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Scaling and patterns in surface fragmentation processes
Physica A 266 (1999) 299–306
Scaling and patterns in surface fragmentation processes I.M. Sokolov ∗ , A. Blumen Theoretische Polymerphysik, Universitat Freiburg, Hermann-Herder-Str.3, D-79104 Freiburg i.Br., Germany
Abstract We model the failure of inhomogeneous surface layers under expansion. The quenched disorder is accounted for by a probability distribution (PD) of local breakdown strengths. The ensuing crack patterns range from percolation-like defect aggregates to well-deÿned polygons. Since no load sharing takes place in surface fragmentation, the e ects governing crack propagation are local: the defects interact only within some correlation length. Fragmentation can be viewed as anisotropic, correlated percolation, where the formation of a new defect depends on the defects already present. Both the overall picture and the scaling of the fragment sizes depend strongly c 1999 Elsevier Science B.V. All rights reserved. on the PD. PACS: 62.20.Mk; 46.30.Nz; 05.40.+j Keywords: Scaling; Surface fragmentation; Load sharing
1. Introduction The statistical description of failure phenomena in disordered, complex systems has drawn recently much attention [1–15]. Apart from the technologically motivated interest in the mechanics of coatings, failure phenomena are part of the large framework of irreversible pattern formation. The mechanical description of fracture often starts from a mesoscopic picture based on ÿnite elements; their mechanical properties vary through the sample, but are, once assigned, ÿxed (quenched) for a given realization. As is well known, the failure patterns are often sequentially created and they depend strongly on the underlaying geometry and on the particular features of the disorder. An important aspect for us here is that the fragmentation of surface layers di ers much from corresponding processes in the bulk; furthermore even the physical questions ∗
Corresponding author. Fax: +49-761-203-5906.
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 6 0 6 - 2
300
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Fig. 1. The model used in the simulations: (a) view of the surface layer; (b) view from the side (vertical cross section).
di er. Thus the interest in the fracture mechanics of bulk materials focusses mainly on failure, i.e. the analysis often ends after the ÿrst crack propagates through the whole sample. In the case of surface layers one is much more interested in fragmentation, i.e. in the collective behavior of a large assembly of cracks. When a surface layer breaks under the deformation of its substrate there appear crack forms which may range (depending on the parameters) from percolation-like, loose aggregates to well-deÿned polygonal parquet patterns [1,10,12]. The particular forms which are created and their sizes depend on the type of the underlying disorder. The peculiarities encountered in the fragmentation of thin coatings are connected with the load redistribution in such surface layers. In most bulk systems load redistribution takes place through load sharing: the overall force acting on any cross section of the system stays constant and is equal to the applied load. In surface fragmentation no such load sharing takes place (the bulk carries most of the load), so that the factors governing crack propagation are local: the defects feel each other’s presence only within a characteristic correlation length, which depends on the elastic properties of the coating. Fragmentation in layers occurs as a kind of anisotropic, correlated percolation, where the probability for the appearance of a new defect depends on whether the defect is formed within an intact area, or near a crack’s tip, or near the internal part of a crack. In what follows we consider a simple, discrete model; we envisage the fragmentation of an inhomogeneous coating covering a substrate under slow expansion. We model the coating through an array of springs and account for its quenched statistical inhomogeneities by assigning to each spring a breakdown threshold taken from a given probability distribution (PD). The adhesion to the bulk is modelled through other springs, which connect the coating to the substrate. We start thus from the model put forward in Ref. [12]. There the coating is viewed as being an array of springs with elastic constants d, forming a triangular lattice, see Fig. 1. The lattice constant (the sidelength of each triangle) is taken to be unity. The ÿlm is attached to the substrate elastically, so that the surface layer can move relative to the bulk; this motional freedom is accounted for by connecting the nodes of the coating and the corresponding sites of the substrate through leaf springs of elastic constant D, see Fig. 1. The surface layer is brittle, so that each spring can break under stress. The value at which a
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particular spring breaks is random, but ÿxed at the start of the fragmentation process (i.e. the disorder is quenched). The PD of breakdown thresholds is a material’s property and is known from the start. As examples, we focus here on breakdown thresholds which are homogeneously distributed in the interval [fmin ; fmin + W ]; for these the PD is p( fb ) = 1=W p( fb ) = 0
