Transient damage spreading and anomalous scaling in mortar crack surfaces

The scaling properties of a post-mortem mortar crack surface are investigated. The root mean square of the height fluctuations is found to obey anomalous scaling properties, but with three exponents, two of them characterizing the local roughness ($\zeta \simeq 0.79$ and ${\zeta }_e\simeq 0.41$) and the third one driving the global roughness $\left({\zeta }_g\simeq 1.60\right)$. The critical exponent $\zeta \simeq 0.79$ is conjectured to reflect damage screening occurring for length scales smaller than the process zone size, while the exponent ${\zeta }_e\simeq 0.41$ characterizes roughness at larger length scales, i.e., at length scales where the material can be considered as linear elastic. Finally, we argue that the global roughness exponent could be material dependent contrary to both local roughness exponents ($\zeta \simeq 0.8$ and ${\zeta }_e\simeq 0.4$) which can be considered as universal.

Figure 1

(Color online) Root-mean-square roughness $\Delta h\left(l\right)$ vs $l$ for two profiles located at the beginning ($x=x_1$, circles) and at the end ($x=x_2$, squares) of the domain of growing roughness. The slopes of both straight lines give estimates of the local roughness exponents $\zeta =0.79$ (for $l⪡\xi$) and ${\zeta }_e=0.41$ (for $l⪢\xi$). All length scales are given in $\mu \text{m}$.

Figure 2

(Color online) Log-log plot of the rms roughness as a function of the distance $x$ to the initial notch. The part between the positions $x_1$ and $x_2$ corresponds to the roughness growth domain. According to Eq. (2), the lower straight line corresponds to the fit of the roughness growth observed at small length scales [i.e., for $l⪡\xi \left(x\right)$] and its slope is equal to 0.24 corresponding to the scaling exponent $\left({\zeta }_g-\zeta \right)/z_x$. The upper straight line is related to the roughness growth at large length scales [i.e., for $l⪢\xi \left(x\right)$] and leads to the estimate of the scaling exponent $\left({\zeta }_g-{\zeta }_e\right)/z_x=0.35$.

Figure 3

(Color online) Modified anomalous scaling function $g_m\left(u\right)$ [Eq. (3)]. The data collapse (corresponding to 110 profiles corresponding to positions $x_1\simeq 400\mu \text{m}$ to $x_2\simeq 5.9\text{mm}$) leads to the exponent values $\zeta =0.79$, ${\zeta }_e=0.41$, ${\zeta }_g=1.60\pm 0.10$, and $z_x=3.4\pm 0.2$.