## No 1 (2017)

**Year:**2017**Articles:**13**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/18**DOI:**https://doi.org/10.15593/perm.mech/2017.1

Expansion of a cylindrical cavity in a compressible elastic-plastic medium

#### Abstract

This paper presents a new problem formulation related to the cylindrical cavity expansion in a compressible elastic-plastic medium. Nonlinear compressibility, finite strains and Mises-Schleicher criterion are taken into account in the problem formulation The feature of the formulation (nonlinear compressibility) allows us to investigate different porous materials and analyze the effect of porosity on problem parameters. We have presented the derivation of the analytical relations that determine the cavity expansion resistance and size of the plastic zone for two classes of materials, such as metals and porous media (soils). These analytical relations are used to obtain qualitative results about the nature and size of strains in the medium. For example, the flowing medium is characterized by an anomalously large zone of plastic strain. We have considered a set of metals, alloys and soft soils within the numerical analysis. On the basis of the numerical analysis, we have obtained the estimated range of values of the most important parameters. For metals, this parameter is the dimensionless bulk modulus . By using the bulk modulus, we have approximated the expansion resistivity, the sizes of the cavity and compaction zone. Since soft soil is the material with nonlinear properties, the dependence of parameters has a more complicated form, namely, the additional effect of the material porosity on the expansion resistance and size of strains. By using the results of numerical calculations we have made a comparative analysis of the porosity effect on the size of the compaction zone in the neighborhood of the cavity for sandy and clay soils. These estimations of the expansion medium resistance of the cylindrical cavity are useful to pose and refine the applied relations with regard to the theory of the dynamic penetration of slender (sharp) impactors.

**PNRPU Mechanics Bulletin**. 2017;(1):5-23

Influence of the state of faults in lithospheric plates on the starting earthquake

#### Abstract

In earlier works the authors have constructed a model of the starting earthquake from the preliminary stage and up to the earthquake itself. The model is based on the boundary problem formulation accepted in a number of works and represents two semi-infinite lithospheric plates with straight boundaries modeled by Kirchhoff plates slowly moving towards each other. The plates are placed on a three-dimensional linearly deformable foundation. It is assumed that the foundation surface is a Conrad boundary separating the top granite and bottom basalt structures of the Earth's crust. For such a boundary problem, we have studied the change in the contact stresses concentration under the lithospheric plates for the cases with and without a distance between the plates and the change in the contact stresses concentration in the convergence zone of the lithospheric plates. When the plates are close to each other, the concentration of stresses in the convergence zone, which is a fault, is singular. According to the experience of solving boundary problems for linear elastic materials, this means the destruction of a zone with such stresses concentration, i.e. the occurrence of an earthquake. For this model, different effects from the impact on the lithospheric plates have been studied. As a result, it was found that the same effects were characteristic for faulting and breaking earthquakes which had actually occurred. The study was carried out for the static problem with vertical external effects on the lithospheric plates assuming no boundary stresses at the ends. The study based on the block element method executed in the present work is aimed at building a more complex model, assuming the presence of stresses, i.e. shearing forces and bending moments, at the ends of the plates, that is, the end fault boundaries. Ratios that take into account the presence of different variants of the mentioned boundary conditions on the banks of the faults were built, and their effect on an earthquake possibility was studied.

**PNRPU Mechanics Bulletin**. 2017;(1):24-38

Numerical investigation of a physically nonlinear problem of the longitudinal bending of the sandwich plate with a transversal-soft core

#### Abstract

In this paper, a numerical investigation of a physically nonlinear problem of the longitudinal bending of an infinitely long sandwich plate with a transversal-soft core is carried out. We assume that in the right face section the edges of the carrier layers are clamped and there is no adhesive joint of the core with the support element, in the left face section the edges of the carrier layers of the plates are hinge supported on a completely rigid in the transverse direction diaphragms, glued with the end section of the core. The problem is considered in the one-dimensional geometrically nonlinear statement. It is assumed that the relationship between the tangential stress and strain shear corresponds to the ideal elastic-plastic models, i.e., the tangential stress modules in the core do not exceed a certain limiting value. This condition means the prevention of the structural failure and corresponds to an account of the physical nonlinearity in the core material by the ideal elastic-plastic model. The generalized statement is formulated as a problem of finding a saddle point of the Lagrange generalized functional. Lagrange functional properties are investigated. Its convexity, lower semicontinuity and coercivity on the basic variables (displacements of the points of the middle surface of the carrier layers), the concavity, upper semicontinuity and anti-coercivity on the Lagrange multipliers (tangential stresses in the core) are established. It made it possible to use the general theory of the existence of saddle points to prove the existence and uniqueness theorem. To solve the problem the two-layer iterative Uzawa method is proposed, each step of which is reduced to the solving of the linear elasticity problem and finding the projection onto the convex closed set. We have established the convergence of the method. By using the software package developed in Matlab environment, the numerical experiments for a model problem have been carried out. The analysis of the results is made. The numerical results correspond to the physical picture.

