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A percolation-type fracture criterion for composites with randomly oriented fibres
theoreticaland applied fracture mechanics Theoretical
and Applied Fracture Mechanics
28 (1997) 95-108
A percolation-type fracture criterion for composites with randomly oriented fibres S.T. Mileiko *, A.K. Stepanov Solid State Physics Institute of the Russian Academy of Sciences, 142 432 Chernogolouka Moscow distr.. Russia
Abstract A fracture model is built up for a solid composed of brittle fibres randomly oriented in the matrix volume. The fracture process includes a stable growth of microcracks caused by fibre breaking under the load and formation of an infinite cluster of the microcracks. Both upper and lower bounds for ultimate stress in a fibre system are found as functions of the fibre volume fraction. The calculation of the ultimate stresses are performed by using the percolation theory and the theory of branching processes. At the present stage of the theory under consideration, only two types of the microcracks are appraised, namely that of a delamination type which corresponds to a weak fibre/matrix interface, and that of a penny shape which corresponds to a strong fibre/matrix interface. A particular solid contains only one type of the microcracks. In both cases, non-linear dependencies of the ultimate composite strength on fibre volume fraction are obtained. 0 1997 Elsevier Science Ltd.
1: Introduction Despite that various fracture processes include a stage of combining damage, such as microcracks, delamination, into a microcrack finalizing the fracture process which means that a percolation nature is inherent to fracture processes, usage of the percolation theory to analyse fracture has not been a common practice. The situation seems to be paradoxical. However, the modem fracture mechanics does certainly need not so much new mathematics as new fundamental ideas and that one, considering a new analytical possibility, thinks on whether it will bring new qualitative conclusions or not. Hence, the first aim of the present paper is to look at possibilities to be opened by application of the percolation theory to fracture mechanics of non-homogeneous solids. Consider composites consisting of high modulus brittle fibres and low modulus matrix. The fibres are randomly oriented in the matrix and their centers are uniformly distributed in the matrix volume. The structural architecture is assumed not to change during the loading. Suppose that fibres start to break under a particular fibre stress and each fibre break results in either the fibre/matrix delamination and formation of a cylindrical microcrack or microcrack entering the surrounding matrix and formation of a penny-shaped microcrack. Both microcracks can either grow in a stable manner or lead to a catastrophic failure of the composite.
??
Corresponding
author. Fax: +7-095-7420142
ext. 2493; e-mail: [email protected].
0167.8442/97/$17.00 6 1997 Elsevier Science Ltd. All rights reserved. PII SO167-8442(97)00034-7
S.7.. Mtleiko. A.K. Stepano~ / 7’hrorericul
96
and Applied Fracture
Mechanics
28 (1997)
95-108
Only stable growth of the microcracks shall be considered, in which process an infinite chain or a cluster is finally formed. The formation of such a cluster can be considered as a limiting state of the composite. A model to be considered here is somewhat simplified.
2. Mechanical
model
To simplify the model, consider the fibres of a constant length. Fibre centers are uniformly distributed in a matrix volume with parameter Ai being the mean number of fibre centers in the unit volume. Assume that A, is sufficiently small to neglect neighboring fibres connections. Fibres are randomly oriented in a matrix volume which means that a fibre unit vector is uniformly distributed on a unit sphere S2. 2.1. Cylindrical nhw,xdx When the fibre/matrix interface strength is sufficiently low, a fibre break yields the formation of a cylindrical microcrack. If the applied load direction coincides with that of a fibre (see Fig. 1) the elastic energy AW, arisen as a result of the fibre break equals approximately to energy AW,, absorbed by the surface of the cylindrical microcrack. Therefore,
Tr:Lg +rrff; -1
AW, =
7
f
2,
AW,, = 27r r, Ly,,
(2)
where 5 is the regular tensile stress in the fibre, Y,,,, is interface energy, E, is the Young’s fibre, L is the fully unloaded fibre length, and I is the recovery length. Taking into account _
modulus
of the
(3) where cy is a coefficient
equal to about 1, -r,C, is the fibre/matrix
interface strength.
Eqs. (l)-(3)
yield
,?‘?
r,=L(C)
=
ad2rfr,,LfL__ Tfrl!
(4)
l-,
where -2
u L =
rf.
2
The expression
-
cl<;<1
(5)
for L will be used provided that
r,
(6) Tfn,
where (of* > is the mean fibre strength. From Eqs. (4)-(6)
the interval of Cr is found: (7)
XT. Mileiko, A.K. Stepanov / Theoretical and Applied Fracture Mechanics 28 (1997) 95-l 08
97
Fig. 1. Schematic of the cylindrical microcrack.
where
wnEf
J
x=
(8)
__
rr
When the applied load direction forms angle 8 with the fibre axes (O-directed fibre), the regular tensile stress in a fibre is [lj-
cO=(T.
(9)
(cos*O - + sin*0 >
where 8 lays changing /c0seI2 For a B-directed
in the interval
(10)
5 fibre,
L,=L,(cr,)
=
instead
of Eqs. (4) and (51, it is found that
ff/m r‘:,
.-
z;‘*
(11)
1 - Ze
and
~. r f
ze = 2 yfm Ef
(c0s2e-~&9)2~~2,
O
1
(12)
2.2, Penny-shaped microcracks When the fibre/matrix interface is sufficiently strong, a fibre break yields formation of a penny-shaped microcrack in the matrix. If the fibre direction coincides with the load direction (see Fig. 2) then neglecting matrix unloading there results
dI.$.l =Tr f
4’ ‘4 E,
(13)
98
XT. Mileiko,
A.K. Strpanou/
Throrrticul
Fig. 2. Schematic
and Applied
Fracture
of the penny-shaped
Mechanics
28 (1997)
95-108
microcrack.
and AW,, = 7~( Kf - r;)
(‘4)
ytm
where R, is the radius of the microcrack
and length 1 is given by Eq. (3). This yields
(15) The consideration
is valid when
(‘6) or
Using Eq. (9) for a o-directed fibre yields radius
R,,,
for penny ((7r/2)
- 8)-directed
microcrack
as
-------R,,,=
Rc,o( 6)
= rf
( 18)
where C values belongs to the interval given by Eq. (17) and t) angle belongs to the interval given by Eq. (IO).
