No 2 (2025)
- Year: 2025
- Articles: 9
- URL: https://ered.pstu.ru/index.php/mechanics/issue/view/458
- DOI: https://doi.org/10.15593/perm.mech/2025.2
An Asymptotic Method for Solving Contact Problems on the Effect of a Half-Strip Stamp on an Anisotropic Composite
Abstract
For the first time, an asymptotic solution has been constructed for the contact problem of friction of a rigid half-strip stamp of an anisotropic multilayer composite material. The relative width of the half-strip is assumed to be a large parameter that determines the asymptotic expansion. The method is based on generalizing the approach to constructing asymptotic solutions for simpler contact problems. Previously, the asymptotic method has been developed to solve the contact problem in case of a strip-stamp with a large relative width. The method proved to be effective because it provided a satisfactory agreement with the solution constructed by the counter asymptotic expansion for the stamps in the form of a strip of a small relative width. In this paper, it is applied in a much more complex and previously unsolved contact problem for a stamp in the form of a semi-infinite strip. The complexity of this problem lies in the fact that in order to apply the asymptotic approach, it is necessary to develop a method for solving two-dimensional Wiener-Hopf equations, which has already been done by the authors and it is already used in this work. Similar problems occur in engineering practice and construction when creating various objects, when developing an electronic element base, in seismology, when assessing the state of seismicity in the transition zone of a mountain range into a plain. By using the existing numerical methods, it is possible to describe the behavior of the concentration of contact stresses at the stamp boundary, and especially at the corner points of the boundary, where the most vulnerable parts of the structure are located. However, it is not possible to construct a complete solution of the distribution of contact stresses under the half-strip stamp together with the features at the boundary, due to the area infiniteness. In this paper, a solution is constructed that correctly reflects the real distribution of the contact stresses under the stamp and aims at an accurate solution with an increasing strip width parameter.
PNRPU Mechanics Bulletin. 2025;(2):5-13
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Features of SH-Wave Propagation in a Bimorphic Piezoelectric/Piezomagnetic Plate Made of Functionally Graded Materials
Abstract
Propagation of shear horizontally polarized surface acoustic waves (SH-SAW) in a composite magnetoelectroelastic plate of inhomogeneous piezoelectric and piezomagnetic layers are studied in a quasi-static approximation. To simulate inhomogeneity of the layers, a two-component model of functionally gradient materials with properties depending on thickness is used (ranging from parameters of the base material to the parameters of the inclusion material). PZT-5H and CoFe2O4 materials are used as a base of the piezoelectric and piezomagnetic layers of the plate. The inclusions of the piezoelectric layer are PZT-based ceramics with different elastic, piezoelectric, and dielectric properties. The inhomogeneity of the piezomagnetic layer models a solid solution of the layer materials in a narrow transition region at the interface. The propagation of SH-SAW in the plate is initiated by the action of a remote source of harmonic oscillations, the mode of which is assumed to be steady. The adhesion conditions are met at the interface of the inhomogeneous layers. On external surfaces freely contacting with vacuum, in the absence of mechanical stresses, four types of electrical and magnetic conditions are considered, depending on which four problems are studied. The solution is constructed in the space of Fourier images by reducing to a system of initial-boundary value Cauchy problems. Matrix representations of the dispersion equations of the problems convenient for the numerical implementation are obtained. By using an example of the problem with electrically short-circuited and magnetically open surface conditions, we investigated how inhomogeneity of the piezoelectric and piezomagnetic layers of the plate influences the behavior of SH-SAW velocities in a wide frequency range. We determined the influence of localization of various types of inclusions of the piezoelectric layer on their behavior. Features of the behavior of SH-SAW velocities in a magnetoelectroelastic plate are described for different characteristics of inhomogeneity at the interface. The obtained results are given in dimensionless parameters and can be of particular interest in the development, design and optimization of new materials for modern micro- and nanosized devices and devices using SH SAW.
PNRPU Mechanics Bulletin. 2025;(2):14-29
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Finding the J-integral for a Crack-Like Defect of a Solid in the Form of a Physical Section in Finite Element Representation
Abstract
A crack-like defect in the form of a physical section with a characteristic thickness in a linearly elastic medium is studied. The thickness of the physical section is considered as a linear parameter. For an external load, the stress-strain state of the neighborhood of the physical section is determined by the finite element method, allowing stress vectors on the free surface to be different from zero. On the basis of the thermomechanical relation, the energy characteristic of the J-integral type, including stress vectors on the free surface in the vicinity of the crack-like defect, is determined in the form of three additive integral summands. The part of the energy characteristic on the end surface of the physical section and the summands on the shores contiguous to the end are isolated. We solved the loading problems of applying normal rupture and transverse shear to the physical section based on the solution in the finite element complex ANSYS for the physical section and the model of representation of the medium on the continuation of the physical section as a layer, which is homogeneous in the thickness distribution of the stress-strain state. We compared the energy response, when the linear parameter tends to zero value, and the values of the J-integral for the crack representation in the form of a mathematical section. The correspondence of the J-integral value for the mathematical section to the considered energy characteristic at a relatively small value of the linear parameter is obtained. At the same time, its part on the end surface, depending on the model under consideration, is more than sixty per cent. For the physical section using the layer model, the closeness of the investigated characteristic to the value of the J-integral for the mathematical section is shown at a significantly lower value of the elastic modulus of the layer material with respect to the basic medium. At the same time, the influence of non-face summands of the energy characteristic is found to decrease.
