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Decay mechanism of fracture induced electromagnetic pulses

A. Rabinovitcha)

Department of Physics, Ben-Gurion University of the Negev, Beer Sheva84105, Israel

and The Deichmann Rock Mechanics Laboratory of the Negev, Beer Sheva, Israel

V. Frid and D. Bahat

Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev,

Beer Sheva 84105, Israel and The Deichmann Rock Mechanics Laboratory of the Negev, Beer Sheva, Israel

J. Goldbaum

Department of Physics, Ben-Gurion University of the Negev, Beer Sheva84105, Israel

and The Deichmann Rock Mechanics Laboratory of the Negev, Beer Sheva, Israel

共Received 20 August 2002; accepted 28 January 2003兲

In this article we consider the rise and fall time 共which were earlier shown experimentally to be the

same兲of electromagnetic radiation 共EMR兲from propagating cracks. This feature is shown

theoretically to be inversely proportional to the pulse frequency

and to the fourth degree of the

absolute temperature. It is shown experimentally that in glass and in glass ceramics, which are not

porous, and in granite, whose porosity is of the order of 5%,

is indeed inversely proportional to

. In chalk, whose porosity is as high as 40%, however, this relation is not observed. We argue that

the latter result is due to the interaction between the cracks which emit the EMR and the pores of

the material and speciﬁcally to the spread of ensuing temperatures of the cracks caused by this

interaction. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1562752兴

I. INTRODUCTION

Electromagnetic radiation 共EMR兲from propagating

cracks was observed by Stepanov in 1933 in fractured KCl

crystals.1In 1973 Misra2detected alternating magnetic ﬁelds

in the form of decaying pulses while fracturing some metals

and alloys in tension. In 1975 Nitsan3measured EMR during

his experiments of fracturing quartz-containing rocks. In the

1980’s and 1990’s a lot of fracture experiments, in which

EMR was detected, were carried out in various materials,

e.g., ionic crystals,4rocks,5–12 ice13,14 and glass.15 Various

tests were

used to obtain fracture by tension and compression, by ap-

plying uniaxial,5,7,8 triaxial,10–12 bending6and impact6meth-

ods. It was found that the EMR intensity depends on the

elastic rigidity of the material fractured12,16 and the new frac-

tured area,17 but is not affected by the fracture mode, i.e.,

tension or shear.17 In spite of all measurements to date, the

origin of EMR is still not completely understood. Existing

models are unable to quantitatively explain the phenomenon.

Rabinovitch, Frid, and Bahat9carried out parametriza-

tion of EMR pulses. They showedthat the shape of an indi-

vidual EMR pulse can be described by the formula

A共t兲⫽

冦

A0sin

共t⫺t0兲

冋

1⫺exp

冉

⫺t⫺t0

冊

册

;t0⬍t⬍T

¯

A0sin

共t⫺t0兲

冋

1⫺exp

冉

⫺T

¯

⫺t0

冊

册

exp

冉

⫺t⫺T

¯

冊

;t⬎T

¯

.共1兲

Here A(t) is the pulse intensity as a function of time, t0is

the time of pulse origin,

is the rise and fall time 共which turn

out to be the same兲,A0is the amplitude of the pulse enve-

lope maximum, T

¯

is the time when the amplitude is maxi-

mal, and

is the pulse frequency. An example of a pulse

with ﬁtted parameters and their errors is shown in Fig. 1. As

can be seen, the ﬁt is quite good with very low errors. This

kind of ﬁt was achieved for all pulses with errors in the

parameters of less than 5% 共except for

, whose errors are

usually ⬃2%, but infrequently can be as large as 30%兲.It

has been found that T⬘⫽T

¯

⫺t0, the time from the pulse ori-

gin to its maximum, is proportional to the crack length, L,

T⬘⫽L/vcr where vcr is the crack velocity, while

is in-

versely proportional to the crack width, b,

⫽

vR

b.共2兲

a兲Author to whom correspondence should be addressed; electronic mail:

avinoam@bgumail.bgu.ac.il

JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 9 1 MAY 2003

50850021-8979/2003/93(9)/5085/6/$20.00 © 2003 American Institute of Physics

Downloaded 12 May 2003 to 132.72.138.1. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

Here vRis the Rayleigh velocity.

