Strain Ellipsoid

Let us refer again to the sphere inscribed within our cube that undergoes homogeneous strain. Giving this sphere axes \( \lambda _1 \), \( \lambda _2 \) and \( \lambda _3 \), we find that these remain orthogonal through the homogeneous deformation and can be used to describe the long, medium, and short dimensions of the resulting ellipsoid. (Figure 1, Several basic types of homogeneous strain imposed on a cube of ideally uniform composition.



In each case, the inscribed circle and ellipse represent cross sections through the strain ellipsoid before and after deformation. The types of strain shown are: (a) uniform extension; (b) uniform flattening; and (c) plane strain. Note that no strain occurs in the intermediate direction.)

Together, these are called the principal strain axes of the strain ellipsoid. Just as stress can be said to exist at every point within a body, so is there a corresponding strain ellipsoid for these points, once deformation has taken place. Thus, comparison between the "before" and "after" shape and axes of the sphere inscribed within our cube provides us with a measure of the amount and type of strain.

It is standard practice for geologists to derive the principal axes of stress by superimposing them on the strain ellipsoid. ( Figure 2, Note that this superposition assumes homogeneous strain, i.e., maximum shortening occurs in the direction of maximum principal stress, and maximum extension in the direction of minimum principal stress).



This is an optimistic simplification (as we have seen, the principal axes of stress and strain coincide only under conditions of homogeneity), but can be very useful. It has, for example, offered considerable insight into basic mechanisms and patterns of deformation-particularly faulting and fracturing-on many scales. This will become increasingly clear in the following two sections on folding and faulting.

Because of their frequent use of structural cross sections, geologists have also found it advantageous to make use of a strain ellipse-essentially a cross section through the strain ellipsoid along the \( \lambda _1 \) \( \lambda _3 \) plane (i.e., the one that involves \( \sigma _1 \) and \( \sigma _3 \)). The justification for this is, again, dependent on the assumption of homogeneous strain. Because of the regional nature of most tectonism, and the layered nature of lithologic sequences, many geologic examples of strain can be considered to approximate plane strain (see part c of Figure 1). In this type of deformation, the intermediate axis remains the diameter of the "original" sphere (the \( \lambda _2 \) axis-- parallel to \( \sigma _2 \)-- in part c of Figure 1), while shortening and stretching occur along the other two axes.

Thus, two dimensions are sufficient to describe the strain at a particular point. If we are ready to accept the assumption of homogeneous strain, the strain ellipse becomes one of our principal indicators for the summation- of-local-strains method. (Figure 3 Hypothetical cross section and diagram to illustrate domains of pure and simple shear in a series of folds that show progressively greater total strain.



The shape of the strain ellipse can be the same for either type of shear and cannot be used to derive detailed strain history.)

As shown by Figure 4, (Example of how the strain ellipse – here constructed from deformed oolites – can be used as a descriptive guide to deformation intensity and orientation.



Shown is a cross section through the south Mountain fold of western Maryland, U.S) the strain ellipse can be an important guide to the general degree and style of deformation.

Some natural materials, such as ooids, spherulites, pebbles, certain fossils, and reduction spots in shales, can be used as qualitative ellipses or, in some cases, ellipsoids. However, because volume changes frequently occur during deformation (especially in carbonates) any quantitative determinations of strain based on such materials must be used with caution.

In principle, any object whose initial shape is known can act as a strain indicator. Such an indicator can be important to the subsurface explorationist, since it may be the only direct evidence available for how much strain has affected the fabric-and thus porosity and permeability-of a lithologic section. In most cases, the degree of tectonic influence on grain texture is fairly apparent from petrographic study. Strain indicators are primarily useful where this may not be clear and where special circumstances warrant mathematical determinations of strain. Specific techniques for measuring finite strain from oolites and spherulites are given by Ramsay (1967) and Ramsay and Huber (1985).

Nearly all deformation in nature is in-homogeneous. Not only do originally planar surfaces become complexly curved, but volume changes that involve both loss and addition of material frequently take place. Because of their pronounced heterogeneity in composition, thickness, and thus strength, rocks do not behave passively during deformation, but adjust in complex ways. Some units become strain-hardened and are able to withstand and transfer greater and greater amounts of stress as deformation progresses; other lithologies, in contrast, are fated to absorb stress by flowage, recrystallization, and the development of secondary fabrics such as cleavage.

Again, despite the dominance of inhomogeneity in nature, both local and regional deformational history can be reconstructed by assuming near-homogeneous strain domains. On a large scale, this often establishes the regional nature of stress and strain. Geologists often estimate a regional strain ellipse based on the orientation of major structural trends. This is called the mean strain ellipse and is often useful in explaining such trends in terms of plate interactions.