Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Research articles
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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- recent Research articles articles1471-2946Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[A diffuse interface model for the analysis of propagating bulges in cylindrical balloons]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2218/20180333?rss=1
With the aim to characterize the formation and propagation of bulges in cylindrical rubber balloons, we carry out an expansion of the nonlinear axisymmetric membrane model assuming slow axial variations. We obtain a diffuse interface model similar to that introduced by van der Waals in the context of liquid–vapour phase transitions. This provides a quantitative basis to the well-known analogy between propagating bulges and phase transitions. The diffuse interface model is amenable to numerical as well as analytical solutions, including linear and nonlinear bifurcation analyses. Comparisons to the original membrane model reveal that the diffuse interface model captures the bulging phenomenon very accurately, even for well-localized phase boundaries.
]]>2018-10-10T00:34:51-07:00info:doi/10.1098/rspa.2018.0333hwp:master-id:royprsa;rspa.2018.03332018-10-10Research articles47422182018033320180333<![CDATA[Asymptotic analysis for dispersion relations and travel times in noise cross-correlations: spherically symmetric case]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2218/20180382?rss=1
The cross-correlations of ambient noise or earthquake codas are massively used in seismic tomography to measure the dispersion curves of surface waves and the travel times of body waves. Such measurements are based on the assumption that these kinematic parameters in the cross-correlations of noise coincide with those in Green's functions. However, the relation between the cross-correlations of noise and Green's functions deserves to be studied more precisely. In this paper, we use the asymptotic analysis to study the dispersion relations of surface waves and the travel times of body waves, and come to the conclusion that for the spherically symmetric Earth model, when the distribution of noise sources is laterally uniform, the dispersion relations of surface waves and the travel times of SH body-wave phases in noise correlations should be exactly the same as those in Green's functions.
]]>2018-10-10T00:34:51-07:00info:doi/10.1098/rspa.2018.0382hwp:master-id:royprsa;rspa.2018.03822018-10-10Research articles47422182018038220180382<![CDATA[Topological data analysis for true step detection in periodic piecewise constant signals]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2218/20180027?rss=1
This paper introduces a simple yet powerful approach based on topological data analysis for detecting true steps in a periodic, piecewise constant (PWC) signal. The signal is a two-state square wave with randomly varying in-between-pulse spacing, subject to spurious steps at the rising or falling edges which we call digital ringing. We use persistent homology to derive mathematical guarantees for the resulting change detection which enables accurate identification and counting of the true pulses. The approach is tested using both synthetic and experimental data obtained using an engine lathe instrumented with a laser tachometer. The described algorithm enables accurate and automatic calculations of the spindle speed without any choice of parameters. The results are compared with the frequency and sequency methods of the Fourier and Walsh–Hadamard transforms, respectively. Both our approach and the Fourier analysis yield comparable results for pulses with regular spacing and digital ringing while the latter causes large errors using the Walsh–Hadamard method. Further, the described approach significantly outperforms the frequency/sequency analyses when the spacing between the peaks is varied. We discuss generalizing the approach to higher dimensional PWC signals, although using this extension remains an interesting question for future research.
]]>2018-10-03T01:50:17-07:00info:doi/10.1098/rspa.2018.0027hwp:master-id:royprsa;rspa.2018.00272018-10-03Research articles47422182018002720180027<![CDATA[Proton tunnelling in hydrogen bonds and its implications in an induced-fit model of enzyme catalysis]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2218/20180037?rss=1
The role of proton tunnelling in biological catalysis is investigated here within the frameworks of quantum information theory and thermodynamics. We consider the quantum correlations generated through two hydrogen bonds between a substrate and a prototypical enzyme that first catalyses the tautomerization of the substrate to move on to a subsequent catalysis, and discuss how the enzyme can derive its catalytic potency from these correlations. In particular, we show that classical changes induced in the binding site of the enzyme spreads the quantum correlations among all of the four hydrogen-bonded atoms thanks to the directionality of hydrogen bonds. If the enzyme rapidly returns to its initial state after the binding stage, the substrate ends in a new transition state corresponding to a quantum superposition. Open quantum system dynamics can then naturally drive the reaction in the forward direction from the major tautomeric form to the minor tautomeric form without needing any additional catalytic activity. We find that in this scenario the enzyme lowers the activation energy so much that there is no energy barrier left in the tautomerization, even if the quantum correlations quickly decay.
]]>2018-10-03T01:50:17-07:00info:doi/10.1098/rspa.2018.0037hwp:master-id:royprsa;rspa.2018.00372018-10-03Research articles47422182018003720180037<![CDATA[Allosteric interactions in a birod model of DNA]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2218/20180136?rss=1
Allosteric interactions between molecules bound to DNA at distant locations have been known for a long time. The phenomenon has been studied via experiments and numerical simulations, but a comprehensive understanding grounded in a theory of DNA elasticity remains a challenge. Here, we quantify allosteric interactions between two entities bound to DNA by using the theory of birods. We recognize that molecules bound to DNA cause local deformations that can be captured in a birod model which consists of two elastic strands interacting via an elastic web representing the basepairs. We show that the displacement field caused by bound entities decays exponentially with distance from the binding site. We compute the interaction energy between two proteins on DNA as a function of distance between them and find that it decays exponentially while oscillating with the periodicity of the double helix, in excellent agreement with experiments. The decay length of the interaction energy can be determined in terms of the mechanical properties of the strands and the webbing in our birod model, and it varies with the GC content of the DNA. Our model provides a framework for viewing allosteric interactions in DNA within the ambit of configurational forces of continuum elasticity.
]]>2018-10-03T01:50:17-07:00info:doi/10.1098/rspa.2018.0136hwp:master-id:royprsa;rspa.2018.01362018-10-03Research articles47422182018013620180136<![CDATA[A Hamiltonian mean field system for the Navier-Stokes equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2218/20180178?rss=1
We use a Hamiltonian interacting particle system to derive a stochastic mean field system whose McKean–Vlasov equation yields the incompressible Navier–Stokes equation. Since the system is Hamiltonian, the particle relabeling symmetry implies a Kelvin Circulation Theorem along stochastic Lagrangian paths. Moreover, issues of energy dissipation are discussed and the model is connected to other approaches in the literature.
]]>2018-10-03T01:50:17-07:00info:doi/10.1098/rspa.2018.0178hwp:master-id:royprsa;rspa.2018.01782018-10-03Research articles47422182018017820180178<![CDATA[Determination of the instantaneous geostrophic flow within the three-dimensional magnetostrophic regime]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2218/20180412?rss=1
In his seminal work, Taylor (1963 Proc. R. Soc. Lond. A274, 274–283. (doi:10.1098/rspa.1963.0130).) argued that the geophysically relevant limit for dynamo action within the outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations. Within this limit, he showed the existence of a necessary condition, now well known as Taylor's constraint, which requires that the cylindrically averaged Lorentz torque must everywhere vanish; magnetic fields that satisfy this condition are termed Taylor states. Taylor further showed that the requirement of this constraint being continuously satisfied through time prescribes the evolution of the geostrophic flow, the cylindrically averaged azimuthal flow. We show that Taylor's original prescription for the geostrophic flow, as satisfying a given second-order ordinary differential equation, is only valid for a small subset of Taylor states. An incomplete treatment of the boundary conditions renders his equation generally incorrect. Here, by taking proper account of the boundaries, we describe a generalization of Taylor's method that enables correct evaluation of the instantaneous geostrophic flow for any three-dimensional Taylor state. We present the first full-sphere examples of geostrophic flows driven by non-axisymmetric Taylor states. Although in axisymmetry the geostrophic flow admits a mild logarithmic singularity on the rotation axis, in the fully three-dimensional case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular.
