Critical Transport and Failure in Continuum Crack Percolation Didier Somette

Critical transport and failure in continuum crack percolation Didier Somette

To cite this version:

Didier Somette. Critical transport and failure in continuum crack percolation. Journal de Physique, 1988, 49 (8), pp.1365-1377. 10.1051/jphys:019880049080136500. jpa-00210817

HAL Id: jpa-00210817 https://hal.archives-ouvertes.fr/jpa-00210817 Submitted on 1 Jan 1988

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Classification Physics Abstracts

05.50 - 05.60 - 05.70J - 62.20M

Critical transport and failure in continuum crack percolation

Didier Somette

Laboratoire de Physique de la Matière Condensée, CNRS UA190, Faculté des Sciences, Parc Valrose, 06034 Nice Cedex, France

(Reçu le 29 octobre 1987, révisé le ler avril 1988, accepté le 14 avril 1988)

Résumé. 2014 On étudie les propriétés de transport et de rupture d’un nouveau modèle de percolation continue (fromage bleu) dans lequel le milieu qui supporte le transport est l’espace situé entre des fissures placées aléatoirement. Le comportement critique de la conductivité électrique, de la perméabilité, des constantes élastiques et des seuils de rupture sont, pour la plupart d’entre eux, différents de ceux obtenus en percolation sur réseau discret et avec le modèle de percolation continue du type gruyère. De plus, on suggère que l’assymétrie de la réponse élastique d’une fissure sous contrainte de compression ou d’extension conduit, pour les propriétés élastiques macroscopiques, à un nouveau seuil de percolation pour une concentration de fissures plus grande que celle donnant la percolation géométrique usuelle. Cela est dû à la présence de configurations en crochets ou déferlements dans le squelette de percolation qui transforme localement la contrainte d’extension macroscopique en une contrainte de compression. En dimension deux, on propose une analogie avec la percolation dirigée. Quand les fissures ont une largeur finie, on retrouve le seuil de percolation géométrique habituel et l’analyse des propriétés de transport est analogue à celle développée par Halperin et al. pour le modèle de gruyère. Pour une plaque mince élastique non contrainte dans un plan, les deux lèvres d’une fissure peuvent localement se recouvrir : cela conduit à l’existence de deux modes de déformation supplémentaires (ouverture et flambage) qui donnent des comportements critiques distincts. On analyse le scénario de rupture fragile de type Griffith dans lequel les fissures s’agrandissent sous l’action de contraintes d’extension. Un nouveau comportement critique est annoncé.

Abstract. 2014 Critical transport and failure properties of a new class of continuum percolation systems (blue cheese model), where the transport medium is the space between randomly placed clefts, are discussed. The critical behaviour of electrical conductivity, fluid permeability, elastic constants and failure thresholds are, for most of them, distinct from their counterparts in both the discrete-lattice and Swiss-cheese continuum percolation models. Furthermore, it is argued that the asymmetric elastic response of a crack submitted to compression or extension leads, for the macroscopic mechanical properties, to a new percolation threshold at a crack concentration higher than that of usual geometric percolation. This is due to the presence of hooks or overhang crack configurations which locally transform the macroscopic applied extensional stress into a compressional stress. In two dimensions, an analogy with directed percolation is suggested. When the cracks have a non-vanishing width, one recovers the usual geometric percolation threshold and the analysis of the transport properties is similar to that of the Swiss cheese model developed by Halperin et al. For elastic sheets which are not constrained to lie in a flat plane, the two edges of a crack may overlap : this leads to the existence of two additional elastic deformation modes (crack opening and buckling) which exhibit distinct critical behaviours. The brittle Griffith failure scenario which corresponds to the growth of a crack is analysed and introduces new critical failure exponents.

1. Introduction. micro-bond strengths in the continuous percolation system. Recently, it was realized that the critical behaviour The purpose of this paper is to introduce a third of electrical conductivity, fluid permeability, elastici- class, the continuous blue-cheese model where the ty modulus [1] and failure threshold [2] was different transport medium is the space between randomly in discrete-lattice percolation and in a class of placed clefts with small or even vanishing thickness continuous percolation models (Swiss-cheese), point- (and not spherical holes as in the Swiss-cheese ing out the existence of at least two universality model). The critical behaviour of the transport classes of transport phenomena in percolation. This quantities and failure thresholds are found, for most is due to the existence of fluctuations [3] in the of them, to be distinct from their counterparts in

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049080136500 1366 both the discrete-lattice and Swiss-cheese continuum of cracks per unit surface (in 2d) or per unit volume percolation models (see Tab. I which summarizes (in 3d) increases, a percolation threshold N c is the results). This stems both from the existence of reached corresponding to the geometrical deconnec- micro-bond fluctuations (as occurs in the Swiss- tion of the sample in multiple fragments. In three cheese model) and from the peculiar topology of the dimensions, the cracks form a percolating structure singly-connected bonds of the blue cheese model at a first threshold N1 strictly smaller than Nc, before (see below). deconnecting the sample in pieces. We are only The relevance of the blue-cheese model is interested in the regime N > N 1. suggested from the crack structure of many mechan- Percolation involving anisotropic objects has been ical [4] (damaged crystals, solids, ceramics, rocks...) studied [7-10] numerically and theoretically and I and natural sytems [5] (arrays of faults in geology in will make use of the known geometrical facts for the relation to oil recovery, Geothermics and ear- analysis of transport and failure properties. Note thquakes [6]...) which often consists in random however that, in the blue cheese model, the transport arrays of micro-cracks of vanishing thicknesses. medium is the space left between the randomly Percolation has qualified as a powerful model of placed empty clefts whereas for previous studies of disordered media for deriving simple and powerful transport properties in overlapping figure stick (or concepts : in a first step, it is natural to simplify the rectangle) percolation models [7], transport is carried difficult problem of the analysis of the transport on the percolating structure of sticks or disks them- properties of a medium deteriorated by cracks in the selves. This is the main difference between this large disorder limit by studying its corresponding model and previous studies. It has important con- percolation version. This is my justification for sequences for the critical electrical and mechanical introducing the blue cheese model. As in other properties which are completely different. percolation models, its simplicity permits to explore One important geometrical result of previous geometrical, transport and rupture properties which studies is that N c is known to decrease as the crack appear either universal or dependent upon specific length I c increases according to Nc - 1 c d [7, 10], details of the systems (see below). resulting in the existence of a quasi-invariant The blue-cheese model is defined as follows. Nc 19 = 5.7 [7-10] independent of lc. The intuitive Empty rectangle holes of length I e in two dimensions idea is that each crack defines a sort of excluded or disk-like holes of diameter Ie in three dimensions volume of the order of l d around it : then the critical with thickness a are randomly distributed in a percolation threshold Nc corresponds to the presence uniform electric or elastic medium. As the number N of a constant average number of cracks in each

