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[Water-limited vegetated ecosystems driven by stochastic rainfall: feedbacks and bimodality]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20170649?rss=1
In arid or semi-arid ecosystems, water availability is one of the primary controls on vegetation growth. When subsurface water resources are unavailable, the vegetation growth is dictated by the rainfall, and the random nature of the rainfall arrivals and quantities induces a probability distribution of soil moisture and vegetation biomass via the coupled dynamic equations of biomass balance and water balance. We have previously obtained an exact solution for these distributions under certain conditions, and shown that the mapping of rainfall variability to observed biomass variability can be successfully applied to a field site. Here, we expand upon our earlier theoretical work to show how the dynamics can give rise to more complicated, bimodal (and multimodal) structures in the biomass distribution when positive feedbacks between growth and water availability are included. We also derive some new analytical results for the crossing properties of this system, which enable us to determine on what time scale the effects of these feedbacks will be felt, and, relatedly, how long the system will take to cross between different modes.
]]>2018-06-20T00:09:52-07:00info:doi/10.1098/rspa.2017.0649hwp:master-id:royprsa;rspa.2017.06492018-06-20Research articles47422142017064920170649<![CDATA[Nematic liquid crystals on curved surfaces: a thin film limit]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20170686?rss=1
We consider a thin film limit of a Landau–de Gennes Q-tensor model. In the limiting process, we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. The main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau–de Gennes model, we consider an L^{2}-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties.
]]>2018-06-20T00:09:52-07:00info:doi/10.1098/rspa.2017.0686hwp:master-id:royprsa;rspa.2017.06862018-06-20Research articles47422142017068620170686<![CDATA[Composite wave models for elastic plates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20180103?rss=1
The long-term challenge of formulating an asymptotically motivated wave theory for elastic plates is addressed. Composite two-dimensional models merging the leading or higher-order parabolic equations for plate bending and the hyperbolic equation for the Rayleigh surface wave are constructed. Analysis of numerical examples shows that the proposed approach is robust not only at low- and high-frequency limits but also over the intermediate frequency range.
]]>2018-06-20T00:09:53-07:00info:doi/10.1098/rspa.2018.0103hwp:master-id:royprsa;rspa.2018.01032018-06-20Research articles47422142018010320180103<![CDATA[Exact eigenstates of a nanometric paraboloidal emitter and field emission quantities]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20170692?rss=1
The progress in field emission theory from its initial Fowler–Nordheim form is centred on the transmission coefficient. For the supply (of electrons) function one still uses the constant value due to a supply of plane-waves states. However, for emitting tips of apex radius of 1–5 nm this is highly questionable. To address this issue, we have solved the Schrödinger equation in a sharp paraboloidally shaped quantum box. The Schrödinger equation is separable in the rotationally parabolic coordinate system and we hence obtain the exact eigenstates of the system. Significant differences from the usual Cartesian geometry are obtained. (1) Both the normally incident and parallel electron fluxes are functions of the angle to the emitter axis and affect the emission angle. (2) The WKB approximation fails for this system. (3) The eigenfunctions of the nanoemitter form a continuum only in one dimension while complete discretization occurs in the other two directions. (4) The parallel electron velocity vanishes at the apex which may explain the recent spot-size measurements in near-field scanning electron microscopy. (5) Competing effects are found as the tip radius decreases to 1 nm: The electric field increases but the total supply function decreases so that possibly an optimum radius exists.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2017.0692hwp:master-id:royprsa;rspa.2017.06922018-06-13Research articles47422142017069220170692<![CDATA[Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20170770?rss=1
We present the first steps of a procedure which discretizes surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Bäcklund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalization of the integrable discrete Tzitzéica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretization of affine spheres in affine differential geometry.
]]>2018-06-13T00:17:57-07:00info:doi/10.1098/rspa.2017.0770hwp:master-id:royprsa;rspa.2017.07702018-06-13Research articles47422142017077020170770<![CDATA[Force appropriation of nonlinear structures]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20170880?rss=1
Nonlinear normal modes (NNMs) are widely used as a tool for developing mathematical models of nonlinear structures and understanding their dynamics. NNMs can be identified experimentally through a phase quadrature condition between the system response and the applied excitation. This paper demonstrates that this commonly used quadrature condition can give results that are significantly different from the true NNM, in particular, when the excitation applied to the system is limited to one input force, as is frequently used in practice. The system studied is a clamped–clamped cross-beam with two closely spaced modes. This paper shows that the regions where the quadrature condition is (in)accurate can be qualitatively captured by analysing transfer of energy between the modes of the system, leading to a discussion of the appropriate number of input forces and their locations across the structure.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2017.0880hwp:master-id:royprsa;rspa.2017.08802018-06-13Research articles47422142017088020170880<![CDATA[An asymptotic higher-order theory for rectangular beams]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20180001?rss=1
A direct asymptotic integration of the full three-dimensional problem of elasticity is employed to derive a consistent governing equation for a beam with the rectangular cross section. The governing equation is consistent in the sense that it has the same long-wave low-frequency behaviour as the exact solution of the original three-dimensional problem. Performance of the new beam equation is illustrated by comparing its predictions against the results of direct finite-element computations. Limiting behaviours for beams with large (and small) aspect ratios, which can be established using classical plate theories, are recovered from the new governing equation to illustrate its consistency and also to illustrate the importance of using plate theories with the correctly refined boundary conditions. The implications for the correct choice of the shear correction factor in Timoshenko's beam theory are also discussed.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2018.0001hwp:master-id:royprsa;rspa.2018.00012018-06-13Research articles47422142018000120180001<![CDATA[Tension-dependent transverse buckles and wrinkles in twisted elastic sheets]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20180062?rss=1
We investigate with experiments the twist-induced transverse buckling instabilities of an elastic sheet of length L, width W and thickness t, that is clamped at two opposite ends while held under a tension T. Above a critical tension T_{} and critical twist angle _{tr}, we find that the sheet buckles with a mode number n≥1 transverse to the axis of twist. Three distinct buckling regimes characterized as clamp-dominated, bendable and stiff are identified, by introducing a bendability length L_{B} and a clamp length L_{C}(<L_{B}). In the stiff regime (L>L_{B}), we find that mode n=1 develops above _{tr}_{S}~(t/W)T^{–1/2}, independent of L. In the bendable regime L_{C}<L<L_{B}, n=1 as well as n>1 occur above trB~t/LT–1/4. Here, we find the wavelength B~LtT–1/4, when n>1. These scalings agree with those derived from a covariant form of the Föppl-von Kármán equations, however, we find that the n=1 mode also occurs over a surprisingly large range of L in the bendable regime. Finally, in the clamp-dominated regime (L<L_{C}), we find that _{tr} is higher compared to _{B} due to additional stiffening induced by the clamped boundary conditions.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2018.0062hwp:master-id:royprsa;rspa.2018.00622018-06-13Research articles47422142018006220180062<![CDATA[The nature of Earth's correlation wavefield: late coda of large earthquakes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20180082?rss=1
The seismic correlation wavefield constructed from the stacked cross-correlograms of the late coda of earthquake signals at stations across the globe provides a wealth of observed pulses as a function of inter-station distance. The interval from 3 to 10 h after the onset of major earthquakes is employed for the period range from 15 to 50 s. The observations can be well matched by synthetic seismograms for a radially stratified Earth. Many of the correlation phases have similar time behaviour to those in the regular wavefield, but others have no correspondence. All such correlation phases can be explained by the interaction of arrivals with a common slowness at the each of the stations being correlated. Using a generalized ray description of the seismic wavefield, the time-distance behaviour of these correlation phases arises from differences in accumulated phase on different propagation paths through the Earth. Distinct arrivals emerge from the correlation field when there are many ways in which combinations of seismic phases can arise with the same difference in propagation legs. The constituents of the late coda are dominated by steeply travelling waves, and in consequence features associated with multiple passages through the whole Earth emerge distinctly, such as high-order multiples of PKIKP.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2018.0082hwp:master-id:royprsa;rspa.2018.00822018-06-13Research articles47422142018008220180082<![CDATA[An algorithm to explore entanglement in small systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20180023?rss=1
A quantum state’s entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels.
