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Failure thresholds in hierarchical and euclidian space by real space renormalization group
Didier Sornette
To cite this version:
Didier Sornette. Failure thresholds in hierarchical and euclidian space by real space renormalization group. Journal de Physique, 1989, 50 (7), pp.745-755. �10.1051/jphys:01989005007074500�. �jpa- 00210952�
Failure thresholds in hierarchical and euclidian space by real
space renormalization group
Didier Sornette
Laboratoire de Physique de la Matière Condensée, CNRS URA 190, Faculté des Sciences, Parc Valrose, 06034 Nice Cedex, France
(Reçu le 30 septembre 1988, révisé le 30 novembre 1988, accepté le 5 décembre 1988)
Résumé. 2014 On présente une analyse du groupe de renormalisation (GR) dans l’espace réel pour les lois d’échelle de la force de rupture en fonction de la taille L et du désordre, dans des espaces
hiérarchiques et euclidiens. Dans les modèles hiérarchiques pour lesquels le GR est exact, on
trouve deux classes d’universalité : aux faibles désordres, le système est « résistant » à la rupture
et sa force croît comme une puissance de sa taille, tandis qu’aux forts désordres, le système a une
force finie qui ne croît pas avec sa taille. Les exposants trouvés ne sont pas universels et
dépendent du détail de la construction hiérarchique et du désordre. Un GR approximé est aussi proposé pour les systèmes euclidiens en dimensions deux et trois. On trouve que le système est toujours resistant dans le sens que sa force à la rupture croît comme une puissance de sa taille avec
un exposant dépendant faiblement du désordre et donc non universel.
Abstract. 2014 The scaling of the failure strength as a function of the size L and the influence of disorder in hierarchical and Euclidian systems are discussed within a real space renormalization group (RG). In hierarchical models for which the RG is exact, two universality classes are found :
for « small disorder », the system is strong and its strength increases as a power of its size whereas for « large disorder », the system has a finite strength which does not increase as L increases.
However, the exponents are not universal and depend upon the detailed hierarchical construction and on disorder. An approximate RSRG is also proposed to treat the case of two and three
dimensional Euclidian systems. We find that the system is always strong in the sense that its strength increases as a powerlaw of its size, with an exponent which is weakly dependent on
disorder and is thus not strictly universal.
Classification
Physics Abstracts
05.50 - 05.70J - 62.20M
1. Introduction.
Rupture phenomena constitute an extreme case of the general problem of transport properties in random media. Many experimental, numerical and theoretical studies are
presently being carried out to unravel their properties [1-6]. A few questions among others
one would like to understand in this field are the following.
1) What is the strength of a system and how does this strength evolve for large systems ? 2) How does the strength of a system depend on disorder ?
3) Are there universal behaviours in the breaking characteristics of a disordered system which are independent of the details of the models ? What are the corresponding universality
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005007074500
classes ? Can we define a renormalization group governing the evolution with scale of the distribution of rupture strength ?
4) What is the relative importance of the initial quenched disorder and the « growth » aspect of the failure ?
Very few results are available at present concerning these different questions. Within some limits, however, exact results are available. In one-dimensional systems (1 D), i.e. in models of links associated in series with randomly distributed breakdown thresholds, failure is then
associated with the statistics of extremes [7-9] : the weakest part of a system submitted to a given load fails first and this corresponds to the macroscopic failure. In general, the strength
of such a system decreases with its size (1) [4].
In the other limit where L links with random rupture thresholds are associated in parallel, a
central limit theorem holds [9] and the system strength increases as the size of the system S - L [5].
Most physical or mechanical systems are more complex and one must consider models where several links share the total load in a more complicated way. Consider for example an
Euclidian lattice of L d bonds with randomly distributed failure thresholds and assume that the stress is applied along the z-direction. In this case, both parallel and series associations are
involved in a complex mixed way. Determining the global rupture properties is an extremely
difficult task since it involves non-local long-range screening and enhancement effects as well
as connectivity effects which cannot be described easily by perturbative or probabilistic approaches.
