Damage cascade in a softening interface

A model describing the damage at an interface which is coupled to an elastic homogeneous block is introduced. Resorting to a real-space renormalization analysis, we show that in the absence of heterogeneity localization proceeds through a cascade of bifurcations which progressively concentrates the damage from the global interface to a narrow region leading to a crack nucleation. The equivalent homogeneous interface behaviour is obtained through this entire cascade, allowing for the analysis of size effects. When random heterogeneities are introduced in the interface, prior to the onset of localization damage proceeds by a sequence of avalanches whose mean size diverges at the first bifurcation point of the homogeneous interface. The large scale features of the bifurcation cascade are preserved, while the details of the late stage are smeared out by the randomness.

A model describing the damage at an interface which is coupled to an elastic homogeneous block is introduced[ Resorting to a real!space renormalization analysis\ we show that in the absence of heterogeneity localization proceeds through a cascade of bifurcations which progressively concentrates the damage from the global interface to a narrow region leading to a crack nucleation[ The equivalent homogeneous interface behaviour is obtained through this entire cascade\ allowing for the analysis of size e}ects[ When random heterogeneities are introduced in the interface\ prior to the onset of localization damage proceeds by a sequence of avalanches whose mean size diverges at the _rst bifurcation point of the homogeneous interface[ The large scale features of the bifurcation cascade are preserved\ while the details of the late stage are smeared out by the randomness. [

0[ Introduction
Progressive failure of quasi!brittle materials can be separated in three di}erent phases ] _rst\ the material response is elastic\ then microcracking appears and these microcracks coalesce eventually in order to form a macro!crack which propagates suddenly[ From the theoretical point of view\ the di.culties involved in the transition between the last two phases are quite important[ In the second phase\ the strain _eld is quasi!homogeneous at a macroscopic scale[ Then\ the strain _eld becomes more and more heterogeneous\ and the strain grows only inside a narrow region[ The subsequent apparition of a discontinuity is often called strain localization in a general sense[ There are in the literature di}erent approaches to the description of this transition[ One of them is the continuum approach\ e[g[ with continuous damage models[ It is based on the description of the average behaviour of the material "see e[g[ Krajcinovic and Lemaitre\ 0876^Laws and Brock! enbrough\ 0876^Lemaitre\ 0881#[ For such continuum models\ the transition is viewed as a bifurcation problem[ When strain localization is due to strain softening\ the tangent sti}ness operator ceases to be de_nite positive[ The partial di}erential equations of equilibrium lose their ellipticity which authorizes discontinuous rate of deformation _elds to develop suddenly[ The inception of strain localization might be for instance depicted under some restrictive assumptions by Hill|s criterion "Hill\ 0848# ] det ðn = H = nŁ 9 "0# where H is the tangent sti}ness operator at the continuum point level and n is the orientation of the localized band[ Another one is the loss of stability at the material level in the sense of the Drucker postulate[ Note that in some well!de_ned cases "associative constitutive laws#\ the loss of uniqueness coincides with the loss of stability in the rheological sense[ The second approach is discrete random modelling[ It is directed towards the description of the study of the material heterogeneities "i[e[ at a scale lower than the representative volume of the material# "see e[g[ Delaplace et al[\ 0885^Fokwa\ 0881#[ Because of heterogeneity\ the usual employed localization criteria in continuum models cannot be used for two reasons mainly ] the _rst one is that the solution is always unique[ To some extent\ the situation for discrete models is the same as the situation for some rate dependent models where bifurcation is not possible and strain localization cannot be viewed as a loss of uniqueness problem anymore "see e[g[ Dudzinski and Molinari\ 0880^Leroy\ 0880#[ The second one is that no tangent operator can be calculated because of the discrete characteristic of the response\ and because of the~uctuations that appear all along the curve[ There lies a subtle di.culty ] consider the dimensionless ratio o a:L\ of the microstructure units a of a discrete model over the system size L\ which characterizes the dis! creteness of the medium[ When o tends to 9\ a continuum description is expected to hold[ As we will see later\ the stressÐstrain response of the system converges towards a smooth law whose tangent operator H can be de_ned[ The latter does provide information on the stability of the structure[ However\ when stability is analysed using actual responses for non!zero o\ it can be shown that the~uctuations in the stressÐstrain responses give rise to a non!di}erentiable law\ and this feature brings some useful additional informations on the approach to the loss of stability[ Elaborating over these notions leads to the useful concept of {avalanches|[ The aim of this paper is to propose a tool to characterize the transition from a homogeneous state of microcracking to a localised one for the discrete models[ This tool should also be applied to any response with~uctuations\ like those met in experiments where dispersions and~uctuations due to material heterogeneity are unavoidable[ Because one cannot deal in this case with a loss of uniqueness\ this tool should be based on stability considerations in the broad sense[ Therefore\ we study the~uctuations that are encountered all along the response of the system[ More precisely\ the avalanche statistics of the~uctuations are analysed[ This formalism is used in many kinds of model "Bak\ 0885^Paczuski et al[\ 0884#\ from biological di}usion up to earthquake response[ All these models have at least one thing in common ] their evolutions are structured around a critical point\ as for the _bre bundle model that we used as a basis for our di}erent proposed models[ For the sake of simplicity\ we will consider a model problem ] the case of a band made of a strain softening "discrete# material assembled in series to an elastic block " Fig[ 0#[ This system is loaded by uniaxial tension\ perpendicular to the direction of the band[ Thus\ the problem of localization will be strictly one directional as the orientation of the localised band is _xed [ Since Fig[ 0[ The model problem[ the band has a _nite width which might become small with respect to the block dimensions\ its response may also be regarded to be the same as that of a softening interface located in between a rigid substrate and an elastic body[ In the _rst part of the paper\ we will recall the analytical results obtained on the _bre bundle model\ also called Daniels model[ We will particularly present the properties of the avalanches distribution in the presence of~uctuations due to the variability of the _bres strength and deal with a simple derived model\ that is a Daniels model and a spring connected in series[ We will look at the evolution of the avalanche properties\ and apply them for detecting the loss of stability[ In order to have a realistic representation of the local mechanical redistribution of the stress _eld when a micro!crack appears\ we will use in the second part a hierarchical model that takes into account redistribution\ i[e[ a non!local load sharing on the surviving _bres when a bond breaks[ Again\ we will look at the evolution of the avalanche properties\ and we will carry out a complete study in terms of stability[ In order to better understand the mechanism of rupture and the apparition of successive bifurcation points\ this model will be compared to the equivalent con! tinuum one[ This model also allows to have direct access to the damage pro_le at the onset of unstable propagation of a macrocrack[

1[ The Daniels model and avalanche statistics
The Daniels model "Daniels\ 0834# albeit simple\ displays an amazingly rich behaviour which is*at least partly*representative of the role of heterogeneity in the mechanical behaviour of some materials[ It is commonly called the _bre bundle model[ N parallel _bres are equally stretched between two rigid beams[ The _bre behaviour is elastic up to a threshold force where the _bre breaks irreversibly[ The sti}ness is the same for all _bres*and thus can be chosen to be unity* but the threshold force t is a random variable characterized by its probability distribution function p"t#\ or its cumulative distribution P"t# Ð t 9 p"t?# dt?[ The advantage of this model is that it is completely solvable analytically[ For instance\ the mean force F\ that is the applied force divided by the total number of _bres N\ vs displacement u is easily obtained as F"u# "0−P"u##u "1# Then\ for a sti}ness of 0 "t u for a single _bre at failure#\ the displacement u varies between 9 Note that similar~uctuations are also encountered experimentally on quasi!brittle heterogeneous materials\ like _bre!reinforced concrete\ but they are usually not described by continuum models[ It is to be noted right away that the amplitude of these~uctuations vanishes as N −0:1 \ and thus considering the limit of an in_nite system size\ N : \ the response of the system converges to the above given mean behaviour[ It is our aim to show that these~uctuations\ albeit of modest amplitude\ are of interest both from an experimental and a theoretical standpoint\ and that some care has to be taken when considering the in_nite size limit[ Since we are interested in stability\ it is important to incorporate in the analysis the boundary and load conditions[ In the following study\ we will consider that the bundle is loaded with a testing machine of known sti}ness k[ Hence\ the analysis will be applied to a bundle connected in series with a spring whose sti}ness is that of the testing machine as shown in Fig[ 2[ This simple system can also be seen as a rough model for the mechanical behaviour of an elastic body "the spring# attached to a rigid substrate through a damageable interface "the _bre bundle#[ The overall displacement "bundle plus spring# will be controlled during the loading sequence[ If the bundle is loaded with a sti} enough testing machine\ only one _bre may break for a constant loading[ However\ if the sti}ness is reduced\ one single failure may induce catastrophically a sequence of failures before reaching a new