Helbig, M.

Seelig, Th.

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Institute of MechanicsMultiscale modeling of failure in ABS materialsMartin Helbig, Thomas Seelig15. International Conference onDeformation, Yield and Fracture of PolymersKerkrade, April 2012Institute of MechanicsKaiserstr. 12D-76131 KarlsruheTel.: +49 (0) 721/ 608-42071Fax: +49 (0) 721/608-47990E-Mail: info@ifm.kit.eduwww.ifm.uni-karlsruhe.deMultiscale modeling of failure in ABS materialsMartin Helbig, Thomas SeeligInstitute of Mechanics, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, GermanySummary: A continuum mechanical description of crazing at different lengthscales is introduced. The model relies on the idea that distributed crazing con-tributes to the overall inelastic strain in a direction determined by maximumprincipal stress. This concept is used in a homogenized model for ABS whichexplicitely accounts for several microstructural parameters. Numerical simula-tions are used to investigate material specific aspects in the behaviour of ABS.IntroductionABS (acrylonitrile-butadiene-styrene), consists of an amorphous glassy matrix (SAN) and rub-ber (butadiene) particles. The enhanced fracture toughness and ductility of ABS, compared toneat SAN, relies on microscopic deformation and damage mechanisms, such as void growth,shear yielding and crazing, see e.g. [1],[2],[9]. Despite numerous experimental studies, manydetails of these micro-mechanisms, their individual contribution to the overall toughness andtheir dependence on micro-structural parameters (e.g. rubber particle size, void-volume fractionand distribution) are still not well understood. Theoretical models and numerical simulationsmay provide a deeper understanding of these issues. In the present work a constitutive modelingapproach for the effect of crazing at different length scales is suggested and analyzed.Continuum modeling of crazingCrazes are localized zones of fibrillated polymer material which form in the direction normalto the maximum principal stress and are able to transfer stress. Under continued loading theirgrowth proceeds until a critical craze thickness, typically of the order of a micron, is reached.Then rupture of the fibrils takes place and the craze turns into a micro-crack. Individual crazeson the level of a homogeneous glassy polymer have been modeled either as discrete cohesivezones, e.g. [7],[10], or using special continuum finite elements as in [8]. In both approachesthe craze (or crack) is constrained by the finite element mesh. As an alternative, a constitutivemodel is presented here which accounts for the essential features of crazing on the continuumlevel.Using the additive decomposition of the rate of deformation tensor D = De +Dc into anelastic and an inelastic part, a hypoelastic relation for the Jaumann rate of the Cauchy stress canbe written as∇σ= E : (D −Dc) where E is the isotropic forth order elasticity tensor. In thepresent work, crazing is considered as the only source of inelasticity (“craze yielding”). Withthe direction n of maximum principal stress at the onset of crazing, the inelastic part of the rateof deformation tensor is – similar to [3] – is governed by the flow ruleDc = ε̇cn⊗ n , ε̇c = ε̇0 exp(AT(σn − σc)), σn = n · σ · n . (1)Here, ε̇c denotes the equivalent visco-plastic strain rate, A and ε̇0 are material parameters, andT is the temperature. Crazing is driven by the resolved normal stress on the craze σn which hasto overcome the craze yield strength σc(εc). The latter is taken a function of the accumulatedinelastic strain, featuring softening upon craze initiation and subsequent rehardening, analogousto the cohesive zone model employed in [7].Analysis of a single crazeThe following example merely serves to illustrate how crazes in the present approach can freelyform in arbitrary directions determined by the loading state, independent of the finite elementmesh. A single craze which forms at a void in a cubic unit cell therefore is considered. Figure 1shows half of the unit cell and indicates by the regions in bright grey where crazing has occuredunder the different types of loading. Figure 1c provides an example of crack formation from acraze zone, modeled by the elimination of finite elements once the local failure criterion of acritical value of inelastic strain is met.a) b) c)Figure 1: Accumulated inelastic (crazing) strain in a cubic unit cell with a spherical void:a) tension parallel to cell edges, b) and c) different amounts of shearHomogenized model for distributed crazing in rubber-toughened materialsExperimental studies, e.g. [6], have shown that inelastic deformation of ABS by multiple craz-ing is accompanied by the formation of many band-like damage zones (as sketched in Fig. 2,left). Similar to individual crazes, yet on a larger length scale and comprising several rubber par-ticles, these zones are oriented normal to the principal loading direction. This suggests that thekinematics of inelastic deformation in (1) is also suitable for the description on the homogenizedcontinuum level of ABS. A constitutive model for ABS that explicitely accounts for the size andvolume fraction of rubber particles via appropriate effective properties is set up in the following.Figure 2: Sketch of localized craze zones in voided polymer matrix (left), single void unit cellwith craze zone used for micromechanical modeling (right)The rubber particles are assumed to cavitate prior to the occurrence of crazing and are consid-ered as voids in the present study. Moreover, despite the oriented nature of the damage zones,the overall