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Subjects   Force

           Stress

           Strain

           Deformation

           Rheology
 Mass:   Dimension: [M]                   Unit: g or kg
 Length: Dimension: [L]                   Unit: cm or m
 Time: Dimension: [T]                     Unit: s
Velocity, v = distance/time = dx/dt
    Change in distance per time)
  v =[L/T] or [LT-1] units: m/s or cm/s

Acceleration (due to gravity): g = velocity/time
 Acceleration is change in velocity per time (dv/dt).
   g = [LT-1 ]/[T] = LT-2, units: m s -2
Force:       F = mass . acceleration
 F = mg F = [M][LT-2]
 units: newton: N = kg m s-2
 A property or action that changes or tends to
 change the state of rest or velocity or direction
 of an object in a straight line

 In the absence of force, a body moves at
 constant velocity, or it stays at rest

 Force is a vector quantity; i.e., has magnitude,
 direction
 Gravitational force
   Acts over large distances and is always attractive
       Ocean tides are due to attraction between Moon & Earth
 Thermally-induced forces
   e.g., due to convection cells in the mantle.
   Produce horizontal forces (move the plates)

 The other three forces act only over short ranges
  (atomic scales). May be attractive or repulsive
    Electromagnetic force
       Interaction between charged particles (electrons)

   Nuclear or strong force
      Holds the nucleus of an atom together.

   Weak force
     Is responsible for radioactivity
 Any part of material experiences two types of
 forces:
   surface & body


 Body Force: Results from action of a field at
 every point within the body
   Is always present
   Could be due to gravity or inertia
      e.g., gravity, magnetic, centrifugal
   Its magnitude is proportional to the mass of the body
 Act on a specific surface area in a body
   Are proportional to the magnitude of the area
   Reflect pull or push of the atoms on one side of a
    surface against the atoms on the other side

      e.g., force of a cue stick that hits a pool ball
 Forces applied on a body do either or both of
 the following:
   Change the velocity of the body
   Result in a shape change of the body


 A given force applied by a sharp object (e.g.,
 needle) has a different effect than a similar force
 applied by a dull object (e.g., peg). Why?

 We need another measure called stress which
 reflect these effects
 A force acting on a small area such as the tip of a sharp nail,
    has a greater intensity than a flat-headed nail!
                     s = [MLT-2] / [L2]=[ML -1T-2]

         s = kg m-1 s-2 pascal (Pa) = newton/m2
   1 bar (non-SI) = 105 Pa ~ 1 atmosphere
   1 kb = 1000 bar = 108 Pa = 100 Mpa
   1Gpa = 109 Pa = 1000 Mpa = 10 kb
   P at core-mantle boundary is ~ 136 Gpa (at 2900 km)
   P at the center of Earth (6371 km) is 364 Gpa
 Tension: Stress acts _|_ to and away from a
 plane
   pulls the rock apart
   forms special fractures called joint
   may lead to increase in volume


 Compression: stress acts _|_ to and toward a
 plane
   squeezes rocks
   may decrease volume


 Shear: acts parallel to a surface
   leads to change in shape
 Force (F) across any of these planes can be resolved into two
  components: Shear stress: Fs , & normal stress: Fn, where:
              Fs = F sin θ         Fn = F cos θ
                        tan θ = Fs/Fn
 Smaller θ means smaller Fs
 Stress on an arbitrarily-oriented plane through a point, is
  not necessarily perpendicular to the that plane


 The stress (s) acting on a plane can be resolved into two
  components:
 Normal stress (sn)
   Component of stress perpendicular to the plane, i.e.,
    parallel to the normal to the plane
 Shear stress (ss) or t
   Components of stress parallel to the plane
   The average overburden pressure (i.e., lithostatic P) at
    the base of a 1 km thick rock column (i.e., z = 1 km),
    with density (r) of 2.5 gr/cm3 is 25 to 30 MPa


