321 research outputs found
Π’Π΅ΠΏΠ»ΠΎΠ²ΠΎΠ΅ ΡΠ°Π·ΡΡΡΠ΅Π½ΠΈΠ΅ ΠΏΠΎΠ»ΠΈΠΌΠ΅ΡΠ½ΡΡ Π²ΠΎΠ»ΠΎΠΊΠΎΠ½ Π² ΡΠ΅ΠΎΡΠΈΠΈ Π²ΡΠ΅ΠΌΠ΅Π½Π½ΠΎΠΉ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠΈ ΠΏΡΠΎΡΠ½ΠΎΡΡΠΈ
Objectives. This study mathematically describes the mutual influence of micro- and macrostages of the process of destruction of polymer materials and determines its main parameters and limiting characteristics. In addition, a relationship is established between molecular constants characterizing the structure of a material and those characterizing its macroscopic characteristics of strength. Finally, theoretical representations of the thermokinetics of the process of thermal destruction of polymer fibers from the standpoint of the kinetic thermofluctuation concept are developed, which makes it possible to predict the thermal durability of a sample under thermal loading.Methods. The structuralβkinetic thermofluctuation theory was used to describe the initial stages of the fracture process and to derive a generalized formula for the rate of crack growth. The mathematical theory of cracks is used to describe the thermally stressed state of a material in the vicinity of an internal circular crack under mechanical and thermal loadings of the sample.Results. A theoretical formula for the full isotherm of durability in the range of mechanical stresses from safe to critical, as well as a theoretical relationship for the time dependence of the strength of polymer fibers under purely thermal loading in the full range of heat loads from safe to critical and at the stage of nonthermal crack growth, is given. The main parameters and limiting characteristics of durability under thermal loading are also indicated.Conclusions. A generalized structuralβkinetic theory of the fracture of polymer fibers under purely thermal action on cracked specimens is presented. The developed theory combines three independent approaches: structuralβkinetic (thermofluctuation theory), mechanical, and thermodynamic. The obtained theoretical relations are of practical interest for the development of methods for localization, intensification, and control of the crack growth kinetics.