Mechanics Of Deformable Solids Research Articles

Compatibility of strains and the three-fold differentiability of the displacement field

The problem of the necessary class of smoothness of solutions to quasi-static problems of deformable solid mechanics in terms of displacements was discussed. It is shown that in order for the equations of compatibility of deformations to become identities when displacements are substituted in them, the existence of some third mixed derivatives of displacements is required. A counterexample for a linearly elastic compressible isotropic elastic medium was given. In this counterexample, the displacement field, being a doubly differentiable solution to the boundary value problem for the system of Lame equations in the entire domain, is not a solution to the displacement problem at all points in this domain.

COMPARATIVE ANALYSIS OF STRESS-STRAIN STATE OF SINGLE-ROOTED TEETH DURING INDIRECT RESTORATION OF THE CROWN WITH STUMP INLAYS OF DIFFERENT DESIGNS

The aim of our study on the biomechanics of the tooth inlay is to select the most efficient design of stump pin inlay for indirect restoration of single-rooted teeth. Materials and methods. To implement the aim of the study, the biomechanics research was conducted in the following four designs of stump inlays: 1 - traditional, 2 - with a single-stage change in the diameter of the pin along its length, 3 - with a two-stage change in the pin along its length, and 4 - with a single-stage change in the diameter of the pin and a cap in the crown of the tooth, covering the end of the root. The mathematical modeling was performed using the common software MSC NASTRAN for finite element analysis. The package, with the help of which the elastic three-dimensional models of the dentition are built and analyzed based on the finite element procedure, determines the displacement of each node of the finite element along the axis system, with three coordinates, normal and tangential stresses, as well as equivalent Huber-Mises stresses, which are calculated according to the well-known formula of deformable solid mechanics Results. The minimum values of equivalent stresses in both the pin inlay and the dentin of the root of the tooth at the level of the apical end of the pin inlay occur when using a pin inlay with a one-stage change in the diameter of the pin and a cap on the crown of the premolar. At the same time, the values of equivalent stress in dentin at the level of the apical end of the inlay pin are closest to the maximum values of equivalent stresses observed on the surface of the root of the tooth (differences do not exceed 10%). The maximum values of equivalent stresses in both the pin inlay and the dentin of the root of the tooth at the level of the apical end of the pin inlay occur when using a pin inlay of conventional design. In this case, the difference in the values of equivalent stresses in the pin compared to the inlay with a one-stage change in the diameter of the pin and a cap is 9%, and in the dentin at the level of the apical end of the inlay pin, the difference in stresses is 10%. The maximum values of equivalent stresses in both the dentin of the root of the tooth to be restored and the pin of the stump inlay occur when applying a traditionally used pin inlay. Among the various pin inlays considered, a stump inlay featuring a single-stage change in the diameter of the pin and a cap on the crown of the tooth to be restored demonstrates the highest efficiency, yielding the lowest equivalent stresses in the dentin of the tooth's root.

On determining residual stresses in a viscoelastic coating of arbitrary thickness formed during multilayer application on the inner surface of the straight pipe wall

Abstract The technological problem of forming a coating of arbitrary thickness on the inner surface of the wall of a straight pipe due to the multilayer application of a material having the property of viscoelasticity is considered. It is assumed that during the forming process, the coating experiences the action of inertia forces due to the relatively intense rotation of the coated wall. To determine the residual stresses in the resulting coating, a non-classical model of deformable solid mechanics, based on the concepts of growing continuum is used. With the help of the developed model, boundary value problems are formulated, the solutions of which make it possible to construct the desired residual stress distributions for arbitrary programs for changing the parameters of the implemented technological process.

Bending of composite timber

Technologies and production widely use composite materials now. “Mechanics of deformable solids” was formed as a science based on the study of materials used in the 19th and 20th centuries. Modern composite materials require new theoretical and experimental studies. Determining the stresses and deformations that occur at the points of contact of the matrix with the fibers is a special problem. Composites with a plastic matrix play an important role in modern technology. These materials successfully cope with cracking and significantly slow down the growth of cracks. In this article, the problem of the stress state of a composite beam with an elastic-plastic matrix and elastic fibers located along the axis of the beam is solved. It is assumed that in the zone of contact of the matrix with the fibers, according to the model of Yu. N. Worknov, a constant tangential stress is realized, less than the yield strength of the fiber. One end of the beam is fixed, and a constant force applied to the center of gravity coinciding with the origin of coordinates acts on the second. It is assumed that at the free boundary of the beam and at the points of contact of the beam with the fibers, the stresses reach the plasticity limit. The problem is solved with the help of conservation laws. This makes it possible to find the stress state at an arbitrary point of the section as a calculation of integrals along the outer boundary of the beam and the boundaries of the matrix and fibers.

