Double-layer composite structures with one load-bearing high-strength layer made of metal, reinforced composite or rigid plastic and other one made of high deformability polymer or elastomer providing high tightness, chemical and corrosion protection are widespread. Due to differences in the mechanical properties and linear thermal expansion coefficient, high level of stresses can occur in the layers of such structures during temperature changes. Considering the long-term nature of loading, the defects probability, the availability of complex stress state in the layers and temperature dependence of the physical and mechanical properties of polymers, the issues of the providing of long-term mechanical strength and failure-free operability of such structures are highly relevant. Object of study in this paper is polymeric pipe combined of two layers made of thermoplastic (HDPE) and fiberglass. The purpose of research is to identify the causes of brittle fracture of thermoplastic layer at low temperatures. The experimental results of cold resistance of HDPE specimens and pipe samples at temperatures down to -50°С are presented in this paper. A comparison with the previous results of calculations of technological residual stresses in these pipes was done. The test method for fragility of HDPE under complex stress state at low temperatures was proposed. As a result of pipe samples testing the process steps which can effect on cold resistance of plastic pipes were identified.

Keywords: stress-strain state, fracture, HDPE, polyethylene, fiberglass, cold-resistance, test, bi-axial loading, combined pipe, long-term strength, technological stress.

Aleksandr N. Anoshkin (Perm, Russian Federation) – Doctor of Technical Sciences, Professor, Head of Science and Innovation, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, e-mail: anoshkin@pstu.ru).

Aleksei B. Pospelov (Perm, Russian Federation) – Chief specialist Scientific and educational center of Aircraft Composite Technology, Perm National Research Polytechnic University (15, Akademik Korolev str., 614990, Perm, Russian Federation, e-mail: pospelov-kt@pstu.ru).

Ravil' M. Iakushev (Perm, Russian Federation) – Ph.D. in Technical Sciences, Head of Laboratory, Institute of Technical Chemistry, Ural Branch of Russian Academy of Sciences (3, Akademik Koroleva str., 614013, Perm, Russian Federation, e-mail: ravilyakushev@yandex.ru).

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Mathematical models of fluid flow through non-deformable or elastically deformable porous media have become widely used. Especially they are common in water resources and oil drilling problems. The proposed hydrodynamic model of fluid filtration through plastically deforming porous skeleton, finds application in the investigation of the rapeseed extrusion extraction process. During this process the porous skeleton undergoes large inelastic deformation, which requires an alternative approach to the material behavior description. The material was presented by a two-component mixture of plastically deformable compressible porous medium saturated by oil. The mixture components were assumed to be nonreactive. According to the polymers extrusion processing theory, the problem was considered in reverse motion, the screw channel was unrolled on the plane, and viscous fluid model was used as governing equation for both mixture components. Further problem formulation was performed in the framework of the Euler motion description approach in a two-dimensional formulation for the plane of screw channel middlesection. Boundary-value problem formulated based on momentum balance and mass conservation equations for each mixture component. The boundary value problem independent variables are the mixture pressure, oil pressure, mixture velocity and oil velocity. The hypothesis of extraction speed proportionality to the oil pressure allows obtaining an approximate analytical solution for a constant filtration and compressibility coefficients.

Keywords: mathematical model, plastically deformed porous medium, extrusion and extraction, filtration, oil, rape seeds.

Sergey D. Anferov (Perm, Russian Federation) – Research Engineer of Laboratory of Thermoplastics Mechanics, Institute of Continuous Media Mechanics UrB RAS (1, Akademik Korolev str., 614013, Perm, Russian Federation, e-mail: anferov@icmm.ru).

Oleg I. Skulskiy (Perm, Russian Federation) – Doctor of Technical Sciences, Leading Researcher of Laboratory of Thermoplastics Mechanics, Institute of Continuous Media Mechanics UrB RAS (1, Akademik Korolev str., 614013, Perm, Russian Federation, e-mail: skul@icmm.ru).

