## No 2 (2023)

**Year:**2023**Articles:**13**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/339**DOI:**https://doi.org/10.15593/perm.mech/2023.2

DYNAMIC STABILITY OF A STRAIGHT PIPE CONVEYING PULSATILE FLOW UNDER THERMAL LOADS

#### Abstract

Pipes conveying fluid are considered as a fundamental dynamical problem in the field of fluid- structure interaction. They are widely used in the petroleum industry, in nuclear engineering, aviation and aerospace, in nanostructures. This article investigates the effect of temperature load on the dynamic stability of a straight pipe conveying pulsatile flow. The fluid velocity is a harmonic function of time. The Galerkin method is applied for the solution of the differential equation of the transverse vibrations of the pipe. The differential equation is reduced to a first-order differential equation system. The system of differential equations is transformed and rewritten in a matrix form. The harmonic function of the fluid velocity allows the Floquet theory to be applied in order to investigate the dynamic stability of the system. The static scheme of the investigated pipe is a beam with restricted horizontal and vertical displacements at both of its ends. A numerical solution for a straight pipe conveying fluid with specified geometric and physical characteristics has been carried out. The temperature load and the constant fluid rate are considered as parameters of the problem. The results show that the temperature load affects the vibrational characteristics of the pipe, as well as its critical velocity.

**PNRPU Mechanics Bulletin**. 2023;(2):5-10

METHOD FOR STUDYING THE POROSITY OF FLUID PHASE SAMPLES BY X-RAY COMPUTED TOMOGRAPHY UNDER UNIAXIAL COMPRESSION

#### Abstract

The development of composite materials and products with a complex internal structure poses questions to develop experimental methods to determine the stress-strain state. Standard experimental methods do not provide a broad picture of the internal changes in the inhomogeneous samples under loading. Moreover, a presence of internal defects, porosity, local buckling can significantly effect on the obtained results. Despite the development of tomography and data processing methods most studies are performed without any external loading on sample. The research presents a methods for studying the porosity of samples under uniaxial compression using an X-ray computed tomography. For this purpose, special equipment that allows loading the sample inside the tomography was made. Additionally, a methodology was developed. The equipment allows not only to transfer the axial compression force, but also to measure the corresponding load value. The equipment was designed to reduce artifactual rays in a place where studied sample mounted. To determine the reference points under loading, a contrasting copper grid was used. A modified Harris detector was used to quantify the displacements. The displacements of the reference points were interpolated to a regular initial grid to estimate the displacements inside the sample. Test samples were designed and manufactured using additive technologies to illustrate the methodology. A series of loadings and tomography scans were carried out for each sample. The tomography data were processed according to the methodology. As a result, the displacement fields of the samples, the values of porosity, volumetric strain and their distribution over the sample for each loading step were obtained.

**PNRPU Mechanics Bulletin**. 2023;(2):11-21

STRESS-STRAIN STATE DURING DRAFT OF WIDE BANDS WITH SHEAR

#### Abstract

Forging processes are traditional methods of metalworking, their application is very extensive and allows the manufacture of metal products for various industries in a wide temperature range. The redistribution of the main acting forces during forming is a necessary condition for the transfer of traditional forging methods to high-tech methods of metal production. The main products of press-forging production are forgings such as pallets and plates. In the present work, the effect of shear forces on the stress-strain state is studied when the friction forces are redistributed on the contact surface and/or the nature of the metal flow changes during upsetting of wide strips. The analysis of the stress state was carried out by the method of slip lines compared with the existing method of settlement without shear. The field of slip lines and the hodograph of velocities for the draft of the strip with a shift are compiled. The stresses and intensity of the shear deformation were estimated by the analytical method. It was revealed that the upsetting of the strip between plane-parallel plates is accompanied by extremely uneven deformation over the section of the workpiece. The stress state is compared with traditional deformation and with superimposed shear deformation. The use of shears made it possible to realize predominantly compressive stresses, which make it possible to eliminate internal defects of the workpiece of foundry origin. The introduction of shear deformations contributes to the intensification of the plastic deformation process over the entire cross section of the strip, the stress values during draft with additional shear increase on average 4-6 times compared to normal draft. The increase in stress occurs due to the development of the intensity of shear deformation, reaching a value of 0.4 per compression.

