## No 3 (2021)

**Year:**2021**Articles:**19**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/144**DOI:**https://doi.org/10.15593/perm.mech/2021.3

The Influence of thickness on residual flexural strength of composite with low-velocity impact damages: Experimental study

#### Abstract

Assessment of the residual strength of composite structural elements that can be subjected to low-velocity impact during operation is still a vital engineering problem. Its solution requires not only the understanding of the basic mechanisms of energy dissipation in composites but, the study of factors affecting the impact resistance of the material. The thickness is one of the main parameters that affects the mechanical behaviour of the composite at low-velocity impacts and, as a consequence, its residual strength. This paper presents the experimental study of the material thickness and impact energy influence on the residual flexural strength of woven glass fibre-reinforced plastic specimens. At the first stage of the study, low-velocity impact tests with different impact energies on GFRP plate specimens with the thickness of 2 mm, 4 mm, and 6 mm were carried out. At the second stage, the beam specimens were cut out from the plate specimens with impact damages, and three-point bending tests were carried out. The dependencies of the residual flexural strength were obtained at various impact energies for all specimen thicknesses. The sensitivity of the Flexure-After-Impact test protocol to the delamination and fibre damages in the composite specimens were assessed.

**PNRPU Mechanics Bulletin**. 2021;(3):6-11

Importance of thermal conductivity and stress level during a phase (hydride) transformation in magnesium

#### Abstract

There are many different systems of an autonomous energy storage including accumulators and storage devices for renewable energy. Systems based on reversible metal hydrogenation have recently been introduced. The selection of metals is based on considerations of temperature and pressure conditions of the hydrogenation/dehydrogenation cycle, as well as the desired storage hydrogen capacity. Magnesium is one of the main challenging metals with respect to these main conditions since having a hydrogen capacity up to 7.6 w.%. For Mg forming MgH2, it was soon established that the size of particles plays a critical role since the kinetics (rate) of hydride formation accelerates when the size of the particles decreases. The present study shows that the overall diameter of the particles is the main characteristic controlling the kinetics of hydride formation because of distinct issues. A distribution of the size entails a strong dispersion transferring the heat of reaction, which characterizes Mg to MgH2 phase transition. Moreover, the formation of MgH2, is accompanied by a great increase of the unit-cell volume, developing noticeable internal stresses within the surface layers of the particles, thus turning to a systematic flaking and a systematic decrease of sizes of the powder particles. The results of the numerical modeling comply with the experimental data. This makes it possible to predict the best size of the initial Mg powder able to achieve fast kinetics during hydrogenation. Furthermore, the present analysis demonstrates the best hydrogenation kinetics, not only when using fine powders, but also when the deviation from the average particle size is minimized.

**PNRPU Mechanics Bulletin**. 2021;(3):12-21

STRESS STATE AND CONDITIONS FOR CRACK INITIATION IN THE ADHESION LAYER OF THE COMPOSITE

#### Abstract

The paper deals with the elastic deformation of a composite consisting of plates connected with an adhesive layer on the basis of the interaction layer concept. The layer thickness is taken as a linear parameter. Under loading by the normal separation in the case of plane deformation, the triaxial stress state of the adhesive layer is taken into account. From the general variational setting, by using the hypothesis of flat sections, the problem is formulated in a differential form. Within the framework of a simplified formulation, an analytical solution is found that agrees with the numerical solution obtained by the finite element method. It is shown that Poisson's ratio of the adhesive layer significantly affects its stress state, in which there is a coincidence of the two principal stresses. For weakly compressible adhesion layers, a stress state close to hydrostatic tension is realized. In order to analyze the stress state of the adhesion layer, which is singular at extremely small values of the linear parameter, it is proposed to use the energy product (EP), local stresses (LS) and local deformations (LD). EP is determined as the product of the specific free energy of the layer and the linear parameter, and LS (LD) is determined as the product of stresses (deformations) and the square root of the linear parameter. It is shown that these characteristics are not singular with respect to small layer thicknesses and do not depend on the linear parameter. It was found that the value to which the EP converges at a fixed external load and the linear parameter tends to zero does not depend on the mechanical properties of the adhesive, and the LS (LD) values depend on the properties of the adhesive. At the critical load of crack initiation in the adhesive, the LD in the direction of the separation is significantly (by several orders of magnitude) higher than the LD in the orthogonal direction. In this case, the LS and LD of separation make the main contribution to the formation of the EP. A method is proposed for determining the critical value of the EP corresponding to crack initiation in the adhesive based on the use of the maximum external load from the experimental R-curves.

