Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences

Published by Royal Society, The
Online ISSN: 1471-2946
We present the results of experimental determination of the heat capacity of the pyrochlore Er2Ti2O7 as a function of temperature (0.35-300 K) and magnetic field (up to 9 T), and for magnetically diluted solid solutions of the general formula (Er1-x Y x )2Ti2O7 (x≤0.471). On either doping or increase of magnetic field, or both, the Néel temperature first shifts to lower temperature until a critical point above which there is no well-defined transition but a Schottky-like anomaly associated with the splitting of the ground state Kramers doublet. By taking into account details of the lattice contribution to the heat capacity, we accurately isolate the magnetic contribution to the heat capacity and hence to the entropy. For pure Er2Ti2O7 and for (Er1-x Y x )2Ti2O7, the magnetic entropy as a function of temperature evolves with two plateaus: the first at [Formula: see text], and the other at [Formula: see text]. When a very high magnetic field is applied, the first plateau is washed out. The influence of dilution at low values is similar to the increase of magnetic field, as we show by examination of the critical temperature versus critical field curve in reduced terms.
Abdominal aortic aneurysms (AAAs) are characterized by significant changes in the architecture of the aortic wall, notably, loss of functional elastin and smooth muscle. Because collagen is the principal remaining load-bearing constituent of the aneurysmal wall, its turnover must play a fundamental role in the natural history of the lesion. Nevertheless, detailed investigations of the effects of different aspects of collagen turnover on AAA development are lacking. A finite-element membrane model of the growth and remodelling of idealized AAAs was thus used to investigate parametrically four of the primary aspects of collagen turnover: rates of production, half-life, deposition stretch (prestretch) and material stiffness. The predicted rates of aneurysmal expansion and spatio-temporal changes in wall thickness, biaxial stresses and maximum collagen fibre stretch at the apex of the lesion depended strongly on all four factors, as did the predicted clinical endpoints (i.e. arrest, progressive expansion or rupture). Collagen turnover also affected the axial expansion, largely due to mechanical changes within the shoulder region of the lesion. We submit, therefore, that assessment of rupture risk could be improved by future experiments that delineate and quantify different aspects of patient-specific collagen turnover and that such understanding could lead to new targeted therapeutics.
(a) Accession dates (first regnal year) obtained by this research for the first eight rulers of Egypt. (b) Intervals between the accession dates as indicated. In both cases (a,b), the marginal posterior density functions are shown with the corresponding 68% and 95% hpd ranges beneath. (Online version in colour.) 
The sequence of events from the Badarian, through the Predynastic (Naqada period) to the establishment of the First Dynasty. The divisions are placed at the medians of each hpd range. (Online version in colour.) 
The Egyptian state was formed prior to the existence of verifiable historical records. Conventional dates for its formation are based on the relative ordering of artefacts. This approach is no longer considered sufficient for cogent historical analysis. Here, we produce an absolute chronology for Early Egypt by combining radiocarbon and archaeological evidence within a Bayesian paradigm. Our data cover the full trajectory of Egyptian state formation and indicate that the process occurred more rapidly than previously thought. We provide a timeline for the First Dynasty of Egypt of generational-scale resolution that concurs with prevailing archaeological analysis and produce a chronometric date for the foundation of Egypt that distinguishes between historical estimates.
Synthetic organic chemists have the power to replicate some of the most intriguing molecules of living nature in the laboratory and apply their developed synthetic strategies and technologies to construct variations of them. Such molecules facilitate biology and medicine, as they often find uses as biological tools and drug candidates for clinical development. In addition, by employing sophisticated catalytic reactions and appropriately designed synthetic processes, they can synthesize not only the molecules of nature and their analogues, but also myriad other organic molecules for potential applications in many areas of science, technology and everyday life. After a short historical introduction, this article focuses on recent advances in the field of organic synthesis with demonstrative examples of total synthesis of complex bioactive molecules, natural or designed, from the author's laboratories, and their impact on chemistry, biology and medicine.
Alternating partial hydrogenation of the interior region of a polycyclic aromatic hydrocarbon gives a finite model system representing systems on the pathway from graphene to the graphane modification of the graphene sheet. Calculations at the DFT and coupled Hartree-Fock levels confirm that sp(2) cycles of bare carbon centres isolated by selective hydrogenation retain the essentially planar geometry and electron delocalization of the annulene that they mimic. Delocalization is diagnosed by the presence of ring currents, as detected by ipsocentric calculation and visualization of the current density induced in the π system by a perpendicular external magnetic field. These induced 'ring' currents have essentially the same sense, strength and orbital origin as in the free hydrocarbon. Subjected to the important experimental proviso of the need for atomic-scale control of hydrogenation, this finding predicts the possibility of writing single, multiple and concentric diatropic and/or paratropic ring currents on the graphene/graphane sheet. The implication is that pathways for free flow of ballistic current can be modelled in the same way.
Propagation of boundary data with a Gaussian profile. The field magnitude |u(x, 0)| (solid line) and its real part Re u(x, 0) (dashed line) evaluated along the positive x-axis (a) and the attained accuracy log 10 |u(x, 0) − ˜ u(x, 0)| (b).
Propagation of a spherical wave restricted to a square aperture and converging to a point on the optical axis. As expected, the diffraction pattern is approximately a scaled version of the function | sinc( x 1 ) sinc( x 2 ) | . We display the magnitude 
Comparison of the magnitude of the field evaluated near the focal point on the x 1 -axis. We display the magnitude 
We describe a fast algorithm to propagate, for any user-specified accuracy, a time-harmonic electromagnetic field between two parallel planes separated by a linear, isotropic and homogeneous medium. The analytical formulation of this problem (ca 1897) requires the evaluation of the so-called Rayleigh-Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is very large, it can be accurately approximated by asymptotic methods. In the large intermediate region of practical interest, where the oscillatory Rayleigh-Sommerfeld kernel must be applied directly, current numerical methods can be highly inaccurate without indicating this fact to the user. In our approach, for any user-specified accuracy ϵ>0, we approximate the kernel by a short sum of Gaussians with complex-valued exponents, and then efficiently apply the result to the input data using the unequally spaced fast Fourier transform. The resulting algorithm has computational complexity [Formula: see text], where we evaluate the solution on an N×N grid of output points given an M×M grid of input samples. Our algorithm maintains its accuracy throughout the computational domain.