An approach to describe the twinning and detwinning processes of the martensitic structure in ferromagnetic alloy with shape memory in magnetic and force fields

In the article, based on the theory of micromagnetism, a special case of the moment theory of elasticity for an anisotropic material and the Hadamard compatibility condition for deformations, a microstructural model of the behavior of the Heusler alloy with shape memory in magnetic and force fields is constructed. The dynamics of the magnetic process is described by the Landau-Lifshitz-Gilbert equation. Using the Galerkin procedure, variational equations corresponding to the differential relations of the magnetic problem are written out and the finite element method is used for numerical realization of these equations. Considering the problems of the moment theory of elasticity we come to a special case of this theory in which, under the action of force and magnetic fields, the tensor of moment stresses initiated only by the magnetic field is zero. Using this result, the general expression for the state equation is constructed that describes, within the framework of small deformations, the behavior of an anisotropic material under force and magnetic influences. The stress tensor in this equation consists of a symmetric part and a skew-symmetric part, depending only on the magnetic field. For the problem of deformable solid mechanics with such a state equation, the Lagrange variational equation was constructed, which made it possible to reduce the requirements for the smoothness of the desired solution by an order of magnitude. The general expression for the state equation has been concretized for an anisotropic material having one axis of symmetry of the fourth order and two axes of symmetry of the second order. The solution of the Hadamard compatibility equation for deformations made it possible to determine the sliding surfaces and sliding directions in a tetragonal crystal cell of a ferromagnetic material in the martensitic state, along which a shift occurs, leading to the appearance and dissipation of a twinned structure. The microstructural model, constructed using all these positions, is used to describe experimental results on the combined action of magnetic and force fields on the Heusler alloy.

Числовий аналіз впливу стрічкових включень на концентрацію напружень в тонких циліндричних і конічних оболонках з прямокутними отворами

Thin-walled plate-shell structures are widely used in various branches of technology and the national economy, in particular in the aerospace industry, the oil and gas industry, power engineering, construction, etc. The continuity of such structures is often disrupted by various inhomogeneities in the form of openings, inclusions, recesses, cracks, etc., which are local stress concentrators. Reducing the concentration of the stresses that develop in the vicinity of such structural inhomogeneities is an important problem in deformable solid mechanics. In particular, a pressing problem in the design of new equipment in modern mechanical engineering is a significant reduction in material consumption and an increase in the service life of cast parts taking into account the use of new materials and technologies. Such parts are responsible for the competitiveness of new equipment for various industries. This paper presents the results of a numerical simulation and analysis of the stress and strain field of thin-walled cylindrical and truncated conical shells with rectangular openings and tape inclusions around them. The material of the inclusions has properties that differ from those of the base material of the shells. The effect of the geometrical and mechanical characteristics of the inclusions on the parameters of the stress and strain field in the vicinity of the openings was studied by varying the elastic modulus of the inclusion material and the inclusion width. For definiteness, the inclusions were assumed to be homogeneous and located in the shell plane. The stress and strain intensity distributions in the zones of local stress concentration were obtained. The numerical results for shells of both shapes were compared with the corresponding results for shells with a circular opening. The study showed that the presence of a “soft” homogeneous tape inclusion helps in reducing the stress concentration around rectangular openings by ~ (21 – 54) % depending on the width of the inclusion and its elastic modulus, both in cylindrical and in conical shells. Unlike shells with a circular opening, in this case the presence of inclusions does not cause the mechanical effect of shifting the stress concentration zone from the contour of the opening to the interface between the materials.

Study of motion stability of a viscoelastic rod

Stability of non-conservatively loaded elastic and inelastic bodies - a classic section of deformable solid mechanics that has been of interest for many years. In this paper, we study the motion stability of a free rod subjected to a constant tracking force on one of its ends. The defining ratio of the rod material is the Kelvin-Voigt model. The solution is presented in the form of an expansion in terms of beam functions. The number of terms of this expansion is substantiated. The values of the critical load in the presence and absence of viscosity are determined. The given analytical results are confirmed by numerical calculations.

