The stresses that cause deformation Understand "stress calculations”

1. The stresses that cause deformation Understand "stress calculations” Spend some time with these calculations to convince yourself that stress on a given plane resolves itself into a single stress tensor. Stress (s) = force/unit area s = F/A

2. Stress Goals 1) Interpret the stresses responsible for deformation. 2) Describe the nature of the forces that cause the stresses. 3) Understand the relations between stress, strain and rock strength. Describing stress and force is a mathematical exercise.

3. Responses to Stresses 1) Folding 2) Brittle faults 3) Ductile shear zones 4) Joints

4. Force Force: changes in the state of rest or motion of a body. Only a force can cause a stationary object to move or change the motion (direction and velocity) of a moving object. force = mass x acceleration, F = ma, mass = density x volume, m =rV, therefore, r = m/V, Weight is the magnitude of the force of gravity (g) acting upon a mass. The newton (N) is the basic (SI) unit of force. 1 newton = 1 kg meter/sec2 1 dyne = 1g cm/sec2 so 1 N = 105 dyne 1 pascal = newton/m2

5. Forces as Vectors Force is a vector - it has magnitude and direction. Vectors can be added and subtracted using vector algebra. We can evaluate vectors in order to determine whether the forces on a body are in balance. Load Force

6. Units of Stress 1 newton = 1 kg meter/sec2 = this is a unit of force 1 pascal = 1 newton/m2 = unit of stress • 1 newton is about 0.224 809 pounds of force • 1 pascal is about 0.020 885 lb/ft2, thus pressure is measured in kPa • 1 kPa = 0.145 lb/in2 • 9.81 Pa is the pressure caused by a depth of 1mm of water

7. Stress on a 2-D plane: • Normal stress act perpendicular to the plane • Shear stress act along the plane. • Normal and shear stresses are perpendicular to one another

8. Stress(s) Stress is force per unit area: s= F/A

9. Relations between F and s (a) Fn and Fs and angle q with top and bottom surface. EF is trace of plane, ABCD is cube with ribs of length AG. Magnitudes of vectors Fs and Fn is function of angle q Fn = F cos q, Fs = F sin q (b) The magnitude of normal and shear stresses is function of angle q and the area, sn= s cos2q ss = s sin2q

10. Stress ellipsoid A point represents the intersection of an infinite number of planes and stresses on these planes describe an ellipse. In 3-dimensions, the ellipsoid is defined by three mutually perpendicular principal stresses (s1]> s2 > s3). These three axes are normal to the principal

11. Stress ellipsoid • What is important about the principal stresses (s1> s2 > s3)? • The axes are perpendicular to each other. • They do not contain shear stresses • The state of stress of any body is described by the orientation and magnitude of the principal stresses.

12. Components of stress • Three normal stresses • Components parallel shear stresses • Reference system x, y, z

13. Geology sign conventions Stresses 1) Normal stress Positive or negative 2) Shear stress Positive or negative Compressive stress is + (positive) Tensional stress is – (negative) Clockwise shear stress is – (negative) Counter clockwise shear stress is + (positive)

14. Stress State • If the 3 principal stresses are equal in magnitude = isotropic stress • Here the state of stress is represented by a sphere, not an ellipsoid. • If the principal stress are unequal in magnitude = anisotropic stress • Here the greatest stress is called s1 • The intermediate stress, s2 and minimum stress is called s3 s1 > s2 > s3 As a geologist, what is it called if all three principal stresses are equal? s1 = s2 = s3

15. Hydrostatic Stress If we calculate stress vectors within a point of a hydrostatic stress field, we find that the stress vectors have the same value. Each stress vector is oriented perpendicular to the plane. All stress vectors are normal vectors, they have no shear stress components. Hydrostatic stress = all principal stresses in a plane are equal in all directions. No shear stresses! Equal stress magnitudes in all directions. Dive into a pool. All stresses have the same values.

16. Lecture outline • Overview of stress • Minimum and maximum stress • Types of stress on a plane • Normal stress • Shear stress • 3. Mean stress • 4. Differential stress • 5. Deviatoric stress • 6. Hydrostatic state of stress • 7. Stress and the Mohr circle • Problem set outline • Apparent dip • Angle between lines • Angle between planes

17. Stress on a dipping plane in the Earth’s crust 2 components Normal stress & Shear stress sn = s cos2q ss = s sin2q Review sign conventions for normal and shear stresses

18. We resolve stress into two components • Normal stress, sn and the component that is parallel to the plane, shear stress, ss • Normal compressive stresses tend to inhibit sliding along the plane and are considered positive if they are compressive. • Normal tensional stresses tend to separate rocks along the plane and values are considered negative. • 3) Shear stresses tend to promote sliding along the plane, labeled positive if its right-lateral shear and negative if its left-lateral shear.

19. Squeeze a block of clay between two planks of wood AB, trace of fracture plane that makes an angle q with 3. The 2-D case is simple, since 2 = 3(atmospheric pressure) Important: What is angle q?

