To achieve the objective of the project three scientific pillars have been formulated (work packages I-III), complemented by a fourth work packages which exploits and connects the scientific and technological advances in the first three work packages.
Work Package I: A framework has been developed for upscaling the pressure difference at the sub-grid scale to the macroscopic scale. The physical implications of different possibilities for the macroscopic interpolation of the pressure at the crack have been highlighted, which is considered to be a major achievement. The derivations and implementations of a power-law (non-Newtonian) fluid as well as of multi-phase fluid flow in a fractured porous medium have been completed, including verification and a significant improvement of the numerical formulation. This has resulted in a much faster convergence.
Work Package II: An isogeometric analysis method using a novel formulation of Locally Refined (LR)-splines and T-splines has been developed, including adaptive hierarchical refinement. It enables capturing a propagating cohesive crack along a predefined path. This breakthrough has enabled the development of a methodology to accurately simulate the propagation of (cohesive) cracks along non-predefined paths, one of the ultimate goals of this work package. Limitations have been identified and quantified. Among them is the difficulty to simulate branching. The use of B-splines which are based on triangles, so-called Powell-Sabin B-splines, removes this restriction. The technology has been developed and implemented successfully, including casting the formulation in a standard finite element framework via Bézier extraction. As an alternative formulation, eXtended isogeometric analysis has been developed and successfully implemented, again cast in a standard finite element data structure via Bézier extraction. Another recent development is that phase-field methods have been investigated, where the discrete crack is distributed over a finite width. They have been built on top of adaptive isogeometric analysis. The efficacy achieved through adaptivity is important since phase-field methods are computationally highly demanding. At present, efforts are devoted to incorporate cohesive-zone models and fluid flow in the formulation. Two different avenues are being pursued, one in which the crack width is discretised as an independent field, which provides knowledge of the crack width necessary for the cohesive-zone crack model and the description of fluid flow within fractures. The other approach relies on reconstructing the crack based on the strain field and the regularisation.
Work Package III: The first step towards a reliability-based description of crack propagation is the implementation of a deterministic crack model. Different from the original plan, a continuum description of the fracture model has been chosen for the deterministic description. Then, the combination with a stochastic approach must be realised.
Work Package IV: The main thrust has been to investigate the issue that computations can divergence when several dissipative mechanisms co-exist. The latter typically holds for earthquakes, where there is energy dissipation in the fault, but also plasticity, and hence energy dissipation, in the surrounding bulk material. Recently, it has been proven rigorously that the destabilising effect of a non-associated flow rule, which is ubiquitous in the description of the inelastic behaviour of rocks and soils, is at the root of these convergence difficulties. Moreover, proper regularisation mechanisms have been identified, and it has been proven that quadratic convergence of the Newton-Raphson method can be restored for inelastic media without a crack, with a stationary crack, and with a propagating crack. As a last addition, the role of fluids and fluid inertia has been examined, with an emphasis on shear fracture propagation.
