Fracture of notched ductile bulk metallic glass bars subjected to tension-torsion: Experiments and simulations

Acta Materialia 168 (2019) 309e320

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Fracture of notched ductile bulk metallic glass bars subjected to tension-torsion: Experiments and simulations Devaraj Raut a, R.L. Narayan b, Yoshihiko Yokoyama c, Parag Tandaiya d, **, Upadrasta Ramamurty b, e, *, 1 a

Boeing Research & Technology, India Center, Bangalore, 560016, India School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, 639798, Singapore Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan d Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India e Structural Materials Department, Institute of Materials Research and Engineering (IMRE), Singapore, 138634, Singapore b c

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 October 2018 Received in revised form 7 February 2019 Accepted 15 February 2019 Available online 20 February 2019

In bulk metallic glasses (BMGs) that are susceptible to shear band mediated plasticity, crack initiation and growth is a strain-controlled process associated with a unique material specific length scale. In notched specimens subjected to pure modes I and II, and mixed mode I/II loading conditions, this length scale determines the extent to which a crack can grow within a dominant shear band before propagating catastrophically. While a similar fracture mechanism is expected in the anti-plane shear mode (mode III) dominant loading condition, there are few studies that have investigated this aspect. In this paper, an attempt is made to understand the crack growth processes and fracture mechanisms under mode III and mixed mode I/III loading conditions by conducting pure torsion and combined tension-torsion experiments on a Zr-based BMG that exhibits significant room temperature plasticity. Specimens with either high or low notch acuity are tested to assess the effects of plastic constraint on the fracture process. Detailed finite element simulations were performed to study the evolution of stress and strain fields within each specimen before the onset of fracture. These are correlated with the fractographic features to determine the fracture criterion and mechanism. Results indicate that the length scale associated with fracture and the mechanism of crack growth are both sensitive to the application of tensile loads. The fracture toughness of BMGs under different modes of loading are evaluated and compared. © 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Bulk metallic glass Tension-torsion Ductile fracture Finite element analysis Shear band Crack propagation

1. Introduction Compared to the volume of research work that was performed on glass forming ability, structure, and plastic deformation characteristics of bulk metallic glasses (BMGs), relatively little effort has been put in understanding their fracture behavior [1e7]. In particular, experimental investigations that provide key insights into the micro-mechanisms of crack initiation and propagation are few and far in between. Nevertheless, the governing fracture criterion and mechanism for ductile BMGs are now well established via detailed * Corresponding author. School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, 639798, Singapore. ** Corresponding author. E-mail addresses: [email protected] (P. Tandaiya), [email protected] (U. Ramamurty). 1 Prof. Upadrasta Ramamurty was an editor of the journal during the review period of the article. To avoid a conflict of interest, Prof. Christopher A. Schuh acted as editor for this manuscript. https://doi.org/10.1016/j.actamat.2019.02.025 1359-6454/© 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

experiments that are coupled with finite element analyses (FEA) of notched BMG specimens subjected pure modes I, II and mixed mode (i.e., combinations of modes I and II) loading conditions [8,9]. In contrast, only one study is available for fracture under pure torsional (or mode III) loading of notched BMG bars [10]. While there are indeed a number of studies performed on unnotched BMG cylindrical bars subjected to torsion, the primary purpose of them was to identify the yield criterion [11e15]. Even the only study on torsion of notched Zr61Ti2Cu25Al12BMG by Song et al. [10] is purely experimental in nature. Given that the visualization of the notch deformation and fracture initiation in cylindrical bars is not possible via side surface observations, as performed in modes I and II experiments, it is imperative that complementary FEA, with the appropriate constitutive models [16e18] incorporated, are performed so as to develop a comprehensive understanding of the fracture process under such complex loading conditions. Keeping the above paucity of data on mode III fracture of BMGs

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in view, we have performed pure torsion as well as tension-torsion experiments, coupled with FEA, on notched Zr-based BMG bars, with the following specific objectives. (a) Understand the notch tip deformation and fracture initiation processes in detail, in particular, the effect of superimposed hydrostatic stress, sh , on the fracture process. The current understanding of fracture in BMGs is that it is relatively easier for crack to initiate in shear-dominant loading conditions that prevail during the in-plane shear failure of mode II specimens vis-
a-vis tension dominant loading that occurs during mode I (or the opening mode); this is the reason for mode II fracture toughness, KIIc, of BMGs being much smaller than mode I fracture toughness, KIc [9]. The rationale for this, from a mechanistic perspective, is that shear bands (SBs), which are the mediators of plastic flow in BMGs, can nucleate and propagate at a much lower applied far-field stress in mode II than in mode I. Crack initiation is aided by sh that develops during plastic deformation ahead of the notch tip and crack growth instability sets in as and when the perturbation in the fluid meniscus within the dominant SB experiences a negative sh gradient [19]. In the current context, one can expect pure torsion (or mode III) loading to be shear dominant and a superimposed tensile stress in a tension-torsion loading could accelerate shear band to shear crack transition easier crack initiation. The validity of such a scenario is examined in this study. (b) The applicability of the fracture initiation criterion, which was identified for modes I and II (as well as mixed modes) of fracture via experiments, fractographic observations, and FEA, for fracture under the mode III condition. If not, identify an appropriate fracture criterion for failure under torsional loads. (c) Compare the deformation and fracture responses of shallow and sharp notched specimens. The former is somewhat similar to that of an unnotched or dog-bone shaped bar (except that it allows to focus on some 'gauge length' of the specimen) whereas the sharper notched specimen is more reflective of a fracture specimen. These changes in the notch acuity also result in differences in the plasticity spread ahead of the notch tip as well as affect sh distributions. Together, these will allow us to shed some more light on fracture in BMGs. (d) A comparison of the fracture toughness values in all the three modes. Whereas KIc and KIIc for some BMGs have already been evaluated and compared, simultaneous evaluation of the mode III fracture toughness, KIIIc, of any BMG was not performed earlier and is done in the present work. While it is reasonable to expect KIIIc to be lower than KIc, given that mode III is shear dominant, whether the out-of-plane tearing mode of fracture be more difficult than mode II is not known yet. The present work critically examines this crucial aspect. From a technological point of view, such knowledge is important because it is one of the common failure modes of cracked bodies and could occur readily in circumferential cracked shafts, pipes and cylinders subjected to torsion. 2. Experiments An as-cast Zr-based amorphous alloy with the composition, Zr70Ni16Cu6Al8 is utilized for the experimental investigations performed in this work. Processing details of this alloy can be found in Ref. [20]. Its relevant mechanical properties are: Young's modulus, E ¼ 69 GPa, Poisson's ratio, n ¼ 0.39, yield strength in compression, scy ¼ 1.65 GPa, tensile yield strength, sty ¼ 1.5 GPa [20,21]. Two sets of circumferentially notched cylindrical specimens, designated as S1 and S2, were fabricated from the alloy stock by turning on a lathe machine. Both S1 and S2 have the same grip section diameter, grip section length, and notch depth of 8, 20, and 2 mm, respectively. While the notch root radius, r0, in S1 is 2 mm, it is 0.5 mm in S2. A schematic illustration of both these specimen types is provided in Fig. S1 of the supplementary information (SI). One specimen from each set is tested under pure torsion and combined tension-torsion