P = rgz                                    [ML -1T-2]
P = (2670 kg m-3)(9.81 m s-2)(103 m)
      = 26192700 kg m-1s-2 (pascal)
                          = 26 MPa
    The geopressure gradient:
       dP/dz  30 MPa/km  0.3 kb/km (kb = 100 MPa)
   i.e. P is  3 kb at a depth of 10 km
 Physical quantities, such as the density or
  temperature of a body, which in no way depend on
  direction
   are expressed as a single number
   e.g., temperature, density, mass
   only have a magnitude (i.e., are a number)
   are tensors of zero-order
 Some physical quantities are fully specified
  by a magnitude and a direction, e.g.:
 Force, velocity, acceleration, and
  displacement

 Vectors:
   relate one scalar to another scalar
   have magnitude and direction
   are tensors of the first-order
   have 1 subscript (e.g., vi) and 21 and 31 components in
    2D and 3D, respectively
 Some physical quantities require nine
  numbers for their full specification (in 3D)
 Stress, strain, and conductivity are examples
  of tensor

 Tensors:
   relate two vectors
   are tensors of second-order
   have 2 subscripts (e.g., sij); and 22 and 32 components
    in 2D and 3D, respectively
 The stress tensor matrix:
       | s11 s12       s13 |
sij = | s21 s22        s23 |
       | s31 s32       s33 |
 Can be simplified by choosing the coordinates so that they
  are parallel to the principal axes of stress:
       | s1      0      0 |
sij = | 0       s2      0 |
       |0       0      s3 |
 In this case, the coordinate planes only carry normal
  stress; i.e., the shear stresses are zero
 The s 1 , s2 , and s 3 are the major, intermediate, and
  minor principal stress, respectively
 s1>s3 ; principal stresses may be tensile or compressive
 A component of deformation dealing with shape
  and volume change
 Distance between some particles changes
 Angle between particle lines may change


 The quantity or magnitude of the strain is given by
  several measure based on change in:
 Length (longitudinal strain) - e
 Angle (angular or shear strain) - 
 Volume (volumetric strain) - ev
 Extension or Elongation, e: change in length per
  length

           e = (l´-lo) / lo = Dl/ lo [dimensionless]

 Where l´ and lo are the final and original lengths of a
  linear object
 Note: Shortening is negative extension (i.e., e < 0)
 e.g., e = - 0.2 represents a shortening of 20%


Example:
 If a belemnite of an original length (lo) of 10 cm is now 12
  cm (i.e., l´=12 cm), the longitudinal strain is positive, and
  e = (12-10)/10 * 100% which gives an extension, e = 20%
 Stretch: s = l´/lo = 1+e = l   [no dimension]

                        X =  l1 = s1
                        Y =  l2 = s2
                        Z =  l3 = s3

 These principal stretches represent the semi-length of
  the principal axes of the strain ellipsoid. For Example:

Given lo = 100 and l´ = 200
Extension: e = (l´-lo)/ lo = (200-100)/100 = 1 or 100%
Stretch: s = 1+e = l´/lo = 200/100 = 2

i.e., The line is stretched twice its original length!
 Gives the change of volume compared with its
  original volume
 Given the original volume is vo, and the final
  volume is v´, then the volumetric stain, ev is:
ev   =(v´-vo)/vo = dv/vo [no dimension]
 Any deformed rock has passed through a whole
 series of deformed states before it finally reached
 its final state of strain

 We only see the final product of this progressive
 deformation (finite state of strain)

 Progressive strain is the summation of small
 incremental distortion or infinitesimal strains
 Incremental strains are the increments of
 distortion that affect a body during deformation

 Finite strain represents the total strain
 experienced by a rock body

 If the increments of strain are a constant volume
 process, the overall mechanism of distortion is
 termed plane strain (i.e., one of the principal
 strains is zero; hence plane, which means 2D)

 Pure shear and simple shear are two end
 members of plane strain
 Distortion during a homogeneous strain leads to
  changes in the relative configuration of particles
   Material lines move to new positions
 In this case, circles (spheres, in 3D) become
  ellipses (ellipsoids), and in general, ellipses
  (ellipsoids) become ellipses (ellipsoids).

