Mode II fracture behavior of a Zr-based bulk metallic glass

ARTICLE IN PRESS

Journal of the Mechanics and Physics of Solids 54 (2006) 2418–2435 www.elsevier.com/locate/jmps

Mode II fracture behavior of a Zr-based bulk metallic glass Katharine M. Floresa, Reinhold H. Dauskardtb, a

Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210-1178, USA b Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305-2205, USA Received 30 June 2005; accepted 4 May 2006

Abstract Shear band formation and fracture are characterized during mode II loading of a Zr-based bulk metallic glass. The measured mode II fracture toughness, KIIc ¼ 7574 MPaOm, exceeds the reported mode I fracture toughness by
4 times, suggesting that normal or mean stresses play a significant role in the deformation process at the crack tip. This effect is explained in light of a mean stress modified free volume model for shear localization in metallic glasses. Thermal imaging of deformation at the mode II crack tip further reveals that shear bands initiate, arrest, and reactivate along the same path, indicating that flow in the shear band leads to permanent changes in the glass structure that retain a memory of the shear band path. The measured temperature increase within the shear band is a fraction of a degree. However, heat dissipation models indicate that the temperature could have exceeded the glass transition temperature for less than 1 ms immediately after the shear band formed. It is shown that this time scale is sufficient for mechanical relaxation slightly above the glass transition temperature. r 2006 Elsevier Ltd. All rights reserved. Keywords: Fracture; Metallic materials; Mechanical testing; Shear band

1. Introduction The deformation and fracture behavior of metallic glasses in thin ribbon and, more recently, bulk form has been studied extensively (Masumoto and Maddin, 1971; Alpas Corresponding author. Tel.: +1 6507250679; fax: +1 6507254034.

E-mail address: [email protected] (R.H. Dauskardt). 0022-5096/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2006.05.003

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et al., 1987; Donovan, 1989; Bruck et al., 1994; Gilbert et al., 1997; Lowhaphandu and Lewandowski, 1998; Flores and Dauskardt, 1999a, b, 2001b; Hess et al., 2005; Sergueeva et al., 2005). It is well known that plasticity is highly localized in shear bands, where the glass viscosity is lower than in the surrounding material, as evidenced by molten droplets and characteristic vein patterns on the failure surface. Several explanations for this localized viscosity decrease have been proposed, including adiabatic heating to above the glass transition temperature and a mean stress induced dilatation of the free volume (Spaepen, 1977; Argon et al., 1985; Liu et al., 1998; Flores and Dauskardt, 2001b). While the formation and propagation of shear bands in tensile loading typically leads to unstable failure, the highly localized stress field ahead of a crack tip may be used to constrain the shear bands in a stable damage zone, allowing detailed study as previously reported for mode I loading (Flores and Dauskardt, 1999a, b). The objective of the present study was to examine such crack tip shear banding and fracture mechanisms associated with pure mode II loading. In the case of brittle materials, the mode II fracture toughness is typically equal to or greater than the mode I toughness (Suresh et al., 1990; Awaji, 1998; Awaji and Kato, 1998). For ductile metals, however, several authors have reported lower mode II toughnesses for a variety of alloys (Aoki et al., 1990; Prasad et al., 1994; Laukkanen et al., 1999). This has been attributed to the different micromechanisms of fracture in brittle and ductile materials (Laukkanen et al., 1999). Fracture in brittle materials is typically stresscontrolled, with the maximum principal or normal crack tip stress controlling the fracture process. For ductile materials, the crack tip fracture process is generally strain-controlled. Increased shear loading contributes to the crack tip plastic strain, lowering the resistance to fracture under mode II loading. While the fracture surfaces of metallic glasses may appear ductile in that they exhibit local softening, it is has not been established that a strain-controlled fracture criterion is valid for these materials. Rather, studies suggest that normal or hydrostatic stresses play a significant role for incipient flow and shear banding processes (Liu et al., 1998; Lowhaphandu et al., 1999; Flores and Dauskardt, 2001b; Wright et al., 2001; Zhang et al., 2003). An examination of the mode II fracture, in which a state of pure in-plane shear exists ahead of the crack tip, is therefore of interest. Questions surrounding the contribution of flow-induced dilatation or adiabatic heating to the viscosity decrease within shear bands have led to numerous examinations of the temperature change associated with flow. In situ high resolution infrared images have been obtained at a crack tip loaded in mode I (Flores and Dauskardt, 1999b) as well as during tensile and fatigue loading of round bars (Yang et al., 2004, 2005). These prior studies revealed very small temperature increases (o1 K) associated with the formation of shear bands. More recent observations using a low melting temperature metal coating suggest a minimum temperature increase of
200 K (Lewandowski and Greer, 2006). Significantly larger temperature increases, on the order of several hundred degrees, have been observed under high strain rate or impact conditions, wherein the energy dissipated and therefore temperature rise are expected to be much larger than under quasi-static conditions (Bruck et al., 1996; Gilbert et al., 1999). In the present study, the first observations of the mode II fracture behavior of a Zr–Ti–Ni–Cu–Be bulk metallic glass are presented. Prior to failure, the temperature increase associated with crack tip shear bands formed under mode II loading was examined in situ using a high resolution infrared imaging system. Plastic flow, again revealed by a small (o1 K) increase in temperature along a shear band in front of the