Π¦Π΅Π»ΠΈ. ΠΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈ ΠΎΠΏΠΈΡΠ°ΡΡ Π²Π·Π°ΠΈΠΌΠ½ΠΎΠ΅ Π²Π»ΠΈΡΠ½ΠΈΠ΅ ΠΌΠΈΠΊΡΠΎ- ΠΈ ΠΌΠ°ΠΊΡΠΎΡΡΠ°Π΄ΠΈΠΉ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠ°Π·ΡΡΡΠ΅Π½ΠΈΡ ΠΏΠΎΠ»ΠΈΠΌΠ΅ΡΠ½ΡΡ
ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ², ΠΎΠΏΡΠ΅Π΄Π΅Π»ΠΈΡΡ Π΅Π³ΠΎ ΠΎΡΠ½ΠΎΠ²Π½ΡΠ΅ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΡ ΠΈ ΠΏΡΠ΅Π΄Π΅Π»ΡΠ½ΡΠ΅ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ, ΡΡΡΠ°Π½ΠΎΠ²ΠΈΡΡ ΡΠ²ΡΠ·Ρ ΠΌΠ΅ΠΆΠ΄Ρ ΠΌΠΎΠ»Π΅ΠΊΡΠ»ΡΡΠ½ΡΠΌΠΈ ΠΊΠΎΠ½ΡΡΠ°Π½ΡΠ°ΠΌΠΈ, Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΠ·ΡΡΡΠΈΠΌΠΈ ΡΡΡΡΠΊΡΡΡΡ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»Π° Ρ ΠΎΠ΄Π½ΠΎΠΉ ΡΡΠΎΡΠΎΠ½Ρ ΠΈ ΠΌΠ°ΠΊΡΠΎΡΠΊΠΎΠΏΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠ°ΠΌΠΈ ΠΏΡΠΎΡΠ½ΠΎΡΡΠΈ Ρ Π΄ΡΡΠ³ΠΎΠΉ. Π Π°Π·ΡΠ°Π±ΠΎΡΠ°ΡΡ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΡ ΡΠ΅ΡΠΌΠΎΠΊΠΈΠ½Π΅ΡΠΈΠΊΠΈ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠ³ΠΎ ΡΠ°Π·ΡΡΡΠ΅Π½ΠΈΡ ΠΏΠΎΠ»ΠΈΠΌΠ΅ΡΠ½ΡΡ
Π²ΠΎΠ»ΠΎΠΊΠΎΠ½ Ρ ΠΏΠΎΠ·ΠΈΡΠΈΠΉ ΠΊΠΈΠ½Π΅ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅ΡΠΌΠΎΡΠ»ΡΠΊΡΡΠ°ΡΠΈΠΎΠ½Π½ΠΎΠΉ ΠΊΠΎΠ½ΡΠ΅ΠΏΡΠΈΠΈ, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠ΅ΠΉ ΠΏΡΠΎΠ³Π½ΠΎΠ·ΠΈΡΠΎΠ²Π°ΡΡ ΡΠ΅ΡΠΌΠΈΡΠ΅ΡΠΊΡΡ Π΄ΠΎΠ»Π³ΠΎΠ²Π΅ΡΠ½ΠΎΡΡΡ ΠΎΠ±ΡΠ°Π·ΡΠ° ΠΏΡΠΈ Π΅Π³ΠΎ ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠΌ Π½Π°Π³ΡΡΠΆΠ΅Π½ΠΈΠΈ.ΠΠ΅ΡΠΎΠ΄Ρ. ΠΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½Π° ΡΡΡΡΠΊΡΡΡΠ½ΠΎ-ΠΊΠΈΠ½Π΅ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΠ΅ΡΠΌΠΎΡΠ»ΡΠΊΡΡΠ°ΡΠΈΠΎΠ½Π½Π°Ρ ΡΠ΅ΠΎΡΠΈΡ Π΄Π»Ρ ΠΎΠΏΠΈΡΠ°Π½ΠΈΡ ΡΠ»Π΅ΠΌΠ΅Π½ΡΠ°ΡΠ½ΠΎΠ³ΠΎ Π°ΠΊΡΠ° ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠ°Π·ΡΡΡΠ΅Π½ΠΈΡ ΠΈ Π²ΡΠ²ΠΎΠ΄Π° ΠΎΠ±ΠΎΠ±ΡΠ΅Π½Π½ΠΎΠΉ ΡΠΎΡΠΌΡΠ»Ρ ΡΠΊΠΎΡΠΎΡΡΠΈ ΡΠΎΡΡΠ° ΡΡΠ΅ΡΠΈΠ½Ρ ΠΈ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΠ΅ΠΎΡΠΈΡ ΡΡΠ΅ΡΠΈΠ½ Π΄Π»Ρ ΠΎΠΏΠΈΡΠ°Π½ΠΈΡ ΡΠ΅ΡΠΌΠΎΠ½Π°ΠΏΡΡΠΆΠ΅Π½Π½ΠΎΠ³ΠΎ ΡΠΎΡΡΠΎΡΠ½ΠΈΡ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»Π° Π² ΠΎΠΊΡΠ΅ΡΡΠ½ΠΎΡΡΠΈ Π²Π½ΡΡΡΠ΅Π½Π½Π΅ΠΉ ΠΊΡΡΠ³ΠΎΠ²ΠΎΠΉ ΡΡΠ΅ΡΠΈΠ½Ρ ΠΏΡΠΈ ΠΌΠ΅Ρ
Π°Π½ΠΈΡΠ΅ΡΠΊΠΎΠΌ ΠΈ ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠΌ Π½Π°Π³ΡΡΠΆΠ΅Π½ΠΈΡΡ
ΠΎΠ±ΡΠ°Π·ΡΠ°.Π Π΅Π·ΡΠ»ΡΡΠ°ΡΡ. ΠΡΠΈΠ²ΠΎΠ΄ΠΈΡΡΡ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΠΎΡΠΌΡΠ»Π° ΠΏΠΎΠ»Π½ΠΎΠΉ ΠΈΠ·ΠΎΡΠ΅ΡΠΌΡ Π΄ΠΎΠ»Π³ΠΎΠ²Π΅ΡΠ½ΠΎΡΡΠΈ Π² ΠΈΠ½ΡΠ΅ΡΠ²Π°Π»Π΅ ΠΌΠ΅Ρ
Π°Π½ΠΈΡΠ΅ΡΠΊΠΈΡ
Π½Π°ΠΏΡΡΠΆΠ΅Π½ΠΈΠΉ ΠΎΡ Π±Π΅Π·ΠΎΠΏΠ°ΡΠ½ΠΎΠ³ΠΎ Π΄ΠΎ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ, Π° ΡΠ°ΠΊΠΆΠ΅ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ ΡΠΎΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΠ΅ Π΄Π»Ρ Π²ΡΠ΅ΠΌΠ΅Π½Π½ΠΎΠΉ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠΈ ΠΏΡΠΎΡΠ½ΠΎΡΡΠΈ ΠΏΠΎΠ»ΠΈΠΌΠ΅ΡΠ½ΡΡ
Π²ΠΎΠ»ΠΎΠΊΠΎΠ½ ΠΏΡΠΈ ΡΠΈΡΡΠΎ ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠΌ Π½Π°Π³ΡΡΠΆΠ΅Π½ΠΈΠΈ Π² ΠΏΠΎΠ»Π½ΠΎΠΌ ΠΈΠ½ΡΠ΅ΡΠ²Π°Π»Π΅ ΡΠ΅ΠΏΠ»ΠΎΠ²ΡΡ
Π½Π°Π³ΡΡΠ·ΠΎΠΊ ΠΎΡ Π±Π΅Π·ΠΎΠΏΠ°ΡΠ½ΠΎΠΉ Π΄ΠΎ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΈ Π½Π° ΡΡΠ°Π΄ΠΈΠΈ Π°ΡΠ΅ΡΠΌΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΡΠΎΡΡΠ° ΡΡΠ΅ΡΠΈΠ½Ρ. Π£ΠΊΠ°Π·Π°Π½Ρ ΠΎΡΠ½ΠΎΠ²Π½ΡΠ΅ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΡ ΠΈ ΠΏΡΠ΅Π΄Π΅Π»ΡΠ½ΡΠ΅ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ Π΄ΠΎΠ»Π³ΠΎΠ²Π΅ΡΠ½ΠΎΡΡΠΈ ΠΏΡΠΈ ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠΌ Π½Π°Π³ΡΡΠΆΠ΅Π½ΠΈΠΈ.ΠΡΠ²ΠΎΠ΄Ρ. ΠΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½Π° ΠΎΠ±ΠΎΠ±ΡΠ΅Π½Π½Π°Ρ ΡΡΡΡΠΊΡΡΡΠ½ΠΎ-ΠΊΠΈΠ½Π΅ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΠ΅ΠΎΡΠΈΡ ΡΠ°Π·ΡΡΡΠ΅Π½ΠΈΡ ΠΏΠΎΠ»ΠΈΠΌΠ΅ΡΠ½ΡΡ
Π²ΠΎΠ»ΠΎΠΊΠΎΠ½ ΠΏΡΠΈ ΡΠΈΡΡΠΎ ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠΌ Π²ΠΎΠ·Π΄Π΅ΠΉΡΡΠ²ΠΈΠΈ Π½Π° ΠΎΠ±ΡΠ°Π·ΡΡ Ρ ΡΡΠ΅ΡΠΈΠ½ΠΎΠΉ. Π Π°Π·Π²ΠΈΡΠ°Ρ ΡΠ΅ΠΎΡΠΈΡ ΠΎΠ±ΡΠ΅Π΄ΠΈΠ½ΡΠ΅Ρ ΡΡΠΈ ΡΠ°ΠΌΠΎΡΡΠΎΡΡΠ΅Π»ΡΠ½ΡΡ
