Mean-field Modelling Research Articles

Enhanced mean-field modelling for impact response of composite laminates incorporating strain rate-dependent matrix behaviour and 3D failure criteria

Enhanced mean-field modelling for impact response of composite laminates incorporating strain rate-dependent matrix behaviour and 3D failure criteria

Cubic elasticity of porous materials produced by additive manufacturing: experimental analyses, numerical and mean-field modelling

Although the elastic properties of porous materials depend mainly on the volume fraction of pores, the details of pore distribution within the material representative volume are also important and may be the subject of optimisation. To study their effect, experimental analyses were performed on samples made of a polymer material with a predefined distribution of spherical voids, but with various porosities due to different pore sizes. Three types of pore distribution with cubic symmetry were considered and the results of experimental analyses were confronted with mean-field estimates and numerical calculations. The mean-field ‘cluster’ model is used in which the mutual interactions between each of the two pores in the predefined volume are considered. As a result, the geometry of pore distribution is reflected in the anisotropic effective properties. The samples were produced using a 3D printing technique and tested in the regime of small strain to assess the elastic stiffness. The digital image correlation method was used to measure material response under compression. As a reference, the solid samples were also 3D printed and tested to evaluate the polymer matrix stiffness. The anisotropy of the elastic response of porous samples related to the arrangement of voids was assessed. Young’s moduli measured for the additively manufactured samples complied satisfactorily with modelling predictions for low and moderate pore sizes, while only qualitatively for larger porosities. Thus, the low-cost additive manufacturing techniques may be considered rather as preliminary tools to prototype porous materials and test mean-field approaches, while for the quantitative and detailed model validation, more accurate additive printing techniques should be considered. Research paves the way for using these computationally efficient models in optimising the microstructure of heterogeneous materials and composites.

Mean-field modelling of precipitation kinetics with a Fokker-Planck equation in the Lifshitz–Slyozov–Wagner space

Mean-field theory has been widely applied in modelling the kinetics of microstructure evolution. Considering a free energy contribution by microstructure entropy, a Fokker-Planck equation for precipitation kinetics is variationally derived from thermodynamic principles. Using dimensionless transformation into the Lifshitz-Slyozov-Wagner space, the Fokker-Planck equation is effectively solved within a fixed radius interval and an explicit Euler scheme, which is computationally advantageous. Through comparisons with the Ni-12.5 Al at. % alloy, it is shown that the Fokker-Planck equation can represent the typical stages of precipitation kinetics, i.e., nucleation, growth, and coarsening.

Spontaneous electric-polarization topology in confined ferroelectric nematics

Topological textures have fascinated people in different areas of physics and technologies. However, the observations are limited in magnetic and solid-state ferroelectric systems. Ferroelectric nematic is the first liquid-state ferroelectric that would carry many possibilities of spatially-distributed polarization fields. Contrary to traditional magnetic or crystalline systems, anisotropic liquid crystal interactions can compete with the polarization counterparts, thereby setting a challenge in understating their interplays and the resultant topologies. Here, we discover chiral polarization meron-like structures, which appear during the emergence and growth of quasi-2D ferroelectric nematic domains. The chirality can emerge spontaneously in polar textures and can be additionally biased by introducing chiral dopants. Such micrometre-scale polarization textures are the modified electric variants of the magnetic merons. Both experimental and an extended mean-field modelling reveal that the polarization strength plays a dedicated role in determining polarization topology, providing a guide for exploring diverse polar textures in strongly-polarized liquid crystals.

Influence of ordering phenomena on carbon segregation in tungsten screw dislocation

Concentration profiles on the different sites of tungsten screw dislocation are calculated as a function of the nominal carbon atom concentration, temperature and dislocation density. Rigid lattice Monte Carlo simulations and associated mean-field modelling are performed based on interaction energies from ab initio calculations. We show that the repulsive interactions induce ordering phenomena that drastically change the concentration in the core of the dislocation and leads to ordered structures among the octahedral sites. We detail the domains of short-range ordering (SRO) and long-range ordering (LRO).

Bifunctional hydroformylation on heterogeneous Rh-WOx pair site catalysts.

