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

TL;DR

This study demonstrates how controlling the mean coordination number in disordered 2D networks influences the brittle-ductile failure transition, revealing a critical point at the isostatic value where failure mode changes.

Contribution

It introduces an experimental approach to tune failure modes in disordered networks via mean coordination number and elucidates the underlying cooperative mechanisms using a pebble game algorithm.

Findings

Transition occurs at mean coordination number $rac{z}{7} = 3$ in large-friction limit.

Brittle failure associated with rigid, spanning clusters.

Ductile failure occurs in floppy networks with nonspanning rigid clusters.

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 $\langle z \rangle$ 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 $\langle z \rangle = 3$ 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.

Figures

Tables

Table 1. Table 1: Forecasting efficiency of the rigid cluster decomposition for different samples. The first column indicates proportion of failed edges that are floppy, the second one lists the proportion of floppy edges in the initial network.

		Floppy Failed	System Floppy
$⟨z⟩$	Loading	Edges ($\%$)	Edges ($\%$)
3.0	Compression	83	57
3.0	Tension	90	57
3.0	Tension	100	53
3.10	Tension	73	45
3.15	Tension	74	44

Full text

Materials as varied as semiconductors oshima_extraordinary_2018 , aqueous foams arif_spontaneous_2012 , polymers jang_ductile-brittle_1984 , metals Tanguy2005 and rocks wong_brittle-ductile_2012 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 roux_rupture_1988 ; alava_role_2008 ; shekhawat_damage_2013 ; curtin_brittle_1990 ; kahng_electrical_1988 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. hanifpour_mechanics_2018 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. driscoll_role_2016 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. zhang_fiber_2017 , 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 bouzid_network_2017 focused on the role of connectivity in a disordered system, by studying the mechanical properties of simulated soft gels, showing that the topology greatly affects stress redistribution and deformation of the branches, resulting in brittle failure of the more homogeneous stiffer gels with higher mean connectivity.

Here, we focus on experimentally-addressing this last question – tuning connectivity within an inherently disordered experimental system – by isolating the effects of the mean coordination number $\langle z\rangle$. We show that this parameter controls access to a variety of behaviors, from diffuse to brittle failure, in experimentally fabricated disordered 2D lattices made of stiff acrylic. This determination is particularly important for a large class of disordered materials: solid lattices or rigid foams composed of disordered beams bonded at their intersections. These materials present an attractive choice for producing lightweight, tunable structures and have recently been developed as (mechanical) metamaterials Paulose7639 ; bertoldi_flexible_2017 ; goodrich_principle_2015 ; reid_auxetic_2018 ; rocks_designing_2017 ; hanifpour_mechanics_2018 . Furthermore, such discrete structures have been commonly used in statistical models of fracture alava_statistical_2006 , as a means to conveniently discretize a material. For example, the random fuse model (RFM) de_arcangelis_random_1985 ; gilabert_random_1987 ; alava_role_2008 , fiber bundle model (FBM) pradhan_failure_2010 ; Kun2006 or elastic springs model alava_role_2008 ; curtin_brittle_1990 ; zhang_fiber_2017 ; nukala_percolation_2004 ; ray_breakdown_2006 descriptions rely on such lattices in which material disorder is introduced. Despite a wealth of models, few fracture experiments to discriminate between the various models have been performed on disordered lattices driscoll_role_2016 ; hanifpour_mechanics_2018 , in part due to the difficulty of creating samples by hand gilabert_random_1987 .

With the advent of laser-cutting technology, we are able to readily produce samples with precise, reproducible properties and thereby perform controlled experiments on the failure behavior of disordered lattices with various connectivities. We focus on the limiting case where disorder is primarily set via the geometry of the lattice, and material fluctuations are intended to be negligible. Rather than diluting broedersz_criticality_2011 ; ellenbroek_rigidity_2015 ; feng_percolation_1984 ; zhang_fiber_2017 ; driscoll_role_2016 and perturbing driscoll_role_2016 a regular lattice (where the underlying crystalline structure is still present), we instead draw on experimentally-generated, inherently disordered materials, as shown in Fig. 1. This approach is inspired by prior studies on numerically-generated packings of jammed frictionless particles driscoll_role_2016 ; goodrich_principle_2015 ; ellenbroek_non-affine_2009 ; ellenbroek_rigidity_2015 . Using such packing-derived networks has the additional benefit of allowing connections to some particular features of jammed packings liu_jamming_2010 ; van_hecke_jamming_2010 . For example, while their shear modulus vanishes at the jamming point, their bulk modulus remains finite ellenbroek_rigidity_2015 , which is rather different from, for example, central-force spring networks where both the bulk and shear modulus vanish at their equivalent jamming point ellenbroek_non-affine_2009 .

