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# 2. Failure Mechanics

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### 2. Failure Mechanics

1. 1. James A. Craig Omega 2011
2. 2.  Concepts of Failure  Tensile Failure  Shear Failure  Failure Criteria  Mohr-Coulomb Failure Criterion
3. 3.  Failure occurs to any solid material when:  Sufficiently large stress is applied.  The material does not return to its original state after stress relief.  Mode of failure depends on:  Stress state  Type and geometry of material  Fatigue makes failure to occur below the stress level.
4. 4. Uniaxial Test  Stress is applied to the end faces of the specimen.  No radial (confining stress)  Also called Unconfined Compression Test.
5. 5. Elastic region Specimen returns to its original state after stress relief. Yield Point Permanent changes beyond this point. Specimen does not return to its original state after removal of stress. Uniaxial compressive strength The peak stress. Ductile region Permanent deformation, but can still support load. Brittle region Ability to withstand stress decreases rapidly as deformation increases.
6. 6. Triaxial Test  In addition to axial stress, confining pressure of different magnitude is applied to the circumference of the cylinder (by a confining oil bath).
7. 7.  Two of the principal stresses are equal.  Process:  Axial & confining loads are increased simultaneously until a prescribed hydrostatic stress level is reached.  Confining pressure is kept constant while axial load increases until failure occurs.
8. 8. Difference in principal stresses is plotted against axial deformation. Specimen can still support load after failure due to high confining pressure. It is called Work Hardening or Strain Hardening.
9. 9. X → abrupt brittle failure Uniaxial test
10. 10.  Tensile failure occurs when the effective tensile stress across some plane is the sample exceeds a critical limit called Tensile Strength.
11. 11.  Tensile failure is caused by the stress concentrations at the edges of thin cracks oriented normal to the direction of the least compressive principal stress.  For isotropic rocks, conditions for failure will always be fulfilled first for the lowest principal stress.   3   3  P  To To = tensile strength (in Pa, atm, psi or bar).
12. 12.  Most sedimentary rocks have a rather low tensile strength, typically only a few MPa or less.  Standard approximation for several applications is that the tensile strength is zero
13. 13.  It occurs when the shear stress along some plane in the sample is too large.
14. 14.  Mohr–Coulomb  Hoek–Brown  Drucker–Prager  Griffith (tensile)
15. 15.   f     So    So = cohesion or inherent shear strength of material (in Pa, atm, psi or bar).  µ = coefficient of internal friction.  Shear stress must overcome the cohesion plus the internal friction in order to produce a macroscopic shear failure.
16. 16. Failure Line Slope = tan    Mohr Circle So A  So cot  tan  If the Mohr’s circle lies below the failure line, the rock does not fail and remains intact.
17. 17.  φ = angle of internal friction. It varies from 0 to 90o (approx. 30o)  A = attraction (in Pa, atm, psi or bar).  β = angle that fulfils the failure criterion. It gives orientation of the failure plane. Varies between 45o and 90o.  At point P:  Angle 2β gives the position of coincidence of Mohr’s circle and the failure line.  Coordinates are given as: 1    1   3  sin 2 2 1 1   1   3   1   3  cos 2 2 2 2  90o    OR   4   2
18. 18.  Co = uniaxial compressive strength (in Pa, atm, psi or bar).
19. 19. 2 So cos  a  Co  2 So tan  1  sin   1  sin   b  tan   1  sin    tan   1 sin    tan   1  1  a  b 3  1  Co   3 tan 2 
20. 20. © Haimson and Song (1995)
21. 21.  Principle of effective stress is introduced, i.e. subtract fluid pressure from the total stress.  Previously:  1  a  b 3  1   1   Pf   Then:  And 3   3   Pf   1  a  b 3 1  sin   2So cos  1   Pf    3   Pf  1  sin   1  sin  
22. 22.  Pore fluid can affect the failure of the rock in 2 ways:  Mechanical effect of pore pressure.  Chemical interactions between the rock and the fluid.
23. 23.  Effect of pore pressure on failure:  Shear stress is unaffected by the pore pressure  Minimum & maximum principal stresses are decreased by the same amount.  Radius of the Mohr circle in unchanged.  Center of the circle has shifted to the left.  Circle moves towards the failure line when the fluid pressure is increased for a material obeying the criterion.