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The problem of optimal braking hard indenter of the inhomogeneous plate during the impact of the normal was first formulated in 1978 (Aptukov V.N.). Results published later, based on the Pontryagin maximum principle was derived criteria for the optimal structure of inhomogeneous plate a minimum weight for different projectile shapes. At the present time this problem in a similar or different formulations studied by various researchers, some examples are presented in this paper.

Variant of the viscous crater formation is implemented for medium velocity of the impact of the little-deformed sharp indenter into plastic target with a medium hardness. For this condition known empiric dependence of the penetration resistance is widely using. The dependence is applied under certain limitations on velocity, thicknesses of the target, shape of the indenter, mechanical characteristics, this fact is confirmed by numerous experiments that is carried out in the Stepanov V.A. laboratory in the Ioffe LPTI (at present Ioffe Physical-Technical Institute of the Russian Academy of Sciences in the St. Petersburg).

We used numerical algorithm and we attempt to refine the problem of optimization taking into account the effect of free surfaces of the plate and friction on resistance to penetration in this paper. Gradual increase of complexity of the model by including new factors helps to approach a more realistic description of the penetration process. This improvement allows to further study the problem in a new improved model.

Method of acicular variations is used to solve the problem. We received final solution of the problem and we formulated criteria for the optimal structure of the target in some cases. We have not received of the analytical solution in other cases, but we presented the results of the numerical calculation. We showed that the inclusion of additional effect theoretically lead to a qualitatively new type of solution compared to previously known solutions in some cases. We derived the algorithm for determining the optimal structure of the slab to the problem of the impact of the cone with n materials.

Keywords: optimal braking; inhomogeneous plate, friction, effect of free surfaces of the plates, method of acicular variations.

Valery N. Aptukov (Perm, Russian Federation) – Doctor of Technical Sciences, Professor, Head of Department of Fundamental Mathematics, Perm State National Research University (15, Bukireva st., 614990, Perm, Russian Federation, e-mail: aptukov@psu.ru)

Artur R. Khasanov (Perm, Russian Federation) – Undergraduate Student of Department of Fundamental Mathematics, Perm State National Research University (15, Bukireva st., 614990, Perm, Russian Federation,
e-mail: artur_raisovich@rambler.ru).

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The paper is devoted to theoretical investigation of the influence of stress state parameters of healed fault zones fragments on the characteristics of their mechanical response under shear deformation in the conditions of nonequiaxial compression. Investigation was based on computer-aided simulation by the movable cellular automaton method. Dimensionless parameter, named the degree of nonequiaxiality of compression was used as the basic parameter of stress state of the medium. This parameter characterizes the ratio of lateral and normal stresses in the plane of deformation. The main objective of the paper was to analyze the dependences of the shear strength, ultimate shear strain and value of volume changing (dilatancy) on the degree of compression nonequiaxiality at the initial stage of activization of the fault zone fragment. It is shown that the degree of compression nonequiaxiality of the medium is an important factor affecting the conditions under which healed fault zone could be activated. Here, the value of the shear stresses acting in a fragment of the medium, as well as corresponding ultimate values of shear strain and dilatancy, at which the fault zone could be activated, are essentially dependent on the ratio and dynamics of change of the local values of some stress tensor invariants. Among them are pressure and stress intensity (von Mises stress). This is due to the fact that these parameters determine the ability to operate in the geological medium of one of a key deformation mechanism that is associated with the formation and evolution of damages at the interface of the structural elements in the block structure medium. In particular, the decrease in the pressure in the medium fragment at relatively low levels of stress intensity can lead to an increase in ultimate shear strain and dilatancy at the beginning stages of activization of the fault zone. At the same time a significant increase in the stress intensity while decreasing the pressure could lead to a decrease in the shear strength of the geomedium.

Keywords: geological medium, fault zone, nonequiaxial compression, shear loading, stress state, stress tensor invariants, shear strength, ultimate strain, dilatancy, slip, computer-aided simulation.

Sergey V. Astafurov (Tomsk, Russian Federation) – Ph.D. in Physical and Mathematical Sciences, Scientific Associate of Laboratory of Computer-Aided Design of Materials, Institute of Strength Physics and Materials Science SB, RAS (2/4, Akademicheskii av., 634055, Tomsk, Russian Federation, e-mail: svastafurov@gmail.com), Scientific Associate of Laboratory of Multiscale Dynamical Analysis of Materials and Constructions, National Research Tomsk State University (36, Lenina av., 634050, Tomsk, Russian Federation,).

