From Unrolled Geometry to Complex Shapes in Equilibrium

Developable Surfaces can be manufactured starting from the unrolled geometry without tearing or stretching. You can use different materials, but you must apply forces to turn the 2D planar shape into the 3D design structure. Our main goal is to define a computational model to simulate and control this process.


Introduction
Developable Surfaces are ruled surfaces with zero Gaussian curvature that can be manufactured starting from the unrolled geometry without overlapping or stretching. Different materials can be adopted but we must always apply forces to turn 2D planar shapes into 3D structures. The materials adopted can be flexible by nature, or you can transform a stiff material into an elastically one by cutting planar sheets. This technique is called "kerf bending". The main goal of our research is to define the forces needed to turn the unrolled shape into the complex surface that we have designed in relation to different materials used. The design of constraints and connections, to achieve the equilibrium conditions of the final structure, is based on these forces. Starting from prototypes, our aim is to define a computational model to simulate the stress state of the structure and to control the design process.

The Research
You can easily generate conical or cylindrical surfaces and define their unrolled shape using traditional methods; on the contrary, it is more difficult to find the unrolled shape of a developable tangential surface. According to differential geometry (Migliari 2009) developable surface can be flattened on a plane without distortion and, in a limited region, without overlapping. The unrolled shape of a tangential surface can be obtained by considering n generatrices and by flattening on the plane the surfaces included between two consecutive generatrices (Capone 2020). According to these theoretical premises, we defined an algorithmic tool that allows you to find the unrolled shape of any tangential surface (Capone 2018).
You can manufacture a tangential developable by cutting the unrolled shape, that is planar, therefore you can use a planar panel of flexible material, or you can transform a rigid material into a flexible one with kerf bending technique (Capone 2019).
This technique is based on subtracting material at some points in the panel to improve its flexibility, thus the ability to deform elastically and return to its original shape when the applied forces are removed. The maximum curvature (minimum radius) that the panel can achieve depends on the material features, the thickness of the panel and kerf pattern used in order to modify the material propriety. In our research we are going to define a tool that allows you to determine the new features of a material in relation to different kerf patterns application. Our goal is to define a relation between the pattern distribution on the panel and the local flexibility (Guzelci 2017). The challenge is to verify the efficiency and accuracy of our computational model and improving it by physical prototypes testing activity. Our tool allows you to distribute the kerf pattern in relation to curvature using the straight isocurves. We are improving the algorithm by introducing the materials physical parameters to simulate structural behaviour and to provide more scientific rigor to the process previously relied on empirical approach. To plan the testing activities, our starting point is based on some specific physical principles: elastic deformation vs. plastic deformation and stiffness vs. flexibility.
Elastic deformation: materials can change their shape when a force is applied and return to their original shape when this force is removed. When the force is applied the particles and the bonds between them are distorted but the internal structure of the material can return to its original form.
Plasic deformation: materials can change their shape when a force is applied, but they do not return to their original shape when the force is removed. Although the internal structure remains a single unit, it has changed permanently, some internal bonds may have been broken, and it cannot easily return to the original form. You must apply an external force to come back to original shape.
All materials are elastic and plastic in a specific range. Therefore, an elastic material can be highly tough (ductile) or can be very brittle. Elasticity is material resistance against permanent deformation (against plastic deformation), while ductility (flexibility) is related to energy that material can absorb during deformation.
In this paper we analyse the case of developable helicoid manufacturing. It is a tangential surface generated by the motion of a straight line (generatrix) tangent to a cylindric helix (Migliari 2009). The horizontal section of this surface with a plane orthogonal to cylinder axis of the helix is an evolute of circle.
One of the main problems to manufacture a 3D surface from a planar panel is to find the 2D shape to cut, define how to turn the 2D shape into a 3D surface and finally, how to constrain the structure to obtain an object in equilibrium.
We know that the unrolled shape of a developable helicoid piece is composed of two concentric circles and two tangents, and the radius of the inner circle is the radius of osculating circle of the helix (Capone 2018).
We tested different configurations based on these geometric properties, different materials and different ways to assemble and constrain the pieces.

