Some fractal models of fracture

$Page 1: Some fractal models of fracture$
Pergamon 1. Me& Phw. Solids, Vol. 45, No. 2, 239-259. 1997 pp.

Copyright 9: 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved

PI1 : SOO22-5096(96)00080-4 0022-5096197 $17.OO+O.U0

SOME FRACTAL MODELS OF FRACTURE*

F. M. BORODICHt

Department of Applied Mathematics and Theoretical Physics, University of Cambridge. Silver Street, Cambridge CB3 9EW, U.K.

(Received 28 September 1995 ; in recised form 22 June 1996)

ABSTRACT

The paper deals with applications of fractal geometry methods to problems of fracture in brittle and quasi- brittle materials Possible ways to construct models, taking into account the fractal properties of the phenomenon, are discussed. It is shown that classical approaches do not work in fractal fracture and lead to the paradoxical conclusion that fractal cracking is impossible. Some new concepts appropriate for fractal fracture are introduced. In particular, the concept of specific energy-absorbing capacity for a unit of a fractal measure of a fractal set is considered. It is shown that fractal properties of a fractal pattern of microcracks can characterize the fracture energy of polyphase materials such as rock, concrete, ceramics. etc., and that fractal properties of the main crack surface can be inessential. (4 1997 Elsevier Science Ltd. All rights reserved

Keywords: A. fracture, A. microcracking, B. ceramic material, B. concrete, B. rock, C. optical microscopy.

1. INTRODUCTION

During the past two decades methods of fractal geometry have attracted wide atten- tion as a tool for describing various non-smooth natural phenomena, in particular in

mechanics and physics of solids [see, e.g. Mandelbrot (1982) and Turcotte (1992)].

For example following the pioneering studies of Mandelbrot et ul. (1984) the fractal

nature of fracture surfaces of real solids has been confirmed by numerous experimental studies. The surfaces of cracks in rock as well as ceramics, concrete and metal have

been found to be fractal [see reviews in Dauskardt et al. (1990), Milman et al. (1994), Mosolov (1993) and (Saouma and Barton (1994)]. It was also shown that patterns of

microcracks and crack size distributions possess self-similar nature (Barenblatt and Botvina, 1986 ; Chudnovsky and Wu, 1992 ; Barenblatt, 1993 ; Botvina et al., 1995).

There arises the following question: is there any connection between the fractal dimension of fracture surfaces and the fracture toughness of the material? So far no unequivocal answer has been found. Studies of ceramics suggest a positive answer (Mecholsky et al., 1989), but in recent experiments with glass and porcelains no

quantitative relationship has been established (Baran et al., 1992).

*The paper was presented at IUTAM Symposium on Nonlinear Analysis of Fracture. University of Cambridge, 3-7 September, 1995.

t Present address: Department of Mathematics, Glasgow Caledonian University, Glasgow G4 OBA. U.K.

239

240 F. M. BORODICH

The terms “fractal”, “fractal dimension” and “fractal geometry” have been used

above without a definition. However, what is fractal? How can we evaluate the fractal dimension of a surface or pattern? Sets with non-integer Hausdorff dimension were

named fractals by Mandelbrot (1982). Note that the Hausdorff dimension may be established for pure mathematical objects only. In applications other definitions of

fractal sets are used. Usually the fractal dimension is understood on the intuitive level as follows. For 6 > 0, let N(6) be the number of cubes or balls of diameter 6 needed

to cover the set S. Then, if N(6) increases like d-D as 6 + 0, one says that S has fractal dimension D. Of course, this definition is not a pure mathematical definition of fractal

objects. We reckon that fracture surfaces of real solids and the process zone (a pattern of microcracks) can be described by methods of fractal geometry. Here we use both fractal mathematical and physical models, namely we describe the following models

of fractal fracture :

(1) solitary fractal crack in brittle materials (a mathematical fractal) ; (2) solitary fractal crack in quasi-brittle materials (a physical fractal) ; (3) mathematical description of fractal fracture network in brittle materials ; (4) physical fractal description of multiple quasi-brittle fracture.

This paper is organized as follows. In Section 2 we give basic definitions of fractal

geometry, consider the difference between the mathematical and physical fractals and

describe practical methods of evaluating fractal dimension in physical applications. We consider some experimental results concerning evaluation of fractal dimensions

of fracture surfaces and patterns. In Section 3 we consider model problems of propagation of a solitary crack in

brittle and quasi-brittle solids. We discuss consequences of postulating that the fracture surface is modelled as a mathematical fractal surface ; in particular we show that

in the framework of fractal geometry the classical approaches do not work. For

example, the Griffith criterion in its classical formulation leads to the paradoxical conclusion that no fractal cracking is possible (Borodich, 1992, 1994a).

