## No 3 (2017)

**Year:**2017**Articles:**13**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/16**DOI:**https://doi.org/10.15593/perm.mech/2017.3

Processing fine magnesium materials. Numerical simulation and experimental analyses

#### Abstract

Currently developing elements for a renewable hydrogen storage and transportation is needed and its demand is rapidly increasing. Metal hydrides are among optimum solutions for hydrogen storage in terms of effectiveness and safety. The promising material which is good for such approach is magnesium and its alloys that can reversibly absorb hydrogen in quantities up to 7.6 % by weight which meets the DOE requirement. At first, determining a fast hydrogen saturation of Mg-based alloys has consisted in mechanical grinding of materials up to delivering the micrometric grain size. Increasing markedly the specific surface of the treated powders by plastic deformation processing leads to delivering very reactive samples contrary to bulk materials which are markedly un-reactive. More recently, great improvements of the H-sorption characteristics were demonstrated efficiently when applying the Equal Channel Angular Pressing (ECAP) treatments to bulk Mg-alloys. The implementation of ECAP process entails workpieces to pass the dedicated matrix formed by two channels crossing at a certain angle, i.e. from 90, 105 up to 120 degrees according to the degree of plasticity of the material. Effectively, the stress level delivered to the material depends on the angle between the channels, the applied pressure on the sample, the friction and (or) counter-pressure effects and obviously the physical and mechanical characteristics of the sample versus temperature. As the dimensions of the workpiece in terms of cross section are not changed, the deformation process can be applied several times successively, with the aim to achieve extremely high degrees of stresses and deformation. So, during such a severe deformation process the achievement of a fine-grained microstructure in the magnesium bulk samples is accompanied by the formation of a high density of defects and overall texture. When considering the ECAP process applied to the magnesium alloys, the numerical simulations were developed to anticipate the mechanical behaviour of the workpieces parallel to experimental characterizations using different methods such as structural and texture analyses. The present article reports on the deformation process Mg-based materials by using the numerical simulation method. By using the numerical LS-Dyna package for spatial deformed states we have successively applied ECAP operations in order to determine the optimum conditions of deformation delivering fine-grained Mg-materials with a high level of internal stresses. The results of the calculations are found in good agreement with the experimental data; and make it possible to use the proposed optimised process and adapted tool to up-scale the effective mass production.

**PNRPU Mechanics Bulletin**. 2017;(3):5-16

Stability of toroidal shell segments at variation of a deflection angle

#### Abstract

The paper presents the study of the stability of panels of steel toroidal thin-walled shell structures with different angles of deviation from the vertical axis. The mathematical model (the model of Timoshenko-Reissner) is geometrically nonlinear and is represented as a functional of the total potential energy of deformation. To reduce the variation problem into solving a system of algebraic equations, the Ritz method was applied with the use of two different types of basis, i.e. trigonometric and polynomial (based on the Legendre polynomials). The process of forming the approximating functions is considered in detail taking into account the symmetry of the toroidal panels. The final system of algebraic equations is nonlinear and solved by Newton method. The solution is made by the Maple 2017. The calculations of segments of toroidal shells are carried out under the action of the external evenly distributed transverse load and the load values with the loss of stability are obtained. The parameter of the large radius was fixed when choosing the variants of constructions for two purposes. The first one is to the covering area of the considered segment of the shell which remained unchanged and the second one is to the small radius which is dependent on the angle of deviation from the vertical axis. In some cases, local stability losses are observed. The effect of the deflection angle from the vertical axis on the values of the stability loss loads and the maximum values of deflections are analyzed. The results obtained for two types of approximation are presented. The calculations showed that both variants of approximation give close results at low loads, but they significantly differ at large loads. The increase in the deflection angle leads to the decrease in value of the critical load, which may be caused by an increase in the surface area of the shell. However, the value of the maximum deflection decreases.

