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Journal of Applied Mechanics and Technical Physics

2021 year, number 5

Using Eigenmoduli and Eigenstates to Evaluate the Possibility of Martensitic Phase Transformations

B. D. Annin1,2, N. I. Ostrosablin1, R. I. Ugryumov1,2
1Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
2Novosibirsk National Research State University, Novosibirsk, 630090, Russia
Keywords: eigenmoduli and eigenstates, shape-memory alloys, elastic moduli, compliances, cubic and hexagonal lattices, specific strain energy

Abstract >>
The possibility of phase transitions (martensitic transformations) in shape-memory alloys is evaluated using the concept of eigenmoduli and eigenstates from the linear theory of elasticity. For alloys with cubic and hexagonal lattices, the matrices of elastic moduli and compl are given and expressions for their eigenmoduli and eigenstates are written. For cubic and hexagonal phases, the specific strain energy is presented as the sum of six independent terms corresponding to six orthogonal eigenstates. It is shown that depending on the ratio of eigenmoduli, there are six types of materials (alloys) with cubic and hexagonal symmetry. The specific strain energies in the cubic and hexagonal phases are compared. If the strain energy is greater in the hexagonal phase than in the cubic phase, the alloy can tend to return to its original state with lower energy. In addition, the strain energies in different phases can be compared using the formulas of the tensors closest in the Euclidean energy norm to cubic and hexagonal tensors. The energies are compared for some values of elastic constants.

DOI: 10.1134/S0021894421050011

Investigation of the Three-Dimensional Helmholtz Equation for a Wedge Using the block Element Method

V. A. Babeshko1,2, O. V. Evdokimova1, O. M. Babeshko2
1Southern Scientific Center, Russian Academy of Sciences, Rostov-on-Don, 344006, Russia
2Kuban State University, Krasnodar, 350040, Russia
Keywords: block element method, boundary-value problem, Helmholtz equation, pseudo-differential equations

Abstract >>
For boundary-value problems, the Helmholtz equations in wedge-shaped domains, it is shown that in packed block elements corresponding to the same boundary-value problem can be combined taking into account the type of boundary conditions, also forming a packed block element. The result is verified using another method. It is shown that in the presence of corner points in the domain in which the boundary-value problem is considered, combining block elements does not involve additional complications. It is found that since the solutions of some boundary-value problems in continuum mechanics and physics can be represented as a combination of solutions of boundary-value problems of the Helmholtz equation, this approach can be used to study more complex boundary-value problems and design materials with mosaic structure.

DOI: 10.1134/S0021894421050023

Indentation of a Rigid Cylinder with a Flat Rough Base into a Thin Viscoelastic Layer

I. G. Goryacheva, A. A. Yakovenko
Ishlinsky Institute for Problems in Mechanics, Moscow, 119526, Russia
Keywords: microrelief, indentation, viscoelastic layer, actual contact area

Abstract >>
Interaction between a thin viscoelastic layer and a rigid cylinder whose contacting end surface is nominally flat but has a microrelief is studied. The microrelief is modeled by a periodic system of axisymmetric indenters. Analytical expressions for the depth of indentation and the actual contact area are obtained using an approach based on consideration of micro- and macroscale levels. The effect of the surface microgeometry of the punch and mechanical properties of the layer on the time dependences of the indentation depth and the actual contact area.

DOI: 10.1134/S0021894421050035

Compensating Role Self-Balanced Stress Fields in Constructing Nonsingular Solutions Using a Non-Euclidean Model of a Continuous Medium for an Incompressible Sphere

M. A. Guzev1,2, W. Liu3, Ch. Qi3, E. P. Riabokon2
1Institute of Applied Mathematics, Far-Eastern Branch, Russian Academy of Sciences, Vladivostok, 690041 Russia
2Perm National Research Polytechnic University, Perm, 614990, Russia
3Beijing University of Civil Engineering and Architecture, Beijing, 100044, China
Keywords: inconsistency condition, non-Euclidean continuous model, singularities, self-balanced stress field

Abstract >>
A self-balanced stress field for an incompressible sphere is constructed based on a non-Euclidean model of a continuous medium. The total stress field is presented as the sum of the elastic and self-balanced fields. The requirement that there are no singular contributions to the stress field leads to the fact that the coefficients at the singularities of the elastic and self-balanced stress fields can be related by a linear transformation, ensuring the removal of singularities. The compensating role of self-balanced stress fields allows one to construct a nonsingular equilibrium stress field for a spherically symmetric state of a continuous medium.