]]>2018-10-03T01:50:17-07:00info:doi/10.1098/rspa.2018.0412hwp:master-id:royprsa;rspa.2018.04122018-10-03Research articles47422182018041220180412<![CDATA[Microstructural patterns with tunable mechanical anisotropy obtained by simulating anisotropic spinodal decomposition]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2218/20180535?rss=1
The generation of mechanical metamaterials with tailored effective properties through carefully engineered microstructures requires avenues to predict optimal microstructural architectures. Phase separation in heterogeneous systems naturally produces complex microstructural patterns whose effective response depends on the underlying process of spinodal decomposition. During this process, anisotropy may arise due to advection, diffusive chemical gradients or crystallographic interface energy, leading to anisotropic patterns with strongly directional effective properties. We explore the link between anisotropic surface energies during spinodal decomposition, the resulting microstructures and, ultimately, the anisotropic elastic moduli of the resulting medium. We simulate the formation of anisotropic patterns within representative volume elements, using recently developed stabilized spectral techniques that circumvent further regularization, and present a powerful alternative to current numerical techniques. The interface morphology of representative phase-separated microstructures is shown to strongly depend on surface anisotropy. The effective elastic moduli of the thus-obtained porous media are identified by periodic homogenization, and directionality is demonstrated through elastic surfaces. Our approach not only improves upon numerical tools to simulate phase separation; it also offers an avenue to generate tailored microstructures with tunable resulting elastic anisotropy.
]]>2018-10-03T01:50:17-07:00info:doi/10.1098/rspa.2018.0535hwp:master-id:royprsa;rspa.2018.05352018-10-03Research articles47422182018053520180535<![CDATA[Planar morphometrics using Teichmüller maps]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20170905?rss=1
Inspired by the question of quantifying wing shape, we propose a computational approach for analysing planar shapes. We first establish a correspondence between the boundaries of two planar shapes with boundary landmarks using geometric functional data analysis and then compute a landmark-matching curvature-guided Teichmüller mapping with uniform quasi-conformal distortion in the bulk. This allows us to analyse the pair-wise difference between the planar shapes and construct a similarity matrix on which we deploy methods from network analysis to cluster shapes. We deploy our method to study a variety of Drosophila wings across species to highlight the phenotypic variation between them, and Lepidoptera wings over time to study the developmental progression of wings. Our approach of combining complex analysis, computation and statistics to quantify, compare and classify planar shapes may be usefully deployed in other biological and physical systems.
]]>2018-09-26T00:05:19-07:00info:doi/10.1098/rspa.2017.0905hwp:master-id:royprsa;rspa.2017.09052018-09-26Research articles47422172017090520170905<![CDATA[An ordinary state-based peridynamic model for the fracture of zigzag graphene sheets]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180019?rss=1
This study develops an ordinary state-based peridynamic coarse-graining (OSPD-CG) model for the investigation of fracture in single-layer graphene sheets (SLGS), in which the peridynamic (PD) parameters are derived through combining the PD model and molecular dynamics (MD) simulations from the fully atomistic system via energy conservation. The fracture failure of pre-cracked SLGS under uniaxial tension is studied using the proposed PD model. And the PD simulation results agree well with those from MD simulations, including the stress–strain relations, the crack propagation patterns and the average crack propagation velocities. The interaction effect between cracks located at the centre and the edge on the crack propagation of the pre-cracked SLGS is discussed in detail. This work shows that the proposed PD model is much more efficient than the MD simulations and, thus, indicates that the PD-based method is applicable to study larger nanoscale systems.
]]>2018-09-26T00:05:19-07:00info:doi/10.1098/rspa.2018.0019hwp:master-id:royprsa;rspa.2018.00192018-09-26Research articles47422172018001920180019<![CDATA[A mathematical and numerical framework for near-field optics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180150?rss=1
This paper is concerned with the inverse problem of reconstructing small and local perturbations of a planar surface using the field interaction between a known plasmonic particle and the planar surface. The aim is to perform a super-resolved reconstruction of these perturbations from shifts in the plasmonic frequencies of the particle-surface system. In order to analyse the interaction between the plasmonic particle and the planar surface, a well-chosen conformal mapping, which transforms the particle-surface system into a coated structure, is used. Then the even Fourier coefficients of the transformed domain are related to the shifts in the plasmonic resonances of the particle-surface system. A direct reconstruction of the perturbations of the planar surface is proposed. Its viability and limitations are documented by numerical examples.
]]>2018-09-26T00:05:19-07:00info:doi/10.1098/rspa.2018.0150hwp:master-id:royprsa;rspa.2018.01502018-09-26Research articles47422172018015020180150<![CDATA[Points of constancy of the periodic linearized Korteweg-deVries equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180160?rss=1
We investigate the points of constancy in the piecewise constant solution profiles of the periodic linearized Korteweg–deVries equation with step function initial data at rational times. The solution formulae are given by certain Weyl sums, and we employ number theoretic techniques, including Kummer sums, in our analysis. These results constitute an initial attempt to understand the complementary phenomenon of ‘fractalization’ at irrational times.
]]>2018-09-26T00:05:19-07:00info:doi/10.1098/rspa.2018.0160hwp:master-id:royprsa;rspa.2018.01602018-09-26Research articles47422172018016020180160<![CDATA[Theoretically optimal forms for very long-span bridges under gravity loading]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20170726?rss=1
Long-span bridges have traditionally employed suspension or cable-stayed forms, comprising vertical pylons and networks of cables supporting a bridge deck. However, the optimality of such forms over very long spans appears never to have been rigorously assessed, and the theoretically optimal form for a given span carrying gravity loading has remained unknown. To address this we here describe a new numerical layout optimization procedure capable of intrinsically modelling the self-weight of the constituent structural elements, and use this to identify the form requiring the minimum volume of material for a given span. The bridge forms identified are complex and differ markedly to traditional suspension and cable-stayed bridge forms. Simplified variants incorporating split pylons are also presented. Although these would still be challenging to construct in practice, a benefit is that they are capable of spanning much greater distances for a given volume of material than traditional suspension and cable-stayed forms employing vertical pylons, particularly when very long spans (e.g. over 2 km) are involved.
]]>2018-09-19T00:05:16-07:00info:doi/10.1098/rspa.2017.0726hwp:master-id:royprsa;rspa.2017.07262018-09-19Research articles47422172017072620170726<![CDATA[Two-dimensional translation-invariant probability distributions: approximations, characterizations and no-go theorems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20170822?rss=1
We study the properties of the set of marginal distributions of infinite translation-invariant systems in the two-dimensional square lattice. In cases where the local variables can only take a small number d of possible values, we completely solve the marginal or membership problem for nearest-neighbours distributions (d = 2, 3) and nearest and next-to-nearest neighbours distributions (d = 2). Remarkably, all these sets form convex polytopes in probability space. This allows us to devise an algorithm to compute the minimum energy per site of any TI Hamiltonian in these scenarios exactly. We also devise a simple algorithm to approximate the minimum energy per site up to arbitrary accuracy for the cases not covered above. For variables of a higher (but finite) dimensionality, we prove two no-go results. To begin, the exact computation of the energy per site of arbitrary TI Hamiltonians with only nearest-neighbour interactions is an undecidable problem. In addition, in scenarios with d≥2947, the boundary of the set of nearest-neighbour marginal distributions contains both flat and smoothly curved surfaces and the set itself is not semi-algebraic. This implies, in particular, that it cannot be characterized via semidefinite programming, even if we allow the input of the programme to include polynomials of nearest-neighbour probabilities.
]]>2018-09-19T00:05:16-07:00info:doi/10.1098/rspa.2017.0822hwp:master-id:royprsa;rspa.2017.08222018-09-19Research articles47422172017082220170822<![CDATA[The Horton-Rogers-Lapwood problem for an inclined porous layer with permeable boundaries]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180021?rss=1
A formulation of the Horton–Rogers–Lapwood problem for a porous layer inclined with respect to the horizontal and characterized by permeable (isobaric) boundary conditions is presented. This formulation allows one to recover the results reported in the literature for the limiting cases of horizontal and vertical layer. It is shown that a threshold inclination angle exists which yields an upper bound to a parametric domain where the critical wavenumber is zero. Within this domain, the critical Darcy–Rayleigh number can be determined analytically. The stability analysis is performed for linear perturbations. The solution is found numerically, for the inclination angles above the threshold, by employing a Runge–Kutta method coupled with the shooting method.
