Introduction to Seismology: Stress

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Introduction to SeismologyStress Strain

• Jesse F. Lawrence

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Preparing for Seismic Waves

• Stress Tensor
• Strain Tensor
• Equations of motion
• Constitutive equation

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Continuum Mechanics

• Continuum Mechanics A branch of physics that
  deals with continuous matter at the infinitesimal
  limit.
• Force per unit volume
• Mass Per unit volume (density)

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Traction

• Traction The vector describes a force applied to
  some finite area (a stress)
• Stress force per unit area
• Pressure force per unit area
• Function of the unit normal vector,

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Stress Traction

• Stress is the force per unit area that the
  traction at the surface exerts on the
  interior.exerts on the
• The traction vector acts on the surface whose
  outward normal is the positive ej direction.
• Three traction vectors
• With components Ti(j)
• Superscript (j) indicates the surface
• Subscript (i) indicates the component.

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Stress Tensor

• The stress tensor gives the traction vector T
  acting on any surface within the medium.
• The traction on surface dS, whose normal n is not
  along a coordinate axis is

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Normal Shear Stresses

• Normal Stress
• Shear Stresses
• Normal stresses are positive outward (expand the
  volume)
• Tension
• Normal stresses are negative inward, (contraction
  of volume)
• Compression

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Torque

• If the stress tensor is not symmetric, then
  torques cause rotations.
• Units
• Force/area
• cgs 1 bar 106 dyn/cm2
• 1 atm 1.01 bars
• SI 1 Pascal (Pa) 1 N/m2
• 10-5 bars

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A Few Quick Notes on Math

• A vector has components. These components can be
  related from one coordinate system to another
  using a transformation matrix, A.
• Similarly, s is a tensor (not just a matrix)
  because it transforms between coordinates
  according to

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Stress Coordinate Systems

• Case 1 A block of material with faces
  perpendicular to the ?1 and ?2 axes is subject
  only to normal stresses s1 and s2, so the stress
  tensor is diagonal
• Case 2 Now rotate the sides of the block
• If ?11 1 and ?22 1, then ??12 -1 and ??21
  -1

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Principle Axes of Stress

• Principle Stress Axes For any non-rotating
  stress state there exists an angle of rotation
  where there are only normal stresses. At this
  coordinate rotation the axes are called the
  principle axes.
• Principle stresses are the normal stresses in
  the directions of the principle axes. These are
  found using the concepts of eigenvalues and
  eigenvectors.
• Principle stress axes n
• Principle stresses ?

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Principle Axes Stresses

• Because we assume only normal axes
• This equation has roots ?m
• In geosciences we define
• If

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Maximum Shear Stress

• The plane of maximum shear stress is located 45?
  from the maximum and minimum normal stress axes,
  and the value of shear stress is
• If the max min stress axes are (1,0,0) and
  (0,0,1), then the planes of maximum shear stress
  are defined by

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Cohesive Strength

• Because of the cohesive strength of rocks,
  fractures often occur at about 25? from the min
  max stress directions rather than at 45?.

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Three types of Stress
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Mean Deviatoric Stress

• Mean Stress
• Deviatoric Stress

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Strain Tensor

• The strain tensor describes the deformation
  resulting from the differential motion within a
  body
• Rotation (?) and deformation (?) (by adding and
  subtracting ?uj/?ui)

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Dilatation

• The sum of diagonal terms of the strain tensor is
  the dilatation (change in volume per unit
  volume)
• For initial volume dx1dx2dx3 the volume after
  deformation is

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Constitutive Equation/Hookes Law

• The elastic moduli, cijkl, describe the
  deformation of a material as a result of a force.
• The stress tensor is symmetric (21 independent
  components)

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Isotropy

• Isotropy Material behaves the same wave
  regardless of orientation
• 2 Constants the lame constants
• ??- No physical Meaning
• ? - Shear Modulus
• Stress

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Bulk Modulus/Incompressibility

• The bulk Modulus is the inverse of
  compressibility
• Compressibility
• Incompressibility
• Stress

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Poissons Ratio Youngs Modulus

• Poissons Ratio the ratio of the contraction
  along one axis to the extension along the axis
  where extension is applied.
• Youngs Modulus the ratio of the tensional
  stress to the resulting extensional strain.

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Relations between Moduli