Period fissioning and other instabilities of stressed elastic membranes

We study the shapes of elastic membranes under the simultaneous exertion of tensile and compressive forces when the translational symmetry along the tension direction is broken. We predict a multitude of novel morphological phases in various regimes of a 2-dimensional parameter space $(\epsilon,\nu)$ that defines the relevant mechanical and geometrical conditions. Theses parameters are, respectively, the ratio between compression and tension, and the wavelength contrast along the tension direction. In particular, our theory associates the repetitive increase of pattern periodicity, recently observed on wrinkled membranes floating on liquid and subject to capillary forces, to the morphology in the regime ($\epsilon \ll 1,\nu \gg 1$) where tension is dominant and the wavelength contrast is large.Comment: 4 pages, 4 figures. submitted to Phys. Rev. Let

Regimes of wrinkling in an indented floating elastic sheet

A thin, elastic sheet floating on the surface of a liquid bath wrinkles when poked at its centre. We study the onset of wrinkling as well as the evolution of the pattern as indentation progresses far beyond the wrinkling threshold. We use tension field theory to describe the macroscopic properties of the deformed film and show that the system passes through a host of different regimes, even while the deflections and strains remain small. We show that the effect of the finite size of the sheet ultimately plays a key role in determining the location of the wrinkle pattern, and obtain scaling relations that characterize the number of wrinkles at threshold and its variation as the indentation progresses. Some of our predictions are confirmed by recent experiments of Ripp \emph{et al.} [arxiv: 1804.02421].Comment: 22 pages, 11 figures, revised versio

Indentation metrology of clamped, ultra-thin elastic sheets

We study the indentation of ultrathin elastic sheets clamped to the edge of a circular hole. This classical setup has received considerable attention lately, being used by various experimental groups as a probe to measure the surface properties and stretching modulus of thin solid films. Despite the apparent simplicity of this method, the geometric nonlinearity inherent in the mechanical response of thin solid objects renders the analysis of the resulting data a nontrivial task. Importantly, the essence of this difficulty is in the geometric coupling between in-plane stress and out-of-plane deformations, and hence is present in the behaviour of Hookean solids even when the slope of the deformed membrane remains small. Here we take a systematic approach to address this problem, using the membrane limit of the F\"{o}ppl-von-K\'{a}rm\'{a}n equations. This approach highlights some of the dangers in the use of approximate formulae in the metrology of solid films, which can introduce large errors; we suggest how such errors may be avoided in performing experiments and analyzing the resulting data

Stress field around arbitrarily shaped cracks in two-dimensional elastic materials

The calculation of the stress field around an arbitrarily shaped crack in an infinite two-dimensional elastic medium is a mathematically daunting problem. With the exception of few exactly soluble crack shapes the available results are based on either perturbative approaches or on combinations of analytic and numerical techniques. We present here a general solution of this problem for any arbitrary crack. Along the way we develop a method to compute the conformal map from the exterior of a circle to the exterior of a line of arbitrary shape, offering it as a superior alternative to the classical Schwartz-Cristoffel transformation. Our calculation results in an accurate estimate of the full stress field and in particular of the stress intensity factors K_I and K_{II} and the T-stress which are essential in the theory of fracture.Comment: 7 pages, 4 figures, submitted for PR

Convergent Calculation of the Asymptotic Dimension of Diffusion Limited Aggregates: Scaling and Renormalization of Small Clusters

Diffusion Limited Aggregation (DLA) is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. We present a convergent calculation of the fractal dimension D of DLA based on a renormalization scheme for the first Laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster. The theory is applicable from very small (2-3 particles) to asymptotically large (n \to \infty) clusters. The computed dimension is D=1.713\pm 0.003

New Algorithm for Parallel Laplacian Growth by Iterated Conformal Maps

We report a new algorithm to generate Laplacian Growth Patterns using iterated conformal maps. The difficulty of growing a complete layer with local width proportional to the gradient of the Laplacian field is overcome. The resulting growth patterns are compared to those obtained by the best algorithms of direct numerical solutions. The fractal dimension of the patterns is discussed.Comment: Sumitted to Phys. Rev. Lett. Further details at http://www.pik-potsdam.de/~ander

Mechanics of large folds in thin interfacial films

A thin film at a liquid interface responds to uniaxial confinement by wrinkling and then by folding; its shape and energy have been computed exactly before self contact. Here, we address the mechanics of large folds, i.e. folds that absorb a length much larger than the wrinkle wavelength. With scaling arguments and numerical simulations, we show that the antisymmetric fold is energetically favorable and can absorb any excess length at zero pressure. Then, motivated by puzzles arising in the comparison of this simple model to experiments on lipid monolayers and capillary rafts, we discuss how to incorporate film weight, self-adhesion and energy dissipation.Comment: 5 pages, 3 figure

On the stabilization of ion sputtered surfaces

The classical theory of ion beam sputtering predicts the instability of a flat surface to uniform ion irradiation at any incidence angle. We relax the assumption of the classical theory that the average surface erosion rate is determined by a Gaussian response function representing the effect of the collision cascade and consider the surface dynamics for other physically-motivated response functions. We show that although instability of flat surfaces at any beam angle results from all Gaussian and a wide class of non-Gaussian erosive response functions, there exist classes of modifications to the response that can have a dramatic effect. In contrast to the classical theory, these types of response render the flat surface linearly stable, while imperceptibly modifying the predicted sputter yield vs. incidence angle. We discuss the possibility that such corrections underlie recent reports of a ``window of stability'' of ion-bombarded surfaces at a range of beam angles for certain ion and surface types, and describe some characteristic aspects of pattern evolution near the transition from unstable to stable dynamics. We point out that careful analysis of the transition regime may provide valuable tests for the consistency of any theory of pattern formation on ion sputtered surfaces

Reconstructing data : evidence-Based Medicine and Evidence-Based Public Health in context

The emergence of Evidence-Based Medicine (EBM) as the gold-standard practice in biomedicine and public health practices represents a significant epistemological turn in modern medicine. The development of Evidence-Based Public Health (EBPH) followed the emergence of Evidence-Based Medicine, as an attempt to ground health policies and interventions on «sound facts». The present paper analyzes the historical and sociological roots of this turn. We evaluate the ethical and social consequences of this transformation, both within the medical profession (the polarization between a medical elite which strengthened its professional status, and a rank and file which experienced a process of «de-professionalization») and in its relationship to the welfare state (the link between the medical elite, EBM, EBPH and the commodification of health care and public health)