Scaling laws of shrinkage induced fragmentation phenomena

TL;DR

This work addresses how slow shrinkage drives fragmentation in thin heterogeneous layers attached to substrates, revealing two distinct phases: a damage phase with cracks but a dominant intact piece, and a fragmentation phase with many small fragments. The authors implement a two-dimensional discrete element model based on a Voronoi tessellation of convex polygons connected by beams and anchored to the substrate, with shrinkage implemented via a decreasing natural length and breakage governed by a combined strain–bending criterion; adhesion is modeled by springs to the substrate. A finite-size scaling analysis uncovers a continuous-transition-like damage-to-fragmentation crossover at a critical damage $d_c$, with scaling forms for the largest fragment mass and the average fragment mass that yield exponents near the 2D bond-percolation values, and with fragment-mass statistics transitioning from a small-$m$ power law ($\tau\approx 2$) to a log-normal distribution in the fragmentation phase. The results suggest universal scaling laws governing shrinkage-induced fragmentation and have implications for interpreting natural crack patterns and designing controlled fragmentation in industrial contexts, while noting limitations due to 2D modeling and finite system sizes.

Abstract

We investigate the shrinkage induced breakup of thin layers of heterogeneous materials attached to a substrate, a ubiquitous natural phenomenon with a wide range of potential applications. Focusing on the evolution of the fragment ensemble, we demonstrate that the system has two distinct phases: damage phase, where the layer is cracked, however, a dominant piece persists retaining the structural integrity of the layer, and a fragmentation phase, where the layer disintegrates into numerous small pieces. Based on finite size scaling we show that the transition between the two phases occurs at a critical damage analogous to continuous phase transitions. At the critical point a fully connected crack network emerges whose structure is controlled by the strength of adhesion to the substrate. In the strong adhesion limit, damage arises from random microcrack nucleation, resembling bond percolation, while weak adhesion facilitates stress concentration and the growth of cracks to large extensions. The critical exponents of the damage to fragmentation transition agree to a reasonable accuracy with those of two-dimensional bond percolation. Our findings provide a novel insight into the mechanism of shrinkage-induced cracking revealing generic scaling laws of the phenomenon.

Figures (11)

• Figure 1: Construction of the discrete element model of restrained shrinking of a thin brittle layer. $(a)$ The layer is discretized using space filling convex polygons obtained by a Voronoi tessellation with controlled disorder. $(b)$ Polygons are connected by beams which exert cohesive forces and are able to break. Successive breaking of neighboring beams results in cracks in the layer. $(c)$ Adhesion is represented by springs connecting the center of polygons to the substrate.
• Figure 2: $(a)$ Average value $\left<F_n\right>$ of the longitudinal force of beams ahead of a crack as a function of the distance $x$ measured from the crack tip. Results are presented for four different values of the adhesion strength $D_s/D_b$. Along the vertical and horizontal axis the data are scaled with the value of the background force $F_n^0$ and of the average polygon size $l_0$, respectively. The continuous lines represent fits with Eq. (\ref{['eq:forcefit']}). The spatial distribution of the longitudinal force is illustrated by the inset where beams are colored according to the value of $F_n^{ij}$. $(b)$ The parameter $A$ controlling the range of load redistribution around crack tips as a function of the adhesion strength $D_s/D_b$. The straight line represents a power law of exponent $\kappa=0.3$.
• Figure 3: Time evolution of shrinking induced cracking of a circular layer of radius $R=120$ at the adhesion strength $D_s/D_b=8\cdot10^{-2}$. First, cracks nucleate at the weakest spots and grow in random directions $(a)$. Growing cracks merge and form a connected network along which the sample falls apart into fragments $(b)$. Further shrinking results in crack nucleation inside fragments which breaks them into two daughter pieces $(c,d)$.
• Figure 4: Snapshots of the cracking layer right after the connected crack network is formed at four different values of the adhesion strength $D_s/D_b$: $(a)$$1$, $(b)$$8\cdot10^{-2}$, $(c)$$8\cdot10^{-3}$, $(d)$$1.6\cdot10^{-3}$.
• Figure 5: $(a)$ The average fragment mass $\left<M_{av}\right>$ normalized by the total system mass $M_0$ as a function of the cumulative damage $d$ for different adhesion strengths $D_s/D_b$. The sharp maximum marks the critical damage $d_c$, where the crack network becomes fully connected, leading to the breakup of the layer into a large number of fragments. $(b)$ The critical damage $d_c$ as a function of the adhesion strength $D_s/D_b$ for three different system sizes $R$.