**PNRPU Mechanics Bulletin**. 2017;(1):39-51

Influence of the gas compressibility in the pump cylinder on longitudinal oscillations of the rod string

#### Abstract

The paper presents a model development related to the longitudinal oscillations of the rod string inthe deep-well pumping unit taking into account the compressibility of the gas-liquid mixture in the cylinder of the pump. The approach which uses the problem formulation in the form of quasi-variational inequalities was applied to solve this problem. The solution of this problem can be reduced to a sequence of non-smooth minimization problems. This approach is quite versatile and can be used for columns in highly deviated wells. In this model, the gas component of the mixture is subject to Boyle's law. This model does not consider the compressibility of the liquid components, the dilution of gas in a liquid and extraction of gas from the fluid. On the basis of the proposed model, we considered the influence of the gas content in the mixture which fills the pump cylinder on the dynagraph of the rod string in the upper section. Besides, different behaviors of the rod string in two cases are considered. In the first case the gas is uniformly distributed (but not dissolved) in a liquid which fills the pump cylinder. In the second case the gas fills a localized volume in the pump cylinder. If the gas is distributed in the pumped mixture, the dynagraph segments which correspond to the compression phase have a weaker inclination than the ones of the extension phase. It can be explained by the fact that the mass of the gas and, thus the compressibility of the mixture in the phase of compression is higher than the one in the extension phase. This fact also explains the weakening of free oscillations which is much more significant for that phase when the plunger moves down, rather than the phase when the plunger moves upward. If the gas occupies the localized volume in the pump cylinder, then it has the same shape both in the compression and extension phases. When the plunger moves downward, no excessive weakening of oscillations is found.

**PNRPU Mechanics Bulletin**. 2017;(1):52-62

Indication of thermoelastic instability of sliding contact using embedded piezoceramic interlayer

#### Abstract

Operation of tribotechnical components and devices is subjected to the instability of the sliding frictional contact, accompanied by an increase in temperature and pressure, which often leads to emergency situations, accidents, etc. In order to prevent such phenomena, a system for monitoring the occurrence of instability is proposed. It is based on an indirect indication of temperature and stress by measuring and analyzing the induction and electric fields in the internal piezoceramic interlayer. In order to model such a monitoring scheme of thermoelastic instability, a non-stationary dynamic contact problem of the thermoelectroelasticity is considered for a rigid body sliding over a half-plane coated by an elastic layer. Friction forces and frictional heat generation at the contact spot are taken into account. Piezoceramic layer is located between the elastic coating and non-deformable base and perfectly bonded to them. The piezoceramic layer is insulated from the thermoelastic coating; its polarization vector is directed along the vertical axis orthogonal to boundaries of the piezoceramic layer. These boundaries have electrodes placed on them with a different supplied voltage. The solution of the problem on the sliding thermoelastic frictional contact is constructed using the Laplace integral transform, which allows to present the basic physical parameters of the problem, such as the temperature, voltage, displacement, induction and electric field strength, the electrostatic potential in the form of contour integrals of the inverse Laplace transform, or as the Laplace convolution. After studying the properties of the integrands and their isolated singular points in the complex plane of the integration variable, the integrals are calculated using the methods of complex analysis. The calculation of the integrals leads to infinite series over the poles of the integrands, which are the eigenvalues of the problem. The solution in this form allows to determine the domains of stable and unstable problem solutions at an infinite time interval. Expressions for the electric current and voltage in the piezoceramic interlayer as well as the expressions for contact temperature and stresses, allow diagnosing the emergence and development of thermoelastic instability of the sliding contact.

