Finite Strain Theory - Displacement

Displacement

A change in the configuration of a continuum body results in a displacement. 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 to a current or deformed configuration (Figure 1).

If after a displacement of the continuum there is a relative displacement between particles, a deformation has occurred. On the other hand, if after displacement of the continuum the relative displacement between particles in the current configuration is zero i.e. the distance between particles remains unchanged, then there is no deformation and a rigid-body displacement is said to have occurred.

The vector joining the positions of a particle in the undeformed configuration and deformed configuration is called the displacement vector in the Lagrangian description, or in the Eulerian description, where and are the unit vectors that define the basis of the material (body-frame) and spatial (lab-frame) coordinate systems, respectively.

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. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field. In general, the displacement field is expressed in terms of the material coordinates as

or in terms of the spatial coordinates as

where are the direction cosines between the material and spatial coordinate systems with unit vectors and, respectively. Thus

and the relationship between and is then given by

Knowing that

then

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in, and the direction cosines become Kronecker deltas, i.e.

Thus, we have

or in terms of the spatial coordinates as