the sample.The dierence in length between the specimen andthe stopper ring determines the maximum strain ex-

perienced by the specimen. Further information onthe use of stopper rings for high strain rate com-

pression testing are given by Ref. [12]. Strains assmall as 0.02 can be achieved using the stopperrings.

The quasi-statically and dynamically tested speci-mens were characterized by optical and scanning

Table 1. Average chemical composition of the tungsten used in the present study

Impurity Fe Mo Ni K Na C O N

Content (p.p.m.) 4 18 2

electron microscopy. The fracture surfaces of thecatastrophically failed samples were further investi-

gated via scanning electron microscopy. Thepolished surfaces were lightly etched using a watersolution of 5% K3Fe(CN)6 and 5% KOH. The

samples were swabbed with the etchant for 515 s.A quantitative analysis of the damage introduced tothe tungsten specimens by mechanical testing was

carried out by means of optical microscopy. Sixmicrographs at a magnification of 100 were takenof the longitudinal section of each sample. Care

was taken to align the specimen loading axis paral-lel to one axis of the micrograph. Only a light etch-ing was applied in order not to alter the crackshape nor to increase the width of the grain bound-

aries which would make the distinction betweengrain-boundary debonding and intact grain bound-aries dicult. Correct etching and observation

enabled distinguishing between grain-boundarycracks and grain boundaries.

3. RESULTS AND DISCUSSION

3.1. Eect of heat treatments on microstructure

The as-received microstructure is shown inFig. 1(a). In the plane perpendicular to the rolling

direction an equiaxed grain structure is evident. Thelongitudinal planes show a strongly elongated struc-ture in the rolling direction. Body-centered-cubic

metals are known to exhibit a h011i crystallographictexture in either the rolling, forging or extrusiondirection. The apparent grain diameter, obtained by

the linear intercept method and multiplying by1.5 [13], is equal to 25 mm. Annealing the specimensat 1750, 2600, and 28008C yielded grain diametersequal to 83.7, 98.4, and 111.0 mm, respectively.Note that the 2600 and 28008C treatments were pre-ceded by the 17508C treatment. The elongation ofthe grains, still present after the 17508C treatment(intercepts are: lx=45.8 mm; ly=43.1 mm;lz=60.4 mm), essentially disappears after the 2600and 28008C treatments. The structure of the ma-terial annealed at 26008C is shown in Fig. 1(b).

3.2. Mechanical characterization

The stressstrain response of the W specimen

subjected to the dierent heat treatments is shownin Fig. 2. The quasi-static curves are solid lines andthe dynamic curves are dashed-dotted (103/s) and

dashed (4103/s) lines. The quasi-static curves ofthe annealed conditions are very similar; the yieldstress is approximately 0.75 GPa and is not a strong

function of grain size. The progressive disappear-ance of the yield phenomenon with increased treat-ment temperature is caused by impurity segregation

to the grain boundaries. Unlocking of the dislo-cations results and causes the more round yieldpoint. The as-received condition exhibits a higheryield stress (01.2 GPa) and a lower work-hardening

rate because it is in a hardened condition, contain-ing a higher dislocation density. The high strain-

rate sensitivity of the as-received material isexpected; the work-hardening rates at low and highstrain rates are similar. This is the classical response

for b.c.c. metals, in which the plastic flow is gov-erned by overcoming of the Peierls barriers. Zerilliand Armstrong [14] describe this behavior well;

Meyers et al. [15] obtained a similar response fortantalum. On the other hand, f.c.c. metals like cop-per show a strain-rate dependence of work

hardening [16]. It should be noted that at highstrain rates, the inherent oscillations due to elasticwave dispersion and lack of stress equilibrium atlow strains make the determination of yield inaccur-

ate. The yield stress increased from 1.2 to 2 GPa forthe as-received material (a 40% increase); thisincrease was from 0.8 to 1.15 GPa for the heat-trea-

ted conditions, a similar fractional increase. Zureket al. [17] tested similar material (tungsten plate)and obtained results that closely match the as-

received material in this investigation: low workhardening and a plastic flow plateau that increasedfrom01.5 GPa at 103/s to 2 GPa at 2 103/s. Thismaterial was, according to Zurek et al. [17], in awork-hardened condition.The failure strain at strain rates on the order of

0103/s of the heat treated tungsten (at 1750, 2600,and 28008C) was found to be approximately 00.07;this is significantly lower than under quasi-staticloading for the corresponding annealing condition.

