University of Groningen

Effects of microstructure on crack tip fields and fracture toughness in PC/ABS polymer blendsSeelig, Thomas; Van der Giessen, Erik

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DOI:10.1007/s10704-007-9117-y

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Int J Fract (2007) 145:205–222DOI 10.1007/s10704-007-9117-y

ORIGINAL PAPER

Effects of microstructure on crack tip fields and fracturetoughness in PC/ABS polymer blends

Thomas Seelig · Erik Van der Giessen

Received: 22 March 2007 / Accepted: 23 August 2007 / Published online: 14 September 2007© Springer Science+Business Media B.V. 2007

Abstract Numerical simulations are performed inorder to gain a better understanding of the effects ofvarious microstructural features and toughening mecha-nisms in amorphous PC/ABS polymer blends. Cracktip loading under global small-scale yielding condi-tions is considered with the blend microstructure expli-citly resolved in the near-tip process zone. Constitutivemodels are employed which account for large visco-plastic deformations, the characteristic softening-rehardening behavior of glassy polymers, as well asthe effect of plastic dilatancy in the ABS phase due torubber particle cavitation. The influence of blend com-position and morphology on the local stress distributionand the development of the plastic zone at a stationarycrack tip are analyzed. Furthermore, crack propaga-tion and the evolution of fracture toughness are studiedusing different cohesive surface models for failure inthe different phases of the blend microstructure.

Keywords Polymer blends · Microstructure · Cracktip fields · Toughening mechanisms · Crack resistancecurves

T. Seelig (B)Fraunhofer-Institute for Mechanics of Materials,Woehlerstrasse 11, Freiburg, 79108, Germanye-mail: see@iwm.fhg.de

E. Van der GiessenMaterials Science Center, University of Groningen,Groningen, 9747 AG, The Netherlands

1 Introduction

Polymeric materials used in technical applications arefrequently composed of different constituents in orderto improve their mechanical performance and otherphysical properties. An important class of materialswhich has received increasing attention in recent yearsare blends of polycarbonate (PC), an amorphous glassythermoplastic, and acrylonitrile-butadiene-styrene(ABS). Since ABS itself is a two-phase material withsmall (butadiene) rubber particles dispersed in a matrixof styrene-acrylonitrile (SAN, also an amorphous glassythermoplastic) PC/ABS is referred to as a ternary blend.Among several other reasons (e.g. better processabi-lity), one purpose of blending PC with ABS is to reducethe severe notch-sensitivity of neat PC and to increasethe fracture toughness. Roughly speaking, this worksbecause ABS is already a so-called rubber-toughenedpolymer where the dispersed rubber particles serve toinitiate energy dissipating microscopic deformationmechanisms at many sites throughout the material. Asan example of the toughening that can be achieved thisway, Kinloch and Young (1983) report an impact frac-ture energy of sharply notched PC specimens which isonly about 8% of the value for bluntly notched spe-cimens while for ABS the respective value for shar-ply notched specimens is still about 60% of that for ablunt notch. In ABS as well as other “classical” rubber-toughened materials, such as high-impact polystyrene(HIPS), the beneficial role of the modifier phase in theintrinsically brittle glassy matrix is fairly well

123

206 T. Seelig, E. Van der Giessen

understood (Bucknall 1977). PC, in contrast, displaysan ambivalent behavior ranging from high ductilityunder unnotched conditions to brittle failure in the pre-sence of a sharp notch. Hence, it is far less clear underwhich circumstances blends of PC and ABS show animproved mechanical performance. Another complica-ting factor is the enriched microstructure of PC/ABSwith the ABS content in the blend, the morphology, andthe rubber content in the ABS being the most importantparameters from a mechanical point of view.

Despite the complex interrelation between these para-meters some qualitative understanding of their effecton the macroscopic deformation and failure behaviorof PC/ABS blends has emerged from a large numberof experimental studies; see e.g. Greco (1996) for areview. A general trend reported in the experimentalliterature is that under tensile loading of unnotchedspecimens the ductility (strain at failure) of PC/ABSdecreases with increasing ABS content because theability of neat PC to undergo large strains and duc-tile neck propagation (“cold drawing”) is hampered bythe presence of the ABS (e.g., Greco et al. 1994; Bala-krishnan and Neelakantan 1998). The dependence ofthe elastic stiffness and the yield stress of PC/ABS onthe ABS content is controlled by the amount of (soft)rubber in the ABS since the SAN matrix of the latteris stiffer and has a higher yield stress than PC (e.g.,Kurauchi and Ohta 1984; Greco et al. 1994; Ishikawa1995). These dependencies can qualitatively be explai-ned from simple rules of mixtures or other microme-chanical considerations (Seelig 2004).

Far more complex, however, is the fracture behaviorof PC/ABS blends as observed e.g., in Charpy impact orSENT tensile tests, and expressed in terms of fractureenergies or crack resistance curves. For instance, Grecoet al. (1994) found a pronounced synergistic effect, i.e.,a fracture toughness of PC/ABS significantly higherthan that of each constituent, for an intermediate rangeof 10–40% ABS in the blend, while this tougheningeffect in turn strongly depends on the ABS type. Thelatter was mainly determined by the rubber content, andthe maximum toughness was observed at around 15%rubber in the ABS. Similar highly non-monotonousvariations of the fracture properties with compositionare reported by other researchers (e.g., Lee et al. 1992;Balakrishnan and Neelakantan 1998; Inberg 2001). Inaccordance with the above mentioned behavior of PC/ABS under unnotched conditions, an enhanced fracturetoughness compared to that of neat PC is only observed

for sufficiently sharp notches. In this situation neat PCtypically fails by the formation and unlimited propa-gation of a single craze initiated by the concentrationof hydrostatic stress ahead of the notch (e.g. Narisawaand Yee 1993). In contrast, the presence of ABS whichis able to undergo volumetric expansion upon cavita-tion of the rubber particles causes a relief of hydrostaticstress (thereby suppressing crazing) and enables shearyielding in the PC. The qualitative picture of this tou-ghening mechanism in PC/ABS, which is confirmed bymicroplastic deformations visible on the fracture sur-face, is generally agreed upon (e.g., Ishikawa and Chiba1990; Lee et al. 1992; Seidler and Grellmann 1993;Greco et al. 1994; Ishikawa 1995; Inberg 2001). Howe-ver, the efficiency of toughening strongly depends onvarious parameters, the rubber content in the ABS pro-bably being the most important one. Another influencearises from the blend morphology which changes fromone with ABS particles (of a few microns diameter)embedded in the PC matrix at low ABS content toa co-continuous (often lamellar) one when the ABScontent is increased above 40% (Greco 1996; Inberg2001). Obviously, the patterns of local plastic deforma-tion (shear yielding) enabled in the PC—and perhaps sothe efficiency of toughening—depend on this morpho-logy. While the individual roles of these microstruc-tural features in the toughening process can hardly beanalyzed from experiments, it is the aim of the presentwork to gain some additional insight and basic unders-tanding from micromechanical models and numericalsimulations.

