## No 2 (2015)

**Year:**2015**Articles:**11**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/25**DOI:**https://doi.org/10.15593/perm.mech/2015.2

Method of numerical calculation of optimal forms of bodies of revolution at movement in soil medium

#### Abstract

The method of calculating the optimal forms of axisymmetric projectiles during motion in soil medium on the base of direct optimization method is developed. The methods of local variations and cyclic by coordinate descent are used. Direct numerical calculations are carried out in an axisymmetric formulation. The form of a optimal body found on the basis of a model of local interaction is taken as an initial approximation. A local interaction model (LIM) is used below, based on the analytical solution of the one-dimensional problem of the expansion of a spherical cavity in a Grigoryan soil medium assuming that the medium behind the shock wave front is incompressible. The main assumptions made in solving the problem of optimizing the shape of axisymmetric bodies within the framework of the LIM, i.e., that a model that is quadratic in the velocity is applicable, the friction is proportional to the pressure and flow over the body occurs without separation, have been confirmed theoretically and experimentally earlier. The applicability of the LIM in describing the penetration of sharp cones has been demonstrated experimentally and theoretically by comparison with the results of numerical calculations in an axisymmetric formulation using a Grigoryan soil medium model. The errors in the LIM in determining the drag forces when applied to blunt bodies have been shown too. In this work the effectiveness of the developed methodology is shown in the problem determining the shape of minimum resistance projectile among the bodies of revolution having a set length and radius of the cross section. Good agreement has been reached between the results for the generatrix of a body of revolution in the form of a parametric Bezier polynomial and a piecewise linear curve. Convergence of successive approximation methods for the solution of a parametric optimization problem is studied. The essential role of two-dimensional flow effects was revealed.

**PNRPU Mechanics Bulletin**. 2015;(2):5-20

Version of the theory of thermoplasticity

#### Abstract

The paper explains the basic concepts and equations of the theory of thermoplasticity belonging to the class of theories of plastic flow in the combined hardening. The tensor strain rate is represented as a sum of tensors of the velocities of elastic and plastic deformations. The elastic deformation follows the generalized Hooke's law distributed to non-isothermal loading. The surface loading is introduced which isotropically widens or narrows and shifts in the process of loading. For the radius of the surface loading the authors formulated evolutionary equation taking into account the additional isotropic hardening under non-proportional loading, also generalized to non-isothermal loading. We have taken the parameter of Kadashevich-Mosolova corresponding to the angle between the velocity vectors of strain and stress as the parameter characterizing the degree of complexity of the process of loading. Surface displacement loading is described based on the model of Novozhilov-Chaboche implying that the total displacement is the sum of the displacements and each displacement has its own evolutionary equation. An earlier analysis of a loop of plastic hysteresis allowed distinguishing three types of micro-strains (displacements) and formulating three types of evolution equations. Here these evolutionary equations are generalized to the non-isothermal loading. To determine the axial velocity of plastic deformation we used associate (gradient) flow law. It became possible to determine expressions for speed of the accumulated plastic strain for hard and soft modes of loading. The authors have formulated conditions of elastic and elastic-plastic conditions. The kinetic equation of damage accumulation is introduced for the description of nonlinear processes of damage accumulation. Here, energies equal to the work of microstresses of the first and second types to the field of plastic deformations are accepted as the energy spent on creating damage in the material. These equations are generalized to the non-isothermal loading. We have highlighted material options completing theory option, formulated the basic experiment and the method of identifying material functions. The paper describes the verification of thermoplasticity theory on a wide range of structural steels and alloys and programs of experimental studies. New results have adequate descriptions within one theory of the following phenomena: - landing loop of plastic hysteresis in nonsymmetrical rigid cyclic loading; - ratcheting of loop plastic hysteresis in nonsymmetrical soft cyclic loading; - the regularities of complex loading as on planar or spatial trajectories; - the effects of additional isotropic hardening under disproportionate (complex) cyclic loading; - the effects of non-linear summation of damage to arbitrary loading processes; - the patterns of non-isothermal loading.

