J. Phys. Chem. B 2000, 104, 3881-3886
3881
Scaling Laws in the Fragmentation of Disordered Layers† Ulrich A. Handge,* Igor M. Sokolov, and Alexander Blumen Theoretical Polymer Physics, UniVersity of Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg im Breisgau, Germany ReceiVed: October 5, 1999; In Final Form: January 27, 2000
In this work we investigate the fragmentation of layers under external forces, model the systems through spring networks, and focus on the interplay between elasticity and disorder. We concentrate on thin layers on supports. Our study shows that spring-network models display universal scaling: After an initial stage, the mean fragment length 〈L〉 scales with the applied strain , 〈L〉 ∝ -R where R only depends on the degree of disorder in the layer and the nonlinearity in the force-elongation relation of the springs. We derive analytical expressions for R and exemplify our findings by numerical simulations. In addition, we compare the theoretical results with available experimental data.
1. Introduction This article is devoted to studying the scaling properties of fragments of membranes or polymers (thin films) which form when their underlying supporting structures are stretched. The thin films are not locally homogeneous in their breakage properties, which introduces a stochastic aspect into the picture. Furthermore, fragmentation occurs sequentially, so that the whole process develops in time. Such a problem is intimately connected to the works of Scher,1-7 both in determining the influence of the disorder on the temporal and geometric aspects of the process and, of the utmost importance, in determining the scaling relations in connecting the microscopic to the macroscopic realms. Thus, in his work with Zallen, Scher, starting from a pragmatic idea of formulating an approximate rule for threshold percolation concentrations for different lattices, has applied the percolation theory to continuous, topologically disordered system.1 In his fundamental works on continuous time random walks (CTRW),2 the basic role of algebraic stepping-time distributions on the scaling properties of photocurrents in amorphous semiconductors was thoroughly investigated; these works emphasize the impact of basic mathematical concepts in describing processes occurring in disordered materials. Recently, Scher studied with co-workers problems of materials' strength and failure, which lead to different regimes of damage.6,7 The model which we consider in the following is concerned with the sequential appearance of damage in thin layers, and as in previous works,8-10 we focus on the role of local inhomogeneities present in the sample. We consider a thin surface layer covering a supporting system. The changes of the lengths of the layer and the support caused by stretching or bending the supporting structure, temperature or solvant changes, etc., often cause multiple cracks in the layer, often producing a mosaic pattern of separated fragments. The statistical geometrical properties of these patterns are fascinating. Moreover, the failure of membranes and polymers on supporting structures is related to the fragmentation of coatings and fibers. A previous study11 on the fragmentation of coatings under uniaxial tension has shown that three distinct stages of frag†
Part of the special issue “Harvey Scher Festschrift.” * Corresponding author. E-mail address: ulrich.handge@ physik.uni-freiburg.de.
mentation exist. The initial stage is characterized by randomly located cracks, where the probability to fail is nearly independent of their position in the sample. In the second stage, fragments break close to their middle (“midpoint cracking”). Finally, debonding sets in, which is closely associated with a shear stress singularity at the edge. In what follows we consider fragmentation in a simple one-dimensional situation and concentrate on the simplest metric characteristic of the pattern, namely, on the mean fragment size. We show that the behavior of this quantity as a function of the support’s elongation exhibits scaling properties which depend strongly on the disorder and on the geometrical properties of the system. The one-dimensional fragmentation model is not only the simplest theoretical approach, it can also be applied to many real situations. Some fragmentation processes are essentially one-dimensional; examples are the fragmentation of thin brittle coatings and fibers under uniaxial tension12-17 or the formation of cracks in colloidal suspensions.18,19 In addition to that, previous studies of two-dimensional network models9,20 have revealed that one-dimensional (1d) models capture the basic features of the process, and many aspects of the two-dimensional (2d) models already occur in 1d models. 2. Scaling in the One-Dimensional Fragmentation The model under study is related to the 2d spring-network model discussed in refs 21-23. Our one-dimensional arrangement is depicted in Figure 1. The surface layer is modeled by N nodes connected by N-1 breakable coil springs of equilibrium length leq. The nodes in the surface layer are attached to the support via leaf springs. Comparing with classical continuum mechanics, the coil springs mimic the tensile elasticity of the coating, whereas the leaf springs represent the shear elasticity of the adhesive layer. Each spring in Figure 1 expands under an acting force; the force-elongation relation for the springs in the surface layer is denoted by fk(uk), where fk is the force of the kth surface layer spring and uk its elongation. The forces of the springs between surface layer and support are denoted by Fk(Vk) with the elongations being Vk. The (homogeneous) expansion of the support is modeled by an increase in the distance between the support’s nodes. Initially, the distance between two neighboring
10.1021/jp9935586 CCC: $19.00 © 2000 American Chemical Society Published on Web 03/18/2000
3882 J. Phys. Chem. B, Vol. 104, No. 16, 2000
Handge et al.
