Out of Plane: Utilizing Auxetic Origami Systems to Optimize Synclastic Curvature by Daniel Vrana

OUT OF PLANE Utilizing Auxetic Origami Systems to Optimize Synclastic Curvature

by Daniel John Vrana September 2015

A thesis submitted to the Faculty of the Graduate School of the University at Buffalo, State University of New York in partial fulfillment of the requirements for the degree of Master of Architecture

School of Architecture and Planning

To my mother, Lynne, and my father, Ken, for always supporting my academic pursuits and giving me the confidence to continue chasing my dreams

[ ii ]

[ Acknowledgements ] I would like to thank my thesis committee, professors, colleagues, and friends for always believing in my abilities, for never accepting mediocrity, and for pushing me as hard as possible over the last six years. Thank you. George Behn Nicholas Bruscia Daniel Fiore Philip Gusmano Omar Khan Robert Miller Christopher Romano Lindsay Romano Aaron Salva Kimberly Sass Michael Tuzzo John Wightman

[ iii ]

[ iv ]

[ Table of Contents ] List of Figures Abstract 01_Introduction 02_Auxetic Geometry 02.00_Introduction 02.01_Micro Scale Application 02.02_Macro Scale Application 02.03_Design Scale Application 02.04_Conclusion 03_Reentrant Hexagon Patterning

vii ix 03 05 05 05 07 09 11 13

03.00_Preliminary Experimentation 03.01_Reentrant Hexagon Manipulation 03.02_Paper Models 03.03_Composite System 03.04_Force Models 03.05_Force Models Translation 03.06_Reentrant Hexagon Conclusions 04_Bistable Origami 04.00_Introduction to Origami 04.01_Bistable Origami 04.02_Preliminary Unit 04.03_Unit Movement 04.04_Deformation 04.05_Scale 04.06_Material Studies 04.07_Bracing the Plane 04.08_Aggregation and Double Curvature 04.09_Removing the Frame 04.10_Optimization 04.11_Creating an Origami Fabric 04.12_Final Study Model 04.13_Final Models 05_Simulation 05.00_Introduction to Simulation 05.01_Initial Simulations 05.02_Single Unit Origami Simulation 05.03_Composite Origami Simulation 06_Conclusion 07_References

13 15 16 23 26 27 30 33 33 34 35 37 37 42 42 46 47 53 54 56 57 60 67 67 69 70 71 75 79

[v]

[ vi ]

[ List of Figures ] fig. 01 fig. 02 fig. 03 fig. 04 fig. 05 fig. 06 fig. 07 fig. 08 fig. 09 fig. 10

[ diagram ] [ diagram ] [ rendering ] [ drawing ] [ photograph ] [ photograph ] [ photograph ] [ drawing ] [ drawing ] [ drawing ]

basic auxetic principles stent geometries (Zheng-Dong and Liu 2011) auxetic stent double curvature (Bock et al. 2012) auxetic apparel (Toronjo 2014) synclstic curvature in PLA 3D printed materials (Bastian 2014) rubberized origami (Trex:Lab 2014) Betatype material (Betatype 2014) reentrant hexagon patterning non uniform reentrant hexagon patterning reentrant hexagon composite drawing (with Nicholas Bruscia)

05 06 07 08 10 10 11 13 13 14

fig. 11 fig. 12 fig. 13 fig. 14 fig. 15 fig. 16 fig. 17 fig. 18 fig. 19 fig. 20 fig. 21 fig. 22 fig. 23 fig. 24 fig. 25 fig. 26 fig. 27 fig. 28 fig. 29 fig. 30 fig. 31 fig. 32 fig. 33 fig. 34 fig. 35 fig. 36 fig. 37 fig. 38 fig. 39

[ screenshot ] [ diagram ] [ drawing ] [ drawing ] [ drawing ] [ photograph ] [ photograph ] [ photograph ] [ photograph ] [ photograph ] [ photograph ] [ photograph ] [ photograph ] [ photograph ] [ diagram ] [ photograph ] [ photograph ] [ photograph ] [ diagram ] [ diagram ] [ diagram ] [ diagram ] [ photograph ] [ diagram ] [ photograph ] [ drawing ] [ drawing ] [ drawing ] [ drawing ]

2D mapping of reentrant hexagon patterning to 3D surfaces reentrant unit reentrant unit variation - length of vertical strut reentrant unit variation - length of reentrant strut reentrant unit variation - measure of reentrant angle reentrant hexagon paper model with connected modules reentrant hexagon paper model with layering reentrant hexagon paper models with ribbons reentrant hexagon paper model with bounded internal angles reentrant hexagon paper model with flat struts volumetric unit paper model volumetric unit with bent strut paper model volumetric unit with bent strut aggregation paper model composite system model overlay broken down reentrant hexagon pattern internally bounded reentrant hexagon tension model reentrant hexagon tension models with compliant core reentrant hexagon elevated force models variations of Ron Resch patterns (Tachi 2013) variations of star-tuck origami (Tachi 2013) bistability graph and square-twist origami square-twist derivation and variations preliminary square-twist unit studies plan time lapse of unit motion plan time lapse of unit motion motion of an individual square-twist unit motion of an aggregation of square-twist units motion of an aggregation of square-twist units original deformation studies

15 16 17 18 19 20 20 20 20 22 22 22 22 24 26 27 28 29 33 34 34 35 36 37 37 38 38 39 41

[ vii ]

fig. 40 fig. 41 fig. 42 fig. 43 fig. 44 fig. 45 fig. 46 fig. 47 fig. 48 fig. 49 fig. 50 fig. 51 fig. 52 fig. 53 fig. 54 fig. 55 fig. 56 fig. 57 fig. 58 fig. 59 fig. 60 fig. 61 fig. 62 fig. 63 fig. 64

[ photograph ] [ photograph ] [ diagram ] [ photograph ] [ photograph ] [ photograph ] [ photograph ] [ diagram ] [ photograph ] [ photograph ] [ photograph ] [ photograph ] [ photograph ] [ diagram ] [ photograph ] [ photograph ] [ photograph ] [ screenshot ] [ screenshot ] [ diagram ] [ screenshot ] [ diagram ] [ screenshot ] [ diagram ] [ screenshot ]

scale studies material studies geometry of rigid bracing for planar quads preliminary aggregation studies hyperbolic paraboloid tension models assembly of plane-to-plane aggregation point, line, and plane hyperbolic studies family breakdown of fabrication panels for the Phare Tower (Bergin 2015) origami fabric study models final origami fabric expansion final study model final study model connections assembly of final models final model crease patterns final model (dome) final model (vault) final model (hyperbolic paraboloid) hinge force in simple plane folding of an individual square-twist unit initial fabric for simulation (dome) final results of digital simulation (dome) initial fabric for simulation (vault) final results of digital simulation (vault) initial fabric for simulation (hyperbolic paraboloid) final results of digital simulation (hyperbolic paraboloid)

[ viii ]

43 44 47 48 50 51 52 55 57 58 59 59 60 61 62 63 64 69 71 72 72 73 73 74 74

[ Abstract ] Auxetic geometry, which can be defined as geometry which expands in a direction perpendicular to a tensile force and contracts in a direction perpendicular to a compressive force, has found application at multiple scales within biology, textiles, aerospace, and the military. However, the kinetic nature of auxetics requires internal movement of the geometry when subjected to external forces. This movement often results in complex joints, making them difficult to fabricate and deploy at a larger, architectural scale. A conceit of this thesis is that geometric characteristics associated with auxetics make them ideal candidates for architectural application, such as their ability to adapt to synclastic (double) curvature and their ability to deform locally due to a stimulus. Out of Plane aims to investigate the potentials of auxetic geometry for architectural application, but also with digital and physical simulations of such applications in order to understand potential fabrication methods as well as design strategies for auxetic systems. With initial questions about the portrayal of auxetic patterning and the implication of the uniformly patterned systems that are typically represented, the potential of three dimensional auxetic origami systems is of specific interest. If the kinetic nature of auxetic geometries was to be exploited in conjunction with their innate ability to respond to local stimuli, it could be hypothesized that an initially uniform pattern could apply to countless different exterior and interior conditions.

[ ix ]

[ Out of Plane ]

[ 01_Introduction ] In an essay entitled “Digital Crafting: Performative Thinking for Material Design” by Mette Ramsgard Thomsen and Martin Tamke, a term ‘digital crafting’ is introduced which is used to describe “...a new engagement in the small scales of material specification, leading to a new practice in which architects and designers become the designers of the materials themselves, as well as of buildings and artefacts.”1 Thomsen and Tamke describe an environment, facilitated through digital means, that can change the way designers think of material practices through an understanding of material performance. In a digital age where design, simulation, analysis, and fabrication are inherently linked, the architect plays a new role in the design of materials for their specific purposes. While this new design of materials can be associated with solving a very specific architectural problem through geometric intervention(s), it can also be thought about in a reversed manner, where geometric characteristics are taken as a potential solution for an architectural problem which is not predetermined. Out of Plane concerns itself with the latter method, deriving characteristics from auxetic geometry that can be assembled to find application within the practice of architecture. Auxetic geometries have the unique characteristic that they expand when stretched in a direction perpendicular to a tensile force, and likewise, contract perpendicular to a compressive force.2 While these geometries are said to be scaleless (applicable to nano, micro, and macro scales), there is a direct conflict with the ability of these materials to be fabricated effectively and with predictability at all scales.3 Auxetic materials have found application in fields such as biology, medicine, aerospace engineering, textiles, and numerous others, but have yet to fully find their way into an architectural application. So the overarching question reveals itself: what is an effective architectural scale and application of auxetic geometry? With the embedded kinetic nature of auxetic geometries and their ability to exhibit both local and global

Thomsen, M. R. and M. Tamke (2014). “Digital Crafting: Performative Thinking for Material Design.”

242-253. 2

Darja, Tatjana, and Alenka, “Auxetic Textiles,” Acta Chim. Slov., no. 60 (2013): 715-723.

Zhang, Hu, Liu, and Xu, “Study of an Auxetic Structure Made of Tubes and Corrugated Sheets,” Physica

Status Solidi B, no. 10 (2013): 1996-2001.

[ 03 ]

characteristics, how can forces be enacted both internally and externally on the geometries which allow them to perform an architectural function? Tangentially, can this kinetic nature and the ability of auxetics to take on multiple forms begin to inform optimization methods in the practice of architecture? The research intends to take a design-based approach in order to use simulation methods, both digital and physical, to realize an application for auxetic materials in architecture. Digital simulation allows auxetics to be studied in their idealized form, relieving material and environmental constraints and strictly studying embedded geometric principles, while physical simulation necessitates ideas about scale and material to be integrated into design. The two methods of working are referred to as â&#x20AC;&#x2DC;simulationâ&#x20AC;&#x2122; since, in both digital and physical experiments, the models are simplified in order to investigate very specific relationships and concepts which attempt to prove the viability of auxetic geometry within an architectural application. This interpretation of simulation also describes the importance of not viewing auxetic geometries as static, but rather, always as dynamic, changing their shape through internal reactions to external stimuli over time.

[ 04 ]

[ 02_Auxetic Geometry ]

02.00_Introduction Poisson’s ratio is defined as the ratio between lateral contractile strength and longitudinal tensile strength, and it is grouped with three other material properties in order to form all of the elastic constants for isotropic materials (Young’s modulus, shear modulus, and bulk modulus). The sign and magnitude of Poisson’s ratio is determined by geometry and deformation mechanisms of the material structure, and when the sign of Poisson’s ratio is negative, the material is considered to be auxetic.4 Materials with a negative Poisson’s ratio exhibit enhanced shear stiffness, increased plane strain fracture toughness, increased indentation resistance, the ability to be applied to synclastic curvature (double curvature), and improved energy absorption properties.2 fig. 01 [ diagram ] basic auxetic principles [ left ] original unit with arrows representing tensile force being put onto the unit [ right ] resulting unit with arrows representing the direction of expansion

02.01_Micro Scale Application While the concept of auxetic geometry is quite simple, thinking of how this geometry may be implemented into real world scenarios can be a bit more complicated. One very specific application for auxetic geometry has been within the medical industry in the form of stents. A stent is ideally made from a material with a low elastic modulus and a high yield stress, two characteristics that are typically used to describe auxetic materials. Auxetic geometry is especially useful in stent applications because it can expand and contract along with the strains from cyclical pressures of blood flowing throughout the body.5

Evans, K. E., and A. Alderson. “Auxetic Materials: Functional Materials and Structures from Lateral Thinking!” Advanced Materials 12.9 (2000): 617-28. Web. Bhullar, S. K., et al. (2013). “Influence of Negative Poisson’s Ratio on Stent Applications.” Advances in Materials 2(3): 42-47.