All aspects of this model except

were experimentally

conﬁrmed.9–12,17 In this article

is considered in detail.

II. EXPERIMENTAL PROCEDURE

A. Experimental equipment and sample preparation

A triaxial load frame 共TerraTeck stiff press model FX-S-

33090, axial pressure up to 450 MPa, conﬁning pressure up

to 70 MPa, stiffness 5⫻109N/m), combined with a closed-

loop servocontrol, was used for the measurement. The servo-

control 共linearity 0.05%兲was used to maintain a constant

axial piston displacement rate. The load was measured by a

sensitive load cell LC-222M 共maximum capacity 220 kN,

linearity 0.5% full scale兲. Acantilever set, consisting of axial

and lateral detectors, was used to measure the sample strains

in three orthogonal directions parallel to the principal

stresses. A magnetic one-loop antenna with a diameter of 3

cm 共EHFP-30 Near Field Probe set, Electro-Metrics Penril

Corporation兲was used for the EMR detection. The signals

were ampliﬁed by means of a low-noise microsignal ampli-

ﬁer 共Mitek Corporation Ltd., frequency range 10 kHz-500

MHz, gain 60⫾0.5 dB, noise level 1.4⫾0.1 dB across the

entire frequency band兲and transferred to a Tecktronix TDS

420 digital storage oscilloscope. The latter was connected to

an IBM PC by means of a general purpose interface bus, so

the signals were stored on the computer hard disk for further

processing. The antenna was placed 2 cm away from the

center of the loaded sample, its normal pointing perpendicu-

lar to the cylinder axis. The EMR was monitored with an

overall sensitivity of 1

V.

To reduce the background noise level, the following

means were employed: 共1兲the measurements were carried

out in a thick-wall steel pressure vessel; 共2兲special radio

frequency ﬁlters were used; 共3兲the ampliﬁer power supply

was independent of the industrial net; 共4兲the antenna was

connected to the oscilloscope via the ampliﬁer by means of

special double-screen cables 共Alpha wire Corporation Ltd.兲.

The samples had a cylindrical shape, with standard

length of 100 mm and standard diameter of 53 mm. The

experiments were carried out on: 1. chalk samples taken

from the Horsha Foundation in the Beer Sheva syncline18 共all

samples were cut from the same layer with the same orien-

tation within the rock兲, 2. Eilat granite,10 3. soda-lime glass

and 4. glass ceramics.10,19 Also included in the analysis are

some of the results obtained in our drilling experiments of

glass and granite.20

B. Measurement of pore radii distribution in chalk

The distribution of pore radii was experimentally mea-

sured in a thin cut of chalk 共20⫻30 mm area, 30

m thick-

ness兲. It was examined under a microscope with an enlarge-

ment of 100. The cut was placed on a table that was moved

by means of a computer program Automatic Point Meter

共steps 300

m in the ‘‘horizontal’’ direction and 600

min

the ‘‘vertical’’ direction兲so that different pieces of the cut

were sequentially seen, and at each measuring instance the

radius of the pore situated at the same point of the image was

measured. Thus, as a result of a large number of measure-

ments, pore-radii distribution was obtained. The measure-

ment was carried out by means of a micrometer with accu-

racy of 5

m, which is actually a ‘‘ruler’’ of 1 mm divided

by 100 equal parts, 10

m each.

A photograph of the cut, where pores can clearly be

seen, is shown in Fig. 2.

III. RISE AND FALL TIME

A. Theoretical considerations

We assume that EMR is originated by surface atomic

共ionic兲vibrations21 caused by bonds rupture, similar to Ray-

leigh waves or surface optical phonons. It has been shown

elsewhere22 that surface waves decay as a result of interac-

tion with bulk phonons. We therefore consider

to be the

relaxation time of such a surface 共Rayleigh-like兲wave,

which interacts with a bulk phonon, leading to the creation of

another bulk phonon 共a three-phonon process兲, and use Eq.

