## Vol 20, No 2 (2016)

**Year:**2016**Articles:**12**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1220

Articles

Professor Dyuis D. Ivlev. Dedication to 85th Birtday

#### Abstract

Dyuis D. Ivlev (1930-2013) is an outstanding scientist in the fields of Continuum Mechanics (theory of Perfect Plasticity and Fracture) and AppliedMathematics. He has much contributed to the mathematical theory of plasticity, especially to study of hyperbolic three-dimensional problems of theperfect plasticity. Dyuis D. Ivlev was born in Chuvashia Republick, Russia, on September 6, 1930. In 1948 he left Chuvashia and after passing examinations entered Moscow State University. He is a Mechanical Engineering graduate(1953) of Moscow State University. In 1953 he continued his research workas a post graduate student of the same university. In 1956 he received PhDin Solid Mechanics from Moscow State University. The title of his PhD dissertation work is Approximate Solution of Elasti-Plastic Problems by thesmall parameter method. Three years later he was awarded DSc (Phys. &Math.) Degree from Moscow State University for his dissertation study Three-Dimensional Problem of the Theory of Perfect Plasticity. Since 1959he has been working as head of the Department of Elasticity and Plasticityof Voronezh State University, then (1966-1970) as Prof. of Bauman StateTechnical University and (1971-1982) as head of the Department of Higher Mathematics of Russian Polytechnical University. In 1982 he returned toChuvashia working in Chuvash State University (until 1993) and ChuvashState Pedagogical University (1993-2013) as head of Department of Mathematical Analysis. Prof. Dyuis D. Ivlev has been a member of National Committee on Theoretical and Applied Mechanics, Scientific Council on Problems of SolidMechanics, Mathematics and Mechanics Expert Council of the Higher Attestation Committee. He is the author of several books on theory of perfectplasticity and its applications and nearly 250 papers on the subject.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):197-219

A non-local problem for a loaded mixed-type equation with a integral operator

#### Abstract

We study the existence and uniqueness of the solution of non-local boundary value problem for the loaded elliptic-hyperbolic equation $$ u_{xx} + \mathop{\mathrm{sgn}} (y) u_{yy} + \frac{1 - \mathop{\mathrm{sgn}} (y)}{2} \sum\limits_{k = 1}^n {R_k}(x, u(x, 0)) = 0 $$ with integral operator $$ {R_k}\bigl(x, u(x, 0)\bigr) = \left\{ \begin{array}{lc} {p_k}(x)D_{x\,\,1}^{ - {\alpha _k}}u(x, 0), & q \le x \le 1,\\[2mm] {r_k}(x)D_{ - 1\,x}^{ - {\beta _k}}u(x, 0), & - 1 \le x \le - q, \end{array} \right. $$ where $$ \begin{array}{l} \displaystyle D_{ax}^{ - {\alpha _k}}f(x) = \frac{1}{{\Gamma ({\alpha _k})}} \int _a^x \frac{f(t)}{(x - t)^{1-{\alpha _k} }}dt, \\ \displaystyle D_{xb}^{ - {\beta _k}}f(x) = \frac{1}{{\Gamma ({\beta _k})}} \int _x^b \frac{f(t)}{(t - x)^{1-{\beta _k}}}dt , \end{array} $$ in double-connected domain $\Omega $, bounded with two lines: $$ \sigma _1:~x^2 + y^2 = 1,\quad \sigma _2:~ x^2 + y^2 = q^2 \quad \text{at $y > 0$,}$$ and characteristics: $$ A_j C_1:~ x + ( - 1)^j y = ( - 1)^{j + 1},\quad B_j C_2:~x + ( - 1)^j y = ( - 1)^{j + 1} \cdot q$$ of the considered equation at $y < 0$, where $0 < q < 1$, $j = 1, 2$; $A_1 ( 1; 0),$ $A_2( - 1; 0)$, $B_1(q; 0)$, $B_2( - q; 0)$, $C_1(0; - 1)$, $C_2(0; - q)$, $\beta _k$, $\alpha _k > 0$. Uniqueness of the solution of investigated problem was proved by an extremum principle for the mixed type equations. Thus we need to prove that, the loaded part of the equation is identically equal to zero if considerate problem is homogeneous. Existence of the solution of the problem was proved by a method of the integral equations, thus the theory of the singular integral equations and Fredholm integral equations of the second kind were widely used.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):220-240

The Cauchy problem for a general hyperbolic differential equation of the n-th order with the nonmultiple characteristics

