## No 4 (2020)

**Year:**2020**Articles:**22**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/97**DOI:**https://doi.org/10.15593/perm.mech/2020.4

On stress-affected kinetics of intermetallic compound growth in the presence of electromigration

#### Abstract

This paper is concerned with the analytical modeling of an intermetallic compound formation in a eutectic tin solder joint on copper interconnects subjected to an electrical current. We propose a model that couples mechanical stresses, chemical reaction, diffusion, temperature, and electromigration. The kinetics of the chemical reaction fronts of the intermetallic phase formation is investigated based on the notion of the chemical affinity tensor within the small strain approximation. It allows incorporating the influence of stresses and strains on the chemical reaction rate and the normal component of the reaction front velocity in a rational manner. Electromigration is introduced into the model as an additional summand in the total flux of the diffusive constituents, which, in turn, also affects the reaction front velocity. In the considered model, the mechanical stresses arise due to the internal strains produced by the chemical transformation and by the thermal expansion. We formulate a model problem for planar reaction fronts. Within this model, the influence of stresses and electromigration on the reaction front kinetics is studied analytically. Based on the Mean-Time-To-Failure (MTTF) criteria, we calculate the critical thickness of the solder joint and estimate the amount of the accumulated vacancies. We introduce a dimensionless parameter, which characterizes the accumulation of vacancies due to electromigration enhanced diffusion. Finally, we discuss the coupling between the accumulated vacancies and Kirkendall void nucleation.

**PNRPU Mechanics Bulletin**. 2020;(4):7-14

Numerical analysis of the spherical bearing geometric configuration with antifriction layer made of different materials

#### Abstract

Requirements to critical elements of transport and logistics systems have been increased due urbanization in the territories of Russia and the world. Bridge bearings, which perceive the vertical and horizontal loads from the bridge span, as well as absorb thermal expansion and contraction, shrinkage, seismic disturbances, etc. refer to such elements. Requirements to strength, durability, wear resistance, operation maintenance-free periods, etc., imposed on the bridge bearing are increasing due to a stable growth of loads on the bridge elements and increase in vehicle fleets. Recently, international and Russian companies have been engaged in development of new polymeric and composite materials, which have improved physical and mechanical, frictional, thermo-mechanical and rheological properties and can be used as a thin layer of sliding bearings bridges. A number of problems are outlined in studying material properties and geometric configuration of bridge bearings in order to rationalize the work of its structure. Three topical problems of solid mechanics are reviewed in the work. This is the identification of qualitative and quantitative patterns of the deformation behavior of modern antifriction polymer and composite materials as relatively thin sliding layers of bridge spherical bearings in order to formulate scientifically grounded recommendations for the selection of the interlayer material regarding the study unit operation. This is an analysis of the influence of the sliding layer material physical and mechanical, frictional, thermomechanical and rheological properties on the structure deformation as a whole and the change in the contact zone parameters, in particular. It is an analysis of the influence of the sliding layer geometric configuration on the structure performance. A significant decrease in the area of full adhesions of the contact surfaces, including up to 0, and the occurrence or increase in the area of the divergence of the contact surfaces (no contact) is observed during frictional contact taking into account the lubrication on the mating surfaces. The surface percentage on which the contact surfaces divergence (no contact) is observed decreases, on average by more than 2 times, if the sliding layer thickness is increased.

**PNRPU Mechanics Bulletin**. 2020;(4):15-26

Research of the asymmetric rolling of workpieces

#### Abstract

The asymmetric rolling process has proven itself well as a way to reduce the pressure on the rolls, reduce the rolling force, and improve the mechanical characteristics of the rolled metal. As the factors providing the asymmetry of the process, the mismatch of the circumferential speeds of the work rolls, the different diameters of the rolls, the coefficients of friction, and others are usually used. Methods that provide a change in the nature of the metal flow due to the action of working elements with a special configuration of the working surface are especially promising. This article presents the results of the study of the stress state, speed and power parameters when rolling a strip in biconical rolls with concave and convex surfaces. Analysis of the results obtained by analytical methods show that the intensity of shear deformation rates along the strip width is 0.36-0.65 s-1, which is impossible to implement when rolling in smooth cylindrical rolls, since there is an intense elongation of grains in the direction of rolling. The occurrence of the intensity of shear deformations creates favorable conditions in the deformation zone to prevent stretching of the structure and to reduce dangerous tensile stresses. The results of the study showed the prevalence of compressive stresses in the deformation zone, which prevent the intensive elongation of grains in the longitudinal direction, reduce tensile stresses and contribute to the leveling of mechanical properties, closing and welding of internal defects. Theoretical dependencies are proposed to calculate the force parameters for asymmetric rolling in biconical rolls. The obtained models of the stress state, velocity hodograph, force characteristics predict the efficiency of using the biconical rolls in cold and hot rolling mills.