Our networks are generated from real, frictional packings majmudar_contact_2005 drawn from experiments on uniaxially compressed quasi-2D bidisperse granular packings of frictional photoelastic disks, created using the techniques described in puckett_equilibrating_2013 ; daniels_photoelastic_2017 ; kollmer_betweenness_2018 ; bililign2018 . In each case, the test structure has a geometry defined by the binary contact network of these disks. Each particle center is set to be a crosslink (node), and each particle contact in the original packing is a beam (edge). The beams are of uniform thickness and connect at the crosslinks. This process is shown in Fig. 1, resulting in a disordered lattice that can be cut from a single acrylic sheet. Once cut, each sample can be exposed to either tensile or compressive tests through to the point of successive failures.

In our experiments, we measure the spatial and temporal pattern of failure as a function of the connectivity of the disordered lattice. We find a transition between ductile and brittle behaviors as a function of the coordination number $\langle z\rangle$. This is characterized by both the force response to compressive or tensile deformation, and the magnitude of the resulting force drops. In the brittle samples, failure develops along narrow, meandering cracks until the sample breaks in two.

We use rigidity theory to interpret changes in these patterns. It has been successfully applied to disordered networks of linear springs moukarzel_stressed_1995 ; jacobs_generic_1995 and to granular packings mapped to a node-spring network with 2 or 3 degrees of freedom at each node ellenbroek_rigidity_2015 ; ellenbroek_non-affine_2009 ; feng_percolation_1985 ; henkes_rigid_2016 . To identify rigid clusters based on the topology of the spring network of frictionless packings, one uses a $(2,3)$ pebble game jacobs_algorithm_1997 . However, our disordered 2D lattices originate from frictional packings, which add particle rotations and tangential forces to the central forces, so that we use a $(3,3)$ pebble game recently developed for the study of frictional granular packings henkes_rigid_2016 .

We associate the central forces to the stretching/compressing of the beams and the tangential ones to the rotations around welded points. Arguing that these deformations dominate the response in our networks, we find that the frictional rigid cluster decomposition allows us to differentiate the response of low- vs. high-connectivity networks and capture the change in failure behavior from ductile to brittle. We are also able to establish that in the presence of both rigid and floppy clusters for networks near the transition point, identified close to $\langle z\rangle_{\mathrm{iso}}=3$, which is the isostatic connectivity of an infinite friction coefficient packing, failures primarily occur on floppy edges located between the rigid clusters.

Methods

Each sample’s geometry is derived from a force chain network observed in an experimental 2D granular material, as shown in Fig. 1(a). The apparatus used to obtain these configurations contains $N=824$ to $890$ bidisperse particles (two distinct radii, 5.5 mm and 7.7 mm, in approximately equal numbers) made from photoelastic Vishay PhotoStress circular disks. The mixture is placed on an air-table to remove basal friction and a cyclic uniaxial loading introduces local contact rearrangements. This allows for the observation of multiple force distributions. We use an open source photoelastic solver jekollmer_pegs:_2018 using the method described in daniels_photoelastic_2017 to obtain the contact network underlying the observed force chains. An example of the binary network obtained by this method is shown in Fig. 1(b).

We characterize each network by its mean coordination number $\langle z\rangle$, which is the mean number of beams intersecting at the crosslinks in each sample. In performing our experiments, we use 8 networks, drawn from 4 different initial granular configurations loaded by different amounts. Two of these networks are obtained using a packing derived from previously-collected data bililign2018 . Due to the lack of experimental data near that connectivity value, the network with $\langle z\rangle\approx 3$ was obtained by pruning a higher connectivity network: contacts with a force below a threshold were discarded, the threshold being set so as to obtain the targeted connectivity. Together, this provides values ranging from $\langle z\rangle=2.4$ to $\langle z\rangle=3.6$, selected to be approximately evenly spaced in the vicinity of $\langle z\rangle_{\mathrm{iso}}=3$.

