Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1993"

Indentation of a power law creeping solid

Abstract: The aim of this paper is to establish a rigorous theoretical basis for interpreting the results of hardness tests on creeping specimens. We investigate the deformation of a creeping half-space with uniaxial stress-strain behaviour ⋵ = ⋵ 0 (σ/σ 0 ) m , which is indented by a rigid punch. Both axisymmetric and plane indenters are considered. The shape of the punch is described by a general expression which includes most indenter profiles of practical importance. Two methods are used to solve the problem. The main results are found using a transformation method suggested by R. Hill. It is shown that the creep indentation problem may be reduced to a form which is independent of the geometry of the punch, and depends only on the material properties through m . The reduced problem consists of a nonlinear elastic half-space, which is indented to a unit depth by a rigid flat punch of unit radius (in the axisymmetric case), or unit semi-width (in the plane case). Exact solutions are given for m = 1 and m = ∞. For m between these two limits, the reduced problem has been solved using the finite element method. The results enable the load on the indenter and the contact radius to be calculated in terms of the indentation depth and rate of penetration. The stress, strain and displacement fields in the half-space may also be deduced. The accuracy of the solution is demonstrated by comparing the results with full-field finite element calculations. The predictions of the theory are shown to be consistent with experimental observations of hardness tests on creeping materials reported in the literature.

Harmonic wavelet analysis

Abstract: A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure w(x) = {exp(i4$\pi $x)-exp(i2$\pi $x)}/i2$\pi $x. This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain). An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.

An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities

Abstract: A set of four tensors corresponding to Eshelby’s tensor in elasticity are obtained for an ellipsoidal inclusion embedded in an infinite piezoelectric medium. These tensors, which describe the elastic, piezoelectric, and dielectric constraint of the matrix, are obtained from W. F. Deeg’s solution to inclusion and inhomogeneity problems in piezoelectric solids. These tensors are then used as the backbone in the development of a micromechanics theory to predict the effective elastic, dielectric, and piezoelectric moduli of particle and fibre reinforced composite materials. The effects of interaction among inhomogeneities at finite concentrations are approximated through the Mori-Tanaka mean field approach. This approach, although widely utilized in the study of uncoupled elastic and dielectric behaviour, has not before been applied to the study of coupled behaviour. To help ensure confidence in the theory, the analytical predictions are proven to be self-consistent, diagonally symmetric, and to exhibit the correct behaviour in the low and high concentration limits. Finally, numerical results are presented to illustrate the effects of the concentration, shape, and material properties of the reinforcement on the effective properties of piezoelectric composites and analytical predictions are shown to result in good agreement with existing experimental data.

Existence and uniqueness of travelling waves for a neural network

Abstract: A one-dimensional scalar neural network with two stable steady states is analysed. It is shown that there exists a unique monotone travelling wave front which joins the two stable states. Some additional properties of the wave such as the direction of its velocity are discussed.

Structural studies of trigonal lamellar particles of gold and silver

Abstract: The results of detailed structural studies of trigonal lamellar particles of both gold and silver are presented. The particles have been characterized both in sol by means of optical spectroscopy and powder X-ray diffraction and ex sol using high resolution electron microscopy in both plan view and profile imaging modes. The results of these studies have indicated that the particles have a trigonal outline and are shortened along a direction to give a plate-like morphology. The presence of small numbers of parallel {111} twin planes has also been confirmed and used to explain the presence of the formally forbidden $\frac{1}{3}${422} reflections observed in plan view. The precise structural requirements for the observation of such reflections has also been confirmed using multislice calculations. Possible growth mechanisms for these particles are also discussed.

Rank one property for derivatives of functions with bounded variation

Abstract: In this paper we introduce a new tool in geometric measure theory and then we apply it to study the rank properties of the derivatives of vector functions with bounded variation.

