Instabilities during tension of thin voided viscoplastic sheets
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Instabilities during Tension of Thin Voided Viscoplastic Sheets HUSSEIN M. ZBIB and ELIAS C. AIFANTIS The purpose of the present communication is to elaborate on the findings of two articles by Chakrabarti and Spretnak la'b and Ghosh2 published in earlier issues of this journal and concerned with the instabilities occurring during straining of viscoplastic sheets. Roughly speaking, there have generally been noted two main types of instabilities during tensile tests of sheet metals. The first, most often observed and commonly known as necking, occurs after the point of maximum load, dP = O, and is accompanied by variations of the cross-sectional area in the form of diffuse necks. It may thus be termed geometric instability, as it is always followed by a drop in load due to the dramatic reduction of the corresponding crosssectional area. The second instability, commonly known as shear banding, can occur before or after the point of maximum load depending on the material and the type of loading. It appears in the form of oblique localized necks or shear bands and is accompanied by negligible changes in cross-section dimensions. More precisely, it occurs at the point of "loss of ellipticity" of the corresponding incremental equilibrium equations as has been shown, for example, by Hill and Hutchinson. 3 In particular, they computed the critical loads associated with both types of instability and determined that for rate-independent plastic materials under tension and plane strain conditions, oblique shear bands develop after necking. If, in addition, a vertex is allowed to exist in the corresponding yield surface (corner theory), Hutchinson and Tvergaard* predicted that shear bands can occur at critical levels of straining considerably lower than those reported above. 3 A similar behavior was noted by Chang and Asaro, 5 who by using a crystal plasticity theory, illustrated that shear bands can occur before necking, in agreement with their experiments. Another deviation from classical plasticity that significantly reduces the critical strain for shear band forHUSSEIN M. ZBIB, Graduate Student, and ELIAS C. AIFANTIS, Professor, are with the Department of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, Houghton, MI 49931. Manuscript submitted March 25, 1985. METALLURGICALTRANSACTIONS A
mation is the plastic dilation which may appear as the macroscopic manifestation of void nucleation and growth. This was shown, among others, by Yamamoto6 and Tvergaard7 who utilized void-dependent constitutive equations to illustrate that voids promote shear localization. As mentioned earlier, the present communication is partly motivated by the work of Chakrabarti and Spretnak ~ who proposed the condition of "maximum true stress", do" = 0, for the onset of shear localization and verified it experimentally for simple tension of thin sheets (AISI 4340 steel). Since the condition do" = 0 occurs after the condition dP = 0, a strict interpretation of Chakrabarti and Spretnak 1 results, would imply that neck
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