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Abstract
A non-standard aspect of phase-field fracture formulations for pressurized cracks is the application of the pressure loading, due to the fact that a direct notion of the fracture surfaces is absent. In this work we study the possibility to apply the pressure loading through a traction boundary condition on a contour of the phase field. Computationally this requires application of a surface-extraction algorithm to obtain a parametrization of the loading boundary. When the phase-field value of the loading contour is chosen adequately, the recovered loading contour resembles that of the sharp fracture problem. The computational scheme used to construct the immersed loading boundary is leveraged to propose a hybrid model. In this hybrid model the solid domain (outside the loading contour) is unaffected by the phase field, while a standard phase-field formulation is used in the fluid domain (inside the loading contour). We present a detailed study and comparison of the varGamma-convergence behavior and mesh convergence behavior of both models using a one-dimensional model problem. The extension of these results to multiple dimensions is also considered.
Highlights
IntroductionOver the last decade phase-field models for fracture [1,2,3,4]—which are closely related to traditional gradient-enhanced damage models [5]—have been successfully applied to a wide range of problems, such as dynamic fracturing [6,7,8,9], large deformation fracturing [10, 11], fracturing of electromechanical materials [12, 13], cohesive fracturing [14, 15], fracturing of thermo-elastic solids [16, 17] and many more
We have studied the possibility of incorporating the loading of pressurized fractures in phase-field fracture formulations as a non-homogeneous Neumann condition over a phase field contour. This approach to modeling the fracture loading is enabled by a bisection-based surface tessellation scheme which was originally developed in the context of immersed finite element simulations
In its simplest form the standard phase-field model for fracture is supplemented with an a-contour integral to incorporate the pressure loading
Summary
Over the last decade phase-field models for fracture [1,2,3,4]—which are closely related to traditional gradient-enhanced damage models [5]—have been successfully applied to a wide range of problems, such as dynamic fracturing [6,7,8,9], large deformation fracturing [10, 11], fracturing of electromechanical materials [12, 13], cohesive fracturing [14, 15], fracturing of thermo-elastic solids [16, 17] and many more. The flexibility of the phase-field framework with respect to the representation of complex fracture patterns is exploited in the context of fluid-driven fracture propagations (see references below) These models are highly relevant in applications concerning flow in porous media such as hydraulic fracturing. The absence of a discrete fracture surface in the formulation impedes direct incorporation of a versatile fluid-flow model and fluid-solid interface conditions, and requires a reconstruction of the fracture opening for the fluid-flow model In this contribution we study the possibility to directly incorporate the sharp fracture surface pressure-loading in a phase-field fracture representation. Based on a model problem we present a detailed analysis of both models, which highlights the superior fracture opening approximation behavior of the hybrid model.
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