The specimens fragmented at a strain of 00.25under quasi-static loading.

3.3. Constitutive modeling

The description of the mechanical behavior of

tungsten is modeled using the ZerilliArmstrong(ZA) and the Mechanical Threshold Stress (MTS)constitutive models.In the ZA model for b.c.c. materials, the depen-

dence of the flow stress s on strain e, strain rate,and temperature T is given by

s C0 C1 expC3T C4T ln _e C5enp: 2The athermal stress C0 takes into account the grainsize through a HallPetch relation,

C0 s0 kd1=2, where s0 is the intrinsic athermalstress contribution, k is a strength coecient and dis the average grain size for the material. C1, C3,

and C4 take the strain-rate and temperature and C5the strain-hardening eects on the flow stress intoaccount. Equation (2) takes into account that b.c.c.

metals do not exhibit a strain-rate and temperaturedependence of the work hardening and grain-boundary strengthening (both terms are additive).

The MTS model uses the basic ingredients of thethermally activated dislocation model for plasticflow and includes the eect of strain rate. It isbased on the dislocation concepts advanced by

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6269

Fig. 1. Longitudinal and transverse sections of: (a) as-received W and (b) W annealed at 26008C.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN6270

Kocks et al. [18] and was originally developed byFollansbee and Kocks [19, 20]. The MTS constitu-

tive model can be expressed in the form [21] forstress s in terms of the athermal (sa) and thermal(st) components. For thermally activated glide pro-cesses, the relation between the applied stress andthe mechanical threshold stress can be expressed as

stm

1

kT

g0mb3ln

_e0_e

1=q1=p s^tm0 St_e,T s^tm0

3

where St is a structure factor which is a function ofthe strain rate and temperature as shown above.

The thermal component of the stress consists of

two parts, one which is intrinsic (si), and whichprovides the rate dependence of the flow stress due

to intrinsic barriers such as the Peierls stress whichdoes not evolve after yielding and the other, (se),which evolves with strain due to dislocation multi-

plication, i.e. work hardening. Hence the constitu-tive law can be rewritten as

sm sam Si_e,T s^im0

Se_e,T s^em04

where m is the temperature dependent shear mod-ulus and m0 the shear modulus at a reference tem-perature. The work hardening is taken into accountin the form of a Voce law. The details of the MTS

model applied to tungsten can be found in Chenand Gray [21].

The temperature T during high-strain-rate adia-

batic deformation is calculated using

T T0 brce0

s dx 5

where b is the fraction of rate of plastic work con-verted to heat which is taken to be a constant, 0.9

in our modeling. T0 is the ambient temperature,

and r and c are the mass density and the specificheat of the material, respectively.

In Fig. 3(a), the flow stresses at a plastic strain of

0.03, determined from the present mechanical test-

ing, are plotted vs the strain rate and compared

with the ZerilliArmstrong model. The parameter

C1 is chosen in a way that the stressstrain rate

curve predicted by equation (2) passes through the

data points of the present quasi-static tests for each

material condition. Under quasi-static conditions

each material yielded by plastic flow as indicated by

the stressstrain curves (Fig. 2). In the case of the

as-received tungsten the flow stress at a strain rate

of 3103/s matches closely the one predicted bythe ZA model. This indicates that the parameters

obtained by Chen and Gray [21] are applicable to

the present material and that under dynamic con-

ditions the as-received tungsten yielded by plastic

flow. The flow stress of the heat-treated condition is

approximately 0.8 GPa at low strain rates. The par-

ameter C1 was modified for the annealed conditions

Fig. 2. Stressstrain curves for W at dierent strain rates: (a) as-received; annealed (b) at 17508C, (c) at26008C, and (d) at 28008C.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6271

and a good fit was obtained. The ZA model par-

ameters for the as-received and the annealed (1750,

2600, and 28008C) tungsten, are given in Table 2.At high strain rates, the heat-treated tungsten at all

annealing conditions (1750, 2600, and 28008C) exhi-

bits flow stresses at a strain of 0.03 well represented

by the ZA constitutive description.

It should be pointed out that, when another

rationale was used to implement the ZA model, a

very poor fit was obtained. This second approach

Fig. 3. Comparison between the observed flow stress dependence on stress rate and constitutive descrip-tion by (a) the ZerilliArmstrong model and (b) the MTS model; parameters from Chen and Gray [21].