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crack, was shown to occur along the same plane in several discrete events during loading. Final mode II failure occurred along the same plane. The mode II toughness exceeded the mode I toughness value by
4 times. The maximum shear band temperature achieved and the stresses required for shear band operation were also estimated in light of a mean stress modified free volume model for flow localization. 2. Experimental To facilitate mode II loading of a crack, edge notched flexure (ENF) fracture mechanics specimens were prepared from a cast Zr41.25Ti13.75Ni10Cu12.5Be22.5 bulk metallic glass plate by wire electrical discharge machining (EDM). The specimen geometry is shown in Fig. 1. The specimen shape is similar to a double cantilever beam (DCB) geometry with half height h ¼ 10 mm, width ¼ 105 mm, and notch length a0 ¼ 15 mm. Specimens were fatigue precracked in mode I using a DCB stress intensity factor, KI, calibration. The mode I stress intensity range, DKI, during precracking was kept fairly low (o4 MPaOm) to maintain a smooth crack path, thus minimizing frictional contact of fracture surface asperities during mode II loading. To prevent crack deflection from the central plane of the specimen during mode II loading, a groove approximately half the total thickness was machined on one face along the crack plane. The remaining thickness of the grooved region was 1.4 mm. The ENF sample was loaded in three-point-bending (outer pin span S ¼ 80 mm), making the central plane (i.e. the crack plane) the plane of maximum shear. Care was taken to locate the outer loading pin on the notch side ahead of the notch tip to ensure proper load transfer across the crack plane. The crack length was measured from this pin, and thus the notch length was ignored in the mode II calculations. The initial mode II crack length, ai, was
14 mm. The stress intensity calibration for the ENF geometry with h=W p0:4 and a=W o0:8 is given by Fett and Munz (1994): pffiffiffiffiffiffi K II ¼ t pa ½1:684ða=hÞ2 þ 1:12154 1=4 , (1) where t is the shear stress on the crack plane and W is half the outer pin span, S, as shown in Fig. 1. The shear stress is given by t¼

3 P , 8 Bmin h

(2)

S a

2h

Fig. 1. Schematic of the end notched flexure (ENF) sample geometry used for mode II fracture toughness measurements. Nominal dimensions were h ¼ 10 mm, S ¼ 2 W ¼ 80 mm, and ai ¼ 14 mm. The sample was fatigue precracked in mode I using the pin holes. A groove on one face ensured that the crack would remain on the shear plane in mode II. Mode II testing was performed in a three point bend configuration, with the crack length, a, measured from the outside loading pin.

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where P is the applied load, and Bmin thickness of the grooved section. Note that for this range of h=W and a=W , t and KII do not explicitly depend on either W or S. All testing was performed on a high resolution electro-servo-hydraulic test system operating under displacement control with a displacement rate of 1 mm/s. Load–displacement data were recorded and crack lengths were monitored optically during the fracture experiment. Thermal imaging was performed with a liquid nitrogen cooled infrared camera (Raytheon Lab Radiance HS, Sierra Olympic Technologies, Inc., Issaquah, WA), with a 64  64 pixel array each imaging a 30  30 mm2 area on the specimen surface with a 1 ms integration time. Images were captured at 1000 Hz. Crack growth was monitored with the thermal imaging camera. The specimen side face was lightly coated with colloidal graphite to lower its reflectivity. During post processing, a thermal image taken immediately prior to the onset of crack tip deformation or crack growth was subtracted from subsequent images to remove background temperature variations. The instrument was carefully calibrated using images obtained from the sample at known temperatures. Since the onset of deformation or fracture was difficult to accurately predict, the frame capture could not be synchronized with incipient deformation. The camera was therefore set to begin capturing images slightly before the deformation or fracture event.

