Influence of loading path on the plastic instability strain in anisotropic plane sheets

Although the plastic instability condition for sheet metals under biaxial stress states has been considered by several investigators in the past, the effect of variation of the stress ratio that generally occurs under actual loading conditions does not seem to have been examined before. The present investigation is therefore concerned with the onset of instability in which the variation of the stress ratio is included in the theoretical framework, under the assumption that the ratio of the two applied loads is held constant during the straining. The sheet metal is assumed to be isotropic in its plane, obeying the customary quadratic yield criterion and its associated flow rule. The results are presented in graphical forms, which bring out the essentials features of the solution to the instability problem. The estimation of the limiting drawing ratio in the deep drawing of cylindrical cups is finally considered as an application of the derived results on plastic instability in plane sheets.