for fmin ¡ fb ¡ fmin + W ;
otherwise :
(1)
As we show, the strength of the disorder is characterized by the parameter = W=fmin . Systems with W=fmin 1 can be viewed as being highly ordered. On the other hand, systems with fmin = 0 contain elements with breakdown thresholds in the vicinity of zero and belong to the case of strong disorder [7–9,12]. At the beginning of the process the surface layer is in equilibrium, and no forces act on it. Let us concentrate on the situation of a quasistatically growing stress. Quasistatically (in equilibrium) the overall force f(ri ) acting on the node vanishes at all times. As a projection on the (x; y)-plane parallel to the substrate on has X di; j (|rj − ri | − r0 )eji + D(Ri − ri ) = 0 : (2) f(ri ) = j
Here the ÿrst sum runs over the nearest neighbors, rj is the position of the jth node, Rj that of the corresponding substrate site, r0 is the equilibrium length of a spring in the absence of stress, and eji =(rj −ri )=|rj −ri | is a unit vector in the spring’s direction. The elastic constants di; j are di; j =d for intact and di; j =0 for broken springs. Note that the elastic behavior of the model is fully determined by the ratio d=D, characterizing the relative p strength of the coupling of the ÿlm to the substrate. We refer to the value of = d=D as being a correlation length: shows at what distance from a failure site (measured in lattice units) the displacements of the neighboring springs still di er considerably from these in a bulk without defects (vide infra). The expansion of the substrate under external forces is modelled by gradual changes in the coordinates Rj = (Xj ; Yj ) of the substrate’s sites: the corresponding coordinates grow with the elapsing time t (here: procedure step) as Xj = (1 + at)Xj; 0 and Yj = (1 + at)Yj; 0 (we consider here isotropic and homogeneous changes). At each step Eq. (2) is solved by using a relaxation procedure, i.e. by solving the related time-dependent equations r˙i = −f(ri ) with an appropriate relaxation constant ¿ 0 [12]. After determining the new positions of the nodes, we check whether these lead to the breaking of some springs. This happens whenever the elastic force f acting on a spring, f = di; j (|rj − ri | − r0 ), attains its predetermined breakdown threshold fi;b j . When this occurs we let the spring break irreversibly, and set its elastic constant to zero. This is followed by additional relaxation steps, in which the new equilibrium positions are calculated; these steps may lead (for a brittle regime of fracture propagation) to the breaking of additional springs. If at a given strain no further springs break, the whole procedure is iterated by increasing the time from t to t + t. By this a new strain increase takes place.
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Fig. 2. Fragmentation patterns on a 150 × 150 lattice with = 20 after 4000 bonds have failed [12]; the PD is Eq. (1), with fmin = 100. The value of W is 15 in (a) and 400 in (b).
In what follows we focus mostly on the emerging 2d crack patterns. As main result we ÿnd that these depend crucially on = W=fmin , the relative strength of the disorder. As an example, we present in Fig. 2 (using the data of [12]) two typical situations, obtained on a 150 × 150 lattice with = 20 after 4000 bonds have failed. Fig. 2a displays for small relative disorder ( = 0:15) a parquet-type crack pattern, whereas Fig. 2b shows for = 4 a loose assembly of defect sites. Furthermore, when we follow how individual cracks develop, we observe that the formation of such patterns also depends on : Thus for small the growth of well-deÿned cracks is the main process leading to fragmentation, while for strongly disordered systems the defects form almost independently of each other, and the visible crack patterns arise from the coalescence of such defects. In both cases the rate-limiting process for fragmentation is the formation of new defects, rather than crack propagation. For weak disorder defect formation (starting form a seed “microcrack”) is followed almost immediately by fast crack propagation and by the mechanical relaxation of the ÿlm near the “banks” of the newly formed crack (this mirrors brittle fracture). This may also be inferred from Fig. 2a, where only very few cracks with free ends are to be found. For strong disorder the coalescence of defects (which leads to the pattern of Fig. 2b), is a spectacular but not particularly decisive feature: most of the defects belong to small clusters, not forming any medium-scaled connected structure. To quantify the characteristic length in such patterns we introduce as in [12] the distance L between two neighboring defects, measured along randomly drawn straight lines (the procedure is then averaged over di erent realizations of such lines). The averaged characteristic length hLi shows a power-law dependence on the elongation R, i.e. hLi ˙ (R)− , where the exponent is 12 for weak disorder and is around 13 in the strongly disordered case: see Fig. 3 where the data for a 200 × 200 system with = 20 are presented [12].
