Probing the Critical Stress Intensity Factor for Slip Transfer across Grain Boundaries by Subgranular Indentation Alfons
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Probing the Critical Stress Intensity Factor for Slip Transfer across Grain Boundaries by Subgranular Indentation Alfonso H.W. Ngan and Y.L. Chiu Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, P.R. CHINA. ABSTRACT By analysing the relevant results in the literature, we have found that, when indentation is made on a subgranular level, the hardness varies roughly inversely with the square root of the distance between the indent and the grain boundary. This effect is analogous to the Hall-Petch effect for macroscopic deformation.
INTRODUCTION It is well-known that the yield strength Y of a polycrystalline material vaires with the grain size d according to the Hall-Petch relation Y = Yo + k y d −1 / 2
(1)
where Yo and ky are constants. The Hall-Petch slope ky is identified as a measure of the ease of slip transmission across grain boundaries. Because of the dependence on the polycrystalline resolution factor and the Taylor factor, ky is an averaged value of all the grain boundaries within the polycrystalline sample. On the other hand, subgranular microhardness indentation has been routinely carried out in the literature to measure the so-called degree of hardening of grain boundaries [1-4]. In this type of experiments, indentation is often performed near a grain boundary in the edge-on position (Fig. 1), and nearly all the results indicate that the closer the point of indentation to the grain boundary, the higher the measured hardness. In this work, we first conjecture that in the subgranular situation, the hardening effect of the grain boundary is describable by a relation similar to eqn. (1), i.e. the measured hardness H is conjectured to be 80
Nb (right)
75
d
Nb (left)
70
Hv
65 60 55 50
Grain 1
Grain 2
45 40 0
0.2
0.4
0.6
0.8
1
1/Sqrt(d) (microns)^-0.5
Figure 1. A bicrystal experiment
Figure 2. Hardness data for niobium bicrystal. 5 gf load. (data from ref. [3])
Q4.10.1
H = H o + k y ' d −1 / 2
(2)
where Ho and ky’ are constants, and d is now the distance of the indent from the grain boundary as shown in Fig. 1. We will check the validity of eqn. (2) using representative results from the literature. We will then develop a mechanistic model to present an interpretation to the meaning of the parameter ky’ in eqn. (2).
ANALYSIS OF EXPERIMENTAL RESULTS We first use the data by Chou et al [2] on a [011] [111] symmetric tilt bicrystal of niobium. In Fig. 2 are plotted their hardness values vs 1 / d , where the open and closed diamonds represent results measured on the “right” and “left” of the grain boundary respectively. The original results show no hardening effect of the grain boundary when d is larger than about 10 µm on both sides of the grain boundary. Thus in Fig. 2, H remains roughly constant when the abscissa is less than ~ 0.3 µm-0.5. When d is smaller than ~10 µm, H increases roughly linearly with d-1/2 in accordance with eqn. (2). From the slope of the ascending portion of Fig. 2, ky’ for this grain boundary is about 0.30 MPam1/2. Another
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