MAT-43

Hall–Petch

Grain-size strengthening.

Reading speed

GoverningHall–Petch

Governing equation

\sigma_y=\sigma_0+k d^{-1/2}

$d \in \left[0.005, 0.5\right]$

where

s0: s0 (MPa)
k: k (MPa√mm)
d: d (mm)
sy: Hall–Petch (MPa)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\sigma_y=\sigma_0+k d^{-1/2}

$d \in \left[0.005, 0.5\right]$

$d = 0.005$ $sy = 59.9$

Along the curve

②

Domain of the variable — blue

d_{\min} = 0.005,\quad d_{\max} = 0.5

$d \in \left[0.005, 0.5\right]$

$d = 0.005$ $d = 0.005$

Along the curve

③

Response curve — green

\sigma_y=\sigma_0+k d^{-1/2}

$d \in \left[0.005, 0.5\right]$

$d = 0.005$ $sy = 59.9$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}sy}{\mathrm{d}d}

$d \in \left[0.005, 0.5\right]$

$d = 0.008$ $sy^{\prime} = -528.9$

Along the curve

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-43 — Hall–Petch) is the form associated with Hall–Petch. Working symbols: $s0$ , $k$ , $d$ $\rightarrow$ $sy$ . Grain-size strengthening. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

sy

from

s0

k

d

in Materials via

\sigma_y=\sigma_0+k d^{-1/2}

Grain-size strengthening. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

s0 = 50.000\,\mathrm{MPa}

k = 0.700\,\mathrm{MPa√mm}

d = 0.040\,\mathrm{mm}

, the governing relation

\sigma_y=\sigma_0+k d^{-1/2}

yields

sy = 53.500\,\mathrm{MPa}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

s0 s050 MPak k0.7 MPa√mmd d0.04 mm

Outputs

Hall–Petch sy53.500 MPa

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Grain-size strengthening. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube