## 2 Modeling and Finite Element Analysis
### Periodic Boundary Conditions and Elastic Property Calculations  
Periodic boundary conditions allow for the modeling of a bulk material using a representative volume element (RVE). In order to do this, the nodes on the boundary of the RVE are constrained so that each boundary node's displacements are matched to those of the corresponding node on the opposite boundary. An RVE constrained in this way has been shown to deform as if it were a single member in a bulk material composed of identical RVEs repeating in all directions [39]. The requirement that nodes be matched with corresponding nodes on opposite faces adds one additional step to the mesh generation, wherein the boundaries are re-seeded with nodes that correspond with each other, and the mesh realigned to these new bounding-nodes. Once the boundary constraints are in place, a displacement is applied to one face, with the remaining faces free of traction. The resulting forces and displacements of the nodes are then calculated using FEA. [PERSON] and [PERSON] [39] demonstrated using Gauss' divergence theorem that the average strain through the RVE can be determined directly from the displacement of the surface (2),  
\[\bar{\epsilon}_{ij}=\frac{1}{V}\int_{V}\epsilon_{ij}dV=\frac{1}{V}\int_{S}(u_{i }n_{j}+u_{j}n_{i})dS \tag{2}\]  
where \(V\) is RVE volume, \(S\) is a boundary surface, \(u_{i}\) is displacement in the \(i\)-direction, \(n_{i}\) is the \(i\)-component of the surface normal, and \(\bar{\epsilon}_{ij}\) is the averaged \(i,j\) component of the strain tensor. (3) shows how this reduces to the conventional strain definition for longitudinal strains in the \(x\)-direction,  
\[\bar{\epsilon}_{xx}=\frac{1}{V}\int_{S}(u_{x}n_{x})dS=\frac{\Delta L_{x}}{L_{ x}} \tag{3}\]  
where \(\bar{\epsilon}_{xx}\) is the average normal component of strain in the \(x\)-direction, \(L_{x}\) and \(\Delta L_{x}\) are the length and elongation in the \(x\)-direction, respectively. Likewise, using the principle of strain energy and external work equivalence, the average stress is shown to be by the applied force divided by the area of the surface [39] (4),  
\[\bar{\sigma}_{xx}=\frac{1}{V}\int_{V}\sigma_{xx}dV=\frac{F_{x}}{A_{x}} \tag{4}\]  
where \(\bar{\sigma}_{xx}\) is the average normal component of stress in the \(x\)-direction, \(F_{x}\) is the normal force resultant summed across the boundary (\(x=L_{x}\)) of the RVE, and \(A_{x}\) is the area of this face. While the local stresses and strains vary throughout the model, these \"average\" properties are the behavior observed in a bulk material composed of these RVEs. The applied strain and predicted stress can then be used to calculate the effective elastic properties of this bulk-structure. Several similar analyses are needed to determine all of the properties of the bulk structure in different directions. Normal strains applied in any of the cardinal  
Figure 3: Example of a solid model of the Syroid structure unit-cell (50% relative density). directions allow for calculation of the corresponding longitudinal modulus and the two corresponding Poisson's ratios. For example, applying a normal displacement \(\Delta L_{x}\) to the boundary (\(x=L_{x}\)) of the RVE causes it to deform such that there is also a length change in the \(y\), and \(z\)-directions (\(\Delta L_{y}\) and \(\Delta L_{z}\)), and nodal resultant forces along the face \(x=L_{x}\) which sum to \(F_{x}\). Dividing \(F_{x}\) by the area of the face yields a resultant stress. These deformations/resultants are used in the standard definition of elastic modulus and Poisson's ratio (5)-(7).