S.P. Timoshenko Institute of Mechanics

In this article, for scalar nonlinear equations with uncertain parameters, a new approach has been proposed for studying the integral funnel (antifunnel) of a family of their solutions. Efficient conditions for a family of solutions to remain in an integral funnel (antifunnel) and conditions for narrowing of the integral funnel were established. The proposed approach is based on a comparison scheme and a set of regularized differential equations. The obtained results contribute to the development of the qualitative theory of uncertain systems and to the expansions of the appropriate applications.

Abstract Within the framework of the three‐dimensional linearized theory of stability of deformable bodies, the plane problem on compression of a piecewise‐homogeneous half‐plane along a defect, which is a frictionless sliding zone located at the rectilinear interface between two rigidly connected media, a semi‐bounded homogeneous body (base) and a homogeneous thin coating layer, have been studied. The initial stage of body fracture is associated with the loss of material stability in a local area near the specified defect. The base and coating materials are considered as highly elastic materials described by an elastic potential with different mechanical characteristics. The boundary value problem, formulated in terms of potential harmonic functions, using Fourier integral transforms, is reduced to an eigenvalue problem for the Fredholm integral equation of the first kind, which is studied numerically by employing Bubnov—Galerkin method. For a compressible material with a harmonic potential, critical values of the load parameters that correspond to local instability have been found. The critical load parameters obtained have also been compared with similar values of the critical parameters corresponding to the near‐surface instability of a piecewise‐homogeneous half‐plane (without a defect) compressed along the interface between the media that are rigidly connected or slide without friction against each other.

The data of chemical composition and content of trace elements in alkaline-ultrabasic rocks of the northwestern region of the Ukrainian Shield are summarized. The occurrences of alkaline rocks of this composition were discovered as results of drilling in four areas (Horodnytska, Hlumchanska, Bolyarkivska, and Hubkivska). They are presented by small hypabyssal intrusions, stem- and dike-like bodies. The rocks are represented by the jacupirangite-melteigite series, which belong to high-Mg (Mg# 0.7–0.8) varieties. Less common are alkaline and subalkaline gabbroids found in thin apophyses and veins near alkaline-ultrabasic bodies. It is assumed that these gabbroids were formed as a result of crust material contamination of the primary alkaline-ultrabasic melts. The alkaline-ultrabasic rocks of all occurrences have high Cr (233–2737 ppm) and Ni (95–1022 ppm), but are unusually depleted in incompatible elements (Nb, Ta, Zr, Hf, REE, Y), with moderate Ba and Sr. Their geochemical features sharply differ from similar rocks, carbonatite and alkaline complexes known as in the Ukrainian Shield and other regions of the world. Despite the primitive composition of alkaline-ultrabasic rocks and anomalous REE depletion, their chondrite-normalized patterns have negative slopes and are similar to differentiated rocks with a prevailing of LREE (La/Yb = 6–16). The petrogenesis of such primitive and trace-element depleted alkaline-ultrabasic rocks is enigmatic. We suppose that these unusual geochemical features are the result of primary melt(s) generation from mantle source and their evolution by a similar way. For the alkaline rocks of the Horodnytska and Hlumchanska intrusions, and possibly the Hubkivska occurrence, the primary melts arose through partial melting of garnet (with amphibole?) peridotites. At the same time, the primary melt for the Bolyarkivska intrusion was generated by partial melting of spinel peridotite, probably in a shallower environment. The anomalous depletion at HFSE and REE can be considered as the result of specific conditions of partial melting of depleted mantle source at low temperature and pressure, when refractory minerals concentrating these elements (rutile, amphibole) remained in the restite, or due to early crystallization and fractionation of concentrating minerals (perovskite, apatite).