MAT-04

Paris crack-growth law

da/dN = C (ΔK)^m in the mid-ΔK regime.

Reading speed

FractureParis–Erdogan 1963ASTM E647

Governing equation

\dfrac{\mathrm{d}a}{\mathrm{d}N}=C(\Delta K)^m

$\Delta K \in \left[2, 80\right]$

where

C: Paris C (mm/cyc, MPa√m) (—)
\Delta K: SIF range (MPa√m)
m: Paris exponent (—)
da/dN: Growth per cycle (mm/cyc)

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

\dfrac{\mathrm{d}a}{\mathrm{d}N}=C(\Delta K)^m

$\Delta K \in \left[2, 80\right]$

$\Delta K = 2$ $da/dN = 8\times 10^{-8}$

Along the curve

②

Domain of the variable — blue

\Delta K_{\min} = 2,\quad \Delta K_{\max} = 80

$\Delta K \in \left[2, 80\right]$

$\Delta K = 2$ $\Delta K = 2$

Along the curve

③

Response curve — green

\dfrac{\mathrm{d}a}{\mathrm{d}N}=C(\Delta K)^m

$\Delta K \in \left[2, 80\right]$

$\Delta K = 2$ $da/dN = 8\times 10^{-8}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}da/dN}{\mathrm{d}\Delta K}

$\Delta K \in \left[2, 80\right]$

$\Delta K = 2.491$ $da/dN^{\prime} = 1.88\times 10^{-7}$

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-04 — Paris crack-growth law) is the form associated with Paris–Erdogan 1963 · ASTM E647. Working symbols: $C$ , $\Delta K$ , $m$ $\rightarrow$ $da/dN$ . Paris and Erdogan collapsed fatigue-crack data onto a power law of the stress-intensity range, the basis of damage-tolerant design.

Purpose

Purpose: compute

da/dN

from

C

\Delta K

m

in Materials via

\dfrac{\mathrm{d}a}{\mathrm{d}N}=C(\Delta K)^m

da/dN = C (ΔK)^m in the mid-ΔK regime. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

C = 1.000e-8\,\mathrm{—}

\Delta K = 15.000\,\mathrm{MPa√m}

m = 3.000\,\mathrm{—}

, the governing relation

\dfrac{\mathrm{d}a}{\mathrm{d}N}=C(\Delta K)^m

yields

da/dN = 3.375e-5\,\mathrm{mm/cyc}

. Mode I, mid-Paris regime, constant amplitude. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Paris C (mm/cyc, MPa√m) C1e-8 —SIF range \Delta K15 MPa√mParis exponent m3 —

Outputs

Growth per cycle da/dN0.000034 mm/cyc

Reading speed

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Narration of this film

Mode I, mid-Paris regime, constant amplitude.

Paris and Erdogan collapsed fatigue-crack data onto a power law of the stress-intensity range, the basis of damage-tolerant design.

Reading speed

Watch on YouTube