Random neighbour model for yielding

We introduce a model for yielding, inspired by fracture models and the failure of a sheared granular medium in which the applied shear is resisted by self-organized force chains. The force chains in the granular medium (GM) are considered as a bundle of fibres of finite strength amongst which stress is randomly redistributed after any other fibre breaks under excessive load. The model provides an exponential distribution of the internal stress and a log-normal shaped distribution of failure stress, in agreement with experimental observations. The model displays critical behaviour which approaches mean field as the number of random neighbours $k$ becomes large and also displays a failure strength which remains finite in the limit of infinite size. From comparison with different models it is argued that this is an effect of uncorrelation. All these macroscopic properties appear statistically stable with respect to the choice of the chains' initial strength distribution. The investigated model is relevant for all systems in which some generic external load or pressure is borne by a number of units, independent of one another except when failure of a unit causes load transfer to some random choice of neighbouring units.