Structural Geology Lectures Series 1.pdf

Structural Geology

Lecture 1

Introduction:

The parameters of structural geology

(constants, conversion factors, etc)

Structural geology boils down to a study of Newton's famous laws of motion as they pertain to the deformation of rocks within the earth. Newton's first law of motion states that "Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed on it". The very presence of faults and folds in the crust suggests that rocks, seemingly at rest, were once subject to forces that changed their original state by motion of one point relative to another. Structural geology is the study of the deformation of rocks. In its simplest form this is a description of present geometries. A study of the motion causing the geometries within rocks is called kinematics. A study of the forces that cause the motion is called

dynamics. The mathematics of structural geology are designed to simplify the study of kinematics and dynamics.

Structural geology is the study of the geometry, kinematics, and dynamics of rock structures. Geometric analysis is the descriptive or qualitative portion of structural geology. This portion of structural geology is as the name implies: A study of the size, shape, and orientation of structures. This portion was covered in classical structural geology courses. However, in this set of lecture notes the study of geometry will be delayed until a good mathematical base is established. In the meantime, many of the lab exercises will be devoted to geometric analysis. One of the most useful tools in geometric analysis is the stereonet which is a qualitative tool that serves the same purpose as vectors within a coordinate system.

Kinematic analysis requires a mathematical base for a rigorous treatment.

Kinematics, as you learned when taking elementary physics, is a mathematical description of the motion of objects. In the case of structural geology kinematics is the description of the path that rocks took during deformation. It is also the mathematical description of the relative position of two infinitesimal points during the deformation of rocks. Two points can change by translating together, rotating around each other, or changing in distance relative to one another. We shall call such a mathematical description deformation mapping.

Dynamics is the study of the forces which caused the deformations studied during kinematic analysis. In the case of structural geology dynamics includes the study of how rocks react to stress. For every stress the rocks respond with a finite strain. In a sense rock structures would not have formed, if rocks had not been subject to a stress. A study of dynamics starts with the fourth lecture.

numerical data and symbols for physical properties. Physics has its table of fundamental and derived physical constants which includes the speed of light (c = 3 x 108 _m/sec),

Avogadro's number (No = 6.02 x 1023 / mole), and universal gas constant (R = 8.23

joules/(mole)(K°). A comparable table involves numerical data which depend on circumstances such as geographic location and are, therefore, not strictly physical constants. Another table consists of a list of common physical properties which are represented by symbols including letters of the Greek alphabet. These numerical data and material properties form an important component of the language of structural geology.

Table of symbols for the structural geologist

Symbol Name Units

ρ density ML-3 σ stress ML-1_T-2 τ shear stress ML-1_T-2 σn normal stress ML-1T-2 ε strain dimensionless [LL-1_] E Young's Modulus ML-1_T-2

ν Poisson's ratio dimensionless

γ Engineering shear strain dimensionless

Pp pore pressure ML-1T-2 φ porosity dimensionless T temperature C° q heat flow JL-2_T-1 κ thermal conductivity JL-1_T-1_C°-1 z depth L

Table of numerical data for the structural geologist

Symbol Name Magnitude

g average gravity at sea level 9.8 m/sec2

ρm mean density of the mantle 4.5 x 103 kg/m2 (4.5 g/cm2)

ρquartz density of quartz 2.65 x 103 kg/m2 (2.65 g/cm2)

Table of conventions for the structural geologist

Name Convention

Principal stresses σ1 > σ2 > σ3

Stresses in the crust

Maximum horizontal stress SH

Minimum horizontal stress Sh

Vertical horizontal stress Sv

Compressional normal stress positive

Tensile normal stress negative

Table of conversion factors for the structural geologist

Stress and pressure

1 atm = 14.5 psi = 1 bar = 106 _dynes/cm2 _{= 10}5 _N/m2 _{= 10}5 _{Pascals (Pa)}

1 MPa = 10 bars = 106 _N/m2

pressure applies to a fluid stress applies to a solid

Table of definitions for the structural geologist

Name Definition

A Component of Principal Stress σii or σi

Any Component of Stress σij

Differential Stress σd = σ1 - σ3

Maximum Shear Stress τ_max = σ1−σ3

2 Lithostatic Stress SH = Sh = Sv Hydrostatic Pressure Pp = Pp = Pp Mean Stress σm = σ₁+σ₂ +σ₃ 3

Deviatoric Stress (3 components) σm - σ1, σm - σ2, σm - σ3

Effective Stress σi - Pp

Table of equations for the structural geologist

Stress and pressure Pp = ρH2Ogz Sv = ρrockgz

if ρrock = 2.5 x 103 kg/m3, g = 9.8 m/sec2, z = 103 m,

then Sv = 2.5 x 105 kg/m-sec2 = 25 MPa/km

Finally, the average geothermal gradient (dT/dz) within the crust of the earth is about 20°C/km where T is temperature and z is depth. This gradient can vary from 10°C/km in a glaucophane-schist terrain to 40°C/km. The low geothermal gradients can occur in the vicinity of crystalline overthrusts where cold crust is depressed. High gradients occur in a region of magmatic intrusion. Heat flow (q) at the surface is an

indication of geothermal gradient provided the thermal conductivity (K) of the crust is known

Structural Geology

Lecture 2

The mathematics of structural geology

(vectors, tensors, Einstein summation)

Scalar - A physical quantity expressed by a single number; a tensor of zero rank. Gradient of a Scalar - The direction at which a quantity increases the fastest.

grad_{φ = ∇φ (2-1)}

Vector - A physical quantity expressed by a three numbers; a tensor of rank one. Coordinate system for a vector: RIGHT HANDED - looking up the x₃ axis from the origin in positive direction, the x₁ axis will rotate into the x₂ axis in 90° in a clockwise direction. The components of a vector, F= F, are:

f_i

[ ]

The mathematical tool necessary to model Newton's laws of motion is the vector. Newton invented calculus to manipulate vectors. Needless to say, quantitative structural geology is based on the use of vector calculus. Remember that a vector is a quantity having both magnitude and direction whereas a scalar has just magnitude. Within this text a vector is represented by variables printed in bold letters (e.g. a). A knowledge of direction implies that a coordinate system is known. For structural geology a rectangular or Cartesian coordinate system represented by the direction of three positive unit vectors i,j, and k will do (e.g. i = 1, etc) (Fig. 2-1). Any vector can then be represented with scalar components f₁, f₂, and f₃ which represent distances along the axes denoted by the

three unit vectors. Every vector in space many be written as a linear combination of unit vectors i, j, andk. The vector f is then

f = f₁i + f₂ j + f₃k (2-3)

where f₁, f₂, and f₃ are components of the vector f. These three components are

independent of the choice of the origin point of the coordinate system. Vectors such as af, bf can be added by summing the individual components so that the resultant vector F is:

F = af + bf = (af₁ + bf₁ )i + (af₂ + bf₂)j + (af₃ + bf₃)k Magnitude of a vector is F = f₁2+ f2₂+ f2₃ (2-4) F2 = f₁2+ f2₂+ f2₃= f_if_i 1 3

∑

F+ G = G + F (2-6)

Vectors obey the associative law:

F+ G + H = F + G

(

)

(

)

Vector times a scalar is a larger vector.

