Growth of a flexible fibre in a deformable ring

We study the equilibrium configurations related to the growth of an elastic fibre in a confining flexible ring. This system represents a paradigm for a variety of biological, medical, and engineering problems. We consider a simplified geometry in which initially the container is a circular ring of radius R. Quasi-static growth is then studied by solving the equilibrium equations as the fibre length l increases, starting from l = 2R. Considering both the fibre and the ring as inextensible and unshearable, we find that beyond a critical length, which depends on the relative bending stiffness, the fibre buckles. Furthermore, as the fibre grows further it folds, distorting the ring until it induces a break in mirror symmetry at l > 2πR. We get that the equilibrium shapes depend only on two dimensionless parameters: the length ratio μ = l/R and the bending stiffnesses ratio κ. These findings are also supported by finite element simulation. Finally we experimentally validate the theoretical results showing a very good quantitative prediction of the observed buckling and folding regimes at variable geometrical parameters.

Packing of elastic rings with friction

We study the deformations of elastic filaments confined within slowly shrinking circular boundaries, under contact forces with friction. We perform computations with a spring-lattice model that deforms like a thin inextensible filament of uniform bending stiffness. Early in the deformation, two lobes of the filament make contact. If the friction coefficient is small enough, one lobe slides inside the other; otherwise, the lobes move together or one lobe bifurcates the other. There follows a sequence of deformations that is a mixture of spiralling and bifurcations, primarily the former with small friction and the latter with large friction. With zero friction, a simple model predicts that the maximum curvature and the total elastic energy scale as the wall radius to the − 3 / 2 and − 2 powers, respectively. With non-zero friction, the elastic energy follows a similar scaling but with a prefactor up to eight times larger, due to delayering and bending with a range of small curvatures. For friction coefficients as large as 1, the deformations are qualitatively similar with and without friction at the outer wall. Above 1, the wall friction case becomes dominated by buckling near the wall.

Delamination of open cylindrical shells from soft and adhesive Winkler's foundation

The interaction between thin elastic films and soft-adhesive foundations has recently gained interest due to technological applications that require control over such objects. Motivated by these applications we investigate the equilibrium configuration of an open cylindrical shell with natural curvature κ and bending modulus B that is adhered to soft and adhesive foundation with stiffness K. We derive an analytical model that predicts the delamination criterion, i.e., the critical natural curvature, κ_{cr}, at which delamination first occurs, and the ultimate shape of the shell. While in the case of a rigid foundation, K→∞, our model recovers the known two-states solution at which the shell either remains completely attached to the substrate or completely detaches from it, on a soft foundation our model predicts the emergence of a new branch of solutions. This branch corresponds to partially adhered shells, where the contact zone between the shell and the substrate is finite and scales as ℓ_{w}∼(B/K)^{1/4}. In addition, we find that the criterion for delamination depends on the total length of the shell along the curved direction, L. While relatively short shells, L∼ℓ_{w}, transform continuously between adhered and delaminated solutions, long shells, L≫ℓ_{w}, transform discontinuously. Notably, our work provides insights into the detachment phenomena of thin elastic sheets from soft and adhesive foundations.

Buckling morphology of an elastic ring confined in an annular channel

This paper studies the buckling morphology transition of an elastic ring confined in an annular channel. Under uniform axial strain, the ring would first form one inward blister and then transit to an "S" shape, but does not induce more blisters due to an energy barrier caused by the annular shape of the channel. In order to overcome the energy barrier, external perturbation is employed and a stable morphology with multiple blisters may be obtained. A theoretical framework is then established to calculate the bifurcation points of the shape transition, which agrees well with finite-element (FEM) simulation results. The diagrams of the stable buckling morphologies with respect to the geometrics of the elastic rings are presented, which may provide useful insights for practical applications, for example, the design of a peristaltic pump.

Delamination of a thin sheet from a soft adhesive Winkler substrate

A uniaxially compressed thin elastic sheet that is resting on a soft adhesive substrate can form a blister, which is a small delaminated region, if the adhesion energy is sufficiently weak. To analyze the equilibrium behavior of this system, we model the substrate as a Winkler or fluid foundation. We develop a complete set of equations for the profile of the sheet at different applied pressures. We show that at the edge of delamination, the height of the sheet is equal to sqrt[2]ℓ_{c}, where ℓ_{c} is the capillary length. We then derive an approximate solution to these equations and utilize them for two applications. First, we determine the phase diagram of the system by analyzing possible transitions from the flat and wrinkled to delaminated states of the sheet. Second, we show that our solution for a blister on a soft foundation converges to the known solution for a blister on a rigid substrate that assumed a discontinuous bending moment at the blister edges. This continuous convergence into a discontinuous state marks the formation of a boundary layer around the point of delamination. The width of this layer relative to the extent length of the blister, ℓ, scales as w/ℓ∼(ℓ_{c}/ℓ_{ec})^{1/2}, where ℓ_{ec} is the elastocapillary length scale. Notably, our findings can provide guidelines for utilizing compression to remove thin biofilms from surfaces and thereby prevent the fouling of the system.

A tale of two nested elastic rings

Elastic rods in contact provide a rich paradigm for understanding shape and deformation in interacting elastic bodies. Here, we consider the problem of determining the static solutions of two nested elastic rings in the plane. If the inner ring is longer than the outer ring, it will buckle creating a space between the two rings. This deformation can be further influenced by either adhesion between the rings or if pressure is applied externally or internally. We obtain an exact solution of this problem when both rings are assumed inextensible and unshearable. Through a variational formulation of the problem, we identify the boundary conditions at the contact point and use the Kirchhoff analogy to give exact solutions of the problems in terms of elliptic functions. The role of both adhesion and pressure is explored.