We obtain an exactification of the Poincaré asymptotic expansion (PAE) of the Hankel integral, [Formula: see text] as [Formula: see text], using the distributional approach of McClure & Wong. We find that, for half-integer orders of the Bessel function, the exactified asymptotic series terminates, so that it gives an exact finite sum representation of the Hankel integral. For other orders, the asymptotic series does not terminate and is generally divergent, but is amenable to superasymptotic summation, i.e. by optimal truncation. For specific examples, we compare the accuracy of the optimally truncated asymptotic series owing to the McClure-Wong distributional method with owing to the Mellin-Barnes integral method. We find that the former is spectacularly more accurate than the latter, by, in some cases, more than 70 orders of magnitude for the same moderate value of b. Moreover, the exactification can lead to a resummation of the PAE when it is exact, with the resummed Poincaré series exhibiting again the same spectacular accuracy. More importantly, the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences.
(a) Diagram of a typical LIAD source. Reproduced with permission from [87]. (b) LIAD signal intensity for various metal foil substrates. Reproduced with permission from [88]. 
Diagram of the ESI process. Reproduced with permission from [102]. 
In many chemical reactions, an activation barrier must be overcome before a chemical transformation can occur. As such, understanding the behaviour of molecules in energetically excited states is critical to understanding the chemical changes that these molecules undergo. Among the most prominent reactions for mankind to understand are chemical changes that occur in our own biological molecules. A notable example is the focus towards understanding the interaction of DNA with ultraviolet radiation and the subsequent chemical changes. However, the interaction of radiation with large biological structures is highly complex, and thus the photochemistry of these systems as a whole is poorly understood. Studying the gas-phase spectroscopy and ultrafast dynamics of the building blocks of these more complex biomolecules offers the tantalizing prospect of providing a scientifically intuitive bottom-up approach, beginning with the study of the subunits of large polymeric biomolecules and monitoring the evolution in photochemistry as the complexity of the molecules is increased. While highly attractive, one of the main challenges of this approach is in transferring large, and in many cases, thermally labile molecules into vacuum. This review discusses the recent advances in cutting-edge experimental methodologies, emerging as excellent candidates for progressing this bottom-up approach.
We investigated free-vibration acoustic resonance (FVAR) of two-dimensional St Venant-Kirchhoff-type hyperelastic materials and revealed the existence and structure of colour symmetry embedded therein. The hyperelastic material is isotropic and frame indifferent and includes geometrical nonlinearity in its constitutive equation. The FVAR state is formulated using the principle of stationary action with a subsidiary condition. Numerical analysis based on the Ritz method revealed the existence of four types of nonlinear FVAR modes associated with the irreducible representations of a linearized system. Projection operation revealed that the FVAR modes can be classified on the basis of a single colour (black or white) and three types of bicolour (black and white) magnetic point groups: [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. These results demonstrate that colour symmetry naturally arises in the finite amplitude nonlinear FVAR modes, and its vibrational symmetries are explained in terms of magnetic point groups rather than the irreducible representations that have been used for linearized systems. We also predicted a grey colour nonlinear FVAR mode which cannot be derived from a linearized system.
Structure of ancestry data. Log–log plot showing distributions of ancestry numbers across several industries and countries. Inset: Zipf plot for US banking ancestry data; the dashed line has a gradient of −2.  
Comparison between the master equation, agent-based model and real data from the surviving population in the US banking sector. (a) The range of ancestry data obtained from the agent-based model over 1000 simulations (grey-shaded area), compared with that for the live population within the real US banking dataset (black circles). (b) A comparison with the master equation result. Both plots are shown as functions of the shifted ancestry variable n A + 1.  
Real business organic growth. The chart shows the actual, organic growth of entities within each acquisition grouping, excluding the effects of GDP growth and balance sheet aggregation from mergers between US banks from 1992 to 2013. For each of the surviving banks in 2013, we aggregated the balance sheets of each of their ancestors in 1992, and indexed by the GDP. As a result, the real organic growth is derived from the ratio between the balance sheets of surviving entities as at 2013, and those of their respective ancestors (GDP linked and aggregated) as at 1992. The scale is relative to GDP such that values > 0, = 0 and < 0 indicate growth greater than, tracking, and less than GDP, respectively. The red bar in the main plot shows the average over those banks with ancestries > 25; the inset shows a refined view of growth for these largest banks, by number of acquisitions. Note that in averaging the data shown in the inset to arrive at the > 25 bucket shown in the main plot, the average is weighted by balance sheet size.  
Understanding the fundamental mechanisms behind the complex landscape of corporate mergers and acquisitions is of crucial importance to economies across the world. Adapting ideas from the fields of complexity and evolutionary dynamics to analyse business ecosystems, we show here that ancestry, i.e. the cumulative sum of historical mergers across all ancestors, is the key characteristic to company mergers and acquisitions. We verify this by comparing an agent-based model to an extensive range of business data, covering the period from the 1830s to the present day and a range of industries and geographies. This seemingly universal mechanism leads to imbalanced business ecosystems, with the emergence of a few very large, but sluggish 'too big to fail' entities, and very small, niche entities, thereby creating a paradigm where a configuration akin to effective oligopoly or monopoly is a likely outcome for free market systems.
In this paper, we provide a unified and self-consistent treatment of a functionally graded material (FGM) microbeam with varying thermal conductivity subjected to non-uniform or uniform temperature field. Specifically, it is our objective to determine the effect of the microscopic size of the beam, the electrostatic gap, the temperature field and material property on the pull-in voltage of the microbeam under different boundary conditions. The non-uniform temperature field is obtained by integrating the steady-state heat conduction equation. The governing equations account for the microbeam size by introducing an internal material length-scale parameter that is based on the modified couple stress theory. Furthermore, it takes into account Casimir and van der Waals forces, and the associated electrostatic force with the first-order fringing field effects. The resulting nonlinear differential equations were converted to a coupled system of algebraic equations using the differential quadrature method. The outcome of our work shows the dramatic effect and dependence of the pull-in voltage of the FGM microbeam upon the temperature field, its gradient for a given boundary condition. Specifically, both uniform and non-uniform thermal loading can actuate the FGM microbeam even without an applied voltage. Our work also reveals that the non-uniform temperature field is more effective than the uniform temperature field in actuating a FGM cantilever-type microbeam. For the clamped-clamped case, care must be taken to account for the effective use of thermal loading in the design of microbeams. It is also observed that uniform thermal loading will lead to a reduction in the pull-in voltage of a FGM microbeam for all the three boundary conditions considered.