PNRPU Mechanics Bulletin. 2025;(2):30-38
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Effects of Terms of High Order in Synthesized Polynomial Series Approximation for Fields Associated with the Crack Tip in Anisotropic Media. Part 2. Accuracy Estimates of Asymptotic Solutions
Abstract
The second part of the article discusses the accuracy of generalized asymptotic series representing the stress and displacement fields associated with the tip of an acute crack in orthotropic materials in the formulation of the plane problem of the theory of anisotropic elasticity. We compare the exact analytical solution to the problem of stretching an infinite anisotropic plane with an inclined central crack obtained using methods of the complex variable theory and an approximate solution found using the power series expansion method. For the first time, the fields of absolute error allowed for the truncation of the asymptotic series on a different number of terms for materials with a cubic symmetry of elastic properties are obtained. The analysis of the fields of the decadic logarithm of absolute errors showed that near the crack tip of all types of combined (mixed) deformation there are geometric points – loci of accuracy, in which the approximate solution almost coincides with the exact solution, which can be used in the interpretation of experimental and computational data obtained in order to determine the coefficients of asymptotic series for stresses and displacements near the tip of a crack or notch. To quantify the errors allowed when truncating the asymptotic series on the th term, a relatively accurate analytical solution obtained on the basis of the theory of the function of a complex variable the -norm is introduced into scrutiny, which gives an opportunity to choose and specify the number of terms of the series necessary to achieve the required accuracy when presenting the asymptotic ansatz with a multi-coefficient asymptotic series, for a wide range of slope angle values of the crack to the vertical axis (the axis of action of the applied tensile load) and the angle, setting the location of the anisotropy axes of the elastic properties of the material.
PNRPU Mechanics Bulletin. 2025;(2):39-56
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Gapped Momentum States and Dispersion Analysis of Mechanical Behavior of Viscoelastic Media
Abstract
The occurrence of gapped momentum states at different scales, characterized by wavenumber intervals with zero frequency values in the dispersion relation, determines qualitative changes in the momentum transfer mechanism during the interaction of collective modes in non-equilibrium critical systems. To describe the formation of gaps in dispersion curves one needs specialized forms of the dispersion relation. The investigation of dispersion relations with the gap in momentum space can facilitate the establishment of universal viscoelastic properties in condensed matter under specific conditions, where fluids exhibit shear elasticity and solids demonstrate flow behavior. The paper focuses on identification of gapped momentum states in the analysis of dispersion relations obtained using viscoelastic models, specifically the Kelvin-Voigt, Maxwell, standard linear solid, and fractional derivative Kelvin-Voigt models. To derive wave equations corresponding to the presented models, a modification of the wave equation for non-decayed transverse waves in solids was performed to account for viscosity and dissipation. Using the plane wave hypothesis, the general form of the dispersion equations was determined for each model, and analytical (numerical) solutions were obtained. Criteria for a qualitative change in the form of the dispersion relations accompanied by the appearance of a gap in momentum space (k-space) have been formulated. Frequency-wavenumber dispersion curves were constructed for various relaxation and retardation times, considering classical viscoelastic models. The phenomenological significance of fractional models for describing the mechanical behavior of polymeric, composite, and biological systems with a broad spectrum of relaxation mechanisms is highlighted. A numerical solution for the fractional derivative Kelvin-Voigt model was obtained for various values of the fractional derivative order. It is shown that the dispersion equations of the fractional derivative Kelvin-Voigt model and the standard linear solid model transform into the dispersion relations of the Kelvin-Voigt and Maxwell models, respectively, under specific conditions, which indicate the adequacy of the derived relations.
PNRPU Mechanics Bulletin. 2025;(2):57-69
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The Mathematical Model of a Built-In Fiber Optic TFBGs Sensor with Tilted Bragg Gratings for Diagnosing a Complex Deformed State in Polymer Composite Structures
Abstract
The paper presents mathematical models of functioning and numerical values of information transfer coefficients for new built-in fibre-optic TFBGs-sensors (Tilted Fiber Bragg Gratings) with tilted Bragg gratings to diagnose a complex stress-strain state inside loaded polymer composite structures. The fiber optic TFBGs sensors have the form of a continuous structured cable system, in which six unidirectional light guides with the built-in Bragg gratings are placed with a fixed mutual hexagonal arrangement in extended continuous cylindrical polymer sensor housing. Different 3D orientations of reflecting surfaces for different light guides were defined through the coordinates of non-planar normals to these surfaces. Numerical modeling of deformation fields in the elements of the fiber optic TFBGs-sensor was carried out for the calculation area composite material/built-in sensor within the linear theory of elasticity. We present color diagrams of distributions for various components of the strain field along the middle cross-section of the calculation area with corresponding simple cases of its macrostrains. Also we give numerical values of strain tensor components averaged over the area of each light guide. Further, values of strain components averaged over the light guides are used to calculate axial strains along non-planar vectors - normals to reflecting surfaces of the tilted Bragg gratings. As a result, numerical values of the desired information transfer coefficients of the fibre-optic TFBGs sensor were found taking into account given orientations of reflecting surfaces of the tilted Bragg gratings of the sensor light guides. Thus, the task of diagnosing a complex deformed state inside a loaded polymer composite structure in a local neighborhood of a built-in fiber optic TFBGs sensor is reduced to solving a system of linear algebraic equations regarding the desired six independent components of the macrodeformation tensor of this neighborhood from the measured spectra of reflections of the optical fibers of the sensor.
