The failure time of samples of heterogeneous materials (wood, ¯berglass) is stud-
ied as a function of the applied stress. It is shown that in these materials the failure time is
predicted with a good accuracy by a model of microcrack nucleation proposed by Pomeau. It
is also shown that the crack growth process presents critical features when the failure time is
approached

Ciliberto, S. (S.)

Garcimartín-Montero, Á. (Ángel)

Guarino, A. (A.)

Montero, Á. (Ángel)

Dadun, University of Navarra

EUROPHYSICS LETTERS 15 August 1999Europhys. Lett., 47 (4), pp. 456-461 (1999)Failure time and microcrack nucleationA. Guarino1, S. Ciliberto1(∗) and A. Garcimart´ın21 Laboratoire de Physique, CNRS UMR5672, Ecole Normale Superieure de Lyon46 alle´e d’Italie, 69364 Lyon, France2 Departamento de F´ısica, Facultad de Ciencias, Universidad de NavarraE-31080 Pamplona, Spain(received 29 March 1999; accepted 28 June 1999)PACS. 62.20Mk – Fatigue, brittleness, fracture, and cracks.PACS. 46.50+a – Fracture mechanics, fatigue and cracks.PACS. 46.65+g – Random phenomena and media.Abstract. – The failure time of samples of heterogeneous materials (wood, fiberglass) is stud-ied as a function of the applied stress. It is shown that in these materials the failure time ispredicted with a good accuracy by a model of microcrack nucleation proposed by Pomeau. Itis also shown that the crack growth process presents critical features when the failure time isapproached.It is very well known that different materials subjected to a constant load may break after acertain time, which is a function of the applied load [1-5]. Many models have been proposed topredict this failure time, but the physical mechanisms remain often unclear [1-6]. Very recentlyPomeau proposed a model [6], which explains quite well the failure time of microcrystals [7]and gels [8] submitted to a constant stress. This model is based on the interesting idea thata nucleation process of microcracks has to take place inside the materials, in order to formthe macroscopic crack. This nucleation process is controlled by an activation law, as thecoalescence of a phase into another in a liquid-solid transition. Based on this prediction [6],Pauchard et al. [7] found that if a constant load is applied to a bidimensional microcrystal, itbreaks after a time τ given by the equation τ = τ0eP20 /P2, where P is the applied pressure,and τ0 and P0 are constants. Bonn et al. [8] found a similar law for gels. Pomeau predictedthat for three-dimensional microscopic systems the lifetime should beτ = τ0e(P0/P )4, (1)where τ0 is a characteristic time and P0 a characteristic pressure, which mainly depend onthe material characteristics, the experimental geometry and temperature. This idea is quiteinteresting and it merits to be checked experimentally in heterogeneous materials, such as(∗) E-mail: cilibe@physique.ens-lyon.frc© EDP Sciencesa. guarino et al.: failure time and microcrack nucleation 457DSTop ViewMSP1P2DSSide ViewEVHPRCrtlM MMMSFig. 1. – Sketch of the apparatus. S is the sample, DS is the inductive displacement sensor (which hasa sensitivity of the order of 1 µm). M are the four wide-band piezoelectric microphones. P = P1−P2is the pressure supported by the sample. P is measured by a differential pressure sensor (sensitivity =0.002 atm) that is not represented here. EV is the electronic valve which controls P via the feedbackcontrol system Ctrl. HPR is the high-pressure air reservoir.fiberglass and wood pannels. Indeed, in two recent papers [9,10], we have shown that in thesematerials the microcracks preceding the main crack form something like a coalescence aroundthe final path of the main crack.Driven by these observation we decided to study the behaviour of these materials as afunction of time and to check whether eq. (1) could be useful in order to predict the samplelife time. To do this we monitor the acoustic emission (AE) released before the final break-upby a sample placed between two chambers between which a pressure difference P is imposed.In fig. 1 a sketch of the apparatus is shown. We have prepared circular wood and fiberglasssamples with a diameter of 22 cm and a thickness of 4 mm (wood) and 2 mm (fiberglass).In our samples the AE consists of ultrasound bursts (events) produced by the formation ofmicrocracks inside the sample. For each AE event, we record the energy ε detected by