Using the new concepts of specific energy absorbing capacity of a fractal surface

we give a resolution of the above paradox and show that the fractal crack propagating in a perfect brittle solid is stable.

In Section 4 we model a tip of a continuous crack as surrounded by a growing cloud of microcracks which we postulate to be a fractal cluster with multiple branches. The crack surface is only one realization of infinitely many. Hence its fractal dimension does not coincide with the fractal dimension of the microcrack cloud. The model is

specific for cracking in polyphase quasi-brittle materials. In our models we use two kinds of self-similarity, namely we suppose that if the

size of the considered area of the fracture is less than some critical size then the growing fracture is self-similar with respect to the group of the homogeneous coordinate dilations, while if the size of the considered area of the fracture is equal to the critical size then the growing fracture is self-similar with respect to the group of the coordinate translations.

We give some estimates for the fracture energy of fractal cracking in brittle materials and quasi-brittle materials. Using these estimates we obtain an explanation for those cases when a positive correlation between the fractal dimension of the fracture surface

Some fractal models of fracture 241

and the fracture energy may be observed. We give also a fractal interpretation of experimental results regarding the behaviour of fracture energy of concrete. It is shown that fractal properties of fractal patterns of microcracks can characterize the fracture energy of polyphase materials such as rock, concrete, ceramics, etc., and that fractal properties of the main crack surface are usually inessential.

3 L. FRACTAL GEOMETRY AND FRACTURE MECHANICS

Now we give some definitions of fractal objects and describe a practical method to evaluate fractal dimension in applications to fracture. We consider also some experimental results concerning evaluation of fractal dimensions of fracture surfaces and pattern of fracture or faults.

2.1. Some basic definitions of fractal geometry

Usually one says that a fractal is a set with a non-integer fractal dimension. As fractal dimension one takes one from numerous dimensions: Hausdorff dimension, Kolmogorov capacity, similarity dimension, etc.

Let 0 be the totality of open balls and G be a cover of a set S. Recall that for s 3 0 the Hausdorff s-measure of a set S is defined as the following limit

mH(S, s) = Jlp+ ini c (diam k’), : S c 1 V, diam V < 6 .

VEG I’EG

Here G is a finite or denumerable subset of 0. It was shown that the value m,,(S. .s) possesses the following property : there exists the value D such that

%, for s < 0. mH (S, s) =

0, for s > D.

The Hausdorff s-measure mH(S,s) possesses also the following property of homogeneity [see, e.g. Falconer (1990)]

mH(k3,s) = A’m,(S,s) foreveryi > 0, (xE/1S)+2 ‘x =x, ES).

The Hausdorff dimension of the set S is defined by

dim,S=D=inf(s:m,(S,s)=Oj=sup(s:m,(S,s)=cc~.

Now we give another definition of the fractal dimension, namely the Kolmogorov capacity which usually is called the box counting dimension. For 6 > 0, let N(6) be the smallest number of E-dimensional balls or cubes of diameter 6 needed to cover the set S. Here E is the Euclidean dimension of the space in which the set is embedded. Consider an s-measure m, of the set, i.e. the following limit

m,(S) = gl$n+ N(6) 8’. (1)

The value D such that

242 F. M. BORODICH

m&f9 = 0, s > D,

00, s < D,

is called the box counting dimension of the set or the box dimension. We say that a set is D-measurable if its measure has a finite positive value m,(S)

for s equal to the fractal dimension D, i.e.

0 < m,(S) = Jjl+ N(6) dD < co. (3)

We call mD(S) the fractal measure of set S. It is easy to check that the fractal measure exhibits the following property of

homogeneity (a scaling law)

m,(S) = ADm,(S), (xEAS)*(AX’x = x1 ES). (4)

Note that if a set S is a subset of 1.S for every ,? > 0 then S is a non-fractal set. Thus, the statement SE AS, V/z > 0, where S is a fractal, should be understood only in a statistical sense. Evidently, the above fractal dimensions of the set do not change under the transformation of homogeneous coordinate dilations.

In this paper we consider only fractal sets such that are D-measurable and invariant (in statistical sense) under the transformation of homogeneous coordinate dilations in an interval between some upper and lower cutoff, i.e. we deal with D-measurable self-similar fractals.

If we consider a natural phenomenon and suppose that it has fractal features then, in order to describe it, we can use one of the following ways :

(1)

(4

we give a mathematical simulation of the phenomenon, i.e. we model the phenomenon as a pure mathematical fractal and apply the strictly mathematical approach ; we consider a “physical fractal”, i.e. we consider that an object obeying fractal laws when the scale varies in an interval between upper and lower cutoff and apply the collection of typical methods of fractal geometry.

