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Fractal dimension of the fractured surface of materials
Ph.~st~a D 38 I Iq$O) _4_-_45 " " " North-Holland. Amsterdam
FIL~.CTA[. DIMENSION OF THE FRACTURED SURFACE OF MATERIALS C.W. L U N G " ~~ and S.Z. Z H A N G ~" " Im(rnaltonal Centre for Theoretwa/Physics, Trieste. hal)" " I~ztcrnanonalCentre/'or ,llater~al.~ Ph.v~ws..4cademta Stnlca, i I00 I Y Shenyang. People's Rcpubh: ,l'Cho~a Fzmdamental Phvsm~ Centre. [ 'mversto' Of Science and Tethnology t~fChma. He.let..4nhuh PeopIe'~ Rcpubhc of China
The fraoal d~menston of the fractured surface of matenals ,s discussed m order to show that the or=gin of the negative corrclauon bet~.een D,. and the toughness hes m the method of fractal dimension measurement with the per,meter-area relation and also m the ph.xsncal mechamsm of crack propagation.
i. Introduction
of fractal dimension measurement and also in the physical mechanism o f crack propagation.
Mandelbrot et al. [I,2] firstly showed that fractured surfaces are fractals in nature and that the fractal dimensions of the surfaces correlate well with the toughness of the materials. The ~alues o f measured fractal d~mension D,: decrease smoothly with an mcrease of the toughaess of materials. One of the present authors [3] suggested that the effective crmcal crack extens,on force calculated from a fractal surface ~ould be larger than that calculated from a flat fracture surface due to the areas of the fractal surfaces being actually larger than flat ones. The fractured surface can be approximately considered as a fractal surface. He analyzed this problem in connection with fracture mechamcs. Mu and Lung [4] investigated lhe relationshi~ between '.he fractal dimension of the fractured surface and the fracture [oughness of materials. l h e x alues of fractal dimens'on decrease linearl} ~ t h an increase of the Iogar,~m ~aluc3 of fracture toughness K u . At the same ~nme man.~ authors haxe p(.mted out [ 2 , 4 - 6 ] that lhe ,:oriOatton b~t,~een Dr, and tough,less ~d)namlc tear energ.,,, fracture toughness, etc ) is a negatuve one ],.c. h~gher toushncss for smaller D,,). Th~s. howc~er. is d~fficuh to explain. The a~m of the presc.nt paper us to sho~ that the ~',r~gnn of the negat~,~e correlation hes m the method "J
• tll ll,~rh',Uf ,31 ~,Cl"h,-Ill ~
" r = , ( I J l . . ,n i"l-i, ~L~.S -
~
%|3r',dc],~r,)1
M1Jrc, n,,, and ! ~ c d c r I C O , r . ' , r . , ~
2. The perimeter-area relation of the Koch island with a finite number of generations The mathematical fractal model might have an infinite number of generations without upper and 16wer h m i t s otherx~ ise coarse graining ofthe initiator or fine graimng of the lowest I~mit of generation would not give the same shape as the original one. But the real;site fractals are always of a finite number ofgenerations. As an example, x~e may consider the triadic Koch island in fig. I. The initiator is a triangle with side length c,,=l. The coastline dimension is D,:.=log4/Iog 3 = 1.2618.
n=O
W=|
n=2
/',,,
...
~_/~_~ "'~____._.~
~,x_)
k__,'~, t
/
I
~'7
'1
L
,?_. 1
~'o = ~
1 ~ ~l =q --i
'3
,1,z
~
3~
Fng I The trnadn.: K c c h ~'r,_land ",tth a trmn.'le m n i a l o r ,~1¢~7.2-,',~ ':~,~ $03 ~0 '(' Elsc~ k'r Soen(c Pubhshers 13 ~, d,N c n h - H" o , I and P!l~,,.n~:s Publ,shm~ D r . qs ,)nl
C i1~ Lmlg and S Z. Zhang 'Fractal d~oten~_too of the I}',~ctured surta,'¢ olotatcrt,ff~
Considering the perimeter-area ( P-A ) relation of the triadic Koch island, the nth generation per,meter P,, and the area .4,, are
P,,/Po=(4/3)". A ,~.,-It,=
at some level nm,,. such that there could be no further structure to be revealed by refining our ruler. ~e ~ould find in units ofF',,, for ~z> ~z...... trig 3 ~.
