The variation of plastic anisotropy during straining
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I.
INTRODUCTION
e t + ew+ e , = 0
[2]
Truzkowski t4] observed changes in r-value with strain for a number of metals and demonstrated that, for some materials, r-value did not decay toward unity with strain but could increase its deviation from unity. Truzkowski attributed the change in r-value to textural changes during deformation. This led to a rebuttal by Hu.tS] Dabrowski, Karp, and Bunge t6] examined variations in crystallographic texture caused by tensile, deep drawing, and biaxial deformation. Based on the assumption of {112} ( l i d glide, they predicted r-value changes. While declines in both longitudinal and transverse r-values were predicted for drawing and biaxial stretching, an increase was predicted for tensile deformation for both longitudinal and transverse directions for an aluminum killed sheet steel. While their model may not be valid, it does indicate that it is difficult to produce a theoretical justification for crystallographic texture changes being responsible for the reported decreases in r-value with increasing tensile strain. For plasticity studies, the basic definition of r-value has been replaced with the instantaneous r-value, r,, which is defined as
r = - e w / ( e t + ew)
[3]
r, = d e w / d e ,
[3a]
Alternatively, making use of the relationship of Eq. [3] it can be expressed in terms of the contraction ratio in the Taylor theory (here referred to as b,)
THE factor which is usually considered to govern the deep drawability of sheet metal is the tendency of the material in the flange of a pressing to increase in thickness and decrease in the dimension perpendicular to the drawing direction. This tendency depends on crystallographic texture of the sheet; however, it is usually measured as a mechanical property of the sheet itself. Measurements can be made by drawing circular cups, but this is time consuming and the extent of thickening during drawing is usually assumed to be represented well by the tendency to thin during a tensile test. Lankfordm proposed a parameter, the plastic strain ratio (the r-value or r) to represent the drawability of sheet metal. It is defined in terms of true strain, e, in the length, l, width, w, and thickness, t, of the gage length of a tensile test piece: r = ew/e,
[1]
Assuming constancy of volume
-ew
= e l [ r / ( 1 + r)]
r-value is dependent on the texture of the metal, which is not symmetrical in the plane of the sheet, and consequently the r-value is dependent on testing direction. The r-value at a given elongation, usually 15 pct (et = 0.140), has been used for many years as a quality control indicator of drawability. More recently, there has been interest in the effect of strain on the plastic strain ratio. H u i2'31 reported that the r-value of steel decayed rapidly toward unity with increasing strain. That is, higher r-values decreased and lower ones increased, with the rate of change of r-value depending on its deviation from unity. While acknowledging that changes in the crystallographic texture occurred with increasing strain, Hu p
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