## No 3 (2016)

**Year:**2016**Articles:**19**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/19**DOI:**https://doi.org/10.15593/perm.mech/2016.3

Surface SH-Waves in Pre-Stressed Piezoelectrics with Functionally Graded

#### Abstract

A model of a ferroelectric structure consisting of a homogeneous piezoactive half-space with an inhomogeneous coating which is either a layer or a packet of homogeneous or functionally graded piezoactive layers is suggested. It is assumed that the half-space, as well as the coating, being piezoelectrics of the hexagonal syngony of a 6 mm class in their intrinsic state, are in the conditions of the action of the initial mechanical stresses. Investigations of dynamic properties of the functionally oriented pre-stressed structures are carried out in Lagrangian rectangular coordinate system. Linearized constitutive relations and motion equations are used. The boundary problem for the system of differential equations in partial derivatives is reduced to the system of ordinary differential equations by means of the operational calculus. In case of the structure homogeneous components, Green’s function is constructed analytically in a closed form on the basis of solving the system of differential equations with constant coefficients. In case of inhomogeneous (functionally graded) components, the system of differential equations with variable coefficients is reduced to the system of Cauchy initial problems by means of a special substitution. In this case, Green’s function is constructed numerically on using Runge-Kutta numerical methods with Merson modification, which allows us to control the error of calculations in an effective way. When constructing Green’s function of the ferroelectric structure with an inhomogeneous coating, we used the matrix approach that allows us to combine analytical and numerical methods of constructing its isolated components. The influence of the type and the value of the initial stresses on the peculiarities of the surface wave propagation in heterostructures is studied. It became possible to determine the conditions under which the action of the initial mechanical stresses leads to the increase of Bleustein-Gulyaev wave velocity with respect to the velocity of the original material, as well as the conditions under which the piezoelectric structure ceases to be a weakly inhomogeneous one.

**PNRPU Mechanics Bulletin**. 2016;(3):7-27

Non-stationary axisymmetric waves in electromagnetoelastic space with a spherical cavity

#### Abstract

We consider the associated non-stationary problem of propagation of axisymmetric disturbances from a spherical cavity in electromagnetoelastic space. It is assumed that the medium is a homogeneous isotropic conductor. Linear equations of motion of an elastic medium are used taking into account the linearized Lorentz forces, as well as Maxwell's equations, together with the linearized generalized law. The initial conditions are zero, at the boundary of the cavity defined displacement and the tangential component of the electric field. The desired functions are arranged in series of Legendre and Gegenbauer polynomials, as well as in series according to a small parameter characterizing the connection of mechanical and electromagnetic fields. Apart from that the applicable Laplace transform in time is used. The result is a recurrence of the small parameter sequence of boundary value problems, the solution of which is represented in the integral form with kernels in the form of volume and surface Green's functions. Images of Green's functions are found in an explicit manner. Their "elastic" part due to the relation between the modified Bessel functions and elementary functions is reduced to the sum of products of rational functions of the parameter of the Laplace transform to the exponent that lets you find exactly the originals using the corresponding theorems of operational calculus. The “Electromagnetic” part of the Green's function is being constructed in a quasi-static approximation. As a result, in the space of the original resolution of the system is became possible to build recurrence equations which allows finding and moving all the components of the electromagnetic field. In calculating its constituent integrals quadrature formulas are used. The examples of computations are provided. The numerical study of the convergence of series in the small parameter is presented.

