Folding and crumpling of thin plates has been studied extensively by many research communities.
Creases in thin plate structures appear sometimes in a controlled (folding) or uncontrolled (crumpling) fashion.
The detailed behaviour of creasing itself and its effect upon the surrounding plate is an emerging research area underpinned, equally, by geometrical and mechanical processes.
The aim of the workshop was to bring together leading local engineers, scientists, physicists and technologists who research the folding/unfolding and/or creasing of thin plates, in order to
present the state-of-art of current understanding in the mechanics of origami
discuss existing and possible applications in engineering, robotics, packaging, architecture and biology (for example)
help identify, establish, and strengthen, a willing scientific community interested in these problems.
Keith Seffen (Cambridge) Paradoxical Creased Shells
Mark Schenk (Bristol) Residual creases : mechanics of partly-(un)folded structures
Frederic Lechenault / Benjamin Thiria (Paris) Origami mechanics: from a
single crease to generalized vertices
In addition to keynote lectures, we would like to hear presentations of on-going work (research students are particularly welcome).
Please give a title for your presentation, if applicable, when registering: note that the language of the workshop is English.
Following this workshop, the Berlin group “3D-structures and codes” will publish a collection of papers, ”On folding”.
Participants to the Paris workshop (Folding and Creasing of Thin Plate Structures) are invited to contribute to this collection.
Here is a call for article.
The papers should be in English, 15-20 pages
A title and a short abstract of the proposed contribution are to be sent before 4th of April.
Decision with respect to which papers will be included in the volume will be given shortly after.
Duncan J, Duncan J, Sowerby R and Levy B (1981), "Folding without distortion: Curved-line Folding of Sheet Metal", Sheet Metal Industries. Vol. 58(7), pp. 527-533.
Abstract: This paper introduces the concept of folding or bending sheet metal along a curved line to produce two separate curved surfaces. Each of these surfaces is developed and is curved without stretching or distortion in the plane of the sheet. In existing sheet-metal practice, when a component contains two or more different developable surfaces these are produced either by forming each surface individually and joining the separate sheets or by folding the curved sheet along a straight line. In such cases, the two separate developable surfaces are joined by bending along a common generator. In the examples presented in this paper sheet metal is folded along a curved line which is not a generator of either surface. This process greatly extends the range of shapes which can be produced from sheet metal without requiring deformation or straining in the sheet. These shapes are of particular interest for components formed from very-high-strength sheet materials; these materials often have limited stretching ability but they can be bent and folded into developable shapes.
BibTeX:
@article{duncan1981b,
author = {J.L. Duncan and J.P. Duncan and R. Sowerby and B.S. Levy},
title = {Folding without distortion: Curved-line Folding of Sheet Metal},
journal = {Sheet Metal Industries},
year = {1981},
volume = {58},
number = {7},
pages = {527-533}
}
Duncan JP and Duncan JL (1982), "Folded Developables", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. Vol. 383(1784), pp. 191-205.
Abstract: A plane, inextensible sheet may be folded or creased along a curved line to produce two connected but distinct developable surfaces. Various theorems applying to this folding process are identified and two special cases investigated. In one, the fold line remains a plane curve during deformation and in the other the dihedral angle at the fold is constant along the curve. Curved-line folding occurs naturally in the collapse of thin-sheet-metal structures composed of developable surfaces. The theorems presented identify the kinematic constraint existing between pairs of developable surfaces connected by curved-line folds and permit the design of sheet-metal products that use these surfaces. This expands considerably the range of engineering products that can be made by folding and bending a single inextensible sheet.
BibTeX:
@article{duncan1982,
author = {J. P. Duncan and J. L. Duncan},
title = {Folded Developables},
journal = {Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences},
year = {1982},
volume = {383},
number = {1784},
pages = {191-205},
doi = {10.1098/rspa.1982.0126}
}
Filipov ET, Tachi T and Paulino GH (2014), "Toward optimization of stiﬀness and ﬂexibility of rigid, ﬂat-foldable origami structures", In 6th International Meeting on Origami in Science, Mathematics and Education (6OSME). University of Tokyo, August 10 -- 13, 2014.