**PNRPU Mechanics Bulletin**. 2020;(4):27-35

Investigation of the Winkler foundation model applicability for describing the contact interaction of elastoplastic shells with a core under external pressure

#### Abstract

The problem of deformation and elastoplastic buckling of shells of revolution with a thick-walled elastic core under combined static and dynamic loading is formulated in a two-dimensional planar formulation based on two approaches: full-scale modeling within continuum mechanics and a simplified formulation based on the hypotheses of the theory of shells of the Timoshenko type and the Winkler foundation. Both approaches allow solving the problems of deformation and stability of non-shallow shells on the basis of Timoshenko's hypotheses, taking into account geometric nonlinearities. The statement from the perspective of continuum mechanics makes it possible to approximate the shell in thickness by a number of layers of finite elements. The constitutive relations are formulated in Lagrange variables using a fixed Cartesian coordinate system as a reference one. Kinematic relations are recorded in the metric of the current state. The elastic-plastic properties of shells are described by the theory of plastic flow with isotropic hardening. The equations of motion follow from the balance of the virtual powers of the work. In the first approach, the contact interaction of a shell and an elastic body is modeled by the conditions of nonpenetration along the normal and free slip along the tangent. The nonpenetration conditions are satisfied only in the active phase of the contact interaction; if the contact is broken, they are replaced by conditions on the free surface. In the second approach, the contact interaction of the elastic core with the shell is modeled by the Winkler foundation. Both approaches allow one to describe the nonlinear subcritical deformation of shells of revolution with an elastic core, to determine the limiting (critical) loads in a wide range of loading rates, taking into account the geometric imperfections of the shape. Using both approaches, a numerical simulation of epy contact interaction problem of an elastoplastic cylindrical shell with a thick-walled elastic core at a quasi-static uniform external pressure is carried out. The study of the influence of the thickness and initial deflection of the shell, as well as the stiffness and thickness of the core, on the value of the critical pressure and the form of buckling has been carried out. Based on these calculations, a conclusion was made about a wide range of applicability of the Winkler foundation model.

**PNRPU Mechanics Bulletin**. 2020;(4):36-48

Analysis of the fracture toughness parameters at the free edge in layered composites

#### Abstract

The problem of deformation and elastoplastic buckling of shells of revolution with a thick-walled elastic core under combined static and dynamic loading is formulated in a two-dimensional planar formulation based on two approaches: full-scale modeling within the framework of continuum mechanics and a simplified formulation based on the hypotheses of the theory of shells of the Timoshenko type and the Winkler foundation. Both approaches allow solving the problems of deformation and stability of non-shallow shells on the basis of Timoshenko's hypotheses, taking into account geometric nonlinearities. The statement from the perspective of continuum mechanics makes it possible to approximate the shell in thickness by a number of layers of finite elements. The constitutive relations are formulated in Lagrange variables using a fixed Cartesian coordinate system as a reference one. Kinematic relations are recorded in the metric of the current state. The elastic-plastic properties of shells are described by the theory of plastic flow with isotropic hardening. The equations of motion follow from the balance of the virtual powers of the work. In the first approach, the contact interaction of a shell and an elastic body is modeled by the conditions of nonpenetration along the normal and free slip along the tangent. The nonpenetration conditions are satisfied only in the active phase of the contact interaction; if the contact is broken, they are replaced by conditions on the free surface. In the second approach, the contact interaction of the elastic core with the shell is modeled by the Winkler foundation. Both approaches allow one to describe the nonlinear subcritical deformation of shells of revolution with an elastic core, to determine the limiting (critical) loads in a wide range of loading rates, taking into account the geometric imperfections of the shape. Using both approaches, a numerical simulation of contact interaction problem of an elastoplastic cylindrical shell with a thick-walled elastic core at a quasi-static uniform external pressure is carried out. The study of the influence of the thickness and initial deflection of the shell, as well as the stiffness and thickness of the core, on the value of the critical pressure and the form of buckling has been carried out. Based on these calculations, a conclusion was made about a wide range of applicability of the Winkler foundation model.