thefour microphones, the place where it was originated, the time at which the event was detectedand the instantaneous pressure and deformation of the sample. The energy is defined as theintegral of the sum of the squared signals. A more detailed description of the experimentalmethods can be found in [9, 10].We first investigate the behaviour of the samples as a function of time when they aresubmitted to a constant load. Our interest is focused on the lifetime of the sample, on thebehaviour of the released acoustic energy near the fracture and on the distributions of energyand time elapsed between two consecutive events. The behaviour of the energy as a functionof time for a system submitted to a constant load has been studied by geologists, but theywere not specially interested in what happened near the fracture [11-13].We first imposed a constant strain to our samples, as has been made for crystals. As strainis fixed, every microcrack leads to a pressure decrease, so the system reaches a stationarystate. This is because a microfracture weakens the material. In the absence of microcraksthe pressure remains constant. One sample was submitted to a large deformation (close tothe fracture) and it did not break after three days. Therefore, at imposed strain, the effectobserved in microcrystals is not valid for heterogeneous materials. On the other hand, if aconstant stress is applied to the system, it will break after a certain time which depends onthe value of the applied pressure. The reason for this is that after every single microcrack thesame load must be endured by the weakened sample, so it becomes more and more unstable.We have submitted several samples to different constant pressures and we have measured thetime until the break-up (the lifetime τ). The values obtained are well fitted by eq. (1), that is458 EUROPHYSICS LETTERSa)τ(s)1/P (atm )4 -40.00 10.00 20.00 30.00 40.00100.001000.0010000.00100000.000.01 0.10 1.00( -t) /τ τ1.000.50.3E/Emaxb)Fig. 2. – Measures on wood samples: a) The time τ needed to break the wood samples under animposed constant pressure P is here plotted as a function of 1/P 4 in a semilog scale. The dashedline represents the solution proposed by Mogi [11] (τ = ae−bP ). The continuous line is the solutionproposed by Pomeau for microcrystals (τ = τ0e(P0/P )4). In the plot τ0 = 50.5 s and P0 = 0.63 atm.Every point is the average of 10 samples. The error bar is the statistical uncertainty. For the fiberglasssamples, we find τ0 = 44.6 s and P0 = 2.91 atm. b) The cumulated energy E, normalized to Emax,as a function of the reduced control parameter τ−tτat the neighborhood of the fracture point (caseof imposed constant pressure). The circles are the average for 9 wood samples. The solid line is thefit E = E0(τ−tτ)−γ . The exponent found, γ = 0.26, does not depend on the value of the imposedpressure. In the case of a constant pressure rate the same law has been found [9,10].the exponential function predicted by Pomeau; the lifetime expression τ = ae−bP proposed byMogi [11], on the other hand, does not conform to our data [14]. In fig. 2a τ is plotted vs. 1P 4in a semilog scale, and a straight line is obtained. Even if the pressure difference is very smallthe sample will eventually break, although the lifetime can be extremely long. For example,using eq. (1) and the best-fit parameters of fig. 2a, one estimate τ ' 5000 s at P = 0.43 atm.Halving the imposed pressure causes τ to become extremely large: τ = 4.4 · 1037 years atP = 0.21 atm).When a constant pressure is applied to the sample, the acoustic emission of the material ismeasured as a function of time. We find that the cumulated acoustic energy E diverges as afunction of the reduced time τ−tτ , specifically E ∝ ( τ−tτ )γ with γ = 0.27 (see fig. 2b). Notably,the exponent γ, found in this experiment with a constant applied pressure, is the same as theone corresponding to the case of constant stress rate [9]. Indeed it has been shown [9,10] thatif a quasi-static constant pressure rate is imposed, that is P = Apt, the sample breaks at acritical pressure Pc and E released by the final crack precursors (microcracks) scales with thereduced pressure or time (time and pressure are proportional) in the following way:E ∝(Pc − PPc)γ=(τ − tτ)γ, (2)where τ = Pc/Ap in this case. Thus it seems that the real control parameter of the failureprocess is time, regardless