**PNRPU Mechanics Bulletin**. 2023;(2):22-28

ON THE REDUCIBILITY OF SOLUTIONS FOR THE GENERALIZED YIELD CRITERION TO SOLUTIONS FOR TRESCA’S YIELD CRITERION UNDER AXIAL SYMMETRY

#### Abstract

Many continuum mechanics models are reduced to simpler models at certain parameter values. However, solutions for the general model may not converge to the corresponding solutions for a simpler model. In the mathematical theory of plasticity, the yield criterion completely determines the material's behavior if the associated plastic flow rule is accepted. In this paper, the reducibility of axisymmetric solutions for the generalized yield criterion to the corresponding solutions for Tresca’s criterion is investigated when the generalized yield condition tends to Tresca’s criterion. It is shown that there is no convergence if the maximum friction law is one of the boundary conditions. In this case, the solutions for both yield criteria are singular. In particular, the quadratic invariant of the strain rate tensor tends to infinity near the friction surface. The strain rate intensity factor controls the magnitude of this invariant in the vicinity of the friction surface. The strain rate intensity factor is involved in some constitutive equations for predicting the evolution of material properties near frictional interfaces in metal forming processes. In this paper, using the solution of a specific boundary value problem, the behavior of this factor is investigated when the generalized yield criterion tends to Tresca’s criterion. It is shown that the strain rate intensity factor continuously changes when the generalized yield criterion deviates from Tresca’s yield criterion. This behavior of the strain rate intensity factor justifies its use in the constitutive equations for the evolution of material properties near friction surfaces.

**PNRPU Mechanics Bulletin**. 2023;(2):29-37

MODELING OF STEADY-STATE LIQUID FILTRATION IN A PIECEWISE INHOMOGENEOUS ELASTIC-POROUS MEDIUM IN THE CLASS OF ALMOST-PERIODIC FUNCTIONS (PLANE PROBLEM)

#### Abstract

When modeling liquid filtration in a porous medium, it is assumed that the filtration coefficient is constant, as a result of which the solution is simplified and reduced to a boundary value problem for the Laplace equation. In this paper, the almost periodic of Bohr analytical solutions of the plane problem of steady-state liquid filtration in an elastic – porous piecewise inhomogeneous domain are constructed using a generalized discrete Fourier transform . The domain is a strip consisting of several layers (strips) with different elastic and filtration parameters. Assuming that the filtration coefficient of an elastic-porous medium depends on the first invariant of the stress tensor, we consider it linearly dependent on the coordinate varying along the bandwidth. The filtration problem is reduced to solving a system of partial differential equations with specified boundary conditions on the upper and lower boundaries of the entire multilayer strip and conditions on the internal lines of the media interface, which in turn is reduced to solving the Cauchy problem for a system of Bessel ordinary differential equations. All solutions in this paper are obtained in the form of absolutely convergent Bohr-Fourier series, the coefficients of which are expressed in terms of given functions. Fluid filtration in a three-layer strip consisting of various light and sufficiently elastic-porous sedimentary and igneous rock layers is modeled. Graphs of the desired mechanical parameters are constructed. Their convergence to boundary conditions and conditions on the interface lines of media is shown. The paper also provides basic information concerning the properties of almost-periodic functions and the generalized discrete Fourier transform necessary for a more detailed understanding of the problem.

**PNRPU Mechanics Bulletin**. 2023;(2):38-46

IDENTIFICATION OF STRESS INTENSITY FACTORS, T-STRESSES AND HIGHER-ORDER COEFFICIENTS OF REGULAR TERMS IN THE WILLIAMS SERIES EXPANSION THROUGH MOLECULAR DYNAMICS SIMULATIONS