ΠΏΠΎΠ΄Ρ
ΠΎΠ΄Π°: ΡΡΡΡΠΊΡΡΡΠ½ΠΎ-ΠΊΠΈΠ½Π΅ΡΠΈΡΠ΅ΡΠΊΠΈΠΉ (ΡΠ΅ΡΠΌΠΎΡΠ»ΡΠΊΡΡΠ°ΡΠΈΠΎΠ½Π½Π°Ρ ΡΠ΅ΠΎΡΠΈΡ), ΠΌΠ΅Ρ
Π°Π½ΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΈ ΡΠ΅ΡΠΌΠΎΠ΄ΠΈΠ½Π°ΠΌΠΈΡΠ΅ΡΠΊΠΈΠΉ. ΠΠΎΠ»ΡΡΠ΅Π½Π½ΡΠ΅ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΠΎΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΡ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΡΡ ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΈΠ½ΡΠ΅ΡΠ΅Ρ Π΄Π»Ρ ΡΠ°Π·ΡΠ°Π±ΠΎΡΠΊΠΈ ΡΠΏΠΎΡΠΎΠ±ΠΎΠ² Π»ΠΎΠΊΠ°Π»ΠΈΠ·Π°ΡΠΈΠΈ, ΠΈΠ½ΡΠ΅Π½ΡΠΈΡΠΈΠΊΠ°ΡΠΈΠΈ ΠΈ ΡΠΏΡΠ°Π²Π»Π΅Π½ΠΈΡ ΠΊΠΈΠ½Π΅ΡΠΈΠΊΠΎΠΉ ΡΠΎΡΡΠ° ΡΡΠ΅ΡΠΈΠ½Ρ
Stable one-dimensional periodic waves in Kerr-type saturable and quadratic nonlinear media
We review the latest progress and properties of the families of bright and
dark one-dimensional periodic waves propagating in saturable Kerr-type and
quadratic nonlinear media. We show how saturation of the nonlinear response
results in appearance of stability (instability) bands in focusing (defocusing)
medium, which is in sharp contrast with the properties of periodic waves in
Kerr media. One of the key results discovered is the stabilization of
multicolor periodic waves in quadratic media. In particular, dark-type waves
are shown to be metastable, while bright-type waves are completely stable in a
broad range of energy flows and material parameters. This yields the first
known example of completely stable periodic wave patterns propagating in
conservative uniform media supporting bright solitons. Such results open the
way to the experimental observation of the corresponding self-sustained
periodic wave patterns.Comment: 29 pages, 10 figure
ΠΠΎΠ΄ΠΈΡΠΈΠΊΠ°ΡΠΈΡ Π·Π°Π΄Π°ΡΠΈ Π.Π. ΠΠΎΠ»ΠΊΠΎΠ²Π° ΠΎ Β«ΡΠΎΡΡΠ΅Π΄ΠΎΡΠΎΡΠ΅Π½Π½ΠΎΠΉ Π΅ΠΌΠΊΠΎΡΡΠΈΒ» ΠΏΡΠΈ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠΈ ΡΠ΅ΡΠΌΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅Π°ΠΊΡΠΈΠΈ Π΄ΠΈΡΠΊΠ° Ρ ΠΊΡΡΠ³ΠΎΠ²ΡΠΌ Π²ΡΡΠ΅Π·ΠΎΠΌ.