**PNRPU Mechanics Bulletin**. 2020;(4):49-59

The Tree-Level Viscoelastic Model: Analysis of the Influence of the Packing Defect Energy to the Response of Materials under Complex Loading

#### Abstract

The development of new and improvement of existing modes of thermomechanical treatment of metals and alloys in present conditions is impossible without development of appropriate mathematical models, that allow determining material characteristics during technological processes. Constitutive equations are the core, the main components that determine the quality of such models. Macrophenomenological theories of plasticity relying on processing the results of experiments on macrosamples, have become widespread as such in solving applied problems of solid mechanics. Taking into account the need to describe the memory of processes, the equations of this class have a complicated mathematical structure, require expensive tests (generally speaking, for complex loading) for each material, due to which they are not universal. In the past 15-20 years, constitutive models based on the introduction of internal state variables, of a multilevel approach, and physical theories of inelasticity (plasticity, viscoplasticity) became very popular. Models of this class are focused on describing the evolving structure (including microstructure), which ultimately determines the physical and mechanical properties of materials and constructions. As the physical mechanisms and their carriers are identical for wide classes of materials, the models of this class have significant versatility, including the prediction of behavior of new, not yet existing materials, to study the physical mechanisms of the occurrence of various effects, observed in macro experiments. Hardening is one of interesting effects observed in experiments on complex (including cyclic) loading (as compared to directional loading) of samples, made of various metals and alloys, arising from a significant evolution of the microstructure. Empirical data analysis made it possible to establish that the tendency to manifest this effect is usually experienced by metals and alloys with a low stacking fault energy (SFE). The paper provides a brief analysis of the experimental work and mathematical models describing the response of a material to complex deformation. It is noted that macrophenomenological theories do not allow one to describe in an explicit form the evolution of the microstructure and the carriers of plastic deformation and hardening mechanisms, thus they do not provide an opportunity to explain the physical reasons for the above effects. The purpose of this work is to develop, study and implement a multilevel elasto-visco-plastic model that allows describing the evolution of crystal lattice defects in materials with different SFE under different thermomechanical processing, different strengthening mechanisms at different structural-scale levels. In the framework of constructing a constitutive model, special attention is paid to the development of a submodel, focused on description of the evolution dislocations and barrier densities on slip systems. Kinetic equations for dislocation densities on slip systems make it possible to analyze the nucleation of dislocations due to the activation of Frank - Read sources, annihilation of dislocations of different signs on one slip system, interaction of split dislocations of intersecting slip systems with the formation of barriers. Relations for the description of hardening are given, taking into account the current density of dislocations and barriers. The general structure of the model and the relationship between the parameters of submodels of different levels are considered. An algorithm and a program of implementing the model were developed, the evolution of dislocation densities on slip systems was analyzed, and the intensity of hardening and the formation of barriers on split dislocations were obtained depending on the type of loading.

**PNRPU Mechanics Bulletin**. 2020;(4):60-73

Investigation of damage accumulation and delamination propagation in polymer composite materials based on two-level fracture models

#### Abstract

The work is devoted to the study of deformation and fracture processes occurring in layered composites under combined loading modes. The aim of the work is numerical analysis of different modes of fracture, which are simultaneously realized in the samples of laminated composite material. Models of laminated composite material with imitation of technological defects in the form of material debonding are constructed. The delamination process is implemented using the virtual crack closure technique. The processes of damage accumulation and fracture of laminated composites are set on the basis of the models for reduction of stiffness properties using the Hashin criterion and Matzemiller model. The models are based on the laws corresponding to brittle and plastic fracture. Several models of fracture and degradation of elastic properties have been compared. A multiscale approach was used to solve the difficulties related to the precise description of the composite's internal structure. The essence of the approach is that the analysis of a laminated composite can be performed on three different scales: macro level, meso level and micro level. At the macro level, an equivalent material is used for which the effective properties are determined by homogenization methods, in particular by the mean field method. The multiscale finite element modeling is implemented, in the course of which macroscopic parameters of material sample at each step depend on characteristics and properties of components at the micro-level. The behavior of two samples of laminated polymer composite material was studied with different configuration of embedded defects under the load of two types: uniaxial compression and torsion, and only uniaxial compression. The influence of internal defects on the processes of damage accumulation and material delamination has been established.