3. Percolation
criterion
In a composite volume under an applied load we observe a set of independent, for each hV,O, Poisson’s assemblages of the crack centers. Parameter Alp,” is the density of crack centers oriented as shown in Fig. 3. The
ST. Mileiko, A.K. Stepanou / Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
99
Fig. 3. A fibre unit vector on a unit sphere S2.
joint density of a pair ( cp, 8) of independent cos 8 477. ’
(
g(rp,O)=
random variables
is
for -n-
0,
(19) for other cp, 8.
The sum of A,,, over all values (rp,
e)
equals to
(20) Assume that the composite fracture corresponds to formation of an infinite cluster of the connected microcracks. Hence, evaluation can be made for the critical value of a, that is 3 * as a function of h, and structural parameters of the composite. 3.1. Lower bound for the ultimate stress To obtain a lower bound for Cr * , construct an infinite cluster as a random tree in a branching process of the Halton-Watson type [2]. Each generation of such a tree produces a random number of the branches independent for each generation. The branching tree becomes infinite when the mean number, M, of the branches generated by a single branch in the previous generation is larger than 1. So the solution of equation M = 1 gives the lower bound for the infinite branching tree. By the definition,
(21) Here pk, k=O, m=
i
1,2,
. . . are the probabilities
to have k branches,
produced
by a single branch, where
k-p,
(22)
k=O
is the mean number of branches root m, for m > 0: 1120
=
1.60
or connections
on a crack. It was found [3] that equation
M = 1 has a single
(23)
XT. Mileiko, A.K. Stepanou / Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
100
Fig. 4. V,,,-region i-axis.
containing
all the centers of ( q. 0 )-directed cylindrical
microcracks
intersecting
a given microcrack
oriented along the
In the terms accepted, each microcrack of the infinite cluster belongs to a definite generation, say to the n-th generation. A number of the branches in the next, (n + l)th, generation is the number of microcracks connected to the original one. 3.1 .I. Cylindrical microcrack Introduce the Cartesian (x, y, z) co-ordinate system associated with a crack. Without loss of generality we assume that the z axis of the crack coincides with the load direction. Since in the vicinity of the lower limit of the infinite cluster, the mean number, m, of intersections of a given crack is small and because rf < L, the intersections of cracks in the same direction can be neglected. Define pk(go, 0) as the Poisson’s probability of the intersections of a given crack by k( cp, 13))directed cracks. Hence,
Pk(P, 0) =
(A,.*.Jvyl,oJ)k k!
exp( - A,,, . Iv,,,l)
(24)
where IV,.,1 = 2r,. L. L,. sin 0
(25)
(see Fig. 4). Let 7rL, sin 0 N, =
rr
(26)
101
S.T. Mileiko, A.K. Siepanov / Theoretical and Applied Fracture Mechanics 28 11997) 95-108
be the number of points on the unit sphere S2 in a qo-plane with directions
2rf
qj=j.tan-'-
L, sin 0
'
j=O, 1 , . . . . N,
(27)
Region Vq,,e and distribution ~~(4p~)~ is acquired to each direction 4~~. A set of directions distribution jj& r3) that is the probability that a given crack is intersected by k cracks from the &( t9) is the probability that k centers of cracks of set (s)~ belong to a set of all regions Poissonian assemblage of cracks centers are independent for each (cp, 8)-direction, probability represented by
=-
i,
((~i)~ forms set (‘pi&. So (V,,,). Since p&e) can be
(~~A,,oL.LZB sin’e)’ *exp( -2nh,,eLL2, sin*e}
(28)
A set of directions (~~pi>~is a subset of the set, (p),, for all cp-directions and a fixed O-direction. Thus, for probability p,(8) that a given microcrack is intersected by k of the O-directed microcracks, there results M-9 Distribution
250)
(29)
r?;&O) is the Poissonian
distribution
with parameter
th,.LZ,.c0s8.sin28=hfl~l
(30)
volume using polynomial distribution for each finite set of characteristic where Iv01 is a characteristic If0 I from the interval 18I < 7r/2 leads to limit of & when the number of IvOI goes to infinity as
volumes.
(31) Note that jYk is the Poissonian
distribution
with parameter
~hfLj_+~~~Osin2Bcosede=Afl~O~
(32)
where If I is a characteristic volume. In other words, jk is the probability that a given crack is intersected by k cracks from set ( qj, 0) = u @( q~~)~.Since set (qj, 0) is a subset of all direction set (rp, 0) = S*, it is found that (3%
Pk >fik Hence, for the mean number,
M, of intersections
on a crack there results
m 2 fi = A,lvl
(34)
At m = m,, from Eq. (23), there are still no infinite clusters of the cracks. So for all F satisfying the inequality A,lv'l
A,lr;;t> Af@C2
(39 region satisfying
to Eq. (35), a lower bound
(36)
S. T. M&i&o. A K. .Srepuno~~/ Theorer~cul urrd Applied
IO?
Fruct~e
Mechanics
2X (1997) 95-108
where
X
(LXX%
ill/7 /
- iT/?
