The Quarterly Journal of Mechanics and Applied Mathematics - recent issues

Surface tension effects on interaction between two fluids near a wall

Vanden-Broeck, J.-M., Smith, F. T. — 2008-04-24

Interaction between two fluids near a fixed solid surface is modelled, with surface tension acting as an important influence on the assumed planar motion. The two fluids are immiscible, incompressible and have small density and viscosity ratios; the heavier more viscous body of fluid is approaching the solid surface and the other fluid is lying as a thin layer in between. In the so-called supercritical range where, for both fluids, inviscid forces dominate over viscous ones, a pair of pressure–shape relations is found which leads to a nonlinear integro-differential equation for the unknown interface shape. Analysis, computation and comparisons are applied to the equation. Travelling-state solutions are found of periodic and non-periodic form, including interesting cases which exhibit parabolic growth of the layer thickness in the far field.

Nonlinear transversely isotropic elastic solids: an alternative representation

Shariff, M. H. B. M. — 2008-04-24

A strain energy function which depends on five independent variables that have immediate physical interpretation is proposed for finite strain deformations of transversely isotropic elastic solids. Three of the five variables (invariants) are the principal stretch ratios and the other two are squares of the dot product between the preferred direction and two principal directions of the right stretch tensor. The set of these five invariants is a minimal integrity basis. A strain energy function, expressed in terms of these invariants, has a symmetry property similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. The formulation is applied to several types of deformations, and in these applications, a mathematical simplicity is highlighted. The proposed model is attractive if principal axes techniques are used in solving boundary-value problems. Experimental advantage is demonstrated by showing that a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed. A specific form of strain energy function can be easily obtained from the general form via a triaxial test. Using series expansions and symmetry, the proposed general strain energy function is refined to some particular forms. Since the principal stretches are the invariants of the strain energy function, the Valanis–Landel form can be easily incorporated into the constitutive equation. The sensitivity of response functions to Cauchy stress data is discussed for both isotropic and transversely isotropic materials. Explicit expressions for the weighted Cauchy response functions are easily obtained since the response function basis is almost mutually orthogonal.

Dynamics of a bridged crack in a discrete lattice

Mishuris, G. S., Movchan, A. B., Slepyan, L. I. — 2008-04-24

The paper addresses a problem of partial fracture of a lattice by a propagating fault modelling a crack bridged by elastic fibres. It is assumed that the strength of bonds within the lattice alternates periodically, so that during the dynamic crack propagation only weaker bonds break, whereas the stronger bonds remain intact. The mathematical problem is reduced to the functional equation of the Wiener–Hopf type, which is solved analytically. The load–crack speed dependence is presented, which also has implications on the stability analysis for the bridged crack propagating within the lattice. In particular, we address the evaluation of the dissipation rate, which is found to be strongly dependent on the crack speed. In this lattice model, our results also cover the case of the supercritical crack speed.

Exact solutions for the evolution of ellipsoidal inclusions in porous media

Buchak, P., Crowdy, D. — 2008-04-24

This paper derives exact mathematical solutions for the time-dependent evolution of a single ellipsoidal inclusion in a porous medium when a linear straining flow is active in the far field. This represents a two-phase free boundary problem. It is shown that the dynamics is such that an initially ellipsoidal inclusion remains ellipsoidal under evolution. The theory of ellipsoidal harmonics is used to determine the system of ordinary differential equations governing the geometrical parameters of the ellipsoidal inclusion.

Fourth-order cartesian tensors: old and new facts, notions and applications

Moakher, M. — 2008-04-24

Fourth-order tensors can be represented in many different ways. For instance, they can be represented as multilinear maps or multilinear forms. It is also possible to describe a fourth-order tensor in a given vector space by a second-order tensor but in another vector space with higher dimension. Such a representation makes the manipulation of fourth-order tensors similar to that of the more familiar second-order tensors. In this paper, we use these three descriptions to discuss the different symmetries of fourth-order tensors, to present the algebra of the space of fourth-order symmetric tensors and to describe different metrics on this space. Isotropic tensors and orientation tensors are presented using these different representations of fourth-order tensors. Applications to elasticity and high angular resolution diffusion imaging are discussed. Finally, we present a systematic and consistent approach to finding the tensor of an even order that best fits in some sense a tensor of higher even order.

