## No 2 (2017)

**Year:**2017**Articles:**13**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/17**DOI:**https://doi.org/10.15593/perm.mech/2017.2

Multiscale modeling and formation analysis of technological interlaminar stresses in thick-walled GFRP rings

#### Abstract

Thick-walled GFRP rings are widely used in high-loaded friction units of working elements of road-building machines as sliding bearings. They must endure high radial loads and work in the condition of a limited lubrication by using the lubricant which is located only in the open pores of the bushing. In large structures of sliding bearings (bearing elements of road-building machines) the thickness of a bushing might be dozens of millimeters. The reality is that it is necessary to use bushings with a rather big thickness to reduce the dynamic load for road-building machines. However, if the thickness of a composite ring increases, it often results in the inter-laminar failure which is caused by high technological (inter-layer) and stretching stresses due to the anisotropic thermal expansion and shrinkage of fabric composites. Furthermore, these stresses can also cause a significant distortion of a thick-walled GFRP ring after machining. In this work, it is assumed that the tension winding of the pre-impregnated fabric on a rigid mandrel leads to a heterogeneous compression of the internal layers and a distortion of the original composite woven structure through the thickness. The threads of the inner layers are curved more than the threads of the outer layers. For this reason, the coefficients of thermal expansion (CTE) of the inner layers happen to be larger than the CTE of the outer layers. Irregular distribution of the CTE gives rise to residual tensile stresses in the transverse (radial) direction. This paper presents the results of dilatometric studies of distributing of the CTE in the thickness of the hoop and axial directions. To determine the volume fraction of filaments and matrix into the threads of GFRP, we used the burn-out method. The observation of slices in the optical microscope allowed us to determine the degree of the curvature of fibres and the thickening of inner ring layers for further 3D modelling. The structure of the woven composite ring was modelled by using impregnated unidirectional (UD microplastic) threads and repeated unit cell. Modulus of elasticity, Poisson's ratios and coefficients of thermal expansion at microscale, macroscale and mesoscale were calculated by FEA (ANSYS Workbench). Thermomechanical properties were used to evaluate the residual technological interlayer stresses in the GFRP sliding bearing busing at the macroscale.

**PNRPU Mechanics Bulletin**. 2017;(2):5-22

Comparative analysis of variants of plasticity theories under cyclic loading

#### Abstract

Based on the results of the experimental investigations of the samples made of stainless steels SS304, 12X18H9 and 12X18H10T subject to a hard (controlled deformations) and soft (controlled stresses) cyclic loadings under uniaxial tension-compression and normal temperature we carried out a comparative analysis of the adequacy of the currently used variants of the plasticity theories which belong to flow theories with a combined hardening. The loads that include the sequences of cyclic and monotonic loads as wells as placing and ratcheting of the plastic hysteresis loop are considered. The loads are applied until fracture, i.e. the emergence of macrocracks with a length of 1 mm. Within the scope of our analysis we have considered the following variants of theories including the models of Korotkih, Bondar and Chaboche. For each model, a set of material functions is given on basis of which the calculations of the kinetics of the stress-strain state of the cyclic loading processes are made. By comparing the results of the calculations and experiments, it has been shown that the description of the cyclic loop placing and ratcheting processes is possible only within the Bondar and Chaboche models with a certain advantage of the modified Bondar model, which enables an adequate description of the nonstationary cyclic loading and nonlinear damage accumulation. It is shown that the modified Bondar model - including the third type of microvoltage (Ono-Wang) and kinetic equations of damage accumulation - provides the most adequate (in comparison with other models) description of the kinetics of the stress-strain state and fracture under nonstationary nonsymmetric cyclic loading processes. It can be recommended for practical computations of the kinetics of the stress-strain state and the design life affected by an arbitrary load action.

**PNRPU Mechanics Bulletin**. 2017;(2):23-44

Modelling the generation of new material surfaces in a composite with an adhesion layer under cohesive destruction

