Stress and strain

Stress and strain have to do with why and how a solid body deforms. Each point within a body under a load will have a set of stresses associated with it, varying in direction, magnitude, and the planes on which they act, according to the intensity of the forces acting within the body at that point. Each point within a deformed body will have a set of strains associated with it that indicate the translation, rotation, dilatation, and distortion experienced by the material at that point during the deformation.

Physics of Rigid Bodies

Stress and strain are concepts that help to explain how and why rocks deform. Strain describes the deformation, and stress pertains to the system of forces that produce it. In considering strain, first it will be helpful to review some aspects of the physics of rigid bodies to appreciate the significance of these concepts.

When dealing with many problems in mechanics, it is common to assume that the bodies involved are perfectly rigid; that is, they do not deform. Such problems usually involve the balancing of forces, or, if forces do not balance, determining the resulting accelerations. Movement of a rigid body can involve translation, rotation, or both, but the individual points within the body do not move relative to one another. In a translation, all points within the body move the same linear distance and in the same direction. In a rotation, all points within a body rotate through the same angle around the same center of the rotation. Any rigid body motion can be described in terms of a translation plus a rotation. Deformation introduces further complications. A volume-conserving, shape-changing deformation is called distortion. A change in volume, without a change in shape, is called dilatation (or dilation). Strain combines all four of these possibilities: translation, rotation, distortion, and dilatation.

Net Strains

If the beginning and ending locations and orientations of a rigid object are known, it is easy and straightforward to determine the net translation and the net rotation of the object and of every point within the object. For example, if an airplane begins in New York facing north as it is loaded and ends up in Madrid facing northwest as it is unloaded, then it can be said to have moved 5,781 kilometers to the east and rotated 45 degrees counterclockwise. Similarly, every piece of luggage on that airplane had a net translation of 5,781 kilometers to the east and a net rotation of 45 degrees counterclockwise. Very little information is needed to determine such net displacements and rotations, but the path that the airplane took is not well represented by them. The plane probably flew along a great circular route, changing its bearings constantly, and it very likely circled a bit after taking off and again before landing. To describe the path of the plane, one would need much more data. These data might consist of a series of translations and rotations taken at one-minute intervals. Each item of luggage, rigidly fixed within the hold of the aircraft, would move through an exactly identical series of translations and rotations. Furthermore, by applying the basic laws of mechanics, one could attribute each acceleration (linear or angular) to the forces resulting from the interplay of the thrust of the engines, the force of gravity, air resistance, prevailing winds, and other relevant factors.

In much the same way that net translations and rotations can be determined by knowing the original and final locations and orientations, net distortions and dilatations can often be determined relatively easily when initial and resultant shapes and volumes are known. Analysis is simplified if the area of study can be divided into subareas such that straight, parallel lines within each area remain straight and parallel after deformation. Such deformation, called homogeneous strain, is often assumed in the study of strain. Under these conditions, initially circular objects deform into ellipses.

Strain Paths

Determining the strain path requires a series of known translations, rotations, distortions, and dilatations; in turn, to tie the strain to the series of forces and stresses that produced it, one needs to know the strain path. Just as there are an infinite number of ways to fly from New York to Madrid, so there are an infinite number of strain paths that could result in identical net strains.

As an indication of the problem, consider a circle one centimeter in radius that deforms into an ellipse with a semimajor axis of two centimeters and a semiminor axis of 0.5 centimeter. Although there is no net dilatation, the deformation may have consisted of stretching in one direction and shrinking in the direction perpendicular to it. Alternatively, this deformation could have been produced entirely by distortion, as can be seen by drawing a circle on the edge of a deck of cards and then moving each card slightly to the right of the one below it. Each card will have two spots on it, one from each side of the circle. Since the distance between the spots on an individual card does not change, and the number of cards does not change, the area inside the resulting ellipse will remain constant. Continuing to deform the deck in this manner (a process called shearing) will result in the ellipse getting longer and thinner.

Stress Measurement

Strains are produced by stresses similar to the way movements of rigid bodies are produced by forces. More specifically, unbalanced forces acting on a rigid object cause it to accelerate. Within the elastic limit, the applied uniaxial stress is proportional to the resulting uniaxial strain (Hooke's law). In brittle rocks, initial faulting occurs at approximately thirty degrees to the greatest compressional stress (Mohr-Coulomb failure criterion). The amount of acceleration can be calculated if the net force and the mass of the object are known. The intensity of the forces acting within a body causes it to deform, and this force intensity is called stress. The units used to measure stress are the same as those used to measure pressure and are given in terms of force per unit area. Data may be presented in terms of atmospheres, pounds per square inch, bars, or similar units. The appropriate unit (based on the International System of Units) is the pascal, defined as one newton per square meter, or one kilogram per meter-second-squared. It is important to note that stress measurements contain an area term, and therefore, they cannot be added, subtracted, or resolved as if they were forces. By multiplying a stress by the area over which it is applied, it can be converted to a force, which can then be treated like any other force. It is customary to resolve it into forces parallel and perpendicular to a plane of interest. Finally, by dividing by the area of this plane, stresses can be obtained once again, yielding the shear stress and normal stress, respectively.