@conference{Filipov2004,
author = {E. T. Filipov and T. Tachi and G. H. Paulino},
title = {Toward optimization of stiﬀness and ﬂexibility of rigid, ﬂat-foldable origami structures},
booktitle = {6th International Meeting on Origami in Science, Mathematics and Education (6OSME)},
year = {2014}
}
Furuya H, Inoue Y and Masuoka T (2005), "Deployment Characteristics of Rotationally Skew Fold Membrane for Spinning Solar Sail", In 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference. Austin, Texas, 18 - 21 April, 2005. (AIAA 2005-2045)
Abstract: A rotationally skew fold membrane for the spinning solar sail is discussed to examine the deployment characteristics. The membrane is characterized by double corrugation fold and is advantageous in the complete folding and compact storage. Spinning experiments with scaled models are performed to investigate the geometrical and deployment characteristics. As the result of the spinning experiments, it is indicated that the rotationally skew fold membrane is completely deployed and there is a minimum spin rate to complately deploy. The fact that the spin-direction wrapped membrane realizes quick deployment is also indicated. To investigate the dynamic characteristics, a non-dimensional similarity parameter derived with the theoretical analysis for one-dimensional Z-fold membrane is expressed by the geometrical parameters as the radius, folding pitch, material properties and spin rate. The theoretical similarity parameter is applied to the results of the spinning experiments and indicates the effect of the folding pitch of the rotationally skew fold membrane. Also the similarity parameter based on the experimental results is introduced.
BibTeX:
@inproceedings{Furuya2005,
author = {Hiroshi Furuya and Yosuke Inoue and Tadashi Masuoka},
title = {Deployment Characteristics of Rotationally Skew Fold Membrane for Spinning Solar Sail},
booktitle = {46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference},
year = {2005},
number = {AIAA 2005-2045}
}
Greschik G and Mikulas M (1996), "On imperfections and stowage creases in aluminum-rigidized inflated cylinders", In 37th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference. Salt Lake City, Utah, 18-19 April, 1996. (AIAA-96-1332)
@inproceedings{Greschik1996,
author = {G. Greschik and M. Mikulas},
title = {On imperfections and stowage creases in aluminum-rigidized inflated cylinders},
booktitle = {37th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference},
year = {1996},
number = {AIAA-96-1332},
doi = {10.2514/6.1996-1332}
}
Guest SD and Pellegrino S (1996), "The Folding of Triangulated Cylinders, Part III: Experiments", ASME Journal of Applied Mechanics. Vol. 63(1), pp. 77-83.
Abstract: This paper describes an experimental investigation of a type of foldable cylindrical structure, first presented in two earlier papers. Three cylinders of this type were designed and manufactured, and were then tested to find the force required to fold them. The results from these tests show some discrepancies with an earlier computational simulation, which was based on a pin-jointed truss model of the cylinders. Possible explanations for these discrepancies are explored, and are then verified by new simulations using computational models that include the effect of hinge stiffness, and the effect of geometric imperfections.
BibTeX:
@article{guest1996,
author = {S. D. Guest and S. Pellegrino},
title = {The Folding of Triangulated Cylinders, Part III: Experiments},
journal = {ASME Journal of Applied Mechanics},
year = {1996},
volume = {63},
number = {1},
pages = {77-83},
doi = {10.1115/1.2787212}
}
Hanna BH, Lund JM, Lang RJ, Magleby SP and Howell LL (2014), "Waterbomb base: a symmetric single-vertex bistable origami mechanism", Smart Materials and Structures. Vol. 23(9), pp. 094009.
Abstract: The origami waterbomb base is a single-vertex bistable origami mechanism that has unique properties which may prove useful in a variety of applications. It also shows promise as a test bed for smart materials and actuation because of its straightforward geometry and multiple phases of motion, ranging from simple to more complex. This study develops a quantitative understanding of the symmetric waterbomb baseʼs kinetic behavior. This is done by completing kinematic and potential energy analyses to understand and predict bistable behavior. A physical prototype is constructed and tested to validate the results of the analyses. Finite element and virtual work analyses based on the prototype are used to explore the locations of the stable equilibrium positions and the force–deflection response. The model results are verified through comparisons to measurements on a physical prototype. The resulting models describe waterbomb base behavior and provide an engineering tool for application development.
