Strain-Stress A PDF

Deformation Compiled By: Prof. Fathy Hassan Deformation Depending on if the structure is (at the scale of observation) behave as rigid or non-rigid bodies. Rigid-body deformation: occurs when the rock mass is intact and does not change size or shape - rotations and translations. Sha...

Deformation Compiled By: Prof. Fathy Hassan Deformation Depending on if the structure is (at the scale of observation) behave as rigid or non-rigid bodies. Rigid-body deformation: occurs when the rock mass is intact and does not change size or shape - rotations and translations. Shape is preserved - all points in the body have the same spatial arrangement (line length and angles) before and after deformation.  Deformation Rigid Body Translation Rigid Body Rotation Strain results from non- rigid body deformation, which is ▪ Change in size - positive or negative dilation. Original ▪ and/or Body Nonrigid Body Deformation by ▪ Change in shape - Distortion distortion. Dilation and distortion will result in changes in line length and angles between Nonrigid Body points. Deformation by Dilation  Deformation ▪ Rigid- and non-rigid body deformation commonly occur together. Movement on faults is normally considered to be a rigid-body motion. ▪ If the faults however are very closely spaced (smaller than the scale of observation) then the deformation is considered penetrative, and therefore it is a non-rigid body deformation. SCALE OF OBSERVATION IS KEY! ▪ Translation: is a rigid body deformation involving movement of the body from one place to another, i.e., change in position ♣ Particles within the body do not change relative position ♣ No rotation or strain are involved ♣ Particle lines do not rotate relative to an external coordinate system ♣ Displacement vectors are straight lines ♣ e.g., passengers in a car, movement of a fault block ▪ During pure translation, a body of rock is displaced in such a way that all points within a body move along parallel paths relative to some external reference frame  Translation Parallel to the Y axis ♣ At the largest scale, tectonic plates are considered to be rigid bodies. ♣ At the smallest scale, individual fractured grains slip on small discontinuities. Again… SCALE OF OBSERVATION IS KEY! Rotation ▪Rotation of a rigid-body occurs around an axis that is either within the body or outside. ▪ We can describe the axis of rotation by it’s trend and plunge (a line), it’s sense of rotation (cw or ccw), and the magnitude of the rotation (in degrees). Non-rigid body deformation Strain ▪ Strain – changes in the shape or size of a rock body caused by stress ▪ How rocks deform ♣ Rocks subjected to stresses greater than their own strength begin to deform usually by folding, flowing, or fracturing. Strain- (A- Dilation) ▪ Dilation is a non-rigid body operation involving a change in volume ▪ Pure dilation: ♣ The overall shape remains the same ♣ Internal points of reference spread apart (+ev) or pack closer (-ev) together ♣ Line lengths between points become uniformly longer or shorter  Strain (B- Distortion) ▪ Distortion is a non-rigid body operation that involves the change in the spacing of points within a body of rock in such a way that the overall shape of the body is altered with or without a change in volume ▪ Changes of points in body relative to each other o Particle lines may rotate relative to an external coordinate system o Translation and spin are both zero o Example: squeezing a paste ▪ In rocks we deal with processes that lead to both movement and distortion. ▪ Pure distortion is a change in shape without any change in area (2D) or volume (3D). o Usually accompanied by a change in line length and angles. o Systematic non-rigid deformation usually produces interesting results: ▪ initially spherical oolites after deformation become elipsoids that embody the full extent of the deformation Strain or Distortion  Strain - General Concept ▪Strain produces dilation (change in size) and distortion (change in shape). ▪ Typically, we simplify our lives by working on structures that exhibit homogeneous deformation, where: ♣ straight lines before, are straight after deformation. ♣ parallel lines before, are parallel after deformation. Heterogeneous Homogeneous Deformation Deformation Unstrained Unstrained Body Body Definitions Homogeneous Strain ▪ Homogeneous Strain - Lines that are straight and parallel before deformation remain straight and Heterogeneous (Inhomogeneous) parallel after Strain deformation. ▪ Inhomogeneous Strain - The landscape is distorted and lines may be broken.  Inhomogeneous Strain ▪ Heterogeneous strain affects non-rigid bodies in an irregular, non-uniform manner and is sometimes referred to as non-homogeneous or inhomogeneous strain ▪ During heterogeneous strain, parallel lines before strain are not parallel after strain ▪ Circles and squares or their three-dimensional counter parts, cubes and spheres, are distorted into complex forms. ▪ Rotational and Irrotational Strain ▪ If the strain axes have the same orientation in the deformed as in undeformed state, we describe the strain as a non-rotational (or irrotational) strain ▪ If the strain axes end up in a rotated position, then the strain is rotational. ▪ An example of a non-rotational strain (Coaxial)is pure shear - it's a pure strain with no dilation of the area of the plane. ▪ An example of a rotational strain (Nom-coaxial) is a simple shear.  Pure Shear Versus simple Shear Pure Shear Simple Shear Coaxial Non-coaxial Non-rotational Rotational Rotational and Irrotational Strain ▪ In theory, any rotational strain can be decomposed into two parts - a pure strain, and a rotation without distortion (rigid body rotation). ▪ In principle, if we only have the initial and final states we cannot tell the difference between a rotational strain like simple shear, and a pure shear followed by a rotation ▪ In practice, if we look at the fabric development in the rock, there may be differences in fabric development depending on whether the rotation was part of the strain history, or if it was applied later. ▪ Notice that in theory, theory and practice are the same; in practice, however, practice and theory are different! ▪Series of strain increments, from the original state, that result in final, finite state of strain ▪A final state of "finite" strain may be reached by a variety of strain paths. ▪ Finite strain is the final state; incremental strains represent steps along the path.  Pure Simple Pure Shear Versus simple Shear  Homogeneous Strain Irrotational Rotational Pure Simple Non- Coaxial Shear Shear Coaxial  Homogeneous Strain Irrotational Rotational Pure Simple Non- Coaxial Shear Shear Coaxial  Strain Ellipse ▪ It is always possible to find three originally mutually perpendicular material lines in the undeformed state that remain mutually perpendicular in the strained state. ▪ These lines, in the deformed state, are parallel to the principal axes of the strain ellipsoid, and are known as the principal axes of strain ▪ However, the length of the material lines parallel to the principal strains have changed during strain! ♣Strain Ellipsoid - Graphical tool that provides a reference object for estimating shape change from an assume Elliptical sections through these are sometimes printed on geologic maps to indicate geologic strain. ♣Made of three mutually perpendicular axes x, y, and z, where X  Y  Z.  Stress Ellipsoid Y Versus Strain X Ellipsoid Z Stress Ellipsoid The Strain Ellipsoid usually has an inverse relationship with the Stress Ellipsoid. X corresponds to σ3. The Relation Between Stress & Strain Than k You