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Acta Materialia 55 (2007) 6541–6552

Mode I crack tip fields in amorphous materials withapplication to metallic glasses

Parag Tandaiya a, R. Narasimhan a, U. Ramamurty b,*

a Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, Indiab Department of Materials Engineering, Indian Institute of Science, Bangalore 560 012, India

Received 17 May 2007; received in revised form 3 August 2007; accepted 6 August 2007Available online 4 October 2007

Abstract

In this work, stationary crack tip fields in amorphous materials such as metallic glasses under mode I loading are studied to under-stand the factors that control crack tip plasticity and in turn impart toughness to those materials. For this purpose, finite element sim-ulations under plane strain, small scale yielding conditions are performed. A continuum elastic–viscoplastic constitutive theory, whichaccounts for pressure sensitivity of plastic flow as well as the localization of plastic strain into discrete shear bands, is employed to rep-resent the material behavior. The influence of internal friction and strain softening on the plastic zone, stress and deformation fields andnotch opening profile is examined. It is found that higher internal friction leads to a larger plastic zone. Also, it enhances the plastic strainahead of the notch tip but leads to a substantial decrease in the opening stress. Thus, it appears that a higher friction parameter promotestoughening of amorphous solids. The shear band patterns within the plastic zone and brittle crack trajectories around the notch rootgenerated from the simulations match qualitatively with those observed in experiments.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Bulk amorphous materials; Metallic glasses; Mode I crack tip fields; Finite element analysis; Shear bands

1. Introduction

Bulk metallic glasses (BMGs) represent a relativelyyoung class of structural materials with an impressive com-bination of mechanical properties. Indeed, extremely highstrengths coupled with reasonable toughness of these mate-rials has led to substantial recent research effort withemphasis on the underlying deformation physics in them[1]. In spite of their intrinsic inability to undergo extensiveplastic straining under tension, some of the metallic glassesdo exhibit good toughness values, whereas others are quitebrittle. Despite the importance of this issue for widespreadstructural usage of metallic glasses, their fracture behaviorhas received relatively little attention. This work wasundertaken to develop detailed understanding of the frac-ture mechanics and mechanisms in amorphous solids, espe-

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.08.017

* Corresponding author. Tel.: +91 80 2293 3241; fax: +91 80 2360 0472.E-mail address: ramu@materials.iisc.ernet.in (U. Ramamurty).

cially the development of the plastic zone ahead of thecrack tip and the material parameters that control it. In thispaper, results of finite element simulations of mode I load-ing of a crack in an amorphous material under plane strain,small scale yielding (SSY) conditions are reported. Theconstitutive behavior is represented by the model devel-oped by Anand and Su [2], which accounts for pressuresensitivity of plastic flow as well as the localization of plas-tic strain into discrete shear bands typically exhibited bymetallic glasses. Although this study pays specific attentionto fracture in amorphous alloys, many of the observationsmade and implications drawn from this study are equallyapplicable to other amorphous materials such as glassypolymers.

2. Background

The inelastic deformation of amorphous materials suchas BMGs, glassy polymers and ceramics is fundamentally

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6542 P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552

different from that of crystalline metals. At a microscopiclevel, for temperatures below the glass transition tempera-ture, the inelastic deformation in metallic glasses occurs bylocal shearing of clusters of atoms, referred to as sheartransformation zones, in regions having high free-volume.This shearing is accompanied by inelastic dilatation thatcauses strain softening [3], which then leads to localizationof plastic strain into discrete shear bands. Similarly, yield-ing in glassy polymers and ceramics is sensitive to hydro-static stress [4,5] and is accompanied by plastic dilatation.Further, the yield stress under uniaxial tension for amor-phous materials is different from that in uniaxial compres-sion (see, for example, [6,7]). It has also been observed that,in these materials, the shear bands depart from the plane ofmaximum shear stress [7].

Several constitutive models have been proposed in theliterature (see e.g. Refs. [6–9]) to describe the yield behaviorof metallic glasses and polymers. Experiments and molecu-lar dynamics simulations on BMGs [7,10] indicate that thenormal as well as shear stress acting on the slip plane influ-ences their yield behavior, which is accurately captured bythe Mohr–Coulomb yield criterion. This is also true ofglassy polymers exhibiting plastic flow by discrete shearbanding, such as polystyrene and polyethylene terephthal-ate [4,6]. In this connection, it must be mentioned that theMohr–Coulomb based constitutive model proposedrecently by Anand and Su (AS) [2] is capable of represent-ing inhomogeneous deformation by discrete shear banding.

Over the last decade, some experimental research hasbeen undertaken to characterize the fracture response ofmetallic glasses. Fracture toughness values ranging fromas low as 2 MPa

ffiffiffiffimp

, which is near to that of brittle silicateglasses, to 80 MPa

ffiffiffiffimp

, which is closer to that of toughstructural alloys, have been reported [11]. The fracture sur-face morphology of ductile metallic glasses is coarse, withdeep vein patterns, whereas that of brittle metallic glassesshows quasi-cleavage features with much finer vein pat-terns [12]. This dual (ductile and brittle) nature of fractureprocesses in metallic glasses is not clearly understood. Oneof the reasons for the variability in the fracture toughnessof metallic glasses is the nature of processes occurring nearthe crack tip such as crack branching and shear banding[13]. The fracture toughness of glassy polymers is alsolow and lies in the range 1–5 MPa

ffiffiffiffimp

[14]. In these mate-rials, processes like crazing and shear banding occur nearthe crack tip which affect the fracture response [15–17]. Itis important to note that all the fracture mechanisms men-tioned above depend strongly on the features of the near-tip stress and deformation fields. These, in turn, are likelyto be influenced by the mechanical characteristics of amor-phous materials like internal friction, softening, plasticdilatation, etc.

