## No 4 (2023)

**Year:**2023**Articles:**13**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/360**DOI:**https://doi.org/10.15593/perm.mech/2023.4

EXACT SOLUTION OF THE PROBLEM OF ACOUSTICS IN AN ARBITRARY MULTILAYER MEDIUM DURING CONTACT INTERACTION WITH A WEDGE SHAPED STAMP

#### Abstract

This paper is the first to study the behavior of the exact solution of the contact problem for a wedge-shaped stamp in terms of shape in an anisotropic layered medium. We consider the contact problem of the action of a wedge-shaped, right-angled, rigid stamp on the surface of a multilayer anisotropic medium. The case of a sharp-angled stamp in terms of some transformation is reduced to the one under consideration. The stamp is assumed to act on a multilayer medium without friction. There may be cases of static and dynamic effects caused by harmonic oscillations of the stamp. The main attention is paid to analyzing the surface behavior of an anisotropic layered medium outside the contact zone. Formulas describing the behavior of the surface in the far zone are constructed and an example of calculating the necessary parameters for their application is given. The considered mixed problem is reduced to solving the two-dimensional Wiener – Hopf integral equation, the Fourier transform of the kernel of which represents the ratio of two analytical functions. The isotropic case of the presence of the ratio of two integer functions in the representation of the kernel has recently been investigated by a universal modeling method, which prompted the transition to the little-studied anisotropic case. In spatial contact problems, the study is carried out by numerical methods that are ineffective for anisotropic media. The exact solution could be constructed only in cases of one-dimensional or integral equations reducible to them. Along with static tasks, the method developed in the article allows studying the acoustic properties of the surface outside the contact zone of the stamp with the medium in the dynamic case, which have little-studied specifics of behavior by sectors. The two-dimensional Wiener-Hopf integral equation solved for the first time can be used in problems of radio wave propagations, in the design of the element base of radio electronics, in the problem of strength in mechanics, and in numerous other important areas.

**PNRPU Mechanics Bulletin**. 2023;(4):5-11

ANALYSIS OF METHODS FOR CONSTRUCTING TRUE DEFORMATION DIAGRAMS OF ELASTOPLASTIC MATERIALS UNDER LARGE DEFORMATIONS

#### Abstract

To study the deformation and strength properties of materials, it is important to use an experimental and computational approach that allows taking into account the ununiaxiality and heterogeneity of the stress-strain state without accepting simplifying hypotheses. True deformation diagrams are constructed using an iterative procedure of updating the strain intensity– stress intensity relation proportionally to the relative difference in the values of axial forces as obtained numerically and experimentally for an inhomogeneous stress-strain state, accounting for necking, up to rupture. The procedure requires multiple solutions of the problem, which is a time-consuming computational task. Two scenarios of analyzing the boundary-value problem are considered. The first scenario involves analyzing the entire direct problem over the whole loading interval; in the second one, the entire loading process is subdivided into several intervals defined by discrete values of an experimentally found generalized displacement–generalized force relation. At each small interval, a deformation diagram is constructed, using a nonlinear extrapolation procedure. At the end of each interval, the difference between the calculated and experimentally determined generalized forces is checked, and the stress intensity value is iteratively updated. The presented numerical studies show that constructing a deformation diagram with accuracy less than 1% according to the first scenario required 5–10 repeated analyses of the direct problem, whereas in the second scenario not more than two direct analyses suffice. Monotonous convergence and computational efficiency of the proposed iterative algorithms are shown for a number of tasks: stretching solid cylindrical rods and bolts M8 with smooth and threaded working parts. Based on the experimental and computational approach, the true deformation diagrams for steels 12H18N10T, 10HSND and St35 up to destruction are determined.

