Global attractors for the coupled suspension bridge system with temperature

What is it about?

This paper deals with the longterm properties of thermo-mechanical dynamics (in the transversal direction) of a string-beam model composed of an extensible thermoelastic beam with hinged ends coupled with a thermoelastic cable with fixed ends. This system is related to the well-known Lazer-McKenna suspension bridge model, in that the spring and the beam are coupled through a continuously distributed system of suspenders, which are assumed to behave as one-sided springs. Hence, the restoring force is proportional to the elongation of the suspenders if they are stretched, and vanishes if they are compressed. Moreover, the beam constitutive equation contains a geometric nonlinear term, as proposed in the fifties by Woinowsky-Krieger, in order to account for a nonlinear dependence of the axial strain on the deformation gradient. This model is quite naive: it neglects torsional effects on the beam and the influence of the side parts and piers deformations. Nevertheless, it is doubly nonlinear and its statics and dynamics are nontrivial. The existence of regular global attractors for the associated solution semigroup is proved for time-independent supplies and any values of the real parameter accounting for he axial force acting in the reference configuration on the beam. .

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Why is it important?

The system is doubly nonlinear. In particular no mechanical dissipation occurs in the equations, since the loss of energy is entirely due to thermal effects. The existence of regular global attractors is obtained without resorting to a bootstrap argument.