Table I. - Comparison between Swiss-cheese and blue-cheese continuum transport and failure percolation exponents refered to the corresponding exponents on the discrete-lattice, a is the width of the cracks in the blue cheese model. a =1= 0 => case of cracks with a finite width, a = 0 => case of cracks with vanishing thickness, a * = 0 a case of cracks with vanishing thickness but whose leaves can overlap in two dimensions, 1 = => buckling deformation mode, G => Griffith mechanism of rupture. The exponents and 2 refer respectively to the energy failure criterion (first criterion of [2b]) and to the voltage or strain failure criterion (second criterion in [2b]). 1367 typical volume lf. This result allows the definition of set of such micro-bonds connecting the upper side to singly-connected micro-bonds to be justified and the the lower side of the page is shown in heavy lines in electrical and mechanical properties of each micro- figure 3a. Each narrow neck i.e. equivalent micro- bond to be treated separately (see below). bond such as that in figure 2 is schematized by the The geometrical critical exponents such as the sign « = » : this allows us to draw the equivalent correlation length exponent v are believed to be the discrete lattice and identify the corresponding con- same as for isotropic objects and discrete systems. nectivity shown in figure 3b. Previous studies of transport properties have been essentially concerned with percolation systems where the transport medium is the connected set of anisotropic objects. For example, in reference [11] the conductivity and elasticity of percolating carbon fibers are studied. In this case, transport is controlled by the percolation structure of the carbon fiber set. Note that in contrast to these studies, I focus here on the connected set complementary to the crack clus- ters. A typical sample structure just below N c is given in figure 1 for d = 2. One observes large undeteriorated regions (of size of the order of lc) separated by narrow necks which have the generic topology

2. the construction shown in - shown in figure Using Fig. 3. A small portion of the system shown in figure 1 : figure 2, one can define a mapping of the continuous the heavy line serves as a guide to the eye and shows the crack model onto a discrete percolation model in a connecting paths. The corresponding schematic represen- way similar to the mapping of the Swiss cheese tation of the discrete lattice structure equivalent to the model to a discrete model [13]. In d = 2, with each continuum crack percolation structure is shown in b : the two-crack configuration, one can associate a straight symbol = indicates the presence of weak micro-bonds with a in 2. path as drawn in figure 2. Each such path defines a topology represented figure micro-bond in a corresponding percolation lattice model. These necks may sometimes by separated by The of this in plaquettes having a polygonal shape of size lc. A generalization mapping three dimensions involves planes and is straightforward. Figure 4 shows the corresponding generic neck which involves a three-crack configuration. As in the Swiss cheese model, a micro-bond is determined from the relative position of d neighbouring objects where d is the space dimension (empty holes in the Swiss cheese and empty cracks in the blue cheese) which almost completely intersect each other. Therefore, two linear cracks in d = 2 suffice to define a micro-bond as shown in figure 2 whereas three intersecting cracks must be used to construct a generic micro- bond in d = 3. Generically, two cracks (1) and (2) intersect with a finite angle. Let us note P12 the line formed by the intersection of (1) and (2). A third Fig. 1. - A typical random crack structure before crack (3) with a random orientation will intersect geometric disconnection in two dimensions obtained by a both (1) and (2) along lines called P13 and P23 which computer with the algorithm described in the text (courtesy of C. Vanneste).

Fig. 2. - Illustration of a generalized Voronoi tesselation for a micro-bond configuration in the blue cheese model: the dashed line schematizes the equivalent discrete-lattice Fig. 4. - Typical topology of weak micro-bonds in three micro-bond. dimensions. The width a of the cracks has been taken zero. 1368

