On the nature of periodic traction boundary conditions in micromechanical FE analyses of unit cells

Well-formulated unit cells used in micromechanical analysis are mostly derived from periodic conditions, which result in boundary conditions for the unit cells relating the displacements and traction on one part of the boundary of the unit cell to those on another. Finite elements are often adopted as the solver of this boundary value problem. Since most FE codes are formulated from a displacement-based variational principle, such as the minimum total potential energy, periodic displacement boundary conditions are essential boundary conditions. There has been no rigorous account available, however, about the nature of the periodic traction boundary for these unit cells. It is the objective of this paper to prove that periodic traction boundary conditions are indeed natural boundary conditions from the minimum total potential energy principle. As a result, imposing them as if they were essential boundary conditions is neither necessary nor correct. Unit cells; Periodic conditions; Traction boundary conditions; Natural boundary conditions, Total potential energy; Variational principle.