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On a Class of Contact Problems in Rock Mechanics

http :// minelab . mred . tuc . gr. On a Class of Contact Problems in Rock Mechanics. Exadaktylos George, Technical University of Crete. Acknowledgements.

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On a Class of Contact Problems in Rock Mechanics

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  1. http://minelab.mred.tuc.gr On a Class of Contact Problems in Rock Mechanics Exadaktylos George, Technical University of Crete

  2. Acknowledgements • We would like to thank the financial support from the EU 5th Framework Project “Integrated tool for in situ characterization of effectiveness and durability of conservation techniques in historical structures” (DIAS) with Contract Number: DIAS-EVK4-CT-2002-00080 http://minelab.mred.tuc.gr

  3. A class of plane contact problems in Rock Mechanics includes: Mixed Boundary Value Problems • The interaction of a thin liner with a circular opening in an elastic, isotropic, homogeneous rock (design of tunnel support) • The indentation of rocks (design of cutting tools (contact stresses), characterization of elasticity of rocks) • The cutting of rocks (design of cutting tools, characterization of strength of rocks)

  4. Why considering these 3 problems simultaneously ? • Because in underground construction the rock excavation precedes every other work (design of proper cutting tools, i.e. picks, discs, drag bits etc. & operational parameters of machine). • Rock cutting gives precious information for the strength of the intact rock whereas indentation for its elasticity at mesoscale (~ 1mm -100 mm). • Support must be manufactured by considering rock deformability and strength.

  5. Problem #1: Elastic interaction of a thinshell in perfect contact with a circular opening • Equilibrium eqn’s for the shell (Kirchhoff-Love) • Add BC’s • Constitutive relations

  6. Method of solution • Kolosov-Muskhelishvili complex variable method 1) 2) 3)

  7. Numerical implementation n=20 Ivanov (1976) System of (8n+4) eqns with (8n+4) unknowns

  8. Comparison with classical analytic solution by Savin (1961) for ‘welded’ elastic ring • Discre-pancy ? Other References: Einstein and Schwartz (1979), Bobet (2001)

  9. Why choosing the Complex Variable technique ? Stress Intensity factors at crack tips: KI, KII

  10. Interaction of 2 straight cracks with supported hole System of (16n+8) eqns with (16n+8) unknowns

  11. Problem #2: Rock indentation by DIAS portable indentor where k is the ‘penetration stiffness’ with dimensions

  12. Elasticity of mtl from back-analysis of indentation test data (analytical solution by Lur’e, 1964) Surface waves Indentation Ø=2.5 mm

  13. Recurrent loading-unloading cycles Ø=2.5 mm More complex σ-ε paths ?

  14. Problem #3: Rock cutting by drilling 3rd generation of DFMS [light instrument with jackleg (like jackhammer)] 2nd generation of DFMS with tripod Ø=5 mm

  15. WOB-Torque measurements • Normal & tangential forces during drilling are linearly constrained Each point is a test with different cutting depth δ

  16. Numerical modeling of rock cutting by drilling(Stavropoulou, 2005) comminution vx Elasto-visco-plastic cutting model (FLAC2D) DIAS EU R&D Project 2003-2005 (http://minelab.mred.tuc.gr/dias)

  17. Comparison of numerical simulations with experimental drilling data (Stavropoulou, 2005) Remark #2: c,φ for numerical modeling estimated from triaxial compression tests in lab (ψ=0o) Remark #1: Initiation of strain localization

  18. An approach to design structures in brittle rock masses: • Elasticity & strength of intact rock (L = .001 – 0.1 m) • - Fast drilling/indentation/ acoustic measurements 1. Excavation • Rock transected by cracks(L=.1 – 100 m) • - LEFM (fast algorithms) for stress analysis, KI,KII,KIII estimations & check of micromech – damage models ! • Stiffness and strength of joint walls (another contact problem) Hoek, Kaiser & Bawden (1995) • Support • DIAS measurements • Modeling

  19. END

  20. System of complex integro-differential eqns Note from the 1st eqn that for the limit of zero relative rigidity or thickness of the shell the radial and tangential stresses vanish Boundary element method

  21. Limit for relative rigidity of the thin shell tending to infinity

  22. Frictional contact of a gently dipping rigid slider with an elastic half-plane • Boundary conditions 1) 2) 3)

  23. Analytical solution (Muskhelishvili, 1963) • Normal force varies proportionally with indentation depth

  24. Graphical illustration of the solution Remark #2 tanφ=0, tanφ=0.5 tanφ=1 tanφ=0, tanφ=0.5 tanφ=1 Remark #1 Remark #3

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