Journal of the
Korean Mathematical Society JKMS

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, 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 beams and arches is studied under the assumptions of the weak and strong damping. The presence of cracks forces weaker regularity results for the arch motion, as compared to the beam case.

Keywords: Shallow arch, beam, subdifferential, cracks, equation of motion

MSC numbers: 47J35, 35Q74, 35D30, 70G75

Supported by: Sudeok Shon is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020 R1I1A1A01065032).