]]>2018-06-13T00:17:57-07:00info:doi/10.1098/rspa.2018.0023hwp:master-id:royprsa;rspa.2018.00232018-06-13Research articles47422142018002320180023<![CDATA[Periodic behaviour in ground-level environmental radioactivity: fingerprints of solar activity?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20180109?rss=1
Long-term data on ground-level environmental radioactivity are analysed, these referring to daily records of the total β activity in air and absorbed dose rate between 1976 and 2016 in Zagreb, Croatia. A Fourier analysis of these two datasets reveals a periodic behaviour with numerous periods ranging from approximately four months to roughly 21 years. Ninety per cent of the periods agree remarkably well with known periodicities in solar activity throughout solar cycles 21–24, which suggests that the observed modulations were most probably caused by variations in the cosmogenic radionuclide production in the upper atmosphere. Hence, one can extract the fine structure of solar activity from environmental radioactivity monitoring data, which implies that it could be possible to supplement the existing information on the exposure of Earth to cosmic rays by analysing results from radioactivity monitoring stations all over the world. These datasets cover Earth densely due to the existence of a global radioecological network and are complementary to those resulting from dedicated cosmic ray detection experiments. Because of that, such analyses might possibly lead to new findings.
]]>2018-05-30T00:09:59-07:00info:doi/10.1098/rspa.2018.0109hwp:master-id:royprsa;rspa.2018.01092018-05-30Research articles47422132018010920180109<![CDATA[Avoiding/inducing dynamic buckling in a thermomechanically coupled plate: a local and global analysis of slow/fast response]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20180206?rss=1
The nonlinear response of a reduced model of an orthotropic single-layered plate with thermomechanical coupling is investigated in the presence of thermal excitations, in addition to mechanical ones. Different issues are addressed via accurate and extended local and global analyses. (i) Assessing the possible occurrence, disappearance or modification of mechanical buckling as a result of thermal aspects; (ii) exploiting global dynamics to unveil the effects of coupling; (iii) highlighting the crucial role played by the slow thermal transient evolution in modifying the fast steady mechanical response; (iv) framing the influence of coupling and underlining the need to use a thermomechanical model to grasp the actual plate dynamics; and (v) getting hints of technical interest as to the outcome robustness with respect to variations in the external/internal thermal parameters.
]]>2018-05-30T00:09:59-07:00info:doi/10.1098/rspa.2018.0206hwp:master-id:royprsa;rspa.2018.02062018-05-30Research articles47422132018020620180206<![CDATA[Revealing new dynamical patterns in a reaction-diffusion model with cyclic competition via a novel computational framework]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170608?rss=1
Understanding how patterns and travelling waves form in chemical and biological reaction–diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction–diffusion models, such as Lotka–Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction–diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.
]]>2018-05-23T00:10:03-07:00info:doi/10.1098/rspa.2017.0608hwp:master-id:royprsa;rspa.2017.06082018-05-23Research articles47422132017060820170608<![CDATA[Quantum stopwatch: how to store time in a quantum memory]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170773?rss=1
Quantum mechanics imposes a fundamental trade-off between the accuracy of time measurements and the size of the systems used as clocks. When the measurements of different time intervals are combined, the errors due to the finite clock size accumulate, resulting in an overall inaccuracy that grows with the complexity of the set-up. Here, we introduce a method that, in principle, eludes the accumulation of errors by coherently transferring information from a quantum clock to a quantum memory of the smallest possible size. Our method could be used to measure the total duration of a sequence of events with enhanced accuracy, and to reduce the amount of quantum communication needed to stabilize clocks in a quantum network.
]]>2018-05-23T00:10:03-07:00info:doi/10.1098/rspa.2017.0773hwp:master-id:royprsa;rspa.2017.07732018-05-23Research articles47422132017077320170773<![CDATA[Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170844?rss=1
We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto–Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM–LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.
]]>2018-05-23T00:10:03-07:00info:doi/10.1098/rspa.2017.0844hwp:master-id:royprsa;rspa.2017.08442018-05-23Research articles47422132017084420170844<![CDATA[Determining erosion rates in Allchar (Macedonia) to revive the lorandite neutrino experiment]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170470?rss=1
^{205}Tl in the lorandite (TiAsS_{2}) mine of Allchar (Majdan, FYR Macedonia) is transformed to ^{205}Pb by cosmic ray reactions with muons and neutrinos. At depths of more than 300 m, muogenic production would be sufficiently low for the 4.3 Ma old lorandite deposit to be used as a natural neutrino detector. Unfortunately, the Allchar deposit currently sits at a depth of only 120 m below the surface, apparently making the lorandite experiment technically infeasible. We here present 25 erosion rate estimates for the Allchar area using in situ produced cosmogenic ^{36}Cl in carbonates and ^{10}Be in alluvial quartz. The new measurements suggest long-term erosion rates of 100–120 m Ma^{–1} in the silicate lithologies that are found at the higher elevations of the Majdanksa River valley, and 200–280 m Ma^{–1} in the underlying marbles and dolomites. These values indicate that the lorandite deposit has spent most of its existence at depths of more than 400 m, sufficient for the neutrinogenic ^{205}Pb component to dominate the muon contribution. Our results suggest that this unique particle physics experiment is theoretically feasible and merits further development.
]]>2018-05-16T00:26:05-07:00info:doi/10.1098/rspa.2017.0470hwp:master-id:royprsa;rspa.2017.04702018-05-16Research articles47422132017047020170470<![CDATA[Multi-vortex crystal lattices in Bose-Einstein condensates with a rotating trap]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170553?rss=1
We consider vortex dynamics in the context of Bose–Einstein condensates (BECs) with a rotating trap, with or without anisotropy. Starting with the Gross–Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.
]]>2018-05-16T00:10:11-07:00info:doi/10.1098/rspa.2017.0553hwp:master-id:royprsa;rspa.2017.05532018-05-16Research articles47422132017055320170553<![CDATA[A mathematical model for fitting and predicting relaxation modulus and simulating viscoelastic responses]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170540?rss=1
We propose a mathematical model for relaxation modulus and its numerical solution. The model formula is extended from sigmoidal function considering nonlinear strain hardening. Its physical meaning can be interpreted by a macroscale elastic network-viscous medium model with only five model parameters in a simpler format than the molecular-chain-based polymer models to represent general solid materials. We also developed a finite-element (FE) framework and robust numerical algorithm to implement this model for simulating responses under both static and dynamic loadings. We validated the model through both experimental data and numerical simulations on a variety of materials including asphalt concrete, polymer, spider silk, hydrogel, agar and bone. By satisfying the second law of thermodynamics in the form of Calusius–Duhem inequality, the model is able to simulate creep and sinusoidal deformation as well as energy dissipation. Compared to the Prony series, the widely used model with a large number of model parameters, the proposed model has improved accuracy in fitting experimental data and prediction stability outside of the experimental range with competitive numerical stability and computation speed. We also present simulation results of nonlinear stress–strain relationships of spider silk and hydrogels, and dynamic responses of a multilayer structure.
