Mean-field Estimates Research Articles

Computational homogenization of sheet molding compound composites based on high fidelity representative volume elements

Sheet molding compound (SMC) composites combine high lightweight potential with excellent formability and are frequently used in industrial applications. To reduce safety factors in dimensioning SMC parts, the influence of processing parameters and stochastic variation of microstructural and physical properties needs to be quantified accurately.Taking into account the inherent three-scale structure of SMC, we improve the microstructure generator of Chen et al. [Compos. Struct. 188, pp. 25–38, 2018] in various respects. Firstly, we consistently rely upon state-of-the-art closure approximations for the fourth order fiber orientation tensor. More precisely, we show that for a planar fiber orientation state, there is an explicit formula for the fast exact closure approximation of Montgomery-Smith et al. [J. Fluid Mech. 680, pp. 321–335, 2011]. Secondly, we exploit the use of quasi-random numbers in sampling the fiber orientation distribution, leading to dramatic improvements in accuracy compared to pseudo-random Monte Carlo sampling. Last but not least, we rely upon fast Fourier transform based methods for rapid computational homogenization.With these methodological improvements at hand, we thoroughly investigate the influence of the mechanical and microstructural parameters on the effective elastic properties of SMC composites, and compare the results to direct numerical simulations on large scale digital volume images and mean-field estimates.

Reaction processes among self-propelled particles.

We study a system of self-propelled disks that perform run-and-tumble motion, where particles can adopt more than one internal state. One of those internal states can be transmitted to other particle if the particle carrying this state maintains physical contact with another particle for a finite period of time. We refer to this process as a reaction process and to the different internal states as particle species, making an analogy to chemical reactions. The studied system may fall into an absorbing phase, where due to the disappearance of one of the particle species no further reaction can occur, or may remain in an active phase where particles constantly react. By combining individual-based simulations and mean-field arguments, we study the dependency of the equilibrium densities of particle species on motility parameters, specifically the active speed v0 and tumbling frequency λ. We find that the equilibrium densities of particle species exhibit two very distinct, non-trivial scaling regimes, with v0 and λ depending on whether the system is in the so-called ballistic or diffusive regime. Our mean-field estimates lead to an effective renormalization of reaction rates that allow building the phase-diagram v0-λ that separates the absorbing and active phases. We find an excellent agreement between numerical simulations and mean-field estimates. This study is a necessary step towards an understanding of phase transitions into absorbing states in active systems and sheds light on the spreading of information/signaling among moving elements.

Droplet formation in a one-dimensional system of attractive spinless fermions

A translation invariant one-dimensional system of spinless fermions with a finite-range attraction experiences a quantum phase transition to a phase-separated state. While being a conventional Luttinger liquid for a small interaction strength, spinless fermions form a droplet with the size smaller than the available one-dimensional volume when the interaction strength exceeds some critical value. A particularly remarkable signature of the droplet formation is the change in the lower edge of the many-body excitation spectrum. In the homogeneous phase, it has a Luttinger-liquid shape of periodic arcs on top of the shallow parabolic dispersion of the center-of-mass. When the interaction strength is increased, the arcs disappear completely as soon as the droplet is formed. We perform an exact diagonalization study of this system with the focus on the signatures of the quantum phase transition and the droplet properties. The one-particle and density-density correlation functions, the pressure, the sound velocity, and the droplet density are examined. The value of the critical interaction strength obtained from numerical data reasonably agrees with a simple mean-field analytical estimate. Due to the boson-fermion correspondence valid in one dimension, our results also hold for hard-core bosons with a finite-range attraction.

Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities

This paper presents a homogenisation-based constitutive model to describe the effective transient diffusion behaviour in heterogeneous media in which there is a large contrast between the phase diffusivities. In this case mobile species can diffuse over long distances through the fast phase in the time scale of diffusion in the slow phase. At macroscopic scale, contrasted phase diffusivities lead to a memory effect that cannot be properly described by classical Fick’s second law. Here we obtain effective governing equations through a two-scale approach for composite materials consisting of a fast matrix and slow inclusions. The micro-macro transition is similar to first-order computational homogenisation, and involves the solution of a transient diffusion boundary-value problem in a Representative Volume Element of the microstructure. Different from computational homogenisation, we propose a semi-analytical mean-field estimate of the composite response based on the exact solution for a single inclusion developed in our previous work [Brassart, L., Stainier, L., 2018. Effective transient behaviour of inclusions in diffusion problems. Z. Angew Math. Mech. 98, 981–998]. A key outcome of the model is that the macroscopic concentration is not one-to-one related to the macroscopic chemical potential, but obeys a local kinetic equation associated with diffusion in the slow phase. The history-dependent macroscopic response admits a representation based on internal variables, enabling efficient time integration. We show that the local chemical kinetics can result in non-Fickian behaviour in macroscale boundary-value problems.