-
(
1/4)sir1%)~
sin’0 cos 6, db, (37)
1 + (e-f;n)\i(hn~fw,
- (cm% - ( 1/4)sin%)‘j
Using Eqs. (36) and (38) for the left hand side of Eq. (35) and Eq. (23) for the right hand side of Eq. (3.5) the result is
3.1.2. Pennyshuped microcruck To get lower bound of ?? * we shall use the same procedure as in the previous section as well as the same definitions. Set the center of a given penny microcrack at the center of Cartesian, (x, J. :)-coordinate system such that the direction of a fibre axes coincides with the z-axis of load u. For 101 < (n/2), there is the need to define the orientation of penny crack in the space. Let the center of a penny crack be the center of the co-ordinate system. A penny crack is a ( cp. H)-directed crack when the vector orthogonal to the intersection of a disk and (x, v) plane forms a 6’angle with the positive direction of the :- axis and its projection on the (s. y) plane forms a 0 angle with the positive direction of the .r-axis. Without loss of generality, to simplify the calculations further let a (cp, @)-directed crack be a square shape of R,,, side length. With such an assumption. probability pa(~, 0) that k(cp. H)-directed cracks intersect a given one is given by Eq. (29), where IV, ,,I = 8 R, Rc,o( R, + R,,,)sin
H
(39)
Here V,.H is a region containing centers of (cp, H )-directed cracks (see Fig. 5). Following the proof scheme of Section 2. I, the probability pn( 0) is found to be the Poissonian distribution with parameter h,IVg/=h,‘8R,R,,H(Ro+R,,H)sinH
(40)
Note that the angle 8 in Eq. (19) is changed by the ((n/2) - @)-angle because the crack plane is orthogonal to the fibre axes direction. The next step to get probability pk is analogous to that in Section 2.1 and that Rk is Poissonian distribution with parameter A~iVl=h,-RR,IiX,,,(R,
+R,,,)sin%d@
(41)
0
Therefore, m =
the mean number
m of intersections
on a crack is
A,(V(
(42)
Following the same procedure, there is the need to find the lower bound for the right hand side of Eq. (41). Using Eqs. (15) and (1 S>, it is found that A,IVI 2 16h,R,
/0
wRf.s sin20 d6 = h,@,@, (T9/2
(43)
XT. Mileiko, A.K. Stepanov / Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
Fig. 5. V,.B region containing all the centers of (cp, 6)-directed the origin of coordinates and being orthogonal to the z-axis.
square-shaped
microcracks
intersecting
a given microcrack
103
with a center at
where
Following the last step of the same procedure (43), there results m0 cr*2
i
drp,
in getting lower bound
Cr * , and so using Eqs. (7), (231, (3.5) and
2/g
1
. A;
219
(45)
3.2. Upper bound To obtain lower bound of si * , by using the theory of branching process, construct the infinite cluster which has no closed cycles of crack surfaces. Actually, the infinite cluster does contain some of them. Take into account the formation of closed cycles to obtain the upper bound for Z * . 3.2.1. Cylindrical microcrack Consider, in the (rp, 8)-space-volume, a (2r,, e&layer of thickness 2r, which forms the angle 8 with the positive direction of load (z-axis). Obviously, when a (2r,, 8)-layer contains an infinite cluster of the
104
S.T. Mileiko, A.K. Stepanor, / Theoretical and Appltrd Fracture Mechanics 28 11997) 95-108
intersecting cracks, then the cluster already exists in the whole (cp, 0)-space-volume but not vice versa. To evaluate the upper bound for 5 * we need to construct the infinite cluster in a (2r,, 0)-layer. Subdivide (2r,, fI )-layer into rectangular cells of a size (\/5/ 1O)L, . (6,’ 1O)L, .2 rr. Centers of these cells form the regular square lattice 2; belonging to the (2r,, 0)-layer. Let p(cp, n/2, 0) be the probability that a (&?/IO)&. <&/lo)&. 2r, cell contains at least one center of (cp. 0 j-directed crack and at least one center of (cp + (7r/2), o&directed crack. If such an event occurs we say that the center of a cell is occupied. Using Poissonian distribution and taking into account rf < L. L,, < L,-there renders
(46) Probability p(rr/2, 0) that a (&/lo>&. cracks, follows from Eqs. (19) and (46)
(&/lo)&.
2r,-cell
contains
at least two centers
of orthogonal
(47) With probability p(rr/2, 0 I, a site of the square lattice Zi associated with a (2r,, @)-layer is occupied independently of other sites of Zi. Let p(rr/2) be the probability that on at least one of the square lattices Zi, (101 < 7r/2) a site of a lattice is occupied independently of other sites of the same lattice. Eq. (47) yields
Now, reduce a problem of occurrence of the infinite cluster of intersecting cracks in some (2r,, 6)-layer to a well-known site problem of the percolation theory on a two-dimensional square lattice Z2 [4]. It has already been mentioned that a site of Zj is occupied with probability p(rr/2, 0) independently of other sites of 2;. The occupation of a site of Z,’ is caused by the intersection of at least two orthogonal cracks in a (&/lo)&. (fi/lO)L, .2r,-layer. The size of a cell is chosen such that two neighboring occupied sites of Zi cause intersecting pairs of the orthogonal cracks and three neighboring occupied sites of Zi cause closed cycles of intersecting cracks (see Fig. 6). It is seen that the infinite cluster of occupied sites on Zi causes the infinite cluster of intersecting cracks in a (2r,, @)-layer. The site problem on Zi yields that for all (T satisfying the inequality
(‘) is the critical probability of the site problem on Z2, there exists an infinite cluster of occupied where pCr of Zl with the probability equal to 1. This means that for all G satisfying the inequality
P f i
1
sites
(50)
> Pi:’
in at least one (2 rf, 8 )-layer, there exists an infinite cluster of intersecting cracks with the probability equal to 1. In other words, the inequality given by Eq. (50) describes a region of the a values in which an infinite cluster of intersecting cracks exists almost surely in a whole space-volume of the composite. To specify a region of i$ values satisfying Eq. (50). the upper bound for p(rr/2) is found. Eqs. (4)-(8) and (48) yield p
f
(
i
I1
-exp(
-A,F,F,i?*)
(51)
XT. Mileiko. A.K. Stepanov/
Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
105
Fig. 6. A fragment of 2’ lattice (marked by points) with occupied sites (marked by + ) when the lattice cell contains at least two centers of two orthogonal cylindrical microcracks.
where tq*
F, = (T$(
F2
=
1 + (q
h2r?(1 + ( d#Er/rry2 ))\/2y,E,/r,(
(cos28 - fsin28)6
l/fTi2 20
1 + ( o/r,)d2y,,E,/r,)-“2)1’2 . cos 0 de
-m/2 1 +((~/(0~*))\11-(1
+ (o/T&)/~)~"~
- (c0s2e-
+sin28)2)2 (52)
Eqs. (49)-(51)
are applied to give
(53)
3.2.2. Penny-shaped microcrack For penny microcracks, a slight change of the proof in Section 2.1 is made. Decompose ( cp, 8 )-space-volume Centers of these cubes form a cubic lattice 2; in a space volume. Let by cubes of side length a0 = (1/6)R,,,. p( ~0, 7r/2, 0) be a probability that a,. a, . a, cube contains at least one center of (q, O&directed crack and at least one center of (cp + (7r/2), O)-directed crack. Then the Poissonian distribution yields
(54)
ST. Mileiko. A.K. Stepanu / Tlwweticul and Applied Fructure Mechanics 28 119971 95-108
106
Fig. 7. A fragment of Z’ lattice. Points 0,.
