## No 1 (2022)

**Year:**2022**Articles:**13**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/156**DOI:**https://doi.org/10.15593/perm.mech/2022.1

Utilization of viscoelatic models with non-linear springs and dashpots in delamination study of multilayered beams

#### Abstract

This paper analyzes delamination of multilayered inhomogeneous beam structure by utilization a non-linear viscoelastic mechanical model. The non-linear time-dependent response is treated by a non-linear spring and a non-linear dashpot connected in series to a system of two linear springs and two linear dashpots. The model is under stress that is a linear function of time. The constitutive law of the model representing a non-linear dependence between stress, strain and time is derived. The main goal of the paper is to obtain a solution of the strain energy release rate for the delamination in the multilayered inhomogeneous beam by applying the non-linear viscoelastic model. Solutions are derived by using the complementary strain energy and by analyzing the balance of the energy with taking into account the non-linear viscoelastic behaviour. For this purpose, the constitutive law of the non-linear viscoelastic mechanical model is used. The solutions are applied to obtain results for multilayered beams with non-linear variation of material properties in longitudinal direction. The influence of different parameters on the time-dependent strain energy release rate is assessed. The study indicates the effectiveness of the viscoelastic mechanical models with non-linear springs and dashpots in time-dependent delamination analyses of multilayered inhomogeneous beam structures.

**PNRPU Mechanics Bulletin**. 2022;(1):5-13

Numerical simulation of brush seal tests

#### Abstract

A numerical analysis of deformation of the brush seal wire package as part of dynamic rotor&stator system of gas turbine engines is describes. The estimated definition of the reaction force of the brush package is based on solving the contact problem of the interaction of bristles with each other, with the rotor and the rear plate of the seal case, taking into account the action of the distributed gas flow pressure. Comparison of the results obtained with the data of well-known estimated and experimental studies. According to the results of calculations, it was noted that the accounting of the spatial contact interaction of the elements of the brush seal significantly affects on reaction force value and, accordingly, on the brush package rigidity. In the presence of a gas pressure drop on the compaction, the results of calculations on a simplified half-analytical model based on the analysis of the deflection of one bristles according to the model of the flat bending of the console straight beam, and the spatial model differ in order. It is necessary to take into account this effect when calculating the wear of the brush seal and the forecast of its effectiveness with the course of operation. A characteristic feature of the dependence of the reaction of the brush package from the magnitude of the rotor displacement is the presence of hysteresis in the "load - unloading" cycle due to friction in the system. The impact of gas flow on the bristles significantly increases the scope of hysteresis loops and changes its shape. On the scale of the loop also strongly affects the coefficient of friction pair "schedules - rotor". The obtained dependences of the reaction of the brush package from the magnitude of the tension / displacement of the rotor (hysteresis loops) allow you to clarify the calculated mathematical model of the brush seal after testing on the stand.

**PNRPU Mechanics Bulletin**. 2022;(1):14-22

Numerical modeling of natural vibrations of coaxial shells partially filled with fluid, taking into account the effects on the free surface

#### Abstract

This work is devoted to the numerical analysis of vertical elastic coaxial cylindrical shells completely or partially filled with a quiescent compressible fluid with account for sloshing of its free surface. The problem is solved in the axisymmetric formulation using a semi-analytical version of the finite element method. The fluid medium is described by the wave equation, which together with the conditions prescribed at its boundaries is reduced to the weak form by the Bubnov - Galerkin method. A mathematical formulation of the dynamic problem for thin-walled structures is developed using the variational principle of virtual displacements and linear theory of thin shells based on Kirchhoff - Love hypotheses. The fluid pressure on the walls of the structure is calculated according to Bernoulli's equation. Sloshing modes caused by gravitational effects on free surface of liquid medium are excluded from the resulting system of equations through the use of the iteration dynamic condensation method. The numerical model has been verified by comparing it with the known data for the case of a single shell partially filled with fluid. The influence of the fluid level on the lowest natural frequencies of the system vibrations at different variants of kinematic boundary conditions for shells (rigid clamping at both edges, cantilever support) and different widths of the annular gap between them has been evaluated. It has been found that for the examined configurations, the level of the fluid in the annular channel has a stronger influence on the frequency spectrum compared to the level of fluid in the cavity of the inner shell due to the fact that vibration frequencies change in a wider range.

