One of the major factors governing the mode of failure in disordered solids is the effective range R over which the stress field is modified following a local rupture event. In a random fiber bundle model, considered as a prototype of disordered solids, we show that the failure mode is nucleation dominated in the large system size limit, as long as R scales slower than L(ζ), with ζ=2/3. For a faster increase in R, the failure properties are dominated by the mean-field critical point, where the damages are uncorrelated in space. In that limit, the precursory avalanches of all sizes are obtained even in the large system size limit. We expect these results to be valid for systems with finite (normalizable) disorder.

The two principal ingredients determining the failure modes of disordered solids are the strength of heterogeneity and the length scale of the region affected in the solid following a local failure. While the latter facilitates damage nucleation, the former leads to diffused damage-the two extreme natures of the failure modes. In this study, using the random fiber bundle model as a prototype for disordered solids, we classify all failure modes that are the results of interplay between these two effects. We obtain scaling criteria for the different modes and propose a general phase diagram that provides a framework for understanding previous theoretical and experimental attempts of interpolation between these modes. As the fiber bundle model is a long-standing model for interpreting various features of stressed disordered solids, the general phase diagram can serve as a guiding principle in anticipating the responses of disordered solids in general.

We report a critical behavior in the breakdown of an equal-load-sharing fiber bundle model at a dispersion δc of the breaking threshold of the fibers. For δ < δc, there is a finite probability P b , that rupturing of the weakest fiber leads to the failure of the entire system. For δ ≥ δc, P b = 0. At δc, P b ∼ L −η , with η ≈ 1/3, where L is the size of the system. As δ → δc, the relaxation time τ diverges obeying the finite-size scaling law: τ ∼ L β (|δ−δc|L α ) with α = β ≈ 1/3. At δc, the system fails, at the critical load, in avalanches (of rupturing fibers) of all sizes s following the distribution P (s) ∼ s −κ , with κ ≈ 1/2. We relate this critical behavior to the brittle to quasibrittle transition in the model. For the local-load-sharing scheme, the system is found to be always brittle for sufficiently large system sizes.

We investigate the effective rheology of two-phase flow in a bundle of parallel capillary tubes carrying two immiscible fluids under an external pressure drop. The diameter of the tubes vary along the length which introduce capillary threshold pressures. We demonstrate through analytical calculations that a transition from a linear Darcy to a non-linear behavior occurs while decreasing the pressure drop ∆P , where the total flow rate Q varies with ∆P with an exponent 2 as Q ∼ ∆P 2 for uniform threshold distribution. The exponent changes when a lower cut-off Pm is introduced in the threshold distribution and in the limit where ∆P approaches Pm, the flow rate scales as Q ∼ (|∆P | − Pm) 3/2 . While considering threshold distribution with a power α, we find that the exponent γ for the non-linear regime vary as γ = α + 1 for Pm = 0 and γ = α + 1/2 for Pm > 0. We provide numerical results in support of our analytical findings. PACS numbers:Understanding the hydrodynamic properties of simultaneous flow of two or more immiscible fluids is essential due its relevance to a wide variety of different systems in industrial, geophysical and medical sectors [1,2]. Different applications, such as bubble generation in microfluidics, blood flow in capillary vessels, catalyst supports used in the automotive industry, transport in fuel cells, oil recovery, ground water management and CO 2 sequestration, deal with the flow of bubble trains in different types of systems, ranging from single capillaries to more complex porous media. The underlying physical mechanisms in multiphase flow are controlled by a number of factors, such as the capillary forces at the interfaces, viscosity contrast between the fluids, wettability and geometry of the system, which make the flow properties different compared to single phase flow. When one immiscible fluid invades a porous medium filled with another fluid, different types of transient flow patterns, namely viscous fingering [3,4], stable displacement [5] and capillary fingering [6] are observed while tuning the physical parameters [7]. These transient flow patterns were modeled by invasion percolation [8] and diffusion limited aggregation (DLA) models [9]. When steady state sets in after the initial instabilities, the flow properties in are characterized by relations between the global quantities such as flow rate, pressure drop and fluid saturation [10,11]. It has been observed theoretically and experimentally that, in the regime where capillary forces compete with the viscous forces, the two-phase flow rate of Newtonian fluids in the steady state no longer obeys the linear Darcy law [12,13] but varies as a power law with the applied pressure drop [14][15][16][17]. Tallakstad et al. [14,15] experimentally measured the exponent of the power law to be close to two (= 1/0.54) in a two-dimensional system and followed this observation up with arguments why the exponent should be two. Rassi et al. [16] found a value for the exponent varying between 2.2 (= 1/0.45) and 3.0 (= 1/0.33) in a three-dimensio...