The Quarterly Journal of Mechanics and Applied Mathematics - recent issues

Resonances of an elastic plate coupled with a compressible confined flow

Bonnet-Ben Dhia, A.-S., Mercier, J.-F. — 2009-04-13

A theoretical study of the resonances of an elastic plate in a compressible flow in a two-dimensional duct is presented. Due to the fluid–structure coupling, a quadratic eigenvalue problem is involved, in which the resonance frequencies k solve the equations (k) = k², where is the eigenvalue of a self-adjoint operator of the form A + kB. In a previous paper, we have proved that a linear eigenvalue problem is recovered if the plate is rigid or the fluid at rest. We focus here on the general problem for which elasticity and flow are jointly present and derive a lower bound for the number of resonances. The expression of this bound, based on the solution of two linear eigenvalue problems, points out that the coupling between elasticity and flow generally reduces the number of resonances. This estimate is validated numerically.

Nonaxisymmetric stokes flow between concentric cones

Hall, O., Hills, C. P., Gilbert, A. D. — 2009-04-13

We study the fully three-dimensional Stokes flow within a geometry consisting of two infinite cones with coincident apices. The Stokes approximation is valid near the apex and we consider the dominant flow features as it is approached. The cones are assumed to be stationary and the flow to be driven by an arbitrary far-field disturbance. We express the flow quantities in terms of eigenfunction expansions and allow for the first time for nonaxisymmetric flow regimes through an azimuthal wave number. The eigenvalue problem is solved numerically for successive wave numbers. Both real and complex sequences of eigenvalues are found, their relative dominance affecting the flow features observed. The implications for the presence of eddy-like structures (analogous to those found in other corner geometries) are discussed and we find that these flow features depend not only upon the internal angles of the two cones but also upon the symmetry of the driving mechanism. For an arbitrary disturbance, the dominant flow mode is not axisymmetric but rather is associated with wave number one and, by breaking axisymmetry, eddies can be avoided in this geometry.

On obtaining effective orthotropic elasticity tensors

Kochetov, M., Slawinski, M. A. — 2009-04-13

We consider the problem of obtaining the effective orthotropic tensor that corresponds to a given generally anisotropic one; herein, by ‘effective’, we mean the closest in the sense of the Frobenius norm, without a priori assuming the orientation of the orthotropic tensor. It is difficult to find the absolute minimum of the distance function since the minimization process is nonlinear, exhibiting several local minima. To find the effective orthotropic tensor, the minimization process must be performed on a three-dimensional manifold SO(3). In the case of monoclinic and transversely isotropic tensors, it can be performed on a two-dimensional sphere, which lends itself to an insightful plot that allows us to guide a numerical method. We use the orientation of the symmetry-plane normal of the effective monoclinic tensor to guide the method and obtain the effective orthotropic tensor—a two-step process.

Circular map for supercavitating flow in a multiply connected domain

Antipov, Y. A., Silvestrov, V. V. — 2009-04-13

A nonlinear free boundary-value problem of supercavitating flow past n + 1 hydrofoils is analyzed. To describe the cavities’ closure mechanism, the Tulin–Terent'ev single-spiral-vortex model is employed. The flow domain is considered as the image of an (n + 1)-connected circular domain. The conformal map is constructed in terms of the solutions to two Riemann–Hilbert problems of the theory of symmetric automorphic functions. One of the problems is homogeneous and its coefficients are continuous functions while the second problem is inhomogeneous and has discontinuous coefficients. The exact solutions to the problems are found by using quasiautomorphic and quasimultiplicative analogs of the Cauchy kernel. The case of a single plate is considered in detail and the numerical results are reported.

Interaction of elastic plate weakened by multiple holes with a set of patches

Zemlyanova, A. Y., Silvestrov, V. V. — 2009-04-13

The problem of interaction of a thin elastic infinite plate weakened by several holes with a system of superimposed patches is considered. Each patch covers completely one or several holes or is located in the exterior of all holes. Some holes may not be covered by patches. The patches are attached to the plate continuously along their boundaries. The patches and the holes can be of arbitrary shape and size. The plate is loaded with given in-plane stresses at infinity and along the boundaries of the holes. All parts of the construction are assumed to be in a generalised plane stressed state. Using special integral representations of complex potentials, the problem is reduced to a system of singular integral equations. Numerical examples on reinforcement of a plate with two or three holes covered by one or two patches are given.

