## No 6 (2023)

**Year:**2023**Articles:**14**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/382**DOI:**https://doi.org/10.15593/perm.mech/2023.6

ENERGY CONDITIONS DETERMINING SIZE RANGE IMPACTING FORMATION OF MAGNESIUM HYDRIDE

#### Abstract

Hydrogen, one of the most abundant elements in nature, is potentially suitable to produce, store and consume clean energy, namely for various industrial uses. The safest technique for hydrogen storage is metal hydride, e.g. as MgH2, with safe facilities to store and transport. A critical indicator of a metal-hydrogen system is the kinetic of hydrogen sorption, which is known particularly low for magnesium. However, under given thermodynamic conditions, the kinetics of magnesium/hydrogen reaction is especially sensitive to the particle size i.e. specific surface, crystallite/grain size i.e. boundary extension and nature of potential additives as catalysts. The present calculations aimed at determining the occurrence of hydride nucleation are based on a new energy ratio, as proposed at first time, which takes into account both physicschemical and mechanical factors. The calculations of the energy ratio are based on the minimum total energy of the system, which correctly reflects the processes occurring during hydride formation in magnesium. The critical size of a nucleus at the phase formation (hydride) in magnesium is controlled by the ratio of the volume and surface area of the emerging component, similarly to a crystallization process from a solution. The influence of the mechanical response of the system to the formation of hydride allows one to propose an interpretation of some phenomena regularly recorded during such experiments, i.e. the influence of special additives and mechanical texture, which lead to the acceleration of hydride formation. The results obtained suggest a mechanism favoring oriented nucleation of the hydride in a textured magnesium matrix thanks to the anisotropy of the elastic characteristics of the newly formed phase.

**PNRPU Mechanics Bulletin**. 2023;(6):5-17

METHOD FOR CALCULATING RESONANCES OF ACOUSTIC STRESSES AT THE BOUNDARIES OF AN ANISOTROPIC LAYER

#### Abstract

The conditions of occurrence of acoustic stress resonances at the boundaries of an anisotropic layer are investigated. In general, under the action of an incident elastic wave, six elastic waves are formed in an anisotropic layer. The total effect of these waves determines the stressstrain state of the layer and is displayed in the spectra of waves scattered by the layer into the environment. The scattering spectra and acoustic stresses were modeled by solving the equations of motion of a continuous medium and the generalized Hooke's law. This system of differential equations is solved with respect to the components of the displacement vector and the stress tensor in the Cartesian coordinate system. The Peano-Becker method of solving a system of differential equations by means of a matrix exponential is used. The components of the displacement vector and the stress tensor at two opposite boundaries of the layer with thickness d are expressed through each other using a sixth-order transfer matrix T = exp(Wd), where matrix W is determined by the parameters of the layer under study. The method of scaling and multiple squaring is used. According to this approach, T = (exp(Wd/m))m. A method for selecting the scaling parameter m is proposed to estimate the errors of truncation and rounding when calculating exp(Wd/m). A guaranteed accuracy and the best efficiency of calculations of all elements of the matrix exponential of the sixth order, in comparison with other known methods, is provided by the use of the method of polynomials of the principal minors of matrix W. The modeling of elastic wave scattering spectra (conversion coefficients) and stress dependences on the angles of incidence for cubic crystal layers is given using the example of indium. The interpretation of resonances of acoustic stresses arising in the crystal layer under the action of a shear wave incident on the crystal is given.