Li and Pan [18] studied the mode I asymptotic, crack tipfields in pressure-sensitive dilatant materials under 2Dplane strain conditions. Jeong et al. [17] constructed theo-retical slip lines in front of a round notch tip. Subramanyaet al. [19] performed a 3D finite element analysis of mode I

crack tip fields under SSY conditions. It should be notedthat all of these studies were carried out within the frame-work of the small strain Drucker–Prager yield criterion.Further, a power law hardening response and a constantdilatancy parameter were assumed. However, as discussedearlier, the mechanical response of amorphous materialsundergoing inhomogeneous plastic flow by shear bandingis better described by a constitutive model such as that pro-posed by AS [2], which incorporates the Mohr–Coulombyield criterion, a variable dilatancy parameter and dilata-tion induced softening. Indeed, AS [2,20] demonstratedthat their model can accurately represent overall featuressuch as load versus displacement response, as well asdetails of shear band formation during the deformationof metallic glasses in a variety of situations such as inden-tation and strip bending. It should be noted that a finitedeformation analysis of crack tip fields in glassy polymerswas conducted by Basu and Van der Giessen [15] using atotally different constitutive model that incorporates soft-ening as well as orientation hardening [21]. However, thismodel is not based on Coulomb yield theory and also doesnot effectively represent pressure dependence of yielding.

In this work, a detailed finite element investigation ofcrack tip fields in amorphous plastic solids under modeI, plane strain, SSY conditions is carried out. The consti-tutive model proposed by AS [2] is used. The influence offriction parameter and flow softening on the plastic zone,stress and deformation fields and notch opening profilesare studied. In order to gain insight into the near-tip pro-cesses like shear banding and crack branching, plastic slipline fields and possible brittle crack trajectories aroundthe notch root are generated. Further, the localizationof plastic strain into discrete shear bands is simulatednumerically by employing a statistical distribution ofcohesion. It is found that a higher value of the frictionparameter leads to a larger plastic zone and higher plasticstrain level ahead of the notch tip, but leads to a substan-tial decrease in the opening stress. The propensity forlocalization of plastic strain into shear bands increaseswith friction parameter and is promoted by softening.The simulated brittle crack trajectories and shear bandpatterns match qualitatively with those observed in exper-iments reported in the literature.

3. Material model

The constitutive equations employed in this study arebased on the finite deformation, Mohr–Coulomb typemodel proposed recently by AS [2]. In this model, a multi-plicative decomposition of the deformation gradient intoelastic and plastic parts is assumed as

F ¼ FeFp ð1ÞThe elastic part of the deformation is assumed to be smalland the relationship between the principal Kirchhoff stres-ses and the principal logarithmic elastic strains is taken tobe governed by the isotropic Hooke’s law [22].

P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552 6543

Following AS [2], it is assumed that plastic flow occursby shearing accompanied by dilatation relative to some slipsystems. For an isotropic material, there are no preferreddirections other than the principal directions of stress(denoted by unit vectors n1, n2 and n3). Accordingly, sixpotential slip systems are defined relative to these principaldirections, with a conjugate pair lying in each of the threeplanes formed by (n1,n3), (n1,n2) and (n2,n3). In each ofthese planes, the slip systems are oriented such that the slipdirections, represented by unit vectors s(a) (where, a denotesthe slip system number), make an angle of � p

4þ /

2

� �(/

being the angle of internal friction), with the maximumprincipal stress direction in that plane. The resolved shearstresses and compressive normal tractions on these poten-tial slip systems are defined as

sðaÞ ¼ sðaÞ � smðaÞ and rðaÞ ¼ �mðaÞ � smðaÞ;

a 2 f1; 2; . . . ; 6g

respectively, where m(a) is the unit slip plane normal for theath slip system and s is the Kirchhoff stress. It can beshown that conjugate slip systems are equivalent to eachother in the sense that they have equal resolved shear stres-ses and compressive normal tractions and thus, as seenfrom Eq. (3) below, develop equal slip rates.

The flow rule is written for the plastic part of the spatialvelocity gradient similar to AS [2] as

lp ¼X6

a¼1

_vðaÞ½ðsðaÞ �mðaÞÞ þ bðmðaÞ �mðaÞÞ�; _vðaÞ P 0; ð2Þ

where, _vðaÞ is the plastic shearing rate and b is the dilatancyfunction. On examining the above equation, it can be seenthat this flow rule accounts for plastic shearing accompa-nied by dilatancy. The viscoplastic law for _vðaÞ is taken to be

_vðaÞ ¼ _v0

sðaÞ

cþ lrðaÞ

� �1=m

P 0; ð3Þ

where, _v0 is a reference plastic shearing rate and m > 0 isthe strain rate sensitivity parameter. The model becomesrate-independent in the limit as m! 0. Further,l = tan/ is the friction parameter and c is a stress-likeinternal variable called the cohesion which represents theyield strength in pure shear. It must be noted that b = lcorresponds to the associated flow rule. The dilatancyparameter b is assumed to be an exponential function ofthe plastic volumetric strain g as

b ¼ b̂ðgÞ ¼ g0

ðe� 1Þ e 1� ggcv

ð Þ � 1n o

; ð4Þ

so that it varies smoothly from g0 to 0 as g changes from 0to gcv.

