## No 3 (2015)

**Year:**2015**Articles:**13**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/24**DOI:**https://doi.org/10.15593/perm.mech/2015.3

Mathematical modeling of development of beyond design basis accident in integral fast-neuron reactors vessel

#### Abstract

The process of nonstationary deformation of construction of the fast reactor with liquid metal coolant under postulated ULOF beyond design basis accident is examined. This type of accident includes core melting caused by disconnection of the main circulation pump arrangements of the primary circuit with the associated failure of the emergency protection. As a result of core melting, the area with a high energy level pressure is created. It is filled with sodium vapor. The progressive expansion of the energy area in the coolant leads to an increase of stress-strain state level of the reactor vessel and may lead to its destruction. The reactor facility must save integrity, provide localization of consequences of beyond design basis accident inside of the pressure vessel and avoid dangerous radiation effects on personnel of nuclear power station and the environment in these conditions. Current Lagrangian formulation is used for the description of coolant motion and structural elements of the reactor. The equation of motion derives from the balance of virtual capacity. Equations of the theory plastic flow are used in physical relations for metals. Deviatoric stress components are assumed to be equal to zero in the coolant; and the equation between hydrostatic pressure and density is taken as a state equation of quasiacoustic type. The contact between the coolant with structural elements of the reactor is simulated by the conditions of non-penetration. The problem solution is based on the method of moment schema of FEM and explicit finite-difference time integration scheme of the “cross” which are implemented in the computing system “Dynamics 3”. The deformation of the fast reactor vessel is investigated numerically in the ULOF beyond design basis accident. The possibility of localizing effects consequences of beyond design basis accident inside of the pressure vessel of the reactor is analyzed.

**PNRPU Mechanics Bulletin**. 2015;(3):5-14

Researching stages of destruction of titanium alloys during highand gigacycle fatigue by analyzing morphology of fracture surfaces

#### Abstract

Fatigue (high- and gigacycle) crack initiation and its propagation in titanium alloys with coarse and fine grain structure are studied by fractography analysis of fracture surfaces. Specimens were loaded using a resonance fatigue machine Shimadzu USF-2000 in gigacycle regime and servohydraulic machine BISS bi-00-100 in high-cycle fatigue regime. Fine grain alloys demonstrated higher fatigue resistance for both HCF and gigacycle fatigue regimes. Fractured specimens were analyzed by interferometer microscope and SEM to improve methods of monitoring damage accumulation during fatigue test and verify the models of fatigue crack kinetics. Fatigue crack initiation occurred from the surface of the sample in high-cycle fatigue regime and bulk of material with a characteristic type of fracture - “fish-eye” in gigacycle fatigue regime. Quantitative fractography is an effective method for investigating the role of the original structural heterogeneity, the accumulation of defects in various scale levels (dislocation ensembles, micropores, microcracks) in the evaluation of the critical conditions for the transition from particulate to the macroscopic fracture. The description of the stages of this transition, including the initiation and propagation of cracks, is the basis for assessing the temporary resource products in terms of fatigue. High resolution profilometry (interferometer-profiler New View 5010) data of fracture surface roughness allowed estimating scale invariance (the Hurst exponent) and establishing the existence of two characteristic areas of damage localization (different values of the Hurst exponent). Area 1 with diameter ~300 µm has the pronounced roughness and is associated with damage localization hotspot. Area 2 shows less amplitude roughness, occupies the rest of fracture surface; and it is considered as the trace of the fatigue crack path corresponding to the Paris kinetics.

**PNRPU Mechanics Bulletin**. 2015;(3):15-24

Experimental research of self-regularities of ceramics fracture under shock wave loading

#### Abstract

A series of UEWE (underwater electrical wire explosion) experiments were carried out to study the fragmentation of ceramic tubes under shock wave (SW) loading caused by explosion of the axial conductor. The amplitude of shock wave loading of the samples in the UEWE setup was regulated by varying the energy of the battery of capacitors charged by high voltage source. Duration of the discharge was 0,3-0,8 µs. The statistical distribution of the tube fragments was determined by gathering fragments from the bottom of the explosion chamber. The mass of the collected fragments was about 98 % of the mass of the original sample. The size of fragments estimated by two methods is “weighing” and “photography”. The “photography” method was based on the form factor of fragments, which was calculated separately. The distributions obtained by both techniques were in good agreement. Since the mass of the tubes was varied, the parameter of the specific energy of the loading pulse was introduced in order to correlate the data. The collected fragments can be divided into two classes: quasi-two dimensional objects (2D) with characteristic fragment size d* is greater or equal to the thickness d of the tube wall and three-dimensional (3D) with characteristic size less than the wall thickness. It was found that the distribution of the 3D objects could be described by a power law, in which the value of the power remains constant and does not depend on the specific energy. Distribution of the 2D fragments obeys an exponential law. It is shown that the inflection point of the fragment distribution curve is shifted toward smaller scales with increasing of the specific energy. The existence of different distributions for the different scale of the fragments is explained.

