Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 2002"

Huygens's clocks

Abstract: The 336–year–old synchronization observations of Christiaan Huygens are re–examined in modern experiments. A simple model of synchronization is proposed.

Towards a coherent philosophy for modelling the environment

Abstract: The predominant philosophy underlying most environmental modelling is a form of pragmatic realism. The limitations of this approach in practical applications are discussed, in particular, in relati...

Normal impact of a liquid drop on a dry surface: model for spreading and receding

Abstract: The normal impact of a liquid drop on a dry solid surface is studied experimentally and theoretically. In this paper a strictly theoretical model is introduced, which predicts the evolution of the drop diameter. The spreading and receding phases of the impact are described by the motion of a rim appearing at the edge of the liquid film (lamella) due to the surface-tension forces. The mass and the momentum equations of the rim are considered, taking into account the effects of inertial, viscous and surface forces, and wettability. Also, simplified approximations for the maximum spreading diameter of the drop and for the velocity of the merging of the rim in the receding phase are obtained. The theoretical predictions agree well with available experimental data.

Non-convex potentials and microstructures in finite-strain plasticity

Abstract: A mathematical model for a finite–strain elastoplastic evolution problem is proposed in which one time–step of an implicit time–discretization leads to generally non–convex minimization problems. The elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations. The results for a single slip–system and von Mises plasticity illustrate that finite–strain elastoplasticity generates reduced problems with non–quasiconvex energy densities and so allows for non–attainment of energy minimizers and microstructures.

The slump origin of the 1998 Papua New Guinea Tsunami

Abstract: The origin of the Papua New Guinea tsunami that killed over 2100 people on 17 July 1998 has remained controversial, as dislocation sources based on the parent earthquake fail to model its extreme run–up amplitude. The generation of tsunamis by submarine mass failure had been considered a rare phenomenon which had aroused virtually no attention in terms of tsunami hazard mitigation. We report on recently acquired high–resolution seismic reflection data which yield new images of a large underwater slump, coincident with photographic and bathymetric evidence of the same feature, suspected of having generated the tsunami. T–phase records from an unblocked hydrophone at Wake Island provide new evidence for the timing of the slump. By merging geological data with hydrodynamic modelling, we reproduce the observed tsunami amplitude and timing in a manner consistent with eyewitness accounts. Submarine mass failure is predicted based on fundamental geological and geotechnical information.

Computation of the linear elastic properties of random porous materials with a wide variety of microstructure

Abstract: A finite-element method is used to study the elastic properties of random three-dimensional porous materials with highly interconnected pores. We show that Young's modulus, E, is practically independent of Poisson's ratio of the solid phase, nu(s), over the entire solid fraction range, and Poisson's ratio, nu, becomes independent of nu(s) as the percolation threshold is approached. We represent this behaviour of nu in a flow diagram. This interesting but approximate behaviour is very similar to the exactly known behaviour in two-dimensional porous materials. In addition, the behaviour of nu versus nu(s) appears to imply that information in the dilute porosity limit can affect behaviour in the percolation threshold limit. We summarize the finite-element results in terms of simple structure-property relations, instead of tables of data, to make it easier to apply the computational results. Without using accurate numerical computations, one is limited to various effective medium theories and rigorous approximations like bounds and expansions. The accuracy of these equations is unknown for general porous media. To verify a particular theory it is important to check that it predicts both isotropic elastic moduli, i.e. prediction of Young's modulus alone is necessary but not sufficient. The subtleties of Poisson's ratio behaviour actually provide a very effective method for showing differences between the theories and demonstrating their ranges of validity. We find that for moderate- to high-porosity materials, none of the analytical theories is accurate and, at present, numerical techniques must be relied upon.

Solution of the generalized Riemann problem for advection–reaction equations

Abstract: We present a method for solving the generalized Riemann problem for partial differential equations of the advection–reaction type. The generalization of the Riemann problem here is twofold. Firstly, the governing equations include nonlinear advection as well as reaction terms and, secondly, the initial condition consists of two arbitrary but infinitely differentiable functions, an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws. The solution procedure, local and valid for sufficiently small times, reduces the solution of the generalized Riemann problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional Riemann problems for homogeneous advection equations for spatial derivatives of the initial conditions. We illustrate the approach via the model advection–reaction equation, the inhomogeneous Burgers equation and the nonlinear shallow–water equations with variable bed elevation.

