Fault anatomy

Fault anatomy

Faults drawn on seismic or geologic sections are usually portrayed as single lines of even thickness. In detail, however, faults are rarely simple surfaces or zones of constant thickness. In fact, most faults are complex structures consisting of a number of structural elements that may be hard to predict. Because of the variations in expression along, as well as between, faults, it is not easy to come up with a simple and general description of a fault. In most cases it makes sense to distinguish between the central fault core or slip surface and the surrounding volume of brittlely deformed wallrock known as the fault damage zone, as illustrated in Figure 8.10.

Simplified anatomy of fault.

The fault core can vary from a simple slip surface with a less than millimeter-thick cataclastic zone through a zone of several slip surfaces to an intensely sheared zone up to several meters wide where only remnants of the primary rock structures are preserved. In crystalline rocks, the fault core can consist of practically non-cohesive fault gouge, where clay minerals have formed at the expense of feldspar and other primary minerals. In other cases, hard and flinty cataclasites constitute the fault core, particularly for faults formed in the lower part of the brittle upper crust. Various types of breccias, cohesive or non-cohesive, are also found in fault cores. In extreme cases, friction causes crystalline rocks to melt locally and temporarily, creating a glassy fault rock known as pseudotachylyte. The classification of fault rocks is shown in heading below.
In soft, sedimentary rocks, fault cores typically consist of non-cohesive smeared-out layers. In some cases, soft layers such as clay and silt may be smeared out to a continuous membrane which, if continuous in three dimensions, may greatly reduce the ability of fluids to cross the fault. In general, the thickness of the fault core shows a positive increase with fault throw, although variations are great even along a single fault within the same lithology. 

The damage zone is characterized by a density of brittle deformation structures that is higher than the background level. It envelops the fault core, which means that it is found in the tip zone as well as on each side of the core. Structures that are found in the damage zone include deformation bands, shear fractures, tensile fractures and stylolites, and Figure below shows an example of how such small-scale structures (deformation bands) only occur close to the fault core, in this case defining a footwall damage zone width of around 15 meters.

Damage zone in the footwall to a normal fault with 150–200 m throw. The footwall damage zone is characterized by a frequency diagram with data collected along the profile line. A fault lens is seen in the upper part of the fault. Entrada Sandstone near Moab, Utah.

The width of the damage zone can vary from layer to layer, but, as with the fault core, there is a positive correlation between fault displacement and damage zone thickness (Figure below a). Logarithmic diagrams such as shown in Figure below are widely used in fault analysis, and straight lines in such diagrams indicate a constant relation between the two plotted parameters. In particular, for data that plot along one of the straight lines in this figure, the ratio between fault displacement D and damage zone thickness DT is the same for any fault size, and the distance between adjacent lines in this figure represents one order of magnitude. Much of the data in Figure below a plot around or above the line D=DT, meaning that the fault displacement is close to or somewhat larger than the damage zone thickness, at least for faults with displacements up to 100 meters. We could use this diagram to estimate throw from damage zone width or vice versa, but the large spread of data (over two orders of magnitude) gives a highly significant uncertainty. 

A similar relationship exists between fault core thickness (CT) and fault displacement (Figure below b). This relationship is constrained by the straight lines D=1000CT and D-10CT, meaning that the fault core is statistically around 1/100 of the fault displacement for faults with displacements up to 100 meters.

(a) Damage zone thickness (DT) (one side of the fault) plotted against displacement (D) for faults in siliciclastic sedimentary rocks. (b) Similar plot for fault core thickness (CT). Note logarithmic axes. Data from several sources.

Layers are commonly deflected (folded) around faults, particularly in faulted sedimentary rocks. The classic term for this behavior is drag, which should be used as a purely descriptive or geometric term. The drag zone can be wider or narrower than the damage zone, and can be completely absent. The distinction between the damage zone and the drag zone is that drag is an expression of ductile fault-related strain, while the damage zone is by definition restricted to brittle deformation. They are both part of the total strain zone associated with faults. In general, soft rocks develop more drag than stiff rocks.

Fault rocks

When fault movements alter the original rock sufficiently it is turned into a brittle fault rock. There are several types of fault rocks, depending on lithology, confining pressure (depth), temperature, fluid pressure, kinematics etc. at the time of faulting. It is useful to distinguish between different types of fault rocks, and to separate them from mylonitic rocks formed in the plastic regime. Sibson (1977) suggested a classification based on his observation that brittle fault rocks are generally non-foliated, while mylonites are well foliated. He further made a distinction between cohesive and non-cohesive fault rocks. Further subclassification was done based on the relative amounts of large clasts and fine-grained matrix. Sibson’s classification is descriptive and works well if we also add that cataclastic fault rocks may show a foliation in some cases. Its relationship to microscopic deformation mechanism is also clear, since mylonites, which result from plastic deformation mechanisms, are clearly separated from cataclastic rocks in the lower part of the diagram. 

Fault breccia is an unconsolidated fault rock consisting of less than 30% matrix. If the matrix fragment ratio is higher, the rock is called a fault gouge. A fault gouge is thus a strongly ground down version of the original rock, but the term is sometimes also used for strongly reworked clay or shale in the core of faults in sedimentary sequences. These unconsolidated fault rocks form in the upper part of the brittle crust. They are conduits of fluid flow in non-porous rocks, but contribute to fault sealing in faulted porous rocks.

Pseudotachylyte consists of dark glass or microcrystalline, dense material. It forms by localized melting of the wall rock during frictional sliding. Pseudotachylyte can show injection veins into the sidewall, chilled margins, inclusions of the host rock and glass structures. It typically occurs as mm- to cm-wide zones that make sharp boundaries with the host rock. Pseudotachylytes form in the upper part of the crust, but can form at large crustal depths in dry parts of the lower crust. 

Crush breccias are characterised by their large fragments. They all have less than 10% matrix and are cohesive and hard rocks. The fragments are glued together by cement (typically quartz or calcite) and/or by microfragments of mineral that have been crushed during faulting.

Cataclasites are distinguished from crush breccias by their lower fragment–matrix ratio. The matrix consists of crushed and ground-down microfragments that form a cohesive and often flinty rock. It takes a certain temperature for the matrix to end up flinty, and most cataclasites are thought to form at 5km depth or more. 

Mylonites, which are not really fault rocks although loosely referred to as such by Sibson, are subdivided based on the amount of large, original grains and recrystallised matrix. Mylonites are well foliated and commonly also lineated and show abundant evidence of plastic deformation mechanisms rather than frictional sliding and grain crushing. They form at greater depths and temperatures than cataclasites and other fault rocks; above 300 C for quartz-rich rocks. The end-member of the mylonite series, blastomylonite, is a mylonite that has recrystallized after the deformation has ceased (postkinematic recrystallization). It therefore shows equant and strain-free grains of approximately equal size under the microscope, with the mylonitic foliation still preserved in hand samples.

