Long-time dynamics and singular limit of a shear beam model

Abstract

This paper is dedicated to studying the long-term dynamics of a beam model known as the Shear beam model (without rotary inertia). Unlike the classical Timoshenko beam model, which combines bending moment and shear force, the Shear beam model has only one wave speed without blow-up at lower frequencies. This distinction has a significant impact on the analysis of long-term dynamic properties. We prove that the Euler–Bernoulli beam equation can be obtained as a singular limit of the Shear beam model when the shear elasticity modulus \(\kappa \) tends to infinity. By introducing a dissipative mechanism in the vertical displacement equation, we prove the existence of a smooth global attractor with finite fractal dimension. Finally, we demonstrate that the global attractor for the Shear beam model converges upper-semicontinuously to the global attractor for the Euler–Bernoulli equation as \(\kappa \rightarrow \infty \).