# Vol 25, No 3 (2019)

**Year:**2019**Articles:**7**URL:**https://journals.ssau.ru/est/issue/view/412

## Full Issue

## Статьи

### TO THE QUESTION OF FRACTIONAL DIFFERENTIATION. PART II

#### Abstract

In the paper the investigation continues with the help of deﬁnition Fourier fractional diﬀerentiation setting in the previous paper "To the question of fractional diﬀerentiation". There were given explicit expressions of a fairly wide class of periodic functions and for functions represented in the form of wavelet decompositions. It was shown that for the class of exponential functions all derivatives with non-integer exponent are equal to zero. The found derivatives have a direct relationship to practical problems and let them use to solve a large class of problems associated with the study of phenomena such as thermal conduction, transmissions, electrical and magnetic susceptibility for a wide range of materials with fractal dimensions.

**Vestnik of Samara University. Natural Science Series**. 2019;25(3):7-11

### INTEGRO-DIFFERENTIAL EQUATIONS EMBODYING POWERS OF A DIFFERENTIAL OPERATOR

#### Abstract

We establish solvability and correctness criteria for two Fredholm type linear integro-diﬀerential operators *B*2*, B*4 encompassing up to second and fourth powers, respectively, of a diﬀerential operator *A*� with aknown inverse *I *= *A*�−1. We also derive explicit solution formulae to corresponding initial and boundary value problems by using the inverse of the diﬀerential operator. The approach is based on the theory of the extensions of linear operators in Banach spaces. Three example problems for ordinary and partial integro-diﬀerential operators are solved.

**Vestnik of Samara University. Natural Science Series**. 2019;25(3):12-21

### PROBLEMS OF DIFFERENTIAL AND TOPOLOGICAL DIAGNOSTICS. PART II. PROBLEM OF DIFFERENTIAL DIAGNOSTICS

#### Abstract

Proposed work is the second in the cycle, therefore, we present the classiﬁcation of malfunctions and introduce the concept of reference malfunctions that can occur in the control system of the object and in the neighborhoods of these malfunctions. The simplest possible approaches to mathematical modeling of malfunctions and their neighborhoods are formulated, and the problem of nondegeneracy of reference malfunctions is discussed in detail. The concept of diagnostic space is introduced, and its mathematical structure is deﬁned. We also prepare the material for the consideration of the problem of diﬀerential diagnostics.

**Vestnik of Samara University. Natural Science Series**. 2019;25(3):22-32

### THE SOLUTION OF CAUCHY PROBLEM FOR THE HYPERBOLIC DIFFERENTIAL EQUATIONS OF THE FOURTH ORDER BY THE RIMAN METHOD

#### Abstract

In the article the Cauchy problem for the one system of the diﬀerential equations of the fourth order is received in the plane of two independent variables. This system of the hyperbolic diﬀerential equations of the fourth order does not contain derivatives less than the fourth order. The regular solution of the Cauchy problem for the system of the hyperbolic diﬀerential equations of the fourth order is explicitly built. The solution of the Cauchy problem for the system of the hyperbolic diﬀerential equations of the fourth order is found by the Riman method. In the paper the matrix of Riman for the system of the hyperbolic diﬀerential equations of the fourth order is constructed also. The matrix of Riman is expressed through hypergeometrical functions of matrix argument.

**Vestnik of Samara University. Natural Science Series**. 2019;25(3):33-38

### STUDYING THE CRACK DISTRIBUTION BY THE MOLECULAR DYNAMICS METHOD IN A COPPER PLATE

#### Abstract

Using the method of molecular dynamics, the process of crack propagation in a single-crystal copper plate with a crack is simulated under the action of mixed loading corresponding to normal separation and transverse shear. A comprehensive study of the inﬂuence of geometric characteristics (model dimensions, crack length), temperature, strain rate and loading mixing parameter on the plate strength, crack growth and direction was carried out. The angles of propagation of a central crack in a copper plate are determined using the molecular dynamics method.

**Vestnik of Samara University. Natural Science Series**. 2019;25(3):39-61

### DETERMINATION OF THE WILLIAMS SERIES EXPANSION’S COEFFICIENTS USING DIGITAL PHOTOELASTICITY METHOD AND FINITE ELEMENT METHOD

#### Abstract

In the present work, photoelastic and ﬁnite element methods have been employed to study the near crack tip ﬁelds in isotropic linear elastic cracked bodies under mixed mode loading. The investigated fracture results have been obtained for a series of cracked specimens by testing plates with two parallel cracks, two inclined parallel cracks, three-point bend specimens, four-point bend specimens, inclined edge crack triangular shape specimens subjected to symmetric three point bend loading. The multi-parameter Williams series expansion is used for the crack tip ﬁeld characterization. Digital photoelasticity method is utilized for determination of the Williams series expansion’s coeﬃcients. The unknown coeﬃcients in the multi-parameter equation are determined using a linear least squares method in an over-deterministic manner. Together with the experimental determination of the fracture mechanics parameters ﬁnite element method is invoked to describe the crack tip stress ﬁeld. Coeﬃcients of higher order terms are either found numerically by ﬁnite element method. A good agreement is found between the numerical and experimental results. The signiﬁcant advantages using multi-parameter equations in the analysis of the stress ﬁeld are shown and the errors that a study with a limited number of terms produce is demonstrated. The comparison with ﬁnite element analysis highlighted the importance and precision of the photoelastic observation for the evaluation of the fracture mechanics parameters. The experimental SIF, T-stress values and coeﬃcients of higher-order terms estimated using the digital photoelasticity method for the extensive series of cracked specimens are compared with ﬁnite element analysis (FEA) results, and are found to be in good agreement.

**Vestnik of Samara University. Natural Science Series**. 2019;25(3):62-82

### VEERING OF SAFFMAN LIFT FORСE AT FLOW PAST SPHERE WITHOUT SEPARATION

#### Abstract

The article shows the characteristics of a sphere placed in a linear shear ﬂow. The Reynolds number ranges from 0.1 to 10, and the dimensionless velocity gradient is 0.1. The coeﬃcients of drag and lift forces do not depend on the change in the distance to the boundaries of the computational domain and the reduction of cell sizes. The results are obtained in the Ansys Fluent package. For small values of the problem parameters, the solution results have a good agreement with the known results. The results conﬁrm the classical view of the Saﬀman lift force: if the relative velocity is positive, there is a lift force toward the higher velocity of the continuous phase. On the other hand, if the relative velocity is negative the lift force is toward the lower velocity of the continuous phase. Between Reynolds numbers from 4 to 5, the Saﬀman lift force reverses direction. This results for the ﬁrst time conﬁrms McLaughlin assumption about negative Saﬀman lift force.

**Vestnik of Samara University. Natural Science Series**. 2019;25(3):83-92