Nearly exact and highly efficient elastic-plastic homogenization and/or direct numerical simulation of low-mass metallic systems with architected cellular microstructures

Additive manufacturing has enabled the fabrication of lightweight materials with intricate cellular architectures. These materials are interesting due to their properties which can be optimized upon the choice of the parent material and the topology of the architecture, making them appropriate for a wide range of applications including lightweight aerospace structures, energy absorption, thermal management, metamaterials, and bioscaffolds. In this paper we present the simplest initial computational framework for the analysis, design, and topology optimization of low-mass metallic systems with architected cellular microstructures. A very efficient elastic-plastic homogenization of a repetitive Representative Volume Element (RVE) of the microlattice is proposed. Each member of the cellular microstructure undergoing large elastic-plastic deformations is modeled using only one nonlinear three-dimensional (3D) beam element with 6 degrees of freedom (DOF) at each of the 2 nodes of the beam. The nonlinear coupling of axial, torsional, and bidirectional-bending deformations is considered for each 3D spatial beam element. The plastic hinge method, with arbitrary locations of the hinges along the beam, is utilized to study the effect of plasticity. We derive an explicit expression for the tangent stiffness matrix of each member of the cellular microstructure using a mixed variational principle in the updated Lagrangian corotational reference frame. To solve the incremental tangent stiffness equations, a newly proposed Newton homotopy method is employed. In contrast to the Newton's method and the Newton-Raphson iteration method, which require the inversion of the Jacobian matrix, our homotopy methods avoid inverting it. We have developed a code called CELLS/LIDS (CELLular Structures/Large Inelastic DeformationS), which provides the capabilities to study the variation of the mechanical properties of the low-mass metallic cellular structures by changing their topology. Thus, due to the efficiency of this method we can employ it for topology optimization design and for impact/energy absorption analyses.