[ 05 ]

Auxetic geometry is incredibly useful in the case of stents because it is highly customizable to the local conditions necessitated by the human body. For instance, typical hexagonal patterning can be intertwined with reentrant hexagon patterning (see 03.01_Reentrant Hexagon Patterning) in order to create portions of the stent which expand under tension while keeping some portions of the stent regular when subjected to any kind of stress (fig. 02).6 fig. 02 [ diagram ] multiple stent geometries and the effect that this geometry has on the auxetic nature of the stent (Zheng-Dong and Liu 2011) [ left ] reentrant hexagon patterning is located only at the edges, and therefore expansion can only happen at the edges [ right ] reentrant hexagon patterning is located only at the center, and therefore expansion can only happen at the center

Along with their ability to be highly customizable, auxetic stents are also useful because of the inherent characteristic of auxetic geometry to be able to accommodate complex shapes and curvatures. Since non-cartesian forms are found in the human body, this trait is directly applicable in auxetic stents. Furthermore, the curvature that auxetic stents are able to achieve can be highly choreographed before they ever enter the human body. By controlling the location of auxetic geometry within the stent, curvature (synclastic or anticlastic) can be precisely calculated during design and fabrication, even before implementation.

Zheng-Dong, Ma, Yuanyuan Liu. â&#x20AC;&#x153;United States Patent: 20110029063 A1 - Auxetic Stentsâ&#x20AC;?, February 3, 2011.

[ 06 ]

While auxetic stents can be extremely precise before they enter the body, the kinetic 1355

S. De Bock et al. / Journal of Biomechanics 45 (2012) 1353–1359

nature of the geometry does not necessarily need to be highly specific as one geometric pattern can assume multiple forms of curvature without breaking. In specific applications, the stent is also allowed to grow in order to fill a larger space. This condition once again relies on the ability of auxetic geometry to adapt locally as it expands. Since Fig. 2. The 3 stent designs used in the study, with a detail of the second order hexahedral mesh; from left to right: C1,C2 and N1.

the wall thickness of a given site within the human body is rarely consistent, auxetic

Table 2 Stent dimensions and mesh size.

stents are useful as they can expand as much or as little as necessary depending on C1 C2 N1

Thickness (mm)

Width (mm)

Number of repetitive L (mm) units over L

Number of elements

0.06 0.06 0.06

0.065 0.065 0.065

5 5 4

15,072 15,072 12,480

10 10 10

local conditions within the host.7

S. De Bock et al. / Journal of Biomechanics 45 (2012) 1353–1359

fig. 03 [ rendering ] auxetic stent (Bock et al. 2012) [ left ] the original unit and its expansion to fill the host

Fig. 4. A schematic view of the calculated parameters used for treatment evaluation; the strut coverage (SC), tortuosity T¼D/L � 1 and the apposition, quantiﬁed by the strut to vessel wall distance.

[ right ] varying synclastic curvature which can be accommodated by auxetic stent

the stented region and D is the straight distance between these two points (Thomas et al., 2005). The tortuosity measure describes the fractional increase of the vessel length, relative to the shortest path between its two endpoints (a straight vessel having a T¼ 0). 3. incomplete strut apposition to the vessel wall, computed as the distance from the external stent surface nodes to the closest vessel surface element. In order Fig. 7. Stent apposition the vessel wall for all of cases. The contourplot distance the vessel the wall, ranging from blue (0 mm) to re totoquantify the amount stent struts that areshows well the apposed, wetocomputed percentage area of stent struts within a threshold distance of the vessel wall, relative to the total stent area. Two thresholds (0.1 and mm) were chosen(Shobayashi et al., 2010); h low0.2 bending stiffness Table 4 to differentiate in the severity of the malapposition, a third, morewith ﬂexible (C2)high design investigated, the low Tortuosity increment, strut coverage and percentage of stent surface with threshold (0.5Values mm) are indicating struts at the aneurysm distances less than the chosen thresholds. averagedstent for each stentthat on are located observed with the open-cell design is not reach all the vessel geometries. dome or at vessel branches.

02.02_Macro Scale Application

strated here, open-cell stents like the N1, are mo the global vessel curvature and tortuosity, and straightening of the vessel. Vessel straightening by has been previously reported, and it could po 3. Results C1 42.9 7 9.8 7.8 7 1.6 56.7 710.1 70.4 7 5.1 83.8 7 1.1 arterial kinking, uneven mechanical strain d C2 26.9 7 11.9 8.7 7 1.6 59.8 72.6 73.4 7 4.6 85.8 7 0.7 other mechanical effects, which might cause in N1 stent. 5.1 7 1.6 11.0 7Strut 1.1 25.0 75.9 56.0 7 6.4 81.8 7 3.4 3.1. coverage Fig. 3. The complete deployment procedure, starting from the undeformed (Hsu et al., 2006). The stent is crimped and bent according to the vessel geometry. Upon release, the The open-cell design does have some well kn super-elastic stent expands to fit the vessel geometry. An exemplary movement of The strut coverage of the aneurysm neck is which visualized ings, appearand in the two wide necked aneu the cylindrical sheath, used to enforce the displacement, is shown in the bottom of quantified in N1 Fig.is5.well While for anfrom undeformed stent, percendesign. The fish scaling of the detected the cases thethe stents show outward prolapse of 8 the figure. stent strut surface area covering a cylindrical vessel the simulations and cantage occur at different locations: aneurysm domeof (i.e. fish-scaling) when situated a of a parent artery, and into the vessel lumen, same diameter is similar for the three designs (8.1%, 8.4% and 8.3% geometry using vmtk (the vascular modeling toolkit; http://www.vmtk.org). in the inner curvature of theand vessel, in the parent of an artery. For the former case, fish-scali for C1,C2 N1protruding respectively), notable regions differences can be To enable analytical computations, the centerline was approximated using circular artery (Fig. arrowhead indicator) of one or more stent cells, whic arcs. The sheath was bent according to these arcs. The radius of curvature was 7—double observed between different stent designs, andincreased differentopening aneurysm at an enlargement of the vessel diameter due to the (wide necked) migration of small coils through the stent cells modified in relation to the analysis time step. In this way the bending of the stent geometries, with clear dissimilarities between wide (A1,A2) and 2004; Akpek et al., 2005). In the latter case, the inw occurred smoothly, helping the convergence of the implicit aneurysm simulation.(Fig. 7A1/N1 and A2/N1, at the aneurysm neck) narrow-necked (A3) aneurysms. The deployment procedure is shown in Fig. 3. struts will likely promote a hemodynamic effect In the wide neck A1 and A2 geometries, thevorable closed-cell designsthat could lead to stent thr flow patterns In these aneurysm geometries, incomplete apposition of the do large not part, cover the aneurysm but for pass the aneurysm in-stent stenosis.with, The presence of struts in the l stent struts is, for the located at the sameneck locations 2.4. Treatment evaluation approximately, a diameter that isbetween the same as the parent vesselsthe placement of the throm obstruct or encumber the different designs, with the largest differences or other endovascular resulting in N1 low strutvisible coverage (7.3/6.5% for C1 and devices. designs originatingdiameter, from fish-scaling of the design, at We evaluated the impact of the implanted stents in the different aneurysm Moreover, bothcan for open and closed designs, q multiple locations. 8.2/7.5% Table 4 summarizes the performance of the hand, the for C2 in A1/A2). On the other N1 stent geometries, focusing on the following, potentially clinically relevant, parameters: incompletefor stent can be one of the pr stents, averaged over the three more easilyaneurysms. adapt to the vessel shape (11.5/11.8% N1apposition in term patency of the stented aneurysm (Ebrahimi A1/A2). The A3, having a narrow neck, allows for all designs to 1. the strut coverage in the aneurysm neck, defined as the percentage in neck area experimental in vitro study reports kinking and i TM cover the aneurysm neck surface (9.5/10.5/9.8% for C1/C2/N1 in A3). that is covered by the expanded stent surface. This is calculated by a view 4. taking Discussion apposition for closed cell design at high cu cut of the vessel at the location of the aneurysm neck, with the view perpendicular (Krischek et al., 2011), and in-vivo studies also sug to the vessel centerline and circumference. The percentage of stent surface in the 3.2.have Tortuosity In this study we used finite element analysis to gain cell designs are more susceptible to malappos aneurysm neck is quantified using an image based pixel count method; insight into the deployment and mechanical behavior of cereet al., 2009; Heller and Malek, 2011a). Contrary to 2. vessel tortuosity T is calculated pre- and post-stenting, as a measure of the brovascular stents and their interaction with the vascular wall. the results in these vessel geometries s The vessel straightening is summarized in Table 3. obtained It can be vessel straightening induced by the stent insertion. It is defined as T¼ L/D � 1 The are used to assess stent meets thebehavior N1 design givesdesign worse apposition values compar clearly observed thathow thisa straightening is stent where L is the length of the (tortuous) centerline from the origin to simulation the end of results primary mechanical requirements, and to quantify possible cell designs. It is possible that these discrepancie adverse effects of stent deployment. The strut coverage of the geometries of the parent artery, that can easily aneurysm neck can be an indirect indicator of coil protrusion and both stent designs, and from the design of the inv can promote aneurysm occlusion by reducing the blood flow in cell stents, which are either overly stiff, and straig the dome. The amount of struts covering the aneurysm neck or sufficiently flexible, and appose well. Furtherm depends not only on stent design, but also on the vessel geometry. discussed fish-scaling of the N1 design also con incomplete apposition. While this dependence on diameter has been previously reported in silicone mock arteries (Krischek et al., 2011), the finite element The presented virtual framework can be used t of adverse effects of stent design, and predict the results take into account the complex geometry of patientspecific aneurysms. In general, for a wide-necked aneurysm, for each specific patient in an acceptable run modern desktop pc). In contrast, when using a pro closed-cell designs adapt less to the aneurysm geometry than open-cell designs. Closed-cell designs can be designed to have a these mechanical effects would not have made a A schematic view of the calculated parameters is shown in Fig. 4.

Tortuosity increment (%)

Strut coverage (%)

S0.1/Sstent (%)

S0.2/Sstent (%)

S0.5/Sstent (%)

At a slightly larger scale, auxetic geometry is being used in textile manufacturing across multiple disciplines. There are two types of auxetic textiles; those that use auxetic fibers in order to produce the textile’s structure and those that use conventional fibers.

One of the pioneers in the area of auxetic textiles is Dr. Patrick Hook, managing director at Auxetix Ltd., who has been working alongside the University of Exeter, Dow Corning

Ltd., and Advanced Fabric Technologies LLC in order to develop auxetic fabrics. One specific fabric which he has been working on is called Zetix

helical-auxetic fiber

technology, which is a blast-mitigation fabric that has been tested and used throughout North America. The fabric uses an auxetic yarn which is able to deform when subjected to the pressures associated with an explosion without tearing. It does this by expanding

its patterning under pressure, allowing thousands of air holes to be created through which the pressures associated with the blast can pass. While allowing air through

Bock, S. De, F. Iannaccone, G. De Santis, M. De Beule, P. Mortier, B. Verhegghe, and P. Segers. “Our

Capricious Vessels: The Influence of Stent Design and Vessel Geometry on the Mechanics of

Intracranial Aneurysm Stent Deployment.” Journal of Biomechanics 45, no. 8 (2012): 1353-359. Darja, Tatjana, and Alenka, “Auxetic Textiles,” Acta Chim. Slov., no. 60 (2013): 715-723.

[ 07 ]

the holes, debris is rejected by the fabric as the holes are not large enough for it to pass through. David O’Keefe of Advanced Fabric Technologies describes ZetixTM, “It is the only material on the market that automatically adjusts its strength and thickness in response to explosive forces. Because it has memory, it returns to its neutral state when the stress is dissipated.” The fabric technology was tested with eight close range grenade explosions and sustained minimal damage.9 While ZetixTM fabric is being used largely in stationary applications, The Fly-Bag project, a proposal for a portable auxetic fabric bag, is being funded by the European Union. The purpose of The Fly-Bag project is to propose a retrofitting solution for blast protection within airplanes since thickening the skins of planes would be extremely expensive and lead to a weigh penalty.10 A single bag would be able to contain thirty pieces of luggage and would be able to contain small explosions and would be able to smother a postblast fire. In tests, a prototype was able to successfully contain five detonations of a small explosive called RDX.11 While the defense industry is capitalizing on the useful characteristics of auxetic textiles for blast protection, other companies are introducing auxetic textiles into everyday applications. Under Armour, Inc. is one of the leading entities in auxetic fabric research fig. 04 [ drawing ] auxetic apparel (Toronjo 2014) [ left ] backpack with impact resistant auxetic fabric [ right ] shin guard with impact resistant patterning

Bealer Rodie, Janet. “The Auxetic Effect.” Textile World. May 2010. Accessed January 30, 2015.

http://www.textileworld.com/.