Metal-catalysed reactions are often hypothesized to proceed on bifunctional active sites, whereby colocalized reactive species facilitate distinct elementary steps in a catalytic cycle1-8. Bifunctional active sites have been established on homogeneous binuclear organometallic catalysts9-11. Empirical evidence exists for bifunctional active sites on supported metal catalysts, for example, at metal-oxide support interfaces2,6,7,12. However, elucidating bifunctional reaction mechanisms on supported metal catalysts is challenging due to the distribution of potential active-site structures, their dynamic reconstruction and required non-mean-field kinetic descriptions7,12,13. We overcome these limitations by synthesizing supported, atomically dispersed rhodium-tungsten oxide (Rh-WOx) pair site catalysts. The relative simplicity of the pair site structure and sufficient description by mean-field modelling enable correlation of the experimental kinetics with first principles-based microkinetic simulations. The Rh-WOx pair sites catalyse ethylene hydroformylation through a bifunctional mechanism involving Rh-assisted WOx reduction, transfer of ethylene from WOx to Rh and H2 dissociation at the Rh-WOx interface. The pair sites exhibited >95% selectivity at a product formation rate of 0.1 gpropanal cm-3 h-1 in gas-phase ethylene hydroformylation. Our results demonstrate that oxide-supported pair sites can enable bifunctional reaction mechanisms with high activity and selectivity for reactions that are performed in industry using homogeneous catalysts.

Ordering and phase separation in multi-principal-element metallic alloys: Contribution from mean-field atomic-scale modelling and simulation

Multi-principal-element alloys (MPEAs) are a wide class of materials currently at the center of numerous investigations. Key-issues with MPEAs are related with controlling the onset of (i) long-range order and (ii) phase separation, both processes being in general detrimental to the properties. While trends (i) and (ii) are strongly dependent on the precise proportions of metallic elements around the reference equiatomic MPEA, the tremendously large composition space makes it a tricky task to heuristically identify optimized MPEAs. In practice, this often leads to undesired second-phase ordered particles. This deficiency emphasizes the relevance of modelling and simulation tools designed to facilitate the exploration of MPEA composition spaces. In this atomic-scale work, we propose a “composition-constrained” point mean-field formalism resting on alloy pair energetics, to investigate trends (i) and (ii). This formalism is currently applicable to a wide panel of such systems built from ~ 30 chemical elements. Its application to several MPEAs along the AlCoCrFeNi → AlCrFeMnNi → AlCrFeMnMo sequence evidences striking differences between these systems, and demonstrates its efficiency to get at-a-glance information about (i) and (ii) features for various MPEAs. The flexibility of the proposed simulation approach makes it easily employable in conjunction with experiments on well-equilibrated MPEA samples, for thermodynamic characterizations of MPEAs.

Coexisting multi-states in catalytic hydrogen oxidation on rhodium

Catalytic hydrogen oxidation on a polycrystalline rhodium foil used as a surface structure library is studied by scanning photoelectron microscopy (SPEM) in the 10−6 mbar pressure range, yielding spatially resolved X-ray photoemission spectroscopy (XPS) measurements. Here we report an observation of a previously unknown coexistence of four different states on adjacent differently oriented domains of the same Rh sample at the exactly same conditions. A catalytically active steady state, a catalytically inactive steady state and multifrequential oscillating states are simultaneously observed. Our results thus demonstrate the general possibility of multi-states in a catalytic reaction. This highly unusual behaviour is explained on the basis of peculiarities of the formation and depletion of subsurface oxygen on differently structured Rh surfaces. The experimental findings are supported by mean-field micro-kinetic modelling. The present observations raise the interdisciplinary question of how self-organising dynamic processes in a heterogeneous system are influenced by the permeability of the borders confining the adjacent regions.

Non-linear mean-field modelling of UD composite laminates accounting for average asymmetric plasticity of the matrix, debonding and progressive failure

As an effective and accurate method for modelling composite materials, mean-field homogenization is still not well studied in modelling non-linear and damage behaviours of UD composites. Investigated micro FE-simulations show that the matrix of UD composites exhibits different average plastic behaviour, named as average asymmetric matrix plasticity (AAMP), when the composite behaves different under shear, longitudinal and transverse loadings. In this study, a non-linear mean-field debonding model (NMFDM) combining a mean-field model and a fibre–matrix interface debonding model, is developed to simulate UD composites under consideration of AAMP, fibre–matrix interface damage and progressive failure. AAMP is considered by using so-called stress mode factor, which is expressed in terms of basic invariants of the matrix deviatoric stress tensor and is used as an indicator for detection of differences in the loading mode. The material behaviour of UD composites with imperfect interface is assumed identical as for perfect interface and stiffness reduced fibres. Progressive failure criteria are established with consideration of fibre breakage and matrix crack for different fibre orientations. As a representative example for the NMFDM, a C30/E201 UD composite is studied. To verify the model, experiments are conducted on polymers, carbon fibres and UD CFRPs. Finally, the model is applied to simulate a perforated CFRP laminate, which shows excellent prediction ability on deformation, debonding and progressive failure.

