## No 3 (2023)

**Year:**2023**Articles:**12**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/341**DOI:**https://doi.org/10.15593/perm.mech/2023.3

MACHINE LEARNING METHOD IN ELASTIC PLATE TOPOLOGY OPTIMIZATION PROBLEMS

#### Abstract

In this article, as an alternative to traditional methods of topology optimization of deformable solids, an approach to topology optimization based on machine learning methods is proposed. The proposed approach significantly reduces the time spent on obtaining the optimal solution and allows to avoid the use of time-consuming finite element calculations at the stage of obtaining the optimal solution. All of time-consuming calculations are performed at the network training stage. A review of the world literature on the application of machine learning methods in the problem of topology optimization of an elastic deformable solid is given. Further, as an example, a square elastic plate is considered, fixed on one side and loaded with a force on the other. For a given plate, a series of topology optimization problems (maximizing stiffness under volume constraints) are solved using the method of moving asymptotes to build a training data set. A comparative study of a neural network with one custom nonlinear layer, based on the nature of the optimal topology, and a three-layer neural network built using the standard PyTorch library features is carried out. The input parameter of the neural network is the force application point, the output parameter is the optimal topology of the plate. The network is trained using the backpropagation method. The quadratic norm of the predicted design variable values deviations compared to the true values form last optimization is minimized during neural network learning. The considered example shows the possibility of applying the approach to other problems that differ in geometry, boundary conditions, etc. The results and unsolved problems of the method are also discussed.

**PNRPU Mechanics Bulletin**. 2023;(3):5-14

MODELING OF PRESTRESSED PLATES WITH MATERIAL INHOMOGENEITY, PERFORATIONS AND INCLUSIONS

#### Abstract

In the present article, we propose the model of in-plane oscillations of inhomogeneous prestressed plates, both solid ones and those containing a set of holes and inclusions made of different materials. We treat the plates’ mechanical properties and the prestress tensor components in the considered 2D problem statement as functions of two coordinates. In order to formulate the boundary value problems of steady-state in-plane vibrations of plates, we employ the general linearized formulation for an elastic body under conditions of an initial stress-strain state. The developed vibration model makes it possible to specify an arbitrary type of prestress state in the plate in the form of analytical dependences, as well as numerically, by solving the corresponding static problem, in which prestresses arise as a result of applying some initial load. To implement the finite element (FE) approach to solving the problems, we formulated the weak problem statement by projecting the original governing equations on the field of test displacements satisfying the essential boundary conditions. To increase the accuracy of calculations for plates with holes and inclusions, the local refinement of FE meshes are used. The proposed approach to calculating plate vibrations is implemented as a software package via FreeFem++. A method for assessing the effect of prestress on dynamic plates’ characteristics under various types of loads is described; a comprehensive analysis is carried out to identify the probing modes, frequency ranges and response pickup areas, most sensitive to the prestress changes, for each of the plates. We systematize and generalize the results obtained during the analysis, give a few practical recommendations on the choice of probing modes for each type of the plates considered, allowing to perform the most efficient schemes for identifying the prestress components.

**PNRPU Mechanics Bulletin**. 2023;(3):15-29

ON THE INFLUENCE OF THE MECHANICAL CHARACTERISTICS OF A THIN ADHESION LAYER ON THE COMPOSITE STRENGTH. PART 2. ELASTIC-plastic DEFORMATION

#### Abstract

The limiting state of a thin adhesive layer at its normal rupture in the plane stressed and plane deformed states is considered. The behavior of the layer is described by an ideal elastoplastic model with the Tresca – Saint-Venant yield condition. The deformation of the layer is carried out by means of consoles operating within the framework of the relations of the Mindlin – Reissner plate theory. In the area of plastic flow of the adhesive layer, the condition of complete plasticity is assumed. The presence of several diagonal components of the stress tensor in the layer, which are related to the cantilever stresses by the equilibrium conditions, is taken into account. On the basis of the problem posed, analytical representations are obtained for the displacement field of cantilevers in the region of conjugation with the layer. On the basis of experimental data on the destruction of adhesive layers with given mechanical properties, the critical values of J-integrals are found depending on the type of the considered plane problem for the layer. It is shown that a decrease in the thickness of the adhesive layer leads to an unlimited growth of deformations in its end zone, however, the values of the J-integrals stabilize. In this case, in the case of a plane deformed state, the length of the plastic zone decreases and the main contribution to the J-integral is made by the energy component. In a plane stressed state, the length of the plastic zone increases, and the dissipative component of the J-integral exceeds the energy one. A significant difference in the critical values of the J-integral is obtained, which is a consequence of the developed plasticity zone in the plane stress state. For plane deformation in extremely thin adhesive layers, taking into account their elastoplastic properties is insignificant and the values of the J-integral can be found within the framework of a linearly elastic model of adhesive behavior or in a model with rigid coupling of mating bodies, which excludes the mechanical properties of the adhesive from consideration.

