Abstract
We study precursors of failure in hierarchical random fuse network models which can be considered as idealizations of hierarchical (bio)materials where fibrous assemblies are held together by multilevel (hierarchical) crosslinks. When such structures are loaded towards failure, the patterns of precursory avalanche activity exhibit generic scale invariance: irrespective of load, precursor activity is characterized by powerlaw avalanche size distributions without apparent cutoff, with powerlaw exponents that decrease continuously with increasing load. This failure behavior and the ensuing superrough crack morphology differ significantly from the findings in nonhierarchical structures.
Introduction
Hierarchical materials are characterized by microstructure features that repeat on different length scales in a selfsimilar fashion. Biological materials provide compelling examples. Collagen, for instance, exhibits a hierarchical fiber organization which at different length scales comprises molecules, microfibrils, fibers, and fiber bundles^{1}. Such complex organization was shown to provide enhanced toughness over assemblies of isolated collagen molecules. Several authors (see e.g.^{2}) have suggested that hierarchical organization may delay or prevent the nucleation and spreading of critical flaws which control failure of nonhierarchical heterogeneous materials^{3,4}. Models of hierarchical materials have mostly used hierarchical generalizations of the wellknown equalloadsharing fiber bundle model (ELSFBM) which is a meanfield model for brittle fracture in disordered materials (see e.g.^{5}). In hierarchical variants, fibers are recursively grouped into bundles and load is assumed to be distributed equally among the intact fibers within each bundle  a salient feature which makes such models amenable to analytical treatment as renormalization arguments can be used to deduce the overall strength^{6} and the statistics of damage accumulation. Hierarchical fiber bundle models have been used in the context of biomaterials (see e.g.^{7}) and also of composites^{8}. A variant which consists in envisaging the structural elements of a hierarchical fiber bundle not as simple fibers but as chainsofbundles does not greatly alter the basic conceptual framework since, at least in the limit of elasticbrittle local constitutive behavior, the properties of a bundle can be inferred from those of the single fibers using standard methods^{9} and those of a chainofbundles then be deduced by weakestlink statistics. Models of this type were introduced for a speculative nanotubespaceelevator cable^{10} and for hierarchical biomaterials^{7}.
Practically all investigations of hierarchical fiber bundles focus on the effective strength of the hierarchical structures, whereas fundamental questions concerning the nature of the failure process (critical behavior vs. subcritical crack nucleationandgrowth) and the concomitant nature and statistics of precursor events have received little attention^{9}. In fact, because of their meanfield nature, ELSFBM and their generalizations are not well suited for investigating spatial patterns of damage accumulation and failure. In the present work we therefore depart from the fiber bundle paradigm. To explore how hierarchical organization affects the precursor activity in the runup to failure and ultimately changes the mode of failure, we formulate for the first time hierarchical generalizations of the wellknown random fuse network (RFN)^{11,12} which, unlike ELSFBM, is known to capture essential features of spatial stress patterns occurring during failure of continuous media such as the r^{−1/2} character of cracktip stress singularities. At the same time we emphasize that RFN models, representing a scalar caricature of tensorial elasticity, can provide a quantitative description of fracture of materials only in exceptional cases, such as tearing of thin sheets loaded in antiplane shear or uniaxial loading of materials with zero Poisson ratio (for an example see^{13}).
Results
Construction and statistical properties of hierarchical fuse networks
Our aim is to generalize fuse network models in such a manner that they can be used as concept models for investigating the impact of hierarchical architecture on the mode of failure of materials, highlighting substantial differences between hierarchical and nonhierarchical materials, and drawing analogies with the behavior of hierarchically architectured systems outside the realm of materials mechanics. To this end, we generalize the RFN model into a hierarchically crosslinked network of breakable fibers of heterogeneously distributed strength, which we denote as Hierarchical Fuse Network (HFN). The construction of such a network is illustrated in Fig. 1. The network consists of interconnected links of unit length and unit conductance (fuses) that are contained between two bus bars, which we visualize as located at the top and the bottom of the network. Through the bus bars, a load on the network is imposed, either in the form of a prescribed voltage between top and bottom bar or in terms of a prescribed total current flowing from top to bottom. The vertical direction is hence referred to as the loading direction (or load paralleldirection), whereas the horizontal direction is referred to as the loadperpendicular direction.