 Strain ellipsoid
   Represents the finite strain at a point (i.e., strain
    tensor)
   Is a concept applicable to any deformation, no matter
    how large in magnitude, in any class of material
 Series of strain increments, from the original state,
  that result in final, finite state of strain

 A final state of quot;finitequot; strain may be reached by a
  variety of strain paths

 Finite strain is the final state; incremental strains
  represent steps along the path
 We can think of the strain ellipse as the product
 of strain acting on a unit circle




 A convenient representation of the shape of the
  strain ellipse is the strain ratio
      Rs = (1+e1)/(1+e3) = S1/S3 = X/Z
 It is equal to the length of the long axis over the
  length of the short axis
 If a line parallel to the radius of a unit circle, makes a
  pre-deformation angle of  with respect to the long
  axis of the strain ellipse (X), it rotates to a new angle
  of ´ after strain

 The coordinates of the end point of the line on the
  strain ellipse (x´, z´) are the coordinates before
  deformation (x, z) times the principal stretches (S1,
  S 3)
.
 3D equivalent - the ellipsoid produced by
 deformation of a unit sphere

 The strain ellipsoids vary from axially symmetric
 elongated shapes –
 cigars and footballs - to
 axially shortened pancakes and cushions
 If the strain axes have the same orientation in the
  deformed as in undeformed state we describe the
  strain as a non-rotational (or irrotational) strain

 If the strain axes end up in a rotated position, then
  the strain is rotational
 An example of a non-rotational strain is pure shear
 - it's a pure strain with no dilation of the area of the
 plane

 An example of a rotational strain is a simple shear
1. Axially symmetric extension

 Extension in one principal direction (l1) and equal shortening
  in all directions at right angles (l2 and l3)
                            l1 > l2 = l3 < 1

 The strain ellipsoid is prolate spheroid or cigar shaped
2. Axially symmetric shortening

 This involves shortening in one principal direction (l3) and
  equal extension in all directions at right angles (l1 and l2 ).
                          l1 = l2 > 1 > l3

 Strain ellipsoid is oblate spheroid or pancake-shaped
The sides of the parallelogram will progressively lengthen as
  deformation proceeds but the top and bottom surfaces neither stretch
  nor shorten. Instead they maintain their original length, which is the
  length of the edge of the original cube
 In contrast to simple shear, pure shear is a three-
  dimensional constant-volume, irrotational, homogeneous
  flattening, which involves either plane strain or general
  strain.

 Lines of particles that are parallel to the principal axes of
  the strain ellipsoid have the same orientation before and
  after deformation
 It does not mean that the principal axes coincided
  in all increments!

 During homogeneous flattening a sphere is
  transformed into a pancake-like shape and a box is
  changed into a tablet or book-like form.
 During pure shear the sides of the cube that are
  parallel to the z-axis are shortened, while the lengths
  of the sides that are parallel to the x-axis increase. In
  contrast, the lengths of the sides of the cube that are
  parallel to the y-axis remain unchanged.
 When such geometrical changes occur during the
  transformation of a rock body to a distorted state then
  the mechanism of distortion is termed plane strain.
 Collective displacements of points in a body
 relative to an external reference frame

 Deformation describes the transformations from
 some initial to some final geometry

 Deformation of a rock body occurs in response to
 a force
   Deformation involves any one or a combination of
    the following four components:

   Ways that rocks respond to stress:
    1. Rigid Body Translation
    2. Rigid Body Rotation
    3. Distortion or Strain
    4. Dilation
.
 Distortion is a non-rigid body operation that involves
 the change in the spacing of points within a body of
 rock in such a way that the overall shape of the body
 is altered with or without a change in volume

 Changes of points in body relative to each other
   Particle lines may rotate relative to an external
    coordinate system
   Translation and spin are both zero
   Example: squeezing a paste


 In rocks we deal with processes that lead to both
 movement and distortion
 Dilation is a non-rigid body operation
 involving a change in volume