In equilibrium, a vesicle that is adhered to a horizontal substrate by a long-range attractive, short-range repulsive force traps a thin layer of fluid beneath it. In the asymptotic limit that this layer is very thin, there are quasi-two-dimensional boundary-layer structures near the edges of the vesicle, where the membrane's shape is governed by a balance between bending and adhesive stresses. These boundary layers are analysed to obtain corrections to simpler models that instead represent the adhesive interaction by a contact potential, thereby resolving apparent discontinuities that arise when such models are used. Composite expansions of the shapes of two-dimensional vesicles are derived. When, in addition, the adhesive interaction is very strong, there is a nested boundary-layer structure for which the adhesive boundary layers match towards sharp corners where bending stresses remain important but adhesive stresses are negligible. Outside these corners, bending stresses are negligible and the vesicle's shape is given approximately by the arc of a circle. Simple composite expansions of the vesicle's shape are derived that account for the shape of the membrane inside these corners.
Geometry of the membrane in contact with the substrate. Undeformed membrane is shown as a dashed line and deformed membrane as a solid curve. Free body diagram of a small segment of the membrane is shown on the right.
In part I of this work, we study adhesionless contact of a long rectangular elastic membrane with a rigid substrate. Our model is based on finite strain theory and is valid for arbitrarily large deformations. Both frictionless and no-slip contact conditions are considered. Exact closed form solutions are obtained for frictionless contact. For small contact, the differences between these two contact conditions are small. However, significant differences occur for large contacts. For example, frictionless contact predicts a maximum pressure (and contact region) beyond which there is no solution; while the no-slip model places no restriction on both quantities. The effect of adhesion will be considered in part II of this work.
Plot of contact length variation with applied pressure for a frictionless membrane for four different adhesion energies. Arrows indicate the path traversed in a loading–unloading cycle. Owing to contact hysteresis, there are four distinct parts of the curve: free inflation (dotted), increasing contact (solid), contact pinning (dashed) and decreasing contact (dashed-dotted). Plots for four different adhesion energies during unloading are shown: ¯ w ad = 0.001, 0.05, 0.5 and 1.5. β p 1 and β p 2 are the pinch-off pressures for the low adhesion cases: ¯ w ad = 0.001 and ¯ w ad = 0.05, respectively. A 1 and B 1 mark the pull-off points for high adhesion cases: ¯ w ad = 0.5 and ¯ w ad = 1.5, respectively. (Online version in colour.)  
(a–b) Frictionless membrane receding contact plots for intermediate separation ¯ d = 0.5. These results are obtained when we inflate to a maximum contact length of ¯ c = 0.9.  
Pull-off parameters for a rectangular membrane. Results are shown for two different initial separation values: ¯ d = 0.5 (thin lines) and ¯ d = 0.9 (thick lines). For each separation, plots are shown for both no-slip (solid) and frictionless (dashed) boundary conditions. (a) Contact length at pull-off and (b) pressure at pull-off.
Critical value of adhesion for onset of bistability as a function of initial separation between membrane and substrate.  
Critical adhesion for transition from pinch-off to pull-off as a function of initial separation.  
In part I of this work, we presented a theory for adhesionless contact of a pressurized neo-Hookean plane-strain membrane to a rigid substrate. Here, we extend our theory to include adhesion using a fracture mechanics approach. This theory is used to study contact hysteresis commonly observed in experiments. Detailed analysis is carried out to highlight the differences between frictionless and no-slip contact. Membrane detachment is found to be strongly dependent on adhesion: for low adhesion, the membrane 'pinches-off', whereas for large adhesions, it detaches unstably at finite contact ('pull-off'). Expressions are derived for the critical adhesion needed for pinch-off to pull-off transition. Above a threshold adhesion, the membrane exhibits bistability, two stable states at zero applied pressure. The condition for bistability for both frictionless and no-slip boundary conditions is obtained explicitly.
The paper deals with unstable aeroelastic modes for aircraft wing model in subsonic, incompressible, inviscid air flow. In recent author's papers asymptotic, spectral and stability analysis of the model has been carried out. The model is governed by a system of two coupled integrodifferential equations and a two-parameter family of boundary conditions modelling action of self-straining actuators. The Laplace transform of the solution is given in terms of the 'generalized resolvent operator', which is a meromorphic operator-valued function of the spectral parameter λ, whose poles are called the aeroelastic modes. The residues at these poles are constructed from the corresponding mode shapes. The spectral characteristics of the model are asymptotically close to the ones of a simpler system, which is called the reduced model. For the reduced model, the following result is shown: for each value of subsonic speed, there exists a radius such that all aeroelastic modes located outside the circle of this radius centred at zero are stable. Unstable modes, whose number is always finite, can occur only inside this 'circle of instability'. Explicit estimate of the 'instability radius' in terms of model parameters is given.
This paper gives a regular vector boundary integral equation for solving the problem of viscous scattering of a pressure wave by a rigid body. Firstly, single-layer viscous potentials and a generalized stress tensor are introduced. Correspondingly, generalized viscous double-layer potentials are defined. By representing the scattered field as a combination of a single-layer viscous potential and a generalized viscous double-layer potential, the problem is reduced to the solution of a vectorial Fredholm integral equation of the second kind. Generally, the vector integral equation is singular. However, there is a particular stress tensor, called pseudostress, which yields a regular integral equation. In this case, the Fredholm alternative applies and permits a direct proof of the existence and uniqueness of the solution. The results presented here provide the foundation for a numerical solution procedure.
The fact that acoustic radiation from a violin at air-cavity resonance is monopolar and can be determined by pure volume change is used to help explain related aspects of violin design evolution. By determining the acoustic conductance of arbitrarily shaped sound holes, it is found that air flow at the perimeter rather than the broader sound-hole area dominates acoustic conductance, and coupling between compressible air within the violin and its elastic structure lowers the Helmholtz resonance frequency from that found for a corresponding rigid instrument by roughly a semitone. As a result of the former, it is found that as sound-hole geometry of the violin's ancestors slowly evolved over centuries from simple circles to complex f-holes, the ratio of inefficient, acoustically inactive to total sound-hole area was decimated, roughly doubling air-resonance power efficiency. F-hole length then slowly increased by roughly 30% across two centuries in the renowned workshops of Amati, Stradivari and Guarneri, favouring instruments with higher air-resonance power, through a corresponding power increase of roughly 60%. By evolution-rate analysis, these changes are found to be consistent with mutations arising within the range of accidental replication fluctuations from craftsmanship limitations with subsequent selection favouring instruments with higher air-resonance power.
