Why do we carry out stress analysis?. It can be critical to retrieve information approximately stress from deformed rocks. First of all, strain analysis gives us an opportunity to discover the kingdom of pressure in a rock and to map out stress versions in a sample, an outcrop or a place. Strain facts are crucial in the mapping and information of shear zones in orogenic belts. Strain measurements can also be used to estimate the quantity of offset across a shear sector. It is possible to extract vital facts from shear zones if stress is known.

In many instances it's far beneficial to know if the strain is planar or three dimensional. If planar, an vital criterion for section balancing is ful?Lled, be it across orogenic zones or extensional basins. The shape of the stress ellipsoid might also include statistics about how the deformation occurred. Oblate (pancake-shaped) pressure in an orogenic setting may, for instance, suggest ?Attening pressure related to gravity-driven fall apart in preference to conventional push-from-in the back of thrusting.

The orientation of the strain ellipsoid is likewise critical, particularly in terms of rock structures. In a shear quarter putting, it could inform us if the deformation was simple shear or not. Strain in folded layers helps us to apprehend fold-forming mechanism(s). Studies of deformed reduction spots in slates provide desirable estimates on how plenty shortening has occurred throughout the foliation in such rocks, and stress markers in sedimentary rocks can now and again permit for reconstruction of unique sedimentary thickness.

### Strain in one measurement

Two elongated belemnites in Jurassic limestone inside the Swiss Alps. The exceptional approaches that the two belemnites have been stretched provide us some -dimensional records about the pressure ?Eld: the upper belemnite has experienced sinistral shear strain at the same time as the lower one has not and have to be close to the most 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 shaped as spherical volumes of bleached (chemically reduced) rock. Their new shapes are elliptical in move-section and oblate (pancake-formed) in 3 dimensions, re?Ecting the tectonic stress 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 maximum commonplace type of stress statistics, and sectional information may be combine to estimate the 3-dimensional pressure ellipsoid.**

#### Changes in angles

Strain can be discovered if we recognise the unique perspective among sets of traces. The authentic angular family members between systems including dikes, foliations and bedding are sometimes discovered in both undeformed and deformed states, i.E. Inside and outside a deformation zone. We can then see how the strain has affected the angular relationships and use this statistics to estimate pressure. In different cases orthogonal strains of symmetry discovered in undeformed fossils which include trilobites, brachiopods and malicious program burrows (perspective with layering) may be used to decide the angular shear in some deformed sedimentary rocks. In widespread, all we need to recognize is the trade in attitude between sets of traces and that there is no stress partitioning due to contrasting mechanical homes of the items with recognize to the enclosing rock.

If the perspective changed into ninety degree in the undeformed state, the alternate in attitude is the local angular shear. The in the beginning orthogonal strains continue to be orthogonal after the deformation, then they should represent the primary strains and for that reason the orientation of the strain ellipsoid. Observations of variously oriented line units therefore deliver data about the stress ellipse or ellipsoid. All we need is a useful technique. Two of the maximum commonplace strategies used to ?Nd pressure from to start with orthogonal traces are known as the Wellman and Breddin strategies, and are supplied within the following sections.

#### The Wellman approach

Wellman?S approach includes creation of the stress ellipse through drawing parallelograms based on the orientation of in the beginning orthogonal pairs of lines. The deformation was produced on a laptop and is a homogeneous easy shear. However, the pressure ellipse itself tells us nothing approximately the degree of coaxiality: the same end result might have been attained via natural 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

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 foremost axes is unknown. In such instances the statistics are plotted with recognize to an arbitrarily drawn reference line. The facts are then moved horizontally at the graph till they ?T one of the curves, and the orientations of the stress axes are then found at the intersections with the horizontal axis (Figure above). In this case a bigger number of records are wished for top results.

#### Elliptical objects and the Rf/f-approach

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 equal ellipticity (Ri) earlier than the deformation starts. The Rf?F diagram to the proper suggests that Ri=2. A natural shear is then added with Rs=1.5 accompanied by way of a pure shear strain of Rs=three. The deformation matrices for those deformations are proven. Note the trade within the distribution of factors in the diagrams to the right. Rs inside the diagrams is the real pressure this is delivered. Modi?Ed from Ramsay and Huber (1983). |

The Rf/f-method became ?Rst brought by John Ramsay in his well-known 1967 textbook and changed into later stepped forward. The technique is illustrated in Figure above. The markers are assumed to have approximately elliptical shapes within the deformed (and undeformed) nation, and that they must display a signi?Cant variant in orientations for the method to paintings.

**The Rf/f-approach handles to begin with non-round markers, but the method calls for a signi?Cant version inside 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 lengthy (X) axis of the stress ellipse is given through the location of the maximum Rf-values. Strain can also be located by way of ?Tting the information to pre-calculated curves for numerous values for Ri and Rs. In practice, such operations are most ef?Ciently carried out by way 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 worry that can stand up is that the preliminary markers may have had a limited variety of orientations. Ideally, the Rf/f-approach requires the elliptical gadgets to be extra or much less randomly oriented previous to deformation. Conglomerates, to which this method usually is applied, generally tend to have clasts with a preferred orientation. This may additionally result in an Rf?F plot wherein simplest part of the curve or cloud is represented. In this situation the maximum and minimal Rf-values may not be representative, and the formulas above may not provide the precise solution and have to get replaced through a laptop primarily based iterative retrodeformation technique wherein X is enter. However, many conglomerates have a few clasts with initially anomalous orientations that permit the use of Rf/f analysis.

#### Center-to-center technique

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 approach, right here verified in Figure above, is based on the idea that round gadgets have a more or less statistically uniform distribution in our section(s). This means that the distances between neighboring particle facilities had been pretty consistent earlier than deformation. The debris should constitute sand grains in nicely-looked after sandstone, pebbles, ooids, dust crack facilities, pillow-lava or pahoe-hoe lava facilities, pluton centers or different objects which can be of similar size and in which the facilities are effortlessly de?Nable. If you are unsure about how carefully your phase complies with this criterion, attempt anyway. If the technique yields a fairly properly-de?Ned ellipse, then the approach works.

The technique itself is straightforward and is illustrated in Figure above: Measure the gap and route from the center of an ellipse to those of its neighbours. Repeat this for all ellipses and graph the gap d' among the facilities and the angles a' among the center tie strains and a reference line. A instantly line takes place if the phase is unstrained, even as a deformed section yields a curve with most (d' max) and minimal values (d' min). The ellipticity of the pressure ellipse is then given through the ratio: Rs =( d' max)/(d' min).

#### The Fry method

The Fry approach performed manually. (a) The centerpoints for the deformed items are transferred to a transparent overlay. A imperative factor (1 at the ?Gure) is de?Ned. (b) The obvious paper is then moved to another of the points (point 2) and the centerpoints are again transferred onto the paper (the overlay ought to now not be turned around). The method is repeated for all the points, and the result (c) is an image of the pressure 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 technique, in addition to the opposite strategies supplied on this phase, outputs -dimensional stress. Three-dimensional pressure is determined via combining pressure estimates from or more sections via the deformed rock extent. If sections can be observed that each include two of the main stress axes, then two sections are suf?Cient. In different cases three or extra sections are wished, and the 3-dimensional pressure have to be calculated via use of a laptop.

### Strain in three dimensions

Three-dimensional strain expressed as ellipses on exceptional sections through a conglomerate. The foliation (XY-plane) and the lineation (X-axis) are annotated. This instance became published in 1888, but what are actually routine strain techniques have been now not evolved until the Nineteen Sixties. |

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 stress, be it in two or three dimensions, the following conditions want to be met:

**The stress have to be homogeneous at the scale of observation, the mechanical homes of the gadgets have to have been just like those in their host rock all through the deformation, and we have to have a fairly correct understanding about the original shape of strain markers.**

The stress have to be homogeneous at the scale of observation, the mechanical homes of the gadgets have to have been just like those in their host rock all through the deformation, and we have to have a fairly correct understanding 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 acquired from deformed conglomerates, plotted in the Flinn diagram. Different pebble sorts show distinctive shapes and ?Nite traces. 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 ultimate factor above also requires interest: the initial form of a deformed object actually in?Uences its postdeformational form. If we don't forget -dimensional items such as sections via oolitic rocks, sandstones or conglomerates, the Rf/f technique mentioned above can handle this sort of uncertainty. It is higher to degree up or extra sections through a deformed rock the usage of this technique than dig out an item and degree its three-dimensional shape. The unmarried object could have an unexpected preliminary shape (conglomerate clasts are seldom perfectly round or elliptical), however by way of combining severa measurements in numerous sections we get a statistical variation that could resolve or lessen this problem.

**Three-dimensional strain is usually located by combining two-dimensional information from numerous in a different way orientated sections.**

There are actually computer applications that can be used to extract 3-dimensional pressure from sectional records. If the sections each include two of the primary strain axes everything turns into easy, and most effective two are strictly wanted (although three could still be properly). Otherwise, strain facts from at the least 3 sections are required.

Quartz or quartzite conglomerates with a quartzite matrix are normally used for strain analyses. The greater similar the mineralogy and grain length of the matrix and the pebbles, the less deformation partitioning and the higher the pressure estimates. A conventional have a look at of deformed quartzite conglomerates is Jake Hossack?S have a look at of the Norwegian Bygdin conglomerate, posted in 1968. Hossack became lucky he observed natural sections alongside the fundamental planes of the strain ellipsoid at each locality. Putting the sectional facts collectively gave the three dimensional country of pressure (stress ellipsoid) for each locality. Hossack located that pressure geometry and intensity varies inside his ?Eld vicinity. He related the stress pattern to static ?Attening below the weight of the overlying Caledonian Jotun Nappe. Although info of his interpretation can be challenged, his work demonstrates how conglomerates can monitor a complicated strain sample that otherwise might had been impossible to map.

Hossack mentioned 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) **