#### Abstract

Molecular dynamics (MD) approach and finite element analysis are enforced for the investigating the stress – strain fields in the proximity of the notch tip in a copper plate with single horizontal and inclined edge notches. The MD simulation embodied in a classical molecular dynamics program Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is aimed at evaluating conventional continuum linear elastic fracture mechanics key parameters, precisely stress intensity factors (SIF), T-stresses and generalized stress intensity factors (higher order factors) of the Max Williams power series expansion (WE) of the stress field adjacent with the notch tip for pure tensile (Mode I), pure shear (Mode II) and mixed mode (combinations of Mode I and Mode II) loadings of the notched specimen in linear elastic isotropic media. The paramount intent of the research is the comparability of continuum and atomistic procedures for the appraisement of the near notch tip fields exploiting the exemplification of one of the widespread cracked configurations. SIFs, T-stresses and higher order amplitude coefficients of the WE for the single-edge notched Cu plane under Mode I and Mixed Mode loadings are estimated by atomistic and finite element modelling. The wide class of the MD computations in LAMMPS is effectuated. The atomistic values of SIFs and amplitude factors of higher order terms of the WE are correlated with the quantities gained from the numeric solutions obtained by finite element method. It is elucidated that the continuum fracture theory properly characterizes failure and the near notch tip fields even at tremendously limited distances of only few nanometers. The angular stress distributions found from atomic modeling are restored and correlated with the angular behaviours of the stresses obtained from continuum linear elastic fracture theory. The juxtaposition is shown to be in reasonable agreement between two approaches.

**PNRPU Mechanics Bulletin**. 2023;(2):47-77

DYNAMICS AND STABILITY OF KAPITSA'S TWO-LINK PENDULUM

#### Abstract

The object of study. The upper upward position of the pendulum subjected to vibration of the pendulum base is known to be stable for some parameters of the base vibration. The research is devoted to dynamics of the model of a two-link inverted pendulum in a general nonlinear formulation. The goal. The boundaries of the parameters of a given base vibration, under which the inverted mode is stable, are assumed to be known. The goal is to find the regions of the initial conditions of the problem, namely, the initial non-small angles of deviation of the pendulum links from the vertical that result in stable oscillation in the inverted position. We intend to reveal the impact of the rod compressibility on the oscillation mode, as well as the influence of the resonance on stability in the framework of more complex formulation of the problem which involves account for small elastic axial deformation in the rods. Methods. By applying the laws of dynamics to the moving elements of the structure, we derive the complete nonlinear system of equations of the pendulum motion in two formulations: (i) for a system with two and (ii) four degrees of freedom, respectively. The equations include the parameter of small base vibration amplitude, which makes it possible to apply the two-scale asymptotic expansion method. The method leads to a system of averaged equations of motion which is convenient for the benchmark study of parameters. Results. The modes and eigenfrequencies of small oscillations of the pendulum are found depending on the dimensionless parameter of the problem. In the nonlinear for-mulation, the maximum deviations of the pendulum links are calculated which ensure a stable solution to the problem for zero initial angular velocities. Depending on the initial phase of vibration of the base, the boundaries of absolute and partial zones of stability of vibrations are obtained. In the absolute zone, stable oscillations are realized for any value of the initial phase of the base vibration. In the partial region, stable oscillation occurs at least for one set of initial condition. The dynamics of the pendulum is compared with and without account for rod the compressibility. The results are presented in the graphs.

**PNRPU Mechanics Bulletin**. 2023;(2):78-87

INFLUENCE OF SURFACE CHARGE ON THE FORMATION OF THE CARBONIZED LAYER RELIEF ON THE POLYMER SURFACE DURING ION-PLASMA TREATMENT

#### Abstract

The origin hypothesis of the wavy relief on the surface of plasma-treated polyure-thane is considered in this paper. It has been suggested that stresses and strains appear due to the eponymous charge accumulated in the near-surface layer. A technique for cal-culating stresses under the condition of a uniform charge distribution is proposed. The constitutive equations of an elastic medium with a distributed charge, based on the law of conservation of energy and thermodynamic inequality, are obtained. The Cauchy stress tensor contains a term that depends on the charge distribution density in the resulting equations. A calculation, showing the dependence of the magnitude of stresses on ener-gy and fluence of implanted ions, has been carried out. According to the proposed mod-el, calculations show that the stresses in the material are high enough to cause a change in the shape of the surface at certain treatment regimes. It is shown that the loss of stabil-ity and, as a consequence, the appearance of waves on the surface of the material is typ-ical for low-modulus polymers. The calculation results are compared with real images, obtained using optical and atomic force microscopes, of the samples surfaces after the treatment. Conclusions about the viability of the proposed hypothesis are drawn.

