Curvature condensation and bifurcation in an elastic shell

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Entry by Helen Wu, AP225 Fall 2010


"Curvature condensation and bifurcation in an elastic shell"

M. Das, A. Vaziri, A. Kudrolli, L. Mahadevan, Physical Review Letters, 98, 014301 (2007).


elastic shell, defects


Figure 2. Figure of the system being studied. (a-d) The deformation of the sheet along the axis of symmetry for the cylinder. (e-h) Colormap representation of the curvature going from blue (low) to red (high).

Elastic systems often have a hierarchy in structural scales, but the process by which defects form is unknown. The researchers study a thin mylar sheet that has been bent into a half-cylindrical elastic shell, then indented the sheet at one edge along the axis of symmetry. The sheet was reconstructed using laser aided tomography.

Results and discussion

Figure 3. (a) The transition between global and local deformation modes is indicated. Filled and open circles are the condensate before and after twinning, respectively. (b) Looking at the Gauss curvature along the axis of symmetry shows a transition at a secondary maximum in curvature.
Figure 4. (a) The location of the condensate as a function of the indentation (normalized) for different thicknesses. Dotted & dashed lines represent saturation. (b) Location of defect near a bifurcation for various thicknesses.

They observed that for small indentations, the deformation was strong in the immediate area of the edge, but the curvature decayed monotonically away from the edge. Past a critical threshold, another maximum in curvature appears on the symmetry axis that looks like a parabolic defect (a "curvature condensate"). Indenting even more results in the defect bifurcating, and farther beyond that point, the shell folds inwards rather than continuing to move defects. There is no dependence on material parameters (the system only has a dimensionless parameter <math>R/t</math> (R=radius, t=thickness) because of the Young's modulus and thickness).

The authors used numerical methods to study the system, minimizing the elastic energy of the shell with an energy density that had components of energy from in-plane deformations and from out-of-plane deformations. Figure 2 shows the system as observed experimentally and the simulation results.

They studied the transition between 2 modes of deformation - global and local - and determined that there is a threshold at which it happens, indicated by an inflection point in the graph of the Gauss curvature <math>\kappa_G(0,y)</math> as a function of the scaled indentation <math>\delta/R</math>. This only happens in 2D systems.

It was also shown that the location where the condensation happens scales with the indentation size until a saturation point (Figure 4a), and that saturation value increases with <math>t</math>. The indentation size necessary for bifurcation, or twinning, was found to go as <math>t^{1/2}</math>. Above the indentation size <math>\delta_{bif}</math> where this happens, the condensates move from the axis of symmetry in a symmetrical manner (Figure 4b).

The curvature condensate was localized along a crescent shape and the width and radius of curvature were similar to results from a previous paper about the size of the defect core when a sheet is bent into a cone shape.

The Donnell-Föppl-von Kármán equations for large deflections of thin, flat plates were successfully applied to this system to represent the deformations.

The paper gives a general, qualitative overview of curvature condensates, but more work still needs to be done to obtain a quantitative result.