]]>2018-09-19T00:05:16-07:00info:doi/10.1098/rspa.2018.0021hwp:master-id:royprsa;rspa.2018.00212018-09-19Research articles47422172018002120180021<![CDATA[Turbulence generation from a stochastic wavelet model]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180093?rss=1
This research presents a new turbulence generation method based on stochastic wavelets and tests various properties of the generated turbulence field in both the homogeneous and inhomogeneous cases. Numerical results indicate that turbulence fields can be generated with much smaller bases in comparison to synthetic Fourier methods while maintaining comparable accuracy. Adaptive generation of inhomogeneous turbulence is achieved by a scale reduction algorithm, which greatly reduces the computation cost and practically introduces no error. The generating formula issued in this research could be adjusted to generate fully inhomogeneous and anisotropic turbulence with given RANS data under divergence-free constraint, which was not achieved previously in similar research. Numerical examples shows that the generated homogeneous and inhomogeneous turbulence are in good agreement with the input data and theoretical results.
]]>2018-09-19T00:05:16-07:00info:doi/10.1098/rspa.2018.0093hwp:master-id:royprsa;rspa.2018.00932018-09-19Research articles47422172018009320180093<![CDATA[The origin of compression influences geometric instabilities in bilayers]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180267?rss=1
Geometric instabilities in bilayered structures control the surface morphology in a wide range of biological and technical systems. Depending on the application, different mechanisms induce compressive stresses in the bilayer. However, the impact of the chosen origin of compression on the critical conditions, post-buckling evolution and higher-order pattern selection remains insufficiently understood. Here, we conduct a numerical study on a finite-element set-up and systematically vary well-known factors contributing to pattern selection under the four main origins of compression: film growth, substrate shrinkage and whole-domain compression with and without pre-stretch. We find that the origin of compression determines the substrate stretch state at the primary instability point and thus significantly affects the critical buckling conditions. Similarly, it leads to different post-buckling evolutions and secondary instability patterns when the load further increases. Our results emphasize that future phase diagrams of geometric instabilities should incorporate not only the film thickness but also the origin of compression. Thoroughly understanding the influence of the origin of compression on geometric instabilities is crucial to solving real-life problems such as the engineering of smart surfaces or the diagnosis of neuronal disorders, which typically involve temporally or spatially combined origins of compression.
]]>2018-09-19T00:05:16-07:00info:doi/10.1098/rspa.2018.0267hwp:master-id:royprsa;rspa.2018.02672018-09-19Research articles47422172018026720180267<![CDATA[A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180231?rss=1
The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in TI, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.
]]>2018-09-19T00:05:16-07:00info:doi/10.1098/rspa.2018.0231hwp:master-id:royprsa;rspa.2018.02312018-09-19Research articles47422172018023120180231<![CDATA[Robust data-driven discovery of governing physical laws with error bars]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180305?rss=1
Discovering governing physical laws from noisy data is a grand challenge in many science and engineering research areas. We present a new approach to data-driven discovery of ordinary differential equations (ODEs) and partial differential equations (PDEs), in explicit or implicit form. We demonstrate our approach on a wide range of problems, including shallow water equations and Navier–Stokes equations. The key idea is to select candidate terms for the underlying equations using dimensional analysis, and to approximate the weights of the terms with error bars using our threshold sparse Bayesian regression. This new algorithm employs Bayesian inference to tune the hyperparameters automatically. Our approach is effective, robust and able to quantify uncertainties by providing an error bar for each discovered candidate equation. The effectiveness of our algorithm is demonstrated through a collection of classical ODEs and PDEs. Numerical experiments demonstrate the robustness of our algorithm with respect to noisy data and its ability to discover various candidate equations with error bars that represent the quantified uncertainties. Detailed comparisons with the sequential threshold least-squares algorithm and the lasso algorithm are studied from noisy time-series measurements and indicate that the proposed method provides more robust and accurate results. In addition, the data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.
]]>2018-09-19T00:05:16-07:00info:doi/10.1098/rspa.2018.0305hwp:master-id:royprsa;rspa.2018.03052018-09-19Research articles47422172018030520180305<![CDATA[Potential flow through a cascade of aerofoils: direct and inverse problems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180065?rss=1
The potential flow through an infinite cascade of aerofoils is considered as both a direct and inverse problem. In each case, a perturbation expansion about a background uniform flow is assumed where the size of the perturbation is comparable to the aspect ratio of the aerofoils. This perturbation must decay far upstream and also satisfy particular edge conditions, including the Kutta condition at each trailing edge. In the direct problem, the flow field through a cascade of aerofoils of known geometry is calculated. This is solved analytically by recasting the situation as a Riemann–Hilbert problem with only imaginary values prescribed on the chords. As the distance between aerofoils is taken to infinity, the solution is seen to converge to a known analytic expression for a single aerofoil. Analytic expressions for the surface velocity, lift and deflection angle are presented as functions of aerofoil geometry, angle of attack and stagger angle; these show good agreement with numerical results. In the inverse problem, the aerofoil geometry is calculated from a prescribed tangential surface velocity along the chords and upstream angle of attack. This is found via the solution of a singular integral equation prescribed on the chords of the aerofoils.
]]>2018-09-12T00:05:14-07:00info:doi/10.1098/rspa.2018.0065hwp:master-id:royprsa;rspa.2018.00652018-09-12Research articles47422172018006520180065<![CDATA[Transient and asymptotic kinetics of mass transfer by coupled surface and grain boundary diffusion in sintering under strain rate control]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180104?rss=1
An original procedure is developed for simulating pore surface evolution during sintering at high strain rate while distinguishing two types of diffusion fluxes: transient surface fluxes governed by short-range curvature gradients and coupled fluxes at surface and grain boundary governed by strain rate. The latter fluxes become dominant asymptotically, i.e. after damping-out of transient fluxes. The procedure aims at allowing the prediction of the strain rate dependence of macroscopic viscosity, a concept which is meaningful only during the asymptotic stage. The problem is addressed in two-dimension. It is shown that the asymptotic solution of the general partial differential equation of the problem can be obtained as the solution of an ordinary differential equation, of which the resolution lends itself to a semi-analytical procedure. An estimate is also proposed for the rate of convergence of the general solution towards the asymptotic solution. The accuracy of the mathematical procedure is validated by a comparison of the evolution of asymptotic profiles and exact profiles calculated fully numerically during densification or expansion of the system. A method is proposed for mapping the conditions of existence of an asymptotic stage. The method can account for the dependence of average grain coordination on relative density.
]]>2018-09-12T00:05:14-07:00info:doi/10.1098/rspa.2018.0104hwp:master-id:royprsa;rspa.2018.01042018-09-12Research articles47422172018010420180104<![CDATA[The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180264?rss=1
We propose a unique way to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimizing a ‘Hilbert–Schmidt distance’ to the original inner product among the entire class of admissible inner products. We prove that either the minimizer exists and is unique or it does not exist at all. In the former case, we derive a system of Euler–Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supported by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.
]]>2018-09-12T00:05:14-07:00info:doi/10.1098/rspa.2018.0264hwp:master-id:royprsa;rspa.2018.02642018-09-12Research articles47422172018026420180264<![CDATA[Bayesian adaptation of chaos representations using variational inference and sampling on geodesics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180285?rss=1
A novel approach is presented for constructing polynomial chaos representations of scalar quantities of interest (QoI) that extends previously developed methods for adaptation in Homogeneous Chaos spaces. In this work, we develop a Bayesian formulation of the problem that characterizes the posterior distributions of the series coefficients and the adaptation rotation matrix acting on the Gaussian input variables. The adaptation matrix is thus construed as a new parameter of the map from input to QoI, estimated through Bayesian inference. For the computation of the coefficients' posterior distribution, we use a variational inference approach that approximates the posterior with a member of the same exponential family as the prior, such that it minimizes a Kullback–Leibler criterion. On the other hand, the posterior distribution of the rotation matrix is explored by employing a Geodesic Monte Carlo sampling approach, consisting of a variation of the Hamiltonian Monte Carlo algorithm for embedded manifolds, in our case, the Stiefel manifold of orthonormal matrices. The performance of our method is demonstrated through a series of numerical examples, including the problem of multiphase flow in heterogeneous porous media.
