Random and correlated roughening in slow fracture by damage nucleation

We address the role of the nature of material disorder in determining the roughness of cracks, which grow by damage nucleation and coalescence ahead of the crack tip. We highlight the role of quenched and annealed disorders in relation to the length scales $d$ and ${\xi }_c$ associated with the disorder and the damage nucleation, respectively. In two related models, one with quenched disorder in which $d\simeq {\xi }_c$, the other with annealed disorder in which $d⪡{\xi }_c$, we find qualitatively different roughening properties for the resulting cracks in two dimensions. The first model results in random cracks with an asymptotic roughening exponent $\zeta \approx 0.5$. The second model shows correlated roughening with $\zeta \approx 0.66$. The reasons for the qualitative difference are rationalized and explained.

Figure 1

(Color online) A typical crack of 500 growth steps in arbitrary units. Note the difference in scales between the ordinate and the abscissa. The inset shows a magnified segment of the crack corresponding to the rectangle. The overall scale of the ordinate of the inset plot is 100.

Figure 2

A log-log plot of the height fluctuations $h\left(r\right)$ of Eq. (1) as a function of the scale $r$. The straight line corresponds to $\zeta =0.57$. A similar measurement using the variable bandwidth rms method [14] yielded $\zeta =0.51$, therefore our estimation in this case is $\zeta =0.54$ (see the text for more details).

Figure 3

(Color online) The hydrostatic tension $P$ [see Eq. (8)], in units of ${\sigma }^{\infty }$, as a function of $\theta$ (in radians) at a fixed radius $r={\xi }_c$ (solid line) and a fit with $K_{\mathit{I}}=125$ (arbitrary units), $\frac{K_{\mathit{II}}}{K_{\mathit{I}}}=-0.51$ (dashed line). The present data was taken from a crack after 300 growth steps, but it is representative.

Figure 4

(Color online) The average roughness exponent $\zeta$ as a function of the number of growth steps $N$ for model $B$. The decrease towards a stationary state with $\zeta \approx 0.5$ is observed.