Asymptotic model of fields in a thin-walled structure with crack-like defects

Zalipaev, V. V., Movchan, A. B., Jones, I. S. — 2009-01-20

The transition from two-dimensional (2D) wave propagation through the square periodic structure in anti-plane shear time-harmonic case to a discretised model of a 2D lattice with masses connected by springs is considered. A model of a defect in the middle part of the thin-walled bridges is presented. As a first part of the asymptotic model, the effective transmission condition in the vicinity of the transverse cut of the thin-walled bridges is discussed. Then, a boundary layer determining the asymptotic expansion of the field near the tip of the crack is constructed. Stress intensity factors are evaluated for deep cracks in the junction regions. The corresponding boundary layer analysis is non-trivial and has not been attempted elsewhere.

Rayleigh waves having generalised lateral dependence

Parker, D. F., Kiselev, A. P. — 2009-01-20

It is shown that the surface-guided elastic waves found by Kiselev for isotropic materials and having displacements depending linearly upon the Cartesian coordinate orthogonal to the sagittal plane may be generalised in many ways. For surface waves on any anisotropic half-space, a simple procedure applied to the displacements within the standard surface wave having dependence eⁱ, where k · x – t and k is the (surface) wave vector, yields displacements depending linearly upon the surface cartesian coordinate orthogonal to the group velocity vector. Moreover, repeated application of this (differentiation) procedure yields a hierarchy of waves having algebraic dependence of successively increasing degree. For isotropic materials, substantial simplification and generalization are possible. Solutions of all algebraic degrees have identical depth dependence. This allows the solutions to be constructed iteratively and motivates a search for general solutions having depth dependence of the normal displacement the same as in the standard surface wave. The procedure gives a new derivation of the solutions found by Achenbach having amplitude of the normal displacement of the surface given by any solution to the two-dimensional Helmholtz equation. Furthermore, exploiting the scale invariance (a property of surface waves on any homogeneous half-space) shows that in every surface-guided disturbance of an elastic half-space, the elevation of the free surface is a solution of the wave equation in two dimensions (the membrane equation). Using the paraxial approximation to the membrane equation, high-frequency Rayleigh waves propagating as narrow beams are described in terms of a scalar Gaussian beam.

Linear invariants of a Cartesian tensor

Ahmad, F., Rashid, M. A. — 2009-01-20

The number of linear invariants under SO(3) as well as SO(2) of a Cartesian tensor of an arbitrary rank is studied. A linear form is defined in terms of elements of a tensor. It is established that the number of linear invariants of a tensor of rank n under SO(3) equals the dimension of the space of isotropic tensors of rank n. Formulas for the number of invariants in the two cases are also derived. For the elasticity tensor, our analysis confirms the results of Norris.

Elastic SH wave propagation in a layered anisotropic plate with interface damage modelled by spring boundary conditions

Bostrom, A., Golub, M. — 2009-01-20

Elastic SH wave propagation in a layered anisotropic plate with interface damage is modelled in several steps. A single interface crack between two half-spaces is first studied and an explicit solution for the crack-opening displacement is obtained at low frequencies. This is then generalised to a random distribution of cracks at the interface and the result is reformulated as a spring boundary condition. As an example of its usefulness, this boundary condition is then used in the derivation of a plate equation by expanding the displacements in power series in the thickness coordinate.

Oscillatory flow about a cylinder pair

Coenen, W., Riley, N. — 2009-01-20

Oscillatory flow about a pair of circular cylinders is considered. The distance between the cylinders can be varied as can the angle that the undisturbed oscillatory flow makes with the line joining the cylinder centres. In common with other fluid flows dominated by oscillatory flow, a time-independent, or steady streaming, motion develops. Attention is focused on the case of high streaming Reynolds numbers and the resulting jets that erupt from the surfaces of the cylinders.

Inverse membrane problems in elasticity

Pathmanathan, P., Chapman, S. J., Gavaghan, D. J. — 2009-01-20

The inverse elasticity problem of determining the undeformed, deflated, configuration of a nonlinear elastic membrane, given the deformed configuration enclosing an incompressible fluid under known pressure, is considered. It is shown that, in practical cases, it is enough to determine only the undeformed metric tensor, and it is also shown how the two- and three-dimensional cases are fundamentally different. For the three-dimensional case, we set up and classify the partial differential equations to be solved, prove existence of an undeformed state given an undeformed metric and study the axisymmetric case in detail.

