Simple and efficient analyses of micro-architected cellular elastic-plastic materials with tubular members

Abstract

In a novel way, nature evolves cellular structures to obtain mechanically efficient materials. Natural cellular materials combine low weight with superior mechanical properties to provide optimum strength and stiffness at low density such as trabecular bone, hornbill bird beaks, and bird wing bones. Inspired by these naturally architected cellular materials, humankind has also developed lightweight cellular materials for a broad range of applications consisting of structural components, energy absorption, heat exchange, and biomaterials. In this paper we present a highly efficient computational method to predict the bulk elastic-plastic homogenized mechanical properties of low-mass metallic systems with architected cellular microstructures. The proposed methodology provides a computational framework for the analysis, design, and topology optimization of such cellular materials. With a view for Direct Numerical Simulation of a cellular solid or structure with millions of cellular members, and considering the plausible deformations in each member, each such member is sought to be modeled by using only one or only a very few nonlinear three-dimensional (3D) beam elements with 6 degrees of freedom (DOF) at each of the 2 nodes of the element, and the nonlinear coupling of axial-torsional-bidirectional bending deformations is considered for each element. The effect of plasticity in each member is included using the mechanism of plastic hinges which may form at any point(s) along the length of each element or member. To make Direct Numerical Simulation of a micro-latticed cellular solid possible for its eventual applications, the tangent stiffness matrix for a spatial beam element undergoing large elastic-plastic deformation is explicitly derived using a Reissner-type mixed variational principle in the co-rotational updated Lagrangian reference frame. In order to avoid the inversion of the Jacobian matrix, a Newton homotopy method is employed to solve the tangent-stiffness equations. We are developing a code called CELLS/LIDS [CELLular Structures/Large Inelastic DeformationS] providing 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 propose to employ it for topology optimization design, and for impact/energy absorption analyses of elastic-plastic micro-cellular structures, and then micro-architecting them for desired elastic-plastic properties.