INSTABILITY IN A DISLOCATION ENSEMBLE AT PLASTIC DEFORMATION IN METALS

A problem related to the development of instability of a homogeneous state in an ensemble of screw dislocations under plastic deformation of metals is considered . The study of the development of instability and structure formation in the dislocation ensemble is carried out on the basis of the method developed for charged particles in plasma and associated with the correlation interaction of electrons and positively charged ions. Accordingly, the screw dislocation ensemble is represented as a system of dislocations with an opposite Burgers vector, i.e., as a plasma-like medium with opposite dislocation charges. The total dislocation charge of the dislocation ensemble is equal to zero due to the law of conservation of the Burgers vector. In this situation, the elastic field of dislocations is “cut off”. The stress field of a single dislocation is shielded by a uniformly distributed dislocation “background” and is characterized by a certain effective potential. It is found that at long distances it decreases exponentially. Therefore, the value in the argument of the falling potential can be considered as the radius of screening of the elastic field of dislocations. It is shown that the screening radius is equal to the correlation radius, which makes it possible to construct a two-particle correlation function and find the energy of the correlation interaction of dislocations. A system of kinetic equations for a dislocation ensemble is formulated, taking into account the elastic and correlation interaction of dislocations, as well as the processes of their generation and annihilation. The criterion of instability of the homogeneous distribution of dislocations for the formulated system of equations is established. The instability criterion is met under the condition that the dislocation density exceeds a certain critical value that depends on the square of the flow stress and material constants (such as the Burgers vector modulus and shear modulus, as well as indirectly, the packing defect energy). In the framework of linear analysis, it is shown that when one system of sliding screw dislocations is taken into account, a one – dimensional periodic dislocation dissipative structure is formed at the moment of instability occurrence, and when multiple sliding is taken into account, solutions appear in the form of various variants of polyhedral lattices (cellular structures). It is established that the characteristic size of the cellular structure coincides with the experimental dependence both qualitatively and quantitatively ( the cell size is proportional to the square root of the dislocation density, and the proportionality coefficient is about ten). It is shown that the origin of spatially inhomogeneous dislocation structures, based on correlation instability, depends mainly on the features of the elastic interaction of dislocations and is not critical to the choice of the mechanisms of their kinetics (i.e., the mechanisms of generation, annihilation, and runoff of dislocations).

ASYMPTOTICS OF CRITICAL LOADS OF A COMPRESSED NARROW ELASTIC PLATE WITH INTERNAL STRESSES

The problem of the asymptotic solution of a modified system of nonlinear Karman equilibrium equations for a longitudinally compressed elongated elastic rectangular plate with internal stresses lying on an elastic base is considered. Internal stresses can be caused by continuously distributed edge dislocations and wedge disclinations, or other sources. The compressive pressure is applied parallel to the long sides of the plate to the two short edges. The boundary conditions are considered: the long edges of the plate are free from loads, and the short edges are freely pinched or movably hinged. A small parameter is introduced, equal to the ratio of the short side of the plate to the long side. The solution of the system – the compressive load, the deflection function, and the stress function – is sought in the form of series expansions over a small parameter. The system of Karman equations with dimensionless variables is reduced to an infinite system of boundary value problems for ordinary differential equations with respect to the coefficients of asymptotic expansions for the critical load, deflection, and stress function. In this case, to meet the boundary conditions, the boundary layer functions are additionally introduced, which are concentrated near the fixed edges and disappear when moving away from them. Boundary value problems for determining the functions of the boundary layer are constructed. It is shown that the main terms of the small parameter expansions for the critical load and deflection are determined from the equilibrium equation of a compressed beam on an elastic base with the boundary conditions of free pinching or movable hinge support of the ends. In this case, the main term of the expansion into a series of the stress function has a fourth order of smallness in the parameter of the relative width of the plate.

