## No 2 (2019)

**Year:**2019**Articles:**18**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/39**DOI:**https://doi.org/10.15593/perm.mech/2019.2

Description of the Effect of AlMg6 Alloy Strength Decrease during Temperature Increase under Dynamic Loading

#### Abstract

The work is devoted to a theoretical and experimental study of mechanisms related to plastic strain localization under dynamic loading. A mathematical model that takes into account structural relaxation and thermal softening is built within the framework of the study on the basis of wide-range constitutive equations. The proposed model is capable to adequately describe the deformation of plastic materials (metals and alloys) in the range of strain rates of 102-104 s-1 and in the temperature range from 0 to 0.7 of the melting point. AMg6 aluminum alloy was chosen as a material under study as it is a promising material for mechanical engineering. However, all the obtained results can be applied to a wide class of materials: metals and alloys. The model parameters were identified using experimental data (stress-strain diagrams) at various strain rates and temperatures. The series of original experiments on punching disc-shaped barriers by a cylindrical projectile was carried out to test the constructed model. During the experiment, the impact velocity and the temperature on the back surface of the barrier were measured at the moment of an intense localization of plastic deformation and fracture. The strain rate in the experiment ranged from 500 to 30,000 s-1. As a result of the numerical studies, it was shown that defects play a critical part in during strain localization for AlMg6 alloy at strain rates of less than 104 s-1. Thermal softening begins to make a significant contribution at strain rates above 104 s-1. This theoretical result is also confirmed by the original experiments on the penetration of barriers.

**PNRPU Mechanics Bulletin**. 2019;(2):5-14

The Investigation of the Initial Stress-Strain State Influence on Mechanical Properties of Viscoelastic Bodies

#### Abstract

We present a new general statement of the problem on steady-state oscillations of a non-uniform viscoelastic body, taking into account the residual stress-strain state. To derive it, we use the theory of complex modules and the correspondence principle which makes it possible to write down the problem for viscoelastic bodies in the same form as the corresponding elasticity problems by replacing the elastic characteristics with the complex functions of vibration frequency. To describe the viscoelastic behavior, we employ a three-parameter model of the standard viscoelastic body consisting of instantaneous and long-term modules, as well as relaxation time. On the basis of the statement proposed, we consider the model problems on vibrations of inhomogeneous viscoelastic rods and pipes taking into account the presence of both pre-stress and residual strain. Within the problem for the viscoelastic rod, we analyze the longitudinal oscillations and consider the pre-stress and residual strain as functions of the longitudinal coordinate. Regarding the problem for the viscoelastic pipe, we consider the plane stress-strain state and assume that the oscillations are excited by the radial load; the mechanical properties of the cylinder and the residual stress-strain state depend on the radial coordinate. The numerical solutions in both cases are based on the shooting method. For both models problems under study, we present and observe the results of the comparative analysis of the effect of pre-stress and residual deformations on the amplitude-frequency characteristics (AFC), based on the computational experiments. We also provide the analysis of such an influence of each factor on the most sensitive frequencies values. The study conducted shows that for both rod and pipe the residual strain fields have a significantly greater effect on the AFC than the pre-stress fields do. This conclusion correlates with the results obtained previously for elastic bodies.

**PNRPU Mechanics Bulletin**. 2019;(2):15-24

Some Features of Monotonic and Cyclic Loadings. Experiment and Modeling

#### Abstract

Having analyzed the experimental studies of samples of 12X18H10T stainless steel with a hard (controlled deformation) deformation process including sequences of monotonic and cyclic loading modes, under uniaxial tension-compression and normal temperature, we found some features and differences of isotropic and anisotropic hardening processes under monotonic and cyclic loads. To describe these features in the framework of the plasticity theory (Bondar model) belonging to the class of flow theories with combined hardening, the criterion of changing the direction of plastic deformation and the memory surface allowing the separation of monotonic and cyclic deformations is introduced in the plastic strain tensor space. For the description of transient processes from monotonic to cyclic and from cyclic to monotonic ones, the evolutionary equations are formulated for the parameters of isotropic and anisotropic hardening. The basic experiment, on the basis of which the material functions are determined, consists of three stages, i.e. cyclic loading, monotonic loading and subsequent cyclic up to destruction. The method of material functions identification according to the results of the basic experiment is given. For stainless 12X18H10T steel, the material functions at room temperature were determined on the basis of the basic experiment and the identification method. The comparison of the calculated and experimental results for the stainless steel under rigid loading, was made and consisted of a sequence of five stages: cyclic, monotonic, cyclic, monotonic and cyclic up to destruction, are given. The calculated and experimental kinetics of the stress-strain state throughout the deformation process are compared. Changes in the range and average stress of the cycle at the stages of cyclic stress are analyzed. At these stages there is a landing hysteresis loop. A reliable agreement between the calculated and experimental results was obtained. A sufficiently adequate description by the theory of the processes of change in the kinetics, range and average stress of a cycle under hard loading suggests the possibility of a more adequate description and processes of soft loading, especially in non-stationary asymmetric loading conditions.