Superexchange Interaction in Magnetic Insulators with Spin Crossover

We present the derivation of a microscopic superexchange Hamiltonian for undoped magnetic insulators with an arbitrary spin. It is established that the sign of the (ferromagnetic or antiferromagnetic) superexchange between magnetic ions in the dn configuration depends on the spin nature of virtual multielectron states dn± 1, viz., low-spin or high-spin partners with S ± 1/2 relative to ground state of the dn configuration with spin S. A macroscopic substantiation is given for the Goodenough–Kanamori rules and simple mean-field estimates connecting the magnetic ordering temperature with the exchange constant. The conventional Anderson superexchange for magnetic materials with spin S = 1/2 and the P/T magnetic phase diagram for ferroborate FeBO3 with spin crossover S = (5/2 ↔ 1/2) at the Fe3+ ion under a high pressure are also reproduced as a test.

Study of the superconducting order parameter in the two-dimensional negative- U Hubbard model by grand-canonical twist-averaged boundary conditions

By using variational Monte Carlo and auxiliary-field quantum Monte Carlo methods, we perform an accurate finite-size scaling of the $s$-wave superconducting order parameter and the pairing correlations for the negative-$U$ Hubbard model at zero temperature in the square lattice. We show that the twist-averaged boundary conditions (TABCs) are extremely important to control finite-size effects and to achieve smooth and accurate extrapolations to the thermodynamic limit. We also show that TABCs is much more efficient in the grand-canonical ensemble rather than in the standard canonical ensemble with fixed number of electrons. The superconducting order parameter as a function of the doping is presented for several values of $|U|/t$ and is found to be significantly smaller than the mean-field BCS estimate already for moderate couplings. This reduction is understood by a variational ansatz able to describe the low-energy behaviour of the superconducting phase, by means of a suitably chosen Jastrow factor including long-range density-density correlations.

Lateral hardness and the scratch resistance of glasses in the Na2O-CaO-SiO2 system

We report on the lateral hardness of glasses from the Na2O-CaO-SiO2 (NCS) system, including CS and NS binaries. Quantitative data are provided on the work of lateral deformation, obtained through instrumented indentation by monitoring the lateral force as a function of normal load. Experiments were conducted in the regime of elastic-plastic deformation, that is, in the absence of microscopic cracking for normal loads of 50–70 mN. The scratch hardness is determined from the work of deformation per deformed volume. Parallel observations of the elastic properties are undertaken so as to reference scratching data to glass composition and structure. We find that structural densification and high rigidity favor higher scratch hardness. For the considered range of chemical compositions, the scratch hardness exhibits an approximately linear correlation with bulk modulus and packing density. It increases in the direction of compositions with higher content of CaO, and decreases with the content of Na2O. Furthermore, we find a strongly linear correlation between scratch hardness and the deviation of experimental compressibility from the mean-field estimate of the Makishima-Mackenzie model. This points to the importance of intermediate-range structural heterogeneity in the scratching process.

Atomistic and mean-field estimates of effective stiffness tensor of nanocrystalline copper

The full elasticity tensor for nano-crystalline copper is derived in molecular simulations by performing numerical tests for a set of generated samples of the polycrystalline material. The results are analysed with respect to the anisotropy degree of the overall stiffness tensor resulting from the limited number of grain orientations and their spatial distribution. The dependence of the overall bulk and shear moduli of an isotropized polycrystal on the average grain diameter is analysed. It is found that while the shear modulus decreases with grain size, the bulk modulus shows negligible dependence on the grain diameter and is close to the bulk modulus of a single crystal. A closed-form mean-field model of effective elastic properties for a bulk nano-grained polycrystal with cubic grains, i.e. made of a material with cubic symmetry, is formulated. In the model all parameters are based on the data for a single crystal and on the averaged grain size without any need for additional fitting. It is shown that the proposed model provides predictions of satisfactory qualitative and quantitative agreement with atomistic simulations.

