Vol 53, No 6 (2018)

On the basis of the developed analytical model of the motion of a deformable Earth relative to the center of mass, the disturbed motion of the Earth’s pole is investigated. Taking into account the influence of the pole tide by linear and nonlinear dissipative terms, as well as disturbances at close to the Chandler frequency, corrections to the frequency and amplitudes of the Chandler oscillations depending on the dissipation coefficients were found in the equations of motion of the Earth’s pole.

The control problem for stepwise changing linear systems of loaded differential equations with given initial, final, and nonseparated (non-local) multipoint intermediate conditions and optimal control with a quality criterion specified for the entire time interval is considered. The necessary and sufficient condition for the complete controllability, the conditions for the existence of programmed control and motion, are formulated. An explicit form of control action is constructed for the control problem, and a method for solving the optimal control problem is proposed.

A quaternion method for the regularization of differential equations of the perturbed spatial restricted three-body problem is developed. It is closely related, from the methodological point of view, to the quaternion method for the regularization of the differential equations of the perturbed spatial three-body problem in Kustaanheimo–Stiefel variables that was earlier proposed by the author of this article.

Various local and global regular quaternion differential equations of the perturbed spatial restricted three-body problem (both circular and non-circular problem) i.e. equations that are regular in the vicinity of the first or second body of finite mass and equations that are regular at the same time both in the neighborhood of the first and second body of finite mass are obtained. The equations are systems of nonlinear nonstationary differential equations of the tenth or eleventh or nineteenth order with respect to the Kustaanheimo–Stiefel variables, their first derivatives, Kepler or total energies, or variables that are Jacobi integration constants in the case of the unperturbed spatial circular restricted three-body problem, as well as with respect to time and auxiliary time variable. The equations obtained allow one to construct different regular algorithms for integrating the differential equations of the perturbed spatial restricted three-body problem.

This study is an extension of [1, 2].

The article deals with the dynamic 2D problem of the theory of elasticity that considers a massive body with a semi-infinite plane crack, the faces of which are subjected to a normal pulsed tensile load symmetric to the crack. When studying the stress state ahead of the crack tip, the singular and regular terms have been found. They have been used to find the pre-fracture (incubation) period under the threshold loads. It has been shown that the regular term taking into account significantly affects the stress state, in particular, the incubation period. In the case of shock impulses above the threshold (overloads), the period of the crack tip movement with a variable speed (“breakthrough” of a crack) is considered as well as the pre-fracture period. To this end, the structural-time criterion for determining the incubation period is generalized to the case of the crack movement with a variable speed. A comparison of the calculated values of increasing the length of cracks after fracture caused by three shock impulses above the threshold with experimental data is made.

The article consideres the loosening of metallic materials and an irreversible change in density during high-temperature creep is considered as damage parameter. On the basis of this parameter and taking into account the law of mass conservation, interrelated equations for creep deformation and damage parameter are proposed. Approximate and exact solutions for creep deformation and continuity parameter are obtained and criteria for long-term strength are formulated. The corresponding theoretical curves are constructed. It is shown that the theoretical relationships for the damage parameter describe the experimental results on the density change during the high-temperature creep of various metallic materials well.

In the linear theory of elasticity, an arbitrary crack is represented as a combination of three: longitudinal shear cracks, transverse shear cracks, and normal tear cracks that do not interact with each other. In the nonlinear theory, for some types of strain energy potentials, a finite longitudinal shear crack necessarily generates a strain in the transverse plane. This article proposes an asymptotic description of the deformed state of a crack in the transverse plane under the action of a finite longitudinal shear in an incompressible material with a Mooney–Rivlin potential, and an assessment is made of the effect of additional deformation on the condition of the start of a crack.