P,,/ Po( n >
i +3. ( i/3):+3.4. + ..3,4"-
243
tim:., ) =
P,,/ P,,( tlmz~.
) •
(2)
( I / 3 )~
where P,-, corresponds to the perimeter of the init=ator. Then. P,, would hold a constant value even under further refining of our ruler and D(n) would have
' • ( 1/3 )-'".
Therefore
the value
D(n)=D(~) ( I -Do)
log e +log c~ log tx + [ ( I -Do)/Do] log ~+0.5 iog.4m( ~ ) " (I) where P,, and .q,, are the perimeter and area of the initiator, respectively; ~= ( I/3 )". From eq. ( 1 ) [6]. we see that D ( n ) drops to a m i n i m u m value as n = 3 at first and then goes up to a limit value 1.2618 as n increases to infinity {fig. 2). The fractal dimension defined by the P-A relation depends on the generation. Th~s means that the apparent value of fractal dimension measured by the P-A relation depends on the length of the yardstick. This example also illustrates that the P-A relation may be expected to hold only in the limit o f small .~ardsticks. This is conststent ~.ith the argumemsgiven by Feder [ 51 and Lung et al. [6] recently. Llnfortunatel.~. the fractal structure of a fractured surface may o n b be o f a few generations. We should pay attention to the characteristic oflhe first several generations. Note that if we were to terminate the construction
D ( !1 > tim.i, ) = D (
(3)
lima,, , ) .
in all physical situations, such a lower cutoff P,,/ P,( rim,, ) will be present, and an upper cutoff will also be present. Accord,rig to Ihese relations.
p
i 'Din|
m. CIt.,,.4 ~t'2
(4)
2 log P,i = D(~, )( 2 log c~,,-I-log-I,,). For the triadic island. ~e can calculate c,... b) eq. (41withP,, t , , a n d D ( l ~ t l ' r o m e f l s . ( l } - ( ~ , : . F,g. 4a gi~es us the P-A relation as a Iunclton of pt. It s h o ~ s that the l o g / ' - t o g ; relau~ n Is not hnear cspccla!l.~ m the first several generations. Wh~ do C. and -I,, data. measured b.~ man.~ author.',. ~ ~th ttw sh~island method I S I M ). look linear? We t h i n k th~s is a false impression. In the sl~t-island method, we usually measure man.,, different sizes of i n i t i a t o r s Itsland.: I. Then
1.9 m
1
1.26 1.2 D ~ ( rl',
1.1 / / \,%
L___ __
.'
_
_
____1
1OO
I'O-- --
1OOO
r~
F=£- 2 T h e rclatton',_htp o f rn.e3~,ured Ira,rt31 d t m e n ' , = o n D-.f tl} '~ ~2151aS I h ¢
[2CllCrdllOrl
i~umbt,:
i ,t
Ftg
~ Tile r c l a t t o n s h t p o l ' l O g t P .
P,,~ ~cr~,u.~l,',~ ~l~t~ ~n ~;n,~(
(C, llSIFUC!~OF1 o f frdch.qi ~1,,. | h e DumbCF ,3f L~cnL'FahOn ,ll|cr ",,,ltit?ll I0~.1 it'.. ' I'.. T, rt~m3l,-l$ Ct~rl.q3t3t .1., ,I • ~,.
C I1" Lung and S Z Zhang
244
0.4
a
~
0.3
02
0.1
/
/
.'
