Table Of ContentAcoustic Emission from Paper Fracture
L.I. Salminen1, A.I. Tolvanen1, and M.J. Alava1
1Helsinki University of Technology, Laboratory of Physics,
P.O.Box 1100, FIN-02015 HUT, Finland
We report tensile failure experiments on paper sheets. The acoustic emission energy and the
3 waiting times between acoustic events follow power-law distributions. This remains true while the
0 strain rateis varied bymore thantwo ordersof magnitude. Theenergy statistics hastheexponent
0 β ∼ 1.25±0.10 and the waiting times the exponent τ ∼ 1.0±0.1, in particular for the energy
2
roughlyindependentofthestrain rate. Theseresultsdonotcomparewellwithfracturemodels, for
n (brittle) disordered media, which as such exhibit criticality. One reason may be residual stresses,
a neglected in most theories.
J
6 PACSnumbers: 62.20.Mk,05.40.-a,81.40Np,62.20.Fe
1
] A simple, common-day statistical physics experiment microcracks interact in crack dynamics, so that some-
n
is to tear a piece of paper into two parts. During this times it is not correct to consider an isolated object like
n
noise is heard [1], evidently originating from damage to the front or a crack tip. From the materials research
-
s the paper or the propagation of a crack. It is easy to viewpoint the dynamics is interesting since it deals with
i
d make the fracture surfaces “rough” as told by the naked the engineering quantities of strength and toughness, or
t. eyewithout anymoresophisticatedanalysis,andthe ex- theresistancetointrinsicflaws. Forinstanceinfiberand
a
perience is that it is difficult to do the tearing so that two-phase ceramic composites one can control these two
m
the cut is at all close to straight. Meanwhile, one can by varying the mixing and constituents.
d- confirmthe existence ofdisorderin the “sample”by just We study fracture in ordinary paper, as a two-
n looking through the sheet against any reasonable light dimensional disordered material, via acoustic emission
o source. The paper looks cloudy, which comes to a large (AE)analysis. Thereleaseofacousticenergyisrelatedin
c degree from the local fluctuations of areal mass density. paper to irreversibledeformation,microcracks,and, per-
[
Thistellsthatelasticandfracturepropertiesvarylocally. haps, to plasticity. A large-scale analogy is earthquakes.
1 Suchatestposestherelatedquestions: howdoesacrack Their energies are described by a power-law probabil-
v developin aninhomogeneousmaterial,andwhatkindof ity distribution, the Gutenberg-Richter-law [10] with an
9
properties does the acoustic emission have? energy exponent β. Cracks in paper may be self-affine
9
2 In fracture there is evidence of scaling properties fa- [11, 12], with a roughness exponent close to that in a
1 miliar from statistical physics. The roughness of cracks, scalar two-dimensionalfracture model, randomfuse net-
0 as measured by e.g. the root mean square height fluctu- works (RFN’s) [13, 14]. Both these are near 2/3, the
3
ations vs. length scale, can often be described by self- one for surfaces of minimal energy ([8, 12, 14]). Paper
0
affine fractal scaling [2, 3, 4]. Another such quantity is provides an example of crack growth in the presence of
/
t the distribution of strength, and its average versus sam- disorder. The probability distributions of the released
a
m ple size. In particular for (quasi)-brittle materials, the energyandthe temporalstatistics, describingdynamical
average strength can follow scaling laws, if the sample aspects, can be comparedto models for two-dimensional
-
d size is varied, which derive from the presence of disorder failure, and to other work on AE.
n
in the material [5, 6, 7]. Here the strain rate ǫ˙ is varied in the tensile test, to
o
Investigating such questions combines materials sci- studypossiblytime-dependenteffectsinthefracturepro-
c
: ence and statistical mechanics. Two fundamental prob- cess. Ourfirstmainconclusionisthatinstrain-controlled
v
lems are, what is the role of disorder in fracture? And, tests power-laws are found in the statistics, with either
i
X how do cracks develop and interact in real materials? nooratmostaweakdependenceonthestrainrate. The
r These join forces in the failure of a notched three- power-laws do not follow the predictions of theoretical
a
dimensional sample, visualized by the advancement of models outlined below, as fuse networks or mean-field
an one-dimensional crack front or line in the presence of ones. It is also apparent that an eventual localization of
disorder [8, 9]. It has an equation of motion, in an en- the crack [15, 16] does not change the microscopic prop-
vironment with varying properties as local strength and erties of the crackling noise, as long as the crack prop-
elasticmodulus. Thetrailleftbytheline-likefrontisthe agation is stable. This implies that paper failure is not
two-dimensional crack surface, which can become rough a “phase transition” with a diverging correlationlength,
(self-affine) if the propagating line-like object develops but leaves open the origins of the observed scalings.