As zerothorder module we define a vertical (loadparallel) link. The firstorder network, which in Fig. 1 consists of four zerothorder modules plus one loadperpendicular crosslink, is referred to as the HFN generator (other forms of generators are explored in the Supplementary Information). A hierarchical network is then constructed recursively as follows: From the first order network, n = 1, we obtain a network of order n = 2 by replacing each zerothorder module with the generator itself. Hence, the second order network consists of four generator (firstorder) modules plus a connecting horizontal crosslink of length 2^{2} − 1. Accordingly, a network of order n + 1 is constructed by replacing, in a network of order n, each module of order n − 1 by a module of order n and connecting the central network by a link of length 2^{n} − 1. A network of order n that has been constructed in this manner represents an anisotropic structure consisting of 2^{n} − 1 loadparallel wires of length 2^{n} which are crosslinked in a hierarchical manner. The quantity L = 2^{n}, which defines the linear dimension of the network, is also referred to as the network size.
In addition to the deterministically constructed HFN (henceforth referred to as DHFN), we consider several randomized variants. To construct these, we impose periodic boundary conditions in the load perpendicular direction on the DHFN, by replacing the central crosslink of length L − 1 by one of length L and closing periodically. A row is defined as a set of loadperpendicular links that share the same vertical position, and a column is defined as a set of loadperpendicular links that share the same horizontal position. Variant networks as illustrated in Fig. 2, top, are then constructed as follows: (i) A network constructed by starting from a DRFN and then first randomly reshuffling the columns and then the resulting rows is denoted as SHFN (first reshuffling the rows and then the columns produces statistically equivalent results). (ii) A network constructed by independently rotating the rows of a DHFN by random integers i ∈ [0…L − 1] across the periodic boundaries is denoted as RHFN. (iii) We take the HFN crosslinks and distribute them randomly over the L^{2} possible crosslinking sites. This process creates a non hierarchical structure with equal degree of crosslinking, which we denote as a reference random fuse network, RRFN.
The crosslink structure of the different HFN variants can be statistically characterized in two manners illustrated in Fig. 2, bottom left. We may focus on the row structure and envisage the network as an assembly of loadperpendicular crosslinks, where the length of a crosslink is understood as the number of horizontally connected elementary links. Alternatively, we may envisage the network as an assembly of loadparallel gaps, where the length of a gap is referred to as the number of vertically adjacent locations where an elementary crosslink is missing. Interestingly, the different network variants differ substantially in the statistics of these elements, see Fig. 2.
For the DHFN, both crosslink lengths n_{cl} (number of horizontally connected crosslinks) and gap lengths n_{gp} (number of vertically adjacent gaps) are powerlaw distributed, \(p({n}_{{\rm{gp}}})\propto p({n}_{{\rm{cl}}})\propto {n}_{{\rm{gp}},{\rm{cl}}}^{\kappa }\) where the recursive construction implies the exponent κ = 3. The random reshuffling of columns and rows which produces a SHFN does not change these powerlaw distributions of crosslink lengths and gap lengths. While the shortlength behavior of the distributions is slightly modified, the powerlaw exponent of the distributions which governs the decay at large scales is unaltered (red circles and connecting red line in Fig. 2). The RHFN possesses by construction the same crosslink statistics as the DHFN since rotating a row across the periodic boundaries does not change the lengths of the connected crosslinks. However, the gap statistics in this case becomes exponential, see Fig. 2. Finally, for the RRFN both the crosslinks and the gaps are exponentially distributed as expected when crosslinks are randomly distributed over the network. We thus have three kinds of networks: Networks with powerlaw distributed gaps and crosslinks (DHFN and SHFN), networks where crosslinks are powerlaw distributed but gaps are exponentially distributed (RHFN) and networks where both links and gaps are exponentially distributed (DHFN).