The thermal reaction of an infinite disk with a circular cut is investigated. The regularities of the thermoelastic strains when the surface of the cut is heated by the principle of Β«the concentrated capacityΒ» are found.ΠΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½Π° ΡΠ΅ΡΠΌΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΠ΅Π°ΠΊΡΠΈΡ Π±Π΅ΡΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ³ΠΎ Π΄ΠΈΡΠΊΠ° Ρ Π²Π½ΡΡΡΠ΅Π½Π½ΠΈΠΌ ΠΊΡΡΠ³ΠΎΠ²ΡΠΌ Π²ΡΡΠ΅Π·ΠΎΠΌ. Π£ΡΡΠ°Π½ΠΎΠ²Π»Π΅Π½Ρ Π·Π°ΠΊΠΎΠ½ΠΎΠΌΠ΅ΡΠ½ΠΎΡΡΠΈ ΡΠ΅ΡΠΌΠΎΡΠΏΡΡΠ³ΠΈΡ
Π½Π°ΠΏΡΡΠΆΠ΅Π½ΠΈΠΉ ΠΏΡΠΈ Π½Π°Π³ΡΠ΅Π²Π΅ ΠΏΠΎΠ²Π΅ΡΡ
Π½ΠΎΡΡΠΈ Π²ΡΡΠ΅Π·Π° ΠΏΠΎ ΠΏΡΠΈΠ½ΡΠΈΠΏΡ Β«ΡΠΎΡΡΠ΅Π΄ΠΎΡΠΎΡΠ΅Π½Π½ΠΎΠΉ Π΅ΠΌΠΊΠΎΡΡΠΈΒ»
ΠΡΡΠ΅ΠΊΡ ΡΠ²ΡΠ·Π°Π½Π½ΠΎΡΡΠΈ Π² Π΄ΠΈΠ½Π°ΠΌΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅ΡΠΌΠΎΡΠΏΡΡΠ³ΠΎΡΡΠΈ ΠΏΠΎΠ»ΠΈΠΌΠ΅ΡΠ½ΡΡ ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ²
Connection effect of deformation and temperature fields was shown to be significant for some polymers..ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ ΡΡΡΠ΅ΠΊΡ ΡΠ²ΡΠ·Π°Π½Π½ΠΎΡΡΠΈ ΠΏΠΎΠ»Π΅ΠΉ Π΄Π΅ΡΠΎΡΠΌΠ°ΡΠΈΠΈ ΠΈ ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΡ ΡΠ²Π»ΡΠ΅ΡΡΡ ΡΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΡΠΌ Π΄Π»Ρ ΡΡΠ΄Π° ΠΏΠΎΠ»ΠΈΠΌΠ΅ΡΠΎΠ²
Π Π½ΠΎΠ²ΠΎΠΌ ΠΏΠΎΠ΄Ρ ΠΎΠ΄Π΅ ΠΏΡΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΠΈ ΠΊΡΠ°Π΅Π²ΡΡ Π·Π°Π΄Π°Ρ ΠΠΈΡΠΈΡ Π»Π΅ ΠΈ ΠΠ΅ΠΉΠΌΠ°Π½Π° Π΄Π»Ρ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΠΠ°ΠΏΠ»Π°ΡΠ°
The new approach of using of Green function method at the decision of edge tasks of Dirichlet and Neumann for Laplace equation is developedΠΒ ΡΡΠ°ΡΡΠ΅ ΡΠ°Π·Π²ΠΈΡ Π½ΠΎΠ²ΡΠΉ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄ Π² ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠΈ ΠΌΠ΅ΡΠΎΠ΄Π° ΡΡΠ½ΠΊΡΠΈΠΉ ΠΡΠΈΠ½Π° ΠΏΡΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΠΈ ΠΊΡΠ°Π΅Π²ΡΡ
Π·Π°Π΄Π°Ρ ΠΠΈΡΠΈΡ
Π»Π΅ ΠΈ ΠΠ΅ΠΉΠΌΠ°Π½Π° Π΄Π»Ρ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΠΠ°ΠΏΠ»Π°ΡΠ° Π½Π° ΠΏΠ»ΠΎΡΠΊΠΎΡΡΠΈ
ΠΠΠΠ«Π ΠΠΠΠΠΠ¬ΠΠ«Π ΠΠ ΠΠΠ‘Π’ΠΠΠΠΠΠΠ― Π Π’ΠΠΠ ΠΠ ΠΠΠΠΠΠΠΠΠ