“Fly-Bag2.” FLY-BAG2. Accessed January 30, 2015. http://www.fly-bag2.eu/.

“Material Benefits.” The Economist. September 07, 2013. Accessed January 30, 2015.

http://www.economist.com/.

[ 08 ]

for the consumer market. Currently, they are developing auxetic fabrics through focusing on two major features: impact resistance and breathability. Impact resistance has to do with the internal geometry of auxetic materials which allows for the shrinking of the geometry when subject to a compression force. For instance, Under Armour, Inc. is developing proposals for auxetic impact resistant backpack components, which would protect the contents of the backpack when dropped or hit, as well as impact resistant shin guards which compress under a local stimulus from a piece of equipment such as a stick or a ball. Breathability is being taken into account through the development of fabrics for the head and body which expand to conform to the body’s natural curves while also expanding the geometry to allow for air to easily pass through the fabric and to the skin.12

02.03_Design Scale Application The last scale will be termed the “design” scale, which refers to a scale at which the auxetic geometry becomes tangible and visible in the final product. While there are many different examples of auxetics being used at this scale, certain examples have been chosen to show the breadth of the research associated with tangible auxetic geometry. The first example of auxetic material research at a design scale comes from Andreas Bastian, and it concerns itself with low cost, 3D printed prosthetics. The project attempts to create meshes (built from common rigid materials such as 3D printed PLA) that are able to deform and achieve synclastic curvature. By printing a single extruded line (0.1 mm) of material from a 3D print head.13 Although the PLA remains brittle, the thinness of a single extruded filament allows some bending to happen before the PLA breaks. The patterning is seen to be auxetic since a tensile force results in the internal geometry expanding in all directions.

Toronjo, Alan. “United States Patent: EP 2702884 A1 - Articles of Apparel Including Auxetic Materials”, March 5, 2014. Bastian, Andreas. “3D Printed Mesostructured Materials.” ANDREAS BASTIAN. March 17, 2014. Accessed November 12, 2014. http://www.andreasbastian.com/.

[ 09 ]

fig. 05 [ photograph ] synclastic curvature in PLA 3D printed materials (Bastian 2014)

Another example of design scale application of auxetic geometry is by a company called Trex:Lab who is researching advanced rubber origami folding. Their process, called OriMetric, is being used to create products that are both aesthetically pleasing and functional. They see potential in their products to become used in sporting equipment, medical stress relief, interior design, and other industrial applications.14 The products are not directly referenced as being auxetic, but through their potential as compressive, protective patterning, their expanding nature, and their ability to take on double curvature (specifically that of the human body), the products are examples of applied auxetic geometry through origami folding. fig. 06 [ photograph ] rubberized origami being created by Trex:Lab (left and center) and its ability to accommodate double curvature (right) (Trex:Lab 2014)

The last example comes from a material research company called Betatype Limited. The overall material research at Betatype Limited does not concern itself only with auxetic materials, but a certain interest in these materials is shown through their body of work. One specific material which exhibits auxetic behavior reveals itself as a

Trex:Lab. Accessed November 15, 2014. http://www.trex-lab.com/.

[ 10 ]

simple, flat plane of material from which a pattern of bumps emerges when stretched horizontally.15 fig. 07 [ photograph ] Betatype material at rest (left) and after it is put into tension (right) (Betatype 2014)

02.04_Conclusion Auxetic geometries can perform at various scales within various disciplines, but their applications have proven to be highly specific. Whereas reentrant hexagon patterning has been used at the micro scale for its ability to expand blood vessels when subjected to force, at a macro and design scale, the patterning has been used for protection from shock. The drastic difference between these applications shows not only that auxetic materials are useful, but also that they are extremely versatile.

Betatype. Accessed November 15, 2014. http://www.betaty.pe/.

[ 11 ]

[ 03_Reentrant Hexagon Patterning ]

03.00_Preliminary Experimentation The reentrant hexagon pattern, a known auxetic pattern, was the focus of the first case study into existing auxetic geometry. During this phase, the patterning was investigated as pure geometry, meaning that it was understood as lines and planes. It did not concern itself with fabrication methods in any way, meaning that the study was attempting to derive a general understanding of auxetic principles as opposed to zooming in and focusing on one very minute detail of the geometry. It was decided that this general understanding could later influence fabrication as opposed to the research becoming entirely about how to fabricate auxetic geometry The case study was meant to derive preliminary understanding about the interactions within the patterning, how these interactions could be controlled, and how the patterns could be globally activated. fig. 08 [ drawing ] reentrant hexagon patterning in compressed state (left) and expanded state (right)

Reentrant hexagon patterning was of interest due to its typical portrayal as a uniform pattern. Initial experimentation investigated what would happen to the expansion and contraction properties of the patterning if they were not applied in a uniform fashion. fig. 09 [ drawing ] initial pattern (dotted grey) and the expanded pattern (solid no fill)

original shape 2d auxetic patterning expanded auxetic patterning when pulled

[ 13 ]

fig. 10 A17

C17 t.Sd1

E17

A17

C17

A16

C16 E16 E16

A16

C16

A15

C15 E15 E15

A15

C15

A14

C14 E14 E14

C14

A14 A13

C13 E13

E13

A13

C13

A12

C12 E12

E12

A12

C12

A11

C11 E11

E11

E10

A11

C11

A10

C10 C10

A10

t.Ed1 E9

A9 t.Cd2

t.Cd1 E8

C7 C7

A8 t.Ed2

E3 A3

C3 C2

E2 A2

C2 C1

E1 A1

C1 C0

F E

[ 14 ]

t.Sd2

[ drawing ] culmination of preliminary reentrant hexagon geometry studies (in conjunction with Nicholas Bruscia)

The expansion of the 2D linework was facilitated through using Kangaroo in Grasshopper for Rhino. By attempting to force the angles within the system to become 90o, an equilibrium was reached when all angles had expanded as much as possible without changing the dimension of any single line (spring). This process helped to understand the way that the varied module size of the reentrant hexagon pattern affected local rates of expansion when the pattern was subjected to a force. 3D studies were also conducted in Kangaroo. Using a similar methodology, 2D linework was mapped to 3D surfaces with varying amounts of curvature. Once again, the angles were all forced to be as close to 90o as possible. While the results were unpredictable and did not reproduce the same result every time the simulation was run, it was useful to visualize the way that the pattern was able to expand in three-dimensional space. fig. 11 [ screenshot ] expansion of linework derived from curved surfaces

03.01_Reentrant Hexagon Manipulation After the initial visualization exercises, the reentrant hexagon geometry was decomposed in order to understand it further. It was broken down into its three major components: the length of the vertical strut, the length of the reentrant strut, and the angle measure of the reentrant angle.16 By manipulating each one of these variables independently, a better understanding of the reentrant hexagon was gained which allowed for a more intuitive approach to modeling, giving a certain amount of predictability to the amount of expansion of an individual unit or aggregation of units.

Yang, L. (2011). Structural Design, Optimization and Application of Three-Dimensional Re-entrant Auxetic

Structures. Aerospace engineering / Mechanical engineering. Ann Arbor, MI, North Carolina

State University. ph.D: 191.

[ 15 ]

fig. 12 [ diagram ] reentrant unit

x = length of vertical strut y = length of reentrant strut z = reentrant angle

03.02_Paper Models While the pure geometric studies were useful in understanding the way that the reentrant hexagon patterning functioned in an ideal case study, they failed to address how they may be fabricated. All of the initial physical models were done using Stonehenge paper, and they attempted to take the linework from computer-generated drawings and translate it to three dimensional constructs. Initial studies were quite simple, being drawn, cut, folded, and glued by hand, remaining uniform as extrusions of 2D linework. Within the construction of the first model, a fabrication issue was immediately exposed. The model was constructed by first creating individual (identical) reentrant hexagon modules. Then, these modules were assembled into a larger aggregation of units (fig. 16). However, where multiple lines could be coincident without a problem in the digital realm, they could no longer be coincident in physical models, resulting in restricted movement of units, not allowing the modules to expand or contract once glued. Through physical modeling, it was determined that the point-to-line relationship of converging edges would become a focus of the study. Another attempt shifted the rows of modules so that they stacked on top of one another to remove issues created by aggregating modules through connecting their faces (fig. 17). Instead of these face-to-face connections, stacking attempted to create edge-toedge connections in order to to mimic the way that the pure geometry studies had been functioning (overlapping lines). While this helped with the connecting of modules in one axis, face-to-face connections within rows were still problematic as they employed the fabrication technique used in the first experiment. This technique once again restricted movement within the individual rows, therefore decreasing the ability of the assembly to function globally.

[ 16 ]

fig. 13 [ drawing ] breaking down reentrant hexagon patterning

variable = length of vertical strut

X=2 Y=1 Z=1 W1 = .93 W2 = 1.19 WE = 128% H1 = 2.42 H2 = 2.8 HE = 116%

X = 1.5 Y=1 Z=1 W1 = .93 W2 = 1.19 WE = 128% H1 = 1.72 H2 = 2.1 HE = 122%

W1 U1

L4 L3

X=1 Y=1 Z=1 W1 = .93 W2 = 1.19 WE = 128% H1 = 1.03 H2 = 1.4 HE = 136%

L2 L1

W1 U1

U3 U2 L2

U4 L4 L3

A A B

B C C

X = .75 Y=1 Z=1 W1 = .93 W2 = 1.19 WE = 128% H1 = .68 H2 = 1.05 HE = 154%

W1 U1

U3 L2

U4 L3

A B

B C

X = .5 Y=1 Z=1 W1 = .93 W2 = 1.19 WE = 128% H1 = .33 H2 = .69 HE = 209%

U2 L1

W1 U1

L2 U2

L4 U4

[ 17 ]

fig. 14 [ drawing ] breaking down reentrant hexagon patterning

A A B

C C

W1 U1

U3 L2

X=1 Y=2 Z=1 W1 = 1.87 W2 = 2.39 WE = 128% H1 = .65 H2 = 1.4 HE = 215%

A A B

B H1 C C

X=1 Y = 1.5 Z=1 W1 = 1.4 W2 = 1.79 WE = 128% H1 = .84 H2 = 1.4 HE = 167%

U3 L2

B H2

H1 C

X=1 Y=1 Z=1 W1 = .93 W2 = 1.19 WE = 128% H1 = 1.03 H2 = 1.4 HE = 136%

W1 U1

U3 U2

B H2

H1 C

X=1 Y = .75 Z=1 W1 = .7 W2 = .9 WE = 128% H1 = 1.12 H2 = 1.4 HE = 125%

W1 U1

U3 U2

L2 L1

L4 L3

B H1 C

X=1 Y = .5 Z=1 W1 = .47 W2 = .6 WE = 128% H1 = 1.21 H2 = 1.4 HE = 116%

W1 U1

U3 U2 L2

U4 L4 L3

[ 18 ]

variable = length of reentrant strut

fig. 15 [ drawing ] breaking down reentrant hexagon patterning

variable = reentrant angle

B H2

W1 U2

X=1 Y=1 Z=2 W1 = 1.16 W2 = 1.19 WE = 103% H1 = 1.53 H2 = 1.4 HE = 92%

W1 U1

X=1 Y=1 Z = 1.5 W1 = 1.16 W2 = 1.19 WE = 103% H1 = 1.26 H2 = 1.4 HE = 111%

X=1 Y=1 Z=1 W1 = .93 W2 = 1.19 WE = 128% H1 = 1.03 H2 = 1.4 HE = 136%

W1 U1

U3 U2

A A

C C

X=1 Y=1 Z = .75 W1 = .74 W2 = 1.19 WE = 161% H1 = .93 H2 = 1.4 HE = 151%

W1 U1

U3 U2 / L2

U4 / L4

A A

X=1 Y=1 Z = .5 W1 = .52 W2 = 1.19 WE = 229% H1 = .86 H2 = 1.4 HE = 163%

W1 U1

U3 L2 U2

L4 U4 L3

[ 19 ]

fig. 16 [ photograph ] original reentrant hexagon paper models with connected modules

fig. 17 [ photograph ]layered reentrant hexagon paper model

fig. 18 [ photograph ] ribbon reentrant hexagon models [ above ] reentrant hexagon modeled with straight ribbon sections [ below ] reentrant hexagon modeled with curved ribbon sections

fig. 19 [ photograph ] reentrant hexagon model with bounded internal angles

[ 20 ]

