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Edited by: José S. Andrade Jr., Federal University of Ceará, Brazil

Reviewed by: Andre P. Vieira, Universidade de São Paulo, Brazil; Lev Shchur, Landau Institute for Theoretical Physics, Russia

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We introduce two versions of a renormalization group scheme for the equal load sharing fiber bundle model. The renormalization group is based on formulating the fiber bundle model in the language of damage mechanics. A central concept is the work performed on the fiber bundle to produce a given damage. The renormalization group conserves this work. In the first version of the renormalization group, we take advantage of ordering the strength of the individual fibers. This procedure, which is the simpler one, cannot be generalized to other fiber bundle models such as the local load sharing one. The second renormalization group scheme based on the physical location of the individual fibers may be generalized to other fiber bundle models.

In an age where computer modeling of fracture and material breakdown is reaching a stage where the systems one may study span from the atomistic level to the continuum level in a single go [

Looking back in history, the study of equilibrium critical phenomena became “well understood” in the late seventies. Central to conquest of this field was the Ising spin model [

There are similarities between equilibrium phenomena and fracture, but also large differences. The similarities come from the development of long-range correlations as the fracture process proceeds in the same way as such correlations develop when approaching a critical point. On the other hand, whereas parameters need to be adjusted to approach criticality in equilibrium systems, in fracture the system approaches this state without the tuning of parameters. The correlations that develop during the fracture process stem from the way the stress field develops. However, they also reflect themselves in e.g., the spatial correlations in the post mortem fracture surfaces [

The fiber bundle model [

where κ is the spring constant. κ is the same for all fibers. Each fiber has a load threshold

when

It is the aim of this paper to construct a real-space renormalization group scheme [

Our goal is to construct a mapping from a fiber bundle containing

In order to construct the real space renormalization group, it is necessary to formulate the ELS fiber bundle model within

It is an important feature of the ELS fiber bundle model that it is infinite dimensional. That is, all fibers interact with all other fibers in exactly the same way. This is in contrast to e.g., the soft clamp fiber bundle model where the closer two fibers are, the more they interact.

Hence, when we in the renormalization group scheme to be presented choose to replace pairs of fibers by a single fiber by going from

However, if the renormalization group scheme we present is to have any bearing on the more complex fiber bundle models such as the LLS and the SC models where the relative position of the fibers

In section 4 we consider how the strength of the fiber bundle evolves under the renormalization group scheme.

The last section 5 contains a discussion of our results.

We will in this section formulate the equal load sharing model in a damage mechanics formulation based on energetic considerations. Damage mechanics is an approach to fracture in the continuum limit where the fractures are represented by a continuous damage parameter. Abaimov [

When the fiber bundle is loaded, the fibers fail according to their thresholds, the weaker before the stronger. We suppose that

where we have defined the

and used Equation (2). The damage parameter

A fundamental equation in what follows is the relation between damage

where the cumulative probability distribution corresponding to the threshold distribution

The cumulative probability gives the probability to find a threshold smaller than or equal to

We are here assuming the limit

The energy dissipated by the failed fibers is given by

The work performed on the system to reach the state (Δ,

The force conjugated to the load Δ is

which is identical Equation (3), as it must.

The damage driving force

and the equilibrium condition is

which when combined with Equation (11) gives

This equation is equivalent to Equation (5) when

We now turn to controlling the force

The corresponding work we find via the Legendre transform,

Combining this equation with Equations (9, 14) we find

We calculate the damage driving force

The equilibrium condition Equation(12) gives

Equation (18) when combined with Equation (5) gives

which is the force-load characteristics of the fiber bundle model. This equation is usually derived using order statistics. We see that the derivation using damage mechanics leads to the same result.

It is interesting to note that the equilibrium condition (Equation 12) can only be satisfied for

If

The renormalization group transformation that we are about to construct will consist of replacing the original fiber bundle containing

We now demand that the total work performed on the system, Equation (9), is kept constant by the renormalization group transformation. That is, we have

As we are here assuming _{N}. Equation (22) is central in what follows. As the number of individual fibers is reduced from

We have just stated that the energy (Equation 9) is to remain constant under the transformation. The energy consists of two parts,

In order for the elastic energy to be constant under the renormalization group transformation, we need to transform the elastic constant. The elastic energy will be constant if keep the load Δ fixed, i.e.,

and we set

so that

when

We keep the load Δ fixed during the transformation.

