Rigid body displacement field
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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.
Proof:
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
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For a rigid body motion, the strain . Therefore,
-
Plugging into the expression for for a homogeneous deformation, we
have
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