Indentation of a floating elastic sheet: geometry versus applied tension

Indentation of an elastic disk on a circular supporting ring

Thin elastic two-dimensional systems under compressive stresses may relieve part of their stretching energy by developing out-of-plane undulations. We investigate experimentally and theoretically the indentation of an elastic disk supported by a circular ring and show that compressive stresses are relieved via two different routes: either developing buckles which are spread over the system or developing a d-cone where deformation is concentrated in a subregion of the system. We characterize the indentation threshold for wrinkles or d-cone existence as a function of aspect ratio.

Biomechanical fracture mechanics of composite layered skin-like materials

Most protective biological tissues are structurally comprised of a stiff and thin outer layer on top of a soft underlying substrate. Examples include mammalian skin, fish scales, crustacean shells, and nut and seed shells. While these composite skin-like tissues are ubiquitous in nature, their mechanics of failure and what potential mechanical advantages their composite structures offer remains unclear. In this work, changes in the puncture mechanics of composite hyperelastic elastomers with differing non-dimensional layer thicknesses are explored. Puncture behavior of these membranes is measured for dull and sharp conical indenters. Membranes with a stiff outer layer of only 1% of the overall composite thickness exhibit a puncture energy comparable to membranes with a stiff outer layer approximately 20 times thicker. This puncture energy, scaled by its flexural capacity, achieves a local maximum when the top layer is approximately 1% of the total membrane, similar to the structure of numerous mammalian species. The mode of failure for these regimes is also investigated. In contrast with puncture directly beneath sharp tips caused by high stress concentrations, a new type of 'coring' type fracture emerges at large indentation depths, resulting from accumulated tensile strain energy along the sides of the divot as the membrane is deformed with a blunt indenter. These results could enhance the durability and robustness of stretchable materials used for products such as surgical gloves, packaging, and flexible electronics.

Gigantic floating leaves occupy a large surface area at an economical material cost

The giant Amazonian waterlily (genus Victoria) produces the largest floating leaves in the plant kingdom. The leaves' notable vasculature has inspired artists, engineers, and architects for centuries. Despite the aesthetic appeal and scale of this botanical enigma, little is known about the mechanics of these extraordinary leaves. For example, how do these leaves achieve gigantic proportions? We show that the geometric form of the leaf is structurally more efficient than those of other smaller species of waterlily. In particular, the spatially varying thickness and regular branching of the primary veins ensures the structural integrity necessary for extensive coverage of the water surface, enabling optimal light capture despite a relatively low leaf biomass. Leaf gigantism in waterlilies may have been driven by selection pressures favoring a large surface area at an economical material cost, for outcompeting other plants in fast-drying ephemeral pools.

Validity of Winkler's mattress model for thin elastomeric layers: beyond Poisson's ratio

Winkler's mattress model is often used as a simplified model to understand how a thin elastic layer, such as a coating, deforms when subject to a distributed normal load: the deformation of the layer is assumed proportional to the applied normal load. This simplicity means that the Winkler model has found a wide range of applications from soft matter to geophysics. However, in the limit of an incompressible elastic layer the model predicts infinite resistance to deformation, and hence breaks down. Since many of the thin layers used in applications are elastomeric, and hence close to incompressible, we consider the question of when the Winkler model is appropriate for such layers. We formally derive a model that interpolates between the Winkler and incompressible limits for thin elastic layers, and illustrate this model by detailed consideration of two example problems: the point-indentation of a coated elastomeric layer and self-sustained lift in soft elastohydrodynamic lubrication. We find that the applicability (or otherwise) of the Winkler model is not determined by the value of the Poisson ratio alone, but by a compressibility parameter that combines the Poisson ratio with a measure of the layer's slenderness, which itself depends on the problem under consideration.

Response of a raft of particles to a local indentation

Interfaces that are coated with a layer of adsorbed particles (particle "rafts") are common in natural and industrial settings. Particle-coated interfaces may be useful in part because the particulate structure can endow the fluid interface with physical properties distinct from molecular surfactants. We study the mechanics of particulate assemblies by measuring the raft's response to indentation in the vertical direction by a flat, circular disc. We measured force (f) vs. indentation depth (δ) and found two linear regions with different slopes. The first linear region started at δ = 0 and persisted over a range of δ much less than the capillary length. In the second linear region, the raft had the same stiffness (df/dδ) as a liquid interface with no particles. Further, we show that, as long as the indenter was larger than a single particle, the azimuthal compression imposed by the interface deformation relaxed through in-plane rearrangement of particles rather than by the radial wrinkles that are characteristic of thin elastic sheets at fluid interfaces. We show how the force-displacement curves and stiffnesses depended on fluid mass densities, interfacial tensions, and indenter radius. For all cases studied, the particle-raft coated interfaces had a stiffness equal to or smaller than that of a bare fluid interface. Although the interfacial particle raft behaved like a pure fluid interface under a wide range of displacements, we show that the raft could nonetheless withstand substantially greater applied force (up to 2×) and greater indentation depth (up to 2.6×), so that the range of reversible behavior was greatly extended. These results improve our understanding of the mechanics of particulate assemblies at interfaces.

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 center. 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.

Static bistability of spherical caps

Depending on its geometry, a spherical shell may exist in one of two stable states without the application of any external force: there are two 'self-equilibrated' states, one natural and the other inside out (or 'everted'). Though this is familiar from everyday life-an umbrella is remarkably stable, yet a contact lens can be easily turned inside out-the precise shell geometries for which bistability is possible are not known. Here, we use experiments and finite-element simulations to determine the threshold between bistability and monostability for shells of different solid angle. We compare these results with the prediction from shallow shell theory, showing that, when appropriately modified, this offers a very good account of bistability even for relatively deep shells. We then investigate the robustness of this bistability against pointwise indentation. We find that indentation provides a continuous route for transition between the two states for shells whose geometry makes them close to the threshold. However, for thinner shells, indentation leads to asymmetrical buckling before snap-through, while also making these shells more 'robust' to snap-through. Our work sheds new light on the robustness of the 'mirror buckling' symmetry of spherical shell caps.

The mechanical equilibrium of soft solids with surface elasticity

Recent experiments have shown that surface stresses in soft materials can have a significant strain-dependence. Here we explore the implications of this surface elasticity to show how, and when, we expect it to arise. We develop the appropriate boundary condition, showing that it simplifies significantly in certain cases. We show that surface elasticity's main role is to stiffen a solid surface's response to in-plane tractions, in particular at length-scales smaller than a characteristic elastocapillary length. We also investigate how surface elasticity affects the Green's-function problem of a line force on a flat, incompressible, linear-elastic substrate. There are significant changes to this solution, especially in that the well-known displacement singularity is regularised. This raises interesting implications for soft phenomena like wetting contact lines, adhesion and friction. Finally, we discuss open questions, future directions, and close ties with existing fields of research.