Evgeny V. Shilko (Tomsk, Russian Federation) – Doctor of Physical and Mathematical Sciences, Leading Researcher of Laboratory of Computer-Aided Design of Materials, Institute of Strength Physics and Materials Science SB, RAS (2/4, Akademicheskii av., 634055, Tomsk, Russian Federation, e-mail: shilko@ispms.tsc.ru), Professor of Physics of Metals Department of National Research Tomsk State University (36, Lenina av., 634050, Tomsk, Russian Federation,).

Sergey G. Psakhie (Tomsk, Russian Federation) – Corresponding Member of RAS, Doctor of Physical and Mathematical Sciences, Professor, Director of Institute of Strength Physics and Materials Science SB, RAS (2/4, Akademicheskii av., 634055, Tomsk, Russian Federation, e-mail: shilko@ispms.tsc.ru).

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This study is about definition of rational geometry of the cable die, which effectively distributes polymer melt flows in channels; estimation of stability boundary flows of materials with different physical and rheological properties for the technological range of modes of processing; calculation and construction of the velocity fields, pressures, temperatures inside the channels of cable die; experimental determination of the dependence of the thickness of imposed layers of insulation and semiconducting materials from the linear velocity of pulling wires and expenses for each channel. In the modeling of the processes was used the cable die for joint overlay three layers of the polymer coating (layer semiconductive conductor screen, insulation, outer semiconductive screen) used in the production of modern electric cables of medium and high voltage. To analyze the processes of heat and mass in terms of stratified flows in channels of cable die real physical processes have been replaced by a mathematical model, which is a system of nonlinear differential equations, reflecting the basic conservation laws. The system was supplemented of boundary conditions and physical and rheological properties of the materials processed. In order to simplify the model offers a number of assumptions that allowed us, in particular, go to the axisymmetric formulation of the problem. To solve the formulated mathematical model the numerical method was applied, namely the finite element method, implemented through the Ansys set of software. Basing on the received results was made more effective geometry of the cable die, eliminating the effects of twist the polymer streams is developed; velocity, pressure and temperature distribution in the channels of the cable die were obtained; possible overheating of the material inside the channels is presented; the effect of certain parameters of overlay multilayer polymeric insulation process on layer thickness was assessed.

Keywords: stratified flow, mathematical modeling, extrusion, rheology, polymers, abnormally viscous liquids, cable die, numerical solution, interface, axisymmetric, laminar.

Maria V. Bachurina (Perm, Russian Federation) – graduate student of Design and Technology in Electrical Engineering, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Fede­ration, e-mail: ktei@pstu.ru).

Alexey V. Kazakov (Perm, Russian Federation) – Ph.D. in Technical Sciences, Assistant Professor of Design and Technology in Electrical Engineering, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, e-mail: ktei@pstu.ru).

Natalia M. Trufanova (Perm, Russian Federation) – Doctor of Technical Sciences, Professor, Head of Design and Technology in Electrical Engineering Department, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, e-mail: ktei@pstu.ru).

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15. Kazakov A.V., Savchenko V.G., Trufanova N.M. Modelirovanie protcessov teplomassoperenosa polimera v golovke ekstrudera s uchetom i bez ucheta zavisimosti viazkosti ot temperatury [Simulation of heat and mass transfer processes in polymer extrusion head with and without temperature dependence of viscosity]. Intellektual’nye sistemy v proizvodstve, 2010, no. 1, pp. 130-134.

16. Kazakov A.V., Trufanova N.M. Chislennye issledovaniya rezhimov stratifitsirovannogo techeniya i metodika upravleniya protsessom ekstruzionnogo nalozheniya mnogosloinoi izolyatsii [Numerical studies of stratified flow regimes and governance process extrusion multilayer insulation overlay]. Izvestiia Tomskogo politekhnicheskogo universiteta, 2012, vol. 320, no. 4, pp. 167-171.