Force f is a vector commonly used in explaining the causes of geological structures. Force is defined as the cause of an acceleration. Newton's second law states that if f is the net force acting on an object of mass m moving with a velocity v, then force (f) equals mass (m) times acceleration (a). This fundamental equation of classical mechanics is an example of a vector multiplied by a scalar to derive another vector:

f = m(dv/dt) = ma. Two common operations on a vector include the dot and cross products, respectively

(Fig 2-2). The dot product of two vectors is the projection of vector A on vector B: A*B = AB cos θ

done by a force. Consider a rock on which a constant force f acts. Let the rock be given a displacement d. Then the work W done during the displacement of the rock is defined as the product of |d| and the component of |f| in the direction of d, that is,

W = |f||d|cos α = f * d.

where α is the angle between f and d.

Scalar product or Dot product of a vector - magnitude of one vector times the

magnitude of projection of 2nd vector on the first vector. This number of a scalar.

F_{⋅ G = F G cos θ (2-8)}

F ⋅ G = f_ig_i= f₁g₁+ f₂g₂+ f₃g_{3 (2-9)}

Repeated suffixes means summation on i. i = 1to 3. If the vectors are orthogonal then

F⋅ G = 0 (2-10)

The cross product of two vectors is the magnitude of A times B times the sine of the angle between then. The direction (u) is perpendicular to the plane of A and B.

A x B = AB sin θ u

where θ is the angle between A and B. According to plate tectonics theory, as

continental plates move over the mantle a moment of force is generated. The moment of force is also known as a torque τ which is a vector quantity (Figure 2-3).

In mechanics the magnitude of the moment τ of a force f about a point Q is defined as the product τ = |f|.d where d is the perpendicular distance between Q and the line of action L of f. If r is the vector from Q to any point A on L, then d = |r|sin γ and

τ = |r||p|.sin γ = |r X p|.

Vector product or Cross product - The magnitude of H is the area of the parallelogram whose sides are F and G.

H= F × G = − F × G = F G sin θ (2-11)

The cross product is not associative F× G

( ) × H ≠ F × G × H( _{) (2-12)}

With each square matrix we associate a number called its determinant. If the matrix is A, then we demote its determinant by det A.

A = f₁f₂f₃ g₁g₂g₃ h₁h₂h₃ ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ det A = f₁f₂f₃ g₁g₂g₃ h₁h₂h_{3 (2-13)} det A = h₁f_g2f3 2g3 + h₂f_g1f3 1g3 + h₃f_g1f2 1g2 _(2-14) where h₃f_g1f2 1g2 = h₃

(

)

Heat flow, q (cal cm-2 sec-1), in homogeneous, isotropic rock is indicated by a thermal gradient, ∂T ∂x_j . q= − κ ∂T_∂x j _(2-16)

where κ is the thermal conductivity.

For Granite: κ = 8 x 10-3 _{cal cm}-1 _sec-1 _°C-1_.

For Steel: κ = 180 x 10-3 _{cal cm}-1 _sec-1 _°C-1_.

Heat flow is parallel to a thermal gradient so it can be expresses in a vector form where for an isotropic homogeneous rock.

q₁= − κ ∂T_∂x 1 _(2-17a) q₂ = − κ ∂T_∂x 2 (2-17b) q₃ = − κ ∂T_∂x 3 (2-17c)

If the rock is anisotropic then the thermal conductivity is not the same in each of

the three directions. If

a_j= ∂T_∂x j _{, then} q₁= κ₁₁a₁+ κ₁₂a₂+ κ₁₃a₃ (2-18a) q₂ = κ 21a1+ κ22a2+ κ23a3 _(2-18b) q₃ = κ 31a1+ κ32a2+ κ33a3 _(2-18c)

where each q is linearly related to a. If we consider heat flow through a cube, then the first subscript relates to the direction of the heat flow and the second subscript relates to the plane on which the heat flow operates. Remember that the ‘1” plane is defined by the 2- and 3- axes of the coordinae system defining the planes

If the temperature gradient is parallel to the x₁ axis, then ∂T ∂x_j ⎡⎢ ⎣ ⎤⎥⎦= ∂T∂x1 , 0, 0 ⎡⎢ ⎣ ⎤⎥_{⎦ (2-19)}

but heat flow still occurs in three directions according to

q₁= κ_{11 ∂x}∂T 1 ⎡⎢ ⎣ ⎤⎥⎦ _(2-20a) q₂ = κ_{21 ∂x}∂T 1 ⎡⎢ ⎣ ⎤⎥⎦ _(2-20b) q₃ = κ_{31 ∂x}∂T 1 ⎡⎢ ⎣ ⎤⎥⎦ _(2-20c)

Because of anisotropy of the rock, there is heat flow in all three directions of the rock even though the gradient was applied parallel to the x₁ axis.

κ_ij is a tensor of second rank represented by the matrix

κ_ij

[ ]

For a temperature gradient which is not parallel to a coordinate axis in an anisotropic material: q_i= κ_{ij ∂x}∂T j ⎡⎢ ⎣ ⎤⎥⎦ j= 1 3

∑

assumes summation over the term. A free suffix must occur once on each side of the equation.

The expression for hear flow in an anisotropic material may be written in its full form as q₁= κ_{11 ∂x}∂T 1 ⎡⎢ ⎣ ⎤⎥⎦+ κ12 ∂T ∂x₂ ⎡⎢ ⎣ ⎤⎥⎦+ κ13 ∂T ∂x₃ ⎡⎢ ⎣ ⎤⎥_{⎦ (2-23a)}

q₂ = κ_{21 ∂x} 1 ⎡⎢ ⎣ ⎤⎥⎦+ κ22⎡⎢⎣∂x2⎤⎥⎦ + κ_{23 ∂x} 3 ⎡⎢ ⎣ ⎤⎥_{⎦ (2-23b)} q₃ = κ_{31 ∂x}∂T 1 ⎡⎢ ⎣ ⎤⎥⎦+ κ32 ∂T ∂x₂ ⎡⎢ ⎣ ⎤⎥⎦+ κ33 ∂T ∂x₃ ⎡⎢ ⎣ ⎤⎥_{⎦ (2-23c)}

Tensor transformation -- The physical quantity, such as thermal conductivity, κij, is the same regardless of the set of axis chosen. This is the chief characteristic of a tensor. Transformation from one coordinate system to another depends on the direction cosine relating the axes of each of the coordinate systems. The transformation process will be discussed in detail during a later lecture.

