## No 1 (2015)

**Year:**2015**Articles:**10**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/26**DOI:**https://doi.org/10.15593/perm.mech/2015.1

Computational and experimental studies of resonance sound-absorbing multilayer structures

#### Abstract

The paper considers a developed mathematical model of adjusting multi-layer sound-absorbing structures in order to get the set absorption frequencies based on experimental frequency results of single-layer sound-absorbing structures. Mesh samples which are actually Helmholtz resonators are considered as sound-absorbing structures that have resonant sound absorption. Resonant frequency of a single-layer or mesh type with different geometrical parameters is defined by the acoustic device "Interferometer". Combining such one-layer sound-absorbing structures and piling it into a multilayer structure leads to the increase of sound absorbing spectral range and of sound absorption coefficient. Mathematical model of the acoustic system in multilayer resonance sound-absorbing structures is built based on acoustomechanical analogue with resonant system, where the number of freedom degrees complies with the number of sound absorbing structures. The model helps to find the characteristics of sound-absorbing structures adjusted to the set ratio of resonance frequencies. Resonant frequencies of multilayer sound-absorbing structures are analytically determined by means of resonance frequencies of single-layer cellular sound-absorbing structures on the basis of the mathematical model developed for the oscillatory system with several degrees of freedom. Experimental verification of the calculation results showed that the difference between the experimental and calculated resonance frequencies of sound-absorbing structures is not more than 3 %. Thus, the proposed computational and experimental technique is experimentally confirmed, and geometrical parameters of the resonant cellular aggregate to create multilayer cellular sound-absorbing structures which can be used in the construction of aircraft engines is created.

**PNRPU Mechanics Bulletin**. 2015;(1):5-20

NUMERICAL MODELLING OF PERFORATED SHELLS STABILITY

#### Abstract

In this paper, the finite element method is applied to investigate the stability of densely perforated cylindrical shells under external pressure. The problem is resolved based on the shell theory with an orthotropic material model. The orthotropic material parameters in the form of softening coefficients are determined from the solution of the problem of deformation of cyclically repeating structural elements under tensile, shear and bending with variable rates of perforation (porosity). The research of the structural elements has been produced using methods of continuum mechanics and Timoshenko-type theory. As a result of these considerations, softening coefficients are obtained for different values of porosity, also the limits of applicability of the shell theory were defined for the similar problems. The comparison of numerical results with the analytical estimates, obtained by Grigolyuk and Filshtinsky [1] is provided. Verification of the numerically obtained orthotropic parameters is based on solving the problem of bending of a band quarter, which has been performed with one row of holes. The problem is solved in the framework of continuum mechanics and the shell theory in conjunction with the structurally orthotropic model with different rates of porosity. It is confirmed that using the finite element method for the structurally orthotropic shell with parameters, determined from the solution of three-dimensional tensile and shear behavior, is applicable to the long waves bending problem. Investigation of the stability of perforated elastic cylindrical shell under external pressure is provided for two boundary conditions based on the shell theory and the structurally orthotropic mode. As a result, the critical pressure value and corresponding buckling modes are obtained depending on the shell length and perforation rates.

**PNRPU Mechanics Bulletin**. 2015;(1):21-30

Analytical method of constructing field of possible texture parameters changes

#### Abstract

The proposed work is focused on polycrystalline materials with cubic symmetry of the lattice, subjected to thermomechanical treatment and having a crystallographic texture. As a subject of study the link of physical and mechanical properties anisotropy with integral characteristics of texture was examined. As integral texture characteristics (texture parameters) the authors selectedaverage values of certain combinations of the direction cosines defining the provisions of the crystallographic axes grain polycrystalline, in the laboratory system of axes. On the basis of the published data analysis a problem of findingborders of distribution of texture parameters that determine the anisotropy of the elastic properties of textured polycrystals was formulated. A vector-matrix algorithm for constructing the required piecewise smooth enclosed surface was proposed. The initial data for the area construction was used as restrictions imposed on the textural parameters, which derived from the condition of positivity of the weight coefficients in the corresponding problem of averaging - determining the weighted average degree of eigenvalues of a textured polycrystal elasticity operator. It was established that the distribution area of textural parameters has a fold symmetry axis and is limited by flat and conical elements. In the analytical form the equations of boundary surface elements and the lines of their intersection were obtained. Validation of the mathematical model for constructing the distribution area of texture parameters was performed by comparison with the empirical data. The empirical data on the evolution of the texture was obtained earlier, by an independent research in rolling of sheets of aluminum alloys at a multistand camp. It was established that the experimentally determined values of texture parameters belong to the trajectory lying inside the found analytical field. This will lead to a targeted control of technological processes in manufacturing items and semi-finished products of metal materials.

