MAT-21

Hall–Petch grain-size strengthening

σy = σ0 + ky / √d. Yield rises as grains shrink.

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StrengtheningHall 1951Petch 1953

Governing equation

\sigma_y=\sigma_0+\dfrac{k_y}{\sqrt{d}}

where

\sigma_0: Friction stress (MPa)
k_y: Hall–Petch slope (MPa√mm)
d: Grain size (µm)
\sigma_y: Yield stress (MPa)

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-21 — Hall–Petch grain-size strengthening) is the form associated with Hall 1951 · Petch 1953. Working symbols: $\sigma_0$ , $k_y$ , $d$ $\rightarrow$ $\sigma_y$ . Dislocation pile-ups at grain boundaries make yield scale as 1/√d. Hall and Petch established the relation independently.

Purpose

Purpose: compute

\sigma_y

from

\sigma_0

k_y

d

in Materials via

\sigma_y=\sigma_0+\dfrac{k_y}{\sqrt{d}}

σy = σ0 + ky / √d. Yield rises as grains shrink. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\sigma_0 = 100.000\,\mathrm{MPa}

k_y = 20.000\,\mathrm{MPa√mm}

d = 30.000\,\mathrm{\mu m}

, the governing relation

\sigma_y=\sigma_0+\dfrac{k_y}{\sqrt{d}}

yields

\sigma_y = 215.47\,\mathrm{MPa}

. A polycrystal, a grain size, a yield bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Friction stress \sigma_0100 MPaHall–Petch slope k_y20 MPa√mmGrain size d30 µm

Outputs

Yield stress \sigma_y215.47 MPa

Reading speed

Watch on YouTube

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Narration of this film

A polycrystal, a grain size, a yield bar.

Dislocation pile-ups at grain boundaries make yield scale as 1/√d. Hall and Petch established the relation independently.

Reading speed

Watch on YouTube