## Geometric study for the transformation of two-dimensional structures into three-dimensional shells (Master thesis)

#### Δαχής, Ιωάννης

4D printing is the process of producing objects in which time, i.e the fourth dimension, is a basic construction parameter together with the three dimensions (x,y,z) of space. In other words, we have materials which, besides their initial form at time t0, have the ability to transform at time t’ into their final form. So we are talking about programmable matter,that is, materials with the ability to program their physical properties. This change from one state to another with the imposition of an external stimulus (heat, voltage, force, etc.) can be likened to phase transition. The aim of the thesis is to create a system that can perform this transition, and a mathematical model that will describe the physical process by giving us all the essential data. The diploma thesis is divided into two parts. The first part includes the essential theoretical background, and the second part the application, that is, an algorithm for an isometry from a flat surface to a double curvature surface using an auxetic structure. The first chapter consists of the basic elements of differential geometry which are the mathematical tools used to describe curvature on curves and surfaces. The main element is the isometry where we perceive the transformation of the surface and Gauss’s theorema egregium that demonstrates that Gauss curvature is associated with isometries. The second chapter describes the polytopes of two and three dimensions, i.e. polygons and polyhedra, which will help the transition from the continuous to the discrete form. The third chapter describes elements of discrete differential geometry in order to use concepts of differential geometry on discrete surfaces such as polyhedra. The fourth chapter analyzes graph theory which is used in the thesis for the analysis of the unfolding, as well as in which place the cuts will be created. The fifth chapter contains some elements of computational geometry that are necessary for the design of curves and surfaces, and a small part of the theory developed for the unfolding of surfaces. The sixth chapter describes the auxetic materials, that is, the structures with negative Poisson ratio and their properties. The seventh chapter consists of some elements of the mechanics of meta-materials that are applied to the thesis, such as shape memory. The eighth chapter presents statics with the use of graphstatic. The goal is to be able to create a form that the load exerted on it can be analyzed with the thurst line so that it can make the system change phase and return to its flat shape. From the ninth chapter begins the second part where the linking of the auxetic structure is initially analyzed using the nodes of the system. In the tenth chapter, the unit cell of the auxetic structure is constructed geometrically and the formula of its deformation is produced. The eleventh chapter presents the phase transition system which consists of the auxetic structure in combination with three springs, and describes the phase transition and the system memory using the system’s dynamic energy. The twelfth chapter describes geometrically the transformation of the system from a flat surface of zero curvature into a vaulted structure with positive Gauss curvature. Given the initial principal distinct curvature, the algorithm calculates the different sizes in all the elements and the distinct curvatures at all the nodes of the system. All the rotation axes of the auxetic structure, the angles and the unknown variables are calculated geometrically. The thirteenth chapter uses graph theory to create the tree of the unfolding. The appendix consists of diagrams and drawings illustrating the results of the algorithm. The final result of the work is: The construction of geometric models that make possible the isometry of a plane on a surface of positive distinct curvature. Creating a system that has the ability to change phase in pre-designed final forms. The property of the auxetic structures to produce positive Gauss curvature surfaces is known and is produced by bending the structures. In the present work, tensile strength is used to produce synclastic surfaces. Also a further analysis of the rotating triangular auxetic structures is made. All the above are used to create an algorithm that, given the size of the original Lo element and the original discrete curvature, has all the necessary geometric dimensions for creating the system. The basic applications of such a construction start from the creation of programmable materials that produce specific final results and the incorporation of the fourth dimension of time into 3D prints. The main feature that transforms the system is geometry, which can be maintained regardless of scale. As a result, the system can find applications from the microscale and the science of materials to the macroscale and structures. Further future research can be carried out to make a more strict mathematical solution and to extend it to a more general framework for more complex forms. Also, the static construction of the algorithm could be developed and combined with the stiffness of the springs, which could analyze the stress required for phase transition of the system.
 Institution and School/Department of submitter: Πανεπιστήμιο Ιωαννίνων. Πολυτεχνική Σχολή. Τμήμα Μηχανικών Επιστήμης Υλικών Subject classification: Mechanics Keywords: Μηχανική,Μετά-υλικά,4-d εκτύπωση,Mechanics,Meta-matelials,4-D printing URI: http://olympias.lib.uoi.gr/jspui/handle/123456789/29506 Appears in Collections: Διατριβές Μεταπτυχιακής Έρευνας (Masters)

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