Gustavo Gioia

Delamination of Compressed Thin Films

In this article, we specifically concern ourselves with the buckling-driven delamination mechanism, whereby a portion of the film buckles away from the substrate, thereby forming a blister (also termed buckle or wrinkle). Blisters may grow by interfacial fracture, a process which, under the appropriate conditions, may result in the catastrophic failure of the component. Blisters are often observed to adopt convoluted-even bizarre shapes and to fold into intricate patterns. A principal objective of this article is to review some recent developments based on the use of direct methods of the calculus of variations which have proven useful for understanding the mechanics of folding of thin films (Ortiz and Gioia, 1994). These developments are reviewed in Section III, which is extracted from the original publication. The remaining sections are devoted to the application of these principles to the problem of predicting the shape of thin-film blisters.

Micromagnetics of very thin films

We determine by a scaling calculation the limiting form of the free energy governing a ferromagnetic film of vanishing thickness. Our theory generalizes Stoner and Wohlfarths results for flat ellipsoids to arbitrary–shaped very thin films.

The morphology and folding patterns of buckling-driven thin-film blisters

Abstract T hin films and coatings in a state of residual compression can, under appropriate conditions, decohere and buckle away from the substrate to form blisters. These blisters are often observed to adopt intricate shapes and to fold into complex patterns. In this paper, such shapes and patterns are given an energetic interpretation, i.e. they follow as energy minimizers. We formulate the energy of the film by recourse to von Karman theory of moderate deflections of a plate. The energy functional has the following key properties : it contains two terms, namely, the membrane and bending energies, the latter being a singular perturbation of the former; and the membrane energy functional is nonconvex and, consequently, its infimum is generally not attained. In keeping with the conventional mathematical treatment of these problems, we construct solutions by a matched asymptotic expansion. The outer solution follows by membrane energy minimization and determines the essential folding pattern of the film. The inner solution is obtained by fitting boundary layers at sharp edges in the membrane solution. The film deflections thus constructed are found to match, in surprising detail, the observed complex folding patterns adopted by delaminated films. In addition, the boundary layer analysis permits one to accord a well-defined line tension to sharp edges in the membrane solution, and, in particular, to the boundary of the blister. This provides a simple device for assessing the configurational stability of some blister morphologies. In particular, the analysis predicts the transition from straight-sided to telephone-cord morphologies at a critical mismatch strain.

Turbulent Friction in Rough Pipes and the Energy Spectrum of the Phenomenological Theory

The classical experiments on turbulent friction in rough pipes were performed by Nikuradse in the 1930s. Seventy years later, they continue to defy theory. Here we model Nikuradses experiments using the phenomenological theory of Kolmogórov, a theory that is widely thought to be applicable only to highly idealized flows. Our results include both the empirical scalings of Blasius and Strickler and are otherwise in minute qualitative agreement with the experiments; they suggest that the phenomenological theory may be relevant to other flows of practical interest; and they unveil the existence of close ties between two milestones of experimental and theoretical turbulence.

Unified model of tectonics and heat transport in a frigid Enceladus.

Recent data from the Cassini spacecraft have revealed that Enceladus, the 500-km-diameter moon of Saturn, has a southern hemisphere with a distinct arrangement of tectonic features, intense heat flux, and geyser-like plumes. How did the tectonic features form? How is the heat transported from depth? To address these questions, we formulate a simple model that couples the mechanics and thermodynamics of Enceladus and gives a unified explanation of the salient tectonic features, the plumes, and the transport of heat from a source at a depth of tens of kilometers to the surface. Our findings imply that tiny, icy moons can develop complex surficial geomorphologies, high heat fluxes, and geyser-like activity even if they do not have hot, liquid, and/or convecting interiors.

The energetics of heterogeneous deformation in open-cell solid foams

Compressed open–cell solid foams frequently exhibit spatially heterogeneous distributions of local stretch. The theoretical aspects of this deformation habit have not been clearly elucidated. Here we propose a simple nonlinear model aimed at illustrating the most salient features of the micromechanics of uniaxially stretched solid foams. Then we study the energetics of the model to show that the stretch heterogeneity observed in experiments stems from the lack of convexity of the governing energy functional, which favours two characteristic values of local stretch. These characteristic values are independent of the applied overall stretch and define two configurational phases of the foam. The predicted stretch distributions correspond to stratified mixtures of the phases; stretching occurs in the form of a phase transformation, by growth of one of the phases at the expense of the other. We also compare the predicted mechanical response with experimental data for a series of foams of different densities and discuss the analogy between the stretching of foams and the liquefaction of van der Waals gases. Lastly, we perform displacement field measurements using the digital image correlation technique and find the results to be in agreement with our predictions.

Folding Energetics in Thin-Film Diaphragms

We perform experiments to elucidate how the folding patterns of thin-film diaphragms subject to in-plane isotropic and anisotropic compressive strains depend on the shape, thickness and size of the diaphragms. We then use a constrained von Kaármaán model to relate the experimental results to the energetics of folding. We show that the differences between the isotropic and the anisotropic cases can be traced back to the structure of the membraneous energy density function. In the isotropic case, we find foldings which satisfy the boundary conditions and minimize the membraneous energy. In the anisotropic case, no such foldings exist, but we are able to construct sequences of increasingly fine foldings which satisfy the boundary conditions and whose membraneous energies converge to the infimum. In both cases, we obtain solutions by allowing bending to select a preferred folding. The solutions compare well with the experimental observations.

Moon patterns, sun patterns, and wave breaking in rotating granular mixtures

Granular materials, such as powders and sand, tend to segregate due to differences in particle properties. When a cylindrical drum is partially filled with particles of different sizes and rotated about its axis, this leads to radial segregation patterns in which the smaller particles concentrate in a radial core near the axis, and the larger particles near the outside walls of the drum. Under certain conditions, undulations in the radial core of smaller particles grow into radial stripes that extend toward the outer walls of the drum in a manner somewhat reminiscent of viscous fingering. The patterns are strongly dependent on the fill level and rotation speed of the drum. These observations can be explained by two spatially disjoint mechanisms: (1) a wave-breaking mechanism that promotes the growth of the stripes and (2) a filtering mechanism that limits the growth of stripes.

Determination of thin-film debonding parameters from telephone-cord measurements

Methods are put forward for the determination of the fracture energy and kinetic coefficient of thin film/substrate interfaces from measurements performed on telephone cord blisters. This work is based on a previously proposed model for the characterization of debonding features in compressed thin films. The advantages of the proposed methods stem from the following facts concerning telephone cord blisters: (i) they are spontaneous debonding features, and therefore more amenable than artificially contrived features to yield realistic debonding parameters; (ii) they are very commonly observed in compressed thin films; (iii) as revealed by our model, their boundaries are characterized by a constant fracture mode mixity, and (iv) the driving force for debonding equals the fracture energy everywhere on their boundaries.

Two-phase densification of cohesive granular aggregates.

When poured into a container, cohesive granular materials form low-density, open granular aggregates. If pressure is applied to these aggregates, they densify by particle rearrangement. Here we introduce experimental and computational results suggesting that densification by particle rearrangement occurs in the form of a phase transition between two configurational phases of the aggregate. Then we show that the energy landscape associated with particle rearrangement is nonconvex and therefore consistent with our interpretation of the experimental and computational results. Our conclusions are relevant to many technological processes and natural phenomena.