Figure 1. One-dimensional model for surface layer fragmentation under uniaxial strain. Here d and D are the Hooke’s constants of the coil springs, respectively the leaf springs (dashed lines), uk respectively Vk are the corresponding elongations. The equilibrium length of the coil springs is denoted by leq. The springs between the surface layer and the support are shifted vertically for better illustration.
nodes is leq. Increasing this distance to leq + ∆l implies for the applied strain that ) ∆l/leq. Because of the stress transfer from the support to the surface layer, all intact springs of the surface layer expand. A failure (break) in the surface layer takes place when the elongation of one of its springs attains its specific breakdown threshold; each such threshold is determined from the start and denoted by ub, so that ub(k) is the threshold of the kth element. Now the formation of cracks in the material is strongly influenced by the existence of small defects, such as flaws and micropores. These defects, which occur, e.g., during the manufacturing process, are randomly distributed. In this study, we take these statistical heterogeneities into account by a probability distribution p(ub) for the failure thresholds of the springs. Each failure threshold ub(k) is chosen randomly at the beginning and then stays fixed, so that we deal with a quenched disorder situation. It is typical for failure and damage phenomena, that crack formation is associated with weak points.24 In fact, the behavior of the threshold distribution p(ub) close to its minimum value umin will turn out to be of much importance. Here we consider the following form for p(ub):
p(ub) )
{
χ(ub - umin)β 0
for umin e ub e umin + W (1) otherwise
where χ ) (β + 1)W-(β + 1) and β g 0. We remark that expression 1 represents a very large class of probability distributions. This form was used in several previous works, since it allows to model both a power-law increase near the lower limit as well as simple rectangular patterns, and it allows us to interpolate nicely between these cases. Furthermore, the Weibull distribution
W (u;β,N,W) ) 1 - exp[-N(u/W)β+1]
(2)
a form of widespread mechanical use, follows directly from eq 1 with umin ) 0: Consider a fragment consisting of N breakable springs. Then in a homogeneous picture the probability PN(u) that the fragment is still intact under the elongation u is PN(u) ) [1 - Fcu(u)]N ) exp[N ln (1 - Fcu(u))], where Fcu(ub) ) ∫u0b p(u˜ )du˜ is the cumulative probability distribution function. For large N, fragments break under small elongations, so that we can expand the logarithmic function. This leads to PN(u) ≈ exp[-N(u/W)β+1] . Consequently, the probability that the fragment of length N fails under the elongation u is
1 - PN(u) ) 1 - exp[-N(u/W)β+1] ) W(u;β,N,W) (3) After the breakage of a spring, the initially intact layer is separated into two fragments. Since we continue to increase
the applied strain, additional fragments appear. In this article we vary the elastic properties of the springs as well as the properties of the support and concentrate on the dependence of the mean fragment length 〈L〉 on the support’s elongation (respectively on time). Our analysis of the evolution of the average fragment length starts with the elongation u(k,L) t uk of the kth surface layer spring under the strain . The elongation can be approximately expressed by a dimensionless function g(x) of the relative position x ) k/L so that
u(k,L) ≈ aLbg(x)
(4)
where the positive powers a and b and the particular form of the function g(x) depend on the properties of the film and the support. For example, for the scalar linear model which is discussed in detail in the next section, this distribution tends to a quadratic parabola, u(k,L) ∝ k(L - k), so that a ) 1, b ) 2, and g(x) ∝ x(1-x) (vide infra). Using the cumulative distribution Fcu(ub) as defined above, we find the following expression for the probability P that a fragment of length L is still intact under the strain :
P ) [1 - Fcu(u(1))][1 - Fcu(u(2))] ×...× [1 - Fcu(u(L -1))] L-1
) exp{
∑ln[1 - Fcu(u(k))]} ≈ exp{- ∫0 Fcu[u(k,L)]dk} k)1 L
(5)
Since the fragmentation of the layer is associated with the breakage of a few weak springs, one has Fcu[u(k)] , 1 , so that ln (1 - z) ≈ -z holds in eq 5. Now we define the critical fragment length Lc() as being the length of a fragment that stays intact with probability 1/2 at strain . Thus, fragments, whose length is considerably smaller than Lc(). are very unlikely to break under this elongation; fragments larger than Lc() are very unlikely to survive. Changing from the summation in k to the integration and inserting P ) 1/2, Equation 5 leads to
