## No 3 (2014)

**Year:**2014**Articles:**11**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/28**DOI:**https://doi.org/10.15593/perm.mech/2014.3

ROTATIONAL DEFORMATION MODE OF POROUS STRUCTURES

#### Abstract

We consider the process of deformation and buckling of flow during compression of the porous structure on the basis of aluminum with a high concentration of pores. The starting material is AMg6 alloy powder having a particle size of less than 1 mm. Compression tests are performed at an average rate of 1 mm / min. Loading at a temperature T = 293 K was carried out to a certain strain, then the test sample was photographed and stayed. The authors analyzed rotary modes of plastic flow and theirrelationship with the beginning of the fracture process. In order to do this, we allocated reference points - the joints and then determined their trajectory in the process of loading. We established the important role of nonuniform rotation associated with strong heterogeneity of the porous structure. Nonuniform rotation in the vicinity of large central pores lead to a collapse of the two neighboring pores and subsequent destruction. Phenomenology process was as follows: inhomogeneous rotation in the vicinity of large central pores lead to a collapse of two large and one medium-sized pores. This rotation "pulled" the displacement of two small pores related to them, etc. The concentration of large pores (3-5 times larger than the average size) was small 10-2; their statistics lied on the so-called "tail of the distribution" areas where statistical methods were ineffective, i.e. limit inequalities and theorems of probability theory were not executed. These areas ultimately determined moment buckling and fracture of the porous structure. Having a small number of large pores reduced the ultimate strain of 10-15 % and power consumption of the structures 25-35 %.

**PNRPU Mechanics Bulletin**. 2014;(3):5-16

MATHEMATICAL MODEL OF MOVEMENT OF A PULSING LAYER OF VISCOUS LIQUID IN THE CHANNEL WITH AN ELASTIC WALL

#### Abstract

The problem of hydroelasticity of the plate forming at wall of the slot-hole channel with a pulsing layer of viscous incompressible liquid at the set harmonious law of a pulsation of pressure at its end face in flat statement is put and analytically solved. The set regional task represents nonlinear related system of the equations of Navier-Stokes for a layer of viscous incompressible liquid and the equation of dynamics of a plate (beam strip). Conditions of sticking of liquid act as regional conditions to impene- trable walls of the channel, a condition of the free expiration of liquid at end faces of the channel and a condition of a hinged supporting of a plate wall of the channel. The complex of dimensionless variables of a considered task is created and small parameters of a task are allocated. As small parameters we have chosen the relative thickness of a layer of liquid and relative amplitude of a deflection of a plate. Considering asymptotic decomposition in the allocated small parameters of a task we have carried out its linearization by a method of indignations. The solution of the linearized task is obtained by a method of the set forms for a mode of the established harmonic oscillations. Thus, proceeding from boundary conditions for a channel plate wall, the form of its deflection is set in the form of ranks on trigonometrical functions from longitudinal coordinate. The law of a deflection of an elastic wall of the channel and dis- tribution of hydrodynamic parameters are found in liquid. We have obtained frequency dependent func- tions of distribution of amplitudes of a deflection and dynamic pressure along the channel and frequency dependent functions of distribution of phase shift of a deflection of a wall and pressure in the channel of rather initial indignation at an end face. On the basis of calculations it is shown that resonant fluctua- tions of an elastic wall of the channel, pressure excited by insignificant pulsations at its end face, can cause essential changes of dynamic pressure and be the main reason of vibration cavitation in liquid.