**PNRPU Mechanics Bulletin**. 2015;(3):25-37

CAD/CAE modelling of mechanical behavior of composite outlet guide vane for aircraft jet engine

#### Abstract

Today the world trend in aircraft engine industry is the replacement of metal alloys to composite materials which allows a significant weight reduction and increase of product performance. The use of composites not only for lightly loaded elements, but for the critical parts such as the fan blades and outlet guide vanes (OGV) is already researched and implemented. Outlet guide vanes (OGV) also called flow straightening vanes, are radially disposed behind the fan to straighten out the airflow to reduce losses in the outer contour of the engine. Design and manufacturing techniques development for composite outlet guide vane is a complex scientific and engineering problem including modeling, simulation and design of composite structure, specifying its design features and reinforcement schemes, choosing materials and optimal manufacturing techniques. Such giants of aviation industry as GeneralElectric, Rolls-Royce, SNECMA are seriously engaged in this problem. Therefore, manufacturing composite outlet guide vane for new domestic PD-14 aviation propulsion, scheduled for installation on the short-range and mid-range jet aircraft MS-21 is a very important problem. It’s easy to show that the use of polymeric composite materials (PCM) instead of metal in OGV can lead to decrease of engine weight for up to 8-10 kg. The goal of this work is the implementation of CAD/CAE technologies for engineering analysis of composite outlet guide vanes of PD-14 engine. The three-dimensional model of OGV for stress-strain analysis was developed in Siemens NX CAD-system. The numerical simulation of mechanical behavior of OGV made of balanced textile CFRP under the static pressure was carried out with ANSYS Workbench software. The technique of static strength prediction was examined. The developed numerical model allows analyzing the effective stress-strain fields in whole volume and its components in each layer separately. The four reinforcement schemes were reviewed. It was shown that the quasi-isotropic reinforcement scheme also called “black aluminum” is an optimal choice for obtaining the high strength and stiffness properties of OGV. The results of numerical simulation confirm the preliminary estimations of the effectiveness of composite materials application in the design of OGV obtained by the authors earlier.

**PNRPU Mechanics Bulletin**. 2015;(3):38-51

Three-wave resonance interaction in elastoplastic solid

#### Abstract

Resonant interaction of longitudinal and transverse waves in elastoplastic solid is considered. The media behavior is described by cross dependencies between the first invariants of stress and strain tensors and second invariants of stress and strain deviators. The dependence on shear strains is quadratic. The aim of this investigation is to establish the nature of the process of energy exchange between modes in the absence of dissipation. It is assumed that the solid is in the plane strain state. Two cases are considered, with prevailing longitudinal and transverse strains respectively. In both cases the solution of systems of nonlinear partial differential equations are searched in the form of traveling harmonic waves with slowly changing amplitudes. Method of averaging by “fast” variables is used taking into consideration only solutions and items with the order of smallness which is below one. It is obtained that in both cases the form of systems of equations for wave amplitudes is the same. Continual analogues for Manley-Rowe relations are obtained. The solution of the system for a stationary state with boundary conditions for strong input and weak signal waves is presented, as well as the diagram qualitatively illustrating the process of a three-wave interaction in case of decay instability (i.e. frequencies and wave numbers of signal and idler waves are in total equal to the frequency of the input wave) and relation determining the distance of effective energy exchange between the interacting waves. The behavior of magnitudes of wave amplitudes depending on frequency ratios, wave number ratios and media properties (density, shear modulus, limit of intensity of shear strains) has been analysed.