**PNRPU Mechanics Bulletin**. 2023;(3):30-42

BASE NEUTRAL SURFASE SELECTION FOR OPTIMAL DESIGN OF STRUCTURALLY-ANISOTROPIC AIRCRAFT PANELS MADE FROM COMPOSITE MATERIALS WITH REFINED BUCKLING THEORY RESTRICTIONS

#### Abstract

The aim of this study is the approach to the optimal design of structurally-anisotropic aircraft bearing surface panels with the restrictions according to the refined buckling theory for the optimal size-weight design implementation. The panels are subjected to the distributed constant compressive loading applied to the edges in the skin plane in the longitudinal direction. The panel contour boundary conditions are assumed to be the particular case with conformable boundary restrictions for the plane problem and problem of bending. The optimal design problem statement and analytical solution are formulated to determine the geometry parameters of a flat rectangular multilayer panel made from composite materials with the eccentric longitudinal and lateral stiffening set being of minimal mass. The equal-buckling condition is the optimal design base. The general bending mode of buckling and multi-wave torsion mode of buckling must have the same occurrence probability while the buckling margin tends to one. The optimal design problem is reduced to the mathematic conditional extremum investigation of the goal weight function with multiple variables using the analytical and numerical methods. New mathematic model relations for the buckling problem investigation of structurally-anisotropic composite panels are presented. The primary scientific novelty of this research is the further development of the thin-walled elastic rib theory related to the contact problem for the skin and rib with an improved rib model. The analytical solution is reduced to determine the displacements of a base neutral surface that may be select arbitrarily. The schematization of the panel as structurally-anisotropic one has been proposed as a design model when the critical forces of total bending mode of buckling are determined. For a multi-wave torsion buckling study, one should use the generalized function techniques for the discrete stringer stiffness. The solution of the strained surface differential equation of an eighth order is designed by a trigonometric series in the closed form. The results of the optimal design with the refined buckling restrictions based on refined buckling analysis calculations offer opportunities to reduce and optimize the weight characteristics of aircraft elements.

**PNRPU Mechanics Bulletin**. 2023;(3):43-52

DEVELOPMENT OF A METHOD FOR DESTROYING THE ROTOR BLADES OF A HIGH-PRESSURE COMPRESSOR AT A GIVEN FREQUENTATION OF ROTATION

#### Abstract

The design of gas turbine engines (GTE) is inextricably linked with a large amount of work on numerical modeling. With the help of numerical modeling, it is possible to predict the behavior of the part when the engine is operating in various modes. In addition, when modeling, it is possible to predict modes, design load, conditions for engineering tests. This paper presents an approach to modeling an engineering test for the breakage of the rotor blade (RB) of a highpressure compressor (HPC) to confirm the impenetrability of the housing. By calculation, the amount of cutting of the RB HPC for breakage at a given rotor speed is determined. The approach is to implement a combination of two factors: 1) the calculation of the blade strength using the deformation criterion prevented the blade breakage at a lower rotational speed, with the necessary maintenance of operation in the design mode close to the ultimate strength; 2) computational and experimental work to determine the resonance allowed to increase the variable component of stresses for blade breakage at a given frequentation of rotation. It was also possible to model the direction of crack growth numerically.

