The Relaxed Energy Density for Isotropic Elastic Membranes

Capillary Deformations of Bendable Films

“…Here, the deformability of the foundation enables the sheet to respond to the exerted capillary force as a thin elastic body -by bending [6] and developing in-plane stresses [7]. Furthermore, in ultrathin sheets where t ∼ 10 nm, in-plane compression relaxes completely through wrinkles whose number diverges as t decreases [8][9][10][11][12]. The "drop-on-afloating-sheet" experiment ( Fig.…”

“…Using the compression-free limit, known as the "relaxed energy" [10] or "tension field theory" [11], we show: (i) The contact angle between the liquid-vapor and liquid-solid interfaces (ϑ + φ in Fig. 1c) deviates from ϑ Y by an amount ∆ϑ…”

“…In the high bendability regime, 1, where only a tiny level of compression can be accommodated, the unstable axisymmetric stress (which is the basis for standard "near threshold" (NT) and post-buckling theories) must be replaced by a compression-free stress field that satisfies: σ θθ ≥ 0 as → 0 (see Fig. 3) [9,10]. For a given α > α c (τ > τ c ), the compression-free stress is the basis for a far from threshold (FT) theory of the wrinkled state [15].…”

The calculus of variations and materials science

“…For special microstructures, including those that are only possible when special relations between lattice parameters hold, see Bhattacharya [30], [31], Ruddock [100]. For interesting explicit examples of quasiconvexifications see Kohn [74], Pipkin [96], [97].…”

Wrinkles in Square Membranes

“…The wrinkling of membranes has attracted much interest in the past, starting from the development of the tension field theory [1]. Simpler formulations and extensions of this theory were later proposed [2][3][4][5][6][7]. All of these formulations, with accompanying numerical solutions [8,9], model the membrane as a no-compression, two-dimensional continuum with negligible bending stiffness.…”