stable position[ This sequence is what we call an {avalanche|[ In the numerical simulations\ one can easily solve for the response of the bundle with an ideally sti} control\ that is\ imposing strictly a prescribed displacement[ From such a response\ one can also compute the response of the bundle under any boundary conditions "including a _nite sti}ness k\ and large viscous damping to avoid interial overshoot#\ as a succession of equilibrium positions[ Let us _rst consider the limit of an in_nite system size\ and substitute a deterministic damageable as obtained above for a uniform distribution of _bre strengths Daniels "0834#[ The stability analysis of this system is quite straightforward "see Baz ³ant and Cedolin\ 0880# ] under a small enough prescribed displacement U of the entire system "interface u plus elastic body v#\ the system has a unique solution\ i[e[ u ð"0¦k#−z"0¦k# 1 −3kUŁ:1 v u"0−u#:k "3# The second!order work of the system is ] where K is the tangent sti}ness of the bundle\ i[e[ dF:du[ k is always positive\ as K is positive in the prepeak part of the bundle behaviour\ and negative in the postpeak part[ The state of the system is stable if d 1 W × 9\ that is equivalent to K × −k[ At the maximum displacement U "0¦k# 1 :3k\ we _nd a bifurcation point\ where the solution is no longer unique[ At this point\ the displacement is u "0¦k#:1 and the tangent sti}ness in the bundle is It is exactly opposite to that of the elastic body[ It corresponds also to the loss of stability\ because the second!order work is zero[ This is a trivial example of a localization point[ If one tries to increase the displacement past U \ then the entire interface fails catastrophically[ Under an idealized controlled displacement\ the system response follows a snap!back branch\ that represents instability "d 1 W ³ 9#[ Note that prior to the critical equilibrium\ strain softening develops in a stable fashion as also pointed out in Baz ³ant|s analysis Let us now come back to the _nite size _bre bundle as the interface[ The response of the bundle is no longer a di}erentiable law such as eqn "2#\ but rather a sequence of linear elastic responses limited by end!points where a _bre breaks[ Because of randomness of the _bre strength\ the response of the system is always unique\ and no bifurcation point could be de_ned[ Let us call "u i \ F i # the sequence of failure displacements and forces\ respectively\ where i indicates the number of broken _bres[ In a _rst approach\ we can repeat the same analysis as previously[ One cannot of course consider that eqn "5# holds at bifurcation as in the previous example because the response of the bundle is no longer di}erentiable[ Nevertheless\ tracing the maximum of the function F"u#¦ku yields a criterion for the onset of unstable behaviour\ and if F"u# is di}erentiable one recovers the previous analysis[ As we saw before\ the response of such a system is just a succession of~uctuations[ It can be analysed through {avalanches|[ With the introduced variables\ we can de_ne an avalanche in the _bre bundle as follows ] an avalanche of size D and direction k\ starting at "u In the continuous case\ prior to the point u \ damage in the bundle is controlled and we can say that the avalanche size is 9[ At u u \ the critical equilibrium state is reached and a single avalanche of size equal to the remaining number of bonds in the bundle is observed[ In the thermodynamic limit\ N : \ this avalanche has a size which diverges to in_nity[ We observe that avalanches do reproduce the result of a standard stability analysis\ with a simple {binary| "9− # avalanche size distribution[ A crucial point is that this analysis is not entirely correct\ in the sense that we have _rst considered the continuum limit for the force displacement curve\ and analysed the avalanches on this mean response[ Most of the information which can be derived from the concept of avalanches has been lost in this procedure[ Taking into account the full random and discrete nature of the model\ Hemmer and Hansen "Hemmer and Hansen\ 0881^Hansen and Hemmer\ 0883# succeeded in determining the analytical solution of the probability distribution n of observing an avalanche of size D\ for any sti}ness k\ starting at any prescribed displacement u[ They found ] n"D\ u\ k# N where F"x# is a scaling function which is constant for small arguments x ð 0\ and drops to zero rapidly for x × 0[ D is the maximum avalanche size[ Moreover\ the exponent −2:1 and −1 appearing in eqn "7# are universal\ in the sense that they do not depend on the chosen distribution This statistical distribution of avalanches can potentially be used as a precursor of the macro! scopic failure point u [ According to Hemmer and Hansen "0881# analysis and in the course of loading the bundle\ we would observe a series of avalanches\ which can be referred to micro! instabilities\ which progressively becomes larger and larger\ up to the point where they diverge and become {macroscopic|[ This signals in particular that some care has to be taken when taking the thermodynamic limit[ Taking _rst the continuous limit of the forceÐdisplacement response\ and analysing its stability\ erases the progressive development of the avalanches\ and hence misses an important feature of the model[ To demonstrate the utility of such an analysis\ we will analyse the~uctuation in the global