Multiscale modeling for planar lattice microstructures with structural elements

Formulations of linear and nonlinear multiscale analyses for media with lattice periodic microstructures based on the homogenization theory are proposed. For continuum media, the conventional homogenization theory leads to boundary value problems of continuum for both micro- and macroscales. However, it is rational to discretize lattice microstructures, such as cellular solids, by frame elements since they consist of slender members. The main difficulty in utilizing structural elements, such as frame elements, for microscale problems is due to the inconsistency between the kinematics assumed for the frame elements and the periodic displacement field for the microscale problem. In order to overcome this difficulty, we propose a formulation that does not employ the periodic microscale displacement, but the total displacement, including the displacement due to uniform deformation as well as periodic deformation, as the independent variable of the microscale problem. Some numerical examples of cellular solids are provided to show both the feasibility and the computational efficiency of the proposed method.