## No 2 (2021)

**Year:**2021**Articles:**18**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/139**DOI:**https://doi.org/10.15593/perm.mech/2021.2

Contradictions in the Plane Contact Problem of the Theory of Elasticity on the Compression of Cylinders in Contact with Parallel Generators

#### Abstract

The inapplicability of the reference formula for determining the convergence of two statically compressed parallel cylinders made of a homogeneous, isotropic and physically linear material has been proved due to a well-known logarithmic feature in the plane classical problem of mechanics of elastic solids. In the special case of the elastic interaction of a cylinder with a half-plane, when one of the radii has an infinite length, it has been found that the convergence also becomes equal to infinity. This paradoxical result contradicts not only the physical and mechanical meaning of the process under study, but also confirms the inadequacy of Flamant model of a simple radial stress state in determining displacements. The authors have proposed an algorithm for eliminating the contradictions based on the solution of Fredholm integral equation of the first kind. In the future, it can be considered as a new fundamental and applied problem of the theory of elasticity, which is of a great importance for a refined assessment of the contact strength and stiffness of the cylindrical parts of load-bearing structures taking into account the general and local deformations (cylindrical rollers, gears, road surfaces, when they are compacted with steel rollers, etc.) on the basis of a flat Flamant calculation scheme considering three stress components and the width of the cylinder contact area previously developed and mathematically approximated by the authors.

**PNRPU Mechanics Bulletin**. 2021;(2):6-11

The Influence of the Friction-Sliding Coefficient of Support Structures and Parameters of Seismic Actions on Reactions and Reliability of Structures with Seismic Protection

#### Abstract

The study is focused on a structure represented by a multimass elastic cantilever rod with dry friction seismic isolation elements in the support part under a horizontal random impact of a seismic type. The paper aims at investigating the seismic reaction and selecting optimal parameters of the seismic isolation system involving random impact characteristics, limit parameters of the structure, and the seismic isolation system. The researches are based on dynamic computations; the impacts and fluctuations of the system are random processes. The dynamic model of the structure with the considered seismic isolation is presented in the form of a cantilever rod with concentrated masses; a system of differential equations describing the displacement of the structure with the seismic-isolating sliding elements at the level of the top of the foundations is compiled; and a seismic impact is modeled in the form of a nonstationary random process. An algorithm is developed to integrate the system of differential equations of motion and to determine the statistical characteristics of the seismic reaction and reliability indicators of the structures with the seismic isolation. A method aimed at evaluating effectiveness of the seismic isolation system and selecting its rational parameters is suggested. We developed the computational dynamic model of the structure with the seismic-isolating sliding elements installed at the top level of the foundations, and elastic and rigid limiters for the movement of the sliding supports. This model is made in the form of a multimass cantilever rod that takes into account the relative movements of the masses and the stops of the system on the movement limiters. The structure’s movement under a seismic impact is described by a system of differential equations that takes into account the conditions of transitions of the structure from the state of sticking to the state of sliding and vice versa. The statistical characteristics of the seismic reaction and the reliability indicators of the structure in the process of vibrations are determined for different values of the maximum acceleration of the ground vibration, the prevailing period of impact, the number of masses in the calculated model and the coefficient of friction-sliding of the support elements. The influence of the impact parameters and the system on the efficiency of the seismic isolation of the structures with sliding elements is estimated. The proposed approach to selecting the optimal parameters of the seismic isolation system can be used as a research method aimed at improving efficiency of systems with different design options for seismic isolation of structures.

**PNRPU Mechanics Bulletin**. 2021;(2):12-23

Numerical Algorithms for Simulation of a Fluid-Filed Fracture Evolution in a Poroelastic Medium

