Session A39: Focus Session: Elasticity and Geometry of Thin Objects

Delamination of thin elastic sheets from soft, sticky substrates

We study the compression of a soft elastic substrate with a thin sheet adhered to its surface. In this situation, it is energetically expensive for the thin sheet to alter its length. Instead, it accommodates its excess length by delamination from the substrate, allowing it to bend out of the plane. Rather than forming a single `blister', however, we observe the formation of several blisters with a characteristic size. Here, we investigate the dependence of this characteristic blister size on the material properties of the system using a combination of experimental and theoretical analyses.   

The frustrating tearing of adhesive tape

When trying to remove adhesive tape, one often only manages to peel off a useless pointy strip: the fracture tips on both sides of the pulled strip seem to attract each-other, and merge in a finite distance. Why don't they repel each other and lead to a continually increasing width of the strip, as one would like to? We will present an experimental and theoretical study of this pinch-off phenomenon in the rupture of peeled adhesive sheets. The cut shapes are very reproducibles, and we will show that the geometry of the peeling fold, where elastic energy is concentrated, plays a major role here.   

Instability of an elastic knot under twist

In a recent paper, we derived a solution to the Kirchhoff equations representing a knotted elastic rod held by a tensile force applied at its ends. This problem has been formulated as the minimization of a curvature energy in the presence of a topological constraint. We extend this analysis to the case of a knot subjected to both a tensile force and a twisting moment. We unveil a striking instability that can be easily reproduced with a piece of computer cord: a simple knot, initially comprising a large loop merging with a localized braid, can be unfolded under applied twist into a symmetric shape resembling the figure of eight. Doing so, it becomes much easier to untie.   

On the statistical physics of folding and crumpling

Unfolding a ball of crumpled paper reveals numerous ridges with a wide distribution of sizes. How can we describe the statistics of sizes and energies? Can we understand this system using the tools of statistical physics? During my talk, I will review the various experimental and theoretical approaches that we used to tackle these questions, which are typical of glasses and granular media.   

The Shape of the Optimal Javelin

To find the shape of a javelin whose vibrations dampen the fastest, we seek to maximize the eigenvalue of the first eigen-mode of a vibrating rod. The problem is related to (and is inspired by) the classical problems of finding the tallest and strongest columns solved by J. B. Keller [1], and J. B. Keller and F. I. Niordson [2]. A 4${}^{\rm th}$ order ODE for the maximizing eigen-mode is readily found, however it is ill-conditioned at the boundaries, and standard numerical methods fails. Using a similarity solution, we ``peel away'' the singularity, and solve the remaining part ``backwards''. The resulting shape has a frequency of vibration 5 times larger than that of the uniform-diameter rod. The method of solution is applicable to other similar problems. For example, we confirm the shape of the tallest column with it. \newline \newline {[1]} The Strongest Column / J. B. Keller ; Arch. Rat. Mech. Anal. 1960 {\bf(5)}, pp. 275--285 \newline {[2]} The Tallest Column / J. B. Keller and F. I. Niordson ; J. Math. Mech. 1966 {\bf (16)}, pp. 433--446   

Impacts on thin elastic sheets

The radial cracks developing from the impact point of a projectile on a windshield are of common experience. We investigate the origin of this phenomenon using thin elastic sheets as an experimental model. A projectile launched at controlled speed impacts a free membrane at rest. A tensile front sets out from the point of impact and propagates radially at the speed of sound. Flexural waves can propagate in the extended area. Specifically, the interaction between the rigid body and the elastic sheet gives birth to a conical flexural shape whose base expands radially at a well defined velocity. During the propagation of both the tensile and flexural fronts, the radial tensile stress field results in a compressive stress in the azimuthal direction, which triggers a buckling instability. That instability is responsible for the formation of radial folds, with a well defined azimuthal wave number. Based on detailed experimental observations and measurements, we propose a model to understand the wave motion and stress field consecutive to the impact; in addition, we provide a prediction for the number of folds selected during the buckling instability as a function of the relevant parameters, including impact velocity.   

Spiraling Cracks in Thin Sheets

A wide kind of everyday-life industrial products come in a thin package that needs to be torn open by the user, and the opening is not always easy. We built a simple setup to study crack propagation in thin sheets coupled with large out-of-plane displacement : A cylindrical tool is inserted in a straight incision in a thin sheet, and is pushed against the sheet perpendicularly to that incision, eventually propagating a crack. When the blunt tool is continually pushed against the lip, we found that the crack follows a very robust spiraling path. Experiments may be interpreted in terms of ``Spira Mirabilis'' (logarithmic spiral). Starting with crack theory argument, we will show that the early behavior of the cut path follows a portion of a logathmic spiral, and that the path tends to another spiral with a different pitch as the crack adds more turns. Our crack experiment illustrates the fact that thin sheets mechanics is deeply connected to geometry, and finally spirals characteristics allow us to measure material crack properties of the thin layer used.   

