Abstract: The mathematical homogenization method (MHM) is generally implemented by finite element method, and its calculating accuracy depends completely on the order of perturbation and finite element，the perturbations in uncoupled form are defined as the multiplications of influence functions and the derivatives of homogenized displacements. The order of exact solutions for influence function depend on the order of pseudo load which is the right term of the governing equation whereas the exact solution of homogenized displacements and its different orders derivatives depend on the external loads; The order of elements depends on the exact solution of influence function and homogenized displacements simultaneously while the order of perturbations depend mainly on the accuracy of different order derivatives of homogenized displacements; For the static problems of periodical composite rod, the exact solutions can be obtained using correct order of MHM and finite element for the static problem of periodic composite rod subjected to different order of load. Then two dimensional(2D) periodical composite are explored similarly，and the clamped boundary con-dition of periodical unite cell have great influence for calculating accuracy of MHM. Finally, the potential en-ergy functional is used to evaluate the accuracy of MHM, and numerical comparisons validate the conclu-sions.

Key words: periodical composites, mathematical homogenization method, perturbation order, element order, potential energy functional