Unbinding slave spins in the Anderson impurity model

We show that a generic single-orbital Anderson impurity model, lacking for instance any kind of particle-hole symmetry, can be exactly mapped without any constraint onto a resonant level model coupled to two Ising variables, which reduce to one if the hybridisation is particle-hole symmetric. The mean-field solution of this model is found to be stable to unphysical spontaneous magnetisation of the impurity, unlike the saddle-point solution in the standard slave-boson representation. Remarkably, the mean-field estimate of the Wilson ratio approaches the exact value $R_\text{W}=2$ in the Kondo regime.

Elastic-plastic properties of metal matrix composites: Validation of mean-field approaches

Abstract Several micromechanical and numerical approaches to estimating the effective properties of heterogeneous media are analyzed. First, micromechanical mean-field estimates of elastic moduli for selected metal matrix composite systems are compared with the results of finite element calculations performed for two simplified unit cells: spherical and cylindrical. Advantages and deficiencies of such numerical verification of analytical homogenization schemes are indicated. Next, predictions of both approaches are compared with available experimental data for two composite systems for tension and compression tests in the elastic-plastic regime using tangent and secant linearization procedures. In the examined range of strain and ceramic volume content, both the Mori-Tanaka averaging scheme and the generalized self-consistent scheme lead to reliable predictions when combined with the tangent linearization, while the use of secant moduli results in a too stiff response. It is also found that the mean-field predictions for a small ceramic volume content are very close to the results obtained from the finite-element analysis of a spherical unit cell.

Micromechanical and numerical analysis of packing and size effects in elastic particulate composites

Abstract Effects of particle packing and size on the overall elastic properties of particulate random composites are analyzed. In order to account for the two effects the mean-field Morphologically Representative Pattern (MRP) approach is employed and an additional interphase surrounding inclusions (coating) is introduced. The analytical mean-field estimates are compared with the results of computational homogenization performed using the finite element (FE) method. Periodic unit cells with cubic crystal-type arrangements and representative volume elements with random distributions of particles are used for verification purposes. The validity of the MRP estimates with respect to the FE results is assessed.

Population-imbalanced lattice fermions near the BCS-BEC crossover: Thermal physics of the breached pair and Fulde-Ferrell-Larkin-Ovchinnikov phases

We study s-wave superconductivity in the two dimensional attractive Hubbard model in an applied magnetic field, assume the extreme Pauli limit, and examine the role of spatial fluctuations in the coupling regime corresponding to BCS-BEC crossover. We use a decomposition of the interaction in terms of an auxiliary pairing field, retain the static mode, and sample the pairing field via Monte Carlo. The method requires iterative solution of the Bogoliubov-de-Gennes (BdG) equations for amplitude and phase fluctuating configurations of the pairing field. We establish the full thermal phase diagram of this strong coupling problem, revealing $T_c$ scales an order of magnitude below the mean field estimate, highlight the spontaneous inhomogeneity in the field induced magnetization, and discover a strong non monotonicity in the temperature dependence of the low energy density of states. We compare our results to the experimental phase diagram of the imbalanced Fermi gas at unitarity. This paper focuses on the magnetized but homogeneous (breached pair) superconducting state, a companion paper deals with the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) regime.

Finite element modeling of periodic polycrystalline aggregates with intergranular cracks

The present study addresses the prediction of the mechanical response of model polycrystalline aggregates in which the anisotropy of individual crystals induces high internal stresses as well as microcracks. A novel procedure is proposed in order to automatically generate finite element (FE) meshes that conform to the polycrystalline microstructure. The meshes obtained are periodic in 3-D and they contain cohesive zone elements along all grain boundaries. The FE model is applied to two materials with significant elastic anisotropy at the crystal level: stainless steel and graphite. A self-consistent equivalent inclusion solution is shown to produce a valid mean-field estimate of the effective stiffness of the polycrystal. This reference solution is used in order to optimize the assignment of lattice orientations to individual grains prior to the FE simulations. It is demonstrated that relying on such model polycrystals increases the accuracy of FE predictions at an affordable computational cost. The influence of internal cracks on the macroscopic response depends on the anisotropy of individual crystals.

