Elena Cherkaev, University of Utah
2023 AWM Research Symposium
Recent Advancements in the Mathematics of Materials Science [Organized by Anna Zemlyanova and Silvia Jimenez Bolanos]

A key feature of quasicrystalline microgeometry is a long-range order in the absence of periodicity. Their structures possess self-similarity on large scales and lack translational symmetry. Quasiperiodic geometries can be modeled using the cut-and-projection method that restricts or projects a periodic function in a higher dimensional space to a lower dimensional subspace cut at an irrational projection angle. Homogenized equations for the effective behavior of a quasiperiodic composite can be derived by cutting and projecting a periodic function in a higher dimensional space. Using equations for the local problem in the higher dimensional space established in the homogenization process, we develop the Stieltjes analytic representation of the effective properties of quasiperiodic materials; this representation determines the spectral characteristics of fields in quasicrystalline composites and used to derive bounds for the effective properties.

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