O2 show the centers of two intersecting orthogonal square-shaped
Hence, following the procedure used in Section 2.1 probability two centers of two orthogonal cracks is ~(5,“)
p(rr/2,
microcracks.
13) that uH. ug u,-cube contains at least
= i -exp[----&“~R~,6sinO)
(55)
A site of the lattice 2: is said to be occupied with the probability p(7r/2, 0) independently of other sites of Zj. Let p(rr/2) be the probability that on at least one of the cubic lattices Zi, 113 < z-/2 a site of a lattice is occupied independently of other sites of the same lattice. Similarly to what has done to obtain Eq. (48), Eq. (55) may be used to yield = 1 - exp(-
p(t)
&&_(=Rz,,
sin Bd@)
(56)
The occupation of a site of Zi is caused by the intersection of at least two orthogonal cracks in a uB. a6 a0 cube of (9, O&space-volume (see Fig. 7). So any two neighboring sites of 2: cause intersecting pairs of orthogonal cracks and so an infinite cluster of occupied sites of Zi bears an infinite cluster of intersecting cracks. From the site problem on Z’ with the critical probability pz:’ [4] 13) <
P CT
4
(57)
it follows that for all (T satisfying p 5, i
the inequality
i? >piB’
(58)
1
there exists an infinite cluster of the occupied satisfying the inequality
sites of Zi
with the probability
equals
to 1. Then for all C
(59)
S.T. Mileiko, A.K. Stepanou / Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
107
on at least one of the cubic lattices 2: and thus in a whole space-volume of the composite, there exists an infinite cluster of intersecting cracks with probability equal to 1. Hence, to specify the interval of C values satisfying EQ. (59), there is the need to find the upper bound for &n/2). From Eqs. (18) and (59) there results
(60)
p where
G,
=
From Eqs. (59)-(62),
G2=&_/“(l+cos2B-$sinz8)sinBdB 0
(61)
a final upper bound for Cr * is obtained:
(62)
4. Discussion First it is seen that both lower and upper bounds for dependencies volume fraction, h,, are of the same type, those are 5 * o[ A, l/2
of critical fibre stress (+ * on mean fibre
(63)
and z* ah,2/9
(64)
Eq. (63) is valid for a weak fibre/matrix interface (see Eqs. (38) and (53)) and Eq. (64) is valid for a strong fibre/matrix interface (see Eqs. (45) and (62)). Eqs. (63) and (64) yield dependencies for ultimate stress of the composites neglecting a contribution of the matrix to the composite strength as Z * a u:/~
(7* a u:j9
(65) (66)
where vt is the fibre volume fraction. Eqs. (65) and (66) can be actually valid at a sufficiently low fibre volume fraction only since the whole theory has been developed for elastically microcracks. Non-linear dependencies similar to those obtained have been observed in numerous publications although most authors preferred to speak about linear dependencies. The validity of the relationships obtained within a rather narrow interval of small fibre volume fractions does not decrease the importance of the results. Actually, short-fibre composites increasing the composite strength at low vf can often determine a maximum composite strength that can be achieved as shown in Fig. 8. The falling part of the u * (v,> curve for short-fibre composites that interrupts the growing part of it is determined by microstructural features of the composite [5]. Still, it is a primary importance to evaluate the dependence for low vf. Then it is seen that the exponent in the final expressions (Eqs. (63) and (64)) strongly depends on the fibre/matrix interface characteristics. Since in the model considered the situation around a fibre break and microcrack was not analysed in strict terms, it was not important to formulate conditions for a particular microcrack to occur. It is the author’s intention to consider formation of the cluster containing both types of the
S.T. Mileiko, A.K. Stepanor / Theoretical and Applied Fracture Mechanics 28 (1997) 95 -108
108
::
a
g- 1.4
8
4
s 01.2
d 5
E '; ml.0 5
0
i 0.8 -
0.6
i
0.00
Fig. 8. Composite
._
.-._M.-
0.05
0.10 Fibre volume
---s
0.15 fraction
0.20
strength versus fibre volume fraction for a C/TiC/Ti
0.25
composite
system [6].
microcrack as a next step of the development of the model. At that step, the conditions mentioned should be important. Also, the development of the model to bring it close to a real situation calls for accounting for statistics of the fibre strength and, hence, the fibre breakage. An important feature of a large family of short-fibre composites, that is the ceramic-matrix family, is similarity in elastic behaviour of the fibre and matrix. Hence, this case is also to be considered.
5. Conclusion The main result of the present work is a revealed possibility to adequately describe the failure behaviour of randomly reinforced fibrous composites by using the percolation theory. A simple mechanical model analysed by the percolation theory yields semi-qualitative strength/fibre-volume-fraction dependencies that can be of importance in optimising a composite microstructure. It is also important that the results of the first work in this direction show clearly what should be done to develop a percolation type theory of the failure of composite microstructure.
Acknowledgements The work was carried out under the support of Russian Foundation
for Basic Research, Project 96-01-0183 1.
References [l] R.M. Christensen, Mechanics of Composite Materials, Wiley Interscience, New York, 1979. [2] B.A. Sevastianov, Branching Processes, Nauka, Moscow, 1971 (in Russian). [3] A.K. Stepanov, V.V. Tvardovsky, A.A. Khvostunkov, The fibre skeleton structure and transport properties of stochastically reinforced composites, Comp. Sci. Tech. 45 (1992) 221-228. [4] H. Kesten, Percolation Theory for Mathematicians, Birkhause, Boston-Basel-Stuttgart, 1982. [5] N.S. Sarkissyan, A.A. Khvostunkov, S.T. Mileiko, A failure model of short fibre composites with metal and ceramic matrices, Appl. Comp. Mater., submitted. [6] S.T. Mileiko, M.V. Gelachov, A.A. Khvostunkov, V.M. Kiiko. D.B. Skvortsov, Short-fibre/titanium-matrix composites, in: Proc. the 10th Intern. Conf. on Composite Materials, vol. 2, Woodheard Publishing Lmd., Canada, 1995, pp. 131-138.
