Envelopes of crack-like surfaces for modeling cavity growth
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where v, is the velocity of the surface in the direction normal to itself. Equation [1] is posed in adimensional units. The surface quantities were divided by A0, defined in Figure 1, and the time quantities by [2]
t~ = kTA~/D~ 8 , % n
where Ds is the self diffusion coefficient of the surface, 6~ the thickness where surface diffusion occurs, % the surface free energy, ~ the atomic volume, T the absolute temperature, and k the Boltzmann's constant. Chuang and Rice 7 obtained the equation of a crack-like surface which, moving with constant velocity v, is a mathematical solution of [1]. The lower half of the crack surface is given by x = a + {g(h) - g[2 sin(O/2)]}/v v3
[3]
where + V ~ - _ hz
Envelopes of Crack-Like Surfaces for Modeling Cavity Growth
[4] h = vV3y + 2 sin(0/2)
L. MARTiNEZ Fracture of materials at high temperatures and low stresses is associated with the diffusive growth of intergranular cavities. The applied stress introduces gradients in chemical potential which create a flux of matter mainly from the cavity surfaces into the grain boundaries. Several authors ]'2'3 have developed models for cavity growth assuming that the process is controlled by grain boundary diffusion. In this case surface diffusion is fast enough to keep the equilibrium shape of the cavities as they grow. Consideration of surface diffusion as an important factor in the cavity growth process has allowed a group of a u t h o r s 4'5'6 to conclude that the cavities should sometimes grow with a crack-like shape. Chuang et al. 6 have developed a theory, based on a solution of the surface diffusion equation 7 which models, implicitly, the cavity growth with a continuous sequence of crack-like surfaces of variable width. This suggests that the actual profile of the cavity as it grows is the envelope of the sequence of crack surfaces. In this paper the work of Chuang et al. is extended with the calculation of the equation for the envelope. The result is an envelope with the form of a notch. The width of the envelope is a function of the applied stress, the grain boundary to cavity surface diffusivity ratio, the distance between cavities, and the capillarity angle. The gradients of curvature on a surface in a solid induce diffusional movements of atoms along the surface in order to eliminate regions of high curvature. 8 The movement of such surfaces is governed by 7 v.-
02K as 2
is the angle formed by the grain boundary and the crack surface at the tip, and a is the position of the crack tip. The upper half of the surface can be found by symmetry. Chuang et al. 6 used these results to analyze the growth of an array of two dimensional cavities uniformly distributed in a plane grain boundary as shown in Figure 1. Initially the cavities are assumed to have the equilibrium lenticular shape with the angle 0 = Cos-l(~b/2%), where Yb is the grain boundary surface free energy. The half distance between cavities is bAo. The diffusional processes in the grain boundary and on the cavity surfaces were coupled by Chuang et al. 6 using continui
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