Network analysis predicts failure of materials and structures

Network analysis deals with the mathematical characterization of structure and topology of networks which are envisaged as sets of “nodes” connected by “edges.” Its concepts do not depend on the physical nature of the entities that constitute a network—be it neurons connected by synapses, cities connected by highways, or humans connected by acquaintance relationships. Despite their general and abstract nature, network analytic concepts are often useful tools to analyze physical processes on networks, which may strongly depend on structural properties of the network and their evolution. Berthier et al. (1) demonstrate this by forecasting failure locations in load-carrying lattice structures based on so-called geodesic edge betweenness centrality (GEBC).

Edge betweenness centrality ranks the importance of edges based on the fraction of shortest paths between any 2 points on a network that traverse a given edge (2), and Berthier et al. (1) demonstrate that a large GEBC value serves as a good predictor for failure locations in disordered beam networks subjected to tensile or compressive loading: They performed an experiment where they successively loaded a network and recorded the sequence of beam failures, while analyzing the concomitant evolution of the centrality pattern. They show that high GEBC correlates strongly with beam failure and achieves prediction scores that are comparable to those obtained from a simple physical model of load redistribution.

To understand how GEBC might relate to failure processes, let us first consider an idealized road network where traffic occurs between any 2 nodes with equal probability, and where drivers always take the shortest route. In this case it is evident that GEBC directly determines the traffic load on any stretch of road, and therefore the likelihood for such a road to clog up and thereby “fail” (3). However, other transport processes do not necessarily follow a shortest-path metric, as the moving entities cannot usually access deterministic information about to decide what path to choose (electrons, lattice vibrations, or diffusing particles do not own a GPS-cum-navigation system). Instead, many transport processes follow a diffusive dynamics, which samples pathways in a stochastic manner. Consider the transport of heat or of electrical charge on a network, both of which follow a diffusion equation given by

$$\frac{d}{dt}\mathbf{u}=-\mathbf{L}\mathbf{u},$$

where the i-th element u_i of the vector u denotes the density of the transported species at node i and the symmetric Laplacian matrix L characterizes the transport properties of the network. [The off-diagonal element −L_ij is the electrical or thermal conductivity of an edge connecting nodes i and j, and the diagonal element L_ii is the total conductivity of all edges connecting to node i (4).] The resulting diffusive dynamics can represented as u(t) = K(t − t₀) u(t₀), where the heat kernel may be formally expressed as K = exp[−L(t − t₀)]; in terms of the eigenvectors w_k and eigenvalues λ_k of the network Laplacian,

$$\mathbf{K}\left(t-t_0\right)={\displaystyle \sum }_k{e}^{-{\lambda }_k(t-t_0)}{\mathbf{w}}_k\otimes {\mathbf{w}}_k.$$

Each matrix element K_ij can be envisaged as a sum of diffusion trajectories along all possible paths connecting nodes i and j with diffusion time (t − t₀); the network propagator K is thus the discrete counterpart of the path-integral formulation of diffusion in a Euclidean continuum (5). Even though all paths are possible in this case, they are not equally weighted: The path integral is dominated by the path of least action in the continuum path integral formulation, which translates into the path of lowest resistance in the electrical or thermal analog. If all edges of the network have identical length and conductivity, this is simply the shortest path. If we now envisage failure as the result of exceeding a critical current, then again edges with highest GEBC are most likely to fail.

This observation can be transferred from the analysis of transport phenomena to the question of mechanical stability. Just as charge transport across a resistor network is governed by the equilibrium of currents at nodes, the transfer of linear and angular momentum across a beam network is governed by the equilibrium of forces and moments: The equations that govern charge transfer (distribution of currents) in a resistor network are of the same fundamental nature as those that govern transfer of linear momentum (distribution of stresses) in a network of beams. The approach of Berthier et al. (1) may therefore be applicable to mechanical failure processes in a wide range of materials and structures, which can be represented as networks transmitting mechanical loads (Fig. 1). Such networks need not, of necessity, consist of networked material beams as present in the structures studied by Berthier et al. (1) or, in a more regular version, in the structure of the Eiffel tower (Fig. 1A) and of many additively manufactured metamaterials (refs. 6 and 7 and Fig. 1B). Load transfer in granular assemblies is controlled by force chain networks (Fig. 1C), which served as templates for the beam networks studied by Berthier et al. (1). Porous structures under load develop, in the approach to failure, stress transfer chains that can be analyzed along very similar lines (8). Random beam, spring, and fuse networks (Fig. 1D) have been studied extensively as generic models for the deformation and failure of disordered materials (9) and have been applied to model fracture of bulk engineering materials such as concrete (10). Thus, network analysis methods are not restricted to systems that have the physical appearance of a network of edges but may be transferred to bulk material properties.

Fig. 1.

Examples of load-carrying networks. (A) Detail of the beam network structure of the Eiffel tower. (B) Additively manufactured metamaterial (acoustic band gap material; for details see ref. 7). (C) Force chain network in a granular material. Reprinted with permission from ref. 1. (D) Propagating crack in a random fuse model of a disordered material; color scale indicates the local stress level [for details see Moretti et al. (15)].

For example, consider a network structure representing a bulk material in which a crack has formed (Fig. 1D). All shortest paths that originally were crossing the crack surface now must be rerouted around the crack and concentrate at the crack tip. At this location, the GEBC value is dramatically increased, and the same is true for the actual stress which exhibits a typical crack-tip singularity. Both GEBC and mechanical analysis indicate the crack tip as the most likely location for failure propagation.

As always, there are caveats and limitations. Nodes located near the boundary of a network have a tendency to show reduced GEBC values as compared to nodes in the bulk. Predictions based upon GEBC may thus systematically underestimate the role of surfaces in damage and failure, which often initiates at surface heterogeneities. It may also be noted that GEBC determination is computationally expensive, with standard algorithms of O(nm) time complexity, where n and m are the numbers of nodes and edges in the network at hand (11). This is not a problem with the comparatively small beam systems studied by Berthier et al. (1) but, given that every beam breakage necessitates recomputation, it could become a serious limitation if one applied the same approach to beam structures used as models for disordered material microstructures (e.g., of concrete), which may easily comprise >10⁵ nodes. Nevertheless, the close analogies between load transmission in materials and structures and transport problems on networks open promising avenues for the application of network theoretical concepts in materials design and materials analysis. The work by Berthier et al. (1) brings centrality measures to the attention of the materials community. Other approaches focus on spectral properties of the Laplacian matrix, using spectral analysis and eigenvector localization to investigate transport properties and damage localization (12, 13) and the relevance of such signatures in materials failure (14). While none of these methods can fully encompass the complex and multiscale nature of real failure phenomena, we believe that the novel perspectives opened by network analysis will provide new and interesting pathways for assessing structural performance and design of high-performance materials.