of the fact that either a constant pressure rate or a constant pressureis applied.To find a general law, which is valid for a time-dependent imposed stress, we intend togeneralize eq. (1) which is valid only for a constant imposed pressure. In the case where thepressure changes with time, it is reasonable to consider the entire history of the load. Thereforea. guarino et al.: failure time and microcrack nucleation 459a)b) c)20000.00 40000.00t (s)0.000.200.400.600.801.00P(atm)Difference of 5%0.000.200.400.600.801.00Pr2000.00 4000.00t (s)0.000.200.400.600.801.00P(atm)0.000.200.400.600.801.00Pr1000.00 2000.00t (s)0.000.200.400.600.801.00P(atm)0.200.400.600.801.00PrDifference of 3%Difference of 12%Fig. 3. – The imposed time-dependent pressure (bold dotted line) is plotted as a function of time in thecase of wood samples. The continuous line is the integral in time of the function f(P ) = 1τ0e−P40 /P4.On the basis of eq. (3) the predicted breaking time τ is obtained when the integral of f(P ) is equal to 1.The horizontal distance between the two vertical dashed lines in each plot represents the differencebetween the predicted and the measured breaking time. In a) a constant pressure has been appliedduring about 700 s, then the load is suppressed and then the same constant load is applied again. Thedifference between the lifetime predicted by eq. (3) and the experimental result is of 3%. b) Here twopressure plateaux of different value are successively applied to the sample. The difference between themeasured and the predicted lifetime is of 5%. In c) an erratic pressure is applied to the sample. Herethe error is of 10%.we consider that1τ0exp[−(P0P)4]is the density of demage per unit time. The certitude of breaking is obtained after a time τsuch that ∫ τ01τ0e−(P0P )4dt = 1, (3)where τ0 and P0 have the previously determined value. Notice that this equation is equivalentto eq. (1) when a constant pressure is applied.To test this, we have applied the load to the sample following different schemes. We havefirst applied successive pressure plateaux in order to check whether memory effects exist. Infig. 3a the pressure applied to the sample is shown as a function of time. A constant load hasbeen applied during a certain time τ1, then the load is suppressed and then the same constantload for a time interval τ2 is applied again. The sample breaks after a loading a time τ1 + τ2which is equal to the time needed if the same load had been applied continuously without theabsence of load during a certain interval. Therefore there is a memory of the load history.460 EUROPHYSICS LETTERS1E-5 1E-4 1E-3 1E-2Rate (atm/sec)100010000BreakingTime(s)Fig. 4. – A load linearly increasing at different rates Ap has been applied to different samples. Themeasured breaking times are plotted as a function of Ap in a log-log scale; circles and squares representthe measures on wood and fiberglass samples, respectively, at T = 300 K. Bold triangles representmeasures on wood samples at T = 380 K. The lines are the lifetime calculated from eq. (3) using thebest-fit values for P0 and τ0. These experiments show that eq. (3) describes well the lifetime of thesamples submitted to a time-dependent pressure.The lifetime formula (eq. (3)) is also valid if different constant loads are applied successively(fig. 3b). This concept can explain the violation of the Kaiser effect in these materials [10].If the load is not constant, the lifetimes resulting from the proposed integral equation arestill in good agreement with experimental data. A load linearly increasing at different rates Aphas been applied to different samples. The measured breaking times are plotted in fig. 4 alongwith a curve showing the values computed from eq. (3). Even if a quasi-static load is appliederratically (fig. 3d), the calculated lifetime agrees with the measured one. These experimentsshow that eq. (3) describes well the lifetime of the samples submitted to a time-dependentpressure.The question is to understand why eqs. (1) and (3) work so well for a three-dimensionalheterogeneous material. Indeed, in the Pomeau formulationP0 = G(η3Y 2kT)1/4, (4)where Y is the Young modulus, T the temperature, K the Boltzmann constant and η thesurface energy of the material under study. G is a geometrical factor which may depend onthe experimental geometry, on defect shape and density.In our experiment, we