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Proceedings of the Royal Society A

Published by The Royal Society
Online ISSN: 1471-2946
Discipline: mathematics, physics, engineering, materials science, computer science, earth science, chemistry, astronomy
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Proceedings A publishes articles across the chemical, computational, Earth, engineering, mathematical, and physical sciences. The articles published are high-quality, original, fundamental articles of interest to a wide range of scientists, and often have long citation half-lives. As well as established disciplines, we encourage emerging and interdisciplinary areas.

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Journal Diversity Statement: The Royal Society’s journals aim to foster inclusive science and scholarship that reflects the disciplinary, geographic and human diversity of the community. Submissions are encouraged and welcomed from all authors, regardless of their characteristics, protected or otherwise. We are committed to equal opportunity and work diligently to mitigate bias in our editorial review processes. We continually work toward identifying and implementing good practices for scientific publishing. (Endorsed by Editor-in-Chief Jane Hillston)

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Recent publications
This paper investigates the mechanical behavior of a bi-layered panel containing many particles in one layer and demonstrates the size effect of particles on the deflection. The inclusion-based boundary element method (iBEM) considers a fully bounded bi-material system. The fundamental solution for two jointed half spaces has been used to acquire elastic fields resulting from source fields over inclusions and boundary-avoiding multi-domain integral along the interface. Eshelby's equivalent inclusion method (EIM) is used to simulate the material mismatch with a continuously distributed eigenstrain field over the equivalent inclusion. The eigenstrain is expanded at the center of the inclusion, which provides tailorable accuracy based on the order of the polynomial of the eigenstrain. As a single-domain approach, the iBEM algorithm is particularly suitable for conducting virtual experiments of bi-layered composites with many defects or reinforcements for both local analysis and homogenization purposes. The maximum deflection of solar panel coupons is studied under uniform vertical loading merged with inhomogeneities of different material properties, dimensions, and volume fractions. The size of defects or reinforcements plays a significant role in the deflection of the panel, even with the same volume fraction, as the substrate is relatively thin.
Ensuring vertical separation is a key means of maintaining safe separation between aircraft in congested airspace. Aircraft trajectories are modelled in the presence of significant epistemic uncertainty, leading to discrepancies between observed trajectories and the predictions of deterministic models, hampering the task of planning to ensure safe separation. In this paper, a probabilistic model is presented, for the purpose of emulating the trajectories of aircraft in climb and bounding the uncertainty of the predicted trajectory. A monotonic, functional representation exploits the spatio-temporal correlations in the radar observations. Through the use of Gaussian process emulators, features that parameterize the climb are mapped directly to functional outputs, providing a fast approximation, while ensuring that the resulting trajectory is monotonic. The model was applied as a probabilistic digital twin for aircraft in climb and baselined against the base of aircraft data, a deterministic model widely used in industry. When applied to an unseen test dataset, the probabilistic model was found to provide a mean prediction that was 20.56% more accurate, as measured by the mean absolute error, with data-driven credible intervals that were9.54% sharper.
We report experimental results from three successful runs of a Bullard-type homopolar disc dynamo. The set-up consisted of a copper disc with radius of 30 cm and thickness of 3 cm which was placed co-axially beneath a flat, multi-arm spiral coil of the same size and connected to it electrically at the centre and along the circumference by sliding liquid-metal contacts. Magnetic field was measured using Hall probes which were fixed on the top face of the coil. We measured also the radial voltage drop across the coil. When the disc rotation rate reached Ω ≈ 7 Hz, the magnetic field increased steeply approaching B0 ≈ 40 mT in the central part of the coil. This field was more than two orders of magnitude stronger than the background magnetic field. In the first two runs, the electromagnetic torque braking the disc in the dynamo regime exceeded the breakdown torque of the electric motor driving the disc. As a result, the motor stalled and the dynamo was interrupted. Stalling did not occur in the third run when the driving frequency was set higher and increased faster. We also propose an extended disc dynamo model which qualitatively reproduces the experimental results by taking into account the background magnetic, transient eddy currents in the disc as well as the non-linearity of the torque produced by the electric motor.
Ferroelectric interfaced negative capacitance field effect transistors are gaining popularity for low power applications; however, as temperature is a constant influencing factor, further study is required to comprehend how these devices are influenced. Through a proposed compact model, this paper analytically investigates the influence of temperature on a ferroelectric interfaced negative capacitance double gate junctionless accumulation mode field effect transistor. This device integrates the benefits of negative capacitance with the junctionless accumulation mode structure. An extensive comparison of the proposed device is made with the existing structure to evaluate the benefits offered by the ferroelectric layer at different temperatures. The Landau–Khalatnikov equation and Pao–Sah integral are employed to obtain the surface potential and drain current model with temperature variation. Various key parameters of the device have been analysed extensively by varying the temperature from 200 to 500 K. It has been found that internal voltage amplification declines as temperature rises, but the sub-threshold swing increases from 46 to 72 mV decade ⁻¹ with an increase in temperature. Additionally, with a progressive rise in temperature, the loss of gain and degradation of gate capacitance are observed.
We analyse the linear stability of a reactive plane Poiseuille flow, where a reactant fluid A overlies another reactant B in a layered fashion within a two-dimensional channel. Both reactants are miscible and have the same viscosity, while upon reaction, they produce either a less or more-viscous product fluid C . The reaction kinetics is of simple A + B → C type, and the production of C occurs across the initial contact line of reactants A and B in a mixed zone of small and finite width. All three fluids have the same density. We demonstrate the effects of various controlling parameters such as the log-mobility ratio, Damköhler number, Schmidt number, Reynolds number, position and thicknesses of the reactive zone on the stability characteristics. We show that a tiny viscosity stratification by the reaction destabilizes the flow at a moderate (10–1000) and even at low Reynolds numbers (0.01–1). The maximum growth occurs for shorter waves than for the Tollmien–Schlichting eigenmode, and the ranges of unstable wavenumbers are wider than that known for non-reactive channel flow systems. In most cases, the instability occurs due to the overlap of the critical layer with the viscosity-stratified layer. Surprisingly for some parameters, it is observed that the reaction can make σ M decrease with increasing Reynolds number.
This work introduces a data-driven control approach for stabilizing high-dimensional dynamical systems from scarce data. The proposed context-aware controller inference approach is based on the observation that controllers need to act locally only on the unstable dynamics to stabilize systems. This means it is sufficient to learn the unstable dynamics alone, which are typically confined to much lower dimensional spaces than the high-dimensional state spaces of all system dynamics and thus few data samples are sufficient to identify them. Numerical experiments demonstrate that context-aware controller inference learns stabilizing controllers from orders of magnitude fewer data samples than traditional data-driven control techniques and variants of reinforcement learning. The experiments further show that the low data requirements of context-aware controller inference are especially beneficial in data-scarce engineering problems with complex physics, for which learning complete system dynamics is often intractable in terms of data and training costs.
We present a computational study on the solid–solid phase transition of a model two-dimensional system between hexagonal and square phases under pressure. The atomistic mechanism of phase nucleation and propagation are determined using solid-state Dimer and nudged elastic band (NEB) methods. The Dimer is applied to identify the saddle configurations and NEB is applied to generate the transition minimum energy path (MEP) using the outputs of Dimer. Both the atomic and cell degrees of freedom are used in saddle search, allowing us to capture the critical nuclei with relatively small supercells. It is found that the phase nucleation in the model material is triggered by the localized shear deformation that comes from the relative shift between two adjacent atomic layers. In addition to the conventional layer-by-layer phase propagation, an interesting defect-assisted low barrier propagation path is identified in the hexagonal to square phase transition. The study demonstrates the significance of using the Dimer method in exploring unknown transition paths without a priori assumption. The results of this study also shed light on phase transition mechanisms of other solid-state and colloidal systems.