20. Mohr Stress Diagram • This give us a useful picture or diagram of the stress equations. • b) They solve stress equations on page 49 (Eqs 3.7 and eq. 3.10) • c) Plot N versus S • d) Rearrange Eqs. 3.7 and 3.10 and square them yields • [sn – ½(s1 + s2]2 + ss2 =[½(s1 – s32 )] • form (x –a)2 + y2 = r2 Important: What is angle q? In Mohr space, we use 2q!

21. Mohr Stress Diagram a) Mohr circle radius = ½(s1 – s3] that is centered on ½(s1 + s3] from the origin. b) The Mohr circle radius, ½(s1 - s2] is the maximum shear stress ss max. c) The stress difference (s1 – s3), called differential stress is indicated by sd.

22. Mohr Stress Diagram Mohr circle: sn on x-axis ss on y-axis. Maximum principal stress (s1) and minimum stress (s3) act on plane P that makes an angle q with the s3 direction. In Mohr space, we plot s1 and s3 on sn-axis These principal stress values are plotted on the sn-axes because they are the normal stresses acting on plane P. The principal stresses always have zero shear stress values (ss = 0).

23. Mohr Stress Diagram Remember, sn,p = ½(s1 + s3] + ½(s1 - s3] cos 2q ss,p = ½(s1 - s3] sin 2q Construct a circle thought points s1 and s3 with 0, the midpoint, at ½(s1 + s3) as the center with radius, ½(s1 - s3]. Now draw a line OP, so that angle POs1 is equal to 2q– confusing step, plot twice the angle q, which is the angle between the plane and s3. Remember we measure 2q from the s1 side on the sn-axis. We can read the values of sn,p along the sn-axis, and ss,p along ss-axis for our plane P.

24. Mohr Stress Diagram When the principal stress magnitudes change w/o differential stress, the Mohr circle moves along the sn-axis without changing ss How is this achieved? Suggest geologic examples?

25. Mohr Stress Diagram When the principal stress magnitudes change w/o differential stress, the Mohr circle moves along the sn-axis without changing ss • Change confining pressure (Pc). Increase air pressure on our clay experiment, or carry the experiment underwater. • Burial of rocks changes confining pressure. Which way along the sn-axis? • Exhumation of rock changes confining pressure. Again, in what direction along the sn-axis?

26. Problem set #1. Handouts in class and go online for additional graph paper in Mohr space

27. Various states of stress Uniaxial compression, two of the three principal stresses are zero. Hydrostatic stress, a single point on the Mohr circle that lies on the x-axis. All normal stresses are the same, and no shear stresses.

28. Various states of stress Triaxial stress, all three principal stresses are different. Biaxial stress, all three principal stresses are non-zero, but two of the principal stresses have the same value. Typical stress ellipse (plane stress).

29. Mean stress and deviatoric stress Because a body’s response to stress, we subdivide the stress into two components, mean and deviatoric stress. Mean stress = [s1 + s2 + s3]/3 or sm In 2-D, [s1 + s3]/2 Deviatoric stress is the difference between the mean stress and total stress. stotal =smean + sdev smeanis often called the hydrostatic component (s1 = s2 = s3)

30. Lithostatic pressure (Pl). For rocks at depth, we use lithostatic pressure. Consider a rock at 3 km depth. Lithostatic pressure F (weight of rock of overlying column). Pl = rxgxh if r (density) = 2700 km/m3, g (gravity) = 9.8 m/s2 and h (depth) is 3000 m, we get: Pl = 2700 x 9.8 x 3000 = 79.4 x 206 Pa ~ 80 Mpa For every km in the Earth’s crust, the lithostatic pressure increases 27 Mpa. The lithostatic pressure is equal in all directions (isotropic stress), [s1 = s2 = s32 ]

31. Lithostatic pressure (Pl). So we divide the rocks state of stress into an isotropic (lithostatic/hydrostatic) and an anisotropic (deviatoric). Isotropic stresses act equally on all directions, resulting in a volume change of the rock – increase water pressure on a human, or air pressure on take-off or landing. Deviatoric stress, changes the shape of the body. The difference between isotropic stress and additional stress from tectonic forcing.

32. Measuring Stress • Present day stress • Difficult to measure • EQ focal mechanisms • Bore-hole breakouts • in situ measurements in situ borehole measurements of sd (s1 –s3) with depth.

33. Stress in the Earth World stress map and topography showing maximum horizontal stress.

34. Stress in the Earth Generalized pattern based on stress trajectories for individual plates.

35. Stress and strength at depth Strength – the ability of a material to support different stress Maximum stress before a rock fails Strength curves: differential stress magnitude versus depth. • Regional with low geothermal gradient • Regional high geothermal gradients Give some geologic examples? This is important and will be on exam 1!

36. Stress and strength at depth

37. Stress and strength at depth