Part VI – June 1968 - Papers - Deformation Theory of Hot Pressing-Yield Criterion

The basic density equation originally dericed ' to predict the increase in density of a compact of spherical particles with the progressive deformation at the points of contact has been further modified to include the yield strength of the material. This has been done by assuming that the contact areas grow to stable sizes under a fixed stress which is equal to three times the yield strength. The final equation has the form: where Do and D me the initial and final bulk densities of the compact, u is the applied pressure, and Y is the yield strength of the material. This equation was tested with the data obtained on spheres of lead, K-Monel, and sapphire. The calculated yield strength t~alues for lead and sapphire are within the range of values reported in the literature. A few of the earliest hot pressing models proposed to explain the mechanism by Murray, Livey, and williams2 and then by McClelland3 are based on a plastic flow mechanism. However, more recent investigations suggest that the overall densification process is a combination of several mechanisms, such as particle rearrangement, fragmentation, plastic flow, and stress-enhanced diffusional creep. While fragmentation and particle rearrangement are considered to be responsible for the densification in the early stages,"475 it has been concluded that the final stages of hot pressing are controlled by stress-enhanced diffusional creep.516 The manner in which the densification takes place, i.e., by fragmentation, particle rearrangement, plastic flow, or stress-enhanced diffusional creep, would depend upon the type of material, the temperature, and the stress level used during the hot-pressing experiments. Metal compacts can be expected to have a much greater contribution from plastic flow than ceramic oxides. Also, plastic flow would be a significant contributing factor to densification at high temperatures and high stresses. Most of these works, directed towards elucidation of densification mechanism, have dealt with kinetics of the process. The results of most of the authors vary from one another and they have proposed either new empirical or semiempirical equations to fit their data. The densification rate was found to vary with the type of the powder, shape and size of the powder, initial packing density of the compact, and a few other factors such as rate of heating, pressure, and so forth. Beyond the initial stages, the densification process has been considered to be as time-dependent flow, controlled by a diffusional process, e.g., Nabarro-Herring creep. Palm our, Bradley, and johnson' have attempted to use modified creep rate equations to interpret the data of densification under hot-pressing conditions. Beyond the initial stages, however, the densification would be controlled by a process depending upon the temperature, pressure, and size of the powders. It is the authors' belief that such densification cannot be exclusively controlled by a single process and so attempts should be made to study some observable phenomenon like microstructure, yield strength, and so forth. The emphasis of this work has been toward studying the densification problem from a more fundamental point of view. Some of the principal variables, like initial packing density, mode of packing, and size of the powders, have been controlled to a great extent. The total strain produced on pressure application (instantaneous) in such a case can be considered to be due to plastic and elastic deformation. The elastic component of the strain can be determined by decreasing the load to the initial value. The strain remaining then can be correlated with the contact areas produced by deformation and the corresponding applied load. In a previous paper,' the possible deformation behavior of spheres in a compact has been theoretically analyzed and experimentally tested. The change in contact area radius a relative to the particle radius R was related to the bulk density and the bulk strain for simple and systematic modes of packing. Tt was found that a density equation relating the above parameters can be represented by: where D and Do are the bulk densities of the compact at any value of a/R and a/R = 0, respectively. This basic equation should hold for any material as it was derived from geometrical considerations alone. An attempt has been made in this work to include the yield strength in the above density equation, so that a knowledge of the properties of any material can be used in predicting the densification behavior during the hot-pressing process. THEORETICAL CONSIDERATTONS The deformation of two spheres in contact under a static load can be compared to the deformation occurring between a hard spherical indentor and the flat face of a softer metal. Tt has been shown theoretically by both ~encky~ and lshlinskyg and experimentally by ~abor" that, for a material incapable of appreciable work hardening, the mean pressure required to produce plastic yielding (for deformation occurring between flat face and a hemispherical indentor) is approximately equal to three times the elastic limit, Y, of the material (in tension or compression experiments). Tabor has further observed that the same relationship is valid in the case of work-hardening materials, if the elastic limit at the edge of the indenta-