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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- recent issues1471-2946Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[Textural equilibrium melt geometries around tetrakaidecahedral grains]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170639?rss=1
In textural equilibrium, partially molten materials minimize the total surface energy bound up in grain boundaries and grain–melt interfaces. Here, numerical calculations of such textural equilibrium geometries are presented for a space-filling tessellation of grains with a tetrakaidecahedral (truncated octahedral) unit cell. Two parameters determine the nature of the geometries: the porosity and the dihedral angle. A variety of distinct melt topologies occur for different combinations of these two parameters, and the boundaries between different topologies have been determined. For small dihedral angles, wetting of grain boundaries occurs once the porosity has exceeded 11%. An exhaustive account is given of the main properties of the geometries: their energy, pressure, mean curvature, contiguity and areas on cross sections and faces. Their effective permeabilities have been calculated, and demonstrate a transition between a quadratic variation with porosity at low porosities to a cubic variation at high porosities.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0639hwp:master-id:royprsa;rspa.2017.06392018-04-11Research articles47422122017063920170639<![CDATA[Computing diffusivities from particle models out of equilibrium]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170694?rss=1
A new method is proposed to numerically extract the diffusivity of a (typically nonlinear) diffusion equation from underlying stochastic particle systems. The proposed strategy requires the system to be in local equilibrium and have Gaussian fluctuations but it is otherwise allowed to undergo arbitrary out-of-equilibrium evolutions. This could be potentially relevant for particle data obtained from experimental applications. The key idea underlying the method is that finite, yet large, particle systems formally obey stochastic partial differential equations of gradient flow type satisfying a fluctuation–dissipation relation. The strategy is here applied to three classic particle models, namely independent random walkers, a zero-range process and a symmetric simple exclusion process in one space dimension, to allow the comparison with analytic solutions.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0694hwp:master-id:royprsa;rspa.2017.06942018-04-11Research articles47422122017069420170694<![CDATA[Stable, high-order computation of impedance-impedance operators for three-dimensional layered medium simulations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170704?rss=1
The faithful modelling of the propagation of linear waves in a layered, periodic structure is of paramount importance in many branches of the applied sciences. In this paper, we present a novel numerical algorithm for the simulation of such problems which is free of the artificial singularities present in related approaches. We advocate for a surface integral formulation which is phrased in terms of impedance–impedance operators that are immune to the Dirichlet eigenvalues which plague the Dirichlet–Neumann operators that appear in classical formulations. We demonstrate a high-order spectral algorithm to simulate these latter operators based upon a high-order perturbation of surfaces methodology which is rapid, robust and highly accurate. We demonstrate the validity and utility of our approach with a sequence of numerical simulations.
]]>2018-04-04T00:05:20-07:00info:doi/10.1098/rspa.2017.0704hwp:master-id:royprsa;rspa.2017.07042018-04-04Research articles47422122017070420170704<![CDATA[Effects of Lewis number on the statistics of the invariants of the velocity gradient tensor and local flow topologies in turbulent premixed flames]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170706?rss=1
The behaviours of the three invariants of the velocity gradient tensor and the resultant local flow topologies in turbulent premixed flames have been analysed using three-dimensional direct numerical simulation data for different values of the characteristic Lewis number ranging from 0.34 to 1.2. The results have been analysed to reveal the statistical behaviours of the invariants and the flow topologies conditional upon the reaction progress variable. The behaviours of the invariants have been explained in terms of the relative strengths of the thermal and mass diffusions, embodied by the influence of the Lewis number on turbulent premixed combustion. Similarly, the behaviours of the flow topologies have been explained in terms not only of the Lewis number but also of the likelihood of the occurrence of individual flow topologies in the different flame regions. Furthermore, the sensitivity of the joint probability density function of the second and third invariants and the joint probability density functions of the mean and Gaussian curvatures to the variation in Lewis number have similarly been examined. Finally, the dependences of the scalar--turbulence interaction term on augmented heat release and of the vortex-stretching term on flame-induced turbulence have been explained in terms of the Lewis number, flow topology and reaction progress variable.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0706hwp:master-id:royprsa;rspa.2017.07062018-04-11Research articles47422122017070620170706<![CDATA[Wrinkles and creases in the bending, unbending and eversion of soft sectors]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170827?rss=1
We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asymptotic solutions for thin sectors. When the deformations are severe enough, the compressed side of the elastic material may buckle and wrinkles could then develop. We analyse, in detail, the onset of this instability for the Mooney–Rivlin strain energy, which covers the cases of the neo-Hookean model in exact nonlinear elasticity and of third-order elastic materials in weakly nonlinear elasticity. In particular, the associated theoretical and numerical treatment allows us to predict the number and wavelength of the wrinkles. Guided by experimental observations, we finally look at the development of creases, which we simulate through advanced finite-element computations. In some cases, the linearized analysis allows us to predict correctly the number and the wavelength of the creases, which turn out to occur only a few per cent of strain earlier than the wrinkles.
]]>2018-04-18T00:29:50-07:00info:doi/10.1098/rspa.2017.0827hwp:master-id:royprsa;rspa.2017.08272018-04-18Research articles47422122017082720170827<![CDATA[Dilatancy induced ductile-brittle transition of shear band in metallic glasses]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170836?rss=1
Dilatancy-generated structural disordering, an inherent feature of metallic glasses (MGs), has been widely accepted as the physical mechanism for the primary origin and structural evolution of shear banding, as well as the resultant shear failure. However, it remains a great challenge to determine, to what degree of dilatation, a shear banding will evolve into a runaway shear failure. In this work, using in situ acoustic emission monitoring, we probe the dilatancy evolution at the different stages of individual shear band in MGs that underwent severely plastic deformation by the controlled cutting technology. A scaling law is revealed that the dilatancy in a shear band is linearly related to its evolution degree. A transition from ductile-to-brittle shear bands is observed, where the formers dominate stable serrated flow, and the latter lead to a runaway instability (catastrophe failure) of serrated flow. To uncover the underlying mechanics, we develop a theoretical model of shear-band evolution dynamics taking into account an atomic-scale deformation process. Our theoretical results agree with the experimental observations, and demonstrate that the atomic-scale volume expansion arises from an intrinsic shear-band evolution dynamics. Importantly, the onset of the ductile–brittle transition of shear banding is controlled by a critical dilatation.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0836hwp:master-id:royprsa;rspa.2017.08362018-04-11Research articles47422122017083620170836<![CDATA[Reflection from a multi-species material and its transmitted effective wavenumber]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170864?rss=1
We formally deduce closed-form expressions for the transmitted effective wavenumber of a material comprising multiple types of inclusions or particles (multi-species), dispersed in a uniform background medium. The expressions, derived here for the first time, are valid for moderate volume fractions and without restriction on the frequency. We show that the multi-species effective wavenumber is not a straightforward extension of expressions for a single species. Comparisons are drawn with state-of-the-art models in acoustics by presenting numerical results for a concrete and a water–oil emulsion in two dimensions. The limit of when one species is much smaller than the other is also discussed and we determine the background medium felt by the larger species in this limit. Surprisingly, we show that the answer is not the intuitive result predicted by self-consistent multiple scattering theories. The derivation presented here applies to the scalar wave equation with cylindrical or spherical inclusions, with any distribution of sizes, densities and wave speeds. The reflection coefficient associated with a halfspace of multi-species cylindrical inclusions is also formally derived.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0864hwp:master-id:royprsa;rspa.2017.08642018-04-11Research articles47422122017086420170864<![CDATA[A geometric framework for dynamics with unilateral constraints and friction, illustrated by an example of self-organized locomotion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170886?rss=1
We present a geometric framework to deal with mechanical systems which have unilateral constraints, and are subject to damping/friction, which cannot be treated within usual classical mechanics. In this new framework, the dynamical evolution of the system takes place on a multidimensional curvilinear polyhedron, and energetics near the corners of the polyhedron leads to qualitative behaviour such as stable entrapment and bifurcation. We illustrate this by an experiment in which dumbbells, placed inside a tilted hollow cylindrical drum that rotates slowly around its axis, climb uphill by forming dynamically stable pairs, seemingly against the pull of gravity.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0886hwp:master-id:royprsa;rspa.2017.08862018-04-11Research articles47422122017088620170886<![CDATA[An upper bound on the particle-laden dependency of shear stresses at solid-fluid interfaces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170332?rss=1
In modern advanced manufacturing processes, such as three-dimensional printing of electronics, fine-scale particles are added to a base fluid yielding a modified fluid. For example, in three-dimensional printing, particle-functionalized inks are created by adding particles to freely flowing solvents forming a mixture, which is then deposited onto a surface, which upon curing yields desirable solid properties, such as thermal conductivity, electrical permittivity and magnetic permeability. However, wear at solid–fluid interfaces within the machinery walls that deliver such particle-laden fluids is typically attributed to the fluid-induced shear stresses, which increase with the volume fraction of added particles. The objective of this work is to develop a rigorous strict upper bound for the tolerable volume fraction of particles that can be added, while remaining below a given stress threshold at a fluid–solid interface. To illustrate the bound’s utility, the expression is applied to a series of classical flow regimes.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0332hwp:master-id:royprsa;rspa.2017.03322018-03-21Research articles47422112017033220170332<![CDATA[Continuum modelling of segregating tridisperse granular chute flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170384?rss=1
Segregation and mixing of size multidisperse granular materials remain challenging problems in many industrial applications. In this paper, we apply a continuum-based model that captures the effects of segregation, diffusion and advection for size tridisperse granular flow in quasi-two-dimensional chute flow. The model uses the kinematics of the flow and other physical parameters such as the diffusion coefficient and the percolation length scale, quantities that can be determined directly from experiment, simulation or theory and that are not arbitrarily adjustable. The predictions from the model are consistent with experimentally validated discrete element method (DEM) simulations over a wide range of flow conditions and particle sizes. The degree of segregation depends on the Péclet number, Pe, defined as the ratio of the segregation rate to the diffusion rate, the relative segregation strength _{ij} between particle species i and j, and a characteristic length L, which is determined by the strength of segregation between smallest and largest particles. A parametric study of particle size, _{ij}, Pe and L demonstrates how particle segregation patterns depend on the interplay of advection, segregation and diffusion. Finally, the segregation pattern is also affected by the velocity profile and the degree of basal slip at the chute surface. The model is applicable to different flow geometries, and should be easily adapted to segregation driven by other particle properties such as density and shape.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0384hwp:master-id:royprsa;rspa.2017.03842018-03-14Research articles47422112017038420170384<![CDATA[A hydrostatic model of the Wirtz pump]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170533?rss=1
