First-Order Transition in the Breakdown of Disordered Media

Abstract

We study the approach to global breakdown in disordered media driven by increasing external forces. We first analyze the problem by mean-field theory, showing that the failure process can be described as a first-order phase transition, similarly to the case of thermally activated fracture in homogeneous media. Then we quantitatively confirm the predictions of the mean-field theory using numerical simulations of discrete models. Widely distributed avalanches and the corresponding meanfield scaling are explained by the long-range nature of elastic interactions. We discuss the analogy of our results to driven disordered first-order transitions and spinodal nucleation in magnetic systems.[ S0031-9007(97) PACS numbers: 05.70. Ln, 62.20.Mk, 64.60.Fr The breakdown of solids under external forces is a longstanding problem that has both theoretical and practical relevance [1]. The first theoretical approach to fracture mechanics, proposed by Griffith [2] more than 75 years ago, is similar to the classical theory of nucleation in firstorder phase transitions [3]. An elastic solid under stress is in a "metastable state" and will decay to the "stable fractured state" by the formation of cracks. Recently it has been shown [4] that the point of zero external stress has the same mathematical properties as the condensation point in gas-liquid first-order transitions. In the language of phase transitions, the stress imposed on the solid plays the role of an external field and cracks are the analog of droplets of a new phase. The classical theory of nucleation is expected to fail close to the limit of stability, the spinodal point [5], and it has been suggested [6-9] that a similar behavior should occur for fracture, for large values of the external stress. Thus, the failure threshold corresponds to the spinodal point of first-order phase transitions.The Griffith theory and related calculations deal with the situation in which fracture is thermally activated and quenched disorder is absent or weak. In many realistic situations, however, the solid is not homogeneous, and disorder, in the form of vacancies or microcracks, strongly affects the nucleation process [8,9]. There are situations, encountered, for example, in material testing, in which the system is driven by an increasing external stress [11] and the time scale of thermal fluctuations is much larger than the time scale induced by the driving. In those cases, the system can be effectively considered as being at zero temperature, so only quenched disorder is relevant. It has been experimentally observed [10][11][12][13][14] that the response (acoustic emission) of stressed disordered media takes place in bursts of widely distributed intensity, indicative of an internal avalanche dynamics.The understanding of the breakdown of disordered systems has considerably progressed due to the use of large scale simulations of lattice models [15]. These models have provided a good description of geometrical and topological properties of cracks, leading to the introduction in this f...

Applicability and non-applicability of equilibrium statistical mechanics to non-thermal damage phenomena

“…This corresponds to the point S in Fig. 3 and as it was suggested by many authors [7,9,14] is in situ a spinodal point of the model. As the constant external stress σ approaches So, we have found that under the EBC of constant stress the FBM exhibits presence of the first order phase transition and spinodal.…”

Crack roughness and avalanche precursors in the random fuse model

“…In previous simulations the exponent resulted to be close to τ = 5/2 , the value expected in the fiber bundle model (FBM) [34,35]. In the FBM, load is redistributed equally in all the fibers, representing thus a sort of mean-field limit of the RFM [32]. The load transfer in the RFM is long-ranged and is thus possible that RFM and FBM display universal behavior [36].…”

“…The qualitative behavior of the avalanche statistics is well understood in FBM, which can be solved exactly representing a mean-field version of the RFM [32]. The FBM can be formulated as a parallel set of fuses, with random breaking threshold, under a constant applied current I.…”

“…In this paper, we follow Ref. [32] , considering only current driven avalanches. The avalanche size is defined as the number of fuses in an avalanche.…”

Statistical analysis of dislocation dynamics during viscoplastic deformation from acoustic emission

“…One central property of first-order phase transitions is the presence of a macroscopic instability (discontinuity ofthe order parameter) atthe transitionpoint. This framework was thus proposed to describe the global breakdown of a system in fracturing phenomena [Andersen et al, 1997;Zapperi et al, 1997a]. However, dislocation-driven viscoplastic deformation during viscoplastic steady state behavior does not lead to a breakdown (ormacroscopic instability) of the system.…”