**PNRPU Mechanics Bulletin**. 2015;(2):21-35

The variational equations of the extended N’th order shell theory and its application to some problems of dynamics

#### Abstract

One of topical problems of thick shells theory consists in an accurate modelling of high-frequency shell vibrations and wave propagation, as well as in the application of developed mathematical models to engineering problems of structural dynamics. Many observed processes of both periodic and non-steady shell dynamics cannot be adequately simulated on the groundwork of most traditional hypothesis-based shell theories. Therefore the engineering practice requires new shell models considering higher degrees of freedom besides the displacement and rotation vectors and taking into account the interference of tangential and transversal motions. Here a variant of the N’th order shell model of I.N. Vekua type is analyzed. The proposed model improves the A. Amosov’s shell theory; it is based on the Lagrange’s formalism of analytical mechanics extended to continuum systems, on the dimensional reduction approach, and on the biorthogonal expansion technique. The shell is finally represented by a material surface framed by a set of generalized coordinates (or field variables) and a scalar generator function - the surficial density of the Lagrangian. The dynamic equations for the shell are formulated as generalized Lagrange equations of the second kind. A propagation of normal waves in a plane elastic layer is considered, the corresponding homogeneous problem of N’th order plate theory is formulated, and its solution analysis is extended. In particular, the investigation of the second longitudinal propagating mode and its modelling by approximated plate theories are expanded. The main feature of the second mode is the “inverse wave” effect, i.e. the opposite signs of the phase and group velocities at small wavenumbers. It was shown earlier that the accurate dispersion curves for both phase frequency and group velocity can be obtained on the basis of the fourth- and fifth-order theories. Using the known solution of the spectral problem of N’th order plate theory the eigenfunctions are obtained. The second propagating mode evaluation at the smallest wavenumbers is modelled, the forms of the second propagating wave are investigated, and the approximation of the waveforms at some wavenumbers by plate theories of various orders is estimated. The accuracy of the approximation of the “inverse wave” by some applied theories of various orders based on the orthogonal expansions of displacement vector is discussed.

**PNRPU Mechanics Bulletin**. 2015;(2):36-59

Research of influence of barrier crest on polymer extrusion process

#### Abstract

In this paper the effect of an additional (barrier) crest on heat and mass transfer processes and phase transitions of polymers in the channel plasticizing extruder is investigated. On the basis of the energy and hydrodynamics equations the mathematical model describing the processes of motion of a solid and a liquid polymer inside the channel as well as undergoing phase transformations is given. To create a mathematical model of melting processes a number of simplifying assumptions was introduced, namely: 1) the process has a stationary character at a constant mass flow rate; 2) helical path is unfolded on the plane and the principle of reverse movement is used; 3) diffusion of heat along the channel is not taken into account; 4) loss of the melt through the main ridge is neglected; 5) the elastic polymer in the melt process is not taken into account. As a result, the process of resin moving and heat exchange in the melting zone of the extruder was modeled by nonclassical heat and mass transfer in a long rectangular channel divided into two by barrier crest (solid phase channel and melt channel) in which the upper wall is moving at a constant speed equal to the peripheral speed of the worm at helical line cutting angle to the channel axis. The resulting model is solved by finite difference method allowing doing the numerical investigation of the dependence of the extrusion process of polymers on the geometric parameters of the barrier screw. The calculation was performed for the screw ME-90 with a number of L / d = 26 at a rate of 78.7 kg/hour. The width of the barrier crest was assumed to be 16 mm, clearance above the barrier crest was 1 mm. Feed material was high density polyethylene. Data analysis led to the conclusion that the presence of additional (barrier) crest increases dissipative heating of the molten resin circulating above it, as well as reduces the amount of molten polymer film on the solid phase, which allows intensifying melting process and improve its homogeneity.

**PNRPU Mechanics Bulletin**. 2015;(2):60-69

Decomposition of systems of equations for continuum mechanics 1. Elasticity, thermoelasticity, poroelasticity

#### Abstract

The work is devoted to the development of decomposition methods for systems of linear partial differential equations that arise in continuum mechanics, in particular, in the theory of elasticity and thermoelasticity and poro-elasticity. These methods are based on the decomposition (splitting) of systems of coupled equations into several independent equations. The decomposition significantly simplifies the qualitative study and interpretation of the most important physical properties related to three-dimensional equations and allows an effective study of their wave and dissipative properties. Moreover in certain cases the decomposition makes it possible to find exact analytical solutions of the corresponding boundary and initial-boundary value problems, and greatly simplifies the application of numerical methods, allowing us to use the appropriate routines for simpler equations and independent subsystems. In the first part of the work various systems of equations, including equations of elasticity theory in the form of Tedone and in the form of Beltrami-Donati-Michell are given, their dynamic generalizations are proposed, and various forms of the equations of classical and hyperbolic thermoelasticity as well as the equations poroelasticity are described. A number of historical facts, which are directly related with the considered questions and weakly reflected in Russian literature,are presented. Various types of decomposition and their generalizations are described. The representation of solutions of dynamical systems of equations resulting from the toroidal-poloidal decomposition, as well as the decompositions of Green-Lamé, Cauchy-Kovalevski-Somigliana, Naghdi-Hsu-Chandrasekharaiah, and Teodorescu types are discussed in details. Special attention is given to their static analogues. A generalization of the representation of Savin for the dynamic equations of elasticity is obtained. The representations in curvilinear coordinates, in particular, the representations of Boussinesq, Timpe, Love, Michell, and Muki types are given. The bibliographical references to the original papers are listed.