**PNRPU Mechanics Bulletin**. 2014;(3):17-35

COMPARISON OF THE RESULTS OF SOLVING THE PROBLEM OF FRACTURE MECHANICS FOR PIPE WITH NON-THROUGH CRACK

#### Abstract

Dangerous conditions to the pipelines can often be caused by different defects which occur in the pipe walls. Non-through surface cracks attract particular interest. Generally these cracks have com- pound front form - in other words, they are polyvalent. Modern methods of nondestructive inspection do not give complete information on the crack front shape with adequate accuracy. In global practice de- fects are approximated with the semielliptical cracks to simplify calculation methods. In this case the defect is considered two-parameter and it is only defined by maximum depth and length.This paper examines a steel pipe, which has been weakened by the semi-elliptical non-through surface crack. The crack is common to the external pipe area and has longitudinal orientation. The pipe is exposed to internal pressure. Fracture mechanics problem is resolved with ANSYS CAE-system. Stress intensity factor values distribution for the crack front points is under analysis. These values were obtained by using invariant J-integral. J-integral values calculation was performed using integration over a region technique. The obtained results are compared with the data published by other authors. These data resulted from the analysis of pipes and cylindrical pressure vessels weakened by non-through cracks. The results of numerical modelling correlate accurately with the existing solutions. Accuracy of the fracture mechanics problem solution can be significantly increased by using regular mesh with mul- tiple finite elements along to the crack front. Fracture mechanics parameters investigation identified presence of edge effect common to the area where the crack front goes to the pipe surface. Edge effect refers to the local maximum values which are much higher than the crack front end points values. These values should be used while investigating crack propagation under variable loading - that is when pul- sations take place in load condition.

**PNRPU Mechanics Bulletin**. 2014;(3):36-49

TWO-DIMENSIONAL (SHELL-TYPE) AND THREE-DIMENSIONAL MODELS FOR ELASTIC THIN-WALLED CYLINDER

#### Abstract

The variant of the classical theory of shells (CTS) built on the basis of Lagrange analytical me- chanics is under analysis. The direct approach to shells as material surfaces, the elements of which are material normals with five degrees of freedom - three translations and two rotations, is used. The sys- tem of equations and boundary conditions is derived from the principle of virtual work with direct tensor calculus. Such approach makes it possible to discard the problems and controversies characteristic of conventional concepts. This paper is aimed at comparing this theory of shells (CTS) with widely known variants, as well as with the solution of the spatial problem.Problems for the thin-walled infinite cylinder have been formulated and solved on the basis of three theories: CTS, the well-known theory of A.L. Goldenweiser and three-dimensional elasticity the- ory. For the shell-based models, we have linear algebraic systems, for the three-dimensional models - the ordinary differential equation (ODE) over the thickness. Exponential solutions of static problems with different variability are built analytically. Numerical solutions using computer mathematics have been found.In comparing exponents of solutions with the boundary load, it was found that for small values of the wave number and the shell thickness, both shell theories agree well with the three-dimensional the- ory. As the wavelength decreases relative to the shell thickness, their uncertainty increases, though the area of CTS applicability has turned to be somewhat wider than that in the theory of A.L. Goldenweiser.According to both theories, the detected displacements of the shell under the load rapidly changing by the coordinates are well coordinated with each other. The coordination with the three- dimensional theory is suitable for small values of wave numbers. The calculations have shown that, under external load having the axial and circumferential components, CTS predicts a normal displace- ment component with a greater accuracy.

**PNRPU Mechanics Bulletin**. 2014;(3):50-70

ABOUT DRAWING OF THE YIELD SURFACE FOR STEEL 45 AND VERIFICATION OF THE POSTULATE OF ISOTROPY ON STRAIGHT-LINE PATHS DURING REPEATEDSIGN-VARIABLE LOADINGS