Mechanical modeling of rubber-toughened polymersis still in a rather early stage and so far has mostlyfocused on two-phase materials, i.e., a glassy matrixcontaining soft rubber particles, with the latter typi-cally being treated as voids (Smit et al. 1999; Steen-brink and Van der Giessen 1999; Socrate and Boyce2000; Pijnenburg and Van der Giessen 2001; Daniels-son et al. 2002; Meijer and Govaert 2003). These stu-dies have utilized cell models of the voided polymersubjected to uniform overall deformation in conjunc-tion with a constitutive model for the deformation beha-vior of the glassy matrix, as the one developed by Boyceet al. (1988), and analyzed the effect of voids on matrixyielding and the reduction of hydrostatic stress as indi-cators of an enhanced toughness. Similarly, Seelig andVan der Giessen (2002) used cell models to investi-gate localized plastic deformations and stress distribu-tions in ternary blends with ABS particles embedded

123

Effects of microstructure on crack tip fields 207

in a PC matrix. Cavitated rubber particles in ABS the-reby were accounted for by describing ABS in a homo-genized manner as a porous glassy polymer. Fracturemechanical models for a stationary crack tip have beenapplied to ABS materials by Pijnenburg et al. (2005),using a variety of ways to represent the microstructureunder crack-tip loading conditions. In order to investi-gate the interaction of crack tip plasticity, crazing andsubsequent crack propagation in homogeneous glassypolymers, Estevez et al. (2000) and Estevez and Van derGiessen (2005) employed a cohesive surface model ofcrazing as developed in (Tijssens et al. 2000).

As a natural continuation of the above approaches,the present work deals with crack tip fields, crack pro-pagation and toughening mechanisms in PC/ABSblends. In Sect. 2 we start with an outline of how theblend microstructure is modeled in order to accountfor essential features of the different constituents. Also,the computational model for the blend material undercrack tip loading conditions is described. Constitutivemodels for the finite strain deformation behavior ofPC and ABS as well as for their failure—treated inthe framework of a cohesive surface methodology—are presented in Sect. 3. These models involve a num-ber of simplifying assumptions and hence the nume-rical results discussed in Sect. 4 are of a qualitativenature, but allow to investigate the separate effects ofblend morphology and composition on the distributionof stress and plastic flow in the vicinity (‘process zone’)of a stationary crack tip. Afterwards, crack propagationis simulated and the influence of the ABS type (in termsof its rubber content) on the fracture toughness of theblend and its evolution in the course of crack advanceis studied. As the present work is a first step in thenumerical investigation of toughening mechanisms inPC/ABS blends, a critical discussion of the employedmodeling concepts as well as suggestions for modelimprovements are given in Sect. 5.

2 Problem formulation

2.1 Blend modeling

The microstructure of a PC/ABS blend as shown by themicrograph in Fig. 1a consists of homogeneous regionsof PC and regions of ABS which itself has a heteroge-neous microstructure with rubber particles embedded

in the SAN matrix. Due to loading of the material, the(dark) rubber particles visible in the micrograph havecavitated and grown to voids (bright). The morphologyof the PC and ABS regions depends on their volumefractions and typical grades may contain up to 50%ABS. At a low ABS content (e.g., 30% and less) theABS prevails as approximately spherical particles ofa few microns diameter in the PC matrix whereas for50/50 blends (micrograph in Fig. 1a) a co-continuousmorphology (often lamellar due to injection moulding)is found with a thickness of the PC and ABS regionsalso of a few microns (Greco 1996; Inberg 2001).

In order to simplify the modeling of the microstruc-ture of PC/ABS blends, the key assumption is madehere that ABS can be described as a homogenized effec-tive medium, so that the PC does not ‘see’ the indivi-dual rubber particles embedded in the SAN matrix ofABS (Fig. 1b). Moreover, since rubber particles cavi-tate at a relatively low stress level and have a stiffnessmuch smaller than that of the surrounding glassy poly-mer, they are represented as voids in the present model(see also Sect. 3.2). Hence, PC/ABS ternary blends aredescribed as a two-phase material consisting of a neat(PC) and a porous (ABS) glassy polymer. The sameapproach was adopted in (Seelig and Van der Giessen2002) where details on the homogenization proceduremay be found; a description of the resulting porousplasticity model for ABS is given in Sect. 3.2. Besidesthe enormous computational effort that would be nee-ded to model the entire microstructure, the compromiseof resolving only the PC and ABS regions by homo-genizing the ABS is also motivated by the fact that inthe present context we are interested in deformationmechanisms on the scale of the PC and ABS regions,e.g., shear banding in the PC enabled by the plasticdilatancy of the ABS under the highly triaxial overallloading as it prevails ahead of a crack tip.

2.2 Crack tip problem

As discussed in the Introduction an improved fracturetoughness of PC/ABS blends compared to that of neatPC is observed only in case of a sufficiently sharp crackwith a crack tip radius rtip much smaller than all otherspecimen dimensions. This is the situation we focuson in this work and it allows to assume ‘small-scaleyielding’ conditions to hold, i.e., all inelastic processesare confined to a small region around the crack tip (the

123

208 T. Seelig, E. Van der Giessen

Fig. 1 (a) Microstructureof real PC/ABS (Inberg2001) and (b) modeling as atwo-phase material of PCand homogenized ABS

^

^

glassy polymerABS = porous

homogenization

particles (= voids)cavitated rubber

SAN matrix

ABS microstructure(a) (b)

PC

ABS

so-called ‘process zone’) outside of which the materialbehaves linear elastically (Fig. 2). The near crack tipfields then are uniquely controlled by the stress inten-sity factor K and numerical modeling can be restictedto the K -field dominated region. The computationalmodels in the present work are two-dimensional andplane strain conditions are considered. Mode I loa-ding is imposed in terms of the respective displace-ments u prescribed as far-field boundary conditions,see e.g., (Lai and Van der Giessen 1997). Inside the pro-cess zone the two-phase blend microstructure is expli-citly resolved and different morphologies—particulateor lamellar—are considered as sketched in Fig. 2. Thistype of modeling is, in terms of length scales, the oppo-site extreme case to that analyzed in (Seelig and Vander Giessen 2002) where PC/ABS blends subjected touniform overall loading were investigated. Finite ele-ment discretizations of the numerically analyzedK -field dominated region and the encompassed pro-cess zone (for the case of a particulate morphology)are shown in Fig. 3. Due to the mode I symmetry only

half of the problem needs to be modeled. Accordingto Fig. 3b the crack tip radius in the model is abouttwo times the ABS particle diameter; via this interrela-tion and a typical ABS particle diameter of about 5µm(Greco 1996) an absolute value of about rtip = 10µmcan be assigned to the crack tip radius in the computa-tional model.

