Finite-time singularities in surface-diffusion instabilities are cured by plasticity

A free material surface which supports surface diffusion becomes unstable when put under external nonhydrostatic stress. Since the chemical potential on a stressed surface is larger inside an indentation, small shape fluctuations develop because material preferentially diffuses out of indentations. When the bulk of the material is purely elastic one expects this instability to run into a finite-time cusp singularity. It is shown here that this singularity is cured by plastic effects in the material, turning the singular solution to a regular crack.

Figure 1

(Color online) A typical half profile of the stressed interface under the action of surface diffusion and plastiticity. To the bare eye the effect of plasticity is not seen here, and one needs to compare elastic and plastic solution in Figs. 2, 3.

Figure 2

(Color online) The tip curvatures of the elastic solution. This solution appears to approach a singularity at $t^{*}=5.37\times {10}^{-4}$.

Figure 3

(Color online) The tip velocity (a) and its first and second time derivatives [(b) and (c), respectively]. We see that the singularity is cured and the velocity decelerates due to the plastic effects.