#### Abstract

The paper deals with the computational framework for the numerical simulation of the three dimensional fluid-filled fracture evolution in a poroelastic medium. The model consists of several groups of equations including the Biot poroelastic model to describe a bulk medium behavior, Reynold’s lubrication equations to describe a flow inside fracture and corresponding bulk/fracture interface conditions. The geometric model of the fracture assumes that it is described as an arbitrary sufficiently smooth surface with a boundary. Main attention is paid to describing numerical algorithms for particular problems (poroelasticity, fracture fluid flow, fracture evolution) as well as an algorithm for the coupled problem solution. An implicit fracture mid-surface representation approach based on the closest point projection operator is a particular feature of the proposed algorithms. Such a representation is used to describe the fracture mid-surface in the poroelastic solver, Reynold’s lubrication equation solver and for simulation of fracture evolutions. The poroelastic solver is based on a special variant of X-FEM algorithms, which uses the closest point representation of the fracture. To solve Reynold’s lubrication equations, which model the fluid flow in fracture, a finite element version of the closet point projection method for PDEs surface is used. As a result, the algorithm for the coupled problem is purely Eulerian and uses the same finite element mesh to solve equations defined in the bulk and on the fracture mid-surface. Finally, we present results of the numerical simulations which demonstrate possibilities of the proposed numerical techniques, in particular, a problem in a media with a heterogeneous distribution of transport, elastic and toughness properties.

**PNRPU Mechanics Bulletin**. 2021;(2):24-35

Healing of Internal Defects in a Compressive Stress Field Using the Plastic Properties of Materials

#### Abstract

Numerical simulation of defect healing process in the field of previously created compressive stresses is performed. Isotropic cylinders with small axisymmetrically located defects are used as samples. The pressure created the initial field of compressive stresses inside the cylinder. The defects were modeled as a small blind closed annular cavity or as a through annular cut located around the cylinder axis. In the first case, a numerical three-dimensional solution is considered. For the second defect, the plane stress state model was used. The problems were solved in both elastic and elastoplastic formulations with an ideally elastoplastic behavior of the material. The external pressure was varied from values significantly lower than the yield strength to the yield strength and (for the first problem) for values slightly exceeding it. Based on the results of the numerical solution, the radial displacements of the cavity sides parallel to the cylinder axis are obtained depending on the external pressure. We found the values of pressure at which the cylindrical surfaces of the void defect were in contact. For the blind cavity, at any external pressure, there were unhealed areas. Healing was assessed by the volume of the material filling the initial cavity at the initial residual stresses. The value of the newly formed contact pressure at a certain value of the compressive stresses was determined by the ratio of the height of the healed area to the cavity height. The evaluation of the healing effect for a through cut in the cylinder was performed by varying the size of the gap formed by the cut between the inner cylinder and the outer ring depending on the applied external pressure. When the gap is completely healed, the values of the maximal contact pressure in the notch zone are determined.

**PNRPU Mechanics Bulletin**. 2021;(2):36-43

The Image-Based Finite Element Evaluation of the Deformed State

#### Abstract

The article presents one of the possible approaches to modeling objects with anisotropic properties based on images of the study area. Data from such images are taken into account when building a numerical model. In this case, material inhomogeneity can be included by integrating the local stiffness matrix of each finite element with a certain weight function. The purpose of the presented work is to develop a finite element for the formation of a computational ensemble and simulation of mechanical behavior taking into account the data of two-dimensional medical images. To implement the proposed approach, we used the assumption that there is a correlation between the values in the image pixels and the elastic properties of the material. Meshing was based on a four-node plane finite element. This approach allows using the quantitative phase or scanning electronic images, as well as computed tomography data. A number of test problems for compression of elementary geometry samples were calculated. The distal part of the rat femur was considered as a model problem. A computed tomography scan of the sample was used to construct a numerical model taking into account the inhomogeneity of the material distribution inside the organ. The distribution field of the nodal displacements based on data obtained from the images of the study area is presented. Within the framework of a model problem, we considered how a computer tomograph resolution influences the quality of the obtained results. For this purpose, calculations were carried out based on compressed input medical images.

