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2014 year, number 1
B. D. Annin^{1,2}, N. I. Ostrosablin^{1}
^{1}Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia ^{2}Novosibirsk State University
Keywords: steady creep, eigen coefficients of anisotropy, eigenstates, transverse isotropy, orthotropy, incompressibility
Abstract >>
The Kelvin approach describing the structure of the generalized Hooke's law is used to analyze the potential model of anisotropic creep of materials. The creep equations of incompressible transversely isotropic, orthotropic materials and those with cubic symmetry are considered. The eigen coefficients of anisotropy and eigen tensors for the anisotropy tensors of these materials are determined.

V. G. Bazhenov, E. G. Gonik, A. I. Kibets, D. V. Shoshin
Research Institute of Mechanics of Lobachevsky Nizhnii Novgorod State niversity, Nizhnii Novgorod, 603950 Russia
Keywords: spherical shell, plastic deformation, critical load, postcritical behavior, finite element method, experiment
Abstract >>
The problem of elastoplastic deformation, buckling, and postcritical behavior of spherical shells is solved using a finite element method and a crosstype explicit scheme of time integration. Stability problems for hemispherical shells under external pressure and compression between rigid plates are considered. The influence of holes and boundary conditions on shell deformation is investigated. It is shown that the calculation results are in good agreement with experimental data.

A. V. Berezin
Blagonravov Institute of Machines Science, Russian Academy of Sciences, Moscow, 101990 Russia
Keywords: fracture, crack, strength
Abstract >>
Cracks arising in dilatant and multimodulus materials (different elasticity moduli for different types of the stress state) are under study. The stress and strain distributions in dilatant multimodulus bodies with cracks, which allow estimating the impact of damage on crack resistance characteristics, are investigated.

V. D. Bondar'
Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: antiplane strain, boundaryvalue problem, type of the equation, load, Legendre transform, nonlinearity
Abstract >>
Antiplane strain in an incompressible cylindrical body is studied within the framework of a nonlinear model. The system of equations is reduced to a secondorder nonlinear equation for the displacement. It is shown that the type of the equation is determined by the strain potential and can be mixed. An example of the strain in a hollow elliptic cylinder is considered, for which a displacement of the screw dislocation type is found and the corresponding stresses and load are studied.

M. P. Bondar'^{1,2}, E. V. Karpov^{1,2}
^{1}Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia ^{2}Novosibirsk State University
Keywords: nanocomposite, mesocomposite, quasidynamic pressing, mechanism of formation, microstructure, mechanical properties
Abstract >>
The mechanism of obtaining highstrength mesocomposite materials with a cellular microstructure is studied. It is shown that the formation of the mesocomposite microstructure is determined by the method of pressing characterized by high strains and strain rates, and also by the mesocomposite composition; the highplasticity matrix and nondeformable particles of the hardening phase, which are not bonded to the mesocomposite matrix, assist in the process of selforganization, i.e., formation of a cellular microstructure. The set of mesocomposite properties is studied to understand the general laws of mesocomposite formation and optimize the composition. A composite with an optimal composition is obtained, which retains high electrical conductivity of copper and sufficiently high plasticity necessary for its effective application; its strength is greater than the base strength by an order of magnitude; moreover, its strength and wear resistance are much higher than those of dispersionhardened alloys obtained by the method of internal oxidation. The results obtained can be used for creating new composite materials.

V. I. Butyrin, V. N. Maksimenko, L. V. Pavshok, B. S. Reznikov
Novosibirsk State Technical University, Novosibirsk, 630092 Russia
Keywords: multilayer shells, composite material, set of loads, strength, optimal weight design, coordinatewise descent method in the unit interval
Abstract >>
The stress–strain state of axisymmetric multilayer shells is analyzed using kinematic and static hypotheses that allow for the transverse shear stresses satisfying the necessary equations of state, continuity conditions at the boundaries between the layers and given boundary conditions. A numerical solution of the problem of the stress–strain state for a multilayer bar is compared with the Lekhnitskii solution (for a cantilever beam loaded by a concentrated force and moment) to asses the applicability of the employed bending equations of multilayer shells. It is shown that these solutions are in good agreement. The problem of the initial fracture of the shells considered is formulated using phenomenological strength criteria for each layer. A coordinatewise descent method in the unit interval is proposed to solve weight optimization problems for multilayer shells of composite materials under combined loading. Regions of safe operating loads and the optimal weight distribution of layer thicknesses are determined for a multilayer bar acted upon by a uniformly distributed load and concentrated force.

