A strain-consistent elastic plate model with surface elasticity

Abstract

A strain-consistent elastic plate model is formulated in which both initial surface tension and the induced residual stress are treated as finite values, and the exactly same strain expressions are consistently employed for both the surface and the bulk plate. Different than most of previous related models which follow the original Gurtin–Murdoch model and include some non-strain displacement gradient terms (which cannot be expressed in terms of the surface infinitesimal strains or the von Karman-type strains) in the surface stress–strain relations, the present model does not include any non-strain displacement gradient terms in the surface stress–strain relations. For a free elastic plate with in-plane movable edges, the present model predicts that initial surface tension exactly cancels out the induced residual compressive stress. On the other hand, if all edges are in-plane immovable, residual stress cannot develop in the plate and the initial surface tension causes a tensile net membrane force. In addition, the present model predicts that initial surface tension reduces the effective bending rigidity of the plate, while this reduction does not depend on Poisson ratio. In particular, self-buckling of a free elastic plate under tensile surface tension cannot occur unless the effective bending rigidity of plate vanishes or becomes negative.