**PNRPU Mechanics Bulletin**. 2021;(2):44-54

The deformation model of the DCB-sample with elastoplastic properties

#### Abstract

The loading of a strip with a crack-like defect according to mode I is considered. In contrast to the classical representation of a crack in the form of a mathematical section, the proposed model defines a crack as a physical cut with a characteristic linear size. The mental continuation of a physical cut in a solid forms an interaction layer (IL). It is important that the stress-strain state of the layer at a finite value of the linear parameter does not introduce a singularity into the crack model. The process of elastoplastic deformation with a constant layer length is considered. We obtained a simplified analytical solution to the problem of deformation of two elastic bodies connected by a thin layer with elastoplastic properties. The dependence of the displacement and stress fields on the length and thickness of the interaction layer has been found. It is shown that, under the classical plasticity condition, the range of variation of the external load leading to a purely elastic behavior is possible only for a finite layer thickness. As the layer thickness tends to zero, as in the Dugdale model, the plasticity region is formed at an arbitrarily small external load. For small layer thicknesses, a local plasticity criterion is proposed, by using which it is possible to distinguish the intervals of the external load variations associated with elastic and plastic deformations. The local plasticity condition, determined by the critical value of the energy product, makes it possible to reflect the stage of elastic deformation at an arbitrarily small finite thickness of the interaction layer. An asymptotic dependence of the external load on the IL thickness and the reduced length of the plastic zone is obtained. At the same time, the separation of the external load into elastic and plastic components is preserved. From the analysis of the experimental data, an estimate of the elastic limit of the energy product for the AV138 adhesive was obtained.

**PNRPU Mechanics Bulletin**. 2021;(2):55-63

An Analytical Approach to Obtaining the Stress Field of a Cylindrical Shell with a Circular Hole under Tension

#### Abstract

The problem of a cylindrical shell with a circular hole under uniaxial tension is considered. The main obstacle of solving this problem is the necessity to find such coefficients in the expansion of the solution into a sum of basis functions, for which this solution satisfies the boundary conditions. The study of the classical works led to understanding that none of the so far proposed approaches can be considered successfully, and the results of these approaches differ, so it is not clear, which results can be used as a basis. In the present paper, a new analytical approach to studying this issue is proposed. It allows expanding the range of applicability of the solution and gives the opportunity for the analytical study of the stress state. The idea consists in expanding each of the basis functions in a Fourier series by dividing the variables, which allows obtaining explicitly an infinite system of algebraic equations for finding coefficients. One of the important steps of this research is that the authors were able to prove which exact equation is a linear combination of the others and exclude, which made it possible to compose a reduced system for finding unknown coefficients. The proposed approach, in contrast to most classical works, does not impose mathematical restrictions on the values of the main parameter characterizing the cylindrical shell. The existing restrictions are of mechanical nature, as larger cutouts require another model. Moreover, the numerical results obtained by the new method are presented in a fairly complete manner and they are compared with the results of the classical works.

**PNRPU Mechanics Bulletin**. 2021;(2):64-75

Research and Modeling of Viscoelastic Behavior of Elastomeric Nanocomposites

#### Abstract

The paper presents results of studying mechanical properties of polymer composites depending on types of filler particles (granular - carbon black, nanodiamonds; layered - graphene plates; fibrous - single-walled nanotubes). These nanofillers differ greatly from each other in their structure and geometry. A significant difference in behavior of nanocomposites was revealed even with little introduction of particles into the elastomer. The highest level of reinforcement of the matrix was obtained when single-wall nanotubes and detonation nanodiamonds were used as fillers. The viscoelastic properties and the Mullins softening effect [1-4] were investigated in experiments performed with material samples subjected to complex uniaxial cyclic deformation. In these experiments, the amplitude of deformations was changed step by step; and at each step a time delay was specified to complete rearrangement processes of the material structure. It was found that a pronounced softening effect after the first cycle of deformation and significant hysteresis losses occur in the material filled with single-walled nanotubes. These characteristics are insignificant for the rest of nanocomposites until elongation increases twofold. In accordance with the obtained results, a new version of the mathematical model to describe properties of the viscoelastic polymer materials was proposed. The constants of the constitutive relations were calculated for each material; the theoretical and experimental load curves were compared. As a result, the introduced model is able to describe the behavior of elastomeric nanocomposites with a high accuracy. Moreover, this model is relatively easy to use, suitable for a wide range of strain rates and stretch ratios and does not require the entire history of deformation as needed for integral models of viscoelasticity.