The finite strain inelastic constitutive models for theindividual phases (PC and ABS) in the process zoneare presented in Sects. 3.1 and 3.2. Outside the processzone where no plasticity takes place effective elasticconstants obtained from homogenizing the PC/ABScould be used. However, micromechanical analyses bySeelig (2004) have shown that for the range of com-position considered here the variation of the effectivestiffness of PC/ABS around that of neat PC is not morethan about ±10%. In view of other, more severe sim-plifications this variation as well as a possible elasticanisotropy in case of lamellar morphologies are neglec-ted in the present study and material parameters for neatPC are used for the material outside the process zone.

Fig. 2 Crack tip modelingin PC/ABS blends undersmall-scale yieldingconditions

u ~ KIr

1 / 2

tip 2

small scale yielding

PC

ABS

r

process zone

123

Effects of microstructure on crack tip fields 209

Fig. 3 Finite elementmeshes of entire crack tipregion (a) and processzone (b)

(a) (b)

2.3 Cohesive zone modeling of failure

The fracture model sketched in Fig. 2 in conjunctionwith the constitutive models to be described in Sects. 3.1and 3.2 will be evaluated in Sect. 4.1 with regard toeffects of the blend microstructure on the situation (e.g.,local fields) at a stationary crack tip. In order to analyzeeffects on the fracture toughness, however, one needsto account for a fracture process. For the simulation ofmode I crack propagation in Sect. 4.2, the mechanicalmodel is supplemented by a cohesive surface along thesymmetry axis inside the process zone as sketched inFig. 4. The cohesive surface is endowed with differentproperties representing failure in the alternating PC andABS regions along the crack path, as will be discussedin Sects. 3.3 and 3.4. Attention will be restricted to thecase of equal volume fractions (50/50) of PC and ABSwhere the real microstructure often displays a lamellarmorphology (see Fig. 1) as sketched in Fig. 4.

3 Constitutive modeling

3.1 Homogeneous glassy polymers

Constitutive models for the large strain visco-plasticdeformation behavior of amorphous glassy polymersare quite well established in the literature; see e.g.,

(Smit et al. 1999), (Gearing and Anand 2004). In thepresent work we employ the model originally deve-loped by Boyce et al. (1988) in the slightly modifiedversion given in (Wu and Van der Giessen 1996). Hereit is adopted to represent the behavior of the PC matrixon the blend level as well as that of the SAN matrix inthe ABS model (Sect. 3.2), though with different setsof material parameters as listed in Table 1.

The theory makes use of the standard additive decom-position of the rate of deformation tensor into its elasticand plastic parts: D = De + Dp. Visco-elastic effectsprior to yield are of minor importance in the presentstudy and are neglected. The small strain elastic res-ponse is governed by Hooke’s law written in rate formas

De = L−1 ∇σ (1)

where∇σ is the Jaumann rate of the Cauchy stress and L

is the standard fourth-order isotropic elasticity tensor.The isochoric visco-plastic strain rate

Dp = γ p

√2τ

σ ′ (2)

is specified in terms of the equivalent plastic shear strainrate γ p = √

Dp · Dp and the deviatoric driving stressσ ′ normalized by the equivalent driving shear stress

τ =√

12 σ ′ · σ ′. The latter serves to determine γ p via

the visco-plastic constitutive equation

Fig. 4 Modeling mode Icrack propagation inco-continuous PC/ABS(50/50) blend

u ~ K r 1/2I

tip

process zonesmall scale yielding

cohesive 2r

PC

d

ABS

surface

123

210 T. Seelig, E. Van der Giessen

Table 1 Material parameters used for PC and SAN at room temperature in the present work

E/s0 ν ss/s0 As0/θ h/s0 α λmax C R/s0 s0 (MPa) γ0 (sec−1)

PC 9.4 0.3 0.79 79.2 5.15 0.08 2.5 0.059 97 2 ·1015

SAN 12.5 0.38 0.79 52.2 12.6 0.25 3.5 0.033 120 1.06 ·108

γ p = γ0 exp

[− As

θ

(1 −

(τ

s

)5/6)]

(3)

where γ0 and A are material parameters, and θ is theabsolute temperature which is constant in the presentanalysis. The shear resistance s in (3) is taken to evolvewith plastic strain according to

s(γ p) = ss + (s0 − ss) exp (−hγ p/ss) + αp (4)

from the initial, athermal yield strength s0 to a satura-tion value ss in order to phenomenologically describethe intrinsic softening of the glassy polymer (Boyceet al. 1988). Furthermore, (4) incorporates the depen-dence of yield on the pressure p = − 1

3 trσ via theconstant pre-factor α. This pressure dependence is dueto a changing molecular mobility and not associatedwith plastic dilatancy of the bulk material, tr Dp = 0.From (2) and the definition of τ it follows that the plas-tic dissipation rate per unit volume of the material isgiven by σ ′ · Dp = √

2τ γ p.The progressive hardening of a glassy polymer after

yield due to stretching and alignment of the molecu-lar network is described by the back stress tensor bincorporated in the driving stress tensor σ ′ = σ ′ −b. Drawing on the analogy with cross-linked rubber(Arruda and Boyce 1993) the principal components ofthe back stress tensor are specified in terms of principalstretches. The back stress model involves two additio-nal material parameters: the initial hardening modulusCR and the limit stretch of the molecular chains λmax

at which the network responds with an infinite stiff-ness and no further yielding is possible. Full details ofthe constitutive model may be found in (Wu and Vander Giessen 1996) along with a convenient numericalintegration scheme.

The response of the constitutive model for glassypolymers under plane strain tension and at constantstrain rate is illustrated in Fig. 5. It is based on the setsof material data for PC and SAN listed in Table 1 whichare adopted from (Boyce et al. 1988) and (Steenbrinkand Van der Giessen 1999). The model well capturescharacteristic features of the behavior of glassy poly-mers such as the intrinsic softening upon yield and the

PCSAN

40

20

00 0 .2 0. 4 0.6 ε

σ

0.8 1 .0

60

80

ε = 0.01sec

[MPa]

120

100

.−1

Fig. 5 Plane strain tension response of PC and SAN in terms oftrue stress vs. logarithmic strain at constant strain rate and roomtemperature, computed from data in Table 1

progressive rehardening. It should be mentioned thatthe values for Young’s modulus in Table 1 are smallerthat those typically given in the literature because here,by neglecting the nonlinear deformation regime prior toyield, they represent the secant moduli correspondingto the yield stress and yield strain of the materials.