 Pure dilation:
   The overall shape remains the same
   Internal points of reference spread apart (+ev) or
    pack closer (-ev) together
   Line lengths between points become uniformly longer
    or shorter
 Though commonly confused with each other, strain
 is only synonymous with deformation if there
 has been distortion without any volume change,
 translation, or rotation

 Strain represents only one of four possible
  components involved in the overall deformation
  of a rock body where it has been transformed
  from its original position, size, and shape to
  some new location and configuration
 Strain describes the changes of points in a body
  relative to each other, or, in other words, the
  distortions a body undergoes
 The reference frame for strain is thus internal
 Originally straight lines remain straight


 Originally parallel lines remain parallel


 Circles (spheres) become ellipses (ellipsoids)
.
Heterogeneous strain affects non-rigid bodies in an
irregular, non-uniform manner and is sometimes referred
to as non-homogeneous or inhomogeneous strain


                                   Leads to distorted
                                     complex forms
 Viscous deformation is a function of time
 This means that strain accumulates over time
 Hence deformation is irreversible, i.e. strain is
   Non-recoverable
   Permanent
• Flow of water is an example of viscous behavior.
• For a constant stress, strain will increase
  linearly with time (with slope: s/h)
• Thus, stress is a function of strain and time!
                        s = he/t
 The terms elastic and plastic describe the nature of
 the material

 Brittle and ductile describe how rocks behave.


 Rocks are both elastic and plastic
 materials, depending on the rate of
 strain and the environmental
 conditions (stress, pressure,
 temperature), and we say that rocks are
 viscoelastic materials.
 Plasticity theory deals with the behavior of a
  solid.
 Plastic strain is continuous - the material
  does not rupture, and the strain is
  irreversible (permanent).
 Occurs above a certain critical stress
       (yield stress = elastic limit)
 where strain is no longer linear with stress

 Plastic strain is shear strain at constant
 volume, and can only be caused by shear
 stress
 Brittle rocks fail by fracture at less
 than 3-5% strain

 Ductile rocks are able to sustain, under
 a given set of conditions, 5-10% strain
 before deformation by fracturing
 Confining pressure, Pc


 Effective confining pressure, Pe
   Pore pressure, Pf is taken into account


 Temperature, T

                   .
 Strain rate, e
– Increasing T increases
  ductility by activating
  crystal-plastic processes
– Increasing T lowers the
  yield stress (maximum
  stress before plastic
  flow), reducing the
  elastic range
– Increasing T lowers the
  ultimate rock strength
 •Ductility: The % of strain that a rock can take
 without fracturing in a macroscopic scale
The time interval it
  takes to accumulate a
  certain amount of
  strain
Change of strain with
  time (change in length
  per length per time).
  Slow strain rate means
  that strain changes
  slowly with time
– How fast change in
  length occurs per unit
  time
 Shear strain rate:

                 .        .
                =2e                 [T-1]

 Typical geological strain rates are on the order of
 10-12 s-1 to 10-15 s-1

 Strain rate of meteorite impact is on the order of
 102 s-1 to 10-4 s-1
 Decreasing strain rate:
   decreases rock strength
   increases ductility


                 .
 Effect of slow e is analogous to increasing T


 Think about pressing vs. hammering a silly putty


 Rocks are weaker at lower strain rates
 Slow deformation allows diffusional crystal-plastic
 processes to more closely keep up with applied
 stress
• Increasing
 confining
 pressure:
• Greater amount of
 strain accumulates
 before failure
• i.e., increases
 ductility
       –increases the viscous component and
       enhances flow
       –resists opening of fractures
          •i.e., decreases elastic strain
• Increasing pore fluid pressure
   – reduces rock strength
   – reduces ductility
    •   The combined reduced ductility and strength
        promotes flow under high pore fluid pressure

    •   Under ‘wet’ conditions, rocks deform more
        readily by flow

  – Increasing pore fluid pressure is analogous to
   decreasing confining pressure
 Rupture Strength (breaking strength)
   Stress necessary to cause rupture at room
    temperature and pressure in short time
    experiments