The next series of models broke down the reentrant hexagon into two components: the vertical struts and a continuous reentrant ribbon (fig. 18). By doing this, the ribbons acted as a single piece which could accordion in and out depending on where stress was placed. The ribbon would then be intersected on alternating sides by the vertical struts. This assembly was able to expand and contract much more freely when subjected to external forces, but still had issues where the vertical struts intersected the ribbon. If they were not glued perfectly, the glue restricted and partially controlled movement within the joint. The last model in the initial group took an individual reentrant hexagon module and examined how different angles could be controlled internally. In order to do this, a larger module was created, and then strips of Stonehenge paper were added to the joints which would restrict the movement (fig. 19). While adding paper to all of the joints proved problematic as the module could not expand or contract, an initial idea about controlling internal forces within the pattern was derived that would persist throughout the entirety of the research. Moving away from simply extruding 2D linework, the next group of paper models attempted to add a three-dimensionality that previous experiments were lacking. These studies did not necessarily evolve from the reentrant hexagon patterning, and introduced new patterns to the preliminary experimentation. The goal of these models was to move away from simple extrusions and examine multi- axis relationships within compositions that contained discrete auxetic members as well as more typical, nonauxetic connecting geometries. The first model began with a simple extrusion of the reentrant hexagon module while incorporating strips to connect the individual modules. Folds were also added to the flat struts in an attempt to have the patterning function in multiple axes (fig. 20). The connecting strips, however, were quite flimsy, making it very difficult to apply any type of stress to the assembly. The second series of models created a number of volumetric units to be tested. Beginning with a reentrant hexagon in elevation, square surfaces were added to the top and bottom, and the reentrant struts were extruded along all the edges of the

[ 21 ]

fig. 20 [ photograph ] reentrant hexagon modules connected with flat strut pieces

fig. 21 [ photograph ] volumetric unit models [ above ] single unit studies [ below ] aggregation of multiple units

fig. 22 [ photograph ] volumetric unit with bent strut connecting top and bottom

fig. 23 [ photograph ] rotational motion of the strut within a larger assembly [ left ] compressed [ right ] expanded

[ 22 ]

square (fig. 21). The units were able to take some stress, but connection was once again a problem as the edge-to-edge connections of the faces needed to be glued along a single, paper thin line. Here, the glue began to impede the performance of the module as it could fill the reentrant angle and restrict its movement. In certain cases, the glue also failed under excessive stress. While the individual module was functional, aggregating the three dimensional units proved to be very difficult. When aggregated in a straight line, the modules still functioned, but when aggregated in multiple directions, they began to lose functionality as movement became more restricted. Leaving the reentrant hexagon patterning behind for a moment, a unit was created which used a square on the top and bottom with a single bent face connecting edgeto-edge. Unlike the reentrant volumetric module, this module used the bent face to connect in a rotational fashion (fig. 22). When put into tension, the volume expanded to a certain point (auxetic) and then contracted (typical material) as the bent face was able to create rotation. This unit was subject to the same edge-to-edge glue issues as previous iterations, but it was used to introduce an idea about rotational motion within an auxetic module. Aggregation of the units happened vertically and functioned similarly to the individual module (fig. 23).

03.03_Composite System The reentrant hexagon case study helped to understand relationships that could occur within the patterning, but the study was still lacking a good solution to the physical connection between members within the system. Digital simulations and typical portrayals of auxetic geometry do not imply nor concern themselves with physical connections, but as soon as application, scale, material, and/or purpose were considered, these connections needed to be invented in order for the pattern not only to exist, but also to function effectively. The focus of the next study was to address this edge-to-edge connection without being overly prescriptive of the flexibility within the overall aggregation. If the node connection between edges was too rigid or only allowed a very limited amount of movements, then it was actually just the design of the node that dictated whether the overall pattern would behave as auxetic, rather than the larger geometry which created the pattern. However, if the node allowed for too much movement, it would be hard to determine whether it was the geometry which was

[ 23 ]

fig. 24 [ photograph ] overlay of the motion from the composite system when put into tension from two sides

[ 24 ]

making the system expand or if it was the weakness of the joint between edges failing which was allowing for expansion. The system was to be composed of a rigid/semi-rigid member used for all struts along with a flexible node connection to connect all members. The node’s flexibility was not a function of its internal mechanisms, but rather, its material composition.

It was

determined that the material palette for the new model would be limited to wood, Stonehenge, and thin gauge aluminum for the struts, and urethane rubber for the nodes. A series of tests were done which tested the ability of the urethane rubber (SmoothOn ReoFlex 40 and Smooth-On ReoFlex 60) to bind to the wood (1/64” plywood), Stonehenge, and aluminum (.05” thick). Molds were created using layered 1/4” clear acrylic, and the shapes of the nodes were varied slightly in order to test the ability of the rubber to be poured into various forms. It was found that the 1/64” plywood bonded the best to the rubber as the aluminum was able to slide out of the cast, and the Stonehenge ripped multiple times during demolding. Due to its ease of pouring, ReoFlex 40 was chosen for the final composite system. In order to increase the ability of the wood to bond to the rubber, small holes were cut into the ends of each wooden strut in order to allow the rubber to bond to itself through the wood. A larger acrylic mold was created which held the wooden struts in place while the rubber was being poured. Once demolded, it was found that the system was able to transfer loads through the nodes, but it was quite flimsy. The system worked well when placed on a flat surface and then activated (put into tension), but when it was held up, the surface was highly subject to forces of gravity which stretched the geometry without any other external force. It was also found that the surface was able to accommodate double curvature with ease (it was able to completely fold in on itself in any direction), but it was difficult to tell whether this is solely due to the weak joints or whether the global geometry was playing a role as well. Local interventions into the surface proved extremely interesting in understanding what happened to the larger aggregation when only a certain portion was activated. As the force resonated away from the local stimulus, the pattern was less affected, and there were certain areas of the system (depending on how large it was and how large the force was) that remained unaffected by the stimulus.

[ 25 ]

03.04_Force Models The composite system model drastically changed the way that the reentrant hexagon pattern was researched as it returned the conversation to one about internal forces within the larger system. While the composite system was able to be put into tension and expanded, it also retained the capacity within each joint to go back to its original form. This partially began to answer a question that had repeatedly come up in conversation: where does the force come from? The auxetic systems which were being investigated were all responsive to a certain stimulus, whether it be internal or external, which allowed them to function in a useful way. Moving away from the composite system and a focus on the joint between edges, an idea came to the forefront about how to control the internal forces within the reentrant hexagon unit. While this had been looked at earlier with paper models being bounded at all of the angles, a slightly different approach was taken this time. If, instead of bounding the angles, one was to bound the distance between the innermost points of the reentrant struts with a flexible material, the flexible material could help to return each individual module to its original state after it was acted upon by an external force. fig. 25 [ diagram ] reentrant hexagon pattern (left) being broken down and bounded at the reentrant strut

The first model used 1/64â&#x20AC;? plywood cut as the main material for the reentrant hexagon pattern. Similar to the techniques used in the paper models, strips of plywood were cut into ribbons (for the reentrant struts of the system) with a pattern of slits which would allow them to bend in either direction. These ribbons were intersected with straight, rigid sections which would serve as the top and bottom of the reentrant hexagon. Built into the straight sections were small hooks which could be connected by small, clear rubber bands. When pulled, the pattern was able to expand slightly, and then the force put into the rubber bands returned the reentrant hexagon pattern to its original state. While the system did partially respond as predicted, the reentrant struts were extremely fragile due to the slit pattern, restricting the amount of force the system could handle before breaking. The rubber bands were also quite stiff, making it difficult to get much movement out of the system even when a large amount of force was introduced.

[ 26 ]

fig. 26 [ photograph ] overall reentrant hexagon system with rubber bands between each of the flat struts within the system (right)

Taking cues from earlier textile research, the model was changed to have a compliant core (flexible core) to replace the reentrant ribbon, and a series of rigid struts to structurally bind the compliant core (see far right fig. 25 - dotted lines are removed, black lines replaced by compliant core, and red lines replaced by rigid strut). Rubber bands were used for the compliant core and varying length 1/16â&#x20AC;? basswood was used for the rigid struts. While the models (fig. 27) appear as a series of regular hexagons as opposed to reentrant hexagons, they are simply meant to analyze the internal forces within the reentrant hexagon pattern - they are meant to be a visual representation of the internal forces within the pattern rather than a visual representation of the actual geometry which is creating those forces. The stressed rubber band (compliant core) is meant to represent the internal stresses of the reentrant strut ribbon while the rigid struts are meant to represent the force existing between the innermost vertices of the reentrant struts. While the exercise was successful in creating a static visualization of these forces, they needed to be more dynamic, taking into account the changing amount of force within members as the pattern was acted upon by external forces. For instance, if the rigid strut had been replaced by a more flexible material, it would have been easier to analyze the way that the two components acted upon each other.

03.05_Force Models Translation An immediate critique of the force model studies was their apparent flatness and two dimensionality. Everything to this point, while three dimensional in some sense, was extremely flat, typically only becoming three dimensional due to an extrusion of certain pieces. The next goal was to get the patterning to not merely be an extrusion but to actually deform out of plane and become three dimensional in its own right.

[ 27 ]

fig. 27 [ photograph ] reentrant hexagon tension models with compliant core [ left ] original grid lines where the compliant core (rubber band) was oriented [ right ] iterations of inserting rigid struts connecting compliant cores

[ 28 ]

fig. 28 [ photograph ] plan and perspective of elevated force models [ 1 series ] rubber band as reentrant ribbon (flexible / elastic) [ 2 series ] steel cable as reentrant ribbon (flexible / nonelastic) [ 3 series ] steel wire as reentrant ribbon (not flexible / nonelastic)

[ 29 ]

The first step in this process was to take the force models studies and elevate them off of a ground plane. While they still remained somewhat flat, the stiffness of the rubber band was now more apparent as it needed to span between two exterior pieces. Once the original rubber band models were duplicated between two planes, they were analyzed. Due to the elastic nature of the compliant core, the reentrant ribbon had stretched when it was bound by the rigid struts. This new length was calculated and used to create new force models which replaced the compliant core with rigid materials. The first material to be incorporated was thin braided steel cable (used for jewelry making). The cable was cut to length and then pinned between two planes. The rigid struts were introduced, putting the reentrant ribbon into tension and preventing it from sagging, but not allowing it to deform out of plane. The reason for this was quite obvious - while the reentrant ribbon was now longer, it still had, inherent within its material properties, the flexibility to take the shape that the original rubber band had taken. Effectively, replacing the rubber band with the same length of flexible, nonelastic material only made the surface more rigid. In the next model, the cable was replaced with slightly thicker steel wire. The wire was not nearly as flexible as the cable, and was cut to the same length as both the elongated rubber band and the steel cable. When the wire was bound between the two planes, it already exhibited a different behavior, splaying apart at the middle because it could not fit between the two planes properly. When bound by the rigid struts, the wire could no longer splay in the XY plane, and was forced in the Z direction. This happened due to the distance horizontally between the wire being bound at the end points as well as intermediate points along the span. The model represented the first time that a model had deformed out of plane due to an internal force.

03.06_Reentrant Hexagon Conclusions The reentrant hexagon pattern served as an extensive case study in order to gain an understanding of the behavior of auxetic geometry as well as to exploit multiple characteristics of these geometries that would frame the next stages of research. While the study began as an investigation into pure geometry, it progressed into a study of the connections within an overall system, different ways to fabricate such a system (at a design scale), and the forces that existed within an individual unit as well as within a larger aggregation.

[ 30 ]

Moving forward, the case study proved worthwhile in framing certain interests associated with auxetic geometry: 1. Local control v. global control Auxetic aggregations were able to respond extremely well to local conditions, but the effect that these local conditions had on the global geometry was of interest as well. What was the extent to which a system could respond to local stimuli? What was the scale at which these local stimuli could intervene? How did the global control of the system effect its ability to respond to local conditions? 2. Internal v. external forces Internal forces always existed within auxetic geometry, but the extent to which they existed vary. They could simply hold the geometry in place against forces of gravity, or they could be a lot more active, repelling most external forces. Was there potential for an auxetic system to be completely enacted by internal forces? How did these internal forces need to be arranged in order to respond to external conditions and forces? 3. Synclastic curvature Auxetic geometry was known to be good at conforming to double curvature of various types and severities. Could an auxetic construct be tuned in order to accommodate certain curvatures at local conditions? What were the potentials for auxetic geometry to be able to solve problems of highly complex curvature in the built environment? 4. Kinetic and dynamic nature of auxetic geometry The most interesting models that were yielded during the reentrant hexagon study were those that were dynamic and were able to move. Static representations were helpful at certain intervals, but for the most part, in order to prove the auxetic nature of a material or geometry, it had to function. How could this function move forward into dynamic forms of representation and design to yield an application for auxetic geometry?