Mechanisms of drag reduction through shape reconfiguration have been extensively studied on model geometries of plates and beams that deform primarily in bending. Adding an origami crease pattern to such plates produces a distinct class of deformation modes, with large shape changes along selected degrees of freedom. Here, we investigate the impact of those creases on reconfiguration processes and on drag, focusing on the waterbomb base as a generic case. When placed in a uniform airflow, this origami unit folds into a compact structure, whose frontal area collapses with increasing flow velocity. It enhances drag reduction to the point that fluid loading eventually ceases to increase with flow speed, reaching an upper limit. We further show that this limit is adjustable through the origami structural parameters: the stiffness and rest angle of the folds, and their pattern. Experimental results, corroborated by a fluid–elastic theoretical model, point to a scenario consistent with the previous literature: reconfiguration is governed by a dimensionless Cauchy number that measures the competition between fluid loading and elastic resistance to deformation, here embodied in creases. This foldable system yet stands out through the rare passive drag-capping lever it provides, a valuable asset for self-protection in strong wind.
An intense explosion in dusty gas is analyzed considering the Beirut ammonium nitrate explosion of 2020 as a case study. The data available from various sources for the Beirut explosion are used to determine the energy released and the possible reduction in damage radius if there were additional dust particles in the atmosphere.The relationship derived in our previous article (Proc. R. Soc. A 476, 2020) between the blast radius and the time of arrival of the shock seems quite accurate, and here the energy released by the explosion and the mass of trinitrotoluene (TNT) equivalent based on energy considerations and blast damage are estimated for dusty gases. These estimates depend on Γ , the ratio of specific heats of the dusty gas mixture. The hydrostatic overpressure and dynamic pressure are calculated from previous results by using the data available. The variation of the three dust parameters in the atmosphere, i.e. (i) mass concentration of dust particles (kp), (ii) specific heat ratio (β), and (iii) density ratio (G), gives the estimated TNT equivalence as ≈0.1 kt to 2.5 kt up to a distance of 500 m. Enhanced decay and reduced pressure is evident for dusty gases with appropriate dust parameters.
Shaded view of the supertetrahedral solid (left) with SOD zeolite topology, which is filled with metal atoms to form the initially unrelaxed ONM (right). The ONM is obtained once the original sacrificial hard template framework is removed. (Online version in colour.)
Free energy profiles (including error bars) of a biased metal atom moving on a material surface, calculated by NPT MetaD simulations for Ni (a-d), Pt (e-h) and Au (i-l) atoms, in four types of environments: flat environment of the internal surface of CRI-ST3 ONM (subfigures a, e and i), corner environment of the internal surface of CRI-ST3 ONM (subfigures b, f and j), bulk metal surface (111) (subfigures c, g and k), and (100) surface (subfigures d, h and l). In the top of the panel there are four snapshots of each environment. We have plotted for each system (metal and environment) the radial distribution function, g(r), between the moving metal atom and the closest neighbour (5.5 Å) in an NPT MD equilibration simulation (when the moving particle is not yet biased). The colours in the g(r) are red, green and blue for Ni, Pt and Au atoms, respectively. The atom colour code for the structural snapshots is the following: in yellow the moving metallic atom, in green the defined nesttype environment, and in blue the rest of the metallic atoms of the corresponding material. For each system, the jumping free energy barriers are shown. (Online version in colour.)
Pore limiting diameters (PLDs) of Pt-based ONMs, for the 17 different topologies investigated. (Online version in colour.)
Ordered Nanoporous Metals (ONMs) form a new family of nanoporous materials composed only of pure metals. The expected impact is considerable from combining the ordered nanopore structure of MOFs, zeolites and carbon schwartzites with the robustness and electronic conductivity of metals. Little is known about their stability and structural features. Here we address these points to provide clues toward their rational synthesis, introducing an automatic atomistic design that uses model building and molecular dynamics structural relaxation, and is validated against the experimentally known ONMs. Analysing the properties of the 10 stable structures out of the 17 studied (14 of which are designed in this work) using four noble metals (Pt, Pd, Au and Ag), we have deciphered some key elements and structural descriptors that provide guidelines for the experimental synthesis of ONMS. The long-lived metastability of the stable ONMs is evidenced by the high free energy landscape, computed via Metadynamic simulations. The new ONMs permit molecular diffusion of various molecules of industrial relevance, increasing the expectation for their use in catalysis, separation, nanofiltration, batteries, fuel cells, etc. Stable low-cost ONMs are predicted using Earth-abundant Ni metal, which maintains the main features of their relative noble metal forms.
Comparison between the window functions defined in terms of the prolate spheroidal wave functions (solid red curves) and those constructed by the eigenfunctions of the Toeplitz matrix B (dashed blue curves).
The amplitude spectra of multiple windows, constructed by the discretized eigenvalue problem. (a) The amplitude spectra of the five optimal window functions. The order of the eigenvalues is denoted as p. (b) The smoothed amplitude spectrum obtained by a weighted average of the five power spectra.
Multiple windows constructed with Gaussian functions, with the (half-)bandwidth B = 20 Hz. (a) Five windows formed by a stack of Gaussian functions, p = {0, 1, . . . , 4}. (b) The weighted average spectrum of these five window functions. Averaging the spectra produced by this set of window functions produces a spectrum with an excellently flat passband.
Multiple windows constructed by Gaussian functions, with the (half-)bandwidth B = 40 Hz. (a) Five windows formed by a group of Gaussian functions. (b) The weighted average spectrum of the five window functions.
Spectral analysis with multiple windows, at a given time t. The original signal segment is x(t, τ ) (the black signal on the top) with the reference time at t = 1.2 s and the (half) length of 0.1 s. The segments formed by five window functions are y p (t, τ ) (the blue signals). The average of five time-frequency spectra (in red) results in a smooth amplitude spectrum |˜y|˜y(t, τ )|.
For a non-stationary seismic signal, time–frequency analysis methods often include a time window function that serves as a weighting function and by which the signal is multiplied to form a segment. The time window function often has the highest weighting coefficient for the central sample of the signal segment. For the rest of the segment, there is no adequate representation in the frequency spectrum. Here I propose to use multiple orthogonal window functions to properly represent the local spectral property in the time–frequency plane and recover the information lost due to time-windowing before applying the Fourier transform. First, I propose to construct multiple window functions directly using a stack of Gaussian functions. The weighted average spectrum of the multiple window functions has a flat passband, which is better than the conventional multiple windows. Taking advantage of the linearity of the Fourier transform, we can apply each window to the analytic signal to generate the instantaneous autocorrelation accordingly and form an averaged instantaneous autocorrelation by a weighted sum before performing the Fourier transform to generate the Wigner–Ville distribution (WVD). This multi-window WVD method successfully represents the local spectrum of the non-stationary seismic signal in the time–frequency plane.
