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Elasticity/Rigid body motions

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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