**PNRPU Mechanics Bulletin**. 2023;(3):53-62

NON-CONTACT DEFORMATION WITH REVERSIBLE SURFACE PLASTIC DEFORMATION

#### Abstract

The article considers non-contact deformation during surface plastic deformation based on the reverse rotation of the deforming tool. Using software for 3D design (Solid work 2019) and computational modeling (Ansys workbench 19.2), calculations were made to determine the size of an elastic-plastic wave depending on the main parameters of reverse surface plastic deformation (SPD) and the physical and mechanical properties of the material. The stress state in elastoplastic waves generated in the direction of feed (A1) and in the direction of the main movement (A2) is also established. It has been established that the linear dimensions of elastoplastic waves reach a maximum at a preload value of t = 0.4 mm. The main parameters of the reversible PPD, which characterize the kinematics of the working tool (reverse speed of the working tool, the speed of the workpiece, the initial angle of the working tool and the amplitude of the angle of the reverse rotation of the working tool) have a significant impact on the change in the size of the elastoplastic wave in the direction of the longitudinal feed and slightly affect the change in the size of the elastic-plastic wave in the direction of the main movement. The change in the stress state of the surface layer is shown depending on the physical and mechanical properties of the material: large sizes of the elastic-plastic wave during elastic-plastic deformation are formed in the material with a reduced yield strength and elastic modulus. It was also found that the larger the size of the elastic-plastic wave, the higher the maximum tensile stresses at their vertices. The resulting stress state of the wave allows us to conclude that maximum tensile stresses are formed at their tops, the value of which reaches 202–271 MPa (2,4–3,2 times less than the ultimate strength of the material), which practically does not cause strength failure hardened surfaces.

**PNRPU Mechanics Bulletin**. 2023;(3):63-74

UNSTEADY THERMOELASTIC DIFFUSION VIBRATIONS OF THE BERNOULLI – EULER BEAM UNDER THE ACTION OF A DISTRIBUTED TRANSVERSE LOAD

#### Abstract

The paper deals with the problem of unsteady vibrations of the Bernoulli – Euler beam, taking into account relaxation of temperature and diffusion processes. The original mathematical model includes a system of equations of non-stationary bending oscilla-tions of the beam taking into account heat and mass transfer, which is obtained from the general model of thermoelastic diffusion for continuum using variational D'Alembert prin-ciple. Based on the obtained equations, the statement of the initial-boundary problem concerning bending of the hinged orthotropic beam, which is under the action of ther-mo-elastic diffusion perturbations distributed on the surface, is formulated. Solutions of the problem of unsteady thermoelastic diffusion vibrations of the beam are sought in integral form. The kernels of integral representations are Green’s functions, for finding of which decompositions into trigonometric Fourier series and Laplace trans-formation over time are used. Laplace transformants of Green's functions are represented through the rational functions of Laplace transformation parameter. Transition into the space of the originals is carried out analytically using deductions and tables of opera-tional calculus. Analytical expressions for Green functions of the problem under consid-eration are obtained. On the example of a simple supported three-component beam made of an alloy of zinc, copper, and aluminum, which is under the influence of mechanical load distributed along the length, the interaction of mechanical, temperature and diffusion fields is inves-tigated. The influence of relaxation effects on the kinetics of heat and mass transfer is analyzed. The solution is presented in analytical form and in the form of graphs of the dependence of the desired fields of movement, temperature increments, and increments of concentration of medium components on time and coordinates. In conclusion, the main conclusions concerning influence of field connectivity and relaxation effects on the stress-strain state and heat and mass transfer in the bendable beam are given.

