## No 3 (2019)

**Year:**2019**Articles:**16**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/40**DOI:**https://doi.org/10.15593/perm.mech/2019.3

Simulation of Metal Powder Bidirectional Compression in a Pressing Tool with a Floating Die

#### Abstract

The process of bidirectional compression of a powder titanium sponge in a pressing tool with a floating die is considered. The paper studies powder screenings of titanium sponge of TG-100 grade according to GOST 17746-96 with a fraction of -3 + 1 mm. A new design of the pressing tool is proposed to implement the process of bidirectional compression. The simultaneous movement of the upper and lower punches relative to the sectional die is achieved by placing the die between two rubber springs. The pressing tool provides an efficient compression due to the implementation of a synchronous vibration-free movement of the lower and upper punches relative to the sectional die, as well as a uniform compression unloading on all sides, which eliminates the occurrence of splitting cracks. Numerical simulation of the process was performed using a finite element analysis. In the simulation, it was assumed that the elements of the pressing tool have properties of an absolutely rigid body. The powder material is considered as a continuous compressed elastoplastic medium with initial isotropic properties. The powdered material yield condition is described by the Modified Drucker-Prager Cap yield model. The rate of plastic deformation is proportional to the voltage at the current moment, the stress state determines the instantaneous increments of plastic deformation components. The associated plastic flow rule is fulfilled. Mooney-Rivlin hyperelastic material model was used as a model of the material (SKU-7L rubber) of the spring elements. As a result of the simulation, the calculated density distribution in the cross-section of the compacted powder workpieces at different compression pressures was obtained. The dependence of the average relative density of the powder workpieces on the compression pressure was investigated. For the compacts obtained at different compression pressures (100 and 500 MPa), a micrographic study of the microstructure of the samples longitudinal sections surfaces was performed. The change in the shape and size of pores in the lower and upper parts of compacts was studied. In order to assess micromechanical properties of the obtained workpieces, kinetic microindentation of thin section surfaces of the longitudinal axial sections was carried out, which allowed determining the microhardness values, the creep characteristic, and the reduced modulus of elasticity. It is shown that the proposed design of the pressing tool allows obtaining workpieces with a more uniform density distribution in comparison with those manufactured using monodirectional compression.

**PNRPU Mechanics Bulletin**. 2019;(3):5-16

The Analysis of residual stresses in layered composites with [0°/90°] layup

#### Abstract

Recently an increased attention has been paid to the deep understanding of the processes caused by residual stresses in laminated composites, specifically, shape distortion and possible product damages at the stage of manufacturing. The presence of residual stresses in a composite material affects the occurrence of such defects as delamination, warping, as well as the appearance of microcracks in a matrix of a material, which has a significant impact on the static and fatigue strength of the product made of composite materials. This paper considers a change in the distribution of the residual stress field in a specimen from the known material AS4/8552-1 with the layers [0°/90°]n in the curing process and the stresses on the free edge after the cut of the sample along the reinforcement direction. The curing model is numerically implemented using ABAQUS finite element modeling package with special user subroutines. The coupled thermal and strength problem under plane strain conditions is solved to model the curing. A significant difference in residual stresses at the free edge before and after the mechanical cutoff is shown. Computed residual stresses are further used to analyze the development of the composite fracture under uniaxial tension. As a result of modelling the loading of the composite up to failure with a consideration of the residual stresses, it is shown that tension leads to cracking of the matrix in layers where the orientation of the reinforcement coincides with the direction of the load application. At the same time, when ignoring residual stresses, the appearance of such defects is not observed. This research shows that residual stresses affect both the strength of the material and the mode of failure.

**PNRPU Mechanics Bulletin**. 2019;(3):17-26

The Assessment of Fatigue Durability and Critical Plane Determination for Multiaxial Cyclic Loading at an Arbitrary Shift of Phases