**PNRPU Mechanics Bulletin**. 2015;(2):70-102

Experimental finding of dynamic deformation fields in metal and composite plates under impact

#### Abstract

The purpose of research is to develop and verify methods of experimental determination of the dynamic strain fields in the study of impact damage and destruction of the plates. An experimental rig that realizes dynamic load of the studied plate at high-speed impact with a projectile and determination of dynamic strain fields on the plate surface is developed. The method is based on digital image correlation procedure in combination with a high-speed video recording that implemented using hardware-software Vic-3D complex. The investigation is performed in two series of experiments with different materials and dimensions of the test plates and with different materials and speed of the projectile. The experimentally obtained fields of the dynamic strain for the high-speed impact of the aluminum plate and a spherical steel projectile and for the carbon fiber plate and ice projectile are presented. The results are presented in the form of time-lapse recording of the tensor component fields of dynamic deformation and time dependences of deformations at certain points of the plate. The implemented deformation rate is up to 1,.5· * 103 sec--1. The state of strain for carbon-fiber composite during high speed destruction is obtained. The reliability of the digital image correlation method results is confirmed by direct measurement of residual deformations in the plate. The described method provides detailed experimental data about the processes of high-speed deformation for metals and composite materials. This data is of interest for experimental verification of deformation and destruction models behavior of materials at high strain rates. In particular, it provides an opportunity to obtain the necessary data for verification of deformation models and failure criteria of deformation under biaxial stress state. This method can be used for mathematical models testing and experimental investigation of the ballistic damage and destruction dependences of critical structural elements in cases of foreign objects damage of aircraft and engines parts or armor piercing.

**PNRPU Mechanics Bulletin**. 2015;(2):103-115

Fractal analysis of fracture surface of aluminum alloy AMg6 under fatigue and dynamic loading

#### Abstract

In this paper we investigated the localization of deformation during high-speed deformation and fatigue crack propagation in gigacycle loading regime in the aluminum-magnesium alloy AMg6. Localization of plastic deformation under conditions closely approximating simple shear was investigated using the shear-compression specimens (SCS) tested on the split-Hopkinson bar device. After the experiment the maintained specimens were subject to microstructure analysis using the NewView-5010 optical interferometer. Fatigue tests were conducted on Shimadzu USF-2000 ultrasonic fatigue testing machine. This machine provides 109-1010 loading cycles with the amplitude from 1 to several dozens of microns and frequency of 20 kHz, which reduces testing time to a few days, as opposed to the classical fatigue testing machines, in which the same number of cycles can be reached only in few years. The New View 5010 interferometer-profiler of high structural resolution (resolution of 0,1 nm) was used as an instrument of qualitative analysis, which provided data allowing us to find correlation between the mechanical properties and scale-invariant characteristics of defective structures formed under dynamic and gigacycle fatigue loading conditions. The authors propose an original form of writing the kinetic equation, which relates the rate of the fatigue crack growth with a change in the stress intensity factor. The scale invariance of defect structures responsible for the formation of the fracture surface relief under gigacycle fatigue loading was found to be related to the power exponent of the Paris law.

**PNRPU Mechanics Bulletin**. 2015;(2):116-126

Features of strain localization in sylvinite under tension

#### Abstract

This work is devoted to investigation of spatio-temporal localization of deformation at the direct uniaxial tensile of sylvinite with using of digital image correlation method and acoustic emission technique. For the implementation of direct tension there was used a special reversing device allowing to convert compressive force into tensile. Samples are concreted into the device before the experiment. Sequentially photographed images of side surfaces of specimens were used to reconstruct the distribution of components of displacement vector field and strain tensor on the side of sylvinite samples under uniaxial quasi-static tension. As result of obtained data analysis it is found that the deformation process occurs as two consistently following forms of spatiotemporal localization: a system of equidistantly located stationary foci of localized deformation and a single stationary dissipative localized structure. Areas of specimen outside the localization zones are in the undeformed state, also as result of uniaxial quasistatic tension of specimen it is established that the zones of localized deformation can be tensile as well as compressive. The transition from one localization form to another occurs in the maximum stress vicinity and is accompanied by a sharp decrease in the concentration parameter. Concentration parameter characterizes the degree of interaction between defects of different scale levels through their elastic fields and can be estimated from the acoustic emission data. Analysis of dependence of the spatial period of localized deformation bands on the size of computational cell showed that localized deformation bands are self-similar structures and process of deformation is localized at the grain boundaries minerals composing the sample.