Table 2. Parameters for the ZerilliArmstrong (ZA) and MTS models

C0 (MPa) C1 (MPa) C3 (/K) C4 (/K) C5 (MPa) n

0 $ 0.002 0.0001 800 0.6

m0 (GPa) D (GPa) T0 (K) k/b3 (MPa/K) sa (MPa) si (MPa) E0i (/s) g0i qi pi

159.5 33.69 1217 0.671 100 % 108 0.142 3/2 1/28

e0e (/s) g0e qe pe y0 (MPa) a g0es ses0 (MPa) e0es (/s)

107 1.6 1.0 2/3 3000 1 1.775 1035 107

$3000 MPa (Chen and Gray), 2800 MPa (as-received), 1600 MPa (annealed). Parameters for the mechanical threshold stress (MTS)model.%3230.2 MPa (Chen and Gray), 2900 MPa (as-received), 1600 MPa (annealed).

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN6272

uses the physically correct assumption that the as-received material is severely work hardened and

that, additionally, there are dierences in grain size.Both work hardening and grain-size dierences canbe incorporated into the ZA equation through the

parameters C0 and C5, respectively. Body-centered-cubic metals do not exhibit a strain-rate and tem-perature dependence of the work hardening and

grain-boundary strengthening [both terms are addi-tive in equation (2)]. The grain-size parameter C0and work-hardening parameters C5 and n were cho-

sen such that the stressstrain rate curve predictedby equation (2) passes through the data points ofthe present quasi-static tests for each material con-dition. In the case of the as-received tungsten the

flow stress at a strain rate of 3 103/s matchedexactly that predicted by the ZA model at both lowand high strain rates, in accordance with Chen and

Gray [21]. The 0.5% flow stress of the annealedconditions is approximately 0.8 GPa at low strainrates. As mentioned, the strain-rate dependence of

the flow stress is not sensitive to grain size and dis-location density; this is equivalent to shifting thecurve of the predicted stress downwards retaining

the other ZA parameters. At high strain ratesannealed conditions exhibited a 0.5% flow stresslower than that predicted by the ZA constitutivedescription. This indicates very clearly that yielding

at high strain rates of the three annealed conditionsis not well represented by the ZA equation underthese assumptions.

The comparison of the experimental results withthe MTS model is shown in Fig. 3(b). For the as-received tungsten, the parameters of Chen and

Gray [21] were used, with only a minor adjustmentin si (from 3230 to 2900 MPa) to take into accountthe slightly lower yield stress observed in the cur-rent experiments. For the annealed conditions, siwas reduced to 1600 MPa to obtain a fit at 103/s.All other parameters by Chen and Gray [21] wereretained and are shown in Table 2. The fit is found

to be excellent both for the as-received andannealed conditions.The results shown in Fig. 3 show very clearly

that both the ZA and MTS models represent theoverall response of the tungsten satisfactorily, andthat the onset of inelastic deformation can be

attributed to dislocation activity for all, as-receivedand annealed, conditions.

3.4. Characterization of damage

We classify damage into microcrack and macro-crack damage. Microcracks have a scale on the

order of the grain size, and macrocracks of the spe-cimen size. The most obvious feature concerningdamage in the heat-treated materials is that it is

governed predominantly by grain-boundary decohe-sion. Figure 4 gives an example of the appearanceof damage within the heat-treated material whensubjected to high strain rates. Longitudinal and

cross sections of a specimen annealed at 26008Cand deformed to a strain of 0.05 at a strain rate of103/s are shown. Damage by macrocracks (markedby arrows 1) parallel to the loading direction is

clear. This failure mode is commonly referred to asaxial splitting [22, 23]. The dark dots (arrow 2) rep-resent grains that were pulled out during polishing.

They had been separated by debonded interfaces.Note that the specimen shown in Fig. 4 did not fail.

The transverse section shows, Fig. 4(a), that macro-cracks form radially, starting at the specimen per-imeter and propagate towards the center. The

center of this section exhibits numerous linkedmicrocracks with no favored orientation.Figure 5 shows representative areas from longi-

tudinal sections for the materials annealed at 1750and 26008C and deformed quasi-statically to astrain of approximately 0.25 (loading direction istop to bottom). The 17508C annealed material,Fig. 5(a), exhibits a preferred orientation in grain-

boundary debonding (microcracks are indicated byarrows), corresponding to the axially texturedmicrostructure. In the equiaxed 26008C annealedcondition, Fig. 5(b), the debonding appears to beless oriented.