DOI: 10.1134/S0021894421050047

Application of Electroconducting Composite Materials for Additional Damping of Smart Systems Based on Piezoelements

V. P. Matveenko, D. A. Oshmarin, N. A. Yurlova
Institute of Continuous Media Mechanics, Ural Branch, Russian Academy of Sciences, Perm, 614013, Russia
Keywords: graphene composites, piezoelectric elements, vibrations, multimodal damping, numerical simulation

Abstract >>
A problem of damping vibrations of a smart structure consisting of elastic and viscoelastic materials and piezoelements with connected shunt circuits is considered. It is proposed to replace the classical resistor in the shunt circuit by an element made of an electroconducting material, in particular, a polymer material filled with graphene nanoparticles. This element plays the role of several resistors with different resistance values, which ensure multimodal damping of vibrations. A mathematical formulation of the problem of forced steady and natural vibrations of smart systems under consideration is provided, as well as results of numerical calculations, which show that graphene-based composites can be used for additional damping of vibrations of smart structures based on piezoelements.

DOI: 10.1134/S0021894421050059

Creation of a Heterogeneous Material Based on the Titanium Alloy and Titanium Boride by the Method of Controlled Laser Processing

V. M. Fomin, T. A. Brusentseva, A. A. Golyshev, A. G. Malilkov, A. V. Mishin, A. M. Orishich, A. A. Filippov
Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
Keywords: selective laser sintering, functionally graded material, titanium alloy, ceramics, heterogeneous system, Young's modulus

Abstract >>
Results of theoretical and experimental investigations of physical and mechanical properties of a heterogeneous material based on the TiB ceramics and VT-6 metallic alloy obtained by means of controlled laser processing are reported. Elastic properties of the heterogeneous structure under analysis are described by the method of conditional moments. Young's modulus of the created heterogeneous material cased on the titanium alloy and titanium boride is measured. The experimental data are found to be in good agreement with the numerical predictions.

DOI: 10.1134/S0021894421050060

Analysis of Elastoplastic Deformation of a Rotating Solid Cylinder under the General Piecewise-Linear Plasticity Condition

A. N. Prokudin, A. A. Burenin
Institute of Mechanical Engineering and Metallurgy, Khabarovsk Federal Research Center, Far-East Branch, Russian Academy of Sciences, Komsomolsk-on-Amur, 681005, Russia
Keywords: rotating cylinder, stresses, strains, elastoplasticity, piecewise-linear condition, exact solution

Abstract >>
Results of the analysis of elastoplastic deformation of a rotating solid cylinder with fixed end faces under monotonic loading by centrifugal forces are reported. The theory of small elastoplastic strains is used in formulating the problem. The general piecewise-linear condition of plasticity and the associated law of the flow are used for calculating the plastic component of the strain. The chosen plasticity condition depends on the parameter that can be considered as a material characteristic. An exact solution of the governing system of equations is derived. Regular features of plastic flow development are found. It is demonstrated that six plasticity domains are formed in the cylinder in the general case; these domains correspond to different ribs and faces of the surface defined by the general piecewise-linear condition. The dependence of the critical velocity of cylinder rotation on the parameter included into the plasticity condition is derived.

DOI: 10.1134/S0021894421050072

Controlling the Properties of a BaxSr1-xTiO3 Ferrelectric Film

V. B. Shirokov1,2, P. E. Timoshenko2, V. V. Kalinchuk1
1Southern Scientific Center, Russian Academy of Sciences, Rostov-on-Don, 344006, Russia
2Southern Federal University, Rostov-on-Don, 344006, Russia
Keywords: ferroelectric film, phenomenological model, thermoelastic deformation, initial deformation, initial stress, surface acoustic waves, electromechanical coupling coefficient

Abstract >>
A method is proposed to control the properties of thin ferroelectric films under forced deformation due to the size mismatch of the crystalline lattices of film and substrate materials and due to the difference of their thermal expansion coefficients. Control is based on additional mechanical deformation of the substrate. A model of a single-crystal BaxSr1-xTiO3 film is studied within the framework of phenomenological theory using the Landau potential. It is shown that additional uniaxial deformation of the substrate in a BaxSr1-xTiO3 film changes the material constants of the film. Abnormal change occurs at deformation values close to the values at which the phase state of the film changes. The generation of surface acoustic waves is studied. The results of modeling simulations indicate the possibility of controlling the excitation of surface acoustic waves in the film-silicon substrate heterostructure.