∫L0 Fcu[u(k,Lc)] dk ) ln 2 c
(6)
Introducing x ) k/Lc and inserting for u(k), eq 4 yields
2Lc W
β+1
∫x1/2 [aLbc()g(x) - umin]β+1 dx ) ln 2
(7)
min
where xmin is defined by aLcbg(xmin) ) umin. From eq 7 we obtain the evolution of the characteristic length Lc(). For a fixed value of β, the degree of disorder in the coating is characterized by ζ ) (W/umin)β+1. Large Lc(umin/W)β+1 values imply Lc . ζ , i.e., umin > 0. Moreover for Lc(umin/W)β+1 large, the integral in eq 7 is small, i.e., aLbc g(1/2) ≈ umin . From this a scaling relation follows: Lc ∝ -a/b. The opposite limiting case considers small values of Lc(umin/W)β+1. Note that this case includes umin ) 0. Taking umin ) 0 leads to Lc(Lbc a)β+1, which yields the scaling law Lc ∝ -a(β+1)/[b(β+1)+1]. In any case, the characteristic length Lc scales with the relative elongation as Lc ∝ -R, where the exponent R is given by
R)
a(β + 1) b(β + 1) + 1
(8)
and, since β g 0, the scaling exponent R lies in the interval
Fragmentation of Disordered Layers
J. Phys. Chem. B, Vol. 104, No. 16, 2000 3883
a a eRe b+1 b
(9)
where the lower bound corresponds to the case of extremely strong disorder (umin ) 0, β ) 0) and the upper bound holds for the perfectly ordered layer (W ) 0 or β f ∞ in eq 1). The fact that the degree of disorder strongly influences the patterns and the kinetics of fragmentation can be also observed in related failure phenomena, see, e.g., ref 25. The fragmentation of the surface layer proceeds as follows: With increasing the value of the critical length decays, so that fragments whose length was initially smaller than the critical value Lc() eventually enter into the critical region (L > Lc()) and break into two segments, having typically lengths around Lc/2. These two newly formed fragments are independent, so that the function g(x) for the parts of the initial segment has (approximately) the same form as for the initial one and is universal. This shows that for such a process Lc is a single characteristic length of the problem, so that, e.g., the mean fragment length 〈L〉 follows the same scaling law as the characteristic length Lc, namely, 〈L〉 ∝ -R, where the bounds for R are given by the inequality 9. For some special models this scaling relation can be proved by solving the corresponding kinetic equation for the fragment length distribution.26 In the next sections we will evaluate the exponents a and b in eq 4 within realistic physical models and will compare the results with experimental data. 3. Scalar Linear Model The first analytical approach to the fragmentation problem that we discuss here is the scalar linear model of a surface layer. This model is related to the shear-lag analysis adapted to the fragmentation of fibers by Cox.27 It applies to a case when the deformation of the support, caused by the mismatch in the elastic properties of the surface layer and the support, can be neglected. Both the surface layer springs and the springs between surface layer and support are considered to be linear elastic elements, so that we have fk ) duk and Fk ) DVk where d and D denote the respective Hooke’s constants. Each node in the surface layer is in equilibrium, so that Dνk ) d(uk - uk-1) holds, with the boundary conditions u0 ) uN ) 0. Moreover, the network arrangement of Figure 1 implies ∆l ) leq ) νk - νk+1 + uk. This yields
d leq ) - (uk+1 - 2uk + uk-1) + uk D
(10)
Going to a continuum picture, we can replace uk+1 - 2uk + uk-1 by d2u(k)/dk2 with u(k) t uk, cf. refs 10 and 28. Hence
leq ) -ξ2
d2u(k) dk2
+ u(k)
(11)
where we set ξ ) (d/D)1/2 . Solving eq 11 leads to8,26
(
u(k) ) leq 1 -
)
cosh [(2k - N)/(2ξ)] cosh [N/(2ξ)]
(12)
The dependence of the elongation on k is strongly influenced by the ratio ξ/N. For N . ξ the elongation u(k) is nearly independent of k and attains the value leq. Only at the boundaries of the surface layer, u(k) decreases exponentially to zero within the length ξ, which is often referred to as a correlation length, a relaxation length or a stress recovery length.8,29-31 For N , ξ, the plateau in the middle of the coating
Figure 2. Fragmentation of thin SiOx coatings on PET substrates under uniaxial tension. Shown is the mean fragment length 〈L〉 versus applied strain in a double-logarithmic plot. Note that 〈L〉 scales with . The scaling exponents and the coating thicknesses are indicated in the Figure. The data are taken from ref 33.