loading conditions. From the point of view of fracture testing, a mode mixity parameter that combines modes I and III loading can be obtained from the finite element (FE) simulations of specimens subjected to pure torsion as well as tension-torsion loading. This plastic modemixity parameter, MP is defined as [22]:


2 szz ðr; F ¼ 0Þ MP ðrÞ ¼ tan1 p sqz ðr; F ¼ 0Þ

(1)

Here, the radial distance, r, is measured from the notch tip in the undeformed configuration, and the angle F from the line in front of the notch tip in the counter-clockwise direction and centered at the notch root tip. Also, szz and sqz are normal stress along the bar axis and tangential shear stress in the notch root plane, respectively, near the notch root tip. Note that MP essentially represents the ratio of normal to shear tractions on the plane ahead of the notch tip with MP ¼ 0 and 1 representing locally modes III and I conditions, respectively. In this study, MP ¼ 0.6 and 0 for specimens subjected to tension-torsion and pure torsion loading, respectively. The tension-torsion and pure torsion tests on the BMG specimens were performed on a servohydraulic axial tension-torsion testing machine. Specimens were knurled at the grip section in order to avoid slippage during testing. Details of the applied angular and axial displacement rates are provided in Table 1. For tension-torsion tests, the axial and angular displacements were applied simultaneously whereas for pure torsion tests, the upper gripping head of the testing machine was allowed to move freely in the axial direction so that no significant axial force can act on the specimen throughout the test. Note that the angular displacement rate in tension-torsion tests is nearly-half of that in pure torsion. This is because specimens loaded in tension-torsion have superimposed tensile displacements in addition to angular displacements. To ensure that the overall material deformation occurs at a similar rate under both loading conditions, the angular rate in the former test is reduced. Also, both the chosen angular displacement rates are well within the prescribed quasi-static loading conditions of BMGs, which rules out the possibility of rate effects influencing the results of the study. All the tests were conducted at room temperature (~300 K), which is well below the glass transition temperature (Tg ¼ 640 K) of the alloy [20]. Finally, the fracture surfaces of the specimens were examined using scanning electron microscopy (SEM) to discern the characteristic surface morphologies and identify the operative failure mechanisms. 3. Finite element analyses (FEA) Deformation behavior of both types of circumferentially notched specimens with different notch acuity and each tested under both pure torsion and tension-torsion loading was analyzed by recourse to FEA. The Anand and Su [17] constitutive model for metallic glasses implemented [18] in ABAQUS/Standard through user material subroutine UMAT was employed in the simulations. This model is a three dimensional, finite strain, multi-axial deformation constitutive model that is developed on the basis of multisurface Mohr-Coulomb based plasticity and accounts for the unique features of plasticity in BMGs such as pressure/normal stress sensitivity of yielding, plastic dilatancy, strain softening and inhomogeneous deformation [4,7,17,19]. To obtain the material parameters for this constitutive model, the value of Young's modulus, E, Poisson's ratio, n, the uniaxial tension and compression stressstrain curves of the BMG tested in our work were obtained from Ref. [20]. By fitting the experimental stress-strain curves with simulated ones, the internal friction parameter, m, initial cohesion, c0, saturation value of cohesion, ccv, saturation free volume, hcv,

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Table 1 Details of loading conditions and loading rates used in experiments on different specimens. Specimen

Loading type

Axial displacement rate (mm/s)

Angular displacement rate ( /s)

