Stress II

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

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

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

force : Fs , & normal force : Fn, where:

Fs = F sin θ Fn = F cos θ

tan θ = Fs/Fn

• Smaller θ means smaller Fs

• Note that if θ =0, Fs=0 and all force is Fn

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

Normal and Shear Stress

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 areaThe 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

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/cm3 is 25 to 30 MPa

P = gz [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:

P/z 30 MPa/km 0.3 kb/km (kb = 100 MPa)

• i.e. P is 3 kb at a depth of 10 km

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

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 30 components in 2D and 3D,

respectively

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 31 components in 2D and 3D,

respectively

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 22 and 32 components in 2D and

3D, respectively

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

Tensors• Tensors are vector processorsA tensor (Tij) such as strain, transforms aninput vector Ii (such as an original particle line) into an output vector, 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|

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 = 1.5O2 = -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|

Cauchy’s Law and Stress TensorCauchy’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 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 lj representing the unit vector perpendicular to the plane (i.e., its normal)

• The stress tensor (ij) 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)

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

Stress tensor

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 | | 0 0 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

Stress Ellipse

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 = (1 + 2 + 3 ) / 3

• Pressure is positive when it is compressive, and negative when it is tensile

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

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