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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[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[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[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[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<![CDATA[Buckling of a stiff thin film on an elastic graded compliant substrate]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170410?rss=1
The buckling of a stiff film on a compliant substrate has attracted much attention due to its wide applications such as thin-film metrology, surface patterning and stretchable electronics. An analytical model is established for the buckling of a stiff thin film on a semi-infinite elastic graded compliant substrate subjected to in-plane compression. The critical compressive strain and buckling wavelength for the sinusoidal mode are obtained analytically for the case with the substrate modulus decaying exponentially. The rigorous finite element analysis (FEA) is performed to validate the analytical model and investigate the postbuckling behaviour of the system. The critical buckling strain for the period-doubling mode is obtained numerically. The influences of various material parameters on the results are investigated. These results are helpful to provide physical insights on the buckling of elastic graded substrate-supported thin film.
]]>2017-12-13T00:05:16-08:00info:doi/10.1098/rspa.2017.0410hwp:master-id:royprsa;rspa.2017.04102017-12-13Research articles47322082017041020170410<![CDATA[On modelling the interaction between two rotating bodies with statistically distributed features: an application to dressing of grinding wheels]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170466?rss=1
The mechanisms of interaction between bodies with statistically arranged features present characteristics common to different abrasive processes, such as dressing of abrasive tools. In contrast with the current empirical approach used to estimate the results of operations based on attritive interactions, the method we present in this paper allows us to predict the output forces and the topography of a simulated grinding wheel for a set of specific operational parameters (speed ratio and radial feed-rate), providing a thorough understanding of the complex mechanisms regulating these processes. In modelling the dressing mechanisms, the abrasive characteristics of both bodies (grain size, geometry, inter-space and protrusion) are first simulated; thus, their interaction is simulated in terms of grain collisions. Exploiting a specifically designed contact/impact evaluation algorithm, the model simulates the collisional effects of the dresser abrasives on the grinding wheel topography (grain fracture/break-out). The method has been tested for the case of a diamond rotary dresser, predicting output forces within less than 10% error and obtaining experimentally validated grinding wheel topographies. The study provides a fundamental understanding of the dressing operation, enabling the improvement of its performance in an industrial scenario, while being of general interest in modelling collision-based processes involving statistically distributed elements.
]]>2017-12-06T00:05:14-08:00info:doi/10.1098/rspa.2017.0466hwp:master-id:royprsa;rspa.2017.04662017-12-06Research articles47322082017046620170466<![CDATA[Sub-wavelength focusing of acoustic waves in bubbly media]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170469?rss=1
The purpose of this paper is to investigate acoustic wave scattering by a large number of bubbles in a liquid at frequencies near the Minnaert resonance frequency. This bubbly media has been exploited in practice to obtain super-focusing of acoustic waves. Using layer potential techniques, we derive the scattering function for a single spherical bubble excited by an incident wave in the low frequency regime. We then propose a point scatterer approximation for N bubbles, and describe several numerical simulations based on this approximation, that demonstrate the possibility of achieving super-focusing using bubbly media.
]]>2017-12-20T00:05:17-08:00info:doi/10.1098/rspa.2017.0469hwp:master-id:royprsa;rspa.2017.04692017-12-20Research articles47322082017046920170469<![CDATA[Continuous dependence on modelling for temperature-dependent bidispersive flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170485?rss=1
We consider a model for flow in a porous medium which has a double porosity structure. There is the usual porosity herein called macro porosity, but in addition, we allow for a porosity due to cracks or fissures in the solid skeleton. The cracks give rise to a micro porosity. The model considered also allows for temperature effects with a single temperature T. This paper analyses three aspects of structural stability. The first establishes continuous dependence of the solution on the interaction coefficient between the velocities associated with the macro and micro porosity. The second analyses continuous dependence on the viscosity coefficients, while the third establishes continuous dependence on the radiation constant when Newton’s law of cooling is involved on the boundary.