A new method to directly print out a solidified electronic circuit through low-melting-point metal ink is proposed. A functional pen with heating capability was fabricated. Several typical thermal properties of the alloy ink Bi35In48.6Sn16Zn0.4 were measured and evaluated. Owing to the specifically selected melting point of the ink, which is slightly higher than room temperature, various electronic devices, graphics or circuits can be manufactured in a short period of time and then rapidly solidified by cooling in the surrounding air. The liquid-solid phase change mechanism of the written lines was experimentally characterized using a scanning electron microscope. In order to determine the matching substrate, wettability between the metal ink Bi35In48.6Sn16Zn0.4 and several materials, including mica plate and silicone rubber, was investigated. The resistance-temperature curve of a printed resistor indicated its potential as a temperature control switch. Furthermore, the measured reflection coefficient of a printed double-diamond antenna accords well with the simulated result. With unique merits such as no pollution, no requirement for encapsulation and easy recycling, the present printing approach is an important supplement to current printed electronics and has enormous practical value in the future.
Analogues of singular value decomposition (SVD), QR, LU and Cholesky factorizations are presented for problems in which the usual discrete matrix is replaced by a 'quasimatrix', continuous in one dimension, or a 'cmatrix', continuous in both dimensions. Two challenges arise: the generalization of the notions of triangular structure and row and column pivoting to continuous variables (required in all cases except the SVD, and far from obvious), and the convergence of the infinite series that define the cmatrix factorizations. Our generalizations of triangularity and pivoting are based on a new notion of a 'triangular quasimatrix'. Concerning convergence of the series, we prove theorems asserting convergence provided the functions involved are sufficiently smooth.
Schematic illustrating the electrokinetic injection of a sample (concentration c * in the reservoir) into a micro-channel.  
Same as in figure 4 except that profiles of all dependent variables are shown but at a fixed time instant, vt/w = 2. The dashed lines are the concentrations in the capillary calculated using equation (4.7). The bottom panel shows the electric conductivity (s) normalized by its downstream value (s 0 ).  
We consider the problem of electromigration of a sample ion (analyte) within a uniform background electrolyte when the confining channel undergoes a sudden contraction. One example of such a situation arises in microfluidics in the electrokinetic injection of the analyte into a micro-capillary from a reservoir of much larger size. Here, the sample concentration propagates as a wave driven by the electric field. The dynamics is governed by the Nerst-Planck-Poisson system of equations for ionic transport. A reduced one-dimensional nonlinear equation, describing the evolution of the sample concentration, is derived. We integrate this equation numerically to obtain the evolution of the wave shape and determine how the injected mass depends on the sample concentration in the reservoir. It is shown that due to the nonlinear coupling of the ionic concentrations and the electric field, the concentration of the injected sample could be substantially less than the concentration of the sample in the reservoir.
An analytic continuation method for obtaining rigorous bounds on the effective complex permittivity ε * of polycrystalline composite materials is developed. It is assumed that the composite consists of many identical anisotropic crystals, each with a unique orientation. The key step in obtaining the bounds involves deriving an integral representation for ε *, which separates parameter information from geometrical information. Forward bounds are then found using knowledge of the single crystal permittivity tensor and mean crystal orientation. Inverse bounds are also developed, which recover information about the mean crystal orientation from ε *. We apply the polycrystalline bounds to sea ice, a critical component of the climate system. Different ice types, which result from different growth conditions, have different crystal orientation and size statistics. These characteristics significantly influence the fluid transport properties of sea ice, which control many geophysical and biogeochemical processes important to the climate and polar ecosystems. Using a two-scale homogenization scheme, where the single crystal tensor is numerically computed, forward bounds for sea ice are obtained and are in excellent agreement with columnar sea ice data. Furthermore, the inverse bounds are also applied to sea ice, helping to lay the groundwork for determining ice type using remote sensing techniques.
Contour method principle: (step A) a two-dimensional body possessing both tension and compressive residual stresses. (Step B) cut surface deforms due to residual stress relaxation. (Step C) applying displacement vector in the opposite direction to a half part as boundary conditions. 
Geometries: (a) the semi-infinite strip (b) the finite rectangle. (Online version in colour.)
Welded aluminium alloy 2024-T351 specimen used for a practical implementation of the analytical contour method. (Online version in colour.)
The contour method is one of the most prevalent destructive techniques for residual stress measurement. Up to now, the method has involved the use of the finite-element (FE) method to determine the residual stresses from the experimental measurements. This paper presents analytical solutions, obtained for a semi-infinite strip and a finite rectangle, which can be used to calculate the residual stresses directly from the measured data; thereby, eliminating the need for an FE approach. The technique is then used to determine the residual stresses in a variable-polarity plasma-arc welded plate and the results show good agreement with independent neutron diffraction measurements.
Interface is moving monotonically to the left. (Online version in colour.) 
Interface is moving to the right after initially moving to the left. (Online version in colour.) 
Interface moving to the right (D 1 = D 2 = 1.0, c * = 0.8). (a) Interface location and (b) interface location with fitted curves (log-log scale). (a) Dashed line, a = 1.0; solid line, a = 0.8. (b) Circles, a = 1.0; dashed line 1.1752 t 0.5024 ; squares, a = 0.8; solid line 1.984 t 0.3349. (Online version in colour.) 
Anomalous diffusion can be characterized by a mean-squared displacement 〈x(2)(t)〉 that is proportional to t(α) where α≠1. A class of one-dimensional moving boundary problems is investigated that involves one or more regions governed by anomalous diffusion, specifically subdiffusion (α<1). A novel numerical method is developed to handle the moving interface as well as the singular history kernel of subdiffusion. Two moving boundary problems are solved: the first involves a subdiffusion region to the one side of an interface and a classical diffusion region to the other. The interface will display non-monotone behaviour. The subdiffusion region will always initially advance until a given time, after which it will always recede. The second problem involves subdiffusion regions to both sides of an interface. The interface here also reverses direction after a given time, with the more subdiffusive region initially advancing and then receding.
This study focuses on the theoretical aspect of topographic scattering induced by a shallow asymmetric V-shaped canyon under plane shear horizontal-wave incidence. An analytical approach, based on the region-matching technique, is applied to derive a rigorous series solution, which is more general than that in a previous study. For the wave functions constrained in two angular directions, a novel form of Graf's addition formula is derived to arbitrarily shift the local coordinate system. Barrier geometry, angle of incidence and wave frequency are taken as the most significant parameters in exploring the topographic effects of localized concave free surfaces on ground motions. Both surface and subsurface motions are presented. Comparisons with previously published results and boundary-element solutions show good agreement. Frequency-domain results indicate that, for the high-frequency case at a low grazing angle (corresponding to the potential case in teleseismic propagation), the high levels of amplified motions occur mostly on the illuminated side of the canyon. When the windward slope is steeper, the peak amplitude values, at least 2.4 times larger than those of free-field responses, tend to increase. Time-domain simulations display how a sequence of scattered waves travel and attenuate at regional distances.