**PNRPU Mechanics Bulletin**. 2021;(3):22-34

Theories of plasticity under complex loading along flat trajectories of deformations

#### Abstract

The paper considers variants of theories of plastic flow with combined hardening, which are widely used in applied calculations of structures. A comparative analysis of the theories under complex loading along flat strain trajectories is carried out and covers the entire range of strain paths from multilink polylines to curved trajectories of constant and variable curvatures. The strain path from medium to large curvatures are considered. The analysis of the research results is carried out in the vector space of A.A. Il’yushin. We consider plane trajectories of deformations in the form of a square, three circles passing the origin of coordinates, and trajectories in an asteroid-like form. The results of the calculations are compared with the results of the experimental studies of the response stress trajectories, scalar and vector properties. Variants of theories are considered: isotropic hardening model; Ishlinsky-Prager-Kadashevich-Novozhilov model (linear kinematic and isotropic hardening); model similar to Ono-Wang's model; the Armstrong-Frederick-Kadashevich model (the Korotkikh model is based on this model); the Shabosh model with three evolutionary Armstrong-Frederick-Kadashevich equations; Themis model based on the invariant theory of plasticity; Bondar model with a three-term structure of the evolutionary equation for kinematic hardening. We give the material parameters (functions) that close the variants of plasticity theories. A satisfactory agreement with the experiment for all deformation trajectories is achieved in calculations based on the models of Ishlinsky-Prager-Kadashevich-Novozhilov, Shabosh and Temis. The difference between the results of calculations and experiments does not exceed 30 %. The best agreement with the experiment is achieved on the basis of the Bondar model with the difference between the results of the calculations and experiments for all trajectories less than 10 %.

**PNRPU Mechanics Bulletin**. 2021;(3):35-47

Elastic diffusion vibrations of an isotropic Kirchhoff-Love plate under an unsteady distributed transverse load

#### Abstract

We investigated an unsteady elastic diffusion vibration of a simply supported rectangular isotropic Kirchhoff-Love plate. The plate is under the action of a distributed transverse load. A model that describes coupled elastic diffusion processes in a multicomponent continuum is used for the mathematical problem formulation. The model is taking into account the diffusion fluxes relaxation. The transverse vibration equations of a rectangular isotropic Kirchhoff-Love plate with diffusion were obtained from the model using the d'Alembert variational principle. The initial-boundary value problem of a freely supported isotropic rectangular plate bending is formulated on the basis of the obtained equations. The plate is under the action of elastic diffusion perturbations distributed over the surface. The problem solution of an unsteady elastic diffusion plate vibration is sought in an integral form. The surface Green's functions are the kernels of the integral representations. To find the Green's functions, we used the Laplace transform in time and the expansion into double trigonometric Fourier series in spatial coordinates. Green's functions in the image domain are represented in the form of rational functions and depend on the Laplace transform parameter. The transition to the original domain is done analytically through residues and tables of operational calculus. The surface Green's function analytical expressions are obtained. As a calculation example, we considered a freely supported elastodiffusive plate under the action of suddenly applied unsteady bending moments distributed over the plate surface. By using a three-component continuum, a numerical study of interactions between unsteady mechanical and diffusion fieldsis done for an isotropic plate. The influence of relaxation effects on the kinetics of mass transfer is investigated. The solution is presented in the analytical form, as well as in the graphs of the displacement fields and concentration increments on time and coordinates. At the end of the publication, the main conclusions are given about the fields coupling effect and the relaxation of diffusion fluxes on the stress-strain state and mass transfer in the plate.

