Theory of work-hardening applied to stages III and IV
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I.
INTRODUCTION
To
the traditional three stages of work-hardening (of which stage I is observed only under special conditions, e.g., single crystals in single glide), stage IV has been added recently. E3J It is characterized by an almost constant low work-hardening rate and its ability to accommodate large strains, as in wire drawing, torsion, or rolling. ~ .4-7] Idealized, the experimentally observed work-hardening coefficient, 0 = d~'/d3", in stages II, III, and IV may be represented ts,61 as in Figure 1, if ~- is the resolved shear stress and 3' the resolved shear strain, or, equivalently, O* = do'/de, with o-the macroscopically applied stress and e the true engineering strain. The terms 0* and 0, cr and ~-, and e and 3' are related via the Taylor factor, M, such that tr = M'r, e = 3"/M, and 0* = M20, with typically M ~ 2.5 to 3. It has been reported [4,5m that for different testing temperatures and for several metals, the hardening rate at the transition from stage III to stage IV, 0,, scales with the stress a t t h e transition, ~'~v, or the stress at which 0 would extrapolate to 0, "rs. In that case, since in stage IV the work-hardening coefficient remains nearly constant, 0w --~ 0t = c~'w --~ c~'s
[ 1]
with 0~v the hardening rate in stage IV and c a constant between 0.05 and 0.1. Equation [1] was substantiated in Reference 6 by means of a table based on various data in the literature, one of these being the study by Langford and Cohen tll on tensile testing of iron wire after drawing to different engineering strains.
As early as 1970, the results obtained by Langford and Cohen tlj were the subject of a theoretical analysis based on the mesh length theory of work-hardening t21 and were found to be in close agreement with that. Now that stage IV has become of current interest, and, in particular, since dislocation behavior has been recognized to be understood most readily in terms of LEDS, rSl it seemed useful to review and expand that analysis in light of this new development. Actually, in the mesh length theory, the recognition that dislocations tend to arrange in mutually stress-screened structures (thereby minimizing their energy, which is the essence of the LEDS concept) has long been held as important.[9.1~ Recently, however, that concept has been considerably refined and elaborated. In the earlier analysis, ~2~ the various geometrical parameters necessary for the quantitative application of the mesh length theory were extracted from the experimental results by Langford and Cohen, ~l] including a large number of original full-size micrographs kindly supplied by these authors. The work is now recognized as a case of stage IV, but initially it was termed stage II behavior ~2j on account of the nearly constant work-hardening coefficient. It was found that similitude operated throughout the strain range investigated; t~,2j i.e., the dislocation structures could be derived from each other through a shrinking in scale inverse to the stress which formed them. The objective of the present paper will be to
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