A Statistical Perspective for Predicting the Strength of Metals - Revisiting the Hall-Petch Relationship using Machine Learning

Aug 5, 2026

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Mechanical properties of polycrystalline metals depend strongly on microstructure, yet existing datasets often lacked sufficient microstructural variability to support robust physics-based models. A probabilistic machine-learning framework is developed to predict flow stress as a function of microstructural feature variations. An extensive database is generated using over one million randomly sampled microstructures, then mixture models and neural networks quantify the flow-stress distribution and feature importance. Results show strong agreement with experiments and verify that, across grain sizes, the conventional Hall-Petch relationship is statistically valid for linking strength to average grain size and its comparative influence.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages50

Reading Time2 h 6 min

AccessFree to download

Abstract

1. Introduction

Grain refinement and size effects

Hall-Petch relationship and governing equations

Existing theoretical models and numerical approaches

Reported inconsistencies in literature parameters

arXiv :2209 .04891v2 [ cond-mat .mtrl-sci ] 12 May 2023 A Statistical Perspective for Predicting the Strength of Metals: Revisiting the Hall-Petch Relationship using Machine Learning Yejun Gua,b,􀀃, Christopher D. Stilesb,c , Jaafar A. El-Awadyb,􀀃 a Institute of High Performance Computing, A*STAR, Singapore, 138632, Singapore b Department of Mechanical Engineering, Johns Hopkins University, Baltimore, 21218, MD, USAc Research and Exploratory Development Department, Johns Hopkins University Applied Physics Laboratory, Laurel 20723, MD, USA Abstract The mechanical properties of a material are intimately related to its microstructure. This is particularly important for predicting mechanical behavior of polycrystalline metals, where microstructural variations dictate the expected material strength. Until now, the lack of microstructural variability in available datasets precluded the development of robust physicsbased theoretical models that account for randomness of microstructures. To address this, we have developed a probabilistic machine learning framework to predict the 􀀍ow stress as a function of variations in the microstructural features. In this framework, we 􀀌rst generated an extensive database of 􀀍ow stress for a set of over a million randomly sampled microstructural features, and then applied a combination of mixture models and neural networks on the generated database to quantify the 􀀍ow stress distribution and the relative importance of microstructural features. The results show excellent agreement with experiments and demonstrate that across a wide range of grain size, the conventional Hall-Petch relationship is statistically valid for correlating the strength to the average grain size and its comparative importance versus other microstructural features. This work demonstrates the power of the machine-learning based probabilistic approach for predicting polycrystalline strength, directly accounting for microstructural variations, resulting in a tool to guide the design of polycrystalline metallic materials with superior strength, and a method for overcoming sparse data limitations. Keywords: Dislocation-mediated Crystal Plasticity, Polycrystal Strength, Size E􀀋ects, Mixture Models, Neural Network 1. Introduction Discovering and designing materials with extraordinary properties is the ultimate goal of materials science. For structural materials, high strength is one of the most important mechanical properties, which can be enhanced by optimizing the microstructures [1, 2] . Among all strengthening mechanisms (grain re􀀌nement, strain hardening, solid solution strengthening, precipitate strengthening, and grain boundary hardening, etc.), grain re􀀌nement is arguably one of the simplest and most e􀀋ective ways to increase the material strength. It is commonly known that the 􀀍ow stress, 􀀛f , at a given macroscopic plastic strain, ", increases when the average grain size, dave , decreases. This size e􀀋ect is usually expressed by an empirical-based power law relationship in the form [3, 4]: 􀀛f (") = 􀀛0 (") + k0 (") 􀀁 dnve ; (1) where 􀀛0 and k0 are 􀀌tting constants that depend on the chemistry, microstructure, and strain, while n is the grain size exponent, and 􀀛0 is approximately the 􀀍ow stress of the coarse-grained and untextured polycrystal. When n = 0:5, Eq. (1) becomes the conventional Hall-Petch relationship, which was postulated by Hall and Petch in the early 1950's [5, 6] . Experimental results of metallic materials generally support the empirical validity of Eq. (1) [7, 8, 9, 4] . It should be noted that here " denotes the plastic strain, which is the subtraction of the total strain and the elastic strain. Under the tensile loading conditions, the elastic strain is 􀀛f (")=E, where E is the Young's modulus. Thus, the total strain, "tot , is related to the plastic strain by "tot = " + 􀀛f (")=E. Many theoretical models have been proposed in literature to explain the physics that may be responsible

FAQs

Why is predicting polycrystalline metal strength challenging with current datasets?

Many available datasets lack microstructural variability, preventing robust physics-based models from accounting for microstructure randomness.

How does the proposed method predict flow stress?

It generates a large database of flow stress from randomly sampled microstructures, then uses mixture models and neural networks to estimate the flow-stress distribution and microstructural feature importance.

What does the study conclude about the conventional Hall-Petch relationship?

Across a wide grain-size range, the conventional Hall-Petch relationship is statistically valid for correlating strength with average grain size and its relative importance versus other features.

Aug 5, 2026

Upload by Hazel