The distribution of the concentrations a k (x) and the acidity function ψ(x) at λ = 200, see (7.2) and (8.1). The fragment of approximation at 0 ≤ x ≤ 0.3. 
The distribution of the concentration at δ = 15. Comparison between the weak solution (solid line) and the numerical solution (dotted line). λ = 30, see (7.2) and (8.1). 
Dependence W(δ, ψ), R k (ψ * ; δ; ψ), R k (ψ k ; δ; ψ) on δ. 
The mathematical model describing the stationary natural pH-gradient arising under the action of an electric field in an aqueous solution of ampholytes is constructed and investigated. The model is a part of a more general model of the isoelectrofocusing process. Investigation is based on the approximation of a weak solution by the piecewise continuous non-smooth functions. The method can be used for solving classes of problems for ordinary differential equations with a small parameter at the highest derivatives and the turning points.
Schematic of a 16 element circular piezo-electric transducer array device. Each element in the array is excited by an individual voltage source. Matching between the transducers and the fluid is achieved using a quarter-wavelength matching layer and the attenuating backing is used to absorb the waves reflected from the back face of the transducers. (Online version in colour.) 
A J 1 Bessel trap generated by a 60 element point-source array with R = 10l. In two of the cases shown, the trap centre distance from the centre of the array is R T < 1/2l(N /pe) ∼ 3.5l and in the other two cases, this condition is violated. 
Force potential U for a polystyrene bead with six traps generated by (b) a 60 element point-source array and (a) a continuous boundary. Potential landscapes with different average trap spacing d are shown. 
FE-based calculation of a force potential U in a 16 element device. (a) Single device excited at the bottom of the array. (b) All the elements excited to generate a trap at R T ∼ 0.937l distance from the centre. 
A feasibility study of a circular ultrasonic array device for acoustic particle manipulation is presented. A general approach based on Green's function is developed to analyse the underlying properties of a circular acoustic array. It allows the size of a controllable device area as a function of the number of array elements to be established and the array excitation required to produce a desired field distribution to be determined. A set of quantitative parameters characterizing the complexity of the pressure landscape is suggested, and relation to the number of array elements is found. Next, a finite-element model of a physically realizable circular piezo-acoustic array device is employed to demonstrate that the trapping capability can be achieved in practice.
Origin of macroscopic stresses in a periodic array of dislocation dipoles. 
Periodic array (string) of dislocation dipoles. Two types of volumes are considered, both spanning exactly one period along the direction of periodicity: (i) V 0 which is free of defects and (ii) V d which contains the dislocation dipole. 
Circular summation scheme. 
Convergence of the average stress ¯ σ in the primary cell to the macroscopic stress Σ determined from the Eshelby solution according to (5.3). The dipoles are characterized by the Burgers vector b = ( √ 3b/2, b/2, 0), the slip-plane angle φ = π/6, and distance d = L/6. The Poisson's ratio of the material is ν = 1 3 . 
It is known that in two-dimensional periodic arrays of dislocations the summation of the periodic image fields is conditionally convergent. This is due to the long-range character of the elastic fields of dislocations. As a result, the stress field obtained for a doubly periodic array of dislocation dipoles may contain a spurious constant stress that depends on the adopted summation scheme. In the present work, we provide, based on micromechanical considerations, a simple physical explanation of the origin of the conditional convergence of lattice sums of image interactions. In this context, the spurious stresses are found in a closed form for an arbitrary elastic anisotropy, and this is achieved without using the stress field of an individual dislocation. An alternative procedure is also developed where the macroscopic spurious stresses are determined using the solution of the Eshelby's inclusion problem.
Inspiration from nature: Bacterial swimming using flagella. Electron microscope image of a salmonella bacterium with several flagella forming a bundle of chiral shape (Source:, where each flagellum is actuated by a base bacterial motor as shown in the inset (Credit: adapted from (File:Flagellum_base_diagram_en.svg)). (Online version in colour.) 
Partially magnetic flexible polymer film where the dark grey areas represent the magnetic portions of the film (L 0 ) and the arrows indicate the direction of the respective magnetization M (see (a)). The chiral shape is formed by a static external magnetic field and the swimming is achieved by subsequently rotating the magnetic field with a rotation axis along the x-axis (see (b)). An animation of bi-directional swimming with a magnetically induced initial chiral shape is included as electronic supplementary material. (See electronic supplementary material at for the animation of (1) unidirectional swimming with a viscosity-induced chiral shape and (2) bi-directional swimming with a magnetically induced chiral shape.) (Online version in colour.) 
Illustration of the twist angle (θ ) variation along the length of a chiral micro-swimmer. The micro-swimmer has a steady-state swimming velocity of magnitude U and is subjected to an external torque that has an angular frequency of ω.
Influence of nonlinearity exponent (n) in the twist angle variation, θ = θ max (x/L) n , on the swimming velocity, and U normalized = U/U n=1. The swimming velocity is maximum when n ≈ 1. The illustrated cross (×) on the micro-swimmer's image represents the location where θ = θ max /2. 
Biomimetic micro-swimmers can be used for various medical applications, such as targeted drug delivery and micro-object (e.g. biological cells) manipulation, in lab-on-a-chip devices. Bacteria swim using a bundle of flagella (flexible hair-like structures) that form a rotating cork-screw of chiral shape. To mimic bacterial swimming, we employ a computational approach to design a bacterial (chirality-induced) swimmer whose chiral shape and rotational velocity can be controlled by an external magnetic field. In our model, we numerically solve the coupled governing equations that describe the system dynamics (i.e. solid mechanics, fluid dynamics and magnetostatics). We explore the swimming response as a function of the characteristic dimensionless parameters and put special emphasis on controlling the swimming direction. Our results provide fundamental physical insight on the chirality-induced propulsion, and it provides guidelines for the design of magnetic bi-directional micro-swimmers.
Transmission curves (real(t) versus normalized frequency) of the multi-layered stack (solid line) and its effective medium at the seventh-order (dotted-dashed line) and 19th-order (dashed line) approximation. (Online version in colour.) 