**PNRPU Mechanics Bulletin**. 2023;(3):75-85

MODELING OF THE INITIATION CONDITIONS OF CRACKS IN A PIPE UNDER PRESSURE OF A HYDROGENOUS MEDIUM

#### Abstract

An actual problem of modern engineering about destruction of a pipeline as a result of influence of hydrogen contained in the transported products is considered. Hydrogen changes the mechanical properties of metal, affecting the stress-strain state of the pipe, which, in turn, affects the distribution of hydrogen in the pipe. The hypotheses about the nature of this relationship accepted in the paper allowed to explain the reason of circum-ferential crack formation in the pipe under the influence of hydrogen. An algorithm for iterative calculation of the stress-strain state of a tube containing a hydrogen-containing mixture inside the tube has been developed. The coupled problem of the theory of elasticity and diffusion in the planar axisymmetric formulation is solved. Since the interaction process of hydrogen and metal is very slow, it is considered in sequential static formulations. First, the Lame-type problem for a tube with the modulus of elasticity depending on the radial coordinate is solved. By the finite difference method the stress and strain fields of the pressurized pipe are found. Further, the concentration of free hydrogen in the pipe caused by its content on the pipe surfaces and its stress state is determined. The accepted hypothesis about the condition of hydrogen atoms embedding into the crystal lattice of metal allows to estimate the influence of hydrogen on mechanical properties of the pipe material at the next stage of the calculation. The calculation of stress and concentration fields is repeated again with already modified mechanical properties. The iteration process is stopped when the stresses in the tube reach critical values according to Mises criterion or when the mechanical properties of the pipe mate-rial stop changing. The calculations show that at some combination of hydrogen concentration and pressure on the pipe wall, zones of plastic deformation arise in the pipe, which can lead to delamination of the material in the circumferential direction. This result is consistent with known experimental data.

**PNRPU Mechanics Bulletin**. 2023;(3):86–96

NATURAL VIBRATIONS AND HYDROELASTIC STABILITY OF A PLATE WITH A PIEZOELECTRIC ELEMENT CONNECTED TO AN EXTERNAL RL-CIRCUIT

#### Abstract

The possibility of passive damping of harmonic vibrations and controlling the stabil-ity boundary of the plate interacting with the flowing fluid is investigated. The key idea behind the applied vibration control method is to connect the piezoelectric element lo-cated on the surface of the structure to an external shunt circuit. The selection of parame-ters for such circuit, providing the highest rate of vibration damping or the maximum change in the critical velocity of the fluid flow, is performed by solving a series of eigen-value problems. Two mathematical formulations are considered. The first formulation is based on the three-dimensional equations of the linear theory of piezoelasticity, and the second one is a simplification of these equations with the aim of using them in conjunc-tion with the theory of thin plates. The dynamics of an ideal fluid in both cases is de-scribed by a wave equation formulated for the perturbation velocity potential. Together with the impermeability condition and the boundary conditions it is transformed to the weak form. The hydrodynamic pressure is calculated by the linearized Bernoulli formula. The developed finite-element algorithms are verified and their computational efficiency is compared. A change in the complex eigenvalues of the electromechanical system is ana-lyzed depending on the resistance and inductance of the electric circuit connected to the piezoelectric element. The values of these parameters providing the best damping of resonant vibrations of a rectangular plate interacting with the flowing liquid have been selected based on the solution of the optimization problem. The numerical studies have shown that the selected values lead to a smaller change in the frequency spectrum of the original system and provide a higher rate of vibration damping as compared to the val-ues found by the known analytical expressions. The numerical investigation has been performed for two variants of boundary conditions set at the edges of the structure. It is demonstrated that the use of a passive electrical circuit cannot affect the loss of stability by divergence, but is able to change the critical flutter velocity by a few percent

**PNRPU Mechanics Bulletin**. 2023;(3):97–113

NUMERICAL AND EXPERIMENTAL STUDY ON CFRP STRUCTURE OPTIMIZATION FOR COEFFICIENT OF THERMAL EXPANSION

#### Abstract

This paper explores the optimization macrostructure to reach a stable low coefficient of thermal expansion αx of a composite with carbon fibers. To limit the search area, a necessary condition for the existence of αx local minima is proposed, expressed in terms of the radii of hyperspheres in the design space of the angular orientation of the layers transformed by the PСA algorithm. The analysis of the structure variants characterized by low αx shows different sustainability to lamina properties variability. Multi-criteria optimization was carried out. The objective functions are expectation E(αx) and variance Var(αx). The analysis of Pareto fronts and probability density functions make it possible to estimate the reachability of the calculated αx under given conditions of lamina properties variability. The reduction variance opportunity of αx distribution by modifying the polymer matrix with MWCNTs under conditions of reinforcing fibers disorientation and lamina properties variability is investigated. The microstructure modification of the polymer composite material allows to reduce the Var(αx) by 91.61 % with a volume ratio of MWCNTs up to 1 %. Requirement thermomechanical properties are reached by determining the orientation of anisotropic layers. Based on the obtained optimal structures, specimens of CFRP with 0, 1 and 2 vol.% MWCNTs were made. Scanning electron microscopy using FE–SEM Hitachi S–5500 was performed to check the uniformity of distribution and compatibility of the epoxy matrix and MWCNTs. The measurement of αx is determined using a TAInstrumentsQ400 thermomechanical analyzer. Measured αx of specimens is in the range from 6.2·10-8 to 1.98·10-7 1/K. The structure optimization approach proposed in this paper makes it possible to obtain a set of solutions with a consistently low αx in the range up to 1·10-7 1/K. The transformation of the design space of the layers’ orientation angles and the limitation of the search area allowed to reduce the range of solutions under consideration by 83.9 %.

