A microstructure-dependent Kirchhoff plate model based on a reformulated strain gradient elasticity theory

Abstract

A new microstructure-dependent non-classical model for Kirchhoff plates is developed by using a reformulated strain gradient elasticity theory that incorporates both the strain gradient and couple stress effects. The equation of motion and the boundary conditions are simultaneously obtained through a variational formulation based on Hamilton’s principle. The new plate model contains one material constant to account for the strain gradient effect and one material length scale parameter to capture the couple stress effect. The newly developed non-classical plate model includes the plate model incorporating the couple stress effect alone and the plate model based on the classical elasticity as two special cases. To illustrate the new model, the buckling, static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived. The numerical results reveal that the presence of the strain gradient and couple stress effects leads to reduced plate deflections, enlarged critical buckling loads and increased natural frequencies. These microstructure effects are significant when the plate is very thin, but they are diminishing as the plate thickness increases. These predicted trends of the size effects at the micron scale agree with those observed experimentally.

Keywords:

• Buckling
• couple stress
• deflection
• free vibration
• Kirchhoff plate
• microstructure
• natural frequency
• size effect
• strain gradient

Acknowledgements

The authors would like to thank Dr. Vinu Unnikrishnan and one anonymous reviewer for their encouragement and helpful comments on an earlier version of the paper.

Additional information

Funding

GZ acknowledges the support by the National Natural Science Foundation of China [Grant # 12002086] and the Fundamental Research Funds for the Central Universities [Grant # 2242020R10027]. CM acknowledges the support by the National Natural Science Foundation of China [Grants # 11872149 and # 11472079].