**PNRPU Mechanics Bulletin**. 2016;(3):28-46

Dynamics of partially saturated poroelastic solids by boundary-element method

#### Abstract

The paper describes the mathematical model for partially saturated porous media based on the Biot’s model with five basic functions to characterize wave process. The mathematical model of the boundary value problem for the three-dimension dynamic theory of poroelasticity is given in Laplace transform. On the basis of the operational calculus theorem about the original integration, the step method of numerical inversion of the Laplace transform is presented. The direct method of boundary integral equations is selected to solve value problems of the three-dimensional dynamic poroelasticity theory, the corresponding boundary integral equation is given. The corresponding matrices of fundamental and singular solutions of the three-dimensional dynamic poroelastic theory are given. A brief description of the boundary-element discretization is presented. Methodological assurance is based on the regularized boundary integral equation usage. The regularized boundary integral equations are written considering the problem of symmetry. The boundary surface of the investigated solid is divided by generalized eight-node quadrangular elements. The consistent elementwise approximation is used. Collocation solution points of the boundary integral equation coincide with the interpolation nodes of unknown boundary functions. To increase the integration accuracy for an element not containing the collocation point, the hierarchical integration algorithm and Gauss integration formulas are applied. Arising discrete analogs are solved by the Gauss-based step process to obtain the values of the boundary functions. Step process is determined by a step algorithm of the numerical Laplace transform inversion. The problem of a unit surface force shock on the free end of the prismatic partially saturated poroelastic solid is considered. Sandstone is selected as the porous material. An analytical solution of the corresponding one-dimensional problem is used for the boundary element model verification. The dependence of computational grid on convergence of the problem solution is investigated; and the impact of step scheme parameter on the solution is examined.

**PNRPU Mechanics Bulletin**. 2016;(3):47-61

Computational and experimental research of the contact debonding process when rigid reinforcement is pressed into concrete

#### Abstract

Flexible fittings with corrugated surfaces are traditionally used to improve the bond and minimize its debonding with concrete. But rigid I-beam reinforcements, the type required for heavy load bearing, are not corrugated. The onset of the debonding process that occurs when rigid I-beam reinforcement was pressed into concrete was established through comprehensive calculations and experimentation. The aim of the paper was to define the parameters and types of fractures in the “steel-concrete” contact zone. The Cohesive Zone Material Model (CZM), provided by ANSYS Workbench software, was used with the bilinear behavior law on the contact layer and described the separation of “steel-concrete" surfaces. A mathematical model of the contact boundary problem was solved by the finite element method. A comparison of calculations and experimentation confirmed that the bond for smooth rigid reinforcement for B35 class concrete is provided by adhesion; and it is best described by the CZM model. Distribution patterns of contact pressure in the contact zone and of shear stresses on the concrete faces and surfaces adjacent to the I-beam were revealed. The relative displacement of concrete when impacted by an external force was measured. Results of this investigation indicate that the mathematical model is consistent and reflects the experimental data.

**PNRPU Mechanics Bulletin**. 2016;(3):62-75

Influence of first and second phase transitions on the form of shock adiabate in condensed bodies

#### Abstract

The crossing range of a shock adiabate and a melting curve has been considered. The melting curve has been presented by Simon equation. The effect of a first kind phase transition on the form of the shock adiabate has been studied as in the case when the shock adiabate of the solid crosses its melting curve. The crossing of the adiabate of solid and the melting curve determines the beginning of melting state. The crossing of the adiabate of a liquid and melting curve determines the state of the melting termination. The crossing of the shock adiabate and the critical isotherme of a date substance is the example of the effect related to the second kind phase transition on the form of the shock adiabate. The generated model illustrates the behavior of low-melting metals: Pb, Bi, Cd and Sn II. The experimental data of the shocked substances has been analyzed. It is noted that the singularities in change of properties vs temperature possess place upon the critical temperature. The model results have been confirmed by the well-known experimental data. The crossing of the shock adiabate of solid and the melting curve produces a discontinuity of the shock adiabate. A derivative of shock velocity on particle velocity in a liquid range decreases. The crossing of the shock adiabate and the critical isotherme produces a break of a relationship of the shock-particle velocities. The singularities in a change of the properties vs the temperature take place whenever the crossing of the shock adiabate and the critical isotherme is present. There following singularities occur: a break curve, a curve bend, a minimum and a maximum.

