Baaijens, F.P.T.

Huyghe, J.M.R.J.

Kraaijeveld, F.

English

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A singularity solution of shear faulting in swelling ionised porous media

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 A singularity solution of shear faulting in swelling ionisedporous mediaCitation for published version (APA):Kraaijeveld, F., Huyghe, J. M. R. J., & Baaijens, F. P. T. (2005). A singularity solution of shear faulting in swellingionised porous media. Poster session presented at Mate Poster Award 2005 : 10th Annual Poster Contest.Document status and date:Published: 01/01/2005Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 202312/mhjA singularity solution of shear faultingin swelling ionised porous mediaF. Kraaijeveld, J.M. Huyghe and F.P.T. BaaijensBiomedical Engineering, Soft Tissue Biomechanics & EngineeringIntroductionShales, clays, gels and biological tissues all endure swellingassociated with ionisation of the porous medium [2.]. We willstudy intervertebral disc herniation, which includes numeri-cal simulations of crack formation and crack propagation dur-ing shrinkage [3.,4.]. Analytical solutions are a must for vali-dation of numerical codes.MethodsShear faultingA dislocation on an existing crack is considered as verifica-tion (fig 1): ujz=0+   ujz=0  = f(x)H(t).Figure 1: Shear fault: f(x) block function.ModelLanir’s biphasic theory describes the saturated porousmedium by assuming incompressible constituents and in-finitely fast ion flow. Perturbation on an equilibrium gives astrongly coupled system (1), with u displacement, f chemi-cal potential and ,  andK physical constants.r2~u+ (  )~rtr()  ~rf = 0;tr()t  Kr2f = 0: (1)Analytical solutionDecoupling the systems of equations by means of stressfunctions S and E [1.] results in equation (2). The systemis solved using linear transformations (Fourier and Laplace).The shear stress is calculated at the crack surface. UsingSimpson rule for inverse Fourier the chemical potential iscomputed. r2S = 0;r2 Et   Kr4E = 0: (2)Numerical simulationCalculations are performed in Abaqus. The mesh is createdfrom two symmetrical meshes, z  0 and z  0. Nodes out-side the cracksurface are tieed and the cracksurface is madea sliding contact to ensure fluid flow.Figure 2: Abaqus mesh with red line denoting crack surfaceResults and DiscussionShear stressAnalytically the shear stress (red) shows a high singularityin x. Simpson’s rule for Fourier integration cannot cope withdislocation at z = 0. (fig 3a). Numerically we have same sin-gularity patern, but high mesh-dependency is seen (fig 3b).abFigure 3: High singularity in shear stress. a) Analytical versusSimpson’s rule. b) Mesh dependency at left crack-tip.Shear stress at t = 0 is the result of initial incompressible be-haviour of the medium. Fluid flow results in relaxation of thestresses. This is consolidation. Shear stress at equilibriumdecreases with decreasing osmotic pressure. (fig 4)Figure 4: Shear stress in crack: consolidation.Chemical potentialThe chemical potential is high at compression areas (xz > 0)and fluid flow is initiated to lower areas (xz < 0) . Chemicalpotential rapidly decreases further from the crack. (fig 5)Same trend was seen numerically.Figure 5: Chemical potential: fluid transportRemarksA quantitative comparison has not yet been performed.References:[1] J.R. Booker, J. of Geophys. Res., 1974[2] J.M. Huyghe et al., Int. J. of Eng. Science, 1997[3] S. Wognum et al., submitted to Spine, 2005[4] Y. Schroeder, submitted to Eur. Spine J., 2005

A singularity solution of shear faulting in swelling ionised porous media

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