Geology 3120 Powerpoint notes available online at:

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1 Geology 3120 Powerpoint notes available online at:

2 Geology The Mohr Stress Diagram σ s 0 2Θ σ n Stress Space

3 Outline Setting up the Problem The Mohr Stress Diagram Mohr-Coulomb Law of Failure Exercises

4 Setting up the Problem Θ is defined as the angle between the plane and the force vector. Clockwise is positive (+).

5 Decomposing Stresses After several trigonometric and algebraic simplifications, the two equations left are

6 Fundamental Stress Equations Normal Stress Shear Stress σ n = (σ 1 + σ 3 ) - (σ 1 - σ 3 )cos 2Θ 2 2 σ s = (σ 1 - σ 3 )sin 2Θ 2

7 Physical and Stress Space σ 1 σ s σ 3 Θ σ3 0 2Θ σ n σ 1 Physical Space Stress Space

8 Conventions - Physical Space Θ is defined as the σ 1 angle between the plane and the σ 1 stress direction. A clockwise direction is σ 3 +Θ σ 3 positive (+). σ 1

9 Conventions - Stress Space σ s (+) 2Θ 0 σ 3 σ 1 σ n ( ) σ n = tension (+) σ n = compression

10 Conventions - Stress Space σ (p) s σ s σ (p ) n, σ (p) s 2Θ 0 σ 3 σ n σ 1 σ n σ (p) n = (σ 1 + σ 3 ) - (σ 1 - σ 3 )cos 2Θ 2 2 σ s (p) = (σ 1 - σ 3 )sin 2Θ 2

11 Conventions - Stress Space σ s σ (p ) n, σ (p) s (σ 1 - σ 3 )sin 2Θ 2 2Θ 0 σ 3 σ 1 σ n

12 Conventions - Stress Space σ s σ (p ) n, σ (p) s 2Θ 0 σ 3 σ 1 σ n (σ 1 + σ 3 ) 2 Mean Stress - center of circle

13 Conventions - Stress Space σ s σ (p ) n, σ (p) s 2Θ 0 σ 3 σ 1 σ n (σ 1 - σ 3 ) 2 Deviatoric Stress - radius of circle

14 Conventions - Stress Space σ s σ (p ) n, σ (p) s 2Θ 0 σ 3 σ 1 σ n (σ 1 - σ 3 ) Differential Stress - diameter of circle

15 Conventions - Stress Space σ s σ (p ) n, σ (p) s 2Θ 0 σ 3 σ 1 σ n (σ 1 - σ 3 )cos 2Θ 2 Difference between mean stress and normal stress on plane

16 Laboratory Experiments in Rock Deformation Deformed marble rock cylinders

17 Repeated Failure Experiments

18 Stress Requirements for Rock Failure

19 Mohr-Coulomb Law of Failure ( ( σ c Y = mx + b σ n σ n φ = angle of internal friction tan φ = coefficient of internal friction [slope; m] σ n = normal stress [X] σ c = critical shear stress required for faulting [Y] σ = cohesive strength [y-intercept; b]

20 Influence of Pore Fluid Pressure Effective Stress Applied Stress p f Pore fluid pressure decreases normal stresses by the fluid pressure amount. Rock can then fail under the Mohr-Coulomb Law.

21 Problems 1 & 2 1. For the maximum and minimum principal stresses of 600 and 200 MPa oriented as a vertical vector and a horizontal, E-W striking vector, respectively, determine the normal and shear stress on a plane oriented N0 E, 45 E. 2. For the stress state in the problem above determine the deviatoric stress and mean stress.

22 Problems 1 & 2 σ 1 W E σ 3 +Θ 1. For the maximum and minimum principal stresses of 600 and 200 MPa oriented as a vertical vector and a horizontal, E-W striking vector, respectively, determine the normal and shear stress on a plane oriented N0 E, 45 E. σ n = 400 Mpa, σ s =200 MPa 2. For the stress state in the problem above determine the deviatoric stress and mean stress. Deviatoric Stress = 200 MPa, Mean Stress = 400 MPa

23 Problem 3 3. Given two planes P1 and P2 oriented where Θ equals 90 (P1) and 45 (P2), P1 has a normal stress of 500 MPa and P2 has a normal stress of 300 MPa and a shear stress of 200 MPa, determine the magnitudes of the principal stresses, the deviatoric stress and the mean stress. Is this stress state more or less likely to produce failure as that in Problem 1?

24 Problem 3 3. Given two planes P1 and P2 oriented where Θ equals 90 (P1) and 45 (P2) degrees, P1 has a normal stress of 500 MPa and P2 has a normal stress of 300 MPa and a shear stress of 200 MPa, determine the magnitudes of the principal stresses, the deviatoric stress and the mean stress. σ 1 = 500 MPa and σ 3 = 100 MPa Deviatoric Stress = 200 MPa Mean Stress = 300 MPa

25 Problem 3 3. Is this stress state more or less likely to produce failure as that in Problem 1? The stress state of Problem 3 is more likely to produce failure than in Problem 1 since the Mohr circle is closer to the failure envelope.

26 References Slide 15 Twiss, R. J. and E. M. Moores, Structural Geology, W. H. Freeman & Co., New York, 532 p., Slides Davis. G. H. and S. J. Reynolds, Structural Geology of Rocks and Regions, 2nd ed., John Wiley & Sons, New York, 776 p., 1996.