The article considers a new class of model representations in the theory of oscillation of systems described by the classical boundary value problems for hyperbolic equations. The peculiarity of the suggested approach consists in the introduction of an additional term into the basic equation of oscillations. This term characterizes the presence of a temperature gradient in the systems. The developed theory is applicable to longitudinal oscillations of a rod, but can be extended just as well to the problem of the vibrations of strings, membranes, shaft torsional oscillations, electromagnetic waves, etc. Numerical experiments showed a significant effect of the temperature field in the rod on the nature of the vibrations and displacements of the rod cross-sections in comparison with classical solutions.Π Π°ΡΡΠΌΠΎΡΡΠ΅Π½ Π½ΠΎΠ²ΡΠΉ ΠΊΠ»Π°ΡΡ ΠΌΠΎΠ΄Π΅Π»ΡΠ½ΡΡ
ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΠΉ Π² ΡΠ΅ΠΎΡΠΈΠΈ ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ ΡΠΈΡΡΠ΅ΠΌ, ΠΎΠΏΠΈΡΡΠ²Π°Π΅ΠΌΡΡ
ΠΊΠ»Π°ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΠΊΡΠ°Π΅Π²ΡΠΌΠΈ Π·Π°Π΄Π°ΡΠ°ΠΌΠΈ Π΄Π»Ρ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ Π³ΠΈΠΏΠ΅ΡΠ±ΠΎΠ»ΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΡΠΈΠΏΠ°. ΠΡΠΎΠ±Π΅Π½Π½ΠΎΡΡΡ ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π½ΠΎΠ³ΠΎ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄Π° Π·Π°ΠΊΠ»ΡΡΠ°Π΅ΡΡΡ Π²ΠΎ Π²Π²Π΅Π΄Π΅Π½ΠΈΠΈ Π² ΠΎΡΠ½ΠΎΠ²Π½ΠΎΠ΅ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠ΅ ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ Π΄ΠΎΠΏΠΎΠ»Π½ΠΈΡΠ΅Π»ΡΠ½ΠΎΠ³ΠΎ ΡΠ»Π°Π³Π°Π΅ΠΌΠΎΠ³ΠΎ, Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΠ·ΡΡΡΠ΅Π³ΠΎ Π½Π°Π»ΠΈΡΠΈΠ΅ Π² ΡΠΈΡΡΠ΅ΠΌΠ΅ Π³ΡΠ°Π΄ΠΈΠ΅Π½ΡΠ° ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΡ. Π Π°Π·Π²ΠΈΡΠ°Ρ ΡΠ΅ΠΎΡΠΈΡ ΠΊΠ°ΡΠ°Π΅ΡΡΡ ΠΏΡΠΎΠ΄ΠΎΠ»ΡΠ½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ ΡΡΠ΅ΡΠΆΠ½Ρ, Π½ΠΎ Ρ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²ΡΠΌ ΡΡΠΏΠ΅Ρ
ΠΎΠΌ ΠΌΠΎΠΆΠ΅Ρ Π±ΡΡΡ ΡΠ°ΡΠΏΡΠΎΡΡΡΠ°Π½Π΅Π½Π° Π½Π° Π·Π°Π΄Π°ΡΠΈ ΠΎ ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΡΡ
ΡΡΡΡΠ½Ρ, ΠΌΠ΅ΠΌΠ±ΡΠ°Π½Ρ, ΠΊΡΡΡΠΈΠ»ΡΠ½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ Π²Π°Π»Π°, ΡΠ»Π΅ΠΊΡΡΠΎΠΌΠ°Π³Π½ΠΈΡΠ½ΡΡ
ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ ΠΈ Ρ.Π΄. ΠΡΠΎΠ²Π΅Π΄Π΅Π½Ρ ΡΠΈΡΠ»Π΅Π½Π½ΡΠ΅ ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½ΡΡ, ΠΏΠΎΠΊΠ°Π·Π°Π²ΡΠΈΠ΅ ΡΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΎΠ΅ Π²Π»ΠΈΡΠ½ΠΈΠ΅ ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ Π² ΡΡΠ΅ΡΠΆΠ½Π΅ Π½Π° Ρ