**PNRPU Mechanics Bulletin**. 2021;(3):48-57

The Applied mechanical and mathematical model of grinding of a solid particle by static crushing

#### Abstract

Now crushers are one of the most common types of crushing equipment using the principle of a mechanical method of material destruction (for example, rollers, jaws, cone crushers, etc.). To provide effective parameters of the crusher, it is necessary to take into account the correlation between the physical and mechanical characteristics of the material (sizes, shapes, strengths, fragility, uniformity, etc.) and the energy parameters of the crusher (operation and power) at the design stage. The existing theories describing the mentioned dependence and relying on different classical hypotheses allow obtaining a very approximate (inaccurate) result. Consequently, it is necessary to develop a detailed theory of crushing capable of an accurate description of the mechanical process of material destructions by working members of the crushers. Thus, the authors have developed the crushing theory as an original solution of a complex constructively nonlinear engineering and technical problem on the static contact of a spherical model of a comminuted brittle substance with absolutely rigid convex-concave surfaces of cylindrical rolls designed for coarse and medium grinding. The theory is based on the classical assumptions of the mechanics of an elastically deformable continuous medium, the fundamental analytical dependences of Hertz-Shtaerman and the Kirpichev-Kick volumetric energy hypothesis. During the quantitative assessment of the bearing capacity of the ball, we used the well-known physical and mathematical problem of Weber on the stress state of a sphere loaded by two equal forces applied at the poles, and the Kulon-Mor’s strength criterion, which describes the process of destruction of a wide class of brittle homogeneous materials. The developed theory of fragmentation has been brought to the design formulas and illustrated with a typical numerical example.

**PNRPU Mechanics Bulletin**. 2021;(3):58-69

Comparison of foam models from regular and irregular arrays of Gibson-Ashby open-cells

#### Abstract

The paper studies effective elastic properties of foam or cellular materials modeled by a set of Gibson-Ashby open cells with regular or irregular structures. Currently, there are many papers that present results of studying cellular materials using theoretical, numerical and experimental methods. However, these papers consider either regular lattices, or a single cell, or representative volume models based not on the Gibson-Ashby models. In this paper, in addition to the regular lattice, irregular structures were numerically studied. A mathematical formulation of the homogenization problem based on the energy equivalence of a foam-like material and on a homogeneous comparison medium is described. Formulations of six boundary value problems are presented. The solutions of these problems allow us to determine a complete set of effective stiffness modules for foams with different types of physical and geometric anisotropies. All stages of the numerical study were implemented in the ANSYS finite element package. Two algorithms for forming solid-state and finite-element models of irregular Gibson-Ashby lattices with small and large porosity are described in detail. As an example, numerical calculations are carried out for polycarbonate foams. The values of the effective elastic modules for regular and irregular lattices and for the Gibson-Ashby analytical model are compared. The results of numerical experiments showed that the Gibson-Ashby model describes the behavior of highly porous materials quite well (for porosity more than 75%), but this model gives a less satisfactory prediction in case of lower porosity. It is noted that for a large number of cells, regular and irregular lattices statistically give similar results for effective modules. However, for individual structures of irregular lattices, especially with strongly differing cells in individual directions, the effective moduli can have significantly different values, and the effective homogeneous medium can have pronounced anisotropic properties. These effects are due to geometric anisotropy and stress concentration in long connecting beams and at the joints of beams of various sizes in highly irregular Gibson-Ashby lattices. Examples of such lattices are given. We analyze the scatter of value for relative modules, which characterizes the anisotropy of such structures.

