Viktoriya Pasternak’s research while affiliated with Volyn National University named after Lesya Ukrainka and other places

The paper proposes a transparent and compact form of constitutive and equilibrium relations for the plane thermoelasticity of quasicrystal solids. The symmetry and positive definiteness of the obtained extended tensors of material constants are studied. An extension of the Stroh formalism is proposed for solving plane problems of thermoelasticity for quasicrystals. It is proved that the eigenvalues of the Stroh eigenvalue problem in the most general case of 3D quasicrystal materials do are purely complex. The relations between the matrices and vectors of phonon–phason elastic and thermoelastic coefficients of the proposed extended Stroh formalism are obtained. A fundamental solution to the plane problem of thermoelasticity of a quasicrystal medium is derived. The asymptotic behavior of physical and mechanical fields near the vertices of objects whose geometry can be modeled by a discontinuity line (cracks, thin inclusions) is studied, and the concepts of the corresponding generalized field (heat flux and phonon–phason stress) intensity factors are introduced. Examples of the influence of heat sources and sinks on an infinite quasicrystal medium containing a rectilinear heated crack are considered.

This scientific work justifies imaging and visualization methods for analyzing heterogeneous PA-1 structures at micro- and nanoscales. It explores a key aspect of studying heterogeneous materials, namely the relationship between their microstructure and macroscopic behavior. Using Smart-EYE software, the microstructure and heterogeneous structure of PA-1 aluminum powders are justified through a range of factors. Among them, the extended functionality of the program allows for detailed analysis of particle sizes, shapes, and distribution, ensuring high accuracy and reliability of the analysis results. The capability for quick and efficient analysis of large volumes of data is also highlighted. Additionally, the software enables visualization of analysis results, simplifying their interpretation. Furthermore, the obtained results based on the histogram of particle size distribution, such as normal distribution, skewness, and modality, help avoid minor data defects and ensure proper interpretation.

This paper presents a study in the field of modelling the dynamics of spherical elements. The results obtained indicate the successful use of the discrete element method (DEM) as a numerical tool for analysing the behaviour of the system studied with the help of spheres. The results are based on the importance of correct consideration of the boundary conditions for the spheres, which determine the key aspects of modelling with the developed three-dimensional model. The developed model solves a number of important tasks, expanding the field of scientific research. Firstly, it allows studying the main parameters of the formation of a heterogeneous medium by analysing the compaction of spherical elements in different media. Next, the three-dimensional model is used to study the process of changing the structure of a heterogeneous medium from a static to an oscillatory state, which allows for a deeper understanding of this process. By modelling the mathematical behaviour of spherical elements under the influence of external and additional factors, a detailed understanding of their dynamics and contact interaction can be obtained. The application of the developed model to analyse the contact interaction of spherical elements in heterogeneous media allows predicting the main parameters of spheres and their heterogeneous environment with a reliable accuracy of up to ±1 %. It should be noted that the results obtained on the basis of the three-dimensional model are effective and indicate a number of practical applications in various fields.

The research work is devoted to the study of the stress-strain state of a structure comprising a cylinder with a sphere using numerical approaches and Green’s functions. The results obtained include the analysis of stress distribution, study of deformations and determination of stress concentration zones. Safety factors are assessed and the influence of boundary conditions on the behaviour of the structure is revealed. The application of numerical methods allowed for a detailed study of the interaction of the sphere, providing an opportunity to analyse the exact properties and assess the influence of various factors in complex structures. It should be noted that the results obtained, which were evaluated taking into account all factors, affect the real system and can be predicted with a deviation error of 1%.

This scientific study considers the results of a computer experiment with heterogeneous elements (spheres) that proved to be of decisive importance during the separation process, namely their degree of activity, mobility and falling. It has been found that a detailed analysis of the Liapunov function indices allows to effectively understand and predict the dynamics of complex dynamical systems. The results obtained indicate significant changes in the physical and mechanical parameters of spherical balls under the influence of various factors and the environment. It was found that a certain accumulation of spheres occurs due to an increase in the time for simulation. It was also found that the key characteristics of the bulk mass of spherical elements significantly depend on the moulding process, surface condition and environmental conditions.

In this paper, the boundary element method (BEM) is investigated and computer simulations are conducted to study the patterns of structure formation of non-isometric elements. The modeling of this study covered various aspects, including shape, radius, angle from the stable radius, porosity, average coordination number, simulation time, component falling force, and electrostatic constant. The simulation results provided important information about the properties and interaction of non-isometric components under different conditions. It was found that the obtained parameters can be effectively predicted for further research. It should also be noted that important processes, such as deformation and material behavior, colloidal aspects, dynamic modeling of the movement of components with complex shapes, and features of nanotechnology, were observed in parallel with computer simulation.

In this scientific work, mathematical modeling of tetrahedron elements in the finite element method is presented, which includes the determination of geometric shape, shape functions, and material properties. Unknown fields such as displacement vectors, strain, and stress tensors are considered. The methodology of applying the principle of virtual work and equilibrium equations is described, allowing the derivation of a system of differential equations to describe the behavior of the tetrahedral element. Integration over the volume and consideration of boundary conditions help reduce the equations to a system of linear algebraic equations for numerical solution using the finite element method. It was found that modeling tetrahedral elements with a specific given radius (for example, R=0.3 mm) involves stages such as geometry determination, element generation, shape function formation, stiffness matrix computation, and solving a system of linear equations. The radius R of tetrahedral elements is taken into account at all stages, ensuring accuracy and reliability in tetrahedra modeling. The research also focuses on the fact that the occurrence of minor errors in iterative processes may result from several factors, including iteration step, the number of iterations, stopping criteria, linear or nonlinear material behavior, solution method selection, the presence of geometric inhomogeneities, and element size.