disappears. This leads to a parabolic u(k) distribution: u(k) ) leqk(N -k)/(2ξ2). Now let us turn to the fragmentation process and consider a fragment of length L, where u(k) is given by eq 12 with N replaced by L. As long as L . ξ, the u(k) distribution displays a plateau shape and therefore the probability to fail is nearly equal for all springs. In this stage, the weakest springs break first, resulting in a crack pattern that consists of randomly located cracks, as also observed in experiments.11,32 Since the average fragment length 〈L〉 decreases monotonically, one finally enters in the regime where one has L . ξ. The crossover is given by L ∼ ξ. In the following, we concentrate on the case L , ξ. For L , ξ the elongation u(k) attains its maximum in the middle of the fragment, so that cracks occur more probably in the middle of existing fragments. This regime has been also observed experimentally.11,32 The u(k) distribution is parabolic: u(k) ) leqk(L - k)/(2ξ2) which exactly corresponds to eq 4 with a ) 1, b ) 2, and g(x) ) x(1 - x)/(2ξ2). Since in a scalar model for a given support deformation the newly formed fragments are independent, the stress distribution in each of them is described by the same equation as in the initial fragment, and all the scaling assumptions leading to eq 9 are fulfilled. Thus, if the scalar linear model can be applied, the fragmentation shows a universal scaling behavior; the mean fragment length scales with the applied strain according to 〈L〉 ∝ -R . From eq 8, R ) (β + 1)/(2β + 3) holds.26 For the linear model, R ranges between 1/3 (strong disorder and β ) 0) and 1/2 (weak disorder, β f ∞). We emphasize that this scaling behavior is not restricted to the one-dimensional model. Simulations of fragmentation in a two-dimensional spring network9 lead to the same scaling behavior and to the same scaling exponents as in 1d. The results of the scalar model apply to several experimental situations. Comparing the theoretical results to experimental findings available from the literature, one is to note that 〈L〉 ∝ -R describes only the intermediate strain range which is rather restricted (about 1 decade). First at very small strains, the stress in the film is too small and no cracks at all appear. When the stress in the film is of the order of the film’s strength, a crossover regime characterized by a rapid growth of the number of cracks sets in. This regime corresponds to the formation of fragments whose size exceeds the correlation length, so that the stress distribution within the fragments is flat and cracks emerge in an uncorrelated manner. On the other hand, at too large strains the number of fragments saturates, mostly due to their debonding from the sublayer. Thus, the amount of data is inevitably rather small. As an example we consider the fragmentation of thin SiOx -coatings on poly(ethylene terephthalate) (PET) substrates33, see Figure 2. The experiments were reported in refs 15 and 16. The
3884 J. Phys. Chem. B, Vol. 104, No. 16, 2000
Figure 3. Average fragment length 〈L〉 versus applied strain for an oxide film on strained aluminum in a double-logarithmic plot. The solid line with slope -0.44 is the result of a least-squares fit. The data are taken from ref 35.
Handge et al.
Figure 5. The function 〈L()〉 in homogeneous systems (W ) 0 in eq 1) for several m values. The parameters are N ) 215, ξ2/N2 ) 0.1, B ) 6 × 104, ub ) 0.008, and leq ) 1. The m values are as indicated.
assume that the F(V) forces obey the function
(| |
F(ν) ) D sgn(ν) ν +
Figure 4. Fragmentation kinetics of paint coatings on a rubber membrane in a double-logarithmic plot. Here l0 is the coating’s length at 0% strain. Note that 〈L〉 scales with . The slope of the dashed lines is -0.71.