[ 31 ]

[ 04_Bistable Origami ]

04.00_Introduction to Origami In an attempt to address all of the ideas outlined from the reentrant hexagon study, research began into basic origami structures. It was determined that through strategic folding of flat sheet material, origami patterns could create auxetic materials that expanded when put into tension. Origami can be defined as “the art of folding a sheet of paper into various forms without stretching, cutting, or gluing other pieces of paper to it.”17 One of the major figures in the research of origami in architecture is Ron Resch, who worked with origami patterns in the 1960s and 1970s. His work focused around finding stable solutions for folded origami patterns. He specifically proposed a series of modular models, or origami tesselations, which could be folded into a piece of paper in different arrangements in order to create multiple forms. Resch also began to translate these concepts into computer programs which could translate 2D curves into 3D fields of origami folds, allowing for visualization of the 3D origami patterns.18 fig. 29 [ diagram ] variations of triangular crease patterns which are derived from original patterns by Ron Resch (Tachi 2013) [ left to right ] star, truncated star, curly star, twist fold [ top to bottom ] crease pattern, folded surface front, folded surface back

Tachi, Tomohiro. “Designing Freeform Origami Tessellations by Generalizing Resch’s Patterns.” Journal of Mechanical Design 135, no. 11 (2013): 111006. doi:10.1115/1.4025389. Schmidt, Petra, and Nicola Stattmann. Unfolded: Paper in Design, Art, Architecture and Industry. Basel: Birkhäuser, 2009.

[ 33 ]

One advantage of origami structures is that they can be created by folding or bending flat sheet materials, and therefore, 2D fabrication of profiles and crease patterns can lead to highly accurate 3D products. Because of this, the fabrication methods used for origami are extremely simple. However, the methods for designing complex crease patterns can be extremely complicated. Another advantage of origami folding is that they can be very good at approximating doubly curved surfaces. Origami is a developable surface by definition, creating double curvature through using a number of developable planes in order to rationalize the surface. fig. 30 [ diagram ] variations of startuck origami tesselations (Tachi 2013) [ left to right ] bell, hyperbolic paraboloid, dome

04.02_Bistable Origami While many different patterns could have been used to study the potential relationship between origami structures and auxetic materials, it was decided that the project was not going to be a study into the geometry of origami folding patterns, but rather how they could be applied. Because of this, a specific pattern was chosen at the outset: the square-twist pattern. This pattern is considered to be bistable, meaning that it has two stable states (expanded and compressed). Traditionally, this pattern is seen e

fig. 31

as state moves away from stable, PE increases at either end

[ diagram ] bistability and square-twist origami [ left ] graph showing the concept of bistability

state 2: unstable state // max PE state 1: stable state // 0 PE

state 3: stable state // 0 PE

[ right ] square-twist pattern in its expanded state (stable state 1, center) and its compressed state (stable state 2, right) x

to have zero degrees of freedom, meaning that it is not foldable, but through using bending formations that are not inherent within the crease pattern, it can be folded. The square-twist pattern is made up of alternating square and rhombus faces, with all

[ 34 ]

as state moves away from stable, PE increases at either end

of the internal edges having a very specific fold direction (mountain or valley). Through complex trigonometry, the necessary internal angles of the rhombus faces can be state 2: unstable state // max PE

determined which will yield a flexible origami unit. If the internal angles do not meet state 1: stable state // 0 PE

state 3: stable state // 0 PE

this criteria, then the origami has zero degrees of freedom and cannot bend out of plane.19 For the purposes of this research, physical modeling replaced trigonometric x

calculations in order to find a suitable angle for the rhombus facets. fig. 32 [ diagram ] square-twist derivation and variations [ top ] translation of a square grid (far left) to a square-twist grid (far right) through deleting alternating squares, rotating, then adding rhombus faces [ bottom ] rotations of the outer squares in the squaretwist pattern (those degree measures which yield foldable systems are highlighted in grey)

04.03_Preliminary Unit Through a series of paper model studies, a preliminary unit was devised. Three different tests were done: alternating square and rhombus faces, alternating rectangle and diamond faces, and alternating square and diamond faces of varying sizes. The goal of the modeling exercise was to determine which crease patterns and angles yielded origami which could be folded (as described above, allowing a degree of freedom to be folded) and which could not. The models that could not be folded at all were dismissed immediately. Those models which could be folded were assessed based on their ability to fold as well as their degree of change from expanded state to compressed state. In those tests where the rhombus or diamond faces were extremely flat (extremely obtuse internal angle in one axis and extremely acute angle in the other axis), folding was possible, but the overall change in dimension from fully expanded to fully compressed was quite low. Because of this, a unit with a slightly larger rhombus face was chosen.

Silverberg, Jesse L., Jun-Hee Na, Arthur A. Evans, Bin Liu, Thomas C. Hull, Christian D. Santangelo,

Robert J. Lang, Ryan C. Hayward, and Itai Cohen. â&#x20AC;&#x153;Origami Structures with a Critical

Transition to Bistability Arising from Hidden Degrees of Freedom.â&#x20AC;? Nature Materials 14, no. 4

(2015): 389-93. doi:10.1038/nmat4232.

[ 35 ]

fig. 33 [ photograph ] series showing the preliminary unit studies [ left to right, top to bottom ] square with rhombus, rectangle with diamond, variables square with diamond (increasing angles), aggregation of multiple

[ 36 ]

Certain patterns (around a 45 degree rotation of the square faces) yielded a unit which was able to fold, but could lock in on itself when it was fully compressed. This trait was seen as undesirable as it would create an origami aggregation which was resistant to expansion once it had reached its compressed state for the first time. The ideal unit needed to remain just as flexible compressing or expanding (the final unit used a 60 degree twist of the external square geometry). fig. 34 [ diagram ] time lapse of the unit motion in plan (fully expanded, left, and fully compressed, right)

fig. 35 [ photograph ] time lapse of the unit motion in plan (fully expanded, left, and fully compressed, right)

04.03_Unit Movement Throughout the process of researching the individual unit, it was necessary to gain an understanding of the way that the unit was moving. Because of the out of plane deformation of the unit, there was a twisting which occurred as the square faces traversed between their two stable states. In order to address the aggregation of units, the individual motion was studied and then an attempt was made at aggregating multiple origami units through the sharing of a single plane. By tracking all of the vertices, it was found that there was a three dimensional motion to some of the vertices, while others followed a straight path in plan as the unit deformed. This finding was crucial to a larger aggregation as points which traveled in a straight path would be much easier to connect than those with a complex three dimensional path of travel. This issue was investigated further in section 04.11_Creating an Origami Fabric.

04.04_Deformation Although the square-twist unit allowed for out of plane deformation through the

[ 37 ]

e7a

e8a

fig. 36

e6a

e6b

e5b

i4a

e7b

e4b

[ drawing ] motion of an individual square-twist unit

e5a

[ upper left ] plan of single unit

i2b

i3b

i3a

i1b

i4b

e3b

[ upper right ] plan progression highlighting the motion of the exterior square faces

e4a

i2a i1a

e8b

e1a e1b

[ lower left ] axonometric of single unit

e2b

e2a

[ lower right ] axonometric progression highlighting the motion of the exterior square faces

e3a

e5a e6a

e4b e5b

e4a

i3a a4

e7a

e3b

e6b

i2b i4a

a3 i1b

i3b a1

i2a

i4b

e7b

e2b a2

e8a

i1a e8b

e3a e1b e2a

e1a

fig. 37 [ drawing ] axonometric of the motion of all vertices within an aggregation of four square-twist units

[ 38 ]

fig. 38 [ drawing ] motion of an aggregation of square-twist units [ top ] plan diagram showing the motion of all vertices within an aggregation of four squaretwist units [ bottom ] plan progression highlighting the motion of all exterior faces in a four unit aggregation

[ 39 ]

addition of a tensile force to a compressed unit, or a compressive force to an expanded unit, creating a composite system that was activated through these same forces was desired. In order to begin investigating the potential of further out of plane deformation within a larger system, tension cables were introduced. These tension cables would be attached to opposite ends of the square-twist unit, and when pulled, would expand the unit to its flat state through out of plane deformation. Taking cues from the reentrant hexagon force models and general auxetic principles, the system introduced two more tension cables which would run through the sides of the unit. When put into tension, these cables would force the unit to compress in on itself, forcing further out of plane deformation (fig. 39). The new system did produce more results as far as deformation in the unit was concerned, but it brought several more issues to the surface. The first issue had to do with the way that deformation was achieved within the unit. The initial tensioning of the unit allowed the origami to function as planar faces which rotated around the crease pattern due to the introduction of a force. However, since the unit was now pre-tensioned, adding another tensile force to its sides forced the unit to deform in unexpected ways. The unit began to show visible bends in faces which had previously been planar, conflicting with the principles of origami. Another issue had to do with the introduction of a frame to the system. The frame was used in order to resist the tensile forces which were being created by the cables, and it served as an intermediary between the force which was being exerted on the system (by hand) and the internal stresses of the unit which allowed and disallowed deformation. The frame was seen as problematic since it added an outside component to an entirely origami system, making it difficult at times to tell whether the frame or the origami was the reason that models were successes or failures, but it remained during early stages as a datum to from which to analyze self-similar units. A third issue had to do with the precision of the system. There was no way in which the tension in the cables was being calculated or measured. This was seen as a problem because as models were being compared to one another, it was impossible to tell whether they were storing or utilizing the same amount of force.

[ 40 ]

fig. 39 [ photograph ] original deformation studies showing original square-twist unit geometry and corresponding tensioned model

[ 41 ]

The idea about deforming the unit was one that would carry through the research, but the issues which came about at this early stage gave rise to new solutions about how the origami would deform, respond to internal and external forces, break away from the bounding box, and how the system could add some level of precision to the amount of force being introduced into the system.

04.05_Scale The original scale of the square-twist unit had been chosen arbitrarily, so the next series of models briefly examined scale. The scale of the unit is important due to the ratio between the strength of the material and the size of the unbraced faces. A proper ratio would allow for out of plane deformation without bending the planar faces. Three different scales were tested, with the original scale serving as the middle size and the control for the exercise. The smallest unit was able to deform, but it was found to be quite rigid due to the small size of the planar faces. These faces seemed to brace one another and require more force in order to achieve any deformation. Two different tests were done with the largest unit in order to analyze its ability to deform. One test pushed the outer tension cables close to the extents of the unit, testing its ability to subtly deform. The other test brought the outer tension cables towards the center of the unit in order to test its ability to respond to high level of compressive force. While the first test was successful, the second test resulted in a highly deformed, highly bent unit. The previously planar faces were too large to brace themselves at all, and therefore, a rather small amount of force resulted in a bent panel. The scale test served to affirm that with the modeling material of Stonehenge paper, the original unit scale had the highest ability to noticeably deform in response to a force without bending across the planes.

