Additive manufacturing has enabled the fabrication of lightweight materials with intricate cellular architectures. These materials become 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 elasticplastic homogenization of a repetitive Representative Volume Element (RVE) of the micro-lattice 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 co-rotational 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 1 This paper is written in honor of Dr. Pedro Marçal, a true pioneer in Computational Mechanics, on the occasion of the celebration of his Lifetime Achievements at ICCES17, June 2017, in Madeira, Portugal 14 Jul 2017 19:09:55 PDT Version 3 Submitted to J. Mech. Mater. Struct. 2 Homotopy methods avoid inverting it. We have developed 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 can employ it for topology optimization design, and for impact/energy absorption analyses.

中文翻译：

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具有结构化细胞微结构的低质量金属系统的近乎精确和高效的弹塑性均质化和/或直接数值模拟

增材制造使制造具有复杂蜂窝结构的轻质材料成为可能。这些材料因其特性而变得有趣，这些特性可以根据母体材料的选择和架构的拓扑结构进行优化，使其适用于广泛的应用，包括轻质航空航天结构、能量吸收、热管理、超材料和生物支架. 在本文中，我们提出了最简单的初始计算框架，用于分析、设计和拓扑优化具有结构化细胞微结构的低质量金属系统。提出了一种非常有效的微晶格重复代表性体积元素 (RVE) 的弹塑性均质化方法。经历大弹塑性变形的蜂窝微结构的每个成员仅使用一个非线性三维 (3D) 梁单元在梁的 2 个节点中的每个节点处具有 6 个自由度 (DOF)。每个 3D 空间梁单元都考虑了轴向、扭转和双向弯曲变形的非线性耦合。塑性铰法，铰链沿梁的任意位置，用于研究塑性的影响。我们使用更新的拉格朗日共转参考系中的混合变分原理推导出细胞微结构每个成员的切线刚度矩阵的明确表达式。为了求解增量切线刚度方程，采用了新提出的牛顿同伦方法。与需要对雅可比矩阵求逆的 Newton 方法和 Newton-Raphson 迭代方法相比，我们的 1 本文是为了纪念计算力学真正的先驱 Pedro Marçal 博士而写的。他在 ICCES17 上的终身成就，2017 年 6 月，葡萄牙马德拉岛 2017 年 7 月 14 日 19:09:55 PDT 第 3 版提交给 J. Mech。母校。结构。2 同伦方法避免反转它。我们开发了一个名为 CELLS/LIDS [CELLular Structures/Large Inelastic DeformationS] 的代码，提供了通过改变拓扑来研究低质量金属蜂窝结构机械性能变化的能力。因此，由于这种方法的效率，我们可以将其用于拓扑优化设计和冲击/能量吸收分析。