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Fig. 3. The mean fragment size hLi as a function of R for fmin = 100 and W = 10 (upper curve) and W = 100 (lower curve) [12]. Note the double-logarithmic scales. The dashed line has the slope − 12 , the dotted line the slope − 13 .
In order to understand these ÿndings for elastic 2d systems we turn to simpler models, where the displacements x are considered to be scalar and the problem is thus mapped on its electrical analogon. Let us consider ÿrst the one-dimensional system, which is amenable to an analytical treatment [8,9]. In a scalar, one-dimensional case, the condition for equilibrium reads xi−1 − 2xi + xi+1 + −2 (Ri − xi ) = 0 ;
(3)
where xi are the scalar displacements (in electrical language: potentials at the nodes) and is the correlation length introduced above. In this case it is possible to calculate the stress distribution within a fragment (it turns out to be parabolical for fragments of length of the order of or less [8,9]) and to derive explicit expressions both for the distribution of failure positions within a given fragment, and also for the distribution of stresses under which fragments of a given length N break [9]. Thus the relative width w of the distribution of crack positions is given by w = [3=(16N )]1=3 . For 1 the fragments break preferentially in the middle, giving rise to a hierarchical pattern; on the other hand for ∼ =1 the fragments can break with ÿnite probability at every site. Also the distribution of elongations R at breakdown is governed by : introducing y = (N 2 =22 )R, Ref. [9] ÿnds that for y ¿ fmin the distribution of y at breakdown obeys " 1=2 3=2 # N y − fmin 2N y − fmin exp − ; PN (y) = fmin 3 fmin being sharply concentrated around fmin , while for strong disorder one has RN (y) = 2N (2N=3W )exp(− 3W y), which is concentrated near zero [8,9]. Thus, the mean value of the elongation under which a fragment of length N breaks is hRi = 22 hyi=N 2 = 22 fmin =N 2 for weak and hRi = 3W2 =N 3 for strong disorder. The inversion of these
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Fig. 4. The relative excess stress at the crack’s tip, F(l), as a function of the crack’s length l for a scalar system with = 10. The dashed line indicates the limiting value F∞ .