**PNRPU Mechanics Bulletin**. 2022;(1):23-35

Analysis of multiaxial fatigue models based on the results of biaxial tests for aluminum alloys

#### Abstract

During operation, modern structures and their parts are subjected to complex cyclic loads. In this case, a complex stress-strain state can arise in products, which can lead to a significant decrease in their fatigue life. In this regard, the development of methods for predicting fatigue fracture of critical structures becomes urgent. The purpose of this work is to test a simple approach for describing multiaxial fatigue datasets presented in the literature and compare it with known multiaxial fatigue models. Within the framework of the study, actual experimental data was considered, containing experimental data on the multiaxial fatigue of aluminum alloys 2024-T3 and 2024-T4. The alloys are actively used in the designs of aircraft, ships and spacecraft. As a result, the Sines model and its modernized version were considered (taking into account the first invariant of the stress tensor with respect to amplitude values). At the same time, an approach was developed to determine the constants of the models based on the results of two variants of training experiments. The paper presents the results in the form of relationships of the predicted and experimental fatigue life, obtained using the Sines model and its modification for two types of training tests. In the final part of the work, a comparative analysis of the Sines model and its modification, Smith-Watson-Topper, Fatemi-Socie and its modification is carried out. The analysis showed that the modified Sines model significantly improves the predictive ability, the proposed model better described the experimental data than the traditional Sines model and the Smith-Watson-Topper approach. At the same time, the descriptive ability of the model is comparable to the Fatemi-Socie one.

**PNRPU Mechanics Bulletin**. 2022;(1):36-44

The method of fictitious discrete models in calculations of composite bodies

#### Abstract

As is known, the calculation of the static strength of elastic composite bodies (CB) is reduced to finding the maximum equivalent stresses for these bodies. The finite element method (FEM) is widely used for the analysis of the stress state of CB. The basic discrete models (BM), which take into account the inhomogeneous structure of bodies in the framework of a micro-approach, have a high dimension. To reduce the dimension of discrete models, multigrid finite elements (MgFE) are effectively used. However, there are BM CB (for example, BM bodies with a micro-homogeneous structure), which have such a high dimension that the implementation of FEM for such BM using MgFE, due to limited computer resources, is difficult. To solve this problem, it is proposed to use fictitious discrete models whose dimensions are less than the dimension of the BM CB. In this paper, we propose a method of fictitious discrete models (MFDM) for calculating the strength of elastic bodies with an inhomogeneous, micro-homogeneous regular structure. The proposed method is implemented using FEM with the use of MgFE and adjusted strength conditions that take into account the error of approximate solutions. The method is based on the position that the solutions that meet the BM CB differ little from the exact ones. The calculation of CB according to MFDM is reduced to the construction and calculation of the strength of fictitious discrete models (FM), which have the following properties. The FM reflects: the shape, characteristic dimensions, attachment, loading and type of the inhomogeneous structure of the CB, and the distribution of elastic modulus corresponding to the BM CB. The FM dimension is less than the BM dimension of the CB. The sequence consisting of FM converges to BM, i.e. the limiting FM coincides with BM. The convergence of such a sequence ensures uniform convergence of the maximum equivalent voltages of the FM to the maximum equivalent voltage of the BM. Two types of FM are considered. The first type of FM consists of scaled discrete models, the second type consists of FM with variable characteristic dimensions. Calculations show that the implementation of FEM for FM using MgFE leads to a large saving of computer resources, which allows the use of MFDM for bodies with a micro-homogeneous regular structure. The calculation of the strength of CB according to MFDM requires times less computer memory than a similar calculation using BM CB, and does not contain a procedure for grinding BM. The use of adjusted strength conditions allows us to use approximate solutions with a large error in the calculations of CB for strength, which leads to an increase in the efficiency of MFDM. The given example of calculating the strength of a beam with an inhomogeneous regular fibrous structure according to MFDM shows its high efficiency.

