Role of the sample thickness in planar crack propagation

We study the effect of the sample thickness in planar crack front propagation in a disordered elastic medium using the random fuse model. We employ different loading conditions and we test their stability with respect to crack growth. We show that the thickness induces characteristic lengths in the stress enhancement factor in front of the crack and in the stress transfer function parallel to the crack. This is reflected by a thickness-dependent crossover scale in the crack front morphology that goes from from multiscaling to self-affine with exponents, in agreement with line depinning models and experiments. Finally, we compute the distribution of crack avalanches, which is shown to depend on the thickness and the loading mode.

Figure 1

The geometry of the model. We consider a cubic lattice of size $L\times L\times H$ made of conducting bonds with a weak plane in the middle where we place fuses with random breaking threshold. A notch is placed in the weak plane at the beginning of the simulation. The voltage drop is either applied between the top and bottom planes (plane loading) or at the left edges of the plates, along the dashed lines (line loading).

Figure 2

The factor $K$ as a function of the crack length, computed numerically for systems with different thickness $H$. Here, $K$ is defined as the voltage drop in the bonds ahead of the crack tip for unit applied voltage. In the top panel, the system is loaded by imposing a constant voltage at the left edges of the system (line loading). In the bottom panel, the constant voltage is imposed on the entire top plate (plane loading).

Figure 3

The phase diagram of the crack under line loading. The stable region is defined by a decrease of $K$ as $a$ is increased, implying that the stress in front of the crack decreases as the crack advances. Under plane loading the crack is always unstable or at most marginally stable (i.e., $K=const$).

Figure 4

The vertical current across the weak plane decays exponentially as a function of the distance from the crack tip for different thicknesses for both line and plane loading. (a) For line loading, the current decays exponentially to zero. A characteristic length $\xi$ can be extracted from the decay of the currents. As shown in the inset, $\xi$ is a linear function of the thickness $H$, but it is larger for line loading (i.e., ${\xi }_{\mathrm{line}}\simeq 3{\xi }_{\mathrm{plane}}$). (b) For plane loading, the current decays exponentially until it reaches a constant value, which decreases as $1/H$ (inset).

Figure 5

The variations in the stress intensity factor following the failure of one bond ahead of the crack as a function of the distance from the bond parallel to the crack direction. This is equivalent to a self-interaction kernel, scaling as $1/y^2$ up to a cutoff length ${\xi }_{\parallel }$. Data for different thicknesses are collapsed by rescaling the distance by a characteristic length ${\xi }_{\parallel }$ whose values are reported in the insets. Results for line loading (a) and plane loading (b).

Figure 6

The avalanche progression as a function of the loading model and the sample thickness. Different avalanches are identified by random colors.

Figure 7

The $q$ moments of the correlation functions for cracks under plane loading for (a) $H=10$ and (b) $H=100$. The dashed line indicates a power law with exponent $\zeta =0.33$.

Figure 8

The accumulated damage as a function of the applied voltage for (a) line loading and (b) plane loading. The staircase character of the curve is a signature of avalanche behavior.

Figure 9

The avalanche size distribution measured under line loading for different values of sample thickness.

Figure 10

The voltage-current curve for the model under plane loading. The inset shows the peak current $I_c$ as a function of the thickness $H$ for plane and line loading.

Figure 11

The stress survival distribution for (a) plane and (b) line loading as a function of the sample thickness. Data are plotted as a function of reduced variables $(\sigma -\langle \sigma \rangle )/\mathcal{S}$ and the corresponding Gaussian distribution is reported for comparison.