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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
tanθ = 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 componentsin 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:
| s 1 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´ = 200Extension: 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 "finite" 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, S3)
.
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:
.
= 2 e.
[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