Vol 27, No 1 (2023)

In this article, we study the analysis related to generalized Clifford algebras $C_n\left(\underline{a}\right)$, where $\underline{a}$ is a non-zero vector. If $\left\{e_1,...,e_n\right\}$ is an orthonormal basis, the multiplication is defined by relations
${e_j}^2=a_j{e}_j-1,$
$e_i{e}_j+e_j{e}_i=a_i{e}_j+a_j{e}_i,$
for $a_j=e_j·\underline{a}$. The case $\underline{a}=\underline{0}$ corresponds to the classical Clifford algebra. We define the Dirac operator as usual by $D=\sum _j{e}_j{\partial }_{x_j}$ and define regular functions as its null solution. We first study the algebraic properties of the algebra. Then we prove the basic formulas for the Dirac operator and study the properties of regular functions.

It is shown that a quantum path integral can be represented as a functional of the unique path that satisfies the principle of least action. The concept of path will be used, which implies the parametric dependence of the coordinates of a point on time x(t), y(t), z(t). On this basis, the material fields, which are identified with a quantum particle, are represented as a continuous set of individual particles, the mechanical motion of which determines the spatial fields of the corresponding physical quantities. The wave function of a stationary state is the complex density of matter field individual particles. The modulus of complex density sets the density of matter normalized in one way or another at a given point in space, and the phase factor determines the result of the superposition of material fields in it. This made it possible to transform the integral equation of quantum evolution to the Lagrange’s representation. By using the description of a quantum harmonic oscillator as an example, this approach is verified. EPRtype experiment is described in detail, and the possibility of the faster-then light communication is proved, as well as the possible rules of thumb of this communication are proposed.

The paper presents the algorithm for computer simulation of polycrystalline material microstructures with explicitly distinguished grain boundaries. The algorithm is based on the procedure of “growing” structure grains from ellipses, the geometric parameters of which can be set according to different laws of statistical distribution. The grain boundaries of the given thickness are formed from the original granular structure by displacing the boundaries inside the grain. The advantage of the presented algorithm is the possibility of obtaining nonlinear grain boundaries of different thicknesses, the width of which can be specified according to various statistical distribution laws. Polycrystalline material microstructure generation results that contain more than 100 structural elements and have a grain boundary fraction of up to 20 % are presented. New data on computer simulation of the deformation process and fracture of simulated granular materials are presented with different ratios of strength characteristics of grains and grain boundaries. It was revealed that, depending on the strength characteristics ratio value, different fracture mechanisms are realized in the material: intercrystalline, transcrystalline, and mixed forms of fracture.

The dynamic problem of non-isothermal and inelastic deformation of flexible shallow multidirectionally reinforced shells is formulated in the frameworks of the refined theory of bending. The temperature is approximated by a 7th order polynomial over the thickness of constructions. The geometric nonlinearity of the problem is modeled by the Karman approximation. The solution of the formulated coupled nonlinear two-dimensional problem is obtained using an explicit numerical scheme. The thermo-elastic-plastic response of fiberglass and metal-composite cylindrical elongated panels with an orthogonal reinforcement structure, loaded frontally with an air blast wave, has been studied. It is shown that, unlike reinforced plates similar in structure and characteristic dimensions, shallow shells under intense short-term loading must be calculated taking into account the occurrence of temperature fields in them. In this case, the refined theory of bending of curved panels should be used instead of the simplified version (the non-classical theory of Ambartsumyan). The temperature increment at separate points of shallow fiberglass shells can reach 14–34 \textcelsius, and in similar metal-composite panels can reach 50–150 \textcelsius. Cylindrical shallow shells are more intensively deformed when they are loaded by an air blast wave from the side of a convex front surface.

The problem of flow around a smooth convex body moving horizontally at a constant subsonic velocity in a stratified atmosphere at rest consisting of an ideal gas is considered. By the condition of the problem, the (vertical) gradient of the Bernoulli function (taking into account the potential energy of a uniform gravity field) in the atmosphere at rest at all altitudes is nonzero (as is the case in the Earth's standard atmosphere at altitudes up to 51 km), and the flight altitude does not exceed a value equal to the square of the body's flight speed divided by twice the acceleration of gravity. The surface of the earth is considered flat. The coordinate system associated with the body is used. The general spatial case is considered (an asymmetric body or a symmetric body at an angle of attack). We use the generally accepted assumption that in some neighborhood of the stagnation streamline (streamline that ends on the body at the forward stagnation point) there is no second stagnation point, the flow parameters in this neighborhood are twice continuously differentiable, and the stagnation point is spreading point (i.e. in some neighborhood of it, all streamlines on the surface of the body start at this point). Based on a rigorous analysis of the Euler equations, it is shown that the existence of a stationary solution to the problem contradicts this generally accepted (but not strictly proven) idea of the stagnation streamline. This property of the solution of the problem is called paradoxical and casts doubt on the existence of the solution.

We consider the creep of a hydrogenated rod made of VT6 (Ti-6Al-4V) titanium alloy with a piecewise constant dependence of the stress on time up to failure. The results of an experimental and theoretical study on the effect of the concentration of previously introduced hydrogen on the creep and long-term strength of tensile rods made of VT6 titanium alloy at a temperature of 600$°$C and constant nominal tensile stresses in the range from 47 to 217 MPa.