We present a detailed analytical and numerical study of the avalanche distributions of the continuous damage fiber bundle model ͑CDFBM͒. Linearly elastic fibers undergo a series of partial failure events which give rise to a gradual degradation of their stiffness. We show that the model reproduces a wide range of mechanical behaviors. We find that macroscopic hardening and plastic responses are characterized by avalanche distributions, which exhibit an algebraic decay with exponents between 5/2 and 2 different from those observed in mean-field fiber bundle models. We also derive analytically the phase diagram of a family of CDFBM which covers a large variety of potential avalanche size distributions. Our results provide a unified view of the statistics of breaking avalanches in fiber bundle models.

Abstract. -We study the effect of strong heterogeneities on the fracture of disordered materials using a fiber bundle model. The bundle is composed of two subsets of fibers, i.e. a fraction 0 ≤ α ≤ 1 of fibers is unbreakable, while the remaining 1 − α fraction is characterized by a distribution of breaking thresholds. Assuming global load sharing, we show analytically that there exists a critical fraction of the components αc which separates two qualitatively different regimes of the system: below αc the burst size distribution is a power law with the usual exponent τ = 5/2, while above αc the exponent switches to a lower value τ = 9/4 and a cutoff function occurs with a diverging characteristic size. Analyzing the macroscopic response of the system we demonstrate that the transition is conditioned to disorder distributions where the constitutive curve has a single maximum and an inflexion point defining a novel universality class of breakdown phenomena.

We study the damage enhanced creep rupture of disordered materials by means of a fiber bundle model. Broken fibers undergo a slow stress relaxation modeled by a Maxwell element whose stress exponent m can vary in a broad range. Under global load sharing we show that due to the strength disorder of fibers, the lifetime t f of the bundle has sample-to-sample fluctuations characterized by a log-normal distribution independent of the type of disorder. We determine the Monkman-Grant relation of the model and establish a relation between the rupture life t f and the characteristic time t m of the intermediate creep regime of the bundle where the minimum strain rate is reached, making possible reliable estimates of t f from short term measurements. Approaching macroscopic failure, the deformation rate has a finite time power law singularity whose exponent is a decreasing function of m. On the microlevel the distribution of waiting times is found to have a power law behavior with m-dependent exponents different below and above the critical load of the bundle. Approaching the critical load from above, the cutoff value of the distributions has a power law divergence whose exponent coincides with the stress exponent of Maxwell elements.

We analyze the failure process of a two-component system with widely different fracture strength in the framework of a fiber bundle model with localized load sharing. A fraction 0 α 1 of the bundle is strong and it is represented by unbreakable fibers, while fibers of the weak component have randomly distributed failure strength. Computer simulations revealed that there exists a critical composition α c which separates two qualitatively different behaviors: Below the critical point, the failure of the bundle is brittle, characterized by an abrupt damage growth within the breakable part of the system. Above α c , however, the macroscopic response becomes ductile, providing stability during the entire breaking process. The transition occurs at an astonishingly low fraction of strong fibers which can have importance for applications. We show that in the ductile phase, the size distribution of breaking bursts has a power law functional form with an exponent μ = 2 followed by an exponential cutoff. In the brittle phase, the power law also prevails but with a higher exponent μ = 9 2 . The transition between the two phases shows analogies to continuous phase transitions. Analyzing the microstructure of the damage, it was found that at the beginning of the fracture process cracks nucleate randomly, while later on growth and coalescence of cracks dominate, which give rise to power law distributed crack sizes.