^{*}

Edited by: Ferenc Kun, University of Debrecen, Hungary

Reviewed by: Mikko Alava, Aalto University, Finland; Bikas K. Chakrabarti, Saha Institute of Nuclear Physics (SINP), India; Takahiro Hatano, The University of Tokyo, Japan

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 consider the Equal-Load-Sharing Fiber Bundle Model as a model for composite materials under stress and derive elastic energy and damage energy as a function of strain. With gradual increase of stress (or strain) the bundle approaches a catastrophic failure point where the elastic energy is always larger than the damage energy. We observe that elastic energy has a maximum that appears after the catastrophic failure point is passed, i.e., in the unstable phase of the system. However, the slope of elastic energy vs. strain curve has a maximum which always appears before the catastrophic failure point and therefore this can be used as a reliable signal of upcoming catastrophic failure. We study this behavior analytically for power-law type and Weibull type distributions of fiber thresholds and compare the results with numerical simulations on a single bundle with large number of fibers.

Accurate prediction of upcoming catastrophic failure events has important and far-reaching consequences. It is a central problem in material science in connection with the durability of composite materials under external stress [

The central question is—when does the catastrophic failure occur? Is there any prior signature that can tell us whether catastrophic failure is imminent? The inherent heterogeneities of the systems and the stress redistribution mechanisms (inhomogeneous in most cases) make things complicated and a concrete theory of the prediction schemes, even in model systems, is still lacking.

In this article, we address this problem (prediction of catastrophic events) in the Fiber Bundle Model (FBM) which has been used as a standard model [

We organize our article as follows: After the brief introduction (section 1), we define the elastic energy and the damage energy in the Fiber Bundle Model in section 2. In sections 3 and 4 we calculate the elastic and damage energies of the model in terms of strain or extension. In several subsections of sections 3 and 4 we explore the theoretical calculations for power-law type and Weibull type distribution of fiber thresholds. Simulation results are presented and numerical results are compared with the theoretical estimates in these sections. We present a general analysis of elastic energy variations and existence of an elastic-energy maximum in section 5. In section 6 we identify the warning sign of catatrophic failure by locating the inflection point. Here, in addition to uniform and Weibull distributions, we choose a mixed threshold distribution and present the numerical results, based on Monte Carlo simulation, to confirm the universality of the behavior in the ELS models. Finally, we keep some discussions in section 7.

The fiber bundle model consists of

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

The fiber bundle model.

The fiber thresholds are drawn from a probability density

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

where we have defined the

When

We will now assume that the stretch Δ is our control parameter. We can construct the energy budget according to continuous damage mechanics [

The damage energy of the failed fibers is given by:

The total energy at stretch Δ and damage

We are going to analyze the energy relations when the bundle is in equilibrium. We know that there is a certain value, Δ = Δ_{c}, beyond which catastrophic failure occurs and the system collapses completely. We are particularly interested in what happens at the failure point. Is there a universal relation between elastic energy and damage energy at the failure point?

When

and

The force on the bundle at a stretch Δ can be written as:

The force must have a maximum at the failure point Δ_{c}, therefore setting

We start with the simplest threshold distribution: the uniform distribution, which is well-known in fiber bundle research [

Now putting Δ_{c} = 1/2 in Equations (8, 9), we get:

and

Therefore, the ratio between damage energy and elastic energy at the failure point (Δ_{c}) is:

Now we move to a general power law type fiber threshold distributions within the range (0, 1),

The cumulative distribution takes the form:

We insert the expressions for

We can calculate the elastic energy and damage energy at the failure point Δ_{c}:

and

Plugging in the value of Δ_{c} (Equation 18) into the above equations for elastic energy and damage energy we end up with the following relation:

Clearly, the ratio depends on the power factor α (

Ratio between damage energy and elastic energy at the failure point Δ_{c} vs. power law exponent α. Single bundle with ^{7} fibers. Dashed line is the theoretical estimate (Equation 21).

It is easy to show that the work done on the system up to the failure point Δ_{c} is equal to the sum of the energies ^{e} and ^{d}. The total work done on the system can be calculated as:

Inserting the expression for

which is the total of elastic energy and damage energy, ^{e} + ^{d} (see Equations 19, 20). In fact, the energy conservation here is analogous to the one in thermodynamics.

It is known that when the extension exceeds the critical value Δ_{c}, the whole bundle collapses via a single avalanche called the final or catastrophic avalanche [_{c} and the upper cutoff level of the fiber thresholds for the distribution in question.

We calculate the damage energy of the final avalanche for power-law type distributions as

It is important to find out whether the damage energy for the catastrophic avalanche has a universal relation with the elastic or damage energies at the failure point. As already mentioned, the bundle has stable (equilibrium) states up to Δ ≤ Δ_{c}. Therefore, if we correlate the final avalanche energy with ^{e} or ^{d} values at Δ_{c}, we can predict the catastrophic power of the final avalanche.