]]>2018-05-16T00:10:11-07:00info:doi/10.1098/rspa.2017.0540hwp:master-id:royprsa;rspa.2017.05402018-05-16Research articles47422132017054020170540<![CDATA[Power packet transferability via symbol propagation matrix]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170552?rss=1
A power packet is a unit of electric power composed of a power pulse and an information tag. In Shannon’s information theory, messages are represented by symbol sequences in a digitized manner. Referring to this formulation, we define symbols in power packetization as a minimum unit of power transferred by a tagged pulse. Here, power is digitized and quantized. In this paper, we consider packetized power in networks for a finite duration, giving symbols and their energies to the networks. A network structure is defined using a graph whose nodes represent routers, sources and destinations. First, we introduce the concept of a symbol propagation matrix (SPM) in which symbols are transferred at links during unit times. Packetized power is described as a network flow in a spatio-temporal structure. Then, we study the problem of selecting an SPM in terms of transferability, that is, the possibility to represent given energies at sources and destinations during the finite duration. To select an SPM, we consider a network flow problem of packetized power. The problem is formulated as an M-convex submodular flow problem which is a solvable generalization of the minimum cost flow problem. Finally, through examples, we verify that this formulation provides reasonable packetized power.
]]>2018-05-16T00:10:11-07:00info:doi/10.1098/rspa.2017.0552hwp:master-id:royprsa;rspa.2017.05522018-05-16Research articles47422132017055220170552<![CDATA[Static bistability of spherical caps]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170910?rss=1
Depending on its geometry, a spherical shell may exist in one of two stable states without the application of any external force: there are two ‘self-equilibrated’ states, one natural and the other inside out (or ‘everted’). Though this is familiar from everyday life—an umbrella is remarkably stable, yet a contact lens can be easily turned inside out—the precise shell geometries for which bistability is possible are not known. Here, we use experiments and finite-element simulations to determine the threshold between bistability and monostability for shells of different solid angle. We compare these results with the prediction from shallow shell theory, showing that, when appropriately modified, this offers a very good account of bistability even for relatively deep shells. We then investigate the robustness of this bistability against pointwise indentation. We find that indentation provides a continuous route for transition between the two states for shells whose geometry makes them close to the threshold. However, for thinner shells, indentation leads to asymmetrical buckling before snap-through, while also making these shells more ‘robust’ to snap-through. Our work sheds new light on the robustness of the ‘mirror buckling’ symmetry of spherical shell caps.
]]>2018-05-16T00:10:11-07:00info:doi/10.1098/rspa.2017.0910hwp:master-id:royprsa;rspa.2017.09102018-05-16Research articles47422132017091020170910<![CDATA[Horizontally isotropic bidispersive thermal convection]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20180018?rss=1
A bidispersive porous material is one which has usual pores but additionally contains a system of micro pores due to cracks or fissures in the solid skeleton. We present general equations for thermal convection in a bidispersive porous medium when the permeabilities, interaction coefficient and thermal conductivity are anisotropic but symmetric tensors. In this case, we show exchange of stabilities holds and fluid movement will commence via stationary convection, and additionally we show the global nonlinear stability threshold is the same as the linear instability one. Attention is then focused on the case where the interaction coefficient and thermal conductivity are isotropic, and the permeability is isotropic in the horizontal directions, although the permeability in the vertical direction is different. The nonlinear stability threshold is calculated in this case and numerical results are presented and discussed in detail.
]]>2018-05-16T00:10:11-07:00info:doi/10.1098/rspa.2018.0018hwp:master-id:royprsa;rspa.2018.00182018-05-16Research articles47422132018001820180018<![CDATA[Energy extraction from vortex-induced vibrations using period-1 rotation of an autoparametric pendulum]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20180086?rss=1
We propose and analyse the feasibility of extracting energy from vortex-induced vibrations using rotating motion of an attached pendulum. The resulting autoparametric pendulum system is studied primarily to understand the effect of pendulum motion on the performance of the harvester which is typically ignored to result in a simple parametric pendulum. We find that rotating motions are possible only for small values of the pendulum mass when compared with the effective mass of the vibrating structure. However, the pendulum motion reduces the basin of attraction as well as the range of system parameters corresponding to the existence of rotary solutions. This significantly alters the harvester performance. By contrast, the evolution of the pendulum coordinates (angular position and velocity) remains largely unaffected by this interaction. Hence, for the purpose of design of controllers to robustly initiate/maintain rotation from arbitrary disturbances, the simplification to a parametric pendulum is reasonable while for the design of the harvester, this exercise is completely unsatisfactory.
]]>2018-05-16T00:10:11-07:00info:doi/10.1098/rspa.2018.0086hwp:master-id:royprsa;rspa.2018.00862018-05-16Research articles47422132018008620180086<![CDATA[Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20180083?rss=1
Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here, we extend these results to periodically or quasi-periodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two dimensional) time-dependent SSMs. A judicious choice of the parametrization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal-form methods.
]]>2018-05-16T00:10:11-07:00info:doi/10.1098/rspa.2018.0083hwp:master-id:royprsa;rspa.2018.00832018-05-16Research articles47422132018008320180083<![CDATA[Could hydrodynamic Rossby waves explain the westward drift?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20180119?rss=1
A novel theory for the origin of the westward drift of the Earth’s magnetic field is proposed, based upon the propagation of hydrodynamic Rossby waves in the liquid outer core. These waves have the obscure property that their crests always progress eastwards—but, for a certain subset, energy can nevertheless be transmitted westwards. In fact, this subset corresponds to sheet-like flow structures, extended in both the axial and radial directions, which are likely to be preferentially excited by convective upwellings in the Earth’s rapidly rotating outer core. To enable their analysis, the quasi-geostrophic (QG) approximation is employed, which assumes horizontal motions to be independent of distance along the rotation axis, yet accounts for variations in the container height (i.e. the slope of the core–mantle boundary). By projecting the momentum equation onto flows of a QG form, a general equation governing their evolution is derived, which is then adapted for the treatment of two initial value problems—in both Cartesian and spherical geometries—which demonstrate the preference for westward energy propagation by the waves in question. The merits of this mechanism as an explanation for westward drift are discussed.