**PNRPU Mechanics Bulletin**. 2019;(2):25-34

Hydroelastic stability of coaxial cylindrical shells made of piezoelectric material

#### Abstract

The paper numerically investigates the dynamic behavior of electroelastic coaxial shells containing a compressible flowing fluid in the annular gap between them. The solution of the problem is carried out using a semi-analytical version of the finite element method. The shells are made of a material with piezoelectric properties, which is polarized in the radial direction. The behavior of the system is studied in the framework of the classical theory based on the Kirchhoff - Love hypotheses and equations of linear electroelasticity. The distribution of the electric potential through the thickness is assumed to be linear. The motion of a compressible non-viscous fluid is described by the wave equation, which, together with the impenetrability conditions and the corresponding boundary conditions, is transformed using the Bubnov - Galerkin method. The pressure exerted by the fluid on the deformable bodies is calculated from the linearized Bernoulli equation. The mathematical formulation of the problem of the thin-walled structure dynamics is based on the variational principle of virtual displacements. The stability estimate is obtained from the calculation and analysis of complex eigenvalues of a coupled system of equations, developed for unknown quantities of elastic and liquid media. The electrical variables are eliminated at the element level and produce an effect on the dynamic characteristics of the structure in the form of added stiffness. The reliability of the obtained results is evaluated by comparing them with the known data for isotropic shells. The estimation of the stability boundaries are carried out for systems with different geometrical dimensions, variants of kinematic boundary conditions (shells with simply supported edges, clamped at both edges and cantilevered) and different annular gap sizes. It has been shown that the critical velocities of the fluid flow and the form of the loss of stability depend on the electric boundary conditions set on the electrode surfaces of the inner and outer shells.

**PNRPU Mechanics Bulletin**. 2019;(2):35-48

The Contact Problem Solution of the Elasticity Theory for Anisotropic Rotation Bodies with Mass Forces

#### Abstract

In this paper presents a developed method aimed to solve contact axisymmetric problems for limited bodies of revolution from transversely isotropic material which are simultaneously under the action of mass forces. The method involves the development of the energy method of boundary states, which is based on the concepts of spaces of internal and boundary states conjugated by isomorphism, which allows us to establish a one-to-one correspondence between the elements of these spaces. The internal state includes stress tensor components, deformation tensor components and displacement vector. The boundary state includes efforts and moving boundary points and mass forces. The isomorphism of the state spaces is proved, which allows finding the internal state to be reduced to the study of a boundary state that is isomorphic to it. The basis is formed based on the general solution of the boundary value problem for a transversely isotropic body of revolution and based on the method of creating basic displacement vectors while determining the state from continuous nonconservative mass forces. The orthogonalization of the state spaces is carried out, where the double internal energy of elastic deformation is used as scalar products in the space of internal states; in the space of boundary states, the work of external and mass forces is used. Finally, the problem of finding a desired state is reduced to solving an infinite system of algebraic equations regarding the Fourier coefficients. The solution of the contact problem without friction in contacting surfaces for a circular in terms of the cylinder is presented. The material of the cylinder is a transversely isotropic siltstone with the anisotropy axis coinciding with the geometric axis of symmetry. Mass forces act on the body, imitating centrifugal forces of inertia and gravity. Mechanical characteristics have analytical polynomial view. Explicit and indirect convergence patterns of the problem solving and graphical visualization of the results are presented.