3. Results Several heating events on a shear band directly ahead of the crack tip were observed under mode II loading prior to fracture of the specimen. Interestingly, these events were not associated with any noticeable compliance change or crack growth in the thermal image. Examples of the temperature distribution and dissipation immediately following two of these heating events are shown in Fig. 2(a–c) and (d–f). The images shown are at 5 ms intervals. All images cover the same 1.92  1.92 mm area on the specimen side face. Heating repeatedly occurred along the same shear band path with a length of
1.2–1.4 mm ahead of the pre-crack tip at an angle of
13.51. This band was also found to be the final path for unstable crack propagation as shown in Fig. 2(g). The maximum temperature rise measured during these repeated heating events was 0.51–0.55 K, while the peak temperature rise measured at catastrophic failure was 17.83 K. The temperature increase was highly localized along the shear band, falling off quickly with distance normal to the shear band. A ‘‘hot spot’’ was apparent in the band. In Fig. 2 (a) and (g), the highest temperature pixel in the ‘‘hot spot’’ was on the left of the image near to the crack tip, while in (d) the highest temperature pixel is in the middle of the heated zone. However, the heated band remains in the same place relative to the initial crack tip, indicating that the crack front has not moved within the resolution of the imaging system. The temperature distribution perpendicular to the band, through the pixel with the maximum temperature rise, is shown in Fig. 3 for the two stable heating events illustrated in Fig. 2(a) and (d). Note the sharp peaks, with temperatures about 23% greater than the neighboring pixels. The mode II fracture toughness, KIIc, determined from the peak load at final fracture of multiple specimens was found to be 7574 MPaOm with no evidence of stable crack growth. The von Mises plastic zone size at this stress intensity is 0.67 mm, a significant fraction of the specimen thickness. Thus, it should be noted that this is not necessarily a plane strain measurement, although small scale yielding requirements were met. The

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Fig. 2. Sequence of thermal images obtained under mode II loading. The subsequent heat dissipation is shown at 5 ms increments. Stable heating events which were not associated with crack growth are shown in (a–c) for KII ¼ 77.3 MPaOm and (d–f) for KII ¼ 78.2 MPaOm. Final failure at KII ¼ 79.5 MPaOm is shown in (g). All images show the same 1.92 mm  1.92 mm area.

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0.75

Temperature Increase, ∆T (K)

Temperature Increase, ∆T (K)

0.75

0.50

0.25

0.00 -0.2 (a)

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-0.1

0

0.1

0.25

0.00 -0.2

0.2

Vertical Distance from "Hot Pixel" (mm)

0.50

(b)

-0.1

0

0.1

0.2

Vertical Distance from "Hot Pixel" (mm)

Fig. 3. Temperature distribution perpendicular to the shear bands shown in Fig. 2(a) and (d), respectively, through the pixel with the peak temperature.

Fig. 4. SEM micrograph of (a) mode I and (b) mode II fracture surfaces. Note that the vein pattern in (b) is smeared in the direction of shear.

corresponding mode II fracture energy, G IIc ¼ K 2IIc =E, is
59 kJ/m2, where E is Young’s modulus. For the present metallic glass E ¼ 96 GPa. An SEM image of the mode II fracture surface is compared with the surface that results from mode I fracture in Fig. 4. In both cases, the characteristic veins patterns are evident. This suggests significant softening occurred at failure. Note that in the case of the mode II failure, the veins are smeared in the direction of shear although some areas more typical of the mode I vein pattern were observed, suggesting that an opening mode component was also active during unstable fracture. Small smeared regions similar to the mode II morphology were also observed in the mode I fatigue precrack area, probably associated with the shear failure of bridging ligaments. 4. Discussion Plastic flow and shear banding in metallic glasses is generally modeled as a shear stress driven diffusive process highly sensitive to the glass free volume (Spaepen, 1977; Argon,

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1979). The pure shear stress on the central (crack) plane ahead of the mode II crack tip provides an ideal configuration to study shear bands. The 1/Or crack tip stress fields, which decay rapidly with distance from the crack tip, tend to localize the formation of shear bands to the crack tip region and stabilize their propagation. In the present study, the repeated heating events observed with the thermal imaging camera prior to final catastrophic fracture are not associated with crack extension, but rather with the emission and reactivation of a shear band from the crack tip. Indeed, an important observation of the present work is that once a shear band has formed and become inactive, the same band is reactivated on further loading and eventually forms the fracture path. Clearly, flow in the shear band leads to permanent changes in the glass structure that retain a memory of the shear band path. The origin of the change in the glass structure is not currently known, but most likely involves quenched in higher free volume and some chemical reordering in the shear band (Pampillo, 1972; Flores et al., 2002; Li et al., 2002a, b; Wright et al., 2003; Kanungo et al., 2004a, b). The measured mode II fracture toughness of 75 MPaOm exceeds the mode I fracture toughness of
15–20 MPaOm by a factor of
4 (Lowhaphandu and Lewandowski, 1998; Flores and Dauskardt, 1999a, b). This result is now discussed in light of the differing crack tip stress fields. A mean stress modified free volume model for flow in metallic glasses is reviewed in Appendix and is used to describe the volume of material subject to shear band formation ahead of the modes I and II crack tips. The extent of heating in the shear band and its influence on flow are also addressed. 4.1. Crack tip stress analysis The maximum shear stress ahead of cracks loaded in modes I and II may be calculated from the well-known asymptotic stress fields and compared with the stresses required for shear localization. Along the crack plane (y ¼ 01), the stresses ahead of a blunted crack tip with radius rI loaded to stress intensity KI in mode I are: KI  r sIxx ¼ pffiffiffiffiffiffiffi 1  I , (3a) 2r 2pr KI  r sIyy ¼ pffiffiffiffiffiffiffi 1 þ I , 2r 2pr

