1 stress ii (1)
TRANSCRIPT
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Stress II
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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
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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
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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
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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
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Normal and Shear Stress
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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
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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
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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
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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
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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
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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
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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
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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|
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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|
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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)
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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
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Stress tensor
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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
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Stress Ellipse
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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
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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
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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