The yield stress of polycrystalline thin films
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In recent experiments it has been shown that the yield stress of polycrystalline thin films depends separately on the film thickness and the grain size. It was also shown that the grain size dependence varies as the reciprocal of the grain size. In this paper an analysis is presented which leads to these results and provides a more detailed understanding of the origins of the observed behavior. Venkatraman and Bravman1 recently showed that the strength of polycrystalline thin films depends on both their grain sizes and their thicknesses. They further showed that these effects are separable in that the yield stress ay can be given by M
+
(1)
where M is a constant, h is the film thickness, and is the component of the yield stress which is grain size dependent. They further found that 1 (2) ay,gba d' where d is the grain size. The purpose of this note is to describe a simple analysis which leads to the results given by Eqs. (1) and (2). Consider a thin film predominantly composed of grains whose boundaries intersect both surfaces of the film. Each grain in such a film is a right polygonal cylinder. A further simplification is to treat each grain as a right circular cylinder of diameter d and height h, as shown in Fig. 1. Consider the slip occurring by formation of dislocations at the top surface of the film, as shown for the grain in Fig. 1. Once a dislocation has swept through the slip plane in the grain, the work done is given by = (Tb)(£d)
(3)
where € is defined in Fig. 1, and is approximately given by h/ sin cf>, b is the Burgers vector of the dislocation, and r is the shear stress given by T =
O"C0S A COS ,
(5)
J. Mater. Res., Vol. 8, No. 2, Feb 1993
http://journals.cambridge.org
where Wside and Wbottom are the energies per unit lengths of the respective dislocation segments. Nix2 has cited Freund3 and Barnett2 to argue that the expression for ^bottom should be
W,bottom
—
b2 -
4TT(1
r>
v)
sin 4>
bhd -
2h sin 0
d)Wd. (10)
For slip to occur, Wnet ^ 0, so that the flow stress is found by setting Waet = 0. Therefore,
O"v
=
Wd sin b cos A cos 4> )\d
sin 0
h V
(11)
which has the form given by Eqs. (1) and (2). The derivation given here yields the specific grain size dependence given by Eq. (2). This dependence is
238 http://journals.cambridge.org
different from the simple Hall-Petch relationship for However, which it might be expected that ay^bad~m. the relationship of Eq. (2) was found to be in better agreement with experiments. Equations (1) and (2) and (11) indicate that poly crystalline films should have both a critical grain size as well as thickness below which yielding is not expected for a given stress. Also, as recently discussed by Sanchez and Arzt,4 the crystallographic texture of a film should affect its yield stress since ay is a function of and A. ACKNOWLEDGMENTS The author would like to thank J.A. Flora and H. J. Frost for useful discussions. This work was supported by the National Science Foundation through Grant No. 900198-DMR.
REFERENCES 1. 2. 3. 4.
R. Venkatraman and
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