We study experimentally the slow growth of a single crack in polycarbonate
films submitted to uniaxial and constant imposed stress. The specificity of
fracture in polycarbonate films is the appearance of flame shaped macroscopic
process zones at the tips of the crack. Supported by an experimental study of
the mechanical properties of polycarbonate films, an analysis of the stress
dependence of the mean ratio between the process zone and crack lengths, during
the crack growth, show a quantitative agreement with the Dugdale-Barenblatt
model of the plastic process zone. We find that the fracture growth curves obey
strong scaling properties that lead to a well defined growth master curve

Barenblatt G. I.

L Vanel

P. P Cortet

S Ciliberto

S Santucci

Santucci S.

Stojimirovic A.

Zhurkov S. N.

English

arXiv

We study experimentally the slow growth of a single crack in
polycarbonate films submitted to uniaxial and constant imposed
stress. The specificity of fracture in polycarbonate films is the
appearance of flame-shaped macroscopic process zones at the tips
of the crack. Supported by an experimental study of the mechanical
properties of polycarbonate films, an analysis of the stress
dependence of the mean ratio between the process zone and crack
lengths, during the crack growth, shows a quantitative agreement
with the Dugdale-Barenblatt model of the plastic process zone. We
find that the fracture growth curves obey strong scaling
properties that lead to a well-defined growth master curve

P. P. Cortet

S. Santucci

L. Vanel

S. Ciliberto

EDP Sciences OAI-PMH repository (1.2.0)

Slow crack growth in polycarbonate films

International audienceWe study experimentally the slow growth of a single crack in polycarbonate films submitted to uniaxial and constant imposed stress. The specificity of fracture in polycarbonate films is the appearance of flame shaped macroscopic process zones at the tips of the crack. Supported by an experimental study of the mechanical properties of polycarbonate films, an analysis of the stress dependence of the mean ratio between the process zone and crack lengths, during the crack growth, show a quantitative agreement with the Dugdale-Barenblatt model of the plastic process zone. We find that the fracture growth curves obey strong scaling properties that lead to a well defined growth master curve