failure displacement for a _nite size system[ Let us call du the distance to u "k# where the _nal avalanche is initiated[ du can be estimated by writing that the maximum avalanche size at this displacement\ D \ allows to increase the displacement u up to u "k#[ Hence if not impossible to derive this result without considering the notion of avalanches[ Finally\ let us also note as a side!remark that in the continuous limit\ the response of the system is continuous but not di}erentiable[ In this case\ one can show that locally\ the~uctuating part of the forceÐdisplacement response becomes self!a.ne with a Hurst exponent of 0:1[ Thus it belongs to the realm of C 0:1 functions\ rather than C 0 as would be needed to apply a criterion such as Because the displacement is constrained to be the same in each _bre\ the response tends to a well!de_ned behaviour\ and thus _nite size e}ects play only a marginal role in the present example[ For instance\ the peak stress\ F\ tends to well de_ned value with~uctuations of order 0:zN[ However\ other systems display a much more signi_cant size e}ect\ which can be tracked back to the cumulative e}ect of the avalanches[ Hence\ in this case\ the concept of avalanches and of their statistical distribution is unavoidable[ A principal~aw of this simple model is the load redistribution when a _bre breaks ] the external force is shared equally between all surviving _bres[ On the other hand\ when a micro!crack appears in a quasi!brittle material\ the stress is redistributed mainly around it\ and the local interaction decreases fast as the distance with the micro!crack increases "typically as a power law of the distance#[ To take into account this e}ect\ we need to introduce this redistribution in the model[ Note that a simple local redistribution\ where the load of a failed _bre is redistributed equally on the nearest surviving _bres\ has been already studied "see e[g[ Harlow and Phoenix\ 0880#[

2[0[ Presentation and properties
For the sake of simplicity\ we are going to focus on the situation where the band of strain softening material is small with respect to the size of the elastic block modelled above by a spring and a rigid bar[ Hence\ we will deal with a softening interface embedded in between a rigid and an elastic substrate[ This description is suited to adhesion\ and can also be seen as a simpli_cation of a 1!D medium since the redistribution process will be constrained to develop in the direction of the interface[ Nevertheless\ this example contains the basic features involved in the transitional behaviour between di}use and localised cracking[ At variance with the previously discussed case\ we would like to incorporate an elastic coupling between the _bres as mediated directly by the elastic body\ i[e[ without the rigid bar which redistributed equally the displacement among the surviving _bres[ In continuum mechanics\ this e}ect could be represented by the elastic Green function of a semi!in_nite plane[ For the convenience of the analysis\ we resort to a di}erent choice based on a hierarchical decomposition of the elastic body[ The structure is the following one ] a block is split up in three sub!blocks " Fig[ 8#[ The two lower blocks are then subdivided in three\ and this recursively down to the lowest level chosen in the discretization[ Each block is described using an elastic uniaxial behaviour[ Thus\ the elastic body can be seen as composed of springs\ connected in parallel and alternatively in series[ At the lowest level\ each _ner block is connected  Figure 09 shows an example of a 2!generation model[ With 09 _bres for each bundle\ a system of generation 01 is for instance made up of 09×1 01−0 19\379 _bres[ The simplicity of the construction allows for the numerical simulation of extremely large sizes\ while still preserving the long range nature of the elastic couplings[ In two dimensions\ all springs have the same sti}ness k at all generations\ but their initial lengths is divided by two as the generation is decreased by one[ At the _rst generation\ however\ the aspect ratio of the element is twice that of all other generations\ and thus the _rst springs in contact with the interface have a sti}ness k:1[ This allows to obtain a global sti}ness for the entire elastic system which is independent of the discretization level as expected "see the Appendix#[ An important point which will be used later concerns the interaction between an intermediate level "say index i#\ with the rest of the medium[ We can compute recursively the sti}ness L i of this structure deprived from one subblock i\ if a force is applied at this level[ We _nd that L i is exactly equal to k\ i[e[ just as if the subsystem was simply connected to the exterior world by a single block "see the Appendix#[ As the generation increases it can be shown that the elastic coupling will be the same as for a continuous model\ i[e[ with an in~uence function scaling with the same power!law as the Green function in an elastic continuum[ In order to reach this result\ one needs to introduce a distance suited to our discretization[ Considering two points along the interface\ we search from the smallest block which contains both points[ If j is the block generation "which could take value between 0 and N\ which is the generation of the entire system#\ the distance is then de_ned as This distance has