# Micromechanics of composites/Proof 10

## Rigid body displacement field

Show that, for a rigid body motion with infinitesimal rotations, the displacement field ${\displaystyle \mathbf {u} (\mathbf {x} )}$  for can be expressed as

${\displaystyle \mathbf {u} (\mathbf {x} )=\mathbf {c} +{\boldsymbol {\omega }}\cdot \mathbf {x} }$

where ${\displaystyle \mathbf {c} }$  is a constant vector and ${\displaystyle {\boldsymbol {\omega }}}$  is the infinitesimal rotation tensor.

Proof:

Note that for a rigid body motion, the strain ${\displaystyle {\boldsymbol {\varepsilon }}}$  is zero. Since

${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }}={\boldsymbol {\nabla }}{\boldsymbol {\theta }}}$

we have a ${\displaystyle {\boldsymbol {\theta }}=}$  constant when ${\displaystyle {\boldsymbol {\varepsilon }}=0}$ , i.e., the rotation is homogeneous.

For a homogeneous deformation, the displacement gradient is independent of ${\displaystyle \mathbf {x} }$ , i.e.,

${\displaystyle {\boldsymbol {\nabla }}\mathbf {u} ={\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}={\boldsymbol {G}}\qquad \leftarrow \qquad {\text{constant}}~.}$

Integrating, we get

${\displaystyle \mathbf {u} (\mathbf {x} )={\boldsymbol {G}}\cdot \mathbf {x} +\mathbf {c} ~.}$

Now the strain and rotation tensors are given by

${\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2}}({\boldsymbol {\nabla }}\mathbf {u} +{\boldsymbol {\nabla }}\mathbf {u} ^{T})={\frac {1}{2}}({\boldsymbol {G}}+{\boldsymbol {G}}^{T})~;~~{\boldsymbol {\omega }}={\frac {1}{2}}({\boldsymbol {\nabla }}\mathbf {u} -{\boldsymbol {\nabla }}\mathbf {u} ^{T})={\frac {1}{2}}({\boldsymbol {G}}-{\boldsymbol {G}}^{T})~.}$

For a rigid body motion, the strain ${\displaystyle {\boldsymbol {\varepsilon }}=0}$ . Therefore,

${\displaystyle {\boldsymbol {G}}=-{\boldsymbol {G}}^{T}\qquad \implies \qquad {\boldsymbol {\omega }}={\boldsymbol {G}}~.}$

Plugging into the expression for ${\displaystyle \mathbf {u} }$  for a homogeneous deformation, we have

${\displaystyle {\mathbf {u} (\mathbf {x} )={\boldsymbol {\omega }}\cdot \mathbf {x} +\mathbf {c} \qquad \square }}$