STUDY OF THE RELATIONSHIP BETWEEN FATIGUE CHARACTERISTICS AND ELASTIC MODULUS OF METASTABLE AUSTENITIC STEELS

This work is devoted to studying the influence of cyclic deformation on the elastic moduli of metastable austenitic steel. The estimated calculation of the modules of the strain-induced martensite and the modules of the matrix of the material (austenite) in the Voigt approximation. The modulus calculations were made according to the ultrasonic wave velocity measurements taking into account changes in the density of the material due to the difference in the volumes of the unit cells of the matrix and the ??-martensite. The volume fraction of ??-martensite and its change during cyclic deformation were determined by the eddy-current method. The influence of the strain cycle amplitude on the increase in the volume fraction of martensite was studied. The dependences of the relative change in Young's modulus, shear modulus, volume compression modulus on the number of loading cycles for metastable steel 12Kh18N10T are given. Relationships between the moduli change and the characteristics of cyclic loading of steel are obtained: the hardening of the material and the energy density in the cycle, used to calculate the service life using the methods of deformable solid mechanics. Studies shown that the critical change in the modules corresponding to the appearance of a microcrack depends on the strain cycle amplitude.The maximum change in the modules of metastable steel 12Kh18N10T was 4.5%; that by an order more than in steels that don't undergo phase transformations. A high degree of correlation between the change in elastic moduli and the fatigue characteristics of metastable austenitic steel is revealed: the hardening of the material and the energy density in the cycle. Correlations between the fatigue life of 12Kh18N10T steel and the change in elastic modulus – shear modulus and Young's modulus are obtained, which makes it possible to assess the state of the studied steel based on acoustic measurements.

Application of the Double Approximation Method for Constructing Stiffness Matrices of Volumetric Finite Elements

Introduction. When numerically solving problems of elasticity theory in a three-dimensional formulation by the finite element method, finite elements (FE) in the form of parallelepipeds, prisms and tetrahedra are used. Regularly, the construction of stiffness matrices of volumetric FE is based on the principle of isoparametricity, which involves the Lagrange polynomials to approximate the geometry and displacements. In computational practice, the most widespread FE are the so-called multilinear isoparametric FE with a linear law of approximation of displacements. The main disadvantage of these elements lies in the “locking” effect when modulating bending deformations. Moreover, the error of the numerical solution increases drastically in the case when the structure, in comparison to conventional deformations, undergoes significant displacements as a rigid whole. Long-term experience in solving problems of deformable solid mechanics by the finite element method has shown that existing volumetric FE have slow convergence, specifically, when modeling bending deformations of plates and shells. This study aims at constructing stiffness matrices of multilinear volumetric FE of increased accuracy allowing for rigid displacements based on the double approximation method.Materials and Methods. The mathematical apparatus of the double approximation method based on the principle of a separate representation of the distribution functions of displacements and deformations inside the element, was used to construct the stiffness matrices of volumetric FE. The storage and processing of the resulting system of equations was implemented in algorithmic terms of sparse matrices. Software development and computational experiments were carried out using the Microsoft Visual Studio 2013 64-bit computing platform and the Intel ® Parallel Studio XE 2019 compiler with the integrated Intel ® Visual Fortran Composer XE 2019 text editor. Visualization of the calculation results was performed using the descriptor graphics of the MATLAB computer mathematics package. A large eight-node SOLID185 CE of the ANSYS Mechanical software complex was used as a test sample.Results. Mathematical tool and software were developed to study the stress-strain state of massive structures under various types of external actions. The authorized application software package was verified on test examples with known analytical solutions. It has been shown that the constructed FE accurately satisfy the basic requirements for finite element modeling of spatial problems of elasticity theory.Discussion and Conclusion. The performed testing of the developed mathematical and program toolkit has shown that the finite elements constructed on the basis of the double approximation method can successfully compete with similar SOLID185 volumetric elements of the ANSYS Mechanical software complex. The proposed elements can be integrated into domestic import-substituting software systems that implement the finite element method in the form of the displacement method.