**PNRPU Mechanics Bulletin**. 2022;(1):45-57

Finite element calculation of an elliptical cylinder in a geometrically nonlinear formulation using the vector form of the interpolation procedure

#### Abstract

The article describes the developed algorithm for calculating the strength parameters of a thin shell in the form of an elliptical cylinder, taking into account shear deformations in a geometrically nonlinear setting. The numerical finite element method (FEM) was used as a research tool. An algorithm for the formation of the stiffness matrix and the column of nodal forces at the loading step was developed using two versions of the interpolation procedure. In the first version, the standard for FEM interpolation of individual components of the step vector of displacement and the components of the step vector of the angles of rotation of the normal was implemented through the nodal values of the corresponding components. In the second version, the developed vector form of the interpolation procedure was used, in which the interpolation expression was written directly for the step vector and the step vector of the normal rotation angles. As a result of the implementation of the vector form of the interpolation procedure, alternative standard interpolation expressions containing the parameters of the used curvilinear coordinate system were obtained. The element of discretization of the elliptical cylinder was a four-node fragment of the middle surface with nodal variable parameters in the form of components of the step vector of displacement, their first derivatives, as well as the component of the step vector of the angles of rotation of the normal. On the example of calculating an elliptical cylinder in a geometrically nonlinear formulation, loaded in the middle of the span with a concentrated force, a comparative analysis of two variants of the interpolation procedure was carried out. It was shown that when calculating elliptical cylinders in a curvilinear coordinate system in a geometrically nonlinear formulation, it is necessary to apply the developed vector form of the interpolation procedure. The use of the standard FEM form of interpolation of the unknown unknowns made it possible to obtain correct results only in the case of a circular cylinder.

**PNRPU Mechanics Bulletin**. 2022;(1):58-71

Mathematical model of oxide film deformation on the surface of a metallic melt in an alternating magnetic field

#### Abstract

We consider a thin oxide film on the surface of a molten metal during induction melting. The alternating electromagnetic field excites eddy currents in the metal volume, which heat it, and Lorentz force, which causes the forced convection of the melt, the heating and the flows in the metal are discussed briefly. The contribution to the mechanical stresses in the film that gives the alternating electromagnetic field, the thermal expansion of the film, and the melt motion are considered in detail. The equations system describing magnetic field diffusion, equations of motion and heat transfer in the melt are written in the axisymmetric formulation. The corresponding boundary conditions are described, the governing dimensionless criteria, which determine the structure and intensity of the melt flow, including those at the surface where the film is located, are given. The film elastic deformation equation is derived from Hooke's law and written in terms of displacement in dimensional and dimensionless forms. On the base of literature review, the values of physical characteristics of the film, which are not available for direct measurement, are proposed. The verification of the mathematical model is given. Possible flows in the melt are calculated, taking into account the action of dynamic and thermal action of the film on the surface. The unambiguous relation of the film stress state with these flows is shown. The influence of the magnetic field diffusion parameter and the Hartmann number, which determine, respectively, the structure and intensity of the forced flow, on the film deformations is demonstrated. The mode map of regimes is constructed that relates the integral deformation of the film to the parameters of the magnetic field and the initial size of the film. It is found that the situations are possible when the film in the stress-strain state does not change its total size and remains in stable equilibrium on the surface of the moving melt. Recommendations for the usage of the presented results are given.

**PNRPU Mechanics Bulletin**. 2022;(1):72-88

ON THE CONVEXITY OF POTENTIAL OF NONLINEAR ELASTIC MEDIA MODEL WITH TENSOR DAMAGE PARAMETER

#### Abstract

For solving problems of the theory of elasticity using nonlinear models, the question of the convexity of the potential used and the proof of the uniqueness of the solution are of decisive importance. The present work is devoted to the determination of conditions for the local strict convexity of the potential for the nonlinear elasticity model, which ensure the uniqueness of the solution to the problem in a sufficiently small neighborhood of the desired solution. The considered model is a generalization of a scalar nonlinear rheological model of brittle solid deformation for the case of the tensor damage parameter, the principal values of which describe the reduction in the cross-sectional area of the material in three orthogonal directions. An additional term of the second order in deformation in the elastic potential makes it possible to describe the dependence of the elastic moduli on the type of the stress-strain state, the dilatancy of the material under shear deformation, as well as the nonlinear deformation response even at low loads. The introduced damage tensor of the second rank makes it possible to describe the damage-induced anisotropy of the elastic properties of the material. Conditions of local strict convexity in the principal axes of the strain tensor are obtained in this work for the general case of misaligned strain and damage tensors. To illustrate the obtained convexity conditions, two special cases of the damage tensor type are considered: a transversely isotropic fractured medium with coaxial strain and damage tensors, and a transversely isotropic medium with obliquely oriented fracturing. For both cases, the dependence of the limiting values of damage on the parameter of the degree of anisotropy is shown. It is shown that in the case of weak damage anisotropy, the potential convexity conditions for the scalar damage parameter give a minorant estimate of the maximum allowable damage for various types of stress-strain state. For obliquely oriented fracturing, the dependences of the maximum permissible damage on the degree of anisotropy, the angle of inclination and the type of stress-strain state are plotted.