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In order to construct a theory that adequately describes the effects of cyclic loadings, it is initially necessary to analyze the experimental plastic loop of a hysteresis stainless steel SS304; and three types of backstresses responsible for the displacement of the center of the surface of loading are specified on this steel. For each type of backstresses we have formulated evolutionary equations on basis of the equations of the theory of plastic flow in the combined hardening. We have allocated the material functions which close the theory,. We have also formulated the basic experiment and method of the material functions identification.

Evaluating the work of different types of backstresses on the field of plastic deformations under cyclic loadings with various magnitude of the deformation up to the experimental values of the number of cycles before failure, it has been obtained that the work of backstresses second type is a universal characteristic of the material. This result made it possible to formulate the kinetic equation of damage accumulation, based on which we have considered the nonlinear processes of damage accumulation. To determine the material functions responsible for the destruction, we have formulated the basic experiment and identification method. The authors have given material functions for stainless steel SS304.

We have investigated the processes of elastic-plastic deformation of stainless steel SS304 with non-stationary hard cyclic loading under block changes of amplitude and mean deformation of the cycle. Also the processes of soft non-stationary and non-symmetric cyclic loading (ratcheting) under block changes of amplitude and mean stress cycle have been examined. The results of calculations are compared with the experimental results.

Computational research of nonlinear processes of damage accumulation and low cycle fatigue of stainless steel SS304 are conducted under symmetric hard cyclic loading both at the constant amplitude of strain and block change of the amplitude of strain. The calculation results show that the scope of deformation reduction leads to increase of the nonlinearity of damage accumulation, while the increase of the deformation scale results in the fact that the accumulation of damages tends to be linear. There is a significant deviation from the rule of linear summation of damages under a satisfactory conformity of calculation results with the experiments.

The paper presents such new results as:

- specifying three types of backstresses responsible for kinematic hardening analyzing the experimental loops of plastic hysteresis;

- establishing the work universality of the second type backstresses under low-cycle and high-cycle fatigue on basis of experimental results analysis;

- constructing the theory of plastic flow under combined hardening and kinetic equations of damage accumulation on the basis of the evolution equations for three types of backstresses;

- identifying the material parameters and verifying the proposed theory.

Keywords: plastic deformation, backstresses, damage accumulation, cyclic loading, low-cycle durability, nonlinear summation of damages.

Valentin S. Bondar (Moscow, Russian Federation) – Doctor of Physical and Mathematical Sciences, Professor, Honored Worker of Science of RF, Head of Department of Theoretical Mechanics, Moscow State University of Mechanical Engineering (MAMI) (38, B.Semenovskaya st., 107023, Moscow, Russian Federation, e-mail: bondar@mami.ru).

Vladimir V. Danshin (Moscow, Russian Federation) – Ph. D. in Physical and Mathematical Sciences, Ass. Professor, Professor of Department of Theoretical Mechanics, Moscow State University of Mechanical Engineering (MAMI) (38, B.Semenovskaya st., 107023, Moscow, Russian Federation, e-mail: tm@mami.ru)

Dmitry A. Makarov (Moscow, Russian Federation) – Ph. D. in Physical and Mathematical Sciences, Аss. Professor, Department of Informational Systems and Distance Technology, Moscow State University of Mechanical Engineering (MAMI) (38, B.Semenovskaya st., 107023, Moscow, Russian Federation, e-mail: makarovda@yandex.ru).

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2. Bondar V.S., Danshin V.V. Plastichnost'. Proporcional'nye i neproporcional'nye nagruzhenija [Plasticity. Proportional and disproportionate loading]. Moscow: Fizmatlit, 2008. 176 p.

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12. Dafalias Y.F., Feigenbaum H.P. Biaxial ratchetting with novel variations of kinematic hardening. Int. J. of Plasticity, 2011, vol. 27, pp. 479-491.