BibTeX:
@article{Hanna2014,
author = {Brandon H Hanna and Jason M Lund and Robert J Lang and Spencer P Magleby and Larry L Howell},
title = {Waterbomb base: a symmetric single-vertex bistable origami mechanism},
journal = {Smart Materials and Structures},
year = {2014},
volume = {23},
number = {9},
pages = {094009},
doi = {10.1088/0964-1726/23/9/094009}
}
Hedgepeth JM, MacNeal RH, Knapp K and MacGillivray CS (1981), "Considerations in the Design of Large Space Structures" (NASA CONTRACTOR REPORT 165744)
@techreport{Hedgepeth1981,
author = {J. M. Hedgepeth and R. H. MacNeal and K. Knapp and C. S. MacGillivray},
title = {Considerations in the Design of Large Space Structures},
year = {1981},
number = {NASA CONTRACTOR REPORT 165744}
}
Huffman DA (1976), "Curvatures and Creases: A Primer on Paper", IEEE Transactions on Computers. Vol. C-25(10), pp. 1010-1019.
Abstract: This paper represents fundamental results about how zero-cruvature (paper) surfaces behave near creases and apices of cones. These entities are natural generalizations of the edges and vertices of piecewise-planar surfaces. Consequently, paper surfaces may furnish a richer and yet still tractable class of surfaces for computer-aided design and computer graphics applications than do polyhedral surfaces.
Major portions of this paper are dedicated to exploring issues of curvature definition, convexity, and concavity, and interrelationships amoong angles associated with creases and generalized vertices and the orientations of associated surfaces in their vicinities. An electrial network representation is suggested in which there flow currents that are analogous to curvature components on the surface.
BibTeX:
@article{huffman1976,
author = {D. A. Huffman},
title = {Curvatures and Creases: A Primer on Paper},
journal = {IEEE Transactions on Computers},
year = {1976},
volume = {C-25},
number = {10},
pages = {1010-1019},
doi = {10.1109/TC.1976.1674542}
}
Kokotsakis A (1933), "Über bewegliche Polyeder", Mathematische Annalen. Vol. 107, pp. 627-647.
Abstract: We investigate the mechanics of thin sheets decorated by noninteracting creases. The system considered here consists of parallel folds connected by elastic panels. We show that the mechanical response of the creased structure is twofold, depending both on the bending deformation of the panels and the hingelike intrinsic response of the crease. We show that a characteristic length scale, defined by the ratio of bending to hinge energies, governs whether the structure’s response consists in angle opening or panel bending when a small load is applied. The existence of this length scale is a building block for future works on origami mechanics.
BibTeX:
@article{Lechenault2014,
author = {Lechenault, F. and Thiria, B. and Adda-Bedia, M.},
title = {Mechanical Response of a Creased Sheet},
journal = {Phys. Rev. Lett.},
year = {2014},
volume = {112},
pages = {244301},
doi = {10.1103/PhysRevLett.112.244301}
}
Lobkovsky A, Gentges S, Li H, Morse D and Witten TA (1995), "Scaling Properties of Stretching Ridges in a Crumpled Elastic Sheet", Science. Vol. 270(5241), pp. 1482-1485.
Abstract: Strong deformation of a sheet of solid material often leads to a crumpled state having sharp points of high curvature. A scaling property governing the crumpled state has been numerically demonstrated by an examination of the ridges joining pairs of sharp points in a range of simple geometries of variable size. As the linear size X increases sufficiently, the deformation energy grows as X1/3 and consists of similar amounts of bending and stretching energy. The deformation energy becomes concentrated in a fraction of the sheet that decreases as X1/3. Despite this concentration, the local strain in the ridge decreases as X2/3. Nearly all the deformation energy in thin, crumpled elastic sheets was found to be concentrated in ridges that obey these scaling laws.