critical fluctuations. However, it is difficult to propose To describe the interactions in an elastic medium
such a model that would match with the experimental withrandomnessincorporatedonestartsfrommean-field
data. For one, simple models tell that the disorder and stress-sharing. Infiberbundlemodels(FBM)theapplied
2
externalforceF(t)issharedbyalltheN(t)fibers,andas neouslyto AEwithatimeresolutionof0.01s. Wemade
the (random) failure threshold of the currently weakest 20identicalrepetitionsforstatistics. The100by100mm
one is reached, the stress σ(t) = F(t)/N(t) is instanta- sampleshadinitialnotches(size15mm)toachievestable
neously distributed among the remaining fibers. As a crack growth. Typical sheet thickness is about 100 µm.
sign of criticality there is a divergent scale, the size of The fracture statistics are not affected by such a notch
a typical avalanche which is made up of all the fibers [24]. For these strain rates the sound velocity timescale
that break down due to a single, slow-time scale stress- is muchfasterthanthat impliedby 1/ǫ˙. Eachindividual
increment. The totalavalanchesize (N) distributionfol- test contributes, at most, 1000 - 2000 events, so we can
lows P(N) ∼ N−5/2 [17, 18]. Thus global load sharing only look at integrated probability distributions and not
fracture resembles a second-order phase transition. in detail at local averages of e.g the acoustic event size
Local stress enhancements can be added to FBM’s vs. ǫ.
e.g. by considering fiber chains in which the stress from Fig. (1) shows an example of two tests under strain-
microcracks, of adjacent missing fibers, is transfered to control. Stress-strain curves have typically three parts:
the ones neighboring the crack. A more catastrophic pre-failure (almost) linearly elastic one, the regime close
growthresults(withanexponentmuchlargerthan5/2), to the maximum stress where the the final crack starts
resembling a first-order phase transition since the elas- to propagate or is formed, and a tail that arises due to
tic modulus has a finite drop at σ /ǫ [17, 19, 20]. A the cohesive properties of paper which allow for stable
c c
finite-dimensional, but scalar, approximation is given crack propagation. The faster the strain rate the less is
by random fuse networks [21]. RFN’s reproduce with the role of the tail. For the smallest strain rate most
strong enough disorder many of the features of mean- of the AE originates from tail (more than 90 %) while
fieldFBM’s,inspiteofincludingthestress-enhancements for the highest the situation is the opposite. Quantities
and shielding from other microcracks [22, 23]. The dy- of interest are the statistical properties before the max-
namics of crack growth is not understood well even in imum, after it, and the integrated totals, in particular
this simple model, including why the cracks seem to theenergydistributions. Thetimeseriesofeventsallows
become self-affine, with a roughness exponent close to to make qualitative observations of the correlations be-
2/3 [14]. The models’ event statistics can be compared tween subsequent events and to draw conclusions about
with experiments by considering the energies. For the the event properties as such. For this rather brittle pa-
RFN, in an event the lost energy is E ∼ ∆G ǫ2, where per grade,and the strainrates used, the elastic modulus
i i i
∆G ∼ N is the change in the “elastic modulus” is independent of ǫ˙, and by AE we are able to detect a
i broken
due to the AE event, at ǫ . Simulations on RFN’s for constant fraction of the elastic energy [24].
i
strongdisorderthatreproducestheFBMexponent(5/2) Fig. (2) shows the scaling of the energy for a fixed
for the avalanche sizes yield the exponent β ∼ 1.8 strain rate. The behavior is power-law -like, with sev-
RFN
and the FBM value is β ∼2 [24]. eral orders of magnitude of scaling. The same exponent
FBM
Experiments on, mostly, three-dimensional systems fits all three different cases: the pre and post maximum
[15, 16, 25, 26, 27] have yielded power-laws for the AE stresscases,andthesumdistribution. Ifthestrainrateis
energy release; the typical exponent β is 1 ... 1.5. One varied the same conclusion holds and the exponent only
idea is that approaching final failure resembles a phase fluctuatesatrandom[24]. Wehaveβ =1.25±0.10,indis-
transition: in the Lyon group’s stress-controlled exper- agreementwiththefiberbundleonesandwiththatfrom
iments indications were seen of a critical energy release the fuse networks, though β is closer than β .
RFN FBM
rate [15]. This means that the AE energy would diverge The practical implication is, that the material can with-
like a power-law in the proximity of a critical point, the stand more damage than expected since β <β . In
models
sample strength σ . Another problem is the nature of paper, the post-maximum events correlate with the ad-
c
the disorder in the material: whether quenched disorder vancement of the final crack. The remaining ligament
is able to give such power-law statistics for the energy length/width contracts, thus the elastic modulus drops
[22, 25]. should (assuming a constant stress state) here be quali-
Normal newsprint paper samples were tested in the tatively related to AE event energies (see also [30]).