Structural geology requires an understanding of vector differentiation, the first aspect of vector calculus. Vector differentiation is best illustrated by considering the description of a curve in space. If a particular vector P is the position vector p(t) joining the origin O of a coordinate system and any point (x,y,z,), then the vector function u(t) defines x,y, and z as functions of t

p(t) = x(t)i + y(t)j + z(t)k

As t changes, the terminal point of u describes a space curve having parametric equations: x = x(t), y = y(t), z = z (t). Then ∆p/∆t = {[p(t + ∆t)] - p(t)}/∆t

Then the limit

dp/dt as ∆t › 0

will be a vector in the direction of the tangent to the space curve at (x,y,z) and is given by

dp dx dy dz __ = __ i + __ j + __ k

dt dt dt dt The derivative of the vector of position p with respect to time t is velocity v.

Let u(t) be the position vector of a moving block of rock P in the earth where t is the time. The u(t) represents the path C of P. We know that the vector

v = dp(t)/dt is tangent to C and, therefore points in the instantaneous direction of motion of P. The derivative of the velocity vector is called the acceleration vector a; thus

a(t) = dv(t)/dt = d2p(t)/d2t

Acceleration is the second derivative of position with time. Newton's second law includes a derivative of the velocity vector v with respect to time dv/dt. Acceleration a is velocity changing with time. Second derivatives are must useful in structural geology. Another useful operation of calculus is the so-called chain rule of differentiation. If w is a differentiable function of x and x is a differentiable function of t, then

dw/dt = dw/dx.dx/dt.

This chain rule can be generalized for a function of two variables such as w = f(x,y) and x = x(t) and y = y(t) then

dw/dt = ∂w/∂x.dx/dt + ∂w/∂y.dy/dt. The operation ∂/∂x is the well known partial derative of a function. If x and y are functions of position within a coordinate system and t is a function of time, then w varies with position and time. If in contrast to the previous example x = x(p,v) and y = y(p,v) then

∂w/∂p = ∂w/∂x.∂x/∂p + ∂w/∂y.∂y/∂p and

∂w/∂v = ∂w/∂x.∂x/∂v + ∂w/∂y.∂y/∂v.

Partial differential equations are time consuming to write so a convention called Einstein summation is used. To illustrate Einstein summation lets consider a property of a rock that relates two vectors. A position within a rock x is related to a displacement u by displacement gradients E. Position and displacment are both vectors with components x₁, x₂, x₃ and u₁, u₂, u₃, respectively. x and are related in the following way:

u₁ = E₁₁x₁ + E₁₂x₂ + E₁₃x₃

u₂ = E₂₁x₁ + E₂₂x₂ + E₂₃x₃

u₃ = E₃₁x₁ + E₃₂x2 + E33x3

Rather than writing out all of these equations we can shorten then in the following manner: 3 u₁ =

Σ

Σ

Σ

Σ

ui = Eijxj

and introduce the Einstein summation convention: When a letter suffix occurs twice in the same term, summation with respect to that suffix is to be automaticaly understood. In

the above example the suffixes can be any letters. i is the free suffix and j is the dummy suffix. By this we mean that j can take any variable

Structural Geology

Lecture 3

Mapping of Points and Displacements during

Deformation

(introduction to kinematic analysis)

In this lecture techniques for kinematic analysis are introduced. The final product is a full accounting for rock strain which is at the heart of kinematic analysis. We can start our analysis with a one-dimensional example of strain. Strain ( ε ) is a dimensionless quantity defined as a change in length (∆l) per unit length (l).

ε = ∆l /l

Strain is actually defined near a point by the limiting process of differential calculus. ε = lim ( ∆l /l)

l ⇒ 0 Now differential quantities can be introduced by letting δx = l and δu = ∆l

δu ∂u ε = lim ___ = ___

δx ⇒ 0 δx ∂x Strain in rocks is mapped by following two points separated by ∆x through a deformation (Fig 3-1). One dimensional strain may be illustrated by drawing a line out from the origin (O) through two points such as P and Q.

∆x + ∆u = Q' - P'

(Fig. 3-1)

If the line is stretched the points P and Q go to P' and Q', respectively. The length from the origin to P is denoted as x whereas the distance between P and Q is ∆x. The stretch from P

to P' is u so that the total distance from P' to the origin is x + u. The stretch of ∆x is ∆u so that the distance between P' and Q' is ∆x + ∆u. The strain of PQ is then

ε = ∆l/l = ∆u/∆x If ∆u = 0.1 and ∆x = 1, then ε = .1/1 = 10%.

Deformation is conveniently separated into three components, of which two are displayed in this one-dimensional analysis. In one dimension there can only be a rigid-body translation and a stretch. The third component shows up in two and three dimensions where there is an additional deformation knows as a rigid rotation were the body spins about an axis. These concepts will be more fully developed as we progress through the elements of strain analysis.

Motion of a Point in a Deformed Body

Figure 3-1 demonstrates that for strain analysis we must keep track of the relative position of two points within a deforming body. However, before looking at the relative motion of two points, it is instructive to consider the absolute motion of one point. Deformation mapping involves the tracing of a material point within a body from its undeformed position to its deformed position. The material point in its undeformed position is specified by a vector X with components X₁, X₂, X_{3 or Xi. In the new or} deformed position the vector x has components x₁, x₂, x_{3 or xi (Fig 3-2). Note that in} Figure 3-2 the undeformed body is shown as a circle, a very common geological shape. In the deformed state that body becomes an ellipse. The point that we are watching is the center of the circle.

NOTE: You will use capital letters for coordinates and coordinate axes in the

unreformed state and lower-case letters for coordinates and coordinate axes in the deformed state. This convention is used largely to conform with Means (1976).

(Fig. 3-2: Until this figure is redrawn xi = X’i)

There two techniques for keeping track of what happened to the center of the undeformed body (i.e., the circle). First, we can develop a series of equations relating the final resting spot of the center of the circle. Or, we can develop a series of equations describing a vector pointing to the final resting spot from the initial position of the center of the circle.

Mapping of Points

One way of looking at the description of the 2-dimensional motion of a particle of rock is through the following functional relationship which is described by the deformation equations. These equations act to locate the final resting place of a point in a deformed body using the location of the point before deformation.

x₁ = f(X₁, X₂)

x₂ = f(X₁, X₂) or

xi = f(Xj) In three dimensions this same relationship is

x₁ = f(X₁, X₂, X₃) x₂ = f(X₁, X₂, X₃) x₃ = f(X₁, X₂, X₃) or xi = f(Xj)

Written in their complete form, the deformation equations are as follows: x₁ = a₀ + a₁.X₁ + a₂.X₂ + a₃.X₃

x₂ = b₀ + b₁.X₁ + b₂.X₂ + a₃.X₃ x₃ = c₀ + c₁.X₁ + c₂.X₂ + c₃.X₃

The constants, a0, b0, and c0 specify the rigid-body translation. If there is no rigid-body

translation, then the deformation equations for a homogeneous deformation may be written in matrix form:

a₁ a₂ a₃ b₁ b₂ b₃ c₁ c₂ c₃

Later we will use a double subscripted variable to represent the nine components of the deformation equations.