(3b)

KI sIzz ¼ 0 in plane stress or sIzz ¼ 2n pffiffiffiffiffiffiffi in plane strain; 2pr

(3c)

where the origin is rI/2 behind the crack tip, so (rrI/2) is the radial distance ahead of the crack tip (Creager and Paris, 1967). We assume this crack tip blunting occurs by homogeneous flow or by the operation of one or more shear bands in the high stresses of the crack tip region. The blunting radius is given by rI ¼

1 K 2I , 4 sy E 0

(4)

where sy is the yield stress,
2 GPa (Bruck et al., 1994). Using the three principal stresses in Eqs. (3a–c), the maximum shear stress, tImax , as a function of position ahead of the crack

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tip may be calculated: tImax ¼ 12ðsImax  sImin Þ,

(5)

where sImax and sImin are the maximum and minimum principal stresses, respectively. Eq. (5) is plotted in Fig. 5 for KI ¼ KIC ¼ 20 MPaOm under plane strain conditions. The only non-zero stress on the plane ahead of the crack tip under mode II loading is sxy (Creager and Paris, 1967): K II  rII  p ffiffiffiffiffiffiffi sII ¼ 1  . (6) xy 2r 2pr The blunted crack tip radius under mode II loading, rII, is not well defined. However, work on other ductile metals reveals that the length scale of crack tip deformation is generally insensitive to loading mode (Aoki et al., 1990). Thus, we obtain an approximation for r using an expression analogous to that for plane strain mode I loading: rII ¼

1 K 2II , 8 ty E 0

(7)

where the shear yield stress, ty, is 1.03 GPa (Bruck et al., 1994). Since there are no normal stresses acting on the plane ahead of the mode II crack tip, Eq. (6) describes the maximum shear stress in mode II, tII max , and is also plotted in Fig. 5 for KII ¼ KIIC ¼ 75 MPaOm. Note that if the entire shear band softens and flows as a unit, then the shear carrying capacity of the shear plane would be greatly reduced below the values proscribed by Eqs. (5) and (6). However, it is unlikely that flow occurs uniformly across the entire shear plane, and more probable that the shear process involves the propagation of a discrete slipped region akin to a dislocation in crystalline materials (Wright et al., 2001). In this case, the stress fields obtained from Eqs. (5) and (6) accurately describe the shear stress along a shear band ahead of a blunted mode I or mode II crack, respectively. 7000 Maximum Shear Stress, τ max (MPa)

Mode I, KI = 20 MPa√m

6000

Mode II, KII = 75 MPa√m Mode II, KII = 20 MPa√m

5000 4000

KII = 75 MPa√m

3000 2000 KII = 20 MPa√m

1000 KI = 20 MPa√m

0

0

20

40

60

80

100

Distance from Crack Tip,r - ρ/2 (µm) Fig. 5. Shear stress distribution on the plane in front of a blunted crack tip under modes I and II loading.

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A comparison of the shear stresses in modes I and II reveals that, within 100 mm of the I crack tip, tII max rises from 0 to exceed tmax by more than 25 times. The significantly larger shear stress required for flow ahead of the mode II crack tip is an indication of the powerful influence of the triaxial stress state on the onset of failure in mode I, as is discussed below. At KII ¼ 75 MPaOm, tII max reaches a peak of
6.5 GPa
6.2 mm from the crack tip due to the 1/Or stress singularity. This is much higher than previous measurements of the shear yield strength in the bulk glass (Bruck et al., 1994). This may be associated with the small sampling volume near the crack tip and the distribution of free volume in the glass. Nanoindentation studies reveal a significant increase in shear strength as the indentation penetration depth decreases (Wright et al., 2001). This size effect is attributed to the decreasing probability of finding regions of the glass with higher free volume, where flow is easier to initiate, as the sampling volume is decreased with decreasing penetration depth. Other studies of the free volume distribution obtained with positron annihilation spectroscopy techniques also suggest that there is a distribution in free volume site sizes, with flow presumably initiating in regions of locally high free volume, although a detailed description of the spatial distribution of free volume in the glass remains an area of active research (Flores et al., 2000; Suh et al., 2003; Kanungo et al., 2004a, b). Given such a distribution, it is reasonable to assume that the crack would progress more easily through regions of high free volume, and would be pinned where the free volume is low. This resulting wavy crack front would then primarily sample low free volume regions, where the flow stress is larger than that observed in the bulk. While the stresses needed to initiate flow are estimated above, the critical shear stress required for continued propagation of the shear band can be obtained by examining where the shear band stops. Note that far from the crack tip the uniform applied shear stress (Eq. (2)) becomes a significant fraction of the asymptotic stress due to the presence of the crack (Eq. (6)). Including this shear stress contribution in the stress calculation, we find that at the tip or end of the shear band shown on the far right side of Fig. 2a the shear stress falls to sxy
1075 MPa, while sxy
1137 MPa at the tip of the shear band shown in Fig. 2d. (Although the shear band path deviates slightly from the precrack plane, both the mean and normal stress across the shear band plane are compressive and small. Further, the shear stress decreases with increasing deviation from the neutral axis. We have therefore calculated the largest value for the shear stress at 01, using the horizontal projection of the shear band length.) This suggests that the shear stress required for continued propagation of a shear band is at least 1075 MPa, similar to the reported shear yield strength (Bruck et al., 1994). We note finally that all of the above mode II stresses might be reduced somewhat due to frictional contact behind the crack tip, which would reduce the mode II stress intensity factor, although as noted below these effects are expected to be minimal. Indeed, even if the mode II stress intensity is reduced to 20 MPaOm, the shear stress still exceeds the reported shear yield strength over a significant distance, as illustrated in Fig. 5. 4.2. Stress state effect on toughness Because of the obvious influence of shear stresses on the flow process in metallic glasses, previous work (Kimura and Masumoto, 1983; Bruck et al., 1994) has suggested that a von Mises criterion may describe the onset of flow localization and failure. If this were the case and we assume that the toughness is associated with the initiation and unstable