the special feature of being ultrametric "i[e[ the same properties than an usual distance\ except for the triangular inequality where d"A\ C# ¾ min"d"A\ B#\ d"B\C##[ With this de_nition\ one can show that under an applied force F on a point of the interface\ the induced displacement v of an other point is v" j# "N−" j−0## F k ¦v 9 "03# where v 9 is the initial displacement of the considered point\ and N−" j−0# is nothing but the number of springs that separate the two points[ By introducing the distance d\ and for large j\ it where A 0:log 1 and B 1 N are constants[ It is exactly the same form that the Green function of a semi!in_nite plane[ The only di}erence is that this in~uence function consists in constant plateaus whose size increases in geometric series[ This is a residual e}ect of the two!fold splitting of each level[

2[1[ Continuous interface
We now proceed by considering _rst the case where each _bre bundle is changed into a damageable element\ with a behaviour law derived from the mean _bre!bundle response\ eqn "2#[ We use the hierarchical construction to relate the interface law to the global "interface plus elastic body# response[ Let us construct a system at generation "n¦0#\ starting from two generation!n subsystems[ These last subsystems are supposed to be described by two forceÐdisplacement relations F "n# 0 "U "n# # and F "n# 1 "U "n# #[ We wish to _nd the global F "n¦0# "U "n¦0# # response[ The two subsystems are subjected to the same displacement\ hence the force is F "n¦0# F "n# 0 "U "n# #¦F "n# 1 "U "n# # "05# The same force also stretches a spring of sti}ness k in series with the blocks\ and thus the global displacement is U "n¦0# U "n# ¦ F "n# 0 "U "n# #¦F "n# 1 "U "n# # k "06# These equations provide a parametric representation of the "n¦0#th generation system as a function of the nth generation[ Let us _rst assume that the interface is homogeneous\ thus in the previous analysis F 0 F 1 [ Hence U "n¦0# "F# U "n# "F:1#¦F:k U "0# "F:1 n #¦"0−1 −n #F:k "07# The _nal equation simply relates the interface displacement to the global one u"F# U "0# "F#[ We note that as n increases\ the global behaviour is nothing but that of the elastic medium because the displacement in the interface represents a vanishing contribution[ We can invert the previous relation to obtain the homogeneous interface law from the global response[ This is a practical tool to compute the equivalent homogeneous interface law when some inhomogeneity exists locally[ From the previous equation and because the interface response is continuous\ the tangent "subscript tg# and secant "subscript sc# sti}nesses of the entire system at generation n can be computed ] where we have used the local interface displacement u to characterize the loading\ F 1 n−0 u"0−u#\ assuming in this formula a homogeneous displacement all along the interface[

2[2[ Bifurcation analysis
For moderate displacements\ and in the absence of randomness in the interface\ every point of the interface undergoes the same damage[ The system response however may cease to be unique at a particular displacement for which the {global| displacement U "n# is maximum[ Let us _rst assume that one half of the interface is subjected to an increasing damage while the other half is elastically unloaded[ Using eqn "06#\ we see that this bifurcation condition is reached when From the expression for the secant and tangent sti}nesses eqn "08#\ we obtain an equation for the displacement\ u u 0 at the interface level for which a _rst bifurcation is encountered ] "2−1o#"0−1o#"0−u#"0−1u#¦1ko"1−1o#"1−2u#¦3k 1 o 1 9 "10# where o 1 0−n [ Focusing on the large system size limit\ we can expand the solution in order of o and obtain the solution as where we have introduced the size of the interface where the damage localises\ l 0 1 n−0 L:1\ to express the result in physical terms[ L refers here to the number of damageable elements "or _bre bundles in the discrete case#[ At this stage\ there is a bifurcation to three possible evolutions ] either damage remains inhomo! geneous "but this solution is unstable# or only one of the two subsystems continues to be damaged while the other is elastically unloaded[ Due to the symmetry of the system these two solutions are identical[ For a large system size\ o : 9\ we note that u 0 tends to 0:1\ i[e[ the interface displacement at peak force[ Bifurcation is however delayed to a larger displacement by a quantity proportional to "k:L#[ The homogeneity in the latter expression can be restored if we consider that the sti}ness of the interface _bres is not unity\ and that the interface is in fact a band of softening material of width h[ The o}set of the _rst bifurcation point is then of order where E bulk is the Young modulus of the elastic block\ and L its size\ E i and n i are the Young modulus and Poisson ratio of the band of width h[ The occurrence of n i comes from the antiplane displacement in the layer[ In this analysis\ we have postulated that the _rst bifurcation mode appeared at the macroscopic scale[ One can perform the same computation for any intermediate level 0 ¾ i ¾ n\ keeping the boundary condition on U "n# [ The only variance with eqn "19# is that the sti}ness k has now to incorporate all the intermediate levels from i to n[ The hierarchical structure allows to compute this sti}ness which remains simply equal to k at