DEFINITION OF STRAIN

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. Strain may be measured by changes in lengths of

lines, changes in angles between lines, changes in shapes of

objects, and changes in volume or area.

There are many measures of strain. The change in the

length of lines can be quantified using several different strain

measures. Extension (e) = (L’–L)/L, where L = original length

of line, and L’ = final length of line. Stretch (S) = L’/L = (1=s).

The quadratic elongation (s. = L’/L)2 = (1=s)2, whereas the

natural or logarithmic strain (s) s = loge (L’/L) = loge (1=s).

The change of angles is typically measured using the angular

shear (s.. 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

(s.. = tan s. Volumetric strain is a measure of the change in

volume of an object, layer, or region. Dilation (d) = (V’–V)/V,

and measures the change of volume, whereas the volume

ratio = V’/V 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 may indicate

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 displacestrain

ment 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 such as 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. In

this case, strain analysis proceeds by measuring the twodimensional

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 orientation 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, it is possible

to 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 the 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 we can use the relationships we derived

above for the change in angles to determine the angular strain

of the sample.

Strain represents the change from the initial to the final

configuration of a body, but really 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 principal axes remain perpendicular,

but some other lines will be lengthening with each increment,

and others will be shortening. There are some

orientations that experience shortening first, and then lengthening.

This leads to some complicated structures in rocks

deformed by simple shear; for instance folds produced by the

shortening, and then extensional structures, such as faults or

pull-apart structures known as boudens, superimposed on the

early contractional structures.

Natural strains in rocks deform initially spherical

objects into ellipsoids with elongate (prolate) or flattened

(oblate) ellipsoids. All natural strains may be represented

graphically on a graph known as a Flinn Diagram, which

plots a = (X/Y) vs. b = (Y/Z). The number k = a–1/b–1. For

k = 0, strain ellipsoids are uniaxial oblate ellipsoids or pancakes.

For 0 > k > 1, deformation is a flattening deformation,

forming an oblate ellipsoid. For k = 1 the deformation

is plane strain if the volume has remained constant. All

simple shear deformations lie on this line. For 1 > k > infinity,

the strain ellipsoids are uniaxial prolate ellipsoids, or

cigar shapes.

See also STRUCTURAL GEOLOGY.

strain analysis See STRAIN.

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