Generic Crack Patterns in Rubber-Modified Polymers underBiaxial Stress States

S. RAMASWAMY, ALAN J. LESSER

National Center for Polymer Research, Polymer Science and Engineering Department, University of Massachusetts-Amherst, Box 345300, Amherst, Massachusetts 01003

Received 3 April 2002; revised 13 May 2003; accepted 14 May 2003

ABSTRACT: The damage mechanisms in three different systems, namely, acrylonitrile-butadiene-styrene, methacrylate-butadiene-styrene modified poly(vinyl chloride), andstyrene-butadiene-styrene have been investigated. The damage was analyzed over arange of biaxial stress states with confocal microscopy and scanning electron micros-copy. The macroscopic yield followed a linear behavior for all the systems in anoctahedral shear stress versus mean stress plot, whereas popular models for this classof materials predicted a nonlinear response. Over a certain range of biaxial stressstates, a damage pattern generic to all the systems was observed. The damage patternconsisted of an array of cracks propagating perpendicular to the direction of themaximum tensile principal stress and arranged itself in a more or less periodic fashion.There was also self-similarity in the patterns at various length scales. Similar patternshave also occurred in several other polymeric systems. The interaction in the ensembleof cracks created seems to lead to stress reduction at the crack tips, thereby limiting thecrack sizes. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2248–2256, 2003Keywords: damage; cracks; crazing; shielding

INTRODUCTION

Glassy polymers can be significantly toughened bythe dispersion of a rubber phase.1–3 The associatedmechanisms of energy absorption and their se-quence have been the subject of great interest overthe last three decades. Some of these micromecha-nisms are brittle in nature (e.g., cavitation,4–6 mi-crocracking,6,7 and crazing8,9), whereas others in-volve more ductile mechanisms (e.g., shear band-ing,6 rubber particle stretching,4,5 and inelastic voidgrowth1,2). These mechanisms occur at differentlength scales, and the relationship between themhas been the subject of great debate. For example,in systems with ductile matrices [e.g., poly(vinylchloride) (PVC) modified with methacryclate-buta-

diene-styrene (MBS) where the matrix (PVC) isductile], cavitation of rubber particles followed byshear yielding of the matrix is commonly ob-served.5,6 It is generally argued that cavitation re-leases the hydrostatic tension in the vicinity of therubber particle, thereby promoting plastic yieldingin the matrix. However, in systems as high-impactpolystyrene (HIPS) and acrylonitrile-butadiene-sty-rene (ABS) where the matrices are more prone tobrittle failure, microcrazing and microcracking8,9

have been the dominating mechanisms of energyabsorption. The addition of rubber particles to brit-tle matrices primarily increases the craze densitybecause the rubber particles act as stress concen-trators under tensile stress, thereby serving as nu-cleation sites for crazes. There are at least two opin-ions on the termination of these crazes. For HIPS,Bucknall1 has proposed that craze growth is termi-nated at the rubber particles themselves, providedthey are large enough. However, Donald andKramer8,9 found that in HIPS and ABS, solid rub-

Correspondence to: Alan J. Lesser (E-mail: ajl@polysci.umass.edu)Journal of Polymer Science: Part B: Polymer Physics, Vol. 41, 2248–2256 (2003)© 2003 Wiley Periodicals, Inc.

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ber particles cavitate upon interacting with a craze.The voids left behind grow under increasing stressand lead to premature failure of the craze. However,occluded particles can accommodate the displace-ments because of crazing by local fibrillation of therubber shell that surrounds each subinclusion,without the formation of large voids. Consequently,the occluded particles postpone craze breakdown.Thus, the damage at this length scale (i.e., that ofthe rubber particles) is highly dependent on thematerial properties of the constituents, particlemorphology, and interfacial properties between dif-ferent phases.

This work investigates the damage mechanismsin three modified systems tested over a range ofbiaxial stress states. The materials examined wereABS, MBS, modified PVC, and styrene-butadiene-styrene (KratonD). ABS and modified PVC havespherical modification in the size range of 0.1–0.2�m, whereas KratonD has a cocontinuous modifica-tion of about 5 nm in size. Although these materialsvary in morphology, a damage pattern generic to allthree systems was observed over a range of biaxialstress states. This pattern occurs at length scalesmuch larger than the size of the imposed inhomoge-nity in the materials. The damage pattern also oc-curred in other materials as reported in the litera-ture. This study also examines the macroscopicyield behavior of the three systems over biaxialstress states. The onset of microscopic damage wasalso detected via in situ light-transmission mea-surements.