all levels[ Therefore\ the localization at generation i appears for a displacement u n−i given by eqn "11# where l n−i 1 i L:1 n−i is to be substituted to l 0 [ Thus\ these modes will occur much later than the _rst one l 0 1 n−0 [ They will however be of interest if we proceed along one of the two symmetric stable branches past the _rst bifurcation point[ The nth generation subsystem where the damage continues to progress will encounter a bifurcation point similar to the previous for u u 1 [ Past this local displacement\ the damage concentrates on one quarter of the system while the rest will be elastically unloaded[ The same analysis can be carried out to any stage down the cascade of bifurcation always concentrating on a stable branch[ The local displacement of the interface on the active part of the interface at the ith bifurcation is given by eqn "13#[ We thus obtain a simple physical picture of the post!localisation regime "localisation is under! stood here as bifurcation# where the damage zone progressively condenses onto a smaller and smaller "{active|# region\ while the rest of the structure is elastically unloaded[

2[3[ Post bifurcation response
An important feature which deserves a particular interest is the equivalent interface law which can be measured past the _rst bifurcation point[ Indeed\ as soon as the damage is no longer homogeneously distributed\ the equivalent homogeneous law is no longer similar to the one of any of the constituents[ However\ if we were to perform the experiment\ without any a priori knowledge of the cascade of bifurcations\ the equivalent homogeneous law is the one we would extract from the loadÐdisplacement curve[ The easiest way to have access to such an equivalent law for a system at generation n is to use the hierarchical nature of the decomposition of the elastic block[ Let us assume that we know this equivalent law for a system of generation n−0\ and express the equivalent law at the next generation[ As discussed above\ the _rst bifurcation point occurs _rst at the largest scale[ Past this _rst point\ one half of the lattice is simply elastically unloaded[ The other half is described by the homogeneous equivalent law[ Let "0:1¦x "n# 0 \ 0:3−y "n# 0 # be the displacementÐforce coordinate of the _rst bifurcation point[ As observed above\ we know that the x "n# 0 form a geometric sequence of ratio 0:1 ] For the y coordinate\ it su.ces to observe that the _rst bifurcation point lies on the homogeneous characteristic\ and hence 0:3−y "n# 0 "0:1¦x "n# 0 #"0:1−x "n# 0 # where we have used the speci_c parabolic form of the {bare| interface law[ Thus\ Any point "0:1¦x\ 0:3−y# of the "n−0# generation equivalent homogeneous interface law is transformed into "0:1¦x?\ 0:3−y?# such that We label the succession of bifurcation points by a subscript j whereas the superscript n refers to the system generation[ Solving for the asymptotic "large n# behaviour of the "x\ y# variables provides Figure "00# shows the sequence of bifurcation points for n 6\ n 8 and n 00 computed exactly\ together with the asymptotic expression shown as a curve[ We observe that only the latest j ¼ n points are not well described by the asymptotic behaviour[ However\ as n increases\ most of the cascade is very accurately described[ To make the above result more explicit\ we note that past the _rst bifurcation point\ the forceÐ displacement relation becomes size!dependent\ but it can be cast in a simple scaling form using the system size L 1 n ] 0 y y "0# where the scaling function is just ] One important point to be noted here is the di}erence of exponents of L which appears in x and y[ As a consequence\ the equivalent homogeneous interface law shows a sudden decay of the force at constant displacement past the peak force[ It is also of interest to consider the scaling of other physical quantities[ In particular\ if we come back to the picture of the _bre bundle at the interface level\ we may introduce another variable Fig[ 00[ The sequence of the bifurcation points "x n j \ y n j #[ The continuous curve shows the asymptotic behaviour obtained from the recurrence relation[ The dotted curves are the real sequence of bifurcation points for a 6!generation "crosses#\ 8!generation "black point# and 00!generation "triangle# system[ Note that just the _rst points are well!described by this relation\ and the snap!back part is not represented[ which is the number N of broken _bres[ The latter is simply related to the displacement as N 1 n u[ Therefore\ we conclude that two consecutive bifurcations are separated by a _xed number of broken _bres[ In fact the displacement in the active region for the consecutive bifurcations increases exponentially fast\ as 1 j \ but simultaneously the active region shrinks also exponentially\ as 1 −j \ so that the product of these two terms which gives the number of failed _bres remains constant[ It is now a simple matter to express the variations of x or y as a function of the number of broken _bres "past the peak force\ where N N p L:1# ] x is linear in "N−N p #\ whereas y grows exponentially fast[ Simultaneously\ the size of the active region is L:1 j \ decreasing exponentially fast with j or equivalently "N−N p # or x j [

2[4[ From dama`e localization to crack nucleation
