Roughening of Fracture Surfaces: The Role of Plastic Deformation

Post mortem analysis of fracture surfaces of ductile and brittle materials on the $\mu \mathrm{m}$-mm and the nm scales, respectively, reveal self-affine cracks with anomalous scaling exponent $\zeta \approx 0.8$ in three dimensions and $\zeta \approx 0.65$ in two dimensions. Attempts to use elasticity theory to explain this result failed, yielding exponent $\zeta \approx 0.5$ up to logarithms. We show that when the cracks propagate via plastic void formations in front of the tip, followed by void coalescence, the void positions are positively correlated to yield exponents higher than 0.5.

Figure 1

The fracture scenario suggested in 12. This scenario had been documented in detail in corrosive glass fracture, and also more recently in the fracture of paper [13]. (Figure courtesy of E. Bouchaud.)

Figure 2

A forward direction profile of the hydrostatic tension $P$ in units of ${\sigma }_{\mathrm{Y}}$. On the crack $P=\sqrt{3}/2$, and it attains a maximum of $\sqrt{3}$ on the yield curve. The threshold line indicates a value of $P_c$ such that $\sqrt{3}/2<P_c<\sqrt{3}$. The typical length ${\xi }_c$ is shown. Other directions exhibit qualitatively similar profiles.

Figure 3

The yield curve (inset) and the probability distribution function on it for a long straight crack. The distribution is symmetric and wide enough to allow for deviations from the forward direction.

Figure 4

A crack that was generated using our model.

Figure 5

Calculation of the anomalous roughening exponent.

Figure 6

The yield curve (inset) and the probability distribution function on it for a straight crack followed by an upturn. The angle $\theta$ is measured relative to the horizontal direction. It is clear that the distribution is skewed in favor of positive angles.