Comparing the expressions for

As

We can get the last relation (Equation 26) by comparing expressions (Equations 24 and 19) directly. These theoretical estimates are compared with numerical simulation results in

Ratio between damage energy of final catastrophic avalanche and damage energy at the failure point Δ_{c} vs. power law exponent α. Single bundle with ^{7} fibers. Solid line is the theoretical estimate (Equation 25).

Ratio between damage energy of final catastrophic avalanche and elastic energy at the failure point Δ_{c} vs. power law exponent α. Single bundle with ^{7} fibers. The dashed line is the theoretical estimate (Equation 26).

Now, if we put α = 0, we get these energy relations for uniform fiber threshold distribution:

And

That means the energy release during final catastrophic avalanche is much bigger than the the elastic energy stored in the system just before failure (final stable state when Δ = Δ_{c}).

It is commonly believed that during catastrophic events like earthquakes, landslides, dam collapses etc., the accumulated elastic energy releases through avalanches [

We now consider the Weibull distribution, which has been used widely in material science [

where

As the force has a maximum at the failure point Δ_{c}, inserting

From the above equation we can easily calculate the critical extension value as:

The elastic energy at the critical extension Δ_{c} is:

and the damage energy is:

Putting,

we get:

This integral is exactly calculable for

For Weibull index _{c} = 1 and the damage energy expression at the failure point takes the form:

Using integration by parts we arrive at the result:

We get the elastic energy at the failure point directly by putting

Therefore, the ratio between damage and elastic energies at the failure point for Weibull distribution with

In

Damage energy and elastic energy vs. extension Δ (up to the failure point Δ_{c}) for Weibull distribution of thresholds with Weibull index ^{7} fibers.

Ratio between damage energy and elastic energy vs. extension Δ (up to the failure point Δ_{c}) for a fiber bundle with Weibull distribution of thresholds. In simulation, we used a single bundle with ^{7} fibers. Circle and triangle are the theoretical estimates (Equations 40, 44) for the ratios at the failure point Δ_{c}.

For Weibull index

Again, using integration by parts we arrive at the result:

We get the elastic energy at the failure point directly by putting

Therefore, the ratio between damage and elastic energies at the failure point for Weibull distribution with

The theoretical estimates of the ratio between damage and elastic energies at the failure point is compared with numerical results in

There are two distinct phases of the system: A stable phase for 0 < Δ ≤ Δ_{c} and an unstable phase for Δ > Δ_{c}. If we plot the elastic energy and damage energy vs. Δ, we see that damage energy always increases with Δ but elastic energy has a maximum at a particular value of Δ, let us call it Δ_{m}. Can we calculate the exact value of Δ_{m} for a given threshold distribution? Is it somehow connected to Δ_{c}? In this section we are going to answer these questions.

We recall the elastic energy expression (Equation 8). If we differentiate the elastic energy with respect to the extension Δ, we get:

Which is 0 at Δ_{m}, with:

If we consider a general power law type distribution ^{α}, within (0, 1), we can write:

For Weibull distribution ^{k}), we can write:

Therefore we can conclude that Δ_{m} is bigger than Δ_{c}, i.e., elastic energy shows a maximum in the unstable phase (_{m} and Δ_{c} is given in the

Force and elastic energies vs. extension Δ for fiber bundles with uniform distribution of thresholds: Comparison with simulation data for a single bundle with ^{7} fibers.

Force and elastic energies vs. extension Δ for fiber bundles with Weibull distribution of thresholds: Comparison with simulation data for a single bundle with ^{7} fibers.

Are there any prior indications of the catastrophic failure (complete failure) of a bundle under stress? In the fiber bundle model, although the elastic energy has a maximum, it appears after the critical extension value, i.e., in the unstable phase of the system. Therefore it can not help us to predict the catastrophic failure point of the system.

However, if we plot ^{e}/^{e}/_{c} (^{e}/_{max}. We will also see whether there is a relation between Δ_{max} and Δ_{c}.

We recall the expression for the derivative of elastic energy with respect to strain of extension (Equation 45). Taking derivative of the equation, we get:

Setting ^{2}^{e}(Δ)/^{2} = 0 at Δ = Δ_{max} we get for a general power law type distribution:

This expression confirms that Δ_{max} < Δ_{c} for α ≥ 0. For a Weibull distribution with index

The solution (of ^{2}^{e}(Δ)/^{2} = 0) with (−) sign is the acceptable solution for the maximum. Hence,

since,

From Equations (50) and (52) we see that the relation between Δ_{max} and Δ_{c} depends on the threshold distributions and we can express Δ_{c} in terms of Δ_{max} with a prefactor as:

for power-law type distributions and

for Weibull distributions. Now can we find a reasonable approximation for the prefactor that is useful for more than one threshold distributions? An intuitive first choice is the result for the uniform distribution with α = 0 gives Δ_{c} = 1.5Δ_{max}. This is a good approximation for α close to 0, as expected, but not for very large α values. This prediction is also exact for a Weibull distribution with _{max} is smaller than the true failure point Δ_{c}. In this case the estimate errs on the side of caution, and the bundle can withstand more than the estimate predicts. A better choice would be to set a prediction-window (1.2 to 1.5) for the prefactor G(α), H(

The prefactor (Equations 54, 55) vs. α,

A more general argument is given in _{max} and Δ_{c}.