On a class of buckling problems in a singularly perturbed domain

Coman, C. D., Bassom, A. P. — 2009-01-20

We consider the buckling of an annular thin elastic plate when it is subjected to uniform in-plane compressive forces on its outer boundary. This geometrical inhomogeneity means that the pre-buckling stress field is nonconstant and, as a consequence, the resulting variable-coefficient eigenproblem is not solvable in closed form. In the limit when the annulus can be regarded as a disk with a small neighbourhood of its centre removed, singular perturbation techniques are used to construct asymptotic approximations for the critical buckling loads. Our results describe both symmetric and asymmetric buckling patterns and show good agreement with some numerical simulations.

Pure currents in foliated waveguides

Zolla, F., Bouchitte, G., Guenneau, S. — 2008-10-23

We give an explicit characterization of obliquely propagating electric modes within a multilayered planar dielectric waveguide of cross-section [0, 1]. This foliated strip, with fixed metallic boundaries at 0 and 1, consists of an integer number N of periodic cells of thickness = 1/N. We are interested in the high-frequency asymptotic analysis of its spectrum. For this, we rescale the wavelength as and derive, using a transfer matrix formalism, that for vanishing the limit spectrum consists of two parts. The first part is a Bloch (or band) spectrum associated with a family of operators acting on Floquet–Bloch eigenfunctions defined on the real line. The second part is a boundary layer spectrum associated with an operator acting on square integrable eigensolutions defined on the positive real line [0, + ) and satisfying a Neumann condition at 0. This latter part is further characterized via a spectral problem on [0, 1] which is supplied with Neumann conditions at both ends. Eventually, we illustrate our discussion by numerical results derived from this auxiliary spectral problem. Moreover, after a suitable rescaling of the field, we prove that the total electromagnetic energy is entirely located on either boundary of the structure. Finally, in the case of transverse propagation, we apply our results to antiplane shear waves propagating within a foliated acoustic waveguide, whose freely vibrating walls at 0 and 1 are shown to support surface waves.

Axisymmetric buckling of a spherical shell embedded in an elastic medium under uniaxial stress at infinity

Jones, G. W., Chapman, S. J., Allwright, D. J. — 2008-10-23

The problem of a thin spherical linearly elastic shell perfectly bonded to an infinite linearly elastic medium is considered. A constant axisymmetric stress field is applied at infinity in the matrix, and the displacement and stress fields in the shell and matrix are evaluated by means of harmonic potential functions. In order to examine the stability of this solution, the buckling problem of a shell which experiences this deformation is considered. Using Koiter's nonlinear shallow shell theory, restricting buckling patterns to those which are axisymmetric and using the Rayleigh–Ritz method by expanding the buckling patterns in an infinite series of Legendre functions, an eigenvalue problem for the coefficients in the infinite series is determined. This system is truncated and solved numerically in order to analyse the behaviour of the shell as it undergoes buckling and to identify the critical buckling stress in two cases, namely, where the shell is hollow and the stress at infinity is either uniaxial or radial.

Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation

Gao, D. Y., Ogden, R. W. — 2008-10-23

Non-convex variational/boundary-value problems are studied using a modified version of the Ericksen bar model in nonlinear elasticity. The strain-energy function is a general fourth-order polynomial in a suitable measure of strain that provides a convenient model for the study of, for example, phase transitions. On the basis of a canonical duality theory, the nonlinear differential equation for the non-convex, non-homogeneous variational problem, here with either mixed or Dirichlet boundary conditions, is converted into an algebraic equation, which can, in principle, be solved to obtain a complete set of solutions. It should be emphasized that one important outcome of the theory is the identification and characterization of the local energy extrema and the global energy minimizer. For the soft loading device criteria for the existence, uniqueness, smoothness and multiplicity of solutions are presented and discussed. The iterative finite-difference method (FDM) is used to illustrate the difficulty of capturing non-smooth solutions with traditional FDMs. The results illustrate the important fact that smooth analytic or numerical solutions of a nonlinear mixed boundary-value problem might not be minimizers of the associated potential variational problem. From a ‘dual’ perspective, the convergence (or non-convergence) of the FDM is explained and numerical examples are provided.

