Displacement field (mechanics)
[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:
where 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
the displacement field is the difference between two body states:
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.
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
, 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:
are the orthonormal unit vectors that define the basis of the spatial (lab frame) coordinate system.
Expressed in terms of the material coordinates, i.e.
is the displacement vector representing rigid-body translation.
The partial derivative of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor
is the material deformation gradient tensor and
In the Eulerian description, the vector extending from a particle
in the undeformed configuration to its location in the deformed configuration is called the displacement vector:
are the unit vectors that define the basis of the material (body-frame) coordinate system.
Expressed in terms of spatial coordinates, i.e.
The spatial derivative, i.e., the partial derivative of the displacement vector with respect to the spatial coordinates, yields the spatial displacement gradient tensor
is the spatial deformation gradient tensor.
are the direction cosines between the material and spatial coordinate systems with unit vectors
It is common to superimpose the coordinate systems for the deformed and undeformed configurations, which results in
, 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:
