Residual stress gradient in a thin film within the dislocation pileup theory

Abstract

A model for predicting the residual stress gradient in a thin film segment is developed on the basis of the theory of dislocation pileups. The initial shear stress within the film is relaxed via the formation of a pileup of screw dislocations against the impenetrable film–substrate interface. Plastic strain is related to the dislocation density, leading to a fundamental equation, which links the residual stress to this density. The distribution of dislocations within the pileup for an arbitrary, non-uniform residual stress profile is derived analytically by applying the force balance condition. This results in a singular integrodifferential equation for the residual stress profile. The equation is solved numerically by a collocation method for various initial stress distributions: constant, linear, parabolic, and exponential functions. The solutions demonstrate that the established residual stress profile strongly depends on the film segment’s thickness-to-width ratio and the initial stress distribution. As this ratio increases, stress relaxation becomes more effective away from the film–substrate interface. In all cases, equilibrium requires a pileup containing dislocations with both positive and negative Burgers vectors. The total number of dislocations and their density distribution vary significantly with the initial stress profile. This model provides a critical step toward more complex models of residual stress formation in constrained material systems, specifically, thin films.