Chapter 3 Rock Mechanics Stress

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Chapter 3Rock MechanicsStress
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Basic Physics

• Force
• that which changes the state of rest or the state
  of motion of a bodyFma
• Stress
• force applied to an areasF/A

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Basic Physics

• Scalar
• Possesses only a magnitude at some point in time
  or space
• Vector
• Possesses both magnitude and direction
• Tensor
• A field of data with magnitudes and directions

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Basic Physics

• Tensors
• Zero-order tensor is a scalar like temperature
  and has only 1 component
• First-order tensor is a vector like wind
  direction and is described by 3 components (time,
  magnitude, direction)
• Second-order tensor relates sets of tensors to
  each other and has 9 components

The number of components may be determined from
3n where n in the order of the tensor
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Basic Physics

• Stress can be
• Tensional - Pulling apart
• Compressional - Pushing together

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Basic Physics

• Stress on a surface can be broken into two vector
  components
• Normal Stress (sn) - acts perpendicular to the
  reference surface
• Shear Stress (t)- acts parallel to the surface

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Basic Physics

• Principal normal stress components (s1, s2, and
  s3)
• These are oriented perpendicular to each other
  and s1 ? s2 ? s3
• Differential stress is the difference between the
  maximum (s1) and the minimum (s3)
• Mean stress is (s1 s2 s3)/3
• If the differential stress exceeds the strength
  of the rock, permanent deformation occurs

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Basic Physics

• Lithostatic state of stress
• Occurs where the normal stress is the same in all
  directions
• Hydrostatic Pressure
• Confining stress acting on a body submerged in
  water
• Lithostatic Pressure
• Confining stress acting on a body under ground

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Stress on a plane

• Horizontal plane
• F ma volume x density x acceleration
• F 104 m3 x 2,750 kg m-3 x 9.8 ms-2
• Plane is 1 x 1 m, A 1 m2
• What is the Stress?

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Stress on a plane

• sF/A
• F (2.7 x 108 kg ms-2)/1m2
• 2.7 x 108 kg m-1s-2 or 2.7 x 108 Pa or 269.5MPa

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Stress on a plane

• Inclined Plane at 45º
• Through the same 1m x 1m space, actually has a
  larger surface area, now 1.41 m2
• Still F 2.7 x 108 kg m s-2
• So sF/A
• s (2.7 x 108 kg m s-2)/1.41 m2
• or 191 MPa
• How does that compare to the stress on the
  horizontal plane?

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Stress on a plane

• Stress can be broken down into components of
  normal and shear stress.
• sn s cos 45º
• 191 MPa x 0.707
• 135 MPa
• t s sin 45º
• 191 MPa x 0.707
• 135 MPa

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Stress Ellipsoid

• A Shear Ellipsoid is a graphical means of showing
  the relationship between the principal stresses
• The axes represent the principle normal stress
  components s1, s2, and s3
• The planes of maximum shear stress are always
  parallel to s2 and at 45º to s1 and s3.

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Triaxial Test Apparatus
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Mohr Circle Diagram

• Created by Otto Mohr, a german engineer, in 1882
• Enables us to determine the normal and shear
  stress across a plane

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Mohr Circle Diagram
t
t, P
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Mohr Circle Diagram
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Mohr Circle Diagram
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Measuring Present-Day Stress
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Stress in the United States
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