This is a comment on J. Schmittbuhl, A. Hansen, and G. G. Batrouni, Phys.
Rev. Lett. 90, 045505 (2003). They offer a reply, in turn.Comment: 1 page, 1 figur

Alava, M. J.

Zapperi, S.

English

arXiv

M. J. Alava

S. Zapperi

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Comment on “Roughness of Interfacial Crack Fronts: Stress-Weighted Percolation in the Damage Zone”

A Comment on the Letter by Jean Schmittbuhl, Alex Hansen, and G. George Batroun, Phys Rev. Lett. 90, 045505 (2003). The authors of the Letter offer a Reply.Peer reviewe

Alava, Mikko J.

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This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Alava, Mikko J. & Zapperi, S.
Title: Comment on Roughness of Interfacial Crack Fronts:
Stress-Weighted Percolation in the Damage Zone
Year: 2004
Version: Final published version
Please cite the original version:
Alava, Mikko J. & Zapperi, S. 2004. Comment on Roughness of Interfacial Crack Fronts:
Stress-Weighted Percolation in the Damage Zone . Physical Review Letters. Volume 92,
Issue 4. 049601/1. ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.92.049601.
Rights: © 2004 American Physical Society (APS). This is the accepted version of the following article: Alava, Mikko
J. & Zapperi, S. 2004. Comment on Roughness of Interfacial Crack Fronts: Stress-Weighted Percolation in
the Damage Zone . Physical Review Letters. Volume 92, Issue 4. 049601/1. ISSN 0031-9007 (printed). DOI:
10.1103/physrevlett.92.049601, which has been published in final form at
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.049601.
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Comment on ‘‘Roughness of Interfacial Crack Fronts:
Stress-Weighted Percolation in the Damage Zone’’
A recent Letter [1], by Schmittbuhl, Hansen, and
Batrouni (SHB) addresses the question of how interfacial
cracks roughen in the presence of disorder. SHB explain
this process by a stress induced gradient percolation
model that takes into account the damage accumulated,
and translates that into a self-affine crack front profile. In
this Comment, we point out that the results presented in
Ref. [1] do not prove self-affinity but rather support self-
similarity of the crack fronts. This result, however, would
be in disagreement with experiments [2].
In the model of SHB the strain gradient induces a
damage profile and a crack front results. As the load is
raised the width of the front W increases approximately
as a power law, and eventually saturates. As in gradi-
ent percolation [3], the saturated width W scales with
the gradient of the damage profile 1=ly as W  ly with
  =1  where  is the correlation exponent of the
underlying percolation problem. Since in Ref. [1] ly  Lx,
where Lx is the lattice size parallel to the front, SHB
combine the initial dynamic scaling with that of the
saturated width into a ‘‘Family-Vicsek’’-like scaling
form WLx; t  Lx ft=Lzx, and conclude that the fronts
are self-affine interfaces. Such an attempt is misleading,
since presenting data in such a form does not imply that
the fronts are self-affine. In gradient percolation  cannot
be interpreted as a roughness exponent [3]: the front is
