## No 1 (2024)

**Year:**2024**Articles:**10**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/397**DOI:**https://doi.org/10.15593/perm.mech/2024.1

MEMORY DEPENDENT RESPONSE IN AN INFINITELY LONG THERMOELASTIC SOLID CIRCULAR CYLINDER

#### Abstract

Memory-dependent derivatives (MDD) have physical meaning, and compared to fractional derivatives, they are more suitable and convenient for temporal remodeling. In this study, the temperature and stress distributions in an infinitely extended generalized thermally elastic solid circular cylinder have been investigated by utilizing the concept of a memory-dependent heat conduction model. The homogeneous, isotropic, infinitely long solid circular cylinder is considered to have a lateral surface that is free of traction and is subjected to a known surrounding temperature. In the domain of the integral Laplace transform, the problem is worked out, and its complex inversion is performed numerically using the Fourier series expansion method. The material properties of copper metal are chosen for the purpose of numerical computation, and the thermoelastic impact of time delay on temperature distribution, displacement distribution, and thermal stresses are represented graphically. Also, time delay's effect on temperature history, displacement history, and thermal heat transfer stress history are shown, respectively. This study could also benefit mathematicians and researchers involved in the development of thermoelasticity, as it accounts for the memory-related derivatives that are useful in explaining the behaviour of a variety of physical processes. The thermal fluctuations captured by various factors with memory-dependent responses are used in engineering applications to realistically design machines or structures.

**PNRPU Mechanics Bulletin**. 2024;(1):5-12

VISCOPLASTIC DEFORMATION OF GRANULAR NICKEL ALLOY AT HIGH TEMPERATURES

#### Abstract

The viscoplastic deformation of a granular nickel alloy during isothermal rolling under high temperature conditions is considered. The stress state of the alloy during rolling is inhomogeneous and multiaxial during repeated deformation with a variable deformation rate. The diagrams of the viscoplastic deformation of the alloy at high temperatures and various deformation rates have a falling (softening) section up to destruction, which is due to a short-term creep during powerful softening. Mathematical modeling of the viscoplastic behavior of the alloy under such conditions is proposed to be carried out on the basis of a variant of the theory of thermoviscoplasticity based on the theory of flow under combined hardening. A variant of the theory of thermoviscoplasticity is generalized to non-isothermal loading and to dependence of the violation process on the deformation rate. The main provisions and equations of the variant of the theory of thermoviscoplasticity are presented. The material parameters that close the version of the theory, the basic experiment and the method of obtaining material parameters are determined. The material parameters of the granular nickel alloy at high temperatures and various deformation rates are obtained. The results of experiments on uniaxial stretching of cylindrical samples made of granular nickel alloy at high temperatures and different deformation rates are presented. Tests with unloading and subsequent loading are also considered. Mathematical modeling of tests of a granular nickel alloy is carried out on the basis of a numerical solution of the Cauchy problem by the Runge-Kutta method of the 4th order of accuracy of the system of equations for a uniaxial stress state under rigid loading obtained on the basis of general equations of a variant of the theory of thermoviscoplasticity. The obtained calculated diagrams of viscoplastic deformation are compared with experimental ones. There is a reliable correspondence between the calculated and experimental results, which indicates the adequacy of the developed version of the theory of thermoviscoplasticity and the method of identification of material parameters.

**PNRPU Mechanics Bulletin**. 2024;(1):13-20

STRESS-STRAIN STATE OF THIN-WALLED PIPE BLANKS DURING CRIMPING IN A CURVED AXISYMMETRIC MATRIX

#### Abstract

The study proposes a computational and analytical model of finding the stress-strain state and force characteristics when crimping thin-walled pipe blanks in a curved axisymmetric matrix. The mathematical model is based on the equilibrium equation of the momentless theory of thin axisymmetric shells taking into account the nonlinear law of plasticity, changes in the wall thickness of the workpiece and contact friction. As a mathematical model of the material, a linear-power approximation of the deformation diagram of an elastic-plastic body is considered taking into account the compressibility of the material. The methodology for estimating the stress-strain state of the workpiece during crimping is constructed using a generalized formulation for an arbitrary curve forming the working contour of the matrix. The basis of the numerical calculation method was the method of variable elasticity parameters, which makes it possible to determine stresses and strains, the thickness distribution in the meridional section, the amount of contact pressure, and also to plot the change in the crimping force depending on the displacement of the point of application of the force relative to the matrix. The distribution of stresses and strains of a thin-walled billet made of aviation aluminum alloy during crimping is calculated using the example of a matrix, the working contour of which is described by a sinusoidal trigonometric function. During crimping, according to the results of the proposed numerical technique, a thickening of the blank wall is noted, as well as an increase in the deforming force when moving the point of application of the force. The proposed mathematical model can be used to calculate the process of crimping workpieces in axisymmetric matrices of complex shapes, with variable curvature, which is especially important in the field of aircraft engineering. The research results make it possible to consider changes in the workpiece’s thickness and physical nonlinearity in the field of plastic strain, when assessing the picture of the stress-strain state.

