Yibin Fu

A new identity for the surface-impedance matrix and its application to the determination of surface-wave speeds

It is well known in surface–wave theory that the secular equation for the surface–wave speedυcan be written as det M = 0 in terms of the surface–impedance matrix M. It is shown in this paper that M satisfies the simple identity (M-iR) T -1 (M+i R T )-Q+ υ 2 I=0 in the usual notation in the Stroh formalism. This identity provides an efficient method for calculating the wave speed of surface waves in unstressed or pre–stressed elastic half–spaces. The method is explained and illustrated by examples. It is also shown that the buckling/wrinkling pre–stress for a pre–stressed elastic half–space can be calculated using the same procedure but with pre–stress playing the role of υ.

Uniqueness of the Surface-Wave Speed: A Proof That Is Independent of the Stroh Formalism

It is well-known in surface-wave theory that the secular equation for the surface-wave speed v can be written as det M = 0 in terms of the surface impedance matrix M. It has recently been shown by the present authors that M satisfies a simple algebraic Riccati equation. It is shown in the present paper that a purely matrix algebraic analysis of this equation suffices to prove that whenever a surface wave exists it is unique.

Effects of imperfections on localized bulging in inflated membrane tubes.

The problem of localized bulging in inflated membrane tubes shares the same features with a variety of other localization problems such as formation of kink bands in fibre-reinforced composites and layered structures. This type of localization is known to be very sensitive to imperfections, but the precise nature of such sensitivity has not so far been quantified. In this paper, we study effects of localized wall thinning/thickening on the onset of localized bulging in inflated membrane tubes as a prototypical example. It is shown that localized wall thinning may reduce the critical pressure or circumferential stretch by an amount of the order of the square root of maximum wall thickness reduction. As a typical example, a 10 per cent maximum wall thinning may reduce the critical circumferential stretch by 19 per cent. This square root law complements the well-known Koiters two-thirds power law for subcritical periodic bifurcations. The relevance of our results to mathematical modelling of aneurysm formation in human arteries is also discussed.

Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness

Abstract For a spherical shell of arbitrary thickness which is subjected to an external hydrostatic pressure, symmetrical buckling takes place at a value of μ 1 which depends on A 1 A 2 and the mode number, where A 1 and A 2 are the undeformed inner and outer radii, and μ 1 is the ratio of the deformed inner radius to the undeformed inner radius. In the large mode number limit, we find that the dependence of μ 1 on A 1 A 2 has a boundary layer structure: it is a constant over almost the entire region of 0 1 A 2 and decreases sharply from this constant value to unity as A 1 A 2 tends to unity (the thin-shell limit). Simple asymptotic expressions for the bifurcation condition are obtained. The classical result for thin shells is recovered directly from the equations of finite elasticity, and an asymptotic critical neutral curve (which envelops the neutral curves corresponding to different mode numbers) is obtained.

A micromechanical model of woven fabric and its application to the analysis of buckling under uniaxial tension: Part 1: The micromechanical model

Out-of-plane buckling can be observed when a piece of woven fabric is subjected to a uniaxial tension along a direction which is not aligned with the warp or weft direction. Similar buckling does not occur either in the paper or thin fabric-reinforced composite plates. We attribute this difference to a unique feature of fabric, namely that the fabric has a micro-weaving structure and fibres in fabric can slide almost freely unlike those in paper or fabric-reinforced composites. In the first part of this paper, we propose a micromechanical model which takes into account this unique feature and modifies the traditional orthotropic continuum model by adding an extra compressive stress field. In the second part, a linear buckling analysis is conducted with the aid of both the traditional orthotropic continuum model and our micromechanical model. We show that the above-mentioned buckling phenomenon is predicted by our micromechanical model and not by the traditional model.

On the imperfection sensitivity of a coated elastic half-space

For the structure of a neo–Hookean surface layer bonded to a neo–Hookean half–space which is subjected to a uniaxial compression, a linear stability analysis shows that the bonded structure is less stable than the half–space if r < 1 and more stable if r > 1, where r is the ratio of the shear modulus of the half–space to that of the layer. When the layer is stiffer than the half–space (r < 1), there exists a critical buckling mode number corresponding to a minimum (critical) compression. In this paper we derive the evolution equation for a single near–critical mode. The coefficient of the cubic nonlinear term in the evolution equation determines whether the bonded structure is sensitive to imperfections and its dependence on r is calculated. It is found that the bonded structure is sensitive to imperfections if 0.575 < r < 1. Some asymptotic results valid in the thin–layer limit are derived and comparisons are made with the classical model for plates on elastic foundations. Participation of sideband modes in the buckling process makes it possible to have localized buckling solutions, but we show that localized buckling solutions are unstable to localized perturbations in the present context.

Characterization and stability of localized bulging/necking in inflated membrane tubes

We consider localised bulging/necking in an inflated hyperelastic membrane tube with closed ends. We first show that the initiation pressure for the onset of localised bulging is simply the limiting pressure in uniform inflation when the axial force is held fixed. We then demonstrate analytically how, as inflation continues, the initial bulge grows continually in diameter until it reaches a critical size and then propagates in both directions. The bulging solution before propagation starts is of the solitary-wave type, whereas the propagating bulging solution is of the kink-wave type. The stability, with respect to axially symmetric perturbations, of both the solitary-wave type and the kinkwave type solutions is studied by computing the Evans function using the compound matrix method. It is found that when the inflation is pressure-controlled, the Evans function has a single non-negative real root and this root tends to zero only when the initiation pressure or the propagation pressure is approached. Thus, the kink-wave type solution is probably stable but the solitary-wave type solution is definitely

The speed of interfacial waves polarized in a symmetry plane

The surface-impedance matrix method is used to study interfacial waves polarized in a plane of symmetry of anisotropic elastic materials. Although the corresponding Stroh polynomial is a quartic, it turns out to be analytically solvable in quite a simple manner. A specific application of the result concerns the calculation of the speed of a Stoneley wave, polarized in the common symmetry plane of two rigidly bonded anisotropic solids. The corresponding algorithm is robust, easy to implement, and gives directly the speed (when the wave exists) for any orientation of the interface plane, normal to the common symmetry plane. Through the examples of the couples (Aluminum)–(Tungsten) and (Carbon/epoxy)–(Douglas pine), some general features of a Stoneley wave speed are verified: the wave does not always exist; it is faster than the slowest Rayleigh wave associated with the separated half-spaces.

WKB Method with Repeated Roots and Its Application to the Buckling Analysis of an Everted Cylindrical Tube

For a linear ordinary differential equation of variable coefficients in which the highest order derivative

Hamiltonian interpretation of the Stroh formalism in anisotropic elasticity

d^ny/dx^n