An analysis of transient cavity growth controlled by grain boundary diffusion
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t + ~g(t)
where g ( t ) are functions which vary from zero to a constant value during the transient. This allows us to compute the amount of cavity growth which can be attributed to the transient. We show that the transient period is always very short compared to the rupture time and that the amount of growth attributable to the transient is more than four orders of magnitude smaller than the total cavity growth at fracture. I. I N T R O D U C T I O N Many structural materials subjected to high temperature creep conditions fail by the growth and coalescence of cavities at their grain boundaries. ~Diffusion controlled mechanisms have been proposed wherein the growth results from atomic transport of material away from the cavity, mainly along the cavity surface and through the adjoining grain boundary. On the cavity surface the atomic flux is driven by gradients in a curvature, and controlled by surface self diffusion, 2 while in the grain boundary the flux is driven by gradients in the normal tractions and controlled by grain boundary self diffusion? Although these two processes occur in sequence and should be coupled, they are often studied separately. Hull and Rimmer, 3 Speight and Harris, 4 Raj and Ashby ~and Raj S have all contributed to the development of the theory of cavity growth controlled by diffusion in the grain boundary. Chuang and Rice, 2 Pharr and Nix 6 and FuentesSamaniego and Nix 7 have studied the case of cavity growth controlled by surface diffusion. Both transient and steady state calculations have been made for these two limiting cases. In the present paper, we extend the work of Raj 5 by evaluating the duration and contribution to growth of the elastic transient associated with grain boundary diffusion controlled cavity growth. Raj analyzed the behavior of preexisting cavities in the grain boundaries of a polycrystalline solid subjected to an applied stress. Initially the polycrystal reacts to the applied stress as a homogeneous linear elastic solid. The diffusional process produces an accumulation of matter in the grain boundary which is absorbed by the neighboring crystals. A part of this is accommodated by a bulk growth L. MARTINEZ is Visiting Scholarfor Department of Materials Scienceand Engineering,Stanford University,Stanford CA 94305, and W. D. NIX is Professorfor Department of Materials Scienceand Engineering, Stanford University. Manuscript submitted March 25, 1980. METALLURGICAL TRANSACTIONS A
of the crystals and the rest by an elastic deformation. This mechanism leads to changes in the elastic stress distribution from the initial state to that which exists in steady state. When the steady state is reached all the matter leaving the cavity is accommodated by bulk growth of the neighboring crystals. Although Raj concluded that the transient may be important because for certain conditions its duration can be measured in hours, he did not establish its relative importance in terms of characteristic times for steady state cavity growth. We have derived an expression for the growth of a
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