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A model for unstable shear crack propagation in pipes containing gas pressure The speed and path of an unstable crack in the wall of a pipe under pressure affects the length of fracture. To learn more about this problem, McClure, Duffy and Eiber have made full-scale tests of the pipe sizes, steels, and pressures employed in natural gas transmission lines under sponsorship of the Pipeline Research Committee. Their work shows that 100 % shear cracks propagate at speeds ranging from 400-800 fps for X-52 and X-60 grades to 800-1200 fps for higher strength quenched and tempered grades. Initially, such failures follow a straight, axial path, but tend to veer into a helical trajectory which greatly reduces the axial and actual path component of velocity and the length of pipe damaged. Surprisingly enough, the line pressure (or hoop stress) seems to have no clearly discernible effect on speed or trajectory; neither were systematic effects assigned to backfill or pipe geometry (the ratio of radius to wall thickness), though the latter was examined only in a limited way covering only the diameters and wall thicknesses in general use for gas transmission pipelines. While the full-scale tests have exposed the main features of crack propagation in pipes, they do not, by themselves, explain the speeds and trajectories observed. Only limited progress has so far been made in this direction. McClure and co-workers have shown analytically that a certain class of elastic waves, which can propagate in cylinders, have speeds and trajectories similar to cleavage cracks. The principles that govern the selection of particular waves was not worked out in detail. It also seems unlikely that elastic waves will interact strongly with the slower moving shear fractures, and no other explanation for the mechanisms controlling shear cracks has been proposed. Yet, the shear mode is important technologically, since shear fractures have been observed to propagate unstably. This paper takes the first step towards assembling a treatment of shear crack propagation from numerical descriptions of the six underlying processes: 29860
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(1) Axial decompression
(2) Bulging
(3) Radial decompression
(4) Local stress and strain intensification
(5) Plastic deformation
(6) Ductile cracking
The treatment is quasi-static; dynamic effects on the pipe wall may well be important but are difficult to handle and have not been included. The descriptions of bulging and stress intensifications are derived for an axial crack and do not consider nonaxial paths. It should therefore be clear that a highly idealized fracture event is being treated with an imperfect model. Thus, while some of the features of the model are in accord with experience, others are apparently contradicted by data from full-scale tests. The treatment of shear fracture described recognizes the 6 processes which are shown schematically as follows:
(I) Axial Decompression
This term refers to the difference between PL, the line pressure before failure, and Pc, the pressure existing near the tip of the propagating crack. The pressure difference arises from the loss of gas through the rupture and depends on the speed of decompression waves in the gas relative to the speed of the crack.
(2) Bulging
The term bulging is used to refer to the radial, out-of-round distortion of the unsupported pipe wall on either side of the crack. As a result of the bulging, the pipe wall develops a radial component of velocity that depends on the extent of the bulging and the forward speed of the crack. This bulging also places a limit on Cs, the steady-state crack length in that portion of the pipe that can still be regarded as a pressure-containing vessel.
(3) Radial Decompression
This process leads to the difference between Pc and the effective pressure Pe acting on the bulging pipe wall. This difference is a consequence of the radial velocity of the bulging wall relative to the velocity of the gas molecules. In the actual case, the difference between Pc and Pe may be insignificant within the steady-state crack length distance; some full-scale data seem to indicate this. However, for the purposes of this analysis, Pc and P~ are viewed as separate parameters.