RESUMO
We investigate a model for fatigue crack growth in which damage accumulation is assumed to follow a power law of the local stress amplitude, a form that can be generically justified on the grounds of the approximately self-similar aspect of microcrack distributions. Our aim is to determine the relation between model ingredients and the Paris exponent governing subcritical crack-growth dynamics at the macroscopic scale, starting from a single small notch propagating along a fixed line. By a series of analytical and numerical calculations, we show that, in the absence of disorder, there is a critical damage-accumulation exponent γ, namely γ_{c}=2, separating two distinct regimes of behavior for the Paris exponent m. For γ>γ_{c}, the Paris exponent is shown to assume the value m=γ, a result that proves robust against the separate introduction of various modifying ingredients. Explicitly, we deal here with (i) the requirement of a minimum stress for damage to occur, (ii) the presence of disorder in local damage thresholds, and (iii) the possibility of crack healing. On the other hand, in the regime γ<γ_{c}, the Paris exponent is seen to be sensitive to the different ingredients added to the model, with rapid healing or a high minimum stress for damage leading to m=2 for all γ<γ_{c}, in contrast with the linear dependence m=6-2γ observed for very long characteristic healing times in the absence of a minimum stress for damage. Upon the introduction of disorder on the local fatigue thresholds, which leads to the possible appearance of multiple cracks along the propagation line, the Paris exponent tends to m≈4 for γâ²2 while retaining the behavior m=γ for γâ³4.
RESUMO
This paper studies the Sznajd model for opinion formation in a population connected through a general network. A master equation describing the time evolution of opinions is presented and solved in a mean-field approximation. Although quite simple, this approximation allows us to capture the most important features regarding the steady states of the model. When spontaneous opinion changes are included, a discontinuous transition from consensus to polarization can be found as the rate of spontaneous change is increased. In this case we show that a hybrid mean-field approach including interactions between second nearest neighbors is necessary to estimate correctly the critical point of the transition. The analytical prediction of the critical point is also compared with numerical simulations in a wide variety of networks, in particular Barabási-Albert networks, finding reasonable agreement despite the strong approximations involved. The same hybrid approach that made it possible to deal with second-order neighbors could just as well be adapted to treat other problems such as epidemic spreading or predator-prey systems.