Deformation, Strengths and Strains to Failure of Polymer Bonded Explosives

Abstract: This paper describes a study of the strength and failure properties of a range of polymer bonded explosives (PBXS). These are composite systems in which small (typically micrometre up to millimetre-sized) explosive crystals are bonded by a polymer (typically 2-10% (by mass)). In PBXS it is important to optimise the mechanical properties, while maintaining a low sensitiveness (i.e. the material is safe to manufacture, store and handle) and high explosiveness (i.e. reacts powerfully to a prescribed stimulus). The Brazilian test, in which a disc-shaped specimen is loaded diametrically, was chosen for the study. The advantages are that relatively small specimens of typically 10 mm diameter and 4 mm thickness can be used, and that the tensile stresses on the central axis are achieved by applying compressive stresses at the anvil so that complicated gripping arrangements are not required. The technique of double-exposure laser speckle photography was chosen to measure the in-plane displacement field. The technique can measure displacements to sub-micrometre accuracy and provide information over the whole specimen surface. These are distinct advantages over strain gauge methods that involve attaching gauges to the specimen and which only give pointwise information. The double-exposure speckle pattern records were interpreted using an automated Young's fringes method. The PBXS were of three explosive types and those based on HMX were studied systematically for two crystal sizes and three different binder materials, of two different weight percents. In general, compositions based on micronized crystals were the strongest. Polishing techniques were developed to study the deformation of the individual crystals, the points of nucleation of failure and the fracture paths through the PBXS. The failure modes are discussed in terms of various theoretical models. The mechanical twinning which was shown in earlier work to be an important failure mode in $\beta $-HMX also takes place in PBXS based on HMX. The general applicability of the techniques developed in this research for other composite systems is emphasized.

Designing Localized Waves

Abstract: In this paper we re-interpret a recently introduced method for obtaining non-separable, localized solutions of homogeneous partial differential equations. This re-interpretation is in the form of a geometrical consideration of the algebraic constraint that the Fourier transforms of such solutions must satisfy in the transform domain (phase space). With this approach we link two classes of localized, non-separable solutions of the homogeneous wave equation, and examine the transform domain characteristic that determines the space-time localization properties of these classes. This characterization allows us to design classes of solutions with better localization properties. In particular, we design and discuss the properties of several novel subluminal and superluminal solutions of the homogeneous wave equation. We also design families of non separable, localized, subluminal and superluminal solutions of the Klein-Gordon equation by using the same technique.

Chaotic classical and half-classical adiabatic reactions: geometric magnetism and deterministic friction

Abstract: We study the dynamics of a heavy (slow) classical system coupled, through its position, to a classical or quantal light (fast) system, and derive the first-order velocity-dependent corrections to the lowest adiabatic approximation for the reaction force on the slow system. If the fast dynamics is classical and chaotic, there are two such first-order forces, corresponding to the antisymmetric and symmetric parts of a tensor given by the time integral of the force–force correlation function of the fast motion for frozen slow coordinates. The antisymmetric part is geometric magnetism, in which the ‘magnetic field’ is the classical limit of the 2-form generating the quantum geometric phase. The symmetric part is deterministic friction, dissipating slow energy into the fast chaos; previously found by Wilkinson, this involves the same correlation function as governs the fluctuations and drift of the adiabatic invariant. In the ‘half-classical’ case where the fast system is quantal with a discrete spectrum of adiabatic states, the only first-order slow force is geometric magnetism; there is no friction. This discordance between classical and quantal fast motion is explained in terms of the clash between the semiclassical and adiabatic limits. A generalization of the classical case is given, where the slow velocity, as well as position, is coupled to the fast motion; to first order, the symplectic form in the lowest-order hamiltonian dynamics is modified.

Compressive failure of notched carbon fibre composites

Abstract: We present an investigation into the compressive fracture properties of carbon fibre/epoxy laminates. Unnotched and notched strengths are reported for a wide range of ‘lay-ups’, and results are interpreted in terms of existing theories of strength. In all cases, we find that compressive failure is governed by plastic microbuckling of the 0° plies. For unnotched laminates, the failure strain is independent of lay-up configuration, suggesting that a critical strain to failure, or maximum strain criterion can predict the failure point adequately. In tests on notched panels, the growth of damage from the edge of a single hole is examined. For notched materials, the failure strength and damage zone size at failure support the prediction of the Soutis, Fleck and Smith model.

Incoherent Imaging of Thin Specimens Using Coherently Scattered Electrons

Abstract: We consider the imaging of phase objects using a scanning transmission electron microscope equipped with a large inner-angle annular detector. We show, contrary to popular expectation, that incoherent imaging theory can be used to describe the imaging process in a plane perpendicular to the optical axis. Interference effects between atoms possessing the same projected coordinates must, however, be considered explicitly.

Permanence in ecological systems with spatial heterogeneity

Abstract: A basic problem in population dynamics is that of finding criteria for the long-term coexistence of interacting species. An important aspect of the problem is determining how coexistence is affected by spatial dispersal and environmental heterogeneity. The object of this paper is to study the problem of coexistence for two interacting species dispersing through a spatially heterogeneous region. We model the population dynamics of the species with a system of two reaction–diffusion equations which we interpret as a semi-dynamical system. We say that the system is permanent if any state with all components positive initially must ultimately enter and remain within a fixed set of positive states that are strictly bounded away from zero in each component. Our analysis produces conditions that can be interpreted in a natural way in terms of environmental conditions and parameters, by combining the dynamic idea of permanence with the static idea of studying geometric problems via eigenvalue estimation.