#### Abstract

The real experience of different structural elements shows that cyclic loading conditions and stress strain state are often quite different from laboratory test conditions such as axial tension-compression, bending or pure torsion. As a rule, the structural elements are subjected to complex (multiaxial) loading conditions during service, that is why multiaxial fatigue criteria should be used for durability estimation of such elements. A present multiaxial criterion allows us to estimate the number of cycles to fracture for specimens or structural element (fatigue life) with an influence of the orientation of the so-call critical plane of fatigue damage accumulation. Analytical solutions were obtained for the present fatigue criterion to determine the orientation of the critical plane of the fatigue damage development under cyclic loading for multiaxial stress states. Cyclic loading with arbitrary shifts of phases for the classical fatigue range (low-cycle and high-cycle fatigue) was considered. Several common cases of the three dimensional cyclic loading, i.e. tension-compression and bending-torsion, are studied. It is shown that at certain values of phase shifts the fatigue durability can be very low even under stress amplitudes that do not lead to fracture in the case of sine phase loading. The comparison with the experimental data and numerical calculations based on the other criterion is made. A disc of a low-pressure stage of a gas-turbine engine subjected to cyclic loading due to centrifugal forces was considered. By using a simplified stress distribution, as well as a stress distribution calculated by the finite element method, the areas of stress concentration and orientation of the critical plane in these zones are determine and the durability of the disk operating was estimated.

**PNRPU Mechanics Bulletin**. 2019;(3):27-36

Inverse coefficient problems in mechanics

#### Abstract

The main statements and methods for studying coefficient inverse problems are presented. The classification of coefficient inverse problems is given depending on the objects being reconstructed. Three classes of tasks are highlighted: finite dimensional tasks, problems on determining one-dimensional functions, problems of multidimensional functions restoring. The main approaches in the study of each type of inverse problems are discussed. Finite-dimensional inverse problems research methods including Prony's method that allows to simplify the solution scheme for a nonlinear inverse problem are described. As an example of a finite-dimensional inverse problem, a method for determining the linear laws of strip inhomogeneity is given. Basic technics for the study of the coefficient inverse problems by the definition of one or several functions in the analysis of steady-state oscillations in various settings are given. In the first setting, the components of the physical fields inside the body are specified as additional information. As an example, the problem of determining the variable Young's modulus of a beam in the analysis of bending vibrations is given. It reduces to a problem for a linear operator equation with a compact operator. As the second typical example of coefficient inverse problem in the first formulation, the problem of determining the variable Lame coefficients when analyzing rectangle oscillations is given. It is reduced to solving the Cauchy problem for a system of partial differential equations of the first order. In the second formulation, the displacement field at the boundary of the body is given in a certain frequency range. It leads to a significantly non-linear incorrect problem. Using the example of the problem of determining the Young's modulus, shear modulus, and density for a functionally gradient cantilever rod of the constant cross section, an iterative process is developed for the analysis of longitudinal, bending and torsional vibrations. At each step, a solution for a system of Fredholm integral equations of the first kind is constructed. The issues of the uniqueness of restoring the desired characteristics are discussed.

**PNRPU Mechanics Bulletin**. 2019;(3):37-47

Construction of Multigrid Finite Elements to Calculate Shells, Plates and Beams Based on Generating Finite Elements

#### Abstract

In practice, composite shells, plates and beams of complex shapes are widely used. Finite element calculations of three-dimensional composite bodies, taking into account their structure and complex shape are reduced to the construction of high dimensional discrete models. To reduce the discrete model dimension, multigrid finite elements (MgFE) are effectively used. When constructing m - grid finite element ( m gFE) m nested grids are used. The fine grid is generated by the base partitioning of m gFE, taking into account its heterogeneous structure and shape. On m - 1 coarse grids, the displacement functions used to reduce the dimension of the base partitioning are determined, which allows one to develop a small dimensional MgFE. The displacement functions and stress state in the MgFE described by the equations of the three-dimensional elasticity problems are represented in local rectangular coordinates. Characteristic properties of MgFE are as follows. When constructing MgFE (without increasing their dimension), arbitrarily small basic partitions can be used arbitrarily closely taking into account the complex inhomogeneous structure and shape of the MgFE, and arbitrarily closely describing the three-dimensional stress state in the MgFE. The present paper proposes a method of generating finite elements (FE) to construct complex-shaped three-dimensional composite multi-grid finite elements (MgFE) of two types. The principal of the generating FE method is as follows. The 1st type MgFE region is obtained by turning a given flat generating single-grid FE (complex shape) around a given axis by a given angle, the 2nd type MgFE region by a parallel moving the generated FE in a given direction to a specified distance. The 1st type MgFEs are used to calculate the composite rotational shells, the 2nd type MgFE are for composite cylindrical complex-shaped shells (with a variable radius of curvature), plates and beams. The advantages of the proposed MgFE are to take into account the complex heterogeneous and micro-heterogeneous body structure and shape, to give rise to small-dimensional discrete models and high accuracy solutions.

