The temporal variations of void growth rates normalized with the respective rate predicted by Hull-Rimmer model, are shown in Figure 4 for all the three metals. Indeed, all the curves collapse into a single distribution indicating that the choice of time scale is appropriate. It is worth mentioning that Raj [7] was the first to suggest the appropriate time scale as given in equation (4). It is interesting to note that Trinkaus [8] and Shewmon and Anderson [9] note that in the case of an isolated cavity along a grain boundary, the stress and displacement field expand around the cavity in a self-similar manner in proportion to the cavity radius, with a ∝ t1/3. They obtained this result assuming rigid grains. In our problem, a wedge of material is introduced ahead of the cavity tip during the transient stage, due to material elasticity. In the work of Trinkaus [8] and Shewmon and Anderson [9] a wedge is introduced because the cavity is isolated (well separated and small). In addition, unlike the Hull and Rimmer model they assume that the grain boundary thickening vanishes at some distance away from the cavity. However, their solution is different from the present in that elasticity and void interactions are not taken into account. 1.00 10.00 100.00 1000.00 0.00 0.10 0.20 0.30 0.40 0.50 Al Cu VFEM / VHull-Rimmer t / τ γ-Fe 0.75 1.00 1.25 1.50 0.00 0.10 0.20 0.30 0.40 0.50 Al Cu VFEM / VHull-Rimmer t / τ γ-Fe Figure 4. The temporal variations of void growth rates normalized with the respective rate predicted by Hull-Rimmer model for all the three metals. CONCLUSIONS The effect of elastic accommodation on the grain boundary diffusion-controlled void growth was analyzed using an axisymmetric unit cell model. In order to accomplish this we have extended the formulation of Needleman and Rice [2] to account for material elasticity. This extension also involved an incremental formulation of the virtual work principle of the boundary value problem involving grain boundary diffusion. The model accounts for void interaction effects. The results of the analyses on void growth rates agree well with the Hull-Rimmer model after the initial transient time for the three different metals considered. During the elastic transient, void growth rates can be several orders of magnitude higher than the steady state growth rate. Using the predictions of finite element analyses for several metals, we demonstrated that the characteristic time is appropriately given by equation (4). Indeed, Raj [7] was the first to suggest a similar form for the characteristic time. It was observed that the transient time is larger for larger a/b values (or larger volume fraction of cavities). Though the elastic transient time may occupy a small portion of the total rupture time, in metallic components experiencing cyclic loading conditions with short hold times, elasticity effects may be important. ACKNOWLEDGMENT This work was supported by Department of Energy, Office of Basic Sciences under Grant No. DEFG02-90ER14135. The authors thank Dr. B. Armaly and R. Price for their support.
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