Elliptic Two-Dimensional Invariant Tori for the Planetary Three-Body Problem

The spatial planetary three-body problem (i.e., one star and two planets, modelled by three massive points, interacting through gravity in a three dimensional space) is considered. It is proved that, near the limiting stable solutions given by the two planets revolving around the star on Keplerian ellipses with small eccentricity and small non-zero mutual inclination, the system affords two-dimensional, elliptic, quasi-periodic solutions, provided the masses of the planets are small enough compared to the mass of the star and provided the osculating Keplerian major semi-axes belong to a two-dimensional set of density close to one.     

Capillary fingering: Percolation and fractal dimension

We present experimental and theoretical results concerning immiscible displacements (drainage) in 2-dimensional permeable media. When capillary forces are predominant, the injected fluid presents very thin fingers and the Representative Elementary Volume concept cannot be used for describing the partial saturations. The purpose of this paper is to show how this classical concept can be replaced by a statistical approach based on fractal geometry.Communication presented at the International Symposium on the Stochastic Approach to Subsurface Flow, Montvillargenne 4–7 June 1985.     

On the vanishing of the additive measures of strain and rotation for finite deformations

Explicit formulae for the finite strain and rotation measures are given, in the cases when either one of the infinitesimal tensors of strain and rotation vanishes. Conversely, when the finite strain or rotation measure vanishes, explicit formulae for the infinitesimal tensors of strain and rotation are also obtained.     

On the representation of the interaction energy between two polar molecules

Summary An attempt has been made to represent the interaction energy between two polar molecules by using an effective relative orientation of the pair of the colliding molecules. Calculations made for gaseous ammonia show fairly satisfactory results.     

Über singuläre Lastfälle in einer linearen Schalentheorie und ihre finite Behandlung

Übersicht Ausgehend von bekannten Fundamentallösungen für Platten bzw. Scheiben wird die Erstellung singulärer Ansatzfunktionen gezeigt, wie sie für finite Näherungsverfahren benötigt werden, die von den Funktionalen der totalen Energie bzw. der komplementären Energie ausgehen. Das Vorgehen wird eingehend an Kreiszylinderschalen erläutert.

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Summary Starting from known fundamental solutions of plates the construction of singular basic estimate functions is shown. These are necessary in finite approximation methods basing on the functionals of total energy or complementary energy. The proceeding is explained in detail in a cylindrical shell analysis.

     

On the Temperature of a Rarefied Gas in Non‐Equilibrium

This paper addresses the problem of the proper definition of temperature of a gas in nonequilibrium. It shows that the mean kinetic energy of the atoms of a rarefied gas is not a good measure for thethermodynamic temperature, because in general it jumps at a wall, and because it is nonmonotone in a onedimensional process of stationary heat conduction. The jump of the kinetic temperature is calculated and found to be about 5K in a rarefied gas. The basis for the calculations is provided by the arguments of extended thermodynamics of 14 moments. An essential tool is the minimax principle of entropy production recently postulated by Struchtrup Weiss [1], because it furnishes one important boundary condition.Sommario. Il lavoro riguarda la corretta definizione della temperatura di un gas in condizioni di nonequilibrio. Si mostra come lenergia cinetica media degli atomi di un gas rarefatto non sia una buona misura della temperatura termodinamica poiché in generale, essa risulta discontinua su una parete e nonmonotona in un processo unidimensionale di conduzione stazionaria del calore. Viene calcolato il salto della temperatura cinetica che risulta pari a circa 5K in un gas rarefatto. La base per il calcolo è fornita dal contesto della termodinamica estesa di 14 momenti. Uno strumento essenziale è rappresentato dal principio di minimax di produzione di entropia recentemente postulato da Struchtrup and Weiss [1], che fornisce unimportante condizione alcontorno.     

Inhomogeneous ‘longitudinal’ plane waves in a deformed elastic material

By definition, a homogeneous isotropic compressible Hadamard material has the property that an infinitesimal longitudinal homogeneous plane wave may propagate in every direction when the material is maintained in a state of arbitrary finite static homogeneous deformation. Here, as regards the wave, homogeneous means that the direction of propagation of the wave is parallel to the direction of eventual attenuation; and longitudinal means that the wave is linearly polarized in a direction parallel to the direction of propagation. In other words, the displacement is of the form u = ncos k(n · x – ct), where n is a real vector. It is seen that the Hadamard material is the most general one for which a longitudinal inhomogeneous plane wave may also propagate in any direction of a predeformed body. Here, inhomogeneous means that the wave is attenuated, in a direction distinct from the direction of propagation; and longitudinal means that the wave is elliptically polarized in the plane containing these two directions, and that the ellipse of polarization is similar and similarly situated to the ellipse for which the real and imaginary parts of the complex wave vector are conjugate semi-diameters. In other words, the displacement is of the form u = {S exp i(S · x – ct)}, where S is a complex vector (or bivector). Then a Generalized Hadamard material is introduced. It is the most general homogeneous isotropic compressible material which allows the propagation of infinitesimal longitudinal inhomogeneous plane circularly polarized waves for all choices of the isotropic directional bivector. Finally, the most general forms of response functions are found for homogeneously deformed isotropic elastic materials in which longitudinal inhomogeneous plane waves may propagate with a circular polarization in each of the two planes of central circular section of the ⁿ -ellipsoid, where is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.     

