Localized Necking of Sheet at Negative Minor Strains
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I.
INTRODUCTION
SHEET materials deforming under multiaxial states of stress, as in sheet metal forming operations, usually fail by localized necking. The current interest in understanding sheet metal formability has led to several theoretical analyses of localized necking based on different criteria. These localized necking criteria include: a localized shear zone along a direction of zero-extension, ~ materials imperfections, 2 the presence of a vertex on the yield surface, 3 and void growth. 4 Localized necking along a direction of zero-extension was originally proposed by Hill. ~Hill's theory predicts that the maximum principal strain e~* prior to localized necking (i.e., the limit strain) has a magnitude of el* -- n at plane strain and increases to el* = (1 + R)n for the uniaxial tension deformation of sheet exhibiting normal anisotropy with a plastic anisotropy parameter R, which is defined as the ratio of the width strain to thickness strain of sheet specimens deformed under uniaxial tension. For plastically isotropic material (R --- 1), the limit strain at uniaxial tension reduces to the well-known el* = 2n expression. Hill's theory, however, does not take into account the strain rate hardening of the material or preexisting imperfection and cannot explain the phenomenon of localized necking under biaxial stretching. Strain localization developed by local weakness of material (imperfection) was first proposed by Marciniak and Kuczynski (M-K) 2 and extended by Sowerby and Duncan 5 as a means of describing localized necking in biaxial stretching when the minor principal strain e2 is positive. The M-K analysis assumes the presence of a material imperfection in the form of a groove or trough. Imposing the same e2 inside and outside the groove while proportional straining is maintained outside the groove, M-K have shown that deformation within the groove occurs at a faster rate than the rest of the sheet. The concentration of strain (e~) within the groove eventually leads to the plane strain condition (de2 = O) K. S. CHAN, formerly with the Department of Metallurgical Engineering, Michigan Technological University, is now Research Engineer with Southwest Research Institute, 6220 Culebra Road, San Antonio, TX 78284. D.A. KOSS is Professor, Department of Metallurgical Engineering, Michigan Technological University, Houghton, MI 49931. A.K. GHOSH is Manager, Metals Processing, Rockwell International Science Center, 1049 Camino Dos Rios, Thousand Oaks, CA 91360. Manuscript submitted January 14, 1983. METALLURGICALTRANSACTIONSA
within the groove and to localized necking. The M-K model is thus able to explain localized necking in biaxial stretching and has been used with reasonable success to calculate forming limit diagrams for A-K steel and 70-30 brass. 6 In an analysis of localized necking in thin sheet, Hutchinson and Neale, 7 Lee and Zaverl, 8 and Thomas and co-workers 9 have also examined the influence of a linear geometric imperfection on the limit strain for the onset of localized necking. Their analyses are
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