#### Abstract

In article are presented results of experimental studies at repeated sign-variable loadings of a thin-walled tubular specimen at stretching compression and torsion. The experiments were implemented on an automated computational and experimental installation for complex loading SN-EVM named A.A. Ilyushin in the vector space of deformation (rigid loading). According to experimental data evaluated the Baushinger effect and magnitude of secondary yield strength at various admissions on residual defor- mation. Given the influence of the admission on residual deformation on radius function and position of the center of spherical yield surface in stress space, used in the plastic-flow theory. With increase in the admission at residual deformation the parameter characterizing Bauschinger effect and the radius func- tion of the yield surface increases, and the displacement of its center decreases. With increasing length of the arc of plastic deformation parameter characterizing the Bauschinger effect decreases and tends to a stationary value. It is established that the radius of spherical yield surface makes temporary de- crease and then increase with increasing length of the arc of the plastic deformation. Some mathemati- cal models of the plastic-flow theory attributed this decrease in the radius of the yield surface not with change of internal structure of a material at meso-level and orientation of microtension, and with an elastic softening initially isotropic body, mistakenly believing in this strain rate negative. For the realized types of experimental path of repeated sign-variable loadings with breaks on 180 degrees verification of the postulate of isotropy by A.A. Ilyushin showed that on scalar and vector properties it is carried very well.

**PNRPU Mechanics Bulletin**. 2014;(3):71-88

IDENTIFICATION METHOD OF GRADIENT MODELS PARAMETERS OF INHOMOGENEOUS STRUCTURES BASED ON DISCRETE ATOMISTIC SIMULATIONS

#### Abstract

This paper considers characteristics, features and corresponding boundary value problems of gradient theories of elasticity. A brief description of one-parametric applied model, which is one of the several variants of the gradient elasticity theories is given here. In relation to that, we represent a con- tinuum gradient model of two-phase composite structures that allow evaluation of the influence of scale parameters on their effective mechanical properties.In identifying the additional physical parameters of gradient elasticity models, a new method is introduced where a comparison of the results of continuum and discrete-atomistic modelling for specific tested heterogeneous structures is made. As a result we suggested a procedure and the respective algorithm defining the additional parameter of applied gradient continuum model of heterogeneous me- dia; and in such procedure, an interphase zone is characterized at the contact surface of a two-phase composite and the scale effects represented by cohesions-interaction fields, which are localized near to the boundaries of contact surfaces. This additional physical parameter of gradient model is found through parameters of potentials, which are used to describe the specific studied structure in the dis- crete-atomistic modelling.To justify and validate the proposed method, a numerical investigation is conducted and com- parison is made between the results of continuum and discrete-atomistic modelling. The examination reveals that a high degree of accuracy of prediction can be provided by the continuum one-parametric gradient theory when describing the effective properties of countable multiple set of two-phase hetero- geneous studied structures, which are formed by atomic substructures with various properties (various parameters of potentials).Finally, it is demonstrated that the identification method of parameters in gradient elasticity theo- ries for heterogeneous structures is well described by Leonard-John potential and Morse potential. Fur- thermore, we consider that when the parameters of potentials are known, the various types of cross interactions of atoms can be treated as ‘ideal’ or ‘damaged’ interactions as per Lorentz-Berthelot’s rule.

**PNRPU Mechanics Bulletin**. 2014;(3):89-112

IMPACT OF EQUAL-CHANNEL ANGULAR PRESSING ON MECHANICAL BEHAVIOR AND MICROSTRUCTURE OF MAGNESIUM ALLOYS

#### Abstract

Preparation of materials for hydrogen saturation as a primary treatment involves its refinement. This allows improving the kinetic parameters of sorption / desorption of hydrogen by increasing the pro- portion of the specific surface of the sample. The efficiency of particle size reduction depends on the mechanical characteristics of the material.We studied the mechanical properties of samples of magnesium and magnesium alloys AZ31 and ZK60 before and after deformation with equal channel angular pressing (ECAP). It is shown that results in deformation of the ECAP are: anisotropy of mechanical properties appearance as well as increased strength, yield strength and modulus decrease. The possible mechanisms for the implemen- tation of microplastic deformation were discussed. It was shown that the ECAP deformation creates the texture in material which lead to anisotropy of mechanical properties.It was found that in the investigated materials, severe plastic deformation, as for example ECAP, is carried out mainly by sliding along the basal planes, which is typical for crystals with hexago- nal close packing with the ratio of the lattice parameters close to ideal (c / a ~ 1.63). The second contri- bution (in magnitude and significance) on the deformation is twinning. Calculation texture formed during ECAP deformation showed that in alloys based on magnesium twinning implemented on the {10-12} plants in the direction of shear <-1011>. This one prevents the occurrence in the deformation process involving other slip systems, namely, sliding on prismatic and pyramidal planes of the crystal lattice.The analysis of the deformation behavior of magnesium and its alloys can be used as the basis for creation of the technology of materials with ultra fine structure for hydrogen storage based on the reversible hydrides.