Fractional heat conduction equation and associated thermal stresses in an infinite solid with spherical cavity

Povstenko, Y. Z. — 2008-10-23

In this work, the temperature distribution and thermal stresses in an infinite medium with a spherical cavity are studied in the framework of a quasi-static uncoupled theory of thermoelasticity based on heat conduction equation with a time fractional derivative of order 0 < ≤ 2. The Caputo fractional derivative is used. As the fractional heat conduction equation in the case 1 ≤ ≤ 2 interpolates the standard heat conduction equation ( = 1) and the wave equation ( = 2), the proposed theory interpolates the classical thermoelasticity and the thermoelasticity without energy dissipation introduced by Green and Naghdi. The solution is obtained using the integral transform technique. Numerical results are illustrated graphically.

Point source excitation of a layered sphere: direct and far-field inverse scattering problems

Tsitsas, N. L., Athanasiadis, C. — 2008-10-23

A layered sphere is excited by a time-harmonic spherical acoustic wave, generated by a point source located either in the interior or in the exterior of the sphere. The sphere's core may be acoustically soft, hard, resistive or penetrable. Significant applications such as radiation from the neuron currents lying inside the human brain and localization and shape reconstruction of buried objects in layered media motivate the investigation of direct and inverse scattering problems involving such types of scatterers and excitations. The exact Green's function is determined by solving the corresponding boundary-value problem, by applying a combination of Sommerfeld's and T-matrix methods. We then introduce the low-frequency assumption and extract from the determined exact Green's function the low-frequency far-field results for a small layered sphere. The spherical wave low-frequency far-fields obtained reduce to those due to plane wave incidence on a layered sphere and also recover as special cases several classic results of the literature concerning the exterior spherical wave excitation of homogeneous spheres. The derived low-frequency far-field expansions are then utilized in order to establish inverse scattering algorithms for the determinations of (i) the sphere's center and the layers’ radii, (ii) the layers’ physical parameters and (iii) the location of the point source. The distance of the point source from the sphere's centre plays a significant role in the development of these algorithms. Several numerical results are included concerning the far-field interactions between the point source and the layered sphere.

The initial development of a jet caused by fluid, body and free surface interaction. part 3. an inclined accelerating plate

Needham, D. J., Chamberlain, P. G., Billingham, J. — 2008-10-23

The free surface and flow field structure generated by the uniform acceleration of a rigid plate, inclined at an angle (0, /2) (/2, ) to the exterior horizontal, as it advances into an initially stationary and horizontal strip of inviscid, incompressible fluid, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating vertical plate, when = /2, as studied in King and Needham (J. Fluid. Mech. 268 (1994)). Particular attention is devoted to the inner region in the vicinity of the intersection point between the plate and the free surface. It emerges that the angle = /2 is a bifurcation point in this local structure. For (0, /2), a weak jet rises up the plate when t = 0⁺, with thickness O(t²) as t -> 0⁺, independent of , with the free surface slope at the plate being as t -> 0⁺; this slope is O(1/ log(t)) as t -> 0⁺ when = /2. However, when (/2, ), the jet becomes significantly stronger, with a highly nonlinear structure, and the thickness now depending on and increasing with , being O(t), where = (1 – /4)^–1. In this case, moreover, a classical solution to the evolution problem is possible only when (/2, _c], where _c 1·791 102·6^°. When = _c, a 120^° corner forms on the free surface when t = 0⁺ at the initial intersection point of the plate and free surface, and convects self-similarly into the inner region for 0 < t << 1. We conjecture that no classical solution exists when t = 0⁺ for plate angles (_c, ). In practice, surface tension allows a solution to exist in some finite neighbourhood of t = 0, a result that will be presented in a later paper.

Electromagnetic fields induced by electric current in an infinite plate with an elliptical hole

Hasebe, N., Jin, X., Keer, L. M., Wang, Q. — 2008-10-23

Analytical solutions to the electromagnetic field in a thin conductive plate with an elliptical hole are derived by means of complex potentials and conformal mapping techniques. The steady-state current field in a thin conductive plate is two dimensional (2D) and is explored by a standard complex variable technique. The current is disturbed around the elliptical hole, and produces a three dimensional magnetic field. In this case, using the complex variable method to solve the real magnetic field can be challenging. The magnetic boundary conditions take different forms for the soft ferromagnetic and the para- or diamagnetic materials under consideration. A simplified analysis taking account of the magnitude of the magnetic permeability of the magnetic material and air surrounding the material is proposed to reduce the magnetic field in a thin plate to 2D calculations. The magnetic field distributions are derived for each material and the equations of the magnetic components at the tip of elliptical hole are presented.