Perimeter analysis of the Von Koch island, application to the evolution of grain boundaries during heating

This paper introduces an analyse of the fractal dimension by Richardson’s method. Two different ways to calculate the fractal dimension are presented with their related calculation errors and applied the Von Koch curves. A Monte-Carlo simulation of the evolution of the grains’ boundaries when heating shows that the interfaces lose their fractal characteristics as reported in experimental work. This result is interpreted by dissipation of the energy during the evolution of the grain boundary

The measurement problem on classical diffusion process: inverse method on stochastic processes

In a high number of diffusive systems, measures are processed to calculate material parameters such as diffusion coefficients, or to verify the accuracy of mathematical models. However, the precision of the parameter determination or of the model relevance depends on the location of the measure itself. The aim of this paper is first to analyse, for a mono-dimensional system, the precision of the measure in relation with its location by an inverse problem algorithm and secondly to examine the physical meaning of the results. Statistical mechanic considerations show that, passing over a time–distance criterion, measurement becomes uncertain whatever the initial conditions. The criterion proves that this chaotic mode is related to the production of anti-entropy at a mesoscopique scale that is in violation to quantum theory about measurement

A numerical method to calculate the Abbott parameters: A wear application

A numerical technique was proposed to plot the Abbott curve and to compute its associated parameters defined by the DIN 4776 and ISO 13565 norms. These parameters were then extended and applied to non-sigmoid Abbott curves. By studying the discretisation errors, we show that a minimum of 200 intercepts, with parabolic interpolations between discretised data profiles, have to be taken into consideration to calculate the parameters as accurately as possible. Experimental profiles were eroded by means of a numerical wear model, and it was shown that the Abbott parameters correlate well with the wear model parameters. Our numerical estimations of Abbott parameters were performed for electro-eroded, tool machined, polished, worn and sandblasted surfaces. Manual measures were compared with our algorithmic method and it was shown that the difference is lower than 1% for Mr1 and Mr2 Abbott parameters, but the numerical technique leads to a lower dispersion

Statistical artefacts in the determination of the fractal dimension by the slit island method

This paper comments upon some statistical aspects of the slit island method which is widely used to calculate the fractal dimension of fractured surfaces or of materials’ features like grain geometry. If a noise is introduced when measuring areas and perimeters of the islands (experimental errors), it is shown that errors are made in the calculation of the fractal dimension and more than a false analytical relation between a physical process parameter and the fractal dimension can be found. Moreover, positive or negative correlation with the same physical process parameter can be obtained whether the regression is performed by plotting the variation of the noisy area versus the noisy perimeter of the considered islands or vice versa. Monte-Carlo simulations confirm the analytical relations obtained under statistical considerations

Multiscale measures of equilibrium on finite dynamic systems

This article presents a new method for the study of the evolution of dynamic systems based on the notion of quantity of information. The system is divided into elementary cells and the quantity of information is studied with respect to the cell size. We have introduced an analogy between quantity of information and entropy, and defined the intrinsic entropy as the entropy of the whole system independent of the size of the cells. It is shown that the intrinsic entropy follows a Gaussian probability density function (PDF) and thereafter, the time needed by the system to reach equilibrium is a random variable. For a finite system, statistical analyses show that this entropy converges to a state of equilibrium and an algorithmic method is proposed to quantify the time needed to reach equilibrium for a given confidence interval level. A Monte-Carlo simulation of diffusion of A* atoms in A is then provided to illustrate the proposed simulation. It follows that the time to reach equilibrium for a constant error probability, te, depends on the number, n, of elementary cells as: te∝n2.22±0.06. For an infinite system size (n infinite), the intrinsic entropy obtained by statistical modelling is a pertinent characteristic number of the system at the equilibrium

Physical Interpretations of the Numerical Instabilities in Diffusion Equations Via Statistical Thermodynamics

The aim of this paper is to analyze the physical meaning of the numerical instabilities of the parabolic partial differential equations when solved by finite differences. Even though the explicit scheme used to solve the equations is physically well posed, mathematical instabilities can occur as a consequence of the iteration errors if the discretisation space and the discretisation time satisfy the stability criterion. To analyze the physical meaning of these instabilities, the system is divided in sub-systems on which a Brownian motion takes place. The Brownian motion has on average some mathematical properties that can be analytically solved using a simple diffusion equation. Thanks to this mesoscopic discretisation, we could prove that for each half sub-cell the equality stability criterion corresponds to an inversion of the particle flux and a decrease in the cell entropy in keeping with time as criterion increases. As a consequence, all stability criteria defined in literature can be used to define a physical continuous 'time-length' frontier on which mesoscopic and microscopic models join

Identification of Local Lubrication Regimes on Textured Surfaces by 3D Roughness Curvature Radius