In an unsteady pulsatile non-Newtonian fluid past a tube with a thin wall layer, the dispersion of a narrow uniform slug of injected solute over a cross-section is examined. At the interface between the mobile fluid phase and the immobile wall phase, both irreversible and reversible reactions have been adopted. The Carreau–Yasuda model is used to describe the fluid’s rheology. The impacts of fluid rheology and reaction parameters on the concentration profiles in the fluid- and wall-phases and the three transport coefficients, viz , the depletion coefficient ( K 0 ) , the convection coefficient ( K 1 ) , the dispersion coefficient ( K 2 ) in the fluid phase are predicted numerically. A considerable shift in the behaviour of K 1 and K 2 with a higher reaction rate may be observed in the transient stage. The axial dispersion of mobile-phase concentration in the unsteady Carreau–Yasuda II fluid model is significantly larger than in Poiseuille and steady Carreau–Yasuda II fluid models, and flow pulsatility on the immobile-phase concentration is prominent upstream at a longer time. In addition, the peak value of the mobile-phase section-mean concentration is consistently lower than in other fluid models. This study could help researchers to understand the drug delivery in blood vessels and pulmonary mechanical ventilation.
Phase portrait of the slow-fast C-R interaction for different values of the mean habitat temperature, i.e., T = 0, 10, 20, 30, and 40 • C. The fold points F(R, C) separate the stable (red) and unstable (green) parts of the critical manifold L, respectively. The dashed line corresponds to the consumer nullcline. Trajectories start on the stable part, move along the fast (double arrows) R direction and then equilibrate slowly (single arrows) at (R * 3 , C * 3 ). The small diagram on the right shows a magnification of the region of the phase portrait demonstrating the fast flow of the trajectories. Parameters: a = 217.59 (m 3 /(g year) −1 ), e = 5.99 × 10 −4 , h = 0.004 (year), m = 0.12 (year −1 ), K = 9 (g m 2 ) (Binzer et al 2012), and = 0.001.
Location of the tipping threshold (singular canard, solid red lines ) and the associated folded saddle equilibrium of the de-singularized system for: (A) α = −0.1, and (B) α = 0.1. All the parameters are same as in the Fig. 4.
Impact of varying rates of warming on the location of the tipping threshold (singular canard) and the associated F S for different rates of warming. The canard trajectories (A) (I), (II), (III), and (IV) correspond to α = −0.09, −0.1, −0.12, and −0.15, respectively, and (B) (I), (II), (III), and (IV) corresponds to α = 0.07, 0.09, 0.1 and 0.12, respectively. All other parameters are same as in the Fig. 3.
In the age of climate warming, comprehension of ecosystems' future is one of the pressing challenges to humanity. While most studies on climate warming focus on the 'magnitude of change' of the Earth's temperature, the 'rate' at which it is increasing cannot be ruled out. Rapid warming has already caused sudden ecosystem transitions at numerous biodiversity hot spots; a mechanistic understanding of such transitions is crucial. Here, we study a slow-fast consumer-resource ecosystem interacting in rapid warming scenarios. Employing geometric singular perturbation theory, we find that while a gradual change in mean temperature may accord population persistence, a critical warming rate can drive the resource's sudden collapse, termed a warming-induced abrupt transition. This further triggers the bottom-up effect, resulting in the extinction of the consumer. The difference between the optimum temperature of the resource's growth rate and the habitat temperature is crucial in deciding the critical rate of warming. Consequently, species inhabiting extreme temperature regions are more susceptible to warming-induced collapse than those within intermediate temperature ranges. We find that stochastic fluctuations in the system can advance warming-induced transitions, and the efficacy of generic early warning signals to anticipate sudden transitions is challenged.
Surface wrinkling in stiff-film/soft-substrate bilayers is a common phenomenon in biological systems and is increasingly being exploited in thin-film technology. While the onset of surface wrinkling in end-compressed bilayers is well understood, questions remain with regards to the evolution of the wrinkling pattern in the intermediate and deep post-wrinkling regimes, especially when the substrate is strongly pre-compressed. Here, we explore the bifurcation landscape of end-compressed bilayers with strongly pre-compressed substrates, using hyperelastic, plane strain finite elements and generalised path-following algorithms. After bifurcating from a flat into a sinusoidally wrinkled state, bilayers undergo further n-tupling bifurcations into stable wrinkling patterns of longer wavelength whose periodicity n = {4,. .. , 8} is a function of overall bilayer length. These five n-tupling wrinkling patterns are shown to be independent localisations of the deformation mode and are accordingly identified as stable 'building blocks' that govern the intermediate post-wrinkling regime. Additional end-shortening into the deep post-wrinkling regime then leads to further period doubling and coalescence of the building blocks. Beyond a certain length threshold, a bilayer can form a combinatorial side-by-side arrangement of the five building blocks. In the limit of an infinitely long bilayer, this leads to the phenomenon known as spatial chaos with the emergence of an infinite set of possible wrinkling patterns. In reality, though, the precise side-by-side arrangement of the building blocks is governed by the initial conditions. We show that the morphological evolution of the wrinkling pattern can be programmed by a judicial placement of manufactured dents in the thin film, creating new manufacturing capabilities for textured bilayers.
For a plane strain wedge indentation problem, detailed comparisons between slip-line field theory (SLFT) solutions, experimental observations and finite-element analysis results are carried out. Several past remaining questions on this problem are explored. The finite-element computations appear to account for the nature of a modified slip-line field solution (Petryk H. 1980 Journal de Mécanique Appliquée4, 255–282) that assumes the existence of an additional fan-shaped slip-line field just above the boundary between yielding and unyielding regions. For obtuse-angled wedges, finite-element computations show that the indentation load could be smaller for a rough wedge than for a smooth wedge. The question that was raised in the literature with respect to the reversed order of required loads for smooth and rough wedges is not necessarily a paradox. It is revealed that with increasing friction coefficient, a distinct deformation mode resembling the dead metal cap solution in the SLFT is established immediately after passing the critical friction coefficient value at which the contact mode theoretically changes from frictional sliding to perfect sticking. The profiles of the bulged-out lip material remain straight in all the finite-element computations for non-hardening materials. Nonlinearity of the lip profiles frequently observed in experiments appears in the presence of strain hardening or strain-rate sensitivity.
In quantum mechanics, the wave function predicts probabilities of possible measurement outcomes, but not which individual outcome is realized in each run of an experiment. This suggests that it describes an ensemble of states with different values of a hidden variable. Here, we analyse this idea with reference to currently known theorems and experiments. We argue that the ψ-ontic/epistemic distinction fails to properly identify ensemble interpretations and propose a more useful definition. We then show that all local ψ-ensemble interpretations which reproduce quantum mechanics violate statistical independence. Theories with this property are commonly referred to as superdeterministic or retrocausal. Finally, we explain how this interpretation helps make sense of some otherwise puzzling phenomena in quantum mechanics, such as the delayed choice experiment, the Elitzur-Vaidman bomb detector and the extended Wigner's friends scenario.
In this paper, we solve the Klein–Gordon oscillator analytically under Lorentz-violating effects defined by a tensor field subject to a Coulomb-type scalar potential. We obtain the bound-state solutions of the quantum system by choosing various electromagnetic field configurations and discuss the effects on the energy profiles and the wave function of these oscillator fields.
