MAT-38

Orowan looping stress

τ = G b / λ. Bypass of impenetrable particles by dislocation loops.

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StrengtheningOrowan 1948

Governing equation

\tau=\dfrac{G b}{\lambda}

$\lambda \in \left[5, 400\right]$

where

G: Shear modulus (GPa)
b: Burgers vector (nm)
\lambda: Particle spacing (nm)
\tau: Orowan stress (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

\tau=\dfrac{G b}{\lambda}

$\lambda \in \left[5, 400\right]$

$\lambda = 5$ $\tau = 4080$

Along the curve

②

Domain of the variable — blue

\lambda_{\min} = 5,\quad \lambda_{\max} = 400

$\lambda \in \left[5, 400\right]$

$\lambda = 5$ $\lambda = 5$

Along the curve

③

Response curve — green

\tau=\dfrac{G b}{\lambda}

$\lambda \in \left[5, 400\right]$

$\lambda = 5$ $\tau = 4080$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\tau}{\mathrm{d}\lambda}

$\lambda \in \left[5, 400\right]$

$\lambda = 7.484$ $\tau^{\prime} = -409.3$

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-38 — Orowan looping stress) is the form associated with Orowan 1948. Working symbols: $G$ , $b$ , $\lambda$ $\rightarrow$ $\tau$ . Orowan: a dislocation bows between particles spaced λ and leaves a loop. The critical stress is ~Gb/λ.

Purpose

Purpose: compute

\tau

from

G

b

\lambda

in Materials via

\tau=\dfrac{G b}{\lambda}

τ = G b / λ. Bypass of impenetrable particles by dislocation loops. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

G = 80.000\,\mathrm{GPa}

b = 0.255\,\mathrm{nm}

\lambda = 80.000\,\mathrm{nm}

, the governing relation

\tau=\dfrac{G b}{\lambda}

yields

\tau = 255.00\,\mathrm{MPa}

. Two particles, a bowed line, a leftover loop. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Shear modulus G80 GPaBurgers vector b0.255 nmParticle spacing \lambda80 nm

Outputs

Orowan stress \tau255.00 MPa

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Narration of this film

Two particles, a bowed line, a leftover loop.

Orowan: a dislocation bows between particles spaced λ and leaves a loop. The critical stress is ~Gb/λ.

Reading speed

Watch on YouTube