Cortet, Pierre-Philippe

Santucci, Stéphane

Vanel, Loïc

Ciliberto, Sergio

HAL-ENS-LYON

Slow crack growth in polycarbonate filmsPierre-Philippe Cortet, Ste´phane Santucci, Lo¨ıc Vanel, Sergio CilibertoTo cite this version:Pierre-Philippe Cortet, Ste´phane Santucci, Lo¨ıc Vanel, Sergio Ciliberto. Slow crack growthin polycarbonate films. EPL, European Physical Society/EDP Sciences/Societa` Italiana diFisica/IOP Publishing, 2005, 71 (2), pp.242. <10.1209/epl/i2005-10077-3>. <ensl-00156838>HAL Id: ensl-00156838https://hal-ens-lyon.archives-ouvertes.fr/ensl-00156838Submitted on 22 Jun 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ensl-00156838, version 1 - 22 Jun 2007Europhysics Letters PREPRINTSlow crack growth in polycarbonate filmsP.P. Cortet, S. Santucci, L. Vanel and S. CilibertoLaboratoire de physique, CNRS UMR 5672, Ecole Normale Supe´rieure de Lyon, 46 alle´ed’Italie, 69364 Lyon Cedex 07, FrancePACS. 62.20.Mk – Fatigue, brittleness, fracture and cracks.PACS. 62.20.Fe – Deformation and plasticity (including yield, ductility, and superplasticity).PACS. 05.70.Fh – Phase transitions: general studies.Abstract. – We study experimentally the slow growth of a single crack in polycarbonate filmssubmitted to uniaxial and constant imposed stress. The specificity of fracture in polycarbonatefilms is the appearance of flame shaped macroscopic process zones at the tips of the crack.Supported by an experimental study of the mechanical properties of polycarbonate films, ananalysis of the stress dependence of the mean ratio between the process zone and crack lengths,during the crack growth, show a quantitative agreement with the Dugdale-Barenblatt model ofthe plastic process zone. We find that the fracture growth curves obey strong scaling propertiesthat lead to a well defined growth master curve.Introduction. – Solids with a single crack usually break at a critical rupture stress.However, experiments [1] show that a given solid submitted to a subcritical stress breaks aftera certain amount of time. Therefore, understanding the mechanisms of subcritical macroscopicfracture growth in solids has become an important goal of fracture physics in order to improvethe resistance of structures to failure. Recent experimental works [2, 3] have shown that sub-critical crack growth in paper can be successfully described by a thermal activation modelfor elastic brittle media. In this letter, we present an experimental study of slow growth of asingle crack in a polycarbonate film which is a highly non-brittle material. We observe thata large flame shaped area, the process zone, forms ahead of each crack tip. We find that thedependence of the process zone length with the applied stress, during the crack growth, isin reasonable quantitative agreement with the Dugdale-Barenblatt model. In that respect,polycarbonate appears to be a good model material to understand the mechanisms of crackgrowth in non-elastic media. We show that the crack growth curve obeys remarkable scalingproperties that are not theoretically understood yet.The experimental setup and the experiment. – The experiment consists in loading 125µmthick isotropic polycarbonate films (height 21cm, length 24cm) with uniaxial and constantimposed stress σ. The polycarbonate films used are made of Bayer Makrofol r© and presentthe properties of bulk material. An initial crack of length ℓi is initiated at the center of eachpolycarbonate sample using calibrated blades of different lengths (from 0.5cm to 3cm). Then,a constant force Fc is applied to the film perpendicularly to the crack direction, so that we getc© EDP Sciences2 EUROPHYSICS LETTERS0 0.02 0.04 0.06 0.08 0.1 0.120123456 x 107strain stress  (N.m−2 )σplat σp Fig. 1 – Stress as a function of the strain for a 125µm thick polycarbonate film loaded with a46.25µm.s−1 strain rate.a mode 1 crack opening type. For more details about the setup see [2, 3]. A high resolutionand high speed camera (Photron Ultima 1024) is used to follow the crack growth.We follow the growth of the single linear fracture and its process zones, under constantapplied stress σ, till the total rupture of the sample. The applied stress σ is chosen suchthat crack growth is slow, i.e., smaller than the critical one, σc, above which a fast crackpropagation occurs.Mechanical properties of polycarbonate films. – In order to characterize the materialin which the crack will grow, we performed some preliminary experiments on polycarbon-ate films, without crack, submitted to uniaxial deformation at a constant deformation rate(46.25µm.s−1). A typical experimental stress-strain plot is presented in figure 1. The polymerfilms show the classical behavior of a plastic material with a quasi-elastic behavior for smallstrains followed by a bell profile and a plateau. The different characteristic values observedon this graph are in good agreement with the ones measured by Lu and Ravi-Chandar in bulkpolycarbonate [4]. We measured the experimental values of the maximum reachable stress,σp = 5.2 107N.m−2, the plastic plateau stress σplat = 4.45 107N.m−2 and the Young modulusfor small strains Y = 194 107N.m−2.The flame shaped process zone ahead of the crack tip. – In each experiment, during theloading phase of the film, a macroscopic flame shaped process zone appears at each tip ofthe crack and grows with the applied stress (cf. figures 2 and 3). This zone was previouslynoticed by Donald and Kramer [5]. In the late loading stage, the fracture may also growa little. Consequently, the real experimental initial condition, obtained when the constantstress σ is reached, is not exactly ℓi. During the imposed stress stage, the process zone andthe fracture are both growing till the final breakdown of the sample in a way that the fracturenever catches up the process zone tip.Inside the process zone, the film is subjected to a thinning which brings its thickness from125µm to about 70µm (measured on post-mortem samples). It is worth noticing that onmicroscopic images (cf. figure 3) one can see in the process zone the presence of striationsquasi-parallel to the fracture front with a wave length of about 22µm. These striations seemto be thickness oscillations of the film. It is still an open question whether this process zone,P.P.Cortet et al.: Slow crack growth in polycarbonate films 3Process zone ℓcrackℓpzFig. 2 – Image of a crack in a polycarbonate film with its macroscopic process zone at each tip; ℓcrackis the crack length and ℓpz is the process zone length from tip to tip.once it has been formed, continues to behave as a visco-plastic zone or as an elastic zonewith an effective macroscopic Young modulus different from the one of the rest of the film.Understanding the mechanical nature of the process zone is the key to reach a model offracture growth in polycarbonate.Dependence of the process zone length on the fracture length. – The experimental relationbetween the process zone length (defined on figure 2), ℓpz, and the crack length, ℓcrack, duringthe crack growth process, is plotted on figure 4a. It seems that this relation is not statisticalbecause all curves for identical experimental conditions are almost identical. The experimentalstress ranges for different ℓi do not overlap, so that it is not possible to plot the curves forthe same ℓi and very different σ. However, it is expected that the ℓpz against ℓcrack curvedepends on the applied stress only.It is important to notice that during an experiment the fracture length grows on timescales much larger than the time scale of relaxation of the process zone shape. This has beenchecked by suddenly forcing the fracture length to grow using a blade during an imposedstress experiment. This action induces a quasi-instantaneous growth of the process zone toits new equilibrium state. So, it is appropriate to think that during the fracture experiments,the process zone is in a quasi-equilibrium state for each given crack length (below the criticalone) and applied load. Therefore, we can analyze these data using an equilibrium model forthe process zone.The relation between ℓpz and ℓcrack has first been theoretically described by the Dugdale-Barenblatt model [6, 7]. Dugdale considers the process zone as an isotropic plastic materialin which the stress is uniformly equal to the plastic yield stress σy. Assuming the non-divergence of the stress at the tip of the process zone, he concludes to a zero stress intensity 0.5 cm 540µm Fig. 3 – On the left: process zone at the tip of a growing crack in a polycarbonate film; on the right:microscopic image showing striations quasi-parallel to the fracture direction in the process zone.4 EUROPHYSICS LETTERSa) b)0 1 2 3 4 5 60123456789ℓcrack(cm)ℓ pz(cm)0 2 4 6 800.511.522.5ℓcrack(cm)ℓ pz/ℓ crackℓi = 3cm Fc = 750Nℓi = 2cm Fc = 850Nℓi = 1.5cm Fc = 900Nℓi = 0.5cm Fc = 1000NFig. 4 – On the left: process zone length as a function of the crack length for three experimentsperformed in the same experimental conditions (ℓi = 1.5cm and Fc = 900N); on the right: ℓpz/ℓcrackratio for four experiments performed with different experimental conditions as a function of the cracklength.factor (SIF) [8] at the tip of the process zone: Ktot = Kel(σ; ℓpz) + K˜(σy; ℓ, ℓpz) = 0, whereKel is the traditional SIF at the tip of a fracture of length ℓpz in an elastic film submittedto σ at its border and K˜ the SIF for a film fractured on a length ℓpz and submitted only toσy on the fracture lips between ℓcrack and ℓpz. Using analytical expressions for these SIFs inthe case of an infinite elastic