Host contact and shedding patterns clarify variation in pathogen exposure and transmission in threatened tortoiseGopherus agassizii: implications for disease modelling and management

Most directly transmitted infections require some form of close contact between infectious and susceptible hosts to spread. Often disease models assume contacts are equal and use mean field estimates of transmission probability for all interactions with infectious hosts. Such methods may inaccurately describe transmission when interactions differ substantially in their ability to cause infection. Understanding this variation in transmission risk may be critical to properly model and manage some infectious diseases. In this study, we investigate how varying exposure and transmission may be key to understanding disease dynamics in the threatened desert tortoise Gopherus agassizii. We created heterogeneity in Mycoplasma agassizii exposure (the putative bacterial agent of a respiratory disease) by varying the duration of interactions between naturally infected and uninfected captive desert tortoises. Using qPCR, we identified new infections and compared models of transmission probability as a function of contact duration and pathogen load. We then examined the contact patterns of a wild tortoise population using proximity loggers to identify heterogeneity in contact duration. The top-ranked model predicting M.agassizii transmission included a dose term defined as the product of the number of days in proximity to an infected host and the infection level of that host. Models predicted low transmission probability for short interactions, unless the infectious host had a high load of M.agassizii: such hosts were predicted to transmit infection at higher rates with any amount of contact. We observed predominantly short-lived interactions in a free-ranging tortoise population and thus, expect transmission patterns in this population to vary considerably with the frequency and duration of high infection levels. Mean field models may misrepresent natural transmission patterns in this and other populations depending on the distribution of high-risk contact and shedding events. Rapid outbreaks in generally solitary species may result from changes to their naturally low-risk contact patterns or due to increases in the frequency of severe infections or super-shedding events - population characteristics that should be further investigated to develop effective management strategies.

Disentangling the influences of mean body size and size structure on ecosystem functioning: an example of nutrient recycling by a non-native crayfish.

Body size is a fundamental functional trait that can be used to forecast individuals' responses to environmental change and their contribution to ecosystem functioning. However, information on the mean and variation of size distributions often confound one another when relating body size to aggregate functioning. Given that size-based metrics are used as indicators of ecosystem status, it is important to identify the specific aspects of size distributions that mediate ecosystem functioning. Our goal was to simultaneously account for the mean, variance, and shape of size distributions when relating body size to aggregate ecosystem functioning. We take advantage of habitat-specific differences in size distributions to estimate nutrient recycling by a non-native crayfish using mean-field and variance-incorporating approaches. Crayfishes often substantially influence ecosystem functioning through their omnivorous role in aquatic food webs. As predicted from Jensen's inequality, considering only the mean body size of crayfish overestimated aggregate effects on ecosystem functioning. This bias declined with mean body size such that mean-field and variance-incorporating estimates of ecosystem functioning were similar for samples at mean body sizes >7.5g. At low mean body size, mean-field bias in ecosystem functioning mismatch predictions from Jensen's inequality, likely because of the increasing skewness of the size distribution. Our findings support the prediction that variance around the mean can alter the relationship between body size and ecosystem functioning, especially at low mean body size. However, methods to account for mean-field bias performed poorly in samples with highly skewed distributions, indicating that changes in the shape of the distribution, in addition to the variance, may confound mean-based estimates of ecosystem functioning. Given that many biological functions scale allometrically, explicitly defining and experimentally or statistically isolating the effects of the mean, variance, and shape of size distributions is necessary to begin generalizing relationships between animal body size and ecosystem functioning.

Self-heterodyne detection of thein situphase of an atomic superconducting quantum interference device

We present theoretical and experimental analysis of an interferometric measurement of the {\it in-situ} phase drop across and current flow through a rotating barrier in a toroidal Bose-Einstein condensate (BEC). This experiment is the atomic analog of the rf-superconducting quantum interference device (SQUID). The phase drop is extracted from a spiral-shaped density profile created by the spatial interference of the expanding toroidal BEC and a reference BEC after release from all trapping potentials. We characterize the interferometer when it contains a single particle, which is initially in a coherent superposition of a torus and reference state, as well as when it contains a many-body state in the mean-field approximation. The single-particle picture is sufficient to explain the origin of the spirals, to relate the phase-drop across the barrier to the geometry of a spiral, and to bound the expansion times for which the {\it in-situ} phase can be accurately determined. Mean-field estimates and numerical simulations show that the inter-atomic interactions shorten the expansion time scales compared to the single-particle case. Finally, we compare the mean-field simulations with our experimental data and confirm that the interferometer indeed accurately measures the {\it in-situ} phase drop.

