Stress ellipsoid pdf

The ellipsoid is actually a sphere in this case. Anisotropic is a stress state where at least one axis has a different magnitude to the other axes. This describes an ellipsoid. Deviatoric stress is the part of the total stress that is left after the mean stress is removed. Deviatoric stress is equivalent to tectonic stress and is the sress responsible for deformation. Deviatoric stress has the symbol sigma dev. Uniaxial stress can be in tension or compression. The sign convention for tension in geology is negative, and compression is positive Pluijm etal, The ellipsoid is 'needle-like'.

Biaxia l stress is where one axis equals zero. The general triaxia l state of stress is where none of the three principle stress axis can be zero. That is,. The Stress Ellipsoid is one description for the state of stress at a point. There is also a mathematical description that defines this state of stress, called the Stress Tensor. We will briefly detail this mode of defining stress.

For a more detailed explanation refer to the reference list. The orientation and magnitude of the state of stress of a body can be defined in terms of its components in a specific Cartesian reference frame. A Cartesian reference frame has three mutually perpendicular coordinate axes, X, Y, and Z. Or in this image X1, X2 and X3. The image shows a stress acting on a plane X1, X2.

Its vector components are also shown on this image. The normal stress component is parallel to X3. The shear stress components are expressed as being parallel to X1 and X2. In three dimensions, we can think of the point of stress as a cube. The stress acting on the three faces perpendicular to the axes of the cube can be resolved into their component parts. For example, the face normal to the X1 axis can be resolved into its normal component, sigma11 and the shear components are sigma12 and sigma This can be done for the face with the normal component sigma22 and sigma This gives us a total of nine stress components.

This is given in the form of a matrix:. We can relate the stress tensor cube to the stress ellipsoid. For example, the stress ellipsoid has three principal planes of stress, where they contain no shear stress components. We can consider this state on the cube. These normal components represent the principal stresses. The matrix would be reduced to:.

Again, this can be visualised as the stress ellipsoid. Strain is the response of rocks to stress. It describes the final shape of a rock in terms of the initial shape. During deformation, a rock will change in size and shape. Deformation describes the complete transformation from the initial to the final geometry of a body, and can be broken down into 4 main components:. Strain specifically relates to the changes of points in a body relative to each other.

There are two types of strain: Homogeneous and Heterogeneous. The way we differentiate between the two is by examining marker lines drawn on an object both before and after deformation. Straight marker lines remain straight after homogeneous deformation. Straight marker lines become curved after heterogenous deformation. Homogeneous strain results when any two portions of a body which were similar in form and orientation before strain, are still similar in form and orientation after the strain.

As a consequence, straight lines remain straight, parallel lines remain parallel, and planar surfaces remain planar. In two dimensions, circles will become ellipses, and in three dimensions spheres will become ellipsoids. Strain is heterogeneous when it varies across the surface of an object. Changes in the size and shape of small parts of the body are proportionately different from place to place. As a consequence, straight lines become curved, planes become curved surfaces, and parallel lines generally do not remain parallel after deformation.

In order to analyse a heterogeneously strained body, we need to break it down into homogeneous portions. We therefore divide a deformed body into volumes that are small enough for the deformation to be described as locally homogeneous. The picture on the left represents the undeformed state. The picture on the right depicts heterogeneous strain on the scale of the whole block. We divide the whole block into small sections, as depicted by the yellow circles.

At the scale of each individual cell containing a yellow circle, the circles have been transformed into ellipses, and the strain at each cell level is now homogeneous. The strain ellipsoid is used as a three dimensional way to represent strain. The strain ellipsoid results from the homogeneous deformation of an imaginary sphere, which represents the undeformed state of a body. In any homogeneously strained, three dimensional body, there will be at least three lines of particles, also known as material lines, that will not rotate relative to each other.

After strain, we will therefore have three material lines that remain perpendicular. These lines define the axes of an ellipsoid, and are known as the principal strain axes. They are referred to as X, Y and Z, where. The axes of the strain ellipsoid will therefore be different in length to the length of the axes in an undeformed sphere.

This difference is a measure of the strain magnitude. The finite strain ellipsoid FSE represents the end product of the homogeneous deformation of an imaginary sphere.