self-similar (i.e., the scaling is isotropic) up to a length
scale W [4] and it is trivially flat on scales beyond .
Self-affinity implies instead that on any length scale
l <  the system rescales anisotropically. Although strain
induced correlations could change the values of the criti-
cal exponents from the standard percolation ones, the
basic picture remains the same.
Figure 1 shows the data of the corresponding Fig. 1
from [1], displaying the broken springs. We also in-
clude the hull of the (damage) gradient percolation clus-
ter and the corresponding solid-on-solid (SOS) interface.
Comparing these two shows that the SOS presentation is
just an artificial projection from the fractal perimeter of
the damage zone which is not self-affine. In particular, we
see that the size of overhangs is of the same order of the
width.We have also studied an effective medium model in
the spirit of Ref. [5] in which the strain profile is com-
puted similarly to Ref. [1], but the damage is replaced by
its average along the transverse direction [6]. This model
is able to reproduce the features of the Family-Vicsek data
collapse of SHB, but the fronts are obviously described by
standard gradient percolation. From our simulations we
find that the gradient ly depends on the elastic constants of
the problem. In Ref. [1] the Green function Gij is normal-
ized so that
P
ijGij=LxLy is constant. Since Ly is kept
constant this amounts to rescaling the elastic constant by
Lx, producing an effective dependence of ly on Lx.
In conclusion, a correct interpretation in the framework
of gradient percolation of the data presented in Ref. [1]
implies that fronts are self-similar rather than self-affine.
Thus the model of Ref. [1] does not explain the roughness
of planar cracks observed experimentally [2].
M. J. Alava1,2 and S. Zapperi2
1Laboratory of Physics
Helsinki University of Technology
FIN-02015 HUT, Finland
2SMC-INFM
Dipartimento di Fisica
Universita` ‘‘La Sapienza’’
Piazzale le A. Moro 2 00185 Roma, Italy
Received 15 May 2003; published 27 January 2004
DOI: 10.1103/PhysRevLett.92.049601
PACS numbers: 62.20.Mk
[1] J. Schmittbuhl, A. Hansen, and G. G. Batrouni, Phys.
Rev. Lett. 90, 045505 (2003).
[2] K. J. Ma˚loy and J. Schmittbuhl, Phys. Rev. Lett. 87,
105502 (2002).
[3] B. Sapoval, M. Rosso, and J. F. Gouyet, J. Phys. Lett.
(Paris) 46, L149 (1985).
[4] A. Baldassarri, G. Gabrielli, and B. Sapoval, Europhys.
Lett. 59, 232 (2002).
[5] S. Zapperi, H. J. Herrmann, and S. Roux, Eur. Phys. J. B
17, 131 (2000).
[6] M. J. Alava and S. Zapperi (to be published).
0 50 100
x
0
10
20
30
40
50
60
70
80
90
100
110
120
y
55 75
5
10
15
20
25
FIG. 1 (color online). The damage reported in Fig. 1 of Ref. [1]
is plotted together with the front perimeter (solid line) and the
SOS approximation (dotted line). As shown in the inset, the
perimeter displays substantial overhangs, whose size is compa-
rable with the width, and it is thus not self-affine.
P H Y S I C A L R E V I E W L E T T E R S week ending30 JANUARY 2004VOLUME 92, NUMBER 4
049601-1 0031-9007=04=92(4)=049601(1)$22.50  2004 The American Physical Society 049601-1