form finite non-vanishing angles with respect to the 2. Electrical transport and failure. first line P 12. This result justifies that the generic configuration is of the form depicted in figure 4. 2.1 ELECTRICAL CONDUCTIVITY. - The typical structure of a weak bond in two dimensions is shown The continuous to discrete mapping allows us to conclude that the blue cheese model has the typical in figure 2. We are interested in the conductance of Nodes-Links-Blobs geometrical structure, well each singly-connected bond which has the shape of a in 2d or the of a brick in known in discrete percolation models [12, 13] : in a plaquette shape polygonal 3d with a size A micro-bond system of size 3 > §, it is a hypercube of size typical 1c. generic jd, with nodes separated typically by the percolation configuration is characterized by a finite angle correlation length § - (N e - N )- v, and the nodes between two neighbouring cracks. Since we are interested in the with it has the same are connected by macro-links (there are of course scaling ð, multi-connected blobs decorating the macro-links). conductance dependence as that of a configuration of two cracks to each other with the Each macro-link can be viewed as consisting in perpendicular same which itself is to the of several sequences of singly connected micro-bonds 6, equivalent problem of total number L1 = (Nc - N )- d with d = 1 [14], in an electric current flowing perpendicular to two collinear cracks separated by a distance 5 in a series with blobs corresponding to multiple paths plate of typical size lc (see Fig. 5a). If Io is the total current connecting the same nodes as depicted in figure 3. Such a continuous to discrete mapping has been used for discussing the transport and failure properties of the continuous Swiss cheese model [1, 2] and is also important in the blue cheese model in order to use the known critical geometrical properties of discrete- lattice percolation. The difference between the transport and failure 5. - for the electrical properties of the discrete-lattice and the blue cheese Fig. Equivalent configuration conductivity and the bond bending elasticity in d = 2 (a ) models now relies, as in the Swiss-cheese percolation = and in d 3 (b ). The cracks are represented by the two model, on the form of the continuous distribution lines separated by 8 in a) and by the plane with a hole of p ( ð) of micro-bond strengths (controlled by the diameter = 8. bottleneck width 5 shown in Figs. 2 and 4). The essential ingredient of the analysis is to recognize that the continuous probability distribution p ( d ) flowing through the hole, j (r) = Io/,7rr is the current approaches a non-zero limit p (0) for 6 - 0+ . This density at large distances r > 5 to the hole. From result can be intuitively visualized as follows. Con- E = u - 1 j, one gets the conductance of the hole sider a plausible algorithm for constructing a blue cheese structure in two dimensions (the three dimensional case is similar) : one first chooses at random the positions of the crack centers with a given density N. Then, the orientation of each crack must which vanishes only logarithmically as I e/ S -+ + oo , be chosen randomly in the interval [0, 7T]. Since the i.e. when the bond thickness 5 goes to zero. This crack centers and their orientation are distributed result has already been reported in [15]. To estimate the we in uniformly, the value 6 = 0 is not a special point and macroscopic conductivity, ignore (as [1]) p (8 ) is finite. This implies that, in the random blue- the resistance of the blobs and approximate the cheese percolation model, very small dss can be conductance of a string of Ll singly connected bonds encountered. The typical minimum value ðmin of by 5 along a string of L1 singly connected bonds is found by using a Griffith type of argument [1] valid x for intermittency in random fields [3] : d6 f0 0 p(,6) This yields a lower bound for G-1, therefore an for G and a lower bound for the is the probability that 5 « and the condition upperbound c’ exponent t (see below). This formula relies on the that less than one event Ll p (8 ) d8 ,1 says structure of the micro-bonds a 0 string-like L, along La macro-link which allows the rules for conductance 6 a has occurred over L1 trials. This allows us to series association to be used. the fact that the identify Using distribution p ( 8 ) discussed in paragraph 1 does not vanish as ð -+> 0, one finds that the sum (3) is not Due to the presence of these very narrow bonds, dominated by the weakest bonds since a micro-bond transport and failure properties depart from their resistance varies less than linearly (in fact only discrete-lattice analog as we now discuss. logarithmically) with ð. Therefore, the macroscopic 1369 conductivity is related to the conductance of a are related typically to the second moment of the macro-link of length 6 by the classical transport distributions. This leads to failure exponents which relationship L = g 2 - d G. This yields are larger in the Swiss cheese continuum percolation model than in the discrete-lattice model [2]. Now, we briefly discuss this effect on electrical failure in the context of the blue cheese model. which is the usual lower bound estimate of the 2.2.1 Weakest bond failure threshold. - Consider a conductivity critical exponent whose values is system of size 3 > § . If a current is injected from one t = 1.3 [1]. This shows that the blue cheese model is side of the hypercube jd, the number of current equivalent in 2d to the discrete lattice model or to carrying paths is (j / 03BE)d - 1. However, in each path, the Swiss cheese model as far as the conductivity there are (j /§ ) macro-links and each macro-link has exponent is concerned. L1 singly connected micro-bonds. Therefore, the In 3d, the generic structure of a micro-bond is number of current carrying singly-connected micro- shown in figure 4 (and Fig. 5b for the equivalent bonds is of the order of R - L 1 (1 / 03BE )d - 1 hole problem giving the scaling in terms of 5). It is (j/ 03BE) ’" L1 (j/ 03BE)d (we neglect the micro-bonds seen that the conductance of such a micro- easily which belong to loops since they are less strained). bond is proportional to that of a volume of size Among these 3t singly-connected micro-bonds, lc almost disconnected in its middle apart from a the weakest neck will fail first. Its width 8min is the narrow neck of width 5 in order to from since, go minimum of all neck widths out of the total number 4 to 5b, one has to rotate the figure figure only in the whole system of size 3 and micro-bond around the line. The conductance of :R = Ll (j / 03BE)d P 12 scales as the micro-bond shown in figure 4 therefore reads : lc -1

E dr . E ~ from the g (8 ) ~ lo s Using lo/ar2 When the is submitted to a current conservation of the total electric current, one obtains system given density j (defined for the undeteriorated system i.e. j d - 1 j = Io is fixed corresponding to a total imposed current 10), the current flowing in each macro-link is The conductance of a micro-link to zero goes of order [15-17] linearly with the neck width d. The conductance of a macro-link of size ) is therefore obtained from equation (3) and is written and the Ohmic loss in it is written [2b]

where gmin is obtained from 8min(1)with equation (2) in d = 2 and equation (4) in d = 3. Assuming that one finds that the macroscopic Using L = ç 2 - d G, rupture occurs if 0 becomes larger than a critical conductivity is given by value Oc (first criterion of Ref. [2b]) leads to a current density threshold j r :

