Buckling and longterm dynamics of a nonlinear model for the extensible beam

Details Buckling and longterm dynamics of a nonlinear model for the extensible beam Buckling and longterm dynamics of a nonlinear model for the extensible beam

2011-02-07

arxiv.org/abs/1102.1259v1

Mathematical and Computer Modelling, Vol. 51 (2010), pp. 833-846

Physics

Mathematical Physics

14 pages, 2 figures

Scientific paper

This work is focused on the longtime behavior of a non linear evolution problem describing the vibrations of an extensible elastic homogeneous beam resting on a viscoelastic foundation with stiffness k>0 and positive damping constant. Buckling of solutions occurs as the axial load exceeds the first critical value, \beta_c, which turns out to increase piecewise-linearly with k. Under hinged boundary conditions and for a general axial load P, the existence of a global attractor, along with its characterization, is proved by exploiting a previous result on the extensible viscoelastic beam. As P<\beta_c, the stability of the straight position is shown for all values of k. But, unlike the case with null stiffness, the exponential decay of the related energy is proved if P<\bar\beta(k), where \bar\beta(k) < \beta_c(k) and the equality holds only for small values of k.

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