**PNRPU Mechanics Bulletin**. 2023;(2):88-97

PECULIARITIES OF SH WAVE PROPAGATION IN A TWO-LAYER STRUCTURE OF INHOMOGENEOUS PIEZOELECTRIC AND DIELECTRIC LAYERS

#### Abstract

An approach to modeling dynamic processes in a semi-infinite composite plate of inhomogeneous piezoelectric and dielectric layers is proposed. When modeling the inhomogeneity of the layers, a two-component model with a functionally gradient change in properties was used, in which the physical parameters of the base material continu-ously change along the thickness up to the inclusion parameters. The material of the piezoelectric layer is a combination of PZT-based piezoceramics with a significant differ-ence in speed characteristics. The possibility of localizing the inhomogeneity both at the outer surface of the plate, and in the middle of the layer or at the interface has been im-plemented. The dielectric layer is made of SiO2, the inhomogeneity of the dielectric layer models the interpenetration of the piezoelectric and the dielectric in a narrow transition region near the interface. The elastic and dielectric moduli of the piezoelectric material located near the interface were considered as parameters of the inclusion material. The outer surfaces of the composite plate are stress-free and electrically short-circuited. The problem of the propagation of surface SH-waves in a composite structure of functionally gradient piezo- and dielectric layers initiated by the action of an infinitely distant source of harmonic oscillations is considered. The solution is constructed in the space of Fou-rier images by reducing to the solution of a system of ordinary differential equations with variable coefficients, which in turn is constructed using the Runge – Kutta – Merson method. The dispersion equation of the problem is presented, the analysis of which made it possible to investigate the influence of the nature, size of the transition region of materials and localization of structural inhomogeneity on the behavior of SAW phase velocities for a wide frequency range. The results obtained are given in dimensionless parameters and may be of particular interest in the development, design and optimization of new materials for micro- and nanoscale devices and devices based on SH SAW with high performance characteristics.

**PNRPU Mechanics Bulletin**. 2023;(2):98-109

CALCULATION OF THE TOOL SHAPE FOR TUBE MIDDLE PART DISTRIBUTION BY RIGID MATRIX

#### Abstract

In modern mechanical engineering in general, and in the aircraft industry in particu-lar, a large number of parts are obtained by sheet stamping. For the manufacture of ele-ments of hydro-gas systems of aircraft, shaping operations are often used, in which liq-uids and rubber-like materials serve as a working medium that transfers the pressing force to a deformable workpiece. In this paper, we consider the process of expanding the middle part of a tubular billet made of titanium alloy OT4-1. The internal pressure on the tubular billet is determined by the action of the working fluid during compression. One of the significant disadvantages of cold sheet forming is the springback of the material after being removed from the tool-ing. Therefore, the shape of the die used for the technological process of expansion must set the proactive shape of the tubular billet, providing the desired residual shape after unloading. To determine such a matrix form, an inverse problem is formulated and solved. The implementation of the method for solving the inverse problem is carried out in the MSC.Marc system. For the axisymmetric shape of the part, a two-dimensional state-ment of the problem is used. In the case of thin-walled structures, the modeling of the shaping process is carried out under plasticity conditions, taking into account small deformations, but large displacements and rotations (general Lagrangian formulation). The solution of the inverse contact problem of shaping is found by the iterative method, which is based on the quasi-static variational principle. This solution algorithm is gener-alized to three-dimensional problems, when the part has a non-axisymmetric shape, in particular, ellipsoidal, tee, etc. As a result of solving the inverse contact problem of shaping a tubular workpiece by the iterative method, the required geometry of the rigid matrix was determined. Compari-son of the numerical results with the conducted full-scale experiment showed a satisfac-tory agreement. Thus, the presented method and its implementation in the CAE system makes it possible to design tooling at the pre-production stage.

**PNRPU Mechanics Bulletin**. 2023;(2):110–117

DISPERSIONAL DEPENDENCES AND PECULIARITIES OF ENERGY TRANSFER BY FLEXIBLE WAVES IN A BEAM LYING ON A GENERALIZED ELASTIC BASE