The Wirtz pump is not only an excellent example of alternative technology, using as it does the kinetic energy of a stream to raise a proportion of its water, but its mathematical modelling also poses several intriguing problems. We give some history of the Wirtz pump and describe its operation. Taking a novel dynamical systems approach, we then derive a discrete mathematical model in the form of a mapping that describes its hydrostatic behaviour. Our model enables us to explain several aspects of the behaviour of the pump as well as to design one that gives approximately maximal, and maximally constant, output pressure.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0533hwp:master-id:royprsa;rspa.2017.05332018-03-21Research articles47422112017053320170533<![CDATA[Resilience of riverbed vegetation to uprooting by flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170547?rss=1
Riverine ecosystem biodiversity is largely maintained by ecogeomorphic processes including vegetation renewal via uprooting and recovery times to flow disturbances. Plant roots thus heavily contribute to engineering resilience to perturbation of such ecosystems. We show that vegetation uprooting by flow occurs as a fatigue-like mechanism, which statistically requires a given exposure time to imposed riverbed flow erosion rates before the plant collapses. We formulate a physically based stochastic model for the actual plant rooting depth and the time-to-uprooting, which allows us to define plant resilience to uprooting for generic time-dependent flow erosion dynamics. This theory shows that plant resilience to uprooting depends on the time-to-uprooting and that root mechanical anchoring acts as a process memory stored within the plant–soil system. The model is validated against measured data of time-to-uprooting of Avena sativa seedlings with various root lengths under different flow conditions. This allows for assessing the natural variance of the uprooting-by-flow process and to compute the prediction entropy, which quantifies the relative importance of the deterministic and the random components affecting the process.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0547hwp:master-id:royprsa;rspa.2017.05472018-03-14Research articles47422112017054720170547<![CDATA[Vibration of carbon nanotubes with defects: order reduction methods]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170555?rss=1
Order reduction methods are widely used to reduce computational effort when calculating the impact of defects on the vibrational properties of nearly periodic structures in engineering applications, such as a gas-turbine bladed disc. However, despite obvious similarities these techniques have not yet been adapted for use in analysing atomic structures with inevitable defects. Two order reduction techniques, modal domain analysis and modified modal domain analysis, are successfully used in this paper to examine the changes in vibrational frequencies, mode shapes and mode localization caused by defects in carbon nanotubes. The defects considered are isotope defects and Stone–Wales defects, though the methods described can be extended to other defects.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0555hwp:master-id:royprsa;rspa.2017.05552018-03-14Research articles47422112017055520170555<![CDATA[Random distributions of initial porosity trigger regular necking patterns at high strain rates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170575?rss=1
At high strain rates, the fragmentation of expanding structures of ductile materials, in general, starts by the localization of plastic deformation in multiple necks. Two distinct mechanisms have been proposed to explain multiple necking and fragmentation process in ductile materials. One view is that the necking pattern is related to the distribution of material properties and defects. The second view is that it is due to the activation of specific instability modes of the structure. Following this, we investigate the emergence of necking patterns in porous ductile bars subjected to dynamic stretching at strain rates varying from 10^{3}s^{–1} to 0.5x10^{5}s^{–1} using finite-element calculations and linear stability analysis. In the calculations, the initial porosity (representative of the material defects) varies randomly along the bar. The computations revealed that, while the random distribution of initial porosity triggers the necking pattern, it barely affects the average neck spacing, especially, at higher strain rates. The average neck spacings obtained from the calculations are in close agreement with the predictions of the linear stability analysis. Our results also reveal that the necking pattern does not begin when the Considère condition is reached but is significantly delayed due to the stabilizing effect of inertia.
]]>2018-03-28T00:05:20-07:00info:doi/10.1098/rspa.2017.0575hwp:master-id:royprsa;rspa.2017.05752018-03-28Research articles47422112017057520170575<![CDATA[Rayleigh wave at the surface of a general anisotropic poroelastic medium: derivation of real secular equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170589?rss=1
A secular equation governs the propagation of Rayleigh wave at the surface of an anisotropic poroelastic medium. In the case of anisotropy with symmetry, this equation is obtained as a real irrational equation. But, in the absence of anisotropic symmetries, this secular equation is obtained as a complex irrational equation. True surface waves in non-dissipative materials decay only with depth. That means, propagation of Rayleigh wave in anisotropic poroelastic solid should be represented by a real phase velocity. In this study, the determinantal system leading to the complex secular equation is manipulated to obtain a transformed equation. Even for arbitrary (triclinic) anisotropy, this transformed equation remains real for the propagation of true surface waves. Such a real secular equation is obtained with the option of boundary pores being opened or sealed. A numerical example is solved to study the existence and propagation of Rayleigh waves in porous media for the top three (i.e. triclinic, monoclinic and orthorhombic) anisotropies.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0589hwp:master-id:royprsa;rspa.2017.05892018-03-21Research articles47422112017058920170589<![CDATA[Localization in semi-infinite herringbone waveguides]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170590?rss=1
The paper includes novel results for the scattering and localization of a time-harmonic flexural wave by a semi-infinite herringbone waveguide of rigid pins embedded within an elastic Kirchhoff plate. The analytical model takes into account the orientation and spacing of the constituent parts of the herringbone system, and incorporates dipole approximations for the case of closely spaced pins. Illustrative examples are provided, together with the predictive theoretical analysis of the localized waveforms.
]]>2018-03-28T00:05:20-07:00info:doi/10.1098/rspa.2017.0590hwp:master-id:royprsa;rspa.2017.05902018-03-28Research articles47422112017059020170590<![CDATA[Effects of geometric nonlinearity in an adhered microbeam for measuring the work of adhesion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170594?rss=1
Design against adhesion in microelectromechanical devices is predicated on the ability to quantify this phenomenon in microsystems. Previous research related the work of adhesion for an adhered microbeam to the beam's unadhered length, and as such, interferometric techniques were developed to measure that length. We propose a new vibration-based technique that can be easily implemented with existing atomic force microscopy tools or similar metrology systems. To make such a technique feasible, we analysed a model of the adhered microbeam using the nonlinear beam theory put forth by Woinowsky–Krieger. We found a new relation between the work of adhesion and the unadhered length; this relation is more accurate than the one by Mastrangelo & Hsu (Mastrangelo & Hsu 1993 J. Microelectromech. S., 2, 44–55. (doi:10.1109/84.232594)) which is commonly used. Then, we derived a closed-form approximate relationship between the microbeam's natural frequency and its unadhered length. Results obtained from this analytical formulation are in good agreement with numerical results from three-dimensional nonlinear finite-element analysis.
]]>2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2017.0594hwp:master-id:royprsa;rspa.2017.05942018-03-07Research articles47422112017059420170594<![CDATA[Nonlinear dynamics of a dispersive anisotropic Kuramoto-Sivashinsky equation in two space dimensions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170687?rss=1
A Kuramoto–Sivashinsky equation in two space dimensions arising in thin film flows is considered on doubly periodic domains. In the absence of dispersive effects, this anisotropic equation admits chaotic solutions for sufficiently large length scales with fully two-dimensional profiles; the one-dimensional dynamics observed for thin domains are structurally unstable as the transverse length increases. We find that, independent of the domain size, the characteristic length scale of the profiles in the streamwise direction is about 10 space units, with that in the transverse direction being approximately three times larger. Numerical computations in the chaotic regime provide an estimate for the radius of the absorbing ball in L2 in terms of the length scales, from which we conclude that the system possesses a finite energy density. We show the property of equipartition of energy among the low Fourier modes, and report the disappearance of the inertial range when solution profiles are two-dimensional. Consideration of the high-frequency modes allows us to compute an estimate for the analytic extensibility of solutions in C2. We also examine the addition of a physically derived third-order dispersion to the problem; this has a destabilizing effect, in the sense of reducing analyticity and increasing amplitude of solutions. However, sufficiently large dispersion may regularize the spatio-temporal chaos to travelling waves. We focus on dispersion where chaotic dynamics persist, and study its effect on the interfacial structures, absorbing ball and properties of the power spectrum.
]]>2018-03-28T00:05:20-07:00info:doi/10.1098/rspa.2017.0687hwp:master-id:royprsa;rspa.2017.06872018-03-28Research articles47422112017068720170687<![CDATA[On the theory of drainage area for regular and non-regular points]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170693?rss=1
The drainage area is an important, non-local property of a landscape, which controls surface and subsurface hydrological fluxes. Its role in numerous ecohydrological and geomorphological applications has given rise to several numerical methods for its computation. However, its theoretical analysis has lagged behind. Only recently, an analytical definition for the specific catchment area was proposed (Gallant & Hutchinson. 2011 Water Resour. Res.47, W05535. (doi:10.1029/2009WR008540)), with the derivation of a differential equation whose validity is limited to regular points of the watershed. Here, we show that such a differential equation can be derived from a continuity equation (Chen et al. 2014 Geomorphology219, 68–86. (doi:10.1016/j.geomorph.2014.04.037)) and extend the theory to critical and singular points both by applying Gauss’s theorem and by means of a dynamical systems approach to define basins of attraction of local surface minima. Simple analytical examples as well as applications to more complex topographic surfaces are examined. The theoretical description of topographic features and properties, such as the drainage area, channel lines and watershed divides, can be broadly adopted to develop and test the numerical algorithms currently used in digital terrain analysis for the computation of the drainage area, as well as for the theoretical analysis of landscape evolution and stability.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0693hwp:master-id:royprsa;rspa.2017.06932018-03-14Research articles47422112017069320170693<![CDATA[Small nanoparticles, surface geometry and contact forces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170723?rss=1
In this molecular dynamics study, we examine the local surface geometric effects of the normal impact force between two approximately spherical nanoparticles that collide in a vacuum. Three types of surface geometries—(i) crystal facets, (ii) sharp edges, and (iii) amorphous surfaces of small nanoparticles with radii R<10 nm—are considered. The impact forces are compared with their macroscopic counterparts described by nonlinear contact forces based on Hertz contact mechanics. In our simulations, edge and amorphous surface contacts with weak surface energy reveal that the average impact forces are in excellent agreement with the Hertz contact force. On the other hand, facet collisions show a linearly increasing force with increasing compression. Our results suggest that the nearly spherical nanoparticles are likely to enable some nonlinear dynamic phenomena, such as breathers and solitary waves observed in granular materials, both originating from the nonlinear contact force.