Stable determination of a crack from boundary measurements

Abstract: We treat the problem of determining a crack inside a conductor when two pairs of current and voltage boundary measurements are given. We prove a theorem of continuous dependence from the data.

High resolution neutron powder diffraction: a case study of the structure of C60

Abstract: High resolution time-of-flight neutron powder diffraction has been used to determine the detailed structure of C$\_{60}$ as a function of temperature. Rapid data collection coupled with high resolution has enabled subtle aspects of the 86 K orientational glass transition and precursor effects of the 260 K order-disorder transition to be observed. This surveying capability complements traditional single crystal methods. The power of the Rietveld method of profile refinement is demonstrated in the elucidation of the detailed crystal structure of the orientationally-ordered low temperature phase and in the evaluation of the departure from isotropic scattering of the C$\_{60}$ molecule in the disordered high temperature phase. The counter-intuitive success in obtaining high-order cubic-harmonic coefficients, albeit to poorer precision than single crystal X-ray measurements, confirms the efficacy of the Rietveld profile refinement method. The collapse of three dimensions of diffraction information on to the one dimension of a high resolution powder diffraction pattern can still lead to an impressive amount of structural information that substantiates the assertion made by W. H. Bragg `the second method [powder diffraction], first used independently by Debye and Hull, can be used when the crystal is in powder, and can, therefore be employed when no single crystal can be obtained of sufficient size. All the spectra of the different planes are thrown together on the same diagram or photograph, and must be disentangled. This is not as difficult as it may seem...'.

Regularity of an elliptic problem with a singular nonlinearity

Abstract: We study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular. We prove that the positive solution is in general Holder-continuous up to the boundary and has even better regularity in some special cases.

Capillary Surfaces: Stability from Families of Equilibria with Application to the Liquid Bridge

Abstract: A method for determining the stability of general static capillary surfaces is illustrated by application to the liquid bridge. Axisymmetric bridges with fixed contact lines under gravity are parametrized by three quantities: bridge length L, bridge volume V, and Bond number B. The method delivers i) stability envelopes in the {L,V,B,} parameter space for constant pressure and constant volume disturbances (recovering classical and generating new results), ii) the number of unstable modes for any equilibrium (state of instability) once the stability of one equilibrium state is known, based on, iii) a demonstration that all known families of equilibria are ultimately connected. The state of instability of an equilibrium shape relative to its neighbors is immediate from a plot of volume V versus pressure p, a "preferred" bifurcation diagram. The preferred diagram method gives stronger results than classical bifurcation theory based on properties of eigenvalues of the Jacobi equation for problems with a variational formulation. Application to other capillary surfaces including drops and nonaxisymmetric shapes is discussed. In addition, motivated by general tangency considerations, an invariant wavenumber classification is introduced and used to label the numerous families of liquid bridge equilibria.

The Continuum Mechanics of Coherent Two-Phase Elastic Solids with Mass Transport

Abstract: We develop a theory for the dynamics of an interface in a two-phase elastic solid with kinetics driven by mass transport and stress. We consider a two-phase system consisting of bulk regions separated by a sharp interface endowed with energy and capable of supporting force. Our discussion is based on balance laws for mass and force in conjunction with a version of the second law-appropriate to a mechanical system out of equilibrium-which we use to develop a suitable constitutive theory for the interface. It is assumed that mass transport is characterized by the bulk diffusion of a single independent species; we neglect mass diffusion within the interface; limit our discussion to a continuous chemical potential and to a coherent interface; neglect the elasticity of the interface; and consider only infinitesimal deformations, neglecting inertia. We show that the field equations and free-boundary conditions can be developed in a simple manner in terms of the diffusion potential and its time derivatives, as opposed to the usual formulation in terms of concentration. Natural consequences of the thermodynamic framework are Lyapunov functions for the resulting evolution problems. This leads to a hierarchy of variational principles that should describe the equilibrium shapes of misfitting particles as well as possible microstructures that might form; these principles are applicable both in the absence and presence of an applied stress.

