Effect of the porosity on the fracture surface roughness of sintered materials: From anisotropic to isotropic self-affine scaling

To unravel how the microstructure affects the fracture surface roughness in heterogeneous brittle solids like rocks or ceramics, we characterized the roughness statistics of postmortem fracture surfaces in homemade materials of adjustable microstructure length scale and porosity, obtained by sintering monodisperse polystyrene beads. Beyond the characteristic size of disorder, the roughness profiles are found to exhibit self-affine scaling features evolving with porosity. Starting from a null value and increasing the porosity, we quantitatively modify the self-affine scaling properties from anisotropic (at low porosity) to isotropic (for porosity >10%).

Figure 1

(a) Geometry and dimensions of the wedge-splitting samples. (b) Evolution sketch of the temperature of the oven, of the sample, and of the force applied during the sintering protocol (see [38] for details). (c) Temperature values at the different stages of the sintering protocol for the different bead sizes.

Figure 2

(a) Sketch of the experimental setup. (b) Typical force-vs-displacement curve.

Figure 3

Microscope visualization of wedge-split fractured surfaces of the sintered-PS-bead samples. The bead diameter $d_0$, wedge speed $V_{\mathrm{wdg}}$, sintering load $F_s$, and corresponding porosity $\Phi$ are as follows.

(a) $d_0=21 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},F_s=8$ t, $\Phi =0\%$.

(b) $d_0=228 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},F_s=8$ t, $\Phi =0\%$.

(c) $d_0=583 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},F_{\mathrm{s}}=8$ t, $\Phi =0\%$.

(d) $d_0=583 \mu \text{m},V_{\mathrm{wdg}}=1.6 \mathrm{nm}{\mathrm{s}}^{-1},F_s=8$ t, $\Phi =0\%$.

(e) $d_0=42 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},F_s=0.1$ t, $\Phi =15\%$.

(f) $d_0=42 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},F_s=1$ t, $\Phi =1.45\%$.

Figure 4

Experimental height profiles along the crack propagation direction of the same samples as in Fig. 3. At the right, the bead shape is given as a reference, using the same abscissa and ordinate scales as in the plots. Since these scales are different, the bead appears distorted. (a) Experiment 2: $d_0=21 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},\Phi =0\%$. (b) Experiment 13: $d_0=228 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},\Phi =0\%$. (c) Experiment 16: $d_0=583 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},\Phi =0\%$. (d) Experiment 14: $d_0=583 \mu \text{m},V_{\mathrm{wdg}}=1.6 \mathrm{nm}{\mathrm{s}}^{-1},\Phi =0\%$. (e) Experiment 21: $d_0=42 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},\Phi =15\%$. (f) Experiment 3: $d_0=42 \mu \text{m},V_{\mathrm{wdg}}=16 \mathrm{nm}{\mathrm{s}}^{-1},\Phi =1\%$

Figure 5

Structure functions $S_x$ of intergranular experiments, along the propagation direction at $\Phi =0\%$. Each symbol corresponds to a given value of $d_0$. Inset: Same distributions made dimensionless by $d_0^2$.

Figure 6

Structure functions normalized by the optimal lengths ${\ell }_x$ and ${\ell }_z$. Measurements performed along (a) $x$ (propagation direction) and (b) $z$ (front crack). Both axes are logarithmic in the main panels and semilogarithmic in the insets. Straight lines in the main panels are power-law fits with exponents reported in Table 3.

Figure 7

Dimensionless structure functions along the propagation direction for different porosities $\Phi$.

Figure 8

(a) Normalized roughness amplitudes in the $x$ and $z$ directions $\sqrt{S\left(d_0\right)}/d_0$ versus porosity $\Phi$. (b) Large-scale roughness exponents ${\zeta }^{+}$ vs $\Phi$. Symbols match the experiments in Fig. 7. Crosses correspond to the averaged exponent values in Table 3.

Figure 9

Scaling anisotropy defined as ${\zeta }_z^{+}/{\zeta }_x^{+}$ as a function of the porosity $\Phi$.

Figure 10

$T$ stress as a function of the crack tip position $x_{\mathrm{tip}}$ obtained by classical finite-element simulations and digital image correlation.