04.06_Material Studies Lasercut and folded Stonehenge paper was used for the initial studies, mostly due to the working knowledge obtained in prior experiments. However, it was necessary to test other materials before accepting paper as the choice modeling material for more advanced studies. Following material studies described below, paper was used carry out the rest of the research. While it was not proposed that the paper would scale up

[ 42 ]

fig. 40 [ photograph ] scale studies [ 1a ] small scale study [ 2a ] original scale study [ 3a ] large scale subtle deformation [ 3b ] large scale drastic deformation 1a

[ 43 ]

fig. 41 [ photograph ] material studies [ 1a - 1b ] .005â&#x20AC;? Duralar studies with variable hole size to relieve internal forces and allow for adherence of secondary material [ 2a - 2b ] ABS 3D print studies with flipped relationship of mountains and valleys

[ 3a - 3d ] .005â&#x20AC;? Duralar and Hydrocal pretensioned then cast

[ 44 ]

past the models, it did imply a certain thinness and flexibility in application that began to inform potential materiality. The thinness of the material was essential as it needed to be able to reach a fully expanded as well as a fully compressed state. At a minimum, an extremely thin instance would necessarily have to occur at the fold lines where two faces of the material were coincident in the collapsed state. It was also recognized, through the use of paper as a medium, that a certain amount of flexibility had to be allowed by the material. As the origami unit traversed between stable states, a minimal amount of bending happened in the planar faces, allowing the unit to slightly twist. The deflection was not significant nor did it result in a plastic deformation of the unit, but it was necessary in order for the unit to function. Three different types of units were tested in order to find the best option for modeling the square-twist kinetic geometry. The first test used .005â&#x20AC;? Duralar (a polyester film which serves as an Acetate alternative for drawing). The Duralar was cut using the lasercutter and then folded along dashed lines (fig. 41). While the material itself was more durable than the original paper tests, it had too much of a memory when it came to the folding process. The plastic nature of the material meant that once it was folded, it was very difficult to return it to a flat, fully expanded state. Next, ABS plastic 3D printing was used in order to create a single unit. By creating intentionally weak spots in the 3D print along crease lines, it was thought that the unit would be able to deform. These weak spots were achieved by printing a single layer of ABS (similar to the Andres Bastian example in 02.03_Design Scale Application) which could be bent to a certain extent before failing. By alternating the sides of the unit that these weak spots were on, mountains and valleys could be predetermined through the output of the 3D printer. The major issue with the printing of the units had to do with the inherent grain of the 3D print. Since the print was achieved by laying down lines of material, the unit was able to bend either parallel or perpendicular to the direction of the grain of the print, but any folds which did not fall directly parallel or perpendicular failed, and the lines of material began to separate. The third test used the same .005â&#x20AC;? Duralar from test 1, but it added another layer of Hydrocal to make the unit a composite. The Hydrocal served as a means by which to store the energy of the unit in a static system. The unit was first put into tension

[ 45 ]

and then deformed through the addition of tension members which forced the unit to compress (as described in the previous sections). It was then sprayed with a light mist of water and dusted with a coat of powdered Hydrocal. This process was repeated about 20 times on either side, slowly building up a coating of Hydrocal on top of the deformed Duralar units. Once dry, the tension cables were cut, and the unit was able to hold its shape do the casting process. The objectives of this material study were quite different from tests 1 and 2 in that the goal was to create a composite unit which could be pre-tensioned, cast in place, then rid of tension cables. This idea resulted from a preliminary conversation about how to remove the bounding box that had been necessary for all previous models. The proposed idea was that the bounding box could be seen as temporary, resisting the initial tensile forces of the cables, and then removed once the unit could sustain its shape. Although the unit did hold its shape, different ideas would surface regarding ways to remove the frame from the system, and therefore, this study was not pursued.

04.07_Bracing the Plane The next problem to address was that of the bending planes during the deformation process, specifically the significant visible bending that occurred in planar faces once the dynamic system had reached a stable state. In order to do this, five different patterns were devised which attempted to brace the planar quads and resist any bending force which was being placed onto them. The first iterations attempted to fully brace the planes with rigid material (1/16â&#x20AC;? and 1/8â&#x20AC;? basswood). These studies were able to maintain the planarity of the quads, but resisted the folding of the units completely. Further studies removed material from the fully braced quad in order to allow for the unit to fold while still being braced. These studies allowed for slightly more flexibility in the unit, but they were still not able to fold completely due to the addition of a thick layer of material. From these studies, it was determined that adding a thick layer to the surface of the units was not a viable solution for bracing them. The thinness of the paper allowed the units to fold and unfold, and anything that impeded this process took away from the effectiveness of the module. Because of this, it was determined that the system

[ 46 ]

fig. 42 [ diagram ] geometry of the rigid braces applied to the planar quads of the squaretwist unit

would remain completely unbraced. Resisting the internal forces of the system through bracing planes was unsatisfactory because it reduced the amount that the internal forces were determining both local and global geometry. The amount of bending in the planar quads was monitored, and any system that exhibited inherent bending would require further distribution of internal forces in order for the bending to be removed.

04.08_Aggregation and Double Curvature To this point, models had contained a maximum aggregation of three square-twist units as the studies had been primarily concerned with the movement and fabrication of the individual unit. The next step was to devise a system which could accommodate a larger number of units and behave as an aggregate whole. Two different systems were created; one introducing units between tension cables as fabricated in prior experiments, and the other implementing an overlap between units made possible by sharing a single tension cable between them. Both larger aggregations performed similarly, allowing for too much bending in the planar quads due to high levels of stress within the system (fig. 43). While out of plane deformation was occurring in the first aggregation models, they remained, similar to the reentrant hexagon models, quite flat. Due to the flat nature of

[ 47 ]

fig. 43 [ photograph ] preliminary aggregation models after tensioning [ top ] individual units are connected between tension cables [ bottom ] overlap is created between units sharing the same tension cable

the frame within which the models were built, the out of plane deformation could only achieve a limited amount of three-dimensionality. These models had already taken the approach of using two parallel axes in order to create the frame for the tension cables, but by taking one of these axes and rotating it vertically 90o, the models could gain substantial three-dimensionality while also attempting to approximate a hyperbolic surface, which is naturally doubly curved. By this simple rotation of one of the axes, the origami could begin to address synclastic curvature once again through a very clear geometric form.

[ 48 ]

Although the hyperbolic paraboloid was used as a base geometry to move away from flat models, it also represented a critical point at which it was decided that the auxetic origami system would be tested for its ability to approximate multiple forms of double curvature. As complex building forms continue to emerge from advanced computational modeling techniques, they often require specialized expertise to be realized. Often, these construction systems are forced to rationalize and simplify the geometry as created by the architect in order to be built. However, if an auxetic origami system could approximate multiple forms of double curvature through the expansion and contraction of units, it could drastically simplify the process of constructing synclastic curvature, resulting in very little simplification of the architectâ&#x20AC;&#x2122;s original form. In addition, the auxetic origami fabric could be derived digitally based on the original form and constructed off site in a highly controlled and precise manner. Through the flexibility of the individual modules, auxetic origami systems could prove to move away from highly specialized computational optimization/ rationalization techniques in favor of an extremely simple system in which the optimization is built into the unit, and therefore the fabric, as opposed to being built into algorithms and software. The first attempt at approximating the hyperbolic form utilized an aggregation of units interspersed between tension cables. The two different methods for creating the hyperbolic form were simple. The first utilized two static axes, one being vertical and one being horizontal. The net of units was tensioned in place with variable lengths of tension cable. The second method was to assemble the net with two horizontal axes, just like previous methods. Each tension cable was exactly the same length and remained somewhat loose between the axes. After the net was in place, one axis was rotated to become vertical, forcing the system into tension. Both models failed, but in different ways: 1) because the amount of tension required to approximate the hyperbolic paraboloid was too much for the paper units to handle as one unit ripped during the tensioning process; 2) because the amount of tension in the system forced the braided cable to snap. This happened at the extremities (along the top and bottom), where the tension was the most. The square-twist unit was partially functioning, but the approximation of the system actually had very little to do with the origami unit that was placed within the tension

[ 49 ]

fig. 44 [ photograph ] hyperbolic paraboloid tension models [ top ] static axes model (note: ripped unit bottom left) [ bottom ] movable axes model (note: torn cable bottom left)

[ 50 ]

cable net. The same results could have happened regardless of the piece that was placed within the net, whether that piece was folded, flexible, or bent. The system needed to really exploit the origami piece and show whether or not the unit itself could approximate double curvature through auxetic expansion and contraction. The tension cables were removed from the system in favor of a system made solely of square-twist origami units. The units were connected in three different configurations: point, line, and plane. By using small eyelets, the paper units were connected into a larger aggregation free of tension cables. They were then draped between two opposing axes in order to approximate a hyperbolic surface. fig. 45 [ photograph ] assembly of the plane-to-plane connection aggregation using Stonehenge paper and small eyelets

The point-to-point connection model retained the most flexibility between units, and therefore approximated the double curvature quite accurately. However, this flexibility was created through a slight amount of bending that happened at the edge of the unit where the point connection was made. There was not enough material to resist the bending force in the system. Therefore, the approximation of the curvature was deceptive as it came largely from a weak connection instead of a functional geometry. The line-to-line connection model was subject to the same issues as the point

[ 51 ]

fig. 46 [ photograph ] point, line, and plane studies approximating a hyperbolic surface [ top ] point-to-point connection [ middle ] line-to-line connection [ bottom ] plane-to-plane connection

[ 52 ]

connection model. Since the units were only connected along a single edge condition, the edges of each unit were subject to bending. Again, while the model approximated the hyperbolic curvature quite nicely, it was largely due to the weakness of the system. The plane-to-plane connection model was more successful since the planes were not able to bend as easily. Connections happened face-to-face, doubling the amount of material at all connection areas thereby stiffening the system. While the stiffness of the system was useful as far as connections were concerned, the connections severely restricted the motion of the individual units and made the aggregation act more as a continuous fabric as opposed to individual units connected to one another. Therefore, the system was not able to approximate the hyperbolic curvature easily, bending in areas of weakness as opposed to folding at fold lines. The plane-to-plane connections were seen as advantageous due to their increased strength in the overall fabric, but the pattern that the units were aggregated in needed to be changed. By changing the pattern of the units, forces could be more effectively distributed to fold lines, allowing each individual origami unit to function on its own as well as in conjunction with its neighbors, controlling global form through subtle local deformation.

04.09_Removing the Frame The frame had proven helpful to resist tensile forces as well as a guide for the shape of the models. However, one of the discoveries derived from the reentrant hexagon studies had been the importance of internal forces to counteract external forces, a behavior necessary to form a true auxetic system. As previously mentioned, this frame served as an intermediary between internal forces and external forces, adding an external constraint to the square-twist origami system. Removing the frame completely was crucial to advance the study of how internal forces within the system to could be tuned to repel and balance the external forces of gravity. In other words, the goal was to achieve an internally activated out of plane deformation which would hold its own weight. This out of plane deformation was to happen locally, creating gradual shifts in the global form. The incremental local shifts would result in a predetermined, doublycurved form.

[ 53 ]

04.10_Optimization In simply achieving double curvature through the use of expanding auxetic origami units, the system would be able to enter into conversation regarding synclastic curvature and its ability to be constructed economically, but this did not set the system apart from the majority of solutions which had already been devised to date for this same problem. Optimization in architecture typically is concerned with two factors: buildability and cost. These two factors are inevitably linked as simplified construction can lead to lower costs and vice versa. Therefore, deriving simple construction methods from complex forms and components as proposed by the architect is essential in order for a project to be considered buildable. Cristiano Ceccato of Zaha Hadid Architects described one method of optimization that is often used within his workflow: “‘Pre-rationalisation’ is also referred to as designing with ‘first principles’, using a set of given geometric rules and methods to produce a solution that is constructible from the outset…a pre-rationalised process embeds the necessary assembly constraints and design logic into its constituent geometric rules, such that only constructible designs can be produced by the system.”20 In this excerpt, Ceccato described a method of inserting optimization into the early stages of schematic design development to insure that all results of early design stages are able to be constructed. This process was exemplified by Zaha Hadid Architects during the Wangjing SOHO building in China where the firm experienced unfamiliar subcontractors and extremely low-tech fabrication tools. Because of this, algorithms were used to break down the doubly-curved panels of the building into a catalog of singly-curved and repetitive panels for fabrication. This process is an extremely common optimization technique within architecture; larger, more complex forms are broken down into smaller, simpler geometries which can be aggregated into the larger system.

Ceccato, Cristiano. “Material Articulation: Computing and Constructing Continuous Differentiation.” Architectural Design 82.2 (2012): 96-103. Print.

[ 54 ]

While Ceccato and Zaha Hadid Architects typically use a system of pre-rationalization, the opposite technique, that of post-rationalization is also a viable option. Collaborators such as ARUP’s Advanced Geometries Unit and Evolute GmbH exist who serve an essential role in rationalizing existing forms created by architects.21 An example of post-rationalization of form can be seen in the example of the Phare Tower by Morphosis architects. The project, a doubly-curved tower set to be the tallest tower in Paris, posed fabrication questions from the outset. The Morphosis design team wanted to maintain the relationships that existed between the diagrid frame, window wall, and exterior skin that existed during the design development stage, but in order to do this exactly, there would have been thousands of unique façade elements, making the budget unreasonable. Custom developed Java software allowed the design team to interface with typical 3D modeling environments. However, this interfacing between programs was able to occur seamlessly, back and forth, which allowed the office to quickly work through iterations that could be examined by the design team to approve the relationships between the three design elements.22 fig. 47 [ diagram ] family breakdown of fabrication panels for the Phare Tower (Bergin 2015)

Ceccato, Cristiano. “The Master-Builder-Geometer.” (2011): 9-14. Web.

Marble, Scott. Digital Workflows in Architecture: Designing Design -- Designing Assembly -- Designing

Industry. Basel: Birkhäuser, 2012. Print.

