and Computational Methods

doi: 10.18698/2309-3684-2020-2-95106

The paper considers the problem of constructing a network of equidistant surveys of coastal waters during the design of marine terminals, as well as for landing amphibious assaults on the coast. A computer algorithm is proposed for constructing equidistant complex irregular curves on the ground. The algorithm is implemented as a computer program. The program is tested on an example of a curve of a real coastal strip.

Валишин А.А., Туманов И.А., Ахунд-заде М.Р. Компьютерное построение сети эквидистант сложных негладких кривых на местности. Математическое моделирование и численные методы. 2020. № 2. с. 95–106.

doi: 10.18698/2309-3684-2018-3-321

The purpose of this study was to evaluate the influence of the inertial effects and their deviation from the same quasi-static results. The role of inertial effects in the problem of thermal shock is studied on the example of a massive body with an internal spherical crack. We study the thermal reaction of an elastic space with an internal spherical crack whose surface, initially stress-free and at a temperature of T0, is instantly heated to a temperature of TC > T0 and then maintained at that temperature. Thermal stress state occurs under different modes of heat exposure, creating heat stroke. The most common in practice, three cases: temperature heating, thermal heating and heating medium. The generalized dynamic thermoelasticity equation for all three cases in rectangular and curvilinear coordinates is obtained. Considered the thermal response of a massive rigid body with internal spiroborate crack. The exact analytical solution of the problem is obtained. Earlier in the works of one of the authors the solution of the dynamic problem in the form of bulky functional structures was obtained, which greatly complicated their practical use. In this paper, we propose a solution to the problem in new classes of functions, which makes the solution more convenient for numerical experiments. A generalized differential relation for dynamic thermoelasticity is proposed, which has an extensive field of practical applications in the study of thermal response to heat stroke of solids of different shapes. It is shown that the component of the radial stress is a spherical elastic wave propagating from the cavity surface into the material. Numerical calculations of dynamic effects are performed and it is shown that the quasi-static interpretation of time problems in the theory of heat stroke does not allow to take into account the basic laws of transient thermoelasticity and inertial effects.

Валишин А.А., Карташов Э.М. Математическое моделирование термических напряжений в твердом теле с внутренней трещиной. Математическое моделирование и численные методы, 2018, № 3, с. 3–21.

doi: 10.18698/2309-3684-2019-3-318

Deformation of solids under the action of non-stationary external mechanical, thermal or other effects is accompanied by the inverse thermodynamic effect of the release of additional heat due to internal friction, i.e.a change in the temperature field. This causes additional deformation, which in turn leads to the release of heat. This effect of the interaction of mechanical and temperature fields is called the connectivity effect. The consequence of this effect is the appearance of heat fluxes leading to an increase in the entropy of the thermodynamic system and thermoelastic energy dissipation. The purpose of the work is to study the influence of the interaction of deformation and temperature fields for different materials. For “classical” materials, such as metals and glass, the thermodynamic effect of the interaction of deformation and temperature fields is insignificant and it is usually neglected in the calculation, design and operation of structures. For some polymer materials such as various polyvinylacetals, this effect is significant; it must be taken into account when creating composite materials on their basis and when designing products and structures of them. A dynamic coupled problem of thermoelasticity for an elastic layer of various structural, consumer and construction materials under rapid application of a normal compressive load to thermally insulated surfaces is considered. It is shown that for glass and steel temperature increasing due to the interaction of the deformation and temperature fields being really negligible is 0.180–0.183 K (or 0.061–0.062 %). For polymers, first of all, from the class of polyvinylacetals, it is substantial, and it can no longer be neglected.

Валишин А.А., Карташов Э.М. Моделирование эффектов связанности в задаче об импульсном нагружении термоупругих сред. Математическое моделирование и численные методы, 2019, № 3, с. 3–18.

doi: 10.18698/2309-3684-2022-1-4262

When designing products made of composite materials intended for use in difficult conditions of inhomogeneous deformations and temperature, it is important to take into account viscoelastic, including spectral and dynamic, properties of the binder and fillers. The article considers dynamic characteristics (complex modulus, complex malleability,their real and imaginary parts, loss angle tangent) and spectral characteristics of relaxation and creep and their dependence on each other. The characteristics mentioned above were found for all known types of creep kernel and relaxation kernel. To find the spectral characteristics, one of the numerical methods of inverting the Laplace transform was used — the method of quadrature formulas with equal coefficients. Algorithms and computer programs for the implementation of this method have been compiled. The obtained graphs are quite accurate (the maximum error of calculations in the average does not exceed 5%), despite the fact that the error is very noticeable in the initial time segments.

Валишин А.А., Тиняев М.А. Моделирование динамических и спектральных вязкоупругих характеристик материалов на основе численного обращения преобразования Лапласа. Математическое моделирование и численные методы, 2022, № 1, с. 42–62.

doi: 10.18698/2309-3684-2020-3-321

When designing products made of composite materials intended for operation in difficult conditions of inhomogeneous deformations and temperatures, it is important to take into account the viscoelastic properties of the binder and fillers. The article analyzes the relationship between relaxation and creep characteristics. All known creep and re-laxation kernels in the literature are considered. The problem of transformation of creep characteristics into relaxation characteristics and vice versa is discussed in de-tail. There is a simple relationship between the creep and relaxation functions in the Laplace image space. However, when returning to the space of the originals, in many cases there are great difficulties in reversing the Laplace transform. Two numerical methods for inverting the Laplace transform are considered: the use of the Fourier series in sine and the method of quadrature formulas. Algorithms and computer programs for realization of these methods are made. It is shown that the operating time of a computer program implementing the Fourier method by sine is almost 2 times less than the operating time of a computer program implementing the quadrature formula method. However, the first method is inferior to the latter method in accuracy of calculations: the relaxation functions and relaxation rates, it is advisable to find the first method, since the computational error is almost indiscernible, and the functions of creep and creep speed, the second way, because for most functions, the result obtained by the second method is much more accurate than the result obtained by the first method. The Gavrillac-Negami creep function and the Gavrillac-Negami creep rate function could not be constructed due to a complex recursive formula for the series coefficients, but using both methods, these functions can still be obtained and compared with each other.

Валишин А.А., Тиняев М.А. Моделирование вязкоупругих характеристик материалов на основе численного обращения преобразования Лапласа. Математическое моделирование и численные методы, 2020, № 3, с. 3–21.