threshold to be found. A third point on the buckling threshold was found using Tube 32 and an even smaller slope was achieved by making measurements in an alternate test apparatus which is very similar to that shown in Fig. 28.1, but features windows for visualizing the buckling process. The compliance of these windows reduces the speed of the pressure waves and increases the impulse for a given peak pressure. The three tubes tested in Fig. 28.10 were identical, so differences in the peak pressure at which buckling occurred are caused only by the change in impulse of the pressure wave. For the shortest impulse tested, the peak pressure required to buckle the tube was about seven times the static buckling threshold, which highlights the importance of inertial effects. The trend observed in these results agrees with the theoretical predictions of [7], which show that for large impulses the dynamic buckling threshold tends towards the static buckling pressure (1.3 MPa in this case), while for small impulses the threshold approaches a vertical asymptote at a particular critical impulse. In Ref. [7], the following semi-empirical estimate of the critical impulse is suggested: Icr ¼1:15 ffiffiffiffiffifi ρE p h2 a (28.13) This formula predicts a critical impulse of about 0.57 kPa-s for the tubes in Fig. 28.10, though the effects of plasticity will become important as this limit is approached. This prediction is not inconsistent with the measurements shown in Fig. 28.10, but additional data points at lower impulses are needed to confirm that the buckling threshold approaches this asymptote. 28.4 Conclusions Dynamic buckling of cylindrical tubes has been studied by submerging them in a thin annulus of water and generating axially-propagating shock waves in the water using a projectile impact facility. Measurements reveal that for low pressures, the response is similar to that of a waterhammer event occurring inside of a pipe, and suitably adapted waterhammer models are capable of adequately predicting the speed of the coupled fluid-solid waves. Elastic buckles are observed at higher pressures, but due to inertial effects these buckles do not fail the tube until the peak pressure is several times greater than the static buckling pressure. At that point, the onset of plastic deformation is found to substantially reduce the tubes’ loadcarrying capacity; however, it appears that plastic deformation does not significantly affect the motion of pressure waves since the leading edge of the pressure wave travels far from the buckle by the time plastic deformation is reached. Finally, these experiments have demonstrated that extruded tubes often feature sinusoidal wall thickness variations around the circumference, and these variations play a critical role in determining the orientation of mode 2 buckles. Acknowledgements This research was supported by the Office of Naval Research DOD MURI on Mechanics and Mechanisms of Impulse Loading, Damage and Failure of Marine Structures and Materials (ONR Grant No. N00014-06-1-0730), program manager Dr. Y. D. S. Rajapakse. Tomohiro Nishiyama of the Japan Patent Office and Prof. Kazuaki Inaba, currently at the Tokyo Institute of Technology, executed the initial Fig. 28.10 Pressure vs. impulse curves for three identical aluminum tubes with radius a ¼22 mmand a=h ¼24:5. The static buckling threshold for these tubes is 1.3 MPa. Tube 47 was tested with an aluminum buffer, Tube 50 with a steel buffer, and Tube 32 in an alternate apparatus, allowing various regions of the pressure-impulse space to be reached. Black x marks points at which plastic deformation first occurred 28 Dynamic Buckling of Submerged Tubes due to Impulsive External Pressure 235
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