BibTeX:
@article{Lobkovsky1995,
author = {A. Lobkovsky and S. Gentges and H. Li and D. Morse and T. A. Witten},
title = {Scaling Properties of Stretching Ridges in a Crumpled Elastic Sheet},
journal = {Science},
year = {1995},
volume = {270},
number = {5241},
pages = {1482-1485},
doi = {10.1126/science.270.5241.1482}
}
MacNeal RH and Robbins WM (1967), "Tensile Properties of a Tape with a Transverse Crease" (ARC-R-241)
@techreport{MacNeal1967,
author = {R. H. MacNeal and W. M. Robbins},
title = {Tensile Properties of a Tape with a Transverse Crease},
year = {1967},
number = {ARC-R-241}
}
Okuizumi N, Muta A and Matsunaga S (2011), "Enhancement of a Spring-mass System Model for Numerical Simulations of Centrifugal Deployment Dynamics of Folded Square Membranes", In 28th International Symposium on Space Technology and Science. Okinawa, Japan, June 5--12, 2011. (2011-c-30)
Abstract: In this paper, the spring-mass system model developed for simple numerical simulations of thin membranes is enhanced by taking into account the properties of buckling and creases and applied to the numerical simulations of centrifugal deployments of folded square membranes. The membranes are small-scale models for solar sail spacecraft “IKAROS”. First the folding and deployment methods are reviewed. Then the formulation of the enhanced spring-mass system model is explained. Numerical simulations of the centrifugal deployments of two kinds of folded square membranes with different crease intervals are performed and the numerical results are compared with the corresponding experimental results. The deployment behaviors are discussed and the validity of the spring-mass system model is examined.
BibTeX:
@inproceedings{Okuizumi2011,
author = {N. Okuizumi and A. Muta and S. Matsunaga},
title = {Enhancement of a Spring-mass System Model for Numerical Simulations of Centrifugal Deployment Dynamics of Folded Square Membranes},
booktitle = {28th International Symposium on Space Technology and Science},
year = {2011},
number = {2011-c-30}
}
Papa A and Pellegrino S (2008), "Systematically Creased Thin-Film Membrane Structures", Journal of Spacecraft and Rockets. Vol. 45(1), pp. 10-18.
Abstract: This paper presents a study of a square membrane, creased according to the Miura-ori folding pattern. When the membrane is allowed to expand from its packaged configuration, it initially expands elastically under zero corner forces. Starting from this naturally expanded configuration, the paper investigates the stress distribution and the load-displacement relationship when in-plane, diagonal loads are applied at the corners. It is found that out-of-plane bending is the main load-carrying mode and, for stress magnitudes typical of current solar-sail designs, the behavior of the membrane remains linear elastic. A simple analytical model, originally proposed for randomly creased membranes, is shown to predict with good accuracy the load-displacement relationship of the corners. It uses physically based and hence directly measurable membrane parameters.
BibTeX:
@article{Papa2008,
author = {A. Papa and S. Pellegrino},
title = {Systematically Creased Thin-Film Membrane Structures},
journal = {Journal of Spacecraft and Rockets},
year = {2008},
volume = {45},
number = {1},
pages = {10--18},
doi = {10.2514/1.18285}
}
Resch R and Christiansen HN (1971), "Kinematic Folded Plate System", In Proceedings of IASS Symposium on Folded Plates and Prismatic Structures. Vienna, Austria
Abstract: This paper consists of two nearly independent parts. The first, written by Professor Resch, describes his creations in the area of foldable plate structures. The second part, written by Professor Christiansen, discusses an analysis system which provides solutions to the elastic and kinematic problems associated with foldable structures.
The first part describes the general notion of kinematic systems, an example of which are kinematic folded-plate systems. The dynamic potential of these kinematic folded plate systes for the architectural designer has recently been realized through computer simulation. The combination of this design system and the computer simulation spells the possible elimination of monotonous serial production in favour of mass production of non-identical shell forms.
Key developments, in the latter part of the paper, describe the theory by which the elastic forces are generated for the large displacement problem and a fold element which is utilized for structural stability. Kinematic solutions are obtained by repeated application of a process involving an elastic analysis, an updating of the nodal coordinates, and a relaxation of fold elements. Also discussed is the implementation of these capabilities in a digital computer system which includes interactive displays.
BibTeX:
@inproceedings{resch1971,
author = {R. Resch and H. N. Christiansen},
editor = {R. Krapfenbauer},
title = {Kinematic Folded Plate System},
booktitle = {Proceedings of IASS Symposium on Folded Plates and Prismatic Structures},
year = {1971}
}
Schenk M, Allwood JM and Guest SD (2011), "Cold Gas-Pressure Folding of Miura-ori Sheets", In International Conference on Technology of Plasticity (ICTP 2011); special issue Steel Research International. Aachen, Germany, September 25-30th, 2011.