**PNRPU Mechanics Bulletin**. 2023;(4):12-22

ELASTIC-PLASTIC BEHAVIOR AND FRACTURE OF STRUCTURES WITH LOSS CONCENTRATORS UNDER CYCLIC LOADS

#### Abstract

The results of experimental and theoretical studies of structures with stress concentrators under cyclic loading are presented. The studies were carried out on cylindrical specimens with an annular undercut under mild cyclic loading. The sample material is bronze alloy BrKh08-Sh. When testing samples using the digital image correlation method, the deformation of the material on the surface of the undercut is measured, which makes it possible to determine the nature of the change in its range from cycle to cycle. For mathematical modeling of elastoplastic behavior and failure of structures with stress concentrators, a variant of the theory of plasticity based on the theory of flow under combined hardening is used. In the chosen plasticity model, a memory surface is introduced that separates the processes of monotonic and cyclic loading. This division allows one to take into account various features of isotropic and anisotropic hardening of the material. Anisotropic hardening is represented as a sum of microstresses of three different types, which make it possible to describe the effects of fitting and stepping out of the elastoplastic hysteresis loop. The plasticity model makes it possible to assess the damaged state of the material based on the kinetic equation of damage accumulation based on the energy principle (the work of microstresses on the field of plastic deformations). The material behavior model is embedded in the finite element software package. Based on the calculation results, cartograms of the accumulated plastic deformation and stress intensity were constructed. Comparison of the results of calculations and experiments on the range of axial deformation, average axial deformation on the surface of the undercut and the cycle of cycles to failure is carried out. It was found that in a structure with a groove with a radius of 0.25 mm in the concentration zone, rigid loading is realized, and with a radius of 1 mm in the concentration zone, soft asymmetric loading with onesided accumulation of deformation (stepping out) is realized. There is a decrease in durability with a decrease in stress concentration due to the different nature of the change in the stressstrain state.

**PNRPU Mechanics Bulletin**. 2023;(4):23-33

INVESTIGATION OF THE ELECTRO-MAGNETO-ELASTIC STATE OF A FINITE MULTIPLY CONNECTED THIN PLATE

#### Abstract

The problem of bending a finite plate with arbitrary holes and cracks is solved with the use of complex potentials of the theory of bending of thin electro-magneto-elastic plates. Moreover, with the help of conformal mappings, expansion of holomorphic functions into the Laurent series or Faber polynomials owing to satisfaction of boundary conditions on the contours of the plate by the generalized least squares method, the problem is reduced to solving an overdetermined system of linear algebraic equations by the method of singular value decompositions. Results of numerical investigations for a circular plate with a circular hole, for a circular plate with an internal or edge crack, for a plate with a two circular internal holes or external recesses are reported. We study how physical and mechanical properties of the plate material and geometric characteristics of holes, cracks and recesses influence the values of the bending moments and moments intensity factors for the crack ends. It is important to consider the piezoproperties of the material on the values of bending moments in the plate. They cannot be neglected in the study of the stressstrain state, that is, it is necessary to solve the problem of electro-magneto-elasticity, and not the problem of the classical theory of bending of an anisotropic plate. Moreover under the electromagnetic field in the piezoelectric plate there are sufficiently large bending moments (hence stresses and deformations), and they can be found only by solving the problem of electromagneto- elasticity. It is determined that a crack in a plate can be considered as an elliptical hole, in which the ratio of the semiaxes is less than 10–3, and in these cases it is possible to calculate the intensity factors of mechanical and electromagnetic moments. We also outline the distances between the contours, which have an insignificant influence of one of them on the stress-strain state around the other and can be neglected.

**PNRPU Mechanics Bulletin**. 2023;(4):34-44

ANALYSIS OF THE STRESS-STRAIN STATE OF ALUMINUM ALLOY D16T UNDER A COMPLEX STRESS STATE ACCOUNTING FOR DAMAGE

#### Abstract

This paper studies the limiting state of D16T aluminum alloy under a complex stress state. Various types of combined loadings by tension, compression, torsion and internal pressure are considered. The determining conditions for many products of modern technologies are combinations of several loading components. The purpose of the study is to determine the effect of the accumulated damage in a material under a complex stress state on the characteristics of the material bearing capacity. A hollow cylindrical sample was chosen for the numerical and experimental studies. The experimental program included various combinations of axial forces, torque, and internal pressure applied to a cylindrical sample. In the numerical study, the exponential and linear-power law of isotropic hardening was used as the law of isotropic hardening. The damage accumulation law Lemaitre was used to determine the damage parameter. The generalized law of damage accumulation Leamitre and the law of isotropic hardening were integrated into the ANSYS finite element complex in the form of a dynamically linked library of custom material for three-dimensional problems. The states of hollow cylindrical samples are investigated. The fields of the stress-strain state, the fields of damage, as well as the values of limit stresses for various types of loading are obtained. Limit state diagrams are constructed taking into account damage accumulation.

