Abstract
In the numerical approximation of phase-field models of fracture in porous media with the finite element method, the problem of numerical locking may occur. The causes can be traced both to the hydraulic and to the mechanical properties of the material. In this work we present a mixed finite element formulation for phase-field modeling of brittle fracture in elastic solids based on a volumetric-deviatoric energy split and its extension to water saturated porous media. For the latter, two alternative mixed formulations are proposed. To be able to use finite elements with linear interpolation for all the field variables, which violates the Ladyzenskaja–Babuska–Brezzi condition, a stabilization technique based on polynomial pressure projections, proposed and tested by previous authors in fluid mechanics and poromechanics, is introduced. We develop an extension of this stabilization to phase-field mixed models of brittle fracture in porous media. Several numerical examples are illustrated, to show the occurrence of different locking phenomena and to compare the solutions obtained with different unstable, stable and stabilized low order finite elements.
Highlights
Porous media are materials characterized by a heterogeneous internal structure consisting of a solid phase, which confers stiffness to the material, and empty spaces, called pores, which may be filled by one or more fluids [26]
In this paper we focused on the problem of the numerical locking, due to a condition of high volumetric stiffness of the solid matrix, that can occur using the finite element (FE) method in the phase-field d and of the water pressure pw(right), at t = 0.06 s: a Q4Pe4sPw4s/Q4, b
We showed how the causes of this state of incompressibility can be traced both to the hydraulic and to the mechanical properties of the material
Summary
Porous media are materials characterized by a heterogeneous internal structure consisting of a solid phase, which confers stiffness to the material, and empty spaces, called pores, which may be filled by one or more fluids [26]. This paper focuses on the study and the development of FE formulations which mitigate the aforementioned locking phenomena For this purpose, we introduce a mixed u− p−d (displacements/pressure/phase-field) formulation for phasefield modeling of fracture in elastic solids and its extension to water saturated porous media. In this work we propose an extension of this stabilization to phase-field mixed models of brittle fracture based on a volumetric-deviatoric energy split This extension stems from the analogy between the equations governing the problem of consolidation in water saturated porous media under undrained conditions and the mixed u − p formulation for incompressible elasticity.
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