This is identical to the usual estimation of the discrete case up to a logarithmic correction. In 3d, t ::::: 1.9 [1]. Contrary to the Swiss cheese model in If rupture occurs following criterion ii) i.e. when the 3d, the fluctuations of the micro-bond strengths do voltage drop over two ends of a micro-bond reaches a critical value one obtains the same of not lead to a different conductivity critical exponent Vc, type but only to a logarithmic correction of the scaling. expression as equation (8) : This stems from the different scaling (4) which must be compared to its analog g ( 8 ) ::::: 8 3/2 for the Swiss cheese model [1]. for criterion ii) . This leads to a scaling form of the first bond rupture 2.2 ELECTRICAL FAILURE. - In rupture phenome- threshold : (Nc - with na, the weakest part of the system fails first and the jr(l)= N)fe effect of disorder is markedly more pronounced than fe= v{-d(d-2)/2+ (d-l)} + (d-2)/2 in transport phenomena [15-18]. The failure threshold is dominated by extreme fluctuations in for criterion i). It is important to realize that these contrast to transport and elastic coefficients which scaling laws are valid as long as the system size 1370

J remains much larger than the percolation length 2.2.2 Macroscopic failure threshold. - An upper- C. bound for the macroscopic failure can be obtained Will the first micro-bond failure lead to a macro- with the current density jr necessary for all the scopic failure ? In reference [18], it is argued that the macro-links to fail. Each macro-link will fail by its first microscopic failure threshold could in general weakest singly-connected micro-bond whose width be distinct from the macroscopic failure threshold Smin scales as L1 -1 (see Eq. (1)), which shows that the for random fuse networks away from the percolation scaling laws of paragraph 2.2.1 are modified by the critical point. In the vicinity of Nc, the same can hold absence of the factor (jl6 )d - 1in d. This leads to true depending on the cases as we now discuss. Using the nodes-links-blobs picture of the percolation structure, one can estimate the failure threshold j r (2) of the second weakest bond of the total system and compare it with jr(I). The following with Fe = 2 v + 1/2 showing a correction equal to argument is developed for the case of a constant 1/2 to the discrete case Fe = 2 v in d = 3. If rupture applied current. Once the first bond has failed, only occurs as the voltage drop over the bond becomes (J/ 03BE)d - 1 macro-links remain and carry the total larger than a critical value Vc (second criterion of current flow jjd- 1. Therefore, each macro-bond will Ref. [2b], one has j, ~- (Nc - N )V(Log I Nc - N I) carry a current slightly larger than that before the in d = 2 and jr ~ (Nc-N) E, for d = 3 with first a factor rupture by equal to {(12013 (61j)d)) - 1 Fe = 2 v + 1. These results are summarized in the (at the basis of this factor is the hypothesis that the table I for comparison with those obtained for the current increase due to the breaking of one macro- discrete-lattice and Swiss cheese models. link is shared homogeneously between all remaining macro-links : this is therefore a kind of mean field 2.2.3 Probability distribution of failure thresholds. argument which neglects possible important local In addition to leading to a different critical failure fluctuations). In competition with this increased exponent (as shown above in paragraphs 2.2.1 and current per macro-link, is the fact that the second 2.2.2), the existence of bond strength fluctuations weakest bond is stronger than the weakest one by a also yields a broad distribution of failure thresholds. factor 8 3 (2)/,B 1 (1) { (1- /j )d)) - 1 since the total This effect is discussed in [19] for the Swiss cheese number of current carrying singly-connected bonds model. A distribution of rupture thresholds stems has decreased by the amount - Ll. Now, let us from the existence of fluctuations 1) of the mesh size assume that the micro-bond conductance is related L of the macro network around the typical size to its width 6 by a powerlaw g ( d ) = d {3 . Then, one 6 and 2) of the individual strength 5 of the singly finds that connected micro-bonds along a macro-link, resulting in large current fluctuations. The form of the failure probability distribution for continuum blue cheese percolation model in the critical region can be derived using the method of reference [19] which rests on the knowledge of the probability distributions of the extremes of the fluctuating variables L If /3 2 for criterion i) and 13 - 1 for criterion ii), and 5 and on the expression of the failure current > which implies that the second weakest r (2) r (1) density jr for a bond of width 5 in a mesh of size L bond will in fail after the first general immediately (obtained from Eq. (2) in d = 2 and Eq. (4) in one. Repeating the argument leads to a macroscopic d = 3). failure as soon as the weakest bond fails. Equa- In two dimensions, j, has a logarithmic depen- tions (2) and (4) show that the blue cheese model dence with respect to d. The corresponding failure belongs to this class. Therefore, macroscopic failure distribution is of the form of the exponential of an in the blue cheese model is given by equations (8) exponential : and (9). In the other cases (13 > 2 for criterion i) and 13 > 1 for criterion ii)), j r (2) ,j r (1) and one observes a succession of increasing failure thresholds ending eventually as a macroscopic failure. Note that for the where c and c’ are two constants. other boundary condition of a given applied voltage, In three dimensions, since the failure threshold the results would be drastically changed since a link scales as a power of dmi., the failure distribution is failure involves the change of the global conductivity controlled by the largest fluctuation field 5 and takes and in general reduces the total current. This leads a Weibull form to a distinction between catastrophic failure at imposed current and controlled failure at imposed voltage. 1371