**PNRPU Mechanics Bulletin**. 2015;(2):127-38

Simulation of stamping workpieces implants process using schemes of severe plastic deformation processes

#### Abstract

Pure titanium is much more biocompatible than the alloys used in medicine but has low strength properties, which hinders its widespread use for the implants manufacturing. Providing the necessary level of strength characteristics requires the use of severe plastic deformation methods (SPD), because traditional methods of metal forming processes do not provide the desired set of properties. Designing the implant geometric model of spine spacer type and forgings for its production was performed. Forging form allows applying an SPD stamping process, which eliminates the need for pre-treatment of the workpiece material before punching. Process steps sequence are proposed as a result of an analysis of the “U-implant” part geometry using stamping deformation schemes corresponding to SPD schemes under plane strain state: sediment workpiece flat punch, three stamping operations of implant forgings that eliminate the preliminary preparation of the microstructure workpiece before punching. Analysis of options for implementing the transition process of plastic deformation using simulation in CAE-systems allowed us to determine the sequence of deformation stages and tool geometry, providing required metal flow. Cumulative degree of deformation during the billet forming is 3.3-7, which provides the necessary change in the structure and strength characteristics of the forging (at 400 °C temperature). The maximum value of the specific efforts in the calculation of the transition stamping does not exceed 160 MPa due to the choice of technological transitions in filling the die cavity draft, which ensures long tool life.

**PNRPU Mechanics Bulletin**. 2015;(2):139-150

Experimental calibration of constitutive model for an amorphous shape memory polymer under large deformations

#### Abstract

The aim of the present study is the experimental calibration of the recent model proposed by the authors which describes the thermomechanical behavior of glassy polymers under large deformations. For such this purpose a series of thermomechanical experiments has been conducted with lightly-linked epoxy specimens, for which large deformations are achievable in the rubbery state. Three experimental series have been carried out on the DMA testing machine in the regime of regulated axial compression load and changed temperature with the axial contraction measuring. The aim of the first experimental series was to prove the elastic potential choice for describing the behavior of the concerned material. For this purpose the samples in rubbery state were deformed under constant loading rate. It was concluded that the Peng-Landel elastic potential could be applied for the accurate description of the epoxy resin deformation process. The second series included several adjusting experiments for the model parameters determination. The samples were heated with the constant temperature rate under the constant compression load. The comparison of results for different rates and load levels is given. Also the procedure of the parameters determination is described. Some parameters were taken as the material constants, for other ones the linear approximation of the dependence on the temperature rate and the load level was made. The third series presents the verification experiment for which a shape memory thermomechanical cycle was chosen. The experimental data were approximated with the numerical model based on the parameters which were found from the adjusting experiments. A good agreement between the experimental and numerical results is shown. A brief review of the most spread experimental schemes for the shape memory cycle is also given in the paper.

**PNRPU Mechanics Bulletin**. 2015;(2):151-163

DYNAMIC AXISYMMETRIC PROBLEM OF ELECTROELASTICITY FOR A RIGIDLY FIXED BIMORPH PLATE

#### Abstract

A nonstationary axisymmetric problem for a circular rigidly fixed bimorph plate consisting of a metal substrate and two piezoelectric elements is studied in this paper. Mechanical vibrations of the structure are made by the action of its end surfaces of electric potential, which is an arbitrary function of the radial coordinate and time. New closed solution is constructed in the framework of electrodynamics in three-dimensional statement by the consistent use of the method of incomplete separation of variables in the form of integral transformations. Consistently Hankel transformation with finite limits on the axial coordinate and generalized finite transformation (FIT) on the radial variable are applied. At each stage of the solution there is a procedure of standardization which allows the appropriate conversion algorithm. The calculated ratio for the components of the displacement vector and the electric field potential allow us to study the variation of the stress-strain state of the bimorph plate. The constructed solution provides an opportunity to make a qualitative and quantitative analysis of connection of electromechanical stress fields in composite laminated electroelastic structures that allow describing the work and finding the geometric characteristics of the typical elements of piezoceramic transducers of resonant and nonresonant classes. Based on the analysis results, it becomes obvious that there is the need for rigidly fixed bimorph systems for excitation of flexural vibrations of the split ring electrodes located on the faces of the piezoceramic plates and for the application of Timoshenko system of equations in applied theory for thin plates taking into account shear deformations. In addition, it became possible to obtain potential change laws, axial-vector components in the tensions and induction of electric field along the thin piezoceramic plate.

**PNRPU Mechanics Bulletin**. 2015;(2):164-178