S1 S1 S2 S2

Pure torsion Tension-torsion Pure torsion Tension-torsion

0 0.001 0 0.001

0.047 0.019 0.047 0.019

dilatancy function (in tension and compression), g0, and the strain softening parameter, b, were determined. Note that the value of the strain rate sensitivity parameter, m, was chosen to be a small value to model the almost rate insensitive response of the BMG at room temperature. The material parameters used in the simulations are listed in Table 2. Finite strain, elasticeplastic FEA of the specimens were carried out to calibrate the energy release rate, J, against the applied torque, T. In order to study the application of tension-torsion on the specimens, three dimensional (3D) specimens were modeled with finite elements. The corresponding finite element meshes used in pure torsion and tension-torsion simulations are displayed in Fig. S2 (a), (b) and S2 (c), (d) of SI for specimens S1 and S2, respectively. The mesh employed in the analyses is well refined at the notch root, with 100 elements along the circumference of the notch root. The mesh for specimen S1 comprises of 33600 elements and 36301 nodes while, the mesh for specimen S2 comprises of 34216 elements and 37044 nodes. The element type C3D8H available in ABAQUS (2012) is used for full specimen modeling in all the simulation studies related to the notched specimen. The hybrid element formulation is used in order to circumvent mesh locking effects due to near-plastic incompressibility [8]. In this work, MP corresponding to specimen S2 subjected to mixed mode loading is evaluated from the above FEA at a normalized radius r/(J/c0) ¼ 4. Two sets of elastic-plastic FEA were performed on these notched specimens. In the first set, the initial cohesion was statistically distributed among the finite elements in order to trigger the formation of shear bands at the notch root of the bar. A Gaussian distribution with a standard deviation of 3% of the mean cohesion value was employed for this purpose. In the second set, an uniform spatial distribution of initial cohesion was assumed. The plastic strain fields and sh obtained from this set of simulations were used to determine the fracture initiation criterion for the BMG examined. 4. Results 4.1. Plastic strain fields in pure torsion and tension-torsion tests The experimental and simulated variations of T as a function of angular displacement, q, for S1 and S2, under pure torsion are plotted in Fig. 1(a) and (c), respectively. For both the specimens, T initially increases linearly with q, before deviating from linearity. Table 2 List of material parameters used in present finite element simulations. Material Parameter

Value

Young's Modulus, E Poisson's ratio, n Internal friction parameter, m Reference strain rate, v_

69 GPa 0.39 0.005 0.001 s1

Strain rate sensitivity parameter, m Initial cohesion, c0 Saturation free volume, hcv Saturation cohesion, ccv Dilatancy function g0 in tension (controls free volume evolution) g0 in compression

0.02 835 MPa 0.0096 635 MPa 0.4 0.015

0

However, T continues to increase with q beyond yield, which is unlike the tensile stress-strain responses of BMGs, wherein the onset of macroscopic yield often coincides with specimen fracture. The peak values of torque, Tmax for both the specimens are similar. S1 fractures abruptly at the peak (q ¼ ~3.5 ) whereas S2 gradually softens first (for up to a q of ~7 ) before a steep drop in T with q is noted (inset of Fig. 1c). The specimen continues to deform with the 'softening' portion of the T-q plot extending to a q of ~34 , at which it eventually fails. The T-q responses for S1 and S2 loaded in combined tension-torsion are displayed in Fig. 1 (b) and (d), respectively. But for the fact that the value of Tmax is slightly higher in tension-torsion than in pure torsion, the experimentally obtained T-q responses of both the specimens are similar to those seen in Fig. 1 (a) and (c), respectively. In all cases, a reasonable agreement between the predicted and measured values of Tmax can be noted. Since the simulations do not incorporate a fracture criterion explicitly, the predicted T-q plots show continued deformation beyond the experimental failure points; in the case of sharp notched S2 specimens the predicted and experimental T-q responses beyond Tmax can also be noted. A close match is also observed between experimental and simulated variations in the axial tensile load, P, with the axial displacement, d, for S1 and S2 subjected to tension-torsion loads (Fig. 2(a) and (b)). Here, both the specimens exhibit nominally linear variations in P with d, until failure. Overall, an excellent match between simulations and experiments, irrespective of the notch acuity or loading condition, validates the constitutive model employed in this work. This also implies that our FE simulations can be used to understand the evolution of the stress and plastic strain fields in both the specimens under different loading conditions, with a reasonable level of confidence. The predicted plastic strain fields in S1 and S2 subjected to pure torsion and tension-torsion are compared in Fig. 3(a)e(d), wherein the longitudinal sectional views of the contour plots of maximum principal logarithmic plastic strain, lnlP1 , at respective Tmax are displayed. (See Fig. S3 (a) to (d) for respective front views.) In all the figures, the contour corresponding to lnlP1 ¼ 0.001 represents the elastic-plastic boundary. From the front view of S1 subjected to pure torsion (Fig. S3(a)), it can be seen that at the outer periphery of the notch root, lnlP1 peaks over an annular ring in the notch root plane that is perpendicular to the loading axis. The longitudinal section view (Fig. 3a) further reveals that this particular contour extends up to some distance from the notch root periphery, x, within the specimen. Similar contours emerge in the front view and longitudinal sectional views of deformed S2 at the onset of failure (see Figs. S3(c) and 3(c)). These represent the shear bands extending on the notch root plane from the periphery, which eventually turn into cracks. The cross-sectional views of the plastic zones, displayed in Fig. 4(a)e(d), show homogeneous concentric plastic strain rings extend from notch root periphery towards the centre of the specimen. A coarse-grained comparison of the strain contour plots of S2 (see Fig. 3(c), (d) and S3(c), (d)) indicates that the diameter of the elastic core and the plastic zone size and shape for both loading conditions are similar. However, on closer examination, several differences can be noted. It appears that the simultaneous application of tensile and torsional loads promotes

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Fig. 1. Torque, T, versus angular displacement, q, responses obtained from experiments and simulations for specimen S1 and S2. Insets in (c) and (d) show the responses until the specimens fracture.

