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Creep of a fracture line in paper peeling J. Koivisto, J. Rosti, and M.J. Alava Helsinki University of Technology, Laboratory of Physics, P.O.Box 1100, FIN-02015 HUT, Finland 7 Theslowmotionofacracklineisstudiedviaanexperimentinwhichsheetsofpaperaresplitinto 0 two halves in a “peel-in-nip” (PIN) geometry under a constant load, in creep. The velocity-force 0 relation is exponential. The dynamics of the fracture line exhibits intermittency, or avalanches, 2 which are studied using acoustic emission. The energy statistics is a power-law, with theexponent n β ∼ 1.8±0.1. Both the waiting times between subsequent events and the displacement of the a fracture line imply complicated stick-slip dynamics. We discuss the correspondence to tensile PIN J testsandothersimilarexperimentsonin-planefractureandthetheoryofcreepforelasticmanifolds. 3 2 PACSnumbers: 62.20.Mk,05.70.Ln,81.40Lm,62.20.Fe ] h The deformation and fracture of materials is a fasci- ics a “depinning transition”. Close to the critical point c e nating topic as one canexplorethe physics evenwithout Kc the line geometryis a self-affinefractalwitha rough- m sophisticated experiments [1]. A piece of paper suffices ness exponent ζ. This is an example of a wide class of - to give ample evidence for the presence of phenomena similar problems, which range from fire fronts in com- t a that need a statistical description. One can tear a sheet bustion to flow fronts in porous media to domain walls t or crumble it, to observe that the response is “intermit- in magnets. The planar crack [11] or the contact line s . tent” and not simply “smooth”. [2, 3, 4, 5]. on a substrate [12] problem has been studied both theo- t a The physics of fracture in a material as paper is gov- retically via renormalization group calculations and nu- m ernedbytwobasicingredients. Thestructureandthemi- mericalsimulations,andviaexperiments. Thereisadis- - croscopic material properties are inhomogeneous, while crepancybetweenthese,inthattheroughnessexponents d n the stress fields follow the laws of elasticity. Typicalsta- are ζtheory ∼ 0.39 and ζexpt ∼ 0.6 [13, 14]. In three- o tistical signatures that show features that emerge from dimensional experiments the in-plane roughness has re- c their interaction are the acoustic emission during a de- cently shown signatures of ζexpt ∼ ζtheory [15]. Imaging [ formationtest and the post-failure properties offracture experiments have shown the existence of avalanches, of 1 surfaces. These are commonly found to be described by localizedintermittentadvancesofthecrackfrontwithan v power-laws, as regards the energies of acoustic events avalanche size s distribution P(s)∼s−1.6···−1.7 [13] 2 (“earthquakes”),theintervalsbetweensubsequentevents Ourworkconsidersthe dynamics ofsuchacrackfront 6 (“Omori’slaw”)[4,6],andthesameistrueforthegeome- during creep, which is done by peeling paper sheets in 5 1 tryofcrackswhereampleevidencepointstowardsaself- a geometry illustrated in Fig. 1 (see also [16]). The 0 affine fractal scaling of the surface fluctuations around creep of elastic lines (or manifolds or domain walls) 7 the mean, which is found in a variety of cases (loading is important since when Keff ≤ Kc thermal fluctua- 0 conditions, materials and so forth) [7, 8, 9]. tions take over for T > 0 [17, 18, 19]. The fluctua- / t The most simple case of fracture - which we also will tions nucleate “avalanches” similarly to what happens a m studyhere-isthepassageofacrack linethroughasam- inzero-temperaturedepinning inthe vicinityofKc. The ple,whenitsmovementisconstrainedonaplane. Inthis avalanches induce a finite velocity vcreep. The advance- - d case,themathematicaldescriptionisgivenbyacrackpo- ment of a crack front might be fundamentally different n sition h(x,t), where x is along the average projection of fromsaya domainwallor a contactline, whichcanfluc- o the crack. The average motion is described by h¯ = vt tuate back and forth around a metastable state: it is c with v the crack velocity. In the situation at hand one easy to see that the crack may in many cases only grow : v has three ingredients: a disordered environment which forward. i X poses obstacles to the line motion, a driving force Kext The form of vcreep(Keff,T) depends on the “energy r (stress intensity factor in fracture mechanics language), landscape”since the currentunderstanding is that creep a and the self-coupling of the fluctuations in h through a takesplacevianucleationeventsoverenergybarriers[17]. long-range elastic kernel [10], which is expected to scale These barriers are related to the roughness exponent ζ as 1/x. and to its origins. The important physics is summarized To describe the line’s physics one needs the language with the creep formula of statistical mechanics. One finds a phase diagram for v ∼exp−(C/mµ) (1) creep v(K ): an immobile crack begins