Computational design of finite strain auxetic metamaterials via topology optimization and nonlinear homogenization

Abstract

A novel computational framework for designing metamaterials with negative Poisson's ratio over a large strain range is presented in this work by combining the density-based topology optimization together with a mixed stress/deformation driven nonlinear homogenization method. A measure of Poisson's ratio based on the macro deformations is proposed, which is further validated through direct numerical simulations. With the consistent optimization formulations based on nonlinear homogenization, auxetic metamaterial designs with respect to different loading orientations and with different unit cell domains are systematically explored. In addition, the extension to multimaterial auxetic metamaterial designs is also considered, and stable optimization formulations are presented to obtain discrete metamaterial topologies under finite strains. Various new auxetic designs are obtained based on the proposed framework. To validate the performance of optimized designs, a multiscale stability analysis is carried out using the Bloch wave method and rank-one convexity check. As demonstrated, short and/or long wavelength instabilities can occur during the loading process, leading to a change of periodicity of the microstructure, which can affect the performance of an optimized design.

Geometric construction of auxetic metamaterials

“…Martinez et al [MSS*19] generate 2D tile geometries with a graduation of mechanical properties computed as Voronoï diagrams of regular lattices under star‐shaped distance functions. Topology optimization allows to generate new design of auxetic metamaterials with prescribed non‐linear properties computationally [WSJ14, ALS14, VCW*17, ZK19, ACN20]. These various approaches underline the growing interest in auxetic structures in the computer graphics community.…”

Mechanical Metamaterials on the Way from Laboratory Scale to Industrial Applications: Challenges for Characterization and Scalability

“…Numerical simulations are applied to screen the parameter space on single unit cell level or are based on networks of unit cells (Figure 4). Variations of parametrization especially can be studied systematically to reduce the experimental work [72,76,77]. In addition to be used to study the influence of design parameters, simulations are also used to complement experimental work.…”

Topology optimization for three‐dimensional elastoplastic architected materials using a path‐dependent adjoint method

“…Researchers have also developed optimization frameworks for creating optimized structures exhibiting viscoelastic creep deformation 40 or structures with designs mitigating damage or buckling by enforcing stress, damage, or buckling constraints 41‐44 . Others have considered the effect of geometric nonlinearities (finite elasticity) on the optimized structures and materials 45‐50 . It is not surprising that these works have inferred that the adoption of nonlinear mechanics greatly impacts the optimized designs providing the loads are large enough to induce nonlinear behavior.…”

“…[41][42][43][44] Others have considered the effect of geometric nonlinearities (finite elasticity) on the optimized structures and materials. [45][46][47][48][49][50] It is not surprising that these works have inferred that the adoption of nonlinear mechanics greatly impacts the optimized designs providing the loads are large enough to induce nonlinear behavior.…”