**PNRPU Mechanics Bulletin**. 2014;(3):113-128

MIXED-MODE LOADING OF THE CRACKED PLATE UNDER PLANE STRESS CONDITIONS

#### Abstract

The paper is devoted to the stress-strain analysis near the crack tip in a power-law material un- der mixed-mode loading. In the paper by the use of the eigenfunction expansion method the stress- strain state near the crack tip under plane stress conditions is found. The type of the mixed - mode loading is specified by the mixity parameter which is varying from 0 to 1. The value of the mixity pa- rameter corresponding to Mode II crack loading is equal to 0 whereas the value corresponding to Mode I crack loading is equal to 1. It is shown that the eigenfunction expansion method results in the nonlinear eigenvalue problem. The numerical solution of the nonlinear eigenvalue problem for all the values of the mixity parameter and for all practically important values of the strain hardening (or creep) exponent is obtained. It is found that the mixed-mode loading of the cracked plate gives rise change of the stress singularity in the vicinity of the crack tip. The mixed - mode loading of the cracked plate results in the new asymptotics of the stress-strain fields which is different from the classical Hutchinson - Rice - Rosengren stress field. The approximate solution of the nonlinear eigenvalue problem is either obtained by the perturbation theory technique (small parameter method). In the framework of the small parameter method the small parameter presenting the difference between the eigenvalue of the nonlinear problem and the undisturbed linear problem is introduced. The analysis carried out shows clearly that the stress singularity in the vicinity of the crack tip is changing under mixed-mode loading in the case of plane stress conditions. The angular distributions of the stress and strain components (eigenvalue functions) in the full range of values of the mixity parameter are given.

**PNRPU Mechanics Bulletin**. 2014;(3):129-162

STOCHASTIC MODELLING OF DEFORMATION PROCESS IN ELASTOPLASTIC COMPOSITES WITH RANDOMLY LOCATED INCLUSIONS USING HIGH ORDER CORRELATION FUNCTIONS

#### Abstract

The aim of this research is to develop the mechanisms of calculation of stress and strain fields statistical characteristics in components of heterogeneous solid media in dependence on microstructural parameters.Multiscale hierarchy of heterogeneous (composite) materials is investigated using the Repre- sentative Volume Element (RVE) concept when parameters of larger scale models are measured or calculated on a smaller scale. Fields of stress, strain and displacements are presented in the form of statistically homogeneous coordinate functions.Analytical expressions for statistical characteristics of structural fields, such as mean values and dispersions, are formed using solution of stochastic boundary value problems and contain structural multipoint correlation functions. The order of the required correlation functions is determined by the solution of stochastic boundary value problem. The boundary value problem is being reduced to inte- gral-differential equation in fluctuations of displacements. The second approximation of the problem solution is obtained. To determine relation between deformation in components and macroscopic de- formation, the iteration procedure was organized.New analytical expressions and numerical results for statistical characteristics of stress and strain fields in components of elastoplastic composite materials are derived with the second approxima- tion of the boundary value problem and correlation functions up to fifth order. 3D models of representa- tive volume of material microstructure with polydisperse ellipsoidal inclusions are synthesized; the mul- tipoint correlation functions up to fifth order have been obtained for them. Numerical results are obtained for porous composites with polydisperse ellipsoidal inclusions in simple shear state of strain for micro- structures with various inclusions volume fraction.

