Rigidity percolation control of the brittle-ductile transition in disordered networks

Abstract

In ordinary solids, material disorder is known to increase the size of the process zone in which stress concentrates at the crack tip, causing a transition from localized to diffuse failure. Here, we report experiments on disordered 2D lattices, derived from frictional particle packings, in which the mean coordination number z of the underlying network provides a similar control. Our experiments show that tuning the connectivity of the network provides access to a range of behaviors from brittle to ductile failure. We elucidate the cooperative origins of this transition using a frictional pebble game algorithm on the original, intact lattices. We find that the transition corresponds to the isostatic value z = 3 in the large-friction limit, with brittle failure occurring for structures vertically spanned by a rigid cluster, and ductile failure for floppy networks containing nonspanning rigid clusters. Furthermore, we find that individual failure events typically occur within the floppy regions separated by the rigid clusters. Materials as varied as semiconductors [1], aqueous foams [2], polymers [3], metals [4] and rocks[5] exhibit a brittle-ductile transition when parameters such as geometry, temperature, pressure, loading rate, or even illumination are varied. Because brittle failure occurs suddenly and unpredictably, often leading to catastrophic effects, it is important to understand which underlying control parameters are able to tune the failure behavior of materials and structures, including those that are heterogeneous and/or hierarchical. Improvements in the prediction of failure modes are essential to the design of optimal properties.Controlled studies of the brittle-ductile transition have been achieved both numerically and experimentally. For example, increasing material disorder has been shown [6-10] to increase the fracture process zone (FPZ) size, resulting in a less concentrated stress at the crick tip. The size of the FPZ eventually diverges for infinite disorder, producing a transition from a brittle narrow crack type failure to percolation-like diffuse behavior. Experimentally, Hanifpour et al. [11] showed that 3D-printed disordered auxetic lattices can fail in a ductile or brittle (with disorder-dependent tensile strength) manner depending on the loading direction. Driscoll et al. [12] demonstrated the key role of material rigidity in experiments on weakly-disordered honeycomb acrylic lattices, and also the key role of connectivity in numerical studies of two different types of spring networks, one of which was derived from numerical realizations of frictionless packings. Numerical studies in Zhang et al.[13], on the other hand, focused on how the nonlinear alignment of springs in a randomly diluted triangular lattice controls the transition at connectivities below what is known as the central force rigidity percolation point. Finally, Bouzid and Del Gado [14] focused on the role of connectivity in a disordered system, by studying the mechanical * corresponding author: Estelle Berthier (ehber...

Forecasting failure locations in 2-dimensional disordered lattices

“…Combined with [36,37], our work provides evidence that betweenness centrality successfully identifies physical properties in both granular packings and lattices that are derived from them. Similarly, analyses inspired by rigidity percolation in granular materials have identified that our disordered lattices undergo a ductile-brittle failure transition as a function of connectivity z 0 , as determined by counting degrees of freedom and constraints [38]. However, we have focused on testing the sensitivity and specificity of our approach in the context of disordered lattices that we generated from force networks in quasi-2D granular packings, and this does not ensure its success for other network structures.…”

“…As was reported in Berthier et al [38], one can control the compressive and tensile failure behaviors of a disordered lattice by tuning the mean degree of its associated network. This control parameter provides a way to create systems with a variety of failure behaviors, ranging from ductile-like to brittle-like failure.…”

“…We conduct experiments on a set of disordered structures that we derived from experimentally-determined force networks in granular materials, as done in Berthier et al [38]. The methodology to create these experimental samples is inspired by the work of [4,9,10,52,53], who performed a similar process numerically.…”

“…It is known that the bulk properties of amorphous solids [38,54] are influenced strongly by z. We use the subscript 0 denote initial networks (i.e., networks before any subsequent modifications from lattice beam failures).…”

“…Note that, as characterized in [38], both tensile and compressive loading of samples with z 0 < z iso will fail from breakages that are well-separated in time and are spatially spread in a sample (i.e., ductile-like failure). By contrast, it was shown in [38] that for z 0 > z iso , a few temporally separated breakages take place before the samples break abruptly and all of the failed beams are localized, forming a narrow crack (i.e., brittle-like failure). Therefore, for the samples with z 0 = 3.35 and z 0 = 3.60, the deterioration of a sample's structure occurs via both small (one to three breakages at a time) and large (more than three simultaneous breakages) failure events.…”