and Applied Fracture Mechanics
28 (1997) 95-108
A percolation-type fracture criterion for composites with randomly oriented fibres S.T. Mileiko *, A.K. Stepanov Solid State Physics Institute of the Russian Academy of Sciences, 142 432 Chernogolouka Moscow distr.. Russia
Abstract A fracture model is built up for a solid composed of brittle fibres randomly oriented in the matrix volume. The fracture process includes a stable growth of microcracks caused by fibre breaking under the load and formation of an infinite cluster of the microcracks. Both upper and lower bounds for ultimate stress in a fibre system are found as functions of the fibre volume fraction. The calculation of the ultimate stresses are performed by using the percolation theory and the theory of branching processes. At the present stage of the theory under consideration, only two types of the microcracks are appraised, namely that of a delamination type which corresponds to a weak fibre/matrix interface, and that of a penny shape which corresponds to a strong fibre/matrix interface. A particular solid contains only one type of the microcracks. In both cases, non-linear dependencies of the ultimate composite strength on fibre volume fraction are obtained. 0 1997 Elsevier Science Ltd.
1: Introduction Despite that various fracture processes include a stage of combining damage, such as microcracks, delamination, into a microcrack finalizing the fracture process which means that a percolation nature is inherent to fracture processes, usage of the percolation theory to analyse fracture has not been a common practice. The situation seems to be paradoxical. However, the modem fracture mechanics does certainly need not so much new mathematics as new fundamental ideas and that one, considering a new analytical possibility, thinks on whether it will bring new qualitative conclusions or not. Hence, the first aim of the present paper is to look at possibilities to be opened by application of the percolation theory to fracture mechanics of non-homogeneous solids. Consider composites consisting of high modulus brittle fibres and low modulus matrix. The fibres are randomly oriented in the matrix and their centers are uniformly distributed in the matrix volume. The structural architecture is assumed not to change during the loading. Suppose that fibres start to break under a particular fibre stress and each fibre break results in either the fibre/matrix delamination and formation of a cylindrical microcrack or microcrack entering the surrounding matrix and formation of a penny-shaped microcrack. Both microcracks can either grow in a stable manner or lead to a catastrophic failure of the composite.
??
Corresponding
author. Fax: +7-095-7420142
ext. 2493; e-mail: [email protected].
0167.8442/97/$17.00 6 1997 Elsevier Science Ltd. All rights reserved. PII SO167-8442(97)00034-7
S.7.. Mtleiko. A.K. Stepano~ / 7’hrorericul
96
and Applied Fracture
Mechanics
28 (1997)
95-108
Only stable growth of the microcracks shall be considered, in which process an infinite chain or a cluster is finally formed. The formation of such a cluster can be considered as a limiting state of the composite. A model to be considered here is somewhat simplified.
2. Mechanical
model
To simplify the model, consider the fibres of a constant length. Fibre centers are uniformly distributed in a matrix volume with parameter Ai being the mean number of fibre centers in the unit volume. Assume that A, is sufficiently small to neglect neighboring fibres connections. Fibres are randomly oriented in a matrix volume which means that a fibre unit vector is uniformly distributed on a unit sphere S2. 2.1. Cylindrical nhw,xdx When the fibre/matrix interface strength is sufficiently low, a fibre break yields the formation of a cylindrical microcrack. If the applied load direction coincides with that of a fibre (see Fig. 1) the elastic energy AW, arisen as a result of the fibre break equals approximately to energy AW,, absorbed by the surface of the cylindrical microcrack. Therefore,
Tr:Lg +rrff; -1
AW, =
7
f
2,
AW,, = 27r r, Ly,,
(2)
where 5 is the regular tensile stress in the fibre, Y,,,, is interface energy, E, is the Young’s fibre, L is the fully unloaded fibre length, and I is the recovery length. Taking into account _
modulus
of the
(3) where cy is a coefficient
equal to about 1, -r,C, is the fibre/matrix
interface strength.
Eqs. (l)-(3)
yield
,?‘?
r,=L(C)
=
ad2rfr,,LfL__ Tfrl!
(4)
l-,
where -2
u L =
rf.
2
The expression
-
cl<;<1
(5)
for L will be used provided that
r,
(6) Tfn,
where (of* > is the mean fibre strength. From Eqs. (4)-(6)
the interval of Cr is found: (7)
XT. Mileiko, A.K. Stepanov / Theoretical and Applied Fracture Mechanics 28 (1997) 95-l 08
97
Fig. 1. Schematic of the cylindrical microcrack.
where
wnEf
J
x=
(8)
__
rr
When the applied load direction forms angle 8 with the fibre axes (O-directed fibre), the regular tensile stress in a fibre is [lj-
cO=(T.
(9)
(cos*O - + sin*0 >
where 8 lays changing /c0seI2 For a B-directed
in the interval
(10)
5 fibre,
L,=L,(cr,)
=
instead
of Eqs. (4) and (51, it is found that
ff/m r‘:,
.-
z;‘*
(11)
1 - Ze
and
~. r f
ze = 2 yfm Ef
(c0s2e-~&9)2~~2,
O
1
(12)
2.2, Penny-shaped microcracks When the fibre/matrix interface is sufficiently strong, a fibre break yields formation of a penny-shaped microcrack in the matrix. If the fibre direction coincides with the load direction (see Fig. 2) then neglecting matrix unloading there results
dI.$.l =Tr f
4’ ‘4 E,
(13)
98
XT. Mileiko,
A.K. Strpanou/
Throrrticul
Fig. 2. Schematic
and Applied
Fracture
of the penny-shaped
Mechanics
28 (1997)
95-108
microcrack.
and AW,, = 7~( Kf - r;)
(‘4)
ytm
where R, is the radius of the microcrack
and length 1 is given by Eq. (3). This yields
(15) The consideration
is valid when
(‘6) or
Using Eq. (9) for a o-directed fibre yields radius
R,,,
for penny ((7r/2)
- 8)-directed
microcrack
as
-------R,,,=
Rc,o( 6)
= rf
( 18)
where C values belongs to the interval given by Eq. (17) and t) angle belongs to the interval given by Eq. (IO).