The evolution of local free volume of amorphous materi-als governs their plastic deformation [3,24]. As assumed byAS [2], the change in free volume of an amorphous material,from its virgin state, is associated with the plastic volumetricstrain g, which is defined as g = ln(Jp), where Jp = det(Fp).Thus, the evolution of g is governed by the relation

_g ¼ trðlpÞ ¼ bX6

a¼1

_vðaÞ: ð5Þ

The cohesion c is assumed to vary smoothly with g accord-ing to the relation

c ¼ ccv þb

ðe� 1Þ e 1� ggcv

ð Þ � 1n o

: ð6Þ

From Eqs. (4)–(6), it can be seen that during the initialstages of plastic flow from a virgin state, g will increasemonotonically and later saturate to an equilibrium levelof gcv, while the cohesion c will decrease smoothly froman initial value of c0 = (b + ccv) to an equilibrium valueccv. The above model has been implemented in the generalpurpose nonlinear finite element code ABAQUS/Standard[25] by writing a user material subroutine UMAT. Theintegration of the constitutive equations is carried outusing a fully implicit backward Euler approach [23].

The objective for the most part of this work is to studythe effects of different material parameters on mode I cracktip fields under small scale yielding conditions by assumingan uniform value of the initial cohesion c0 for the entiredomain. However, in Section 6, the initial cohesion valueis statistically distributed among the elements to numeri-cally simulate strain localization behavior observed inamorphous metals.

4. Modeling and analysis aspects

4.1. Modeling details

A 2D plane strain boundary layer (SSY) formulation isused in the present study. For this purpose, a large circulardomain containing a notch along one of its radii is consid-ered (see Fig. 1a). The radius of the outer boundary (R0) ischosen to be 400 times the initial notch root diameter b0, sothat the plastic zone is well contained within the boundary.In the undeformed configuration, the center of curvature Oof the notch coincides with the center of the circulardomain. A set of cartesian coordinates (X1,X2) is estab-lished with origin at O (see Fig. 1b). Due to symmetry aris-ing because of mode I loading, only the upper-half plane ismodeled with finite elements, as shown in Fig. 1a. Appro-priate symmetry conditions are imposed on the line aheadof the notch tip and traction-free boundary conditions areassumed on the notch surface. The mesh employed in theanalysis is well refined near the notch tip, with 16 elementsplaced along the semi-circumference of the notch (seeFig. 1b), in order to accurately capture the steep strain gra-dients and notch blunting. The size of the smallest elementin the radial direction is about 1.4 · 10�4R0. The meshcomprises of a total of 3352 isoparametric quadrilateralelements and 3463 nodes.

The in-plane displacement components based on theleading term of the mode I elastic crack tip field [26] areprescribed on the outer boundary of the domain. The load-ing is applied in steps by gradually increasing the mode I

R0

X2

X1O

Fig. 1. The finite element mesh used in the simulations showing (a) the fulldomain that was modeled and (b) enlarged view of the region near thenotch tip.

0.04

0.06

0.08 μ = 0.00μ = 0.05μ = 0.15

x 2/ (

KI

/ c0)

2

6544 P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552

stress intensity factor KI. In order to alleviate mesh lockingeffects due to near-plastic incompressibility, the hybrid ele-ment formulation available in ABAQUS [25] is employedin the simulations.

4.2. Choice of material parameters

In order to understand how the constitutive behaviorinfluences the crack tip fields in amorphous plastic solids,a parametric study is conducted. A total of four sets ofmaterial combinations are considered, as listed in Table1. These sets differ mainly in material parameters l and b

which govern the frictional response as well as pressuresensitivity of yielding and softening respectively. The cho-sen values of these parameters given in Table 1 are repre-sentative of different amorphous plastic solids such asmetallic glasses and glassy polymers [4,7,8].

In all cases, the values of E/c0 and Poisson’s ratio aretaken to be 108.5 and 0.36 respectively [11]. The presentstudy is restricted to nearly rate-independent material

Table 1Material combinations considered in the numerical simulations

Material combination E/c0 m l g0 b/c0

A 108.5 0.36 0.00 0.4 0.085B 108.5 0.36 0.05 0.4 0.085C 108.5 0.36 0.15 0.4 0.085D 108.5 0.36 0.15 0.4 0.000

response by assuming a small value of 0.02 for the strainrate sensitivity exponent m. The stress variations are pre-sented in terms of Cauchy stress components normalizedby the initial value of the cohesion c0, which is a commonstress-like quantity for all cases. It must be noted thatmaterial combination A (see Table 1) represents a Tresca-type yield behavior, whereas B and C are typical of anamorphous metal response with an increasing frictionparameter l. They were chosen so that a comparisonbetween them brings out the effect of l on the crack tipfields. The material combination D involves plastic dilata-tion but exhibits no softening after the initial yielding (i.e. itis elastic–perfectly plastic).

5. Results and discussion

The results obtained from the numerical simulations arediscussed in this section.

