Critical Behaviour of the Local Load Sharing Fiber Bundle Model
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A generalised, history independent, Local Load Sharing Fiber Bundle Model, describing damage propagation in materials as failing fibers under under external load is studied. If the fibers' thresholds $t$ are drawn from an exponential probability distribution p(t) with a cut-off parameter t0, the model exhibits a transition when adjusting this cut-off parameter. The transition results in a change of behaviour in the damage spreading from being similar to the Equal Load Sharing Fiber Bundle Model, to be dominated by invasion percolation. By investigating this transition, it has been shown that it is indeed a critical phase transition, and the critical cut-off parameter has been determined to tc = 0.61pm0.02. The correlation length exponent of the transition was attempted estimated by the use of the Gradient Method, but did not succeed as the gradient in $t_0$ induced a forced localisation, independent of the critical cut-off. However, by determining an order parameter for the model, its scaling relation to the system size $L$, and extrapolating it to infinity, the critical exponent beta/nu = 1.5pm0.1 was calculated.