]]>2018-03-21T01:44:55-07:00info:doi/10.1098/rspa.2017.0723hwp:master-id:royprsa;rspa.2017.07232018-03-21Research articles47422112017072320170723<![CDATA[Magic angles for fibrous incompressible elastic materials]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170728?rss=1
In the analysis of the mechanical behaviour of fibre-reinforced incompressible elastic bodies, there is a special angle of orientation of the fibres which leads to a particular mechanical response. This angle has been called a ‘magic angle’ due to its appearance as if by magic in many different aspects of the mechanics of fibrous solids including several examples in biology. It occurs most commonly not only in structural elements composed of circular cylindrical tubes or cylinders reinforced by helically wound fibres but also in flat thin sheets reinforced by fibres in the plane. The occurrence of such a special angle was classically demonstrated using a simple purely geometric analysis in the context of a lattice composed of a single family of helically wound inextensible fibres. Recently, the magic angle concept has been discussed in the framework of nonlinear hyperelasticity for anisotropic materials with detailed constitutive modelling. Our purpose here is to describe some other contexts in which the magic angle occurs starting from earlier work in a special theory of linear elasticity for inextensible fibres and proceeding to relatively accessible models of hyperelasticity. We discuss the role of the magic angle in the quasi-isotropic mechanical response of fibre-reinforced composites as well as the implications for material instability.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0728hwp:master-id:royprsa;rspa.2017.07282018-03-21Research articles47422112017072820170728<![CDATA[A canonical form of the equation of motion of linear dynamical systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170809?rss=1
The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.
]]>2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2017.0809hwp:master-id:royprsa;rspa.2017.08092018-03-07Research articles47422112017080920170809<![CDATA[Post-buckling of a pressured biopolymer spherical shell with the mode interaction]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170834?rss=1
Imperfection sensitivity is essential for mechanical behaviour of biopolymer shells characterized by high geometric heterogeneity. The present work studies initial post-buckling and imperfection sensitivity of a pressured biopolymer spherical shell based on non-axisymmetric buckling modes and associated mode interaction. Our results indicate that for biopolymer spherical shells with moderate radius-to-thickness ratio (say, less than 30) and smaller effective bending thickness (say, less than 0.2 times average shell thickness), the imperfection sensitivity predicted based on the axisymmetric mode without the mode interaction is close to the present results based on non-axisymmetric modes with the mode interaction with a small (typically, less than 10%) relative errors. However, for biopolymer spherical shells with larger effective bending thickness, the maximum load an imperfect shell can sustain predicted by the present non-axisymmetric analysis can be significantly (typically, around 30%) lower than those predicted based on the axisymmetric mode without the mode interaction. In such cases, a more accurate non-axisymmetric analysis with the mode interaction, as given in the present work, is required for imperfection sensitivity of pressured buckling of biopolymer spherical shells. Finally, the implications of the present study to two specific types of biopolymer spherical shells (viral capsids and ultrasound contrast agents) are discussed.
]]>2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2017.0834hwp:master-id:royprsa;rspa.2017.08342018-03-07Research articles47422112017083420170834<![CDATA[Manipulating nanoparticle transport within blood flow through external forces: an exemplar of mechanics in nanomedicine]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170845?rss=1
A large number of nanoparticles (NPs) have been raised for diverse biomedical applications and some of them have shown great potential in treatment and imaging of diseases. Design of NPs is essential for delivery efficacy due to a number of biophysical barriers, which prevents the circulation of NPs in vascular flow and their accumulation at tumour sites. The physiochemical properties of NPs, so-called ‘4S’ parameters, such as size, shape, stiffness and surface functionalization, play crucial roles in their life journey to be delivered to tumour sites. NPs can be modified in various ways to extend their blood circulation time and avoid their clearance by phagocytosis, and efficiently diffuse into tumour cells. However, it is difficult to overcome these barriers simultaneously by a simple combination of ‘4S’ parameters for NPs. At this moment, external triggerings are necessary to guide the movement of NPs, which include light, ultrasound, magnetic field, electrical field and chemical interaction. The delivery system can be constructed to be sensitive to these external stimuli which can reduce the non-specific toxicity and improve the efficacy of the drug-delivery system. From a mechanics point of view, we discuss how different forces play their roles in the margination of NPs in blood flow and tumour microvasculature.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0845hwp:master-id:royprsa;rspa.2017.08452018-03-21Review articles47422112017084520170845<![CDATA[Stochastic isotropic hyperelastic materials: constitutive calibration and model selection]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170858?rss=1
Biological and synthetic materials often exhibit intrinsic variability in their elastic responses under large strains, owing to microstructural inhomogeneity or when elastic data are extracted from viscoelastic mechanical tests. For these materials, although hyperelastic models calibrated to mean data are useful, stochastic representations accounting also for data dispersion carry extra information about the variability of material properties found in practical applications. We combine finite elasticity and information theories to construct homogeneous isotropic hyperelastic models with random field parameters calibrated to discrete mean values and standard deviations of either the stress–strain function or the nonlinear shear modulus, which is a function of the deformation, estimated from experimental tests. These quantities can take on different values, corresponding to possible outcomes of the experiments. As multiple models can be derived that adequately represent the observed phenomena, we apply Occam’s razor by providing an explicit criterion for model selection based on Bayesian statistics. We then employ this criterion to select a model among competing models calibrated to experimental data for rubber and brain tissue under single or multiaxial loads.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0858hwp:master-id:royprsa;rspa.2017.08582018-03-14Research articles47422112017085820170858<![CDATA[DC dynamic pull-in instability of a dielectric elastomer balloon: an energy-based approach]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170900?rss=1
This paper reports an energy-based method for the dynamic pull-in instability analysis of a spherical dielectric elastomer (DE) balloon subjected to a quasi-statically applied inflation pressure and a Heaviside step voltage across the balloon wall. The proposed technique relies on establishing the energy balance at the point of maximum stretch in an oscillation cycle, followed by the imposition of an instability condition for extracting the threshold parameters. The material models of the Ogden family are employed for describing the hyperelasticity of the balloon. The accuracy of the critical dynamic pull-in parameters is established by examining the saddle-node bifurcation in the transient response of the balloon obtained by integrating numerically the equation of motion, derived using the Euler–Lagrange equation. The parametric study brings out the effect of inflation pressure on the onset of the pull-in instability in the DE balloon. A quantitative comparison between the static and dynamic pull-in parameters at four different levels of the inflation pressure is presented. The results indicate that the dynamic pull-in instability gets triggered at electric fields that are lower than those corresponding to the static instability. The results of the present investigation can find potential use in the design and development of the balloon actuators subjected to transient loading. The method developed is versatile and can be used in the dynamic instability analysis of other conservative systems of interest.
]]>2018-03-28T00:05:20-07:00info:doi/10.1098/rspa.2017.0900hwp:master-id:royprsa;rspa.2017.09002018-03-28Research articles47422112017090020170900<![CDATA[Reviewers in 2017]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20180091?rss=1
2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2018.0091hwp:master-id:royprsa;rspa.2018.00912018-03-07Editorial47422112018009120180091<![CDATA[Correction to 'The ductile/brittle transition provides the critical test for materials failure theory]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20180114?rss=1
2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2018.0114hwp:master-id:royprsa;rspa.2018.01142018-03-21Correction47422112018011420180114<![CDATA[A time-domain method for prediction of noise radiated from supersonic rotating sources in a moving medium]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170089?rss=1
This paper presents a time-domain method for noise prediction of supersonic rotating sources in a moving medium. The proposed approach can be interpreted as an extensive time-domain solution for the convected permeable Ffowcs Williams and Hawkings equation, which is capable of avoiding the Doppler singularity. The solution requires special treatment for construction of the emission surface. The derived formula can explicitly and efficiently account for subsonic uniform constant flow effects on radiated noise. Implementation of the methodology is realized through the Isom thickness noise case and high-speed impulsive noise prediction from helicopter rotors.
]]>2018-02-14T00:05:26-08:00info:doi/10.1098/rspa.2017.0089hwp:master-id:royprsa;rspa.2017.00892018-02-14Research articles47422102017008920170089<![CDATA[Numerical study of the Kadomtsev-Petviashvili equation and dispersive shock waves]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170458?rss=1
A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev–Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.
]]>2018-02-14T00:30:26-08:00info:doi/10.1098/rspa.2017.0458hwp:master-id:royprsa;rspa.2017.04582018-02-14Research articles47422102017045820170458<![CDATA[Analysis and control of the dynamical response of a higher order drifting oscillator]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170500?rss=1
This paper studies a position feedback control strategy for controlling a higher order drifting oscillator which could be used in modelling vibro-impact drilling. Special attention is given to two control issues, eliminating bistability and suppressing chaos, which may cause inefficient and unstable drilling. Numerical continuation methods implemented via the continuation platform COCO are adopted to investigate the dynamical response of the system. Our analyses show that the proposed controller is capable of eliminating coexisting attractors and mitigating chaotic behaviour of the system, providing that its feedback control gain is chosen properly. Our investigations also reveal that, when the slider’s property modelling the drilled formation changes, the rate of penetration for the controlled drilling can be significantly improved.
]]>2018-02-21T00:47:25-08:00info:doi/10.1098/rspa.2017.0500hwp:master-id:royprsa;rspa.2017.05002018-02-21Research articles47422102017050020170500<![CDATA[Basis and regularity properties of (p,q)-trigonometric functions and the decay of (p,q)-Fourier coefficients]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170548?rss=1
The basis and regularity properties of the generalized trigonometric functions sinp,q and cosp,q are investigated. Upper bounds for the Fourier coefficients of these functions are given. Conditions are obtained under which the functions cosp,q generate a basis of every Lebesgue space L^{r}(0,1) with 1 < r < ; when q is the conjugate of p, it is sufficient to require that p[p_{1},p_{2}], where p_{1}<2 and p_{2}>2 are calculable numbers. A comparison is made of the speed of decay of the Fourier sine coefficients of a function in Lebesgue and Lorentz sequence spaces with that of the corresponding coefficients with respect to the functions sinp,q.These results sharpen previously known ones.
]]>2018-02-07T00:05:21-08:00info:doi/10.1098/rspa.2017.0548hwp:master-id:royprsa;rspa.2017.05482018-02-07Research articles47422102017054820170548<![CDATA[Intrinsic stochastic differential equations as jets]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170559?rss=1
We explain how Itô stochastic differential equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We also show how jets can be used to derive graphical representations of Itô SDEs, and we show how jets can be used to derive the differential operators associated with SDEs in a coordinate-free manner. We relate jets to vector flows, giving a geometric interpretation of the Itô–Stratonovich transformation. We show how percentiles can be used to give an alternative coordinate-free interpretation of the coefficients of one-dimensional SDEs. We relate this to the jet approach. This allows us to interpret the coefficients of SDEs in terms of ‘fan diagrams’. In particular, the median of an SDE solution is associated with the drift of the SDE in Stratonovich form for small times.