**PNRPU Mechanics Bulletin**. 2023;(4):45-53

STUDYING GRAPHENE ELECTROMECHANICAL BEHAVIOR BASED ON THE ELASTIC PLATES MOMENT-MEMBRANE THEORY

#### Abstract

Graphene being a two-dimensional nanomaterial created a new research area – its use as a functional element in various modern technology nanodevices. To open the way for such an application, it is necessary, first of all, to study the mechanical properties and its behavior under various conditions and learn how to manipulate them in a controlled way. All this necessitates an adequate graphene deformation simulations, the construction of an appropriate continuum theory that takes into account scale effects, microstructure and graphene crystal lattice physical parameters. This work aims at studying a rectangular graphene sheet in an electric field. For graphene deformations, the application of the transverse bending model of the elastic plates of the moment- membrane linear theory is justified. Forces of the electrical origin are modeled by means of a normal load applied to the front plate surface. A study was made of the influence of boundary conditions, the gap between the plate and the gate, the potential difference, and linear dimensions on the graphene sheet static transverse bending. The problem of graphene sheet natural vibrations is also solved. The graphene sheet vibration frequency lies in the GHz range. In the clamped boundary conditions case, the natural oscillations frequency is much higher than in the hinged supported case. Even at small deflections, a change in the constant voltage value has a significant effect on the graphene sheet natural frequency.

**PNRPU Mechanics Bulletin**. 2023;(4):54-67

MODELING OF AXIAL COMPRESSION OF V95/10% SIC ALUMINUM MATRIX COMPOSITE UNDER NON-STATIONARY THERMOMECHANICAL CONDITIONS

#### Abstract

Intensive deformation is necessary to obtain products made of aluminum-matrix composite materials (AMCM) with the required level of mechanical properties. To model the deformation behavior in non-stationary conditions of the thermo-deformation treatment, the identification of the AMCM model remains an urgent task. The use of the Johnson-Cook plasticity model is one of the approaches to describing the material fluidity. This paper aims at studying an AMCM made of granulated high-strength aluminum alloy V95 of the Al–Zn–Mg–Cu system, reinforced with SiC particles 10 % by weight. We investigate how non-stationary thermomechanical (pressure on the workpiece and heating temperature) deformation conditions influence the true deformation and deformation rate of the composite material, as well as identify the material model and verify its application to study the shape changes under certain pressure and temperature ranges. The precipitation process is studied under uniaxial compression of the sintered cylindrical samples of AMCM in the range of the initial pressures of 5.65–7.81 MPa when heated to 510, 530 and 550 °C. In this range, the dependences of the degree of deformation and the average deformation rate for the process are obtained. Identification of the rheological model of the material was carried out. A mode of the preliminary thermomechanical processing is proposed and a prototype is manufactured at an initial pressure of 6.7 MPa on the workpiece and heated to 550 °C in 84 minutes. The above mode provided a relatively uniform filling of the stamp cavities with the composite material. To confirm the possibility of applying the results of the parametric identification of the material model, we simulated the prototype manufacturing process.

**PNRPU Mechanics Bulletin**. 2023;(4):68-76

ESTIMATING PARAMETERS OF PERMISSIBLE DEFECTS IN STRUCTURAL FIBERGLASS BASED ON THEORY OF CRITICAL DISTANCES

#### Abstract

In the process of manufacturing products from composite materials, many defects can occur: cracks, chips, scratches, dents, impact defects, air macro inclusions, and others. Such defects can significantly reduce both the static and fatigue resistance of structures. The purpose of this work is to determine the size of defects that do not affect the strength characteristics of products made of STEF composite material using the point and linear approaches of the theory of critical distances. In the course of the work, a series of tensile tests were carried out on flat specimens of STEF structural fiberglass for electrical purposes. In addition to the experiment, numerical simulation of the tensile processes of these specimens was also carried out. The studied specimens were strips without stress concentrators and with a concentrator in the form of V-shaped notches with different rounding radii at the concentrator top and notch depth. The results obtained were used to determine the material constants according to the theory of critical distances. In this case, two approaches of the theory of critical distances were used, i.e. linear and point ones. To analyze the experimental results, finite element models were built using the ANSYS software package; and numerical simulation was carried out, which resulted in the obtained linearized maximum principal stresses on the central line passing through the top of the stress concentrator. Based on the results of the work, the values of the critical distances of the composite were determined, obtained by using the point and linear methods. On the basis of the data obtained, the sizes of permissible defects in the studied fiberglass were established, which do not affect the strength characteristics of the material. The results obtained can be used to predict the strength characteristics of real products with a complex geometry, as well as to diagnose damaged structural elements.

