1. 1 Structural Geology Force and Stress - Mohr Diagrams, Mean and Deviatoric Stress, and the Stress Tensor Lecture 6 – Spring 2016

2. Graphical Analysis Another way of analyzing stress can be done using graphical methods to solve for σ n and σ s The technique is called a Mohr’s Circle Diagram for Stress Named after Christian Otto Mohr (1835-1918), a German civil engineer 2

3. Mohr’s Circle Diagram A Cartesian (i.e. X-Y) plot of σ s versus σ n that graphically solves the equations for normal and shear stress acting on a plane within a stressed body σ n is plotted on the horizontal axis, and σ s on the vertical axis The following slides show the construction of a Mohr’s Circle 3

4. 4 Mohr Circle Construction 1 The construction for a plane P that makes an angle θ with the σ 3 direction is as follows:  A. Lay off a distance equal to σ 3 from the origin in the positive direction of σ n Mark the point, and label it σ 3 Figure 3.8, text

5. 5 Mohr Circle Construction 2 B. Lay off a distance equal to σ 1 in the positive direction of σ n  Mark the point, and label it σ 1 C. Construct a circle through points σ 1 and σ 3  The center of the circle is located at ½ ( σ 1 + σ 3 ), and the radius is ½ ( σ 1 - σ 3 ) Figure 3.8, text

6. 6 Mohr Circle Construction 3 D. Draw radius OP such that  PO σ 1 equals 2 θ  We plot twice the angle between the plane and the σ 3 axis, but in a counterclockwise sense  (Other conventions exist; be careful if you read similar material in other literature) The diagram is now complete Figure 3.8, text

7. 7 Normal and Shear Stress Components We can read the value of σ n, P along the σ n axis (the σ n is often referred to simply as the σ axis, and the value of σ s, P along the σ s axis, which is also called the τ axis Figure 3.8, text

8. 8 Information from Mohr Diagrams Examining the completed drawing, figure 3_8, we see that:  σ n,P = ½ ( σ 1 + σ 3 ) + ½ ( σ 1 - σ 3 ) C cos2 θ and  σ s,P = ½ ( σ 1 - σ 3 ) C sin2 θ Note: cos2 θ = - sin ( 2 θ – 90º) sin 2 θ = cos ( 2 θ - 90º)

9. 9 More Information from Mohr Diagrams We can also see that there are two planes oriented at  θ and its complement,  (90 E - θ ) which have equal values of σ s but different values of σ n (Points P 1 and P 2 ) Figure 3.9, text

10. 10 More Information from Mohr Diagrams There are also two planes with equal values of σ n but with shear stresses of opposite sign (Points P 2 and P 3 ) Figure 3.9, text

11. 11 Plane Orientations and Points  For every orientation of a plane as defined by the  θ there is a corresponding point on the circle  The coordinates of the point give the normal and shear stresses on the plane If we do a tension experiment ( σ 1 = σ 2 = 0, and σ 3 < 0) the center of the circle will be on the negative side of the origin

12. 12 Stress Difference We can also see that the shear stress will be at a maximum when  θ = 45 E Then 2 θ = 90 E, and σ s = ½ ( σ 1 - σ 3 ) We can use the term stress difference ( σ d ) for ( σ 1 - σ 3 ) Thus,  σ d = 2 σ s

13. 13 3-D Mohr Diagrams Three dimensional Mohr circles are plotted in a similar fashion, except now we must plot σ 1, σ 2 and σ 3 along the σ n axis Figure 3-11 shows an example There are three individual circles, ( σ 1 - σ 2 ), ( σ 1 - σ 3 ), and ( σ 2 - σ 3 )

14. 14 Triaxial Stress Figure 3_11a shows the triaxial case, when no value of the principal stress equals zero

15. 15 Biaxial Stress Figure 3_11b shows the biaxial case, when one principal stress value = 0

16. 16 Uniaxial Compression Uniaxial compression, where σ 1 > 0, and σ 2 = σ 3 = 0, is shown in Figure 3_11c

17. 17 Hydrostatic Stress The last case, figure 3_11d, is for hydrostatic stress Since all three principal stresses are equal, the diagram reduces to a point

18. 18 Mean Stress σ m = ( σ 1 + σ 2 + σ 3 )/3 σ total = σ m + σ dev Mean stress is often referred to as hydrostatic component of stress, since hydrostatic stress is equal in all directions Another name is the hydrostatic pressure. Hydrostatic stress is isotropic Deep within the earth, we use the term lithostatic pressure, denoted P l, for the isostatic component

19. 19 Load Pressure Formula As we have seen,  P l = ρgh At depth, the lithostatic stress is usually orders of magnitude greater than anisotropic differential stresses Thus, deviatoric stress, which is anisotropic, might seem to be of little consequence

20. 20 Deviatoric Stress σ total = σ m + σ dev Deviatoric stress deforms the body, and is responsible for a shape and volume change in the rock In structural geology, it is common to measure the shape change of a body This is the strain

21. 21 Tensor Rank We can represent the stress in terms of a second-order tensor, as has previously been indicated The rank indicates the number of subscripts the quantity has Each subscript ranges in value from one to three, since there are three physical directions

22. 22 σ ij Second-order stress tensor

23. 23 Decomposition in Mean and Deviatoric Stress This notation can be used for decomposition into mean and deviatoric stress, as follows:

24. 24 Deviatoric Component When decomposed in this way, we see that the shear stresses appear only in the deviatoric component A change in reference frame (a rotation) will change the components of the stress tensor, but such changes are much easier to handle in tensor notation than using ellipsoids

25. 25 Stress Trajectories One method of representing the stress field is to plot the position of a selected stress vector, such as σ 1, at a number of points, and then connect the heads of the vectors This gives a series of lines, called stress trajectories A second set of lines representing another principal stress vector can also be drawn

26. 26 Stress Trajectory Image

27. Determination of Stress Differential stress may be determined in a number of ways Determining the absolute stress is much harder All methods of measuring differential stress are applicable only to the upper crust, yet we are often interested in the lower crust, or mantle of the earth 27

28. Time Stress measurements also apply only to today Determining stress states in the past (paleostress) is limited to the analysis of fault and fracture date, and to microstructural studies (changes in grain size, twinning of susceptible minerals) The fault data is limited to restricted to the upper crust, were we can measure fault displacements Microstructural data can be obtained using various remote sensing measurements, such as electrical currents generated by squeezing olivine crystals in the upper mantle (piezoelectric measurements) 28

29. 29 Global Stress Summary Figure 3.15a in text

30. 30 Plate Stress Trajectories Figure 3.15b in text

31. Stress Limits Differential stress cannot increase without bounds  Modern stress measurements often give results in the 50-150 MPa range  If differential stress increases too much, we discover the strength of the rock  Strength is the ability of a material to support differential stress.  If the differential stress exceeds the strength, the rock will fail 31

32. Failure Failure may occur in one of two ways:  A. Rupture (brittle behavior)  B. Flow (plastic behavior) 32

33. 33 Effect of Geothermal Gradient Region of low geothermal gradient (Precambrian Shield) Region of high geothermal gradient (continental rift region)