An observational assessment of the influence of mesoscale and submesoscale heterogeneity on ocean biogeochemical reactions

AbstractNumerous observations demonstrate that considerable spatial variability exists in components of the marine planktonic ecosystem at the mesoscale and submesoscale (100 km–1 km). The causes and consequences of physical processes at these scales (“eddy advection”) influencing biogeochemistry have received much attention. Less studied, the nonlinear nature of most ecological and biogeochemical interactions means that such spatial variability has consequences for regional estimates of processes including primary production and grazing, independent of the physical processes. This effect has been termed “eddy reactions.” Models remain our most powerful tools for extrapolating hypotheses for biogeochemistry to global scales and to permit future projections. The spatial resolution of most climate and global biogeochemical models means that processes at the mesoscale and submesoscale are poorly resolved. Modeling work has previously suggested that the neglected eddy reactions may be almost as large as the mean field estimates in some cases. This study seeks to quantify the relative size of eddy and mean reactions observationally, using in situ and satellite data. For primary production, grazing, and zooplankton mortality the eddy reactions are between 7% and 15% of the mean reactions. These should be regarded as preliminary estimates to encourage further observational estimates and not taken as a justification for ignoring eddy reactions. Compared to modeling estimates, there are inconsistencies in the relative magnitude of eddy reactions and in correlations which are a major control on their magnitude. One possibility is that models exhibit much stronger spatial correlations than are found in reality, effectively amplifying the magnitude of eddy reactions.

Investigation of the effect of aggregates' morphology on concrete creep properties by numerical simulations

Prestress losses due to creep of concrete is a matter of interest for long-term operations of nuclear power plants containment buildings. Experimental studies by Granger (1995) have shown that concretes with similar formulations have different creep behaviors. The aim of this paper is to numerically investigate the effect of size distribution and shape of elastic inclusions on the long-term creep of concrete. Several microstructures with prescribed size distribution and spherical or polyhedral shape of inclusions are generated. By using the 3D numerical homogenization procedure for viscoelastic microstructures proposed by Šmilauer and Bažant (2010), it is shown that the size distribution and shape of inclusions have no measurable influence on the overall creep behavior. Moreover, a mean-field estimate provides close predictions. An Interfacial Transition Zone was introduced according to the model of Nadeau (2003). It is shown that this feature of concrete's microstructure can explain differences between creep behaviors.

Hysteresis in Two-Dimensional Liquid Crystal Models

We make a numerical study of hysteresis loop shapes within a generalized two-dimensional Random Anisotropy Nematic (RAN) model at zero temperature. The hysteresis loops appear on cycling a static external ordering field. Ordering in these systems is history dependent and involves interplay between the internal coupling constantJ, the anisotropy random fieldD, and the ordering external fieldH. Here the external field is represented by a traceless tensor, analogous to extension-type fields in continuum mechanics. The calculations use both a mean field approach and full lattice simulations. Our analysis suggests the existence of two qualitatively different solutions, which we denote assymmetricandsymmetry breaking. For the set of parameters explored, only the symmetric solutions are stable. Both approaches yield qualitatively similar hysteresis curves, which are manifested either by single or double loops. But the quantitative differences indicate that mean field estimates are only of limited predictive value.

Effect of an external magnetic field on the nematic-isotropic phase transition in mesogenic systems of uniaxial and biaxial molecules: a Monte Carlo study.

We determine the nematic-isotropic coexistence curve terminating at the critical point in a temperature-external field phase diagram for nematic liquid crystals with positive diamagnetic anisotropy, where the molecules are either perfectly uniaxial or biaxial using computer simulation of a lattice model. The coexistence curve is much steeper than that predicted by the standard Landau-de Gennes and Maier-Saupe mean-field theories. For the uniaxial system the critical magnetic field is estimated to be one order of magnitude lower than the mean-field estimate but of the same order of magnitude as the experimental measurement. Our study shows that molecular biaxiality could reduce the critical field strength significantly.