The two-dimensional connectivity of
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I.
INTRODUCTION
A classic problem in analyzing microstructures is to convert from a typical two-dimensional metallographic cross section to a true representation of the underlying threedimensional microstructure. For problems such as grain size analysis and inclusion sizing there are straightforward conversions.~ A similar problem exists with liquid phase sintered materials where the microstructure connectivity is a concern. The connectivity determines the rigidity of the compact during liquid phase sintering, which in turn determines the densification rate and shape distortion. 2'3 Likewise, the connectivity affects the properties such as the thermal expansion coefficient and electrical conductivity. Furthermore, recent fracture observations on tungsten heavy alloys (W-Ni-Fe) have shown the tungsten-tungsten contacts are weak and fail at lower strains than the rest of the microstructure: Indeed, in modeling the failure process it appears that crack propagation under slow strain rates is determined by the spacing of these failed tungsten-tungsten contacts) Accordingly, it is possible to link the microstructure to the tensile properties. To obtain such a link it is necessary to determine the connectivity of the tungsten grains from twodimensional metallographic observations. The metallographic cross section through a liquid phase sintered material is random; thus, the intersection of features such as intergranular contacts is also random. Figure 1 shows a typical two-dimensional metallographic section through a tungsten heavy alloy. The microstructure is composed of tungsten grains and a penetrating matrix phase composed of a tungsten-nickel-iron alloy. During sintering the tungsten grains were solid and the matrix alloy was liquid, thus, in referring to this microstructure it is common to denote the two phases as solid and liquid, although both are solid at temperatures below approximately 1440 ~ For this microstructure, the question is what is the true connectivity of the solid phase? Niemi et al. 6 have previously considered this problem for liquid phase sintered materials using experimental measurements on various materials. They concluded that a single curve could represent the variation in the mean contacts per grain (in two-dimensions) v s the volume fraction of solid phase. As shown in Figure 2 the R.M. GERMAN is Professor, Department of Materials Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590. Manuscript submitted June 16, 1986. METALLURGICALTRANSACTIONSA
Fig. 1- - Optical micrograph of a liquid phase sintered tungsten heavy alloy (88 wt pct W-8.4 wt pct Ni-3.6 wt pct Fe) sintered at 1480 ~ showing the near spherical grains in contact.
dihedral angle determines the equilibrium size of the contact between grains; thus, the probability of intersection by a random sectioning plane will increase with dihedral angle. Accordingly, there should be a variation in the number of observed contacts per grain dependent on the number of actual three-dimensional contacts, which relates to the packi
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