There are some special cases for deformation which are important in simplifying geological analysis. In three dimensions one example of a homogeneous deformation can be represented by the following equations:

x₁ = a₀ + a₁.X₁ + a₂.X₂ x₂ = b₀ + b₁.X₁ + b₂.X₂

x₃ = X₃

The reason that this is called a homogeneous deformation is that all points x'i are linearly related to points xi. This is an also example of deformation in plane strain where all motion is parallel to the plane normal to the x₃ axis. (Plane strain is further discussed with regard to faulting in a future lecture). The point (0,0,0) goes to the point (a₀,b₀,0) which defines the rigid body translation. Points along the direction x₁ where x₂ = 0 go to the point

x₁ = a₀ + a₁.X₁

x₂ = b₀ + b₁.X₁

In contrast a non-homogeneous deformation the points x'i are related to xi in a nonlinear manner. The following is one example of such a deformation in plane strain

x₁ = a₀ + a₁.X₁ + a₂.X₂ + a₃X₁2

x₂ = b₀ + b₁.X₁ + b₂.X₂ + b₃X₁2

matrix to specify our deformation equations: 1 −1 0

2 2 0

0 0 1

The three equations mapping the deformation might be the following provided it is understood that the rigid body translation (a0 = 5) is not specified by the matrix given

above (Fig. 3-3):

x₁ = 5 + X₁ - X₂ x₂ = 2X₁ + 2X₂ x₃ = X₃

Figure 3-3 is a mapping of three points of an undeformed body to its deformed shape. In this example you can think of the undeformed body as the outline of a shark’s tooth. You will see that the deformation actually ‘sharpened’ the shark’s tooth. All of this was done using the deformation equations as given above.

(Fig 3-3)

Displacement Vector

The motion of the material point from Xi to xi is also described by a displacement uo or (uo)i. uo ,which is a displacement vector, is nothing more than the final position, x, minus the initial position, X. In Figure 3-3 the motion of the corner’s of the shark’s tooth was shown by two vectors. The vectors may be expressed in vector components

(u₀)₁ = x₁ - X₁ and (u₀)₂ = x₂ - X₂ or (uo)i = xi - Xi or u₀ = x - X

These equations are known as the displacement equations. The displacement (uo)i of the point Xi represents a major part of the motion of all points within a rock body. The motion (uo)i is called rigid-body translation because it does not describe the motion of particles or rock relative to each other but rather specifies that all particles follow the same path. In this brief introduction we have only specified how individual points move during

deformation. We have not yet considered the relative motion of the points.

The relative motion of the points describes the shape change of the shark’s tooth. The change in shape of the rock is predicted by the displacement gradients which will be developed in the next lecture.

Structural Geology

Lecture 4

Displacement and Deformation Equations

(kinematic analyses)

Kinematics, as you learned when taking elementary physics, is a mathematical description of the motion of objects. In the case of structural geology kinematics is the description of the path that rocks took during deformation. It is also the mathematical description of the relative position of two infinitesimal points during the deformation of rocks. Two points can change by translating together, rotating around each other, or changing in distance relative to one another. We shall call such a mathematical description deformation mapping.

(Fig. 4-1: Until this figure is redrawn and xi = X’i)

The deformation gradient

If during rigid-body translation the particles of rock move relative to each other we must devise other equations to account for their relative motion. To the undeformed state we can attach a line segment dX whose components are dXi. The study of deformation is concerned with the change in orientation and length of dX as the point at X is moved by deformation to the point at x. We say that the vector dXi is both stretched and rotated to become the new vector dxi. To account for the relative motion of particles within the rock, we consider how the motion ui of any vector dX differs from the motion (uo)i of the vector X (Fig. 4-1). If

then xi + ∆xi = f( Xj + ∆Xj ) By Taylor's expansion f x

( )

( )

∑

(

)

∂xi/∂Xj are coefficients called the deformation gradient and are a function of the location

within the rock (Xi). Referring back to the deformation of the shark’s tooth as discussed in lecture #3: ∂x₁ ∂x_j = 1 −1 0 2 2 0 0 0 1

The displacement of the vector ui is the summation of (uo)i plus Eijdx where Eij (a displacement gradient) is a function of position within the rock.

Eij = ∂ui/∂xj The three equations mapping the deformation might be the following provided it is understood that the rigid body translation (a0 = 5) is not specified by the deformation

gradients (Fig. 4-4):

x₁ = 5 + X₁ - X₂ x₂ = 2X₁ + 2X₂ x₃ = X₃

In geology the displacement equations prove to be more useful than the deformation equations. However, for the purpose of an introduction to strain analysis a brief

introduction of the deformation equations is instructive. Now it is time to consider the characteristics of the displacement equations.

Strain is represented in the same manner as the deformation of shapes which was presented in the previous lecture. Lets examine the problem of representing strain in three dimensions using the coordinate system x1,x2,x3. We define a point P(x1,x2,x3)

between P and P' is uP where

uP = ux1' i + ux2' j + ux3' k.

(Fig. 4-2)

As is the case for one dimension strain in rocks is mapped by following two points, so we need an additional point Q which is displaced to Q'. Its displacement vector is then

uQ = ux1" i + ux2" j + ux3" k.

Now let A be the vector from P to Q and A' be the vector from P' to Q' (Fig. 4-3). Here strain is the difference between A and A' which is represented as

δA = A - A'.

δA/δx is the rate of change of displacement with position within the rock. By vector analysis we have uP + A' = A + uQ and δA = uQ - uP

A function u is said to be analytic at a point P if it can be represented by a power series of powers of P - a with a radius of convergence R > 0. Every analytic function can be represented by a power series. Because u is an analytic function we can express uQ in

terms of uP by employing a Taylor series expansion of uQ about P. Remember that a

Taylor expansion takes the following form:

f(z) = f(a) + f'(a).(z - a) + (1/2!)f''(a).(z - a)2 _{+ ...}

This is a series centered at a. So the expansion of uQ about uP is

uQ = uP + (∂u/∂X1)P∆X1 + (∂u/∂X2)P∆X2 +

(∂u/∂X3)P∆X3 + ...

where ∆X1 = A1 (i.e. the components of A). Here higher order terms may be

neglected because of there small value.

Using the summation of indices as discussed in the appendix uQ = uP + (∂u/∂Xj)PAj j = 1,2,3 Now δA = (∂u/∂Xj)PAj By convection (δA)i = δAi and (∂u/∂Xj)i = (∂ui/∂Xj). So

δAi = (∂ui/∂Xj).Aj.

∂ui/∂Xj is called the displacement gradient. These scalar quantities are the components

of the displacement equations which take the following form:

u₁= ∂u1 ∂X₁X1+ ∂u₁ ∂X₂ X2+ ∂u₁ ∂X₃X3

u₂ = ∂X₁X1+∂X₂ X2+∂X₃ X3 u₃= ∂u3 ∂X₁X1 + ∂u₃ ∂X₂ X2+ ∂u₃ ∂X₃X3

As a quick aside the derivation of the displacement gradient can also be illustrated using pages 94 and 95 of Nye (1957). In the three dimensional case displacement ui, are linearly related to the independent variables, in this case vectors dxj, by simultaneous equations u_{1 = (u0})₁ + E₁₁dX₁ + E₁₂dX₂ + E₁₃dX₃ u₂ = (u₀)₂ + E₂₁dX₁ + E₂₂dX₂ + E₂₃dX₃ u₃ = (u₀)₃ + E₃₁dX₁ + E₃₂dX₂ + E₃₃dX₃ or ui = (uo)i + EijdXj.

