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
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