Micromechanics of composites/Proof 10

From Wikiversity
Jump to navigation Jump to search

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 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