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

$\dot{𝑭}=\frac{\partial }{\partial t}[𝑭(𝐗,t)]=\frac{\partial }{\partial t}(\frac{\partial }{\partial 𝐗}[𝐱(𝐗,t)])=\frac{\partial }{\partial 𝐗}(\frac{\partial }{\partial t}[𝐱(𝐗,t)])=\frac{\partial \dot{𝐱}}{\partial 𝐗}={\mathrm{\nabla }}_0\dot{𝐱}.$

The average time rate of the deformation gradient is defined as

$⟨\dot{𝑭}⟩:=\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\text{dV}.$

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

$⟨\dot{𝑭}⟩=\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\dot{𝐱}\otimes 𝐍\text{dA}=\frac{1}{V}\underset{\partial \mathrm{\Omega }}{\int }(\dot{𝐱}\otimes 𝐧)\cdot 𝑭\text{da}.$

The velocity gradient ($𝒍$) is given by

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

where $𝐯(𝐱)$ is the velocity.

The average velocity gradient in a RVE is defined as

$\overline{𝒍}:=⟨\dot{𝑭}⟩\cdot ⟨𝑭{⟩}^{-1}.$

Note that $\overline{𝒍}=⟨𝒍⟩=⟨\dot{𝑭}\cdot {𝑭}^{-1}⟩$ only if $𝑭=1$.

Template:Subpage navbar