]]>2018-02-14T00:30:26-08:00info:doi/10.1098/rspa.2017.0559hwp:master-id:royprsa;rspa.2017.05592018-02-14Research articles47422102017055920170559<![CDATA[Nematic director fields and topographies of solid shells of revolution]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170566?rss=1
We solve the forward and inverse problems associated with the transformation of flat sheets with circularly symmetric director fields to surfaces of revolution with non-trivial topography, including Gaussian curvature, without a stretch elastic cost. We deal with systems slender enough to have a small bend energy cost. Shape change is induced by light or heat causing contraction along a non-uniform director field in the plane of an initially flat nematic sheet. The forward problem is, given a director distribution, what shape is induced? Along the way, we determine the Gaussian curvature and the evolution with induced mechanical deformation of the director field and of material curves in the surface (proto-radii) that will become radii in the final surface. The inverse problem is, given a target shape, what director field does one need to specify? Analytic examples of director fields are fully calculated that will, for specific deformations, yield catenoids and paraboloids of revolution. The general prescription is given in terms of an integral equation and yields a method that is generally applicable to surfaces of revolution.
]]>2018-02-21T00:47:25-08:00info:doi/10.1098/rspa.2017.0566hwp:master-id:royprsa;rspa.2017.05662018-02-21Research articles47422102017056620170566<![CDATA[Texture evolution and mechanical behaviour of irradiated face-centred cubic metals]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170604?rss=1
A physically based theoretical model is proposed to investigate the mechanical behaviour and crystallographic texture evolution of irradiated face-centred cubic metals. This model is capable of capturing the main features of irradiated polycrystalline materials including irradiation hardening, post-yield softening and plasticity localization. Numerical results show a good agreement with experimental data for both unirradiated and irradiated stress–strain relationships. The study of crystallographic texture reveals that the initial randomly distributed texture of unirradiated metals under tensile loading can evolve into a mixture of [111] and [100] textures. Regarding the irradiated case, crystallographic texture develops in a different way, and an extra part of [110] texture evolves into [100] and [111] textures. Thus, [100] and [111] textures become dominant more quickly compared with those of the unirradiated case for the reason that [100] and [111]-oriented crystals have higher strength, and their plastic deformation behaviours are more active than other oriented crystals. It can be concluded that irradiation-induced defects can affect both the mechanical behaviour and texture evolution of metals, both of which are closely related to irradiation hardening.
]]>2018-02-28T00:05:38-08:00info:doi/10.1098/rspa.2017.0604hwp:master-id:royprsa;rspa.2017.06042018-02-28Research articles47422102017060420170604<![CDATA[A generalized sound extrapolation method for turbulent flows]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170614?rss=1
Sound extrapolation methods are often used to compute acoustic far-field directivities using near-field flow data in aeroacoustics applications. The results may be erroneous if the volume integrals are neglected (to save computational cost), while non-acoustic fluctuations are collected on the integration surfaces. In this work, we develop a new sound extrapolation method based on an acoustic analogy using Taylor’s hypothesis (Taylor 1938 Proc. R. Soc. Lon. A164, 476–490. (doi:10.1098/rspa.1938.0032)). Typically, a convection operator is used to filter out the acoustically inefficient components in the turbulent flows, and an acoustics dominant indirect variable Dcp' is solved. The sound pressure p' at the far field is computed from Dcp' based on the asymptotic properties of the Green’s function. Validations results for benchmark problems with well-defined sources match well with the exact solutions. For aeroacoustics applications: the sound predictions by the aerofoil–gust interaction are close to those by an earlier method specially developed to remove the effect of vortical fluctuations (Zhong & Zhang 2017 J. Fluid Mech.820, 424–450. (doi:10.1017/jfm.2017.219)); for the case of vortex shedding noise from a cylinder, the off-body predictions by the proposed method match well with the on-body Ffowcs-Williams and Hawkings result; different integration surfaces yield close predictions (of both spectra and far-field directivities) for a co-flowing jet case using an established direct numerical simulation database. The results suggest that the method may be a potential candidate for sound projection in aeroacoustics applications.
]]>2018-02-14T00:05:26-08:00info:doi/10.1098/rspa.2017.0614hwp:master-id:royprsa;rspa.2017.06142018-02-14Research articles47422102017061420170614<![CDATA[Globular cluster formation and evolution in the context of cosmological galaxy assembly: open questions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170616?rss=1
We discuss some of the key open questions regarding the formation and evolution of globular clusters (GCs) during galaxy formation and assembly within a cosmological framework. The current state of the art for both observations and simulations is described, and we briefly mention directions for future research. The oldest GCs have ages greater than or equal to 12.5 Gyr and formed around the time of reionization. Resolved colour-magnitude diagrams of Milky Way GCs and direct imaging of lensed proto-GCs at z~6 with the James Webb Space Telescope (JWST) promise further insight. GCs are known to host multiple populations of stars with variations in their chemical abundances. Recently, such multiple populations have been detected in ~2 Gyr old compact, massive star clusters. This suggests a common, single pathway for the formation of GCs at high and low redshift. The shape of the initial mass function for GCs remains unknown; however, for massive galaxies a power-law mass function is favoured. Significant progress has been made recently modelling GC formation in the context of galaxy formation, with success in reproducing many of the observed GC-galaxy scaling relations.
]]>2018-02-14T00:05:27-08:00info:doi/10.1098/rspa.2017.0616hwp:master-id:royprsa;rspa.2017.06162018-02-14Review articles47422102017061620170616<![CDATA[Mobile assemblies of Bennett linkages from four-crease origami patterns]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170621?rss=1
This paper deals with constructing mobile assemblies of Bennett linkages inspired by four-crease origami patterns. A transition technique has been proposed by taking the thick-panel form of an origami pattern as an intermediate bridge. A zero-thickness rigid origami pattern and its thick-panel form share the same sector angles and folding behaviours, while the thick-panel origami and the mobile assembly of linkages are kinematically equivalent with differences only in link profiles. Applying this transition technique to typical four-crease origami patterns, we have found that the Miura-ori and graded Miura-ori patterns lead to assemblies of Bennett linkages with identical link lengths. The supplementary-type origami patterns with different mountain–valley crease assignments correspond to different types of Bennett linkage assemblies with negative link lengths. And the identical linkage-type origami pattern generates a new mobile assembly. Hence, the transition technique offers a novel approach to constructing mobile assemblies of spatial linkages from origami patterns.
]]>2018-02-07T00:05:20-08:00info:doi/10.1098/rspa.2017.0621hwp:master-id:royprsa;rspa.2017.06212018-02-07Research articles47422102017062120170621<![CDATA[Shear flow dynamics in the Beris-Edwards model of nematic liquid crystals]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170673?rss=1
We consider the Beris-Edwards model describing nematic liquid crystal dynamics and restrict it to a shear flow and spatially homogeneous situation. We analyse the dynamics focusing on the effect of the flow. We show that in the co-rotational case one has gradient dynamics, up to a periodic eigenframe rotation, while in the non-co-rotational case we identify the short- and long-time regimes of the dynamics. We express these in terms of the physical variables and compare with the predictions of other models of liquid crystal dynamics.
]]>2018-02-14T00:05:27-08:00info:doi/10.1098/rspa.2017.0673hwp:master-id:royprsa;rspa.2017.06732018-02-14Research articles47422102017067320170673<![CDATA[Pairwise approximation for SIR-type network epidemics with non-Markovian recovery]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170695?rss=1
We present the generalized mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic partial differential equation (PDE) system, where the population of infective nodes and links are structured by age since infection. We show that the PDE system can be reduced to a system of integro-differential equations, which is analysed analytically and numerically. We investigate the asymptotic behaviour of the generalized model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model, we recover known results for the exponentially distributed and fixed recovery time cases. For gamma- and uniformly distributed infectious periods, new pairwise models are derived. Theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation. A major benefit of the generalized pairwise model lies in approximating the time evolution of the epidemic.
]]>2018-02-21T00:47:25-08:00info:doi/10.1098/rspa.2017.0695hwp:master-id:royprsa;rspa.2017.06952018-02-21Research articles47422102017069520170695<![CDATA[On the homogenization of the acoustic wave propagation in perforated ducts of finite length for an inviscid and a viscous model]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170708?rss=1
The direct numerical simulation of the acoustic wave propagation in multiperforated absorbers with hundreds or thousands of tiny openings would result in a huge number of basis functions to resolve the microstructure. One is, however, primarily interested in effective and so homogenized transmission and absorption properties and how they are influenced by microstructure and its endpoints. For this, we introduce the surface homogenization that asymptotically decomposes the solution in a macroscopic part, a boundary layer corrector close to the interface and a near-field part close to its ends. The effective transmission and absorption properties are expressed by transmission conditions for the macroscopic solution on an infinitely thin interface and corner conditions at its endpoints to ensure the correct singular behaviour, which are intrinsic to the microstructure. We study and give details on the computation of the effective parameters for an inviscid and a viscous model and show their dependence on geometrical properties of the microstructure for the example of Helmholtz equation. Numerical experiments indicate that with the obtained macroscopic solution representation one can achieve an high accuracy for low and high porosities as well as for viscous boundary conditions while using only a small number of basis functions.
]]>2018-02-28T00:05:42-08:00info:doi/10.1098/rspa.2017.0708hwp:master-id:royprsa;rspa.2017.07082018-02-28Research articles47422102017070820170708<![CDATA[A dislocation-based model for twin growth within and across grains]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170709?rss=1
A computational method is presented for representing twins via two-dimensional dislocation statics in an isotropic elastic solid. The method is compared with analytical approximations of twin shape and is used to study how twins evolve within grains subjected to an arbitrary external shear stress. Twin transfer across grains is then studied using the same computational method. The dislocation-based model for twin growth gives the following dependencies: twin thickness increases linearly with grain size and external stress, and increases substantially as the grain is able to traverse multiple grain boundaries with low misorientation angles; the model also predicts that twin transfer becomes less prominent across grain boundaries with high misorientation angles. These predictions are consistent with experimentally measured extension twin growth in magnesium polycrystals. This study suggests that representing twins via discrete dislocations provides a physically reasonable approximation of twinning that can be naturally incorporated into existing dislocation statics and dynamics codes.