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propagation of a shear band ahead of the crack tip, then the mode II fracture toughness might be expected to be lower than the mode I fracture toughness, since the stress field ahead of the crack tip in mode II is pure in-plane shear. Rather, we find that the mode II fracture toughness of the bulk metallic glass alloy used in this study is
4 times larger than the mode I fracture toughness of
15–20 MPaOm (Lowhaphandu and Lewandowski, 1998; Flores and Dauskardt, 1999b). Clearly, some of the increased mode II fracture energy may be attributed to the formation of a frictional contact zone behind the crack tip. However, in the present study, the fatigue precrack was produced at low precracking loads and the resulting fracture surface was relatively smooth, thus minimizing any frictional contact processes. Further, note that a similarly large mode II contribution to the plastic energy dissipation during the propagation of shear cracks in Ni78Si10B12 metallic glass ribbons (45.3 kJ/m2) has been reported, although that data depended on the thickness of the ribbons (Alpas et al., 1987). Alternatively, it is reasonable to expect that the initiation of flow and fracture may be influenced by the mean stress or the stress normal to the failure plane, since the shear bands associated with plasticity are areas of increased free volume. Several studies have noted an effect of tensile versus compressive loading on the angle of the failure plane, and attributed this to sensitivity to normal stresses, described by the Mohr–Coulomb criterion (Donovan, 1989; Liu et al., 1998; Lowhaphandu et al., 1999; Wright et al., 2001; Lewandowski and Lowhaphandu, 2002; Lund and Schuh, 2003; Zhang et al., 2003). The hypothesis is that tensile stresses cause a local free volume increase, decreasing the shear stress needed to initiate flow. The fact that the mode II fracture toughness of the bulk metallic glass alloy is so much larger than the mode I fracture toughness indeed suggests that normal stresses play a significant role in flow and failure. Spaepen (1977) and Steif et al. (1982) presented a free volume model for flow localization in metallic glasses which describes flow as a diffusional process in which an applied shear stress biases the atomic jump direction, giving rise to the plastic shear strain. More recently, this model has been modified to include the influence of mean stresses on the initial free volume distribution and therefore the onset of flow (Flores and Dauskardt, 2001b). The mean stress modification is reviewed and updated in Appendix, and will now be applied to determine the influence of the triaxial stress state ahead of a mode I crack on the shear stress required for flow. Under plane strain conditions, the mean stress, sm, on the plane ahead of a blunted mode I crack tip is KI sIm ¼ 2 pffiffiffiffiffiffiffi ð1 þ nÞ. 2pr

(8)

(Recall that the origin is rI/2 behind the blunted crack tip, making the mean stress finite at the crack tip.) Using this mean stress as a function of position and the free volume model described in Appendix and in the reference by Flores and Dauskardt (2001b), we have determined the shear stress required for flow localization, tlocal, as a function of the distance from the crack tip. This is compared with the maximum applied shear stress, tImax , at KI ¼ KIC ¼ 20 MPaOm in Fig. 6. The mean stress lowers tlocal considerably over a range of
34 mm ahead of the crack tip. Beyond 34 mm, tlocal begins to increase as the mean stress falls below 1240 MPa. Due to this mean stress induced minima in the localization stress, tImax exceeds tlocal over a range of 4.3 mm from the crack tip. This provides a measure of the

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5000 Mode I, KI = 20 MPa√m Localization shear stress

Shear Stress (MPa)

4000

3000

2000

1000

0

0

5

10

15

20

Distance from Crack Tip, r - ρ/2 (µm) Fig. 6. Comparison of shear stress distribution on the plane in front of a blunted mode I crack tip with the shear stress required for localized flow in a shear band, according to a mean stress modified free volume model.