Free convective boundary-layer flow in a heat-generating porous medium: similarity solutions

Merkin, J. H. — 2008-04-24

The free convection boundary-layer flow on a vertical surface in a porous medium with local heat generation proportional to (T – T)^p, where T is the local temperature and T is the ambient temperature, is considered when there are power-law variations in either the wall temperature or the wall heat flux which enables the equations to be reduced to similarity form. When the wall temperature is prescribed, solutions are found for p ≤ 2 and p ≥ p_c (p_c = 10.673) with a saddle-node bifurcation at p = p_c and two solution branches for p > p_c. When the wall heat flux is prescribed, solutions are found only for p < 2. The special case p = 2 is considered and the limiting forms as p -> 2 and p -> are obtained and compared with the solutions obtained from solving the similarity equations numerically

Electromagnetic fields in the presence of an infinite dielectric wedge: the phased line source excitation case

Salem, M. A., Kamel, A. H. — 2008-04-24

Electromagnetic fields, excited by an electric phased line source in the presence of an infinite dielectric wedge, are determined by application of the Kontorovich–Lebedev transform. The Maxwell's equations together with the conditions of continuity of the tangential field components at the material interfaces are formulated as a vector boundary-value problem. By representing the field components as Kontorovich–Lebedev integrals, the problem is reduced to a system of singular integral equations for the unknown spectral functions. We construct numerical solutions to those equations that permit fields evaluation for values of the wedge refractive index, not necessarily close to unity, and for arbitrary positioned source and observer. Numerical results showing the influence of a wedge presence on the directivity of a phased line source are presented and verified through finite-difference frequency-domain simulations.

Matrix Wiener-Hopf approximation for a partially clamped plate

Abrahams, I. D., Davis, A. M. J., Llewellyn Smith, S. G. — 2008-04-24

This article examines the classic problem of deflection of a thin elastic plate subjected to static or dynamic normal loading. The plate is infinite in extent in one coordinate direction and finite in the other. On one infinite edge, the plate is clamped, and on the other the plate has mixed boundary conditions, clamped on a semi-infinite part of the edge and free on the remaining half. The boundary-value problem is reduced to a Wiener–Hopf equation, but it is of matrix form belonging to a class for which no exact solution technique is known. An explicit approximate solution, in general accurate to any specified degree, is obtained by a recent method which employs Padé approximants. Numerical results are presented for the plate deflection, and these exhibit convergence to the exact solution as the order of the approximant is increased.

On the buckling of elastic plates

Balmforth, N. J., Craster, R. V., Slim, A. C. — 2008-04-24

The Föppl-von Kármán equations are used to explore the onset of linear instability and the subsequent nonlinear development of buckling patterns in a flat elastic plate due to an imposed shear or body force such as gravity. Experimental results are also presented for a clamped and sheared sheet of Neoprene rubber and these compare favourably with theory.

Buckling of an axisymmetric vesicle under compression: the effects of resistance to shear

Preston, S. P., Jensen, O. E., Richardson, G. — 2008-01-17

We consider the axisymmetric deformation of an initially spherical, porous vesicle with incompressible membrane having finite resistance to in-plane shearing, as the vesicle is compressed between parallel plates. We adopt a thin-shell balance-of-forces formulation in which the mechanical properties of the membrane are described by a single dimensionless parameter, C, which is the ratio of the membrane's resistance to shearing to its resistance to bending. This results in a novel free-boundary problem which we solve numerically to obtain vesicle shapes as a function of plate separation, h. For small deformations, the vesicle contacts each plate over a small circular area. At a critical value of plate separation, h_TC, there is a transcritical bifurcation from which a new branch of solutions emerges, representing buckled vesicles which contact each plate along a circular curve. For the values of C investigated, we find that the transcritical bifurcation is subcritical and that there is a further saddle-node bifurcation (fold) along the branch of buckled solutions at h = h_SN (where h_SN > h_TC). The resulting bifurcation structure is commensurate with a hysteresis loop in which a sudden transition from an unbuckled solution to a buckled one occurs as h is decreased through h_TC and a further sudden transition, this time from a buckled solution to an unbuckled one, occurs as h is increased through h_SN. We find that h_SN and h_TC increase with C, that is, vesicles that resist shear are more prone to buckling.