As an example of a fractal curve we can consider the classic von Koch curve [Fig. l(a)]. However, as pointed out by Baiant (1995a, b) the von Koch curve does not allow kinematic separation of surfaces, because of the possibility of recessive segments of fracture path. Therefore, such a curve is a physically impossible model of fractal fracture. As examples of fractal curves which allow kinematic separation we can consider the graphs of fractal functions b, and b2 [Fig. 1 (b) and (c)]

b,(x;p) = xb,(x;p), b,(x;p) = x*b,(x;P),

b,(x;p) = xD-’ .f ~+~~*‘(l -cospnx)+A,, n= pm

(5)

where p > 1, 1 < D < 2, and A, is a constant. They are a particular case of the so- called parametric-homogeneous functions introduced by Borodich (1994b) [see also Borodich (1995)]. Both functions strictly satisfy the following equation

Some fractal models of fracture

(a)

(b)

Fig. I. Examples of fractal curves: (a) the von Koch curve; (b) the graph of the fractal parametric- homogeneous function b, as an example of a self-similar fractal crack; (c) the graph of the fractal parametric-homogenous function h, as an example of a self-affine fractal crack. Both graphs h, and b2 have the same Hausdorff dimension as the Weierstrass-Mandelbrot function, whose box-dimension is D. They

were drawn using (5) for]] = 1.5 ; D = 1.5 ; A, = -6.33998.

244 F. M. BORODICH

hi(PkX;P) =Pkdbd(X;P), kez

for d = 1 and d = 2, respectively, and have the same Hausdorff dimension as the Weierstrass-Mandelbrot function, whose box-dimension is D.

If we consider a curve as a mathematical fractal then we have the following properties :

(1) the length .ZF of a fractal curve does not converge but, instead, increases as a power function with exponent D

ZF N s-N(6), N(6) 2: (Z&y,

where _Y,, is the length of the measured interval along the straight line and 6 is a current measuring size.

(2) a fractal curve has an infinite set of vertices. Moreover, it can be continuous everywhere and differentiable nowhere.

For such a curve we cannot use, at least in the usual sense, such a common notion as a normal. Thus it is impossible to use, for a solid with a fractal crack, the classical formulation of a mixed boundary value problem.

From (1) and (3) we have that the dimension of a fractal curve is less than or equal to 2, i.e. 1 < D < 2. Similarly, the dimension of a fractal surface is less than or equal to 3, i.e. 2 < D < 3. If we use the fractional part D* of the fractal dimension, 0 < D* < 1, then we can write D = 1+ D* in the case of a fractal curve and D = 2 + D* in the case of a fractal surface.

In this paper we will consider only fractal curves and sets of intervals (cuts) making up a fractal, therefore the dimension of the fractal measure m,(.YP,) is

[m,(_Yo)] = LltD*,

where L denotes the dimension of length and LYO is the length of the projection of the fractal curve onto the x-axis. Using (3) and (4) we can obtain for a fractal curve

where _P, is the length of the projection of the extended fractal curve onto the x-axis. It seems to us that if the fracture process is imagined as a fractal one then it is

natural to attribute physical quantities to the fractal measure m, of the considered mathematical model, rather than to the infinite length of the fractal curve, or to the infinite area of the fractal surface (Borodich, 1992, 1994a). The idea was also discussed by Barenblatt with regard to some biological problems (Barenblatt, 1987 ; Barenblatt and Monin, 1983).

We will apply the models of the mathematical fractals to problems of brittle fracture and the physical fractals to problems of quasi-brittle fracture.

2.2. A practical method to define fractal dimension

There are a lot of methods for practical estimation of fractal dimension of a considered object. There follow two examples of such methods for objects in a plane I%*.

Some fractai modeis of fracture

Fig. 2. Box counting method.

245

The box counting method was often employed for practical estimation of fractal

dimension of an object in a plane. To obtain the value D of box dimension we discretize the considered region into squares with the size length S (Fig. 2). Then we

count the number N(S) of squares of size 6 which have intersections with the con-

sidered pattern. If the object has fractal properties then repeating this procedure for different values of 6 the following relation is obtained

and we estimate D from the slope of linear growth of ln(N(6)) against ln(6). The Richardson method was often employed for practical estimation of fractal

dimension of curves and profiles of vertical sections of fractured surfaces. To get the value D of the dimension we measure the profile length 2, by the measuring size 6.

If the object has the fractal properties then repeating this procedure for different values of 6 the following relation is obtained

and we estimate D from the slope of linear growth of ln(Y,) against ln(6). Using the above methods and analogous ones the following experimental results

shown in Tables 1 and 2 were obtained. The analysis of published experimental data in Tables 1 and 2 shows that usually

it is possible to give the following estimations I .04 < D < 1.33 [the Sakellariou et al. (I 99 1) results for rock specimens are out of this interval only] and 1.47 < C < 1.79.

