A random neighbour model for yielding

We introduce a model for yielding, inspired by fracture models andthe failure of a sheared granular medium in which the applied shear is resistedby self-organized force chains. The force chains in the granular medium (GM)are considered as a bundle of elastic brittle fibres of finite failure strength.Upon the failure of any fibre, its load is randomly redistributed amongst theremaining fibres. The model provides an exponential distribution of the internalstress and a log-normal shaped distribution of global failure stress, in agreementwith experimental observations. The model displays critical behaviour whichapproaches mean field as the number of random neighbours k becomes largeand also displays a failure strength which remains finite in the limit of infinitesize. From comparison with different models it is argued that this is an effect ofnon-correlation. All of these macroscopic properties appear statistically stablewith respect to the choice of the chains' initial failure strength distribution. Themodel investigated is relevant for all systems in which some generic external loador pressure is borne by a number of units, independent of one another exceptwhen failure of a unit causes load transfer to some random choice of neighbouringunits.