**PNRPU Mechanics Bulletin**. 2021;(3):70-83

Finite Element Model Updating Method of Dynamic Systems

#### Abstract

The finite element model updating method of dynamical systems based on results of modal tests is proposed. The purpose of updating is to change eigenspectrum. The method alters a stiffness matrix by adding an updating finite element model created on the nodes of the intial one with respect to the existing links between the linear degrees of freedom. The stiffnesses of the updating elements are utilized as the updating parameters to be defined. The objective function equals to the least square weighted sum of residuals between the target, which were determined experimentally, and current values of modal stiffnesses. The iterative solution process is carried out. At each iteration step the conjugate gradient method is applied to solve the unconstrained minimization problem. The modeshapes, which were calculated as the result of solving the generalized eigenvalue problem at the previous iteration step, are employed to calculate the current modal stiffnesses. The method does not have a limit to a size of matrices and keeps their sparsity and symmetry. It provides the model updating of selected regions of a structure and step-by-step model updating of predefined groups of eigenfrequencies. Moreover, geometrical features of a structure, such as the presence of the symmetry planes and structurally identical elements, may be taken into account. The method is implemented into a program and verified by the example of the free dynamically-scaled model of Tu-204. In order to perform the ground vibration testing, the model was suspended with a low-rigidity flexible support. The finite element model made of solid elements has been updated on the basis of the six experimentally determined sets of eigenfrequencies. The target frequencies from each set have been achieved with a high level of accuracy.

**PNRPU Mechanics Bulletin**. 2021;(3):84-95

Simulation of impact failure of tubular samples made of composite material, depending on the loading rate

#### Abstract

In this paper, a model that takes into account the dependence of strength properties on the damage rate is considered for modelling of the impact failure of composite materials. The constants of the material model are determined on the basis of experimental diagrams to compression and shear impact loading of unidirectional composite, which exhibit a nonlinear dependence on the strain rate. The proposed model is implemented to the Abaqus finite element modeling software for the case of a three-dimensional stress state. As an example of numerical modeling, we consider tubular composite specimens made of carbon fiber with an epoxy matrix and layers of different orientations, which are commonly used for determining of characteristics of impact energy absorption. Diagrams of impact loading of the considered tubular specimens are obtained. The influence of the chamfer (taper of the cross-section) on the edge of a tubular specimen on the impact loading diagram at the initial stage of the process, which serves as the initiator of crushing, is studied. The proposed approach allows us to estimate the magnitude of the peak of the amplitude associated with the crushing of the chamfer. In addition, at the initial stage of loading, maximum strain rates occur, which entails the hardening of the material, also expressed in the loading diagram as an increase in the amplitude. If the analysis neglects the effects associated with the strain hardening of the material and the geometry of the chamfer, the result may be expressed in an underestimation of the reaction, especially at the initial stage of the process. The absence of the described effects in the model of composite structures may lead to significantly incorrect results with complete failure with zero reaction. The developed approach is effective in the design and testing of damping elements made of composite material with properties that are sensitive to the loading rate.

**PNRPU Mechanics Bulletin**. 2021;(3):96-102

Frequency characteristics of built-in fiber-optic PEL-sensor to diagnose complex harmonic deformations within the polymer composite structure

#### Abstract

Fundamentals of operation of an optical fiber piezoelectroluminescent (PEL) sensor inside a polymer composite structure at its cyclic loading are considered. The optical fiber PEL-sensor is considered as part of the composite/sensor electromechanical system taking into account the presence of anisotropy, piezoactivity and Maxwell-Wagner relaxation of the electric fields of the sensor elements. The purpose of the optical fiber PEL-sensor is to diagnose the inhomogeneous complex volumetric deformed state of a long cylindrical area (a neighborhood along the built-in linear sensor) inside a cyclically loaded composite structure. A numerical model has been developed to solve the 3D related boundary value problem of electric elasticity for a representative fragment of the system composite/sensor in the ANSYS package. The numerical modeling of deformation and electric harmonic fields inside the representative fragment was carried out; in particular, distributions of amplitudes of these fields in elements of the structure of the optical fiber PEL sensor were found. The resonant modes are revealed, and the analysis is given of regularities of frequency dependences for the real and imaginary parts of controlling and informative transfer coefficients of the built-in fiber-optic PEL-sensor in the composite/sensor system. Additionally, graphs of frequency dependencies of tangents of mechanical loss angles for various cases of deformation of the composite/sensor system are given. Damping of the composite/sensor system is carried out as a result of the conversion of some part of the mechanical energy (transmitted from the composite to the sensor during their joint deformation) into Joule heat by the fiber-optic PEL sensor with a subsequent dispersion. The latter is caused by the direct piezoelectric effect and Maxwell-Wagner relaxation of electric fields in the sensor elements. The frequency range of deformation of the composite/sensor system is set, in which the passive vibration damping mode is most effectively implemented. It is numerically confirmed that for the extreme high-frequency case of deformation of the composite/sensor system, relaxation processes are not implemented and, as a result, solutions for the controlling and informative transfer coefficients of the PEL-sensor practically coincide with previously obtained numerical solutions that did not take into account the electrical conductivity of the sensor structure elements.