Points where links are mutually connected are referred to as nodes; a network of size L has L(L − 1) nodes. Once the network morphology is established, we assign to each link a critical current: The link connecting nodes k, l fails once the current I_{kl} flowing through this link exceeds the critical value t_{kl}. Stochastic material heterogeneity is mimicked by taking the thresholds t_{kl} to be independent random variables which we assume to be uniformly distributed between 0 and 1, representing an assembly of highly unreliable elements. Other critical current distributions yield qualitatively similar results, see Supplementary Information.
Behavior under load
The networks can be loaded by adjusting the voltage difference V between the bus bars to maintain a fixed total current I (load control), or vice versa (voltage control). Except where explicitely noted, in the following we present results for the case of load control. The voltage V_{k} at node k represents a displacementlike variable, while the currents I_{kl} flowing between nodes represent stresslike variables. The equilibrium equations for this scalar model of elasticity result from Kirchhoff’s node law, imposing that the algebraic sum of all forces (currents) at a node must be zero. We follow the standard loading protocol for quasistatic RFN simulations^{12} (see Methods section). Under load control, the external load (the imposed current) is increased to the precise level where the first link breaks and then kept fixed while link failure leads to load redistribution which may trigger further failures: damage accumulates through bursts of local failures (avalanches). The number of failures occurring as a consequence of internal load redistribution at fixed total current defines the avalanche size s. Subsequent to an avalanche the load is again increased to induce link breaking, and this is repeated until global failure disconnects the network.
Figure 3a shows average currentvoltage characteristics for the HFN (voltage control). Comparison between the different simulated network variants demonstrates that RHFNs possess the highest peak current (which corresponds to the failure current in current control), followed by the reference RFN, DHFN and SHFN. While the peak currents for all morphologies are of the same order of magnitude, the crack patterns are significantly different between DHFN and SHFN on the one hand, and RHFN and reference RFN on the other hand. We show in Fig. 3b a typical crack profile for a DHFN close to failure together with a crack profile for a reference RFN. The RFN crack profiles exhibit typical selfaffine shapes as studied extensively in the literature on RFN models (see e.g.^{14}). The crack shape in both DHFN and SHFN is qualitatively different. In these networks the hierarchical structure with a power law distribution of vertical gaps imposes wide discontinuous jumps in the crack profile which are visually reminiscent of crack profiles encountered e.g. in bone^{15}.
Avalanche statistics
To understand the differences between DHFN and SHFN on the one hand, and RRFN and RHFN on the other hand, we study the size distributions of avalanches of link breakings that occur prior to global system failure. We resolve these distributions with respect to the applied load (current): the loading curve is subdivided into load value intervals and avalanche size distributions are computed separately for each interval. For nonhierarchical RFN, the statistics of precursors to failure is well established: avalanche activity in the runup to failure is characterized by truncated powerlaw distributions of avalanche sizes of the form
with a fixed exponent τ and a cutoff that increases with load and diverges at the point of failure^{14,16}. More realistic spring or beam models^{17,18} yield similar results. The same picture can also be found in our own simulations of RRFN where the lateral crosslinks between the load carrying fibers are located randomly to create a nonhierarchical reference structure, see Fig. 4 top right, where the avalanche size distributions can be well fitted by Eq. 1 with τ = 2.3 and 1/σ = 1.95. For comparison, ref.^{14} reports values of τ ≈ 2 and 1/σ ≈ 1.4 with a weak dependence on lattice morphology. While these exponent values differ from the meanfield values τ = 1.5 and σ = 1, the avalanche size distributions are of the same type as for ELSFBM. A similar picture emerges from simulations of RHFN as shown in Fig. 4 bottom right. In the case of DHFN and SHFN, the picture is completely different as avalanche sizes are distributed as power laws with continuously varying exponents throughout the loading curve without an apparent cutoff. The distributions cannot be meaningfully fitted by Eq. 1 but are well represented by modified Pareto distributions,
where now the exponent τ decreases with increasing load I in an approximately linear manner (Fig. 4, lefthand side). Only at the peak current the distributions for HFN and RFN approach each other, as in the former case the cutoff diverges while for the HFN the exponent of the scale free distribution approaches the asymptotic value τ = 2.3 that also characterizes the random reference network. We may thus conclude that, while RFN exhibit a kind of criticallike behavior which is scale free only at the point of failure, in DHFN and SHFN such scale free behavior is a robust, intrinsic feature of the dynamics as the avalanche size distributions have powerlaw characteristics without cutoff even far away from the peak load.