]]>2018-09-12T00:05:14-07:00info:doi/10.1098/rspa.2018.0285hwp:master-id:royprsa;rspa.2018.02852018-09-12Research articles47422172018028520180285<![CDATA[Acousto-thermo-mechanical deformation of hydrogels coupled with chemical diffusion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180293?rss=1
We develop an acousto-thermo-mechanical theory for nonlinear (large) deformation of temperature-sensitive hydrogels subjected to temperature and ultrasonic inputs, with diffusion mass transport driven by osmotic pressure accounted for. On the basis of the strain energy due to network stretching, the mixing energy of polymers and small molecules, the Cauchy stress of the deformed hydrogel can be obtained. The acoustic radiation stress generated by the ultrasonic inputs is incorporated into the Cauchy stress to give the constitutive equations of the acousto-thermal-mechanical hydrogel. The mixing energy contains an interaction parameter as a function of temperature and polymer concentration so that hydrogel deformation is temperature dependent. By employing the incompressible condition of polymers and molecules, both the temperature and acoustic radiation stress contribute to osmotic pressure, inducing hydrogel swelling (or shrinking). Specifically, for a temperature-sensitive hydrogel layer immersed in solvent, its acoustic-triggered large deformation is comprehensively analysed under different boundary conditions (e.g. free swelling, uniaxial constraint and biaxial constraint).
]]>2018-09-12T00:05:14-07:00info:doi/10.1098/rspa.2018.0293hwp:master-id:royprsa;rspa.2018.02932018-09-12Research articles47422172018029320180293<![CDATA[Multiple input control strategies for robust and adaptive climate engineering in a low-order 3-box model]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180447?rss=1
A low-order 3-box energy balance model for the climate system is employed with a multivariable control scheme for the evaluation of new robust and adaptive climate engineering strategies using solar radiation management. The climate engineering measures are deployed in three boxes thus representing northern, southern and central bands. It is shown that, through heat transport between the boxes, it is possible to effect a degree of latitudinal control through the reduction of insolation. The approach employed consists of a closed-loop system with an adaptive controller, where the required control intervention is estimated under the RCP4.5 radiative scenario. Through the online estimation of the controller parameters, adaptive control can overcome key issues related to uncertainties of the climate model, the external radiative forcing and the dynamics of the actuator used. In fact, the use of adaptive control offers a robust means of dealing with unforeseeable abrupt perturbations, as well as the parametrization of the model considered, to counteract the RCP4.5 scenario, while still providing bounds on stability and control performance. Moreover, applying multivariable control theory also allows the formal controllability and observability of the system to be investigated in order to identify all feasible control strategies.
]]>2018-09-12T00:05:14-07:00info:doi/10.1098/rspa.2018.0447hwp:master-id:royprsa;rspa.2018.04472018-09-12Research articles47422172018044720180447<![CDATA[A doubly anharmonic oscillator in an induced electric dipole system]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20170881?rss=1
The behaviour of the interaction of the induced electric dipole moment of an atom with a uniform magnetic field and a non-uniform electric field are investigated in a rotating reference frame. An interesting aspect of this interaction is that it gives rise to an analogue of a spinless particle subject to the doubly anharmonic oscillator. Then, it is shown that analytical solutions to the Schrödinger equation can be obtained. Another point raised is that the quantum effects on the induced electric dipole moment can be observed if the uniform magnetic field possesses a discrete set of values.
]]>2018-09-05T00:46:40-07:00info:doi/10.1098/rspa.2017.0881hwp:master-id:royprsa;rspa.2017.08812018-09-05Research articles47422172017088120170881<![CDATA[A systematic mixed-integer differential evolution approach for water network operational optimization]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20170879?rss=1
The operational management of potable water distribution networks presents a great challenge to water utilities, as reflected by the complex interplay of a wide range of multidimensional and nonlinear factors across the water value chain including the network physical structure and characteristics, operational requirements, water consumption profiles and the structure of energy tariffs. Nevertheless, both continuous and discrete actuation variables can be involved in governing the water network, which makes optimizing such networks a mixed-integer and highly constrained decision-making problem. As such, there is a need to situate the problem holistically, factoring in multidimensional considerations, with a goal of minimizing water operational costs. This paper, therefore, proposes a systematic optimization methodology for (near) real-time operation of water networks, where the operational strategy can be dynamically updated using a model-based predictive control scheme with little human intervention. The hydraulic model of the network of interest is thereby integrated and successively simulated with different trial strategies as part of the optimization process. A novel adapted mixed-integer differential evolution (DE) algorithm is particularly designed to deal with the discrete-continuous actuation variables involved in the network. Simulation results on a pilot water network confirm the effectiveness of the proposed methodology and the superiority of the proposed mixed-integer DE in comparison with genetic algorithms. It also suggests that 23.69% cost savings can be achieved compared with the water utility's current operational strategy, if adaptive pricing is adopted for all the pumping stations.
]]>2018-09-05T00:46:40-07:00info:doi/10.1098/rspa.2017.0879hwp:master-id:royprsa;rspa.2017.08792018-09-05Research articles47422172017087920170879<![CDATA[The effect of root exudates on rhizosphere water dynamics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180149?rss=1
Most water and nutrients essential for plant growth travel across a thin zone of soil at the interface between roots and soil, termed the rhizosphere. Chemicals exuded by plant roots can alter the fluid properties, such as viscosity, of the water phase, potentially with impacts on plant productivity and stress tolerance. In this paper, we study the effects of plant exudates on the macroscale properties of water movement in soil. Our starting point is a microscale description of two fluid flow and exudate diffusion in a periodic geometry composed from a regular repetition of a unit cell. Using multiscale homogenization theory, we derive a coupled set of equations that describe the movement of air and water, and the diffusion of plant exudates on the macroscale. These equations are parametrized by a set of cell problems that capture the flow behaviour. The mathematical steps are validated by comparing the resulting homogenized equations to the original pore scale equations, and we show that the difference between the two models is 7% for eight cells. The resulting equations provide a computationally efficient method to study plant–soil interactions. This will increase our ability to predict how contrasting root exudation patterns may influence crop uptake of water and nutrients.
]]>2018-09-05T00:46:40-07:00info:doi/10.1098/rspa.2018.0149hwp:master-id:royprsa;rspa.2018.01492018-09-05Research articles47422172018014920180149<![CDATA[Nonlinear Bloch waves and balance between hardening and softening dispersion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180173?rss=1
The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unit-cell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size.
]]>2018-09-05T00:46:40-07:00info:doi/10.1098/rspa.2018.0173hwp:master-id:royprsa;rspa.2018.01732018-09-05Research articles47422172018017320180173<![CDATA[Asymmetry of the atomic-level stress tensor in homogeneous and inhomogeneous materials]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2217/20180155?rss=1
The stress tensor is described as a symmetric tensor in all classical continuum mechanics theories and in most existing statistical mechanics formulations. In this work, we examine the theoretical origins of the symmetry of the stress tensor and identify the assumptions and misinterpretations that lead to its symmetric property. We then make a direct measurement of the stress tensor in molecular dynamics simulations of four different material systems using the physical definition of stress as force per unit area acting on surface elements. Simulation results demonstrate that the stress tensor is asymmetric near dislocation cores, phase boundaries, holes and even in homogeneous material under a shear loading. In addition, the atomic virial stress and Hardy stress formulae are shown to significantly underestimate the stress tensor in regions of stress concentration.
]]>2018-09-05T00:46:40-07:00info:doi/10.1098/rspa.2018.0155hwp:master-id:royprsa;rspa.2018.01552018-09-05Research articles47422172018015520180155<![CDATA[Optimized multi-shot imaging inspection design]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20170319?rss=1
The inspection of complex-shaped components, such as those enabled by additive manufacturing, is a major challenge in industrial quality assurance. A frequently adopted approach to volumetric non-destructive evaluation is X-ray computed tomography, but this has major drawbacks. Two-dimensional radiography can overcome some of these problems, but does not generally provide an inspection that is as capable. Moreover, designing a detailed inspection for a complex-shaped component is a labour-intensive task, requiring significant expert input. In response, a computational framework for optimizing the data acquisition for an image-based inspection modality has been devised. The initial objective is to advance the capabilities of radiography, but the algorithm is, in principle, also applicable to alternative types of imaging. The algorithm exploits available prior information about the inspection and simulations of the inspection modality to allow the determination of the optimal inspection configuration, including specifically component poses with respect to the imaging system. As an intermediate output, spatial maps of inspection performance are computed, for understanding spatially varying limits of detection. Key areas of innovation concern the defect detectability evaluation for arbitrarily complex indications and the creation of an application-specific optimization algorithm. Initial trials of the algorithm are presented, with good results.
