The influence of surface stress and surface-induced internal residual stresses on the size-dependent behaviors of Kirchhoff microplate

  • Y. M. Yue
  • K. Y. XuEmail author
  • Z. Q. Tan
  • W. J. Wang
  • D. Wang


The present paper develops a size-dependent Kirchhoff microplate model with surface effects by using simplified strain gradient elasticity theory and surface elasticity theory. This new model is able to capture size-dependent behaviors and surface effects. The most noticeable difference of the proposed model from the existing plate models about microplates is that not only strain gradient and surface stress are taken into account, but also the surface-induced internal residual stresses are considered. Based on whether the plates having surface-induced internal residual stress or not, their governing equations have distinct differences. An extended Kantorovich method is employed to provide approximate closed-form solution for the rectangular microplate with simply supported boundary conditions. For the microplate with biaxial initial residual surface stress, the numerical results reflect that when the simply supported microplates do not have surface-induced internal residual stresses, internal length scale and biaxial surface residual stress have significant influence on the static bending of the microplates. However, when the simply supported microplates have nonzero surface-induced internal residual stresses, the effects of internal length scale and biaxial surface residual stress become very weak. It indicates that the effect of surface-induced internal residual stresses can counteract most of the effects of internal length scale and surface residual stress. This work provides a more general model for the analysis of microplate problems.


Strain gradient Kirchhoff microplate Surface stress Surface-induced internal residual stress 



We give our sincere thanks to China Scholarship Council (CSC), National Natural Science Foundation of China (No.11072138), and Natural Science Foundation of Shanghai (No. 15ZR1416100).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Y. M. Yue
    • 1
  • K. Y. Xu
    • 2
    Email author
  • Z. Q. Tan
    • 3
  • W. J. Wang
    • 4
  • D. Wang
    • 5
  1. 1.Department of Engineering MechanicsShijiazhuang Tiedao UniversityShijiazhuangChina
  2. 2.Department of Mechanics, Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  3. 3.School of Mechanical EngineeringChangzhou UniversityChangzhouChina
  4. 4.Department of Mechanical EngineeringUniversity of AlbertaEdmontonCanada
  5. 5.HCIG New-Energy Co., LtdShijiazhuangChina

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