EXPERIMENTAL

The materials used in this study are summarizedin Table 1. Three different rubber-modified sys-tems were tested. In one of them, ABS had spher-ical rubber particles in the size range of 0.1–0.2�m. The refractive indices of the matrix and therubber were matched by the addition of a small

amount of poly(methyl methacrylate), thus ren-dering the material transparent. The morphologyof ABS is shown in the transmission electron mi-croscopy (TEM) image in Figure 1(A). The secondsystem was PVC modified with MBS at threedifferent concentrations (5, 10, and 15%). PVC-MBS also had spherical modification in the sizerange of 0.1–0.15 �m [Fig. 1(C,D)]. The thirdmaterial was transparent KratonD 1401P, a sty-rene-butadiene-styrene triblock copolymer with acocontinuous modification [5–6 nm in size, Fig.1(B)]. The transparency of these materials wasexploited in the multiaxial tests (described subse-quently) with a laser to detect the onset of micro-scopic damage.

The specimens used in the multiaxial testswere in the form of thin-walled hollow cylinders.Hollow cylinders of ABS were compression-molded with a displacement mold preheated to220 °C. Twelve grams of polymer were placed inthe mold, which was closed with a press. The

Table 1. Materials

Component Product Name Manufacturer

PVC Geon 334 The Geon Co.MBS modifier Polaroid BTA 733 Rohm and HassProcessing aid Polaroid K120ND Rohm and HassHeat stabilizer Advastab TM-181 Morton International, Inc.Lubricant Myverol 18-06 Quest InternationalABS Lustran 266 Bayer Corp.KratonD–SBS copolymer Kraton D 1401 Kraton Polymers

Figure 1. Rubber particle distributions: (A) ABS, (B)KratonD, (C) PVC-5% MBS, and (D) PVC-15% MBS.

GENERIC CRACK PATTERNS 2249

mold was left to air cool to room temperature for2 h while still under compression. The mold wasthen opened and the hollow cylinder removed.KratonD tubes were made by a similar procedure

at 190 °C. PVC was dry-blended with three dif-ferent concentrations (5, 10, and 15%) of MBS (aheat stabilizer, processing aid, and lubricant werealso used; see Table 2) and compounded by roll-milling at 180 °C for 10 min. The resulting sheets,approximately 0.2 mm thick, were then cut andcompression-molded at 180 °C to make hollowcylinders.

All cylinders were machined down with a latheto create a thinner gauge as shown in Figure 2(A).This process ensured that the yielding initiated inthe thinner region and provided a location for insitu light-transmission measurements (describedin the next section). A detailed discussion of thismethod can be found elsewhere.10,11

Table 2. MBS Modified PVC Formulations Showingthe Weight Percentage of Each Component Added

MBS PVC LubricantHeat

StabilizerProcessing

Aid

5 90 2 2 110 85 2 2 115 80 2 2 1

Figure 2. (A) Sample geometry, (B) transmitted laser intensity versus stress, and (C)test setup.

2250 RAMASWAMY AND LESSER

Biaxial Tests and in situ Light-TransmissionMeasurements

The materials were tested in different biaxialstress states with a tension torsion hydraulic In-stron modified to apply internal pressure to thehollow cylinder.12,13 Simultaneous loading andapplication of internal pressure led to the follow-ing stress state in the hollow cylinder:

�1 � � L�Dt� � �pD

4t ��2 � �pD

2t � (1)

where �1 is the axial principal stress, �2 is thecircumferential principal stress, L is the axialload, p is the applied internal pressure, D is themean cylinder diameter, and t is the wall thick-ness. By controlling L and p, various biaxialstress states can be achieved. The stress in theradial direction is negligible because the thick-ness of the tube is much smaller as comparedwith its diameter (D/t � 20). The octahedralstrain rate was held constant for each materialover the range of stress states. This was done in apseudo-strain-controlled mode with Young’s mod-ulus and Poisson ratio as the inputs. The experi-mental setup is depicted in Figure 1(C).

A laser beam (� � 633 nm, beam diameter: 3mm2) was passed through the thinner section ofthe initially transparent cylinder, and the inten-sity of the transmitted light was detected with a100-mm2 silicon detector. The transmitted inten-sity, which remained constant at low levels ofstress, dropped when critical conditions for cavi-tation or void formation were reached [onset ofmicroscopic damage: Fig. 2(B)]. Thus, the onset ofmicroscopic damage was detected with the laser.The tests were stopped at the onset of microscopicdamage, defined as a 10% drop in the intensity ofthe transmitted light, and the samples were thenanalyzed for damage.