]]>2018-02-28T00:05:42-08:00info:doi/10.1098/rspa.2017.0709hwp:master-id:royprsa;rspa.2017.07092018-02-28Research articles47422102017070920170709<![CDATA[The electron density function of the Hückel (tight-binding) model]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170721?rss=1
The Hückel (tight-binding) molecular orbital (HMO) method has found many applications in the chemistry of alternant conjugated molecules, such as polycyclic aromatic hydrocarbons (PAHs), fullerenes and graphene-like molecules, as well as in solid-state physics. In this paper, we found analytical expressions for the electron density matrix of the HMO method in terms of odd-powers of its Hamiltonian. We prove that the HMO density matrix induces an embedding of a molecule into a high-dimensional Euclidean space in which the separation between the atoms scales very well with the bond lengths of PAHs. We extend our approach to describe a quasi-correlated tight-binding model, which quantifies the number of unpaired electrons and the distribution of effectively unpaired electrons. In this case, we found that the corresponding density matrices induce embedding of the molecules into high-dimensional Euclidean spheres where the separation between the atoms contains information about the spin–spin repulsion between them. Using our approach, we found an analytic expression which explains the bond length alternation in polyenes inside the HMO framework. We also found that spin–spin interaction explains the alternation of distances between pairs of atoms separated by two bonds in conjugated molecules.
]]>2018-02-14T00:05:27-08:00info:doi/10.1098/rspa.2017.0721hwp:master-id:royprsa;rspa.2017.07212018-02-14Research articles47422102017072120170721<![CDATA[Equal-area criterion in power systems revisited]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170733?rss=1
The classic equal-area criterion (EAC) is of key importance in power system analysis, and provides a powerful, pictorial and quantitative means of analysing transient stability (i.e. the system's ability to maintain stable operation when subjected to a large disturbance). Based on the traditional EAC, it is common sense in engineering that there is a critical cleaning time (CCT); namely, a power system is stable (unstable) if a fault is cleared before (after) this CCT. We regard this form of CCT as bipartite. In this paper, we revisit the EAC theory and, surprisingly, find different kinds of transient stability behaviour. Based on these analyses, we discover that the bipartite CCT is only one type among four major types, and, actually, the forms of CCT can be diversified. In particular, under some circumstances, a system may have no CCT or show a periodic CCT. Our theoretical analysis is verified by numerical simulations in a single-machine-infinite-bus system and also in multi-machine systems. Thus, our study provides a panoramic framework for diverse transient stability behaviour in power systems and also may have a significant impact on applications of multi-stability in various other systems, such as neuroscience, climatology or photonics.
]]>2018-02-07T00:05:19-08:00info:doi/10.1098/rspa.2017.0733hwp:master-id:royprsa;rspa.2017.07332018-02-07Research articles47422102017073320170733<![CDATA[Analytic analysis of auxetic metamaterials through analogy with rigid link systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170753?rss=1
In recent years, many structural motifs have been designed with the aim of creating auxetic metamaterials. One area of particular interest in this subject is the creation of auxetic material properties through elastic instability. Such metamaterials switch from conventional behaviour to an auxetic response for loads greater than some threshold value. This paper develops a novel methodology in the analysis of auxetic metamaterials which exhibit elastic instability through analogy with rigid link lattice systems. The results of our analytic approach are confirmed by finite-element simulations for both the onset of elastic instability and post-buckling behaviour including Poisson’s ratio. The method gives insight into the relationships between mechanisms within lattices and their mechanical behaviour; as such, it has the potential to allow existing knowledge of rigid link lattices with auxetic paths to be used in the design of future buckling-induced auxetic metamaterials.
]]>2018-02-21T00:47:25-08:00info:doi/10.1098/rspa.2017.0753hwp:master-id:royprsa;rspa.2017.07532018-02-21Research articles47422102017075320170753<![CDATA[A family of wave equations with some remarkable properties]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170763?rss=1
We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found for two members of the family investigated. For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair, we construct a Miura-type transformation relating the original equation to the Korteweg–de Vries (KdV) equation. This transformation, on the other hand, enables us to obtain solutions of the equation from the kernel of a Schrödinger operator with potential parametrized by the solutions of the KdV equation. In particular, this allows us to exhibit a kink solution to the completely integrable equation from the 1-soliton solution of the KdV equation. Finally, peakon-type solutions are also found for a certain choice of the parameters, although for this particular case the equation is reduced to a homogeneous second-order nonlinear evolution equation.
]]>2018-02-14T00:05:27-08:00info:doi/10.1098/rspa.2017.0763hwp:master-id:royprsa;rspa.2017.07632018-02-14Research articles47422102017076320170763<![CDATA[How the formation of amyloid plaques and neurofibrillary tangles may be related: a mathematical modelling study]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170777?rss=1
We develop a mathematical model that enables us to investigate possible mechanisms by which two primary markers of Alzheimer's disease (AD), extracellular amyloid plaques and intracellular tangles, may be related. Our model investigates the possibility that the decay of anterograde axonal transport of amyloid precursor protein (APP), caused by toxic tau aggregates, leads to decreased APP transport towards the synapse and APP accumulation in the soma. The developed model thus couples three processes: (i) slow axonal transport of tau, (ii) tau misfolding and agglomeration, which we simulated by using the Finke–Watzky model and (iii) fast axonal transport of APP. Because the timescale for tau agglomeration is much larger than that for tau transport, we suggest using the quasi-steady-state approximation for formulating and solving the governing equations for these three processes. Our results suggest that misfolded tau most likely accumulates in the beginning of the axon. The analysis of APP transport suggests that APP will also likely accumulate in the beginning of the axon, causing an increased APP concentration in this region, which could be interpreted as a ‘traffic jam’. The APP flux towards the synapse is significantly reduced by tau misfolding, but not due to the APP traffic jam, which can be viewed as a symptom, but rather due to the reduced affinity of kinesin-1 motors to APP-transporting vesicles.
]]>2018-02-07T00:05:20-08:00info:doi/10.1098/rspa.2017.0777hwp:master-id:royprsa;rspa.2017.07772018-02-07Research articles47422102017077720170777<![CDATA[Rogue periodic waves of the focusing nonlinear Schrödinger equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170814?rss=1
Rogue periodic waves stand for rogue waves on a periodic background. The nonlinear Schrödinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov–Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine’s breather). The magnification factor of the rogue periodic waves is computed as a function of the elliptic modulus. Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.
]]>2018-02-21T00:47:25-08:00info:doi/10.1098/rspa.2017.0814hwp:master-id:royprsa;rspa.2017.08142018-02-21Research articles47422102017081420170814<![CDATA[The ductile/brittle transition provides the critical test for materials failure theory]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170817?rss=1
It is reasoned that any materials failure theory that claims generality must give full account of ductile versus brittle failure behaviour. Any such proposed theory especially must admit the capability to generate the ductile/brittle transition. A derivation of the failure surface orientations from a particular isotropic materials failure theory reveals that uniaxial tension has its ductile/brittle transition at T/C = 1/2, where T and C are the uniaxial strengths. Between this information and the corresponding ductile/brittle transition in uniaxial compression it becomes possible to derive the functional form for the fully three-dimensional ductile/brittle transition. These same general steps of verification must be fulfilled for any other candidate general failure theory.
]]>2018-02-07T00:05:20-08:00info:doi/10.1098/rspa.2017.0817hwp:master-id:royprsa;rspa.2017.08172018-02-07Research articles47422102017081720170817<![CDATA[Computational logic: its origins and applications]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170872?rss=1
Computational logic is the use of computers to establish facts in a logical formalism. Originating in nineteenth century attempts to understand the nature of mathematical reasoning, the subject now comprises a wide variety of formalisms, techniques and technologies. One strand of work follows the ‘logic for computable functions (LCF) approach’ pioneered by Robin Milner, where proofs can be constructed interactively or with the help of users’ code (which does not compromise correctness). A refinement of LCF, called Isabelle, retains these advantages while providing flexibility in the choice of logical formalism and much stronger automation. The main application of these techniques has been to prove the correctness of hardware and software systems, but increasingly researchers have been applying them to mathematics itself.
]]>2018-02-28T00:05:38-08:00info:doi/10.1098/rspa.2017.0872hwp:master-id:royprsa;rspa.2017.08722018-02-28Perspective47422102017087220170872<![CDATA[Energy approach to brittle fracture in strain-gradient modelling]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170878?rss=1
In this paper, we exploit some results in the theory of irreversible phenomena to address the study of quasi-static brittle fracture propagation in a two-dimensional isotropic continuum. The elastic strain energy density of the body has been assumed to be geometrically nonlinear and to depend on the strain gradient. Such generalized continua often arise in the description of microstructured media. These materials possess an intrinsic length scale, which determines the size of internal boundary layers. In particular, the non-locality conferred by this internal length scale avoids the concentration of deformations, which is usually observed when dealing with local models and which leads to mesh dependency. A scalar Lagrangian damage field, ranging from zero to one, is introduced to describe the internal state of structural degradation of the material. Standard Lamé and second-gradient elastic coefficients are all assumed to decrease as damage increases and to be locally zero if the value attained by damage is one. This last situation is associated with crack formation and/or propagation. Numerical solutions of the model are provided in the case of an obliquely notched rectangular specimen subjected to monotonous tensile and shear loading tests, and brittle fracture propagation is discussed.
]]>2018-02-28T00:05:42-08:00info:doi/10.1098/rspa.2017.0878hwp:master-id:royprsa;rspa.2017.08782018-02-28Research articles47422102017087820170878<![CDATA[Stability of barotropic vortex strip on a rotating sphere]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170883?rss=1
We study the stability of a barotropic vortex strip on a rotating sphere, as a simple model of jet streams. The flow is approximated by a piecewise-continuous vorticity distribution by zonal bands of uniform vorticity. The linear stability analysis shows that the vortex strip becomes stable as the strip widens or the rotation speed increases. When the vorticity constants in the upper and the lower regions of the vortex strip have the same positive value, the inner flow region of the vortex strip becomes the most unstable. However, when the upper and the lower vorticity constants in the polar regions have different signs, a complex pattern of instability is found, depending on the wavenumber of perturbations, and interestingly, a boundary far away from the vortex strip can be unstable. We also compute the nonlinear evolution of the vortex strip on the rotating sphere and compare with the linear stability analysis. When the width of the vortex strip is small, we observe a good agreement in the growth rate of perturbation at an early time, and the eigenvector corresponding to the unstable eigenvalue coincides with the most unstable part of the flow. We demonstrate that a large structure of rolling-up vortex cores appears in the vortex strip after a long-time evolution. Furthermore, the geophysical relevance of the model to jet streams of Jupiter, Saturn and Earth is examined.
