Brittle to quasibrittle transition in a compound fiber bundle

TL;DR

This paper investigates the transition from brittle to quasibrittle failure in a fiber bundle model with bimodal strength distribution, identifying critical parameters that govern the transition.

Contribution

It introduces a fiber bundle model with a bimodal strength distribution and analytically determines the critical width for brittle to quasibrittle transition.

Findings

Identified the critical width $d_c(s,p)$ for the transition.

Confirmed the transition through numerical simulations.

Provided insights into how distribution parameters affect failure behavior.

Abstract

The brittle to quasibrittle transition has been studied for a compound of two different kinds of fibrous materials, having distinct difference in their breaking strengths under the framework of the fiber bundle model. A random fiber bundle model has been devised with a bimodal distribution of the breaking strengths of the individual fibers. The bimodal distribution is assumed to be consisting of two symmetrically placed rectangular probability distributions of strengths $p$ and $1 - p$, each of width $d$, and separated by a gap $2s$. Different properties of the transition have been studied varying these three parameters and using the well known equal load sharing dynamics. Our study exhibits a brittle to quasibrittle transition at the critical width $d_c(s,p) = p(1/2 - s)/(1 + p)$ confirmed by our numerical results.

Equations

$$P_c(b)=\{\begin{array}{ll}p(b-1/2+s+d)/d,& \text{for block 1}\\ p+(1-p)(b-1/2-s)/d,& \text{for block 2}.\end{array}$$

$${\sigma }_0(\sigma )=\sigma (1-P_c(\sigma )).$$

$${\sigma }_0(\sigma )=\{\begin{array}{ll}[1-p(\sigma -1/2+s+d)/d]\sigma & \text{for block 1}\\ [(1-p)-(1-p)(\sigma -1/2-s)/d]\sigma & \text{for block 2}.\end{array}$$

$$\sigma_{c}^{\alpha}(N)=\text{max}\Big{\{}b_{(1)}^{\alpha},\frac{N-1}{N}b_{(2)}^{\alpha},\frac{N-2}{N}b_{(3)}^{\alpha},...,\frac{1}{N}b_{(N)}^{\alpha}\Big{\}}.$$

$$P_c(b)=(b-1/2-s)/d$$

$${\sigma }_0(\sigma )=\sigma (1-(\sigma -1/2-s)/d).$$

$${\sigma }_{0c}(d,s)=d/2+(1/2+s)/2.$$

$$d_c/2+(1/2+s)/2=1/2+s$$

$$⟨\Delta (d,N)⟩={\Sigma }_j{\Delta }_j^2(d,N)/{\Sigma }_k{\Delta }_k(d,N)$$

$${\sigma }_{0c}(d,s)=d[1+(1/2-s-d)/d]/2.$$

$$d_c=(1/2-s)/2.$$

$$d_c(N)-d_c=B{N}^{-1/\nu }.$$

$${\sigma }_{0c}(d,s,p)=d[1+p(1/2-s-d)/d]/(2p).$$

$$d_c(s,p)=p(1/2-s)/(1+p).$$

Full text

I I. Introduction

Understanding the behavior and properties of different materials subjected to applied external stress is important for their useful applications as well as for the prevention of their mechanical failures. The fiber bundle model (FBM) is a simple framework to study such breakdown processes where the material is in the form of a bundle of thin fibers. This model was introduced by Pierce Pierce to study the strength of cotton yarns which was then further extended from the point of view of statistical physics by Daniels (Daniels, ). Moreover, the catastrophic global failures in FBMs have also been studied in the context of earthquakes, traffic systems etc. RoyHatano ; Pradhan0 ; KunZapperi ; Pradhan1 ; HansenBook ; Pradhan2 ; Herrmann ; Chakrabarti ; Sornette ; Sahimi ; Bhattacharya . In this paper we have studied the statistical properties of FBM when the bundle consists of a mixture of two different types of fibers, e.g., cotton and nylon.