coatings were evaporated by a PVD (physical vapor deposition) technique for the coatings with 30-56 nm thickness, whereas the 20 nm thick coating was prepared by PECVD (plasma enhanced physical vapor deposition). Figure 2 presents linear least-squares fits to the data plotted on logarithmic scales. The fits lead to R values between 1/3 and 1/2, i.e., within the range predicted by the theory. Although the data points are rather scarce and their statistical scatter is large, one readily infers that the exponent R typically grows with the thickness of the layer. This also confirms our theoretical results, since independent experiments have shown that thicker coatings are generally less disordered than thinner ones.34 Moreover, the 20 nm coating with a larger R value than the 30 nm coating was prepared by a PECVD technique, which produces in general more homogeneous coatings than PVD. Another example is provided by Figure 3, where we replot the data of Edeleanu and Law35 on fracture of oxide films on strained aluminum. Here a least-squares fit in the range 2-20% leads to R ) 0.44 ( 0.04, also within the range predicted by our model. We note that the available data are rather scarce, and their statistics is rather poor so that additional experiments of fragmentation of thin brittle coatings are welcome. 4. Scalar Nonlinear Model Although the predictions of the scalar linear model are well supported by experimental data on thin inorganic coatings, some experiments on polymeric coatings show scaling behavior of 〈L()〉 with scaling exponents around R ) 0.7,36 see Figure 4, lying significantly outside the interval allowed for by the linear theory. Since polymers belong to a class of materials which display nonlinear stress-strain relations, we extend our analysis to include nonlinear force-elongation dependencies. Here we take the force-elongation relations for the springs between the surface layer and the support to be nonlinear. We
)
|ν|m B
(13)
where D, B, and m are positive parameters (m > 1). One may interpret the function F(V) as “stress hardening”. The formal structure of eq 13 also appears in related situations, e.g., in the discussion of stress and strain near a crack tip in a “strainhardening” material, see refs 37-39. For nonlinear springs it is convenient to start from a differential equation for the elongations V(k) t Vk. The geometrical construction in Figure 1 leads to ∆l ) leq ) νk - νk+1 + uk. Subtracting from this equation the corresponding one for k - 1 yields, with fk ) duk,
1 νk+1 -2νk + νk - 1 ) uk -uk - 1 ) (fk - fk -1) d
(14)
Since the kth layer node is in equilibrium, we have Fk ) fk fk-1 with f0 ) fN ) 0. Inserting this relation into eq 14 leads to (in the continuum picture):
ν''(k) )
F(k) d
(15)
From f0 ) fN ) 0, respectively u0 ) uN ) 0 and ∆l ) leq ) νk - νk+1 + uk one has as boundary conditions ν'(1/2) ) ν'(N + 1/ ) ) -l . 2 eq In ref 40, the elongations u(k) in the late stages of fragmentation are discussed. Here we only report the results of this study; the reader is referred to ref 40 for the detailed analysis. In the late fragmentation stages, u(k) is given by
u(k) ≈ mLm+1gm(x)
(16)
where we set x ) k/L and gm(x) ) (1 - |2x - 1|m+1)lmeq/[ξ2B(m + 1)2m+1]. Increasing m from 1 to ∞ changes the shape of u(k) from a parabola (m ) 1) to a function with a wide plateau (m f ∞). Thus, the overall u(k) behavior corresponds to eq 4 with a ) m and b ) m + 1 and we are again led to the scaling relation 〈L〉 ∝ -R . For nonlinear force-elongation relations the scaling exponent R ranges between m/(m + 2) and m/(m + 1). These two limits correspond to strongly and to weakly disordered coatings, respectively. Numerical simulations of the fragmentation process using the one-dimensional model with N ) 215 indeed show that the mean fragment length scales according to our analysis. Figure 5 shows 〈L〉 versus the applied strain in a double-logarithmic plot for a homogeneous system (ub(k) ) const). The parameters are ub ) 0.008, ξ2/N2 ) 0.1, and B ) 6 × 104. The m values range between 3 and 25 and are indicated in the Figure. Figure 5 reveals the algebraic decay of 〈L〉. In conclusion, the scaling relation 〈L〉 ∝ -R also holds for nonlinear models. Note that a
Fragmentation of Disordered Layers
J. Phys. Chem. B, Vol. 104, No. 16, 2000 3885 dimensionless function f(η). Therefore, the force Fmax is given by
Fmax ) 2Lhσ∞
Figure 6. A narrow fragment of length L on a sublayer of width 2W, see text for details.
