Measurement and prediction of plastic anisotropy in deep-drawing steels
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I.
INTRODUCTION
THE prediction
of R-values using the series expansion method is generally carried out within the framework of the Taylor model, t~'2] In this paper, methods based on the Sachs-Kochendrrfer and intermediate "relaxed constraint" grain interaction models are developed and employed in order to improve the accuracy of predictions, as has been done for the simulation of deformation textures (see Reference 3 for a review). Measurements are made on 15 commercial drawing-quality low-carbon steels and compared with the predictions, so that the most appropriate hypotheses for the description of bcc polycrystal deformation can be selected. Yield stress anisotropies and yield surface sections were also calculated. It is shown how the different hypotheses affect yield surface shape and can thus influence formability predictions such as forming limit diagrams, t4]
II.
GRAIN INTERACTION MODELS F O R R-VALUE C A L C U L A T I O N S
IN R O L L E D S T E E L S H E E T S
A. Description of Plastic Deformation in Bcc Crystals In contrast to the case of fcc plasticity, calculations on bcc materials involve questions regarding the slip systems that are considered to be operating and the values of critical resolved shear stress (CRSS) that should be associated with each of these systems, tSj Although there is general agreement that (111) is the slip direction, {112} and {123} slip planes have been reported, in addition to the {110}. To avoid the complexities associated with such a large number of crystallographic planes (48 for the 3 sets), "pencil glide" models have also been used, which
involve analytic solutions to the stress-strain response of bcc crystals. [6.7]In such models, all planes containing the (111) slip direction are admissible. However, both experimental and theoretical studies support the view that the crystallographic description of glide on {110} and {112} planes is sufficient for the description of plastic flow in bcc crystals. I81 This is the approach employed in the present work. Another feature of bcc deformation is the existence of asymmetric slip on the {112} planes. On each of these, glide in the twinning (or soft) direction is easier than in the antitwinning (or hard) one. Moreover, there is no reason for the CRSS ~- to be the same on the {110} and {112} planes. So, the following ratios as =
T{112}twinning T{II0}
a. =
T{112}antitwinning
[1]
T{II0}
are employed below to characterize the parameters of plastic flow. Their importance has been pointed out in some simulations of bcc deformation textures, t91 It has been shown that there is a limited range for mixed slip on the {110} and {112} planes: V ~ / 2 = 0.866 < as -< aH < 2 / V ~ = 1.155. tl~ Experimental values reported in the literature include as ~ 1.0 and an ~ 1.04 to 1.14 for decarburized Fe single crystals, t81 From experiments on Fe-3 wt pct Si single crystals, Orlans-Joliet et al. [Hj obtained values of as ~ 0.93 and aH ~ 0.96. Also, it has been pointed out that interstitial elements appear to make slip more difficult on the
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