Π°ΡΠ°ΠΊΡΠ΅Ρ ΠΊΠΎΠ»Π΅Π±Π°Π½ΠΈΠΉ ΠΈ ΡΠΌΠ΅ΡΠ΅Π½ΠΈΠΉ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΡΡΠ΅ΡΠΆΠ½Ρ ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ ΠΊΠ»Π°ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡΠΌΠΈ
ΠΠΠΠΠΠ’ΠΠ§ΠΠ‘ΠΠΠ Π ΠΠ¨ΠΠΠΠ― ΠΠΠΠΠ ΠΠΠΠΠ§ΠΠ‘ΠΠΠ₯ ΠΠΠΠΠΠΠ ΠΠΠ‘Π’ΠΠ¦ΠΠΠΠΠ ΠΠΠ Π’ΠΠΠΠΠΠ ΠΠΠΠΠΠΠ‘Π’Π
Practically important problems of non-stationary heat conduction for hyperbolic transport models are considered. An analytical approach based on contour integration of operational solutions of hyperbolic models is developed. This leads to new integral relationships convenient for numerical experiments. The equivalence of new functional constructions and known analytical solutions of this class of problems is shown. On the basis of the obtained relations, the wave character of the nonstationary thermal conductivity is described taking into account the finite velocity of heat propagation. The jumps at the front of the heat wave are calculated. The proposed approach gives effective results when studying the thermal reaction to heating or cooling regions bounded from within by a flat surface, either a cylindrical cavity or a spherical surface.Π Π°ΡΡΠΌΠΎΡΡΠ΅Π½Ρ ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΈ Π²Π°ΠΆΠ½ΡΠ΅ Π·Π°Π΄Π°ΡΠΈ Π½Π΅ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΎΠΉ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈ Π΄Π»Ρ Π³ΠΈΠΏΠ΅ΡΠ±ΠΎΠ»ΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ ΠΏΠ΅ΡΠ΅Π½ΠΎΡΠ°. Π Π°Π·Π²ΠΈΡ Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄, ΠΎΡΠ½ΠΎΠ²Π°Π½Π½ΡΠΉ Π½Π° ΠΊΠΎΠ½ΡΡΡΠ½ΠΎΠΌ ΠΈΠ½ΡΠ΅Π³ΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Π³ΠΈΠΏΠ΅ΡΠ±ΠΎΠ»ΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ, ΠΏΡΠΈΠ²ΠΎΠ΄ΡΡΠΈΠΉ ΠΊ Π½ΠΎΠ²ΡΠΌ ΠΈΠ½ΡΠ΅Π³ΡΠ°Π»ΡΠ½ΡΠΌ ΡΠΎΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΡΠΌ, ΡΠ΄ΠΎΠ±Π½ΡΠΌ Π΄Π»Ρ ΡΠΈΡΠ»Π΅Π½Π½ΡΡ
ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½ΡΠΎΠ². ΠΠΎΠΊΠ°Π·Π°Π½Π° ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½ΠΎΡΡΡ Π½ΠΎΠ²ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΎΠ½Π°Π»ΡΠ½ΡΡ
ΠΊΠΎΠ½ΡΡΡΡΠΊΡΠΈΠΉ ΠΈ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΡ
Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Π΄Π°Π½Π½ΠΎΠ³ΠΎ ΠΊΠ»Π°ΡΡΠ° Π·Π°Π΄Π°Ρ. ΠΠ° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΡΡ
ΡΠΎΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΠΉ ΠΎΠΏΠΈΡΠ°Π½ Π²ΠΎΠ»Π½ΠΎΠ²ΠΎΠΉ Ρ
Π°ΡΠ°ΠΊΡΠ΅Ρ Π½Π΅ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΎΠΉ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈ Ρ ΡΡΠ΅ΡΠΎΠΌ ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠΉ ΡΠΊΠΎΡΠΎΡΡΠΈ ΡΠ°ΡΠΏΡΠΎΡΡΡΠ°Π½Π΅Π½ΠΈΡ ΡΠ΅ΠΏΠ»ΠΎΡΡ; ΡΠ°ΡΡΡΠΈΡΠ°Π½Ρ ΡΠΊΠ°ΡΠΊΠΈ Π½Π° ΡΡΠΎΠ½ΡΠ΅ ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠΉ Π²ΠΎΠ»Π½Ρ. ΠΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π½ΡΠΉ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄ Π΄Π°Π΅Ρ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΡΠ΅ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΡ ΠΏΡΠΈ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠΈ ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠΉ ΡΠ΅Π°ΠΊΡΠΈΠΈ Π½Π° Π½Π°Π³ΡΠ΅Π² ΠΈΠ»ΠΈ ΠΎΡ
Π»Π°ΠΆΠ΄Π΅Π½ΠΈΠ΅ ΠΎΠ±Π»Π°ΡΡΠ΅ΠΉ, ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½Π½ΡΡ
ΠΈΠ·Π½ΡΡΡΠΈ ΠΏΠ»ΠΎΡΠΊΠΎΠΉ ΠΏΠΎΠ²Π΅ΡΡ
Π½ΠΎΡΡΡΡ, Π»ΠΈΠ±ΠΎ ΡΠΈΠ»ΠΈΠ½Π΄ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΏΠΎΠ»ΠΎΡΡΡΡ, Π»ΠΈΠ±ΠΎ ΡΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΏΠΎΠ²Π΅ΡΡ
Π½ΠΎΡΡΡΡ
ΠΡΠ°Π΅Π²ΡΠ΅ Π·Π°Π΄Π°ΡΠΈ Π΄Π»Ρ Π³ΠΈΠΏΠ΅ΡΠ±ΠΎΠ»ΠΈΡΠ΅ΡΠΊΠΈΡ ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ ΠΏΠ΅ΡΠ΅Π½ΠΎΡΠ°
The new representation of transfer edge tasks for the equations of hyperbolic type is developedΠΒ ΡΡΠ°ΡΡΠ΅ ΡΠ°Π·Π²ΠΈΡΡ Π½ΠΎΠ²ΡΠ΅ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΡ ΠΎ ΠΊΡΠ°Π΅Π²ΡΡ
Π·Π°Π΄Π°ΡΠ°Ρ
ΠΏΠ΅ΡΠ΅Π½ΠΎΡΠ° Π΄Π»Ρ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ Π³ΠΈΠΏΠ΅ΡΠ±ΠΎΠ»ΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΡΠΈΠΏΠ°
ΠΡΠ΅Π½ΠΊΠΈ Π΄Π»Ρ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ ΠΏΠ°ΡΠ°Π±ΠΎΠ»ΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΡΠΈΠΏΠ° Π² Π½Π΅ΡΠΈΠ»ΠΈΠ½Π΄ΡΠΈΡΠ΅ΡΠΊΠΈΡ ΠΎΠ±Π»Π°ΡΡΡΡ
An estimation method was developed for solutions of boundary problems of thermal conductivity of the generalized type.Π Π°Π·Π²ΠΈΡ ΠΌΠ΅ΡΠΎΠ΄ ΠΎΡΠ΅Π½ΠΎΠΊ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΊΡΠ°Π΅Π²ΡΡ
Π·Π°Π΄Π°Ρ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½Π½ΠΎΠ³ΠΎ ΡΠΈΠΏΠ°
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