**PNRPU Mechanics Bulletin**. 2020;(4):74-85

Numerical simulation of deformation and fracture of metal-matrix composites with considering residual stresses

#### Abstract

Thermomechanical behavior of metal-matrix composite materials is investigated. Boron carbide B4C and high-strength aluminum alloy 6061-T6 are used as strengthening particle and matrix materials, respectively. Microstructure of the metal-matrix composite takes into account the complex shape of particles explicitly. Isotropic elastoplastic and elastic-brittle models were used to simulate the mechanical response of the aluminum matrix and ceramic particles, respectively. To investigate the crack initiation and propagation in ceramic particles, a Huber type fracture criterion was chosen that takes into account the type of the local stress state in ceramic materials: bulk tension or compression. The composite material with a single particle of both the really observed in the experiment and ideally round shapes is considered. The influence of the residual thermal stresses arising during cooling of the composite material from the temperature of aluminum recrystallization to the room temperature on the character of plastic strain localization in the aluminum matrix and fracture of carbide particles and on the macroscopic strength of the composite under external tension or compression is studied numerically. Two-dimensional dynamic boundary value problems in the plane-stress and plane-strain formulations were solved numerically by the finite element method using the Explicit module of the Abaqus software package. VUMAT subroutine procedures incorporating the constitutive models were developed and integrated into the Abaqus solver. Based on the results of the numerical simulation, it was concluded that the residual thermal stresses arising during cooling lead to the change in the mechanism of the particle fracture from in-particle cracking to debonding and increase the strength of the composite subjected to tension after the cooling.

**PNRPU Mechanics Bulletin**. 2020;(4):86-96

On a modeling stress-controlled surface growth of solids

#### Abstract

Various processes are associated with the surface growth of solids, such as biological growth, formation of surfaces, processes accompanying additive technologies. Experiments show that the growth process of living and non-living matter can be controlled by external influences, including mechanical ones. This paper presents a surface growth model based on the expression for the configurational force obtained from the fundamental balances of mass, momentum and energy, and the second law of thermodynamics in the form of the Clausius-Duhem inequality. It is shown that the configurational force is the normal component of the tensor, called the surface growth tensor, which controls the processes of growth and adaptation to external mechanical loads. A kinetic equation in the form of the dependence of the growth rate on the growth tensor is formulated. A solid body is considered, in which a volumetric supply and subsequent diffusion of matter to the growth boundary occur. On the surface of the body, the transformation of one substance into another occurs, resulting in surface growth or resorption of the body. The surface growth process depends on the stress-strain state of the body and the concentration of the diffusing matter. In the process of growth, stresses and deformations change, affecting the configurational force and the rate of the matter supply, which also affects the configurational force. In addition, the model takes into account the growth strains that can occur in new layers of the material and affect the growth velocity. Thus, there is a coupled problem including the description of the supply, diffusion and growth processes and determination of the stress-strain state. The model was used for the problems of surface growth of various bodies under various loading conditions.

**PNRPU Mechanics Bulletin**. 2020;(4):97-106

Inelastic behavior and destruction of materials under isothermal and non-isothermal, simple and complex loads

#### Abstract

The paper deals with mathematical modeling of inelastic behavior and destruction of structural materials (steels and alloys) under simple, complex, isothermal and non-isothermal loads in repeated and long-term exposures to thermomechanical loads. The modeling is carried out on the basis of the applied theory of inelasticity, which belongs to the class of flow theories in combined hardening. The main provisions are formulated and a summary of the main equations of the applied theory of inelasticity is given. The material functions closing the applied theory of inelasticity are determined, and the connection of the defining functions with the material ones is given. Further, the results of some original experimental studies are considered, which are compared with the results of calculations based on the applied theory of inelasticity. In all studies, inelastic deformation is performed under conditions repeated and long-term exposures to thermomechanical loads. Inelastic deformation of AL-25 aluminum alloy samples under uniaxial tension-compression under both isothermal and non-isothermal cyclic loading is considered. Inelastic deformation under complex loading along the two-link polyline deformation paths with different deformation rates under high temperature conditions is studied on tubular 30HGSA alloy samples. Inelastic deformation of tubular stainless steel 304 samples under complex loading at elevated temperatures is considered. Soft cyclic loading is performed along two-link stress trajectories with different fracture angles. At the end of the links of the stress trajectory, exposure is carried out for 8 hours. The results of the calculations based on various theories used in the calculations are analyzed. Inelastic deformation and destruction of samples made of 12X18N9 stainless steel under rigid cyclic deformation under both isothermal and non-isothermal loads is considered. The duration of the loading cycle is 4 minutes, which allowed the effects of healing and embrittlement to appear at a high temperature. There is a significant difference (much higher) in the number of cycles to failure in common-phase and anti-phase modes of changes in force strain and temperature.

