A non-autonomous variational problem describing a nonlinear Timoshenko beam

TL;DR

This paper investigates a complex variational problem modeling a nonlinear Timoshenko beam, proving existence and properties of minimizers in a setting involving different regularity spaces, relevant in elastostatics and dynamics.

Contribution

It introduces a novel variational framework with mixed regularity spaces for nonlinear Timoshenko beams and establishes existence and qualitative properties of minimizers.

Findings

Existence of global minimizers proven.

Qualitative properties of minimizers analyzed.

Under additional conditions, local minimizers' existence and regularity established.

Abstract

We study the non-autonomous variational problem: \begin{equation*} \inf_{(\phi,\theta)} \bigg\{\int_0^1 \bigg(\frac{k}{2}\phi'^2 + \frac{(\phi-\theta)^2}{2}-V(x,\theta)\bigg)\text{d}x\bigg\} \end{equation*} where $k>0$, $V$ is a bounded continuous function, $(\phi,\theta)\in H^1([0,1])\times L^2([0,1])$ and $\phi(0)=0$ in the sense of traces. The peculiarity of the problem is its setting in the product of spaces of different regularity order. Problems with this form arise in elastostatics, when studying the equilibria of a nonlinear Timoshenko beam under distributed load, and in classical dynamics of coupled particles in time-depending external fields. We prove the existence and qualitative properties of global minimizers and study, under additional assumptions on $V$, the existence and regularity of local minimizers.