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The class of polyhedral coherent risk measures that are used in making decisions under uncertainty is investigated. Operations are introduced on the measures of this class, and properties of these measures are studied. The problems of portfolio optimization based on the profitability-risk ratio for such risk measures are proved to be reducible to the corresponding linear programming problems.

A nonparametric index approach, used to estimate systems with two types of outputs, is considered. Efficiency indices are introduced for such systems, and methods for their calculation are discussed.

An approach is considered to how to allow for the interaction between an ellipsoidal heterogeneity (inclusion) and an elliptic crack in an elastic medium. Using the superposition of perturbed stress states, the boundary conditions are satisfied on the ellipsoidal surface by the method of equivalent inclusion and on the crack surface by the least-squares method. A numerical analysis is carried out. Typical mechanical effects are revealed. In the calculations, the stress state near the ellipsoidal heterogeneity is approximated by a polynomial of the second degree in Cartesian coordinates, whereas the load on the crack surface is simulated by a polynomial of the fourth degree in Cartesian coordinates. In particular cases, the results are in good agreement with the data obtained by other authors

A nonuniform thermal effect on the surface of an elliptic crack in a transversally isotropic medium is studied. Use is made of the previous result on the analogy between elastic and thermoelastic problems for an arbitrary smooth flat crack in isotropic and transversally isotropic materials. A numerical analysis is performed to study how the stress intensity factors depend on the parameters and orientation of the crack

Eshelby's equivalent-inclusion method is extended to a finite number of arbitrarily oriented anisotropic ellipsoidal inclusions in an elastic isotropic matrix under polynomial mechanical loading and heating. The interaction of two identical and two different triaxial ellipsoidal inclusions in an elastic medium is studied as numerical examples. In special cases, the results are compared with those obtained by other authors

Optimization problems with criterion functions and admissible sets symmetric with respect to some transformations are considered. Conditions are studied which guarantee that solutions of these problems exist on a set of points fixed with respect to the initial transformation for continuous problems and on a set of quasifixed points for integer problems.

Abstract: The stress structure inside an ellipsoidal inhomogeneity (inclusion) is studied for polynomial force and temperature interactions of arbitrary order in an elastic isotropic medium. It is shown that with perfect mechanical and thermal contact between the phases, the stressed state within the inhomogeneity is also a polynomial in the cartesian coordinates whose order equals the maximum order of the polynomials for the external force and temperature fields.

Abstract: An interrelationship between the stress intensity factors for plane cracks in isotropic and transversally isotropic media is found for symmetric and obliquely symmetric force and symmetric temperature loadings. It is shown that the stress intensity factors for a plane crack in a transversally isotropic material can be obtained using standard formulas for the transition from the stress intensity factor for the corresponding problem in the case of an isotropic material. As examples, stress intensity factors are found for a circular crack, for a periodic system of circular cracks in a transversally isotropic material under force and temperature loading, etc.

The stress intensity factors (SIFs) are evaluated for flat elliptical cracks located in a transversally isotropic material (cracks are assumed perpendicular to the transtropy axis) under an arbitrary load and symmetric temperature. The SIFs for an elliptical crack in a transversally isotropic medium are determined using the formulas (derived by the author in his previous studies) of transition from an isotropic to transversally isotropic material and the relative problem for an isotropic medium. It is proved that these formulas can be employed for an arbitrary homogeneous transversally isotropic material (no matter whether the roots of some characteristic equation of the material are real or complex) with an arbitrary flat crack or a system of coplanar flat cracks, including elliptical ones, under an arbitrary load and symmetric temperature. A transversally isotropic material with two coplanar elliptical cracks is considered as an illustrative example. The dependences of the SIFs on the parameters of cracks and their arrangement at a decreasing temperature are presented.

The possibility exists for the full-strength state of the juncture between an isotropic medium and an orthotropic inclusion when the phases are in ideal contact and the stresses are tensile (compressive). It is found that an ellipsoid, whose semiaxes depend heavily on the elastic characteristics of the matrix and inclusion and the parameters of external loading, is such a form. A system of transcendental equations is obtained to search for the unknown semi-axes. Previous data confirm the results obtained. Numerical investigations are performed, and interrelations are established between geometrical parameters and loading conditions and the properties of the phases for transversally isotropic and orthotropic inclusions.

This paper presents a comprehensive study of SiH4 doping of GaN grown by metal-organic chemical vapour deposition (MOCVD) on sapphire substrates. The silicon incorporation appeared to be virtually independent of the growth temperature and a 1:1 relationship between the carrier concentration and the input mol fraction silane was found. Optical microscopy was used to study the morphology of the layers where photoluminescence provided a measure of the optical quality. The electrical properties were determined with Hall measurements and a comparison was made with ellipsometry measurements. The relationship between the free carrier concentration and the ellipsometric data made it possible to use ellipsometry as an alternative, quick, contactless and non-destructive technique to determine the free electron concentration in GaN layers doped with silicon.

