Published 6 issues per year
ISSN Print: 1543-1649
ISSN Online: 1940-4352
Indexed in
Variational Principle and Mechanical Computation for Energy Bands of Periodic Materials
ABSTRACT
Based on the Bloch theorem and tight-binding theory, a variational principle is applied to analyze the energy bands of crystals. The stiffness matrix used in the finite element method (FEM) is introduced for the expression of the energy of the unit cell of the crystal, and thus the coordinate transformation technique in FEM is applied in the assembly of the total energy and the stiffness matrix of the crystal. The periodical boundary conditions are given, and the energy bands of the three-dimensional crystal are computed. Using the dynamic substructure model and introducing the dual variables, the energy band analysis of the free vibration of the atomic chain is transformed into symplectic eigenvalue problems. The potential energy and mixed energy are computed by combining segments recursively until the shortest periodical length of the chain is assembled. Finally, the pass-band eigenvalues of the energy bands are calculated using the Wittrick-Williams algorithm. The numerical results are given to illustrate the potential of the theory and algorithm developed.
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Zhang H.W., Yao Z., Wang J.B., Zhong W.X., Phonon dispersion analysis of carbon nanotubes based on inter-belt model and symplectic solution method, International Journal of Solids and Structures, 44, 20, 2007. Crossref