DOI: 10.1134/S0021894421050084

Viscometric Flow of Elastoplastic Material Heated by Wall Friction

A. S. Begun1,2, L. V. Kovtanyuk1
1Institute of Automation and Control Processes, Far Eastern Scientific Center, Siberian Branch, Russian Academy of Sciences, Vladivostok, 690041, Russia
2Vladivostok State University of Economics and Service, Vladivostok, 690014, Russia
Keywords: elasticity, viscosity, plasticity, viscometric flow, large deformations, thermoplasticity

Abstract >>
A mathematical model of large deformations is used to solve a coupled boundary-value problem about the deformation of an elastic-viscoplastic material in a cylindrical viscometer with account for its heating due to wall friction. The deformation of a material enclosed between rigid surfaces due to the rotation of an inner cylindrical surface at a variable velocity is investigated. It is taken into account that a yield point depends on temperature. The motion of elastoplastic boundaries is described. Stresses, strains, and temperature in a thermoelastic deformation region and in a flow region both during the development of the flow and during its deceleration, including stopping, unloading, and cooling, are calculated. Residual stresses and deformations are determined.

DOI: 10.1134/S0021894421050096

Simulating an Impact on Composite Damping Elements

A. N. Fedorenko1, B. N. Fedulov2, E. V. Lomakin1,3
1Center for Design, Manufacturing Technology, and Materials, Skolkovo Institute of Science and Technology, Moscow, 121205, Russia
2Lomonosov Moscow State University, Moscow, 119991, Russia
3Moscow Aviation Institute, Moscow, 125993, Russia
Keywords: composite materials, fracture, damage parameters, shock loads, damping

Abstract >>
A model of dynamic deformation and fracture of composite materials is been developed. This model accounts for the significant nonlinearity of shock loading diagrams with hardening, which depends on strain rate. An approach is used in which the dependence of ultimate strength on damage parameters and their variation rate is introduced in the form of constitutive relations. The proposed relations are similar to those of the Johnson-Cook model, but stresses are expressed via damage parameters and their variation rate rather than in terms of plastic deformations and the variation rate of plastic deformations. On the basis of the developed model, the impact fracture of a tubular profile made of a composite material based on carbon fiber and a polymer binder are numerically simulated. The influence of the orientation of unidirectional layers of a composite on specific absorption energy is investigated.

DOI: 10.1134/S0021894421050102

Projectile Impact onto a Membrane

G. P. Cherepanov
New York Academy of Sciences, New York, USA
Keywords: projectile, armor, critical kinetic energy of the projectile

Abstract >>
A problem of the projectile impact onto a membrane is studied under the following assumptions: the armor shell is a membrane with a zero flexural stiffness and a large Young's modulus under tension; the frontal part of the projectile is shaped as a paraboloid; the projectile impact onto the membrane is considered in a quasi-static formulation; the membrane is broken when a certain limiting surface tension is reached at the paraboloid vertex. It is found that the armor shell ensures safety if the kinetic energy of the projectile does not exceed some maximum value for this armor determined theoretically or experimentally.

DOI: 10.1134/S0021894421050114

Numerical Study of Wave Propagation in Nonlinear Dissipative Material

Yu. M. Volchkov1,2, I. O. Bogulskii3,4
1Novosibirsk National Research State University, Novosibirsk, 630090 Russia
2Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
3Siberian Federal University, Krasnoyarsk, 660041, Russia
4Krasnoyarsk State Agrarian University, Krasnoyarsk, 660049, Russia
Keywords: metal rubber, damping, numerical modeling, nonlinear elasticity, deformation theory

Abstract >>
We have investigated a class of materials whose experimental stress-strain behavior does not allow them to be considered plastic or elastic. They are not elastic since unloading occurs along a curve significantly different from the loading curve; nor are they plastic since in a full loading-unloading cycle, residual deformations are absent. As such materials we investigate so-called metal rubbers - materials made from twisted wire pressed into an almost homogeneous body. The propagation of shock waves in these materials is studied using a one-dimensional model.