**PNRPU Mechanics Bulletin**. 2023;(6):18-28

FATIGUE SENSITIVITY OF GFRP UNDER PROPORTIONAL CYCLIC TENSION WITH TORSION

#### Abstract

The constructions made of composite materials are subjected to cyclic, dynamic, vibration and other loads in operations related to damage accumulation and degradation of progressive mechanical characteristics. Thus it is important to conduct experimental and theoretical studies of the combined impact effects on the change in the mechanical characteristics of the material. In this case, it is necessary to take into account the complex stress state realized in the structures. This work deals with an experimental study of the degradation patterns of the stiffness characteristics of fiberglass tubular specimens obtained by continuous winding as fatigue damage accumulates due to biaxial proportional cyclic loading. The methodological aspects of realization of biaxial loading are considered. Quasi-static and fatigue tests were performed on specimens with different winding angles under uniaxial tension, torsion, and proportional tension, i.e. torsion with three different ratios of the normal and shear stress tensor components. The presence of a decreasing region in the torsional load diagrams has been revealed. Strength surfaces are constructed. By using the approximation of the fatigue sensitivity curves previously developed by the authors, we process the experimental data on the decrease in the dynamic modulus of elasticity as the number of cycles of exposure increases. The high descriptive ability of the developed model and the low values of the variation coefficients of the calculated parameters were noted. Nonmonotonic dependences of model parameters on the type of stress state are revealed. A significant influence of the winding angle on the fatigue sensitivity of the composite has been found. We have made the conclusion about necessity of taking into account the decrease of mechanical characteristics of materials in calculations of constructions and rationality of further experimental researches for verification of earlier developed models.

**PNRPU Mechanics Bulletin**. 2023;(6):29-40

METHOD OF ESTIMATING PLASTIC DEFORMATION DISTRIBUTIONS AT THE FATIGUE CRACK TIP BASED ON THE SOLUTION OF LINEAR ELASTICITY THEORY

#### Abstract

In this paper, an experimental study of the strain fields at the fatigue crack tip was carried out. The strain fields were measured by an optical camera based on the digital image using the correlation method. Images were recorded using a Basler acA2440-75uc optical camera with a TC23007 OptoEngineering lens to achieve a spatial resolution of at least 3 μm. Recording frequency was 100 Hz. The possibility of using the solution of the linear singularity problem of elasticity theory to estimate the distribution of plastic strain at the fatigue crack tip was shown. Mechanical tests of uniaxial cyclic deformation with simultaneous registration of the strain field at the crack tip of different lengths were carried out on flat specimens of titanium alloys Ti Grade 2, Ti-1.1Al- 0.9Mn, Ti Grade 9. The specimens were loosened by means of a lateral semicircular notch in order to localize the crack. The solution of the problem of a specimen with a notch in the elastic formulation was carried out numerically in the finite element modelling package Comsol Myltiphysics. The peculiarity of the work is the use of the hypothesis of the functional relationship between real deformations and the elastic solution and the value of the secant modulus of the material to estimate the plastic deformation at the crack tip. The size of the zone of intense plastic deformations at the fatigue crack tip for different crack lengths was determined experimentally and numerically. By comparing the calculated and experimental data we showed the possibility of using the proposed dependence to estimate the distribution of the plastic strain field at the crack tip. The results obtained allow the analysis of the irreversible strain fields at the crack tip for mixed mode loading.

**PNRPU Mechanics Bulletin**. 2023;(6):41-49

ALLOY AGING AS A MULTISCALE EFFECT WITHIN THE NANOCOMPOSITE THEORY

#### Abstract

Within the theory of finely dispersed nanocomposites, the dependence of the effective Young's modulus on the absolute size of the reinforcing particles is obtained. Two cases of controlling/changing the effective Young's modulus at a constant relative volume fraction of reinforcing particles are considered. The first is the disintegration of reinforcing particles into smaller ones, followed by diffusion throughout the volume of the matrix. In this case, the effective modulus of the nanocomposite increases. The second one is the agglomeration of reinforcing particles into larger ones. In this case, the effective modulus of the nanocomposite decreases. These patterns seem to be universal and independent of heat treatment technology. It can be assumed that the agglomeration or decomposition of the reinforcing particles depends on the choice of a heat treatment technology for a nanocomposite. It is important to emphasize that the selected heat treatment technology is to be such that during the heat treatment no phase transitions occur either in the material of the reinforcing particles or in the matrix material. It is necessary to eliminate the appearance of phase transitions, since the new phase represents a field of defects, in particular, the field of substitutional dislocations. For such processes, the gradient theory of a defect-free medium is no longer valid. It is necessary to build models of defective environments that are more complex. Therefore, this article does not consider the criteria for choosing a heat treatment technology. The question remains open that, along with the gradient generalization of the theory of composites, a nonlinear generalization is possible. Indeed, unlike ceramics, which retain physical linearity almost until destruction, metal composites exhibit plasticity over a large range of deformations. However, generalization to physical nonlinearity, and even more so to plasticity, is complicated by the fact that there is still no generally accepted theory for constructing a stress-strain curve even for homogeneous materials.

