A rigid deformation has the form
where are fixed material points and is an orthogonal (rotation) tensor.
The strain tensors in this case are given by
Hence the infinitesimal strain tensor does not measure the correct strain when there are large rotations though the finite strain tensor can.
Rigid displacements involve motions in which there are no strains.
Properties of rigid displacement fields
If is a rigid displacement field, then the strain field corresponding to is zero.
If the displacement is rigid we have
Infinitesimal Rigid Displacement[edit | edit source]
An infinitesimal rigid displacement is given by
where is a skew tensor.
Rigid body displacement field[edit | edit source]
Show that, for a rigid body motion with infinitesimal rotations, the
displacement field for can be expressed as
where is a constant vector and is the infinitesimal
Note that for a rigid body motion, the strain is zero. Since
we have a constant when , i.e., the rotation is
For a homogeneous deformation, the displacement gradient is
independent of , i.e.,
Integrating, we get
Now the strain and rotation tensors are given by
For a rigid body motion, the strain . Therefore,
Plugging into the expression for for a homogeneous deformation, we