A class of one dimensional periodic microstructures exhibiting effective Timoshenko Beam behavior

¹ Université de Toulon, Institut de Mathématiques de Toulon, Toulon, France
² Università degli Studi di Sassari, DADU, Alghero, Italy
³ Università degli Studi dell’Aquila, M&MoCS, L’Aquila, Italy

^* Corresponding author: seppecher@imath.fr

Received: 15 March 2023
Accepted: 15 June 2023

Abstract

We study, from a variational viewpoint, the asymptotic behavior of a planar beam with a periodic wavy shape when the amplitude and the wavelength of the shape tend to zero. We assume that the beam behaves, at the microscopic level, as a compressible Euler–Bernoulli beam and that the material properties have the same period as the geometry. We allow for distributed or concentrated bending compliance and for a non-quadratic extensional energy. The macroscopic Γ-limit that we obtain corresponds to a non-linear model of Timoshenko type.