Histories of large-scale horizontal and vertical lithosphere motion hold important information on mantle convection. Here, we compare continent-scale hiatus maps as a proxy for mantle flow induced dynamic topography and plate motion variations in the Atlantic and Indo-Australian realms since the Upper Jurassic, finding they frequently correlate, except when plate boundary forces may play a significant role. This correlation agrees with descriptions of asthenosphere flow beneath tectonic plates in terms of Poiseuille/Couette flow, as it explicitly relates plate motion changes, induced by evolving basal shear forces, to non-isostatic vertical motion of the lithosphere. Our analysis reveals a timescale, on the order of a geological series, between the occurrence of continent-scale hiatus and plate motion changes. This is consistent with the presence of a weak upper mantle. It also shows a spatial scale for interregional hiatus, on the order of 2000–3000 km in diameter, which can be linked by fluid dynamic analysis to active upper mantle flow regions. Our results suggest future studies should pursue large-scale horizontal and vertical lithosphere motion in combination, to track the expressions of past mantle flow. Such studies would provide powerful constraints for adjoint-based geodynamic inverse models of past mantle convection.
Individual components of dislocations and disclinations on a Kirchhoff-Love shell. (Online version in colour.)
The reference, current and the natural configurations of the shell mid-surface.
The deformation w and stress fields (σ 11 , σ 22 and σ 12 ) for an isolated straight edge dislocation with the defect line along the Y = 1 line. (Online version in colour.)
Conservation of α ρ1 and α ρ2 . (Online version in colour.)
A general framework is developed to study the deformation and stress response in Föppl–von Kármán shallow shells for a given distribution of defects, such as dislocations, disclinations and interstitials, and metric anomalies, such as thermal and growth strains. The theory includes dislocations and disclinations whose defect lines can both pierce the two-dimensional surface and lie within the surface. An essential aspect of the theory is the derivation of strain incompatibility relations for stretching and bending strains with incompatibility sources in terms of the various defect and metric anomaly densities. The incompatibility relations are combined with balance laws and constitutive assumptions to obtain the inhomogeneous Föppl–von Kármán equations for shallow shells. Several boundary value problems are posed, and solved numerically, by first considering only dislocations and then disclinations coupled with growth strains.
We provide an analytical formulation to model the propagation of elastic waves in a homogeneous half-space supporting an array of thin plates. The technique provides the displacement field obtained from the interaction between an incident wave generated by a harmonic source and the scattered fields induced by the flexural motion of the plates. The scattered field generated by each plate is calculated using an ad-hoc set of Green’s functions. The interaction between the incident field and the scattered fields is modelled through a multiple scattering formulation. Owing to the introduction of the multiple scattering formalism, the proposed technique can handle a generic set of plates arbitrarily arranged on the half-space surface. The method is validated via comparison with finite element simulations considering Rayleigh waves interacting with a single and a collection of thin plates. Our framework can be used to investigate the interaction of vertically polarized surface waves and flexural resonators in different engineering contexts, from the design of novel surface acoustic wave devices to the interpretation of urban vibration problems.
(a) Shape buckling of a green algae (Acetabularia schenckii) [2]. (b) Dendrobium helix wavy morphology of orchid petals (image from (c) Edge wrinkles in a floating lotus leaf upon growth [3]. (d) A free swimming sea slug (Hexabranchus Sanguineus) (image from (e) Morphology of a bacterial biofilm (E. coli AR3110) [4]. (f ) Transverse wrinkles in a polyethylene sheet under uniaxial tension [5]. (g) Transversal waviness on adhesive strips resulting from instabilities during twisting [6]. (h) Dynamic buckling of an elastic annulus [7]. (i) Dielectric elastomer film during pull-in wrinkling [8]. (j) Epidermal sensor [9]. (k) Solar cell [10]. (l) Three-dimensional integrated electric circuit [11]. (m) Micro light-emitting diode [12]. (Online version in colour.)
(a) Inelastic behaviour of stretched thin films: a film remains flat upon first loading (left panel) but wrinkles appear upon cyclic loading (right panel). The residual strain is about ε = 0.08. (b) Maximum wrinkling amplitude |w max | versus applied strain ε, for sheets with β = 1.4. There is no wrinkling during the first loading [126]. (c) Schematic of stability boundary for a purely hyperelastic shell. (d) Deflection of the neighbouring peaks (solid curves) and the valleys (stars) versus stretching strain upon loading (red curve and star) and unloading (blue curve and star). Inset indicates the peak and valley of wrinkles on cross section (z = L/2). (e) Evolutions of deformations in soft shells in the axial loading-unloading process. The shell exhibits global bending deformation with no surface wrinkle, while wrinkles appear upon unloading due to Mullins effect. Geometric parameters: α = 800, β = 2 and κ = 0.0012 [83].
Wrinkling of thin films under tension is omnipresent in nature and modern industry, a phenomenon which has aroused considerable attention during the past two decades because of its intricate nonlinear behaviors and intriguing morphology changes. Here, we review recent advancements in the mechanics of tension-induced film wrinkling and restabilization, by identifying three major stages of its progress: small-strain (<5%) wrinkling of stiff sheets, finite-strain (up to 30%) wrinkling and restabilization (isola-center bifurcation) of soft films, and the effects of curved configurations and material properties on pattern formation. Growing demand for fundamental understanding, quantitative prediction, and precise tracking of secondary bifurcation transitions in morphology evolution of thin films helps to advance finite-strain plate/shell theories and sophisticated modeling methods. This progress not only promotes our insightful understanding of complex instability behavior, but also reveals novel phenomena and sheds light on developing wrinkle-tunable membrane structures and functional surfaces.
Collision of eigenvalues at = c for k = 1 and ξ = 0.25. (a) KP-ILW-I and (b) KP-Whitham-II.
We study transverse stability and instability of one-dimensional small-amplitude periodic travelling waves of a generalized Kadomtsev-Petviashvili equation with respect to two-dimensional perturbations, which are either periodic or square-integrable in the direction of the propagation of the underlying one-dimensional wave and periodic in the transverse direction. We obtain transverse instability results in KP-fKdV, KP-ILW and KP-Whitham equations. Moreover, assuming the spectral stability of one-dimensional wave with respect to one-dimensional square-integrable periodic perturbations, we obtain transverse stability results in the aforementioned equations.
(a) Sub-face (black) for a p-cage with P = 10 and a hole with Q = 4. (b) Structure of the flattened sub-faces hole. (c) Close-up view of face-edges contributing to a hole and depicting the angles α between the face hole edges (segmented dotted lines) and the angles γ between the face hole edges and the sub-face hole edge (black bold line). Note i = γ i + β i . (Online version in colour.)
Following the discovery of a nearly symmetric protein cage, we introduce the new mathematical concept of a near-miss polyhedral cage (p-cage) as an assembly of nearly regular polygons with holes between them. We then introduce the concept of the connectivity-invariant p-cage and show that they are related to the symmetry of uniform polyhedra. We use this relation, combined with a numerical optimization method, to characterize some classes of near-miss connectivity-invariant p-cages with a deformation below 10% and faces with up to 17 edges.
Accurate predictions of basal melt rates on ice shelves are necessary for precise projections of the future behaviour of ice sheets. The computational expense associated with completely resolving the cavity circulation using an ocean model makes this approach unfeasible for multi-century simulations, and parametrizations of melt rates are required. At present, some of the most advanced melt rate parametrizations are based on a one-dimensional approximation to the melt rate that emerges from the theory of subglacial plumes applied to ice shelves with constant basal slopes and uniform ambient ocean conditions; in this work, we present an asymptotic analysis of the corresponding equations in which non-constant basal slopes and typical ambient conditions are imposed. This analysis exploits the small aspect ratio of ice shelf bases, the relatively weak thermal driving and the relative slenderness of the region separating warm, salty water at depth and cold, fresh water at the surface in the ambient ocean. We construct an approximation to the melt rate that is based on this analysis, which shows good agreement with numerical solutions in a wide variety of cases, suggesting a path towards improved predictions of basal melt rates in ice-sheet models.