machinedirection[29]onamodeIlaboratorytestingma- In an experiment with a varying strain rate ǫ˙ both
chineoftypeMTS400/M.Duetothe lackofconstraints the event durations and the waiting time between any
the samples could have out-of-plane deformations, and two events may (consider Fig. (1)) depend on ǫ˙. If the
none of the three fracture modes (I, II, III) is excluded crackdynamicsbecomes“fast”theneventstakeplaceon
on the microscopic level. The deformation rates ǫ˙ var- timescales set by the sound velocity. This establishes a
ied between 0.1 %/min and 100 %/min. The AE system timescale t =∆x/v where ∆x is the spatial separation
s s
consistedofapiezoelectrictransducer,arectifyingampli- oftwoeventsandthesoundvelocityv ∼2×103m/s. In
s
fierandcontinuousdata-acquisition. Thetime-resolution a sample of linear size 0.1 m this results in a maximum
of the measurements was 10 µs and the data-acquisition t ∼ 10−4s. In the failure of brittle carbon foams the
s
was free of deadtime. The stress was measured simulta- eventual critical crack growth is dictated by such fast
3
events [16]. giesfollowpower-law-likestatistics,notallthequantities
In quasi-static fracture models the dynamics of cracks do so. Also, ii) the temporal behavior hints of compli-
is assumed such that the stress field is equilibrated in- cated time-dependent phenomena not directly relatedto
finitelyfastduringmicrofractureevents,betweenfurther the fast relaxation of stress. A possible candidate is the
adiabatic increases of strain or stress. Thus the only viscoelastic nature of the wood fibers in paper, but note
time-dependence of any temporal statistics is in the av- that the macroscopic stress-strain behavior remains al-
erage time interval, between AE events, proportional to most linearly elastic while AE events already occur. We
1/ǫ˙. For the FBM model there are no correlations be- suspect the gradual release of internal stresses plays a
tween the waiting times and event size or durations, ex- role, perhaps due to frictionalpull-out of fibers from the
ceptforthetrendthattheaveragewaitingtimedecreases network. Finally, iii) the power-laws as obtained are off
as σ is approached. It is ’critical’ and thus the energy those predicted by simple fracture models. It remains
c
releaseratefollowsapower-lawclosetoσ . Waitingtime to be seen whether these models can be tailored closer
c
results do not generallyexistfor morecomplicated mod- to such tensile experiments [33]. One suggestion is that
els[31]. Onecancomparetotheexperimentalsignatures thedynamicsofenergyreleaseduringtheeventsfollowsa
of the intervals between AE events and the integrated differentcoursefromthemodelrules,inparticularfinite-
energy release rate integral R dǫE . As noted above rate dynamics allows for stress overshoots [34, 35]. The
event
there is some evidence - from stress-controlled experi- relationofthe acousticemissionto why cracksgetrough
ments - for a critical behavior for this quantity [15]. may be indirect. The latter could relate to the develop-
Fig. (3) demonstrates waiting time τ distributions for ment of pre-failure plastic deformation.
different strain rates. There is a clear power-law, whose
We acknowledgesupportby the AcademyofFinland’s
exponent (τ) remains roughly the same for all the strain
Center of Excellence program, and discussions with K.
rates [32]. Importantly, this is true for the post-events
Niskanen, J. Weiss, and S. Zapperi.
regardless of the origin (before/after σ ) of the major-
c
ity of the AE energy. For the pre-events there might be
some evidence of the exponent increasing with ǫ˙. In the
time-seriesof events,thosewith long durationsaresepa-
rated,ontheaveragebyshorterintervalsfromtheneigh-
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4
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FIG. 1: The acoustic data series (crosses and circles) and
E64, 066131 (2001).
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[31] An exception is F. Tzschichholz and H.J. Herrmann,
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Phys. Rev.E51, 1961 (1995).
curve, which is more important for ǫ˙ 0.5 [%/min] (solid line
[32] Notice that thelargest values of τ are still much smaller
and crosses) than 5 [%/min] (dashed line and circles).
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5
100
10-1
10-2
10-3
10-4
P(E)10-5
10-6
10-7 pre fracture events
10-8 palols et vferanctsture events
10-9 lsq fit to all events, slope -1.30
10-11000 101 102 103 104 105 106 107 108 109
AE Energy E [arb. units]
FIG.2: Thethree(post/pre/total)energydistributions,ǫ˙1.0
[%/min](forwhichthepost/pre-contributionstoAEenergy
are roughly thesame).
100
10-1 0.1 %/min (slope -0.99)
5 %/min (slope -1.09)
10-2 10 %/min (slope -1.25)
τ)10-3
P(10-4
10-5
10-6
101-70-5 10-4 10-3τ [s]10-2 10-1 100
FIG. 3: Plots of the event interval τ (waiting time) distribu-
tions, for threestrain rates.
104
N,pre
τ >2, pre
N, post
τ >2, post
-5∆t) [1 s]103
> (
<tsilent102
101
1 10 100
∆t, Event length [10-5 s]
FIG. 4: The waiting time as a function of event length ∆t, ǫ˙
1.0 [%/min].
6
107
0.1 %/min
106 0.5 %/min
s] 1 %/min
nit105 10 %/min
u
arb. 104
E[int 103
AE, 102
101
1000 0.2 0.4 0.6
Strain ε [%]
FIG. 5: Increase of the integrated AEenergy vs. strain ǫ.