relations provides us with the characteristic length Lc of fragments which are just about to break under R; we ÿnd Lc ˙ (R)−1=2 for weak and Lc ˙ (R)−1=3 for strong disorder. These 1d ÿndings reproduce thus the situation encountered by us in 2d. We now turn to highlighting the transition from crack propagation to coalescence of defects in 2d in terms of a two-dimensional scalar model. In this case we consider a large (inÿnite) portion of a square lattice, where the displacements (potentials) are given by X (xj − xi ) + −2 (Ri − xi ) = 0 (4) j
(the sum runs over the nearest neighbors of i, which are connected with i via intact bonds). Now we introduce a crack of length l along the horizontal x-axis, by cutting the l vertical bonds connecting the sites (j; 0) with (j; 1) for −l6j60. We calculate the distribution of stresses (currents) at the corresponding (j; 0)-(j; 1) bonds for j ¿ 0, using the same relaxation approach as in the vectorial case discussed above. Here we ÿnd, parallel to the situation in the bulk [16], that the stress attains its maximal value ftip at the tip of the crack (i.e. for j = 1); ftip is considerably higher than the stress f0 far within an intact area. This stress enhancement depends on the crack length and can be characterized by the function F(l) = (ftip − f0 )=f0 . Now, there are di erences in the situation for bulk vs. for layer fragmentation. √ In the two-dimensional scalar bulk model of Ref. [16] one ÿnds that F(l) ≈ A l and that F(l) grows with l in an unbounded manner. Contrary to this behavior, we ÿnd that the maximal stress at the tip of a crack in a surface layer does not increase indeÿnitely, but that F(l) attains a ÿnite limiting value F∞ (), which depends on . This behavior is shown in Fig. 4 for = 10. In Fig. 5 we show the local stress distribution in the bonds of an intact “bridge” between two semi-inÿnite cracks, placed symmetrically along the x-axis to the left of −l, and to the right of l. From Fig. 5 one infers readily that the stress
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Fig. 5. The stress distribution in a bridge between two linear cracks in the scalar model of Fig. 4. Note that the stress at the tips is considerably enhanced only when the distance between the cracks gets to be smaller than .
within the “bridge” gets to be considerably enhanced only when the distance 2l is of the order of ; this shows that the interaction between the cracks is local and that no load sharing takes place. These ÿndings allow us to discuss the transition from crack propagation to defect coalescence. In physically interesting situations the number of broken bonds is small compared to the total number of bonds. Thus for weak disorder (fmin ¿ 0) a new crack nucleates if the local stress in the interior of a layer fragment exceeds fmin . Then, due to the stress enhancement discussed above, this crack may propagate in brittle fashion through the layer until it reaches the boundary of the fragment. On the other hand, in 2d whenever W=fmin ¿ 1 + F∞ () even a very long crack will stop eventually by the encounter of strong bonds; then the formation of new defects by breaking weaker bonds will compete successfully with crack propagation. The last condition is of current occurrence for strong disorder, and it may be even weakened when the size of the remaining fragments in the sample drops below . All this emphasizes the fundamental role played by the strength of the disorder on pattern formation in fragmentation. We conclude by summarizing our ÿndings. We studied a model for the fragmentation of surface layers under quasistatical, slowly increasing strain. We analyzed the pattern of cracks and the dependence of the fragment sizes on the strain. We showed that the mode of fragmentation depends on the disorder’s strength, deÿned to be = W=fmin . Thus, for weak disorder the system breaks through brittle crack propagation, whereas for strong disorder point defects are created independently of each other, and then coalesce to form cracks. The characteristic size Lc of the broken fragment decreases with the elongation R of the substrate as Lc ∼ (R)−1=2 for weak and Lc ∼ (R)−1=3 for strong disorder. We explained these ÿndings in the framework of scalar models both in 1d and also in 2d, and showed how the di erent fragmentation modes emerge from the interplay between the disorder and the geometric features of the problem.
