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MAT-02

Hall–Petch grain-size strengthening

σy = σ0 + ky / √d.

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StrengtheningHall 1951Petch 1953

Governing equation

σy=σ0+kyd\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-02 — Hall–Petch grain-size strengthening) is the form associated with Hall 1951 · Petch 1953. Working symbols: σ0\sigma_0, kyk_y, dd →\rightarrow σy\sigma_y. Dislocation pile-ups at grain boundaries make yield scale as 1/√d. Hall and Petch established the relation independently.

Purpose

Purpose: compute σy\sigma_y from σ0\sigma_0, kyk_y, dd in Materials via σy=σ0+kyd\sigma_y=\sigma_0+\dfrac{k_y}{\sqrt{d}} σy = σ0 + ky / √d. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given σ0=100.000 MPa\sigma_0 = 100.000\,\mathrm{MPa}, ky=20.000 MPa√mmk_y = 20.000\,\mathrm{MPa√mm}, d=30.000 μmd = 30.000\,\mathrm{\mu m}, the governing relation σy=σ0+kyd\sigma_y=\sigma_0+\dfrac{k_y}{\sqrt{d}} yields σy=215.470 MPa\sigma_y = 215.470\,\mathrm{MPa}. Equiaxed polycrystal, no twinning-dominated regime. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Yield stress \sigma_y215.470 MPa
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MAT-02 · curve
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Narration of this film

Equiaxed polycrystal, no twinning-dominated regime.

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

Reading speed

Watch on YouTube