Effects of plastic anisotropy and yield surface shape on sheet metal stretchability
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I.
INTRODUCTION
S H E E T metals subjected to biaxial stretching usually fail by localized necking. Hence, the stretchability of sheet metals depends on the material's resistance to localized necking and, in particular, upon material factors which delay the onset of such a plastic instability. The beneficial effects of strain hardening and strain rate hardening on stretchability are well known; both of these effects increase the forming limit strains of sheet metals, which are usually defined in terms of the maximum principal strains (el and E'2) attainable from sheets prior to the onset of localized necking. ~.2It is well known that a high degree of plastic anisotropy as represented by a large R-value, which is the ratio of width strain to thickness strain in a uniaxial tensile specimen, promotes formability in drawing. Recent stretch forming studies on strongly textured Ti-alloy sheets indicate that a large R-value also increases the resistance to localized necking in stretching conditions involving negative minor strains. 3'4 The experimental results, 3'4 a theoretical analysis s which assumes localized necking being initiated from an imperfection aligned along Hill's direction of zeroextension 6 and a reexamination 5 of Hill's theory, 6 all indicate that in the 62 < 0 regime, the enhancement of the forming limit strains by the R-value can be understood in terms of a critical thickness strain criterion for localized necking. According to this criterion, e* increases with the R-value as the result of increasing difficulties in attaining a critical thickness strain (e*) at a higher R-value. The effects of the R-value on biaxial stretching involving positive minor strains, on the other hand, remain inconclusive because of difficulties in separating the effects of plastic anisotropy from those of strain hardening and strain rate hardening in sheets with relatively similar, low R-values 1 (R < 2) and of difficulties in determining whether localized necking takes place prior to fracture in sheets with high R-values .4 Theoretical efforts to elucidate the effects of the R-value are complicated by the fact that the calculated FLCs, in addition to strain hardening and strain-rate harden-
K. S. CHAN is Research Engineer, Department of Materials Sciences, Southwest Research Institute, 6220 Culebra Road, P.O. Drawer 28510, San Antonio, TX, 78284. Manuscript submitted March 13, 1984. METALLURGICAL TRANSACTIONS A
ing, also depend on the choice of the yield function 7'8 (for example, quadratic v s nonquadratic Hill yield functions), the plasticity theory 9'1~ (flow vs deformation or vertex theories of plasticity), and on whether localized necking originates from material imperfections, 12 instability resulting from the presence of a vertex on the yield surface, 9 or void growth.~3 Since Hill's direction of zero-extension6 does not exist when 62 2> 0 , neither Hill's theory nor the critical thickness strain criterion for localized necking is applicable under these biaxial stretching conditions. The hypothesis that localized
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