**PNRPU Mechanics Bulletin**. 2015;(1):31-42

Theory of inelasticity without loading surface and associated flow rule

#### Abstract

Based on the analysis of experimental hysteresis loops (cyclic diagram) of steel 40Х16Н9Г2С three sections are allocated describing the different behavior of the stress, i.e., three types of stresses. For each type of stress the authors have formulated appropriate evolutionary equations characterizing anisotropic hardening. To describe isotropic hardening, evolutionary equation for the parameter saturation stresses of the second type is introduced. In case of additional isotropic hardening under non-proportional cyclic loading we consider the stress saturation parameter of the second type dependent on the measure of disproportionality (complexity) of the loading process. To describe the phenomena of ratcheting under non-symmetric soft cyclic loading we consider the parameter, which is an evolutionary equation for the stress of the first type to be dependent on the accumulated strain. The stress deviator is defined as the sum of stress deviators of three types. To describe the nonlinear process of damage accumulation, the authors introduced the kinetic equation based on the energy principle, where energy equal to the work of stress of the second type on the deformations field is taken as energy spent on creating damage in the material. Kinetic equation for damage caused by stress of the first type on the field deformations is introduced under nonsymmetrical soft cyclic loading in case of ratcheting of the hysteresis loop. It became possible to assign material functions, close theory, formulate the basic experiment and the method of material functions identification. We have presented material functions of steel 40Х16Н9Г2С and the results of verification of the theory under proportional (simple) hard cyclic loading and disproportionate (complex) loading sweep deformations in the form of concentric circles with a common center at the origin of coordinates. Five turns of the trajectory, from the trajectory of the greater curvature to the average trajectory curvature, are considered. There is a reliable agreement between the results of calculations and experiments.

**PNRPU Mechanics Bulletin**. 2015;(1):43-57

Mathematical modelling of dynamics plastic forming of aft fairing in missile transformed during flight

#### Abstract

This article considers an innovative idea of decreasing the aerodynamic drag of missile by an inflight forming of an aft fairing of preferable geometry on the basis of plastic deformation of corrugated thin-walled shell by gasdynamic influence. The mathematical description of the interfaced impulse gas-dynamic and deformation processes of forming aft fairing in flight is given with the thermal effects. The algorithm for the numerical simulation of these co-occurring processes is presented. In the numerical experiment the authors proved preferred technical solutions that realize the principle of the flight transformation. The adequacy of the results obtained by the developed research instruments confirmed by comparing the numerical simulation data with the data of natural experiments. The numerical experiment preferred variants of the technical solutions implementing the principle of flight transformation are justified. In particular, it is shown that a membrane with a coaxial arrangement of the corrugations is more preferable than the radial arrangement. In this case we have managed to present the version of the mounting membrane with the best streamlined form during deployment of the membrane. The paper proposed two ways to improve the fairing: by changing the geometric parameters of the original thin-walled diaphragm; and due to variation of the deforming impacts. The best form is obtained when creating a combined force, which first of all transforms the central part and then the peripheral region of the corrugated shell. Also, the work shows a perspective direction based on the formation of the shell by a special gas jet. In general, on the basis of the research it became possible to prove the principal possibility of obtaining the specified geometrical form on board the missile by pulsed deformation of corrugated shell of the aft fairing, which is optimal from the viewpoint of reducing aerodynamic drag.

