## Asymptotic structures & edge-wrinkling of thin elastic plates

#### Dr Ciprian Coman (National Physical Laboratory)

Conference Room B, Schuster Building (2nd floor),

Edge-wrinkling represents a particular type of bifurcation instability encountered in thin elastic plates and shells subjected to non-compressive loads. Unlike traditional buckling which is the result of compressive external loads, the distribution of membrane stresses prior to the onset of edge-wrinkling has a mixed character in which one of the principal stresses becomes partly compressive in some region of the plate/shell. Typically, such regions tend to be close to a boundary or an edge, hence leading to a local deformation pattern concentrated near those areas for sufficiently strong external loads.

In this lecture I shall discuss the asymptotic solution of a long-standing open problem involving a weakly clamped circular elastic plate
subjected to uniform transverse pressure. It was shown by Bromberg (1956) that the nonlinear axisymmetric solution describing the deformation experienced by this configuration does not represent an energy-minimum configuration for sufficiently strong transverse pressures. An earlier attempt to describe edge-wrinkling numerically within a similar context was made by Panov & Feodosiev (1948), whose Galerkin-type solution was later found to be fairly inaccurate because of the ad-hoc assumptions made for the radial deflection of the plate (i.e., the mode shape). In recent times, Adams
(1993) made another attempt at describing numerically edge-wrinkling in a similar configuration by replacing the uniform pressure with a concentrated load at the centre of the plate; unfortunately, his radial eigenmodes contain a spurious oscillatory component that has no physical basis.

The bifurcation equations for the aforementioned scenarios will be presented in two alternative formulations, which will be subsequently used to obtain
full numerical solutions that illustrate the behaviour of the edge-wrinkling pattern. In the case of a very thin plate (which can be also regarded as an
elastic thin-film) a number of asymptotic parameters present in this problem make it possible to obtain a fairly detailed picture of the structure of the critical
wrinkling pressure on various other parameters that feature in the bifurcation equations. The role of bending stiffness in the pre-bifurcation state and a number of related aspects will be covered as well, if time permits.

This is joint work with Andrew P. Bassom (Tasmania) and Dr. Miccal Matthews (Western Australia).