Effect of disorder on shrinkage-induced fragmentation of a thin brittle layer

We investigate the effect of the amount of disorder on the shrinkage-induced cracking of a thin brittle layer attached to a substrate. Based on a discrete element model we study how the dynamics of cracking and the size of fragments evolve when the amount of disorder is varied. In the model a thin layer is discretized on a random lattice of Voronoi polygons attached to a substrate. Two sources of disorder are considered: structural disorder captured by the local variation of the stiffness and strength disorder represented by the random strength of cohesive elements between polygons. Increasing the amount of strength disorder, our calculations reveal a transition from a cellular crack pattern, generated by the sequential branching and merging of cracks, to a disordered ensemble of cracks where the merging of randomly nucleated microcracks dominate. In the limit of low disorder, the statistics of fragment size is described by a log-normal distribution; however, in the limit of high disorder, a power-law distribution is obtained.

Figure 1

Demonstration of the model construction for a small system of size $L=30$: the brittle layer is discretized by means of randomly shaped convex polygons (a). Polygons along the sample perimeter are highlighted by red color. (b) Neighboring polygons are coupled by elastic beams which form a triangular lattice. The polygons are attached to the substrate by spring elements (not shown). (c) The subsequent breaking of beams gives rise to the formation of cracks.

Figure 2

Deformation of a single beam in its local coordinate system. The displacement vectors ${\vec{u}}_i,{\vec{u}}_j$ and the bending angles ${\mathrm{\Theta }}^i,{\mathrm{\Theta }}^j$ are illustrated on the left and right figures, respectively.

Figure 3

Probability distribution $p\left({\varepsilon }_{th}\right)$ of the breaking thresholds ${\varepsilon }_{th}$ of the stretching breaking mode. Uniform distribution is used where the amount of disorder can be controlled by varying the width $W_{\varepsilon }$. In the figure two distributions of different width are illustrated. The same functional form is considered also for the bending thresholds ${\mathrm{\Theta }}_{\text{th}}$.

Figure 4

Time evolution of a system at $\delta =0$. The beams are colored according to their longitudinal strain $\varepsilon$ so that yellow beams are nearly undeformed, while pink and red colors stand for growing elongation and the black beams tend to break. (Blue is the background color of polygons.) Since the strength disorder is zero, only a small amount of random crack nucleation occurs at early stages of the shrinking process (a). As the shrinkage strain $\varepsilon$ increases a connected crack network builds up and the sample falls apart into a large number of pieces (b, c). The fragments undergo further breaking events resulting in usually two daughter pieces (d).

Figure 5

Demonstration of the primary (a, b) and secondary (c, d) fragmentation mechanisms. The branching-merging scenario of propagating cracks leads to the formation of primary fragments. The tip of the propagating crack in (a) at both ends splits into two sub-branches, which then undergo further splitting events. Fragments are created by the merging of nearby subbranches (b). As shrinking proceeds, in the highly stressed interior of fragments a crack emerges and splits the fragment into two pieces (c, d).

Figure 6

Snapshot of the system at four different values of strength disorder $\delta =0.0,0.30,0.65,0.80$, for (a), (b), (c), and (d), respectively, obtained at the shrinkage strain where the fully connected crack network is formed.

Figure 7

The fraction of broken beams $n_b\left(\varepsilon \right)$ as a function of strain $\varepsilon$ for several values of the amount of disorder $\delta$, from left to right $\delta =1.0,0.9,0.8,0.7,0.6,0.5,0.4,0.3,0.25,0.2,0.15,0.1,0.05,0.0$.

Figure 8

The threshold strain $\varepsilon$ where cracking sets on in the layer varying the amount of strength disorder in the entire range $0\le \delta \le 1$.

Figure 9

Main panel: Maximum of $M_{\text{av}}$ normalized by the total mass $m_{\text{tot}}$ of the system as a function of the disorder amplitude on a double logarithmic plot. The straight line represents a power law of exponent 0.54. Inset: The average fragment mass $M_{\text{av}}$ as a function of strain $\varepsilon$ for several values of the disorder amplitude $\delta$.

Figure 10

Mass distribution of fragments $p\left(m\right)$ measured at different strain values $\varepsilon$ in the fragmentation phase at zero strength disorder $\delta =0.0$. The corresponding critical point of fragmentation is ${\varepsilon }_c=0.017$. The fragment mass $m$ is made dimensionless by division with the average mass of single polygons $m_0$.

Figure 11

Data collapse of the mass distributions of fragments obtained by rescaling the two axis of Fig. 10 with the arithmetic average of fragment masses. The continuous line represents a fit with the log-normal distribution Eq. (19). Note the logarithmic scale on the horizontal axis.

Figure 12

Mass distribution of fragments at high strength disorder $\delta =0.8$ for several different $\varepsilon$ values. The straight line represents a power law of exponent $\tau =2.25$. The fragment mass $m$ is made dimensionless by division with the average mass of single polygons $m_0$.