5.1. Effect of friction parameter

The results of simulations for material combinations A,B and C (see Table 1) are compared in this subsection inorder to understand the effect of varying the frictionparameter (which, in turn, affects the pressure sensitivityof yielding) on the notch tip stress and deformation fields.Fig. 2 shows the plastic zones for three different values offriction parameter l = 0, 0.05 and 0.15, plotted in notchtip coordinates normalized by (KI/c0)2. The plastic zoneshape and size are expected to remain invariant withrespect to these normalized coordinates, once the notchhas blunted adequately. Here, the plastic zone boundaryis defined by the curve where the maximum principal loga-rithmic plastic strain ln kp

1 has a very small value of 0.001.

0 0.025 0.05 0.0750

0.02

x1/ (KI / c0)2

Fig. 2. Plastic zones plotted using normalized notch tip coordinates fordifferent values of friction parameter.

r/(J/c0)

σh/c

0

μ = 0.00μ = 0.05μ = 0.15

0 1 2 3 4 50

1

2

3

r/(J/c0)

σh/c

0

θ = 45°

θ = 0°

μ = 0.00μ = 0.05μ = 0.15

0 1 2 3 4 50

1

2

3

Fig. 4. Radial distribution of normalized hydrostatic stress at (a) h = 0�and (b) h = 45� for different values of friction parameter.

P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552 6545

It can be seen from Fig. 2 that the plastic zone sizeincreases with l. In particular, the extent of the plastic zoneahead of the notch tip doubles when l is increased from 0to 0.15. Further, the location of the maximum extent ofplastic zone rotates forward with respect to the notch lineand increases from 0.05 (KI/c0)2 to 0.07 (KI/c0)2. Xi et al.[27] have found a correlation showing direct correspon-dence between the fracture toughness and plastic processzone size for various metallic glasses. In view of this, theincreased plastic zone size with l suggests the enhancedfracture toughness of metallic glasses. It can be noticedfrom Fig. 2 that the propensity for a serrated plastic zoneboundary increases with l. This could be interpreted as atendency to form the shear bands which have beenobserved in experiments [13] on amorphous metals (seealso Section 6). It must be noted that metallic glasses haveextremely high yield strength and low to moderate fracturetoughness values [11], resulting in small plastic zone sizes inan absolute sense.

Fig. 3 shows the effect of varying the friction parameter lon the distribution of the opening stress, normalized by theinitial cohesion c0, ahead of the notch tip. Here, the radialdistance r is measured from the notch tip in the undeformedconfiguration and is normalized by (J/c0), where c0 is theyield strength in pure shear and J is the energy release rate.For small scale yielding conditions, the J integral is calcu-lated from KI using the relation J ¼ K2

I ð1� m2Þ=E. It canbe seen from Fig. 3 that an increase in l leads to a dramaticdrop in rhh ahead of the notch tip. The peak stress drops byabout 27% with an increase in l from 0 to 0.15, accompa-nied by a shift in the radial location from the notch tip atwhich it occurs, from 1.25 to 2.3 J/c0. The reduction inrhh with increasing l is significant for r/(J/c0) < 3, beyondwhich it tends to diminish. A similar trend can be seen inthe radial distribution of hydrostatic stress, rh, ahead ofthe notch tip (see Fig. 4a). The peak value of rh drops byabout 21% with an increase in l from 0 to 0.15. Fig. 4b

0 1 2 3 4 50

1

2

3

4

5

r/(J/c0)

σ θθ/c

0

μ = 0.00μ = 0.05μ = 0.15

Fig. 3. Radial distribution of normalized opening stress at h = 0� fordifferent values of friction parameter.

shows the radial distribution of normalized hydrostaticstress at h = 45�. Here, the angle h is measured from the lineahead of the notch tip and centered at the notch tip in theundeformed configuration. It can be seen that higher lresults in a significant and uniform drop (by about 18%)in rh up to fairly large radial distances (r/(J/c0) 6 5) fromthe notch tip.

The above trends are qualitatively similar to thosereported by Li and Pan [18] and Subramanya et al. [19]on the role of pressure sensitivity index from small strainanalysis of crack tip fields in materials obeying the Druc-ker–Prager yield condition. By contrast, the present analysisis based on a Mohr–Coulomb type plasticity model thatincludes the effect of the third invariant J3 of the deviatoricstress (see Section 3) and also incorporates finite geometrychanges. Flores and Dauskardt [28] have shown, from theirexperimental results, that failure in a Zr-based bulk metallicglass is associated with a critical tensile hydrostatic stress. Inview of this, the decrease in the opening stress and hydro-static stress levels near the notch tip with increasing l

6546 P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552

suggests a reduced tendency to brittle cracking or increasedresistance to fracture.

In Figs. 5a–d, the angular distributions of normalizedradial, tangential, shear and hydrostatic stresses aroundthe notch tip are displayed at r/(J/c0) = 1.5, for three dif-ferent values of l. It can be observed from Fig. 5a that ahigher l leads to a uniform drop in rrr at all anglesaround the notch tip. Also, Fig. 5b shows that there isa dramatic decrease in rhh at h = 0� which diminisheswith h. This drop in rhh is noticeable in the angularrange h 6 75�. The effect of increase in l on rrh is simi-lar. It is interesting to note from Figs. 5a and b that, asl increases, the difference (rhh � rrr) in the region aheadof the notch tip diminishes. This implies that the stressstate ahead of the notch tends towards pure hydrostatictension as l increases even though the level of rh itselfdrops (see Fig. 5d). This could have important implica-tions on the fracture behavior of the material. For exam-ple, the shape assumed by a circular void as it grows inthe vicinity of a notch tip (i.e. whether it would remain

0 60 120 1800

1

2

3

θ (deg.)

