Envelope enrichment method for homogenization of non-periodic structures

The homogenization of composite materials is critical for accurately predicting their mechanical performance, particularly when complex reinforcement arrangements are involved. Microstructural characteristics exert a substantial influence on the composite's properties during practical applications. A widely adopted approach for analyzing composites is the numerical simulation of a Representative Volume Element (RVE). Although this method is well-established for periodic RVEs, it encounters difficulties when applied to non-periodic meshes, which complicates the imposition of classical boundary conditions on nodes exhibiting varying properties. To address this challenge, we propose a novel methodology that involves modeling an envelope surrounding the RVE, to which periodic boundary conditions are applied. By defining the envelope as a homogeneous material, stress transmission to the RVE is facilitated. The stiffness tensor of the envelope is updated iteratively through a homogenization process, ultimately converging to the effective properties of the RVE. The method is validated on a non-periodic arrangement of spherical inclusions embedded within a matrix. Convergence is observed in the different cases studied within ten iterations and the results are found within the Voigt and Reuss bound.