**PNRPU Mechanics Bulletin**. 2019;(2):49-62

Research methods of structural and phase transformations in nanomaterials deformed under pressure

#### Abstract

The comparative analysis is given on the main research methods of the structural and phase transformations proceeding in nanostructures of metals and alloys during plastic deformation under pressure. It is shown that an adequate description of changes in structure of compressed materials under deformation is impossible without the use of continual models of the linear, planar and dot defects making a basis of any nanostructure. Within the theory of irreversible deformations based on the continual model of Debye and Gryunayzen's approach volume properties of dislocations, their congestions and intercrystalline borders are investigated. It is shown that dislocations have to have the excess volume which size is defined by the asymmetry of potentials of interatomic interactions in relation to stretching and compression of materials. The data confirming a considerable influence of an excess volume on the speed of processes of a diffusive mass transfer along dislocation lines are provided. Also it is shown that the excess volume of dislocation congestions significantly depends not only on volume properties of individual dislocations but also on a structure of congestions. The received results are applied to the analysis of the problems arising at a research of effects of increase in plasticity of materials under pressure. It is shown that the squeezing pressure can promote increase in speed of processes of a relaxation of internal tension and suppress processes of concentration of tension in places of origin of the centers of destruction of materials. However, it does not interfere with developments of a lack of adhesion, and at rather low temperatures a condition of hydrostatic compression can lead to acceleration of processes of cavitation. The methods describing the deformation interaction of dot defects in chemically non-uniform materials are considered. The analysis is given regarding shortcomings of the existing microscopic and continual theories applied to the description of volume properties of dot defects in the non-uniform environments corresponding to nanostructures of metals and alloys. The models describe not local deformation interaction of dot defects in continual environments with any properties of anisotropy.

**PNRPU Mechanics Bulletin**. 2019;(2):63-85

An asymptotic Solution of the Hypersingular Boundary Integral Equation Simulating Wave Scattering by the Interface Strip-Like Crack

#### Abstract

The ultrasonic non-destructive testing is widely used in different civil and engineering applications as one of the most effective and convenient method of structural health monitoring. It is necessary to have a reliable mathematical model simulating scattering caused by defects and inhomogeneities in order to apply effective ultrasonic methods. Modern composite materials used in manufacturing have a laminated structure; therefore it is important to detect damages occurrence located between two materials. Scattering caused by interface cracks can be investigated using the boundary integral equation (BIE), the method which is analytically oriented. The unknown function of the crack opening displacement in the BIE is expanded in terms of orthogonal polynomials. Then the integral equation is projected onto a set of polynomials. Regularization of the hypersingular BIE using the Bubnov-Galerkin scheme is obtained through a repeated integration on the crack faces. This paper uses the BIE method to derive an asymptotic solution describing the elastic wave diffraction by the strip-like crack located at the interface between two dissimilar elastic half-spaces. The Fourier transformation of Green's matrix is applied to obtain a scattered field. Asymptotic representations of the equation kernel around zero and at infinity are derived with the assumption that the crack size is much less than a wavelength of an incident wave. The Bubnov-Galerkin scheme is used to obtain the frequency dependent asymptotic solution of BIE which has a wider accuracy frequency range than the existing quasi-static solution. A good agreement of the derived asymptotic solution with the numerical solution is shown for different materials of the considered structure. The asymptotic solution allows increasing the BIE method potency by reducing the computational cost of integrals. It can also be used to describe dynamic damaged interfaces in the Bostrom-Wickham model's term.

**PNRPU Mechanics Bulletin**. 2019;(2):86-99

Plastic stress intensity factor in fracture mechanics

#### Abstract

Application of fracture mechanics approaches and criterions for nonlinear deformation under complex stress state is concerned with special features. The influence of a complex stress state is realized through the plastic zone at the crack tip. In this case a nonlinear analysis of the stress-strain state in experimental specimens and elements of structures is needed. A generalization of the implemented approaches to solving problems of computational and experimental fracture mechanics on the base of a single parameter in the form of the plastic stress intensity factor (SIF) is presented. Plastic SIF takes into account the nonlinear behavior of the material, loading conditions, the stress state and the in-plane and out-of-plane constraint effects. The finite element analysis results presented in this paper show prospects for application of the plastic SIF as a nonlinear fracture resistance parameter that take into account loading conditions, stress state, influence of geometry of cracked bodies and the material mechanical properties in the temperature range. It is demonstrated that plastic SIF is a one-parameter approach that allows obtaining fracture resistance material properties under static loading. Efficiency of the plastic SIF implementation for interpretation of the fatigue crack growth data under complex stress state and mixed mode loading is shown. An application of plastic SIF for residual life prediction of the power steam turbine disk with a surface flaw is demonstrated.