#### Abstract

The dynamics of a Bernoulli – Euler beam lying on an elastic foundation is consid-ered. A generalized model of an elastic foundation is selected, which includes two inde-pendent bedding coefficients: the stiffness of the foundation for tensile-compression deformation and for shear deformation. Unlike the classical elastic foundation model (Winkler's model), the generalized model takes into account the distribution capacity of the soil, i.e. its property to settle not only under the loaded area, under the foundation, but also near it. The beam is considered to be infinite. Such idealization is permissible if optimal damping devices are located on its boundaries, that is, the parameters of the boundary fixation are such that the perturbations falling on it will not be reflected. This makes it possible to consider the beam model without taking into account the boundary conditions, and consider vibrations propagating along the beam as traveling bending waves. The influence of a two-constant elastic foundation on the parameters of a bending wave propagating in a beam is studied. It is shown that with an increase in the shear stiffness of the elastic base, waves with the same wavenumber (i.e., waves of the same length) will have a higher frequency, higher phase and group velocities. For the system under consideration, the energy transfer equation is written in divergent form. It is shown that the average rate of energy transfer is equal to the group velocity of the flexural wave. The equality of these velocities serves as an additional factor indicating the internal physical consistency of the model of bending vibrations of a beam lying on a general-ized elastic foundation.

**PNRPU Mechanics Bulletin**. 2023;(2):118–125

DESCRIPTION OF THE EFFECT OF SOFTENING IN ELASTOMERIC COMPOSITES OBSERVED IN TESTS WITH INCREASING STRAIN AMPLITUDE

#### Abstract

This paper presents the results of complex uniaxial mechanical tests conducted to study polymer nanocomposites with SKU-PPL-2102 prepolymer as a matrix. The unfilled elastomer is a structurally heterogeneous material – it has solid domain structures and therefore can be considered as a nanocomposite exhibiting complex mechanical behav-ior. Tests were carried out on the samples prepared of polyurethane without filler and with 0.5 phr of few-layer graphene filler under conditions of increasing deformation cy-cles and long time-delays between changes in the direction of gripper motion. These tests made it possible to monitor the degree of softening in the material subjected to loading and the increase of dissipative losses at different elongation ratios. It was found that, even at low filler content, the mechanical properties of the material changed signifi-cantly. In addition, note that the viscoelastic properties of the materials’ samples manifest themselves insignificantly (i.e., the "trained" materials can be considered elastic with a certain degree of accuracy) during repeated deformations. An elastic potential that is based on the notion of the effective behavior of loaded polymer chains was used to describe the elastic properties of the material taking into account the Mullins softening effect. The viscoelastic behavior of the nanocomposite under study was described in the framework of the previously described thermodynamic model. The Mullins softening effect was taken into account in both elastic and dissipative terms of the Cauchy stress tensor. The obtained data demonstrate that the consideration of this effect has a considerable effect on the dissipative component of the stress tensor.

**PNRPU Mechanics Bulletin**. 2023;(2):126–132

MECHANICAL BEHAVIOR OF POLYURETHANE COMPOSITES WITH NANOSIZED FILLERS

#### Abstract

This paper reports the results of a study on the structure and properties of elasto-meric nanocomposites based on polyurethane, which is of special interest due to its heterogeneous structure. So, the unfilled polyurethane also can be considered as a nanocomposite with complex mechanical behavior. Our study was performed on unfilled polyurethane and polyurethane filled with carbon particles of various morphologies: 1) few-layer graphene; 2) multi-walled carbon nanotubes; 3) diamond charge (detonation nanodiamonds). The content of filler in composites was 0.5 and 4 phr. Analysis of the mechanical behavior of the materials under consideration was car-ried out on the basis of the results of classical uniaxial tensile tests and cyclic experi-ments with increasing amplitude of deformation. The results obtained demonstrated that the incorporation of even a small filler amount in a polyurethane matrix leads to a signifi-cant change in the mechanical properties of the material. First of all, the stiffness of the material decreases in all cases. Secondly, the rupture (tensile failure) of the filled material significantly increases (almost in all cases) as compared to the unfilled polyurethane. The true stresses in some materials also increase at the time of sample’s rupture. A series of tearing tests were performed to gain a deep insight into the peculiarities of the failure mechanisms of materials. It was established that an increase in the macro-fracture of the unfilled material (until its rupture) is insignificant. In the materials rein-forced with nanofillers, the growth of macrofractures takes a longer time, and they prop-agate to a much greater extent. Numerical calculations were carried out to give an explanation for the reduction in the polyurethane stiffness and the macrofracture growth retardation when the filler is added to polyurethane. It was hypothesized that a soft interfacial layer is formed near the particle surface. Furthermore, the finite element model proposed here can be used to explain the growth of deformations at the moment of rupture for filled polyurethane sys-tems.

**PNRPU Mechanics Bulletin**. 2023;(2):133–141