Dynamics of Fractures in Quenched Disordered Media

We introduce a model for fractures in quenched disordered media. This model has a deterministic extremal dynamics, driven by the energy function of a network of springs (Born Hamiltonian). The breakdown is the result of the cooperation between the external field and the quenched disorder. This model can be considered as describing the low temperature limit for crack propagation in solids. To describe the memory effects in this dynamics, and then to study the resistance properties of the system we realized some numerical simulations of the model. The model exhibits interesting geometric and dynamical properties, with a strong reduction of the fractal dimension of the clusters and of their backbone, with respect to the case in which thermal fluctuations dominate. This result can be explained by a recently introduced theoretical tool as a screening enhancement due to memory effects induced by the quenched disorder.Comment: 7 pages, 9 Postscript figures, uses revtex psfig.sty, to be published on Phys. Rev.

Generalized Dielectric Breakdown Model

We propose a generalized version of the Dielectric Breakdown Model (DBM) for generic breakdown processes. It interpolates between the standard DBM and its analog with quenched disorder, as a temperature like parameter is varied. The physics of other well known fractal growth phenomena as Invasion Percolation and the Eden model are also recovered for some particular parameter values. The competition between different growing mechanisms leads to new non-trivial effects and allows us to better describe real growth phenomena. Detailed numerical and theoretical analysis are performed to study the interplay between the elementary mechanisms. In particular, we observe a continuously changing fractal dimension as temperature is varied, and report an evidence of a novel phase transition at zero temperature in absence of an external driving field; the temperature acts as a relevant parameter for the ``self-organized'' invasion percolation fixed point. This permits us to obtain new insight into the connections between self-organization and standard phase transitions.Comment: Submitted to PR

Laplacian Fractal Growth in Media with Quenched Disorder

We analyze the combined effect of a Laplacian field and quenched disorder for the generation of fractal structures with a study, both numerical and theoretical, of the quenched dielectric breakdown model (QDBM). The growth dynamics is shown to evolve from the avalanches of invasion percolation (IP) to the smooth growth of Laplacian fractals, i. e. diffusion limited aggregation (DLA) and the dielectric breakdown model (DBM). The fractal dimension is strongly reduced with respect to both DBM and IP, due to the combined effect of memory and field screening. This implies a specific relation between the fractal dimension of the breakdown structures (dielectric or mechanical) and the microscopic properties of disordered materials.Comment: 11 pages Latex (revtex), 3 postscript figures included. Submitted to PR

Statistical properties of fractures in damaged materials

We introduce a model for the dynamics of mud cracking in the limit of of extremely thin layers. In this model the growth of fracture proceeds by selecting the part of the material with the smallest (quenched) breaking threshold. In addition, weakening affects the area of the sample neighbour to the crack. Due to the simplicity of the model, it is possible to derive some analytical results. In particular, we find that the total time to break down the sample grows with the dimension L of the lattice as L^2 even though the percolating cluster has a non trivial fractal dimension. Furthermore, we obtain a formula for the mean weakening with time of the whole sample.Comment: 5 pages, 4 figures, to be published in Europhysics Letter

Experimental analysis of the performance of fractal stirrers for impinging jets heat transfer enhancement

Paper presented to the 10th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Florida, 14-16 July 2014.A new passive method for the heat transfer enhancement of circular impinging jets is proposed and tested. The method is based on enhancing the mainstream turbulence of impinging jets using square fractal grids, i.e. a grid with a square pattern repeated at increasingly smaller scales. Fractal grids can generate much higher turbulence intensity than regular grids under the same inflow conditions and with similar blockage ratio, at the expense of a slightly larger pressure drop. An experimental investigation on the heat transfer enhancement achieved by impinging jets with fractal turbulence promoters is carried out. The heated-thin foil technique is implemented to measure the spatial distribution of the Nusselt number on the target plate. The heat transfer rates of impinging jets with a regular grid and a fractal grid insert are compared to that of a jet without any turbulator under the same condition of power input. A parametric study on the effect of the Reynolds number, the nozzle-to-plate distance and the position of the insert within the nozzle is carried out. The results show that a fractal turbulence promoter can provide a significant heat transfer enhancement for relatively small nozzle-to-plate separation (at distance equal to 2 diameters 63% increase with respect to the circular jet at the stagnation point, and 25% if averaged over an area of radius equal to 1 nozzle diameter; respectively, against 9% and 6% of the regular grid in the same conditions of power input).dc201

Non-perturbative renormalization of the KPZ growth dynamics

We introduce a non-perturbative renormalization approach which identifies stable fixed points in any dimension for the Kardar-Parisi-Zhang dynamics of rough surfaces. The usual limitations of real space methods to deal with anisotropic (self-affine) scaling are overcome with an indirect functional renormalization. The roughness exponent $\alpha$ is computed for dimensions $d=1$ to 8 and it results to be in very good agreement with the available simulations. No evidence is found for an upper critical dimension. We discuss how the present approach can be extended to other self-affine problems.Comment: 4 pages, 2 figures. To appear in Phys. Rev. Let

Levy-Nearest-Neighbors Bak-Sneppen Model

We study a random neighbor version of the Bak-Sneppen model, where "nearest neighbors" are chosen according to a probability distribution decaying as a power-law of the distance from the active site, P(x) \sim |x-x_{ac }|^{-\omega}. All the exponents characterizing the self-organized critical state of this model depend on the exponent \omega. As \omega tends to 1 we recover the usual random nearest neighbor version of the model. The pattern of results obtained for a range of values of \omega is also compatible with the results of simulations of the original BS model in high dimensions. Moreover, our results suggest a critical dimension d_c=6 for the Bak-Sneppen model, in contrast with previous claims.Comment: To appear on Phys. Rev. E, Rapid Communication

Diffusion limited aggregation as a Markovian process. Part I: bond-sticking conditions

Cylindrical lattice Diffusion Limited Aggregation (DLA), with a narrow width N, is solved using a Markovian matrix method. This matrix contains the probabilities that the front moves from one configuration to another at each growth step, calculated exactly by solving the Laplace equation and using the proper normalization. The method is applied for a series of approximations, which include only a finite number of rows near the front. The matrix is then used to find the weights of the steady state growing configurations and the rate of approaching this steady state stage. The former are then used to find the average upward growth probability, the average steady-state density and the fractal dimensionality of the aggregate, which is extrapolated to a value near 1.64.Comment: 24 pages, 20 figure

Two-dimensional Granular Gas of Inelastic Spheres with Multiplicative Driving

We study a two-dimensional granular gas of inelastic spheres subject to multiplicative driving proportional to a power $|v(\vec{x})|^{\delta}$ of the local particle velocity $v(\vec{x})$. The steady state properties of the model are examined for different values of $\delta$, and compared with the homogeneous case $\delta=0$. A driving linearly proportional to $v(\vec{x})$ seems to reproduce some experimental observations which could not be reproduced by a homogeneous driving. Furthermore, we obtain that the system can be homogenized even for strong dissipation, if a driving inversely proportional toComment: 4 pages, 5 figures (accepted as Phys. Rev. Lett.

Discretized Diffusion Processes

We study the properties of the ``Rigid Laplacian'' operator, that is we consider solutions of the Laplacian equation in the presence of fixed truncation errors. The dynamics of convergence to the correct analytical solution displays the presence of a metastable set of numerical solutions, whose presence can be related to granularity. We provide some scaling analysis in order to determine the value of the exponents characterizing the process. We believe that this prototype model is also suitable to provide an explanation of the widespread presence of power-law in social and economic system where information and decision diffuse, with errors and delay from agent to agent.Comment: 4 pages 5 figure, to be published in PR