Size scaling and bursting activity in thermally activated breakdown of fiber bundles

Maximum entropy approach to reliability

The aging process is a common phenomenon in engineering, biological, and physical systems. The hazard rate function, which characterizes the aging process, is a fundamental quantity in the disciplines of reliability, failure, and risk analysis. However, it is difficult to determine the entire hazard function accurately with limited observation data when the degradation mechanism is not fully understood. Inspired by the seminal work pioneered by Jaynes [Phys. Rev. 106, 620 (1956)PHRVAO0031-899X10.1103/PhysRev.106.620], this study develops an approach based on the principle of maximum entropy. In particular, the time-dependent hazard rate function can be established using limited observation data in a rational manner. It is shown that the developed approach is capable of constructing and interpreting many typical hazard rate curves observed in practice, such as the bathtub curve, the upside down bathtub, and so on. The developed approach is applied to model a classical single function system and a numerical example is used to demonstrate the method. In addition its extension to a more general multifunction system is presented. Depending on the interaction between different functions of the system, two cases, namely reducible and irreducible, are discussed in detail. A multifunction electrical system is used for demonstration.

Blending stiffness and strength disorder can stabilize fracture

Quasibrittle behavior, where macroscopic failure is preceded by stable damaging and intensive cracking activity, is a desired feature of materials because it makes fracture predictable. Based on a fiber-bundle model with global load sharing we show that blending strength and stiffness disorder of material elements leads to the stabilization of fracture, i.e., samples that are brittle when one source of disorder is present become quasibrittle as a consequence of blending. We derive a condition of quasibrittle behavior in terms of the joint distribution of the two sources of disorder. Breaking bursts have a power-law size distribution of exponent 5/2 without any crossover to a lower exponent when the amount of disorder is gradually decreased. The results have practical relevance for the design of materials to increase the safety of constructions.

Acoustic Emission from Breaking a Bamboo Chopstick

The acoustic emission from breaking a bamboo chopstick or a bundle of spaghetti is found to exhibit similar behavior as the famous seismic laws of Gutenberg and Richter, Omori, and Båth. By the use of a force-sensing detector, we establish a positive correlation between the statistics of sound intensity and the magnitude of a tremor. We also manage to derive these laws analytically without invoking the concept of a phase transition, self-organized criticality, or fractal. Our model is deterministic and relies on the existence of a structured cross section, either fibrous or layered. This success at explaining the power-law behavior supports the proposal that geometry is sometimes more important than mechanics.

Kinetic Monte Carlo algorithm for thermally induced breakdown of fiber bundles

Fiber bundle models are one of the most fundamental modeling approaches for the investigation of the fracture of heterogeneous materials being able to capture a broad spectrum of damage mechanisms, loading conditions, and types of load sharing. In the framework of the fiber bundle model we introduce a kinetic Monte Carlo algorithm to investigate the thermally induced creep rupture of materials occurring under a constant external load. We demonstrate that the method overcomes several limitations of previous techniques and provides an efficient numerical framework at any load and temperature values. We show for both equal and localized load sharing that the computational time does not depend on the temperature; it is solely determined by the external load and the system size. In the limit of low load where the lifetime of the system diverges, the computational time saturates to a constant value. The method takes into account the secondary failures induced by subsequent load redistributions after breaking events, with the additional advantage that breaking avalanches always start from a single broken fiber.

Creep rupture as a non-homogeneous Poissonian process

Creep rupture of heterogeneous materials occurring under constant sub-critical external loads is responsible for the collapse of engineering constructions and for natural catastrophes. Acoustic monitoring of crackling bursts provides microscopic insight into the failure process. Based on a fiber bundle model, we show that the accelerating bursting activity when approaching failure can be described by the Omori law. For long range load redistribution the time series of bursts proved to be a non-homogeneous Poissonian process with power law distributed burst sizes and waiting times. We demonstrate that limitations of experiments such as finite detection threshold and time resolution have striking effects on the characteristic exponents, which have to be taken into account when comparing model calculations with experiments. Recording events solely within the Omori time to failure the size distribution of bursts has a crossover to a lower exponent which is promising for forecasting the imminent catastrophic failure.