Fractal dtmen$ton qf the fractured sur.t'acc of mate.al.~
= 2 log r, (r,=Po,IP,). because the initiators are normal triangles. We may easily choose the values of Po,/ P,-, such that. in fig. 4b. data I' ( n = I ). 2' ( n = 2 ) . and 3' (n = 3 ) can be obtained. A linear relationship seems to be obtained by combining them w~tla the original points I ( n = l ) , 2 ( n = 2 ) , and 3 ( n = 3 ) . it seems questionable to determine the fractal dimension with the slope of this line because the reasonable lines would be o,] I. oz2 2 and oL33 in fig. 4a. We think that the fractal dimension measured with the P -A relation is not the real fractai dimension Do o f the fractured surfaces and that it is one of the origins of the negative correlation between measured Dm and the Iou[,nffe~,-~of materials. One o f the present authors and his coworker measured D m with different lengths of yardsticks. For sufficiently small yardstick length, a positive correlation between Dm and K~c was observed [6]. The best way to measure the fractal dimension of fractured surfaces may be the relation [ 8 ]
r
fJ
I I
I I /
/
I t
/
'~,
~3
0.5
08
b 0.4
0.3
A°
L(~ ) = L , ~ ~-r~'
/ /
0.2
.,~ 2'
We may measure tne total length o f crack propagation with different lengths of)ardsticks. Then D,j can be obtained b;
1 x ./'. 2 /' O.1
" i I
;
O
D,,= I-Io~.[Ll~),L,,l/Iog~
t" ,,,,'
/
(6)
.
(7)
I
0.3
0.5
'1
3. The micromechanism of fracture p. '.-~g J The penmeler-ar:a relation as a IunctJon of ,.
log i P,../P, ) = log ( P., / P,,, I + log ( P,,,/P,:, )
=1o~[ P. ( , ' ~ ] + l o g r .
(5)
log( 4...'.-[, ) = log(.4,, .' L,. ) + Iog{..-I,,..,'.-/,,) =log[ i.,.(n ) ] + 2 log r,. ~ here P,-,,and .-I,~,are the ~alues of the perimeter and area ofthe m m a t o r o f t h e tth isla,;d, respecovei,~. We nottce that log(P,, /P,.,,) and Iog(.4. /.-t,:,,~ are func~:or, s of pC and that !og(.4,:,,,,'.4,,) = 21oglP,,./P,-,I
Are there any other reasons thai ihe correlation between D,-, and Kt¢ is negative, other than the SIM of fractal dimension measurement? One of the present authors analyzed the critical crack exlension force with the fractal model for the ime:-granular fracture was found. In this paper we anal3ze the distance be~ e e n t'~.o large inclusions and lhe number of grains over the dislance. We show lhat in this case the c o l relat,on bet~een D,, and fraclure Ioughness could be a negative one. Let us suppose that r,e~ segmems of microcracks of grain size ~',ere sta~crimposcd on the preceding
C. II Lung and S.Z. Zhang ,' Fractal donenston tfflhe fractured 5urfdt'e ~tmatertals
_4.~
Table I The ~anatton of fractal d,menston of D,., ~ tth lhe change of segment r, umber A"and the angle ~,,, ~,(degl
90 100 120 140
,~r 2
3
4
5
6
7
8
2.0000 1.6247 1.2619 i.0986
1.3652 ~.2631 1.1292 I 0526
1.3333 1.2380 I.II58 1.0470
1.2549 1.1862 1.0932 1.0384
1.2398 1.1747 1.0873 1.0360
1.2091 1.1538 1.0778 1.0323
1.2000 11470 1.0743 1.0308
larger segment due to large inclusions. The fraclal dimension may be calculated as [ 7 ]
Do.. =log(N)/log[Nsin(~o/2 ) ]. 2 s i n - ' ( IIN ,/2)
N=even,
(8)
1.1833 1.1355 1.0690 10278
boundaries, inclustons, second phase particles, etc. E,;i- :rimental results are the total effects o f many elerr :ntary processes. We should take care to select the mare factor. Even then, it seems hopeful to use fractals to characterize fractured surfaces with which material parameters can be correlated.
Dot, =log(N)/Iog[ I + (N : - ! ) sin:(~/2) ]'"' 2sin-I[I/(N+l)l':]<~o<~t,
Acknowledgements
N=odd.
