Stress II

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Stress II
Cauchy formula Consider a small cubic element of
rock extracted from the earth, and imagine a
plane boundary with outward normal, n, and area,
?A cutting through this element - so it is
reduced to a triangular element with sides 1 and
2.
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The force components acting on sides 1 and 2
are 1. Note that 2. Replacing 2 in 1
gives 3.
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Force balance leads to Rearranging the
above This is equivalent to where tj is
the traction acting on ni.
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Stress II
Principal stresses We have learned that the
stress tensor is symmetric. A property of
symmetric matrices is that they may be
diagonaliszd. The transformation from the
non-diagonal to the diagonal tensor requires
transformation of the coordinate system. The axes
of the new coordinate system are the principal
axes, and the diagonal elements of the tensor are
referred to as the principal stresses. Note
that the shear stresses along the principal axes
are equal to zero.
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Using principal stresses to calculate shear and
normal stresses on a given plane Adding
vectors in directions parallel and normal to the
plane in question
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This is equivalent to Substituting the
following trigonometric identities gives
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The above equation defines a circle with a center
on the horizontal axes at (?1 ? 3)/2, and a
radius that is equal to (? 1 - ?
3)/2. (?1 ? 3)/2 is the mean
stress. (?1 - ? 3)/2 is the deviatoric
stress. ??is the angle between ?1 and the
normal to the plane - positive when measured
counter-clockwise from ?1 .
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Stress II
Note that for a given stress tensor, the mean
stress is independent of the plane in question,
that is We can thus write the stress tensor
as a sum of the mean stress field and the
deviatoric stress field
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• Note that
• Shear stresses equal to zero at ?0 and 90
  degrees.
• Maximum shear stress is equal to (? 1 - ? 3)/2
  at ?45 degrees.
• The shear stresses along the principal
  directions are equal to zero.
• The principal axes are orthogonal.

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Stress II
Mohr circle in 3D A single Mohr circle describes
the variation of shear and normal stress along a
principal plane (a plane that contains 2
principal axes). The representation of a 3D
state of stress is obtained by the superposition
of three Mohr circles, as follows The
state of stress on planes that are not
perpendicular to a principal plane fall within
the shaded area.
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Mohr circles and the state of stress Uniaxial
stress Only one non-zero principal stress. For
example Biaxial stress One principal
stress equals zero, the other two do not. For
example
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Mohr circles and the state of stress Triaxial
stress All principal stresses are non-zero. For
example Axial stress Two of the three
principal stresses are equal. For example
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Pressure This is a special state of stress in
which the shear stress is equal to zero, i.e.
?1 ?2 ?3. Question How does this state of
stress plot on a Mohr diagram? It is useful to
consider two pressures Lithostatic and
hydrostatic. Lithostatic pressure The stress
equals the weight of the overlying column of
rock. In the absence of tectonic forces or
fluids, the state of stress would be lithostatic.
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Stress II
Pressure Hydrostatic pressure The stress equals
the weight of a column of water.
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Role of fluid pressure and effective stress Pore
fluid Is the fluid within the pores. Pore
pressure Is the pressure within the pore
fluid. Usually the fluid is water, but it can
also be oil or gas. In a granular medium, the
pore pressure acts to reduce the contact between
the grains.
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Effective stress The effective stress tensor
is Question Is pressure a vector or a
scalar?
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Effective stress The effect of pore pressure
increase (for example, due to water pumping) is
to lower the effective stress. Graphically, this
may be illustrated as follows
P