Abstract
We introduce a beam network model for hierarchically patterned materials. In these materials, load-parallel gaps intercept stress transmission in the load perpendicular direction in such a manner that damage is confined within hierarchically nested, load-carrying ‘modules’. We describe the morphological characteristics of such materials in terms of deterministically constructed, hierarchical beam network (DHBN) models and randomized variants thereof. We then use these models to analyse the process of damage accumulation (characterized by the locations and timings of beam breakages prior to global failures, and the concomitant avalanche statistics) and of global failure. We demonstrate that, irrespective of the degree of local disorder, failure of hierarchically (micro)structured materials is characterized by diffuse local damage nucleation which ultimately percolates on the network, but never by stress-driven propagation of a critical crack. Failure of non hierarchical reference networks, on the other hand, is characterized by the sequence of damage nucleation, crack formation and crack propagation. These differences are apparent at low and intermediate degrees of material disorder but disappear in very strongly disordered materials where the local failure strengths exhibit extreme scatter. We furthermore demonstrate that, independent of material disorder, the different modes of failure lead to significant differences in fracture surface morphology.
Highlights
Hierarchical materials consist of microstructural features which have themselves internalstructures, forming self-similar repeating patterns at different scales
Typical patterns of damage growth are depicted in Fig. 3 for random beam network (RBN) and DHBN of size L = 1024 where beam thresholds are Weibull distributed with shape parameters β = 4.0 and β = 0.5
In case of non-hierarchical materials at low to intermediate disorder as illustrated by the failure sequence of a RBN with β = 4 in Fig. 3a, fracture occurs by nucleation and propagation of a crack which becomes critical at the system’s peak load, and is propagated by crack-tip stress concentrations until it spans the entire system
Summary
Hierarchical materials consist of microstructural features which have themselves internal (micro)structures, forming self-similar repeating patterns at different scales. The ensuing system of equations can be considered a scalar version of the equations of force balance in mechanics Such models capture effects of load concentration at crack tips, i.e., the associated stress singularity and the long-range decay of the crack-tip stress field, and have been widely used in statistical studies of fracture of disordered media (Alava et al 2006; Nukala et al 2005). Unlike the previously mentioned concept models, such models, which we consider here in the quasi-static (non-inertial) limit, can be quantitatively parametrized to capture the key features of fracture of real materials as a multi-scale process: The existence of local failure thresholds which reflect properties of the local material microstructure, the existence of an internal length above which the material can be described as a continuum, and the coupling of different material elements by long-range stress fields that emerge in response to local failure, and that are related to strains by tensorial constitutive relations such that general loadings and complex geometries, which in general lead to locally multi-axial stress states, can be adequately captured. The remainder of the paper is organized as follows: In Sect. 2 (method) we briefly introduce the BNM (technical details are given in the appendix), in Sect. 3 we show results of fracture simulations of this model and analyse the results in terms of fracture precursors, fracture/failure mode and fracture surface morphology, and in Sect. 4 we provide a general discussion of our results in the wider context of failure of disordered media
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