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The paper is concerned with studying the dynamic characteristics of constructions and their response to vibrations initiated by weak natural environmental actions (microseismic, wind and other impacts). Such investigations require special equipment capable of recording the development of dynamic processes. Since the structures are constantly exposed to natural impacts of winds and seismic noises, it is reasonable to conduct continued observations or monitoring. The study focuses on the dynamic behavior of structures supported on pile foundations under natural outside impacts, among which the wind action and microseismic actions of different mechanisms, for example, cars are most common to urban environment. The investigation is based on the analysis of eigenfrequency, which is one of the most informative dynamic parameters for estimating the current state of constructions. Thus, a variation in the spectrum of eigenvalues is indicative of a change in the stiffness properties of the structure elements due to damage accumulation and also suggests the appearance of changes in the soil and contact pressure between the soil and foundation. In this paper, a continuous monitoring technique is used to analyze the frequency spectrum of the structure, to determine its eigenfrequencies and their seasonal changes and to perform numerical simulations allowing an adequate description of seasonal impacts on the structure frequency. The lowest eigenfreqiencies of the structure have been determined from the numerical experiment on natural frequencies of the structure-pile foundation-soil system, in which coupled springs are used to model the interactions between the pile foundation and soil. It has been found that seasonal changes in eigenfrequencies of structures supported on pile foundations are caused by frost penetration into the soil, which affects the stiffness properties of the soil-pile foundation-structure system.

Keywords: structural health monitoring, eigenfrequency, velocimeter, wavelet spectrum, Fourier spectrum, numerical simulation, pile foundation.

Valeriy V. Korepanov (Perm, Russian Federation) – Ph. D. in Physical and Mathematical Sciences, Research associate, Department of Complex Problems of Mechanics of Deformable Bodies, Institute of Continuous Media Mechanics, Ural Branch of RAS (1, Akademik Korolev st., 614013, Perm, Russian Federation, e-mail: kvv@icmm.ru).

Roman V. Tsvetkov (Perm, Russian Federation) – Ph. D. in Technical Sciences, Research associate, Department of Complex Problems of Mechanics of Deformable Bodies, Institute of Continuous Media Mechanics, Ural Branch of RAS (1, Akademik Korolev st., 614013, Perm, Russian Federation, e-mail: flower@icmm.ru).

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6. Kapustian N.R., Dykhovichnaya N.A. Seismicheskii monitoring vetrovykh kolebanii vysotnykh zdanii [Seismic monitoring of wind-driven vibrations of high-rise buildings]. Monitoring nebezopasnykh geologi­ches­kikh protsessov i ekologicheskogo sostoianiia sredy; UB of RAS. Ekaterinburg-Perm, 2005. P. 225.

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8. Tsvetkov R.V., Shardakov I.N. Avtomatizirovannaia sistema izmereniia neravnomernosti osadok sooruzheniia [Automated system for measurement of structure settlement non-uniformity]. Vestnik Volgograd­skogo gosudarstvennogo arkhitekturno-stroitelnogo universiteta. Stroitelstvo i arkhitektura, 2008, vol. 10 (29), pp. 128-134.

9. Sharadakov I.N., Matveenko V.P., Tsvetkov R.V, Golotina L.A. Sistema monitoringa zdaniia, nakhodiashchegosia pod deistviem vozmushchenii ot ego fundamenta [System for monitoring the construction under the action of foundation oscillations]. Patent RF no. 2378457 2010.

10. Tsvetkov R.V., Shardakov I.N. Modelirovanie deformatsionnykh protsessov v sisteme «gruntovoe-osnovanie-fundament-zdanie» pri nalichii karstovykh iavlenii [Modeling of deformation processes in the soil-foundation construction system under karstic conditions]. Computational continuum mechanics, 2010, vol. 3, no. 3, pp. 102-116.

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13. Martins N., Caetano E., Diord S., Magalhaes F., Cunha A. Dynamic monitoring of a stadium suspension roof: Wind and temperature influence on modal parameters and structural response. Engineering Structures, 2014, vol. 59, pp. 80-94.

14. Yuen K.V., Kuok S.C. Ambient interference in long-term monitoring of buildings. Engineering Structures, 2010, vol. 32, pp. 2379-2386.

