Asymptotic Stability Analysis for Sheet Metal Forming—Part I: Theory

In this paper is presented a general methodology for predicting puckering instabilities in sheet metal forming applications. A novel approach is introduced which does not use shell theory approximations. The starting point is Hill’s stability functional for a three-dimensional rate-independent stressed solid which is modified for contact. By using a multiple scale asymptotic technique with respect to the small dimensionless thickness parameter ε, one can derive the two-dimensional version of the stability functional which is accurate up to $Oε4,$ thus taking into account bending effects. Loss of positive definiteness of this functional indicates possibility of a puckering instability in a sheet metal forming problem with a known stress and deformation state. An advantage of the proposed method is that the puckering investigation is independent of the algorithm used for calculating the deformed state of the sheet. [S0021-8936(00)00804-7]