**PNRPU Mechanics Bulletin**. 2015;(1):58-75

Multilevel models of crystal plasticity and viscoplasticity in multiphase polycrystalline materials

#### Abstract

The paper considers the problem of modelling the processes of inelastic deformation of polycrystalline materials with regard to microstructure, physical mechanisms and their carriers. At present physical approach based on an explicit introduction of variables (responsible for these mechanisms) into the mathematical model is widely spread. It is necessary to consider deeper scale levels than a macroscopic one for description of such models. That is why this type of model can be considered as a multilevel model. Classification features in multilevel models based on the physical theory of plasticity may be: hypothesis transition between scale levels, the number of the scale levels involved in the consideration and physical model lying in the low level. The fact that physical laws of meso- and microlevel are quite universal, this class of models can be used to analyze a wide range of materials and processes, and their scope is constantly increasing. To be more exact, these are multiphase materials, both widely used and newly developed and implemented in production, for example, steel and titanium alloys. Analysis of meso-and microstructure shows the dependence of the response of the material at the macro level on its current state. The peculiarity of such materials is a high degree of heterogeneity of stress and strain fields arising due to the physical heterogeneity of the individual phases of polycrystal. The present paper provides a review of multilevel models of plasticity theory based on the explicit consideration of carriers and mechanisms of inelastic deformation. The review contains different aspects of application of various modifications of multilevel physical models for description of behavior of multiphase materials that are widely used in industry. Special attention is paid to critical analysis of models.

**PNRPU Mechanics Bulletin**. 2015;(1):76-105

Deformation processes of elastoplastic material with defects under electrodynamic loading

#### Abstract

The processes occurring in the metallic samples under the impact of electrical current of high density are considered. The processes occurring in the vicinity of microdefects in the form of flat cracks under the action of electric current are studied. The dynamic problem is solved numerically for a representative element of the material with crack. The problem is solved in two stages using finite elements method. At the first stage, we have studied the thermal electrodynamic problem in order to obtain the temperature distribution and the regions of phase transformations in the material. Regions of the phase transformations (melting and evaporation of the material) are cross-calculated without the explicit allocation of the phase boundaries. At the second stage, we have solved a coupled unsteady thermomechanical problem of deformation of the heated elastoplastic sample taking account of the initial temperature field distribution in the material obtained at the first stage at different moments of time. Additionally, quasistatic thermomechanical problem was solved in order to obtain the displacement field (residual strain) after temperature equalization in the material. The influence of the size and orientation of microcracks on the localization of the electromagnetic field in the region of the defect is examined. The calculations on the base of the proposed model show that the current density at the tips of the microcracks may by an order exceed the current density applied to the sample. Numerical modelling has shown, that large gradients of electromagnetic field and current arise in the vicinity of the microdefects,, which leads to intensive heating, melting and evaporation of the metal in the tips of the microcracks. The melted material flows into the microcrack under the action of thermal stresses. At the same time the metal starts to evaporate. The shores of the microcracks converge. All these processes lead to a "healing" of defects.

**PNRPU Mechanics Bulletin**. 2015;(1):106-120

Singularities of contact problems for systems of strings and beams with weakly restrained elements

#### Abstract

In known contact problems for strings and beams, contact elastic elements are restrained in such a way that each of them can be in the equilibrium for the arbitrary applied load. However, other types of restraint are possible where the above-mentioned condition does not hold for one of the elements; this element can be called weakly restrained. At the same time, the system as a whole can be in the equilibrium for the wide set of external loads. The systems of this type have some singularities in the contact problem statement, in the proof of the uniqueness of solution and in constructing the analytical solution. The singularities are as follows: the special condition for the weakly restrained element equilibrium, the additional unknown parameter describing the indefinite part of the weakly restrained element displacement, the necessity to prove the uniqueness of not only the contact forces but also this parameter, the expansion of the set of allowable contact loads, the exclusion of zero external loads. These singularities are considered for two following examples: 1) two strings; the first one has the free edges; 2) two beams with the gap between them; the first one has the hinted bearing and the free edge. The modification of the usual plan of consideration of contact problems for strings and beams is proposed. This modification keeps the main ideas of the usual plan and gives the opportunity to take the weakly restrained elements into account. For each example, the uniqueness of the solution of the contact problem is proved and the analytical solution is built.