3.2 Homogenized ABS model

Experimental (e.g. Ramaswamy and Lesser 2002) aswell as numerical (Pijnenburg et al. 2005) studies haveshown that caviation of the rubber particles in ABSclose to a crack tip takes place in an early stage of loa-ding and at stress states well below the yield (or crazing)stress. Once cavitated, the stiffness of these particlescan be neglected compared to that of the surroundingmatrix and therefore the rubber particles in the presentwork are treated as voids from the beginning on. Theoverall behavior of ABS can then be approximated bythat of porous SAN. The SAN itself, being the thermo-plastic matrix phase in ABS, is described by the consti-tutive model given in the previous section. The isotropic

123

Effects of microstructure on crack tip fields 211

0% porosity (neat SAN)60

40

20

00 0.1 0.4

30% porosity

porous SAN (periodic)porous SAN (random) ABS model

10% porosity

0.30.2

τ [MPa]

γ

(a)

0 0.05 0.1

60

40

20

0

porous SAN (random) porous SAN (periodic)

ABS model

0.15

30% porosity

10% porosity

[MPa]m

ε

σ

v

(b)

Fig. 6 Overall response of RVE and homogenized ABS model to simple shear (a) and equi-biaxial strain (b)

elasticity tensor for the porous material is now givenin terms of effective elastic constants E∗( f ), ν∗( f )

depending on the porosity f . Respective expressionscan be found, e.g., in (Pijnenburg and Van der Giessen2001). With the initial value f0 representing the rubbercontent in ABS, the porosity may evolve in the course ofdeformation according to f = (1 − f )Dp

kk due to voidgrowth in the plastically incompressible SAN matrix.Hence, macroscopic yielding of the porous materialwill involve plastic dilatancy under hydrostatic stress.For this, we adopt the phenomenological macroscopicyield function

� ≡ 1

2σ ′ · σ ′ + a f b

0 σ 2m − [(1 − f )τc]2 (5)

which exhibits a quadratic dependence on the devia-toric and hydrostatic stress. The parameters a and bexpress the influence of hydrostatic (mean) stress σm

and are fitted to values of a ≈ 1 and b ≈ 0.7 from calcu-lations with a representative volume element of voidedSAN. The parameter c is a function of f0 determinedfrom micromechanical considerations (Seelig and Vander Giessen 2002) as c ≈ (1 + √

f0)−1. The equiva-

lent driving stress τ in the SAN matrix phase due tothe stress σ acting on the porous material is determi-ned from the condition � = 0. The plastic strain rateis determined via the (normality) flow rule

Dp = ∂�

∂ σ(6)

where the multiplier is computed from the conditionthat the plastic work rate per unit deformed volume ofthe porous material equals that in the matrix:

σ · Dp = (1 − f )√

2τ γ p . (7)

With the equivalent driving stress τ solved from (5), theequivalent plastic strain rate γ p obtained from (3) nowrepresents the ‘effective’ visco-plastic behavior of theentire matrix phase. Due to the highly heterogeneousplastic flow in a porous glassy polymer the intrinsicsoftening of the matrix is evened out in the overall res-ponse as supported by large scale simulations by Smitet al. (1999). To incorporate this effect in the homo-genized model, intrinsic softening is taken to decreasewith increasing porosity. Full details on this modelingof ABS as a porous glassy polymer may be found in(Seelig and Van der Giessen, 2002).

Figure 6 shows the response of the homogenizedABS model in comparison to the overall response ofunit cell computations of a representative volumeelement with a SAN matrix and different void arran-gements, under macroscopic simple shear and equi-biaxial strain. For values of the porosity in the relevantrange, the homogenized ABS model captures the elasticstiffness, the yield point, as well as the post-yield beha-vior of a porous glassy polymer to an acceptable degreeof accuracy. Also shown in Fig. 6a is the response ofneat SAN, i.e. ABS with 0% rubber content (porosity).The decrease of the elastic stiffness and yield strengthof ABS with increasing rubber content featured by thepresent model is in good qualitative agreement withexperimental findings (Ishikawa 1995). It should bementioned that the volumetric expansion enforced byhighly triaxial loading (Fig. 6b) leads to an overall sof-tening response due to void growth in the post-yieldregime.

123

212 T. Seelig, E. Van der Giessen

3.3 Crazing in PC

Failure of neat glassy polymers (here PC) typically pro-ceeds by the formation and propagation of a craze inwhich the polymeric material is drawn into numerousfibrils (of a few nanometers thickness) of highly stret-ched and oriented molecules (e.g., Narisawa and Yee1993; Haward and Young 1997; Estevez et al. 2000).These fibrils connect the two craze–bulk interfaces andare able to transfer stress until they rupture and thecraze locally turns into a crack, typically at a criticalcraze width (depending on the material) of the orderof a micron or below. Representing the effect of stress-carrying fibrils between the separating craze-bulk inter-faces in the course of craze widening by a rate-dependenttraction–separation law, Tijssens et al. (2000) have deve-loped a cohesive surface model for crazing. This modelwas subsequently applied by Estevez et al. (2000, 2005)to investigate the competition between bulk plasticity(shear yielding) and crazing in homogeneous glassypolymers; it is also employed in the present work tomodel failure by crazing in the PC regions of PC/ABSblends. The crazing cohesive model recognizes threestages: craze initiation, craze widening and craze break-down, which will now be presented briefly.

Craze initiation in glassy polymers is mainly gover-ned by hydrostatic stress and a number of different cri-teria may be found in the literature (e.g., Kinloch andYoung 1983; Narisawa and Yee 1993). Based on expe-riments, Sternstein and coworkers proposed a criterionwhich states that craze initiation takes place when themaximum principal stress Tn reaches a critical valueσ cr

n which is a decreasing function of hydrostatic stressσm (Estevez et al. 2000):

Tn = σ crn (σm) ≡ σm − A

2s0 + B

6

s20

σm. (8)

Here, s0 is the athermal yield strength of the bulk poly-mer (Table 1). Through the direction of maximum prin-cipal stress the above criterion also determines the crazeorientation which, however, is fixed in the present study.The relation (8) is shown in Fig. 7a along with theinfluence of the parameter B and displays the afo-rementioned strong influence of hydrostatic stress oncraze initiation. Since the criterion (8) appears to beonly meaningful in the range where the critical tensilestress is a decreasing function of hydrostatic stress, itis beyond that range continued with constant valuesindicated by the dotted part of the curves in Fig. 7a.

The relation between the traction Tn normal to thecraze and the separation �n of the craze-bulk interfacesis written in the following elastic visco-plastic rate form(Tijssens et al. 2000):

Tn = kn(�n − �c

n

). (9)

Prior to craze initiation the visco-plastic craze wide-ning rate �c

n vanishes and the elastic stiffness kn is alarge, purely artificial ‘penalty’ parameter. After ini-tiation kn reflects the instantaneous elastic stiffness ofthe craze matter. For craze widening upon initiation, avisco-plastic relation analogous to (3) has been propo-sed by Tijssens et al. (2000)

�cn = �0n exp

[− Acσc

θ

(1 − Tn

σc

)](10)

where �0n, Ac and σc are material constants and θ isthe (here constant) absolute temperature. Hence, crazewidening at a constant rate is assumed to take place ata constant stress (e.g. Tn = σc at �c

n = �0n). Valuesfor these parameters which cannot be directly relatedto their bulk counterparts in (3) are fairly unclear, andassuming them to be constant is a severe simplifica-tion. The reason for the fundamental difference fromthe bulk behavior is that complex processes of disen-tanglement and molecular chain-scission, necessary forthe drawing of fibrils, take place in the so-called ‘activezone’ at the craze-bulk interface (Tijssens et al. 2000;Estevez et al. 2000). These processes locally change thecharacteristics of the molecular network, e.g. its harde-ning behavior, and may introduce a rate-dependenceby their own. Following heuristic arguments in (Este-vez et al. 2000) the values of material parameters givenin Table 2 are assumed in the present work.

