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Benjamin Davidovitch
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, University of Massachusetts

Thin solid sheets exhibit a rich variety of patterns under generic confinements and featureless distribution of forces. This morphological diversity reflects the coupling between geometry and mechanics in elastic sheets, a problem whose foundations had been set many years ago. This field has seen recently a surge of research activity, driven by studies that demonstrated its relevance for morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, and by developing material applications at ever decreasing scales that enable the mechanical control of tiny structures. A major theoretical challenge posed by this progress is the development of a formalism that explains the basic mechanisms for pattern formation in thin sheets under various loadings and confinements. The hurdle here is associated with the highly nonlinear nature and the geometric complexity of the problem when the sheet thickness becomes exceedingly small. This complexity requires the development of new concepts and methods, beyond traditionally-used ones.

During 2010-2011, the ACS-PRF grant supported theoretical progress in this front by Davidovitch and his group at UMass Amherst and collaborators in Santiago, Paris and Oxford. The supported research contributed to our basic understanding of wrinkling patterns and their coexistence with other types of deformations in thin sheets. The ACS-PRF grant provided support for three supported projects:

**A prototypical model for tensional wrinkling:**

Wrinkling is typically defined as a periodically buckled pattern formed upon uniaxial confinement. The behavior of wrinkling in one-dimensional (1D) set-ups, is simple and quite well understood. However, in 2D set-ups, where the stress field is nonuniform, the evolution of wrinkling patterns away from threshold is highly nontrivial. A ”simplest yet nontrivial” model for 2D wrinkles, often called the “Lame’ problem”, was addressed in a recent theoretical work (Davidovitch et al, PNAS 2011). In this set-up, an annular sheet is subjected to planar, axisymmetric tensile forces that may give azimuthal (hoop) compression, and to the consequent formation of radial wrinkles. This theory identified two generic (dimensionless) parameters, termed *“bendability”* and *“confinement”*, which govern the transition of wrinkling patterns from near-threshold (NT) to far-from-threshold (FFT) conditions. Importantly, it was shown that the wrinkling patterns of ultra-thin sheets are typically in the high-bendability, far-from-threshold regime, characterized by an asymptotically vanishing compressive stress and hence significant deviation of the stress field from its pre-buckling value. Notably, since FFT wrinkles reflect a strong variation of the pre-buckling stress distribution, they are characterized by an extent and wavelength that are markedly different from NT wrinkles. Building on previous contributions, this recent work provided a quantitative framework for understanding the behavior of wrinkling patterns upon variation of the control parameters of the system.

**Sheets on curved substrate:**

A familiar everyday experience is the frustrating attempt, doomed to fail, to (physically) place or (mathematically) map a sheet onto a curved substrate. This could is easily demonstrated when placing a sticker on a curved bumper – where the sticker develops blisters and folds, or by mapping earth on a 2D piece of paper – where some latitudes and meridians must be distorted.

A physical realization of this basic phenomenon is a recent experiment, by Menon’s group at UMass Amherst, where a circular ultrathin sheet is paced in a liquid drop. Upon a gradual increase of the drop’s curvature (by increasing the Laplace’s pressure inside the drop), the sheet deforms in a fascinating way: first forming radial wrinkles near the perimeter, which are then replaced by sharp folds (crumples). This beautiful experiment allows us to infer the basic mechanisms by which an elastic sheet accommodates a uniform (Gaussian) curvature that is imposed on it. In a recent work (King, Schroll, Davidovitch, and Menon, submitted 2011), we showed how the concepts of *“morphologically-relevant”* parameters (a generalized version of the aforementioned bendability and confinement parameters) allows us to understand the morphological transitions experienced by the sheet as distinct instabilities that spontaneously break the axial symmetry of this system. In a work in progress (Buchak and Davidovitch, in preparation) we develop an effective model that will provide us quantitative tools to analyze these transitions and compare to other known mechanical and thermodynamic phase transitions.

**Elastic building blocks for thin sheets:**

Upon confinement, sheets become sometimes sharply crumpled, where the elastic stress is focused in narrow regions (called “vertices” and “ridges”), and at other times deform smoothly, where the stress is “diffuse” throughout the sheet. A natural and important question is what determines whether a sheet responds to confining forces by focusing or diffusing the elastic stress. In a recent paper (Schroll, Katifori, and Davidovitch, PRL 2011) we addressed this question by constructing the “simplest curtain” model: a variant of Euler buckling of a semi-infinite sheet, on which a “3-buckle” shape is imposed at the free edge. This boundary-forcing provides the simplest, weakest way to “inject” so-called Gaussian curvature into a buckled sheet, which forbids the formation a stress-free developable shape. While certain boundary conditions led to the formation of purely-diffuse or purely-focused shapes, we found that under generic conditions, the sheet splits into distinct, well separated domains of “diffuse-stress” that coexist with stress-focusing domains.

This interesting observation motivate an idea to develop a *“building blocks” formalism*, for an efficient computational analysis of the highly-convoluted shapes of e.g. crushed metal foils. In this formalism, deformed sheets are described through coexisting wrinkling-type domains (which can be described through a set of nonlinear ordinary differential equations), and stress-focusing domains (vertices, ridges). A central challenge for further development of this idea into an efficient computational formalism is the derivation of effective (and generic) “stitching conditions” between the two domain types.