In FBM, a set of $N$ fibers is placed in parallel to one another. These fibers are imagined to be thin elastic massless strings suspended vertically, clamped at the lower end and supported rigidly at the top. An external load $F$ is applied to the entire bundle at the bottom to stretch the fibers. Every fiber $i$ has a distinct breaking threshold $b_{i}$ of its own whose value is randomly drawn from a probability density function $P(b)$.

In this model, the failure process follows a stress conservative dynamics. Under the applied stress, each fiber elongates linearly obeying the Hooke’s law. For simplicity, the Young’s modulus for every fiber is assumed to be equal to unity. On increasing the load the stress acting through the $i$-th fiber reaches its breaking threshold $b_{i}$, beyond which it fails. In general, when a fiber fails, it releases the stress that was acting through it. The released stress then gets distributed equally among all the remaining intact fibers. This procedure is referred as the Equal load sharing (ELS) dynamics. Consequently, the stresses acting through the intact fibers are increased which may result in the failure of additional fibers though the external load $F$ has been maintained to the same value. Thus, quite often the failure of only a single fiber triggers a cascade of fiber failures, known as the ‘avalanche’. The avalanche terminates when a stable state is reached where there is no more fiber failure since each intact fiber has its breaking threshold larger than the externally applied load per fiber. When $F$ is enhanced quasi-statically, the complete failure of the bundle takes place in a sequence of such successive avalanches Hidalgo ; Kloster ; Pradhan4 ; Hansen .

In general, fiber bundles are classified into two different categories depending on how the entire bundle fails. If the failure of the weakest fiber, having the smallest breaking threshold, results in a huge avalanche which leads to the failure of the entire bundle, the bundle is referred as ‘brittle’. On the other hand, if more than one avalanche are needed to break all the fibers, the fiber bundle is called ‘quasibrittle’. Whether a fiber bundle would exhibit brittle or quasibrittle type failure, is determined by the characteristics of the probability density function $P(b)$. Recently, transitions from the brittle to quasibrittle states have been studied for FBMs with linear and non-linear elastic fibers Subhadeep ; Roy3 ; Kovacs ; SRoy .

We have studied here such transitions for the compound FBMs. Compound materials with mixtures of different kinds of fibers are very important in industrial applications. For example, good quality fabrics are produced using mixtures of cotton and nylon fibers. Breaking thresholds of fibers of these materials are widely different. Keeping this in mind, one therefore likes to ask what changes in the properties of a fiber bundle take place when the bundle is made of two different types of fibers with different sets of breaking thresholds. This prompts us to study the properties of a compound fiber bundle model when the failure thresholds of the individual fibers assume a bimodal distribution. A fiber bundle model with discontinuous distribution of breaking thresholds in a different form had been studied in the literature to consider the breakdown properties AmitDutta ; AmitDutta1 . It has been shown that introduction of a lower cutoff in the breaking threshold distribution of the fibers affects properties of fiber bundle model Pradhan3 ; Raischel .

In section 2, we describe the characteristics of the bimodal distribution and determine the critical threshold of the compound fiber bundle. In section 3, we describe the brittle to quasibrittle transition in this model of the fiber bundle. Finally, we summarize our work and conclude in section 4.

II II. Bimodal distribution and the breaking threshold

We consider a fiber bundle having $N$ fibers whose breaking thresholds $\{b_{i},i=1,\ldots,N\}$ are drawn from a bimodal distribution. This distribution is a combination of two uniform distributions of width $d$, symmetrically placed about the midpoint $b=1/2$ of the $b$ axis, and are separated by an amount of $2s$ as shown in Fig. 1. The first and the second blocks are extended over the regions $1/2-s-d\leq b\leq 1/2-s$ and $1/2+s\leq b\leq 1/2+s+d$ respectively. The probability that the breaking threshold of a randomly selected fiber is in the first and the second block are denoted by $p$ and $1-p$ respectively. On the other hand, the probability that its breaking threshold lies between $b$ and $b+db$ is denoted by $P(b)db$. Similarly, the cumulative probability $P_{c}(b)$ that the fiber has strength less than $b$ is given by:

[TABLE]

In an arbitrary intermediate stable state the average value of the external load per fiber $\sigma_{0}(\sigma)$ is given by Pradhan1 ; Pradhan2 ; Roy

[TABLE]

For our compound fiber bundle we have

[TABLE]

This variation of $\sigma_{0}(\sigma)$ against $\sigma$ has been displayed in Fig. 2(a) for a specific set of values of the parameters $s$ = 0.1, $d$ = 0.4 and for five different values of the first block probability $p$ = 0, 0.4, 0.6, 0.8, and 1. When $p=0$, then all the fibers are in the second block which implies that all the breaking threshold values are confined between $1/2+s$ and 1. Here, the system is always observed to be brittle and this behavior is evident from the plot in Fig. 2(a) which shows that $\sigma_{0}(\sigma)$ is always a decreasing function of $\sigma$. As the value of $p$ increases the system becomes quasibrittle as can be seen from the variations of $\sigma_{0}(\sigma)$. In Fig. 2(b) the same variation is plotted for the specific set $s$ = 0.1, $p$ = 0.5 and for four different values of $d$ = 0.1, 0.2, 0.3 and 0.4. Each plot has two regions, one for the first block and the other for the second block. In both the figures $\sigma_{0}(\sigma)$ varied linearly with $\sigma$ between the two blocks. Here also, as the value of $d$ is increased the system can be seen to change from a brittle phase to a quasibrittle phase. The critical load per fiber of the system is always the maximum value of $\sigma_{0}(\sigma)$ for all cases.

We first study the critical load per fiber $\sigma_{c}$ for the global failure of the fiber bundle for different values of the parameters $s,d$ and $p$. In particular, we consider the case where the value of the parameter $s$ is fixed at 0.1 and using different values of $p$, we vary the value of the block width $d$ (Smith1982, ; McCartney1983, ). For the estimation of $\sigma_{c}$ numerically we follow the method of (Smith1981, ). We first arrange the breaking thresholds of a particular bundle $\alpha$ in an increasing order such that $b_{(1)}^{\alpha}<b_{(2)}^{\alpha}<b_{(3)}^{\alpha}<...<b_{(N)}^{\alpha}$. Then, the critical load per fiber $\sigma_{c}^{\alpha}(N)$ for a particular bundle $N$ can be calculated as

[TABLE]

This critical load is then averaged over a considerably large number of configurations to get $<\sigma_{c}^{\alpha}(N)>=\sigma_{c}(N)$. We assume that it converges to $\sigma_{c}(\infty)\equiv\sigma_{c}$ in the asymptotic limit according to $\sigma_{c}(N)=\sigma_{c}+AN^{-1/\nu}$ where $A$ is a constant and $1/\nu$ is a finite size correction exponent. For a sufficiently large $N$ the correction term becomes negligible. In Fig. 3 we have plotted eleven different sets of data for different values of first block probability $p$ tuned between 0 and 1 at equal intervals of 0.1 for $N=2^{18}$. Since this is a considerably large number we expect that this behavior would hold for asymptotically large bundle sizes as well. For small values of $p$ the $\sigma_{c}$ remains constant in the entire range of variation of $d$. For example, $p=0$ implies all fibers have breaking thresholds larger than $1/2+s$ and the weakest fiber will always have the value $1/2+s$. Since for all values of $d$ the system is always brittle, when the external load per fiber is raised to 0.6, the weakest fiber fails and this leads to a cascade of fiber failures resulting in the break down of the entire fiber bundle. For this reason, the fiber bundle is said to be brittle for this set of parameter values, independent of the block width $d$. As $p$ is increased the $\sigma_{c}$ decreases and for $p\geq 0.3$ the values are not constant any more. This is because as the number of fibers in the first block increases it lowers the critical load of the system. The same process has also been carried out for $s$ = 0, 0.2, 0.3 and 0.4. The dashed line has the equation $\sigma_{c}(N,s,p)=1/2-s-d$ which means that the set of $(s,p,d)$ for which a $\sigma_{c}(N,s,p)$ falls on that line is a brittle system.