Investigation and mean-field modelling of microstructural mechanisms driving the tensile properties of dual-phase steels

A hybrid composite medium-field (Hy-MFC) model was developed to predict the tensile properties of dual-phase steels under monotonic loading based on physical parameters of the microstructure (phase fraction, chemical composition, and grain size of each phase). The Hy-MFC model is intended to be applicable to a wide range of fully ferritic to fully martensitic steels, particularly for alloy design and production-line monitoring. Accounting for the prior austenitic grain size as well as the chemical composition of martensite in the model resulted in good agreement between the modelling and experimental data for the investigated industrial and ternary steels with various martensite fractions. In addition, electron backscatter diffraction monitoring performed during tensile tests allowed to understand the different interactions necessary to reproduce the macroscopic hardening of dual-phase steels. In particular, a hybrid scaling transition law was proposed to reproduce the strain-hardening rate for small deformations for bi-percolant microstructures.

Reversible catalysis.

We describe as 'reversible' a bidirectional catalyst that allows a reaction to proceed at a significant rate in response to even a small departure from equilibrium, resulting in fast and energy-efficient chemical transformation. Examining the relation between reaction rate and thermodynamic driving force is the basis of electrochemical investigations of redox reactions, which can be catalysed by metallic surfaces and biological or synthetic molecular catalysts. This relation has also been discussed in the context of biological energy transduction, regarding the function of biological molecular machines that harness chemical reactions to do mechanical work. This Perspective describes mean-field kinetic modelling of these three types of systems - surface catalysts, molecular catalysts of redox reactions and molecular machines - with the goal of unifying concepts in these different fields. We emphasize that reversibility should be distinguished from other figures of merit, such as rate or directionality, before its design principles can be identified and used to engineer synthetic catalysts.

On the role of solute drag in reconciling laboratory and natural constraints on olivine grain growth kinetics

SUMMARY We investigate the effect of solute drag on the grain growth (GG) kinetics in olivine-rich rocks through full field and mean field modelling. Considering a drag force exerted by impurities on grain boundary migration allows reconciling laboratory and natural constraints on olivine GG kinetics. Solute drag is implemented in a full field level-set framework and in a mean field model that explicitly accounts for a grain size distribution. After calibration of the mean field model on full field results, both models are able to both reproduce laboratory GG kinetics and predict grain sizes consistent with observations in peridotite xenoliths from different geological contexts.

Mean field and full field modelling of microstructure evolution and phase transformations during hot forming and cooling of low carbon steels

Mean field and full field modelling of microstructure evolution and phase transformations during hot forming and cooling of low carbon steels

Impact of household quarantine on SARS-Cov-2 infection in mainland China: A mean-field modelling approach.

The novel coronavirus, named SARS-Cov-2, has raged in mainland China for more than three months, and it causes a huge threat to people's health and economic development. In order to curb the SARS-Cov-2 prevalence, the Chinese government enacted a series of containment strategies including household quarantine, traffic restriction, city lockdowns etc. Indeed, the pandemic has been effectively mitigated, but the global transmission is not still optimistic. Evaluating such control measures in detail plays an important role in limiting SARS-Cov-2 spread for public health decision and policymakers. In this paper, based on the cumulative numbers of confirmed cases and deaths of SARS-Cov-2 infection, from January 31st to March 31st, announced by the National Health Commission of the People's Republic of China, we established a mean-field model, considering the substantial contact change under some restrictive measures, to study the dynamics of SARS-Cov-2 infection in mainland China. By the Metropolis-Hastings (M-H) algorithm of Markov Chain Monte Carlo numerical method, our model provided a good fitting to the overall trends of SARS-Cov-2 infections and discovers the transmission heterogeneities by some extreme containment strategies to some extent. The basic reproduction number was approximated to be 2.05 (95% CI [1.35,2.87]); the hospitalized cases arrived at the peak of 29766 (95% CI [29743,29868]) on February 7th (95% CI [Feb.6th, Feb.8th]). Importantly, we identified that the highest risk group of SARS-Cov-2 was the family of four, which has the biggest probability of degree distributions at such node, suggesting that contact patterns play an important role in curtailing the disease spread.

Stellar dynamos with solar and antisolar differential rotations: Implications to magnetic cycles of slowly rotating stars

ABSTRACT Simulations of magnetohydrodynamics convection in slowly rotating stars predict antisolar differential rotation (DR) in which the equator rotates slower than poles. This antisolar DR in the usual αΩ dynamo model does not produce polarity reversal. Thus, the features of large-scale magnetic fields in slowly rotating stars are expected to be different than stars having solar-like DR. In this study, we perform mean-field kinematic dynamo modelling of different stars at different rotation periods. We consider antisolar DR for the stars having rotation period larger than 30 d and solar-like DR otherwise. We show that with particular α profiles, the dynamo model produces magnetic cycles with polarity reversals even with the antisolar DR provided, the DR is quenched when the toroidal field grows considerably high and there is a sufficiently strong α for the generation of toroidal field. Due to the antisolar DR, the model produces an abrupt increase of magnetic field exactly when the DR profile is changed from solar-like to antisolar. This enhancement of magnetic field is in good agreement with the stellar observational data as well as some global convection simulations. In the solar-like DR branch, with the decreasing rotation period, we find the magnetic field strength increases while the cycle period shortens. Both of these trends are in general agreement with observations. Our study provides additional support for the possible existence of antisolar DR in slowly rotating stars and the presence of unusually enhanced magnetic fields and possibly cycles that are prone to production of superflare.