**PNRPU Mechanics Bulletin**. 2019;(2):100-115

Mixed Form Equations for Ribbed Shells of a General Type And Their Solutions

#### Abstract

The paper examines general shells supported from the incurvity side by a cross-sectional ribbing directed parallel to coordinate lines. Ribs’ position on a shell is set using ordinary bar graph functions so that the rib and shell contact is arranged along the strip. A mean shell surface shall be considered as a coordinate surface. Geometrical nonlinearity and transverse shears are considered; and the shell is considered to be shallow. Forces are expressed via stress function in the mid-surface of the shell in such a way that the first two equilibrium equations are fulfilled identically. Shell deformation is expressed via this function. Introduction of ribs by means of ordinary bar graph functions does not cause difficulties for expression of deformations using forces with the consequent insertion to moments, since ordinary bar graph functions may be also used in denominator, this is not applicable for delta-function (when positions of narrow ribs are set using delta-functions). Mixed equations are established starting from the minimum of shell energy deformation functional. At that, except for equilibrium equations, the variational procedure allows obtaining the third equation of strain compatibility in a shell mid-surface for ribbed shells too. Curvature and torsion change functions are registered in the same way as for Kirchhoff-Love model considering transverse shears. Mixed form equations are given for ribbed shells of the general form and for the Kirchhoff-Love model. For ribbed shallow shells, an algorithm for their solution has been developed and the results of calculating their stability for a different number of reinforcing ribs are given.

**PNRPU Mechanics Bulletin**. 2019;(2):116-134

Analysis of Dynamic Characteristics of a Contact Interaction of Solids Using Computational Software

#### Abstract

The reliability of connections of machine parts and mechanisms is laid at the stage of engineering design. Studies show that more than 80% of failures of machineries and mechanisms are caused the processes occurring in the zone of contact parts. Therefore, modern engineering design is difficult to imagine without tools for solving contact problems. From the point of the mechanics of contact interaction, a connection of machine parts is a complex technical system. Therefore, to determine performance indicators, the main research method of complex systems are used, i.e. the method of mathematical modeling. The method of calculating the contact approach of elastic-plastic smooth bodies should organically be combined with the approach of applying the classical contact problems of the theory of elasticity and plasticity in strength and rigidity calculations in machine-building. And in relation to the study of rough surfaces, an organic combination with the developed theories of contacting rough surfaces is necessary. Therefore, the development of a computational model of inelastic deformation of materials is one of the fundamental problems of modern engineering. To reduce the time for the process of calculating the dynamic characteristics, a computational-software complex was created in which a model for calculating static and dynamic parameters (proximity, voltages, and amplitudes of contact oscillations) needed to describe and predict the operation of conditionally fixed connections of machine parts at stages was laid at the stage of engineering design. When creating a computational software package aimed at evaluating the dynamic characteristics of a mechanical contact in an elastoplastic dissipative contact, factors determining the state of the object under study were identified. The main factors include physicomechanical properties of the contacting bodies, geometric characteristics of surfaces, and external conditions. The purpose of this work is to analyze the dynamic characteristics of the contact interaction of solids beyond the elastic limit in the normal direction of the external load to the contact plane, which were obtained using a computational software package evaluating the dynamic characteristics of a mechanical contact in an elastoplastic dissipative contact. The presented theoretical results allow us to estimate both the quality of the software itself and the physical and mathematical model embedded in the calculation and software complex.