]]>2018-02-28T00:05:42-08:00info:doi/10.1098/rspa.2017.0883hwp:master-id:royprsa;rspa.2017.08832018-02-28Research articles47422102017088320170883<![CDATA[Modelling resonant arrays of the Helmholtz type in the time domain]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2210/20170894?rss=1
We present a model based on a two-scale asymptotic analysis for resonant arrays of the Helmholtz type, with resonators open at a single extremity (standard resonators) or open at both extremities (double-sided resonators). The effective behaviour of such arrays is that of a homogeneous anisotropic slab replacing the cavity region, associated with transmission, or jump, conditions for the acoustic pressure and for the normal velocity across the region of the necks. The coefficients entering in the effective wave equation are simply related to the fraction of air in the periodic cell of the array. Those entering in the jump conditions are related to near field effects in the vicinity of the necks and they encapsulate the effects of their geometry. The effective problem, which accounts for the coupling of the resonators with the surrounding air, is written in the time domain which allows us to question the equation of energy conservation. This is of practical importance if the numerical implementations of the effective problem in the time domain is sought.
]]>2018-02-28T00:05:42-08:00info:doi/10.1098/rspa.2017.0894hwp:master-id:royprsa;rspa.2017.08942018-02-28Research articles47422102017089420170894<![CDATA[Qualitative models and experimental investigation of chaotic NOR gates and set/reset flip-flops]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170111?rss=1
It has been observed through experiments and SPICE simulations that logical circuits based upon Chua’s circuit exhibit complex dynamical behaviour. This behaviour can be used to design analogues of more complex logic families and some properties can be exploited for electronics applications. Some of these circuits have been modelled as systems of ordinary differential equations. However, as the number of components in newer circuits increases so does the complexity. This renders continuous dynamical systems models impractical and necessitates new modelling techniques. In recent years, some discrete dynamical models have been developed using various simplifying assumptions. To create a robust modelling framework for chaotic logical circuits, we developed both deterministic and stochastic discrete dynamical models, which exploit the natural recurrence behaviour, for two chaotic NOR gates and a chaotic set/reset flip-flop. This work presents a complete applied mathematical investigation of logical circuits. Experiments on our own designs of the above circuits are modelled and the models are rigorously analysed and simulated showing surprisingly close qualitative agreement with the experiments. Furthermore, the models are designed to accommodate dynamics of similarly designed circuits. This will allow researchers to develop ever more complex chaotic logical circuits with a simple modelling framework.
]]>2018-01-31T00:05:41-08:00info:doi/10.1098/rspa.2017.0111hwp:master-id:royprsa;rspa.2017.01112018-01-31Research articles47422092017011120170111<![CDATA[Determination of macro-scale soil properties from pore-scale structures: model derivation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170141?rss=1
In this paper, we use homogenization to derive a set of macro-scale poro-elastic equations for soils composed of rigid solid particles, air-filled pore space and a poro-elastic mixed phase. We consider the derivation in the limit of large deformation and show that by solving representative problems on the micro-scale we can parametrize the macro-scale equations. To validate the homogenization procedure, we compare the predictions of the homogenized equations with those of the full equations for a range of different geometries and material properties. We show that the results differ by 2% for all cases considered. The success of the homogenization scheme means that it can be used to determine the macro-scale poro-elastic properties of soils from the underlying structure. Hence, it will prove a valuable tool in both characterization and optimization.
]]>2018-01-31T00:05:40-08:00info:doi/10.1098/rspa.2017.0141hwp:master-id:royprsa;rspa.2017.01412018-01-31Research articles47422092017014120170141<![CDATA[Dynamics of flexural gravity waves: from sea ice to Hawking radiation and analogue gravity]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170223?rss=1
The propagation of flexural gravity waves, routinely used to model wave interaction with sea ice, is studied, including the effect of compression and current. A number of significant and surprising properties are shown to exist. The occurrence of blocking above a critical value of compression is illustrated. This is analogous to propagation of surface gravity waves in the presence of opposing current and light wave propagation in the curved space–time near a black hole, therefore providing a novel system for studying analogue gravity. Between the blocking and buckling limit of the compressive force, the dispersion relation possesses three positive real roots, contrary to an earlier observation of having a single positive real root. Negative energy waves, in which the phase and group velocity point in opposite directions, are also shown to exist. In the presence of an opposing current and certain critical ranges of compressive force, the second blocking point shifts from the positive to the negative branch of the dispersion relation. Such a shift is known as the Hawking effect from the analogous behaviour in the theory of relativity which leads to Hawking radiation. The theory we develop is illustrated with simulations of linear waves in the time domain.
]]>2018-01-17T01:23:48-08:00info:doi/10.1098/rspa.2017.0223hwp:master-id:royprsa;rspa.2017.02232018-01-17Research articles47422092017022320170223<![CDATA[Matter rogue waves for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170276?rss=1
To show the existence and properties of matter rogue waves in an F=1 spinor Bose–Einstein condensate (BEC), we work on the three-component Gross–Pitaevskii (GP) equations. Via the Darboux-dressing transformation, we obtain a family of rational solutions describing the extreme events, i.e. rogue waves. This family of solutions includes bright–dark–bright and bright–bright–bright rogue waves. The algebraic construction depends on Lax matrices and their Jordan form. The conditions for the existence of rogue wave solutions in an F=1 spinor BEC are discussed. For the three-component GP equations, if there is modulation instability, it is of baseband type only, confirming our analytic conditions. The energy transfers between the waves are discussed.
]]>2018-01-03T00:05:18-08:00info:doi/10.1098/rspa.2017.0276hwp:master-id:royprsa;rspa.2017.02762018-01-03Research articles47422092017027620170276<![CDATA[Regular approximate factorization of a class of matrix-function with an unstable set of partial indices]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170279?rss=1
From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk.XIII, 3–72. (Russian).), it is well known that the set of partial indices of a non-singular matrix function may change depending on the properties of the original matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original matrix function, there exists another matrix function possessing a different set of partial indices. As a result, the factorization of matrix functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing matrix functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting matrix function, allow to construct another family of matrix functions with the same origin that preserves the non-stable partial indices and is close to the original set of the matrix functions.
]]>2018-01-17T00:05:26-08:00info:doi/10.1098/rspa.2017.0279hwp:master-id:royprsa;rspa.2017.02792018-01-17Research articles47422092017027920170279<![CDATA[Dynamics of non-holonomic systems with stochastic transport]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170479?rss=1
This paper formulates a variational approach for treating observational uncertainty and/or computational model errors as stochastic transport in dynamical systems governed by action principles under non-holonomic constraints. For this purpose, we derive, analyse and numerically study the example of an unbalanced spherical ball rolling under gravity along a stochastic path. Our approach uses the Hamilton–Pontryagin variational principle, constrained by a stochastic rolling condition, which we show is equivalent to the corresponding stochastic Lagrange–d’Alembert principle. In the example of the rolling ball, the stochasticity represents uncertainty in the observation and/or error in the computational simulation of the angular velocity of rolling. The influence of the stochasticity on the deterministically conserved quantities is investigated both analytically and numerically. Our approach applies to a wide variety of stochastic, non-holonomically constrained systems, because it preserves the mathematical properties inherited from the variational principle.
]]>2018-01-10T00:05:18-08:00info:doi/10.1098/rspa.2017.0479hwp:master-id:royprsa;rspa.2017.04792018-01-10Research articles47422092017047920170479<![CDATA[Quantum algorithm for multivariate polynomial interpolation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170480?rss=1
How many quantum queries are required to determine the coefficients of a degree-d polynomial in n variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields Fq, R and C. We show that kC and 2kC queries suffice to achieve probability 1 for C and R, respectively, where kC=(1/(n+1))(n+dd) except for d=2 and four other special cases. For Fq, we show that (d/(n+d))(n+dd) queries suffice to achieve probability approaching 1 for large field order q. The classical query complexity of this problem is (n+dd) , so our result provides a speed-up by a factor of n+1, (n+1)/2 and (n+d)/d for C, R and Fq, respectively. Thus, we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of 2. For the case of Fq, we conjecture that 2kC queries also suffice to achieve probability approaching 1 for large field order q, although we leave this as an open problem.
]]>2018-01-17T00:05:26-08:00info:doi/10.1098/rspa.2017.0480hwp:master-id:royprsa;rspa.2017.04802018-01-17Research articles47422092017048020170480<![CDATA[Finite indentation of highly curved elastic shells]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170482?rss=1
Experimentally measuring the elastic properties of thin biological surfaces is non-trivial, particularly when they are curved. One technique that may be used is the indentation of a thin sheet of material by a rigid indenter, while measuring the applied force and displacement. This gives immediate information on the fracture strength of the material (from the force required to puncture), but it is also theoretically possible to determine the elastic properties by comparing the resulting force–displacement curves with a mathematical model. Existing mathematical studies generally assume that the elastic surface is initially flat, which is often not the case for biological membranes. We previously outlined a theory for the indentation of curved isotropic, incompressible, hyperelastic membranes (with no bending stiffness) which breaks down for highly curved surfaces, as the entire membrane becomes wrinkled. Here, we introduce the effect of bending stiffness, ensuring that energy is required to change the shell shape without stretching, and find that commonly neglected terms in the shell equilibrium equation must be included. The theory presented here allows for the estimation of shape- and size-independent elastic properties of highly curved surfaces via indentation experiments, and is particularly relevant for biological surfaces.
]]>2018-01-24T00:05:35-08:00info:doi/10.1098/rspa.2017.0482hwp:master-id:royprsa;rspa.2017.04822018-01-24Research articles47422092017048220170482<![CDATA[Parameter-robustness analysis for a biochemical oscillator model describing the social-behaviour transition phase of myxobacteria]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170499?rss=1
We develop a tool based on bifurcation analysis for parameter-robustness analysis for a class of oscillators and, in particular, examine a biochemical oscillator that describes the transition phase between social behaviours of myxobacteria. Myxobacteria are a particular group of soil bacteria that have two dogmatically different types of social behaviour: when food is abundant they live fairly isolated forming swarms, but when food is scarce, they aggregate into a multicellular organism. In the transition between the two types of behaviours, spatial wave patterns are produced, which is generally believed to be regulated by a certain biochemical clock that controls the direction of myxobacteria’s motion. We provide a detailed analysis of such a clock and show that, for the proposed model, there exists some interval in parameter space where the behaviour is robust, i.e. the system behaves similarly for all parameter values. In more mathematical terms, we show the existence and convergence of trajectories to a limit cycle, and provide estimates of the parameter under which such a behaviour occurs. In addition, we show that the reported convergence result is robust, in the sense that any small change in the parameters leads to the same qualitative behaviour of the solution.