To render these networks as physical samples, we laser cut acrylic plastic plates of thickness $h=3.17$ mm, and approximately $30\times 30$ cm2. In the sample, each contact force of the granular system corresponds to a beam of width $1.4$ mm. The beams intersect at crosslinks (particle centers), and are of variable length depending on the radii of the two contacting particles. Thus, the sample has beams with width:length ratios of $0.15$ to $0.2$; shorter beams are slightly stiffer than the longer ones, for constant elastic modulus $E\approx 3$ GPa. In addition, some material disorder remains intrinsic to the bulk material, and some defects are presumably created during the laser-cutting process.

We test each sample in displacement-controlled experiments using an Instron 5940 Single Column system. Compression experiments are conducted with a displacement rate of $1.5$ mm/min, and tensile experiments at $1.0$ mm/min; both loading rates allow for well separated failure events or avalanches. In compression, the structure is constrained between two transparent acrylic plates to prevent out-of-plane buckling. To analyze the deformation and failure behaviors, we measure the macroscopic force $F$ as a function of the imposed displacement $\Delta$ and record each experiment using a Nikon D850 digital camera, at a frame rate ranging from 24 to 60 fps. During the course of each experiment, fracture events occur within the sample and sequentially deteriorate its structure through the sudden breaking of one or more beams. Each event appears as a drop in the macroscopic response, as shown in Fig. 2(a). The corresponding failure locations are determined from examining the image taken at the corresponding time.

Results

Macroscopic behavior

Several typical macroscopic force-displacement responses measured are shown in Fig. 2(a), for three different samples tested in compression. For low applied displacement $\Delta$, there is an initial elastic regime in which deformations are principally accommodated by beam-bending at the nodes. At a displacement $\Delta^{1}$ and corresponding force $F^{1}$, the first beam breaks, as shown by the green square in Fig. 2(a) on the curve taken from the $\langle z\rangle=3.6$ sample. Each such event triggers an abrupt force drop $\delta F$. After one or a few such failure events (typically small, i.e. involving one beam breakage), the system reaches a peak force $F^{p}$, indicated by the red dot in Fig. 2(a). From this point onward, the response intermittently switches between two behaviors: elastic deformation, and dissipative failure events. This intermittency continues until a critical loading $\Delta^{f}$ is reached, marked by a blue diamond, at which the failures have laterally spanned the entire system, such that there is no set of beams connecting the top and bottom boundaries. We end all tests at this point.

This general behavior is observed for all samples, but the specifics are $\langle z\rangle$-dependent. As shown in Fig. 2(a), networks with higher coordination number are stiffer, as previously observed in simulations ellenbroek_non-affine_2009 ; ellenbroek_rigidity_2015 . The peak force $F^{p}$ increases with $\langle z\rangle$, as plotted in Fig. 2(b) for both compression and tension experiments. Finally, lower-$\langle z\rangle$ networks exhibit a different type of intermittency: many small events are observable before failure, rather than a few large events, and some of the force drops correspond to sudden sliding of beams put in contact due to the high compression.

To characterize the failure behavior we consider 4 quantities: the loading $\Delta^{1}$, its corresponding force $F^{1}$, the loading at failure $\Delta^{f}$, and the maximum observed force drop $\mathrm{max}(\delta F)$. Each is shown in Fig. 3 as a function of $\langle z\rangle$, for both compression (solid blue lines) and tension (dashed red lines). In compression, we observe that as $\langle z\rangle$ increases, the applied displacement $\Delta^{1}$ decreases, i.e. the first failure event occurs at a lower displacement, while the force at that events rises. The displacement $\Delta^{f}$ at which there is system-spanning failure is also significantly lower for larger $\langle z\rangle$. Finally, for $\langle z\rangle\leq 3$, there is progressive damaging of the samples, with typically one or two beams involved in a failure avalanche. For $\langle z\rangle>3$, there are also large (involving more than 3 beams breakage), catastrophic events, associated with large drops of force. This is captured by the evolution of the maximum force drop $\mathrm{max}(\delta F)$ with $\langle z\rangle$.