#### Abstract

In the paper the problem of Cauchy is considered for the hyperbolic differential equation of the n-th order with the nonmultiple characteristics. The Cauchy problem is considered for the hyperbolic differential equation of the third order with the nonmultiple characteristics for example. The analogue of D'Alembert formula is obtained as a solution of the Cauchy problem for the hyperbolic differential equation of the third order with the nonmultiple characteristics. The regular solution of the Cauchy problem for the hyperbolic differential equation of the forth order with the nonmultiple characteristics is constructed in an explicit form. The regular solution of the Cauchy problem for the $n$-th order hyperbolic differential equation with the nonmultiple characteristics is constructed in an explicit form. The analogue of D'Alembert formula is obtained as a solution of this problem also. The existence and uniqueness theorem for the regular solution of the Cauchy problem for the $n$-th order hyperbolic differential equation with the nonmultiple characteristics is formulated as the result of the research.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):241-248

A problem on longitudinal vibration of a bar with elastic fixing

#### Abstract

In this paper, we study longitudinal vibration in a thick short bar fixed by point forces and springs. For mathematical model we consider a boundary value problem with dynamical boundary conditions for a forth order partial differential equation. The choice of this model depends on a necessity to take into account the result of a transverse strain. It was shown by Rayleigh that neglect of a transverse strain leads to an error. This is confirmed by modern nonlocal theory of vibration. We prove existence of orthogonal with load eigenfunctions and derive representation of them. Established properties of eigenfunctions make possible using the separation of variables method and finding a unique solution of the problem.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):249-258

On one nonlocal problem for the Euler-Darboux equation

#### Abstract

The boundary value problem with displacement is determined for the generalized Euler-Darboux equation in the field representing the first quadrant. This problem, unlike previous productions, specifies two conditions, connect integrals and fractional derivatives from the values of the sought solution in the boundary points. On the line of singularity of the coefficients of the equations the matching conditions continuous with respect to the solution and its normal derivation are considered. The authors took for the basis of solving the earlier obtained by themselves the Cauchy problem solution of the special class due to the integral representations of one of the specified functions acquired simple form both for positive and for negative values of Euler-Darboux equation parameter. The nonlocal problem set by the authors is reduced to the system of Volterra integral equations with unpacked operators, the only solution which is given explicitly in the corresponding class of functions. From the above the uniqueness of the solution of nonlocal problem follows. The existence is proved by the direct verification. This reasoning allowed us to obtain the solution of nonlocal problem in the explicit form both for the positive and for the negative values of Euler-Darboux equation parameter.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):259-275

A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation

#### Abstract

In this paper, we consider a problem for a one-dimensional hyperbolic equation with nonlocal integral condition of the second kind. Uniqueness and existence of a generalized solution are proved. In order to prove this statement we suggest a new approach. The main idea of it is that given nonlocal integral condition is equivalent with a different condition, nonlocal as well but this new condition enables us to derive a priori estimates of a required solution in Sobolev space. By means of derived estimates we show that a sequence of approximate solutions constructed by Galerkin procedure is bounded in Sobolev space. This fact implies the existence of weakly convergent subsequence. Finally, we show that the limit of extracted subsequence is the required solution to the problem.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):276-289

Experimental research of residual stresses kinetics in the hardened hollow cylindrical specimens of D16T alloy at the axial tension under the creep conditions

#### Abstract

We study experimentally the effect of the axial tension load on the residual stresses relaxation in the surface-hardened hollow cylindrical specimens of D16T aluminium alloy at a temperature of 125 ℃. The surface is hardened by the air shot-peening. We describe the testing machine and the routine of experiment. The experimental curves of hardened specimens creep under the axial loads 353, 385, 406.2, 420 MPa and test duration of 100-160 hours are obtained. The axial and circumferential residual stresses after the hardening and the creep at the given temperature and load conditions are constructed by the method of circles and strips. The significant qualitative and quantitative changes of residual stresses take place under the tension load $\bar \sigma$ in comparison with the thermal exposure (heat exposal with no load). The relaxation of residual stresses is essentially independent of the thermal exposure. In contrast, the loading leads to the significant residual stresses relaxation and to the changes in the distribution type. The axial and circumferential residual stresses evolve from the compressive to the tension with the increase of the axial tension load. Also the depth of residual stresses location changes with the increase of the axial tension load from the 600 microns in the original state after the air shot-peening to the 250-300 microns after the creep under the given loading. It is very important for the engineering applications to take into account the described behaviours of the residual stresses in the hardened specimens of D16T alloy when predicting the characteristics of endurance of the surface-hardened details operate under the elevated temperatures.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):290-305

A nonlinear boundary integral equations method for the solving of quasistatic elastic contact problem with Coulomb friction