Biot poroelasticity of a chemically active shale

Abstract: The Biot theory of poroelasticity relates the strain e of a porous material to changes of the applied stress σ and of the pore pressure p . Additional osmotic effects are present in some rocks, such as shales. This paper modifies the thermodynamic arguments used by Biot in order to include the chemical potentials μ r of all the chemical species within the pore fluid. In the limit in which salt is unable to move into or out of the shale, the deformation depends only on the chemical potential μ w of the water and the applied stress. In the limit of a chemically inert rock, the standard Biot analysis is obtained, and the pore pressure p is again the important variable. Real shales lie somewhere between these two limits.

Two representations of a conditioned superprocess

Abstract: We consider a class of measure-valued Markov processes constructed by taking a superprocess over some underlying Markov process and conditioning it to stay alive forever. We obtain two representations of such a process. The first representation is in terms of an “immortal particle” that moves around according to the underlying Markov process and throws off pieces of mass, which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess. As a consequence of this representation, we show that the tail σ-field of the conditioned superprocess is trivial if the tail σ-field of the underlying process is trivial. The second representation is analogous to one obtained by LeGall in the unconditioned case. It represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process. This representation is useful for studying the “transience” and “recurrence” properties of the closed support process.

A pendulum theorem

Abstract: We consider N linked pendulums which are inverted and balanced on top of one another, and establish a general theorem which shows how they may be stabilized by small vertical oscillations of the support.

An analytical expression for the rate of binary condensational particle growth

Abstract: Analytical and semianalytical expressions for the growth rate of binary droplets in supersaturated vapour mixtures are derived. The analytical approach is appropriate for continuum regime droplets, the semianalytical approach allows for corrections needed in the transition regime and the Kelvin effect. Both models are based on the assumption of constant and uniform droplet composition, which depends only on a mass flux ratio. The predictions given by the models are compared to the results calculated according to a quasisteady numerical model for water- n -propanol droplets. The approximate models provides reasonable estimations for the growth rate, for the droplet temperature and for the droplet composition when the vapour activities are relatively low, i. e. when the sum of the activities does not strongly exceed unity.

On the effective viscoelastic moduli of two-phase media. I. Rigorous bounds on the complex bulk modulus

Abstract: The dynamic response of isotropic composites of two viscoelastic isotropic phases mixed in fixed proportions is considered in the frequency range where the acoustic wavelength is much larger than the inhomogeneities. The effective bulk-modulus bounds of Hashin-Shtrikman-Walpole are extended to viscoelasticity in this quasi-static regime, where the properties of the isotropic composite can be described by complex bulk and shear moduli. The effective bulk modulus is shown to be constrained to a lens-shaped region of the complex plane bounded by the outermost pair of four circular arcs (three circular arcs in the case of two-dimensional elasticity). This is proved using a new variational principle for viscoelasticity together with two established techniques for deriving bounds on effective moduli, namely the translation method and the Hashin-Shtrikman method. In this application the Hashin-Shtrikman method needs to be generalized to allow the reference tensor to have an associated quasiconvex energy. Microstructures are identified which have bulk-moduli that correspond to various points on each of the circular arcs. Thus these microstructures have extremal viscoelastic behaviour when the associated arc forms one of the outermost pair. The bounds and the extremal microstructures are similar to those obtained for the complex dielectric constant, but the methods used here are entirely different.

Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle

Abstract: Time-harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle. The corresponding electromagnetic transmission problem is reduced to a single integral equation over S for a single unknown tangential vector field, where S is the interface between the obstacle and the surrounding medium. In fact, several different integral equations are derived and analysed, including two previously-known equations due to E. Marx and J. R. Mautz, and two new singular integral equations. Mautz's equation is shown to be uniquely solvable at all frequencies. A new uniquely solvable singular integral equation is also found. The paper also includes a review of methods using pairs of coupled integral equations over S. It is these methods that are usually used in practice, although single integral equations seem to offer some computational advantages.

General edge asymptotics of solutions of second-order elliptic boundary value problems II

Abstract: This is the second of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears trie phenomenon of “crossing” of the exponents of singularities. In Part I, we introduced for the Dirichlet problem appropriate combinations of the simple tensor product singularities.In this second part, we extend the results of Part I to general non-homogeneous boundary conditions. Moreover, we show how these combinations of singularities appear in a natural way as sections of an analytic vector bundle above the edge. In the case when the interior operator is the Laplacian, we give a simpler expression of the combined singular functions, involving divided differences of powers of a complex variable describing the coordinates in the normal plane to the edge.