This paper proposes a new method of 3D roughness peaks curvature radius calculation and its application to tribological contact analysis as a characteristic signature of tribological contact. This method is introduced through the classical approach of calculation of radius of asperity in 2D. Actually, the proposed approach provides a generalization of Nowicki's method [ ], depending on horizontal lines intercepting the studied profile. Here, the basic idea consists in intercepting the rough surface by a horizontal plane and to calculate the cross section area without including “islands into islands”, i.e. the small peaks enclosed in bigger ones. Then, taking into account the maximal value of the height amplitude of the roughness included into this area, an appropriate algorithm is proposed, without requiring the classical hypothesis of derivability, which may be unstable when applied to engineering surfaces. This methodology is validated on simulated surfaces, and applied to engineering surfaces created experimentally, with a laboratory aluminium strip drawing process. The regions of the textured and lubricated specimens surface are analysed, and the results gives interesting prospects to qualitatively identify the local lubrication regimes: regions with high curvature radii correspond to severe contact (boundary/mixed lubrication regime) while regions with low curvature radii correspond to hydrodynamic lubrication regime

Vers la construction d'un patrimoine des surfaces topographiques

Dans la quasi-totalité des études de morphologie de surface, nous sommes souvent en terrain inconnu. L'opérateur, en fonction de son degré d'expertise, propose une interprétation physique de la création de la topographie locale ou globale de la surface investiguée. Ses conclusions sont certes basées sur l'historique de formation de la surface, mais surtout sur une connaissance de quelques morphologies types qu'il a rencontrées, connaissance jamais formalisée. On imagine alors la richesse si l'observateur disposait alors d'un ensemble de surface étudiées, répertoriées par une description précise, qui correspond au type de morphologie observée... Quelle aide à l'interprétation, voire même à la compréhension des mécanismes de création. De même, si une surface rencontrée dans la nature possède des propriétés voulues, existe-t-il un moyen de la réaliser ? Existe-t-il une surface morphologiquement proche fabriquée par l'humain ? Si la réponse est positive, comment le savoir ? De nouveau, une classification des surfaces existantes, proche de celle observée et extraite d'une base de données, permettrait potentiellement de fournir ces surfaces. De nombreuses industries structurent des surfaces, volontairement ou involontairement, mais comment assurer leurs diffusions dans les diverses communautés ? De nouveau, leurs archivages dans une base de données introduiraient alors un brassage interdisciplinaire dans les modes de structuration de surface. Il est certain que les propriétés physiques de la surface contribuent fortement à sa fonctionnalité et ne se limitent pas qu'à des données morphologiques. Le choix donc d'une surface fonctionnelle ne peut se faire qu'avec des mesures topographiques et le choix optimal doit inclure les propriétés de surfaces. Mais a-t-on besoin d'une précision forte sur les propriétés physiques (ou plus précisément de mesurer ses propriétés sur la surface elle-même) dans le but unique de faire un choix d'une gamme de surfaces admissibles avec un objectif voulu ? Cette réponse a déjà été formulée par Mike Ashby pour choisir une classe de matériaux optimale pour répondre à une fonctionnalité recherchée en optimisant une fonctionnelle avec des possibilités d'introduire des contraintes supplémentaires sur des propriétés des matériaux. Il est certain que les caractéristiques des matériaux sont assez floues (comme pour les polymères) mais suffisamment fines vis-à-vis de la gamme importante des matériaux investigués. En reformulant la méthodologie Ashby pour les surfaces fonctionnelles, il deviendra possible dans un futur raisonnable, de proposer une méthode pour choisir une surface rugueuse qui permette de répondre à une fonctionnalité et de son mode d'obtention. Nous pourrions répondre alors à la question suivante : quel est l'ensemble des surfaces existantes ayant une fonctionnalité recherchée (résistance thermique forte, brillance élevée, résistance à l'usure, hydrophobe...) qui permet de minimiser un ou plusieurs critères (coût, effet environnement, masse...) avec des contraintes fixées (corrosion, inflammabilité, rigidité, etc....)

Fractal Dimension of Grain Boundary during Heating. Comparison between Images Analyses and Monte Carlo Simulation

There are few articles that mention fractal dimension in grain growth mechanism. Some authors build a simplified analytic model showing that initial fractal dimension of grain boundary has an influence on interface modification velocity. Nevertheless they postulate the relation L = c s(1−Δ) where L is the grain length, c is a constant, s is grain size and Δ the fractal dimension. The aims of this paper is to experimentally analyze by image analysis the fractal dimension of A5 aluminum sheet grain boundaries during heating and to simulate their evolution by a Monte Carlo method to validate experimental data.. It is shown by Monte-Carlo simulation and confirmed experimentally that the grain growth process decreases the fractal dimension of grain border. It can be concluded that it is very hazardous to build a model of grain growth without including the effect of grain’s morphology. The macroscopic fractal morphology of the grain structure could then be used to validate microscopic relation between Monte Carlo Steps time and real time

Multiscale functional analysis of wear: A fractal model of the grinding process

International audienceIn this paper, we propose to create a fractal function defined by an infinite series to model worn surfaces obtained by a grinding process. In this series, each elementary term characterizes a wear process at a given scale. This series is only defined by two parameters: an amplitude parameter and the fractal dimension. This model is tested on worn profiles obtained by using different grinding paper grades and roughness is assessed by tactile profilometry. Then an inverse method is developed to obtain simulated profiles that present the same morphology as the experimental ones. The results from this study prove that our method allows simulation of profiles with elementary functions that characterize the wear process