cubic nonlinearity leads to the range 3/5 e R e 3/4, which encompasses the experimentally observed value of R ≈ 0.7, see ref 36. Such a cubic nonlinearity may stem from a quartic potential, as appears in the Taylor expansion of interatomic potentials in the theory of crystal lattices, and is necessary on general grounds in order to maintain stability41. 5. Linear Elastic Model in Two Dimensions The scalar linear model predicts a quadratic dependence of the maximal stress (attained in the middle of the fragment) on the fragment’s length and a linear dependence on the support’s elongation. These dependencies lead to the scaling relation 〈L〉 ∝ -R, with R lying between 1/3 and 1/2, i.e., 1/3 e R e 1/2. Taking into account possible nonlinearities shifts this inequality in the direction of higher R values. As already mentioned, scalar models do not take into account the possible strong deformation of the sublayer, i.e., they apply to cases in which the surface layer is rather soft. On the other hand, if the surface film (although thin or narrow) is stiff compared to the sublayer, these deformations can be considerable and must be taken into account. In what follows we consider such a situation, which might apply to the experimental conditions of ref 36, where a rather stiff paint covers a thick and broad but soft rubber band. Such a situation is typified by two cases: having a perfectly adhering narrow strip on a broad sheet or having a thin film on a thick sublayer. Here we consider the first case and start from a single fragment of length L glued to a sublayer of width 2W, see Figure 6. Under the elongation of the support, its portion in the immediate vicinity of the fragment stays practically unstretched and is thus essentially stress free. The typical linear dimension of this region is of the order of the fragment’s size L. We now consider the longitudinal stress distribution in a cross section of a system passing through the middle of the surface layer (vertical line AB in Figure 6) and present a simple scaling argument to obtain the values of a and b in eq 4 for this case. Let us focus on a single fragment on a stretched sublayer. The overall force balance in the system implies that the integral of the horizontal component of the stress σxx along the line AB is equal to the overall force F∞ acting on the elastic sublayer, i.e.,
∫
h σxx(l)nxdl ) F∞ ) 2σ∞Wh
(17)
Here l denotes the coordinate along the AB line in Figure 6 and h is the thickness of the support. Since the parts of the sublayer immediately beneath the coating are essentially stress free the corresponding load is fully carried by the film. The force in the middle of the coating is maximal and is given by Fmax ) h ∫[σ∞ - σxx(l)]nxdl. Note that within the theory of linear elasticity σxx(l) is proportional to σ∞ and thus to . Moreover, since for L , W the value of L is the only characteristic scale of the problem, the value of σxx(l) can only depend on the dimensionless position η ) l/L; hence, σxx(l) ) σ∞f(η) with a
∫-∞0 [1 - f(η)]dη
(18)
A similar form holds for u, the local elongation of the fragment. This expression leads to the values a ) 1 and b ) 1 in eq 4, since σ∞ ∝ . These considerations, which imply a ) 1 and b ) 1, yield 1/ e R e 1, see inequality 9. This is in accordance with 2 experimental findings: Recent experiments of C. Beauvais, E. Kolb, and E. Cle´ment have shown that the better the quality of the coating, the higher is the experimentally determined R, so that R values as high as R ≈ 0.9 can be observed under certain conditions. This finding supports the analysis of such fragmentation patterns within the framework of the linear elastic model. Our scaling considerations on the maximal stress in a single fragment are also supported by an exact calculation in the case of a single fragment on a symmetric infinite plane support. In this case the problem reduces to the known elastic problem of a rigid elliptic inclusion in an infinite elastic medium, which can be solved based on Airy stress functions, which satisfy a biharmonic equation (see ref 42 for a detailed discussion). The solution follows by the conformal mapping of the twodimensional problem of a rigid inclusion on a circular inclusion of unit radius; here the rigid inclusion is taken to be parallel to the applied stress. In the complex plane, z ) x + iy, the mapping is given by
z ) ω(ζ) ) R(ζ + 1/ζ)
(19)
with R ) L/4. This complex ζ should not be mistaken for the degree of disorder ζ introduced after eq 7. The solution in the ζ plane is easily parametrized in polar coordinates, ζ ) Feiϑ and reads
{
σϑϑ - iσFϑ ) σ∞R Φ(ζ) + Φ(ζ) -
ζ2
[ω(ζ) Φ'(ζ) +
F2ω'(ζ)
ω'(ζ)Ψ(ζ)]
}
(20)