**PNRPU Mechanics Bulletin**. 2021;(2):76-87

A Comparison of Experimental and Computation Results of Finding Effective Characteristics of Elastic Properties of Polymer Layered Composites from Carbon and Glass Fabrics

#### Abstract

Anisotropy of mechanical properties of the entire material and each of its layers is characteristic for polymer layered composite materials, as well as the fact that production processes of the composite material and parts from it are often combined in time. In this case, the elastic properties and strength of the material will be different not only in the thickness of the part, but also at each point. All this leads to a complication of the design process, which is due to the need to determine the elastic properties and strength of the polymer layered composite materials, taking into account the structure of the entire material and each of its layers. This work aims at evaluating the existing computational methods of finding effective characteristics of elastic properties by comparing computation results obtained by various methods with each other, as well as with the experimental results related to elastic properties of polymer layered composite materials from carbon and glass fabrics. We estimated the computational methods of finding effective characteristics of the elastic properties of composites based on the experimental results of finding the characteristics of the elastic properties of polymer layered composite materials made of carbon and glass fabrics, differing in density and type of weaving. The experimental values of the effective characteristics of elastic properties were determined as a result of standard tensile tests of laboratory specimens. As a result of the study, it was found that all the considered models and methods give consistent results when calculating the longitudinal modulus of elasticity E 11, the results of calculating shear modulus E 33 and shear moduli G 12 and G 23 are less consistent for all the considered materials. The comparison of the results of the experimental studies and computations showed that the Chamis model and the bridge model are better than other models to predict the values of the longitudinal elastic modulus.

**PNRPU Mechanics Bulletin**. 2021;(2):88-105

Vibrations of a Cantilevered Thick Plate

#### Abstract

It has been for the first time that an analytical solution to the problem of free vibrations of a cantilevered thick orthotropic plate is presented. This problem is quite cumbersome for using the exact methods of the theory of elasticity; therefore, methods based on the variational approach were developed to solve it. The paper suggests using the superposition method to construct a general solution of the vibration equations of a plate in the series form of particular solutions obtained with the help of a variables separation. The particular solutions of one of the coordinates are built in the form of trigonometric functions of a special type (modified trigonometric system). The constructed solution, in contrast to the solutions known in the literature on the basis of the variational approach, accurately satisfies the equations of vibrations. The use of a modified trigonometric system of functions makes it possible to obtain uniform formulas for even and odd vibration shapes and to reduce the quantity of boundary conditions on the plate sides from twelve to nine ones, while five of the nine boundary conditions are also accurately satisfied. The structure of the presented solution on the plate boundary is such that, each of the kinematic or force characteristics of the plate is represented as a sum of two series, i.e. a trigonometric series and a series in hyperbolic functions. Remaining boundary conditions make it possible to obtain an infinite system of linear algebraic equations with respect to the unknown coefficients of the series representing the solution. The convergence of the solution by the reduction method of the infinite system is investigated numerically. Examples of the numerical implementation are given; numerical studies of the spectrum of natural frequencies of the cantilevered thick plate were carried out based on the obtained solution, both with varying elastic characteristics of the material and with varying geometric parameters.

**PNRPU Mechanics Bulletin**. 2021;(2):106-117

The Computational Method of Estimating Vibration Stress Levels for GTE Compressor Blades

#### Abstract

At present, the process of designing a GTE involves a large amount of computational modeling. With the help of computational modeling, it is possible to predict a behavior of an engine part during engine operations before conducting experimental studies. For example, the numerical dynamic behavior analysis of compressor blades and prediction of dynamic stress levels during fluctuations in free modes are urgent problems. A high level of dynamic stress in the compressor blades in resonant modes can break a blade and stop an engine. In this paper, we propose a simple vibration stress estimation method for the compressor blades based on the calculation of natural frequencies and vibration forms. The method is based on a comparative analysis and scaling of stresses by the value of the total potential or kinetic energy. This estimation method is valid for local changes in the blade geometry, which do not lead to changes in the natural frequencies and vibration forms of the blades, assuming that the geometry change does not change the level of the aerodynamic excitation of the blade or its damping. At the stage of development or revision of the blade, a large number of variants of the blade geometry needs to be analyzed in order to reduce dynamic stresses. The proposed vibration stress estimation method has shown its high efficiency in developing and refining the geometry of the compressor blade. The vibration stress estimation method was tested using the rotor blade of a high-pressure compressor. As a result of the experimental study of the rotor blade, a high level of vibration stresses exceeding the permissible level was found for natural frequencies and vibration forms. To reduce the vibration stresses, measures were proposed to modify the geometry of the blade. For the modified blade geometry, the vibration stress estimation was performed with a prediction of the vibration stress values based on the manifested vibration forms. In order to verify the estimated vibration stress change, an experimental study of the modified blade was conducted. The vibration stress estimation method for the compressor blades was successfully verified.