**PNRPU Mechanics Bulletin**. 2016;(3):76-96

Applicability of Reversed Experiment Technique to Determine Dynamic Characteristics of Saturated Soils

#### Abstract

The problem of determining the penetration resistance of the flat end projectile into dry and water-saturated sandy soils with the use of the reversed experiment technique is studied in the paper. In the experiment, the container with soil strikes the end of the measuring rod, and the resistance force is determined based on the strain gauge readings on the surface of the rod at a distance from its end. The propagation of pulse compression resulting from the impact of the rod flat end against the container with the water-saturated and dry sandy soils. The compressibility of the soil is described with Hugoniots, and the shear properties of the soil are described with a rational dependence of yield strength of the pressure. Hugoniots of the soils with different water saturation were obtained with the use of a multi-component model of the medium. The dependences of the resistance force of the cylindrical projectile on the time in dry, wet and water-saturated sandy soils were obtained numerically. In wet soils, the the non-stationary phase of power pulse was shorter than in dry soils. The analysis of errors was carried out in determining the forces acting on the projectile, from the values of the strain pulse on the surface of the rod. The effect of geometric dispersion during propagation along the rod pulse compression with a wavelength comparable to the radius of the cylinder was numerically shown. In order to restore the pulse at the end of the rod according to its values on the surface at a distance from the load point, the modified techniques were used after adjusting for variance and with an additional view of the uneven deformation distribution over the cross section of the rod. The distortions of the restored pulse were pointed out, the dependence of errors in determining the maximum value of the duration on the non-stationary part of the initial pulse was received. The accuracy of determining the quasi-stationary value of the resistance implementation after the introduction of amendments related to the dispersion both in dry and in water-saturated soils.

**PNRPU Mechanics Bulletin**. 2016;(3):97-107

Yarn-level modelling of woven and unidirectional thermoplastic composite materials under ballistic impact

#### Abstract

Composite materials made of high-strength fibres (for example, aramid or UHMWPE) are extensively used in such protective structures as bulletproof vests, helmets, etc. Many researchers have carried out numerical simulations of ballistic impact on composite laminates applying continuum, multiscale and mesoscale approaches. The continuum approach requires a little computational time but cannot catch all features of composite panel or fabric plies behaviour during high-velocity impact. Thus, using the mesoscale and multiscale models has recently been increased. In this paper, mesoscale approach was used to simulate a 6.35 mm steel ball impact on two types of hot-pressed thermoplastic composites with LS-DYNA finite-element code. The first type of the composite panel is made of aramid fabric KV110P (plane weave structure) with LDPE matrix. The second one was Dyneema® HB80 UD laminate. The proposed models of the real-sized panels were based on the combination of shell (for yarns) and solid (for resin) elements with common nodes to reduce an overall number of contacts and CPU time. The yarn-level modelling allowed using simple material models and fracture criteria. The models reflect the main failure modes in the real panels including the fracture of fibres, delamination, fabric/matrix debonding, yarns pull-out, etc. The experimentally obtained ballistic curves were used to validate results of the numerical simulations.

**PNRPU Mechanics Bulletin**. 2016;(3):108-119

Static and dynamic analysis for 3D problems of linear magneto-electro-elasticity using BEM

#### Abstract

Magneto-electro-elastic materials have drawn increasing attention due to their magnetic-electric-mechanical coupling effect. They have the ability to convert the energy from one type to another and have a wide range of technical applications. This paper presents a Laplace domain direct boundary element formulation for static and transient dynamic problems of three-dimensional linear magneto-electro-elasticity. The standard contracted notation is used to express the coupled problem in the elastic-like fashion. The formulation is based on the displacement boundary integral equation. The Laplace transformed generalized fundamental solution is represented as a sum of singular and regular parts. Dynamic part is expressed as the surface integral over a half of a unit sphere and singular static part as an integral over a unit circumference. Classical collocation scheme is employed along with the mixed boundary elements for spatial discretization. The boundary is discretized with quadratic quadrilateral elements. Generalized displacements and tractions are approximated by linear and constant shape functions in each boundary element. In order to accelerate the integration process, regular dynamic parts of the fundamental solutions and their spatial derivatives are interpolated over a boundary element. Time domain solutions are retrieved via a numerical inversion technique. Two numerical examples are presented: static behaviour of the rectangular prism under prescribed tension and transient response of the unit cube under uniform uniaxial impact loading. A convergence study is presented for the dynamic problem and excellent agreement with the analytical solution is achieved for the static problem.

