Zur Prandtlschen Membrananalogie der Torsion

Summary Ludwig Prandtl's membrane analogy reduces the problem of torsion, which involves the theory of elasticity, to that of calculating a membrane, which relates to structural statics; this enables a solution to be found by means of the known methods applied in structural statics.

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Es sei mir gestattet, meinen Glückwunsch an Prof.J. Ackeret durch diesen erg?nzenden Hinweis auf eine bekannte Arbeit seines auch von ihm verehrten LehrersLudwig Prandtl darzubringen.     

弹塑性运动裂纹的动力位错解

本文基于细观观点,用线排列动力位错模拟运动裂纹,通过叠加方法处理弹塑性运动裂纹问题,求出了各种情形下的动力张开位移,从而给出了运动裂纹的一个弹塑性失稳判据,并就Ⅰ,Ⅱ,Ⅲ型运动裂纹展开进行了讨论.     

A theory of dislocations and disclinations in elastic plates

The problem of the stress state of a thin elastic plate, containing dislocations and disclinations, is considered using Kirchhoff's theory. The problem of the equilibrium of a multiply connected plate with Volterra dislocations with specified characteristics is formulated. The problem of the flexure of an annular slab resulting from a screw dislocation and a twisting disclination is solved. The solutions of problems of concentrated (isolated) dislocations and disclinations in an unbounded plate as well as the dipoles of dislocations and disinclinations are found. It is shown that a screw dislocation in a thin plate is equivalent to the superposition of two orthogonal dipoles of torsional disclinations. By taking the limit from a discrete set of defects to their continuous distribution, a theory of thin plates with distributed dislocations and disclinations is constructed. Solutions of problems of the flexure of circular and elliptic plates with continuously distributed disclinations are obtained. An analogy is established between the problem of the flexure of a plate with defects and the plane problem of the theory of elasticity with mass forces, and also between a plane problem with dislocations and disclinations and the problem of the flexure of a plate with specified distributed loads.     

A Three-dimensional Continuum Theory of Dislocation Plasticity - Modelling and Application to a Composite

The increasing demand for materials with well defined microstructure, which is accompanied by the advancing miniaturization of devices calls for physically motivated, dislocation-based continuum theories of plasticity. Only recently rigorous techniques have been developed for performing meaningful averages over systems of moving, curved dislocations, yielding evolution equations for a higher order dislocation density tensor. Our continuum dislocation theory allows for generalizing the planar system towards a three-dimensional system, where dislocations may have arbitrary orientation and curvature. With the inclusion of curvature, the theory naturally takes into account a deformation-induced increase in the overall dislocation density without having to invoke ad-hoc assumptions about dislocation sources. A numerical implementation and some benchmark tests of this continuum theory for dislocation dynamics has already been discussed in the literature. In this paper, we apply this continuum theory to composite materials, where we analyze a plastically deforming matrix with an elastic inclusion. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Torsion of prismatic elastic bodies containing screw dislocations

The problem of the stressed state of a prismatic anisotropic rod containing screw dislocations, the axes of which are parallel to the rod axis, is considered. Such defects may arise during the growth of filamentary crystals (metal “whiskers”), and may also exist in multiply connected cylindrical structures. The torsion of an anisotropic elastic bar with a multiply connected cross-section is investigated initially, assuming that the stresses and strains are single-valued but dispensing with the requirement that the warping function should be single-valued. The boundary-value problem is formulated in terms of the Prandtl stress function, which, unlike the warping function, is single-valued in a multiply connected region. A variational formulation of the boundary-value problem for the stress function is given. From the variational principle obtained a torsion boundary-value problem is formulated when there are lumped or continuously distributed dislocations. A modification of the membrane analogy for the torsion problem is proposed which takes into account the presence of dislocations. General theorems of the theory of the torsion of a rod containing dislocations are formulated. An effective formula is derived for the angle of torsion of a bar due to a specified dislocation distribution. Problems on dislocations in a thin-walled rod and a rectangular anisotropic bar are solved.     

The Wholeness Axioms and V=HOD

If the Wholeness Axiom wa is itself consistent, then it is consistent with v=hod. A consequence of the proof is that the various Wholeness Axioms are not all equivalent. Additionally, the theory zfc+wa is finitely axiomatizable. Received: 16 February 1999 / Revised version: 1 June 1999     

The minimum shift design problem

The min-Shift Design problem (MSD) is an important scheduling problem that needs to be solved in many industrial contexts. The issue is to find a minimum number of shifts and the number of employees to be assigned to these shifts in order to minimize the deviation from workforce requirements. Our research considers both theoretical and practical aspects of the min-Shift Design problem. This problem is closely related to the minimum edge-cost flow problem (MECF), a network flow variant that has many applications beyond shift scheduling. We show that MSD reduces to a special case of MECF and, exploiting this reduction, we prove a logarithmic hardness of approximation lower bound for MSD. On the basis of these results, we propose a hybrid heuristic for the problem, which relies on a greedy heuristic followed by a local search algorithm. The greedy part is based on the network flow analogy, and the local search algorithm makes use of multiple neighborhood relations. An experimental analysis on structured random instances shows that the hybrid heuristic clearly outperforms our previous commercial implementation. Furthermore, it highlights the respective merits of the composing heuristics for different performance parameters.     

