Surface-elastic control of screw-dislocation equilibria in a finite elastic film: A spectral bifurcation analysis

The equilibrium behavior of a straight screw dislocation in an isotropic elastic film bounded by two identical Gurtin–Murdoch surfaces is investigated. The anti-plane boundary-value problem is formulated in Fourier space, where surface elasticity is represented by a wavenumber-dependent reflection factor. Repeated reflections between the two surfaces lead to an exact integral representation of the image force that satisfies the required reflection symmetry and recovers the classical traction-free film in the limit of vanishing surface elasticity. The equilibrium structure is governed by the ratio of the surface-elastic length to the film thickness. Analysis of the mid-plane force gradient identifies a unique critical value, β c = 0 . 701365 , where β c denotes the critical value of the dimensionless surface-elastic parameter β = ℓ s H at which the mid-plane equilibrium changes stability. Below β c , the mid-plane equilibrium is unstable, and two symmetric stable off-center equilibria exist; at β = β c , these equilibria merge with the central state, which becomes the unique stable equilibrium for β > β c . Near criticality, the off-center branches exhibit square-root scaling, while the associated energy barrier vanishes quadratically. A separate weak-surface-elasticity analysis shows that the stable off-center equilibria approach the film boundaries on the intrinsic surface-elastic length scale. Surface elasticity can therefore change both the number and stability of screw-dislocation equilibria in a finite film.