**PNRPU Mechanics Bulletin**. 2017;(3):17-36

Variational method for non-classical problems of mechanics with constraints based on finite elements approximations and local variations

#### Abstract

This paper is devoted to the solution of non-classical variational problems of mechanics consisting in the minimization of the integral-type functional under various constraints. As constraints we consider various conditions for unknown function, which minimizes the optimized functional. The considered constraints include various dependences of the unknown function on space variables. It is supposed that the minimized functional is integral and includes the dependence both on the unknown function and its partial derivatives. We suppose that the inclusion of the inequality constraints on the unknown function corresponds to the contact conditions arising when deformable bodies interact and when these bodies (medium) contact rigid obstacles. The type of the relevant arising conditions characterizes the considered minimization problem of the functional with local constraints imposed at separate points of the definition domain as the non-classical problem of calculus of variations. In order to solve the considered non-classical problem of calculus of variations we applied a new approach based on finite element approximations (Galerkin type approximation) and procedures of local variations. The original domain aiming to determine the minimizing functional and unknown varied function are decomposed into separate small sub domains (cells of domain). The unknown function is given in the nodes, and the approximation of the function is given with the help of the shape function. It is supposed that the shape functions belong to the space of Sobolev functions differentiable with the quadratic integrable, and the system of basic functions is polynomial and has a small definition domain. The problem constraints are transformed within the framework of introduced finite element approximations. The additive functional of the problem is approximated by the integrals for the cells from the entire domain. Then we consider the problem, formulated as the problem of finding the nodes values satisfying the arising non-classical double constraints on the unknown function and minimizing the optimized functional. The presented variational algorithm is affected by the successive approximations. After choosing the initial approximation satisfying the constraints, each iteration acts as a local variation of the unknown solution consistently for all the nodes and performs the minimization of the optimized functional. In this context we do not violate geometric (contact) constraints and reduce the integral sum over the domain of the varied point. When the local variation process in all the cells is completed and the updated version of the solution is constructed, the process is repeated until a complete convergence is achieved, with a gradual decrease in the variation step and the necessary refinement of the finite element mesh. Thus we provide the solution of the considered variational problem. As an example we present the problem solution related to the torsion of the elastic-plastic bar. The solution of this problem has been obtained for different cross-sectional areas for various angles of torsion using proposed method. The computed results, which are in agreement with the experimental results, are shown for the plastic distributions domains.

**PNRPU Mechanics Bulletin**. 2017;(3):37-52

Plasticity of materials under proportional and nonproportional cyclic loading

#### Abstract

The mathematical modeling of elastoplastic deformation of materials under proportional (simple) and nonproportional (complex) cyclic loading is considered. In this paper we consider a very simple version of the theory of plasticity, which is a particular version of the theory of inelasticity. The version of the theory of plasticity refers to the class of single-reversed flow theories with a combined hardening. The applicability range of the version of the theory of plasticity is limited to small deformations of initially isotropic metals at the temperatures without phase transformations and strain rates, when dynamic and rheological effects can be neglected. Materials with the additional isotropic hardening effect under nonproportional cyclic loading are considered. The results of computational and experimental studies of the elastoplastic deformation and fatigue destruction of materials are presented for various nonproportional cyclic loadings. Based on the analysis of the experimental results related to the material deformation under rigid asymmetrical cyclic loading, the principle of the symmetry of cyclic properties is formulated. Corollaries are given from the principle of symmetry for soft asymmetric cyclic loads. The calculated and experimental results of investigating the asymmetric cyclic loading are given. An adequate description of the processes of complex loading, the effects of the additional isotropic hardening and ratcheting, as well as the destruction processes within the framework of a single, rather simple version of the theory of plasticity are the obvious advantages of the mathematical modeling under consideration. Meanwhile the number of the material functions (in this case 14 parameters and 1 function) is much less than the number of the material functions and parameters that finalize modern theories. In addition, the basic experiment and the method of identifying the material functions of the considered version of the theory of plasticity are clearly defined and are fairly simple and easily implemented. The comparison of the calculation and experimental results indicate the adequacy of the proposed mathematical modeling.

**PNRPU Mechanics Bulletin**. 2017;(3):53-74

The numerical model of dynamic mechanical behavior of brittle materials based on the concept of the kinetic theory of strength

#### Abstract

A model of the dynamic mechanical behavior of brittle materials based on the ideas of the kinetic theory of strength is developed. The proposed model is a generalization of the classical "quasi-static" Nikolaevsky plasticity model (non-associated flow law with the plasticity criterion in the form of Mises-Schleicher) to the strain rate interval corresponding to the dynamic loading. In contrast to the traditional approach to constructing dynamic models, in which the dependence of the model parameters on the strain rate is specified, the proposed model suggests to use the relaxation time and time of fracture as the key parameters. The presented model allows taking into account the change in the strength and rheological properties of brittle materials with an increase in the loading rate. This ensures a correct transition from the quasi-static regime of loading to the dynamic one in the range of strain rates within 10-3 < <103 s-1. Within the framework of the proposed model it is assumed that there exist the experimental data about the dependences of the strength and rheological characteristics of the material on the times of different scales discontinuities nucleation. However, in view of the complexity of obtaining this information, we propose a way of obtaining the estimates of these dependencies by transforming the dependences of the mechanical properties on the strain rate that can be gained with standard tests. The developed dynamic model can be implemented within various Lagrangian numerical methods using an explicit integration scheme (including the finite element and discrete element methods) and is relevant for solving a new class of applied problems related to natural and technogenic dynamic impacts to structures of artificial building materials, including concretes, ceramic elements of structures and natural rock materials.