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This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Alava, Mikko J. & Zapperi, S.Title: Comment on Roughness of Interfacial Crack Fronts:Stress-Weighted Percolation in the Damage ZoneYear: 2004Version: Final published versionPlease cite the original version:Alava, Mikko J. & Zapperi, S. 2004. Comment on Roughness of Interfacial Crack Fronts:Stress-Weighted Percolation in the Damage Zone . Physical Review Letters. Volume 92,Issue 4. 049601/1. ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.92.049601.Rights: © 2004 American Physical Society (APS). This is the accepted version of the following article: Alava, MikkoJ. & Zapperi, S. 2004. Comment on Roughness of Interfacial Crack Fronts: Stress-Weighted Percolation inthe Damage Zone . Physical Review Letters. Volume 92, Issue 4. 049601/1. ISSN 0031-9007 (printed). DOI:10.1103/physrevlett.92.049601, which has been published in final form athttp://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.049601.All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)Comment on ‘‘Roughness of Interfacial Crack Fronts:Stress-Weighted Percolation in the Damage Zone’’A recent Letter [1], by Schmittbuhl, Hansen, andBatrouni (SHB) addresses the question of how interfacialcracks roughen in the presence of disorder. SHB explainthis process by a stress induced gradient percolationmodel that takes into account the damage accumulated,and translates that into a self-affine crack front profile. Inthis Comment, we point out that the results presented inRef. [1] do not prove self-affinity but rather support self-similarity of the crack fronts. This result, however, wouldbe in disagreement with experiments [2].In the model of SHB the strain gradient induces adamage profile and a crack front results. As the load israised the width of the front W increases approximatelyas a power law, and eventually saturates. As in gradi-ent percolation [3], the saturated width W scales withthe gradient of the damage profile 1=ly as W  ly with  =1  where  is the correlation exponent of theunderlying percolation problem. Since in Ref. [1] ly  Lx,where Lx is the lattice size parallel to the front, SHBcombine the initial dynamic scaling with that of thesaturated width into a ‘‘Family-Vicsek’’-like scalingform WLx; t  Lx ft=Lzx, and conclude that the frontsare self-affine interfaces. Such an attempt is misleading,since presenting data in such a form does not imply thatthe fronts are self-affine. In gradient percolation  cannotbe interpreted as a roughness exponent [3]: the front isself-similar (i.e., the scaling is isotropic) up to a lengthscale W [4] and it is trivially flat on scales beyond .Self-affinity implies instead that on any length scalel <  the system rescales anisotropically. Although straininduced correlations could change the values of the criti-cal exponents from the standard percolation ones, thebasic picture remains the same.Figure 1 shows the data of the corresponding Fig. 1from [1], displaying the broken springs. We also in-clude the hull of the (damage) gradient percolation clus-ter and the corresponding solid-on-solid (SOS) interface.Comparing these two shows that the SOS presentation isjust an artificial projection from the fractal perimeter ofthe damage zone which is not self-affine. In particular, wesee that the size of overhangs is of the same order of thewidth.We have also studied an effective medium model inthe spirit of Ref. [5] in which the strain profile is com-puted similarly to Ref. [1], but the damage is replaced byits average along the transverse direction [6]. This modelis able to reproduce the features of the Family-Vicsek datacollapse of SHB, but the fronts are obviously described bystandard gradient percolation. From our simulations wefind that the gradient ly depends on the elastic constants ofthe problem. In Ref. [1] the Green function Gij is normal-ized so thatPijGij=LxLy is constant. Since Ly is keptconstant this amounts to rescaling the elastic constant byLx, producing an effective dependence of ly on Lx.In conclusion, a correct interpretation in the frameworkof gradient percolation of the data presented in Ref. [1]implies that fronts are self-similar rather than self-affine.Thus the model of Ref. [1] does not explain the roughnessof planar cracks observed experimentally [2].M. J. Alava1,2 and S. Zapperi21Laboratory of PhysicsHelsinki University of TechnologyFIN-02015 HUT, Finland2SMC-INFMDipartimento di FisicaUniversita` ‘‘La Sapienza’’Piazzale le A. Moro 2 00185 Roma, ItalyReceived 15 May 2003; published 27 January 2004DOI: 10.1103/PhysRevLett.92.049601PACS numbers: 62.20.Mk[1] J. Schmittbuhl, A. Hansen, and G. G. Batrouni, Phys.Rev. Lett. 90, 045505 (2003).[2] K. J. Ma˚loy and J. Schmittbuhl, Phys. Rev. Lett. 87,105502 (2002).[3] B. Sapoval, M. Rosso, and J. F. Gouyet, J. Phys. Lett.(Paris) 46, L149 (1985).[4] A. Baldassarri, G. Gabrielli, and B. Sapoval, Europhys.Lett. 59, 232 (2002).[5] S. Zapperi, H. J. Herrmann, and S. Roux, Eur. Phys. J. B17, 131 (2000).[6] M. J. Alava and S. Zapperi (to be published).0 50 100x0102030405060708090100110120y55 75510152025FIG. 1 (color online). The damage reported in Fig. 1 of Ref. [1]is plotted together with the front perimeter (solid line) and theSOS approximation (dotted line). As shown in the inset, theperimeter displays substantial overhangs, whose size is compa-rable with the width, and it is thus not self-affine.P H Y S I C A L R E V I E W L E T T E R S week ending30 JANUARY 2004VOLUME 92, NUMBER 4049601-1 0031-9007=04=92(4)=049601(1)$22.50  2004 The American Physical Society 049601-1

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