]]>2018-05-16T00:10:11-07:00info:doi/10.1098/rspa.2018.0119hwp:master-id:royprsa;rspa.2018.01192018-05-16Research articles47422132018011920180119<![CDATA[On the dynamic homogenization of periodic media: Willis approach versus two-scale paradigm]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170638?rss=1
When considering an effective, i.e. homogenized description of waves in periodic media that transcends the usual quasi-static approximation, there are generally two schools of thought: (i) the two-scale approach that is prevalent in mathematics and (ii) the Willis’ homogenization framework that has been gaining popularity in engineering and physical sciences. Notwithstanding a mounting body of literature on the two competing paradigms, a clear understanding of their relationship is still lacking. In this study, we deploy an effective impedance of the scalar wave equation as a lens for comparison and establish a low-frequency, long-wavelength dispersive expansion of the Willis’ effective model, including terms up to the second order. Despite the intuitive expectation that such obtained effective impedance coincides with its two-scale counterpart, we find that the two descriptions differ by a modulation factor which is, up to the second order, expressible as a polynomial in frequency and wavenumber. We track down this inconsistency to the fact that the two-scale expansion is commonly restricted to the free-wave solutions and thus fails to account for the body source term which, as it turns out, must also be homogenized—by the reciprocal of the featured modulation factor. In the analysis, we also (i) reformulate for generality the Willis’ effective description in terms of the eigenfunction approach, and (ii) obtain the corresponding modulation factor for dipole body sources, which may be relevant to some recent efforts to manipulate waves in metamaterials.
]]>2018-05-09T00:15:38-07:00info:doi/10.1098/rspa.2017.0638hwp:master-id:royprsa;rspa.2017.06382018-05-09Research articles47422132017063820170638<![CDATA[The correct and unusual coordinate transformation rules for electromagnetic quadrupoles]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170652?rss=1
Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat space–time due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates-free manner. This is particularly useful given their complicated coordinate transformation.
]]>2018-05-09T02:05:01-07:00info:doi/10.1098/rspa.2017.0652hwp:master-id:royprsa;rspa.2017.06522018-05-09Research articles47422132017065220170652<![CDATA[Stochastic modelling of urban structure]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170700?rss=1
The building of mathematical and computer models of cities has a long history. The core elements are models of flows (spatial interaction) and the dynamics of structural evolution. In this article, we develop a stochastic model of urban structure to formally account for uncertainty arising from less predictable events. Standard practice has been to calibrate the spatial interaction models independently and to explore the dynamics through simulation. We present two significant results that will be transformative for both elements. First, we represent the structural variables through a single potential function and develop stochastic differential equations to model the evolution. Second, we show that the parameters of the spatial interaction model can be estimated from the structure alone, independently of flow data, using the Bayesian inferential framework. The posterior distribution is doubly intractable and poses significant computational challenges that we overcome using Markov chain Monte Carlo methods. We demonstrate our methodology with a case study on the London, UK, retail system.
]]>2018-05-09T00:15:38-07:00info:doi/10.1098/rspa.2017.0700hwp:master-id:royprsa;rspa.2017.07002018-05-09Research articles47422132017070020170700<![CDATA[Stability transitions of an axially moving string subjected to a distributed follower force]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170779?rss=1
The transverse vibrations of an axially moving string that is subjected to a distributed follower force are examined here. This model provides an insight into the complex dynamics of seemingly simpler systems such as silicon wafer cutting using wire saws, and aerial or marine towing, where a relatively long flexible structure is dragged through fluid. The equation of motion is derived and it includes the axial variation in the tension that arises due to acceleration and the follower force. As the exact analytical solution of this equation is difficult to determine, the approximate closed-form modal solution of a non-travelling counterpart of the system is obtained using the asymptotic technique, which is then used as a basis to obtain the numerical solution for the axially moving string. The effect of the follower force and viscous dissipation on the eigenstructure of the system is investigated. Mathematical operations such as the Hermite form and the Routh–Hurwitz criterion are applied to the characteristic polynomial to investigate the dynamic behaviour of these modes. The semi-analytical approach presented explains the ‘mathematical’ instability (in the absence of damping) that arises when both axial transport and follower force are simultaneously present. An unusual transition of the dynamic behaviour from the stable to the overdamped and then directly to the unstable regime is observed.
]]>2018-05-09T00:15:38-07:00info:doi/10.1098/rspa.2017.0779hwp:master-id:royprsa;rspa.2017.07792018-05-09Research articles47422132017077920170779<![CDATA[Simultaneity of centres in Zq-equivariant systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170811?rss=1
We study the simultaneous existence of centres for two families of planar Zq-equivariant systems. First, we give a short review about Zq-equivariant systems. Next, we present the necessary and sufficient conditions for the simultaneous existence of centres for a Z2-equivariant cubic system and for a Z2-equivariant quintic system.
]]>2018-05-09T00:15:38-07:00info:doi/10.1098/rspa.2017.0811hwp:master-id:royprsa;rspa.2017.08112018-05-09Research articles47422132017081120170811<![CDATA[Surface instability of elastic half-spaces by using the energy method]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170854?rss=1
Finding the complete set of stability conditions of an elastic half-space has been an open problem ever since Biot (Biot 1963 Appl. Sci. Res.12, 168–182 (doi:10.1007/BF03184638)) first studied the surface instability of half-spaces by seeking solutions of the incremental equilibrium equations. Towards solving this problem, a method based on the energy stability criterion is developed in the present work. A variational problem of minimizing the elastic energy associated with a half-space is formulated. The second variation condition is derived and is converted to an eigenvalue problem. For a half-space of neo-Hookean materials, the eigenvalue problem is solved, which leads to complete descriptions of stability and instability regions in the deformation space.
]]>2018-05-09T00:15:38-07:00info:doi/10.1098/rspa.2017.0854hwp:master-id:royprsa;rspa.2017.08542018-05-09Research articles47422132017085420170854<![CDATA[New variational and multisymplectic formulations of the Euler-Poincare equation on the Virasoro-Bott group using the inverse map]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20180052?rss=1
We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler–Poincaré equations defined on the Virasoro–Bott group, by using the inverse map (also called ‘back-to-labels’ map). This family contains as special cases the well-known Korteweg–de Vries, Camassa–Holm and Hunter–Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with 2-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.
]]>2018-05-09T00:10:05-07:00info:doi/10.1098/rspa.2018.0052hwp:master-id:royprsa;rspa.2018.00522018-05-09Research articles47422132018005220180052<![CDATA[Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20180050?rss=1
We consider the reflection–transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we show that such reflectionless modes can be characterized as eigenfunctions of an original non-self-adjoint spectral problem. To select ingoing waves on one side of the obstacle and outgoing waves on the other side, we use complex scalings (or perfectly matched layers) with imaginary parts of different signs. We prove that the real eigenvalues of the obtained spectrum correspond either to trapped modes (or bound states in the continuum) or to reflectionless modes. Interestingly, complex eigenvalues also contain useful information on weak reflection cases. When the geometry has certain symmetries, the new spectral problem enters the class of PT-symmetric problems.
]]>2018-05-09T00:15:38-07:00info:doi/10.1098/rspa.2018.0050hwp:master-id:royprsa;rspa.2018.00502018-05-09Research articles47422132018005020180050<![CDATA[Time periodic solutions of one-dimensional forced Kirchhoff equations with x-dependent coefficients]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2213/20170620?rss=1
In this paper, we consider the one-dimensional Kirchhoff equation with x-dependent coefficients under Dirichlet boundary conditions, which models the forced vibrations of a clamped inhomogeneous string in the presence of a time periodic external forcing with period 2/ and amplitude . By using the Nash–Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period 2/ and order . Such results hold for parameters (,) in a positive measure Cantor set that has asymptotically full measure as the amplitude goes to zero.
