Static and Dynamic Responses of One-Dimensional Hexagonal Piezoelectric Quasicrystal Microbeams Resting on Elastic Foundations.

At the microscale, the interplay between size effects and boundary constraints strongly influences the mechanical behavior of quasicrystal structures. To elucidate this interplay, we investigate the static and dynamic responses of one-dimensional (1D) hexagonal piezoelectric quasicrystal Timoshenko microbeams. Our model integrates the modified couple stress theory with a Winkler-Pasternak two-parameter elastic foundation. Under open-circuit conditions, the electric potential is generated solely by mechanical deformation through piezoelectric coupling. Using Hamilton's principle, we establish a non-classical beam theory framework for piezoelectric quasicrystals that simultaneously incorporates size effects, piezoelectric coupling, and foundation constraints. A normalized system contact stiffness is introduced to quantify the coupling between size effects and the foundation, providing a possible reference for evaluating foundation parameters. Fourier series solutions reveal that increasing the material length scale parameter markedly enhances both bending stiffness and the free vibration frequencies, particularly at small scales. The elastic foundation further suppresses static deformation and elevates all natural frequencies. Notably, the phason field exhibits a non-classical, non-monotonic response under foundation constraints. Increasing the Pasternak shear stiffness not only reduces the overall phason deflection but also triggers a qualitative reversal of the size sensitivity. The apparent critical Pasternak stiffness required for this transition decreases systematically as the Winkler spring stiffness increases, indicating a synergistic effect between the two foundation parameters. Detailed scanning around the critical interval further reveals a transitional regime in which the phason deflection first rises and then declines, signifying a competitive handover between two driving mechanisms within a narrow stiffness band. Unlike existing size-dependent quasicrystal beam models, the present formulation simultaneously incorporates a two-parameter elastic foundation and piezoelectric coupling in a Timoshenko microbeam, and further reveals a non-monotonic phason response governed by a critical energy partition ratio. These findings provide theoretical insight into the complex multi-field coupling in quasicrystals, and the open-circuit formulation may be particularly relevant for high-frequency or electrically insulated configurations.