**PNRPU Mechanics Bulletin**. 2017;(1):63-84

Strength computational and experimental analysis of turbojet fan case under the impact of a torn blade

#### Abstract

The paper considers the problem of holding a torn blade of a turbojet fan to ensure safety for passengers and crews. It is concerned with one of the main trends in designing fans satisfying this requirement. The experimental and computational analysis of strength for several fan case designs is given. A difference between the calculation results and experimental data for metals with different plasticity is presented. We outlined and described the well-known procedures of calculating metal armoured protection, specified its common disadvantage which is neglecting the material ultimate strain. We suggested the calculation procedure for the metal case which takes the material ultimate strain in account including the boundary value of the material ultimate strain dividing metals on rigid and flexible ones with the calculated formulas of the armoured wall. A good convergence is revealed between the calculation data based on the proposed correlations and the experimental results for Ti and Al alloys. We described the structural design, analyzed strength and the rate of impact absorption by a separated blade using a specially developed goffered hood (a cover) made of the high-strength cloth which can be installed on any fan case made of either metal or polymer composite materials (PCM) without principal changes in the structural configuration. The calculated advantages of using the developed goffered hood design in comparison with the conventional one have been shown. The efficiency has been estimated with respect to applying aluminum alloys as a production material for a fan case being an alternative to the titanium ones. Experimental equipment for testing full-size cases is proposed. The disadvantages of the testing equipment for model cases have been outlined. We analyzed the problem of PCM application for a load-carrying case which is cut by fan blades when one blade is torn away. The engineering estimation of using PCM in a fan case and the solutions aiming to increase the ballistic resistance of PCM cases have been presented.

**PNRPU Mechanics Bulletin**. 2017;(1):85-103

Experimental studies of the high temperature influence on strength and deformation properties of combined glass organoplastics

#### Abstract

The investigation of high and low temperature influence on the mechanical properties and failure mechanisms of composite materials is an important problem, as well as the determination of the temperature dependence between elastic and strength characteristics for fiber composites which are used in critical structures. The study is focuses on combined composite materials which are based on a combined glass organofabrics with two types of matrixes (epoxy, polyimide). The aim of the present study has been to investigate the degradation of mechanical characteristics of composite materials at higher temperatures. The research has been carried out using a universal electromechanical system Instron 5882 which includes a thermal chamber and non-contact video extensometer. The high temperature studies at 500 °C have been conducted using the servo hydraulic test system Instron 8850 with a high temperature chamber. The paper presents the results of the experimental studies related to the high temperature influence on the mechanical properties of composites based on glass organofabrics with two types of matrixes (epoxy, polyimide) under tension and bearing. Mechanical tensile and bearing tests at normal and high (100 °C, 250 °C and 500 °C) temperatures were carried out. The postcritical deformation stage has been determined. The mechanical properties (elastic modulus, strength limit and bearing strength) of the composites have been found. The stress-strain diagrams for composite specimens have been built. The temperature degradation of mechanical properties for composites has been analyzed.

**PNRPU Mechanics Bulletin**. 2017;(1):104-117

Numerical and experimental study of composite bulkhead partition strength with in-situ x-ray monitoring

#### Abstract

The current work presents a qualitative and quantitative assessment of microdamage occurring in the structure as a result of different loading tests by the example of the "U"-shaped bulkhead segment made of carbon fiber reinforced plastic (CFRP) using microfocus X-radiography. The resin transfer molding (RTM) technology was used to manufacture this structure. X-ray images with the fixed tensile force were made during the sample tests. These images were analyzed in order to reveal a delamination as the basic damage type occurring in the tested part. The areas of interlayer cracks localization and their estimated sizes were identified at the various load levels. It was noted that the cracks were completely closed after unloading; and it was difficult to identify them by any non-destructive testing methods. The three-dimensional computer model was created to carry out the stress-strain analysis of the sample. A detailed ply-by-ply analysis allowed us to estimate the interlaminar normal and shear stresses which determine the structural failure. The numerical simulation of this problem was carried out using the finite element method (FEM) with ANSYS Workbench software. Due to a high dimensionality of the FE model, the high-performance computing complex was used. The numerical simulations results with regard to the mechanical tests of the bulkhead segment were compared with the X-ray radiography results, and a good qualitative correlation was found. Thus, it was shown that the areas of maximal interlaminar stresses are localized in the part’s bending areas. The difference between the experimental and numerical values of the bearing strength was about 23% but it could be corrected after testing a series of similar samples and refining the material interlaminar gap strength.