**PNRPU Mechanics Bulletin**. 2019;(2):135-142

Dynamics of a Viscoelastic Plate Carrying Concentrated Mass with Account of Physical Nonlinearity of Material. Part 1. Mathematical Model, Solution Method and Computational Algorithm

#### Abstract

In dynamic calculations of thin-walled structures, an account of nonlinear viscoelastic properties of material plays an important role in a reliable assessment of the strength capability of structures. In this regard, in the mechanics of a deformable rigid body, much attention is paid to the description of nonlinear material properties and the methods to solve specific problems for various thin-walled structures under static and dynamic loads. Thin-walled structures such as plates and shells often play the role of a bearing surface, to which lining, fasteners, various instrument assemblies and other structural elements are attached. In dynamic calculation, the attached elements having an inertial character are considered as additional mass rigidly connected to the systems and concentrated in points. The effect of concentrated mass is introduced using the Dirac delta function. In this paper, a mathematical model has been constructed, a solution method has been proposed, and a computational algorithm has been developed for the problem of oscillations of a viscoelastic plate carrying concentrated mass, with account of physically nonlinear strain of material under different conditions of fixing the plate contours within the Kirchhoff-Love hypothesis. The physical relationship between stresses and strains, with account of nonlinearity, is taken in the form of the Boltzmann-Volterra integral model, where the weakly singular Koltunov-Rzhanitsyn kernel is taken in calculations as the relaxation kernel. Discretization on spatial variables has been conducted by the Bubnov-Galerkin method, and non-decaying systems of integro-differential equations (IDE) with respect to time function of the problem have been obtained in a general case. To solve the IDE, a numerical method was proposed based on the use of quadrature formulas, which eliminate the features in the relaxation kernel. A unified computational algorithm to determine the deflection of a viscoelastic plate with concentrated masses has been developed.

**PNRPU Mechanics Bulletin**. 2019;(2):143-153

Deformation model of a five-layer panel with a hard filler

#### Abstract

The system of differential equations and natural boundary conditions was obtained based on the variational problem solution. The system describes deformation of a five-layer isotropic panel with a solid filler under transverse shear loaded with forces acting both in a transverse direction and over the panel contour. The differential equation system includes three equations. The first two equations describe deformation due to loads applied to the panel contour. The third equation describes the panel deformation due to the regularly distributed transverse loading. The system of boundary conditions includes conditions at the panel edges and at its corners. The differential equation system solution in transitions for the case of a regularly distributed load at pin-edge fixing corresponds to the solution of double trigonometric sequences. As for the forces regularly distributed on the panel contour, it is represented in the form of linear functions of these forces. As an example confirming the applicability of the proposed approach, the verification of the finite-element model of a five-layer panel with the use of the obtained analytical solution has been conducted. It has been demonstrated that for an agreement of the analytical and finite-element solution results it is required to superpose in the finite-element model the datum surface and median surface of the panel. The verified finite-element model can be used for examination of the structures referred to the biostructures class, which found a wide application in various branches of industry. The analytical model application is extended to the design definition stage; the application of the verified finite-element model is extended to the to the design experimental work for creation of five-layer panels with a solid filler.

**PNRPU Mechanics Bulletin**. 2019;(2):154-162

Filtration in Fluid-Saturated Poro-Plastic MaterialS during Lateral Extrusion

#### Abstract

It is known that non-compact materials (porous, powdery, with defects in continuity) are much less resistant to shear than to hydrostatic compression. The effect of dilatancy in such media causes a change in density during shear deformation. For compact materials, a process of lateral extrusion (or equal-channel angular pressing) is known. The ECAP realizes a stress state close to a pure shear in the deformation zone (as opposed, for example, to direct extrusion, where a simple shear is realized). It can be expected that ECAP process for non-compact materials is less energy consuming and leads to a more intensive consolidation of the frame material than hydrostatic compression. In particular, ECAP can be considered as one of the methods of extracting the fluid from a porous medium (oils from vegetable raw materials, water from soils, etc.). This paper is devoted to modeling of such processes. We consider the plane problem of stationary plastic deforming of a material in the region of junction of slot channels. The cross section of the deformation region is an annular sector. We assume that the material is irreversibly compressible and obeys the elliptic Green type yield condition. One of the walls of the channel is permeable to fluid. We consider the fluid filtration in the pores. We use a number of model assumptions. The motion of the frame material particles is a flat azimuthal in a cylindrical coordinate system. The mechanical characteristics of the material vary slightly in accordance with small changes in density. The intraporous pressure is small in compare with the stressed state of the skeleton material and does not have a significant effect on the process of plastic flow. A rigid-plastic analysis was performed and the exact solution of the mechanical part of the problem was obtained. The exact solution of the fluid filtration problem in case of a constant filtration coefficient is obtained. The solution is the intraporous pressure field. When using these results one can determine the two-dimensional vector field of the fluid velocity and the total discharge of flow. In case of a non-constant filtration coefficient, the problem is reduced to integrating the boundary value problem of anisotropic thermal conductivity with a special case of anisotropy, for which a number of exact solutions are known.

