Eliciting conditional and unconditional rank correlations from conditional probabilities

Causes of uncertainties may be interrelated and may introduce dependencies. Ignoring these dependencies may lead to large errors. A number of graphical models in probability theory such as dependence trees, vines and (continuous) Bayesian belief nets [Cooke RM. Markov and entropy properties of tree and vine-dependent variables. In: Proceedings of the ASA section on Bayesian statistical science, 1997; Kurowicka D, Cooke RM. Distribution-free continuous Bayesian belief nets. In: Proceedings of mathematical methods in reliability conference, 2004; Bedford TJ, Cooke RM. Vines—a new graphical model for dependent random variables. Ann Stat 2002; 30(4):1031–68; Kurowicka D, Cooke RM. Uncertainty analysis with high dimensional dependence modelling. New York: Wiley; 2006; Hanea AM, et al. Hybrid methods for quantifying and analyzing Bayesian belief nets. In: Proceedings of the 2005 ENBIS5 conference, 2005; Shachter RD, Kenley CR. Gaussian influence diagrams. Manage Sci 1998; 35(5) [15].] have been developed to capture dependencies between random variables. The input for these models are various marginal distributions and dependence information, usually in the form of conditional rank correlations. Often expert elicitation is required. This paper focuses on dependence representation, and dependence elicitation. The techniques presented are illustrated with an application from aviation safety.     

Mechanics of curvilinear electronics and optoelectronics

Recent advances in materials, mechanics and fabrication strategies have successfully realized high-performance stretchable electronic and optoelectronic systems that are capable of conformally wrapping onto complex, soft, curvilinear surfaces. This capability enables lots of novel applications that were not possible through conventional electronics that is rigid and flat in nature. In this review, we summarize the advances in mechanics of curvilinear electronics and optoelectronics. Three typical forms of curvilinear systems will be discussed, including fixed hemispherical, tunable hemispherical and general curvilinear shapes. These mechanics models and analyses have shown to be very powerful in design and optimization of such systems, and can also provide important insights to the development of other related systems that combine rigid elements with soft substrates.     

Debonding of rigid curvilinear inclusions in longitudinal shear deformation

The problem of two symmetrically placed interface cracks at rigid curvilinear inclusions under longitudinal shear deformation is considered. A solution valid for arbitrary inclusion shapes is found. It depends on a parameter β describing the cracks. For $\text{β = e}{}^{\mathrm{i\alpha }}$ where α is an angle, the cracks lie in the interface. For β real and greater than unity, we have two radial cracks emanating from a curvilinear cavity. The solution for $\text{β = 1}$ corresponds to a completely debonded inclusion.Examples of elliptic, square with rounded corners, and rectangular inclusions are worked out in detail. It is shown that the crack tip stress intensity factor becomes infinite for interface cracks terminating at cusps and corners. This phenomenon is attributed to the change in the nature of the singularity as the crack tip approaches a cusp or corner. The singularity is three-quarter power at a cusp and two-thirds power at a corner of a rectangular inclusion. Finally, the application of the results to composite materials is indicated.     

Antiplane problem of curvilinear cracks in bonded dissimilar materials

A general solution to the antiplane problem of curvilinear cracks in bonded dissimilar materials is provided. The analysis is based upon the Hilbert problem formulation and the technique of analytical continuation. To illustrate the use of the present approach, detailed results are given for a single circular-arc crack lying along the interface between dissimilar materials. The expressions of the complex potentials are derived explicitly in both the unit disk and the surrounding medium. Both the stress intensity factors and contact stress are provided in an explicit form and the former are verified by comparison with existing ones. The effect of material and geometrical parameters upon the contact stress and stress intensity factors has also been discussed and shown in graphic form.     

Toward the problem of static deformations in curvilinear rods

The article presents expressions for deformation tensor components, potential energy, differential equations, and boundary conditions as applied to a curvilinear rod with axis curved along a circle of given radius and subject to the effect of concentrated tensile and bending loads. Analytical solutions of three boundary value problems on bending and tension of a cantilever rod with axially constant cross section are given under some simplifying assumptions. Characteristics of the stress-strain state of curvilinear rods are noted for various methods of loading.Translated from Problemy Prochnosti, No. 6, pp. 84–89, June, 1991.     