3. SOLITARY FRACTAL CRACK IN BRITTLE SOLIDS

The problem of fracture for a solitary fractal crack is only a model problem because the natural fracture surfaces are not usually self-similar fractals. However, this model is very helpful for consideration of fracture fractal patterns.

246 F. M. BORODICH

Table 1. Fractal dimensions D of fracture surfaces for main cracks or,fbults

Material Fractional part of

fractal dimension D* Authors

Steel alloys Steel alloys Aluminium alloys Porcelain Ceramics Concrete Concrete Rock Limestone Sandstone Earth’s crust (San Andreas fault) :

Entire Southeast segments

0.1-0.28 0.040.26 0.1 l-0.30 0.160.33 0.09-0.33 0.04-0.26

0.071-0.165 0.117-0.525

0.20-0.24 0.27-0.33

0.29-0.33 0.18-0.28

Mandelbrot et al., 1984 Dauskardt et al., 1990 Bouchaud et al., 1990 Baran rt al., 1992 Mecholsky et al., 1989 Saouma et al., 1990 Saouma and Barton, 1994 Sakellariou et al., 199 1 Zhao et al., 1993 Zhao et al., 1993 Okubo and Aki, 1987, Aviles et al., 1987

Table 2. Fractal dimensions C offracture networks or,fault patterns

Material Fractional part

fractal dimension C* Authors

Uncemented quartz sand 0.74f0.05 Earth’s crust (Japan) 0.60.7 Marble 0.47-0.57 Dolomite marble 0.560.73

Sornette et al., 1990, Davy et al., 1990 Hirata, 1989 Zhao et al., 1993 Chelidze et al., 1994

3.1. Theoretical description ojljiacture processes

Now we will recall some concepts of classical fracture mechanics. Usually, fracture mechanics deals with an isolated, ideal atomically sharp crack, and is based on the classic Griffith approach (Griffith, 1920).

Consider a linear elastic specimen with a straight-through planar cut of an initial length 21 [Fig. 3(a)]. Griffith established a criterion for crack growth by estimating the

energy cost for creating a cut in a solid under a uniaxial tensile stress (i perpendicular to the cut.

To estimate the energy reduction after cutting, we will use the so-called force lines method (Yokobori, 1978; Kershtein et al., 1989; Baiant and Kazemi, 1990).

When the specimen has no cut, the field of force lines is straight and uniform [Fig. 3(b)]. The elastic energy U,, stored in a unit volume of the specimen, is equal to

U, = 0’/(2E,), (6)

where E, is the elastic modulus, equal to the Young modulus E for plane stress and to E/( 1 -v’) for plane strain (v is the Poisson ratio).


0 (a) 0 (b) 0 (c) Fig. 3. Schematic illustrations of a specimen loaded under a tensile stress : (a) the specimen with a central crack ; (b) the field of force lines in the specimen without a crack ; (c) the field of force lines in the specimen

with a cut and the domain of relaxed stress.

After cutting, the field of force lines is sharply heterogeneous in the neighbourhood of the cut tips. The cut relaxes the stress in a domain, whose shape can be approximated

as an ellipse (Yokobori, 1978) or a rhombus (Kershtein et ul., 1989; Baiant and

Kazemi, 1990). The area A of the domain is estimated as

A = k,l’, (7)

where k, is a constant [Fig. 3(c)]. Suppose that the total release energy AU is pro-

portional to the area A. Using (6) and (7) we have

AU = -k,I’tU, = -k,I’to2/(2E,),

where t is the specimen thickness.

(8)

According to Griffith, one of the most important parameters characterizing the resistance of materials to cracking is the so-called specific surface energy y. Physically,

this quantity gives a measure of the elastic energy used on forming a unit of a new

solid surface. In the overwhelming majority of papers the specific surface energy 7 is

regarded as a universal constant of the material. Using the quantity y, it is possible to calculate the surface energy TI of the cut as

n = 4yrl.

Substituting (8) and (9) in the Griffith criterion

d(Il+ ACT)/dl = 0,

we obtain the value of critical stress (T, when the crack begins to propagate.

(9)

(10)

CJ~ = ,,/4E, y/(k, I). (11)

Exact calculations show that k, = 271 in the case of a smooth cut. Formula (11) leads to the conclusion that the crack state for cr = gC is unstable

because the critical stress decreases for a propagating crack.

248 F. M. BORODICH

3.2. Paradox of fractal fracture in brittle solids and its resolution (a mathematical

fractal model)

Now we present some consequences of the proposed fractal character of fracture, in particular we show that classical approaches do not work in fractal geometry and introduce some appropriate new concepts for describing fractal fracture.

