Elasticity/Rigid body motions

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Rigid body motions[edit | edit source]

Rigid Deformation[edit | edit source]

A rigid deformation has the form

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




The strain tensors in this case are given by



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

Rigid Displacement[edit | edit source]

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[edit | edit source]

If the displacement is rigid we have

Infinitesimal Rigid Displacement[edit | edit source]

An infinitesimal rigid displacement is given by

where is a skew tensor.

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