Vol 20, No 2 (2016)

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.

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.

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.

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.

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.

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.