PNRPU Mechanics Bulletin. 2025;(2):70-83
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GREEN’S FUNCTION METHOD IN THE INVESTIGATION OF DYNAMIC STABILITY OF A FLUID-CONVEYING PIPE
Abstract
Fluid-conveying pipes represent a fundamental dynamic problem within the realm of fluid-structure interaction. They find extensive applications in various industries, including petroleum, nuclear engineering, aviation, aerospace, and nanostructures. This paper applies the Green’s function method to solve the stability problem of a fluid-conveying pipe, hinged at both ends and supported by intermediate linear-elastic supports. The objective is to examine the influence of the number and rigidity of these supports on the critical fluid velocity, which is the velocity at which the pipe loses stability. A numerical solution was performed for a straight pipe conveying fluid with specified geometric and physical characteristics, where the number and rigidity of the elastic supports were considered as parameters. The numerical analysis presented herein includes graphs illustrating the dependence of the critical fluid velocity on the number of elastic supports for varying support rigidities. These results reveal that the elastic supports affect both the vibrational characteristics and the critical velocity of the conveyed fluid. The solution results are compared with those obtained using one of the most widely employed methods for analyzing the dynamic stability of pipe systems (Transfer Matrix Method – TMM). A good agreement between the results is observed. The paper aims at presenting a method for obtaining the exact solution to the differential equation governing the lateral displacements of a pipe system. This paper discusses the authors' perceived pros and cons of the Green's function method in comparison to the most popular methods for the dynamic investigation of fluid-conveying pipes.
PNRPU Mechanics Bulletin. 2025;(2):84-90
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Experimental Study of the GFRP Perforation Stochasticity Near the Ballistic Limit
Abstract
The work aims at experimentally analyzing features of the impact interaction of a steel spherical projectile with glass fibre-reinforced plastic (GFRP) specimens with the thicknesses of 4 mm, 6 mm, and 7.3 mm at the velocities near the corresponding ballistic limits. The experimental study was carried out in two stages. At the first stage, ballistic curves, estimations of V50 and limit perforation and non-perforation velocities were obtained based on the results of the first series of impact tests using the Lambert-Jonas approximation. At the second stage, a series of tests was carried out for GFRP specimens of each thickness, when the initial projectile velocity was selected so that it fell into the zone of mixed results to obtain the perforation frequency curves. Based on the results of more than 300 experiments, it was established that the perforation frequency curves for GFRP specimens with the thicknesses of 6 mm and 7.3 mm can be obtained using the normal distribution law. Also it was found that the ratio of the width of the zone of mixed results to the corresponding V50 estimation for two thicknesses of specimens was about 4%, which is significantly less than the scatter of the strength characteristics of GFRP specimens obtained during the static tests.
PNRPU Mechanics Bulletin. 2025;(2):91-99
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Equilibrium State of a Straight-Line Internal Crack Near the Corner Point of an Elastic Region Reinforced along the Contour
Abstract
The problem of plane deformation of an elastic isotropic wedge-shaped region weakened by an internal defect in the form of a crack is considered. The boundaries of the investigated region are supported by a thin flexible coating. The boundary conditions determining the influence of the coating are modeled by special relations based on the asymptotic analysis of the problem solution for the strip. The adequacy of the mathematical model of the coating is verified by a series of numerical experiments in the authors' previous studies. The problem solution was carried out by the method of integral transformations. The Mellin transformation allowed reducing the problem to a system of ordinary differential equations of the second order. Its general solution is constructed. To determine the unknown coefficients in the found general solution, a system of linear algebraic equations is obtained. The conjugation condition on the crack location line allowed us to obtain a singular integral equation with a Cauchy kernel, which is characteristic of the problems of stress concentration at the crack ends in the plane formulation. Its numerical solution was carried out, providing the possibility of calculating the values of the normal stress intensity factors at the crack ends. The concept of the indicator of the coating restraining effect is introduced, its behavior for coatings with different parameters is studied, the influence of physical and mechanical characteristics of the coating. We investigated its thickness and stiffness, as well as the crack size, its location relative to the corner point of the area, the angle of opening of the wedge-shaped area on the crack opening. The assessment of how the coatings influence the stress-strain state of sections of products weakened by stress concentration zones contributes to the development of new design approaches to the structural component of products, allowing us to enhance the strength and wear resistance of machine parts and elements of building structures.
PNRPU Mechanics Bulletin. 2025;(2):100-110
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