Interaction Between Two Localized Wrinkle Patterns

A drop of water placed on the surface of a freely floating ultrathin polymer film produces a radial wrinkling pattern due to the capillary force it exerts on the film. ~We have previously characterized [1] the number N and length L of the wrinkles. We now study the interaction between two such localized wrinkling patterns each induced by one drop of water. The patterns distort, and radial symmetry about each drop is lost, with the wrinkles extending further along the line between the drops. When the drops are brought closer, a single long wrinkle forms along this axis. We use the distance at which this connecting wrinkle appears to quantify the range of the interaction between the wrinkles. We will present data for this interaction length as a function of other length scales in the experiment. \newline 1. Full reference here. Science 317, 650(2007)   

Pattern transformation triggered by deformation.

Periodic elastomeric cellular solids are subjected to uniaxial compression and novel transformations of the patterned structures are found upon reaching a critical value of applied load. The results of a numerical investigation reveal that the pattern switch is triggered by a reversible elastic instability. Excellent quantitative agreement between numerical and experimental results is found and the transformations are found to be remarkably uniform across the samples. Moreover the phenomenon is found to be robust for a range of soft solids including rubber and jelly. *Joint work with M.C. Boyce, K. Bertoldi and S. Deschanel, MIT.   

Granular Silo collapse: an experimental study

We present an experimental work that develop some basic insight into the pre-buckling behavior and the buckling transition toward plastic collapse of a granular silo. We study different patterns of deformation generated on thin paper cylindrical shells during granular discharge. We study the collapse threshold for different bed height, flow rates and grain sizes. We compare the patterns that appear during the discharge of spherical beads, with those obtained in the axially compressed cylindrical shells. When the height of the granular column is close to the collapse threshold, we describe a ladder like pattern that rises around the cylinder surface in a spiral path of diamond shaped localizations, and develops into a plastic collapsing fold that grows around the collapsing silo.   

Sudden ridge collapse in the stress relaxation of thin crumpled polymer films

Uniform compression of thin crumpled sheets subjected to a constant weight has been shown to exhibit a remarkably wide range of scaling behaviour, covering up to five orders of magnitude [1], i.e. time scales from seconds to weeks. We demonstrate that this scaling behaviour is not smooth, but rather interrupted by sudden changes in height of the uniformly compressed crumple, which we attribute to sudden ridge collapses. The height of the discontinuous jumps due to sudden ridge collapse during the compression process increases with increasing thickness of the polymer film. This is attributed to the fact that thick films exhibit a smaller defect density, but increased defect length. Interestingly, when plotting the time laps between successive ridge collapses as a function of time, the data collapses to a single line for all film thicknesses, with a slope of d$\Delta $t/dt=1 over a scaling regime of four orders of magnitude. Possible explanations will be discussed. [1] K. Matan, R.B. Williams, T.A. Witten, S.R. Nagel, Phys. Rev. Lett., 88, (2002), 076101.   

Geometry, mechanics and statistical physics in crumpled structures

There's been a recent surge of interest in the study of low-dimensional packed elastic manifolds. In fact, the simple act of crumpling a piece of paper does require the simultaneous interaction of many fascinating mechanisms. These include energy condensation from large length scales to small singular structures, topological self-avoidance and complex phase space landscapes reminiscent of frustration in the context of glassy systems. We will present a numerical experiment modeling the folding of an elastic rod (1D) restricted to a shrinking 2D space. The confinement is obtained by preparing an initially disordered elastic line embedded in a quadratic potential. Varying the strength of this confining potential shows that many metastable states can be observed. We are interested in a statistical analysis of the emerging folded patterns. We will discuss the relevance of our results with recent theoretical models (inspired by the free-volume theory of Edwards in the context of granular matter) and recent experiments of crumpled paper. \newline Some references: L. Bou\'e {\it et al}, PRL {\bf 97} (2006) 166104, L. Bou\'e and E. Katzav EPL {\bf 80} (2007) 54002, E. Katzav, M. Adda-Bedia and A. Boudaoud PNAS {\bf 103} (2006) 18900-18904.   

Inside a Ball of Crumpled Aluminum Foil

We have studied the three-dimensional geometry of a crumpled sheet via x-ray CT scans. We crumple circular sheets of aluminum with thicknesses of 30--50$\mu m$ and diameter 100000$\mu m$ into spherical balls of diameter 15000$\mu m$ to 20000$\mu m$. We then perform CT scans with a resolution of 6$\mu m^3$/voxel. This allows us to fully resolve the conformation of the sheet. We use the reconstructed CT images to determine the mass distribution inside the crumpled ball. We also report on a box-counting analysis to assess the fractal nature of the mass distribution.