**PNRPU Mechanics Bulletin**. 2017;(1):118-133

Serrated yielding: crystal viscoplastic models

#### Abstract

Serrated yielding as a phenomenon of plastic deformation instability is found in a variety of metals and alloys in some range of strain rates and temperatures under different loading conditions. Theoretical and experimental studiesreveal the fact that the properties of polycrystalline materials on the macrolevel during inelastic deformation are significantly defined by a condition of the evolving meso- and microstructure of the material. It is necessary to create models which describe plastic instability and microstructural self-organization process that can ultimately lead to aspontaneous appearance of strain localization. The first part of the review (Trusov P.V., ChechulinaЕ.А. Serrated yielding: Physical mechanisms, experimental dates, macrophenomenological models. PNRPU Mechanics Bulletin . 2014. No. 3. - Pp. 186-232. doi:10.15593/perm.mech/2014.3.10) considersthe works describing physical mechanisms, experimental studies of serrated yielding and the best known macrophenomenological models. However, a correct description of plastic deformation, its heterogeneity and account for all its attendant real physical mechanisms need to study the material behavior not on the macrolevel, but on a lower meso- and microstructural levels. Plastic deformationis associated with a non-uniform motion of elementary carriers of plastic flow, i.e. dislocations (in time and space). Dislocations and dislocation barriers (Lomer-Cottrell, immobile dislocations) and nondislocation barriers (clouds of impurity atoms, solid particles, solids inclusionof the secondary phase etc.) make it possible to describe the multilevel model at meso- and microlevels. In this part of the review some of the theoretical works based on the crystal plasticity are analyzed,thus describing deformation features of alloys in a wide range of temperatures and strain rates in which diffusion processes have a significant influence on the materials behavior. Special attention is paid to describingthe Portevin-Le Chatelier (PLC) effect; the majority of the authors think that it appears due to the interaction of dislocations with impurity atoms.

**PNRPU Mechanics Bulletin**. 2017;(1):134-163

Dynamic axisymmetric problem of a direct piezoeffect for a round bimorph plate

#### Abstract

The dynamic axisymmetric problem for a round bimorph structure consisting of the metal substrate and the axially polarized piezoceramic plate is considered. Its bending modes are caused by the action of the mechanical loading (normal tension) on the side surface; the loading is a function of the radial coordinate and time. The rigid and hinge supports of the plate are considered. The initial design relations are formulated for the piezoceramic material with the hexagonal crystal lattice of the 6 mm class. In order to solve the problem of the theory of electrodynamics in a three-dimensional formulation, the finite integral Hankel transformations along the axial coordinate and a generalized transformation (FIT) over the radial variable is used. At each stage, the standardization is carried out which allows implementing an appropriate transformation algorithm. In the first case the boundary conditions are presented in a mixed form; and in the second case they are homogeneous by introducing auxiliary functions. This approach allows gaining precise (within the used models) calculated ratios in a most general form. The built closed solution allows defining the frequency range of the axisymmetric oscillations, the stress-strain state and the nature of the changing induced electric field of the bimorph plate. This makes it possible to establish the conventional solutions of the designed devices, determine a way of fixing the electrical signal, pick up all the geometrical and physical characteristics of the typical elements of the piezoceramic transducers Also, the developed solution allows solving the problems of the elasticity and electroelasticity theory for circular thick and thin plates with an arbitrary number of layers under most general loading conditions without the use of kinematic hypotheses.

**PNRPU Mechanics Bulletin**. 2017;(1):164-180

A stochastic model for the residual stresses calculation in the surface-hardened hollow cylinder under creep

#### Abstract

We suggest a new stochastic model to calculate the residual stresses in the surface-hardened hollow cylinder under the creep conditions with axial tension. A set of experimental studies was carried outto determine the creep strain distribution along the hardened specimen made of D16T alloy at a temperature of 125 C degrees under four modes of tension stresses: 353; 385; 406.2; 420 MPa. We found an essential variation of the creep strains in the local sections which strains may differ from each other by three or four times. The phenomenological stochastic creep equations are suggested; the compliance of these equations with the experimental data is examined. The computational method for solving the stochastic boundary value problem for the surface-hardened products is developed and realized for the hollow cylinder made of D16T alloy under axial tension. The relaxation of residual stresses in the hollow cylinder is theoretically studied. The longitudinal statistical estimates for the fields of residual stresses under the creep are obtained. We have shown the confidential intervals for the components of the residual stresses tensor of the specimen under creep at different time moments. It was experimentally study how the tension load affect the residual stresses relaxation in the hollow cylindrical specimen made of D16T alloy. Final experimental diagrams for axial and circumferential components of the residual stresses tensor under creep for 83-166 hours have been given. The theoretical and experimental studies made it possible to make a conclusion about an almost full relaxation of the residual stresses in the cylindrical specimens because of creep at tension conditions during 100-160 hours. The applicability of the rings and strips method for the experimental data variation under the creep and intensive relaxation of residual stresses is discussed.

