Research Articles

Two-Dimensional Mode I Crack Propagation in Saturated Ionized Porous Media Using Partition of Unity Finite Elements

[+] Author and Article Information
F. Kraaijeveld

e-mail: famkekraaijeveld@gmail.com

J. M. Huyghe

Associate Professor
e-mail: j.m.r.huyghe@tue.nl
Department of Biomedical Engineering,
Eindhoven University of Technology,
Eindhoven, 5600MBThe Netherlands

J. J. C. Remmers

Assistant Professor
e-mail: j.j.c.remmers@tue.nl

R. de Borst

e-mail: r.d.borst@tue.nl
Department of Mechanical Engineering,
Eindhoven University of Technology,
Eindhoven, 5600MBThe Netherlands

1Corresponding author.

Manuscript received December 5, 2010; final manuscript received April 10, 2012; accepted manuscript posted October 25, 2012; published online February 6, 2013. Assoc. Editor: Younane Abousleiman.

J. Appl. Mech 80(2), 020907 (Feb 06, 2013) (12 pages) Paper No: JAM-10-1442; doi: 10.1115/1.4007904 History: Received December 05, 2010; Revised April 10, 2012; Accepted October 25, 2012

Shales, clays, hydrogels, and tissues swell and shrink under changing osmotic conditions, which may lead to failure. The relationship between the presence of cracks and fluid flow has had little attention. The relationship between failure and osmotic conditions has had even less attention. The aim of this research is to study the effect of osmotic conditions on propagating discontinuities under different types of loads for saturated ionized porous media using the finite element method (FEM). Discontinuous functions are introduced in the shape functions of the FEM by partition of unity method, independently of the underlying mesh. Damage ahead of the crack-tip is introduced by a cohesive zone model. Tensile loading of a crack in an osmoelastic medium results in opening of the crack and high pressure gradients between the crack and the formation. The fluid flow in the crack is approximated by Couette flow. Results show that failure behavior depends highly on the load, permeability, (osmotic) prestress and the stiffness of the material. In some cases it is seen that when the crack propagation initiates, fluid is attracted to the crack-tip from the crack rather than from the surrounding medium causing the crack to close. The results show reasonable mesh-independent crack propagation for materials with a high stiffness. Stepwise crack propagation through the medium is seen due to consolidation, i.e., crack propagation alternates with pauses in which the fluid redistributes. This physical phenomenon challenges the numerical scheme. Furthermore, propagation is shown to depend on the osmotic prestressing of the medium. This mechanism may explain the tears observed in intervertebral disks as degeneration progresses.