3. Percolation
criterion
In a composite volume under an applied load we observe a set of independent, for each hV,O, Poisson’s assemblages of the crack centers. Parameter Alp,” is the density of crack centers oriented as shown in Fig. 3. The
ST. Mileiko, A.K. Stepanou / Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
99
Fig. 3. A fibre unit vector on a unit sphere S2.
joint density of a pair ( cp, 8) of independent cos 8 477. ’
(
g(rp,O)=
random variables
is
for -n-
0,
(19) for other cp, 8.
The sum of A,,, over all values (rp,
e)
equals to
(20) Assume that the composite fracture corresponds to formation of an infinite cluster of the connected microcracks. Hence, evaluation can be made for the critical value of a, that is 3 * as a function of h, and structural parameters of the composite. 3.1. Lower bound for the ultimate stress To obtain a lower bound for Cr * , construct an infinite cluster as a random tree in a branching process of the Halton-Watson type [2]. Each generation of such a tree produces a random number of the branches independent for each generation. The branching tree becomes infinite when the mean number, M, of the branches generated by a single branch in the previous generation is larger than 1. So the solution of equation M = 1 gives the lower bound for the infinite branching tree. By the definition,
(21) Here pk, k=O, m=
i
1,2,
. . . are the probabilities
to have k branches,
produced
by a single branch, where
k-p,
(22)
k=O
is the mean number of branches root m, for m > 0: 1120
=
1.60
or connections
on a crack. It was found [3] that equation
M = 1 has a single
(23)
XT. Mileiko, A.K. Stepanou / Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
100
Fig. 4. V,,,-region i-axis.
containing
all the centers of ( q. 0 )-directed cylindrical
microcracks
intersecting
a given microcrack
oriented along the
In the terms accepted, each microcrack of the infinite cluster belongs to a definite generation, say to the n-th generation. A number of the branches in the next, (n + l)th, generation is the number of microcracks connected to the original one. 3.1 .I. Cylindrical microcrack Introduce the Cartesian (x, y, z) co-ordinate system associated with a crack. Without loss of generality we assume that the z axis of the crack coincides with the load direction. Since in the vicinity of the lower limit of the infinite cluster, the mean number, m, of intersections of a given crack is small and because rf < L, the intersections of cracks in the same direction can be neglected. Define pk(go, 0) as the Poisson’s probability of the intersections of a given crack by k( cp, 13))directed cracks. Hence,
Pk(P, 0) =
(A,.*.Jvyl,oJ)k k!
exp( - A,,, . Iv,,,l)
(24)
where IV,.,1 = 2r,. L. L,. sin 0
(25)
(see Fig. 4). Let 7rL, sin 0 N, =
rr
(26)
101
S.T. Mileiko, A.K. Siepanov / Theoretical and Applied Fracture Mechanics 28 11997) 95-108
be the number of points on the unit sphere S2 in a qo-plane with directions
2rf
qj=j.tan-'-
L, sin 0
'
j=O, 1 , . . . . N,
(27)
Region Vq,,e and distribution ~~(4p~)~ is acquired to each direction 4~~. A set of directions distribution jj& r3) that is the probability that a given crack is intersected by k cracks from the &( t9) is the probability that k centers of cracks of set (s)~ belong to a set of all regions Poissonian assemblage of cracks centers are independent for each (cp, 8)-direction, probability represented by
=-
i,
((~i)~ forms set (‘pi&. So (V,,,). Since p&e) can be
(~~A,,oL.LZB sin’e)’ *exp( -2nh,,eLL2, sin*e}
(28)
A set of directions (~~pi>~is a subset of the set, (p),, for all cp-directions and a fixed O-direction. Thus, for probability p,(8) that a given microcrack is intersected by k of the O-directed microcracks, there results M-9 Distribution
250)
(29)
r?;&O) is the Poissonian
distribution
with parameter
th,.LZ,.c0s8.sin28=hfl~l
(30)
volume using polynomial distribution for each finite set of characteristic where Iv01 is a characteristic If0 I from the interval 18I < 7r/2 leads to limit of & when the number of IvOI goes to infinity as
volumes.
(31) Note that jYk is the Poissonian
distribution
with parameter
~hfLj_+~~~Osin2Bcosede=Afl~O~
(32)
where If I is a characteristic volume. In other words, jk is the probability that a given crack is intersected by k cracks from set ( qj, 0) = u @( q~~)~.Since set (qj, 0) is a subset of all direction set (rp, 0) = S*, it is found that (3%
Pk >fik Hence, for the mean number,
M, of intersections
on a crack there results
m 2 fi = A,lvl
(34)
At m = m,, from Eq. (23), there are still no infinite clusters of the cracks. So for all F satisfying the inequality A,lv'l
A,lr;;t> Af@C2
(39 region satisfying
to Eq. (35), a lower bound
(36)
S. T. M&i&o. A K. .Srepuno~~/ Theorer~cul urrd Applied
IO?
Fruct~e
Mechanics
2X (1997) 95-108
where
X
(LXX%
ill/7 /
- iT/?
-
(
1/4)sir1%)~
sin’0 cos 6, db, (37)
1 + (e-f;n)\i(hn~fw,
- (cm% - ( 1/4)sin%)‘j
Using Eqs. (36) and (38) for the left hand side of Eq. (35) and Eq. (23) for the right hand side of Eq. (3.5) the result is
3.1.2. Pennyshuped microcruck To get lower bound of ?? * we shall use the same procedure as in the previous section as well as the same definitions. Set the center of a given penny microcrack at the center of Cartesian, (x, J. :)-coordinate system such that the direction of a fibre axes coincides with the z-axis of load u. For 101 < (n/2), there is the need to define the orientation of penny crack in the space. Let the center of a penny crack be the center of the co-ordinate system. A penny crack is a ( cp. H)-directed crack when the vector orthogonal to the intersection of a disk and (x, v) plane forms a 6’angle with the positive direction of the :- axis and its projection on the (s. y) plane forms a 0 angle with the positive direction of the .r-axis. Without loss of generality, to simplify the calculations further let a (cp, @)-directed crack be a square shape of R,,, side length. With such an assumption. probability pa(~, 0) that k(cp. H)-directed cracks intersect a given one is given by Eq. (29), where IV, ,,I = 8 R, Rc,o( R, + R,,,)sin
H
(39)
Here V,.H is a region containing centers of (cp, H )-directed cracks (see Fig. 5). Following the proof scheme of Section 2. I, the probability pn( 0) is found to be the Poissonian distribution with parameter h,IVg/=h,‘8R,R,,H(Ro+R,,H)sinH
(40)
Note that the angle 8 in Eq. (19) is changed by the ((n/2) - @)-angle because the crack plane is orthogonal to the fibre axes direction. The next step to get probability pk is analogous to that in Section 2.1 and that Rk is Poissonian distribution with parameter A~iVl=h,-RR,IiX,,,(R,
+R,,,)sin%d@
(41)
0
Therefore, m =
the mean number
m of intersections
on a crack is
A,(V(
(42)
Following the same procedure, there is the need to find the lower bound for the right hand side of Eq. (41). Using Eqs. (15) and (1 S>, it is found that A,IVI 2 16h,R,
/0
wRf.s sin20 d6 = h,@,@, (T9/2
(43)
XT. Mileiko, A.K. Stepanov / Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
Fig. 5. V,.B region containing all the centers of (cp, 6)-directed the origin of coordinates and being orthogonal to the z-axis.
