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A theory of electromigration damage in thin films by grain boundary grooving processes is presented. A bicrystal film configuration is assumed and profile changes caused by the applied electric field are calculated. The grain boundary constraint converts mass transport along the surface into equivalent thinning changes normal to this direction. Grooving processes are consistent with damage manifestations, such as grain boundary cracks and voids, as well as thinning; specific grooving mechanisms due to the effects of the variable surface curvature and temperature are considered separately and the kinetics of these damage modes are derived. While the latter mechanism yields damage times in agreement with experiment, the effects of variable surface curvature considerably overestimate the film lifetime. This suggests that accelerated thinning due to localized heating, lack of adhesion, and locally enhanced current densities, may be quite important in cases where gross heating effects and temperature gradients are small. The kinetics of the resistance variation in a piecewise straight conductor, grooving by surface curvature effects, is derived and compared with experiment.