The typical response of the cohesive law at a constantwidening rate is shown in Fig. 7b. Craze initiation herehas been assumed to take place at Tn = 100 MPa whe-reas the widening resistance in (10) has a value ofσc ≈ 90 MPa (Table 2). Since Tn > σc at craze ini-tiation, the visco-plastic widening rate �c

n computedfrom (10) initially is very large and via (9) causes thesharp drop of Tn seen in Fig. 7b. Thereby �c

n decreasesuntil it reaches the value of the (here prescribed) totalwidening rate �n , and Tn according to (9) attains aconstant value. This plateau value is somewhat lowerthan σc since �n < �0n has been chosen here. Finally,when the craze has widened to a critical value �c cr

n thecohesive traction Tn is rapidly (stepwise in numericalsimulations) decreased to zero to describe breakdown

123

Effects of microstructure on crack tip fields 213

Fig. 7 (a) Craze initiationcriterion and (b)traction-separation law forcrazing in PC at a constantwidening rate�n/�cr

n = 10/s

00.80

1

2

0.4 os

B

m

o

σ

snT(a)

[MPa]

0

craze initiation

40

80

∆∆

Tn

n n

0cr

(b)

Table 2 Material parameters used in cohesive surface model for crazing in PC at room temperature in the present work

A B σc/s0 Acσc/θ �0n/�c crn (sec−1) kn�c cr

n (MPa) �c crn /rtip

0.7 2.8 0.9 44 103 103 0.2

of the craze. It has to be noted that the cohesive tractionTn after craze initiation depends on the widening rate�n and therefore the (specific) work of separation, i.e.the area under the curve in Fig. 7b, is in general notconstant.

As mentioned in (Haward and Young 1997) crazingis less likely to occur in material that has already under-gone plastic deformation because of the stretching ofthe molecular network. It is therefore assumed here thatcrazing can only take place if the maximum principalstretch is less than 2, i.e., 80% of the limit stretch λmax

of PC according to Table 1. If this value is reached priorto crazing at some point along the cohesive surface, thematerial there is considered to fail by brittle ruptureonce it has fully locked and a critical tensile stress of150 MPa is reached. This strength value, which can beestimated from the molecular structure of PC (Seitz1993) corresponds to the fully locked regime in Fig. 5.Brittle rupture (at zero length of decohesion) is treatednumerically by releasing the traction to zero in a smallnumber of time steps.

3.4 Failure of ABS

In contrast to neat glassy polymers (e.g., PC above)where failure takes place by a single craze it appearsrather unclear how to set up a cohesive zone model forfracture in ABS. The reason is that failure of ABS istypically preceded by the formation and coalescenceof multiple crazes between the rubber particles. Sincemultiple crazing often also contributes to the inelastic

bulk deformation of ABS it is hardly possible to uni-quely separate the bulk behavior from the (cohesive)fracture process zone behavior. For simplicity, a cohe-sive zone model of the same general structure as that forPC is adopted here, yet with properties correspondingto the bulk deformation behavior of ABS. Obviously,appropriate parameter values are fairly unclear and anumber of assumptions and estimates from heuristicconsiderations therefore have to be made. The cohe-sive strength σc entering the rate-dependent traction-separation law analogous to (10) is taken to scale withthe area fraction of the stress-carrying ligament bet-ween voids (times the yield strength ss of SAN, Table 1).For the 2D case of cylindrical voids to which also theporous plasticity model for bulk ABS has been fitted(Sect. 3.2) this means that σc( f0) ∼ 1 − √

f0. Initia-tion of the separation process is assumed to take placeat a critical value T cr

n ( f0) of the normal traction on thecohesive surface which depends on the porosity in asimilar fashion as σc( f0). Since the deformation beha-vior of ABS according to Sect. 3.2 as well as the cohe-sive law for separation display a plateau-like behavior(see Figs. 6 and 8) these values have to be picked care-fully to guarantee a continuous transition from merebulk deformation to separation. The values used in thesimulations are listed in Table 3. A reasonable estimatefor the critical separation �c cr

n at which the cohesivetraction Tn decreases to zero appears to be of the orderof the rubber particle size. Whether or not it should betaken to depend on the rubber content is not clear; hereit is assumed constant and half the value chosen for PC,

123

214 T. Seelig, E. Van der Giessen

Table 3 Values used in cohesive zone model for ABS

f0 T crn (MPa) σc (MPa)

0.1 65 70

0.15 60 65

0.25 45 55

0.4 30 45

0.5 20 35

∆

40

20

00

[MPa]

n

Tn

ABS (40%)

ABS (25%)

ABS (15% rubber)

Fig. 8 Traction-separation law for failure of ABS at a constantwidening rate �n/�cr

n = 10/s

i.e. �c crn (ABS) = 0.5�c cr

n (PC). All other parametersin the cohesive relation (10), governing essentially thedependence on rate and temperature, are for simpli-city taken equal to those in the cohesive zone modelfor PC. The response of the cohesive zone model forABS at a constant separation rate is shown in Fig. 8 fordifferent values of the rubber content. Due to the ratedependence the traction levels at initiation and duringseparation need not be the same as indicated by the‘overshoot’ at initiation.

As an alternative to the above model, one could fullymap the transition to failure into the bulk deformationbehavior and simply release (at zero length of decohe-sion, numerically in several steps) the tractions on thecohesive surface once a critical strain has been attai-ned in the adjacent bulk material. In the present blendmodel, however, this approach would cause a conflictwith the cohesive zone model of the neighboring PCwhich has a finite length of decohesion.

4 Results

In the following the fracture mechanical models sket-ched in Figs. 2 and 4 are evaluated numerically. Loa-ding is prescribed in terms of the applied stress intensityfactor K I which is normalized as K = K I /s0

√rtip with

the yield strength s0 = 97 MPa of PC (Table 1) and the

crack tip radius rtip. A constant loading rate ˙K = 1/secis chosen in all simulations. With rtip ≈ 10µm thiscorresponds to K ≈ 0.3 MPa

√m/sec which is well in

the range where isothermal conditions (assumed in thepresent work) are likely to prevail as has been analyzedby Estevez et al. (2005).

4.1 Crack tip fields at a stationary crack tip

The toughness of a material arises from energy dis-sipation in the various mechanisms but in laboratorytests they cannot be aportioned. The emphasis in thepresent section is on the effect of blend microstructureon energy dissipation prior to fracture and the under-lying development of a plastic zone. The propensityof fracture initiation is monitored in terms of localstresses responsible for triggering crazing. Differentmorphologies—particulate and lamellar —as they occurin real PC/ABS blends are considered.