**PNRPU Mechanics Bulletin**. 2015;(3):52-62

Analysis of fragmentation of plates of synthetic mineral alloys under impact of ball high velocity by visualization technique

#### Abstract

The object of research are synthetic mineral alloys. Their structure comprises a 95 % crystalline and 5 % amorphous phase components, these materials have strong dissipative properties and strong perspectives for use as body armor components. To realize the potential of the dissipative properties of these materials must be a detailed study of the deformation behavior at high shock loads. The goal of work is to study the destruction and fragmentation of synthetic mineral alloys at high velocity impact by imaging technique. For experiments was used pneumatic gun length 3950 mm and a steel ball diameter of 23.8 mm, the speed of the ball was 233-234 m/s. For visualization was used a complex of video recording, consisting of a high-speed camera Photron Fastcam SA5 and lighting equipment. Shooting speed was 50,000 frames / sec. Experiments was made with differing material samples: synthetic mineral based alloys gabbro and basalt. In results was found stages of damage and fragmentation as per-frame images, quantitative analysis of the data registration process speed collision and kinetics of cracks (thickness, disclosure) during the fragmentation. These data allowed to establish qualitative characteristics of fragmentation of synthetic mineral alloys of the type quasi-brittle materials. This information is of interest to develop improved varieties of synthetic mineral alloys and protective structures of them. Vizualization techniques can be used to practice the following materials science research and experimental study of the laws of ballistic damage and destruction of critical structural elements of synthetic mineral alloys.

**PNRPU Mechanics Bulletin**. 2015;(3):63-73

Dimensionless parameters in the theory of reinforced shells

#### Abstract

In this paper a variant of the dimensionless parameters for a wide range of shell structures is proposed. For shallow shells of have a rectangular form, the dimensionless parameters are used for a long time, as for the general form of shells; there is no single form of dimensionless ratios for each type of shell, because Lame parameters differ not only in values, but also in dimensions. Therefore, the work shows the dimensionless relations for deformations, stresses, forces, moments and functional of total potential energy of deformation. The relations consider geometric nonlinearity, transverse shifts, orthotropy of material and the introduction of ribs on the structural anisotropy of the method in accordance with their shear and torsional rigidity. The authors show a further approach to solving the strength and stability tasks of different types of shells in the dimensionless parameters. Some methods for solving nonlinear problems of stability do not look quite correct, when using the dimensional parameters (e.g., a technique, based on the method of continuation of solution for the best parameter). In dimensionless parameters, all calculations are beyond doubt. The calculations of some shell structures are provided in dimensionless and dimensional parameters and their consistency is shown. The proposed approach allows one to obtain values for the calculation of a series of such shells, which is more convenient to optimize the choice of design parameters. We have examined some of the differences in critical loads, obtained in dimensionless and dimensional solution of the problem.

**PNRPU Mechanics Bulletin**. 2015;(3):74-94

Decomposition of systems of equations for continuum mechanics 2. General results and applications

#### Abstract

In the second part of work presents the general decomposition methods for systems of linear partial differential equations that arise in continuum mechanics, in particular, in the theory of elasticity and thermoelasticity and poroelasticity. A systematic approach to the decomposition of the equations of continuum mechanics is proposed. Asymmetrical and symmetrical decomposition methods for various classes of three-dimensional linear (and model nonlinear) systems of equations arising in the theory of elasticity, thermoelasticity, and thermoviscoelasticity, the mechanics of viscous and viscoelastic incompressible and compressible barotropic gas are described. These methods are based on the decomposition of systems of coupled equations into several simpler independent equations and the use of two stream functions. It is shown that in the absence of body forces any solution of considered steady and unsteady three-dimensional systems is expressed in terms of solutions of two independent equations. The methods of direct decomposition that do not require expansion of the right hand side of the equations into the components are proposed. A generalization of the considered methods to the decomposition of higher orders systems of equations, as well as to special classes of model nonlinear equations are obtained. The examples of the decomposition of specific systems are given. Formulas and split equations given in the work significantly simplify the qualitative study and the interpretation of the most important physical properties of a wide class of coupled systems of equations for continuum mechanics and allow studying their wave and dissipative properties. These results can be used for the exact integration of linear systems of mechanics, as well as for testing of numerical methods for nonlinear equations of continuum mechanics.