**PNRPU Mechanics Bulletin**. 2020;(4):107-119

Damage development under very-high-cycle fatigue regime

#### Abstract

A method of testing metallic materials using piezoelectric elements under very-high-cycle fatigue (VHCF) regime is described in the paper. The scheme and structure of a high-frequency fatigue machine are discussed. For the experiments, the first mode of resonant longitudinal vibrations is calculated, corresponding to the natural frequency of the cylindrical corset sample. The results of some experiments on very-high-cycle fatigue fracture of VT3-1 titanium alloy samples at various stress ratio coefficients are presented. Mathematical modeling of the process of fatigue damages development in VHCF is carried out. For this, a bimodal representation of the fatigue curve is used. This bimodal representation contains two branches: the left branch corresponds to the classical low-cycle and high-cycle fatigue modes, and the right branch describes the very-high-cycle fatigue regime. On this basis, a kinetic model was built, associated with the known criterion of multiaxial fatigue fracture SWT, which contains a mechanism associated with the development of normal opening microcracks. This kinetic model of damage development can be used to calculate various modes of fatigue failure from low-cycle to very-high-cycle fatigue. On the basis of this model, a numerical method of solving the evolutionary equation for the damage function has been developed. The fatigue fracture of the material in this model corresponds to the degradation of its elastic moduli due to an increase of the damage function. Calculations of the crack-like zones development for fatigue fracture in the titanium corset specimens at different stress ratio coefficients for cyclic loading, used in the tests on a piezoelectric machine, was fulfilled. To test the performance of the model, a comparison between the experimental and calculated fatigue curves in the region of the VHCF was carried out.

**PNRPU Mechanics Bulletin**. 2020;(4):120-129

Estimating of mechanical stresses, plastic deformations and damage by means of acoustic anisotropy

#### Abstract

Acoustic anisotropy is a consequence of anisotropy of the mechanical characteristics of a solid. In metals, it is associated with microstructural anisotropy of mechanical characteristics, internal mechanical stresses and strains, including residual stresses and plastic deformations. Sensors measuring acoustic anisotropy do not require complex preparations of a metal surface, therefore it is easy to measure which makes it possible for measurement results to be used to quantify stresses and strains in metals based on the magnitude of phase shifts of the shear wave velocities of the orthogonal polarization. Acoustic anisotropy is one of the manifestations of the phenomenon of changes in the elastic properties of an acoustic medium caused by mechanical stresses and deformation (acoustoelastic effect). This makes it possible to use the effect of acoustic anisotropy for the development of quantitative methods of acoustic tensometric measurements, as well as methods of non-destructive testing, which enables effective quality controls and diagnostics of the residual life of structures and machine parts. The article describes the history of the discovery and theoretical substantiation of the acoustoelastic effect and the quantitative relationship of acoustic anisotropy with stresses and deformations, starting with the pioneering works of the twentieth century. The way of forming the theory based on nonlinear mechanics of continuous media is shown. The third part of the article is concerned with an overview of the current state of research. An analysis is presented of experimental works on the measurement of acoustic anisotropy in low- and high-carbon steels, aluminum alloys, as well as in composites and other structural materials. Special attention is paid to a review of studies on the relationship between acoustic anisotropy and plastic deformations and the applicability limitations of the acoustic method. It also provides a list of the main applied results related to the measurement and use of acoustic anisotropy to control the blades of compressors and gas turbine engines, pipe steels, welded joints, etc. A review is given of the main publications on system analysis and generalization of theoretical and experimental scientific results obtained by domestic and foreign researchers in the field of studying the acoustic anisotropy of metallic structural materials under conditions of uniaxial and complex stress states, plastic deformation, thermomechanical loading and fatigue fracture is given.

**PNRPU Mechanics Bulletin**. 2020;(4):130-151

On the specifics of behavior of the sandwich plate composite facing layers under local loading

#### Abstract

The problem of a four-point bending of sandwich plates with external layers of a fiber reinforced plastic is considered, the results of numerical and experimental studies are presented. It is shown that in the vicinity of the loading roller, which exerts a local effect on the external layer, there is a strong decrease in the transverse shear secant modulus of fiber reinforced plastic with an increase in the transverse shear strains. The numerical solution of the problem of the plate bending is carried out in a physically and geometrically nonlinear formulation using various relations of the finite element method, two variants of geometrically nonlinear kinematic relations of the equations of the elasticity theory and different variants of the loading process parameter. Along with the classical nonlinear relations, the problem solutions are also constructed on the basis of consistent relations between strains and displacements, the use of which allows one to avoid the appearance occurrence of false bifurcation points. The results of the numerical calculations obtained with different methods and use of different ratios are presented, the analysis of which showed their small difference. It was revealed that the stability loss and failure of the external layers of a sandwich plate occurs due to the stability loss in the nonclassical transverse-shear mode. To determine the ultimate load, which is accompanied with a loss of strength of the external loaded layer, the Tsai-Wu criterion was used. A comparative analysis of the behavior of the plate external layers at different thicknesses and different diameters of the loading roller is carried out. It is shown that the ultimate load is practically not affected by the roller diameter, while the load at which the external layer loses its stability, is very sensitive to a change in its value.