4.1.1 Effect of morphology

The superior fracture toughness of PC/ABS blends com-pared to neat PC in the presence of sharp notches orpre-cracks is commonly ascribed to the occurrence ofmassive plastic deformation enabled by the ABS andto the suppression of crazing in the PC (e.g., Lee et al.1992; Greco et al. 1994; Inberg 2001). The latter is aconsequence of a relief of hydrostatic stress caused bythe dilation of the ABS. To illustrate how these mecha-nisms act in the framework of the present model, Fig. 9shows the distribution of equivalent plastic strain rateγ p (left) and hydrostatic stress σm (right) in the vicinityof a crack tip in neat PC (top) and two PC/ABS blendseach containing 30% ABS with 10% rubber (center andbottom). In the center of Fig. 9 a morphology with dis-persed ABS particles is considered as it is found for realblends with 30% ABS. The ‘lamellar’ morphology ofelongated regions of PC and ABS (bottom) typicallyprevails in the range of approximately equal contentsof both phases (e.g. Greco et al. 1994; Inberg 2001),

123

Effects of microstructure on crack tip fields 215

neat PC neat PC

PC/ABS PC/ABS

0.0400.0360.0320.0280.0240.0200.0160.0120.0080.0040.0000.000

.γ p

[sec ]−1

PC/ABS

908478726660544842363024181260

[MPa]σm

PC/ABS

Fig. 9 Equivalent plastic strain rate (left) and hydrostatic stress(right) at stationary crack tip in neat PC (top) and PC/ABS blends(30% ABS, 10% rubber in ABS) of particulate (center) and lamel-

lar (bottom) morphology at normalized stress intensity factorK = 2.25

but is considered here anyway in order to gain someinsight in the qualitative sensitivity to morphology.

As already mentioned earlier, plastic deformation ata notch under plane strain conditions in neat PC takesplace by the formation of a pair of shear bands whichintersect at some distance ahead of the notch (Narisawaand Yee 1993; Lai and Van der Giessen 1997; Este-vez et al. 2000; Gearing and Anand 2004). Close tothis intersection a concentration of hydrostatic stressappears (Fig. 9 top) which can lead to the initiation of asingle craze as a precursor of brittle failure. Obviously,

plastic flow and hydrostatic stress are more delocalizedin the presence of ABS (center and bottom of Fig. 9).In case of the particulate morphology, yielding is seento take place inside the ABS particles as well as inthe PC matrix. The particles close to the crack tip dis-play pronounced volumetric expansion. In the micro-structure with ABS layers perpendicular to the crack(Fig. 9 bottom), plastic flow in both phases spreadssuccessively along the layers. For both morphologiesthe plastic zone in the PC/ABS blends is significantlylarger than in neat PC. At the same time, the values

123

216 T. Seelig, E. Van der Giessen

Fig. 10 Effect ofmorphology of PC/ABSblends on (a) peakhydrostatic stress in the PCphase and (b) energydissipation (per unitthickness); both blendscontain 30% ABS with 10%rubber

Wdiss /s r tip2

0

lamellar ABS

particulate

neat PC

ABS

0.8 1.2 1.6 2 2.430

40

50

70

80

100

K

[MPa]σmmax(a)

lamellar ABS

particulate

neat PC

0.8 1.2 1.60

2 2.4 K

ABS90

60

0.4

0.2

0.6

(b)

of hydrostatic stress in the blends are lower and lessconcentrated than in neat PC. Multiple stress peaks arevisible in the PC matrix, especially in case of the par-ticulate microstructure, and if these would initiate cra-zing it would take place more distributed than in neatPC. Both, the energy dissipation associated with plas-tic flow and the reduction and delocalization of hydro-static stress are indicative of a higher toughness of thePC/ABS blends; some more quantitative results are pre-sented below.

The amount of volumetric expansion of the ABSenabled by its porosity (rubber) can qualitatively beseen from the distribution of hydrostatic stress. For bothmorphologies, σm in the ABS phase at locations furtheraway from the crack tip is higher than in the surroun-ding PC matrix because of the higher bulk modulus ofABS at low (initial) porosity. Closer to the crack tipthe opposite holds, since the capacity of ABS to carryhydrostatic stress there has strongly decreased due tothe increase of the porosity.

The peak value of hydrostatic stress, σmaxm , (being

critical for craze initiation) found in the PC matrixthroughout the process zone and the dissipated workWdiss = ∫ t

0

∫V σ · DpdV dt can be considered as tou-

ghening indicators and are traced in the course of loa-ding. In correspondence to Fig. 9, the effect of differentmorphologies of PC/ABS blends on σmax

m and on Wdiss

(per unit thickness in the present plane strain problemand normalized by s0r2

tip) is depicted in Fig. 10. Tosmooth out local effects resulting only from the par-ticular arrangement of the ABS close to the crack tip,averaging over three different realizations of the micro-structure is performed for each of the two morpho-logies. Also shown in Fig. 10 is the response of neatPC. Obviously, local peak values of hydrostatic stress

in the PC are more affected by the blend morphologythan the (global) energy dissipation. The latter is nearlythe same for both PC/ABS blends and, as expected,significantly higher than for neat PC. In the particulatemicrostructure the hydrostatic stress still attains valuesof about 90 MPa which according to the literature (e.g.Kinloch and Young 1983; Narisawa and Yee 1993) maycause craze initiation in PC. However, peak hydrosta-tic stresses then prevail at several locations betweenthe ABS particles (Fig. 9, center) and eventual failurewould probably occur less localized than in neat PC.The values of hydrostatic stress in case of the lamel-lar morphology suggest that crazing is suppressed. Onehas to bear in mind, however, that local stress concen-trations in the PC induced by possible localized failuremechanisms inside the ABS (see discussion in Sect. 5)are not accounted for in the present model since it treatsABS as a homogeneous porous medium.

4.1.2 Effect of rubber content in ABS

The importance of plastic dilatancy of the ABS for tou-ghening is illustrated by comparing the above results tothe behavior of a PC/SAN blend, i.e., 0% rubber, whereboth phases are plastically incompressible. Figure 11shows the distribution of plastic strain rate γ p (a) andhydrostatic stress σm (b) for the same 30% particle mor-phology as in the center of Fig. 9. It is clearly seen thatthe localization of plastic flow in shear bands and theconcentration of hydrostatic stress in one strong peakfound in neat PC (Fig. 9, top) is only slightly disturbedby the presence of SAN. Enlargement of the plasticzone and delocalization of hydrostatic stress accompli-shed by the ABS (Fig. 9, center and bottom) can—at

123

Effects of microstructure on crack tip fields 217

908478726660544842363024181260

[MPa]σm

0.0400.0360.0320.0280.0240.0200.0160.0120.0080.0040.0000.000

.γ p

[sec ]−1

PC/SANPC/SAN

(a) (b)

Fig. 11 Distribution of equivalent plastic strain rate (a) and hydrostatic stress (b) in PC/SAN blend (i.e., 0% rubber in ABS) with 30%SAN particles at K = 2.25

Fig. 12 Effect of rubbercontent in ABS on (a) peakhydrostatic stress in the PCphase and (b) energydissipation (per unitthickness); 30% ABSparticles. The •’s mark foreach case the initiation ofcrazing when this isassumed to occur when thehydrostatic stress reaches avalue of 90 MPa

Wdiss /s rtip2

0

K0.8 1.2 1.6 2 2.4

20 % rubber 10 % rubber 0 % rubber neat PC

40

50

60

70

80

90

100

maxσm [MPa](a) (b)

K0.8 1.2 1.6 2 2.40

20 % rubber10 % rubber 0 % rubber neat PC

0.6

0.4

0.2

least according to the present model—not be achievedby blending PC with SAN.

