Structural Geology 3443 Ch. 2 Kinematic Analysis

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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Department of Geology University of Texas at
Arlington
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Some definitions Kinematics The description of
the motion of point(s) in space. The point can be
the center of mass of an object or all the points
that make up an object Coordinate system The
reference frame to which point(s) are referred.
May be local, global, Solar, Galactic etc.
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Types of motion Translation Rotation Deformation
- distortion Deformation - dilation
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Structural GeologyCh. 2 Kinematic Analysis
Kinematics is a branch of Analytic geometry, and
displacement of points are described by the
Transformation Equations (text pg 85-93) X
a10 a11X a12Y a13Z Y a20 a21X a22Y
a23Z Z a30 a31X a32Y a33Z In two
dimensions these reduce to X a10 a11X
a12Y Y a20 a21X a22Y
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
These equations mathematically describe all the
displacement shown on the right. X a10 a11X
a12Y Y a20 a21X a22Y
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Rigid Translation X 0.5 1.0X 0Y Y
0.5 0X 1.0Y
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Displacement Vectors On the graph, draw
displacement vectors from the initial point to
the deformed point. How do you describe a vector
mathematically (analytically)? In words?
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Displacement Vectors What are some geologic
examples of rigid body translation vectors?
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Rigid Body Rotation X 0 Cos(q) X Sin(q) Y
Y 0 - Sin(q) X Cos(q) Y
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Structural Geology (3443)Ch. 2 Kinematic
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Rigid Body Rotation What are some geological
examples of rigid body rotation?
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Dilation (volume Change X 0 X 0 Y Y
0 0 X Y
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Dilation (volume Change X 0 X 0 Y Y
0 0 X Y What are some geologic examples
of volume change?
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Strain (Distortion) Pure Shear (no rigid
rotation, no volume change) X 0 a11X 0 Y
Y 0 0 X (1/a11)Y
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Strain (Distortion) Simple Shear (Includes rigid
rotation, no volume change) X 0 1 X 0 Y
Y 0 Tan(y) X 1Y
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Strain (distortion) The transformation equations
describe the displacement of individual points,
and refers mathematically to an infinitesimally
small volume of material in a continuum. In
general, the deformation changes from one
infinitesimally small volume to another.
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Strain (distortion) If the strain (distortion) is
the same over a large volume, it is called
homogeneous strain. Otherwise heterogeneous Defor
med worm tubes (right)
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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• Strain Definitions
• Two components of strain measurements
• Linear (Normal) strain
• Shear Strain

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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• Strain Definitions
• Linear (Normal) strain has several different
  definitions
• Engineering Strain e (l lo)/lo
• Stretch S l/lo e 1
• Quadratic Elongation l (l/lo)2 S2 (e
  1)2
• This is called Lagrangian Strain because it is
  referred to lo, the undeformed state as opposed
  to the deformed length (Eulerian strain).

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Structural Geology (3443)Ch. 2 Kinematic
Analysis
On the graph for Pure shear, calculate all three
measures of strain for lines parallel to the x
and y axes and for the diagonal.
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Strain Definitions The amount of horizontal
linear strain in crustal rocks can be estimated
from cross sections. Calculate engineering
strain and stretch for each cross section
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
Strain Definitions Shear Strain and Angular
Strain Angular strain (y) is the angular
deflection of two lines that were perpendicular
before deformation. If the deflection is
clockwise it is positive (). Counterclockwise is
negative (-). Shear strain (g) is the tangent of
that angle whether or - g Tan (y)
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Structural Geology (3443)Ch. 2 Kinematic
Analysis
g Tan (y) On the graph for simple shear,
calculate the angular strain and shear strain for
lines parallel to the x and y axes.
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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• Relationship between linear strain and
  transformation equations
• Neglecting translation
• X a11X a12Y
• Y a21X a22Y
• And
• L2 X2 Y2
• Lo2 X2 Y2
• So,
• l (L/Lo)2 (a112 a212)X2 2(a11a12
  a21a22)XY (a122 a222)Y2/X2 Y2

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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• Continuing,
• l (L/Lo)2 (a112 a212)X2 2(a11a12
  a21a22)XY (a122 a222)Y2/X2 Y2
• Dividing by X2, substituting Tan(q) Y/X (Where
  q is angle between X axis and line Lo) and Using
  trig identities, Can derive
• 1/2(a112 a212 a122 a222) (a11a12
  a21a22)Sin2q
• 1/2(a112 a212 - a122 - a222)Cos2q

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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• 1/2(a112 a212 a122 a222) (a11a12
  a21a22)Sin2q 1/2(a112 a212 - a122 -
  a222)Cos2q
• For simple shear (no rigid rotation), a21 a12
  0
• So,
• 1/2(a112 a222) 1/2(a112 - a222)Cos2q

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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• 1/2(a112 a222)
• 1/2(a112 - a222)Cos2q
• What is l when q 0 and 90?
• If l1 is the maximum strain and l2 is the minimum
  strain, correlate those strains with a112and a222

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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• So,
• 1/2(l1 l2)
• 1/2(l1 l2) Cos2q
• Where q is the angle between l1 and the line
  being strained.
• If l1 1.78 and l2 .5625 What is l for the
  diagonal line?

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Structural Geology (3443)Ch. 2 Kinematic
Analysis
The equivalent Equation for shear strain is g
(l1 l2)/2(l1l2 )1/2 Sin2q Where q is the
angle between l1 and the line being
strained. If l1 1.78 and l2 .5625 What is
g for the diagonal line?
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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• The Mohr Diagram for Strain
• Before Computers, it was important to find
  graphical solutions to messy equations. In 1882,
  Otto Mohr noted that these equations for strain
  were like parametric equations for circles and
  ellipses. He noted the similarity of
• 1/2(l1 l2) 1/2(l1 l2) Cos2q
• (l1 l2)/2(l1l2 )1/2 Sin2q
• To the equations for an ellipse
• X C R1 Cos(a)
• Y R2 Sin(a)

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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• The Mohr Diagram for Strain
• 1/2(l1 l2) 1/2(l1 l2) Cos2q
• (l1 l2)/2(l1l2 )1/2 Sin2q
• X C R1 Cos(a)
• Y R2 Sin(a)
• C 1/2(l1 l2)
• R1 1/2(l1 l2)
• R2 (l1 l2)/2(l1l2 )1/2
• a 2q

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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• The Mohr Diagram for Strain becomes a circle if
  there is no volume change l1 1/l2
• 1/2(l1 l2) 1/2(l1 l2) Cos2q
• (l1 l2)/2(l1l2 )1/2 Sin2q
• Then,
• 1/2(l1 l2) Sin2q

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If l1 1.78 and l2 .5625 What is g for q
30, 45, 60, and 90?
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Structural Geology (3443)Ch. 2 Kinematic
Analysis

• Incremental/Infinitesimal Strain,
  Finite/accumulated Strain, and Strain Path.
• Incremental /infinitesimal Strain represents a
  tiny amount of strain (The equations can be
  simplified)
• Finite/accumulated strain means a significant
  amount of strain has occurred. It is the strain
  that has accumulated from many small increments
  superposed on each other.
• Strain Path, is knowledge of all the increments
  that make up the finite strain. Not usually known.