The physical picture which arises from this analytic solution is of particular interest\ since it is one of the rare situations where some insight can be obtained past the _rst bifurcation [ We have seen that the interface degradation process consists in a progressive condensation of the damaging region from the structure scale down to the basic constitutive unit[ At the end of this _rst cascade\ exactly one of the smallest size interface element is totally broken[ This naturally forms the initiation stage for a crack propagation regime[ Unfortunately\ following the crack propagation is of little interest in our model\ which is then too much sensitive to the detail of the hierarchical decomposition to pretend any possible comparison with reality[ However\ up to the crack nucleation\ we believe that the hierarchical interface model is a faithful description of continuum model\ yet simple enough to be amenable to an analytic solution[ An interesting feature is to be noted at the crack nucleation stage ] the progressive condensation of the damage can be read back from the damage pro_le along the interface[ Indeed\ the damage D of the interface is a simple linear function of the maximum displacement ever encountered by a homogeneous domain\ that varies between 9 and 0[ Tracing backward the damage in the active region\ we obtain that the damage D at a distance d from the crack nucleation point is where we have used the earlier de_ned distance ðeqn "02#Ł[ Thus\ the cascade of bifurcation leads to a rather unusual damage pro_le ahead of the crack[ If we de_ne a {process zone| as a damage zone ahead of a crack\ we would conclude that the process zone is of in_nite extent[ However\ this is important to note that the damage decreases very fast with the distance\ i[e[ as an inverse law[ Then this process zone seems more similar than the classical one\ that is a quasi!con_ned damage zone of _nite length ahead of a crack[

3[ Discrete interface model with redistribution
We have already underlined the importance of the notion of avalanches for a disordered _bre bundle[ In the interface model\ basically the results of Hemmer and Hansen still hold[ The same statistics is expected in this case[ The only variant comes from the boundary conditions[ We have seen that an elastic coupling to the _bre bundle has the major e}ect of moving the interface displacement ðde_ned in eqn "3#Ł at which the avalanche size diverges[ In the interface case\ the elastic coupling is a little more complex\ and thus the point of divergence for avalanches requires some discussion[ Let us consider a block at generation n[ This block is subjected to an imposed displacement through a device of sti}ness k "n# [ The avalanches which are meaningful at this level are those which are constructed from the global forceÐdisplacement characteristic at generation n with a slope −k[ We would like to relate those avalanches to the one computed at the previous generation[ We have seen above how to relate the forceÐdisplacement relations from one generation to the next[ This provides a simple equivalent sti}ness of the loading device k "n−0# to be considered at the "n−0#th generation[ where the function H is shown in Fig[ 01[ Iterating the previous transformation allows to compute the elastic coupling to be considered directly at the interface level[ The function H has two _xed points\ k 9 and k k:1[ The _rst one is attractive\ whereas the second one is repulsive[ In order to better understand what is the practical measuring of the slope\ let us consider a system consisting in a few elements[ If we are looking for the _rst bifurcation point\ that is equivalent to the divergence of the avalanche sizes\ we have to consider avalanches with a sti}ness equal to k k:1 for just one bundle response[ Because we are far from the _xed points\ we use the relation eqn "21# to obtain this slope[ Figure 02 illustrates this point for an 00!generation system[ Therefore\ for most values of k n \ the equivalent sti}ness to be considered at the interface level k "0# tends to 9 as the system size tends to in_nity[ This means in practice that for most boundary conditions\ the avalanches should be analysed at the interface level with an elastic coupling which tends to 9\ i[e[ under a constant force condition[ This is precisely what has been shown in the previous analysis\ where we considered k "n# : \ i[e[ a constant displacement imposed on the entire elastic domain\ and we have retrieved that the _rst bifurcation occurred for a displacement at the interface level which approached the apex of the forceÐdisplacement curve "u 0:1#[ The slight delay in this displacement resulted from the last iterations of the function F[ Indeed\ for k : 9\ H"k# ¼ k:1\ and thus\ k "i−0# ¼ k "i# :1 for i ð n[ We observed that the _rst bifurcation in a generation n system occurred at points u "0# 0:1¦B1 −n where B is a constant\ and thus the tangent sti}ness du:dF"u u "0## B1 0−n is indeed a geometric series of ratio 0:1[ The existence of the unstable _xed point k k:1 can also easily be understood ] if we invert the relation eqn "21#\ we can relate the larger scale sti}ness to the lower one\ through the function inverse of H[ In this case the _xed points remain obviously identical but their attractive or repulsive character is turned to the opposite[ This means that the sti}ness of the entire system tends to k:1 as n increases to in_nity[ The k:1 is nothing but the sti}ness of the