MECHANICAL MODELING OF QUASI-CRYSTALS AT THE MACRO- AND NANO-LEVELS: A STATE-OF-THE-ART REVIEW

This work is a state-of-the-art review on the mechanical modeling of macro-and nano-heterogeneous quasi-crystals (QCs), and specifically on elastic, piezoelectric and magneto-electro-elastic (MEE) materials that containing cracks, holes and inclusions.QCs exhibit unique properties, including high stiffness and brittle behavior under both static and dynamic loading, making them promising for advanced technological applications.The paper first summarizes available fundamental solutions and half-space Green's functions in elastostatics and elastodynamics.These provide the foundation for modeling the mechanical response of QCs using advanced computational techniques such as the boundary element method (BEM), meshless methods, Green's function methods and the method of fundamental solutions, all of which serve as a basis for numerical solutions involving various mechanical models.Next, models based on Bak's theory are discussed, highlighting the macro-and nano-scale approaches and including the Gurtin and Murdoch surface elasticity model for nanoscale heterogeneities.The review systematically discusses elastostatic and elastodynamic formulations across different material classes and defect configurations, providing an overview of current modeling capabilities.Finally, open questions and challenges are identified, offering guidance for future research aimed at bridging the gap between theoretical models and practical applications of QCs in high-tech industries.