**PNRPU Mechanics Bulletin**. 2023;(4):77-86

BUILT-IN FIBER-OPTIC MECHANOPHOTOLUMINESCENT SENSOR OF COMPLEX DEFORMED STATE FOR MONITORING VIBRATIONS OF POLYMER COMPOSITE STRUCTURES

#### Abstract

A mathematical model of an embedded fiber-optic mechanical (elastic) photoluminescent (MFL) sensor of a complex stressed-deformed state for monitoring vibrations of polymer compo- site structures has been developed. The sensor includes one or more light guides doped with many spherical MFL nanoparticles (uniformly distributed over the volume of the light guide) of the "core/shell" type. The latter is an elastomechanoluminescent (EML) core with a photoluminescent (FL) shell. Here the EML effect is the light output of the material with its elastic (non-destructive) deformations. The FL-shell of each capsulated particle transforms the informative "internal" ML- radiation of the core into an "external" informative FL-light flux within the light guide. The resulting value of the FL-light flux from all particles is recorded at the output of each light guide. An addi- tional function of the shell is the localization (within the boundaries of each particle) of the infor- mation glow of the EML-core, which, as a result, improves the spatial resolution of the sensor to diagnose significantly heterogeneous (along the length of the sensor) deformation fields. The MFL-sensor is designed to diagnose the components of the harmonic macrodeformation ampli- tude tensor of the local composite region under consideration, i.e. the vicinity of the built-in sen- sor based on the measurement results of informative photoluminescent FL-light fluxes at the outputs from the light guides of the sensor. Control and adjustment of the output (in the working end "input/output" of the sensor) and recorded informative FL-light fluxes is carried out by using a variable input control light flux, in particular, the same for all light guides of the sensor. It was found that in case of using the single "quartz/MFL particle" light guide (pressure sensor), the desired "spectrum" of pressure amplitudes (the density function of the distribution of amplitude values along the longitudinal axis of the sensor) is a solution of the Fredholm integral equation of the 1st kind based on the results of measurements (at the output from the light guide) of the in- formative resulting FL-light flux as a function of the control incoming (ML) light flux flow. The results of the numerical modeling are obtained for the dependence of the light FL flux value on the control ML flux for cases of uniform and non-uniform (but with a "uniform" spectrum) distribu- tions of the diagnosed pressure amplitude value along the sensor length.

**PNRPU Mechanics Bulletin**. 2023;(4):87-100

INVESTIGATION OF THE STABILITY OF FBG SENSOR MEASUREMENTS UNDER VARIOUS CLIMATIC CONDITIONS

#### Abstract

The sensing elements used to measure various parameters, including strain, are not only expected to ensure the reliability of the measured values, but also the stability of the measurements over a long period of time when exposed to various environmental conditions. The paper presents the results of studying the stability of the readings of point fiber-optic sensors based on a fiber Bragg grating under various climatic conditions. During the study, strain of the optical fiber was measured at various levels of temperature and relative humidity, under an external load using a fiber Bragg grating inscribed in the core of the optical fiber. For the experiments, a special setup was created that allows fixing an optical fiber with fiber Bragg grating and applying an external load in the form of a suspended load. The duration of the experiments ranged from 550 to 900 hours. Humidity and temperature ranges correspond to the most common values at which fiber-optics sensors are operated. The data analysis showed that for the fiber-optic sensors the readings do not change significantly during the time at different relative humidity and air temperature, as well as at loads corresponding to 50 and 70 % of the maximum load for an optical fiber with a fiber Bragg grating inscribed by the phase mask method. This result indicates the possibility of the effective use of fiber-optic sensors based on a fiber Bragg grating for long-term measurements of strain in the ambient temperature range from – 40 to + 80 °C and relative humidity from 5 to 95 %.

**PNRPU Mechanics Bulletin**. 2023;(4):101–109

EXPERIMENTAL AND THEORETICAL INVESTIGATIONS OF STRUCTURAL MECHANISMS AND PLASTIC STRAIN LOCALIZATION EFFECTS IN ALMG6 ALLOY UNDER DYNAMIC LOADING

