A quasi-cotinuum appoach for modeling fracture in disordered networked materials: Can small world architectures save the day?

The skeleton of many natural and artificial structures may be abstracted as networks of nonlinearly interacting elements. Examples include rubber, gels, soft tissues, and lattice materials. Understanding the multiscale nature of deformation and failure of networked structures hold key for uncovering origins of fragility in many complex systems including biological tissues and enables designing novel materials. However, these processes are intrinsically multiscale and for large scale structures it is computationally prohibitive to adopt a full discrete approach.
Here, we introduce a new adaptive numerical algorithm for solving polymer networks, the building blocks in many biological and engineering systems, using an extended version of the Quasi-Continuum (QC) method. In regions of high interest, for example near defects or cracks, each polymer chain is idealized using the worm like chain model. Away from these imperfections, the network structure is computationally homogenized, using Hill-Mandell’s principle, to yield an anisotropic material tensor consistent with the underlying network structure. Dynamic adaptivity provides a seamless transition across the two models. Overall, the proposed method provides a multi-resolution capability by retaining explicit representation of small scale heterogeneities and topological features, where they matter near the crach tips, while still accurately accounting for bulk elasticity and loading. We illustrate the efficiency of the method by applying it to study the fracture of large scale polymer network problems as realized in experiments on hydrogels. We further apply the method to test the influence of network topology on its fracture resistance and demonstrate that networks with small-world architectures, balancing clustering and avergae path length, may lead to an optimium fracture toughness. We discuss the implications of our findings for the analysis and design of tough networks.