A numerical study of submarine-landslide-generated waves and run-up

Abstract: A mathematical model is derived to describe the generation and propagation of water waves by a submarine landslide. The model consists of a depth–integrated continuity equation and momentum equations, in which the ground movement is the forcing function. These equations include full nonlinear, but weak frequency–dispersion, effects. The model is capable of describing wave propagation from relatively deep water to shallow water. Simplified models for waves generated by small seafloor displacement or creeping ground movement are also presented. A numerical algorithm is developed for the general fully nonlinear model. Comparisons are made with a boundary integral equation method model, and a deep–water limit for the depth–integrated model is determined in terms of a characteristic side length of the submarine mass. The importance of nonlinearity and frequency dispersion in the wave–generation region and on the shoreline movement is discussed.

The rheology of a bubbly liquid

Abstract: A semiempirical constitutive model for the visco-elastic rheology of bubble suspensions with gas volume fractions < 0.5 and small deformations (Ca 1) is developed. The model has its theoretical fou...

A proof of the Gamma test

Abstract: From a dataset of input–output vectors, the Gamma test estimates the variance of the noise on an output modulo any smooth model with bounded partial derivatives. We present a proof of the Gamma test under fairly weak hypotheses.

Multidimensional solutions of time-fractional diffusion-wave equations

Abstract: Green's functions and propagator functions for the timefractional generalized diffusionwave equation in one and three dimensions are expressed in terms of Wright functions. In two dimensions they a...

Quasi–static studies of the deformation and failure of β–HMX based polymer bonded explosives

Abstract: This paper describes research into the low strain rate mechanical properties of polymer bonded explosives (PBXs) and follows on from that presented by Palmer and co-workers in 1993. PBXs are highly filled composite materials comprised of crystals of a secondary explosive supported in ca .5–10% (by mass) polymeric binder. In general, the modulus of the binder is 10 5 times lower than that of the crystalline explosive, resulting in the behaviour of the composite being heavily influenced by the properties of the binder despite the low concentration present. The Brazilian test, in which a disc of material is loaded diametrically in compression, has been used to generate tensile failure in the materials studied. Three methods of microscopy have been used to examine the nature of failure in two UK compositions and one US material (PBX 9501). Pre– and post–failure optical and electron microscopy examinations of the materials have been undertaken to gain a greater understanding of the role of the microstructure and this has been aided by the use of an environmental scanning electron microscope to follow real-time failure in the PBXs. Failure in all three compositions has been observed to start around the edges of larger filler particles perpendicular to the direction of tensile strain. Compositions with rubbery binders have been observed forming binder filaments which bridge the crack walls, while clean crystal faces are observed on the larger particles. In a composition with a nitrocellulose-based binder, hair-like features of nitrocellulose have been seen sticking out from the failure surfaces and the rougher crack walls.

The Structure of the Multiverse

Abstract: The structure of the multiverse can be understood by analysing the ways in which information can flow in it. We may distinguish between quantum and classical information processing. In any region where the latter occurs — which includes not only classical computation but also all measurements and decoherent processes — the multiverse contains an ensemble of causally autonomous systems, each of which resembles a classical physical system. However, even in those regions, the multiverse has additional structure.

Anti-self-dual four-manifolds with a parallel real spinor

Abstract: Antiselfdual metrics in the ( ) signature that admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourthorder integrable partial differ...

Pauli's theorem and quantum canonical pairs: the consistency of a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with non–empty point spectrum

Abstract: In single Hilbert space, Pauli's wellknown theorem implies that the existence of a selfadjoint time operator canonically conjugate to a given Hamiltonian requires the Hamiltonian to possess complet...

