Effect of Grain anisotropy on limit strains in biaxial stretching: part i. influence of sheet thickness and grain size i
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THE classical theories of tensile instability and necking failure are based on continuum concepts which ignore the influences of material inhomogeneities on a microscopic scale. Hill's analysis I of localized necking in biaxially stretched sheets refers to an ideally homogeneous strain-hardening material which obeys a Mises-type yield criterion. It provides useful predictions of the behavior of polycrystalline metals subjected to plane stress loading when the ratio of principal strains in the sheet plane p = ~z/~ < 0 but it implies that localized necking will not occur when p > 0. Two distinctly different adaptations of continuum plasticity theory have been suggested in order to accommodate the fact that real materials do fail by localized necking in biaxial stretching. The approach proposed by St0ren and Rice2 explores the consequences of departures from smooth Mises-type yield surfaces. In particular, the destabilizing effects of yield-surface vertices which might arise as a consequence of the crystallographic character of the flow processes and the effects of pressure-sensitive yielding which can develop as a result of internal void formation have been examined? The other approach, due to Marciniak and Kuczynski,4 (MK), recognizes that strain inhomogeneities can develop from local regions of weakness in the material. MK analyzed the process of strain localization at a dimensional inhomogeneity in the form of a long trough perpendicular to the direction of major strain, el, the severity of the defect, f , being defined simply in terms D. V. WILSON and W. T. ROBERTS are in the Department of Industrial Metallurgy, University of Birmingham, United Kingdom. P. M. B. RODRIGUES is now with Swiss Aluminum Ltd., Neuhausen, Switzerland. Manuscript submitted November 13, 1980.
of the ratio of initial thickness in the trough to that of the surrounding material. The idea that limit strains in biaxially stretched sheets of commercial alloys might be predicted by a MK-type of analysis, using an "effective weakness parameter" f ' to characterize the effects of inhomogeneity in a given material, is attractive. Experimental studies have shown that necking failure in sheets can often be associated with microstructural inhomogeneities and, with a given microstructure, limit strains decrease with decreasing sheet thickness within the range commonly used in sheet formingA~ However, the MK analysis in its initial form does not provide a reliable prediction of the p-dependency of the biaxial limit strains of most commercial sheets of good quality. Important features of the long groove model are that material within the groove is constrained by surrounding material in only one direction in the sheet plane and that, as strain concentration develops, there is a drift in strain state towards plane strain within the groove whilst proportional straining is maintained outside the groove. When f is reasonably small these features lead to limit strains which increase steeply with increasing P in the range 0 to 1. When the important inhomogenei
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