We investigate a high-order homogenization (HOH) algorithm for periodic multi-layered stacks. The mathematical tool of choice is a transfer matrix method. Expressions for effective permeability, permittivity and magnetoelectric coupling are explored by frequency power expansions. On the physical side, this HOH uncovers a magnetoelectric coupling effect (odd-order approximation) and artificial magnetism (even-order approximation) in moderate contrast photonic crystals. Comparing the effective parameters' expressions of a stack with three layers against that of a stack with two layers, we note that the magnetoelectric coupling effect vanishes while the artificial magnetism can still be achieved in a centre-symmetric periodic structure. Furthermore, we numerically check the effective parameters through the dispersion law and transmission property of a stack with two dielectric layers against that of an effective bianisotropic medium: they are in good agreement throughout the low-frequency (acoustic) band until the first stop band, where the analyticity of the logarithm function of the transfer matrix ([Formula: see text]) breaks down.
Rescaled square of the terminal velocity of a self-assembling cylinder as a function of N −2. As described in the text, the squared velocity scales linearly with N −2. The dotted line is the theoretical prediction (4.8) with the prefactor α modified by the efficiency E (see the main text). The symbols used to encode different sphere sizes and magnetizations are as described in the legend. (Online version in colour.) 
Spherical neodymium-iron-boron magnets are permanent magnets that can be assembled into a variety of structures owing to their high magnetic strength. A one-dimensional chain of these magnets responds to mechanical loadings in a manner reminiscent of an elastic rod. We investigate the macroscopic mechanical properties of assemblies of ferromagnetic spheres by considering chains, rings and chiral cylinders of magnets. Based on energy estimates and simple experiments, we introduce an effective magnetic bending stiffness for a chain of magnets and show that, used in conjunction with classic results for elastic rods, it provides excellent estimates for the buckling and vibration dynamics of magnetic chains. We then use this estimate to understand the dynamic self-assembly of a cylinder from an initially straight chain of magnets.
This paper discusses the science-policy interface, emphasizing the role of evidence and scientific assessments. It then presents the key findings from the UK National Ecosystem Assessment (NEA), which provided much of the evidence for the Natural Environment White Paper for England as a case study. It also influenced the development of the biodiversity strategy for England. The NEA demonstrates the importance of a multi-disciplinary team of experts to prepare and peer review assessments and the importance of input from funding agencies and relevant stakeholder groups in co-designing and reviewing. Much of the text and all of the figures in the NEA section are taken from the Synthesis Report of the NEA, which I drafted as co-chair of the NEA.().
We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations, we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking.
Three-dimensional sketch of bifurcation surface.
(Colour online) Cross-sections of the bifurcation surface at constant positive values of the bifurcation parameter μ (shown in top left of panels). The black regions correspond to the OS, while the light and grey regions (green and and blue online) correspond to 1 or 2 stable TWs, respectively. The outer circles represent the cuts through the sphere μ 2 + h 2 2 + h 3 2 = 1 (where 
(Colour online) Walker case: temporal profiles of ϕ0(τ ) (left) and x * (τ ) (right). The dark curves (blue online) represent the explicit solutions, while the light lines (green online) show the average precession and propagation.
(Colour online) Average velocity of DW as a function of applied current, for different values of the nonadiabatic damping parameter β. Fixed parameter values: k2 = 0.1, α = 0.1.
We develop a systematic asymptotic description for domain wall motion in one-dimensional magnetic nanowires under the influence of small applied magnetic fields and currents and small material anisotropy. The magnetization dynamics, as governed by the Landau-Lifshitz-Gilbert equation, is investigated via a perturbation expansion. We compute leading-order behaviour, propagation velocities and first-order corrections of both travelling waves and oscillatory solutions, and find bifurcations between these two types of solutions. This treatment provides a sound mathematical foundation for numerous results in the literature obtained through more ad hoc arguments.
Plots of 
The dispersion branches, for the antisymmetric mode, ( Ω = κ as dotted line) are shown in ( a ) for cylinders with radii of 0.95 (solid line) and 0.99 (bold solid line). The asymptotics are shown as dashed lines. In ( b ), the antisymmetric eigensolution U 0 is shown for r = 0.95. (Online version in colour.) 
An effective surface equation, that encapsulates the detail of a microstructure, is developed to model microstructured surfaces. The equations deduced accurately reproduce a key feature of surface wave phenomena, created by periodic geometry, that are commonly called Rayleigh-Bloch waves, but which also go under other names, for example, spoof surface plasmon polaritons in photonics. Several illustrative examples are considered and it is shown that the theory extends to similar waves that propagate along gratings. Line source excitation is considered, and an implicit long-scale wavelength is identified and compared with full numerical simulations. We also investigate non-periodic situations where a long-scale geometrical variation in the structure is introduced and show that localized defect states emerge which the asymptotic theory explains.
We consider the initial-boundary-value problem for the cubic nonlinear Schrödinger equation, formulated on a half-line with inhomogeneous Robin boundary data. We study traditionally important problems of the theory of nonlinear partial differential equations, such as the global-in-time existence of solutions to the initial-boundary-value problem and the asymptotic behaviour of solutions for large time.
Metamaterial and photonic crystal structures are central to modern optics and are typically created from multiple elementary repeating cells. We demonstrate how one replaces such structures asymptotically by a continuum, and therefore by a set of equations, that captures the behaviour of potentially high-frequency waves propagating through a periodic medium. The high-frequency homogenization that we use recovers the classical homogenization coefficients in the low-frequency long-wavelength limit. The theory is specifically developed in electromagnetics for two-dimensional square lattices where every cell contains an arbitrary hole with Neumann boundary conditions at its surface and implemented numerically for cylinders and split-ring resonators. Illustrative numerical examples include lensing via all-angle negative refraction, as well as omni-directive antenna, endoscope and cloaking effects. We also highlight the importance of choosing the correct Brillouin zone and the potential of missing interesting physical effects depending upon the path chosen.
The magnitude of the atomic-level displacement discrepancy between CAC and MD simulation results of the 5 (210) model with different d (the smallest distance between the atomic/continuum interface and the GB: (a) 0.0 Å; (b) 7.0 Å; (c) 8.73 Å; (d) 12.22 Å; (e) 26.2 Å. (Online version in colour.)  
Averaged tensile stress–strain curves for the four different GB models. (Online version in colour.)  
(a–d) Snapshots of CAC results showing the effect of the crack–GB interaction at the applied strain of 4.52%. (Online version in colour.)  
Snapshots of strain sequences of the 5 (310) model. The marked structure units showing the effect of the crack–GB interaction strain: (a) 4.16%; (b) 4.21%; (c) 4.29%; (d) 4.44%. (Online version in colour.)  
