ES2202

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Deformation:

Deformation is the transformation from an initial to a final geometry by means of rigid body translation, rigid body rotation, strain (distortion) and/or volume change.

Components of deformation:

Translation- moves every particle in the rock in the same direction and the same distance, and its displacement field consists of parallel vectors of equal length.

Rotation- is here taken to mean rigid rotation of the entire deformed rock volume that is being studied.

Strain- Any change in shape, with or without change in volume, is referred to as strain, and it implies that particles in a rock have changed positions relative to each other.

Volume change- also referred to as dilation, is commonly considered to be a special type of strain, called volumetric strain. However, it is useful to keep this type of deformation separate if possible.

How Rocks Deform Depend on:

• Temperature

• Pressure (rock or lithostatic pressure)

• Pore fluid pressure

• Composition of rocks (e.g. quartzite vs. limestone)

• Force applied

• Rate of deformation

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We will think in terms of particle positions (or vectors) and see how particles change positions during deformation. If (x, y) is the original position of a particle, then the new position will be denoted (x0,y0). For homogeneous deformation in two dimensions (i.e. in a section) we have that

x0 = D11x + D12y & y0 = D21x + D22y

These equations can be written in terms of matrices and position vectors as

x0 = Dx

The matrix D is called the deformation matrix, and the equation describes a linear transformation or a homogeneous deformation. There is a corresponding or inverse matrix D^-1 (where the matrix product DD^-1=I and I is the identity matrix) that represents the reciprocal or inverse deformation. D^-1 reverses the deformation imposed by D: x = D^-1x0

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One-dimensional strain:

Elongation : e= (l-l0)/l0, where l0 and l are the lengths of the line before and after deformation.

natural elongation e = ln(e)

Stretching : s =1+e = l/l0

Quadratic elongation, lambda = s^2

Natural strain = ln(s)

Strain in two dimensions

Angular shear, shi, (change in angle between two originally perpendicular lines in a deformed medium).

Shear strain, gamma = tan(shi)

Strain is said to be plane (two-dimensional) where there is no length change along the Y-axis, while three-dimensional strain implies a length change along X, Yand Z.

Uniform extension, also referred to as axially symmetric extension, is a state of strain where stretching in X is compensated for by equal shortening in the plane orthogonal to X. Uniform flattening (axially symmetric flattening) is the opposite, with shortening in a direction Z compensated for by identical stretching in all directions perpendicular to Z.

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The strain ellipse is the ellipse that describes the amount of elongation in any direction in a plane of homogeneous deformation. The strain ellipse is the ellipse that describes the amount of elongation in any direction in a plane of homogeneous deformation. The two axes have lengths 1+e1 and 1+e2, and the ratio R=X/Y or (1+e1)/(1+e2) describes the ellipticity or eccentricity of the ellipse and thus the strain that it represents. For a circle (no strain), R=1

The finite spatial change in shape that is connected with deformation is completelydescribed by the strain ellipsoid.

Locus of all possible surface stresses that act on planes of all orientations through a point.

Principal Stresses are the axes of the stress ellipse, and are the maximum (σ1) and minimum (σ2) of all normal stresses on the planes.

σij: i is a plane on which stress is acting; j is an axis parallel to stress.

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(Body Force: They act on each particle of rock in a volume, and its magnitude is proportional to the mass within that volume.

Surface Force: Forces that result from the action of one body / part of a body on another across a shared surface. They act on a specific surface of a body.

Traction: Intensity of Force = F/A

Of the four stress components in the (x,z) coordinate system only three are independent: σxx , σxz (= -σzx),, σzz

4 scalar components in 2D - second rank tensor

Rank of a tensor determines the number of scalar required to describe the tensor completely

Component of a tensor quantity = Dimension of physical space (d) ^rank (r)

To define state of stress :

• Three independent components of stress • Magnitudes of principal stresses + angle specifying the orientation of the principal axes relative to the reference axes.

Failure: Rock is unable to support stress increase without permanent deformation .

Brittle Failure – Rock breaks to form continuous fractures resulting in the loss of cohesion.

Ductile Failure – Material deforms permanently without losing cohesion

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Hydrostatic pressure (water): σ1 = σ2 = σ 3 = p.

All principal stresses are compressive; no shear stresses exist on any plane.

Mohr circle reduces to a point on the normal stress axis

Lithostatic pressure (rock): σ1 = σ2 = σ 3 = p.

Deviatoric stress (shape change only): Deviatoric normal stress on a plane is defined by subtracting the mean normal stress from any normal stress component, including in particular, the principal stresses. Deviatoric shear stresses are unchanged from their original values. σ dev = σ n - σ m

Differential stress (strength of a rock) = σ(dif) = σ 1 – σ3 (difference between maximum & minimum principal stresses that causes a rock to fracture or flow)

Strain – A shape change or volume loss (size change) that is a response to an applied stress.

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Coaxial deformation implies that lines along the principal strain axes have the same orientation as they had in the undeformed state.

Pure shear (Figure 2.8b) is a perfect coaxial deformation. This means that a marker that is parallel to one of the principalaxeshasnotrotatedaway fromitsinitialposition.

constant volume plane strain s1>1; s2=1, s3<1 ... Shortening in one direction gets balanced by extension in other direction.

For non-coaxial deformations, the orientations of the principal strain axes are different for different amounts of strain, while for coaxial deformations they always point in the same directions (same orientation, different lengths).

Simple shear (Figure 2.8) is a special type of constant volume plane strain deformation. There is no stretching or shortening of lines or movement of particles in the third direction. it is a non-coaxial deformation, meaning that lines parallel to the principal strain axes have rotated away from their initial positions.

Constant volume plane strain. s1>1; s2=1, s3<1

Between pure shear and simple shear is a spectrum of planar deformations, commonly referred to as subsimple shear (general shear). Subsimple shearcan be considered as a mix of pure and simple shear because the internal rotation involved is less than for simple shear.

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Progressive deformation:

Nonrigid motion of a body that carries the body from its initial undeformed state to its final deformed state.

Strain states through which the body passes during progressive deformation is strain path .

Progressive Pure Shear Irrotational Strain - Principal axes do not rotate w.r.t. coordinate axes (irrotational progressive deformation)

-All other lines rotate toward X (max. principal direction).

- Lines that rotate most rapidly are 45° to the principal axes of instantaneous Strain ellipse.

--Principal axes of finite strain always parallel to/coaxial with principal axis of instantaneous strain – coaxial deformation.

- Coaxial deformation & irrotational strain are not synonymous.

A non-coaxial deformation history impliesthat the orientation of the progressive strain ellipsoid is different at any two points in time during the deformation, Another characteristic feature is that lines that are parallel to ISA or the principal strain axes rotate during deformation.

During a coaxial deformation history the orientation of the strain ellipsoid is constant throughout the course of the deformation and lines parallel to ISA do not rotate.

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In the two white fields, lines are continuously being stretched, and we can call them the fields of instantaneous stretching. Similarly, lines in the two yellow fields continuously experience shortening and we call them the fields of instantaneous shortening. The borderlines between these fields are the lines of no stretching or shortening. In three dimensions the fields become volumes and the borderlines become surfaces of no stretching or shortening. These fields remain constant during the deformation history while lines may rotate from one field to the next.

Progressive Simple Shear

--Principal axes do rotate

– rotational deformation.

-- Principal axes of finite strain rotate w.r.t. those of instantaneous strain (noncoaxial progressive deformation)

--Principal axes of finite strain move through the rock volume during deformation