Plastic instability criteria for necking of bars and ballooning of tubes

Plastic-instability criteria applicable to the necking of bars under tension and to the ballooning of thin-wall tubes under internal pressure were derived from basic geometrical considerations. In the case of bars under tension, plastic instability prevails if the percentage rate of decrease of the cross-sectional area in the (potential) necking region is greater than that in the bulk of the bar. When the loading characteristics and constitutive equation were taken into account, an instability criterion was deduced in terms of the stress, strain, strain rate, temperature and material properties. This criterion was shown to be reducible to the classical Considere condition for non-rate-sensitive materials. For rate-sensitive materials under isothermal conditions, a simple relationship among the strain, the strain-hardening and strain-rate-sensitivity parameters was also obtained. In the case of thin-wall tubes under internal pressure (with or without imposed axial loading), plastic instability prevails if the percentage rate of increase of the diameter (or equivalently decrease of wall thickness) in the (potential) ballooning region is greater than that in the bulk of the tube. An instability criterion in terms of the axial strain rate and the axial derivatives of the hoop strain and hoop strain rate was first deduced. Then the loading characteristics, the constitutive equation, the thin-wall approximation, and the Prandtl-Reuss flow rules were taken into consideration. This resulted in a further statement of the criterion in terms of the state of strain, the material properties, and a ratio of the imposed axial stress to the circumferential stress. As in the case of necking of bars, the role of the hardening parameter is clear: i.e., a larger hardening parameter implies a more stable material and vice versa