A derivation is given of two-scale models that are able to describe deformation and flow in a fluid-saturated and progressively fracturing porous medium. From the micromechanics of the flow in the cavity, identities are derived that couple the local momentum and the mass balances to the governing equations for a fluid-saturated porous medium, which are assumed to hold on the macroscopic scale. By exploiting the partition-of-unity property of the finite element shape functions, the position and direction of the fracture is independent from the underlying discretisation. The finite element equations are derived for this two-scale approach and integrated over time. The resulting discrete equations are nonlinear due to the cohesive crack model and the nonlinearity of the coupling terms. A consistent linearisation is given for use within a Newton—Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach.
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References
Terzaghi K. Theoretical Soil Mechanics. Wiley, New York, 1943
Biot MA. Mechanics of Incremental Deformations. Wiley, Chichester, 1965
Lewis RW, Schrefler BA. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, Second Edition. Wiley, Chichester, 1998
Snijders H, Huyghe JM, Janssen JD. Triphasic finite element model for swelling porous media. International Journal for Numerical Methods in Fluids 1997; 20: 1039–1046
Huyghe JM, Janssen JD. Quadriphasic mechanics of swelling incompressible media. International Journal of Engineering Science 1997; 35: 793–802
Van Loon R, Huyghe JM, Wijlaars MW, Baaijens FPT. 3D FE implementation of an incompressible quadriphasic mixture model. International Journal for Numerical Methods in Engineering 2003; 57: 1243–1258
Jouanna P, Abellan MA. Generalized approach to heterogeneous media. In Modern Issues in Non-Saturated Soils. Gens A, Jouanna P, Schrefler B (eds). Springer, Wien, New York, 1995; 1–128
de Borst R, Réthoré J, Abellan MA. A numerical approach for arbitrary cracks in a fluid-saturated medium. Archive of Applied Mechanics 2006; 75: 595–606
Réthoré J, de Borst R, Abellan MA. A discrete model for the dynamic propagation of shear bands in a fluid-saturated medium. International Journal for Numerical and Analytical Methods in Geomechanics 2007; 31: 347–370
Réthoré J, de Borst R, Abellan MA. A two-scale approach for fluid flow in fractured porous media. International Journal for Numerical Methods in Engineering 2007; 71: 780–800
Réthoré de Borst R, Abellan MA. A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks. Computational Mechanics 2008; 42: 227–238
Babuska I, Melenk JM. The partition of unity method. International Journal for Numerical Methods in Engineering 1997; 40: 727–758
Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 1999; 45: 601–620
Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46: 131– 150
Wells GN, Sluys LJ. Discontinuous analysis of softening solids under impact loading. International Journal for Numerical and Analytical Methods in Geomechanics 2001; 25: 691–709
Wells GN, Sluys LJ, de Borst R. Simulating the propagation of displacement discontinuities in a regularized strain-softening medium. International Journal for Numerical Methods in Engineering 2002; 53: 1235–1256
Wells GN, de Borst R, Sluys LJ. A consistent geometrically non-linear approach for delamination. International Journal for Numerical Methods in Engineering 2002; 54: 1333–1355
Remmers JJC, de Borst R, Needleman A. A cohesive segments method for the simulation of crack growth. Computational Mechanics 2003; 31: 69–77
Samaniego E, Belytschko T. Continuum—discontinuum modelling of shear bands. International Journal for Numerical Methods in Engineering 2005; 62: 1857–1872
Réthoré J, Gravouil A, Combescure A. An energy-conserving scheme for dynamic crack growth using the extended finite element method. International Journal for Numerical Methods in Engineering 2005; 63: 631–659
Réthoré J, Gravouil A, Combescure A. A combined space—time extended finite element method. International Journal for Numerical Methods in Engineering 2005; 64: 260–284
Areias PMA, Belytschko T. Two-scale shear band evolution by local partition of unity. International Journal for Numerical Methods in Engineering 2006; 66: 878–910
de Borst R, Remmers JJC, Needleman A. Mesh-independent numerical representations of cohesive-zone models. Engineering Fracture Mechanics 2006; 173: 160–177
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de Borst, R., Rethore, J., Abellan, MA. (2009). A Precis Of Two-Scale Approaches For Fracture In Porous Media. In: de Borst, R., Sadowski, T. (eds) Lecture Notes on Composite Materials. Solid Mechanics And Its Applications, vol 154. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8772-1_5
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