This is similar to the results already discussed for the important and should be considered in geological Swiss cheese model [19]. situations since the opening mode I is unlikely to occur on a large scale in the earth because of the 3. Critical permeability. large compressional stresses created by gravity. A simple method of stress analysis in elastic solids Permeability also exhibits critical behaviour. How- with many random cracks has recently been pro- ever, in this paper we study the permeability, not of posed based on a superposition technique and the the crack structures, as would seem natural for ideas of self-consistency applied on the average example in cracks occurring in geological situations, tractions on individual cracks [20]. This method is but of the complementary medium. Therefore one essentially a mean field theory and gives reasonable must envision that the cracks are barriers to the flow results far from the critical percolation point. On the of a fluid as in a labyrinth structure made of thin long contrary, this paper focuses on the very high disorder walls placed in the impenetrable randomly transport limit modeled by the percolation threshold. In this medium. This system can serve as a model of certain case, another type of is called for and is rock structures in which analysis loosely packed impermeable described below. rocks have very anisotropic shapes. An essential feature of crack percolation is the The fluid permeability k is defined as the flux-rate existence of an asymmetry of the response of individ- of fluid flow through the space between the random ual cracks submitted to an externally applied stress. barriers under a unit macroscopic pressure gradient. If the stress is tensile, the crack cavity opens up and The permeability of a micro-bond of diameter takes a finite volume leading to a significant soften- 5 is to the proportional corresponding conductivity ing of the elastic behaviour of the system. If the g (6) multiplied by its cross-section S = S d - 1. There- stress is compressive, the two edges of the cracks are fore, in 2d, k (6 ) = d /Log (lc/ S ). The macroscopic pressed close together and the system responds as in permeability is found to vanish as N - according Nc the absence of cracks. This can be summarized by a to very asymmetric (stress-strain) characteristic as depicted in figure 6. leading to

where e = t -- 1.3 is equal to the permeability exponent of the discrete-lattice case. In 2d, the blue cheese permeability exhibits only logarithmic correc- tions with respect to the discrete case. In 3d, k( d) ~ 8 3. This yields k - 1 (N c - N )g Fig. 6. - Schematic form of the stress-strain characteristic of a single crack in an homogeneous medium which with e = e + 2 giving a correction to the discrete outlines the large asymmetry of its response with respect case equal to 2 (see Tab. I for comparison with the to compression or tension. Swiss cheese model).

4. Mechanical properties. This feature leads to new behaviours in the of deteriorated 4.1 ELASTIC PROPERTIES. macroscopic elasticity system by cracks. Under compression, the system responds as 4.1.1 Directed percolation behaviour. - The mech- if undeteriorated and no critical scaling behaviour is anical properties of system deteriorated by cracks expected. Under a macroscopic tensile stress, one are strikingly different from those of systems could think that the elastic constant should vanish at weakened by empty holes as in the Swiss cheese the geomerical percolation threshold Nc. However, model [1]. this is not true [21], since not all the cracks will be First, three distinct deformation modes may be submitted to a local tensile stress but some of them defined [27]. Mode I corresponds to an elongation, will feel locally a compressional stress. An example mode II to a shear and mode III to a flexure and of a configuration where this effect occurs is pre- twisting in beams. The analogy is loose since a crack sented in figure 7. This leads to the curious fact that submitted to a parallel shear will exhibit both even at the geometrical percolation threshold where deformation mode I at its tips and mode II on its a connected path of cracks cuts the system in several sides. In the following, we will only consider defor- pieces, the system still resists against a macroscopic mation mode I and leave the study of the role of tensile stress. This is due to the existence of hook- modes II and III for future work. These modes are like or overhang configurations schematized in fig- 1372

and choose Ox from left to right. Then draw on each crack an arrow having a projection on Ox in the sense of the chosen direction. Now, starting from the left, try to find a path going from left to right without overhangs or hooks. This problem is in close analogy with a continuous version of the directed percolation model. In this model, one starts from ordinary bond percolation on a discrete lattice and introduces a Fig. 7. - Typical hook or overhang configuration in a direction to the problem by drawing arrows on the percolating path. The part of the crack presenting a bonds which follow the main direction. The question compressional stress is drawn in heavy line. is then to find a connecting path which follows the arrows [23]. Therefore, at the directed percolation threshold N = N DP, there exists a connected set of ure 7 : the of the backbone submitted to a portion cracks without hooks or overhangs. Under tension, is There- local compression depicted in heavy line. this set will deconnect the system in two without any fore, the geometrical percolation threshold N c is not elastic resistance. This shows that N c D is equal to the critical for the tensile elastic of the blue properties directed percolation threshold N c Dp. For a pure cheese model. as the crack increa- However, density shear, the mechanical threshold should also be given ses above the distance between hooks Nc, typical by the directed percolation threshold : one must now increases until one reaches a new threshold when N c D choose the Ox axis making an angle 7T /4 with a set of cracks which deconnects the appears sample respect to the shear. This does not change the into at least two and whose structure pieces does not percolation threshold due to the isotropy of the contain any hook or overhang configurations. In the random crack structure. of this one a critical vicinity threshold, expects One can now use the properties which are known behaviour for the extensional elastic constant which about directed percolation for predicting its elastic should vanish as properties. The directed percolation threshold N DP is known to be larger than the usual percolation threshold : for example in the discrete square bond lattice, the directed threshold fraction is The general description of this new threshold and percolation the corresponding scaling laws is difficult in three around 0.64 compared to the geometrical threshold at a fraction of bonds to dimensions especially since it is related to the physics occurring occupied equal of oriented random surfaces [22]. 0.5 [21]. Directed percolation is characterized by two correlation which are the In two dimensions however, it is easy to see that lengths 611 and 03BE 1. distance between in the direction N c D corresponds to the directed percolation threshold typical overhangs Ox and in the direction perpendicular to Ox, respect- [23]. To prove this assertion, let us define arbitrarily ively. They diverge as N - N DP(- ) as an oriented line Ox as one of the two directions (left to right or right to left) along a line perpendicular to the macroscopic tensile stress as shown in figure 8. To be concrete, let us assume a vertical tensile stress