Fig. 2. Comparison of tensile load, P, versus axial displacement, d, curves obtained from experiment and simulation (a) S1 specimen under tension-torsion loading. (b) S2 specimen under tension-torsion loading.

localization of strain on two additional planes in S1. These two planes are equidistant from the notch root shear plane and lie on either side of it (see Fig. 3(b) and Fig. S3(b)). To understand the deformation and failure behavior in greater detail, the evolution of plasticity in S2 specimens subjected to

either pure torsion or tension-torsion loading are compared in Figs. S4 and S5, which display the snapshots extracted at progressively increasing T. The movies of the plasticity evolution are also given in SI. From Fig. S4(a), the nucleation of a (nearly) half-penny shaped plastic deformation region at the notch root periphery,

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Fig. 3. Contour plots of maximum principal logarithmic plastic strain, lnlP1 , in longitudinal section view for specimen S1and S2 subjected to pure torsion and tension-torsion loading at Tmax.

possibly aided by the fluctuations in the shear strength of the material, in S2 specimen subjected to pure torsion can be seen. In the next instance (Fig. S4(b)), the plastic zone spreads circumferentially until an annular region of plastic zone with uniform width is obtained. Only then, the plastic zone moves inwards, i.e., towards the center of the specimen. In Figs. S4(c) and (d), independent nucleation of several intensely plastic regions (color coded with red) can be noted. As seen from the subsequent and successive snapshots, their growth, as T increases, is more circumferential than radial. Thus, it appears that the pure torsional load favors the perturbations in plasticity to spread around the notch circumference on the shear plane first. Supplementary video related to this article can be found at https://doi.org/10.1016/j.actamat.2019.02.025. The superposition of tensile load over torsion alters the above situation in a considerable manner, as illustrated via the snapshots obtained on specimen S2 subjected to tension-torsion loading (Figs. S5(a)e(f)). Here, the development of plasticity is not uniform around the periphery. Instead, the three plastic regions that nucleate simultaneously at the notch root, grow inwards with equal alacrity as they spread circumferentially. P The line variations of lnl1 in the notch root plane at Tmax are plotted as a function of the distance, x, for all the specimens in P Fig. 5. In all cases, lnl1 is maximum at x ¼ 0 mm and continuously decreases towards the interior of the specimen. The peak values of P lnl1 in S2 are substantially larger than the corresponding values in S1. This is consistent with the fact that S2 has a sharper notch tip than S1 and hence can accumulate larger plastic strains due to greater stress concentration. It is noted that lnlP1 reduces to zero at x of ~1.0 mm in S1 and ~1.5 mm in S2 when subjected to pure torsion; the corresponding distances are ~1.2 mm in S1 and ~1.4 mm in S2 subjected to tension-torsion. Owing to the high lnlP1 , the gradient of

it with x in S2 is substantially larger than that seen in S1, for both the types of loading. Furthermore, the gradients in S1 are constant, whereas it increases slightly closer to the notch root in S2. These differences in the nature of the plastic strain distributions are directly related to r0 of the two specimens. Note that the unnotched sections of the specimens deform elastically. Therefore, plastic flow in the circumferential direction is constrained; a smaller r0 leads to a larger plastic constraint. As a result, specimen S2, which has lower r0 than specimen S1, will plastically deform to a higher value of lnlP1 in the radial direction. A change in the loading mode from pure torsion to tension-torsion not only reduces the peak values of lnlP1 , but also leads to a reduction in the strain gradients within the notched sections of both S1 and S2. These are a direct consequence of the elevated levels of triaxiality in the notched ligament induced by the superimposed tensile load. 4.2. Fractography Fractographs of S1 and S2 subjected to pure torsion are shown in Figs. 6 and 7, respectively, while the fractographs of the two specimens subjected to tension-torsion loading are displayed in Figs. 8 and 9. In each of these figures, enlarged views of two locations marked A and B are included as insets along with the full view of the fracture surface; A is located at the specimen periphery while B is at the center of the specimen. Broadly, a reasonable correspondence between these fractographs and the respective contour plots of lnlP1 , displayed in Fig. 4(a) and (b), and 4(c) and (d), can be noted. In specimen S1, that was subjected to pure torsion, a relatively smooth annular region with a width of ~230 mm and extending from the periphery of the specimen can be seen (Fig. 6). Higher magnification imaging of this region, displayed in inset A, shows that this region consists of a network of smeared vein and ridge

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Fig. 4. Contour plots of maximum principal logarithmic plastic strain, lnlP1 , on the notch root cross section for specimens S1 and S2 subjected to pure torsion and tension-torsion loading at Tmax.

P

Fig. 5. Variation of maximum principal logarithmic plastic strain, ln l1 , along the notch root cross-section for specimens S1 and S2 subjected to pure torsion and tension-torsion loading at Tmax.