σ rr/c

0

μ = 0.00μ = 0.05μ = 0.15

0 60 120 180

0

0.5

1

1.5

θ (deg.)

σ rθ/c

0

μ = 0.00μ = 0.05μ = 0.15

Fig. 5. Angular distributions of (a) radial, (b) tangential, (c) shear and (d) hyddifferent values of friction parameter.

circular or become oblate or prolate) depends on the nat-ure of the stress state [29,30].

Fig. 6 shows the angular distribution of ln kp1 at r/(J/

c0) = 1.5. It can be seen that at angles h 6 90� the plasticstrain level is strongly enhanced with increasing l. Thus,the peak plastic strain increases by about 17.5% with achange in l from 0 to 0.15 and is accompanied by a shiftin its angular location towards the front of the notch tip.As the level of l increases, many distinct serrations in thenear-tip angular distribution of plastic strain can beobserved in the forward sector ahead of the notch tip (upto about h = 90�). This corroborates with similar featuresperceived on the leading boundary of the plastic zone inFig. 2. As already mentioned, it suggests a tendency forplastic strains to localize into discrete shear bands, whichhave been observed in experiments on both metallic glassesand amorphous polymers [13,17]. This issue will be takenup in subsequent sections.

It is interesting to compare the near-tip angular distribu-tion of plastic strain shown in Fig. 6 with that reported in

0 60 120 1800

1

2

3

4

5

θ (deg.)

σ θθ/c

0

μ = 0.00μ = 0.05μ = 0.15

0 60 120 1800

1

2

3

θ (deg.)

σ h/c0

μ = 0.00μ = 0.05μ = 0.15

rostatic stresses normalized by c0 around the notch tip at r/(J/c0) = 1.5 for

0 60 120 1800

0.02

0.04

0.06

θ (deg.)

lnλ 1p

μ = 0.00μ = 0.05μ = 0.15

Fig. 6. Comparison of angular distribution of maximum principallogarithmic plastic strain at r/(J/c0) = 1.5 for different values of frictionparameter.

4Case A : μ = 0.00; SofteningCase C : μ = 0.15; Softening

P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552 6547

Ref. [19] from 3D, small strain analysis using the Drucker–Prager model in the mid-plane of a plate. Subramanyaet al.’s results show more pronounced elevation in the peakplastic strain with pressure sensitivity index. The peak plas-tic strain in their plots occurs at a value of h between 75�and 90�, which is similar to the curves presented inFig. 6. However, since their computations are based on asmall strain formulation and pertain to a sharp crack (i.e.ignoring blunting effects), very little plastic strain developsdirectly ahead of the tip, irrespective of the pressure sensi-tivity index. By contrast, the present results show a largeraccumulation of plastic strain in the region ahead of thetip, especially with an increase in the friction parameter l(see Fig. 6). This leads to an appreciable growth in the plas-tic zone size ahead of the notch tip, as observed fromFig. 2.

In Fig. 7, the notch opening profiles at a fixed level ofnormalized energy release rate, J/(c0b0) = 7.22, pertainingto three different values of l are compared. The distancesare normalized by the initial notch root diameter b0. Allthree cases show significant blunting of the notch by a fac-

4 2 0 2 40

2

4

x1/b

0

x 2/b0

μ = 0.00μ = 0.05μ = 0.15

Fig. 7. Notch opening profiles corresponding to J/(c0b0) = 7.22 fordifferent values of friction parameter.

tor of three to four owing to strong influence of plasticdeformation. It can be observed from this figure that thenotch opening profile is enhanced with increasing l. Also,a prominent hump can be observed in the notch profile,especially for high l, which is characteristic of non-harden-ing materials having a high yield strain.

In Fig. 8, the notch opening displacement (b � b0) nor-malized by b0 is plotted against the normalized energyrelease rate J/(c0b0). Here, b is the current notch widthcalculated at the points which are located above andbelow the center of curvature of the notch in the unde-formed configuration. It can be observed from this figurethat in all cases there is an almost linear relation betweennotch opening displacement and energy release rate J.Also, as already noted, the notch opening enhances withl at a given level of J. It is well known that the cracktip opening displacement scales with J where the scalingfactor depends on the strain hardening exponent and ini-tial yield strain [31]. The present results suggest that foramorphous plastic solids, it also depends on the frictionparameter l. Thus, the ratio (b � b0)/(J/c0), calculatedfrom the curves presented in Fig. 8 for cases A and Cincreases from 0.42 to 0.52 as l changes from 0 to 0.15.The larger notch opening displacement with increase inl corroborates with the enhancement in the near-tip plas-tic strain as well as the normalized plastic zone size withincrease in friction parameter.