The tested samples were polished along asmaller thickness with a polishing wheel. Thepolished sections were then observed under a Bio-Rad MRC-600/1000 confocal microscope operat-ing in reflection mode. Voids and cracks presentin the sample scattered light and appeared asbright regions against a dark background in thereflected image. A set of two-dimensional imageswas obtained at different depths in a sample. Thesamples were also investigated under a conven-tional optical microscope.

RESULTS AND DISCUSSION

For each of the systems, two sets of experimentswere performed over a range of biaxial stressstates ranging from uniaxial compression to equi-biaxial tension. In one set, hollow cylinders weretaken to macroscopic yield (defined by a zero slopein the stress vs strain curve), and the onset ofmicroscopic damage (defined by a 10% drop intransmitted light intensity) was recorded. In asecond set of experiments, samples were taken tothe onset of microscopic damage and the testingwas stopped. The results of these two sets of ex-periments are shown together in a plot of octahe-dral shear stress �oct versus mean stress �m inFigure 3(A–C) for ABS, PVC-10% MBS, and Kra-tonD, respectively. The macroscopic behavior forPVC at other concentrations of MBS (5 and 15%)was also linear, as reported by Crawford andLesser.10

Macroscopic Yield Behavior

The macroscopic yield points are represented asfilled circles in Figure 3(A–C). The macroscopicyield locus for each of the three systems is linearin a plot of �oct versus �m. They can be describedwith Sternstein’s model14 as follows:

�yoct��yo

oct���m (2)

The values of parameters �yooct and � are listed

in Table 3 for the three materials.This linear behavior differed from Lazzeri’s

and Bucknall’s15 nonlinear model for the macro-scopic yield for cavitated polymeric systems, amore detailed evaluation of which has been per-formed in refs. 10 and 11. Interestingly, even forKratonD that has a cocontinuous modification,the macroscopic yield locus was more or less lin-ear. However, the interaction between the hydro-static and shear stresses was nearly double thanin the other systems tested.

Onset of Microscopic Damage Behavior

The onset of microscopic damage is plotted ashollow circles in Figure 3(A–C). Different trendsare observed in each of the systems. For ABS, theonset of microscopic damage is linear under pos-itive mean stress as seen in Figure 3(A). It can bedescribed by eq 2 with �wo

oct � 23 MPa and �� 0.89. For PVC-10% MBS, the onset microscopicdamage occurred at constant octahedral shear

GENERIC CRACK PATTERNS 2251

stress �wooct � 12 MPa. Crawford and Lesser10 have

reported that this constant value does not changesignificantly with the concentration of MBS. Themicroscopic damage behavior differs from the con-stant positive mean-stress criterion predicted byexisting energy-balance-based models for cavita-tion4,15 (these apply only to spherically modifiedsystems). For KratonD, however, the onset of mi-croscopic damage occurs very close to the yield

Figure 3. Macroscopic yield and onset of microscopic damage behavior in (a) ABS, (b)PVC-10% MBS, and (c) KratonD.

Table 3. Macroscopic Yield Behavior — Parametersin eq 9

System �yooct (MPa) �

ABS 23 0.13PVC-10%MBS 26 0.19Kraton D 1401 P 14 0.35

2252 RAMASWAMY AND LESSER

point except in the case of equi-biaxial tension. Nodefinitive trend is observed. In all three systems,microscopic damage occurred only under positivemean stress. In stress states with negative meanstress, the intensity of transmitted light did notchange until macroscopic yield.

Although the macroscopic yield in the threesystems was linear [as plotted in Fig. 3(A–B)], theonset of microscopic damage behavior varied dra-matically from one system to another. It is rea-sonable to anticipate that the associated damagemechanisms in the materials may also be differ-ent.

Microscopic Damage in the Materials

To capture the early stages of microscopic dam-age, some tests were stopped at the onset of mi-croscopic damage. These samples were then ana-lyzed with microscopy techniques as described inthe Experimental, and the observations are asfollows.

In stress states with negative mean stress [Fig.4(A)], void formation was not observed until mac-roscopic yield in all the systems. The intensity ofthe transmitted laser light remained essentiallyconstant until macroscopic yield was reached.

In stress states with positive mean stress [Fig.4(B)], the microscopic damage in the materialswas in the form of an ensemble of cracks. Forinstance, Figure 5 portrays the damage in ABStested in a biaxial stress state where the stress inone direction is twice the stress in the other. Thecracks propagate perpendicular to the direction ofthe higher principal stress. This is also the case

with PVC-10% MBS and KratonD. In any generalbiaxial stress state with positive mean stress, thecracks propagate perpendicular to the direction ofmaximum applied tensile principal stress.

