Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 2012"

Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

Abstract: We study the existence of positive solutions for the nonlinear Schrodinger equation with the fractional LaplacianFurthermore, we analyse the regularity, decay and symmetry properties of these solutions.

General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation

Abstract: General high-order rogue waves in the nonlinear Schrodinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the general N -th order rogue waves contain N −1 free irreducible complex parameters. In addition, the specific rogue waves obtained by Akhmediev et al. (Akhmediev et al. 2009 Phys. Rev. E 80 , 026601 (doi:10.1103/PhysRevE.80.026601)) correspond to special choices of these free parameters, and they have the highest peak amplitudes among all rogue waves of the same order. If other values of these free parameters are taken, however, these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns.

Geostatistics of Extremes

Abstract: We describe a prototype approach to flexible modelling for maxima observed at sites in a spatial domain, based on fitting of max-stable processes derived from underlying Gaussian random fields. The models we propose have generalized extreme-value marginal distributions throughout the spatial domain, consistent with statistical theory for maxima in simpler cases, and can incorporate both geostatistical correlation functions and random set components. Parameter estimation and fitting are performed through composite likelihood inference applied to observations from pairs of sites, with occurrence times of maxima taken into account if desired, and competing models are compared using appropriate information criteria. Diagnostics for lack of model fit are based on maxima from groups of sites. The approach is illustrated using annual maximum temperatures in Switzerland, with risk analysis proposed using simulations from the fitted max-stable model. Drawbacks and possible developments of the approach are discussed.

From Wrinkles to Creases in Elastomers: The Instability and Imperfection-Sensitivity of Wrinkling

Abstract: The stability of the wrinkling mode experienced by a compressed half-space of neo-Hookean material is investigated using analytical and numerical methods to study the post-bifurcation behaviour of periodic solutions. It is shown that wrinkling is highly unstable owing to the nonlinear interaction among the multiple modes associated with the critical compressive state. Concomitantly, wrinkling is sensitive to exceedingly small initial imperfections that significantly reduce the compressive strain at which the instability occurs. The study provides insight into the connection between wrinkling and an alternative surface mode, the finite amplitude crease or sulcus. The shape of the critical combination of wrinkling modes has the form of an incipient crease, and a tiny initial imperfection can trigger a wrinkling instability that collapses into a crease.

Tin-containing silicates: structure-activity relations

Abstract: The selective conversion of biomass-derived substrates is one of the major challenges facing the chemical industry. Recently, stannosilicates have been employed as highly active and selective Lewis acid catalysts for a number of industrially relevant reactions. In the present work, four different stannosilicates have been investigated: Sn-BEA, Sn-MFI, Sn-MCM-41 and Sn-SBA-15. When comparing the properties of tin sites in the structures, substantial differences are observed. Sn-beta displays the highest Lewis acid strength, as measured by probe molecule studies using infrared spectroscopy, which gives it a significantly higher activity at low temperatures than the other structures investigated. Furthermore, the increased acid strength translates into large differences in selectivity between the catalysts, thus demonstrating the influence of the structure on the active site, and pointing the way forward for tailoring the active site to the desired reaction.

Revisiting the rocking block: closed-form solutions and similarity laws

Abstract: In this paper, the dynamic response of the rocking block subjected to base excitation is revisited. The goal is to offer new closed-form solutions and original similarity laws that shed light on the fundamental aspects of the rocking block. The focus is on the transient dynamics of the rocking block under finite-duration excitations. An alternative way to describe the response of the rocking block, informative of the behaviour of rocking structures under excitations of different intensity, is offered. In the process, limitations of standard dimensional analysis, related to the orientations of the involved physical quantities, are revealed. The proposed dimensionless and orientationless groups condense the response and offer a lucid depiction of the rocking phenomenon. When expressed in the appropriate dimensionless–orientationless groups, the rocking response becomes perfectly self-similar for slender blocks (within the small rotations range) and practically self-similar for non-slender blocks (larger rotations). Using this formulation, the nonlinear and non-smooth rocking response to pulse-type ground motion can be directly determined, and need only be scaled by the intensity and frequency of the excitation.