The amount of damage accumulated within thespecimens tested dynamically is much higher thanfor the ones deformed quasi-statically, at the same

plastic strain. For the quasi-statically tested speci-mens, the shape of the stressstrain curves and thepresence of low damage after straining above 0.20

indicates a typical ductile process, as shown byBilello [24]. Figure 6 shows the as-received W after

dynamic testing; a mixture of intergranular andtransgranular (arrows) fracture characterizes the as-received material. In quasi-static testing, the as-

received material did not exhibit microcrackdamage up to a strain of 0.20. The 26008C annealedspecimen deformed to a strain of 0.052 at a strain

rate of 4103/s as shown in Fig. 6(b); mainly inter-granular cracks characterize the annealed material.An axial macrocrack is indicated by arrow 1. The

right-hand side of the image shows misalignedmicrocracks which are frequently related to twin-

grain-boundary intersections (arrows 2). Optical mi-croscopy at higher magnification reveals that besidegrain-boundary failure, twin formation is associated

with transgranular fracture.The Appendix A provides a quantitative analysis

of crack orientation and density for the variousconditions and strain rates. These results show in aclear fashion that (a) the damage by micro- and

macrocracking increases with increasing strain andstrain rate and (b) that the angular distribution ofcracks is initially random (microcraks) and becomes

progressively more aligned with the loading axis asthe cracks grow or coalesce (macrocracks).

Figure 7(a) (from W annealed at 26008C andtested at 103/s) shows a twin intersection that nucle-ates a microcrack from the twin tips (arrow 1).

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6273

Argon and Maloof [25] also observed this in tung-sten [011] single crystals and Edmondson [26] in[011] in iron. They concluded that a crack could be

formed when the twin intersection line lies in ah001i cleavage plane; in this case the shear due tothe crossing twin can be accommodated by nuclea-tion of cleavage cracks. Propagation of these cracksrequires further deformation, since the driving ten-

sile stress is exhausted after the crack is initiated.Figure 7(a) also shows a twin crossing the entire

grain and connecting the opposite grain boundaries(arrows). At the twin-grain boundary intersectionsgrain-boundary failure occurred. In this case, it is

obvious that the twin initiated microcracking and it

is unlikely that both cracks preceded the twin in

such a configuration [27]. The micrograph in

Fig. 7(b) shows cracks within twinned areas, indi-

cated by 3. These cracks may have been nucleated

during twin formation to relieve the tensile stresses

within the twin. These cracks were also observed by

Subhash et al. [8] in h011i textured polycrystallinetungsten. Argon and Maloof [25] concluded that

these cracks are initiated at the twin boundaries

and can extend over the twinned region due to

further deformation.

Fig. 4. W annealed at 26008C, deformed to a strain of 5% (_e=103/s): (a) cross section and (b) longi-tudinal section.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN6274

The tips of two twins in Fig. 7(b) intersect with

two planar transgranular cracks, marked by arrows

4. Subhash et al. [8] observed this also in h011i tex-tured tungsten. It supports the conclusion that the

cracks initiated twinning. This is indicated by the

dierence in contrast of the twinned and untwinned

area. Subhash et al. [8] and Argon and Maloof [25]

proposed that if the twins were formed first, the

Fig. 5. Annealed W quasi-statically tested to025% strain: (a) 17508C; (b) 26008C.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6275

cracks would have entered the twins and the con-

trast across the crack surfaces would be the same.

The present result confirms their proposal. The

above observations of twin-transgranular crack in-

teraction are very rare compared to the frequency

of grain-boundary cracking not related to twinning.

Fig. 6. (a) Longitudinal section of as-received W, deformed to a strain of 9.4% (_e=3 103/s); (b) longi-tudinal section of W annealed at 26008C, deformed to 5.2% (_e=4 103/s).