V. I. Van'ko, E. S. Perelygina
Bauman Moscow State Technical University, Moscow, 105005 Russia
Keywords: buckling, hinged bar, initial deflection, arbitrary (Ïƒ–Îµ) diagram
Abstract >>
Buckling of a bar of an elastoplastic material is studied. It is shown that for any (σ–ε) diagram of the bar material, the limit load (the longitudinal external force) in dimensionless variables that the bar can withstand does not exceed the current bending stiffness of the most loaded (in terms of the bending moment) section.

I. N. Vasserman, V. P. Matveenko, I. N. Shardakov, A. P. Shestakov
Institute of Mechanics of Continuous Media, Ural Branch, Russian Academy of Sciences, Perm', 614013 Russia
Keywords: magneticresonance imaging, excitable media, singledomain model, ionic model, splitting method
Abstract >>
Based on a singledomain model of myocardial conduction, isotropic and anisotropic finite element models of the myocardium are developed allowing excitation wave propagation to be studied. The Aliev–Panfilov phenomenological equations were used as the relations between the transmembrane current and the transmembrane potential. Interaction of an additional source of initial excitation with an excitation wave that passed and the spread of the excitation wave are studied using heart tomograms. A numerical solution is obtained using a splitting algorithm that allows the nonlinear boundaryvalue problem to be reduced to a sequence of simpler problems: ordinary differential equations and linear boundaryvalue problems in partial derivatives.

Yu. M. Volchkov^{1,2}
^{1}Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia ^{2}Novosibirsk State University
Keywords: orthotropic material, plates, cylindrical bending, Legendre polynomials
Abstract >>
Based on approximations of solutions of elasticity theory equations by Legendre polynomial segments, differential equations for bending of orthotropic plates are constructed. In contrast to equations constructed with the use of kinematic and force hypotheses, the order of these differential equations is independent of the type of conditions on front surfaces. The matrices of the constructed equations depend on the type of boundary conditions. An analytical solution is given for the system of equations in the case with normal and shear stresses being specified on the upper and lower front surfaces.

R. V. Goldstein, S. E. Aleksandrov
Ishlinskii Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526 Russia
Keywords: elastoplastic body, large deformations, residual stress and strain, dependence of elastic moduli on accumulated strain
Abstract >>
A semianalytical solution of the problem of compression of a tube by an outer pressure with subsequent unloading is obtained. The effect of the relationship between the shear modulus and plastic strain on the residual stress and strain was evaluated using experimental data, according to which at an accumulated plastic strain of 0.25, the shear modulus decreases by 20%. It is found that despite the significant decrease in the shear modulus, its dependence on the accumulated strain has no significant effect on the residual strain. The effect of this dependence is manifested mainly in the distribution of the residual radial stress, but, in this case, too, it is extremely weak. The obtained general solution can be used to evaluate the effect of the relationship between the shear modulus and the accumulated plastic strain on the residual stress and strain for other materials.

I. G. Goryacheva, M. M. Gubenko, Yu. Yu. Makhovskaya
Ishlinskii Institute of Mechanics, Russia Academy of Sciences, Moscow, 117526 Russia
Keywords: contact interaction, adhesion, friction, viscoelasticity
Abstract >>
The problem of sliding of a spherical indenter on a viscoelastic foundation is solved in a quasistatic formulation taking account the forces of adhesive attraction which are considered different at the entrance to and exit from the contact region due to changes in the surface properties during the interaction. It is found that the contact characteristics and the frictional force due to the imperfect elasticity of the foundation depend on the surface and bulk properties of the materials of the interacting bodies and the interaction conditions (load, velocity, etc.).

