The shrinkage of grain-boundary voids under pressure
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20/6/03
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The Shrinkage of Grain-Boundary Voids under Pressure HUA WANG and ZHONGHUA LI The shrinkage rate of the grain-boundary void under pressure is developed. The rate process is controlled by grain-boundary diffusion under the combined action of void surface energy, grain-boundary energy, and elastic energy stored in the solid. A critical load parameter that controls the void shape is introduced, below which the void will be in an equilibrium state and will be stable during shrinking. The equilibrium shape of the shrinking grain-boundary void is obtained. Then, an explicit expression for the shrinkage rate of the grain boundary void is derived as a function of the applied pressure, void shape parameter, and relevant material parameters.
I. INTRODUCTION
NUMEROUS investigations in crack healing at high temperature,[1,2,3] and other similar physical processes, such as sintering[4,5] and diffusion bonding,[6–9] have shown that the strength recovers partially or completely in damaged materials. The crack healing procedure observed by experiments and numerical simulations[10–13] shows four stages: (1) pinching and splitting of initial cracks; (2) the formation of pore channels, (3) break up of the pore channels subject to Rayleigh instabilities, leading to the formation of discrete voids both on grain boundary and within grain; and (4) void shrinkage. Many efforts have been made to investigate the void shrinkage or growth on the planar bonding interface or on the planar grain-boundary interface.[14,15,16] Some void shrinkage models[6,8,17] were developed from powder sintering models[4] and others [7,18] were derived from void growth models,[19,20,21] treating the void shrinkage as negative void growth. In some of the previous works, attention has been paid mainly to the rate of void growth (or shrinkage) under the condition that the voids maintain their spherical shape during evolution.[15,21,22] However, Takahashi et al.[23] noted that a void changes its shape continuously toward a stationary shape under pressure during shrinkage and the shrinkage process depends on the geometrical parameters such as void spacing, void height, and void width. It is also believed that the shrinkage rate may effectively be promoted by increasing the externally applied pressure. However, the pressure cannot be increased too high. Otherwise, it will collapse to a slit. As shown in Figure 1 for the two-dimensional version of this problem, imagine a void perturbed from its circular shape, say, an ellipse, both elastic and surface energies drive the atoms to diffuse on the void surface, but in the opposite directions. Let k be the curvature of, and w the elastic energy density on, the void surface. Because kA kB, the surface energy strives to move atoms from B to A and restore the circular symmetry; because wA wB, the elastic energy strives to move atoms from A to B and to amplify the asymmetry. Let r0 be the radius of the initial circular void, gs the surface energy, and E the Young’s modHUA WANG, Doctor, and ZHO
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