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Acknowledgements We are thankful to Ulrich Handge and Dr. Eric ClÃement for discussions. The support of the German-Israeli Foundation (GIF), of the DFG through SFB428 and of the Fonds der Chemischen Industrie is gratefully acknowledged. References [1] J. Walker, Sci. Am. 255 (1986) 178. [2] Y. Leterrier, Y. Wyser, J.-A.E. Ma nson, J. Hilborn, J. Adhesion 44 (1994) 213. [3] T. Walmann, A. Malthe-SHrenssen, J. Feder, T. JHssang, P. Meakin, H.H. Hardy, Phys. Rev. Lett. 77 (1996) 5393. [4] P. Meakin, in: H.J. Herrmann, S. Roux (Eds.), Statistical Models for the Fracture of Disordered Media, North-Holland, Amsterdam, 1990. [5] P. Meakin, Science 252 (1991) 226. [6] P. Meakin, A.T. Skjeltorp, Adv. Phys. 42 (1993) 1. [7] H. Colina, L. de Arcangelis, S. Roux, Phys. Rev. 48 (1993) 3666. [8] O. Morgenstern, I.M. Sokolov, A. Blumen, J. Phys. A 26 (1993) 4521. [9] O. Morgenstern, I.M. Sokolov, A. Blumen, Europhys. Lett. 22 (1993) 487. [10] A. Groisman, E. Kaplan, Europhys. Lett. 25 (1994) 415. [11] I.M. Sokolov, O. Morgenstern, A. Blumen, Macromol. Symp. 81 (1994) 235. [12] T. Hornig, I.M. Sokolov, A. Blumen, Phys. Rev. E 54 (1996) 4293. [13] K.M. Crosby, R.M. Bradley, Phys. Rev. E 55 (1997) 6084. [14] J.V. Andersen, D. Sornette, K.-t. Leung, Phys. Rev. Lett. 78 (1997) 2140. [15] U. Handge, I.M. Sokolov, A. Blumen, Europhys. Lett. 40 (1997) 275. [16] P.M. Duxbury, P.L. Leath, P.D. Beal, Phys. Rev. B 36 (1987) 367.
Scaling and patterns in surface fragmentation processes I.M. Sokolov ∗ , A. Blumen Theoretische Polymerphysik, Universitat Freiburg, Hermann-Herder-Str.3, D-79104 Freiburg i.Br., Germany
Abstract We model the failure of inhomogeneous surface layers under expansion. The quenched disorder is accounted for by a probability distribution (PD) of local breakdown strengths. The ensuing crack patterns range from percolation-like defect aggregates to well-deÿned polygons. Since no load sharing takes place in surface fragmentation, the e ects governing crack propagation are local: the defects interact only within some correlation length. Fragmentation can be viewed as anisotropic, correlated percolation, where the formation of a new defect depends on the defects already present. Both the overall picture and the scaling of the fragment sizes depend strongly c 1999 Elsevier Science B.V. All rights reserved. on the PD. PACS: 62.20.Mk; 46.30.Nz; 05.40.+j Keywords: Scaling; Surface fragmentation; Load sharing
1. Introduction The statistical description of failure phenomena in disordered, complex systems has drawn recently much attention [1–15]. Apart from the technologically motivated interest in the mechanics of coatings, failure phenomena are part of the large framework of irreversible pattern formation. The mechanical description of fracture often starts from a mesoscopic picture based on ÿnite elements; their mechanical properties vary through the sample, but are, once assigned, ÿxed (quenched) for a given realization. As is well known, the failure patterns are often sequentially created and they depend strongly on the underlaying geometry and on the particular features of the disorder. An important aspect for us here is that the fragmentation of surface layers di ers much from corresponding processes in the bulk; furthermore even the physical questions ∗
Corresponding author. Fax: +49-761-203-5906.
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 6 0 6 - 2
300
I.M. Sokolov, A. Blumen / Physica A 266 (1999) 299–306
Fig. 1. The model used in the simulations: (a) view of the surface layer; (b) view from the side (vertical cross section).
