Cooperative effect of load and disorder in thermally activated rupture of a 2d random fuse network

A random fuse network, or equivalently a 2d spring network with quenched disorder, is submitted to a constant load and thermal noise, and studied by numerical simulations. Rupture is thermally activated and the lifetime follows an Arrhenius law where the energy barrier is reduced by disorder. Due to the non-homogenous distribution of forces from stress concentration at microcracks’ tips, spatial correlations between rupture events appear, but they do not affect the energy barrier’s dependence on disorder, they affect only the coupling between disorder and the applied load. PACS numbers: 61.43.-j, 05.10.-a, 05.70.Ln, 62.20.Mk The effect of quenched disorder on dynamics is a recurring problem in many physical systems with elastic interactions. The motion of vortex lines in supraconductors, chargedensity waves in Bragg glasses, magnetic domains walls, or contact lines in wetting show a competition between elastic interactions and pinning by disorder [1, 2]. While many studies have focused on systems driven above a critical depininng threshold, an important issue remains to understand the sub-critical regime, when thermally activated creep motion occurs [3, 4]. Rupture in disordered brittle solids falls in the same class of problems [5]. Elastic interactions tend to make a crack propagate in a straight direction while disorder creates roughness [6] or causes spatially diffuse damage [7, 8]. In the sub-critical rupture regime, a very important quantity for safety reasons is the lifetime, i.e. the mean time for a sample to break under a prescribed load. The lifetime follows an Arrehnius law [9, 10], but thermal noise is generally too small compared to recent theoretical estimates of the energy barrier [11, 12, 13] to explain experimental observations in heterogenous materials [14, 15, 16]. For athermal systems, disorder actually reduces the energy barrier and can be seen as an effective temperature [17]. In order to clarify the role of disorder in thermal systems, one-dimensional Thermal and Disordered Fiber Bundles Cooperative effect of load and disorder in thermally activated rupture 2 Models (1d-TDFBM) have been introduced to model the thermally activated rupture of an heterogenous material submitted to a constant external load [18, 19, 20, 21, 22]. The TDFBM considers an elastic system in equilibrium at constant temperature where statistical force fluctuations occur in time due to thermal noise. This is very different from previous thermal random fuse models [23] where rupture results from an increase in fuse temperature due to dissipation through a generalized Joule effect until reaching a critical melting temperature. In a TDFBM, elastic energy is the equivalent of dissipation in the thermal random fuse model but does not cause rupture when the system is at mechanical equilibrium; rupture is caused by elastic force fluctuations analogous to Nyquist noise. One problem with the 1d-TDFBM investigated up to now is that the load is shared equally among all the unbroken fibers. This is not a realistic load-sharing rule for experimental geometries where elasticity cause stress concentration at microcracks’ tips and lead to a non-uniform redistribution of stress. In this letter, we show that spatial correlations between rupture events in 2d do not affect the dependence of the energy barrier on disorder but only the coupling between disorder and the applied load. First, we discuss briefly results obtained by several authors [18, 19, 20, 21, 22] on the 1d-TDFBM. The system considered is made of a set of N parallel fibers, each carrying an initial force f0 and behaving as a linear elastic spring with unity stiffness. Each fiber j can carry a maximum force f (j) c before it breaks. Quenched disorder is introduced in the system by distributing thresholds f (j) c according to a gaussian distribution of mean < f (j) c >= 1 and variance Td; for each fiber, the value f (j) c is a time-independent constant. Contrary to the case of non-thermal 1d-DFBM where the system evolves due to a progressive increase in total current, we consider that the total force applied to the 1d-TDFBM is kept constant. Dynamics is introduced in the system by introducing fluctuations in spring forces due to thermal noise. We write fj the average force on fiber j. The fluctuations in force δfj that occur in time on fiber j are assumed to follow a gaussian probability distribution with 0 mean value and variance T , where T represents the thermodynamical temperature in unit of square force. When the total force on a fiber fj + δfj is larger than the threshold f (j) c , the fiber breaks. The remaining fibers share equally the total force: this is a so-called democratic model. The bundle will break completely as soon as the average force on each fiber exceeds the breaking threshold. Roux has shown that the mean time to break the first fiber follows an Arrhenius law where disorder acts as an additive temperature [18]: τ ∼ exp ( (1− f0) 2 2(T + Td) )