#### Abstract

This paper is concerned with substantiating one of the mechanisms of plastic strain localization under high rate loading associated with structural transitions in the defect structure of materials. For this purpose, a series of experiments were carried out to study the localization of plastic strain in skewed specimens of the AMg6 alloy subjected to loading in a split Hopkinson pressure bar. The temperature fields generated during the plastic deformation tests designed to identify the characteristic stages of strain localization were investigated "in-situ" using a high-speed infrared camera CEDIP Silver 450M. The values of temperatures in the strain localization zone indicate that in the AMg6 alloy under the implemented loading conditions the mechanism of strain localization caused by thermoplastic instability is not realized. Structure analysis of dynamically loaded specimens was carried out using the Olympus GX- 51 optical microscope and FEI PHENOM G2 ProX scanning electron microscope. It supports the structure-dependent regularities of the strain localization mechanism under dynamic loading. The experimental results of dynamic loading with a subsequent investigation of the temperature fields, the structural studies with an optical and electron microscope, as well as the data of the numerical modeling considering the kinetics of the mesodefect accumulation in the material suggest that one of the mechanisms of plastic strain localization in the AMg6 alloy under realized loading conditions is caused by structural transitions in the defect structure of the material.

**PNRPU Mechanics Bulletin**. 2023;(4):110–120

BERNOULLI HYPOTHESES IN THE PROBLEM OF BENDING A MECHANICALLY INCOMPRESSIBLE BEAM

#### Abstract

The incompressibility condition for an isotropic linearly elastic material seriously restricts the application of the classical hypotheses of the beam bending theory formulated by Bernoulli for small deformations and displacements. At the same time, it is assumed that such a strong kinematic condition as the condition of immutability of volume must be unconditionally fulfilled. The term “mechanical incompressibility” implies the impact on the beam exclusively of a force load, but with thermal action on it, the deformation of the volume change is a function of temperature. Nevertheless, in both of these cases, the condition of mechanical incompressibility may conflict with the classical hypotheses of beam bending, which may lead to the problem degeneration. Therefore, before solving any problem for mechanically incompressible materials, it is necessary to check all hypotheses used and sufficiently justified for conventional materials for compliance with the kinematic condition of volume immutability. In case of inconsistency, it is necessary to build a calculation model based on other hypotheses that do not contradict incompressibility, which will not lead to a serious complication of the tasks being solved. For a bent beam, the Bernoulli model is used, the basis of which is the kinematic hypothesis of a straight normal (the transverse segment after deformation remains straight, orthogonal to the curved axis of the beam and the distances between the points of the segment remain unchanged) and the force hypothesis of the non-compressibility of the beam fibers in the transverse direction. Each of the above hypotheses should be checked for compliance with the condition of immutability of the beam volume when exposed to the surface force bending load. The consideration of transverse deformations is relevant for low-modulus materials and especially for materials with a low shear modulus in the transverse direction. Incompressible materials, as a rule, belong to low-modulus, but this is not their property that is decisive in the analysis of Bernoulli hypotheses.

**PNRPU Mechanics Bulletin**. 2023;(4):121–129

APPLICATION OF THE FINITE ELEMENT METHOD IN COMBINATION WITH THE CONTACT LAYER METHOD TO DETERMINE THE STRESS-STRAIN STATE OF MULTILAYER BEAMS

#### Abstract

The purpose of this article is to develop a method for calculating multilayer beams using the finite element method in combination with the contact layer method. The contact layer is an elastic anisotropic medium, consisting of rigid short bars, working only in tension-compression in the vertical direction and shear. The contact layers model the connections through which the layers of multilayer beams interact. To determine the stress-strain state of the beam, it is represented as a set of beam finite elements (FE) of each layer connected by finite elements of contact layers. As beam elements, modified FEs are used, in which horizontal displacements along the upper and lower edges, as well as deflection, act as degrees of freedom in the node. An example of the calculation of a three-layer beam hinged at the ends under the action of a uniformly distributed load is presented. The outer layers of the beam are made of carbon fiber, and the middle layer is made of the syntactic one based on glass spheres. The calculation is performed with and without taking into account the deformations of the transverse shear of the layers. The meshing of the beam along the length into finite elements is assumed to be non-uniform with a thickening in the near-support zone in order to be able to catch the edge effects. The solution is implemented in the MATLAB environment. As a result of the calculation, it was found that there is a range of change in which the stiffness of the contact layers does not have a noticeable effect on the deflections of the structure. For the considered example, a significant difference was revealed in the values of maximum displacements, as well as in the character of the diagrams of bending moments and shear forces in the outer layers when calculating with and without taking into account transverse shear deformations. At the same time, transverse shear deformations do not have a noticeable effect on the stresses in the contact layers.

**PNRPU Mechanics Bulletin**. 2023;(4):130-139