Let us use the above Griffith arguments for the case when the crack growing from the cut has a fractal trajectory, i.e. for a mathematical fractal. We will assume that there is some length A* such that

_5?r N 6(_Y,/6) = _Y’, N(6) -(S!?,J6) for 6 2 A*,

i.e. fractal features of the crack are displayed for 6 < A*. We will call the maximal distance H, between the x-direction and the fractal crack trajectory crack-thickness.

After loading the crack advances to 2(l+a), where a is the crack growth during loading, i.e. a is the length of the projection of the fractal crack onto the x-axis. We consider the case when a < A,. If the fractal crack-thickness H, satisfies the condition H, << l+a, then we obtain from (8) that

AU = -k, to’(l+a)2/(2E,), (12)

i.e. the amount of energy released during fractal cracking is the same as for the smooth cut.

The above force lines method was applied to the fractal cracking problem by Borodich (1994). Independently this method was applied to the problem by Baiant (1995a).

The total surface energy fI of the initial cut and of the advanced crack is equal to

II(l+a) = II(l) +n(a).

However we cannot use (9) for II(a) in the case of fractal cracking. Indeed, if we suppose y to be constant and the crack surface to be fractal, then using (4) we obtain for the surface energy II(a)

II(a) = 2yt6(a/6)D + a3 as 6 + O+.

Thus the Griffith criterion in the classical formulation leads to the paradoxical conclusion that fractal cracking is impossible. Therefore, we should introduce some new concepts to consider fractal cracks.

Borodich (1992) proposed the introduction of the so-called specific energy absorbing capacity of a fractal surface p(D*). Physically, /?(D*) gives the amount of elastic energy spent on forming a unit of the fractal measure m,.

The dimension of p(D*) is

[B(D*)] = FL/L’+‘*,

where F is the dimension of force. Using the /?(D*) and the D-measurability of fractals we obtain for some fixed value

of fractal advance a = _YO

II = 2tP(D*)m,(Z,) < a3.


Using the scaling properties of the fractal measure (4) where 2 = u/JZO, we have for

the surface energy

II(a) = 2t~(D*)m&z,)(a/sP”)D < @z.

If we rewrite the Griffith criterion (10) as

(13)

$(II+A(i) = 0

then, using (4) and (13) we obtain

28(0*)Dm,(~~)(a/~,)“-‘~~’ -k,al(/+a)/E, = 0.

Thus we obtain the value of critical stress CJ~

We see that the fractal crack propagated in a perfect brittle solid is stable. Indeed, a<

grows with a. If A* CC a then we have

n(/+a> = ~(I)+2t~(D*)m,(P’,) (2)” (t); AU = -k,ta’(Z+a)‘/(2E,).

The above estimation of ac was obtained by Borodich (1994a). Later Baiant ( 1995a.

b, c) obtained a similar formula using the asymptotic expansion technique. If one evaluates this estimation by the force line method as very rough, then we

note that anyway the amount of energy U is finite and the area of a mathematical fractal surface is infinite. Then the use of the term 1’ leads to the above paradox.

3.3. Fracture energy of afractal crack in quasi-brittle solids (aphysicalfractal model)

The fracture energy G, is defined as the energy absorbed by the specimen per unit of crack advance [see, e.g. Inghels et al. (1990) and Brameshuber and Hilsdorf (1990)]

1 i;(w)

GF =f--- i3a ’

where a is crack growth during loading and (IV) is the average of the total amount

of energy absorbed by the crack. However, we are not able to measure and record the increasing crack length smoothly enough in an experiment. Thus, the amount GF is usually calculated as the average value covering several millimeters increments of crack extension [see, e.g. Inghels et al. (1990) and Brameshuber and Hilsdorf (1990)].

Let us take into account the dissipation of volume energy, which takes place in

250 F. M. BORODICH

Fig. 4. Schematic illustrations of a solitary fractal crack and the layer of plastic deformation: S, is the smallest measuring size which has influence on the area of the layer, 6, is the measuring size for the crack

and the layer.

quasi-brittle materials. We will assume that the surface of a fractal crack is enclosed within a thin layer of width 2h, where inelastic deformations take place. We take as the layer of plastic deformation the zone consisting of points whose distance from the crack trajectory is less than h (Fig. 4).

Let us suppose that the width h is much smaller than the fractal crack-thickness H,, i.e. h cc He.

Evidently, there is such a small measuring size 6,, that the plastic zone area is independent on the fractal crack roughness of the order 6,.

