## Abstract

The problem of damage in heterogeneous materials has received particular attention in recent years. The numerical models currently used in the simulation of damage require an internal length that is not currently related to a characteristic length of the material components. However, understanding damage regarding the size of the heterogeneities of the material is of crucial importance, particularly in civil engineering. The Fiber Bundle Model has been widely used to qualitatively address the issue of damage in heterogeneous media by studying the statistics of failure events during damage. The so-called ZIP model derives from Fiber Bundle Model to mimic crack propagation. In this work, a spatial correlation of tensile strength of fibers is added to the ZIP model to highlight the role of heterogeneity size in statistics of failure events during crack propagation. The addition of spatial correlation into the ZIP model modifies the distribution of failure events. Indeed, for a simulated material without spatial correlation, failure events follow two regimes. By adding a spatial correlation to the material, a transitional regime appears. The influence of spatial correlation on fiber rupture avalanche strongly depends on the ratio between the sizes of the shapes of the stress field and of the heterogeneities.

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## Acknowledgements

The Laboratoire 3SR, the LabEx Tec 21 (Investissements d’Avenir—Grant agreement ANR-11-LABX-0030) and the PolyNat Carnot Institute (Investissements d’Avenir—Grant Agreement No. ANR-16-CARN-0025-01).

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## Appendix: Numerical convergence study of avalanches statistics

### Appendix: Numerical convergence study of avalanches statistics

### Numerical convergence of Karhunen–Loève decomposition

The convergence study has been carrying out on dimensionless avalanche distribution as presented in Fig. 6, with \(N=5\times 10^5\), \(\xi =\) 10,000 and \(b=10\). For each point of avalanche distribution, its residual depending on *M* is calculated and represented on the following graph: *M* and \(M'\) terms in Karhunen–Loève decomposition such as \(M=10M\). The dimensionless avalanche distribution with *M* terms in Karhunen–Loève decomposition is the set of points \((X(\varDelta ),Y_M(\varDelta ))\) as:

with \(\varDelta \) the size of avalanche counted by number of broken fibers, \(\xi \) the characteristic length of the deformation field of the beam and \(p(\varDelta )\) the probability of an avalanche of size \(\varDelta \).

The residual \(Res_M(\varDelta )\) of the avalanche distribution is defined as:

In Fig. 6, the residual of the dimensionless avalanche distribution is plotted for different values of *M*. With \(M=10^6\), the residual is under 0.1 for all the range of \(\varDelta \) calculated. Thus, we consider that simulation is converged depending the number of terms for \(M=10^6\).

The number *M* seems very large, but given that the autocorrelation size *b* (\(=10\)) is so small in front of the size of the simulation i.e. the number of fibers *N* (\(=5\times 10^5\)) and the Karhunen–Loève is a spectral decomposition into modes with increasing wavenumber, the decomposition has to go up to modes which are of the size of *b*. Then, the KL decomposition has to go up to high frequencies to capture all the spatial correlation between tensile strength of fibers.

### Influence of the number N of fibers

A similar convergence study depending on *N* has been conducted on dimensionless avalanche distribution, with \(\xi =10^5\) and \(b=10\). Two simulations have been carried out at \(N=5\times 10^5\) and \(N=5\times 10^6\) considering respectively \(M=10^6\) and \(M=10^7\), numbers of terms necessary to get the numerical convergence depending the Karhunen–Loève decomposition for each case.

The residual is calculated as the absolute difference between the two dimensionless avalanche distributions simulated:

On Fig. 7 is plotted the residual of the dimensionless avalanche distribution for \(N=5\times 10^5\). The residual is under 0.1 except for the two largest avalanche sizes. Thus, we consider that the results are independent of the number of fibers taking \(N=5\times 10^5\).

To sum up, from the numerical convergence study \(N=5\times 10^5\) and \(M=10^6\) are chosen to have a convergence up to the first decimal of values of the dimensionless avalanche distribution.

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Villette, F., Baroth, J., Dufour, F. *et al.* Influence of material heterogeneities on crack propagation statistics using a Fiber Bundle Model.
*Int J Fract* **221, **87–100 (2020). https://doi.org/10.1007/s10704-019-00409-2

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### Keywords

- Crack propagation
- Heterogeneous materials
- Numerical modelling
- Fiber Bundle Model
- Spatial correlation
- Karhunen–Loève decomposition