sampling volume required to find a large enough free volume site to initiate flow and failure under mode I loading. In contrast, under mode II loading where the mean stress is 0 on the crack plane, we have already noted that the crack tip shear stress exceeds the nominal shear yield strength over a considerable distance. It is also notable that thermal imaging of specimens loaded in mode I did not reveal any indication of shear band activation and arrest without measurable crack growth, nor has this phenomenon been reported elsewhere except in situations where considerable crack branching shielded the crack tip and changed the local stress distribution (Flores and Dauskardt, 1999a, b) or during uniaxial loading of smooth tensile bars (Yang et al., 2004). Other authors have reported the spontaneous formation of nanovoids in shear bands formed in tension, but not under other loading modes (Li et al., 2002a, b; Jiang and Atzmon, 2003; Wright et al., 2003). Such larger scale flow defects would tend to grow under tensile mean stresses, significantly decreasing the stability of shear bands subjected to the triaxial stress state of the mode I crack and leading to catastrophic growth at lower loads than in mode II. 4.3. Shear band heating The temperature increases measured during the shear band activation events shown in Fig. 2(a) and (d) were obtained at some time after the initiation of the shear band. In order to more accurately examine the role of heating in shear band formation, the peak temperature at initiation is estimated by examining the heat dissipation behavior following the activation of the shear band heat source. We model the shear band as a planar slab instantaneous heat source of thickness 2b centered at y ¼ 0 in an infinite body. The slab is given an initial temperature change of DT0, which then decays through heat conduction to

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the surrounding material. The temperature change, DT, as a function of position and time is given by Carslaw and Jaeger (1959): " # DT 0 by bþy erf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ erf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , DTðy; tÞ ¼ (9) 2 2 aðt  t0 Þ 2 aðt  t0 Þ where a is the thermal diffusivity of the material ða ¼ k=rcp ¼ 1:274  106 m2 =sÞ, (t–t0) represents the time after activation of the source, t is the thermal image capture time, and t0 provides an offset time between source activation and the capture of the first thermal image. While this solution is exact, after a sufficiently long time it approaches the solution for a planar heat source of strength Q placed at y ¼ 0 (Carslaw and Jaeger, 1959):   Q y2 DTðy; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  . (10) 4aðt  t0 Þ 2 paðt  t0 Þ

0.6

0.6

0.5

0.5

Temperature Change, ∆T (K)

Temperature Change, ∆T (K)

The quantity of heat liberated per unit area of the plane is given by Qrcp. The solution given by Eq. (9) approaches that given by Eq. (10) at y ¼ b after diffusion time t ¼ b2 =2a; that is, after sufficient time has passed to allow the heat to diffuse through the thickness of the slab considered in Eq. (9). Using Eq. (10) with fitting parameters Q and t0, we may fit the temperature change as a function of time after heat source activation for various choices of position y within the hottest pixel. The results are shown in Fig. 7. For yo15 mm (half the height of a pixel), the resulting Q and t0 are relatively insensitive to the choice of position for the fit, with Q decreasing by less than 0.06% and t0 decreasing by less than 10% over the range 0oyo15 mm. The time offset, t0, between initiation and the capture of the first image for the shear band shown in Fig. 2(d–f) is
0.983 ms, less than the time between images. For the shear band shown in Fig. 2(a–c), the time offset is
2.68 ms, longer than the time elapsed between images. Q is also slightly larger for this sequence. However, note that in Fig. 2(a), there is a secondary heat source near the hottest pixel, below the plane of the main shear

0.4 0.3 0.2 0.1

0.4 0.3 0.2 0.1

@ y = 100 nm, Q =1.0903x 10-4 mK, t0 = 2.6787 ms

0 (a)

0

5

10 Time, t (ms)

@ y =100 nm: Q =6.4686x 10-5 mK,t0 =0.98286 ms

0

15 (b)

0

2

4

6 8 Time, t (ms)

10

12

Fig. 7. Transient heat dissipation behavior of shear band events shown in Fig. 2(a–c) and (d–f), respectively, at the point of the maximum temperature rise.

