<p>The present research focuses on the longitudinal vibration analysis of a microbar resting on an elastic foundation, incorporating deformable (spring-based) boundary conditions. The material is modeled as functionally graded in two spatial directions. The governing equation for the axial vibration of the microbar is derived using the Hamilton’s principle and the Aifantis strain gradient theory, which captures small-scale effects. Using the resulting governing and force boundary equations, an eigenvalue problem is constructed, and the roots of this problem yield the axial vibration frequencies. The main novelty of this study lies in the development of a unified and generalized formulation that simultaneously accounts for bi-directional grading, deformable boundary conditions, elastic foundation interaction, and strain gradient elasticity within a single eigenvalue framework. Unlike most existing studies, which often assume idealized boundary conditions, the proposed approach allows general boundary conditions to be modeled simply by adjusting spring parameters. The model is validated through comparisons and convergence studies, and the effects of material gradation, scale parameter, elastic foundation, geometric properties, and spring stiffness are thoroughly examined.</p>

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Longitudinal vibrations of bi-directionally functionally graded microbars with deformable boundaries

  • Murat Akpınar,
  • Büşra Uzun,
  • Mustafa Özgür Yaylı

摘要

The present research focuses on the longitudinal vibration analysis of a microbar resting on an elastic foundation, incorporating deformable (spring-based) boundary conditions. The material is modeled as functionally graded in two spatial directions. The governing equation for the axial vibration of the microbar is derived using the Hamilton’s principle and the Aifantis strain gradient theory, which captures small-scale effects. Using the resulting governing and force boundary equations, an eigenvalue problem is constructed, and the roots of this problem yield the axial vibration frequencies. The main novelty of this study lies in the development of a unified and generalized formulation that simultaneously accounts for bi-directional grading, deformable boundary conditions, elastic foundation interaction, and strain gradient elasticity within a single eigenvalue framework. Unlike most existing studies, which often assume idealized boundary conditions, the proposed approach allows general boundary conditions to be modeled simply by adjusting spring parameters. The model is validated through comparisons and convergence studies, and the effects of material gradation, scale parameter, elastic foundation, geometric properties, and spring stiffness are thoroughly examined.