The energy dissipated by the failed fibers,

When

where we have ordered the thresholds, _{(1)} ≤ _{(2)} ≤ ⋯ ≤ _{(N−1)} ≤ _{(N)} and

These ordered thresholds are averaged over an ensemble. That is, we have

A fundamental result in order statistics is that the average frequency of the

in the limit when

In order to complete the renormalization group transformation, we need to define the threshold transformation

so that Equation (28) is fulfilled. There is no unique way to do this. We will in the following present two different transformations. The first one, which we call the

We write the sum in Equation (28) as

Hence, we define the

where as in Equation (29) the index

Equations (23, 24, 26, and 33) define the order space renormalization group transformation, fulfilling Equation (22).

Using Equation (30), we have

when

Combining this expression with renormalization group transformation Equation (33), gives

where _{(2j)} is the 2

We show in Figure ^{6} samples, each having ^{12} fibers, as we reiterate the order space renormalization group transformation. The ^{12} thresholds for the initial system were generated from a flat distribution on the unit interval. As we see from the figure, the distribution does not change as the renormalization group transformation is iterated. In fact, the formulation in Equations (32–36) are quite general in nature and the results should not depend on the type of thresold distribution. We have produced numerical results (Figure ^{α}, with α = 1. It is clear from Figure

Evolution of threshold distribution for ^{6} fiber bundles, each containing ^{12} fibers when repeating the order space renormalization group transformation. The initial threshold distribution was uniform on the unit interval. We have added a constant factor to

Evolution of threshold distribution for ^{4} fiber bundles, each containing ^{12} fibers when repeating the order space renormalization group transformation. The initial threshold distribution was non-uniform (linearly increasing) on the unit interval. We have added a constant factor to

The renormalization group transformation in terms of the parameters of the model, Equations (23), (24), and (33), defines the flow in parameters space,

We will now study the flow of the parameters (Δ, κ,

For the uniform distribution on the unit interval, i.e.,

We will in the following assume this threshold distribution for simplicity.

We assume that _{N}(_{N}(_{N}(_{N}(

which for the uniform distribution on the unit interval gives

For a given load Δ, we then have

giving rise to the flow diagram shown in Figure

Flow in parameter space under the renormalization group transformation (37) projected onto the (κ, ^{15}, and the data are averaged over 10^{6} samples. The renormalization group is iterated 15 times so that the last bundle contains one fiber. The threshold distribution was uniform on the unit interval.

We show in Figure

The force-load curve as we iterate the order-space renormalization group. The initial number of fibers was ^{15} and the uniform distribution on the unit interval was assumed. Averages were taken over 10^{6} samples.

Combined with Equation (40), this gives

and the corresponding peak stress is

This is illustrated in Figure

^{15} and the uniform distribution on the unit interval was assumed. Averages were taken over 10^{6} samples.

The order space renormalization group we have defined and explored in section 3.1 is tailored for the ELS fiber bundle model since the physical position of the fibers do not matter. Hence, for the renormalization group procedure to be generalizable to more complex models than the ELS fiber bundle model, the LLS fiber bundle model being an example, we group

We assume the fibers to be placed along a one-dimensional line. They are numbered from 1 to

We follow the same procedure as for the order space renormalization group except that the group together of pairs of fibers are now in real space rather than in order space. Fiber number _{i}. The renormalization group transformation of the thresholds then becomes

at the individual sample level. As in the order space renormalization group, the work performed on the fiber bundle is conserved, see Equation (22). For an

In contrast to the order space renormalization group, the threshold distribution is _{c}, _{c})]. In Figure _{c}. Since the work is conserved as for the order space renormalization group, we will have that

Evolution of the cumulative threshold probability under the real space renormalization group iteration is shown where lines represent the ^{15}. A uniform threshold distribution on the unit interval is assumed, so that the cumulative probability is ^{6} samples.