III III. Brittle to quasibrittle transition

To describe the brittle to quasibrittle transition we analyze the following three quantities, namely: (i) the fraction $f(d,N)$ of fibers broken before the last avalanche, (ii) the average number $\langle M(d,N)\rangle$ of avalanches required for the complete failure of the bundle, and (iii) the average size $\langle\Delta(d,N)\rangle$ of the avalanches (Roy3, ). Variations of these quantities have been studied against the width parameter $d$ over its entire range. In particular, we have estimated the critical value $d_{c}$ of the width that demarcates the brittle phase of the bundle from its quasibrittle phase.

III.1 A. Case $p=0$

First we consider the case when only the right block exists, i.e., $p=0$ Pradhan3 . This implies that the breaking thresholds of all the fibers in the bundle are selected in the second block. In this case, the cumulative probability $P_{c}(b)$ reduces to

[TABLE]

and Eqn. (3) becomes

[TABLE]

At the breaking point of the bundle, Eqn. (6) has a maximum at $\sigma=\sigma_{0c}(d,s)$ and it is calculated to be

[TABLE]

In this situation the value of $\sigma_{0c}(d,s)$ is equal to the minimum value of the breaking thresholds of the bundle, i.e., $1/2+s$. Thus,

[TABLE]

which gives $d_{c}=1/2+s$, the critical point. This result implies that for a system with all the fibers in the second block, the bundle will always be brittle and no transition can be observed from a brittle to a quasibrittle phase. When $s=0$, the critical width $d_{c}=1/2$. This result is consistent with the observations of Pradhan3 ; Raischel . In general, for one block uniform distribution within the limits of $(b_{min},b_{max})$, the condition for brittleness is: $b_{min}>b_{max}/2$ where, $b_{min}$ and $b_{max}$ are the lower and upper bounds of the breaking threshold distribution.

Numerically, we study the fraction $f(d,N)$ of fibers broken before the last avalanche against the block width $d$ for four different sizes of the fiber bundle Roy2 . By definition, $f(d,N)$ is identically zero when the bundle is completely brittle and non-zero when it is quasibrittle. Fig. 4(a) exhibits the variation of $f(d,N)$ against $d$. As the bundle size $N$ increases, the larger portion of the curve coincides with the $d$ axis. Therefore, the minimal value $d_{c}(N)$ of $d$ where $f(d,N)$ is non-zero increases and approaches the value of 1/2. This implies that over the entire range of the width parameter $d$, the bundle is in the brittle phase.

The variation of the scaled average number $\langle M(d,N)\rangle/N$ of avalanches required for the complete failure of the bundle has been plotted against the block width $d$ and is shown in Fig. 4(b). This quantity is also seen to be increasingly smaller with increasing value of the bundle size $N$ indicating the absence of any transition in the system.

The size of an avalanche $\Delta(d,N)$ is measured by the number of fibers failed during the avalanche. Following the method of Kun et. al. Kun we define the average size of the avalanches, excluding the last avalanche. The average avalanche size $\langle\Delta(d,N)\rangle$ is defined as the ratio of second moment to the first moment of the avalanche sizes, as

[TABLE]

where both the summation indices $j$ and $k$ run over all avalanches except the last avalanche. This quantity $\langle\Delta(d,N)\rangle$ has been plotted in Fig. 4(c) that has no maximum for any value of $d$ which proves that for this particular case there is no transition. This result is expected because the case $s=0$ and $p=0$ implies that all the fibers are in second block where the bundle always remains in a brittle phase.

III.2 B. Case $p=1$

The case with $p=1$ implies that all the fibers in the bundle have their breaking thresholds drawn from the first block. For this case

[TABLE]

On equating $\sigma_{0c}(d,s)$ to the value of the lowest breaking threshold for this case $1/2-s-d$ we get the transition point as

[TABLE]

Thus, when all the fibers in the bundle are in the first block then the value of $d_{c}$ decreases linearly with increase in $s$.

