A Class of One Dimensional Periodic Microstructures Exhibiting Effective Timoshenko Beam Behavior

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 I -limit that we obtain corresponds to a non-linear model of Timoshenko type.