Sheet Necking—I. Validity of Plane Stress Assumptions of the Long-Wavelength Approximation

Abstract

Two special solutions are used to illustrate the errors involved in the analysis of sheet necking from invoking plane stress assumptions. The first is an exact solution for the growth-rate of a small amplitude, sinusoidal thickness variation in a sheet of material characterized by $\dot \in$ = ασⁿ in simple tension. The plane stress assumptions become accurate when the ratio of variation wavelength to average thickness exceeds four and otherwise lead to overestimates of the actual growth-rate. When this same ratio is approximately unity the relative size of the thickness variation decays with increasing deformation—an effect not predicted by a plane stress analysis. The second special solution is obtained using a perturbation expansion with the nonlinear long-wavelength solution (i.e., the plane stress solution) as the lowest order contribution. In this way explicit corrections to the plane stress solution are obtained. Selected comparisons with fully nonlinear finite element calculations are made.

The material in this three-part paper was presented orally in Session II under the title “Constitutive Relations for Sheet Metal” and Session IV under the title “Sheet Necking: Influence of Constitutive Theory and Strain-Rate Dependence”.