Vol 25, No 1 (2021)

We study the existence and uniqueness of solution of the non-local problem for the parabolic-hyperbolic type equation with non linear loaded term involving Caputo derivative
\[
f(x) = \begin{cases}
{u_{xx}}-_CD_{0y}^\alpha u+a_1(x)u^{p_1}(x,0), & y > 0,
\\
{u_{xx}}-{u_{yy}}+a_2(x)u^{p_2}(x,0), & y<0,
\\
\end{cases}
\]
where
\[
{}_CD_{0y}^{\alpha }f(y) = \frac{1}{{\Gamma (1-\alpha )}}\int_0^y {(y - t)}^{-\alpha}f'(t)\,dt, \quad 0 < \alpha < 1,
\]
\(a_i(x)\) are given functions, \(p_i\), \(\alpha=\mathrm{const}\), besides \(p_i>0\) \((i=1,2)\), \(0 < \alpha < 1\) in the domain \(\Omega\) bounded with segments:
\[
A_1 A_2 = \{ (x,y): x = 1, 0 < y < h\},\quad B_1 B_2 = \{ (x,y): x = 0, 0 < y < h\},
\]
\[
B_2 A_2 = \{ (x,y): y = h, 0 < x < 1\}
\]
at the \(y > 0\), and characteristics:
\[
A_1C: x - y = 1,\quad B_1C: x + y = 0
\]
of the considered equation at \(y < 0\), where \(A_1 (1, 0)\), \(A_2 (1, h)\), \(B_1( 0, 0)\), \(B_2( 0, h)\), and \(C(1/2, -1/2)\).
Uniqueness of solution of the investigated problem was proved by an integral of energy. The existence of solution of the problem was proved by the method of integral equations. The theory of the second kind Fredholm type integral equations and the successive approximations method were widely used. We notice, that boundary value problems for the mixed type equations of fractional order with non linear loaded term have not been investigated.

We investigate the Tricomi boundary value problem for a differential-difference leading-lagging equation of mixed type with non-Carleman deviations in all in all arguments of the required function. A reduction is applied to a mixed-type equation without deviations. Symmetric pairwise commutative matrices of the coefficients of the equation are used. The theorems of uniqueness and existence are proved. The problem is unambiguously solvable.

The plane stress state of a uniformly piecewise-homogeneous plane obtained by alternately joining two dissimilar strips is considered, which along the lines of joints of dissimilar strips is weakened by a periodic system of two semi-infinite interfacial cracks and is deformed using normal loads applied to the crack banks. The basic cell of the problem in the form of a two-component strip is considered and, using the generalized Fourier transform, a governing system of equations for the problem is obtained in the form of one singular integral equation of the second kind for a complex combination of contact stresses in the junction zone of the strips.

As a special case, tending the height of the strips to infinity, the governing equation of the problem for a two-component plane of two dissimilar half-planes with two semi-infinite interfacial cracks is obtained and its exact solution is constructed. The governing equation for the stated problem is also obtained in the form of one singular integral equation of the first kind with respect to normal contact stresses in another particular case, when all strips are made of the same material, i.e. in the case of a homogeneous plane, a weakened periodic system of parallel, two semi-infinite cracks.

In the general case, the behavior of the unknown function at the end points of the integration interval is determined and the solution of the problem by the numerical-analytical method of mechanical quadratures is reduced to solving a system of algebraic equations. Simple formulas are obtained to determine the intensity factors, the Cherepanov–Rice integral and crack opening. A numerical calculation has been performed. Regularities of changes in contact stresses and the Cherepanov–Rice integral at the endpoints of cracks are revealed, depending on the elastic characteristics of heterogeneous strips and the geometric parameters of the problem.

The paper considers a number of theoretical models for describing longitudinal vibrations of a rod. The most simple and common is based on the wave equation. Next comes the model that takes into account the lateral displacement (Rayleigh correction). Bishop’s model is considered to be more perfect, taking into account both transverse displacement and shear deformation. It would seem that the more perfect the theoretical model, the better it should agree with the experimental data. Nevertheless, when compared with the actually determined experimental spectrum of longitudinal vibrations of the rod on a large base of natural frequencies, it turns out that this is not entirely true. Moreover, the most complex Bishop’s model turns out to be a relative loser. The comparisons were made for a bar with small annular grooves that simulate surface defects, which is considered as a stepped bar. The questions of refinement with the help of experimentally found frequencies of the velocity of longitudinal waves and Poisson's ratio of the rod material are also touched upon.

The article deals with the construction of a mathematical model of the underwater shock wave pulse based on the results of the experiment and numerical and analytical scientific research. The results of the development and comparative analysis of various numerical methods for nonlinear estimation of the parameters of this model are presented. A numerical method is proposed for estimating the pulse energy of a shock wave based on the experimental results in the form of an overpressure waveform both over an infinite period of time and at a given pulse duration. The results of testing the developed numerical methods for mathematical modeling of the underwater shock wave pulse when processing the results of the experiment at the explosion of model charge are presented. The reliability and efficiency of the computational algorithms and numerical methods of nonlinear estimation presented in this paper is confirmed by the results of numerical and analytical studies and mathematical models constructed on the basis of experimental data.

The problem of a pulsed high-energy electromagnetic field action on an edge crack in a thin plate, reproducing the pioneering experiment of Soviet scientists on the destruction of the crack tip by a strong electromagnetic field, is considered. The numerical simulation is based on the proposed electrothermomechanical model of a short-pulse high-energy electromagnetic field (HEMF) action on a material with a crack. The model takes the phase transformations (melting and evaporation) of the material occurring in the vicinity of defects and the corresponding changes in the rheology of the material in the areas of these transformations into account, as well as the possibility of electric current flowing between the free surfaces of the crack (breakdown due to electron emission). All physical and mechanical properties of the material are considered temperature-dependent. The model equations are coupled and solved together on a moving finite element grid using the arbitrary Euler–Lagrangian method. The processes of localization of the current density and temperature fields, phase transformations (melting and evaporation) at the crack tip, autoelectronic and thermoelectronic emissions between free crack surfaces, and the effect of these processes on crack healing are investigated. The simulation results are compared with the available experimental data on the pulse field action on the edge crack in the plate. The average metal heating rate, temperature gradients and time forming of the crater obtained in the vicinity of the crack tip are in good quantitative agreement with the experimental data. Away from the crack, as well as on the crack sides away from the tip, the temperature rises slightly. The process of modeling the electromagnetic field action, similar to the experiment, was accompanied by melting at the crack tip, as well as metal evaporation. Thus, under the considered current action, a crater is formed at the crack tip, which prevents the further spread of the crack, leading to its healing. It was not possible to obtain similar results using the previously proposed models.