]]>2017-12-20T00:05:17-08:00info:doi/10.1098/rspa.2017.0485hwp:master-id:royprsa;rspa.2017.04852017-12-20Research articles47322082017048520170485<![CDATA[Predicting acoustic relaxation absorption in gas mixtures for extraction of composition relaxation contributions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170496?rss=1
The existing molecular relaxation models based on both parallel relaxation theory and series relaxation theory cannot extract the contributions of gas compositions to acoustic relaxation absorption in mixtures. In this paper, we propose an analytical model to predict acoustic relaxation absorption and clarify composition relaxation contributions based on the rate-determining energy transfer processes in molecular relaxation in excitable gases. By combining parallel and series relaxation theory, the proposed model suggests that the vibration-translation process of the lowest vibrational mode in each composition provides the primary deexcitation path of the relaxation energy, and the rate-determining vibration–vibration processes between the lowest mode and others dominate the coupling energy transfer between different modes. Thus, each gas composition contributes directly one single relaxation process to the molecular relaxation in mixture, which can be illustrated by the decomposed acoustic relaxation absorption spectrum of the single relaxation process. The proposed model is validated by simulation results in good agreement with experimental data such as N_{2}, O_{2}, CO_{2}, CH_{4} and their mixtures.
]]>2017-12-20T00:05:17-08:00info:doi/10.1098/rspa.2017.0496hwp:master-id:royprsa;rspa.2017.04962017-12-20Research articles47322082017049620170496<![CDATA[The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170596?rss=1
The patterns of fringes produced by an interferometer have long been important testbeds for our best contemporary theories of physics. Historically, interference has been used to contrast quantum mechanics with classical physics, but recently experiments have been performed that test quantum theory against even more exotic alternatives. A physically motivated family of theories are those where the state space of a two-level system is given by a sphere of arbitrary dimension. This includes classical bits, and real, complex and quaternionic quantum theory. In this paper, we consider relativity of simultaneity (i.e. that observers may disagree about the order of events at different locations) as applied to a two-armed interferometer, and show that this forbids most interference phenomena more complicated than those of complex quantum theory. If interference must depend on some relational property of the setting (such as path difference), then relativity of simultaneity will limit state spaces to standard complex quantum theory, or a subspace thereof. If this relational assumption is relaxed, we find one additional theory compatible with relativity of simultaneity: quaternionic quantum theory. Our results have consequences for current laboratory interference experiments: they have to be designed carefully to avoid rendering beyond-quantum effects invisible by relativity of simultaneity.
]]>2017-12-06T00:05:18-08:00info:doi/10.1098/rspa.2017.0596hwp:master-id:royprsa;rspa.2017.05962017-12-06Research articles47322082017059620170596<![CDATA[On the characterization of vortex configurations in the steady rotating Bose-Einstein condensates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170602?rss=1
Motivated by experiments in atomic Bose–Einstein condensates (BECs), we compare predictions of a system of ordinary differential equations (ODEs) for dynamics of one and two individual vortices in the rotating BECs with those of the Gross–Pitaevskii mean-field model written as a partial differential equation (PDE). In particular, we characterize orbitally stable vortex configurations in a symmetric harmonic trap due to a cubic repulsive interaction and a steady rotation. The ODE system is analysed in detail and the PDE model is approximated numerically. Good agreement between the two models is established in the semi-classical (Thomas–Fermi) limit that corresponds to the BECs at large values of the chemical potential.
]]>2017-12-06T00:05:13-08:00info:doi/10.1098/rspa.2017.0602hwp:master-id:royprsa;rspa.2017.06022017-12-06Research articles47322082017060220170602<![CDATA[An instability of the standard model of cosmology creates the anomalous acceleration without dark energy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20160887?rss=1
We identify the condition for smoothness at the centre of spherically symmetric solutions of Einstein’s original equations without the cosmological constant or dark energy. We use this to derive a universal phase portrait which describes general, smooth, spherically symmetric solutions near the centre of symmetry when the pressure p=0. In this phase portrait, the critical k=0 Friedmann space–time appears as a saddle rest point which is unstable to spherical perturbations. This raises the question as to whether the Friedmann space–time is observable by redshift versus luminosity measurements looking outwards from any point. The unstable manifold of the saddle rest point corresponding to Friedmann describes the evolution of local uniformly expanding space–times whose accelerations closely mimic the effects of dark energy. A unique simple wave perturbation from the radiation epoch is shown to trigger the instability, match the accelerations of dark energy up to second order and distinguish the theory from dark energy at third order. In this sense, anomalous accelerations are not only consistent with Einstein’s original theory of general relativity, but are a prediction of it without the cosmological constant or dark energy.