 Fundamental Strength
   Stress at which a material is able to withstand,
    regardless of time, under given conditions of T,
    P and presence of fluids without fracturing or
    deforming continuously

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force Stress strain deformation

  • 1. Subjects Force Stress Strain Deformation Rheology
  • 2.
  • 3.  Mass: Dimension: [M] Unit: g or kg  Length: Dimension: [L] Unit: cm or m  Time: Dimension: [T] Unit: s Velocity, v = distance/time = dx/dt  Change in distance per time) v =[L/T] or [LT-1] units: m/s or cm/s Acceleration (due to gravity): g = velocity/time  Acceleration is change in velocity per time (dv/dt). g = [LT-1 ]/[T] = LT-2, units: m s -2 Force: F = mass . acceleration  F = mg F = [M][LT-2]  units: newton: N = kg m s-2
  • 4.  A property or action that changes or tends to change the state of rest or velocity or direction of an object in a straight line  In the absence of force, a body moves at constant velocity, or it stays at rest  Force is a vector quantity; i.e., has magnitude, direction
  • 5.  Gravitational force  Acts over large distances and is always attractive  Ocean tides are due to attraction between Moon & Earth  Thermally-induced forces  e.g., due to convection cells in the mantle.  Produce horizontal forces (move the plates)  The other three forces act only over short ranges (atomic scales). May be attractive or repulsive  Electromagnetic force  Interaction between charged particles (electrons)  Nuclear or strong force  Holds the nucleus of an atom together.  Weak force  Is responsible for radioactivity
  • 6.  Any part of material experiences two types of forces:  surface & body  Body Force: Results from action of a field at every point within the body  Is always present  Could be due to gravity or inertia  e.g., gravity, magnetic, centrifugal  Its magnitude is proportional to the mass of the body
  • 7.  Act on a specific surface area in a body  Are proportional to the magnitude of the area  Reflect pull or push of the atoms on one side of a surface against the atoms on the other side  e.g., force of a cue stick that hits a pool ball
  • 8.  Forces applied on a body do either or both of the following:  Change the velocity of the body  Result in a shape change of the body  A given force applied by a sharp object (e.g., needle) has a different effect than a similar force applied by a dull object (e.g., peg). Why?  We need another measure called stress which reflect these effects
  • 9.
  • 10.
  • 11.  A force acting on a small area such as the tip of a sharp nail, has a greater intensity than a flat-headed nail! s = [MLT-2] / [L2]=[ML -1T-2] s = kg m-1 s-2 pascal (Pa) = newton/m2  1 bar (non-SI) = 105 Pa ~ 1 atmosphere  1 kb = 1000 bar = 108 Pa = 100 Mpa  1Gpa = 109 Pa = 1000 Mpa = 10 kb  P at core-mantle boundary is ~ 136 Gpa (at 2900 km)  P at the center of Earth (6371 km) is 364 Gpa
  • 12.  Tension: Stress acts _|_ to and away from a plane  pulls the rock apart  forms special fractures called joint  may lead to increase in volume  Compression: stress acts _|_ to and toward a plane  squeezes rocks  may decrease volume  Shear: acts parallel to a surface  leads to change in shape
  • 13.  Force (F) across any of these planes can be resolved into two components: Shear stress: Fs , & normal stress: Fn, where: Fs = F sin θ Fn = F cos θ tan θ = Fs/Fn  Smaller θ means smaller Fs
  • 14.  Stress on an arbitrarily-oriented plane through a point, is not necessarily perpendicular to the that plane  The stress (s) acting on a plane can be resolved into two components:  Normal stress (sn)  Component of stress perpendicular to the plane, i.e., parallel to the normal to the plane  Shear stress (ss) or t  Components of stress parallel to the plane
  • 15.
  • 16. The average overburden pressure (i.e., lithostatic P) at the base of a 1 km thick rock column (i.e., z = 1 km), with density (r) of 2.5 gr/cm3 is 25 to 30 MPa P = rgz [ML -1T-2] P = (2670 kg m-3)(9.81 m s-2)(103 m) = 26192700 kg m-1s-2 (pascal) = 26 MPa   The geopressure gradient: dP/dz  30 MPa/km  0.3 kb/km (kb = 100 MPa)  i.e. P is  3 kb at a depth of 10 km
