Topology Optimized Architectures with Programmable Poisson's Ratio over Large Deformations

Abstract

Structural materials are used in myriad applications, including aerospace, automotive, biomedical, and acoustics. Most materials have positive or zero Poisson's ratio, with cork serving as a well-known example of the latter type of behavior. The Poisson's ratio describes the relative amount a given material contracts transversally when stretched axially. Recently, artificial materials that exhibit a negative Poisson's ratio have been introduced. [1][2][3] These auxetic materials expand transversally when axially stretched, seemingly defying the fundamental laws of nature. [1][2][3] They exhibit enhanced mechanical properties, such as shear resistance, [4,5] indentation resistance [6][7][8][9] and extraordinary damping properties, [10] making them well suited for targeted applications. To date, several types of auxetic materials have been introduced [2,3,[11][12][13][14][15][16][17][18][19][20] . However, current embodiments suffer from two primary limitations: 2(1) they only exhibit the desired response over a narrow range of strains (less than a few %) and (2) they are difficult to manufacture in a scalable manner. [17,[21][22][23][24][25] While recent structures (e.g. chiral honeycombs, [14] tilting square structures [24] or Bucklicrystals [19] ), for specific values of Poisson's ratio, exhibit near constant values over large strains, they are either not generalizable to other Poisson's ratio values or they exhibit low effective stiffness and/or must be pre-stressed to yield the desired performance.Here, we combine topology optimization to programmably design their architecture with 3D printing to digitally fabricate the designs and validate against the numerically predicted behavior. Specifically, we create a new class of architected materials with programmable Poisson's ratios between -0.8 and 0.8 that display a nearly constant Poisson's ratio over large deformations of up to 20% or more. Figure 1 shows two representative examples of microstructures designed using topology optimization. [26] The linear model that is applied by existing design methods ( Figure 1a) assumes small deformations. By contrast, an emerging approach (Figure1b), described in detail in a recent study [26] by Sigmund and coworkers, uses a geometrically nonlinear model and includes a requirement of a constant prescribed Poisson's ratio when straining the material. While both examples are designed to have a Poisson's ratio of -0.8, the performance of the linearly designed material rapidly deteriorates when the material is strained more than a few percent ( Figure 1c).Mathematically, the optimization goal is defined as minimizing the error between the actual and the pre-defined value of Poisson's ratio over a range of discrete, nominal strain values up to 20%. [26] To ensure scalable fabrication of these architectures, several geometric constraints are imposed on the topology optimization design problem. A requirement of uniform structural features is implemented as a combination of imposing a minimum [27,28] and a maximum length scale. Th...

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