where the primes denote the derivative and the overbars the complex conjugation. The auxiliary functions Φ and Ψ are given by Φ(ζ) ) φ'(ζ)/ω'(ζ) and Ψ(ζ) ) ψ'(ζ)/ω'(ζ), where φ(ζ) ) ζ/4 - 1/(4κζ) and ψ(ζ) ) -ζ/2 + [κ/ζ - 2ζ/(ζ2 -1)]/4 - (ζ2 + 1)/[4κζ(ζ2 - 1)]. For a plane stress condition (this is the case of an inclusion glued to a thin broad support strip) the value of κ is κ ) (3 - V)/(1 + V), where V is the Poisson ratio. Using this expression one can calculate both σϑϑ(F,π/2) corresponding to σxx along the x axis, and σFϑ(1,ϑ), corresponding to the tangential stress component τ(x) along the line. Note that, due to the balance of forces, the value of Fmax can also be 0 expressed as Fmax ) h ∫-L/2 τ(x)dx. The integration here gives 2 Fmax ) Lhσ∞(1 + κ) /8κ. In many cases one has V ≈ 1/3; with this value κ ≈ 2 and thus Fmax ) 0.56Lhσ∞ follow. Note that the expression eq 20 has simple pole singularities at ζ ) (1. Since the relation between ζ and z obtained via inverting eq 19 reads ζ ) z/2R ( [(z/2R)2 - 1]1/2, the poles in the ζ plane correspond to a square-root singularity of shear stress near the ends of the fragment. Such a singularity also appears when considering the stress enhancement near a crack’s tip, see ref 42. We note that the case of a thin film covering the surface of a thick sublayer (the one-sided case) implies a more difficult
3886 J. Phys. Chem. B, Vol. 104, No. 16, 2000 theoretical analysis due to the bending of the film during the support’s elongation. Numerical simulations (to be reported elsewhere) show that the behavior of the maximal stress in the middle of the fragment is qualitatively the same as for the rigid inclusion. The situation in the 2d linear elastic model discussed here differs considerably from both variants of the 1d scalar models, in the sense that now the pieces formed after the failure of an initially intact larger fragment cannot behave independently, since they interact with each other via the elastic support. We can consider the situation in terms of energy release. A newly formed narrow crack in the surface layer practically does not release energy, since the amount of energy stored in the film is negligible compared to that stored in the sublayer. To release a considerable amount of elastic energy one needs to form a wide enough portion of free sublayer surface. This implies delamination, which turns out to be inevitable for large energy release. Now delamination is caused by the inverse-square-root singularity of the shear stress at the fragment’s tip, as discussed above, which supports a mode II-crack growth in the glue between the fragment and its support, and leads to freeing a portion of the support’s surface, see refs 12 and 43. Our preliminary simulations show that the stress distribution near the midpoints of newly formed fragments starts to resemble that of free fragments, if the debonding length d exceeds a tenth of the initial fragment size L. The global cracking-delamination process is very complicated and leads to interesting phenomena, such as correlations between the width of the crack opening, the time of crack formation and the size of the neighboring fragments, all phenomena which lay outside of the scope of the present paper. Conclusions To study scaling relations in the fragmentation of layers on supporting structures under uniaxial tension, we have analyzed different network models with random elements. Assuming that u, the local elongation in a fragment, scales with the applied strain and with the fragment’s length L, i.e., u ∝ aLb, we have determined that the mean fragment length 〈L〉 obeys a scaling law: 〈L〉 ∝ -R. The power-law exponent R ranges between a/(b + 1) and a/b. Its specific value depends on the behavior of the distribution of failure thresholds for low threshold values and on the degree of nonlinearity in the (shear) stress-strain relation. We have exemplified these findings using a 1d scalar linear model, a 1d scalar nonlinear model, and a 2d linear elastic model in which we incorporated the thickness of the support. The analysis of these models reveals that they all fulfill u ∝ aLb, and hence, the average fragment length obeys the scaling law 〈L〉 ∝ -R. Data of fragmentation tests using (a) SiOx coatings on PET substrates, (b) oxide films on aluminum, and (c) paint material on rubber membranes can be understood in the framework of these models; we find that (a) and (b) are well reproduced by the scalar linear model, while (c) is in line with the 1d nonlinear model and the 2d model. Acknowledgment. The authors thank Prof. J. -A.E. Månson and Dr. Y. Leterrier for valuable discussions and for providing us with the data of their experiments. We also appreciate
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