#### Abstract

The paper is concerned with the subcritical elastoplastic deformation of the trilaminar composite and the delamination accompanied by the fracture of the adhesive layer. The problem is reduced to the system of two variational equilibrium conditions with respect to the velocity fields of the bonded layers by means of averaging the stress components in the adhesive layer thought its thickness. When we solve the elastoplastic problem in terms of subcritical deformation, the δ-area is distinguished where the fracture criterion is reached. The distribution of load (node forces) which affects the body from the δ-area is determined by resolving the pre-critical deformation problem with the known motion law of the δ-area boundary. As the next step, we consider the changes in the body’s stress-strain state during the fracture of the δ-area. We solve the elastoplastic problem under simple unloading of the body’s δ-surface and remaining the external load which correspond to the beginning of the destruction process. During the δ-unloading, the formation of new plastic areas, partial unloading and reaching the destruction criterion are possible. As a result, the body’s stress-strain state in the moment when the local unloading begins differs from its state when the δ-unloading ends. This constitutes the principal distinction from the common procedure of “killing the element” when the element rigidity (after reaching the fracture criterion) is supposed to be close to null. Herewith the body state outside the deleted element is considered to be unchangeable; and the generation of the unloading zones and additional loading zones (after the element is excluded) is not considered. In case of linear plasticity, the solution of the problem with the ruptured area under the fixed external load coincides with the δ-unloading solution by virtue of the solution uniqueness and the principle of superposition. However, the solution of the elastoplastic problem for the body with the ruptured area under simple loading will not coincide with the δ-unloading solution. The paper presents the solutions of the composite delamination problems which illustrate the simple δ-unloading method both in the linear elastic as well as in the elastoplastic formulations.

**PNRPU Mechanics Bulletin**. 2017;(2):45-59

Boundary value problem of nonsymmetrical deformation of the cylindrical vessel with liquid in the thermal field

#### Abstract

We framed a precise solution of the nonsymmetrical boundary value problem of the elasticity theory for a cylindrical vessel with liquid placed in the thermal field. The thermoelastic problem is unlinked, i.e. at first we solve the thermal conductivity equation, and then the linear problem of the elasticity theory for a circular cylinder in displacements. It should be noted that until the present time there were no precise solutions of nonsymmetrical problems of the elasticity theory in the cylindrical coordinate system with a consideration of the thermal field. It is explained by the complexity of the system of resolvent equations, such as high order, variable coefficients. The authors of the article managed to form integrable combinations of resolvent equations in this work, at first by taking no account and then considering the thermal fields. For this purpose an additional equation related to a volumetric deformation was introduced into the system of resolvent equations instead of the relator connecting the volumetric deformation with the movement of the cylinder points. When we took into account the heat conduction equation, we managed to gain the equation which had been obtained earlier without the consideration of the thermal elements. As a result, the problem was brought to a successive solution of each equation separately. Since the additional equation was obtained by the derivation of the rest of the equations, the order of the resolvent equations system became higher which resulted in «excess» constants of the integration. The authors proved that the use of the replaced correlation between the volumetric deformation and displacements as an additional condition eliminated this disadvantage. We formed a precise solution of the boundary value problem for the cylindrical vessel with liquid upon the condition of the linear dependence of temperature and displacements of the cylinder along its axis. The numerical example was considered where the temperature of the external side area of the cylinder is changed in the circumferential direction.

**PNRPU Mechanics Bulletin**. 2017;(2):60-77

NONLINEAR EQUILIBRIUM EQUATIONS OF THE CONICAL SHELL STIFFENED BY A DISCRETE SET OF FRAMES

#### Abstract

Studying the stress state of stiffened thin shells is one of important issues of solid mechanics. The simplified methods of computing the stiffened shells based on the models that use the concept of "smoothing” do not always give satisfactory results. Therefore, it is relevant to develop and investigate the computational methods for such shells; and it is in line with considering the discreteness of the position of the stiffening set of frames and identifying the characteristics of stress-strain states that are generated by them. In order to take into account the discreteness of the location of the set of frames in case of a fully operating skin, we "joint" the solutions for the shell and the set of frames, as well as used the variational and finite element methods. A number of works have recently appeared where authors suggest considering the discreteness of the stiffen set by recording the variable stiffness of the system using the Dirac delta function. The problem is reduced to equations with singular coefficients. The conical shell which is stiffened with a discrete set of frames is a discretecontinuous system which combines the continual element, i.e. - the shell itself and discrete components, i.e. -frames. This system is considered by means of generalized functions as an "integrated" shell of a non-homogeneous orthotropic generalized material, i.e. as a shell with a variable stiffness. The paper presents the mathematical model of the deformation of the stiffened conical shell. The derivation of the nonlinear equilibrium equations of the shell are supported by a discrete set of frames using vector analysis. Also the geometrical aspect of the problem is considered here. When considering the physical aspects, we provide the elasticity equations for the shell and obtain the equations of the frame elasticity.

