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Influence of material heterogeneities on crack propagation statistics using a Fiber Bundle Model

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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References

  1. Baxevanis T, Dufour F, Pijaudier-Cabot G (2006) Interface crack propagation in porous and time-dependent materials analyzed with discrete models. Int J Fracture 141(3–4):561–571

    Article  Google Scholar 

  2. Bazant ZP (1984) Size effect in blunt fracture: concrete, rock, metal. J Eng Mech 110(4):518–535

    Article  Google Scholar 

  3. Bazant ZP (1994) Nonlocal damage theory based on micromechanics of crack interactions. J Eng Mech 120(3):593–617

    Article  Google Scholar 

  4. Bazant ZP, Pijaudier-Cabot G (1989) Measurement of characteristic length of nonlocal continuum. J Eng Mech 115(4):755–767

    Article  Google Scholar 

  5. Bazant ZP, Vořechovský M, Novák D (2007) Asymptotic prediction of energetic-statistical size effect from deterministic finite-element solutions. J Eng Mech 133(2):153–162

    Article  Google Scholar 

  6. Daniels H (1945) The statistical theory of the strength of bundles of threads. Proc R Soc Lond A183:405435

    Google Scholar 

  7. Delaplace A, Roux S (1999) Damage cascade in a softening interface. Int J Solids Struct 36(1972):91–125

    Google Scholar 

  8. Delaplace A, Roux S, Pijaudier-Cabot G (2001) Avalanche statistics of interface crack propagation in fiber bundle model: characterization of cohesive crack. J Eng Mech 9399(July):646–652

    Article  Google Scholar 

  9. Ghanem RG, Spanos PD (1997) Spectral techniques for stochastic finite elements. Arch Comput Methods Eng 4(1):63–100

    Article  Google Scholar 

  10. Giry C, Dufour F, Mazars J (2011) Stress-based nonlocal damage model. Int J Solids Struct 48(25–26):3431–3443

    Article  Google Scholar 

  11. Harlow D, Phoenix S (1991) Approximations for the strength distribution and size effect in an idealized lattice model of material breakdown. J Mech Phys Solids 39(2):173–200

    Article  Google Scholar 

  12. Hemmer PC, Hansen A (1992) The distribution of simultaneous fiber failures in fiber bundles. J Appl Mech 59(4):909

    Article  Google Scholar 

  13. Krasnoshlyk V, Roscoat SRd, Dumont PJJ, Isaksson P (2018) Influence of the local mass density variation on the fracture behavior of fiber network materials. Int J Solids Struct 138:236–244

    Article  Google Scholar 

  14. Kun F, Nagy S (2008) Damage process of a fiber bundle with a strain gradient. Phys Rev E Stat Nonlinear Soft Matter Phys 77(1):016608

    Article  Google Scholar 

  15. La Borderie C, Lawrence C, Menou A (2007) Approche mésoscopique du comportement du béton. Revue Européenne de Génie Civil 11(4):407–421

    Article  Google Scholar 

  16. Loève M (1977) Probability theory, vol 45. Springer, London

    Google Scholar 

  17. Lu YL, Elsworth D, Wang LG (2013) Microcrack-based coupled damage and flow modeling of fracturing evolution in permeable brittle rocks. Comput Geotech 49:226–244

    Article  Google Scholar 

  18. Manouchehrian A, Cai M (2016) Influence of material heterogeneity on failure intensity in unstable rock failure. Comput Geotech 71:237–246

    Article  Google Scholar 

  19. Mazars J, Pijaudier-Cabot G, Saouridis C (1991) Size effect and continuous damage in cementitious materials. Int J Fracture 51(2):159–173

    CAS  Google Scholar 

  20. Niskanen K, Alava M, Seppala E, Astrom J (1999) Fracture energy in fibre and bond failure. J Pulp Paper Sci 25(5):167–169

    CAS  Google Scholar 

  21. Niskanen K, Kettunen H, and Yu Y (2001) Damage width: a measure of the size of fracture process zone. In: The science of papermaking, 12th fundamental research symp, 2(September 2001):1467–1482

  22. Peerlings RH, De Borst R, Brekelmans WA, De Vree JH (1996) Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39(19):3391–3403

    Article  Google Scholar 

  23. Peirce F T (1926) 32X Tensile tests for cotton yarns: V. The weakest link theorems on the strength of long and of composite specimens. J Text Inst Trans 17(7):T355–T368

    CAS  Article  Google Scholar 

  24. Phoenix SL (1975) Probabilistic inter-fiber dependence and the asymptotic strength distribution of classic fiber bundles. Int J Eng Sci 13(3):287–304

    Article  Google Scholar 

  25. Phoenix S, Taylor HM (1973) The asymptotic strength distribution of a general fiber bundle. Appl Probab Trust 5:200–216

    Article  Google Scholar 

  26. Pijaudier CG, Bazant ZP (1987) Non local damage theory. J Eng Mech 113(10):1512–1533

    Article  Google Scholar 

  27. Poh LH, Sun G (2017) Localizing gradient damage model with decreasing interactions. Int J Numer Methods Eng 110(6):503–522

    Article  Google Scholar 

  28. Pradhan S, Hansen A, Hemmer PC (2005) Crossover behavior in burst avalanches: signature of imminent failure. Phys Rev Lett 95(12):125501

    Article  Google Scholar 

  29. Riggio M, Sandak J, Franke S (2015) Application of imaging techniques for detection of defects, damage and decay in timber structures on-site. Construct Build Mater 101:1241–1252

    Article  Google Scholar 

  30. Smith RL, Phoenix SL (1981) Asymptotic distributions for the failure of fibrous materials under series-parallel structure and equal load-sharing. J Appl Mech Trans ASME 48(1):75–82

    Article  Google Scholar 

  31. Sudret B, Kiureghian aD (2000) Stochastic finite element methods and reliability: a state-of-the-art report. Technical Report November

  32. Tang SB, Huang RQ, Tang CA, Liang ZZ, Heap MJ (2017) The failure processes analysis of rock slope using numerical modelling techniques. Eng Fail Anal 79(June):999–1016

    Article  Google Scholar 

  33. Vandoren B, Simone A (2018) Modeling and simulation of quasi-brittle failure with continuous anisotropic stress-based gradient-enhanced damage models. Comput Methods Appl Mech Eng 332:644–685

    Article  Google Scholar 

  34. Zietlow WK, Labuz JF (1998) Measurement of the intrinsic process zone in rock using acoustic emission. Int J Rock Mech Min Sci 35(3):291–299

    Article  Google Scholar 

Download references

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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Correspondence to Julien Baroth.

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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:

$$\begin{aligned} X(\varDelta )=log(\frac{\varDelta }{\xi }); \quad Y_M(\varDelta )=log(\xi ^{1.5} p(\varDelta )) \end{aligned}$$
(13)

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:

$$\begin{aligned} Res_M (\varDelta )=|Y_{M'}(\varDelta )-Y_M(\varDelta )| \end{aligned}$$
(14)

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:

$$\begin{aligned} Res_N (\varDelta )=|Y_{N=5\times 10^6}(\varDelta )-Y_{N=5\times 10^5}(\varDelta )| \end{aligned}$$
(15)

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