square-shaped
microcracks
intersecting
a given microcrack
103
with a center at
where
Following the last step of the same procedure (43), there results m0 cr*2
i
drp,
in getting lower bound
Cr * , and so using Eqs. (7), (231, (3.5) and
2/g
1
. A;
219
(45)
3.2. Upper bound To obtain lower bound of si * , by using the theory of branching process, construct the infinite cluster which has no closed cycles of crack surfaces. Actually, the infinite cluster does contain some of them. Take into account the formation of closed cycles to obtain the upper bound for Z * . 3.2.1. Cylindrical microcrack Consider, in the (rp, 8)-space-volume, a (2r,, e&layer of thickness 2r, which forms the angle 8 with the positive direction of load (z-axis). Obviously, when a (2r,, 8)-layer contains an infinite cluster of the
104
S.T. Mileiko, A.K. Stepanor, / Theoretical and Appltrd Fracture Mechanics 28 11997) 95-108
intersecting cracks, then the cluster already exists in the whole (cp, 0)-space-volume but not vice versa. To evaluate the upper bound for 5 * we need to construct the infinite cluster in a (2r,, 0)-layer. Subdivide (2r,, fI )-layer into rectangular cells of a size (\/5/ 1O)L, . (6,’ 1O)L, .2 rr. Centers of these cells form the regular square lattice 2; belonging to the (2r,, 0)-layer. Let p(cp, n/2, 0) be the probability that a (&?/IO)&. <&/lo)&. 2r, cell contains at least one center of (cp. 0 j-directed crack and at least one center of (cp + (7r/2), o&directed crack. If such an event occurs we say that the center of a cell is occupied. Using Poissonian distribution and taking into account rf < L. L,, < L,-there renders
(46) Probability p(rr/2, 0) that a (&/lo>&. cracks, follows from Eqs. (19) and (46)
(&/lo)&.
2r,-cell
contains
at least two centers
of orthogonal
(47) With probability p(rr/2, 0 I, a site of the square lattice Zi associated with a (2r,, @)-layer is occupied independently of other sites of Zi. Let p(rr/2) be the probability that on at least one of the square lattices Zi, (101 < 7r/2) a site of a lattice is occupied independently of other sites of the same lattice. Eq. (47) yields
Now, reduce a problem of occurrence of the infinite cluster of intersecting cracks in some (2r,, 6)-layer to a well-known site problem of the percolation theory on a two-dimensional square lattice Z2 [4]. It has already been mentioned that a site of Zj is occupied with probability p(rr/2, 0) independently of other sites of 2;. The occupation of a site of Z,’ is caused by the intersection of at least two orthogonal cracks in a (&/lo)&. (fi/lO)L, .2r,-layer. The size of a cell is chosen such that two neighboring occupied sites of Zi cause intersecting pairs of the orthogonal cracks and three neighboring occupied sites of Zi cause closed cycles of intersecting cracks (see Fig. 6). It is seen that the infinite cluster of occupied sites on Zi causes the infinite cluster of intersecting cracks in a (2r,, @)-layer. The site problem on Zi yields that for all (T satisfying the inequality
(‘) is the critical probability of the site problem on Z2, there exists an infinite cluster of occupied where pCr of Zl with the probability equal to 1. This means that for all G satisfying the inequality
P f i
1
sites
(50)
> Pi:’
in at least one (2 rf, 8 )-layer, there exists an infinite cluster of intersecting cracks with the probability equal to 1. In other words, the inequality given by Eq. (50) describes a region of the a values in which an infinite cluster of intersecting cracks exists almost surely in a whole space-volume of the composite. To specify a region of i$ values satisfying Eq. (50). the upper bound for p(rr/2) is found. Eqs. (4)-(8) and (48) yield p
f
(
i
I1
-exp(
-A,F,F,i?*)
(51)
XT. Mileiko. A.K. Stepanov/
Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
105
Fig. 6. A fragment of 2’ lattice (marked by points) with occupied sites (marked by + ) when the lattice cell contains at least two centers of two orthogonal cylindrical microcracks.
where tq*
F, = (T$(
F2
=
1 + (q
h2r?(1 + ( d#Er/rry2 ))\/2y,E,/r,(
(cos28 - fsin28)6
l/fTi2 20
1 + ( o/r,)d2y,,E,/r,)-“2)1’2 . cos 0 de
-m/2 1 +((~/(0~*))\11-(1
+ (o/T&)/~)~"~
- (c0s2e-
+sin28)2)2 (52)
Eqs. (49)-(51)
are applied to give
(53)
3.2.2. Penny-shaped microcrack For penny microcracks, a slight change of the proof in Section 2.1 is made. Decompose ( cp, 8 )-space-volume Centers of these cubes form a cubic lattice 2; in a space volume. Let by cubes of side length a0 = (1/6)R,,,. p( ~0, 7r/2, 0) be a probability that a,. a, . a, cube contains at least one center of (q, O&directed crack and at least one center of (cp + (7r/2), O)-directed crack. Then the Poissonian distribution yields
(54)
ST. Mileiko. A.K. Stepanu / Tlwweticul and Applied Fructure Mechanics 28 119971 95-108
106
Fig. 7. A fragment of Z’ lattice. Points 0,.