found P0 = 0.62 atm for wood, which has Y = 1.8 · 108 N/m2, andP0 = 2.91 atm for fiberglass, which has Y = 1010 N/m2. Thus the ratio between the values ofP0 found for the two materials is closed to the ratio of the square root of their Young moduli.In contrast temperature does not seem to have a strong influence on τ . In fact we changedtemperature from 300 K to 380 K, which is a temperature range where the other parameters,Y and η, do not change too much. For this temperature jump one would expect a change inτ of about 50% for the smallest pressure and of about 100% for the largest pressure. Lookingat fig. 4 we do not notice any change of τ within experimental errors which are about 10%.In order to maintain the change of τ within 10% for a temperature jump of 80 K one has toassume that the effective temperature of the system is about 3000 K. Notice that this claima. guarino et al.: failure time and microcrack nucleation 461is independent of the exact value of the other parameters and G.These observations seem to indicate that the nucleation process of microcraks is activatedby a noise much larger than the thermal one. Such a large noise can be probably producedby the internal random distribution of the defects in the heterogeneous materials that weused in our experiments. This internal random distribution of material defects evolves in timebecause of the appearance of new microcracks and the deformation of the sample. Thereforethis internal and time-dependent disorder of the material could actually be the mechanismthat activates the microcrak coalescence and play the role of a very high-temperature. Similarconclusions about a disorder-induced high temperature have been reached in other disorderedsystems [15]. This is an important point that merits to be deeply explored. Simple numericalsimulations which we performed in fuse networks seem to confirm this hypothesis.As a conclusion we have shown that a model based on the nucleation of microcracks is inagreement with experimental data of failure of two heterogeneous materials. This model seemsto be quite general because it also explains the failure of gels [8] and microcrystals [7]. It will becertainly interesting to test it in other materials. However many questions remain open. Thefirst one concerns the high temperature of the system. The second is related to the behaviourof acoustic energy close to the failure time. Indeed if AE is considered as a susceptibility [16] itis not easy to put together the observed critical divergency with a nucleation process. Probablythe standard phase transition description can be only partially applied to failure because ofthe intrinsic irreversibility of the crack formation. All of these are very interesting aspects ofthis problem which certainly merit to be clarified in the future.REFERENCES[1] Atkinson B. K., Introduction to Fracture Mechanics (Academic Press) 1989.[2] Kaninnen F. and Popelar C. H., Advanced Fracture Mechanics (Oxford University Press) 1985.[3] Lawn B., Fracture of Brittle Solids (Cambridge University Press) 1993.[4] Shah S. P., Toughening Mechanisms in Quasi Brittle Materials (Kluwer Accademic Pub-lisher) 1991.[5] Dieter G., Mechanical Metallurgic (McGraw-Hill) 1988.[6] Pomeau Y., C. R. Acad. Sci. Paris, Ser. II, 314 (1992) 553 (in French).[7] Pauchard L. and Meunier J., Phys. Rev. Lett., 70 (1993) 3565.[8] Bonn D., Kellay H., Prochnow M., Ben-Djemiaa K. and Meunier J., Sciences, 280 (1998)265.[9] Garcimart´ın A., Guarino A., Bellon L. and Ciliberto S., Phys. Rev. Lett., 79 (1997) 3202.[10] Guarino A., Garcimart´ın A. and Ciliberto S., Eur. Phys. J. B, 6 (1998) 13.[11] Mogi K., Bull. Earth Res. Inst. Tokyo, 40 (1962) 125.[12] Stead D. and Szczepanik Z., in Fifth Conference on Acoustic Emission/Microseismic Activityin Geologic Structures and Materials (Pennsylvania State University) 1991, p. 97.[13] Maes C. Van Moffaert A., Frederix H. and Strauven H., Phys. Rev. B , 57 (1998) 4987.[14] Several functions with many free parameters have been proposed [1-5] in order to fit the de-pendence of the sample life time on the applied stress. Among the fitting functions, which aresupported by a microscopic physical mechanism, we show in fig. 2a the two functions which givethe best fit of our data.[15] Sollich P., Phys. Rev. E, 58 (1998) 738.[16] Zapperi S., Vespignani A. and Stanley H. E., Nature, 338 (1997) 658.