In ^{7}) of fibers and the agreement is convincing. We have used Monte Carlo technique to generate uncorrelated fiber thresholds that follow a particular statistical distributions (uniform and Weibull distributions). It is obvious that in simulations we can measure energy values in the stable phase only.

Now we choose a mixed fiber threshold distribution. Can we see similar signature (maximum of ^{e}/_{c}) as we have seen in previous section? The chosen distribution is a mixture of uniform distribution and Weibull distribution (^{e}/_{c} and Δ_{max} is somewhere in between the respective Δ_{max} values for uniform and Weibull threshold distributions—as expected intuitively. If we express the Δ_{c} value in terms of Δ_{max}, the prefactor is well inside the prediction-window, shown in

A mixture of uniform and Weibull (

Force and elastic energies vs. extension Δ for a fiber bundle (10^{7} fibers) with a mixed distribution of thresholds.

The Fiber Bundle Model has been used as a standard model for studying stress-induced fracturing in composite materials. In the Equal-Load-Sharing version of the model, all intact fibers share the load equally. In this work we have chosen the ELS models and we have studied the energy budget of the model for the entire failure process, starting from intact bundle up to the catastrophic failure point where the bundle collapses completely. Following the standard definition of elastic and damage energies from continuous damage mechanics framework, we have calculated the energy relations at the failure points for different types of fiber threshold distributions (power law type and Weibull type). At the critical or catastrophic failure point, the elastic energy is always larger than the total damage energy. Another important observation is that the elastic energy variation has a distinct peak before the catastrophic failure point. Also, the energy-release during final catastrophic event is much bigger than the elastic energy stored in the system at the failure point (see section 3 and section 4). Our simulation results on a single bundle with large numbers (10^{7}) of fibers, show perfect agreement with the theoretical estimates. We have chosen a

These observations can form the basis of a prediction scheme by finding the correlation between the position (strain or stretch level) of elastic energy variation peak and the actual failure point. The main concern is to find a direct relation between the elastic energy inflection point (Δ_{max}) and the failure point (Δ_{c}). We found that Δ_{c} and Δ_{max} are related through a prefactor that depends on the exponent of the threshold distribution. Then we tried to find a window of the prefactor (Equations 54 and 55) that can cover a wide range of threshold distributions (_{c} = prefactor · Δ_{max} and the approximate prefactor window is 1.2 to 1.5 (

Our observations in this work have already opened up some scientific questions and challenges: what happens for Local-Load-Sharing (LLS) models [

During rock-fracturing experiments [

Our next article will resolve some of these issues –we are now working on energy budget of LLS models.

The datasets generated in this study are available on request to the corresponding author.

SP: theory, simulation, data analysis, and plotting. JK: theory (Appendix part). AH: 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.

This work was partly supported by the Research Council of Norway through its Centers of Excellence funding scheme, project number 262644.

As stated in section 2, the elastic energy in the system at extension Δ is:

where

The critical extension Δ_{c} where the bundle collapses is hence given by:

Numerical data seems to suggest that ^{e} > ^{d} at the critical point for most threshold distributions. Let us try to prove this analytically by investigating the difference between elastic and damage energy:

The derivative of this energy difference is:

We can now express the energy difference in terms of the forces acting on the fiber bundle. We integrate this expression to find:

by partial integration. In particular, this gives the result:

at the critical point. Since σ_{c} = max σ(Δ), we see that _{diff}(Δ_{c}) > 0 for all threshold distributions. The only exception possible is a threshold distribution with a constant force σ(Δ) = σ_{c}. But this results in a lower cut-off Δ_{0} = Δ_{c} > 0 (for the threshold distribution to be normalizable), and then

First, rewrite (Equation 3) as:

This definition of _{m}, which is given by:

i.e.,

Comparing this expression to Equation (8) allows us to find a relation between Δ_{c} and Δ_{m}. We investigate the function

It is clear from Equation (12) that for a threshold distribution with only a single maximum in the load curve, _{m}) = 1/2 must occur in the unstable phase, i.e., Δ_{m} > Δ_{c}.

The elastic energy maximum occurs after the critical point and is hence unsuitable as a predictor for failure. But what about the maximum of the _{max}? As stated in the section 6:

Setting this second derivative to zero and rearranging terms gives the equation:

To investigate the relation between Δ_{max} and Δ_{c}, we combine this with the relations ^{2}σ/^{2} = − 2_{max} to get:

Let's once again assume that we are working with a threshold distribution that has only a single maximum in its load curve. Then ^{2}σ/^{2} < − 2/Δ · _{max} in the stable phase, i.e., Δ_{max} < Δ_{c}.

This is a general (but weak) condition that is sufficient, but not necessary, for Δ_{max} < Δ_{c}.