In the Euclidian lattice case, a simplification also occurs in the « infinite disordered limit »
(see Ref. [3] for a precise definition of what this means) where the failure is completely
controlled by disorder. Within this limit, failure becomes a purely geometrical problem
connected to percolation. In real systems, characterized by a finite disorder, the initial failure
behaviour ressembles that of the infinite disordered model but there is a transition to a regime
where the stress enhancement on large cracks largely controls the failure. The « infinite disorder » limit is typical for the beginning of a rupture process (local microcracks), whereas
the small disorder limit is reached at the end of the rupture sequence with the development of
a single macrocrack.
In this paper, we present new exact results on the failure properties in hierarchical models.
These systems can be considered as the limit of Euclidian lattices of dimension d -+ + 00. An
interesting phase diagram is found as a function of disorder. For small disorder
(mu 2), the system is strong and its strength increases as a power of its size whereas for large
disorder (m 2), the system has a finite strength which does not increase as L increases. This
corresponds to the existence of a non-trivial unstable fixed point of the real space renormalization group (RSRG) which is exact for hierarchical lattices.
An approximate RSRG is also proposed to treat the case of two- and three-dimensional Euclidian systems. We find that the system is strong in the sense that its strength increases as a powerlaw of its size, with an exponent which is weakly dependent on disorder. This is
qualitatively in agreement with recent numerical results [1].
2. Failure of hierarchical lattices.
Let us consider the diamond lattice constructed by repeated iteration shown in figure la. One
can also consider other variants shown in figures 1b and lc. The failure properties of this
(1) for example as a powerlaw S - L -l/m for failure thresholds of single links distributed with the Weibull distribution p (x ) = 1- F (x ) = exp { - (xl À /" } of order m, where p (x ) is the probability that rupture in a link occurs at a stress value larger than x.
Fig. 1. - Basic step of the construction of a hierarchical lattice. Models a) and b) do not differ in their
global statistical failure properties.
system can be solved recursively since at each iteration, a link at the n-th generation is replaced by a system involving association in series and in parallel of a finite number b of bonds of the n + 1-th generation (b = 4 for case la and 1b and b = 9 for the case lc of Fig. 1).
Suppose we have stopped the iterative construction of the hierarchical lattice of figure la at
the N-th generation. The lattice contains therefore L = 2N links in parallel starting from the
upper and lower nodes. Suppose a stress S is applied at the upper and lower nodes and note
s = S/2N the stress on each link. Due to their hierarchical association, we can solve for the failure probability distribution for one element supporting the stress s’ (= 2 s ) which is made
of b (= 4 ) bonds (for the case of Figs. 2a and b). Then, the process can be repeated since at
the next level of the hierarchical lattice, the link at the n-th generation becomes one of the b
bonds of a link at the n-1-th generation. After n = N successive iterations, we obtain a very
simple system made of a single link whose strength is obtained from N iterations of a recursion relation (1) which is written below. The strength probability distribution of this single
« renormalized » link is exactly that of the hierarchical lattice made of 2N links.
Let us denote pn(s) the probability that the link does not break under s, i.e. that rupture in
a link at the n-th generation occurs for a stress larger than s. Then, we have the exact
recursion relation, for the system of figure la,
The first term of the r.h.s. of equation (1) is the probability that all four bonds do not break under s/2 and the second term is the probability that two bonds along a line hold under s while one of the bonds of the other line has failed under s/2. This recursion relation (1) is a mapping in the space of the probability distribution functions p (s ). From a given function p (s ), equation (1) gives a new function p’ (s ). In principle, the space in which the renormalization flow occurs is a space of functions and has therefore infinite dimensionality of
order of infinity N 2 (Aleph 2), instead of being a space of coupling parameters of finite
dimensionality as is the case in usual critical phenomena. Such a situation is rather unusual but has also been found to occur for example in functional renormalization flow of surface
phase transition [10]. Starting from a given p (s ) as shown in figure 2, the iteration of equation
. (2) leads to a sequence of renormalized p’ (s ). Practically, this procedure can only be implemented numerically. Figure 2 shows the two possibilities for the renormalization of
p(s) :
i) p (s ) is renormalized to the attractive fixed point pl (s ) = 1 shown in figure 2a ; ii) p (s ) is attracted to a non-trivial fixed point p * (s ) of the form of a step-function shown in figure 2b. The arguments which justify these results are the following.