]]>2018-01-24T00:05:35-08:00info:doi/10.1098/rspa.2017.0499hwp:master-id:royprsa;rspa.2017.04992018-01-24Research articles47422092017049920170499<![CDATA[Dynamics and locomotion of flexible foils in a frictional environment]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170503?rss=1
Over the past few decades, oscillating flexible foils have been used to study the physics of organismal propulsion in different fluid environments. Here, we extend this work to a study of flexible foils in a frictional environment. When the foil is oscillated by heaving at one end but is not free to locomote, the dynamics change from periodic to non-periodic and chaotic as the heaving amplitude increases or the bending rigidity decreases. For friction coefficients lying in a certain range, the transition passes through a sequence of N-periodic and asymmetric states before reaching chaotic dynamics. Resonant peaks are damped and shifted by friction and large heaving amplitudes, leading to bistable states. When the foil is free to locomote, the horizontal motion smoothes the resonant behaviours. For moderate frictional coefficients, steady but slow locomotion is obtained. For large transverse friction and small tangential friction corresponding to wheeled snake robots, faster locomotion is obtained. Travelling wave motions arise spontaneously, and move with horizontal speeds that scale as transverse friction coefficient to the power 1/4 and input power that scales as the transverse friction coefficient to the power 5/12. These scalings are consistent with a boundary layer form of the solutions near the foil’s leading edge.
]]>2018-01-17T00:05:26-08:00info:doi/10.1098/rspa.2017.0503hwp:master-id:royprsa;rspa.2017.05032018-01-17Research articles47422092017050320170503<![CDATA[Assessing the features of extreme smog in China and the differentiated treatment strategy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170511?rss=1
Extreme smog can have potentially harmful effects on human health, the economy and daily life. However, the average (mean) values do not provide strategically useful information on the hazard analysis and control of extreme smog. This article investigates China's smog extremes by applying extreme value analysis to hourly PM2.5 data from 2014 to 2016 obtained from monitoring stations across China. By fitting a generalized extreme value (GEV) distribution to exceedances over a station-specific extreme smog level at each monitoring location, all study stations are grouped into eight different categories based on the estimated mean and shape parameter values of fitted GEV distributions. The extreme features characterized by the mean of the fitted extreme value distribution, the maximum frequency and the tail index of extreme smog at each location are analysed. These features can provide useful information for central/local government to conduct differentiated treatments in cities within different categories and conduct similar prevention goals and control strategies among those cities belonging to the same category in a range of areas. Furthermore, hazardous hours, breaking probability and the 1-year return level of each station are demonstrated by category, based on which the future control and reduction targets of extreme smog are proposed for the cities of Beijing, Tianjin and Hebei as an example.
]]>2018-01-24T00:05:35-08:00info:doi/10.1098/rspa.2017.0511hwp:master-id:royprsa;rspa.2017.05112018-01-24Research articles47422092017051120170511<![CDATA[On the approximate solutions of fragmentation equations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170541?rss=1
A numerical model based on the finite volume scheme is proposed to approximate the binary breakage problems. Initially, it is considered that the particle fragments are characterized by a single property, i.e. particle’s volume. We then investigate the extension of the proposed model for solving breakage problems considering two properties of particles. The efficiency to estimate the different moments with good accuracy and simple extension for multi-variable problems are the key features of the proposed method. Moreover, the mathematical convergence analysis is performed for one-dimensional problems. All mathematical findings and numerical results are validated over several test problems. For numerical validation, we propose the extension of Bourgade & Filbet (2008 Math. Comput.77, 851–882. (doi:10.1090/S0025-5718-07-02054-6)) model for solving two-dimensional pure breakage problems. In this aspect, numerical treatment of the two-dimensional binary breakage models using finite volume methods can be treated to be the first instance in the literature.
]]>2018-01-17T01:53:20-08:00info:doi/10.1098/rspa.2017.0541hwp:master-id:royprsa;rspa.2017.05412018-01-17Research articles47422092017054120170541<![CDATA[Quantum machine learning: a classical perspective]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170551?rss=1
Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning (ML) techniques to impressive results in regression, classification, data generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets is motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed up classical ML algorithms. Here we review the literature in quantum ML and discuss perspectives for a mixed readership of classical ML and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in ML are identified as promising directions for the field. Practical questions, such as how to upload classical data into quantum form, will also be addressed.
]]>2018-01-17T00:05:29-08:00info:doi/10.1098/rspa.2017.0551hwp:master-id:royprsa;rspa.2017.05512018-01-17Review articles47422092017055120170551<![CDATA[A nanoscale perspective on the effects of transverse microprestress on drying creep of nanoporous solids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170570?rss=1
The Pickett effect describes the excess non-additive strain developed during drying of a nanoporous solid material under creep. One explanation for its origins, developed using micromechanical models, is the progressive relaxation of internally developed microprestress. However, these models have not explicitly considered the effects of this microprestress on nanoscale energy barriers that govern the relative motion and displacement between nanopore walls during deformation. Here, we evaluate the nanoscale effects of transverse microprestresses on the drying creep behaviour of a nanoscale slit pore using coarse-grained molecular dynamics. We find that the underlying energy barrier depends exponentially on the transverse microprestress, which is attributed to changes in the effective viscosity and degree of nanoconfinement of molecules in the water interlayer. Specifically, as the transverse microprestress is relaxed (i.e. its magnitude decreases), the activation energy barrier is reduced, thereby leading to an acceleration of the creep behaviour and a stronger Pickett effect. Based on our simulation results, we introduce a new microprestress-dependent energy term into our existing Arrhenius model, which describes the relative displacement of pore walls as a function of the underlying activation energy barriers. Our findings further verify the existing micromechanical theories for the origin of the Pickett effect and establish a quantitative relationship between the transverse microprestress and the intensity of the Pickett effect.
]]>2018-01-17T03:24:44-08:00info:doi/10.1098/rspa.2017.0570hwp:master-id:royprsa;rspa.2017.05702018-01-17Research articles47422092017057020170570<![CDATA[On compressible and piezo-viscous flow in thin porous media]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170601?rss=1
In this paper, we study flow through thin porous media as in, e.g. seals or fractures. It is often useful to know the permeability of such systems. In the context of incompressible and iso-viscous fluids, the permeability is the constant of proportionality relating the total flow through the media to the pressure drop. In this work, we show that it is also relevant to define a constant permeability when compressible and/or piezo-viscous fluids are considered. More precisely, we show that the corresponding nonlinear equation describing the flow of any compressible and piezo-viscous fluid can be transformed into a single linear equation. Indeed, this linear equation is the same as the one describing the flow of an incompressible and iso-viscous fluid. By this transformation, the total flow can be expressed as the product of the permeability and a nonlinear function of pressure, which represents a generalized pressure drop.
]]>2018-01-03T00:05:18-08:00info:doi/10.1098/rspa.2017.0601hwp:master-id:royprsa;rspa.2017.06012018-01-03Research articles47422092017060120170601<![CDATA[Discrete-to-continuum modelling of weakly interacting incommensurate two-dimensional lattices]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170612?rss=1
In this paper, we derive a continuum variational model for a two-dimensional deformable lattice of atoms interacting with a two-dimensional rigid lattice. The starting point is a discrete atomistic model for the two lattices which are assumed to have slightly different lattice parameters and, possibly, a small relative rotation. This is a prototypical example of a three-dimensional system consisting of a graphene sheet suspended over a substrate. We use a discrete-to-continuum procedure to obtain the continuum model which recovers both qualitatively and quantitatively the behaviour observed in the corresponding discrete model. The continuum model predicts that the deformable lattice develops a network of domain walls characterized by large shearing, stretching and bending deformation that accommodates the misalignment and/or mismatch between the deformable and rigid lattices. Two integer-valued parameters, which can be identified with the components of a Burgers vector, describe the mismatch between the lattices and determine the geometry and the details of the deformation associated with the domain walls.
]]>2018-01-31T00:05:43-08:00info:doi/10.1098/rspa.2017.0612hwp:master-id:royprsa;rspa.2017.06122018-01-31Research articles47422092017061220170612<![CDATA[Dynamics of lumps and dark-dark solitons in the multi-component long-wave-short-wave resonance interaction system]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170627?rss=1
General semi-rational solutions of an integrable multi-component (2+1)-dimensional long-wave–short-wave resonance interaction system comprising multiple short waves and a single long wave are obtained by employing the bilinear method. These solutions describe the interactions between various types of solutions, including line rogue waves, lumps, breathers and dark solitons. We only focus on the dynamical behaviours of the interactions between lumps and dark solitons in this paper. Our detailed study reveals two different types of excitation phenomena: fusion and fission. It is shown that the fundamental (simplest) semi-rational solutions can exhibit fission of a dark soliton into a lump and a dark soliton or fusion of one lump and one dark soliton into a dark soliton. The non-fundamental semi-rational solutions are further classified into three subclasses: higher-order, multi- and mixed-type semi-rational solutions. The higher-order semi-rational solutions show the process of annihilation (production) of two or more lumps into (from) one dark soliton. The multi-semi-rational solutions describe N(N≥2) lumps annihilating into or producing from N-dark solitons. The mixed-type semi-rational solutions are a hybrid of higher-order semi-rational solutions and multi-semi-rational solutions. For the mixed-type semi-rational solutions, we demonstrate an interesting dynamical behaviour that is characterized by partial suppression or creation of lumps from the dark solitons.
]]>2018-01-17T01:23:48-08:00info:doi/10.1098/rspa.2017.0627hwp:master-id:royprsa;rspa.2017.06272018-01-17Research articles47422092017062720170627<![CDATA[Hybrid integral transforms for flow development in ducts partially filled with porous media]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170637?rss=1
A hybrid numerical–analytical solution is developed for laminar flow development in a parallel plate duct partially filled with porous media. The integral transform method is employed in combination with a single domain reformulation strategy for representing the heterogeneous media within the channel. A novel eigenfunction expansion basis is proposed, including abrupt spatial variations of physical properties due to the domain transitions. The introduction of the new basis allows for a solution with similar convergence rates as in previous applications with simpler formulations, as demonstrated through a careful convergence analysis of the expansions. The inherent automatic error control characteristic of the integral transforms approach then provides benchmark results for the developing velocity profile. Moreover, a physical analysis further verifies the consistency of both the proposed expansion and the mixed symbolic–numerical code developed. A detailed verification with a finite-element commercial code is also performed.