**PNRPU Mechanics Bulletin**. 2019;(2):163-171

Experimental Studies of Mechanical Properties of Implants for Plasty of Hernial Defects

#### Abstract

The success of modern hernioplasty is associated with the introduction of synthetic endoprostheses (mesh implants) from various polymers. However, a significant number of complications arising, including those involving the erroneous application of an implant, make it necessary to get a deeper understanding of mechanisms of not only biological but also mechanical behaviors of structures of this type During the work, a technique is proposed to assess deformation properties of mesh implants made based on foam and polyester. These implants are used in surgical operations with non-stretching hernioplasty, which are now most common. As a result, tests were conducted to study the deformation of the mesh implant type SPMM, TCM, Reperen, TEC under the influence of abdominal pressure under quasi-static conditions. In order to conduct the study we completed the development of the proposed methodology. The data of the deformation dependence on intra-abdominal pressure were obtained. As a result, a research methodology and a mathematical model have been developed that describe the mechanical behavior of the mesh implant installed in the abdominal cavity under conditions of non-stretching hernioplasty, which is under the influence of intra-abdominal pressure. The analysis of mechanical behavior for various sizes of working areas of a reticular implant under the conditions of various values of intraperitoneal pressures in a range from low (2 kPa) to high (20 kPa) pressures is carried out. The obtained results allow an assessment of the mechanical behavior of implants and their applicability for a clinical case depending on defect sizes in living tissues and expected intra-abdominal pressures.

**PNRPU Mechanics Bulletin**. 2019;(2):172-180

Studying of the influence of 3D wire deposition process parameters on the formation of residual deformations

#### Abstract

Additive technologies make it possible to manufacture products using a layer-by-layer synthesis, thus obtaining products of complex shapes. When solving a complex problem of numerical modeling of additive technological processes, it is necessary to describe various thermomechanical phenomena with high accuracy. The most effective in this regard is the use of a combination of capabilities of specialized software systems and the development of unique algorithms for them, taking into account the maximum possible number of process parameters. The paper considers the calculation algorithm of non-stationary temperature fields and stress-strain state of the structure in the process of its creation by 3D welding of wire materials developed and implemented using the ANSYS Mechanical package in the APDL language. In particular, this model takes into account the non-stationary radiant transfer of thermal energy of the welding arc to the surface of the product. The review highlights three of the most commonly used methods for modeling material deposition, i.e. the so-called element birth, sleeping element (quiet element) and hybrid activation (hybrid activation). It is shown that in order to ensure greater efficiency of calculations, it is necessary to use the principle in which successive steps of melting and even layers are grouped together for a subsequent simultaneous activationn. In the presented model, the task is divided into the boundary problem of non-stationary thermal conductivity and the boundary problem of thermomechanics of the stress-strain state, which are uncoupled. To solve them we applied the technology of killing and the subsequent birthing of a part of the material, implemented with the ANSYS package. Verification of the developed numerical algorithm aimed at solving the three-dimensional problem of the arc surfacing of wire materials was carried out by a comparison with the results of the experimental tests on D16 aluminum alloy samples. To describe the elastoplastic behavior of the alloy, the BISO model of bilinear isotropic plasticity with a temperature dependence of yield strength was used. A good consistency of the calculated data with the experiment was shown. The effect was studied on the level of the residual distortion of such process parameters as exposure time to the next layer, the motion path of the slicer, ambient temperatures. It is shown that the latter parameter is the most effective way to reduce residual shape deviations, but requires high thermal stability of equipment and accuracy of arc energy control.

