Divergence instabilities of nonuniformly prestressed travelling webs

The phenomenon of edge-buckling in an axially moving stretched thin elastic web is described as a nonstandard singularly perturbed bifurcation problem, which is then explored through the application of matched asymptotic techniques. Previous numerical work recently reported in the literature is re-evaluated in this context by approaching it through the lens of asymptotic simplifications. This allows us to identify two distinct regimes characterised by qualitative differences in the corresponding eigen-deformations; some simple approximate formulae for the critical eigenvalues are also proposed. The obtained analytical results capture the intricate relationship between the critical speeds, the background tension, and other relevant physical and geometric parameters that feature in the mathematical model

Wrinkling structures at the rim of an initially stratched circular thin plate subjected to transverse pressure

Short-wavelength wrinkles that appear on an initially stretched thin elastic plate under transverse loading are examined. As the degree of loading is increased so wrinkles appear and their structure at the onset of buckling takes on one of three distinct forms depending on the size of the imposed stretching. With relatively little stretching, the wrinkles sit off the rim of the plate at a location which is not known a priori, but which is determined via a set of consistency conditions. These take the form of constraints on the solutions of certain coupled nonlinear differential equations that are solved numerically. As the degree of stretching grows, so an asymptotic solution of the consistency conditions is possible which heralds the structure that governs a second regime. Now the wrinkle sits next to the rim where its detailed structure can be described by the solution of suitably scaled Airy equations. In each of these first two regimes the Föppl–von Kármán bifurcation equations remain coupled, but as the initial stretching becomes yet stronger the governing equations separate. Further use of singular-perturbation arguments allows us to identify the wavelength wrinkle which is likely to be preferred in practice

On the asymptotic reduction of a bifurcation equation for edge-buckling instabilities

Weakly clamped uniformly stretched thin elastic plates can experience edge buckling when subjected to a transverse pressure. This situation is revisited here for a circular plate, under the assumption of finite rotations and negligible bending stiffness in the pre-buckling range. The eigenproblem describing this instability is formulated in terms of two singularly perturbed fourth-order differential equations involving the non-dimensional bending stiffness ε>0. By using an extension of the asymptotic reduction technique proposed by Coman and Haughton (Acta Mech 55:179–200, 2006), these equations are formally reduced to a simple second-order ordinary differential equation in the limit ε→0+. It is further shown that the predictions of this reduced problem are in excellent agreement with the direct numerical simulations of the original bifurcation equations

Eigen-transitions in cantilever cylindrical shells subjected to vertical edge loads

A thin cantilever cylindrical shell subjected to a transverse shear force at the free end can experience two distinct modes of buckling, depending on its relative thickness and length. If the former parameter is fixed then a short cylinder buckles in a diffuse manner, while the eigenmodal deformation of a moderately long shell is localised, both axially and circumferentially, near its fixed end. Donnelltype buckling equations for cylindrical shells are here coupled with a non-symmetric membrane basic state to produce a linear boundary-value problem that is shown to capture the transition between the aforementioned instability modes. The main interest lies in exploring the approximate asymptotic separation of the independent variables in the corresponding stability equations, when the eigen-deformation is doubly localised. Comparisons with direct numerical simulations of the full buckling problem provide further insight into the accuracy and limitations of our approximations