Eij are nine numbers in a 3 x 3 matrix that describe all motion within a rock that is not rigid-body translation. The terms Eij are called displacement gradients. As pointed out in the previous lecture a term with two suffixes relates two vectors. In this example the vectors are the displacement u and the position vector dXi. The Einstein summation convention as introduced in Lecture 2 applies to these equations for displacement.

Displacement and deformation gradients can be distinguished by the nature of the information that they impart. The displacement equations indicate the direction of motion of particles during the deformation. The vector u (u₁, u₂, u₃) points to the final position of the particle from the initial position of the particle given the initial coordinate system (X₁, X₂, X_{3). The displacement gradients (∂ui/∂xj) are a description of how the} movement of the initial point varies with position in the rock. In contrast, the

deformation equations map the final position of a particle of rock (x₁, x₂, x₃ ) in terms of the initial coordinate system (X₁, X₂, X₃ ).

Deformation is conveniently separated into three components, of which two are displayed in this one-dimensional analysis. In one dimension there can only be a rigid-body translation and a stretch. The third component shows up in two and three dimensions where there is an additional deformation knows as a rigid rotation were the body spins about an axis. These concepts will be more fully developed as we progress through the elements of strain analysis. In terms of the displacement gradients the stretch is denoted as

εij = 1_/

and the rigid-body rotation is denoted as

ωij = 1_/

Structural Geology

Lecture 5

Geological Deformation

(a first look at ‘real’ strain)

Having introduced the nature of deformation and displacement equations in the previous lectures it is time to consider three of the more famous examples of geological deformation. In the following examples the problems are specified in terms of both the displacement and deformation equations.

Problem #1 The first deformation to occur in the history of a sedimentary rock is

overburden compaction. This is represented by flattening in the vertical direction with no deformation in the horizontal directions. The following equations

represent overburden compaction as shown in Figure 5-1. Displacement equations: u₁ = 0X₁ + 0X₂ + 0X₃ u₂ = 0X₁ + 0X₂ + 0X₃ u₃ = 0X₁ + 0X₂ - 0.5X₃ Deformation equations: x₁ = 1X₁ + 0X₂ + 0X₃ x₂ = 0X₁ + 1X₂ + 0X₃ x₃ = 0X₁ + 0X₂ + 0.5X₃ (Fig. 5-1)

Problem 2#: If the initial body were turned on its side this deformation could represent a tectonic compaction. The Martinsburg shales near Harrisburg, Pennsylvania, have been isoclinally folded. Beds which were once flat-lying are now standing on end. Assuming that the shales were deformed with no internal strain we wish to represent this motion. The following equations represent the rigid rotation of a block by 90° as shown in Figure 5-2.

Displacement equations (A vector from the old point to its new location): u₁ = 0X₁ + 0X₂ + 0X₃

u₂ = 0X₁ - 1X₂ - 1X₃ u₃ = 0X₁ + 1X₂ - 1X₃

Deformation equations (A map of the new point given the coordinates of its old position):

x₁ = 1X₁ + 0X₂ + 0X₃ x₂ = 0X₁ + 0X₂ - 1X₃ x₃ = 0X₁ + 1X₂ + 0X₃

These equations are represented in the following figure:

(Fig. 5-2)

For an example in Problem #2, the initial point (1,1,1) is mapped to the new point (1, -1, 1). The vector between these two points is u = (0, -2, 0).

behavior of a fault zone subject to simple shear. In higher grade terrains these fault zones are simply called shear zones. Simple shear is illustrated by the deformation of a unit cube in Fig. 5-3.

Displacement equations (A vector from the old point to its new location): u₁ = 0X₁ + 0X₂ + 0X₃

u₂ = 0X₁ + 0X₂ + 0.5X₃ u₃ = 0X₁ + 0X₂ + 0X₃

Deformation equations (A map of the new point given the coordinates of its old position):

x₁ = 1X₁ + 0X₂ + 0X₃ x₂ = 0X₁ + 1X₂ + 0.5X₃ x₃ = 0X₁ + 0X₂ + 1X₃

These equations are represented in the following figure:

Structural Geology

Lecture 6

A Second Derivation of the Displacement Gradient

(another look at strain)

Three examples of deformation mapping were given in the previous lecture. The deformation can be represented in two manners. The displacement vector u was mapped in one case leading to a displacement gradient. In the other case the new position of each particle was mapped based on the initial position of the particle leading to a deformation gradient. In this lecture we will show that in 2 and 3 dimensions deformation consists of rotations as well as stretches.

We start by taking another look at strain with some simple definitions such as a change in length of line per unit length of line.

ε = ∆l/l

This is equivalent to a stretch. A formal definition of shear strain (γ) is the change in angle (ψ) between two initially perpendicular lines (Fig. 6-1).

γ = tan ψ

(Fig. 6-1)

A second measure of shear strain is the tensor shear strain which is half the tangent of the change in angle between initially perpendicular lines.

represented by line rotations. This gives the first indication that the strain can be separated into a rotational and irrotational component. We will deal in more detail with these concepts in the next lecture.

The concept of strain in one dimension uses l0 to indicate the initial length of a

line and l1 the final length of a line. We will start with the following definitions

ε = ∆l/l0 (elongation)

S = l1/l0 = (1 + ε) (stretch)

λ = (l⁄/l0)2 = (1 + ε)2 (quadratic elongation)

ε = dl/l0 (infinitesimal strain)

ε = ∆l/l0 (small increment of strain)

l1

ε =

∫

l0

(natural strain)

Now let's take another look at the displacement gradients by looking at how a rectangular element at P is distorted as shown in Figure 6-2.

The meaning of u1 and u2 is as shown above. Stretch of PQ1 parallel to the 1 axis

is:

∆u₁ = ∂u1

∂x₁∆x1 = e11∆x1 Anticlockwise rotation of PQ1 is given as

∆u₂ = ∂u2

∂x₁ ∆x1 = e21∆x1 The angle through which PQ1 turns is:

tanθ = ∆u2

∆x₁+ ∆u₁ Because ∆u1 << ∆x1:

θ = ∆u2

∆x₁ = e21

For no distortion the following situation holds:

Structural Geology

Lecture 7

Irrotational vs. Rotational Strain

(another look at strain)

The rotational strain tensor, e_ij, is invariant with respect to arbitrary, superposed shifts of the point of origin of the coordinate system. We will discuss the implications of this property in detail when looking at the stress tensor. For the time, we must accept the fact that any tensor property is independent of the position and orientation of the

coordinate system.