]]>2018-05-02T00:10:07-07:00info:doi/10.1098/rspa.2017.0620hwp:master-id:royprsa;rspa.2017.06202018-05-02Research articles47422132017062020170620<![CDATA[Mean field dynamics of some open quantum systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170856?rss=1
We consider a large number N of quantum particles coupled via a mean field interaction to another quantum system (reservoir). Our main result is an expansion for the averages of observables, both of the particles and of the reservoir, in inverse powers of N. The analysis is based directly on the Dyson series expansion of the propagator. We analyse the dynamics, in the limit N->, of observables of a fixed number n of particles, of extensive particle observables and their fluctuations, as well as of reservoir observables. We illustrate our results on the infinite mode Dicke model and on various energy-conserving models.
]]>2018-04-25T00:10:10-07:00info:doi/10.1098/rspa.2017.0856hwp:master-id:royprsa;rspa.2017.08562018-04-25Research articles47422122017085620170856<![CDATA[Inverse design of an isotropic suspended Kirchhoff rod: theoretical and numerical results on the uniqueness of the natural shape]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170837?rss=1
Solving the equations for Kirchhoff elastic rods has been widely explored for decades in mathematics, physics and computer science, with significant applications in the modelling of thin flexible structures such as DNA, hair or climbing plants. As demonstrated in previous experimental and theoretical studies, the natural curvature plays an important role in the equilibrium shape of a Kirchhoff rod, even in the simple case where the rod is isotropic and suspended under gravity. In this paper, we investigate the reverse problem: can we characterize the natural curvature of a suspended isotropic rod, given an equilibrium curve? We prove that although there exists an infinite number of natural curvatures that are compatible with the prescribed equilibrium, they are all equivalent in the sense that they correspond to a unique natural shape for the rod. This natural shape can be computed efficiently by solving in sequence three linear initial value problems, starting from any framing of the input curve. We provide several numerical experiments to illustrate this uniqueness result, and finally discuss its potential impact on non-invasive parameter estimation and inverse design of thin elastic rods.
]]>2018-04-25T00:10:10-07:00info:doi/10.1098/rspa.2017.0837hwp:master-id:royprsa;rspa.2017.08372018-04-25Research articles47422122017083720170837<![CDATA[Wrinkles and creases in the bending, unbending and eversion of soft sectors]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170827?rss=1
We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asymptotic solutions for thin sectors. When the deformations are severe enough, the compressed side of the elastic material may buckle and wrinkles could then develop. We analyse, in detail, the onset of this instability for the Mooney–Rivlin strain energy, which covers the cases of the neo-Hookean model in exact nonlinear elasticity and of third-order elastic materials in weakly nonlinear elasticity. In particular, the associated theoretical and numerical treatment allows us to predict the number and wavelength of the wrinkles. Guided by experimental observations, we finally look at the development of creases, which we simulate through advanced finite-element computations. In some cases, the linearized analysis allows us to predict correctly the number and the wavelength of the creases, which turn out to occur only a few per cent of strain earlier than the wrinkles.
]]>2018-04-18T00:29:50-07:00info:doi/10.1098/rspa.2017.0827hwp:master-id:royprsa;rspa.2017.08272018-04-18Research articles47422122017082720170827<![CDATA[Textural equilibrium melt geometries around tetrakaidecahedral grains]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170639?rss=1
In textural equilibrium, partially molten materials minimize the total surface energy bound up in grain boundaries and grain–melt interfaces. Here, numerical calculations of such textural equilibrium geometries are presented for a space-filling tessellation of grains with a tetrakaidecahedral (truncated octahedral) unit cell. Two parameters determine the nature of the geometries: the porosity and the dihedral angle. A variety of distinct melt topologies occur for different combinations of these two parameters, and the boundaries between different topologies have been determined. For small dihedral angles, wetting of grain boundaries occurs once the porosity has exceeded 11%. An exhaustive account is given of the main properties of the geometries: their energy, pressure, mean curvature, contiguity and areas on cross sections and faces. Their effective permeabilities have been calculated, and demonstrate a transition between a quadratic variation with porosity at low porosities to a cubic variation at high porosities.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0639hwp:master-id:royprsa;rspa.2017.06392018-04-11Research articles47422122017063920170639<![CDATA[Computing diffusivities from particle models out of equilibrium]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170694?rss=1
A new method is proposed to numerically extract the diffusivity of a (typically nonlinear) diffusion equation from underlying stochastic particle systems. The proposed strategy requires the system to be in local equilibrium and have Gaussian fluctuations but it is otherwise allowed to undergo arbitrary out-of-equilibrium evolutions. This could be potentially relevant for particle data obtained from experimental applications. The key idea underlying the method is that finite, yet large, particle systems formally obey stochastic partial differential equations of gradient flow type satisfying a fluctuation–dissipation relation. The strategy is here applied to three classic particle models, namely independent random walkers, a zero-range process and a symmetric simple exclusion process in one space dimension, to allow the comparison with analytic solutions.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0694hwp:master-id:royprsa;rspa.2017.06942018-04-11Research articles47422122017069420170694<![CDATA[Effects of Lewis number on the statistics of the invariants of the velocity gradient tensor and local flow topologies in turbulent premixed flames]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170706?rss=1
The behaviours of the three invariants of the velocity gradient tensor and the resultant local flow topologies in turbulent premixed flames have been analysed using three-dimensional direct numerical simulation data for different values of the characteristic Lewis number ranging from 0.34 to 1.2. The results have been analysed to reveal the statistical behaviours of the invariants and the flow topologies conditional upon the reaction progress variable. The behaviours of the invariants have been explained in terms of the relative strengths of the thermal and mass diffusions, embodied by the influence of the Lewis number on turbulent premixed combustion. Similarly, the behaviours of the flow topologies have been explained in terms not only of the Lewis number but also of the likelihood of the occurrence of individual flow topologies in the different flame regions. Furthermore, the sensitivity of the joint probability density function of the second and third invariants and the joint probability density functions of the mean and Gaussian curvatures to the variation in Lewis number have similarly been examined. Finally, the dependences of the scalar--turbulence interaction term on augmented heat release and of the vortex-stretching term on flame-induced turbulence have been explained in terms of the Lewis number, flow topology and reaction progress variable.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0706hwp:master-id:royprsa;rspa.2017.07062018-04-11Research articles47422122017070620170706<![CDATA[Reflection from a multi-species material and its transmitted effective wavenumber]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170864?rss=1
We formally deduce closed-form expressions for the transmitted effective wavenumber of a material comprising multiple types of inclusions or particles (multi-species), dispersed in a uniform background medium. The expressions, derived here for the first time, are valid for moderate volume fractions and without restriction on the frequency. We show that the multi-species effective wavenumber is not a straightforward extension of expressions for a single species. Comparisons are drawn with state-of-the-art models in acoustics by presenting numerical results for a concrete and a water–oil emulsion in two dimensions. The limit of when one species is much smaller than the other is also discussed and we determine the background medium felt by the larger species in this limit. Surprisingly, we show that the answer is not the intuitive result predicted by self-consistent multiple scattering theories. The derivation presented here applies to the scalar wave equation with cylindrical or spherical inclusions, with any distribution of sizes, densities and wave speeds. The reflection coefficient associated with a halfspace of multi-species cylindrical inclusions is also formally derived.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0864hwp:master-id:royprsa;rspa.2017.08642018-04-11Research articles47422122017086420170864<![CDATA[A geometric framework for dynamics with unilateral constraints and friction, illustrated by an example of self-organized locomotion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170886?rss=1
We present a geometric framework to deal with mechanical systems which have unilateral constraints, and are subject to damping/friction, which cannot be treated within usual classical mechanics. In this new framework, the dynamical evolution of the system takes place on a multidimensional curvilinear polyhedron, and energetics near the corners of the polyhedron leads to qualitative behaviour such as stable entrapment and bifurcation. We illustrate this by an experiment in which dumbbells, placed inside a tilted hollow cylindrical drum that rotates slowly around its axis, climb uphill by forming dynamically stable pairs, seemingly against the pull of gravity.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0886hwp:master-id:royprsa;rspa.2017.08862018-04-11Research articles47422122017088620170886<![CDATA[Dilatancy induced ductile-brittle transition of shear band in metallic glasses]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170836?rss=1
Dilatancy-generated structural disordering, an inherent feature of metallic glasses (MGs), has been widely accepted as the physical mechanism for the primary origin and structural evolution of shear banding, as well as the resultant shear failure. However, it remains a great challenge to determine, to what degree of dilatation, a shear banding will evolve into a runaway shear failure. In this work, using in situ acoustic emission monitoring, we probe the dilatancy evolution at the different stages of individual shear band in MGs that underwent severely plastic deformation by the controlled cutting technology. A scaling law is revealed that the dilatancy in a shear band is linearly related to its evolution degree. A transition from ductile-to-brittle shear bands is observed, where the formers dominate stable serrated flow, and the latter lead to a runaway instability (catastrophe failure) of serrated flow. To uncover the underlying mechanics, we develop a theoretical model of shear-band evolution dynamics taking into account an atomic-scale deformation process. Our theoretical results agree with the experimental observations, and demonstrate that the atomic-scale volume expansion arises from an intrinsic shear-band evolution dynamics. Importantly, the onset of the ductile–brittle transition of shear banding is controlled by a critical dilatation.