**PNRPU Mechanics Bulletin**. 2023;(6):50-56

METHOD FOR CALCULATING SPHERICAL DOMES FOR STRENGTH AND BUCKLING

#### Abstract

The paper considers new applications of the models, algorithms, software and methods developed by the authors to study shell structures of spherical shells (domes). For this type of structures, a method has been proposed to bypass the singularity at the top of the dome by choosing modified approximating functions. The mathematical model is geometrically nonlinear; it takes into account transverse shears, and is presented as a functional of the total potential strain energy. To reduce the variational problem to solving a system of algebraic equations, the Ritz method was used. The resulting system is solved by the method of continuing the solution using the best parameter with an adaptive mesh selection. The algorithm is implemented in the Maple analytical computing environment. A steel dome was estimated using different methods of border fixing, the values of the critical buckling load and the limit stress load were obtained. A graph of the load – deflection relationship and the deflection fields in the subcritical and supercritical stages were constructed. Fields are shown in the local and global Cartesian coordinate systems. The convergence of the Ritz method in terms of the critical load value is demonstrated. The methodology was verified by comparing the solution to the test problem with the known solution obtained by E.I. Grigolyuk and E.A. Lopanitsyn. The comparison results demonstrate the reliability of the data obtained. It was revealed that for the dome under consideration, the loss of strength occurs much earlier than the buckling, and therefore it can be recommended to select a steel grade with a higher yield strength for its design. A simply support border condition in this case gives a higher value of the maximum permissible load.

**PNRPU Mechanics Bulletin**. 2023;(6):57-67

NUMERICAL SIMULATION OF THE INTERACTION BETWEEN A WAVE IN HYDROGEN AND A BARRIER IN A MODEL CHANNEL

#### Abstract

The paper considers the influence of physical and mechanical characteristics (PMC) of materials of the constructions to propagate acoustic waves in gas at a model channel. The research of the influence of PMC materials, pipelines, in particular, on the propagation of wave processes is associated with the problem of noise that arises during the transportation of natural gas and hydrogen- containing mixtures. The problem of noise is especially relevant given the forecasts for the development of the hydrogen energy transportation and storage industry. Modeling of acoustic processes is often associated with the sources of occurrence and propagation in the simulated environment. In this case, the possible occurrence of the resonance phenomena or processes of attenuation of acoustic waves in the dynamic gas-structure system are not taken into account. The boundary value problem is formulated in a bidirectional interaction statement (2-way Fluid-Structure Interaction or 2FSI) between the deformable structure and the hydrogen flow. Predicting the behavior of the pipeline structure in a model representation under the influence of gas during transportation will make it possible to select the optimal PMC material to reduce the acoustic impact both inside and outside the channel. The research presented in this work is carried out using the ANSYS engineering analysis system, which allows modeling the processes under consideration in the 2FSI statement. The paper analyzes the behavior of a wave generated by a unimodal sound source interacting with a barrier clamped in a rectangular pipe. The main results of the research are presented in the form of dependencies of the pressure amplitude versus time at characteristic points; the dependence of displacement versus time of the model barriers made of different PMC materials; the dependencies of changes in pressure and displacement for different working fluids.