**PNRPU Mechanics Bulletin**. 2016;(3):120-130

Stress-strain state near the wedge top with rigidly fastened sides

#### Abstract

Deformable bodies containing wedge form elements with rigidly fastened sides under temperature loading are investigated. Stress-strain state research method based on identifying a singular point with a representative volume of the body is offered. This approach (in contrast to the commonly used asymptotic methods) makes it possible to formulate essential restrictions at а singular point. It is shown that typically the number of restrictions in the singular point is redundant (larger than usual at the body surface). This situation causes a new (compared to classical) formulation of the problem of solid mechanics contained at a singular point. The investigation of restrictions for the composite wedge with rigidly fastened sides in the vicinity of its top is done. Combinations of material and geometric parameters of construction elements that lead to various variants for problem formulation in solid mechanics are revealed. The critical values of set parameters at which the stress at the singular point increases indefinitely are identified. Load parameters conditions under which a singular point ceases to show singular behavior are formulated. Stress distributions problem near the top of composite wedge with 180 degrees vertex angle under the temperature loading is solved by the iterative numerical-analytical method. The comparison of the solutions obtained by the iterative technique and the classical finite element method is performed. It is shown that the iterative solution matches with all the singular point definable restrictions. Outside a singular point small neighborhood it matches with classical method decisions. But the classical asymptotic solution of the finite element method in the singular point small neighborhood cannot be declared allowed, since it does not satisfy the constraints formulated for such points. This makes it possible to evaluate the region near the critical point, which has no correct asymptotic solution. The typical size of such region is of five to ten characteristic size of the representative volume of the deformable body material. When material parameters approximate to the critical combination, the stress components demonstrate the singular character. The greatest stress value is reached not at the singular point, but at its proximate neighborhood.

**PNRPU Mechanics Bulletin**. 2016;(3):131-147

Load-bearing capacity of ice weakened plates of curvilinear shape with variable thickness

#### Abstract

A method is developed for determining the load-bearing capacity of weakened ice plates, which are modeled by an ideal rigid-plastic plate located on an incompressible foundation. The plate is simply supported or clamped on arbitrary piece-wise smooth curvilinear external contour. The central part of the plate contains a free opening with an arbitrary contour. The thickness of the plate decreases when approaching the boundary of the opening. The plate is subject to a load distributed locally around the opening in the region of an arbitrary shape. The load is an arbitrary function of coordinates. The property of ice with different a resistance in tension and compression is taken into account. The solution is made based on the principle of virtual work. Two variants of kinematically admissible deformations are considered in dependence on the geometric parameters of the plate. In both deformation schemes, the central part of the plate (under loading) moves in the direction of the load; while the area near the external contour (due to the foundation incompressibility) moves in the opposite direction. An orthogonal curvilinear coordinate system associated with curvilinear external contour of the plate is considered. In this system, it is convenient to calculate double integrals describing the solution of the problem. Analytical expressions for the limit loads are obtained. Two integral characteristics of the load are determined; and it is shown that in case the plate is affected by differently distributed surface loads (in which these two characteristics coincide) the plate will have the same limit load. А simply supported and clamped plate shaped as an ellipse with a linear function of thickness under the action of several types of local surface loads is considered as an example. The proposed method allows calculating the load-bearing capacity of weakened curvilinear ice plates on an incompressible foundation and estimating the possibility to increase the load-bearing capacity by increasing the loaded area and by redistributing the load on the area of loading.