**PNRPU Mechanics Bulletin**. 2014;(3):163-185

SERRATED YIELDING: PHYSICAL MECHANISMS, EXPERIMENTAL DATES, MACRO- PHENOMENOLOGICAL MODELS

#### Abstract

Discontinuous deformation as a phenomenon of plastic deformation instability, observes for a variety ductile materials in some range of strain rates and temperature. It is known fact that the tem- perature and strain rate are the most important parameters of the inelastic deformation. For the majority of polycrystalline materials, in the absence of phase transitions, temperature increase and decrease of strain rate leads to a reduction in plastic deformation resistance. At the same time for the majority of alloys in some ranges of temperatures and strain rates observed inverse dependence of the flow stress. Most researchers considered that the main cause of the anomalous behavior is a diffusion process and dislocation-impurity interaction. Portevin-Le Chatelier effect is the best known manifestation of the influ- ence of diffusion processes on the behavior of deformable material. Establishing the ranges of impacts which implement discontinuous yielding and eliminate them from the technological regimes is a very urgent problem today.Various methods and approaches based on mathematical modeling, are most preferred for the analysis of discontinuous yielding, determine the effective processing conditions and design of new materials. Experimental methods for studying this phenomenon resource intensive and applicable only for existing materials. Construction of mathematical models which accurately reproduce investigated processes is impossible without studying the available empirical information to establish the leading physical mechanisms.In the first part of the review considered the works devoted the description of the physical mechanisms and experimental studies of serrated yielding. The main mechanism is considered the interactions between mobile dislocations and diffusing solute atoms. Three main types of Portevin-Le Chatelier effects has been allocated based on the experimental data of uniaxial loading, in real experi- ments could experience different combination of these three types. Different approaches and models (macro phenomenological, structure-mechanical, physical) used for the theoretical description of the discontinuous deformation, in the present review we analyze only phenomenological models.

**PNRPU Mechanics Bulletin**. 2014;(3):186-223

FEATURES OF BEHAVIOR OF LOW-MODULUS VISCOELASTIC POLYMER COMPOSITES UNDER CHANGING STRAIN AMPLITUDE OF LOW-FREQUENCY COMPONENT OF BIHARMONIC LOAD

#### Abstract

Uniaxial monoharmonic (one-frequency) tests under different values of pre-static strain, strain amplitude and uniaxial biharmonic (two-frequency) tests under different values of strain amplitudes of first (low-frequency) and second (high-frequency) harmonics were conducted on samples of low- modulus viscoelastic polymer composite. Complex operators method was used to describe behavior of a viscoelastic material under harmonic loads. Dynamic deformation properties of the composite material (loss angle, dynamic modulus under monoharmonic tests; loss angle, dynamic modulus first and second harmonics under biharmonic tests) were determined using special methods. Dependencies of dynamic modulus on pre-static strain under different values of strain amplitude of monoharmonic load were con- structed. And also dependencies of dynamic modules and loss angles low- and high-frequency compo- nents (harmonics) of biharmonic load on strain amplitude low-frequency harmonic under different values of strain amplitude high-frequency harmonic were constructed. It was shown that pre-static strain under monoharmonic tests corresponds to low-frequency harmonic under biharmonic tests (dependence of dynamic modulus on pre-static strain under one-frequency tests similar to dependence of dynamic modulus of second harmonic on strain amplitude of first harmonic under two-frequency tests). Compari- son of dynamic modules and loss angles of tested material under two-frequency (biharmonic) and re- lated one-frequency (monoharmonic) loads was conducted. We determined strain amplitude ratio of low- and high-frequency components in which the value of dynamic modulus of low-frequency compo- nent of biharmonic load does not differ from the value of dynamic modulus under monoharmonic load.

**PNRPU Mechanics Bulletin**. 2014;(3):223-251