to move at a critical ext value K of K such that for K > K v > 0. This whichstatesthatthecreepvelocityisrelatedtothedriv- c eff c transition between a “mobile line” and a pinned line is ing force, m (as for mass, see Fig. 1 again),. There is a commonly called in non-equilibrium statistical mechan- creep exponent µ, whichdepends onthe interactionsand 2 dimensionofthemovingobject(aline). Oneexpectsthe ment, and the typical pair of values for the environment scaling is 49 rH and 27 oC (though we also have data for other combinations). α−2+2ζ µ=θ/ν = (2) Our main result is given in Figure 4, where we show α−ζ the v(m) vs. 1/m characteristics. There are four main in d dimensions. (d = 1 for a line, d = 2 for a domain datasets depicted. These differ slightly in the typical wallinsayabulkmagnet). θ,ν,andζ denotetheenergy temperature and humidity (for set 2 44 % rH instead of fluctuation, correlationlength, and roughness exponents about 49 % rH for the others). These all imply an ex- relevantfortheproblem,andexponentrelationsareusu- ponential behavior, and by fits to the data sets we can ally postulated between these three. They exponents all infer a creep exponent ν =1.0±0.1. This lends itself to depend on α, the k-space decay exponent of the elas- twodifferentinterpretations: eitherthecreepisinthe1d tic kernel. For long range elasticity, one would assume random field (RF) domain wall universality class, since α = 1 whereas for the so-called local case α = 2 is ex- with α = 2 ν = 1 implies ζ = 1, the roughness expo- pected. Expressed as above, the question of the value nent of the RF universality class at very small external of µ boils down to the values of ζ and α - what is the fields/forces. This assumes local elasticity being domi- effective roughness at hand, and what is the elasticity of nant. If we would then assume that ζ takes the value the line like? ζ ∼0.6...0.7,of the avalancheregime,this wouldimply The fundamental formula of Eq. (2) has been con- screened interactions with α ∼ 1.4. The thirdr possibil- firmed in the particular case of 1+1-dimensional do- ity is to use non-local line elasticity, with α = 1. Then main walls [20], and further experimental studies exist Eq. 2 indicates that ζ ∼ 1/3. This is exactly the equi- [21]. Our main results on fracture line creep concernthe librium exponent of lines with long-range elasticity [26]. velocity-force-relationv(m),andonthepictureofthedy- The velocity in the case of “paper” is influenced by the namicsthatensues,inparticularfromAcousticEmission humidity: this is clearin our case. Meanwhile,note that (AE)dataoftheavalanchesandtheirdynamics. Wefind the temperature variation is insignificant here. anexponentialv(m)anddiscussits interpretations. The The avalanche behavior is illustrated by Fig. 4a, by dynamics shows signatures of intriguing but weak corre- the way of the AE event energy distribution P(E). We lations, andwe discuss the similarities and differences to presenttwokindsofavalanchedata: integratedover1ms the tensile case. windows(Ei whereiisanintegertimeindexforCD)and In Figure 1 we show the apparatus [22]. The failure extracted from single avalanches. The former obviously line can be located along ridge, in center of the the Y- sums over all the avalanches during the 1 ms duration shaped construction formed by the unpeeled part of the (if more than one are present). The data agrees rather sheet (below) and the two parts separated by the ad- wellregardlessofthemassmandtheductility(CD,MD) vancing line. Diagnostics consist of an Omron Z4D-F04 with the scalingP(E)∼E−β withβ =1.7±0.1. This β laser distance sensor for the displacement, and a stan- is close to the one observed in the corresponding tensile dard plate-like piezoelectric sensor [22]. It is attached to experiment (depicted in the Fig. 4a) [22]. The simplest the setup inside one of the rolls visible in Fig.1, and the interpretationwouldbethatonceanavalancheiscreated signalisfilteredandamplifiedusingstandardtechniques. through a thermal fluctuation, it follows a deterministic Thedataacquisitioncardhasfourchannelsat312.5kHz course. This is similar to the predictions of recent the- per channel. We finally threshold the AE data. The dis- ories, which indicate the presence of a small nucleation placementdata is as expected highly correlatedwith the scale,andthat the avalanchesshouldbe as atthe depin- corresponding AE, but the latter turns out to be much ning critical point [18, 19, 24]. less noise-free and thus convenient to study. For paper, We have studied the temporal statistics and correla- weuseperfectlystandardcopypaper,withanarealmass tionsusingtheAEtimeseries,inparticularthewindowed 2 or basis weight of 80 g/m . Industrial paper has two one, and the direct displacement vs. time -signal (note principal directions, called the “Cross” and “Machine” that the AE has better accuracy in the time domain). Directions (MD/CD) [23]. The deformation characteris- The waiting times τ, the silent intervals between either tics are much more ductile in CD than in MD, but the two avalanches or two windows with E > 0 are shown i fracture stress is higher in MD. We tested a number of inFig. 