We recall that the network with $\langle z\rangle=3.0$ was artificially obtained by removing low-force contacts from the initial, higher-connectivity network. Yet, no peculiar behavior is noticeable from the results: the samples follow the same trends as the others. In tension, similar behaviors are obtained, but the data is noisier as a function of $\langle z\rangle$ and the samples are less resilient than in compression: premature failure occurs more frequently.

Spatial patterns of failure

We find that the spatial pattern of the individual failed beams varies significantly with $\langle z\rangle$, as illustrated in Fig. 4 for the two extreme values of $\langle z\rangle=2.4$ and $\langle z\rangle=3.6$, under both compression and tension. The initial disordered lattice is shown in gray, with each beam that failed during the entire experiment colored as a thick bar. For low-connectivity lattices ($\langle z\rangle<3$), failures are rather spread throughout the sample, while for highly connected networks ($\langle z\rangle\geq 3$) the failed edges are localized, forming a narrow meandering (especially in tension) crack. While this general trend is not affected by the type of loading, we nonetheless observe that the failure locations are not identical in the two cases. From repeating the same experiment with “identical” samples, we observe that this can be partly attributed to the sample-to-sample variation in the material disorder, both inherent and introduced during cutting.

To characterize these spatial patterns, we define a crack width $w$ as follow: we compute the eigenvalues of the covariance matrix of the cloud of broken points, which we then scale such that the corresponding ellipse, centered on the centroid of the broken edges, encloses $70\%$ (darker ellipses in Fig. 5(a)) to $100\%$ (lighter ellipses) of the broken edges. The crack width is defined as the mean value of the scaled smallest eigenvalues (minor axis of the ellipses). In the case of a meandering crack, we use two or more ellipses to cover the straight pieces (see Fig. 5a) and take the mean as the crack width. When no particular orientation is noticeable, as in the top row of Fig. 4, one ellipse is used.

As suggested by the images in Fig. 4, $w$, normalized by the smallest beam dimension $\ell$, decreases as a function of $\langle z\rangle$ (see Fig. 5(b)). A quantitatively similar trend is observed for both compression and tension experiments.

Therefore, we identify two types of behavior depending on the connectivity of the lattices. For samples with $\langle z\rangle<3$, failure events progress continuously in the system, with a rather broad spatial distribution of failure events. We refer to this behavior as ductile. As $\langle z\rangle$ increases, the characteristic parameters (loading at first and last failure event, maximum force drop, crack width) evolve continuously. For $\langle z\rangle>3$, a combination of small and large localized catastrophic failure events contribute to the failure of the system via the spanning of a narrow crack. For simplicity, we will refer to this more brittle behavior as “brittle”, to contrast it with the more ductile behavior observed for lower connectivity.

Frictional pebble game

To better understand how this transition emerges, we apply rigidity theory to a mathematical representation inspired by the granular origins of our samples. In creating the lattices, we start from frictional particle packings and essentially “glue” the disks together, so that we have effectively increased the friction coefficient to $\mu=\infty$. We have chosen our pebble game parameters to take these permanent attachments into account.

The mean coordination at which the rigidity transition occurs can be estimated by Maxwell constraint counting maxwell_l._1864 for frictionless granular packings and using a generalized isostaticity criterion introduced by Shundyak et al. shundyak_force_2007 for frictional packings. To go beyond mean-field, one can perform a rigid cluster decomposition and identify the location of the rigidity transition: this occurs when at least one rigid cluster spans the system. For frictionless packings, Laman’s theorem laman_graphs_1970 can be implemented using what is known as a $(2,3)$ pebble game jacobs_algorithm_1997 , where $2$ represents the number of degrees of freedom per particle (translational) and $3$ represents the number of trivial global degrees of freedom. To do so, the packings are mapped to a constraint network where nodes (particle centers) are connected by a bond if their associated particles are in contact ellenbroek_rigidity_2015 ; ellenbroek_non-affine_2009 ; feng_percolation_1985 ; henkes_rigid_2016 .

