Continuum models for stretching- and bending-dominated periodic trusses undergoing finite deformations
Advances in additive manufacturing across scales have enabled the creation of random, periodic, or hierarchical truss networks containing millions and more of individual truss members. In order to significantly reduce computational costs while accurately capturing the dominant deformation mechanisms, we introduce a simple yet powerful homogenized continuum description of truss lattices, which is based on applying a multi-lattice Cauchy-Born rule to a representative unit cell (RUC). Beam theory applied at the level of the RUC introduces rotational degrees of freedom and leads to a generalized continuum model that depends on the effective deformation gradients, rotation, and curvature on the macroscale. While affinely deforming the RUC is shown to produce excellent results for simple Bravais lattices, a multi-lattice extension is required for general and especially bending-dominated lattices, which cannot be described by a pure Taylor expansion of the RUC deformation; the importance of the multi-lattice concept is demonstrated through analytical examples. The resulting method is a beneficial compromise between inefficient FE^2 techniques and micropolar theories with limited applicability. By implementing the model within a finite element framework, we solve and report several benchmark tests in 2D to illustrate the accuracy and efficiency of the model, which comes with only a small fraction of the computational costs associated with the full, discrete truss calculation. By using a corotational beam description, we also capture finite beam rotations. We further demonstrate that a second-gradient homogenization formulation is beneficial in examples involving localization, providing higher local accuracy at the RUC level than a first-gradient scheme, while affecting the global response only marginally.