Session Z28: Focus Session: Wrinkling

Wrinkling patterns of thin sheets glued to a negative curvature surface

Gauss's Theorema Egregium provides an intimate connection between the metric and the Gaussian curvature of a surface. If a thin sheet is adhered to a substrate with a negative Gaussian curvature it will experience stress due to the curvature-driven change of its metric. In the inextensible limit any changes of metric are not possible and the sheet will relieve the stress by locally deforming via wrinkles or folds. Using geometric arguments and numerical simulations of a non-linear elastic model we analyse the wrinkling pattern as a function of the shape of the adhering substrate.   

Drag Control through Wrinkling on Curved Surfaces

We present the results of an experimental investigation on the wrinkling of positively curved surfaces and explore their use towards drag reduction applications. In our precision model experiments we make use of rapid prototyping techniques to cast samples with custom geometry and material properties out of silicone-based rubbers. Our structures consist of a thin stiff shell that is chemically bonded to a thicker soft substrate. The substrate contains a spherical cavity that can be depressurized, under controlled volume conditions, to compress the ensemble structure. Under this compressive loading, the initially smooth outer-shell develops complex wrinkling patterns. We systematically characterize and quantify the morphology of the various patterns and study the phase diagram of the system. We consider both geometric and material quantities in the parameter space. Moreover, since the wrinkling patterns can be actuated dynamically using a pressure signal, we systematically characterize the aerodynamic behavior of our structures in the context of fluid drag reduction. An added advantage of our novel mechanism is that it allows for both dynamic switching and tuning of the surface morphology, thereby opening paths for drag control.   

Folds and crease from wrinkles

We present stability and post bifurcation analyses of free-surface deformation from wrinkles to folds and creases, caused by lateral compression of a neo-Hookean material with varying elastic modulus with depth from the free surface. The post-bifurcation behavior of the wrinkle mode is investigated by high order perturbation as well as finite element analyses. We show that there is a critical strain beyond which the initial wrinkle mode is unstable. Using the finite element software ABAQUS, we reveal other deformation mode that may emerge due to the nonlinear bifurcation of the material surface. Bifurcation chart is constructed and shows that localized modes such as crease and fold may emerge depending on the geometric and material properties.   

Mechanics of Graded Wrinkles

Shark skin is known for its anti-fouling and self-cleaning properties. In attempts to mimic this pattern for getting similar properties, different surface patterns such as Sharklet and wrinkles have been previously introduced. Wrinkled patterns have gained importance in applications such as microfluidics, wetting and adhesion. Through buckling of a thin film of stiff material on a substrate of softer material, and maintaining symmetric geometries, ordered wrinkled patterns can be created. However, it can be shown that using the same principle, by changing the geometry of the surface, the dimensions of the wrinkles can be altered. This alteration turns ordered wrinkles into graded wrinkles which have more resemblance to shark skin than the ordered wrinkles, maintaining the same wave length while each wave having different amplitude. Here using finite element models, experiments and analytical solutions, the relations between different geometries and the resulting patterns were investigated.   

Wrinkling in Cellular Structured Composites

Many structured composites found in nature possess undulating and wrinkled interfacial layers that regulate mechanical, chemical, acoustic, adhesive, thermal, electrical and optical functions of the material. This research focused on the formation of wrinkling patterns in cellular structured composites and the effect of the wrinkling pattern on the overall structural response. The cellular composites consisted of stiffer interfacial layers constructing a network submerged in a soft matrix. Analytical and finite element models were developed to capture various aspects of the wrinkling mechanism. The characteristics of the undulation patterns and the instability modes were investigated as functions of model geometry and material composition. Mechanical experiments were designed to further explore the modeling results. The cellular composite samples were fabricated by using different types of elastomers and by varying the geometry and the material properties. The experimental and numerical results were consistent with the analytical predictions. The results in this research improve understanding of the mechanisms governing the undulation pattern formation in cellular composites and can be used to enable on-demand tunability of different functions to provide, among others, active control of wave propagation, mechanical stiffness and deformation, and material swelling and growth.   

Numerical Simulation of the Combined Bending, Stretching, and Wrinkling of Thin Sheets

A two-dimensional theory of plates and shells derived from three-dimensional finite elasticity is presented. The approach is based on a systematic small thickness expansion of the exact three-dimensional strain energy density of the plate or shell. The theory involves the small thickness explicitly and accounts for both bending and stretching in a unified framework. Thus, wrinkling instabilities in thin sheets are accommodated as a natural outgrowth of the model. The plate model is demonstrated numerically via a specially designed finite difference code utilizing the method of dynamic relaxation. The code is used to simulate several equilibrium deformations of thin sheets and plates undergoing finite deformation with wrinkling.   