Nomenclature for the rotational strain tensor e_ij is as follows:

e_ij =

e₁₁e₁₂e₁₃ e₂₁e₂₂e₂₃ e₃₁e₃₂e₃₃

The first subscript, identifying the axis (i.e. direction) of a displacement

component u_i, is permuted from 1 to 3 in each of the columns, the second, identifying the face (i.e. the surface normal to an axis) whose center is displaced by the (u)_j, is permuted from 1 to 3 in each row.

We wish to make a distinction between infinitesimal strains and finite strains. An infinitesimal strain is one for which the final positions are very close to the initial

positions of adjacent particles. Infinitesimal strain theory is used in the theory of elasticity where strains are very small. While infinitesimal strains are independent of actual path of displacement, this is not the case for finite strains. Strain evident in outcrop is finite.

The rotational strain tensor, e_ij, applies to infinitesimal strains and is a general (or asymmetric) second rank tensor and can be expressed as the sum to a symmetric and an antisymmetric tensor

e_ij = ε_ij + ω_ij

where

ε_ij = 1₂

₍

₎

ω_ij = 1₂

₍

₎

Here is ε_ij symmetric and ω_ij is antisymmetric. The symmetric component of the

infinitesimal rotational strain tensor, e_ij, involves only dilatation (the change in volume) and distortion (the change in shape). Because of symmetry, the component, ε_ij , consists of only six independent variables. The component, ε_ij , can be regarded as strain proper, or irrotational strain. The antisymmetric tensor, ω_ij , has only three independent

components and involves only the rotational component of the strain tensor.

With these rather simple definitions the distinctions between simple shear and pure shear become clear. In a previous lecture we learned that the displacement equations for a pure shear involving overburden compaction were as follows:

u₁ = 0x₁ + 0x₂ + 0x₃ u₂ = 0x₁ + 0x₂ + 0x₃ u₃ = 0x₁ + 0x₂ - 0.005x₃

Note that this time I have indicated a very small overburden compaction. This was done because I am now dealing with infinitesimal (i.e. very small strains). The deformation gradient matrix for these equations is

∂u_i ∂x_j =

0 0 0

0 0 0

0 0 − 0. 005

The rotational strain tensor, e_ij, is

e_ij =

0 0 0

0 0 0

0 0 − 0. 005

For infinitesimal strain the rotational strain tensor and the deformation gradient are the same. The irrotational strain tensor, ε_ij, is

ε_ij =

0 0 0

0 0 0

0 0 − 0. 005

The rotational component of strain is

ω_ij = 0 0 00 0 0 0 0 0

For pure shear there is a remarkable similarity between the rotational strain tensor and the irrotational strain tensor,. This similarity disappears for the case of simple shear.

Again recall that displacement equations which were previously discussed for the same of simple shear as might be encountered within a fault zone. For infinitesimal strain these equations were

u₁ = 0x₁ + 0x₂ + 0x₃ u₂ = 0x₁ + 0x₂ + 0.005x₃ u₃ = 0x₁ + 0x₂ + 0x₃

The deformation gradient matrix and the rotational strain tensor for these equations is ∂u_i

∂x_j = eij=

0 0 0

0 0 0. 005

0 0 0

The irrotational strain tensor, ε_ij, is

ε_ij =

0 0 0

0 0 0. 025

0 0. 025 0

The rotational component of strain is

ω_ij = 00 00 0. 0250 0 _{− 0. 025 0}

The distinction between pure shear and simple shear is further clarified by considering the principal strain axes. If the directions of the principal axes of strain do not change as a result of displacement, then that deformation is termed irrotational strain. In the case of simple shear the principal axes of strain always differ depending on the amount of shear. The difference defines the rotational component of strain which is known as rotational strain. It is important to note that although simple shear is a

rotational deformation, there has been no actual rotation in space within a fault zone. The development of rotational strain does not necessarily imply that the body has to spin physically around some axis. Because of this lack of real rotation of within a fault zone, some like to refer to the rotation as an internal rotation. This type of rotation is in

contrast to our example of external rotation where bedded sediments being turned on end as given by the displacement equations below.

u₁ = 0X₁ + 0X₂ + 0X₃ u₂ = 0X₁ - 1X₂ - 1X₃ u₃ = 0X₁ + 1X₂ - 1X₃

Note that in the case of external rotation we are dealing with finite strain where the components of the deformation gradient matrix are large.

Structural Geology

Lecture 8

Strain Markers

(Strain Analysis)

Strain in rocks is measured using objects of known initial shape. The initial shapes can vary from round crinoid columnals to the irregular shape of breccia fragments. In two dimensions initially round objects include scolithus tubes, crinoid columnals, reduction spots, vesicules, concretions, and oolites. Initially elliptical objects include congolmerate pebbles. Fossils are usually irregular in shape but some such as leaves, brachiopods and trilobites may have a bilateral symmetry. More complicated shapes include the spiral of the ammonite, cavities of the coral, belemnites, and the branches of the graptolite. Other markers are deposited with centers at uniform distances from their nearest neighbors. In this lecture we are going to consider the analysis of four situations where rock strain can be inferred from the shape or position of deformed markers.

The least complicated strain analysis is the measure the elliptical shape of initially circular objects. This approached was used to map strain over 45,000 km2 _{of the}

Appalachian Plateau. The Devonian Catskill Delta of the Appalaachian Plateau contains many beds in which crinoid columnals parallel the bedding plane. Their elliptical shape on pavement surfaces testifies to the deformation of the Appalachian Plateau. The state of strain parallel to bedding in these outcrops is represented by three components: principal strains, ε₁ and ε₂, and the orientation of the strain ellipse as measured using one of the principal strain axes relative to north. In the case of the Plateau the strike of the long axes of the ellipse relative to north is designated as θ. Strain on the plateau may have a

rotational component but this was impossible to measure.

The actual measurements consisted of two numbers: the axial ratio and the orientation of the long axes. The axial ratio is a measure of ellipticity (R):

R = (1 + ε1 )/(1 + ε₂ )

In a paper published in Geology, Engelder and Engelder1 _{(1977) presented the a map}

showing the strike of θ and the value of R which varied between 1.1 and 1.2. The values of ε₁ and ε₂ are not possible to measure directly nor can they be calculated from the above equation because the strain on the Plateau is a volume loss strain.

The second type of deformed object that proves to be a useful strain marker is the ellipse with an initial ellipticity Ri. Upon deformation the shape of the final ellipse Rf is a function of the orientation and ratio of the initial ellipse ralative to the deformation (Figure 8-1). In the deformed state the orientation of the long axis of the ellipse relative to some marker is _{Φ. Data on Rf and Φ can be graphed to form Rf /Φ plots. These plots can then}

1 _{- As a matter of general interest, Prof. Engelder was a research scientist at Columbia}

university at the time this paper was written. His brother, Richard, was then a graduate student at Penn State.

be compared with standard Rf /Φ reference curves for different values fo initial ellipticity Ri and the strain ellipse Rs. The Rf /Φ plots have two shapes depending on whether Ri > Rs or Ri < Rs. In the former case the data envelope is symmetric about the orientation of the long axis of the strain ellipse and shows maximum and minimum Rf values. In the latter case the data envelope is closed and the data points shown a limited range of orientations defining the fluctuation F.