]]>2018-08-29T00:05:15-07:00info:doi/10.1098/rspa.2017.0319hwp:master-id:royprsa;rspa.2017.03192018-08-29Research articles47422162017031920170319<![CDATA[Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painleve VI]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20180080?rss=1
We present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral. The Schwarz–Christoffel accessory parameter problem, relevant when all sides have zero curvature, is also captured within our approach. The method exploits the isomonodromic tau function associated with the Painlevé VI equation. Recently, these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the Young diagrams. After showing how to extract the monodromy data associated with the target domain, we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem. The viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides.
]]>2018-08-29T00:05:15-07:00info:doi/10.1098/rspa.2018.0080hwp:master-id:royprsa;rspa.2018.00802018-08-29Research articles47422162018008020180080<![CDATA[Buckling of a spinning elastic cylinder: linear, weakly nonlinear and post-buckling analyses]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20180242?rss=1
An elastic cylinder spinning about a rigid axis buckles beyond a critical angular velocity, by an instability driven by the centrifugal force. This instability and the competition between the different buckling modes are investigated using analytical calculations in the linear and weakly nonlinear regimes, complemented by numerical simulations in the fully post-buckled regime. The weakly nonlinear analysis is carried out for a generic incompressible hyperelastic material. The key role played by the quadratic term in the expansion of the strain energy density is pointed out: this term has a strong effect on both the nature of the bifurcation, which can switch from supercritical to subcritical, and the buckling amplitude. Given an arbitrary hyperelastic material, an equivalent shear modulus is proposed, allowing the main features of the instability to be captured by an equivalent neo-Hookean model.
]]>2018-08-29T00:05:15-07:00info:doi/10.1098/rspa.2018.0242hwp:master-id:royprsa;rspa.2018.02422018-08-29Research articles47422162018024220180242<![CDATA[Neutron reflectometry with the Multi-Blade 10B-based detector]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20180266?rss=1
The Multi-Blade is a boron-10-based gaseous detector developed for neutron reflectometry instruments at the European Spallation Source in Sweden. The main challenges for neutron reflectometry detectors are the instantaneous counting rate and spatial resolution. The Multi-Blade has been tested on the CRISP reflectometer at the ISIS Neutron and Muon Source in the UK. A campaign of scientific measurements has been performed to study the Multi-Blade response in real instrumental conditions. The results of these tests are discussed in this paper.
]]>2018-08-22T00:05:16-07:00info:doi/10.1098/rspa.2018.0266hwp:master-id:royprsa;rspa.2018.02662018-08-22Research articles47422162018026620180266<![CDATA[Interfacial load transfer mechanisms in carbon nanotube-polymer nanocomposites]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20170705?rss=1
Carbon nanotubes (CNTs) are highly promising for strength reinforcement in polymer nanocomposites, but conflicting interfacial properties have been reported by single nanotube pull-out experiments. Here, we report the interfacial load transfer mechanisms during pull-out of CNTs from PMMA matrices, using massively- parallel molecular dynamics simulations. We show that the pull-out forces associated with non-bonded interactions between CNT and PMMA are generally small, and are weakly-dependent on the embedment length of the nanotube. These pull-out forces do not significantly increase with the presence of Stone Wales or vacancy defects along the nanotube. In contrast, low-density distribution of cross-links along the CNT-PMMA interface increases the pull-out forces by an order of magnitude. At each cross-linked site, mechanical unfolding and pull-out of single or pair polymer chain(s) attached to the individual cross-link bonds result in substantial interfacial strengthening and toughening, while contributing to interfacial slip between CNT and PMMA. Our interfacial shear-slip model shows that the interfacial loads are evenly-distributed among the finite number of cross-link bonds at low cross-link densities or for nanotubes with short embedment lengths. At higher cross-link densities or for nanotubes with longer embedment lengths, a no-slip zone now develops where shear-lag effects become important. Implications of these results, in the context of recent nanotube pull-out experiments, are discussed.
]]>2018-08-08T00:05:15-07:00info:doi/10.1098/rspa.2017.0705hwp:master-id:royprsa;rspa.2017.07052018-08-08Research articles47422162017070520170705<![CDATA[Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20180063?rss=1
We consider two ‘comprehensive’ modelling approaches for engineering fabrics. We distinguish the two approaches using the terms ‘semi-discrete’ and ‘continuum’, reflecting their natures. We demonstrate a fitting procedure, used to identify the constitutive parameters of the continuum model from predictions of the semi-discrete model, the parameters of which are in turn fitted to experimental data. We, then, check the effectiveness of the continuum model by verifying the correspondence between semi-discrete and continuum model predictions using test cases not previously used in the identification process. Predictions of both modelling approaches are compared against full-field experimental kinematic data, obtained using stereoscopic digital image correlation techniques, and also with measured force data. Being a reduced order model and being implemented in an implicit rather than an explicit finite-element code, the continuum model requires significantly less computational power than the semi-discrete model and could therefore be used to more efficiently explore the mechanical response of engineering fabrics.
]]>2018-08-08T00:05:15-07:00info:doi/10.1098/rspa.2018.0063hwp:master-id:royprsa;rspa.2018.00632018-08-08Research articles47422162018006320180063<![CDATA[On the geometrically exact low-order modelling of a flexible beam: formulation and numerical tests]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20180423?rss=1
This paper proposes a low-order geometrically exact flexible beam formulation based on the utilization of generic beam shape functions to approximate distributed kinematic properties of the deformed structure. The proposed nonlinear beam shapes approach is in contrast to the majority of geometrically nonlinear treatments in the literature in which element-based—and hence high-order—discretizations are adopted. The kinematic quantities approximated specifically pertain to shear and extensional gradients as well as local orientation parameters based on an arbitrary set of globally referenced attitude parameters. In developing the dynamic equations of motion, an Euler angle parametrization is selected as it is found to yield fast computational performance. The resulting dynamic formulation is closed using an example shape function set satisfying the single generic kinematic constraint. The formulation is demonstrated via its application to the modelling of a series of static and dynamic test cases of both simple and non-prismatic structures; the simulated results are verified using MSC Nastran and an element-based intrinsic beam formulation. Through these examples, it is shown that the nonlinear beam shapes approach is able to accurately capture the beam behaviour with a very minimal number of system states.
]]>2018-08-08T00:05:15-07:00info:doi/10.1098/rspa.2018.0423hwp:master-id:royprsa;rspa.2018.04232018-08-08Research articles47422162018042320180423<![CDATA[Radon signals in geological (natural) geogas and in a simultaneous enhanced confined mode simulation experiment]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20170787?rss=1
An enhanced confined mode (ECM) radon simulation experiment, tested in the laboratory in Jerusalem, was relocated to a subsurface geophysical observatory located 400 km apart, at a depth of 150 m and with a stable temperature. Five gamma sensors are placed around the ECM canister and lead shielding minimizes the influence of natural local gamma radiation. Simultaneous measurement of the geological radon and from radon in the ECM system indicates that the temporal variation of gamma radiation from radon in the ECM system contains annual, multi-day and daily signals, that correspond to signals in the local geological radon. This implies that a common external driver influences the radiation pattern of the geological radon and from radon inside the ECM canister. Once activated at BGO the typical variation pattern of the experimental system occurring in the laboratory changed to that occurring at the observatory. This is interpreted to indicate that the overall style of the temporal patterns of radiation from radon is site dependent. The outcome of this investigation conforms and further substantiates the recent suggestion that a component in solar radiation is driving the annual and daily periodic components in the variation of radon. New geophysical research potential is indicated.