**PNRPU Mechanics Bulletin**. 2021;(3):103-116

Mathematical modeling of relaxation of residual stresses in thin-walled pipelines in the delivery state and after bilateral surface hardening at creep

#### Abstract

A mathematical model of reconstruction of the fields of residual stresses and plastic deformations in thin-walled cylindrical tubes in the delivery state and after bilateral surface plastic hardening was developed. It included a method for identification of the model parameters using the example of thin-walled mm tubes made of X18N10T steel based on experimental data for the axial and circumferential components of the residual stress tensor for samples in the delivery state and after bilateral mechanical ultrasonic hardening. The adequacy of the mathematical model for the reconstruction of residual stresses in the thin-walled tubes made of X18N10T steel was verified by the experimental data in the state of delivery and after bilateral surface plastic hardening, taking into account the anisotropy of the distribution of plastic deformation after hardening in the axial and circumferential directions. A method for calculating the two-way relaxation of residual stresses on the outer and inner surfaces of thin-walled tubes under creep conditions was developed based on the generalization of the corresponding method for unilateral hardening. The relaxation process in the thin-walled tubes made of 08X18N9 steel (an early analogue of X18N10T steel) under conditions of thermal exposure, axial tension, internal pressure and the combined action of axial tension and internal pressure was investigated on the basis of the constructed phenomenological creep theory for this steel. A detailed analysis of the kinetics of the fields of residual stresses during creep in the thin-walled samples in the delivery state and after bilateral hardening at different times was performed. It is shown that under these conditions, almost a complete relaxation of residual technological stresses occurs both in the samples in the delivery state and after bilateral surface plastic deformation within 50 hours.

**PNRPU Mechanics Bulletin**. 2021;(3):117-128

Identification of gyroscopic forces in the oscillatory system of a Coriolis flowmeter

#### Abstract

The phase difference between the oscillations of the Coriolis flowmeter (CFM) arms is the main experimentally observed parameter during measurements of liquid flow rates in pipelines. Usually, steady-state oscillations and known dependences between the flow rate and the measured phase shift are assumed. However, these conditions are met with a sufficient accuracy only for homogeneous and single-phase flows. For inhomogeneous and multiphase flows, the correction of measurements is necessary. This correction in most cases is empirical. However, to improve the methodology of Coriolis flowmeter measurements, more detailed information about flow-tube interactions is needed. The experimental obtaining of such data is expensive and laborious. On the other hand, this data can be acquired during numerical experiments on the CFM virtual prototype. However, to effectively simulate liquid flows, it is necessary to separate the contribution of gyroscopic and dissipative forces to the experimentally observed signal (phase shift). This problem is complicated by the fact that gyroscopic forces are not uniformly distributed along the length of the tube, and the model for dissipative forces is not sufficiently developed yet. In this work, gyroscopic forces were separated by the 3D finite element modeling of steady-state oscillations of a tube with the ideal (inviscid) liquid. We discussed the usage of the simulation results in a simplified discrete model. It is shown that the magnitude of the phase shift recorded by the flowmeter depends both on the features of the distribution of gyroscopic forces and on the elastic coupling of the natural vibrations of the elastic tube caused by the fluid flow. The influence of the tube shape on the experimentally observed phase shift was investigated. For the tube shapes considered in the work, the difference in the phase shift for the displacements of the sections of the installation of the recording coils reaches nearly 5 times. The parameters of both gyroscopic and elastic coupling depend on the shape of the tube, and a change in the shape of the tube can increase the gyroscopic coupling and decrease the elastic one, and vice versa. The creation of a simplified discrete model of the flowmeter based on the results of the 3D finite element calculations is discussed. The quantitative estimates of the integral parameters of the oscillatory system of the CFM are carried out, allowing one to compare both the magnitude of the gyroscopic forces arising during the flow of the liquid and the degree of conformity of the tube shape to the special requirements for the oscillatory system of the CFM.