]]>2017-11-22T00:05:31-08:00info:doi/10.1098/rspa.2016.0887hwp:master-id:royprsa;rspa.2016.08872017-11-22Research articles47322072016088720160887<![CDATA[Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170018?rss=1
A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow–fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for non-zero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary conditions. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition, and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow–fast spatial dynamical system.
]]>2017-11-08T00:05:22-08:00info:doi/10.1098/rspa.2017.0018hwp:master-id:royprsa;rspa.2017.00182017-11-08Research articles47322072017001820170018<![CDATA[On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170104?rss=1
In a recent article (Jentzen et al. 2016 Commun. Math. Sci.14, 1477–1500 (doi:10.4310/CMS.2016.v14.n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d{4,5,...}, there exist d-dimensional stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two (d=2) and three (d=3) space dimensions.
]]>2017-11-29T00:05:34-08:00info:doi/10.1098/rspa.2017.0104hwp:master-id:royprsa;rspa.2017.01042017-11-29Research articles47322072017010420170104<![CDATA[Fluid flow in porous media using image-based modelling to parametrize Richards' equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170178?rss=1
The parameters in Richards' equation are usually calculated from experimentally measured values of the soil–water characteristic curve and saturated hydraulic conductivity. The complex pore structures that often occur in porous media complicate such parametrization due to hysteresis between wetting and drying and the effects of tortuosity. Rather than estimate the parameters in Richards' equation from these indirect measurements, image-based modelling is used to investigate the relationship between the pore structure and the parameters. A three-dimensional, X-ray computed tomography image stack of a soil sample with voxel resolution of 6 μm has been used to create a computational mesh. The Cahn–Hilliard–Stokes equations for two-fluid flow, in this case water and air, were applied to this mesh and solved using the finite-element method in COMSOL Multiphysics. The upscaled parameters in Richards' equation are then obtained via homogenization. The effect on the soil–water retention curve due to three different contact angles, 0°, 20° and 60°, was also investigated. The results show that the pore structure affects the properties of the flow on the large scale, and different contact angles can change the parameters for Richards' equation.
]]>2017-11-22T00:05:35-08:00info:doi/10.1098/rspa.2017.0178hwp:master-id:royprsa;rspa.2017.01782017-11-22Research articles47322072017017820170178<![CDATA[Three-dimensional quasi-periodic shifted Green function throughout the spectrum, including Wood anomalies]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170242?rss=1
This work, part II in a series, presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called ‘Wood anomaly frequencies’. At these frequencies, there is a grazing Rayleigh wave, and the quasi-periodic Green function ceases to exist. We present a modification of the Green function by adding two types of terms to its lattice sum. The first type are transversely shifted Green functions with coefficients that annihilate the growth in the original lattice sum and yield algebraic convergence. The second type are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on this new Green function for scattering by doubly periodic surfaces at and around Wood frequencies. We believe this is the first solver able to handle Wood frequencies for doubly periodic scattering problems in three dimensions. We demonstrate the method by applying it to acoustic scattering.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0242hwp:master-id:royprsa;rspa.2017.02422017-11-01Research articles47322072017024220170242<![CDATA[Geometric description of a discrete power function associated with the sixth Painleve equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170312?rss=1
In this paper, we consider the discrete power function associated with the sixth Painlevé equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with W~(3A1(1)) symmetry. By constructing the action of W~(3A1(1)) as a subgroup of W~(D4(1)), i.e. the symmetry group of P_{VI}, we show how to relate W~(D4(1)) to the symmetry group of the lattice. Moreover, by using translations in W~(3A1(1)), we explain the odd–even structure appearing in previously known explicit formulae in terms of the function.