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band. This small heating event may be associated with another shear band or small crack and is visible two images (2 ms) prior to the main shear band event. It accounts for the slight error in the offset and the increase in Q. The time offset for both sequences is much longer than the time required for heat to diffuse across any reasonable shear band thickness, indicating that the error associated with using the simplified solution given by Eq. (10) rather than the full solution given by Eq. (9) is very small. Further, it suggests that while the spatial resolution of the thermal images was limited by the pixel size, sufficient time elapsed prior to the first image to ‘‘blur’’ the heat distribution over this area. Within 1 ms, heat diffuses more than 50 mm, more than the width of a single 30 mm pixel. The average temperature of the ‘‘hot’’ pixel thus provides a reasonable estimate for the actual peak within the pixel. In fact, because of this significant time offset, it proved impossible to use Eq. (9) to obtain both the initial temperature rise, DT0, and the shear band thickness, 2b. However, by comparing nonlinear regression fits to both Eqs. (9) and (10), the product DT02b was found to consistently equal Q. Eq. (10) may be rewritten as (Priorier and Geiger, 1994):   DT 0 Dy y2 DTðy; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  , (11) 4aðt  t0 Þ 2 paðt  t0 Þ where Dy is the vanishingly small thickness of the plane source. Again, because of the relatively large t0, we found that Dy is a good approximation for shear band thickness 2b. Thus, given Q and a reasonable guess for the thickness of the shear band, we can estimate the temperature rise within the shear band at initiation. Using Q ¼ 6.47  105 m K, obtained from the shear band shown in Fig. 2(d–f), we find that for the temperature to rise above the glass transition of the alloy (Tg
625 K), the shear band thickness would have to be less than 200 nm. This is consistent with the estimates for shear band thickness (Masumoto and Maddin, 1971; Li et al., 2001) and recent measures of heating from shear bands using low melting point coatings (Lewandowski and Greer, 2006). We note, however, that dilatation effects and even shear disordering may lower the temperature needed for flow below Tg. If the shear band is indeed thin enough so that the peak temperature reaches Tg, the heating must be relatively short lived, since within 1 ms the temperature change is less than one degree. To address the duration of the temperature rise, the mechanical relaxation time associated with flow is considered below. The peak temperature increase at failure of 17.83 K was similar to that previously measured under mode I loading (Flores and Dauskardt, 1999b, 2001a). Under mode I loading, heat dissipation models were employed to estimate the temperature increase during fracture (
139.5 K), which was less than an order of magnitude higher than the peak temperature measured slightly after the initiation of crack growth (
22.5 K). The lack of thermal images after unstable crack growth precludes a similar analysis of the heat dissipation behavior in the present work; however, similar increases in temperature consistent with the locally softened final fracture appearance are anticipated (Fig. 4b). 4.4. Shear band displacements We may use the amount of heat liberated during shear band propagation to obtain an estimate of the shear displacement associated with the formation of a single shear band at

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the mode II crack tip. We assume that the shear displacement increases the crack length by Da, thus causing a decrease in the strain energy of the system, U: heat Uj  U aþDa j ¼ Qrcp ¼ a , area BDa

(12)

where B is the specimen thickness and U is evaluated for a crack length of a and a+Da, with no measured change in the applied loads. The expression on the right-hand side of Eq. (12) is simply the negative of the change in the crack driving force, G, associated with an increase in crack length of Da. Thus, Qrcp ¼ GjaþDa  Gja .

(13)

Using Q ¼ 6.47  105 m K, and calculating the change in crack length required for all of the strain energy change to be released as heat, we obtain Da ¼
2.7 mm. This is consistent with the shear offsets measured in compression experiments (Wright et al., 2001), indicating that Eq. (10) is a reasonable model for the transient heat dissipation after a shear band event. 4.5. Time scales for flow The analysis above assumes that all of the heat is instantaneously deposited on the entire shear band plane. Physically, however, flow requires some finite amount of time, and during this time heat is continually conducted away from the shear band, thus lowering the temperature within the band. This leads us now to consider the time scales involved with flow in the shear band. The capture rate from the thermal images (1000 Hz) provides an estimate for the maximum time allotted for relaxation processes in these experiments. The mechanical relaxation time, t, in the vicinity of Tg for this bulk metallic glass alloy has been directly measured using dynamic and transient mechanical techniques (Suh et al., 2002; Suh and Dauskardt, 2002). While the relaxation times typically associated with the glass transition temperature are much longer than the time between thermal images (
10 s compared to the 1 ms image capture rate), the relaxation time drops dramatically with increasing temperature above Tg. An estimate for the relaxation time for such viscous behavior is given by Z t , m

(14)

where Z is the viscosity and m the shear modulus. The viscosity of the supercooled liquid has been determined and fit to the Vogel–Fulcher–Tammann (VFT) relation (Waniuk et al., 1998): ZðTÞ ¼ Z0 exp

D T 0 T  T0

(15)

with Z0 ¼ 4  105 Pa s, D* ¼ 18.5, and T0 ¼ 412.5 K. For the relaxation time to be less than 1 ms, Eqs. (14) and (15) indicate that the temperature must rise to at least 690 K, consistent with estimates based on measured relaxation times (Suh and Dauskardt, 2002). Again using the value of Q ¼ 6.47  105 m K, this implies that the shear band is less than 165 nm thick.