[ 55 ]

In the case of the auxetic origami system, inherent within the square-twist module, (and any auxetic geometry), is a dynamic behavior that is observed as a global response to an applied force. This kinetic nature allows the geometry to expand and contract with a high amount of local control. Therefore, the auxetic geometry can bridge multiple scales without changing proportion. This makes auxetic geometry a relevant topic of discussion in relation to current optimization techniques. If the same auxetic module could be aggregated into a larger fabric and then expanded in a highly specific manner due to the local conditions of a complex surface, it may be possible that a single, kinetic auxetic module could replace multiple families, if not all families, within a fabrication family such as that of the Wangjing SOHO.

04.11_Creating an Origami Fabric Taking the composition of the origami fabric more seriously, identical square-twist units were cut and assembled in different ways, ranging from point connections to line connections to plane connections (fig. 48). There were multiple criteria upon which the study fabrics were evaluated. The first of these criteria, and the most important, was the ability of the fabric to expand and contract. It needed to be able to be completely closed and completely opened in order for it to be successful. Another criterium against which the fabrics were judged was the strength and type of connections they used. As derived from the hyperbolic studies, a plane-to-plane connection was the most desirable since it reduced the bendability of the planar faces. The last criteria upon which the models were based was their overall rigidity and retention of planar quads through expansion and contraction. This was important because the fabric had to be rigid enough to resist bending forces within the planes while being flexible enough to allow for local intervention as well as expansion and contraction. The final fabric was composed of two different units. The two units had the same 2D profile as cut on the lasercutter with the only difference between them being a reversal of the mountain and valley folds. The units were connected with a plane-to-plane connection in a checkerboard pattern. This checkerboard pattern in conjunction with the alternating crease patterns allowed for the fabric to be easily folded and expanded.

[ 56 ]

fig. 48 [ photograph ] numerous study models used to find a suitable origami fabric

04.12_Final Study Model Once the final auxetic square-twist fabric was created, it was necessary to activate the fabric internally in order to test its ability to create doubly curved surfaces. The first study model used a fabric that was composed of 25 units. The fabric began in its compressed state, and then it was fully expanded by hand. Once it was expanded, thin gauge steel rod was added within each unit to bind the distance between two opposite edges. The goal of this exercise was to locally control the size of individual units in order to create a subtle dome shape. However, it was found that, although the steel members did control the size of each unit, the surface remained flat. It gained rigidity since the units were no longer able to expand and contract, but it lacked a critical piece: the local connection from unit to unit. Without this connection being controlled, the forces created by controlling the size of each unit were disseminated into the void space between units. Therefore, instead of deforming out of plane, the shapes of these voids between units were forced to change, resulting in the overall surface staying flat.

[ 57 ]

fig. 49 [ photograph ] series showing the final fabric and its ability to expand from its compressed state

[ 58 ]

fig. 50 [ photograph ] of final study model showing steel wire connections within individual units

fig. 51 [ photograph ] zoom in of connections between units connecting opposite faces

[ 59 ]

04.13_Final Models With the new knowledge gained from the final study model, members were added which would control the size of voids between units. The final models attempted to approximate three geometries: dome, vault, and hyperbolic paraboloid. By approximating these three geometries with identical auxetic fabrics, the viability of the system could be tested. All of the final models began with a 7 unit by 7 unit auxetic square-twist fabric. The fabric was built in its expanded form using plane-to-plane connections. It was then compressed to prove its kinetic nature. The fabric was expanded to its maximum dimension, and then tension members were placed in the voids to compress the fabric. Once the fabric was compressed, compression members were added to individual units in order to force them to expand. Similar to the reentrant hexagon study models, this expansion was forced locally in areas where the unit was not able to expand in the XY plane due to the force being placed on them by the tension members. Therefore, the units gradually expanded out of plane in the Z axis, resulting in double curvature. fig. 52 [ photograph ] assembly of the final models

[ 60 ]

fig. 53 [ diagram ] crease patterns (red = mountain, blue = valley), internal compression members (green), and tension members between units (orange) [ top left ] dome layout [ top right ] vault layout [ bottom ] hyperbolic paraboloid layout (double sided)

The final models were able to address very specific criteria: 1. Local Control v. Global Control The local interventions of compression and tension members to the system both within individual units and spanning between units was able to gradually change the global form of the fabric. By starting with identical fabrics and adding constraints to them in different ways, local control was able to directly affect global form. 2. Internal v. External Forces Forces internal to the system (compression and tension members) were able to control the overall form of the system. By not adding tension and compression members everywhere, the system was able to respond in different ways to the distribution of internal forces. There are “rigid units” (those bounded by tension and compression members), “semi-rigid units” (those only bounded by one or two members), and “flexible units” (those which were not bound by any members). The flexible members were seen as crucial to the overall functioning of the system as they allowed for expansion and contraction to occur through the kinetic nature of the component. These units were

[ 61 ]

fig. 54 [ photograph ] final model (dome) [ top ] plan showing connections within and between units [ center ] perspective showing double curvature [ bottom ] side view showing double curvature

[ 62 ]

fig. 55 [ photograph ] of final model (vault) [ top ] plan showing connections within and between units [ center ] perspective showing double curvature [ bottom ] side view showing double curvature

[ 63 ]

fig. 56 [ photograph ] of final model (hyperbolic paraboloid) [ top ] plan showing connections within and between units [ center ] perspective showing double curvature [ bottom ] side view showing double curvature

[ 64 ]

able to freely bend in order to accommodate the out of plane deformation of the rigid and semi-rigid units. External forces of gravity were resisted by the defined internal forces of the system, allowing the forms to hold their form. A level of precision was gained by controlling the size of the tension and compression members. 3. Synclastic Curvature The three different forms of curvature proved the viability of the system to adapt to synclastic curvature of varying types. Through changing the location and forces within internal members (length of wires), the system changed its curvature, including having multiple points of inflection. The hyperbolic paraboloid proved that the system could change its curvature by changing the side that the tension and compression members were on. 4. Kinetic and Dynamic Nature of Auxetic Geometry The kinetic nature of the fabric was an important part of qualifying the system as an auxetic system. Through applying tension to the system, it expanded in the XY plane. It was also defined as an auxetic system through the local interventions that occurred. Through putting a unit into compression, the system was able to shrink locally, and likewise, putting a unit into tension forced a local growth. As previously mentioned, the kinetic nature of individual components was crucial to the way that internal forces were distributed and redistributed throughout the system. 5. Removing the Frame Through adding rigid members to the system and containing the forces, the system was able to exist without the use of a frame. Origami systems have been examined for their ability to appeal to complex curvature, but typically, these applications require a frame which controls the edge conditions of the origami. By removing the frame and controlling the forces internally, the auxetic fabric system pushes away the notion that a flexible origami systemâ&#x20AC;&#x2122;s shape must be dictated by an external frame while also introducing the concept of a self-contained and self-supporting origami system. 6. Optimization By using an identical fabric, the three models showed that there is a certain optimization inherent within auxetic kinetic systems. Since the units within the fabric could expand

[ 65 ]

and contract, curvature could be achieved locally and globally. The extent that these fabrics are able to approximate complex surfaces is dependent upon the scale of the individual units.

[ 66 ]

[ 05_Simulation ]

05.00_Introduction to Simulation One of the major problems associated with origami geometries is how to accurately predict the way that they will function. Here, computational methods prove very useful. As opposed to an experimental approach that cannot predict the outcome of an auxetic fabric which is activated internally, digital simulation can begin to inform the origami systems and be integrated into the design process from the start. Digital simulation maintains itself as a scaleless entity. On-screen geometry can be viewed at a micro or macro scale quite easily, with no differentiation in modeling techniques. Digital simulation, at its very core, is dictated by mathematical principles, principles which have no clear objective.23 The value of digital simulation comes in the ability to study not only geometry, as dictated by math, but also to study complex behaviors in relation to time. These complex behaviors can be analyzed in order to feed back into the design process. The current work of figures such as Sean Ahlquist, Moritz Fleischmann, and Achim Menges, just to name a few, attempts to tackle the inherent issues found in the relationships between physical and digital simulation. For these designers, the solutions lie in the ability to “design methods that can actively couple the digital simulation with the analog methods for building and physical structure.”24 They attempt to explain a process whereby physical and digital simulation are brought together through the physical traits of material becoming embedded within digital systems. While they view this problem under an architectural guise, the issue is very much one that has been the center of engineering practice for many years, that is, how to simulate the behavior and scale of specific materials within complex structural systems. This begins to set up digital and physical simulation, in their case, as a somewhat linear process. There are interactions between physical simulations of material studies which begin to feed back into the digital simulations, eventually becoming the core of the

Burry, Jane and Mark Burry. The New Mathematics of Architecture (New York: Thames & Hudson, 2010),

1-13. 24

Ahlquist, Sean and Moritz Fleischmann. “Elemental Methods for Integrated Architectures:

Experimentation with Design Processes for Cable Net Structures.” International Journal of

Architectural Computing, issue 4, volume 6: 453-475.

[ 67 ]

behavioral constraint placed on structural members. The material studies lead directly to on-screen form generation utilizing developed material constraints, and the final test comes in the form of a built simulation which is analyzed to test its fitness to the computationally-derived form. Here, the material and fabrication constraints become engrained within the very fabric of the algorithms dictating form generation in as close to a 1:1 relationship as can be drawn. It is worth mentioning that the purpose of the digital simulations in this case is not to analyze and output precise data, but rather to understand the behavior and patterns inherent within material systems.25 Simulation was used at the end of the research to understand the complex behaviors and patterns that emerge from origami systems. It was also seen as a way to experiment quickly with the origami as opposed to time intensive physical simulations. The Kangaroo physics engine for Grasshopper in Rhino3D was chosen to facilitate the simulations. Kangaroo is a collection of algorithms that allow the computer to simulate particle dynamics, a branch of Newtonian mechanics concerned with forces and their effects on motion.

Newtonian mechanics take into account extremely

simple relationships, but they allow an accurate modeling of a vast range of physical phenomena. A particle is used which has a position (always occurring in 3D with an X,Y, and Z coordinate), a velocity (vector), and a mass (scalar). Particle dynamics uses Newtonâ&#x20AC;&#x2122;s Second Law in order to describe the movement of a particle through space (Force = Mass x Acceleration). Force, and more specifically net force, is the overall vector that is acting on an individual particle. In Kangaroo, numerous forces can be combined into one in order to simulate more complex physical relationships. Mass represents how difficult a particle is to move. Since particles do not have volume, the particle represents the center of mass of a given body. Lastly, acceleration, a vector, is the rate of change of velocity. If acceleration can be calculated, then velocity, the rate of change of position, can be calculated as well which is how particle dynamics are able to calculate the position of a given particle in space.

Menges, Achim. â&#x20AC;&#x153;Integral Formation and Materialisation,â&#x20AC;? in Computational Design Thinking, ed. Achim Menges and Sean Ahlquist (United Kingdom, John Wiley & Sons Ltd, 2011), 198-210.

[ 68 ]

With these three very simple components, Kangaroo iteratively calculates the position of a particle in space, then recalculates the forces on the particle, recalculates the acceleration of the particle, and calculates the new position of a particle. This process continues until the particle is at equilibrium.26

05.01_Initial Simulations The first simulations attempted to gain control over basic auxetic geometry. The Origami component within Kangaroo was examined as a possibility to control the folding of a rectangular surface with one crease line in it (fig. 57). The problem with the component was that it required very specific angle measures which would be applied to the specified mountain and valley curves. In the square-twist auxetic system designed as part of this research, this would be nearly impossible to predict for each individual crease, especially if the geometry changed locally across the global surface. In order to avoid this, the crease lines needed to be given a direction, but not a specific angle measure. Two parallel edges were taken from the rectangular surface, and another parallel line was used as the crease. The two edge curves were subject to a CurvePull force which forced the edges of each rectangular face to move. By making all of the springs in the system extremely stiff and adding a Hinge force along the crease line with a very small amount of strength, the crease line was given a direction it was supposed to bend, but the low amount of strength in the force meant that the pulling of the edges would dictate the final angle measure between the two surfaces. The test yielded a new way to control mountain and valley crease patterns without having a final angle measure as a parameter. fig. 57 [ screenshot ] creating a weak hinge force in order to control mountains and valleys [ left ] original surface with projected edges (red) [ right ] original surface (light grey) and new bent surface (cyan)

Piker, Daniel. â&#x20AC;&#x153;Kangaroo: Physics and Form-Finding with Grasshopper.â&#x20AC;? Lecture, Kangaroo Webinar, July 22, 2015.