DOI: 10.1134/S0021894421050126

Dynamic Tension of a Rod Made of an Ideally Rigid-Plastic Material

D. V. Georgievskii
Lomonosov Moscow State University, Moscow, 119991, Russia
Keywords: ideal plasticity, yield stress, rod, tension, neck, quasistatic, dynamics, strain rate, stress, asymptotic expansions

Abstract >>
A stress-strain state arising under dynamic tension of a homogeneous rod made of an incompressible ideally rigid-plastic material that satisfies the Mises-Hencky criterion is investigated. The possibility of thickening or thinning of the rod along its length is taken into account in an axisymmetric formulation, which makes it possible to simulate the formation and development of a neck. Three dimensionless time functions are introduced, one of which is a small geometric parameter, namely the ratio of an average radius to half the rod length. The ratios of the orders of smallness of the other two dimensionless functions to the small geometric parameter determine the influence of inertial terms in equations of motion on the stress and strain rate distribution. These ratios may vary at different time intervals, which determines one or another dynamic tension regimes. Two such characteristic regimes are revealed: one of them depends on a velocity at which the end sections move away from each other, and the other one depends on their acceleration. For the second regime, an asymptotic integration based analysis makes it possible to find the stress-strain state parameters, in which case this state is an “inertial correction”' with respect to a quasistatic state in the rod with a cylindrical lateral surface.

DOI: 10.1134/S0021894421050138

Solving the Problems of Mechanics of Cylindrical Anisogrid Lattice Bodies of Spacecrafts

A. V. Lopatin, V. V. Moskvichev, A. E. Burov
Federal Research Center for Information and Computational Technologies, Krasnoyarsk, 660049, Russia
Keywords: spacecraft, composite materials, anisogrid structure, cylindrical lattice shell, analytical and numerical methods for calculating the shell

Abstract >>
Results of solutions to problems of determining the parameters of anisogrid lattice structures of spacecraft bodies are presented. These problems include analyzing the deformability of a body loaded with transverse inertial forces, determining the fundamental frequency of transverse vibrations of a cantilevered body, studying the deformability of a body loaded with axial compressive force, and analyzing the deformability of a body with an internal fuel tank loaded with transverse inertial forces. The problems under consideration are solved using a continuous model of the lattice structure of a cylindrical body and the finite-element method.

DOI: 10.1134/S002189442105014X

Variational Formulation of Coupled Hydrodynamic Problems

S. A. Lurie1,2, P. A. Belov1
1Institute of Applied Mechanics, Russian Academy of Sciences, Moscow, 125040, Russia
2Lomonosov Moscow State University, Moscow, 119991, Russia
Keywords: variational models, non-integrable variational forms, irreversible processes, spatial-temporal continuum, coupled hydrodynamics and heat transfer, Darcy equations, Brinkman equations, heat balance equations

Abstract >>
A method is proposed for constructing variational models of continuous media for reversible and irreversible processes based on the generalized Hamilton-Ostrogradskii principle, which reduces to the principle of steadiness for a non-integrable variational form of a spatial-temporal continuum. For dissipative processes, the corresponding linear variational form is constructed as a sum of variation of the Lagrangian of the reversible part and a linear combination of dissipation channels of physically nonlinear processes. Examples of using the variational approach to the description of hydrodynamic models are considered. The corresponding variational models of the Darcy hydrodynamics, linear Navier-Stokes hydrodynamics, Brinkman hydrodynamics, gradient hydrodynamics, and some generalization of the classical nonlinear Navier-Stokes hydrodynamics are constructed. For modeling irreversible processes of hydrodynamics with allowance for coupling of deformation with the associated physical processes of heat transfer, it is proposed to use variational formalism for the spatial-temporal continuum, where the spatial and temporal processes are considered simultaneously and consistently because the normalized time is a coordinate.

DOI: 10.1134/S0021894421050151

Laser Welding of Disparate Materials Based on Thermally Hardened Aluminum Alloys

A. G. Malikov1, A. M. Orishich1, I. E. Vitoshkin1, E. V. Karpov2, A. I. Ancharovc3,4
1Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
2Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
3Institute of Solid State Chemistry and Mechanochemistry, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630128, Russia
4Budker Institute of Nuclear Physics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia
Keywords: laser welding, disparate materials, aluminum-lithium alloy, strength, structural-phase composition

Abstract >>
Results of an experimental study of the strength characteristics and structural-phase composition of a nondetachable connection of disparate materials based on thermally hardened aluminum alloys D16T of the Al-4,4Cu-1,5Mg system and 1420 of the Al-5,2Mg-2,1Li system obtained by butt-end laser welding are reported. The phase composition of the welded joint is studied with the use of synchrotron radiation by the method of transmission diffraction. The welded joint microstructure is considered by optical microscopy. It is demonstrated that heat treatment (quenching and artificial aging) allows one to improve the chemical properties of the welded joint.