Effective properties of composite materials containing voids

Abstract: Recent results of theoretical and practical importance prove that the two-dimensional (in-plane) effective (average) Young’s modulus for an isotropic elastic material containing voids is independent of the Poisson’s ratio of the matrix material. This result is true regardless of the shape and morphology of the voids so long as isotropy is maintained. The present work uses this proof to obtain explicit analytical forms for the effective Young’s modulus property, forms which simplify greatly because of this characteristic. In some cases, the optimal morphology for the voids can be identified, giving the shapes of the voids, at fixed volume, that maximize the effective Young’s modulus in the two-dimensional situation. Recognizing that two-dimensional isotropy is a subset of three-dimensional transversely isotropic media, it is shown in this more general case that three of the five properties are independent of Poisson’s ratio, leaving only two that depend upon it. For three-dimensionally isotropic composite media containing voids, it is shown that a somewhat comparable situation exists whereby the three-dimensional Young’s modulus is insensitive to variations in Poisson’s ratio, v m , over the range 0 ≤ v m ≤ ½, although the same is not true for negative values of v m . This further extends the practical usefulness of the two-dimensional result to three-dimensional conditions for realistic values of v m .

Transport Properties of a Three-Phase Composite Material: The Square Array of Coated Cylinders

Abstract: The analytic properties of the effective dielectric constant of a class of three-phase composite materials are studied. Specifically, we investigate the effective dielectric constant of a periodic array of coated cylinders, as a function of the core dielectric constant ( ϵ c ) and the shell dielectric constant ( ϵ s ), while keeping the matrix dielectric constant ( ϵ b ) fixed. We show that when ϵ s = – ϵ c , the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant ϵ c and radius equal to the outer radius of the original coated cylinder. We also show that when ϵ s = – 1, the composite has exactly the same effective dielectric constant as a periodic array of solid cylinders with dielectric constant ϵ c , and radius exceeding the shell radius. We explore the location of poles and zeros of the three-phase effective dielectric constant in the ( ϵ s , ϵ c ) plane. The lines ϵ s = – 1 and ϵ s + ϵ c = 0 are loci of essential singularities. We also comment on the behaviour of the effective dielectric constant in the neighbourhood of the two special points ( ϵ s , ϵ c ) = (0,0) and ( ϵ s , ϵ c ) = ( - 1 , + 1 ).

Skeletal Point Stresses in Circumferentially Notched Tension Bars Undergoing Tertiary Creep Modelled with Physically Based Constitutive Equations

Abstract: Constitutive equations are proposed in which the stress level dependence of creep rate is described by a sinh function, and two damage state-parameters are used to model the tertiary softening caused by: (i) grain boundary cavity nucleation and growth, and (ii) the multiplication of mobile dislocations. These constitutive equations are applicable to polycrystalline nickel-base superalloys and are used together with a continuum damage mechanics finite element based solver, DAMAGE XX, to study the behaviour of axisymmetrically notched tension bars and simulate the complex stress states that may be encountered at geometrical stress-raisers in high temperature components. Numerical studies of such bars show that their behaviour can be accurately represented in terms of a ‘skeletal effective stress’ located at a point within the notch throat, and the stress state at this point. It is shown that this conclusion is valid not only for those materials that fail by grain boundary cavitation alone, but also for materials such as superalloys where grain boundary cavitation is accompanied by mobile dislocation multiplication.

Exact Results in the Micromechanics of Fibrous Piezoelectric Composites Exhibiting Pyroelectricity

Abstract: Piezoelectric fibrous composites of two, three and four phases are considered. The phase boundaries are cylindrical but otherwise the microgeometry is totally arbitrary. The constituents are transversely isotropic, and exhibit pyroelectricity. Exact relations are derived between the local fields arising under a uniform electromechanical loading and a uniform temperature change in the piezoelectric composite. For given overall material symmetry, exact connections are obtained among the effective elastic, piezoelectric and dielectric constants of two- and three- phase systems. It is also shown that the effective thermal stress and pyroelectric coefficients can be expressed in terms of the effective elastic, piezoelectric, dielectric constants and constituent properties in two-, three- and four-phase composites.

Operators preserving orthogonality are isometries

Abstract: We prove that every operator preserving orthogonality in a real Banach space is an isometry multiplied by a constant.

Weak lower semicontinuity of polyconvex integrals

Abstract: Multiple integrals with polyconvex integrands are studied on the class of all sense-preserving diffeomorphisms from W1,p(Ω, Rn) where Ω is an open subset of Rn. They are proved to be sequentially weakly lower semicontinuous if 1 < p = n –1. An example is presented showing that a similar result is not valid if p