Scaling laws in fracture (original) (raw)
Scaling Laws of Stress and Strain in Brittle Fracture
arXiv (Cornell University), 2005
A numerical realization of an elastic beam lattice is used to obtain scaling exponents relevant to the extent of damage within the controlled, catastrophic and total regimes of mode-I brittle fracture. The relative fraction of damage at the onset of catastrophic rupture approaches a fixed value in the continuum limit. This enables disorder in a real material to be quantified through its relationship with random samples generated on the computer.
Scale-invariant disorder in fracture and related breakdown phenomena
Physical Review B, 1991
We introduce and discuss the concept of scale-invariant disorder in connection with breakdown and fracture models of disordered brittle materials. We show that in the case of quenched-disorder models where the local breaking thresholds are randomly sampled, only two numbers determine the scaling properties of the models. These numbers characterize the behavior of the distribution of thresholds close to zero and to infinity. We review brieAy some results obtained in the literature and show how they fit into this framework. Finally, we address the case of an annealed disorder, and show via a mapping onto a quenched-disorder model, that our analysis is also valid there.
Fracture Strength of Disordered Media: Universality, Interactions, and Tail Asymptotics
Physical Review Letters, 2012
We study the asymptotic properties of fracture strength distributions of disordered elastic media by a combination of renormalization group, extreme value theory, and numerical simulation. We investigate the validity of the 'weakest-link hypothesis' in the presence of realistic long-ranged interactions in the random fuse model. Numerical simulations indicate that the fracture strength is well described by the Duxbury-Leath-Beale (DLB) distribution which is shown to flow asymptotically to the Gumbel distribution. We explore the relation between the extreme value distributions and the DLB type asymptotic distributions, and show that the universal extreme value forms may not be appropriate to describe the non-universal low-strength tail. PACS numbers: 62.20.mj,62.20.mm,62.20.mt,64.60.ae,64.60.Q-It has been known for centuries that larger bodies have lower fracture strength. The traditional explanation of this size effect is the 'weakest link' hypothesis: the sample is envisaged as a set of non-interacting sub-volumes with different failure thresholds, and its strength is determined by the failure of the weakest region. If the sub-volume threshold distribution has a power law tail near zero then the strength distribution can be shown to converge to the universal Weibull distribution for large sample sizes [1], an early application of extreme value theory (EVT) [2].
Spatial scaling in fracture propagation in dilute systems
Physica A: Statistical Mechanics and its Applications, 1996
The geometry of fracture patterns in a dilute elastic network is explored using molecular dynamics simulation. The network in two dimensions is subjected to a uniform strain which drives the fracture to develop by the growth and coalescence of the vacancy clusters in the network. For strong dilution, it has been shown earlier that there exists a characteristic time t c at which a dynamical transition occurs with a power law divergence (with the exponent z) of the average cluster size. Close to t c , the growth of the clusters is scaleinvariant in time and satis es a dynamical scaling law. This paper shows that the cluster growth near t c also exhibits spatial scaling in addition to the temporal scaling. As fracture develops with time, the connectivity length of the clusters increses and diverges at t c as (t c t) , with = 0:83 0:06. As a result of the scale-invariant growth, the vacancy clusters attain a fractal structure at t c with an e ective dimensionality d f 1:85 0:05. These values are independent (within the limit of statistical error) of the concentration (provided it is su ciently high) with which the network is diluted to begin with. Moreover, the values are very di erent from the corresponding values in qualitatively similar phenomena suggesting a di erent universality class of the problem. The values of and d f supports the scaling relation z = d f with the value of z obtained before.
International Journal of Fracture, 2006
We analyze scaling and localization phenomena in the fracture of a random central-force spring lattice model with strong disorder by means of computer simulation. We investigate the statistical and topological properties of the developing damage pattern and the scaling behaviour of the threshold. Our observations show that from the beginning and up to the point of maximum stress, damage develops in a uniform manner, qualitatively like in a percolating lattice, but numerically different from random percolation. Beyond the maximum-stress point localization and anisotropy come into play, resulting in final crack formation. The fraction of broken bonds at which the lattice fails, as well as the strain corresponding to failure, scale with the lattice size via power laws. The roughness of the final crack scales as a power law of the crack length over three decades of lengthscale.
Scaling of fracture systems in geological media
Reviews of Geophysics, 2001
Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hydrocarbon reservoir management, and earthquake hazard assessment. Relevant publications are therefore spread widely through the literature. Although it is recognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law exponents and fractal dimensions from observations, al-though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these techniques and suggest guidelines for the accurate and objective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after critical appraisal of published studies, to show a wide variation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The physical causes of power law scaling and variation in exponents and fractal dimensions are still poorly understood.
Fracture precursors in disordered systems
Europhysics Letters (EPL), 2004
A two-dimensional lattice model with bond disorder is used to investigate the fracture behaviour under stress-controlled conditions. Although the cumulative energy of precursors does not diverge at the critical point, its derivative with respect to the control parameter (reduced stress) exhibits a singular behaviour. Our results are nevertheless compatible with previous experimental findings, if one restricts the comparison to the (limited) range accessible in the experiment. A power-law avalanche distribution is also found with an exponent close to the experimental values. PACS numbers: 46.50.+a, 62.20.Mk, 05.70.Ln Fractures are very complex phenomena which involve a wide range of spatial and sometimes temporal scales. Accordingly, the development of a general theory is quite an ambitious goal, since it is not even clear whether a continuous coarse-grained description makes sense; additionally, for the very same reason, realistic simulations are almost unfeasible. However, such difficulties have not prevented making progress on several aspects of fracture dynamics such as propagation velocity, roughness, or the failure time under a constant stress [1, 2, 3]. In this paper we are interested in studying the development of the socalled precursors, microcracks preceding the macroscopic fracture in a brittle disordered environment. Some recent experiments suggest that we are in the presence of a critical phenomenon, although the accuracy is not yet high enough not only to discuss its universality properties, but also to assess the order of the transition.
Scaling behaviour of damage in the fracture of two-dimensional elastic beam lattices
Europhysics Letters (EPL), 2007
A numerical realization of an elastic beam lattice is used to obtain scaling exponents relevant to the extent of damage within the controlled, catastrophic and total regimes of mode-I brittle fracture. The relative fraction of damage at the onset of catastrophic rupture approaches a fixed value in the continuum limit. This enables disorder in a real material to be quantified through its relationship with random samples generated on the computer.
Phase transitions and correlations in fracture processes where disorder and stress compete
Physical Review Research, 2020
We study the effect of the competition between disorder and stress enhancement in fracture processes using the local load sharing fiber bundle model, a model that hovers on the border between analytical tractability and numerical accessibility. We implement a disorder distribution with one adjustable parameter. The model undergoes a localization transition as a function of this parameter. We identify an order parameter for this transition and find that the system is in the localized phase over a finite range of values of the parameter bounded by a transition to the non-localized phase on both sides. The transition is first order at the lower transition and second order at the upper transition. The critical exponents characterizing the second order transition are close to those characterizing the percolation transition. We determine the spatiotemporal correlation function in the localized phase. It is characterized by two power laws as in invasion percolation. We find exponents that are consistent with the values found in that problem.