Furthermore, apart from being oriented in oneparticular direction, there appears to be a patternassociated with the ensemble of cracks. For in-stance, Figure 6 displays the damage in ABS un-der a 7:2 biaxial stress state. This is a series ofconfocal microscope images taken one below an-other at intervals of 5 �m in the sample. Thecascading pattern is seen in many of these imagesas indicated by the arrows. In fact, all three con-figurations just described exist in Figure 5 forABS. A complex combination of these three con-figurations also appears in PVC-10% MBS andKratonD in Figure 7. Thus, the crack patterns

Figure 4. (A) Biaxial stress states with negative mean stress and (B) biaxial stressstates with positive mean stress.

Figure 5. Damage in ABS tested in a biaxial stressstate; cracks grow perpendicular to the direction of themaximum applied principal stress.

GENERIC CRACK PATTERNS 2253

seem to be generic to the three systems over arange of biaxial stress states.

Similar patterns have also been reported byJones and Lesser16 for isotactic polypropylene intheir study of fatigue-induced damage in the ma-terial [Fig. 8(A)]. Here, cascading arrays of crazescan be observed. Other modified systems asHIPS8 and PC modified with PS17 also producedthe same kind of features. For example, Figure8(B) is an optical micrograph of the crazes inHIPS (under uniaxial tension) investigated byDonald and Kramer.8 Miscible blends of PS andPPO18 also contain analogous craze patterns un-der uniaxial tension [Fig. 8(C)]. The collinear,parallel, and cascading pattern is evident in Fig-ure 8.

Thus, the cascading crack–craze pattern is ge-neric over a range of materials that vary dramat-ically in the nature of the imposed inhomogenity:rubber particle size, shape, and concentration.They also repeat at different length scales (Figs.5–8). Their universality suggests that the ob-served configurations might be energetically fa-vorable patterns that evolve at a length scaleintermediate between the microscopic damage

(material dependent) and macroscopic yield.Therefore, any micromechanical model that pre-dicts a material response must consider the effectof these mesoscale mechanisms, which can betreated within the realm of continuum mechan-ics.

Useful insight into the evolution of these crackpatterns may be obtained through a better under-standing of their mutual interaction. First, con-sider the nature of stress distribution around anisolated crack subjected to a mode I field (unitnormal stress), although such considerations areby no means rigorous because the field can besignificantly distorted by the presence of othercracks.19 In Figure 9 there are regions of compres-sive and tensile stresses around the crack. In thecompressive zone, the stresses are lower than thefar field stress, whereas in the tensile region thestresses are higher. Now consider two possibletypes of interactions in the simplest case of twocracks with overlapping stress fields. In a col-linear configuration [as in Fig. 10(A)], the tensilestress zones overlap leading to an amplificationtype of interaction, which results in increased

Figure 7. 1, KratonD and 2, PVC 15% MBS axialtension; the collinear, parallel, and cascading patternsare visible.

Figure 6. ABS biaxial stress state. X–Y images takenat intervals of 5 �m in the z direction show relativeorientation of the cracks with depth. The cascadingpattern is evident.

2254 RAMASWAMY AND LESSER

stress intensities at the crack tips19 (as comparedwith a single crack). However, a parallel configu-ration [as in Fig. 10(B)] results in reduction of

stress intensities at the crack tips, known as ashielding type of interaction.19 Further, in Figure9 the region of compressive stress is substantiallylarger than the region of tensile stress: small andintense amplification zones (near crack tips) andmore-diffused shielding zones. As a consequence,the (amplifying) collinear interactions become no-ticeable only at small spacings between cracks,whereas the (shielding) stacked interactions ofparallel cracks have a longer range.19

On the basis of the aforementioned discussion,one can possibly provide a qualitative explanationof interactions in the array of cracks seen in Fig-ures 5–8. The shielding effect may be dominantbecause the distance between the cracks (in thedirection of the maximum applied tensile stress)is much smaller than the ligaments between col-linear cracks. With the cracks interacting in ashielding manner, nucleation in a different regionmight be more favorable than crack propagation.

Figure 8. (A) Crazes in isotactic polypropylene (thecascading pattern is evident), (B) crazes in HIPS, and(C) PS-PPO where tensile stress was applied perpen-dicular to the cracks. Collinear, parallel, and cascadingpatterns can be observed. (From A. M. Donald and E. J.Kramer, J Mater Sci, 1982, 17, 2351; D. L.Wilfong, etal., In Advances in Chemistry Series 211; Paul, D. R.;Sperling, L. H., Eds.; American Chemical Society:Washington, DC, 1986; p 273, �2003 John Wiley &Sons, Inc., reproduced by permission.)