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN6276

Fig. 7. (a) Crack nucleation at twin intersection (1) and at twin-grain boundary intersection (2); (b)microcracks between twin boundaries (3), and twin nucleation at microcracks (4); W annealed at

26008C.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6277

It is concluded that twin initiation does not signifi-

cantly alter damage evolution. Figure 8(a) is a SEMmicrograph showing the intergranular fracture sur-face of a 26008C specimen deformed to 0.05 strainat a rate of 103/s. The bonds between the adjacent

grains seem to be mainly intact. Figure 8(b) shows

the fracture surface of a 26008C specimen,deformed at a strain rate of 4103/s. Several twinsappear at the grain boundaries (arrows). The twin-ning incidence is higher than at 103/s, Fig. 8(a).

Fig. 8. Fracture surface of 26008C annealed W, deformed (a) at 103/s and (b) at 4 103/s.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN6278

4. ANALYSIS

4.1. Damage mechanisms

The quantitative analysis presented in the

Appendix A clearly shows that (a) damage bycracking increases with increasing strain (at a con-stant strain rate) and strain rate (at a constant

strain) and (b) that the angular distribution ofcracks becomes progressively more aligned with theloading axis as the cracks grow or connect. Thus

microcracks (which have a random orientation)gradually evolve into axial splitting (which isaligned with the compression axis).

The experimental observations presented inSection 3.4 indicate that there are three competingdeformation mechanisms in tungsten: slip, twinning,and grain-boundary decohesion. Due to the strain-

rate sensitivity of the flow stress, twinning andgrain-boundary decohesion are the favored mechan-isms under high-strain-rate conditions. The

annealed conditions exhibited grain-boundary deco-hesion after the onset of plastic flow, due to grain-boundary segregation and associated grain-bound-

ary embrittlement. At high strain rates, the localtensile stresses near the initially present flawsexceeds the grain-boundary cohesive strength at lowplastic strains and can lead to the initiation of inter-

granular brittle fracture immediately after plasticyielding.Figure 9 shows the stressstrain curves for the

material annealed at 17508C at two strain rates:103 and 4 103/s. Also indicated in this figure is aband bound by sic and sfc. These stresses mark theinitiation and finish, respectively, of grain-boundarydecohesion, and are shown as strain-rate insensitivein Fig. 9. This is a reasonable assumption, because

decohesion occurs with a minimum of plastic defor-

mation and thermal activation should play only a

minor role. Figure 9 shows in a very clear fashion

that decohesion starts immediately, upon plastic de-

formation, at 4103/s, whereas it initiates ated00.145 at 103/s. It is possible to develop ex-pressions that describe, in a quantitative fashion,

the temperature and strain-rate sensitivities of the

various mechanisms; this is done, in a simplified

fashion, below.

The grain-boundary decohesion stress decreases

with grain size. The eect of increasing the grain

size on the grain-boundary decohesion stress is to

increase the impurity concentration (P, S, and O) at

the grain boundaries. Due to the low solubility for

interstitial atoms in tungsten, the total amount of

embrittling grain-boundary segregates is expected to

be nearly the same in each heat treatment con-

dition. The grain-boundary impurity concentration

cGB in mol/unit area then is directly proportional to

the grain-boundary area per unit volume. Thus [13]

cGB n pd3

6

where n is the impurity content in mol/unit volume

and d is the grain size. In accordance with Seah [28],

a diminution in the grain-boundary decohesion

stress DsGBcGB is caused by an increase in cGB.For simplicity, it is assumed that DsGB increaseslinearly with cGB. The grain-boundary decohesion

stress sGB can be then described as

sGB sic DsGBcGB and sGB sic kd 7where sic is the decohesion stress of a cleanboundary and k is a constant. The slope k is given

by the total amount of embrittling segregates and

their embrittling activity.

Fig. 9. Schematic description of interactions between plastic deformation and damage at dierent strainrates by means of stressstrain curves of the 17508C W: sic, initiation of damage; sfc, catastrophic fail-

ure of specimen.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6279

It is known that both the flow and twinning stres-ses, sy and sT, can be described by HallPetch re-lationships as a function of grain size [29, 30]:

sy s0 kSd1=2 8

sT s0T kTd1=2 9where s0 and s0T are the frictional stresses for slipand twinning, respectively, and d is the grain size.The HallPetch slope for twinning, kT, is known tobe considerably higher than the one for slip, kS.