In frictional packings ($\mu>0$), each particle has 2 translational and one rotational degrees of freedom. The contacts between particles provide two constraints—one for the central (normal) force and one for the tangential force, except for contacts at the Coulomb threshold where normal and tangential forces are coupled, leading to only one independent constraint. To take that into account, the $(2,3)$ pebble game was recently extended henkes_rigid_2016 to a $(3,3)$ pebble game. It accommodates the additional rotational degree of freedom and the constraint network contains both single (sliding) bonds or double (frictional) bonds.

From the gluing approach used to build our sample, one concludes that playing a $(3,3)$ pebble game on a constraint network with all double bonds ($\mu=\infty$) maps out the rigid clusters. However, in frictional granular packings, local stresses will eventually be accommodated by contact slipping, and ultimately rearrangements. To account for accommodation in the stiff lattices, we make the following assumption based on experimental observations. We distinguish between contacts in the disordered lattice forming linear chains (only two contacts connect at each nodes), and those that intersect at nodes with multiple other contacts. Since transverse forces can easily bend linear chains (unlike bonds having each node with degree $>2$, i.e. short links), we map each chain to a unique single bond. With this approximation, the constraint network has now changed from all double bonds to both single and double bonds, depending on the local coordination number.

Finally, to match the experimental boundary conditions, we introduce one particle representing a confining wall at the top and bottom of the system and connect them with single constraint bonds to the corresponding boundary nodes.

.1 Rigidity transition determines ductile vs. brittle failure

Fig. 6 shows sample results from applying the described pebble game algorithm on three intact disordered lattices. Each double (single) bond is indicated by a thick (thin) line. All identified rigid clusters are marked as colored subregions. For samples with $\langle z\rangle<3$, the analysis reveals predominantly-floppy networks, with only a few small isolated rigid clusters (see Fig. 6(a)). For samples with $\langle z\rangle>3$, a rigid cluster percolates to the boundaries.

To highlight the connection between rigidity transition and the brittle-ductile transition, in Fig. 7 we compare the $\langle z\rangle$-evolution of the loading at failure ($\Delta^{f}$) and crack width ($w$), with the evolution of the vertical size of the largest rigid cluster ($L_{RC}$). Both $\Delta_{f}$ and $w$ decrease with $\langle z\rangle$. For connectivity values $\langle z\rangle>3$, a brittle behavior is reached. The largest cluster size evolves in the opposite manner: it increases until reaching the system size when $\langle z\rangle>3$. Therefore, samples for which a clear ductile failure behavior is identified are the ones for which there is a percolating floppy region with non-spanning rigid clusters. In contrast, brittle-like failure occurs at large $\langle z\rangle$, in systems for which there is a percolating rigid cluster.

Finally, the geometry with $\langle z\rangle=3.0$ is of particular interest as it presents a floppy matrix but also large embedded rigid clusters. For this structure, the failure pattern was shown to exhibit narrow, crack-like, brittle features, while the failure process and characteristic loadings were more like the ductile samples. This mix of behaviors is echoed by the pebble game rigidity analysis below. Therefore, from the failure observation and rigidity analysis, the transition point is identified as close to $\langle z\rangle\approx 3$, the infinite friction isostatic point $\langle z\rangle_{\mathrm{iso}}$.

.2 Rigidity signatures in the failure locations

For samples for which mesoscale rigid regions are identified ($\langle z\rangle=3.0-3.15$), the rigid cluster decomposition additionally provides insights into the locations at which beams fail. For these networks, we isolate floppy vs. rigid regions. Since each beam failure corresponds to creating a new network by removing a bond from the previous, less damaged network, we re-apply the pebble game again after each failure event. Since the pebble game is highly non-local, a failure on one part of the lattice can cause a rigid edge in another part of the lattice to become floppy. Nonetheless, only a few edges evolve from rigid to floppy so that we here show the rigid clusters determined on the intact network.

The rigid cluster decompositions are shown on Fig. 8 for the intact structure of degree $\langle z\rangle=3.0$ tested in (a) compression and (b) tension. The failure locations, which vary between samples and loadings directions, are numbered by their temporal ordering and indicated by ellipses. For the samples shown in Fig. 8 one (in panel $a$) and two (in panel $b$) failed bonds were initially rigid, but became floppy after a failure event elsewhere. The slight differences in geometry result from beams that were broken during manufacturing, but these minor variations do not affect the rigid cluster decompositions which remain identical.