**PNRPU Mechanics Bulletin**. 2019;(3):48-57

Stability of chemical reaction fronts in the vicinity of a blocking state

#### Abstract

In the current work we consider a chemical reaction localized on the sharp interface between a solid and a diffusive constituent. The driving force for the reaction front propagation is a normal component of the chemical affinity tensor. The configuration of the deformed state where the driving force is zero corresponds to the reaction blocking state. The paper is aimed at studying this configuration. The utilized approach to analyze the stability of the equilibrium interphase between two deformable domains involves a linearized analysis of the kinetic equation for the perturbed interface. Previously this method was applied for an interface stability analysis in the case of phase transformations. The advantage of this method is that it also shows how the loss of stability occurs. An analytical solution for the perturbed kinetic equation is only possible for some simple configurations. Numerical procedures were applied in order to solve these types of problems. As an example, we used the numerical procedure, and the problem of the chemical reaction front propagation in a hollow cylinder is solved. For the unstable configuration we analyzed a stress state in the vicinity of the reaction blocking.

**PNRPU Mechanics Bulletin**. 2019;(3):58-64

The Modeling of the Edge Fatigue Crack Growth under High Frequency Loading

#### Abstract

Multiple fracture surface investigations of real aviation industry products have proved that low-amplitude high-frequency oscillations can lead to an unexpected failure of aviation elements and other structures. High-frequency loading is the reason for this, as it produces a large number of loading cycles that often exceed an investigated area of a material fatigue behavior. This new fatigue failure mode (very-high-cycle fatigue) requires a special experimental study with the use of both experimental and mathematical modeling. This paper is focused on a numerical determination of stress intensity factors (SIF) in the specimen with the edge notch specimen loaded by harmonic high frequency displacements of small amplitudes. The numerical calculations were performed for the case of loading with frequency close to its natural frequency. A dimensionless adjusting function was determined for SIF that takes into account a change in modal characteristics of the resonance system (the specimen with rectilinear edge notch) due to crack propagation. The obtained equation was used to model the position of the curvilinear fatigue crack front. A general scheme of a piezoelectric fatigue testing machine is introduced with a technique of tensile-compression fatigue tests on the titanium crack growth specimen with an edge notch in the range of very high cycle fatigue. The analysis of the fracture surfaces with the identification of the front stop lines and mathematical modeling of the crack front evolution under high-frequency loading are carried out. The results of mathematical modeling are comparing with experimental data obtained during high-frequency fatigue tests on a piezoelectric fatigue testing machine. The numerical calculations have shown that this approach allows us to qualitatively and quantitatively simulate the evolution of the edge fatigue crack with a curvilinear front under very-high-cycle fatigue (high-frequency) loading mode.

**PNRPU Mechanics Bulletin**. 2019;(3):65-74

Numerical Simulation of the Phase Transition Control in Torsion of a Hollow Cylinder Made of Heusler Alloy

#### Abstract

The paper discusses ferromagnetic alloys with shape memory (Heusler alloys) undergoing a phase transformation from the high-temperature cubic phase (austenite) to the low-temperature tetragonal phase (martensite) in the ferromagnetic state. In these alloys, significant macroscopic strains are generated during the direct temperature phase transition from the austenitic to the martensitic state due to the application of mechanical stresses. Since the critical temperatures of the process depend on the magnetic field and stress fields, in such alloys it is possible to control the austenite - martensite phase transformation process by applying a magnetic field. In this work, within the framework of finite deformations a model has been presented to describe the process of controlling the forward (austenite - martensite) and reverse (martensite - austenite) phase transformation by applying a magnetic field in ferromagnetic polycrystalline materials with shape memory under the action of external force, thermal and magnetic fields. Since the magnetic field affects the strain of the material, which in turn changes the magnetic field, a connected boundary value problem has been constructed. The problem of azimuthal torsion of a long hollow cylinder (plane deformation) from a Heusler alloy has been solved numerically by the finite element method using the step-by-step loading procedure. The accumulation of phase strains occurred during the forward phase transformation in a cylinder, the outer surface of which was previously twisted in the azimuthal direction with respect to a fixed inner. In this case, the value of the accumulated phase strains depended on the boundary conditions (force or kinematic), that produced twisting the outer surface of the cylinder. The complete removal of the accumulated strains and the unwinding of the outer surface of the cylinder occurred both when the sample was heated in the temperature range of the reverse phase transformation in the absence of a magnetic field, and at a constant temperature exceeding this interval, when the magnetic field applied previously in the martensitic state was removed.