**PNRPU Mechanics Bulletin**. 2023;(6):68-77

X-RAY COMPUTED TOMOGRAPHY CHARACTERIZATION OF 3D-PRINTED ACRYLONITRILE BUTADIENE STYRENE AND POLYETHERETHERKETONE COMPOSITES SUBJECTED TO LASER SHOCK PEENING

#### Abstract

The paper deals with X-ray computed tomography results of a polyetheretherketone (PEEK) and a short carbon fibre reinforced by acrylonitrile butadiene styrene (ABS+CF). Individual carbon fibres and 3D printing defects (consolidated structures of interconnected fibres and densified resin clumps) are detected on an initial microstructure of the ABS+CF composite. The carbon fibres and the consolidated structures are densely packed with uniform sub-horizontal locations throughout in a sample volume. In the PEEK samples, the process-induced defects during the composite manufacturing process are visualised as tubular structures of a densified resin with internal voids. Significant changes in the structure of both composites are observed after five times pulsed laser shock peening. In case of a single pulse exposure and a surface treatment, no microstructural changes occur. In a test mode without a protective layer, a material evaporation to a depth of 0.3 mm and a structural degradation of the PEEK samples takes place, while the process-induced interlayer voids do not close. A single consolidated area with a porous spongy structure occurs due to melting of the carbon fibres in the ABS+CF composite. The results show that the laser shock peening has a significant effect on the surface microstructure. It is therefore necessary to carry out further experiments to select the optimum laser shock peening parameters and a protective layer material to eliminate the process-induced defects and improve the strength properties of the composites.

**PNRPU Mechanics Bulletin**. 2023;(6):78-90

INFLUENCE OF VARIABLE YOUNG’S MODULUS ON RESIDUAL STRESSES INDUCED BY ROTATIONAL AUTOFRETTAGE OF A TUBE WITH FIXED ENDS

#### Abstract

Autofrettage processes are designed to strengthen hollow cylindrical and spherical parts and usually consist of one load-unload cycle. At the first stage, the workpiece is loaded to cause either partial or complete plastic deformations. During unloading, residual compressive stresses are formed in the vicinity of the inner surface of a part. The present work is devoted to a theoretical study of the process of rotational autofrettage of a hollow cylinder with fixed ends. The formulation of the problem is based on the theory of infinitesimal elastoplastic deformations, the Tresca plasticity condition and the flow rule associated with it. It is assumed that at the loading stage the cylinder material follows the linear-exponential law of isotropic hardening, and when unloaded it behaves as purely elastic body. The effect of a decrease in Young's modulus during unloading as a result of preliminary plastic deformation and its influence on residual stresses caused by rotational autofrettage of the cylinder are studied. To quantitatively describe the variation in Young's modulus, an exponential model with saturation is used. For the load stage, an exact analytical solution is obtained based on the Lambert W-function. Calculation of residual stresses in the cylinder is performed using the Runge-Kutta method. As an example, materials with significant decrease in Young's modulus are considered, namely aluminum alloy AA6022, steel DP980 and manganese steel. It has been established that taking into account the variable Young's modulus can lead to a significant reduction in the calculated level of residual stresses. This effect is especially important for the calculation of thick-walled cylinders and fairly high autofrettage velocities.

**PNRPU Mechanics Bulletin**. 2023;(6):91-103

LOCAL DAMAGES IN THE HIP JOINT ENDOPROSTHESIS FROM THE С/С COMPOSITE DURING OVERLOADS

#### Abstract

The paper investigates the process of local damages in the hip joint endoprosthesis (HJ) made of unidirectional carbon-carbon composite material (C/C composite) with pyrolytic carbon (PС) matrix. A mathematical model of deformation of the endoprosthesis from the C/C composite has been developed taking into account the processes of local damage. These damages are possible due to overloads, which may be caused by accidental circumstances during human movements. The developed model is a synthesis of an algorithmic model that takes into account the heterogeneity of the pyrocarbon matrix and composite, and an engineering computational model of the biomechanical endoprosthesis-femur system. The matrix algorithm solves the stochastic boundary-value problem of finding mesostresses in PС grains taking into account possible damages. The result of this algorithm is the probability distribution densities for meso-stresses in PС crystallites and the properties of the damaged matrix. The results of calculations based on the engineering model are the fields of macrostrains and macrostresses. At each step of loading of the endoprosthesis, the state of the matrix is monitored and the effective modules of the carbon composite are changed. This is implemented by a continuous exchange of data between the two algorithms, the recalculation of the properties of the composite, which are the input data for the engineering model. The continuous change in the effective properties of the C/C composite during deformation is replaced by a stepwise change. To do this, the volume of the endoprosthesis was divided into areas in which the properties become variable, starting with a certain loading step. The areas of change were determined based on the distribution patterns of macrodeformation fields. A nonlinear loading diagram of the endoprosthesis is constructed taking into account the damage. It is shown that the destruction of the carbon part of the prosthesis begins with local damage, which gradually engulfs the neighboring areas. Damage occurs when the standard load exceeds 1740 Newtons. The maximum force response of the prosthesis to an external load is equal to 2004 newtons. The deformation of the prosthesis at the stage of a critical reduction in load-bearing capacity exceeds the deformation at standard load by 16 %. The high reliability of the considered variant of the endoprosthesis was confirmed, the absence of catostrophic sharp decreases in load-bearing capacity under a significant excess of standard loads was confirmed.

