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The MLPG applied to porous materials with variable stiffness and permeability

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Abstract

Two-dimensional (2-d) and axisymmetric consolidation problems are treated with a meshless local Petrov–Galerkin approach. The porous continuum is modeled with Biot’s theory, where the solid displacements and the pore pressure are chosen as unknowns (u-p-formulation). These unknowns are approximated with independent spatial discretizations using the moving least-squares scheme. The method is validated by a comparison with an 1-d analytical solution. Studies with graded material data show that a variable permeability has a strong influence on the consolidation process. The example of a disturbed zone around a borehole shows as well the importance of graded material data.

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References

  1. Terzaghi K (1925) Erdbaumechanik auf bodenphysikalischer Grundlage. Franz Deuticke, Leipzig

    MATH  Google Scholar 

  2. Biot MA (1935) Le problème de la consolidation des matières argileuses sous une charge. Ann Soc Sci Brux B55:110–113

    Google Scholar 

  3. Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12:155–164

    Article  MATH  ADS  Google Scholar 

  4. Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 26:182–185

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Biot MA (1956) General solutions of the equations of elasticity and consolidation for a porous material. J Appl Mech Trans ASME 78:91–96

    MathSciNet  Google Scholar 

  6. Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27:240–253

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Biot MA (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33:1482–1498

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Selvadurai APS (1996) Mechanics of poroelastic media. Kluwer Academic, Dordrecht

    Book  MATH  Google Scholar 

  9. Detournay E, Cheng AHD (1993) Fundamentals of poroelasticity, volume II of comprehensive rock engineering: principles, practice and projects, chapter 5. Pergamon Press, Oxford, pp 113–171

  10. Schanz M (2009) Poroelastodynamics: linear models, analytical solutions, and numerical methods. Appl Mech Rev 62:030803–030815

    Article  ADS  Google Scholar 

  11. Zienkiewicz OC, Shiomi T (1984) Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution. Int J Numer Anal Methods Geomech 8:71–96

    Article  MATH  Google Scholar 

  12. Bonnet G (1987) Basic singular solutions for a poroelastic medium in the dynamic range. J Acoust Soc Am 82:1758–1762

    Article  ADS  Google Scholar 

  13. Selvadurai APS (2007) The analytical method in geomechanics. Appl Mech Rev 60:87–106

    Article  ADS  Google Scholar 

  14. Lewis RW, Schrefler BA (1998) The finite element method in the static and dynamic deformation and consolidation of porous media, 2nd edn. Wiley, Chichester

    MATH  Google Scholar 

  15. Zienkiewicz OC, Chan AHC, Pastor M, Schrefler BA, Shiomi T (1999) Computational geomechanics with special reference to earthquake engineering. Wiley, Chichester

    MATH  Google Scholar 

  16. Li X, Han X, Pastor M (2003) An iterative stabilized fractional step algorithm for finite element analysis in saturated soil dynamics. Comput Methods Appl Mech Eng 192:3845–3859

    Article  MATH  ADS  Google Scholar 

  17. Soares D Jr (2008) A time-domain FEM approach based on implicit Green’s functions for the dynamic analysis of porous media. Comput Methods Appl Mech Eng 197:4645–4652

    Article  MATH  ADS  Google Scholar 

  18. Ehlers W, Bluhm J (eds) (1998) Porous media—theory, experiments and numerical applications. Springer, Berlin

    Google Scholar 

  19. Babuska I (1973) The finite element method with Lagrange multiplier. Numer Math 20:179–192

    Article  MATH  MathSciNet  Google Scholar 

  20. Brezzi F (1974) On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO 8-R2:129–151

    MathSciNet  Google Scholar 

  21. Huang MS, Zienkiewicz OC (1998) New unconditionally stable staggered solution procedures for coupled soil-pore fluid dynamic problems. Int J Numer Methods Eng 43:1029–1052