^{15}. Averages were taken over 10^{6} samples.

which gives _{N}(Δ_{N}(_{N}(_{c} is a symmetry point with half the thresholds smaller and half the thresholds larger than _{c}. Hence, we have _{c}) = 1/2. This is what is seen in Figure

We observe numerically that

where

with

We show in Figure

Evolution of the force-load curve under the real space renormalization group. The starting point was 10^{6} fiber bundles each containing ^{15} fibers. The threshold distribution was uniform on the unit interval. The dotted curve Δ(1−Δ) is the initial force-load curve.

Evolution of the peak-stress and peak-strain under the real space renormalization group. The starting point was 10^{6} fiber bundles each containing ^{15} fibers. The threshold distribution was uniform on the unit interval.

The damage parameter _{c} and _{c} is unstable. For the uniform threshold distribution on the unit interval, we have _{c} = 1/2.

The flow in (Δ, κ, ^{15} having uniform fiber strength distribution on the unit interval. Averages were taken over 10^{3} samples.

An important aspect of the renormalization group is how fluctuations are handled. Here we consider the avalanche distribution [

Evolution of the histogram of avalanche sizes as the real space renormalization group is iterated. The initial threshold distribution was uniform on the unit interval. The initial number of fibers was ^{14} and averages were taken over 10^{5} samples.

We plot in Figure ^{5} fibers (we have started with a bundle of 2^{14} fibers).

Evolution of the largest avalanche occurring before complete failure as the real space renormalization group is iterated. The initial threshold distribution was uniform on the unit interval. The initial number of fibers was ^{14} and averages were taken over 10^{5} samples.

Now we are going to compare the

where α≥0. When α = 0, we have the uniform distribution. We will study the initial and final fiber bundle strength at imposed load Δ under the order or real space renormalization scheme. The force on a bundle at load Δ is

By solving

Since the work performed on the fiber bundle is conserved by the renormalization group, at the final renormalization step (when

which gives the final strength of the bundle (with

Figure

Initial and final strengths of the fiber bundle vs. power law index (α) under both the order space and the real space renormalization group schemes. For α = 0, the distribution is reduced to the uniform distribution and the initial and final strength values match well with our numerical results.

We see that the final strength of a bundle has gone up as a result of the renormalization group scheme ! Our renormalization group transform consists of replacing pair of fibers in the original bundle by single fibers. To keep the energy constant under the trasformation, the strength of the new fiber is chosen according to Equation (33), where the final strength of single fiber is always greater than the average strength of the two initial fibers - and this is the origin of the strength enhancement in the final renormalized strength of the bundle.

We have in this paper introduced a renormalization group for the equal load sharing fiber bundle model based on formulating the fiber bundle in the context of damage mechanics. The idea behind the damage mechanics formulation is to introduce a continuous damage variable so that the binary nature of the single fibers is no longer in focus. In this way, we are able to group together the fibers belonging to a given fiber bundle into smaller fiber bundles and map the parameters of the larger fiber bundle onto the smaller bundles. A central concept in this mapping is the conservation of the work applied to the fiber bundle to create a certain level of damage, Equation (9). This work is kept invariant under the renormalization group procedure.

We have presented two versions of the renormalization group. In the

The problem with the order space renormalization group scheme is that there is no obvious way to generalize it to other fiber bundle models such as the local load sharing model. Rather than grouping together the fibers according to their strength, we may group them together according to their locations, hence defining the

In order to formulate a real space renormalization group scheme for fiber bundle models with non-trivial stress redistribution, such as the

We have in this paper only presented the renormalization group schemes themselves together with a number of their properties. We have not attempted to implement the renormalization group on more complex fiber bundle models. We do see a strong potential in the use of the renormalization group as a tool to investigate the fiber bundle models, in particular in connection with fluctuations (see Figure

SP: Numerical Simulations, data analysis, plotting and some theory; AH: Theory; PR: Theory. Equal contributions in discussions and writing the article.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank Martin Hendrick, Jonas T. Kjellstadli and Laurent Ponson for interesting discussions. This work was partly supported by the Research Council of Norway through its Centers of Excellence funding scheme, project number 262644.