We consider a ring network of theta neurons with non-local homogeneous coupling. We analyse the corresponding continuum evolution equation, analytically describing all possible steady states and their stability. By considering a number of different parameter sets, we determine the typical bifurcation scenarios of the network, and put on a rigorous footing some previously observed numerical results.
Tie transparency, which may provide more unbiased information to deepen mutual understanding, thus builds trust and prompts cooperation in social networks. Little is known, however, about social connections’ transparency. We introduce knowable degree (KD) to characterize the transparency of a social tie, defined as the number of other entities who perceive the tie. We design a two-phase experiment to collect KDs in a network of 155 students. We find that structural property and node attribute significantly predict tie transparency. Meanwhile, we also find there almost always exist a few covert ties due to non-reciprocity. Furthermore, we focus on exploring the boundary of scopes of perception and evaluating individuals’ perceptual capability. We describe the two degrees of perception phenomenon that people can generally catch the relationships between their 2-neighbours at most. We propose a generic quantitative model to recognize high-capability perceivers, who are found more sociable and enjoy exploring the social context as well.
Using a perturbation method with the aid of MATLAB® Symbolic Toolbox, a new set of Stokes wave solutions, up to the fifth order, are derived. The new solutions are expressed in terms of free surface profile, velocity potential and wave celerity (or frequency dispersion relation). The solutions in the deep water limit are also deduced and discussed. The new solutions are compared with the existing solutions up to the fifth order. Differences appear among the existing and the present solutions, starting at the third order. The causes for the differences are identified. The characteristic differences were highlighted in the figures for high order solutions of Stokes waves. Numerical examples are provided to quantify the differences between the new fifth-order solutions and Fenton’s, and Skjelbreia and Hendrickson’s solutions.
In this paper, we show how to efficiently achieve thermal cloaking from a computational standpoint in several virtual scenarios by controlling a distribution of active heat sources. We frame this problem in the setting of PDE-constrained optimization, where the reference field is the solution of the time-dependent heat equation in the absence of the object to cloak. The optimal control problem then aims at actuating the space–time control field so that the thermal field outside the obstacle is indistinguishable from the reference field. In particular, we consider multiple scenarios where material’s thermal diffusivity, source intensity and obstacle’s temperature are allowed to vary within a user-defined range. To tackle the thermal cloaking problem in a rapid and reliable way, we rely on a parametrized reduced order model built through the reduced basis method, thus entailing huge computational speedups compared with a high-fidelity, full-order model exploiting the finite-element method while dealing both with complex target shapes and disconnected control domains.
Illustration of the geometry. In (b) a plan view showing incident wave heading with respect to the internal cylinder structure.
The instantaneous free surface elevation computed under shallow water theory for a/h = 4, d/h = 1 and D/h = 2 with β = 0 • and kh = 1.
A graphical illustration of location of imaginary roots (circles) for representative examples of h/d = 2, t = π/4
The variation of the modulus of first 10 imaginary roots, ˜ µd, as a function of t ∈ [0, π/2] for h/d = 4, Kd = 0.5. At t = 0, t = π/2 the roots coincide with the roots of dispersion relation for a fluid depth h, d (respectively).
This paper describes the solution to the problem of scattering of plane incident waves on water of constant depth by a bottom mounted circular cylinder, extending partially through the depth, which has an internal structure comprised of closely-spaced thin vertical barriers between which fluid is allowed to flow. The problem solved under full depth dependent linearised water wave theory using an effective medium equation to describe the fluid motion in cylinder and effective boundary conditions to match that flow to the fluid region outside the cylinder. The theory is compared with a shallow water approximation based on the recent work of Marangos & Porter (2021) and with an accurate computation of an exact representation of the geometry using discrete set of plates. Other results highlight the resonant directional lensing effects of this type of cylindrical plate array device.
Maximum mechanical stress and maximum WSS in function of the scaffold porosity (adapted from [53]). (Online version in colour.)
Bone injuries or defects that require invasive surgical treatment are a serious clinical issue, particularly when it comes to treatment success and effectiveness. Accordingly, bone tissue engineering (BTE) has been researching the use of computational fluid dynamics (CFD) analysis tools to assist in designing optimal scaffolds that better promote bone growth and repair. This paper aims to offer a comprehensive review of recent studies that use CFD analysis in BTE. The mechanical and fluidic properties of a given scaffold are coupled to each other via the scaffold architecture, meaning an optimization of one may negatively affect the other. For example, designs that improve scaffold permeability normally result in a decreased average wall shear stress. Linked with these findings, it appears there are very few studies in this area that state a specific application for their scaffolds and those that do are focused on in vitro bioreactor environments. Finally, this review also demonstrates a scarcity of studies that combine CFD with optimization methods to improve scaffold design. This highlights an important direction of research for the development of the next generation of BTE scaffolds.
Schematic illustration depicting the forward and inverse problem relationship for a stretched elastic plate with randomized stiffness properties. The forward finite-element model inputs (causalities) are shown as the non-homogeneous stiffness properties while the model output is the displacement field. By contrast, the inverse model aims to estimate the stiffness properties given the displacement field. (Online version in colour.)
An example of vision-based bolt loosening detection where image (a) and (b) are images of a bolted connection taken at different inspection periods. Two loosened steel bolts are shown in the blow-up figures with counterclockwise rotations in their bolt heads. Using a series of image processing techniques, the differential features caused by the bolt loosening can be identified in c). Detailed discussion can be found in [215].
Examples of EIT and piezoresistive inversion applied to self-sensing nanocomposites. (a) A soft carbon nanofibre/polyurethane (CNF/PU) is deformed by rigid, non-conductive indentors [274]. EIT is then used to image the deformation-induced conductivity changes, and piezoresistive inversion is used to recover the displacement field (multiplied by a factor of 5 for ease of visibility). (b) A hard CNF/epoxy is loaded in tension with a stress raiser at its centre [275]. EIT is again used to image the conductivity change. Lastly, piezoresistive inversion is used to recover the underlying displacement field. With knowledge of the material's elastic properties, strains and stresses can be spatially mapped. The first principal stress of the guage section is shown here along with comparison with a traditional FEM solution for validation. (Online version in colour.)
The field of structural engineering is vast, spanning areas from the design of new infrastructure to the assessment of existing infrastructure. From the onset, traditional entry-level university courses teach students to analyse structural responses given data including external forces, geometry, member sizes, restraint, etc.—characterizing a forward problem (structural causalities → structural response). Shortly thereafter, junior engineers are introduced to structural design where they aim to, for example, select an appropriate structural form for members based on design criteria, which is the inverse of what they previously learned. Similar inverse realizations also hold true in structural health monitoring and a number of structural engineering sub-fields (response → structural causalities). In this light, we aim to demonstrate that many structural engineering sub-fields may be fundamentally or partially viewed as inverse problems and thus benefit via the rich and established methodologies from the inverse problems community. To this end, we conclude that the future of inverse problems in structural engineering is inexorably linked to engineering education and machine learning developments.
