# Vol 24, No 3 (2020)

## Full Issue

### On a micropolar theory of growing solids

#### Abstract

The present paper is devoted to the problem of boundary conditions formulation in the growing micropolar solid mechanics. The static equations of the micropolar continuum in terms of relative tensors (pseudotensors) are derived due to virtual work principle for a solid of constant staff. The constitutive quadratic form of the elastic potential (treated as an absolute scalar) for a linear hemitropic micropolar solid is presented and discussed. The constitutive equations for symmetric and antisymmetric parts of force and couple stress tensors are given. The final forms of the static equations for the hemitropic micropolar continuum in terms of displacements and microrotations rates are obtained including the case of growing processes. A transformation of the equilibrium equations is proposed to obtain boundary conditions on the propagating growing surface in terms of relative tensors in the form of differential constraints. Those are valid for a wide range of materials and metamaterials. The algebra of rational relative invariants is intensively used for deriving the constitutive relations on the growing surface. Systems of joint algebraic rational relative invariants for force, couple stress tensors and also unit normal and tangent vectors to propagating growing surface are obtained, including systems of invariants sensitive to mirror reflections and 3D-space inversions.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):424-444
424-444

### The steady-state creep of long membrane in a rigid matrix at a variable transverse pressure

Lokoshchenko A.M., Teraud W.V., Shevarova E.A.

#### Abstract

The problem of the steady-state creep of a long rectangular membrane in constrained conditions inside a rigid matrix is investigated with a piecewise constant dependence of the transverse pressure $q$ on time $t$. The problem considers a long matrix of rectangular cross section, in which the ratio of its height to width is not less than 0.5. As an example, the creep of the membrane is investigated with a single change in the magnitude of the transverse pressure over time. Three variants of the contact conditions of the membrane and the matrix are considered: perfect sliding, adhesion and sliding taking friction into account. In this paper, four stages of membrane deformation were investigated. At the first stage (elastic deformation), the membrane, flat in the initial state, under the action of pressure, instantaneously is deformed elastically, acquiring the form of an open circular cylindrical shell with a central angle $2\alpha _1$. At the second stage, the membrane is deformed under steady-state creep conditions up to the moment when the side walls of the matrix touch. The third stage ends when the membrane touches the transverse wall of the matrix. In the fourth stage, the membrane is in contact with the matrix on the transverse and lateral sides. The analysis is carried out until the time of almost complete adherence of the membrane to the matrix, at which the ratio of the radius of the membrane near the corners of the matrix to the initial width of the membrane is 0.005. For the third and fourth stages, the friction force of the membrane on the matrix walls is additionally taken into account. The dependences of the thickness of various parts of the membrane on time and on the intensity of stresses in the membrane on time are obtained. In relation to this formulation of the problem, deviations from the rule of summing the partial times of filling the matrix are considered.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):445-468
445-468

### Properties of the strain rate sensitivity function produced by the linear viscoelasticity theory and existence of its maximum with respect to strain and strain rate

Khokhlov A.V.