di er. Thus the interest in the fracture mechanics of bulk materials focusses mainly on failure, i.e. the analysis often ends after the ÿrst crack propagates through the whole sample. In the case of surface layers one is much more interested in fragmentation, i.e. in the collective behavior of a large assembly of cracks. When a surface layer breaks under the deformation of its substrate there appear crack forms which may range (depending on the parameters) from percolation-like, loose aggregates to well-deÿned polygonal parquet patterns [1,10,12]. The particular forms which are created and their sizes depend on the type of the underlying disorder. The peculiarities encountered in the fragmentation of thin coatings are connected with the load redistribution in such surface layers. In most bulk systems load redistribution takes place through load sharing: the overall force acting on any cross section of the system stays constant and is equal to the applied load. In surface fragmentation no such load sharing takes place (the bulk carries most of the load), so that the factors governing crack propagation are local: the defects feel each other’s presence only within a characteristic correlation length, which depends on the elastic properties of the coating. Fragmentation in layers occurs as a kind of anisotropic, correlated percolation, where the probability for the appearance of a new defect depends on whether the defect is formed within an intact area, or near a crack’s tip, or near the internal part of a crack. In what follows we consider a simple, discrete model; we envisage the fragmentation of an inhomogeneous coating covering a substrate under slow expansion. We model the coating through an array of springs and account for its quenched statistical inhomogeneities by assigning to each spring a breakdown threshold taken from a given probability distribution (PD). The adhesion to the bulk is modelled through other springs, which connect the coating to the substrate. We start thus from the model put forward in Ref. [12]. There the coating is viewed as being an array of springs with elastic constants d, forming a triangular lattice, see Fig. 1. The lattice constant (the sidelength of each triangle) is taken to be unity. The ÿlm is attached to the substrate elastically, so that the surface layer can move relative to the bulk; this motional freedom is accounted for by connecting the nodes of the coating and the corresponding sites of the substrate through leaf springs of elastic constant D, see Fig. 1. The surface layer is brittle, so that each spring can break under stress. The value at which a
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particular spring breaks is random, but ÿxed at the start of the fragmentation process (i.e. the disorder is quenched). The PD of breakdown thresholds is a material’s property and is known from the start. As examples, we focus here on breakdown thresholds which are homogeneously distributed in the interval [fmin ; fmin + W ]; for these the PD is p( fb ) = 1=W p( fb ) = 0
for fmin ¡ fb ¡ fmin + W ;
otherwise :
(1)
As we show, the strength of the disorder is characterized by the parameter = W=fmin . Systems with W=fmin 1 can be viewed as being highly ordered. On the other hand, systems with fmin = 0 contain elements with breakdown thresholds in the vicinity of zero and belong to the case of strong disorder [7–9,12]. At the beginning of the process the surface layer is in equilibrium, and no forces act on it. Let us concentrate on the situation of a quasistatically growing stress. Quasistatically (in equilibrium) the overall force f(ri ) acting on the node vanishes at all times. As a projection on the (x; y)-plane parallel to the substrate on has X di; j (|rj − ri | − r0 )eji + D(Ri − ri ) = 0 : (2) f(ri ) = j
Here the ÿrst sum runs over the nearest neighbors, rj is the position of the jth node, Rj that of the corresponding substrate site, r0 is the equilibrium length of a spring in the absence of stress, and eji =(rj −ri )=|rj −ri | is a unit vector in the spring’s direction. The elastic constants di; j are di; j =d for intact and di; j =0 for broken springs. Note that the elastic behavior of the model is fully determined by the ratio d=D, characterizing the relative p strength of the coupling of the ÿlm to the substrate. We refer to the value of = d=D as being a correlation length: shows at what distance from a failure site (measured in lattice units) the displacements of the neighboring springs still di er considerably from these in a bulk without defects (vide infra). The expansion of the substrate under external forces is modelled by gradual changes in the coordinates Rj = (Xj ; Yj ) of the substrate’s sites: the corresponding coordinates grow with the elapsing time t (here: procedure step) as Xj = (1 + at)Xj; 0 and Yj = (1 + at)Yj; 0 (we consider here isotropic and homogeneous changes). At each step Eq. (2) is solved by using a relaxation procedure, i.e. by solving the related time-dependent equations r˙i = −f(ri ) with an appropriate relaxation constant ¿ 0 [12]. After determining the new positions of the nodes, we check whether these lead to the breaking of some springs. This happens whenever the elastic force f acting on a spring, f = di; j (|rj − ri | − r0 ), attains its predetermined breakdown threshold fi;b j . When this occurs we let the spring break irreversibly, and set its elastic constant to zero. This is followed by additional relaxation steps, in which the new equilibrium positions are calculated; these steps may lead (for a brittle regime of fracture propagation) to the breaking of additional springs. If at a given strain no further springs break, the whole procedure is iterated by increasing the time from t to t + t. By this a new strain increase takes place.
