Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing steepest descent with respect to (an)isotropic surface energy. It has been proposed that this motion could be computed by solving the analogous evolution law
using a crystalline approximation to the surface energy. We present the first convergence analysis for a numerical scheme of this type. Our treatment is restricted to one dimensional surfaces (curves in the plane) which are graphs. In this context, the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension.
This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit which compresses the film. Blistering permits the film to expand locall
y, reducing the elastic energy of the system. It is natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the {it elastic deformation} of the film and substrate and the {it geometry} of the blistered region. It involves three small parameters: the {it nondimensionalized thickness} of the film, the {it compliance ratio} of the film/substrate pair and the {it mismatch strain}. In formulating the problem, we use a small-slope (Foppl-von Karman) approximation for the elastic energy of the film, and a local approximation for the elastic energy of the substrate. For a 1D version of the problem, we obtain matching upper and lower bounds on the minimum energy, in the sense that both bounds have the same scaling behavior with respect to the small parameters. For a 2D version of the problem, our results are less complete. Our upper and lower bounds only match in their scaling with respect to the nondimensionalized thickness, not in the dependence on the compliance ratio and the mismatch strain. The upper bound considers a 2D lattice of blisters, and uses ideas from the literature on the folding or crumpling of a confined elastic sheet. Our main 2D result is that in a certain parameter regime, the elastic energy of this lattice is significantly lower than that of a few large blisters.
A body of literature has developed concerning cloaking by anomalous localized resonance. The mathematical heart of the matter involves the behavior of a divergence-form elliptic equation in the plane, $ ablacdot (a(x) abla u(x)) = f(x)$. The complex-
valued coefficient has a matrix-shell-core geometry, with real part equal to 1 in the matrix and the core, and -1 in the shell; one is interested in understanding the resonant behavior of the solution as the imaginary part of $a(x)$ decreases to zero (so that ellipticity is lost). Most analytical work in this area has relied on separation of variables, and has therefore been restricted to radial geometries. We introduce a new approach based on a pair of dual variational principles, and apply it to some non-radial examples. In our examples, as in the radial setting, the spatial location of the source $f$ plays a crucial role in determining whether or not resonance occurs.
It is well known that an elastic sheet loaded in tension will wrinkle and that the length scale of the wrinkles tends to zero with vanishing thickness of the sheet [Cerda and Mahadevan, Phys. Rev. Lett. 90, 074302 (2003)]. We give the first mathemati
cally rigorous analysis of such a problem. Since our methods require an explicit understanding of the underlying (convex) relaxed problem, we focus on the wrinkling of an annular sheet loaded in the radial direction [Davidovitch et al., PNAS 108 (2011), no. 45]. Our main achievement is identification of the scaling law of the minimum energy as the thickness of the sheet tends to zero. This requires proving an upper bound and a lower bound that scale the same way. We prove both bounds first in a simplified Kirchhoff-Love setting and then in the nonlinear three-dimensional setting. To obtain the optimal upper bound, we need to adjust a naive construction (one family of wrinkles superimposed on a planar deformation) by introducing a cascade of wrinkles. The lower bound is more subtle, since it must be ansatz-free.