**PNRPU Mechanics Bulletin**. 2017;(2):78-98

On healing metal damages using high-energy pulsed electromagnetic field

#### Abstract

The concentration of the field takes place on the structural defects of the material, if it is affected by the electromagnetic field. In particular, it initiates electrical, thermal and mechanical processes in the vicinity of micro-defects (cracks, pores, inclusions, etc.). The transformation and interaction of defects are investigated in the article, e.g. the flat intergranular micro-cracks with linear dimensions of the order of 10 microns. These processes occur in the material when the metal samples are treated with a high-energy pulsed electromagnetic field which induces a short pulse of a high density electrical current in the material. The study uses the numerical coupled model related to the impact of the high-energy electromagnetic field on the pre-damaged thermal elastoplastic material with defects. This model considers the metal’s melting and evaporation, as well as the dependence of its physical and mechanical properties on temperature. The system of equations is solved numerically using the finite elements method on adaptive lattices using the alternative method of Euler-Lagrange. The simulation shows that the treatment by the short pulse of current results in the welding of the crack and healing of the micro-defects. The healing occurs due to a simultaneous reduction of length, ejection of the molten metal into the cracks and closing of micro-crack edges. It leads to the fact that the edges of the crack start to contact the jet stream of the molten material, and, finally, the stream becomes completely jammed by the crack’s edges. Meanwhile the volume of the micro-cracks starts to decrease in time. In this paper, the material healing and damage parameters are introduced for the macroscopic description of the healing process. The parameter of healing is determined as a relation of the micro-crack’s change of volume to the initial micro-crack’s volume at a time when the material is affected by the electromagnetic field. The damage (porosity) is understood as a ratio of the micro-crack’s volume at a time to the volume of the representative element. The healing of micro-cracks increases the material’s healing parameter and reduces its damage. The paper studies the changes in the material’s healing and damage parameters depending on time under the action of the current pulse. The issues of selecting the preferred regions of integration in modeling the considered processes are researched. It is investigated how the distance between the micro-cracks and their mutual arrangement influence the healing and damage over time. The simulation of the considered processes in the entire investigated range of distances between the defects (or, for any initial damage equivalently) have shown that the dependences of the healing and damage on the time will not be different, no matter if we calculate these dependences in the regions of integration consisting of one or several representative elements. The arrangement of micro-cracks relative to each other and the distance between them do not affect the dependences of the healing and damage on the time under the current pulse. These changes are affected by the value of the initial damage only. The dependences of healing and damage on time will be practically the same for all different mutual arrangements of micro-cracks provided that the initial damages are equal for these different mutual arrangements of defects. Based on the simulation results, the approximate piecewise-linear dependences of healing and damage on time and the initial damage are obtained. It is clear that until a certain moment all the micro-cracks in the material (regardless of the initial damage) are not healed or damaged when they are affected by the current. After this moment, the process of micro-cracks’ healing starts. Meanwhile, under the action of the current, the material’s damage decreases over time at a constant rate (independent of the initial damage), while the healing increases over time at a rate inversely proportional to the initial damage of the material.

**PNRPU Mechanics Bulletin**. 2017;(2):99-124

A shock loading on a bar with a central crack

#### Abstract

Linear thermoelasticity is studied of a plane regular truss formed by four families of The paper is concerned with the problem of calculating the time dependence of the stress intensity factor for a plane-strain bar with a stationary central crack caused by the opening mode. A uniformly distributed load has been immediately imposed on the basis of the bar and remained unchanged later. The model of the crack with cohesive forces distributed by Barenblatt’s postulates is used. In this case the stress intensity factor is a result of the calculation of the released energy that is determined by the cohesive forces. The solution is found with a new numerical method that is an adaptation of the method of lines to dynamic fracture mechanics problems. The Crank-Nicolson implicit finite-difference scheme is used for time integration. Boundary problems arising at each step of time integration are solved by the finite element method. The special cohesive finite elements are used, so that the solution of the problem could satisfy Barenblatt’s postulates. Previously these elements were used to solve quasi-static nonlinear fracture mechanics problems. By introducing the additional degrees of freedom of the nodes lying on the crack line, it becomes possible to ensure a smooth closing of the crack edges at its tip; and it is equivalent to the absence of the singularity stress and strain fields at its tip. The cohesive forces are calculated as the constraints. Their field of action (cohesive zone) is localized within the finite element which is adjacent to the tip of the crack. Thus, the smaller the finite element mesh is, the better it satisfies the requirement of Barenblatt’s theory. This requirement concerns the length of the cohesive zone which is small compared to the length of the crack. The stated problem is called Chen’s problem and had earlier been solved by researchers that used different methods. The proximity of the obtained results makes it possible to consider Chen’s problem as a test; and its solution obtained by the developed method agrees well with the data of other researchers.

