Micromechanics of composites/Average deformation gradient in a RVE

Average Deformation Gradient in a RVE

The average deformation gradient is defined as

${ \langle \boldsymbol{F}\rangle := \cfrac{1}{V_0}\int_{\Omega_0} \boldsymbol{F}~\text{dV} }$

where $V_0$ is the volume in the reference configuration.

We can express the average deformation gradient in terms of surface quantities by using the divergence theorem. Thus,

$\langle \boldsymbol{F}\rangle = \cfrac{1}{V_0}\int_{\Omega_0} \boldsymbol{F}~\text{dV} = \cfrac{1}{V_0}\int_{\Omega_0} \boldsymbol{\nabla}_0~ \mathbf{x}~\text{dV} = \cfrac{1}{V_0} \int_{\partial{\Omega}_0} \mathbf{x}\otimes\mathbf{N}~\text{dA} = \cfrac{1}{V_0} \int_{\partial{\Omega}_0} (\mathbf{X}+\mathbf{u})\otimes\mathbf{N}~\text{dA}$

where $\mathbf{N}$ is the unit outward normal to the reference surface $\partial{\Omega}_0$ and $\mathbf{u}(\mathbf{X}) = \mathbf{x} - \mathbf{X}$ is the displacement.

The surface integral can be converted into an integral over the deformed surface using Nanson's formula for areas:

$\text{d}\mathbf{a} = \det(\boldsymbol{F})~\boldsymbol{F}^{-T}~\text{d}\mathbf{A}\qquad\equiv\qquad \mathbf{n}~\text{da} = \det(\boldsymbol{F})~\boldsymbol{F}^{-T}\cdot\mathbf{N}~\text{dA} \quad \implies \quad \cfrac{1}{\det{\boldsymbol{F}}}~\boldsymbol{F}^T\cdot\mathbf{n}~\text{da} = \mathbf{N}~\text{dA}$

where $\text{da}$ is an element of area on the deformed surface, $\mathbf{n}$ is the outward normal to the deformed surface, and $\text{dA}$ is an element of area on the reference surface.

The conservation of mass gives us

$J := \det(\boldsymbol{F}) = \cfrac{\rho_0}{\rho} = \cfrac{V}{V_0}~.$

Therefore,

$\mathbf{x}\otimes\mathbf{N}~\text{dA} = \mathbf{x}\otimes(\mathbf{N}~\text{dA}) = \mathbf{x}\otimes\left(\cfrac{V_0}{V}~\boldsymbol{F}^T\cdot\mathbf{n}~\text{da}\right) = \left(\cfrac{V_0}{V}\right)~\mathbf{x}\otimes(\boldsymbol{F}^T\cdot\mathbf{n})~\text{da}$

Plugging into the surface integral, we have

$\langle \boldsymbol{F} \rangle = \cfrac{1}{V_0} \int_{\partial \Omega_0} \mathbf{x}\otimes\mathbf{N}~\text{dA} = \cfrac{1}{V_0} \int_{\partial \Omega} \left[ \left(\cfrac{V_0}{V}\right)~\mathbf{x}\otimes(\boldsymbol{F}^T\cdot\mathbf{n}) \right]~\text{da} = \cfrac{1}{V} \int_{\partial \Omega } \mathbf{x}\otimes(\boldsymbol{F}^T\cdot\mathbf{n})~\text{da} ~.$

Using the identity $\mathbf{a}\otimes(\boldsymbol{A}\cdot\mathbf{b}) = (\mathbf{a}\otimes\mathbf{b})\cdot\boldsymbol{A}^T$ (see Appendix), we get

$\langle \boldsymbol{F}\rangle = \cfrac{1}{V} \int_{\partial{\Omega}} (\mathbf{x}\otimes\mathbf{n})\cdot\boldsymbol{F}~\text{da} ~.$

Therefore, the average deformation gradient in surface integral form can be written as

${ \langle \boldsymbol{F}\rangle = \cfrac{1}{V_0} \int_{\partial{\Omega}_0} \mathbf{x}\otimes\mathbf{N}~\text{dA} = \cfrac{1}{V} \int_{\partial{\Omega}} (\mathbf{x}\otimes\mathbf{n})\cdot\boldsymbol{F}~\text{da} ~. }$

Note that there are three more conditions to be satisfied for the average deformation gradient to behave like a macro variable, i.e.,

$\det\langle \boldsymbol{F}\rangle > 0 ~;~~ \langle \boldsymbol{F}\rangle^{-1} = \langle \boldsymbol{F}^{-1 \rangle} ~;~~ V = V_0 \det\langle \boldsymbol{F}\rangle ~.$

These considerations and their detailed exploration can be found in Costanzo et al.(2005).