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)
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
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)
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
