Topology optimization of dissipative metamaterials at finite strains based on nonlinear homogenization
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RESEARCH PAPER
Topology optimization of dissipative metamaterials at finite strains based on nonlinear homogenization Guodong Zhang1 · Kapil Khandelwal1 Received: 17 October 2019 / Revised: 22 February 2020 / Accepted: 3 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This study presents a novel computational framework for designing optimal dissipative (damping) metamaterials under time-dependent loading conditions at finite deformations. In this framework, finite strain computational homogenization is integrated with a density-based multimaterial topology optimization. In addition, a thermodynamically consistent finite strain viscoelasticity model is incorporated together with an analytical path-dependent sensitivity analysis. Optimization formulations with and without stiffness and mass constraints are considered, and various new damping metamaterial designs are obtained that combine soft viscoelastic and stiff hyperelastic material phases. Multiscale stability analysis using the Bloch wave analysis and rank-1 convexity checks is also carried out to investigate stability of the optimized designs. Stability analyses demonstrate that the inclusion of voids or soft material phases can make a metamaterial more prone to lose micro and macro-stability. Furthermore, the concept of tunable metamaterials is explored wherein metamaterial’s response is steered towards a stable deformation path by tailoring the design with a preselected micro buckling mode. Keywords Dissipative metamaterials · Multimaterial topology optimization · Viscoelasticity · Hyperelasticity · Nonlinear homogenization · Multiscale stability
1 Introduction Mechanical metamaterials with tailored functionalities have received considerable attention in recent years due to the unprecedented progress in additive manufacturing technologies (Gibson et al. 2014; Gao et al. 2015). In essence, these metamaterials are obtained by carefully designing the underlying material microstructure, and the exotic properties of these metamaterials are dependent on the tailored material microstructures rather than on the material’s chemical constitution. While many metamaterials have been obtained by experimental design and trial-and-error methods (Surjadi et al. 2019), advanced computational methods based on multiscale mechanics and mathematical optimization can provide a rigorous framework for designing such metamaterials (Sigmund 1994). This study is concerned Responsible Editor: Juli´an Andr´es Norato Kapil Khandelwal
[email protected] 1
Department of Civil & Environmental Engineering & Earth Sciences, University of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, IN 46556, USA
with the design of optimized metamaterials with desirable damping and stiffness properties under finite strains using nonlinear homogenization and topology optimization methods. Homogenization theories, starting with the pioneering work of Hill (1972) and Mandel (1972), provide a rigorous mathematical framework for predicting the effective properties of co
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