5.2. Effect of softening

In this section, comparison of results for material com-binations C (which shows softening behavior) and D(which exhibits elastic–perfectly plastic response) is pre-sented in order to understand the influence of softeningresponse on the notch tip stress and deformation fields. Itis found that softening leads to a decrease in all stress com-ponents around the notch tip. For example, in Fig. 9 the

0 2 4 6 80

1

2

3

J/(c0 b

0)

(b_ b

0)/b 0

Case D : μ = 0.15; Perfectly plastic

Fig. 8. Variation of notch opening displacement with respect to J/(c0b0)for different material combinations.

0 60 120 1800

1

2

3

θ (deg.)

σ h/c0

Case D (Perfectly plastic)Case C (Softening)

Fig. 9. Angular distribution of normalized hydrostatic stress around thenotch tip at r/(J/c0) = 1.5 for material combinations C and D.

0 60 120 1800

0.01

0.02

0.03

θ (deg.)

lnλ 1p

Case D (Perfectly plastic)Case C (Softening)

Fig. 10. Angular distribution of ln kp1 at r/(J/c0) = 3 for material combi-

nations C and D.

6548 P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552

angular distribution of normalized hydrostatic stressaround the notch tip at r/(J/c0) = 1.5 is displayed for thetwo cases. It can be seen from this figure that softeningleads to a perceptible reduction in rh near the notch tipat all angles h 6 75�. In particular, the hydrostatic stressahead of the notch tip (h = 0�) decreases by about 12%due to presence of softening.

Fig. 10 shows the angular distribution of ln kp1 at r/(J/

c0) = 3 for material combinations C and D. It can beobserved from this figure that, although the angular loca-tion of the peak plastic strain is unaffected by softening,its magnitude is increased. In particular, the plastic strainlevels around the notch tip for 30� 6 h 6 90� are consider-ably enhanced (by up to about 11%). Another striking fea-ture of this figure is the appearance of sharp serrations inthe angular distribution of plastic strain when the materialexhibits softening. It must be noted that a similar trend isseen in Fig. 6 with increase in l. This suggests that the pro-pensity for localization of plastic strain into discrete shearbands increases with l and is also enhanced by the presenceof softening in the material response.

The effect of softening on the evolution of notch openingdisplacement with the normalized energy release rate canbe seen from Fig. 8 by comparing the curves pertainingto cases C and D. It can be observed from this figure thatthe notch opening displacement varies linearly with J andthat softening results in about 7% higher notch bluntingat any J. This result is consistent with the fact that soften-ing promotes the plastic strain levels around the notch tip,as seen in Fig. 10.

5.3. Slip line fields around the notch

Strain softening exhibited by amorphous materials suchas metallic glasses and glassy polymers, is invariablyaccompanied by localization of plastic strain into shearbands [13,17]. Crack propagation upon accumulation of

a critical amount of plastic strain after continued shearingalong these discrete bands is believed to be one of the fail-ure modes in metallic glasses [32]. Hence, in order to gain abetter understanding of such a ductile failure mechanism,the plastic slip line fields around the notch are studied inthis subsection.

Fig. 11a shows the slip line fields prevailing around theblunted notch for material combination C correspondingto normalized energy release rate of J/(c0b0) = 0.8. In thisfigure, the slip lines generated using the stress fieldsobtained from the finite element solution are plotted inthe deformed configuration and the distances are normal-ized by the initial notch root diameter b0. Here, and in sim-ilar plots shown subsequently, the trajectories in the lowerhalf are obtained from those in the upper-half by reflectionabout x2 = 0 line for complete visualization. Under 2Dplane strain conditions, there are two slip systems in thex1–x2 plane which are symmetrically disposed at anglesof � p

4þ /

2

� �about the maximum in-plane principal stress

direction, where / = tan�1l (see Section 3). The two fam-ilies of slip lines, indicated by solid and dashed-dotted linesin Fig. 11a, represent the trajectories of these directions inthe x1–x2 plane. In this figure, the slip line network isenclosed within a curve (indicated by the dotted line) whichrepresents the elastic–plastic boundary. For clarity, anenlarged view of the slip line network close to the notchand entirely inside the zone of large plastic deformationis shown in Fig. 11b.

The included angle between the two families of sliplines emanating from the notch surface in Figs. 11a andb is 81.5�, unlike isotropic pressure insensitive plastic sol-ids, where the slip lines under plane strain conditionscoincide with the maximum shear directions and are thusorthogonal. The slip line fields for a rigid-perfectly plastic,von Mises material [26] and a pressure-sensitive dilatantmaterial obeying the Drucker–Prager yield condition [18]predict intense straining above and below a sharp crack

-6 -4 -2 0 2 4 6 -6

-4

-2

0

2

4

6

x1/b

0

x 2/b0

-1 0 1 2 3 4 5-3

-2

-1

0

1

2

3

x1/b

0

x 2/b0

Fig. 11. Slip line fields around the blunted notch for material combinationC corresponding to J/(c0b0) = 0.8. (a) Full field view and (b) enlarged viewnear the notch tip.

-3 -2 -1 0 1 2 3 4 5 -4

-3

-2

-1

0

1

2

3

4

x1/b

0

x 2/b0

Fig. 12. Trajectories of normal to the maximum principal stress directionin front of the notch, for material combination C corresponding to J/(c0b0) = 0.2.

