Theses Doctoral

# Enhanced continuum damage modeling of mechanical failure in ice and rocks

Mobasher, Mostafa

Modeling fracture in geomaterials is essential to the understanding of many physical phenomenon which may posses natural hazards e.g. landslides, faults and iceberg calving or man-made processes e.g. hydraulic fracture and excavations. Continuum Damage Mechanics (CDM) models the crack as a solid region with a degraded stiffness. This continuum definition of cracks in CDM allows more feasible coupling with other forms of material non-linearity and eliminates the need to track complicated crack geometry. Using CDM to analyze fracture for the modeling of fracture in geomaterials encounters several challenges e.g.: 1) the need to model the multiple physical processes occurring in geomaterials, typically: coupled fluid flow and solid deformation, 2) the need to consider non-local damage and transport in order to capture the underlying long range interactions and achieve mesh-independent finite element solutions and 3) the elevated computational cost associated with non-linear mixed finite element formulations.
The research presented in this thesis aims at improving the CDM formulations for modeling fracture geomaterials. This research can be divided into three main parts. The first is the introduction of a novel non-local damage transport formulation for modeling fracture in poroelastic media. The mathematical basis of the formulation are derived from thermodynamic equilibrium that considers non-local processes and homogenization principles. The non-local damage transport model leads to two additional regularization equations, one for non-local damage and the other for non-local transport which is reduced to non-local permeability. We consider two options for the implementation of the derived non-local transport damage model. The first option is the four-field formulation which extends the (u/P) formulation widely used in poroelasticity to include the non-local damage and transport phenomena. The second option is the three-field formulation, which is based on the coupling of the regularization equations under the assumptions of similar damage and permeability length scales and similar driving local stress/strain for the evolution of the damage and permeability. The three-field formulation is computationally cheaper but it degrades the physical modeling capabilities of the model. For each of these formulations, a non-linear mixed-finite element solution is developed and the Jacobian matrix is derived analytically. The developed formulations are used in the analysis of hydraulic fracture and consolidation examples.
In the second part, a novel approach for CDM modeling of hydraulic fracture of glaciers is pretended. The presence of water-filled crevasses is known to increase the penetration depth of crevasses and this has been hypothesized to play an important role controlling iceberg calving rate. Here, we develop a continuum damage-based poro-mechanics formulation that enables the simulation of water-filled basal and/or surface crevasse propagation. The formulation incorporates a scalar isotropic damage variable into a Maxwell-type viscoelastic constitutive model for glacial ice and the effect of the water pressure on fracture propagation using the concept of effective solid stress. We illustrate the model by simulating quasi-static hydro-fracture in idealized rectangular slabs of ice in contact with the ocean. Our results indicate that water-filled basal crevasses only propagate when the water pressure is sufficiently large and that the interaction between simultaneously propagating water-filled surface and basal crevasses can have a mutually positive influence leading to deeper crevasse propagation which can critically affect glacial stability.
In the third part, we propose a coupled Boundary Element Method (BEM) and Finite Element Method (FEM) for modeling localized damage growth in structures. BEM offers the flexibility of modeling large domains efficiently while the nonlinear damage growth is accurately accounted by a local FEM mesh. An integral-type nonlocal continuum damage mechanics with adapting FEM mesh is used to model multiple damage zones and follow their propagation in the structure. Strong form coupling, BEM hosted, is achieved using Lagrange multipliers. Since the non-linearity is isolated in the FEM part of the system of equations, the system size is reduced using Schur complement approach, then, the solution is obtained by a monolithic Newton method that is used to solve both domains simultaneously. The method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature.

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