**PNRPU Mechanics Bulletin**. 2023;(3):114–123

FINITE-STRAIN ELASTIC-PLASTIC TORSION: ANALYTICAL AND FEM MODELING FOR NONMONOTONICALLY HARDENING POLYMERS

#### Abstract

Polymeric materials, depending on the structure and chemical composition, exhibit various types of isotropic strain hardening. In particular, in the range of plastic deformation on the "true strain – true stress" curve, there may be a descending section of softening caused by the weakening of intermolecular bonds. This softening region is further replaced by a power-law hardening. The laws of deformation of materials can be established from simple experiments, one of which is often torsion. For torsion of thin-walled cylindrical specimens, the stress-strain state is practically uniform; therefore, such experiments are easy to interpret. However, stability problems arise at large deformations of thin-walled specimens. For solid cylindrical specimens, the stress state is inhomogeneous; interpretation of such experiments is possible on the basis of FEM modeling or using exact or approximate analytical solutions of the corresponding initial-boundary value problems of mechanics. In the present study, an exact analytical solution of the elastic-plastic problem of torsion of a cylindrical sample is presented, which is valid for an arbitrary law of isotropic hardening. The multiplicative decomposition is utilized as the kinematics of elastic-plastic deformation. The non-linear elastic properties of the material are described by the Mooney – Rivlin model. In the plasticity condition, the Tresca equivalent stress is used, which makes it possible to obtain a closed solution. The integral characteristics of the process (torque and axial force that represents a second order effect) are calculated. The analytical results are compared with the results of numerical simulations in MSC.Marc, as well as with the available experimental data. The analytical solution for torque corresponds very closely to the numerical solution obtained by the finite element method. Also, the curves of axial force coincide satisfactorily. At moderate strains, the analytical solution accurately describes the experimental results.

**PNRPU Mechanics Bulletin**. 2023;(3):124–136

REGULARITIES OF FATIGUE FAILURE TYPICAL COMPOSITE FLANGE

#### Abstract

One of the most important tasks in the development of aircraft structures from polymer composite materials is to ensure fatigue strength. It is known that the behaviour under cyclic loading of parts and standard samples is not the same. The problem of transferring the results of standard tests to a full-scale product is especially acute due to the variety of reinforcement schemes, the influence of technological and design factors. This forces us to test full-scale products or structurally similar elements. This work is devoted to the study of the patterns of fatigue failure of a typical element of shell aircraft structures – an L-shaped flange made of laminated carbon fibre. A method of fatigue testing of the critical zone of the flange has been developed. It reproduces the operational conditions of flange loading. The fatigue loading of the sample is carried out by the inertia forces of the load attached to it. The tests were carried out with resonant harmonic vibrations on a vibration stand in a bending shape. The experimental setup ensures constant maintenance of the resonant mode. The experimental setup ensures continuous monitoring of deformation, the resonant frequency of vibrations of the sample and the temperature field on its surface. The characteristic patterns of the development of fatigue damage of the flange are established. The main mechanism of destruction is the appearance and development of delaminations in the flange and in the area of its connection with the shell. Fatigue failure is accompanied by a drop in the resonant frequency of vibrations of the sample, due to a decrease in its rigidity. The typical dependences of the resonant frequency drop on the relative fatigue time reflect an abrupt change in the stiffness of the sample. The established regularities of the fatigue damage accumulation process are confirmed by the analysis of the thermal state of the sample during testing, which changes as a result of its self-heating during cyclic loading. The experimental data can be used in the development of the design of flanges made of CFRP and in the development of models for predicting their fatigue life.

**PNRPU Mechanics Bulletin**. 2023;(3):137–145