**PNRPU Mechanics Bulletin**. 2020;(4):152-164

On modeling of airflow in human lungs: constitutive relations to describe deformation of porous medium

#### Abstract

Within the framework of a multilevel mathematical model to describe the evolution of functional disorders of the human organism under the influence of environment factors, a mathematical model of the "meso-level" of the human respiratory system is developed. The article is deals with the development of the meso-level model - the formulation of a constitutive model to describe the airflow in a porous lung medium. Human lungs filled with small airways and alveoli, with air contained in them, are modeled by an elastically deformable saturated porous medium enclosed in an internal chamber with varying volume (movable walls). Conceptual and mathematical statements are presented. Air movement in the deformable porous medium of lungs is described by ratios of the mechanics of deformable solid body and filtration theory. As an element of this sub-model an analytical solution is obtained for an auxiliary geometrically linear problem of the all-round compression of an elastic thin-walled hollow sphere filled with air to determine the rate of mean stress of the two-phase medium of the lungs, taking into account the interaction between the lung tissue and the air contained in the lungs. To confirm the hypothesis on the acceptability of a linear solution of an auxiliary problem for large deformations, a similar problem was numerically solved in a geometrically nonlinear formulation. The results show that the obtained analytical solution is in satisfactory agreement with the solution of a similar problem in a nonlinear formulation both for calm and deep breathing, which indicates the possibility of using the former in the construction of the considered sub-model.

**PNRPU Mechanics Bulletin**. 2020;(4):165-174

Modeling of deformation and fracture of porous media taking into account their morphological composition

#### Abstract

This paper investigates the mechanical behavior and fracture of porous materials with an aluminum matrix. The purpose of the work was to create numerical models of failure of representative volume elements of such materials and to reveal the dependences of the nature of the failure processes on their structural morphology. Representative volume elements of these materials are random non-uniform structures of closed-cell and open-cell types. To create three-dimensional geometric models of the closed-cell structures, methods of sequential synthesis the possibility of their mutual intersection were used. For creation of models of interpenetrating structures of the open-cell type, methods based on the analytical determination of surfaces separating the two phases are used. In this paper, three approaches to fracture mechanics of representative volume elements of porous materials were studied and implemented. The first approach is an implementation of the elastic model and damage accumulation based on elastic properties degradation in accordance with the criterion of maximum stresses with reduction of the stiffness matrix coefficients in individual elements. The second approach is an implementation of the same model, but with removal of the failed elements. The third approach is based on the Johnson-Cook elastic plastic behavior and fracture model. Numerical modeling of the representative volumes was carried out with finite element analysis using each of the above approaches. The influence of the internal structure of the representative volumes of the porous materials on the processes of deformation and failure was studied on the example of several structures of open-cell and closed-cell types. The influence of stress concentrators on the distribution of stresses in representative volumes and character of their subsequent failure has been studied.

**PNRPU Mechanics Bulletin**. 2020;(4):175-187

Study Smart-layer effect on the physical and mechanical characteristics of the samples from polymer composite materials under quasi-static loading

#### Abstract

Currently, developments of the so-called Smart-constructions are relevant as they enable a real-time monitoring of changes in required values. Smart designs are widely used in the construction, automotive and aerospace industries. Technologies of creating products from polymer composite materials make it possible to introduce various sensors directly into the structure of a material, thereby create systems monitoring the state of structures. The most recommended for such implementation are fiber-optic sensors, which have a number of advantages over other sensors (luminescent, strain gauge, piezoelectric ones). However, when introducing the fiber-optic sensors, there is a number of difficulties, which are primarily associated with fragility of the optical fiber and lead to the breakdown of fiber-optic lines. As a result, it is necessary to develop a Smart-layer that will protect the optical fiber leads and will not significantly change the physical and mechanical characteristics. This paper aims to determine the stiffness and strength characteristics of samples made of polymer composite materials: reference samples, samples with embedded fiber-optic sensors, samples with embedded Smart-layers. In this work, a Smart-layer is understood as a coating that protects the fiber-optic sensors at the stage of implementation into a structure. The paper considers the following configurations of the Smart-layer: polymer reinforced mesh, polyamide and polyurethane layer. We analyzed and compared the influence of the embedded optical fiber and various configurations of the Smart-layer in the composite structure on the physicomechanical characteristics of the samples obtained under quasi-static loading (tension, compression, and interlayer shear). For a more detailed analysis of using the fiber-optic sensors and various configurations of the Smart-layer, the corresponding loads were simulated to assess their mechanical behavior. Based on the obtained physical and mechanical characteristics, a specific configuration of the Smart-layer was selected and justified for further researches.

