Elastic properties of a disordered continuum of regular fractal structure: Self-consistent approach and finite-element method simulations

Macromolecular structures, as well as aggregation of filler in polymer-based composites, often may be described properly as fractals. Scaling behavior of the elastic moduli of a modeled fractal, the Sierpinski carpet, was the subject of this study. Sheng and Tao 1] and Patlazhan 2] found that, in the case of voids in on elastic host, axial and shear moduli exhibit distinct scaling dependencies on the size of the system. Nevertheless, it is widely accepted that moduli of random isotropic fractals (percolation clusters) scale with the same exponents. Explanation of the discrepancy is one of the main targets of the paper. The self-consistent approach and position space renormalization group technique (PSRG) have been applied for this goal. The mapping, corresponding to PSRG, was constructed numerically using the finite-element method (FEM) in the cases of voids and rigid inclusions. The self-consistent approach gives scaling behavior with exponents of values of about 0.11, independent of the modulus and type of inclusion, at developed stages of the fractal. It has been shown that mappings of PSRG on the plane, for two ratios of three independent moduli, have stable fixed points. This means that different elastic moduli exhibit scaling behavior with the same exponents (0.29 for voids and 0.17 for rigid squares) for developed fractal structure. The discrepancy in the exponent values obtained in the previous simulations is caused by the analysis of the initial stages of the structure. We believe that analogous results are valid for the wide class of self-similar fractals, and the dimension is the main parameter that governs the exponents and fixed point values.