Micromechanics of composites/Average velocity gradient in a RVE

Average velocity gradient in a RVE

The time rate of the deformation gradient is given by

${\dot {\boldsymbol {F}}}={\frac {\partial }{\partial t}}[{\boldsymbol {F}}(\mathbf {X} ,t)]={\frac {\partial }{\partial t}}\left({\frac {\partial }{\partial \mathbf {X} }}[\mathbf {x} (\mathbf {X} ,t)]\right)={\frac {\partial }{\partial \mathbf {X} }}\left({\frac {\partial }{\partial t}}[\mathbf {x} (\mathbf {X} ,t)]\right)={\frac {\partial {\dot {\mathbf {x} }}}{\partial \mathbf {X} }}={\boldsymbol {\nabla }}_{0}~{\dot {\mathbf {x} }}~.$

The average time rate of the deformation gradient is defined as

${\langle {\dot {\boldsymbol {F}}}\rangle :={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}~{\text{dV}}~.}$

Following the same procedure as in the previous section, we can show that

${\langle {\dot {\boldsymbol {F}}}\rangle ={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes \mathbf {N} ~{\text{dA}}={\cfrac {1}{V}}\int _{\partial {\Omega }}({\dot {\mathbf {x} }}\otimes \mathbf {n} )\cdot {\boldsymbol {F}}~{\text{da}}~.}$

The velocity gradient (${\boldsymbol {l}}$) is given by

${\boldsymbol {l}}={\boldsymbol {\nabla }}\mathbf {v} ={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}$

where $\mathbf {v} (\mathbf {x} )$ is the velocity.

The average velocity gradient in a RVE is defined as

${{\overline {\boldsymbol {l}}}:=\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {F}}\rangle ^{-1}~.}$

Note that ${\overline {\boldsymbol {l}}}=\langle {\boldsymbol {l}}\rangle =\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}\rangle$ only if ${\boldsymbol {F}}={\boldsymbol {\mathit {1}}}$.

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Micromechanics of composites

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