elastic behavior of ABS is here for simplicity taken isotropic and represented byeffective elastic constants via analytical functions of the porosity. A cubic single void unit cell(Fig. 2, right) is employed to establish proper scaling relations between the inelastic deforma-tion behavior and the microstructure of ABS. Key microstructural parameters, thereby, are therubber content (porosity) f , the rubber particle size (radius) r, and the maximum craze width∆ccrit at which local failure takes place. From the porosity f =43πr3/b3 and the assump-tion that at each void a single craze forms (see Fig. 2, right), the average spacing of crazes isb(r, f) = r (4π/3f)1/3. With the current craze width ∆c, the overall inelastic strain rate due todistributed crazing isε̇c =∆̇cb+ ∆c=∆̇c∆ccrit(r∆ccrit(4π3f)1/3+∆c∆ccrit)−1, (2)where for normalization purposes the maximum craze width ∆ccrit is introduced. On the otherhand, the driving stress for craze growth (as a function of macroscopic loading Σ and porosity)is obtained from unit cell equilibrium considerations asσn = n ·Σ · n(1− π(3f4π)2/3)−1. (3)The (normalized) craze widening rate ∆̇c/∆ccrit in (2) is, analogous to (1), governed by an expo-nential dependence on the driving stress σn(f) and the craze yield strength σc(∆c/∆ccrit). Theinelastic part of the rate of deformation tensor on the macroscopic level of homogenized ABSDc = ε̇c(f, r/∆ccrit,∆c/∆ccrit,Σ)n⊗ n (4)hence explicitely depends on the rubber content, the rubber particle size and the amount ofcrazing. Failure takes place when a critical craze width ∆c = ∆ccrit is reached.The effect of an increasing amount of distributed crazing on the overall stiffness of ABS isalso accounted for in the model. Therefore, the effective elastic constants are (besides theirdependence on f ) taken to decrease monotonically with increasing ∆c/b.Calibration of the macroscopic modelIn order to calibrate the material model, uniaxial tensile tests were performed on a commercialABS material with unknown composition. The rubber content was estimated to f≈0.2 from themeasured elastic stiffness and the known Young’s modulus of the SAN matrix. Then the yieldstrength relation σc(∆c/∆ccrit) was fitted so that the overall response of the model for f ≈ 0.2agrees with the experimental stress-strain curve (Fig. 3a). Due to the micromechanically basedexplicit dependence on f , the thus calibrated model captures the influence of the rubber contenton the behavior of ABS in a reasonable manner. The predicted decrease of the yield strengthand the increase of failure strain with increasing rubber content in Fig. 3a is well known fromthe literatur, e.g. [4].The strain rate dependence in the model is calibrated by adjusting the reference strain rateε̇0 in (1) to experimental data. The comparison of stress-strain curves over four decades of thestrain rate in Fig. 3b shows a good agreement between experimental data and model response.a) b)Figure 3: Comparison of experimental true stress vs. log. strain curves with model response:a) effect of rubber content (at ε̇ = 10−3s−1), b) effect of strain rate (at f=0.2)Tensile tests under loading and unloading conditions show a decreasing slope with increasinginelastic deformation, hence a damage evolution takes place (Fig. 4). By adjusting the effectiveelastic stiffness as a function of craze damage ∆c/b, this behavior is well described by the modelas illustrated in Fig. 4. Not yet captured by the model is the hysteresis in the experiment (dashedline in Fig. 4).Figure 4: Comparison of experimental true stress vs. log. strain curve and model response forloading and unloading testPlastic zone in notched specimenThe enhanced fracture toughness of ABS corresponds to the formation of a large stress-whitened(‘plastic’) zone at a crack tip or notch (Fig. 5a). In a previous study [5], void growth was con-sidered as the dominant damage mechanism in ABS which led to an unrealistic shape of theplastic zone. As an alternative, we here apply the above presented model for distributed crazingto the situation of a notched tensile specimen. Finite element simulations are performed under2D plane strain conditions. Figure 5b shows the computed plastic zone which displays the char-acteristic elongated shape observed in real ABS (Fig. 5a).a) b)Figure 5: Plastic zone at notch in SENT specimen: a) experiment [9], b) simulationAcknowledgmentFinancial support of this work by the German Science Foundation (DFG) under grant no. SE872/5-2 is gratefully acknowledged. We would also like to thank Prof. Arild H. Clausen forsupport with the experiments performed at SIMLab (NTNU Trondheim) and the DAAD forfunding the visit to SIMLab.References[1] Bernal, C.R., Frontini, P.M., Sforza, M., Bibbo, M.A., J. Appl. Pol. Sci., 1995, 58, 1-10[2] Bucknall, C.B., Toughened Plastics. Applied Science, London, 1977[3] Gearing, B.P., Anand, L., Int. J. Solids Structures 2004, 41, 3125-3150[4] Ishikawa, M., Polymer, 1995, 36, 2203-2210[5] Pijnenburg, K.G.W., Seelig, Th., Van der Giessen, E., Eur. J. Mech. A/Solids, 2005, 24,740-756[6] Ramaswamy, S., Lesser, A.J., Polymer 2002, 43, 3743-3752[7] Seelig, Th., Van der Giessen, E., Comp. Mat. Sci. 2009, 45, 725-728[8] Socrate, S., Boyce, M.C., Lazzeri, A., Mech. Mater., 2001, 33, 155-175[9] Steenbrink, A.C., Shaker PhD thesis, TU Delft, 1998[10] Tijssens, M.G.A., Van der Giessen, E., Sluys, L.J., Mech. Mater. 2000, 32, 19-35

Multiscale modeling of failure in ABS materials

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Multiscale modeling of failure in ABS materials

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