1. Stress II

2. Stress as a Vector - Traction Force has variable magnitudes in different directions (i.e., it’s a vector) Area has constant magnitude with direction (a scalar):  Stress acting on a plane is a vector  = F/A or  = F. 1/A A traction is a vector quantity, and, as a result, it has both magnitude and direction These properties allow a geologist to manipulate tractions following the principles of vector algebra Like traction, a force is a vector quantity and can be manipulated following the same mathematical principals

3. Stress and Traction Stress can more accurately be termed "traction." A traction is a force per unit area acting on a specified surface This more accurate and encompassing definition of "stress" elevates stress beyond being a mere vector, to an entity that cannot be described by a single pair of measurements (i.e. magnitude and orientation) "Stress" strictly speaking, refers to the whole collection of tractions acting on each and every plane of every conceivable orientation passing through a discrete point in a body at a given instant of time

4. Normal and Shear Force Many planes can pass through a point in a rock body Force (F) across any of these planes can be resolved into two components: Shear stress: F s, & normal stress: F n, where: F s = F sin θ F n = F cos θ tan θ = F s /F n Smaller θ means smaller F s Note that if θ =0, F s =0 and all force is F n

5. Normal and Shear Stress Stress on an arbitrarily-oriented plane through a point, is not necessarily perpendicular to the that plane The stress (  acting on a plane can be resolved into two components: Normal stress (  n ) Component of stress perpendicular to the plane, i.e., parallel to the normal to the plane Shear stress (  s ) or  Components of stress parallel to the plane

6. Normal and Shear Stress

7. Stress is the intensity of force Stress is Force per unit area  = lim  F/  A when  A →0 A given force produces a large stress when applied on a small area! A given force produces a large stress when applied on a small area! The same force produces a small stress when applied on a larger area The same force produces a small stress when applied on a larger area The state of stress at a point is anisotropic: Stress varies on different planes with different orientation

8. Geopressure Gradient  P/  z The average overburden pressure (i.e., lithostatic P) at the base of a 1 km thick rock column (i.e., z = 1 km), with density (  ) of 2.5 gr/cm 3 is 25 to 30 MPa P =  gz [ML -1 T -2 ] P = (2670 kg m -3 )(9.81 m s -2 )(10 3 m) = 26192700 kg m -1 s -2 (pascal) = 26 MPa  The geopressure gradient:  P/  z  30 MPa/km  0.3 kb/km (kb = 100 MPa) i.e. P is  3 kb at a depth of 10 km

9. Types of Stress Tension: Stress acts  to and away from a plane pulls the rock apart forms special fractures called joint may lead to increase in volume Compression: stress acts  to and toward a plane squeezes rocks may decrease volume Shear: acts || to a surface leads to change in shape

10. Scalars Physical quantities, such as the density or temperature of a body, which in no way depend on direction are expressed as a single number e.g., temperature, density, mass only have a magnitude (i.e., are a number) are tensors of zero-order have 0 subscript and 2 0 and 3 0 components in 2D and 3D, respectively

11. Vectors Some physical quantities are fully specified by a magnitude and a direction, e.g.: Force, velocity, acceleration, and displacement Vectors: relate one scalar to another scalar have magnitude and direction are tensors of the first-order have 1 subscript (e.g., v i ) and 2 1 and 3 1 components in 2D and 3D, respectively

12. Tensors Some physical quantities require nine numbers for their full specification (in 3D) Stress, strain, and conductivity are examples of tensor Tensors: relate two vectors are tensors of second-order have 2 subscripts (e.g.,  ij ); and 2 2 and 3 2 components in 2D and 3D, respectively

13. Stress at a Point - Tensor To discuss stress on a randomly oriented plane we must consider the three-dimensional case of stress The magnitudes of the  n and  s vary as a function of the orientation of the plane In 3D, each shear stress,  s is further resolved into two components parallel to each of the 2D Cartesian coordinates in that plane

14. Tensors Tensors are vector processors A tensor ( T ij ) such as strain, transforms an input vector I i (such as an original particle line) into an output vector, O i (final particle line): O i =T ij I i ( Cauchy’s eqn.) e.g., wind tensor changing the initial velocity vector of a boat into a final velocity vector! |O 1 | | a b | |I 1 | |O 2 | = | c d | |I 2 |

15. Example (O i =T ij I i ) Let I i = (1,1) i.e, I 1 =1; I 2 =1 and the stress T ij be given by:|1.5 0| |-0.5 1| The input vector I i is transformed into the output vector(O i ) (NOTE: O i =T ij I i ) | O 1 |=| 1.5 0||I 1 | = |1.5 0||1| | O 2 | | -0.5 1||I 1 | |-0.5 1||1| Which gives: O 1 = 1.5I 1 + 0I 2 = 1.5 + 0 = 1.5 O 2 = -0.5I 1 + 1I 2 = -0.5 +1 = 0.5 i.e., the output vector O i =(1.5, 0.5) or: O 1 = 1.5 or |1.5| O 2 = 0.5|0.5|

16. Cauchy’s Law and Stress Tensor Cauchy’s Law: P i = σ ij l j (I & j can be 1, 2, or 3) P 1, P 2, and P 3 are tractions on the plane parallel to the three coordinate axes, and l 1, l 2, and l 3 are equal to cos , cos , cos  direction cosines of the pole to the plane w.r.t. the coordinate axes, respectively For every plane passing through a point, there is a unique vector l j representing the unit vector perpendicular to the plane (i.e., its normal) The stress tensor (  ij ) linearly relates or associates an output vector p i (traction vector on a given plane) with a particular input vector l j (i.e., with a plane of given orientation)

17. Stress tensor In the yz (or 23) plane, normal to the x (or 1) axis: the normal stress is  xx and the shear stresses are:  xy and  xz In the xz (or 13) plane, normal to the y (or 2) axis: the normal stress is  yy and the shear stresses are:  yx and  yz In the xy (or 12) plane, normal to the z (or 3) axis: the normal stress is  zz and the shear stresses are:  zx and  zy Thus, we have a total of 9 components for a stress acting on a extremely small cube at a point |  xx  xy  xz |  ij = |  yx  yy  yz | |  zx  zy  zz | Thus, stress is a tensor quantity

18. Stress tensor

19. Principal Stresses The stress tensor matrix: |  11  12  13 |  ij =|  21  22  23 | |  31  32  33 | Can be simplified by choosing the coordinates so that they are parallel to the principal axes of stress: |  1 0 0 |  ij =| 0  2 0 | | 00  3 | In this case, the coordinate planes only carry normal stress; i.e., the shear stresses are zero The  1,  2, and  3 are the major, intermediate, and minor principal stress, respectively  1 >  3 ; principal stresses may be tensile or compressive

20. Stress Ellipse

21. State of Stress Isotropic stress (Pressure) The 3D stresses are equal in magnitude in all directions; like the radii of a sphere The magnitude of pressure is equal to the mean of the principal stresses The mean stress or hydrostatic component of stress: P = (  1 +  2 +  3 ) / 3 Pressure is positive when it is compressive, and negative when it is tensile

22. Pressure Leads to Dilation Dilation (+e v & -e v ) Volume change; no shape change involved We will discuss dilation when we define strain e v =(v´-v o )/v o =  v/v o [no dimension] Where v ´ & v o are final & original volumes, respectively

23. Isotropic Pressure Fluids (liquids/gases) such as magma or water, are stressed equally in all directions Examples of isotropic pressure: hydrostatic, lithostatic, atmospheric All of these are pressures (P) due to the column of water, rock, or air, with thickness z and density  ; g is the acceleration due to gravity: P =  gz