**PNRPU Mechanics Bulletin**. 2017;(3):75-99

Direct and inverse problems of pipeline bending

#### Abstract

The paper considers the static linear pipe bending at river crossings and ravines under the gravity of the pipe and transported fluid. It is assumed that the parts of the pipeline on either sides of the sagging part are embedded into the ground with same properties. A simple model of the ground elasticity is used which entails a replacement of its distributed spring system with a certain rigidity in the longitudinal and transverse directions of the pipeline. The fluid speed is not considered. The internal pressure difference exerts an unbalanced lateral force directed towards the convexity of the curved axial line. The impact on the axial extension bend of the pipe produced by the axially symmetric deformation is also taken into consideration. The direct problem is to determine the deflection for the given size, stiffness and strength characteristics of the pipe and ground. To simplify the problem, the case of a high internal pressure and shallow pipe occurrence in the ground is considered. The dependence of bending is determined based on the stiffness of the ground and pipeline, as well as on the internal pressure. The pressure growth results in the bending increase. In particular, the critical value of the internal pressure is determined when the bending increases without limit in the linear problem. The inverse problem is to determine the relative stiffness of the ground under the instrumental determination of the pipeline bending or deformation of its outer fibers. It is achieved using the method of the pipeline additional loading by the concentrated power and determination of an appropriate instrumental bending or deformation. In particular, the loading and corresponding measurements are carried out at the midpoint of the pipeline span. The critical value of the relative ground stiffness is determined, below which the bending starts to increase without limit.

**PNRPU Mechanics Bulletin**. 2017;(3):100-112

Thermomechanical boundary problems for a cylinder and sphere made of shape memory alloy

#### Abstract

The paper presents the solutions of the coupled boundary value problems for a thick-walled cylinder and a sphere made of shape memory alloy (SMA). Their material undergoes a direct thermoelastic martensitic phase transformation upon cooling under a constant internal pressure. We have considered slow cooling processes in which the temperature distribution over the material at each instant of time can be considered as uniform. The results are compared with the analytical solutions of the same problems previously obtained within the assumption of a uniform distribution over the material of the martensitic volume part parameter. A quasistatic motion is simulated in the material of the fronts of the onset and completion of the phase transition, as well as the redistribution of stresses and strains along the cross section associated with these motions. It is established that a direct phase transformation begins on the inner surface of the shells. The phase transition zone is quite fast (compared to the growth of the phase composition parameter at the points of the inner surface) propagating along the radial coordinate to the outer surface. After this, the phase transition develops over the entire thickness, the magnitude of the martensitic volume part parameter q being a weakly decreasing function of the radial coordinate (the difference in the values q on the inner and outer surfaces is a small fraction of the maximum q value equal to 1). The completion of the phase transition is observed for the first time on the inner surface, after which the boundary of the phase transition end rapidly moves through the thickness from the inner surface to the outer surface. The intensity of stresses and the annular stress in the process of the phase transition vary non-monotonically and in different ways for the inner and outer surfaces. On the inner surface, these stresses have the maximum values at the points of the beginning and the end of the phase transition; and the minimum values take place at an intermediate point. On the contrary, in case of the external surface, the minimum values of stresses are observed at the beginning and the end of the phase transition, while the maximum value take place at the intermediate point of the process.