patterns, which are thin finger-like features that extend from each edge of the larger vein patterns and have no specific orientation. However, the vein patterns appear to be aligned with the direction of shear and make an angle of ~45 with the specimen periphery. These features are typically associated with stable crack extension

in BMGs. Further away from the specimen periphery, the morphology of the fracture surface becomes gradually rough. Coarse, dimple-like features that are characteristic of unstable and catastrophic crack propagation in BMGs are prominent in regions that are closer to the centre of the specimen (inset B in Fig. 6). In the inset A of Fig. 7 (specimen S2 subjected to pure torsion), an almost featureless annular region, having a width of ~400 mm, is seen along the periphery of the specimen. This region, containing concentric and shallow groove markings, is the shear sliding zone (SSZ), which forms as a result of mutual sliding between two mating surfaces under shear stress. In the interior of this annulus, two additional features can be observed. Near the inner boundary of SSZ and up to a distance of ~100 mm, shallow vein patterns are observed. However, close to the centre of the specimen, as shown in inset B, the fracture surface is significantly rough and contains dimple-like patterns. It is noteworthy that SSZ is not present in the fracture surface of specimen S1 subjected to pure torsion. However, it has a significantly larger zone of vein patterns compared to that of S2. This, in turn, suggests that although both specimens finally undergo catastrophic failure, the mechanism of crack initiation and growth in them may be different. The fracture morphology of specimen S1 subjected to tensiontorsion is distinctly different. In inset A of Fig. 8, a 30 mm wide, featureless and flat annular zone is observed at the periphery of the specimen. Unlike the featureless zone observed in specimen S2 that was subjected to pure torsion, no regular concentric groove patterns could be discerned in this specimen. Additionally, along the inner boundary of this annular zone, a continuous ridge is seen. From this continuous ridge, several smaller ridges originate and

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Fig. 6. SEM fractographs of specimen S1 subjected to pure torsion loading. Detailed view of the region in box A showing the smooth region with vein patterns from the periphery towards the inside of the specimen. High magnification view of the region in box B showing dimple like patterns on the fracture surface.

Fig. 7. SEM fractographs of specimen S2 subjected to pure torsion loading. Detailed view of the region in box A showing the smooth region from periphery of notch root to inside. High magnification view of the region in box B shows dimple like patterns coexisting with vein patterns on the fracture surface.

extend radially towards the center of the specimen. Further away from the periphery of the specimen, the ridges branch and meander in random directions. Additionally, at some regions, starlike patterns, which consist of a smooth featureless circular core surrounded by radiating ridges, are also seen. All the abovementioned features are superimposed over relatively smooth regions in the fracture surface. Fracture features close to the centre of the specimen, as can be seen in inset B of Fig. 8, are much rougher and non-coplanar. However, instead of the dimple-like pattern observed in specimens subjected to pure torsion, the morphology resembles rough hackle, which is typically observed when the material experiences ductile tearing. It is noteworthy that some of these fracture features have been previously observed in both notched and unnotched specimens that were subjected to tensile loads (more on this later) [23,24]. Additionally, star-shaped features are also commonly observed in the fracture surface of uniaxial tensile specimens [20]. In the fracture surface of specimen S2 that was subjected to

tension-torsion (see Fig. 9), a rough, ~200 mm wide annular region consisting of dimple like patterns forms along the specimen periphery. Additionally, discontinuous patches of SSZ and ridge-like features can also be seen in some portions of the annulus, indicating that the crack growth in this specimen is not necessarily annular. In the region situated between the inner boundary of this annulus and the center of the specimen, the morphology of the fracture surface changes to that of rough hackle. Finally, a combination of vein patterns and ridges appear at the center of the fractured specimen, as shown in inset B. The diverse nature of fracture features observed above indicates that the precise nature of the fracture mechanism may be sensitive to the geometry of the specimen and the type of loading. In particular, the observations suggest that the fracture mechanism under tension-torsion and pure torsion in the S2 specimens could be significantly different. Nevertheless, fracture in all the specimens is ductile, since no brittle fracture features, such as mirrormist morphologies, are observed on the fracture surface [25].

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Fig. 8. SEM fractographs of specimen S1 subjected to tension-torsion loading. Detailed view of the region in box A showing shear offset of 30 mm and vein patterns emerging from the boundary of the offset zone. Detailed view of the region in box B showing catastrophic fracture region.

Fig. 9. SEM fractographs of specimen S2 subjected to tension torsion loading. Detailed view of the region in box A. High magnification view of the region in box B showing vein patterns on the fracture surface.

5. Discussion 5.1. Fracture in modes I and II As mentioned already, ductile fracture in BMGs has hitherto been studied only in the cases of pure modes I, II, and their mixed mode loading conditions. Tandaiya et al. [9,16] have conducted systematic experiments and complementary simulations on symmetric and asymmetric notched rectangular four point bend (4 PB) specimens to understand the mechanism of fracture as well as identify the fracture criterion under those conditions. To ascertain whether the same criterion and mechanism also govern the fracture of the BMG utilized in the current study, we performed modes I and II tests on 4 PB specimens whose geometries are similar to those utilized by Tandaiya et al. Results and analyses of these experiments lead us to similar conclusions with regards to the fracture processes in the current BMG under modes I and II, which is as follows. (These similarities are the reason for not including these

results in the main text.) Under both modes of loading, an incipient crack grows inside the dominant SB (see Fig. S6 (a), (b) and S7 (a), (b)). Since the material within an SB is ‘liquid-like’, the fracture features can be described a mechanism that is akin to those seen in the fracture of liquids [26,27]. A typical fracture surface of a mode I specimen is shown in Fig. S8. The smooth featureless zone that extends from the notch front is the ‘notch blunting zone’. This zone forms from the intersection of SBs with the notch root. This is followed by a relatively smooth region that contains ridges oriented parallel to the crack growth direction. Each ridge region forms when the notch surface, behaving like a fluid meniscus, breaks into fingers and grows within the SBs in the presence of a positive hysh drostatic stress gradient, ddx . Following this region containing ridges and vein patterns, the fracture surface becomes markedly rough and contains either dimples or rough hackle. While the ridge patterns were not observed in fracture surfaces of specimens loaded in pure mode II as the cracked surfaces rub against each other while sliding past each other [16], they also contain three