共13兲of King and Sheard23 to characterize the process. The

rate of occurrence of the process per unit time, or its transi-

tion probability, is given by the golden rule formula24 Pi

f

⫽2

/ប

兩

具

f

兩

H3

兩

i

典

兩

2

␦

(Ef⫺Ei), where the initial and the ﬁnal

states are

兩

i

典

⫽

兩

nR,nb1,nb2

典

,

兩

f

典

⫽

兩

nR⫺1,nb1⫺1,nb2⫹1

典

,

where H3is the time dependent anharmonic part of the crys-

FIG. 1. An experimental EMR pulse emitted from chalk and its numerical

ﬁt. The ﬁtted parameters are: A⫽0.041⫾0.002 mV,

⫽3⫻10⫺7⫾0.27

⫻10⫺8s, t0⫽⫺2.683⫻10⫺7⫾9⫻10⫺10 s, T

¯

⫽3.4⫻10⫺7⫾2⫻10⫺8s,

⫽6.59⫻107⫾1⫻105s⫺1.

FIG. 2. Microscope photograph of a thin section of chalk; the real width of

the picture is 1.2 mm. Large white spots are plankton skeletons 共calcite兲;

black spots are pores 共the one shown by arrow is 84

m in diameter兲and

partly iron oxide; gray structure is micrite, or cryptocrystals 共very ﬁne grains

of carbonate, less than 10

m兲; small white spots are microsparite, ﬁne to

medium size grains 共about 10–50

m兲of calcite.

5086 J. Appl. Phys., Vol. 93, No. 9, 1 May 2003 Rabinovitch

et al.

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tal Hamiltonian, nR,nb1and nb2are the numbers of surface

phonons, and initial and ﬁnal bulk phonons, respectively. Ei

and Efare the initial and the ﬁnal energies of the three-

phonon system, so that Ef⫺Ei⫽ប(

b2⫺

R⫺

b1), where

the

-sdenote the related frequencies. The relaxation time is

obtained from the golden rule formula 共1/

being propor-

tional to the transition probability兲using the explicit expres-

sion for H3关Eq. 共6.47兲in Srivastava22兴, the displacement

ﬁeld due to the surface modes written in second quantized

notation 关Eq. 共8.26兲op. cit.22兴, and integrating over the states

of the initial and ﬁnal bulk phonons b1 and b2; one obtains

关Eq. 共8.36兲op. cit.22兴

1

R⬀

RT4

3vR

2,共3兲

where Tis the temperature,

is the material density, and vR

共as mentioned before兲is the Rayleigh wave velocity. The

proportionality coefﬁcient contains data of bulk phonons and

of the crystallographic orientation and is considered to be

constant for the same material. Note that Tis the local tem-

perature at the crack tip. It is much higher than room tem-

perature. 共See, e.g., Ref. 25.兲

In a developing crack, a surface wave propagates along

the crack surfaces.21 Its emitted EMR frequency is the same

as that of the oscillating ions of the crack sides,9

. There-

fore we equate

⫽

R.

B. Experimental results and analysis

We have analyzed EMR pulses of chalk, granite, glass

and glass ceramics by ﬁtting them to Eq. 共1兲and deriving

their parameters 共in particular

and

兲. Figure 3共a兲shows

the dependence of

on

for glass, glass ceramics and gran-

ite. The slopes on a logarithmic scale are within ⫺1⫾0.1,

with R2⫽0.96, 0.85 and 0.91, respectively, agreeing with the

theoretical prediction 关Eq. 共3兲兴. For chalk, however, an ‘‘ef-

fective’’ slope of ⫺0.7 is obtained 关Fig. 3共b兲兴. It cannot be

attributed to the low accuracy of the ﬁtting parameters: the

error for

ranges between 2% and 共seldom兲30%, which is

quite small in a logarithmic scale, while the error for

does

not exceed a few percent, and is usually only a fraction of a

percent. We would like to attribute this ‘‘change of slope’’to

the spread in temperatures of the small cracks there.

The distribution of (

)⫺1/4, which according to Eq. 共3兲

should be proportional to the absolute temperature, for small

and large crack widths in granite is shown in Figs. 4共a兲and

4共b兲, respectively. We denote (

)⫺1/4 values henceforth as

‘‘temperatures.’’ The mean values of the crack temperatures

in relative units are, respectively, 0.67 and 0.65 for small and

large cracks, which are considered to be the same within the

accuracy for temperature values 共10%兲. Similar distributions

for chalk are shown in Figs. 5共a兲and 5共b兲. Here, however,

the mean values are different, being 0.47 for small crack

widths and 0.67 for large ones. As mentioned, the error for

is not more than a few percent, mostly a fraction of a percent,

while for

it ranges between 2% and 30%. Thus the error of

(

)⫺1/4 may be estimated as ⬃10%, yielding temperatures

of 0.47⫾0.05 for small cracks and 0.67⫾0.07 for large

cracks.