Here, we have assumed the angles between two adjacent segments to be equal. We also know that the value ofthe angle t0 depends en the grain configurations, in this case, the fiactal dimension Do ma3 decrease with an increase o f l h e segment number N (tabE'- ~ I. Decreasing either the test temperature .c.~ !~1e tempering temperature would produce maler~als ~h,ch have a high yield strength. High strength materials have a smaller critical crack length for propagation due to their low Ktc. It may induce more smaller cracks (inclusions) to propagate This makes the crack propagation betv, een two small inclusions easier. Then, the segment number N decreases, and the fractal dimension of the fractured surface mcreas:s. We therefore come ~o the conclusic, n that lht ~ractal dimension of fractured surfaces increases with *,he ~,J~l,.l~d~l~
~.t k I l d l , . I U l ~
LUUI~,IIIlI~33.
I Ilt.
'~.J~z~..~Lu'~.
o*.
t~ een D,, and .~,~,: is lherefore ne~.a~i~.e Real !'raclure processes 3re 4.-"~ :?.c.m~..~e~:.?6 ~ ~ ~de variety of mechanisms play rele,.aal ro!es: gz,zia
One of the authors ( C.W.L. ) would like to thank Professor Abdus Salam, the lnternatmnal A~omw Energy Agency and e,--,-, Internatio,al Centre for Theoretical Ph.vsics. Tr~esle. and Professor S. Lv.ndqx,'.s'~ for h~. advice and encovragemenl on ot~r "~ork on fraclals at Tnes'.e.
References [ I ] B B. Mandelbrot. The Fmcta Geome',~, of Nature ~Freeman. San Franosco. 19~3) pp. 25, 2'L 459. [ 2 ] B.B. Mandelbr?L D.E. Passoja end A.J. Pauliay, Naiure 30~. 4 I o~qa ~ 72 i.
[.,)C. ~V Lung ,n F,'aclals m Pn~s~c~, L. Plelronero a~d E Tosa'tl. ed~ : No.~-,-Holla~O. -km:,4erdam. I ~St, ;. t~ I S°
i~..i ~'V¢.Loo,~-_n_~:,:..
l~It: Pn.~s...V,e~ B3S{I~.~I
IJ-,.¢~.
FIL~.CTA[. DIMENSION OF THE FRACTURED SURFACE OF MATERIALS C.W. L U N G " ~~ and S.Z. Z H A N G ~" " Im(rnaltonal Centre for Theoretwa/Physics, Trieste. hal)" " I~ztcrnanonalCentre/'or ,llater~al.~ Ph.v~ws..4cademta Stnlca, i I00 I Y Shenyang. People's Rcpubh: ,l'Cho~a Fzmdamental Phvsm~ Centre. [ 'mversto' Of Science and Tethnology t~fChma. He.let..4nhuh PeopIe'~ Rcpubhc of China
The fraoal d~menston of the fractured surface of matenals ,s discussed m order to show that the or=gin of the negative corrclauon bet~.een D,. and the toughness hes m the method of fractal dimension measurement with the per,meter-area relation and also m the ph.xsncal mechamsm of crack propagation.
i. Introduction
of fractal dimension measurement and also in the physical mechanism o f crack propagation.
