Interlocking-induced stiffness in stochastically microcracked materials beyond the transport percolation threshold

We study the mechanical behavior of two-dimensional, stochastically microcracked continua in the range of crack densities close to, and above, the transport percolation threshold. We show that these materials retain stiffness up to crack densities much larger than the transport percolation threshold due to topological interlocking of sample subdomains. Even with a linear constitutive law for the continuum, the mechanical behavior becomes nonlinear in the range of crack densities bounded by the transport and stiffness percolation thresholds. The effect is due to the fractal nature of the fragmentation process and is not linked to the roughness of individual cracks.

Figure 1

Crack pattern in a model with $f=0.67$ fraction of cracked boundaries. The cracked boundaries are shown in black and the uncracked boundaries are not shown. This model contains a single percolated path (red thick line).

Figure 2

Variation of the measure of overhangs, $\overline{m}-1$, with $f$, for $f>f_{c1}$ and for two model sizes, $L/\overline{d}=31.6$ and ${10}^3$. The inset shows that $\overline{m}-1\sim {\left(f_{c2}-f\right)}^2$ in the vicinity of $f_{c2}$. Bars represent standard deviation based on 100 replicas and, for $L/\overline{d}={10}^3$ and $f\ge 0.69$, are of the size of the symbols.

Figure 3

Variation of the normalized effective stiffness with $f$. The vertical dashed line indicates the transport percolation threshold, ${f_c}_1$. The horizontal dashed line indicates the largest stiffness of a model with at least one percolated path with no interlocking. The continuous line shows the prediction of the model in [24] valid in the limit $f\to 0$. The shade indicated the range of $f$ in which interlocking controls the mechanical behavior. The shape and color of symbols indicates the three regimes: nonpercolated (blue circles), percolated with interlocking (black filled circles), and percolated without interlocking (red open diamonds). The inset shows the power law variation of the effective stiffness in the vicinity of ${f_c}_2=0.78$. All data obtained with model with $L/\overline{d}={10}^3$.

Figure 4

Variation of the normalized effective stiffness with the geometric parameter $\overline{m}-1$ indicating the presence of overhangs and hence interlocking. The inset shows the variation of function $g$ defined in text with the model size, with different symbols indicating different $f$ values. The line of slope–1 was added to guide the eye.

Figure 5

Uniaxial tension stress-strain curves for 2D samples with $f=0.75$ (continuous line) and 0.63 (dashed line) and for a 3D sample of itacolumite (symbols). The vertical axis is normalized with $E_{\mathrm{eff}}$ (Fig. 3) and, for the itacolumite, with the slope of the asymptote at larger strains.