A laterally confined thin elastic sheet lying on a liquid substrate displays regular undulations, called wrinkles, characterized by a spatially extended energy distribution and a well-defined wavelength λ. As the confinement increases, the deformation energy is progressively localized into a single narrow fold. An exact solution for the deformation of an infinite sheet was previously found, indicating that wrinkles in an infinite sheet are unstable against localization for arbitrarily small confinement. We present an extension of the theory to sheets of finite length L, accounting for the experimentally observed wrinkle-to-fold transition. We derive an exact solution for the periodic deformation in the wrinkled state, and an approximate solution for the localized, folded state. We find that a second-order transition between these two states occurs at a critical confinement F = λ 2 /L.

The existing theory of incompatible elastic sheets uses the deviation of the surface metric from a reference metric to define the strain tensor [Efrati et al., J. Mech. Phys. Solids 57, 762 (2009)]. For a class of simple axisymmetric problems we examine an alternative formulation, defining the strain based on deviations of distances (rather than distances squared) from their rest values. While the two formulations converge in the limit of small slopes and in the limit of an incompressible sheet, for other cases they are found not to be equivalent. The alternative formulation offers several features which are absent in the existing theory. (a) In the case of planar deformations of flat incompatible sheets, it yields linear, exactly solvable, equations of equilibrium. (b) When reduced to uniaxial (one-dimensional) deformations, it coincides with the theory of extensible elastica; in particular, for a uniaxially bent sheet it yields an unstrained cylindrical configuration. (c) It gives a simple criterion determining whether an isometric immersion of an incompatible sheet is at mechanical equilibrium with respect to normal forces. For a reference metric of constant positive Gaussian curvature, a spherical cap is found to satisfy this criterion except in an arbitrarily narrow boundary layer.

Kirchhoff's kinetic analogy relates the deformation of an incompressible elastic rod to the classical dynamics of rigid body rotation. We extend the analogy to compressible filaments and find that the extension is similar to the introduction of relativistic effects into the dynamical system. The extended analogy reveals a surprising symmetry in the deformations of compressible elastica. In addition, we use known results for the buckling of compressible elastica to derive the explicit solution for the motion of a relativistic nonlinear pendulum. We discuss cases where the extended Kirchhoff analogy may be useful for the study of other soft matter systems.Analogies to dynamical problems have been used to simplify the physics of various condensed-matter systems, ranging from the deformation of elastic bodies to the order-parameter profile across an interface between coexisting phases. A particularly well known example is Kirchhoff's kinetic analogy [1]. In this theory the threedimensional (3D) deformation of a slender elastic rod is reduced to the bending deformation of an incompressible curve, representing the mid-axis of the rod. This problem, in turn, is analogous to the dynamics of a rigid body rotating about a fixed point, where the distance along the curve and its local curvature are analogous, respectively, to time and angular velocity. When the filament is confined to a two-dimensional (2D) plane (the celebrated Euler elastica [2]), the equation of equilibrium coincides with the equation of motion of a physical pendulum [1,19].In the examples above the elastic system was reduced to an indefinitely thin, incompressible body, whose equilibrium shape follows the trajectory of a classical dynamical system. In the present work we show that relaxing the incompressibility constraint introduces terms akin to relativistic corrections to classical dynamics. Within this analogy, the compression modulus, Y , plays the role of the relativistic particle's rest mass, and the bendability parameter (Y /B) 1/2 ≡ h −1 , where B is the bending modulus, is analogous to the speed of light. The limit of an incompressible rod (h → 0) corresponds to the nonrelativistic limit.Despite the relevance to real systems, including compressible fluid membranes [4], there have not been many studies of compressible elastica (see [5] and references therein). Following these works, we consider the 2D deformation of a compressible filament, represented by a planar curve of relaxed length L. The same model applies to thin elastic sheets, as well as fluid membranes [6], provided that they are deformed along a single direction. The deformation away from the flat, stress-free * ozzoshri@tau.ac.il

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A uniaxially compressed thin elastic sheet that is resting on a soft adhesive substrate can form a blister, which is a small delaminated region, if the adhesion energy is sufficiently weak. To analyze the equilibrium behavior of this system, we model the substrate as a Winkler or fluid foundation. We develop a complete set of equations for the profile of the sheet at different applied pressures. We show that at the edge of delamination, the height of the sheet is equal to sqrt[2]ℓ_{c}, where ℓ_{c} is the capillary length. We then derive an approximate solution to these equations and utilize them for two applications. First, we determine the phase diagram of the system by analyzing possible transitions from the flat and wrinkled to delaminated states of the sheet. Second, we show that our solution for a blister on a soft foundation converges to the known solution for a blister on a rigid substrate that assumed a discontinuous bending moment at the blister edges. This continuous convergence into a discontinuous state marks the formation of a boundary layer around the point of delamination. The width of this layer relative to the extent length of the blister, ℓ, scales as w/ℓ∼(ℓ_{c}/ℓ_{ec})^{1/2}, where ℓ_{ec} is the elastocapillary length scale. Notably, our findings can provide guidelines for utilizing compression to remove thin biofilms from surfaces and thereby prevent the fouling of the system.