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Urban, J. P., and Roberts, S., 2003, “Degeneration of the Intervertebral Disc,” Arthritis Res. Ther., 5(3), pp. 120–130. [CrossRef] [PubMed]
Adams, M. A., and Roughley, P. J., 2006, “What Is Intervertebral Disc Degeneration, and What Causes It?” Spine, 31(18), pp. 2151–2161. [CrossRef] [PubMed]
Hutton, W. C., Ganey, T. M., Elmer, W. A., Kozlowska, E., Ugbo, J. L., Doh, E. S., and T. E. Whitesides, J., 2000, “Does Long-Term Compressive Loading on the Intervertebral Disc Cause Degeneration?” Spine, 25(23), pp. 2993–3004. [CrossRef] [PubMed]
Simunic, D. I., Broom, N. D., and Robertson, P. A., 2001. “Biomechanical Factors Influencing Nuclear Disruption of the Intervertebral Disc,” Spine, 26(11), pp. 1223–1230. [CrossRef] [PubMed]
Simunic, D. I., Robertson, P. A., and Broom, N. D., 2004, “Mechanically Induced Disruption of the Healthy Bovine Intervertebral Disc,” Spine, 29(9), pp. 972–978. [CrossRef] [PubMed]
Callaghan, J. P., and McGill, S. M., 2001, “Intervertebral Disc Herniation: Studies on a Porcine Model Exposed to Highly Repetitive Flexion/Extension Motion With Compressive Force,” Clin. Biomech., 16(1), pp. 28–37. [CrossRef]
Yu, C. Y., Tsai, K. H., Hu, W. P., Lin, R. M., Song, H. W., and Chang, G. L., 2003, “Geometric and Morphological Changes of the Intervertebral Disc Under Fatigue Testing,,” Clin. Biomech., 18(6), pp. S3–S9. [CrossRef]
Battie, M. C., Videman, T., and Parent, E., 2004, “Lumbar Disc Degeneration: Epidemiology and Genetic Influences,” Spine, 29(23), pp. 2679–2690. [CrossRef] [PubMed]
Videman, T., and Battie, M. C., 1999, “The Influence of Occupation on Lumbar Degeneration,” Spine, 24(11), pp. 1164–1168. [CrossRef] [PubMed]
Wognum, S., Huyghe, J. M., and Baaijens, F. P., 2006, “Influence of Osmotic Pressure Changes on the Opening of Existing Cracks in 2 Intervertebral Disc Models,” Spine, 31(16), pp. 1783–1788. [CrossRef] [PubMed]
Huang, N. C., and Russell, S. G., 1985, “Hydraulic Fracturing of a Saturated Porous Medium—I: General Theory,” Theoret. Appl. Fract. Mech., 4(3), pp. 201–213. [CrossRef]
Rice, J. R., and Cleary, M. P., 1976, “Some Basic Stress Diffusion Solutions for Fluid-Saturated Elastic Porous Media With Compressible Constituents,” Rev. Geophys. Space Phys., 14(2), pp. 227–241. [CrossRef]
Detournay, E., and Cheng, A. H. D., 1991, “Plane Strain Analysis of a Stationary Hydraulic Fracture in a Poroelastic Medium,” Int. J. Solids Struct., 27(13), pp. 1645–1662. [CrossRef]
Emerman, S. H., Turcotte, D. L., and Spence, D. A., 1986, “Transport of Magma and Hydrothermal Solutions by Laminar and Turbulent Fluid Fracture,” Phys. Earth Planet. Inter., 41(4), pp. 249–259. [CrossRef]
Boone, T. J., Ingraffea, A. R., and Roegiers, J. C., 1991, “Simulation of Hydraulic Fracture Propagation in Poroelastic Rock With Application to Stress Measurement Techniques,” International J. Rock Mech., 28(1), pp. 1–14. [CrossRef]
Detournay, E., and Garagash, D. I., 2003 “The Near-Tip Region of a Fluid-Driven Fracture Propagating in a Permeable Elastic Solid,” J. Fluid Mech., 494, pp. 1–32. [CrossRef]
Kfoury, M., Ababou, R., Noetinger, B., and Quintard, M., 2006, “Upscaling Fractured Heterogeneous Media: Permeability and Mass Exchange Coefficient,” ASME J. Appl. Mech., 73(1), pp. 41–46. [CrossRef]
Dormieux, L., Kondo, D., and Ulm, F. J., 2006, “A Micromechanical Analysis of Damage Propagation in Fluid-Saturated Cracked Media,” Comptes Rendus Mecanique, 334(7), pp. 440–446. [CrossRef]
Dugdale, D. S., 1960, “Yielding of Steel Sheets Containing Slits,” J. Mech. Phys. Solid., 8(2), pp. 100–104. [CrossRef]
Barenblatt, G. I., 1962. “The Mathematical Theory of Equilibrium Cracks in Brittle Fracture,” Adv. Appl. Mech., (7), pp. 55–129. [CrossRef]
Simoni, L., and Secchi, S., 2003, “Cohesive Fracture Mechanics for a Multi-Phase Porous Medium,”. Eng. Computat., 20(5–6), pp. 675–698. [CrossRef]
Schrefler, B. A., Secchi, S., and Simoni, L., 2006, “On Adaptive Refinement Techniques in Multi-Field Problems Including Cohesive Fracture,” Comput. Meth. Appl. Mech. Eng., 195(4–6), pp. 444–461. [CrossRef]
Secchi, S., Simoni, L., and Schrefler, B. A., 2007, “Mesh Adaptation and Transfer Schemes for Discrete Fracture Propagation in Porous Materials,” Int. J. Numer. Anal. Meth. Geomech., 31(2), pp. 331–345. [CrossRef]
Babuska, I., and Melenk, J. M., 1997, “The Partition of Unity Method,” Int. J. Numer. Meth. Eng., 40(4), pp. 727–758. [CrossRef]
Belytschko, T., and Black, T., 1999, “Elastic Crack Growth in Finite Elements With Minimal Remeshing,” Int. J. Numer. Meth. Eng., 45(5), pp. 601–620. [CrossRef]
Wells, G. N., 2001, “Discontinuous Modelling of Strain Localisation and Failure,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
Moes, N., and Belytschko, T., 2002, “Extended Finite Element Method for Cohesive Crack Growth,” Eng. Fract. Mech., 69(7), pp. 813–833. [CrossRef]
Remmers, J. J. C., de Borst, R., and Needleman, A., 2003, “A Cohesive Segments Method for the Simulation of Crack Growth,” Comput. Mech., 31(1–2), pp. 69–77. [CrossRef]
Larsson, R., Runesson, K., and Ottosen, N. S., 1993, “Discontinuous Displacement Approximation for Capturing Plastic Localization,” Int. J. Numer. Meth. Eng., 36(12), pp. 2087–2105. [CrossRef]
Larsson, J., and Larsson, R., 2000, “Localization Analysis of a Fluid-Saturated Elastoplastic Porous Medium Using Regularized Discontinuities,” Mech. Co.-Frict. Mater., 5(7), pp. 565–582. [CrossRef]
Armero, F., and Callari, C., 1999, “An Analysis of Strong Discontinuities in a Saturated Poro-Plastic Solid,” Int. J. Numer. Meth. Eng., 46(10), pp. 1673–1698. [CrossRef]
Roels, S., Moonen, P., Proft, K. D., and Carmeliet, J., 2006, “A Coupled Discrete-Continuum Approach to Simulate Moisture Effects on Damage Processes in Porous Materials,” Comput. Meth. Appl. Mech. Eng., 195(52), pp. 7139–7153. [CrossRef]
Al-Khoury, R., and Sluys, L. J., 2007, “A Computational Model for Fracturing Porous Media,” Int. J. Numer. Meth. Eng., 70(4), pp. 423–444. [CrossRef]
Rethore, J., de Borst, R., and Abellan, M. A., 2007, “A Two-Scale Approach for Fluid Flow in Fractured Porous Media,” Int. J. Numer. Meth. Eng., 71(7), pp. 780–800. [CrossRef]
de Borst, R., 2008, “Challenges in Computational Materials Science: Multiple Scales, Multi-Physics and Evolving Discontinuities,” Comput. Mater. Sci., 43(1), pp. 1–15. [CrossRef]
Lanir, Y., 1987, “Biorheology and Fluid Flux in Swelling Tissues. 1. Bicomponent Theory for Small Deformations, Including Concentration Effects,” Biorheology, 24(2), pp. 173–187. [PubMed]
Camacho, G. T., and Ortiz, M., 1996, “Computational Modelling of Impact Damage in Brittle Materials,” Int. J. Solids Struct., 33(20–22), pp. 2899–2938. [CrossRef]
Vermeer, P. A., and Verruijt, A., 1981, “An Accuracy Condition for Consolidation by Finite Elements,” Int. J. Numer. Anal. Meth. Geomech., 5(1), pp. 1–14. [CrossRef]
Remmers, J. J. C., 2006, “Discontinuities in Materials and Structures. A Unifying Computational Approach,” Ph.D. thesis, University of Technology Delft, The Netherlands.
Remmers, J. J. C., de Borst, R., and Needleman, A., 2008, “The Simulation of Dynamic Crack Propagation Using the Cohesive Segments Method,” J. Mech. Phys. Solids, 56(1), pp. 70–92. [CrossRef]
Kraaijeveld, F., Huyghe, J. M., Remmers, J. J. C., de Borst, R., and Baaijens, F. P. T., 2013, “Shear Fracture in Osmoelastic Saturated Porous Media: A Mesh-Independent Model,” Eng. Fract. Mech. (submitted).
Zienkiewicz, O. C., Qu, S., Taylor, R. L., and Nakazawa, S., 1986, “The Patch Test for Mixed Formulations,” Int. J. Numer. Meth. Eng., 23(10), pp. 1873–1883. [CrossRef]