L. A. Igumnov, A. V. Amenitskii, A. A. Belov, S. Yu. Litvinchuk, A. N. Petrov
Research Institute of Mechanics of Lobachevsky Nizhnii Novgorod State University, Nizhnii Novgorod, 603950 Russia
Keywords: method of boundary integral equations, viscoelasticity, porous elastic bodies, inversion of the Laplace transform
Abstract >>
This paper presents the results of mathematical and discrete modeling of linear dynamics problems for threedimensional viscoelastic and porous elastic bodies. The employed methods and approaches are based on formulating boundary integral equations solved using boundary elements. The model of a standard viscoelastic body is employed as the viscoelastic model. The properties of porous elastic materials are described using the full Biot model with four basic functions. Examples of numerical solutions of the problems are compared with known results of solutions.

E. V. Karpov, A. Yu. Larichkin
Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: complex loading, constrained torsion, lowcycle fatigue, Plexiglas, “frozen” highly elastic strain, fracture, strain localization
Abstract >>
Experimental studies of deformation and fracture of rods made of Plexiglas (PMMA) under complex loading (quasistatic and cyclic torsion under axial compression) are performed. The existence of a range of critical values of axial stress, within which the rod is fractured by the torque, is established. The localization of “frozen” highly elastic strains in constrained cyclic torsion is revealed, and the conditions of its occurrence are determined. The effect of axial stress on strain localization and fracture and the influence of cyclic torsion on the loss of stability of a rod under axial compression are shown.

S. N. Korobeynikov^{1,2}, N. G. Torshenov^{1,2}, I. V. Lyubashevskaya^{1,2}, A. Yu. Larichkin^{1,2}, E. V. Chunikhina^{1,2}
^{1}Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia ^{2}Novosibirsk State University
Keywords: cylindrical shell, zirconium alloy, buckling, creep
Abstract >>
Experiments were performed to study the deformation and buckling of axially compressed circular cylindrical shells of Zr2.5Nb zirconium alloy under creep conditions. Computer simulation using the MSC.Marc 2012 software was conducted by stepbystep integration of the equations of quasistatic deformation of thin shells using Norton's law of steady creep. The results of the experiment and computer simulation show that the buckling modes are a combination of axisymmetric bulges located near one end or both ends of the shell and axisymmetric buckling modes with the formation of three or four waves in the circumferential direction. A comparison is made of the time dependences of the axial strain of the shells obtained in the experiment and by computer simulation. It is shown that for large axial compressive stresses, these dependences are in satisfactory agreement. For lower values of these stresses, the difference between the theoretical and experimental dependences is greater.

A. M. Lokoshchenko
Institute of Mechanics at the Lomonosov Moscow State University, Moscow, 119992 Russia
Keywords: creep, longterm strength, damage, tension, compression, kinetic parameters, vibrocreep, criterial approach, kinetic approach, shells, shrinkage, membranes
Abstract >>
Basic results of experimental and theoretical research of creep processes and longterm strength of metals obtained by researchers of the Institute of Mechanics at the Lomonosov Moscow State University are presented. These results further develop and refine the kinetic theory of creep and longduration strength proposed by Yu. N. Rabotnov. Some problems arising in formulating various types of kinetic equations and describing experimental data for materials that can be considered as statically homogeneous materials (in studying the process of deformation and rupture of such materials, there is no need to study the evolution of individual cracks) are considered. The main specific features of metal creep models at constant and variable stresses, in uniaxial and complex stress states, and with allowance for one or two damage parameters are described. Criterial and kinetic approaches used to determine longterm strength under conditions of a complex stress state are considered. Methods of modeling the metal behavior in an aggressive medium are described. A possibility of using these models for solving engineering problems is demonstrated.