(Fig. 8-1)

The center-to-center technique allows the assessment of the bulk strain of a rock. This is commonly known as the Fry technique. Sometimes rigid inclusions such as fossils do not deform uniformly with the matrix of the rock. The best technique for determining the behavior of the matrix is the measure the distance between neighboring grains. Figure 8-2 shows an example where the distance and azimuth between various grains has been measured. From the distance/azimuth plot the ratio of the strain ellipse and its orientation can be determined. The Fry technique may also be used to take advantage of the

characteristic center-to-center distance of the deformed matrix of a rock.

The most beautiful and, hence, popular strain markers are the trilobites and brachiopods with lines of symmetry. If the line of symmetry lies parallel to a principal strain direction, then the angular shear strain of the principal lines in the fossil are zero. In this orientation the deformed fossil is still in a symmetrical form. If the initial lines of symmetry are not parallel to principal strain directions the lines of symmetry appear to shear during deformation. Final shape of the brachiopod or trilobite is an oblique form. The oblique forms can be either right or left handed depending upon the deflection of the symmetry axis. The symmetrical forms can appear in a narrow or broad form depending the initial orientation of the fossil relative to the shortening direction (Figure 8-3).

(Fig. 8-2)

The initial shape of the fossil can be characterized in terms of its length to breadth: r₀ = l0/b₀

The strain of the rock can be determined by using the ratio of the final length and breadth. In the narrow form

rn = ln/bn = (l0R)/b₀

and in the broad form

rb = lb/bb = (l0R)/b₀

Even though the original shape ratio r‚ is unknown we can calculate the strain ellipse and r‚ from the following equations.

R = (rn/rb)1/2

(Fig. 8-3)

The strain ellipsoid represents strain in three dimensions where the three axes of finite strain are known as (1 + ε₁) > (1 + ε₂ ) > (1 + ε₃). The longest extension is in the ε₁ direction (Fig. 8-4). The principal strain ratios are defined as

Rxy = (1 + ε1 )/(1 + ε₂)

and

Ryz = (1 + ε1 )/(1 + ε_3).

The plot of Rxy versus Ryz became known as the Flinn Graph (Fig. 18-4). Flinn suggested the parameter k to describe the position of the strain ellipsoid in the Flinn Graph.

Rxy - 1 k = ________

Ryz - 1

If k is greater than one then the strain ellipsoid is in the form of a cigar whereas if k is less than one the strain ellipsoid is in the form of a pancake.

Structural Geology

Lecture 9

Properties of Tensors

(Tensor Analysis)

Einstein summation convention -- Here the Einstein summation convention applies and the components _{σ ij relate two vectors fi and lj in a linear manner.}

The components _{σij form a 3 x 3 matrix:}

σ₁₁ σ₁₂ σ₁₃

σ₂₁ σ₂₂ σ₂₃

σ₃₁ σ₃₂ σ₃₃

In the study of vector analysis we have introduced three sorts of quantities: the scalar; the vector; and then a nine component quantity which relates two three-component vectors. These three quantities are all called tensors where:

Tensor of zero rank = Scalar -- This is a single number unrelated to any axes of reference.

Tensor of first rank = Vector -- This is specified by three numbers or components, each of which is associated with one of the axes of reference.

Tensor of second rank = (e.g. stress) -- This is specificed by nine numbers, or components, each of which is associated with a pair of axes (taken in a particular order).

Recall that the 2 dimensional Mohr's diagram described stresses associated with a pair of axes. In this sense the Mohr diagram is a visual representation of one-third of the second rank stress tensor. A tensor has the unique property that it can be transformed without losing its value. By this we mean that a tensor has

conponents representing a physical quantity which retains its identity regardless of which coordinate system (e.g. rectangular, cylindrical, etc.) is used or how the axes of that coordinate system may be rotated. The physical quantity is the same regardless of the direction from which it is viewed.

Tensor transformation -- The physical quantity, such as thermal conductivity, κij, is the same regardless of the set of axis chosen. This is the chief characteristic of a tensor. Transformation from one coordinate system to another depends on the direction cosine relating the axes of each of the coordinate systems.

aij are the components of the transformation matrix where a12 is the cosine of the

angle between x₁ of the old coordinate system and x₂' of the new coordinate system. The complete transformation matrix is

x₁' x₂' x₃' x₁ x₂ x₃ a₁₁a₁₂ a₁₃ a₂₁a₂₂ a₂₃ a₃₁a₃₂ a₃₃ (9-1) The first sugscript in the a's refers to the 'new' axes and the second to the 'old'.

There are only three independent direction cosines of the nine given in the transformation matrix. The coordinates of a point (x₁,x₂,x₃) can also be transformed to the coordinates (x₁',x₂',x₃') using direction cosines in the same manner (Fig. 9-1) x₁' = a₁₁x₁ + a₁₂x₂ + a₁₃x₃ x₂' = a₂₁x₁ + a₂₂x₂ + a₂₃x₃ x₃' = a₃₁x₁ + a₃₂x₂ + a₃₃x₃. (9-2) Figure 9-1

Likewise the equations for the transformation of a vector from one coordinate system to another resemble the equations for transforming a point.

(9-3) where p'

i are the components of the vector in the new coordinate system and pj are

the components of the vector in the old coordinate system. Here the free suffix is i and the dummy suffix, j, is together in the right-hand term. Written in its full form equation 3-2 becomes

p'₁= a

11p1+ a12p2 + a13p3

p'₂ = a₂₁p₁+ a₂₂p₂+ a₂₃p₃

p'₃ = a

31p1+ a32p2+ a33p3 (9-4)

We can reverse the transformation process to find the old vector in terms of the new vector

p_i= a_jip'_j

(9-5) Here the dummy suffix, j, is separated by the free suffix, i.

Relations between direction cosines - Since each row of the array (3-1) represents three direction cosines of a straight line with respect to a coordinate system, x₁, x₂, x₃, the following equations hold true

a₁₁2 _{+ a} 12 2 _{+ a} 13 2 _{= 1} a2_{21+ a} 22 2 _{+ a} 23 2 _{= 1} a2_{31+ a} 32 2 _{+ a} 33 2 _{= 1} (9-6) In a condensed version equation 3-5 is written

a_ika_jk = 1, if i= j

(9-7) Since eachpair of rows of the array 3-1 represents the direction cosines of two

lines at right angles, we have a situation comparable to a scalar product of two vectors at right angles.

a_ika_jk = 0, if i≠ j

(9-8) We can combine the orthogonality relations into one equation

a_ika_jk = δ_ij= 1

(

)

(

)

⎨

where the symbol, δ_ij, is called the Kronecker delta.

