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)