Order reduction of fracture mechanics in porous microstructures: A multiscale computing framework

Abstract

We present a multiscale framework to systematically reduce the order of fracture-mechanics computations in complex porous microstructures captured by a pore-scale image (e.g., X-ray μCT). The framework allows formulating approximations that are more accurate than classical discrete element methods (DEM) and faster than high-fidelity direct numerical simulation (DNS) techniques, such as phase-field models (PFM). We propose two such approximations called the pore-level multiscale method (PLMM) and the image-based discrete element method (iDEM), the latter a generalization of classical DEM wherein the solid geometry is not simplified. Unlike iDEM, PLMM is equipped with the ability to iteratively control errors away from, but not at, the cracks. We refer to the uncorrected approximation as M0, and that with ω corrections as Mω. Both methods are based on the hypotheses that (H1) cracks nucleate at geometric constrictions of the solid, and (H2) bending/torsion moments at such constrictions are negligible. iDEM additionally assumes that (H3) the geometric enlargements of the solid are rigid. Through a series of benchmarks, performed on complex geometries subject to hydraulic, tensile, shear, and compressive loading, we show that H1 is largely valid under non-compressive loading, and H2 degrades the most under shear loading when bending/torsion moments dominate. H3 has two consequences: (1) M0 predictions can be reproduced by iDEM at a fraction of the cost, (2) but to compute them, iDEM’s parameters (spring constants) must be calibrated to reference data, which may not be available. Moreover, the calibrated parameters are non-unique and depend on the reference data chosen. The foregoing benchmarks are validated against two well-known PFMs, in which inherent uncertainties and deficiencies are also identified, especially under compressive loading. These are attributed to the so-called energy splits used. Lastly, we derive and validate closed-form expressions for the tensile and shear strengths of the solid used by PLMM/iDEM and the length-scale parameter and fracture toughness in PFM. This work provides fundamental insights into how porous materials fail, and how accurate approximations based on such understanding can be constructed and refined in the future.