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. This reformulated strain gradient elasticity theory evolves from Form I of Mindlin’s strain gradient elasticity theory and the main advantage of this theory is the simultaneous incorporation of the strain gradient and couple stress effects using just one material length scale parameter for each effect. A variational formulation based on Hamilton’s principle is also employed, which leads to the simultaneous determination of the equations of motion and the complete boundary conditions and provides a unified treatment of the strain gradient and couple stress effects. 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 current non-classical plate model reduces to its classical elasticity-based counterpart when the strain gradient and couple stress effects are both suppressed. In addition, the newly developed plate model includes the models considering the strain gradient effect only or the couple stress effect alone as two special cases. To illustrate the new model, three boundary value problems dealing with the buckling, static bending and free vibration of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived and compared with the corresponding results based on the modified couple stress theory and classical continuum theory. For the buckling problem, the numerical results reveal that the critical buckling load predicted by the current non-classical or the modified couple stress theory model is always higher than that by the classical model, and the difference is significant when the plate is sufficiently thin. For the static bending problem, it is observed that the maximum deflection predicted by the current or modified couple stress theory model is always smaller than that predicted by the classical model, and the difference is significant when the plate thickness is very small but diminishes as thickness increases. For the free vibration problem, it is found that the first natural frequency predicted by the current or modified couple stress theory model is always higher than that predicted by the classical model. In addition, the natural frequency of the plate is also influenced by the microstructure-dependent strain gradient and couple stress effects, which is significant for very thin plates.These predicted trends of the size effects at the micron scale agree with those observed experimentally.


A Non-Classical Kirchhoff Plate Model Incorporating the Couple Stress and Strain Gradient Effets

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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