#### Abstract

Three-dimensional quasistatic contact problem of two linearly elastic bodies’ interaction with Coulomb friction taken into account is considered. The boundary conditions of the problem have been simplified by the modification of the Coulomb’s law of friction. This modification is based on the introducing of a delay in normal contact tractions that bound tangent contact tractions in the Coulomb’s law of friction expressions. At this statement the problem is reduced to a sequence of similar systems of nonlinear integral equations describing bodies’ interaction at each step of loading. A method for an approximate solution of the integral equations system corresponded to each step of loading is applied. This method consists of system regularization, discretization of regularized system and iterative process application for solving the discretized system. A numerical solution of a contact problem of an elastic sphere with an elastic half-space interaction under increasing and subsequently decreasing normal compressive force has been obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):306-327

A numerical method for the determination of parameters of the strain softening creep model

#### Abstract

The trends of decreasing of weight of machines and increasing of their quality, and intention to the fullest use of mechanic properties of materials demand the development of numerical methods for analysis of the stress-strain state of materials undo the terms of creep. The article discusses the development of new numerical method for determining the parameters of the strain softening creep model. The base of new method is generalized regression model, which was built on the basis of difference equations for describing the creep. The relations between coefficients of difference equation and parameters of the strain softening creep model allow reduce the problem of parametric identification to an iterative procedure for RMS of coefficients of regression model, which is linear. The approbation of numerical method with five creep curves of aluminum alloy is accomplished. The approbation confirms scientific credibility of built relations and efficiency of new numerical method.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):328-341

On one method for solving transient heat conduction problems with asymmetric boundary conditions

#### Abstract

Using additional boundary conditions and additional required function in integral method of heat-transfer we obtain approximate analytical solution of transient heat conduction problem for an infinite plate with asymmetric boundary conditions of the first kind. This solution has a simple form of trigonometric polynomial with coefficients exponentially stabilizing in time. With the increase in the count of terms of a polynomial the obtained solution is approaching the exact solution. The introduction of a time-dependent additional required function, setting in the one (point) of the boundary points, allows to reduce solving of differential equation in partial derivatives to integration of ordinary differential equation. The additional boundary conditions are found in the form that the required solution would implement the additional boundary conditions and that implementation would be equivalent to executing the original differential equation in boundary points. In this article it is noted that the execution of the original equation at the boundaries of the area only (via the implementation of the additional boundary conditions) leads to the execution of the original equation also inside that area. The absence of direct integration of the original equation on the spatial variable allows to apply this method to solving the nonlinear boundary value problems with variable initial conditions and variable physical properties of the environment, etc.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):342-353

Numerical integration of the boundary value problems for the second order nonlinear ordinary differential equations of an arbitrary structure using an iterative procedure

#### Abstract

An iterative procedure for numerical integration of boundary-value problems for nonlinear ordinary differential equations of the second order of arbitrary structure is suggested. The initial differential equation by algebraic transformation can be written as a linear inhomogeneous differential equation of the second order with constant coefficients; the right part of which is represented as a linear combination of the derivatives of the required function up to the second order and a differential equation of arbitrary structure under study. Taylor polynomials were used in the construction of the difference boundary value problem. This allowed to abandon the approximation of derivatives by finite differences. The degree of Taylor polynomials can be chosen as any natural number greater than or equal to two. Obtained inhomogeneous linear differential equation has three arbitrary coefficients. It is shown that the coefficient at the initial differential equations of any structure on the right side of the obtained non-homogeneous linear differential equation is associated with the convergence of the iterative procedure; and the coefficients at the derivatives of the required function affect the stability of difference boundary value problem at each iteration. The values of coefficients at the derivatives of the required function which ensure the stability of difference boundary value problem regardless of the type of the initial equation are theoretically set up. Numerical experiment showed that the coefficient providing the convergence of the iterative procedure depends on the type of the initial differential equation. Numerical experiments showed that the increase in the degree of the Taylor polynomial reduces the error between the exact and the obtained approximate solutions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):354-365

On the wave dynamics in damaged shells interacting with the volume of the cavitating liquid

#### Abstract

We study the details of shock front propagation in the system of deformable medium (shells) with damages and two-phase liquid with gas or steam bubbles. We develop the models for the nonlinear processes of media interacting taking into account the phase transformations in liquid and the damaging kinetics of deformable medium. The destruction of deformable medium is considered as the evolution of microdamages or spherical pores, taking as the gas bubbles similarly with the cavitating liquid. The aggregation of the bubbles at the viscoplastic flow cases the macrofracture forming. We formulate the nonlinear boundary value problem of the multiphase medium dynamics, that includes the equations of the phase interaction and phase transformations. The solution of the problem is based on the decomposition method (an expansion in the processes), finite difference method and finite element method. The results presented are of interest for the practical applications.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(2):366-386