On Korn's first inequality with non-constant coefficients

Abstract: In this paper we prove a Korn-type inequality with non-constant coefficients which arises from applications in elasto-plasticity at large deformations. More precisely, let Ω ⊂ R3 be a bounded Lipschitz domain and let Γ ⊂ ∂Ω be a smooth part of the boundary with non-vanishing two-dimensional Lebesgue measure. Define and let be given with det Fp(x) ≥ μ+ > 0. Moreover, suppose that Rot . Then Clearly, this result generalizes the classical Korn's first inequality which is just our result with Fp = 11. With slight modifications, we are also able to treat forms of the type

Multi–dimensional solutions of space–time–fractional diffusion equations

Abstract: Green9s functions and propagators for the multi–dimensional anisotropic space–time fractional diffusion equation are obtained in integral form. Uniqueness of the solutions is studied in the framework of abstract Volterra equations. Unimodality of propagator functions is demonstrated in the centrally symmetric case. The transition from diffusion to wave propagation is discussed. Numerical examples are constructed.

From the discrete to the continuous coagulation–fragmentation equations

Abstract: The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

The oscillatory pipe flow with arbitrary wall injection

Abstract: The linearized NavierStokes equations play a central role in describing the unsteady motion of a viscous fluid inside a porous tube. Asymptotic solutions of these equations have been found and here...

Null–space construction using cofactors from a screw–algebra context

Abstract: This paper develops an approach for constructing the null space N ( A ) of a linear system of homogeneous equations using the cofactors of an augmented coefficient matrix A . The relationship between the row space R ( A T ) and null space is exploited by introducing an augmenting vector which is linearly independent of the row space and dependent on the null space. The resultant null space is shown to be a vector of cofactors of the augmenting row of the coefficient matrix and is invariant. This provides a straightforward solution to a linear system of homogeneous equations without going through Gauss-Seidel elimination. The approach is derived from a onedimensional null space and is extended to a multidimensional one by partitioning the coefficient matrix and consequently constructing a set of ( n − m ) null–space vectors based on cofactors. Examples are given and accuracy is compared with Gauss–Seidel elimination. The approach is further used in a screw–algebra context with a simple procedure to obtain a system of reciprocal screws representing a set of constraint wrenches from a set of twists of freedom, in the form of a linear system of homogeneous equations in R 6 . The paper provides rigorous proofs and applications in both linear algebra and advanced kinematics.

Stability chart for the delayed Mathieu equation

Abstract: In the space of system parameters, the closedform stability chart is determined for the delayed Mathieu equation defined as (t)(cost)x(t) bx(t2). This stability chart makes the connection between t...

Long-wave forcing by the breaking of random gravity waves on a beach

Abstract: This paper presents new laboratory data on long-wave (surf-beat) forcing by the random breaking of shorter gravity water waves on a plane beach. The data include incident and outgoing wave amplitudes, together with shoreline oscillation amplitudes at long-wave frequencies, from which the correlation between forced long waves and short-wave groups is examined. A detailed analysis of the cross-shore structure of the long-wave motion is presented, and the observations are critically compared with existing theories for two-dimensional surf-beat generation. The surf beat shows a strong dependency on normalized surf-zone width, consistent with long-wave forcing by a time-varying breakpoint, with little evidence of the release and reflection of incident bound long waves for the random-wave simulations considered. The seaward-propagating long waves show a positive correlation with incident short-wave groups and are linearly dependent on short-wave amplitude. The phase relationship between the incident bound long waves and radiated free long waves is also consistent with breakpoint forcing. In combination with previous work, the present data suggest that the breakpoint variability may be the dominant forcing mechanism during conditions with steep incident short waves.

Identification of elastic-plastic material parameters from pyramidal indentation of thin films

Abstract: The indentation experiment is a popular method for the investigation of mechanical properties of thin films. By application of conventional methods, the hardness and the stiffness of the film material can be determined by limiting the indentation depth to well below the film thickness so that the substrate effects can be eliminated. In this work a new method is proposed, which allows for a determination of the reduced modulus as well as the nonlinear hardening behaviour of both the film and substrate materials. To this end, comparable deep indentations are made on the film/substrate composite to obtain sufficient information on the mechanical properties of both materials. The inverse problem is solved by training neural networks on the basis of finite–element simulations using only the easily measurable hardness and stiffness behaviour as input data. It is shown that the neural networks are very robust against noise in the load and depth. The identification of the material parameters of aluminium films on different substrates results in a significant increase in yield stress and initial work–hardening rate for a reduction of the film thickness from 1.5 to 0.5 µm, while the elastic modulus and the extent of work hardening remain constant.