The rubber content in ABS is reported to be a para-meter of utmost importance for toughening in real PC/ABS blends (Greco 1996). Its effect in the frameworkof the present model is depicted in Fig. 12. The micro-structure considered here consists of 30% dispersedparticles (see center of Fig. 9). Again the response ofneat PC is shown for comparison. Peak values of hydro-static stress found in the PC matrix decrease with increa-sing rubber content in ABS (10% and 20%) while forthe extreme case of a PC/SAN blend (0% rubber, asshown in Fig. 11) they clearly exceed those prevailingin neat PC. Energy dissipation in the PC/SAN blend isvery low and comparable to that in neat PC whereasit is significantly enhanced by the presence of someamount of rubber (porosity) in the ABS (Fig. 12b).

One way to crudely interpret the present results withregard to the fracture initiation toughness is to look

at the onset of failure which can be associated withcraze initiation in the PC; according to Narisawa andYee (1993) this occurs at a critical hydrostatic stressof about 90 MPa. The intersection of the correspon-ding horizontal line with the different curves in Fig. 12ayields the critical loading stages in terms of K at whichthe individual materials would fail (indicated by •). Therelated amounts of dissipated energy can then be obtai-ned from Fig. 12b. Comparing the values for PC/ABSwith 20% rubber, neat PC and PC/SAN leads to theratio PC/ABS : PC : PC/SAN ≈ 10 : 2 : 1. Thoughnot directly comparable with the situation consideredhere because of the different loading conditions, expe-rimental results by Kurauchi and Ohta (1984) should bementioned who found a similar ordering in the impactfracture energy with values for PC/ABS about ten timeshigher than those for PC/SAN.

As an alternative to the peak hydrostatic stress σmaxm

considered in the present work as an indicator for craze

123

218 T. Seelig, E. Van der Giessen

0.50.450.40.350.30.250.20.150.10.050.010

0.50.450.40.350.30.250.20.150.10.050.010

pγ

pγ

(b)

(c) (d)

(a)

Fig. 13 Crack propagation and plastic zone evolution in PC/ABS (50/50) blends with 15% rubber (a, b) and 40% rubber (c, d)in the ABS

initiation, one may evaluate Sternstein’s σ crn (σm)

criterion (8) which varies with K in a similar way asσmax

m (Seelig et al. 2001). Common to both criteria isthe decreasing slope of the variation of the critical stresswith increasing K which makes it difficult to trace theonset of crazing. It therefore might be interesting tolook instead at strain-based craze initiation criteria asare discussed e.g., in (Kinloch and Young 1983); yetmuch less data are available in terms of craze initiationstrain than in terms of stress.

4.2 Crack growth

The model employed in the previous section is nowextended to allow for crack propagation as sketched inFig. 4. Here we consider only the case of a co-continuous

morphology with equal volume fractions of PC andABS, and focus on the effect of the rubber contentin ABS. The blend microstructure with layers of bothphases perpendicular to the initial crack is resolvedinside the process zone where plastic flow and crackpropagation are expected to take place. The elastic (far-field) region outside the process zone is, for simplicityagain, described using the isotropic elastic constantsof neat PC. The error made by ignoring the proper(here anisotropic) overall elastic behavior of PC/ABSis believed to be small compared to uncertainties inmodeling deformation and failure in the process zone.The same constitutive models for PC and ABS as in theprevious sections are employed here, i.e., ABS is againdescribed as a porous plastic medium with the porosityreflecting the rubber content. Failure and crack pro-pagation are considered only along the symmetry axis

123

Effects of microstructure on crack tip fields 219

of the problem where a cohesive surface is introdu-ced (see Fig. 4) with alternating properties for the PCand ABS layers as specified in Sects. 3.3 and 3.4. Sincethe morphology and the ABS content are fixed in thepresent model, the rubber content (porosity) in the ABSis the only remaining “free” microstructural parameterwith its effect on the fracture process to be investigated.Blends with a rubber content in the ABS varying bet-ween 15% and 50% are being considered. As in theprevious section, loading is imposed via far-field boun-dary conditions determined by the normalized mode Istress intensity factor K = K I /s0

√rtip. The lamel-

lar morphology of real (50/50) PC/ABS blends typi-cally shows a layer thickness of approximately 2–5µm(Inberg 2001). From the relative length scales visiblein Fig. 13 this means that the crack tip radius rtip in thecomputational model is on the order of 10µm (as inSect. 4.1, see also Sect. 2).

Figure 13 shows snap-shots of the evolving plasticzone, in terms of the distribution of accumulated plasticstrain, as the crack propagates. The figures on the left(a, c) refer to the onset of crack propagation, i.e., thefirst occurrence of cohesive zone failure (‘breakdown’)while the figures on the right (b, d) show the situation atan amount of crack advance of about 6rtip. The perfor-mance of the two blends, with rubber contents of 15%and 40%, differs significantly in the amount of plas-tic deformation. In the case of the low rubber content(Fig. 13a and b) the lateral extension of the plastic zonedecreases with increasing crack length. In contrast, theplastic zone in the blend with a large rubber content(Fig. 13c and d) is larger after some amount of crackgrowth (right) than at crack initiation (left). The plasticzone size tends to a stationary width in the ABS layerswhich is significantly larger than the initial crack tipradius. Massive plastic deformation of PC and ABScan be seen along the fracture surface (Fig. 13d).

