The presence of heterogeneously distributed mineral grain boundaries introduces a characteristic length-scale leading to a size-dependent strength and toughness of brittle rocks. This study focuses on fracture nucleation and quasi-static propagation in three-point bending experiments in notched beams. We employed a variational approach to fracture based on the phase-field regularization that accounts for the interactions between the heterogeneously distributed grains and the fracture. Firstly, we describe the mathematical model and its capabilities. We then provide a series of numerical simulations varying the ratio of grain to interface fracture energy and compare them against available experimental results to replicate the sample size dependency.


Brittle rocks are known to exhibit size dependent strength, transitioning from a size-insensitive regime governed by the material’s strength (small-scale yielding) to a power-law size-dependent regime following the linear elastic fracture mechanics (LEFM) asymptote. In granite, as in many rocks in general, the characteristics of the granular assembly control the macroscopic structural response (Barton 1982; Tarokh and Fakhimi, 2014). The grain arrangement determines the microstructure, while the volume fraction composition of the grains controls the strength of individual grains and their interfaces (Tarokh et al., 2017). Several studies have tackled the problem by solving explicitly the inhomogeneous structure of granite with a common approach that relies on the Distinct Element Method -DEM (Lisjak et al., 2013; Hofmann et al., 2015; Peng et al., 2017; Zhou et al., 2019). The DEM method implies a modeling choice that affects the overall framework: i) either the particles are defined as the grains in the assembly (Lisjak et al., 2013), which constrains the fractures to propagate exclusively at the grain interfaces; ii) either the grains are themselves split into several forming particles to tackle inter-grain fracturing (Peng et al., 2017; Zhou et al., 2019), which then requires the rather arbitrary choice of the inter-grain particle size and their contact properties. To overcome these deficiencies, we propose a variational approach based on a phase-field (PF) method that takes into account the discontinuities at the grain interfaces (Bourdin et al., 2000). The PF method with discontinuities has been successfully applied to solve problems regarding material interfaces (Yoshioka et al. 2021), the interaction of hydro-fractures with natural fractures (Lepillier et al., 2020) and the propagation fractures with residual frictional strength (Fei and Choo, 2020). The PF method represents the granular assembly at the continuous scale allowing the propagation of both intra- and inter-granular fractures without the necessity to assume a sub-granular particle distribution.

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