Investigating the stochastic dispersion of 2D engineered frame structures under symmetry of variability

Additive manufacturing has enabled the construction of increasingly complex mechanical structures. However, the variability of mechanical properties may be higher than that of conventionally manufactured structures. Typically, the computational cost of the numerical modeling of such structures considerably increases when variability is considered. In deterministic analyses of periodic structures, the dispersion diagrams obtained for the first Brillouin zone (FBZ) can be used to predict attenuation bands for any direction of propagation. This can be further simplified considering only the contour of the irreducible Brillouin zone (IBZ) if the unit cell presents symmetries. The objective of the current investigation is to present evidence that, similarly to what occurs in deterministic cases, the stochastic results obtained by scanning only the IBZ contour of the proposed two-dimensional unit cell under 4-fold rotational symmetry of statistical variability coincides with statistical results obtained scanning the FBZ. This is not a direct result, because each individual unit cell sample is asymmetric. We show that, under symmetry of variability statistics, the stochastic results computed for supercells and for finite metastructures consisting of a finite number of cells also coincide with the results for the IBZ contour of the unit cell. This result is important, as it dramatically reduces the computation cost of the stochastic analysis of such structures.