David Bourne

Energy Bounds for a Compressed Elastic Film on a Substrate

We study pattern formation in a compressed elastic film which delaminates from a substrate. Our key tool is the determination of rigorous upper and lower bounds on the minimum value of a suitable energy functional. The energy consists of two parts, describing the two main physical effects. The first part represents the elastic energy of the film, which is approximated using the von Kármán plate theory. The second part represents the fracture or delamination energy, which is approximated using the Griffith model of fracture. A simpler model containing the first term alone was previously studied with similar methods by several authors, assuming that the delaminated region is fixed. We include the fracture term, transforming the elastic minimisation into a free boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination. After rescaling, the energy depends on only two parameters: the rescaled film thickness,

Centroidal power diagrams, Lloyd's algorithm and applications to optimal location problems

{\sigma }

Optimality of the triangular lattice for a particle system with Wasserstein interaction

σ, and a measure of the bonding strength between the film and substrate,

A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction. Part II: Analysis of Convergence

{\gamma }

Rotational symmetry vs. axisymmetry in shell theory

γ. We prove upper bounds on the minimum energy of the form