Extended Dummy Node Rule for analysis and characterization of pin-jointed periodic cellular materials: An approach based on Bloch-wave method

Micro-truss lattice materials are a class of hybrid periodic cellular solids that consist of a combination of solids and voids. The material is partitioned into cells structured in a given cell topology tessellated to form an almost unbounded micro-truss framework. The Bloch-wave method is one technique that can describe the propagation of a wave function over a periodic infinite lattice. It demonstrated the effectiveness of modeling the static and dynamic responses of lattices of various topologies. However, the Bloch-wave method presents limitations when applied to unit cell topologies whose structural elements extend across their envelopes to adjacent unit cells, as a given unit cell may not contain the full nodal parameters and periodicity information necessary to describe the kinematic and static wave propagations across the periodic lattice. The first part of this paper presents the Dummy Node Rule (DNR), which overcomes this limitation. The rule introduces dummy nodes at the intersection points between cell envelopes and elements extending between the adjacent unit cells, which are initially treated as part of the finite unit cell structure. Then, the rule establishes mathematical relationships between the static and kinematic wave functions propagating across dummy nodes and those propagating across the connected cell elements. The second part of the paper describes DNR for specific applications such as (a) the development of static and kinematic systems of the unit cell finite structure, which aids in the determinacy analysis of periodic lattice structures, and (b) the development of the Cauchy-Born kinematic boundary condition that is used to homogenize the kinematic response of the infinite periodic structure to an assumed macroscopic strain field, which in turn, forms the effective properties of the lattice material. Furthermore, the last part of the paper shows examples where the scheme is applied for the stiffness characterization of selected lattice topologies.