[ 69 ]

While the CurvePull method had worked for the simple rectangular surface, it would not work to simulate the square-twist origami. The three dimensional movement of the single unit of square-twist origami was too complex, not to mention the movement of an entire auxetic origami fabric. Because of this, a new method was devised. The technique of creating a weak Hinge force to control mountains and valleys was kept, but the method for controlling the bending was changed. Instead of using CurvePull, new springs were created which had a rest length equivalent to the length between vertices at rest in the square-twist origami. When the lengths of these springs were changed, the origami was forced to fold in order to accommodate the shrinking springs connecting opposing vertices.

05.02_Single Unit Origami Simulation With the method of simulation figured out, the simulation of a single square-twist origami unit was to be tested next before the entire fabric was simulated. The variable length springs were connected across the origami geometry in its completely flat, expanded state. They were set to the actual length between these points and then activated. When the spring lengths were then adjusted with a slider, the behavior of the unit became quite unpredictable. This was due to the fact that, since the spring length was consistent with a flat geometry, the Hinge force was not being activated properly at the outset of the simulation, and mountains and valleys did not exist since the geometry remained flat. In order to properly activate the Hinge force at the outset of the simulation, the initial spring length between the opposite vertices needed to be slightly shortened. This allowed the mountains and valleys to be activated properly, and when the spring was then shortened, the origami unit was able to fold the same as its physical counterpart. The expansion and contraction of the digital unit did have their limits. As the unit approached either stable state (completely expanded or completely contracted), the Hinge force once again began to fail, confusing mountains and valleys in the crease pattern. Due to time constraints, this characteristic of the digital simulation was seen as acceptable, and careful attention was paid not to come too close to either completely stable state.

[ 70 ]

fig. 58 [ screenshot ] folding of an individual square-twist origami unit (red = mountain fold, blue = valley fold, green = spring connection)

05.03_Composite Origami Simulation The transition between single unit origami simulation and the larger fabric simulations was quite simple. The larger fabrics required the addition of variable length springs between the square-twist units. The fabrics were able to simulate in a similar way to their physical counterparts, with the end results of the three simulations resembling the dome, vault, and hyperbolic physical studies. The results at the time of the completion of the thesis solely showed that physics simulations of auxetic origami fabric was feasible as a means by which to approximate physical findings. The method described has its limitations as it does not have the ability to create a highly specific global form, but rather, only has the ability to control local conditions in an effort to approximate a global idea. Further studies should look into the ability for physics-based simulations to find the local conditions within an auxetic origami fabric that could yield to a given global geometry. This way of working would be extremely useful, especially with the proposed application of the system. If the system is to serve as a means by which to rationalize complex surfacing with an optimized auxetic fabric, the fabric must be able to approximate a given geometry.

[ 71 ]

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00 1.00

1.00

edge lengths, mountains (red), valleys (blue), original springs (dotted with black dots), and actuated springs (solid with white dots) are shown

1.00

[ diagram ] initial origami fabric used for physical simulation of dome iteration

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00 1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

fig. 59

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00 1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

fig. 60 [ screenshot ] final results of digital simulations of squaretwist origami dome iteration

[ 72 ]

fig. 62 [ screenshot ] final results of digital simulations of squaretwist origami vault iteration

[ 73 ]

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00 1.00

1.00

1.00 1.00

1.00

edge lengths, mountains (red), valleys (blue), original springs (dotted with black dots), and actuated springs (solid with white dots) are shown

1.00

[ diagram ] initial origami fabric used for physical simulation of vault iteration

1.00

fig. 61

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00 1.00

1.00

edge lengths, mountains (red), valleys (blue), original springs (dotted with black dots), and actuated springs (solid with white dots) are shown

1.00

[ diagram ] initial origami fabric used for physical simulation of hyperbolic paraboloid iteration

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00 1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

fig. 63

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00 1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

1.00 1.00

1.00

fig. 64 [ screenshot ] final results of digital simulations of squaretwist origami hyperbolic paraboloid iteration

[ 74 ]

[ 06_Conclusion ]

Auxetic geometries, or those geometries that exhibit negative Poissonâ&#x20AC;&#x2122;s ratio, have the unique characteristic that they expand when stretched in a direction perpendicular to a tensile force, and likewise, contract perpendicular to a compressive force. With this inherent kinetic nature, certain applications for auxetic geometries have been found in fields such as biology, medicine, aerospace engineering, and textiles. While these fields have found uses for auxetic geometry at nano and micro scales, architecture has yet to find an application for which auxetic geometry would both be useful and economical. Taking this question on, Out of Plane situates the kinetic scaling of auxetic origami as a tool for optimization of complex surfacing while using a minimal amount of unique two dimensional profiles. Through an in-depth case study of reentrant hexagon patterning, ideas were derived about out of plane deformation and its role in creating doublycurved surfaces with high levels of local control. In an effort to study the potential of auxetic systems within the discipline of architecture rather than studying pure auxetic geometry, a bistable square-twist unit was adopted as the geometry to be studied, but other geometries may serve the task of rationalization through more effective means. Out of Plane was able to find success in many ways. The development of a kinetic auxetic fabric from a single two-dimensional profile with varying crease patterns demonstrated how the square-twist unit could accommodate three unique forms. By changing the location and strength of internal tension and compression members, there was a high level of local control that dictated whether an area would achieve positive curvature, negative curvature, or no curvature at all. The surfaces created contained different kinds of curvature, points of inflection, as well as apex moments which were all handled through the expansion and contraction of the origami units. The system was able to be simulated digitally as well, proving that it could become a digital tool as well as an empirical physical modeling exercise and implying its ability to be used as a rationalization tool for existing doubly-curved surfaces. While some of the larger ideas were quite successful, the research has yet to propose and test full-scale materiality and scalar context. A critique of the work at the time of the thesis defense, is the lack for a direct proposal for how this system could work to produce architectural space. . In addition, the proposed workflow addition of digital

[ 75 ]

means to the current physical workflow was not effective in pushing the research forward. Rather, it served merely as a means to represent the physical process of the folding origami and to show that there was a certain level of control over the system in the digital realm. This critique of the work helped to bring larger architectural issues back into the conversation about the work, many of which had been put off during the final weeks of production. In future research into utilizing auxetic origami systems to optimize synclastic curvature, there will be three main focuses: Geometry: Instead of taking the bistable square-twist origami unit as a given geometry, new geometries will be explored that examine the potential for profiles and crease patterns to be completely uniform across the surfacing. This would further prove the effectiveness of auxetic origami as a rationalization tool for complex surfacing. Scale and Application: Moving out of â&#x20AC;&#x153;designâ&#x20AC;? scale and into actual scale and materiality is essential to prove the viability of the system. Auxetic geometries are theorized and portrayed as scaleless, being applicable as long as certain relationships are not disturbed, but this principle only exists through pure auxetic geometry, not real-world application. Fabrication techniques, material, and scale for full scale application will need to be devised in conjunction with the development of the system. Simulation: While simulation was useful for representation and was able to create predictable results throughout the course of the thesis, a development of a parametric simulation tool that is able to rationalize existing surfaces is very important to integrate into the system. While empirical methods were used throughout the course of the research to intuitively derive form, more precise methods are needed digitally which can influence the physical construction and implementation of auxetic origami systems. There exists a need for simple and economical means of optimization within the built environment today. With this prevalent issue in mind, Out of Plane proposes the

[ 76 ]

ability of auxetic geometry to take on complex rationalization. Early questioning of auxetic geometry prodded at why auxetic geometry is typically portrayed as regular and uniform. It asked if there was a way to create irregular auxetic patterns that would respond in different ways. The short answer was that auxetic systems could be applied to irregular geometries. But perhaps the most interesting finding from the research was that was that the problem of pure and functional auxetic geometry was initially misunderstood as a detriment to its ability to scale, but in later phases was accepted as the opposite. In the end, a conceit of the thesis states that the uniformity with which auxetic geometries are portrayed holds great potential as far as optimization is concerned; uniform modules within an auxetic aggregation are only uniform if they exist without force. When force is introduced, these once uniform modules are able to deform through rotational expansion and contraction in order to appeal to regular and irregular geometries alike.

[ 77 ]

[ 07_References ]

Ahlquist, Sean and Moritz Fleischmann. “Elemental Methods for Integrated

Architectures: Experimentation with Design Processes for Cable Net

Structures.” International Journal of Architectural Computing, issue 4,

volume 6: 453-475.

Bastian, Andreas. “3D Printed Mesostructured Materials.” ANDREAS BASTIAN. March

17, 2014. Accessed November 12, 2014. http://www.andreasbastian.com/.

Bealer Rodie, Janet. “The Auxetic Effect.” Textile World. May 2010. Accessed January

30, 2015. http://www.textileworld.com/.

Bergin, Michael S. “Trends in Generative Design.” Archinect Blog. March 31, 2012.

Accessed May 05, 2015.

http://archinect.com/archlab/trends-in-generative-design.

Betatype. Accessed November 15, 2014. http://www.betaty.pe/. Bhullar, S. K., et al. (2013). “Influence of Negative Poisson’s Ratio on Stent

Applications.” Advances in Materials 2(3): 42-47.

Bock, S. De, F. Iannaccone, G. De Santis, M. De Beule, P. Mortier, B. Verhegghe, and P.

Segers. “Our Capricious Vessels: The Influence of Stent Design and Vessel

Geometry on the Mechanics of Intracranial Aneurysm Stent Deployment.”

Journal of Biomechanics 45, no. 8 (2012): 1353-359.

Burry, Jane and Mark Burry. The New Mathematics of Architecture (New York: Thames

& Hudson, 2010), 1-13.

Ceccato, Cristiano. “Material Articulation: Computing and Constructing Continuous

Differentiation.” Architectural Design 82.2 (2012): 96-103. Print.

Ceccato, Cristiano. “The Master-Builder-Geometer.” (2011): 9-14. Web.

[ 79 ]

Darja, Tatjana, and Alenka, “Auxetic Textiles,” Acta Chim. Slov., no. 60 (2013):

715-723.

Evans, K. E., and A. Alderson. “Auxetic Materials: Functional Materials and Structures

from Lateral Thinking!” Advanced Materials 12.9 (2000): 617-28. Web.

“Fly-Bag2.” FLY-BAG2. Accessed January 30, 2015. http://www.fly-bag2.eu/. Marble, Scott. Digital Workflows in Architecture: Designing Design -- Designing

Assembly -- Designing Industry. Basel: Birkhäuser, 2012. Print.

“Material Benefits.” The Economist. September 07, 2013. Accessed January 30,

2015. http://www.economist.com/.

Menges, Achim. “Integral Formation and Materialisation,” in Computational Design

Thinking, ed. Achim Menges and Sean Ahlquist (United Kingdom, John Wiley

& Sons Ltd, 2011), 198-210.

Piker, Daniel. “Kangaroo: Physics and Form-Finding with Grasshopper.” Lecture,

Kangaroo Webinar, July 22, 2015.

Schmidt, Petra, and Nicola Stattmann. Unfolded: Paper in Design, Art, Architecture

and Industry. Basel: Birkhäuser, 2009.

Silverberg, Jesse L., Jun-Hee Na, Arthur A. Evans, Bin Liu, Thomas C. Hull, Christian

D. Santangelo, Robert J. Lang, Ryan C. Hayward, and Itai Cohen. “Origami

Structures with a Critical Transition to Bistability Arising from Hidden Degrees

of Freedom.” Nature Materials 14, no. 4 (2015): 389-93.

doi:10.1038/nmat4232.

Tachi, Tomohiro. “Designing Freeform Origami Tessellations by Generalizing Resch’s

Patterns.” Journal of Mechanical Design 135, no. 11 (2013): 111006.

doi:10.1115/1.4025389.

[ 80 ]

Thomsen, M. R. and M. Tamke (2014). “Digital Crafting: Performative Thinking for

Material Design.” 242-253.

Toronjo, Alan. “United States Patent: EP 2702884 A1 - Articles of Apparel Including

Auxetic Materials”, March 5, 2014.

Trex:Lab. Accessed November 15, 2014. http://www.trex-lab.com/. Yang, L. (2011). Structural Design, Optimization and Application of Three-Dimensional

Re-entrant Auxetic Structures. Aerospace engineering / Mechanical

engineering. Ann Arbor, MI, North Carolina State University. ph.D: 191.

Zhang, Hu, Liu, and Xu, “Study of an Auxetic Structure Made of Tubes and Corrugated

Sheets,” Physica Status Solidi B, no. 10 (2013): 1996-2001.

Zheng-Dong, Ma, Yuanyuan Liu. “United States Patent: 20110029063 A1 - Auxetic

Stents”, February 3, 2011.

[ 81 ]