Time evolution of damage due to environmentally assisted aging in a fiber bundle model

Damage growth in composite materials is a complex process which is of interest in many fields of science and engineering. We consider this problem in a fiber bundle model where fibers undergo an aging process due to the accumulation of damage driven by the locally acting stress in a chemically active environment. By subjecting the bundle to a constant external load, fibers fail either when the load on them exceeds their individual intrinsic strength or when the accumulated internal damage exceeds a random threshold. We analyze the time evolution of the breaking process under low external loads where aging of fibers dominates. In the mean field limit, we show analytically that the aging system continuously accelerates in a way which can be characterized by an inverse power law of the event rate with a singularity that defines a failure time. The exponent is not universal; it depends on the details of the aging process. For localized load sharing, a more complex damage process emerges which is dominated by distinct spatial regions of the system with different degrees of stress concentration. Analytical calculations revealed that the final acceleration to global failure is preceded by a stationary accumulation of damage. When the disorder is strong, the accelerating phase has the same functional behavior as in the mean field limit. The analytical results are verified by computer simulations.

Force-induced breakdown of flexible polymerized membrane

We consider the fracture of a free-standing two-dimensional (2D) elastic-brittle network to be used as protective coating subject to constant tensile stress applied on its rim. Using a molecular-dynamics simulation with a Langevin thermostat, we investigate the scission and recombination of bonds, and the formation of cracks in the 2D graphenelike hexagonal sheet for different pulling force f and temperature T. We find that bond rupture occurs almost always at the sheet periphery, and the first mean breakage time <τ> of bonds decays with membrane size as <τ> ∝N(-β), where β≈0.50±0.03 and N denotes the number of atoms in the membrane. The probability distribution of bond scission times t is given by a Poisson function W(t)∝t(1/3)exp(-t/<τ>). The mean failure time <τ(r)> necessary to rip off the sheet declines with growing size N as a power law <τ(r)>∝N(-φ(f)). We also find <τ(r)>∝exp(ΔU(0)/k(B)T), where the nucleation barrier for crack formation ΔU(0)∝f(-2), in agreement with Griffith's theory. <τ(r)> displays an Arrhenian dependence of <τ(r)> on temperature T. Our results indicate a rapid increase in crack spreading velocity with growing external tension f.

Microstructure of damage in thermally activated fracture of Lennard-Jones systems

We investigate the effect of thermal fluctuations on the critical stress and the microstructure of damage preceding macroscopic fracture of Lennard-Jones solids under a constant external load. Based on molecular dynamics simulations of notched specimens at finite temperatures, we show that the crystalline structure gets distorted ahead of the crack in the secondary creep regime. The damage profile characterizing the spatial distribution of lattice distortions is well described by an exponential form. The characteristic length of the exponential form provides the scale of damage, which is found to be an increasing function of the temperature: At low temperatures, damage is strongly localized to the crack tip, while at high temperatures, damage extends to a broader range, leading to more efficient relaxation of overloads. As a consequence, the stress intensity factor decreases with increasing temperature. The final macroscopic failure of the system occurs suddenly and is initiated by the creation of vacancies and voids. The creep strength exhibits inverse square root scaling with the notch size corrected by the extension of the process zone.

Statistical model with two order parameters for ductile and soft fiber bundles in nanoscience and biomaterials

Traditional fiber bundles models (FBMs) have been an effective tool to understand brittle heterogeneous systems. However, fiber bundles in modern nano- and bioapplications demand a new generation of FBM capturing more complex deformation processes in addition to damage. In the context of loose bundle systems and with reference to time-independent plasticity and soft biomaterials, we formulate a generalized statistical model for ductile fracture and nonlinear elastic problems capable of handling more simultaneous deformation mechanisms by means of two order parameters (as opposed to one). As the first rational FBM for coupled damage problems, it may be the cornerstone for advanced statistical models of heterogeneous systems in nanoscience and materials design, especially to explore hierarchical and bio-inspired concepts in the arena of nanobiotechnology. Applicative examples are provided for illustrative purposes at last, discussing issues in inverse analysis (i.e., nonlinear elastic polymer fiber and ductile Cu submicron bars arrays) and direct design (i.e., strength prediction).