]]>2017-11-22T00:05:35-08:00info:doi/10.1098/rspa.2017.0312hwp:master-id:royprsa;rspa.2017.03122017-11-22Research articles47322072017031220170312<![CDATA[A post Gurney quantum mechanical perspective on the electrolysis of water: ion neutralization in solution]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170371?rss=1
Electron fluxes crossing the interface between a metallic conductor and an aqueous environment are important in many fields; hydrogen production, environmental scanning tunnelling microscopy, scanning electrochemical microscopy being some of them. Gurney (Gurney 1931 Proc. R. Soc. Lond.134, 137 (doi:10.1098/rspa.1931.0187)) provided in 1931 a scheme for tunnelling during electrolysis and outlined conditions for it to occur. We measure the low-voltage current flows between gold electrodes in pure water and use the time-dependent behaviour at voltage switch-on and switch-off to evaluate the relative contribution to the steady current arising from tunnelling of electrons between the electrodes and ions in solution and from the neutralization of ions adsorbed onto the electrode surface. We ascribe the larger current contribution to quantum tunnelling of electrons to and from ions in solution near the electrodes. We refine Gurney's barrier scheme to include solvated electron states and quantify energy differences using updated information. We show that Gurney's conditions would prevent the current flow at low voltages we observe but outline how the ideas of Marcus (Marcus 1956 J. Chem. Phys.24, 966–978 (doi:10.1063/1.1742723)) concerning solvation fluctuations enable the condition to be relaxed. We derive an average barrier tunnelling model and a multiple pathways tunnelling model and compare predictions with measurements of the steady-state current–voltage relation. The tunnelling barrier was found to be wide and low in agreement with other experimental studies. Applications as a biosensing mechanism are discussed that exploit the fast tunnelling pathways along molecules in solution.
]]>2017-11-08T00:05:22-08:00info:doi/10.1098/rspa.2017.0371hwp:master-id:royprsa;rspa.2017.03712017-11-08Research articles47322072017037120170371<![CDATA[Rolling resistance of shallow granular deformation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170375?rss=1
Experiments are conducted to measure the resistance experienced by light cylinders rolling over flat beds of granular media. Sand and glass spheres are used for the beds. The trajectories of the rolling cylinders are determined through optical tracking, and velocity and acceleration data are inferred through fits to these trajectories. The rolling resistance is dominated by a velocity-independent component, but a velocity-dependent drag exceeding the expected strength of air drag is also observed. The results are compared to a theoretical model based on a cohesionless Mohr–Coulomb rheology for a granular medium in the presence of gravity. The model idealizes the flow pattern underneath the rolling cylinder as a plastically deforming zone in front of a rigidly rotating plug attached to the cylinder, as proposed previously for cylinders rolling on perfectly cohesive plastic media. The leading-order, rate-independent rolling resistance observed experimentally is well reproduced by the model predictions.
]]>2017-11-29T00:05:34-08:00info:doi/10.1098/rspa.2017.0375hwp:master-id:royprsa;rspa.2017.03752017-11-29Research articles47322072017037520170375<![CDATA[Force direction patterns promote whole body stability even in hip-flexed walking, but not upper body stability in human upright walking]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170404?rss=1
Directing the ground reaction forces to a focal point above the centre of mass of the whole body promotes whole body stability in human and animal gaits similar to a physical pendulum. Here we show that this is the case in human hip-flexed walking as well. For all upper body orientations (upright, 25°, 50°, maximum), the focal point was well above the centre of mass of the whole body, suggesting its general relevance for walking. Deviations of the forces' lines of action from the focal point increased with upper body inclination from 25 to 43 mm root mean square deviation (RMSD). With respect to the upper body in upright gait, the resulting force also passed near a focal point (17 mm RMSD between the net forces' lines of action and focal point), but this point was 18 cm below its centre of mass. While this behaviour mimics an unstable inverted pendulum, it leads to resulting torques of alternating sign in accordance with periodic upper body motion and probably provides for low metabolic cost of upright gait by keeping hip torques small. Stabilization of the upper body is a consequence of other mechanisms, e.g. hip reflexes or muscle preflexes.