P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552 6549

tip and not directly in front of it. In the latter, the slip linegrid is formed by two non-orthogonal families of charac-teristic lines and the angular extents of the constant stresssectors in front of the tip and adjacent to the crack facevary with the pressure sensitivity index of the material.By contrast, the slip line field in Fig. 11a suggests largeplastic strains immediately ahead of the blunting notchtip due to large geometry changes. It resembles the theo-retical slip line field around a notch constructed by Jeonget al. [17] for a pressure sensitive and dilatant material(see Fig. 6 of their paper). Also, the dashed-dotted familyof slip lines in the upper half and the solid family of slip

lines in the lower half of Fig. 11a look similar to the shearband patterns observed around a crack in experimentsconducted by Flores and Dauskardt [13] using SEN(T),Vitreloy 1 metallic glass fracture specimen (see Fig. 3 oftheir paper).

5.4. Brittle crack trajectories in front of the notch

Under macroscopic tension, amorphous materials likebulk metallic glasses and glassy polymers such as PMMAare quasi-brittle because they exhibit very little plasticdeformation. Hence, it is important to examine the prob-able crack paths if failure in these materials occurs due tobrittle fracture. Since fracture in a brittle solid is con-trolled by the maximum normal tensile stress acting onthe fracture plane, it is reasonable to assume that crackgrowth should be orthogonal to the direction of most ten-sile principal stress [33]. Such trajectories have been suc-cessfully employed to trace the paths of contact-inducedtransverse cracks in brittle coatings on soft substrates byLee et al. [34]. The above trajectories corresponding tomaterial combination C are plotted in Fig. 12 at a valueof J/(c0b0) = 0.2 when the notch blunting is not substan-tial (b/b0 = 1.13).

From Fig. 12, it can be seen that the crack trajectoriesbecome asymptotic to the x2 = 0 line, which is a principalstress direction owing to mode I symmetry. Also, planestangent to the notch surface, which is traction-free, repre-sent principal planes containing the maximum principalstress direction. Thus, the crack trajectories originateorthogonally from the notch surface. Another interestingfeature that can be observed is that the trajectories, after

6550 P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552

starting orthogonal to the notch surface, rise and reach apeak height at a distance of about 1.5 times the currentnotch width in front of the notch tip and later becomeasymptotic to the x2 = 0 line. These trajectories resemblethe crack traces observed in front of a notch tip in experi-ments conducted on a Zr-based metallic glass by Lowhap-handu and Lewandowski [35] (see Fig. 3 of their paper)using three-point bend specimens and by Fujita et al. [36]using compact tension specimens. From these experimentalpictures and the nature of crack trajectories in Fig. 12, oneis led to conclude the following about the brittle fracturemechanism in metallic glasses. Several microcracks simul-taneously originate orthogonally from the notch surface.With increasing load, these microcracks propagate alongtrajectories similar to those shown in Fig. 12. At somestage, they merge together at a finite distance in front ofthe notch tip to form a single macroscopic mode I crackwhich subsequently causes brittle catastrophic failure ofthe specimen.

6. Simulation of shear band pattern around the notch

All the results discussed in the previous section wereobtained by assuming the same initial value of the cohesionfor the entire domain. However, at temperatures well belowtheir glass transition temperature, amorphous solids suchas metallic glasses exhibit inhomogeneous deformationwhen subjected to high stresses [37]. The inhomogeneousdeformation in metallic glasses is characterized by anincrease in local free volume at some discrete locations fol-lowed by localization of plastic strain in those regionsresulting in the formation of narrow shear bands. In orderto numerically simulate this inhomogeneous deformationbehavior at the continuum scale, the initial value of thecohesion is statistically varied among all elements in themesh following AS [2]. This leads to an inhomogeneouslydistributed plastic volumetric strain (which, in the presentconstitutive model, is considered to be equivalent to thechange in free volume from the virgin state) and providesa sufficient number of nucleation sites for the shear bandsto form (see AS [2]).

In this section, the results of a simulation in which theinitial value of the cohesion is varied among all elementsusing a normal distribution with standard deviation of3% of the mean value c0 are presented. The other propertiesused in this simulation correspond to material combinationC (see Table 1). In Fig. 13, contour plots of maximum prin-cipal logarithmic plastic strain, ln kp

1, near the notch are dis-played in the deformed configuration corresponding todifferent stages of loading.

Figs. 13a–c show the development of shear bands nearthe notch at normalized energy release rate J/(c0b0) of0.2, 0.45 and 0.8, respectively. In Fig. 13d, an enlargedview of the region around the notch of Fig. 13c is shownfor clarity. In these figures, the lowest contour ofln kp

1 ¼ 0:001 gives a representation of the elastic–plasticboundary. Thus, the plastic zone which is confined to

the region just in front of the notch at J/(c0b0) = 0.2(see Fig. 13a) grows with increasing load and assumesa close to self-similar shape at J/(c0b0) = 0.8 (seeFig. 13c), which roughly corresponds to that depictedin Fig. 2 for l = 0.15. It should be noted from Figs.13a–c that plastic deformation progresses by way of for-mation of new shear bands and propagation of existingones. The shear bands resemble discrete fingers projectingout from the notch, leading to the elastic–plastic bound-ary having a serrated nature (see Figs. 13b and c). Thisfeature of the elastic–plastic boundary can also beobserved to a lesser extent in Fig. 2. Such plastic zonescaused by inhomogeneous flow along discrete bands havealso been reported by Lai and Van der Giessen [21] (seeFig. 8 of their paper) employing an entirely different con-stitutive model for amorphous polymers which exhibitsstrain softening.

