1. Geologic Structures and Deformation Regimes - Supplements
Contents
1S.1.1 Force, Stress, Strain and Deformation
1S.1.2 Reference Frame and Transformation
We have already used the words stress, strain, and deformation without definition, because these are common English words and most people have an intuitive grasp of what they mean. Stress presumably has something to do with pushing and pulling, and strain and deformation have something to do with bending, breaking, stretching, or squashing. But in standard English, stress and strain are often used interchangeably; for example, advertisements for aspirin talk about “the stress and strain of everyday life.” In structural geology, however, these terms have more exact meanings, so right from the start we want to clarify their usage (and avoid headaches).
The stress (σ) acting on a plane is the force per unit area of the plane (σ = F/area). We will see in Chapter 2 that, when referring to the stress at a point in a body, a more complicated definition is needed. Deformation refers to changes in shape, position or orientation of a body resulting from the application of a differential stress. The latter means a state in which the magnitude of stress is not the same in all directions. Deformation consists of three components (Figure S1.1):
(1) a rotation, which is the pivoting of a body around a fixed axis,
(2) a translation, which is a change in the position of a body; and
(3) a strain (or distortion), which is a change in shape of a body.
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FIGURE S1.1. The components of deformation, before and after:
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To visualize a strain, consider the test crash of a car that is rapidly approaching a brick wall. The car and the wall attempt to occupy the same space at the same time, with variable success. Since the structural integrity of the car is less than that of the wall, the push between car and wall squashed the car, thereby resulting in a strain. In homogeneous strain, the strain exhibited at one point in the body is the same as the strain at all other points in the body. Cars are designed so that strain is heterogeneous, meaning that the strain is not equal throughout the body, and the passengers are protected from some of the impact.
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FIGURE S1.2. The translational and rotational components of deformation shown schematically along a fault. (a) A translated fault block; (b) a rotated fault block in the hanging wall. |
What about translation and rotation? Geologically, a rigid body of rock that has moved along a fault plane has been translated relative to the opposing side of the fault (Figure S1.2a),while a fault block in which strata become inclined relative to horizontal strata on the opposing wall of the fault has been rotated (Figure S1.2b). Translation and rotation occur at all scales, as emphasized by work in paleomagnetism, which demonstrates that continental blocks (or, terranes) have traveled and rotated as a consequence of displacement on major faults.
In order to describe deformation, it is necessary to define a reference frame. The reference frame used in structural geology is loosely called the undeformed state. We can’t know whether a rock body has been moved or distorted unless we know where it originally was and what its original shape was. Ideally, if we know both the original and final positions of an array of points in a body of rock, we can describe a deformation with mathematical precision by defining a coordinate transformation. For example, in Figure S1.3a, four points (labeled m, n, o, and p) define a square in a Cartesian coordinate system. If the square is sheared by stresses acting on the top and bottom surfaces, as indicated by the arrows, and moved from its original location, it changes into a parallelogram that is displaced from the origin (Figure S1.3b). The deformation can be described by saying that points m, n, o, and p moved to points m′, n′, o′, and p′, respectively. In other words, coordinates of all four corners of the square have been transformed. If you are mathematically inclined, you will have recognized that coordinate transformation is a mathematical concept in tensor algebra, but we won’t get into that here.
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FIGURE S1.3. Deformation represented as a coordinate transformation. Points m, n, o, and p move to new positions m′, n′, o′, and p′. Note that circle transforms into ellipse. |
In many real circumstances, we don’t have an external reference frame, so we can only partially describe the deformation of a body. For example, at an isolated outcrop we may be able to describe strain—say, because of the presence of deformed fossils—but we have no absolute record of translations or rotations. Then we talk about relative displacement and relative rotation. A flat-lying bed of Paleozoic limestone in the Midcontinent region of the United States was at one time below sea level and, because of plate motion, it was formed at a different latitude than today, but we can’t immediately characterize these movements. If, however, we see a fault offset a limestone bed by 2 meters, we say that one side of the fault has moved 2 m relative to the other side.
Many books explore field and geometric analysis, and computer and geophysical applications in structural geology. The following selection is based on our own usage, which is by no means complete.
Groshong, R. H., 1999. 3-D structural geology: A practical guide to surface and subsurface map interpretation. Springer Verlag.
Lisle, R. J., 1996. Geological structures and maps: A practical guide (2nd edition). Oxford: Butterworth- Heinemann.
Marshak, S., and Mitra, G., 1988. Basic methods of structural geology. Englewood Cliffs: Prentice Hall.
McClay, K., 1987. The mapping of geological structures. Geological society of London handbook. Berkshire: Open University Press.
Rowland, S. M., and Duebendorfer, E. M., 1994. Structural analysis and synthesis (2nd edition). Boston: Blackwell.
Collinson, J. D., and Thompson, D. B., 1989. Sedimentary structures (2nd edition). Unwin Hyman: London.
Jackson, M. P. A., and Talbot, C. J., 1994. Advances in salt tectonics. In Hancock, P. L., ed., Continental deformation. Pergamon Press, pp. 159–179.
Melosh, H. J., 1996. Impact cratering: A geologic process. Oxford University Press: New York.
Paterson, S. R., Vernon, R. H., and Tobisch, O. T., 1989. A review of criteria for the identification of magmatic and tectonic foliations in granitoids. Journal of Structural Geology, 11, 349–363.
Selley, R. C., 1988. Applied sedimentology. Academic Press: New York.
Shrock, R. R., 1948. Sequence in layered rocks. McGraw-Hill: New York.
Worrall, D. M., and Snelson, S., 1989. Evolution of the northern Gulf of Mexico, with emphasis on Cenozoic growth faulting and the role of salt. In The geology of North America—an overview, v. A, pp. 97–138. Geological Society of America.