Elasticity/Rigid body motions

Rigid body motions

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Rigid Deformation

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A rigid deformation has the form

 

where   are fixed material points and   is an orthogonal (rotation) tensor.

Therefore

 

and

 .

The strain tensors in this case are given by

 

but

 .

Hence the infinitesimal strain tensor does not measure the correct strain when there are large rotations though the finite strain tensor can.

Rigid Displacement

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Rigid displacements involve motions in which there are no strains.

Properties of rigid displacement fields

If   is a rigid displacement field, then the strain field corresponding to   is zero.

Finite Rigid Displacement

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If the displacement is rigid we have

 

Infinitesimal Rigid Displacement

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An infinitesimal rigid displacement is given by

 

where   is a skew tensor.

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