#### Abstract

Strain rate sensitivity of stress-strain curves family generated by the Boltzmann–Volterra linear viscoelasticity constitutive equation (with an arbitrary relaxation modulus) under uni-axial loadings at constant strain rates is studied analytically as the function of strain and strain rate. The general expression for strain rate sensitivity index is derived and analyzed assuming relaxation modulus being arbitrary. Dependence of the strain rate sensitivity index on strain and strain rate and on relaxation modulus qualitative characteristics is examined, conditions for its monotonicity and for existence of extrema, the lower and the upper bounds and the limit values of the strain rate sensitivity as strain rate tends to zero or to infinity are studied. It is found out that (within the framework of the linear viscoelasticity) the strain rate sensitivity index which is, generally speaking, the function of two independent variables (namely strain and strain rate), depends on the single argument only that is the ratio of strain to strain rate. So defined function of one real variable is termed the strain rate sensitivity function and it may be regarded as a material function. The explicit integral expression (and the two-sided bound) for relaxation modulus in terms of strain rate sensitivity function is derived which enables one to restore relaxation modulus assuming a strain rate sensitivity function is given. The strain rate sensitivity function is represented as a linear function of ratio of tangent modulus to secant modulus of a stress-strain curve at any fixed constant strain rate and can be evaluated in such a way using experimental data. It is proved that the strain rate sensitivity value is confined in the interval from zero to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) whatever strain and strain rate magnitudes. It is found out that the linear theory can reproduce increasing or decreasing or non-monotone dependences of strain rate sensitivity on strain rate (for any fixed strain) and it can provide existence of local maximum or minimum or several extrema as well without any complex restrictions on the relaxation modulus. General properties and peculiarities of the theoretic strain rate sensitivity function are illustrated by the examination of the classical regular and singular rheological models (consisting of two, three or four spring and dashpot elements) and fractional models. Namely, the Maxwell, Kelvin–Voigt, standard linear solid, Zener, anti-Zener, Burgers, anti-Burgers, Scott–Blair, fractional Kelvin–Voigt models and their parallel connections are considered. The carried out analysis let us to conclude that the linear viscoelasticity theory (supplied with common relaxation function which are non-exotic from any point of view) is able to produce high values of strain rate sensitivity index close to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) and to provide existence of the strain rate sensitivity index maximum with respect to strain rate. Thus, it is able to simulate qualitatively existence of a flexure point on log-log graph of stress dependence on strain rate and its sigmoid shape which is one of the most distinctive features of superplastic deformation regime observed in numerous materials tests.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):469-505
469-505

### Modeling of viscoelastoplastic deformation of flexible shallow shells with spatial-reinforcements structures

Yankovskii A.P.

#### Abstract

Based on the procedure of time steps, a mathematical model of the viscoelastoplastic behavior of shallow shells with spatial reinforcement structures is constructed. Plastic deformation of the components of the composition is described by flow theory with isotropic hardening; viscoelastic deformation by the equations of the Maxwell–Boltzmann model. The possible weakened resistance of composite curved panels to transverse shear is taken into account in the framework of the hypotheses of Reddy's theory, and the geometric nonlinearity of the problem is taken into account in the Karman approximation. The solution of the formulated initial-boundary value problem is constructed using an explicit numerical scheme of the “cross” type. The elastoplastic and viscoelastoplastic flexural dynamic behavior of “flat” and spatially reinforced fiberglass cylindrical panels under the action of explosive loads has been investigated. Using the example of relatively thin composite structures, it is shown that, depending on which of the front surface (convex or concave), a load is applied, replacing the traditional “flat” reinforcement structure with a spatial one can lead to both an increase and a decrease in the residual deflection. However, in both cases, such a replacement can significantly reduce the intensity of residual deformations of the binder material and fibers of some families. It was demonstrated that the amplitudes of oscillations of curved composite panels in the neighborhood of the initial moment of time significantly exceed the maximum absolute values of the residual deflections. In this case, the residual deflections are rather complicated. It is shown that the calculations carried out within the framework of the elastoplastic deformation theory of the composition components do not even allow an approximate the magnitude determination of the residual deformations of the materials making up the composition.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):506-527
506-527

### Convective layered flows of a vertically whirling viscous incompressible fluid. Temperature field investigation

Burmasheva N.V., Prosviryakov E.Y.

#### Abstract

The paper discusses a class of exact solutions of the Oberbeck–Boussinesq equations suitable for describing three-dimensional nonlinear layered flows of a vertically swirling viscous incompressible fluid. An inhomogeneous distribution of the velocity field (there is a dependence of the field components on the horizontal coordinates) generates a vertical swirl in the fluid without external rotation (excluding Coriolis acceleration). Setting the linearly distributed heat field and the field of shear stresses at the boundaries of the flow region is one of the reasons inducing convection in a viscous incompressible fluid. The main attention is paid to the study of the properties of the temperature field. The effect of vertical twist on the distribution of isolines of this field is studied. It is shown that the homogeneous component of the temperature field can be stratified into several zones relative to the reference value, and the number of such zones does not exceed nine. The inclusion of inhomogeneous components of the temperature field can only decrease this number. It is also demonstrated that the class discussed in the paper allows one to generalize the previously obtained results on modeling convective flows of viscous incompressible fluids.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):528-541
528-541

### A divergence-free method of collocations and least squares for the computation of incompressible fluid flows and its efficient implementation

Vorozhtsov E.V., Shapeev V.P.