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distinct regions that closely resemble the three regions identified in mode I fracture specimens. As mentioned earlier, according to the ductile fracture criterion of BMGs, which was successfully applied for fracture under mode I, II and mixed mode (I and II) loading conditions, fracture occurs when ln lP1 exceeds a critical value of strain, εc over some critical sh length, lc beyond which ddx becomes negative as fluid meniscus instability (FMI) sets in Refs. [19,28]. Both lc and εc are determined with the combined use of post-mortem fractography and FEA, which is illustrated for the mode I fracture tests of the present study. An SEM fractograph obtained from the fractured specimen is shown in Fig. S8. From it, an lc of ~262 ± 13 mm, which corresponds to the distance from the notch tip to the point where the ridges that form as a result of the FMI mechanism disappear. Next, at the radius from the notch tip equal to the value of lc, the values of lnlP1 from FEA of the fracture specimen are examined. Given that the crack grows within a dominant SB, it was then determined that εc ~10.4%. The same procedure is employed to determine lc and εc for mode II specimens. However, in this study, lc could not be determined from post-mortem fractography as the specimen was crushed at the loading points just before catastrophic crack propagation could occur (see Figs. S9 and S10). Nevertheless, from SEM images of the notch tip after the specimen got crushed, it was determined that the crack had grown to a length of ~258 ± 10 mm, before undergoing out-of-plane deflection (see Fig. S10). Assuming that this length corresponds to lc and applying it to the simulated strain contours at the notch tip for the mode II specimen in Fig. S7, a εc ~9.8% is obtained, which is close to that noted for mode I specimen. 5.2. Effect of notch geometry and loading conditions on sh Given sh 's role in the stable crack growth mechanism, we examined its development in the four different geometry-loading combinations examined in this work next. In Fig. 10, the computed variations of sh at Tmax in the notch root plane of specimens S1 and S2 subjected to either pure torsion or tension-torsion are plotted as a function of x. For both the loading conditions, sh in specimen S2 is always greater than that in S1. This is due to the smaller r0 in the former which elevates the stress triaxiality inside the notched ligament. In S1, sh is five times higher at the notch root periphery when subjected to tension-torsion than under pure torsion, which again is due to an increase in triaxiality caused by the sh addition of a tensile load in the former. Importantly, ddx is positive at

Fig. 10. Variation of hydrostatic stress along the notch root cross-section for specimen S1 and S2 subjected to pure torsion and tension-torsion loading at Tmax.

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the notch root for S1 irrespective of the loading mode and for S2 subjected to tension-torsion. Values of bph, which are the distances from the notch root at which sh reaches a maximum, are listed in Table 3. It appears that the widths of the regions containing ridges observed in the fractographs, lc, which are also listed in Table 3, are in reasonable agreement with bph extracted from FEA. The exception is S2 subjected to pure torsion; in this case, sh is maximum at sh the notch root itself, and steadily decreases till x ¼ ±1 mm. Since ddx is negative, FMI cannot be sustained at the notch root periphery. This explains the absence of ridges in the annular region along the periphery of the fracture surface of specimen S2 in Fig. 7 (see inset A). At distances closer to the centre of the specimen, there are some sh local fluctuations in ddx . Even a slightly positive value of the gradient can promote the appearance of some vein patterns and dimpled features in the fracture surface during catastrophic crack growth. The sh vs: x plots displayed in Fig. 10 also help in rationalizing other subtle fractographic observations as following. For S1 subjected to pure torsion, the low magnitude of sh also supports the notion that ridges should be oriented along the direction of shear, which is indeed found to be the case. The observation of a second peak in sh at x of ~1 mm suggests that nucleation and growth of voids aided by a positive sh can occur; this inference is corroborated by the dimpled appearance of the fracture surface at the centre of the specimen (inset B in Fig. 6). For S1 subjected to tension-torsion, sh does not decrease markedly further away from the periphery of the notch root, but in fact, it remains relatively high even close to the centre of the specimen. This is possibly the reason for the ridge patterns being oriented nearly perpendicular to the direction of shear displacement. Furthermore, since the fluctuations in sh with x are not prominent, dimple patterns are not observed on the fracture surface. On the basis of the above observations, we can conclude that the operative fracture mechanism in S1 is FMI, irrespective of the loading conditions [19,27]. FMI is also the mechanism that dictates fracture in S2, but only under tension-torsion loading. It cannot be used to describe fracture processes in S2 specimen subjected to sh pure torsion, because ddx is negative at x ¼ 0 itself. From the discussion of the results, presented so far, it is clear that the FMI mechanism governs fracture in S1 subjected to pure torsion and tension-torsion as well as in S2 subjected to tensiontorsion. This mechanism also explains certain fracture features in a consistent manner. For instance, the ridges on the fracture surface are remnants of the unstable meniscus growth in the direction of the positive hydrostatic stress gradient, which gives it a finger like appearance. Similarly, the morphologies of different cavities on the fracture surface can be rationalized. The liquid-like nature of the material within the shear band makes it prone to cavitation, which is aided by the presence of tensile principal stresses. When the shear stresses dominate, oblong cavity growth within the shear plane would occur. On the other hand, high hydrostatic tension aids in out of plane cavity growth that results in the observation of vein like patterns. The observation of some additional fracture features like the star shaped patterns (see box A in Figs. 8 and 9) can be attributed to the local fluctuations in the suction gradient around the growing cavities, which, in turn, can result in the operation of Taylor's FMI and formation of fingers around the cavity. A potential mechanism for the observed featureless annular ring in the fracture surface of the S2 specimen subjected to pure torsion (Fig. 7) is the following. The absence of a positive hydrostatic stress gradient prevents the fluid meniscus from growing inside the shear band. Instead, the continued rubbing of the crack surfaces forces the fluid like layer within the shear band to spread uniformly inwards, leading to the smooth annular ring.

