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