Асимптотическое поведение решения осесимметричной динамической задачи теории упругости для трансверсально-изотропного сферического слоя малой толщины

Abstract

In this paper, we study the axisymmetric dynamic problem of the theory of elasticity for the transversely isotropic spherical layer of small thickness that does not contain any of the poles 0 and π. It is assumed that the lateral surface of the sphere is free of stresses, and boundary conditions are set on conical sections. Using the method of asymptotic integration of equations of the theory of elasticity, the dynamic problem of this theory is analyzed for the transversely isotropic spherical layer as the thin-walled parameter tends to zero. A possible form of wave formation in the transversely isotropic spherical layer has been studied depending on the frequency of the influencing forces. Homogeneous solutions are constructed and their classification is given. Asymptotic expansions of the homogeneous solutions are obtained, which make possible to calculate the stress-strain state for various values of the frequency of the influencing forces. It is shown that for the high-frequency oscillations in the first term of the asymptotics, the dispersion equation coincides with the well-known Rayleigh-Lamb equation for the elastic band. In the general case of loading on the sphere using the Hamilton variational principle, the boundary-value problem is reduced to the solving infinite systems of linear algebraic equations.