]]>2017-11-08T00:05:22-08:00info:doi/10.1098/rspa.2017.0404hwp:master-id:royprsa;rspa.2017.04042017-11-08Research articles47322072017040420170404<![CDATA[Age distribution dynamics with stochastic jumps in mortality]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170451?rss=1
While deterministic age distribution models have been extensively studied and applied in various disciplines, little work has been devoted to understanding the role of stochasticity in birth and mortality terms. In this paper, we analyse a stochastic M’Kendrick–von Foerster equation in which jumps in mortality represent intense losses of population due to external events. We present explicit solutions for the probability density functions of the age distribution and the total population and for the temporal dynamics of their moments. We also derive the dynamics of the mean age of the population and its harmonic mean. The framework is then used to calculate the age distribution of salt in the soil root zone, where the accumulation of salt by atmospheric deposition is counteracted by plant uptake and by jump losses due to percolation events.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0451hwp:master-id:royprsa;rspa.2017.04512017-11-01Research articles47322072017045120170451<![CDATA[Swelling-induced wrinkling in layered gel beams]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170454?rss=1
Gels are widely employed in smart mechanical devices and biomedical applications. Swelling-induced bending actuation can be obtained by means of a simple bilayer gel beam. We show that this system can also exhibit wrinkling patterns of potential interest for structural morphing and sensing. We study swelling-induced wrinkling at the extrados of a bilayer gel beam with the softer layer on top. The bent configuration at finite strain is recovered first and, starting from it, a linear perturbation analysis is performed. We delimit the zone corresponding to wrinkling modes in a parameter plane encompassing a mechanical stiffness ratio and a geometric top layer to total height ratio. Interestingly, we observe that surface instability precedes and envelopes wrinkling modes of finite wavelength. Finally, we discuss the effect of changes in stiffness and of the Flory–Huggins parameters on the size of the wrinkling domain.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0454hwp:master-id:royprsa;rspa.2017.04542017-11-01Research articles47322072017045420170454<![CDATA[Local fields and effective conductivity tensor of ellipsoidal particle composite with anisotropic constituents]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170472?rss=1
An accurate semi-analytical solution of the conductivity problem for a composite with anisotropic matrix and arbitrarily oriented anisotropic ellipsoidal inhomogeneities has been obtained. The developed approach combines the superposition principle with the multipole expansion of perturbation fields of inhomogeneities in terms of ellipsoidal harmonics and reduces the boundary value problem to an infinite system of linear algebraic equations for the induced multipole moments of inhomogeneities. A complete full-field solution is obtained for the multi-particle models comprising inhomogeneities of diverse shape, size, orientation and properties which enables an adequate account for the microstructure parameters. The solution is valid for the general-type anisotropy of constituents and arbitrary orientation of the orthotropy axes. The effective conductivity tensor of the particulate composite with anisotropic constituents is evaluated in the framework of the generalized Maxwell homogenization scheme. Application of the developed method to composites with imperfect ellipsoidal interfaces is straightforward. Their incorporation yields probably the most general model of a composite that may be considered in the framework of analytical approach.
]]>2017-11-29T00:05:34-08:00info:doi/10.1098/rspa.2017.0472hwp:master-id:royprsa;rspa.2017.04722017-11-29Research articles47322072017047220170472<![CDATA[On the buckling of an elastic holey column]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170477?rss=1
We report the results of a numerical and theoretical study of buckling in elastic columns containing a line of holes. Buckling is a common failure mode of elastic columns under compression, found over scales ranging from metres in buildings and aircraft to tens of nanometers in DNA. This failure usually occurs through lateral buckling, described for slender columns by Euler’s theory. When the column is perforated with a regular line of holes, a new buckling mode arises, in which adjacent holes collapse in orthogonal directions. In this paper, we firstly elucidate how this alternate hole buckling mode coexists and interacts with classical Euler buckling modes, using finite-element numerical calculations with bifurcation tracking. We show how the preferred buckling mode is selected by the geometry, and discuss the roles of localized (hole-scale) and global (column-scale) buckling. Secondly, we develop a novel predictive model for the buckling of columns perforated with large holes. This model is derived without arbitrary fitting parameters, and quantitatively predicts the critical strain for buckling. We extend the model to sheets perforated with a regular array of circular holes and use it to provide quantitative predictions of their buckling.
