Article
Design of Auxetic Structures via Mathematical Optimization
University of ErlangenNuremberg, Institute of Advanced Materials and Processes (ZMP), Dr.MackStr. 81, 90762 Fürth, Germany.
Advanced Materials (Impact Factor: 17.49). 06/2011; 23(2223):26504. DOI: 10.1002/adma.201004090 Source: PubMed
ABSTRACT
The optimization and manufacturing of an auxetic structure is presented. An inverse homogenization method is used to obtain the optimized geometry shown in the figure. The resulting structure is then produced using selective electron beam melting. The numerically predicted properties are experimentally verified.

 "Such materials would have various effective properties according to the mechanisms of their internal geometries. Innovative materials have indeed been developed by integrating mathematical and numerical design methods and AM technologies; e.g., tissue engineering a bone scaffold having compatibility with human bone in terms of stiffness and permeability [3] [4] and developing materials with a negative Poisson's ratio [5] [6]. "
Article: Porous composite with negative thermal expansion obtained by photopolymer additive manufacturing
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ABSTRACT: Additive manufacturing (AM) could be a novel method of fabricating composite and porous materials having various effective performances based on mechanisms of their internal geometries. Materials fabricated by AM could rapidly be used in industrial application since they could easily be embedded in the target part employing the same AM process used for the bulk material. Furthermore, multimaterial AM has greater potential than usual singlematerial AM in producing materials with effective properties. Negative thermal expansion is a representative effective material property realized by designing a composite made of two materials with different coefficients of thermal expansion. In this study, we developed a porous composite having planar negative thermal expansion by employing multimaterial photopolymer AM. After measurement of the physical properties of bulk photopolymers, the internal geometry was designed by topology optimization, which is the most effective structural optimization in terms of both minimizing thermal stress and maximizing stiffness. The designed structure was converted to a threedimensional stereolithography (STL) model, which is a native digital format of AM, and assembled as a test piece. The thermal expansions of the specimens were measured using a laser scanning dilatometer. Negative thermal expansion corresponding to less than −1 × 10−4 K−1 was observed for each test piece of the N = 3 experiment. 
 "However, the obtained designs are not always manufacturable due to the thin hinge connections in the microstructure. Topology optimization can also be applied to decrease Poisson's ratio between two of the principal directions (Sigmund et al., 1998; Schwerdtfeger et al., 2011) for anisotropic materials. "
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ABSTRACT: We present a method to design manufacturable extremal elastic materials. Extremal materials can possess interesting properties such as a negative Poisson’s ratio. The effective properties of the obtained microstructures are shown to be close to the theoretical limit given by mathematical bounds, and the deviations are due to the imposed manufacturing constraints. The designs are generated using topology optimization. Due to high resolution and the imposed robustness requirement they are manufacturable without any need for postprocessing. This has been validated by the manufacturing of an isotropic material with a Poisson’s ratio of ν=0.5ν=0.5 and a bulk modulus of 0.2% times the solid base material’s bulk modulus.  [Show abstract] [Hide abstract]
ABSTRACT: Materials and microstructures with specific configurations are able to have negative Poisson’s ratio. This paper proposes a topology optimization methodology of frame structures to design a planar periodic structure that exhibits negative Poisson’s ratio. Provided that beam section of each existing member is chosen from a set of some given candidates, we can reduce the topology optimization problem to a mixedinteger linear programming problem. Since the proposed approach treats frame structures and stress constraints are rigorously addressed, the optimal solution contains no hinge region. A heuristic method with local search is used to solve largescale problems. Numerical examples and fabrication test demonstrate that planar periodic frame structures exhibiting negative Poisson’s ratio can be successfully obtained by the proposed method.
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