Microstructural Stability of Stressed Lamellar Eutectics
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INTRODUCTION
Directionally solidified eutectics are excellent candidate materials for several high temperature applications which demand extremes of strength, toughness and creep and fatigue resistance. However, an important consideration for these materials is the thermal stability of the rod or plate-like morphology under operating conditions. Cline[ 1] demonstrated that the rod-like microstructure displays a surface energy induced Rayleigh instability and consequently the rods tend to decompose into spheres, following exposure to high temperatures for long times, leading to an overall degradation of the composite properties. It has also been reported that stresses can accelerate this instability [2]. These stresses can originate from either misfit strains, such as coherency strains or differential thermal expansion strains, or from the substantial loads typically applied under operating conditions. In this work, we examine the conditions under which stresses, both misfit and externally applied, lead to microstructural instabilities. As opposed to rod eutectics, lamellar eutectics have a plate morphology which do not exhibit a Rayleigh or surface energy induced instability. Consequently, the lamellar eutectic composite is an ideal system to investigate the effect of stresses on thermal stability without the complication of surface energy induced instabilities. Therefore, in this paper, we theoretically examine the effects of both misfit generated stresses as well as external stresses on the microstructural stability of the plate-like morphology in a lamellar eutectic composite. 2. MATHEMATICAL FORMULATION AND METHOD OF SOLUTION In order to model a lamellar eutectic composite, we idealize the microstructure by considering a periodic array of plates within a uniform matrix as shown in Fig. 1. The eigenstrain tensor associated with the misfitting plates are assumed to be isotropic with components E* = Eii, which corresponds to a uniform volume dilatation. The plate and the matrix are assumed to be elastically isotropic. In addition, the composite is subjected to an uniform far field strain cx = P°, in order to simulate externally applied stresses. To make the analyses tractable, the Poisson's ratios of the film and the substrate are assumed to be equal (v = 1/3). The relevant length scales in the problem include the plate thickness (h) and the 445
Mat. Res. Soc. Symp. Proc. Vol. 398 ©1996 Materials Research Society
interplate spacing (H). Further, in this paper, we focus our attention on the case where the interplate spacing is very large (i.e., h/H -* oo). The effect of large plate volume fraction on the microstructural stability is discussed in detail in ref.[3].
T Y
MATRIX
H
I
-*
E
Fig.1 Schematic of an uniaxially strained lamellar composite with the plates periodically arranged along the y direction. The plate-matrix interface shape is described by Eqn. 1. We now examine the stability of the flat plate-matrix interfaces by slightly perturbing the plate shape Y = ±(h/2) + 8 cos(21rx/ X)
(1)
where Y d
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