Size-Dependent Behavior of Piezoelectric Circular Ultrathin Films With Consideration of Surface Effects for NEMS Applications

A strain-consistent elastic plate model with surface elasticity

“…Some controversial results have been reported in the literature regarding the effect of the initial surface tension σ 0 on the effective bending rigidity [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. For example, some works does not predict any contribution of the initial surface tension σ 0 to the effective bending rigidity [6,10,11], although other works predict either a positive [7] or a negative contribution [5,8,9,14,15].…”

“…Most of the previous related models [6][7][8][9][10][11] treated the surface tension σ 0 as a finite value but the σ 0 -induced residual stresses as infinitesimal, and actually adopted finite strain expression (which contain not only infinitesimal strains but also extra displacement gradients) for the surface and infinitesimal strain expression for the bulk plate. In the present strain-consistent plate model, different than the previous models [6][7][8][9][10][11], both the initial surface tension σ 0 and the σ 0 -induced residual strains/stresses are treated as finite values and the above strain expressions (6) of von Karman type will be consistently employed for both the surface and the bulk plate.…”

“…In the present strain-consistent plate model, different than the previous models [6][7][8][9][10][11], both the initial surface tension σ 0 and the σ 0 -induced residual strains/stresses are treated as finite values and the above strain expressions (6) of von Karman type will be consistently employed for both the surface and the bulk plate. Now let us consider the bulk elastic strain energy and the surface strain energy.…”

“…Here, it must be highlighted that the third and fourth terms on LHS of the linearized equation (19) are kept simply because the residual strains are now treated as finite values in the present model. Obviously, the third and fourth terms on LHS of (19) would have been deleted in the linearized equation (19) as nonlinear terms if the bulk residual strains are considered as infinitesimal, as assumed in the original GM model and all previous related models [6][7][8][9][10][11]. This feature distinguishes the present model from the original GM model and all previous related models.…”

“…In connection with this inconsistence between linearized infinitesimal bulk strains and second-order finite surface strains in the GM model, several controversial results have been seen in the recent literature on elastic plate models with surface elasticity. For example, in some of the recent works (such as [6][7][8][9][10][11]), the surface tension σ 0 -induced residual (compressive) stress has not been included in the linearized governing equations even for an elastic plate with in-plane movable edges, which led to a conclusion that tensile surface tension σ 0 increases natural frequencies of small-amplitude vibration of the plate. On the other hand, in some other works such as [12,13], the surface tension σ 0 has not been included although the surface tension σ 0 -induced residual (compressive) stress has been included in the linearized governing equations, which leads to a conclusion that the surface tension σ 0 -induced residual compressive stress can cause self-buckling of an extremely thin free elastic plate.…”