**PNRPU Mechanics Bulletin**. 2015;(1):121-129

Investigation of surface plastic hardening anisotropy influence on residual stresses distribution in hollow and solid cylindrical specimens

#### Abstract

For the calculation of residual stresses and plastic strains in hollow and solid surface-hardened cylindrical specimens we suggest the mathematical models, which take into account both the cases of hardening leading to the isotropy (hydraulic shot blasting procedure) and to the anisotropy (roller burnishing process) of plastic strains in the surface layer. The introduced mathematical model has a hardening anisotropy parameter which ties the axial and circumferential components of the residual plastic strains tensor. We use the determined axial and/or circumferential components of the residual stresses tensor as the input information. Also we use the following assumptions: smallness of the off-diagonal elements of the residual plastic strains tensor and residual stresses tensor, plastic incompressibility of material, absence of the secondary plastic strains of the material in the compression area of the surface layer. The boundary value problems of the hardened layer stress-strain state estimation after the hardening for the hollow and solid cylindrical specimens are solved and the solution is given in the paper. We give the method for the mathematical model parameter identification under the condition of self-equilibrated residual stresses and the method for the experimental determination of residual stresses by the circles and strips approach. Also, the cycle of the hardening experiments for the hollow and solid specimens from the 40Kh steel (having different proportions of internal and outer diameters) in the roller burnishing process and hydraulic shot blasting modes was performed with the determination of the residual stresses. The mathematical model adequacy is verified through the comparisons with experimental data; good agreement of the calculated and experimental data is demonstrated. The calculated numerical values of the hardening anisotropy parameter are given. The fact that the procedure of surface anisotropic hardening (roller burnishing process) leads to the strong layering of the epures of axial and circumferential residual stresses in depth of the hardened layer is established. Contrastingly, the case of the isotropic hardening (hydraulic shot blasting) leads to almost coinciding epures. Using the experimental and calculated data we have shown that the hardening of tool roller type leads to the larger absolute values of the compressing stresses, which by more than 30% exceeds the stresses after the hardening of hydraulic shot blasting type for the specimens of the same geometry. For the main results of the research we have given the necessary data in the table and also we have presented the epures for the residual stresses distribution.

**PNRPU Mechanics Bulletin**. 2015;(1):130-147

A two-level method for calculation of microstress on reinforced plates with circular hole in case of extension normal to principal direction

#### Abstract

The stress concentration must often be examined at two levels while analyzing the stress condition of composite materials. The macroconcentration depends on the presence of holes, notches and other local areas of a construction. Typical dimensions of macroconcentration distribution areas are of the order of 0,01-0,1 m. Macroconcentration analysis is performed using the models of homogeneous material. Microstress concentration occurs in structurally inhomogeneous composites due to the structural heterogeneity of the composite. The sizes of concentration areas in regular structures are defined by the sizes of periodically recurring areas. In fibrous composites, such areas have the size of approximately 0,0001 m or less. This makes it necessary to use a two-level approach for the analysis of the stress concentration in the construction of composite materials. The aim of the present study was to compute the stress concentration in unidirectional reinforced composite plate with circular hole with respect to the volume ratio of the component materials in composite. The contour of the circular hole and its dependency on the structure of plates was calculated in order to study the behaviors of macro- and microstresses. The boundary conditions at a large distance from the hole are pressure, uniformly distributed on the plate. Also this problem is analyzed with the finite element method by package ANSYS. Macroconcentration is defined based on the solution of the plane problem of elasticity theory of the orthotropic material by the virtue of functions of a complex variable. The finite element method was used to investigate the stress distribution at microlevel. Boundary conditions that model the state of the specified two-dimensional representative cell in the composite structure were established. The results demonstrated the macro- and microstresses and behavior of the orthotropic plate with a circular hole calculated for two different structures.

**PNRPU Mechanics Bulletin**. 2015;(1):148-157