Failure time and microcrack nucleation

Universidad de Navarra

Deposito Adademico Digital Universidad De Navarra

EUROPHYSICS LETTERS 15 August 1999
Europhys. Lett., 47 (4), pp. 456-461 (1999)
Failure time and microcrack nucleation
A. Guarino1, S. Ciliberto1(∗) and A. Garcimart´ın2
1 Laboratoire de Physique, CNRS UMR5672, Ecole Normale Superieure de Lyon
46 alle´e d’Italie, 69364 Lyon, France
2 Departamento de F´ısica, Facultad de Ciencias, Universidad de Navarra
E-31080 Pamplona, Spain
(received 29 March 1999; accepted 28 June 1999)
PACS. 62.20Mk – Fatigue, brittleness, fracture, and cracks.
PACS. 46.50+a – Fracture mechanics, fatigue and cracks.
PACS. 46.65+g – Random phenomena and media.
Abstract. – The failure time of samples of heterogeneous materials (wood, fiberglass) is stud-
ied as a function of the applied stress. It is shown that in these materials the failure time is
predicted with a good accuracy by a model of microcrack nucleation proposed by Pomeau. It
is also shown that the crack growth process presents critical features when the failure time is
approached.
It is very well known that different materials subjected to a constant load may break after a
certain time, which is a function of the applied load [1-5]. Many models have been proposed to
predict this failure time, but the physical mechanisms remain often unclear [1-6]. Very recently
Pomeau proposed a model [6], which explains quite well the failure time of microcrystals [7]
and gels [8] submitted to a constant stress. This model is based on the interesting idea that
a nucleation process of microcracks has to take place inside the materials, in order to form
the macroscopic crack. This nucleation process is controlled by an activation law, as the
coalescence of a phase into another in a liquid-solid transition. Based on this prediction [6],
Pauchard et al. [7] found that if a constant load is applied to a bidimensional microcrystal, it
breaks after a time τ given by the equation τ = τ0eP
2
0 /P
2
, where P is the applied pressure,
and τ0 and P0 are constants. Bonn et al. [8] found a similar law for gels. Pomeau predicted
that for three-dimensional microscopic systems the lifetime should be
τ = τ0e(P0/P )
4
, (1)
where τ0 is a characteristic time and P0 a characteristic pressure, which mainly depend on
the material characteristics, the experimental geometry and temperature. This idea is quite
interesting and it merits to be checked experimentally in heterogeneous materials, such as
(∗) E-mail: cilibe@physique.ens-lyon.fr
c© EDP Sciences
a. guarino et al.: failure time and microcrack nucleation 457
DS
Top View
M
SP1
P
2
DS
Side View
EV
HPR
Crtl
M M
M
M
S
Fig. 1. – Sketch of the apparatus. S is the sample, DS is the inductive displacement sensor (which has
a sensitivity of the order of 1 µm). M are the four wide-band piezoelectric microphones. P = P1−P2
is the pressure supported by the sample. P is measured by a differential pressure sensor (sensitivity =
0.002 atm) that is not represented here. EV is the electronic valve which controls P via the feedback
control system Ctrl. HPR is the high-pressure air reservoir.
fiberglass and wood pannels. Indeed, in two recent papers [9,10], we have shown that in these
materials the microcracks preceding the main crack form something like a coalescence around
the final path of the main crack.
Driven by these observation we decided to study the behaviour of these materials as a
function of time and to check whether eq. (1) could be useful in order to predict the sample
life time. To do this we monitor the acoustic emission (AE) released before the final break-up
by a sample placed between two chambers between which a pressure difference P is imposed.
In fig. 1 a sketch of the apparatus is shown. We have prepared circular wood and fiberglass
samples with a diameter of 22 cm and a thickness of 4 mm (wood) and 2 mm (fiberglass).