sample, Dugdale finds a proportionnality dependence of ℓpz onℓcrack: ℓpz/ℓcrack = 1/ cos(π2σ/σy). This model was successfully compared with experimentaldata in metals [6]. However, for many polymers, it does not predict the correct order ofmagnitude for the process zone size if one uses the yield stress constant of the material as theDugdale stress constant [9–11]. On figure 4b, it appears clearly that the ℓpz/ℓcrack ratio isnot constant during a constant stress experiment, so that the ℓpz against ℓcrack relation is nota proportionality relation.To make a more quantitative comparison with the model, we represent, on figure 5, theℓpz/ℓcrack ratio as a function of the applied stress σ for four experiments performed withdifferent applied stresses. The dispersion of the data for each experiment corresponds to theimportant discrepancy from a proportionality law between the two lengths. For each of thefour experiments, the mean value of the ℓpz/ℓcrack ratio is superimposed on the data. Fittingthese mean values against σ with the Dugdale-Barenblatt law, ℓpz/ℓcrack = 1/ cos(π2σ/σy),leads to an estimate of the Dugdale stress constant: σy = 5.2 107N.m−2. It is striking howthe value found here for σy is close to the plastic peak stress σp of the rheology curve of figure1. So, even if the Dugdale-Barenblatt model does not predict the non-linear dependence ofℓpz on ℓcrack, it predicts the correct order of magnitude for the ratio ℓpz/ℓcrack, as a functionof σ, using a plastic stress constant σy that corresponds very well to the maximum reachablestress σp obtained from mechanical tests (cf. figure 1). We insist that it is not usually the casefor polymers. We believe that the reason it works reasonably well is that, for polycarbonate,plasticity remains confined close to the crack tip while for many other polymers plasticity ismuch more diffuse.Regarding the non-proportionality of the curves in figure 4a, finite size corrections to theDugdale-Barenblatt model lead to an opposite curvature [11] to the one observed experimen-P.P.Cortet et al.: Slow crack growth in polycarbonate films 50 1 2 3 411.522.53Experimental data Mean value σ ( 107N.m−2)ℓ pz/ℓ crack1/ cos(pi2σ5.2 107)Fig. 5 – ℓpz/ℓcrack ratio as a function of the applied stress σ for four experiments performed withdifferent experimental conditions and the fit of its mean values (the larger points correspond to themean value of the data for each σ) by the Dugdale-Barenblatt law.tally. An explanation for the experimental curvature, at the beginning or at the end of theexperiment, could be dynamical effects that bring the process zone out of equilibrium. How-ever, for the main part of the data, crack growth is slow and process zone equilibrium modelsneed to be improved to explain the observed non-linearity.Breaking time and crack growth curves. – The measured breaking times for a givenexperimental condition (ℓi, Fc) are statistical. Actually, we commonly observe a factor fivebetween the smallest and the largest breaking times.Typical growth curves of the fracture and process zone are shown on figure 6a. Both curvesshow a quite similar smooth shape. We observe, once the loading phase is finished, at thebeginning of the constant stress phase, large velocities of the fracture and the process zonetips which decrease till reaching quasi-constant values before increasing back dramaticallytill the final rupture. So, the first part of the growth curves in the constant stress phasecorresponds to a deceleration of the fracture tips. We think this phenomenon is caused bydelays in deformation accumulated during the loading phase. In other words, this decelerationcorresponds to the time needed by the viscous effects to relax all those delays to quasi-equilibrium. The final acceleration corresponds to a transition to fast crack dynamics.We can point out some important scaling properties of these growth curves. Actually, byredefining the origin of time and length using the coordinates of the inflexion point (ℓx andtx) of the fracture curve (cf. figure 6b) and rescaling the time by the time left to break τ(τ = trupture − tx), all the growth curves corresponding to identical experimental conditionsfall on a master curve (cf. figure 7a). By introducing a rescaling factor on the lengths, we canas well make all growth curves, for different experimental conditions, fall on the same mastercurve. This rescaling property implies that the dynamics of the final part (the part after theinflexion point) of the crack growth follows:t = τ(σ, ℓi, realization) g(ℓ− ℓxλ)(1)where g is the functional form of the master curve. λ and ℓx are a priori functions of σ andℓi. In this formula, all the statistics is included in τ . No conclusive study of λ(ℓi, σ) has been6 EUROPHYSICS LETTERSa) b)0 2 4 6 