In this paper, we present the development of a concurrent atomistic-continuum (CAC) methodology for simulation of the grain boundary (GB) structures and their interaction with other defects in ionic materials. Simulation results show that the CAC simulation allows a smooth passage of cracks through the atomistic-continuum interface without the need for additional constitutive rules or special numerical treatment; both the atomic-scale structures and the energies of the four different [001] tilt GBs in bi-crystal strontium titanate obtained by CAC compare well with those obtained by existing experiments and density function theory calculations. Although 98.4% of the degrees of freedom of the simulated atomistic system have been eliminated in a coarsely meshed finite-element region, the CAC results, including the stress-strain responses, the GB-crack interaction mechanisms and the effect of the interaction on the fracture strength, are comparable with that of all-atom molecular dynamics simulation results. In addition, CAC simulation results show that the GB-crack interaction has a significant effect on the fracture behaviour of bi-crystal strontium titanate; not only the misorientation angle but also the atomic-level details of the GB structure influence the effect of the GB on impeding crack propagation.
Resonance-free region in theorem 2.1 (grey-shaded), approximate resonance strings of Rittby et al. (circles) and Abramov et al. (squares and star). (Online version in colour.) 
Resonance approximations computed by Rittby et al. (circles), Korsch et al. (asterisks), Abramov et al. (squares and star) and analytic bound from theorem 2.1. (Online version in colour.) 
Rotated box of side length l k /2 excluding approximate resonance ± k , k = 16 + ,. .. , 28 + , from Korsch et al. marked by asterisk. (Online version in colour.) 
Excluded approximate resonances of Korsch et al. (asterisks) surrounded by respective excluding boxes, together with approximate resonances of Rittby et al. (circles), Abramov et al. (star and square) and analytic exclosure (grey-shaded) in the λ-plane. (a) Approximate resonances 0 + ,. .. , 38 + of Rittby et al., 0 + ,. .. , 29 − of Korsch et al. and excluding boxes for approximate resonances 16 + ,. .. , 28 + by Korsch et al. (b) Zoom into region marked by dashed line in (a) showing the approximate resonances 16 + , 17 − , 18 + with respective excluding boxes. (Online version in colour.) 
In this paper, we confirm, with absolute certainty, a conjecture on a certain oscillatory behaviour of higher auto-ionizing resonances of atoms and molecules beyond a threshold. These results not only definitely settle a more than 30 year old controversy in Rittby et al. (1981 Phys. Rev. A 24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) and Korsch et al. (1982 Phys. Rev. A 26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)), but also provide new and reliable information on the threshold. Our interval-arithmetic-based method allows one, for the first time, to enclose and to exclude resonances with guaranteed certainty. The efficiency of our approach is demonstrated by the fact that we are able to show that the approximations in Rittby et al. (1981 Phys. Rev. A 24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) do lie near true resonances, whereas the approximations of higher resonances in Korsch et al. (1982 Phys. Rev. A 26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)) do not, and further that there exist two new pairs of resonances as suggested in Abramov et al. (2001 J. Phys. A 34, 57–72 (doi:10.1088/0305-4470/34/1/304)).
For a sufficiently slender axially symmetric body placed in a uniform stream, only convectively unstable modes are found in previous experiments. This work imposes theoretically and computationally a pair of most unstable helical modes, symmetrically and asymmetrically. The Reynolds stress modification of the developing laminar mean wake flow is modified into an elliptic-like cross section for symmetrical forcing; the consequences of unequal upstream amplitudes are also explored. Energy-transfer mechanisms between the mean flow and the relevant dominant modes and between the modes through 'triad interactions' are studied. The results from dynamical considerations provide the physical understanding of the generation of a standing wave mode at twice the azimuthal wavenumber; it is necessary that the wave envelopes of participating modes, including that of the mean flow, overlap in their spatial development, which is a necessary supplement to kinematical conditions for such interactions to take place effectively. Standing wave motions, which are otherwise only found naturally in wakes behind blunt-trailing-edge axisymmetric bodies, can be rendered present through appropriate forcing and nonlinear interactions behind very slender axisymmetric bodies.
The analysis of soil water partitioning in seasonally dry climates necessarily requires careful consideration of the periodic climatic forcing at the intra-annual timescale in addition to daily scale variabilities. Here, we introduce three new extensions to a stochastic soil moisture model which yields seasonal evolution of soil moisture and relevant hydrological fluxes. These approximations allow seasonal climatic forcings (e.g. rainfall and potential evapotranspiration) to be fully resolved, extending the analysis of soil water partitioning to account explicitly for the seasonal amplitude and the phase difference between the climatic forcings. The results provide accurate descriptions of probabilistic soil moisture dynamics under seasonal climates without requiring extensive numerical simulations. We also find that the transfer of soil moisture between the wet to the dry season is responsible for hysteresis in the hydrological response, showing asymmetrical trajectories in the mean soil moisture and in the transient Budyko's curves during the 'dry-down' versus the 'rewetting' phases of the year. Furthermore, in some dry climates where rainfall and potential evapotranspiration are in-phase, annual evapotranspiration can be shown to increase because of inter-seasonal soil moisture transfer, highlighting the importance of soil water storage in the seasonal context.
A hybrid beam with a gel layer bonded on the top of an elastic non-swellable substrate has been commonly adopted to make various sensors and actuators. Usually, different models need to be developed for the hybrid beam when different gels are used in the system. In this article, based on the generalized ideal elastomeric gel model, we formulate a unified relationship between the swelling of hydrogels and the bending curvature of the elastic beam, which is independent of specific swelling mechanisms of gels. We further illustrate that the equations derived in the article can be used to validate the ideal elastomeric gel model and measure the elasticity of polymer networks of the gels.
Solid lines represent the two-dimensional view of the top, middle and bottom surfaces for α = 0.03 (a) and α = 0.3 (b), β = 0.05, and h/b = 2. A wrinkling pattern is clearly evidenced, involving only the top beam, at α = 0.03. (Online version in colour.) 
We describe swelling-driven curving in originally straight and non-homogeneous beams. We present and verify a structural model of swollen beams, based on a new point of view adopted to describe swelling-induced deformation processes in bilayered gel beams, that is based on the split of the swelling-induced deformation of the beam at equilibrium into two components, both depending on the elastic properties of the gel. The method allows us to: (i) determine beam stretching and curving, once assigned the characteristics of the solvent bath and of the non-homogeneous beam, and (ii) estimate the characteristics of non-homogeneous flat gel beams in such a way as to obtain, under free-swelling conditions, three-dimensional shapes. The study was pursued by means of analytical, semi-analytical and numerical tools; excellent agreement of the outcomes of the different techniques was found, thus confirming the strength of the method.