Charge separation of plasma flow in curvilinear magnetic field

The energy spectra of ions occurring in various charged states in a flow of titanium plasma in a steady-state vacuum arc discharge have been studied. It is established that, during the motion of this flow through a plasma transport system based on a curvilinear magnetic field, ions with different charges are spatially separated. As a result, at the system output, ions with greater charges are concentrated in the inner part of the plasma flow, so that the average charge of ions in this region is higher than that in the outer part of the flow.     

A system of curvilinear cracks in an isotropic elastic half-plane

The plane elasticity problem of an arbitrary system of curvilinear cracks in an isotropic elastic half-plane bonded to another half-plane consisting of a different isotropic elastic material is formulated by using the complex variable technique and reduced to a complex Cauchy type singular integral equation along the cracks. The special cases of a half-plane with a stress-free boundary and of a periodic array of curvilinear cracks are also treated. The numerical techniques available for the solution of complex Cauchy type singular integral equations are presented and a discussion on them is made. Finally, three applications to some special cases of straight or circular-arc-shaped cracks are made.

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Résumé Le problème de l'élasticité plane d'un système arbitraire de fissures curvilignes dans un demi-plan élastique isotrope joint a un autre demi-plan constitué d'un autre matériel élastique isotrope est formulé en utilisant la technique des variables complexes et reduit a une équation intégrale singulière complexe de type Cauchy au long des fissures. Les cas particuliers d'un demi-plan avec une limite libre de contraintes et d'une série périodique de fissures curvilignes sont aussi étudiés. Les techniques numériques disponsibles pour la solution d'équations intégrales singulières complexes de type Cauchy sont présentées et une discussion sur elles est faite. Finalement, trois applications à quelques cas particuliers de fissures droites ou ayant la forme d'un arc de circle sont faites.

     

Array of periodic curvilinear cracks in an infinite isotropic medium

Summary The problem of an array of periodic curvilinear cracks in an infinite isotropic medium under conditions of generalized plane stress or plane strain is reduced, by using the method of complex potentials of Muskhelishvili [1], to a complex Cauchy-type singular integral equation on one of the cracks, which can be further numerically solved by reduction to a system of two real Cauchytype singular integral equations and application of the Lobatto-Chebyshev method of numerical solution of such equations. Applications to the cases of arrays of straight or arc-shaped cracks are also given.

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Anordnung von periodisch gekrümmten Rissen in einem unendlichen, isotropen Medium.

Zusammenfassung Das Problem der Anordnung periodisch gekrümmter Risse in einem unendlichen, isotropen Medium wird untersucht. Der ebene Spannungs-bzw. Dehnungszustand wird für einen der Risse mit Hilfe komplexer Potentiale des Muskhelishvilischen Verfahrens in eine komplexe Cauchysche singuläre Integralgleichung reduziert. Die numerische Lösung des Problems erfolgt durch eine weitere Reduktion in ein System mit zwei Cauchyschen singulären Integralgleichungen unter Anwendung des Lobatto-Chebyshevschen Verfahrens. Beispiele für den Fall der Anordnung gerader wie auch bogenförmiger Risse werden angegeben.

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Propagation of cracks near an arbitrary curvilinear hole

An approximate method of deriving a function mapping the exterior of a curvilinear hole with two curvilinear cracks on the exterior of a unit circle is proposed. The stress state near a curvilinear hole and two rectilinear cracks in a plane field was studied for the case of planes in uniaxial and omnidirectional tension. Simple relations were obtained for the determination of critical loads required to start crack propagation.     

Hydrodynamics and heat transfer in curvilinear channels of rectangular cross section

Experimental results of a study of hydraulic resistance and heat transfer in helical channels of rectangular cross section for laminar and turbulent flows in the isothermal and nonisothermal regimes are given.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 6, pp. 994–1000, June, 1978.