Stress II

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: Fs , & normal stress:Fn, where: Fs = F sin θFn = F cos θ tanθ = Fs/Fn • Smaller θ means smaller Fs • Note that if θ =0, Fs=0 and all force is Fn

5. Normal and Shear Stress • Stress on an arbitrarily-oriented plane through a point, is not necessarily perpendicular to the that plane • The stress (s) acting on a plane can be resolved into two components: • Normal stress (sn) • Component of stress perpendicular to the plane, i.e., parallel to the normal to the plane • Shear stress (ss) 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 s = lim dF/dA when dA →0 • 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 state of stress at a point is anisotropic: • Stress varies on different planes with different orientation

8. Geopressure Gradient dP/dz • 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 (r) of 2.5 gr/cm3 is 25 to 30 MPa P = rgz [ML -1T-2] P = (2670 kg m-3)(9.81 m s-2)(103 m) = 26192700 kg m-1s-2 (pascal) = 26 MPa • The geopressure gradient: dP/dz  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 20 and 30components 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., vi) and 21 and 31components 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., sij); and 22 and 32components 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 sn and ss vary as a function of the orientation of the plane • In 3D, each shear stress, ss 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 (Tij) such as strain, transforms an input vectorIi (such as an original particle line) into an outputvector, Oi (final particle line): Oi=Tij Ii (Cauchy’s eqn.) e.g., wind tensor changing the initial velocity vector of a boat into a final velocity vector! |O1| |a b||I1| |O2| = |c d||I2|

15. Example (Oi=TijIi ) • Let Ii = (1,1) i.e, I1=1; I2=1 and the stress Tij be given by: |1.5 0| |-0.5 1| • The input vector Ii is transformed into the output vector(Oi) (NOTE: Oi=TijIi) | O1|=| 1.5 0||I1| = |1.5 0||1| | O2| | -0.5 1||I1| |-0.5 1||1| • Which gives: O1 = 1.5I1 + 0I2 = 1.5 + 0 = 1.5 O2 = -0.5I1 + 1I2 = -0.5 +1 = 0.5 • i.e., the output vector Oi=(1.5, 0.5) or: O1 = 1.5 or |1.5| O2 = 0.5 |0.5|

16. Cauchy’s Law and Stress Tensor Cauchy’s Law: Pi= σijlj(I & j can be 1, 2, or 3) • P1, P2, and P3 are tractions on the plane parallel to the three coordinate axes, and • l1, l2, and l3 are equal to cosa, cosb , cosg • 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 lj representing the unit vector perpendicular to the plane (i.e., its normal) • The stress tensor (sij) linearly relates or associates an output vector pi (traction vector on a given plane) with a particular input vector lj (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 sxx and the shear stresses are: sxy and sxz • In the xz (or 13) plane, normal to the y (or 2) axis: the normal stress is syy and the shear stresses are: syx and syz • In the xy (or 12) plane, normal to the z (or 3) axis: the normal stress is szz and the shear stresses are: szx and szy • Thus, we have a total of 9 components for a stress acting on a extremely small cube at a point |sxxsxysxz | sij = |syxsyysyz | |szxszyszz | • Thus, stress is a tensor quantity

18. Stress tensor

19. Principal Stresses • The stress tensor matrix: | s11s12s13 | sij = | s21s22s23 | | s31s32s33 | • Can be simplified by choosing the coordinates so that they are parallel to the principal axes of stress: | s 1 0 0 | sij = | 0 s2 0 | | 0 0 s3 | • In this case, the coordinate planes only carry normal stress; i.e., the shear stresses are zero • The s 1 , s2 , and s 3 are the major, intermediate, and minor principal stress, respectively • s1>s3 ; principal stresses may be tensile or compressive

20. Stress Ellipse

21. State of StressIsotropic 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 = (s1 + s2 + s3 ) / 3 • Pressure is positive when it is compressive, and negative when it is tensile

22. Pressure Leads to Dilation • Dilation (+ev & -ev) • Volume change; no shape change involved • We will discuss dilation when we define strain ev=(v´-vo)/vo = v/vo[no dimension] • Where v´ & vo 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 r; g is the acceleration due to gravity: P = rgz