Näherungstheorien für die Berechnung von Strömungsgrenzschichten

Summary Calculations of turbulent boundary layers in a compressible flow with heat transfer at the wall require the knowledge of empirical laws for the wall-friction and dissipation valid under these general conditions. An approximate theory is developed which involves a generalization of the empirical wall-friction and dissipation laws well known for the incompressible case to be valid up to about Mach number 10 and arbitrary heat transfer conditions. Universal functions occurring in this theory have been evaluated so that practical calculations may be performed in a short time. Comparisons of theoretical results with experiments show good agreement. The fundamental idea of deriving averaged boundary layer conditions may also be applied to the complete equations ofNavier andStokes, thus creating a fairly simple method of checking or improving the results ofPrandtl's simplified theory, e.g., in cases, where the conditionsδp/δy=0 and high Reynolds numbers are no longer fulfilled. This approximate theory, however, has so far been advanced to a preliminary state only.      

The Clebsch transformation and its capabilities towards fluid and solid mechanics

In fluid dynamics, Clebsch made use of the representation for the velocity field in terms of three potentials Φ, α, β in order to construct a first integral of the equations of motion in case of an inviscid flow with vortices. Apart from this, he received a self-adjoint form of the equations allowing for deriving them from a variational formulation. In latter times the Clebsch transformation has been applied to different physical problems, for instance to baroclinic flow, Maxwell equations in classical electrodynamics [1], in Magnetohydrodynamics and even quantum theory within the context of a quantization of vortex tubes. Viscous flow, however, has not yet been formulated in terms of Clebsch variables to our best knowledge. It is the aim of this paper to demonstrate how Clebsch variables can be applied to viscous flow on the one hand, leading to a first integral of Navier-Stokes equations as a first example. As a second example, solid mechanics is considered: by making use of an analogy between vortices in fluid flow on the one hand and dislocations in crystals on the other hand, a dynamic theory of dislocations can be established by using a certain modification of the Clebsch transformation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the carleman's model for the boltzmann equation and its generalizations

Summary T. Carleman [1] showed that a certain 2 x 2 hyperbolic, semilinear system of partial differential equations has many features in common with the integro-differential equation ofBoltzmann. We produce here a global existence theory for theCauchy problem for this system and its generalizations. The main result is derived by exploiting the fact that the solution of a certainRiccati equation depends isotonically (with respect to a suitable partial order) on the forcing term and the initial data. Research supported by the National Science Foundation, Grant NSF G-21053.     

Geometries involving affine connections and general linear connections

Summary Part I of this paper is concerned with the theory of differential invariants of a symmetric affine connection and a general linear connection of theK?nig type. Part II deals with a geometry in which the components of the affine connection areChristoffel symbols, and the general linear connection is of a special sort. This section can be considered in part as an extension of theEinstein-Mayer (5) geometry,     

Duale Optimierungsaufgaben und Sattelpunktsätze

Zusammenfassung Es wird der formale Zusammenhang zwischen der Existenz eines Sattelpunktes und einem Paar dualer Optimierungsaufgaben angegeben. Dabei werden Voraussetzungen wie konvex oder konkav nicht benötigt. Die Theorie ist so aufgezogen, daß dieFenchelsche Theorie der konjugierten Funktionen gut in diesen Rahmen eingebaut werden kann. An vielen Beispielen wird der Zusammenhang mit bekannten Dualitätssätzen dargestellt.

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Summary We treat the formal relations between the existence of a saddle-point and a pair of dual programming problems without using assumptions like convex or concave. The theory is built in such a way, thatFenchels theory of conjugate functions fits very well in its frame. The relationship between the theory and other well known duality theorems are explained by many examples.

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Vorgel. v.:W. Krelle.     

Aerodynamische Kräfte auf schlanke Rotationskörper

Summary Comparison between measurements of the normal force of fiveslender bodies of revolution with different base areas in supersonic flow and the simple airshiphull theory ofMunk or the more accurate linearised theory ofpotential flow show considerable discrepancy not only at large, but also at small angles of attack.To get better agreement with the experimental results, it is shown to be more important to estimate theeffects of viscosity than to consider higher orders of potential flow theory.

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Vorgetragen am Astronautischen Kongress 1953 in Zürich, zufolge Abwesenheit des Verfassers durch HerrnB. Chaix, Institut für Aerodynamik der ETH.     

Bimeasures and their integral extensions

Summary The object of this paper is to motivate three papers byM. Morse andW. Transue, which will presently appear, where the Authors will lay the foundations of a general theory of C-bimeasures and their integral extensions. To Mauro Picone on his 70^th birth day.     