Figure 6 shows the distribution 共histogram兲of log(1/

)

关where 1/

should be proportional to crack widths, Eq. 共2兲兴

FIG. 3. Rise and fall time

as a function of frequency

: the points are the

experimental results of individual pulses, and the line is the power law

ﬁtting result: 共a兲for glass 共triangles兲, ﬁtting

⫽1.87

⫺0.95 共short-dashed

line兲; for glass ceramics 共squares兲, ﬁtting

⫽28.1

⫺1.1;共solid line兲for

granite (x’s), ﬁtting

⫽2.23

⫺0.93 共long-dashed line兲;共b兲for chalk

共circles兲, ﬁtting

⫽1.87

⫺0.7.

FIG. 4. Frequency counts of temperatures (

)⫺1/4 for cracks in granite: 共a兲

short cracks; 共b兲long cracks.

5087J. Appl. Phys., Vol. 93, No. 9, 1 May 2003 Rabinovitch

et al.

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for small and large cracks. We denote by small cracks those

cracks which have

values larger than 2⫻107s⫺1关e.g.,

Fig. 3共b兲兴. By large cracks we denote those with

values

smaller than 2⫻107s⫺1. If one assumes the Rayleigh veloc-

ity in chalk to be26 1200 m/s, then the calculated small crack

widths in chalk 关by Eq. 共2兲兴 range between 52.5 and 130

m.

For large cracks the corresponding range is from 130

mto

1.8 cm.

IV. DISCUSSION

It has been shown that the temperature of dynamically

propagating cracks rises.25 The actual temperature rise de-

pends on crack velocity and on material properties.27 The

spread of experimental points in Fig. 3共a兲can be attributed to

a spread in crack velocities that causes a spread in tempera-

tures. Since we assume that 1/

⬃

T4, the dependence on

temperature is very strong. However, cracks in glass and in

glass ceramics 共which are not porous兲and in granite 共the

porosity of which is about 5%兲lead to almost uniform dis-

tribution of crack temperatures around their mean 共see Fig.

4兲, resulting in the almost exact

⫽const relation of Fig.

3共a兲.In chalk, however,

⫽const. The ‘‘temperature distri-

bution’’ 共histograms兲for small and large cracks are given in

Figs. 5共a兲and 5共b兲. It can be seen that temperatures of small

cracks are signiﬁcantly lower than those of large cracks.

We assume that this temperature difference in chalk is

due to the interaction between the propagating cracks and the

existing material pores. Chalk is the only material of the four

analyzed that has large porosity 共of about 40%兲. The pore

radii in chalk range28 between 0.05 and 100

m. The results

of our measurements 共see Sec. IIB兲, together with their ex-

ponential ﬁtting, are given in Fig. 7. Note that although the

range of observed pore sizes was between 2 and 210

m,

their three-dimensional 共3D兲ﬁtting, being exponential, in-

cludes all pores, also those of lower sizes 关see Eqs. 共5兲and

共6兲below兴. The cracks in chalk encounter one or more pores

during their development. Thus crack heat can be partially

spent on raising the temperature of the air inside the pores; in

addition, air from the pore can enter the crack and expand

adiabatically, which process could also lower the latter’s

temperature. Although all cracks in chalk are cooled down in

this way, small sized cracks’temperatures are more strongly

inﬂuenced by these processes than large cracks, since the

ﬁnal temperature depends on the ratio between the pores’

volume and the volume of the crack. The greater this ratio,

the stronger is the temperature decrease. Note that the only

EMR emitting objects are the cracks and the inﬂuence of the

pores is only a passive one, namely changing the temperature

of these cracks. While the volume of the interacting pores is

proportional to the area of the crack 共as will be shown pres-

ently兲, the crack volume is proportional both to the crack

area and to its aperture; the latter, however, depends on crack

length in the form of a power law29

⫽kx

␣

,共4兲

FIG. 5. Frequency counts of temperatures (

)⫺1/4 for cracks in chalk

共columns兲and calculated 共solid line兲using Eq. 共A19兲:共a兲for short cracks;

共b兲for long cracks; crack aperture is assumed to behave as ⬃kx

␣

, where x

is the crack half width; the parameters obtained are Tp⫽0.32, Tcr⫽0.9,

␣

⫽0.22, k⫽25

m1⫺

␣

, both for 共a兲and 共b兲. Parameters found by best ﬁt.