Upper and lower limits on the stability of calving glaciers from the yield strength envelope of ice

Abstract: Observations indicate that substantial changes in the dynamics of marine-terminating ice sheets and glaciers are tightly coupled to calving-induced changes in the terminus position. However, the calving process itself remains poorly understood and is not well parametrized in current numerical ice sheet models. In this study, we address this uncertainty by deriving plausible upper and lower limits for the maximum stable ice thickness at the calving face of marine-terminating glaciers, using two complementary models. The first model assumes that a combination of tensile and shear failure can render the ice cliff near the terminus unstable and/or enable pre-existing crevasses to intersect. A direct consequence of this model is that thick glaciers must terminate in deep water to stabilize the calving front, yielding a predicted maximum ice cliff height that increases with increasing water depth, consistent with observations culled from glaciers in West Greenland, Antarctica, Svalbard and Alaska. The second model considers an analogous lower limit derived by assuming that the ice is already fractured and fractures are lubricated by pore pressure. In this model, a floating ice tongue can only form when the ice entering the terminus region is relatively intact with few pre-existing, deeply penetrating crevasses.

Electrostatics of two charged conducting spheres

Abstract: We prove that two charged conducting spheres will almost always attract each other at close approach, even when they have like charges. The one exception is when the two spheres have the same charge ratio that they would obtain by being brought into contact. In this case, they repel, and we derive an analytical expression for the force at contact, for any size ratio, generalizing a force formula for equal spheres obtained by Kelvin in 1853. We also give the electrostatic energy of two arbitrarily charged spheres, and its analytical forms at large and small separations. Expressions are derived for the surface charge densities of the two spheres. Attraction occurs between two positively charged spheres because of mutual polarization: one of the spheres obtains a negatively charged region (neighbouring the other sphere).

An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process

Abstract: We analyse the strong approximation of the Cox–Ingersoll–Ross (CIR) process in the regime where the process does not hit zero by a positivity preserving drift-implicit Euler-type method. As an error criterion, we use the p th mean of the maximum distance between the CIR process and its approximation on a finite time interval. We show that under mild assumptions on the parameters of the CIR process, the proposed method attains, up to a logarithmic term, the convergence of order 1/2. This agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations with Lipschitz coefficients, despite the fact that the CIR process has a non-Lipschitz diffusion coefficient.

Vibration and buckling analysis of a piezoelectric nanoplate considering surface effects and in-plane constraints

Abstract: This work investigates the surface effects on the vibration and buckling behaviour of a simply supported piezoelectric nanoplate (PNP) by using a modified Kirchhoff plate model. Two kinds of in-plane constraints are defined for the PNP, and the surface effects are accounted in the modified plate theory through the surface piezoelectricity model and the generalized Young–Laplace equations. Simulation results show that the influence of surface effects on the plate resonant frequency depends on the in-plane constraints significantly. For the PNP with different in-plane constraints, the effects of the applied electric potential, the mode number, the plate aspect ratio and the plate thickness on the resonant frequency are examined with consideration of the surface effects. The possible mechanical buckling of the PNP is also studied, and it is found that the surface effects on the critical electric voltage for buckling are sensitive to the plate thickness and aspect ratio. Our results also reveal that there exists a critical transition point at which the combined surface effects on the critical electric voltage may vanish under certain conditions.

Features for damage detection with insensitivity to environmental and operational variations

Abstract: This paper explores and compares the application of three different approaches to the data normalization problem in structural health monitoring (SHM), which concerns the removal of confounding trends induced by varying operational conditions from a measured structural response that correlates with damage. The methodologies for singling out or creating damage-sensitive features that are insensitive to environmental influences explored here include cointegration, outlier analysis and an approach relying on principal component analysis. The application of cointegration is a new idea for SHM from the field of econometrics, and this is the first work in which it has been comprehensively applied to an SHM problem. Results when applying cointegration are compared with results from the more familiar outlier analysis and an approach that uses minor principal components. The ability of these methods for removing the effects of environmental/operational variations from damage-sensitive features is demonstrated and compared with benchmark data from the Brite-Euram project DAMASCOS (BE97 4213), which was collected from a Lamb-wave inspection of a composite panel subject to temperature variations in an environmental chamber.

Double twinning mechanisms in magnesium alloys via dissociation of lattice dislocations

Abstract: In this work, we propose dislocation mechanisms for the formation of double twin structures in hexagonal close packed (HCP) crystals through the nucleation of secondary twins within primary twin do...