**PNRPU Mechanics Bulletin**. 2016;(3):148-163

Structural geometrical transitions under dynamic loading of materials

#### Abstract

It is known that the destruction of solids, including brittle and quasi-brittle ones, in the field of external forces is preceded by the appearance of a certain density of crystal structure defects. Collective movements of such defects in the cooperative interaction with the structure of the material at different scale structural levels (from nano- to macro-) define the process of destruction. In fact, the destruction is the final stage of plastic deformation of solids. Over the past two or three decades, it was found that the profile and surface of dynamically destructible materials are fractal objects. In studies which had been carried out aimed at the possible use of the fractal dimension as a characteristic that allows linking the various parameters of the processes of destruction and the dynamic properties of the materials. Laboratory samples of the three alloys used for pipeline ship fittings were exposed to impact tensile; and then structural studies were carried out of the destroyed specimens. The experiments were performed according to the Kolsky method using a split Hopkinson bar (SHB) at strain rates from 103 to 3·103 s-1. Properties were investigated of the 3M titanium, as well as of 08Kh18N10T stainless steel and BrAZhNMts bronze. We obtained the dynamic stress-strain curves, strength properties and the limit characteristics of plasticity. It was found that the samples destruction was preceded by the acts of microplastic deformation within the activation volume not exceeding the volume of the material grain. Under high-speed loading the damage of material is implemented consistently with the participation of ensembles of the crystal structure defects due to accumulation and changing of their spatial organization. As а parameter for searching the correlations between the strain rate, the type of fracture and the mechanism of structural arrangement it is offered to use the fractal dimension of the contour in the fracture surface of the dynamically loaded material specimen. Also, the possibility of using the fractal dimension for the ranking of material properties was demonstrated.

**PNRPU Mechanics Bulletin**. 2016;(3):164-174

A numerical study of stress-strain state evolution in structurally inhomogeneous materials subjected to uniaxial loading

#### Abstract

Describing structurally inhomogeneous materials as complex hierarchical systems allows deriving a consistent pattern of stress-strain state related to history dependence of damage evolution. In this work we use a random microstructure subvolume to describe a methodology of a numerical study of stress-strain response evolution of a structurally inhomogeneous material subjected to uniaxial tension and compression. The study involves micro- and macrolevel and takes both internal structure and rheological properties of material constituents into account. We use a particle reinforced metal matrix composite with a 99.8% pure aluminum matrix and silicon carbide reinforcing particles. Particles are considered to have an irregular prism shape. The geometric structure of a composite subvolume on the microlevel is modeled by the piecewise-homogenous medium. The medium consists of particle model volumes surrounded by a matrix model volume. To take a surrounding material into account, we introduce an additional buffer layer with averaged macromechanical properties of the composite. A microlevel computational model based on the above assumptions complies to the macrolevel representative volume of the composite with the microstructure fragment in the geometric center. Simulating the model loading behavior allows studying a stress-strain evolution of the random microstructure subvolume and describing it. Boundary conditions in the microlevel model are imposed in the way to represent the macrolevel strain in a point of material. The strain is obtained from macrolevel simulations. A buffer layer is used to improve accuracy of transferring the stress strain state from the macrolevel to the microlevel. The rheological properties of a matrix and buffer layer are taken into account by assigning experimentally obtained strain-hardening curves of the pure aluminum and composite. The matrix material is modeled by an elastoplastic medium with isotropic hardening. The buffer layer is assigned to have isotropic elastoviscoplastic properties. The silicon carbide particle material is considered to be isotropic linear elastic. A finite element discretization is generated with the aid of an in-house software. The software implements special techniques to generate three-dimensional model volumes of inhomogeneous materials with a complex internal structure. The numerical simulation allowed obtaining data on the evolution of the stress tensor components and strain increment tensor components in finite element mesh nodes. Contrary to homogenous macrolevel stress and strain fields emerging in loading simulations with the quasi-homogenous model of the composite material, computations yield peculiar heterogeneous stress-strain state of the microstructure subvolume. We describe features of the stress concentration area emergence and the local plastic strain regions development. We depict strain dependence of stress stiffness coefficient fields and Lode-Nadai coefficient fields. The statistical sampling of such microstructure subvolumes followed by a numerical study adhering to the computational model allows generalizing modeling results and deriving general laws of the stress-strain state evolution of the material on the microlevel.