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Table 3 Details of critical distance, strain and fracture toughness data obtained from different specimens under various loading conditions.

Mode I Mode II Mode III Mixed-Mode Mode III Mixed-Mode (I and III) (Mp ¼ 0.6)

S1 S1 S2 S2

-

Pure torsion tension torsion Pure torsion tension torsion

Experimental critical length, lc (mm)

Critical strain (corresponding to experimental distance) εc (%)

Hydrostatic stress peak distance from notch tip, bph (mm)

Critical strain at bph, εc (%)

Fracture toughness, Jc (N/mm)

262 ± 13 258 ± 10

10.4 9.8

322 315

9.6 8.8

228 ± 9 225 ± 11 400 ± 15 273 ± 8

10.1 5.7 38.5 32.1

230 236 0 210

10.0 5.6 e 33.7

135 17.5 ±2.12 e e 45 25

5.3. Characteristic lengths and strains Next, the critical parameters, lc and εc, which capture the fracture criterion are evaluated for specimens S1 and S2 subjected to torsion and tension-torsion and are examined. At the fracture initiation in these specimens, the crack is expected to grow radially inside the circumferential SB. On revisiting the post-mortem fractography in Section 4.2, it can be noted that only S1 exhibits distinct stable crack growth features such as ridges and vein patterns all around the notch root periphery, which transitions to the rough hackle (or coarse dimpled region) at the onset of unstable crack growth, for both the loading conditions examined. On this basis, lc is taken as the radial distance between the notch root periphery and point where the morphology of the fracture surface changes markedly. Thus, lc for specimen S1 subjected to pure torsion was estimated to be 228 ± 9 mm whereas it is 225 ± 11 mm when loaded in tension-torsion. On mapping these values of lc on the corresponding line profiles of lnlP1 shown in Fig. 5, it was found that respective εc are ~10.1% and 5.7%. The values of lc for S1 are similar between pure torsion and tension-torsion, but are lower by ~13% of those estimated from modes I and II fracture experiments. While εc estimated for S1 subjected to pure torsion is in excellent agreement, εc for S1 subjected to tension-torsion is substantially smaller. In this context, it is worth noting that S1 is a blunt notched specimen and more akin to that of the gage-section of a commonly used 'dog-bone' shaped tensile specimen, but with an hour glass shape. This has the following implications. The stress concentration in this specimen is far lower than that in S2, which implies that the plastic envelope will be more spread out in the specimen. As a result, the magnitude of the local plastic strain in the blunt notch will be low when failure occurs. Therefore, the value of εc for S1 subjected to tension-torsion does not need to match with those of sharp notched modes I and II specimens. Turning to S2, the estimated values of lc are 400 ± 15 and 273 ± 8 mm for pure torsion and tension-torsion loaded specimens, respectively. These correspond to distances from the notch root at which the fracture morphology transitions from dimple like morphology or smooth grooved surface to the rough hackle region. Values of εc at these lc are 38.5% and 32.1%, respectively. The high value of εc in S2 subjected to tension-torsion may be rationalized, especially since lc is similar to that seen in modes I and II specimens, by considering that the local strains could be higher due to extra torsional shear experienced in this relatively sharper notched specimen. However, for S2 subjected to pure torsion, both εc and lc are substantially larger, which indicates that the ductile fracture criterion, in its present form, may not be applicable to pure mode III specimens. In order to explore an alternate ductile fracture criterion, attention is focused on the effective displacement/separation, l, across the dominant shear band at the point of fracture initiation, which is defined as the following:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !2 u u u 2 ut n t l¼ þ c c

dn

dt

(2)

Here, un and ut are the normal and tangential displacements/ separations across the dominant shear band at the notch root periphery of the specimen, dcn and dct are the respective critical values at fracture initiation purely under normal and tangential separations across the dominant shear band. This discussion is restricted to only S2 because of the sharper notch it contains as compared to S1, and hence is closer to a genuine fracture specimen. The proposed criterion states that fracture initiates in the specimen when the effective separation l reaches a critical value of 1. The values of un and ut at the point of fracture initiation for S2 specimen are determined from the results of the FE simulation at Tmax. For this purpose, the difference in normal and tangential displacements of two nodes located just above and below the dominant shear band at the notch root periphery are considered. The values of un for pure torsion and tension-torsion cases at Tmax are found to be 0.45 and 2.57 mm, respectively. Similarly, the values of ut for pure torsion and tension-torsion cases at Tmax are 58 and 39 mm, respectively. It turns out that the above described critical effective separation criterion is able to predict the point of fracture initiation for S2 specimen under both pure torsion and tensiontorsion loading cases with the values dcn ¼ 3.5 mm and dct ¼ 58.5 mm. However, it should be mentioned that some more experiments are needed to further verify the applicability of the above criterion in fracture specimens with different mode mixities. Also, the critical effective separation criterion is similar in spirit and consistent with the critical strain criterion discussed earlier because, physically, both are based on the attainment of a critical deformation within the shear band. More fundamental studies based on atomistic simulations of fracture initiation within the shear band, under different combinations of normal and tangential stresses acting on it, are needed to further understand the fracture initiation in BMGs. 5.4. Comparison of notch fracture toughness in different loading modes The calibration curves of energy release rate, J, for the mode I and mode II specimens are plotted together as a function of the load, P, in Fig. S11. In the pure mode I specimen, J, rises steeply with increasing P, whereas it is more gradual for the pure mode II specimen. From the experimentally measured values of the fracture initiation load, Pc, the critical energy release rate (notched toughness), Jc, in pure modes I and II are obtained as 135 and 17.5 ± 2.12 N/mm, respectively. A substantially smaller Jc in pure mode II is due to the easier initiation of SBs under this loading mode [16,19,29]. Variations of J with T for S2 specimen under pure mode III and mixed mode I/III loading conditions are displayed in