REPRESENTATIVE VOLUME AND EFFECTIVE MATERIAL CHARACTERISTICS OF PERIODIC AND STATISTICALLY UNIFORMLY REINFORCED FIBER COMPOSITES

In the deformable solid mechanics, there are concepts associated with continuum points (displacements, relative elongations, shifts) and a set of continuum points – an elementary volume (mass, energy, stresses). The role of such volume in the mechanics of composite materials is played by the representative volume element (RVE).This concept was first introduced by R. Hill (1963). Modern authors use the W.J. Drugan, J.R. Willis (1996) formulation. Based on the analysis of the RVE concept, we formulated its essential features: RVE is the minimum possible sample for numerical tests to determine the effective material parameters of the composite; under any RVE loading, its macroscopic stress-strain state is uniform. Its significance for the mechanics of composite materials is revealed: the existence of RVE for a composite is a criterion for applying the effective modulus theory to the analysis of its stress-strain state; the dehomogenization of a stressed-state composite material at a point is a solution to the micromechanics problem of the RVE stress-strain state determination; the characteristic size of RVE limits the size of the sampling grid in the numerical study. An iterative algorithm for constructing a representative volume of a periodic structure composite and its effective material thermoelastic characteristics is proposed. It is shown that the geometric shape of such a composition is a rectangular parallelepiped. The RVE construction algorithm for periodic compositions is extended to the composites statistically uniformly reinforced with continuous fibers. A method for modeling such materials with a following regular structure is suggested described: in the section perpendicular to the fibers, fiber centers should be located at the vertices of regular triangles. Examples of constructing RVE and thermoelastic material characteristics of specific compositions are given. The calculation results are compared with the data obtained using certified software products.

Комп’ютерне моделювання впливу кільцевих включень на концентрацію напружень в тонких циліндричних і конічних оболонках з круговими отворами

Shell structures are used in various industries, such the aerospace industry, the oil and gas industry, power engineering, mechanical engineering, construction, etc. Due to their design or manufacturing features, their integrity may be disrupted by the presence of various openings, around which local stresses develop. Finding ways to reduce stress concentrations around openings is an important problem in deformable solid mechanics. This paper presents the results of a computer simulation and a finite-element analysis of the stress and strain field of thin-walled cylindrical and truncated conical shells with circular openings in the presence of annular inclusions around them made of a material whose properties differ from the main material of the shells. The effect of the elastic modulus of an inclusion and its geometric parameters on the stress and strain concentration in the vicinity of the openings was studied. Several inclusion materials and inclusion widths were considered. An annular inclusion made of a homogeneous material and located in the shell plane was considered. Stress and strain intensity distributions in the local stress concentration zones were calculated. A comparative analysis of the results obtained for cylindrical and conical shells was carried out. The study showed that the presence of a “soft” homogeneous annular inclusion makes it possible to reduce the stress concentration around the opening by ~13–35% depending on the inclusion width and elastic modulus both for a cylindrical and a conical shell. Certain combinations of the geometric and mechanical parameters of the inclusion give rise to a “mechanical” effect, which consists in shifting the stress concentration zone from the opening edge to the inclusion – shell material interface. For conical shells, due to their geometric features, a “conical” effect occurs: the stresses increase not only in the vicinity of the opening-weakened zone, but also near the cone basis.

Calculation of parameters of the space tethered systems

The work carried out an analytical search for restrictions on the parameters of a space transport tether system for stable operation in orbital conditions. The main function of the considered system is load transportation from the massive “big” satellite to the small spacecraft (counterweight), along a kevlar stretchable tether. Such systems make it possible to move payloads between orbits or spacecraft, and “clean” the operational orbit of space debris without significant fuel costs. The analytical model of the system taking into account wave processes in the tetheris developed. The tether was modeled as a flexible elastic thread using the methods of deformable solid mechanics. Also processes of releasing the transported load in the lower part of the tether and its launching into the Earth atmosphere were considered.

Grid-Characteristic Numerical Method on an Irregular Grid with Extending the Interpolation Stencil

A grid-characteristic numerical method for solving a multidimensional transport equation on an unstructured grid with an order higher than one is proposed; this method does not use auxiliary points on edges and faces. The avoidance of auxiliary points on edges and faces simplifies the topology of the computational grid during its motion, which is important for solving dynamic problems of mechanics of deformable solids. To increase the approximation order, an analog of the grid stencil extension implemented for an unstructured grid is used. Results of testing the proposed numerical scheme for continuously differentiable, continuous, discontinuous solutions are presented.