]]>2018-08-01T00:05:26-07:00info:doi/10.1098/rspa.2017.0787hwp:master-id:royprsa;rspa.2017.07872018-08-01Research articles47422162017078720170787<![CDATA[Analytical estimation of non-local deformation-mediated magneto-electric coupling in soft composites]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20170803?rss=1
For a long time, the search for magneto-electric materials concentrated on multi-ferroics and hard-matter composites. By contrast, rather recently the exploitation of strain-mediated magneto-electric (ME) coupling in soft composites was proposed. The basic idea behind this approach is to combine the magneto- and electro-mechanical responses of composites consisting of a soft matrix carrying magnetic inclusions. Despite that such composites are straightforward to manufacture and have cheap constituents, they did not gain much attention up to now. In this contribution, we demonstrate that ME coupling induced by finite deformations could be of significant magnitude. Our approach relies on shape effects as a special non-local phenomenon in magneto- and electro-elasticity. Based on that we characterize an up to now overlooked ME coupling mechanism which purely relies on these shape effects in soft-matter-based magnetic and electric media. While soft magnetic media are commonly realized as composites, the coupling effect to be highlighted exists independently of the origin of a body's magnetic and electric properties. We show that the magnitude of the effect is indeed significant and, among ellipsoidal bodies, most pronounced for those of spherical to moderately prolate shape. Finite-element simulations are performed to assess the quality of the analytical predictions.
]]>2018-08-01T00:05:26-07:00info:doi/10.1098/rspa.2017.0803hwp:master-id:royprsa;rspa.2017.08032018-08-01Research articles47422162017080320170803<![CDATA[Small-amplitude static periodic patterns at a fluid-ferrofluid interface]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20180038?rss=1
We establish the existence of static doubly periodic patterns (in particular rolls, squares and hexagons) on the free surface of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetization law. A novel formulation of the ferrohydrostatic equations in terms of Dirichlet–Neumann operators for nonlinear elliptic boundary-value problems is presented. We demonstrate the analyticity of these operators in suitable function spaces and solve the ferrohydrostatic problem using an analytic version of Crandall–Rabinowitz local bifurcation theory. Criteria are derived for the bifurcations to be sub-, super- or transcritical with respect to a dimensionless physical parameter.
]]>2018-08-01T00:05:26-07:00info:doi/10.1098/rspa.2018.0038hwp:master-id:royprsa;rspa.2018.00382018-08-01Research articles47422162018003820180038<![CDATA[Dispersive shock waves governed by the Whitham equation and their stability]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20180278?rss=1
Dispersive shock waves (DSWs), also termed undular bores in fluid mechanics, governed by the non-local Whitham equation are studied in order to investigate short wavelength effects that lead to peaked and cusped waves within the DSW. This is done by combining the weak nonlinearity of the Korteweg–de Vries equation with full linear dispersion relations. The dispersion relations considered are those for surface gravity waves, the intermediate long wave equation and a model dispersion relation introduced by Whitham to investigate the 120° peaked Stokes wave of highest amplitude. A dispersive shock fitting method is used to find the leading (solitary wave) and trailing (linear wave) edges of the DSW. This method is found to produce results in excellent agreement with numerical solutions up until the lead solitary wave of the DSW reaches its highest amplitude. Numerical solutions show that the DSWs for the water wave and Whitham peaking kernels become modulationally unstable and evolve into multi-phase wavetrains after a critical amplitude which is just below the DSW of maximum amplitude.
]]>2018-08-01T00:05:26-07:00info:doi/10.1098/rspa.2018.0278hwp:master-id:royprsa;rspa.2018.02782018-08-01Research articles47422162018027820180278<![CDATA[From arteries to boreholes: transient response of a poroelastic cylinder to fluid injection]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2216/20180284?rss=1
The radially outward flow of fluid through a porous medium occurs in many practical problems, from transport across vascular walls to the pressurization of boreholes in the subsurface. When the driving pressure is non-negligible relative to the stiffness of the solid structure, the poromechanical coupling between the fluid and the solid can control both the steady state and the transient mechanics of the system. Very large pressures or very soft materials lead to large deformations of the solid skeleton, which introduce kinematic and constitutive nonlinearity that can have a non-trivial impact on these mechanics. Here, we study the transient response of a poroelastic cylinder to sudden fluid injection. We consider the impacts of kinematic and constitutive nonlinearity, both separately and in combination, and we highlight the central role of driving method in the evolution of the response. We show that the various facets of nonlinearity may either accelerate or decelerate the transient response relative to linear poroelasticity, depending on the boundary conditions and the initial geometry, and that an imposed fluid pressure leads to a much faster response than an imposed fluid flux.
]]>2018-08-01T00:05:26-07:00info:doi/10.1098/rspa.2018.0284hwp:master-id:royprsa;rspa.2018.02842018-08-01Research articles47422162018028420180284<![CDATA[Elastic waves in periodically heterogeneous two-dimensional media: locally periodic and anti-periodic modes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170908?rss=1
Propagation of anti-plane waves through a discrete square lattice and through a continuous fibrous medium is studied. In the long-wave limit, for periodically heterogeneous structures the solution can be periodic or anti-periodic across the unit cell. It is shown that combining periodicity and anti-periodicity conditions in different directions of the translational symmetry allows one to detect different types of modes that do not arise in the purely periodic case. Such modes may be interpreted as counterparts of non-classical waves appearing in phenomenological theories. Dispersion diagrams of the discrete square lattice are evaluated in a closed analytical from. Dispersion properties of the fibrous medium are determined using Floquet–Bloch theory and Fourier series approximations. Influence of a viscous damping is taken into account.
]]>2018-07-25T00:05:18-07:00info:doi/10.1098/rspa.2017.0908hwp:master-id:royprsa;rspa.2017.09082018-07-25Research articles47422152017090820170908<![CDATA[Interfacial waveforms in chiral lattices with gyroscopic spinners]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20180132?rss=1
We demonstrate a new method of achieving topologically protected states in an elastic hexagonal system of trusses by attaching gyroscopic spinners, which bring chirality to the system. Dispersive features of this medium are investigated in detail, and it is shown that one can manipulate the locations of stop-bands and Dirac points by tuning the parameters of the spinners. We show that, in the proximity of such points, uni-directional interfacial waveforms can be created in an inhomogeneous lattice and the direction of such waveforms can be controlled. The effect of inserting additional soft internal links into the system, which is thus transformed into a heterogeneous triangular lattice, is also investigated, as the hexagonal lattice represents the limit case of the heterogeneous triangular lattice with soft links. This work introduces a new perspective in the design of periodic media possessing non-trivial topological features.
]]>2018-07-25T00:05:18-07:00info:doi/10.1098/rspa.2018.0132hwp:master-id:royprsa;rspa.2018.01322018-07-25Research articles47422152018013220180132<![CDATA[Quantifying plasma immersion ion implantation of insulating surfaces in a dielectric barrier discharge: how to control the dose]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20180263?rss=1
The plasma physics of dielectric barrier discharges (DBD) for carrying out ion implantation in insulators is investigated. A hollow cathode DBD excited by high-voltage pulses is suitable for ion bombardment of the surfaces of insulating tubing, porous material, particles and sheets. Plasma immersion ion implantation of insulating surfaces is useful for many applications in medicine and engineering. The ion bombardment of glass is useful for cleaning and surface modification. The ion implantation of polymers creates radicals that are able to bind molecules to their surfaces for applications in medical procedures and diagnostics. A wire diagnostic probe and optical emission spectroscopy are used for experimental work. A theory based on mutual capacitance is developed to convert data from the probe to give implanted charge as a function of pressure, voltage and pulse duration. We find the operating conditions that allow for charge to be implanted and those that achieve the highest implanted charge.
]]>2018-07-25T00:05:18-07:00info:doi/10.1098/rspa.2018.0263hwp:master-id:royprsa;rspa.2018.02632018-07-25Research articles47422152018026320180263<![CDATA[The underwater resonant airbag: a new wave energy converter]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170192?rss=1
The time-domain simulation follows the heaving of the conical float in waves and calculates the bag shape, ballast motion, adiabatic air pressure and the flow through the turbine. There are two independent oscillators, the float with its resonance and the bag/ballast with its resonance. The coupling of the two oscillators gives rise to a wide band response with two peaks in the capture width each reaching the theoretical /2. In this new wave energy converter, apart from the turbine, there are no mechanical moving parts, no joints nor pistons, no end stops nor sliding seals, no flaps nor one-way valves. The expected life of the airtight flexible bag remains to be determined, but potential manufacturers are optimistic.