**PNRPU Mechanics Bulletin**. 2021;(2):118-131

On the Theory of Elasticity of Microinhomogeneous Media with Account for Stochastic Changes in the Connectivity of Constituent Components

#### Abstract

The paper proposes a mathematical model aimed at calculating the effective elastic moduli of a micro-inhomogeneous two-component isotropic composite material, which components are connected randomly depending on the level of their relative volumetric contents. A stochastic equation is formulated for the connectivity parameter of the constituent components, according to which, with an increase in the volumetric content of the filler, individual inclusions build the structures of the matrix mixture in the form of interpenetrating frameworks, and then turn into a new binding matrix with individual inclusions from the material of the rest of the old matrix. The algorithm for the numerical solution of this stochastic differential equation is constructed in accordance with the Euler-Maruyama method. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for the random connectivity function of the constituent components of the composite material. A variant of the method aimed at calculating the mathematical expectation of a random connectivity function of the constituent components has been developed and the corresponding differential equation has been obtained for it. It is shown that the numerical solution of this equation and the average value of the production factor function calculated for all realizations of stochastic trajectories give close numerical values. New macroscopic constitutive relations are found for microinhomogeneous materials with a variable microstructure and their effective elastic moduli are calculated. It is noted that the formulas for these effective elastic moduli generalize the known results for isotropic composite materials. The values of the effective elastic moduli, constructed according to the expressions obtained in the paper, lie within the Khashin-Shtrikman range for the lower and upper bounds of the effective elastic moduli of the composite materials. The numerical analysis of the developed models showed a good agreement with the known experimental data.

**PNRPU Mechanics Bulletin**. 2021;(2):132-143

Modeling the Mechanical Response of Auxetic Metamaterials to Dynamic Effects

#### Abstract

The paper investigates the mechanical response of a 3D auxetic structure created on the basis of a unit cell with pre-buckled structural elements to dynamic loading. The aim of the work is to study deformations of the auxetic structure made of an alpha titanium alloy during uniaxial compression at 100 m/s, to evaluate dissipative properties of the structure during high-speed deformation, and to estimate the characteristic time of the metamaterial’s compaction with a relative density of 0.0115. The numerical simulation of the metamaterial at effective strain rate of 2000 1/s has been performed using LS DYNA solver. To describe the mechanical behavior of the titanium alloy in frame elements, we use a model of an elastic-plastic damaged medium, which takes into account the strain rate sensitivity of the plastic flow, temperature changes due to dissipative effects, and the effect of the stress state triaxiality parameter on nucleation and growth of structural damages. The numerical studies have shown that the auxetic effect in the studied metamaterial is retained under high-rate elastoplastic deformation. At a speed of the uniaxial compression of 100 m/s, deformation in the volume of the metamaterial proceeds nonuniformly. Under dynamic loading of the considered auxetic metamaterial, the deformation and fracture modes depend not only on the parameters of the cell geometry, but also on the mechanical behavior of the framework material, as well as on the relative density. This makes it possible to control the deformations of the cells under mechanical stress. Layers of compacted cells are formed near the dynamic loading surface. The instability of the cells of the auxetic metamaterials increases the absorbed energy. The calculated value of the specific absorbed energy under dynamic uniaxial compression reaches 3.4 kJ/kg, and is comparable with the values for frame structures made of Ti-6Al-4V with an equivalent specific mass density. The results indicate the possibility of creating protective structures using auxetic cellular structures on the base of the pre-buckled elements of the rolled metal.