Mandelbrot et al. [I,2] firstly showed that fractured surfaces are fractals in nature and that the fractal dimensions of the surfaces correlate well with the toughness of the materials. The ~alues o f measured fractal d~mension D,: decrease smoothly with an mcrease of the toughaess of materials. One of the present authors [3] suggested that the effective crmcal crack extens,on force calculated from a fractal surface ~ould be larger than that calculated from a flat fracture surface due to the areas of the fractal surfaces being actually larger than flat ones. The fractured surface can be approximately considered as a fractal surface. He analyzed this problem in connection with fracture mechamcs. Mu and Lung [4] investigated lhe relationshi~ between '.he fractal dimension of the fractured surface and the fracture [oughness of materials. l h e x alues of fractal dimens'on decrease linearl} ~ t h an increase of the Iogar,~m ~aluc3 of fracture toughness K u . At the same ~nme man.~ authors haxe p(.mted out [ 2 , 4 - 6 ] that lhe ,:oriOatton b~t,~een Dr, and tough,less ~d)namlc tear energ.,,, fracture toughness, etc ) is a negatuve one ],.c. h~gher toushncss for smaller D,,). Th~s. howc~er. is d~fficuh to explain. The a~m of the presc.nt paper us to sho~ that the ~',r~gnn of the negat~,~e correlation hes m the method "J
• tll ll,~rh',Uf ,31 ~,Cl"h,-Ill ~
" r = , ( I J l . . ,n i"l-i, ~L~.S -
~
%|3r',dc],~r,)1
M1Jrc, n,,, and ! ~ c d c r I C O , r . ' , r . , ~
2. The perimeter-area relation of the Koch island with a finite number of generations The mathematical fractal model might have an infinite number of generations without upper and 16wer h m i t s otherx~ ise coarse graining ofthe initiator or fine graimng of the lowest I~mit of generation would not give the same shape as the original one. But the real;site fractals are always of a finite number ofgenerations. As an example, x~e may consider the triadic Koch island in fig. I. The initiator is a triangle with side length c,,=l. The coastline dimension is D,:.=log4/Iog 3 = 1.2618.
n=O
W=|
n=2
/',,,
...
~_/~_~ "'~____._.~
~,x_)
k__,'~, t
/
I
~'7
'1
L
,?_. 1
~'o = ~
1 ~ ~l =q --i
'3
,1,z
~
3~
Fng I The trnadn.: K c c h ~'r,_land ",tth a trmn.'le m n i a l o r ,~1¢~7.2-,',~ ':~,~ $03 ~0 '(' Elsc~ k'r Soen(c Pubhshers 13 ~, d,N c n h - H" o , I and P!l~,,.n~:s Publ,shm~ D r . qs ,)nl
C i1~ Lmlg and S Z. Zhang 'Fractal d~oten~_too of the I}',~ctured surta,'¢ olotatcrt,ff~
Considering the perimeter-area ( P-A ) relation of the triadic Koch island, the nth generation per,meter P,, and the area .4,, are
P,,/Po=(4/3)". A ,~.,-It,=
at some level nm,,. such that there could be no further structure to be revealed by refining our ruler. ~e ~ould find in units ofF',,, for ~z> ~z...... trig 3 ~.
P,,/ Po( n >
i +3. ( i/3):+3.4. + ..3,4"-
243
tim:., ) =
P,,/ P,,( tlmz~.
) •
(2)
( I / 3 )~
where P,-, corresponds to the perimeter of the init=ator. Then. P,, would hold a constant value even under further refining of our ruler and D(n) would have
' • ( 1/3 )-'".
Therefore
the value
D(n)=D(~) ( I -Do)
log e +log c~ log tx + [ ( I -Do)/Do] log ~+0.5 iog.4m( ~ ) " (I) where P,, and .q,, are the perimeter and area of the initiator, respectively; ~= ( I/3 )". From eq. ( 1 ) [6]. we see that D ( n ) drops to a m i n i m u m value as n = 3 at first and then goes up to a limit value 1.2618 as n increases to infinity {fig. 2). The fractal dimension defined by the P-A relation depends on the generation. Th~s means that the apparent value of fractal dimension measured by the P-A relation depends on the length of the yardstick. This example also illustrates that the P-A relation may be expected to hold only in the limit o f small .~ardsticks. This is conststent ~.ith the argumemsgiven by Feder [ 51 and Lung et al. [6] recently. Llnfortunatel.~. the fractal structure of a fractured surface may o n b be o f a few generations. We should pay attention to the characteristic oflhe first several generations. Note that if we were to terminate the construction
D ( !1 > tim.i, ) = D (
(3)
lima,, , ) .
in all physical situations, such a lower cutoff P,,/ P,( rim,, ) will be present, and an upper cutoff will also be present. Accord,rig to Ihese relations.