]]>2018-07-18T00:09:44-07:00info:doi/10.1098/rspa.2017.0192hwp:master-id:royprsa;rspa.2017.01922018-07-18Research articles47422152017019220170192<![CDATA[Shock waves in media with negative electromagnetic susceptibilities]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170522?rss=1
The Fock–Kemmer theory of precursor shock wave speeds in classical field theory is extended to the case of media with negative electromagnetic susceptibilities.
]]>2018-07-18T00:09:44-07:00info:doi/10.1098/rspa.2017.0522hwp:master-id:royprsa;rspa.2017.05222018-07-18Research articles47422152017052220170522<![CDATA[Effects of anisotropy and regime of diffusion on the measurement of lattice diffusion coefficient of hydrogen in metals]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170677?rss=1
The lattice diffusion coefficient of hydrogen in metals is commonly measured by permeation tests. Such tests assume isotropic diffusivity with a set-up which measures permeation flux only along one direction. The measured values of the lattice diffusion coefficient are strongly influenced by the trapping of hydrogen at microstructural defects. These factors lead to highly inaccurate determination of the lattice diffusion coefficient, more so in an anisotropic medium. In this work, we present a three-dimensional (3D) diffusion equation in non-dimensional form for an anisotropic medium with source and sink terms which account for detrapping and trapping of hydrogen. The concentration of hydrogen at lattice and trap sites is assumed to be in a local equilibrium. An initial boundary value problem of the permeation test is formulated and the governing partial differential equation is implemented in a 3D finite-element code. The influence of anisotropic diffusivity on the measurement of lattice diffusion coefficient is shown by numerical simulations. Asymptotic analysis of the governing equation reveals that the lattice diffusion coefficient can only be measured in certain regimes when performing permeation tests at varying temperatures. The nonlinear behaviour of Arrhenius plots of diffusion coefficient versus inverse of temperature due to trapping is analytically and numerically predicted.
]]>2018-07-18T00:09:44-07:00info:doi/10.1098/rspa.2017.0677hwp:master-id:royprsa;rspa.2017.06772018-07-18Research articles47422152017067720170677<![CDATA[Shapes of river networks]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20180081?rss=1
River network scaling laws describe how their shape varies with their size. However, the regional variation of this size-dependence remains poorly understood. Here we show that river network scaling laws vary systematically with the climatic aridity index. We find that arid basins do not change their proportions with size, while humid basins do. To explore why, we study an aspect ratio L_{}/L_{||} between basin width L_{} and basin length L_{||}. We find that the aspect ratio exhibits a dependence on climate and argue that this can be understood as a structural consequence of the confluence angle. We then find that, in humid basins, the aspect ratio decreases with basin size, which we attribute to a common hydrogeological hierarchy. Our results offer an explanation of the variability in network scaling exponents and suggest that the absence of self-similarity in humid basins can be understood as a morphological expression of subsurface processes.
]]>2018-07-18T00:09:44-07:00info:doi/10.1098/rspa.2018.0081hwp:master-id:royprsa;rspa.2018.00812018-07-18Research articles47422152018008120180081<![CDATA[Negative and positive stiffness in auxetic magneto-mechanical metamaterials]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20180003?rss=1
This work discusses the concept of allowing the control of the stiffness of a particular class of re-entrant auxetic magneto-mechanical metamaterials through the introduction of magnets to the system. It is shown, through experimental testing backed up by a theoretical model, that the appropriate insertion of magnets in such a system will alter its stiffness, possibly even making it exhibit ‘negative stiffness’. This leads to a completely different behaviour of the structure in terms of stability. It is also reported that the investigated mechanical metamaterials may exhibit both negative stiffness and negative Poisson's ratio at the same time. Moreover, it is shown that the effect which magnets have on the stiffness of the system may be fine-tuned upon replacing magnets with electromagnets. Such systems have the potential to be used in a wide range of practical applications such as vibration damping devices where achieving a negative stiffness is of fundamental importance.
]]>2018-07-18T00:09:44-07:00info:doi/10.1098/rspa.2018.0003hwp:master-id:royprsa;rspa.2018.00032018-07-18Research articles47422152018000320180003<![CDATA[Bayesian inference of spreading processes on networks]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20180129?rss=1
Infectious diseases are studied to understand their spreading mechanisms, to evaluate control strategies and to predict the risk and course of future outbreaks. Because people only interact with few other individuals, and the structure of these interactions influence spreading processes, the pairwise relationships between individuals can be usefully represented by a network. Although the underlying transmission processes are different, the network approach can be used to study the spread of pathogens in a contact network or the spread of rumours in a social network. We study simulated simple and complex epidemics on synthetic networks and on two empirical networks, a social/contact network in an Indian village and an online social network. Our goal is to learn simultaneously the spreading process parameters and the first infected node, given a fixed network structure and the observed state of nodes at several time points. Our inference scheme is based on approximate Bayesian computation, a likelihood-free inference technique. Our method is agnostic about the network topology and the spreading process. It generally performs well and, somewhat counter-intuitively, the inference problem appears to be easier on more heterogeneous network topologies, which enhances its future applicability to real-world settings where few networks have homogeneous topologies.
]]>2018-07-18T00:09:44-07:00info:doi/10.1098/rspa.2018.0129hwp:master-id:royprsa;rspa.2018.01292018-07-18Research articles47422152018012920180129<![CDATA[The n-component nonlinear Schrödinger equations: dark-bright mixed N- and high-order solitons and breathers, and dynamics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170688?rss=1
The general n-component nonlinear Schrödinger equations are systematically investigated with the aid of the Darboux transformation method and its extension. Firstly, we explore the condition of the existence for dark–bright mixed soliton solutions and derive an explicit formula of dark–bright mixed multi-soliton solutions in terms of the determinant. Secondly, we present the formula of dark–bright mixed high-order semi-rational solitons, and predict their general Nth-order wave structures. Thirdly, we investigate the wing-shaped structures of breather. Finally, we perform the numerical simulations for some representative solitons to study their dynamical behaviours.
]]>2018-07-11T00:09:41-07:00info:doi/10.1098/rspa.2017.0688hwp:master-id:royprsa;rspa.2017.06882018-07-11Research articles47422152017068820170688<![CDATA[Effect of surface bending and stress on the transmission of line force to an elastic substrate]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170775?rss=1
For a broad class of soft materials their surface stress can strongly influence mechanical behaviour. For example, a line force applied to the surface of an elastic substrate is locally supported by surface stress over an elasto-capillary length lc (surface stress/elastic modulus). Surface stress regularizes the otherwise highly singular stress and strain fields. However, surface such as lipid bilayer interfaces can also resist deformation by bending. This has not been studied either by experiments or theories. We analyse a theoretical model of the response of a half-space to a line force when the surface carries both a stress and resistance to bending. We find that surface bending further regularizes the singular fields. The local stress field near the line load can be separated into three regions. Region 1 occupies distances from the line load smaller than an elasto-capillary bending length lb (bending stiffness/elastic modulus to the 1/3 power) where surface bending dominates and the elastic stress and strains are continuous. Region 2 occupies intermediate distances between lb and lc ( > lb) where surface stress dominates. At distances larger than lc we retrieve the classical elasticity solution. The size of region 2 depends on =lc/lb and vanishes for small lc.