In d = 2, vll = 1.7334 ± 0.0001 and v 1. = 1.0972 ± 0.0001 [21, 23] which should be compared to v = 4/3. These values which are determined for discrete systems should remain valid in continuous directed percolation due to universality in the geometrical scaling. Within the directed percolation analogy, one can estimate the critical behaviour of the macroscopic elastic constant with an argument of the type node- link-blob adapted to the directed percolation model. Below the directed threshold and from Fig. 8. - Definition of the directed continuous percolation percolation problem for the suggested analogy with the crack elasticity the definition of yl and 6, , there is one hook or Forces are in two dimensions. The upper path corresponds to a overhang configuration per area 611 03BE 1.. system below the directed percolation threshold for which applied on each hook via a typical arm of a lever overhangs appear which are related to the segments 03BE~ I The corresponding macroscopic elastic oriented in the wrong direction. The lower path corres- modulus reads Y; 1 ,2 II / 1. leading to ponds to a system above the directed percolation threshold. 1373

Note that due to the existence of two length scales (d-2 ) 03BEII and 6, , the usual factor 6- relating the macro-link elastic constant to the macroscopic effec- tive elastic modulus is changed into 03BE1/03BE in two dimensions. One therefore obtains Ym::- (N c DP - N )rD with Fig. 10. - Magnified view of the small portion of the neck which suffers the bending deformation in the finite crack width mode deformation depicted in figure 9. Equation (19) gives (as usual in this type of argument) a lower bound for TD. It should be compared to lower bound of o--o 3.2 the for the elastic modulus In equation (20) we assume that for 5 a the exponent in usual vectorial percolation [4] whose bending strain is supported by the weak part of the values rather turn to be T = 3.96 ± 0.04 [4]. bond of thickness 8. We can estimate the bending Note that this argument can be used to compute a rigidity of the flawed system with the classical lower bound for the conductivity in a directed formula used in [1] electrical problem such as in diode lattices : one obtains the directed electric exponent tD = vp - v 1 and therefore TD = tD + 2 v jj which is similar to the relation between T and t (T = t + 2 v ) in ordinary percolation. One only gets a logarithmic correction to the discrete The scaling (19) may be modified outside a narrow result Ym ~- (Nc - N)f. critical region if the shear modulus Yshear is small In three dimensions, the microscopic bond bend- compared to the compression/extension modulus ing mechanism also controls the macroscopic elastici- In the of an a shear Y com. vicinity overhang, large ty when the crack widths a are finite. The bending should and over the rotational appear superimpose constant crosses over from y = Y8 4/a for 6 a to deformation of the hook. This implies a subdominant y = yl c 3 for 6 > a. Therefore, the shear stress modu- scalar correction v||p - v_L to the scaling which lus is f = f + 3 with f = dv + 1 for din.--a, i.e. should be important as long as the effect of the lever sufficiently near the percolation threshold. If arm 03BE||2 does not win over the ratio Ycom/Yshear. d min :> a, f = f. As N increases towards Nc, one 4.1.2 Usual percolation : case of cracks which have a should first observe the discrete-lattice shear modu- thickness a - When the cracks have a finite (B a ). lus exponent and finally cross over to f = f + 3 in a finite thickness a, the preceding anisotropy of the narrow critical region (Nc - N -- deformation under compressional or extensional a/I c). stresses at least for small disappears sufficiently 4.1.3 Out of plane two dimensional elasticity deformations. The of the elastic analysis properties - In two dimensions, another regime in then follows that for the Swiss (a d ). closely developed which the two edges of a crack can go out of the cheese model. In this case, the elastic percolation plane can be relevant depending on the experimental threshold is defined in 1. Under a Nc paragraph conditions. In the elastic plate is kept between two stress, the micro-bonds tend to accom- macroscopic rigid plates, the elastic properties of the system are modate the strains in the form of a bending defor- described in paragraph 4.1.1 or paragraph 4.1.2. mation concentrated in the thin portion of width a However, if the plate can buckle in the third which shows a view of a neck (see Fig. 9), magnified direction, different elastic properties appear. In this between two cracks. regime, two modes of elastic deformation (crack opening and buckling) can be distinguished and lead to distinct critical behaviours. Let us analyse them in turn and discuss their domain of validity.

Crack opening. - In two dimensions, in-plane deformations can only occur via the opening or overlapping of cracks or bending of the micro- Fig. 9. - Bending mode for cracks having a finite width a. bonds. In the case where the width a of the cracks is very small (smaller than the smallest bond width 8), the of a micro-bond involves a crack The corresponding bending constant of a micro- bending opening a as shown in bond in two dimensions is that of a plate shown in or overlapping by given angle 0 figure 10 and is written figure 11. Note that the ability for the crack edges to overlap suppresses the elastic asymmetry of the individual crack shown in figure 6. 1374

