# Multiscale and Failure Analysis of Periodic Lattice Structures
## 5 Examples and Discussion  
Two-dimensional lattice structures were studied for both their displacements and induced stresses. The first structure is a cantilever beam with a transverse force at its end, as shown in Figure 6. The ratio of the beam length to thickness, \(L/H\), varied from 20 to 80. The cantilever beam was made of a lattice structure with the unit cell as shown in Figure 3 or Figure 4. That is, the first structure had a rectangular shape of lattices, and the second structure had both rectangular shapes of lattices along with two diagonal internal members. The homogenization technique was used to determine the effective material properties of the lattice structures. Then, the effective material properties were used to determine the tip deflection of the cantilever beam using the analytical solution, which is  
\[\delta_{tip}=\frac{PL^{3}}{3E_{eff}I} \tag{15}\]  
where \(\delta_{tip}\) is the tip deflection of the cantilever beam, \(P\) is the applied force at the tip, \(L\) is the length of the beam, \(E_{eff}\) is the homogenized elastic modulus, and \(I\) is the second moment of inertia of the cross-section of the homogenized beam. The tip deflection of the homogenized beam was compared to the tip deflection of the original lattice structure, which was solved using the finite element method with beam elements for every lattice member. Table 1 compares the tip deflection between the homogenized model and the discrete lattice structure model for different cantilever lengths and geometric dimensions of lattice members, with the unit cell shown in Figure 3 without diagonal lattice members. The unit cell has a square shape with \(w/l=1\). Equation (3) was used for the homogenized model's effective modulus in Equation (15). The discrete model included every lattice member in the analysis. The table shows the ratio of the tip deflection of the homogenized model to that of the discrete lattice structure model. If the upscaling process is reliable, the ratio of the tip deflection must be very close to 1. \begin{table}
\begin{tabular}{c c c c c} \hline \(L/H\) & \(H/I\) & \(t/I\) & **No. of Unit Cells in Beam \({}^{*}\)** & **Ratio of Tip Deflection \({}^{**}\)** & **Ratio of Maximum Stress *** \\ \hline
20 & 8 & 0.05 & 240 \(\times\) 12 & 0.951 & 0.868 \\
20 & 16 & 0.05 & 320 \(\times\) 16 & 0.972 & 0.897 \\
20 & 4 & 0.10 & 80 \(\times\) 4 & 0.962 & 0.913 \\
20 & 8 & 0.10 & 160 \(\times\) 8 & 0.994 & 0.937 \\
20 & 4 & 0.20 & 80 \(\times\) 4 & 1.081 & 1.033 \\
20 & 8 & 0.20 & 160 \(\times\) 8 & 1.024 & 1.022 \\ \hline \end{tabular}
\end{table} 
Table 1: Results of the cantilever made of the square unit cell (\(w/l=1\)) of Figure 3.  
Figure 6: Cantilever beam with a force at the tip. The tip deflection of the homogenized model was very close to that of the discrete lattice model, as shown in Table 1. The result also showed that the homogenized model is closer to the discrete model as the former contains more unit cells. In other words, as more lattice members are smeared into the homogenized model, its deflection is closer to that of the discrete model. This was expected because the discrete model becomes homogenized as the number of lattice members approaches infinity. Thus, the results confirm the convergence of the upscaling process, i.e., the homogenization model. Overall, the difference between the discrete and homogenized models was less than 5%, except for one case with 8%. The latter case had a smaller number of repeated unit cells in the homogenized model with \(t/l\) = 0.2. As the ratio \(t/l\) becomes larger, the representative beam length in the unit cell becomes shorter, which results in a larger error. However, such a corrective action was not conducted here to reduce the error because that was not the major objective of the study. As the shape of the unit cell was changed from square to rectangle, the tip deflection of the cantilever beam was compared in Table 2 between the homogenized and discrete models. Both deflections agreed well, too.