elastic body computed in the preceding section[ This shows that the conditions for bifurcation become independent of the global boundary conditions as the system size diverge[ It also underlines the fact that in order to observe the cascade of bifurcation\ one should use an active control on the loading conditions\ with the ability to decrease the loading fast compared to the typical time needed to fracture the _bres\ or to redistribute the load to the _bres[ This imposes some severe constraints on the monitoring of the experiment[ A possible way to build this control might be to use the acoustic emission during loading[ Let us note that the notion of avalanche allows to understand naturally the cascading process[ Indeed\ we have seen that for a subblock embedded inside the entire structure the e}ective sti}ness of the surrounding medium amounts to k "instead of k:1 if all other subblocks of the same generation are subjected to the same displacement#[ Therefore\ at the bifurcation point where the damage is localized in a subblock of generation i\ the two subblocks at generation i−0 are still stable\ i[e[ the maximum avalanche size in each of these two blocks is _nite[ Hence the damage will be shared between the two subblocks up to the next bifurcation point[ This argument also allows to estimate the validity of the cascade once the~uctuations due to the random nature of the _bre bundles are taken into account[ As the size of the active region\ l\ decreases\ the force~uctuation increases as l −0:1 \ and the proportion of broken _bres displays ã uctuation of order l −0:2 [ Comparisons of the level of~uctuations with the increment of force\ displacement or number of broken bonds\ show that the late stage of the process "l small enough# are dominated by the~uctuations\ but in contrast\ the early stage is well de_ned[ Therefore\ we anticipate that the _rst steps of the cascade may be correctly described by the above homogeneous situation\ whereas the more mature stage may be scrambled by the presence of disorder[ A representation of the location of the _bres that break under the loading gives a good physical idea of the cascade phenomena " Fig[ 03#[ The bifurcation cascade observed during the failure is not the usual idea of the failure of a joint ] such a failure generally occurs catastrophically\ and then the _rst idea is to think that it is due to a critical~aw[ It is important to note that our model follows the same catastrophic behaviour if we consider the global loadÐdisplacement response[ Hence\ Fig[ 04 shows the global response of the model for three di}erent generation systems[ As the generation increases\ the behaviour becomes more and more elastic brittle\ as expected for a joint failure[ But if we consider just the interface response\ we _nd e}ectively the previous behaviour with the bifurcation cascade[ Finally\ in spite of their di}erent description\ we see that the heterogeneous discrete system " for large enough sizes# and the homogeneous one have the same post!peak behaviour ] , The relation F vs u is similar until the apparition of the _rst crack[ , The onset of localisation appears at the same time " Fig[ 02#[ , The damage cascade is observed in both cases[

4[ Conclusion
For continuous models\ the localization is well!de_ned\ using some criteria based on the loss of uniqueness or the study of the tangent sti}ness operator[ For discrete models\ the localization could not be de_ned in such a manner[ The solution is always unique\ and no tangent could be calculated on the response because of the~uctuations that are superimposed[ In some well!known cases\ the loss of uniqueness coincides with the loss of stability\ where a bifurcation point is encountered[ In the _rst part\ we show that the study of avalanche statistics allows to detect this point[ Precisely\ the divergence of the avalanche sizes could be directly compared to the loss of stability in a continuous model[ After de_ning this equivalence\ we propose\ as an application\ to study a damage interface coupled with an elastic block[ For the sake of simplicity\ the interface is chosen to be thin\ then the damage propagates only in the interface direction[ We are interested particularly in the unstable path\ that is very di.cult to observe with continuous models[ The discrete model that we use is a hierarchical model\ that has a good representation of the Green in~uence function in an elastic continuum[ Our conclusions are the following ones ] We _rst propose to establish the recurrence relation between the sti}ness of a i!generation structure\ K i \ and a "i¦0#!generation one\ K i¦0 [ Per de_nition of the sti}ness\ the external force F is ] We search the expression of K i¦0 \ as a function of K i and k[ Using the hierarchical structure of the block\ we can write and thus\ By identifying with eqn "22#\ we obtain the recurrence relation ] The stable _xed point of this recurrence relation is Then\ choosing the value K 0 k:1 "i[e[ the sti}ness of the _rst springs in contact with the interface#\ the global sti}ness of the hierarchical structure is independent of the discretization level as expected[ We now give the sti}ness L i of a n!generation structure deprived from one subblock i\ if a force is applied at this level[ Again using the hierarchical structure of the block leads to a simple recurrence ] The stable _xed point is thus