6. Programa 2013: activitats prioritàries
6.1. Conservació i tractament físic del territori
Normal (σn) and shear (σs) components of stress on a real or imaginary plane are important for the understanding of theories of development of faults and joints. The equations of normal and shear stresses in terms of principal stresses (σ1-σ2, σ1-σ3 or σ2 -σ3 space) are most useful in understanding the basic concept of stress. Finally, pictorial representation of how normal and shear stresses vary with the change in orientation of the plane with respect to principal stresses is very illustrative; this we do with the help of a diagram called Mohr circle.
The equations for the normal and shear stresses can be derived from force-balance problems, which assumes that if a body is in equilibrium (i.e., it does not move or spin) then all the forces in any one direction sum to zero. Note that we should always balance forces and not stresses. We should first find the forces acting in any particular direction, determine stresses in terms of the forces, and then derive the expressions for normal and shear stresses.
Consider a prismatic body in the earth with two of the bounding planes oriented parallel to the maximum and minimum force vectors (Fig.4.1). Another bounding plane P has an area A and whose orientation may be specified by the angle θ made by the perpendicular to the plane with maximum (F1) force direction. Therefore, the areas of the left and right faces of the prism are Asinθ and Acosθ, respectively. Resolution of the magnitudes (F1 and F2) of forces acting on the three bounding planes is shown in Fig.
4.1. Resolved forces are parallel and perpendicular to the plane P.
Forces trying to push the prismatic body upward must be equal to the forces trying to push the prismatic body downward, if the body has to remain in equilibrium.
Therefore, from the balance of forces acting across the plane P (Fig. 4.1):
F1n + F3n = F1 cosθ + F3 sinθ
or, Fn = F1 cosθ + F3 sinθ (4.1)
And, from balance of forces acting parallel to the plane P we obtain:
F1s + F3 cosθ = F3s + F1 sinθ
Figure 4.1. A prismatic body bound by planes parallel to F1 and F3 directions, and a plane P with area A and whose normal is inclined at θ to the F1 direction.
As the body is at rest, forces acting in any one direction must sum to zero. Force trying to move the prism from left to right is exactly balanced by the force acting from right to left and force trying to move the body from top to bottom is balanced by the force acting from bottom to top. Resolution of magnitudes of F1
and F3 forces on the three bounding surfaces are shown on the figure.
We now calculate the stresses in terms forces from the relation, stress = force / area:
σn = Fn / A σs = Fs / A
or, Fn = A σn Fs = A σs (4.3)
and
σ1 = F1 / (A cosθ) σ3 = F3 / (A sinθ)
or F1 = σ1 A cosθ F3 = σ3 A sinθ (4.4)
Replacing (eq. 4.3) and (eq. 4.4) in (eq. 4.1) and (eq. 4.2), we obtain
σn = σ1 cos2θ + σ3 sin2θ (4.5)
σs = σ1 sinθ cosθ + σ3 sinθ cosθ (4.6)
Using the identities cos2θ = (1 + cos2θ)/2, sin2θ = (1 - cos2θ)/2 and sinθ cosθ = sin2θ in eqs. 4.5 and 4.6, and collecting terms we obtain
σn = ½ (σ1 + σ3) + ½ (σ1 - σ3) cos2θ (4.7) σ3)/2, respectively. This circle is known as the Mohr circle for stress.
σs, bars
Figure 4.2. Construction of a Mohr circle. (a) A very small rectangular block showing state of stress and orientations of two planes on which normal and shear stresses are to be determined using Mohr circle. (b) Mohr circle for the state of stress as shown in (a). See text for details.
Let us consider a very small rectangular block in which there is a plane P inclined at 70° from the horizontal. σ1 is vertical and has a magnitude of 325 bars, and σ3 is horizontal with a magnitude of 75 bars. A line perpendicular to plane P makes 70° with σ1 direction, i.e., θ is 70° (Fig. 4.2a). We want to show pictorially the magnitudes of σn and σs on this plane; we can, of course, calculate the same using equations 4.7 and 4.8. A pair of orthogonal coordinate axes is drawn with σn and σs
as horizontal and vertical axes, respectively (Fig. 4.2b). The two axes are graduated in units of bars with same scale on both the axes. On the positive side of σn, points are located at 325 and 75 bars and marked as σ1 and σ3, respectively. We locate a point half way between σ1 and σ3, i.e., at 200 bars, i.e., (σ1 + σ3)/2. Taking this point as center we draw a circle that passes through both σ1 and σ3. The resultant circle is Mohr circle of stress for a state of stress with magnitudes of σ1 and σ3 at 325 and 75 bars, respectively. We find a point P on the Mohr circle such that a line drawn from P to the center makes an angle of 140° (i.e., 2θ) from the positive end of the σn axis.