Kertész line of thermally activated breakdown phenomena

Based on a fiber bundle model we substantially extend the phase-transition analogy of thermally activated breakdown of homogeneous materials. We show that the competition of breaking due to stress enhancement and due to thermal fluctuations leads to an astonishing complexity of the phase space of the system: varying the load and the temperature a phase boundary emerges, separating a Griffith-type regime of abrupt failure analogous to first-order phase transitions from disorder dominated fracture where a spanning cluster of cracks emerges. We demonstrate that the phase boundary is the Kertész line of the system along which thermally activated fracture appears as a continuous phase transition analogous to percolation. The Kertész line has technological relevance setting the boundary of safe operation for construction components under high thermal loads.

Time-dependent fiber bundles with local load sharing. II. General Weibull fibers

Fiber bundle models (FBMs) are useful tools in understanding failure processes in a variety of material systems. While the fibers and load sharing assumptions are easily described, FBM analysis is typically difficult. Monte Carlo methods are also hampered by the severe computational demands of large bundle sizes, which overwhelm just as behavior relevant to real materials starts to emerge. For large size scales, interest continues in idealized FBMs that assume either equal load sharing (ELS) or local load sharing (LLS) among fibers, rules that reflect features of real load redistribution in elastic lattices. The present work focuses on a one-dimensional bundle of N fibers under LLS where life consumption in a fiber follows a power law in its load, with exponent rho , and integrated over time. This life consumption function is further embodied in a functional form resulting in a Weibull distribution for lifetime under constant fiber stress and with Weibull exponent, beta. Thus the failure rate of a fiber depends on its past load history, except for beta=1 . We develop asymptotic results validated by Monte Carlo simulation using a computational algorithm developed in our previous work [Phys. Rev. E 63, 021507 (2001)] that greatly increases the size, N , of treatable bundles (e.g., 10(6) fibers in 10(3) realizations). In particular, our algorithm is O(N ln N) in contrast with former algorithms which were O(N2) making this investigation possible. Regimes are found for (beta,rho) pairs that yield contrasting behavior for large N. For rho>1 and large N, brittle weakest volume behavior emerges in terms of characteristic elements (groupings of fibers) derived from critical cluster formation, and the lifetime eventually goes to zero as N-->infinity , unlike ELS, which yields a finite limiting mean. For 1/2<or=rho<or=1 , however, LLS has remarkably similar behavior to ELS (appearing to be virtually identical for rho=1 ) with an asymptotic Gaussian lifetime distribution and a finite limiting mean for large N. The coefficient of variation follows a power law in increasing N but, except for rho=1, the value of the negative exponent is clearly less than 1/2 unlike in ELS bundles where the exponent remains 1/2 for 1/2<rho<or=1. For sufficiently small values 0<rho1, a transition occurs, depending on beta , whereby LLS bundle lifetimes become dominated by a few long-lived fibers. Thus the bundle lifetime appears to approximately follow an extreme-value distribution for the longest lived of a parallel group of independent elements, which applies exactly to rho=0. The lower the value of beta , the higher the transition value of rho , below which such extreme-value behavior occurs. No evidence was found for limiting Gaussian behavior for rho>1 but with 0<beta(rho+1)<1, as might be conjectured from quasistatic bundle models where beta(rho+1) mimics the Weibull exponent for fiber strength.

Fiber bundle model with stick-slip dynamics

We propose a generic model to describe the mechanical response and failure of systems which undergo a series of stick-slip events when subjected to an external load. We model the system as a bundle of fibers, where single fibers can gradually increase their relaxed length with a stick-slip mechanism activated by the increasing load. We determine the constitutive equation of the system and show by analytical calculations that on the macroscale a plastic response emerges followed by a hardening or softening regime. Releasing the load, an irreversible permanent deformation occurs which depends on the properties of sliding events. For quenched and annealed disorder of the failure thresholds the same qualitative behavior is found, however, in the annealed case the plastic regime is more pronounced.