]]>2018-01-10T00:05:18-08:00info:doi/10.1098/rspa.2017.0637hwp:master-id:royprsa;rspa.2017.06372018-01-10Research articles47422092017063720170637<![CDATA[Lightweight steel tidal power barrages with minimal environmental impact: application to the Severn Barrage]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170653?rss=1
For tidal power barrages, a breast-shot water wheel, with a hydraulic transmission, has significant advantages over a conventional Kaplan turbine. It is better suited to combined operations with pumping that maintain the tidal range upstream of the barrage (important in reducing the environmental impact), and is much less harmful to fish. It also does not require tapered entry and exit ducts, making the barrage much smaller and lighter, so that it can conveniently be built in steel. For the case of the Severn Estuary, UK, it is shown that a barrage at Porlock would generate an annual average power of 4 GW (i.e. 35 TWh yr^{–1}), maintain the existing tidal ranges upstream of it and reduce the tidal ranges downstream of it by only about 10%. The weight of steel required, in relation to the annual average power generated, compares very favourably with a recent offshore wind farm.
]]>2018-01-31T00:05:40-08:00info:doi/10.1098/rspa.2017.0653hwp:master-id:royprsa;rspa.2017.06532018-01-31Research articles47422092017065320170653<![CDATA[Electromagnetic pulses, localized and causal]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170655?rss=1
We show that pulse solutions of the wave equation can be expressed as time Fourier superpositions of scalar monochromatic beam wave functions (solutions of the Helmholtz equation). This formulation is shown to be equivalent to Bateman's integral expression for solutions of the wave equation, for axially symmetric solutions. A closed-form one-parameter solution of the wave equation, containing no backward-propagating parts, is constructed from a beam which is the tight-focus limit of two families of beams. Application is made to transverse electric and transverse magnetic pulses, with evaluation of the energy, momentum and angular momentum for a pulse based on the general localized and causal form. Such pulses can be represented as superpositions of photons. Explicit total energy and total momentum values are given for the one-parameter closed-form pulse.
]]>2018-01-17T01:23:48-08:00info:doi/10.1098/rspa.2017.0655hwp:master-id:royprsa;rspa.2017.06552018-01-17Research articles47422092017065520170655<![CDATA[Waves and fluid-solid interaction in stented blood vessels]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170670?rss=1
This paper focuses on the modelling of fluid–structure interaction and wave propagation problems in a stented artery. Reflection of waves in blood vessels is well documented in the literature, but it has always been linked to a strong variation in geometry, such as the branching of vessels. The aim of this work is to detect the possibility of wave reflection in a stented artery due to the repetitive pattern of the stents. The investigation of wave propagation and possible blockages under time-harmonic conditions is complemented with numerical simulations in the transient regime.
]]>2018-01-17T01:23:48-08:00info:doi/10.1098/rspa.2017.0670hwp:master-id:royprsa;rspa.2017.06702018-01-17Research articles47422092017067020170670<![CDATA[Variational formulation for dissipative continua and an incremental J-integral]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170674?rss=1
Our aim is to rationally formulate a proper variational principle for dissipative (viscoplastic) solids in the presence of inertia forces. As a first step, a consistent linearization of the governing nonlinear partial differential equations (PDEs) is carried out. An additional set of complementary (adjoint) equations is then formed to recover an underlying variational structure for the augmented system of linearized balance laws. This makes it possible to introduce an incremental Lagrangian such that the linearized PDEs, including the complementary equations, become the Euler–Lagrange equations. Continuous groups of symmetries of the linearized PDEs are computed and an analysis is undertaken to identify the variational groups of symmetries of the linearized dissipative system. Application of Noether’s theorem leads to the conservation laws (conserved currents) of motion corresponding to the variational symmetries. As a specific outcome, we exploit translational symmetries of the functional in the material space and recover, via Noether’s theorem, an incremental J-integral for viscoplastic solids in the presence of inertia forces. Numerical demonstrations are provided through a two-dimensional plane strain numerical simulation of a compact tension specimen of annealed mild steel under dynamic loading.
]]>2018-01-17T00:05:29-08:00info:doi/10.1098/rspa.2017.0674hwp:master-id:royprsa;rspa.2017.06742018-01-17Research articles47422092017067420170674<![CDATA[Does flutter prevent drag reduction by reconfiguration?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170678?rss=1
The static reconfiguration of flexible beams exposed to transverse flows is classically known to reduce the drag these structures have to withstand. But the more a structure bends, the more parallel to the flow it becomes, and flexible beams in axial flows are prone to a flutter instability that is responsible for large inertial forces that drastically increase their drag. It is, therefore, unclear whether flexibility would still alleviate, or on the contrary enhance, the drag when flapping occurs on a reconfiguring structure. In this article, we perform numerical simulations based on reduced-order models to demonstrate that the additional drag induced by the flapping motion is almost never significant enough to offset the drag reduction due to reconfiguration. Isolated and brief snapping events may transiently raise the drag above that of a rigid structure in the particular case of heavy, moderately slender beams. But apart from these short peak events, the drag force remains otherwise always significantly reduced in comparison with a rigid structure.
]]>2018-01-03T00:05:18-08:00info:doi/10.1098/rspa.2017.0678hwp:master-id:royprsa;rspa.2017.06782018-01-03Research articles47422092017067820170678<![CDATA[Bifurcation of elastic solids with sliding interfaces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170681?rss=1
Lubricated sliding contact between soft solids is an interesting topic in biomechanics and for the design of small-scale engineering devices. As a model of this mechanical set-up, two elastic nonlinear solids are considered jointed through a frictionless and bilateral surface, so that continuity of the normal component of the Cauchy traction holds across the surface, but the tangential component is null. Moreover, the displacement can develop only in a way that the bodies in contact do neither detach, nor overlap. Surprisingly, this finite strain problem has not been correctly formulated until now, so this formulation is the objective of the present paper. The incremental equations are shown to be non-trivial and different from previously (and erroneously) employed conditions. In particular, an exclusion condition for bifurcation is derived to show that previous formulations based on frictionless contact or ‘spring-type’ interfacial conditions are not able to predict bifurcations in tension, while experiments—one of which, ad hoc designed, is reported—show that these bifurcations are a reality and become possible when the correct sliding interface model is used. The presented results introduce a methodology for the determination of bifurcations and instabilities occurring during lubricated sliding between soft bodies in contact.
]]>2018-01-10T00:05:18-08:00info:doi/10.1098/rspa.2017.0681hwp:master-id:royprsa;rspa.2017.06812018-01-10Research articles47422092017068120170681<![CDATA[Computational tameness of classical non-causal models]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170698?rss=1
We show that the computational power of the non-causal circuit model, i.e. the circuit model where the assumption of a global causal order is replaced by the assumption of logical consistency, is completely characterized by the complexity class UPcoUP. An example of a problem in that class is factorization. Our result implies that classical deterministic closed timelike curves (CTCs) cannot efficiently solve problems that lie outside of that class. Thus, in stark contrast to other CTC models, these CTCs cannot efficiently solve NP-complete problems, unless NP=UPcoUP=coNP, which lets their existence in nature appear less implausible. This result gives a new characterization of UPcoUP in terms of fixed points.
]]>2018-01-10T00:05:18-08:00info:doi/10.1098/rspa.2017.0698hwp:master-id:royprsa;rspa.2017.06982018-01-10Research articles47422092017069820170698<![CDATA[A nonlinear theory for fibre-reinforced magneto-elastic rods]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170703?rss=1
We derive a model for the finite motion of a fibre-reinforced magneto-elastic rod. The reinforcing particles are assumed weakly and uniformly magnetized, rigid and firmly embedded into the elastomeric matrix. We deduce closed-form expressions of the quasi-static motion of the rod in terms of the external magnetic field and of the body forces. The dependences of the motion on the shape of the inclusions, their orientation, their anisotropic magnetic properties and the Young modulus of the matrix are analysed and discussed. Two case studies are presented, in which the rod is used as an actuator suspended in a cantilever configuration. This work can foster new applications in the field of soft-actuators.
]]>2018-01-17T01:23:48-08:00info:doi/10.1098/rspa.2017.0703hwp:master-id:royprsa;rspa.2017.07032018-01-17Research articles47422092017070320170703<![CDATA[Biological growth in bodies with incoherent interfaces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170716?rss=1
A general theory of thermodynamically consistent biomechanical–biochemical growth in a body, considering mass addition in the bulk and at an incoherent interface, is developed. The incoherency arises due to incompatibility of growth and elastic distortion tensors at the interface. The incoherent interface therefore acts as an additional source of internal stress besides allowing for rich growth kinematics. All the biochemicals in the model are essentially represented in terms of nutrient concentration fields, in the bulk and at the interface. A nutrient balance law is postulated which, combined with mechanical balances and kinetic laws, yields an initial-boundary-value problem coupling the evolution of bulk and interfacial growth, on the one hand, and the evolution of growth and nutrient concentration on the other. The problem is solved, and discussed in detail, for two distinct examples: annual ring formation during tree growth and healing of cutaneous wounds in animals.
]]>2018-01-24T00:05:35-08:00info:doi/10.1098/rspa.2017.0716hwp:master-id:royprsa;rspa.2017.07162018-01-24Research articles47422092017071620170716<![CDATA[The role of frictional contact of constituent blocks on the stability of masonry domes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2209/20170740?rss=1
The observation of old construction works confirms that masonry domes can withstand tensile hoop stresses, at least up to a certain level. Here, such tensile resistance, rather than a priori assumed as a property of the bulk material, is attributed to the contact forces that are developed at the interfaces between interlocked blocks under normal pressure, specified by Coulomb’s friction law. According to this rationale, the aspect ratio of the blocks, as well as the bond pattern, becomes of fundamental importance. To investigate the complex assembly of blocks, supposed rigid, we present a non-smooth contact dynamic analysis, implemented in a custom software based on the Project Chrono C++ framework and complemented with parametric-design interfaces for pre- and post-processing complex geometries. Through this advanced tool, we investigate the role of frictional forces resisting hoop stresses in the stability of domes, either circular or oval, under static and dynamic loading, focusing, in particular, on the structural role played by the underlying drum and the surmounting tiburium.
]]>2018-01-31T00:05:40-08:00info:doi/10.1098/rspa.2017.0740hwp:master-id:royprsa;rspa.2017.07402018-01-31Research articles47422092017074020170740