]]>2017-11-15T00:05:25-08:00info:doi/10.1098/rspa.2017.0477hwp:master-id:royprsa;rspa.2017.04772017-11-15Research articles47322072017047720170477<![CDATA[Bidispersive vertical convection]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170481?rss=1
A bidispersive porous material is one which has usual pores but additionally contains a system of micro pores. We consider a fluid-saturated bidispersive porous medium in the vertical layer x(–1/2,1/2) with gravity in the –z (downward) direction. The walls of the layer are maintained at different constant temperatures. A suitable Rayleigh number is defined and we derive a global stability threshold below which no instability may arise. We additionally show that the porous layer is stable for all Rayleigh numbers provided the initial temperature gradient is bounded in a precise sense.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0481hwp:master-id:royprsa;rspa.2017.04812017-11-01Research articles47322072017048120170481<![CDATA[Analysing causal structures with entropy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170483?rss=1
A central question for causal inference is to decide whether a set of correlations fits a given causal structure. In general, this decision problem is computationally infeasible and hence several approaches have emerged that look for certificates of compatibility. Here, we review several such approaches based on entropy. We bring together the key aspects of these entropic techniques with unified terminology, filling several gaps and establishing new connections, all illustrated with examples. We consider cases where unobserved causes are classical, quantum and post-quantum, and discuss what entropic analyses tell us about the difference. This difference has applications to quantum cryptography, where it can be crucial to eliminate the possibility of classical causes. We discuss the achievements and limitations of the entropic approach in comparison to other techniques and point out the main open problems.
]]>2017-11-01T03:11:33-07:00info:doi/10.1098/rspa.2017.0483hwp:master-id:royprsa;rspa.2017.04832017-11-01Review articles47322072017048320170483<![CDATA[Multi-scale mechanics of granular solids from grain-resolved X-ray measurements]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170491?rss=1
This work discusses an experimental technique for studying the mechanics of three-dimensional (3D) granular solids. The approach combines 3D X-ray diffraction and X-ray computed tomography to measure grain-resolved strains, kinematics and contact fabric in the bulk of a granular solid, from which continuum strains, grain stresses, interparticle forces and coarse-grained elasto-plastic moduli can be determined. We demonstrate the experimental approach and analysis of selected results on a sample of 1099 stiff, frictional grains undergoing multiple uniaxial compression cycles. We investigate the inter-particle force network, elasto-plastic moduli and associated length scales, reversibility of mechanical responses during cyclic loading, the statistics of microscopic responses and microstructure–property relationships. This work serves to highlight both the fundamental insight into granular mechanics that is furnished by combined X-ray measurements and describes future directions in the field of granular materials that can be pursued with such approaches.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0491hwp:master-id:royprsa;rspa.2017.04912017-11-01Research articles47322072017049120170491<![CDATA[The grasshopper problem]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170494?rss=1
We introduce and physically motivate the following problem in geometric combinatorics, originally inspired by analysing Bell inequalities. A grasshopper lands at a random point on a planar lawn of area 1. It then jumps once, a fixed distance d, in a random direction. What shape should the lawn be to maximize the chance that the grasshopper remains on the lawn after jumping? We show that, perhaps surprisingly, a disc-shaped lawn is not optimal for any d>0. We investigate further by introducing a spin model whose ground state corresponds to the solution of a discrete version of the grasshopper problem. Simulated annealing and parallel tempering searches are consistent with the hypothesis that, for d<^{–1/2}, the optimal lawn resembles a cogwheel with n cogs, where the integer n is close to (arcsin(d/2))–1. We find transitions to other shapes for d–1/2.
