Micromechanics of composites/Average deformation gradient in a RVE

Average deformation gradient in a RVE

The average deformation gradient is defined as

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

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,

$⟨𝑭⟩=\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }𝑭\text{dV}=\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }{\mathrm{\nabla }}_0𝐱\text{dV}=\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }𝐱\otimes 𝐍\text{dA}=\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }(𝐗+𝐮)\otimes 𝐍\text{dA}$

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

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

$\text{d}𝐚=\det (𝑭){𝑭}^{-T}\text{d}𝐀\equiv 𝐧\text{da}=\det (𝑭){𝑭}^{-T}\cdot 𝐍\text{dA}⟹\frac{1}{\det 𝑭}{𝑭}^T\cdot 𝐧\text{da}=𝐍\text{dA}$

where $\text{da}$ is an element of area on the deformed surface, $𝐧$ 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

$J:=\det (𝑭)=\frac{{\rho }_0}{\rho }=\frac{V}{V_0}.$

Therefore,

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

Plugging into the surface integral, we have

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

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

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

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

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

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

$\det ⟨𝑭⟩>0;⟨𝑭{⟩}^{-1}=⟨{𝑭}^{-1⟩};V=V_0\det ⟨𝑭⟩.$

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

Template:Subpage navbar