Comparison of Unit Cell Geometry for Bloch Wave Analysis in Two Dimensional Periodic Beam Structures
Date of Award
8-2018
Document Type
Thesis
Degree Name
Master of Science (MS)
Department
Mechanical Engineering
Committee Member
Dr. Lonny L. Thompson, Committee Chair
Committee Member
Dr. Gang Li
Committee Member
Dr. Huijuan Zhao
Abstract
The wave propagation behavior for one-dimensional rods, beams, and two-dimensional periodic lattice structures are studied using Bloch wave finite element analysis. Dispersion relations relating wave vector components and frequency are obtained by enforcing periodic conditions on a unit cell and solving an eigenvalue problem.
The one-dimensional Bloch wave finite element analysis is performed for continuous rod and beam structures treated as periodic structures with repeating unit cells in order to validate the frequency-wavenumber dispersion relationships obtained with exact solutions. In the case of the rod structure, the frequency-wavenumber relation is linear with a constant wave speed, whereas for the beam structure, the frequency-wavenumber relation is nonlinear and manifests dispersive behavior. For the beam structure, both classical Bernoulli-Euler beam theory and Timoshenko beam theory which includes transverse shear deformation and mass rotary inertia effects are compared. Results from the Bloch wave finite element analysis are shown to converge to the exact solutions with mesh refinement.
For two-dimensional Bloch wave analysis, both periodic rectangular grid lattices and hexagonal honeycomb structures are considered for both in-plane and out-of- plane bending free-wave propagation. For the rectangular grid lattice, there is only one unique choice of unit cell and basis vectors for Bloch wave analysis. Results for this case display expected anisotropic dispersion behavior with wave direction verified with results in the literature.
For hexagonal honeycomb structures, the periodic unit cell used for Bloch wave analysis is not unique. In the literature, truncated rectangular unit cells with rectangular basis, and different rhombic unit cells in skew coordinates with wave analysis in contra-variant basis directions have been used to study frequency response from Bloch wave analysis. Rhombic unit cell with contra variant basis is scaled and transformed to a rectangular basis. The frequency-wavenumber relationship for truncated hexagonal unit cell is compared to the frequency-wavenumber relationship of rhombic unit cell in contra variant basis. Both in-plane and out of plane wave propagation analysis is performed.
Recommended Citation
Marneni, Likitha, "Comparison of Unit Cell Geometry for Bloch Wave Analysis in Two Dimensional Periodic Beam Structures" (2018). All Theses. 2926.
https://open.clemson.edu/all_theses/2926