The two-scale convergence method applied to generalized Besicovitch spaces

Abstract: The twoscale convergence method has proved to be a very useful tool for dealing with periodic homogenization problems. In the present paper we develop this theory to generalized Besicovitch spaces,...

Quasi–static studies of the deformation and failure of PBX 9501

Abstract: This paper examines the influence of microstructure on the quasi–static failure of PBX 9501, a polymer–bonded explosive (PBX) manufactured for the Los Alamos National Laboratory in America. Optical microscopy has been used to examine qualitatively cracked and pristine material. Consequent on the manufacturing process, the explosive crystals display angular features and natural facets. In addition, considerable growth twinning, internal defects and voidage has been observed. These defects are found significantly to alter the failure path. In common with other PBXs, failure paths tend to run around the long straight edges of the explosive filler and avoid regions of fine filler and binder. Explosive crystals were found to fracture due either to cracks propagating from another region or internal defects. These observations are confirmed by the use of high–resolution moire interferometry. This sensitive optical technique allows the deformation of the sample to be measured up to and including the point of failure. By taking white–light micrographs that are in exact registration with the measured displacement maps, the influence of the underlying microstructure can be seen.

Vibrations of a shallow cable with a viscous damper

Abstract: The optimal tuning and effect in terms of modal damping of a viscous damper mounted near the end of a shallow cable are investigated. The damping properties of free vibrations are extracted from the complex wavenumber. The full solution for the lower modes is evaluated numerically, and an explicit and rather accurate analytical approximation is obtained, generalizing recent results for a taut cable. It is found that the effect of the damper on the nearly antisymmetric modes is independent of the sag and the stiffness parameter. In contrast, the nearly symmetric modes develop regions of reduced motion near the ends, with increasing cable stiffness, and this reduces the effect of the viscous damper. Explicit results are obtained for the modal damping ratio and for optimal tuning of the damper.

The size effect in foams and its theoretical and numerical investigation

Abstract: Experimental data for foams lead to different values of the elastic moduli depending on the performed test, i.e. compression and tension tests give a different set of parameters than shear and bending tests. This may be explained by the size effect, which depends on the microstructure of the foams. Thus, in this paper, the behaviour of foams is investigated on the basis of both microscopic and macroscopic mechanical models. The microscopic approach is based on a lattice beam model. The solution of this model shows that the boundary–layer effect is strongly local but allows for the explanation of the size effect. Furthermore, the size effect can be included in the macroscopic continuum model by application of a Cosserat formulation. The extended continuum model allows for the independent fit of material parameters to different load cases, i.e. to compression and shear. The solution of the macroscopic Cosserat model permits a relation of the internal length–scale to the average cell size of the microstructure.

Asymptotic moments of near–neighbour distance distributions

Abstract: Let C be a compact convex body in R⊃ m and consider a set of points selected at random from C according to some well–behaved sampling distribution. We obtain an asymptotic expression for the positive moments of the k th near–neighbour distance distribution as the number of points increases to infinity.

On Green's function for a three-dimensional exponentially graded elastic solid

Abstract: The problem of a point force acting in an unbounded, three–dimensional, isotropic elastic solid is considered. Kelvin solved this problem for homogeneous materials. Here, the material is inhomogeneous; it is ‘functionally graded’. Specifically, the solid is ‘exponentially graded’, which means that the Lame moduli vary exponentially in a given fixed direction. The solution for the Green9s function is obtained by Fourier transforms, and consists of a singular part, given by the Kelvin solution, plus a non–singular remainder. This grading term is not obtained in simple closed form, but as the sum of single integrals over finite intervals of modified Bessel functions, and double integrals over finite regions of elementary functions. Knowledge of this new fundamental solution for graded materials permits the development of boundary–integral methods for these technologically important inhomogeneous solids.

Supercoiling of DNA plasmids: mechanics of the generalized ply

Abstract: In this paper we address the mechanics of ply formation in DNA supercoils. We extend the variable ply formulation of Coleman & Swigon to include end loads, and the derived constitutive relations of...