In a specific case, the result of $d_{c}=1/4$ from Eqn. (11) for $s=0$ and $p=1$ has been verified numerically. The probability $P_{b}(d,N)$ that a randomly selected sample of the fiber bundle is brittle has been plotted against $d$ in Fig. 5(a) for four different bundle sizes. The critical width $d_{c}(N)$ for a specific bundle size $N$ is defined as the minimum value of $d$ for which $P_{b}(d,N)$ vanishes. We assume the transition to be continuous and that it follows the usual finite size scaling relations. The estimated values of $d_{c}(N)$ are assumed to converge to their asymptotic value $d_{c}$ as:

[TABLE]

To estimate the asymptotic value $d_{c}$ and the exponent $\nu$ we have plotted the $d_{c}(N)$ against $N^{-1/\nu}$ in Fig. 5(b). The precise value of the exponent is tuned so that we get the best straight line with minimal fitting error. Our best estimate from this plot are $d_{c}=0.246$ and $1/\nu=0.308$.

The critical width $d_{c}(N)$ has also been estimated from the statistics of avalanche sizes. The average size of the avalanches $\langle\Delta(d,N)\rangle$ given by Eqn. (9) has been studied and plotted against $d$ in Fig. 6(a) for four different values of the bundle sizes. It is seen that for every bundle size the curve has a maximum at a certain value of $d$ which we assume as the second definition of $d_{c}(N)$. A finite size scaling of the data turned out to be very nice when we plotted $\langle\Delta(d,N)\rangle N^{0.659}$ against $[d-d_{c}(N)]N^{0.333}$. We use the $d_{c}(N)$ values estimated from the peak positions and in Fig. 6(b) all four curves fall very closely on one another. In Fig. 6(c) we again plot $d_{c}(N)-d_{c}$ against $N^{-1/\nu}$ and tune $\nu$ to get the best fitted straight line. Our results are $d_{c}=0.250$ and $1/\nu=0.331$.

III.3 C. Case $0<p<1$

We have further observed that for other intermediate values of the first block probability parameter $p$, again with the separation parameter $s=0$, there exists non-trivial phase transitions from the brittle to the quasibrittle phases. For example, in a particular case of $p=0.2$, we have again studied the same quantities, namely the fraction $f(d,N)$ of fibers broken before the last avalanche, the average number $\langle M(d,N)\rangle$ of avalanches required for the complete failure of the bundle and the average size $\langle\Delta(d,N)\rangle$ of the avalanches Danku . These quantities have been plotted against the block width $d$ in Figs. 7(a), 7(b) and 8(a) where $d$ has been tuned from 0 to $1/2$ at the interval of 0.0001. It is again observed that $\langle\Delta(d,N)\rangle$ exhibits a maximum at $d_{c}(N)$ and grows with the bundle size indicating a phase transition. Finally, in Fig. 8(b) we have plotted the scaled variable $\langle\Delta(d,N)\rangle N^{0.651}$ against $[d-d_{c}(N)]N^{0.340}$ which exhibits a good collapse of data. The value of $d_{c}$ for large $N$ have been estimated by extrapolating the bundle sizes against $N^{-0.325}$ over $N=2^{14},2^{16}$ and $2^{18}$.

Similarly, the calculation for $d_{c}(s,p)$ for $s=0$ has been repeated for the other values of $p$ in the range $0<p\leq 1$ at the interval of 0.1 and plotted in Fig. 9. Moreover, four other sets of data of $d_{c}(s,p)$ against $p$ for $s=$ 0.1, 0.2, 0.3 and 0.4 have been plotted in the same Fig. 9. For a particular value of $s$, $d_{c}$ is observed to increase with increase in $p$. This is because as the number of fibers in the first block increases, the load accumulated during the breaking process increases leading to the increase in the possibility of breaking the weakest fiber from the second block.