Understanding and Controlling Morphological Transitions of Wrinkles

The ability to generate micron and sub-micron structures across extensive lengths on soft materials surfaces is critical for numerous technologies, yet current fabrication methods do not provide cost-effective solutions for these diverging demands. In Nature, elastic instabilities often are used to produce materials structures on small scales from simple building blocks to achieve necessary performance on larger, macroscopic size scales. We present an overview of our efforts to understand and use elastic instabilities, such as wrinkling and folding, to define surface structures with advantageous properties. In particular, we address two questions related to morphological transitions: the roles of overstress and curvature on selecting the specific wrinkle morphology created under equibiaxial stress conditions; and non-linear transitions, including wrinkle-to-fold, and the suppression of such transitions to achieve high-aspect ratio wrinkle structures. The lessons described provide new insight into the physics of these complex material deformations while also introducing scalable methods that are expected to help transfer elastic instabilities into current technologies.   

Transition from wrinkles to crumples in an elastic sheet

A circular sheet confined to a surface of increasing curvature initially breaks azimuthal symmetry creating a finite pattern of radial wrinkles along its perimeter. At larger curvature, sharp crumpled features emerge and dominate the shape. Using optical profilometry, we study the transition from wrinkling to crumpling of a polystyrene sheet floating on a drop of glycerol by measuring the spatial distribution of curvatures of the sheet as a function of drop curvature. We observe that collisions of neighboring wrinkles at their tips generate cusps. These cusps subsequently sharpen and merge to produce large crumpled features, around which gaussian curvature focuses. Surprisingly, the stress field in the central, unwrinkled portion is not sensitive to the appearance of crumpled features. The transition shows little hysteresis and is smooth with respect to measured quantities.   

Wrapping a sphere: stress relaxation by wrinkling

The low energy deformations of thin elastic sheets are isometries because these incur no stretching energy while the cost of bending is small. Since there is no isometric map of a flat sheet, i.e. a developable surface, onto the surface of a sphere, it is natural to suspect that any such map must cost finite stretching energy. However, I will show that there are an enormous number of almost isometric mappings which approximate a sphere with arbitrary accuracy and with arbitrarily small stretching energy. I will construct an example using multiscale analysis of a radial wrinkle pattern in a thin elastic sheet bent over a sphere. These techniques could be applied to other wrinkling problems and to problems connected to developable surfaces, e.g. textures in smectic liquid crystals.   

Wrinkling of Inhomogeneously Strained Thin Polymer Films

Wrinkles occur due to a mechanical instability when sufficient strain is applied to an incompressible thin film attached to a deformable substrate. For wrinkles made with a polymer film supported on a soft elastomer, the amplitude is directly proportional to the wavelength and the square root of the applied strain. This dependence has been confirmed with ideal substrates where the global strain is homogeneously distributed, but the influence of strain inhomogeneity has not been considered previously. We use the contact line wrinkling technique to prepare polystyrene thin films with periodic regions of different wrinkle amplitudes, hence strains, on soft substrates. The surfaces with inhomogeneous wrinkle amplitudes and directions approach a homogeneous structure upon the application of sufficiently large strains. The surface becomes homogeneous at a relatively small strain due to the growth rate difference between pre-wrinkles and new wrinkles. Moreover, we find the pre-wrinkled region starts strain localizing prior to the initially flat region. We derive relationships to describe these processes, providing fundamental knowledge of the wrinkling mechanism.   

Hydrostatic and Flow Measurements on Wrinkled Membrane Walls

In this study, we investigate structural properties of wrinkled silicon nitride (SiN) membranes, under both hydrostatic perturbations and flow conditions, through surface profile measurements. Rectangular SiN membranes with linear dimensions of $15$~mm $\times~ 1.5$~mm $\times~ 1~\mu$m are fabricated on a 500$-\mu$m-thick silicon substrate using standard lithography techniques. These thin, initially flat, tension-dominated membranes are wrinkled by bending the silicon substrate. The wrinkled membranes are subsequently incorporated as walls into rectangular micro-channels, which allow both hydrostatic and flow measurements. The structural response of the wrinkles to hydrostatic pressure provides a measure of the various energy scales in the problem. Flow experiments show that the elastic properties and the structural undulations on a compliant membrane completely dominate the flow, possibly providing drag reduction. These measurements pave the way for building and using compliant walls for drag reduction in micro-channels.