Surface creasing of soft elastic continua as a Kosterlitz-Thouless transition

Abstract

Harnessing a model from composite materials science, we show how point-like cusped surface features arise as quasi-particle excitations, termed "ghost fibers", on the surface of a homogeneous soft elastic material. These deformations appear above a critical compressive strain at which ghost fiber dipoles unbind, analogous to vortices in the Kosterlitz-Thouless transition. Finite-length creases can be described in the same framework. Our predictions for crease surface profiles and onset strain agree with previous experiments and simulations, and further experimental tests are proposed.PACS numbers: 46.35.+z, 62.20.mq, 68.35.Rh, 82.60.Nh, 87.15.Zg Cusped inward folds known as creases form on compressed surfaces of a variety of soft elastic materials [1], including natural rubber [2,3], polymer gels [4-6], silicone elastomers [7][8][9][10][11][12], starchy foods [13,14], and the developing mammalian brain [11,12,15,16]. In the latter context, creases are called "sulci". Unlike the long-wavelength buckling of a compressed beam, or the smooth sinusoidal wrinkles observed on the skin of drying fruit or a tensioned elastic sheet [17][18][19], creases are sharply localized in both their elastic deformation and stresses, thereby defying a linear perturbation analysis [3,7,13,20,21]. Owing to this difficulty, numerical minimization of a nonlinear neo-Hookean energy functional has become the standard theoretical tool for investigating the onset of creases [7, 11-13, 15, 20-22]. A central claim in much of this work is that creasing is a fundamentally new, nonlinear instability with no scale [20,21]. Experimental work has also studied the growth of pre-existing long creases, describing these behaviors in analogy to crack propagation [8,9].Here we develop a new quasi-particle framework for shear stress focusing in surface-compressed solids, assuming planar geometry and neglecting surface tension. We apply our theory to the creasing instability, obtaining a markedly different picture than [20,21]. We find evidence that (i) creasing onset maps to the Kosterlitz-Thouless (KT) transition [23], (ii) nonlinear elasticity is needed only within a small region analogous to a vortex core, and (iii) compression-induced shear strain fluctuations set the fundamental, microscopic lengthscale in the problem. Our theory makes contact with experimental and simulation results on critical strain, surface profiles, and crease patterns. In particular, we obtain a universal critical compressive plane strain above which creases emerge, in reasonable agreement with the measured value of 35% [5,13,21]. Finally, the theory points to a set of minimal physical ingredients for creasing, and suggests a possible unification with ridging (formation of localized surface protrusions) [24], and dimple crystallization [25,26].Our point of departure from prior work is to consider a distinct regime of zero-length creases, qualitatively similar to those observed in [6][7][8], immediately upon nucleation, and those in [15], as the critical point is approached f...

Superdomain dynamics in ferroelectric-ferroelastic films: Switching, jamming, and relaxation

“…[47] Very recently it has been pointed out that surface creases in elastic films are analogous to Kosterlitz-Thouless instabilities. [48] Other authors have shown [49,50] that Kosterlitz-Thouless melting within n-vertex Potts model descriptions requires vertices with n>4 and does not arise from the threefold and fourfold vertices considered by Srolovitz and Scott [51] and characteristic of BaTiO3, PbTiO3, and PZT films considered here. We should note that our analysis is supported by a recent work by Sigov who suggested that defects in ferroelectrics can be correlated and comply with the KosterlitzThouless model.…”