J. Korean Math. Soc.
Published online July 26, 2022
Copyright © The Korean Mathematical Society.
Semion Gutman, Junhong Ha, and Sudeok Shon
University of Oklahoma, Korea University of Technology and Education
We develop a rigorous mathematical framework for studying dynamic behavior of cracked beams and shallow arches. The governing equations are derived from the first principles using the Extended Hamilton's Principle, and stated in terms of the subdifferentials of the bending and the axial potential energies. The existence and the uniqueness of the solutions is established under various conditions. The corresponding mathematical tools dealing with vector-valued functions are comprehensively developed. The motion of the beams and arches is studied under the assumptions of the weak and strong damping. The presence of the cracks forces a weaker regularity results for the arch motion, as compared to the beam case.
Keywords: Shallow arch, beam, subdifferential, cracks, eigenvalues and eigenfunctions
MSC numbers: 47J35, 35Q74, 35D30, 70G75
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