The role of the network structure
In order to understand the origin of this robust scale free behavior, we note that hierarchical modular organization has been known to produce generic scale invariant behavior in systems apparently unrelated to materials mechanics. Models of activity propagation in both real and computer generated mappings of the human brain, in particular, have produced similar avalanche size distributions with continuously varying, nonuniversal exponents^{19}. Powerlaw distributed avalanche sizes are believed to be a direct consequence of the morphology of the brain networks, which are organized into a hierarchy of modules of exponentially increasing size yet exponentially decreasing number. Thus, scale free dynamic patterns are a consequence of scale invariant hierarchical organization of the underlying network, a consideration that holds for processes as varied as activity propagation and percolation, and is backed by renormalization results^{20}. In such structures, critical points marking phase transitions may be replaced by extended criticallike regions (“Griffiths phases”) as discussed by Moretti and Muñoz^{19} and here for the first time observed in the context of mechanical breakdown.
To understand how hierarchical organization ensures the scale free statistics of precursor activity, we compare the behavior of the different network variants. The behavior of the DHFN and the randomly reshuffled SHFN is essentially the same: in both cases we observe powerlaw avalanche size distributions with an exponent τ that decreases towards the value at failure, τ = 2.2, as an approximately linear function of the current I. At large avalanche sizes, the distributions are very clean power laws. At small sizes, deviations show up which can be characterized by a Pareto scale parameter s_{0} that goes to zero in a linear manner as the current approaches the critical value I_{p} (Fig. 4 lefthand side and insets). Networks with differing generators and differing threshold current statistics show similar behavior (see Supplementary Information). Differences between DHFN and SHFN concern only the numerical values of τ and s_{0}, which are both smaller for the SHFN but approach common values at failure. The behavior of the RHFN is qualitatively different from the hierarchical networks but identical to that of a reference network with completely random crosslinks. In both cases, one finds the same truncated powerlaw distributions with exponent τ = 2.2 and a cutoff that diverges as the current approaches I_{p}. Since the RHFN has the same distribution of crosslink lengths as the DHFN but the same exponential distribution of gap sizes as the random reference network, we can safely conclude that the robust scalefree behavior of the avalanche statistics in the hierarchical networks results from the scalefree gap size distribution, both in deterministic network models (DHFN) and in more realistic randomized variants (SHFN). This expectation is in line with the fracture pattern of a DHFN in Fig. 3b, which demonstrates that the final crack is deflected on all scales by the vertical gaps which interrupt stress transmission at the crack tip, leading to a superrough crack morphology. This qualitative idea is borne out by a quantitative analysis of the distribution of vertical deflections Δy of the crack which is characterized by truncated power laws, \(p({\rm{\Delta }}y)\propto {\rm{\Delta }}{y}^{\theta }\varphi ({\rm{\Delta }}y/L)\) where the cutoff is given by the system size, as shown in Fig. 5. The observed exponent θ = 1.75 differs from the value θ′ = 2 for the gap size distribution along a horizontal line, indicating nontrivial dynamics as stress concentrations at the tip of the emergent crack interact with the network morphology. RRFN and RHFN, on the other hand, exhibit an exponential distribution of Δy with an average deflection that is slightly larger than the mean gap size.
