Materials science

Boundary value problem

Bending

Classical mechanics

Surface (topology)

Bending of plates

Solid mechanics

Deflection (physics)

Physics

Mathematics

Elasticity (physics)

Microstructure

Plate theory

Mathematical analysis

Geometry

Length scale

Composite material

Surface energy

Mechanics

Acta Mechanica

A new non-classical model for circular Kirchhoff plates subjected to axisymmetric loading is developed using a modified couple stress theory, a surface elasticity theory and Hamilton’s principle. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation. The new plate model contains a material length scale parameter to capture the microstructure effect and three surface elasticity constants to describe the surface energy effect. The current non-classical plate model includes the plate models considering the microstructure influence or the surface energy effect alone as special cases and recovers the classical elasticity-based Kirchhoff plate model when both the microstructure and surface energy effects are suppressed. To demonstrate the new model, the static bending problem of a clamped solid circular Kirchhoff plate subjected to a uniform normal load is analytically solved by directly applying the general formulas derived. The numerical results reveal that the deflection of the clamped circular plate predicted by the current Kirchhoff plate model is smaller than that predicted by the classical elasticity-based model, but the former approaches the latter with the increase of the plate thickness. It is also found that the difference between the plate deflection predicted by the model incorporating the surface energy effect and that predicted by the classical model is significant when the plate thickness is sufficiently small. However, the difference is diminishing as the plate thickness increases. These predicted size effects at the micron scale agree with the general trends observed in experiments.

A non-classical model for circular Kirchhoff plates incorporating microstructure and surface energy effects