Abstract: Folding sheets from flat sheet materials into 3D surfaces provides a way to form textured sheets with a deep relief, without stretching the base material. Manufacturing can therefore be done using only low-energy bending operations along the fold lines. An important challenge to be overcome in the manufacturing process is the significant in-plane biaxial contraction during the folding process. A novel manufacturing process is herein introduced, which uses cold gas-pressure to fold the sheets and requires a minimum of initial tooling. Calculations were done to determine the required forming pressure to fold an example folded sheet, a Miura-ori sheet, and were compared with trials.
BibTeX:
@inproceedings{Schenk2011,
author = {M. Schenk and Julian M. Allwood and Simon D. Guest},
title = {Cold Gas-Pressure Folding of Miura-ori Sheets},
booktitle = {International Conference on Technology of Plasticity (ICTP 2011); special issue Steel Research International},
year = {2011}
}
Schenk M and Guest SD (2011), "Origami Folding: A Structural Engineering Approach", In Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education (5OSME). , pp. 293-305. CRC Press.
Abstract: In this paper we present a novel engineering application of Origami, using it for both the flexibility and the rigidity the folding patterns provide. The proposed Folded Textured Sheets have several interesting mechanical properties. The folding patterns are modelled as a pin-jointed framework, which allows the use of established structural engineering methods to gain insight into the kinematics of the folded sheet. The kinematic analysis can be naturally developed into a stiffness matrix approach; by studying its softest eigenmodes, important deformations of a partially folded sheet can be found, which aids in the understanding of Origami sheets for engineering applications.
BibTeX:
@incollection{schenk2010,
author = {Mark Schenk and Simon D. Guest},
editor = {Patsy Wang-Iverson and Robert J. Lang and Mark YIM},
title = {Origami Folding: A Structural Engineering Approach},
booktitle = {Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education (5OSME)},
publisher = {CRC Press},
year = {2011},
pages = {293-305}
}
Schenk M and Guest SD (2013), "Geometry of Miura-folded metamaterials", Proceedings of the National Academy of Sciences. Vol. 110(9), pp. 3276-3281.
Abstract: This paper describes two folded metamaterials based on the Miura-ori fold pattern. The structural mechanics of these metamaterials are dominated by the kinematics of the folding, which only depends on the geometry and therefore is scale-independent. First, a folded shell structure is introduced, where the fold pattern provides a negative Poisson’s ratio for in-plane deformations and a positive Poisson’s ratio for out-of-plane bending. Second, a cellular metamaterial is described based on a stacking of individual folded layers, where the folding kinematics are compatible between layers. Additional freedom in the design of the metamaterial can be achieved by varying the fold pattern within each layer.
BibTeX:
@article{Schenk2013,
author = {Schenk, Mark and Guest, Simon D.},
title = {Geometry of Miura-folded metamaterials},
journal = {Proceedings of the National Academy of Sciences},
year = {2013},
volume = {110},
number = {9},
pages = {3276-3281},
doi = {10.1073/pnas.1217998110}
}
Silverberg JL, Evans AA, McLeod L, Hayward RC, Hull T, Santangelo CD and Cohen I (2014), "Using origami design principles to fold reprogrammable mechanical metamaterials", Science. Vol. 345(6197), pp. 647-650.
Abstract: Although broadly admired for its aesthetic qualities, the art of origami is now being recognized also as a framework for mechanical metamaterial design. Working with the Miura-ori tessellation, we find that each unit cell of this crease pattern is mechanically bistable, and by switching between states, the compressive modulus of the overall structure can be rationally and reversibly tuned. By virtue of their interactions, these mechanically stable lattice defects also lead to emergent crystallographic structures such as vacancies, dislocations, and grain boundaries. Each of these structures comes from an arrangement of reversible folds, highlighting a connection between mechanical metamaterials and programmable matter. Given origami’s scale-free geometric character, this framework for metamaterial design can be directly transferred to milli-, micro-, and nanometer-size systems.