FIG. 6. Frequency counts for log(

⫺1)(

⫺1is proportional to crack widths,

b⫽

vR/

) in chalk.

FIG. 7. Measured pore radii distribution in chalk, ﬁtted to P⬘(r)

⫽a2rexp(⫺ar); best ﬁt obtained for a⫽0.106⫾0.004

m⫺1(R2⫽0.98).

5088 J. Appl. Phys., Vol. 93, No. 9, 1 May 2003 Rabinovitch

et al.

where is the aperture and xis the crack half width 共mea-

sured cracks had approximately equal widths and lengths兲.

Crack volume is thus proportional to x2⫹

␣

and small cracks

are cooled more than large ones when intersecting pores.

The distribution of pore radii in the sample volume can

be approximated by a negative exponential

P共r兲⫽aexp共⫺ar兲.共5兲

However the distribution of pore radii that cross a chosen

plane does not have the same form. Simmons30 showed that

the cracks’ distribution obtained by means of a thin section

needs corrections in order to get a 3D distribution. These

corrections are included here 共see the Appendix兲. The distri-

bution of pore radii that cross a chosen plane is thus given by

P⬘共r兲⫽a2rexp共⫺ar兲,共6兲

which ﬁts the experimentally measured pore distribution

quite well 共Fig. 7兲.

To show the self-consistency of our assumptions, we

have theoretically derived the probability distribution of ﬁnal

temperatures 共see the Appendix兲. The comparison of the de-

rived distribution with the experimental result is carried out

by ﬁtting the following parameters of the former: the power

␣

and the proportionality coefﬁcient kof Eq. 共4兲, and the

initial temperatures of the pore and of the crack, Tpand Tcr .

The best ﬁtting parameters both for small and large cracks

are

␣

⫽0.22, Tp⫽0.32, Tcr⫽0.9, and k⫽25

m1⫺

␣

. The ex-

perimental and the calculated 关Eq. 共A19兲兴 distributions of

ﬁnal temperatures for small and large cracks are shown in

Figs. 5共a兲and 5共b兲. The agreement is adequate.

V. SUMMARY

The rise and fall time

is theoretically shown to be

inversely proportional to the pulse frequency

and to the

fourth degree of temperature T. Experimentally for glass and

glass ceramics, which are not porous, and for granite, whose

porosity is as low as 5%, the

⬃

⫺1relation is shown to

hold while for chalk, whose porosity is of the order of 40%,

this relation is not fulﬁlled. We conjecture that the latter fact

is due to the temperature difference between small and large

cracks: the temperature of cracks drops when encountering

pores since heat emanating at the crack tip is spent on heat-

ing both the crack and the air in the pore; and the tempera-

ture decrease is larger for small cracks. Results show ad-

equate agreement between experimental and calculated

values.

ACKNOWLEDGMENTS

The authors are thankful to E. Shimshilashvili and Pro-

fessor V. Samoilov for their help in conducting microscopic

measurements, and to Professor Thieberger and B. Gold-

baum for useful discussions. This research No. 98/02-01 was

supported by the Israel Science Foundation.

APPENDIX: CALCULATION OF DISTRIBUTION OF

CRACK TEMPERATURES AFTER THEIR

INTERACTION WITH PORES

The temperature of a crack after encountering pore共s兲is

assumed to be given by the heat exchange between the air in

the crack ‘‘volume’’ and the pores

T⫽Tp⫹Vcr

Vcr⫹Vpores 共Tcr⫺Tp兲,共A1兲

where Tcr and Tpare the initial temperatures of the crack and

the pores, and Vcr and Vpores are, respectively, their volumes.

Our assumptions are: 共1兲pores are homogeneously distrib-

uted in the sample and have a spherical shape; 共2兲cracks are

rectangular of dimensions of 2x⫻2x⫻;共3兲the crack aper-

ture depends on the crack width in the form of a power law

关Eq. 共4兲兴;共4兲all the cracks and all the pores have identical

initial temperatures Tcr and Tp, respectively; 共5兲porosity is

taken as 40% for all samples.