**PNRPU Mechanics Bulletin**. 2019;(3):75-87

About Electrical Conductivity of the Epoxy Matrix with Carbon Nanotubes

#### Abstract

Polymeric materials on the basis of epoxy resins are widely applied in production of various devices and structures. In real work electrical conductivity of epoxy matrixes modified by carbon nanotubes under continuous current in the course of their polymerization at a constant tension is experimentally investigated. The relevance of the research is bound to an adequate theoretical model of orientation ordering of carbon nanotubes for a liquid polymeric matrix for an electric field. The description of the pilot unit and measurement technique is provided. The paper presents the measurement results of volt-ampere characteristics of specimens of composite materials with various percentages (by weight) of carbon nanotubes, which prove that the conductivity has an ohmic character. Stabilization times of the current proceeding through the specimens after the beginning of polymerization at a constant tension are defined. According to established values of current it is shown that the conductivity of the specimens of epoxy matrixes through which current at polymerization proceeded is higher than the conductivity of specimens through which current didn't proceed. At the same time with an increase of nanotubes concentration, there comes an increase of difference of the specimens’ conductivity. The research of dependence of the specimen conductivity on temperature in a day after the beginning of polymerization of the composite material is conducted. It is established that when heating to 90 ͦ C, the conductivity of the specimen imposed with the current during polymerization decreases to values of the specimen conductivity without current. The research is aimed at studying how to make solidification technologies of composite materials faster for production of large parts in space applications. The received results can be also used for development of prospective manufacturing techniques of composite materials with the given electrophysical and mechanical characteristics using electric fields of the corresponding configuration in the course of their polymerization.

**PNRPU Mechanics Bulletin**. 2019;(3):88-93

Simulation of the interaction fullerite C60 with the substrate solids

#### Abstract

The widespread use of carbon in various branches of engineering and instrumentation has led to a large amount of research related to its nanoscale allotropic forms - graphene, nanotubes, fullerene, and fullerite. In this article, we studied the process of interaction of fullerite C60 with a solid substrate in order to establish the conditions under which fullerite itself or its fullerenes are deposited on the surface of the substrate. The application of the research can be the creation of new wear-resistant coatings. The study of the interaction of fullerite C60 with a solid substrate was carried out at various parameters of the system. The study has the following conditions: system temperatures - 300, 700, 1150 K; the speed of motion of fullerite is 0.005, 0.01, 0.02 Å/fs; fullerite orientation relative to the substrate. The velocity vector of fullerite C60 was directed along the normal to the upper surface of the substrate. The orientation of fullerite determined by what part it will interact with the substrate - “face”, “edge” or “top”. An iron crystal Fe (100) was simulated as a substrate. To carry out computer simulation, the LAMMPS software package was used, which uses molecular dynamics methods. The results of the research are the revealed patterns of behavior of fullerite C60 in general, and fullerenes in its composition in particular, upon contact with the substrate. For example, it was found that with an increase in the rate of fullerite C60, the number of fullerenes deposited on the substrate decreases. In addition, the orientation of fullerite C60 relative to the substrate has a significant impact on both the process of their interaction and the behavior of settled fullerenes.

**PNRPU Mechanics Bulletin**. 2019;(3):94-103

Buckling Problem of Composite Thin-Walled Structures with Properties Dependent on Loading Types

#### Abstract

Experimental studies of deformation of laminated composite materials often show a complex dependence of stiffness and strength characteristics on the type of loading. The most obvious examples are the difference of elastic moduli in tension and compression, nonlinear shear diagrams. It is possible to take into account such effects in applications only with the use of nonlinear elasticity models. Such models should not contradict fundamental physical principles and, in addition, should not require too complex experiments for validation. In this paper, we consider and numerically implement a model of nonlinear elasticity based on the use of the triaxiality parameter to describe the type of the material stress state. For a numerical implementation, the finite element modelling software Abaqus with user-defined subroutines was used. As a demonstration example, the applied problem of compression of a thin-walled composite cylindrical shell supported by stiffeners is considered. During loading, the shell exhibits an unstable behavior and stress state in its various areas can change drastically. For example, a change from the prevailing tension to compression and vice versa. In this case, damage to the shell material is not observed until a complete structure failure due to the loss of stability of stiffeners, which allows considering only the elastic model for this problem. The use of linear elasticity theory to model such a critical behavior of the structure leads to inaccurate results even at the initial stage of loading, while the proposed model shows a good consistency of the loading diagram, as well as the shape and magnitude of the shell deflections with the experiment.