The coordinates of point P are 80.3 and 104.2 bars, which are the σs and σn, respectively. The plane Q (Fig. 4.2a) inclined at 20° (θ = 70°) from horizontal has σs
and σn values of 80.3 and 295.8 bars, respectively. So it seems that every point on Mohr circle represents a plane and the coordinates of the point equal normal and shear stresses associated with the plane. Therefore, the circle is the loci of infinite number of points, each of which represents stress on a plane whose orientation is specified by θ.
Fig. 4.3 shows how the different terms in equations 4.7 and 4.8 are related to coordinates of point P on the Mohr circle. Figs. 4.2, 4.3 are drawn for the principal plane of stress containing σ1-σ3; this is two dimensional. We can also draw Mohr circles for σ1-σ2 and σ2-σ3 principal planes. The three Mohr circles in Fig. 4.4 represent three-dimensional state of stress. The largest, intermediate and smallest circles represent Mohr circles for σ1-σ3, σ1-σ2 and σ2-σ3 surfaces, respectively.
+σs
σs σn
−σn +σn
−σs
2θ σ1
σ3
1/2(σ −σ ) 1 3 cos2θ
1/2(σ −σ )1 3 1/2(σ −σ )1 3 sin2θ σ = [(σ + σ )/2 + (σ − σ )/2] n 1 3 1 3 cos2θ
P
1/2(σ σ )1+ 3
Figure 4.3. Mohr circle showing significance different terms in equations 4.7 and 4.8.
σs
−σs
σn
−σn σ3 σ2 σ1
σ σ1- 3 σ σ1- 2 σ σ2- 3
Figure 4.4. Mohr circles for state of stress in three dimensions. The three circles represent three principal planes of stress.
Sign conventions for the Mohr diagram are as follows: compressive normal stresses and shear stresses represented by counterclockwise or sinistral pair of arrows are positive and tensile normal stresses and shear stresses represented by clockwise or dextral pair of arrows are negative (Figs. 4.2, 4.3); θ is positive for planes whose normal can be located towards counterclockwise direction from σ1, otherwise θ is negative (Fig.
4.5).
Plane P the largest absolute magnitude of shearing stress, which is equal to the radius of the Mohr circle [or, (σ1 - σ3)/2 for σ1-σ3 principal plane]. Shear stresses for planes oriented perpendicular to any of the principal stress directions are zero. It may sound logical to presume that shear fractures (i.e., faults) should form at ± 45° to σ1. As we will see (section 7) shear fractures usually develop at angles less than this.
σs
Figure 4.6. Maximum shearing stress is possible on planes oriented at ± 45° (2θ
= ± 90°) to σ1 axis.
Different classes of two-dimensional state of stress at a point are shown in Fig.
4.7. In hydrostatic tension (lithostatic tension for rocks), stress across all planes is tensile and equal and the Mohr circle is a point on the negative side of the σn axis (Fig. 4.7a). If the stress across all planes is compressive and equal, the state of stress may be called hydrostatic compression and the Mohr circle is a point on the positive side of the σn axis (Fig. 4.7b). The term “hydrostatic” refers to the stress experienced by a fluid at rest. However, this term is also widely used to describe similar state of
stress in solid. Both the principal stresses can be either positive or negative and the states of stress are general tension (Fig. 4.7c) or general compression (Fig. 4.7d), respectively. In uniaxial tension only one principal stress is non-zero and it is tensile (Fig. 4.7e), whereas in uniaxial compression only one principal stress is non-zero and it is compressive (Fig. 4.7f). In many states of stress one principal stress is tensile and the other principal stress is compressive (Fig. 4.7g,h,i). Pure shear stress is a special class of state of stress where the two principal stresses have the same magnitude but opposite sign (Fig. 4.7i). In such cases, planes of maximum shear stress are also planes of pure shear, i.e., normal stresses across these planes are zero.