4b. Itis interestingto note howthe distributions samples for both directions,with strips of width 30mm. P(τ)∼τα changewith the appliedforce. Forlargedriv- The weightusedforthe creeprangesfrom380g(CD) to ingforcesitappearsthatthere are(perhaps)twopower- 533 g (MD), and the resulting data has upto to tens of lawlikeregimes: onewithanexponentα=1.3,andmore thousandsofAE events(avalanches)per test(Fig.3). It clearlyone for largeτ with α∼2 which is also found for isunforunatelynotpossibletoinferthecriticaldepinning m small. The general form for m large resembles also m > m . The mechanical (and creep) properties of that of similar tensile tests where α∼1 is found. In the c used paper depend on the temperature and humidity. In our tensile case, there is a typical time/lengthscale (arising setup both remain at constant levels during an experi- from paper structure) visible in tensile crackline tests, 3 which might here also be related to the change in the [2] J.P. Sethna, K.A. Dahmen, and C.R. Myers, Nature slope of P(τ). (London) 410, 242 (2001), one can also “crumble” and The dynamics of the line exhibits stick-slip: the in- listen; P.A.HouleandJ.P. Sethna,Phys.Rev.E54, 278 (1996). tegrated velocity depends on the window length under [3] J. Kertész, V. K. Horvath,and F. Weber, Fractals 1, 67 consideration. This is already evident from the P(E) (1993). and extends to longer timescales than what P(E) im- [4] L.I.Salminen,A.I.Tolvanen,andM.J.Alava,Phys.Rev. pliesviadurationsofevents. A“stick-slip”exponentcan Lett. 89, 185503 (2002). be extracted also from the displacement data as well as [5] S. Santucci, L. Vanel, and S. Ciliberto Phys. Rev. Lett. integrated from the AE data. It appears that a relation 93, 095505 (2004). P(∆h) ∼ ∆h−1.7 arises. This implies the presence of [6] A.Guarino,A.Garcimartin,andS.Ciliberto,Eur.Phys. J.B6,13(1998);A.Garcimartin,A.Guarino,L.Bellon, correlations, which we next study by the energy release and S.Ciliberto, Phys. Rev.Lett 79, 3202 (1997). rate T: the duration of the active time T as measured [7] B. B.Mandelbrot, D.E.Passoja, andA.J.Paullay, Na- as subsequent windows where Ei > 0. It can be seen as ture (London) 308, 721 (1984). illustrated in Figure 5 that P(T) ∼ T−2.7, so that on a [8] E. Bouchaud,J. Phys.Cond. Mat. 9, 4319 (1997). millisecondscale(muchslowerthanavalanchedurations, [9] L. Ponson, D. Bonamy, and E. Bouchaud, Phys. 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Salminen et al., Europhys.Lett. 73, 55 (2006). a statistical physics problem. Acknowledgments: The [23] M.J.AlavaandK.J.Niskanen,Rep.Prog.Phys.69,669 (2006). Center of Excellence program of the Academy of Fin- [24] A.B. Kolton, A. Rosso, T. Giamarchi, and W. Krauth land is thanked for financial support as well as KCL Ltd Phys. Rev.Lett. 94, 057001 (2006) and the funding of TEKES,Finnish Funding Agency for [25] J. Weiss and D. Marsan, Science 299, 89 (2003). Technology and Innovation. R. Wathén, J. Lohi, and K. [26] P. Le Doussal, K.J. Wiese, and P. Chauve, Phys. Rev. Niskanencontributedbothwithmoralandpracticalsup- E69, 026112 (2004). port. L. Ponson is acknowledgedfor a crucial reminder. [1] M.J.Alava,P.N.N.K.Nukala,andS.Zapperi,Adv.Phys. 54, 347 (2006). 4 FIG. 1: Schematic view of the experimental setup (rolls, paper, AE sensor, camera, weight), and two closer views of the geometry. 1e+07 w) [arb. units] 1e+06 windo100000 m s E(in 1 10000 1000 50 100 150 200 t [s] FIG.2: Anexampleoftheacousticorstick-slipactivity(Ei> 0) during one single creep test. The energy Ei is integrated after thresholding over1 ms windows. 5 100 10 m/s] 1 m m) [ 1/ 0.1 v( 0.01 0.0021 0.0022 0.0023 0.0024 0.0025 0.0026 0.0027 0.0028 1/m [1/g] FIG. 3: The creep velocity vs. the inverse of the applied force or mass. The line indicates an exponential decay (v ∼ expa/m). The four sets (circle,rectangle,diamond,triangle) differ such that the 2nd one was done under lower relative humidity,rH % 43). 1 1e-05 tensile 380 g o] 410 g p. t 441050 gg (CD) E) [pro1e-10 444123500 ggg (((CCCDDD))) P( 478 g (MD) 502 g (MD) 518 g (MD) 1e-15 533 g (MD) 1 100 10000 1e+06 1e+08 E [arb. units] 1 1e-05 p. to] t3e8n0s igle τP() [pro1e-10 444411010505 gggg ((CCDD)) 420 g (CD) 440 g (CD) 1e-15 487 g (MD) 502 g (MD) 518 g (MD) 533 g (MD) 1e-06 0.0001 0.01 1 100 τ [s] FIG. 4: a): The probability distributions of the acoustic events for various cases. b): pots of the event interval (wait- ingtime)distributions,fordifferentmasssesm. Thedatasets havebeen shifted for clarity. 6 1 0.01 T)0.0001 P( 1e-06 1e-08 1 100 T FIG.5: TheP(T)ofdurationsofactivetimesT (fordefinition of T see text) The line has slope -2.7.