Mean-field modelling of magnetic nanoparticles: The effect of particle size and shape on the Curie temperature

A Heisenberg mean-field model is used to study the effect of size and shape on the Curie temperature of magnetic nanoparticles. Simple cubic, body-centered cubic, and magnetite nanoparticles are modelled as spheres, cubes, and needlelike particles. The Curie temperatures of particles of different shape, but with the same crystal structure and smallest dimension $d$, are found to differ. The range in the value of the Curie temperature between particles of different shape, $\mathrm{\ensuremath{\Delta}}{T}_{C}$, is found to be $\ensuremath{\sim}20%$ of the bulk value of ${T}_{C}$ in particles where $dl10$ atoms. As particle size increases, the value of $\mathrm{\ensuremath{\Delta}}{T}_{C}$ reduces rapidly and becomes negligible above a threshold size. This threshold size differs between systems and is controlled predominantly by crystal structure. All systems were fit to the finite-size scaling equation, with values of the scaling exponent $\ensuremath{\nu}$ found to lie between 0.46 and 0.55, in good agreement with the expected value of $\ensuremath{\nu}=0.5$. No trend in the value of $\ensuremath{\nu}$ due to shape was found.

Evolutionary, mean-field and pressure-resistance game modelling of networks security

The recently developed mean-field game models of corruption and bot-net defence in cyber-security, the evolutionary game approach to inspection and corruption, and the pressure-resistance game element, can be combined under an extended model of interaction of large number of indistinguishable small players against a major player, with focus on the study of security and crime prevention. In this paper we introduce such a general framework for complex interaction in network structures of many players, that incorporates individual decision making inside the environment (the mean-field game component), binary interaction (the evolutionary game component), and the interference of a principal player (the pressure-resistance game component). To perform concrete calculations with this overall complicated model we work in three basic asymptotic regimes; fast execution of personal decisions, small rates of binary interactions, and small payoff discounting in time. By this approach we construct a class of solutions having the so-called turnpike property.

Evaluation of using distribution functions for mean field modelling of multiphase steels

Abstract Selection and development of distribution functions (descriptors) for the complex phase steels is described in the paper. The objective was to design the multiphase microstructure with smooth gradients of microstructural features, which will prevent deterioration of local ductility of dual phase steels during stretch-forming process. Sensitivity analysis of descriptors with respect to microstructural parameters is described. Identification of descriptors based on experimental data was performed. The descriptors of complex phase microstructure were defined in the form of distribution functions.

Mean-field modelling of the intermetallic precipitate phases during heat treatment and additive manufacture of Inconel 718

A multi-phase, multi-component mean-field model has been developed for simulating the intermetallic precipitation kinetics in Inconel 718. The aim of this work is to develop predictive capability to aid in process optimisation and explore precipitation kinetics during additive manufacturing (AM). The model has been calibrated to available experimental data, and then applied to predict precipitation kinetics during typical solid solution treatment and aging operations, and during AM. It is shown that a Computer Coupling of Phase Diagrams and Thermochemistry (CALPHAD) based modelling approach provides a unified particle growth rate which can capture the growth, coarsening and dissolution of γ′, γ* and δ precipitates under relevant heat treatment conditions. To apply the model to AM, finite element simulations of a simple rectangular build have been carried out, using a property switching method to simulate the material deposition. The component level simulation provides the thermal fields to calculate precipitation kinetics during deposition, also allowing for the examination of the heat affected zone in the substrate. The modelling approach can capture the repeated nucleation and dissolution of precipitates that occurs during AM. The model shows good agreement with experimental data when applied to predicting precipitation kinetics during heat treatment.

Optimal consensus control of the Cucker-Smale model

We study the numerical realisation of optimal consensus control laws for agent-based models. For a nonlinear multi-agent system of Cucker-Smale type, consensus control is cast as a dynamic optimisation problem for which we derive first-order necessary optimality conditions. In the case of a smooth penalisation of the control energy, the optimality system is numerically approximated via a gradient-descent method. For sparsity promoting, non-smooth l1-norm control penalisations, the optimal controllers are realised by means of heuristic methods. For an increasing number of agents, we discuss the approximation of the consensus control problem by following a mean-field modelling approach.