An efficient anti-aliasing spectral continuous window shifting technique for PIV

A new sub-pixel correlation peak locating algorithm for PIV analysis is introduced. The method is theoretically consistent with the method of continuously shifting interrogation sub-windows by fractional displacements, which has proven to be an effective way to reduce the bias error associated with integer pixel aliasing, or peak-locking. However the proposed algorithm performs continuous window shifting in the spatial frequency domain using the shift property of the Fourier transform, thus it is equivalent to interpolating the original digital image with the Fourier transform reconstruction. Synthetic and real PIV images are used to test the new algorithms performance relative to that of traditional (non-iterative) peak-finding methods and other peak-locking reduction algorithms, such as the continuous window shifting technique. The resultant bias error of the proposed algorithm is smaller (by an order of magnitude in some cases), and importantly, the periodic nature of the bias error, the characteristic signature of peak-locking, is eliminated as long as the discrete particle images have been sampled at a rate greater than the Nyquist sampling frequency. Moreover, this new algorithm is shown to be computationally efficient and it converges faster than the competing algorithms.     

New trends in the analysis of masonry structures

The modern theory of masonry structures has been set up on the hypothesis of no-tension behaviour, with the aim of offering a reference model, independent of materials and building techniques employed. This hypothesis gives rise to inequalities which have to be satisfied by the stress tensor components and, as a dual aspect, to the kinematic behaviour characteristics of media which can be classified as lying between solids and fluids: the structure of the masonry material consists of particles reacting elastically only when in contact. An examination of the plane-stress problem leads us to define, within the prescribed domain under admissible loads, three different subdomains with null, regular, or non-regular principal stress tensors, respectively. As the boundaries of such subdomains are not known a priori, the problem can be classified as a free boundary value problem. The analysis concerns mainly the subdomains where the stress tensor is non-regular; and a non-regularity condition det =0 is added to the equilibrium equations. This condition makes the stress problem isostatic and leads to a violation of Saint-Venant's compliance conditions on strains. Hence there is a need to introduce a strain tensor, not related to the stress tensor, which can be decomposed into an extensional component and a shearing component; we prove that such strains, of the class ^c, are similar to those of the theory of plastic flow. From the point of view of computational analysis the anelastic strains are considered as given distortions; they are computed by means of the Haar-Kármán principle, modified for computational purposes by an idea of Prager and Hodge.

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Sommario La moderna teoria delle strutture murarie, fondata sulla rigorosa non reagenza a trazione del materiale, ha lo scopo di fornire un modello di riferimento indipendente sia dalle caratteristiche del materiale sia dalle techniche costruttive impiegate. L'ipotesi di non reagenza a trazione si traduce in disuguaglianze che le componenti del tensore di stress devono verificare; dualmente il comportamento caratteristico cinematico può esser classificato di confine, come del resto la stessa statica, tra solidi e fluidi: la struttura ipotizzata del materiale muratura consiste di particelle che reagiscono solo se sono in contatto. L'esame del problema piano porta a definire all'interno del dominio di definizione tre differenti tipi di sub-regioni in cui lo stress è nullo, canonico, o singolare. Poiché le frontiere di queste sub-regioni non sono note a priori il problema può anche essere classificato di frontiera libera. L'analisi concerne fondamentalmente la sub-regione in cui il tensore è non regolare, perché deve verificare anche la condizione det =0. Ciò rende isostatico il problema e conduce anche alla violazione della condizione di integrabilità delle deformazioni. Questo passaggio può essere superato introducendo un tensore di deformazioni a tensioni nulle che si può decomporre in una componente estensionale ed in una componente di scorrimento; si dimostra che queste deformazioni sono equivalenti a quelle che intervengono nella Teoria del flusso plastico. Dal punto di vista computazionale le deformazioni anelastiche sono considerate come distorsioni impresse determinate attraverso il principio di Haar-Kármán modificato, per le techniche computazionali, su idee di Prager e Hodge.

