Analysis for inversion load and energy absorption of a circular tube

Abstract

The basic theoretical approach in the mechanics of external or inside-to-outside inversion of circular tubes is examined and extended by the introduction of two additional and previously unperceived sub-structural mechanisms. It is shown that these refinements when applied to published results for ductile aluminium tubing inverted, in a non-frictional process, furnish very encouraging correlations. Since this analysis is made on an ‘engineering plastic strain’ basis a supplementary calculation is made employing ‘pure plastic strain’ or ‘natural strain’ in the predominant sub-structural processes. The resulting comparisons have shown that for rigid/plastic material characteristics the simplification derived in the use of engineering strain does not introduce serious errors in the evaluation of the inversion load. Here it might be mentioned that in many of the cited papers on external inversion, although the initial equations are set up in pure plastic strain terms, frequently the ensuing analysis is simplified by expressing the logarithmic strain as a series and eliminating all but the first term. This reverts the analysis to an engineering strain type. Hence the work described in this connection validates these previously adopted simplifications. In view of the satisfactory correlations achieved by introducing the above theoretical refinements for the external inversion process, the same procedure has also been applied to the case of internal or outside-to-inside tube inversion. Published analysis of the internal inversion process has previously neglected the pronounced thickening effect clearly demonstrated in all experimental results. (Tube-wall thickness change is not at all evident during external inversion and in fact this also assists in simplifying that analysis). In the present approach the effects of friction and work hardening in a compressive die process of internal inversion have been included in the manner already deduced in other research but in addition a further sub-structural process explaining and quantifying the associated thickening of the tube wall has been formulated. Here it should be mentioned that some uncertainty exists in relation to the application of strain hardening based on average plastic strains. However, since the four published experimental results available have been made with materials exhibiting strain hardening characteristics, the total newly developed and previously evolved analyses have been combined to enable some correlation to be made. The predicted internal inversion loads for two separate sets of results in three different materials are in very reasonable agreement with experimental values.