**PNRPU Mechanics Bulletin**. 2024;(1):21-30

ON IDENTIFICATION OF PARAMETERS OF A NONLINEAR CONSOLIDATION MODEL OF SANDY SOIL

#### Abstract

Previously, the authors of this article formulated a physically and geometrically nonlinear formulation of the problem of porous fluid-saturated medium deformation during fluid filtration (consolidation problem) in terms of the rate of solid phase displacement and the change in pore pressure in differential and variational forms. The developed consolidation model takes into account changes in the porosity and permeability of the medium during deformation. Deformation-type constitutive relations are used in the model. The developed consolidation model can be used to simulate non-stationary processes in the soil, for example, the formation of ruts and unevenness of dirt roads, as well as to calculate the uneven settlement of engineering structures. This work is devoted to the experimental determination of the deformation and strength properties of water-saturated sandy soils, which is the next stage in the consolidation process simulation. The results of the experimental determination of the bulk and shear properties of sandy soil using the ASIS automated complex (OOO NPP “Geotek”) are presented. The studies were carried out on three quartz sands of various grain sizes. To determine the volumetric moduli of dry and water-saturated sandy soils, compression tests were carried out under a continuously growing vertical load at a constant strain rate. The experiments were carried out for various strain rates in the range from 3·10–6 to 3·10–3 s–1. According to the experimental results, the bulk properties do not depend on the strain rate in the specified range. Deformation and strength shear characteristics of sandy soils were determined by the method of multiplanar shear, approximating a simple shear. The tests were carried out under a kinematically applied shear load with a given constant strain rate according to the scheme of consolidated-drained shear. The dependences of the deformation and strength properties of coarse and fine quartz sands on the shear strain rate in the range from 2·10–4 tо 4·10–3 s–1 were studied. Increasing, decreasing and nonmonotonic dependences of the internal friction angle on the shear strain rate were obtained for dry and water-saturated sands of various grain sizes. For water-saturated sands, the maximum spread in the values of the internal friction angle for different strain rates does not exceed 7 %. A technique has been developed for recalculating the obtained properties and experimental dependences into the parameters of the proposed sandy soil consolidation model.

**PNRPU Mechanics Bulletin**. 2024;(1):31-44

VARIATIONAL-DIFFERENCE SOLUTION OF DEFORMATION AND BUCKLING PROBLEMS OF ELASTOPLASTIC SHELLS OF REVOLUTION WITH ELASTIC FILLER UNDER COMBINED QUASI-STATIC AND DYNAMIC AXISYMMETRIC LOADINGS

#### Abstract

The paper suggests a formulation and method for a numerical solution of deformation and buckling of elastoplastic shells of revolution with elastic filler under quasi-static and dynamic loadings. The problem is solved in a two-dimensional plane or generalized axisymmetric formulation with torsion. The governing system of equations is written in a Cartesian or cylindrical coordinate system. Modeling of deformation of an elastic-plastic shell is carried out based on the hypotheses of the theory of shells of the Timoshenko type, taking into account geometric nonlinearities. Kinematic relations are written in velocities and formulated in the metric of the current state. The elastoplastic properties of the shell are described by the flow theory with nonlinear isotropic hardening. Filler modeling is based on continuum mechanics hypotheses. The filler material is assumed to be linearly elastic. The variational equations of motion of structural elements (both shells and filler) are reduced from the three-dimensional equation of the balance of virtual powers of the work of continuum mechanics taking into account the accepted hypotheses of the theory of shells or a flat deformed state or generalized axisymmetric deformation with torsion. The modeling of the contact interaction between the shell and the filler is based on the condition of nonpenetration along the normal and slippage along the tangential. The finite-difference method and an explicit time integration scheme of the cross type are used to solve the defining system of equations. Approbation of the technique was carried out on the problem of buckling of a steel cylindrical shell with an elastic filler under quasi-static and dynamic compression by an external pressure that linearly increases with time. The results of the numerical study are compared with calculations performed using two other approaches developed earlier by the authors. The first approach is based on full-scale modeling of the process of deformation of the shell and filler within the framework of continuum mechanics. In the second approach, a simplified formulation is used, in which the deformation of the shell is modeled according to the hypotheses of the theory of non-sloping shells of the Timoshenko type taking into account geometric nonlinearities, and the filler is modeled according to the Winkler foundation hypothesis. The developed approaches make it possible to model the nonlinear subcritical deformation of shells of revolution with an elastic filler, to determine the ultimate (critical) loads in a wide range of loading rates taking into account geometric shape imperfections, to study buckling in axisymmetric and non-axisymmetric shapes under dynamic and quasi-static combined loadings in plane and axisymmetric deformations.