Porosity

is the ratio between the volume of all the

pores to the total sample volume. Let us ascribe to a pore of

volume Va volume Vunit , so that

⫽V/Vunit ; the radius of

this volume is

⫽r

冑

3

.共A2兲

Then the concentration of the pores 共number of pores per

unit volume兲in the sample is given by

n⫽1

Vunit

⫽

V.共A3兲

Pore radii distribution is taken as 共see above兲,

P共r兲⫽aexp共⫺ar兲共A4兲

and the concentration of pores with radii between rand r

⫹dr is

n共r兲dr⫽nP共r兲dr.共A5兲

The number of pores of radius rthat intersect a chosen plane

共per unit area兲is equal to the number of pores of radius r,

whose centers are situated in a layer of thickness 2raround

the plane, per unit area

n⬘共r兲⫽2rn共r兲⫽2rnP共r兲⫽2narexp共⫺ar兲⫽n⬘P⬘共r兲,

共A6兲

where n⬘is the total number of pores that cross the chosen

plane 共per unit area兲

n⬘⫽

冕

0

⬁n⬘共r兲dr⫽2n

a.共A7兲

Thus the probability distribution of radii of pores that cross

the plane is

P⬘共r兲⫽a2rexp共⫺ar兲.共A8兲

When a pore of radius r, the center of which is situated

at a distance yfrom the plane, cuts the latter, the area ‘‘oc-

cupied’’by it in the plane is the area that is cut from it by the

volume of radius

关Eq. 共A2兲兴 ascribed to the pore

S共r,y兲⫽

共

2⫺y2兲⫽

共r2

⫺2/3⫺y2兲.共A9兲

5089J. Appl. Phys., Vol. 93, No. 9, 1 May 2003 Rabinovitch

et al.

The average area of the plane that is occupied by a pore of

radius ris

S

¯

共r兲⫽

冕

0

rS共r,y兲P共y兲dy⫽

冕

0

r

共r2

⫺2/3⫺y2兲dy

r

⫽

r2

冉

1

2/3 ⫺1

3

冊

⫽C

r2,

共A10兲

where the probability distribution, P(y), of the pore center

from the plane is assumed to be homogeneous, i.e.,

P共y兲dy⫽dy

r共A11兲

and

C⫽

⫺2/3⫺1

3.共A12兲

Hence the number of pores of average radius r, which inter-

act with a crack of width 2x,is

N共r兲⫽4x2

S

¯

共r兲

共A13兲

and the volume of these encountered pores is

Vpores共r兲⫽V共r兲N共r兲⫽4

3

r34x2

C

r2⫽16rx2

3C.共A14兲

The crack volume 关Eq. 共A1兲兴 is

Vcr⫽4x2⫽4kx2⫹

␣

共A15兲

and thus the ﬁnal temperature after crack–pores interaction

is

T⫽Tp⫹k

k⫹4r

3Cx

␣

共Tcr⫺Tp兲

⫽Tp⫹k

k⫹C1r

x

␣

共Tcr⫺Tp兲.共A16兲

Now we have to calculate the probability distribution of the

ﬁnal temperatures based on the following results: a. Tis only

a function of z⫽r/x

␣

共all other parameters are assumed to be

constant兲; b. the distribution of pores radii is given by P(r)

共Fig. 7兲and c. the distribution of crack half widths is given

by P1(x) for both short and long cracks 关Figs. 6共a兲and 6共b兲兴.

We ﬁrst calculate the distribution of z31

P共z兲⫽P

冉

r

x

␣

冊

⫽兺

ixi

␣

P共xi

␣

z兲P1共xi

␣

兲⌬xi

␣

⫽兺

i关a2xi

2

␣

zexp共⫺axi

␣

z兲P1共xi兲⌬x兴.共A17兲

Now from the relations

z⫽k

C1

Tcr⫺T

T⫺Tp;dz

dT⫽⫺ k

C1

Tcr⫺Tp

共T⫺Tp兲2共A18兲

we obtain the temperature distribution

P共T兲⫽P关z共T兲兴

冏

dz

dT

冏

⫽a2

冉

k

C1

冊

2Tcr⫺T

T⫺Tp

Tcr⫺Tp

共T⫺Tp兲2

⫻兺

i

冋

xi

2

␣

exp

冉

⫺axi

␣

k

C1

Tcr⫺T

T⫺Tp

冊

P共xi兲⌬xi

册

.

