Nonlinear micromorphic Timoshenko beam modeling and vibration analysis of microstructured beams

Generalized continuum theories can describe the mechanical behavior of microstructured materials more accurately than the classical Cauchy theory. In this manuscript, a micromorphic beam theory is developed for the efficient multiscale analysis of the linear and nonlinear deformation and vibration behavior of metamaterial beams. The proposed approach extends the conventional nonlinear Timoshenko beam theory by including three additional independent degrees of freedom, which allow to accurately capture four distinct microstrains for stretch, bending, and two types of shear behavior at the microscale level. The novel beam model is able to capture size effects and can accurately describe beams with only few unit cells through the thickness direction. However, consisting of 3 macro and 3 micro degrees of freedom, it is much more efficient than 2D or 3D micromorphic continuum models. It is demonstrated that the micromorphic material parameters can be identified from comparison studies with representative volume elements of the microstructure. For the numerical discretization of the governing equations for static deformations as well as vibrations, the differential quadrature method is employed here. The presented numerical examples show the accuracy of the method in obtaining deflections, linear eigenfrequencies, and nonlinear frequency responses for metamaterial beams with weakly separated macro and micro scales.