**PNRPU Mechanics Bulletin**. 2016;(3):175-187

Non-stationary problems for elastic half-plane with moving point of changing boundary conditions

#### Abstract

The paper presents a method of solving plane unsteady problems for an elastic half-space with a mobile boundary related to changing the boundary conditions of a mixed type. The movement of the half-space is described with two wave equations in terms of elastic potentials. The initial conditions are assumed to be zero. It became possible to obtain an explicit solution of the problem in an integral form by using integral relations for a normal displacement of the half-space boundary in the form of two-dimensional convolution of stress with the influence function (arising from the principle of superposition), properties of the convolution in two variables, and of the theory of generalized functions. At the same time, this solution is based on the method of splitting the functions of influence, according to which it is represented as a product of two factors which satisfy the necessary conditions. Thus, in order to obtain final results it is required to carry out the factorization of the influence function which has the desired properties. The analysis of the Fourier and Laplace transformation of the influence function revealed the presence of two simple poles and four branch points. Getting the desired factorization influence function is based on the representation of its transformation as a product of two multipliers; each of them contains only one critical point. In case when critical points are simple, the separation is performed by using a simple factorization, while the branch points are separated with the help of Cauchy-type integrals. The described method allows obtaining the required factorization of the influence function in any typical speed range of the separating point: sub Rayleigh, subsonic, transonic and supersonic ones. As a result, it became possible to obtain the explicit integral formulas which allow solving the problem. They allow defining the unknown displacements and stresses at any speed range of the moving separating point of the boundary conditions. Asymptotic representations of stresses and displacements are found in the neighborhood of boundary conditions change.

**PNRPU Mechanics Bulletin**. 2016;(3):188-206

Finite element analysis of biaxial cyclic tensile loading of elasto-plastic plate with central elliptical hole

#### Abstract

Elements of structures which work in real conditions are quite often affected by variable temperatures and loadings. Nowadays the growth of interest related to the asymptotic behavior of inelastic structures subjected to cyclic loading leads to development of direct and incremental methods of stabilized state determining. If loadings vary and the body deforms elastically, then its durability is defined by fatigue characteristics of materials; destruction comes after a large number of cycles. If the body experiences elasto-plastic deformation, at loadings below the limit, the achievement of a dangerous state at a rather small number of cycles is possible. Thus it is necessary to distinguish two cases. The first case occurs when destruction comes due to the alternation of plastic deformations with different signs (for example, after plastic stretching there is plastic compression, etc.); it is cyclic plasticity (plastic or low-cyclic fatigue). The second case occurs when plastic deformations do notchange signs, but grow with each cycle (the progressing deformation - ratcheting). It leads to the inadmissible accumulation of plastic deformations. Thestudy results present finite-element (FEM) calculations of the asymptotical behavior of an elastoplastic plate with the central elliptic hole under the biaxial cyclic loading for three different materials. Incremental cyclic loading of the sample with the stress concentrator (the elliptic central hole) is performed in the multifunctional finite-element package SIMULIA Abaqus. The ranges of loads found for the shakedown, cyclic plasticity and ratcheting are presented. The obtained results are generalized and analyzed. Convenient normalization is suggested. At the expense of the chosen normalization all computed results corresponding to separate materials stay within one common curve with a minimum scattering of points. The convenience of the generalized chart consists of a possibility to receive the asymptotical behavior of the inelastic structure for those materials which computer calculations were not made.