A thin rivulet of perfectly wetting fluid subject to a longitudinal surface shear stress

Sullivan, J. M., Wilson, S. K., Duffy, B. R. — 2008-01-17

The lubrication approximation is used to obtain a complete description of the steady unidirectional flow of a thin rivulet of perfectly wetting fluid on an inclined substrate subject to a prescribed uniform longitudinal surface shear stress. The quasi-steady stability of such a rivulet is analysed, and the conditions under which it is energetically favourable for such a rivulet to split into one or more subrivulets are determined.

Eshelby's conjecture in finite plane elastostatics

Kim, C. I., Vasudevan, M., Schiavone, P. — 2008-01-17

We consider an inhomogeneity–matrix system from a particular class of compressible hyper- elastic materials of harmonic type undergoing finiteplane deformations. We discuss the analogue of Eshelby's conjecture for this class of materials. Specifically, we examine whether the stress distribution inside an inhomogeneity is uniform if and only if the inhomogeneity is elliptic when the system is subjected to uniform remote stress.

Linear waves on Roseau's two-parameter beach profile: a generalized near-shore wave amplification formula

Ehrenmark, U. — 2008-01-17

Roseau's exact solution for infinitesimal waves on a two-parameter (, a) convex beach profile is here revisited using an alternative solution approach. Unlike Roseau's original development, the revised method appears to yield a description of the solution which (in terms of integral structure hierarchy) is similar to that of the classical plane beach problem. Thus, computation can be undertaken at the same level of simplicity pointing the way for the model (and its three-parameter infinite channel extension) to become more widely used, for example, in validation and calibration studies of the various mild-slope equations that are currently the topic of much study. It is shown that the plane beach represents the limiting case as the parameter a-> and the solution developed is amenable to exact examination of the shoreline wave amplification factor. A formula is established which is shown to reduce to the classical result developed for plane beaches and indicates that for each value of (the beach angle at shore) the amplification has a peak for a certain finite value of far-field depth (a) and that this peak is about 10% greater than that predicted in a plane beach theory with a similar slope. It is shown also that the restriction noted by Roseau on the acute nature of the angle can be relaxed in the present solution to include the case of an overhanging cliff. Results for this case indicate some similarity to those for a plane beach but only for sufficiently large values of a.

Embedding formulae for diffraction by non-parallel slits

Skelton, E. A., Craster, R. V., Shanin, A. V. — 2008-01-17

Embedding formulae for diffraction theory encode the diffraction coefficients for some given wave incidence on a scatterer in terms of the directivity from a single or reduced number of scattering problems. If one deduces the relation between these directivities, then the resulting formulae enable rapid computations and allow one to concentrate computational resources accordingly. Unfortunately, the range of applicability of embedding formulae is currently rather restricted. In this article, we demonstrate how embedding is applied to plane-wave scattering by non-parallel strips or slits. Primarily, we concentrate upon the problem of a line crack, or strip, inclined to a flat infinite surface and we derive and implement the embedding formula. Various other generalizations are possible given these formulae and we outline them.

Resonances of an elastic plate in a compressible confined fluid

Bonnet-Ben Dhia, A.-S., Mercier, J.-F. — 2007-11-13

We present a theoretical study of the resonances of a fluid–structure problem, an elastic plate placed in a duct in the presence of a compressible fluid. The case of a rigid plate has been largely studied. Acoustic resonances are then associated to resonant modes trapped by the plate. Due to the elasticity of the plate, we need to solve a quadratic eigenvalue problem in which the resonance frequencies k solve the equations (k) = k², where are the eigenvalues of a self-adjoint operator of the form A + kB. First, we show how to study the eigenvalues located below the essential spectrum by using the min–max principle. Then, we study the fixed-point equations. We establish sufficient conditions on the characteristics of the plate and of the fluid to ensure the existence of resonances. Such conditions are validated numerically.

