Stress II. Stress as a Vector - Traction Force has variable magnitudes in different directions...
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Transcript of Stress II. Stress as a Vector - Traction Force has variable magnitudes in different directions...
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 stress: Fs , & normal stress: 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
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 A given force produces a large stress when applied on a small area!applied on a small area!
The same force produces a small stress when The same force produces a small stress when applied on a larger areaapplied 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 an
input 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
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
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