# Hall–Petch Description of the Necking Point Stress

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dislocation Density Based Constitutive Modeling of the Hall–Petch Behavior

## 3. The Necking Stress Locus as a Basis for an H–P Analysis

## 4. An H–P Dependence for ${\sigma}_{N}$ of Nickel and Titanium—A Compilation of Experimental Results

^{1/2}, which is in fair agreement with what is typically reported for Ni [14]; the Pearson’s $r$ value for the entire dataset approximation is 0.95.

^{1/2}, is only slightly higher than the value of ${K}_{0.2}^{HP}$ corresponding to the 0.2% proof strain. Considering the significant scatter of experimental data, it is fair to say that both values are practically indistinguishable, and as a first order approximation, a near equality ${K}_{0.2}^{HP}\approx {K}_{N}^{HP}$ holds.

## 5. Discussion

^{1/2}for a specific sub-set (plotted as a dashed linear regression line) is almost triple the slope of the linear regression line for the whole dataset (137 MPa × μm

^{1/2}). Furthermore, both these values are remarkably different from the H–P coefficient of 173 MPa × μm

^{1/2}derived from the data presented by Figueiredo and Langdon [69] for ultrafine grained Grade 2 Ti. The results reported by Khamsuk et al. [71] for fine grain Al1100, with a different processing history, corroborate the above statement that the H–P coefficient is sensitive to the microstructure. These authors showed that within approximately the same range of grain sizes, the slope of the H–P diagrams varied by a factor of five—from 28 MPa × μm

^{1/2}for cold-rolled and annealed samples to 58 MPa × μm

^{1/2}for those produced by torsion, and 139 MPa × μm

^{1/2}for the material fabricated by accumulated roll bonding.

## 6. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Hall–Petch plots showing the grain size dependence of (

**a**) the yield stress ${\sigma}_{0.2}$ and (

**b**) the true stress at maximum load ${\sigma}_{N}$ for nickel polycrystals. The linear regression line for ${\sigma}_{0.2}$ presented in (

**a**) is reintroduced in (

**b**) for easier comparison. The references for the datapoints collected from different studies by Di Leo et al. (2019) are given in [14]; other points are adapted from [45,46,47,48,49].

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Vinogradov, A.; Estrin, Y. Hall–Petch Description of the Necking Point Stress. *Metals* **2023**, *13*, 690.
https://doi.org/10.3390/met13040690

**AMA Style**

Vinogradov A, Estrin Y. Hall–Petch Description of the Necking Point Stress. *Metals*. 2023; 13(4):690.
https://doi.org/10.3390/met13040690

**Chicago/Turabian Style**

Vinogradov, Alexey, and Yuri Estrin. 2023. "Hall–Petch Description of the Necking Point Stress" *Metals* 13, no. 4: 690.
https://doi.org/10.3390/met13040690