The generalised dynamical problem of thermoelasticity for the hollow sphere
Journal Title: Вісник Одеської державної академії будівництва та архітектури - Year 2018, Vol 1, Issue 72
Abstract
The generalized dynamical problem of thermoelasticity for the hollow sphere has been considered. It is supposed that the propagation of the temperature is symmetric with respect to the center of the sphere. The temperature and stresses are equal to zero at initial time. Radial displacements and the heat flow are given at surfaces of the sphere. Due to complexity of the statement of such boundary-value problems the most of the published solutions have been obtained as a result of using approximate methods to expand these solutions in the form of series for small and large times. In this work the closed exact solution of the given problem has been received. In introducing thermoelastic potentials which are connected with the temperature and the displacement (quasistatic and dynamical potentials) the problem is reduced to the solution of the system of two differential equations of the second and the third order in partial derivatives with respect to the time and the space coordinate. To improve the convergence of series at the boundary of the sphere potentials are represented as the sum of two functions. One of these functions is the solution of the system of equations with homogeneous conditions and the other is with nonhomogeneous conditions. Applying the finite integral transform on the space coordinate we come to the system of ordinary differential equations with respect to time potentials. The eigenfunctions of the corresponding Sturm – Liouvill problem are the kernels of the finite integral transforms. The asymptotic formula for eigenvalues depending on the parameter which tends to infinity is obtained. Laplace transform is used to get the solution of derived equations. It is analyzed the special case when the harmonic displacement at the surface is given and the heat flow equals zero. It was shown if the coupling constant tends to zero, then the temperature diminishes to zero. In this case the temperature disturbances can be large is the coupling constant is however small and the wave number is close to the eigenvalue of the corresponding Sturm – Liouvill problem (resonance conditions).
Authors and Affiliations
V. N. , Gavdzinski, E. V. , Maltseva
Keywords
Related Articles
- EP ID EP510817
- DOI -
- Views 100
- Downloads 0
Determination of the lattice optimal geometric shape in the combined arch system for circular plan buildings
The methodology of determining the optimal geometric shape of the lattice in the combined arch system was developed under conditions of tensile capacity of all elements of the flexible lattice and the minimum material co...
Analysis of the state of surface water sources in the Dnipro and Dniester river basins and the ways of solving the problem of providing the population with good water
The modern water-ecological situation in Ukraine is characterized. The classification of surface water by the quality of water is given. The analysis of the state of Ukraine's water sources in the Dnipro and Dniester riv...
Structural calculation approaches to the astimation of the technical state of constructions of monolithic buildings
The traditionally carried out evaluation of the technical state of monolithic reinforced concrete constructions can be qualitatively complemented by the use of the offered structural calculation approach. The essence of...
Bearing capacity of damaged flexible reinforced concrete columns
The aim of the study is to develop a calculation model for determining the residual bearing capacity of compressed damaged reinforced concrete elements pre-modeled simulated damage and 1 without damage were manufactured...
The temperature limitations for cargo vehicles on Ukraine's motorways
In time of summer heat the temperature of road surface chemise can exceed bitum softening point. In this case cargo vehicles’ intensive current traffic results in road’s tearing away and rutting. During cold time the roa...