From Eqn. (3), one obtains the maximum value of $\sigma_{0c}(d,s,p)$ for a quasibrittle state for the first block as

[TABLE]

When a bundle is in a brittle state, the critical load per fiber $\sigma_{0c}(d,s,p)$ at the breaking point of the bundle should be equal to the value of the weakest fiber in it i.e. ($1/2-s-d$). Therefore, on solving for $d$ we get

[TABLE]

Substituting the values of $s=0$ and $p=1/2$ one gets back the well established result of the critical width $d_{c}=1/6$ Subhadeep . The cases $p=0$ and $p=1$ gives back the results of the limiting cases discussed in previous sections. The Eqn. (14) has been plotted in Fig. 9 for $s=0,0.1,0.2,0.3$ and $0.4$ along with the numerical results.

Next, we studied the average avalanche size $\langle\Delta(d,N)\rangle$ against $d$. The average avalanche size for cases when $s\neq 0.0$ behaves differently depending on the fraction of fibers $p$ in the first block. This quantity is plotted in Fig. 10(a) for $s$ = 0.1 and for different values of $p$ = 0.4,0.5 and 0.6. For small values of $d$ for $s=0.1$ and $p=0.4$ the value of $\langle\Delta(d,N)\rangle$ is vanishingly small followed by a discontinuous jump leading to a plateau. Then it’s value sharply decreases as $d$ is increased further. Similar curves are observed for $p$ = 0.5 as well. However, no plateau is observed for $p$ = 0.6. On the other hand for a much smaller value of $\langle\Delta(d,N)\rangle$, a small peak is observed for the same plot as shown in Fig. 10(b). Such behavior is observed till $p$ = 0.7 after which only the small peaks remain and the large peaks vanish.

The formation of the plateau region occurs because when $d$ is small, the bundle is in the brittle regime and all the fibers from the first block break in either one or only a few avalanches. Thus the average avalanche size $\langle\Delta(d,N)\rangle$ excluding the last avalanche remains constant as the total load released by these broken fibers is not enough to break even the weakest fiber in the second block. At the edge of the plateau the number of avalanches increase significantly and thus the value of $\langle\Delta(d,N)\rangle$ is seen to fall sharply. In this case, we define the critical width $d_{c}$ to be located at the end of the plateau instead of the beginning. This is because even though more than one avalanche is required to break all the fibers in the first block, the breakdown is rapid and the number of such avalanches is very small.

The value of the width $d$ at the right edge of the plateau where $\langle\Delta(d,N)\rangle$ sharply decreases is considered to be the critical width $d_{c}(N)$ for the system size $N$. The numerical values obtained have been plotted in Fig. 9. For $s$ = 0.1 and $p$ = 0.6 and 0.7, two significant peaks have been observed. The value of $d$ at which the smaller peak occurs as shown in Fig. 10(b) has been defined as the $d_{c}(N)$ for these cases. This is because the small peak indicates that a considerable number of avalanches occur of small sizes as $p$ is large enough and $d$ is not too small. All the values of $d_{c}(N)$ obtained through the above mentioned method have been observed to match very closely with the analytical result obtained in Eqn. (14).

IV IV. Summary

To summarize, we have studied the brittle to quasibrittle transition in a compound fiber bundle model characterized by bimodal distribution of fiber breaking thresholds. We have observed that the critical load per fiber for the failure of the bundle strongly depends on all the three parameters, namely, the width $d$ of the blocks, the separation $s$ between the blocks and the probability $p$ of the first block. We have parameterized such a transition using three different quantities, namely: (i) the average fraction of fibers broken before the last avalanche, (ii) the average number of avalanches required for the complete breakdown of a fiber bundle and (iii) the average avalanche size excluding the last avalanche. In addition, we could formulate a general expression for the critical width $d_{c}(s,p)$ of the phase transition analytically and have verified it by the numerical analysis.

We thank Mr. Sumanta Kundu very much for many helpful discussions.