**PNRPU Mechanics Bulletin**. 2017;(2):125-135

Thermoelasticity of a plane regular truss With orthogonal structure

#### Abstract

Linear thermoelasticity is studied of a plane regular truss formed by four families of straight homogeneous rods. Rods of two of them are mutually orthogonal and form rectangular cells that are repeated in two perpendicular directions. The other two families combine differently oriented diagonal rods of the cells. All the rods are only in tension-compression and their elastic lines belong to the same plane. Adjacent rods are rigidly connected together in the truss nodes, i.e. the points of intersection of the elastic lines of mutually orthogonal rods. The regularity of the truss implies the invariance of the thermoelastic and geometrical parameters of the rods within the same family. External actions on the truss belong to its plane and generally include the external nodal forces, the linear axial forces of the rods and their non-uniform heating. The rigorous linear thermoelasticity of the truss is constructed by the gluing method. According to this method, the truss was split into rods and nodes, i.e. the elements of the truss. The given external actions were applied to the isolated elements as well as the forces of interaction with their neighbors. Then the analytical study of thermoelastic rods with geometric conditions of the conjugation of the adjacent elements and the analysis of the equilibrium of the nodes were carried out. The theory was formulated in terms of nodal displacements and total elongations and internal initial forces of the rods. All these variables are the functions of two integer arguments used for numbering of the elements of the truss. The complete closed system of equations of the truss thermoelasticity is represented by geometric and physical relationships, the equilibrium equations of the nodes and the compatibility equations of the total elongations of the rods. Alternative formulations of the discrete boundary-value problems are presented with their help. An application of the theory is illustrated by the exact analytical solution of the problem of thermoelastic deformation of the truss without internal nodes.

**PNRPU Mechanics Bulletin**. 2017;(2):136-152

Modeling of the cross-section ovality of single crystal nickel-based superalloy samples under tension

#### Abstract

The problem of uniaxial tension of a cylindrical body from a single-crystal material with a cubic symmetry is considered. The loss of the initial circular shape of the cross section of a cylindrical specimen under tension or compression in conditions of misalignment of the sample axis with the anisotropy axes is one of the characteristic external manifestations of the mechanical properties anisotropy. The problem is solved in elastic, elasto-plastic and elasto-visco-plastic formulations. Both analytical and numerical (three-dimensional finite element modeling) methods are used to solve boundary value problems. The dimensionless ovality parameter is introduced as a ratio of the difference between the largest and smallest diameters to the smallest ones aiming to estimate the change of the cross-section shape. The results of the calculations are compared with the experimental data concerning the high-temperature creep of the single-crystal nickel-based superalloy VZhM4. The deviation effect of the sample axis orientation from the anisotropy axes on the cross-section shape under loading and unloading is investigated. Both micromechanical (crystallographic) and phenomenological models are used to solve inelastic problems; and later the obtained results are compared. Taking into account the geometric nonlinearity in the solution of the elasto-plastic boundary value problem makes it possible to describe the evolution of the neck formation process which can be non-axisymmetric for anisotropic materials. The obtained results indicate the need to abandon the measurement methods when performing the mechanical tests of anisotropic materials under uniaxial tension related to recording the change in the cross-section size in order to determine the axial deformation. The ovality can be directly used to estimate the axial strain, and also to compensate the missing information about the crystallographic orientation of the sample.

**PNRPU Mechanics Bulletin**. 2017;(2):153-177

Studying the energy distribution of the dynamic influences of road transport on the layers of nonrigid pavements

#### Abstract

The paper deals with studying the distribution of the scattered (dissipating) mechanical energy transferred to the pavement cover when it is used by road transport. For the numerical simulation of the energy transfer process we improved the analytical model of the dynamic stress-strain state of the multilayered half-space by introducing the moving coordinate system. The energy distribution was studied for three road structures with different strengths. For each structure we obtained the amplitude-time characteristics of stress and strain on the surface of the coating layers, on the base and subgrade soil which have been used to build the dynamic hysteresis loops. We analyzed the areas of the dynamic hysteresis loops on the surface of the coating layers as well as the base and subgrade soil, which made it possible to reveal the qualitative and quantitative dependences of the density distribution of energy dissipation in the pavement layers. It is found that when the solidity of the road structure increases, the energy density which is dissipated on its surface decreases. The rate of the energy attenuation of wave fields in road structures which has been generated by the impact of the design load vary significantly depending both on the solidity of the pavement and various types of its structural layers. In this case the greatest difference is determined by the material properties which are used as the base layer (reinforced, unreinforced). Based on the studies, a new approach of evaluating the design service life of nonrigid road structures in terms of the energy transferred on its surface during its entire service life has been proposed.