**PNRPU Mechanics Bulletin**. 2019;(3):104-111

Simulation of residual stresses during pulsed thermo-force surface hardening

#### Abstract

The paper deals with solving the mechanic connected problems applying to residual stresses under non-stationary thermal effects. The research is aimed at the electromechanical processing technology in the application to titanium pseudo-α-alloys which change their phase composition due to the martensitic phase transition under local thermal effects on the surface layer. The paper provides a mathematical formulation, considered features and methods for solving a thermostable contact problem inclusive of the phase transformations occurring during high-speed cooling. The main stages of building the necessary defining relations are shown. Relevant correlations of the plastic flow theory in the velocity form in the limits of the isotropic-translational hardening model are given, issues of integrating these correlations are considered. The technique determining the non-stationary contact zone of an absolutely rigid stamp and a deformable half-space is shown. The main linearization stages of the variational equation used are considered separately. Within the developed algorithm, a series of computational experiments were made simulating the temperature-force effect in Ti6Al2V titanium pseudo-α-alloy as applied to the technology of the pulsed thermal-force surface hardening. It is established that the electromechanical surface treatment of titanium alloys leads to the formation of discretely structured regions of residual stresses in the surface layer, which is connected to one of the following factors: the heat source (sinusoid), and on the other hand, the martensitic structure which is discrete. The significant role of the deformational effect on the material when residual stresses formation is shown. In particular, it has been established that in the case of an increasing loading on the tool from 10 N to 250 N, the value of the tensile residual stresses decreases by 3 times.

**PNRPU Mechanics Bulletin**. 2019;(3):112-124

A Review of some non-traditional applications of the engineering theory of high-speed penetration

#### Abstract

In this review, the traditional tasks of high-speed penetration are defined as tasks aimed at describing the movement of penetrators in monolithic barriers and determining integral characteristics of penetration, such as the depth of penetration into a semi-infinite shield or the ballistic limit when penetrating the barrier of finite thickness. Most of the papers on penetration mechanics dedicated to such problems are based on experimental and numerical methods; many of them are based on the use of analytical methods. Tasks that do not fall into this category will be referred to as non-traditional; this review is dedicated to them, with the main emphasis on tasks for the study of which the use of analytical methods or the potential and feasibility of using them has been characteristic. These are the tasks of penetration into non-monotonic barriers (multilayer layers with adjacent layers or with air gaps between them); the task of optimizing the shape of the strikers; the task of modeling and optimizing the "active" penetration when controlling the movement of the penetrator (the use of impactor with jet thruster to increase the penetration depth with retaining the integrity of the penetrator when getting soil samples from the surface of the planets or when delivering explosives to the object of destruction; the use of artillery for driving piles); the effectiveness analysis of segmented impactors (impactors with spaced elements); and others. An important feature of the review is the desire to highlight the characteristic methodological features of the implemented approaches, which are often hidden behind the list of research results. The material presented in the review covers promising areas of research and designed to facilitate orientation in the relevant topics, in particular, to select a revevant topic of the thesis, which is of both theoretical and practical interest.

**PNRPU Mechanics Bulletin**. 2019;(3):125-139

The Structure of Interfacial Regions in Polymer Nanocomposites

#### Abstract

The paper analyzes the dependence of characteristics of the interface areas between the filler and the metal-polymer nanocomposite on structural parameters of the metal-polymer nanocomposite obtained by cross-linking two polymers (epoxy resin ED-20 and butadiene styrene rubber - BSR) and the filler, i.e. copper nanoparticles. To calculate the volume fraction of the interphase region, φмф, we used the previously established expressions for the fractal dimension of the surface of the filler particles, dH, and the diameter of the filler particle aggregates, Dag. The polymer under study is considered as a natural nanocomposite using cluster models to describe the peculiarities of its amorphous state structure. According to this model, a mesh polymer consists of two components - a loosely packed matrix and nanoclusters, the latter plays the role of a nano-filler, and a loosely packed matrix which plays the role of a natural nanocomposite matrix. On the basis of the fractal estimates of the surface of nanoclusters, the dependence of the volume fraction of the loosely packed matrix on the composite temperature has been found. It has been shown that the dimensional effect of nanoclusters is identical to the corresponding effect of the dispersed nano-filler in artificial polymeric nanocomposites, namely, the reduction of the number of statistical segments in one cluster and the radius of the clusters increases the degree of amplification (modulus of elasticity) of the natural nanocomposite. In addition, for the mesh polymer under consideration, there are no interphase regions structurally different from the loosely packed matrix, since the calculations of the relative volume fractions of the interphase regions and the loosely packed matrix, respectively, show a good convergence within a given error of the initial data.

