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1. Structural form-finding of Auxetic Materials using Graphic Statics

   Hablicsek, Márton; Akbarzadeh, Masoud (October 2021, International Association of Shell and Spatial Structures)

   The Auxetic materials are structural systems with a negative Poisson’s ratio. Such materials show unexpected behavior when subjected to uni-axial compression or tension forces. For instance, they expand perpendicular to the direction of an applied compressive force. This behavior is the result of their internal structural geometry. These materials, with their unique behavior, have recently found many applications in the fields of sensors, medical devices, sport wears, and aerospace. Thus, there is a lot of relevant research in the artificial design of auxetic metamaterials and the prediction of their behavior [2]. Since the behavior of these materials heavily relies on the geometry of their internal structure, the geometry-based methods of structural design, known as graphic statics, are very well suited to derive their geometry or describe their behavior. Nevertheless, graphic statics has never been used in the design of such materials. For the first time, this paper provides an introduction to the use of graphic statics in the design and form-finding of auxetic metamaterials. The paper explains multiple equilibrium states of various auxetic structures using algebraic formulations of 2d/3d graphic statics [1, 3]. Moreover, it sheds light on the geometric behavior of auxetic materials by changing the force diagram of graphic statics. Therefore, it suggests a novel approach in predicting the changes in the geometry of the material under various loading conditions by controlling the force equilibrium geometrically. 

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2. Strut‐Based Cellular to Shellular Funicular Materials

   https://doi.org/10.1002/adfm.202109725  

   Akbari, Mostafa; Mirabolghasemi, Armin; Bolhassani, Mohammad; Akbarzadeh, Abdolhamid; Akbarzadeh, Masoud (January 2022, Advanced Functional Materials)

   Abstract Owing to the fact that effective properties of low‐density cellular solids heavily rely on their underlying architecture, a variety of explicit and implicit techniques exists for designing cellular geometries. However, most of these techniques fail to present a correlation among architecture, internal forces, and effective properties. This paper introduces an alternative design strategy based on the static equilibrium of forces, equilibrium of polyhedral frames, and reciprocity of form and force. This novel approach reveals a geometric relationship among the truss system architecture, topological dual, and equilibrium of forces on the basis of 3D graphic statics. This technique is adapted to devise periodic strut‐based cellular architectures under certain boundary conditions and they are manipulated to construct shell‐based (shellular) cells with a variety of mechanical properties. By treating the materialized unit cells as representative volume elements (RVE), multiscale homogenization is used to investigate their effective linear elastic properties. Validated by experimental tests on 3D printed funicular materials, it is shown that by manipulating the RVE topology using the proposed methodology, alternative strut materialization schemes, and rational addition of bracing struts, cellular mechanical metamaterials can be systematically architected to demonstrate properties ranging from bending‐ to stretching‐dominated, realize metafluidic behavior, or create novel hybrid shellulars. 

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3. Generative Design of Statistically Self-Similar Mechanical Structures

   https://doi.org/10.1115/DETC2023-117063  

   Hill, Noah; Ebert, Matthew; Maurice, Mena; Krishnamurthy, Vinayak (August 2023, American Society of Mechanical Engineers)

   Abstract We present a novel methodology to generate mechanical structures based on the idea of fractal geometry as described by the chaos game. Chaos game is an iterative method that generates self-similar point-sets in the limiting case within a polygonal domain. By computing Voronoi tessellations on these point-sets, our method generates mechanical structures that adopts the self-similarity of the point-sets resulting in fractal distribution of local stiffness. The motivation behind our approach comes from the observation that a typical generative structural design workflow requires the ability to generate families of structures that possess shared behavioral (e.g. thermal, mechanical, etc.) characteristics making each structure distinct but feasible. However, the generation of the alternatives, almost always, requires solving an inverse structural problem which is both conceptually and computationally challenging. The objective of our work is to develop and investigate a forward-design methodology for generating families of structures that, while not identical, exhibit similar mechanical behavior in a statistical sense. To this end, the central hypothesis of our work is that structures generated using the chaos game can generate families of self-similar structures that, while not identical, exhibit similar mechanical behavior in a statistical sense. Furthermore, each family is uniquely identifiable from the parameters of the chaos game, namely, the polygonal domain, fractional distance, and number of samples. We present a systematic study of these self-similar structures through modal analysis and demonstrate a preliminary confirmation of our hypothesis. 

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4. From design to the fabrication of shellular funicular structures

   Akbari, M.; Lu, Y.; Akbarzadeh, M. (September 2021, Proceedings of the Association for Computer-Aided Design in Architecture (ACADIA))

   Shellular Funicular Structures (SFSs) are single-layer, two-manifold structures with anticlastic curvature, designed in the context of graphic statics. They are considered as efficient structures applicable to many functions on different scales. Due to their complex geometry, design and fabrication of SFSs are quite challenging, limiting their application in large scales. Furthermore, designing these structures for a predefined boundary condition, control, and manipulation of their geometry are not easy tasks. Moreover, fabricating these geometries is mostly possible using additive manufacturing techniques, requiring a lot of support in the printing process. Cellular funicular structures (CFSs) as strut-based spatial structures can be easily designed and manipulated in the context of graphic statics. This paper introduces a computational algorithm for translating a Cellular Funicular Structure (CFS) to a Shellular Funicular Structure (SFS). Furthermore, it explains a fabrication method to build the structure out of a flat sheet of material using the origami/ kirigami technique as an ideal choice because of its accessibility, processibility, low cost, and applicability to large scales. The paper concludes by displaying a design and fabricated structure using this technique. 

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5. Geometrically non-linear topology optimization via geometry projection

   https://doi.org/10.1016/j.cma.2024.117636  

   Hu, Jingyu; Wallin, Mathias; Ristinmaa, Matti; Norato, JA; Liu, Shutian (February 2025, Computer Methods in Applied Mechanics and Engineering)

   Geometry projection-based topology optimization has attracted a great deal of attention because it enables the design of structures consisting of a combination of geometric primitives and simplifies the integration with computer-aided design (CAD) systems. While the approach has undergone substantial development under the assumption of linear theory, it remains to be developed for non-linear hyperelastic problems. In this study, a geometrically non-linear explicit topology optimization approach is proposed in the framework of the geometry projection method. The energy transition strategy is adopted to mitigate excessive distortion in low-stiffness regions that might cause the equilibrium iterations to diverge. A neo-Hookean hyperelastic strain energy potential is used to model the material behavior. Design sensitivities of the functions passed to the gradient-based optimizer are detailed and verified. The proposed method is used to solve benchmark problems for which the output displacement in a compliant mechanism is maximized and the structural compliance is minimized. 

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