**PNRPU Mechanics Bulletin**. 2016;(3):207-221

On delamination of a stripe along the boundary between two elastic layers Part 3. Exact analytical solution for a particular case

#### Abstract

The analytical solution has been obtained for a homogeneous problem of elasticity in plane strain setting of semi-infinite crack separating two isotropic layers of equal thicknesses and different but related with a particular restriction (corresponding to vanishing of the second Dundurs parameter) elastic constants. The problem under consideration is a particular case (allowing us to obtain the exact analytical solution) of the general problem on semi-infinite crack separating two isotropic layers of different thickness and arbitrary combination of elastic constants posed. The solutions for the cases of normal and shear crack were obtained under the assumption of the possibility to neglect cross-terms related to the influence of the normal stresses on the shear displacements and the shear stresses on the normal displacements. In this part of the work the solution for the case in question has been derived by means of Laplace transformation; and reducing it to a homogeneous Riemann-Hilbert problem with the matrix coefficient. The restriction imposed on elastic constants and the demand of equal thicknesses are determined by the used method allowing the factorization of the matrix coefficient. The asymptotical expression has been derived for the relative displacements of the crack faces far from its tip, corresponding to a beam deflection with the boundary condition of the type of generalized elastic clamping. i.e. the proportionality of the displacement and angle of rotation of the clamping point to the total vector and bending moment of the applied load by means of the matrix of coefficients of compliance. The analytical expressions for these coefficients have been obtained. The asymptotical expression for the stress field near the crack tip (stress intensity factor and energy release rate) has also been derived.

**PNRPU Mechanics Bulletin**. 2016;(3):222-240

Normal and shear stresses estimation in deformed metals based on infrared thermography data

#### Abstract

The work is dedicated to the development and implementation of numerical-experimental method of evaluation of the stress and strain components on the base of infrared thermography. The infrared thermography is a non-contact method of visualization and measurement of temperature fields of objects. It could be used as a method of non-destructive testing. A program complex which could be used for the evaluation of the individual components of stress and strain is developed on the base of the solution of the boundary value problem and data of the sample temperature change caused by the thermoelastic effect. In order to verify the proposed method, a series of experiments on quasi-static tensile of specimens from structural steel 8X18H10 and titanium alloy VT1-0 with the stress concentrators were carried out. As a result, it is shown that in contrast to the similar approaches (e.g., TSA-Thermal Stress Analysis) the method allows us to obtain additional information about stress-deformed state of the material and to conduct a more detailed assessment of the degree of the critical state of the structure. The methodology of the proposed method is based on the experimental measurement of the first invariant of the stress tensor using the infrared scanning technique and its subsequent recalculation to determine boundary conditions. This allows us to identify all stress components at any point in the considered area of specimens or construction on the base of the numerical solution of the corresponding boundary value problem. Special feature of the developed approach is a small computational cost for the determination of the stress components, which allows using this technique in the analysis of a wide class of engineering structures in real time.

**PNRPU Mechanics Bulletin**. 2016;(3):241-251

Experimental study of deformation properties of a package of woven metal mesh under dynamic and quasi-static stressing

#### Abstract

The multilayer gas-permeable bags of woven metal meshes are a promising damping element that protects the structure from shock and explosive effects. Due to the developed interface, the packages may select a significant proportion of the explosion energy of hot products and reduce the intensity of shock waves. Mesh bags are constructively formed by a free deposition of layers on each other while maintaining the directions of threads, so the packages can be considered as a highly porous deformable medium having orthotropic properties. Experimental studies have been made on deformation and strength properties of structurally orthotropic packages of woven metal grids under static and dynamic stressing. Mesh Package resists compression normal to the layers of the grid, it is the first orthotropy axis, and resists the stretching along the directions of threads, it is the other orthotropy axis. For the impact stretching in the plane of the layers we used an analogue of Nicholas circuit which is a modification of the Kolskiy method. In the split Hopkinson bars we made longitudinal grooves which house and fix the test specimen. Shock stretching is carried out in the wave of stretching which emerges in the bars as a result of reflection from the free end of the primary compression wave that passes the junction of rods without deforming the sample. Static stretching of grid samples is performed using Zwick testing machines. Tests were carried out for samples with different pitch and weave meshes and a different number of layers. Experiments have determined the necessary dimensions of the working part of the sample. It is shown that the stress-strain diagram in tension along the threads in the plane of the layers and the compressive normal to the layers of the grid for all loading conditions are non-linear and irreversible, but also show a significant dependence on the strain rate. In the quasi-static tensile packages of grids in the direction of the filaments is significantly affected by their prior compression normal to the layers of mesh. Under dynamic stretching this effect is much weaker.

