A rigid deformation has the form
![{\displaystyle {\boldsymbol {\varphi }}(\mathbf {X} )=\mathbf {X} _{1}+{\boldsymbol {Q}}\bullet [\mathbf {X} -\mathbf {X} _{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/247ac9b15f18173f46c0f6880e40a57c793b2a30)
where
are fixed material points and
is an orthogonal (rotation) tensor.
Therefore
![{\displaystyle {\boldsymbol {F}}={\boldsymbol {Q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f142cc6e7f6d72ff57acfcfe027a8ee7251ec6)
and
.
The strain tensors in this case are given by
![{\displaystyle {\boldsymbol {E}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b0d6363b26ea6d15166fb295275eae5e75134a)
but
.
Hence the infinitesimal strain tensor does not measure the correct strain when there are large rotations though the finite strain tensor can.
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.
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If the displacement is rigid we have
![{\displaystyle {\begin{aligned}\mathbf {u} (\mathbf {X} )&=\mathbf {X} _{1}+{\boldsymbol {\nabla }}\mathbf {u} \bullet [\mathbf {X} -\mathbf {X} _{0}]+{\boldsymbol {1}}[\mathbf {X} -\mathbf {X} _{0}]-\mathbf {X} \\&=(\mathbf {X} _{1}-\mathbf {X} _{0})+{\boldsymbol {\nabla }}\mathbf {u} \bullet [\mathbf {X} -\mathbf {X} _{0}]\\&=\mathbf {u} _{0}+{\boldsymbol {\nabla }}\mathbf {u} \bullet [\mathbf {X} -\mathbf {X} _{0}]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10e0f83b51737877c3a9f3d801e8188858138ea0)
An infinitesimal rigid displacement is given by
![{\displaystyle \mathbf {u} (\mathbf {X} )=\mathbf {u} _{0}+{\boldsymbol {W}}\bullet [\mathbf {X} -\mathbf {X} _{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cabd539a85b0b02f56e528f6d2fe948baa44874f)
where
is a skew tensor.
Show that, for a rigid body motion with infinitesimal rotations, the
displacement field
for can be expressed as
![{\displaystyle \mathbf {u} (\mathbf {x} )=\mathbf {c} +{\boldsymbol {\omega }}\cdot \mathbf {x} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c22b59efe7ae3ddfac631197163f2aa4cb6ce00)
where
is a constant vector and
is the infinitesimal
rotation tensor.
Proof:
Note that for a rigid body motion, the strain
is zero. Since
![{\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }}={\boldsymbol {\nabla }}{\boldsymbol {\theta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9f1c4346f8feca9b3dcd9fc17c208b12fdcaf4)
we have a
constant when
, i.e., the rotation is
homogeneous.
For a homogeneous deformation, the displacement gradient is
independent of
, i.e.,
![{\displaystyle {\boldsymbol {\nabla }}\mathbf {u} ={\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}={\boldsymbol {G}}\qquad \leftarrow \qquad {\text{constant}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8731ff788015c6ea0c98c4de59e382feac0830)
Integrating, we get
![{\displaystyle \mathbf {u} (\mathbf {x} )={\boldsymbol {G}}\cdot \mathbf {x} +\mathbf {c} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70a8c1fae1723449c9647f52529b7625ab5374de)
Now the strain and rotation tensors are given by
![{\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2}}({\boldsymbol {\nabla }}\mathbf {u} +{\boldsymbol {\nabla }}\mathbf {u} ^{T})={\frac {1}{2}}({\boldsymbol {G}}+{\boldsymbol {G}}^{T})~;~~{\boldsymbol {\omega }}={\frac {1}{2}}({\boldsymbol {\nabla }}\mathbf {u} -{\boldsymbol {\nabla }}\mathbf {u} ^{T})={\frac {1}{2}}({\boldsymbol {G}}-{\boldsymbol {G}}^{T})~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5384dfdfaf20429f66cb1ec814fd55a57c5aa434)
For a rigid body motion, the strain
. Therefore,
![{\displaystyle {\boldsymbol {G}}=-{\boldsymbol {G}}^{T}\qquad \implies \qquad {\boldsymbol {\omega }}={\boldsymbol {G}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5bcb5d03ac7cd5980049bbee97c4a7468303f96)
Plugging into the expression for
for a homogeneous deformation, we
have
![{\displaystyle {\mathbf {u} (\mathbf {x} )={\boldsymbol {\omega }}\cdot \mathbf {x} +\mathbf {c} \qquad \square }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97488d6e57398cfe51a420a6476e47039cc43b24)