Consider first the limit of a small load s. The flow for s - 0 i.e. p (s) - 1, i.e. near the fixed point pl (s ) = 1, is controlled by the derivative ap’ lap of the flow equation (1) :
Fig. 2. - Schematic representation of the renormalization flow of the failure probability distribution.
Case a) p (s ) is uniformly renormalized to the attracting fixed point pl (s ) = 1. This behaviour is obtained in the « weak disorder » regime (m -- 2 for model la of Fig. 1). This case corresponds to p(s) : exp {- (s/ h )2 } . Case b) p (s ) is renormalized to the non-trivial fixed point p * (s ). This
behaviour is obtained in the « strong disorder » regime (m 2 for model la of Fig. 1). This case corresponds to p (s ) > exp (SIX )2} .
We observe that 8p’18p 0+ as s --+ 0. This means that we are always in a situation where p (s ) = 1 is attracting for s sufficiently small. Now, the difference appears from the manner in which the tail of p (s) at large s is renormalized.
Consider therefore the behaviour of the flow at large s, i.e. for the tail of the distribution
p (s). From equation (1), pn -1(s)/pn(s) is controlled by the first term of equation (1) i.e.
{ p n (s/2 ) } 4/p n (s ) for small pn(s) i. e . large s. If the tail of pn(s) is a powerlaw p’(s) - s-x with x > 0, then p"- 1(s)lp"(s) - (161s’)x is less than one for x 0. This means
that the tail of the renormalized p (s ) will converge to zero. A more interesting class of
function is the Weibull distribution pn(s) -- exp {- (s/ A )m } which describes empirically a variety of real materials and also satisfies interesting self-consistent conditions [9]. We use its
from for large s. Then, equation (1) yields
distinct behaviours occur.
1) The tail of p(s) decreases faster than exp {- (si À )2} .
Then, by iteration of equation (3), p (s ) increases and eventually reaches for n - + oo the attracting fixed point pl (s ) = 1 for s = 0, + oo. Since the portion of p (s ) in the neighbourhood
of s = 0 also converges to 1, p (s ) converges as a whole to the fixed point pl (s ) = 1. Note that the function po (s ) = 0 for s = 0, ..., + oo is also a fixed point but it is unstable for functions whose tail decreases faster than exp{- (si À )2} . The signification of this result is the
following. Suppose a large stress S is applied to the system corresponding to a very low but
non zero value of p (s ). As one applies the recursion law (3), p n(s ) increases and after an
infinite number of iterations (if N is infinite) it is attracted to the fixed point pl (s ) = 1. This reasoning can go further and allows one to obtain the scaling of the strength of a system of finite size L = 2N, , for the class of functions whose behaviour at large s is
Since the functional form (4) is preserved at large s under the RSRG (1) which gives
the number of iterations n necessary to escape from the neighborhood of 0, starting from a
very small value p (s ) to reach the region where p = 1 is such that
This yields 2n (- - 2) _ Log {[P (s) ]-1}. Using the form of the conserved Weibull distribution,
this gives the typical strength of the system S/A - 2N(m-2)/m. Noting L = 2N as the number of branches, we obtain
with e =1- 2/m. The strength of the system increases as a power of L. Note that within the limit where m --+ + 00, we recover £ - 1, since this corresponds to the limit of vanishing
disorder [11]. This behaviour is somewhat similar to that obtained for an association of L links in parallel where each link is itself made of L2 bonds associated in series [4, 5].