**PNRPU Mechanics Bulletin**. 2023;(6):104-114

ESTIMATION OF RESIDUAL STRESSES IN BIMETALLIC PLUNGER PUMP CYLINDERS AFTER THERMAL DEFORMATION

#### Abstract

Plunger pumps used in oil production are made of long hollow bimetallic cylinders. These components are thermo-mechanically treated to improve strength and other physical and mechanical properties. These operations result in residual stresses within the parts, which can lead to positive, undesirable and unacceptable changes in the geometry. In the present work, we consider the problem of choosing the optimal machining modes. Estimations of residual stresses in the whole product take too much time, so it was decided to use small rings, which are representative for each particular pipe. In view of complexity or impossibility of applying the existing methods, the authors have designed a novel technique to estimate the level of residual stresses. For this purpose, we formulated and solved this problem within the theory of elasticity. An analytical solution, which makes it possible to find the level of stresses depending on the experimental measurements when cutting the rings, has been obtained. Three different steels were chosen. i.e. 38Cr2MoAl, 15Cr5Mo, 12Cr18Ni10Ti. Based on operating conditions, four optimization criteria for the heat treatment have been produced: the minimum level of residual stresses in the pipe; the minimum difference between stresses in the shell and liner; the minimum change in the pipe radius after the treatment; the highest value of adhesion between the liner and the shell. The obtained results have been analyzed based on the above four criteria. We revealed the optimum and intolerable modes of thermo-mechanical processing, which enabled undesirable changes in products. The required degree of deformation and temperature of the post-deformation heating have been found for each steel under study. As a result, recommendations for industrial enterprises have been drawn up.

**PNRPU Mechanics Bulletin**. 2023;(6):115-123

SIMULATING THE STRESS-STRAIN STATE DURING LASER METAL DEPOSITION TO DETERMINE THE FINAL WARPING OF PRODUCTS

#### Abstract

Additive technologies, including laser powder deposition (a repair technology), enable a sequential deposition of powder layers. This process involves large temperature gradients and technological residual stresses, which can lead to shape violations and change mechanical and operational characteristics of products. To control and prevent residual deformations in the hardfacing body, it makes sense to carry out finite element modeling of the laser powder hardfacing using layer-by-layer activation technology or adding new finite elements to the surface of the hardfacing model. The Element Birth/Death method is the most suitable method for this problem. In this case the elements for the material to be created are deactivated (so not included in the solution area), and then gradually revived and included in the solution area. The material is built up discretely. At each sub-stage of the calculation, corresponding to the revival of the next sub-domain of dead elements, the coupled problem of thermal conductivity and solid mechanics is solved, and thus the result of the solution of the previous sub-stage serves as the initial conditions for the next one. A mathematical model and an algorithm for modeling warping during the deposition are developed, and calculations for the deposition of cylindrical specimens are carried out. During the calculations, the multilinear MISO plasticity model for the sample material and the BISO bilinear plasticity model for the filler powder were used. We verified the model based on the optical control results of changes in the geometry of the experimental samples after the deposition had been carried out. The error in warpage calculation did not exceed 5%.