**PNRPU Mechanics Bulletin**. 2022;(1):89-101

Calculation of VAT and assessment of the strength of a segmented cylindrical shell made of composite materials with metal inserts

#### Abstract

The paper considers a cylindrical segmented shell - a pressure cylinder made by winding from unidirectional roving. The shell has a different thickness and reinforcement scheme for segments, and metal embedded elements on the front and rear ends of the shell for fixing the covers. The problem of calculating the stress-strain state and estimating the strength of the shell under loading by internal pressure is posed. To solve the problem, a two-level numerical mathematical model of the shell is developed. At the first level, the shell is modeled using the effective characteristics of the composite material by segments. At the second level, the deformation of individual segments of the shell is modeled - the front and rear joint nodes with metal embedded elements and a pin-pin connection with an explicit description of the segment reinforcement scheme. The boundary conditions for the segment deformation problem are determined as a result of calculating the shell VAT using the first-level model. The developed model is compared with the results of bench tests of the shell for the effect of internal pressure. Based on the results of solving the problem for the shell segments, the analysis of interlayer stresses in the contact zone of the composite shell - metal embedded was performed and estimates of the structural strength were obtained.

**PNRPU Mechanics Bulletin**. 2022;(1):102-114

Calculation of the thermal conductivity coefficient of nanocrystals

#### Abstract

Mathematical modeling methods are powerful tools in the design of various types of nanosystems and the analysis of processes taking place in them. Note that the main tasks of mathematical modeling in nanoscale systems are: the formation of nanoelements, the interaction of individual elements of the nanosystem, the determination of the structure of the isolated nanosystem in dynamic states, the calculation of the parameters of the nanosystem during its interaction with the environment, the calculation of the macro parameters of the nanosystem. With this work, we continue the consistent presentation of theoretical foundations, methods of modeling, and results of calculations of macro characteristics of nanosystems, based on the work on modeling of processes of formation and structure of different nanosystems. This paper provides the physical basis as well as numerical methods for calculating the thermal conductivity of homogeneous nanosystems. Computer simulation of the calculation of the thermal conductivity coefficient of silicon-based nanocrystals by molecular dynamics was carried out in the LAMMPS software complex. Equations describing multi-particle MEAM potentials are considered. The problem of determining the thermal conductivity coefficient has been solved in several steps. In molecular dynamic calculations, the value of the thermal conductivity coefficient can be calculated in various ways. This work uses the Green - Kubo formalism, which associates the autocorrelation function of heat flow with the coefficient of thermal conductivity. Temperature dependencies of thermal conductivity coefficient for silicon-based materials are determined. Calculations of thermophysical characteristics of homogeneous silicon-based nanosystems were made. The curves of the temperature dependence of thermal conductivity coefficient for systems of different dimensions are presented. Data obtained using MEAM potentials were compared with experimental data. It was found that the nature of the curves and the values obtained during the simulation are well consistent with the experimental data.

**PNRPU Mechanics Bulletin**. 2022;(1):115-122

Evolution of the grain structure of metals and alloys under severe plastic deformation: continuum models

#### Abstract

Products made of metals and alloys with a fine-grained structure, which have high physical, mechanical and operational characteristics, are becoming increasingly in demand in many technical and technological fields. The most common and efficient technologies for the production of parts from this class of materials are various processes of severe plastic deformation (SPD) (in general practice, at low homologous temperatures). At the same time, to achieve high degrees of deformation, a significant part of metals and alloys require increased processing temperatures, undergo significant heating during deformation processes, which may be accompanied by changes in the grain and subgrain structure due to recovery and recrystallization processes. An empirical approach to the development of SPD modes that ensure the formation of the necessary grain structure requires huge time and financial costs, and therefore considerable attention is paid by researchers in the field of solid mechanics and metal forming to approaches and methods of mathematical modeling. In connection with the foregoing, the number of publications on this subject has been intensively growing in recent years. The currently known models differ significantly in their approaches, the depth of penetration into the physics of processes, and the scope of consideration. The proposed brief review is focused on a qualitative analysis of works on this topic, a preliminary classification of existing models according to their purpose, versatility, and functionality. It seems possible to single out the two most common approaches to describing the change in the grain structure in the processes of thermomechanical processing of metals and alloys: continual (in most cases, single-level) and multilevel, based on the introduction of internal variables and physical theories. This review is devoted to the consideration of publications focused on the first of these approaches. Until now, the most common are macrophenomenological continuum models based on the analysis of experimental data conducted both in laboratories on macrosamples and in real production conditions. Models of this class are usually formulated in the form of operator relations over field quantities, they are relatively easy to implement due to their easy “embedding” into widely used commercial software packages, but they require significant costs for experiments to identify models, they are characterized by a low degree of versatility. Continuous theories are relatively less common, but still often used. These theories are based on the description of physical mechanisms and the evolution of the structure of metals and alloys in terms of continuum variables.