Geometry
1. Two pieces of helicoid with a common helix (external border); 2. One piece of helicoid between two helixes; 3. one piece of helicoid with horizontal section (evolute of circle).

Materials
Material A-elastic deformation: Polypropylene 1 mm.
Material B -plastic deformation: metal sheet 0.5 mm. Material C -stiff material transformed in a flexible material: plywood 3 mm. We performed the same tests with 3 mm plexiglass and plywood panels to define the different features of materials. The tests consisted of applying different kerf pattern to a rectangular piece in order to evaluate the maximum curvature before breaking point. The results were used in design process. The challenge is to define a tool that allows you to simulate the structural behaviour in order to apply the right cutting pattern in relation to the curvature of the designed shape, cutting only where needed (Fig. 1).

Assembly
Our aim is to test different methodologies for complex shapes manufacturing based on form-finding and bending-active principles (Lienhard 2014) that allows you to turn a 2D shape into 3D structure.
We focused on some specific topics to address: 1. Determining the forces that must be applied to transform the 2D shape into 3D structure (kinetic process); 2. Determining the relationship between material properties and reversible deformation process in order to obtain a shape in equilibrium; 3. Designing connection points and needed constraints to obtain a shape in equilibrium.
Working with physical models allows you instantaneous and intuitive changes the input variables to obtain an immediate result. We tested the same configuration using different materials to evaluate the different applied forces that we can use to turn the unrolled geometry into a complex surface.
Starting from the tests results we are defining an algorithm model that you can use to simulate the kinetic feature of the system and to select the best kerf pattern to apply to improve the elastic behaviour in relation to material and curvature.

Testing
Starting from the simplest tangential surface, the developable helicoid, we manufactured several prototypes using different materials, different techniques and different scale.
Prototype 1: Polypropylene 1 mm-elastic. We connected two pieces of two opposite helicoids using hinges, we tested two different sizes in order to evaluate where to apply the loads (Fig. 2) and different solutions in order to design the physical constrains, therefore, where to place struts and tie-rods.
Prototype 2: sheet metal 0.5 mm-plastic We applied forces in order to turn the 2D shape into a piece of developable helicoid.
In this case the material is plastic, that means it can change its shape when a system of forces is applied, but it does not return to its original shape when the force is removed. Therefore, it holds the shape until the system of forces is applied again (Fig. 3).
Prototype 3: plywood panels 3 mm-stiff material transformed in flexible material. We connected two pieces of opposite helicoids using hinges, as prototype 1. In this case the original material is stiff and we improved its flexibility by using the kerf bending technique. The goal is to obtain a flexible material, not too elastic, in order to easily turn the 2D shape into 3D configuration and lock the structure in position with simple connections (Fig. 4).
Prototype 4: different materials. We connected two pieces of opposite helicoids using different materials: Polypropylene 1 mm panels and plywood 3 mm panels. We connected the two pieces with hinges, as prototype 1, using the same two rigid elements to lock the shape. The surfaces are subjected to stresses during the bending process that cause some breaks for materials different elastic behaviours (Fig. 5).
Prototype 5: different geometries. In this first cardboard prototype the design of the connection is based on geometric features. We are going to test different geometric configuration (Fig. 6).

Conclusion
Conclusion from test results analysis.
Elastic material: we do not know exactly what the best way to constrain the structure is in order to lock the shape.
Plastic material: it is very easy to lock the shape. Stiff material transformed into flexible material: there are some fabrication problems to solve in relation to the distance between the cuts.
Different materials: there are some breaks because we did not simulate stress conditions.
In relation to these test results analysis our aim is: 1. Defining a computational model to simulate the kinetic process in order to determinate the needed forces to design constraints and connections to obtain a structure in equilibrium; 2. Defining a computational model to determine the maximum curvature (Fallavollita 2012) of a generical tangential surface in order to distribute kerf bending pattern, to control the shaping process, to design connection points and constraints.
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