Dynamical mean-field theory for a spring-block model of fracture

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Abstract
In a recently proposed spring-block model of fracture, it was found that growth of domains of positive and negative components of the stress field before cracking sets in, was crucial for the pattern formation of the cracks. In this paper a mean-field theory is proposed to describe the dynamic behavior of the stress field in the spring-block model. Mean-field site and pair approximations are made for the system before cracking sets in. The single-site mean-field approximation gives steady-state densities of positive, negative, and zero components of the stress field, in quantitative agreement with the spring-block model. The pair approximation gives densities of the components of the stress field which are in close agreement with the simulation of the spring model, and predicts domain growth of the positive and negative components of the stress field as seen in the spring-block model.