An empirical model for hot isostatic pressing of metal powders
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I.
INTRODUCTION
THE process
of hot pressing of metal powders plays an important role in powder metallurgy. It is a fairly complicated process which involves thermomechanical material response properties as well as microstructural effects. The basic process involves heating the powder to a temperature below the melting point and also the application of pressure. The presence of a nonhydrostatic component of applied stress ( e . g . , compaction by uniaxial or triaxial stress instead of pressure) adds further complexity. Various mathematical models which have been proposed to describe these processes are described in review papers by Ramqvist,~ James, 2 Spriggs and Dutta, 3Wilkenson and Ashby, 4 and others. References to earlier work are also given in a recent paper by Swinkels, Wilkinson, Arzt, and Ashby, 5in which a new model is proposed and some experimental data on hot isostatic pressing of lead, tin, and PMMA are presented. In the present paper, a new mathematical model is proposed to describe time dependent pressurization and densification of metal powders at constant temperature. It is a f'LrStorder ordinary differential equation in the pressure and an appropriate measure of densification, and it incorporates an instantaneous response law, an equilibrium response law, and a creep response law. A major departure from previous theories of hot isostatic pressing is the recognition that materials which exhibit creep will also, under appropriate circumstances, exhibit stress relaxation. Thus, creep models may not be adequate to describe the response to rapid pressure variation, such as the 30 seconds pressure rise in the experiments of Swinkels et a l . 5 Accordingly, the governing differential equation includes a rate of pressure term. Another feature of the model is that a single equation describes the entire compaction process. The model is based, to some extent, on previous models and on micromechanical (hollow sphere) analysis, but the development is largely empirical, based on the experimental data of Swinkels et a l . 5 These excellent data support the assumed linear forms of the instantaneous and equilibrium
M.M. CARROLL is Shell Distinguished Professor, Department of Mechanical Engineering, University of California, Berkeley, CA 94720. Manuscript submitted March 5, 1985. METALLURGICALTRANSACTIONS A
response laws and the experimental creep law, and allow determination of the parameters in the model. The resulting agreement between the experimental data and theoretical creep curves for lead and tin at 100 ~ shows that the proposed ordinary differential equation models the full range of creep compaction for these materials.
II.
BACKGROUND
A . Static C o m p a c t i o n
Several empirical equations have been proposed to describe rate independent pressure compaction of powders, and also of porous metals. The most widely used equation is one proposed by Konopicky6 and by Shapiro and Kolthoff,7 which has the form 1 P =A +B l n ~ 1-D
(D ---D0),
[1]
where P is the applied pressure, D is the relative dens
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