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Analysis and simulations of a contact problem for a nonlinear dynamic beam with a crack. (English) Zbl 1248.74032
Summary: This work studies and simulates the dynamics of a Gao beam that may come in contact with a reactive foundation and which has an evolving symmetric edge crack. The problem is set as a variational inequality coupled with a differential equation for the evolution of the damage variable, which measures the severity of the crack. The existence of a local weak solution for the model is proved using approximations, result for abstract problems in Hilbert spaces, and a priori estimates. A finite-element scheme for the problem is proposed, and representative simulations presented. In particular, the dependence of the beam’s vibration frequency on the crack’s position and size is depicted.

MSC:
74M15 Contact in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74R10 Brittle fracture
74H20 Existence of solutions of dynamical problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
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