Let us introduce the effective energy absorbing capacity Gr of an elementary piece 6, of fractal crack, 6, < 6,. In other words, we view the crack as a cluster of elementary crack particles of length 6, clinging to each other. For such a physical fractal model, the scaling relation can be rewritten as

_YF N 6*(_PO/&JD for 6, CC _YO.

Note that this cluster consists of a solitary branch only. Using (13) we obtain

(14)

Gf = 2tB(D*)m,(6,)6- ’ * + G,,

where G, is the energy absorbed by plastic deformation while the crack grows by the length 6, along the fractal trajectory. Usually 2tg(D*),,(6*) 6; ’ << G,, i.e. Gr 2: G,.

The dimensions of G, and G, are

[G,] = F, [G,] = F.

Analogously to the case of brittle cracking we can write the following formula for the average of the total amount (IV’) of absorbed energy for a fractal crack growing in a quasi-brittle material from a cut or a notch

(w> ‘y G&Y&). (15)

Then using (14) and that for x > A* the fractal picture is repeated, i.e. each piece of the crack of length A* absorbs the same value of energy, we obtain

Some fractal models of fracture

(w>- G,~~,(.Y/&)“. b, < s < A*,

G,6,(A,/6,)“(x/A,), A* < s.

251

(16)

From (16) we have the following formula for the fracture energy

GF = t.- ’ d(W) G,( 1 + ~*)(.u/&+J”* ‘t, 0, < .Y < A*,

d.u G,( 1 + 0*)(A&S,)“*,‘r. A* < .v. (17)

Evidently, this formula is valid for .Y > 6, only, i.e. when the fractal crack is already

formed. Since the “mass” of the fractal curve is non-uniform along the s-direction. the real value of GF(x) is a stochastic function and the above G, is the average of the function GF(x).

We have shown some realizations of the above idea that physical quantities adopted

in fractal models can have non-traditional dimensions. In fracture mechanics (in applications to a solitary fractal crack model) this idea was first introduced by

Borodich (1992). Afterwards, this idea was discussed by various authors, e.g. Borodich (1994), Carpinteri (1994) and Baiant (1995~). Note that Carpinteri (1994) called the

term /?(O*) as renormalized fracture energy 9: and Baiant (1995~) used the term fractal fracture energy Gn. Indeed, for t = 1 we can write G,, = /I(D*)~I~)(~*) 6i” in

the case of brittle fracture (G, = 0) and Gfl = G, SkP1’ in the case of quasi-brittle fracture. Then from (16) we obtain the following formula

(W>jt - G,,.Y”. ii, < .Y < A*

used by Baiant (I 995~).

4. GROWTH OF A FRACTAL CLOUD OF MICROCRACKS

Now we imagine that a tip of a continuous crack is shielded by a growing cloud of microcracks, which is described as a fractal cluster with multiple branches (Fig. 5).

We suppose that if the size &! of the considered area of the fracture pattern is less than some critical A* then the growing cloud is self-similar with respect to the group

of the homogeneous coordinate dilations. This concept reflects the observation that during the process of cloud growth even though the dimensional parameters, such as

the statistically average microcrack size and the mean distance between microcracks. change, dimensionless statistical properties of the cloud remain the same over the wide range of scales (Barenblatt and Botvina, 1983, 1986; Barenblatt, 1993).

We assume that the fractal dimension of the microcrack cloud is equal to C and is constant during the cloud growth. We have seen above that usually. it is possible to give the following estimations 1.47 < C < 1.79.

We assume also that for %? > A* the fractal picture is repeated, i.e. the growing cloud is self-similar with respect to the group of the coordinate translations. This concept reflects the observation that for some size of the microcrack cloud it begins to grow without increasing of the crack-thickness and each piece of the cluster of the size A* absorbs the same value of energy.

F. M. BORODICH

* \ ’ 2mm ’ c;

J

/

Fig. 5. Successive stages in the development of microcrack pattern during the fracture of double cantilever beam test specimen of dolomite marble [after Nolen-Hoeksema and Gordon (1987)]. The load is given in

percent of the fracture load, A* > 1 cm.

4.1. Mathematical description ofjiiactalfracture network in brittle materials

Let us introduce the concept of specific energy absorbing capacity of a fractal measure of the microcrack cloud p(C*), where C* is the fractional part of C.

Similar to the B(I)*) the value j?(C*) has the dimension

[p(C*)] = FL/L2+‘*

and gives the amount of elastic energy spent on forming a unit of the fractal measure

MB). Using property of homogeneity (scaling law) of the fractal measure (4) we obtain

me(S) = ACm,(S), (xEX$)o(L’x = x, ES), (18)

i.e.

m&J&) = mc(~I)(%/WI)C’.