     

Retrieval of Spacewise Dependent Hydraulic Properties of Anisotropic Rocks from Transient Flow Experiments

An inverse finite-difference method (FDM) is developed to characterise the spatially dependent components of the hydraulic conductivity tensor together with the specific storage for anisotropic materials, using experimental data generated from the direct FDM. This simulated surface data serves as additional information to a genetic algorithm (GA) optimisation procedure, using a modified least squares functional, that minimises the difference between the experimental data and the FDM-predicted boundary pressure and/or average hydraulic flux measurements based on current hydraulic conductivity tensor and specific storage estimates.     

Disappearing Wakes and Dispersion in Numerically Simulated Flows Through Tube Bundles

Flow through a staggered array or bundle of parallel rigid cylinders of diameter D is computed with the help of a three-dimensional direct numerical simulation (DNS) at various values of Reynolds number between 50 and 6000. Two different spacings L of the tubes, i.e. L/D= 2 and L/D= 3, have been considered. When Re 500 the flow is laminar. In that case the converging flow between a pair of adjacent cylinders brings the oppositely signed vorticity at the two edges of the wake closer together behind the upstream cylinder so that the vorticity decreases quickly due to cancellation by diffusion. At Re 6000, when the flow is highly turbulent, the wake vorticity disappears rather by turbulent diffusion. This disappearance of the wakes in the closely packed flows (i.e. L/D 2) causes the mean flow in a cell, which consists of the region around a single cylinder, to be effectively independent of that in other cells. Another consequence is that the mean velocity field can be very well approximated by potential flow except in a thin boundary layer along the cylinder and a short wake behind it. The results have been applied to the transport of scalars in closely packed arrays. As in other complex flows, the dispersion of the scalars is dominated by the divergence and convergence of the streamlines around the cylinder rather than by the wake turbulence. Approximate expressions are derived for this topologically influenced dispersion in terms of the geometry of the array. The fact when most of the flow in the array can be approximated by a potential flow, allows us to introduce a fast approximate calculation method to compute the dispersion.     

Development of a test machine and method for galling studies

A test method for evaluating galling resistance in tubular connections by using the established evaluation technique known as the up and down or Bruceton method was studied. A unique test machine that would closely simulate the actual field conditions was designed and constructed. By using the Taguchi technique, the relative effects of rotational speed, roughness, and axial load on the galling resistance were also studied and discussed.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

On near-wall turbulence-generating events in a turbulent boundary layer on a riblet surface

The boundary layer over a drag reducing riblet surface is investigated using hot-wire anemometry and flow visualisation. The concept of a riblet sublayer is introduced, and a definition is proposed in terms of a region of reduced turbulence energy production formed near the wall by the addition of riblets. The hot wire records are examined using a modified form of quadrant analysis, and results obtained over plain and riblet surfaces are compared. Close to the wall, the addition of riblets produces a marked reduction in the occurrence of ejection (2nd quadrant) events. A corresponding increase in the incidence of sweep (4th quadrant) events is accompanied by the development of a strong tendency toward a preferred event duration, and a preferred interval between events. These changes diminish rapidly with distance from the surface, becoming almost undetectable beyondy ⁺=40. They are discussed in the light of flow visualisation results, and interpreted in terms of mechanisms associated with the interaction between the riblets and the inner boundary layer flow structures. A conceptual model of the flow mechanisms in the riblet sublayer is proposed.     

Finite Elastic Deformations of a Tyre Modelled as an Ideal Fibre-Reinforced Shell

Vehicle tyres are anisotropic inhomogeneous fibre-reinforced shells which undergo finite elastic deformations. Calculation of their stress and deformation fields is a difficult task and is normally performed using the finite element technique. In this paper an attempt is made to provide an approximate analysis of the deformation field modelling the tyre as an ideal fibre-reinforced material. Radial-ply tyres are reinforced by a belt of fibres running around the wheel in the circumferential direction under the tread of the tyre. A second set of fibres lies in each radial cross-section, of the tyre and runs from the bead wire which seats against one wheel rim to the bead wire at the other wheel rim. We shall assume each radial cross-section of the tyre is in a state of plane strain and is formed from an arch of fibre-reinforced composite material which is reinforced in the hoop direction. This composite is assumed to be an ideal material which is inextensible in the fibre-direction and is incompressible. The plane-strain deformations of this section are examined and then used to analyse the deformation of the tyre as a whole.     

A General Method for Estimating Dynamic Parameters of Spatial Mechanisms

Dynamic equations of motion require a large number of parameters for each element of the system. These can include for each part their mass, location of center of mass, moment of inertia, spring stiffnesses and damping coefficients. This paper presents a technique for estimating these parameters in spatial mechanisms using any joint type, based on measurements of displacements, velocities and accelerations and of external forces and torques, for the purpose of building accurate multibody models of mechanical systems. A form of the equations of spatial motion is derived, which is linear in the dynamic parameters and based on multibody simulation code methodologies. Singular value decomposition is used to find the essential parameter set, and minimum parameter set. It is shown that a simulation of a four-bar mechanism (with spherical, universal, and revolute joints) and based on the estimated parameters gives accurate response.     