#### Abstract

The problem of the acceleration of the iterative process of numerical solution by the collocation and least squares (CLS) method of boundary value problems for partial differential equations is considered. For its solution, it is proposed to apply simultaneously three ways to accelerate the iterative process: preconditioner, multigrid algorithm, and Krylov method. A method for finding the optimal values of the parameters of the two-parameter preconditioner is proposed. The use of the found preconditioner significantly accelerates the iterative process. The influence on the iterative process of all three ways of its acceleration is investigated: each separately, and also at their combined application. The application of the algorithm using Krylov subspaces gives the greatest contribution. The combined use of all three ways to speed up the iteration process of solving boundary value problems for two-dimensional Navier-Stokes equations has reduced the CPU time up to 362 times as compared with the case when only one of them, the preconditioner, was applied.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):542-573
542-573

### Well-posedness of a mixed type problem for the multidimensional hyperbolic-parabolic equation

Aldashev S.A.

#### Abstract

We consider the modeling multidimensional hyperbolic-parabolic equation in the cylindrical area of Euclidean space and formulate the mixed problem with non-homogeneous boundary conditions for it. We show the unique solvability of the problem for the class of continuously differentiable functions and give a way to construct its explicit classical solution.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):574-582
574-582

### Dynamic thermal stability of heated geometrically irregular cylindrical shell under the influence of a periodic temporal coordinate load

Belostochny G.N., Myltcina O.A.

#### Abstract

In the framework of a Love type model, a geometrically irregular isotropic shallow cylindrical shell is considered, based on a strict continuum-shell-rib model. It is assumed that the geometrically irregular shell is heated to a constant temperature $\theta_0$, two opposite edges are exposed to a tangential load periodic in time coordinate, the amplitude and frequency of which are known ($p(t)=p_0 \cos \vartheta t$). The problem of determining the regions of dynamic instability of a thermoelastic system is reduced to considering a singular system of three differential equations of dynamic thermal stability of a geometrically irregular shell in displacements containing a term with tangential forces in the Brian form. These forces arising in the shell during its heating are preliminarily determined on the basis of closed solutions of the singular system of differential equations of the momentless thermoelasticity of the geometrically irregular shell. The specific initialized system of equations is transformed to the Mathieu equations, which are written in terms of the classical athermal theory of smooth plates containing corrections for geometric parameters — curvature, relative height of the reinforcing elements, their number, and temperature. The first three regions of dynamic instability of a geometrically irregular shell are determined. A quantitative analysis of the influence of the geometric parameters of the elastic system and temperature on the configuration of the regions of dynamic instability and the magnitude of the excitation coefficient is carried out.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):583-594
583-594

### On the solution of one problem of deformation of rod systems that does not satisfy the Hadamard conditions by the simple iteration method

Struzhanov V.V., Korkin A.V.

#### Abstract

A rod system under the action of a quasi-statically increasing tensile tension is considered. The load is carried out according to soft and hard schemes. One of the rods of the system has the property of deformation softening, that is, its tension diagram has a branch falling to zero. As a result, the equilibrium equations do not satisfy the Hadamard conditions. The system has several equilibrium positions, including unstable ones. The application of the simple iterations method is shown to determine the parameters of all possible equilibrium positions and their stability when solving these equations that do not satisfy the Hadamard conditions.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):595-603
595-603

### A problem with dynamical boundary condition for a one-dimensional hyperbolic equation

Beylin A.B., Pulkina L.S.

#### Abstract

In this paper, we consider a problem with dynamical boundary conditions for a hyperbolic equation. The dynamical boundary condition is a convenient method to take into account the presence of certain damper when fixing the end of a string or a beam.Problems with dynamical boundary conditions containing first-order derivatives with respect to both space and time variables are not self-ajoint, that complicates solution by spectral analysis.However, these difficulties can be overcome by a method proposed in the paper. The main tool to prove the existence of the unique weak solution to the problem is the priori estimatesin Sobolev spaces. As a particular example of the wave equation is considered.The exact solution of a problem with dynamical condition is obtained.
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2020;24(3):407-423
407-423