Floppy edges represent about half of the total edges of the system for the samples with $\langle z\rangle=3-3.15$. Failure events predominantly occur ($73-100\%$ of them) on the floppy edges separating rigid clusters. A physical explanation would be that floppy edges are freer to bend further, and thereby fail. In Table 1, we tabulate the proportion of failed edges that were marked as floppy for each sample, and the proportion of floppy edges of each intact system. These results indicate that in samples where both rigid and floppy clusters are present, the failure locations are not randomly distributed in the system. Instead, fractures seem to principally occur within the floppy regions determined by a rigidity analysis. Interestingly, because failures are occurring at locations outside of the rigid clusters, the clusters themselves do not evolve significantly as the damage progresses. On the other hand, because networks with connectivity $\langle z\rangle\gg 3$ have homogeneous-like structures that consist of a large rigid cluster, no information can be extracted for the failure locations. Indeed, breakages almost always occur on rigid bonds and a detailed analysis would be required to further understand the failure behavior of these networks. Similarly, for $\langle z\rangle\ll 3$, there is a large floppy matrix, and breakages occur primarily on floppy bonds.

Discussion and Conclusions

To study the failure locations and interpret the brittle-ductile transition, we have here considered a constraint-counting game that is rooted in how the samples were obtained. Yet, other games could be envisaged, in particular if one decides to focus on capturing the physical properties of the samples and assume that the network does not retain any information about the particle packing from which it was derived. In that case, in assigning the degrees of freedom and building the constraints network, one must account that the lattices studied here consist of welded beams, which can be mechanically described as a spring network composed of two-body springs and three-body, or angular, springs between any two beams. The latter have been shown to stabilize networks below the central-force isostatic point feng_percolation_1984 ; feng_percolation_1984-1 ; kantor_elastic_1984 ; das_redundancy_2012 . In fact, an effective medium theory in Das et al. das_redundancy_2012 has shown that the onset of rigidity percolation for welded beams correlates with the onset of connectivity percolation, which was initially argued in Kantor and Webman kantor_elastic_1984 .

In this paper, we have shown that low- and high-connectivity networks differ dramatically in their response to uniaxial loading. Under both tension and compression loadings, a ductile-like response is present in low-connectivity samples, and a more brittle response is observed in high-connectivity samples. This behavior is consistently manifested in a number of quantifiable ways, from maximum load (larger for higher connectivity) to the width of cracks (diffuse for low-connectivity, narrow and localized for high connectivity).

Furthermore, we find that an adaptation of the frictional pebble game provides quantitative insight into how these two classes of disordered lattices fail. In particular, low- and high-connectivity networks both have rigid and floppy sub-regions and the size of the rigid clusters correlates with the observation of the ductile vs. brittle behavior. Low-connectivity networks are dominated by floppy regions in which many failure locations are possible, while high-connectivity networks are dominated by rigid clusters where one failure can trigger a cascade of nearby failures within rigid clusters. Intriguingly, we additionally observe that in samples with both rigid and (sometimes newly-created) floppy regions, the edge failures are more likely to occur along the floppy regions separating rigid clusters.

This observation can be tested in numerical simulations for a variety of networks. A systematic study would permit testing its validity for structures of geometry constructed from different origins, i.e. not only from visible force-chains in frictional packings. Identifying the possible failure locations also raises the possibility of designing disordered structures with pre-determined failure behaviors. Finally, a possible connection between the failure locations observed in the lattices and contact rearrangements or force-chain buckling in the granular packings remains to be investigated.

Acknowledgements: We are grateful to support from the James S. McDonnell Foundation, National Science Foundation Grant DMR-1206808, and DMR-1507938.

Author contributions: J.E.K. performed the granular experiments and extracted the force chain networks; E.B. created the lattices and performed, processed, and analyzed the experiments; S.E.H, J.M.S. wrote the pebble-game code; K.L., E.B. performed the pebble-game analysis; E.B., J.M.S, S.E.H, and K.D. connected the theoretical and experimental results; E.B., J.M.S, S.E.H, and K.D. wrote the manuscript.

Competing interests:. The authors have no competing interests to declare.

Materials & Correspondence: Estelle Berthier (ehberthi@ncsu.edu)