**PNRPU Mechanics Bulletin**. 2021;(3):129-140

ABOUT THE YIELD PLATEAU UNDER SIGN-VARIABLE LOADING

#### Abstract

During tension of samples of low-carbon steels and some other plastic materials, the sharp yield point and the yield plateau are fixed on the deformation diagram. In this case, Chernov-Luders bands appear on the surface of the sample in the local section, which then propagate along the tension axis. The physical nature of the sharp yield point is established: the drop in the load after reaching the (upper) yield limit occurs as a result of dislocations being an extraction out of the cloud of the embedded atoms and vacancies (Cottrell's cloud). At the occurrence of the sharp yield point , the plastic strain is limited to a small area. When the sample deformation increases, the plasticity zone (the yield plateau is marked on the diagram at this time) expands, and the stress-strain state in this zone becomes almost homogeneous, if we do not consider its boundary with the elastic region. At the end of the yield plateau, the entire sample experiences a uniform plastic deformation, excluding its ends (galtels). From this point on, the strain diagram shows the of material hardening; presumably, this hardening occurred from the very beginning, but was hidden under the yield plateau. This is evidenced by the resulting strain-induced anisotropy. After unloading the sample, when the plastic strain front (in the form of Chernov-Luders bands) has not yet passed through the entire sample, and the Bauschinger's effect is observed with the subsequent change in the stress sign. The fact that after the occurrence of the sharp yield point , the plastic strain is not localized in a certain volume, similar to what occurs during the neck formation, but spreads along the sample, serves as proof of the material hardening due to plastic deformation immediately after the load falls. Therefore, if the plastically deformable part of the sample did not have a hardening of the material due to the plastic strain growth, it would not be able to spread to the elastic part of the sample. This paper presents the experimental data of the sign-variable torsion of the thin-walled tubular samples of steel 45 in the annealed state. The deformation atdiagram of the occurrence and development of the yield plateau in distinct sections along the length of the test sample is obtained. The material hardening diagram hidden under the yield plateau is reconstructed using the well-known Masing's principle.

**PNRPU Mechanics Bulletin**. 2021;(3):141-153

The Experimental study of plastic strain localization in the AMg6 alloy under various types of dynamic loads

#### Abstract

This study is concerned with substantiation of one of the mechanisms of plastic strain localization under high rate loading associated with discontinuous processes in the defect structure of materials. To this end, a series of experiments were carried out to study the localization of plastic strain in specimens of the AMg6 alloy subject to high rate loads on a split Hopkinson-Kolsky bar and during target perforation tests. The temperature fields generated during the plastic deformation tests aimed 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 subject to dynamic loads the mechanism of strain localization caused by thermoplastic instability is not realized. After testing of specially-shaped specimens, their surface relief was explored using an optical interferometer-profilometer NewView-5010 with a subsequent processing of the obtained 3D data of the deformation relief and computation of the scale invariant (Hurst exponent) and the spatial scale of the region displaying a correlated behavior of mesodefects. The results of experimental studies on dynamic loading with a subsequent investigation of the temperature fields, the study of the surface relief of deformed specimens suggest that one of the mechanisms of plastic strain localization in the AMg6 alloy under the realized loading conditions is caused by jump processes in the defect structure of the material.