]]>2018-07-11T00:09:41-07:00info:doi/10.1098/rspa.2017.0775hwp:master-id:royprsa;rspa.2017.07752018-07-11Research articles47422152017077520170775<![CDATA[Determination of macro-scale soil properties from pore scale structures: image-based modelling of poroelastic structures]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170745?rss=1
We show how a combination of X-ray computed tomography (X-CT) and image-based modelling can be used to calculate the effect of moisture content and compaction on the macroscopic structural properties of soil. Our method is based on the equations derived in Daly & Roose (2018 Proc. R. Soc. A474, 20170141. (doi:10.1098/rspa.2017.0141)), which we have extended so they can be directly applied to the segmented images obtained from X-CT. We assume that the soils are composed of air-filled pore space, solid mineral grains and a mixed phase composed of both clay particles and water. We considered three different initial soil treatments, composed of two different compaction levels and two different moisture contents. We found that the effective properties of the soils were unaffected by compaction over the range tested in this paper. However, changing the moisture content significantly altered the hydraulic and mechanical properties of the soils. A key strength of this method is that it enables the optimization or even design of soils composed from different constituents, with specific mechanical and hydraulic properties.
]]>2018-07-11T00:09:41-07:00info:doi/10.1098/rspa.2017.0745hwp:master-id:royprsa;rspa.2017.07452018-07-11Research articles47422152017074520170745<![CDATA[Three-dimensional visualization and quantification of the fracture mechanisms in sparse fibre networks using multiscale X-ray microtomography]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20180175?rss=1
The structural changes that are induced by the initiation and the propagation of a crack in a low-density paper (LDP) were studied using single edge-notched fracture tests that were imaged under an optical microscope or in laboratory or synchrotron X-ray microtomographs. The two-dimensional optical images were used to analyse the links between the mesoscale structural variations of LDP and the crack path. Medium-resolution X-ray three-dimensional images were used to analyse the variations in the thickness and local porosity of samples as well as their displacement field that were induced by the LDP fracture. High-resolution three-dimensional images showed that these mesostructural variations were accompanied by complex fibre and bond deformation mechanisms that were, for the first time, in situ imaged. These mechanisms occurred in the fracture process zone that developed ahead of the crack tip before the crack path became distinct and visible. They were at the origin of the aforementioned thickness variations that developed more particularly along the crack path. They eventually led to fibre–fibre bond detachment phenomena and crack propagation through the fibrous network. These results can be used to enhance the current structural and mechanical models for the prediction of the fracture behaviour of papers.
]]>2018-07-11T00:09:41-07:00info:doi/10.1098/rspa.2018.0175hwp:master-id:royprsa;rspa.2018.01752018-07-11Research articles47422152018017520180175<![CDATA[Large numbers of explanatory variables: a probabilistic assessment]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170631?rss=1
Recently, Cox and Battey (2017 Proc. Natl Acad. Sci. USA114, 8592–8595 (doi:10.1073/pnas.1703764114)) outlined a procedure for regression analysis when there are a small number of study individuals and a large number of potential explanatory variables, but relatively few of the latter have a real effect. The present paper reports more formal statistical properties. The results are intended primarily to guide the choice of key tuning parameters.
]]>2018-07-04T00:09:48-07:00info:doi/10.1098/rspa.2017.0631hwp:master-id:royprsa;rspa.2017.06312018-07-04Research articles47422152017063120170631<![CDATA[We-T classification of diesel fuel droplet impact regimes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170759?rss=1
A combined experimental and computational investigation of micrometric diesel droplets impacting on a heated aluminium substrate is presented. Dual view high-speed imaging has been employed to visualize the evolution of the impact process at various conditions. The parameters investigated include wall-surface temperature ranging from 140 to 400°C, impact Weber and Reynolds numbers of 19–490 and 141–827, respectively, and ambient pressure of 1 and 2 bar. Six possible post-impact regimes were identified, termed as Stick, Splash, Partial-Rebound, Rebound, Breakup-Rebound and Breakup-Stick, and plotted on the We-T map. Additionally, the temporal variation of the apparent dynamic contact angle and spreading factor have been determined as a function of the impact Weber number and surface temperature. Numerical simulations have also been performed using a two-phase flow model with interface capturing, phase-change and variable physical properties. Increased surface temperature resulted to increased maximum spreading diameter and induced quicker and stronger recoiling behaviour, mostly attributed to the change of liquid viscosity.
]]>2018-07-04T00:09:48-07:00info:doi/10.1098/rspa.2017.0759hwp:master-id:royprsa;rspa.2017.07592018-07-04Research articles47422152017075920170759<![CDATA[A generalized model for dynamic deformation and failure of clamped beam under impact loading]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170799?rss=1
A mathematical model representing the dynamic behaviour (both plastic deformation and fracture) of a clamped beam under impact loading is developed. Indentation at the impact point, transverse deformation, formation and propagation of plastic hinge, arrest of plastic hinge leading to plastic work concentration and finally failure are the physical processes which constitute the basis of the derived model. The effect of imperfection is also incorporated. Imperfection is considered in the form of a -shaped notch located at the impact point, at supports or at both. The distinct feature of the present formulation is that it accommodates different possible deformation and failure modes in a single model. Final plastic deformation, time histories of different field variables and failure modes as predicted from the derived model are found to be in good agreement with the corresponding experimental and numerical results. This model provides a quick understanding of the dynamic behaviour of beam under impact and also the effect of various underlying parameters which may be useful for forming design provisions for impact-resistant structures.
]]>2018-07-04T00:18:17-07:00info:doi/10.1098/rspa.2017.0799hwp:master-id:royprsa;rspa.2017.07992018-07-04Research articles47422152017079920170799<![CDATA[An elliptical liquid inclusion in an infinite elastic plane]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20170813?rss=1
Beyond recent related literature, which focused on spherical incompressible liquid inclusions, the present work studies an elliptical compressible liquid inclusion in an infinite elastic plane under static remote mechanical loading. Here, it is assumed that the change of pressure inside the liquid inclusion is linearly related to the change of inclusion volume with the bulk modulus of the liquid as the proportionality coefficient. Also, the role of the liquid surface tension on the solid–liquid interface is examined especially when the size of the liquid inclusion is comparable to or smaller than the elastocapillary length. Our results show that both the surface tension and the change of liquid pressure have a significant effect on reducing the stress concentration factor at the endpoints of an elliptical liquid inclusion. In addition, the pressure change inside the liquid inclusion is studied when a uniaxial remote stress is applied perpendicular or parallel to the major axis of the elliptical liquid inclusion. In particular, the effective plane-strain Young's modulus of a solid–liquid composite containing circular liquid inclusions predicted by the present model is linearly related to the volume fraction of the liquid inclusions, in reasonable agreement with existing experimental data.
]]>2018-07-04T00:09:48-07:00info:doi/10.1098/rspa.2017.0813hwp:master-id:royprsa;rspa.2017.08132018-07-04Research articles47422152017081320170813<![CDATA[Extraordinary interactions between light and matter determined by anomalous weak values]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2215/20180030?rss=1
Some predictions regarding pre- and post-selected states are far-reaching, thereby requiring validation with standard quantum measurements in addition to the customary weak measurements used so far, as well as other advanced techniques. We go further pursuing this goal, proposing two thought experiments which incorporate novel yet feasible validation methods of unconventional light-matter interactions. An excited atom traverses a Mach–Zehnder interferometer (MZI) under a special combination of pre- and post-selection. In the first experiment, photons emitted by the superposed atom, after being hit by two laser beams, are individually counted. Despite the interaction having definitely taken place, as revealed by the atom becoming ground, the numbers of photons emitted from each arm of the MZI are predicted, at the ensemble level, to be different from those expected with standard stimulated emission. In the second experiment, the atom spontaneously emits a photon while still in the MZI. This photon later serves as a strong measurement of the atom's energy upon hitting a photographic plate. The experiment is repeated to enable an interference effect of the emitted photons. Interestingly, the latter gives the appearance that the photons have been emitted by the atom from a position much farther from the two MZI arms L and R, as if in a ‘phantom arm’ R'. Nevertheless, their time of arrival is similar to that of photons coming from L and R. These experiments also emphasize the key role of anomalous weak values in determining light–matter interactions. In fact, they present a straightforward realization of an entity we term counter-particles, namely pre- and post-selected states acting as if they have negative physical variables such as mass and energy. The novel verification methods we suggest for testing these predictions resemble weak measurements in some aspects, yet result from definite atomic transitions verified by the detected photons.
]]>2018-07-04T00:09:48-07:00info:doi/10.1098/rspa.2018.0030hwp:master-id:royprsa;rspa.2018.00302018-07-04Research articles47422152018003020180030