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The electroelastic problem of the stress state of an electroelastic transversely isotropic medium in which a rigid disk-shaped inclusion is displaced under a load non-centrally applied to its surface is solved exactly. Closed analytical expressions for displacements and stress distributions in the piezoelectric material below the inclusion are derived. The solutions of the static electroelastic equations for a transversely isotropic body represented in terms of harmonic functions are used. The problem is reduced to pair integral equations. Expressions for the stress state of a purely elastic transversely isotropic material with a disk-shaped inclusion under non-central load on its surface the follow from the obtained analytical formulas as partial cases. The effect of the coupling of the mechanical and electric fields on the stress distribution below the rigid inclusion under non-central loading is studied numerically. [ABSTRACT FROM AUTHOR]

Polyhedral coherent risk measures and their worst-case constructions with respect to the ambiguity set are considered. For the case of the discrete distribution and polyhedral ambiguity set, calculating such risk measures reduces to linear programming problems. The distributionally robust portfolio optimization problems based on the reward-risk ratio using worst-case constructions with respect to the polyhedral ambiguity set for these risk measures and average return are analyzed. They are reduced to the appropriate linear programming problems. [ABSTRACT FROM AUTHOR]

Based on a rigorous mathematical statement that takes into account the coupling of the mechanical and electric fields, the problem of the compression of two electroelastic transversely isotropic half-spaces is solved. One of the half-spaces contains an inclined near-surface notch of elliptical cross-section. The analytical solution of the problem is found by representing the solution of the static equations of electroelasticity in terms of harmonic functions and by reducing the boundary-value problem to an integro-differential equation with an unknown domain of integration. As a partial case, the parameters of the contact of two elastic transversely isotropic half-spaces (if one of them has a notch of elliptical cross-section), as well as the parameters of the contact interaction of two electroelastic half-spaces, one of which contains a an axisymmetric notch, are obtained from the analytical expressions. Numerical results are obtained. The effect of the electroelastic properties of the half-spaces, the geometrical parameters of the notch, and loading on the contact interaction as well as the closure of the gap between the bodies is studied. [ABSTRACT FROM AUTHOR]

The properties of risk measures in the form of infimal convolution are analyzed. The dual representation of such measures, their subdifferential, extremum conditions, representation for optimization and use in constraints are described. The results of the study are demonstrated by examples of known risk measures of such structure. This allows systematization of the available results and facilitates a potential search for new variants of risk measures. [ABSTRACT FROM AUTHOR]

Properties of the apparatus of polyhedral coherent risk measures, its relationship with problems of robust and distributionally robust optimization, as well as its application under uncertainty are described. Problems of calculating robust structures of polyhedral coherent risk measures and their minimization, which are reduced to the corresponding linear programming problems, are considered. [ABSTRACT FROM AUTHOR]

The problem of the stress state in an orthotropic piezoelectric body with a triaxial ellipsoidal inclusion under homogeneous force and electric loads is considered. The problem is solved by the Eshelby method of equivalent inclusion generalized to the case of a piezoelectric orthotropic space. The approach is validated against the example of a spheroidal cavity in a transversely isotropic material (the axis of revolution coincides with the symmetry axis) for which the exact solution is known. The stress distribution over the surface of the ellipsoidal cavity subject to tension is analyzed numerically. [ABSTRACT FROM AUTHOR]

Polyhedral coherent risk measures are extended to the case of imprecise scenario estimates of random variables. Optimization problems under uncertainty are considered that cover a wide class of stochastic programming and robust optimization problems. It is shown how they are reduced to linear programming problems in the linear case. Problems of portfolio optimization by the reward-to-risk ratio are considered. [ABSTRACT FROM AUTHOR]

The contact interaction of two compressed electroelastic transversely isotropic half-spaces with different properties is studied. One of the half-spaces has an axisymmetric notch of special shape. The contact plane coincides with the plane of isotropy of the half-spaces. Explicit formulas for the contact pressure, displacements, and the size of the gap between piezoelectric half-spaces are derived. These contact characteristics for transversely isotropic and isotropic elastic half-spaces are obtained as special cases [ABSTRACT FROM AUTHOR]

We consider a stressed state problem of a piezoelectric medium containing an arbitrarily oriented spheroidal inclusion under uniform mechanical and electrical loads. The problem has been solved by using a generalized Eshelby method of equivalent inclusion for the case of piezoceramic material. Testing of the approach for the case of a spheroidal cavity (when the cavity rotation axis coincides with the material polarization axis), for which an exact solution of the problem exists, confirms its high efficiency. Numerical investigations have been carried out, and the stress distribution along the surface of an arbitrarily oriented spheroidal cavity has been studied. [ABSTRACT FROM AUTHOR]

The elastic stress state in a piezoelectric body with an arbitrarily oriented elliptic crack under mechanical and electric loads is analyzed. The solution is obtained using triple Fourier transform and the Fourier-transformed Green’s function for an unbounded piezoelastic body. Solving the problem for the case of a crack lying in the isotropy plane, for which there is an exact solution, demonstrates that the approach is highly efficient. The distribution of the stress intensity factors along the front of a crack in a piezoelectric body under uniform mechanical loading is analyzed numerically for different orientations of the crack [ABSTRACT FROM AUTHOR]

The paper gives explicit expressions of the elastic T-stress components T I, T II, and T III for an elliptic crack in an unbounded body under uniform pressure and bending and expressions of all the T-stress components for parabolic and tunnel cracks under uniform loading. These formulas are derived by analyzing the asymptotic behavior of the stress components near the crack front using special harmonic functions. The dependence of the T-stresses on Poisson’s ratio, semiaxes and parametric angle of the elliptic crack is studied. The expressions of T I, T II, and T III for a penny-shaped crack under arbitrary uniform pressure and bending follow as a special case from the respective expressions for an elliptic crack [ABSTRACT FROM AUTHOR]

The stress state of an elastic orthotropic medium with an arbitrarily oriented triaxial ellipsoidal inclusion is analyzed. A solution is obtained using the triple Fourier transform and the Fourier-transformed Green’s function for an infinite anisotropic medium. The high efficiency of the approach is demonstrated by solving the problem for a transversely isotropic material with a spheroidal cavity for which the exact solution is known. A numerical analysis is conducted to study the stress distribution over the surface of the inclusion with different orientations in the orthotropic space. It is revealed that in some cases the orientation of the inclusion has a strong effect on the stress concentration [ABSTRACT FROM AUTHOR]

This paper is a study into the interaction of two triaxial ellipsoidal cavities whose surfaces are under different pressures with an elliptic crack in an infinite elastic medium. The stress state in the elastic space is represented by a superposition of perturbed states due to the presence and interaction of the cavities and the crack. The exact solution of the problem is constructed by using a modified method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for the elliptic crack. A numerical analysis is carried out to find how the geometry of the cavities and the crack, the distance between them, and the pressure on their surfaces affect the stress intensity factors [ABSTRACT FROM AUTHOR]

The interaction of an elastic ellipsoidal inclusion with an elliptic crack in an infinite elastic medium under triaxial loading is analyzed. The stress state in the elastic space is represented as a superposition of the principal state and perturbed states, which are due to the presence and interaction of the inclusion and the crack. The analytical solution of the problem is found using the method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for an elliptic crack. The effect of triaxial loading on the stress intensity factors is analyzed [ABSTRACT FROM AUTHOR]