Implications of an analysis of the shear transformation of a zone of an oblate spheroid for dislocation theory, grain boundary sliding and superplasticity
Abstract
Grain boundary sliding models of superplastic flow take the elementary event to be the shear of an oblate-spheroidal unit embedded in a solid matrix, whose rate follows transition state theory. The mechanics of this unit, and the assumptions linking it to dislocations and macroscopic flow, have not been stated explicitly. Here the unit is analysed as an Eshelby inclusion, a continuum, mathematical treatment without using physical concepts. The analysis gives closed-form shear and volumetric energy coefficients ($0.4456$ and $0.8889$ for aspect ratio 1/2 and Poisson ratio 1/3) and shows that the shear stress of the unit is a force dipole, decaying as r−3 and changing sign with direction, unlike a dislocation. The inputs, assumptions and proposals needed to connect this unit of shear to dislocation theory and superplastic flow are then stated and classed as experimentally validated, physically realistic or fitting constants. Under three such proposals, a row of about four units acquires the Peierls-Nabarro profile of a full dislocation, and two to three units that of a Shockley partial; the Peierls and Frank-Read stresses and the stacking-fault energy then follow from experimental inputs. With the unit energy as the free energy of activation and strains fitted to superplastic data, the flow law reproduces all 52 measured strain rates of a Cu alloy within an order of magnitude. Molecular-dynamics calculations for Cu outline the extension to dislocation emission, mesoscopic grain boundary sliding and the inverse Hall-Petch effect. This gives phenomenological models of superplastic flow a mechanical basis.
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BibTeX
@misc{linda2026implications,
title = {{Implications of an analysis of the shear transformation of a zone of an oblate spheroid for dislocation theory, grain boundary sliding and superplasticity}},
author = {Albert Linda and K. A. Padmanabhan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.29061},
eprint = {2606.29061},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.29061},
}
RIS
TY - UNPB
TI - Implications of an analysis of the shear transformation of a zone of an oblate spheroid for dislocation theory, grain boundary sliding and superplasticity
AU - Albert Linda
AU - K. A. Padmanabhan
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2606.29061
UR - https://arxiv.org/abs/2606.29061
AB - Grain boundary sliding models of superplastic flow take the elementary event to be the shear of an oblate-spheroidal unit embedded in a solid matrix, whose rate follows transition state theory. The mechanics of this unit, and the assumptions linking it to dislocations and macroscopic flow, have not been stated explicitly. Here the unit is analysed as an Eshelby inclusion, a continuum, mathematical treatment without using physical concepts. The analysis gives closed-form shear and volumetric energy coefficients ($0.4456$ and $0.8889$ for aspect ratio 1/2 and Poisson ratio 1/3) and shows that the shear stress of the unit is a force dipole, decaying as r^−3 and changing sign with direction, unlike a dislocation. The inputs, assumptions and proposals needed to connect this unit of shear to dislocation theory and superplastic flow are then stated and classed as experimentally validated, physically realistic or fitting constants. Under three such proposals, a row of about four units acquires the Peierls-Nabarro profile of a full dislocation, and two to three units that of a Shockley partial; the Peierls and Frank-Read stresses and the stacking-fault energy then follow from experimental inputs. With the unit energy as the free energy of activation and strains fitted to superplastic data, the flow law reproduces all 52 measured strain rates of a Cu alloy within an order of magnitude. Molecular-dynamics calculations for Cu outline the extension to dislocation emission, mesoscopic grain boundary sliding and the inverse Hall-Petch effect. This gives phenomenological models of superplastic flow a mechanical basis.
ER -