Periodic Folding of Thin Sheets

Mahadevan, L.; Keller, Joseph B.

doi:10.1137/s0036139994263410

Cited by 17 publications

(18 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this respect, it seems to have a similar origin as the folding mechanism reported in sliding wear (11), although the resulting flow here is significantly different-there is actual material removal, and folds of very large amplitude occur. Fold patterns observed in the sinuous flow show remarkable resemblance with other well-studied folding phenomena in geophysics (12), thin films (13), fluid flow (14,15), and non-Newtonian plastics (16).…”

supporting

confidence: 61%

Sinuous flow in metals

Yeung

Viswanathan

Compton

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

Annealed metals are surprisingly difficult to cut, involving high forces and an unusually thick "chip." This anomaly has long been explained, based on ex situ observations, using a model of smooth plastic flow with uniform shear to describe material removal by chip formation. Here we show that this phenomenon is actually the result of a fundamentally different collective deformation mode-sinuous flow. Using in situ imaging, we find that chip formation occurs via large-amplitude folding, triggered by surface undulations of a characteristic size. The resulting fold patterns resemble those observed in geophysics and complex fluids. Our observations establish sinuous flow as another mesoscopic deformation mode, alongside mechanisms such as kinking and shear banding. Additionally, by suppressing the triggering surface undulations, sinuous flow can be eliminated, resulting in a drastic reduction of cutting forces. We demonstrate this suppression quite simply by the application of common marking ink on the free surface of the workpiece material before the cutting. Alternatively, prehardening a thin surface layer of the workpiece material shows similar results. Besides obvious implications to industrial machining and surface generation processes, our results also help unify a number of disparate observations in the cutting of metals, including the so-called Rehbinder effect.folding | plasticity | metal cutting | instability | deformation

show abstract

supporting

confidence: 61%

Sinuous flow in metals

Yeung

Viswanathan

Compton

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

show abstract

“…As expected, the cross section of a nanotube for a broad range of the values of their parameters is similar to the fold found in Ref. 7.…”

Section: Discussionsupporting

confidence: 88%

“…͑1͒ would still hold, but the stress F y would become a function of s and F y Ј would equal the weight per unit length along s. The equations of the elastica ͑1͒ and ͑7͒ have been applied to different elastic materials taking the weight into account. 3,4,7 I will mention three applications that show how the equations of the elastica help to determine a physical parameter or understand the shape of a folded material. The equations of the elastica for an initially horizontal lamina with one end clamped were solved numerically to check the classic Peirce's cantilever test for the bending rigidity of textiles.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of the shape of a sheet of paper when two opposite edges are joined

Colom

2006

American Journal of Physics

View full text Add to dashboard Cite

Exact non-Hookean scaling of cylindrically bent elastic sheets and the large-amplitude pendulum Am. J. Phys. 79, 657 (2011) The Cramster Conclusion Phys. Teach. 49, 291 (2011) Strain in layered zinc blende and wurtzite semiconductor structures grown along arbitrary crystallographic directions Am.The profiles of the edge of rectangular sheets of paper folded so as to join opposite edges were measured and compared with the profiles calculated with the theory of elasticity. I show how the calculation can be accomplished by using the aspect ratio of the loops as the only input parameter.

show abstract

“…Inspection of the energy shows that a natural length scale in the problem arises from the balance of gravitational and bending energies, the ''gravity length'' ᐉ g ϭ (B͞hg) 1/3 (11). For the rubber sheet (Fig.…”

Section: Ds(1ϫcos ͞Sin ␤) [5]mentioning

confidence: 99%

The elements of draping

Cerda

Mahadevan

Pasini

2004

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

View full text Add to dashboard Cite

We consider the gravity-induced draping of a 3D object with a naturally flat, isotropic elastic sheet. As the size of the sheet increases, we observe the appearance of new folded structures of increasing complexity that arise because of the competition between elasticity and gravity. We analyze some of the simpler 3D structures by determining their shape and analyzing their response and stability and show that these structures can easily switch between a number of metastable configurations. For more complex draperies, we derive scaling laws for the appearance and disappearance of new length scales. Our results are consistent with commonplace observations of drapes and complement large-scale computations of draping by providing benchmarks. They also yield a qualitative guide to fashion design and virtual reality animation. T he couturier drapes the 3D human body with a 2D fabric, working hard to subvert the relentless force of gravity to her cause by using a combination of cuts, folds, and tucks to transform a featureless textile into a piece of art. Indeed, the depiction of drapery, in the form of a carelessly thrown shawl on one's knee, is an important theme in Renaissance art, in both sculpture and sketching (1). Modern art has found another expression for the aesthetics of drapery in the carefully orchestrated wrapping of an entire building (2, 3). From a scientific viewpoint, drapery affords a common example of the complex patterns that arise from simple causes (Fig. 1a). In particular, drapery involves the large elastic (and reversible) deformations of naturally thin flat sheets, e a subject with comparatively recent theoretical origins going back to the early 20th century, when the first models valid for moderate deformations were formulated (4). In the last two or three decades, various geometrically exact formulations that go beyond the approximate theories have been put forward (5); these have also been the subject of large-scale computational approaches (6, 7). However, as is clearly evident in Fig. 1a, the fabric deformations in draping are highly inhomogeneous and result in the strong localization of strain in the neighborhood of points (8), thus making the computations difficult. Additionally, it is a matter of common experience that there are many local equilibria that the drapery is equally comfortable in; for example, the number of pleats in a skirt or sari can be easily modified to suit the wearer, suggesting a certain degeneracy among the solutions. To complement these quantitative computational approaches, which are in their nascent stages, here we approach the problem of drapery from a slightly different perspective by using a combination of exact analysis and scaling based on experimental observations of a variety of draping patterns to understand each of the above issues directly and thus provide a set of benchmarks while serving as a qualitative guide to the complexity of draping.Visually dissecting the drape of a complex, relatively rigid surface, we see that it is constituted mostly of f...

show abstract

Periodic Folding of Thin Sheets

Cited by 17 publications

References 6 publications

Sinuous flow in metals

Sinuous flow in metals

Analysis of the shape of a sheet of paper when two opposite edges are joined

The elements of draping

Contact Info

Product

Resources

About