In our samples the AE consists of ultrasound bursts (events) produced by the formation of
microcracks inside the sample. For each AE event, we record the energy ε detected by the
four microphones, the place where it was originated, the time at which the event was detected
and the instantaneous pressure and deformation of the sample. The energy is defined as the
integral of the sum of the squared signals. A more detailed description of the experimental
methods can be found in [9, 10].
We first investigate the behaviour of the samples as a function of time when they are
submitted to a constant load. Our interest is focused on the lifetime of the sample, on the
behaviour of the released acoustic energy near the fracture and on the distributions of energy
and time elapsed between two consecutive events. The behaviour of the energy as a function
of time for a system submitted to a constant load has been studied by geologists, but they
were not specially interested in what happened near the fracture [11-13].
We first imposed a constant strain to our samples, as has been made for crystals. As strain
is fixed, every microcrack leads to a pressure decrease, so the system reaches a stationary
state. This is because a microfracture weakens the material. In the absence of microcraks
the pressure remains constant. One sample was submitted to a large deformation (close to
the fracture) and it did not break after three days. Therefore, at imposed strain, the effect
observed in microcrystals is not valid for heterogeneous materials. On the other hand, if a
constant stress is applied to the system, it will break after a certain time which depends on
the value of the applied pressure. The reason for this is that after every single microcrack the
same load must be endured by the weakened sample, so it becomes more and more unstable.
We have submitted several samples to different constant pressures and we have measured the
time until the break-up (the lifetime τ). The values obtained are well fitted by eq. (1), that is
458 EUROPHYSICS LETTERS
a)
τ
(s
)
1/P (atm )
4 -4
0.00 10.00 20.00 30.00 40.00
100.00
1000.00
10000.00
100000.00
0.01 0.10 1.00
( -t) /τ τ
1.00
0.5
0.3
E
/E
m
a
x
b)
Fig. 2. – Measures on wood samples: a) The time τ needed to break the wood samples under an
imposed constant pressure P is here plotted as a function of 1/P 4 in a semilog scale. The dashed
line represents the solution proposed by Mogi [11] (τ = ae−bP ). The continuous line is the solution
proposed by Pomeau for microcrystals (τ = τ0e
(P0/P )
4
). In the plot τ0 = 50.5 s and P0 = 0.63 atm.
Every point is the average of 10 samples. The error bar is the statistical uncertainty. For the fiberglass
samples, we find τ0 = 44.6 s and P0 = 2.91 atm. b) The cumulated energy E, normalized to Emax,
as a function of the reduced control parameter τ−t
τ
at the neighborhood of the fracture point (case
of imposed constant pressure). The circles are the average for 9 wood samples. The solid line is the
fit E = E0(
τ−t
τ
)−γ . The exponent found, γ = 0.26, does not depend on the value of the imposed
pressure. In the case of a constant pressure rate the same law has been found [9,10].
the exponential function predicted by Pomeau; the lifetime expression τ = ae−bP proposed by
Mogi [11], on the other hand, does not conform to our data [14]. In fig. 2a τ is plotted vs. 1P 4
in a semilog scale, and a straight line is obtained. Even if the pressure difference is very small
the sample will eventually break, although the lifetime can be extremely long. For example,
using eq. (1) and the best-fit parameters of fig. 2a, one estimate τ ' 5000 s at P = 0.43 atm.
Halving the imposed pressure causes τ to become extremely large: τ = 4.4 · 1037 years at
P = 0.21 atm).
When a constant pressure is applied to the sample, the acoustic emission of the material is
measured as a function of time. We find that the cumulated acoustic energy E diverges as a
function of the reduced time τ−tτ , specifically E ∝ ( τ−tτ )γ with γ = 0.27 (see fig. 2b). Notably,
the exponent γ, found in this experiment with a constant applied pressure, is the same as the
one corresponding to the case of constant stress rate [9]. Indeed it has been shown [9,10] that
if a quasi-static constant pressure rate is imposed, that is P = Apt, the sample breaks at a
critical pressure Pc and E released by the final crack precursors (microcracks) scales with the
reduced pressure or time (time and pressure are proportional) in the following way:
E ∝
(
Pc − P
Pc
)γ
=
(
τ − t
τ
)γ
, (2)
where τ = Pc/Ap in this case. Thus it seems that the real control parameter of the failure
process is time, regardless of the fact that either a constant pressure rate or a constant pressure
is applied.
