Analysis and simulations of a nonlinear elastic dynamic beam. (English) Zbl 1261.35093
Summary: A model for the dynamics of a Gao elastic nonlinear beam, which is subject to a horizontal traction at one end, is studied. In particular, the buckling behavior of the beam is investigated. Existence and uniqueness of the local weak solution is established using truncation, approximations, a priori estimates, and results for evolution problems. An explicit finite differences numerical algorithm for the problem is presented. Results of representative simulations are depicted in the cases when the oscillations are about a buckled state, and when the horizontal traction oscillates between compression and tension. The numerical results exhibit a buckling behavior with a complicated dependence on the amplitude and frequency of oscillating horizontal tractions.
MSC:
| 35L86 | Unilateral problems for nonlinear hyperbolic equations and variational inequalities with nonlinear hyperbolic operators |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 49J40 | Variational inequalities |
| 74H20 | Existence of solutions of dynamical problems in solid mechanics |
| 74M15 | Contact in solid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
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