Complete 1D continuum model for a pantographic beam by asymptotic homogenization from discrete elements with shear deformation measure

Starting from a discrete model of a pantographic beam based on beam elements with 6 DOF in 2D space, 2 translational and 1 rotational for each end node, and proposing an energy quadratic in elongation, shear, bending and torsion deformation measures, we find by asymptotic homogenization a model of a complete 1D second gradient continuum. Compared to previous models, the discrete model of the present work includes shear deformation by considering the beam element's end nodes relative orientation. The resulting continuum model differs in a stiffness coefficient stemming from the shear deformation, but it coincides in the limit of shear stiff elements.