The nonlinear dynamics of coupled P T -symmetric excitations and Toda-like vibrations on a one-dimensional lattice are studied analytically and elucidated graphically. The nonlinear exciton-phonon system as the whole is shown to be integrable in the Lax sense inasmuch as it admits the zero-curvature representation supported by the auxiliary linear problem of third order. Inspired by this fact, we have developed in detail the Darboux–Bäcklund integration technique appropriate to generate a higher-rank crop solution by dressing a lower-rank (supposedly known) seed solution. In the framework of this approach, we have found a rather non-trivial four-component analytical solution exhibiting the crossover between the monopole and dipole regimes in the spatial distribution of intra-site excitations. This effect is inseparable from the pronounced mutual influence between the interacting subsystems in the form of specific nonlinear superposition of two essentially distinct types of travelling waves. We have established the criterion of monopole-dipole transition based upon the interplay between the localization parameter of Toda mode and the inter-subsystem coupling parameter.
Muography uses muons naturally produced in the interactions between cosmic rays and atmosphere for imaging and characterization of density differences and time-sequential changes in solid (e.g. rocks) and liquid (e.g. melts±dissolved gases) materials in scales from tens of metres to up to a few kilometres. In addition to being useful in discovering the secrets of the pyramids, ore prospecting and surveillance of nuclear sites, muography successfully images the internal structure of volcanoes. Several field campaigns have demonstrated that muography can image density changes relating to magma ascent and descent, magma flow rate, magma degassing, the shape of the magma body, an empty conduit diameter, hydrothermal activity and major fault lines. In addition, muography is applied for long-term volcano monitoring in a few selected volcanoes around the world. We propose using muography in volcano monitoring in conjunction with other existing techniques for predicting volcanic hazards. This approach can provide an early indication of a possible future eruption and potentially the first estimate of its scale by producing direct evidence of magma ascent through its conduit in real time. Knowing these issues as early as possible buy critically important time for those responsible for the local alarm and evacuation protocols.
Time series of (a) horizontal trajectory with the best fitting trend lines and (b) pure surge oscillation of a free sheet (L = 3, m = 0.1, D = 1) under the action of cnoidal waves (λ/L = 3) of different heights H without current.
shows time series of the horizontal trajectory of a freely floating sheet, its horizontal
Time series of (a) horizontal trajectory, (b) horizontal velocity, and (c) horizontal force for a free sheet (L = 3, m = 0.1, D = 1) under the action of cnoidal waves (H = 0.2) of different lengths λ without current.
Interaction of a free sheet (L = 3, m = 0.1, D = 1) with a cnoidal wave (λ/L = 5, H = 0.2) with and without current: (a) time series of horizontal trajectory of the sheet; (b,c,d) vectors and dimensionless magnitude of fluid particles velocity at the moment of time t/T = 9.5. Vertical dashed line indicates the initial position of sheet's leading edge.
The drift motion of a freely floating deformable ice sheet in shallow water subjected to incident nonlinear waves and uniform current is studied by use of the Green–Naghdi theory for the fluid motion and the thin plate theory for an elastic sheet. The nonlinear wave- and current-induced forces are obtained by integrating the hydrodynamic pressure around the body. The oscillations and translational motion of the sheet are then determined by substituting the flow- induced forces into the equation of motion of the body. The resulting governing equations, boundary and matching conditions are solved in two dimensions with a finite difference technique. The surge and drift motions of the sheet are analysed in a broad range of body parameters and various wave-current conditions. It is demonstrated that wavelength to sheet length ratio plays an important role in the drift response of the floating sheet, while the sheet mass and rigidity have comparatively less impact. It is also observed that while the presence of the ambient current changes the drift speed significantly (almost linearly), it has little to no effect on its oscillations. However, under the same ambient current, the drift speed changes remarkably by the wave period (or wavelength).
Institutions can provide incentives to enhance cooperation in a population where this behaviour is infrequent. This process is costly, and it is thus important to optimize the overall spending. This problem can be mathematically formulated as a multi-objective optimization problem where one wishes to minimize the cost of providing incentives while ensuring a minimum level of cooperation, sustained over time. In this paper, we provide a rigorous analysis of this optimization problem, in a finite population and stochastic setting, studying both pair-wise and multi-player cooperation dilemmas. We prove the regularity of the cost functions for providing incentives over time, characterize their asymptotic limits (infinite population size, weak selection and large selection) and show exactly when reward or punishment is more cost efficient. We show that these cost functions exhibit a phase transition phenomena when the intensity of selection varies. By determining the critical threshold of this phase transition, we provide exact calculations for the optimal cost of incentive, for any given intensity of selection. Numerical simulations are also provided to demonstrate analytical observations. Overall, our analysis provides for the first time a selection-dependent calculation of the optimal cost of institutional incentives (for both reward and punishment) that guarantees a minimum level of cooperation over time. It is of crucial importance for real-world applications of institutional incentives since intensity of selection is often found to be non-extreme and specific for a given population.
Cover image of Proc. R. Soc. A, vol 477, issue 2253, September 2021 ( For more details, see the article "Nematic liquid crystalline elastomers are aeolotropic materials" by L.A. Mihai, H. Wang, J. Guilleminot & A. Goriely ( or (
The exponential and Cayley maps on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe–Kaas and generalized- α schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed-form relations along with the relevant proofs, including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized- α scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.
Continuum models describing ideal nematic solids are widely used in theoretical studies of liquid crystal elastomers. However, experiments on nematic elastomers show a type of anisotropic response that is not predicted by the ideal models. Therefore, their description requires an additional term coupling elastic and nematic responses, to account for aeolotropic effects. In order to better understand the observed elastic response of liquid crystal elastomers, we analyse theoretically and computationally different stretch and shear deformations. We then compare the elastic moduli in the infinitesimal elastic strain limit obtained from the molecular dynamics simulations with the ones derived theoretically, and show that they are better explained by including nematic order effects within the continuum framework.
We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a C^2-smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations are fixed at a portion of the boundary. Put otherwise, if the tangent vectors of the new surface are obtained locally by only rotating the given tangent vectors, and if these rotations have a rotation axis which coincides everywhere with the normal of the initial surface, then the two surfaces are equal provided they coincide at a portion of the boundary. In the language of differential geometry of surfaces, we show that any isometry which leaves normals invariant and which coincides with the given surface at a portion of the boundary is the identity mapping.
Mechanical topological insulators have enabled a myriad of unprecedented characteristics that are otherwise not conceivable in traditional periodic structures. While rich in dynamics, new developments in the domain of mechanical topological systems are hindered by their inherent inability to exhibit negative elastic or inertial couplings owing to the inevitable loss of dynamical stability. The aim of this paper is, therefore, to remedy this challenge by introducing a class of architected inertial metamaterials (AIMs) as a platform for designing mechanical lattices with novel topological and dispersion traits. We show that carefully coupling elastically supported masses via moment-free rigid linkages invokes a dynamically stable negative inertial coupling, which is essential for topological classes in need of such negative interconnection. The potential of the proposed AIMs is demonstrated via three examples: (i) a mechanical analogue of Majorana edge states, (ii) a square diatomic AIM that can sustain the quantum valley Hall effect (classically arising in hexagonal lattices), and (iii) a square tetratomic AIM with topological corner modes. We envision that the presented framework will pave the way for a plethora of robust topological mechanical systems
In this paper, we explore the mega riverbed-patterns, whose longitudinal and vertical length dimensions scale with a few channel widths and the flow depth, respectively. We perform the stability analyses from both linear and weakly nonlinear perspectives by considering a steady-uniform flow in an erodible straight channel comprising a uniform sediment size. The mathematical framework stands on the dynamic coupling between the depth-averaged flow model and the particle transport model including both bedload and suspended load via the Exner equation, which drives the pattern formation. From the linear perspective, we employ the standard linearization technique by superimposing the periodic perturbations on the undisturbed system to find the dispersion relationship. From the weakly nonlinear perspective, we apply the centre-manifold-projection technique, where the fast dynamics of stable modes is projected on the slow dynamics of weakly unstable modes to obtain the Stuart-Landau equation for the amplitude dynamics. We examine the marginal stability, growth rate and amplitude of patterns for a given quintet formed by the channel aspect ratio, wavenumber of patterns, shear Reynolds number, Shields number and relative roughness number. This study highlights the sensitivity of pattern formation to the key parameters and shows how the classical results can be reconstructed on the parameter space.