**PNRPU Mechanics Bulletin**. 2022;(1):123-155

Two-dimensional model of grain boundary diffusion and oxidation

#### Abstract

The grinding of the structure of materials is accompanied by a change in the physical and mechanical properties. This occurs largely due to the accumulation of energy and defects in the structure, which activates the diffusion of impurities contained in the material. The increase in the number of grain boundaries and joints can cause the inelastic behavior of the material, its additional chemical activation. For some metals and alloys this leads to strengthening, while for others it leads to rapid degradation of mechanical properties. Grain boundary diffusion in such materials is the main mechanism of transport of alloying components or harmful impurities, so its study is important. This paper presents a two-dimensional model of grain boundary diffusion in a material with an explicit structure assignment. The model takes into account the presence of chemical transformations that can determine the corrosion mechanisms under operating conditions. For simplicity of calculation the material structure is taken symmetrical and contains two phases: grains and a boundary phase. Diffusion and kinetic parameters of the phases may differ. The model is represented in dimensionless form so that the distances between neighboring grains or the widths of the boundary phase are the same and the grain sizes can vary. Depending on the ratio of phase sizes we can speak about micro- and nanocrystalline structure. The problem was solved numerically using implicit difference scheme and coordinate splitting. Diffusion and kinetic parameters, which are close to the parameters of oxygen grain boundary diffusion in titanium and titanium oxidation, respectively, were taken for the calculations. Integral concentrations reflect the dynamics (kinetics) of oxygen and oxides accumulation over the calculation area. Results showing the role of changes in the oxidation kinetics due to changes in the reaction constants in the phases and the phase size ratio are presented. The model can be useful for assessing the degree of influence of grain boundary diffusion on the oxidation process and the accompanying change in properties, as well as for setting up appropriate experiments.

**PNRPU Mechanics Bulletin**. 2022;(1):156-166

The Simulation of the Nonlinear Consolidation of Porous Media

#### Abstract

In this paper, the general formulation of the problem of coupled deformation of a porous deformable medium with a fluid flowing through the pores is formulated, mathematically investigated and numerically implemented within the framework of physical and geometric nonlinearity. We present the formulation of the problem in velocities of solid phase displacements and the rate of pore pressure change in differential and variational forms. A phenomenological approach was used to formulate the mechanical model. The equations of the coupled consolidation model were derived from the general conservation laws of continuum mechanics using spatial averaging over a representative volume element. The consolidation model took into account the change in the porosity and permeability of the medium during deformation. The equations of filtration and porosity change, originally presented in Euler approach, were reformulated in Lagrangian coordinates of the solid phase using the relative fluid velocity according to ALE (Arbitrary Lagrangian - Eulerian) approach. The Gâteaux differentiation technique was used to linearize the variational equilibrium equations. For spatial discretization of the saddle system of equations, the finite element method (FEM) was used: quadratic serendipity elements for approximating the equilibrium equations and Brick type elements for approximating the filtration equation. To solve the system of equilibrium and filtration equations, a generalization of the implicit scheme with internal iterations at each time step by the Uzawa method was used. The results of numerical simulation of elastoplastic deformation of a water-saturated soil under load with fluid outflow are presented. To simulate the constitutive relations of elastoplastic deformation of soil under short-term loads, a generalization of S.S. Grigoryan's model to large deformations is proposed. The calculations were carried out in our own program code. The developed consolidation model can be used to simulate the formation of tracking ruts and unevenness of natural roads, as well as to calculate the uneven settlement of engineering structures.

**PNRPU Mechanics Bulletin**. 2022;(1):167-176