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5. Conclusions The mode II fracture toughness, KIIc, of a Zr-based bulk metallic glass was measured and found to be 7574 MPaOm in repeated experiments. The mode II toughness was considerably larger than the previously reported mode I toughness. A mean stress modified free volume flow model was employed to rationalize the effect of the tensile mean stress ahead of a mode I crack on reducing the mode I toughness. Prior to final fracture, the repeated operation of a shear band was observed using thermal imaging ahead of the mode II crack tip. An important observation of the present work was that once a shear band had formed and become inactive, the same band was reactivated on further loading and eventually formed the fracture path. Flow in the shear band therefore appears to lead to permanent changes in the glass structure that retain a memory of the shear band path. Using an analysis of the stresses ahead of the blunted crack tip, the shear stress required to initiate the shear band was estimated to be
6.5 GPa. The shear band ceased propagating when the shear stresses dropped to
1.1 GPa. The surprisingly high shear stress required to initiate the shear band was rationalized in terms of the small sampling volume ahead of the crack tip. The temperature increase measured on the side face of the specimen revealed small temperature increases (o1 K) prior to fracture. By examining the heat dissipation behavior, estimates for the amount of heat evolved within the shear band were obtained, which in turn provided a measure of the temperature rise within the band, given a reasonable band thickness. It was shown that for a shear band thickness of less than 200 nm, the temperature within the band could have reached the glass transition temperature, suggesting that the softening within the shear band may be associated with heating but over very short time scales. Acknowledgements This work was supported by a grant from the Boeing Corporation and by the MRSEC program of the National Science Foundation under award No. DMR-0080065. This work was also supported in part by a National Science Foundation CAREER award under NSF award No. DMR-0449651. Appendix. Review of mean stress modified free volume model for flow Spaepen (1977) and Steif et al. (1982) presented a free volume model for flow localization in metallic glasses. This model combines a shear stress induced disordering process in which atoms squeeze into available free volume sites, increasing the free volume, and a diffusion controlled reordering process in which the surrounding atomic cage collapses down on the newly open site, decreasing the free volume. At the softening (localization) stress, the free volume creation rate exceeds the annihilation rate, and the model predicts a sudden net increase in free volume. For applied shear stresses sufficiently large such that the free volume creation rate is greater than the annihilation rate, the net rate of change of the free volume, vf, is given by Steif et al. (1982):        ag v 2ag kT DG m tO 1 1  cosh v_f ¼ v f exp  , (A.1) exp  2kT nD vf S kT vf

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where v* is the critical free volume required for an atomic jump (
0.8 O), nD is the number of jumps required to annihilate v*, DGm is the activation energy for an atomic jump, t is the applied shear stress, O is the atomic volume, f is the jump frequency (
Debye frequency), k is Boltzman’s constant, T is the temperature (
300 K in these experiments), ag is a geometrical factor on the order of 1, S is a material constant given by ð2=3Þmðð1 þ nÞ=ð1  nÞÞ, n is Poisson’s ratio; and m is the shear modulus. Within the shear band, we assume that the metallic glass behaves homogeneously. For a solid undergoing homogeneous shearing       ag v  DG m tO t_ , (A.2) exp  g_ ¼ þ 2f exp  sinh 2kT m vf kT where g_ is the (constant) strain rate, and t_ the rate of change of the applied stress (Steif et al., 1982). Eqs. (A.1) and (A.2) may be solved numerically to determine the stress–strain behavior for a glass with initial average free volume vi. More recently, this free volume model for flow has been modified to include the effect of mean stress induced dilatations on the initial free volume distribution, which in turn influences the onset of softening and strain localization (Flores and Dauskardt, 2001b). It was shown that for stress states with compressive mean stresses, the initial average free volume can be assumed to be  sm  vi ¼ vc0 1 þ , (A.3a) B where vc0 is the initial free volume with no superimposed mean stress and B the bulk modulus. Under tensile mean stress conditions, the initial average free volume takes the form  sm  sm vi ¼ vt0 1 þ , (A.3b) þO B B where now vt0 is the initial free volume when sm ¼ 0.1 Ideally, vc0 ¼ vt0 and the transition from Eqs. (A.3a) to (A.3b) occurs at stress state sm =seff ¼ 0. However, Eqs. (A.3a) and (A.3b) were derived using a continuum elastic model to calculate volume changes at an atomistic level, and as such vc0 and vt0 are treated as fitting parameters. We find that to obtain a good fit to experimental data, vc0 4vt0 , and the transition from Eqs. (A.3a) to (A.3b) occurs at some small but positive stress state. Nevertheless, this mean stress modified free volume model effectively predicts the onset of failure over a range of stress states from sm =seff ¼ 0:33 (uniaxial compression) to sm/seff ¼ 1.15. Updated fits to the experimental data presented in the reference by Flores and Dauskardt (2001b) have been obtained using an activation energy value for mechanical relaxation via local atomic position adjustments in this alloy at low temperatures (Suh and Dauskardt, 2002). Using DGm ¼ 0.1 eV and a strain rate of 104 s1 to obtain a nondimensionalized strain rate g_ =½f expðDGm =kTÞ ¼ 7  1016 , failure is predicted when the normalized free volume vc0 =ag v ¼ 0:026 for sm/seffo0.5 (Eq. (A.3a)) and vt0 =ag v ¼ 0:017 for sm/seff40.5 (Eq. (A.3b)). Both of these values for the unstressed initial free volume are in good agreement with experimental results for a similar glass (Lambert and Flores, 2005). 1

The term v0 ðsm =BÞ was previously neglected as small compared to Oðsm =BÞ, but is now included for completeness.

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