In this paper, we explore the mega riverbed-patterns, whose longitudinal and vertical length dimensions scale with a few channel widths and the flow depth, respectively. We perform the stability analyses from both linear and weakly nonlinear perspectives by considering a steady-uniform flow in an erodible straight channel comprising a uniform sediment size. The mathematical framework stands on the dynamic coupling between the depth-averaged flow model and the particle transport model including both bedload and suspended load via the Exner equation, which drives the pattern formation. From the linear perspective, we employ the standard linearization technique by superimposing the periodic perturbations on the undisturbed system to find the dispersion relationship. From the weakly nonlinear perspective, we apply the centre-manifold-projection technique, where the fast dynamics of stable modes is projected on the slow dynamics of weakly unstable modes to obtain the Stuart-Landau equation for the amplitude dynamics. We examine the marginal stability, growth rate and amplitude of patterns for a given quintet formed by the channel aspect ratio, wavenumber of patterns, shear Reynolds number, Shields number and relative roughness number. This study highlights the sensitivity of pattern formation to the key parameters and shows how the classical results can be reconstructed on the parameter space.
Comparison of the theoretical predictions of dimensionless wavelength of patterns with the experimental data. The solid line represents the line of perfect agreement. The dashed lines bound the ±20% error band. (Online version in colour.)
Comparison of the theoretical predictions of dimensionless amplitude of patterns with the experimental data. The solid line represents the line of perfect agreement. The dashed lines bound the ±20% error band. (Online version in colour.)
In this paper, we explore the mega riverbed-patterns, whose longitudinal and vertical length dimensions scale with a few channel widths and the flow depth, respectively. We perform the stability analyses from both linear and weakly nonlinear perspectives by considering a steady-uniform flow in an erodible straight channel comprising a uniform sediment size. The mathematical framework stands on the dynamic coupling between the depth-averaged flow model and the particle transport model including both bedload and suspended load via the Exner equation, which drives the pattern formation. From the linear perspective, we employ the standard linearization technique by superimposing the periodic perturbations on the undisturbed system to find the dispersion relationship. From the weakly nonlinear perspective, we apply the centre-manifold-projection technique, where the fast dynamics of stable modes is projected on the slow dynamics of weakly unstable modes to obtain the Stuart-Landau equation for the amplitude dynamics. We examine the marginal stability, growth rate and amplitude of patterns for a given quintet formed by the channel aspect ratio, wavenumber of patterns, shear Reynolds number, Shields number and relative roughness number. This study highlights the sensitivity of pattern formation to the key parameters and shows how the classical results can be reconstructed on the parameter space.
The hyperbolic paraboloid (hypar) form has been widely used in long-span roof structures and the subject of much research under out-of-plane loading. However, the behaviour of hypars under in-plane loading has been less keenly studied and there is no suitable guidance for their design in current codes of practice. A nonlinear analytical model treating the hypar as a deliberate imperfection applied to a flat plate is presented. A Rayleigh–Ritz formulation using appropriate shape functions is developed and the resulting equations are solved using numerical continuation techniques. The results are verified with nonlinear finite element models, showing good correlation across a range of thicknesses and degrees of initial curvature. Key modal contributions that influence the behaviour of the hypar are identified, providing insight into the nonlinear behaviour of hypars subject to in-plane shear. The main differences in behaviour between the flat plate and the hypar panel are shown to be most prevalent in the early stages of loading, where the influence of the initial geometry is at its greatest.
Targeted energy transfer (TET) represents the phenomenon where energy in a primary system is irreversibly transferred to a nonlinear energy sink (NES). This only occurs when the initial energy in the primary system is above a critical level. There is a natural asymmetry in the system due to the desire for the NES to be much smaller than the primary structure it is protecting. This asymmetry is also essential from an energy transfer perspective. To explore how the essential asymmetry is related to TET, this work interprets the realization of TET from a symmetry breaking perspective. This is achieved by introducing a symmetrized model with respect to the generically asymmetric original system. Firstly a classic example, which consists of a linear primary system and a nonlinearizable NES, is studied. The backbone curve topology that is necessary to realize TET is explored and it is demonstrated how this topology evolves from the symmetric case. This example is then extended to a more general case, accounting for nonlinearity in the primary system and linear stiffness in the NES. Exploring the symmetry-breaking effect on the backbone curve topologies, enables the regions in the NES parameter space that lead to TET to be identified.
In this article, we show that significant deviations from the classical quasi-steady models of droplet evaporation can arise solely due to transient effects in the gas phase. The problem of fully transient evaporation of a single droplet in an infinite atmosphere is solved in a generalized, dimensionless framework with explicitly stated assumptions. The differences between the classical quasi-steady and fully transient models are quantified for a wide range of the 10-dimensional input domain and a robust predictive tool to rapidly quantify this difference is reported. In extreme cases, the classical quasi-steady model can overpredict the droplet lifetime by 80%. This overprediction increases when the energy required to bring the droplet into equilibrium with its environment becomes small compared with the energy required to cool the space around the droplet and therefore establish the quasi-steady temperature field. In the general case, it is shown that two transient regimes emerge when a droplet is suddenly immersed into an atmosphere. Initially, the droplet vaporizes faster than classical models predict since the surrounding gas takes time to cool and to saturate with vapour. Towards the end of its life, the droplet vaporizes slower than expected since the region of cold vapour established in the early stages of evaporation remains and insulates the droplet.
The classification of four game classes on í µí°· í µí±Ÿ − í µí°· í µí±” plane (−1 ≤ í µí°· í µí±Ÿ , í µí°· í µí±” ≤ 1). The first, second, third, and fourth quadrants chronologically depict PD ( í µí°· í µí±Ÿ > 0, í µí°· í µí±” > 0 ), CH( í µí°· í µí±Ÿ < 0, í µí°· í µí±” > 0), TR (í µí°· í µí±Ÿ < 0, í µí°· í µí±” < 0), and SH (í µí°· í µí±Ÿ > 0, í µí°· í µí±” < 0).