Über die Zulässigkeit des Verfahrens von Galerkin bei linearen,nichtselbstadjungierten Eigenwertproblemen

Summary In this report it is proved that the method ofGalerkin remains valid for a certain class of linear, non-self-adjoint eigenvalue problems. This class contains all eigenvalue problems which arise from originally self-adjoint problems by addition of a linear differential expression which destroys the former self-adjointness.Helge von Koch's theory of infinite determinants is used in deriving the above result.     

Eine Ungleichung von van der Corput und Kemperman

The tool of van der Corput's difference theorem in the theory of uniform distribution is his so-called fundamental inequality.Kemperman showed that even the non-constructive proofs of the difference theorem byBass, Bertrandias andCigler implicitly use a more general form of van der Corput's fundamental inequality. In this article, the inequality which constitutes the basis of the difference theorem will be proved under a very general setting, applications will be demonstrated in connection with the uniform distribution of products of linear forms and a quantitative version of the difference theorem, i. e. an estimation of discrepancies, will be derived.     

A simple proof of a general duality theorem of convex programming

Summary The first part of this note presents concisely and partially proves in logical terms the relations betweenUzawa's andKuhn andTucker's equivalence theorems of nonlinear programming. In the second part we give simple mathematical proofs of two lemmata linking theKuhn-Tucker conditions and dual solutions and use them to establish a nonlinear duality theorem of considerable generality.

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Zusammenfassung Im ersten Abschnitt dieses Beitrags demonstrieren wir in kurzer, symbolischlogischer Form die Relationen zwischenUzawas undKuhn undTuckers Äquivalenzsätzen des nichtlinearen Programmierens. Im zweiten Teil geben wir mathematische Beweise zweier Lemmata, die dieKuhn-Tucker-Bedingungen und duale Lösungen in Zusammenhang bringen, und benützen sie und die logische Struktur des ersten Teils, um ein nichtlineares Dualitätstheorem von besonderer Allgemeinheit zu beweisen.

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The author is indebted toP. Van Moeseke for teaching him programming theory some time ago, and toD. Bent for helpful comments and criticism.

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Vorgel. v.:G. Tintner.     

Zur Theorie elastisch-plastischer Wellen in dünnen Stäben

Summary The simple classical theory of elastic-plastic loading waves in thin rods due to,v. Kármán, Taylor, andRakhmatulin is generalized, according to a proposal ofMazzoleni, by taking into account the time dependent variation of the cross sectional area. Thus, an essential difference arises between tensional and compressional waves. Furthermore, the theory becomes applicable to the limiting case of ideally plastic materials which had to be excluded so far. Here, only compressional waves are possible. The ratio of the plastic and the elastic wave velocities is approximated by , whereY denotes the uniaxial yield stress andE is Young's modulus.

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Der Aufsatz stellt ein Teilergebnis des von der Deutschen Forschungsgemeinschaft (DFG) geförderten Vorhabens Li 115/2 dar.     

Anwendung derWienerschen Filtertheorie auf Probleme der Produktion und Lagerhaltung

Zusammenfassung DieWienersche Filtertheorie wird auf einfache Probleme der Produktion und Lagerhaltung angewandt. Diese Theorie gestattet, im Gegensatz zu den sonst verwendeten regelungstechnischen Verfahren, die Berechnung der kostenoptimalen Produktionspolitik eines vorgegebenen Modells. Die Einschränkungen, die dasWiener-Verfahren der Menge der regelungstechnisch unter-suchbaren Modelle auferlegt, werden diskutiert, und es wird gezeigt. daß sich mit diesem Verfahren ein Großteil der ökonomisch relevanten Modelle erfassen läßt.

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Summary TheWiener Filtering Theory is applied to simple problems of production and inventory. In contrast to approaches using servo-mechanical methods, this theory allows the calculation of the optimal production policy of a given model. The restrictions caused by theWiener-approach on the set of models being investigable by servo-mechanical methods are discussed. It is shown that this approach may be applied to many of the economically relevant models.

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Vorgel. v.:F. Ferschl.     

Le flux d'énergie en mécanique: Application aux turbo-machines

Summary The concept of flux of energy, first introduced in electrodynamics byJ. H. Poynting, can be extended to the domain of fluid-mechanics. It conduces to recognize in the flux of energy for a continuous portion of matter two parts: the first one, which can be called ‘transported’, corresponds to the energy localized in the matter and carried with it as a consequence of its displacement; the other results from the work developed by the stresses and can be called ‘transmitted’. The expression of this flux of energy in the general case of variable motion of a compressible fluid introduces the concept of enthalpy. Its application to an extension ofHelmholtz's theorem on the conservation of vorticity in a perfect fluid, according which bound vortexes have a definite relative motion across the fluid, contributes to explain the mechanism by which, even in the case of grids with a large pitch: chord ratio (Kaplan turbines, screw propellers), the energy of all parts of fluid is finally transmitted to the moving blades.