Theoretical/Computational Studies of Fractals in Materials

Abstract

1. The industrial importance of adhesives is constantly increasing; yet it is difficult to systematize the vast amount of practical knowledge, which has accumulated covering chemistry, interfacial physics, and mechanics. We attempt to bridge the gap between polymer science and fracture mechanics.

2. We develop a relation for fracture strength of a porous solid or random network near the percolation threshold.

3. A new view point for the dislocation velocity exponent based on the iteration model of dislocation regeneration process is established. Results indicate that there is a relationship between dislocation velocity exponent and dislocation regeneration dynamic bifurcation with the characteristics in plastic deformation process of materials.

4. The spiralling self avoiding lattice walk in a random environment is studied. The average size of an N-step walk is asymptotically proportional to N logN with a coefficient which increase with disorder.

A phase transition appears after scaling the temperature appropriately with system size. In the low-temperature phase the walk segment occupy a few low energy positions while in the high temperature phase they are effectively free. An analogy with the Random Energy Model is pointed out. The model also allows for a spinglass interpretation. The pair varies chaotically with temperature both above and below the critical point.

5. An attempt is made to model the self-organizing dynamics of low dimensional magnetic domain patterns with magnetic bubble traps. Asymptotic forms for distribution of avalanche sizes and lifetimes are found analytically. They are power laws with exponential damping factors.

6. Complex ground states appearing in classical gas models and their behavior in the presence of thermal motions are discussed. We base our model on the nonperiodic tilings of the plane and Ising type model with a fractal ground state.

7. We model transport properties of natural porous media at low saturation of wetting phase i.e. when the total wetting phase saturation is the sum of thin films and pendular structure inventories. 8. A differential equation for diffusion in isotropic arid homogenous fractal structures is derived within the context of fractional calculus. The asymptotic behavior of the probability density function is obtained exactly. Also modeling is done of the transport properties of natural porous media at low saturation of a wetting phase.

9. As a by product of this project two students have been enrolled for their Ph.Ds in 1999 with the following Theses Topics: - i. Fractal study of Some Models related to Physical and Biological Systems. ii. Fractal study of the Dynamics of Fracture Mechanics.

10. Three M. Phil. Students R. Ahmed, R. Mazhar, and N. Hussain completed their M. Phil. theses as a by product of this Project in 1998.

11. A book entitled “Fractals" has been accepted for publication by the National Book Foundation Islamabad. (Copy of Acceptance letter is attached).

12. The following Monographs have been sent to the Publishers against each title.

i. Fractals and Multi Fractal Structures in Kinetic Critical Phenomena: Bureau of Scientific Publication, Pretoria, South Africa. (1999).

ii. Random Walks with Memory: Ellis Horwood Ltd., Chichester, West Sussex, UK. (l999).

Iii. Fractals in Condensed Matter Physics: An Introduction: World Scientific Publication, Singapore. (1999).