Validation of propositions 3.1 (panel (a)) and 3.2 (panel (b)) for the case TR+PD under weak selection intensity. In this case, both TR (panel (a-ii)) and PD (panel (b-ii)) yield í µí±¥ í µí±Ží µí± í µí± * ≈ 0.5. If 1/(1 + í µí¼ƒ) > 0.5) (left panels), then í µí±¥ í µí±Ží µí± í µí± * < 1/(1 + í µí¼ƒ). Hence, according to proposition 3.1, (í µí±¥ í µí±Ží µí± í µí± * , 0) is the only possible stable point. On the other hand, if 1/(1 + í µí¼ƒ) < 0.5) (right panels), í µí±¥ í µí±Ží µí± í µí± * > 1/(1 + í µí¼ƒ) which inevitably suggests that (í µí±¥ í µí±Ží µí± í µí± * , 0) is the only possible stable point (proposition 3.2). All panels share the common payoff matrix parameters, í µí°· í µí±” 0 = −0.5, í µí°· í µí±Ÿ 0 = −0.6, í µí°· í µí±” 1 = 0.8, and í µí°· í µí±Ÿ 1 = 0.5 with í µí¼– = 0.1. The initial system state is set as (í µí±¥ 0 , í µí±› 0 ) = (0.1,0.9).
Validation of propositions 3.1 (panel (a)) and 3.2 (panel (b)) for the case TR+PD under strong selection. Panel (a-i) illustrates the variation of stability as a function of aspiration level (í µí°´íµí°´í µí° ¶ = í µí°´íµí°´í µí°· = í µí°´). If 1/(1 + í µí¼ƒ) > 0.5) (left panels) then there are two possible stable equilibria, either (1/(1 + í µí¼ƒ), í µí±› * ) (orange coloured in panel (a-i)) or (í µí±¥ í µí±Ží µí± í µí± * , 0) (blue coloured in panel (a-i)) depending upon whether í µí±¥ í µí±Ží µí± í µí± * > 1/(1 + í µí¼ƒ) or í µí±¥ í µí±Ží µí± í µí± * < 1/(1 + í µí¼ƒ) at í µí±› = 0 (panel (a-ii)). If 1/(1 + í µí¼ƒ) < 0.5) (right panels) then there are two possible stable equilibria, either (1/(1 + í µí¼ƒ), í µí±› * ) (orange coloured in panel (b-i)) or (í µí±¥ í µí±Ží µí± í µí± * , 1) (green coloured in panel (b-i)) depending upon whether í µí±¥ í µí±Ží µí± í µí± * < 1/(1 + í µí¼ƒ) or í µí±¥ í µí±Ží µí± í µí± * > 1/(1 + í µí¼ƒ) at í µí±› = 1 (panel (b-ii)). The vertical dashed lines in all panels depict the threshold aspiration levels where the stability changes. All panels share the common payoff matrix parameters, í µí°· í µí±” 0 = −0.5, í µí°· í µí±Ÿ 0 = −0.6, í µí°· í µí±” 1 = 0.8, and í µí°· í µí±Ÿ 1 = 0.5 with í µí¼– = 0.1. The initial system state is set as (í µí±¥ 0 , í µí±› 0 ) = (0.1,0.9).
Possible evolutionary outcomes under aspiration dynamics ((a) weak and (b) strong selection) corresponding to propositions 3.1 and 3.2 for the case TR+PD. Panels (a, b) show variation of stability as a function of í µí¼ƒ with two different aspiration levels. The bifurcation of the stability occurs at í µí¼ƒ = 1 (vertical dashed line). Phase planes (í µí±¥ vs. í µí±›) in panels (c) and (d) confirm the global stability of the fixed points. Panels (c) and (d) presume the aspiration level í µí°´=µí°´= 0.8. All panels share the common parameters í µí°· í µí±” 0 = −0.5, í µí°· í µí±Ÿ 0 = −0.6, í µí°· í µí±” 1 = 0.8, í µí°· í µí±Ÿ 1 = 0.5, and í µí¼– = 0.1. The initial system state for panels (a, b) is set as (í µí±¥ 0 , í µí±› 0 ) = (0.1,0.9).
Evolutionary outcomes for asymmetric aspirations (í µí°´íµí°´í µí° ¶ ≠ í µí°´íµí°´í µí°· ) corresponding to the case TR+PD under weak (upper panel) and strong (lower panel) selection. The cases for í µí°´íµí°´í µí° ¶ < í µí°´íµí°´í µí°· and í µí°´íµí°´í µí° ¶ > í µí°´íµí°´í µí°· yield the similar outcome with weak selection (panels (a, b)) which is different in the strong selection (panels (c, d)) case. Unlike panels (a, b), the threshold level (vertical dashed lines) of í µí¼ƒ does not appear along í µí¼ƒ = 1 under strong selection. For í µí°´íµí°´í µí° ¶ < í µí°´íµí°´í µí°· (í µí°´íµí°´í µí° ¶ = 0.1, í µí°´íµí°´í µí°· = 0.6), the level goes below 1 (panel (c)), while for í µí°´íµí°´í µí° ¶ > í µí°´íµí°´í µí°· (í µí°´íµí°´í µí° ¶ = 0.6, í µí°´íµí°´í µí°· = 0.1), the level goes beyond 1 (panel (d)). Possible stable points are (í µí±¥ í µí±Ží µí± í µí± * , 0) (blue region), ( 1 1+í µí¼ƒ , í µí±› * ) (orange coloured region), or (í µí±¥ í µí±Ží µí± í µí± * , 1) (green region). All panels share the common parameters í µí°· í µí±” 0 = −0.5, í µí°· í µí±Ÿ 0 = −0.6, í µí°· í µí±” 1 = 0.8, í µí°· í µí±Ÿ 1 = 0.5, and í µí¼– = 0.1. The initial system state is set as (í µí±¥ 0 , í µí±› 0 ) = (0.1,0.9).
Feedback-evolving games characterize the interplay between the evolution of strategies and environments. Rich dynamics have been derived for such games under the premise of the replicator equation, which unveils persistent oscillations between cooperation and defection. Besides replicator dynamics, here we have employed aspiration dynamics, in which individuals, instead of comparing payoffs with opposite strategies, assess their payoffs by self-evaluation to update strategies. We start with a brief review of feedback-evolving games with replicator dynamics and then comprehensively discuss such games with aspiration dynamics. Interestingly, the tenacious cycles, as perceived in replicator dynamics, cannot be observed in aspiration dynamics. Our analysis reveals that a parameter θ —which depicts the strength of cooperation in enhancing the environment—plays a pivotal role in comprehending the dynamics. In particular, with the symmetric aspiration level, if replete and depleted states, respectively, experience Prisoner's Dilemma and Trivial games, the rich environment is achievable only when θ > 1. The case θ < 1 never allows us to reach the replete state, even with a higher cooperation level. Furthermore, if cooperators aspire less than defectors, then the enhanced state can be achieved with a relatively lower θ value compared with the opposite scenario because too much expectation from cooperation can be less beneficial.
In this work, we demonstrate that three-dimensional chiral mechanical metamaterials are able to self-twist and control their global rotation. We also discuss the possibility of adjusting the extent of the global rotation manifested by the system in a programmable manner. In addition, we show that the effect of the global rotation can be observed both for small systems composed of a single structural unit as well as more complex structures incorporating several structural elements connected to each other. Finally, it is discussed that the results presented in this work are very promising from the point of view of potential applications such as satellites or telescopes in space, where appropriately designed mechanical metamaterials could be used for the attitude control as well as other systems where the control of the rotational motion is required.
Journal metrics
£1700/ $2380/ €2040
Article Processing Charges (APC)
Acceptance rate
34 days
Submission to first decision
3.479 (2021)
Journal Impact Factor™
0.658 (2021)
Immediacy Index
0.015 (2020)
4.5 (2020)
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