Buckling. - If the macroscopic 2d-plate is not rigidly clamped, it can become preferable for the elastic plate not only to have the crack edges to overlap slightly but even to buckle out of the plane in order to relax the elastic stresses as in figure 12. In other words, the major part of the elastic energy is Fig. 11. - Crack opening deformation mode in two found in the bending and twisting deformation dimensions when crack edges overlapping is allowed : modes. If this is permitted, buckling will always figure 11a corresponds to the case when the horizontal allow the horizontal crack in the T configuration to crack is to remain undeformed whereas supposed relax to its optimum shape. One has therefore to llb the case when both cracks are figure represents the dislocation case : the strain deformed. analyse only in-plane will be efficiently screened by the buckling deformation of the plate due to the coupling between in- plane strain and transverse deformation [25]. This If in the of an external the presence stress, screening of the elastic stress by buckling has been of the T of a micro- horizontal crack configuration estimated in [26]. The diverging expression bond Fig. 2) did not bend as in figure 11 a, the (see Yb 2 Log l c/ z is replaced by a curvature energy elastic E stored in that bond would corre- energy independent of the size lc of the form spond to that of a disclination of angle aperture (or charge) s = 0. We refer to reference [24] for the definition of the elastic energy and quote the result leading to a bending coefficient y = YE 3 corresponding to a plate of thickness E. The independence of y with respect to 5 shows that the buckling relaxes very for a single disclination in a plate of size L where Y is efficiently the stresses which tend to accumulate in the Young elastic modulus and z is the solid mesh the weak bonds and, as a result, the weak bonds are size. We take L = l c since strains are screened at no longer weak in the asymptotic critical regime scales larger than 1 c due to the presence of other N - Nc. The shear modulus and failure exponent cracks. Due to the independence of E with respect to are therefore equal to their discrete counterparts. 5, no correction to the discrete-lattice percolation case is expected. The reason is that, as 6 -> 0, micro- bonds do not weaken in this mode of deformation. However, usually there are no constraints on the deformation of the cracks and in particular, the horizontal crack of the T configuration does bend in general so as to minimize the elastic stress induced

by the aperture 0 (as shown in Fig.11b). This Fig. 12. - Schematic representation of the buckling of a corresponds to the existence of two disclinations of plate in the presence of a pair of oppositely charged opposite apertures or charges ± 6 at a distance disclinations forming a dislocation. We have used the 5 from one another forming a dislocation of analogy between figure 2 and figure 5a for replacing the Burgers vector b -- d0. In this case, the elastic two almost intersecting cracks problem by a two collinear cracks The is such that the in- energy reads [24] configuration. buckling plane elastic strain is completely transformed in bending energy.

The corresponding bending coefficient is I’ = Y82Logle/z. Knowing y (8 ), one can follow Note also that, if buckling is allowed, it will always reference [1] and compute the macroscopic shear occur as long as 52 s Log 1,,Iz :::. E3 i. e. for suffi- modulus Y m = g 2 - d K with the expression ciently thin plates. Otherwise there is a cross-over from a discrete exponent regime for Nc 2013 N not too small to the planar crack opening exponent regime.

4.2 MECHANICAL FAILURE.

= 1 where This with + - gives Ym = (N, - N )f f f 4.2.1 Brittle Griffith failure (d = 2 and d = 3). In f = dv + 1 = 3.7 is the shear modulus exponent of a system containing cracks, brittle rupture following the discrete case. In this deformation mode, the the Griffith mechanism [27] will occur under tensile fluctuations of the micro-bond strengths lead to an stress (mode I [27]). This rupture scenario is ex- important correction for the macroscopic elastic pected to be prominent for cracks with vanishing exponent compared to the discrete-lattice case. thickness. In this failure mechanism, the concen- 1375 tration of the stresses at the apex of a crack tends to but one can rely on the heuristic argument (checked open the two edges and lenghten the crack, thus with the exact result of the 2d problem) that most of leading to failure. This mechanism must be con- the elastic energy is stored in the vicinity of the disk. sidered when the tips of the cracks are sharp and no Therefore p. uS 2 should be of the order of Y-1 F2 screening due for example to the presence of round- leading to p . u o--o Y- 1 F 25-2. Writing the Griffith ing or of a cavity occurs at the tips of the cracks. criterion p. u 2 s leads to a stress failure threshold The Griffith criterion states that a crack growth is exponent F. = 3 v + 1. More generally, in arbitrary governed by a balance between the mechanical dimension, one has F. = d v + (d - 1 )/2 with a energy released and the fracture surface energy correction (d - 1 )/2 to the failure exponent of the spent as the crack propagates. More precisely, a discrete-lattice case. crack will grow if the virtual work of the stress field Note that it is not clear whether this failure on the crack, deplaced on a distance equal to the threshold for the weakest crack’s local configuration strain, is greater than the surface energy necessary to coincides with the macroscopic failure threshold, create the new solid-void interfaces. In the blue due to complex screening and enhancement interac- cheese model, one can argue that the typical generic tions. Further attention should be given to this configuration for the weakest part of the system difficult problem. involves one crack in close proximity to a border 4.2.2 Bond flexion failure : case of cracks with finite other as shown in 2. This (created by cracks) figure thickness. - This to the case treated in is similar to the of two collinear cracks corresponds problem paragraph 4.1.2 for the elasticity. The analysis separated by a distance d. closely follows reference [2b]. Bending failure will In the d = 2 case, one can use the exact mapping occur on the weakest bond and is characterized by a to the problem of determining the stress and strain rupture threshold given by distributions in a semi-infinite plane submitted to a force F homogeneously applied on the upper perime- ter over a finite width . 5 [28]. The expressions of with = 2 v + 1/2 for the critical elastic the stress p and strain u along the axis x are [28] Fm energy criterion in 2d and Fm = 2 v + 1 for the critical strain criterion in 2d since the bending constant scales as 8 (for 6 a) as given by equation (12). In 3d, the failure exponent is Fm = 3 v + 2 for the The virtual work entering the Griffith criterion is critical elastic energy criterion and Fm = 3 v + 4 for proportional to the product p . u = y-l F2 S - 1. This the critical strain criterion since the bending constant result can be reobtained heuristically by writing that y scales as 8 4 in the critical region ( 8 a ). Outside 5 almost all the elastic energy is stored on the length this critical region (8 a ), one recovers the discre- separating the two cracks. Therefore, p . u5 must be te-lattice exponents. of the order of y-l F2 which recovers p. u = two-dimensional Y-1 F2 S -1. The instability of the crack occurs when 4.2.3 Out of plane rupture. p. u becomes equal to the surface tension energy Crack opening. - In this case, the failure problem 2 s of the solid-void interface. This yields the failure can be analysed with the tools of reference [2]. 2d force F r = S 1/2. This result can be obtained directly mechanical failure is dominated by the weakest for the two-crack configuration of figure 2 using the bond. If a stress a is applied at the boundary, the techniques of reference [29] : this confirms the force F supported by a macro-link is F -- 0-6 (d- 1) analogy between the problems posed by figures 2 and the torque M which is transmitted in the macro- and 5a for d = 2 and between figures 4 and 5b for link is [17] M = Fg = ugd. d = 3. Assuming that rupture will occur if the bending Since the force F results from the torques exerted elastic energy E = ’Y - 1 M2:=::::: 8 - 2 u 2 g 2 d in that on a scale 6, F .le = Mg = (TE2, this leads to bond becomes larger than a threshold value Er (first F = (TE2/lc. Therefore, the stress failure threshold criterion of [2b]), one obtains the failure threshold is