In addition to the powerlaw behaviour (6), we can also obtain the full scaling of the stress-
deformation characteristics (S, £L ). Iterating equation (5) and with the scaling (6), we obtain
the asymptotic form of the cumulative failure probability distribution
pL (S ) is the probability that the system of size L does not break under S. Noting
S = Ls, the stress which applies on each remaining bond under S is just
LSILPL (S) = SIPL (S). The global strain EL is thus L times this effective stress acting on each remaining bond. This yields
which is easily inverted and gives the leading behaviour of the stress-strain characteristics :
We thus obtain a scaling law of the form
which is similar to the numerical results of reference [1].
Furthermore, the number n of broken bonds under a given stress S is L (1 - pL (S)) in the
limit of large L. For S/AL ’ « 1, using equation (7), L(l - PL(S» takes the simple form
which is of the form
as suggested by reference [1]. For hierarchical systems, the exponent y takes its mean field
value y = 1.
2) Now consider the other case where p (s) decreases more slowly than exp {- (s / A )2} at
large s. In this case, we have seen that for large s, p (s ) is attracted by the fixed point po (s ) = 0. On the other hand, we have seen that, by equation (2), p(s) - 1 for
s -+ O. By continuity, we expect the existence of a new unstable fixed point 0 p * 1 such
that for infinite recursions,
This expectation - is born out by considering, for example, the subspace of Weibull
distributions. The typical form of p’ (p ) is depicted in figure 3. The limiting probability failure
distribution is shown in figure 2b. In this case, the strength of the system does not increase as L = 2N increases but tends to a finite value s *, such that p (s * ) = p *, of the order of the
strength À of each individual link.
In the vicinity of the fixed point p *, we have
with y = 3p’/9p( *. Starting in the vicinity of p * with p - p * o, after n iterations, p’ will escape towards 1. The value n = Log (p - p * )-1/Log y gives the number of iterations necessary to escape to 1. This corresponds to a system size e = 2" - (p - p * )- v with
v = Log 2/Log y. The correlation length § is the typical size of the system which does not fail under a stress s corresponding to a given value p (s ) in the neighborhood of p *. It diverges as p - p * with a critical exponent vwhich depends on the class of distribution failure thresholds.
One can also interpret e as the number of broken links in the same bundle of an infinite hierarchical system.
3) Let us now analyse the intermediate case m = 2 where p (s ) behaves asymptotically as
In this case, ap’ lap - 1 + as p - 0. This means that po (s ) = 0 is a repulsive fixed point and
that p (s ) will converge to pl (s ) = 1 for all s. But the speed of convergence will be very slow.
As shown in figure 3, this is due to the fact that the flow curve is tangent to the main bissectrix in the vicinity of p = 0. Starting from a given small p, the flow equation gives p’ - p == 2 p Z. Considering n iterations, we can go to the continuous limit for large n and
obtain the equation governing the escape of p towards 1 :
The solution of equation (16) is N - (2 p )-1. With the form (15), this gives a strength
since L = 2N. This is a very slow increase of the strength compared to the power law given by equation (6).
Figure 4 summarizes all the results obtained so far on a « phase diagram » (p, m ). For a
failure distribution with Weibull order m 2, there exist three fixed points 0 and 1, which are stable, and p * (m ) which is unstable. The line of unstable fixed point p * (m ) goes to zero at
m = 2 with an essential singularity as
For failure distribution whose order m is larger than 2, the 0 fixed point which has merged
with the unstable fixed point p * becomes unstable.