15. Tsvetkov R.V., Shardakov I.N. Otsenka vibratsionnykh protsessov v sisteme «gruntovoe osnovanie-fundament-zdanie» pri monitoringe konstruktsii zdaniia [Estimation of vibrational processes in the soil-foundation- construction system during monitoring of the construction state] // XVII Zimniaia shkola po mekhanike sploshnykh sred (XVII Winter School on continuum mechanics), Perm, February 28 – March 3, 2011. Perm-Ekaterin­burg, 2011. P. 326.

16. Matveenko V.P., Fedorova V.A., Shardakov I.N. Theoretical Justification of the Possibility of Constructing a Fiber-Optic Earth Surface Deformation Monitoring System. Mechanics of Solids, 2013, vol. 48, no. 5, pp. 520-524.

At this paper the algorithm of the subprogram for solving a two-point boundary value problem for system of the nonlinear differential equations of the first order is presented. The new algorithm of the subprogram named KLPALG united in itself the main ideas of the subprograms BVPFD (DD14AD, PASVA3) and PASSIN realizing the technique of continuation of solution by parameter. Besides, the generalized results of works of authors in a problem of nonlinear dynamic deformation of a thin spatial curvilinear rod calculated by its differential model are presented.

The unknown functions in the equations of motion are calculated at discrete mesh points. The methods of direct integration allow us to express time derivatives by the current coordinates and coordinates and velocities calculated in the previous time steps. The first derivative of coordinate is replaced by finite difference; boundary conditions are added. The obtained system of nonlinear algebraic equations is solved by Newton method with the step length control of the convergence conditions. The Jacobi matrix of this system is of the block-tridiagonal structure which lends itself to efficient LU-decomposition. This decoupling of the Jacobi matrix allows you to quickly solve the corresponding system of linear algebraic equations of the big sizes. If the condition of convergence of Newton's method gives too small step, then used the technique of continuation of the solution on the parameter a (pseudo arc-length). As soon as the system of nonlinear equations is solved, to refine the nodal values of the calculated functions we use the deferred correction method. This method subtracts from the received solution the mistakes made by the approximation derived by the method of finite differences in the initial phase of the numerical solution. Thus obtained numerical solution is of accuracy appointed by user. This method is implemented in KLPALG subroutine which algorithm is presented in this paper.

Keywords: nonlinear boundary value problem, thin curvilinear rod, differential model, numerical solution, KLPALG, BVPFD, DD14AD, PASVA3, PASSIN.

Nikolay V. Pustovoy (Novosibirsk, Russian Federation) – Doctor of Technical Sciences, Professor, Head of the Department of Aircraft Strength, rector of Novosibirsk State Technical University (20, K. Marks av., 630073, Novosibirsk, Russian Federation, e-mail: rector@nstu.ru).

Vladimir E. Levin (Novosibirsk, Russian Federation) – Doctor of Technical Sciences, Professor, Deputy head of the Department of Aircraft Strength of Novosibirsk State Technical University (20, K. Marks av., 630073, Novosibirsk, Russian Federation, e-mail: levin@craft.nstu.ru).

Dmitry A. Krasnorutskiy (Novosibirsk, Russian Federation) – Ph. D. in Technical Sciences, Associate Professor of the Department of Aircraft Strength of Novosibirsk State Technical University (20, K. Marks av., 630073, Novosibirsk, Russian Federation, e-mail: krasnorutskiy@corp.nstu.ru).

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This paper presents experimental studies of the mechanical properties of technical plasticine – which is a composite material consisting of a matrix (a mixture of wax and oils) and particulate filler (talc, clay, pigments) – under tension, compression, shear and penetration of a spherical indenter. At a constant strain rate (tension, compression) “stress – strain” diagrams have been obtained and characterized by a small elastic zone and yield strains up to 15–20 %. On the basis of experimental data power law for dependence of yield strength vs the strain rate in the range of 0.0004 ... 80 s^–1 have been obtained. Thus, the deformation of technical plasticine can be described by viscoelastic Norton-type model with a serial connection of elastic and viscous elements.