**PNRPU Mechanics Bulletin**. 2019;(2):181-194

Modeling of Strands Formation in Elastomeric Composites

#### Abstract

Filling of rubbers with active fillers significantly improves their strength and deformation properties. One of the possible explanations of this phenomenon is presented in this article. It is based on a well-known fact that for large deformations of the elastomer binder filled in the gaps between neighboring filler particles is in the stress-strain state close to uniaxial tension. In this case, most of the polymer chains are oriented along the axis connecting the centers of inclusions. The paper suggests that the strength of the matrix in such a state (due to orientation) should be higher in comparison with other possible states with the same strain intensity. An dequate strength criterion was developed to account for this effect. The results of simulating the elastomeric binder destruction around two absolutely solid spherical inclusions are presented. A model of an incompressible hyperelastic material whose properties are given by a neo-Hookean potential was used to describe the properties of an elastomeric matrix. In the framework of computer experiments it was shown that when the system is deformed, the binder breaks should appear not in the gap between the filler particles, but at some distance from it. The elastic bond between the inclusions remains. A polymeric fiber (nanostrand) is formed between the inclusions, capable of withstanding higher tensile loads. It is known that layers with other physical-mechanical properties can be formed near the filler particles. Solutions are obtained for the problems in which the matrix in the gaps between the filler particles has a higher modulus to evaluate the possible influence of such layers. It is that this factor has virtually no effect on the emergence and formation of strands.

**PNRPU Mechanics Bulletin**. 2019;(2):195-202

Method of calculation of elastic effective properties of two-phase polydisperse media using multipoint statistical descriptors and the integral equations technique

#### Abstract

The aim of this study is to develop a new analytical approach to calculation of effective properties of elastic heterogeneous media based on multipoint approximations of solutions of stochastic boundary value problems. Prediction of macroscopic properties of heterogeneous media is associated with the need for a reliable description of their microstructural behavior, including the interaction between individual components. A number of analytical and numerical approaches have been developed to evaluate the effective properties of structurally inhomogeneous media. However none of them makes it possible to calculate the effective properties of such media with an absolute accuracy. One of the main limitations is imposed by taking into account the features of the microstructure of the medium, such as orientation, size, shape and distribution of inclusions, as well as the features of influence of the matrix on inclusions. In this paper, multipoint approximations of solutions of boundary value problems are used to calculate the effective characteristics, in which it is necessary to employ higher-order correlation functions, which allows to take into account the multiparticle interaction of microstructural elements with a higher extent. Analytical expressions for the calculation of the effective properties of structurally inhomogeneous media using multipoint higher-order approximations of solutions of stochastic boundary value problems in elastic formulations are obtained. A numerical comparison of the calculation results of relative effective characteristics of porous inhomogeneous polydisperse media with spherical inclusions of different volume fractions is performed. For the numerical solution of the integral-differential equations, a global adaptive strategy is applied in conjunction with the multidimensional integration rule and the IMT variable transformation rule to handle the singularity of the Green’s function. Some conclusions are made on the effectiveness and limitations of the proposed approach.

**PNRPU Mechanics Bulletin**. 2019;(2):203-214

Modeling the Dynamic Behavior of Spatially Reinforced Plates of Nonlinear Elastic Materials within the Refined Bending Theory

#### Abstract

Refined mathematical models are constructed for the flexible deformation of spatially reinforced flexible plates of nonlinear elastic materials of the composition components. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The obtained equations allow determining the deformed state of such structures with different degrees of accuracy taking into account their possible weak resistance to transverse shear. As a special case, based on these equations the relations of the traditional non-classical Reddy theory are obtained. The solution of the initial boundary value problem is obtained using an explicit numerical scheme of the “cross” type. It is shown that in a general case of plates an explicit numerical scheme cannot be developed for all spatial structures of reinforcement. The dynamic behavior is investigated for the flat and spatially reinforced plates of different shapes and relative thickness under the action of an air blast wave. It is shown that in the case of a strongly denominated anisotropy for relatively thick rectangular plates, the replacement of a flat structure with a spatial reinforcement structure reduces deflections modulo several tens of percent, and reduces the intensity of deformation of the components of the composition by several times. The reduction of the anisotropy degree of the composition and the relative thickness of the plates leads to a weakening of the effect of replacing the flat reinforcement structure on the spatial structure. This effect does not occur in annular plates with a rigid inner insert. On the contrary, the replacement of the flat reinforcement structure with the spatial structure leads to the deterioration of dynamic characteristics of such structures even of their relatively large thickness. It is shown that the dynamic behavior of plates calculated according to the Reddy theory significantly differ from the calculations according to the refined theory, especially when comparing the strain states of the components of the composition.

**PNRPU Mechanics Bulletin**. 2019;(2):215-228