**PNRPU Mechanics Bulletin**. 2017;(3):113-128

Calculation and design of hybrid wooden beams

#### Abstract

The problem of determining the stress-strain state and designing hybrid wooden beams is considered. In general, the bar is a rod consisting of several layers. In principle, the number of layers is not unlimited. Each layer can be made of different materials. The geometry of the cross section of a layer may vary a lot. The cross-section of the rod can be either constant or variable in length. The rod experiences a straight transverse bending with stretching-compression. The physical nonlinearity, as well as the different material resistance to stretching and compression are taken into account. The deformations and displacements of the rod are considered to be small values, and it allows one to write the equilibrium equations for the undeformed state. We accept a valid theory of Bernoulli flat sections and a simplified expression for the curvature of a plane curve. The determination of the stress-strain state of the rod is reduced to solving a system of two nonlinear algebraic equations of the third order. To solve it, the load is divided into a number of steps in accordance with the loading history, which allows to linearize the system of resolving equations. Much attention is paid to the design of hybrid beams. Both parametric and functional designs are considered. Based on the calculation examples, it is shown that the use of the criterion of the equal strength in combination with the principles of hybrid design allows to significantly reduce the material consumption of the structure. Also a strong effect of different materials resistance during stretching and compression is shown on the transverse dimensions of the bars obtained during the design. The possibility of latent destruction is demonstrated, when the limiting state is reached in the inner layers of the beam.

**PNRPU Mechanics Bulletin**. 2017;(3):129-152

ANSYS numerical modeling of electroelastic fields in the piezoelectro luminescent fiber-optical sensor diagnosing the composite volume deformed state

#### Abstract

The paper is concerned with the developed numerical 3D model of the piezoelectro luminescent fibre-optical sensor aiming to diagnose the volume deformed state in a composite structure using ANSYS finite element analysis. In general, the model is a parallelepiped with the sensor fragment located on the central axis. The sensor is the sector-compound layered cylinder made from the central optical fiber with electroluminescent, piezoelectric layers and an external uniform elastic buffer layer. The electroluminescent and piezoelectric layers of the sensor are divided by the radial-longitudinal borders, which are shared by both layers, into the geometrically equal six "measuring elements" in the form of the cylindrical two-layer sectors. The directions of the volume polarization for piezoelectric phases and frequencies of lights for electroluminescent phases are different in various sectors. The piezoelectric phases of all the six sectors are represented by one and the same transversal-isotropic polymeric piezoelectric PVDF material but with different acoplanar directions of volume polarization. The translucent "internal" thin cylindrical operating electrode is located between the optical fiber and electroluminescent layer, and the "external" operating electrode is located between the piezoelectric and buffer layers of the sensor. The properties of the parallelepiped are equated to the transversal-isotropic properties of the unidirectional fibrous fibreglass; various simple monoaxial or shear deformations of the parallelepiped are set through the corresponding displacements of points of his sides. The numerical modeling of non-uniform coupled electroelastic fields in the sensor fragment elements is made. The sensor is implemented in the deformed composite volume of fibrous fibreglass and the operating voltage acts on his electrodes. The numerical values of the informative and operating coefficients of the sensor are calculated; these coefficients are necessary to diagnose the components of deformation tensors on macro- and microlevels of the composite.

**PNRPU Mechanics Bulletin**. 2017;(3):153-166

Modeling of dynamic bending of a rigid-plastic reinforced layered curvilinear plate with a supported circular hole under explosive loads

#### Abstract

A general method is developed to calculate the dynamic behavior of rigid-plastic hybrid laminated composite plates with an arbitrary piecewise-smooth free outer contour and a circular inner simply supported or clamped hole. The plates are under a uniformly distributed, short-term dynamic load of a high intensity explosive type. The plates are hybrid fibrous-laminated with the distribution of layers symmetrically relative to the middle surface. In each layer the reinforcing fibres are placed in radial, circumferential and angular directions. The structural model of the reinforced layer with one-dimensional stress states in the fibres is used. Different schemes of dynamic deformation of the plates are possible. In case when the loads slightly exceed the ultimate values, the plates are deformed in the form of ruled surfaces rotating around the supported contour. At high load amplitudes, the plastic hinge in the form of a circle can be formed in the inner region of the plates. On the basis of the virtual power principle together with the principle of d'Alembert for each of the schemes of motion, the dynamic deformation equations are obtained and the conditions for their implementation are analyzed. Analytical expressions are obtained for the evaluation of limit loads, time of deformation and final deflections of the plates. Numerical examples are given for square plates with a supported circular hole and for annular plates. It is shown that the change in the reinforcement parameters significantly affects both the carrying capacity of plates and the final deflections. The proposed solutions can be used in the design of reinforced metal-composite curvilinear flat structural elements with a supported circular hole.

