Effective property calculation and its numerical implementation of spatially graded plate structures based on asymptotic homogenization

Spatially graded heterogeneous plate structures that are mapped from periodic plates based on mapping func-tions are attracting increasing attention due to their excellent mechanical performances and high tailorability, and homogenization approaches are a powerful tool for their efficient numerical analysis. In this work, based on asymptotic homogenization, a unified framework of the effective properties prediction and the corresponding unit cell problems is first theoretically established for spatially-varying plates with arbitrarily-shaped micro-structures, which are distinct from those of periodic plates due to the Jacobian of mapping functions. Moreover, the FE formulation for efficient numerical implementation are also proposed for not only solid elements, but also shell and beam elements, where numerical treatment details in stiffness matrix formulation and periodic boundary conditions are elaborated, so that highly efficient effective stiffness computation can be achieved for thin-walled heterogeneous plate structures. At last, unit cells under different Jacobian matrices, which are related to different kinds of geometric deformation patterns, and two spatially graded plates are analyzed to corroborate the correctness and efficiency of the proposed method.