]]>2017-11-22T00:05:34-08:00info:doi/10.1098/rspa.2017.0494hwp:master-id:royprsa;rspa.2017.04942017-11-22Research articles47322072017049420170494<![CDATA[On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170502?rss=1
The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point. Centre manifold reduction and normal form calculations are performed to investigate the long-term dynamics of the cutting process. The normal form of the four-dimensional centre subsystem is derived analytically, and the possible topologies in the infinite-dimensional phase space of the system are revealed. It is shown that bistable parameter regions exist where unstable periodic and, in certain cases, unstable quasi-periodic motions coexist with the equilibrium. Taking into account the non-smoothness caused by loss of contact between the tool and the workpiece, the boundary of the bistable region is also derived analytically. The results are verified by numerical continuation. The possibility of (transient) chaotic motions in the global non-smooth dynamics is shown.
]]>2017-11-15T00:05:25-08:00info:doi/10.1098/rspa.2017.0502hwp:master-id:royprsa;rspa.2017.05022017-11-15Research articles47322072017050220170502<![CDATA[Advection and Taylor-Aris dispersion in rivulet flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170524?rss=1
Motivated by the need for a better understanding of the transport of solutes in microfluidic flows with free surfaces, the advection and dispersion of a passive solute in steady unidirectional flow of a thin uniform rivulet on an inclined planar substrate driven by gravity and/or a uniform longitudinal surface shear stress are analysed. Firstly, we describe the short-time advection of both an initially semi-infinite and an initially finite slug of solute of uniform concentration. Secondly, we describe the long-time Taylor–Aris dispersion of an initially finite slug of solute. In particular, we obtain the general expression for the effective diffusivity for Taylor–Aris dispersion in such a rivulet, and discuss in detail its different interpretations in the special case of a rivulet on a vertical substrate.
]]>2017-11-15T00:05:27-08:00info:doi/10.1098/rspa.2017.0524hwp:master-id:royprsa;rspa.2017.05242017-11-15Research articles47322072017052420170524<![CDATA[Symmetric rearrangement of groundwater-fed streams]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170539?rss=1
Streams shape landscapes through headward growth and lateral migration. When these streams are primarily fed by groundwater, recent work suggests that their tips advance to maximize the symmetry of the local Laplacian field associated with groundwater flow. We explore the extent to which such forcing is responsible for the lateral migration of streams by studying two features of groundwater-fed streams in Bristol, Florida: their confluence angle near junctions and their curvature. First, we find that, while streams asymptotically form a 72° angle near their tips, they simultaneously exhibit a wide 120° confluence angle within approximately 10 m of their junctions. We show that this wide angle maximizes the symmetry of the groundwater field near the junction. Second, we argue that streams migrate laterally within valleys and present a new spectral analysis method to relate planform curvature to the surrounding groundwater field. Our results suggest that streams migrate laterally in response to fluxes from the surrounding groundwater table, providing evidence of a new mechanism that complements Laplacian growth at their tips.
]]>2017-11-08T00:59:42-08:00info:doi/10.1098/rspa.2017.0539hwp:master-id:royprsa;rspa.2017.05392017-11-08Research articles47322072017053920170539<![CDATA[Lagrangian averaging with geodesic mean]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170558?rss=1
This paper revisits the derivation of the Lagrangian averaged Euler (LAE), or Euler-α equations in the light of an intrinsic definition of the averaged flow map as the geodesic mean on the volume-preserving diffeomorphism group. Under the additional assumption that first-order fluctuations are statistically isotropic and transported by the mean flow as a vector field, averaging of the kinetic energy Lagrangian of an ideal fluid yields the LAE Lagrangian. The derivation presented here assumes a Euclidean spatial domain without boundaries.