BibTeX:
@article{Silverberg2014,
author = {Silverberg, Jesse L. and Evans, Arthur A. and McLeod, Lauren and Hayward, Ryan C. and Hull, Thomas and Santangelo, Christian D. and Cohen, Itai},
title = {Using origami design principles to fold reprogrammable mechanical metamaterials},
journal = {Science},
year = {2014},
volume = {345},
number = {6197},
pages = {647-650},
doi = {10.1126/science.1252876}
}
Silverberg JL, Na J-H, Evans AA, Liu B, Hull TC, Santangelo CD, Lang RJ, Hayward RC and Cohen I (2015), "Origami structures with a critical transition to bistability arising from hidden degrees of freedom", Nature Materials. Vol. 14, pp. 389–-39.
Abstract: Origami is used beyond purely aesthetic pursuits to design responsive and customizable mechanical metamaterials 1, 2, 3, 4, 5, 6, 7, 8. However, a generalized physical understanding of origami remains elusive, owing to the challenge of determining whether local kinematic constraints are globally compatible and to an incomplete understanding of how the folded sheet’s material properties contribute to the overall mechanical response9, 10, 11, 12, 13, 14. Here, we show that the traditional square twist, whose crease pattern has zero degrees of freedom (DOF) and therefore should not be foldable, can nevertheless be folded by accessing bending deformations that are not explicit in the crease pattern. These hidden bending DOF are separated from the crease DOF by an energy gap that gives rise to a geometrically driven critical bifurcation between mono- and bistability. Noting its potential utility for fabricating mechanical switches, we use a temperature-responsive polymer-gel version of the square twist to demonstrate hysteretic folding dynamics at the sub-millimetre scale.
BibTeX:
@article{Silverberg2015,
author = {Jesse L. Silverberg and Jun-Hee Na and Arthur A. Evans and Bin Liu and Thomas C. Hull and Christian D. Santangelo and Robert J. Lang and Ryan C. Hayward and Itai Cohen},
title = {Origami structures with a critical transition to bistability arising from hidden degrees of freedom},
journal = {Nature Materials},
year = {2015},
volume = {14},
pages = {389–-39},
doi = {10.1038/nmat4232}
}
Stachel H (2010), "A kinematic approach to Kokotsakis meshes", Computer Aided Geometric Design. Vol. 27(6), pp. 428 - 437.
Abstract: A Kokotsakis mesh is a polyhedral structure consisting of an n-sided central polygon surrounded by a belt of quadrangles or triangles in the following way: Each side ai of is shared by an adjacent polygon , and the relative motion between cyclically consecutive neighbor polygons is a spherical coupler motion. Hence, each vertex of is the meeting point of four faces. In the case n=3 the mesh is part of an octahedron.
These structures with rigid faces and variable dihedral angles were first studied in the thirties of the last century. However, in the last years there was a renaissance: The question under which conditions such meshes are infinitesimally or continuously flexible gained high actuality in discrete differential geometry. The goal of this paper is to revisit the well-known continuously flexible examples (Bricard, Graf, Sauer, Kokotsakis) from the kinematic point of view and to extend their list by a new family.
BibTeX:
@article{stachel2010,
author = {Hellmuth Stachel},
title = {A kinematic approach to Kokotsakis meshes},
journal = {Computer Aided Geometric Design},
year = {2010},
volume = {27},
number = {6},
pages = {428 - 437},
doi = {10.1016/j.cagd.2010.05.002}
}
Tachi T (2009), "Generalization of Rigid Foldable Quadrilateral Mesh Origami", Journal of the International Association for Shell and Spatial Structures. Vol. 50(3), pp. 173-179.
Abstract: In general, a quadrilateral-mesh surface does not enable a continuous rigid motion because an overconstrained system, in which the number of constraints around degree-4 vertices (three for each vertex) exceeds the number of variables (the number of hinges), is constructed. However, it is known that the developable double corrugation surface, known as Miura-ori, produces a rigid deployment mechanism. The rigid-foldability of Miura-ori is due to the singularity in its pattern, where a single vertex is repeated. We generalize the geometric condition for enabling rigid motion in general quadrilateral-mesh origami without simple repeating symmetry. To ensure the existence of a finite motion, we derive the identity of functions from the formula for degree-4 single-vertex origami. This yields a variety of unexplored generalized shapes of quadrilateral-mesh origami that preserve one-DOF finite rigid-foldability in addition to developability and flat-foldability.