p
i 'Din|
m. CIt.,,.4 ~t'2
(4)
2 log P,i = D(~, )( 2 log c~,,-I-log-I,,). For the triadic island. ~e can calculate c,... b) eq. (41withP,, t , , a n d D ( l ~ t l ' r o m e f l s . ( l } - ( ~ , : . F,g. 4a gi~es us the P-A relation as a Iunclton of pt. It s h o ~ s that the l o g / ' - t o g ; relau~ n Is not hnear cspccla!l.~ m the first several generations. Wh~ do C. and -I,, data. measured b.~ man.~ author.',. ~ ~th ttw sh~island method I S I M ). look linear? We t h i n k th~s is a false impression. In the sl~t-island method, we usually measure man.,, different sizes of i n i t i a t o r s Itsland.: I. Then
1.9 m
1
1.26 1.2 D ~ ( rl',
1.1 / / \,%
L___ __
.'
_
_
____1
1OO
I'O-- --
1OOO
r~
F=£- 2 T h e rclatton',_htp o f rn.e3~,ured Ira,rt31 d t m e n ' , = o n D-.f tl} '~ ~2151aS I h ¢
[2CllCrdllOrl
i~umbt,:
i ,t
Ftg
~ Tile r c l a t t o n s h t p o l ' l O g t P .
P,,~ ~cr~,u.~l,',~ ~l~t~ ~n ~;n,~(
(C, llSIFUC!~OF1 o f frdch.qi ~1,,. | h e DumbCF ,3f L~cnL'FahOn ,ll|cr ",,,ltit?ll I0~.1 it'.. ' I'.. T, rt~m3l,-l$ Ct~rl.q3t3t .1., ,I • ~,.
C I1" Lung and S Z Zhang
244
0.4
a
~
0.3
02
0.1
/
/
.'
Fractal dtmen$ton qf the fractured sur.t'acc of mate.al.~
= 2 log r, (r,=Po,IP,). because the initiators are normal triangles. We may easily choose the values of Po,/ P,-, such that. in fig. 4b. data I' ( n = I ). 2' ( n = 2 ) . and 3' (n = 3 ) can be obtained. A linear relationship seems to be obtained by combining them w~tla the original points I ( n = l ) , 2 ( n = 2 ) , and 3 ( n = 3 ) . it seems questionable to determine the fractal dimension with the slope of this line because the reasonable lines would be o,] I. oz2 2 and oL33 in fig. 4a. We think that the fractal dimension measured with the P -A relation is not the real fractai dimension Do o f the fractured surfaces and that it is one of the origins of the negative correlation between measured Dm and the Iou[,nffe~,-~of materials. One o f the present authors and his coworker measured D m with different lengths of yardsticks. For sufficiently small yardstick length, a positive correlation between Dm and K~c was observed [6]. The best way to measure the fractal dimension of fractured surfaces may be the relation [ 8 ]
r
fJ
I I
I I /
/
I t
/
'~,
~3
0.5
08
b 0.4
0.3
A°
L(~ ) = L , ~ ~-r~'
/ /
0.2
.,~ 2'
We may measure tne total length o f crack propagation with different lengths of)ardsticks. Then D,j can be obtained b;
1 x ./'. 2 /' O.1
" i I
;
O
D,,= I-Io~.[Ll~),L,,l/Iog~
t" ,,,,'
/
(6)
.
(7)
I
0.3
0.5
'1
3. The micromechanism of fracture p. '.-~g J The penmeler-ar:a relation as a IunctJon of ,.
log i P,../P, ) = log ( P., / P,,, I + log ( P,,,/P,:, )
=1o~[ P. ( , ' ~ ] + l o g r .
