Before considering displacement, the state before deformation must be defined. It is a state in which the coordinates of all points are known and described by the function:

• ${\displaystyle {\vec {R}}_{0}}$  is a placement vector
• ${\displaystyle \Omega }$  are all the points of the body
• ${\displaystyle P}$  are all the points in the space in which the body is present

Most often it is a state of the body in which no forces are applied.

Then given any other state of this body in which coordinates of all its points are described as ${\displaystyle {\vec {R}}_{1}}$  the displacement field is the difference between two body states:

The displacement of a body has two components: a rigid-body displacement and a deformation.

• A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size.
• Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration ${\displaystyle \kappa _{0}({\mathcal {B}})}$  to a current or deformed configuration ${\displaystyle \kappa _{t}({\mathcal {B}})}$  (Figure 1).

A change in the configuration of a continuum body can be described by a displacement field. A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. The distance between any two particles changes if and only if deformation has occurred. If displacement occurs without deformation, then it is a rigid-body displacement.

Two types of displacement gradient tensor may be defined, following the Lagrangian and Eulerian specifications. The displacement of particles indexed by variable i may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration ${\displaystyle P_{i}}$  and deformed configuration ${\displaystyle p_{i}}$  is called the displacement vector, ${\displaystyle p_{i}-P_{i}}$ , denoted ${\displaystyle u_{i}}$  or ${\displaystyle U_{i}}$  below.

Material coordinates (Lagrangian description) edit

Using ${\displaystyle \mathbf {X} }$  in place of ${\displaystyle P_{i}}$  and ${\displaystyle \mathbf {x} }$  in place of ${\displaystyle p_{i}\,\!}$ , both of which are vectors from the origin of the coordinate system to each respective point, we have the Lagrangian description of the displacement vector:

Expressed in terms of the material coordinates, i.e. ${\displaystyle \mathbf {u} }$  as a function of ${\displaystyle \mathbf {X} }$ , the displacement field is:

The partial derivative of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor ${\displaystyle \nabla _{\mathbf {X} }\mathbf {u} \,\!}$ . Thus we have,

Spatial coordinates (Eulerian description) edit

In the Eulerian description, the vector extending from a particle ${\displaystyle P}$  in the undeformed configuration to its location in the deformed configuration is called the displacement vector:

Expressed in terms of spatial coordinates, i.e. ${\displaystyle \mathbf {U} }$  as a function of ${\displaystyle \mathbf {x} }$ , the displacement field is:

The spatial derivative, i.e., the partial derivative of the displacement vector with respect to the spatial coordinates, yields the spatial displacement gradient tensor ${\displaystyle \nabla _{\mathbf {x} }\mathbf {U} \,\!}$ . Thus we have,

Relationship between the material and spatial coordinate systems edit

${\displaystyle \alpha _{Ji}}$  are the direction cosines between the material and spatial coordinate systems with unit vectors ${\displaystyle \mathbf {E} _{J}}$  and ${\displaystyle \mathbf {e} _{i}\,\!}$ , respectively. Thus

The relationship between ${\displaystyle u_{i}}$  and ${\displaystyle U_{J}}$  is then given by

Knowing that

Combining the coordinate systems of deformed and undeformed configurations edit

It is common to superimpose the coordinate systems for the deformed and undeformed configurations, which results in ${\displaystyle \mathbf {b} =0\,\!}$ , and the direction cosines become Kronecker deltas, i.e.,

Thus in material (undeformed) coordinates, the displacement may be expressed as:

And in spatial (deformed) coordinates, the displacement may be expressed as:

• Stress
• Strain