**PNRPU Mechanics Bulletin**. 2017;(1):181-207

The possibilities and limitations of the homogenized description of inelastic behavior of brittle porous materials under constrained conditions

#### Abstract

The paper presents the results of studying inelastic deformation and fracture of microscopic regions of porous brittle materials under triaxial compression using the movable cellular automaton method. The study is focused on analyzing the applicability of the classical macroscopic criteria of plasticity and strength for the integral description of the mechanical response of microscopic representative volumes. We analyzed the dependence of the parameters of the integral mechanical response of the microscopic regions on the value of porosity, spatial distribution of pores and strength of the material in the skeleton walls. Main stages of the process of damage accumulation and growth in the skeleton walls and its manifestation in the integral inelastic response of the considered microscopic region are investigated by the example of the axial compression of the samples under constant lateral pressure. It is shown that the increase in the value of the lateral pressure leads to a change in the fracture type of the porous material from the elastic-brittle fracture to the formation of the shear localization zone (shear band) and then (at higher lateral pressures) to the spatially distributed cataclastic flow. The characteristic threshold values of the lateral pressure at which the failure mechanism starts to change, mostly depend on the sensitivity of the skeleton wall strength to the local pressure. The analysis of the simulation results showed that the conventional plasticity conditions (criteria) which take into account the contribution of the local average stress in the linear approximation, are able to adequately describe the response of the microscopic representative volumes of brittle porous materials under constrained conditions only from the beginning of inelastic deformation to the stage of the formation of the system of non-interacting relatively short cracks. It is important to note that the softening of the representative microvolumes of brittle porous materials under constrained conditions is concerned not with the loss of integrity of the sample, but with subsequent processes of forming the localized shear bands and pore collapse in the already fragmented material. This gives reason to believe that the experimentally determined strength of the constrained sample as the maximum resistance force may be significantly overestimated in comparison to the true value (corresponding to the fragmentation of the material). It is established that the conditions for the loss of integrity of brittle porous materials under constrained conditions are adequately described using the "linear" failure criteria with the parameters determined not by standard uniaxial compression/tension tests but by multiaxial compression tests.

**PNRPU Mechanics Bulletin**. 2017;(1):208-232

Nonlinear and coupled thermal effects during finite element simulation of contact thermo-force surface hardening

#### Abstract

The article considers the problem setting of the coupled thermal-structural problem arising in the process of modelling the surface hardening using electromechanical treatment (EMT). A mathematical model is presented with respect to forming the structure of metallic alloys with EMT based on a joint analysis of the calculated data on the dynamics of temperature fields and continuous cooling transformation diagramby using the Ti6Al2V titanium alloy as an example. Kinetics of the martensitic transition during EMT is described using Koistinen-Marburger equation. An algorithm of solving the thermal problem based on the finite element method in the weak Galerkin form is given. The calculated area is approximated by Zienkiewicz type infinite elements. Based on a series of computational experiments, the influence of the magnitude of the time step on the accuracy of the problem solution was investigated. The significance of the considered coupled and nonlinear effects which are specific for high-speed high-temperature thermal processes are analyzed, e.g. the change of the thermophysical properties of the metal, latent heat of phase transitions, thermal radiation and dependence between the metal physical properties and temperature. Two methods are used to consider the effects aiming to solve the thermo-structural problems: a totally coupled non-linear problem using the direct iteration method of Picard in a combination with the relaxation formulae for the convergence acceleration; the quasilinear solution when the values of the nonlinear terms are calculated on the basis of the temperature distribution obtained in the previous time step. Based on the series of numerical experiments, we analyzed the unsteady temperature fields, as well as the distribution of structural domains after EMT in the titanium pseudo-alpha-alloys.

**PNRPU Mechanics Bulletin**. 2017;(1):233-250