Displacement field (mechanics)

A displacement field is an assignment of displacement vectors for all points in a region or body that is displaced from one state to another.[1][2] A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position. For example, a displacement field may be used to describe the effects of deformation on a solid body.

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:

\( {\displaystyle {\vec {R}}_{0}:\Omega \rightarrow P} \)

where

\( {\displaystyle {\vec {R}}_{0}}\) is a placement vector
\( \Omega \)are all the points of the body
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:

\( {\displaystyle {\vec {u}}={\vec {R}}_{1}-{\vec {R}}_{0}} \)

where

\( {\vec {u}} \) is a displacement field, which for each point of the body specifies a displacement vector.