]]>2018-04-11T00:05:14-07:00info:doi/10.1098/rspa.2017.0836hwp:master-id:royprsa;rspa.2017.08362018-04-11Research articles47422122017083620170836<![CDATA[Stable, high-order computation of impedance-impedance operators for three-dimensional layered medium simulations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2212/20170704?rss=1
The faithful modelling of the propagation of linear waves in a layered, periodic structure is of paramount importance in many branches of the applied sciences. In this paper, we present a novel numerical algorithm for the simulation of such problems which is free of the artificial singularities present in related approaches. We advocate for a surface integral formulation which is phrased in terms of impedance–impedance operators that are immune to the Dirichlet eigenvalues which plague the Dirichlet–Neumann operators that appear in classical formulations. We demonstrate a high-order spectral algorithm to simulate these latter operators based upon a high-order perturbation of surfaces methodology which is rapid, robust and highly accurate. We demonstrate the validity and utility of our approach with a sequence of numerical simulations.
]]>2018-04-04T00:05:20-07:00info:doi/10.1098/rspa.2017.0704hwp:master-id:royprsa;rspa.2017.07042018-04-04Research articles47422122017070420170704<![CDATA[Localization in semi-infinite herringbone waveguides]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170590?rss=1
The paper includes novel results for the scattering and localization of a time-harmonic flexural wave by a semi-infinite herringbone waveguide of rigid pins embedded within an elastic Kirchhoff plate. The analytical model takes into account the orientation and spacing of the constituent parts of the herringbone system, and incorporates dipole approximations for the case of closely spaced pins. Illustrative examples are provided, together with the predictive theoretical analysis of the localized waveforms.
]]>2018-03-28T00:05:20-07:00info:doi/10.1098/rspa.2017.0590hwp:master-id:royprsa;rspa.2017.05902018-03-28Research articles47422112017059020170590<![CDATA[Random distributions of initial porosity trigger regular necking patterns at high strain rates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170575?rss=1
At high strain rates, the fragmentation of expanding structures of ductile materials, in general, starts by the localization of plastic deformation in multiple necks. Two distinct mechanisms have been proposed to explain multiple necking and fragmentation process in ductile materials. One view is that the necking pattern is related to the distribution of material properties and defects. The second view is that it is due to the activation of specific instability modes of the structure. Following this, we investigate the emergence of necking patterns in porous ductile bars subjected to dynamic stretching at strain rates varying from 10^{3}s^{–1} to 0.5x10^{5}s^{–1} using finite-element calculations and linear stability analysis. In the calculations, the initial porosity (representative of the material defects) varies randomly along the bar. The computations revealed that, while the random distribution of initial porosity triggers the necking pattern, it barely affects the average neck spacing, especially, at higher strain rates. The average neck spacings obtained from the calculations are in close agreement with the predictions of the linear stability analysis. Our results also reveal that the necking pattern does not begin when the Considère condition is reached but is significantly delayed due to the stabilizing effect of inertia.
]]>2018-03-28T00:05:20-07:00info:doi/10.1098/rspa.2017.0575hwp:master-id:royprsa;rspa.2017.05752018-03-28Research articles47422112017057520170575<![CDATA[Nonlinear dynamics of a dispersive anisotropic Kuramoto-Sivashinsky equation in two space dimensions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170687?rss=1
A Kuramoto–Sivashinsky equation in two space dimensions arising in thin film flows is considered on doubly periodic domains. In the absence of dispersive effects, this anisotropic equation admits chaotic solutions for sufficiently large length scales with fully two-dimensional profiles; the one-dimensional dynamics observed for thin domains are structurally unstable as the transverse length increases. We find that, independent of the domain size, the characteristic length scale of the profiles in the streamwise direction is about 10 space units, with that in the transverse direction being approximately three times larger. Numerical computations in the chaotic regime provide an estimate for the radius of the absorbing ball in L2 in terms of the length scales, from which we conclude that the system possesses a finite energy density. We show the property of equipartition of energy among the low Fourier modes, and report the disappearance of the inertial range when solution profiles are two-dimensional. Consideration of the high-frequency modes allows us to compute an estimate for the analytic extensibility of solutions in C2. We also examine the addition of a physically derived third-order dispersion to the problem; this has a destabilizing effect, in the sense of reducing analyticity and increasing amplitude of solutions. However, sufficiently large dispersion may regularize the spatio-temporal chaos to travelling waves. We focus on dispersion where chaotic dynamics persist, and study its effect on the interfacial structures, absorbing ball and properties of the power spectrum.
]]>2018-03-28T00:05:20-07:00info:doi/10.1098/rspa.2017.0687hwp:master-id:royprsa;rspa.2017.06872018-03-28Research articles47422112017068720170687<![CDATA[DC dynamic pull-in instability of a dielectric elastomer balloon: an energy-based approach]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170900?rss=1
This paper reports an energy-based method for the dynamic pull-in instability analysis of a spherical dielectric elastomer (DE) balloon subjected to a quasi-statically applied inflation pressure and a Heaviside step voltage across the balloon wall. The proposed technique relies on establishing the energy balance at the point of maximum stretch in an oscillation cycle, followed by the imposition of an instability condition for extracting the threshold parameters. The material models of the Ogden family are employed for describing the hyperelasticity of the balloon. The accuracy of the critical dynamic pull-in parameters is established by examining the saddle-node bifurcation in the transient response of the balloon obtained by integrating numerically the equation of motion, derived using the Euler–Lagrange equation. The parametric study brings out the effect of inflation pressure on the onset of the pull-in instability in the DE balloon. A quantitative comparison between the static and dynamic pull-in parameters at four different levels of the inflation pressure is presented. The results indicate that the dynamic pull-in instability gets triggered at electric fields that are lower than those corresponding to the static instability. The results of the present investigation can find potential use in the design and development of the balloon actuators subjected to transient loading. The method developed is versatile and can be used in the dynamic instability analysis of other conservative systems of interest.