To find a general law, which is valid for a time-dependent imposed stress, we intend to
generalize eq. (1) which is valid only for a constant imposed pressure. In the case where the
pressure changes with time, it is reasonable to consider the entire history of the load. Therefore
a. guarino et al.: failure time and microcrack nucleation 459
a)
b) c)
20000.00 40000.00
t (s)
0.00
0.20
0.40
0.60
0.80
1.00
P
(a
tm
)
Difference of 5%
0.00
0.20
0.40
0.60
0.80
1.00
Pr
2000.00 4000.00
t (s)
0.00
0.20
0.40
0.60
0.80
1.00
P
(a
tm
)
0.00
0.20
0.40
0.60
0.80
1.00
Pr
1000.00 2000.00
t (s)
0.00
0.20
0.40
0.60
0.80
1.00
P
(a
tm
)
0.20
0.40
0.60
0.80
1.00
Pr
Difference of 3%
Difference of 12%
Fig. 3. – The imposed time-dependent pressure (bold dotted line) is plotted as a function of time in the
case of wood samples. The continuous line is the integral in time of the function f(P ) = 1
τ0
e−P
4
0 /P
4
.
On the basis of eq. (3) the predicted breaking time τ is obtained when the integral of f(P ) is equal to 1.
The horizontal distance between the two vertical dashed lines in each plot represents the difference
between the predicted and the measured breaking time. In a) a constant pressure has been applied
during about 700 s, then the load is suppressed and then the same constant load is applied again. The
difference between the lifetime predicted by eq. (3) and the experimental result is of 3%. b) Here two
pressure plateaux of different value are successively applied to the sample. The difference between the
measured and the predicted lifetime is of 5%. In c) an erratic pressure is applied to the sample. Here
the error is of 10%.
we consider that
1
τ0
exp
[
−
(
P0
P
)4]
is the density of demage per unit time. The certitude of breaking is obtained after a time τ
such that ∫ τ
0
1
τ0
e−(
P0
P )
4
dt = 1, (3)
where τ0 and P0 have the previously determined value. Notice that this equation is equivalent
to eq. (1) when a constant pressure is applied.
To test this, we have applied the load to the sample following different schemes. We have
first applied successive pressure plateaux in order to check whether memory effects exist. In
fig. 3a the pressure applied to the sample is shown as a function of time. A constant load has
been applied during a certain time τ1, then the load is suppressed and then the same constant
load for a time interval τ2 is applied again. The sample breaks after a loading a time τ1 + τ2
which is equal to the time needed if the same load had been applied continuously without the
absence of load during a certain interval. Therefore there is a memory of the load history.
460 EUROPHYSICS LETTERS
1E-5 1E-4 1E-3 1E-2
Rate (atm/sec)
1000
10000
B
re
a
k
in
g
T
im
e
(s
)
Fig. 4. – A load linearly increasing at different rates Ap has been applied to different samples. The
measured breaking times are plotted as a function of Ap in a log-log scale; circles and squares represent
the measures on wood and fiberglass samples, respectively, at T = 300 K. Bold triangles represent
measures on wood samples at T = 380 K. The lines are the lifetime calculated from eq. (3) using the
best-fit values for P0 and τ0. These experiments show that eq. (3) describes well the lifetime of the
samples submitted to a time-dependent pressure.
The lifetime formula (eq. (3)) is also valid if different constant loads are applied successively
(fig. 3b). This concept can explain the violation of the Kaiser effect in these materials [10].
If the load is not constant, the lifetimes resulting from the proposed integral equation are
still in good agreement with experimental data. A load linearly increasing at different rates Ap
has been applied to different samples. The measured breaking times are plotted in fig. 4 along
with a curve showing the values computed from eq. (3). Even if a quasi-static load is applied
erratically (fig. 3d), the calculated lifetime agrees with the measured one. These experiments
show that eq. (3) describes well the lifetime of the samples submitted to a time-dependent
pressure.