Discussion
We have proposed a simple model of stress redistribution and failure in a model material with a hierarchical microstructure. Analogously to heterogeneous materials that lack multilayer hierarchical organization, damage accumulation proceeds intermittently in the form of avalanches, which are broadly distributed in size. We observe however that in the hierarchical case this phenomenology cannot be interpreted as critical behavior in the vicinity of a continuous phase transition, as paradigmatically implemented in fiber bundle models with equal load sharing. Avalanches with powerlaw distributions without apparent cutoff are observed generically, i.e. for any value of the applied load. Avalanche exponents vary continuously, suggesting that the concept of universality class cannot be invoked. We argue that failure patterns, as well as deformation/load patterns, arise naturally from the hierarchical microstructure of the deforming medium, which is scale invariant by construction. The fracture patterns reflect the same scale invariance and strongly differ from the selfaffine crack morphologies generally observed in nonhierarchical random fuse networks^{14}. Fracture occurs not by nucleationandpropagation of a critical crack as typical of nonhierarchical or RRFN (see Supplementary Video 1), but by coalescence of multiple, widely separated flaws as crack propagation is interrupted by the presence of hierarchically distributed gaps on all scales (see Supplementary Video 2). This intrinsic tendency of the hierarchicalmodular microstructure to localize damage reflects the generic capability of hierarchical networks to localize activity patterns, reported for a wide range of biological^{19,21,22} and even information processing networks with hierarchical microstructure^{23}. Further work is needed to systematically quantify how the scalefree dynamics of damage accumulation and the ensuing crack profiles relate to the parameters governing the scale invariant microstructures (exponents of the distribution of link and gap sizes), which can be “tuned” by changing the number of horizontal and vertical links in the DHFN generator. This tuning capability may represent the ultimate advantage of hierarchical microstructures, as recently suggested in the context of friction^{24}.
The results obtained here represent first steps towards a qualitative understanding of failure processes in hierarchically organized materials. For a quantitative description of failure in complex biomaterials, which combine hierarchical morphology with a composite microstructure containing multiple phases, it will be necessary to go beyond the simplified caricature of load transfer in terms of a scalar load variable that is inherent in fuse models and move towards models that allow for a fully tensorial description of deformation and failure of hierarchically organized multiphase composites.
Methods
Simulations are conducted using the standard quasistatic method for the Random Fuse Model^{12}. At every iteration, both current and voltage are tuned to the lowest values which ensure that one link is broken. This is obtained as follows: (i) a fixed external V = 1 is applied to one of the buses, while the other is kept at V = 0 (ii) voltages at each node k are computed solving Kirchhoff’s node law, (iii) currents I_{kl} at each link kl are computed using Ohm’s law, (iv) the link with the maximum I_{kl}/t_{kl} is removed (t_{kl} being the random threshold assigned to the link), (v) the global values of V and I are adjusted by the factor t_{kl}/I_{kl} (which yields the failure of link kl), and are recorded in the I − V curve. The resulting quasistatic I − V curve allows one to extract information both for current and voltagecontrol loading schemes. In the case of currentcontrol, the size of an avalanche is defined as the number of links that fail without any further increase in the applied load. Avalanche statistics data, as in Fig. 4, are obtained by subdividing the interval of applied currents into subintervals. Avalanche size distributions are computed separately for each subinterval.
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Acknowledgements
We acknowledge funding by DFG under grants No. Za 1719, Mo 3049/11 and Mo 3049/31. M.Z. also acknowledges support by the Chinese State Administration of Foreign Expert Affairs under Grant No MS2016XNJT044. We acknowledge support by Deutsche Forschungsgemeinschaft and FriedrichAlexanderUniversität ErlangenNürnberg (FAU) within the funding programme Open Access Publishing.
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P.M., B.D. and N.E. performed the numerical simulations. P.M. and M.Z. conducted the statistical analysis. M.Z. and P.M. wrote the manuscript.
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Moretti, P., Dietemann, B., Esfandiary, N. et al. Avalanche precursors of failure in hierarchical fuse networks. Sci Rep 8, 12090 (2018). https://doi.org/10.1038/s4159801830539x
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