## No 2 (2024)

**Year:**2024**Articles:**10**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/411**DOI:**https://doi.org/10.15593/perm.mech/2024.2

ABOUT ONE METHOD OF STUDYING THE COEFFICIENT OF THERMAL EXPANSION OF POLYMERS

#### Abstract

The problem of studying the deformation response of film samples made of UV-curable polymers to temperature changes is considered. It is difficult to obtain massive samples of these polymers due to the peculiarities of their polymerization. This paper suggests a methodology for tests that allows us to determine the temperature dependence of the coefficient of linear thermal expansion (CTE) in a wide range, including the relaxation transition. As measuring equipment was used a dynamic mechanical analyzer TA Instruments Q800 DMA with liquid nitrogen cooling system GCA, which allows varying temperature and its rate of change in wide ranges, controlling and measuring forces and displacements with high accuracy. The proposed approaches are applicable to any film samples and provide an opportunity to establish functional dependences of CTE not only on temperature, but also on its rate of change. At the same time, in contrast to traditional methods, the described procedures allow obtaining correct data under the conditions of relaxation transitions taking into account the influence of the temperature change rate on these processes. Considerable attention is paid to calibrate the measuring equipment, in particular, the measurement and compensation of the temperature deformation of the tooling. The measurements results are compared with known literature sources, manufacturers data, the results of measurements of samples with known characteristics and with the results obtained on a horizontal dilatometer. It is shown that the proposed method of measurements is correct and has a number of advantages over traditional methods of research when working with film samples or in cases where it is necessary to take the effect of the rate of temperature change on CTE into account.

**PNRPU Mechanics Bulletin**. 2024;(2):5-15

INFLUENCE OF ADDITIVE MANUFACTURING TECHNOLOGY OF TOPOLOGICALLY OPTIMIZED PRODUCTS FROM PHOTOPOLYMER RESINS ON THE ANISOTROPY OF THEIR MECHANICAL PROPERTIES

#### Abstract

This article discusses the problem of ensuring strength and minimum mass in the design of products manufactured using additive technologies. The authors investigated the possibility of using topological optimization in computer-aided design systems to create an optimized model with the necessary strength at a minimum weight. As part of the work, the topological optimization of the bracket, the production of its samples by additive manufacturing methods and strength tests were carried out. The optimal values were found via finite element analysis using the SolidWorks and ANSYS software. The calculation results show that the optimized model retains about 20% of the original mass and has the necessary mechanical characteristics. In particular, the excess safety margin is reduced by 2.5 times, which is acceptable for this bracket. The subsequent verification of the models is carried out through destruction tests of the products manufactured using additive technologies, i.e. the method of filament deposition and stereolithography. To account for the anisotropy of the material, a series of samples oriented at different angles to the direction of construction was made. The tests were carried out on a test bench for simultaneous biaxial stretching, which corresponds to the design loads on the bracket. An increase in the tensile load on the sample was carried out before its destruction. In the course of the work, the presence of anisotropy of mechanical properties was revealed, and optimization results in various software packages were studied. The results of strength tests allow us to draw two conclusions. Firstly, due to the anisotropy of the material, the strength properties significantly depend on the orientation of the bracket during additive manufacturing. Secondly, the bracket, optimized by means of the SolidWorks software package, generally showed the best strength properties for various orientations during manufacturing. Also, which is quite expected, the samples obtained by stereolithography showed less anisotropy than the samples obtained by the method of filament deposition. In conclusion, it is noted that the use of additive technologies to create optimized forms requires considering printing technologies and anisotropy of properties, as well as the choice of appropriate software.