**PNRPU Mechanics Bulletin**. 2021;(2):144-152

Extreme Clusters of Grains in Random Microstructures of Polycrystals

#### Abstract

It was experimentally observed that in polycrystalline materials under low macro loading of the specimen the first sites of failure initiation take place in the specific clusters of few grains. In some grains of these extreme clusters, the local (meso-) strains and stresses are high enough to cause first damages or plastic slips. In the stochastic microstructure of polycrystals, the formation of an extreme cluster is random and rare. Nevertheless, they govern the failure process initiation and can severely affect the reliability of polycrystalline machine parts. It is time and resource consuming to search and investigate extreme clusters on the real specimens of polycrystalline materials experimentally. A theoretical tool is desirable. Here we present the powerful computational method to look for extreme clusters, to investigate their possible patterns, and to evaluate the absolute maximums of local strains/stresses that can be achieved in these clusters. The experimentally observed clusters consist of few (3-4) preferably oriented neighboring grains or even of one big supergrain. The strain and stress bursts arise due to an interaction of the grains. One can expect that in bigger clusters, larger local bursts of fields can be generated. We found the typical forms of the extreme clusters (small and big) in four different polycrystals with grains of a weak and strong anisotropy for the case of uniaxial tension. In all regarded cases, the extreme clusters have the forms of the symmetrical patterns. In big clusters of highly anisotropic grains, the maximum of mesostrain exceeds the macrostrain by several times. In clusters of weakly anisotropic grains, the local strain concentration is rather moderate (tens of percents).

**PNRPU Mechanics Bulletin**. 2021;(2):153-166

Modeling the Effective Conductive Properties of Polymer Nanocomposites with a Random Arrangement of Graphene Oxide Particles

#### Abstract

Сomposite materials are widely used in various industrial sectors, for example, in the aviation, marine and automotive industries, civil engineering and others. Methods based on measuring the electrical conductivity of a composite material have been actively developed to detect internal damage in polymer composite materials, such as matrix cracking, delamination, and other types of defects, which make it possible to monitor a composite’s state during its entire service life. Polymers are often used as matrices in composite materials. However, almost always pure polymers are dielectrics. The addition of nanofillers, such as graphene and its derivatives, has been successfully used to create conductive composites based on insulating polymers. The final properties of nanomodified composites can be influenced by many factors, including the type and intrinsic properties of nanoscale objects, their dispersion in the polymer matrix, and interphase interactions. The work deals with modeling of effective electric conductive properties of the representative volume elements of nanoscale composites based on a polymer matrix with graphene oxide particles distributed in it. In particular, methods for evaluating effective, electrically conductive properties have been studied, finite element modelling of representative volumes of polymer matrices with graphene oxide particles have been performed, and the influence of the tunneling effect and the orientation of inclusions on the conductive properties of materials have been investigated. The possibility of using models of resistive strain gauges operating on the principle of the tunneling effect is studied. Based on the finite-element modeling and graph theory tools, we created approaches for estimating changes in the conductive properties of the representative volume elements of a nanomodified matrix subjected to mechanical loading.

**PNRPU Mechanics Bulletin**. 2021;(2):167-180

The nonstationary thermoelectric elasticity problem for a long piezoceramic cylinder

#### Abstract

А new closed-loop solution for the coupled nonstationary problem of thermoelectric elasticity is designed for a long piezoceramic radially polarized cylinder. The case of the nonstationary load acting on its inner cylindrical surface is considered as a function of temperature change at a given law of the convection heat exchange on the outer face wall (boundary conditions of heat conductivity of the 1st and 3rd types). Electrodynamic cylinder surfaces are connected to a measuring device with a high input resistance (electric idling). We investigate the problem where the rate of the temperature load changes does not affect the inertial characteristics of the elastic system. It makes it possible to expand the initial linear computational relations with the equilibrium, electrostatics and heat conductivity equations with respect to the radial component of the displacement vector, electric potential as well as the function of temperature field changes. Hyperbolic LS-theory of the thermal conductivity is used in the computations. The problem is solved with a generalized method of biorthogonal finite integral transformation based on a multicomponent ratio of eigen functions of two homogeneous boundary value problems. The structural algorithm of this approach allows identifying a conjugated operator, without which it is impossible to solve non-self-conjugated linear problems in mathematical physics. The resulted computational relations make it possible to determine the stress-strain state, temperature and electric fields induced in the piezoceramic element under an arbitrary external temperature effect. By connecting the electroelastic system to the measuring tool, we can find voltage. Firstly, the analysis of the numerical results allows identifying the rate of the temperature load changes, at which it is necessary to use the hyperbolic theory of thermal conductivity. Secondly, it allows determining the physical characteristics of the piezoceramic material for the case when the rate of changing the body volume leads to a redistribution of the temperature field. The developed computational algorithm can be used to design non-resonant piezoelectric temperature sensors.