Except for lithostatic tension (Fig. 4.7a), all other classes of stress are possible in the earth.
a b
c e g f d
σs
−σs
σn
−σn
h i
Figure 4.7. Mohr circles for various types of state of stress in two dimensions.
Any non-hydrostatic state of stress, either in two-dimensions or in three-dimensions, can be decomposed into two parts: a mean stress (σm) and a stress deviator or deviatoric stress (σd). The mean stress is the average of the principal stresses:
σm = (σ1 + σ2)/2 in two dimensions, and σm = (σ1 + σ2 + σ3)/3 in three dimensions.
The deviatoric stress is defined as:
σd = σn - σm
and is a measure of how much the normal stress in any direction deviates from the mean or hydrostatic stress. Along three principal stress directions we have three principal deviatoric stresses whose magnitudes are given by:
σ1d = σ1 - σm
σ2d = σ2 - σm
σ3d = σ3 - σm
Note that sum of the right hand terms in the above equations is zero. For example, a state of stress with the values of three principal stresses 750, 1050, and 1560 bars can be thought of as combining a mean stress of 1120 bars and three deviatoric stresses of –370, -70, and 440 bars. Mean stress in two-dimension is a state of hydrostatic tension or compression and locates the center of the Mohr circle;
deviatoric stress is a pure shear stress. Given mean stress and deviatoric stress we can construct the Mohr circle.
The deviatoric stress is responsible for distortion (or strain) in a body. The distortion may be elastic (i.e., reversible) or plastic (i.e., permanent). Most deformations are result of differential stress rather than the absolute magnitudes of principal stresses, except for dilation. Differential stress is the difference between the magnitudes of maximum and minimum principal stresses (σ1 - σ3). The mean stress can be thought of as hydrostatic (or isotropic) part of the stress system and causes only volumetric changes (dilation) in the material. Mean stress also controls the strength of materials. For example, fracturing is inhibited with increasing mean stress.
5. Strain
5.1. Deformation
When force is applied to a rock body, the particles within the rock body are displaced and the rock is said to be deformed. Two types of deformations are usually recognized (Fig. 5.1).
• Homogeneous deformation: If particles arranged in straight lines in an undeformed rock remain so after deformation, then the deformation is called homogeneous. Another definition of homogeneous deformation is that all parallel lines of particles remain parallel after deformation.
• Inhomogeneous deformation: In this type of deformation, straight lines of particles become curved after deformation and parallel lines of particles loose their parallelism after deformation.
(a) (b) (c)
Figure 5.1. Homogeneous and inhomogeneous deformations. (a) Original undeformed grid. (b) Homogeneous deformation wherein straight lines remain straight and parallel lines remain parallel after deformation. (c) If straight lines become curved and parallel lines loose their parallelism, the deformation is inhomogeneous.
Mother Nature does not draw a grid, as in Fig. 5.1a, before deforming a rock for our convenience! In some cases, however, Nature preserves features that can be used to determine type and amount of deformation. For example, the branches in plant fossil
Neuroteris are approximately straight and parallel before deformation (Fig. 5.2a). The straightness and parallelism may be preserved (Fig. 5.2b) or destroyed (Fig. 5.2b) after deformation indicating homogeneous or inhomogeneous nature of deformation, respectively.
(a)
(b)
(c)
Figure 5.2. Undeformed (a) plant fossil Neuroteris showing homogeneous (b) and inhomogeneous (c) deformations
Four independent geometric processes contribute to the total displacement of particles during deformation:
• Rigid-body translation is the movement of the entire body through space in such a way that the shape does not change. The movement vectors for all the particles in any external coordinate system have the same orientation and magnitude (Fig. 5.3a).
• Rigid-body rotation also involves movement without any change in shape.
However, in this case the body rotates about a single point, which is fixed with respect to an external reference frame (Fig. 5.3b).
• Distortion produces change in the shape of the body due to movement of particles with respect to each other (Fig. 5.3c).
• Volume change, as the term implies, change in volume of the body without any change in shape. Volume change is also called dilation, although volume change can be either positive or negative (Fig. 5.3d).