Figure 9. Stress field (�yy) around an isolated crackin an infinite plate subjected to unit tensile loading.The lines represent contours of constant stress. Theshielding zone (�yy � 1) is more diffuse than the regionof stress amplification (�yy � 1).

Figure 10. (A) Collinear cracks—amplification, (B)parallel cracks—shielding, and (C) offset cracks—am-plification or shielding.

GENERIC CRACK PATTERNS 2255

Thus, this kind of interaction can limit crack sizesand postpone their growth.

The above argument is based on stress inten-sities at the crack tips. The stress intensities arelocal parameters that describe the scale of thestresses near a crack tip. For a body with a singlecrack, the stress intensity (K) is related to rate ofchange of energy, or equivalently, the energy re-lease rate G of the system as follows

G � KI2/E� (3)

where E� � E for plane stress, E� � E/(1 � 2) forplane strain (E is Young’s modulus and is Pois-son’s ratio), and subscript I corresponds to mode Iloading. Crack growth is governed by a local pa-rameter Kth—a threshold stress intensity factor,which is equivalent to a threshold energy releaserate (Gth) of the body. However, in the case ofmultiple cracks, the relation between stress in-tensities and the energy release rate is far fromobvious. The propagation of a crack tip will notonly be affected by the stress intensity factor butalso by the stiffness of the material surroundingit, both of which are dependent on crack sizes andrelative crack locations.

CONCLUSIONS

The macroscopic yield of ABS, PVC-MBS, andKratonD followed a Sternstein-type behavior. Theonset of microscopic damage for all three systemsis a function of both mean stress and octahedralshear stress. In ABS, the onset stress droppedlinearly with mean stress, whereas in PVC-MBSmicroscopic damage initiated at a constant octa-hedral shear stress. No microscopic damage wasobserved under conditions of negative meanstress.

In biaxial stress states with positive meanstress, a damage pattern generic to all the mate-rials was observed. The damage pattern was inthe form of cracks that propagated perpendicularto the maximum applied tensile principal stress.The cracks appeared to be arranged in a periodicmanner with one crack following another. Thispattern, which repeated itself at different lengthscales, was independent of the nature of the mod-

ification and morphology of the system. The inter-action in the ensemble of cracks created seemedto limit the crack sizes, and this could be one ofshielding, which results in stress reduction at thecrack tips.

REFERENCES AND NOTES

1. Bucknall, C. B. In The Physics of Glassy Polymers;Haward, R. N.; Young, R. J., Eds.; Chapman &Hall: London, 1997; Chapter 8, p 363.

2. Bucknall, C. B. In Toughened Plastics; Applied Sci-ence: London, 1977, pp 182–210.

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7. Argon, A. S.; Cohen, R. E. J Mater Sci 1994, 176,79.

8. Donald, A. M.; Kramer, E. J. J Mater Sci 1982, 17,2351.

9. Donald, A. M.; Kramer, E. J. J Appl Polym Sci1982, 27, 3729.

10. Crawford, E.; Lesser, A. J. Polymer 2000, 41, 5865.11. Ramaswamy, S.; Lesser, A. J. Polymer, 2002, 43,

3743–3752.12. Gurson, A. L. J Eng Mater Technol Trans ASME

1977, 99, 2.13. Sultan, J.; McGarry, F. J Polym Eng Sci 1973, 13,

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Chem Soc Div Polym Chem) 1969, 10, 1117.15. Lazzeri, A.; Bucknall, C. B. J Mater Sci 1993, 28,

6799.16. Jones, N. A.; Lesser, A. J. J Polym Sci Part B:

Polym Phys 1998, 36, 2751.17. Groeninckx, G.; Chandra, S.; Berghmans, H.;

Smets, G. In Advances in Chemistry Series 176;Cooper, S. L.; Estes, G. M., Eds.; American Chem-ical Society: Washington, DC, 1979; p 337.

18. Wilfong, D. L.; Hiltner, A.; Baer, E. In Advances inChemistry Series 211; Paul, D. R.; Sperling, L. H.,Eds.; American Chemical Society: Washington,DC, 1986; p 273.

19. Kachanov, M. In Advances in Applied Mechanics;Hutchinson, J.; Wu, T., Eds.; Academic: New York,1993; p 259.

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