Vol 51, No 2 (2016)

We consider equations and algorithms describing the operation of strapdown inertial navigation systems (SINS) intended for determining the inertial attitude parameters (the Rodrigues–Hamilton (Euler) parameters) and the apparent velocity of a moving object. The construction of these equations and algorithms is based on the Kotelnikov–Study transference principle, Hamiltonian quaternions and Clifford biquaternions, and differential equations in four-dimensional (quaternion and biquaternion) orthogonal operators.

We consider gradient models of elasticity which permit taking into account the characteristic scale parameters of the material. We prove the Papkovich–Neuber theorems, which determine the general form of the gradient solution and the structure of scale effects. We derive the Eshelby integral formula for the gradient moduli of elasticity, which plays the role of the closing equation in the self-consistent three-phase method. In the gradient theory of deformations, we consider the fundamental Eshelby–Christensen problem of determining the effective elastic properties of dispersed composites with spherical inclusions; the exact solution of this problem for classical models was obtained in 1976.

This paper is the first to present the exact analytical solution of the Eshelby–Christensen problem for the gradient theory, which permits estimating the influence of scale effects on the stress state and the effective properties of the dispersed composites under study.We also analyze the influence of scale factors.

An approximate solution describing the compression of an axisymmetric layer ofmaterial on a rigid mandrel under the equations of the creep theory is constructed. The constitutive equation is introduced so that the equivalent stress tends to a finite value as the equivalent strain rate tends to infinity. Such a constitutive equation leads to a qualitatively different asymptotic behavior of the solution near the mandrel surface, on which the maximum friction law is satisfied, compared with the well-known solution for the creep model based on the power-law relationship between the equivalent stress and the equivalent strain rate. It is shown that the solution existence depends on the value of one of the parameters contained in the constitutive equations. If the solution exists, then the equivalent strain rate tends to infinity as the maximum friction surface is approached, and the qualitative asymptotic behavior of the solution depends on the value of the same parameter. There is a range of variation of this parameter for which the qualitative behavior of the equivalent strain rate near the maximum friction surface coincides with the behavior of the same variable in ideally rigid-plastic solutions.

We develop a technique for calculating the plastic strain and fracture toughness fields of a material by solving dynamical 3D problems of determining the stress-strain state in the elastoplastic statement with possible unloading of the material taken into account. The numerical solution was obtained by a finite difference scheme applied to the three-point shock bending tests of parallelepiped-shaped bars made of different materials with plane crack-notches in the middle. The fracture toughness coefficient was determined for reactor steel. The numerically calculated stress tensor components, mean stresses, the Odquist parameter characterizing the accumulated plastic strain, and the fracture toughness are illustrated by graphs.

A spherical tank, being perfect as far as weight is concerned, is used in spacecraft, where the thin-walled elements (shells) are united by frames. Obviously, local actions on the shell and hence the stress concentration in the shell cannot be avoided. Attempts to make weight structure of the spacecraft perfect inevitably decrease the safetymargin of the components, which is possible only if the stress-strain state of the components is determined with a controlled error. A mathematical model of shell deformation mechanics is proposed for this purpose, and its linear differential equations are obtained with an error that does not exceed the error of Kirchhoff assumptions in the theory of shells. The algorithm for solving these equations contains procedures for estimating the convergence of the Fourier series and the series of the hypergeometric function with a prescribed error, and the problem can be solved analytically.