Solving the Problem of Elasticity for a Layer with N Cylindrical Embedded Supports

The main goal of deformable solid mechanics is to determine the stress–strain state of parts, structural elements, and their connections. The most accurate results of calculations of this state allow us to optimize design objects. However, not all models can be solved using exact methods. One such model is the problem of a layer with cylindrical embedded supports that are parallel to each other and the layer boundaries. In this work, the supports are represented by cylindrical cavities with zero displacements set on them. The layer is considered in Cartesian coordinates, and the cavities are in cylindrical coordinates. To solve the problem, the Lamé equation is used, where the basic solutions between different coordinate systems are linked using the generalized Fourier method. By satisfying the boundary conditions and linking different coordinate systems, a system of infinite linear algebraic equations is created. For numerical realization, the method of reduction is used to find the unknowns. The numerical analysis has shown that the boundary conditions are fulfilled with high accuracy, and the physical pattern of the stress distribution and the comparison with results of similar studies indicate the accuracy of the obtained results. The proposed method for calculating the stress–strain state can be applied to the calculation of structures whose model is a layer with cylindrical embedded supports. The numerical results of the work make it possible to predetermine the geometric parameters of the model to be designed.

The Study Influence Analysis of the Mathematical Model Choice for Describing Polymer Behavior.

The article considered the three types of description of the material behavior model: elastic, elastic-plastic, and viscoelastic. The problem is considered in the framework of deformable solid mechanics. The paper considers the possibility of describing modern polymeric and composite materials used as antifriction sliding layers in the viscoelasticity framework. A numerical procedure for finding the coefficients to describe the viscoelastic material behavior using the Prony model has been implemented. Numerical results and experimental data are compared. The model problem of spherical indenter penetration into polymer half-space is realized. The influence of the system discretization on the numerical solution is analyzed. The influence of the polymer behavior description in static and dynamic problem formulations is analyzed.

INVERSE PROBLEM FOR A TIMOSHENKO BEAM WITH AN ADDITIONAL VISCOELASTIC SUPPORT UNDER NONSTATIONARY DEFORMATION

The non-stationary loading of a mechanical system consisting of a beam hinged at the edges and an additional support installed in the span of the beam is considered. The deformation of the beam is modeled on the basis of Timoshenko's hypotheses, taking into account the influence of rotatory inertia and shear. The deformation of the beam is described by a system of partial differential equations, which is solved analytically by means of expansion of the unknown functions into the relevant Fourier series and further use of the Laplace integral transformation. It is assumed that the additional support has linear-elastic and linear-viscous components, and the displacements coincide at the point where the additional support is connected to the beam. The reaction between the beam and the additional support is replaced by an external unknown concentrated force applied to the beam, which varies in time. The law of time variation of this unknown reaction is determined by solving the Volterra integral equation. The inverse problem of deformable solid mechanics is solved, that is, it is assumed that the deflection at a point of the beam with the additional support is known, whereas the law of time variation of the external impulse load causing the deflection is unknown. The application point of the external load and the point of the additional support connection are considered to be known and do not change in the process of deformation (when obtaining the solution of the problem it was supposed that these could be any points of the beam except for its ends). The described inverse problem is reduced to a system of two Volterra integral equations of the first kind with regard to the unknowns of the external disturbing load and reaction between the plate and the additional support, which is solved by analytical and numerical method. Analytical relations and calculation results for specific numerical parameters are given. The results obtained in this work can be used for indirect measurement of impulse and shock loads acting on beams with additional supports, for which not only elastic but also linear-viscous characteristics are taken into account.