    Article  MATH  MathSciNet  Google Scholar 

  22. Chen J, Dargush GF (1995) Boundary element method for dynamic poroelastic and thermoelastic analyses. Int J Solids Struct 32:2257–2278

    Article  MATH  Google Scholar 

  23. Senjuntichai T, Rajapakse RKND (1994) Dynamic Green’s functions of homogeneous poroelastic half-plane. J Eng Mech ASCE 120:2381–2404

    Article  Google Scholar 

  24. Senjuntichai T, Mani S, Rajapakse RKND (2006) Vertical vibration of an embedded rigid foundation in a poroelastic soil. Soil Dyn Earthq Eng J 26:626–636

    Article  Google Scholar 

  25. Norris AN (1994) Dynamic Green’s functions in anisotropic piezoelectric, thermoelastic and poroelastic solids. Proc R Soc Lond A 447:175–188

    Article  MATH  ADS  Google Scholar 

  26. Park KH, Banerjee PK (2006) A simple BEM formulation for poroelasticity via particular integrals. Int J Solids Struct 43:3613–3625

    Article  MATH  Google Scholar 

  27. Dargush GF, Banerjee PK (1991) A boundary element method for axisymmetric soil consolidation. Int J Solids Struct 28:897–915

    Article  MATH  Google Scholar 

  28. Pan E (1999) Green’s function in layered poroelastic half-spaces. Int J Numer Anal Methods Geomech 23:1631–1653

    Article  MATH  Google Scholar 

  29. Cheng AHD (1997) Material coefficients of anisotropic poroelasticity. Int J Rock Mech Min Sci 34:199–205

    Article  Google Scholar 

  30. Vrettos Ch (2008) Green’s functions for vertical point load on an elastic half-space with depth-degrading stiffness. Eng Anal Bound Elem 32:1037–1045

    Article  MATH  Google Scholar 

  31. Ai ZY, Cheng YC, Zeng WZ, Wu C (2013) 3D consolidation of multilayered porous medium with anisotropic permeability and compressible pore fluid. Meccanica 48:491–499

    Article  MathSciNet  Google Scholar 

  32. Soares D, Sladek V, Sladek J (2012) Modified meshless local Petrov–Galerkin formulations for elastodynamics. Int J Numer Methods Eng 90:1508–1528

    Article  MATH  MathSciNet  Google Scholar 

  33. Belytschko T, Krogauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods; an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47

    Article  MATH  ADS  Google Scholar 

  34. Atluri SN (2004) The meshless method (MLPG) for domain and BIE discretizations. Tech Science Press, Forsyth

    Google Scholar 

  35. Sladek J, Stanak P, Han ZD, Sladek V, Atluri SN (2013) Applications of the MLPG method in engineering and sciences: a review. Comput Model Eng Sci 92:423–475

    Google Scholar 

  36. Zhu T, Zhang JD, Atluri SN (1998) A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput Mech 21:223–235

    Article  MATH  MathSciNet  Google Scholar 

  37. Atluri SN, Sladek J, Sladek V, Zhu T (2000) The local boundary integral equation (LBIE) and its meshless implementation for linear elasticity. Comput Mech 25:180–198

    Article  MATH  Google Scholar 

  38. Sladek J, Sladek V, Atluri SN (2000) Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties. Comput Mech 24:456–462

    Article  MATH  MathSciNet  Google Scholar 

  39. Sladek J, Sladek V, Atluri SN (2004) Meshless local Petrov–Galerkin method in anisotropic elasticity. Comput Model Eng Sci 6:477–489

    MATH  MathSciNet  Google Scholar 

  40. Bergamaschi L (2009) An efficient parallel MLPG method for poroelastic models. Comput Model Eng Sci 29:191–215

    MathSciNet  Google Scholar 

  41. Wang JG, Liu GR, Lin P (2002) Numerical analysis of Biot’s consolidation process by radial point interpolation method. Int J Solids Struct 39:1557–1573