S. T. Mileiko
Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, 142432 Russia
Keywords: metal matrix composites, creep, creeprupture strength, hightemperature strength, fracture toughness, heat resistance, oxide fiber, nickel superalloys, molybdenum
Abstract >>
This paper presents an overview of studies of creep–one of the most important characteristics of hightemperature materials. A structural model of creep and methods of accelerated creep tests are considered, and the effect of structural parameters on the creep resistance is studied. The problem of fracture toughness of metal matrix composites is analyzed. The possibility of designing composites with the necessary balance of hightemperature strength, fracture toughness, and oxidation resistance (heat resistance), in particular, composites operated at temperatures of 1400–1600 °C is demonstrated by the example of a molybdenum oxide composite.

Yu. V. Nemirovsky
Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
Keywords: plates, creep, bending, reinforced structures, ageing theory, fracture time
Abstract >>
Based on the ageing theory in Rabotnov's formulation, a unified approach to studying the creep of plates clamped over the contour and having different reinforcement structures is developed. A wide class of straightline and curvilinear reinforcement structures is considered in the Cartesian and polar coordinate systems. Equations of increasing deflections due to creep and equations for determining the ultimate admissible service times are derived for these structures.

A. N. Polilov
Blagonravov Institute of Machine Science, Russian Academy of Sciences, Moscow, 101990 Russia
Keywords: strength, stress concentration, fiber composite, hole, crack, scale effect
Abstract >>
Various models of subcritical fracture that make it possible to explain the significant decrease in the effective stress concentration factor compared with the theoretical value are described for fiber composites with a polymer matrix. The necessity of introducing a material parameter with the dimension of length to describe the scale effect of strength is substantiated. The biomechanical principle of stress concentration reduction by the use of curved fibers flowing around the hole is formulated.

V. L. Prisekin, N. V. Pustovoi, G. I. Rastorguev
Novosibirsk State Technical University, Novosibirsk, 630073 Russia
Keywords: airplane, wing, modeling, system of force control, test program
Abstract >>
The efficiency of algorithms for automatic control of forces prescribed in the test program is analyzed by an example of a largeaspectratio wing considered as a beam loaded by aerodynamic and inertial forces.

V. P. Radchenko, M. N. Saushkin, S. V. Gorbunov
Samara State Technical University, Samara, 443100 Russia
Keywords: kinetic equations, metal materials, plasticity, creep, softening, longterm strength, viscoelastic material, Lyapunov stability, third stage of creep, computational algorithm stability
Abstract >>
Energytype kinetic equations of inelastic rheological deformation are proposed in which the elastic, plastic, and creep strains are the additive components of the total strain, and the damage parameter is taken into account. A model of viscoelastic material with a creep kernel of exponential type is considered. The Lyapunov stability of solutions under constant stress is studied. The stability range of the solutions of the differential equations of the mathematical model corresponding to asymptotically bounded creep is established. It is shown that the instability range of the solutions corresponds to the onset of the third stage of creep. The relationship is determined between the Lyapunov stability of the solutions and the stability of the computational algorithm for the numerical solution of the system of equations. The proposed model is experimentally verified. It is shown that the calculated and experimental data are in good agreement.

G. P. Cherepanov
New York Academy of Sciences, New York, USA
Keywords: theory of rolling, Coulomb problem, rolling law, rolling moment, rolling friction coefficient, normal mode of rolling
Abstract >>
A theory of rolling of round bodies in the normal mode with adhesion conditions satisfied on the entire contact area is proposed. This theory refines the classical Coulomb's theory of rolling in which the rolling moment is directly proportional to the pressing force (e.g., the weight of the rolling body). The rolling moment of cylinders is found to be directly proportional to the pressing force raised to a power of 3/2, and the rolling moment of balls and tori is proportional to the pressing force raised to a power of 4/3. It is shown that the normal mode of uniform rolling can only be provided for a certain ratio of the elastic constants of the materials of the round body and the base forming an ideal pair. The Coulomb problem is solved for the cases of rolling of an elastic cylinder over an elastic halfspace, of an elastic ball over an elastic halfspace, of an elastic torus over an elastic halfspace, and of a cylinder and ball over a tightly stretched membrane. The rolling law is derived for such cases. The rolling friction coefficients, the rolling moment, and the rolling friction force are calculated.

