A work-hardening model of the lower yield strength of discontinuously yielding alloys: Ni 3 Al and mild steel
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This note shows that a model suggested by Schulson et al.1 to explain an unusual dependence of the lower yield strength of Ni3Al on the grain size, viz. (grain size)" 08 , can account also for the usual or (grain size)"05 dependence of the lower yield point of mild steel on the grain size. Also, the note comments on a recent paper by Takeyama and Liu,2 who considered the effect of grain size on the low-temperature yield strength of Ni3Al and other alloys. The model is based on work-hardening and the view that the lower yield strength of discontinuously yielding alloys, which is the stress to propagate a Liiders band (Fig. 1), is equivalent to the stress to continue plastic flow through a matrix which has been strained by an amount equal to the Luders strain. Since the Liiders strain increases with decreasing grain size, the attendant strain hardening increases as well. It follows, therefore, that the finer the grain size, the higher the yield strength. The model is conceptually similar to one presented by Conrad,3 who explained the grain size dependence of the lower yield point in iron and steel in terms of the long-range interaction of dislocations whose density increased with increasing strain. The lower yield strength may be given by the relationship:
aGbpf
x = xp = pd
(4)
where the subscript, p , denotes planar slip and where fi is a constant of order unity. On the other hand, for materials like mild steel which exhibit wavy slip, the slip distance is assumed to be set by the average two-dimensional spacing between dislocations. In these cases:
(5)
x = xm = p-Lm
(1)
where cr0 is a measure of the lattice resistance to slip, a is a dimensionless constant, G is the shear modulus, and b is the Burgers vector; pL is the density of dislocations within the Luders band and may be related to the Luders strain, eL, through the relationship: PL
= -7=
bx
(2)
where m is a parameter which converts normal strain to shear strain, and x is the average distance of dislocation slip. In turn, the Luders strain may be related to the grain size, d, through the phenomenological expression:1 eL = \d~"
(3)
where X and q are constants. The slip distance, 3c, for materials like Ni3Al which exhibit planar slip is assumed to be set by the grain size. In these cases:
a)
On sabbatical leave from Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire.
470
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FIG. 1. Schematic sketches of a true stress (cr)-true plastic strain (ep) curve for materials which yield discontinuously and of the gauge section of a test specimen showing four different stages of Luders band propagation.
J. Mater. Res., Vol. 4, No. 3, May/Jun 1989
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where the subscript, a>, denotes wavy slip. Upon combining Eqs. (1) through (5), two expressions are obtained. For materials in which slip is planar: (6)
= o* + For materials in which slip is wavy:
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