Force, Traction, and Stress

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Force, Traction, and Stress

• Partial summary of Chapter 6 of Fundamentals of
  Structural Geology by Pollard and Fletcher 2006
• NOTE This Powerpoint presentation is prepared
  by Hassan Babaie for his teaching purposes, and
  does not fully cover the chapter in the
  book!Students are advised to read the entire
  chapter in the book.
• URL http//www.gsu.edu/geohab/pages/geol4013/le
  ctures.htm

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Acceleration

• A force, F, acting on a particle with mass, m,
  may lead to a linear acceleration, a, of that
  particle in the direction that the force acts
  (Fig. 6.1a)
• In the case of rigid body, the resultant torque,
  ?, about the axis z, due to the force, f, acting
  at position r, leads to an angular acceleration
  (Fig. 6.1b)

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Fig. 6.1a-d Force, traction, and stress are
illustrated as a) force vector, F, acting on a
particle of mass, m b) traction vector, t(n),
acting on surface element with outward normal n
c) stress tensor acting on volume element with
edges parallel to coordinate axes.
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Traction

• Traction, t(n), is a measure of limit of force
  per unit area acting at a point on the surface of
  a body as the surface shrinks toward zero about
  the point (Fig. 6.1c)
• n is the outward unit normal to the surface
• the surface could be the exterior boundary of a
  rock mass or an imaginary surface in the rock mass

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Stress Tensor

• Tractions can be calculated on all six planes of
  a cube about a point, as the volume of the cube
  shrinks toward zero
• The components of traction on each side of the
  cube define the components of the stress on the
  cube wrt the coordinate system
• The collection of all of these tractions is the
  stress tensor (Fig. 6.1 d)

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Components of the Stress Tensor

• Normal stress components of the stress tensor
  acting perpendicular to the sides of the cube
• Shear stress ?ij where ij
• i and j can take values 1, 2, or 3 (or x, y, z)
• Shear stress components of stress tensor acting
  parallel to the sides of the cube
• Shear stress ?ij where i?j
• i and j can take values 1, 2, or 3 (or x, y, z)

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Fig. 6.1a-d Force, traction, and stress are
illustrated as a) force vector, F, acting on a
particle of mass, m b) traction vector, t(n),
acting on surface element with outward normal n
c) stress tensor acting on volume element with
edges parallel to coordinate axes.
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• NOTE
• Force vector acts at a point mass
• Torque vector acts about an axis
• Traction vector acts on a surface element
• Stress tensor (i.e., a set of forces per unit
  area) acts on a volume element

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Relation of Structures to Stress

• Rock structure is a result of deformation (e.g.,
  strain, translation, rotation, flow), which is
  related to the stresses acting within the rock
  mass and the tractions acting on its surface
• Traction and stress vary with position and time
  i.e., are field quantities with spatial and
  temporal variations (next slide)
• An understanding of the possible variation of
  these field quantities in the Earth is of
  fundamental importance to structural geologists

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Chapter 6 Frontispiece Photoelastic image of
maximum shear stress contours in grains of model
rock. Stress is concentrated at grain contacts.
Inset Photoelastic image of three circular
disks with point contact loads. Gallagher et
al., 1974)
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Chapter 6 Frontispiece-b Photoelastic image of
maximum shear stress contours in three circular
disks with point contact loads.(Gallagher et
al., 1974)
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Chapter 6 Frontispiece-c Image of nearly
circular grains with point contact loads that
have produced fractures. Gallagher et al.,
(1974)
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Chapter 6 Frontispiece-d Image of grains with
concentrated contact loads that have produced
fractures.Gallagher et al., (1974)
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Classes of Force in Deformable Solids and Fluids

• Body forces act on volume elements within the
  rock mass
• These act at a distance rather than through the
  direct contact of two objects
• e.g., forces due to gravity and magnetic
  attraction
• Surface forces at on surface elements, either
  the actual surfaces of the rock, or imaginary
  surfaces within it
• These are due to the direct contact of one object
  with another

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Body Force

• See Fig. 6.2
• b is the body force per unit mass acting on an
  infinitesimal volume element dV
• The resultant of all body forces acting on the
  finite volume, ?V, is
• ??V ?bdV ex ??V ?bxdV ey ??V ?bydV ez ??V
  ?bzdV
• Where is the mass density and bi are the
  Cartesian components of b
• The resultant body force is the vector sum of the
  body forces acting on all infinitesimal elements
  within the finite volume ?v

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• Both ? and body force may vary with the spatial
  coordinates
• If the volumetric element is a rectangular cube
  with sides parallel to the coordinate axes, the
  volume integral is a triple integral for
  appropriate ranges of x, y, and x axes (i.e.,
  z1-z2, y1-y2, and x1-x2)
• ?z1?y1?x1 ?bdxdydz

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• To consider the body forces at an arbitrary point
  P in a deforming body or flowing fluid we assume
  the body is a material continuum
• We consider a sequence of finite volumes (?vi),
  each with its own resultant body force (?fi) and
  average density (?i) and each containing the
  point (see Fig. 6.2)
• We start with a large cube and reduce its size
  notice that the volume of the last cube
  approaches zero as n?? and the volume approaches
  the point
• Body force per unit mass at the point in the
  material continuum is
• b limitn??(?fn/?n?vn)

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Fig. 6.2a-b a) Body force, b, per unit mass
acting on infinitesimal volume element, dV,
within a finite volume. b) Sequence of finite
volumes with resultant body forces
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