Instantaneous and residual stresses developed in hot isostatic pressing of metals and ceramics
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I.
INTRODUCTION
MOSTcurrent models for the hot isostatic pressing (HIP) of powders assume an isostatic stress state within the powder itself, that is, a pure pressure with no deviatoric component. Even when the remote stress state is a pure pressure, deviatoric stresses can appear for a number of reasons, among them constraint due to the can, nonuniformity of the initial powder packing, and nonuniformity of temperature. The deviatoric stresses cause the sample to change its shape as it densities, with undesirable consequences: difticulty in preform design and residual stresses which degrade strength and can cause cracking. A number of research groups 0,2,31 are now seeking ways of analyzing this problem. They differ in their approach, some ul seeking empirical constitutive laws which are titted to experimental data, others t2'3] relying on micromechanical models which include information about mechanism. In this paper, we present an analysis of one aspect of this problem, that of the internal stresses caused by the appearance of temperature gradients when a powder compact, under pressure, is heated too fast.
II.
SIMPLIFIED MODEL
The detailed calculations of differential and residual stresses as presented in Section III tend, by their complexity, to obscure the physical processes we are trying to model. These are bought out more clearly by the following simple, admittedly overidealized, example. Consider a spherical compact of powder of radius Ro. The surface is heated, and almost immediately, a pressure is applied. The hot surface has a lower yield strength (O'ys) than the cold material of the core (tryc). The hot surface layer densities completely to give a dense skin of thickness AR (=Ro - R3, completely enclosing the
W.-B. LI, Associate Professor, is with the Department of Engineering Materials, University of Lule~, S-95187 Luleh, Sweden. K.E. EASTERLING, Professor, is with the School of Engineering, University of Exeter, Exeter, United Kingdom. M.F. ASHBY, Professor, is with the Department of Engineering, University of Cambridge, Cambridge, United Kingdom. Manuscript submitted December 8, 1989. METALLURGICALTRANSACTIONSA
colder, not-yet-densified core, as shown in Figure 1. A stress, P~, acts radially between the skin and the core. For simplicity, consider the case when AR -< Ro. The skin now acts like a can containing the remaining powder. Any further densification requires a compressive yielding of the skin compatible with a volume shrinkage (i.e., densification) of the core. The circumferential stress in the skin, tro, is given by Ri
Cro = ( e - Pi) 2AR
[ 1]
The skin will yield if o"o exceeds Crys, the yield strength of the skin. For either the von Mises or Tresca yield criteria, yield requires that 2~Ro-ys P
-
Pi -
[2]
- Ri
The pressure on the powder in the core, P , is thus less than the applied pressure, P, by the amount 2ARtryJRi (Eq. [2]); hence, as the densification front of thickness AR sweeps inward, the difference between P and P~ increases, and the remaining powder densities more and
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