**PNRPU Mechanics Bulletin**. 2019;(3):140-148

The Phenomenological Model of Hyper Viscoelasticity of Elastomers with a Constant Orientation of the Main Stress Axes in the Accompanying Frame of Reference

#### Abstract

The Bergström-Boyce hyper-viscoelasticity model, which is based on considerations of the microstructure of rubber-like materials and uses the multiplicative decomposition of the total strain gradient, is considered and analyzed. Special attention is paid to the choice of the single-valuedness condition, which determines the rotation of the intermediate (relaxed) configuration and ensures the uniqueness of the multiplicative decomposition of the gradient total strain to the gradient of elastic deformations and the gradient of viscous deformations. To verify the validity of the statement that this choice is not essential, we found a solution of the test problem of simple shear according to the Bergström-Boyce model is obtained for the three most frequently used single-valuedness conditions. The results of numerical calculations showed a significant discrepancy for dynamic stresses and a less significant discrepancy for total stresses. To separate the permissible single-valuedness conditions from the physically unacceptable single-valuedness conditions, it is proposed to use the principle of the material frame-indifference. For this purpose, of we studied the transformation of the elastic deformations and viscous deformations gradients when replacing the reference system, to which there is no single point of view in the scientific literature. For the most frequently proposed and used single-valuedness conditions, it is shown which of them do not depend on the reference system choice. Since the list of such kind of allowable ratios can be expanded many times, the choice a single-valuedness condition should be solved in the same way as it is done, e.g. for elastic potentials: a proper formulation and conducting experimental studies, or a theoretical study of the material microstructure. A phenomenological model of hyper-viscoelasticity is proposed, based on a one-dimensional Kelvin-Poynting model and limited to the case when the main axes of stresses and strains (full, elastic and viscous ones) coincide and do not change their orientation relative to the material lines (fibers). This ensures the uniqueness of the corresponding multiplicative decomposition. In order to expand the range of strain rates when describing experimental data, the dependence of the viscosity coefficient on the second invariant of the right Cauchy-Green viscous strain measure in the power law of the apparent viscosity of the Reiner-Rivlin model is taken into account, which generalizes the corresponding dependence of the Bergström-Boyce model. The developed mathematical model of hyper-viscoelasticity of rubber-like materials is intended for the stress-strain state calculation of highly elastic shells of rotation under symmetric loading.

**PNRPU Mechanics Bulletin**. 2019;(3):149-165

The Stressed state of the double-layer rectangular plate under shift. The Simplified two-dimensional model

#### Abstract

A considerable attention is given to studying the stressed state of adhesive joints due to the structure’s bearing capacity, which is usually determined by the strength of the connections, where the stressed state is irregular. Most of the existing mathematical models of joints are one-dimensional, i.e. they imply an even distribution of stresses across the width of joints. However, there are constructions for which the classical models are not applicable. In order to calculate the stressed state of such joints it’s necessary to take into consideration the irregularity of stresses not only in length but also in width of joints. To solve such problems, a simplified two-dimensional model of the overlapping adhesive joint of the rectangular plates is proposed. The simplification consists in the fact that the displacements of the layers are considered only along one of the axes. The model is actually a two-dimensional generalization of the Volkersen classical connection model. The stresses are considered even distributed over the layer thickness and the adhesive layer works only on a shift. These simplifications allowed us to obtain an analytical solution to the studied problem. The problem of the stressed state of the adhesive jointing of the two rectangular plates is solved, one of which is rigidly fixed along one side, and the second one is loaded with an uneven shear load on the opposite side. The problem is reduced to a system of second-order differential equations in partial derivatives with respect to longitudinal displacements of the two carrier (outer) layers. The solution is constructed using the method of separating the variables, in the form of a functional series consisting of the Eigen functions of the spectral problem. The boundary conditions at the unloaded ends are accurately satisfied. The satisfaction of the boundary conditions on the sides leads to a system of linear equations for the unknown coefficients of the functional series. The convergence of the obtained solution is proved. The model problem is solved and the numerical results are compared with the results of the calculations performed using the finite element method. It is shown that the proposed approach has a sufficient accuracy for engineering tasks.

**PNRPU Mechanics Bulletin**. 2019;(3):166-174