EVALUATION OF THE RESOURCE CHARACTERISTICS OF POLYCRYSTALLINE STRUCTURAL ALLOYS UNDER CYCLIC THERMOMECHANICAL LOADING

The main physical regularities of complex thermoviscoplastic deformation and accumulation of damage in structural materials (metals and their alloys) under various modes of cyclic combined thermomechanical loading and mathematical models of these processes are considered. A mathematical model of the mechanics of a damaged medium has been developed, which makes it possible to simulate the cyclic viscoelastoplastic behavior and determine the resource characteristics of polycrystalline structural alloys under the combined action of degradation mechanisms that combine material fatigue and creep. The model is based on the joint integration of equations describing the kinetics of the stress-strain state and damage accumulation processes. The final relation to the model is the strength criterion, the fulfillment of which corresponds to the formation of a macrocrack. The plasticity equations are based on the basic principles of the flow theory. To describe the creep process in the stress space, a family of equipotential creep surfaces of the corresponding radius and having a common center is introduced. The relationship between the creep equations and the thermoplasticity equations describing “instantaneous” plastic deformations is carried out at the loading stage through the stress deviator and the corresponding algorithm for determining and at the loading stage by means of certain relationships between “temporary” and “instantaneous” scalar and tensor quantities. At the stage of development of damage scattered throughout the volume, the effect of damage on the physical and mechanical characteristics of the material is observed. This influence can be taken into account by introducing effective stresses. In the general case, stresses, plastic strains, and creep strains are determined by integrating the thermal creep equations by the four-point Runge-Kutta method with correction of the stress deviator and subsequent determination of stresses according to the thermoplasticity equations, taking into account the average creep strain rate at a new time. The relationships that simulate the accumulation of damage are based on the energy approach to determining the resource characteristics. The kinetics of fatigue damage accumulation is based on the introduction of a scalar parameter of damage to a structural material and a unified model form for representing the degradation mechanism under fatigue and creep conditions. The influence of scattered damage on the physical and mechanical characteristics of the material is taken into account by introducing effective stresses. The results of numerical simulation of cyclic thermoplastic deformation and accumulation of fatigue damage in heat-resistant alloys (Haynes188) under combined thermomechanical loading are presented. Particular attention is paid to the issues of modeling the processes of cyclic thermoplastic deformation and the accumulation of fatigue damage for complex deformation processes accompanied by the rotation of the main areas of stress and strain tensors.

RELATIVE PORE VOLUME UNDER INDENTATION OF POROUS MATERIALS

Investigation of the function of the relative volume of pores under the load action is carried out on the base of the solution of the static contact problem of the indentation of a layer made of a material with voids or unfilled pores. A rigid strip indenter with a flat base is pressed into a porous layer that is adhered to a non-deformable base along the lower boundary. The formulated 3D problem of the indentation of a porous layer is reduced to solving the plane contact problem of the indentation of a porous strip. The plane contact problem is reduced to solving an integral equation for unknown contact stresses, the solution of which is constructed by the method of successive approximations in the form of an asymptotic expansion in the dimensionless parameter of the problem. The obtained contact stresses and the force acting on the indenter made it possible to study the influence of the nonclassical moduli of the layer porous material (the connectivity modulus and pore rigidity modulus) on the main contact characteristics and on the distribution of the function of the relative pore volume. The connectivity modulus increase leads to an increase in the compliance of the layer porous material, the pore rigidity modulus increase leads to an increase in the rigidity of the layer porous material. The maximum value of the distribution function of the relative pore volume in the porous material of the layer is achieved under the indenter base centre, regardless of the change in the porous material non-classical moduli.

NUMERICAL ANALYSIS OF THE DYNAMICS OF THREE-DIMENSIONAL ANISOTROPIC BODIES BASED ON NON-CLASSICAL BOUNDARY INTEGRAL EQUATIONS

The application of non-classical approach of the boundary integral equation method in combination with the integral Laplace transform in time to anisotropic elastic wave modeling is considered. In contrast to the classical approach of the boundary integral equation method which is successfully implemented for solving three-dimensional isotropic problems of the dynamic theory of elasticity, viscoelasticity and poroelasticity, the alternative nonclassical formulation of the boundary integral equations method is presented that employs regular Fredholm integral equations of the first kind (integral equations on a plane wave). The construction of such boundary integral equations is based on the structure of the dynamic fundamental solution. The approach employs the explicit boundary integral equations. The inverse Laplace transform is constructed numerically by the Durbin method. A numerical solution of the dynamic problem of anisotropic elasticity theory based on the boundary integral equations method in a nonclassical formulation is presented. The boundary element scheme of the boundary integral equations method is built on the basis of a regular integral equation of the first kind. The problem is solved in anisotropic formulation for the load acting along the normal in the form of the Heaviside function on the cube face weakened by a cubic cavity. The obtained boundary element solutions are compared with finite element solutions. Numerical results prove the efficiency of using boundary integral equations on a single plane wave in solving three-dimensional anisotropic dynamic problems of elasticity theory. The convergence of boundary element solutions is studied on three schemes of surface discretization. The achieved calculation accuracy is not inferior to the accuracy of boundary element schemes for classical boundary integral equations. Boundary element analysis of solutions for a cube with and without a cavity is carried out.

INFLUENCE OF INCREASED TEMPERATURE ON THE DYNAMIC PROPERTIES OF ASPEN

As a damping material in the structures of containers for the transportation of hazardous materials, along with plastic metals, fiber-claydite concrete and synthetic foams, it is proposed to use wood of different species. Since containers are transported in different climate regimes, there is an urgent need to study the properties of wood at elevated temperatures. The paper presents the results of dynamic tests of aspen under uniaxial compression under conditions of temperature increased to +60°C. The tests were carried out according to the Kolsky method on a Hopkinson split-bar setup. To study the anisotropy of properties, aspen samples were made and tested by cutting samples along and across the direction of the grains. As a result of processing the experimental data, dynamic stress-strain curves were obtained. According to the experimental data, there are determined the stresses at which the integrity of the samples were violated. The mean values of the moduli of deformation in the active loading regions of stress-strain curves are also presented. The highest slope of the load sections and the highest breaking stresses were observed for the specimens when loaded along the grains, and the smallest values of these parameters were noted when loaded across the grains. For specimens loaded along grains at strain rates above 1500 s–1, after reaching the limiting stress values, a decrease (relaxation) of stresses is observed with increasing deformations. For specimens loaded across the grains, an almost horizontal section the diagrams of deforming or even with some strengthening is more typical. The effect of elevated temperature on the strength and deformation properties of aspen is estimated. There is a tendency towards some decrease in the diagrams at a temperature of +60 °C in comparison with the diagrams at room temperature. In this case, both the moduli in the loading and unloading sections and the limiting (breaking) stresses decrease. The obtained features of the behavior of aspen specimens at elevated temperatures should be taken into account when modeling deforming wood.