Analogously to the above case of a solitary crack we can write the following formula


for the total amount H of energy absorbed by a fractal pattern of microcracks growing

from a cut or a notch in a brittle material

l-l- 2t8(c*)m,(~“>(~/~‘o)C, 2 < A*,

2tP(C*)m,(~“)(A*/Wo)‘(~iA*). A, < 9. (19)

Note that in (19) we have used both kinds of self-similarity of the microcrack cloud growth.

4.2. A model of multiple quasi-brittle fracture

Let us give some estimates for the fracture energy of a fractal pattern of microcracks in quasi-brittle materials.

We will suppose that the width of the plastic layer h is small with respect to the average distance between microcracks. Evidently, we can cover the fracture pattern

(the cluster of microcracks) by squared boxed of some small measuring size 6,.

Let us introduce the effective energy absorbing capacity Gr of an elementary piece of size 6, of the fractal cluster. As above the dimension of Gr is

[G,] = F.

Using the property of homogeneity (scaling law) of the fractal measure (4), analogously to the above case of cracking (15), we can write the following formula for

the total amount (IV) of absorbed energy of the pattern

(W>- G, &+@/S,>“, 6, < W < A,,

Gfs,(A,/s,)“(:a/A*), A* < W.

From (20) we have the following formula for the average value of fracture energy

,WO Gr(1 +C*)(&?/&J”/r, 6, < W < A*, PN Gr = t- do

’ Gr(1 +C*)(A,/&)“*/t, A, -=I W (21)

The graph of G, is given in Fig. 6(a).

5. DISCUSSION AND CONCLUSION

It is still unclear what is the best way to describe the growing pattern of fracture. We have above supposed that the fracture develops as a self-similar fractal. As a consequence of the supposition we obtained the formula (21) for the average of the stochastic function of fracture energy which is in qualitative agreement (Fig. 6) with

experimental results for concrete (Brameshuber and Hilsdorf, 1990). There is no mathematical difference if we consider the fracture as a self-similar

solitary crack or as a self-similar pattern. Indeed, the formula (21) for a fractal pattern is identical to the formula (17) obtained in the model problem of solitary fractal crack propagation. However, experimental studies show that fracture surfaces are usually not self-similar fractals but self-affine fractals (Milman et al., 1994; Schmittbuhl et

F. M. BORODICH

GF -t Gf

( l+ p 4 1 c*

l

R

I K

I I I I

0 100.0 200.0 300.0 400.0

Crack advance [mm] Fig. 6. Relations between the relative fracture energy and crack advance: (a) a schematic graph of theoretical relation (21) : C’ = 0.5, A, = 25 6, ; (b) the experimental relation in a three-point test specimen

of concrete (after Brameshuber and Hilsdorf, 1990).

al., 1994). One can find an extended discussion concerning applications to fracture mechanics of self-similar and self-affine models in Baiant (1995a, b).

The self-similar and self-affine fractals reflect the difference between the Hausdorff

dimension and the Kolmogorov one. In analyzing the surface topography one con- siders a fractured surface as a graph of a functionJlx>, where x = (x,, x2). Usually it reckons (Milman et al., 1994; Schmittbuhl et al., 1994) that the self-similar fractal is invariant (statistically) under homogeneous (isotropic) dilation, namely when

while the self-affine fractal is invariant under quasi-homogeneous (anisotropic) dilation, namely, if


Here a is some scaling exponent. If we look at the definitions of self-affine and self-

similar fractals and compare them with the definition of homogeneous functions then

we see that the former are homogeneous functions of degree a and 1, respectively.

It is possible to show that usually quasi-homogeneous dilations are the particular cases of Lipschitz homeomorphisms (Borodich, 1994b, 1995). The Hausdorff dimen-

sion of a set S does not changed under the action of the Lipschitz homeomorphism L, i.e.

dim, S = dim, L(S);

but the Kolmogorov capacity can change. Usually the profile of a vertical section of a real fractured surface scales differently in the plane of fracture and in the vertical

direction. Thus, if we estimate the fractal dimension of a fracture surface using the

Richardson method in application to the anisotropic enlarged profile of the surface then we must realize that these estimations can be wrong.

The above examples of parametric-homogeneous functions (5) (Borodich, 1994b. 1995) show that the trend of a function is not necessarily connected to fractal dimension. Indeed, both function 6, and b, have the same Hausdorff dimension. However. if we consider the trend of the functions then we get that b, is a self-affine function

with degree u = 2 and b, is a self-similar function. Thus, the common statement, that

the scaling exponent a of self-affine fractals is always closely related to the fractal dimension, is wrong.