Under the tensile and shear strain of the technical plasticine over 3…5 % it begins to rapidly accumulate scattered microdamages, which does not allow using shear test as a test for determination of the yield stress like in the known methods related to metals. The yield stresses under tension and compression are close at the same strain rates.

A linear diagram “force – penetration depth” down to a depth of 3 mm are obtained at penetration of a spherical indenter with a diameter of 43 mm in a plasticine block of 75 mm thick. Thus, the plasticine hardness is constant at a certain strain rate. The authors have managed to get dynamic hardness as the energy of a falling body, divided by the volume of cavity in the plasticine. Static and dynamic penetration is a promising method for the study of plastic properties of materials because of its simplicity. However it is necessary to establish the correlation coefficient relating hardness and yield stress. For the considered material such ratio is 0.24 at the frictionless condition.

Keywords: technical plasticine, plasticity, viscosity, tension, compression, shear, strain rate, diagram of deformation, yield stress, hardness.

Sergey B. Sapozhnikov (Chelyabinsk, Russian Federation) – Doctor of Technical Sciences, Professor, Department of Applied Mechanics, Dynamics and Strength of Machines, South Ural State University (76, Lenin av., Chelyabinsk, Russian Federation, 454080, e-mail: ssb@susu.ac.ru).

Anastasia V. Ignatova (Chelyabinsk, Russian Federation) – Ph.D. student, Department of Applied Mechanics, Dynamics and Strength of Machines, South Ural State University (76, Lenin av., Chelyabinsk, Russian Federation, 454080, e-mail: ign.nastya@gmail.com).

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Hooke’s law (in a modern tensor form considering different types of material anisotropy, finite or velocity formulation) is widely used in solid mechanics including physical and/or geometrical nonlinear problems. In the recent decades it has also been used in the majority of multilevel models oriented on describing inelastic deformation in mono- and polycrystalline materials. As a rule, in this case Hooke’s law is written using symmetrical measures of stress and strain state that are determined in terms of actual, intermediate (unloaded) or reference configuration. For a material that is elastic according to Green, the elastic potential presence naturally leads to the symmetry of elastic four-valent tensor П in the first and second pare of indices, П_ijkl = П_klij. However tensor symmetry in the first and second pair of indices is explained only due to the accepted and established agreement in solid mechanics related to symmetry of stresses and strains tensors. It is worth mentioning that the initial Hooke’s law written for uniaxial loading obviously had nothing to do with the symmetry of properties. The specified agreement made it possible to reduce the number of experiments necessary to find tensor elastic properties; and it is especially important for materials studies with an a priory low or unknown symmetry. Stress tensor symmetry results from law of conservation of angular momentum without distributed volume and surface moments. The neglection of the distributed surface moments is based on a hypothesis that two parts of the body interact with distributed forces, which can be put in to the stresses vector on each surface element. This hypothesis again is based on an idea that there is no correlation of distributed surface loadings on any material area element. It is worth stating that already in 1887 V. Voigt suggested to abandon this idea and put the distributed effects of one body part on the other one on any surface element into stresses vector and distributed moments vector. The specified suggestion is in a full compliance with the method related to putting a random system of forces into the principal vector and principal moment (this method is used in theoretical (classical) mechanics). The problem of a simple shear shows that Hooke’s (symmetrical) law leads to the incompliance of the stress state (found with the law in its conventional formulation) and part of boundary conditions. We have considered Hooke’s law which is oriented on application of asymmetrical measures of stresses and strains and elastic properties tensor with symmetry only in a pair of indices. Asymmetrical Cauchy tensor is used as a stress measure, gradient of displacement velocity (displacement velocities with respect to a stiff moving coordinates which is in charge for a rigid displacement of volume element) – as strain velocity measure; all of them do not depend on the reference coordinate. A type of tensor of elastic properties in Hooke’s law oriented on asymmetric measures is proposed.

Keywords: asymmetric tensors of stresses and strains, modified Hooke’s law.

Peter V. Trusov (Perm, Russian Federation) – Doctor of Physical and Mathematical Sciences, Professor, Head of Department of Mathematical Modelling of Systems and Processes, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation,
e-mail: tpv@matmod.pstu.ac.ru).

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