Crystal plasticity analysis of micro-deformation, lattice rotation and geometrically necessary dislocation density

Abstract: A gradient-enhanced crystal plasticity model is presented that explicitly accounts for the evolution of the densities of geometrically necessary dislocations (GNDs) on individual slip systems of deforming crystals. The GND densities are fully coupled with the crystal slip rule. Application of the model to two distinct and technologically important crystal types, namely hcp Ti and ccp Ni, is given. For the hcp crystals, slip is permitted with a-type slip directions on basal, prismatic and pyramidal planes and c + a-type slip directions on pyramidal planes. First, a single crystal under four-point bending is simulated as the uniform strain gradient expected in the central span provides a good validation of the code. Then, uniaxial deformation of a model near-a Ti polycrystal has been analysed. The resulting distributions of GND densities that develop on the various slip system types have been compared with independent experimental observations. The model predicts that GND density on the c + a systems is approximately an order of magnitude lower than that for a-type systems in agreement with experiment. For the ccp case, slip is considered to take place on the {111} slip systems. Thermal loading of a single-crystal nickel alloy sample containing carbide particles of size approximately 30mm has been analysed. Detailed comparisons are presented between model predictions and results of high-resolution electron backscatter diffraction (EBSD) measurements of the micro-deformations, lattice rotations, curvatures and GND densities local to the nickel–carbide interface. Qualitatively, good agreement is achieved between the coupled and decoupled model elastic strains with the EBSD measurements, but lattice rotations and GND densities are quantitatively well predicted by the coupled crystal model but are less well captured by the decoupled model. The GND coupling is found to lead to reduced lattice rotations and plastic strains in the region of highest heterogeneity close to the Ni matrix/particle interface, which is in agreement with the experimental measurements. The results presented provide objective evidence of the effectiveness of gradient-enhanced crystal plasticity finite element analysis and demonstrate that GND coupling is required in order to capture strains and lattice rotations in regions of high heterogeneity.

Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems

Abstract: Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.

Manipulation of particles in two dimensions using phase controllable ultrasonic standing waves

Abstract: The ability to manipulate dense micrometre-scale objects in fluids is of interest to biosciences with a view to improving analysis techniques and enabling tissue engineering. A method of trapping micrometre-scale particles and manipulating them on a two-dimensional plane is proposed and demonstrated. Phase-controlled counter-propagating waves are used to generate ultrasonic standing waves with arbitrary nodal positions. The acoustic radiation force drives dense particles to pressure nodes. It is shown analytically that a series of point-like traps can be produced in a two-dimensional plane using two orthogonal pairs of counter-propagating waves. These traps can be manipulated by appropriate adjustment of the relative phases. Four 5 MHz transducers (designed to minimize reflection) are used as sources of counter-propagating waves in a water-filled cavity. Polystyrene beads of 10 μm diameter are trapped and manipulated. The relationship between trapped particle positions and the relative phases of the four transducers is measured and shown to agree with analytically derived expressions. The force available is measured by determining the response to a sudden change in field and found to be 30 pN, for a 30 Vpp input, which is in agreement with the predictions of models of the system. A scalable fabrication approach to producing devices is demonstrated.

A viscoplastic theory of shape memory polymer fibres with application to self-healing materials

Abstract: The difficulty in healing structural damage is that most existing schemes need external help to bring the fractured surfaces in contact before healing can occur. To facilitate the existing schemes to heal macroscopic cracks, we envision that the cracked surfaces can be brought in contact through constrained shape recovery of a shape memory polymer (SMP) fibre-reinforced grid skeleton that is embedded in thermoset polymer matrix, similar to stitch a cut in the human skin by suture. In this study, we show that polyurethane SMP fibres can be hardened through cyclic cold-drawing programming, which makes them suitable for reinforcement and healing in thermoset polymer composites. We characterized the microstructure of the SMP fibres, which provides fundamental understanding of the effect of programming on the degree of crystallinity and molecular orientation. Then, a micromechanical multiscale viscoplastic theory is developed to predict the thermomechanical behaviours of the SMP fibres, including the cyclic hardening and stress recovery responses. The proposed theory takes into account the stress-induced crystallization process and the evolution of the morphological texture based on the applied stresses. The cyclic loading and the thermomechanical responses of the SMP fibres confirm the capabilities of the proposed model in capturing these phenomena.

Folding wrinkles of a thin stiff layer on a soft substrate

Abstract: We present the mechanics of folding surface-layer wrinkles on a soft substrate, i.e. inter-touching of neighbouring wrinkle surfaces without forming a cusp. Upon laterally compressing a stiff layer attached on a finite-elastic substrate, certain material nonlinearities trigger a number of bifurcation processes to form multi-mode wrinkle clusters. Some of these clusters eventually develop into folded wrinkles. The first bifurcation of the multi-mode wrinkles is investigated by a perturbation analysis of the surface-layer buckling on a pre-stretched neo-Hookean substrate. The post-buckling equilibrium configurations of the wrinkles are then trailed experimentally and computationally until the wrinkles are folded. The folding process is observed at various stages of wrinkling, by sectioning 20–80 nm thick gold films deposited on a polydimethylsiloxane substrate at a stretch ratio of 2.1. Comparison between the experimental observation and the finite-element analysis shows that the Ogden model deformation of the substrate coupled with asymmetric bending of the film predicts the folding process closely. In contrast, if the bending stiffness of the film is symmetric or the substrate follows the neo-Hookean behaviour, then the wrinkles are hardly folded. The wrinkle folding is applicable to construction of long parallel nano/micro-channels and control of exposing functional surface areas.