    Article  MATH  Google Scholar 

  42. Wang WD, Wang JG, Wang ZL, Nogami T (2007) An unequal-order radial interpolation meshless method for Biot’s consolidation theory. Comput Geotech 34:61–70

    Article  Google Scholar 

  43. Wang ZL, Li YC (2006) Analysis of factors influencing the solution of the consolidation problem by using an element-free Galerkin method. Comput Geosci 32:624–631

    Article  ADS  Google Scholar 

  44. Wang JG, Xie H, Leung CF (2009) A local boundary integral-based meshless method for Biot’s consolidation problem. Eng Anal Bound Elem 33:35–42

    Article  MATH  MathSciNet  Google Scholar 

  45. Soares D Jr (2010) Dynamic analysis of porous media considering unequal phase discretization by meshless local Petrov–Galerkin formulations. Comput Model Eng Sci 61:177–200

    MATH  MathSciNet  Google Scholar 

  46. Soares D Jr, Sladek V, Sladek J, Zmindak M, Medvecky S (2012) Porous media analysis by modified MLPG formulations. Comput Mater Contin 27:101–126

    Google Scholar 

  47. Khoshghalb A, Khalili N (2013) A meshfree method for fully coupled analysis of flow and deformation in unsaturated porous media. Int J Numer Anal Methods Geomech 37:716–743

    Article  Google Scholar 

  48. Rice JR, Cleary MP (1976) Some basic stress-diffusion solutions for fluid saturated elastic porous media with compressible constituents. Rev Geophys Space Phys 14:227–241

    Article  ADS  Google Scholar 

  49. Rajapakse RKND, Gross D (1996) Traction and contact problems for an anisotropic medium with a cylindrical borehole. Int J Solids Struct 33:2193–2211

    Article  MATH  Google Scholar 

  50. Kaewjuea W (2010) Poroelastic solutions for borehole and cylinder. PhD Thesis, Chulalongkorn University

  51. Khoshghalb A, Khalili N, Selvadurai APS (2011) A three-point time discretization technique for parabolic differential equations. Int J Numer Anal Methods Geomech 35:406–418

    Article  MATH  Google Scholar 

  52. Wen PH, Aliabadi MH (2008) An improved meshless collocation method for elastostatic and elastodynamic problems. Commun Numer Methods Eng 24:635–651

    Article  MATH  MathSciNet  Google Scholar 

  53. Lai X, Cai M, Ren F, Xie M, Esaki T (2006) Assessment of rock mass characteristics and the excavation disturbed zone in the Lingxin Coal Mine beneath and Xitian River, China. Int J Rock Mech Min Sci 43:572–581

    Article  Google Scholar 

  54. Kwon S, Lee CS, Cho SJ, Jeon SW, Cho WJ (2009) An investigation of the excavation damaged zone at the KAERI underground research tunnel. Tunn Undergr Space Technol 24:1–13

    Article  Google Scholar 

  55. Kim YK, Kingsbury HB (1979) Dynamic characterization of poroelastic materials. Exp Mech 19:252–258

    Article  Google Scholar 

Download references

Acknowledgments

The first two authors acknowledge the support by the Slovak Science and Technology Assistance Agency registered under number APVV-0014-10 and APVV-0032-10.

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Authors and Affiliations

  1. Institution of Construction and Architecture, Slovak Academy of Sciences, 84503, Bratislava, Slovakia

    J. Sladek & V. Sladek

  2. Institute of Applied Mechanics, Graz University of Technology, Graz, Austria

    M. Schanz

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  1. J. Sladek
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  2. V. Sladek
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  3. M. Schanz
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Correspondence to M. Schanz.

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Sladek, J., Sladek, V. & Schanz, M. The MLPG applied to porous materials with variable stiffness and permeability. Meccanica 49, 2359–2373 (2014). https://doi.org/10.1007/s11012-014-0004-0

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