strain is a measure of the relationship between size and shape of a body before and after deformation. strain is one component of a deformation, a term that includes a description of the collective displacement of all points in a body. Deformation consists of three components: rigid body rotation, a rigid body translation, and a distortion known as strain. strain is typically the only visible component of deformation, manifest as distorted objects, layers, or geometric constructs.
There are many measures of strain: changes in lengths of lines, changes in angles between lines, changes in shapes of objects, and changes in volume or area. The change in the length of lines can be quantified using several different strain measures.
where L is the original length of line, and L' is the final length of line;
The quadratic elongation
= (1 -e)2, whereas the natural or logarithmic strain is expressed as: (e)
The change of angles is typically measured using the angular shear (y angular shear), which is the change in the angle between two lines that were initially perpendicular. More commonly, structural geologists measure angular strain using the tangent of the angular shear, known as the shear strain:
Volumetric strain is a measure of the change in volume of an object, layer, or region. Dilation (5) measures the change in volume:
whereas the volume ratio
measures the ratio of the volume after and before deformation.
strains may be homogeneous or heterogeneous. Heterogeneous strains are extremely difficult to analyze, so the structural geologist interested in determining strain typically focuses on homogeneous domains with the heterogeneous strain field. In contrast, the geologist interested in tectonic problems involving large-scale translation and rotation often finds it necessary to focus on zones of discontinuity in the homogeneous strain field, as these are often sites of faults and high strain zones along which mountain belts and orogens have been transported. For homogeneous strains, the following five general principles hold true: straight lines remain straight and flat planes remain flat; parallel lines remain parallel and are extended or contracted by the same amount; perpendicular lines do not remain perpendicular unless they are oriented parallel to the principal strain axes; circular markers are deformed into ellipses; finally, there is one special initial ellipsoid that becomes a sphere when deformed. When these conditions are met, the strain field is homogeneous, and strain analysis of deformed objects indicates the strain of the whole body.
structural geologists often find it important to measure the strain in deformed rocks in order to reconstruct the history of mountain belts, to determine the amount of displacement across a fault or shear zone, or to accurately delineate the distribution of an ore body—this process is called strain analysis. To measure strain in deformed rocks, the geologist searches for features that had initial shapes that are known and can be quantified, such as spheres (circles), linear objects, or objects like fossils that had initial angles between lines that are known. In most cases, geologists cannot directly see the three-dimensional shape of deformed objects in rocks. strain analysis proceeds by measuring the two-dimensional shapes of the objects on several different planes at angles to each other. The deformed shapes are graphically or algebraically fitted together to get the three-dimensional shape of the deformed object and ultimately the three-dimensional shape and orienta-
Ooids in limestone showing a small amount of deformation. Sample is 1.3 inches (3.3 cm) tall. (Dirk Wiersma/Photo Researchers, Inc.)
tion of the strain ellipsoid. The strain ellipsoid has major, intermediate, and minor axes of X, Y, and Z, parallel to the principal axes of strain.
structural geologists interested in determining the strain of a body search for appropriate objects to measure the strain. Initially spherical objects prove to be among the most suitable for estimating strain. Any homogeneous deformation transforms an initial sphere into an ellipsoid whose principal axes are parallel to the principal strains, and whose lengths are proportional to the principal stretches s1, s2, and s3. using elliptical markers that were originally circular, one can immediately tell the orientation of the principal strains on that surface and their relative magnitudes. However, the true values of the strains are not immediately apparent, because the original volumes are not typically known. strain markers in rocks that serve as particularly good recorders of strain and approximate initially circular or spherical shapes include conglomerate clasts, ooids, reduction spots in slates, certain fossils, and accretionary lapilli.
Angular strain is often measured using the change in angles of bilaterally symmetric fossils, or igneous dikes cutting across shear zones. Many fossils, such as trilobites and clams, are bilaterally symmetric, so if line of symmetry can be found, similar points on opposite sides of the plane of symmetry can be joined, and the change in the angles from the initially right angles in the undeformed fossils can be constructed. When the fossils are deformed, the right angles are also deformed, and the same relationships derived above for change in angles can be used to determine the angular strain of the sample.
strain represents the change from the initial to the final configuration of a body, but it tells us very little about the path the body took to get to the final shape, known as the deformation path. The strain represents the combination of all events that occurred, but they are by no means unique. Fortunately, rocks have a memory, and there are many small-scale structures and textures in the rocks that tell us much about where the rock has been, or what its deformation path was. one of the most important attributes of the strain path to determine is whether the principal strain axes were parallel between each successive strain increment or not. A coaxial deformation is one in which the principal axes of strain are parallel with each successive increment. A noncoaxial deformation is one in which the principal axes of strain rotate with respect to the material during deformation.
Two special geometric cases of strain history are pure shear and simple shear. A pure shear is a coaxial strain (with no change in volume). simple shear is analogous to sliding a deck of cards over itself and is a two-dimensional noncoaxial rotational strain, with constant volume and no flattening perpendicular to the plane of slip.
In simple shear, the principal axes rotate in a regular manner. The principal strain axes start out at 45° to the shear plane, and strain s1 rotates into parallelism with it at very high (infinite) strains. The
Microscopic view in polarized light of mylonite. Mica rich layers are dark, while quartz layers are colorful, with polygonal shapes of individual grains. (Dirk Wiersma/Photo Researchers, Inc.)
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