On magnetoelastic problems of a plane with an arbitrarily shaped hole under stress and displacement boundary conditions

Hasebe, N., Wang, X. F., Nakanishi, H. — 2007-11-13

An analytical method for the static plane problem of magnetoelasticity is developed for an infinite plane containing a hole of arbitrary shape under stress and displacement boundary conditions in a primary uniform magnetic field. The magnetic field influences the elastic field by introducing a body force called the Lorentz ponderomotive force in the equilibrium equations. The body force can be further described in a form relating with the electromagnetic stress tensor. The complex variable method in conjunction with the rational mapping function technique is used in the analysis for both magnetic field and mechanical field. Governing equations and boundary conditions are expressed in terms of complex functions. Complex magnetic potential and stress functions are obtained using Cauchy integrals for the paramagnetic and soft ferromagnetic materials, respectively. The distributions of magnetic field and the stress components are shown for certain directions of primary magnetic fields in an infinite plane with a square hole, as an example. It is found that the stress distributions for the two types of materials are identical despite the difference of magnetic fields. The extreme cases of a free and a fixed hole reduced to a crack and a rigid fibre, respectively, are also investigated. The stress intensity factors at the tips of crack and rigid fibre are computed, and their variation for certain directions of primary magnetic field is shown.

On the coupled theory of thermo-magnetoelectroelasticity

Aouadi, M. — 2007-11-13

Within the context of the coupled theory of thermo-magnetoelectroelasticity, we derive some variational principles which fully characterize the solution of the boundary-initial-value problem. Then we establish a reciprocity relation using a new method of proof, which involves two thermoelastic processes at different instants. We show that this relation can be used to obtain reciprocity and uniqueness theorems. The reciprocity theorem avoids both the use of the Laplace transform and the incorporation of initial conditions into the equations of motion. The uniqueness theorem is derived without making restrictions on the positive definiteness of the elastic moduli or the conductivity tensor. There are also no restrictions on piezoelectric moduli, piezomagnetic moduli and thermal coupling coefficients other than symmetry conditions. The results obtained are applicable for some special cases which can be deduced from our model.

Two-parameter asymptotic approximations in the analysis of a thin solid fixed on a small part of its boundary

Zalipaev, V. V., Movchan, A. B., Jones, I. S. — 2007-11-13

Planar elasticity problems are considered for thin domains fixed along a small part of the end region boundary. The analysis involves two small parameters: the normalized thickness of the body and the normalized length of the fixed part of the boundary. The aim of the paper is to derive an asymptotic approximation of the solution to a boundary-value problem in such a domain and, in particular, analyze the ‘effective boundary conditions’, which occur for the leading-order terms of the asymptotics. We include applications for problems of both anti-plane shear and plane strain elasticity.

Modeling weakly nonlinear acoustic wave propagation

Christov, I., Christov, C. I., Jordan, P. M. — 2007-11-13

Three weakly nonlinear models of lossless, compressible fluid flow—a straightforward weakly nonlinear equation (WNE), the inviscid Kuznetsov equation (IKE) and the Lighthill–Westervelt equation (LWE)—are derived from first principles and their relationship to each other is established. Through a numerical study of the blow-up of acceleration waves, the weakly nonlinear equations are compared to the ‘exact’ Euler equations, and the ranges of applicability of the approximate models are assessed. By reformulating these equations as hyperbolic systems of conservation laws, we are able to employ a Godunov-type finite-difference scheme to obtain numerical solutions of the approximate models for times beyond the instant of blow-up (that is, shock formation), for both density and velocity boundary conditions. Our study reveals that the straightforward WNE gives the best results, followed by the IKE, with the LWE's performance being the poorest overall.

Two-dimensional liquid column and liquid droplet impact on a solid wedge

Wu, G. X. — 2007-11-13

The hydrodynamic impact due to a two-dimensional (2D) liquid column or a 2D liquid droplet hitting on a solid wedge is analysed. The problem is solved using the complex velocity potential together with the boundary element method. A stretched coordinate system is used, which is defined through the ratio of the normal Cartesian coordinate system to an appropriately chosen time-varying length scale. Numerical simulations are first made for impact by a liquid wedge. The results from the time-domain method are found to be in a good agreement with the similarity solution. Simulations are also made for impact by an elliptic droplet. A condition for bisection of the droplet is introduced, which is found to provide stable and converged results.