**PNRPU Mechanics Bulletin**. 2023;(6):124-134

ANALYSIS OF THE INFLUENCE OF ALLOYING ADDITIONS ON THE STRUCTURE OF AL-LI ALLOYS AND DEFORMATION MECHANISMS UNDER SUPERPLASTIC CONDITIONS (AN ANALYTICAL REVIEW)

#### Abstract

The use of structural superplasticity is promising in the development of production technologies with complex shapes and improved physical, mechanical and operational characteristics. Deformation in the superplasticity regime is characterized by reduced (compared to conventional plastic processing) loads on tools and decreased number of finishing operations. It seems preferable to use the superplasticity regime at relatively moderate homologous temperatures (less than 0.7) and high strain rates (on the order of 10–2 s–1), in which the equiaxed grain shape can be preserved with an insignificant change in its size. Under these conditions, staged (bell-shaped) tension curves are observed in experiments on uniaxial tension with access to the structural superplasticity regime for many alloys preliminarily prepared by severe plastic deformations. The latter is associated with the action and interaction of various physical mechanisms, the change in their roles during the deformation and evolution of defective material structures. The above factors are influenced by the initial temperature and strain rate conditions and characteristics of material structures after the pretreatment, in particular, grain shapes and sizes, fraction of high-angle boundaries, degree of recrystallization of the structure, presence of alloying additives that can form various phases in materials. This review attempts to systematize experimental data on superplasticity of aluminum alloys 1420 and 1421 with a focus on the main characteristics of material structures before and during the superplastic deformation tests, as well as its effect on the acting mechanisms. This will make it possible to form a more complete understanding the physical nature of deformation with a transition to structural superplasticity regimes for aluminum alloys and to develop a scenario for the action and interaction of mechanisms taking the influence of the evolving material structure into account. The above will be the concept basis for development the multilevel constitutive models of inelastic deformations of alloys to describe the material structure evolution and change in deformation regimes, which is necessary to improve superplastic forming technologies.

**PNRPU Mechanics Bulletin**. 2023;(6):135-157

REFINED MODEL OF THERMOELASTIC-PLASTIC DYNAMIC DEFORMATION OF FLEXIBLE REINFORCED CYLINDRICAL SHELLS

#### Abstract

Within the refined theory of bending, a coupled initial-boundary value problem of thermoelastic- plastic deformation of flexible circular cylindrical shells with arbitrary reinforcement structures is formulated. The tangential displacements of the shell points and the temperature along the thickness of the structures are approximated by high-order polynomials. This makes it possible to take into account, with varying degrees of accuracy, the weak resistance of fibrous sheaths to transverse shear and to calculate wave processes in them. From the obtained two-dimensional equations of the refined theory, in the first approximation, the relations of the traditional non-classical Ambartsumian theory are obtained. The geometric nonlinearity is modeled in the Karman approximation. The inelastic deformation of the components of the composition is described by the relations of the theory of flow with isotropic hardening. In this case, the loading functions of the materials of the composition phases depend not only on the strengthening parameter, but also on the temperature. For the numerical solution of the formulated nonlinear coupled two-dimensional thermomechanical problem, an explicit scheme of time steps is used. We studied the axisymmetric elastic-plastic deformation of flexible long cylindrical shells, which are reinforced in the circumferential and axial directions. Fiberglass and metal-composite structures from the inner front surface are loaded with pressure, which corresponds to the action of an air blast wave. It is shown that for an adequate calculation of temperature fields in the structures under consideration, it is advisable to approximate the temperature over their thickness with a 7th order polynomial. It has been demonstrated that at some points fiberglass shells can additionally heat up for a short time by only 10…11 C, so the thermal response can be disregarded in their calculations. Metal-composite structures can additionally heat up by more than 40 C. However, for their calculation it is also possible to use the model of elastoplastic deformation of the materials of the composition components. It is shown that when studying the dynamic inelastic behavior of both fiberglass and metalcomposite cylindrical shells, it is advisable to use the refined theory of their bending, rather than its simplest version, the Ambartsumian theory.

**PNRPU Mechanics Bulletin**. 2023;(6):158-169