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

TL;DR

This paper analyzes the long-term behavior and stability of vibrations in an extensible elastic beam on a viscoelastic foundation, identifying conditions for buckling, stability, and energy decay.

Contribution

It establishes the existence of a global attractor for the nonlinear beam model and characterizes stability and decay properties depending on axial load and stiffness.

Findings

Buckling occurs when axial load exceeds a critical value .

Global attractor exists for the nonlinear beam system.

Energy decays exponentially when axial load is below a certain threshold.

Abstract

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.