805001000150020002500fracture processzone   Loadingphase  constantstress  phase   i x tx trupture t(s)ℓℓlength(cm)−1 0 1 2 3−2000−1000010002000300040005000ℓcrack − ℓx(cm)t(s)Fig. 6 – On the left: time as a function of both the crack and process zone lengths for an imposedstress experiment (ℓi = 1.5cm, Fc = 900N); on the right: time as a function of ℓcrack − ℓx for fourexperiments performed in the same experimental conditions (ℓi = 1.5cm, Fc = 900N) with a changein time and length origins.possible due to the limited ranges of ℓi and Fc experimentally accessible.In searching for an analytical expression for g, we first tried to use previously establishedmodels. Santucci et al. [2, 12] developed a thermal activation model that can explain sub-critical single crack growth in elastic brittle films. This model that has been successfullyfaced with experimental data for slow crack growth in paper [2, 3], predicts an exponentialgrowth law: t = τ[1− exp(− ℓ−ℓiξ)]. On figure 7b, we can see that this thermal activation lawa) b)0 0.5 1 1.5 2 2.5 3 3.500.20.40.60.81ℓcrack − ℓx(cm)t/τ0 0.5 1 1.5 2 2.5 3 3.500.20.40.60.81Experimental data                        Chudnovsky modelℓcrack − ℓx(cm)t/τ = 1− exp[−1.3 (ℓcrack − ℓx)]t/τ = erf[1.07 (ℓcrack − ℓx)]t/τFig. 7 – On the left: time rescaled by the time left to rupture as a function of ℓcrack − ℓx forfour experiments performed in the same experimental conditions (ℓi = 1.5cm, Fc = 900N, sameexperiments as for figure 6b) with a change in time and length origins; on the right: time rescaledby the time left to rupture as a function of ℓcrack − ℓx for an imposed stress experiment (ℓi = 1.5cm,Fc = 900N) and fits by the Chudnovsky model, the thermal activation model and an error functionadjusting the initial velocity.P.P.Cortet et al.: Slow crack growth in polycarbonate films 7(dashed line) does not work with experimental data for polycarbonate. Obviously, it is not asurprise since polycarbonate is far from being simply an elastic medium at least because ofthe macroscopic plastic process zone. There are very few models giving an equation of motionfor crack growth in viscoplastic media. A tentative model is the one of Chudnovsky [13] thathas been developed first for stick-slip crack growth description in polyethylene films. Thismodel introduces an aging process to describe the creep of the process zone. By using theDugdale relation for process zone size, Chudnovsky reached a continuous expression for thecrack velocity as a function of the elastic stress intensity factor. The curve obtained by anumerical integration of this equation is shown on figure 7b (dotted line). It also does notfit the experimental data well. Without any theoretical arguments, it appears that the errorfunction is a quite good guess candidate for an analytical description of g as we can see onthe data fit of figure 7b (solid line).Conclusion. – The experimental study of the process zone of polycarbonate film cracksallowed us to analyze the relevance of the Dugdale-Barenblatt model. We have shown that themean ratio between the process zone and crack lengths, as a function of the stress, agrees withthis model. We have also found a characteristic growth curve for crack in polycarbonate filmswhich cannot be explained by recent models based either on thermally activated rupture oron surface energy aging due to viscoplastic flow. It is important to introduce a more precisedescription of the rupture mechanisms in the process zone in order to model crack growthin polycarbonate. One direction is to consider the general framework of viscoplastic ruptureintroduced by Kaminskii [14, 15]. Work is in progress to test the applicability of this modelto crack growth in polycarbonate.REFERENCES[1] Zhurkov S. N., Int. J. Frac. Mech., 1 (1965) 311.[2] Santucci S., Vanel L., Ciliberto S., Phys. Rev. Lett., 93 (2004) 095505.[3] Santucci S., Cortet P. P., Vanel L., Ciliberto S., to appear.[4] Lu J., Ravi-Chandar K., Int. J. Solids. Structures, 36 (1999) 391.[5] Donald A. M., Kramer E. J., J. Mater. Sci., 16 (1981) 2967.[6] Dugdale D. S., J. Mech. Phys. Solids, 8 (1960) 100.[7] Barenblatt G. I., Adv. Appl. Mech., 7 (1962) 55.[8] At a distance r from a crack tip, the local stress behaves as K/√r, where K is the stress intensityfactor.[9] Stojimirovic A., Kadota K., Chudnovsky A., J. Appl. Polym. Sci., 46 (1992) 1051.[10] Haddaoui N., Chudnovsky A., Moet A., Polymer, 27 (1986) 1377.[11] Stojimirovic A., Chudnovsky A., Int. J. Frac., 57 (1992) 281.[12] Santucci S. et al., Europhys. Lett., 62 (2003) 320.[13] Chudnovsky A., Shulkin Y., Int. J. Frac., 97 (1999) 83.[14] Kaminskii A. A., Sov. Appl. Mech., 15 (1979) 1078.[15] Kaminskii A. A., Int. Appl. Mech., 40 (2004) 829.

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