**PNRPU Mechanics Bulletin**. 2024;(2):16-23

COMPUTATIONAL-EXPERIMENTAL METHOD FOR STRESS-STRAIN CURVE CONSTRUCTING UNDER CONDITIONS OF INHOMOGENEOUS STRESS FIELDS

#### Abstract

In order to ensure reliability of structures, it is necessary to study equilibrium damage accumulation which initiates softening zones in solids. It is considered appropriate to use postcritical deformation models when conducting strength analysis of structures. However, obtaining complete stress-strain curves in standard tests is difficult due to the strain localization in the form of a neck. At the same time, the use of true stresses taking into account changes in a specimen cross section is incorrect due to the implementation of a complex stress state. In this regard, it is necessary to develop computational and experimental methods for constructing material stress-strain curves under conditions of inhomogeneous stress fields. In this case it seems reasonable to use data on strain fields on the body’s surface, which can be obtained, for example, using non-contact optical video systems. The paper presents the computational-experimental method for constructing a stress-strain curve under conditions of inhomogeneous stress fields. An elastoplastic model of an isotropic material is considered. The initial data of the method are two elastic constants, the yield stress value, the load diagram of a body with a stress concentrator, and the maximum values of strain intensity corresponding to various states. The developed method was tested by a numerical simulation of deforming an hourglass specimen and a plate with a stress concentrator. Stress-strain curves with and without a yield plateau are considered. The results demonstrate a high agreement between the initially specified and reconstructed stress-strain curves. A conclusion is made about the rationality of using the developed method when constructing stress-strain diagrams and necessity for its modernization to describe the postcritical deformation stage.

**PNRPU Mechanics Bulletin**. 2024;(2):24-32

Large Deflections, Loss of Stability and Over-Critical Behavior of Sloping Panels and Arches of Variable Thickness on an Elastic Base

#### Abstract

The problem of large deflections of a flat arch in its plane (or an infinitely long panel) loaded with transverse loads is considered. A variational approach has been applied to solve it. The resolving nonlinear equations are reduced to finding the deflection and longitudinal force, which is considered constant along the length of the arch due to its flatness. An approximate solution method is proposed by decomposing the displacements into a Fourier series. The peculiarity of the approach used is that in order to pass the limit points, it is not necessary to use special algorithms such as the continuation method by parameter. It also allows you to trace the process of supercritical deformation of the arch. The proposed approach allows us to consider problems for arches of variable thicknesses, located on elastic supports, on an elastic base with a variable bed coefficient and various loads. Therefore, it is also convenient in problems of finding, for example, the optimal thickness distribution under restrictions on critical loads, on rigidity stiffness and on maximum compression or tension stresses. The results of numerical calculations are presented. The convergence is studied of the solution depending on the number of terms of the series, into which the desired deflection decomposes. A good agreement with the known analytical results is obtained earlier by solving the equilibrium equations of the arch element and panel in the case of simple types of loading. At the same time, even in more complex cases of loading and supercritical bending of an arch without an elastic base, the results are obtained when holding three, four and five members of the Fourier series, the maximum deflections and critical loads differed by no more than 2.5%. On the basis of numerical experiments, the features of the arch behavior caused by the rearrangement of geometry during loading are revealed. The processes of stability loss of the arch and its supercritical behavior are investigated. The effect of symmetric deformation in the case of kinematic loading is found, namely, it is revealed that at a certain value of the concentrated force, an infinite number of equilibrium forms are possible. Arches on an elastic base, as well as with variable thickness are considered (the results are presented in the form of load–displacement diagrams and in the form of pictures of deformed arch shapes). An interesting effect has been revealed based on numerical experiments. It turned out that the increase in thickness when moving away from the supports does not change the nature of the load–displacement diagram, i.e., with some load, a slap occurs. On the contrary, a decrease in thickness when moving away from the supports leads to a flattening of the diagram, and after a certain thickness value in the center, its further decrease leads to the fact that the arch does not occur.

**PNRPU Mechanics Bulletin**. 2024;(2):33-41

DEFORMATION OF A THIN FILM AFTER CONTACT LOSS WITH A CYLINDRICAL BASE, WITH ACCOUNTING FOR THE INFLUENCE OF ITS CURVATURE, COMPLIANCE, AND ACTION OF TRANSVERSE FORCES; DELAMINATION LOCATED IN THE CIRCUMFERENTIAL DIRECTION

#### Abstract

Delamination of a coating from a cylindrical substrate under compressive stresses is considered. The solution of the problem was obtained within the theory of cylindrical shells. The Mushtari-Donnell-Vlasov equations were used as the equilibrium equations, and the conditions of the generalized elastic clamped type were used as the boundary conditions. Expressions are obtained for the coating displacement components and the energy release rate during delamination along the rectilinear and curvilinear delamination boundaries. The profiles of the exfoliated section of the coating were obtained. The role of the substrate compliance and its curvature on the values of the energy release rate and the angle of rotation at the clamping point, as well as in the change in the shape of the coating profile, is revealed. It is shown that with an increase in the compliance of the substrate, as well as the value of (positive) curvature, the value of the energy release rate and the value of the normal component of the displacement of the coating increase significantly. It has been found that at sufficiently large values of the substrate curvature, a corrugation effect occurs in the profile of the delaminated section of the coating, which results in the local extrema in the graph of the dependence of the energy release rate on the delamination length. This effect was not observed in the delamination located in the longitudinal direction [1]. It is also shown that for sufficiently compliant substrates there is a certain critical delamination width, for which it becomes energetically more favorable for delamination to develop in the circumferential direction than in the axial one. As the positive curvature and compliance of the substrate increase, the value of the critical width decreases. An alternative criterion for the critical delamination width is also considered, which consists in studying the value of the angle of rotation at the embedding point, which corresponds to the overlap of the coating and substrate surfaces at an angle value of zero.

