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

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This study introduces a novel two-scale asymptotic analysis for periodic thin plates, overcoming limitations of the Asymptotic Homogenization Method (AHM) in bending. Numerical experiments confirm the method

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Area of Science:

  • Solid Mechanics
  • Computational Mechanics
  • Materials Science

Background:

  • Homogenization methods are efficient for periodic structures.
  • The Asymptotic Homogenization Method (AHM) has limitations for periodic plates in bending.
  • Existing methods struggle with the reduced periodicity in the bending direction.

Purpose of the Study:

  • To develop a two-scale asymptotic analysis technique for the bending of periodic thin plates.
  • To adapt homogenization methods for plate structures with bending deformation.
  • To address the limitations of AHM in analyzing periodic plate bending.

Main Methods:

  • Applied a two-scale asymptotic analysis technique.
  • Utilized Kirchhoff plate theory, neglecting normal strains along the thickness.
  • Transformed the 3D problem into a 2D problem governed by a fourth-order PDE with periodic coefficients.
  • Verified well-posedness using the Lax-Milgram theorem.
  • Proved the AHM solution's reasonability via two-scale convergence.

Main Results:

  • Developed a new homogenization method for periodic thin plates under bending.
  • Mathematically verified the well-posedness and accuracy of the proposed method.
  • Demonstrated the method's availability and precision through numerical experiments.
  • Successfully adapted homogenization for plate bending problems.

Conclusions:

  • The proposed two-scale asymptotic analysis is effective for homogenizing periodic Kirchhoff plates.
  • The method overcomes the limitations of traditional AHM in bending scenarios.
  • Numerical results validate the accuracy and applicability of the developed technique.