(a) (b) (c)
(d) (e)
Figure 5.3. Trilobite Phillipsia Geometric processes leading to different types of deformations, as shown by trilobite Phillipsia. (a) Rigid-body translation. (b) Rigid-body rotation. (c) Distortion, i.e., change in shape. (d) Dilation, i.e., volume change. (e) A combination of all the four types of deformation.
Note that the descriptions of distortion and volume change do not require an external reference frame. The four processes are not mutually exclusive – Fig. 5.3e shows a deformation which includes all the four processes. Distortion and dilation together make up strain, which involves movement of particles relative to each other. In a more quantitative sense, strain is a mathematical description that relates the size and shape of a body before and after deformation. A rock body may undergo rigid-body movement, either translation or rotation or both, but it is almost impossible to determine the exact amount of rigid-body movement. However, the strain in a rock can be precisely determined if the rock contains objects of known, original shape and/or size (e.g., Figs. 5.2, 5.3. Note that deformation and distortion (and strain) are not exactly the same although it is not uncommon to find them used interchangeably in the literature. Further it is important to remember that translation/rotation of a rock body may or may not be accompanied by internal strain, i.e., distortion/dilation.
There are several ways strain can be measured but all of them involve measurement of some kind of change from an initial undeformed to a final deformed state. The changes that are generally measured are changes in lines, angles and volume.
5.2 Change in line length
Changes in line lengths, called longitudinal strain, can be measured in different ways, viz., extension (e), stretch (S), quadratic elongation (λ), and logarithmic or natural strain (ε). If Li is the initial undeformed length of a line, Ld is the final deformed length of the same line, and ∆L is the change in the length of the line then (Fig. 5.4):
e = (Ld – Li) / Li = ∆L / Li
S = Ld / Li = (1 + e) λ = (Ld / Li)2 = (1 + e)2
ε = loge (Ld / Li) = loge (1 + e)
Elongation can be either positive or negative depending on whether a line has extended or shortened. Stretch is always positive whether a line has extended or shortened. It has a value of 1.0 if there is no change in length of a line, S < 1.0 for shortening and S > 1.0 for extension. All the four parameters for longitudinal strain are dimensionless. They are not independent, if we know one we can calculate the others;
which one to use for a particular problem depends entirely on convenience. However, logarithmic strain is realistic for several reasons. For example, if one line contracts to half of its original length and another line expands twice its original length, the elongations of the lines are 0.5 and 1.0, respectively. For the same deformed lines logarithmic strains are -loge2 and + loge2. For very large shortening of a line, elongation tends towards –1.0, but the logarithmic strain approaches -∝. The stress-strain curves for isotropic materials are straight (i.e., linear) if logarithmic strain is used.
In nature we almost always measure Ld from deformed linear objects, such as boudinaged quartz vein (Fig. 5.4b). We can put the boudins back into their original position and determine Li, assuming no volume change. Note that in Fig. 5.4a the length of the line has decreased (shortened) but the line in Fig. 5.4b has increased (extended). In both the cases the change in the length of the line is 2.46 mm but parameters describing longitudinal strain are different.
Undeformed (length Li)
Deformed (length Ld)
∆L
(a)
Deformed (length Ld)
Undeformed (length Li) (b)
∆L
2 cm
Figure 5.4. Longitudinal strain. (a) A line with initial length Li = 9.15 cm has shortened to 6.69 (= Ld). The elongation (e), stretch (S), quadratic elongation (λ), and natural strain (ε) are -0.27, 0.73, 0.53 and -0.31, respectively. (b) A more practical scenario wherein boudinaged quartz vein can be used to determine both Li and Ld. The e, S, λ and ε are 0.37, 1.37, 1.87 and 0.31, respectively.
5.3 Change in angle
Shear strain is a measure of change in angle between two originally perpendicular lines. Consider a rectangle ABCD, which after deformation becomes a parallelogram A’B’CD (Fig. 5.5). The angle between lines AD and CD has changed from 90° (∠ADC) before deformation to α (∠A’DC) after deformation. We can state the shear strain in two different ways (Fig. 5.5):
• Angular shear (ψ): This gives the change in angle, i.e., ψ = 90° - α.