共A19兲

A comparison of Eq. 共A19兲with the distribution of tempera-

tures 共or rather (

)⫺1/4) evaluated from the EMR measure-

ments for both the small and the large cracks is shown in

Fig. 5.

1A. A. Urusovskaja, Sov. Phys. Usp. 11, 631 共1969兲.

2A. Misra, Nature 共London兲254, 133 共1975兲.

3U. Nitsan, Geophys. Res. Lett. 4, 333 共1977兲.

4N. G. Khatiashvili and M. E. Perel’man, Phys. Earth Planet. Inter. 57,169

共1989兲.

5J. W. Warwick, C. Stoker, and T. R. Meyer, J. Geophys. Res. 87共NB4兲,

2851 共1982兲.

6T. Ogawa, K. Oike, and T. Miura, J. Geophys. Res. 90共ND4兲, 6245 共1985兲.

7I. Yamada, K. Masuda, and H. Mizutani, Phys. Earth Planet. Inter. 57,157

共1989兲.

8A. Rabinovitch, D. Bahat, and V. Frid, Int. J. Fract. 71, R33 共1995兲.

9A. Rabinovitch, V. Frid, and D. Bahat, Philos. Mag. Lett. 5,289共1998兲.

10A. Rabinovitch, V. Frid, and D. Bahat, Philos. Mag. Lett. 79,195共1999兲.

11 A. Rabinovitch, V. Frid, D. Bahat, and J. Goldbaum, Int. J. Rock Mech.

Mining Sci. 37, 1149 共2000兲.

12V. Frid, A. Rabinovitch, and D. Bahat, Philos. Mag. Lett. 79,79共1999兲.

13D. A. Fifolt, V. F. Petrenko, and E. M. Schulson, Philos. Mag. B 67,289

共1993兲.

14S. G. O’Keefe and D. V. Thiel, Phys. Earth Planet. Inter. 89,127共1995兲.

15M. Miroshnichenko and V. Kuksenko, Sov. Phys. Solid State 22,895

共1980兲.

16N. Khatiashvili, Izv. Akad. Nauk SSSR, Ser. Fiz. 20,656共1984兲.

17V. Frid, D. Bahat, J. Goldbaum, and A. Rabinovitch, Israel J. Earth Sci.

49,9共2000兲.

18D. Bahat, Tectonofractography 共Springer, Heidelberg, 1991兲.

19D. Bahat, V. Frid, A. Rabinovitch, and V. Palchik, Int. J. Fract. 116,179

共2002兲.

20J. Goldbaum, V. Frid, A. Rabinovitch, and D. Bahat, Int. J. Fract. 111,15

共2001兲.

21V. Frid, A. Rabinovitch, and D. Bahat 共unpublished兲.

22G. P. Srivastava, 1990, The Physics of Phonons 共Hilger, Bristol, 1990兲.

23P. J. King and F. W. Sheard, Proc. R. Soc. London, Ser. A 320,175共1970兲.

24See, e.g., L. I. Schiff, Quantum Mechanics, 3rd ed. 共McGraw–Hill, Lon-

don, 1986兲.

25K. N. G. Fuller, P. G. Fox, and J. E. Field, 1975, Proc. R. Soc. London,

Ser. A 341, 537 共1975兲.

26D. Bahat, A. Rabinovitch, and V. Frid, J. Struct. Geol. 23, 1531 共2001兲.

27C. Yatomi, Eng. Fract. Mech. 14,759共1981兲.

28Y. Gueguen and V. Palciauskas, Introduction to the Physics of Rocks 共Prin-

ceton University Press, Princeton, 1994兲.

29T. Walmann, A. Malthe-Sørenssen, J. Feder, T. Jøssang, P. Meakin, and H.

H. Hardy, Phys. Rev. Lett. 77, 5393 共1996兲.

30G. Simmons, T. Todd, and W. S. Baldridge, Am. J. Sci. 275,318共1975兲.

31B. Harris, Theory of Probability 共Addison–Wesley, London, 1996兲.

5090 J. Appl. Phys., Vol. 93, No. 9, 1 May 2003 Rabinovitch

et al.