**PNRPU Mechanics Bulletin**. 2020;(4):188-200

Propagation of non-stationary antisymmetric kinematic perturbations from a spherical cavity in Cosserat medium

#### Abstract

We consider a space filled with a linearly elastic Cosserat medium with a spherical cavity under given nonstationary antisymmetric surface perturbations, which are understood as the corresponding analogue of classical antiplane deformations. The motion of a medium is described by a system of three equations with respect to nonzero components of the displacement vector and potentials of the rotation field, written in a spherical coordinate system with the origin at its center of the cavity. The initial conditions are assumed to be zero. To solve the problem, we use decomposition of functions to Legendre and Gegenbauer polynomials, as well as the Laplace transform in time. As a result, the problem is reduced to independent systems of ordinary differential equations with the Laplace operator for the coefficients of the series. A statement about the structure of the general solution of this system is formulated. Images of the series coefficients are presented in the form of linear combinations of boundary conditions with coefficients - transformants of surface influence functions, the explicit formulas for which include the Bessel functions of a half-integer index. Due to the complexity of these expressions, to determine the originals in the linear approximation, the method of a small parameter is used, which is taken as a coefficient characterizing the relationship between the displacement and rotation fields. Then, taking into account the connection between the Bessel functions and elementary functions, the images are written in the form of linear combinations of exponentials with coefficients - rational functions of the transformation parameter. The further procedure for inverting the Laplace transform is carried out using residues. It is shown that there are three wave fronts corresponding to a shear wave modified with allowance for free rotation and two rotation waves. Examples of calculations for a granular composite of aluminum shot in an epoxy matrix are presented.

**PNRPU Mechanics Bulletin**. 2020;(4):201-210

Structural elements based on the metamaterials

#### Abstract

The article discusses an approach to creating structural elements by forming periodic structures in the structure, developed based on the results of topological optimization. In the article, a metamaterial is understood as a structure with a complex internal periodic organization of strength elements, the details of which are significantly smaller than the typical dimensions of the final structural product. In this paper, the analysis is devoted to panels with a filler based on periodic structures to achieve the required mechanical characteristics. The transition from the results of the topological optimization is carried out on the basis of engineering analysis, taking into account the particularities of loading, fastening and operational effects on the structure. The use of topology optimization makes it possible to determine the distribution density of periodic structures in the material and to shorten the design cycle of a conventionally optimal design. As a first step solution, authors consider panels based on sandwich panels with the pyramidal fillers. Their application is considered in the aircraft, shipbuilding and construction industries. As basic technological solutions, efficient technologies are proposed - laser radiation sources and a high degree of automation. With these technologies, efficiency and costs of testing and certification of manufacturing are reduced in comparison to the standard approaches, when results of the topology optimization are made using expensive additive manufacturing. The proposed elements make it possible to reduce the metal consumption while achieving the same rigidity and strength of the structure. Another advantage of the proposed structures is their modularity and the ability to optimize the panel filling density without significantly changing the manufacturing process and design. As an application, we considered the possibility of creating a large-span panel for civil constructions, which is characterized by high specific loads with a significant span length (20 m).

**PNRPU Mechanics Bulletin**. 2020;(4):211-219

Finite element method with account of singularity for mixed mode cracks

#### Abstract

The paper deals with obtaining an analytical solution for stiffness matrix coefficients at a crack tip area for mixed mode cracks in plane strain conditions. The numerical study is focused on an infinite plate with a straight-through central crack under mixed loading. Analytical solutions are obtained as kinematic boundary conditions for plane strain. We analyzed distribution features of the stress-strain state fields and stress intensity coefficients at the top of the crack area, determined using the finite element method taking into account the singularity. The analytical formulas are obtained which set the kinematic conditions for a general and special case of loading a plate with a defect in the elastic setting for the case of plane deformation. The comparative analysis of the numerical results is presented for two cases of forming the design diagram of the top of the crack: the traditional method of creating a mathematical cut and the finite element method taking into account the singularity. The advantage of using the finite element method considering the singularity is found. We used an example of a plate with a through straight rectilinear central crack with the equal biaxial tension to show that setting the boundary conditions at the top of the crack taking into account the singularity allows one to significantly reduce dimensions of a calculation scheme of the finite element method and keep the calculation accuracy. It is concluded that such a formulation can be applied in an elastic-plastic formulation. The comparison between the classical finite element solution and finite element with singularity is presented. The convenience of the finite element method with singular boundary conditions is demonstrated.