In this paper, we study flexural vibrations of two thin beams that are coupled through an otherwise quiescent viscous fluid. While most of the research has focused on isolated beams immersed in placid fluids, inertial and viscous hydrodynamic coupling is ubiquitous across a multitude of engineering and natural systems comprising arrays of flexible structures. In these cases, the distributed hydrodynamic loading experienced by each oscillating structure is not only related to its absolute motion but is also influenced by its relative motion with respect to the neighbouring structures. Here, we focus on linear vibrations of two identical beams for low Knudsen, Keulegan-Carpenter and squeeze numbers. Thus, we describe the fluid flow using unsteady Stokes hydrodynamics and we propose a boundary integral formulation to compute pertinent hydrodynamic functions to study the fluid effect. We validate the proposed theoretical approach through experiments on centimetre-size compliant cantilevers that are subjected to underwater base-excitation. We consider different geometric arrangements, beam interdistances and excitation frequencies to ascertain the model accuracy in terms of the relevant non-dimensional parameters.
In this paper, we examine the folding behaviour of Tachi-Miura polyhedron (TMP) bellows made of paper, which is known as a rigid-foldable structure, and construct a theoretical model to predict the mechanical energy associated with the compression of TMP bellows, which is compared with the experimentally measured energy, resulting in the gap between the mechanical work by the compression force and the bending energy distributed along all the crease lines. The extended Hamilton's principle is applied to explain the gap which is considered to be energy dissipation in the mechanical behaviour of TMP bellows.
Understanding how humans walk on a surface that can move might provide insights into, for instance, whether walking humans prioritize energy use or stability. Here, motivated by the famous human-driven oscillations observed in the London Millennium Bridge, we introduce a minimal mathematical model of a biped, walking on a platform (bridge or treadmill) capable of lateral movement. This biped model consists of a point-mass upper body with legs that can exert force and perform mechanical work on the upper body. Using numerical optimization, we obtain energy-optimal walking motions for this biped, deriving the periodic body and platform motions that minimize a simple metabolic energy cost. When the platform has an externally imposed sinusoidal displacement of appropriate frequency and amplitude, we predict that body motion entrained to platform motion consumes less energy than walking on a fixed surface. When the platform has finite inertia, a mass- spring-damper with similar parameters to the Millennium Bridge, we show that the optimal biped walking motion sustains a large lateral platform oscillation when sufficiently many people walk on the bridge. Here, the biped model reduces walking metabolic cost by storing and recovering energy from the platform, demonstrating energy benefits for two features observed for walking on the Millennium Bridge: crowd synchrony and large lateral oscillations.
The in-plane behaviour of tetrachiral lattices should be characterized by bi-dimensional orthotropic material owing to the existence of two orthogonal axes of rotational symmetry. Moreover, the constitutive model must also represent the chirality inherent in the lattices. To this end, a bi-dimensional orthotropic chiral micropolar model is developed based on the theory of irreducible orthogonal tensor decomposition. The obtained constitutive tensors display a hierarchy structure depending on the symmetry of the underlying microstructure. Eight additional material constants, in addition to five for the hemitropic case, are introduced to characterize the anisotropy under Z 2 invariance. The developed continuum model is then applied to a tetrachiral lattice, and the material constants of the continuum model are analytically derived by a homogenization process. By comparing with numerical simulations for the discrete lattice, it is found that the proposed continuum model can correctly characterize the static and wave properties of the tetrachiral lattice.
Kinematic solution for motions of the NLG. The projections show how overcentre angle Θ ov varies as a function of (a) shock strut angle θ 1 and (b) locklink angle θ 4. The light grey part of the curve indicates solutions beyond the stops. Eight NLG positions are indicated in panels (c1)-(c8). Rotational joints are denoted by black dots, and the triangles indicate the landing gear attachment points. Panels (c1) and (c6) show the deployed and retracted landing gear, respectively.
Actuator force schedules during a retraction cycle as a function of normalized time.  
This paper discusses the insights that a bifurcation analysis can provide when designing mechanisms. A model, in the form of a set of coupled steady-state equations, can be derived to describe the mechanism. Solutions to this model can be traced through the mechanism's state versus parameter space via numerical continuation, under the simultaneous variation of one or more parameters. With this approach, crucial features in the response surface, such as bifurcation points, can be identified. By numerically continuing these points in the appropriate parameter space, the resulting bifurcation diagram can be used to guide parameter selection and optimization. In this paper, we demonstrate the potential of this technique by considering an aircraft nose landing gear, with a novel locking strategy that uses a combined uplock/downlock mechanism. The landing gear is locked when in the retracted or deployed states. Transitions between these locked states and the unlocked state (where the landing gear is a mechanism) are shown to depend upon the positions of two fold point bifurcations. By performing a two-parameter continuation, the critical points are traced to identify operational boundaries. Following the variation of the fold points through parameter space, a minimum spring stiffness is identified that enables the landing gear to be locked in the retracted state. The bifurcation analysis also shows that the unlocking of a retracted landing gear should use an unlock force measure, rather than a position indicator, to de-couple the effects of the retraction and locking actuators. Overall, the study demonstrates that bifurcation analysis can enhance the understanding of the influence of design choices over a wide operating range where nonlinearity is significant.
We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol's equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.
Compression of a stiff film on a soft substrate may lead to surface wrinkling when the compressive strain reaches a critical value. Further compression may cause a wrinkling-folding transition, and the sinusoidal wrinkling mode can then give way to a period-doubling bifurcation. The onset of the primary bifurcation has been well understood, but a quantitative understanding of the secondary bifurcation remains elusive. Our theoretical analysis of the branching of surface patterns reveals that the wrinkling-folding transition depends on the wrinkling strain and the prestrain in the substrate. A characteristic strain in the substrate is adopted to determine the correlation among the critical strain of the period-doubling mode, the wrinkling strain and the prestrain in an explicit form. A careful examination of the total potential energy of the system reveals that beyond the critical strain of period-doubling, the sinusoidal wrinkling mode has a higher potential energy in comparison with the period-doubling mode. The critical strain of the period-doubling mode strongly depends on the deformation state of the hyperelastic solid, indicating that the nonlinear deformation behaviour of the substrate plays a key role here. The results reported here on the one hand provide a quantitative understanding of the wrinkling-folding transition observed in natural and synthetic material systems and on the other hand pave the way to control the wrinkling mode transition by regulating the strain state in the substrate.
Top-cited authors
Kevin Kendall
  • University of Birmingham
Mike Ashby
  • University of Cambridge
John Frederick Nye
  • University of Bristol
Rainer K Sachs
  • University of California, Berkeley
R. C. Mcphedran
  • The University of Sydney