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Fig. 1

Representation of three types of prestress: free swelling (a) in both directions, (b) in x-direction and (c) in y-direction

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Fig. 2

Schematic representation of the fluid flow at the crack surface with parameter s the distance along the crack

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Fig. 3

Schematic representation of the traction forces at the crack surface

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Fig. 8

Delamination in case of prestress in both directions after 950 time increments (i.e., 9.5e − 2 mm displacement of top boundary). Distribution of (a) chemical potential in (N/mm2) and (b) flow in x-direction in (mm/s).

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Fig. 9

Comparison of the effect mesh and time refinement for delamination at a point just ahead of the initial crack (dx=13.71 mm)

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Fig. 4

Normalized distribution of the exponential cohesive law for tensile loading related traction forces and displacement

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Fig. 5

The mesh and boundary conditions for delamination consisting of 575 elements. Material is pulled at the top and bottom on the left and is on the right in contact with a filter.

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Fig. 13

Distribution for the osmolarity test after 300 increments (i.e., a change in chemical potential of -0.96 MPa) just before damage initiation for prestress in both directions for (a) of chemical potential and (b) of stress component σe,yy

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Fig. 6

The mesh and boundary conditions for pull test: material is pulled at the top with bottom fixed and at sides in contact with a filter

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Fig. 7

The mesh and boundary conditions for the osmolarity test: the material is fixed at bottom, right and top and is in contact with a filter at these sides

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Fig. 10

Distribution of chemical potential in (N/mm2) for delamination in case of prestress in both directions and μΓf=0 after 1100 increments (i.e., displacement of top boundary of du=1.1e − 2 mm) for (a) chemical potential and (b) flow in x-direction

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Fig. 11

Distribution for pull test of (a) chemical potential and (b) stress component σe,yy for the pull test after 1010 increments (i.e., 5.05e − 2mm of pull displacement) in case of prestress in both directions, including cohesive zone

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Fig. 12

Comparison for the pull test of the effect of prestress on crack mouth opening displacement (CMOD), chemical potential and tangential flow in the crack at dx=3.5 mm

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Fig. 14

Distribution at the crack-tip for osmolarity test of flow for prestress in both directions after 306 increments (i.e., a change in chemical potential of -0.98 MPa) just after damage initiation (a) in x-directions and (b) y-direction

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Fig. 15

Evolution for the osmolarity test in plane of crack of chemical potential, tangential flow and crack mouth opening (CMOD) after 130 increments, Δμf=-0.4MPa




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