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Fig. 2. Fragmentation patterns on a 150 × 150 lattice with = 20 after 4000 bonds have failed [12]; the PD is Eq. (1), with fmin = 100. The value of W is 15 in (a) and 400 in (b).
In what follows we focus mostly on the emerging 2d crack patterns. As main result we ÿnd that these depend crucially on = W=fmin , the relative strength of the disorder. As an example, we present in Fig. 2 (using the data of [12]) two typical situations, obtained on a 150 × 150 lattice with = 20 after 4000 bonds have failed. Fig. 2a displays for small relative disorder ( = 0:15) a parquet-type crack pattern, whereas Fig. 2b shows for = 4 a loose assembly of defect sites. Furthermore, when we follow how individual cracks develop, we observe that the formation of such patterns also depends on : Thus for small the growth of well-deÿned cracks is the main process leading to fragmentation, while for strongly disordered systems the defects form almost independently of each other, and the visible crack patterns arise from the coalescence of such defects. In both cases the rate-limiting process for fragmentation is the formation of new defects, rather than crack propagation. For weak disorder defect formation (starting form a seed “microcrack”) is followed almost immediately by fast crack propagation and by the mechanical relaxation of the ÿlm near the “banks” of the newly formed crack (this mirrors brittle fracture). This may also be inferred from Fig. 2a, where only very few cracks with free ends are to be found. For strong disorder the coalescence of defects (which leads to the pattern of Fig. 2b), is a spectacular but not particularly decisive feature: most of the defects belong to small clusters, not forming any medium-scaled connected structure. To quantify the characteristic length in such patterns we introduce as in [12] the distance L between two neighboring defects, measured along randomly drawn straight lines (the procedure is then averaged over di erent realizations of such lines). The averaged characteristic length hLi shows a power-law dependence on the elongation R, i.e. hLi ˙ (R)− , where the exponent is 12 for weak disorder and is around 13 in the strongly disordered case: see Fig. 3 where the data for a 200 × 200 system with = 20 are presented [12].
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Fig. 3. The mean fragment size hLi as a function of R for fmin = 100 and W = 10 (upper curve) and W = 100 (lower curve) [12]. Note the double-logarithmic scales. The dashed line has the slope − 12 , the dotted line the slope − 13 .
In order to understand these ÿndings for elastic 2d systems we turn to simpler models, where the displacements x are considered to be scalar and the problem is thus mapped on its electrical analogon. Let us consider ÿrst the one-dimensional system, which is amenable to an analytical treatment [8,9]. In a scalar, one-dimensional case, the condition for equilibrium reads xi−1 − 2xi + xi+1 + −2 (Ri − xi ) = 0 ;
(3)
where xi are the scalar displacements (in electrical language: potentials at the nodes) and is the correlation length introduced above. In this case it is possible to calculate the stress distribution within a fragment (it turns out to be parabolical for fragments of length of the order of or less [8,9]) and to derive explicit expressions both for the distribution of failure positions within a given fragment, and also for the distribution of stresses under which fragments of a given length N break [9]. Thus the relative width w of the distribution of crack positions is given by w = [3=(16N )]1=3 . For 1 the fragments break preferentially in the middle, giving rise to a hierarchical pattern; on the other hand for ∼ =1 the fragments can break with ÿnite probability at every site. Also the distribution of elongations R at breakdown is governed by : introducing y = (N 2 =22 )R, Ref. [9] ÿnds that for y ¿ fmin the distribution of y at breakdown obeys " 1=2 3=2 # N y − fmin 2N y − fmin exp − ; PN (y) = fmin 3 fmin being sharply concentrated around fmin , while for strong disorder one has RN (y) = 2N (2N=3W )exp(− 3W y), which is concentrated near zero [8,9]. Thus, the mean value of the elongation under which a fragment of length N breaks is hRi = 22 hyi=N 2 = 22 fmin =N 2 for weak and hRi = 3W2 =N 3 for strong disorder. The inversion of these
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Fig. 4. The relative excess stress at the crack’s tip, F(l), as a function of the crack’s length l for a scalar system with = 10. The dashed line indicates the limiting value F∞ .