For b.c.c. metals, Armstrong and Worthington [31]report 2

Fig. 10. (a) Axial wing cracks nucleated from shear displacement of the grain boundary inclined tothe loading axis; (b) SEM micrograph showing the grain-boundary voids on the fracture surface of a

sample annealed at 26008C.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6281

Fig. 11. (a) Schematic of wing-crack formation from the tips of a tilted sliding crack. (b) KI acting at awing-crack tip vs normalized crack length L, for two crack geometries and frictions m= 0.1, 0.3,

and 0.5.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN6282

Poissons ratio (n) of tungsten are 414 GPa and0.29, respectively.In the case of annealed tungsten material, the

sliding on microcracks [e.g. Fig. 10(a)] presumably

takes place during the elastic region of the stressstrain curve. However, it appears that damage bynucleation of wing cracks along the weak grain

boundaries does not appear until the stress levelreaches s= 1200 MPa. The quasi-static stressstrain curve [Figs 2(b)(d)] exhibits considerablehardening during early stages of deformation typi-cal of annealed materials. Also, note that the ma-

terial yields around 0.75 GPa and does not developdamage until strains of approximately 0.25 and hashardened to a stress level of 1.2 GPa. In the case of

dynamic stressstrain response, it is seen that thematerial yields at around 1.2 GPa and develops

considerable damage even at very small strains of0.02 (e.g. Fig. 9). Based on these observations, forour analysis using the results from the wing-crack

model, we will assume that the wing cracks nucleatewhen the applied stress level reaches 1200 MPa.Due to the predominantly intergranular fracture

in the 1750 and 26008C annealed tungsten, thegrains need to be mapped in a plane in order to

apply the above model. The grain-size estimation(Section 3.1) revealed an equiaxed grain structurefor the 26008C and a 36% grain anisotropy (in theloading direction) for the 17508C annealed tungsten.In the case of the 26008C annealed tungsten, thecalculated spherical grain diameter d was set as the

diameter of the average two-dimensional grain. Inthe case of the 17508C annealed material, the cracklength 2a is determined to be 36.2 mm and y and cto be 52.38. This enables the utilization of thewing-crack equations to grain-boundary debond-

ing. In the case of the 26008C annealed material,the crack length 2a is determined to be 50 mm, andy and c to be 608. The coecient of friction Z fortungsten is taken to be 0.1, 0.3, and 0.5, which aretypical sliding friction coecients for hardmetals [33]. The values for 2a, c, and y wereinserted into equation (10) to calculate the criticalmode I stress intensity KIc at the tip of a represen-

tative sliding crack at crack initiation. The criticalstress intensity factor KIc for the annealed materialcan be calculated from equation (10), using the

above values for a, Z, c, y, and s1. They are shownin Fig. 11(b). These values are 1.5 MPa m1/2 for the17508C and 0.66 MPa m1/2 for the 26008C annealedmaterial for L=0 and m=0.5. The fracture tough-ness for the intergranular failure appears to be

much smaller than transgranular failure in tungsten,typically 50 MPa m1/2. The cohesive energies of thegrain boundaries computed using equation (11) are

5 and 0.96 J/m for the 1750 and 26008C annealedmaterials, respectively. The magnitude of cohesiveenergies supports the concept of increasing em-

brittlement with annealing temperature discussed inSection 4.1. The grain boundaries in the 26008C

annealed material are weaker than those in the17508C annealed material.It should be noted that the process of damage ac-

cumulation occurs gradually, meaning that a distri-bution of voids and grain boundaries does exist

which require dierent critical stresses to becomeactivated. The above model assumes that in bothmaterials damage initiates at a constant stress level

equal to 1200 MPa as macroscopically indicated bythe dynamic stressstrain curves. But it is reason-able to assume that within the 26008C material thenumber of debonded grain boundaries or voids islarger than in the 17508C annealed tungsten andthat damage is initiated at a lower stress level. Thisis confirmed by the observation of more embrittled

grain boundaries in the 26008C annealed material.