Second rank tensor -- In general, if a property T relates tow vectors P = [p₁,p₂,p₃] and Q = [q₁,q₂,q₃] in such a way that

p₁= T₁₁q₁+ T₁₂q₂+ T₁₃q₃ p₂ = T₂₁q₁+ T₂₂q₂+ T₂₃q₃

p₃ = T₃₁q₁+ T₃₂q₂+ T₃₃q_{3 (9-10)}

where the components, T₁₁, T₁₂, ...are constants, then T₁₁, T₁₂, ...are said to form a second-rand tensor.

T₁₁ T₁₂ T₁₃ T₂₁ T₂₂ T₂₃ T₃₁ T₃₂ T₃₃ ⎡ ⎢ ⎢ ⎢⎣ ⎤ ⎥ ⎥ ⎥⎦ (9-11)

The values of the coefficients T₁₁, T₁₂, ...depend on the orientation of the coordinate axes x₁, x₂, x₃. Now suppose we choose a new set of coordinate axes x₁', x₂', x₃' related to the old axes by the direction cosines, a_ij. If so, the vectors P and Q have new components p_i' and q_i'. Next we find P' in terms of Q' the series of equations shown below (The arrow means "in terms of").

P'→ P → Q → Q' (9-12)

The following equations permit the transformation p'_i= a_ikp_k (9-13) p_k = T_klq_{l (9-14)} q_l= a_jlq'_j (9-15) Substituting p'_i= a ikTklajlq'j _(9-16) or p'_i= T ij ' _q j ' (9-17) So

T'_{ij = a}

ikajlTkl _(9-18)

This is the Transformation Law for a tensor of second rank. Written out in full form each of its nine equations are rather long. The general equation is

T'_{ij = a}

i1aj1T11+ ai1aj2T12+ ai1aj3T13+ ai2aj1T21+

a_i2a_j2T₂₂ + a_{i 2}a_j3T₂₃ + a_i3a_j1T₃₁+ a_{i 3}a_j2T₃₂ + a_i3a_j3T₃₃

(9-19) Just as the components pi and q_i transform to p_i' and q_i', when the axes are

changed, the nine coefficients T_ij transform to the nine coefficients T'_ij. In this example i and j are free suffixes and k and l are dummy suffixes. A second rank tensor must transform.

At this point it is important to appreciate the characteristics of a stress vector (f) which is nothing more that the stress tensor (σ_ij) expressed as a vector acting on a surface whose normal is the vector l.

f_i= σ_ijl_j

The principal axes of a tensor - The equation for the surface of a quadric (i.e., a second degree surface such as an ellipsoid or hyperboloid) is written

S_ijx_ix_j= 1

(9-20) where S_ij are coefficients. Performing the summation with respect to i and j

equation is written in full by assuming the symmetric condition which is that S_ij = S_ji.

S₁₁x₁2_{+ S}

22x22+ S33x23 + 2S12x1x2+ 2S31x3x1+ 2S23x2x3= 1

(9-21) The coefficients of equation 3-17 transform like the components of a symmetrical

2nd rank tensor.

S'_{ij = a}

ikajlSkl _(9-22)

The theory of the transformation of a symmetrical 2nd rank tensor is thus identical with the theory of th transformation of a quadric. Thus, the surface represented by equation 3-17 is called the representation quadric of a tensor, S_ij. An important property of a quadric is the possession of principal axes. These are three directions at right angles such that, when the general quadric is referred to them as axes, its equation takes the simpler form

S₁₁x₁2_{+ S}

22x22+ S33x23 = 1 _(9-23)

When we refer thje general tensor to principal axes, we have reduced the tensor components to the matrix

S₁ 0 0 0 S₂ 0 0 0 S₃ ⎡ ⎢ ⎢ ⎢⎣ ⎤ ⎥ ⎥ ⎥⎦ (9-24)

where S₁, S₂, S₃ are called the principal components of the tensor [S_ij]. The equation for a standard ellipsoid is written

x2 a2 + y2 b2 + z 2 c2 = 1 _(9-25)

In this equation the semi-axes of the representation quadric are the lengths 1 S₁, 1 S₂, 1 S₃ .

We now define a property of a principal axis as the radius vector which parallels the normal to the quadric surface. The radius of the quadric is specified by S_ijx_j. The normal to the surface of the quadric is specified by the vector, x_i. The condition that the radius vector and the normal are parallel is satisfied provided that the corresponding components are proportional. This condition is satisfied by the equation

S_ijx_j= λx_i

(9-26) where λ is a constant. Equation 3-23 is three homogeneous linear equations in the

variables x_i.

S₁₁x₁+ S₁₂x₂+ S₁₃x₃= λx₁ S₂₁x₁+ S₂₂x₂+ S₂₃x₃= λ x₂

S₃₁x₁+ S₃₂x₂+ S₃₃x₃= λ x_{3 (9-27)}

The trivial solution to these equations is that x_i = 0. If so, these equations would be worthless. For a non trivial solution the determinant of the components of these three equations must be zero.

S₁₁ − λ S₁₂ S₁₃ S₂₁ S₂₂ − λ S₂₃ S₃₁ S₃₂ S₃₃ − λ = 0 (9-28) or S₁₁− λ

(

)

[

(

)

(

)

]

[

(

)

]

[

(

)

]

give the three possible values of λ that ensure that equations 3-24 have non-zero solution. Those values of λ are also called eigenvalues or characteristic values of the matrix S. Each of the roots defines a direction in which the radius vector of the quadric is parallel to the normal, that is, the direction of one of the principal axes. These directions are called eigenvectors or characteristic vectors of S. The direction of the principal axes are found by substituting, say λ', into equation 3-24. These three equations may be solved for the ratios x₁': x₂': x₃'. Knowing the ratios we can solve for the direction cosines of the unit vector by

u'_i= x'_i x' 2₁ + x' 2_{2 + x} 3 ' 2 (9-30)

Structural Geology

Lecture 10

An Introduction to Stress

(Tensor Analysis)

Hooke's Law

When a body changes shape we say that the body is strained. Strain, a

deformation, is measured as the ratio of the change in shape of a body to its initial shape. Previous figures illustrates strain using a long rod with an initial length (L). When the rod is stretched the additional length (i.e., displacement) of the rod is ∆L. Strain (ε) is ∆L divided by L. If the rod is deformed elastically, it will return to the length, L, when the forces are removed from the end of the body.

Another aspect of elasticity which we will examine is the relationship between the amount of force and the amount of strain. Figure 10-1 shows an unloaded spring and the same spring loaded with one, two, and three weights, respectively. The spring gets longer in equal increments with the addition of comparable increments of weight. Remember that a weight is a force. Relationship between change in force and change in stain is characteristic of an elastic material.

ELASTICITY OF A SPRING

Figure 10-1

It is accurate to imagine that the atomic forces which govern the atomic distances in a lattice act just like springs which return to their natural shape regardless of whether they were subject to a push or a pull. The law that governs the action of a spring is called Hooke's Law. We can write the equation for the Hooke's Law of a spring as follows:

force = spring constant X displacement