**PNRPU Mechanics Bulletin**. 2017;(2):178-194

Computing orthotropic constructions using the variation method based on three-dimensional functions with final carriers

#### Abstract

At the moment the finite element method (FEM) is often used to compute complex orthotropic thin-walled constructions including thin-walled orthotropic shells. As a rule, one of two approaches is used to make computations using this method. In the first approach the simplifying hypothesis (for example Tymoshenko's hypothesis) is used in which the distribution of stress along the thickness of a thin-walled construction is neglected which reduces the dimension of a task. The second approach uses the ratio of the three-dimensional theory of elasticity without the use of the simplifying hypotheses. In this presented method which is very similar to the FEM, the ratio of the three-dimensional elasticity theory without the simplifying hypotheses is also used for the computations. The paper presents the variation method aiming to determine the stress-strain state of three-dimensional elastic constructions based on the use of the approximating functions with final carriers having an arbitrary degree of approximation [1]. The three-dimensional approximating functions mentioned before are used to compute the orthotropic constructions in this paper. The same approximating functions are used in the papers [2, 3] for the computation of shells in which the resolving equations are obtained on the basis of the simplifying hypothesis. In a general view, the method is based on the use of the curvilinear system of coordinates that does it quite universal. It is shown that the same approximations can be used to compute the three-dimensional autotrophic constructions and orthotropic shells. It is noted that the computation can be efficiently made not only by thickening the lattice but by increasing the order of the approximating functions. The reliability of the suggested method is confirmed by the presented numerical results which fit well with the known solutions.

**PNRPU Mechanics Bulletin**. 2017;(2):195-207

Mathematical model and experimental studies of behavior of viscoelastic filled polymers under two-frequency loadings

#### Abstract

Highly-filled polymer composites are widely used in critical structures in aerospace and other industries. Such structures have to endure complex harmonic loadings. Thus it is important to develop methods of experimental studies and determine deformation properties of highly-filled polymer materials, as well as computational methods for structures working in extreme conditions. The aim of this research is to develop methods of conducting the dynamic experiment, determine viscoelastic parameters of highly-filled polymer composites under stationary two-frequency loadings and find the mathematical model to calculate the stress-strain state of viscoelastic aerospace structures. Linear and nonlinear integral representations of stress and strain for mechanical behavior description of the viscoelastic materials are presented. The general description of a method to mathematically model the nonlinear viscoelastic behavior was accomplished by Volterra using an earlier representation developed by Frechet (Volterra-Frechet integral series). By using the Volterra-Frechet integral series we presented the nonlinear mathematical model based on complex parameters to describe the viscoelastic material behavior under stationary two-frequency loadings (under various values of frequencies and amplitudes). This mathematical model was analyzed and compared to an earlier model with some assumptions in linear dependencies of viscoelastic parameters on strain amplitudes and without hysteresis loop distortions (harmonics distortions) under deformation. We suggest using polynomials to describe the dependencies of viscoelastic parameters on frequency and temperature by using the time-temperature superposition. The two-frequency (dual-frequency) experiments to determine these polynomials are conducted. After the experimental data has been processed, the dependencies of the viscoelastic parameters on frequency and temperature are determined. These results allow developing an optimal experimental design, determining mathematical model constants and assessing the influence of the viscoelastic parameters on the description accuracy of the material behavior under complex harmonic loadings.

**PNRPU Mechanics Bulletin**. 2017;(2):208-225

Modeling the dynamics of reinforced shallow shells made of nonlinear elastic materials

#### Abstract

The problem of dynamic behavior of flexible reinforced shallow shells made of nonlinear elastic materials of the composition phases is formulated. The geometric nonlinearity of the problem considered in the Karman approximation, and the weakened resistance to shear is taken into account in the framework of non-classical Reddy theory. The numerical integration of the formulated initial-boundary value problem is carried out on the basis of the method of steps in time with the involvement of an explicit "cross" scheme. Specific calculations of the dynamic behavior are carried out for relatively thin and thick shallow spherical shells and plates having an annular shape in plan and with the rigid inner insertion which are under the pressure caused by the blast air wave. Thin-walled structures are clamped on the outer edge and axially reinforced by logarithmic spirals in the plan. The influence of the angles of reinforcement on the flexibility and stress-strain state in the materials of the phase of the composition of elastic plates and shallow shells is studied. It is obtained that on the set of considered structures of reinforcement, the reinforcement in the radial (meridional) directions is rational, as this structure provides the minimal flexibility and the minimal stressed state in the material of the binder matrix of compositions. It is shown that because of the geometric and physical nonlinearity of the studied problem, the dynamic response of composite shallow shells essentially depends on which front surface (convex or concave) the overpressure of the explosive type is applied.

**PNRPU Mechanics Bulletin**. 2017;(2):226-245