A nonlinear elastic beam system with inelastic contact constraints. (English) Zbl 1076.74029

Russell, David L.; White, Luther W.

Appl. Math. Optimization 46, No. 2-3, 291-312 (2002).

Summary: We study freely propagating inertial, i.e., unforced, waves, in an elastic beam constrained so that all motion takes place above and on a flat, rigid support surface, subject to a gravitational force and a compressive longitudinal load. Contact between the beam and the support surface is assumed to be completely inelastic. A nonlinear beam model is used, incorporating a quartic extension of the familiar quadratic potential energy functional for the standard Euler-Bernoulli model. After briefly reviewing the rationale for the model and some of its properties, as developed in earlier articles, we present existence and uniqueness results for the constrained system obtained with the use of a “penalty function” approach involving the addition of a “uni-directional friction” dissipative term, active only when the constraint is violated, to the unconstrained system.

MSC:

74J30	Nonlinear waves in solid mechanics
74K10	Rods (beams, columns, shafts, arches, rings, etc.)
74H20	Existence of solutions of dynamical problems in solid mechanics
74H25	Uniqueness of solutions of dynamical problems in solid mechanics
74M15	Contact in solid mechanics

Keywords:

Nonlinear beam; Supported beam; Buckling; Traveling waves; quartic potential energy functional; existence; uniqueness; penalty function