**PNRPU Mechanics Bulletin**. 2015;(3):95-113

Carrying capacity and optimization of three-layer reinforced concrete annular plate, supported on the internal contour

#### Abstract

Within the model of an ideal rigid-plastic body the exact solution is obtained for the problem of bending of three-layer reinforced concrete annular plates having different angular structure reinforcement at the top and bottom layers. The middle layer of the plate is made of concrete. The plates are hinge supported along the annular contour located within the area of the plate. External and internal contours of the plates are free. The plates are under load uniformly distributed over the surface of the plate. The condition of plasticity for the main moments, based on a structural model of the composite, has the form of a rectangle of type Johansen condition. It is taken into account that the strength of concrete in tension is much less than in compression. It is shown that there are four schemes of limit deformation of the plate, depending on the location of the internal support. The conditions of implementation are defined for all schemes. The main moments and the velocities of the deflections of the plate are defined at different locations of the internal support. The simple analytic expressions are obtained for the limit load. The optimal location of support is determined. The optimal support is such a support , at which the plate has a maximum limit load. It is shown that the optimal position of the support corresponds to the formation of plastic hinge on it. The problem is solved to determine the optimal thickness of the top layer of the plate corresponding to the maximum limit load for a given total thickness of the reinforced layers. It is shown that the location of the support affects the optimal thickness ratio of the upper and lower layers. Numerical examples are given for different structures of reinforcement.

**PNRPU Mechanics Bulletin**. 2015;(3):114-132

A grid method quantifying deformed Mg-alloys by Equal-Channel Angular Pressing

#### Abstract

Among the optimum solutions for hydrogen storage in terms of effectiveness and safety are metal hydrides. Magnesium and its alloys can reversibly absorb hydrogen in large amounts, so fulfilling the DOE's requirements and making those materials attractive for applications. At first, determining a fast hydrogen saturation of Mg-based alloys was consisted in grinding the materials up to micrometric grain size. Increasing markedly the specific surface of the treated powders by plastic deformation processing leads to delivering very reactive samples. As well, huge improvement of H-sorption characteristics of bulk Mg-alloys was shown to be efficient under Equal Channel Angular Pressing (ECAP) treatments. During such a Severe Deformation Process, achievement of a fine grained microstructure in bulk samples is accompanied by the formation of defects and overall texture. The main achievements expected from the application of ECAP treatments to Mg-rich alloys are: - formation of ultra-fine microstructures with high angle boundaries, which especially drastically changes the mechanical characteristics of the alloy, - volume homogenization of the microstructure for the best final stability of the hydrogenation properties of the refined material. Since in most cases a two or even more ECAP passes should be applied to deliver the highly reacting materials, the operating temperature must be adjusted in terms of ductile to fragile characteristics in order to avoid irreversible cracking of the bulk sample. After the application of the ECAP process, the resulting deformation was characterized using different methods such as conventional structural analysis and mechanical engineering followed by numerical simulations. The present article reports on the deformation process of Mg-based by using a grid evaluation method.

**PNRPU Mechanics Bulletin**. 2015;(3):133-145

A set of tests for studying the effect of the stress state on ultimate metal plasticity at high temperature

#### Abstract

A set of tests that can be used tostudy the effect of the stress state on ultimate metal plasticity at high temperature is proposed. Ultimate plasticity can be characterized quantitatively by the strain accumulated to failure. Dimensionless invariant parameters, such as the triaxiality parameter k and the Lode-Nadai parameter μσ, are used as a characteristic of the stress state. A set of the parameters k and μσ unambiguously characterizes the stress state under plastic deformation. Being dimensionless, these parameters can be used to compare the stress state of materials with different levels of strength properties. The triaxiality parameter k characterizes the level of normal stresses. If k > 0, then normal tensile stresses dominate. If k < 0, then normal compressive stresses dominate. The Lode-Nadai parameter μσ characterizes the type of the stress state. The value of μσ = +1/-1 corresponds to the stress state of axisymmetric compression/tension, and μσ = 0 under the plane stress state. The set of tests consists of conventional tension tests of cylindrical specimens, tension tests of notched cylindrical specimens, newly developed tests of “bell” specimens and tests of a thick-walled cup with a thinned bottom. This types of tests has allowed us to investigate ultimate plasticity under monotonic deformation in the range of parameters -1,2 < k < 1,2 and -1< μσ <+1 without applying high pressure technique. The stress-strain state under tests was evaluated by the results of mathematical simulation by means of the finite element method in ANSYS. The tests performed on 01570 aluminium alloy specimens. The test temperature was 360оC. The shear strain rate was H = 0,1; 0,3; 0,5 s-1. The fracture locus of the 01570 aluminium alloy has been obtained from the test results. The fracture locus can be used in damage mechanics models to predict alloy fracture under metal forming processes.