**PNRPU Mechanics Bulletin**. 2021;(3):154-162

The Numerical simulation of composite sandwich cylindrical shells reinforced with circular frames under local loadings

#### Abstract

The paper considers a woven fabric (based on fiberglass) composite structure in the form of a foam core sandwich cylindrical shell reinforced with circular frames, under the action of local loads applied to the frames. A technique is described for obtaining a numerical solution (with the confirmed reliability) to the problem of the stress-strain state of this type of structure using two alternative computational models, one of which is based on the numerical integration method, and the other one is based on the finite element method. The first model is developed by adopting a scheme, in which the frames are considered as short cylindrical shells obeying the single normal hypothesis. The foam core sandwich sections connected to them are considered within the framework of the zig-zag theory of soft core sandwich shells based on the assumption of incompressibility of the core along thickness. In this case, the corresponding calculation problem is formulated in the form of systems of algebraic and differential (in partial derivatives) equations for each of the shell sections being considered, which are supplemented by kinematic and force conditions at their joints, as well as boundary conditions at the left and right ends of the structure. The problem solving for each harmonic number is reduced to solving a set of boundary value problems for systems of 8 and 12 linear ordinary differential equations of the first order, related by conditions at the specified joints and using the procedure for expanding the parameters of the stress-strain state and applied loads into Fourier series in the circumferential direction. The solution algorithm is constructed using the numerical integration procedure in the orthogonal sweep version combined with the displacement method procedure (to satisfy the conditions at the joints). The declared finite element model is built within the ABAQUS software package using S4 shell elements (for composite layers) and C3D20 volume elements (for frames and core). The formed model, when setting deliberately overestimated values of the corresponding elastic modules, can implement a situation close to fulfilling the set of hypotheses adopted when constructing the first model. Having fixed in such a situation the consistency of the calculation results on the basis of the alternative computational models constructed in this way and thereby confirming the reliability of the obtained numerical solution, we carry out the transition to the calculation using the real values of the mentioned modules and the analysis of their influence on the stress-strain state of the investigated sandwich shell. The presented example of the calculation of a composite sandwich cylindrical structure, one of the frames of which is under the action of two local axial loads, demonstrates the possibilities of the adopted modeling method.

**PNRPU Mechanics Bulletin**. 2021;(3):163-174

Effect of plastic compressibility on the strain rate intensity factor in compression of a material layer between parallel plates

#### Abstract

Solutions of boundary value problems for several rigid plastic models may be singular. In particular, some components of the strain rate tensor and the quadratic invariant of the strain rate tensor may approach infinity in the vicinity of certain surfaces. Some models for predicting the high gradient of material properties near frictional surfaces in metal forming processes are based on such behavior of the velocity field. The coefficient of the leading singular term in a series expansion of the quadratic invariant of the strain rate tensor in the vicinity of frictional interfaces is called the strain rate intensity factor. The magnitude of the second invariant of the strain rate tensor is controlled by this factor that depends on geometric parameters of the boundary value problem and parameters involved in the material model. The present paper deals with the effect of plastic compressibility of the material that obeys the double shearing model on the strain rate intensity factor in compression of a plastic layer between two parallel plates. The system of equations comprising the equilibrium equations and constitutive equations is hyperbolic. It is assumed that the surface of the contact between the plates and the deforming material is an envelope of characteristics. An analytic solution of the boundary value problem is found under plane strain deformation. End effects near the free surface of the layer and its center are ignored. The dependence of the strain rate intensity factor on parameters of the boundary value problem including the parameter that controls plastic compressibility is found. In case of the plastically incompressible material, the solution coincides with the available solution.

