The application of the LEFM to fatigue crack growth, through the range of stress intensity factor, varies from simple substitution into the relevant equations (see the theory card), to rather complex. The examples in this section aim to illustrate this variety. This first problem is straightforward and shows a typical application in failure analysis. It should take perhaps 20 minutes to complete.

a) A long pipe has an outer diameter (OD) of 90 mm, an inner diameter (ID) of 70 mm and works at a pressure (p) of 40 MPa. Valve failure downstream in the pipe caused a pressure surge which burst the pipe. Examination of the fracture surface revealed a metallurgical defect at the inner surface of the pipe which was semi-elliptical in shape with a depth of 1.6 mm and a surface length of 4.5 mm. This flaw was orientated perpendicular to the hoop stress in the pipe.

What pressure would have caused this failure?

Note that the formula for hoop stress from thin walled theory is p(ID)/2t, while from thick walled theory it is p[(L^{2} + 1)/(L^{2} - 1)] where t is the wall thickness of the pipe and L is the ratio OD/ID. Which formula would you use and why?

The plane strain fracture toughness of the pipe alloy is 25 MPa m^{½}, and the geometry correction factor can be found from the graph below.

b) A new pipe was manufactured from the same alloy and subjected to NDT prior to installation. This showed that the pipe contained a similarly orientated flaw, 1.5 mm deep, but with a semicircular shape.

Assuming normal operating conditions, i.e. no pressure surges and a daily evacuation to zero pressure, will the pipe last for its desired lifetime of 30 years?

Assume that the geometry correction factor has a constant value of 0.7 in this second part of the question and note that a fatigue crack growth rate of 6.25x10^{-8} mm/cycle corresponds to an applied delta K value of 10 MPa m^{½}. The Paris law exponent m is 4.