The effect of ABS rubber content on the fracturetoughness is shown in Fig. 14 in terms of so-calledR-curves. Blends with a relatively low rubber content(here 15% and 25%) fail in a brittle manner, i.e., thecrack resistance does not increase with crack propaga-tion after initiation—in accordance with the lack of apronounced plastic zone seen in the 15% rubber blendin Fig. 13a and b. In contrast, the build-up of a largeplastic zone (see Fig. 13c and d) in blends with a higherrubber content in the ABS (here 40% and 50%) leadsto an increase of the fracture toughness in the course ofcrack growth. Neat PC, as discussed before and inclu-

∆a/r tip

crack advance

65432101.4

1.6

1.8

2neat PC

50% rubber in ABS

40%

25%15%

/s r tip01/2K

frac

ture

toug

hnes

s

I

Fig. 14 Crack resistance curves for neat PC and co-continuousPC/ABS (50/50) blends with different rubber content in ABS

∆at a = 6 rtip

W /s r tip2

0frac

0.2

0

0.4

0.6

0.8

1.0

rubber content in ABS (%)

PC/ABS blend

neat PC

at initiation

10 20 30 40 50

frac

ture

ene

rgy

Fig. 15 Fracture energy (per unit thickness) vs. rubber contentin ABS at two stages of crack propagation

ded in Fig. 14 for reference, displays brittle failure; yet,its toughness is higher than the initiation toughness ofthe PC/ABS blends considered here. Hence, the initia-tion of fracture taking place somewhere ahead of thenotch root (Fig. 13a and c) in the present blend modelis promoted by the presence of the ABS layers. Crackinitiation starts earlier, i.e., at a lower load level, incase of blends with a larger amount of rubber (softerABS) which, however, subsequently display anR-curve behavior.

The total fracture energy, i.e., the work dissipatedin the course of the fracture process, consists of thework of separation (expended in the cohesive zone)plus the dissipation in the bulk (plastic zone): Wfrac =Wsep (cohes. zone) + Wdiss (bulk). In fracture testingof polymers (e.g., IZOD), Wfrac is often considered asa global measure of toughness. Figure 15 shows, for

123

220 T. Seelig, E. Van der Giessen

two stages of the fracture process corresponding to thesnap-shots in Fig. 13, the fracture energy as a functionof the rubber content in ABS computed from the presentblend model. Obviously, the rubber content in ABShas a stronger effect on the energy absorption duringfracture than at crack initiation. This agrees with theobservation made from Fig. 13 that the amount of rub-ber in ABS determines, at least in the framework ofthe present model, whether massive energy dissipationin a growing plastic zone takes place or not. A ‘syner-gistic effect’ in terms of an optimal rubber content, asreported in part of the experimental literature (see Intro-duction), however, cannot be detected from the presentsimulations.

5 Discussion and conclusions

The present study is concerned with the fracture pro-cesses in amorphous thermoplastic PC/ABS blends. Inorder to investigate micromechanisms and microstruc-tural effects in these materials, the situation at a cracktip has been modeled by resolving the blend microstruc-ture around the crack tip and by accounting for failureof the different phases of the heterogeneous material.Thereby a number of assumptions have been made anda critical assessment of these assumptions with respectto the overall response predicted in terms of the fracturetoughness seems to be pertinent.

Results so far have been presented in terms of nor-malized stress intensity factors and normalized work offracture. Absolute values of these quantities are obtai-ned if an absolute value is assigned to the crack tipradius rtip. The crack tip radius in the present compu-tational model is fixed by its size relative to the ABSparticles and layers (Figs. 3b and 13). With the latterknown from micrographs, we obtain a typical value ofrtip ≈ 10 µm, as mentioned in Sect. 2.2. According toFig. 14 and with s0 ≈ 100 MPa this yields a fracture ini-tiation toughness (K I c) of about 0.63 MPa

√m for neat

PC and about 0.5 MPa√

m for PC/ABS. These valuesare significantly lower than experimental data repor-ted in the literature, i.e. K I c ≈ 2 MPa

√m for neat PC

(e.g., Kinloch and Young 1983) and K I c ≈ 1.5 MPa√

mfor PC/ABS (Seidler and Grellmann 1993), but areconsistent with the tendency that the initiation tough-ness of PC/ABS is roughly 75% below that of neatPC. The origin of the quantitative difference, however,is not obvious. One possible, and in fact likely, rea-

son may be that the crack tip radius of about 10 µm inthe present model is smaller than that in the fracturetests underlying the above experimental data (Inbergand Gaymans 2002). Modeling a larger crack tip withrespect to the microstructure (particle size), however,would enormously increase the numerical expense dueto the discretization of the correspondingly larger pro-cess zone. Another reason might be that the criticalstress values in the local failure initiation criteria havebeen chosen too small in the present study. A morethorough investigation of these issues will be subjectof future work.

The results presented in Sect. 4.2 show an increa-sing fracture toughness of PC/ABS with increasing rub-ber content. So, an issue to be commented on is thesynergistic effect, i.e., the optimal toughness for someintermediate range of composition, observed in someexperimental studies but not reproduced in the simu-lations. A simple explanation is that the situations forwhich such a synergistic effect is reported are not captu-red by the present simulations. In (Greco et al. 1994) asynergistic effect was observed only for blends withparticulate ABS (about 30%) whereas crack growthsimulations here are performed only for a lamellar mor-phology. The synergistic effect reported by Inberg(2001) for a lamellar morphology was ascribed to mas-sive delamination of the PC and ABS layers. Incor-poration of this additional failure mechanism, e.g., bycohesive surfaces, would introduce additional materialparameters for which reliable data are presently notavailable.

A further difference between experiments and simu-lations is due to the loading rates. While the presentsimulations are performed at a low rate of loading andunder isothermal conditions the experiments by Grecoet al. (1994) and by Inberg (2001) showing synergisticeffects were done at rather high loading rates. Underthese conditions the significant temperature rise due toadiabatic heating (Inberg 2001) may play an impor-tant role and affect the measured toughness. Modelinghigh rate fracture processes in future analyses requiresto take these effects into account as done for homoge-neous glassy polymers in (Estevez et al. 2005).

A benefit of micromechanical models is that theyyield information about quantities not accessible tomeasurements, such as, for instance, the amount ofenergy dissipated in the individual phases of a heteroge-neous material such as PC/ABS. A common view in theexperimental literature is that toughening in PC/ABS is

123

Effects of microstructure on crack tip fields 221

(a) (b) (c) (d)PC

ABS

PC

ABS

PC

ABSABS

PC

Fig. 16 Sketch of undeformed PC/ABS blend (a) and mechanisms causing a relief of hydrostatic stress: (b) rubber particle cavitationand void growth in ABS, (c) crazing in ABS, (d) debonding along interface

mainly due to plastic (shear) yielding in the PC enabledby the volumetric expansion and the relief of hydro-static stress accomplished by the ABS. The presentwork is based on the assumption that void growth fromcavitated rubber particles inside the ABS is the domi-nant mechanism for this. Probably as a consequence ofthis assumption, simulations predict a large portion ofthe total energy dissipation to take place in the ABSby plastic deformation of its matrix (SAN). However,as sketched in Fig. 16, other mechanisms such as cra-zing inside the ABS or interface debonding may like-wise accomplish the volumetric expansion necessaryfor yielding of neighboring PC regions under overalltriaxial loading conditions. Indeed, each of the threemechanisms sketched in Fig. 16b–d is found in realABS and, for instance, interface debonding has beensuggested to play an important role in toughening of co-continuous PC/ABS blends (Inberg 2001). Thus, moremicromechanical studies are needed (or at least help-ful) to better understand the competition and the condi-tions for the predominance of the different microme-chanisms; this is a subject of ongoing work (Seelig andVan der Giessen 2006).

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