The question is to understand why eqs. (1) and (3) work so well for a three-dimensional
heterogeneous material. Indeed, in the Pomeau formulation
P0 = G
(
η3Y 2
kT
)1/4
, (4)
where Y is the Young modulus, T the temperature, K the Boltzmann constant and η the
surface energy of the material under study. G is a geometrical factor which may depend on
the experimental geometry, on defect shape and density.
In our experiment, we found P0 = 0.62 atm for wood, which has Y = 1.8 · 108 N/m2, and
P0 = 2.91 atm for fiberglass, which has Y = 1010 N/m
2. Thus the ratio between the values of
P0 found for the two materials is closed to the ratio of the square root of their Young moduli.
In contrast temperature does not seem to have a strong influence on τ . In fact we changed
temperature from 300 K to 380 K, which is a temperature range where the other parameters,
Y and η, do not change too much. For this temperature jump one would expect a change in
τ of about 50% for the smallest pressure and of about 100% for the largest pressure. Looking
at fig. 4 we do not notice any change of τ within experimental errors which are about 10%.
In order to maintain the change of τ within 10% for a temperature jump of 80 K one has to
assume that the effective temperature of the system is about 3000 K. Notice that this claim
a. guarino et al.: failure time and microcrack nucleation 461
is independent of the exact value of the other parameters and G.
These observations seem to indicate that the nucleation process of microcraks is activated
by a noise much larger than the thermal one. Such a large noise can be probably produced
by the internal random distribution of the defects in the heterogeneous materials that we
used in our experiments. This internal random distribution of material defects evolves in time
because of the appearance of new microcracks and the deformation of the sample. Therefore
this internal and time-dependent disorder of the material could actually be the mechanism
that activates the microcrak coalescence and play the role of a very high-temperature. Similar
conclusions about a disorder-induced high temperature have been reached in other disordered
systems [15]. This is an important point that merits to be deeply explored. Simple numerical
simulations which we performed in fuse networks seem to confirm this hypothesis.
As a conclusion we have shown that a model based on the nucleation of microcracks is in
agreement with experimental data of failure of two heterogeneous materials. This model seems
to be quite general because it also explains the failure of gels [8] and microcrystals [7]. It will be
certainly interesting to test it in other materials. However many questions remain open. The
first one concerns the high temperature of the system. The second is related to the behaviour
of acoustic energy close to the failure time. Indeed if AE is considered as a susceptibility [16] it
is not easy to put together the observed critical divergency with a nucleation process. Probably
the standard phase transition description can be only partially applied to failure because of
the intrinsic irreversibility of the crack formation. All of these are very interesting aspects of
this problem which certainly merit to be clarified in the future.
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[4] Shah S. P., Toughening Mechanisms in Quasi Brittle Materials (Kluwer Accademic Pub-
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[5] Dieter G., Mechanical Metallurgic (McGraw-Hill) 1988.
[6] Pomeau Y., C. R. Acad. Sci. Paris, Ser. II, 314 (1992) 553 (in French).
[7] Pauchard L. and Meunier J., Phys. Rev. Lett., 70 (1993) 3565.
[8] Bonn D., Kellay H., Prochnow M., Ben-Djemiaa K. and Meunier J., Sciences, 280 (1998)
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[10] Guarino A., Garcimart´ın A. and Ciliberto S., Eur. Phys. J. B, 6 (1998) 13.
[11] Mogi K., Bull. Earth Res. Inst. Tokyo, 40 (1962) 125.
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[14] Several functions with many free parameters have been proposed [1-5] in order to fit the de-
pendence of the sample life time on the applied stress. Among the fitting functions, which are
supported by a microscopic physical mechanism, we show in fig. 2a the two functions which give
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[15] Sollich P., Phys. Rev. E, 58 (1998) 738.
[16] Zapperi S., Vespignani A. and Stanley H. E., Nature, 338 (1997) 658.


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Failure time and microcrack nucleation

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