  • 17.  Physical quantities, such as the density or temperature of a body, which in no way depend on direction  are expressed as a single number  e.g., temperature, density, mass  only have a magnitude (i.e., are a number)  are tensors of zero-order
  • 18.  Some physical quantities are fully specified by a magnitude and a direction, e.g.:  Force, velocity, acceleration, and displacement  Vectors:  relate one scalar to another scalar  have magnitude and direction  are tensors of the first-order  have 1 subscript (e.g., vi) and 21 and 31 components in 2D and 3D, respectively
  • 19.  Some physical quantities require nine numbers for their full specification (in 3D)  Stress, strain, and conductivity are examples of tensor  Tensors:  relate two vectors  are tensors of second-order  have 2 subscripts (e.g., sij); and 22 and 32 components in 2D and 3D, respectively
  • 20.
  • 21.  The stress tensor matrix: | s11 s12 s13 | sij = | s21 s22 s23 | | s31 s32 s33 |  Can be simplified by choosing the coordinates so that they are parallel to the principal axes of stress: | s1 0 0 | sij = | 0 s2 0 | |0 0 s3 |  In this case, the coordinate planes only carry normal stress; i.e., the shear stresses are zero  The s 1 , s2 , and s 3 are the major, intermediate, and minor principal stress, respectively  s1>s3 ; principal stresses may be tensile or compressive
  • 22.
  • 23.
  • 24.  A component of deformation dealing with shape and volume change  Distance between some particles changes  Angle between particle lines may change  The quantity or magnitude of the strain is given by several measure based on change in:  Length (longitudinal strain) - e  Angle (angular or shear strain) -   Volume (volumetric strain) - ev
  • 25.
  • 26.  Extension or Elongation, e: change in length per length e = (l´-lo) / lo = Dl/ lo [dimensionless]  Where l´ and lo are the final and original lengths of a linear object  Note: Shortening is negative extension (i.e., e < 0)  e.g., e = - 0.2 represents a shortening of 20% Example:  If a belemnite of an original length (lo) of 10 cm is now 12 cm (i.e., l´=12 cm), the longitudinal strain is positive, and e = (12-10)/10 * 100% which gives an extension, e = 20%
  • 27.
  • 28.  Stretch: s = l´/lo = 1+e = l [no dimension] X =  l1 = s1 Y =  l2 = s2 Z =  l3 = s3  These principal stretches represent the semi-length of the principal axes of the strain ellipsoid. For Example: Given lo = 100 and l´ = 200 Extension: e = (l´-lo)/ lo = (200-100)/100 = 1 or 100% Stretch: s = 1+e = l´/lo = 200/100 = 2 i.e., The line is stretched twice its original length!
  • 29.  Gives the change of volume compared with its original volume  Given the original volume is vo, and the final volume is v´, then the volumetric stain, ev is: ev =(v´-vo)/vo = dv/vo [no dimension]
  • 30.
  • 31.
  • 32.  Any deformed rock has passed through a whole series of deformed states before it finally reached its final state of strain  We only see the final product of this progressive deformation (finite state of strain)  Progressive strain is the summation of small incremental distortion or infinitesimal strains
  • 33.  Incremental strains are the increments of distortion that affect a body during deformation  Finite strain represents the total strain experienced by a rock body  If the increments of strain are a constant volume process, the overall mechanism of distortion is termed plane strain (i.e., one of the principal strains is zero; hence plane, which means 2D)  Pure shear and simple shear are two end members of plane strain
  • 34.  Distortion during a homogeneous strain leads to changes in the relative configuration of particles  Material lines move to new positions  In this case, circles (spheres, in 3D) become ellipses (ellipsoids), and in general, ellipses (ellipsoids) become ellipses (ellipsoids).  Strain ellipsoid  Represents the finite strain at a point (i.e., strain tensor)  Is a concept applicable to any deformation, no matter how large in magnitude, in any class of material
  • 35.  Series of strain increments, from the original state, that result in final, finite state of strain  A final state of quot;finitequot; strain may be reached by a variety of strain paths  Finite strain is the final state; incremental strains represent steps along the path
  • 36.
  • 37.
  • 38.