**PNRPU Mechanics Bulletin**. 2024;(2):42-52

MODELING OF PHOTOPOLYMER CURING

#### Abstract

The paper considers the process of curing a photopolymer material. The effect of light on a photopolymer material triggers a reaction that leads to the conversion of polymer chains, which leads to the release of heat and an increase of temperature, solidification or an increase of stiffness of the material, and is also accompanied by a volumetric shrinkage. Such processes cause the distortion of the initial shape of the material. With nonuniform irradiation of the material, the processes are initiated with different intensities and a certain delay causing residual stresses. Stereolithography technology has become widespread in industry, in which the material is irradiated in certain areas, the so-called masks, after which the uncured material is removed. Modern photopolymer 3d printers are focused on this effect, which cures the material in layers with various masks. In 3D printing, the curing of the upper layer is accompanied by a higher shrinkage relative to the lower one, which by this time has a higher degree of curing leading to residual stresses. Thus, each new layer in a produced part initiates a gradual accumulation of residual stresses. As a result, there is a distortion of the originally planned shape of the product and a loss of strength characteristics. Residual stresses realized during the printing process can exceed the strength of the material, which often leads to a rapid increase in cracks and damage of the printed structure. This study proposes a model of a photopolymer material and an algorithm for modeling curing. It considers the process of stereolithography based on the action of a movable laser. A comparison with the experiment is given.

**PNRPU Mechanics Bulletin**. 2024;(2):53-58

THE LIMITING STATE OF A STEEL STRUCTURE UNDER EXTREME THERMOMECHANICAL LOADINGS

#### Abstract

Using numerical methods, the problem of determining the strength and limiting state of a steel shell structure under thermomechanical loading is solved. The operating stresses are determined by solving a physically nonlinear boundary value problem for a shell of revolution. The classical theory of shells, based on the Kirchhoff – Love hypotheses, and the method of integrating shell equations with discrete S.K. Godunov orthogonalization are used. By integrating a system of ordinary differential equations at each point of the shell, the meridional and circumferential stresses and the corresponding deformations are calculated. When taking into account the plastic deformation of the material, the boundary value problem becomes nonlinear. The relationship between stress and strain is linearized by the method of additional strains. A limiting state criterion for thin-walled structures is proposed. In the absence of the necessary parameters for the material of construction, interpolation and extrapolation of the experimental data based on neural networks is used. The method uses the example of a muffle, which is a revolution shell structure loaded with an internal excess pressure of a hydrogen-containing gas and a non-stationary thermal field. The muffle is designed for high-temperature annealing of the electrolytic steel, and is made of non-heat-resistant St3 steel, its mechanical properties have not been sufficiently studied at temperatures above 500 °C. However, the operating temperature of the muffle can reach more than 1000 °C. Under the influence of such a thermal load, noticeable residual deformations are formed in the muffle structure and the muffle may lose its load-bearing capacity. For thermomechanical loads, a maximum temperature of 1000 °C is determined at which the limit state occurs and the operation of the muffle is not permissible. A satisfactory agreement was obtained with the actual muffle temperature during operations of 1100 °C, at which the muffle loses its load-bearing capacity.

