Residual stress gradient in a thin film within the dislocation pile-up 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 pile-ups. The initial shear stress within the film is relaxed via the formation of a pile-up 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 pile-up for an arbitrary, non-uniform residual stress profile is derived analytically by applying the force balance condition. This results in a singular integro-differential 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 pile-up 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 towards more complex models of residual stress formation in constrained material systems, specifically, thin films.
Versions
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BibTeX
@misc{druzhinin2026residual,
title = {{Residual stress gradient in a thin film within the dislocation pile-up theory}},
author = {A. V. Druzhinin and C. Cancellieri},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.12007},
eprint = {2606.12007},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12007},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Residual stress gradient in a thin film within the dislocation pile-up theory
AU - A. V. Druzhinin
AU - C. Cancellieri
PY - 2026
DA - 2026/06/10
PB - arXiv
DO - 10.48550/arxiv.2606.12007
UR - https://arxiv.org/abs/2606.12007
AB - A model for predicting the residual stress gradient in a thin film segment is developed on the basis of the theory of dislocation pile-ups. The initial shear stress within the film is relaxed via the formation of a pile-up 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 pile-up for an arbitrary, non-uniform residual stress profile is derived analytically by applying the force balance condition. This results in a singular integro-differential 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 pile-up 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 towards more complex models of residual stress formation in constrained material systems, specifically, thin films.
ER -