**PNRPU Mechanics Bulletin**. 2015;(3):146-164

Modeling of elastic behavior of multicomponent composite materials based on approximate solution of stochastic boundary value problems

#### Abstract

This work is devoted to development and application of the statistical mechanics approaches to studying microstructural behavior of multi-component composites. The object of the study is heterogeneous materials consisting of more than two components. The aim is to develop analytical tools for the analysis of the microstructural stress and strain fields in multicomponent media taking into account geometrical and mechanical properties of components and basied on calculation of the statistical characteristics of the local fields of stress and strain in components. Research of the microstructure behavior of components of composites is based on the concept of representative volume elements. It is assumed that the components are homogeneous and isotropic. Information about the internal morphology of the representative volume is formalized by means of correlation functions of various orders. Characteristics of the deformation processes are the statistical moments (statistics) of stress and strain fields in the components of the material. Analytical expressions for the statistical characteristics of the local stress and strain fields are obtained using the solution of the boundary problem of elasticity theory in a stochastic formulation. The boundary value problem is solved using the Green's functions method for the elastic medium. The developed analytical model takes into account both the geometrical parameters of the microstructure and mechanical properties of components. For the first time expressions for the moments of the first and second order of local stress fields for the multicomponent materials were derived. Case studies of composites with titanium (Ti) matrix reinforced with randomly distributed particles of silicon carbide (SiC) were investigated. Analysis of the influence of microstructural parameters on the behavior of each phase separately was performed. Numerical results for the statistics of stress and strain fields are presented.

**PNRPU Mechanics Bulletin**. 2015;(3):165-181

About geometrically nonlinear constitutive relations for elastic material

#### Abstract

In the field of solid mechanics it is often necessary to use constitutive (physical) relations in the rate form, for example, for formulation of the boundary value problem in the rate form with contact conditions when a contact area is a priori unknown and changes in a deformation process. The paper considers some questions of formulating geometrically nonlinear constitutive relations of elastic material in the rate form and the relationship of these equations and the constitutive relations in the finite form. In many existing elastic and elasto-plastic models Hooke's law (written in terms of the actual configuration) is used as the constitutive relation. As a rule, the rate measure of the strain is deformation rate tensor (the symmetrical part of velocity gradient) and the stress rate measure is some objective derivative (convective or corotational) of weighted Kirchhoff stress tensor. The use of the convective derivative leads to certain difficulties, for example, analysis of evolution stress state in a deformable basis is complicated; consequently, convective derivatives are excluded from consideration. Using a stress tensor corotational derivative instead of the material derivative (derivative by time) allows satisfying the principle of material indifference (the principle of independence of constitutive relation from the choice of reference frame), however a choice of the derivative type can be implemented by many ways. An arbitrary selection of stress corotational derivative leads to undesirable effects: stress oscillations for the simple shear monotonic loading (for example, for Jaumann derivative), “not closed” stress trajectories and nonzero stress work for a closed deformation trajectory. In papers of A. Meyers, H. Xiao, O. Bruhns corotational derivative (with logarithmic spin or logspin) was proposed.The derivative of the right Hencky's strain tensor is exactly equal to deformation rate tensor. When using this derivative in the constitutive relation the described effects are missing, on the basis of this remarked logspin authors state the logspin exclusivity and recommend it only for using in rate form constitutive relations. The article shows full compliance between Hooke's law in the rate and finite form under conditions: existence of the material basis in which properties of the body (in this case - elastic) remain unchanged and the use of the same type corotational derivative for stress and strain measures. Computational experiments for illustration of the proved assertion about the equivalence of different Hooke's law forms were conducted: various measures of strain state and their corotational derivatives are considered; a kinematic loading in a closed cycle (in the space of deformations) is applied. It is shown for isotropic elastic materials that stress trajectories are closed. When using work-conjugacy stress and strain measures the dissipation energy is absent. Thus it is possible to question the exclusive choice of logarithmic spin and the above-mentioned commonly used measures of stress-strain state. In formulating and solving the problems of solid mechanics there should be an opportunity to use different stress-strain measures and their objective derivatives, the choice of measures and constitutive relations should be justified from the point of physical analysis of the process.

**PNRPU Mechanics Bulletin**. 2015;(3):182-200