relations provides us with the characteristic length Lc of fragments which are just about to break under R; we ÿnd Lc ˙ (R)−1=2 for weak and Lc ˙ (R)−1=3 for strong disorder. These 1d ÿndings reproduce thus the situation encountered by us in 2d. We now turn to highlighting the transition from crack propagation to coalescence of defects in 2d in terms of a two-dimensional scalar model. In this case we consider a large (inÿnite) portion of a square lattice, where the displacements (potentials) are given by X (xj − xi ) + −2 (Ri − xi ) = 0 (4) j
(the sum runs over the nearest neighbors of i, which are connected with i via intact bonds). Now we introduce a crack of length l along the horizontal x-axis, by cutting the l vertical bonds connecting the sites (j; 0) with (j; 1) for −l6j60. We calculate the distribution of stresses (currents) at the corresponding (j; 0)-(j; 1) bonds for j ¿ 0, using the same relaxation approach as in the vectorial case discussed above. Here we ÿnd, parallel to the situation in the bulk [16], that the stress attains its maximal value ftip at the tip of the crack (i.e. for j = 1); ftip is considerably higher than the stress f0 far within an intact area. This stress enhancement depends on the crack length and can be characterized by the function F(l) = (ftip − f0 )=f0 . Now, there are di erences in the situation for bulk vs. for layer fragmentation. √ In the two-dimensional scalar bulk model of Ref. [16] one ÿnds that F(l) ≈ A l and that F(l) grows with l in an unbounded manner. Contrary to this behavior, we ÿnd that the maximal stress at the tip of a crack in a surface layer does not increase indeÿnitely, but that F(l) attains a ÿnite limiting value F∞ (), which depends on . This behavior is shown in Fig. 4 for = 10. In Fig. 5 we show the local stress distribution in the bonds of an intact “bridge” between two semi-inÿnite cracks, placed symmetrically along the x-axis to the left of −l, and to the right of l. From Fig. 5 one infers readily that the stress
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Fig. 5. The stress distribution in a bridge between two linear cracks in the scalar model of Fig. 4. Note that the stress at the tips is considerably enhanced only when the distance between the cracks gets to be smaller than .
within the “bridge” gets to be considerably enhanced only when the distance 2l is of the order of ; this shows that the interaction between the cracks is local and that no load sharing takes place. These ÿndings allow us to discuss the transition from crack propagation to defect coalescence. In physically interesting situations the number of broken bonds is small compared to the total number of bonds. Thus for weak disorder (fmin ¿ 0) a new crack nucleates if the local stress in the interior of a layer fragment exceeds fmin . Then, due to the stress enhancement discussed above, this crack may propagate in brittle fashion through the layer until it reaches the boundary of the fragment. On the other hand, in 2d whenever W=fmin ¿ 1 + F∞ () even a very long crack will stop eventually by the encounter of strong bonds; then the formation of new defects by breaking weaker bonds will compete successfully with crack propagation. The last condition is of current occurrence for strong disorder, and it may be even weakened when the size of the remaining fragments in the sample drops below . All this emphasizes the fundamental role played by the strength of the disorder on pattern formation in fragmentation. We conclude by summarizing our ÿndings. We studied a model for the fragmentation of surface layers under quasistatical, slowly increasing strain. We analyzed the pattern of cracks and the dependence of the fragment sizes on the strain. We showed that the mode of fragmentation depends on the disorder’s strength, deÿned to be = W=fmin . Thus, for weak disorder the system breaks through brittle crack propagation, whereas for strong disorder point defects are created independently of each other, and then coalesce to form cracks. The characteristic size Lc of the broken fragment decreases with the elongation R of the substrate as Lc ∼ (R)−1=2 for weak and Lc ∼ (R)−1=3 for strong disorder. We explained these ÿndings in the framework of scalar models both in 1d and also in 2d, and showed how the di erent fragmentation modes emerge from the interplay between the disorder and the geometric features of the problem.
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