QUASIHARMONIC BENDING WAVE, DISTRIBUTING IN THE BALK OF TIMOSHENKO, LYING ON A NONLINEAR ELASTIC BASE

In this paper, we consider the modulation instability of a quasiharmonic flexural wave propagating in a homogeneous beam fixed on a nonlinear elastic foundation. The dynamic behavior of the beam is determined by Timoshenko's theory. Timoshenko's model, refining the technical theory of rod bending, assumes that the crosssections remain flat, but not perpendicular to the deformable midline of the rod; normal stresses on sites parallel to the axis are zero; the inertial components associated with the rotation of the cross sections are taken into account. The uniqueness of the model lies in the fact that, allowing a good description of many processes occurring in real structures, it remains quite simple, accessible for analytical research. The system of equations describing the bending vibrations of the beam is reduced to one nonlinear fourthorder equation for the transverse displacements of the beam particles. The nonlinear Schrödinger equation, one of the basic equations of nonlinear wave dynamics, is obtained by the method of many scales. Regions of modulation instability are determined according to the Lighthill criterion. It is shown hot the boundaries of these areas shift when the parameters characterizing the elastic properties of the beam material and the nonlinearity of the base change. Nonlinear stationary envelope waves are considered. An equation that generalizes the Duffing equation, which contains two additional terms in negative powers (first and third), is obtained and qualitatively analyzed. Solutions of the Schr?dinger equation in the form of envelope solitons are found and the dependences of their main parameters (amplitude, width) on the parameters of the system are analyzed. The dynamics of the points of intersection of the amplitudes and widths of "light" solitons in the case of soft nonlinearity of the base is shown within the region of modulation instability.

INVESTIGATION OF HYPERELASTIC SOFT SHELLS NONSTATIONARY DYNAMICS PROBLEMS SOLUTION FEATURES

Results of hyperelastic soft shells nonlinear axisymmetric dynamic deforming problems solution algorithm testing are represented in the work. Equations of motion are given in vector-matrix form. For the nonlinear initial-boundary value problem solution an algorithm which lies in reduction of the system of partial differential equations of motion to the system of ordinary differential equations with the help of lines method is developed. At this finite-difference approximation of partial time derivatives is used. The system of ordinary differential equations obtained as a result of this approximation is solved using parameter differentiation method at each time step. The algorithm testing results are represented for the case of pressure uniformly distributed along the meridian of the shell and linearly increasing in time. Three types of elastic potential characterizing shell material are considered: Neo-hookean, Mooney – Rivlin and Yeoh. Features of numerical realization of the algorithm used are pointed out. These features are connected both with the properties of soft shells deforming equations system and with the features of the algorithm itself. The results are compared with analytical solution of the problem considered. Solution behavior at critical pressure value is investigated. Formulations and conclusions given in analytical studies of the problem are clarified. Applicability of the used algorithm to solution of the problems of soft shells dynamic deforming in the range of displacements several times greater than initial dimensions of the shell and deformations much greater than unity is shown. The numerical solution of the initial boundary value problem of nonstationary dynamic deformation of the soft shell is obtained without assumptions about the limitation of displacements and deformations. The results of the calculations are in good agreement with the results of analytical studies of the test problem.

CREEP OF A THICKWALLED QUASILINEAR VISCOELASTIC TUBE UNDER A CONSTANT EXTERNAL AND INTERNAL PRESSURE

We consider the creep problem for a quasilinear viscoelastic model of a thickwalled tube, loaded with constant internal and external pressure; the material is supposed to be incompressible. An exact solution to this problem was received by one of the authors in previous papers, assuming the state of a tube to be plain deformation; hereby we study properties of this solution for arbitrary material functions of quasilinear viscoelasticity constitutive relation. A criterion of stress stationarity is derived; the stress field of a thickwalled tube under a constant pressure evolves in time in the case of unbounded creep function and arbitrary nonlinearity function, except some particular types. The monotonicity of stress field components is studied: the radial stress monotonicity depends only on internal and external pressure values (for internal pressure, greater than an external one, it is negative and increases in radii). For other stress components, there are derived sufficient conditions of monotonicity. For an exponential nonlinearity function and unbounded creep function, a creep curve is determined to be concave up at the initial moment, and concave down during prolonged observation; the creep curve of a bipower nonlinearity function model may change its convexity. The stressstrain state of a model with a bounded creep function is proved to be bounded.