BibTeX:
@article{tachi2009,
author = {Tomohiro Tachi},
title = {Generalization of Rigid Foldable Quadrilateral Mesh Origami},
journal = {Journal of the International Association for Shell and Spatial Structures},
year = {2009},
volume = {50},
number = {3},
pages = {173-179}
}
Waitukaitis S, Menaut R, Chen BG-g and van Hecke M (2015), "Origami Multistability: From Single Vertices to Metasheets", Phys. Rev. Lett.. Vol. 114, pp. 055503.
Abstract: We show that the simplest building blocks of origami-based materials—rigid, degree-four vertices—are generically multistable. The existence of two distinct branches of folding motion emerging from the flat state suggests at least bistability, but we show how nonlinearities in the folding motions allow generic vertex geometries to have as many as five stable states. In special geometries with collinear folds and symmetry, more branches emerge leading to as many as six stable states. Tuning the fold energy parameters, we show how monostability is also possible. Finally, we show how to program the stability features of a single vertex into a periodic fold tessellation. The resulting metasheets provide a previously unanticipated functionality—tunable and switchable shape and size via multistability.
BibTeX:
@article{Waitukaitis2015,
author = {Waitukaitis, Scott and Menaut, Rémi and Chen, Bryan Gin-ge and van Hecke, Martin},
title = {Origami Multistability: From Single Vertices to Metasheets},
journal = {Phys. Rev. Lett.},
year = {2015},
volume = {114},
pages = {055503},
doi = {10.1103/PhysRevLett.114.055503}
}
Wei ZY, Guo ZV, Dudte L, Liang HY and Mahadevan L (), "Geometric Mechanics of Periodic Pleated Origami", Physical Review Letters. Vol. 110(21), pp. 215501.
Abstract: Origami structures are mechanical metamaterials with properties that arise almost exclusively from the geometry of the constituent folds and the constraint of piecewise isometric deformations. Here we characterize the geometry and planar and nonplanar effective elastic response of a simple periodically folded Miura-ori structure, which is composed of identical unit cells of mountain and valley folds with four-coordinated ridges, defined completely by two angles and two lengths. We show that the in-plane and out-of-plane Poisson’s ratios are equal in magnitude, but opposite in sign, independent of material properties. Furthermore, we show that effective bending stiffness of the unit cell is singular, allowing us to characterize the two-dimensional deformation of a plate in terms of a one-dimensional theory. Finally, we solve the inverse design problem of determining the geometric parameters for the optimal geometric and mechanical response of these extreme structures.
BibTeX:
@article{Wei2009,
author = {Z. Y. Wei and Z. V. Guo and L. Dudte and H. Y. Liang and L. Mahadevan},
title = {Geometric Mechanics of Periodic Pleated Origami},
journal = {Physical Review Letters},
volume = {110},
number = {21},
pages = {215501},
doi = {10.1103/PhysRevLett.110.215501}
}
Yasuda H, Yein T, Tachi T, Miura K and Taya M (2013), "Folding behaviour of Tachi-Miura polyhedron bellows", Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 469(2159) The Royal Society.
Abstract: In this paper, we examine the folding behaviour of TachitextendashMiura polyhedron (TMP) bellows made of paper, which is known as a rigid-foldable structure, and construct a theoretical model to predict the mechanical energy associated with the compression of TMP bellows, which is compared with the experimentally measured energy, resulting in the gap between the mechanical work by the compression force and the bending energy distributed along all the crease lines. The extended Hamilton's principle is applied to explain the gap which is considered to be energy dissipation in the mechanical behaviour of TMP bellows.
BibTeX:
@article{Yasuda2013,
author = {Yasuda, Hiromi and Yein, Thu and Tachi, Tomohiro and Miura, Koryo and Taya, Minoru},
title = {Folding behaviour of Tachi-Miura polyhedron bellows},
journal = {Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences},
publisher = {The Royal Society},
year = {2013},
volume = {469},
number = {2159},
doi = {10.1098/rspa.2013.0351}
}