# 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 ?Nal geometry through inflexible frame translation, rigid body rotation, stress (distortion) and/or extent trade.

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

The displacement ?Eld may be decomposed into various additives, depending on the reason of the decomposition. The conventional manner of decomposing it is with the aid of setting apart rigid body deformation inside the form of inflexible translation and rotation from trade in form and volume. In Fig. 2 the interpretation aspect is proven in (b), the rotation element in (c) and the relaxation (the strain) in (d). Let us have a better observe those expressions.

#### Translation

 Fig. Three. The Jotun Nappe in the Scandinavian Caledonides seems to were transported more than three hundred km to the southeast, primarily based on recuperation and the orientation of lineations. The displacement vectors are indicated, but the quantity of rigid rotation across the vertical axis is unknown. The amount of strain is usually concentrated to the base.

Translation moves each particle in the rock in the equal direction and the same distance, and its displacement ?Eld consists of parallel vectors of equal duration. Translations can be giant, as an example in which thrust nappes (detached slices of rocks) have been transported several tens or masses of kilometres. The Jotun Nappe (Fig. Three) is an example from the Scandinavian Caledonides. In this example maximum of the deformation is rigid translation. We do not know the precise orientation of this nappe previous to the onset of deformation, so we cannot estimate the inflexible rotation (see underneath), but ?Eld observations screen that the alternate in form, or pressure, is largely con?Ned to the lower elements. The total deformation consequently consists of a large translation element, an unknown but probable small inflexible rotation aspect and a stress element localised to the bottom of the nappe.

On a smaller scale, rock additives (mineral grains, layers or fault blocks) can be translated along slip planes or planar faults with none internal alternate 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 whole tectonic plate generally occur about vertical axes. Fault blocks in extensional settings, however, may additionally rotate round horizontal axes, and small-scale rotations may additionally occur about any axis.

#### Strain

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

Any change in form, without or with trade in extent, is referred to as strain, and it implies that particles in a rock have changed positions relative to every different.

A rock volume can be transported (translated) and turned around rigidly in any manner and sequence, however we are able to by no means be able to inform just from searching on the rock itself. All we can see in the ?Eld or in samples is stress, and perhaps the manner that stress has accrued. Consider your lunch bag. You can deliver it to school or work, which includes lots of rotation and translation, however you cannot see this deformation at once. It might be that your lunch bag has been squeezed on your manner to high school ? You could inform by means of evaluating it with what it looked like before you left home. If a person else organized your lunch and placed it on your bag, you will use your expertise of ways a lunch bag should be shaped to estimate the strain (change in shape) worried.

The ultimate factor may be very applicable, due to the fact with very few exceptions, we've got no longer visible the deformed rock in its undeformed state. We then need to use our knowledge of what such rocks normally look like while unstrained. For example, if we ?Nd strained ooliths or discount spots in the rock, we may assume them to were round (round in cross-phase) within 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 device need to be selected. Standing on a dock looking a huge deliver entering or departing can provide the influence that the dock, no longer the ship, is transferring. Unconsciously, the reference system receives ?Xed to the ship, and the relaxation of the arena actions via translation relative to the ship. While that is captivating, it isn't always a completely useful desire of reference. Rock deformation should additionally be taken into consideration in the body of some reference coordinate system, and it must be chosen with care to maintain the level of complexity down.

We always want a reference frame while coping with displacements and kinematics.

It is frequently beneficial to orient the coordinate machine alongside critical geologic structures. This might be the base of a thrust nappe, a plate boundary or a nearby shear sector. In many cases we need to eliminate translation and inflexible rotation. In the case of shear zones we typically area two axes parallel to the shear zone with the 0.33 being perpendicular to the sector. If we're inquisitive about the deformation in the shear zone as a whole, the beginning may be ?Xed to the margin of the zone. If we're interested in what goes on round any given particle inside the sector we can ?Glue? The origin to a particle in the zone (nonetheless parallel/perpendicular to the shear quarter obstacles). In both cases translation and rigid rotation of the shear sector are eliminated, because the coordinate system rotates and interprets together with the shear quarter. There is nothing incorrect with a coordinate gadget that is oblique to the shear quarter boundaries, however visually and mathematically it makes matters greater complicated.

### Deformation: indifferent from history

Deformation is the distinction between the deformed and undeformed states. It tells us not anything approximately what truely took place in the course of the deformation history.

A given stress might also have accrued in an in?Nite variety of ways.

Imagine a worn-out student (or professor for that rely) who falls asleep in a ship while ?Shing on the ocean or a lake. The pupil knows in which she or he changed into while falling asleep, and shortly ?Gures out the brand new vicinity while waking up,however the precise course that currents and winds have taken the boat is unknown. The scholar best knows the position of the boat before and after the nap, and may evaluate the stress (change in shape) of the boat (optimistically 0). One can map the deformation, but no longer 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 traces remain instantly, parallel strains stay parallel, and identically shaped and oriented objects can also be identically formed and orientated 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 searching at the deformation matrix, the point made in Fig. 5 must be emphasised:

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 conventional example is the increase in pressure normally seen from the margin toward the centre of a shear zone. The stress is heterogeneous on this scale, however can be subdivided into thinner factors or zones in which strain is about homogeneous. Another instance is shown in Fig. 6, in which a rock extent is penetrated by faults. On a huge scale, the deformation can be considered homogeneous because the discontinuities represented via the faults are relatively small. On a smaller scale, however, those discontinuities grow to be greater apparent, and the deformation need to be considered heterogeneous.

Credits: Haakon Fossen (Structural Geology)