Credits: Haakon Fossen (Structural Geology)

Deformation bands and fractures in porous rocks

Deformation bands

Rocks respond to stress in the brittle regime by forming extension fractures and shear fractures (slip surfaces). Such fractures are sharp and mechanically weak discontinuities, and thus prone to reactivation during renewed stress build-up. At least this is how non-porous and low-porosity rocks respond. In highly porous rocks and sediments, brittle deformation is expressed by related, although different, deformation structures referred to as deformation bands.

Kinematic classification of deformation bands and their relationship to fractures in low-porosity and non-porous rocks. T, thickness; D, displacement.

Deformation bands are mm-thick zones of localised compaction, shear and/or dilation in deformed porous rocks. Figure above  shows how deformation bands kinematically relate to fractures in non-porous and low-porosity rocks, but there are good reasons why deformation bands should be distinguished from ordinary fractures. One is that they are thicker and at the same time exhibit smaller shear displacements than regular slip surfaces of comparable length (Figure (a) below). This has led to the term tabular discontinuities, as opposed to sharp discontinuities for fractures. Another is that, while cohesion is lost or reduced across regular fractures, most deformation bands maintain or even show increased cohesion. Furthermore, there is a strong tendency for deformation bands to represent low permeability tabular objects in otherwise highly permeable rocks. This permeability reduction is related to collapse of pore space, as seen in the band from Sinai portrayed in Figure (b) below. In contrast, most regular fractures increase permeability, particularly in low-permeability and impermeable rocks. This distinction is particularly important to petroleum geologists and hydrogeologists concerned with fluid flow in reservoir rocks. The strain hardening that occurs during the formation of many deformation bands also makes them different from fractures, which are associated with softening. 

(a) Cataclastic deformation band in porous Navajo Sandstone. The thickness of the band seems to vary with grain size, and the shear offset is less than 1 cm (the coin is 1.8 cm in diameter).

(b) Cataclastic deformation band in outcrop (left) and thin section (right) in the Nubian Sandstone, Sinai. Note the extensive crushing of grains and reduction of porosity (pore space is blue in the thin section). Width of bands 1 mm.

The difference between brittle fracturing of nonporous and porous rocks lies in the fact that porous rocks have a pore volume that can be utilised during grain reorganisation. The pore space allows for effective rolling and sliding of grains. Even if grains are crushed, grain fragments can be organised into nearby pore space.

The kinematic freedom associated with pore space allows the special class of structures called deformation bands to form.

What is a deformation band?

How do deformation bands differ from regular fractures in non-porous rocks? Here are some characteristics of deformation bands: 

• Deformation bands are restricted to highly porous granular media, notably porous sandstone.
• A shear deformation band is a wider zone of deformation than regular shear fractures of comparable displacement.
• Deformation bands do not develop large offsets. Even 100 m long deformation bands seldom have offsets in excess of a few centimetres, while shear fractures of the same length tend to show meter-scale displacement. 
• Deformation bands occur as single structures, as clusters, or in zones associated with slip surfaces (faulted deformation bands). This is related to the way that faults form in porous rocks by faulting of deformation band zones.

Types of deformation bands 

Similar to fractures, deformation bands can be classified in a kinematic framework, where shear (deformation)bands, dilation bands and compaction bands form the end members (1st Figure). It is also of interest to identify the mechanisms operative during the formation of deformation bands. Deformation mechanisms depend on internal and external conditions such as mineralogy, grain size, grain shape, grain sorting, cementation, porosity, state of stress etc., and different mechanisms produce bands with different petrophysical properties. Thus, a classification of deformation bands based on deformation processes is particularly useful if permeability and fluid flow is an issue. The most important mechanisms are:

• Granular flow (grain boundary sliding and grain rotation) 
• Cataclasis (grain fracturing) 
• Phyllosilicate smearing 
• Dissolution and cementation 

The different types of deformation bands, distinguished by dominant deformation mechanism.

Deformation bands are named after their characteristic deformation mechanism, as shown in Figure above.

Brittle deformation mechanisms. Granular flow is common during shallow deformation of porous rocks
and sediments, while cataclastic flow occurs during deformation of well-consolidated sedimentary rocks and non-porous rocks.  

Disaggregation bands develop by shear-related disaggregation of grains by means of grain rolling, grain boundary sliding and breaking of grain bonding cements; the process that we called particulate or granular flow (Figure above a). Disaggregation bands are commonly found in sand and poorly consolidated sandstones and form the “faults” produced in most sandbox experiments. Disaggregation bands can be almost invisible in clean sandstones, but may be detected where they cross and offset laminae (Figure below). Their true offsets are typically a few centimeters and their thickness varies with grain size. Fine-grained sand(stones) develop up to 1 mm thick bands, whereas coarser-grained sand (stones) host single bands that may be at least 5 mm thick. 

Macroscopically, disaggregation bands are ductile shear zones where sand laminae can be traced continuously through the band. Most pure and well-sorted quartz-sand deposits are already compacted to the extent that the initial stage of shearing involves some dilation (dilation bands), although continued shear-related grain reorganization may reduce the porosity at a later point.

Right-dipping compaction bands overprinting left-dipping soft-sedimentary disaggregation bands (almost invisible). The sandstone is very porous except for thin layers, where compaction bands are absent. Hence, the compaction bands only formed in very high porosity sandstone. Thin section photo shows that the compaction is assisted by dissolution and some grain fracture. Navajo Sandstone, southern Utah.

Phyllosilicate bands (also called framework phyllosilicate bands) form in sand(stone) where the content of platy minerals exceeds about 10–15%. They can be considered as a special type of disaggregation band where platy

minerals promote grain sliding. Clay minerals tend to mix with other mineral grains in the band while coarser phyllosilicate grains align to form a local fabric within the bands due to shear-induced rotation. Phyllosilicate bands are easy to detect, as the aligned phyllosilicates give the band a distinct color or fabric that may be reminiscent of phyllosilicate-rich laminae in the host rock.

If the phyllosilicate content of the rock changes across bedding or lamina interfaces, a deformation band may change from an almost invisible disaggregation band to a phyllosilicate band. Where clay is the dominant platy mineral, the band is a fine-grained, low-porosity zone that can accumulate offsets that exceed the few centimeters exhibited by other types of deformation bands. This is related to the smearing effect of the platy minerals along phyllosilicate bands that apparentlycounteracts any strain hardening resulting from interlocking of grains. 