with Fm = 2 v + 1 larger than the discrete result = 2 v one. with Fm = 2 v + 1/2 larger than the discrete case by Fm by 1/2. If rupture occurs when the angle 0 by which the neck is bent is than a threshold value In 3d, one has to evaluate the concentration of the larger criterion of one has = 2 v + 2. elastic energy due the presence of the two cracks. 0c, (second [2b]), F. Using the analogy between the topology of figures 4 Buckling failure. - The screening of the strain by and 5b, it is sufficient to estimate the product buckling is so efficient that one recovers the discrete- p . u for a disk of diameter 5. This is not an easy task lattice exponents. 1376

4.2.4 Failure distribution. - Due to fluctuations in properties to the different geometrical limits in the the bond strengths, the failure thresholds are distri- different models ; buted according to a probability distribution taking a 2) table I summarizes the results of [1, 2] and this of Weibull form when the expression the stress work. A striking fact is the wealth of exponents in a of width 5 is failure threshold of micro-bond of the elasticity and mechanical failure corresponding to form the different modes of deformation. What will be the mechanical failure exponent in a given experimental set-up ? The answer is not simple and depends The analysis is similar to that developed for the sensitively on the precise conditions which apply to electrical failure in paragraph 2.2 and is not re- the system under study. In 2d, if out of plane peated. Generally, the appearance of a Weibull buckling is forbidden (by clamping the plate between failure distribution can be tracked back to the to rigid plates) but crack edges overlapping is existence of a powerlaw distribution of micro-bond allowed as in paragraph 4.1.3 and if the widths of the In the case of the blue cheese strengths. percolation cracks are vanishingly small (a B min), crack open- model, the powerlaws are created by the flat distri- ing rupture will dominate. When buckling is permit- bution of bond-widths in the presence of the power- ted, one recovers the discrete-lattice exponent. law dependence of the failure stress with respect to In 3d, Griffith rupture dominates for thin cracks the micro-bond strengths. (a S min ) but leaves room for micro-bond bending failure for cracks with a finite thickness. 5. Concluding remarks. This study therefore demonstrates that, in systems the Griffith failure mechanism An analysis of the transport and failure properties of damaged by cracks, will not occur. This is in contrast with the a new class of percolation model (the blue print of always brittle failure mechanism when the system stick or disk percolation [7]) for systems deteriorated occurring by thin cracks has been presented. Several defor- is damaged by a single crack. In the presence of mation modes have been discussed. Of particular many cracks, bond bending failure must be considered is also the of importance is the asymmetry of the stress-strain (this suggested by analysis [30] the in This characteristic of a single crack. This leads to a new experiments reported [31]). problem another of the appearance of a percolation problem which in two dimensions corres- provides example novel behaviour characteristic of ponds to a continuous version of the directed qualitative large disorder ; percolation model. In three dimensions, one has to analyse the connectivity properties of random disks 3) the different critical exponents summarized in or plaquettes, a problem related to the statistical the table should be observed in a narrow interval properties of random surfaces [22]. This difficult N - Nc as suggested by recent experimental and problem is left for future work. numerical works [31, 16]. This implies facing the Finally, let us note the following points : difficult problem of finite size effects. Note that the cracks must be really distributed at random in order 1) the results of this paper and of references [1, 2] to create the narrow necks which are so crucial for can be contrasted with the exponents for geometrical the distinction between continuum and discrete properties such as v which have been confirmed systems. See for for a careful discussion numerically to be identical for both discrete and example [32] of the numerous that must be continuum systems The difference between experimental pitfalls [13]. avoided the critical continuum and discrete-lattice exponents stems from transport. These predictions are currently being checked in the fact that transport and failure properties are our on model with sheets of influenced by the weak bonds of the distribution of laboratory experiments different materials which are deteriorated cracks. bonds The difference between the Swiss- by strengths. This work will be cheese and blue-cheese continuum percolation reported subsequently. models stems from the different topology of the micro-bonds of the percolating structure. The blue- Acknowledgments. cheese model can be viewed as another way for My interest in this problem was stimulated by taking the limit of a discrete-lattice mesh size z going discussions with A. Sornette. I am grateful to S. to zero. In the Swiss-cheese model, zlb --> 0 (where Roux and C. Vanneste for helpful and stimulating b is the spherical hole radius) in contrast to the blue- remarks and for a critical reading of the manuscript. cheese model where z I l c ->. 0 and the discrete case I thank the unknown referees for judicious remarks where blz -- lclz:, 1. We can thus trace back the and for pointing out an error in the first version of non-universality of the transport and failure critical the derivation of the weakest bond failure threshold. 1377

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