# Multiscale and Failure Analysis of Periodic Lattice Structures
## 5 Examples and Discussion
The next example was the unit cell shown in Figure 4, which had two diagonal lattice members. The comparison of the tip deflection between the homogenized and discrete models is provided in Table 3. Equation (7) was used for the homogenized model. The deflection of the homogenized model was close to that of the discrete model for both square and rectangular shapes of unit cells. For the downscaling process to determine the maximum stresses at the actual lattice member, the displacements were obtained from the homogenized model at the top, left corner of the cantilever, as shown in Figure 7. The displacements at the four corners of the square or rectangular shape with the unit cell size were determined from the homogenized model. Those displacements are equivalent to the group \(U_{1}\) in Equation (10), which is used to determine the group \(U_{2}\) in that equation. Once both \(U_{1}\) and \(U_{2}\) are computed, Equations (8) and (11) are used to determine the stresses at the lattice members. The  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(L\)/\(H\) & \(H\)/\(l\) & \(w\)/\(l\) & \(t\)/\(l\) & \(No. of Unit Cells in Beam\) & Ratio of Tip Deflection & Ratio of Maximum Stress \\ \hline
40 & 4 & 0.5 & 0.04 & 80 \(\times\) 4 & 0.936 & 0.903 \\
40 & 4 & 0.5 & 0.05 & 80 \(\times\) 4 & 0.997 & 0.949 \\
40 & 4 & 0.5 & 0.10 & 80 \(\times\) 4 & 1.092 & 1.072 \\
80 & 4 & 0.25 & 0.02 & 80 \(\times\) 4 & 0.962 & 0.929 \\
80 & 4 &maximum stress occurs at the top lattice member of the unit cell for the cantilever beam with a tip load. Tables 1-3 show the comparison of the maximum stress in the homogenized model to that of the discrete lattice structure model. Overall, the downscaling process yielded lattice stresses very close to those computed directly from the finite element analysis of the discrete lattice structure. The accuracy of the stress from the multiscale analysis was very comparable to that of the displacement. This suggests that both upscaling and downscaling processes are reliable to extract the actual solution of the discrete lattice structure with much less computational time. The next example was a clamped beam at both ends with a force at the center, as sketched in Figure 8. The lattice structure had the unit cell, as shown in Figure 4. That is, the structure had both diagonal lattice members. Table 4 shows the comparison of the deflection at the center of the homogenized model to that of the discrete model. The homogenized model gave very comparable deflections to the discrete model. However, the accuracy of the homogenized model was better for the cantilever beam than the clamped beam at both ends. The maximum stress was also compared for the beam clamped at both ends between the multiscale analysis and the discrete finite element analysis of the lattice structure. Table 4 shows the comparison. Both results compare well. The accuracy of the stress prediction depends on the accuracy of the homogenized model because the displacements of the homogenized model are used for the downscaling process to compute the stresses in actual lattice members. Table 4 also suggests that the multiscale analysis resulted in the stresses as accurate as the displacements. This means the downscaling process is very reliable for determining the stresses in the lattice members. Figure 8: Clamped beam at both ends with a force at the center. Figure 7: Selected unit cell location of the cantilever beam for the maximum lattice stresses. The last set of studies was to predict the failure load of lattice structures using the multiscale technique and the failure criterion as presented in previous sections. The lattice structures were made of unit cells with diagonal members, and they were subjected to three-point bending loads. The length of every lattice structure was 140 mm, and the span length between the two simple supports was 100 mm. Every unit cell had a dimension of 5 mm \(\times\) 5 mm. The heights of the lattice structures varied from 20 mm to 40 mm with an increment of 10 mm.