A Model for yielding in anisotropic metals
- PDF / 769,729 Bytes
- 8 Pages / 594 x 774 pts Page_size
- 61 Downloads / 226 Views
-
1
o'0 corresponds to the stress at which the second and third terms are equal; K1 is similar to the Hall-Petch constant, but is lower in magnitude; K2 is a constant which depends on the average difference in the matrix of elastic constants between adjacent grains and on grain shape. DL and Ds are the upper and lower grain size limits over which elastic interaction takes place, and the third term is zero for d = DL or d = Ds. The constants m and n define, respectively, the limit of the grain over which elastic interaction takes place and the fraction of the grain contributing elastic interaction to the surrounding grains. In effect, [1 - ( d / D D m] [(d/Ds)" - 1] constitutes the volume fraction of the grain boundary region over which elastic interaction assists slip. The equation, with different values of the constants, has been shown to fit O-y d-~/2 data over a wide range of grain sizes for Ni, 22 Zn, 23 and modified 1010 steel. 24 Volume fraction of grain boundary deformation zone data for/3-Ti-Mn 8 are shown to fit predictions as well. The constants 0"0, K1, DL, Ds, m, and n can all be determined experimentally, thus leading to a determination of K2. -
I.
-
INTRODUCTION
THE original work by Hall ~and Petch 2 which revealed the linear relationship O'y :
or 0 "~- K y d -1/2
[1]
_'
al"
between yield stress, try, and the reciprocal square root of grain size, d, has led to a sustained investigation of the relationship between yield, flow, and cleavage strengths and grain size in a variety of metals. These efforts have most recently been summarized by Armstrong 3 and Hansen 4 and Thompson. 5 These studies have been concerned with the resistance to flow which is expressed by Eq. [1]. There is, however, a growing body of evidence which suggests that elastic interactions, which occur in anisotropic metals, assist yielding and must, therefore, be taken into account in any expression for yield strength. This evidence will be reviewed and an equation will be developed which incorporates elastic assistance to yielding.
II. EVIDENCE OF ELASTIC INTERACTIONS IN ANISOTROPIC P O L Y C R Y S T A L L I N E M E T A L S
'
I
I 1 I
, ) Ir
m
---
[E~
I
1
. .r _
Fig. 1 - - H o o k and Hirth 5 shear incompatible bicrystal stresses.
A . Slip E n h a n c e m e n t 1. Bicrystals
Hook and Hirth 6 studied slip in bicrystals of Fe-3 pct Si, bicrystals which were elastically incompatible in shear, parallel to the boundary (Figure 1). This elastic incompatibility produced shear stresses, which added to the applied stress and produced {110}(111) slip at the boundary. This slip would not normally have been expected in a single HAROLD MARGOLIN, Professor, and ZH1RUI WANG and TZIKANG CHEN, Graduate Students, are with the Department of Physical and Engineering Metallurgy, Polytechnic Institute of New York, Brooklyn, NY 11201. Manuscript submitted December 17, 1984. METALLURGICAL TRANSACTIONS A
crystal with the same orientation as one of the bicrystal components. 2. Polycrystals a. Grain B o u n d a r y Slip. Slip nucleation
Data Loading...