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noted. This could again be due to the normal tensile stresses preventing rubbing of the shear fractured planes. Importantly, as the FEA show (see Fig. S5), the superimposed tensile load makes the perturbations in the plastic flow fields grow inwards, which, in turn, allows for an early crack initiation. This could be another reason for the observed reduction. Furthermore, in the mixed mode (I and III) with Mp ¼ 0.6 case, the evolution of damage mechanisms inside the shear band could get accelerated by the normal stress component on the shear band plane, leading to faster conversion of shear band into crack resulting in lower fracture toughness than in pure mode III loading case. 6. Summary and conclusions

Fig. 11. Calibration of energy release rate, J, with torque, T, based on elasticeplastic geometrically non-linear finite element simulation of pure torsion and tensiontorsion of S2 specimen. Open circles marked on the curve indicate the fracture initiation point.

Fig. 11. (Specimen S1 was not considered as it contains too blunt a notch to be approximated as a crack. For a notch to be considered as a sharp crack, the solutions obtained for the J-based stress fields ahead of the notch tip should be valid in the domain over which the plastic zone spreads within the ligament [30]. While this condition is met in specimen S2, the same is not true for specimen S1.) The calibration is performed by assuming four virtual cracks around the circumference of the bar at the notch root only for the deeply notched S2 specimen. For each of these virtual cracks, J is evaluated over several domains. In Fig. 11, the average value of the domain independent J over all the virtual cracks is displayed. From these J-T calibration curves, Jc values, which correspond to experimentally measured Tmax, were extracted and listed in Table 3. From it, we note that Jc in pure mode III is intermediate to that estimated for either pure modes I and II. Since pure mode III loading is also shear dominant–and hence favors the nucleation of SBs at a much lower load as compared to mode I loading, Jc in pure mode III being lower than that in pure mode I is in accordance with the expectation. The following factors prevent us from a direct comparison of the estimated Jc values in modes II and III. Chen et al. [31] reported that mode I fracture toughness, Kq, decreases monotonically as r0 is reduced before reaching a minimum at a critical r0,c, where after it becomes invariant. For Zr-based BMGs, r0,c is reported as ~100 mm. In the context of the current study, the value of Jc, obtained in mode I can be considered independent of r0, as r0 (~30 mm) in those specimens is much smaller than r0,c. For pure mode II specimens also r0 is ~30 mm. Moreover, fracture, which occurs in the dominant shear band, is preceded by substantial sharpening of the one half of the notch root [9]. Thus, Jc obtained under mode II loading can be considered as independent of r0. The same cannot be said, however, for the value of Jc obtained in mode III as in this case r0 (500 mm) is large whereas the value of r0,c for pure mode III is yet unknown. In the light of this, a direct comparison of Jc values obtained under modes II and III cannot be made. More fracture tests under pure torsion loading with sharper notches are essential for such a comparison. A substantial reduction in Jc when the loading condition changes from pure mode III to mixed mode I/III, i.e., tension-torsion, can be

In summary, experiments and complementary FE simulations were performed on shallow and sharp notched ductile BMG specimens subjected to pure torsion and combined tension-torsion loading, with the objective of understanding fracture of BMGs in the anti-plane shear mode. A good match between simulations and experimental results allows systematic investigation of the elastoplastic strain fields and the hydrostatic stress fields within the specimens before the onset of fracture. The combination of these results and the fractographic observations leads us to the following conclusions. a) Pure shear loading causes the plastic strain to localize within the dominant circumferential shear band; an increase in the notch acuity enhances the localization as a result of the higher plastic constraint in the circumferential direction for sharp notches. Although, the addition of tensile loads changes the magnitude of the plastic strain localization, it has a minimal effect on its distribution. Shear crack initiates inside the circumferential shear band. b) The fracture toughness measured under shear dominant modes is lower than that in mode I; toughness in combined mode I/III loading is lower than that in pure mode III loading because the latter case involves energy dissipation via frictional rubbing of the crack faces and damage mechanisms inside the shear band get promoted due to the tensile normal stress in the mixed mode (I and III) case. c) In specimens with low notch acuity, the FMI mechanism can explain the onset of stable crack growth and fracture instability irrespective of the loading conditions. Under pure shear, initial crack growth of sharp notched specimens is governed by shear sliding as the hydrostatic stress gradient is negative near the notch tip. However, final fracture instability still occurs due to the growth of a fluid meniscus due to fluctuations in the hydrostatic stress gradient inside the uncracked ligament. Acknowledgments Parag Tandaiya would like to acknowledge the financial support provided by IRCC, IIT Bombay through an initiation grant (no. RD/ 0512-IRCCSH0-021.) Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.actamat.2019.02.025. References [1] A.L. Greer, Y.Q. Cheng, E. Ma, Shear bands in metallic glasses, Mater. Sci. Eng. R Rep. 74 (2013) 71e132, https://doi.org/10.1016/j.mser.2013.04.001. [2] M.M. Trexler, N.N. Thadhani, Mechanical properties of bulk metallic glasses,

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