]]>2018-03-28T00:05:20-07:00info:doi/10.1098/rspa.2017.0900hwp:master-id:royprsa;rspa.2017.09002018-03-28Research articles47422112017090020170900<![CDATA[An upper bound on the particle-laden dependency of shear stresses at solid-fluid interfaces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170332?rss=1
In modern advanced manufacturing processes, such as three-dimensional printing of electronics, fine-scale particles are added to a base fluid yielding a modified fluid. For example, in three-dimensional printing, particle-functionalized inks are created by adding particles to freely flowing solvents forming a mixture, which is then deposited onto a surface, which upon curing yields desirable solid properties, such as thermal conductivity, electrical permittivity and magnetic permeability. However, wear at solid–fluid interfaces within the machinery walls that deliver such particle-laden fluids is typically attributed to the fluid-induced shear stresses, which increase with the volume fraction of added particles. The objective of this work is to develop a rigorous strict upper bound for the tolerable volume fraction of particles that can be added, while remaining below a given stress threshold at a fluid–solid interface. To illustrate the bound’s utility, the expression is applied to a series of classical flow regimes.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0332hwp:master-id:royprsa;rspa.2017.03322018-03-21Research articles47422112017033220170332<![CDATA[Rayleigh wave at the surface of a general anisotropic poroelastic medium: derivation of real secular equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170589?rss=1
A secular equation governs the propagation of Rayleigh wave at the surface of an anisotropic poroelastic medium. In the case of anisotropy with symmetry, this equation is obtained as a real irrational equation. But, in the absence of anisotropic symmetries, this secular equation is obtained as a complex irrational equation. True surface waves in non-dissipative materials decay only with depth. That means, propagation of Rayleigh wave in anisotropic poroelastic solid should be represented by a real phase velocity. In this study, the determinantal system leading to the complex secular equation is manipulated to obtain a transformed equation. Even for arbitrary (triclinic) anisotropy, this transformed equation remains real for the propagation of true surface waves. Such a real secular equation is obtained with the option of boundary pores being opened or sealed. A numerical example is solved to study the existence and propagation of Rayleigh waves in porous media for the top three (i.e. triclinic, monoclinic and orthorhombic) anisotropies.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0589hwp:master-id:royprsa;rspa.2017.05892018-03-21Research articles47422112017058920170589<![CDATA[A hydrostatic model of the Wirtz pump]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170533?rss=1
The Wirtz pump is not only an excellent example of alternative technology, using as it does the kinetic energy of a stream to raise a proportion of its water, but its mathematical modelling also poses several intriguing problems. We give some history of the Wirtz pump and describe its operation. Taking a novel dynamical systems approach, we then derive a discrete mathematical model in the form of a mapping that describes its hydrostatic behaviour. Our model enables us to explain several aspects of the behaviour of the pump as well as to design one that gives approximately maximal, and maximally constant, output pressure.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0533hwp:master-id:royprsa;rspa.2017.05332018-03-21Research articles47422112017053320170533<![CDATA[Magic angles for fibrous incompressible elastic materials]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170728?rss=1
In the analysis of the mechanical behaviour of fibre-reinforced incompressible elastic bodies, there is a special angle of orientation of the fibres which leads to a particular mechanical response. This angle has been called a ‘magic angle’ due to its appearance as if by magic in many different aspects of the mechanics of fibrous solids including several examples in biology. It occurs most commonly not only in structural elements composed of circular cylindrical tubes or cylinders reinforced by helically wound fibres but also in flat thin sheets reinforced by fibres in the plane. The occurrence of such a special angle was classically demonstrated using a simple purely geometric analysis in the context of a lattice composed of a single family of helically wound inextensible fibres. Recently, the magic angle concept has been discussed in the framework of nonlinear hyperelasticity for anisotropic materials with detailed constitutive modelling. Our purpose here is to describe some other contexts in which the magic angle occurs starting from earlier work in a special theory of linear elasticity for inextensible fibres and proceeding to relatively accessible models of hyperelasticity. We discuss the role of the magic angle in the quasi-isotropic mechanical response of fibre-reinforced composites as well as the implications for material instability.
]]>2018-03-21T00:05:26-07:00info:doi/10.1098/rspa.2017.0728hwp:master-id:royprsa;rspa.2017.07282018-03-21Research articles47422112017072820170728<![CDATA[Small nanoparticles, surface geometry and contact forces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170723?rss=1
In this molecular dynamics study, we examine the local surface geometric effects of the normal impact force between two approximately spherical nanoparticles that collide in a vacuum. Three types of surface geometries—(i) crystal facets, (ii) sharp edges, and (iii) amorphous surfaces of small nanoparticles with radii R<10 nm—are considered. The impact forces are compared with their macroscopic counterparts described by nonlinear contact forces based on Hertz contact mechanics. In our simulations, edge and amorphous surface contacts with weak surface energy reveal that the average impact forces are in excellent agreement with the Hertz contact force. On the other hand, facet collisions show a linearly increasing force with increasing compression. Our results suggest that the nearly spherical nanoparticles are likely to enable some nonlinear dynamic phenomena, such as breathers and solitary waves observed in granular materials, both originating from the nonlinear contact force.
