Two-scale convergence analysis and numerical simulation for periodic Kirchhoff plates

Abstract

Homogenization analysis methods provide a high-efficiency tool to address periodic structures. However, the popular Asymptotic Homogenization Method (AHM) cannot be directly applied to the homogenization of periodic plate structures due to less periodicity in the bending deformation direction. In this paper, we propose a two-scale asymptotic analysis technique to cope with the bending problem of periodic thin plates. By ignoring the normal strains along the thickness direction and using the Kirchhoff plate theory, the three-dimensional structure problem is transformed as a two-dimensional periodic plate problem by a fourth-order partial differential equation with periodic coefficients. Besides, the well-posedness analysis of the PDE is verified by the Lax-Milgram theorem, and the reasonability of the two-scale asymptotic expansion solution by the AHM is mathematically verified through the proof of two-scale convergence. Finally, numerical experiments verify the availability and accuracy of the proposed homogenization method for periodic Kirchhoff plates.

Keywords

• Kirchhoff plate
• Asymptotic analysis
• Partial differential equation
• Periodic
• Two-scale convergence method