The time rate of the deformation gradient is given by
![{\displaystyle {\dot {\boldsymbol {F}}}={\frac {\partial }{\partial t}}[{\boldsymbol {F}}(\mathbf {X} ,t)]={\frac {\partial }{\partial t}}\left({\frac {\partial }{\partial \mathbf {X} }}[\mathbf {x} (\mathbf {X} ,t)]\right)={\frac {\partial }{\partial \mathbf {X} }}\left({\frac {\partial }{\partial t}}[\mathbf {x} (\mathbf {X} ,t)]\right)={\frac {\partial {\dot {\mathbf {x} }}}{\partial \mathbf {X} }}={\boldsymbol {\nabla }}_{0}~{\dot {\mathbf {x} }}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d0dc06cba676e496f3b6ad1c34cafe6d42f958)
The average time rate of the deformation gradient is defined as
![{\displaystyle {\langle {\dot {\boldsymbol {F}}}\rangle :={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}~{\text{dV}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d7a9c332b2ca93d4b754f0f956741873f930c8)
Following the same procedure as in the previous section, we can show
that
![{\displaystyle {\langle {\dot {\boldsymbol {F}}}\rangle ={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes \mathbf {N} ~{\text{dA}}={\cfrac {1}{V}}\int _{\partial {\Omega }}({\dot {\mathbf {x} }}\otimes \mathbf {n} )\cdot {\boldsymbol {F}}~{\text{da}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/243d9e6cbf61f0ca99d16e430f232d52c04284d0)
The velocity gradient (
) is given by
![{\displaystyle {\boldsymbol {l}}={\boldsymbol {\nabla }}\mathbf {v} ={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/127fc7fc81f1794dae518caf5a3e73839c6f4e94)
where
is the velocity.
The average velocity gradient in a RVE is defined as
![{\displaystyle {{\overline {\boldsymbol {l}}}:=\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {F}}\rangle ^{-1}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68acad6ef1a178930f7befba8def7f4f78194d67)
Note that
only if
.