**PNRPU Mechanics Bulletin**. 2021;(2):181-190

The Refined Model of the Elastic-Plastic Dynamic Behavior of Reinforced Curved Panels Sensitive to Strain Rate

#### Abstract

The initial-boundary value problem of dynamic elastic-viscoplastic deformation of flexible curved panels (shallow shells) with plane -cross and spatial reinforcement structures is formulated. The inelastic behavior of the materials of the composition components is described by the constitutive equations of the theory of plastic flow with isotropic hardening, and their sensitivity to strain rate is taken into account. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The used kinematic and dynamic two-dimensional relations and the corresponding boundary conditions make it possible to describe, with varying degrees of accuracy, the mechanical bending behavior of shallow composite shells. This takes into account the possible weak resistance of such reinforced panels to transverse shears. In the first approximation, the used two-dimensional equations, the initial and boundary conditions degenerate into the relations of the traditional non-classical Ambartsumyan theory. For the numerical integration of the formulated nonlinear dynamic problem, an algorithm of time steps is applied, based on the use of an explicit scheme of the cross type. The elastoplastic and elastic-viscoplastic behavior of the reinforced cylindrical shallow shells under transverse dynamic loads generated by an air blast wave is investigated. Metal-composite and fiberglass thin-walled constructions are considered. It is shown that the refusal to take into account the dependence of the plastic properties of the components of the composition on the rate of their deformation does not allow adequately describing the inelastic dynamic behavior of both metal-composite and fiberglass shallow shells. It is shown that in the calculations of even relatively thin reinforced cylindrical panels (with a relative thickness of 1/50), the use of the Ambartsumyan theory leads to completely unacceptable results in comparison with the refined bending theory. It has been demonstrated that even for relatively thin curved fiberglass panels, replacing the traditional flat -cross reinforcement structure with a spatial structure with obliquely laid fiber families can significantly reduce not only the intensity of deformations in the binder, but also the maximum deflection values in modulus. For metal-composite shallow shells with a weakly expressed anisotropy of the composition, the positive effect of the indicated replacement of reinforcement structures is practically not manifested.

**PNRPU Mechanics Bulletin**. 2021;(2):191-202

The Crack Growth in the Imitation Model of a GTE Turbine Disk under Operating Loading Conditions

#### Abstract

The paper presents the experimental results of growing surface cracks in the turbine disk of a gas turbine engine (GTE) under cyclic tension at room and elevated temperatures. The geometry of the imitation model of the GTE turbine disk with a stress concentration zone in the form of a bolt hole was justified. In order to ensure the similarity of the initial damage of the imitation model and the GTE turbine disc in the plane of symmetry of the stress concentration zone, a semi-elliptical notch was made. The loading conditions of the imitation model were developed based on results of a comparative stress-strain state (SSS) analysis of the stress concentration zone of the imitation model and the GTE turbine disc. As a result of the fatigue test of the imitation model at room and elevated temperatures, the experimental positions and sizes of the crack fronts with respect to the drop potential signal on the crack edges were obtained. The fixed positions and sizes of the crack fronts were used as the basis for the numerical calculation of the fracture resistance parameters. For the numerical studies, ten three-dimensional finite element models with different positions and sizes of the crack fronts were considered. The numerical calculation results based on the finite element method were used to determine the distributions of the elastic stress intensity factors along each crack front. The crack growth rate characteristics both on the free surface and at the deepest point of the crack front were obtained at room and elevated temperature conditions. A technique for the automation tests that simulate the block-type loading of the disk material at elevated temperatures was proposed.

**PNRPU Mechanics Bulletin**. 2021;(2):203-217