  • 39.  We can think of the strain ellipse as the product of strain acting on a unit circle  A convenient representation of the shape of the strain ellipse is the strain ratio Rs = (1+e1)/(1+e3) = S1/S3 = X/Z  It is equal to the length of the long axis over the length of the short axis
  • 40.  If a line parallel to the radius of a unit circle, makes a pre-deformation angle of  with respect to the long axis of the strain ellipse (X), it rotates to a new angle of ´ after strain  The coordinates of the end point of the line on the strain ellipse (x´, z´) are the coordinates before deformation (x, z) times the principal stretches (S1, S 3)
  • 41.
  • 42. .
  • 43.  3D equivalent - the ellipsoid produced by deformation of a unit sphere  The strain ellipsoids vary from axially symmetric elongated shapes – cigars and footballs - to axially shortened pancakes and cushions
  • 44.  If the strain axes have the same orientation in the deformed as in undeformed state we describe the strain as a non-rotational (or irrotational) strain  If the strain axes end up in a rotated position, then the strain is rotational
  • 45.  An example of a non-rotational strain is pure shear - it's a pure strain with no dilation of the area of the plane  An example of a rotational strain is a simple shear
  • 46.
  • 47.
  • 48.
  • 49. 1. Axially symmetric extension  Extension in one principal direction (l1) and equal shortening in all directions at right angles (l2 and l3) l1 > l2 = l3 < 1  The strain ellipsoid is prolate spheroid or cigar shaped 2. Axially symmetric shortening  This involves shortening in one principal direction (l3) and equal extension in all directions at right angles (l1 and l2 ). l1 = l2 > 1 > l3  Strain ellipsoid is oblate spheroid or pancake-shaped
  • 50. The sides of the parallelogram will progressively lengthen as deformation proceeds but the top and bottom surfaces neither stretch nor shorten. Instead they maintain their original length, which is the length of the edge of the original cube
  • 51.  In contrast to simple shear, pure shear is a three- dimensional constant-volume, irrotational, homogeneous flattening, which involves either plane strain or general strain.  Lines of particles that are parallel to the principal axes of the strain ellipsoid have the same orientation before and after deformation  It does not mean that the principal axes coincided in all increments!  During homogeneous flattening a sphere is transformed into a pancake-like shape and a box is changed into a tablet or book-like form.
  • 52.  During pure shear the sides of the cube that are parallel to the z-axis are shortened, while the lengths of the sides that are parallel to the x-axis increase. In contrast, the lengths of the sides of the cube that are parallel to the y-axis remain unchanged.  When such geometrical changes occur during the transformation of a rock body to a distorted state then the mechanism of distortion is termed plane strain.
  • 53.
  • 54.  Collective displacements of points in a body relative to an external reference frame  Deformation describes the transformations from some initial to some final geometry  Deformation of a rock body occurs in response to a force
  • 55. Deformation involves any one or a combination of the following four components:  Ways that rocks respond to stress: 1. Rigid Body Translation 2. Rigid Body Rotation 3. Distortion or Strain 4. Dilation
  • 56. .
  • 57.
  • 58.  Distortion is a non-rigid body operation that involves the change in the spacing of points within a body of rock in such a way that the overall shape of the body is altered with or without a change in volume  Changes of points in body relative to each other  Particle lines may rotate relative to an external coordinate system  Translation and spin are both zero  Example: squeezing a paste  In rocks we deal with processes that lead to both movement and distortion
  • 59.
  • 60.  Dilation is a non-rigid body operation involving a change in volume  Pure dilation:  The overall shape remains the same  Internal points of reference spread apart (+ev) or pack closer (-ev) together  Line lengths between points become uniformly longer or shorter
  • 61.
