The 20x2 mm centre-cracked strip of the previous examples is loaded by a stress which varies cyclically between 150 and 250 MPa. The material's Paris index is 4.4, and a stress intensity range of 5.7 MPa√m causes a crack growth rate of 1 mm/Mc.
Integrate the Paris equation ( 5) to determine the number of cycles required for a crack to grow from 0.2 to 2, and from 2 to 20 mm.
Repeat, with a more realistic provision for instability.

The integral of the basic Paris equation ( 5) is ( 5b) with the last term negligible ( a_c --> ∞ ).
With normalised crack size limits of 0.01 and 0.1, the integral ( 5b) may be evaluated numerically, noting that   w = 10 mm or 0.01 m and that each term in { } is dimensionless :-

	{ 1 ∗ ΔN12 / 10 } x { ( 250 -150) √( 0.010 π ) / 5.7 }4.4	=	∫0.010.1 ( ( cos πα/2 )/α )2.2 dα
	{ ( mm/Mc) Mc / mm } { MPa . √m) / MPa√m }	=	195.7     -->   ΔN12 = 13 Mc
	In the second case, with limits of 0.1 and 1, the integral is	=	10.25     -->   ΔN12 = 0.70 Mc

Despite advancing 18 mm in the latter case, compared to only 1.8 mm in the former, the period of the advance has been reduced by a factor of about 20. This reflects the slow growth initially.

A prior R6 analysis of this model, with a peak stress of 250 MPa, has shown that the critical normalised crack size is 0.63, with corresponding elastic configuration factor of 1.35.
So, applying ( 5b) to the first case, the RHS becomes ;     195.7   - ( 1.35 √ 0.63 ) ^-4.4 x ( 0.1 - 0.01)
Clearly the integral and hence the period is unaffected by the modification - because the period lies wholly within stage II.

For the second case, the upper limit must be reduced to the critical 0.63. The RHS of ( 5b) becomes :
∫_0.1^0.63( ( cos πα/2 )/α ) ^2.2 dα - ( 1.35 √0.63 ) ^-4.4 x ( 0.63 - 0.1) = 10.2 - 0.39     -->     ΔN₁₂   =   0.67 Mc

The introduction of the more realistic upper bound has not made much difference to the period - simply because the advance is intrinsically so rapid in the last phase of failure.
In practice, closed form integration is usually impossible; graphical techniques may be used.