**PNRPU Mechanics Bulletin**. 2024;(1):45-57

THERMOMECHANICAL COUPLED MODEL OF COATING SYNTHESIS ON A SUBSTRATE

#### Abstract

Among modern combined laser and electron-beam technologies, special attention is paid to those in which composition formation takes place directly in the process of product creation or coating synthesis. In the present work, a coupled model of coating synthesis on a substrate is constructed. When building the model, a sequential transition was carried out from a three-dimensional model of the coating synthesis process on the substrate to a one-dimensional model, which is useful for qualitative analysis. The one-dimensional model takes into account the main physical features of the physical and chemical processes during the synthesis, as well as the coupled nature of the heat transfer and deformation at the same time taking into account the differences in thermophysical and mechanical properties of different materials. When constructing the intermediate analytical solution, it is assumed that the system "substrate-coating" is in a plane stress state. As a result, explicit expressions for the components of stress and strain tensors connected with changes in temperature and composition are obtained. With the help of the obtained analytical solution, the thermokinetic part of the problem is modified and reduced to a more convenient form. Further, the experience accumulated in the field of macrokinetics is used, which allows us to model the processes of creating new materials (e.g., intermetallic or metal matrix composites) in modern technologies from the point of view of controlling the processes of phase formation in the reaction zone by a moving external source. The transition to dimensionless variables revealed complexes and parameters representing relations of characteristic scales of different processes. The parametric study of the model allowed us to establish interesting qualitative effects. It is demonstrated that the quasi-stationary regime is accompanied by physical and chemical processes in the region, which the laser beam had left, due to the heat accumulated in the materials. It is shown that the coupled nature of different processes significantly affects the dynamics of synthesis and the parameters of the quasi-stationary regime.

**PNRPU Mechanics Bulletin**. 2024;(1):58-74

CALCULATION OF STATIC DEFORMATION OF AXISYMMETRIC SHELLS OF ROTATION WITH DIFFERENTIAL MODEL

#### Abstract

In the paper differential equations of static geometrically nonlinear deformation of axisymmetric shell of rotation are obtained. The resolving functions are projections of vectors in the global coordinate system. The equations allow describing any geometry of meridian (breaks, curvature jumps), large deformations, changing of shell thicknesses during deformation, also cross shears characteristic for thick shells. For the numerical solution, the approach based on the finite difference method is applied, which is realized in the own software package for the calculation of the mechanics of spatial rod systems – DARSYS. The calculations of test problems of the internal pressure inflation of cylindrical, spherical, elliptical, conical shells, as well as a combined conicalcylindrical shell with a meridian break are presented. Graphs of convergence of displacements at the reference points as a function of mesh density and under load variation are given, and deformed meridian configurations are plotted. The solutions obtained in ANSYS by different finite elements of Shell type were used as a reference for comparison. APDL scripts for parametric calculations of the test problems are given in the text of the paper. The proposed approach to the calculation of static deformation of shells of rotation has shown good agreement with finite element modeling in ANSYS (including thick shells) and in the future will be extended to the modeling of dynamic deformation and the possibility of solving coupled problems of interaction with liquid or gas. The given equations of the axisymmetric shell are a special case of the general equations, the development and application of which are beyond the scope of this paper, and the obtained solution results are the first stage of testing the developed complex approach to the calculation of static and dynamic deformation of shells, alternative to finite-element modeling.

**PNRPU Mechanics Bulletin**. 2024;(1):75-95

COMPACT ANALYTICAL MODEL FOR ELECTROMAGNETIC DISK-SHAPED UNDERWATER TRANSDUCER

#### Abstract

This paper proposes a mathematical model for a disk-type pulsed electrodynamic transducer operating in the low-frequency range. A well-known architecture of an electromagnetic acoustic radiator with a helical coil and a conducting disk is studied. In the work, the equations of the electromechanical system in the form of the Lagrange-Maxwell equations are built with the use of the Green's functions of a plane axisymmetric acoustic problem to estimate the reaction force of the fluid. A comparison is made between the results of a numerical solution of the obtained equations and direct numerical calculation in the COMSOL finite element analysis software. The resulting model shows good qualitative agreement with the results of finite element calculations while allowing calculations with a variation for all main model parameters required to design the transducer.