O2 show the centers of two intersecting orthogonal square-shaped
Hence, following the procedure used in Section 2.1 probability two centers of two orthogonal cracks is ~(5,“)
p(rr/2,
microcracks.
13) that uH. ug u,-cube contains at least
= i -exp[----&“~R~,6sinO)
(55)
A site of the lattice 2: is said to be occupied with the probability p(7r/2, 0) independently of other sites of Zj. Let p(rr/2) be the probability that on at least one of the cubic lattices Zi, 113 < z-/2 a site of a lattice is occupied independently of other sites of the same lattice. Similarly to what has done to obtain Eq. (48), Eq. (55) may be used to yield = 1 - exp(-
p(t)
&&_(=Rz,,
sin Bd@)
(56)
The occupation of a site of Zi is caused by the intersection of at least two orthogonal cracks in a uB. a6 a0 cube of (9, O&space-volume (see Fig. 7). So any two neighboring sites of 2: cause intersecting pairs of orthogonal cracks and so an infinite cluster of occupied sites of Zi bears an infinite cluster of intersecting cracks. From the site problem on Z’ with the critical probability pz:’ [4] 13) <
P CT
4
(57)
it follows that for all (T satisfying p 5, i
the inequality
i? >piB’
(58)
1
there exists an infinite cluster of the occupied satisfying the inequality
sites of Zi
with the probability
equals
to 1. Then for all C
(59)
S.T. Mileiko, A.K. Stepanou / Theoretical and Applied Fracture Mechanics 28 (1997) 95-108
107
on at least one of the cubic lattices 2: and thus in a whole space-volume of the composite, there exists an infinite cluster of intersecting cracks with probability equal to 1. Hence, to specify the interval of C values satisfying EQ. (59), there is the need to find the upper bound for &n/2). From Eqs. (18) and (59) there results
(60)
p where
G,
=
From Eqs. (59)-(62),
G2=&_/“(l+cos2B-$sinz8)sinBdB 0
(61)
a final upper bound for Cr * is obtained:
(62)
4. Discussion First it is seen that both lower and upper bounds for dependencies volume fraction, h,, are of the same type, those are 5 * o[ A, l/2
of critical fibre stress (+ * on mean fibre
(63)
and z* ah,2/9
(64)
Eq. (63) is valid for a weak fibre/matrix interface (see Eqs. (38) and (53)) and Eq. (64) is valid for a strong fibre/matrix interface (see Eqs. (45) and (62)). Eqs. (63) and (64) yield dependencies for ultimate stress of the composites neglecting a contribution of the matrix to the composite strength as Z * a u:/~
(7* a u:j9
(65) (66)
where vt is the fibre volume fraction. Eqs. (65) and (66) can be actually valid at a sufficiently low fibre volume fraction only since the whole theory has been developed for elastically microcracks. Non-linear dependencies similar to those obtained have been observed in numerous publications although most authors preferred to speak about linear dependencies. The validity of the relationships obtained within a rather narrow interval of small fibre volume fractions does not decrease the importance of the results. Actually, short-fibre composites increasing the composite strength at low vf can often determine a maximum composite strength that can be achieved as shown in Fig. 8. The falling part of the u * (v,> curve for short-fibre composites that interrupts the growing part of it is determined by microstructural features of the composite [5]. Still, it is a primary importance to evaluate the dependence for low vf. Then it is seen that the exponent in the final expressions (Eqs. (63) and (64)) strongly depends on the fibre/matrix interface characteristics. Since in the model considered the situation around a fibre break and microcrack was not analysed in strict terms, it was not important to formulate conditions for a particular microcrack to occur. It is the author’s intention to consider formation of the cluster containing both types of the
S.T. Mileiko, A.K. Stepanor / Theoretical and Applied Fracture Mechanics 28 (1997) 95 -108
108
::
a
g- 1.4
8
4
s 01.2
d 5
E '; ml.0 5
0
i 0.8 -
0.6
i
0.00
Fig. 8. Composite
._
.-._M.-
0.05
0.10 Fibre volume
---s
0.15 fraction
0.20
strength versus fibre volume fraction for a C/TiC/Ti
0.25
composite
system [6].
microcrack as a next step of the development of the model. At that step, the conditions mentioned should be important. Also, the development of the model to bring it close to a real situation calls for accounting for statistics of the fibre strength and, hence, the fibre breakage. An important feature of a large family of short-fibre composites, that is the ceramic-matrix family, is similarity in elastic behaviour of the fibre and matrix. Hence, this case is also to be considered.
5. Conclusion The main result of the present work is a revealed possibility to adequately describe the failure behaviour of randomly reinforced fibrous composites by using the percolation theory. A simple mechanical model analysed by the percolation theory yields semi-qualitative strength/fibre-volume-fraction dependencies that can be of importance in optimising a composite microstructure. It is also important that the results of the first work in this direction show clearly what should be done to develop a percolation type theory of the failure of composite microstructure.
Acknowledgements The work was carried out under the support of Russian Foundation
for Basic Research, Project 96-01-0183 1.
References [l] R.M. Christensen, Mechanics of Composite Materials, Wiley Interscience, New York, 1979. [2] B.A. Sevastianov, Branching Processes, Nauka, Moscow, 1971 (in Russian). [3] A.K. Stepanov, V.V. Tvardovsky, A.A. Khvostunkov, The fibre skeleton structure and transport properties of stochastically reinforced composites, Comp. Sci. Tech. 45 (1992) 221-228. [4] H. Kesten, Percolation Theory for Mathematicians, Birkhause, Boston-Basel-Stuttgart, 1982. [5] N.S. Sarkissyan, A.A. Khvostunkov, S.T. Mileiko, A failure model of short fibre composites with metal and ceramic matrices, Appl. Comp. Mater., submitted. [6] S.T. Mileiko, M.V. Gelachov, A.A. Khvostunkov, V.M. Kiiko. D.B. Skvortsov, Short-fibre/titanium-matrix composites, in: Proc. the 10th Intern. Conf. on Composite Materials, vol. 2, Woodheard Publishing Lmd., Canada, 1995, pp. 131-138.