**PNRPU Mechanics Bulletin**. 2024;(2):59-68

MATHEMATICAL MODELING OF THE DYNAMICS OF THE AEROELASTIC "PIPELINE – PRESSURE SENSOR" SYSTEM

#### Abstract

This paper proposes mathematical models of mechanical systems "pipeline - pressure sensor" designed to control the pressure of the working medium in the combustion chambers of engines. In such systems, to mitigate the impact of vibration accelerations and high temperatures, the sensor is connected to the engine using a pipeline and is located at some distance from it. The movement of the working medium is described by linear models of fluid and gas mechanics. To describe the dynamics of an elastic sensitive element, linear models of the mechanics of a solid deformable body are used. Based on linear differential equations with partial derivatives, mathematical formulations of problems are proposed that correspond to three-dimensional models of pressure measurement systems in gas-liquid media for some pipeline cross-sectional shapes, namely, for a pipeline with a rectangular cross-section, with a section in the form of a sector and in the form of a ring. By introducing integral characteristics, the solution of problems is reduced to studying one-dimensional models. Equations have been obtained that make it possible to determine the pressure of the working medium in the combustion chamber at each moment of time by the value of deformation of the sensitive element of the sensor. Analytical and numerical-analytical methods for solving the corresponding initial-boundary value problems for systems of differential equations are proposed. In the analytical approach, the solution of the problem is reduced to solving a differential equation with a deviating argument. The numerical-analytical study of the problem is based on the application of the Galerkin method. Also, a numerical experiment was carried out and examples of calculating the deformation of the sensitive element of the sensor in the case of rigid fastening when specifying specific values of the mechanical parameters of the system are presented, also during setting the law of change in the excess pressure of the working medium in the engine.

**PNRPU Mechanics Bulletin**. 2024;(2):

Multilevel Model for Describing Martensitic Transformation: Formation of the Polyhedral Martensite Structure

#### Abstract

Multilevel models of inelastic deformation that take into account microstructure evolution are promising for technology development for creating functional material structures with optimal performance characteristics. The paper discusses the mathematical formulation of a direct multilevel model to describe the inelastic deformation of a polycrystal representative volume (analogous to a macrosample), taking into account the formation and the martensitic structure evolution during the transformation process. The model considers three structural-scale levels. At the macro level, the boundary value problem is solved, the fields of stresses, strains and other model variables are determined. At mesolevel-I, a homogeneous original austenite grain is considered, in which a martensitic transition occurs due to external influences. For a detailed description of the material response at the grain level, an auxiliary scale level is introduced into consideration, i.e. mesolevel-II. At this level, the geometric features of the martensite packet formation are explicitly studied. An original method has been developed to describe martensite polyhedral structure, the construction of which is carried out when the new phase volume fraction in the austenite grain reaches a critical value. A packet description as union of polyhedra consisting of thin plates allows one to introduce the geometric characteristics of the structural elements into the model, in particular, plates and packet boundaries, linear dimensions, volumes etc., and supplement them with crystallographic orientations. The resulting geometric characteristics of martensite package with subsequent processing is transferred to an individual grain level. This makes it possible to take into account the mechanisms of deformation and hardening that occur during the interaction of phases. The results are presented of polyhedral martensite packet structure formations in AISI 304 steel in numerical experiments on uniaxial deformation at room temperature and strain rate 10–5s–1.

**PNRPU Mechanics Bulletin**. 2024;(2):

MODELING THE DEFORMATION PROCESS OF AN UNDERWATER GAS PIPELINE UNDER THE EXPLOSIVE LOADING

#### Abstract

The process of deformation of an underwater two-layer gas pipeline during the explosion of a nearby octogen charge is simulated. For modeling, a specially developed proprietary software package is used to solve three-dimensional dynamic problems of interaction of elastoplastic structures with compressible media, based on a single Godunov scheme of increased accuracy for calculating the joint motion of gas, liquid, and elastoplastic media. The package uses an Eulerian-Lagrangian approach with explicit identification of moving contact surfaces between different media. For each environment, three types of computational grids are used. These are Lagrangian surface meshes in the form of a continuous set of triangles for specifying the initial geometry of bodies and accompanying them during the calculation process, as well as two types of three-dimensional volumetric meshes automatically generated during the calculation process. The charge, which has a spherical shape, is initiated at its center. To describe the process of propagation of steady-state detonation, the hydrodynamic theory of detonation is used. Shock waves generated during an explosion in the surrounding liquid interact with a fragment of a two-layer pipeline and a solid bottom. Wave processes are analyzed both in the steel pipe and in the concrete shell that weighs it down. Loads on the pipeline are estimated depending on the distance to the charge and the position of the pipeline relative to the bottom. The possible destruction of both steel and concrete weight shells in areas of tensile deformations that are formed in places of the maximum bending of the pipeline is shown. It is shown that the proximity of the bottom can significantly enhance the impact of explosive loading due to the action of shock waves reflected from the bottom.

**PNRPU Mechanics Bulletin**. 2024;(2):97-104