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
σ = F/A
or
σ = 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
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
5.
Normal and Shear Stress
• Stress on an arbitrarily-oriented plane through a point, is not necessarily
perpendicular to the that plane
• The stress (σ) acting on a plane can be resolved into two components:
• Normal stress (σ n)
– Component of stress perpendicular to the plane, i.e., parallel to
the normal to the plane
• Shear stress (σ s) or τ
– Components of stress parallel to the plane
7.
Stress is the intensity of force
– Stress is Force per unit area
σ = lim δF/δA when δA →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 δP/δz
•
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 (ρ) of 2.5 gr/cm3 is 25 to 30 MPa
P = ρgz
[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:
δP/δz ≅ 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 30 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., vi) and 21 and 31 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., σ ij); and 22 and 32 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 σ n and σ s vary as a function of the
orientation of the plane
• In 3D, each shear stress, σs 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 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|
|O2|
|a
= |c
b||I1|
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
| O2 | | -0.5
0||I1| = |1.5
1||I1|
|-0.5
0||1|
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|
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 cosα, cosβ , cosγ
– 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 (σ 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)
17.
Stress tensor
• In the yz (or 23) plane, normal to the x (or 1) axis: the normal stress is σxx and the
shear stresses are: σxy and σxz
• In the xz (or 13) plane, normal to the y (or 2) axis: the normal stress is σ yy and the
shear stresses are: σ yx and σ yz
• In the xy (or 12) plane, normal to the z (or 3) axis: the normal stress is σzz and the
shear stresses are: σzx and σzy
• Thus, we have a total of 9 components for a stress acting on a extremely small
cube at a point
|σ xx
σ xy
σ xz |
σ ij =
|σ yx
σ yy
σ yz |
|σ zx
σ zy
σ zz |
• Thus, stress is a tensor quantity
19.
Principal Stresses
• The stress tensor matrix:
| σ 11 σ 12
σ 13 |
σ ij = | σ 21 σ 22
σ 23 |
| σ 31 σ 32
σ 33 |
• Can be simplified by choosing the coordinates so that they are parallel to the
principal axes of stress:
| σ1
0
0 |
σ ij = | 0
σ2
0 |
|0
0
σ3 |
• In this case, the coordinate planes only carry normal stress; i.e., the shear stresses
are zero
• The σ 1 , σ 2 , and σ 3 are the major, intermediate, and minor principal stress,
respectively
• σ 1>σ 3 ; principal stresses may be tensile or compressive
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 = (σ 1 + σ 2 + σ 3 ) / 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 = δv/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 ρ; g is the acceleration due to gravity:
P = ρgz
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