**PNRPU Mechanics Bulletin**. 2024;(1):96-104

MODELING OF HIGH-RATE HARDENING OF A POLYMER COMPOSITE MATERIAL UNDER LOADING ALONG THE REINFORCEMENT DIRECTION

#### Abstract

Modeling the high-rate deformation of composite structures is of great interest in the industry. Moreover, some processes such as accidents, explosions and possible impact issues require analysis of composite materials at significantly high deformation rates. The paper considers the possibility of developing a model of deformation of a composite material based on a polymer matrix and carbon fiber taking into account high-rate hardening. A feature of the study is the development of a model that takes into account a wide range of deformation rates from static to several thousand reverse seconds. Thus, tests were carried out with special equipment and samples that allow us to obtain data with such high loading speeds. The model is based on an approach considering the use of damage parameters, the so-called class of models with progressive degradation. The main innovative part of the chosen model is the formalization of the rate of deformation on the material through the damage parameter, that is, the rate of change in damage values is considered. This approach makes it possible to make constitutive relations based only on the damage parameters, which modify the stiffness and strength characteristics of composites, which greatly simplifies the modeling and analysis of material deformation.

**PNRPU Mechanics Bulletin**. 2024;(1):105-111

FLOW CURVES AND STRESS-STRAIN CURVES GENERATED BY A NONLINEAR MODEL FOR SHEAR FLOW OF THIXOTROPIC VISCOELASTIC MEDIA ACCOUNTING FOR STRUCTURE EVOLUTION

#### Abstract

A systematic analytical study of the mathematical properties of the nonlinear shear flow model of thixotropic viscoelastic-plastic media is continued. It takes into account the mutual influence of а deformation process and structure evolution (the kinetics of the formation and destruction of intermolecular bonds and associates of macromolecules). The model is reduced to the system of two nonlinear differential equations for the dimensionless stress and the degree of structuredness (i.e. cross-links density and so on). Assuming six material parameters and an (increasing) material function governing the model are arbitrary, the phase portrait of the system is analytically studied in the vicinity of its single equilibrium point. Basic properties of flow curves and stress-strain curves with constant shear rate generated by the model are examined. Thus, the analysis of the model ability to describe the behavior of both liquid-like media and solid-like (thickening, hardening, solidifying) viscoelastic-plastic media has been started: the effects of strain-rate and strain hardening, relaxation, creep, recovery, etc. The stress-strain curves dependence on the shear deformation (monotonicity, convexity, instantaneous modulus, tangent modulus evolution), on the shear rate and initial structuredness and on the material parameters and function of the model (in particular, the parameters that control the effect of structuredness on viscosity and shear modulus and the influence stress on the rate of destruction of the structure) has been studied. It is proved that stress-strain curves can be both increasing and have sections of decrease, resembling a yield-drop, and damped oscillations; that all stress-strain curves have horizontal asymptotes (steady flow stress), monotonically dependent on shear rate, and the flow stress strictly increases with increasing shear rate; that their instantaneous shear modulus, on the contrary, depends on the initial structuredness, but does not depend on shear rate. Under certain restrictions on the material parameters, the model is also capable to provide a bilinear form of stress-strain curves, which is intrinsic for an ideal elastoplastic model, but with strain rate sensitivity. It has been established that the family of stress-strain curves does not have to be increasing function of initial structuredness or shear rate: in a certain range of shear rates, in which the equilibrium point is a “mature” focus and pronounced oscillations of stress-strain curves are observed, it is possible that stress-strain curves with different shear rates may interweave with each other. It is studied how structuredness changes in the process of deformation depending on shear rate, stress, material parameters and material function of the model. The initial structuredness affects only the initial arc of stress-strain curves, but does not affect their asymptotes and the steady value of the structuredness, which monotonically decreases with increasing shear rate. A variety of scenarios of structuredness behavior over time (in particular, the observed sharp collapse of the structuredness when critical stress values are reached) generates a number of unusual effects (unusual properties) in comparison with typical properties stress-strain curves of structurally stable materials.

**PNRPU Mechanics Bulletin**. 2024;(1):112-143