**PNRPU Mechanics Bulletin**. 2020;(4):220-236

Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I

#### Abstract

This study aims at obtaining coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending by the use of the digital photoelasticity method. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give a more accurate estimation of the near-crack-tip stress, strain and displacement fields and extend the domain of validity for the Williams power series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. By means of the developed tool, the fringe patterns that contain the whole field stress information in terms of the difference in principal stresses (isochromatics) are captured as a digital image, which is processed for quantitative evaluations. The developed tool allows us to find points that belong to isochromatic fringes with the minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of the stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. The iterative procedure of the over-deterministic method is utilized to find the higher order terms of the Williams series expansion. The procedure is based on the consistent correction of the coefficients of the Williams series expansion. The first fifteen coefficients are obtained. The experimentally obtained coefficients are used for the reconstruction of the isochromatic fringe pattern in the vicinity of the crack tip. The comparison of the theoretically reconstructed and experimental isochromatic fringe patterns shows that the coefficients of the Williams series expansion have a good match.

**PNRPU Mechanics Bulletin**. 2020;(4):237-249

The influence of surface effects in the problem of theory of elasticity for a circular hole in a half-plane

#### Abstract

Surface effects are important for modeling structures, such as nanofilms, nanoporous materials, and other nanoscale constructions. In the current study, we consider the problem of the theory of elasticity - the problem of a half-plane containing a circular hole, stretched by constant stresses applied at infinity, and take into account surface effects such as surface elasticity and surface stresses. The problem solution has been obtained by expanding the Fourier series with the variables written in the bipolar coordinate system (which simplifies the problem solution because one of the coordinates becomes a constant on the hole contour), where the stress components are expressed through a bi-harmonic stress function. The parametric coefficients involved in the solution, namely in the Fourier series, are determined in order to satisfy the boundary conditions on the hole contour. To solve the problem, in addition to the equations of the theory of elasticity, the equations of surface elasticity were used, in particular, by applying the generalized Young-Laplace’s law and the Shuttleworth’s law; the surface stress on the hole contour has been calculated directly. Using recurrence relations for the stress components at the boundary, stress concentration values have been obtained. The resulting expressions can be considered as a generalized solution of the problem in case of the classical elasticity. The stress concentrations are compared for the cases with and without taking into account surface effects at various points on the hole contour. The contribution caused by the surface effects depending on the relative distance between the hole and the half-plane boundary is studied. It is shown that despite a quite simple geometry, owing to the fairly small distance between the hole and the half-plane boundary, the stress concentration with and without taking into account the surface stress are significantly different from each other, due to the significant contribution of surface effects.

**PNRPU Mechanics Bulletin**. 2020;(4):250-259

Dynamic characteristics of three-layer beams with load-bearing layers made of alumino-glass plastic

#### Abstract

Aluminum-fiber-reinforced plastics (GLARE) are promising aviation materials with increased characteristics of specific stiffness and strength, fatigue strength, impact resistance, and residual strength after impact. Nowadays, GLARE are used for making elements of the fuselage of the long-haul passenger aircraft Airbus A380, as well as some other elements of aircraft structures. The results of experimental studies of eigenfrequencies and damping coefficients of three-layer beams made with face sheets of five-layer GLARE and with a core made of polyimide foam are presented. The tests were performed using the method of free bending vibrations of cantilever samples. The dynamic parameters of the three-layer beams were calculated based on the analysis of amplitude-frequency characteristics obtained by the fast Fourier transform method. Mechanical characteristics of GLARE and filler samples are preliminary determined in static and dynamic tests. The damping factor of the core is determined by the dynamic mechanical analysis method. The core shear modulus is determined by measuring the flexural rigidity of manufactured three-layer beams in a quasi-static three-point bending test. Based on a comparison of the design data and the results of the experiments, it is shown that in dynamic tests, the flexural rigidity of three-layer specimens is reduced in comparison with the estimated values, which may be due to the peculiarities of changing the characteristics of the foam core under dynamic loading. The value of the damping factor of GLARE samples was ~0.02, the foamed core was ~0.08 and three-layer beams were ~0.067 in the range of vibration frequency up to 60 Hz.

**PNRPU Mechanics Bulletin**. 2020;(4):260-270