• Shear strain (γ): This represents displacement (distance x) of a particle at a distance y from a particle that does not move. From Fig. 5.5:
γ= tanψ = x/y or, x = y tanψ = y γ if y is unit distance then x = γ = tanψ
Shear strain can be determined if appropriate markers are present in rocks. For example, we can determine shear strain from deformed trilobite Phillipsia because of inherent bilateral symmetry (Fig. 5.6). Similarly, well-preserved worm burrows or mud cracks and stratification surfaces can be used to determine shear strain.
The original perpendicular line (e.g., line AD in Fig. 5.5) may move either in a clockwise direction or in an anticlockwise direction with respect to the original
orientation. Clockwise and anticlockwise shear strains are given negative and positive signs, respectively.
α ψ
x
y
A B
C D
A’ B’
Figure 5.5. Shear strain illustrated by a rectangle ABCD that has changed into a parallelogram A’B’CD after deformation. ∠ADA’ or ∠BCB’ is the angular shear (ψ). Shear strain, γ = tanψ = x/y. If y is of unit length, γ = x.
α ψ
(a) (b)
Figure 5.6. Trilobite Phillipsia shows shear strain. Morphology of Phillipsia is such that two mutually perpendicular imaginary lines can be drawn (a). These lines can be used to determine shear strain from a deformed fossil.
5.4 Change in volume
Volumetric strain or change in volume during deformation is called dilation (∆). If Vi is the initial volume and Vd is the volume after deformation and ∆V is change in volume after deformation, then dilation is given by:
∆ = (Vd - Vi) / Vi = ∆V / Vi
Although to dilate is to enlarge, dilation can have positive (i.e., enlarge) or negative (i.e., contract) values. In two dimensions, we can only determine change in area.
5.5 Strain ellipse and ellipsoid
Strain ellipse (in 2D) and ellipsoid (in 3D) are elegant ways to depict homogeneous deformation of a body as a whole. If particles lying on the periphery of a circle are subjected to homogeneous deformation, the particles will trace an ellipse after deformation. This ellipse is called a strain ellipse. In three dimensions, a sphere in the undeformed state turns into an ellipsoid in the deformed state.
Let us look at the strain ellipse from a different viewpoint (Fig. 5.7). A circle describes a collection of straight lines with equal length but of different orientations, all passing through one point, which is the centre of the circle. Each of the lines connects two particles on either side of the circle. During deformation particles will move with respect to each other and the length of the lines will change. Stress ellipse (or ellipsoid in 3D) describes a collection of straight lines in deformed state, all passing through the same point, which is the centre of the ellipse. Obviously, there is no stress ellipse for inhomogeneous deformation because straight lines do not remain straight.
x x
z
z
λ3 λ1
1 1
(a)
(b)
Figure 5.7. Strain ellipse. See text for discussion.
If the radius of the initial circle is taken to be of unit length, the major and minor axes of the ellipse can be represented by √γ1 (= 1+e1) and √γ3 (= 1+e3), where √γ1 and √γ3
are maximum and minimum elongations (Fig. 5.7). So, the equation of strain ellipse centered at origin is,
x2/γ1 + z2/γ3 = 1
Similarly, the equation of strain ellipsoid is
x2/γ1 + y2/γ2 + z2/γ3 = 1
where, γ1 > γ2 > γ3. The three elongations directions are usually taken parallel to x, y, and z co-ordinate axes, respectively. Plane strain is a type of deformation where γ2 = 1, i.e., along the intermediate axis of the strain ellipsoid there has not been any shortening or elongation.
5.6 Finite and infinitesimal strain
When we look at a diastrophic structure in nature, such as a fold or a distorted fossil, we know that the rock has undergone some amount of strain. However, it is important to remember that the strain that we may observe and measure in rocks did not develop instantaneously but accumulated in small increments over a period of time. This is because, like most natural processes, deformations are also very slow.
Therefore, we observe the end product of a series of deformed states and straining of rocks should be considered as progressive deformation. The final state of strain is called finite strain and small incremental strains are known as infinitesimal strains. It is possible that a line that shows finite extension may have undergone shortening at some
Therefore, we observe the end product of a series of deformed states and straining of rocks should be considered as progressive deformation. The final state of strain is called finite strain and small incremental strains are known as infinitesimal strains. It is possible that a line that shows finite extension may have undergone shortening at some