The value C used above in the models is an average value only because the fractal dimension of a fracture pattern in polyphase materials is a function of external load

(Borodich and Fradkin, 1995). Therefore, the fracture patterns do not change in a pure self-similar way (homogeneous dilation). Thus. the hypothesis, which treats the

growing fracture pattern as self-similar with respect to the group of the homogeneous

coordinate dilations (Barenblatt and Botvina, 1986; Barenblatt, 1993), is not absol- utely valid. However, the value C becomes almost constant when the main crack begins to propagate due to coalescence of microcracks (Zhao et al., 1993 ; Borodich

and Fradkin, 1995). It seems to us that we can split the hypothesis of self-similarity (Barenblatt and

Botvina, 1986 ; Barenblatt, 1993) in two : (1) the hypothesis of self-similarity of defect (microcrack or fault) pattern with respect to the group of the homogeneous coordinate

dilations ; (2) the hypothesis of universality of distribution for defect sizes

N/N, = @(l/l,),

where 0 is some universal function, N* is the maximal defect frequency and I, is the corresponding defect size. The last Barenblatt-Botvina distribution is not a power law (Fig. 7). We can describe the function 0 as a parabola in common logarithmic

coordinates, i.e.

Ig@(l/l,) = -Al’, ( = lg(l/l,), or N/N, =(1,‘1,) 4’g(“*‘,

where A is a positive fitness parameter. The experiments (Barenblatt, 1993) show that usually 0.1 < 5 < 10. A fracture pattern can satisfy the similarity hypothesis 2 (the Barenblatt-Botvina distribution) even if it is lacking the similarity hypothesis 1.

256

10 .

N

N*

01 .

0.01

F. M. BORODICH

01 . 10 . 10.0

1

L Fig. 7. The universal defect size distribution for various materials under various types of loading: 1, polycrystalline copper, fatigue; 2, brass, static tension ; 3, iron, creep; 4, steel 347, creep; 5, steel 304,

creep; 6, rocks [after Barenblatt (1993)] and a fitting curve Ig(N/N,) = -Ig’(l/l,).

It is possible to construct more sophisticated models of multiple cracking in which we can imagine that a tip of a main crack and the process zone propagate discretely, i.e. step by step. The self-similarity of these models can be described using parametric- homogeneous functions (Borodich, 1995). We intend to continue our studies in this direction. Note that log-periodicity (Blumenfeld and Ball, 1991 ; Anifrani et al., 1995; Newman et al., 1995) is a particular case of parametric-homogeneity.

Now let us give an explanation why recent studies come to different conclusions regarding the correlation between the fractal dimension of the fracture surface and the fracture energy. If fracture surface exhibits fractal features and 6, << H, or h << H*, then we can expect that (17) is valid and we may see the correlation. If h - H, or 6, - A* then fracture energy is mainly related to the work done within the non- fractal plastic zone of the crack and the correlation is negative. Experimental studies show (Dauskard et al., 1990; Bouchaud et al., 1990) that for metals A, = 0.1 mm,

Some fractal models of fracture 25-l

i.e. A* 6 h. Thus, the fractal properties of fracture surface do not have to be correlated with toughness of metals.

The experimental studies show (Nolen-Hoeksema and Gordon, 1987 ; Zhao ef al.. 1993) that for polyphase materials such as rock, concrete, ceramics, etc., A* - 1 cm and therefore (21) can be valid [note that Okubo and Aki (1987) showed that for geological faults A* - 100-1000 m]. The above estimations show that the fractal properties of the pattern of microcracks can characterize the fracture energy of the polyphase materials when the fractal properties of the main crack surface are inessential. Thus, the evaluation of the dimension C of the fracture pattern is more

important for fracture mechanics than the evaluation of the dimension D of the

fracture surface. The above consideration shows that the following conclusions can be made.

(1) It is necessary to realize the difference between mathematical and physical fractals. When constructing a model it is necessary to determine clearly which kind of fractals are used and what the fractal behaviour boundaries are in the model.

(2) Fractal dimension of a fracture surface can correlate with fracture energy only in materials with extremely narrow plastic zones. Fractal dimension of the microcrack patterns can correlate with the fracture energy of the polyphase materials.

(3) The use of fractal geometry language in models implies the use of appropriate terms [e.g. B(C*) concept] which can have unusual physical dimension).

(4) Not all self-similar and scaling properties of fracture can be described by methods of fractal geometry (e.g. the Barenblatt-Botvina distribution, log- periodicity).

ACKNOWLEDGEMENTS

I am grateful to Professor J. R. Willis for discussing the problems considered in this paper and the hospitality shown at the University of Cambridge.

Thanks are also due to the Royal Society for funding a visit to DAMTP (University of Cambridge) during which this work was completed, and for assistance in allowing me to attend the IUTAM Symposium on Nonlinear Analysis of Fracture.

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Some fractal models of fracture