**PNRPU Mechanics Bulletin**. 2016;(3):252-262

Evaluation of service life of heat-resistant alloys under cyclic thermomechanical loading

#### Abstract

The problem of evaluating strength and service life of critical engineering facilities, whose operating properties are characterized by multi-parametric non-stationary thermomechanical effects is discussed in the paper. The main degradation mechanisms of structural materials (metals and alloys) specific to these objects are considered. The main requirements to the mathematical models of fatigue damage accumulation are formulated. The basic physical laws of complex thermoplastic deformation and fatigue damage accumulation in structural materials (metals and their alloys) for various modes of combined thermomechanical loading and their main difference from isothermal fatigue processes are considered. In the modern mechanics of damaged media (MDM), a mathematical model is developed which describes the processes of thermoplastic cyclic deformation and fatigue damage accumulation in structural alloys under multiaxial disproportionate modes of combined thermomechanical loading. A MDM model consists of three interrelated parts: defining relations of thermal plasticity accounting for their dependence on failure process, evolution equations of fatigue damage accumulation and a strength criterion of damaged material. It is shown that for certain parameters of equations of cyclic thermal plasticity, using a single point on the experimental fatigue curve, the parameters of evolution equations of damage accumulation can be determined which are used for a high-accuracy computational reconstruction of low-cycle fatigue curves for various complex deformation trajectories. The results of numerical modeling of thermoplastic cyclic deformation and fatigue damage accumulation in heat-resistant alloys (Haynes188) under combined thermomechanical loading are presented. Particular attention is paid to the issues of modeling the process of thermoplastic cyclic deformation and fatigue damage accumulation for complex processes of deformation accompanied by the rotation of the main sites of stress and strain tensors.

**PNRPU Mechanics Bulletin**. 2016;(3):263-281

Graph model of three-dimensional elastic solids in Cartesian Coordinates

#### Abstract

The theory of graphs represents an unsophisticated section of mathematics with a wide range of applications. It is based on the simple ideas and elements such as points and lines. The theory of graphs builds a rich diversity of forms from that, providing efficient tools for construction of models and means of solution to a wide range of problems. The method of a numerical analysis of the mechanical fields in the deformable body, based on the graph model of an elastic medium in the form of the directed graph, is considered. According to the method applied the elastic medium along coordinate planes divides into separate elements. In line with this notion we establish an elementary cell configuration, a subgraph of an element, by installing hypothetical meters on an element of a solid. Derivation of cell equations, which is based on conversion of an element to a cell, relies on an invariant. We use the deformation energy as the invariant. A procedure to determine parameters of the elementary cell is described. The graph of a whole body is built following the same rule as in an elementary cell. With the use of a unit cell having 24 degrees of freedom, the strain field is approximated by linear polynomials (with corresponds to approximated of the displacement fields by quadratic polynomials). The standard finite-element method requires 60 degrees of freedom (elements with 20 nodes) for the same purpose. The proposed graphical approach thus reduces the number of equations that describe the model. Kirchhoff’s laws (apex and contour) realized in the analyzer are shown to correspond to equations of equilibrium and strain compatibility in the elastic body The equations are of no use when determining the stress-strained state of the body in the explicit form with its model.

**PNRPU Mechanics Bulletin**. 2016;(3):282-303