At a low value of J/(c0b0) = 0.2, a dense network of mul-tiple shear bands of both families can be seen in front of thenotch in Fig. 13a. The two families of shear bands close tothe notch tip can also be clearly seen in Fig. 13d at a higherload level with an included angle of less that 90� betweenthem. This pattern of shear bands is similar to the slip linefields in front of the notch predicted from the simulationwith uniform initial cohesion (Fig. 11b). It also resemblesthat observed in experiments on amorphous polymersreported by Jeong et al. [17] (see Figs. 2 and 3 of theirpaper). It can be noticed that the level of plastic strain isgreatest in shear bands which are very near to the notchsurface and diminishes rapidly with increasing distancefrom it. The intense plastic straining in a few bands maycause ductile shear failure along these bands, leading tothe formation of macroscopic cracks [32]. Finally, it shouldbe noted that the shear bands away from the notch root inthe outer reaches of the plastic zone (see Fig. 13c) are sim-ilar to those observed experimentally for Vitreloy 1 metallicglass by Flores and Dauskardt [13] (see Fig. 3 of theirpaper).

An interesting feature that can be observed fromFig. 13d is the blunting of the notch surface into a shapewith several vertices when the initial cohesion is statisticallydistributed. These vertices form at locations on the notchsurface where the two families of shear bands intersect eachother and may be interpreted as shear offsets or surfacesteps. Such waviness on the notch surface has also beenreported by Lai and Van der Giessen [21] in their simula-tions for amorphous polymers. Finally, it should be men-tioned that, although there are qualitative similaritiesbetween the predictions of the present simulations andthe experimental observations reported by Jeong et al.[17], as well as the above computational study [21], the roleof progressive hardening at large plastic strains arising dueto orientation of molecular chains in some polymers [38]has not been assessed in this work. This feature is consid-ered in Refs. [15,21], wherein a different constitutive modelbased on statistical distribution of molecular chains isemployed.

Fig. 13. Contours of maximum principal logarithmic plastic strain showing the development of shear bands for material combination C when the initialcohesion c0 is statistically distributed over the mesh, corresponding to (a) J/(c0b0) = 0.2, (b) J/(c0b0) = 0.45, (c) J/(c0b0) = 0.8. (d) Enlarged view of (c) nearthe notch tip.

P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552 6551

7. Conclusions

In this work, a finite strain Mohr–Coulomb based con-stitutive model, with provision for discrete slip systems(defined locally with respect to the principal stress direc-tions), was employed to study the notch tip stress anddeformation fields in amorphous solids such as metallicglasses under mode I, plane strain, small scale yielding con-ditions. The following are some of the important conclu-sions that can be drawn from the simulations performedin this work.

An increase in the value of friction parameter l increasesthe plastic zone size ahead of the notch tip and causes it to

rotate forward. Also, the plastic zone boundary shows ser-rated features and the notch tip opening displacement isenhanced. A higher l also leads to a severe decrease inopening stress ahead of the notch tip and a reduction inhydrostatic stress at all angles around the notch tip. Onthe other hand, it also results in a strong increase in theplastic strain accumulation ahead of the notch tip. Theseobservations, in combination with the one-to-one corre-spondence between plastic zone size and toughness estab-lished earlier in metallic glasses [27], suggests thatenhanced internal friction could lead to improved fracturetoughness of the amorphous material. Although softeningresults in the reduction in the levels of all stress components,

6552 P. Tandaiya et al. / Acta Materialia 55 (2007) 6541–6552

it enhances the plastic strains around the notch, which isalso reflected in the enlarged notch opening profile. Thepropensity for localization of plastic strain into discreteshear bands increases with l and is further enhanced inthe presence of softening.

The synthetic slip line field constructed around theblunted notch suggests large plastic strains immediatelyahead of the notch tip and resembles that reported fromexperiments on an amorphous polymer [17]. The predictedbrittle crack trajectories rise orthogonal to the notch sur-face and after reaching a peak height become asymptoticto the line directly ahead of the notch tip, which suggestscoalescence of microcracks ahead of the notch tip as seenin experiments on metallic glasses [35,36]. The resultingshear band patterns generated from simulations with statis-tical distribution of initial cohesion are similar to the syn-thetic slip line field predicted from the stress distributionaround the notch as well as to those observed in experi-ments [13,17].

In summary, the present numerical study on crack tipfields in amorphous materials such as metallic glasses andnon-crystalline polymers suggests a competition betweencrack tip plasticity through shear banding and brittle frac-ture by microcracking. Higher friction parameter sup-presses the latter by decreasing the opening stress levelsnear the notch tip. On the other hand, it promotes cracktip plasticity through shear banding. The discrete plasticzone in the form of shear bands, which is aided by soften-ing, may increase the resistance to fracture by dissipationof energy in the shear bands. This is especially true if multi-ple shear bands form, each accommodating moderate plas-tic strains. Eventually, when extremely large plastic strainsare accumulated into a few shear bands, ductile fracturemay occur within them [32]. Also, the large plastic strainsassociated with higher l could give rise to alternate ductilefracture mechanisms such as growth and coalescence ofmicrovoids (e.g. around rubber particles in rubber modifiedepoxies) [29].

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