A Justification of the Timoshenko Beam Model through $\boldsymbol\Gamma$-Convergence

TL;DR

This paper rigorously justifies the Timoshenko beam model as an accurate approximation of 3D linear elasticity for beam-like structures using $ extGamma$-convergence, providing a solid mathematical foundation.

Contribution

It establishes a formal $ extGamma$-convergence proof that connects 3D elasticity to the Timoshenko beam model, clarifying the model's validity.

Findings

The energy functionals for 3D elasticity converge to the Timoshenko beam energy functional.

The Timoshenko model accurately captures the behavior of beam-like bodies in the linear-elastic regime.

The proof provides a rigorous mathematical basis for using the Timoshenko model in engineering applications.

Abstract

We validate the Timoshenko beam model as an approximation of the linear-elasticity model of a three-dimensional beam-like body. Our validation is achieved within the framework of $\Gamma$-convergence theory, in two steps: firstly, we construct a suitable sequence of energy functionals; secondly, we show that this sequence $\Gamma$-converges to a functional representing the energy of a Timoshenko beam.