It does not give us any information about the strain history, or the mode of shear, that is, whether the shear was pure or simple. A rock does not undergo ductile deformation instantaneously. The strain states that a rock will progressively pass through, to reach its final deformed state, is known as the strain path. In nature, we generally see rocks in their final strained state, and we must deduce the initial undeformed state.

The strain path gives us a description of the intermediate stages during the process of straining. A co-axial strain path is where the strain axes are parallel to the same material lines throughout the straining. Therefore, the principal axes of the strain ellipsoid do not rotate during deformation.

A non-coaxial strain path is where the strain axes are parallel to different material lines during each infinitesimal increment of straining. Therefore, the principal axes of the strain ellipsoid move through the material during deformation. At any given increment of strain, the principal axes lie in a different physical part of the deforming material. Incremental and Finite Strain. Most of the structures we see in nature reflect the total strain. The total strain is also referred to as the finite strain.

It is independent of the intermediate steps. Finite strain can be the same for two rocks having different strain paths. Incremental strain is the increment of distortion whose sum led to finite strain. The finite strain is the sum of all of the incremental strains. The axes of the ISE do not change in orientation or magnitude from one increment to the next.

The incremental strain ellipse is at the bottom and does not change. As already seen the FSE does not give us any information regarding the strain history. The FSE can develop according to the two end members of strain regime: pure shear and simple shear. During pure shear, the principal axes of the FSE do not rotate during deformation. During simple shear, the magnitude and orientation of the axes of the FSE will change.

The rotation of the axes will give us information relating to the kinematic of the deformation. Often in nature both strain regimes are present with one being dominant over the other and therefore dominating the appearance of foliation, shear zones and features such as clasts and rigid objects. Together they are called general shear, with symmetric patterns indicating dominant pure shear and asymmetric patterns indicating dominant simple shear.

We have seen that the process of deformation can be defined in terms of stress and strain. The strain path, paleostress or history of deformation can therefore sometimes be extrapolated by looking at the end product.

Actual physical features are used to explore these ideas, and the distortion with time examined. Indications of stress and strain. All the theoretical models so far discussed have the undeformed state represented as a perfect sphere with all stresses being equal.

This is in fact impossible to find in nature, but a few features do come close. Ooides in limestone, conglomerate stones river worn , vesicles and amygdules in basalt flows and reduction zones in shales are all examples of naturally occuring features which approximate a sphere van Pluijn.

Depositational conditions must be taken into account and any process which can deform the sphere other than stress and strain. Basically anything with a set known shape before deformation can be used as long as it is homogenous with its groundmass, meaning they have the same approximate densitys and internal strength.

Garnet clasts in mica shists are not. They tend to disassociate and remineralise with pressure and form pressure shadows or windows, they dont deform homogenously within themselves. These then in pure shear form pinched in sections on either side of the clast and are symmetric.

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Stress and strain ellipsoid 1. Introduction Stress Strain Deformation cept of stress and strain ellipsoids Stress ellipsoid Strain ellipsoid tion between stress and strain ellipsoid Stress and strain ellipsoids 3. The aim of structural geologists is to analyse the deformation of rock masses. Stress and strain of the rock geometrically represented by constructing the stress and strain ellipsoids.

Stress on a rock body is defined as a force applied over an area to cause the deformation. Stress: Tensional compression Shear 5. Stress: Tensional compression Shear 6. Stress: Tensional compression Shear 7. Strain: Strain may be defined as response of rock to the stress, which can be depends on temperature pressure and mineralogical composition of rock. Deformation: Changes in the size, shape, position of the body with respect to the original position of the body is known as deformation.

DuctileBrittle Geologist deals with the rocks and minerals, so the amount of change in the original shape of the mineral is measure of deformation. But since the minerals are of mixed shape the concept of stress and strain ellipsoids put forth.

Greatest stress Minimum stress S1 S2 S3 Compression Tension Shear The Flinn diagram describes two main types of strain ellipsoids, cigar and pancake. Stress and strain ellipsoids— Rupture The position of the fracture planes and their orientation in field, are very important, because only from such data the direction of actual deformative forces , the orientation of stress and strain ellipsoids can be derived.

Compression fractures Intermediate strain axis S2 S1 S1 S2 S3 Normal fault Stress and strain ellipsoids -faults Thrust fault S2 S1 S2 Strike-slip fault Billings , structural geology, 3rd edition Prentice-hall of india private Ltd, PP, PP,46 NayyarSaab Jan.

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