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.

---

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.

     

Existence and Uniqueness of Time-Periodic Physically Reasonable Navier-Stokes Flow Past a Body

Let be a three-dimensional exterior domain of class C^2,, 0<<1. Assume that a Navier-Stokes liquid is moving in under the action of a body force F that is time-periodic of period T, and that the velocity of the liquid is zero at spatial infinity. In this paper we show that, if F satisfies suitable conditions, and its norm, in appropriate function spaces, is sufficiently small, there is at least one time-periodic strong solution. Furthermore, the velocity field v of such a solution decays to zero for large |x| as |x|^–1 and its spatial gradient decays as |x|^–2, both uniformly in time. In addition, the pressure p decays like |x|^–2 and its gradient like |x|^–3, for almost all t[0,T]. In the special case where F is time-independent, these solutions are also time-independent and coincide with Finns physically reasonable solutions [4]. Moreover, we show that our time-periodic solutions are unique in a very large class, namely, the class of time-periodic weak solutions satisfying the energy inequality and with corresponding pressure fields verifying mild summability conditions in ×[0,T].     

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 ‘roller-coaster’ experiments for nonlinear oscillators

We consider the dynamics of roller-coaster type experimental models used as analog devices for nonlinear oscillators. It is shown how to chose the shape of the track in order to achieve a desired oscillator equation, in terms of the are length coordinate or its projection onto the horizontal. Explicit calculations are carried out for the linear oscillator, the so-called escape equation, the two-well Duffing oscillator, and the pendulum.     

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.     

Ši’lnikov Chaos in the Generalized Lorenz Canonical Form of Dynamical Systems

This paper studies the generalized Lorenz canonical form of dynamical systems introduced by elikovský and Chen [International Journal of Bifurcation and Chaos 12(8), 2002, 1789]. It proves the existence of a heteroclinic orbit of the canonical form and the convergence of the corresponding series expansion. The ilnikov criterion along with some technical conditions guarantee that the canonical form has Smale horseshoes and horseshoe chaos. As a consequence, it also proves that both the classical Lorenz system and the Chen system have ilnikov chaos. When the system is changed into another ordinary differential equation through a nonsingular one-parameter linear transformation, the exact range of existence of ilnikov chaos with respect to the parameter can be specified. Numerical simulation verifies the theoretical results and analysis.     

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.     

Unsteady Viscous Flows about Bodies: Vorticity Release and Forces

The force (drag and lift) exerted on a body moving in a viscous fluid is expressed via the free and bound vorticity moments, and the role of vortex shedding is discussed. The formulation encompasses classical, inviscid flows, and leads to efficient computational methods. Numerical results for a few prototype flows are presented.     

Analysis of the USBM wettability test

In this paper, we analyse the capillary pressure curves obtained by the centrifuge method in order to perform the USBM wettability test. The physical displacement mechanisms present both in the porous plate and in the centrifuge method, are described for different cases of wettability of the pore surface.The wetting fluid is defined as the fluid being at the lower pressure while displacing the other fluid, and this displacement is defined as imbibition. On the other hand, the process in which the fluid under the lower pressure is the displaced fluid is defined as drainage. The capillary pressure is defined as the positive pressure difference between the two fluids. By adhering to these definitions, there is a unique and consistent terminology for the same physical process: the displacement of oil by water in an oil wet system and the displacement of water by oil in an water wet system are both designated as drainage.An important result is that the centrifuge method is limited to the determination of drainage capillary pressure curves for strongly oil or water wet samples. There is no capillary equilibrium possible when a water wet sample is centrifuged under water because the wetting phase is under higher pressure than the nonwetting phase; the resulting forced imbibition curve should not be called a capillary pressure curve. For samples with bicontinuous fractional wettability, the curves obtained by the centrifuge method correspond to combination displacement, i.e. a combination of equilibrium drainage and forced imbibition coupled with blob mobilisation.     

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.

---

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.

     

Brittle to Quasi-Brittle Transition

Abstract. The present study focuses on the kinetic and non-deterministic aspects of the brittle to quasi-brittle transition. A solid is approximated by a lattice formed by the interacting continuum particles and the evolution of damage is estimated using particle dynamics. The onset of transition is measured by the rate of the change of correlation length. The proposed method is illustrated on the examples of creep rupture, strain localization and dynamic expansion of a circular void in a brittle plate.Sommario. Viene posta l'attenzione sugli aspetti cinetici e non deterministici della transizione dal comportamento fragile a quello quasi-fragile. Un solido viene approssimato da un reticolo formato da particelle interagenti e l'evoluzione del danno viene stimata tramite la dinamica delle particelle. L'inizio della transizione viene misurato tramite la variazione della lunghezza di correlazione. Il metodo proposto viene illustrato su esempi di rottura per creep, localizzazione della deformazione e l'espansione di un foro circolare in una piastra fragile.     