]]>2018-03-21T01:44:55-07:00info:doi/10.1098/rspa.2017.0723hwp:master-id:royprsa;rspa.2017.07232018-03-21Research articles47422112017072320170723<![CDATA[Continuum modelling of segregating tridisperse granular chute flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170384?rss=1
Segregation and mixing of size multidisperse granular materials remain challenging problems in many industrial applications. In this paper, we apply a continuum-based model that captures the effects of segregation, diffusion and advection for size tridisperse granular flow in quasi-two-dimensional chute flow. The model uses the kinematics of the flow and other physical parameters such as the diffusion coefficient and the percolation length scale, quantities that can be determined directly from experiment, simulation or theory and that are not arbitrarily adjustable. The predictions from the model are consistent with experimentally validated discrete element method (DEM) simulations over a wide range of flow conditions and particle sizes. The degree of segregation depends on the Péclet number, Pe, defined as the ratio of the segregation rate to the diffusion rate, the relative segregation strength _{ij} between particle species i and j, and a characteristic length L, which is determined by the strength of segregation between smallest and largest particles. A parametric study of particle size, _{ij}, Pe and L demonstrates how particle segregation patterns depend on the interplay of advection, segregation and diffusion. Finally, the segregation pattern is also affected by the velocity profile and the degree of basal slip at the chute surface. The model is applicable to different flow geometries, and should be easily adapted to segregation driven by other particle properties such as density and shape.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0384hwp:master-id:royprsa;rspa.2017.03842018-03-14Research articles47422112017038420170384<![CDATA[Resilience of riverbed vegetation to uprooting by flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170547?rss=1
Riverine ecosystem biodiversity is largely maintained by ecogeomorphic processes including vegetation renewal via uprooting and recovery times to flow disturbances. Plant roots thus heavily contribute to engineering resilience to perturbation of such ecosystems. We show that vegetation uprooting by flow occurs as a fatigue-like mechanism, which statistically requires a given exposure time to imposed riverbed flow erosion rates before the plant collapses. We formulate a physically based stochastic model for the actual plant rooting depth and the time-to-uprooting, which allows us to define plant resilience to uprooting for generic time-dependent flow erosion dynamics. This theory shows that plant resilience to uprooting depends on the time-to-uprooting and that root mechanical anchoring acts as a process memory stored within the plant–soil system. The model is validated against measured data of time-to-uprooting of Avena sativa seedlings with various root lengths under different flow conditions. This allows for assessing the natural variance of the uprooting-by-flow process and to compute the prediction entropy, which quantifies the relative importance of the deterministic and the random components affecting the process.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0547hwp:master-id:royprsa;rspa.2017.05472018-03-14Research articles47422112017054720170547<![CDATA[Vibration of carbon nanotubes with defects: order reduction methods]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170555?rss=1
Order reduction methods are widely used to reduce computational effort when calculating the impact of defects on the vibrational properties of nearly periodic structures in engineering applications, such as a gas-turbine bladed disc. However, despite obvious similarities these techniques have not yet been adapted for use in analysing atomic structures with inevitable defects. Two order reduction techniques, modal domain analysis and modified modal domain analysis, are successfully used in this paper to examine the changes in vibrational frequencies, mode shapes and mode localization caused by defects in carbon nanotubes. The defects considered are isotope defects and Stone–Wales defects, though the methods described can be extended to other defects.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0555hwp:master-id:royprsa;rspa.2017.05552018-03-14Research articles47422112017055520170555<![CDATA[On the theory of drainage area for regular and non-regular points]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170693?rss=1
The drainage area is an important, non-local property of a landscape, which controls surface and subsurface hydrological fluxes. Its role in numerous ecohydrological and geomorphological applications has given rise to several numerical methods for its computation. However, its theoretical analysis has lagged behind. Only recently, an analytical definition for the specific catchment area was proposed (Gallant & Hutchinson. 2011 Water Resour. Res.47, W05535. (doi:10.1029/2009WR008540)), with the derivation of a differential equation whose validity is limited to regular points of the watershed. Here, we show that such a differential equation can be derived from a continuity equation (Chen et al. 2014 Geomorphology219, 68–86. (doi:10.1016/j.geomorph.2014.04.037)) and extend the theory to critical and singular points both by applying Gauss’s theorem and by means of a dynamical systems approach to define basins of attraction of local surface minima. Simple analytical examples as well as applications to more complex topographic surfaces are examined. The theoretical description of topographic features and properties, such as the drainage area, channel lines and watershed divides, can be broadly adopted to develop and test the numerical algorithms currently used in digital terrain analysis for the computation of the drainage area, as well as for the theoretical analysis of landscape evolution and stability.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0693hwp:master-id:royprsa;rspa.2017.06932018-03-14Research articles47422112017069320170693<![CDATA[Stochastic isotropic hyperelastic materials: constitutive calibration and model selection]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170858?rss=1
Biological and synthetic materials often exhibit intrinsic variability in their elastic responses under large strains, owing to microstructural inhomogeneity or when elastic data are extracted from viscoelastic mechanical tests. For these materials, although hyperelastic models calibrated to mean data are useful, stochastic representations accounting also for data dispersion carry extra information about the variability of material properties found in practical applications. We combine finite elasticity and information theories to construct homogeneous isotropic hyperelastic models with random field parameters calibrated to discrete mean values and standard deviations of either the stress–strain function or the nonlinear shear modulus, which is a function of the deformation, estimated from experimental tests. These quantities can take on different values, corresponding to possible outcomes of the experiments. As multiple models can be derived that adequately represent the observed phenomena, we apply Occam’s razor by providing an explicit criterion for model selection based on Bayesian statistics. We then employ this criterion to select a model among competing models calibrated to experimental data for rubber and brain tissue under single or multiaxial loads.
]]>2018-03-14T00:05:16-07:00info:doi/10.1098/rspa.2017.0858hwp:master-id:royprsa;rspa.2017.08582018-03-14Research articles47422112017085820170858<![CDATA[Effects of geometric nonlinearity in an adhered microbeam for measuring the work of adhesion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170594?rss=1
Design against adhesion in microelectromechanical devices is predicated on the ability to quantify this phenomenon in microsystems. Previous research related the work of adhesion for an adhered microbeam to the beam's unadhered length, and as such, interferometric techniques were developed to measure that length. We propose a new vibration-based technique that can be easily implemented with existing atomic force microscopy tools or similar metrology systems. To make such a technique feasible, we analysed a model of the adhered microbeam using the nonlinear beam theory put forth by Woinowsky–Krieger. We found a new relation between the work of adhesion and the unadhered length; this relation is more accurate than the one by Mastrangelo & Hsu (Mastrangelo & Hsu 1993 J. Microelectromech. S., 2, 44–55. (doi:10.1109/84.232594)) which is commonly used. Then, we derived a closed-form approximate relationship between the microbeam's natural frequency and its unadhered length. Results obtained from this analytical formulation are in good agreement with numerical results from three-dimensional nonlinear finite-element analysis.
]]>2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2017.0594hwp:master-id:royprsa;rspa.2017.05942018-03-07Research articles47422112017059420170594<![CDATA[A canonical form of the equation of motion of linear dynamical systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170809?rss=1
The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.
]]>2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2017.0809hwp:master-id:royprsa;rspa.2017.08092018-03-07Research articles47422112017080920170809<![CDATA[Post-buckling of a pressured biopolymer spherical shell with the mode interaction]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2211/20170834?rss=1
Imperfection sensitivity is essential for mechanical behaviour of biopolymer shells characterized by high geometric heterogeneity. The present work studies initial post-buckling and imperfection sensitivity of a pressured biopolymer spherical shell based on non-axisymmetric buckling modes and associated mode interaction. Our results indicate that for biopolymer spherical shells with moderate radius-to-thickness ratio (say, less than 30) and smaller effective bending thickness (say, less than 0.2 times average shell thickness), the imperfection sensitivity predicted based on the axisymmetric mode without the mode interaction is close to the present results based on non-axisymmetric modes with the mode interaction with a small (typically, less than 10%) relative errors. However, for biopolymer spherical shells with larger effective bending thickness, the maximum load an imperfect shell can sustain predicted by the present non-axisymmetric analysis can be significantly (typically, around 30%) lower than those predicted based on the axisymmetric mode without the mode interaction. In such cases, a more accurate non-axisymmetric analysis with the mode interaction, as given in the present work, is required for imperfection sensitivity of pressured buckling of biopolymer spherical shells. Finally, the implications of the present study to two specific types of biopolymer spherical shells (viral capsids and ultrasound contrast agents) are discussed.
]]>2018-03-07T00:05:22-08:00info:doi/10.1098/rspa.2017.0834hwp:master-id:royprsa;rspa.2017.08342018-03-07Research articles47422112017083420170834