4.3. Failure originating from grain-boundary voids

A second mechanism for axial grain-boundaryfailure under high-strain-rate compression is due to

pre-existing grain-boundary voids, acting as stressconcentrators. Scanning electron microscopy of the1750 and 26008C annealed material shows that pre-existing voids were formed predominantly at thesegrain-boundary triple points, see Fig. 10(b). Thesevoids are sources for initiation of grain-boundary

failure, acting as stress concentrators. Figure 12(a)shows a spherical void schematically. Based on elas-tic theory, Goodier [34] formulated a solution forthe hoop (circumferential) stress acting at the north

and south pole of a spherical void in a uniaxialstress field as

syyy 0,p;r a 32

1 57 5 s11

1

2s1 12

where r is the radial coordinate of the void, a is its

radius, and n is Poissons ratio. Thus, the maximumtensile stress syy acts at the poles of the void withrespect to the direction of s. The maximum tensilestress is roughly half of the far-field applied stressfor a Poissons ratio of 0.3 which is a typical valuefor metals. If tension cracks are initiated they will

grow in the loading direction, as indicated inFig. 12(a).Sammis and Ashby [35] developed an expression

for the mode I stress intensity acting at the tip of a

crack of length l emanating from the poles of avoid of radius a situated in a field of uniaxial com-pression [Fig. 12(a)]:

KI L 01=2 0:621 L 04:1 s1

pap 13

where L is the normalized crack length and equalto l/a. From Fig. 10(b) the radius a of a representa-tive grain-boundary void within the material

annealed at 26008C is estimated to be 0.5 mm. Thestress intensity acting at the tips of initiated crackscan be calculated by means of equation (13). Figure12(b) shows the stress intensity caused by a void of

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6283

Fig. 12. (a) Schematic of axial crack initiation by spherical void; (b) stress intensity acting at the tips ofan axial crack, nucleated by a grain-boundary void as a function of the normalized crack length L.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN6284

radius a= 0.5 mm as a function of the normalizedcrack length L. The applied compressive stress is

taken as s1=1200 MPa, close to the macroscopicyield strength under high-strain-rate conditions, seeFigs 2(b) and (c). The stress intensity at crack in-

itiation is much lower than that achieved due to thedisplacement of a sliding crack shown in Fig. 12(b).Furthermore, KI decreases faster for increasing

crack length when caused by a void than by a tiltedsliding crack. It appears that a void of radius0.5 mm is not capable of causing the completedebond of the aected grain-boundary facet. TheSEM micrograph in Fig. 10(b) shows numerousvoids of similar radius aligned at grain-boundarytriple points. Overlapping of the stress fields can

cause an increase in stress intensity that, eventually,can cause debonding of the aected grain bound-ary. Therefore, the distribution of these grain-

boundary voids seems to be very important.Randomly distributed, their eect on grain-bound-ary failure does not appear to be critical; however,

agglomerations of voids can be seen as serioussources for intergranular fracture.

4.4. Damage evolution

The two fracture mechanisms discussed abovedeal with the initiation of damage. The cracks aretreated as a dilute distribution, with no interaction

between them. Figure 4 shows that cracks interactwhen a particular amount of damage is accumu-lated. Individual grain-boundary debonds coalesce

to mainly axially oriented macrocracks (Fig. 4). Thegrowth behavior of cracks is influenced by their in-teractions. Figure 6(b) shows a longitudinal section

of a 26008C sample deformed to a strain of 0.052 ata rate of 4103/s. An axial macrocrack is indicatedby arrow 1. Note that the grain shape is not ran-domly distributed. The macrocrack travels along

boundaries of elongated grains. A possible modefor the macrocrack formation is that several alignedcracks grow and coalesce by grain-boundary

debonding between them. An initially discrete axialrow of microcracks becomes gradually more con-tinuous and divides the specimen into parallel col-

umns. Additional axially linked cracks will cause aloss in strength of the specimen at a remainingload, yielding crack propagation to catastrophic

failure [23]. This mechanism may result in the lowamount of grain-boundary fracture in the vicinityof axial macrocracks, which is frequently observedby metallography [Figs 4 and 6(b)].

5. SUMMARY AND CONCLUSIONS

The results of the present investigation are sum-marized below:

1. The failure mode in all heat-treated material ispredominantly of intergranular character. Theas-received tungsten exhibits a larger fraction of

transgranular damage that is governed by cracks

tilted from the loading and texture orientation.2. The flow stress (by slip) has a strain-rate sensi-

tivity that is well described by the MTS constitu-

tive equation; grain-boundary decohesion seemsto have a much lower strain-rate sensitivity.