On the dynamic behaviour of a flexible beam carrying a moving mass

The behaviour of a system containing a mass traveling on a cantilever beam is considered. The mass is induced to move by an applied force as opposed to the case which has been considered in most literature where the position of the moving mass is assumed to be known and independent of the motion of the beam. Furthermore, the system to be discussed has the unique characteristic that the motions of the mass and the beam are coupled. The mathematical model of the system includes two coupled nonlinear integral/partial differential equations which are impossible to solve analytically and are difficult to solve numerically in their original form. As a remedy, the solution is discretized into space and time functions and the equations of motion are reduced to a set of ordinary differential equations. The shape function is chosen so that it satisfies the boundary conditions of the beam as well as the transient conditions imposed by the traveling mass. This choice of the shape function, which considers the mass-beam interaction, provides an improvement over the conventional method of using a simple cantilever beam mode shapes.The ordinary differential equations of motion using the improved shaped functions, are solved numerically to obtain the dynamic behaviour of the system. The results illustrate the validity of the model, and demonstrate the advantages of the improved model to the un-improved equations.     

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.     

Upscaling in Subsurface Transport Using Cluster Statistics of Percolation

Transport/flow problems in soils have been treated in random resistor network representations (RRNs). Two lines of argument can be used to justify such a representation. Solute transport at the pore-space level may probably be treated using a system of linear, first-order differential equations describing inter-pore probability fluxes. This equation is equivalent to a random impedance network representation. Alternatively, Darcys law with spatially variable hydraulic conductivity is equivalent to an RRN. Darcys law for the hydraulic conductivity is applicable at sufficiently low pressure head in saturated soils, but only for steady-state flow in unsaturated soils. The result given here will have two contributions, one of which is universal to any linear conductance problem, i.e., requires only the applicability of Darcys (or Ohms) law. The second contribution depends on the actual distribution of linear conductances appropriate. Although nonlinear effects in RRNs (including changes in resistance values resulting from current, analogous to changes in matric potential resulting from flow) have been treated within the framework of percolation theory, the theoretical development lags the corresponding development of the linear theory, which is, in principle, on a solid foundation. In practice, calculations of the nonlinear conductivity in relatively (compared with soils) well characterized solid-state systems such as amorphous or impure semiconductors, do not agree with each other or with experiment. In semiconductors, however, experiments do at least appear consistent with each other.In the limit of infinite system size the transport properties of a sufficiently inhomogeneous medium are best calculated through application of critical rate analysis with the system resistivity related to the critical (percolating) resistance value, R_c. Here well-known cluster statistics of percolation theory are used to derive the variability, W (R,x) in the smallest maximal resistance, R of a path spanning a volume x³ as well as to find the dependence of the mean value of the conductivity, (x). The functional form of the cluster statistics is a product of a power of cluster size, and a scaling function, either exponential or Gaussian, but which, in either case, cuts off cluster sizes at a finite value for any maximal resistance other than R_c. Either form leads to a maximum in W (R,x) at R=R_c. When the exponential form of the cluster statistics is used, and when individual resistors are exponential functions of random variables (as in stochastic treatments of the unsaturated zone by the McLaughlin group [see Graham and MacLaughlin (1991), or the series of papers by Yeh et al. (1985, 1995), etc.], or as is known for hopping conduction in condensed matter physics), then W (R,x) has a power law decay in R/R_c (or R_c/R, the power being an increasing function of x. If the statistics of the individual resistors are given by power law functions of random variables (as in Poiseiulles Law), then an exponential decay in R for W (R,x) is obtained with decay constant an increasing function of x. Use, instead, of the Gaussian cluster statistics alters the case of power law decay in R to an approximate power, with the value of the power a function of both R and x.     

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.     

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.     

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.     

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.     

Transport of Three-Phase Hyper-Saline Brines in Porous Media: Examples

This paper presents a number of applications of a new code which can simulate the transport of high temperature three-phase (gas, liquid, solid) hyper-saline fluids in a porous medium. The examples presented demonstrate that multiple phase changes occur as the fluid state evolves across the H₂O–NaCl phase diagram. Multi-phase flows occur in a variety of situations, including a horizontal domain with fluid withdrawal, a vertical counter-flowing salt-pipe, and a horizontal domain with saturation shocks and expansion waves. The code is also used to simulate heat, water and salt flows in a large scale model (10s of km) of the Taupo Volcanic Zone, New Zealand.