These factors can be demonstrated with a simple case in which a cube, one square meter on a side, has two forces acting on it in the vertical direction: One force of ten newtons is pushing down on the top, another force of ten newtons is pushing up on the bottom. The forces balance, so there will be no acceleration. Any horizontal plane within this cube will have an area of one square meter. It is subject to stresses of ten pascals, perpendicular to the plane, acting on each side of it. A diagonal plane through this cube, cutting the cube in half from one edge to the other, has an area of 1.414 square meters. The component of the vertical downward force acting perpendicular to this plane (the normal force) will be 7.071 newtons, and another component of the vertical downward force acting parallel to this plane (the shear force) will be 7.071 newtons. When these forces are divided by the area over which they act, it is apparent that there will be a normal stress of five pascals and a shear stress of five pascals acting on the upper surface of this plane.

Similar stresses can be shown to exist on the lower surface of the plane. Planes with different orientations will have other combinations of normal stresses and shear stresses, even though the forces responsible for those stresses remain the same. There are equations to manipulate the general situation, which give the normal and shear stresses acting on any plane as functions of the size and directions of the boundary forces. These result in what is called the stress ellipse, in two dimensions, or the stress ellipsoid in three dimensions. A graphical way of representing these equations (and the equations for strain) was developed by Otto Mohr in 1882 and is now called Mohr's circle.

Modes of Deformation

The results obtained previously may be compared with those for a hydrostatic condition, where stresses are the same in every direction. If stresses of ten pascals are acting on all six sides of the cube, no matter which plane one considers inside the cube, there will be normal stresses of ten pascals acting on each side of it. The stress ellipses and ellipsoids one might construct will be circles and spheres, and there will be no shear stresses anywhere.

Different modes of deformation are favored by different combinations of stresses. Movement on a fault plane, for example, is favored by low normal stresses and high shear stresses on that plane. Through the simple analysis described previously, it becomes clear that faulting is much more likely to occur along diagonal planes than along horizontal ones.

Stress and Strain Fields

The study of stresses often involves determining the stress field in a particular area, either at present or at some time in the past. After some simplifying assumptions are made concerning the geometry, mechanical properties, and boundary stresses of the area, a model is constructed that will indicate certain aspects of the stress field. Sometimes the model can be a physical one, produced from photoelastic plastic, for example. Such models can display the magnitudes of shear stresses when viewed appropriately with polarized light. More often, though, the model is constructed on a computer, and the stress ellipses are calculated for points of interest throughout the area. If geological stress indicators exist, such as the igneous sheet intrusions called dikes, the results of the model can be compared with the observed indicators, and the model can be adjusted until it fits the observations as closely as possible.

Determining at least parts of the strain field is in some ways more direct. Objects are sought in those rocks of the area that have net strains that can be determined. Frequently the distortion experienced by such an object can be easily observed and measured. A fossil that is elliptical but is known from its appearance in other areas to have been circular when it was alive provides a simple example. Strain ellipses showing the distortion of such objects can be constructed by measuring the shapes of these objects in the field.

These ellipses can then be plotted on a map. (A map with ellipses on it is a way of representing a tensor field—for example, a stress field or a strain field.) Most of the time, however, the initial size, location, and orientation of the objects are not known. One cannot tell whether a particular fossil is small because it never grew very large or whether it was once large and became smaller by deformation. If all the deformation occurred within a limited period of time, this map will represent the distortion part of the net strain field for that deformation. When similar data obtained from rocks deformed at different times are combined, a partial strain path can be obtained. The effects of more recent deformations are removed from the effects of earlier ones to isolate the earlier distortions.

The next step is to determine the stresses responsible for each increment of strain observed. To do so, it is necessary to know how each of the rocks responded to stress. Such data on mechanical behavior come from studies of experimental rock deformation. With these data, estimates can be made of the stress field present at different times in the history of the area. Finally, all this work can be applied to the known geological history of the area, permitting quantitative assessments of the various forces thought to have been active in the past.

Applications for Structural Geology

The stresses in a body are a function of the geometry of the body and the distribution of loads acting on it and within it. Determining the distribution of stresses is usually considered to be an exercise in statics, a branch of mechanical engineering, but it also plays a significant role in the earth sciences. Earthquakes occur when rock fails suddenly. Mine collapses, landslides, and dam failures are other catastrophes that occur when stress exceeds the strength of the material involved.

When a load is placed on a solid, the distribution of stresses within that solid is usually uneven. If the solid deforms, the deformation will usually also be uneven. To measure such deformation, one examines strain, which includes movements and changes in size and shape. As with stress, strain usually varies throughout the region being deformed.

A structural geologist is often concerned with determining how a region of the crust of the earth became deformed, and then why it deformed that way. Rocks often contain objects that are presently deformed but whose original shapes are known; such strain indicators include fossils, raindrop impressions, and bubbles. Using these indicators, a geologist seeks to reconstruct the strain field that existed at some time in the past. With enough indicators, along with dates for each, it may be possible to construct a strain history for the area in question. The next step is to guess, using the known mechanical behavior of the rocks involved, what the stresses were that produced the reconstructed strains. A final goal might be to seek causes for those stresses in terms of a larger picture of the earth's history, perhaps involving plate tectonics.

Principal Terms

dilatation: the change in the area or volume of a body; also known as dilation

distortion: the change in shape of a body

normal stress: that component of the stress on a plane that is acting in a direction perpendicular to the plane

rotation: the change in orientation of a body

shear stress: that component of the stress on a plane that is acting in a direction parallel to the plane

translation: the movement of a body from one point to another

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