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MAT-28

Taylor dislocation forest

τ = α G b √ρ. Forest hardening from dislocation density.

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StrengtheningTaylor 1934

Governing equation

τ=αGbρ\tau=\alpha G b\sqrt{\rho}

ρ∈[1×1010,1×1016]\rho \in \left[1\times 10^{10}, 1\times 10^{16}\right]

where

\alpha
Taylor α (—)
G
Shear modulus (GPa)
b
Burgers vector (nm)
\rho
Dislocation density (1/m²)
\tau
Flow stress (MPa)

The figure, part by part

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MAT-28 · curve
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①

Governing law — red

τ=αGbρ\tau=\alpha G b\sqrt{\rho}

ρ∈[1×1010,1×1016]\rho \in \left[1\times 10^{10}, 1\times 10^{16}\right]

ρ=1×1010\rho = 1\times 10^{10}τ=0.612\tau = 0.612

②

Domain of the variable — blue

ρmin⁡=1×1010,ρmax⁡=1×1016\rho_{\min} = 1\times 10^{10},\quad \rho_{\max} = 1\times 10^{16}

ρ∈[1×1010,1×1016]\rho \in \left[1\times 10^{10}, 1\times 10^{16}\right]

ρ=1×1010\rho = 1\times 10^{10}ρ=1×1010\rho = 1\times 10^{10}

③

Response curve — green

τ=αGbρ\tau=\alpha G b\sqrt{\rho}

ρ∈[1×1010,1×1016]\rho \in \left[1\times 10^{10}, 1\times 10^{16}\right]

ρ=1×1010\rho = 1\times 10^{10}τ=0.612\tau = 0.612

④

Rate of change — amber

dτdρ\frac{\mathrm{d}\tau}{\mathrm{d}\rho}

ρ∈[1×1010,1×1016]\rho \in \left[1\times 10^{10}, 1\times 10^{16}\right]

ρ=6.29×1013\rho = 6.29\times 10^{13}τ′=5.41×10−13\tau^{\prime} = 5.41\times 10^{-13}

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-28 — Taylor dislocation forest) is the form associated with Taylor 1934. Working symbols: α\alpha, GG, bb, ρ\rho →\rightarrow τ\tau. Taylor related flow stress to the mean spacing 1/√ρ of a dislocation forest, times G b.

Purpose

Purpose: compute τ\tau from α\alpha, GG, bb, ρ\rho in Materials via τ=αGbρ\tau=\alpha G b\sqrt{\rho} τ = α G b √ρ. Forest hardening from dislocation density. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given α=0.300 —\alpha = 0.300\,\mathrm{—}, G=80.000 GPaG = 80.000\,\mathrm{GPa}, b=0.255 nmb = 0.255\,\mathrm{nm}, ρ=1.000e+12 1/m2\rho = 1.000e+12\,\mathrm{1/m^{2}}, the governing relation τ=αGbρ\tau=\alpha G b\sqrt{\rho} yields τ=6.12 MPa\tau = 6.12\,\mathrm{MPa}. A tangle of lines, a density ρ, a shear tick. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Flow stress \tau6.12 MPa
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MAT-28 · curve
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Narration of this film

A tangle of lines, a density ρ, a shear tick.

Taylor related flow stress to the mean spacing 1/√ρ of a dislocation forest, times G b.

Reading speed

Watch on YouTube