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)
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