Metal–organic frameworks of the MIL-101 family as heterogeneous single-site catalysts

Abstract: In this short review paper, we survey our recent findings in the catalytic applications of mesoporous metal–organic frameworks of the MIL-101 family (Fe- and Cr-MIL-101) and demonstrate their potential in two types of liquid-phase processes: (i) selective oxidation of hydrocarbons with green oxidants—O2 and tert -butyl hydroperoxide—and (ii) coupling reaction of organic oxides with CO2. A comparison with conventional single-site catalysts is made with special attention to issues of the catalyst's resistance to metal leaching and the nature of catalysis.

Time scales for dynamical relaxation to the Born rule

Abstract: We illustrate through explicit numerical calculations how the Born rule probability densities of non-relativistic quantum mechanics emerge naturally from the particle dynamics of de BroglieBohm pil...

Nonlinear pre-stress for cloaking from antiplane elastic waves

Abstract: A theory is presented showing that cloaking of objects from antiplane elastic waves can be achieved by elastic pre-stress of a neo-Hookean nonlinear elastic material. This approach would appear to eliminate the requirement of metamaterials with inhomogeneous anisotropic shear moduli and density. Waves in the pre-stressed medium are bent around the cloaked region by inducing inhomogeneous stress fields via pre-stress. The equation governing antiplane waves in the pre-stressed medium is equivalent to the antiplane equation in an unstressed medium with inhomogeneous and anisotropic shear modulus and isotropic scalar mass density. Note however that these properties are induced naturally by the pre-stress. Since the magnitude of pre-stress can be altered at will, this enables objects of varying size and shape to be cloaked by placing them inside the fluid-filled deformed cavity region.

The thickness of the falling film of liquid around a Taylor bubble

Abstract: We present the results of laboratory experiments that quantify the physical controls on the thickness of the falling film of liquid around a Taylor bubble, when liquidgas interfacial tension can be...

Three-dimensional cellular structures with negative Poisson's ratio and negative compressibility properties

Abstract: A three-dimensional cellular system that may be made to exhibit some very unusual but highly useful mechanical properties, including negative Poisson’s ratio (auxetic), zero Poisson’s ratio, negative linear and negative area compressibility, is proposed and discussed. It is shown that such behaviour is scale-independent and may be obtained from particular conformations of this highly versatile system. This model may be used to explain the auxetic behaviour in auxetic foams and in other related cellular systems; such materials are widely known for their superior performance in various practical applications. It may also be used as a blueprint for the design and manufacture of new man-made multifunctional systems, including auxetic and negative compressibility systems, which can be made to have tailor-made mechanical properties.

Multi-stable composite twisting structure for morphing applications

Abstract: Conventional shape-changing engineering structures use discrete parts articulated around a number of linkages. Each part carries the loads, and the articulations provide the degrees of freedom of the system, leading to heavy and complex mechanisms. Consequently, there has been increased interest in morphing structures over the past decade owing to their potential to combine the conflicting requirements of strength, flexibility and low mass. This article presents a novel type of morphing structure capable of large deformations, simply consisting of two pre-stressed flanges joined to introduce two stable configurations. The bistability is analysed through a simple analytical model, predicting the positions of the stable and unstable states for different design parameters and material properties. Good correlation is found between experimental results, finite-element modelling and predictions from the analytical model for one particular example. A wide range of design parameters and material properties is also analytically investigated, yielding a remarkable structure with zero stiffness along the twisting axis.

On the calculation of an attenuation coefficient for transects of ice-covered ocean

Abstract: Exponential attenuation of ocean surface waves in ice-covered regions of the polar seas is modelled in a two-dimensional, linear setting, assuming that the sea ice behaves as a thin-elastic plate. Attenuation is produced by natural features in the ice cover, with three types considered: floes, cracks and pressure ridges. An inelastic damping parameterization is also incorporated. Efficient methods for obtaining an attenuation coefficient for each class of feature, involving an investigation of wave interaction theory and averaging methods, are sought. It is found that (i) the attenuation produced by long floes can be obtained from the scattering properties of a single ice edge; and (ii) wave interaction theory in ice-covered regions requires evanescent and damped-propagating motions to be included when scattering sources are relatively nearby. Implications for the integration of this model into an oceanic general circulation model are also discussed.