]]>2017-11-15T00:43:18-08:00info:doi/10.1098/rspa.2017.0558hwp:master-id:royprsa;rspa.2017.05582017-11-15Research articles47322072017055820170558<![CDATA[Nonlinear mode interaction in equal-leg angle struts susceptible to cellular buckling]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170583?rss=1
A variational model that describes the interactive buckling of a thin-walled equal-leg angle strut under pure axial compression is presented. A formulation combining the Rayleigh–Ritz method and continuous displacement functions is used to derive a system of differential and integral equilibrium equations for the structural component. Solving the equations using numerical continuation reveals progressive cellular buckling (or snaking) arising from the nonlinear interaction between the weak-axis flexural buckling mode and the strong-axis flexural–torsional buckling mode for the first time—the resulting behaviour being highly unstable. Physical experiments conducted on 10 cold-formed steel specimens are presented and the results show good agreement with the variational model.
]]>2017-11-01T07:43:36-07:00info:doi/10.1098/rspa.2017.0583hwp:master-id:royprsa;rspa.2017.05832017-11-01Research articles47322072017058320170583<![CDATA[How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170607?rss=1
The mechanical response of a homogeneous isotropic linearly elastic material can be fully characterized by two physical constants, the Young’s modulus and the Poisson’s ratio, which can be derived by simple tensile experiments. Any other linear elastic parameter can be obtained from these two constants. By contrast, the physical responses of nonlinear elastic materials are generally described by parameters which are scalar functions of the deformation, and their particular choice is not always clear. Here, we review in a unified theoretical framework several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests. These parameters represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. Universal relations between certain of these parameters are further established, and then used to quantify nonlinear elastic responses in several hyperelastic models for rubber, soft tissue and foams. The general parameters identified here can also be viewed as a flexible basis for coupling elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales.
]]>2017-11-29T01:11:16-08:00info:doi/10.1098/rspa.2017.0607hwp:master-id:royprsa;rspa.2017.06072017-11-29Review articles47322072017060720170607<![CDATA[Theory of energy harvesting from heartbeat including the effects of pleural cavity and respiration]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170615?rss=1
Self-powered implantable devices with flexible energy harvesters are of significant interest due to their potential to solve the problem of limited battery life and surgical replacement. The flexible electronic devices made of piezoelectric materials have been employed to harvest energy from the motion of biological organs. Experimental measurements show that the output voltage of the device mounted on porcine left ventricle in chest closed environment decreases significantly compared to the case of chest open. A restricted-space deformation model is proposed to predict the impeding effect of pleural cavity, surrounding tissues, as well as respiration on the efficiency of energy harvesting from heartbeat using flexible piezoelectric devices. The analytical solution is verified by comparing theoretical predictions to experimental measurements. A simple scaling law is established to analyse the intrinsic correlations between the normalized output power and the combined system parameters, i.e. the normalized permitted space and normalized electrical load. The results may provide guidelines for optimization of in vivo energy harvesting from heartbeat or the motions of other biological organs using flexible piezoelectric energy harvesters.
]]>2017-11-22T00:05:34-08:00info:doi/10.1098/rspa.2017.0615hwp:master-id:royprsa;rspa.2017.06152017-11-22Research articles47322072017061520170615<![CDATA[Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170636?rss=1
A nonlinear two-dimensional (2D) continuum with a latent internal structure is introduced as a coarse model of a plane network of beams which, in turn, is assumed as a model of a pantographic structure made up by two families of equispaced beams, superimposed and connected by pivots. The deformation measures of the beams of the network and that of the 2D body are introduced and the former are expressed in terms of the latter by making some kinematical assumptions. The expressions for the strain and kinetic energy densities of the network are then introduced and given in terms of the kinematic quantities of the 2D continuum. To account for the modelling abilities of the 2D continuum in the linear range, the eigenmode and eigenfrequencies of a given specimen are determined. The buckling and post-buckling behaviour of the same specimen, subjected to two different loading conditions are analysed as tests in the nonlinear range. The problems have been solved numerically by means of the COMSOL Multiphysics finite element software.
]]>2017-11-15T03:22:36-08:00info:doi/10.1098/rspa.2017.0636hwp:master-id:royprsa;rspa.2017.06362017-11-15Research articles47322072017063620170636