If the clay content of the host rock is high enough (more than 40%), the deformation band turns into a clay smear. Clay smears typically show striations and classify as slip surfaces rather than deformation bands. Examples of deformation bands turning into clay smears as they leave sandstone layers are common.

Cataclastic bands form where mechanical grain breaking is significant (Figure b). These are the classic deformation bands first described by Atilla Aydin from the Colorado Plateau in the western USA. He noted that many cataclastic bands consist of a central cataclastic core contained within a mantle of (usually) compacted or gently fractured grains. The core is most obvious and is characterized by grain size reduction, angular grains and significant pore space collapse (Figure b). The crushing of grains results in extensive grain interlocking, which promotes strain hardening. Strain hardening may explain the small shear displacements observed on cataclastic deformation bands (3–4 cm). Some cataclastic bands are pure compaction bands (Figure above), while most are shear bands with some compaction across them. 

Cataclastic bands occur most frequently in sandstones that have been deformed at depths of about 1.5–3 km, although evidence of cataclasis is also reported from deformation bands deformed at shallower depths. Comparison suggests that shallowly formed cataclastic deformation bands show less intensive cataclasis than those formed at 1.5–3 km depth. 

Cementation and dissolution of quartz and other minerals may occur preferentially in deformation bands where diagenetic minerals grow on the fresh surfaces formed during grain crushing and/or grain boundary sliding. Such preferential growth of quartz is generally seen in deformation bands in sandstones buried to more than 2–3 km depth (>90 C) and can occur long after the formation of the bands.

Influence on fluid flow 

Very dense cluster of cataclastic deformation bands in the Entrada Sandstone, Utah.

Deformation bands form a common constituent of porous oil, gas and water reservoirs, where they occur as single bands, cluster zones or in fault damage zones. Although they are unlikely to form seals that can hold significant hydrocarbon columns over geologic time, they can influence fluid flow in some cases. Their ability to do so depends on their internal permeability structure and thickness or frequency. Clearly, the zone of cataclastic deformation bands shown in Figure above will have a far greater influence on fluid flow than the single cataclastic band shown in Figure a or b at the top.

Cataclastic deformation bands show the most significant permeability reductions.

Deformation band permeability is governed by the deformation mechanisms operative during their formation, which again depends on a number of lithological and physical factors. In general, disaggregation bands show little porosity and permeability reduction, while phyllosilicate and, particularly, cataclastic bands show permeability reductions up to several orders of magnitude. Deformation bands are thin, so the number of deformation bands (their cumulative thickness) is important when their role in a petroleum reservoir is to be evaluated. 

Conjugate (simultaneous and oppositely dipping) sets of cataclastic deformation bands in sandstone. Note the positive relief of the deformation bands due to grain crushing and cementation. The bands fade away downward into the more fine-grained and less-sorted unit. Entrada Sandstone, Utah.

Also important are their continuity, variation in porosity/permeability and orientation. Many show significant variations in permeability along strike and dip due to variations in amount of cataclasis, compaction or phyllosilicate smearing. Deformation bands tend to define sets with preferred orientation (Figure above), for instance in damage zones, and this anisotropy can influence the fluid flow in a petroleum reservoir, for example during water injection. All of these factors make it difficult to evaluate the effect of deformation bands in reservoirs, and each reservoir must be evaluated individually according to local parameters such as time and depth of deformation, burial and cementation history, mineralogy, sedimentary facies and more.

The influence of deformation bands on petroleum or groundwater production depends on the permeability contrast, cumulative thickness, orientations, continuity and connectivity.

What type of structure forms, where and when? 

Given the various types of deformation bands and their different effects on fluid flow, it is important to understand the underlying conditions that control when and where they form. A number of factors are influential, including burial depth, tectonic environment (state of stress) and host rock properties, such as degree of lithification, mineralogy, grain size, sorting and grain shape. Some of these factors, particularly mineralogy, grain size, rounding, grain shape and sorting, are more or less constant for a given sedimentary rock layer. They may, however, vary from layer to layer, which is why rapid changes in deformation band development may be seen from one layer to the next. 

Other factors, such as porosity, permeability, confining pressure, stress state and cementation, are likely to change with time. The result may be that early deformation bands are different from those formed at later stages in the same porous rock layer, for example at deeper burial depths. Hence, the sequence of deformation structures in a given rock layer reflects the physical changes that the sediment has experienced throughout its history of burial, lithification and uplift. 

Different types of deformation bands form at different stages during burial. Extension fractures (Mode I fractures) are most likely to form during uplift. 

To illustrate a typical structural development of sedimentary rocks that go through burial and then uplift, we use the diagram and add characteristic structures (Figure above). The earliest forming deformation bands in sandstones are typically disaggregation bands or phyllosilicate bands. Such structures form at low confining pressures (shallow burial) when forces across grain contact surfaces are low and grain bindings are weak, and are therefore indicated at shallow levels in Figures above and figure at the end. Many early disaggregation bands are related to local, gravity-controlled deformation such as local shale diapirism, underlying salt movement, gravitational sliding and glaciotectonics. 

Cataclastic deformation bands can occur in poorly lithified layers of pure sand at shallow burial depths, but are much more common in sandstones deformed at 1–3 km depth. Factors promoting shallow-burial cataclasis include small grain contact areas, i.e. good sorting and well-rounded grains, the presence of feldspar or

other non-platy minerals with cleavage and lower hardness than quartz, and weak lithic fragments. Quartz, for instance, seldom develops transgranular fractures under low confining pressure, but may fracture by flaking or spalling. At deeper depths, extensive cataclasis is promoted by high grain contact stresses. Abundant examples of cataclastic deformation bands are found in the Jurassic sandstones of the Colorado Plateau, where the age relation between early disaggregation bands and later cataclastic bands is very consistent (Figure above).

When a sandstone becomes cohesive and loses porosity during lithification (left side of Figure above), deformation occurs by crack propagation instead of pore space collapse, and slip surfaces, joints and mineral-filled fractures form directly without any precursory formation of deformation bands. This is why late, overprinting structures are almost invariably slip surfaces, joints and mineral-filled fractures. Slip surfaces can also form by faulting of low-porosity deformation band zones at any burial depth. 

Joints and veins typically postdate both disaggregation bands and cataclastic bands in sandstones. The transition from deformation banding to jointing may occur as porosity is reduced, notably through quartz dissolution and precipitation. Since the effect of such diagenetically controlled strengthening may vary locally, deformation bands and joints may develop simultaneously in different parts of a sandstone layer, but the general pattern is deformation bands first, then faulted deformation bands (slip surface formation) and finally joints (tensile fractures in Figure above) and perhaps faulted joints. 

The latest fractures in uplifted sandstones tend to form extensive and regionally mappable joint sets generated or at least influenced by removal of overburden and cooling during regional uplift. Such joints are pronounced where sandstones have been uplifted and exposed, such as on the Colorado Plateau, but are unlikely to be developed in subsurface petroleum reservoirs unexposed to significant uplift. It therefore appears that knowing the burial/uplift history of a basin in relation to the timing of deformation events is very useful when considering the type of structures present in, say, a sandstone reservoir. Conversely, examination of the type of deformation structure present also gives information about deformation depth and other conditions at the time of deformation.

Tentative illustration of how different deformation band types relate to phyllosilicate content and depth. Many other factors influence the boundaries outlined in this diagram, and the boundaries should be considered as uncertain.

Credits: Haakon Fossen (Structural Geology)

Why perform strain analysis?

Why perform strain analysis?

Why do we perform strain analysis?. It can be important to retrieve information about strain from deformed rocks. First of all, strain analysis gives us an opportunity to explore the state of strain in a rock and to map out strain variations in a sample, an outcrop or a region. Strain data are important in the mapping and understanding of shear zones in orogenic belts. Strain measurements can also be used to estimate the amount of offset across a shear zone. It is possible to extract important information from shear zones if strain is known. 

In many cases it is useful to know if the strain is planar or three dimensional. If planar, an important criterion for section balancing is fulfilled, be it across orogenic zones or extensional basins. The shape of the strain ellipsoid may also contain information about how the deformation occurred. Oblate (pancake-shaped) strain in an orogenic setting may, for example, indicate flattening strain related to gravity-driven collapse rather than classic push-from-behind thrusting. 

The orientation of the strain ellipsoid is also important, particularly in relation to rock structures. In a shear zone setting, it may tell us if the deformation was simple shear or not. Strain in folded layers helps us to understand fold-forming mechanism(s). Studies of deformed reduction spots in slates give good estimates on how much shortening has occurred across the foliation in such rocks, and strain markers in sedimentary rocks can sometimes allow for reconstruction of original sedimentary thickness. 

Strain in one dimension

Two elongated belemnites in Jurassic limestone in the Swiss Alps. The different ways that the two belemnites have been stretched give us some two-dimensional information about the strain field: the upper belemnite has experienced sinistral shear strain while the lower one has not and must be close to the maximum stretching direction.

One-dimensional strain analyses are concerned with changes in length and therefore the simplest form of strain analysis we have. If we can reconstruct the original length of an object or linear structure we can also calculate the amount of stretching or shortening in that direction. Objects revealing the state of strain in a deformed rock are known as strain markers. Examples of strain markers indicating change in length are boudinaged dikes or layers, and minerals or linear fossils such as belemnites or graptolites that have been elongated, such as the stretched Swiss belemnites shown in Figure above. Or it could be a layer shortened by folding. It could even be a faulted reference horizon on a geologic or seismic profile. The horizon may be stretched by normal faults or shortened by reverse faults, and the overall strain is referred to as brittle strain. One-dimensional strain is revealed when the horizon, fossil, mineral or dike is restored to its pre-deformational state.

Strain in two dimensions

Reduction spots in Welsh slate. The light spots formed as spherical volumes of bleached (chemically reduced) rock. Their new shapes are elliptical in cross-section and oblate (pancake-shaped) in three dimensions, reflecting the tectonic strain in these slates.

In two-dimensional strain analyses we look for sections that have objects of known initial shape or contain linear markers with a variety of orientations (Figure first). Strained reduction spots of the type shown in Figure above are perfect, because they tend to have spherical shapes where they are undeformed. There are also many other types of objects that can be used, such as sections through conglomerates, breccias, corals, reduction spots, oolites, vesicles, pillow lavas (Figure below), columnar basalt, plutons and so on. Two-dimensional strain can also be calculated from one-dimensional data that represent different directions in the same section. A typical example would be dikes with different orientations that show different amounts of extension.

Section through a deformed Ordovician pahoe-hoe lava. The elliptical shapes were originally more circular, and Hans Reusch, who made the sketch in the 1880s, understood that they had been flattened during deformation. The Rf/f, center-to-center, and Fry methods would all be worth trying in this case.

Strain extracted from sections is the most common type of strain data, and sectional data can be combine to estimate the three-dimensional strain ellipsoid.

Changes in angles 

Strain can be found if we know the original angle between sets of lines. The original angular relations between structures such as dikes, foliations and bedding are sometimes found in both undeformed and deformed states, i.e. outside and inside a deformation zone. We can then see how the strain has affected the angular relationships and use this information to estimate strain. In other cases orthogonal lines of symmetry found in undeformed fossils such as trilobites, brachiopods and worm burrows (angle with layering) can be used to determine the angular shear in some deformed sedimentary rocks. In general, all we need to know is the change in angle between sets of lines and that there is no strain partitioning due to contrasting mechanical properties of the objects with respect to the enclosing rock.

If the angle was 90 degree in the undeformed state, the change in angle is the local angular shear. The two originally orthogonal lines remain orthogonal after the deformation, then they must represent the principal strains and thus the orientation of the strain ellipsoid. Observations of variously oriented line sets thus give information about the strain ellipse or ellipsoid. All we need is a useful method. Two of the most common methods used to find strain from initially orthogonal lines are known as the Wellman and Breddin methods, and are presented in the following sections.

The Wellman method 

Wellman’s method involves construction of the strain ellipse by drawing parallelograms based on the orientation of originally orthogonal pairs of lines. The deformation was produced on a computer and is a homogeneous simple shear. However, the strain ellipse itself tells us nothing about the degree of coaxiality: the same result could have been attained by pure shear.

This method dates back to 1962 and is a geometric construction for finding strain in two dimensions (in a section). It is typically demonstrated on fossils with orthogonal lines of symmetry in the undeformed state. In Figure above a we use the hinge and symmetry lines of brachiopods. A line of reference must be drawn (with arbitrary orientation) and pairs of lines that were orthogonal in the unstrained state are identified. The reference line must have two defined endpoints, named A and B in Figure above b. A pair of lines is then drawn parallel to both the hinge line and symmetry line for each fossil, so that they intersect at the endpoints of the reference line. The other points of intersection are marked (numbered 1–6 in Figure above b, c). If the rock is unstrained, the lines will define rectangles. If there is a strain involved, they will define parallelograms. To find the strain ellipse, simply fit an ellipse to the numbered corners of the parallelograms (Figure above c). If no ellipse can be fitted to the corner points of the rectangles the strain is heterogeneous or, alternatively, the measurement or assumption of initial orthogonality is false. The challenge with this method is, of course, to find enough fossils or other features with initially orthogonal lines typically 6–10 are needed.

The Breddin graph 

The data from the previous figure plotted in a Breddin graph. The data points are close to the curve for R=2.5.

We have already stated that the angular shear depends on the orientation of the principal strains: the closer the deformed orthogonal lines are to the principal strains, the lower the angular shear. This fact is utilized in a method first published by Hans Breddin in 1956 in German (with some errors). It is based on the graph shown in Figure above, where the angular shear changes with orientation and strain magnitude R. Input are the angular shears and the orientations of the sheared line pairs with respect to the principal strains. These data are plotted in the so-called Breddin graph and the R-value (ellipticity of the strain ellipse) is found by inspection (Figure above). This method may work even for only one or two observations. 

In many cases the orientation of the principal axes is unknown. In such cases the data are plotted with respect to an arbitrarily drawn reference line. The data are then moved horizontally on the graph until they fit one of the curves, and the orientations of the strain axes are then found at the intersections with the horizontal axis (Figure above). In this case a larger number of data are needed for good results.

Elliptical objects and the Rf/f-method 

Objects with initial circular (in sections) or spherical (in three dimensions) geometry are relatively uncommon, but do occur. Reduction spots and ooliths perhaps form the most perfect spherical shapes in sedimentary rocks. When deformed homogeneously, they are transformed into ellipses and ellipsoids that reflect the local finite strain. Conglomerates are perhaps more common and contain clasts that reflect the finite strain. In contrast to oolites and reduction spots, few pebbles or cobbles in a conglomerate are spherical in the undeformed state. This will of course influence their shape in the deformed state and causes a challenge in strain analyses. However, the clasts tend to have their long axes in a spectrum of orientations in the undeformed state, in which case methods such as the Rf/f-method may be able to take the initial shape factor into account.

The Rf/f method illustrated. The ellipses have the same ellipticity (Ri) before the deformation starts. The Rf–f diagram to the right indicates that Ri=2. A pure shear is then added with Rs=1.5 followed by a pure shear strain of Rs=3. The deformation matrices for these two deformations are shown. Note the change in the distribution of points in the diagrams to the right. Rs in the diagrams is the actual strain that is added. Modified from Ramsay and Huber (1983).

The Rf/f-method was first introduced by John Ramsay in his well-known 1967 textbook and was later improved. The method is illustrated in Figure above. The markers are assumed to have approximately elliptical shapes in the deformed (and undeformed) state, and they must show a significant variation in orientations for the method to work.

The Rf/f-method handles initially non-spherical markers, but the method requires a significant variation in the orientations of their long axes.

The ellipticity (X/Y) in the undeformed (initial) state is called Ri. In our example (Figure above) Ri=2. After a strain Rs the markers exhibit new shapes. The new shapes are different and depend on the initial orientation of the elliptical markers. The new (final) ellipticity for each deformation marker is called Rf and the spectrum of Rf-values is plotted against their orientations, or more specifically against the angle f' between the long axis of the ellipse and a reference line (horizontal in Figure above). In our example we have applied two increments of pure shear to a series of ellipses with different orientations. All the ellipses have the same initial shape Ri=2, and they plot along a vertical line in the upper right diagram in Figure above. Ellipse 1 is oriented with its long axis along the minimum principal strain axis, and it is converted into an ellipse that shows less strain (lower Rf-value) than the true strain ellipse (Rs). Ellipse 7, on the other hand, is oriented with its long axis parallel to the long axis of the strain ellipse, and the two ellipticities are added. This leads to an ellipticity that is higher than Rs. When Rs=3, the true strain Rs is located somewhere between the shape represented by ellipses 1 and 7, as seen in Figure above (lower right diagram). 

For Rs=1.5 we still have ellipses with the full spectrum of orientations ( 90 to 90 ; see middle diagram in Figure above), while for Rs=3 there is a much more limited spectrum of orientations (lower graph in Figure above). The scatter in orientation is called the fluctuation F. An important change happens when ellipse 1, which has its long axis along the Z-axis of the strain ellipsoid, passes the shape of a circle (Rs=Ri,) and starts to develop an ellipse whose long axis is parallel to X. This happens when Rs=2, and for larger strains the data points define a circular shape. Inside this shape is the strain Rs that we are interested in. But where exactly is Rs? A simple average of the maximum and minimum Rf-values would depend on the original distribution of orientations. Even if the initial distribution is random, the average R-value would be too high, as high values tend to be over represented (Figure above, lower graph). 

To find Rs we have to treat the cases where Rs >Ri and Rs <Ri separately. In the latter case, which is represented by the middle graph in Figure above, we have the following expressions for the maximum and minimum value for Rf:

In both cases the orientation of the long (X) axis of the strain ellipse is given by the location of the maximum Rf-values. Strain could also be found by fitting the data to pre-calculated curves for various values for Ri and Rs. In practice, such operations are most efficiently done by means of computer programs.

The example shown in Figure above and discussed above is idealized in the sense that all the undeformed elliptical markers have identical ellipticity. What if this were not the case, i.e. some markers were more elliptical than others? Then the data would not have defined a nice curve but a cloud of points in the Rf/f-diagram. Maximum and minimum Rf-values could still be found and strain could be calculated using the equations above. The only change in the equation is that Ri now represents the maximum ellipticity present in the undeformed state. 

Another complication that may arise is that the initial markers may have had a restricted range of orientations. Ideally, the Rf/f-method requires the elliptical objects to be more or less randomly oriented prior to deformation. Conglomerates, to which this method commonly is applied, tend to have clasts with a preferred orientation. This may result in an Rf–f plot in which only a part of the curve or cloud is represented. In this case the maximum and minimum Rf-values may not be representative, and the formulas above may not give the correct answer and must be replaced by a computer based iterative retrodeformation method where X is input. However, many conglomerates have a few clasts with initially anomalous orientations that allow the use of Rf/f analysis.

Center-to-center method 

The center-to-center method. Straight lines are drawn between neighbouring object centers. The length of each line (d') and the angle (a') that they make with a reference line are plotted in the diagram. The data define a curve that has a maximum at X and a minimum at the Y-value of the strain ellipse, and where Rs= X/Y.

This method, here demonstrated in Figure above, is based on the assumption that circular objects have a more or less statistically uniform distribution in our section(s). This means that the distances between neighboring particle centers were fairly constant before deformation. The particles could represent sand grains in well-sorted sandstone, pebbles, ooids, mud crack centers, pillow-lava or pahoe-hoe lava centers, pluton centers or other objects that are of similar size and where the centers are easily definable. If you are uncertain about how closely your section complies with this criterion, try anyway. If the method yields a reasonably well-defined ellipse, then the method works.

The method itself is simple and is illustrated in Figure above: Measure the distance and direction from the center of an ellipse to those of its neighbours. Repeat this for all ellipses and graph the distance d' between the centers and the angles a' between the center tie lines and a reference line. A straight line occurs if the section is unstrained, while a deformed section yields a curve with maximum (d' max) and minimum values (d' min). The ellipticity of the strain ellipse is then given by the ratio: Rs =( d' max)/(d' min).

The Fry method

The Fry method performed manually. (a) The centerpoints for the deformed objects are transferred to a transparent overlay. A central point (1 on the figure) is defined. (b) The transparent paper is then moved to another of the points (point 2) and the centerpoints are again transferred onto the paper (the overlay must not be rotated). The procedure is repeated for all of the points, and the result (c) is an image of the strain ellipsoid (shape and orientation). Based on Ramsay and Huber (1983).

A quicker and visually more attractive method for finding two-dimensional strain was developed by Norman Fry at the end of the 1970s. This method, illustrated in Figure above, is based on the center-to-center method and is most easily dealt with using one of several available computer programs. It can be done manually by placing a tracing overlay with a coordinate origin and pair of reference axes on top of a sketch or picture of the section. The origin is placed on a particle center and the centers of all other particles (not just the neighbours) are marked on the tracing paper. The tracing paper is then moved, without rotating the paper with respect to the section, so that the origin covers a second particle center, and the centers of all other particles are again marked on the tracing paper. This procedure is repeated until the area of interest has been covered. For objects with a more or less uniform distribution the result will be a visual representation of the strain ellipse.The ellipse is the void area in the middle, defined by the point cloud around it (Figure above c). 

The Fry method, as well as the other methods presented in this section, outputs two-dimensional strain. Three-dimensional strain is found by combining strain estimates from two or more sections through the deformed rock volume. If sections can be found that each contain two of the principal strain axes, then two sections are sufficient. In other cases three or more sections are needed, and the three-dimensional strain must be calculated by use of a computer.

Strain in three dimensions

Three-dimensional strain expressed as ellipses on different sections through a conglomerate. The foliation (XY-plane) and the lineation (X-axis) are annotated. This illustration was published in 1888, but what are now routine strain methods were not developed until the 1960s.

A complete strain analysis is three-dimensional. Three dimensional strain data are presented in the Flinn diagram or similar diagrams that describe the shape of the strain ellipsoid, also known as the strain geometry. In addition, the orientation of the principal strains can be presented by means of stereographic nets. Direct field observations of three-dimensional strain are rare. In almost all cases, analysis is based on two-dimensional strain observations from two or more sections at the same locality (Figure above). A well-known example of three-dimensional strain analysis from deformed conglomerates is presented in below heading. 

In order to quantify ductile strain, be it in two or three dimensions, the following conditions need to be met:

The strain must be homogeneous at the scale of observation, the mechanical properties of the objects must have been similar to those of their host rock during the deformation, and we must have a reasonably good knowledge about the original shape of strain markers.

The strain must be homogeneous at the scale of observation, the mechanical properties of the objects must have been similar to those of their host rock during the deformation, and we must have a reasonably good knowledge about the original shape of strain markers.

The second point is an important one. For ductile rocks it means that the object and its surroundings must have had the same competence or viscosity. Otherwise the strain recorded by the object would be different from that of its surroundings. This effect is one of several types of strain partitioning, where the overall strain is distributed unevenly in terms of intensity and/or geometry in a rock volume. As an example, we mark a perfect circle on a piece of clay before flattening it between two walls. The circle transforms passively into an ellipse that reveals the two-dimensional strain if the deformation is homogeneous. If we embed a coloured sphere of the same clay, then it would again deform along with the rest of the clay, revealing the three-dimensional strain involved. However, if we put a stiff marble in the clay the result is quite different. The marble remains unstrained while the clay around it becomes more intensely and heterogeneously strained than in the previous case. In fact, it causes a more heterogeneous strain pattern to appear. Strain markers with the same mechanical properties as the surroundings are called passive strain markers because they deform passively along with their surroundings. Those that have anomalous mechanical properties respond differently than the surrounding medium to the overall deformation, and such markers are called active strain markers. 

Strain obtained from deformed conglomerates, plotted in the Flinn diagram. Different pebble types show different shapes and finite strains. Polymict conglomerate of the Utslettefjell Formation, Stord, southwest Norway. 

An example of data from active strain markers is shown in Figure above. These data were collected from a deformed polymictic conglomerate where three-dimensional strain has been estimated from different clast types in the same rock and at the same locality. Clearly, the different clast types have recorded different amounts of strain. Competent (stiff) granitic clasts are less strained than less competent greenstone clasts. This is seen using the fact that strain intensity generally increases with increasing distance from the origin in Flinn space. But there is another interesting thing to note from this figure: It seems that competent clasts plot higher in the Flinn diagram (Figure. above) than incompetent(“soft”) clasts, meaning that competent clasts take on a more prolate shape. Hence, not only strain intensity but also strain geometry may vary according to the mechanical properties of strain markers. 

The way that the different markers behave depends on factors such as their mineralogy, preexisting fabric, grain size, water content and temperature-pressure conditions at the time of deformation. In the case of Figure above, the temperature-pressure regime is that of lower to middle greenschist facies. At higher temperatures, quartz-rich rocks are more likely to behave as “soft” objects, and the relative positions of clast types in Flinn space are expected to change. 

The last point above also requires attention: the initial shape of a deformed object clearly influences its postdeformational shape. If we consider two-dimensional objects such as sections through oolitic rocks, sandstones or conglomerates, the Rf/f method discussed above can handle this type of uncertainty. It is better to measure up two or more sections through a deformed rock using this method than dig out an object and measure its three-dimensional shape. The single object could have an unexpected initial shape (conglomerate clasts are seldom perfectly spherical or elliptical), but by combining numerous measurements in several sections we get a statistical variation that can solve or reduce this problem.

Three-dimensional strain is usually found by combining two-dimensional data from several differently oriented sections.

There are now computer programs that can be used to extract three-dimensional strain from sectional data. If the sections each contain two of the principal strain axes everything becomes easy, and only two are strictly needed (although three would still be good). Otherwise, strain data from at least three sections are required.

Deformed quartzite conglomerates

Quartz or quartzite conglomerates with a quartzite matrix are commonly used for strain analyses. The more similar the mineralogy and grain size of the matrix and the pebbles, the less deformation partitioning and the better the strain estimates. A classic study of deformed quartzite conglomerates is Jake Hossack’s study of the Norwegian Bygdin conglomerate, published in 1968. Hossack was fortunate he found natural sections along the principal planes of the strain ellipsoid at each locality. Putting the sectional data together gave the three dimensional state of strain (strain ellipsoid) for each locality. Hossack found that strain geometry and intensity varies within his field area. He related the strain pattern to static flattening under the weight of the overlying Caledonian Jotun Nappe. Although details of his interpretation may be challenged, his work demonstrates how conglomerates can reveal a complicated strain pattern that otherwise would have been impossible to map. 

Hossack noted the following sources of error:

• Inaccuracy connected with data collection (sections not being perfectly parallel to the principal planes of strain and measuring errors).  
• Variations in pebble composition.  
• The pre-deformational shape and orientation of the pebbles.  
• Viscosity contrasts between clasts and matrix.  
• Volume changes related to the deformation (pressure solution).  
• The possibility of multiple deformation events.

Credits: Haakon Fossen (Structural Geology)

Siccar Point - the world's most important geological site and the birthplace of modern geology

Siccar Point is world-famous as the most important unconformity described by James Hutton (1726-1797) in support of his world-changing ideas on the origin and age of the Earth.

James Hutton unconformity with annotations - Siccar Point 

In 1788, James Hutton first discovered Siccar Point, and understood its significance. It is by far the most spectacular of several unconformities that he discovered in Scotland, and very important in helping Hutton to explain his ideas about the processes of the Earth.At Siccar Point, gently sloping strata of 370-million-year-old Famennian Late Devonian Old Red Sandstone and a basal layer of conglomerate overlie near vertical layers of 435-million-year-old lower Silurian Llandovery Epoch greywacke, with an interval of around 65 million years.

Standing on the angular unconformity at Siccar Point (click to enlarge). Photo: Chris Rowan, 2009

As above, with annotations. Photo: Chris Rowan, 2009

Hutton used Siccar Point to demonstrate the cycle of deposition, folding, erosion and further deposition that the unconformity represents. He understood the implication of unconformities in the evidence that they provided for the enormity of geological time and the antiquity of planet Earth, in contrast to the biblical teaching of the creation of the Earth. 

   

How the unconformity at Siccar Point formed.

At this range, it is easy to spot that the contact between the two units is sharp, but it is not completely flat. Furthermore, the lowest part of the overlying Old Red Sandstone contains fragments of rock that are considerably larger than sand; some are at least as large as your fist, and many of the fragments in this basal conglomerate are bits of the underlying Silurian greywacke. These are all signs that the greywackes were exposed at the surface, being eroded, for a considerable period of time before the Old Red Sandstone was laid down on top of them.

The irregular topography and basal conglomerate show that this is an erosional contact. Photo: Chris Rowan, 2009

The Siccar Point which is a rocky promontory in the county of Berwickshire on the east coast of Scotland.

What is deformation?

The term deformation is, like several other structural geology terms, used in different ways by different people and under different circumstances. In most cases, particularly in the field, the term refers to the distortion (strain) that is expressed in a (deformed) rock. This is also what the word literally means: a change in form or shape. However, rock masses can be translated or rotated as rigid units during deformation, without any internal change in shape. For instance, fault blocks can move during deformation without accumulating any internal distortion. Many structural geologists want to include such rigid displacements in the term deformation, and we refer to them as rigid body deformation, as opposed to non-rigid body deformation (strain or distortion).

Deformation is the transformation from an initial to a final geometry by means of rigid body translation, rigid body rotation, strain (distortion) and/or volume change.

Fig. 1. Displacement field and particle paths for rigid translation and rotation, and strain resulting from simple shear, subsimple shear and pure shear. Particle paths trace the actual motion of individual particles in the deforming rock, while displacement vectors simply connect the initial and final positions. Hence, displacement vectors can be constructed from particle paths, but not the other way around.

It is useful to think of a rock or rock unit in terms of a continuum of particles. Deformation relates the positions of particles before and after the deformation history, and the positions of points before and after deformation can be connected with vectors. These vectors are called displacement vectors, and a field of such vectors is referred to as the displacement field. Displacement vectors, such as those displayed in the central column of Fig. 1, do not tell us how the particles moved during the deformation history they merely link the undeformed and deformed states. The actual path that each particle follows during the deformation history is referred to as a particle path, and for the deformations shown in Fig. 1 the paths are shown in the right column (green arrows). When specifically referring to the progressive changes that take place during deformation, terms such as deformation history or progressive deformation should be used.

Components of deformation

Fig. 2. (a) The total deformation of an object (square with an internal circle). Arrows in (a) are displacement vectors connecting initial and final particle positions. Arrows in (b)–(e) are particle paths. (b, c) Translation and rotation components of the deformation shown in (a). (d) The strain component. A new coordinate system (x, y) is introduced (d). This internal system eliminates the translation and rotation (b, c) and makes it easier to reveal the strain component, which is here produced by a simple shear (e).

 

The displacement field can be decomposed into various components, depending on the purpose of the decomposition. The classic way of decomposing it is by separating rigid body deformation in the form of rigid translation and rotation from change in shape and volume. In Fig. 2 the translation component is shown in (b), the rotation component in (c) and the rest (the strain) in (d). Let us have a closer look at these expressions.

Translation

Fig. 3. The Jotun Nappe in the Scandinavian Caledonides seems to have been transported more than 300 km to the southeast, based on restoration and the orientation of lineations. The displacement vectors are indicated, but the amount of rigid rotation around the vertical axis is unknown. The amount of strain is generally concentrated to the base.

Translation moves every particle in the rock in the same direction and the same distance, and its displacement field consists of parallel vectors of equal length. Translations can be considerable, for instance where thrust nappes (detached slices of rocks) have been transported several tens or hundreds of kilometres. The Jotun Nappe (Fig. 3) is an example from the Scandinavian Caledonides. In this case most of the deformation is rigid translation. We do not know the exact orientation of this nappe prior to the onset of deformation, so we cannot estimate the rigid rotation (see below), but field observations reveal that the change in shape, or strain, is largely confined to the lower parts. The total deformation thus consists of a huge translation component, an unknown but possibly small rigid rotation component and a strain component localised to the base of the nappe.

On a smaller scale, rock components (mineral grains, layers or fault blocks) may be translated along slip planes or planar faults without any internal change in shape.

Rotation

Rotation is here taken to mean rigid rotation of the entire deformed rock volume that is being studied. It should not be confused with the rotation of the (imaginary) axes of the strain ellipse during progressive deformation. Rigid rotation involves a uniform physical rotation of a rock volume (such as a shear zone) relative to an external coordinate system.

Large-scale rotations of a major thrust nappe or entire tectonic plate typically occur about vertical axes. Fault blocks in extensional settings, on the other hand, may rotate around horizontal axes, and small-scale rotations may occur about any axis.

Strain 

Strain or distortion is non-rigid deformation and relatively simple to define:

Any change in shape, with or without change in volume, is referred to as strain, and it implies that particles in a rock have changed positions relative to each other.

A rock volume can be transported (translated) and rotated rigidly in any way and sequence, but we will never be able to tell just from looking at the rock itself. All we can see in the field or in samples is strain, and perhaps the way that strain has accumulated. Consider your lunch bag. You can bring it to school or work, which involves a lot of rotation and translation, but you cannot see this deformation directly. It could be that your lunch bag has been squeezed on your way to school – you can tell by comparing it with what it looked like before you left home. If someone else prepared your lunch and put it in your bag, you would use your knowledge of how a lunch bag should be shaped to estimate the strain (change in shape) involved. 

The last point is very relevant, because with very few exceptions, we have not seen the deformed rock in its undeformed state. We then have to use our knowledge of what such rocks typically look like when unstrained. For example, if we find strained ooliths or reduction spots in the rock, we may expect them to have been spherical (circular in cross-section) in the undeformed state.

Volume change

Even if the shape of a rock volume is unchanged, it may have shrunk or expanded. We therefore have to add volume change (area change in two dimensions) for a complete description of deformation. Volume change, also referred to as dilation, is commonly considered to be a special type of strain, called volumetric strain. However, it is useful to keep this type of deformation separate if possible.

System of reference

For studies of deformation, a reference or coordinate system must be chosen. Standing on a dock watching a big ship entering or departing can give the impression that the dock, not the ship, is moving. Unconsciously, the reference system gets fixed to the ship, and the rest of the world moves by translation relative to the ship. While this is fascinating, it is not a very useful choice of reference. Rock deformation must also be considered in the frame of some reference coordinate system, and it must be chosen with care to keep the level of complexity down.

We always need a reference frame when dealing with displacements and kinematics.

It is often useful to orient the coordinate system along important geologic structures. This could be the base of a thrust nappe, a plate boundary or a local shear zone. In many cases we want to eliminate translation and rigid rotation. In the case of shear zones we normally place two axes parallel to the shear zone with the third being perpendicular to the zone. If we are interested in the deformation in the shear zone as a whole, the origin could be fixed to the margin of the zone. If we are interested in what is going on around any given particle in the zone we can “glue” the origin to a particle within the zone (still parallel/perpendicular to the shear zone boundaries). In both cases translation and rigid rotation of the shear zone are eliminated, because the coordinate system rotates and translates along with the shear zone. There is nothing wrong with a coordinate system that is oblique to the shear zone boundaries, but visually and mathematically it makes things more complicated.

Deformation: detached from history

Deformation is the difference between the deformed and undeformed states. It tells us nothing about what actually happened during the deformation history.

A given strain may have accumulated in an infinite number of ways.

Imagine a tired student (or professor for that matter) who falls asleep in a boat while fishing on the sea or a lake. The student knows where he or she was when falling asleep, and soon figures out the new location when waking up,but the exact path that currents and winds have taken the boat is unknown. The student only knows the position of the boat before and after the nap, and can evaluate the strain (change in shape) of the boat (hopefully zero). One can map the deformation, but not the deformation history. 

Let us also consider particle flow: Students walking from one lecture hall to another may follow infinitely many paths (the different paths may take longer or shorter time, but deformation itself does not involve time). All the lecturer knows, busy between classes, is that the students have moved from one lecture hall to the other. Their history is unknown to the lecturer (although he or she may have some theories based on cups of hot coffee etc.). In a similar way, rock particles may move along a variety of paths from the undeformed to the deformed state. One difference between rock particles and individual students is of course that students are free to move on an individual basis, while rock particles, such as mineral grains in a rock, are “glued” to one another in a solid continuum and cannot operate freely.

Homogeneous and heterogeneous deformation

Where the deformation applied to a rock volume is identical throughout that volume, the deformation is homogeneous. Rigid rotation and translation by definition are homogenous, so it is always strain and volume or area change that can be heterogeneous. Thus homogeneous deformation and homogeneous strain are equivalent expressions. 

Fig. 4. Homogeneous deformations of a rock with brachiopods, reduction spots, ammonites and dikes. Two different deformations are shown (pure and simple shear). Note that the brachiopods that are differently oriented before deformation obtain different shapes.

For homogeneous deformation, originally straight and parallel lines will be straight and parallel also after the deformation, as demonstrated in Fig. 4. Further, the strain and volume/area change will be constant throughout the volume of rock under consideration. If not, then the deformation is heterogeneous (inhomogeneous). This means that two objects with identical initial shape and orientation will end up having identical shape and orientation after the deformation. Note, however, that the initial shape and orientation in general will differ from the final shape and orientation. If two objects have identical shapes but different orientations before deformation, then they will generally have different shapes after deformation even if the deformation is homogeneous. An example is the deformed brachiopods in Fig. 4. The difference reflects the strain imposed on the rock.

Homogeneous deformation: Straight lines remain straight, parallel lines remain parallel, and identically shaped and oriented objects will also be identically shaped and oriented after the deformation.

A circle will be converted into an ellipse during homogeneous deformation, where the ellipticity (ratio between the long and short axes of the ellipse) will depend on the type and intensity of the deformation. Mathematically, this is identical to saying that homogeneous deformation is a linear transformation. Homogeneous deformation can therefore be described by a set of first-order equations (three in three dimensions) or, more simply, by a transformation matrix referred to as the deformation matrix. 

Fig. 5. A regular grid in undeformed and deformed state. The overall strain is heterogeneous, so that some of the straight lines have become curved. However, in a restricted portion of the grid, the strain is homogeneous. In this case the strain is also homogeneous at the scale of a grid cell.

Before looking at the deformation matrix, the point made in Fig. 5 must be emphasized:

A deformation that is homogeneous on one scale may be considered heterogeneous on a different scale.

Fig. 6. Discrete or discontinuous deformation can be approximated as continuous and even homogeneous in some cases. In this sense the concept of strain can also be applied to brittle deformation (brittle strain). The success of doing so depends on the scale of observation.

A classic example is the increase in strain typically seen from the margin toward the centre of a shear zone. The strain is heterogeneous on this scale, but can be subdivided into thinner elements or zones in which strain is approximately homogeneous. Another example is shown in Fig. 6, where a rock volume is penetrated by faults. On a large scale, the deformation may be considered homogeneous because the discontinuities represented by the faults are relatively small. On a smaller scale, however, those discontinuities become more apparent, and the deformation must be considered heterogeneous.

Credits: Haakon Fossen (Structural Geology)