DOI: 10.1134/S0021894421050163

Analysis of Bending of Composite Plates with Account for the Difference in resistance to Tension and Compression

I. E. Petrakov, V. M. Sadovskii, O. V. Sadovskaya
Institute of Computational Modelling, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, 660036, Russia
Keywords: fiber composite, technical theory of plates, multimodulus elasticity

Abstract >>
The finite element method is used to develop a computational algorithm for solving a limited class of problems on the bending of composite plates reinforced with systems of unidirectional high-strength fibers. It is assumed that a neutral plane exists in the region of the plate and behaves similarly to a flexible nondeformable membrane. The displacements of the plate in the longitudinal direction are linear in thickness. In the case of fiber composites of different modulus with different elastic properties under tension and compression, the neutral plane, generally speaking, does not coincide with the median plane. The problem of minimizing the elastic energy functional in accordance with the Lagrange variational principle yields a fourth-order elliptic differential equation for the deflection. The bending stiffnesses of the plate included in the coefficients of the equation are calculated with account for the fact that the elastic characteristics of the reinforcing fibers under tension and compression are significantly different. The numerical solution of the equation is obtained via the finite element method with the help of a Bell triangular element. The paper presents computational results for the bending of rectangular laminated plates in which the fibers are laid in different directions.

DOI: 10.1134/S0021894421050175

Relaxation of Residual Stress in a Surface-Hardened Prismatic Sample in Creep under Biaxial Loading

V. P. Radchenko, T. I. Berbasova, D. M. Shishkin
Samara State Technical University, Samara, 443100, Russia
Keywords: prismatic sample, surface plastic deformation, residual stresses, creep, relaxation, biaxial loading

Abstract >>
A mathematical model has been developed for calculating the relaxation of residual stresses in a surface-hardened prismatic sample in creep under biaxial loading. The calculation results were verified against experimental data for a surface-hardened sample of EP742 alloy after ultrasonic hardening under creep conditions at a temperature of 650oC for 100 h. A detailed theoretical analysis of the influence of the type of stress state on the relaxation of residual stresses during thermal exposure (temperature exposure in the absence of mechanical stress) and under biaxial loading at a constant intensity of external stresses was performed. It is shown that under uniform tension in a plane stress state, the relaxation of residual stresses slows down, and under compression, the relaxation rate increases compared to the case of thermal exposure.

DOI: 10.1134/S0021894421050187

Numerical Simulation of Evolution of Magnetic Microstructure in Heusler Alloys

A. A. Rogovoi, O. S. Stolbova, O. V. Stolbov
Institute of Continuous Media Mechanics, Ural Branch, Russian Academy of Science, Perm, 614018, Russia
Keywords: Heusler alloy, micromagnetism, magnetic domains, variational formulation, finite element method

Abstract >>
A microstructural model of behavior of ferromagnetic material (Heusler alloy) in a magnetic field is constructed within the framework of the theory of micromagnetism. The process dynamics is described by the Landau-Lifshitz-Hilbert equation. The Galerkin procedure is used to assign variational equations to differential relations. A “Christmas Tree” type martensitic structure (twinned version of martensite) with magnetic domains arranged at an angle of 180oC is considered. The twin boundaries act as 90-degree magnetic domain walls. The evolution of this magnetic structure is investigated, namely the motion and interaction of the 180-degree walls of this magnetic domain in the presence of an external magnetic field applied in different directions. The finite element method simulates the formation of these walls and the magnetization vector distribution in them.

DOI: 10.1134/S0021894421050199

Group Analysis of the Ideal Plasticity Equations

S. I. Senashov1, O. V. Gomonova1, O. N. Cherepanova2
1Reshetnev Siberian State University of Science and Technology, Krasnoyarsk, 660037, Russia
2Siberian Federal University, Krasnoyarsk, 660041, Russia
Keywords: ideal plasticity, exact solutions, conservation laws, elastoplastic problem

Abstract >>
The problem of constructing exact solutions of the von Mises three-dimensional plasticity equations based on the group of continuous transformations admitted by the equations (Annin's problem). New classes of solutions of the three-dimensional plasticity equations are given. The problem of compression of an elastoplastic material layer by rigid plates is solved. In this case, the material obeys the exponential plasticity condition, proposed by Annin.

DOI: 10.1134/S0021894421050205