A homogenization technique for materials with periodic lacunar fractal microstructure

This paper proposes an original homogenization approach for materials characterized by a periodic lacunar microstructure with fractal geometry. The geometrical complexity of these materials creates major difficulties for the application of conventional homogenization methods to evaluate their effective properties. As the fractal level increases, the number of degrees of freedom in the numerical models grows sharply, making the computations very expensive and, beyond a certain level, inaccessible. To address this challenge, an indirect homogenization technique was developed on the basis of a classical direct homogenization technique (periodic homogenization by the average method), by embedding into this homogenization process the iterative process of generating fractal geometries following the iterated function systems principle. The two techniques were applied to two-dimensional extruded geometries and to a fully three-dimensional case. For iteration levels where both techniques could be compared, their results were very close. In addition, the indirect technique is substantially more efficient, requiring lower computational cost in terms of time and memory, and enabling the study of iteration levels that cannot be reached with the direct one. Finally, when the iteration level tends to infinity the theoretical and numerical convergence values of the homogenized elasticity tensors are equal. This provides direct proof of the quality of the method.