Note that the special value m = 2 of the Weibull parameter depends on the precise
Fig. 3. - Schematic representation of the mapping p’ (p ) given by equation (1) showing the different
regimes. Case a) for the strong disorder regime (m 2 i.e. p(s) > exp {- (si À )2} ), case b) for the
weak disorder regime (m > 2 i.e. p (s) exp {- (s/À )2} ) and case c) in the intermediate case
(m = 2 i. e. p (s) - exp (slk
Fig. 4. - Phase diagram (p, m ) summarizing the different regimes as a function of the disorder which is
parametrized by the Weibull parameter m. For a hierarchical system with a Weibull distribution of order
m of bond strengths, a given total applied stress S corresponds to a given initial value of p, the
probability that the system holds under S. Then, for this m and this p, which determine a starting point
in the (p ; m ) diagram, the RSRG corresponds to a flow in the direction of the arrows. Thus, for
m > 2, any finite applied stress corresponding to a non-zero p renormalizes into p =1 in the limit of
large systems. The system is therefore strong since its strength increases with its size, as discussed in the
text. On the contrary, for m 2, two regimes are possible depending on the initial value of p and
therefore on the total applied stress S. If the initial p is above the continuous line, we recover the previous behaviour. If the initial p is below the continuous curve, the RSRG flow attracts the point
towards p = 0 meaning that the system has a finite strength which does not increases with the size of the system.
construction of the hierarchical model. As an illustration, if we consider the model of figure lc, the leading behaviour of the renormalization flow for large s reads
which replaces the first term of the r.h.s of equation (1). This leads to
C = 1 - (Log 6/Log 3 ) m-1 for m > Log 6/Log 3 1.63. This critical value for m in this
model lc) is different from the value m = 2 obtained with the model la). The difference is
easily traced back to the different ramification in the two models.
Smalley et al. [14] have previously studied a similar problem. However, they focus on a very
peculiar hierarchical model of failure, a « fractal tree », with the additional built-in hypothesis
that a load supported by a given broken bond is redistributed locally. Furthermore, they do
not study the full renormalization of the failure distribution since they restrain their functional space to Weibull distribution, a hypothesis which is not in general justified under these
conditions. They only find the (m 2) type of behaviour. Note that the hierarchical model studied here does not suffer from these restricted assumptions.
3. RSRG of failure in euclidien lattices.
The real space renormalization group is known to be non exact in this case. Its main difficulty
is that it is not in general possible to control the accuracy of the approximations [12-13].
However, it has the interesting feature of allowing the computation of physical quantities
even far from a critical point [12]. The RSRG applied to hierarchical spaces in section 2, has instructed us that it is possible to determine exactly the global strength of a system under
certain conditions even if the failure should be considered in general a highly non-linear growth process. Another example is also provided by systems made of links associated in
parallel [5]. Thus, our justification for using RSRG techniques is that we are considering the global failure threshold and not the « growth » aspects of the failure. As discussed in section 1, failure usually exhibits two regimes. The initial failure regime is essentially
controlled by disorder. This regime eventually gives place to a « growth » regime controlled by interactions, screening and enhancement effects. We propose that the global scaling of the strength of a system can be found from the analysis of the first regime controlled by disorder
which is therefore amenable to RSRG techniques. In other words, the scaling of the global
failure threshold should be controlled by the end of the « infinite disordered » regime. Of
course, we cannot tell anything about the second regime and its spatial rupture pattern.
Let us consider the scalar problem on a square lattice tilted with an angle Tr/4 with respect
to the horizontal so that all bonds have equal current i in the undeteriorated lattice. Consider the transformation which changes from figures 5a to 5b, by some operation of « decimation » of the links within the large square. As we have learned in section 2, it is the behaviour of the
flow at small p which determines the scaling of the failure strength. We are thus interested in the limit of small p (i ) i.e. large stress i.
We could follow the procedure developed in the previous discussion of hierarchical models and relate the probability that the partial lattice of figure 5a sustains a current i per bond without global breaking, which is p (i )12 + higher order corrections, to the probability that
the lattice of figure 5b does not break, which is p’ (i’ )4 with i’ = 2 i. Equating the two expressions, one obtains the following approximate flow equation
Using the same arguments as for the hierarchical lattices in section 2, this yields a total strength scaling as equation (6), where L is the linear lattice size and e = 1 - (Log 3 /Log 2 )/m. The magnitude of (is significantly less than that determined
numerically in [1] : for example, for m = 2, 03B6 = 0.21 ; m = 3, C = 0.47 ; m = 4, e = 0.60 ;
m = 5, e = 0.68. It is possible to improve the transformation slightly by going to arbitrary b