Linear theory of micropolar media with internal nonlocal potential interactions

AbstractThis paper proposes a mathematical model that can be used to quantify the effects of a polarized electric field on the mechanical properties of a polar dielectric in the framework of mechanics of deformable solids. The physical essence of the phenomenon is known. However, its direct use in practical calculations may cause difficulties due to the complexity of the composition and structure of materials used in the design and manufacture of specific products whose operation regimes are based on polarization effects. A continuum description of the polarization phenomenon makes it possible to circumvent these difficulties.The basis of the generalized model is a micropolar linearly elastic medium. The generalization consists in taking into account the initial stress state of the material, as well as cross effects under the mutual influence of translational and rotational deformations, the possibility to calculate the characteristics of micropolar material properties based on a special nonlocal model of micropolar media with internal nonlocal potential interactions. It takes into account translational and rotational potential interactions of particles of a continuous medium at finite distances. The kinematic characteristics of the material are the same as those used by the traditional model of a micropolar elastic medium ‐ the gradients of translational displacements of the medium's particles, their rotations independent of the medium's rotation and rotations of the particles relative to the medium. Generalized forces performing work on these generalized displacements are volumetrically and surface‐distributed forces and moments. To determine the values of the parameters of the interparticle potentials of the nonlocal model, a method is proposed using data from experiments to determine the material constants of isotropic linearly elastic materials and the dispersion law for plane acoustic waves of high frequency. The paper considers and quantifies the effects of anisotropy in an initially isotropic material, as well as anharmonic effects that cause dielectric heating by a periodically changing polarizing field. The results are compared qualitatively and quantitatively with known experimental data and physical notions of the phenomena in question.

The stress state of the rock mass with spherical cavity

One of the main hypotheses accepted in the mechanics of deformable solids is the assumption of the homogeneity of materials. This means that all mechanical characteristics of the material (modulus of elasticity, Poisson’s ratio, yield strength, relaxation parameters, etc.) are constant over the volume of the body, in other words, these characteristics are constants. This hypothesis makes it possible not to take into account the natural inhomogeneity of materials at the microlevel - the presence of various fractions in composite materials (concrete, fiberglass, etc.), crystal lattice defects, etc. Examples can be given when various physical phenomena (temperature field, radiation exposure, explosive impact, etc.) lead to a change in the mechanical characteristics along the body. These changes can be quite significant. So, for example, in the presence of high-gradient temperature fields, the deformation characteristics of materials at different points of the body can change dozens of times. Thus, when calculating and designing structures, it is necessary to take into account such macro heterogeneity, since it leads to a significant change in the stress-strain state of bodies. This article considers the problem associated with the continuous inhomogeneity of materials. It means such a heterogeneity that arose in the process of creating an underground cavity with the help of an explosion. In contrast to the classical mechanics of a deformable solid body, the problems of which are reduced to differential equations with constant coefficients, in the mechanics of continuously inhomogeneous bodies we deal with equations with variable coefficients, which greatly complicates their solution. In this case, depending on the type of inhomogeneity functions—functions that describe the change in mechanical characteristics along the coordinates—differential equations turn out to be significantly different.

Учет разупрочнения вблизи свободной поверхности в прямой модели физической теории пластичности

Current industrial demands require the improvement of existing technologies and the creation of new ones that allow the production of parts and structures with advanced performance properties. Es-pecially important are the issues of design and analysis of thermomechanical treatment of metals and al-loys by intensive inelastic deformation methods. The study of manufacturing processes for miniaturized parts deserves special attention. The boundary value problems arising in this case belong to the class of physically and geometrically nonlinear problems of the mechanics of deformable solids, for the solution of which it is necessary to develop appropriate mathematical models. Most classical models are based on the macrophenomenological theory of elastoplasticity. However, in the processes under consideration, significant structural changes occur at the meso- and microscale that have a significant influence on the physical and mechanical characteristics of the processed materials and the performance characteristics of the products. These changes are not described by the mentioned theories. An effective approach to de-scribe these processes is to use multilevel models of crystal plasticity. Despite the tendency to miniaturize products, the parameters of such models are usually determined from the results of experiments on mac-rosamples. The question arises whether such parameters are valid for the analysis and modeling of real structures with typical length scales below 500 μm, where the internal and external boundaries of crystal-lites play a special role. In this study, the influence of the free surface on the mechanical properties of single crystalline and polycrystalline materials is considered. Modification of the basic direct crystal plas-ticity model of elastoviscoplasticity to account for the softening of slip systems near free boundaries due to easier exit of dislocations to the surface is proposed.

Моделирование напряженно-деформированного состояния кристалла, выращенного методом Бриджмена

The thermal stress in single crystal grown by Bridgman method is studied. The corresponding quasi-static problem of deformable solid mechanics is considered in axisymmetric case. The governing equations are approximated by finite element method. The effect of boundary conditions and material parameters on the stress-strain state of the single crystal is studied numerically.