A Micromorphic Beam Theory for Beams with Elongated Microstructures

A novel micromorphic beam theory that considers the exact shape and size of the beam's microstructure is developed. The new theory complements the beam theories that are based on the classical mechanics by modeling the shape and size of the beam's microstructure. This theory models the beam with a microstructure that has shape and size and exhibits microstrains that are independent of the beam's macroscopic strains. This theory postulates six independent degrees of freedom to describe the axial and transverse displacements and the axial and shear microstrains of the beam. The detailed variational formulation of the beam theory is provided based on the reduced micromorphic model. For the first time, the displacement and microstrain fields of beams with elongated microstructures are developed. In addition, six material constants are defined to fully describe the beam's microscopic and macroscopic stiffnesses, and two length scale parameters are used to capture the beam size effect. A case study of clamped-clamped beams is analytically solved to show the influence of the beam's microstructural stiffness and size on its mechanical deformation. The developed micromorphic beam theory would find many important applications including the mechanics of advanced beams such as meta-, phononic, and photonic beams.