Deformation
The deformation of a thin straight rod into a closed loop. The length of the rod remains almost unchanged during the deformation, which indicates that the strain is small. In this particular case of bending, displacements associated with rigid translations and rotations of material elements in the rod are much greater than displacements associated with straining
In SI base units	m
Dimension	${\displaystyle {\mathsf {L}}}$

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In physics and continuum mechanics, deformation is the change in the shape or size of an object. It has dimension of length with SI unit of metre (m). It is quantified as the residual displacement of particles in a non-rigid body, from an initial configuration to a final configuration, excluding the body's average translation and rotation (its rigid transformation).^[1] A configuration is a set containing the positions of all particles of the body.

A deformation can occur because of external loads,^[2] intrinsic activity (e.g. muscle contraction), body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc.

In a continuous body, a deformation field results from a stress field due to applied forces or because of some changes in the conditions of the body. The relation between stress and strain (relative deformation) is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials.

Deformations which cease to exist after the stress field is removed are termed as elastic deformation. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations may remain, and these exist even after stresses have been removed. One type of irreversible deformation is plastic deformation, which occurs in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level. Another type of irreversible deformation is viscous deformation, which is the irreversible part of viscoelastic deformation. In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material.

Definition and formulation

Deformation is the change in the metric properties of a continuous body, meaning that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement. If none of the curves changes length, it is said that a rigid body displacement occurred.

It is convenient to identify a reference configuration or initial geometric state of the continuum body which all subsequent configurations are referenced from. The reference configuration need not be one the body actually will ever occupy. Often, the configuration at t = 0 is considered the reference configuration, κ₀(B). The configuration at the current time t is the current configuration.

For deformation analysis, the reference configuration is identified as undeformed configuration, and the current configuration as deformed configuration. Additionally, time is not considered when analyzing deformation, thus the sequence of configurations between the undeformed and deformed configurations are of no interest.

The components X_i of the position vector X of a particle in the reference configuration, taken with respect to the reference coordinate system, are called the material or reference coordinates. On the other hand, the components x_i of the position vector x of a particle in the deformed configuration, taken with respect to the spatial coordinate system of reference, are called the spatial coordinates

There are two methods for analysing the deformation of a continuum. One description is made in terms of the material or referential coordinates, called material description or Lagrangian description. A second description of deformation is made in terms of the spatial coordinates it is called the spatial description or Eulerian description.

There is continuity during deformation of a continuum body in the sense that:

• The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
• The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.

Affine deformation

An affine deformation is a deformation that can be completely described by an affine transformation. Such a transformation is composed of a linear transformation (such as rotation, shear, extension and compression) and a rigid body translation. Affine deformations are also called homogeneous deformations.^[3]

Therefore, an affine deformation has the form

$${\displaystyle \mathbf {x} (\mathbf {X} ,t)={\boldsymbol {F}}(t)\cdot \mathbf {X} +\mathbf {c} (t)}$$

$${\displaystyle {\begin{bmatrix}x_{1}(X_{1},X_{2},X_{3},t)\\x_{2}(X_{1},X_{2},X_{3},t)\\x_{3}(X_{1},X_{2},X_{3},t)\end{bmatrix}}={\begin{bmatrix}F_{11}(t)&F_{12}(t)&F_{13}(t)\\F_{21}(t)&F_{22}(t)&F_{23}(t)\\F_{31}(t)&F_{32}(t)&F_{33}(t)\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}+{\begin{bmatrix}c_{1}(t)\\c_{2}(t)\\c_{3}(t)\end{bmatrix}}}$$

The above deformation becomes non-affine or inhomogeneous if F = F(X,t) or c = c(X,t).

Rigid body motion

A rigid body motion is a special affine deformation that does not involve any shear, extension or compression. The transformation matrix F is proper orthogonal in order to allow rotations but no reflections.

A rigid body motion can be described by

$${\displaystyle \mathbf {x} (\mathbf {X} ,t)={\boldsymbol {Q}}(t)\cdot \mathbf {X} +\mathbf {c} (t)}$$

$${\displaystyle {\boldsymbol {Q}}\cdot {\boldsymbol {Q}}^{T}={\boldsymbol {Q}}^{T}\cdot {\boldsymbol {Q}}={\boldsymbol {\mathit {1}}}}$$