Micromechanics of composites/Proof 10

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

 

For a rigid body motion, the strain  . Therefore,

 

Plugging into the expression for   for a homogeneous deformation, we have