An Iterative Method of Point Mapping under Cell Reference for the Global Analysis: Theory and a Multiscale Reference Technique

In iterative method of Point Mapping under Cell Reference, a cell co-ordinate system, called cell reference, is built to identify the subregions (cells) in the state space. When the cell reference is equipped with the so-called characteristic functions, it can work as an inspector or a recorder to derive the local dynamics of the subregions from the information provided by the trajectories passing through them. This method can retain the accuracy of the Point Mapping Method but greatly reduce the computational work. In this paper, the theoretic basis for this method is first discussed and a multiscale reference technique is then devised which can select an optimal cell reference and make the method more practicable. Finally, an example for application is presented. It is shown that the present method cannot only accurately and efficiently depict the basins of attraction of a dynamical system but also potentially detect other characteristics of the system.     

A Direct Method for the Identification of the Permeability Field Based on Flux Assimilation by a Discrete Analog of Darcy's Law

Inverse models to determine the permeability are generally based on existing forward models for the pressure. The permeabilities are adapted in such a way that the calculated pressures match the specified pressures in a number of points. To assimilate a priori knowledge about the flux, we introduce the flux assimilation method, which is based on the vector potential–pressure formulation of Darcy's law. Thanks to an unconventional discretization technique – the edge-based face element method – not only the specified pressures, but also specified information about the flux density can easily be assimilated. A relatively simple, but insightful analytical example illustrates the potential of this method.     

Analytic Solution to a Problem of Seepage in a Chequer-Board Porous Massif

A study is made of steady two-dimensional seepage in a porous massif composed by a double-periodic system of white and black chequers of arbitrary conductivity. Rigorous matching of Darcy's flows in zones of different conductivity is accomplished. Using the methods of complex analysis, explicit formulae for specific discharge are derived. Stream lines, travel times, and effective conductivity are evaluated. Deflection of marked particles from the natural direction of imposed gradient and stretching of prescribed composition of these particles enables the elucidation of the phenomena of transversal and longitudinal dispersion. A model of pure advection is related with the classical one-dimensional vective dispersion equation by selection of dispersivity which minimizes the difference between the breakthrough curves calculated from the two models.     

Face-Centered and Volume-Centered Discrete Analogs of the Exterior Differential Equations Governing Porous Medium Flow I: Theory

This paper presents a physics-oriented approach to approximate the continuum equations governing porous media flow by discrete analogs. To that end, the continuity equation and Darcys law are reformulated using exterior differential forms. This way the derivation of a system of algebraic equations (the discrete analog) on a finite-volume mesh can be accomplished by simple and elegant translation rules. In the discrete analog the information about the conductivities of the porous medium and the metric of the mesh are represented in one matrix: the discrete dual. The discrete dual of the block-centered finite difference method is presented first. Since this method has limited applicability with respect to anisotropy and non-rectangular grid blocks, the finite element dual is introduced as an alternative. Application of a domain decomposition technique yields the face-centered finite element method. Since calculations based on pressures in volume centers are sometimes preferable, a volume-centered approximation of the face-centered approximation is presented too.     

Numerische Untersuchung des elastisch-plastischen Verhaltens eines Fadenverbundwerkstoffes mit metallischer Matrix Eine Fallstudie

Übersicht Die Fehlervorhersage eines einachsigen Fadenverbundwerkstoffes mit metallischer Matrix erfordert die Kenntnis des nichtlinearen elastisch-plastischen Verhaltens in mikroskopischer Abmessung. Als Fallstudie wurde ein spezieller Verbundwerkstoff mit einem FEM-Programm untersucht. Im plastischen Bereich wurde das klassische, v. Mises Potential' mit dem neuen Übergangsfließpotential unter Berücksichtigung der plastischen Volumendehnung verglichen. Unter transversaler Normalbelastung zeigte der Verbund deutliche Unterschiede in der Zunahme und der Ausdehnung der lokalen plastischen Zonen. Alle kritischen Verformungsstadien wurden von dem Übergangsfließpotential bei geringerer Belastung erreicht.

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Numerical investigation of the elastic-plastic behaviour of a fibre-reinforced composite with a metallic matrix

Summary Failure prediction of unidirectional fibre-reinforced composite with a metallic matrix needs knowledge about the nonlinear elastic-plastic behaviour in a microscopic scale. A specific composite was investigated using a FEM-program as a case study. In the plastic range the classical v. Mises Potential was compared with the new Transition Flow Potential, taking into account the plastic volume dilatation. Subjected to transverse normal loading the composite showed evident differences in the increase and the spread of locally plastic regions. All critical deformation states were reached by the Transition Flow Potential at lower loading.