**PNRPU Mechanics Bulletin**. 2021;(3):175-181

Algorithm for topology optimization of a structure made of anisotropic material with consideration of the reinforcement orientation parameters

#### Abstract

Advanced design methods offer not only a number of methods of modeling structures but also quite profound optimization approaches. The topology optimization is one of such advantageous methods. There are several implementations of this type of structural analysis, but the most common option is to deal with the material density as a generalized parameter. The development of 3D printing technologies increases the interest in algorithms of this type. However, it mostly refers to printing with metal or plastic, which implies the material isotropy. Advances in continuous fiber printing systems and automatic placement machines using composite tapes make it possible for engineers to control not only a material’s position, but also to choose the local orientation of reinforcing elements. Such systems require optimization algorithms taking into account additional parameters characterizing the material anisotropy. In this case the traditional problem of the topology optimization should first be modified for the model of an orthotropic material with restrictions on the value of the angle of rotation of the reinforcement lines along the print path. In this paper, an idea of such a topology optimization algorithm is proposed, which is implemented using the Abaqus finite element modeling facilities, and examples of solving typical problems are considered.

**PNRPU Mechanics Bulletin**. 2021;(3):182-189

Simulation of the hot isostatic pressing process

#### Abstract

This work, models the compaction of a dispersed body under the conditions of a hot isostatic pressing (HIP) cycle using the example of the manufacture of compacts from VZh159ID powder and Inconel alloy 718. For the research, VZh159ID powder of fraction -70 + 25 μm, bulk density of 3.77 g/cm3 (4.83 g/cm3 after tapping), fluidity of 2.3 g/s, specific surface area of 446 cm2/g, and average particle size was used according to Fisher 16 microns, as well as Inconel alloy 718 powder of fraction -315 + 25 microns, bulk density 3.84 ... 4.58 g/cm3 (4.52 ... 5.24 gcm3 after tapping), fluidity 1.58 ... 1.90 g/s, specific surface 330 ... 376 cm2/g and average particle size according to Fischer 19.0 ... 19.5 microns. Before the HIP cycle, the powder backfills underwent thermal degassing in vacuum, since powders with such a high specific surface are subject to rapid gas sorption. Gases on the surface of the powder body as a result of the HIP cycle can form non-metallic inclusions that reduce the properties of the compact. In the microstructure of compacts after HIP, there is no network of residual boundaries from granules (PPBs-Prior Particle Boundaries), which indicates an effective technology of vacuum degassing of the powder. Simulation of the compaction process was carried out according to the modernized equation of E. Ryshkevich, constants b were selected for the materials considered. The results of the experiments of interrupting the HIP cycle and data on the strength of the samples at high temperatures obtained by selective laser sintering were used as the initial data for modeling. The proposed modeling method is quite simple (does not require experiments on an interrupted HIP cycle) due to the shown possibility of experimentally determining the strength characteristics of alloys at elevated temperatures on samples obtained by selective laser sintering. The analysis of the obtained microstructures (estimation of porosity) of the samples after HIP, having different density values, shows a good agreement of the proposed model with the real process of compaction in the gasostatic extruder.

**PNRPU Mechanics Bulletin**. 2021;(3):190-198

Crack growth in rotor P2M steel at elevated temperature

#### Abstract

The paper presents a comprehensive computational and experimental study of the crack growth rate during the creep-fatigue interaction in compact specimens of P2M steel at a temperature of 550 °C. The theoretical part of the study consisted in the formulation of the fracture resistance parameters through the classical and new constitutive equations of the cracked body state taking into account the accumulation of damage. Numerical calculations included the determination of the stress-strain state fields for the conditions of elasticity, plasticity, and creep, as well as the distributions of nonlinear stress intensity factors and the C*-integral along the crack length and along the crack front for each tested compact specimen. The interpretation of the experimental results for specimens of the same shape in plan but different thicknesses is given in terms of elastic and nonlinear stress intensity factors, taking into account the accumulation of creep damage. It was found that the crack growth rate under the fatigue and creep interaction increases monotonically as the crack size increases in comparison with harmonic fatigue by an order of magnitude or more on specimens of the same geometry. Taking into account the damage through the creep stress intensity factors causes differences in the cyclic fracture diagrams. The superposition of fatigue and creep contributions in terms of load holding time shows an order of magnitude increase in the total crack growth rate compared to the interpretation of the experimental data in terms of pure creep.

**PNRPU Mechanics Bulletin**. 2021;(3):199-210