Alexander S. Balankin

Physics of fracture and mechanics of self-affine cracks

Abstract The physics associated with self-affine crack formation and propagation is discussed. Some novel concepts are suggested for the mechanics of self-affine cracks. These concepts are employed to model the crack face morphology and, in turn, to solve various problems with self-affine cracks. It is shown that linear elastic fracture mechanics (LEFM) is a special case of self-affine crack mechanics and should be used only in length scales larger than the self-affine correlation length. The theoretical results are confirmed by available experimental data. It is emphasized that the ASTM standards for test pieces for fracture toughness measurements must be completed by the specification of absolute specimen sizes which should be larger than the self-affine correlation length for the fracture surface roughness.

Effective degrees of freedom of a random walk on a fractal.

We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space F(ν) equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in F(ν) is inferred. The Laplacian operator in F(ν) is constructed. This allows us to map physical problems on fractals into the corresponding problems in F(ν). In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.

Entropic rigidity of a crumpling network in a randomly folded thin sheet

We have studied experimentally and theoretically the response of randomly folded hyperelastic and elastoplastic sheets on the uniaxial compression loading and the statistical properties of crumpling networks. The results of these studies reveal that the mechanical behavior of randomly folded sheets in the one-dimensional stress state is governed by the shape dependence of the crumpling network entropy. Following up on the original ideas by Edwards for granular materials, we derive an explicit force-compression relationship which precisely fits the experimental data for randomly folded matter. Experimental data also indicate that the entropic rigidity modulus scales as the power of the mass density of the folded ball with universal scaling exponent.

Probabilistic mechanics of self–affine cracks in paper sheets

Crack mechanics in single–notched sheets of paper is analysed by an approach that combines the concept of virtual crack trajectory ensemble with self–affine crack mechanics formalism. The crack initiation criterion is determined for three kinds of paper. The probability distributions of crack roughness and strength parameters are established. A probabilistic model of self–affine crack propagation in paper is proposed.

Models of self-affine cracks in brittle and ductile materials

Crack faces in solids are modelled as non-standard patterns whose standard parts are self-affine fractals. The morphological and mechanical features of brittle and ductile fracture are discussed. A concept of equivalent traction is employed to obtain the asymptotic stress fields near the crack tip, which in turn used to derive the relations between nanofracture and macrofracture toughness for brittle and ductile materials.

Scaling dynamics of seismic activity fluctuations

We study the dynamics of the seismic activity in Mexico within a framework of dynamic scaling approach to time series fluctuations, recently suggested by Balankin (Phys. Rev. E, 76 (2007) 056120). We found that the relative seismic activity and the long-sampled fluctuations of seismic activity both display a self-affine invariance within a wide range of consecutive seismic evens. Furthermore, we found that the long-sampled fluctuations of seismic activity obey the dynamic scaling ansatz analogous to the Family-Vicsek dynamic scaling ansatz in the theory of kinetic roughening of moving interfaces. These findings imply that the records of recurrent seismic events possess hidden, long-term correlations associated with the scaling dynamics of seismic activity fluctuations.

A new statistical distribution function for self-affine crack roughness parameters

The statistical topography of rupture lines in paper is studied. A new distribution function is introduced to characterize the crack roughness statistics. This function is employed to fit experimental data on the crack-Iength and crackwidth sample distributions for four kinds of paper. In all cases, the chi-squared test for the goodness of fit yields a significance level better than 0.97. The physical implications of experimental results are discussed.

The concept of multifractal elasticity

A new type of elasticity of random (multifractal) structures is suggested. A closed system of constitutive equations is obtained on the basis of two proposed phenomenological laws of reversible deformations of multifractal structures. The results may be used for predictions of the mechanical behavior of materials with multifractal microstructure, as well as for the estimation of the metric, information, and correlation dimensions using experimental data on the elastic behavior of materials with random microstructure.

The effect of fracture surface morphology on the crack mechanics in a brittle material

In conclusion we emphasize that for test pieces of characteristic specimen size >> self affine correlation length, the measured toughness and the critical energy release rate will be size independent. Hence the ASTM standards should be complicated by the dictation of the absolute specimen sizes. Such dictation may be formulated on the basis of special research to this problem, which include fractographic investigations together with standard mechanical tests

Fractal Analysis of EEG Signals in the Brain of Epileptic Rats, with and without Biocompatible Implanted Neuroreservoirs

Current epilepsy rates in Mexico are 4% (SERSAME-Health Ministry), of which 80% correspond to Temporal Lobe Epilepsy (TLE). Antiepileptic drug administration is systemic, meaning that 90% of the active agent is lost between administration and delivery to the epileptic focus in the brain. Severe toxic secondary effects may occur as a result. The present study is aimed at developing an alternative antiepileptic drug delivery system. In this study, a sol-gel nanostructured titania device, in which valproic acid (VPA) has been encapsulated. This is a nanoparticulate device, which is biocompatible with brain tissue. Stereotactic surgery was used to implant the reservoirs in the temporal lobe of Wistar rats, using chemical kindling, which was used to induce epilepsy. The reservoir was designed to release the drug at a constant rate over a period of at least one year. A functional study was performed on the efficiency of drug delivery in order to evaluate the effect on spontaneous and induced neuron electrical activity. A new discovery, which is presented here, shows that in the case of damaged brain tissue, as is the case in epilepsy, the accumulation of red globules, oxygen transportation results in the formation of calcium carbonate crystals which surround the epileptic focus. Because these crystals have a specific polarization, we propose to characterize their influence on the EEG using statistical methods. The electrical activity was measured by electroencephalography using 5 healthy rats without and 5 rats with an implanted VPA/device. Cerebral signals describe the complex behavior of the brain dynamics as a function of time. Fractal algorithms are sensitive to fluctuations and lead to the analysis and characterization of this kind of complex phenomena. A systematic study of these EEG’s was made in order to observe the variation of signals during seizures and on the controlled rate of release of VPA. We have estimated the Hurst exponent (H) to measure long range-dependence. Preliminary results show that for the control group, signal behavior is persistent (H>0.5), while for the epileptic group antipersistency was observed (H<0.5), with variations due seizure stages. During the protection period using VPA, preliminary results show that values tend to reach original behavior, as the crisis is stabilized.