**PNRPU Mechanics Bulletin**. 2017;(3):167-187

STATE EQUATIONS OF THE DRAPH MODEL OF THREE-DIMENSIONAL ELASTIC SOLIDS IN CARTESIAN COORDINATES

#### Abstract

The paper considers the numerical analysis method of mechanical fields in deformable bodies based on the graph model of an elastic medium in the form of the directed graph. According to the applied method the elastic medium along the coordinate planes is divided into separate elements. In line with this notion we have established an elementary cell configuration, a subgraph of the element by installing hypothetical meters on the solid’s element. The derivation of the cell equations which is based on the element conversion into the cell relies on an invariant. The deformation energy is the invariant. The whole body’s graph is built following the same rule as in the elementary cell. Apart from the opportunity to present the system configuration and mutual connections of its variables, the graphs allow to conduct complex mathematical transformations deliberately in the frameworks of the definite algorithm. The matrices which present several structural elements of the graph and the equations which describe the elementary cells both contribute to deriving the constitutive equations of the intact body. There are matrices that set such structural elements of the graph as cycles, paths and chords. The graph of a whole body is built by following the same rule as in the elementary cell. The method is based on transforming the generalized coordinates of a solid body separated into pieces to a system of generalized coordinates of the initial solid body. The nonsingular and mutually inverse matrices do this transform. The specific nature of the graph model lies in the possibility to construct these matrices with no need for their numerical inversion. The derivation of the defining system of equations is based on the use of vertex and contour of Kirchhoff’s laws for graphs, and the properties of the constructed square transformation matrices. The potentials of the graph method are illustrated by solving a test example.

**PNRPU Mechanics Bulletin**. 2017;(3):188-202

Modeling of elastic-plastic deformation of work material along multielement piecewise zig-zag linear trajectories

#### Abstract

Mathematical simulation of 45 steel elastic-plastic deformation processes which takes place in Ilyushin's vector space and goes along a flat multielement trajectory. Each trajectory consists of four rectilinear piecewise broken segments with its simultaneous stretching or compression with torsion. In the simulation, a mathematical model of the theory of processes for plane trajectories with refined approximations of the process functionalities was used which contains all the necessary parameters for the complex loading for plane trajectories. The basic equations of the mathematical model are reduced to the Cauchy problem, the Runge-Kutta method of the accuracy fourth order was used for the numerical solution and the calculation results. A technique is given for determining the material parameters of approximations in the expressions for the functionals by processing the experimental data on basic trajectories of the "displaced fan" type of two-link trajectories. The experimental test on the SN-EVM testing complex has been carried out in mechanical laboratories in Tver State Technical University. Cylinder thin-walled shells from steel 45 in the condition of delivery were used as physical models for researching on the SN-EVM testing complex. The results of the theoretical numerical calculations are compared with the experimental data to assess the validity of the mathematical models of the processes under complex loading for these types of trajectories of deformation. The offered mathematical model of the theory of processes yields adequate results which are correspond well to the experimental data on material scalar and vector properties. It confirms the reliability of the settlement data which is sufficient for practical issues and accuracy of the constructed approximations of processes’ functionalities related to the used mathematical model of the theory of processes.

**PNRPU Mechanics Bulletin**. 2017;(3):203-215

Finite-element model validation and its execution algorithm

#### Abstract

The paper presents the finite element (FE) model validation. The definition of the term is given as a sequence of operations on a construction and its FE model to get the model reflecting the stress-strain behavior, and/or the dynamic properties of the construction as accurate as only possible. In this study, the validation was illustrated by a FE model of an antenna which has a complex divergent structure. Eigenmodes were selected as responses for FE model validation. Special matrices which are calculated using the FE method program are analyzed to define whether a sensor placement is optimal. Sensors are utilized to get the structural responses during structure tests. This analysis shows dominant eigenmodes that correspond with the deformation of the considerable amount of construction mass, location of the sensors which are placed in the nodes having the maximum values of kinetic energy fractions and the excitation points of all the required structure eigenmodes. According to the previous analysis, the location of the sensors (e.g. accelerometers) and excitation points are chosen. Mathematically, the selection of the sensor positions is accompanied with a reduction of the global FE model matrices to the nodes (degrees of freedom) where the sensors will potentially be placed. The next step is to confirm that the location of the sensors is optimal by means of special correlation matrices. They are employed to compare eigenvectors in the selected nodes of the base and reduced FE models. The certain values of the elements of the correlation matrices are the confirmation that the sensors are located in the optimal way. Further, structural tests are conducted by utilizing the results of the previous analysis. Afterwards, the correlation matrices are calculated again to compare the base finite element model of eigenvectors in the previously chosen nodes (points) with the reduced ones. If the correlation degree of the eigenvectors is high, the base FE model is considered as validated. If the correlation degree of the eigenvectors is low, the base FE model must be updated.

**PNRPU Mechanics Bulletin**. 2017;(3):216-233