Abstract: The feasibility of controlling the stress intensity factors for plane cracks of arbitrary form (distributed in a plane perpendicular to isotropy axis in a transversely isotropic material) which are subjected to symmetric mechanical loading by heating the material is demonstrated by using the congruence theorem and making an analogy between isotropic and transversely isotropic materials based on the theory of thermoelasticity. It is shown that a thermal load which fully compensates for the mechanical load can be created within the range in which the material behaves linearly. If it turns out to be technically impossible to create the necessary thermal load, a simpler temperature regime can be chosen that will “cancel” the mechanical force field with a certain factor of safety.

Abstract: An established thermoelastic analogy for transversally isotropic media and the Hopf principle of the maximum are used to prove a comparison theorem for estimating the stress for plane cracks of complex shape (in the place perpendicular to the plane of isotropy) in the presence of force and temperature loading. This approach makes it possible to construct simple bounding contours and provide two-sided estimates forK 1. The effectiveness of this approach is demonstrated on specific examples.

A new tangential cone for the graph of a set-valued mapping is defined. Local solvability and Buligan-differentiability of the set-valued mapping are considered. An upper convex approximation for the marginal function is derived. The new cone is compared with known tangential cones.

Abstract: Under prescribed thermoelastic stresses and known properties of the matrix and the inclusion in an elastic medium with an inhomogeneity, we find the shape of inhomogeneity that leads to an extremal value of the thermoelastic energy. From the necessary conditions for an extremum of the thermoelastic energy functional we find a condition for seeking the interface. For the case of isotropic cornponents and under loads of stretching (compression) type and uniform heating of the medium the shape of the inclusion can be found explicitly within a certain range of initial parameter values. The results of numerical study are presented and analyzed. One table.

Abstract: We consider the inverse problem consisting of determining the unknown shape of an elastic imperfection contained in a thin plate from the condition of equal strength in the stressed state along the phase interface surface. It is shown that such a state is attained in the case of an elliptic imperfection whose shape depends on the values of the applied moments and the mechanical properties of the component phases. It is established that for the geometry found for the imperfection the sum of the moments is constant and the second invariant of the deviator of the stress tensor is superharmonic over the entire plate. Numerical computations are carried out. In special cases the results obtained coincide with known data. One figure. Bibliography: 5 titles.

We examine the properties of the stressed state of the thin intermediate zone separating a nonhomogeneous inclusion from the matrix. Explicit expressions are given for the components of the stress tensor in the thin coating depending on the properties of the phases and the load parameters.

Two problems of statistical thermoelasticity are considered for an isotropic medium with spherical inhomogeneity. It is assumed that a medium with an inclusion is subjected to piecewise-uniform heating or the action of a homogeneous heat flux. As accurate solutions of the problems are constructed for various types of thermal and mechanical action at the phase interface.

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This article examines a problem of static thermoelasticity for an isotropic medium with an ellipsoidal cavity. The orientation of the cavity relative to the heat flow is assumed to be arbitrary. The solution of the problem is constructed in closed form and is expressed through incomplete elliptic integrals of the first and second kinds. In cases when the ellipsoid degenerates into a spheroid or sphere, the stress-strain state can be written in elementary functions of the ellipsoidal coordinate.

We study the structure of the stresses inside an ellipsoidal irregularity. We obtain a result that generalizes the theorem on polynomial conservation for thermoelastic problems under homogeneous and linear external stress and temperature fields when there is perfect thermal and mechanical contact on the phase interface. We show by a specific example how significantly the stress structure inside the irregularity changes when the form of interaction on the boundary surface changes.

Abstract: We consider an elastic isotropic material containing a fibrous inclusion or a void subject to loads. We study the possibility of increasing the strength of the material using “control action.” As examples of such actions we consider an additional external load, application of pressure in the cavities (voids), and variation of the temperature of uniform heating of the medium with the inclusions. Studies are carried out for two-dimensional problems of linear elasticity theory. From the class of all possible actions we single out a subclass leading to states of equal strength along the phase interface. We study the efficiency of these types of action and show that in certain cases they lead to a manifold increase in the strength of the material. We give the results of numerical studies.

Conclusions Dry ball heated to 60°C reduces the pressure during ramming to a third. Preheating the dry ball of the stated composition should be done to a temperature not exceeding 90°C.

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