3. Slip, twinning, and grain-boundary cracking arecompeting deformation mechanisms, the formerbeing favored under quasi-static and the latter

two under high-strain-rate conditions. As a con-sequence, the material undergoes a ductile-to-brittle transition as the strain rate is increased

from 103 to 103/s. This is seen to be due to thestrong strain-rate sensitivity of the flow stressexhibited by tungsten typical of b.c.c. metals. At

high strain rates the higher values of axial stresscause sliding of the microflaws, leading to in-

itiation of intergranular fracture resulting inaxial splitting at small strain levels. The totaldensity of grain-boundary debonds at constant

strain and strain rate increases with heat treat-ment temperature. This is due to a correspond-ing increase in concentration of embrittling

grain-boundary segregates.4. Failure in all tested material at high strain rates

is seen to be governed by propagation of axialcracks which link the end surfaces of the cylind-rical specimens. In particular, at a strain rate of

103/s the area close to these major cracks seemsto be free of further damage.

5. Optical microscopy reveals that twins were onlyformed in the heat-treated material and only atstrain rates of 103 and 4103/s. The density oftwins increases with strain rate and with strain.Twins are always related to grain-boundarydebonds. Twintwin intersections often yield

transgranular microcracks. The latter fracture isalso observed within twinned areas. It is shown

that twinning is seldom related to cracks that areprone to yield catastrophic failure of the respect-ive sample. Thus, twinning does not play a

major role in damage evolution. SEM fracturesurface observations show that no slip bandsintersect with debonded grain boundaries that

are related to major cracks. Fracture initiation athigh strain rates therefore is seen to be initiatedby failure of weak grain boundaries or stress

concentrations caused by voids located at thegrain boundaries.

6. Two mechanisms were observed for damage in-itiation: (a) sliding cracks from debonded grainboundaries which nucleate wing cracks close to

the loading direction; (b) microcrack nucleationdue to stress concentrations created by pre-exist-ing grain-boundary voids which is frequently

observed in the material heat treated at 26008C.The predictions of both damage mechanisms

support the idea of increasing grain-boundarybrittleness with higher heat treatment tempera-tures accompanied by a decrease in grain-bound-

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6285

ary fracture toughness. Application of the wing-crack mechanism enables the estimation of the

cohesive energy of the grain boundaries.

AcknowledgementsThis research was partially supportedby the U.S. Army Research Oce under the URIProgram on Ultradynamic Performance of Materials(Contract: ARO DAAL 03-92-G-0108) and the U.S. ArmyResearch Laboratory through W. Bruchley. This work ispart of an exchange program coordinated at theUniversity of Karlsruhe (TH) by O. Vohringer.Appreciation is extended to K. S. Vecchio, D. Harach,and A. Rohatgi for performing the quasi-static com-pression tests, to A. J. Strutt for help in metallography, toJ. Hodowany and D. Owen for performing the high-strain-rate compression tests, and to H.-C. B. Chen, Y.-J.Chen, and J. Shih for helpful discussions. The 17508C heattreatments were carried out at Sandia NationalLaboratories through the kind help of A. Romig Jr andM. Karnowsky. The constructive criticism by the referee isgreatly appreciated.

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APPENDIX A

Quantitative damage analysis

Concerning the damage quantification, only the inter-granular grain-boundary failure was taken into account,since the amount of transgranular damage was negligibleat all strain rates for the annealed conditions. Since the as-received W exhibited a large transgranular contribution tothe accumulated damage, it will not be presented here.Damage was initiated by single grain-boundary debond-ings of the length of one grain-boundary facet or less andpropagated by coalescing of grain-boundary debondings.As characteristic feature of damage, the total number ofcomplete debonded grain-boundary facets and those thatwere not separated over the whole length of the facet wererecorded by counting from the micrographs. The cracklength is normalized to the length of a grain-boundaryfacet. The following classification was used: L

facet; L 1 a debonded facet. The following rangeswere established: 0

Fig. A2. Damage in W annealed at 17508C; showing increase in damage, for increasing strain ratesfrom 103 to 4 103/s at constant strains.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN6288

Fig. A3. Damage in W annealed at 26008C; showing a profound increase in damage, for increasingstrain rates from 103 to 103/s (note great dierence in strains: 26% at 103s1 and 25% at 103s1.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN 6289

Fig. A4. Damage in W annealed at 26008C; showing increase in damage, for increasing strain ratesfrom 103 to 4 103/s at constant strains.

DUMMER et al.: STRAIN RATE EFFECTS IN TUNGSTEN6290