MAT-07

Griffith brittle-fracture stress

σf = √(2 E γ /(π a)) for a through crack.

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FractureGriffith 1921

Governing equation

\sigma_f=\sqrt{\dfrac{2E\gamma_s}{\pi a}}

$a \in \left[0.01, 20\right]$

where

E: Modulus (GPa)
\gamma_s: Surface energy (J/m²)
a: Half-crack length (mm)
\sigma_f: Fracture 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

\sigma_f=\sqrt{\dfrac{2E\gamma_s}{\pi a}}

$a \in \left[0.01, 20\right]$

$a = 0.01$ $\sigma_f = 66.76$

Along the curve

②

Domain of the variable — blue

a_{\min} = 0.01,\quad a_{\max} = 20

$a \in \left[0.01, 20\right]$

$a = 0.01$ $a = 0.01$

Along the curve

③

Response curve — green

\sigma_f=\sqrt{\dfrac{2E\gamma_s}{\pi a}}

$a \in \left[0.01, 20\right]$

$a = 0.01$ $\sigma_f = 66.76$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma_f}{\mathrm{d}a}

$a \in \left[0.01, 20\right]$

$a = 0.136$ $\sigma_f^{\prime} = -213.6$

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-07 — Griffith brittle-fracture stress) is the form associated with Griffith 1921. Working symbols: $E$ , $\gamma_s$ , $a$ $\rightarrow$ $\sigma_f$ . Griffith equated the elastic energy released by extending a crack to the surface energy created, obtaining 1/√a strength.

Purpose

Purpose: compute

\sigma_f

from

E

\gamma_s

a

in Materials via

\sigma_f=\sqrt{\dfrac{2E\gamma_s}{\pi a}}

σf = √(2 E γ /(π a)) for a through crack. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

E = 70.000\,\mathrm{GPa}

\gamma_s = 1.000\,\mathrm{J/m^{2}}

a = 0.500\,\mathrm{mm}

, the governing relation

\sigma_f=\sqrt{\dfrac{2E\gamma_s}{\pi a}}

yields

\sigma_f = 9.441\,\mathrm{MPa}

. Infinite plate, through crack 2a, plane stress, brittle. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Modulus E70 GPaSurface energy \gamma_s1 J/m²Half-crack length a0.5 mm

Outputs

Fracture stress \sigma_f9.441 MPa

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Narration of this film

Infinite plate, through crack 2a, plane stress, brittle.

Griffith equated the elastic energy released by extending a crack to the surface energy created, obtaining 1/√a strength.

Reading speed

Watch on YouTube