Micromechanics of composites/Proof 10
Rigid body displacement field edit
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 rotation tensor.
Note that for a rigid body motion, the strain is zero. Since
we have a constant when , i.e., the rotation is homogeneous.
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 have