Statistical approach to normalization of feature vectors and clustering of mixed datasets

Abstract: Normalization of feature vectors of datasets is widely used in a number of fields of data mining, in particular in cluster analysis, where it is used to prevent features with large numerical values...

On the auxetic properties of generic rotating rigid triangles

Abstract: Materials having a negative Poisson9s ratio (auxetic) get fatter rather than thinner when uniaxially stretched. This phenomenon has been often explained through models that describe how particular geometric features in the micro or nanostructure of the material deform when subjected to uniaxial loads. Here, a new model based on scalene rigid triangles rotate relative to each other will be presented and analysed. It is shown that this model can afford a very wide range of Poisson9s ratio values, the sign and magnitude of which depends on the shape of the triangles and the angles between them. This new model has the advantage that it is very generic and may be potentially used to describe the properties in various types of materials, including auxetic foams and their relative surface density. Specific applications of this model, such as a blueprint for a system that can exhibit temperature-dependent Poisson9s ratios, are also discussed.

Quantification of undersea gas leaks from carbon capture and storage facilities, from pipelines and from methane seeps, by their acoustic emissions

Abstract: In recent years, because of the importance of leak detection from carbon capture and storage facilities and the need to monitor methane seeps and undersea gas pipelines, there has been an increased requirement for methods of detecting bubbles released from the seabed into the water column. If undetected and uncorrected, such leaks can generate huge financial and environmental losses. This paper describes a theory by which the passive acoustic signals detected by a hydrophone array can be used to quantify gas leakage, providing a practical (as opposed to research), passive and remote detection system which can monitor over a period of years using simple instrumentation. The sensitivity in detecting and quantifying the flux of gas is shown to exceed by more than two orders of magnitude the sensitivity of the current model-based techniques used commercially for gas leaks from large, long pipelines.

A theory of anisotropic healing and damage mechanics of materials

Abstract: Self-healing smart materials have emerged into the research arena and have been deployed in industrial and biomedical applications, in which the modelling techniques and predicting schemes are crucial for designers to optimize these smart materials. In practice, plastic deformation is coupled with damage and healing in these systems, which necessitates a coupled formulation for characterization. The thermodynamics of inelastic deformation, damage and healing processes are incorporated here to establish the coupled constitutive equations for healing materials. This thermodynamic consistent formulation provides the designers with the ability to predict the irregular inelastic deformation of glassy polymers and damage and healing patterns for a highly anisotropic self-healing system. Moreover, the lack of a physically consistent method to measure and calibrate the healing process in the literature is addressed here. Within the continuum damage mechanics (CDM) framework, the physics of damage and healing processes is used to introduce the healing effect into the CDM concept and a set of two new anisotropic damage–healing variables are derived. These novel damage–healing variables together with the proposed thermodynamic consistent coupled theory constitute a well-structured method for accurately predicting the degradation and healing mechanisms in material systems. The inelastic and damage response for a shape memory polymer-based self-healing system is captured herein. While the healing experimental results are limited in the literature, the proposed theory provides the mathematical competency to capture the most nonlinear responses.

Overall dynamic properties of three-dimensional periodic elastic composites

Abstract:  This article presents a method for the homogenization of three-dimensional periodic elastic composites. It allows for the evaluation of the averaged overall frequencydependent dynamic material constitutive tensors relating the averaged dynamic field variable tensors of velocity, strain, stress and linear momentum. Although the form of the dynamic constitutive relation for three-dimensional elastodynamic wave propagation has been known, this is the first time that explicit calculations of the effective parameters (for three dimensions) are presented. We show that for three-dimensional periodic composites, the overall compliance (stiffness) tensor, as produced directly by our formulation, is Hermitian, regardless of whether the corresponding unit cell is geometrically or materially symmetric. Overall, mass density is shown to be a tensor and, like the overall compliance tensor, always Hermitian. The average strain and linear momentum tensors are, however, coupled, and the coupling tensors are shown to be each others’ Hermitian transpose. Finally, we present a numerical example of a three-dimensional periodic composite composed of elastic cubes periodically distributed in an elastic matrix. The presented results corroborate the predictions of the theoretical treatment illustrating the frequency dependence of the constitutive parameters. We also show that the effective properties calculated in this paper satisfy the dispersion relation of the composite.

Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids

Abstract: Transformation elasticity, by analogy with transformation acoustics and optics, converts material domains without altering wave properties, thereby enabling cloaking and related effects. By noting ...