(5)
log( 4...'.-[, ) = log(.4,, .' L,. ) + Iog{..-I,,..,'.-/,,) =log[ i.,.(n ) ] + 2 log r,. ~ here P,-,,and .-I,~,are the ~alues of the perimeter and area ofthe m m a t o r o f t h e tth isla,;d, respecovei,~. We nottce that log(P,, /P,.,,) and Iog(.4. /.-t,:,,~ are func~:or, s of pC and that !og(.4,:,,,,'.4,,) = 21oglP,,./P,-,I
Are there any other reasons thai ihe correlation between D,-, and Kt¢ is negative, other than the SIM of fractal dimension measurement? One of the present authors analyzed the critical crack exlension force with the fractal model for the ime:-granular fracture was found. In this paper we anal3ze the distance be~ e e n t'~.o large inclusions and lhe number of grains over the dislance. We show lhat in this case the c o l relat,on bet~een D,, and fraclure Ioughness could be a negative one. Let us suppose that r,e~ segmems of microcracks of grain size ~',ere sta~crimposcd on the preceding
C. II Lung and S.Z. Zhang ,' Fractal donenston tfflhe fractured 5urfdt'e ~tmatertals
_4.~
Table I The ~anatton of fractal d,menston of D,., ~ tth lhe change of segment r, umber A"and the angle ~,,, ~,(degl
90 100 120 140
,~r 2
3
4
5
6
7
8
2.0000 1.6247 1.2619 i.0986
1.3652 ~.2631 1.1292 I 0526
1.3333 1.2380 I.II58 1.0470
1.2549 1.1862 1.0932 1.0384
1.2398 1.1747 1.0873 1.0360
1.2091 1.1538 1.0778 1.0323
1.2000 11470 1.0743 1.0308
larger segment due to large inclusions. The fraclal dimension may be calculated as [ 7 ]
Do.. =log(N)/log[Nsin(~o/2 ) ]. 2 s i n - ' ( IIN ,/2)
N=even,
(8)
1.1833 1.1355 1.0690 10278
boundaries, inclustons, second phase particles, etc. E,;i- :rimental results are the total effects o f many elerr :ntary processes. We should take care to select the mare factor. Even then, it seems hopeful to use fractals to characterize fractured surfaces with which material parameters can be correlated.
Dot, =log(N)/Iog[ I + (N : - ! ) sin:(~/2) ]'"' 2sin-I[I/(N+l)l':]<~o<~t,
Acknowledgements
N=odd.
Here, we have assumed the angles between two adjacent segments to be equal. We also know that the value ofthe angle t0 depends en the grain configurations, in this case, the fiactal dimension Do ma3 decrease with an increase o f l h e segment number N (tabE'- ~ I. Decreasing either the test temperature .c.~ !~1e tempering temperature would produce maler~als ~h,ch have a high yield strength. High strength materials have a smaller critical crack length for propagation due to their low Ktc. It may induce more smaller cracks (inclusions) to propagate This makes the crack propagation betv, een two small inclusions easier. Then, the segment number N decreases, and the fractal dimension of the fractured surface mcreas:s. We therefore come ~o the conclusic, n that lht ~ractal dimension of fractured surfaces increases with *,he ~,J~l,.l~d~l~
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t~ een D,, and .~,~,: is lherefore ne~.a~i~.e Real !'raclure processes 3re 4.-"~ :?.c.m~..~e~:.?6 ~ ~ ~de variety of mechanisms play rele,.aal ro!es: gz,zia
One of the authors ( C.W.L. ) would like to thank Professor Abdus Salam, the lnternatmnal A~omw Energy Agency and e,--,-, Internatio,al Centre for Theoretical Ph.vsics. Tr~esle. and Professor S. Lv.ndqx,'.s'~ for h~. advice and encovragemenl on ot~r "~ork on fraclals at Tnes'.e.
References [ I ] B B. Mandelbrot. The Fmcta Geome',~, of Nature ~Freeman. San Franosco. 19~3) pp. 25, 2'L 459. [ 2 ] B.B. Mandelbr?L D.E. Passoja end A.J. Pauliay, Naiure 30~. 4 I o~qa ~ 72 i.
[.,)C. ~V Lung ,n F,'aclals m Pn~s~c~, L. Plelronero a~d E Tosa'tl. ed~ : No.~-,-Holla~O. -km:,4erdam. I ~St, ;. t~ I S°
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