  • 62.  Though commonly confused with each other, strain is only synonymous with deformation if there has been distortion without any volume change, translation, or rotation  Strain represents only one of four possible components involved in the overall deformation of a rock body where it has been transformed from its original position, size, and shape to some new location and configuration  Strain describes the changes of points in a body relative to each other, or, in other words, the distortions a body undergoes  The reference frame for strain is thus internal
  • 63.
  • 64.  Originally straight lines remain straight  Originally parallel lines remain parallel  Circles (spheres) become ellipses (ellipsoids)
  • 65. .
  • 66.
  • 67. Heterogeneous strain affects non-rigid bodies in an irregular, non-uniform manner and is sometimes referred to as non-homogeneous or inhomogeneous strain Leads to distorted complex forms
  • 68.
  • 69.
  • 70.  Viscous deformation is a function of time  This means that strain accumulates over time  Hence deformation is irreversible, i.e. strain is  Non-recoverable  Permanent • Flow of water is an example of viscous behavior. • For a constant stress, strain will increase linearly with time (with slope: s/h) • Thus, stress is a function of strain and time! s = he/t
  • 71.  The terms elastic and plastic describe the nature of the material  Brittle and ductile describe how rocks behave.  Rocks are both elastic and plastic materials, depending on the rate of strain and the environmental conditions (stress, pressure, temperature), and we say that rocks are viscoelastic materials.
  • 72.  Plasticity theory deals with the behavior of a solid.  Plastic strain is continuous - the material does not rupture, and the strain is irreversible (permanent).  Occurs above a certain critical stress (yield stress = elastic limit)  where strain is no longer linear with stress  Plastic strain is shear strain at constant volume, and can only be caused by shear stress
  • 73.  Brittle rocks fail by fracture at less than 3-5% strain  Ductile rocks are able to sustain, under a given set of conditions, 5-10% strain before deformation by fracturing
  • 74.  Confining pressure, Pc  Effective confining pressure, Pe  Pore pressure, Pf is taken into account  Temperature, T .  Strain rate, e
  • 75. – Increasing T increases ductility by activating crystal-plastic processes – Increasing T lowers the yield stress (maximum stress before plastic flow), reducing the elastic range – Increasing T lowers the ultimate rock strength •Ductility: The % of strain that a rock can take without fracturing in a macroscopic scale
  • 76. The time interval it takes to accumulate a certain amount of strain Change of strain with time (change in length per length per time). Slow strain rate means that strain changes slowly with time – How fast change in length occurs per unit time
  • 77.  Shear strain rate: . .  =2e [T-1]  Typical geological strain rates are on the order of 10-12 s-1 to 10-15 s-1  Strain rate of meteorite impact is on the order of 102 s-1 to 10-4 s-1
  • 78.  Decreasing strain rate:  decreases rock strength  increases ductility .  Effect of slow e is analogous to increasing T  Think about pressing vs. hammering a silly putty  Rocks are weaker at lower strain rates  Slow deformation allows diffusional crystal-plastic processes to more closely keep up with applied stress
  • 79. • Increasing confining pressure: • Greater amount of strain accumulates before failure • i.e., increases ductility –increases the viscous component and enhances flow –resists opening of fractures •i.e., decreases elastic strain
  • 80. • Increasing pore fluid pressure – reduces rock strength – reduces ductility • The combined reduced ductility and strength promotes flow under high pore fluid pressure • Under ‘wet’ conditions, rocks deform more readily by flow – Increasing pore fluid pressure is analogous to decreasing confining pressure
  • 81.  Rupture Strength (breaking strength)  Stress necessary to cause rupture at room temperature and pressure in short time experiments  Fundamental Strength  Stress at which a material is able to withstand, regardless of time, under given conditions of T, P and presence of fluids without fracturing or deforming continuously