\subsection{Stress Homogenization}

The homogenized Cauchy stress, $\langle\boldsymbol{\sigma}\rangle$, is derived from the definition of the spatial average of the microscale stress of the deformable blocks of the DEM simulation, $\boldsymbol{\sigma}^m$, over the homogenization domain $\Omega^{h}$. 
\begin{equation}
\langle\boldsymbol{\sigma}\rangle=\frac{1}{A^{h}}\int_{\Omega^{h}}\boldsymbol{\sigma}^m dA\label{eqn:stress1}
\end{equation}


%As previously mentioned, there exists a necessary distinction between the block subdomain and the void subdomain when dealing with stress. Here, the integration over the homogenization domain can be decomposed into two seperate integrations over the block subdomain and the void subdomain: 

%\begin{equation}
%\langle\boldsymbol{\sigma}\rangle=\frac{1}{A^{h}}\left[\int_{\Omega^{b}}\boldsymbol{\sigma}dA+\int_{\Omega^{v}}\boldsymbol{\sigma}dA\right]\label{eqn:stress2}
%\end{equation}

%This distinction is made due to the fact that in a purely mechanical model, there isn't any pressure or stress being retained in the void space, resulting in a negligable contribution to the overall stress state. This allows for the void subdomain integration term to be dropped from the formulation: 

%\begin{equation}
%\langle\boldsymbol{\sigma}\rangle=\frac{1}{A^{h}}\int_{\Omega^{b}}\boldsymbol{\sigma}dA\label{eqn:stress2a}
%\end{equation}

When each deformable block is discretized by constant stress triangles (elements/zones), the integration of the stress over each deformable block subdomain can be written as a summation in the form of a spatially weighted average of the zone stresses. Let $\boldsymbol{\sigma}_{IJ}^{m}$ denote the stress in zone $J$ of deformable block $I$, $N_{I}^{z}$ denote the number of zones within block $I$, $A_{IJ}$ denote area of zone $J$ in block $I$, and $N^{b}$ denote the number of deformable blocks. The homogenized stress is then defined as 

\begin{equation}
\langle\boldsymbol{\sigma}\rangle=\frac{1}{A^{h}}\sum_{I=1}^{N^{b}}\sum_{J=1}^{N_{I}^{z}}\boldsymbol{\sigma}_{IJ}^{m}A_{IJ}\label{eqn:stress4}
\end{equation}

In our implementation, (\ref{eqn:stress4}) is incorporated into the homogenization module.