Singularities and Microstructure in thin elastic sheets

The variational problem (GC) for m = 2, d = 3 shows concentration effects for appropriate boundary conditions. This has been investigated experimentally, by asymptotic analysis [CM98,Lob96] and by numerical simulations [Lob96,KW97]. Two distinct types of singular structures - point-like vertex singularities [CM98] are at the ends of line-like ridge singularities have been identified [BAP97,Lob96,LW97]. These singularities are analogous to the points and the creases in crumpled paper. Asymptotic analysis suggests that, for small $\epsilon$, most of the energy is concentrated on the ridges [Lob96].

Singularities with Additional Structure

Figure. 1 is a schematic representation of a ridge. In the limit $\epsilon$ $\rightarrow$ 0, the ridge becomes a line across which there is a constant jump in the gradient Du. However, for any nonzero $\epsilon$, the ridge has additional structure along it's length. Asymptotic analysis [Lob96] and scaling arguments [WL93,LGL⁺95] suggest that the width of the region over which the gradient changes, as well as the region over which the energy concentrates, are not uniform along the length of the ridge. Rather, the width of this region is given by d (x) $\lesssim$ C$\epsilon^{{1/3}}_{}$x^2/3, where x is the distance to the closest endpoint of the ridge [Lob96]. This is illustrated in Fig. 1. We are close to proving that that

Theorem 1   For the minimal ridge boundary conditions,

c$$\epsilon^{{5/3}}_{}$$L^1/3 $$\leq$$ $$\mathcal {E}$$^$\epsilon$[u^$\epsilon$] $$\leq$$ C$$\epsilon^{{5/3}}_{}$$L^1/3,

where L is the length of the ridge. Further, the energy concentrates on a region whose width is given by c$\epsilon^{{1/3}}_{}$x^2/3 $\leq$ d (x) $\leq$ C max($\epsilon$,$\epsilon^{{1/3}}_{}$x^1/3L^1/3), where x is the distance from the (closest) endpoint of the ridge.

Figure 1: A minimal ridge.

Fig. 1 depicts a thin elastic sheet making a a ``minimal ridge''. The boundary conditions are given by a frame (the thick solid lines) bent through an angle. The sheet is essentially flat outside the region bounded by the two dashed curves, and the bulk of the energy is concentrated in this region.

The nontrivial structure along the length of the ridge singularity is reflected in the scaling for the energy (L^1/3 instead of L) and the scaling for d (x) with an explicit dependence on x. Note also that the scaling result for d (x) is not optimal, and it doesn't agree with the predictions of the scaling analysis. The following questions arise naturally from this result:

Problem 1   Can one obtain a rigorous scaling result for d (x) that is sharp, i.e., can one get upper and lower bounds that only differ by a multiplicative constant?

Problem 2   Identify and analyze other situations (e.g. different values of m and d) where extended singularities have additional structure.

Oscillations and Multiple Scale Solutions

Minimizing sequences for (GC) can also develop very fine scale oscillations [JS01]. Ben Belgacem et al. have shown that [BBCDM00], for a compressed thin film ( m = 2, d = 3 in (GC))

Theorem 2   The energy of the minimizer satisfies

c$$\lambda^{{3/2}}_{}$$$$\epsilon$$L $$\leq$$ $$\mathcal {E}$$^$\epsilon$[u^$\epsilon$] $$\leq$$ C$$\lambda^{{3/2}}_{}$$$$\epsilon$$L,

where L is a typical length scale of the domain, and $\lambda$ is the compression factor. A construction for the upper bound strongly suggests that the minimizers develop an infinitely branched network with oscillations on increasingly finer scales as $\epsilon$ $\rightarrow$ 0.

Oscillatory solutions to variational problems can be crudely classified as follows:

1. Minimizing sequences have oscillations on one (or a finite number) of scales tending to zero as $\epsilon$ $\rightarrow$ 0, e.g. in the regularized Bolza problem [AM01].

2. Minimizing sequences develop oscillations on an infinite number of scales as $\epsilon$ $\rightarrow$ 0, e.g. Self-similar branching near an Austenite-Martensite boundary [KM94,Con00].

Oscillations on a single scale can be characterized by Young measures [You69,KP91] generated by minimizing sequences. This approach can be extended to two scale oscillations by considering Young measures on an appropriate space of patterns [AM01].

The latter class of solutions are related to Gromov's theory of convex integration [Gro86,Spr98] that addresses questions in geometric rigidity and under-determined PDE.

Global Transitions

We can crudely classify the different ways in which a minimizing sequence fails to converge strongly as in Table 1.

Table 1: Examples displaying different scenarios by which minimizing sequences fail to converge strongly.

	Concentration	Oscillations
No additional structure	Vertices in crumpled sheets	Regularized Bolza Problem
Additional structure	Ridges in crumpled sheets	Delamination of Thin films

Interestingly Theorems 2 and 1 imply that the crumpling problem ( m = 2, d = 3) has solutions that are of fundamentally different characters depending on boundary conditions. This is relevant to the important question of whether singularities are indeed robust and universal ``local'' phenomena independent of the precise form of boundary conditions. Since the crumpling problem and the related Bolza problem display the entire range of the behaviors in Table 1, we propose to investigate the following:

Problem 3   Using the generalized crumpling problem as a model system, clarify the role of boundary conditions in selecting solutions to variational problems.

With a one parameter family, we can also smoothly interpolate between boundary conditions that give multiple scale oscillations, and those that give energy concentration on ridges^0.1. This suggests the following intriguing possibility:

Problem 4   Using one parameter families of boundary conditions, investigate the transitions between the different scenarios in Table. 1 for variational problems.